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\section{Introduction}
The classification of finite dimensional algebras over an algebraically closed field $k$ up to derived equivalence is a crucial problem in representation theory. It has a complete answer for algebras of global dimension 1 (see \cite[Corollary~4.8]{Happel87}): Two finite dimensional $k$-algebras $\Lambda=kQ$ and $\Lambda'=kQ'$ are derived equivalent if and only if one can pass from the quiver $Q$ to the quiver $Q'$ by a sequence of reflections (as introduced in \cite{BGP}). Therefore it is possible to decide when two hereditary algebras are derived equivalent by simple combinatorial means. The aim of this paper is to apply results of \cite{AO10} in order to generalize this result to certain algebras of global dimension~2.
The notion of reflection of an acyclic quiver has been generalized by Fomin and Zelevinsky \cite{FZ1} to the notion of mutation in their definition of cluster algebras. Since then, categorical interpretations of the mutation have been discovered via 2-Calabi-Yau triangulated categories. These created a link between cluster algebras and representation theory. First, in~\cite{BMRRT}, cluster categories $\mathcal{C}_Q$ associated to acyclic quivers $Q$ were defined as the orbit categories $\mathcal{D}^b(kQ)/\mathbb{S}_2$, where $\mathbb{S}_2=\mathbb{S}[-2]$ is the second desuspension of the Serre functor $\mathbb{S}$ of the bounded derived category $\mathcal{D}^b(kQ)$. This notion has been generalized in \cite{Ami09} to algebras of global dimension two. In this case the generalized cluster category is defined to be the triangulated hull in the sense of \cite{Kel05} of the orbit category $\mathcal{D}^b\Lambda/\mathbb{S}_2$.
\medskip
In this paper, we study more explicitly the derived equivalence classes of algebras of global dimension 2 which are of acyclic cluster type, that is algebras whose generalized cluster category is equivalent to some cluster category $\mathcal{C}_Q$ where $Q$ is an acyclic quiver. We strongly use the results and the techniques of \cite{AO10}. In particular we use the notion of graded mutation of a graded quiver with potential which is a refinement of the notion of mutation of a quiver with potential introduced in~\cite{DWZ}: Associated to an algebra $\Lambda$ there is a graded Jacobian algebra $\overline{\Lambda}$~\cite{Kel09} whose degree zero subalgebra is $\Lambda$. Graded mutation explains how to mutate such graded Jacobian algebras. Then from \cite{AO10} we deduce an analogue of the result for algebras of global dimension 1.
\begin{thma}[see Theorem~\ref{thm_gradingQ_derivedeq}]
Let $\Lambda_1$ and $\Lambda_2$ be two finite dimensional algebras of global dimension $2$. Assume that $\Lambda_1$ is of acyclic cluster type. Then the algebras $\Lambda_1$ and $\Lambda_2$ are derived equivalent if and only if one can pass from $\overline{\Lambda}_1$ to $\overline{\Lambda}_2$ using a sequence of graded mutations.
\end{thma}
The setup is especially nice when the algebras are of tree cluster type.
\begin{thma}[Corollary~\ref{cor_tree}]
Let $Q$ be an acyclic quiver whose underlying graph is a tree. If $\Lambda$ is an algebra of global dimension 2 of cluster type $Q$, then it is derived equivalent to $kQ$.
\end{thma}
To get a complete understanding of algebras of tame acyclic cluster type, in the rest of the paper, we focus on the algebras of cluster type $\widetilde{A}_{p,q}$. To such an algebra, using graded mutation, we associate an integer that we call weight, which is zero when $\Lambda$ is hereditary. We prove that two algebras of cluster type $\widetilde{A}_{p,q}$ are derived equivalent if and only if they have the same weight (Theorem~\ref{derivedeqiffwegal}). Then a result of \cite{AO10} which shows that two cluster equivalent algebras are graded derived equivalent permits us to compute explicitly the shape of the Auslander-Reiten quiver of the derived category.
\begin{thma}[Corollary~\ref{shapeARquiver}]
Let $\Lambda$ be an algebra of cluster type $\widetilde{A}_{p,q}$ and of weight $w\neq 0$. Then the algebra $\Lambda$ is representation finite and not piecewise hereditary. The Auslander-Reiten quiver of $\mathcal{D}^b(\Lambda)$ has exactly $3|w|$ connected components:
\begin{itemize}
\item $|w|$ components of type $\mathbb{Z} A_\infty^\infty$;
\item $2|w|$ components of type $\mathbb{Z} A_\infty$.
\end{itemize}
\end{thma}
Finally we use the explicit description of the cluster-tilted algebras of type $\widetilde{A}_{p,q}$ of \cite{Bas09} to deduce a description of all algebras of cluster type $\widetilde{A}_{p,q}$ in term of quivers with relations.
\medskip
The paper is organized as follows: Section \ref{section background} is devoted to recalling results on generalized cluster categories, on Jacobian algebras, and on 2-Calabi-Yau categories. In Section \ref{section tree case}, we apply the results of \cite{AO10} to algebras of acyclic cluster type. The special case of algebras of cluster type $\widetilde{A}_{p,q}$ is treated in the last three sections. We introduce the notion of weight and prove that it is an invariant of the derived equivalence class of the algebra in Section \ref{section derived atilde}. We compute the shape of the AR-quiver of the derived category in Section~\ref{section ARquiver}, and we finally describe these algebras explicitly in Section \ref{section explicit}. We end the paper by giving (as an example) the complete classification for $\widetilde{A}_{2,2}$.
\subsection*{Acknowledgment}
The authors would like to thank an anonymous referee for his valuable comments and for pointing us out a mistake in a previous version of this paper.
\section{Background}\label{section background}
Throughout this paper, $k$ denotes an algebraically closed field. All categories appearing are $k$-categories, and all functors are $k$-linear. By an algebra we mean an associative unitary basic $k$-algebra. For an algebra $\Lambda$ we denote by $\mod \Lambda$ the category of finitely generated right modules over $\Lambda$. We denote by $D$ the standard duality ${\sf Hom }_k(-, k) \colon (\mod k)^{\rm op} \to \mod k$.
\subsection{Cluster-tilting objects and mutation in $2$-Calabi-Yau categories}
\begin{dfa}[Iyama]
Let $\mathcal{T}$ be a Krull-Schmidt triangulated category, with finite dimensional ${\sf Hom }$-spaces (${\sf Hom }$-finite for short) and $2$-Calabi-Yau, that is there is a functorial isomorphism ${\sf Hom }_\mathcal{T}(X,Y[2])\cong D{\sf Hom }_\mathcal{T}(Y,X)$ for all objects $X$ and $Y$ in $\mathcal{T}$. An object $T$ is called \emph{cluster-tilting} if
$${\sf add \hspace{.02in} }(T) =\{X\in \mathcal{T} \mid {\sf Hom }_\mathcal{T}(X,T[1])=0\},$$
where ${\sf add \hspace{.02in} }(T)\subset \mathcal{T}$ is the additive closure of $T$.
\end{dfa}
The endomorphism algebra of a cluster-tilting object $T\in \mathcal{T}$ is called a \emph{$2$-Calabi-Yau-tilted algebra}.
\begin{thma}[{\cite[Theorem 5.3]{IY}}]\label{IyamaYoshino}
Let $T$ be a basic cluster-tilting object in a $2$-Calabi-Yau triangulated category $\mathcal{T}$. Let $T_i$ be an indecomposable summand of $T\cong T_i\oplus T'$. Then there exists a unique (up to isomorphism) object $T_i^*$ not isomorphic to $T_i$, such that $\mu_{T_i}(T):=T'\oplus T^*_i$ is a basic cluster-tilting object in $\mathcal{T}$. Moreover $T^*_i$ is indecomposable, and there exist triangles in $\mathcal{T}$ \[\xymatrix{T_i\ar[r] ^u & U\ar[r] & T^*_i\ar[r] & T_i[1]}\textrm{ and }\xymatrix{T^*_i\ar[r] & U'\ar[r]^{u'} & T_i\ar[r] & T^*_i[1]}\] where $u$ (resp.\ $u'$) is a minimal left (resp.\ right) ${\sf add \hspace{.02in} }(T')$-approximation.
\end{thma}
\subsection{Generalized cluster categories}
Let $\Lambda$ be a finite dimensional $k$-algebra of global dimension
at most $2$. We denote by $\mathcal{D}^b(\Lambda)$ the bounded derived category
of finitely generated $\Lambda$-modules. It has a Serre functor that
we denote $\mathbb{S}$. We denote by $\mathbb{S}_2$ the composition
$\mathbb{S}[-2]$.
The generalized cluster category $\mathcal{C}_\Lambda$ of $\Lambda$ has been defined in \cite{Ami09} as
the triangulated hull of the orbit category $\mathcal{D}^b(\Lambda)/\mathbb{S}_2$ (see \cite{Kel05} or \cite[Section~7]{AO10} for more details on triangulated hulls). We
will denote by $\pi_\Lambda$ the triangle functor
$$\xymatrix{\pi_\Lambda\colon \mathcal{D}^b(\Lambda)\ar@{->>}[r] &
\mathcal{D}^b(\Lambda)/\mathbb{S}_2\ar@{^(->}[r] & \mathcal{C}_\Lambda}$$
We set $\overline{\Lambda}:={\sf End }_\mathcal{C}(\pi \Lambda)=\bigoplus_{p\in \mathbb{Z}}{\sf Hom }_\mathcal{D}(\Lambda,\mathbb{S}_2^{-p}\Lambda).$ By definition this algebra is naturally endowed with a $\mathbb{Z}$-grading.
If $\Lambda=kQ$ is the path algebra of an acyclic quiver, then the cluster category $\mathcal{C}_\Lambda=\mathcal{C}_Q$ has been introduced in \cite{BMRRT}. We call it \emph{acyclic cluster category} in this paper.
\begin{dfa}
A finite dimensional $k$-algebra $\Lambda$ of global dimension $\leq 2$ is said to be \emph{$\tau_2$-finite} if the algebra ${\sf End }_\mathcal{C}(\pi \Lambda)=\bigoplus_{p\in \mathbb{Z}}{\sf Hom }_\mathcal{D}(\Lambda,\mathbb{S}_2^{-p}\Lambda)$ is finite dimensional.
\end{dfa}
\begin{thma}[{\cite[Theorem~4.10]{Ami09}}]\label{clustertilting}
Let $\Lambda$ be a finite dimensional algebra of global dimension
$\leq 2$ which is $\tau_2$-finite. Then $\mathcal{C}_\Lambda$ is a ${\sf Hom }$-finite,
$2$-Calabi-Yau category and the object $\pi(\Lambda)\in \mathcal{C}_\Lambda$ is a cluster-tilting object.
\end{thma}
\subsection{Jacobian algebras and 2-Calabi-Yau-tilted algebras}
Quivers with potentials and the associated Jacobian algebras have been studied in \cite{DWZ}.
Let $Q$ be a finite quiver. For each arrow $a$ in $Q$, the \emph{cyclic derivative} $\partial_a$ with
respect to $a$ is the unique linear map
$$\partial_a \colon kQ\rightarrow kQ$$
which sends a path $p$ to the sum $\sum_{p=uav}vu$ taken over all decompositions of the
path $p$ (where $u$ and $v$ are possibly idempotent elements $e_i$ associated to a vertex $i$).
A \emph{potential} on $Q$ is any (possibly infinite) linear combination $W$ of cycles
in $Q$. The associated Jacobian algebra is
$${\sf Jac}(Q,W):=k\hat{Q}/\langle \partial_aW\mid a\in Q_1\rangle,$$
where $k\hat{Q}$ is the completed path algebra (that is the completion of the path algebra $kQ$ at the ideal generated by the arrows of $Q$) and $\langle\partial_aW\mid a\in Q_1\rangle$ is the closure of the ideal generated by $\partial_a W$ for $a\in Q_1$.
Associated with any quiver with potential $(Q,W)$, a cluster category $\mathcal{C}(Q,W)$ is constructed in \cite{Ami09}. This construction uses the notion of Ginzburg dg algebras. We refer the reader to \cite{Ami09} for explicit details. When the associated Jacobian algebra is finite dimensional, the category $\mathcal{C}(Q,W)$ is $2$-Calabi-Yau and endowed with a canonical cluster-tilting object $T_{(Q,W)}$ whose endomorphism algebra is isomorphic to ${\sf Jac}(Q,W)$. The next result gives a link between cluster categories associated with algebra of global dimension $\leq 2$ and cluster categories associated with quiver with potential.
\begin{thma}[{\cite[Theorem 6.12 \textit{a)}]{Kel09}}] \label{keller}
Let $\Lambda=kQ/I$ be a $\tau_2$-finite algebra of global dimension $\leq 2$, such that $I$
is generated by a finite minimal set of relations $\{ r_i \}$. (By this we mean that the set $\{r_i\}$ is the disjoint union of sets representing a basis of the ${\sf Ext }_{\Lambda}^2$-space between any two simple $\Lambda$-modules.) The relation
$r_i$ starts at the vertex $s(r_i)$ and ends at the vertex
$t(r_i)$.
Then there is a triangle equivalence:
$$\mathcal{C}_{\Lambda}\cong \mathcal{C}(\overline{Q},W),$$ where the quiver $\overline{Q}$ is the quiver $Q$ with additional
arrows $a_i \colon t(r_i)\rightarrow s(r_i)$, and the potential $W$ is
$\sum_i a_ir_i$. This equivalence sends the cluster-tilting object $\pi(\Lambda)$ on the cluster-tilting object $T_{(\overline{Q},W)}$.
As a consequence we have an isomorphism of algebras:
$${\sf End }_\mathcal{C}(\pi \Lambda) \cong {\sf Jac}(\overline{Q},W).$$ \end{thma}
\begin{dfa}
A potential $W$ on a quiver $Q$ is said to be \emph{rigid} if any cycle $p$ of $Q$ is cyclically equivalent to an element in the Jacobian ideal $\langle \partial_aW\mid a\in Q_1\rangle$.
\end{dfa}
By \cite{DWZ} rigidity is stable under mutation, and it implies that the Gabriel quiver of the Jacobian algebra has no loops nor 2-cycles.
We end this section with two results linking the mutation of quivers with potential and mutation of cluster-tilting objects in cluster categories.
The first result links the mutation of cluster-tilting objects in the acyclic cluster category and the mutation of rigid quivers with potential defined in \cite{DWZ}.
Acyclic cluster categories are equivalent to stable categories of some Frobenius categories associated to a certain reduced expression in the corresponding Coxeter group \cite[Theorem II.3.4]{BIRSc}. By \cite[Corollary 6.7]{BIRSm} these categories are all \emph{liftable} (see \cite[Section 5]{BIRSm} for definition). Therefore \cite[Corollary 5.4(b)]{BIRSm} implies the following.
\begin{thma}[Buan-Iyama-Reiten-Smith] \label{birs}
Let $\Delta$ be an acyclic quiver, and $T$ be a basic cluster-tilting object in the cluster category $\mathcal{C}_{\Delta}$. Assume that there exists a quiver with rigid potential $(Q,W)$ with an isomorphism
\[ {\sf End }_{\mathcal{C}_{\Delta}}(T)\cong {\sf Jac}(Q,W). \]
Let $i$ be a vertex of $Q$ and denote by $T_i$ the indecomposable summand of $T\cong T_i\oplus T'$ corresponding to $i$. Then there is an isomorphism
\[ {\sf End }_{\mathcal{C}_{\Delta}}(\mu_{T_i}(T))\cong {\sf Jac}(\mu_i(Q,W)), \]
where $\mu_{T_i}(T)$ is defined in Theorem~\ref{IyamaYoshino} and where $\mu_i(Q,W)$ is the mutation at $i$ of $(Q,W)$ as defined in \cite{DWZ} (see also Subsection~\ref{subsection graded mutation} for definition).
\end{thma}
The second result \cite[Theorem 3.2]{KY11} gives, for two quivers with potential linked by a mutation, an equivalence between the associated cluster categories.
\begin{thma}[Keller-Yang]\label{kelleryang}
Let $(Q,W)$ be a quiver with rigid potential whose Jacobian algebra is finite dimensional, and $i\in Q_0$ a vertex. Then there exists a triangle equivalence $$\mathcal{C}(\mu_i(Q,W))\cong \mathcal{C} (Q,W)$$ sending the cluster-tilting object $T_{\mu_i(Q,W)}\in \mathcal{C}(\mu_i(Q,W))$ onto the cluster-tilting object $\mu_{T_i}(T_{(Q,W)})\in \mathcal{C}(Q,W)$, where $T_i$ is the indecomposable summand of $T_{(Q,W)}$ associated with the vertex $i$ of $Q$.
As a consequence we get an isomorphism of algebras
$${\sf End }_{\mathcal{C}(Q,W)}(\mu_{T_i}(T_{(Q,W)})\cong {\sf Jac} (\mu_i(Q,W)).$$
\end{thma}
For a sequence $s=(i_1,\ldots, i_r)$ of vertices of $Q$, we denote by $\mu_s$ the composition $\mu_{i_r}\circ\mu_{i_{r-1}}\circ\cdots\circ \mu_{i_1}$ and by $\mu_{s^-}$ the composition $\mu_{i_1}\circ\mu_{i_{2}}\circ\cdots\circ \mu_{i_r}$.
For a bijection between $\{1,\ldots, n\}$ and the indecomposable summands of a basic cluster-tilting object $T$, we obtain a natural bijection between $\{1,\ldots, n\}$ and the indecomposable summands of $\mu_i(T)$. Therefore, once such bijection is fixed, we will also use the notations $\mu_s$ and $\mu_{s^-}$ for mutating cluster-tilting objects.
\subsection{Classical results on acyclic cluster categories}
In this paper we are interested in algebras of global dimension $\leq 2$ whose generalized cluster category is equivalent to the cluster category associated to an acyclic quiver. Since we will strongly make use of them, we now recall some results for acyclic cluster categories.
The following theorem follows from a result by Happel and Unger \cite{HU03}.
\begin{thma}[{\cite[Prop. 3.5]{BMRRT}} -- see also \cite{Hub08}] \label{connectivity}
Let $Q$ be an acyclic quiver, and let $T$ be a cluster-tilting object of $\mathcal{C}_Q$. Then there exists a sequence of mutations linking the cluster-tilting object $T$ to the canonical cluster-tilting object $\pi_Q(kQ)$. In other words, the cluster-tilting graph of the acyclic cluster category is connected.
\end{thma}
A consequence of this theorem together with Theorem~\ref{birs} is that the endomorphism algebra of a cluster-tilting object in an acyclic cluster category is always a Jacobian algebra of a rigid QP.
Another special feature of acyclic cluster categories is that they can be recognized by the quivers of their cluster-tilting objects.
\begin{thma}[\cite{KR08}] \label{recognition}
Let $\mathcal{C}$ be an algebraic triangulated category, which is ${\sf Hom }$-finite and $2$-Calabi-Yau. If there exists a cluster-tilting object $T\in\mathcal{C}$ such that ${\sf End }_\mathcal{C}(T)\cong kQ$, where $Q$ is an acyclic quiver, then there exists a triangle equivalence $\mathcal{C}\cong \mathcal{C}_Q$ sending $T$ onto $\pi_Q(kQ)$.
As a consequence there is a triangle equivalence $\mathcal{C}(Q,0)\to \mathcal{C}_Q$ sending $T_{(Q,0)}$ onto $\pi_Q(kQ)$.
\end{thma}
Note that an analogue of these results is not known for generalized cluster categories.
An algebra of the form ${\sf End }_{\mathcal{C}_Q}(T')$, where $T'$ is a cluster-tilting object in $\mathcal{C}_Q$, is called a \emph{cluster-tilted algebra of type Q}.
\section{Derived equivalent algebras of tree cluster type}\label{section tree case}
In this section we investigate algebras of global dimension at most 2 whose generalized cluster category is a cluster category associated to a tree. We will strongly use results from~\cite{AO10}.
We start with a definition.
\begin{dfa}(\cite[Def. 5.1]{AO10})
Two algebras $\Lambda_1$ and $\Lambda_2$ of global dimension $\leq 2$ which are $\tau_2$-finite are said to be \emph{cluster equivalent} if there exists a triangle equivalence $\mathcal{C}_{\Lambda_1}\rightarrow \mathcal{C}_{\Lambda_2}$ between their generalized cluster categories.
If $\Lambda$ is cluster equivalent to $kQ$ where $Q$ is an acyclic quiver, we will say that $\Lambda$ is of \emph{cluster type $Q$}.
\end{dfa}
Two derived equivalent algebras of global dimension $\leq 2$ are cluster equivalent (\cite[Cor.~7.16]{AO10}). Hence, if the underlying graph of $Q$ is a tree, then the class of algebras of cluster type $Q$ does not depend on the orientation of $Q$.
From the results of Section~\ref{section background} we deduce the following characterization of algebras of acyclic cluster type.
\begin{cora}\label{recognitioncor}
Let $\Lambda=kQ/I$ be a $\tau_2$-finite algebra of global dimension $\leq 2$. Let $(\overline{Q},W)$ the associated quiver with potential defined in Theorem~\ref{keller}. Then $\Lambda$ is of acyclic cluster type $\Delta$ if and only if there exists a sequence of mutation $s=i_1,\ldots, i_l$ such that $(\Delta,0)=\mu_s(\overline{Q},W)$.
In this case, $(\overline{Q},W)$ is a rigid quiver with potential, and there exists a triangle equivalence $f:\mathcal{C}_{\Lambda}\to\mathcal{C}_\Delta$ sending $\pi_\Lambda(\Lambda)$ to $\mu_{s^-}(\pi_\Delta(k\Delta))$.
\end{cora}
\begin{proof}
By Theorems \ref{keller}, \ref{kelleryang}, and \ref{recognition}, if $(\Delta,0)=\mu_s(\overline{Q},W)$, then we have equivalences
$$\mathcal{C}_{\Lambda}\cong \mathcal{C}(\overline{Q},W)\cong \mathcal{C}(\Delta,0)\cong \mathcal{C}_\Delta$$ sending $\pi_\Lambda(\Lambda)$ to $\mu_{s^-}(\pi_\Delta(k\Delta))$.
Conversely assume that there exists an equivalence $f:\mathcal{C}_{\Lambda}\cong \mathcal{C}_\Delta$. Then by Theorem~\ref{connectivity} there exists a sequence of mutations $s$ such that $\pi_\Delta(k\Delta)\cong \mu_s f(\pi_\Lambda(\Lambda))$. So by Theorem~\ref{birs} we have ${\sf Jac}(\mu_{s}(\overline{Q},W))\cong {\sf Jac}(\Delta,0)=k\Delta$. Since the quiver with potential $\mu_s(\overline{Q},W)$ is reduced, we necessarily have $\mu_s(\overline{Q},W)=(\Delta,0)$.
\end{proof}
\subsection{Graded equivalence and derived equivalence}
Cluster equivalence is strongly related with graded equivalence. In this subsection, we will recall some results shown in \cite{AO10}.
Let $A=\bigoplus_{p\in \mathbb{Z}}A^p$ be $\mathbb{Z}$-graded algebra. We denote by $d$ the degree map sending any homogeneous element of $A$ to its degree, and by ${\sf gr \hspace{.02in} } A$ the category of finite generated graded modules over $A$. For a graded module $M=\bigoplus_{p\in\mathbb{Z}}M^p$, we denote by $M\langle q\rangle$ the graded module $\bigoplus_{p\in\mathbb{Z}}M^{p+q}$ (that is, the degree $p$ part of $M\langle q \rangle$ is $M^{p+q}$). The locally bounded subcategory
\[\cov{A}{d}:={\sf add \hspace{.02in} }\{A\langle p\rangle \mid p\in\mathbb{Z}\} \subseteq {\sf gr \hspace{.02in} } A\]
is called the \emph{$\mathbb{Z}$-covering} of $A$.
\begin{dfa}
Let $A_1$ and $A_2$ be two $\mathbb{Z}$-graded algebras. Assume that $A_1$ and $A_2$ are isomorphic as algebras. We will say that $A_1$ and $A_2$ are \emph{graded equivalent} (and write $A_1\underset{{\sf gr \hspace{.02in} }}{\sim} A_2$) if there exist $r_i\in \mathbb{Z}$ and an isomorphism of $\mathbb{Z}$-graded algebras \[ \xymatrix{A_2\ar[r]^-\sim_-\mathbb{Z} & \bigoplus_{p\in \mathbb{Z}}{\sf Hom }_{\cov{A_1}{\mathbb{Z}}}(\bigoplus_{i=1}^nP_i\langle r_i\rangle, \bigoplus_{i=1}^nP_i\langle r_i+p\rangle)}\]
where $A_1=\bigoplus_{i=1}^nP_i$ is a decomposition of $A_1$ into indecomposable graded projective modules. This is equivalent to the fact that the coverings $\cov{A_1}{d_1}$ and $\cov{A_2}{d_2}$ are equivalent as categories with $\mathbb{Z}$-action.
\end{dfa}
The link between cluster equivalent algebras and graded equivalent algebras is given by the following result.
\begin{thma}[{\cite[Thm 5.8]{AO10}}]\label{AO_derivedeq_gradedeq_thm}
Let $\Lambda_1$ and $\Lambda_2$ be two $\tau_2$-finite algebras of global dimension $\leq 2$. For $i=1,2$, denote by $\mathcal{D}_i$ the bounded derived category of $\Lambda_i$, by $\mathcal{C}_i$ its cluster category, and by $\pi_i$ the natural functor $\mathcal{D}_i \to \mathcal{C}_i$. Assume there is $T \in \mathcal{D}_1$ such that $\pi_1(T)$ is basic cluster-tilting in $\mathcal{C}_1$, and
\begin{enumerate}
\item there is an isomorphism $\xymatrix{{\sf End }_{\mathcal{C}_1}(\pi_1 T)\ar[r]^-\sim & {\sf End }_{\mathcal{C}_2}(\pi_2\Lambda_2)}$
\item this isomorphism can be chosen in such a way that the two $\mathbb{Z}$-gradings defined on $\overline{\Lambda}_2$, given respectively by
$$\bigoplus_{q\in \mathbb{Z}}{\sf Hom }_{\mathcal{D}_2}(\Lambda_2,\mathbb{S}_2^{-q}\Lambda_2)\textrm{ and } \bigoplus_{p\in \mathbb{Z}}{\sf Hom }_{\mathcal{D}_1}(T, \mathbb{S}_2^{-p}T),$$
are equivalent.
\end{enumerate}
Then the algebras $\Lambda_1$ and $\Lambda_2$ are derived equivalent, and hence cluster equivalent.
\end{thma}
Note that the functor $\mathbb{S}_2^{-1}$ acts on the subcategory $\pi_1^{-1}\pi_1(T)={\sf add \hspace{.02in} }\{\mathbb{S}_2^{-q}T, q\in \mathbb{Z}\}\subset \mathcal{D}^b(\Lambda_1)$. As a category with $\mathbb{Z}$-action, it is equivalent to the $\mathbb{Z}$-covering of the graded algebra ${\sf End }_{\mathcal{C}_1}(\pi_1 T)$. Therefore, by the above remark, the graded algebras ${\sf End }_{\mathcal{C}_1}(\pi_1 T)$ and ${\sf End }_{\mathcal{C}_2}(\pi_2 \Lambda_2)$ are graded equivalent if and only if there is an equivalence between the additive categories $\pi_1^{-1}\pi_1(T)$ and $\pi_2^{-1}\pi_2(\Lambda_2)$ as categories with $\mathbb{S}_2$-action.
\subsection{Mutation of a graded quiver with potential}\label{subsection graded mutation}
In order to apply Theorem~\ref{AO_derivedeq_gradedeq_thm} we make use of a tool: the left (or right) mutation of a graded quiver with potential which extends the Derksen-Weyman-Zelevinsky mutation of a quiver with potential \cite{DWZ}. All definitions and results described in this subsection are proved in \cite[Section 6]{AO10}.
\begin{dfa}[{\cite{AO10}}]
Let $(Q,W,d)$ be a $\mathbb{Z}$-graded quiver with potential homogeneous
of degree 1 (graded QP for short). Let $i\in Q_0$ be a vertex, such that there are neither loops
nor 2-cycles incident to $i$. We define $\mu_i^{{\rm L}}(Q,W,d)$, the \emph{left graded mutation of $(Q,W,d)$ at vertex $i$}, as the reduction of the graded QP $(Q',W',d')$. The quiver $Q'$ is defined as in
\cite{DWZ} as follows:
\begin{itemize}
\item for any subquiver $\xymatrix{u\ar[r]^a &i\ar[r]^b & v}$, with
$i$, $u$, and $v$ pairwise distinct vertices, we add an arrow
$[ba]\colon u\rightarrow v$;
\item we replace all arrows $a$ incident with $i$ by an arrow $a^*$ in
the opposite direction.
\end{itemize}
The potential $W'$ is also defined as in \cite{DWZ} by the sum $[W]+W^*$ where $[W]$ is formed
from the
potential $W$ replacing all compositions $ba$ through the vertex $i$
by $[ba]$, where $W^*$ is the sum $\sum a^* b^* [ba]$.
The new degree $d'$ is defined as follows:
\begin{itemize}
\item $d'(a)=d(a)$ if $a$ is an arrow of $Q$ and $Q'$;
\item $d'([ba])=d(b)+d(a)$ if $ba$ is a composition passing through
$i$;
\item $d'(a^*)=-d(a)+1$ if the target of $a$ is $i$;
\item $d'(b^*)=-d(b)$ if the source of $b$ is $i$.
\end{itemize}
\end{dfa}
One can check that this operation is compatible with the reduction of a QP (see \cite{AO10}). It is possible to define the right graded mutation $\mu_i^{{\rm R}}$ by interchanging target and source in the last two items in the definition. We have the following.
\begin{lema}[\cite{AO10}] \label{leftrightmutation}
Let $(Q,W,d)$ be a graded quiver with potential. Then we have an isomorphism of $\mathbb{Z}$-graded algebras \[{\sf Jac}(Q,W,d)\underset{\mathbb{Z}}{\cong}{\sf Jac}(\mu_i^{{\rm R}}\mu_i^{{\rm L}}(Q,W,d)).\]
\end{lema}
Moreover, this mutation preserves graded equivalence. More precisely, we have the following.
\begin{prop}[\cite{AO10}]\label{prop_gradedeq_mutation}
Let $(Q,W,d_1)$ and $(Q,W,d_2)$ be two $\mathbb{Z}$-graded QP such that the graded Jacobian algebras ${\sf Jac}(Q,W,d_1)$ and ${\sf Jac}(Q,W,d_2)$ are graded equivalent. Then the graded Jacobian algebras ${\sf Jac}(\mu_i^{{\rm L}}(Q,W,d_1))$ and ${\sf Jac}(\mu_i^{{\rm L}}(Q,W,d_2))$ are graded equivalent.
\end{prop}
\begin{dfa}[\cite{AO10}]
Let $\Lambda$ be a $\tau_2$-finite algebra of global dimension $\leq2$. Let $T=T_1 \oplus\cdots \oplus T_n$ be an object in $\mathcal{D}^b(\Lambda)$ such that $\pi(T)$ is a (basic) cluster-tilting object in $\mathcal{C}_\Lambda$. Let $T_i$ be an indecomposable summand of $T\cong T'\oplus T_i$. Define $T_i^{{\rm L}}$ as the cone in $\mathcal{D}^b(\Lambda)$ of the minimal left ${\sf add \hspace{.02in} }\{\mathbb{S}_2^p T',p\in\mathbb{Z}\}$-approximation $u\colon T_i\rightarrow B$ of $T_i$. We denote by $\mu_i^{{\rm L}}(T)$ the object $T_i^{{\rm L}}\oplus T'$ and call it the \emph{left mutation of $T$ at $T_i$}.
\end{dfa}
It is immediate to check that we have $\pi(\mu_i^{{\rm L}}(T))=\mu_i(\pi(T))$, thus $\pi(\mu_i^{{\rm L}}(T))$ is a cluster-tilting object in $\mathcal{C}_\Lambda$.
This (left) mutation in the derived category is reflected by the graded (left) mutation of graded QP in the following sense.
\begin{thma} \label{Zgradedbirs}
Let $\Lambda$ be an algebra of acyclic cluster type and $T\in\mathcal{D}^b(\Lambda)$ as above. Assume that there
exist a $\mathbb{Z}$-graded QP $(Q,W,d)$ with rigid potential homogeneous of degree 1 such that we have an
isomorphism of graded algebras
$$\xymatrix{\bigoplus_{p\in\mathbb{Z}}{\sf Hom }_{\mathcal{D}}(T,\mathbb{S}_2^{-p} T)\ar[r]^-\sim_-{\mathbb{Z}} & {\sf Jac}(Q,W,d).}$$
Let $i\in Q_0$ and $T_i$ be the associated indecomposable summand of $T\cong T_i\oplus T'$ .
Then there is an isomorphism of $\mathbb{Z}$-graded algebras
$$\xymatrix{\bigoplus_{p\in
\mathbb{Z}}{\sf Hom }_\mathcal{D}(T'\oplus T^{{\rm L}}_i,\mathbb{S}_2^{-p}(T'\oplus T^{{\rm L}}_i))\ar[r]^-\sim_-{\mathbb{Z}}& {\sf Jac}(\mu_i^{{\rm L}}(Q,W,d)).}$$
\end{thma}
\begin{proof} By Theorem \ref{birs}, we already have an isomorphism between the algebras
$$\bigoplus_{p\in
\mathbb{Z}}{\sf Hom }_\mathcal{D}(T'\oplus T^{{\rm L}}_i,\mathbb{S}_2^{-p}(T'\oplus T^{{\rm L}}_i))\cong {\sf End }_{\mathcal{C}_\Lambda}(\mu_{\pi(T_i)}(\pi T))$$ and ${\sf Jac}(\mu_i^{{\rm L}}(Q,W))$ forgetting the grading. The only thing to check is that it is compatible with the grading. The proof of this fact is the same as the one of Theorem 6.10 in \cite{AO10}.\end{proof}
\begin{dfa}\label{defWgrading}
Let $(Q,W)$ be a reduced quiver with potential. A grading $d$ on $Q$ will be said to be a \emph{$W$-grading} if
\begin{itemize}
\item for all arrows $a$ in $Q_1$, $d(a)\in \{0,1\}$;
\item the potential $W$ is homogeneous of degree $1$;
\item the set of relations $\{\partial_aW \mid d(a)=1\}$ is linearly independent (in particular for all $a\in Q_1$ such that $d(a)=1$, we have $\partial_aW\neq 0$).
\end{itemize}
\end{dfa}
Theorem~\ref{Zgradedbirs} is particularly useful with the following result.
\begin{prop}[\cite{AO10}] \label{propWgrading}
Let $\Lambda=kQ_{\Lambda}/I$ be an algebra of global dimension $\leq 2$ which is $\tau_2$-finite and of acyclic cluster type. Then there exists a rigid quiver with potential $(\overline{Q},W)$ and a $W$-grading $d$ such that we have an isomorphism of graded algebras \[\overline{\Lambda}\underset{\mathbb{Z}}{\cong} {\sf Jac}(\overline{Q},W,d).\]
\end{prop}
The quiver with potential is given by Theorem~\ref{keller}; the arrows of $Q_{\Lambda}$ have
degree zero and the arrows $a_i$ corresponding to the relations have
degree 1. The rigidity comes from Corollary \ref{recognitioncor}.
\subsection{Application to acyclic cluster type}
Let $\Lambda$ be an algebra of cluster type $Q$, where $Q$ is an acyclic quiver. By Corollary \ref{recognitioncor} there exists a sequence of mutations $s$ such that $\mu_s(Q_{\overline{\Lambda}}, W)=(Q,0)$, where $(Q_{\overline{\Lambda}},W)$ is the quiver with potential associated with $\Lambda$ in Theorem \ref{keller}.
\begin{dfa} The map $d_s\colon Q\rightarrow \mathbb{Z}$ is called \emph{a grading induced by $\Lambda$} if it satisfies $$\mu^{{\rm L}}_s(Q_{\overline{\Lambda}},W,d)=(Q,0,d_s),$$
where $s$ is a sequence of mutations as in Corollary \ref{recognitioncor}.
\end{dfa}
\begin{prop}\label{uniqueness_induced_grading}
Let $d_s$ and $d_t$ be two gradings induced by $\Lambda$ on $Q$. If $\mu_s(\pi_{\Lambda} (\Lambda)) \cong \mu_t(\pi_{\Lambda} (\Lambda))$ then $d_s$ and $d_t$ are equivalent up to automorphism of $Q$.
\end{prop}
\begin{proof}
Let
$\mu^{{\rm L}}_s \Lambda \cong T_1 \oplus \cdots \oplus T_n$
be a decomposition of $\mu^{{\rm L}}_s \Lambda$ into indecomposable summands. Then we have
\[ \mu^{{\rm L}}_t \Lambda \cong \mathbb{S}_2^{r_1} T_1 \oplus \cdots \oplus \mathbb{S}_2^{r_n} T_n \]
for certain $r_i \in \mathbb{Z}$, since $\pi_{\Lambda} \mu^{{\rm L}}_s \Lambda = \mu_s \pi_{\Lambda} \Lambda \cong \mu_t \pi_{\Lambda} \Lambda = \pi_{\Lambda} \mu^{{\rm L}}_t \Lambda$. It follows that the algebras
\[ \bigoplus_{p \in \mathbb{Z}} {\sf Hom }_{\mathcal{D}^b(\Lambda)}(\mu^{{\rm L}}_s \Lambda, \mathbb{S}_2^{-p} \mu^{{\rm L}}_s \Lambda) \qquad \text{and} \qquad \bigoplus_{p \in \mathbb{Z}} {\sf Hom }_{\mathcal{D}^b(\Lambda)}(\mu^{{\rm L}}_t \Lambda, \mathbb{S}_2^{-p} \mu^{{\rm L}}_t \Lambda) \]
are graded equivalent. Since, by Theorem~\ref{Zgradedbirs}, these graded algebras are isomorphic to ${\sf Jac}(Q, 0, d_s)$ and ${\sf Jac}(Q, 0, d_t)$, respectively, it follows that the gradings $d_s$ and $d_t$ are equivalent up to automorphisms of $Q$.
\end{proof}
\begin{rema}\label{counter example}
The induced grading depends in general on the choice of a cluster-tilting object in $\mathcal{C}_{\Lambda}$ with hereditary endomorphism ring, as shown in the following example. Let $\Lambda$ be the algebra presented by the quiver
\[ Q =
\begin{tikzpicture}[baseline=13pt]
\node (A) at (0,0) {$1$};
\node (B) at (1,1) {$2$};
\node (C) at (2,0) {$3$};
\node (D) at (3,1) {$4$};
\node (E) at (4,0) {$5$};
\draw [->] (A) -- node [above left=-2pt] {$\scriptstyle \alpha$} (B);
\draw [->] (B) -- node [above right=-2pt] {$\scriptstyle \beta$} (C);
\draw [->] (C) -- (A);
\draw [->] (D) -- (C);
\draw [->] (E) -- (C);
\draw [->] (E) -- (D);
\end{tikzpicture}
\text{ with relation } \beta \alpha = 0. \]
Adding an arrow of degree $1$ for the relation, one obtains a graded quiver with potential $(\overline{Q}, W, d)$. Then it is easy to check that $\mu^{{\rm L}}_2 (\overline{Q}, W, d)$ is given by the graded quiver
\[ \begin{tikzpicture}
\node (A) at (0,0) {$1$};
\node (B) at (1,1) {$2$};
\node (C) at (2,0) {$3$};
\node (D) at (3,1) {$4$};
\node (E) at (4,0) {$5$};
\draw [->] (B) -- node [fill=white, inner sep=0pt] {$\scriptstyle 1$} (A);
\draw [->] (C) -- node [fill=white, inner sep=0pt] {$\scriptstyle 0$} (B);
\draw [->] (C) -- node [fill=white, inner sep=0pt] {$\scriptstyle 0$} (A);
\draw [->] (D) -- node [fill=white, inner sep=0pt] {$\scriptstyle 0$} (C);
\draw [->] (E) -- node [fill=white, inner sep=0pt] {$\scriptstyle 0$} (C);
\draw [->] (E) -- node [fill=white, inner sep=0pt] {$\scriptstyle 0$} (D);
\end{tikzpicture} \]
On the other hand $\mu^{{\rm L}}_2 \mu^{{\rm L}}_1 \mu^{{\rm L}}_4 \mu^{{\rm L}}_5 \mu^{{\rm L}}_2 (\overline{Q}, W, d)$ is given by the graded quiver
\[ \begin{tikzpicture}[baseline]
\node (A) at (0,0) {$4$};
\node (B) at (1,1) {$5$};
\node (C) at (2,0) {$3$};
\node (D) at (3,1) {$1$};
\node (E) at (4,0) {$2$};
\draw [->] (B) -- node [fill=white, inner sep=0pt] {$\scriptstyle 0$} (A);
\draw [->] (C) -- node [fill=white, inner sep=0pt] {$\scriptstyle 0$} (B);
\draw [->] (C) -- node [fill=white, inner sep=0pt] {$\scriptstyle 0$} (A);
\draw [->] (D) -- node [fill=white, inner sep=0pt] {$\scriptstyle 1$} (C);
\draw [->] (E) -- node [fill=white, inner sep=0pt] {$\scriptstyle 1$} (C);
\draw [->] (E) -- node [fill=white, inner sep=0pt] {$\scriptstyle 1$} (D);
\end{tikzpicture} \]
These gradings are not equivalent.
\end{rema}
Using the previous results we can prove the following.
\begin{thma}\label{thm_gradingQ_derivedeq}
Let $\Lambda_1$ and $\Lambda_2$ be two algebras of cluster type $Q$, where $Q$ is an acyclic quiver. Then the following are equivalent.
\begin{enumerate}
\item There exist equivalent gradings $d_{s_i}$ induced by $\Lambda_i$ on $Q$ for $i \in \{1,2\}$.
\item There exists a derived equivalence $\mathcal{D}^b(\Lambda_1)\rightarrow \mathcal{D}^b(\Lambda_2)$.
\end{enumerate}
\end{thma}
\begin{proof}
For $i=1,2$ denote by $(Q_i,W_i,\partial_i)$ the graded QP associated to $\overline{\Lambda}_i$.
$(1)\Rightarrow (2)$ By assumption, we have a graded equivalence \[{\sf Jac}(\mu_{s_1}^{{\rm L}}(Q_1,W_1,\partial_1))\underset{{\sf gr \hspace{.02in} }}{\sim}{\sf Jac}(\mu_{s_2}^{{\rm L}}(Q_2,W_2,\partial_2)).\] Then by Proposition~\ref{prop_gradedeq_mutation} and Lemma~\ref{leftrightmutation}, we immediately get that \[{\sf Jac}(\mu_{s_2^-}^{{\rm R}}\mu_{s_1}^{{\rm L}}(Q_1,W_1,\partial_1))\underset{{\sf gr \hspace{.02in} }}{\sim} {\sf Jac}(Q_2,W_2,\partial_2).\]
Now denote by $T\in \mathcal{D}^b(\Lambda_1)$ the object $\mu_{s_2^-}^{{\rm R}}\mu_{s_1}^{{\rm L}}(\Lambda_1)$. By Theorem~\ref{Zgradedbirs} we have an isomorphism of $\mathbb{Z}$-graded algebras
\[ \bigoplus_{p\in\mathbb{Z}}{\sf Hom }_{\mathcal{D}^b(\Lambda_1)}(T,\mathbb{S}_2^{-p}T)\underset{\mathbb{Z}}{\cong} {\sf Jac}(\mu_{s_2^-}^{{\rm R}}\mu_{s_1}^{{\rm L}}(Q_1,W_1,\partial_1))\]
Therefore, we get the result by Theorem~\ref{AO_derivedeq_gradedeq_thm}.
$(2)\Rightarrow (1)$ Assume that $\Lambda_1$ and $\Lambda_2$ are derived equivalent. Then there exists a tilting complex $T$ in $\mathcal{D}^b(\Lambda_2)$ with ${\sf End }_{\mathcal{D}^b(\Lambda_2)}(T)\cong \Lambda_1$. The derived equivalence induced by $T$ gives rise to a commutative diagram \[\xymatrix{\mathcal{D}^b(\Lambda_1)\ar[d]_{\pi_1}\ar[rr]^{-\overset{\boldmath{L}}{\ten}_{\Lambda_1}T}_\sim && \mathcal{D}^b(\Lambda_2)\ar[d]^{\pi_2}\\ \mathcal{C}_{\Lambda_1}\ar[rr]_f^{\sim}&& \mathcal{C}_{\Lambda_2}}\]
Since the cluster category $\mathcal{C}_{\Lambda_2}$ is acyclic, there exists a sequence of mutations $s$ such that $\mu_s(\pi_2\Lambda_2)=\pi_2(T)=f(\pi_1\Lambda_1).$ Denote by $T'\in \mathcal{D}^b(\Lambda_2)$ the object $T':=\mu_s^{{\rm L}}(\Lambda_2).$ Then we have $\pi_2(T)=\pi_2(T')$, thus we have an equivalence of categories commuting with $\mathbb{S}_2$
\[{\sf add \hspace{.02in} }\{\mathbb{S}_2^pT \mid p\in\mathbb{Z}\}\cong {\sf add \hspace{.02in} }\{\mathbb{S}_2^p T'\mid p\in\mathbb{Z}\}.\] This exactly means that there is a graded equivalence
\[\bigoplus_{p\in\mathbb{Z}}{\sf Hom }_{\mathcal{D}^b(\Lambda_2)}(T,\mathbb{S}_2^{-p}T)\underset{{\sf gr \hspace{.02in} }}{\sim}\bigoplus_{p\in\mathbb{Z}}{\sf Hom }_{\mathcal{D}^b(\Lambda_2)}(T',\mathbb{S}_2^{-p}T').\]
Since $-\overset{\boldmath{L}}{\ten}_{\Lambda_1}T$ is an equivalence, the left term is isomorphic as $\mathbb{Z}$-graded algebra to ${\sf Jac}(Q_1,W_1,\partial_1)$ and by Theorem~\ref{Zgradedbirs} the right term is isomorphic to the $\mathbb{Z}$-graded algebra ${\sf Jac}(\mu_s^{{\rm L}}(Q_2,W_2, \partial_2))$. Therefore we have \begin{equation}\label{eqiso} {\sf Jac}(\mu_s^{{\rm L}}(Q_2,W_2,\partial_2))\underset{{\sf gr \hspace{.02in} }}{\sim} {\sf Jac}(Q_1,W_1,\partial_1).\end{equation}
Now let $s'$ be a sequence such that the cluster-tilting object $T'':=\mu_{s'}(f\pi_1\Lambda_1)$ has endomorphism algebra isomorphic to $kQ$. Then we have \[\mu_{s'}(Q_1,W_1)=(Q,0)\textrm{ and } \mu_{s'}\mu_{s}(Q_2,W_2)=(Q,0).\]
Now let $d_1$ and $d_2$ be the gradings on $Q$ such that we have
\[\mu_{s'}^{{\rm L}}(Q_1,W_1,\partial_1)=(Q,0,d_1)\textrm{ and } \mu^{{\rm L}}_{s'}\mu^{{\rm L}}_{s}(Q_2,W_2,\partial_2)=(Q,0,d_2).\] By \eqref{eqiso} and Proposition~\ref{prop_gradedeq_mutation}, we get
\[ {\sf Jac}(Q,0,d_1)\underset{{\sf gr \hspace{.02in} }}{\sim} {\sf Jac}(Q, 0,d_2), \]
that is the gradings $d_1$ and $d_2$ are equivalent.
\end{proof}
\begin{cora}\label{cor_tree}
If $Q$ is a tree, and if $\Lambda$ is of cluster type $Q$, then $\Lambda$ is derived equivalent to $kQ$.
\end{cora}
\begin{proof}
By Theorem~\ref{thm_gradingQ_derivedeq}, it is enough to observe that all gradings on $kQ$ are equivalent. We consider the map
\[ \begin{array}{rcl}\mathbb{Z}^{Q_0} & \rightarrow & \mathbb{Z}^{Q_1} \\ (r_i)_{i\in Q_0} & \mapsto & (r_{t(a)}-r_{s(a)})_{a\in Q_1}\end{array}. \]
Then the grading on $Q$ which are equivalent to the trivial grading are the image of this map. Moreover, using the fact that $Q$ is a tree, one easily sees that the map is surjective.
\end{proof}
\section{Derived equivalence classes of algebras of cluster type $\widetilde{A}_{p,q}$}\label{section derived atilde}
In this section we study the algebras of cluster type $\widetilde{A}_{p,q}$, i.e.\ all algebras which are cluster equivalent to the path algebra $H$ of the acyclic quiver $Q_H$:
\[\scalebox{.7}{
\begin{tikzpicture}[>=stealth,scale=1.6]
\node (P1) at (0,0) {$1$};
\node (P2) at (1,1) {$2$};
\node (P3) at (2,1) {$3$};
\node (P4) at (3,1){};
\node (P5) at (5.5,1){};
\node (P6) at (7,1) {$p$};
\node (P7) at (8,0) {$p+1$};
\node (P8) at (7,-1) {$p+2$};
\node (P9) at (5.5,-1) {};
\node (P10) at (2.5,-1) {};
\node (P11) at (1,-1) {$p+q$};
\draw [->] (P1) -- node [swap,yshift=3mm] {$a_1$}(P2);
\draw [->] (P2) -- node [swap,yshift=2mm] {$a_2$}(P3);
\draw [->] (P3) -- node [swap,yshift=2mm] {$a_3$}(P4);
\draw [->] (P5) --node [swap,yshift=2mm] {$a_{p-1}$} (P6);
\draw [->] (P6) -- node [swap,yshift=3mm] {$a_p$}(P7);
\draw [->] (P8) -- node [swap,yshift=-3mm,xshift=2mm] {$b_1$}(P7);
\draw [->] (P9) -- node [swap,yshift=-2.5mm] {$b_2$}(P8);
\draw [->] (P11) -- node [swap,yshift=-2.5mm] {$b_{q-1}$}(P10);
\draw [->] (P1) -- node [swap,yshift=-3mm] {$b_q$}(P11);
\draw [loosely dotted, thick] (P4) -- (P5);
\draw [loosely dotted, thick] (P9) --(P10);
\end{tikzpicture}}
\]
Since $Q_H$ is not a tree, these algebras need not be derived equivalent. In this section we introduce an invariant of an algebra of cluster type $Q_H$ which determines its class of derived equivalence.
\subsection{The weight of an algebra of cluster type $\widetilde{A}_{p,q}$}
\begin{dfa}
Let $d$ be a $\mathbb{Z}$-grading on $Q_H$. We define the weight of the grading $d$ by
\[w(H,d):=\sum_{i=1}^pd(a_i)-\sum_{i=1}^qd(b_i)\]
\end{dfa}
\begin{lema}\label{gradedeqandw}
Let $d_1$ and $d_2$ be two $\mathbb{Z}$-gradings on $H$. Then the following are equivalent:
\begin{enumerate}
\item $(H,d_1)$ and $(H,d_2)$ are graded equivalent;
\item $w(H,d_1)=w(H,d_2)$.
\end{enumerate}
\end{lema}
\begin{proof}
We consider the map
$$\begin{array}{rrcl}\phi\colon &\mathbb{Z}^{(Q_H)_0} & \rightarrow & \mathbb{Z}^{(Q_H)_1} \\& (r_i)_{i\in (Q_H)_0} & \mapsto & (r_{t(a)}-r_{s(a)})_{a\in (Q_H)_1}\end{array}.$$
The gradings $d_1$ and $d_2$ are equivalent if and only if $d_1-d_2$ is in the image of $\phi$.
It is straight-forward to check that the sequence of $\mathbb{Z}$-modules
\[ \mathbb{Z}^{(Q_H)_0} \overset{\phi}{\rightarrow} \mathbb{Z}^{(Q_H)_1} \overset{w}{\rightarrow} \mathbb{Z} \rightarrow 0 \]
is exact (using the fact that $Q$ is of type $\widetilde{A}$). Now the claim of the lemma follows.
\end{proof}
This result allows us to define the following.
\begin{dfa}
Let $\Lambda$ be an algebra of global dimension at most 2 and of cluster type $\widetilde{A}_{p,q}$. Let $s$ be a sequence of mutations such that the endomorphism ring of $\mu_s(\pi_{\Lambda}(\Lambda))$ is $kQ_H$ (such a sequence exists by Corollary \ref{recognitioncor}), and let $d_s$ be the corresponding grading on $Q_H$ induced by $\Lambda$. Then define the \emph{weight} of $\Lambda$ by
\[w(\Lambda,s):=w(d_s)=\sum_{i=1}^pd_s(a_i)-\sum_{i=1}^qd_s(b_i).\]
\end{dfa}
If $p>q$, the weight $w(\Lambda,s)$ is well defined since there is no automorphism of $Q_H$.
\subsection{The case $p>q$}
Before proving the main theorem, we prove the following technical result.
\begin{lema} \label{lem.w_well-def}
The weight $w(\Lambda,s)$ does not depend on the choice of the sequence of mutations $s$.
\end{lema}
\begin{proof}
Let $s$ and $t$ be two sequences of mutations such that the endomorphism rings of $\mu_s( \pi_{\Lambda}(\Lambda))$ and of $\mu_t(\pi_{\Lambda} (\Lambda))$ are both isomorphic to $kQ_H$. Then there is an automorphism of the translation quiver $\mathbb{Z} Q_H$, which forms a component of the Auslander-Reiten quiver of the common cluster category, mapping $\mu_s( \pi_{\Lambda}(\Lambda))$ to $\mu_t(\pi_{\Lambda} (\Lambda))$.
Since $p\neq q$, one can check that the automorphism group of the translation quiver $\mathbb{Z} Q_H$ is generated by 2 elements given by the slices
\begin{align*}
T_1 & =\tau^{-1}(e_1H\oplus\cdots\oplus e_pH)\oplus e_{p+q}H\oplus\cdots \oplus e_{p+1}H\text{, and} \\
T_2 & =e_2H\oplus\cdots\oplus e_{p+1}H\oplus \tau^{-1}(e_1H\oplus e_{p+q}H\oplus e_{p+q-1}H\oplus\cdots\oplus e_{p+2}H)\text{, respectively,}
\end{align*}
where $e_i$ is the primitive idempotent of $H=kQ$ associated to the vertex $i$ for any $i=1,\ldots, p+q$.
Then we have
$$T_1=\mu^{{\rm L}}_p\mu^{{\rm L}}_{p-1}\ldots\mu_1^{{\rm L}}(H) \text{ and } T_2=\mu_{p+2}^{{\rm L}}\mu_{p+3}^{{\rm L}}\ldots\mu_{p+q}^{{\rm L}}\mu_{1}^{{\rm L}}(H).$$
It is enough to check that if $d$ is a grading on $Q_H$, then the weights of the graded quivers $\mu^{{\rm L}}_p\mu^{{\rm L}}_{p-1}\ldots\mu_1^{{\rm L}}(Q_H,d)$ and $\mu_{p+2}^{{\rm L}}\mu_{p+3}^{{\rm L}}\ldots\mu_{p+q}^{{\rm L}}\mu_{1}^{{\rm L}}(Q_H,d)$ are equal to $w(d)$. We do this for $\mu^{{\rm L}}_p\mu^{{\rm L}}_{p-1}\ldots\mu_1^{{\rm L}}(Q_H,d)$, the other one is similar.
First note that in the sequence of mutations $\mu^{{\rm L}}_p\mu^{{\rm L}}_{p-1}\ldots\mu_1^{{\rm L}}$, we mutate at a source at each step. Therefore the left graded mutation consists in reversing the arrows and assigning the opposite of the degree. After the sequence of mutations, arrows $a_1,\ldots,a_{p-1}$ have been reversed twice, arrows $a_p$ and $b_q$ have been reversed once, and $b_1,\ldots, b_q$ have not been reversed. Hence the graded quiver $\mu^{{\rm L}}_p\mu^{{\rm L}}_{p-1}\ldots\mu_1^{{\rm L}}(Q_H,d)$ is the following:
\[\scalebox{.7}{
\begin{tikzpicture}[>=stealth,scale=1.6]
\node (P1) at (0,0) {$1$};
\node (P2) at (1,1) {$2$};
\node (P3) at (2,1) {$3$};
\node (P4) at (3,1){};
\node (P5) at (5.5,1){};
\node (P6) at (7,1) {$p$};
\node (P7) at (8,0) {$p+1$};
\node (P8) at (7,-1) {$p+2$};
\node (P9) at (5.5,-1) {};
\node (P10) at (2.5,-1) {};
\node (P11) at (1,-1) {$p+q$};
\draw [->] (P1) -- node [swap,xshift=-3mm,yshift=3mm] {$d(a_1)$}(P2);
\draw [->] (P2) -- node [swap,yshift=3mm] {$d(a_2)$}(P3);
\draw [->] (P3) -- node [swap,yshift=3mm] {$d(a_3)$}(P4);
\draw [->] (P5) --node [swap,yshift=3mm] {$d(a_{p-1})$} (P6);
\draw [<-] (P6) -- node [swap,xshift=5mm,yshift=3mm] {$-d(a_p)$}(P7);
\draw [->] (P8) -- node [swap,yshift=-3mm,xshift=3mm] {$d(b_1)$}(P7);
\draw [->] (P9) -- node [swap,yshift=-3mm] {$d(b_2)$}(P8);
\draw [->] (P11) -- node [swap,yshift=-3mm] {$d(b_{q-1})$}(P10);
\draw [<-] (P1) -- node [swap,yshift=-3mm,xshift=-5mm] {$-d(b_q)$}(P11);
\draw [loosely dotted, thick] (P4) -- (P5);
\draw [loosely dotted, thick] (P9) --(P10);
\end{tikzpicture}}
\]
Hence the weight of this grading is $(-d(b_q)+\sum_{i=1}^{p-1}d(a_i))-(\sum_{j=1}^{q-1}d(b_j)-d(a_p))=w(d).$
\end{proof}
This lemma together with Lemma \ref{gradedeqandw} shows that situation in Remark \ref{counter example} does not occur for a quiver of type $\widetilde{A}_{p,q}$.
Hence we may, in the sequel, refer to the weight $w(\Lambda)$ without specifying a sequence of mutations.
\begin{thma}\label{derivedeqiffwegal}
Let $\Lambda_1$ and $\Lambda_2$ be two algebras of cluster type $\widetilde{A}_{p,q}$ with $p>q$. Then there is a derived equivalence between $\Lambda_1$ and $\Lambda_2$ if and only if $w(\Lambda_1)=w(\Lambda_2)$.
\end{thma}
\begin{proof}
Let $\delta_1$ (resp.\ $\delta_2$) be a grading on $Q_H$ induced by $\Lambda_1$ (resp.\ $\Lambda_2$).
By Lemma~\ref{gradedeqandw}, $\delta_1$ and $\delta_2$ are equivalent if and only if the corresponding weights $w(\Lambda_1)$ and $w(\Lambda_2)$ coincide. Since, by Lemma~\ref{lem.w_well-def}, the weights are independent of the choice of a sequence of mutation, it follows that also the gradings $\delta_1$ and $\delta_2$ are independent of this choice up to graded equivalence. Now the claim follows from Theorem~\ref{thm_gradingQ_derivedeq}.
\end{proof}
\begin{cora}
Let $\Lambda$ be an algebra of cluster type $\widetilde{A}_{p,q}$. Then $\Lambda$ is piecewise hereditary if and only if $w(\Lambda)=0$.
\end{cora}
\begin{proof}
Let us treat the case where $p>q$.
The `if'-part of the assertion is a direct consequence of Theorem~\ref{derivedeqiffwegal} applied for $\Lambda_1=\Lambda$ and $\Lambda_2=H$.
The algebra $\Lambda$ is piecewise hereditary if and only if the derived category $\mathcal{D}^b(\Lambda)$ is equivalent to $\mathcal{D}^b(\mathcal{H})$ for some hereditary category $\mathcal{H}$. Therefore it implies that the generalized cluster category $\mathcal{C}_\Lambda$ is equivalent to the cluster category $\mathcal{C}_\mathcal{H}$. Since $\Lambda$ is of cluster type $\widetilde{A}_{p,q}$, we have $\mathcal{C}_\mathcal{H}\cong \mathcal{C}_{\widetilde{A}_{p,q}}$. Hence we get $\mathcal{D}^b(\mathcal{H})\cong \mathcal{D}^b(\widetilde{A}_{p,q})$. Therefore, by Theorem~\ref{derivedeqiffwegal}, we have $w(\Lambda)=0$.
The case $p=q$ is a direct consequence of the result below.
\end{proof}
\subsection{The case $p=q$}
In the case $p=q$ there is a unique non-trivial automorphism of $Q_H$ fixing the vertices $1$ and $p+1$ and interchanging the vertices $i$ and $2p+2-i$ for $i=2,\ldots,p$.
This automorphism induces a derived equivalence between algebras $\Lambda_1$ and $\Lambda_2$ of cluster type $\widetilde{A}_{p,p}$ such that $w(\Lambda_1)=-w(\Lambda_2)$. Therefore we obtain the following result, whose proof is the same as Theorem~\ref{derivedeqiffwegal}.
\begin{thma}\label{theorem case p=q}
Let $\Lambda_1$ and $\Lambda_2$ be two algebras of cluster type $\widetilde{A}_{p,p}$. Then there is a derived equivalence between $\Lambda_1$ and $\Lambda_2$ if and only if $|w(\Lambda_1)|=|w(\Lambda_2)|$.
\end{thma}
\section{The Auslander-Reiten quiver of the derived category of an algebra of cluster type $\widetilde{A}_{p,q}$}\label{section ARquiver}
Let $\Lambda$ be an algebra of cluster type $\widetilde{A}_{p,q}$ and of weight $w\neq 0$. In this section we compute the Auslander-Reiten quiver of the derived category $\mathcal{D}^b(\Lambda)$. In order to do that, we use some results of \cite[Section 8]{AO10}.
\subsection{Graded derived equivalence}
Let $\Lambda$ be an algebra of cluster type $\widetilde{A}_{p,q}$. Then, by Proposition~\ref{propWgrading}, there exists a graded quiver with reduced potential $(\overline{Q},W,d)$ such that we have an isomorphism of $\mathbb{Z}$-graded algebras $\overline{\Lambda}\underset{\mathbb{Z}}{\cong}{\sf Jac}(\overline{Q},W,d)$. Let $\partial$ be a grading induced by $\Lambda$ on $H$ where $Q_H$ is the following quiver
\[\scalebox{1}{
\begin{tikzpicture}[scale=1,>=stealth]
\node (P1) at (0,0) {$1$};
\node (P2) at (1,1) {$2$};
\node (P3) at (2,1) {$3$};
\node (P4) at (3,1){};
\node (P5) at (5.5,1){};
\node (P6) at (7,1) {$p$};
\node (P7) at (8,0) {$p+1$};
\node (P8) at (7,-1) {$p+2$};
\node (P9) at (5.5,-1) {};
\node (P10) at (2.5,-1) {};
\node (P11) at (1,-1) {$p+q$};
\draw [->] (P1) -- (P2);
\draw [->] (P2) -- (P3);
\draw [->] (P3) -- (P4);
\draw [->] (P5) -- (P6);
\draw [->] (P6) -- (P7);
\draw [->] (P8) -- (P7);
\draw [->] (P9) -- (P8);
\draw [->] (P11) -- (P10);
\draw [->] (P1) -- (P11);
\draw [loosely dotted, thick] (P4) -- (P5);
\draw [loosely dotted, thick] (P9) --(P10);
\end{tikzpicture}}.\]
That is, there exists a sequence of mutations such that $\mu_s(\overline{Q},W,d)=(Q_H,0,\partial)$. Now define the $\mathbb{Z}^2$-graded quiver with reduced potential $(Q',W',(d',\delta))$ by the following
\[ (Q',W',(d',\delta)):=\mu_{s^-}^{{\rm R}}(Q_H,0,(\partial,0)).\]
By Lemma~\ref{leftrightmutation}, we have an isomorphism of $\mathbb{Z}$-graded algebras ${\sf Jac}(Q',W',d')\underset{\mathbb{Z}}{\cong}{\sf Jac}(\overline{Q},W,d)$. Since the potentials $W$ and $W'$ are reduced, the quivers $\overline{Q}$ and $Q'$ are isomorphic. By definition of $\overline{Q}$ and $d$, the quiver $Q_\Lambda$ of the algebra $\Lambda$ is the subquiver of $\overline{Q}$ satisfying $(Q_\Lambda)_0=\overline{Q}_0$ and $(Q_{\Lambda})_1=\{a\in\overline{Q}_1\mid d(a)=0\}$. Moreover we have an isomorphism $\Lambda\cong kQ_\Lambda/\langle \partial_aW, a\in\overline{Q}_1, d(a)=1\rangle$. Since we have an isomorphism of $\mathbb{Z}$-graded algebras ${\sf Jac}(Q',W',d')\underset{\mathbb{Z}}{\cong}{\sf Jac}(\overline{Q},W,d)$, we get an isomorphism $\Lambda\cong kQ_\Lambda/\langle \partial_aW', a\in Q'_1, d'(a)=1\rangle$. The grading $\delta$, which is a grading on $Q'\cong \overline{Q}$, restricts on a grading on $Q_\Lambda$. The grading $\delta$ makes $W'$ homogeneous of degree $1$. Hence the relations $\partial_aW', a\in Q'_1$ with $d'(a)=1$ are homogeneous with respect to the degree $\delta$. Consequently $\delta$ yields a grading on the algebra $\Lambda$ that we still denote by $\delta$.
\medskip
Then we have the following direct consequence of \cite[Theorem~8.7]{AO10}.
\begin{thma}[\cite{AO10}] \label{AO_gradedderivedeq}
In the setup above, there exists a triangle equivalence \[\xymatrix{\mathcal{D}^b(\cov{\Lambda}{\delta})\ar[r]^-F_\sim & \mathcal{D}^b(\cov{H}{\partial})}\] Moreover we have an isomorphism of triangle functors $F\circ\langle 1\rangle_{\delta}\cong \mathbb{S}_2^{-1}\circ\langle -1\rangle_{\partial}\circ F$.
\end{thma}
\begin{rema}
Note that in this situation, the compatibility condition defined in \cite[Definition~8.5]{AO10} is automatically satisfied since $\overline{Q}$ is mutation acyclic and since $W$ is rigid. Indeed, two $\mathbb{Z}$-gradings on a quiver induce a $\mathbb{Z}^2$-grading on it. But in general two $\mathbb{Z}$-gradings on an algebra do not give rise to a $\mathbb{Z}^2$-grading on it.
\end{rema}
Theorem~\ref{AO_gradedderivedeq} implies the following result which will be very useful to compute explicitly the Auslander-Reiten quiver of $\mathcal{D}^b(\Lambda)$.
\begin{cora}\label{k-equivalence}
There exists a $k$-linear equivalence \[\mathcal{D}^b( \Lambda)\cong \mathcal{D}^b(\cov{H}{\partial}))/\mathbb{S}_2\langle1\rangle_\partial\]
\end{cora}
\begin{proof}
By Theorem~\ref{AO_gradedderivedeq} we deduce that there is a $k$-linear equivalence between the orbit categories \[ \mathcal{D}^b(\cov{\Lambda}{\delta})/\langle 1\rangle_\delta\cong \mathcal{D}^b(\cov{H}{\partial})/\mathbb{S}_2\langle1\rangle_\partial.\]
Now we can use the following result due to Keller
\begin{thma}[\cite{Kel05}]
Let $Q_H$ be an acyclic quiver and $\partial$ be a $\mathbb{Z}$-grading on $Q_H$. Let $F:=-\overset{\boldmath{L}}{\ten}_{\cov{H}{\partial}} X$ be an auto-equivalence of $\mathcal{D}^b(\cov{H}{\partial})$ for some object $X\in\mathcal{D}(\cov{H}{\partial})$. Assume that \begin{enumerate}
\item For each indecomposable $U\in\mod\cov{H}{\partial}$, the set $\{i\in\mathbb{Z}\mid F^iU\in\mod\cov{H}{\partial}\}$ is finite.
\item There exists $N\geq 0$ such that for each indecomposable $X\in\mathcal{D}^b(\cov{H}{\partial})$, there exists $0\leq n\leq N$ and $i\in\mathbb{Z}$ with $F^iX[-n]\in\mod\cov{H}{\partial}$.
\end{enumerate}
Then the orbit category $\mathcal{D}^b(\cov{H}{\partial})/F$ admits a natural triangulated structure such that the projection functor $\mathcal{D}^b(\cov{H}{\partial})\rightarrow\mathcal{D}^b(\cov{H}{\partial})/F$ is triangulated.
\end{thma}
It is already shown in \cite{BMRRT} that the functor $\mathbb{S}_2$ satisfies conditions (1) and (2). Since the functor $\langle 1\rangle_\partial$ is an auto-equivalence of $\mod\cov{H}{\partial}$, then the functor $\mathbb{S}_2\langle 1\rangle_{\partial}$ clearly satisfies again conditions (1) and (2). Thus the orbit category $(\mathcal{D}^b(\cov {\Lambda}{\delta})/\langle 1\rangle_\delta)\cong \mathcal{D}^b(\cov{H}{\partial})/\mathbb{S}_2\langle1\rangle_\partial$ is triangulated and the natural functor \[ \xymatrix{\mathcal{D}^b(\cov{\Lambda}{\delta})\ar[r] & \mathcal{D}^b(\cov{\Lambda}{\delta})/\langle 1\rangle_\delta}\] is a triangle functor.
By \cite[Corollary~7.14]{AO10}, the derived category $\mathcal{D}^b(\Lambda)$ is equivalent to the triangulated hull $(\mathcal{D}^b(\cov{\Lambda}{\delta})/\langle 1\rangle_\delta)_\Delta $. Hence we have the following commutative diagram \[\xymatrix{\mathcal{D}^b(\cov{\Lambda}{\delta})\ar[r]^-A\ar@/_.5cm/[rr]_-C & \mathcal{D}^b(\cov{\Lambda}{\delta})/\langle 1 \rangle\ar@{^(->}[r]^-B & \mathcal{D}^b(\Lambda)}\] where $A$ and $C$ are triangle functors. Therefore $B$ commutes with the shift and sends triangles to triangles. Hence the fully faithful functor $B$ is a $k$-equivalence \[\mathcal{D}^b(\cov{\Lambda}{\delta}/\langle 1\rangle_\delta)\cong \mathcal{D}^b(\cov{H}{\partial})/\mathbb{S}_2\langle1\rangle_\partial. \qedhere\]
\end{proof}
\subsection{The shape of the Auslander-Reiten quiver of $\mathcal{D}^b(\Lambda)$}
In this subsection we use Corollary~\ref{k-equivalence} to compute explicitly the shape of the Auslander-Reiten quiver of the derived category of an algebra of cluster type $\widetilde{A}_{p,q}$.
Throughout this subsection $\Lambda$ is an algebra of cluster type $\widetilde{A}_{p,q}$ and of weight $w\neq 0$. Let $\partial$ be a grading induced by $\Lambda$ on $Q_H$. By Lemma~\ref{gradedeqandw} we can assume that $\partial$ is the following grading.
\[\scalebox{.7}{
\begin{tikzpicture}[scale=2,>=stealth]
\node (P1) at (0,0) {$1$};
\node (P2) at (1,1) {$2$};
\node (P3) at (2,1) {$3$};
\node (P4) at (3,1){};
\node (P5) at (5.5,1){};
\node (P6) at (7,1) {$p$};
\node (P7) at (8,0) {$p+1$};
\node (P8) at (7,-1) {$p+2$};
\node (P9) at (5.5,-1) {};
\node (P10) at (2.5,-1) {};
\node (P11) at (1,-1) {$p+q$};
\draw [->] (P1) -- node [fill=white,inner sep=.5mm]{\small{0}} (P2);
\draw [->] (P2) -- node [fill=white,inner sep=.5mm]{\small{0}} (P3);
\draw [->] (P3) -- node [fill=white,inner sep=.5mm]{\small{0}} (P4);
\draw [->] (P5) --node [fill=white,inner sep=.5mm]{\small{0}} (P6);
\draw [->] (P6) --node [fill=white,inner sep=.5mm]{\small{$w$}} (P7);
\draw [->] (P8) --node [fill=white,inner sep=.5mm]{\small{0}} (P7);
\draw [->] (P9) --node [fill=white,inner sep=.5mm]{\small{0}} (P8);
\draw [->] (P11) -- node [fill=white,inner sep=.5mm]{\small{0}} (P10);
\draw [->] (P1) -- node [fill=white,inner sep=.5mm]{\small{0}} (P11);
\draw [loosely dotted, thick] (P4) -- (P5);
\draw [loosely dotted, thick] (P9) --(P10);
\end{tikzpicture}}
\]
By Corollary~\ref{k-equivalence} we have a $k$-linear equivalence \[\mathcal{D}^b(\Lambda)\cong \mathcal{D}^b(\cov{H}{\partial})/\mathbb{S}_2\langle1\rangle_\partial.\]
The quiver of the subcategory $\cov{H}{\partial}={\sf add \hspace{.02in} }\{H\langle p\rangle_{\partial}, p\in\mathbb{Z}\}\subset {\sf gr \hspace{.02in} } H$ consists of $|w|$ connected components of type $A_\infty^\infty$ with the following orientation.
\[\scalebox{.5}{
\begin{tikzpicture}[scale=.7,>=stealth]
\node (P1) at (0,0) {$1$};
\node (P2) at (2,1) {$2$};
\node (P3) at (4,2) {$3$};
\node (P4) at (6,3) {};
\node (P5) at (8,4) {};
\node (P6) at (10,5) {$p$};
\node (P7) at (12,6) {$p+1$};
\node (P8) at (10,7) {}; \node (P8') at (8,8) {};
\node (P9) at (2,-1) {$p+q$};
\node (P10) at (4,-2) {};
\node (P11) at (6,-3) {};
\node (P12) at (8,-4) {$p+1$};
\node (Q1) at (-4,-10) {$1$};
\node (Q2) at (-2,-9) {$2$};
\node (Q3) at (0,-8) {$3$};
\node (Q4) at (2,-7) {};
\node (Q5) at (4,-6) {};
\node (Q6) at (6,-5) {$p$};
\node (Q7) at (-2,-11) {};\node (Q7') at (0,-12) {};
\draw [->] (P1)--(P2);
\draw [->] (P2)--(P3);
\draw [->] (P3)--(P4);
\draw [loosely dotted, thick] (P4)--(P5);
\draw [->] (P5)--(P6);
\draw [->] (P6)--(P7);
\draw [->] (P8)--(P7);
\draw [->] (P1)--(P9);
\draw [->] (P9)--(P10);
\draw [loosely dotted, thick] (P10)--(P11);
\draw [->] (P11)--(P12);
\draw [->] (Q6)--(P12);
\draw [->] (Q1)--(Q2);
\draw [->] (Q2)--(Q3);
\draw [->] (Q3)--(Q4);
\draw [loosely dotted, thick] (Q4)--(Q5);
\draw [->] (Q5)--(Q6);
\draw [->] (Q1)--(Q7);
\draw [loosely dotted, thick](Q7')--(Q7);
\draw [loosely dotted, thick] (P8)--(P8');
\end{tikzpicture}}
\]
Then it is not hard to compute the Auslander-Reiten quiver of the module category $\mod\cov{H}{\partial}$. It has $4|w|$ connected components:
$\mathcal{P}_{i}$, $\mathcal{R}^p_{i}$ , $\mathcal{R}^q_{i}$, and $\mathcal{I}_i$ with $i\in\mathbb{Z}/w\mathbb{Z}$
satisfying $\mathcal{P}_{i}=\mathcal{P}_{0}\langle i\rangle$, $\mathcal{R}^p_{i}=\mathcal{R}^p_{0}\langle i\rangle$,
$\mathcal{R}^q_{i}=\mathcal{R}^q_{0}\langle i\rangle$ and $\mathcal{I}_i=\mathcal{I}_0\langle i \rangle$ for any $i\in\mathbb{Z}/w\mathbb{Z}.$
The component $\mathcal{P}}\newcommand{\Rr}{\mathcal{R}_0$ is of type $\mathbb{N}A_\infty^\infty$ and contains the indecomposable projective modules $P_j\langle 0 \rangle$ for $j=1,\ldots, p+q$. Its shape is described in Figure~\ref{figureP_0}.
\begin{figure}
\[ \scalebox{.5}{
\begin{tikzpicture}
\node (P1+0) at (0,0) {$P_1\langle 0 \rangle$};
\foreach \i in {1,...,4} \node (P1+\i) at (4 * \i, 0) {};
\node (P2+0) at (2,1) {$P_2\langle 0 \rangle$};
\foreach \i in {1,2,3} \node (P2+\i) at (2 + 4 * \i, 1) {};
\node (P3+0) at (4,2) {$P_3\langle 0 \rangle$};
\foreach \i in {1,2,3} \node (P3+\i) at (4 + 4 * \i, 2) {};
\node (P4+0) at (6,3) {};
\foreach \i in {1,2} \node (P4+\i) at (6 + 4 * \i, 3) {};
\node (P5+0) at (8,4) {};
\foreach \i in {1,2} \node (P5+\i) at (8 + 4 * \i, 4) {};
\node (P6+0) at (10,5) {$P_p\langle 0 \rangle$};
\node (P6+1) at (14, 5) {};
\node (P7+0) at (12,6) {$P_{p+1}\langle w\rangle$};
\node (P7+1) at (16, 6) {};
\node (P8+0) at (10,7) {};
\node (P8+1) at (14, 7) {};
\node (P9+0) at (2,-1) {$P_{p+q}\langle 0 \rangle$};
\foreach \i in {1,2,3} \node (P9+\i) at (2 + 4 * \i, -1) {};
\node (P10+0) at (4,-2) {};
\foreach \i in {1,2,3} \node (P10+\i) at (4 + 4 * \i, -2) {};
\node (P11+0) at (6,-3) {};
\foreach \i in {1,2} \node (P11+\i) at (6 + 4 * \i, -3) {};
\node (P12+0) at (8,-4) {$P_{p+1}\langle 0 \rangle$};
\foreach \i in {1,2} \node (P12+\i) at (8 + 4 * \i, -4) {};
\node (Q6+0) at (6,-5) {$P_p\langle -w\rangle$};
\foreach \i in {1,2} \node (Q6+\i) at (6 + 4 * \i, -5) {};
\foreach \i in {0,1} \draw [ultra thick, loosely dotted] (9 + 4 * \i, 7.5) -- (P8+\i);
\foreach \i in {0,1} \draw [ultra thick, loosely dotted] (P8+\i) -- (11 + 4 * \i, 7.5);
\foreach \i in {0,1} \draw [->] (P8+\i) -- (P7+\i);
\draw [->] (P7+0) -- (P8+1); \draw [ultra thick, loosely dotted] (P7+1) -- (17,6.5);
\foreach \i in {0,1} \draw [->] (P6+\i) -- (P7+\i);
\draw [->] (P7+0) -- (P6+1); \draw [ultra thick, loosely dotted] (P7+1) -- (17,5.5);
\foreach \i in {0,1} \draw [->] (P5+\i) -- (P6+\i);
\foreach \i/\j in {0/1,1/2} \draw [->] (P6+\i) -- (P5+\j);
\draw [ultra thick, loosely dotted] (P5+2) -- (17,4.5);
\foreach \i in {0,1,2} \draw [ultra thick, loosely dotted] (P4+\i) -- (P5+\i);
\foreach \i in {0,1,2} \draw [->] (P3+\i) -- (P4+\i);
\foreach \i/\j in {0/1,1/2,2/3} \draw [->] (P4+\i) -- (P3+\j);
\draw [ultra thick, loosely dotted] (P3+3) -- (17,2.5);
\foreach \i in {0,...,3} \draw [->] (P2+\i) -- (P3+\i);
\foreach \i/\j in {0/1,1/2,2/3} \draw [->] (P3+\i) -- (P2+\j);
\draw [ultra thick, loosely dotted] (P3+3) -- (17,1.5);
\foreach \i in {0,...,3} \draw [->] (P1+\i) -- (P2+\i);
\foreach \i/\j in {0/1,1/2,2/3,3/4} \draw [->] (P2+\i) -- (P1+\j);
\draw [ultra thick, loosely dotted] (P1+4) -- (17,.5);
\foreach \i in {0,...,3} \draw [->] (P1+\i) -- (P9+\i);
\foreach \i/\j in {0/1,1/2,2/3,3/4} \draw [->] (P9+\i) -- (P1+\j);
\draw [ultra thick, loosely dotted] (P1+4) -- (17,-.5);
\foreach \i in {0,...,3} \draw [->] (P9+\i) -- (P10+\i);
\foreach \i/\j in {0/1,1/2,2/3} \draw [->] (P10+\i) -- (P9+\j);
\draw [ultra thick, loosely dotted] (P10+3) -- (17,-1.5);
\foreach \i in {0,1,2} \draw [ultra thick, loosely dotted] (P10+\i) -- (P11+\i);
\draw [ultra thick, loosely dotted] (P10+3) -- (17,-2.5);
\foreach \i in {0,1,2} \draw [->] (P11+\i) -- (P12+\i);
\foreach \i/\j in {0/1,1/2} \draw [->] (P12+\i) -- (P11+\j);
\draw [ultra thick, loosely dotted] (P12+2) -- (17,-3.5);
\foreach \i in {0,1,2} \draw [->] (Q6+\i) -- (P12+\i);
\foreach \i/\j in {0/1,1/2} \draw [->] (P12+\i) -- (Q6+\j);
\draw [ultra thick, loosely dotted] (P12+2) -- (17,-4.5);
\foreach \i in {0,1,2} \draw [ultra thick, loosely dotted] (5 + 4 * \i, -5.5) -- (Q6+\i);
\foreach \i in {0,1,2} \draw [ultra thick, loosely dotted] (Q6+\i) -- (7 + 4 * \i, -5.5);
\end{tikzpicture} } \]
\caption{\label{figureP_0}Shape of the component $\mathcal{P}}\newcommand{\Rr}{\mathcal{R}_{0}$ of the Auslander-Reiten quiver of $\mod(\cov{H}{\partial})$. } \end{figure}
The component $\mathcal{I}_0$ is of type $(-\mathbb{N})A^\infty_\infty$ and contains the indecomposable injective modules $I_j\langle 0\rangle$ for $j=1,\ldots,p+q$. Its shape is described in Figure~\ref{figureI_0}.
\begin{figure}
\[
\scalebox{.5}{
\begin{tikzpicture}[scale=.9,>=stealth]
\node (P0-0) at (4,0) {$I_1\langle 0 \rangle$};
\node (P0-1) at (0,0){};
\node (P1-0) at (6,1) {$I_2\langle 0 \rangle$};
\node (P1-1) at (2,1){};
\node (P2-0) at (8,2) {$I_3\langle 0 \rangle$};
\foreach \i in {1,2} \node (P2-\i) at (8-4 * \i, 2) {};
\foreach \i in {0,1,2} \node (P3-\i) at (10-4 * \i, 3) {};
\foreach \i in {0,...,3} \node (P4-\i) at (12-4 * \i, 4) {};
\node (P5-0) at (14,5) {$I_p\langle 0 \rangle$};
\foreach \i in {1,2,3} \node (P5-\i) at (14-4 * \i, 5) {};
\node (P6-0) at (16,6) {$I_{p+1}\langle w\rangle$};
\foreach \i in {1,...,4} \node (P6-\i) at (16-4 * \i, 6) {};
\foreach \i in {0,...,3} \node (P7-\i) at (14-4 * \i, 7) {};
\node (P-1-0) at (6,-1) {$I_{p+q}\langle 0 \rangle$};
\node (P-1-1) at (2,-1){};
\foreach \i in {0,1,2} \node (P-2-\i) at (8-4*\i,-2) {};
\foreach \i in {0,1,2} \node (P-3-\i) at (10-4*\i,-3) {};
\node (P-4-0) at (12,-4) {$I_{p+1}\langle 0 \rangle$};
\foreach\i in {1,2,3} \node (P-4-\i) at (12-4*\i,-4) {};
\node (P-5-0) at (10,-5) {$I_p\langle -w\rangle$};
\foreach \i in {1,2} \node (P-5-\i) at (10-4*\i,-5) {};
\foreach \i in {0,1} \draw[->] (P0-\i)--(P1-\i);
\draw[->](P1-1)--(P0-0);
\foreach \i in {0,1} \draw[->] (P1-\i)--(P2-\i);
\foreach \i/\j in {2/1,1/0} \draw [->] (P2-\i) -- (P1-\j);
\foreach \i in {0,1,2} \draw[->] (P2-\i)--(P3-\i);
\foreach \i/\j in {2/1,1/0} \draw [->] (P3-\i) -- (P2-\j);
\foreach \i in {0,1,2} \draw[ultra thick, loosely dotted] (P3-\i)--(P4-\i);
\foreach \i in {0,1,2,3} \draw[->] (P4-\i)--(P5-\i);
\foreach \i/\j in {3/2,2/1,1/0} \draw[->](P5-\i)--(P4-\j);
\foreach \i in {0,...,3} \draw[->] (P5-\i)--(P6-\i);
\foreach \i/\j in {4/3,3/2,2/1,1/0} \draw [->] (P6-\i) -- (P5-\j);
\foreach \i in {0,...,3} \draw[<-] (P6-\i)--(P7-\i);
\foreach \i/\j in {4/3,3/2,2/1,1/0} \draw [<-] (P7-\j) -- (P6-\i);
\foreach \i in {0,1} \draw [->] (P0-\i)--(P-1-\i);
\draw[->] (P-1-1)--(P0-0);
\foreach \i/\j in {1/0,2/1} \draw[->] (P-2-\i)--(P-1-\j);
\foreach \i in {0,1} \draw [->] (P-1-\i)--(P-2-\i);
\foreach \i in {0,1,2} \draw[ultra thick, loosely dotted] (P-2-\i)--(P-3-\i);
\foreach \i in {0,1,2} \draw[->] (P-3-\i)--(P-4-\i);
\foreach \i/\j in {1/0,2/1,3/2} \draw[->] (P-4-\i)--(P-3-\j);
\foreach \i/\j in {1/0,2/1,3/2} \draw[->] (P-4-\i)--(P-5-\j);
\foreach \i in {0,1,2} \draw[<-] (P-4-\i)--(P-5-\i);
\foreach \i in {1,...,3} \draw[ultra thick, loosely dotted] (P7-\i)--(15-4*\i,7.5);
\foreach \i in {0,...,3} \draw[ultra thick, loosely dotted] (P7-\i)--(13-4*\i,7.5);
\foreach \i in {1,2} \draw[ultra thick, loosely dotted] (P-5-\i)--(11-4*\i,-5.5);
\foreach \i in {0,...,2} \draw[ultra thick, loosely dotted] (P-5-\i)--(9-4*\i,-5.5);
\foreach \i in {-4,0,2,4,6} \draw[ultra thick, loosely dotted] (0,\i)--(-1,\i-0.5);
\foreach \i in {-4,-2,0,2,4,6} \draw[ultra thick, loosely dotted] (0,\i)--(-1,\i+0.5);
\end{tikzpicture}}\]
\caption{\label{figureI_0}Shape of the component $\mathcal{I}_{0}$ of the Auslander-Reiten quiver of $\mod(\cov{H}{\partial})$. }
\end{figure}
The components $\Rr^p_0$ and $\Rr^q_0$ are of type $\mathbb{Z}A_\infty$ and contain the regular modules. The shape of the connected components $\mathcal{R}^p_{0}$ and $\mathcal{R}_{0}^q$ are described in Figures~\ref{figureRp_0} and~\ref{figureRq_0}. In these figures, $S_i\langle m\rangle$ is the $m$-degree-shift of the simple module associated with the vertex $i$, the object $M\langle 0 \rangle$ is defined by the exact sequence \[\xymatrix{P_{p}\langle -w\rangle\ar[r]^{a_p} & P_{p+1}\langle 0 \rangle\ar[r]& M\langle 0 \rangle\ar[r] & 0},\] and the object $N\langle 0 \rangle $ is defined by the exact sequence
\[\xymatrix{P_{p+2}\langle 0\rangle\ar[r]^{b_{q}} & P_{p+1}\langle 0 \rangle\ar[r]& N\langle 0 \rangle\ar[r] & 0}.\]
\begin{figure}
\[\scalebox{.5}{
\begin{tikzpicture}[scale=1,>=stealth]
\node (P1) at (0,0) {$S_2\langle 0\rangle$};
\node (P2) at (2,0) {$S_3\langle 0 \rangle$};
\node (P3) at (4,0) {};
\node (P4) at (8,0) {};
\node (P5) at (10,0) {$S_p\langle 0\rangle$};
\node (P6) at (12,0) {$M\langle w\rangle$};
\node (P7) at (14,0) {$S_2\langle w\rangle$};
\node (Q1) at (0,2) {};
\node (Q2) at (2,2) {};
\node (Q3) at (4,2) {};
\node (Q4) at (8,2) {};
\node (Q5) at (10,2) {};
\node (Q6) at (12,2) {};
\node (Q7) at (14,2) {};
\node (R1) at (-1,1) {};
\node (R2) at (1,1) {};
\node (R3) at (3,1) {};
\node (R4) at (9,1) {};
\node (R5) at (11,1) {};
\node (R6) at (13,1) {};
\node (R7) at (15,1) {};
\draw [->] (R1)--(Q1);
\draw [->] (R1)--(P1);
\draw [->] (R2)--(Q2);
\draw [->] (R2)--(P2);
\draw [->] (R3)--(Q3);
\draw [->] (R3)--(P3);
\draw [->] (R4)--(Q5);
\draw [->] (R4)--(P5);
\draw [->] (R5)--(Q6);
\draw [->] (R5)--(P6);
\draw [->] (R6)--(P7);
\draw [->] (R6)--(Q7);
\draw [->] (P1)--(R2);
\draw [->] (Q1)--(R2);
\draw [->] (P2)--(R3);
\draw [->] (Q2)--(R3);
\draw [->] (P4)--(R4);
\draw [->] (P5)--(R5);
\draw [->] (P6)--(R6);
\draw [->] (P7)--(R7);
\draw [->] (Q4)--(R4);
\draw [->] (Q5)--(R5);
\draw [->] (Q6)--(R6);
\draw [->] (Q7)--(R7);
\draw [loosely dotted, ultra thick] (P3)--(P4);
\draw [loosely dotted, ultra thick] (Q3)--(Q4);
\draw [loosely dotted, ultra thick] (-3,1)--(R1);
\draw [loosely dotted, ultra thick] (0,4)--(Q1);
\draw [loosely dotted, ultra thick] (17,1)--(R7);
\draw [loosely dotted, ultra thick] (14,4)--(Q7);
\end{tikzpicture}}
\]\caption{\label{figureRp_0} Shape of the connected component $\mathcal{R}^p_{0}$.}
\end{figure}
\begin{figure}
\[\scalebox{.5}{
\begin{tikzpicture}[scale=1,>=stealth]
\node (P1) at (0,0) {$S_{p+q}\langle 0\rangle$};
\node (P2) at (2,0) {$S_{p+q-1}\langle 0 \rangle$};
\node (P3) at (4,0) {};
\node (P4) at (8,0) {};
\node (P5) at (10,0) {$S_{p+2}\langle 0\rangle$};
\node (P6) at (12,0) {$N\langle 0 \rangle$};
\node (P7) at (14,0) {$S_{p+q}\langle -w\rangle$};
\node (Q1) at (0,2) {};
\node (Q2) at (2,2) {};
\node (Q3) at (4,2) {};
\node (Q4) at (8,2) {};
\node (Q5) at (10,2) {};
\node (Q6) at (12,2) {};
\node (Q7) at (14,2) {};
\node (R1) at (-1,1) {};
\node (R2) at (1,1) {};
\node (R3) at (3,1) {};
\node (R4) at (9,1) {};
\node (R5) at (11,1) {};
\node (R6) at (13,1) {};
\node (R7) at (15,1) {};
\draw [->] (R1)--(Q1);
\draw [->] (R1)--(P1);
\draw [->] (R2)--(Q2);
\draw [->] (R2)--(P2);
\draw [->] (R3)--(Q3);
\draw [->] (R3)--(P3);
\draw [->] (R4)--(Q5);
\draw [->] (R4)--(P5);
\draw [->] (R5)--(Q6);
\draw [->] (R5)--(P6);
\draw [->] (R6)--(P7);
\draw [->] (R6)--(Q7);
\draw [->] (P1)--(R2);
\draw [->] (Q1)--(R2);
\draw [->] (P2)--(R3);
\draw [->] (Q2)--(R3);
\draw [->] (P4)--(R4);
\draw [->] (P5)--(R5);
\draw [->] (P6)--(R6);
\draw [->] (P7)--(R7);
\draw [->] (Q4)--(R4);
\draw [->] (Q5)--(R5);
\draw [->] (Q6)--(R6);
\draw [->] (Q7)--(R7);
\draw [loosely dotted, ultra thick] (P3)--(P4);
\draw [loosely dotted, ultra thick] (Q3)--(Q4);
\draw [loosely dotted, ultra thick] (-3,1)--(R1);
\draw [loosely dotted, ultra thick] (0,4)--(Q1);
\draw [loosely dotted, ultra thick] (17,1)--(R7);
\draw [loosely dotted, ultra thick] (14,4)--(Q7);
\end{tikzpicture}}
\]\caption{\label{figureRq_0}Shape of the connected component $\mathcal{R}^q_{0}$.}
\end{figure}
Under the forgetful functor $\mod\cov{H}{\partial}\rightarrow\mod H$, the components $\mathcal{P}}\newcommand{\Rr}{\mathcal{R}_i$ are sent to $\mathcal{P}}\newcommand{\Rr}{\mathcal{R}$ the preprojective component of $\mod H$, the components $\mathcal{I}_i$ are sent to the preinjective component $\mathcal{I}$, the components $\mathcal{R}^p_j$ are sent to the exceptional tube of rank $p$ and the components $\mathcal{R}^q_j$ are sent to the exceptional tube of rank $q$. The indecomposable $H$-modules lying in the homogeneous tubes are not gradable if $w\neq 0$, therefore the description above is complete.
\medskip
Since $\mod\cov{H}{\partial}$ is a hereditary category, one can easily deduce the shape of the Auslander-Reiten quiver of $\mathcal{D}^b(\cov{H}{\partial})$. It has three kinds of components
$\mathcal{P}_{(i,n)}$, $\mathcal{R}^p_{(i,n)}$, and $\mathcal{R}^q_{(i,n)}$, with $i\in\mathbb{Z}/w\mathbb{Z}$ and $n\in\mathbb{Z}$. The component $\mathcal{P}}\newcommand{\Rr}{\mathcal{R}_{(0,0)}$ contains $\mathcal{P}}\newcommand{\Rr}{\mathcal{R}_0$ and $\mathcal{I}_0[-1]$ and is described in Figure~\ref{figureP_00}. The component $\mathcal{R}^p_{(0,0)}$ (resp.\ $\mathcal{R}^q_{(0,0)}$) is $\mathcal{R}^p_{0}$ (resp.\ $\mathcal{R}^q_{0}$). Moreover we have $\mathcal{P}_{(i,n)}=\mathcal{P}_{(0,0)}\langle i\rangle [n]$, $\mathcal{R}^p_{(i,n)}=\mathcal{R}^p_{(0,0)}\langle i \rangle [n]$, and $\mathcal{R}^q_{(i,n)}=\mathcal{R}^q_{(0,0)}\langle i\rangle [n]$ for $i\in\mathbb{Z}/w\mathbb{Z}$ and $n\in\mathbb{Z}$.
\begin{figure}
\[
\scalebox{.5}{
\begin{tikzpicture}[scale=.9,>=stealth]
\foreach \i in {-5,-3,-1,1,3,5,7} \node (-3-\i) at (-6,\i){};
\node (-2-0) at (-4,0) {$I_1\langle 0 \rangle[-1]$};
\foreach \i in {-4,-2,2,4,6} \node (-2-\i) at (-4,\i){};
\node (-1-1) at (-2,1) {$I_2\langle 0 \rangle[-1]$};\node (-1--1) at (-2,-1) {$I_{p+q}\langle 0 \rangle[-1]$};
\foreach \i in {-5,-3,3,5,7} \node (-1-\i) at (-2,\i){};
\node (0-0) at (0,0) {$P_1\langle 0 \rangle$};\node (0-2) at (0,2) {$I_3\langle 0 \rangle[-1]$};
\foreach \i in {-4,-2,4,6} \node (0-\i) at (0,\i){};
\node (1-1) at (2,1) {$P_2\langle 0 \rangle$};\node (1--5) at (2,-5) {$I_p\langle -w\rangle[-1]$};\node (1--1) at (2,-1) {$P_{p+q}\langle 0 \rangle$};
\foreach \i in {-3,3,5,7} \node (1-\i) at (2,\i){};
\node (2-2) at (4,2) {$P_3\langle 0 \rangle$};\node (2--4) at (4,-4) {$I_{p+1}\langle 0 \rangle[-1]$};
\foreach \i in {-2,0,4,6} \node (2-\i) at (4,\i){};
\node (3-5) at (6,5) {$I_p\langle 0 \rangle[-1]$};\node (3--5) at (6,-5) {$P_p\langle -w\rangle$};
\foreach \i in {-3,-1,1,3,7} \node (3-\i) at (6,\i){};
\node (4-6) at (8,6) {$I_{p+1}\langle w\rangle[-1]$};\node (4--4) at (8,-4) {$P_{p+1}\langle 0 \rangle$};
\foreach \i in {-2,0,2,4} \node (4-\i) at (8,\i){};
\node (5-5) at (10,5) {$P_p\langle 0 \rangle$};
\foreach \i in {-5,-3,-1,1,3,7} \node (5-\i) at (10,\i){};
\node (6-6) at (12,6) {$P_{p+1}\langle w\rangle$};
\foreach \i in {-4,-2,0,2,4} \node (6-\i) at (12,\i){};
\foreach \i in {-5,-3,-1,1,3,5,7} \node (7-\i) at (14,\i){};
\foreach \i in {-5,-3,-1,1,3,5,7} \node (\i--7) at (2*\i,-7){};
\foreach \i in {-4,-2,0,2,4,6}\node (\i--6) at (2*\i, -6){};
\foreach \k/\l in {-2/-1,0/1,2/3,4/5,6/7} \foreach \i/\j in {-6/-5,-4/-3,-2/-1,0/1,2/3,4/5,6/7} \draw[->] (\k-\i)--(\l-\j);
\foreach \k/\l in {-2/-1,0/1,2/3,4/5,6/7} \foreach \i/\j in {-6/-7,-4/-5,0/-1,2/1,6/5} \draw[->] (\k-\i)--(\l-\j);
\foreach \k/\l in {-2/-3,0/-1,2/1,4/3,6/5} \foreach \i/\j in {-6/-5,-4/-3,-2/-1,0/1,2/3,4/5,6/7} \draw[<-] (\k-\i)--(\l-\j);
\foreach \k/\l in {-2/-3,0/-1,2/1,4/3,6/5} \foreach \i/\j in {-6/-7,-4/-5,0/-1,2/1, 6/5} \draw[<-] (\k-\i)--(\l-\j);
\foreach \k/\l in {-2/-3,0/-1,2/1,4/3,6/5} \draw[ultra thick, loosely dotted] (\k-4)--(\l-3);
\foreach \k/\l in {-2/-1,0/1,2/3,4/5,6/7}\draw[ultra thick, loosely dotted] (\k--2)--(\l--3);
\foreach \i in {-7,-5,-1,1,3,5,7} \draw[ultra thick, loosely dotted] (14,\i)--(15,\i+0.5);
\foreach \i in {-7,-5,-3,-1,1,3,5,7} \draw[ultra thick, loosely dotted] (14,\i)--(15,\i-0.5);
\foreach \i in {-7,-5,-3,-1,1,5,7} \draw[ultra thick, loosely dotted] (-6,\i)--(-7,\i+0.5);
\foreach \i in {-7,-5,-3,-1,1,3,5,7} \draw[ultra thick, loosely dotted] (-6,\i)--(-7,\i-0.5);
\foreach \i in {-3,-1,1,3,5,7} \draw[ultra thick, loosely dotted] (2*\i,7)--(2*\i+1,7.5);
\foreach \i in {-3,-1,1,3,5,7} \draw[ultra thick, loosely dotted] (2*\i,7)--(2*\i-1,7.5);
\foreach \i in {-3,-1,1,3,5,7} \draw[ultra thick, loosely dotted] (2*\i,-7)--(2*\i+1,-7.5);
\foreach \i in {-3,-1,1,3,5,7} \draw[ultra thick, loosely dotted] (2*\i,-7)--(2*\i-1,-7.5);
\end{tikzpicture}}\]
\caption{\label{figureP_00}Shape of the component $\mathcal{P}}\newcommand{\Rr}{\mathcal{R}_{(0,0)}$ of the Auslander-Reiten quiver of $\mathcal{D}^b(\cov{H}{\partial})$. }\end{figure}
The morphisms satisfy the following:
\[\begin{array}{lll}{\sf Hom }(\mathcal{P}}\newcommand{\Rr}{\mathcal{R}_{(i,n)},\mathcal{P}}\newcommand{\Rr}{\mathcal{R}_{(j,m)})\neq 0&\textrm{ if and only if} & i=j \textrm{ and } m \in \{n,n+1\} \\
{\sf Hom }(\mathcal{P}}\newcommand{\Rr}{\mathcal{R}_{(i,n)},\Rr^{p,q}_{(j,m)})\neq 0 &\textrm{ if and only if} & i=j \textrm{ and } m=n\\
{\sf Hom }(\Rr^{p,q}_{(i,n)},\mathcal{P}}\newcommand{\Rr}{\mathcal{R}_{(j,m)})\neq 0 &\textrm{ if and only if} & i=j \textrm{ and } m=n+1\\
{\sf Hom }(\Rr^{p,q}_{(i,n)},\Rr^{p,q}_{(j,m)})\neq 0 &\textrm{ if and only if} & i=j \textrm{ and } m \in \{n,n+1\}
\end{array}.\]
\begin{figure}
\[\scalebox{1}{\begin{tikzpicture}[scale=.3,>=stealth]
\draw [loosely dotted, thick] (0,0) -- (5,0) -- (5,6) -- (0,6) -- cycle;
\draw [loosely dotted, thick] (0,5)--(-1,5)--(-1,-1)--(4,-1)--(4,0);
\draw [loosely dotted, thick] (-1,3)--(-3,3)--(-3,-3)--(2,-3)--(2,-1);
\node (P0) at (0,-5) {$\mathcal{P}_{(i,0)}$};
\draw [loosely dotted, thick] (15,0)--(15,6)--(10,6)--(10,0);
\draw (10,0)--(15,0);
\draw [loosely dotted, thick] (10,5)--(9,5)--(9,-1);
\draw [loosely dotted, thick] (14,-1)--(14,0);
\draw (9,-1)--(14,-1);
\draw [loosely dotted, thick] (9,3)--(7,3)--(7,-3);
\draw [loosely dotted, thick] (12,-3)--(12,-1);
\draw (7,-3)--(12,-3);
\node (P2) at (10,-5) {$\mathcal{R}_{(i,0)}^{p}$};
\draw [loosely dotted, thick] (24,0)--(24,6)--(19,6)--(19,0);
\draw (19,0)--(24,0);
\draw [loosely dotted, thick] (19,5)--(18,5)--(18,-1);
\draw [loosely dotted, thick] (23,-1)--(23,0);
\draw (18,-1)--(23,-1);
\draw [loosely dotted, thick] (18,3)--(16,3)--(16,-3);
\draw [loosely dotted, thick] (21,-3)--(21,-1);
\draw (16,-3)--(21,-3);
\node (P1) at (19,-5) {$\mathcal{R}_{(i,0)}^q$};
\draw [loosely dotted, thick] (29,0) --(34,0)--(34,6)--(29,6)--(29,0);
\draw [loosely dotted, thick] (29,5)--(28,5)--(28,-1)--(33,-1)--(33,0);
\draw [loosely dotted, thick] (28,3)--(26,3)--(26,-3)--(31,-3)--(31,-1);
\node (P4) at (29,-5) {$\mathcal{P}_{(i,1)}$};
\draw [loosely dotted, thick] (44,0)--(44,6)--(39,6)--(39,0);
\draw (39,0)--(44,0);
\draw [loosely dotted, thick] (39,5)--(38,5)--(38,-1);
\draw [loosely dotted, thick] (43,-1)--(43,0);
\draw (38,-1)--(43,-1);
\draw [loosely dotted, thick] (38,3)--(36,3)--(36,-3);
\draw [loosely dotted, thick] (41,-3)--(41,-1);
\draw (36,-3)--(41,-3);
\node at (39,-5) {$\mathcal{R}_{(i,1)}^p$};
\draw [loosely dotted, thick] (-5,2)--(-8,2);
\draw [loosely dotted, thick] (45,2)--(48,2);
\end{tikzpicture}}\]\caption{\label{figureARquiver}Auslander-Reiten quiver of $\mathcal{D}^b(\cov{H}{\partial})$ } \end{figure}
\bigskip
Now we can compute the Auslander-Reiten quiver of the orbit category $\mathcal{D}^b(\cov{H}{\partial})/\mathbb{S}_2\langle 1 \rangle$. We look at the action of the functor $\mathbb{S}_2\langle 1\rangle$ on the connected components of the AR-quiver of $\mathcal{D}^b(\cov{H}{\partial})$. The functor $\mathbb{S}_2\langle 1 \rangle$ acts on the set $\{ \mathcal{P}}\newcommand{\Rr}{\mathcal{R}_{(i,n)}, i\in\mathbb{Z}/w\mathbb{Z}, n\in\mathbb{Z}\}$ by \[ \mathbb{S}_2(\mathcal{P}_{(i,n)})\langle 1\rangle = \tau ( \mathcal{P}_{(i,n)})[-1]\langle 1\rangle =\mathcal{P}_{(i,n)}[-1]\langle 1\rangle = \mathcal{P}_{(i+1,n-1)} \]
Then there are exactly $|w|$ orbits which are $(\mathbb{S}_2\langle 1 \rangle)^\mathbb{Z} \cdot \mathcal{P}}\newcommand{\Rr}{\mathcal{R}_{(i,0)}$ for $i\in\mathbb{Z}/w\mathbb{Z}$.
Similarly $\mathbb{S}_2\langle 1 \rangle$ acts on the sets $\{ \mathcal{R}^p_{(i,n)}, i\in\mathbb{Z}/w\mathbb{Z}, n\in\mathbb{Z}\}$ and $\{ \mathcal{R}^q_{(i,n)}, i\in\mathbb{Z}/w\mathbb{Z}, n\in\mathbb{Z}\}$ by \[ \mathbb{S}_2(\mathcal{R}^p_{(i,n)})\langle 1\rangle = \mathcal{R}^p_{(i+1,n-1)}\ \textrm{and}\ \mathbb{S}_2(\mathcal{R}^q_{(i,n)})\langle 1\rangle = \mathcal{R}^q_{(i+1,n-1)}\]
Therefore we get $2|w|$ orbits which are $(\mathbb{S}_2\langle 1 \rangle)^\mathbb{Z}\cdot\mathcal{R}^p_{(i,0)}$ and $(\mathbb{S}_2\langle 1 \rangle)^\mathbb{Z}\cdot\mathcal{R}^q_{(i,0)}$ for $i\in\mathbb{Z}/w\mathbb{Z}$.
Finally the shape of the AR-quiver of $\mathcal{D}^b(\cov{H}{\partial})/\mathbb{S}_2\langle 1 \rangle$ is described in Figure~\ref{figureARquiver2}
\begin{figure}
\[ \begin{tikzpicture}[>=stealth,scale=.3]
\draw [loosely dotted, thick] (0,0) --(5,0)--(5,6)--(0,6)--(0,0);
\draw [loosely dotted, thick] (0,5)--(-1,5)--(-1,-1)--(4,-1)--(4,0);
\draw [loosely dotted, thick] (-1,3)--(-3,3)--(-3,-3)--(2,-3)--(2,-1);
\node (P0) at (0,-5) {$\mathcal{P}_{(i,0)}$};
\draw [loosely dotted, thick] (15,0)--(15,6)--(10,6)--(10,0);
\draw (10,0)--(15,0);
\draw [loosely dotted, thick] (10,5)--(9,5)--(9,-1);
\draw [loosely dotted, thick] (14,-1) -- (14,0);
\draw (9,-1)--(14,-1);
\draw [loosely dotted, thick] (9,3)--(7,3)--(7,-3);
\draw [loosely dotted, thick] (12,-3)--(12,-1);
\draw (7,-3)--(12,-3);
\node (P2) at (10,-5) {$\mathcal{R}_{(i,0)}^q$};
\draw [loosely dotted, thick] (24,0)--(24,6)--(19,6)--(19,0);
\draw (19,0)--(24,0);
\draw [loosely dotted, thick] (19,5)--(18,5)--(18,-1);
\draw [loosely dotted, thick] (23,-1)--(23,0);
\draw (18,-1)--(23,-1);
\draw [loosely dotted, thick] (18,3)--(16,3)--(16,-3);
\draw [loosely dotted, thick] (21,-3)--(21,-1);
\draw (16,-3)--(21,-3);
\node (P1) at (19,-5) {$\mathcal{R}_{(i,0)}^q$};
\end{tikzpicture} \]
\caption{Auslander-Reiten quiver of the category $\mathcal{D}^b\Lambda$, where $\Lambda$ is of cluster type $\widetilde{A}_{p,q}$ and of weight $w>0$.} \label{figureARquiver2}
\end{figure}
This description can be summarized in the following.
\begin{cora}\label{shapeARquiver}
Let $\Lambda$ be an algebra of cluster type $\widetilde{A}_{p,q}$ and of weight $w\neq 0$. Then the Auslander-Reiten quiver of $\mathcal{D}^b(\Lambda)$ has exactly $3|w|$ connected components:
\begin{itemize}
\item $|w|$ components of type $\mathbb{Z} A_\infty^\infty$;
\item $2|w|$ components of type $\mathbb{Z} A_\infty$.
\end{itemize}
\end{cora}
\subsection{Consequences of the description of the AR-quiver}
From this description we also get the following consequences.
\begin{cora}
Let $\Lambda$ be an algebra of cluster type $\widetilde{A}_{p,q}$ and of weight $w\neq 0$. Then $\Lambda$ is representation-finite.
\end{cora}
\begin{proof}
The algebra $\Lambda$ is derived discrete in the sense of \cite{Vos01}. Hence it is representation-finite.
\end{proof}
\begin{rema}
Let $\Lambda$ be an algebra of cluster type $\widetilde{A}_{p,q}$ and of weight $w\neq 0$. Then the category $\mathcal{D}^b(\Lambda)$ is \emph{locally fractionally Calabi-Yau} of dimensions $\frac{p-2w}{p-w}$ and $\frac{q+2w}{q+w}$ in the sense that there exists objects $X$ and $Y$ such that there are isomorphisms $\mathbb{S}^{p-w}X\cong X[p-2w]$ and $\mathbb{S}^{q+w}Y\cong Y[q+2w]$. Such algebras are studied in \cite[Section~6]{AO}.
\end{rema}
From the previous section, we also deduce a result for the image of the derived category in the cluster category.
\begin{cora}
Let $\Lambda$ be an algebra of cluster type $\widetilde{A}_{p,q}$ and of weight $w\neq 0$. Then the quiver of the orbit category $\mathcal{D}^b(\Lambda)/\mathbb{S}_2$ has 3 connected components which are of the forms $\mathbb{Z}\widetilde{A}_{p,q}$, $\mathbb{Z} A_\infty/(\tau^p)$ and $\mathbb{Z} A_\infty/(\tau^q)$.
\end{cora}
\begin{proof}
From Corollary~\ref{k-equivalence} we have the following diagram
\[\xymatrix{\mathcal{D}^b(\cov{\Lambda}{\delta}) \ar[rr]^\sim \ar[d] && \mathcal{D}^b(\cov{H}{\partial})\ar[d] \\ \mathcal{D}^b(\Lambda)\ar[d]\ar[rr]^-\sim && \mathcal{D}^b(\cov{H}{\partial})/\mathbb{S}_2\langle 1\rangle\ar[d] \\ \mathcal{D}^b(\Lambda)/\mathbb{S}_2\ar@{.>}[rr] && (\mathcal{D}^b(\cov{H}{\partial})/\mathbb{S}_2 \langle 1 \rangle)/\mathbb{S}_2 }\]
The upper functor is a triangle functor, hence it commutes with $\mathbb{S}_2$. Therefore, the $k$-equivalence of Corollary~\ref{k-equivalence} commutes with $\mathbb{S}_2$. Using this we deduce that we have a $k$-equivalence
\[\xymatrix{ \mathcal{D}^b(\Lambda)/\mathbb{S}_2\ar[rr]^-\sim && (\mathcal{D}^b(\cov{H}{\partial})/\mathbb{S}_2 \langle 1 \rangle)/\mathbb{S}_2 }\]
Therefore the AR quiver of $ \mathcal{D}^b(\Lambda)/\mathbb{S}_2$ is the same as the AR quiver of the orbit category $(\mathcal{D}^b(\cov{H}{\partial})/\mathbb{S}_2 \langle 1 \rangle)/\mathbb{S}_2 $. Note that since $\mathbb{S}_2\cong \langle -1 \rangle$ in $(\mathcal{D}^b(\cov{H}{\partial})/\mathbb{S}_2 \langle 1 \rangle)$, we just have to understand the action of $\langle 1\rangle$ in the category $(\mathcal{D}^b(\cov{H}{\partial})/\mathbb{S}_2 \langle 1 \rangle)$.
We know that the category $(\mathcal{D}^b(\cov{H}{\partial})/\mathbb{S}_2 \langle 1 \rangle)$ has $3|w|$ components which are \[\mathcal{P}}\newcommand{\Rr}{\mathcal{R}_{(i,0)}, \mathcal{R}^p_{(i,0)}\ \textrm{and } \mathcal{R}_{(i,0)}^q, \textrm{for } i\in\mathbb{Z}/w\mathbb{Z}\]
Moreover we have the equality \[ \mathcal{P}}\newcommand{\Rr}{\mathcal{R}_{(i,0)}\langle 1 \rangle = \mathcal{P}}\newcommand{\Rr}{\mathcal{R}_{(i+1,0)}, \mathcal{R}^p_{(i,0)}\langle 1 \rangle = \mathcal{R}^p_{(i+1,0)},\ \textrm{and } \mathcal{R}^q_{(i,0)}\langle 1 \rangle = \mathcal{R}^q_{(i+1,0)}\] Thus for each connected component $\Gamma$ of the AR quiver of $(\mathcal{D}^b(\cov{H}{\partial})/\mathbb{S}_2 \langle 1 \rangle)$ we have $ \Gamma\langle w \rangle = \Gamma$. Now it is easy to check that for $X\in \mathcal{R}_{(i,0)}$ we have $X\langle w\rangle\cong \tau ^{-p}X$.
The action of $\langle w\rangle$ on the component $\mathcal{P}}\newcommand{\Rr}{\mathcal{R}_{(0,0)}$ makes $\mathcal{P}}\newcommand{\Rr}{\mathcal{R}_{(0,0)}/\langle w \rangle$ isomorphic to $\mathbb{Z} \widetilde{A}_{p,q}$. Therefore we get the result.
\end{proof}
\section{Explicit description of algebras of cluster type $\widetilde{A}_{p,q}$}\label{section explicit}
In this section, we explicitly describe (in terms of quivers with relations) the algebras of cluster type $\widetilde{A}_{p,q}$, and give an easy method for computing the weight of such an algebra. The strategy consists in describing first the cluster-tilted algebras of type $\widetilde{A}_{p,q}$, following \cite{Bas09}, and then showing that all algebras of cluster type $\widetilde{A}_{p,q}$ can been seen as the degree zero part of a cluster-tilted algebra of type $\widetilde{A}_{p,q}$ with an appropriate grading. We first start by describing the algebras of cluster type $A_n$ using a similar approach.
\subsection{Algebras of cluster type $A_n$}
The aim of this section is to describe all algebras of global dimension~$\leq 2$ which are of cluster type $A_n$. By Corollary~\ref{cor_tree}, we already know that they are the algebras of global dimension~$\leq 2$ derived equivalent to $kQ$, where $Q$ is a quiver of type $A_n$. A description of iterated-tilted algebras of type $A_n$ was done by Assem \cite{Ass82}. Here we use other techniques, based on further developments in cluster-tilting theory.
We start with some definitions.
\begin{dfa}
Let $Q$ be a quiver. A cycle $a_1\ldots a_r$ in $Q$ is called \emph{irreducible} if for all $1\leq i\neq j\leq r$ we have $s(a_i)\neq s(a_j)$. All cycles which are not irreducible are called \emph{reducible}.
\end{dfa}
\begin{dfa}
For $n\geq 1$, we define the class $\mathcal{M}^A_n$ of quivers $Q$ that satisfy the following:
\begin{itemize}
\item they have $n$ vertices;
\item all non-trivial irreducible cycles are oriented and of length 3;
\item a vertex has valency at most 4;
\item if a vertex has valency 4, then two of its adjacent arrows belong to one 3-cycle, and the other two belong to another 3-cycle;
\item if a vertex has valency 3, then two of its adjacent arrows belong to a 3-cycle, and the third arrow does not belong to any 3-cycle.
\end{itemize}
The set $\mathcal{M}^A$ denotes the union of all $\mathcal{M}^A_n$.
For a quiver $Q$ in $\mathcal{M}_A$ we denote by $W_Q$ the sum of all oriented 3-cycles. This is a potential in the sense of \cite{DWZ}.
\end{dfa}
\begin{thma}[\cite{BV08}] \label{dagfinn}
Let $\Gamma$ be a basic finite dimensional algebra. Then $\Gamma$ is a cluster-tilted algebra of type $A_n$ if and only if there exists a quiver $Q$ in $\mathcal{M}^A_n$ such that we have an isomorphism $\Gamma\cong {\sf Jac}(Q,W_Q).$
\end{thma}
The main result of this subsection is the following.
\begin{thma}\label{algebraclustertypeA}
Let $Q$ be a quiver in $\mathcal{M}^A_n$, and let $d$ be a $W_Q$-grading (cf Definition~\ref{defWgrading}). It yields a grading on the Jacobian algebra $B:={\sf Jac}(Q,W)$. Denote by $\Lambda:=B_0$ its degree zero part. Then $\Lambda$ is an algebra of global dimension $\leq 2$ and of cluster type $A_n$. Moreover each basic algebra of global dimension $\leq 2$ and of cluster type $A_n$ is isomorphic to such a $\Lambda$.
\end{thma}
\begin{proof}
We divide the proof into several steps for the convenience of the reader.
\medskip
\textit{Step 1: The global dimension of the algebra $\Lambda$ is at most 2.}
\smallskip
Denote by $Q^{(0)}$ the subquiver of $Q$ which is defined by $Q^{(0)}_0:=Q_0$ and $Q^{(0)}_1:=\{a\in Q_1, d(a)=0\}$. Then we have $$\Lambda\cong kQ^{(0)}/\langle \partial_aW_Q, d(a)=1\rangle.$$
Note that since $W$ is a sum of 3-cycles, the ideal of relations $\langle \partial_aW_Q, d(a)=1\rangle$ is contained in $kQ^{(0)}_2$, that is every relation is of length 2.
Let $i\in Q_0$ be a vertex and $e_i$ be the associated primitive idempotent of $\Lambda$. Then the projective resolution of the simple $\Lambda$-module associated to $i$ is given by
$$\xymatrix{\bigoplus_{b,s(b)=i,d(b)=1} e_{t(b)}\Lambda\ar[rr]^{(f_{a,b})} && \bigoplus_{a,t(a)=i, d(a)=0}e_{s(a)}\Lambda \ar[r] & e_i\Lambda\ar[r] & S_i\ar[r] & 0}$$
Let $b\colon i\rightarrow l$ be an arrow of degree 1, and $a\colon j\rightarrow i$ such that $f_{a,b}\neq 0$. Then there exists a 3-cycle:
$$\xymatrix@-.5cm{ & j\ar[dr]^a & \\ l\ar[ur]^c & & i\ar[ll]^b}$$ and the map $f_{a,b}\colon e_l\Lambda\rightarrow e_j\Lambda$ is induced by $c$. The arrow $c$ does not belong to another 3-cycle, therefore there is no relation between some predecessor of $l$ and $j$. Hence the map $f_{a,b}\colon e_l\Lambda\rightarrow e_j\Lambda$ is a monomorphism.
Let $a$ be an arrow with $t(a)=i$ and of degree $0$. Then $a$ belongs to at most one cycle. Therefore there exists at most one arrow $b$ of degree 1 with $s(b)=i$ such that $f_{a,b}$ does not vanish. Therefore the map $$\xymatrix{\bigoplus_{b,s(b)=i,d(b)=1} e_{t(b)}\Lambda\ar[rr]^{(f_{a,b})} && \bigoplus_{a,t(a)=i, d(a)=0}e_{s(a)}\Lambda}$$ is injective, and the projective dimension of $S_i$ is at most 2.
\medskip
\textit{Step 2: We have an isomorphism ${\sf Jac}(Q,W_Q)\cong {\sf End }_{\mathcal{C}_\Lambda}(\pi\Lambda)$.}
\smallskip
Since $d$ is a $W_Q$-grading the set $\{\partial_aW_Q, d(a)=1\}$ is a set of minimal relations for $\Lambda$. Therefore by Theorem \ref{keller}, we have $\overline{Q}^{(0)}=Q$ and $W_\Lambda=W_Q$ and hence Step 2.
\medskip
\textit{Step 3: There is a triangle equivalence $\mathcal{C}_\Lambda\cong \mathcal{C}_{A_n}$}
\smallskip
By Theorems~\ref{dagfinn}, \ref{connectivity}, and \ref{birs} there exists a sequence of mutation $s$ such that $\mu_s(Q,W_Q)=(Q_H,0)$. Therefore we are done by Corollary~\ref{recognitioncor}.
\medskip
\textit{Step 4: Each algebra of cluster type $A_n$ and of global dimension $\leq 2$ is isomorphic to the degree zero part of a graded Jacobian algebra ${\sf Jac}(Q,W_Q,d)$ where $Q\in\mathcal{M}_n^A$ and $d$ is a $W_Q$-grading.}
\smallskip
Let $\Lambda\cong kQ_\Lambda/I$ be an algebra of global dimension $\leq 2$ and of cluster type $A_n$. Denote by $f\colon \mathcal{C}_\Lambda\rightarrow \mathcal{C}_Q$ the triangle equivalence. By Proposition~\ref{propWgrading} there exists a graded QP $(Q_{\overline{\Lambda}},W,d)$, where $W$ is rigid and $d$ is a $W$-grading, such that we have \[\overline{\Lambda}\underset{\mathbb{Z}}{\cong} {\sf Jac}(Q_{\overline{\Lambda}},W,d).\] Now the object $f(\pi_\Lambda(\Lambda))$ is a cluster-tilting object in $\mathcal{C}_{A_n}$. Thus by Theorem~\ref{dagfinn}, there exists $Q\in\mathcal{M}^A_n$ such that we have an isomorphism \[\overline{\Lambda}\cong {\sf Jac}(Q,W_Q).\]
It is clear that we have $Q=Q_{\overline{\Lambda}}$. We then conclude using the following lemma.
\end{proof}
\begin{lema}\label{uniqueness_rigid_potential_A}
Let $(Q,W,d)$ be a graded QP where $Q\in\mathcal{M}^A_n$ and $W$ is rigid. Then $d$ makes $W_Q$ homogeneous of degree $1$ and we have an isomorphism of $\mathbb{Z}$-graded algebras
$${\sf Jac}(Q,W_Q,d)\underset{\mathbb{Z}}{\cong}{\sf Jac}(Q,W,d).$$
\end{lema}
\begin{proof}
Denote by $C_1,\ldots,C_l$ the oriented 3-cycles of $Q$, so that we have $W_Q=\sum_{i=1}^r C_i$. One easily checks that the irreducible oriented cycles of the quiver $Q$ are exactly the $C_i$'s (up to cyclic equivalence). Therefore we can assume that \[W=\sum_{i=1}^r\lambda_i C_i + \textrm{extra terms}\] where the extra terms are reducible. If there exists $i$ with $\lambda_i= 0$ then the cycle $C_i$ and all the cycles cyclically equivalent to it are not in the Jacobian ideal since there are no cycle of length $\leq 2$ in the quiver $Q$. Therefore since $W$ is rigid, we have $\lambda_i\neq 0$ for all $i=1,\ldots,r$. Now the existence of the grading $d$ implies that $d(C_i)=1$ for all $i=1,\ldots,r$, so any reducible cycle is of degree at least 2. Hence the extra terms in the potential have to be zero and we have $W=\sum_{i=1}^r\lambda_i C_i$ with $\lambda_i\neq 0$.
\end{proof}
\begin{rema}
Note that Theorem~\ref{algebraclustertypeA} is not true for the other Dynkin types. Let $(Q,d)$ be the following graded quiver
\[\xymatrix@-.2cm{& 2\ar[dr]|1^(.6)b& \\ 1\ar[ur]|0^(.6)a\ar[dr]|1_(.4){a'}&& 4\ar[ll]|0^(.6)c\\ &3\ar[ur]|0_(.4){b'}},\] and $W:=cba+cb'a'$. The Jacobian algebra ${\sf Jac}(Q,W)$ is a cluster-tilted algebra of type $D_4$. The grading $d$ is a $W$-grading but the degree zero part of the graded algebra ${\sf Jac}(Q,W,d)$ is an iterated-tilted algebra of type $A_4$ and of global dimension 3, so it cannot be of cluster type $D_4$.
\end{rema}
\subsection{Cluster-tilted algebras of type $\widetilde{A}_{p,q}$}
\begin{dfa}[\cite{Bas09}]
For $p\geq q\geq 1$, we define the class $\mathcal{M}^{\widetilde{A}}_{p,q}$ of quivers $Q$ that satisfy the following conditions:
\begin{itemize}
\item $Q$ has $p+q$ vertices;
\item there exist integers $1\leq p_1<p_2< \ldots < p_r\leq p$ and $1\leq q_1<q_2<\ldots<q_r\leq q$ such that $Q$ contains precisely one full subquiver $C$ which is a non-oriented cycle of type $(p_1,q_1,p_2-p_1,q_2-q_1,\dots, p_r-p_{r-1},q_r-q_{r-1})$ (that is $C$ is the composition of $p_1$ arrows going in one direction with $q_1$ arrows going in the other direction, with $p_2-p_1$ arrows going the first direction, etc...). We denote by $a_1,\ldots, a_{p_r}$ the arrows of $C$ going in one direction and we call them the \emph{$p$-arrows}. We denote by $b_1,\ldots, b_{q_r}$ the arrows going in the opposite direction and we call them the \emph{$q$-arrows}.
\item each arrow connecting $C$ to a vertex not in $C$ is in exactly one 3-cycle of $Q$ of the form \[\scalebox{.7}{\xymatrix{&u(\alpha)\ar[dr]^{\alpha''}&\\ t(\alpha)\ar[ur]^{\alpha'} && s(\alpha)\ar[ll]_{\alpha}}}\] where $\alpha$ is in $C$. We denote by $u(\alpha$) the connecting vertex. It has valency at most 4. When its valency is 4, the adjacent arrows which are not $\alpha'$ and $\alpha''$ belong to exactly one 3-cycle, and when it has valency 3, the third arrow does not belong to any 3-cycle. Moreover, the subquiver containing $C$, $\alpha'$ and $\alpha''$ is a full subquiver of $Q$. Hence we cannot have $u(\alpha)=u(\beta)$ for $\alpha\neq \beta$.
\item the full subquiver of $Q$ whose vertices are not in $C$ is a disjoint union of quivers $Q^\alpha\in\mathcal{M}^A$ where $\alpha$ is an arrow of the non-oriented cycle. The quiver $Q^\alpha$ is empty if there is no $3$-cycle containing $\alpha$, and the quiver $Q^\alpha$ contains the vertex $u(\alpha)$ if there is a 3-cycle
\[\scalebox{.7}{\xymatrix{&u(\alpha)\ar[dr]^{\alpha''}&\\ t(\alpha)\ar[ur]^{\alpha'} && s(\alpha)\ar[ll]_{\alpha}}}\]
\begin{figure}
\[\scalebox{.7}{
\begin{tikzpicture}[>=stealth,scale=1]
\node (P1) at (-6,0){$.$};
\node (P2) at (-5.6,2){$.$};
\node (P3) at (-4.8,4){$.$};
\node (P4) at (-2,5.6){$.$};
\node (P5) at (0,6){$.$};
\node (P6) at (2,5.6){$.$};
\node (P7) at (4.8,4){$.$};
\node (P8) at (5.6,2){$.$};
\node (P9) at (6,0){$.$};
\node (P10) at (0,-6){$.$};
\node (P11) at (-2,-5.6){$.$};
\node (P12) at (-4,-4.8){$.$};
\node (P13) at (-5.6,-2){$.$};
\node (P14) at (-7.2,3.6) {$u(a_2)$};
\node (P15) at (1.4,8) {$u(b_2)$};
\node (P16) at (7,3.8) {$u(b_{q_1})$};
\node (P17) at (-10,3.4) {$Q^{a_2}$};
\node (P18) at (1.4,9.2) {$Q^{b_1}$};
\node (P19) at (9.6,3.8) {$Q^{b_{q_1}}$};
\draw [->] (P1) -- node [swap,xshift=3mm] {$a_1$}(P2);
\draw [->] (P2) -- node [swap,xshift=3mm, yshift=-1mm] {$a_2$}(P3);
\draw [->] (P4) -- node [swap,xshift=-1mm,yshift=-3mm] {$a_{p_1}$}(P5);
\draw [->] (P6) -- node [swap,yshift=-2mm] {$b_1$}(P5);
\draw [->] (P8) -- node [swap,xshift=-2mm] {$b_{q_1}$}(P7);
\draw [->] (P8) -- node [swap,xshift=5mm] {$a_{1_1+1}$}(P9);
\draw [->] (P10) -- node [swap,xshift=2mm,yshift=2mm] {$a_{p_r}$}(P11);
\draw [->] (P12) -- node [swap,xshift=-2mm,yshift=-2mm] {$b_{q_{r-1}+1}$}(P11);
\draw [->] (P1) -- node [swap,xshift=3mm] {$b_{q_r}$}(P13);
\draw [->] (P3) -- (P14);
\draw [->] (P14) -- (P2);
\draw [->] (P5) -- (P15);
\draw [->] (P15) -- (P6);
\draw [->] (P7) -- (P16);
\draw [->] (P16) -- (P8);
\draw (-9,3.4) [dotted] ellipse (3cm and 2cm);
\draw (1.4,9.2) [dotted] ellipse (3cm and 2cm);
\draw (9,3.8) [dotted] ellipse (3cm and 2cm);
\draw [loosely dotted, thick] (P3) .. controls (-4,4.8) and (-3,5.4) .. (P4);
\draw [loosely dotted, thick] (P6) .. controls (3.4,5.2) .. (P7);
\draw [loosely dotted, thick] (P9) .. controls (6,-2) and (3,-6) .. (P10);
\draw [loosely dotted, thick] (P12) .. controls (-5.4,-3) .. (P13);
\end{tikzpicture}}
\]\caption{\label{figuretildeA}Shape of a quiver in $\mathcal{M}^{\widetilde{A}}_{p,q}$. }
\end{figure}
\item We have the equalities $$p= \sum_{l=1}^{p_r} \sharp Q_0^{a_l} + p_r \quad \textrm{and}\quad q=\sum_{l=1}^{q_r} \sharp Q_0^{b_l} + q_r.$$
\end{itemize}
For a quiver $Q$ in $\mathcal{M}^{\widetilde{A}}_{p,q}$ we denote by $W_Q$ the sum of all oriented 3-cycles. More precisely we define \[ W_Q:=\sum_{\alpha \in C}(\alpha''\alpha'\alpha+W_{Q^\alpha})\] where $C$ is the non-oriented cycle of $Q$. This is a rigid potential in the sense of \cite{DWZ}.
\end{dfa}
\begin{thma}[\cite{Bas09}] \label{bastian}
Let $\Gamma$ be a finite dimensional algebra. Then $\Gamma$ is a cluster-tilted algebra of type $\widetilde{A}_{p,q}$ if and only if there exists a quiver $Q$ in $\mathcal{M}^{\widetilde{A}}_{p,q}$ such that we have an isomorphism $$\Gamma\cong {\sf Jac}(Q,W_Q).$$
\end{thma}
\subsection{Algebras of cluster type $\widetilde{A}_{p,q}$}
We have the same kind of result as for the $A_n$ case.
\begin{thma}\label{algebraclustertypetildeA}
Let $Q$ be a quiver in $\mathcal{M}^{\widetilde{A}}_{p,q}$, and let $d$ be $W_Q$-grading. It yields a grading on the Jacobian algebra $B:={\sf Jac}(Q,W_Q)$. Denote by $\Lambda:=B_0$ its degree zero part. Then $\Lambda$ is an algebra of global dimension $\leq 2$ and of cluster type $\widetilde{A}_{p,q}$. Moreover each basic algebra of global dimension $\leq 2$ and of cluster type $\widetilde{A}_{p,q}$ is isomorphic to such a $\Lambda$.
\end{thma}
The proof of the first assertion is exactly the same as in the proof of Theorem \ref{algebraclustertypeA} (Steps~1, 2, and 3). For the proof of the second assertion, we will need the following.
\begin{lema} \label{lemma_compatibility}
Let $(Q,W,d)$ be a graded quiver with reduced potential such that:
\begin{itemize}
\item the quiver $Q$ is in $\mathcal{M}^{\widetilde{A}}_{p,q}$,
\item the potential $W$ is rigid,
\item the grading $d$ is a $W$-grading.
\end{itemize}
Then there exists an algebra isomorphism $\varphi\colon k\hat{Q}\rightarrow k\hat{Q}$ (where $k\hat{Q}$ is the completion of the path algebra $kQ$) such that $\varphi$ is the identity on the vertices, and such that $\varphi(W_Q)$ is cyclically equivalent to $W$. Moreover there exists a $W_Q$-grading $d'$ on $Q$ such that $\varphi\colon (k\hat{Q},d')\rightarrow (k\hat{Q},d)$ is an isomorphism of graded algebras.
\end{lema}
\begin{rema}
This means that the graded QP $(Q,W,d)$ and $(Q,W_Q,d')$ are graded right equivalent in the sense of~\cite[Def. 6.3]{AO10}.
\end{rema}
\begin{proof}
We denote by $C_1,\ldots, C_l$ the oriented cycles such that $W_Q=\sum_{i=1}^lC_i$, by $a_1,\ldots, a_{p_r}$ the $p$-arrows, and by $b_1,\ldots,b_{q_r}$ the $q$-arrows of $Q$. The $C_i$ are irreducible cycles of $Q$, but contrary to the $A_n$-case, there might be other irreducible cycles in the quiver $Q$. We treat here the most complicated case.
\medskip \noindent
\emph{Assume that for all $i=1, \ldots, p_r$, and all $j=1,\ldots, q_r$ we have $Q^{a_i}\neq \varnothing$ and $Q^{b_j}\neq \varnothing$.}
\smallskip
Denote by $C_a$ a cycle containing exactly once the arrows $a_i, i=1,\ldots, p_r$ and $b'_j,b''_j, j=1,\ldots,q_r$ and by $C_b$ a cycle containing exactly once the arrows $b_j, j=1,\ldots, q_r$ and $a'_i,a''_i, i=1,\ldots,p_r$. Then one can check that the irreducible cycles of $Q$ are $C_a$, $C_b$ and the $C_i$'s up to cyclic equivalence. Therefore we can write \[
W=\sum_{i=1}^{l}\lambda_i C_i +\alpha C_a+ \beta C_b +\textrm{ extra terms},\]
where $\lambda_i,\alpha,\beta\in k$ and the extra terms are linear combinations of reducible cycles.
\medskip\noindent
First we show that we can assume $\lambda_i\neq 0$ for all $i=1,\ldots, l$.
If $p_r+q_r\geq 3$, then we have immediately $\lambda_i\neq 0$ for $i=1,\ldots, l$ by the rigidity of $W$.
Assume $p_r=q_r=1$. Then $Q$ is of this form:
\[ \scalebox{.6}{
\begin{tikzpicture}[scale=1,>=stealth]
\node (P1) at (0,0) {$.$};
\node (P2) at (4,0) {$.$};
\node (P3) at (2,2) {$.$};
\draw [->] (0.2,0.1) -- node [swap,yshift=2mm] {$a_1$} (3.8,0.1);
\draw [->] (P2) -- node [swap,xshift=3mm] {$a'_1$} (P3);
\draw [->] (P3) -- node [swap, xshift=-3mm] {$a''_1$} (P1);
\node (P4) at (2,3) {$Q^{a_1}$};
\draw [dotted] (2,3) ellipse (2.8cm and 1.6cm);
\node (P5) at (2,-2){$.$};
\node (P6) at (2,-3) {$Q^{b_1}$};
\draw [->] (0.2,-0.1) -- node [swap,yshift=-2mm] {$b_1$} (3.8,-0.1);
\draw [->] (P2) -- node [swap,xshift=3mm] {$b'_1$} (P5);
\draw [->] (P5) -- node [swap, xshift=-3mm] {$b''_1$} (P1);
\draw [dotted] (2,-3) ellipse (2.8cm and 1.6cm);
\end{tikzpicture}}
\]
We can assume (up to renumbering) that $C_1$ is cyclically equivalent to $a_1a_1'a_1''$ and $C_2$ is cyclically equivalent to $b_1b_1'b_1''$, and we have $C_a=a_1b_1'b_1''$ and $C_b=b_1a'_1a''_1$. Then it is easy to see that the rigidity of $W$ implies that $\lambda_i\neq 0$ for $i=3,\ldots,l$ and that $\lambda_1\lambda_2-\alpha\beta\neq 0$. Therefore, up to the automorphism of $Q$ exchanging $a_1$ and $b_1$ we can assume that $\lambda_1\lambda_2\neq 0$.
\medskip
Since the potential $W$ is homogeneous of degree 1 we have \begin{equation}\label{eqdeg1}
d(a_i)+d(a'_i)+d(a''_i)=d(b_j)+d(b_j')+d(b_j'')= 1 \quad \textrm{ for all } i=1,\ldots, p_r,\ j=1,\ldots, q_r\end{equation}
By definition we have \begin{equation} \label{eqdeg2} d(C_a)=\sum_{i=1}^{p_r}d(a_i)+ \sum_{j=1}^{q_r} (d(b'_j)+d(b''_j))\quad\textrm{and}\quad d(C_b)=\sum_{j=1}^{q_r}d(b_j)+ \sum_{i=1}^{p_r} (d(a'_i)+d(a''_i))\end{equation}
Hence combining \eqref{eqdeg1} and \eqref{eqdeg2} we get \begin{equation}\label{eqdeg3} d(C_a)+d(C_b)=p_r+q_r.\end{equation} Using \eqref{eqdeg3} and the fact that $d(C_a)$ and $d(C_b)$ are non-negative (since $d$ is a map $Q_1\rightarrow\{0,1\}$), we divide the proof into 4 subcases.
\medskip
\noindent
\emph{Case 1: $d(C_a)\geq 2$ and $d(C_b)\geq 2$.}
\smallskip
In this case, since $W$ is homogeneous of degree $1$, we have $\alpha=\beta=0$, and there is no extra term in the potential $W$. For $i=1,\ldots, l$ we denote by $c_i$ the arrow such that $C_i=c_i''c'_ic_i$. Then we define $\varphi$ on $Q_1$ by
\[ \varphi(x)=\left\{\begin{array}{cc}
\lambda_i c_i & \ \textrm{if } x=c_i \\
x & \textrm{otherwise}
\end{array}\right. \]
It is then clear that $\varphi$ is an isomorphism of the graded algebra $(kQ,d)$ sending $W_Q$ onto $W$.
\medskip
\noindent
\emph{Case 2: $d(C_a)=1$ and $d(C_b)\geq 2$.}
\smallskip
In this case, since $W$ is homogeneous of degree 1, we have $\beta=0$ and there is no extra term in the potential $W$. Up to cyclic equivalence and renumbering we can assume that $C_1=b''_1b'_1b_1$ and $C_a=b_1''b_1'C'_a$. For $i=2,\ldots, l$ we denote by $c_i$ the arrow such that $C_i=c_i''c'_ic_i$. Now we define $\varphi$ on $Q_1$ by
\[ \varphi(x)=\left\{\begin{array}{cl} \lambda_1 b_1 +\alpha C'_a & \ \textrm{if } x=b_1\\
\lambda_i c_i & \ \textrm{if } x=c_i , \textrm{ and }i\geq 2\\
x & \textrm{otherwise}
\end{array}\right. \]
Since $C_a=b_1''b_1'C'_a$ and $C_1=b''_1b'_1b_1$ are oriented cycles, the path $C'_a$ has the same source and the same target as $b_1$, thus $\varphi$ is an algebra morphism. Moreover, since $\lambda_i\neq 0$, $\varphi$ is an isomorphism of the completion $k\hat{Q}$.
Now, we have
\[ d(C'_a)=d(C_a)-d(b'_1)-d(b''_1)= 1 -d(b'_1)-d(b''_1)=d(b_1).\]
Therefore $\varphi$ is an isomorphism of the graded algebra $(k\hat{Q},d)$ sending $W_Q$ onto $W$.
\medskip
\noindent
\emph{Case 3: $d(C_a)=1$ and $d(C_b)=1$.}
\smallskip
From \eqref{eqdeg3} we automatically have $p_r=q_r=1$. All the cycles $C_1=a''_1a'_1a_1$, $C_2=b''_1b'_1b_1$, $C_b=a''_1a'_1b_1$, $C_a=b_1''b'_1a_1$, $C_3,\ldots,C_l$ are homogeneous of degree 1. For $i=3,\ldots, l$ we denote by $c_i$ the arrow such that $C_i=c_i''c'_ic_i$. Since $W$ is rigid we have $\lambda_1\lambda_2-\alpha\beta\neq 0$. And since the grading $d$ makes $W$ homogeneous of degree 1, there is no extra term in the potential $W$ and we have $d(a_1)=d(b_1)$. Then we can define:
\[ \varphi(x)=\left\{\begin{array}{cc}
\lambda_1 a_1+\beta b_1 & \ \textrm{if } x=a_1 \\
\lambda_2 b_1+ \alpha a_1 &\ \textrm{if } x=b_1 \\
\lambda_i c_i & \ \textrm{if } x=c_i , \textrm{ and }i\geq 3\\
x & \textrm{otherwise}
\end{array}\right. \]
Since $\lambda_1\lambda_2-\alpha\beta=0$, this algebra morphism is an isomorphism. Moreover, since $d(a_1)=d(b_1)$, $\varphi\colon (kQ,d)\rightarrow (kQ,d)$ is an isomorphism of graded algebras. By construction it sends $W_Q$ onto $W$.
\medskip
\noindent
\emph{Case 4: $d(C_a)=0$.}
\smallskip
From \eqref{eqdeg3}, we have $d(C_b)\geq 2$. Hence, since $W$ is homogeneous of degree 1, we have $\alpha=\beta=0$. In this case, since the degree of $C_a$ is 0, the oriented cycles of the quiver $Q$ which are homogeneous of degree 1 are cyclically equivalent to something of the form $C_iC_a^n$ where $n\in\mathbb{N}$. Thus we can write up to cyclic equivalence:
\[ W=\sum_{i=1}^lC_iP_i(C_a),\]
where $P_i\in k[\![X]\!]$ is a power series with constant term $\lambda_i\neq 0$. For each $i=1,\ldots, l$ we write $C_i=c''_ic'_ic_i$ and we define
\[ \varphi(x)=\left\{\begin{array}{cc}
c_iP_i(C_a)& \ \textrm{if } x=c_i \\
x & \textrm{otherwise}
\end{array}\right. \]
Then $\varphi$ is an automorphism of the completion $k\hat{Q}$ since $\lambda_i\neq 0$ for all $i=1,\ldots, l$. Since $d(C_a)=0$, this automorphism is an automorphism of the graded algebra $(k\hat{Q},d)$.
\medskip
The other cases, namely when \emph{there exists $1\leq i \leq p_r$ and $1\leq j\leq q_r$ such that $Q^{a_i}= \varnothing$ and $Q^{b_j}=\varnothing$}, and \emph{for all $i=1,\ldots p_r$, we have $Q^{a_i}\neq \varnothing$ and there exists $j$ such that $Q^{b_j}= \varnothing$} are simpler since in these cases there are less irreducible cycles. The proof is left to the reader.
\end{proof}
\begin{rema}
\begin{enumerate}
\item
Note that the first part of this lemma can be deduced directly from \cite{DWZ}. Indeed since $W$ is rigid the quiver with potential $(Q,W)$ is right equivalent in the sense of \cite{DWZ} to $(Q,W_Q)$ (there exists a sequence $s$ such that $\mu_s(Q)$ is acyclic, therefore $\mu_s(Q,W)$ is right equivalent to $\mu_s(Q,W_Q)$). By definition, this implies that there exists an automorphism of completed path algebras $\varphi\colon k\hat{Q} \rightarrow k\hat{Q}$ which is the identity on the vertices, and such that $\varphi(W_Q)$ is cyclically equivalent to $W$. However, we have proved this lemma constructing explicitly the isomorphism $\varphi$.
\item In the case $p_r=q_r=1$, it might happen that $\lambda_1\lambda_2=0$. Then the automorphism $\varphi$ constructed above will exchange the arrows $a_1$ and $b_1$. The degree map $d'$ will satisfy $d'(a_1)=d(b_1)$ and $d'(b_1)=d(a_1)$. This is the only case where $d$ and $d'$ are not the same.
\end{enumerate}
\end{rema}
From Lemma~\ref{lemma_compatibility} we deduce the following result, which is a restatement of the second part of Theorem~\ref{algebraclustertypetildeA} and finishes the proof of Theorem~\ref{algebraclustertypetildeA}.
\begin{cora}
Let $\Lambda$ be an algebra of global dimension at most $2$ and of cluster type $\widetilde{A}_{p,q}$. Then there exists a quiver $Q\in \mathcal{M}_{p,q}^{\widetilde{A}}$ and a $W_Q$-grading $d'$ such that $\Lambda$ is isomorphic to the degree zero part of ${\sf Jac}(Q,W_Q,d)$.
\end{cora}
\begin{proof}
By Proposition~\ref{propWgrading}, there exists a reduced graded quiver with potential $(Q,W,d)$, such that $\overline{\Lambda}\underset{\mathbb{Z}}{\sim} {\sf Jac}(Q,W,d)$. Moreover, $W$ is rigid and $d$ is a $W$-grading. By Theorem~\ref{bastian}, the quiver $Q$ is in $\mathcal{M}_{p,q}^{\widetilde{A}}$. By Lemma~\ref{lemma_compatibility}, there exists a $W_Q$-grading $d'$ such that we have ${\sf Jac}(Q,W,d)\underset{\mathbb{Z}}{\sim}{\sf Jac}(Q,W_Q,d')$. Therefore $\Lambda$ is isomorphic to the degree zero part of ${\sf Jac}(Q,W_Q,d')$.
\end{proof}
\begin{rema}
\begin{itemize}
\item[(1)] This corollary implies that an algebra $\Lambda$ of cluster type $\widetilde{A}_{p,q}$ is always isomorphic to an algebra of the form $kQ_\Lambda/I$, where the relations are paths of length~$2$.
\item[(2)] This corollary gives a description of the iterated tilted algebras of global dimension $\leq 2$ of type $\widetilde{A}_{p,q}$. A description of all iterated tilted algebras (not distinguishing with respect to their global dimension) of type $\widetilde{A}_{p,q}$ has been given in \cite{AS87}.
\end{itemize}
\end{rema}
\begin{cora}
There are only finitely many algebras (up to Morita equivalence) of global dimension $\leq 2$ and of cluster type $\widetilde{A}_{p,q}$.
\end{cora}
\begin{proof}
There are only finitely many quivers in the set $\mathcal{M}^{\widetilde{A}}_{p,q}$. And given $Q\in\mathcal{M}^{\widetilde{A}}_{p,q}$ there are finitely many $W_Q$-gradings.
\end{proof}
\subsection{An alternative description of the weight}
In this subsection, we give an explicit way to compute the weight of an algebra of cluster type $\widetilde{A}_{p,q}$.
\begin{dfa}
Let $Q$ be a quiver in $\mathcal{M}_{p,q}^{\widetilde{A}}$ and $d$ be a $\mathbb{Z}$-grading on $Q$ such that the potential $W_Q$ is homogeneous of degree 1. Then let $a_1,\ldots,a_{p_r}$ be the $p$-arrows and $b_1,\ldots,b_{q_r}$ be the $q$-arrows of $Q$. The \emph{weight} of the graded quiver $(Q,d)$ is defined to be
$$w(Q,W_Q,d):=\sum_{l=1}^{p_r}d(a_l)-\sum_{l=1}^{q_r}d(b_l).$$
\end{dfa}
The aim of the subsection is to show the following.
\begin{prop}\label{prop alternative description weight}
Let $Q\in\mathcal{M}_{p,q}^{\widetilde{A}}$ and $d$ be a $W_Q$ grading. Let $\Lambda$ be the degree zero part of the graded algebra ${\sf Jac}(Q,W_Q,d)$. Then $\Lambda$ is an algebra of global dimension at most 2 and of cluster type $\widetilde{A}_{p,q}$ and we have
$$w(\Lambda)=w(Q,W_Q,d).$$
\end{prop}
\begin{proof}
The first part of the statement follows from Theorem~\ref{algebraclustertypetildeA}.
Let $s$ be a a sequence of mutation such that $\mu_s(Q,W_Q)=(H,0)$ where $Q_H$ is an acyclic quiver of type $\widetilde{A}_{p,q}$. Define a grading $\partial$ on $Q_H$ by $\mu_s^{{\rm L}}(Q,W_Q,d)=(H,0,\partial)$. By definition we have $w(\Lambda)=w(H,0,\partial)$. Hence the proof of the proposition comes directly from the following technical lemma.
\end{proof}
\begin{lema}\label{winvariantmutation}
Let $Q$ be a quiver in $\mathcal{M}_{p,q}^{\widetilde{A}}$ and $d$ be a $\mathbb{Z}$-grading on $Q$ such that $W_Q$ is homogeneous of degree 1. Let $i$ be a vertex of $Q$. Then we have
$$\xymatrix{w(\mu_i^{{\rm L}}(Q,W_Q,d))=w(Q,W_Q,d)}.$$
\end{lema}
\begin{proof}
We define the grading $d'$ on the quiver $\mu_i(Q)$ by $\mu_i^{{\rm L}}(Q,W,d)=(\mu_i(Q,W),d')$.
We distinguish the following different cases.
\medskip
\textit{Case 1: There exists $\alpha$ in the non-oriented cycle $C$ of $Q$ such that $i\in Q^\alpha$ and $i\neq u(\alpha)$.}
\smallskip
\noindent
Then the vertices adjacent to $i$ are not in the non-oriented cycle. Therefore the mutation of $Q$ at $i$ does not affect the non-oriented cycle, and so the weight clearly remains the same.
\medskip
\textit{Case 2: $i=u(\alpha)$ for some arrow $\alpha$ which is on the non-oriented cycle $C$.}
\smallskip
\noindent
Assume that $\alpha$ is a $p$-arrow.
There exists a 3-cycle $\alpha''\alpha'\alpha$ which is a summand of $W$.
\[\scalebox{.7}{
\begin{tikzpicture}[scale=1]
\node (P1) at (0,0) {$s(\alpha)$};
\node (P2) at (4,0) {$t(\alpha)$};
\node (P3) at (2,2) {$i=u(\alpha)$};
\draw [->] (P1) -- node [swap,yshift=-2mm] {$\alpha$} (P2);
\draw [->] (P2) -- node [swap,xshift=4mm] {$\alpha'$} (P3);
\draw [->] (P3) -- node [swap, xshift=-3mm] {$\alpha''$} (P1);
\node (P4) at (2,3) {$Q^\alpha$};
\draw [dotted] (2,3) ellipse (2.8cm and 1.6cm);
\node (P5) at (8,0) {$s(\alpha)$};
\node (P6) at (12,0) {$t(\alpha)$};
\node (P7) at (10,2) {$i=u(\alpha)$};
\draw [->] (P7) -- node [xshift=-3.5mm,yshift=-2.5mm] {$(\alpha')^*$} (P6);
\draw [->] (P5) -- node [xshift=+3.5mm,yshift=-2.5mm] {$(\alpha'')^*$} (P7);
\node (P8) at (8,2){$.$};
\node (P9) at (12,2){$.$};
\draw [dotted] [->] (P8) -- (P5);
\draw [dotted] [->] (P7) -- (P8);
\draw [dotted] [->] (P6) -- (P9);
\draw [dotted] [->] (P9) -- (P7);
\draw [dotted] (7.5,2.5) ellipse (1.5cm and 1cm);
\draw [dotted] (12.5,2.5) ellipse (1.5cm and 1cm);
\end{tikzpicture}}
\]
The new arrows $(\alpha')^*$ and $(\alpha'')^*$ become $p$-arrows in the non-oriented cycle. Therefore we have
$$\begin{array}{rcl} w(\mu_i^{{\rm L}}(Q,W,d)) &= &w(Q,W,d)-d(\alpha) +d'((\alpha')^*)+d'((\alpha'')^*)\\ &= & w(Q,W,d)-d(\alpha) -d(\alpha')-d(\alpha'')+1\\ &= & w(Q,W,d).\end{array}$$ The last equality holds since $\alpha''\alpha'\alpha$ is a summand in the potential $W$, and hence we have $d(\alpha)+d(\alpha')+d(\alpha'')=1$.
\medskip
\textit{Case 3: $i$ is on the non-oriented cycle $C$ between two $p$-arrows.}
\smallskip
\noindent
Then the quiver $Q$ locally looks like
\[\scalebox{.6}{
\begin{tikzpicture}[scale=1,>=stealth]
\node (P1) at (0,0) {$s(a_t)$};
\node (P2) at (4,0) {$i$};
\node (P3) at (2,2){$.$};
\node (P5) at (8,0) {$t(a_t)$};
\node (P4) at (6,2){$.$};
\draw [->] (P1) -- node [swap,yshift=-2mm] {$a_t$} (P2);
\draw [->] (P2) -- node [swap,yshift=-2mm] {$a_{t+1}$} (P5);
\draw [->] [dotted] (P3)--(P1);
\draw [->] [dotted] (P2)--(P3);
\draw [->] [dotted] (P5)--(P4);
\draw [->] [dotted] (P4)--(P2);
\draw [dotted] (2,2.4) ellipse (1.8cm and 1.2cm);
\draw [dotted] (6,2.4) ellipse (1.8cm and 1.2cm);
\node (Q1) at (12,0) {$s(a_t)$};
\node (Q2) at (16,1) {$i$};
\node (Q3) at (14,3){$.$};
\node (Q5) at (20,0) {$t(a_t)$};
\node (Q4) at (18,3){$.$};
\draw [->] (Q2) -- node [swap,yshift=2mm] {$a_t^*$} (Q1);
\draw [->] (Q5) -- node [swap,yshift=2mm] {$a_{t+1}^*$} (Q2);
\draw [->] [dotted] (Q3) -- (Q2);
\draw [->] [dotted] (Q5) --(Q2);
\draw [->] [dotted] (Q2)--(Q4);
\draw [->][dotted] (Q4)--(Q3);
\draw [dotted] (16,4) ellipse (4cm and 2cm);
\draw [->] (Q1)--node [swap, yshift=-3mm] {$[a_{t+1}a_t]$}(Q5) ;
\end{tikzpicture}}
\]
The arrows $a_t$ and $a_{t+1}$ are replaced by the new $p$-arrow $[a_{t+1}a_t]$ in the non-oriented cycle. Hence we have
$$\begin{array}{rcl} w(\mu_i^{{\rm L}}(Q,W,d))&=& w(Q,W,d)-d(a_t)-d(a_{t+1})+ d'([a_{t+1}a_t])\\ & = & w(Q,W,d)\end{array}.$$
The case where $i$ is between two $q$ arrows is similar.
\medskip
\textit{Case 4: $i$ is a sink of the non-oriented cycle $C$.}
\smallskip
\noindent
Assume that $i$ is the target of the $p$-arrow $a_l$ and the target of the $q$-arrow $b_t$.
\[\scalebox{.6}{
\begin{tikzpicture}[scale=1,>=stealth]
\node (P1) at (0,0) {$s(a_l)$};
\node (P2) at (4,0) {$i$};
\node (P3) at (2,2){$.$};
\node (P5) at (8,0) {$s(b_t)$};
\node (P4) at (6,2){$.$};
\draw [->] (P1) -- node [swap,yshift=-2mm] {$a_l$} (P2);
\draw [<-] (P2) -- node [swap,yshift=-2mm] {$b_{t}$} (P5);
\draw [->] [dotted] (P3)--(P1);
\draw [->] [dotted] (P2)--(P3);
\draw [->] [dotted] (P2)--(P4);
\draw [->] [dotted] (P4)--(P5);
\draw [dotted] (2,2.4) ellipse (1.8cm and 1.2cm);
\draw [dotted] (6,2.4) ellipse (1.8cm and 1.2cm);
\node (Q1) at (12,0) {$s(a_l)$};
\node (Q2) at (16,1) {$i$};
\node (Q3) at (14,3){$.$};
\node (Q5) at (20,0) {$s(b_t)$};
\node (Q4) at (18,3){$.$};
\draw [->] (Q2) -- node [swap,yshift=2mm] {$a_l^*$} (Q1);
\draw [->] (Q2) -- node [swap,yshift=2mm] {$b_t^*$} (Q5);
\draw [->] [dotted] (Q3) -- (Q2);
\draw [->] [dotted] (Q4) --(Q2);
\draw [->] [dotted] (Q1)--(Q4);
\draw [->][dotted] (Q5)--(Q3);
\draw [dotted] (14,3.4) ellipse (1.8cm and 1.2cm);
\draw [dotted] (18,3.4) ellipse (1.8cm and 1.2cm);
\end{tikzpicture}}
\]
Then the arrows $a_l$ and $b_t$ are replaced by the arrows $a_l^*$ and $b_t^*$ and we have
$$\begin{array}{rcl} w(\mu_i^{{\rm L}}(Q,W,d)) & = & w(Q,W,d)-d(a_l)+d(b_t)+d'(b_t^*)-d'(a_l^*)\\ & = & w(Q,W,d)-d(a_l)+d(b_t)+(-d(b_t)+1)-(-d(a_l)+1)\\ &=&w(Q,W,d).\end{array}$$
The case where $i$ is the source of one $p$-arrow and of one $q$-arrow is similar.
\end{proof}
This alternative description of the weight gives us the following consequences.
\begin{cora}
The number of derived equivalence classes of algebras of cluster type $\widetilde{A}_{p,q}$ is $[\frac{p}{2}]+[\frac{q}{2}] +1$ if $p\neq q$ and $[\frac{p}{2}]+1$ if $p=q$.
\end{cora}
\begin{proof}
Let $Q$ be a quiver in $\mathcal{M}_{p,q}^{\widetilde{A}}$. It is clear from the definition that $w$ is maximal when the $W_Q$-grading satisfies $d(a_l)=1$ for $l=1,\ldots,p_r$ and $d(b_j)=0$ for $j=1,\ldots, q_r$. In this case the weight is equal to $p_r=p-\sum_{l=1}^{p_r}\sharp Q^{a_l}_0$. But since $d$ is a $W_Q$-grading, $d(a_l)=1$ implies that the quiver $Q^{a_l}$ is non-empty. Then we have $w=p_r= p- \sum_{l=1}^{p_r}\sharp Q^{a_l}_0\leq p-p_r$. Thus we have $w\leq [\frac{p}{2}]$.
For the same reason, we have $w\geq -[\frac{q}{2}]$. Now it is easy to see that all values $-[\frac{q}{2}],- [\frac{q}{2}]+1,\ldots ,[\frac{p}{2}]$ can occur.
\end{proof}
\begin{cora}\label{corollary cycle}
Let $\Lambda$ be an algebra of cluster type $\widetilde{A}_{p,q}$, which is not piecewise hereditary. Then there exists a tilting object $T$ in $\mathcal{D}^b(\Lambda)$ such that the quiver of ${\sf End }_{\mathcal{D}^b(\Lambda)}(T)$ has an oriented cycle.
\end{cora}
\begin{proof}
Let $-\frac{q}{2}\leq w\leq \frac{p}{2}$ be a non-zero integer. We construct an algebra $B$ of cluster type $\widetilde{A}_{p,q}$ of weight $w$ such that $Q_B$ has an oriented cycle. Without loss of generality we can assume $w>0$. We define $B$ as follows.
\[\scalebox{.8}{
\begin{tikzpicture}[scale=1.3,>=stealth]
\node (P1) at (0,0) {$1$};
\node (P2) at (1,1) {$2$};
\node(P3) at (2,0) {$3$};
\node(P4) at (3,1){$4$};
\node(P5) at (4,0){};
\node (P6) at (6,0){};
\node (P7) at (7,1) {$2w$};
\node (P8) at (8,0){$2w+1$};
\node (P9) at (8.5,1){};
\node (P10) at (10.5,1){};
\node (P11) at (12,1){$p+1$};
\node (P12) at (7,-1){$p+2$};
\node (P13) at (5.5,-1){};
\node(P14) at (1,-1){$p+q$};
\node(P15) at (2.5,-1){};
\draw [->] (P2)--(P1);
\draw[->] (P3)--(P2);
\draw[->] (P4)--(P3);
\draw[->] (P5)--(P4);
\draw[loosely dotted, thick] (P5)--(P6);
\draw[->] (P7)--(P6);
\draw[->] (P8)--(P7);
\draw[->] (P7)--(P9);
\draw[loosely dotted, thick] (P9)--(P10);
\draw[->] (P10)--(P11);
\draw[->] (P12)--(P8);
\draw[->] (P13)--(P12);
\draw[loosely dotted, thick] (P15)--(P13);
\draw[->] (P14)--(P15);
\draw[->](P1)--(P14);
\draw [loosely dotted, thick] (P1) .. controls (1,.8) .. (P3);
\draw [loosely dotted, thick] (P3) .. controls (3,.8) .. (P5);
\draw [loosely dotted, thick] (P6) .. controls (7,.8) .. (P8);
\end{tikzpicture}}\]
It is clear that $B$ is the degree 0 part of the Jacobian algebra ${\sf Jac}(Q,W_Q,d)$ with the graded quiver
\[\scalebox{.8}{
\begin{tikzpicture}[scale=1.3,>=stealth]
\node (P1) at (0,0) {$1$};
\node (P2) at (1,1) {$2$};
\node(P3) at (2,0) {$3$};
\node(P4) at (3,1){$4$};
\node(P5) at (4,0){};
\node (P6) at (6,0){};
\node (P7) at (7,1) {$2w$};
\node (P8) at (8,0){$2w+1$};
\node (P9) at (8.5,1){};
\node (P10) at (10.5,1){};
\node (P11) at (12,1){$p+1$};
\node (P12) at (7,-1){$p+2$};
\node (P13) at (5.5,-1){};
\node(P14) at (1,-1){$p+q$};
\node(P15) at (2.5,-1){};
\draw [->] (P2)-- node [fill=white,inner sep=.5mm]{\small{0}}(P1);
\draw[->] (P3)-- node [fill=white,inner sep=.5mm]{\small{0}}(P2);
\draw[->] (P4)-- node [fill=white,inner sep=.5mm]{\small{0}}(P3);
\draw[->] (P5)-- node [fill=white,inner sep=.5mm]{\small{0}}(P4);
\draw[loosely dotted, thick] (P5)--(P6);
\draw[->] (P7)-- node [fill=white,inner sep=.5mm]{\small{0}}(P6);
\draw[->] (P8)-- node [fill=white,inner sep=.5mm]{\small{0}}(P7);
\draw[->] (P7)-- node [fill=white,inner sep=.5mm]{\small{0}}(P9);
\draw[loosely dotted, thick] (P9)--(P10);
\draw[->] (P10)-- node [fill=white,inner sep=.5mm]{\small{0}}(P11);
\draw[->] (P12)-- node [fill=white,inner sep=.5mm]{\small{0}}(P8);
\draw[->] (P13)-- node [fill=white,inner sep=.5mm]{\small{0}}(P12);
\draw[loosely dotted, thick] (P15)--(P13);
\draw[->] (P14)-- node [fill=white,inner sep=.5mm]{\small{0}}(P15);
\draw[->](P1)-- node [fill=white,inner sep=.5mm]{\small{0}}(P14);
\draw [->](P1) -- node [fill=white,inner sep=.5mm]{\small{1}}(P3);
\draw [->](P3)-- node [fill=white,inner sep=.5mm]{\small{1}}(P5);
\draw [->] (P6)-- node [fill=white,inner sep=.5mm]{\small{1}} (P8);
\end{tikzpicture}}\]
Then by Theorem~\ref{derivedeqiffwegal} this algebra $B$ is the endomorphism algebra of some tilting complex $T\in\mathcal{D}^b(\Lambda)$.
\end{proof}
\begin{cora}
Let $\Lambda$ be an algebra of cluster type $\widetilde{A}_{p,q}$ and of weight $w$, then the Coxeter polynomial of $\Lambda$ is \[ X^{p+q}-(-1)^wX^{p-w}-(-1)^wX^{q+w}+1.\]
\end{cora}
\begin{proof}
By definition the Coxeter matrix is the matrix of the automorphism $\tau$ at the level of the Grothendieck group $K_0(\mathcal{D}^b(\Lambda))$ in the basis $\{[S_i] \mid 1 \leq i \leq p+q\}$ consisting of the representatives of the simples. The Coxeter polynomial $C(X)$ is its characteristic polynomial.
The result is already known for $w=0$. Without loss of generality we can assume that $0< w\leq \frac{p}{2}$.
By the above results we can assume that $\Lambda$ is given by the following quiver with relations:
\[\scalebox{.8}{
\begin{tikzpicture}[scale=1.3,>=stealth]
\node (P1) at (0,0) {$1$};
\node (P2) at (1,1) {$2$};
\node(P3) at (2,0) {$3$};
\node(P4) at (3,1){$4$};
\node(P5) at (4,0){};
\node (P6) at (6,0){};
\node (P7) at (7,1) {$2w$};
\node (P8) at (8,0){$2w+1$};
\node (P9) at (8.5,1){$2w+2$};
\node (P10) at (10.5,1){};
\node (P11) at (12,1){$p+1$};
\node (P12) at (7,-1){$p+2$};
\node (P13) at (5.5,-1){};
\node(P14) at (1,-1){$p+q$};
\node(P15) at (2.5,-1){};
\draw [->] (P2)--(P1);
\draw[->] (P3)--(P2);
\draw[->] (P4)--(P3);
\draw[->] (P5)--(P4);
\draw[loosely dotted, thick] (P5)--(P6);
\draw[->] (P7)--(P6);
\draw[->] (P8)--(P7);
\draw[->] (P7)--(P9);
\draw[loosely dotted, thick,draw=red] (P9)--(P10);
\draw[->] (P10)--(P11);
\draw[->] (P12)--(P8);
\draw[->] (P13)--(P12);
\draw[loosely dotted, thick] (P15)--(P13);
\draw[->] (P14)--(P15);
\draw[->](P1)--(P14);
\draw [loosely dotted, thick] (P1) .. controls (1,.8) .. (P3);
\draw [loosely dotted, thick] (P3) .. controls (3,.8) .. (P5);
\draw [loosely dotted, thick] (P6) .. controls (7,.8) .. (P8);
\node (P9+) at (10,1) {};
\draw [->,draw=blue] (P9) -- (P9+);
\draw [loosely dotted, thick,draw=blue] (P9+) -- (P10);
\end{tikzpicture}}\]
Then for $0\leq j\leq w-1$, the projective resolution of $S_{2j+1}$ is given by
\[\xymatrix{0\ar[r] & P_{2j+3}\ar[r] & P_{2j+2}\ar[r] & P_{2j+1}\ar[r] & S_{2j+1}\ar[r] & 0}.\] Thus one easily checks the following
\begin{equation}\label{eq1} \textrm{for } 0\leq j\leq w-1\quad \tau [S_{2j+1}]=[S_{2j+3}[1]]=-[S_{2j+3}] \quad\textrm{in } K_0(\mathcal{D}^b(\Lambda)).\end{equation}
We also have the projective resolutions
\[ \xymatrix{0\ar[r] & P_{p+2}\ar[r] & P_{2w+1}\ar[r] & S_{2w+1}\ar[r] & 0} \quad\textrm{and }\] \[\textrm{for }p+2\leq j\leq p+q \quad\xymatrix{0\ar[r] & P_{j+1}\ar[r] & P_{j}\ar[r] & S_{j}\ar[r] & 0},\]
where we use the convention $p+q+1=1$.
Hence we have
\begin{equation}\label{eq2} \tau [S_{2w+1}]=[S_{p+2}] \quad\textrm{and for } p+2\leq j \leq p+q \quad \tau[S_{j}]=[S_{j+1}]\quad\textrm{in } K_0(\mathcal{D}^b(\Lambda)).\end{equation}
Combining \eqref{eq1} and \eqref{eq2} we get
\[ \tau^{q+w}[S_1]=(-1)^w\tau^q[S_{2w+1}]=(-1)^w[S_1]\]
Similarly we have
\begin{equation}\label{eq3} \textrm{for } 2w+3\leq j\leq p+1\quad \tau[S_j]=[S_{j-1}]\quad\textrm{in } K_0(\mathcal{D}^b(\Lambda)).\end{equation}
Now we have to separate the case where $w=1$. Assume $w\geq 2$ then we have
\begin{equation}\label{eq4} \tau[S_{2w+2}]=[P_{2w-2}]\quad\textrm{in } K_0(\mathcal{D}^b(\Lambda)).\end{equation}
\begin{equation}\label{eq5} \textrm{for }2\leq j\leq w-1 \quad \tau[P_{2j}]=-[I_{2j}]=-[P_{2j-2}] \quad\textrm{in } K_0(\mathcal{D}^b(\Lambda)).\end{equation}
Hence if $p+1\geq 2w+2$, then we have $I_2\cong P_{p+1}$ and we get the following equalities in $K_0(\mathcal{D}^b(\Lambda))$:
\begin{align*} \tau^{p-w}[P_2] & =-\tau^{p-w-1}[P_{p+1}]=\tau^{p-w+2}[S_{p+1}] & \\
& =\tau^{w-1}\tau^{p-2w-1}[S_{p+1}]=\tau^{w-1}[S_{2w+2}] & \textrm{by }\eqref{eq3}
\\ & =\tau^{w-2}[P_{2w-2}] &\textrm{by }\eqref{eq4} \\ &=(-1)^{w}[P_2] &\textrm{by }\eqref{eq5} \end{align*}
If $p+1=2w+1$, we have $I_2\cong P_{2w}$, and we also get $\tau^{p-w}[P_2]=(-1)^w[P_2]$.
Assume that $w$ is odd. Then one can checks that the set
\[\{[S_{2j+1}], 0\leq j \leq w\}\cup\{[S_j], 2w+2\leq j\leq p+q\}\cup \{ [P_{2j}], 1\leq j\leq w-1\}\cup \{[I_2]\}\]
is a basis of $K_0(\mathcal{D}^b(\Lambda))$. Therefore the Coxeter matrix is diagonalizable and $C(X)=(X^{q+w}+1)(X^{p-w}+1)$.
Assume that $w$ is even. Then the set
\[\{[S_{2j+1}], 0\leq j \leq w\}\cup\{[S_j], 2w+2\leq j\leq p+q\}\cup \{ [P_{2j}], 1\leq j\leq w-1\}\]
is linearly independent in $K_0(\mathcal{D}^b(\Lambda)$ and we have the relation:
\[ [I_2]- \sum_{j=2w+2}^{p+1}[S_j] -\sum_{j=1}^{w-1}(-1)^j[P_{2j}]=\sum_{j=p+2}^{p+q}[S_j]+\sum_{j=1}^{w}(-1)^j[S_{2j+1}]\]
This element is an eigenvector of the eigenvalue $1$.
Hence
\[ \frac{(X^{q+w}-1)(X^{p-w}-1)}{(X-1)} \]
divides the Coxeter polynomial $C(X)$. Since the degree of $C(X)$ is $p+q$, and since we know that both the leading coefficient and the absolut term of $C(X)$ are $1$, it follows that
\[ C(X) = \frac{(X^{q+w}-1)(X^{p-w}-1)}{(X-1)} \cdot (X-1) = (X^{q+w}-1)(X^{p-w}-1). \]
For the case $w=1$, we introduce the notations for $4\leq j\leq p$:
\[M_j:={\sf Ker } (I_2\rightarrow I_{j+1}\oplus I_{j+1})\cong {\sf Coker } (P_2\rightarrow P_j\oplus P_j); \] \[ M_3:={\sf Ker } (I_2\rightarrow I_{4}\oplus I_{4})\cong P_2;\quad \textrm{and }M_{p+1}:={\sf Coker } (P_2\rightarrow P_{p+1}\oplus P_{p+1})\cong I_2.\]
Then we have
\begin{equation}\label{eq6} \textrm{for } 4\leq j\leq p+1\quad \tau[M_j]=[M_{j-1}] \end{equation}
Therefore we get the equalities in $K_0(\mathcal{D}^b(\Lambda)):$
\[ \tau^{p-1}[P_2]=-\tau^{p-2}[I_2]=\tau^{p-2}[M_{p+1}]=-[M_3]=-[P_2]\]
Finally it is easy to see that the set
\[ \{[M_j], 3\leq j\leq p+1\}\cup \{ [S_1], [S_3]\}\cup \{[S_j], p+2\leq j\leq p+q\}\]
is a basis of $\mathbb{Q}\otimes_{\mathbb{Z}}K_0(\mathcal{D}^b(\Lambda))$. This finishes the proof.
\end{proof}
We end this subsection by giving a result linking the weight of an algebra of cluster type $\widetilde{A}_{p,q}$ with the set of cluster-tilting objects of $\mathcal{C}_{\widetilde{A}_{p,q}}$ coming from tilting complexes in $\mathcal{D}^b(\Lambda)$. We start with some notation.
Let $\Lambda$ be an algebra of global dimension at most 2 which is $\tau_2$-finite. We define a subset $\mathcal{T}_{\Lambda}$ of the set of tilting complexes of $\mathcal{D}^b(\Lambda)$ by \[ \mathcal{T}_\Lambda:=\{ T\in\mathcal{D}^b(\Lambda) \textrm{ tilting }\mid \ \textrm{gldim} ({\sf End }_{\mathcal{D}^b(\Lambda)}(T))\leq 2 \}\]
By Theorem~\ref{clustertilting}, the set $\pi_{\Lambda}(\mathcal{T}_{\Lambda}) \subset \mathcal{C}_\Lambda$ is a subset of the set of cluster-tilting objects of $\mathcal{C}_{\Lambda}$. Moreover, if $\Lambda_1$ and $\Lambda_2$ are derived equivalent, they are cluster equivalent and we clearly have $\pi_{\Lambda_1}(\mathcal{T}_{\Lambda_1})=\pi_{\Lambda_2}(\mathcal{T}_{\Lambda_2})$.
We prove here the converse in the case of algebras of cluster type $\widetilde{A}_{p,q}$, and compare the sets $\pi_{\Lambda_1}(\mathcal{T}_{\Lambda_1})$ and $\pi_{\Lambda_2}(\mathcal{T}_{\Lambda_2})$ when $w(\Lambda_1)\neq w(\Lambda_2)$.
\begin{prop}
Let $\Lambda_1$ and $\Lambda_2$ be two algebras of global dimension 2 and of cluster type $\widetilde{A}_{p,q}$. Then we have
\begin{itemize}
\item $0\leq w(\Lambda_1)<w(\Lambda_2) \Rightarrow \pi_2(\mathcal{T}_{\Lambda_2})\subsetneq \pi_1(\mathcal{T}_{\Lambda_1})$
\item $w(\Lambda_2)<w(\Lambda_1)\leq 0 \Rightarrow \pi_2(\mathcal{T}_{\Lambda_2})\subsetneq \pi_1(\mathcal{T}_{\Lambda_1})$
\item $w(\Lambda_1)w(\Lambda_2)<0 \Rightarrow \pi_1(\mathcal{T}_{\Lambda_1}) \setminus \pi_2(\mathcal{T}_{\Lambda_2}) \neq \emptyset \neq \pi_2(\mathcal{T}_{\Lambda_2}) \setminus \pi_1(\mathcal{T}_{\Lambda_1})$
\end{itemize}
\end{prop}
\begin{proof}
We first show the inclusion in the first claim.
Without loss of generality, we can assume that $w(\Lambda_2)=w(\Lambda_1)+1>0$. It is enough to show that $\pi_2(\Lambda_2)\in\pi_1(\mathcal{T}_{\Lambda_1})$.
We denote by $H$ some hereditary algebra of type $\widetilde{A}_{p,q}$ and by $\pi_2$ a triangle functor $\pi_2 \colon \mathcal{D}^b(\Lambda_2)\rightarrow \mathcal{C}_H$.
Let $(Q,W_{Q},d)$ be a graded quiver with potential such that we have isomorphisms of $\mathbb{Z}$-graded algebras ${\sf End }_{\mathcal{C}_H}(\pi_2\Lambda_2)\cong {\sf Jac}(Q,W_{Q},d)$. Since $w(\Lambda_2)\geq 1$, there exists a $p$-arrow $a_i\in Q$ such that $d(a_i)=1$. Since $d$ is a $W_{Q}$-grading, the subquiver $Q^{a_i}$ is not empty. More precisely, there exists arrows $a_i'$ and $a_i''$ such that $a_ia_i'a_i''$ is an oriented triangle in $Q$.
Define a new degree $d'$ on $Q$ by:
\[ d'(x)=\left\{\begin{array}{cc}
0 & \ \textrm{if } x=a_i \\
1& \textrm{if } x=a'_i\\
d(x) & \textrm{otherwise}
\end{array}\right. \]
It is immediate to see that $d'$ is a $W_{Q}$-grading. Define the algebra $\Lambda_3$ as the degree 0 part of the graded Jacobian algebra ${\sf Jac}(Q,W_{Q},d')$. By Theorem~\ref{algebraclustertypetildeA}, it is an algebra of global dimension 2 which is of cluster type $\widetilde{A}_{p,q}$, and by Corollary~\ref{recognitioncor} we can assume $\pi_2(\Lambda_2)=\pi_3(\Lambda_3)$ where $\pi_3$ is a triangle functor $\pi_3 \colon \mathcal{D}^b(\Lambda_3)\rightarrow \mathcal{C}_{\Lambda_3}$. Moreover by Proposition~\ref{prop alternative description weight}, we have $w(\Lambda_3)=w(\Lambda_2)-1=w(\Lambda_1)$. Therefore by Theorem~\ref{derivedeqiffwegal} the algebra $\Lambda_3$ is derived equivalent to $\Lambda_1$. The image of $\Lambda_3$ through this equivalence is clearly an object $X$ in $\mathcal{T}_{\Lambda_1}$ which satisfies $\pi_1(X)\cong \pi_2(\Lambda_2)$.
For any $0\leq w \leq [\frac{p}{2}]$, one can easily construct a quiver $Q$ such that $Q$ admits a $W_Q$-grading of weight $w$ but no $W_Q$-grading of weight $w+1$. Therefore the inclusion is strict.
The second point holds by symmetry.
For the third point, it is enough to see that for any $w>0$ the quiver $Q$ constructed in the proof of Corollary~\ref{corollary cycle} satisfies the following: there exists a $W_Q$-grading of weight $w$ and for any $w'<0$ there is no $W_Q$-grading of weight $w'$. The argument for $w<0$ holds by symmetry.
\end{proof}
We end this subsection by asking the following intriguing questions:
\begin{Question}
\begin{itemize}
\item
Let $\Lambda_1$ and $\Lambda_2$ be $\tau_2$-finite algebras of global dimension $\leq 2$ which are cluster equivalent. Do we have the implication
\[ \pi_1(\mathcal{T}_{\Lambda_1})=\pi_2(\mathcal{T}_{\Lambda_2})\Rightarrow \mathcal{D}^b(\Lambda_1)\cong\mathcal{D}^b( \Lambda_2) \ ?\]
\item
Let $\Lambda$ be a $\tau_2$-finite algebra of global dimension $\leq 2$. Does the following implication hold?
\[ \pi_{\Lambda}(\mathcal{T}_\Lambda)=\{ X\in \mathcal{C}_{\Lambda} \mid X \textrm{ cluster-tilting }\}\Rightarrow \Lambda \textrm{ piecewise hereditary }\]
\end{itemize}
\end{Question}
\subsection{Example}
In this subsection we compute explicitly all basic algebras (up to isomorphism) of global dimension $\leq 2$ and of cluster type $\widetilde{A}_{2,2}$ and organize them according to their derived equivalence classes.
The strategy consists in first describing all quivers (up to isomorphism of quivers) which are in the set $\mathcal{M}^{\widetilde{A}}_{2,2}$. In our case an easy computation or \cite{Keller_Java} shows that there are only 4 different quivers in $\mathcal{M}^{\widetilde{A}}_{2,2}$ which are:
\[\scalebox{.8}{
\begin{tikzpicture}[scale=1.3,>=stealth]
\node (A1) at (0,0) {$.$};
\node (A2) at (1,1){$.$};
\node (A3) at (2,0){$.$};
\node (A4) at (1,-1){$.$};
\draw [->] (A1)--(A2);
\draw [->] (A2)--(A3);
\draw [->] (A1)--(A4);
\draw [->] (A4)--(A3);
\node (B1) at (4,0) {$.$};
\node (B2) at (5,1){$.$};
\node (B3) at (6,0){$.$};
\node (B4) at (5,-1){$.$};
\draw [->] (B1)--(B2);
\draw [->] (B3)--(B2);
\draw [->] (B1)--(B4);
\draw [->] (B3)--(B4);
\node (C1) at (8,0) {$.$};
\node (C2) at (9,1){$.$};
\node (C3) at (10,0){$.$};
\node (C4) at (9,-1){$.$};
\draw [->] (C1)--(C2);
\draw [->] (C2)--(C3);
\draw [->] (C3)--(C4);
\draw [->] (C4)--(C1);
\draw [->] (C1)--(C3);
\node (D1) at (12,0) {$.$};
\node (D2) at (13,1){$.$};
\node (D3) at (14,0){$.$};
\node (D4) at (13,-1){$.$};
\draw [->] (D2)--(D1);
\draw[->] (D3)--(D2);
\draw [->] (D3)--(D4);
\draw [->] (D4)--(D1);
\draw [->] (12.3,.05)--(13.7,.05);
\draw [->] (12.3,-.05)--(13.7,-.05);
\end{tikzpicture}}\]
Note that since $p=q$ there is an isomorphism between the two quivers corresponding to $p_r=2, q_r=1$ and $p_r=1,q_r=2$
\[\scalebox{.8}{
\begin{tikzpicture}[scale=1.3,>=stealth] \node (C1) at (8,0) {$.$};
\node (C2) at (9,1){$.$};
\node (C3) at (10,0){$.$};
\node (C4) at (9,-1){$.$};
\draw [->] (C1)--(C2);
\draw [->] (C2)--(C3);
\draw [->] (C3)--(C4);
\draw [->] (C4)--(C1);
\draw [->] (C1)--(C3);
\node (D1) at (12,0) {$.$};
\node (D2) at (13,1){$.$};
\node (D3) at (14,0){$.$};
\node (D4) at (13,-1){$.$};
\draw [<-] (D1)--(D2);
\draw [<-] (D2)--(D3);
\draw [<-] (D3)--(D4);
\draw [<-] (D4)--(D1);
\draw [<-] (D1)--(D3);
\end{tikzpicture}}\]
Then one can easily check that there are 11 graded quivers $(Q,d)$ with $Q\in\mathcal{M}^{\widetilde{A}}_{p,q}$ and $d$ a $W_Q$-grading up to isomorphism of graded quiver. Therefore there are exactly 11 non-isomorphic algebras of global dimension 2 and of cluster type $\widetilde{A}_{2,2}$.
The only possible weights are $|w|=0$ or $|w|=1$. Therefore these eleven algebras are divided into two derived equivalence classes.
There are 8 algebras of global dimension $\leq 2$ which are derived equivalent to $\widetilde{A}_{2,2}$:
\[\scalebox{.8}{
\begin{tikzpicture}[scale=1.3,>=stealth]
\node (A1) at (0,0) {$.$};
\node (A2) at (1,1){$.$};
\node (A3) at (2,0){$.$};
\node (A4) at (1,-1){$.$};
\draw [->] (A1)--(A2);
\draw [->] (A2)--(A3);
\draw [->] (A1)--(A4);
\draw [->] (A4)--(A3);
\node (B1) at (4,0) {$.$};
\node (B2) at (5,1){$.$};
\node (B3) at (6,0){$.$};
\node (B4) at (5,-1){$.$};
\draw [->] (B1)--(B2);
\draw [->] (B3)--(B2);
\draw [->] (B1)--(B4);
\draw [->] (B3)--(B4);
\node (C1) at (8,0) {$.$};
\node (C2) at (9,1){$.$};
\node (C3) at (10,0){$.$};
\node (C4) at (9,-1){$.$};
\draw [->] (C1)--(C2);
\draw [->] (C2)--(C3);
\draw [->] (C3)--(C4);
\draw [loosely dotted,thick] (C4)..controls (9.5,-0.25)..(C1);
\draw [->] (C1)--(C3);
\node (D1) at (12,0) {$.$};
\node (D2) at (13,1){$.$};
\node (D3) at (14,0){$.$};
\node (D4) at (13,-1){$.$};
\draw[->] (D1)--(D2);
\draw[->] (D2)--(D3);
\draw[->] (D1)--(D3);
\draw[->] (D4)--(D1);
\draw [loosely dotted, thick] (D3)..controls (12.5,-.25)..(D4);
\node (E1) at (0,-4) {$.$};
\node (E2) at (1,-3){$.$};
\node (E3) at (2,-4){$.$};
\node (E4) at (1,-5){$.$};
\draw[->] (0.2,-3.95)--(1.8,-3.95);
\draw[->] (0.2,-4.05)--(1.8,-4.05);
\draw[->] (E3)--(E2);
\draw[->] (E3)--(E4);
\draw [loosely dotted, thick] (E1)..controls(1.5,-3.75)..(E2);
\draw [loosely dotted, thick] (E1)..controls(1.5,-4.25)..(E4);
\node (F1) at (4,-4) {$.$};
\node (F2) at (5,-3){$.$};
\node (F3) at (6,-4){$.$};
\node (F4) at (5,-5){$.$};
\draw[->] (4.2,-3.95)--(5.8,-3.95);
\draw[->] (4.2,-4.05)--(5.8,-4.05);
\draw[->] (F3)--(F2);
\draw[->] (F4)--(F1);
\draw [loosely dotted, thick] (F1)..controls(5.5,-3.75)..(F2);
\draw [loosely dotted, thick] (F4)..controls(4.5,-4.25)..(F3);
\node (G1) at (8,-4) {$.$};
\node (G2) at (9,-3){$.$};
\node (G3) at (10,-4){$.$};
\node (G4) at (9,-5){$.$};
\draw[->] (8.2,-3.95)--(9.8,-3.95);
\draw[->] (8.2,-4.05)--(9.8,-4.05);
\draw[->] (G2)--(G1);
\draw[->] (G4)--(G1);
\draw [loosely dotted, thick] (G2)..controls (8.5,-3.75)..(G3);
\draw [loosely dotted, thick] (G4)..controls(8.5,-4.25)..(G3);
\node (H1) at (12,-4) {$.$};
\node (H2) at (13,-3){$.$};
\node (H3) at (14,-4){$.$};
\node (H4) at (13,-5){$.$};
\draw[->] (H2)--(H1);
\draw[->] (H3)--(H2);
\draw[->] (H4)--(H1);
\draw[->] (H3)--(H4);
\draw [loosely dotted, thick] (H1)..controls (13,-3.25)..(H3);
\draw [loosely dotted, thick] (H1)..controls (13,-4.75)..(H3);
\end{tikzpicture}}\]
There are 3 algebras of global dimension $\leq 2$ which are of cluster type $\widetilde{A}_{2,2}$ and not derived equivalent to $\widetilde{A}_{2,2}$. They are all derived equivalent to each other and not piecewise hereditary:
\[\scalebox{.8}{
\begin{tikzpicture}[scale=1.3,>=stealth]
\node (A1) at (0,0) {$.$};
\node (A2) at (1,1){$.$};
\node (A3) at (2,0){$.$};
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{
"timestamp": "2012-03-13T01:02:56",
"yymm": "1009",
"arxiv_id": "1009.4065",
"language": "en",
"url": "https://arxiv.org/abs/1009.4065"
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|
\section{Introduction}\label{sec1}
Throughout this paper, we will work over an algebraically closed
field $k$ of characteristic $0$.
The problem of classifying all Hopf algebras of dimension $d$, where
$d$ factorizes in a simple way, attracts many mathematicians'
interest. It is also a question posed by Andruskiewitsch
\cite[Question 6.2]{Andruskiewitsch}. As a pioneer, Zhu \cite{Zhu}
proved that a Hopf algebra of prime dimension over $k$ is a group
algebra. Several years later, a series of papers
\cite{Etingof,Gelaki,Masuoka,Masuoka2} proved that semisimple Hopf
algebras of dimension $p^2$ or $pq$ over $k$ are trivial, where
$p,q$ are distinct prime numbers. That is, they are isomorphic to a
group algebra or to a dual group algebra. Quite recently, Etingof et
al \cite{Etingof2} completed the classification of semisimple Hopf
algebras of dimension $pq^2$ and $pqr$, where $p,q,r$ are distinct
prime numbers. The results in \cite{Etingof2} showed that all these
Hopf algebras can be constructed from group algebras and their duals
by means of extensions.
In this paper, we study the structure of semisimple Hopf algebras of
dimension $p^2q^2$, where $p,q$ are prime numbers with $p^4<q$. As
an application, we also study the structure of semisimple Hopf
algebras of dimension $4q^2$, where $q$ is a prime number.
The paper is organized as follows. In Section \ref{sec2}, we recall
the definitions and basic properties of semisolvability, characters
and Radford's biproducts, respectively.
In Section \ref{sec3}, we study the structure of semisimple Hopf
algebras of dimension $p^2q^2$, where $p,q$ are prime numbers with
$p^4<q$. By checking the order of $G(H^*)$, we prove that if
$|G(H^*)|=p,pq,q^2$ or $pq^2$ then $H$ is not simple as a Hopf
algebra and is semisolvable, in the sense of \cite{Montgomery}; if
$|G(H^*)|=p^2$ or $p^2q$ then $H$ is either semisolvable or
isomorphic to a Radford's biproduct $R\# kG$, where $kG$ is the
group algebra of group $G$ of order $p^2$, $R$ is a semisimple
Yetter-Drinfeld Hopf algebra in ${}^{kG}_{kG}\mathcal{YD}$ of
dimension $q^2$. The possibility that $|G(H^*)|=1$ and $q$ can be
discarded. In particular, we prove that if $p$ does not divide $q-1$
and $q+1$, then $H$ is necessarily semisolvable.
In Section \ref{sec4}, we study the structure of semisimple Hopf
algebras of dimension $4q^2$, where $q$ is a prime number. In view
of the results in Section \ref{sec3}, we discuss the cases that
$q=3,5,7,11$ and $13$.
Throughout this paper, all modules and comodules are left modules
and left comodules, and moreover they are finite-dimensional over
$k$. $\otimes$, ${\rm dim}$ mean $\otimes _k$, ${\rm dim}_k$,
respectively. For two positive integers $m$ and $n$, $gcd(m,n)$
denotes the greatest common divisor of $m,n$. Our references for the
theory of Hopf algebras are \cite{Montgomery2} or \cite{Sweedler}.
The notation for Hopf algebras is standard. For example, the group
of group-like elements in $H$ is denoted by $G(H)$.
\section{Preliminaries}\label{sec2}
\subsection{Semisolvability}\label{sec2-1}
Let $H$ be a finite-dimensional Hopf algebra over $k$. A Hopf
subalgebra $A\subseteq H$ is called normal if $h_1AS(h_2)\subseteq
A$ and $S(h_1)Ah_2\subseteq A$, for all $h\in H$. If $H$ does not
contain proper normal Hopf subalgebras then it is called simple. The
notion of simplicity is self-dual, that is, $H$ is simple if and
only if $H^*$ is simple.
Let $q:H\to B$ be a Hopf algebra map and consider the subspaces of
coinvariants
$$H^{coq}=\{h\in H|(id\otimes q)\Delta(h)=h\otimes 1\}, \mbox{and\,}$$
$$^{coq}\!H=\{h\in H|(q\otimes id)\Delta(h)=1\otimes h\}.$$
Then $H^{coq}$ (respectively, $^{coq}H$) is a left (respectively,
right) coideal subalgebra of $H$. Moreover, we have
$${\rm dim}H ={\rm dim}H^{coq}{\rm dim}q(H) ={\rm dim}{}^{coq}H{\rm dim}q(H).$$
The left coideal subalgebra $H^{coq}$ is stable under the left
adjoint action of $H$. Moreover $H^{coq} ={}^{coq}H$ if and only if
$H^{coq}$ is a (normal) Hopf subalgebra of $H$. If this is the case,
we shall say that the map $q:H\to B$ is normal. See \cite{Schneider}
for more details.
The following lemma comes from \cite[Section 1.3]{Natale4}.
\begin{lem}\label{lem5}
Let $q:H\to B$ be a Hopf epimorphism and $A$ a Hopf subalgebra of
$H$ such that $A\subseteq H^{coq}$. Then ${\rm dim}A$ divides ${\rm
dim}H^{coq}$.
\end{lem}
The notions of upper and lower semisolvability for
finite-dimensional Hopf algebras have been introduced in
\cite{Montgomery}, as generalizations of the notion of solvability
for finite groups. By definition, $H$ is called lower semisolvable
if there exists a chain of Hopf subalgebras
$$H_{n+1} = k\subseteq
H_{n}\subseteq\cdots \subseteq H_1 = H$$ such that $H_{i+1}$ is a
normal Hopf subalgebra of $H_i$, for all $i$, and all quotients
$H_{i}/H_{i}H^+_{i+1}$ are trivial. Dually, $H$ is called upper
semisolvable if there exists a chain of quotient Hopf algebras
$$H_{(0)} =
H\xrightarrow{\pi_1}H_{(1)}\xrightarrow{\pi_2}\cdots\xrightarrow{\pi_n}H(n)
= k$$ such that each of the maps $H_{(i-1)}\xrightarrow{\pi_i}
H_{(i)}$ is normal, and all $H_{(i-1)}^{co\pi_i}$ are trivial.
By \cite[Corollary 3.3]{Montgomery}, we have that $H$ is upper
semisolvable if and only if $H^*$ is lower semisolvable. If this is
the case, then $H$ can be obtained from group algebras and their
duals by means of (a finite number of) extensions. For the
definition of the extension of Hopf algebras, the reader is directed
to \cite[Definition 1.3]{Masuoka3}.
Recall that a semisimple Hopf algebra $H$ is called of Frobenius
type if the dimensions of the simple $H$-modules divide the
dimension of $H$. Kaplansky conjectured that every
finite-dimensional semisimple Hopf algebra is of Frobenius type
\cite[Appendix 2]{Kaplansky}. It is still an open problem. Recently,
many examples show that a positive answer to Kaplansky's conjecture
would be very helpful in the classification problem. For example, in
case that ${\rm dim}H$ is a product of two distinct prime numbers,
Gelaki and Westreich \cite{Gelaki} proved that if $H$ and $H^*$ are
of Frobenius type then $H$ is trivial.
The following result is not explicitly stated in \cite{Etingof2}. We
give a proof for completeness.
\begin{lem}\label{lem1}
Let $H$ be a semisimple Hopf algebra of dimension $p^mq^n$, where
$p,q$ are distinct prime numbers and $m,n$ are non-negative integer.
Then $H$ is of Frobenius type and $H$ has a non-trivial
$1$-dimensional representation.
\end{lem}
The proof of Lemma \ref{lem1} involves some definitions and
properties from fusion categories. We refer the reader to
\cite{Etingof2} and references therein for basic results on fusion
category.
\begin{proof}
Let $Rep(H)$ be the category of representations of $H$. By
\cite[Theorem 1.6]{Etingof2}, $Rep(H)$ is a solvable fusion
category. Comparing \cite[Definition 1.1]{Etingof2} with
\cite[Definition 1.2]{Etingof2}, we find out that $Rep(H)$ is also
weakly group-theoretical. The first statement then follows from
\cite[Theorem 1.5]{Etingof2}. The second statement directly follows
from \cite[Proposition 9.9]{Etingof2}.
\end{proof}
\begin{lem}\label{lem2}
Let $H$ be a semisimple Hopf algebra of dimension $p^2q^2$, where
$p<q$ are prime numbers. If $H$ has a Hopf subalgebra $K$ of
dimension $pq^2$ then $H$ is lower semisolvable.
\end{lem}
\begin{proof}
Since the index of $K$ in $H$ is $p$ which is the smallest prime
number dividing ${\rm dim}H$, the result in \cite{Kobayashi} shows
that $K$ is a normal Hopf algebra of $H$. Since the dimension of the
quotient $H/HK^+$ is $p$, the result in \cite{Zhu} shows that it is
trivial.
Since $K^*$ is also a semisimple Hopf algebra (see \cite{Larson}),
Lemma \ref{lem1} and \cite[Theorem 5.4.1]{Natale3} show that $K$ has
a proper normal Hopf subalgebra $L$ of dimension $p,q,pq$ or $q^2$.
The results in \cite{Etingof,Gelaki,Masuoka,Masuoka2} (mentioned in
Section \ref{sec1}) show that $L$ and $K/KL^+$ are both trivial.
Hence, we have a chain of Hopf subalgebras $k\subseteq L\subseteq
K\subseteq H$, which satisfies the definition of lower
semisolvability.
\end{proof}
\subsection{Characters}\label{sec2-2}
Throughout this section, $H$ will be a semisimple Hopf algebra over
$k$.
Let $V$ be an $H$-module. The character of $V$ is the element
$\chi=\chi_V\in H^*$ defined by $\langle\chi,h\rangle={\rm Tr}_V(h)$
for all $h\in H$. The degree of $\chi$ is defined to be the integer
${\rm deg}\chi=\chi(1)={\rm dim}V$. We shall use $X_t$ to denote the
set of all irreducible characters of $H$ of degree $t$. If $U$ is
another $H$-module, we have $$\chi_{U\otimes
V}=\chi_U\chi_V,\quad\chi_{V^*}=S(\chi_V),$$ where $S$ is the
antipode of $H^*$.
Hence, the irreducible characters, namely, the characters of the
simple $H$-modules, span a subalgebra $R(H)$ of $H^*$, which is
called the character algebra of $H$. By \cite[Lemma 2]{Zhu}, $R(H)$
is semisimple. The antipode $S$ induces an anti-algebra involution
$*: R(H)\to R(H)$, given by $\chi\to\chi^*:=S(\chi)$. The character
of the trivial $H$-module is the counit $\varepsilon$.
The properties of $R(H)$ have been intensively studied in
\cite{Nichols}. We recall some of them here, and will use them
freely in this paper. See also \cite[Section 1.2]{Natale4}.
Let $\chi_U,\chi_V\in R(H)$ be the characters of the $H$-modules $U$
and $V$, respectively. The integer $m(\chi_U,\chi_V)={\rm
dimHom}_H(U,V)$ is defined to the the multiplicity of $U$ in $V$.
This can be extended to a bilinear form $m:R(H)\times R(H)\to k$.
Let $\widehat{H}$ denote the set of irreducible characters of $H$.
Then $\widehat{H}$ is a basis of $R(H)$. If $\chi\in R(H)$, we may
write $\chi=\sum_{\alpha\in \widehat{H}}m(\alpha,\chi)\alpha$. Let
$\chi,\psi,\omega\in R(H)$. Then
$m(\chi,\psi\omega)=m(\psi^*,\omega\chi^*)=m(\psi,\chi\omega^*)$ and
$m(\chi,\psi)=m(\chi^*,\psi^*)$. See \cite[Theorem 9]{Nichols}.
For each group-like element $g$ in $G(H^*)$, we have
$m(g,\chi\psi)=1$, if $\psi=\chi^*g$ and $0$ otherwise for all
$\chi,\psi\in \widehat{H}$. In particular, $m(g,\chi\psi)=0$ if
$deg(\chi)\neq deg(\psi)$. Let $\chi\in \widehat{H}$. Then for any
group-like element $g$ in $G(H^*)$, $m(g,\chi\chi^{*})>0$ if and
only if $m(g,\chi\chi^{*})= 1$ if and only if $g\chi=\chi$. The set
of such group-like elements forms a subgroup of $G(H^*)$, of order
at most $({\rm deg}(\chi))^2$. See \cite[Theorem 10]{Nichols}.
Denote this subgroup by $G[\chi]$. In particular, we have
$$\chi\chi^*=\sum_{g\in G[\chi]}g+\sum_{\alpha\in \widehat{H},{\rm
deg}\alpha>1}m(\alpha,\chi\chi^*)\alpha.$$
The following result can be found in \cite[Lemma 2.2.2]{Natale1}.
\begin{lem}\label{lem3}
Let $\chi\in \widehat{H}$ be an irreducible character of $H$. Then
(1)\,The order of $G[\chi]$ divides $({\rm deg}\chi)^2$.
(2)\,The order of $G(H^*)$ divides $n({\rm deg}\chi)^2$, where $n$ is the
number of non-isomorphic simple $H$-modules of dimension ${\rm deg}\chi$.
\end{lem}
Let $1=d_1,d_2,\cdots,d_s$, $n_1,n_2,\cdots,n_s$ be positive
integers, with $d_1<d_2<\cdots<d_s$. $H$ is said to be of type
$(d_1,n_1;\cdots;d_s,n_s)$ as an algebra if $d_1,d_2,\cdots,d_s$ are
the dimensions of the simple $H$-modules and $n_i$ is the number of
the non-isomorphic simple $H$-modules of dimension $d_i$. That is,
as an algebra, $H$ is isomorphic to a direct product of full matrix
algebras $$H\cong k^{(n_1)}\times
\prod_{i=2}^{s}M_{d_i}(k)^{(n_i)}.$$
If $H^*$ is of type $(d_1,n_1;\cdots;d_s,n_s)$ as an algebra, then
$H$ is said to be of type $(d_1,n_1;\cdots;d_s,n_s)$ as a coalgebra.
A subalgebra $A$ of $R(H)$ is called a standard subalgebra if $A$ is
spanned by irreducible characters of $H$. Let $X$ be a subset of
$\widehat{H}$. Then $X$ spans a standard subalgebra of $R(H)$ if and
only if the product of characters in $X$ decomposes as a sum of
characters in $X$. There is a bijection between $*$-invariant
standard subalgebras of $R(H)$ and quotient Hopf algebras of $H$.
See \cite[Theorem 6]{Nichols}.
\begin{lem}\label{lem4}
Let $G$ be a non-trivial subgroup of $G(H^*)$. If $G[\chi_t]=G$ for
every $\chi_t\in X_t$, then $\chi_t\chi_t'$ is not irreducible for
all $\chi_t,\chi_t'\in X_t$.
\end{lem}
\begin{proof}
This is a consequence of \cite[Lemma 2.4.1]{Natale4}.
\end{proof}
\subsection{Radford's biproduct}\label{sec2-3} In what follows, we
briefly summarize results from \cite{Radford}. Let $A$ be a
semisimple Hopf algebra and let ${}^A_A\mathcal{YD}$ denote the
braided category of Yetter-Drinfeld modules over $A$. Let $R$ be a
semisimple Yetter-Drinfeld Hopf algebra in ${}^A_A\mathcal{YD}$.
Denote by $\rho :R\to A\otimes R$, $\rho (a)=a_{-1} \otimes a_0 $,
and $\cdot :A\otimes R\to R$, the coaction and action of $A$ on $R$,
respectively. We shall use the notation $\Delta (a)=a^1\otimes a^2$
and $S_R $ for the comultiplication and the antipode of $R$,
respectively.
Since $R$ is in particular a module algebra over $A$, we can form
the smash product (see \cite[Definition 4.1.3]{Montgomery}). This is
an algebra with underlying vector space $R\otimes A$, multiplication
is given by $$(a\otimes g)(b\otimes h)=a(g_1 \cdot b)\otimes g_2 h,
\mbox{\;for all\;}g,h\in A,a,b\in R,$$ and unit $1=1_R\otimes1_A$.
Since $R$ is also a comodule coalgebra over $A$, we can dually form
the smash coproduct. This is a coalgebra with underlying vector
space $R\otimes A$, comultiplication is given by $$\Delta (a\otimes
g)=a^1\otimes (a^2)_{-1} g_1 \otimes (a^2)_0 \otimes g_2
,\mbox{\;for all\;}h\in A,a\in R, $$ and counit
$\varepsilon_R\otimes\varepsilon_A$.
As observed by D. E. Radford (see \cite[Theorem 1]{Radford}), the
Yetter-Drinfeld condition assures that $R\otimes A$ becomes a Hopf
algebra with these structures. This Hopf algebra is called the
Radford's biproduct of $R$ and $A$. We denote this Hopf algebra by $
R\#A$ and write $a\# g=a\otimes g$ for all $g\in A,a\in R$. Its
antipode is given by
$$S(a\# g)=(1\# S(a_{-1} g))(S_R (a_0 )\# 1),\mbox{\;for
all\;}g\in A,a\in R.$$
A biproduct $R\#A$ as described above is characterized by the
following property(see \cite[Theorem 3]{Radford}): suppose that $H$
is a finite-dimensional Hopf algebra endowed with Hopf algebra maps
$\iota:A\to H$ and $\pi:H\to A$ such that $\pi \iota:A\to A$ is an
isomorphism. Then the subalgebra $R= H^{co\pi}$ has a natural
structure of Yetter-Drinfeld Hopf algebra over $A$ such that the
multiplication map $R\#A\to H$ induces an isomorphism of Hopf
algebras.
The following theorem is a direct consequence of \cite[Lemma
4.1.9]{Natale4}. We give the proof for the sake of completeness.
\begin{thm}\label{prop1}
Let $H$ be a semisimple Hopf algebra of dimension $p^2q^2$, where
$p,q$ are distinct prime numbers. If $gcd(|G(H)|,|G(H^*)|)=p^2$,
then $H\cong R\#kG$ is a biproduct, where $kG$ is the group algebra
of group $G$ of order $p^2$, $R$ is a semisimple Yetter-Drinfeld
Hopf algebra in $^{kG}_{kG}\mathcal{YD}$ of dimension $q^2$.
\end{thm}
\begin{proof}
By assumption and Sylow Theorem, $G(H^*)$ has a subgroup $K$ of
order $p^2$. Considering the Hopf algebra map $q: H\to (kK)^*$
obtained by transposing the inclusion $kK\subseteq H^*$, we have
that ${\rm dim}H^{coq}=q^2$. Again by assumption and Sylow Theorem,
$G(H)$ also has a subgroup $G$ of order $p^2$. If there exists an
element $1\neq g\in G$ such that $g$ appears in $H^{coq}$, then
$k\langle g\rangle\subseteq H^{coq}$ since $H^{coq}$ is a subalgebra
of $H$. But this contradicts Lemma \ref{lem5} since ${\rm
dim}k\langle g\rangle$ does not divide ${\rm dim}H^{coq}$.
Therefore, $H^{coq}\cap kG=k1$. This means that the restriction
$q|_{kG}$ is injective, and hence $q|_{kG}: kG\to (kK)^*$ is an
isomorphism. Finally, from the discussion above, we know that
$H\cong R\# kG$ is a biproduct, where $R=H^{coq}$.
\end{proof}
\section{Semisimple Hopf algebras of dimension $p^2q^2$}\label{sec3}
Let $p,q$ be distinct prime numbers with $p^4<q$, and $H$ a
semisimple Hopf algebra of dimension $p^2q^2$. By Nichols-Zoeller
Theorem \cite{Nichols2}, the order of $G(H^*)$ divides ${\rm dim}H$.
Moreover, $|G(H^*)|\neq1$ by Lemma \ref{lem1}. Again by Lemma
\ref{lem1}, the dimension of a simple $H$-module can only be
$1,p,p^2$ or $q$. Let $a,b,c$ be the number of non-isomorphic simple
$H$-modules of dimension $p,p^2$ and $q$, respectively. It follows
that we have an equation $p^2q^2=|G(H^*)|+ap^2+bp^4+cq^2$. In
particular, if $|G(H^*)|=p^2q^2$ then $H$ is a dual group algebra.
The proof of the following lemma is direct.
\begin{lem}\label{lem6}
The irreducible characters of degree $1, p$ and $p^2$ span a
standard subalgebra of $R(H)$ corresponding to a quotient Hopf
algebra $\overline{H}$ of $H$ of dimension $|G(H^*)|+ap^2+bp^4$. In
particular, $|G(H^*)|$ divides ${\rm dim}\overline{H}$ and
$|G(H^*)|+ap^2+bp^4$ divides ${\rm dim}H$.
\end{lem}
\begin{lem}\label{lem7}
If $|G(H^*)|=p$ or $pq$, then $H$ is upper semisolvable.
\end{lem}
\begin{proof}
First, $c\neq0$, since otherwise we get the contradiction $p^2\mid
p$.
Consider the quotient Hopf algebra $\overline{H}$ from Lemma
\ref{lem6}. Then $p\mid{\rm dim}\overline{H}$ and since $c\neq0$,
then ${\rm dim}\overline{H} < p^2q^2$. Therefore ${\rm
dim}\overline{H} = p, pq, p^2q, pq^2$ or $p^2$. Moreover, ${\rm
dim}\overline{H}\neq p^2$, since otherwise
$(\overline{H})^*\subseteq kG(H^*)$ by \cite{Masuoka2}, but $p^2
={\rm dim}\overline{H}$ does not divide $|G(H^*)|= p$ or $pq$.
The possibilities ${\rm dim}\overline{H}= p, pq$ or $p^2q$ lead,
respectively to the contradictions $p^2q^2= p+cq^2$,
$p^2q^2=pq+cq^2$ and $p^2q^2=p^2q+cq^2$. Hence these are also
discarded, and therefore ${\rm dim}\overline{H}=pq^2$. This implies
that $H$ is upper semisolvable, by Lemma \ref{lem2}.
\end{proof}
\begin{lem}\label{lem8}
$|G(H^*)|\neq q$.
\end{lem}
\begin{proof}
Suppose on the contrary that $|G(H^*)|= q$. By Lemma \ref{lem6},
${\rm dim}\overline{H}=q+ap^2+bp^4$. On the other hand, the product
of irreducible characters of $\overline{H}$ of degree $>1$ cannot
contain nontrivial characters of degree $1$, by Lemma \ref{lem3}(1).
If $a\neq0$ or $b \neq0$, this would imply $p^2=1+mp$ or $p^4=1+mp$
for some positive integer $m$, which is impossible. Therefore
$a=b=0$. So we have $p^2q^2=q+cq^2$, which is a contradiction.
\end{proof}
\begin{lem}\label{lem9}
If $|G(H^*)|=q^2$, then $H$ is upper semisolvable.
\end{lem}
\begin{proof}
A similar argument as in Lemma \ref{lem8} shows that $a=b=0$. Hence,
$H$ is of type $(1,q^2;q,p^2-1)$ as an algebra. Equivalently, $H^*$
is of type $(1,q^2;q,p^2-1)$ as a coalgebra. The group $G(H^*)$,
being abelian, acts by left multiplication on the set $X_q$. The set
$X_q$ is a union of orbits which have length $1,q$ or $q^2$. Since
$|X_q|=p^2-1$ is less than $q$, every orbit has length $1$. That is,
$G[\chi_q]=G(H^*)$ for all $\chi_q\in X_q$. Let $g\in G(H^*)$ and
$\chi_q\in X_q$. Then $g\chi_q=\chi_q$ and
$g^{-1}\chi_q^*=\chi_q^*$. This means that $g\chi_q=\chi_qg=\chi_q$.
Let $C_i (i=1,\cdots,p^2-1)$ be the non-isomorphic $q^2$-dimensional
simple subcoalgebra of $H^*$. Then $gC_i=C_i=C_ig$ for all $g\in
G(H^*)$. By \cite[Proposition 3.2.6]{Natale4}, $G(H^*)$ is normal in
$k[C_1,\cdots,C_{p^2-1}]$, where $k[C_1,\cdots,C_{p^2-1}]$ denotes
the subalgebra generated by $C_1,\cdots,C_{p^2-1}$. It is a Hopf
subalgebra of $H^*$ containing $G(H^*)$. Counting dimension, we
know $k[C_1,\cdots,C_{p^2-1}]=H^*$. Since $kG(H^*)$ is a group
algebra and the quotient $H^*/H^*(kG(H^*))^+$ is trivial (see
\cite{Masuoka2}), $H^*$ is lower semisolvable. Hence, $H$ is upper
semisolvable.
\end{proof}
From the discussion above, the following lemma is obvious.
\begin{lem}
If $|G(H^*)|=pq^2$, then $H$ is upper semisolvable.
\end{lem}
\begin{lem}\label{lem11}
If $|G(H^*)|=p^2$ or $p^2q$ then $H$ is either semisolvable or
isomorphic to a Radford's biproduct $R\# kG$, where $kG$ is the
group algebra of group $G$ of order $p^2$, $R$ is a semisimple
Yetter-Drinfeld Hopf algebra in ${}^{kG}_{kG}\mathcal{YD}$ of
dimension $q^2$.
\end{lem}
\begin{proof}
This is a corollary of Lemma \ref{lem2}.
\end{proof}
We are now in a position to give the main theorem.
\begin{thm}\label{thm1}
Let $H$ be a semisimple Hopf algebra of dimension $p^2q^2$, where
$p,q$ are prime numbers with $p^4<q$. Then $H$ is either
semisolvable or isomorphic to a Radford's biproduct $R\# kG$, where
$kG$ is the group algebra of group $G$ of order $p^2$, $R$ is a
semisimple Yetter-Drinfeld Hopf algebra in
${}^{kG}_{kG}\mathcal{YD}$ of dimension $q^2$.
\end{thm}
In analogy with the situations for finite groups, it is enough for
many applications to know that a Hopf algebra is semisolvable. Under
certain restrictions on $p$ and $q$, we can obtain a more precise
result.
\begin{cor}
If $p$ does not divide $q-1$ or $q+1$, then $H$ is semisolvable.
\end{cor}
\begin{proof}
It suffices to consider the case that the order of $G(H)$ and
$G(H^*)$ are $p^2$ or $p^2q$, and $H$ is a biproduct. Let $q: H\to
(kK)^*$ be the projection in Theorem \ref{prop1}. Then we have that
${\rm dim}H^{coq}=q^2$. We then consider the decomposition of
$H^{coq}$ as a coideal of $H$. Let $c$ be the number of
non-isomorphic irreducible left coideals of $H$ of dimension $q$. If
$|G(H)|=p^2$ then $c=0$, otherwise Lemma \ref{lem3} (2) shows that
$cq^2\geq p^2q^2$, a contradiction. If $|G(H)|=p^2q$ then $c=0$ by a
similar argument. Hence, by Lemma \ref{lem5}, there are $2$ possible
decompositions of $H^{coq}$ as a coideal of $H$:
$$H^{coq}=k1\oplus\sum_iV_i\oplus\sum_jW_j,\mbox{\,or\,}H^{coq}=kG\oplus\sum_iV_i\oplus\sum_jW_j,$$
where $V_i$ is an irreducible left coideal of $H$ of dimension $p$,
$W_i$ is an irreducible left coideal of $H$ of dimension $p^2$ and
$G$ is a subgroup of $G(H)$ of order $q$. Counting dimensions on
both sides, we have $q^2=1+mp$ or $q^2=q+np$ for some positive
integers $m,n$. This contradicts the assumption that $p$ does not
divide $q-1$ and $q+1$.
\end{proof}
As an immediate consequence of the discussions in this section, we
have the following corollary.
\begin{cor}
If $H$ is simple as a Hopf algebra then $H$ is isomorphic to a
Radford's biproduct $R\# kG$, where $kG$ is the group algebra of
group $G$ of order $p^2$, $R$ is a semisimple Yetter-Drinfeld Hopf
algebra in ${}^{kG}_{kG}\mathcal{YD}$ of dimension $q^2$.
\end{cor}
The following example was pointed out to the author by the anonymous
referee.
\begin{ex}
In fact, examples of nontrivial semisimple Hopf algebras of
dimension $p^2q^2$ which are Radford's biproducts in such a way, and
are simple as Hopf algebras do exists. A construction of such
examples as twisting deformations of certain groups appears in
\cite[Remark 4.6]{Galindo}.
\end{ex}
\section{Semisimple Hopf algebras of dimension $4q^2$}\label{sec4}
Let $q$ be a prime number, and $H$ a semisimple Hopf algebra of
dimension $4q^2$. In this section, we discuss the structure of $H$.
By Theorem \ref{thm1}, it suffices to consider the cases that
$q=3,5,7,11$ and $13$. By Nichols-Zoeller Theorem and Lemma
\ref{lem1}, the order of $G(H^*)$ is $2,4,q,q^2,2q,4q$,$2q^2$ or
$4q^2$. Moreover, if $|G(H^*)|=4q^2$ then $H$ is a dual group
algebra. The dimension of a simple $H$-module can only be $1,2,4$ or
$q$. Let $a,b,c$ be the number of non-isomorphic simple $H$-module
of dimension $2,4$ and $q$, respectively. Then we have
$4q^2=|G(H^*)|+4a+16b+cq^2$. In particular, if $c\neq 0$ then
$c=1,2$ or $3$. By \cite[Chapter 8]{Natale4}, if ${\rm dim}H=36$
then $H$ is upper semisolvable or lower semisolvable. Therefore, we
may assume that $q=5,7,11$ or $13$ in the followings.
\begin{lem}\label{lem12}
If $|G(H^*)|=2$ then $H$ is upper semisolvable.
\end{lem}
\begin{proof}
We first note that $c\neq 0$, otherwise $4q^2=2+4a+16b$ will give
rise to a contradiction $2(q^2-a-4b)=1$. That is, $c=1,2$ or $3$.
We then consider the case that $a\neq 0$. Let $\chi_2\in X_2$. Since
$H$ does not have irreducible characters of degree $3$, we have
$G[\chi_2]=G(H^*)$. Then a similar argument as in Lemma \ref{lem7}
shows that $G(H^*)\cup X_2$ spans a standard subalgebra of $R(H)$.
Hence, $H$ has a quotient Hopf algebra of dimension $2+4a$, and
$2+4a$ divides $4q^2$. Since $q$ is odd, $2+4a$ can not be $q$ and
$q^2$. If $2+4a=4q^2$ then $c=0$, a contradiction. If $2+4a=4q$ then
$1=2(q-a)$, a contradiction. If $2+4a=2q$, then a direct check, for
$q=5,7,11,13$ and $c=1,2,3$, shows that $b$ is not a integer, a
contradiction. Hence, $2+4a=2q^2$ and $H$ has a quotient Hopf
algebra of dimension $2q^2$. Therefore, $H$ is upper semisolvable by
Lemma \ref{lem2}.
Finally, we consider the case that $a=0$. In this case,
$4q^2=2+16b+cq^2$. A direct check, for $q=5,7,11,13$ and $c=1,2,3$,
shows that above equation holds true only when $b=3,q=5,c=2$ or
$b=6,q=7,c=2$ or $b=15,q=11,c=2$ or $b=21,q=13,c=2$. That is, $H$ is
of type $(1,2;4,3;5,2)$, $(1,2;4,6;7,2)$, $(1,2;4,15;11,2)$ or
$(1,2;4,21;13,2)$ as an algebra. We shall prove that all these can
not happen.
Suppose on the contrary that $H$ is of type $(1,2;4,3;5,2)$ as an
algebra. Let $\chi_4\in X_4$ and $G(H^*)=\{\varepsilon,g\}$. Then
there must exist $\chi_5\in X_5$ such that $1\leq
m(\chi_5,\chi_4\chi_4^*)\leq 3$.
If $m(\chi_5,\chi_4\chi_4^*)=3$ then $m(\chi_4,\chi_5\chi_4)=3$.
This implies that $\chi_5\chi_4=3\chi_4+\chi_4'+\chi_4''$, where
$\chi_4\neq \chi_4',\chi_4\neq\chi_4''\in X_4$. In case $\chi_4'\neq
\chi_4''$, we have
$m(\chi_4',\chi_5\chi_4)=m(\chi_5,\chi_4'\chi_4^*)=1$. This implies
that $\chi_4'\chi_4^*=\chi_5+\varphi$, where $m(\chi_5,\varphi)=0$
and $deg\varphi=11$. Since $\chi_4'\neq\chi_4$, $\varepsilon$ can
not appear in $\varphi$. From the introduction in Section
\ref{sec2-2}, we know that the multiplicity of $g$ in $\varphi$ is
less than $2$. Hence, $\varphi=2\chi_5'+g$, where $\chi_5\neq
\chi_5'\in X_5$. From $m(g,\chi_4'\chi_4^*)=m(\chi_4',g\chi_4)=1$,
we have $g\chi_4=\chi_4'$. Hence,
$\chi_4'\chi_4^*=g\chi_4\chi_4^*=g+\chi_5+2\chi_5'$. This means that
$\chi_4\chi_4^*=\varepsilon+g\chi_5+2g\chi_5'=\varepsilon+\chi_5'+2\chi_5$.
In the second equality, we use the fact that $g\chi_5=\chi_5'$ which
is deduced from the fact that
$G[\chi_5]=G[\chi_5']=\{\varepsilon\}$. This contradicts the
assumption that $m(\chi_5,\chi_4\chi_4^*)=3$. In case
$\chi_4'=\chi_4''$, we have
$m(\chi_4',\chi_5\chi_4)=m(\chi_5,\chi_4'\chi_4^*)=2$. This implies
that $\chi_4'\chi_4^*=2\chi_5+\chi_5'+g$. A similar argument shows
that it is also a contradiction.
If $m(\chi_5,\chi_4\chi_4^*)=2$ then $m(\chi_4,\chi_5\chi_4)=2$.
This implies that $\chi_5\chi_4=2\chi_4+2\chi_4'+\chi_4''$, where
$\chi_4\neq \chi_4',\chi_4\neq\chi_4''\in X_4$. In case $\chi_4'=
\chi_4''$, we have
$m(\chi_4',\chi_5\chi_4)=m(\chi_5,\chi_4'\chi_4^*)=3$. This implies
that $\chi_4'\chi_4^*=3\chi_5+g$. Then
$1=m(g,\chi_4'\chi_4^*)=m(\chi_4',g\chi_4)$ implies that
$g\chi_4=\chi_4'$. Hence,
$\chi_4'\chi_4^*=g\chi_4\chi_4^*=g+3\chi_5$. This means that
$\chi_4\chi_4^*=\varepsilon+3g\chi_5=\varepsilon+3\chi_5'$. This
contradicts the assumption that $m(\chi_5,\chi_4\chi_4^*)=2$. In
case $\chi_4'\neq \chi_4''$, we have
$m(\chi_4',\chi_5\chi_4)=m(\chi_5,\chi_4'\chi_4^*)=2$. This implies
that $\chi_4'\chi_4^*=2\chi_5+\chi_5'+g$, where $\chi_5\neq
\chi_5'\in X_5$. A similar argument shows that
$\chi_4\chi_4^*=\varepsilon+2\chi_5'+\chi_5$. This also contradicts
the assumption.
If $m(\chi_5,\chi_4\chi_4^*)=1$ then
$\chi_4\chi_4^*=\varepsilon+\chi_5+2\chi_5'$, where
$\chi_5\neq\chi_5'\in X_5$. In this case,
$m(\chi_5',\chi_4\chi_4^*)=2$. From the discussion above, we know it
is impossible.
Suppose on the contrary that $H$ is of type $(1,2;4,6;7,2)$ as an
algebra. Let $\chi_4\in X_4$ and $G(H^*)=\{\varepsilon,g\}$. Then
there must exist $\chi_7\in X_7$ such that $1\leq
m(\chi_7,\chi_4\chi_4^*)\leq 2$.
If $m(\chi_7,\chi_4\chi_4^*)=1$ then $m(\chi_4,\chi_7\chi_4)=1$.
This implies that $\chi_7\chi_4=\chi_4+\varphi$, where
$m(\chi_4,\varphi)=0$ and $deg\varphi=24$. A direct check shows that
there is no irreducible character of degree $7$ in $\varphi$ and
there exists $\chi_4\neq\chi_4'\in X_4$ such that
$m(\chi_4',\chi_7\chi_4)=2$. Then $m(\chi_7,\chi_4'\chi_4^*)=2$
implies that $\chi_4'\chi_4^*=2\chi_7+\psi$, where $deg\psi=2$.
Since $\chi_4\neq \chi_4'$, $\varepsilon$ does not appear in the
decomposition of $\psi$. Hence, $\psi$ is irreducible or a sum of
$2$ copies of $g$. It is impossible.
If $m(\chi_7,\chi_4\chi_4^*)=2$ then $m(\chi_4,\chi_7\chi_4)=2$.
This implies that $\chi_7\chi_4=2\chi_4+\varphi$, where
$m(\chi_4,\varphi)=0$ and $deg\varphi=20$. From the discussion
above, we know there does not exist $\chi_4\neq\chi_4'\in X_4$ such
that $m(\chi_4',\chi_7\chi_4)=2$. Then we have
$\chi_7\chi_4=2\chi_4+\sum_{i=1}^5\varphi_i$, where
$\{\chi_4,\varphi_1,\cdots,\varphi_5\}=X_4$. From
$m(\varphi_i,\chi_7\chi_4)=m(\chi_7,\varphi_i\chi_4^*)=1$, we have
$\varphi_i\chi_4^*=\chi_7+\psi_i$, where $deg\psi_i=9$ and
$m(\chi_7,\psi_i)=0$. It is clear that $m(g,\psi_i)=1$ for all $i$.
Then $m(g,\varphi_i\chi_4^*)=m(\varphi_i,g\chi_4)$ implies that
$\varphi_i=g\chi_4$. This means that $\varphi_1=\cdots=\varphi_5$, a
contradiction.
Suppose on the contrary that $H$ is of type $(1,2;4,15;11,2)$ as an
algebra. Let $\chi_4\in X_4$ and $G(H^*)=\{\varepsilon,g\}$. Then
$\chi_4\chi_4^*=\varepsilon+\chi_{11}+\varphi_1$, where
$\chi_{11}\in X_{11}$ and $\varphi_1\in X_4$. From
$m(\chi_{11},\chi_4\chi_4^*)=m(\chi_4,\chi_{11}\chi_4)=1$, we have
$\chi_{11}\chi_4=\chi_4+\varphi$, where $m(\chi_4,\varphi)=0$ and
$deg\varphi=40$. A direct check shows that there exists
$\chi_4\neq\chi_4'\in X_4$ such that
$m(\chi_4',\chi_{11}\chi_4)=m(\chi_{11},\chi_4'\chi_4^*)=1$. This
means that $\chi_4'\chi_4^*=\chi_{11}+\varphi_2+g$, where
$\varphi_2\in X_4$. Then $m(g,\chi_4'\chi_4^*)=m(\chi_4',g\chi_4)$
implies that $\chi_4'=g\chi_4$. Hence,
$\chi_4'\chi_4^*=g\chi_4\chi_4^*=\chi_{11}+\varphi_2+g$ implies that
$\chi_4\chi_4^*=g\chi_{11}+g\varphi_2+\varepsilon$. On the other
hand, $\chi_4\chi_4^*=\varepsilon+\chi_{11}+\varphi_1$. Hence,
$\chi_{11}=g\chi_{11}$. This means that $g$ appears in the
decomposition of $\chi_{11}\chi_{11}^*$, and hence
$G[\chi_{11}]=G(H^*)$. This contradicts the fact that the order of
$G[\chi_{11}]$ divides $121$ (See Lemma \ref{lem3}).
Suppose on the contrary that $H$ is of type $(1,2;4,21;13,2)$ as an
algebra. Let $\chi_4\in X_4$. Then the decomposition of
$\chi_4\chi_4^*$ gives a contradiction.
\end{proof}
\begin{lem}\label{lem13}
$|G(H^*)|\neq q$.
\end{lem}
\begin{proof}
Suppose on the contrary that $|G(H^*)|= q$. If $a\neq 0$ we then
take $\chi_2\in X_2$. Since $G[\chi_2]$ is a subgroup of $G(H^*)$
and the order of $G[\chi_2]$ divides $4$ by Lemma \ref{lem3}(1),
$G[\chi_2]=\{\varepsilon\}$ is trivial. This is a contradiction
since $H$ does not have irreducible characters of degree $3$. Hence,
$a=0$ and $4q^2=q+16b+cq^2$. A direct check, for $q=5,7,11,13$ and
$c=0,1,2,3$, shows that above equation holds true only when
$b=22,q=11,c=1$. That is, $H$ is of type $(1,11;4,22;11,1)$ as an
algebra. We shall prove that it is impossible.
Suppose on the contrary that $H$ is of type $(1,11;4,22;11,1)$ as an
algebra. Let $\chi$ be the unique irreducible character of degree
$11$ and $g$ the generator of $G(H^*)$. Then $g\chi=\chi$ and hence
$G[\chi]=G(H^*)$. If $$\chi\chi^*=\chi^2=\sum_{i=1}^{11}g^i+10\chi$$
then $G(H^*)\cup X_{11}$ spans a standard subalgebra of $R(H)$.
Hence, $H$ has a quotient Hopf algebra of dimension $132$. By
Nichols-Zoeller Theorem, it is impossible.
Therefore, there exists $\chi_4\in X_4$ such that
$m(\chi_4,\chi^2)=n\geq1$. From
$m(\chi,\chi_4\chi)=m(\chi,\chi\chi_4^*)=n$, we have
$\chi\chi_4^*\stackrel{(1)}{=}n\chi+\varphi$, where
$deg\varphi=44-11n$ and $m(\chi,\varphi)=0$. On the other hand,
$\chi_4^*\chi_4=\varepsilon+\chi_4'+\chi$, where $\chi_4'\in X_4$.
From $m(\chi,\chi_4^*\chi_4)=m(\chi_4^*,\chi\chi_4^*)=1$, we have
$\chi\chi_4^*\stackrel{(2)}{=}\chi_4^*+\psi$, where $deg\psi=40$ and
$m(\chi_4^*,\psi)=0$. A direct check shows that $\chi$ does not
appear in the decomposition of $\psi$. Hence, $(1)$ and $(2)$ give
rise to a contradiction.
\end{proof}
\begin{lem}\label{lem14}
If $|G(H^*)|=q^2$ then $H$ is upper semisolvable.
\end{lem}
\begin{proof}
A similar argument as in Lemma \ref{lem13} shows that $a=0$, and
hence $3q^2=16b+cq^2$. A direct check, for $c=0,1,2,3$, shows that
$H$ is of type $(1,q^2;q,3)$ as an algebra. The result then follows
from a similar argument as in Lemma \ref{lem9}.
\end{proof}
\begin{lem}\label{lem14}
If $|G(H^*)|=2q$ then $H$ is upper semisolvable.
\end{lem}
\begin{proof}
If $a\neq 0$ then $q^2$ does not divide $a$, otherwise $4a\geq
4q^2$, a contradiction. Then, by Lemma \ref{lem6}, we have that
$2q+4a$ divides $4q^2$. A direct check shows that $2q+4a$ can not be
$q^2$, $4q^2$ and $4q$. Hence, $2q+4a=2q^2$ and $H$ has a quotient
Hopf algebra of dimension $2q^2$. So, $H$ is upper semisolvable by
Lemma \ref{lem2}.
If $a=0$ then $4q^2=2q+16b+cq^2$. A direct check, for $q=5,7,11,13$
and $c=0,1,2,3$, shows that this can not happen.
\end{proof}
The following lemma is obvious.
\begin{lem}
If $|G(H^*)|=2q^2$ then $H$ is upper semisolvable.
\end{lem}
\begin{lem}\label{lem15}
If $|G(H^*)|=4$ or $4q$ then $H$ is either semisolvable or
isomorphic to a Radford's biproduct $R\# kG$, where $kG$ is the
group algebra of group $G$ of order $4$, $R$ is a semisimple
Yetter-Drinfeld Hopf algebra in ${}^{kG}_{kG}\mathcal{YD}$ of
dimension $q^2$.
\end{lem}
\begin{proof}
This is a corollary of Lemma \ref{lem2}.
\end{proof}
Now we reach the main result in this section.
\begin{thm}\label{thm2}
Let $q$ be a prime number, and $H$ a semisimple Hopf algebra of
dimension $4q^2$. Then $H$ is either semisolvable or isomorphic to a
Radford's biproduct $R\# kG$, where $kG$ is the group algebra of
group $G$ of order $4$, $R$ is a semisimple Yetter-Drinfeld Hopf
algebra in ${}^{kG}_{kG}\mathcal{YD}$ of dimension $q^2$.
\end{thm}
\textbf{Acknowledgments:}\quad The author would like to thank the
referee for his/her valuable comments and suggestions, in particular
for providing a new proof of Lemma \ref{lem7}, which shortens the
original version. This work was partially supported by the FNS of
CHINA (10771183) and Docurate foundation of Ministry of Education of
CHINA (200811170001).
|
{
"timestamp": "2010-11-23T02:01:16",
"yymm": "1009",
"arxiv_id": "1009.3541",
"language": "en",
"url": "https://arxiv.org/abs/1009.3541"
}
|
\section{Introduction}
A fractional power $x^r$ of a string~$x$ is defined as
$x^r=xxx\dots xxy$ where $y$ is a prefix of $x$ and
$|x^r|=r|x|$. (We assume that $r>1$ is a fraction with
denominator $|x|$.)
One may ask whether there exists an infinite sequence of letters
that does not contain fractional powers $x^r$ with large $r$ and
long $x$. More precisely, for a given alphabet size $a$, a given
integer $l$ and a given real $\alpha$ one may ask whether there
exists an infinite sequence of letters that does not contain
fractional powers $x^r$ with $r>\alpha$ and $|x|\ge l$.
For $\alpha=1$ the answer is evidently negative (each string $x$
is a fractional power $x^1$). On the other hand, it is easy to
see that for any $a\ge 2$ and $l\ge 1$ the answer is positive if
$\alpha$ is large enough (there exists a binary sequence that
does not contain factors $x^3$). The threshold value that
separates negative and positive answers is denoted by $R(a,l)$
in~\cite{longpowerfree}; the authors note that $1<R(a,l)\le 2$
and compute exact values of $R(a,l)$ for some pairs~$(a,l)$.
Evidently, $R(a,l)$ decreases when $a$ or $l$ increase.
To get a lower bound for $R(a,l)$, let us apply the pigeonhole
principle to $a+1$ letters at positions $0,l,2l,\ldots,al$. Two
of them should be equal and this creates a fractional power
$x^r$ where $|x|\ge l$ and $r\le 1+1/la$ (this power starts and
ends with a letter that appears twice). Therefore,
$$
R(a,l)\ge 1+\frac{1}{la}.
$$
Francesca Fiorenzi, Pascal Ochem and Elise Vaslet
in~\cite{vaslet} gave stronger lower bounds and also some upper
bounds for~$R(a,l)$. In particular, they proved that
$$
1+\frac{1}{1+\left\lfloor\frac{3l+2}{4}(a-1)\right\rfloor}\le
R(a,l)\le 1+\frac{2\ln l}{l\ln\lambda}+O\left(\frac{1}{l}\right),
$$
where $\lambda=\frac{(a-1)+\sqrt{(a-1)(a+3)}}{2}$ and a constant
in~$O$ may depend on~$a$ but not on~$l$.
In this paper we use Lov\'asz local lemma to prove a stronger
upper bound for~$R(a,l)$. Our upper bound differs from the lower
bound only by a constant:
$$
R(a,l)\le 1+\frac{c}{la}
$$
for some $c$ and for all $a\ge 2$, $l\ge 1$.
\section{Kolmogorov complexity of subsequences}
We present the proof using the notion of \emph{Kolmogorov
complexity} (also called \emph{algorithmic complexity} or
\emph{description complexity}). We refer the reader
to~\cite{livitan} or~\cite{shen-lecture-notes} for the
definition and basic properties of Kolmogorov complexity.
For an infinite sequence~$\omega$ and finite set $X\subset
\mathbb{N}$ let $\omega(X)$ be a string of length $\#X$ formed
by $\omega_i$ with $i\in X$ (in the same order as in $\omega$).
We use the following result from~\cite{csr-2007} that guarantees
the existence of a sequence $\omega$ such that strings
$\omega(X)$ have high Kolmogorov complexity for all simple~$X$:
\begin{theorem}\label{subsequence1}
%
Let $\alpha$ be a positive real number less than~$1$. There
exists a binary sequence $\omega$ and an integer $N$ such that
for any finite set $X$ of cardinality at least $N$ the
inequality
$$
\K(X,\omega(X)|t)\ge\alpha\#X
$$
holds for some $t\in A$.
%
\end{theorem}
Here $\K(X,\omega(X)|t)$ is conditional Kolmogorov complexity
of a pair $(X,\omega(X))$ relative to~$t$.
We need a slightly more general version of this result (for any
alphabet size):
\begin{theorem}\label{subsequencea}
%
Let $a\ge 2$ be an integer. Let $\alpha$ be a positive real less
than~$1$. There exists a sequence $\omega$ in $a$-letters
alphabet and an integer $N$ such that for any finite set $X$ of
cardinality at least $N$ the inequality
$$
\K(X,\omega(X)|t)\ge\alpha\#X\log a
$$
holds for some $t\in X$.
%
\end{theorem}
\begin{proof}
%
Theorem~\ref{subsequencea} can be proven using exactly
the same argument as in~\cite{csr-2007} (Lovasz local lemma
technique). It can also be formally derived from
Theorem~\ref{subsequence1} as follows: we encode $a$ letters of
the alphabet by bit blocks of some length $t$ (large enough).
This encoding is not bijective (several blocks encode the same
letter) but is chosen in such a way that all letters have almost
the same number of encodings (about $2^t/a$). Then we take a
sequence from Theorem~\ref{subsequence1}, split it into $t$-bit
blocks and replace these blocks by corresponding letters. If some
subsequence formed by the letters is simple, then the
corresponding bit subsequence is simple, too. (Technically we
should change $\alpha$ slightly to compensate for ``boundary
effects''.)
%
\end{proof}
\iffalse
\proofbegin
%
We fix the numbers $a$, $\alpha$ the alphabet $A$ ($|A|=a$) and
some constants $m\in\mathbb{N}$ and $\alpha'\in(0,1)$ that will be defined
later. Then we construct a binary sequence~$\tau$ by the
Theorem~\ref{subsequence1} and cut it in $m$-bit blocks.
Let us fix some map~$\phi$ from $m$-bit blocks to the alphabet~$A$
with almost the same inverse images sizes (inverse images sizes
may differ by~$1$ for different letters). Then we apply the map~$\phi$
for the blocks of~$\tau$ and get the sequence~$\omega$ under the alphabet~$A$.
Let $X$ be a finite set. We should find $t\in X$ such that
$\K(X,\omega(X)|t)\ge\alpha\#X\log a$. We denote by $Y$
the union of blocks with indexes in~$X$ ($\#Y=n\#X$). By the
proposition of Theorem~\ref{subsequence1} if $\#Y$ is enough
large then there exists $s\in Y$ such that
$\K(\tau(Y),Y|s)\ge\alpha'\#Y=\alpha'm\#X$.
Now assume that $t$ is a number of block contained~$s$ (clearly
$t\in X$). $m$~is a constant, so $\K(t|s)=O(1)$ and
$\K(Y|X)=O(1)$. The string $\tau(Y)$ could be constructed by
applying the reversed map~$\phi$ for each letter in $\tau(X)$
but the reversed map is not single valued: size of reversed
image is about $\frac{2^m}{a}$ (it differs less that by~$1$ from
the value). Therefore
$$
\K(\tau(Y)|\omega(X))\le\#X\log\left(\frac{2^m}{a}+1\right)+O(1)\le\#X\left(m-\log a+\frac{a}{2^m}\right).
$$
Composing these inequalities we get
$$
\K(\omega(X),X|t)\ge\#X\left(\log a-m(1-\alpha')-\frac{a}{2^m}\right)-O(\log\#X).
$$
At last we have to define the constants. We make $m$ enough large such
that $\frac{a}{2^m}<\frac{1}{2}(1-\alpha)\log a$, then we make $\alpha'$
enough close to~$1$ such that $m(1-\alpha')<\frac{1}{2}(1-\alpha)\log a$
and then we choose enough large~$N$ such that for any $X$ of cardinality
at least $N$ we get $\K(X,\omega(X)|t)\ge\alpha\#X\log a$.
\proofend
\fi
\section{Weak upper bound}
To illustrate the technique, we first prove a simple
generalization of a result obtained by
Berk~\cite{repetitionfree} and provide an upper bound for
$R(a,l)$ that is weaker that our final bound:
\begin{theorem}
%
For every $a\ge 2$ and every real number $b\in(1,a)$ there exists a
number~$N$ and a sequence~$\omega$ in $a$-letters alphabet such
that for every $n\ge N$ the distance between any two different
occurrences of the same substring of length~$n$ in $\omega$ is at
least~$b^n$.
%
\end{theorem}
\begin{proof}
%
Construct a sequence~$\omega$ using Theorem~\ref{subsequencea}
with $\alpha$ close enough to~$1$.
Let $I$ and $J$ ($|I|=|J|=n$) be different intervals where the
same substring of length $n$ occurs in~$\omega$. Let $X=I\cup
J$. Then $n<\#X\le 2n$ (intervals $I$ and $J$ are not
necessarily disjoint) and the first $n$ letters of $\omega(X)$
are equal to the last $n$ letters of $\omega(X)$. It is easy
to see that the string $\omega(X)$ is determined
by its first $\#X-n$ letters, $n$ and $\#X$, so
$\K(\omega(X))\le(\#X-n)\log a+O(\log n)$.
Assume $t\in X$. Then $X$ is determined by $t$, the number $n$,
the distance between $I$ and $J$ and the ordinal number of $t$
in $X$. So if the distance between $I$ and $J$ is less than
$b^n$ then $\K(\omega(X),X|t)\le(|X|-n)\log a+ n\log b+O(\log
n)\le\alpha|X|\log n$ for large enough~$n$ and $\alpha$ that is
close enough to~$1$ (because $\log b<\log a$). This contradicts
the inequality of Theorem~\ref{subsequencea}. Therefore
sequence~$\omega$ does not contain a pair of different
occurrences of the same substring of sufficiently large length~$n$
with distance between them less than $b^n$.
%
\end{proof}
In particular, for every integer $a\ge 2$, every real number
$b\in(1,a)$ and for large enough $l$ the following inequality
holds:
$$
R(a,l)<1+\frac{\log_b l}{l}.
$$
\section{The final upper bound}
In the weak upper bound we used the same sequence for all values
of~$l$. And now we need different sequences for different values
of~$l$ but we want the constant~$c$ to be the same. To achieve
this goal we use the following ``$l$-uniform'' version of
Theorem~\ref{subsequence1}.
\begin{theorem}\label{subsequencel}
%
Let $\alpha$ be a positive real number less than~$1$. There
exists an integer $N$ such that for every integer~$l$ there
exists a binary sequence $\omega$ that has the following
property: for every finite set $X$ of cardinality at least $N$
the inequality
$$
\K(X,\omega(X)|t,l)\ge\alpha\#X
$$
holds for some $t\in A$.
%
\end{theorem}
Note that $\omega$ may depend on~$l$ while $N$ is the same for
all values of~$l$. (If we allowed $N$ to be dependent on $l$,
this would be a standard relativization of
Theorem~\ref{subsequence1}.)
\begin{proof}
%
Theorem~\ref{subsequencel} can be proven in the same way as
Theorem~\ref{subsequence1}. And it can also be formally derived
from it: if a sequence~$\tau$ and a number~$N$ satisfy the
requirements of Theorem~\ref{subsequence1} and $z:\mathbb{N}^2\to\mathbb{N}$
is a computable bijection, then the sequence $i\mapsto
\omega_i=\tau_{z(i,l)}$ and the same number~$N$ satisfy the
requirements of Theorem~\ref{subsequencel} for the integer~$l$.
(The bijection adds $O(1)$-term, but this can be compensated by
a small change in $\alpha$: the statement is true for every
$\alpha<1$.)
%
\end{proof}
Now we can start proving the upper bound.
\begin{theorem}
%
There exists a constant~$c$ such that for any $a\ge 2$ and $l\ge 1$ the
following inequality holds:
$$
1+\frac{1}{al}\le R(a,l)\le 1+\frac{c}{al}.
$$
%
\end{theorem}
\begin{proof}
The lower bound is easy (as shown in the introduction). Let us
prove the upper bound. Let as assume first that $a=2$ (the
general case can be reduced to this special one).
Consider a sequence~$\omega$ satisfying the requirements of
Theorem~\ref{subsequencel} for some $\alpha>\frac{1}{2}$. Then
the required sequence with long fractional powers will be
constructed as
$$
\tau_i=\omega_{f(i)}
$$
for some mapping $f:\mathbb{N}\to\mathbb{N}$.
At first let us define $f$ at the first $l$ integers (the value
of integer constant~$m$ will be chosen later):
%
\begin{itemize}
\item[1.] $f(i)=i\bmod m$ for $i<l$ and $(i\bmod m)\neq m-1$ (we say that
these indexes have rank~$1$).
\item[2.] $f(mi+m-1)=(m-1)+(i\bmod m)$ for $mi+m-1<l$ and $(i\bmod m)\neq m-1$
(we say that these indexes have rank~$2$).
\item[3.] $f(m^2i+m^2-1)=2(m-1)+(i\bmod m)$ for $m^2i+m^2-1<l$ and $(i\bmod m)\neq m-1$
(we say that these indexes have rank~$3$).
\item[] (And so on until $f$ is defined at all first $l$ integers.)
\end{itemize}
Then we define $f$ on other blocks of $l$ integers in the same
way but using fresh bits each time. So if
$f(\{0,1,\dots,l-1\})=\{0,1,\dots,L-1\}$ then $f(i+jl)=f(i)+jL$.
Suppose the sequence $\tau_i=\omega_{f(i)}$ contains some
fractional power $xyx$ with $|xy|\ge l$ and the exponent
$\displaystyle \frac{|xyx|}{|xy|}\ge 1+\frac{c}{2l}$. Without
loss of generality we can assume that the exponent
$1+\frac{c}{2l}$ is not greater than~$2$ (otherwise the
statement of the theorem follows from the existence of a binary
sequence, called Thue-Morse sequence, that does not contain any
fractional power with exponent greater than~$2$,
see~\cite{thue-1},~\cite{thue-2}). Also we can assume that
$c>2m$ (increasing $c$, we make our task easier). So
$l\ge\frac{c}{2}>m$ and $|x|\ge\frac{c}{2l}|xy|>m$.
First we consider the case when both occurrences of~$x$ in $xyx$
lie entirely in some blocks of size~$l$ (in two different
blocks, because $|xy|\ge l$). Denote by $n$ the number of $l$-sized
blocks between these two occurrences of $x$ and denote by $k$ the integer
number that satisfies the inequality $m^{k-1}\le|x|<m^k$.
Then $m^k>\frac{c}{2}n$ and $k\ge 2$ (because $|x|\ge\frac{c}{2l}|xy|>m$).
Let us denote by $I$ and $J$ the sets of values of $f$ for the
first and second occurrences of $x$ (respectively) whose rank is
not greater than~$k$ (obviously there is at most $1$ index in
each of these occurrences of~$x$ whose rank is greater
than~$k$). The sets $I$ and $J$ are disjoint because these
occurrences of~$x$ lies in the different $l$-sized blocks.
Assume $Z=I\cup J$, then for some $t\in Z$ we have
$\K(Z,\omega(Z)|t,l)\ge\alpha\#Z$ by the statement of
Theorem~\ref{subsequencel} (we need here that $m>N+1$ since
$\#Z$ should be greater than~$N$).
Obviously,
$$
\frac{1}{2}\#Z=\#I+O(1)=\#J+O(1)=(k-1)(m-1)+\frac{|x|}{m^{k-1}}+O(1).
$$
The set $Z$ is determined by $t$, $l$, $m$, $n$, $k$, $|x|$ and
the start/end positions for the two occurrences of the word~$x$
modulo $m^k$ (and one bit saying whether $t$ belongs to the
first occurrence of $x$ or to the second one). So $\K(Z\mid
t,l)\le\log n+O(\log(m^k))=O(k\log m)$ (since
$m^k>\frac{c}{2}n$). We can also calculate $\omega(Z)$ if
$\omega(I)$ is given (we need at most one extra bit for
calculating the entire string~$x$). Therefore
$$
O(k\log m)+\frac{1}{2}\#Z\ge\alpha\#Z,
$$
but $\alpha>\frac{1}{2}$ and $\#Z\ge 2(k-1)(m-1)+O(1)\ge
k(m-1)+O(1)$. So $k(m-1)<O(k\log m)$ that is a contradiction if
$m$ is large enough. (Recall that the choice of $m$ was
postponed.)
Consider now the general case for the position of the two occurrences
of~$x$. If length of $x$ is not large, i.e. $|x|\le l$, we can reduce
this case to the previous one by splitting $x$ into parts and choosing the
largest part (we must multiply the constant~$c$ by $3$).
Now let $x$ be longer than the block size ($|x|>l$). We can assume that
there is no $l$-sized block that intersects both occurrences of $x$
(in the other case we also split the word~$x$ in parts).
Let us denote by $I$ and $J$ the sets of values of $f$ in the
first and second occurrences of $x$ respectively. The sets
$I$ and $J$ are disjoint. Assume $Z=I\cup J$. Then for some
$t\in Z$ we have $\K(Z,\omega(Z)|t,l)\ge\alpha\#Z$.
The set $Z$ is determined by $t$, $l$, $m$ and the relative
start/end positions of the two occurrence of the word~$x$ with
respect to the one of the preimages of $t$ (for example, the
first one). So $\K(Z\mid t,l)\le\log|xy|+O(\log l)=O(\log|x|)$
(since $|x|\ge l$ and $|x|\ge\frac{c}{2l}|xy|$). To compute
$\omega(Z)$, it is enough to know at most a half of it
($\omega(I)$ or $\omega(J)$, whichever is smaller). Therefore
$$
O(\log|x|)+\frac{1}{2}\#Z\ge\alpha\#Z,
$$
but $\alpha>\frac{1}{2}$ and
$\#Z=\Omega\left(\frac{|x|}{l}(m-1)\log_m l\right)=
\Omega\left((\log|x|)\frac{m-1}{\log m}\right)$
(here we use that $|x|>l>m$ and $\frac{|x|}{\log|x|}\ge\frac{l}{\log l}$).
That is a contradiction if $m$ is large enough.
This finishes the proof for $a=2$.
Assume now that $a\ge 6$ and $a$ is even. Let $\omega$ be the
sequence constructed for binary alphabet and
$l'=\frac{a-2}{2}l$. To get the required sequence~$\nu$ we will
color the terms of~$\omega$ into $\frac{a}{2}$ colors: the
$i$-th block of size~$l$ gets color $i\bmod\frac{a}{2}$. Then
the size of the alphabet of sequence~$\nu$ (whose terms are now
$\langle$bit, color$\rangle$ pairs) equals to $a$ and $\nu$ does
not contain fractional powers $z^p$ with $|z|\ge\frac{a-2}{2}l$
and $p\ge 1+\frac{c}{(a-2)l}$. And obviously $\nu$ does not
contain any fractional powers $z^p$ with
$l\le|z|\le\frac{a-2}{2}l$ (because it does not contain pairs of
equal letters at these distances).
Therefore $R(a,l)\le 1+\frac{c}{(a-2)l}$ if $a\ge 6$ and $a$ is
even, and $R(2,l)\le 1+\frac{c}{2l}$.
To prove the theorem for arbitrary $a$ it remains to note that
that $R(a,l)$ is decreasing in $a$, so $R(a,l)\le
1+\frac{3c}{al}$ for every $a\ge 2$, $l\ge 1$.
%
\end{proof}
\section{Acknowledgements}
The author is grateful to Gregory Kucherov who explained this
problem to the author and suggested to apply the Kolmogorov
complexity technique to it, and to Anna Frid who encouraged the
author to write down the proofs.
|
{
"timestamp": "2010-12-02T02:02:51",
"yymm": "1009",
"arxiv_id": "1009.4454",
"language": "en",
"url": "https://arxiv.org/abs/1009.4454"
}
|
\section{Introduction}
The Gribov-Zwanziger theory \cite{Gribov:1977wm,Zwanziger:1988jt,Zwanziger:1989mf} arises from the Landau gauge Faddeev-Popov action when the domain of integration in the Euclidean functional integral is restricted to the so called Gribov region $\Omega$\footnote{The Gribov region $\Omega$ is defined as the set of gauge field configurations which obey the Landau gauge condition and for which the Faddeev-Popov operator is strictly positive, namely $\Omega=\{A^a_\mu, \; \partial_\mu A^a_\mu=0, \; -\partial_\mu (\partial_\mu \delta^{ab} + gf^{acb}A^c_\mu) > 0 \}$.}, whose boundary $\partial \Omega$ is known as the first Gribov horizon, such a restriction being needed in order to account for the phenomenon of the Gribov copies. So far, the Gribov-Zwanziger theory has been proven to be renormalizable \cite{Zwanziger:1988jt,Zwanziger:1989mf,Maggiore:1993wq,Dudal:2005na,Dudal:2010fq}, while providing a mechanism for gluon confinement, as displayed by the two-point gluon correlation function
\begin{equation}
\langle A^a_\mu(k) A^b_\nu(-k) \rangle = \delta^{ab} \frac{k^2}{k^4+\gamma^4} \left( \delta_{\mu\nu} -\frac{k_\mu k_\nu}{k^2} \right) \;, \label{paai}
\end{equation}
which exhibits complex poles. As such, it does not correspond to the propagation of a physical particle. The parameter $\gamma$ stands for the Gribov mass parameter. It is not a free parameter, being determined in a self-consitent way as a function of the gauge coupling constant $g$ through a gap equation, called the horizon condition \cite{Zwanziger:1988jt,Zwanziger:1989mf}.\\\\Several efforts have been undertaken in the last years \cite{Dudal:2007cw,Dudal:2008sp,Gracey:2006dr,Baulieu:2008fy,Dudal:2009xh,Sorella:2009vt,Huber:2009tx,Baulieu:2009ha,Zwanziger:2010iz} to achieve a better understanding of the Gribov-Zwanziger theory and of its relationship with confinement. Though, there still exist aspects of the theory which remain to be unraveled. Let us quote, for example, the issues of the BRST symmetry and of the construction of a set of local operators enabling us to make contact with the spectrum of Yang-Mills theories. \\\\In this work we address the issue of the BRST symmetry. We point out that the soft breaking of the BRST symmetry exhibited by the Gribov-Zwanziger action can be converted into a linear breaking by introducing a set of BRST quartets of auxiliary fields. \\\\This observation has far-reaching consequences. We underline that, unlike a soft breaking quadratic in the fields, a linear breaking turns out to be compatible with the Quantum Action Principle \cite{Piguet:1995er}. The linearly broken BRST symmetry can be thus directly converted into a set of useful Slavnov-Taylor identities. Therefore, the quantum aspects of the Gribov-Zwanziger theory can be analyzed by means of the cohomology of a local nilpotent operator. In particular, both the characterization of the invariant counterterms and the renormalization of local gauge invariant composite operators can be achieved through the identification of cohomology classes of the BRST operator, for which powerful mathematical tools are available \cite{Piguet:1995er}. \\\\ Although we shall present our results within the context of the Gribov-Zwanziger action, it is worth emphasizing that the mechanism of converting the soft quadratic BRST breaking into a linear breaking equally applies to the so-called refined Gribov Zwanziger action (RGZ) introduced in \cite{Dudal:2007cw,Dudal:2008sp}. The RGZ action takes into account additional nonperturbative effects related to the existence of dimension two condensates, see \cite{Dudal:2008sp} for a detailed discussion. These condensates modify in a nontrivial way the infrared behavior of the gluon and ghost propagators. For example, the RGZ gluon propagator turns out to be \cite{Dudal:2008sp}
\begin{equation}
\langle A^a_\mu(k) A^b_\nu(-k) \rangle_{RGZ} = \delta^{ab} \frac{k^2+M^2}{k^4+(m^2+M^2)k^2 + m^2M^2+ 2g^2N\gamma^4} \left( \delta_{\mu\nu} -\frac{k_\mu k_\nu}{k^2} \right) \;, \label{rgz}
\end{equation}
where the mass parameters $M^2,m^2$ are dynamical parameters related to the nonvanishing dimension two condensates $\langle {\bar \varphi^{ab}_\mu \varphi^{ab}_\mu} -{\bar \omega^{ab}_\mu \omega^{ab}_\mu} \rangle$ and $\langle A^a_{\mu}A^a_\mu \rangle$ \cite{Dudal:2008sp}. Unlike the Gribov propagator, eq.\eqref{paai}, expression \eqref{rgz} does not vanish at zero momentum, while still providing a violation of the positivity \cite{Dudal:2008sp}, thus accounting for gluon confinement. The infrared behavior of the gluon and ghost propagators stemming from the RGZ theory is in good agreement with the recent numerical simulations at large volume \cite{Cucchieri:2007rg,Cucchieri:2008fc,Bornyakov:2009ug}. In particular, as observed in \cite{Dudal:2010tf}, expression \eqref{rgz} provides an accurate fit of the gluon propagator up to $k\approx 1.5 GeV$.
\\\\The present work is organized as follows. In Sect.2. a short survey on the Gribov-Zwanziger action is given. In Sect.3 we show how the soft breaking can be converted into a linear breaking upon introduction of BRST quartets. Sect.4 is devoted to the derivation of the Slavnov-Taylor identities as well as of the large set of additional Ward identities fulfilled by the new formulation of the Gribov-Zwanziger theory.
\section{A short survey on the Gribov-Zwanziger theory}
The action of the Gribov-Zwanziger theory is given by the following expression:
\begin{eqnarray}
S_{GZ} & = & \int d^4 x \left( \frac1{4} F_{\mu\nu}^{a}F_{\mu\nu}^{a} + ib^a\partial_{\mu}A^a_\mu + {\bar c}^a \partial_\mu D_{\mu}^{ab} c^b \right) \nonumber \\
{\ }{\ }{\ } & + &\int d^4x \left( - {\bar \varphi}^{ac}_\mu \partial_\nu D_{\nu}^{ab} \varphi^{bc}_{\mu} + {\bar \omega}^{ac}_\mu \partial_\nu D_{\nu}^{ab} \omega^{bc}_{\mu} + g f^{amb} (\partial_{\nu}{\bar \omega}^{ac}_{\mu}) (D^{mp}_{\nu} c^p)\varphi^{bc}_{\mu} \right) \nonumber \\
{\ }{\ }{\ } & +& \int d^4x\left(\gamma^2\,g\,f^{abc}A_\mu^{a}(\varphi_\mu^{bc}-\bar{\varphi}_\mu^{bc})-d(N^2-1)\gamma^4 \right) \nonumber \\
& = &\frac1{4}\int d^4 x F_{\mu\nu}^{a}F_{\mu\nu}^{a} + s\int d^4 x \left(\bar{c}^a\partial_{\mu} A^a_{\mu} - \bar{\omega}_\mu^{ac}\partial_\nu D^{ab}_\nu \varphi_{\mu}^{bc}\right) + S_{\gamma} \;, \label{GZact}
\end{eqnarray}
with
\begin{align}
S_{\gamma}&=\int d^4x\left(\gamma^2\,g\,f^{abc}A_\mu^{a}(\varphi_\mu^{bc}-\bar{\varphi}_\mu^{bc})-d(N^2-1)\gamma^4 \right) \;, \label{BRSbreak}
\end{align}
where $N$ is the number of colors and $d=4$ the space-time dimension. The Gribov parameter $\gamma^2$ is determined in a self-consistent way by the horizon condition \cite{Zwanziger:1988jt,Zwanziger:1989mf}, which reads
\begin{align}
\frac{\partial {\cal E}_{vac}}{\partial \gamma^2}=0 \;, \label{geq}
\end{align}
where $ {\cal E}_{vac} $ is the vacuum energy
\begin{align}
e^{- {\cal E}_{vac} } = \int [d\Phi] \; e^{-S_{GZ}} \;, \label{vc}
\end{align}
and $[d\Phi]$ stands for the functional integration over all fields appearing in $S_{GZ}$.
In the absence of the term $S_{\gamma}$, the action \eqref{GZact} enjoys the nilpotent BRST symmetry
\begin{align}
sA_\mu^{a} &= - D^{ab}_\nu c^b = -( \partial_\mu \delta^{ab} + g f^{acb} A^c_\mu)c^b\;,\nonumber \\
sc^a &= \frac{g}{2} f^{acb}c^b c^c\;,\nonumber\\
s\bar{c}^a &= ib^a\;,\qquad sb^a=0\;,\nonumber\\
s\bar{\omega}_\mu^{ab} &= \bar{\varphi}_\mu^{ab}\;,\qquad s\bar{\varphi}_\mu^{ab}=0\;,\nonumber\\
s\varphi_\mu^{ab} &= \omega_\mu^{ab}\;,\qquad s\omega_\mu^{ab}=0\;, \label{BRS}
\end{align}
and
\begin{equation}
s \int d^4 x \left(\frac1{4} F_{\mu\nu}^{a}F_{\mu\nu}^{a} + s \left(\bar{c}^a\partial_{\mu} A^a_{\mu} - \bar{\omega}_\mu^{ac}\partial_\nu D^{ab}_\nu \varphi_{\mu}^{bc}\right) \right) =0 \;.
\end{equation}
The Gribov-Zwanziger action is, however, not left invariant by the BRST transformations, eqs.\eqref{BRS}, which are broken by the term $S_{\gamma}$, namely
\begin{align}
sS_{GZ} = sS_{\gamma} = \gamma^2\,\int d^4x\left(-g\,f^{abc}(D^{ad}_\mu c^d) (\varphi_\mu^{bc}-\bar{\varphi}_\mu^{bc}) +g\,f^{abc}A_\mu^{a}\omega_\mu^{bc}\right)\;. \label{BRSactbreak}
\end{align}
Notice that the breaking term, being of dimension two in the fields, is a soft breaking. Though, as the breaking is quadratic in the fields, {\it i.e.} it is a composite field operator, eq.\eqref{BRSactbreak} cannot be directly translated into a suitable set of Slavnov-Taylor identities.
\section{Converting the soft breaking into a linear breaking}
As already mentioned, it turns out that the Gribov-Zwanziger action, eq.(\ref{GZact}), admits an equivalent representation in terms of a new set of fields leading to a nilpotent BRST symmetry which is only linearly broken. There are several ways to achieve a formulation for which the BRST symmetry is linearly broken, all relying on the introduction of a set of BRST quartets of auxiliary fields. Here, we shall present the minimal formulation in which only a pair of BRST quartets is needed. The equivalent action is given by the following expression
\begin{align}
S_{GZ}^{lin} &= \frac1{4}\int d^4 x \,F_{\mu\nu}^{a}F_{\mu\nu}^{a} + s\int d^4 x \left(\bar{c}^a\partial_{\mu} A^a_{\mu} - \bar{\omega}_\mu^{ac}\partial_\nu D^{ab}_\nu \varphi_{\mu}^{bc}\right)\nonumber\\
&+ s\int d^4 x \left(g f^{abc} \bar{\mathcal{C}}_{\mu\nu}^{cd} A^a_{\mu}\varphi_{\nu}^{bd} - \bar{\mathcal{C}}_{\mu\nu}^{ab}\eta_{\mu\nu}^{ab}
-\bar{\mathcal{C}}_{\mu\nu}^{ab}\bar{\eta}_{\mu\nu}^{ab}
-gf^{abc} \bar{\eta}_{\mu\nu}^{cd} A^a_{\mu}\bar{\omega}_{\nu}^{bd}
-\bar{\rho}_{\mu\nu}^{ab}\bar{\eta}_{\mu\nu}^{ab}\right)\nonumber\\
& +\int d^4 x \left(\gamma^2 \eta_{\mu\nu}^{ab} \delta^{ab} \delta_{\mu\nu}
+\gamma^2 \bar{\lambda}_{\mu\nu}^{ab} \delta^{ab} \delta_{\mu\nu}\right)\;, \label{GZactlinear}
\end{align}
where $(\bar{\mathcal{C}}_{\mu\nu}^{ab}, \lambda_{\mu\nu}^{ab}, \eta_{\mu\nu}^{ab}, \mathcal{C}_{\mu\nu}^{ab})$ and $(\bar{\rho}_{\mu\nu}^{ab}, \bar{\lambda}_{\mu\nu}^{ab}, \bar{\eta}_{\mu\nu}^{ab}, \rho_{\mu\nu}^{ab})$ are two BRST quartets of auxiliary fields, namely
\begin{align}
s\bar{\mathcal{C}}_{\mu\nu}^{ab} &= \lambda_{\mu\nu}^{ab}\;,\qquad s\lambda_{\mu\nu}^{ab} = 0\;,\qquad
s\eta_{\mu\nu}^{ab} = \mathcal{C}_{\mu\nu}^{ab}\;,\qquad s\mathcal{C}_{\mu\nu}^{ab} = 0\;,\nonumber\\
s\bar{\rho}_{\mu\nu}^{ab} &= \bar{\lambda}_{\mu\nu}^{ab}\;,\qquad s\bar{\lambda}_{\mu\nu}^{ab} = 0\;,\qquad
s\bar{\eta}_{\mu\nu}^{ab} = \rho_{\mu\nu}^{ab}\;,\qquad s\rho_{\mu\nu}^{ab} = 0\;. \label{BRS2}
\end{align}
The fields $(\lambda_{\mu\nu}^{ab},\eta_{\mu\nu}^{ab})$ and $(\bar{\lambda}_{\mu\nu}^{ab}, \bar{\eta}_{\mu\nu}^{ab})$ are commuting fields, while $(\bar{\mathcal{C}}_{\mu\nu}^{ab}, \mathcal{C}_{\mu\nu}^{ab})$, $( \bar{\rho}_{\mu\nu}^{ab}, \rho_{\mu\nu}^{ab})$ are anticommuting. Each of these fields has $16(N^2-1)^2$ components\footnote{The color indices $(a,b)$ run from $1$ to $N^2-1$, while $(\mu,\nu)$ from $1$ to $4$. }. Moreover, the fields $(\bar{\mathcal{C}}_{\mu\nu}^{ab}, \bar{\rho}_{\mu\nu}^{ab})$ have ghost number $-1$, and $({\mathcal{C}}_{\mu\nu}^{ab}, {\rho}_{\mu\nu}^{ab})$ ghost number $1$.
It is easily checked that, unlike expression \eqref{BRSactbreak}, the BRST symmetry defined by eqs.(\ref{BRS}) and by eqs.(\ref{BRS2}) is now linearly broken, {\it i.e.} the resulting breaking term is linear in the fields
\begin{align}
sS_{GZ}^{lin} = \gamma^2\,\int d^4x\,\delta^{ab} \delta_{\mu\nu}\mathcal{C}_{\mu\nu}^{ab}\;. \label{BRSactbreak2}
\end{align}
In order to prove the equivalence between the two formulations, eq.(\ref{GZact}) and eq.(\ref{GZactlinear}), we first give the explicit expression of $S_{GZ}^{lin}$
\begin{eqnarray}
S_{GZ}^{lin} &=& \frac{1}{4}\int d^4 x \; F_{\mu\nu}^{a}F_{\mu\nu}^{a} + s\int d^4 x \left(\bar{c}^a\partial A^a_{\mu} - \bar{\omega}_\mu^{ac}\partial_\nu D^{ab}_\nu \varphi_{\mu}^{bc}\right)\nonumber\\
&&+\int d^4 x \left(g f^{abc} \lambda_{\mu\nu}^{cd} A^a_{\mu}\varphi_{\nu}^{bd}
-gf^{abc}\bar{\eta}^{cd}_{\mu\nu}\left( A^{a}_{\mu}\bar\varphi^{bd}_{\nu}-(D^{ap}_{\mu}c^{p})\bar\omega^{bd}_{\nu}\right)
-\lambda^{ab}_{\mu\nu}\bar\eta^{ab}_{\mu\nu}\right)\nonumber\\
&&+\int d^4 x\left(\bar{\mathcal{C}}^{cd}_{\mu\nu}\left(\mathcal{C}^{cd}_{\mu\nu}+\rho^{cd}_{\mu\nu}
-gf^{abc}A^{a}_{\mu}\omega^{bd}_{\nu}
+gf^{abc}(D^{ap}_{\mu}c^{p})\varphi^{bd}_{\nu}\right)
-\rho^{cd}_{\mu\nu}(\bar\rho^{cd}_{\mu\nu}
+gf^{abc}A^{a}_{\mu}\bar\omega^{bd}_{\nu})\right)\nonumber\\
&&+\int d^4 x\left(\bar\lambda^{ab}_{\mu\nu}(\gamma^{2}\delta^{ab}\delta_{\mu\nu}-\bar\eta^{ab}_{\mu\nu})
+\eta^{ab}_{\mu\nu}(\gamma^{2}\delta^{ab}\delta_{\mu\nu}-\lambda^{ab}_{\mu\nu})\right)\;.
\label{GZactlinear2}
\end{eqnarray}
We observe now that the new fields $(\lambda_{\mu\nu}^{ab},\eta_{\mu\nu}^{ab})$ and $(\bar{\lambda}_{\mu\nu}^{ab}, \bar{\eta}_{\mu\nu}^{ab})$ can be eliminated in an algebraic way by using their equations of motion. In addition, the anticommuting fields $(\bar{\mathcal{C}}_{\mu\nu}^{ab}, \mathcal{C}_{\mu\nu}^{ab})$, $( \bar{\rho}_{\mu\nu}^{ab}, \rho_{\mu\nu}^{ab})$ can be decoupled by suitable field redefinitions. Let us show how this works at the level of the partition function. As it is apparent from expression \eqref{GZactlinear2}, the fields
$\eta_{\mu\nu}^{ab}$ and $\bar{\lambda}_{\mu\nu}^{ab}$ are Lagrange multipliers. In the path integral formulation they constrain the fields $\lambda_{\mu\nu}^{ab}$ and $\bar{\eta}_{\mu\nu}^{ab}$ to take a constant value. In fact
\begin{align}
\int [d\Xi] \; e^{-S_{GZ}^{lin}} &= \int [d\tilde{\Xi}] [d\eta][d\bar{\eta}][d\lambda][d\bar{\lambda}]\; e^{-\tilde{S}_{GZ}^{lin}
- \int d^4 x\left(\bar\lambda^{ab}_{\mu\nu}(\gamma^{2}\delta^{ab}\delta_{\mu\nu}-\bar\eta^{ab}_{\mu\nu})
+\eta^{ab}_{\mu\nu}(\gamma^{2}\delta^{ab}\delta_{\mu\nu}-\lambda^{ab}_{\mu\nu})\right)}\nonumber\\
&= \int [d\tilde{\Xi}][d\bar\eta][d{\lambda}] \delta\left(\lambda_{\mu\nu}^{ab} - \gamma^2 \delta^{ab} \delta_{\mu\nu}\right)\delta\left(\bar{\eta}_{\mu\nu}^{ab} - \gamma^2 \delta^{ab} \delta_{\mu\nu}\right)\; e^{-\tilde{S}_{GZ}^{lin}
\;, \label{partition-delta}
\end{align}
where
\begin{eqnarray}
\tilde{S}_{GZ}^{lin} &=& \frac{1}{4}\int d^4 x\; F_{\mu\nu}^{a}F_{\mu\nu}^{a} + s\int d^4 x \left(\bar{c}^a\partial A^a_{\mu} - \bar{\omega}_\mu^{ac}\partial_\nu D^{ab}_\nu \varphi_{\mu}^{bc}\right)\nonumber\\
&&+\int d^4 x \left(g f^{abc} \lambda_{\mu\nu}^{cd} A^a_{\mu}\varphi_{\nu}^{bd}
-gf^{abc}\bar{\eta}^{cd}_{\mu\nu} \left( A^{a}_{\mu}\bar\varphi^{bd}_{\nu}-(D^{ap}_{\mu}c^{p})\bar\omega^{bd}_{\nu}\right)
-\lambda^{ab}_{\mu\nu}\bar\eta^{ab}_{\mu\nu}\right)\nonumber\\
&&+\int d^4 x\left(\bar{\mathcal{C}}^{cd}_{\mu\nu}\left( \mathcal{C}^{cd}_{\mu\nu}+\rho^{cd}_{\mu\nu}
-gf^{abc}A^{a}_{\mu}\omega^{bd}_{\nu}
+gf^{abc}(D^{ap}_{\mu}c^{p})\varphi^{bd}_{\nu} \right)
-\rho^{cd}_{\mu\nu}(\bar\rho^{cd}_{\mu\nu}
+gf^{abc}A^{a}_{\mu}\bar\omega^{bd}_{\nu})\right)\; \label{GZactlinear3}
\end{eqnarray}
Performing the integration over $\lambda_{\mu\nu}^{ab}$ and $\bar{\eta}_{\mu\nu}^{ab}$, it follows
\begin{equation}
\int [d\Xi] \; e^{-S_{GZ}^{lin}} = \int [d\tilde{\Xi}]\; e^{-{\hat{S}}_{GZ}^{lin}} \;, \label{eqa1}
\end{equation}
with
\begin{equation}
{\hat{S}}_{GZ}^{lin} = \tilde{S}_{GZ}^{lin} \big|_{ \lambda_{\mu\nu}^{ab} = \gamma^2 \delta^{ab} \delta_{\mu\nu} ; \bar{\eta}_{\mu\nu}^{ab} = \gamma^2 \delta^{ab} \delta_{\mu\nu}} \;, \label{eqa2}
\end{equation}
where $\Xi$ is a shorthand notation to denote all fields appearing in $S_{GZ}^{lin}$, while $\tilde{\Xi}$ refers to all fields of ${\hat{S}}_{GZ}^{lin}$, {\it i.e.} it does not contain $\eta,\bar{\eta},\lambda,\bar{\lambda}$. \\\\The action ${\hat{S}}_{GZ}^{lin}$, eq.\eqref{eqa2}, takes the form
\begin{eqnarray}
{\hat{S}}_{GZ}^{lin} &=& S_{GZ}
+\int d^4 x\;\gamma^{2}gf^{abc}(D^{ad}_{\mu}c^{d})\bar\omega^{bc}_{\mu}\nonumber\\
&+&\int d^4 x\;\bar{\mathcal{C}}^{cd}_{\mu\nu}\left( \mathcal{C}^{cd}_{\mu\nu}+\rho^{cd}_{\mu\nu}
-gf^{abc}A^{a}_{\mu}\omega^{bd}_{\nu}
+gf^{abc}(D^{ap}_{\mu}c^{p})\varphi^{bd}_{\nu} \right) \nonumber\\
&-&\int d^4 x\;\rho^{cd}_{\mu\nu}(\bar\rho^{cd}_{\mu\nu}
+gf^{abc}A^{a}_{\mu}\bar\omega^{bd}_{\nu})\;. \label{GZequiv}
\end{eqnarray}
We proceed now by first redefining the variables ${\mathcal{C}}_{\mu\nu}^{cd}, \bar\rho^{cd}_{\nu}$ as
\begin{align}
\tilde{\mathcal{C}}_{\mu\nu}^{cd} &\equiv \mathcal{C}_{\mu\nu}^{cd}
+\rho^{cd}_{\mu\nu}
-gf^{abc}A^{a}_{\mu}\omega^{bd}_{\nu}
+gf^{abc}(D^{ap}_{\mu}c^{p})\varphi^{bd}_{\nu}\;, \nonumber\\
\tilde{\bar\rho}_{\mu\nu}^{cd} &\equiv \bar\rho^{cd}_{\nu}
+gf^{abc}A^{a}_{\mu}\bar\omega^{bd}_{\nu}\;, \label{GZequiv2a}
\end{align}
which has the effect of decoupling them from the action. Further, we eliminate the term $\gamma^{2}gf^{abc}(D^{ad}_{\mu}c^{d})\bar\omega^{bc}_{\mu}$ in expression \eqref{GZequiv} by redefining the field $\omega^{ab}_{\mu}$ as\footnote{It is useful to pint out that the redefinition \eqref{GZequiv2b}, albeit nonlocal due to the presence of $ [(\partial\cdot{D})^{-1}]$, is perfectly allowed within the Gribov region $\Omega$, in which the operator $-\partial_\mu D_\mu$ is strictly positive.}
\begin{align}
\tilde{\omega}^{bc}_{\mu}&\equiv \omega^{bc}_{\mu}+[(\partial\cdot{D})^{-1}]^{bd}\left( \gamma^{2}gf^{dec}D^{ep}_{\mu}c^{p} \right)\;,\nonumber\\
\label{GZequiv2b}
\end{align}
As the redefinitions \eqref{GZequiv2a}, \eqref{GZequiv2b} have unity Jacobian, it follows
\begin{align}
\int [d\tilde{\Xi}]\; e^{{\hat{S}}_{GZ}^{lin} } &= \int [d\Phi][d\bar{\mathcal{C}}][d\tilde{\mathcal{C}}][d\tilde{\bar{\rho}}][d{\rho}] e^{-S_{GZ} - \int d^4 x\; \left(\bar{\mathcal{C}}_{\mu\nu}^{ab} \tilde{\mathcal{C}}_{\mu\nu}^{ab} + \tilde{\bar{\rho}}_{\mu\nu}^{ab}{\rho}_{\mu\nu}^{ab} \right) } \nonumber\\
&= {\cal N}\int [d\Phi] e^{-S_{GZ} } \;, \label{GZequiv3}
\end{align}
where ${\cal N}$ is a constant factor. Expression (\ref{GZequiv3}) shows thus the equivalence between $S_{GZ}$, given by eq.(\ref{GZact}) and $S_{GZ}^{lin}$, given by eq.(\ref{GZactlinear}).
\section{The Ward identities}
The linearly broken BRST symmetry, eq.\eqref{BRSactbreak2}, can be directly converted into a useful set of Slavnov-Taylor identities. This stems from the fact that an equation of the type of \eqref{BRSactbreak2} turns out to be compatible with the Quantum Action Principle \cite{Piguet:1995er}. In order to derive the Slavnov-Taylor identities, it is useful to follow \cite{Zwanziger:1988jt,Zwanziger:1989mf,Maggiore:1993wq,Dudal:2005na,Dudal:2010fq} and introduce a multi-index notation
\begin{eqnarray}
\left( \varphi_{\mu}^{ab}, \bar{\varphi}_{\mu}^{ab}, \omega_{\mu}^{ab}, \bar{\omega}_{\mu}^{ab} \right) &= \left( \varphi_{i}^{a}, \bar{\varphi}_{i}^{a}, \omega_{i}^{a}, \bar{\omega}_{i}^{a} \right)\;, \nonumber\\
\left( \bar{\mathcal{C}}_{\mu\nu}^{ab}, \lambda_{\mu\nu}^{ab}, \eta_{\mu\nu}^{ab}, \mathcal{C}_{\mu\nu}^{ab} \right) &= \left(\bar{\mathcal{C}}_{\mu i}^{a}, \lambda_{\mu i}^{a}, \eta_{\mu i}^{a}, \mathcal{C}_{\mu i}^{a} \right)\;, \nonumber\\
\left( \bar{\rho}_{\mu\nu}^{ab}, \bar{\lambda}_{\mu\nu}^{ab}, \bar{\eta}_{\mu\nu}^{ab}, \rho_{\mu\nu}^{ab} \right) &= \left(\bar{\rho}_{\mu i}^{a}, \bar{\lambda}_{\mu i}^{a}, \bar{\eta}_{\mu i}^{a}, \rho_{\mu i}^{a} \right) \;, \label{multiindex}
\end{eqnarray}
where $i = 1,...,f$, with $f=d(N^2-1)$. Therefore
\begin{eqnarray}
S_{GZ}^{lin} &=& \int d^4 x \left( \frac1{4} F_{\mu\nu}^{a}F_{\mu\nu}^{a} + ib^a\partial_{\mu}A^a_\mu + {\bar c}^a \partial_\mu D_{\mu}^{ab} c^b \right) \nonumber \\
{\ }{\ }{\ } & + &\int d^4x \left( - {\bar \varphi}^{a}_i \partial_\nu D_{\nu}^{ab} \varphi^{b}_i + {\bar \omega}^{a}_i \partial_\nu D_{\nu}^{ab} \omega^{b}_i + g f^{amb} (\partial_{\nu}{\bar \omega}^{a}_i) (D^{mp}_{\nu}c^p) \varphi^{b}_i \right) \nonumber \\
&+& \int d^4 x \left( g f^{abc} \lambda_{\mu i}^{c} A^a_{\mu}\varphi_i^{b}
-gf^{abc}\bar{\eta}^{c}_{\mu i}\left( A^{a}_{\mu}\bar\varphi^{b}_{i}-(D^{ap}_{\mu}c^{p})\bar\omega^{b}_{i} \right)
-\lambda^{a}_{\mu i}\bar\eta^{a}_{\mu i} \right) \nonumber\\
&+&\int d^4 x \left( {\bar{\mathcal{C}}^{c}_{\mu i}} \left( {\mathcal{C}}^{c}_{\mu i}+\rho^{c}_{\mu i}
-gf^{abc}A^{a}_{\mu}\omega^{b}_{i}
+gf^{abc}(D^{ap}_{\mu}c^{p}) \varphi^{b}_{i} \right)
-\rho^{c}_{\mu i}(\bar\rho^{c}_{\mu i}
+gf^{abc}A^{a}_{\mu}\bar\omega^{b}_{i}) \right) \nonumber\\
&+&\int d^4 x\left(\bar\lambda^{a}_{\mu i}(\gamma^{2}\delta^{ab}\delta_{\mu\nu} \delta^{i}_{\nu b}-\bar\eta^{a}_{\mu i})
+\eta^{a}_{\mu i}(\gamma^{2}\delta^{ab}\delta_{\mu\nu}\delta^{i}_{\nu b}-\lambda^{a}_{\mu i})\right)\;.
\label{GZactlinear2_i}
\end{eqnarray}
Introducing thus two BRST invariant external sources $(\Omega^a_\mu, L^a)$ coupled to the nonlinear BRST variations of the fields $(A^a_\mu,c^a)$
\begin{align}
\Sigma_{GZ}^{lin} = S_{GZ}^{lin} + \int d^4 x\; \left( - \Omega_{\mu}^{a} D^{ab}_\mu c^b + \frac{g}{2} f^{acb}L^a c^b c^c\right) \;, \label{quant-GZlinear}
\end{align}
it follows that the action $\Sigma_{GZ}^{lin}$ fulfills the linearly broken Slavnov-Taylor identities
\begin{align}
\mathcal{S}(\Sigma_{GZ}^{lin}) = \gamma^2\,\int d^4x\;\delta^{ab} \delta_{\mu\nu}\delta_{b\nu}^i \mathcal{C}_{\mu i}^{a} \label{st1}
\end{align}
where
\begin{align}
\mathcal{S}(\Sigma_{GZ}^{lin})&= \int d^{4}x\,\biggl( \frac{\delta\Sigma_{GZ}^{lin}}{\delta A_{\mu}^a}\frac{\delta\Sigma_{GZ}^{lin}}{\delta \Omega_{\mu}^a} + \frac{\delta\Sigma_{GZ}^{lin}}{\delta L^a}\frac{\delta\Sigma_{GZ}^{lin}}{\delta c^a}+ ib^a\frac{\delta\Sigma_{GZ}^{lin}}{\delta \bar{c}^a}
+ \lambda_{\mu i}^a\frac{\delta\Sigma_{GZ}^{lin}}{\delta \bar{\mathcal{C}}_{\mu i}^a}+ \mathcal{C}_{\mu i}^a\frac{\delta\Sigma_{GZ}^{lin}}{\delta \eta_{\mu i}^a} \nonumber\\
&+ \bar{\lambda}_{\mu i}^a\frac{\delta\Sigma_{GZ}^{lin}}{\delta \bar{\rho}_{\mu i}^a} + \rho_{\mu i}^a\frac{\delta\Sigma_{GZ}^{lin}}{\delta \bar{\eta}_{\mu i}^a} + \bar{\varphi}^{a}_{i}\frac{\delta\Sigma}{\delta\bar{\omega}^{a}_{i}}
+ \omega^{a}_{i}\frac{\delta\Sigma}{\delta\varphi^{a}_{i}}\biggr) \, \label{sti}
\end{align}
From eq.\eqref{st1} it follows that the linearized Slavnov-Taylor operator, defined as
\begin{eqnarray}
{\cal B}_{\Sigma_{GZ}^{lin}} & = & \int d^{4}x\, \biggl( \frac{\delta\Sigma_{GZ}^{lin}}{\delta A_{\mu}^a}\frac{\delta }{\delta \Omega_{\mu}^a} + \frac{\delta\Sigma_{GZ}^{lin}}{\delta \Omega_{\mu}^a}\frac{\delta }{\delta A_{\mu}^a}
+ \frac{\delta\Sigma_{GZ}^{lin}}{\delta L^a}\frac{\delta }{\delta c^a} +
\frac{\delta\Sigma_{GZ}^{lin}}{\delta c^a}\frac{\delta }{\delta L^a}
+ ib^a\frac{\delta }{\delta \bar{c}^a} \nonumber \\
&{\ }{\ }{\ }{\ }{\ }{\ }& + \lambda_{\mu i}^a\frac{\delta}{\delta \bar{\mathcal{C}}_{\mu i}^a}+ \mathcal{C}_{\mu i}^a\frac{\delta}{\delta \eta_{\mu i}^a}+ \bar{\lambda}_{\mu i}^a\frac{\delta}{\delta \bar{\rho}_{\mu i}^a} + \rho_{\mu i}^a\frac{\delta}{\delta \bar{\eta}_{\mu i}^a}+ \bar{\varphi}^{a}_{i}\frac{\delta}{\delta\bar{\omega}^{a}_{i}}
+ \omega^{a}_{i}\frac{\delta}{\delta\varphi^{a}_{i}} \biggr) \, \label{linsti}
\end{eqnarray}
is nilpotent, {\it i.e.}
\begin{equation}
{\cal B}_{\Sigma_{GZ}^{lin}} {\cal B}_{\Sigma_{GZ}^{lin}}= 0 \;. \label{nilp}
\end{equation}
We remind here that, according to the framework of the algebraic renormalization \cite{Piguet:1995er}, the invariant counterterms needed to renormalize the theory correspond to the cohomology of the linearized operator ${\cal B}_{\Sigma_{GZ}^{lin}}$ in the space of the integrated local polynomials in the fields with dimensions bounded by four. \\\\In addition to the Slavnov-Taylor identities, eq.\eqref{sti}, the action ${\Sigma_{GZ}^{lin}}$ fulfills a rather large set of additional Ward identities, which we enlist below:
\begin{enumerate}
\item The equations of motion of the fields $b^a$, $\bar{c}^a$, $\mathcal{C}_{\mu i}^{a}$, $\bar\rho_{\mu i}^{a}$, $\eta_{\mu i}^{a}$ and $\bar{\lambda}_{\mu i}^{a}$
\begin{equation}
\frac{\delta \Sigma_{GZ}^{lin}}{\delta b^a} = i \partial_{\mu} A_{\mu}^a \;, \qquad
\frac{\delta \Sigma_{GZ}^{lin}}{\delta \bar{c}^a} + \partial_{\mu}\frac{\delta \Sigma_{GZ}^{lin}}{\delta \Omega_{\mu}^a} = 0 \;, \label{eqmotion1}
\end{equation}
\begin{equation}
\frac{\delta \Sigma_{GZ}^{lin}}{\delta \eta_{\mu i}^{a}} = -\lambda_{\mu i}^{a} + \gamma^2 \delta^{ab} \delta_{\mu\nu}\delta_{b\nu}^i \;, \qquad
\frac{\delta \Sigma_{GZ}^{lin}}{\delta \bar{\lambda}_{\mu i}^{a}} = -\bar{\eta}_{\mu i}^{a} + \gamma^2 \delta^{ab} \delta_{\mu\nu}\delta_{b\nu}^i \;, \label{eqmotion2}
\end{equation}
\begin{equation}
\frac{\delta \Sigma_{GZ}^{lin}}{\delta \mathcal{C}_{\mu i}^{a}} = -\bar{\mathcal{C}}_{\mu i}^{a} \;, \qquad
\frac{\delta \Sigma_{GZ}^{lin}}{\delta \bar\rho_{\mu i}^{a}} = {\rho}_{\mu i}^{a} \;. \label{eqmotion3}
\end{equation}
Notice that all breakings appearing in eqs.\eqref{eqmotion1}, \eqref{eqmotion2}, \eqref{eqmotion3} are linear in the fields.
\item The Ward identity for the Gribov parameter $\gamma$, namely
\begin{align}
\frac{\partial \Sigma_{GZ}^{lin}}{\partial \gamma^2} = \int d^4x\;\delta^{ab} \delta_{\mu\nu}\delta_{b\nu}^i (\eta_{\mu i}^{a}+\bar{\lambda}_{\mu i}^{a}) \;. \label{nonrenorm}
\end{align}
Again, this identity exhibits a linear breaking. We underline that the Ward identity \eqref{nonrenorm} has a very special role, as it enables us to control the dependence of the invariant counterterms from the Gribov parameter $\gamma$. In particular, this equation provides a simple understanding of the nonrenormalization properties enjoyed by the Gribov parameter, as already reported in \cite{Zwanziger:1988jt,Zwanziger:1989mf,Maggiore:1993wq,Dudal:2005na,Dudal:2010fq}.
\item The local, linearly broken, equation of motion of $\bar{\varphi}_i^a$
\begin{align}
\frac{\delta \Sigma_{GZ}^{lin}}{\delta \bar{\varphi}_i^a}
+\partial_{\mu}\frac{\delta \Sigma_{GZ}^{lin}}{\delta \lambda_{\mu i}^a}
+gf^{abc}A_{\mu}^b\frac{\delta \Sigma_{GZ}^{lin}}{\delta \bar{\lambda}_{\mu i}^c}
= \Delta^{ai}_{\bar{\varphi}} \;, \label{eqmotion-barphi}
\end{align}
where
\begin{align}
\Delta^{ai}_{\bar{\varphi}} = -\partial^2 \varphi_i^a - \partial_{\nu}\eta_{\nu i}^a - \partial_{\mu}\bar{\eta}_{\mu i}^a - g\gamma^2 f^{abc}A_{\mu}^c\delta_{b\mu}^i\;. \label{delta-barphi}
\end{align}
\item The local, linearly broken, equation of motion of $\bar{\omega}_i^a$
\begin{align}
\frac{\delta \Sigma_{GZ}^{lin}}{\delta \bar{\omega}_i^a}
+\partial_{\mu}\frac{\delta \Sigma_{GZ}^{lin}}{\delta \bar{\mathcal{C}}_{\mu i}^a}
+gf^{abc}A^{b}_{\mu}\frac{\delta\Sigma_{GZ}^{lin}}{\delta\bar\rho^{c}_{\mu i}}
+ gf^{abc}\left(\frac{\delta\Sigma_{GZ}^{lin}}{\delta\bar\lambda^{b}_{\mu i}}-\gamma^{2}\delta^{i}_{\mu b}\right)
\frac{\delta\Sigma_{GZ}^{lin}}{\delta\Omega^{c}_{\mu}}
= \Delta^{ai}_{\bar{\omega}} \;, \label{eqmotion-baromega}
\end{align}
where
\begin{align}
\Delta^{ai}_{\bar{\omega}} = \partial^2 \omega_i^a + \partial_{\mu}\mathcal{C}_{\mu i}^a
+\partial_{\mu}\rho^{a}_{\mu i} \;. \label{delta-baromega}
\end{align}
\item The local, linearly broken, equation of motion of $\varphi_i^a$
\begin{align}
\frac{\delta \Sigma_{GZ}^{lin}}{\delta \varphi_i^a}
-\partial_{\mu}\frac{\delta\Sigma_{GZ}^{lin}}{\delta\bar\eta^{a}_{\mu i}}
-igf^{abc}\bar\varphi^{b}_{i}\frac{\delta\Sigma_{GZ}^{lin}}{\delta b^{c}}
+gf^{abc}\bar\omega^{b}_{i}\frac{\delta\Sigma_{GZ}^{lin}}{\delta\bar{c}^{c}}
-gf^{abc}A^{b}_{\mu} \frac{\delta\Sigma_{GZ}^{lin}}{\delta\eta^{c}_{\mu i}}
-gf^{acm} \frac{\delta\Sigma_{GZ}^{lin}}{\delta\mathcal{C}^{c}_{\mu i}}
\frac{\delta\Sigma_{GZ}^{lin}}{\delta\Omega_{\mu}^m}
= \Delta^{ai}_\varphi \;, \label{eqmotion-phi}
\end{align}
where
\begin{align}
\Delta^{ai}_\varphi =-\partial^{2}\bar\varphi^{a}_{i}
+\partial_{\mu}\lambda^{a}_{\mu i}
+\partial_{\mu}\bar\lambda^{a}_{\mu i}
-\gamma^2gf^{abc}A_{\mu}^b\delta_{c\mu}^i\;. \label{delta-phi}
\end{align}
\item The local, linearly broken, equation of motion of $\omega_i^a$
\begin{align}
\frac{\delta \Sigma_{GZ}^{lin}}{\delta \omega_i^a}
-\partial_{\mu}\frac{\delta\Sigma_{GZ}^{lin}}{\delta\rho^{a}_{\mu i}}
-igf^{abc}\bar{\omega}_i^b\frac{\delta \Sigma_{GZ}^{lin}}{\delta b^c}
-gf^{abc}A_{\mu}^b\frac{\delta \Sigma_{GZ}^{lin}}{\delta {\mathcal{C}}_{\mu i}^c} = \Delta^{ai}_{\omega}\;, \label{eqmotion-omega}
\end{align}
where
\begin{align}
\Delta^{ai}_{\omega}=-\partial^{2}\bar\omega^{a}_{i}+\partial_{\mu}\bar\rho^{a}_{\mu i} + \partial_{\mu} \bar{\mathcal{C}}^a_{\mu i} \;.
\end{align}
\item The integrated Ward identity
\begin{align}
\int d^4x\; \left(c^a\frac{\delta \Sigma_{GZ}^{lin}}{\delta \omega_i^a} - \bar{\omega}_i^a\frac{\delta \Sigma_{GZ}^{lin}}{\delta \bar{c}^a} + \frac{\delta\Sigma_{GZ}^{lin}}{\delta \mathcal{C}_{\mu i}^c}\frac{\delta\Sigma_{GZ}^{lin}}{\delta \Omega_{\mu}^c} -\partial_{\mu}c^c \frac{\delta\Sigma_{GZ}^{lin}}{\delta \mathcal{C}_{\mu i}^c}\right)= 0\;. \label{ward-ident}
\end{align}
\item The ghost equation \cite{Piguet:1995er,Blasi:1990xz}
\begin{align}
\mathcal{G}^{a}(\Sigma_{GZ}^{lin})=\Delta^{a}_{c}\;,
\end{align}
where
\begin{align}
\mathcal{G}^{a}&=\int d^{4}x\biggl[\frac{\delta}{\delta{c}^{a}}
+gf^{abc}\biggl(-i\bar{c}^{b}\frac{\delta}{\delta b^{c}}
+\bar\omega^{b}_{i}\frac{\delta}{\delta\bar\varphi^{c}_{i}}
+\varphi^{b}_{i}\frac{\delta}{\delta\omega^{c}_{i}}
+\bar\eta^{b}_{\mu i}\frac{\delta}{\delta\rho^{c}_{i}}
+\bar{\mathcal{C}}^{b}_{\mu i}\frac{\delta}{\delta\lambda^{c}_{\mu i}}
+\bar\rho^{b}_{\mu i}\frac{\delta}{\delta\bar\lambda^{c}_{\mu i}}
+\eta^{b}_{\mu i}\frac{\delta}{\delta\mathcal{C}^{c}_{\mu i}}\biggr)\biggr]\;,
\end{align}
and $\Delta^{a}_{c}$ is a linear breaking
\begin{align}
\Delta^{a}_{c}=\int d^{4}x\,gf^{abc}(\Omega^{b}_{\mu}A^{c}_{\mu}
-L^{b}c^{c}
-\gamma^{2}\delta^{i}_{c\mu}\bar\rho^{b}_{\mu i})\;.
\end{align}
\item The linearly broken identity
\begin{equation}
{\cal N}_{ij} \left(\Sigma_{GZ}^{lin}\right) = -\gamma^2 \int d^4x\; \delta^{ab} \delta_{\mu\nu} \delta^{j}_{\nu b} \left( {\bar \rho}^a_{\mu i} + \bar{\mathcal{C}}^{a}_{\mu i} \right) \;, \label{nij}
\end{equation}
and
\begin{equation}
{\cal N}_{ij} = \int d^4x \left( - {\bar \omega}^a_i \frac{\delta}{\delta {\bar \varphi}^a_j} + {\varphi}^a_j \frac{\delta}{\delta {\omega}^a_i} +{\bar \eta}^a_{\mu j} \frac{\delta}{\delta {\rho}^a_{\mu i}}
-{\bar{\mathcal{C}}}^a_{\mu i} \frac{\delta}{\delta {\lambda}^a_{\mu j}}
- ({\bar{\rho}^a_{\mu i}}+ {\bar{\mathcal{C}}}^a_{\mu i} )\frac{\delta}{\delta {\bar{\lambda}}^a_{\mu j}}
+( {\eta}^a_{\mu j} + {\bar \eta}^a_{\mu j}) \frac{\delta}{\delta {\cal C}^a_{\mu i}}
\right) \label{nijop}
\end{equation}
\item The linearly broken global symmetry $U(f)$
\begin{align}
\mathcal{Q}_{ij}(\Sigma_{GZ}^{lin})= \gamma^2 \delta^{ab} \delta_{\mu\nu}\int d^4x\;(\delta_{b\nu}^j \eta_{\mu i}^{a}-\delta_{b\nu}^i \bar{\lambda}_{\mu j}^{a}) \;,
\end{align}
where
\begin{align}
\mathcal{Q}_{ij}&=\int d^4x\ \left(\varphi_i^a\frac{\delta }{\delta \varphi_j^a}
-\bar{\varphi}_j^a\frac{\delta }{\delta \bar{\varphi}_i^a}
+\omega_i^a\frac{\delta }{\delta \omega_j^a}
-\bar{\omega}_i^a\frac{\delta}{\delta \bar{\omega}_j^a}
+\mathcal{C}_{\mu i}^a\frac{\delta }{\delta \mathcal{C}_{\mu j}^a}
-\bar{\mathcal{C}}_{\mu j}^a\frac{\delta }{\delta \bar{\mathcal{C}}_{\mu i}^a}
\right.\nonumber\\
&\left.
+\rho_{\mu i}^a\frac{\delta}{\delta \rho_{\mu j}^a}
-\bar{\rho}_{\mu j}^a\frac{\delta }{\delta \bar{\rho}_{\mu i}^a}
+\eta_{\mu i}^a\frac{\delta}{\delta \eta_{\mu j}^a}
-\lambda_{\mu j}^a\frac{\delta }{\delta \lambda_{\mu i}^a}
+\bar{\eta}_{\mu i}^a\frac{\delta}{\delta \bar{\eta}_{\mu j}^a}
-\bar{\lambda}_{\mu j}^a\frac{\delta }{\delta \bar{\lambda}_{\mu i}^a}
\right)\;,
\label{uf}
\end{align}
\item The linearly broken rigid identity
\begin{equation}
{\cal W}^a (\Sigma_{GZ}^{lin}) = \gamma^2 f^{abc} \int d^4x\; \left( {\bar \lambda}^{bc}_{\mu\mu} + {\eta}^{bc}_{\mu\mu} \right) \;, \label{color}
\end{equation}
with
\begin{eqnarray}
{\cal W}^a &=& \int d^4x\; f^{abc}\Bigl( A^b_\mu \frac{\delta}{\delta A^c_\mu} +\Omega^b_\mu \frac{\delta}{\delta \Omega^c_\mu} +c^b \frac{\delta}{\delta c^c} +L^b \frac{\delta}{\delta L^c} +{\bar c}^b \frac{\delta}{\delta {\bar c}^c} +b^b \frac{\delta}{\delta b^c} + {\bar \omega}^b_i \frac{\delta}{\delta {\bar \omega}^c_i} + { \omega}^b_i \frac{\delta}{\delta {\omega}^c_i} + {\bar \varphi}^b_i \frac{\delta}{\delta {\bar \varphi}^c_i} \nonumber \\
&+& {\varphi}^b_i \frac{\delta}{\delta {\varphi}^c_i} +{\bar \eta}^b_{\mu i} \frac{\delta}{\delta {\bar \eta}^c_{\mu i} }
+{ \eta}^b_{\mu i} \frac{\delta}{\delta { \eta}^c_{\mu i} } +{\bar{\mathcal{C}}}^b_{\mu i} \frac{\delta}{\delta {\bar{\mathcal{C}}}^c_{\mu i} } + {{\mathcal{C}}}^b_{\mu i} \frac{\delta}{\delta {{\mathcal{C}}}^c_{\mu i} }
+{\bar{\rho}}^b_{\mu i} \frac{\delta}{\delta {\bar{
\rho}}^c_{\mu i} }
+{\rho}^b_{\mu i} \frac{\delta}{\delta {\rho}^c_{\mu i} }
+{\bar{\lambda}}^b_{\mu i} \frac{\delta}{\delta {\bar{
\lambda}}^c_{\mu i} }
+{\lambda}^b_{\mu i} \frac{\delta}{\delta {\lambda}^c_{\mu i} }
\Bigl) \nonumber \\
\label{colorop}
\end{eqnarray}
\end{enumerate}
\section{Conclusion}
In this work the issue of the BRST symmetry in the Gribov-Zwanziger theory has been addressed. We have pointed out that the soft breaking of the BRST symmetry exhibited by the Gribov-Zwanziger action can be converted into a linear breaking upon introduction of a set of BRST quartets of auxiliary fields. Due to its compatibility with the Quantum Action Principle \cite{Piguet:1995er}, the linearly broken BRST symmetry gives rise to suitable Slavnov-Taylor identities, as summarized by eq.\eqref{st1}. The renormalization aspects of the theory can thus be addressed by looking at the cohomology of the nilpotent linearized operator ${\cal B}_{\Sigma_{GZ}^{lin}}$, eq.\eqref{linsti}. \\\\Although the details of the renormalizability of the Gribov-Zwanziger theory in the new set of variables will be reported in a more detailed work,
we believe that the present observation might improve our current understanding of the issue of the BRST symmetry in the presence of the Gribov horizon.
\section*{Acknowledgments}
The Conselho Nacional de Desenvolvimento Cient\'{\i}fico e
Tecnol\'{o}gico (CNPq-Brazil), the Faperj, Funda{\c{c}}{\~{a}}o de
Amparo {\`{a}} Pesquisa do Estado do Rio de Janeiro, the Latin
American Center for Physics (CLAF), the SR2-UERJ, the
Coordena{\c{c}}{\~{a}}o de Aperfei{\c{c}}oamento de Pessoal de
N{\'{\i}}vel Superior (CAPES) are gratefully acknowledged.
|
{
"timestamp": "2010-10-22T02:03:03",
"yymm": "1009",
"arxiv_id": "1009.4135",
"language": "en",
"url": "https://arxiv.org/abs/1009.4135"
}
|
\section{Introduction}
The first major event in the Universe after the
big bang and the inflation stage is the creation of light elements
\citep{wfh67}
including D, He-3 and Li, which are of importance for our everyday life or considered as energy source on the Earth in the future. The combination of neutron and proton
to form deuterium is the first step in the nuclear reaction chain. The big
bang nucleosynthesis (BBN) starts
when the temperature of the Universe is about $T\sim 0.1$MeV (redshift $z\sim
4\times
10^8$) and the deuterium
destroying high
energy photons have become rare. BBN proceeds swiftly after this deuterium
bottleneck is overcome and is over by $T\sim 0.01$MeV. The next major event
in the Universe is that of the recombination of primordial plasma
(\cite{zks68};\cite{peebles68}) to form
neutral atoms starting with helium III to helium II recombination at $4500\lesssim z \lesssim 7000$ and
ending with hydrogen recombination at $800 \lesssim z \lesssim 1800$. In this paper we will
explore a unique connection between these two major events, the
recombination of beryllium ${}^7$Be and its decay to lithium ${}^7\rm{Li}$ and
in the process also mention a doublet of narrow lines in the cosmic neutrino
background. This unique event brings together the MeV scale nuclear
physics of nucleosynthesis and eV scale atomic physics of
recombination. We will also find that the very low energy atomic physics of
hyperfine transitions in ${}^7\rm{Be}$ has a surprisingly strong influence on the
decay properties of ${}^7\rm{Be}$ usually associated with MeV scale physics. This
problem of beryllium to lithium conversion is one of the last
problems not investigated in detail so far
connected with the primordial nucleosynthesis and the process of
recombination of ions and electrons in the radiation field of the cosmic
microwave background (CMB). This problem is very
attractive because it demonstrates a very rare case in astrophysics when
the atomic processes and even the populations of the hyperfine structure
influences nuclear processes.
The cosmological parameters used for numerical calculations are
Hubble constant $H_0=73\rm{km/s/Mpc}$, matter fraction $\Omega_m=0.24$,
CMB temperature $T_{CMB}=2.726K$, baryon
fraction
$\Omega_b=0.043$, cosmological constant $\Omega_{\Lambda}=0.76$.
\section{Decay of ${}^7\rm{Be}$}
At the end of BBN the abundances of light elements relative to hydrogen
are \citep{serpico2004}: $X_{{}^4\rm{He}}=6\times 10^{-2}$,
$X_{{}^2\rm{H}}=2.5\times 10^{-5}$, $X_{{}^3\rm{He}}=10^{-5}$,
$X_{{}^3\rm{H}}\sim 10^{-7}$, $X_{{}^7\rm{Be}}\sim 10^{-10}$,
$X_{{}^7\rm{Li}}\sim 10^{-11}$, $X_{{}^6\rm{Li}}\sim 10^{-14}$ and
$X_{\rm{M}} \lesssim 10^{-15}$, where $X_{\rm{M}}$ is the abundance of
all heavier elements, $X_i=n_i/n_{\rm{H}}$, $n_i$ is the abundance of
species $i$ and $n_{\rm{H}}$ is the number density of hydrogen nuclei.
Note that about $\sim 90\%$ of the primordial ${}^7\rm{Li}$ comes from
${}^7\rm{Be}$ and only $\sim 10\%$ of ${}^7\rm{Li}$ is produced directly
in BBN, this division is however sensitive to the baryon to photon ratio
$\eta$ \citep{bbnreview}.
${}^7\rm{Be}$ in its fully ionized state in the early Universe is
stable. Fully ionized beryllium can capture electrons from the plasma at
the high densities in the stellar interiors \citep{bahcall}. However the
electron capture rate for the fully ionized beryllium from the
plasma is completely negligible in the early
Universe due to the low electron density. Once
recombined with an electron beryllium decays by
electron capture through the following reactions
\begin{align}
{}^7\rm{Be}+ e^- \rightarrow {}^7\rm{Li}+ \nu_e \label{r1}\\
{}^7\rm{Be}+ e^- \rightarrow {}^7\rm{Li}^{\ast}+ \nu_e,\hspace{4 pt}
{}^7\rm{Li}^{\ast}\rightarrow {}^7\rm{Li} + \gamma,\label{r2}
\end{align}
where ${}^7\rm{Li}^{\ast}$ is the excited state of lithium nucleus with
energy $477.6$KeV above the ground state.
The Q-value of the reaction is
$861.8$KeV \citep{tilley2002}. The laboratory value for the half life of
${}^7\rm{Be}$ is 53.2 days. The branching ratio is $10.44\%$ for reaction \ref{r2}.
These laboratory values have been calculated and measured for neutral ${}^7\rm{Be}$ with four
electrons.
The ${}^7\rm{Be}$ in the primordial plasma will however decay as soon as
it acquires a single electron. Therefore we are interested in hydrogen-like
${}^7\rm{Be}^{3+}$. The electrons in
the $2s$ shell of neutral ${}^7\rm{Be}$ have a $\sim 1\%$ effect on the decay rate as shown by
experiments involving $\rm{Be}^{2+}(\rm{OH_2})_4$ as well as other chemical
compositions \citep{exp1} and we
will ignore the influence of 2s shell electrons in the discussions below. Helium-like ${}^7\rm{Be}^{2+}$ has practically the same lifetime and
branching ratio as the neutral ${}^7\rm{Be}$. The change in the decay rate and the
branching ratio from helium-like ${}^7\rm{Be}^{2+}$ to hydrogen-like ${}^7\rm{Be}^{3+}$ depends strongly on the spin temperature of the hyperfine structure of hydrogen-like ${}^7\rm{Be}^{3+}$ ions.
This statement is easy to understand. We will make simple estimates of
decay rates based
on the calculations by \cite{becalc}. ${}^7\rm{Li}^{\ast}$ has a nuclear spin of
$I=1/2$ which is different from the nuclear spin of $I=3/2$ of ${}^7\rm{Li}$ and
${}^7\rm{Be}$. The two initial states available for ${}^7\rm{Be}^{3+}$ are with total angular
momentum $F^+=2$ and $F^-=1$. The two final states available to
${}^7\rm{Li}^{\ast}+\nu_e$ are $F=0,1$. Thus
the angular momentum can only be conserved for the reaction \ref{r2} from
the initial state
$F^-=1$ of ${}^7\rm{Be}^{3+}$ to the final state $F=1$ of ${}^7\rm{Li}^{\ast}$. The reaction is
suppressed for the other initial hyperfine state with total angular momentum
$F^+=2$. Correspondingly for reaction \ref{r1}, where the nuclear spin does
not change in the reaction, the two available final states have the same
total angular momentum as the initial state, $F=1,2$, and the reactions from both
the initial hyperfine states are possible.
Millielectronvolt hyperfine splitting dictates the branching ratio for
these two nuclear reactions. In hydrogen-like ${}^7\rm{Be}^{3+}$ ion the population
levels of the two hyperfine states (or equivalently the spin temperature) are
determined by the rates of radiative decay and pumping due to well known physical processes.
Our calculation is simplified because pumping by the CMB radiation field
and due to electron collisions is much faster than the decay rate and
cosmological time, i.e. the spin temperature should be close to the CMB
temperature $T$ equal at that time to the electron temperature. The energy
of the hyperfine splitting is much smaller than the temperature at that
time. This means that the spin temperature is very high compared to the
hyperfine splitting and the hyperfine structure sublevels are populated according to their statistical weights.
The Lyman-$\alpha$ resonant scattering \citep{field,vars} with rate
$P_{\alpha} \sim 4\pi \int d\nu \sigma_{\nu}I_{\nu}/(h\nu) \sim
100\rm{s}^{-1}$ at $z\sim 30000$, spin changing collisions with electrons with rate
$\sim n_e \kappa_{10} \sim 10^{-4}\rm{s}^{-1}$ \citep{furn} and stimulated
emission and absorption of CMB at hyperfine transition with rate $\sim B_{21}I_{\nu}\sim
4\times 10^{-6}\rm{s}^{-1}$ are all faster than the decay rate $\lambda \sim
7\times 10^{-8}\rm{s}^{-1}$ and would make the spin temperature equal to the
matter/radiation temperature $T$. Above $\sigma_{\nu}$ is the absorption
cross section for Lyman$-\alpha$ photons including the line profile from Doppler broadening. $I_{\nu}$
is the background CMB intensity. It corresponds to Wien region of the black
body spectrum for the Lyman-$\alpha$
photons and Rayleigh-Jeans region for the hyperfine transition
radiation. $B_{21}$ is the Einstein B coefficient for stimulated emission. $n_e$ is the electron number density and $\kappa_{10}$ is the
spin change cross section. $\kappa_{10}\propto Z^{-2}$ for hydrogenic
ions \citep{amg89}, where $Z$ is the nuclear charge. There is also a mild
energy dependence of the cross section at temperatures of interest of $\sim
1/T^{1/2}$. The spin change cross section for hydrogen-electron
collisions is of the
order $\sim 10^{-9}\rm{cm}^3\rm{s}^{-1}$ at $T\sim 10^5$K (\cite{smith};\cite{furn}) and thus for ${}^7\rm{Be}^{3+}$ it will be
$\sim 10^{-10}\rm{cm}^3\rm{s}^{-1}$.
Thus Lyman-$\alpha$ scattering, Ionization and recombination, Collisions with electrons and stimulated emission and radiation of CMB and
maintains $T_{spin}=T$. These rates are plotted in Figure \ref{fig1} along
with the expansion rate defined by the Hubble parameter $H(z)$. It is interesting to note that a typical hydrogenic beryllium atom
would have flipped its electron spin a billion times due to Lyman-$\alpha$
scattering and recombined and ionized $100$ times before finally being able
to decay. The energy difference between
the two hyperfine states is $10^{-4}\rm{eV}<< T$. Thus the hyperfine states
would always be distributed according to their statistical
weights.
\begin{figure}[h]
\epsffile{fig1.eps}
\caption{Comparison of different processes influencing beryllium to lithium
decay. It is interesting to note that a typical hydrogenic beryllium atom
would have flipped its electron spin a billion times due to Lyman-$\alpha$
scattering and recombined and ionized $100$ times before finally being able
to decay.}
\label{fig1}
\end{figure}
\begin{figure}[h]
\epsffile{fig2_v2.eps}
\caption{Effect of hyperfine splitting of ${}^7\rm{Be}^{3+}$ on the nuclear
reactions. The half-life given above corresponds to the high redshift universe
when the spin temperature is maintained at a high value by collisions
with electrons. The half-life would be different if the populations
of hyperfine levels of hydrogenic ${}^7\rm{Be}^{3+}$ are
not distributed according to their statistical weights.}
\label{fig2}
\end{figure}
\begin{figure}[h]
\epsffile{fig3_v2.eps}
\caption{Decay of helium-like ${}^7\rm{Be}^{2+}$.}
\label{fig3}
\end{figure}
In helium-like ${}^7\rm{Be}^{2+}$ (or Li-like and neutral ${}^7\rm{Be}$) spin directions of two $1s$
shell
electrons is always opposite and the problem radically differs from the
problem in the hydrogen-like ion. Thus
there is no hyperfine splitting in the case of helium-like ions, but the relative
direction of the spins of the captured electron and nucleus is of great
importance. We can write the probabilities of the two reactions in helium-like
ions in terms of the corresponding probabilities of the hydrogen-like ions
in the two hyperfine states using the recent theoretical calculations of \cite{becalc}
which have the support of experiments involving hydrogen-like and helium-like ${}^{142}\rm{Pm}$ ions \citep{beexp}. Then we
can use these relations and the relative populations of the hyperfine structure sub-levels in
the hydrogen-like ions to find the formula for the probabilities of both channels of
interest. The reactions for hydrogen-like ${}^7\rm{Be}^{3+}$ and helium-like ${}^7\rm{Be}^{2+}$ are summarized in Figures
\ref{fig2} and \ref{fig3}.
For reaction \ref{r1} the initial and final spin of the nuclei are same, $I=3/2$ and
the electron capture rates of ${}^7\rm{Be}^{2+}$ and ${}^7\rm{Be}^{3+}$ are related by \citep{becalc}
\begin{equation}\label{eq3}
\lambda_{{}^7\rm{Be}^{2+}}=\left(\frac{2F^++1}{2I+1}\right)\lambda^+ + \left(\frac{2F^-+1}{2I+1}\right)\lambda^-,
\end{equation}
where $\lambda^{\pm}$ are the decay rates from two
hyperfine states of ${}^7\rm{Be}^{3+}$. If $f^{\pm}$ are the fraction of atoms in the
two hyperfine states, the net decay rate is given by
\begin{align}\label{eq4}
\lambda_{{}^7\rm{Be}^{3+}}&=f^+ \lambda^{+}
+f^- \lambda^{-} \nonumber\\
&=\left(\frac{2F^++1}{2(2I+1)}\right)\lambda^+ +
\left(\frac{2F^-+1}{2(2I+1)}\right)\lambda^-\nonumber\\
&=\lambda_{{}^7\rm{Be}^{2+}}/2,
\end{align}
where we used the fact that $(2F^++1)+(2F^-+1)=2(2I+1)$. The factor of two
can be understood as follows. In the helium-like atom both
channels corresponding to $\lambda^+$ and $\lambda^-$ are available
simultaneously which are added according to the statistical weights. For
the hydrogen-like atom also for high spin temperature we add the
probabilities of two channels according to the statistical weights. But for
the helium-like atom the capture rate gets multiplied by a factor of two
corresponding to the fact that any of the available electrons can be
captured through any of the available channels. Thus the factor of two is
the result of the fact that there are two electrons available for
helium-like atom and that for high spin temperature for the hydrogen-like
atom we add the capture rates
according to their statistical weights. This factor would, in general,
be different from two if the atoms in the hyperfine state were not
distributed according to their statistical weights, for example at low
temperatures comparable to the energy difference between the two states.
Since the statistically averaged rate coefficients relevant for our
calculation for the helium-like and hydrogen-like atoms
differ by a just factor of two, we do not need the individual rates for
$\lambda^+$ and $\lambda^-$ and can use the experimental result for the
helium-like/neutral ${}^7\rm{Be}$.
For reaction \ref{r2}
however the ${}^7\rm{Li}^{\ast}$ has a spin of $1/2$ and only capture from $F^-$
state is allowed. The above formula is valid with $\lambda^+=0$ and for the
net
decay rate we have the same result, $\lambda_{{}^7\rm{Be}^{3+}}^{\ast}=\lambda_{{}^7\rm{Be}^{2+}}^{\ast}/2$. We are using ${}^{\ast}$ for the reaction rates of reaction
\ref{r2} while no ${}^{\ast}$ indicates the reaction rates for reaction
\ref{r1}. Using the laboratory value of
$\lambda_{\rm{lab}}=\ln{2}/53.2\rm{days}$ for the total decay rate
of ${}^7\rm{Be}^{2+}$, we get net decay rate
$\lambda_{\rm{bbn}}=0.5\lambda_{\rm{lab}}$ with the same branching ratio,
$10.4\%$ of the decays following reaction \ref{r2}. The half life of
hydrogen-like ${}^7\rm{Be}^{3+}$ with the hyperfine states distributed according to
their statistical weights is thus equal to 106.4 days. Coincidently this is the
same result we would have got from the simple considerations of the electron
density at the nucleus. We must emphasize here that the exact life time of
${}^7\rm{Be}^{3+}$ is not very important for us as long as it is much shorter than
the cosmological time at the epoch of beryllium to lithium conversion and
the duration of ${}^7\rm{Be}^{3+}$ recombination. All
the width of the two neutrino lines comes from the kinetics of recombination.
In particular we have ignored the electron-electron interaction in the
helium-like ${}^7\rm{Be}^{2+}$, the effect of which on our calculation of time of
${}^7\rm{Be}$ decay and the neutrino spectrum would be $\sim few \%$.
\section{Recombination of ${}^7\rm{Be}$}
Now we can write down the kinetic equations for recombination of ${}^7\rm{Be}^{4+}$.
\begin{align}
\frac{dX_{{}^7\rm{Be}^{4+}}}{dz} & =
\frac{1}{H(z)(1+z)}\left[n_e(z)X_{{}^7\rm{Be}^{4+}}\alpha_{{}^7\rm{Be}^{4+}} -
\beta_{{}^7\rm{Be}^{3+}}X_{{}^7\rm{Be}^{3+}}\right]\\
\frac{dX_{{}^7\rm{Be}^{3+}}}{dz} & =
\frac{1}{H(z)(1+z)}\left[-n_e(z)X_{{}^7\rm{Be}^{4+}}\alpha_{{}^7\rm{Be}^{4+}} +
\beta_{{}^7\rm{Be}^{3+}}X_{{}^7\rm{Be}^{3+}}+\lambda_{\rm{bbn}}X_{{}^7\rm{Be}^{3+}}\right]\\
X_{{}^7\rm{Be}} & = X_{{}^7\rm{Li}}^{decay}+X_{{}^7\rm{Be}^{4+}}+X_{{}^7\rm{Be}^{3+}},
\end{align}
where $n_e(z)$ is the number density of electrons at redshift $z$, $H(z)$ is the Hubble
parameter, $\alpha_{{}^7\rm{Be}^{4+}}$ is the total recombination coefficient
including recombination to the ground state. $X_{{}^7\rm{Li}}^{decay}$ is the lithium fraction coming from decay of
beryllium and excludes the lithium produced during BBN.
Because of the extremely low
number density of beryllium, the number of ionizing photons
released during direct recombination to the ground state is negligible
(about 10 orders of magnitude less)
compared to the ionizing photons already present in the background
radiation and can be neglected. $\beta_{{}^7\rm{Be}^{3+}}$ is the total ionization coefficient
which is related to $\alpha_{{}^7\rm{Be}^{4+}}$ by the condition that Saha equation
must be satisfied in equilibrium. Thus
\begin{equation}
\beta_{{}^7\rm{Be}^{3+}}=\alpha_{{}^7\rm{Be}^{4+}}
\frac{(2\pi
m_e k_B T)^{3/2}}{(2\pi \hbar)^3}e^{\frac{-\chi_{Be}}{k_BT}}.
\end{equation}
Note that the 2-photon decay rate from $2s$ state of ${}^7\rm{Be}^{3+}$ is $3.4\times
10^4 \rm{s}^{-1}$ \citep{goldman}. This is much faster than the same rate
for hydrogen $\sim 8 \rm{s}^{-1}$ due to the higher charge of the nucleus. More importantly it is much faster than the ionization rate
$\beta \sim 6\times 10^{-3} \rm{s}^{-1}$ at $z \sim 30000$ and the electron capture rate
$\lambda_{bbn}=7.5\times 10^{-8}\rm{s}^{-1}$, $t_{1/2}=106.4$days. Thus $2s$ level must be included in
the total recombination coefficient. The number of Ly$\alpha$ photons
produced would be of order $X_{{}^7\rm{Be}}$ and can be neglected. Thus
equations for beryllium recombination are much simpler than that of
hydrogen and helium recombination.
$\alpha_{{}^7\rm{Be}^{4+}}$ is the total recombination coefficient including direct
recombinations to the ground state and is
given by \citep{ppb91}
\begin{equation}
\alpha_{{}^7\rm{Be}^{4+}} = 10^{-13}Z\frac{at^b}{1+ct^d}\hspace{4 pt} \rm{cm}^3\rm{s}^{-1},
\end{equation}
where $t=(T/10^4K)/Z^2$, $a=5.596$, $b=-0.6038$, $c=0.3436$ and $d=0.4479$
and $Z$ is the nuclear charge..
Figure \ref{fig4} shows the result of integrating the recombination
equations for beryllium. For comparison the Saha equilibrium solution for
beryllium recombination is also shown. The beryllium to lithium conversion
occurs significantly earlier at $z=30000$ than the $z=25000$ value
predicted by the
Saha solution. The reason of the difference is connected with the short
decay time of recombined ${}^7\rm{Be}^{3+}$. In the Saha equation we follow the
balance between the recombination and photoionization but a typical atom
has recombined and ionized many times even though the net recombination in
equilibrium may be small. In reality
due to the decay of beryllium on a time scale much shorter than the
cosmological time the equilibrium is
never established.
\begin{figure}[h]
\epsffile{fig4.eps}
\caption{Lithium number density as a fraction of total
beryllium and lithium number density from decay of beryllium. For
comparison thermal equilibrium results using Saha equation are also
plotted.}
\label{fig4}
\end{figure}
\section{Energy release and narrow doublet of neutrino lines}
Every decay of beryllium to lithium is accompanied by emission of an
electron neutrino with $89.6\%$ of them having energy of $861.8$KeV and
nuclear recoil energy of $56.5$eV. $10.4\%$ of the decays go to excited
state of lithium, electron neutrino gets $384.2$KeV and nuclear recoil has
energy of only $11.2$eV which is slightly larger than the thermal energy
$T=7eV$ of the plasma at
redshift $z=30000$. The recoil velocity of $v/c=5.85\times 10^{-5}$ will result in a natural line width of $56$eV due
to Doppler shift for
the $477.6$KeV photon emitted when lithium relaxes to ground state almost
instantaneously ($T_{1/2}=73$fs). These
photons will down scatter by Compton scattering with electrons giving most of their energy
to the plasma due to recoil effect \citep{zs1969}. The energy transfer by Compton scattering will become
inefficient when the photons reach the critical energy of
$m_eH(z)/n_e\sigma_T|_{z=30000} \sim 80$eV when the energy transfer rate
becomes less than the expansion rate (\cite{bd90}). This will leave a small distortion in
the high energy part of CMB. The energy transfered to plasma will
additionally cause a $y-$type distortion of $\sim
(0.104E_{\gamma}/T)X_{{}^7\rm{Be}}\eta\sim 10^{-16}$. The neutrinos however will
free stream to us and we can calculate the neutrino spectrum today. This
is plotted in Figure \ref{fig5} including both $861.8$KeV neutrinos and
$384.2$KeV neutrinos,
\begin{equation}
-\frac{dn_{\nu}}{dE}(E)=n_{H0}\left[\frac{0.896(1+z)^2}{861.8\rm{KeV}}\frac{dX_{{}^7\rm{Li}}^{decay}}{dz}|_{1+z=\frac{861.8\rm{KeV}}{E}}+\frac{0.104(1+z)^2}{384.2\rm{KeV}}\frac{dX_{{}^7\rm{Li}}^{decay}}{dz}|_{1+z=\frac{384.2\rm{KeV}}{E}}\right],
\end{equation}
where $n_{H0}$ is the density of hydrogen nuclei at $z=0$. The full width at
half maximum (FWHM) for the first line is $2.3$eV and the central energy is
$29.5$eV. For the second line the FWHM is $1$eV with a central energy of
$13.1$eV. The line width at half maximum is $\delta E/E =7.8\%$. The width and asymmetric line profile of these lines is defined by the
kinetics of recombination.
\begin{figure}[h]
\epsffile{fig5.eps}
\caption{Neutrino spectrum from beryllium decay in the early
Universe. There are two lines corresponding to two electron capture decay
branches. For comparison equilibrium results (marked Saha) are also
plotted. The FWHM for high energy line is $2.3$eV with a central frequency of
$29.5$eV. For second line FWHM is $1$eV
central frequency is $13.1$eV. The line width at half maximum is $\delta
E/E =7.8\%$. The width and asymmetric line profile of these lines is defined by the
kinetics of recombination.}
\label{fig5}
\end{figure}
\section{Comparison with other sources of neutrinos and energy release before
recombination}
At $T\sim 0.1\rm{MeV}$ annihilation of electrons and positrons makes the biggest
contribution to cosmic neutrinos, however these neutrinos have a much
broader thermal spectrum and form part of the low energy cosmic neutrino
background (CNB). Decay of neutrons and other nuclear reactions during BBN
also contribute neutrinos, but due to high redshift $z\sim 10^8$ these
neutrinos are also today redshifted to the sub-eV energies, $\sim
1\rm{MeV}/10^8\sim 0.01\rm{eV}$. The spectrum of these neutrinos would be
broad since they are emitted over a period when redshift changes by a
factor of $\sim 2$. Energy released
during BBN is also quickly thermalized and does not lead to spectral
distortions. Decay of tritium created during BBN occurs at $z\sim 2.5\times
10^5$ corresponding to half life of $12.32$ years. Decay of tritium to
helium-3 results in an electron with average energy of $Q_e=5.7$KeV and an
antineutrino with average energy $\sim 12.9$KeV. The antineutrino spectrum would
be
broad as they decay over a period corresponding to the age of the
Universe at that time and these antineutrinos would have an average energy
of $\sim 0.05 eV$ today assuming they have zero mass. About $10\%$ of tritium
has decayed by $z=6.35\times 10^5$ and $90\%$ by z=$1.35\times 10^5$
leading to the width of the neutrino spectrum of $\delta E/E\sim
1.4$. This broadening is comparable to the intrinsic width of the neutrino spectrum
$\delta E/E\sim E_{max}/E_{avg}\sim 18.6/12.9=1.4$. Thus non-thermal neutrinos from all sources
before
recombination other than ${}^7\rm{Be}$ decay would have a spectrum much broader than those from ${}^7\rm{Be}$
decay and would have much lower energy. These low energy neutrinos would
be non-relativistic today in more than one mass eigenstates \citep{pdg}
compared to relativistic neutrinos from ${}^7\rm{Be}$ decay.
The
energy released into the plasma from tritium decay in the form of energetic electrons would
result in a chemical potential of $\mu \sim (Q_e/T)(n_{{}^3H}/n_{\gamma}) \sim
10^{-15}$ in the CMB. This is about the same amount of entropy generated in
the beryllium decay much later.
\section{Conclusions}
Lithium-7 observed today was originally produced as beryllium-7 during
primordial nucleosynthesis \citep{wfh67}. Although half life of beryllium atoms is very short,
it has to wait until the beginning of recombination epoch to decay. We have
calculated the exact redshift when this happens. We have also estimated the
effect of energy release during the decay on the cosmic microwave
background. In addition the neutrinos produced during the decay give rise
to unique narrow lines in the cosmic neutrino background.
These lines are too weak to be observable today. We should mention that
the detection of the much more numerous but lower energy CNB
neutrinos is currently being discussed \citep{neutrino}. The recombination and decay of
beryllium has nevertheless theoretical significance. It marks the end of
primordial nucleosynthesis and the beginning of recombination.
\section*{Acknowledgements}
Rishi Khatri would like to thank Prof. Brian Fields for informing him about
beryllium decay in the early Universe and discussions on the subject.
|
{
"timestamp": "2011-06-01T02:03:38",
"yymm": "1009",
"arxiv_id": "1009.3932",
"language": "en",
"url": "https://arxiv.org/abs/1009.3932"
}
|
\section{Introduction}
The \textit{proceedings} are the records of a conference.
ACM seeks to give these conference by-products a uniform,
high-quality appearance. To do this, ACM has some rigid
requirements for the format of the proceedings documents: there
is a specified format (balanced double columns), a specified
set of fonts (Arial or Helvetica and Times Roman) in
certain specified sizes (for instance, 9 point for body copy),
a specified live area (18 $\times$ 23.5 cm [7" $\times$ 9.25"]) centered on
the page, specified size of margins (2.54cm [1"] top and
bottom and 1.9cm [.75"] left and right; specified column width
(8.45cm [3.33"]) and gutter size (.083cm [.33"]).
The good news is, with only a handful of manual
settings\footnote{Two of these, the {\texttt{\char'134 numberofauthors}}
and {\texttt{\char'134 alignauthor}} commands, you have
already used; another, {\texttt{\char'134 balancecolumns}}, will
be used in your very last run of \LaTeX\ to ensure
balanced column heights on the last page.}, the \LaTeX\ document
class file handles all of this for you.
The remainder of this document is concerned with showing, in
the context of an ``actual'' document, the \LaTeX\ commands
specifically available for denoting the structure of a
proceedings paper, rather than with giving rigorous descriptions
or explanations of such commands.
\section{The {\secit Body} of The Paper}
Typically, the body of a paper is organized
into a hierarchical structure, with numbered or unnumbered
headings for sections, subsections, sub-subsections, and even
smaller sections. The command \texttt{{\char'134}section} that
precedes this paragraph is part of such a
hierarchy.\footnote{This is the second footnote. It
starts a series of three footnotes that add nothing
informational, but just give an idea of how footnotes work
and look. It is a wordy one, just so you see
how a longish one plays out.} \LaTeX\ handles the numbering
and placement of these headings for you, when you use
the appropriate heading commands around the titles
of the headings. If you want a sub-subsection or
smaller part to be unnumbered in your output, simply append an
asterisk to the command name. Examples of both
numbered and unnumbered headings will appear throughout the
balance of this sample document.
Because the entire article is contained in
the \textbf{document} environment, you can indicate the
start of a new paragraph with a blank line in your
input file; that is why this sentence forms a separate paragraph.
\subsection{Type Changes and {\subsecit Special} Characters}
We have already seen several typeface changes in this sample. You
can indicate italicized words or phrases in your text with
the command \texttt{{\char'134}textit}; emboldening with the
command \texttt{{\char'134}textbf}
and typewriter-style (for instance, for computer code) with
\texttt{{\char'134}texttt}. But remember, you do not
have to indicate typestyle changes when such changes are
part of the \textit{structural} elements of your
article; for instance, the heading of this subsection will
be in a sans serif\footnote{A third footnote, here.
Let's make this a rather short one to
see how it looks.} typeface, but that is handled by the
document class file. Take care with the use
of\footnote{A fourth, and last, footnote.}
the curly braces in typeface changes; they mark
the beginning and end of
the text that is to be in the different typeface.
You can use whatever symbols, accented characters, or
non-English characters you need anywhere in your document;
you can find a complete list of what is
available in the \textit{\LaTeX\
User's Guide}\cite{Lamport:LaTeX}.
\subsection{Math Equations}
You may want to display math equations in three distinct styles:
inline, numbered or non-numbered display. Each of
the three are discussed in the next sections.
\subsubsection{Inline (In-text) Equations}
A formula that appears in the running text is called an
inline or in-text formula. It is produced by the
\textbf{math} environment, which can be
invoked with the usual \texttt{{\char'134}begin. . .{\char'134}end}
construction or with the short form \texttt{\$. . .\$}. You
can use any of the symbols and structures,
from $\alpha$ to $\omega$, available in
\LaTeX\cite{Lamport:LaTeX}; this section will simply show a
few examples of in-text equations in context. Notice how
this equation: \begin{math}\lim_{n\rightarrow \infty}x=0\end{math},
set here in in-line math style, looks slightly different when
set in display style. (See next section).
\subsubsection{Display Equations}
A numbered display equation -- one set off by vertical space
from the text and centered horizontally -- is produced
by the \textbf{equation} environment. An unnumbered display
equation is produced by the \textbf{displaymath} environment.
Again, in either environment, you can use any of the symbols
and structures available in \LaTeX; this section will just
give a couple of examples of display equations in context.
First, consider the equation, shown as an inline equation above:
\begin{equation}\lim_{n\rightarrow \infty}x=0\end{equation}
Notice how it is formatted somewhat differently in
the \textbf{displaymath}
environment. Now, we'll enter an unnumbered equation:
\begin{displaymath}\sum_{i=0}^{\infty} x + 1\end{displaymath}
and follow it with another numbered equation:
\begin{equation}\sum_{i=0}^{\infty}x_i=\int_{0}^{\pi+2} f\end{equation}
just to demonstrate \LaTeX's able handling of numbering.
\subsection{Citations}
Citations to articles \cite{bowman:reasoning, clark:pct, braams:babel, herlihy:methodology},
conference
proceedings \cite{clark:pct} or books \cite{salas:calculus, Lamport:LaTeX} listed
in the Bibliography section of your
article will occur throughout the text of your article.
You should use BibTeX to automatically produce this bibliography;
you simply need to insert one of several citation commands with
a key of the item cited in the proper location in
the \texttt{.tex} file \cite{Lamport:LaTeX}.
The key is a short reference you invent to uniquely
identify each work; in this sample document, the key is
the first author's surname and a
word from the title. This identifying key is included
with each item in the \texttt{.bib} file for your article.
The details of the construction of the \texttt{.bib} file
are beyond the scope of this sample document, but more
information can be found in the \textit{Author's Guide},
and exhaustive details in the \textit{\LaTeX\ User's
Guide}\cite{Lamport:LaTeX}.
This article shows only the plainest form
of the citation command, using \texttt{{\char'134}cite}.
This is what is stipulated in the SIGS style specifications.
No other citation format is endorsed.
\subsection{Tables}
Because tables cannot be split across pages, the best
placement for them is typically the top of the page
nearest their initial cite. To
ensure this proper ``floating'' placement of tables, use the
environment \textbf{table} to enclose the table's contents and
the table caption. The contents of the table itself must go
in the \textbf{tabular} environment, to
be aligned properly in rows and columns, with the desired
horizontal and vertical rules. Again, detailed instructions
on \textbf{tabular} material
is found in the \textit{\LaTeX\ User's Guide}.
Immediately following this sentence is the point at which
Table 1 is included in the input file; compare the
placement of the table here with the table in the printed
dvi output of this document.
\begin{table}
\centering
\caption{Frequency of Special Characters}
\begin{tabular}{|c|c|l|} \hline
Non-English or Math&Frequency&Comments\\ \hline
\O & 1 in 1,000& For Swedish names\\ \hline
$\pi$ & 1 in 5& Common in math\\ \hline
\$ & 4 in 5 & Used in business\\ \hline
$\Psi^2_1$ & 1 in 40,000& Unexplained usage\\
\hline\end{tabular}
\end{table}
To set a wider table, which takes up the whole width of
the page's live area, use the environment
\textbf{table*} to enclose the table's contents and
the table caption. As with a single-column table, this wide
table will ``float" to a location deemed more desirable.
Immediately following this sentence is the point at which
Table 2 is included in the input file; again, it is
instructive to compare the placement of the
table here with the table in the printed dvi
output of this document.
\begin{table*}
\centering
\caption{Some Typical Commands}
\begin{tabular}{|c|c|l|} \hline
Command&A Number&Comments\\ \hline
\texttt{{\char'134}alignauthor} & 100& Author alignment\\ \hline
\texttt{{\char'134}numberofauthors}& 200& Author enumeration\\ \hline
\texttt{{\char'134}table}& 300 & For tables\\ \hline
\texttt{{\char'134}table*}& 400& For wider tables\\ \hline\end{tabular}
\end{table*}
\subsection{Figures}
Like tables, figures cannot be split across pages; the
best placement for them
is typically the top or the bottom of the page nearest
their initial cite. To ensure this proper ``floating'' placement
of figures, use the environment
\textbf{figure} to enclose the figure and its caption.
This sample document contains examples of \textbf{.eps}
and \textbf{.ps} files to be displayable with \LaTeX. More
details on each of these is found in the \textit{Author's Guide}.
\begin{figure}
\centering
\epsfig{file=fly.eps}
\caption{A sample black and white graphic (.eps format).}
\end{figure}
\begin{figure}
\centering
\epsfig{file=fly.eps, height=1in, width=1in}
\caption{A sample black and white graphic (.eps format)
that has been resized with the \texttt{epsfig} command.}
\end{figure}
As was the case with tables, you may want a figure
that spans two columns. To do this, and still to
ensure proper ``floating'' placement of tables, use the environment
\textbf{figure*} to enclose the figure and its caption.
Note that either {\textbf{.ps}} or {\textbf{.eps}} formats are
used; use
the \texttt{{\char'134}epsfig} or \texttt{{\char'134}psfig}
commands as appropriate for the different file types.
\subsection{Theorem-like Constructs}
Other common constructs that may occur in your article are
the forms for logical constructs like theorems, axioms,
corollaries and proofs. There are
two forms, one produced by the
command \texttt{{\char'134}newtheorem} and the
other by the command \texttt{{\char'134}newdef}; perhaps
the clearest and easiest way to distinguish them is
to compare the two in the output of this sample document:
This uses the \textbf{theorem} environment, created by
the\linebreak\texttt{{\char'134}newtheorem} command:
\newtheorem{theorem}{Theorem}
\begin{theorem}
Let $f$ be continuous on $[a,b]$. If $G$ is
an antiderivative for $f$ on $[a,b]$, then
\begin{displaymath}\int^b_af(t)dt = G(b) - G(a).\end{displaymath}
\end{theorem}
The other uses the \textbf{definition} environment, created
by the \texttt{{\char'134}newdef} command:
\newdef{definition}{Definition}
\begin{definition}
If $z$ is irrational, then by $e^z$ we mean the
unique number which has
logarithm $z$: \begin{displaymath}{\log e^z = z}\end{displaymath}
\end{definition}
\begin{figure}
\centering
\psfig{file=rosette.ps, height=1in, width=1in,}
\caption{A sample black and white graphic (.ps format) that has
been resized with the \texttt{psfig} command.}
\end{figure}
Two lists of constructs that use one of these
forms is given in the
\textit{Author's Guidelines}.
\begin{figure*}
\centering
\epsfig{file=flies.eps}
\caption{A sample black and white graphic (.eps format)
that needs to span two columns of text.}
\end{figure*}
and don't forget to end the environment with
{figure*}, not {figure}!
There is one other similar construct environment, which is
already set up
for you; i.e. you must \textit{not} use
a \texttt{{\char'134}newdef} command to
create it: the \textbf{proof} environment. Here
is a example of its use:
\begin{proof}
Suppose on the contrary there exists a real number $L$ such that
\begin{displaymath}
\lim_{x\rightarrow\infty} \frac{f(x)}{g(x)} = L.
\end{displaymath}
Then
\begin{displaymath}
l=\lim_{x\rightarrow c} f(x)
= \lim_{x\rightarrow c}
\left[ g{x} \cdot \frac{f(x)}{g(x)} \right ]
= \lim_{x\rightarrow c} g(x) \cdot \lim_{x\rightarrow c}
\frac{f(x)}{g(x)} = 0\cdot L = 0,
\end{displaymath}
which contradicts our assumption that $l\neq 0$.
\end{proof}
Complete rules about using these environments and using the
two different creation commands are in the
\textit{Author's Guide}; please consult it for more
detailed instructions. If you need to use another construct,
not listed therein, which you want to have the same
formatting as the Theorem
or the Definition\cite{salas:calculus} shown above,
use the \texttt{{\char'134}newtheorem} or the
\texttt{{\char'134}newdef} command,
respectively, to create it.
\subsection*{A {\secit Caveat} for the \TeX\ Expert}
Because you have just been given permission to
use the \texttt{{\char'134}newdef} command to create a
new form, you might think you can
use \TeX's \texttt{{\char'134}def} to create a
new command: \textit{Please refrain from doing this!}
Remember that your \LaTeX\ source code is primarily intended
to create camera-ready copy, but may be converted
to other forms -- e.g. HTML. If you inadvertently omit
some or all of the \texttt{{\char'134}def}s recompilation will
be, to say the least, problematic.
\section{Conclusions}
This paragraph will end the body of this sample document.
Remember that you might still have Acknowledgments or
Appendices; brief samples of these
follow. There is still the Bibliography to deal with; and
we will make a disclaimer about that here: with the exception
of the reference to the \LaTeX\ book, the citations in
this paper are to articles which have nothing to
do with the present subject and are used as
examples only.
\section{Acknowledgments}
This section is optional; it is a location for you
to acknowledge grants, funding, editing assistance and
what have you. In the present case, for example, the
authors would like to thank Gerald Murray of ACM for
his help in codifying this \textit{Author's Guide}
and the \textbf{.cls} and \textbf{.tex} files that it describes.
\bibliographystyle{abbrv}
\section{Proofs}
\begin{proof}[Proof of \autoref{lem:elem}]
Suppose $B=(s_1, \cdots, s_k)$, using the definition of $u$ we have:
\begin{align}
u(B|A) & =
\sum_{j=1}^k u(s_j | A \bot B_{[1, j-1]}) \label{eq:dis:bound:p}
\end{align}
The sum on the right hand side of \eqref{eq:dis:bound:p} consist of $k$ terms, so there should be at least
one term which is above or equal to the average of the terms. That means there should be an index $1 \le j'
\le k$ such that \eqref{eq:dis:bound:jp} holds.
\begin{align}
u(s_{j'}|A\bot B_{[1, j'-1]}) & \ge \frac{1}{k} u(B|A) \label{eq:dis:bound:jp}\\
u(s_{j'}|A) & \ge \frac{1}{|B|} u(B|A) \label{eq:dis:bound:jpd}
\end{align}
Combining \eqref{eq:dis:bound:jp} with \autoref{cond:ncomp} because $A \prec A \bot B_{[1, j'-1]}$ we get
\eqref{eq:dis:bound:jpd} which completes the proof.
\end{proof}
\begin{proof}[Proof of \autoref{thm:dis}]
According to \autoref{lem:elem} we argue that for any $H=(s_1, \cdots, s_T)$ and $\alpha$ for which
\eqref{eq:thm:dis} holds, \eqref{eq:dis:elem} must also hold.
\begin{align}
u(s_i | H_{[1, i-1]}) & \ge \frac{\alpha}{T} u(O | H_{[1, i-1]}) \label{eq:dis:elem} \\
u(s_i | H_{[1, i-1]}) & \ge \frac{\alpha}{T} (u(O \bot H_{[1, i-1]})-u(H_{[1, i-1]})) \label{eq:dis:elem1} \\
u(s_i | H_{[1, i-1]}) & \ge \frac{\alpha}{T} (u(O)-u(H_{[1, i-1]})) \label{eq:dis:elem2} \\
u(H_{[1, i]})-u(H_{[1, i-1]}) & \ge \frac{\alpha}{T} (u(O)-u(H_{[1, i-1]})) \label{eq:dis:elem3} \\
u(H_{[1, i]}) & \ge \frac{\alpha}{T}u(O) + (1-\frac{\alpha}{T}) u(H_{[1, i-1]}) \label{eq:dis:elem:fin}
\end{align}
In order to derive \eqref{eq:dis:elem2} from \eqref{eq:dis:elem1} we have used \autoref{cond:mono} to infer
that $u(O \bot H_{[1, i-1]}) \ge u(O)$.
\begin{align}
u(H_{[1, T]}) & \ge \left(1-(1-\frac{\alpha}{T})^T\right) u(O) \\
u(H) & \ge \left(1-\left((1-\frac{\alpha}{T})^\frac{T}{\alpha}\right)^\alpha \right) u(O)\\
u(H) & \ge \left(1-\frac{1}{e^\alpha}\right) u(O) \label{eq:dis:final}
\end{align}
Notice that \eqref{eq:dis:elem:fin} defines a recurrence relation which can be solved to get
\eqref{eq:dis:final} which completes the proof.
\end{proof}
\begin{proof}[Proof of \autoref{lem:dv}]
Suppose $B=((s_1, \Delta t_1), \cdots, (s_k, \Delta t_k))$ and let $B^i = ((s_1,\Delta t_1), \cdots, (s_i,
\Delta t_i))$. Using the definition of $u$ and \eqref{cor:con:udots} we have:
\begin{equation}
\label{eq:con:dv:sum}
u(B | A) = \sum_{i=1}^k \int_0^{\Delta t_i} \dot{u}_{s_i}(x|A \bot B^{i-1}) dx
\end{equation}
We argue that there should be some $1 \le i \le k$ for which there exist some $\delta \in [0, \Delta t_i)$
such that $\dot{u}_{s_i}(\delta |A \bot B^{i-1}) \ge \frac{1}{|B|}u(B|A)$ otherwise that means the term
inside the integral on the right hand side of \eqref{eq:con:dv:sum} is always less than $\frac{1}{|B|}u(B|A)$
which means the sum of the integrals would be less that $u(B|A)$ which contradicts the \eqref{eq:con:dv:sum}.
Suppose for $i'$ and $\delta'$ \eqref{eq:con:dv:B} holds.
\begin{align}
\dot{u}_{s_{i'}}(\delta' |A \bot B^{i'-1}) & \ge \frac{1}{|B|}u(B|A) \label{eq:con:dv:B}\\
\dot{u}_{s_{i'}}(\delta' |A) & \ge \frac{1}{|B|}u(B|A) \label{eq:con:dv:delta}\\
\dot{u}_{s_{i'}}(0|A) & \ge \frac{1}{|B|}u(B|A) \label{eq:con:dv:0}
\end{align}
We can infer \eqref{eq:con:dv:delta} from \eqref{eq:con:dv:B} by using \autoref{lem:udot:ncomp}. Applying
\autoref{cor:con:dec} to that we get \eqref{eq:con:dv:0} which completes the proof.
\end{proof}
\begin{proof}[Proof of \autoref{thm:con}]
Using \autoref{lem:dv} we have \eqref{eq:thm:con:dv}. Combining that with \eqref{eq:thm:con} we get
\eqref{eq:thm:con:ncomp}. Using the definition of marginal values and using \autoref{cond:ncomp} we get
\eqref{eq:thm:con:de} which is a differential equation.
\begin{align}
& \forall t \in [0, T) \ \exists s \in S : \ \dot{u}_s(0|H_{[0,t)}) \ge \frac{1}{|O|} u(O|H_{[0,t)}) \label{eq:thm:con:dv} \\
& \forall t \in [0, T) : \ \frac{d}{dt}u(H_{[0,t)}) \ge \frac{\alpha}{T} u(O|H_{[0,t)}) \label{eq:thm:con:ncomp} \\
& \forall t \in [0, T) : \ \frac{d}{dt}u(H_{[0,t)}) \ge \frac{\alpha}{T} (u(O \bot H_{[0,t)})-u(H_{[0,t)})) \\
& \forall t \in [0, T) : \ \frac{d}{dt}u(H_{[0,t)}) \ge \frac{\alpha}{T} (u(O)-u(H_{[0,t)})) \label{eq:thm:con:de}
\end{align}
We can rephrase the \eqref{eq:thm:con:de} as \eqref{eq:thm:con:de2} and solve it to get
\eqref{eq:thm:con:fin0}.
\begin{align}
u(H_{[0,t)})+\frac{T}{\alpha} \frac{d}{dt}u(H_{[0,t)}) & \ge u(O) \label{eq:thm:con:de2} \\
\frac{d}{dt}\left(\frac{T}{\alpha} e^{\frac{\alpha}{T}t}u(H_{[0,t)})\right)
& \ge \frac{T}{\alpha} e^{\frac{\alpha}{T}t} u(O) \\
\int_0^x \frac{d}{dt}\left(\frac{T}{\alpha} e^{\frac{\alpha}{T}t}u(H_{[0,t)})\right) dt
& \ge \int_0^x e^{\frac{\alpha}{T}t} u(O) dt \\
\frac{T}{\alpha} e^{\frac{\alpha}{T}x}u(H_{[0,x)}) & \ge \frac{T}{\alpha} (e^{\frac{\alpha}{T}x}-1) u(O) \\
u(H_{[0,x)}) & \ge (1-\frac{1}{e^{\frac{\alpha}{T}x}}) u(O) \label{eq:thm:con:fin0} \\
u(H) & \ge (1-\frac{1}{e^\alpha}) u(O) \label{eq:thm:con:fin1}
\end{align}
Setting $x=T$ in \eqref{eq:thm:con:fin0} we get \eqref{eq:thm:con:fin1} which completes the proof.
\end{proof}
\begin{algorithm}[h]
\caption{Query Rewriting Algorithm \label{alg:rew}}
$H \leftarrow \emptyset$ \;
\For{$i \in M$}{
$ B_i \leftarrow \textrm{budget of ad $i$}$ \;
}
$\Bv \leftarrow (B_1, \cdots, B_m)$ \;
\While{$N \neq \emptyset$}{
$\Delta' \leftarrow 0$ \;
\tcp{find the best partial allocation to append}
\For{$j \in N$}{
$Y_j \leftarrow \emptyset$ \;
\tcp{find the best $k$ rewrites greedily}
\For{$w=1, \cdots, k$}{
$\delta' \leftarrow 0$ \;
\For{$r \in R \backslash Y_j$}{
$\delta \leftarrow u((j, Y_j\cup \{r\}, \Bv)|H)$ \;
\If{$\delta' < \delta$}{
$\delta ' \leftarrow \delta$ \;
$r' \leftarrow r$ \;
}
}
$Y_j \leftarrow Y_j \cup \{r'\}$ \;
}
\tcp{compute the marginal utility of adding $(j, Y_j)$}
$\Delta' \leftarrow u((j, Y_j, \Bv)|H)$ \;
\If{$\Delta' < \Delta$}{
$\Delta' \leftarrow \Delta$ \;
$j' \leftarrow j$ \;
}
}
Define $\Bv^{j'}$ to be exactly equal to how much of
the budgets are used by $(j, Y_j\cup \{r\}, \Bv)$ when appended to $H$ \;
$H \leftarrow H \bot (j', Y_{j'}, \Bv^{j'})$ \;
$N \leftarrow N \backslash \{j'\}$ \;
$\Bv \leftarrow \Bv-\Bv^{j'}$ \;
}
\end{algorithm}
\section{Introduction}
Submodularity over set functions is an important concept in combinatorial optimization. Many classical
discrete problems with greedy algorithm belong to this class. In economics, as well, the submodularity
property has received considerable attention since it captures the notion of decreasing marginal utilities.
Therefore, discovering the characteristics of submodular functions is of great interest. Most of the
instances in this class are NP-complete though. As a result, most of the work in the literature has been
focused on designing efficient approximation algorithms for these problems and the greedy approach has always
been a natural choice leading to simple implementations. Many have studied the behavior of greedy algorithms
on maximizing non-decreasing, submodular functions and it has been shown \cite{NWF78, NW78, W82} that greedy
algorithms achieve remarkably good approximations for maximizing non-decreasing submodular functions subject
to some constraints. We should note that however, in all the previous work, the submodularity property has
been defined on set functions. On the other hand there are some maximization problems that are defined on a
sequence rather than a set in which the order of elements matters. There are also problems that are defined
over continuous sequences. In this paper, we define the notion of \emph{Sequence Submodularity} for functions
defined over sequences (both continuous and discrete sequences). We will show that if the objective sequence
function is \emph{Sequence Submodular}, \emph{Non Decreasing} and in case of continuous sequences
\emph{Differentiable} then a greedy approach achieves $(1- \frac{1}{e})$ of the optimal solution for
maximizing the objective function under knapsack constraints (i.e., when there is a limit on the length of
the sequence). At the end, we present an application of this framework to internet advertising and more
specifically to the online ad allocation and query rewriting problem.
\section{Related Work}
\paragraph{Submodularity and greedy algorithms}
Submodularity has been studied in more depth in recent years due to its applications to combinatorial auctions (e.g.,
the submodular welfare problem \cite{LLN06, KLMM05}), generalized assignment problems \cite{FGMS06}, etc.
The greedy approach is a natural tool to solve maximization problems with a submodular objective function. Nemhauser
and Wolsey \cite{NW78} showed that greedy approach gives an $\frac{e}{e-1}$-approximation for maximizing a
non-decreasing submodular function over a uniform matroid. Nemhauser, Wolsey, and Fisher \cite{NWF78} considered this
problem over the independence system. They showed that if the independence system is the intersection of $M$ matroids,
the greedy algorithm gives an $M+1$ approximation. Recently, Goundan and Schulz \cite{GS07} generalized both these
results and showed that if an $\alpha$-approximate incremental oracle is available, then the greedy solution is a
$e^{1/\alpha}/(e^{1/\alpha}-1)$ approximation for maximizing a non-decreasing submodular functions over a uniform
matroid and an $\alpha M + 1$ approximation for the intersection of $M$ matroids. Feige et al in \cite{FMV07}, gave a
general framework for solving the non-monotone submodular problems.
\paragraph{Online allocation problem}
There is a considerable amount of literature on adword auctions considering different variations for improving web and
paid search results in the economics and computer science community. The adword auction problem that is considered in
this paper, is the online allocation problem. In the online allocation problem, the goal is to decide which ads to show
for each incoming query so that the the obtained profit from the advertisers is maximized. Several \cite{MSVV07,
LPSV07} papers have studied this problem. Mehta et al \cite{MSVV07} presented a deterministic algorithm with the
competitive ratio of $(1 - \frac{1}{e})$ in the worst case model. It can be shown that the competitive ratio for the
greedy algorithm is $\frac{1}{2}$ in the worst case analysis. Later, Goel et al \cite{GM08} showed that the competitive
ratio of the greedy approach in the random permutation model as well as the i.i.d model is $(1 - \frac{1}{e})$ and in
fact, the analysis is tight. Their proof is partly based on the techniques used in \cite{KVV90} for the online
bipartite matching problem. The offline variant of ad allocation has been studied in \cite{AM04, FGMS06}). It has also
been shown that the problem is NP-complete with the best known approximation factor of $(1 - \frac{1}{e})$ and the
results still hold even if bids are not very small compared to budgets. (If the bids are very small compared to budgets
the solution obtained based on LP rounding has an approximation factor very close to $1$.)
\section{Our Contribution}
In the previous works, the submodularity property is defined only on functions over sets. Nevertheless, there
are problems in which the goal is to choose a sequence of actions to maximize some utility function defined
over that sequence. In some of these problems, the order of actions matters. Also, sometimes, the actions are
continuous and each action is used for some specified duration. Such problems cannot be modeled using a
submodular set function. Throughout the rest of this paper, we define and characterize the conditions that
are necessary for sequence functions so that we can obtain the same conclusions about the behavior of a
greedy approach over this class of functions. A series of operations with the property that each operation is
performed for some specified duration can be seen as a continuous sequence. What we will show is that if a
sequence function has the three properties of being ``non-decreasing'', ``Sequence Submodular'' and
``differentiable'', a greedy approach always achieves a solution that is at least $(1 - \frac{1}{e})$ of the
optimal solution for the maximization problem subject to a constraint on the maximum length of the solution
sequence. As an example, we show that the online ad allocation problem with a fixed distribution of keywords
over time can be modeled as maximizing a continuous non-decreasing submodular sequence function for which we
can guarantee that the greedy approach achieves at least $(1- \frac{1}{e})$ of the optimal and also for the
problem of query rewriting as explained in \autoref{sec:rew} we achieve a $1-\frac{1}{e^{1-\frac{1}{e}}}
\approx 0.47$ approximation improving upon the $\frac{1}{4}$ approximation of \cite{MCKW08}.
\section{Conclusion}
In this paper, we defined the notion of \emph{submodularity} for
functions over sequences and then showed that if a sequence function
is submodular and non-decreasing, the approximation ratio of greedy
algorithm for maximizing such a function subject to a maximum length
constraint on the solution sequence is $(1 - \frac{1}{e})$. As an
example, we modeled the online ad allocation problem in this
framework implying that a greedy approach achieves a $1-\frac{1}{e}$
approximation assuming that the distribution of queries over
keywords do not change over time (i.e., queries are i.i.d random
variables).
\section{Model}
\label{sec:model}
Here we define the notation that we will use throughout the rest of this paper:
\begin{description}
\item[Discrete Sequence:] Let $S$ be a finite set. Any $A=(s_1, \cdots, s_k)$ where $k \in \mathds{N}\cup \{0\}$
and $s_i \in S$, is called a discrete sequence of elements of $S$ ($k=0$ is the empty sequence). We also denote the set
of all finite discrete sequences of $S$ by $\mathds{H}^D(S)$ which is formally defined as:
\begin{equation}
\mathds{H}^D(S) = \{A = (s_1, \cdots, s_k) | k \in \mathds{N}\cup\{0\} , s_i \in S \}
\end{equation}
Notice that a discrete sequence actually defines a discrete function from $\{1, \cdots, k\}$ to $S$ and any such
discrete function can be represented using a discrete sequence. We denote the value of the function defined by discrete
sequence $A$ at point $x$ by $A(x)$.
\item[Continuous Sequence:] Let $S$ be a finite set. Any $A=((s_1, \Delta t_1),
\cdots, (s_k, \Delta t_k))$ where $k \in \mathds{N}\cup\{0\}$ and $a_i \in S$ and $\Delta t_i \in \mathds{R}^+$, is
called a finite continuous sequence of elements of $S$. We also denote the set of all finite continuous sequences of
$S$ by $\mathds{H}^C(S)$ which is formally defined as:
\begin{equation}
\begin{split}
\mathds{H}^C(S) & = \{A = ((s_1, \Delta t_1), \cdots, (s_k, \Delta t_k)) | \nonumber \\
& \quad k \in \mathds{N}\cup\{0\}, a_i \in S, \Delta t_i \in \mathds{R}^+ \} \\
\end{split}
\end{equation}
Notice that a continuous sequence actually defines a function from $[0, \sum_{i=1}^k \Delta t_i)$ to $S$ in which any
$x \in [\sum_{j=1}^{i-1} \Delta t_j, \sum_{j=1}^i \Delta t_j)$ is mapped to $s_i$. Also notice that any function from
$[0, T)$ to $S$ in which the output changes a finite number of times when the input changes continuously from $0$ to
$T$ can also be represented using a finite continuous sequence. We denote the value of the function defined by
continuous sequence $A$ at point $x$ by $A(x)$.
\item[Sequence Function:] Let $S$ be a finite set. Any function $u : \mathds{H}^D(S) \rightarrow \mathds{R}$ is
called a sequence function (discrete). Also, any function $u : \mathds{H}^C(S) \rightarrow \mathds{R}$ is called a
sequence function (continuous).
\item[Length of a Sequence:] We denote the length of a sequence $A$ by $|A|$ which we define next. For any discrete
sequence $A=(s_1, \cdots, s_k)$ we define $|A|=k$. For any continuous sequence $A=((s_1, \Delta t_1), \cdots, (s_k,
\Delta t_k))$ we define $|A| = \sum_{i=1}^k \Delta t_i$.
\item[Equivalence of Sequences:] We say two sequences $A$ and $B$ are equivalent and denote that by $A \equiv B$
if they represent the same sequence that is if and only if they have the same length and their corresponding functions
have the same value at every point in their domain. The formal definition is given next.
If $A$ and $B$ are two discrete sequences, then $A \equiv B$ if and only if $|A| = |B|$ and for $\forall i \in \{1,
\cdots, |A|\} : A(i) = B(i)$.
If $A$ and $B$ are two continuous sequences, then $A \equiv B$ if and only if $|A| = |B|$ and $\forall x \in [0, |A|) :
A(x) = B(x)$.
\item[Concatenation of Sequences:] We denote the concatenation of two sequences $A$ and $B$ by $A \bot B$.
\item[Refinement of a Sequence:] We denote the portion of a discrete sequence $A$ in $[x,y]$ by $A_{[x,y]}$
and also the portion of a continuous sequence $A$ in $[x,y)$ by $A_{[x,y)}$ which we formally define as the following.
For a discrete sequence $A =(s_1, \cdots, s_k)$, if the intersection of $[1,k]$ and $[x,y]$ is empty we define
$A_{[x,y]}$ to be the empty sequence. Otherwise suppose $[f,l]$ is the intersection of the two, then we define
$A_{[x,y]}=(s_f, \cdots, s_l)$.
For a continuous sequence $A=((s_1, \Delta t_1), \cdots, (s_k, \Delta t_k))$, if the intersection of $[0,|A|)$ and
$[x,y)$ is empty we define $A_{[x,y)}$ to be the empty sequence. Otherwise suppose $[f,l)$ is their intersection then
we define:
\begin{align}
A_{[x,y)} & = ((s_p, \Delta t_p-\delta), (s_{p+1}, \Delta t_{p+1}), \cdots \nonumber \\
& \quad \cdots, (s_{q-1}, \Delta t_{q-1}), (s_q, \Delta t_q-\delta'))
\end{align}
where $q,l \in \mathds{N}$ and $\delta,\delta' \in \mathds{R}^+\cup\{0\}$ are chosen such that:
\begin{align}
\sum_{i=1}^{p-1} \Delta t_i & \le f < \sum_{i=1}^p \Delta t_i \\
\sum_{i=1}^{q-1} \Delta t_i & < l \le \sum_{i=1}^q \Delta t_i \\
\delta & = f-\sum_{i=1}^{p-1} \Delta t_i \\
\delta' & = \sum_{i=1}^q \Delta t_i-l
\end{align}
\item[Domination of Sequences:] We say sequence $A$ is dominated by sequence $B$ and we show that by $A \prec B$
if we can cut out parts of $B$ to get $A$. Next we give a formal definition.
If $A$ and $B$ are discrete sequences then $A \prec B$ if and only if $A$ is a subsequence of $B$.
If $A$ and $B$ are continuous sequences then $A \prec B$ if and only if there exist $m \in \mathds{N}, 0 \leq x_1 < x_2
< \cdots < x_{2m} \leq |B|$ such that:
\begin{equation}
A \equiv B_{[x_1, x_2)} \bot \cdots \bot B_{[x_{2m-1}, x_{2m})}
\end{equation}
\item[Marginal Value of a Sequence Function:] For a sequence function $u : \mathds{H}(S) \rightarrow \mathds{R}$ we
define $u(B|A) = u(A \bot B)-u(A)$ where $A,B \in \mathds{H}(S)$.
\end{description}
Throughout this paper we will use the $\emptyset$ to denote the empty sequence. We will also use $\mathds{H}(S)$
instead of $\mathds{H}^C(S)$ and $\mathds{H}^D(S)$ when a proposition applies to both discrete sequences as well as
continuous sequences.
\section{Submodular Non-decreasing Sequence Functions}
\label{sec:conds}
In this section we define the class of submodular non-decreasing sequence functions. In the next sections we provide a
greedy heuristic for maximizing such functions subject to a given maximum length for the solution sequence.
Let $S$ be a finite set and $u : \mathds{H}(S) \rightarrow \mathds{R}$ be a sequence function. We define the following
conditions:
\begin{condition}[Non-Decreasing]
\label{cond:mono}%
A sequence function $u$ is \emph{non-decreasing} if:
\begin{align}
\label{eq:mono}%
& \forall A, B \in \mathds{H}(S) : A \prec B \Rightarrow u(A) \le u(B) \\
& u(\emptyset) = 0
\end{align}
\end{condition}
\begin{condition}[Sequence-Submodularity]
\label{cond:ncomp}%
A sequence function $u$ is \emph{sequence-submodular} if:
\begin{equation}
\label{eq:ncomp}%
\forall A, B, C \in \mathds{H}(S) : A \prec B \Rightarrow u(C|A) \ge u(C|B)
\end{equation}
\end{condition}
\begin{condition}[Differentiability]
\label{cond:cont}
This condition only applies to continuous sequence functions. Note that we use the term \emph{``continuous sequence
function''} to signify that the argument to the function is a continuous sequence and not the function itself, however
the \emph{differentiability} condition that we define next is a property of the function. A continuous sequence
function $u : \mathds{H}^C(S) \rightarrow \mathds{R}$ satisfies the differentiability condition if for any $A \in
\mathds{H}^C(S)$, $u(A_{[0, t)})$ is continuous and differentiable with a continuous derivative with respect to $t$ for
$t \in [0, \infty)$ except that at a finite number of points it may have different left and right derivatives and thus
a non-continuous derivative.
\end{condition}
\section{Greedy Heuristic (Discrete)}
\label{sec:dis}
Here we provide a greedy heuristic for maximizing non-decreasing submodular sequence functions (discrete). Let $S$ be a
finite set and $u : \mathds{H}^D(S) \rightarrow \mathds{R}$ be non-decreasing submodular sequence function. Consider
the problem of finding a sequence $H \in \mathds{H}^D(S)$ that maximizes $u$ subject to $|H| \le T$ for a given $T \in
\mathds{N}$. Also suppose that $O \in \mathds{H}^D(S)$ where $O=(r_1, \cdots, r_T)$ is the optimal solution to this
problem.
\begin{lemma}
\label{lem:elem}
For any $A,B \in \mathds{H}^D(S)$ there exist $s \in S$ such that $u(s|A) \ge \frac{1}{|B|}u(B|A)$
\end{lemma}
All of the proofs are in the appendix when omitted.
We use the \autoref{lem:elem} to prove the following theorem:
\begin{theorem}
\label{thm:dis}
For sequence $H \in \mathds{H}^D(S)$ where $H=(s_1, \cdots, s_T)$ and $\alpha \in [0, 1]$ if:
\begin{equation}
\label{eq:thm:dis}
\forall i \in \{1, \cdots, T\}, \forall s \in S : u(s_i|H_{[1,i-1]}) \ge \alpha~u(s|H_{[1,i-1]})
\end{equation}
then:
\begin{equation}
\frac{u(H)}{u(O)} \ge 1-\frac{1}{e^\alpha}
\end{equation}
\end{theorem}
The condition of \autoref{thm:dis} is simply saying that $H = (s_1, \cdots, s_T)$ should be chosen by choosing each
$s_i$ locally such that $p(s_i|H_{[1,i-1]})$ is at least $\alpha$ times its optimal local maximum. Setting $\alpha=1$
means we can compute the locally optimal $s_i$ conditioned on $s_1, \cdots, s_{i-1}$. Based on the previous intuition
we present the greedy algorithm \ref{alg:discrete} to find $H$.
\begin{algorithm}
\caption{Greedy algorithm for the discrete case \label{alg:discrete}}
$H^0 \leftarrow \emptyset$ \;
\For{$i=1$ \KwTo $T$}{
find $s_i$ that maximizes $u(s_i | H^{i-1})$ \nllabel{alg:discrete:max}\;
$H^i \leftarrow H^{i-1} \bot s_i$ \;
}
$H \leftarrow H^T$ \;
\end{algorithm}
The greedy algorithm \autoref{alg:discrete} starts with an empty sequence $H^0$ and then builds the complete sequence
by finding at iteration $i$ the $s_i$ that gives the highest increase in the value of $u$ when appended to the end of
the current sequence or more formally the $s_i$ that maximizes $u(s_i|H^{i-1})$ (or equivalently maximizes
$u(H^{i-1}\bot s_i)$). Also note that in \autoref{alg:discrete}, at the step where we find $s_i$ that maximizes $u(s_i
| H^{i-1})$ . We may not be able to find the locally optimal $s_i$ and instead we may only be able to find $s_i$ for
which $u(s_i|H^{i-1})$ is at least $\alpha$ times its locally optimal maximum.
\begin{theorem}
\label{thm:dis:fin}
For any non-decreasing submodular function $u$ and any given $T \in \mathds{N}$, the greedy algorithm
\ref{alg:discrete} can be used to find a sequence that produces a value of $u$ which is at least $1-\frac{1}{e^\alpha}$
times of the optimal. In particular if we can locally find the optimal at each iteration the resulting sequence gives a
value of $u$ which is at least $1-\frac{1}{e}$ of the global optimal.
\end{theorem}
\begin{proof}
The proof of \autoref{thm:dis:fin} trivially follows from \autoref{thm:dis} and \autoref{alg:discrete}.
\end{proof}
\section{Greedy Heuristic (Continuous)}
\label{sec:con}
In this section we provide an equivalent of the greedy heuristic of \autoref{sec:dis} for the continuous version. Let
$S$ be a finite set and $u : \mathds{H}^C(S) \rightarrow \mathds{R}$ be differentiable non-decreasing submodular
sequence function. Consider the problem of finding a continuous sequence $H \in \mathds{H}^C(S)$ that maximizes $u$
subject to $|H| \le T$ for a given $T \in \mathds{R}^+$. Also suppose that $O \in \mathds{H}^C(S)$ where $O=((r_1,
\Delta w_1), \cdots, (r_k, \Delta w'_k))$ is the optimal solution.
We define $\dot{u}_s(\delta|A)$ where $s \in S$, $\delta \in \mathds{R}^+$ and $A \in \mathds{H}$ as the following:
\begin{align}
\label{eq:con:udot}
\dot{u}_s(\delta|A) & = \frac{d}{d \delta}u((s, \delta)|A) \\
& = \frac{d}{d \delta}\left(u(A \bot (s, \delta)) - u(A)\right) \\
& = \frac{d}{d \delta}u(A \bot (s, \delta)) \label{eq:con:udot:last}
\end{align}
We also define $\dot{u}_s(\delta|A)$ at $\delta=0$ as the following:
\begin{equation}
\label{eq:con:udot0}
\dot{u}_s(0|A) = \lim_{\delta \rightarrow 0^+} \dot{u}_s(\delta|A)
\end{equation}
Note that \eqref{eq:con:udot} is always defined because we are assuming that $u$ satisfies the \autoref{cond:cont} and
\eqref{eq:con:udot:last} can be written as $\frac{d}{d \delta}u((A \bot (s, \infty))_{[0,|A|+\delta)})$. Also note that
according to \autoref{cond:cont} $\dot{u}_s$ is a continuous function over $\mathds{R}^+$ except at a finite number of
points.
\begin{corollary}
\label{cor:con:udots}
For any $A \in \mathds{H}^C$ like $A=((s_1, \Delta t_1), \cdots \Br, (s_k, \Delta t_k))$ let $A^i = ((s_1,\Delta t_1),
\cdots, (s_i, \Delta t_i))$ then all of the following hold:
\begin{align}
u((s,\delta)|A) & = \int_0^\delta \dot{u}_s(x|A) dx \label{eq:con:int} \\
u((s,\delta_2)|A\bot(s,\delta_1)) & = \int_{\delta_1}^{\delta_2} \dot{u}_s(x|A) dx \label{eq:con:int2} \\
u(A) & = \sum_{i=1}^k \int_0^{\Delta t_i} \dot{u}_{s_i}(x|A^{i-1}) dx \label{eq:con:sum}
\end{align}
\end{corollary}
\begin{proof}
\eqref{eq:con:int} and \eqref{eq:con:int2} trivially follow from \eqref{eq:con:udot} and \eqref{eq:con:sum} follows
from the definition of marginal values.
\end{proof}
\begin{lemma}
\label{lem:udot:ncomp}
For any $A,B \in \mathds{H}^C$ such that $A \prec B$ and any $s \in S$, we have $\dot{u}_s(\delta|A) \ge
\dot{u}_s(\delta|B)$ for any $\delta \in \mathds{R}^+\cup \{0\}$ except at a finite number of points.
\end{lemma}
\begin{proof}
The proof is by contradiction. Suppose there are $A,B \in \mathds{H}^C$ such that $A \prec B$ and $s \in S$ and $\delta
\in \mathds{R}^+$ for which $\dot{u}_s(\delta|A) < \dot{u}_s(\delta|B)$. If either $\dot{u}_s(\delta|A)$ or
$\dot{u}_s(\delta|B)$ is non-continuous at $\delta$ then this is one of the finite number of points that are exceptions
in \autoref{lem:udot:ncomp}. Otherwise since they are both continuous at $\delta$ there should be a small neighborhood
around $\delta$ in which $\dot{u}_s(\delta|B)$ is greater than $\dot{u}_s(\delta|A)$. More formally:
\begin{equation}
\label{eq:udot:ncomp}
\exists \epsilon \in \mathds{R}^+ , \forall x \in [\delta-\epsilon,\delta+\epsilon] : \dot{u}_s(x|A) < \dot{u}_s(x|B)
\end{equation}
Now we show that \eqref{eq:udot:ncomp} can never happen:
\begin{align}
u((s,\epsilon)|A \bot (s,\delta-\epsilon)) & = \int_{\delta-\epsilon}^\delta \dot{u}_s(x|A) \\
u((s,\epsilon)|A \bot (s,\delta-\epsilon)) & < \int_{\delta-\epsilon}^\delta \dot{u}_s(x|B) \\
u((s,\epsilon)|A \bot (s,\delta-\epsilon)) & < u((s,\epsilon)|B \bot (s,\delta-\epsilon)) \label{eq:udot:ncomp:contradiction}
\end{align}
Notice that $A \bot (s,\delta-\epsilon) \prec B \bot (s,\delta-\epsilon)$ and therefore
\eqref{eq:udot:ncomp:contradiction} contradicts \autoref{cond:ncomp} which says $u$ is a submodular sequence function.
It shows that our assumption of $\dot{u}_s(\delta|A) < \dot{u}_s(\delta|B)$ leads to contradiction which completes the
proof.
\end{proof}
\begin{corollary}
\label{cor:con:dec}
For any $A \in \mathds{H}^C(S)$, and any $\delta \in [0, \infty)$, $\dot{u}_s(\delta|A)$ is a monotonically
non-increasing function in $\delta$. That is $\delta_1 < \delta_2 \Rightarrow \dot{u}_s(\delta_1|A) \ge
\dot{u}_s(\delta_2|A)$.
\end{corollary}
\begin{proof}
The proof is similar to the proof of \autoref{lem:udot:ncomp}.
\end{proof}
The following lemma in the equivalent of \autoref{lem:elem} for the continuous case.
\begin{lemma}
\label{lem:dv}
For any $A,B \in \mathds{H}^C(S)$ there exist $s \in S$ such that $\dot{u}_s(0|A) \ge \frac{1}{|B|}u(B|A)$
\end{lemma}
Next we present our main result for this section.
\begin{theorem}
\label{thm:con}
For any sequence $H \in \mathds{H}^C(S)$ where $H=((s_1, \Delta t_1), \cdots, (s_k, \Delta t_k))$ and $|H|=T$ and
$\alpha \in [0, 1]$, if:
\begin{equation}
\label{eq:thm:con}
\forall t \in [0, T), \forall s \in S : \frac{d}{dt}u(H_{[0,t)}) \ge \alpha~\dot{u}_s(0|H_{[0,t)})
\end{equation}
then:
\begin{equation}
\frac{u(H)}{u(O)} \ge 1-\frac{1}{e^\alpha}
\end{equation}
\end{theorem}
The condition of \autoref{thm:con} is simply saying that $H$ should be chosen such that at each point $t \in [0, T)$,
the derivative of $u$ is at least $\alpha$ times its optimal local maximum. Setting $\alpha=1$ means at each $t \in [0,
T)$ we can find the best $s \in S$ conditioned on $H_{[0, t)}$. Based on the previous intuition we present a generic
greedy algorithm \ref{alg:cont} to find $H$. This algorithm in general may not terminate, however if it terminates, for
the resulting $H$, $u(H)$ will be at least $(1-\frac{1}{e^\alpha})$ times the optimal. In general there can be other
ways for finding such an $H$ for each specific problem as we will show one such example later in this paper.
\begin{algorithm}
\caption{Greedy algorithm for the continuous case \label{alg:cont}}
$t \leftarrow 0$ \;
$i \leftarrow 1$ \;
$H^0 \leftarrow \emptyset$ \;
\While{$t < T$}{
find $(s_i, \Delta t_i)$ such that $\forall s \in S ~ \forall \delta \in [0, \Delta t_i):
\ \dot{u}_{s_i}(0|H^{i-1}\bot(s_i, \delta)) \ge \alpha~\dot{u}_s(0|H^{i-1}\bot(s_i,\delta))$ \;
$H^i \leftarrow H^{i-1} \bot (s_i, \Delta t_i)$ \;
$t \leftarrow t+\Delta t_i$ \;
$i \leftarrow i+1$ \;
}
$H \leftarrow H^{i-1}$ \;
\end{algorithm}
In algorithm \autoref{alg:cont} in the main loop we need an \emph{Incremental Oracle} that is specific to each problem.
As we mentioned before, there might be other ways for finding a sequence $H$ that satisfies the condition of
\autoref{thm:con} and as long as it satisfies that condition we have the $1-\frac{1}{e^\alpha}$ guarantee.
\section{Online ad allocation problem}
\label{sec:ad} The motivation for online ad allocation problem is the keyword based ad auctions. In these
auctions, advertiser submits to the search engine his bid for each keyword plus his total budget. Based on
these information, search engine should decide which ads to show for each keyword. The objective of online ad
allocation is to find a way to perform this allocation with maximum revenue for the search engine. Assuming
that for each query, the search engine can show $d$ ads simultaneously, the online ad allocation problem can
be defined as follows: We have $m$ ads and $n$ distinct keywords (query types). Let $M$ be the set of ads and
$N$ the set of query types. Let $p_{ij}$ be the expected payment of the advertiser to the search engine for
showing ad $i$ for a query of type $j$. The expected payment could be computed based on the click-through
rate of the ad, the relevance of the ad to the keyword, the bid of the advertiser for that keyword and
possibly other parameters. Also each ad $i$ has a budget $B_i$. The goal is to assign incoming queries to ads
as they arrive in such a way that maximizes the profit of the search engine in a given time period. Here we
make the assumption that the types of the incoming queries are i.i.d random variables drawn from a fixed but
possibly unknown distribution $q_j$ where $q_j$ is the probability of a query being of type $j$ ($\sum_j q_j
= 1$). Also we assume that the expected payments ($p_{ij}$) are small compared to budgets ($B_i$). Note that
in a sequence of $r$ queries the expected number of queries of type $j$ is $rq_j$. We would like to express
this as a function of time so we define a virtual time based on the number of queries that have arrived so
far. In terms of our virtual time the expected number of queries arriving in a period of length $\Delta t$ of
type $j$ is $\Delta t q_j$. Throughout the rest of this section we will omit the word ``virtual'' and always
use ``time'' to refer to virtual time unless explicitly stated otherwise. Also let $T$ be the end of the time
period in terms of the virtual time. So the problem is ti find an allocation that maximizes the revenue of
the search engine in time $[0,T)$.
Consider the offline version of the problem in which we know the queries in advance. We could solve the
problem using LP rounding and get a solution close to optimal (with the approximation ratio very close to $1$
assuming that $p_{ij} \ll B_i$). Now consider the online version of the problem in which we knew the
distribution $q_j$. Again we could use LP rounding to get a solution with the expected value very close to
optimal expected value. There are two problems however with the online version. The first one is that we
cannot use LP if we do not know the distributions and the second one is that due to the huge size of the
input it is not possible to use LP rounding in practice for this problem.
Next we consider the greedy algorithm and we show that its expected performance is at least $1-\frac{1}{e}$
of the optimal. The important advantages of the greedy algorithm are that it does not depend on the
distribution of the queries and it is easy and fast to compute in real time even with huge input data. As
such it is being used in practice \cite{GM08}.
We define a \emph{``configuration''} as a mapping of query types to ads such that each query type is mapped
to at most $d$ ads. Let $S$ be the set of all possible configurations. We can now represent any allocation of
ads to queries over time $[0, T)$ by a continuous sequence $H=((s_1, \Delta t_1), \cdots, (s_k, \Delta t_k))$
where $s_i \in S$, $\Delta t_i \in \mathds{R}^+$, $k \in \mathds{N}$ and $|H| = T$ which means \emph{``Use
each configuration $s_p$ (in order) for a duration of $\Delta t_p$ for $p \in \{1, \cdots, k\}$''}. We call
$H$ an \emph{``Allocation Strategy''}.
Let $u(H)$ be the expected utility of the search engine for using an allocation strategy $H$. Note that for
any given sequence of queries we can say based on $H$ exactly which ads are displayed for each incoming query
and so we can directly compute the utility of the search engine. Next we show that $u$ is a \emph{Submodular
Non-decreasing Sequence Function} and so using a greedy algorithm yields an allocation that is at least
$1-\frac{1}{e}$ of the optimal. First we explain how the greedy algorithm works.
At any point in time, the greedy method chooses the best configuration as follows: For each query type $j$
map it to the $d$ ads with highest $p_{ij}$ among those that have not exhausted their budgets yet and denote
them by $Q_j(s)$. Let $r(s)$ be the expected revenue rate of such a configuration $s$. We can write $r(s)$ as
follows:
\begin{align}
r(s) & = \sum_{j \in N} q_j \sum_{i \in Q_j(s)} p_{ij}
\end{align}
Note that the revenue of the search engine for using configuration $s$ for a short period of length $\Delta
t$ assuming that none of the ads exhaust their budget during that time is given by $r(s) \Delta t$.
The greedy algorithm works as follows: Choose the best configuration (the one with maximum $r(s)$) as
explained above by assigning query type $j$ to the $d$ ads with highest $p_{ij}$ among those that have not
exhausted their budgets yet. Keep that configuration until at least one of the ads runs out of budget. Then
recompute the best configuration and switch to it. It is easy to see that the derivative of $u(H_{[0,t)})$
with respect to $t$ is $r(s)$ where $s = H(t)$ is the configuration that is active at time $t$ in $H$. That
is because $\frac{d}{dt}u(H_{[0,t)}) = \dot{u}_s(H_{[0,t)})$ is exactly the rate at which the search engine
is accumulating profit at time $t$ which is $r(s)$ and for all other $s' \in S$ we have $r(s') \leq r(s)$ at
time $t$. That also means that our greedy algorithm satisfies the requirement of the incremental oracle in
\autoref{alg:cont} as the current configuration always has a higher revenue rate than all the other
configurations. Also note that we may need to change the configuration only when an ad runs out of budget
which means the total number of configuration changes is no more than $m$. The only thing that remains to be
shown is that the utility function $u$ is a \emph{Submodular Non-decreasing Sequence Functions} which we
prove next.
\begin{lemma}
\label{lemma:admonotone} The utility function of online ad allocation problem satisfies \autoref{cond:mono}.
In particular, consider the allocation strategies $A,B \in \mathds{H}$ and assume that $A \prec B$. The
remaining budget of each ad at the end of using $B$ is less than or equal to its remaining budget in $A$.
\end{lemma}
\begin{proof}
Consider the allocation strategies $A,B \in \mathds{H}$ and assume that $A \prec B$. We argue that the profit
extracted from each ad in $B$ is at least as much as the profit extracted from each ad in sequence $A$.
We partition the ads into two categories:
\begin{itemize}
\item
Ads that have no budget left after running sequence $B$.
\item
Ads that still have budget after running sequence $B$.
\end{itemize}
In the former case, sequence $B$ extracted the maximum possible budget from the ad. So for this set of ads,
our claim holds.
For the ads that belong to the second category, we know that they still have budget available. Consider an ad
$i$ that belongs to this category. We will show that the profit extracted by $B$ from this ad is at least as
much as the profit extracted by $A$.
Consider the configuration $s \in S$ that is active in $B$ for a total time of $\Delta t$. For all queries of
type $j$ that arrive during that time and any ad $i$ that is allocated to them by configuration $s$, we know
that the profit extracted from budget of ad $i$ by those queries is $\Delta t q_j$ because ad $i$ never ran
out of budget. Since $A \prec B$, configuration $s$ is either not present in $A$ or was used in $A$ for less
total time than $B$ and so the total profit extracted from ad $i$ in $A$ is no more than the profit extracted
from ad $i$ in $B$.
Since for both categories the expected profit extracted by $B$ from each ad is higher than or equal to the
profit extracted by $A$ from that ad, we can conclude that the non-decreasing property holds.
\end{proof}
Next, we show that \autoref{cond:ncomp} holds as well.
\begin{lemma}
\label{lemma:adncomp} Online ad allocation problem satisfies \autoref{cond:ncomp}.
\end{lemma}
\begin{proof}
Consider the allocation strategies $A,B,C \in \mathds{H}$ and assume that $A \prec B$. First of all, based on
\autoref{lemma:admonotone}, we know that the remaining budget of each ad after $A$ is less than or equal to
its remaining budget after $B$. It is also easy to see that the contribution of each ad to $u(C|B)$ or
$u(C|A)$ is equal to the difference in its budget before and after using the $C$. Now, consider using the
allocation strategy $A$ first followed by $C$. Again we partition the ads into two categories:
\begin{itemize}
\item Ads that have exhausted all of their budget after running $A\bot C$.
\item Ads that still have budget after running $A \bot C$.
\end{itemize}
The contribution of the ads in the first category to $u(C|B)$ is no more than their contribution to $u(C|A)$
because they had equal or more remaining budget after using $A$ than after using $B$ and they have
contributed all of their remaining budget to $u(C|A)$.
Now consider the ads that belong to the second category. By the same reasoning as we did for the proof of
\autoref{lemma:admonotone} we conclude that $C$ has has extracted profit from those ads at full rate since
they did not run out of budget. So their contribution to $u(C|A)$ and $u(C|B)$ is equal.
\end{proof}
Finally, we can show that \autoref{cond:cont} is also met. Notice that the derivative of the utility function
is a step function that changes its value only when either in the sequence there is change of configuration
or when some ad runs out of budget. The utility function is therefore differentiable and its derivative is
continuous except on the endpoints of each piece. The total number of pieces is bounded by the number of ads
which is finite. Therefore we conclude that the utility function is differentiable and its derivative is
continuous except at a finite number of points.
Using the above properties, we conclude that the approximation ratio of greedy algorithm that will select the
best configuration at each point of time is $(1 -\frac{1}{e})$.
\section{Query Rewriting}
\label{sec:rew} Query rewriting is a common mechanism used in information retrieval to improve the relevance
of the returned result. This method also has been used in the search advertising. When dealing with large
data sets, handling queries in real time might not be possible if we need to access a large portion of the
data set so some filtering might be required. Query rewriting tries to do the filtering while preserving the
quality as much as possible. At the high level, query rewriting outputs a list of rewrites relevant to the
original query. In this section we focus on the specific query rewriting problem related to search
advertising which is defined in \cite{MCKW08} and is explained next.
We are given a set $M$ of ads and a set $N$ of query types (keywords). Also for each ad $i$ and query $j$
define $p_{ij}$, $B_i$ and $q_j$ as in \autoref{sec:ad}. We are also given a set $R$ of rewrites. Each
rewrite $r \in R$ is associated with a small subset of the ads which we denote by $W_r$ and is also given to
us. The goal is to associate each query type with at most $k$ rewrites so that later the ad-allocator only
considers the ads that are associated with the rewrites of each query type in order to find the best ads to
show for incoming queries of that type. Suppose that $Y_j$ is the set of rewrites associated with query $j$.
Formally, the ad-allocator will then only consider the ads like $i$ such that $i \in \bigcup_{r \in Y_j} W_r$
to find the best $d$ ads to show for queries of type $j$. The problem is how to find the sets $Y_j$ so as to
maximize the maximum profit that can be extracted by the ad allocator.
Next we give a greedy algorithm that gives a $1-{\frac{1}{e}^{1-\frac{1}{e}}} \approx 0.47$ approximation
which improves the previous $0.25$ approximation given in \cite{MCKW08}.
We define a \emph{``partial allocation''} as a tuple of the form $(j, Y_j, \Bv^j)$ where $\Bv^j = (B_{1j},
\cdots, B_{mj})$ is a vector of budgets in which $B_{ij}$ is the maximum budget that we allow the ad
allocator to extract from ad $i$ for displaying the ad $i$ for query type $j$. $Y_j$ is the set of rewrites
for query type $j$. Note that any solution of the the query rewriting problem and the corresponding
allocation problem can be written as a sequence of the following form:
\begin{align}
H & = (j_1, Y_{j_1}, \Bv^{j_1}), \cdots, (j_n, Y_{j_n}, \Bv^{j_n})
\end{align}
We now define a utility function $u(H)$ on the sequences of the above form as follows:
\begin{definition}[$u(H)$]
\label{def:rew} Initialize $u$ to $0$ and set each of the $B_i$ to the total budget of ad $i$. For each of
the partial allocation tuples in order do the following. Suppose the current tuple is $(j, Y_j, \Bv^j)$. Set
the current budget limit of each ad $i$ to the minimum of $B_i$ and $B_{ij}$. Also suppose that $p_{ij} =0 $
for all ads that are not associated with any of rewrites in the $Y_j$. Also ignore all the queries of type
other than $j$. Use the greedy algorithm of \autoref{sec:ad} to solve the assignment problem for only the
queries of type $j$ considering the current budget limits. Notice that the greedy algorithm is optimal when
we have only one query type. After computing the allocation, update each $B_i$ to reflect how much of its
budget has been used by $j$ and then proceed to the next tuple in the sequence. Let $\Bv(H)$ denote the
vector of remaining budgets at the end of this process (we will use this notation later). Suppose $OPT=(1,
Y_1, \Bv^1), \cdots, (n, Y_n, \Bv^n)$ is the sequence in which $Y_j$'s are the optimal rewrites and $\Bv^j$
is the vector of budgets used by ad $j$ in the corresponding optimal allocation. Clearly $u(OPT)$ is equal to
the total utility of the search engine for the optimal solution.
\end{definition}
\begin{lemma}
\label{lemma:rew_sub} $u(H)$ as defined in \autoref{def:rew} is a non-decreasing submodular sequence
function.
\end{lemma}
\begin{proof}[Proof Sketch:]
We only give the sketch of the proof as it is very similar to the proof of \autoref{lemma:admonotone} and
\autoref{lemma:adncomp}. Again we separate the ads to two groups. Those who have exhausted their budget at
the end of computing $u$ using the \autoref{def:rew} and those who still have budget left. We can then verify
that for any two sequence of partial allocations $A$ and $B$ such that $A \prec B$, all of the ads that are
in the first group after computing the $u(B)$ using \autoref{def:rew} have made their maximum contribution to
$u(B)$ and so cannot contribute more to $u(A)$. For all the other ads we can show that their contributions to
$u(A)$ and $u(B)$ are equal. The submodularity property follow in the same was as for
\autoref{lemma:adncomp}.
\end{proof}
In order to be able to use the greedy algorithm of \autoref{alg:discrete} to approximate the OPT we need an
oracle that can find the best partial allocation $(j, Y_j, \Bv^j)$ to be appended to the current sequence.
The marginal utility of adding a partial allocation $(j, Y_j, \Bv^j)$ is a non-decreasing submodular function
in terms of $Y_j$ with the constrain that $|Y_j|=k$. Therefor for each $j$ we can get a $1-\frac{1}{e}$
approximation by using a greedy algorithm start from an empty $Y_j$ and add the rewrite the increases the
marginal utility the most until $k$ rewrites have been added. We then select among all possible query types
$j$ the one for which $(j, Y_j, \Bv^j)$ has the highest marginal utility and append that to the current
sequence of partial allocation. Since we are approximating the best $(j, Y_j, \Bv^j)$ within a factor of
$1-\frac{1}{e}$, based on \autoref{thm:dis:fin} the approximation ratio of the overall algorithm is
$1-\frac{1}{e^{1-\frac{1}{e}}} \approx 0.47$. The complete algorithm is described in \autoref{alg:rew}.
|
{
"timestamp": "2010-09-22T02:02:53",
"yymm": "1009",
"arxiv_id": "1009.4153",
"language": "en",
"url": "https://arxiv.org/abs/1009.4153"
}
|
\section{Introduction}
Grammars are mostly used to describe languages, like programming languages, in order to parse them. In this paper, we are not interested in the parsing problem of a language. On the contrary, the objective is to use a grammar from which the word parsing problem is known to be hard (as in NP), or even better, undecidable. Indeed, if one wants to do some metamorphism through the use of a grammar, one may want to avoid grammars for which techniques to build practical word recognizers of a language are known.
Van Wijngaarden grammars are different than the one which fall in Chomsky\rq{}s classification. Their writing is particular, and above all, their production process is quite different than the grammars in Chomsky\rq{}s hierarchy. We will see that these grammars may be used as \lq\lq{}code translators\rq\rq{}. They indeed have some rules which allow them to be very expressive.
\section{Metamorphism vs. Polymorphism}
The difference between polymorphism and metamorphism is often not very clear in people's mind, so we describe it quickly in this section.
\subsection{Polymorphism}
Polymorphism first appeared to counter the detection scheme of AV companies which was, and still is for a main part, based on signature matching. The aim of virus writers was to write a virus whose signature would change each time it evolves. In order to do so, the virus body is encrypted by an encryption function and it is decrypted by its decryptor at the runtime. The key used to encrypt each copy of the virus is changed, so that each copy has a different body (Figure \ref{polymorphism}).
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.4\textwidth]{polymorphismnb.png}
\caption{Two files infected by the same virus.}
\label{polymorphism}
\end{center}
\end{figure}
Another technique that can be used is to apply a different encryption scheme for each copy of the code. Of course such a technique alone is not enough to evade signature detection as it only shifts the problem, the decryptor being a good candidate for a signature. To resolve this, the decryption routine has to be changed too between each copy of the virus. To do so, virus writers include a mutation engine, which is also encrypted during the propagation process, and which is used to randomly generate a new decryption routine so it is different from copy to copy (\makebox{Figure \ref{polymorphism2}}).
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.4\textwidth]{polymorphism2nb.png}
\caption {Two files infected by the same virus.}
\label{polymorphism2}
\end{center}
\end{figure}
While in the first case the decryption routine can be used as a signature, this is not the case in the second one. Indeed, the decryption routine changes from mutation to mutation, thanks to the engine. \\
The mutation engine cannot be used as a signature neither, because it is a part of the body, thus it is encrypted. The propagation process can be summed up in five \makebox{steps :}
\begin{itemize}
\item The decryption routine decrypts the encrypted body;
\item The body is executed;
\item The code calls the mutation engine (which is decrypted at this stage) to transform the decryption routine;
\item The code and the mutation engine are encrypted;
\item The transformed decryption routine and the new encrypted body are then appended onto a new program.
\end{itemize}
\subsection{Metamorphism}
Metamorphism differs from polymorphism in the fact that there is no use of a decryption routine, because there is no encryption process. In other words, while a polymorphic code has to decrypt itself before it can be executed, a metamorphic one is executed directly.
Indeed, a metamorphic engine can be seen as a \lq\lq{}semantic translator\rq\rq{}. The idea is to rewrite a given code into another syntactically different, yet semantically equivalent one (Figure \ref{metamorphism}).
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.35\textwidth]{metamorphism.png}
\caption{Four equivalent codes.}
\label{metamorphism}
\end{center}
\end{figure}
Different techniques can be used to build an efficient metamorphic engine. Among these techniques we can observe :
\begin{itemize}
\item Junk code insertion :
a junk code is a code that is useless for the main code to perform its task.
\item Variable renaming :
the variables used between different versions of the code are different.
\item Control flow modifications :
some instructions are independent from each other, and so, can be swapped. Otherwise instructions can be shuffled and linked by jumps.
\end{itemize}
\section{Grammars}
In this section, we recall what formal grammars are and the link they have with languages.
\subsection{What is a grammar}
\begin{definition}
Let $\Sigma$ be a finite set of symbols called alphabet. A formal grammar G is defined by the \makebox{4-tuple} \makebox{$G = (V{}_{N}, V{}_{T}, S, P)$} \makebox{where :}
\begin{itemize}
\setlength{\itemsep}{0 pt}
\item $V_{N}$ is a finite set of \emph{non-terminal} symbols, \makebox{$V_{N} \cap \Sigma^{\ast} = \emptyset$};
\item $V_{T}$ is a finite set of \emph{terminal} symbols, \makebox{$V{}_{N} \cap V{}_{T} = \emptyset$};
\item S $\in V{}_{N}$ is the \emph{starting symbol} of the grammar;
\item P $\subseteq (V{}_{T}\cup V{}_{N})^{\ast}\times(V{}_{T}\cup V{}_{N})^{\ast}$ is a set of \emph{production rules}.
\end{itemize}
\end{definition}
Basically, a grammar can be seen as a set of rewriting rules over an alphabet. An alphabet is a finite set of symbols (like \lq{}$a$\rq{}, \lq{}$b$\rq{}). We distinguish two sets of symbols. The first one is the set of \emph{non-terminal} symbols, and the second is the set of \emph{terminal} symbols. Non-terminal symbols are symbols which are used to be replaced by the right-hand side of a production rule. On the contrary, terminal symbols are symbols which cannot be modified by a rule. Of course, the two sets are disjoint. A rewriting rule is a rule which defines how a given sequence of symbols can be rewritten into another sequence of symbols. A special symbol, called the start symbol, is used to specify where the rewriting must start. This particular symbol belongs to the set of non-terminal symbols. We then note $G = (N,T,S,P)$ to define the grammar $G$ composed of the set of non-terminal symbols $N$, the set of terminal symbols $T$, the starting symbol $S$, and the set of production (rewriting) rules $P$.
\begin{definition}
Let $G = (V{}_{N}, V{}_{T}, S, P)$ be a formal grammar. The language described by G is L(G) = $\lbrace x \in \Sigma^{\ast} \mid S \rightarrow^{\ast} x \rbrace$ .
\end{definition}
Grammars are used to describe languages. A language is a set of words, each word being a sequence of symbols. A word may or may not have a meaning nor a structure. For instance, the grammar \makebox{$G$ $=$ $(\lbrace S\rbrace$, $\lbrace a\rbrace$, $S$, $\lbrace S \rightarrow aS$; $S \rightarrow a\rbrace)$} (here \makebox{$N = \lbrace S\rbrace$}, \makebox{$T = \lbrace a\rbrace$}, \makebox{$S = S$}, and \makebox{$P = \lbrace S \rightarrow aS; S \rightarrow a\rbrace)$} describes the language \makebox{$L(G) = \lbrace a^{n} \mid n \geq 1 \rbrace$} (i.e.\ the words \lq{}$a$\rq{}, \lq{}$aa$\rq{}, \lq{}$aaa$\rq{}, $\ldots$). There exist different forms which are used to represent grammars. For convenience, we will write the production rules of a grammar as \makebox{follows :}
\begin{IEEEeqnarray*}{s}
S $\rightarrow$ aS \\
S $\rightarrow$ a
\end{IEEEeqnarray*}
When some rules share the same left-hand side, as it is the case here, we can shrink the different alternatives in one rule, separated by a \lq{}$\mid$\rq{} :
\begin{IEEEeqnarray*}{s}
S $\rightarrow$ aS $\mid$ a
\end{IEEEeqnarray*}
To generate a word from these rules one proceeds as follows : start from the starting symbol and replace it by one of its alternatives. Then two cases have to be considered :
\begin{itemize}
\item[-] either a sequence of symbols of the produced sentential form matches the left-hand side of a rule;
\item[-] either it is not the case and, if the sentential form does not contain any non-terminal symbols, it is a word of the language described by the grammar.
\end{itemize}
Whenever a sequence of symbols matches the left-hand side of a rule, it is replaced by one of the alternatives of the rule, and the process goes on until no more match is found.
As an example, take the above rule. The starting word is \lq{}$S$\rq{}. Suppose that \lq{}$S$\rq{} produces the sentential form \lq{}$a$\rq{}. As \lq{}$a$\rq{} does not match any left-hand side of the rules at our disposal, and as it is a terminal symbol, it is a word of the language. Now suppose that \lq{}$S$\rq{} produces the sentential form \lq{}$aS$\rq{}. The non-terminal \lq{}$S$\rq{} in \lq{}$aS$\rq{} matches the left-hand side of one of the rules, so we replace it by one of its alternatives : \lq{}$a$\rq{} or \lq{}$aS$\rq{}. We thus obtain the sentential forms \lq{}$aa$\rq{} or \lq{}$aaS$\rq{}. Hence, the words generated by the above rule are : $a$, $aa$, $aaa$, \makebox{$aaaa$, $\ldots$}
Now, if we use some x86 instructions as the terminal symbols, we can write rules which will generate x86 instructions sequences \cite{Filiol07-2,DBLP:journals/virology/Zbitskiy09}. From a given sequence of instructions, it is easy to write a grammar which will generate it. For instance, the instruction sequence :
\begin{IEEEeqnarray*}{s}
mov eax, key\\
xor [ ebx ], eax\\
inc ebx
\end{IEEEeqnarray*}
can be generated by the following production rules :
\begin{IEEEeqnarray*}{s}
$S \rightarrow$ mov eax, key $T$\\
$T \rightarrow$ xor [ ebx ], eax $U$\\
$U \rightarrow$ inc ebx $V$
\end{IEEEeqnarray*}
The instruction sequence is thus represented by the sequence of non-terminal symbols S $\rightarrow$ T $\rightarrow$ U $\rightarrow$ V, the non-terminal S being rewritten into the sentential form \lq\lq{}mov eax, key $T$\rq\rq{}, which is then rewritten into the sentential form \lq\lq{}mov eax, key\ \ xor [ ebx ], eax $U$\rq\rq{}, etc\ldots
Once the production rules are defined, one may want to generate an equivalent sequence of instructions. It is rather easy :
\begin{IEEEeqnarray*}{ss}
$S \rightarrow$ & mov eax, key $T$ $\mid$ push key; pop eax $T$\\
$T \rightarrow$ & xor [ ebx ], eax $U$ $\mid$ mov ecx, [ ebx ]; \\
& and ecx, eax; not ecx; or [ ebx ], eax; \\ & and [ ebx ], ecx $U$ \\
$U \rightarrow$ & inc ebx $V$ $\mid$ add ebx, 1 $V$
\end{IEEEeqnarray*}
The production rules now generate 8 $(2\times2\times2)$ different sequences, each of them acting the same. In a same manner, one may want to add some junk code. This can be done by adding a new non-terminal which generates \lq\lq{}useless\rq\rq{} instructions\footnote{Care must be taken on the place where to add these instructions, as they may modify some flags which are check later, e.g.\ by a $jcc$ instruction.} :
\begin{IEEEeqnarray*}{ss}
$S \rightarrow$ & $G$ mov eax, key $T$ $\mid$ $G$ push key; pop eax $T$\\
& \ldots \\
$G \rightarrow$ & add edx, 1; dec edx $\mid$ push eax; add esp, 4
\end{IEEEeqnarray*}
For this example, the addition of the rule $G$, which is composed of only two alternatives, increases the number of instruction sequences that can be generated to 216 $(6\times6\times6)$. This number can be made infinite pretty easily, by adding alternatives which generate only junk code for example, like :
\begin{IEEEeqnarray*}{ss}
$S \rightarrow$ & $G$ $S$ $\mid$ mov eax, key $T$ $\mid$ push key; pop eax $T$\\
& or else \\
$G \rightarrow$ & $G$ $G$ $\mid$ add edx, 1; dec edx $\mid$ push eax; add esp, 4
\end{IEEEeqnarray*}
\subsection{Classification of grammars}
Chomsky provided a well-known classification of grammars \cite{Cho56}. He defined four main types, from \makebox{\emph{Type 0}} to \emph{Type 3}, each type defining a set of languages, each of them being a subset of the set described by any lower numbered grammar. In other words, \emph{Type 0} are the most general grammars, while \emph{Type 3} are the most restrictive. Among these grammars, \emph{Type 2}, also called context-free grammars, are the most popular. They describe context-free languages. Most of the programming languages are described by such grammars. The rules of \emph{Type 2} grammars have the following \makebox{form :}
\begin{center}
U $\rightarrow$ V
\end{center}
Where U is a single non-terminal symbol, and V belongs to $(N\times T)^{\ast}$.
In other words, U can be rewritten as a possibly empty sequence of terminal and non-terminal symbols. The name context-free comes from the fact that the left-hand side of a rewriting rule is a single non-terminal, so the rewriting does not depend of what may be next to it in a sentential form, unlike in \emph{Type 0} and \emph{Type 1} grammars.
We have the relation \emph{Type 0} $\supset$ \emph{Type 1} $\supset$ \emph{Type 2} $\supset$ \emph{Type 3}. Thus, \emph{Type 0} grammars can define all the languages that are definable by \emph{Type 1, Type 2} or \emph{Type 3} grammars.
\section{Van Wijngaarden grammars}
\subsection{Context-sensitivity restrictions}
Context-sensitive languages are more complex than context free languages because one part of the string may \lq\lq{}interact\rq\rq{} with the structure of the other parts of the string. Once a non-terminal symbol has been produced in a sentential form in a context-free grammar, its further development is independent of the rest of the sentential form, whereas a non-terminal symbol in a sentential form of a context-sensitive grammar has to look at its neighbours, on its left and on its right, to see what are the production rules that are allowed for it. So a context-free grammar cannot express some \lq\lq{}long-range\rq\rq{} relations.
Yet, these relations are often valuable, as they make possible some fundamental properties of words to be described (like the only use of variables that have been declared). Programming languages are usually context-sensitive. For example a user is usually not allowed to use a variable that has not been created. So as it is not possible to express such properties through a context-free grammar, a solution, which is used most of the time, is to describe the structure of the correct words by a context-free grammar. The properties are checked by a separate program after that the word has been recognize by the grammar (though it may not belong to the \lq\lq{}real\rq\rq{} language). However, this solution is not very satisfactory as the interest of using a grammar is to have a (formal) description of all the properties of the language.
One can ask why a context-sensitive grammar is not used to describe the language. Actually this would pose some problems. Indeed, in general, context-sensitive languages cannot be parsed efficiently. Moreover, even though context-sensitive grammars have the power to express some long-ranged relations in a sentential form, they don't do it in a way that is easily understandable.
Also it would make sense that after having written a grammar for $a^{n}b^{n}c^{n}$, the writing of $a^{n}b^{n}c^{n}d^{n}$ would work the same way. But this is not the case : the grammar for $a^{n}b^{n}c^{n}d^{n}$ is more complex. The reason behind that is that to express a long-range relation, informations have to flow through the sentential form, thanks to the non-terminal symbols (which look at their neighbours to rewrite a sentential form into another). Thus it requires almost all rules to know something about almost all the other rules.
Several grammar forms which make these relations more readable and easier to construct have been created. Among them are Van Wijngaarden grammars.
\subsection{VW grammar definition}
Basically, a VW grammar can be seen as the composition of two context-free grammars (that is why such grammars are also called two-level grammars). The first context-free grammar is used to generate a set of terminal symbols which will act as non-terminals for the second context-free grammar.
Before going further, a few terms have to be introduced.
\begin{itemize}
\item A $protonotion$ is a sequence of small syntactic marks ;
\item A $metanotion$ is a sequence of big syntactic marks which is defined in a metarule ;
\item A $hypernotion$ is a possibly empty sequence of metanotions and protonotions ;
\item A $metarule$ defines a metanotion as a possibly empty sequence of hypernotions ;
\item A $hyperrule$ defines a sequence of hypernotions as another sequence of hypernotions, separated by a comma. Actually, they represent a possibly infinite set of production \makebox{rules ;}
\item A VW grammar is defined by a set of metarules (or metaproduction rules) and a set of hyperrules ;
\item Whenever a metanotion appears more than once in a hyperrule, each of its occurrence have to be replaced consistently throughout the rule. This is called the \emph{Uniform Replacement Rule}.
\end{itemize}
\begin{definition}\cite{1781176}
A Van Wijngaarden grammar is a grammar G = ( M, V, N, T, $R_{M}$, $R_{V}$, S ) \makebox{with :}
\begin{IEEEeqnarray*}{sts}
$M$ & : & a finite set of metanotions \\
$V$ & : & a finite set of metaterminals, $M \cap V = \emptyset$ \\
$N$ & : & a finite set of hypernotions, $N \subseteq ( M\cap V)^{+} )$ \\
$T$ & : & a finite set of terminals \\
$R_{M}$ & : & a finite set of metarules, $X \rightarrow Y$ with \\
&& $X \in M$, $Y \in ( M\cap V)^{\ast}$ such that for all \\ && $W \in M$, $(M, V, W, R_{M})$ is a context-free \\ && grammar \\
$R_{V}$ & : & a finite set of hyperrules \\
\IEEEeqnarraymulticol{3}{s}{$S \in N$ : the starting symbol }
\end{IEEEeqnarray*}
\end{definition}
The first set of rules are the metarules. They represent a modified grammar in which the non-terminals are replaced by metanotions, and the terminals are replaced by protonotions. The second set of rules are the hyperrules. They represent some possibly infinite set of production rules.
In order to make a distinction between the metarules, the hyperrules, and the production rules, the production symbol is changed. Instead of the symbol \lq{}$\rightarrow$\rq{} we use \lq{}$::$\rq{} for the metarules and \lq{}$:$\rq{} for the hyperrules. To separate the different alternatives of a rule, the symbol \lq{}$;$\rq{} is used instead of \lq{}$\mid$\rq{}. In metarules, members are separated by a blank, and in hyperrules, by a comma. The metanotions have to be chosen wisely, so that any sequence of metanotions is not also a different sequence of metanotions. For instance, if we have a metanotion \emph{X} and a metanotion \emph{Y}, then the metanotion \emph{XY} should be avoided as it would induce some ambiguity.
To make it clearer, here is a VW grammar which describes the language \makebox{$L = \{ a^n b^n c^n\mid n >= 1 \}$} (i.e.\ $abc$, $aabbcc$, $aaabbbccc$, \ldots) :
\begin{IEEEeqnarray*}{us}
N &:: i N; i.\\
A &:: a; b; c. \medskip \\
S & : aN, bN, cN.\\
AiN & : A symbol, AN.\\
Ai & : A symbol.
\end{IEEEeqnarray*}
The first two rules are the metarules, and the last three are the hyperrules. The metanotions are $N$ and $A$. The hypernotions are $AiN$, $Ai$, $A$, $AN$, $aN$, $bN$, and $cN$.
In the definition of a VW grammar, a member is a terminal symbol if it ends in $symbol$ (like \lq{}b symbol\rq{} for the terminal symbol \lq{}b\rq{}), otherwise it is a non-terminal. So, here the rule \lq\lq{}Ai : A symbol.\rq\rq{} produces the terminal symbols $a, b$ and $c$.
The metanotion $N$ produces an infinite set of $i$. The $i$'s act as a counter for the number of letters to be produced. Indeed, as we said, the hypernotions describe a possibly infinite set of production rules. For instance here, the rule \lq\lq{}AiN : A symbol, AN.\rq\rq{} actually produces the \makebox{rules :}
\begin{IEEEeqnarray*}{us}
aii & : a symbol, ai. \\
aiii & : a symbol, aii. \\
\vspace{5pt}
& \hspace{15pt} etc$\ldots$ \\
bii & : b symbol, bi.\\
biii & : b symbol, bii.\\
\vspace{5pt}
& \hspace{15pt} etc$\ldots$ \\
cii & : c symbol, ci.\\
ciii & : c symbol, cii.\\
& \hspace{15pt} etc$\ldots$
\end{IEEEeqnarray*}
To obtain these sets, the metanotion $A$ is replaced consistently by all the words it can generate. Here these are $a$, $b$ and $c$. So we obtain the following three rules :
\begin{IEEEeqnarray*}{us}
aiN & : a symbol, aN.\\
biN & : b symbol, bN.\\
ciN & : c symbol, cN.
\end{IEEEeqnarray*}
Then the same thing is done with the metanotion N. As it generates the infinite language \makebox{$L(N) = \lbrace i^{n} \mid n \geq 1 \rbrace$} (i.e. \lq{}$i$\rq{}, \lq{}$ii$\rq{}, \lq{}$iii$\rq{}...), we obtain the above three sets of infinite production rules.
\subsection{Place in Chomsky's hierarchy}
By construction, Van Wijngaarden grammars do not belong to any category of Chomsky\rq{s} classification. However, one can compare the expressive power of a Van Wijngaarden grammar and the different types of Chomsky\rq{}s hierarchy. In terms of expressive power, they are in fact equivalent to \emph{Type 0} grammars. In a sense, they are even more powerful than \emph{Type 0} grammars since they can handle infinite symbols sets. For instance, as shown in Figure \ref{infiniteset}, a Van Wijngaarden grammar can produce the \makebox{set :}
\begin{center}$S\ =\ \lbrace\ t_{1}^{n}\cdots t_{k}^{n} \mid n \geq 0,\ k > 0,\ t_{1} \cdots t_{k}$ are different symbols $\rbrace$\end{center}
A \emph{Type 0} grammar cannot generate this set since its number of (terminal) symbols is infinite.
\begin{figure}[h]
\begin{IEEEeqnarray*}{us}
N & :: n N; $\varepsilon$. \\
C & :: i ; i C. \\
\\
S & : N\ i tail. \\
N C tail\ & : N, C, N C i tail ; $\varepsilon$. \\
N n C & : C symbol, N C. \\
C & : $\varepsilon$.
\end{IEEEeqnarray*}
\caption{A grammar handling an infinite alphabet}
\label{infiniteset}
\end{figure}
Sintzoff \cite{sintzoff} showed that there exists a Van Wijngaarden grammar for every semi-thue system\footnote{A semi-thue system is a string rewriting system. It is equivalent to Chomsky\rq{}s \emph{Type 0} grammars.}. Van Wijngaarden \cite{681877} showed that a Van Wijngaarden grammar can simulate a \emph{Turing Machine}. Thus, both proved that these grammars are at least as powerful as \emph{Type 0} grammars (i.e.\ that they are Turing complete). As a consequence, parsing of these grammars is undecidable in general.
On a side note, it is to be noted that, if the first set of rules, i.e.\ the metarules, does not generate an infinite language, then the Van Wijngaarden grammar is equivalent to a standard context-free grammar.
Indeed, if the language generated by a metarule is finite, one can write as much production rules as there is words in the language, and the consistent
substitution can be \lq\lq{}emulated\rq\rq{} by the addition of rules which produce only one sentence. For instance the grammar :
\begin{IEEEeqnarray*}{us}
S $\rightarrow$ & P1 BODY P2 $\mid$ P3 BODY P4\\
P1 $\rightarrow$ & (\\
P2 $\rightarrow$ & )\\
P3 $\rightarrow$ & <\\
P4 $\rightarrow$ & >
\end{IEEEeqnarray*}
ensures that the opening bracket matches the ending one. By increasing the number of rules of the grammar, we can express more and more context-sensitive conditions. It follows that if we have an infinite collection of context-free grammar rules, we can express any number of context-sensitive conditions, and so we can achieve full context-sensitivity. As said in the beginning of this section, this is the idea behind Van Wijngaarden grammars : a VW grammar can be seen as the composition of two context-free grammars. The first context-free grammar is used to generate a language which can in turn be described by the second context-free grammar. Nonetheless, as mentioned in the previous section, it is possible to produce every words of the languages they may describe.
\subsection{VW grammars and word generation}
Dick Grune \cite{2356} made a program which can produce all the sentences of a Van Wijngaarden grammar. The program reads a grammar on its input, and then the generation of the words starts. If the input\rq{s} grammar describes an infinite language, then an infinite number of words will be produced. We modified some parts of this program in order to implement our mutation engine, and we have written a VW grammar based on the x86 instructions set.
It is not possible to generate the words of a Van Wijngaarden grammar in the same way that those of a context-free grammar are. Indeed, to generate a terminal production for a context-free language, we start from the start symbol. Intermediate results of a production (sentential forms) are stored in a queue. To rewrite a sentential form, we consider initially the first sentential form in the queue. Then, we search for a sequence of symbols which match the left-hand side of a production rule. If such a match is found, the sentential form is replaced by all its alternatives by making as much copies as the number of alternatives, and each copy is appended at the end of the queue. If no match is found, it means the sentential form is a terminal production.
This process cannot be applied to Van Wijngaarden grammars, as there may be an infinite number of left-hand side resulting from a same hyperrule. Actually, it would require us to scan all the possible left-hand side of the hyperrule, so you may have to look at an infinite number of left-hand side to know if there is a possible match. In theory this takes an infinite amount of time, but a solution to this problem can be found. The main issue comes from the fact that a metanotion can generate an infinite language (i.e. an infinite number of words). What we want to do is to find the terminal productions of the metanotions which are in the left-hand side of the hyperrule so that, after substitution, it corresponds to the sentential form. So, we want to parse the sentential form in accordance to the \lq\lq{}metagrammar\rq\rq{}, with the left-hand side of the hyperrule as the starting form. When the parsing is done, we can deduce which are the terminal productions that have to be used to match the sentential form. As the metagrammar is a context-free language, it can be parsed efficiently. So the problem can be solved. Thus, with this mechanism a member is considered to be a terminal symbol if no match is found in the left-hand side of the hyperrules. So it is not needed to append the symbol \lq\lq{}symbol\rq\rq{} at the end of a member to make it a terminal symbol.
Now, we know how to produce words from a VW grammar. We know too that VW grammars can handle context-sensitivity. So now we want to write rules which transform one sentential form into another one, while preserving its semantic (its context's information).
In order to do so, we modified a little the mechanism of the grammar : the word we want to transform is used as the starting word, and we do not try to parse it. In fact, a sort of parsing is handled by the way the production process works. Moreover, we use a random generator during the production process, to enable the production to randomly generate any word of the language described by the grammar.
As an example take these \makebox{metarules :}
\begin{IEEEeqnarray*}{us}
N :: & 0; 1; 2;$\ldots$; 9; 0N; 1N;$\ldots$; 9N.\\
HEX :: & N; a; b; $\ldots$ ; f; a HEX; b HEX; \\
& $\ldots$; f HEX.\\
ADR :: & 0xN.\\
NUM :: & ADR; HEX.\\
INST :: & mov; push; pop.\\
REG :: & eax; ebx; edx.\\
STACK :: & esp.\\
REGS :: & STACK; REG.\\
REGNUM :: & REGS; NUM.\\
MEM :: & [ REGS ]; [ ADR ].\\
COMMA :: & \lq{},\rq{}.
\end{IEEEeqnarray*}
The metanotion $NUM$ represents an address or an hexadecimal number. The metanotion $INST$ represents three instructions (mov, push and pop). And so on.. \\
The hyperrules :
\begin{IEEEeqnarray*}{us}
mov REGS CO&MMA REGNUM : \\
& move REGNUM in REGS. \\
\IEEEeqnarraymulticol{2}{s}{push REGNUM :} \\
& save REGNUM. \\
\IEEEeqnarraymulticol{2}{s}{pop REGS :} \\
& restore REGS.
\end{IEEEeqnarray*}
modify an instruction into a readable sentence. For example the word \lq\lq{}mov eax, 0\rq\rq{} will be replaced by \lq\lq{}move 0 in eax\rq\rq{}, because of the first hyperrule.\\
We can add hyperrules which will transform these sentence into other equivalent sentence(s) :
\begin{IEEEeqnarray*}{ss}
move REG&NUM in MEM : \\
& mov, MEM, COMMA, REGNUM; \\
\IEEEeqnarraymulticol{2}{s}{move REGNUM in REGS :} \\
& mov, REGS, COMMA, REGNUM; \\
& save REGNUM, restore REGS. \\
\IEEEeqnarraymulticol{2}{s}{save REGNUM : push, REGNUM;} \\
& subtract 4 from esp, move \\
& REGNUM in [ esp ]. \\
\IEEEeqnarraymulticol{2}{s}{restore REGS : pop, REGS;} \\
& move [ esp ] in REGS, ADD 4 to esp. \\
& $\ldots$
\end{IEEEeqnarray*}
Now the sentential form obtained before (\lq\lq{}move 0 in eax\rq\rq{}) can be replaced by either \lq\lq{}mov, eax, \lq{},\rq{}, 0\rq\rq{} or by \lq\lq{}save 0, restore eax\rq\rq{}. If the first alternative is selected, then the generation will stop. Indeed, the sentential form is composed of \lq\lq{}mov\rq\rq{}, \lq\lq{}eax\rq\rq{}, \lq\lq{} \lq{},\rq{} \rq\rq{} and \lq\lq{}0\rq\rq{}, and none of these words match a left-hand side of a hyperrule. On the other side, if the second alternative is selected then the generation continues, and both parts of the sentential form, \lq\lq{}save 0\rq\rq{} and \lq\lq{}restore eax\rq\rq{}, can be replaced independently from each other. Thus, the sentential form \lq\lq{}save 0\rq\rq{} can be replaced by
\lq\lq{}push, 0\rq\rq{} (so the generation stops) or by \lq\lq{}subtract 4 from esp, move 0 in [ esp ]\rq\rq{}, etc.
The metarules used above can be more sophisticated so they generate an infinite set of instructions, and so the hyperrules generate an infinite number of production rules. Hence we can have a (infinite) rewriting system handling an infinite number of instructions.
\section{\emph{K}-ary viruses}
\subsection{What is a \emph{K}-ary viruses}
\begin{definition}[\cite{DBLP:journals/virology/Filiol07}]
A \emph{K}-ary virus is composed of a family of \emph{k} files (some of which may not be executable), whose union constitutes a computer virus and performs an offensive action that is equivalent of that of a true virus. Such a code is said sequential if the \emph{k} constituent parts are acting strictly one after the another. It is said parallel if the \emph{k} parts executes simultaneously.
\end{definition}
The interest of combined virus lies in the fact that the viral information is split in various parts, which taken separately can have a non-malicious behaviour. Because of this separation of the viral information, we are out of the scope of Cohen's model. His model supposes that a virus is made of a unique sequence of symbols, which is not the case with combined viruses.
Two main classes of \emph{K}-ary viruses have been identified \cite{DBLP:journals/virology/Filiol07} :
\begin{itemize}
\item Class 1 codes. These are the codes that work sequentially. \\
This class is composed of three subclasses :
\begin{itemize}
\item Subclass A. Each code refers or contains a reference to the others. Thus, the detection of one of these codes leads to the detections of all of the others.
\item Subclass B. None of the codes refers of contains a reference to the others. Thus, detecting one code does not affect the other codes. The detected code can be replaced by another code.
\item Subclass C. The dependence of the code is directed. Thus detecting one code does not affect the codes which are before it in the sequential execution.
\end{itemize}
\item Class 2 codes. These are the codes that work in parallel.
This class is composed of the same three subclasses as the class 1.
\end{itemize}
\subsection{Van Wijngaarden representation}
The power of a \emph{K}-ary virus lies in the fact that it is split in several parts. Thus, one can see a \emph{K}-ary virus as a distributed program whose global action is the same as that of a virus. If we look at this type of program from the point of view of formal grammars, we can feel that such a program can be described by them.
\begin{definition}
Let $\mathnormal{x_{1}, x_{2}}$ be two files, and $\mathnormal{v}$ $\in$ $\mathnormal{L(G_{v})}$ a virus. We define the relation $\mathnormal{R_{v}}$ by
\begin{center}$\mathnormal{x_{1} R_{v} x_{2} \Leftrightarrow \lbrace \exists \omega \in ( x_{1} \oplus x_{2} ) \mid \omega \in L(G_{v}) \rbrace}$ \end{center}
\end{definition}
The $\oplus$ operator is a selection function, whose result is a set of words over its input. The idea is that is does a selection of some parts of its inputs to extract a word from them, and if one of the results is in the language generated by $G_{v}$ then its inputs form a \emph{K}-ary virus.
The different parts of a \emph{K}-ary virus can each be described separately by a grammar. If we put all these parts together, we have the description of the virus as a whole. Thus a Van Wijngaarden grammar can be used to define \emph{K}-ary virus. The starting symbol produces all the parts of the \emph{K}-ary virus, then the different parts are recognized by some hyperrules of the grammar. The consistent substitution allows some informations to be shared between each parts while they are created. As an example, for a \emph{K}-ary virus with \emph{K}=3, the rules would look like :
\begin{IEEEeqnarray*}{us}
S : PAR&T1 INFOS, PART2 INFOS, PART3 INFOS\\
PART1 &INFOS : \\
VW-&Grammar of PART1 knowing INFOS\\
PART2 &INFOS : \\
VW-&Grammar of PART2 knowing INFOS\\
PART3 &INFOS : \\
VW-&Grammar of PART3 knowing INFOS\\
\end{IEEEeqnarray*}
Once the combined virus is produced (that is, that we have different files that contains the elements of the virus) each part may mutate on its own. While \emph{K}-ary malware have been formally defined \cite{DBLP:journals/virology/Filiol07} and their detection addressed, our approach enables to formalize the automatic generation of \emph{K}-ary malware while providing a constructive proof.
\section{Conclusion}
Van Wijngaarden grammars are very powerful, and can be easily understood by a human. The power of these grammars comes from the two context-free grammars that are jointly used, coupled to the uniform replacement rule which allows context-sensitive conditions to be expressed.
It is thus possible to handle undecidable problems suitable to design undetectable malwarse in a far easier way than considering formal grammars of class 0 directly.
\emph{K}-ary virus have been defined through the use of a Van Wijngaarden grammar. The main idea is that the alternatives of the starting symbol are actually themselves the starting symbol of a grammar, describing each part (file) that the virus is composed of. This formal definition produces a constructive method to generate those codes automatically.
\section*{Acknowledgement}
The author would like to thank Eric Filiol for his fruitful discussions about formal grammars, his active support to this work and all the people at the Operational Cryptology and Virology lab for the wonderful, stimulating and friendly environment they generate.
\bibliographystyle{alpha}
|
{
"timestamp": "2010-09-22T02:01:12",
"yymm": "1009",
"arxiv_id": "1009.4012",
"language": "en",
"url": "https://arxiv.org/abs/1009.4012"
}
|
\section{Experimental}
Single crystals of \Mn\ and \Co\ were grown from the melt.
During the single crystal growth process of \Mn\ special care was
taken to keep manganese in divalent state. This was achieved by
the use of high growth temperatures and the renouncement of melt
solvents. Using the top seeded growth technique and low cooling
rate (0.08~K/h), ruby-red transparent single crystals were
obtained starting from a growth temperature of 1574~K.\cite{becker07}
Single crystals of \Co\ were obtained from
Na$_2$W$_2$O$_7$ melt solution with a small surplus of WO$_3$ (molar ratio CoWO$_4$ : Na$_2$W$_2$O$_7$ : WO$_3$ = 1 : 2 : 0.5),
starting at 1363~K and applying a cooling rate of~3 K/h.
The XAS spectra were collected at the Dragon Beamline of the
National Synchrotron Radiation Research Center in Hsinchu, Taiwan.
The samples were cleaved \emph{in-situ} in an ultra-high vacuum
chamber with pressures in the $10^{-10}$ mbar range, guaranteeing
the high surface quality required to perform bulk-representative
XAS studies in the total electron yield mode. The degree of linear
polarization was $99\pm 1$\%, with an energy resolution of
approximately 0.3~eV. The spectra were collected at room
temperature in the paramagnetic phase of both compounds. Single
crystals of MnO and CoO were measured simultaneously in a separate
chamber as references for \Mn\ and \Co, respectively.
\section{Spectroscopic results}
\begin{figure}[t]
\includegraphics[angle=0,width=9.2cm]{figure1.eps}
\caption[]{(color online)
Top panel: experimental and theoretical Mn \led\ XAS spectra of \Mn
with the $\mathbf{E}$ vector of the light parallel to the
$\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ crystallographic axes.
The spectrum of MnO is included as reference.
Bottom panels: a close-up revealing the polarization dependence
of the spectra.} \label{MnL}
\end{figure}
Fig.~\ref{MnL} shows the room temperature Mn-$L_{2,3}$ XAS spectra
of \Mn\ taken with the $\mathbf{E}$ vector of the light parallel
to the $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$
crystallographic axes. The spectrum of a MnO single crystal is also
included for reference purposes. The spectra are dominated by the
Mn $2p$ core-hole spin-orbit coupling which splits the spectrum
roughly in two parts, namely the $L_{3}$ ($h\nu \approx$ 638-645~eV)
and $L_{2}$ ($h\nu \approx$ 649-654~eV) white line regions.
The line shape strongly depends on the multiplet structure given
by the Mn 3$d$-3$d$ and 2$p$-3$d$ Coulomb and exchange
interactions, as well as by the local crystal fields and the
hybridization with the O 2$p$ ligands. Unique to soft XAS is that
the dipole selection rules are very sensitive in determining which
of the 2$p^{5}$3$d^{n+1}$ final states can be reached and with
what intensity, starting from a particular 2$p^{6}$3$d^{n}$
initial state ($n=5$ for Mn$^{2+}$) \cite{degroot94a,tanaka94a}.
This makes the technique extremely sensitive to the symmetry of
the initial state, i.e., the valence and the crystal field states
of the ions.
Comparing the \Mn\ spectra with the MnO, one can immediately
observe that the spectral features are similar with the $L_3$ main peaks at
identical
energies. This indicates directly that the Mn ions in \Mn\ are also
in the high-spin Mn$^{2+}$ electronic configuration as the Mn ions
in MnO. Yet, the clear differences between the spectra of \Mn\ and
MnO, as seen e.g. at the low and high energy shoulders of the $L_3$ white line,
provide a hint to a crystal field level scheme different from
$O_h$ for the \Mn\ system.
A small but clear polarization dependence for \Mn\ was found, as
can be seen in the bottom panels of Fig.~\ref{MnL}, where a
close-up is shown of the spectra taken with the three different
polarizations. The intensities as well as the energy positions of
several peaks vary with the polarization of the incoming light.
This polarization dependence is yet another indication that the
local symmetry of the Mn ions in \Mn\ is lower than $O_h$.
\begin{figure}[t]
\includegraphics[angle=0,width=8cm]{figure2.eps}
\caption[]{(color online) Experimental and theoretical Co \led\ XAS spectra of \Co.
The spectrum of multi-domain CoO single crystal is included as reference.} \label{CoL}
\end{figure}
Fig.~\ref{CoL} depicts the Co-$L_{2,3}$ XAS spectra of \Co\ taken
with the $\mathbf{E}$ vector of the light parallel to the
$\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ crystallographic axes,
together with the spectrum of a multi-domain CoO single crystal as
reference. Following the same argumentation as given above for the
Mn \led\ edge, from the strong similarities between the \Co\ and
the CoO spectra we can conclude that \Co\ contains Co$^{2+}$ ions
with a $3d^7$ high-spin configuration like in CoO. In comparing
the \Co\ with the \Mn, one observes that the polarization dependence
is much larger. In the following we will discuss the different
origins of the linear polarization dependence for the two compounds
and its implications for their very different magnetic properties.
\section{Local electronic structure}
To interpret and understand the spectral line shapes and their
polarization dependence, we have performed simulations of the
atomic-like $2p^{6}3d^{n} \rightarrow 2p^{5}3d^{n+1}$ ($n=5$ for
Mn$^{2+}$ and $n=7$ for Co$^{2+}$) transitions using the
well-proven configuration-interaction cluster
model.\cite{tanaka94a,degroot94a,thole97a} Within this method we have
treated the Mn or Co ion within an MnO$_6$ or CoO$_6$ cluster,
respectively, which includes the full atomic multiplet theory and
the local effects of the solid. It accounts for the intra-atomic
$3d$--$3d$ and $2p$--$3d$ Coulomb interactions, the atomic $2p$
and $3d$ spin-orbit couplings, the local crystal field, and the
O~$2p$--Mn~$3d$ or O~$2p$--Co~$3d$ hybridization. This
hybridization is taken into account by adding the
$3d^{n+1}\underline{L}$ and $3d^{n+2}\underline{L}^{2}$ etc.
states to the starting $3d^{n}$ configuration, where
$\underline{L}$ denotes a hole in the O $p$ ligands.
\begin{figure}[t]
\includegraphics[angle=0,width=8cm]{figure3.eps}
\caption[]{(color online) The structure of \Mn\ and \Co.
For the calculations of the XAS spectra, a local coordinate system
($x$, $y$, $z$) was introduced, in which the axes point roughly
along the Mn/Co-O bonds. The global crystallographic coordinates
system can be transformed into the local coordinate system by rotating
along $b$ by $\approx 34^\circ$ and then along the $c$ axis by $45^\circ$
(see text for more details).} \label{structure}
\end{figure}
To facilitate the setup and the interpretation of the
calculations, a local coordinate system was introduced, with its
axes ($x$, $y$, $z$) pointing roughly along the Mn-O and Co-O
bonds. To transform the global coordinates ($a$, $b$, $c$) into
the local system, first a rotation by 33.9$^\circ$ along $b$ is
applied, resulting in a new system with ($a^\prime$, $b^\prime$,
$c^\prime$). This system is then rotated by 45$^\circ$ along the
\emph{old} axis $c$, transforming ($a^\prime$, $b^\prime$,
$c^\prime$) into ($x$, $y$, $z$). The local coordinate system is
shown in Fig.~\ref{structure}. The coordinates of the oxygen ions
as well as the Mn-O and Co-O distances are summarized in
Table~\ref{ltab}. The crystal structure has been taken from
Ref.~\onlinecite{lautenschlaeger93}. There are two distorted octahedra in
the unit cell and they are connected to each other by inversion,
thereby contributing to the polarization dependence in the same
manner.
\begin{table}
\begin{tabular}{ccccccc}
\hline
MnWO$_4$ \\
\hline O & \hspace{3mm}x\hspace{3mm} & \hspace{3mm}y\hspace{3mm} &
\hspace{3mm}z\hspace{3mm} & Mn-O dist. &
\hspace{3mm}$pd\sigma$\hspace{3mm} & \hspace{3mm}
$pd\pi$\hspace{3mm} \\
1 & 0.25 & 0.24 & 2.15 & 2.18 & -1.18 & 0.54 \\
2 & 0.25 & 0.24 & -2.15 & 2.18 & -1.18 & 0.54 \\
3 & 0.31 & -2.07 & -0.02 & 2.09 & -1.36 & 0.63 \\
4 & -2.06 & 0.36 & 0.02 & 2.09 & -1.36 & 0.63 \\
5 & 0.31 & 2.24 & -0.08 & 2.27 & -1.02 & 0.47 \\
6 & 2.25 & 0.25 & 0.08 & 2.27 & -1.02 & 0.47 \\
& \\
\hline
CoWO$_4$ \\
\hline
O & x & y & z & Co-O dist. & $pd\sigma$ & $pd\pi$ \\
1 & 0.19 & 0.19 & 2.10 & 2.12 & -1.07 & 0.49 \\
2 & 0.19 & 0.19 & -2.10 & 2.12 & -1.07 & 0.49 \\
3 & 0.24 & -2.02 & -0.04 & 2.03 & -1.24 & 0.57 \\
4 & -2.02 & 0.25 & 0.04 & 2.03 & -1.24 & 0.57 \\
5 & 0.24 & 2.14 & -0.03 & 2.16 & -1.01 & 0.47 \\
6 & 2.14 & 0.23 & 0.03 & 2.16 & -1.01 & 0.47 \\
\end{tabular}
\caption[]{Coordination of the \Mnz\ and \Coz\ ions in their
distorted oxygen octahedra. All oxygen positions are given in \AA\ in the
local coordinate system specified in fig.~\ref{structure}, where
the numbering of the atoms is also defined. The hybridization
strengths $pd\sigma$ and $pd\pi$ are in units of eV.} \label{ltab}
\end{table}
The Mn-O and Co-O bond lengths were used to estimate the
hybridization strength using Harrison's
description,\cite{Harrison} resulting in values for $pd\sigma$ and
$pd\pi$ as shown in Table~\ref{ltab}. The Slater-Koster
formalism\cite{slater54} provides the angular dependence of the
hybridization strength. Values for the crystal fields were tuned
to find the best match to the experimental spectra. Parameters for
the multipole part of the Coulomb interactions were given by the
Hartree-Fock values,\cite{tanaka94a} while the monopole parts
($U_{dd}$, $U_{pd}$) were estimated from photoemission experiments
on MnO\cite{bocquet92a} and previous work on CoO.\cite{csiszar05a}
The simulations were carried out using the program XTLS~8.3,\cite{tanaka94a}
and the parameters used are listed in Refs.~\onlinecite{Mn2para, Co2para}.
The calculated polarization-dependent spectra for the Mn \led\
edge of \Mn\ are plotted in Fig.~\ref{MnL}. One can observe that
the general line shape of the experimental spectra is very well
reproduced. Equally important, also the small energy shifts in the
peak positions as well as the small variations in the peak
intensities as a function of polarization can all be simulated.
This indicates that we have been able to capture the local
electronic structure of the Mn ion in \Mn\ with great accuracy. We
now will look into this in more detail.
\begin{figure}[t]
\includegraphics[angle=0,width=9cm]{figure4.eps}
\caption[]{Energy level diagrams for (a,b) Mn$^+$ ($3d^6$) and
(c,d) \Coz\ clusters, excluding and including the $3d$ spin-orbit
coupling $\zeta$. Only the states up to 1~eV are shown.}
\label{elevel}
\end{figure}
The ground state of a Mn$^{2+}$ ion in $O_h$ symmetry is $^6A_1$,
and this will not split or change upon lowering the crystal
symmetry. The electronic charge distribution of this high-spin
half-filled shell ion remains spherical. The presence of lower
symmetry crystal fields will then show up most clearly when an
electron is added (or removed) from the valence shell. To
illustrate such a situation, we depict in the left panel of
Fig.~\ref{elevel} the total energy level diagram of a Mn$^+$
($3d^6$) ion using the crystal and ligand field parameters which
reproduce best the Mn \led\ spectra of \Mn\ as described above.
One can readily observe a splitting of about 0.7 eV between states
with the added electron in the \tg\ vs. the \eg\ orbitals. Upon
switching off the $3d$ spin-orbit interaction, see
Fig.~\ref{elevel} (a), one can also identify an intra-\tg\
splitting of about 65 and 95~meV as well as an intra-\eg\
splitting of 50 meV. All of the orbitals are of mixed type as a
result of the low local symmetry. The two orbitals lowest in
energy are approximately $1/\sqrt{2}(xz+yz)$ and $1/\sqrt{2}(xz-yz)$, resembling
the $xz$ and $yz$ orbitals, rotated by $\pm$45$^\circ$ along $z$.
This is a result of the local $S_2$ symmetry axis of the MnO$_6$
cluster, which lies in the $xy$ plane with an angle of roughly
45$^\circ$ to the oxygen bonds.
Also in the Mn \led\ XAS process one extra electron is added into
the valence shell. The lowest lying peak at 639.6~eV, see Fig.~\ref{MnL},
involves the excitation of the $2p$ core
electron into the \tg\ orbitals. Different polarizations access
different orbitals with different probabilities. For the 640.6 eV
peak, the polarization dependence is reflected in terms of shifts
in its energy position. If one could have measured the
polarization dependence along the local coordinates of one Mn ion,
one may expect to observe peak shifts of 65 and 95~meV associated
with the intra~\tg\ splittings shown in Fig.~\ref{elevel}. Yet,
having two Mn ions in the unit cell, and measuring along the
global $a$, $b$, and $c$ crystallographic directions, the shifts
between the effectively composite peaks become reduced. In the
experiments and in the simulations we are left with shifts of
about 20 and 30 meV as can be seen in the middle bottom panel of
Fig.~\ref{MnL}. For the 639.6 eV peak, the shifts due to variation
of the polarization are more difficult to quantify since there is
also a variation in the intensity of the peaks. The latter is
apparently caused by the rather complicated multiplet effects
involving the $2p$ core hole, as demonstrated by the excellence of
the match by the simulations, see the left bottom panel of
Fig.~\ref{MnL}. These intensity variations are yet quite small,
and \textit{integrated} over the entire \led\ range, would have
been even identical to zero if the non-cubic hybridization effects
were absent.
The simulations for the polarization dependent spectra for the Co
\led\ edge of \Co\ are shown in Fig.~\ref{CoL}. Like in the Mn
case, we have been able to achieve a satisfying fit to the
experimental spectra: all features are well reproduced,
including the strong polarization dependence. We will now analyze
the local electronic structure of the Co ion on the basis of the
parameters used in these simulations.
For the Co$^{2+}$ ions, the ground state of the $3d^7$
configuration in $O_h$ symmetry is $^4T_1$ ($\approx
t_{2g}^5e_g^2$). Unlike for the $^6A_1$ of the Mn, this $^4T_1$
state will be split upon going to lower crystal symmetry. The
degeneracy in the orbital part, i.e. in the $t_{2g}$ sub-shell,
will be lifted. This can be seen from the total energy level
diagram shown in the right panel of Fig.~\ref{elevel}. The
calculation with the $3d$ spin-orbit interaction switched off, see
Fig.~\ref{elevel} (c), reveals the presence of three low-lying
quartets within the first 0.1 eV. In an one-electron language,
each of them would correspond to the hole occupying one of the
three \tg orbitals. The intra-\tg\ splittings are 40 meV and 80
meV. These values are quite close to those of the \Mn\ case
reflecting the similar crystal structure. The importance of the
lifted degeneracy is that the orbital occupation will no longer be
isotropic, i.e. the Co ion charge distribution will be highly
non-spherical. This then explains the strong polarization
dependence in the absorption spectra.
It is also important to notice how the low-lying states of \Coz\
are influenced by the $3d$ spin-orbit coupling. The energy level
diagram in Fig.~\ref{elevel} (d) reveals that there is indeed a
large amount of mixing due to spin-orbit interaction. The
spin-orbit coupling constant of $\zeta=66$ meV is of the same
order of magnitude as the splittings in the \tg\ subshell. All
this has direct consequences on the magnetism, as will be
discussed in the next section.
\begin{figure}
\includegraphics[angle=0,width=6cm]{figure5.eps}
\caption[]{(color online) Calculated magnetic anisotropy of \Coz\ in \Co.
(a) Three-dimensional representation of the magnetic anisotropy in
the unit cell. The surface depicts the expectation value of the
magnetic moment $M_\parallel$ pointing parallel to a small
exchange field applied to the ion. (b) Two-dimensional cut of
$M_\parallel$ in the plane of easy and intermediate magnetic axis
(corresponding approximately to the $xy$ plane in local
coordinates). (c) Cut through the plane of hard and intermediate
magnetic axis.}
\label{dir}
\end{figure}
\section{Single-Ion anisotropy}
From the results of the simulation, we find that the $3d$
spin-orbit interaction has a major influence on the energies and
nature of the low lying states of the \Co\ system. The magnetic moment will then not
only have a spin contribution but also an appreciable orbital one.
This results in a strong coupling of the direction of magnetic moment to the crystal structure.
We can make an estimate of the single ion anisotropy of the Co$^{2+}$
ion. For that we use the same parameters with which we
have reproduced the XAS spectra and the polarization dependence
therein, and we apply a small exchange field of $\mu_B
H_{ex}=10^{-6}$ eV ($H_{ex}\approx 200$ Oe) to align the magnetic
moment. Here we would like to note that the results hardly changes
with increasing the strength of the field, i.e. they are robust
even beyond $\mu_B H_{ex}=10^{-2}$ eV.
Fig.~\ref{dir} shows the expectation value of the magnetization
parallel to the applied exchange field $M_\parallel$ as a
three-dimensional representation and in a polar plot, the latter
in local coordinates. It can be seen that the single-ion
anisotropy is of easy-axis type, with the axis lying along $\phi=$
135$^\circ$ at $\theta=90^\circ$ in the local coordinate system.
In the direction of the easy axis, the moment is completely
directed along the magnetic field, and the perpendicular component
of the magnetization vanishes. The easy axis is also perpendicular
to the $S_2$ symmetry axis of the distorted octahedron, reflecting the
orbital character of the two lower lying ligand field levels. The
total magnetic moments of the hard and easy axes are 1.2~$\mu_B$
and 3.0~$\mu_B$, respectively, with an orbital contribution of 0.1~$\mu_B$
and 0.9~$\mu_B$.
The easy axis of the total moment
coincides directly with that of the orbital moment, which is also
pointed out earlier by Bruno.\cite{bruno89} It has also been
reported that for low symmetries, the orbital and total moment
could differ in their behavior due to the magnetic dipole moment
$T_z$.\cite{vanderlaan98} In the case of \Co, however, with the
symmetry of the CoO$_6$ groups being not too far from cubic, this $T_z$ is around
$-0.01$ eV, and does not influence the anisotropy significantly.
In \Co, the single-ion anisotropy is strong enough to lead to an
effective pinning of the spins in the magnetically ordered phase.
Isotropic superexchange will be the dominating magnetic
interaction, although in addition anisotropic exchange is
possible, as the bonds lack inversion centers. But the
Dzyaloshinskii-Moriya (DM) term
$\mathbf{D}\cdot(\mathbf{S_1}\times\mathbf{S_2})$, that favors
canted magnetic moments, is small compared to the energy involved
in the magnetic anisotropy of \Co, reflected by the large difference
in moment of more than 1~$\mu_B$. The superexchange will then
minimize the energy in the magnetically ordered phase by aligning
the moments along the easy axis. The easy axis is identical for
both CoO$_6$ clusters in the unit cell, and thus the single-ion
anisotropy itself will not lead to canting between the magnetic
moments of the \Coz\ ions. The result is a collinear
antiferromagnet with an easy axis approximately in the $ac$ plane
$\approx 40^\circ$ off from $a$. This direction is illustrated in
Fig.~\ref{dir} and is close to the one found in the experiment by
neutron diffraction. \cite{weitzel77a}
Returning to \Mn\ case: to the leading order, the local spherical
$^6A_1$ ground state does not produce an orbital moment. The spin
is then free to point in any direction. This is reproduced in the
calculation, where the response of the magnetic moment to an
exchange field is practically isotropic. Yet, in second order, the
spin-orbit coupling does affect the Mn-O hopping. This produces a
non-vanishing moment perpendicular to the exchange field, albeit
in the order of 10$^{-2}$~$\mu_B$, being much lower than in the
case of the spin-orbit active \Co. Consequently, the DM term is on
the same energy scale as the anisotropy, and spin canting can
easily occur. The propagation direction of the spiral magnetic
order is in the \emph{local} $xy$ plane,
meaning that the second order process leads to a hard axis along
$z$. This demonstrates the importance of the single-ion anisotropy
in this system. The form of the anisotropy, however, depends
strongly on the magnetic exchange. This explains that even three
different magnetically ordered phases exist in the material at low
temperatures.\cite{ehrenberg97} Additionally to the hard axis, the easy plane also
has two principal axes for the magnetic anisotropy with a small
difference between the two. This, in combination with the subtle
dependency of the anisotropy on the exchange field, makes
incommensurabilities likely to appear.
\section{Conclusion}
The single-ion anisotropies extracted from our soft x-ray
absorption data and the corresponding full-multiplet calculations
provide a natural explanation for the magnetic ordering phenomena
occurring in \Co\ and \Mn. In \Co, the spin-orbit coupling and a
non-vanishing orbital moment overwhelm the anisotropic Dzyaloshinskii-Moriya exchange
and cause the formation of a collinear ordering of the moments.
Contrary to that, although being isostructural, \Mn\ shows only a
small magnetic anisotropy which is comparable in energy to the Dzyaloshinskii-Moriya
interactions. This will cant the spins in such a way that the
spiral propagates perpendicular to the direction of smallest
magnetic moment, allowing a modulated spin spiral which causes the
ferroelectricity in the material.
\section{Acknowledgements}
We gratefully acknowledge the NSRRC staff for providing us with
beamtime. The research in Cologne is supported by the Deutsche
Forschungsgemeinschaft through SFB 608. N. H. is also supported by
the Bonn-Cologne Graduate School of Physics and Astronomy. We are
grateful for discussions with M.~W.~Haverkort and D.~I.~Khomskii.
|
{
"timestamp": "2010-09-23T02:01:51",
"yymm": "1009",
"arxiv_id": "1009.4338",
"language": "en",
"url": "https://arxiv.org/abs/1009.4338"
}
|
\section{Introduction}
The concept of an acoustic analogy was introduced by Lighthill \cite{1}.
This idea offered him the possibility to derive important results on the
generation of aerodynamic noise without relying on expansions or perturbation
schemes. This is especially important, as the acoustic energy generated by an
unsteady flow field represents usually only a minute part of the energy flux
occurring in the flow and errors which are small compared to the flow
quantities may be very large if one compares them with the sound quantities.
The conditions under which perturbation schemes based on small sound
level or on small flow Mach number may be used, have been clarified in later
researches \cite{1a,2a,3a}, although there still remain open questions.
Lighthill derived without any approximations an exactly valid equation, which
admits an acoustic interpretation. He showed that every compressible flow
fulfills an inhomogeneous wave equation with a quadrupole type source
distribution. As the inhomogeneous wave equation describes the generation and
propagation of sound in an ideal acoustic medium at rest, he had related
aerodynamic noise to sound waves in the ideal acoustic medium. This relation
is exactly true, but it relates the effects of convection and refraction by a
steady basic flow to a source distribution in an acoustic medium without flow
and this is difficult to visualize. One is therefore inclined to look for a
relation to an ideal acoustic medium in motion.
Here we will identify the operator which describes the propagation of sound in
irrotational, isentropic flow and we will base the analogy on that operator.
This of course does not mean, that vortices or entropy inhomogeneities are
excluded. It is very similar to the situation found in Lighthill's analogy.
There, flows were exluded from the medium and they occurred as sources. Here,
vortices and entropy inhomogeneities are excluded and they occur as sources.
The equation is therefore very well suited to study the sound radiation from
vortices which are convected in an irrotational flow. The acoustic variable
which we use is the stagnation enthalpy. This variable has been used with
aeroacoustic applications in mind before \cite{5a,5b}. The equations
considered before differ however from that derived here, but compare
\cite{6a}. The operator which replaces the wave operator here is a
self-adjoint one, even if the flow field is completely arbitrary. It is not
necessary that it is irrotational or that it fulfills the basic eqations of
fluid dynamics. The self-adjointness leads to a reciprocity principle which is
valid for an exchange of source and observer. It also implies the existence of
a variational principle from which the basic equation of the analogy can be
derived. This principle is the simplest generalization of the variational
principle for the wave equation, partial derivatives with respect to time are
replaced by material derivatives, which seems natural if one requires
Galilei-invariance. From the variational principle one can then conclude the
validity of an energy theorem. The main drawback is of course the lack of a
simple explicit expression for the Green's function. Notice however, that for
certain important situations a Green's function to first order has been
determined in \cite{10a}. An alternative would of course be a numerical
solution. To check its feasibility, a general purpose PDE-solver for a PC has
been applied to calculate the sound generated by a two-dimensional vortex
convected along a circular cylinder.
\section{The acoustic analogy}
\subsection{Preliminaries}
Lighthill based his theory of aerodynamic sound generation on an eqaution,
which he obtained by cross-differentiation from the Euler equations and from
the continuity equation
\begin{eqnarray*}
\abl{{\bf \rho v}}t + \nabla\cdot \rho {\bf v\, v} + \nabla p & = & 0 \\
\abl{\rho}t + \nabla \cdot \rho {\bf v} & = 0,
\end{eqnarray*}
namely
\begin{equation}
\Delta p -\frac1{a^2} \frac{\partial^2 p}{\partial t^2} +
\nabla\cdot\nabla\cdot \rho {\bf v\; v} = 0. \label{gl1}
\end{equation}
Here $\rho$ denotes the density, $p$ the pressure, $a$ the speed of sound, and
${\bf v}$ the particle velocity. This equation is valid for isentropic flow if
losses and temporal variations of the speed of sound can be ignored. Temporal
averages can be subtracted, therefore one may assume that only fluctuating
quantities are contained in eq.(\ref{gl1}). No linearization has been
performed in the derivation. Therefore eq.(\ref{gl1}) is valid under very
general conditions. Often, especially for low Mach number flows, one neglects
the acoustic contributions in the double divergence in eq.(\ref{gl1}) and
considers this term as a known source term for the sound generation. Then one
describes the sound field as a wave obeying the wave equation, i.e. as a sound
field propagating in a non-moving medium. Convection and refraction effects
are then neglected. Eq.(\ref{gl1}) is however generally true and these effects
are in principle contained in eq.(\ref{gl1}). Attempts to extract them have
been made by separating the velocity into a mean and fluctuating part and
thereby to obtain these effects. Although it should be possible to describe
mean flow effects in this way, it has been felt that a more apropriate
description should be obtained through a modification of eq.(\ref{gl1}).
Instead of the wave operator, a "convected wave operator" which describes the
propagation of sound in a moving medium seems more apropriate. One then
rewrites all fluid quantities as a superposition
\[ {\bf v} \rightarrow {\bf v}_0 + {\bf v}, p \rightarrow p_0 + p, \dots \]
of "nonacoustic" and "acoustic" variables and rewrites the basic equations
in terms of these variables. If one neglects contributions which are nonlinear
in the non-subscribed variables, one obtains the acoustic equations. It is
well known, that one can derive from this system one "convected wave equation"
for unidirectional (in $x$-direction) shear flow or for potential flows,
namely
\[
{\cal L}_{\rm shear}\, p = \frac{1}{a_0^2} \subabl{} \nabla \cdot a_0^2\nabla p
- \frac{1}{a_0^2}\frac{D^3 p}{D\,t^3} -
2 (\nabla u_0) \cdot\nabla \,\abl{p}{x} =0 \quad {\rm with} \quad
\frac{D}{D\, t}=\frac{\partial}{\partial t} + {\bf v} \cdot \nabla
\]
for unidirectional and with an acoustic potential $\phi$
\[
{\cal L}_{\rm pot}\, \phi =
\nabla\cdot (\rho_0 \nabla \phi - \frac{\rho_0}{a_0^2}\subabl{\phi}{\bf v}_0)
- \abl{\,}{t}\frac{\rho_0}{a_0^2} \subabl{\phi} = 0\]
for irrotational flows. For aerodynamic noise one should not neglect the
nonlinear terms. It is however possible, very similar to Lighthill's approach,
to collect the linear and nonlinear terms and to derive an inhomogeneous wave
equation. It is even possible -- and it has been done e.~g. by Tam and Auriault
\cite{LiTa} -- to follow this approach in the full continuity and Euler
equations and derive an inhomogeneous linear system of equations. Then one
could use for ${\bf v}_0$, $p_0$, etc. the temporal mean values. The equations
which are obtained by this method are much more complicated than the wave
equation and are usually solvable only numerically. Furthermore they are
rather different from the equations usually studied in mathematical physics
and little is known about existence and uniqueness of solutions. It is however
known, that the linear parts of the equations agree with the stability
equations and as many flows are unstable, one has to expect, that the
equations will be unstable. Of course, exponentially growing solutions are
physically excluded -- at least for longer times -- and therefore they do not
occur in the correct solutions. Small errors will however produce these
instabilities and provisions have to be made to limit their growth. How these
provisions influence the sound obtained from these calculations is difficult
to assess.
Here we follow a different strategy which is related to the propagation of
sound waves in potential flows, but differs from the above described method
significantly. We do not separate the velocity into an irrotational part and a
remainder but follow a path used by Howe \cite{5a}. He observed that
Bernoulli's equation states that the stagnation enthalpy differs from the
potential only by a sign and by a time derivative. For irrotational sound
waves one could therefore use the stagnation enthalpy instead of the
potential. The stagnation enthalpy is however defined also for rotational
flows and could be used as a variable of an acoustic analogy for arbitrary
flows. This is what we will do. We will derive a convected wave equation for
the stagnation enthalpy. We will discuss its main properties and show that it
possesses many of the formal properties of the ordinary wave operator. An
important example is an energy conservation theorem with an energy density
which is positive for subsonic flows. This excludes instabilities, the
influence of small errors remains small. The provisions which are necessary in
many other analogies to limit the troublesome growth of instabilities are not
necessary here.
\subsection{Basic Relations}
To obtain the equations
for the acoustic analogy including convection effects, one starts from the
Euler equation for compressible flow. Crocco's form of these equations reads
\begin{equation} \frac{\partial {\bf v}}{\partial t} + \nabla B = -{\bf L},
\qquad {\bf L} = {\mmitb \omega} \times {\bf v} - T \nabla s, \quad
{\mmitb \omega} = \curl {\bf v}.\label{a1}
\end{equation}
$B$ denotes the total
enthalpy $B=h+\frac 12 {\bf v}^2$ with the enthalpy $h$ and the velocity
${\bf v}$. $T$ is the temperature und $s$ the entropy. From the energy theorem
one finds for the total enthalpy
\begin{equation} \frac{D\, B}{D\, t} = \frac1{\rho}
\frac{\partial p}{\partial t}\label{a2}
\end{equation}
If one writes $ d \rho = a^{-2} d\,p + \rho_sd\,s $ where $\rho_s$ denotes
the derivative of the density with respect to the entropy, one gets from the
continuity equation
\begin{equation} \frac{\rho}{a^2}\Bigl(\frac{D\, B}{D\, t}\Bigr) +
\dvg {\bf w} = - \rho_s \frac{\partial s}{\partial t}=q_s, \label{a3}
\end{equation}
where ${\bf w}$ denotes the mass flux. If one multiplies
Crocco's vortex theorem (\ref{a1}) with the density $\rho$ one obtains for the
mass flux ${\bf w}$ the equation
\begin{equation}
\frac{\partial {\bf w}}{\partial t} -
\frac{\rho {\bf v}}{a^2} \frac{D\, B}{D\, t} + \rho \nabla B =
-\rho {\bf L} + q_s {\bf v} = {\bf K}. \label{a4}
\end{equation}
It is easy to eliminate the mass flux ${\bf w}$ from these equations. One then
obtains an equation, which is linear in $B$
\begin{equation} {\cal L}\,B =
\nabla \cdot (\rho \nabla B - \frac{\rho {\bf v}}{a^2} \frac{D\, B}{D\, t}) -
\frac{\partial}{\partial t} \frac{\rho}{a^2}\frac{D\, B}{D\, t} = -\dvg \rho
{\bf L}+\frac {\partial q_s}{\partial t} + \dvg q_s{\bf v} = q_{tot}.
\label{a5}
\end{equation}
The operator ${\cal L}$ obtained here agrees completetely with the operator
${\cal L}_{\rm pot}$ given in the previous section for the propagation of
irrotational sound waves. If one inserts the sources from the equations
(\ref{a3}) und (\ref{a4}), one finds
\begin{equation} q_{tot} =
\left(\frac{\partial}{\partial t}\rho_s \frac{\partial}{\partial t}+ \dvg
\rho_s {\bf v} \frac{\partial}{\partial t} + \dvg \rho T \nabla \right)s +
\dvg \rho {\bf v}\times \omega. \label{aq}
\end{equation}
The sources are linear expressions in the vorticity vector and the entropy. In
this analogy one may think of the sound as being generated from vorticity and
entropy inhomogeneities.
With acoustical applications in mind, the total enthalpy was first used by Howe
\cite{5a} and recently proposed also by Doak \cite{5b}. A comparison with
their equations shows, that the equation for $B$ is not uniquely determined.
Howe's convected wave operator is
\[ {\cal L}_{\rm Howe}\, B = \Delta B -\frac1{a^2}\subabl{{\bf v}}\cdot\nabla B
-\subabl{\;}\frac1{a^2}\subabl{B} \]
and Doak's
\[{\cal L}_{\rm Doak}\, B = \Delta B -\frac1{a^2}
\Bigg[\frac{\partial^2 B}{\partial t^2} + \Bigg(2{\bf v}\abl{\;}{t} +
{\mmitb \omega} \times {\bf v} +T\nabla s - 2 \nabla h\Bigg)\nabla B +
{\bf v}{\bf v}\cdot\nabla\cdot\nabla B\Bigg]. \]
The right hand sides of these equations differ from the right hand side
of eq.(\ref{a5}). We will restrict ourselves to eq. (\ref{a5}). Notice however
that the principle parts -- i.e. those terms which contain second derivatives
of $B$ -- of the three convected wave operators agree. This means that they
agree in the high frequency limit of geometric acoustics and agree with the
well known geometric acoustic theory.
A certain simplification of eq. (\ref{a5}) is possible if one requires
that $\rho$ and ${\bf v}$ fulfill the continuity equation. One then gets
\begin{equation}
{\cal L}\,B =\nabla \cdot (\rho \nabla B )
- \rho \frac{D}{D t} \frac{1}{a^2}\frac{D\, B}{D\, t}.
\label{a5a}\end{equation}
The equation (\ref{a5}) or (\ref{a5a}) is a generalization of the wave
equation. It reduces to the wave equation if one assumes in (\ref{a5}) ${\bf
v} =0$ and if one assumes further, that $\rho$ and $a^2$ are constant. The
equation agrees then with Lighthill's equation in the form of Powell
\cite{4a}. It seems to be a rather complicated equation. Considering a
variational principle we will however see, that equation (\ref{a5},\ref{a5a})
is actually the simplest equation which contains a flow velocity.
A simplification of the source is possible for an ideal gas. For an
ideal gas $\rho$ is the product of two functions which depend only on $p$
and on $s$. Then one has $\rho_s = \rho f(s)$ with some function $f(s)$ and
one obtains for the sources of eq. (\ref{a5})
\[q_{tot}=-\dvg \rho {\bf L}+\rho \frac{D}{Dt}f(s)\frac{\partial s}{\partial t}
= - \dvg \rho {\bf L}-
\rho \frac{\partial {\bf v}}{\partial t}\cdot f(s) \nabla s
\]
if one makes use of the entropy conservation law.
The equation (\ref{a3}) and (\ref{a4}) rsp. (\ref{a5}) are now considered as
the basic equations of the acoustic analogy. They agree formally with the
linearized equations which describe small perturbations of a steady potential
flow. The zeroth order equations are then given by $B_0=0$ (because of the
Bernoulli equation) and $\dvg {\bf w}_0 =0$, i.e. the zeroth order versions of
the eqs. (\ref{a3}) and (\ref{a4}). The first order eqs. of (\ref{a3}) and
(\ref{a4}) are then obtained, if the fields $\rho$, $a$, and ${\bf v}$ are
replaced by their zero order values. Notice that $\rho {\bf v}$ in eq.
(\ref{a4}) then becomes $\rho_0 {\bf v}_0$ and it differs from ${\bf w}$ which
becomes ${\bf w}_1$. In that sense the left hand side of the eqs. (\ref{a3})
and (\ref{a4}) rsp. (\ref{a5}) are considered as equations which describe
sound propagation in potential flows. This is very similar to Lighthill's
interpretation of the wave equation as an equation which describes sound
propagation in a medium at rest. This interpretation is valid only if $\rho$,
$a$, and ${\bf v}$ are steady fields which fulfill the equations for
compressible irrotational flow. We will however not require this, as ${\cal
L}$ is also well defined without this assumption and there are
no advantages in assuming it. The eqs. (\ref{a3}) and (\ref{a4}) rsp.
(\ref{a5}) are exactly valid identities. They are considered in the following
as a system of linear partial differential equations for the variables $B$ and
${\bf w}$. This implies also that the right hand sides of the equations
(\ref{a3}) and (\ref{a4}) rsp. (\ref{a5}) are considered as the sources of the
sound. They are related to vortices and entropy inhomogeneities. This seems
reasonable if one wants to study the generation and propagation of sound in a
potential flow. Then the flow consists of a superposition of an irrotational
steady part and an unsteady part. Often one will superpose these two
contributions linearly. We will not require that. For the steady flow one has
a constant value of the total enthalpy $B$. One may then assume $B$ to be
zero. $B$ is then solely related to the unsteady part and is small if this
part is small. In irrotational flow there is a potential $\Phi$. This obeys
the Bernoulli equation
\begin{equation}
\Phi_t + B = 0 \label{a6}
\end{equation}
$B$ then differs in regions where the flow is irrotational from the temporal
derivative of the potential only by its sign, it is however -- contrary to the
acoustic potential defined everywhere.
This equation (\ref{a5}) was originally derived in \cite{6a}. Let us now
derive its main properties. The first important point is
that ${\cal L}$ is formally self-adjoint. This follows from the
fact that one has for arbitrary functions $B$ and $\tilde B$ the identity
\begin{equation}
\tilde B {\cal L} B - B {\cal L} \tilde B =
\frac{\partial l_0}{\partial t} + \frac{\partial l_i}{\partial x^i}
\label{b0} \end{equation}
with
\begin{equation}
l_0=-\frac{\rho}{a^2}\Bigl(\tilde B \frac{D\, B}{D\, t} -
B \frac{D\, \tilde B}{D\, t} \Bigr),\quad
l_i=\rho \Bigl(\tilde B \frac{\partial B}{\partial x^i} -
B \frac{\partial \tilde B}{\partial x^i} \Bigr)+l_0 v_i . \label{b01}
\end{equation}
This equation is easy to check. It implies also that one has for a scalar
product $(f,g)$ defined by $(f,g)=\int f\, g \, d^3xdt$ the relation
\begin{equation}
(\tilde B,{\cal L}B) = (B,{\cal L}\tilde B) \label{a7}
\end{equation}
if $B$ and $\tilde B$ vanish on the boundary of the integration region or
decay sufficiently rapidly at infinity.
\subsection{Reciprocity}
One may derive from the symmetry in eq. (\ref{a7}) a reciprocity relation. To
be specific let $G({\bf x},t,{\bf y},t')$ be the Green's function associated
with ${\cal L}$, i.e.
\begin{equation}
{\cal L} G({\bf x},t,{\bf y},t') =-\delta({\bf x}-{\bf y})\, \delta(t-t'),
\quad {\rm with} \quad G({\bf x},t,{\bf y},t') =0 {\quad} {\rm for} \quad
t<t', \label{b1}
\end{equation}
where we have assumed that $G$ is causal, i.e. it vanishes for all times
before the source is switched on, which occurs at $t=t'$. There exists also an
advanced Green's function $G_{\rm adv}$ with
\begin{equation}
{\cal L} G_{\rm adv}({\bf x},t,{\bf y},t') =-\delta({\bf x}-{\bf y})\,
\delta(t-t'), \quad {\rm with} \quad G_{\rm adv}({\bf x},t,{\bf y},t') =0
{\quad} {\rm for} \quad
t>t'. \label{b2}
\end{equation}
It is easy to see that eq. (\ref{a5}) is invariant with respect to time
reversal $t \rightarrow -t$ if at the same time the sign of the velocity is
reversed. Therefore time reversal transforms a Green's function into a Green's
function. As the time reversal interchanges the inequalities $t>t'$and $t>t'$
one has
\begin{equation}
G({\bf x},-t,{\bf y},-t';-{\bf v}({\bf x},-t)) =
G_{\rm adv}({\bf x},t,{\bf y},t';{\bf v}({\bf x},t)) \label{b3}
\end{equation}
where we have added for clarity the function ${\bf v}({\bf x},t)$ to the list
of arguments of the Green's function. In addition the functions $\rho({\bf
x},t)$ and $a({\bf x},t)$ have to be replaced by $\rho({\bf x},-t)$
and $a({\bf x},-t)$.
One may now apply eq.(\ref{a7}) with
\[ B =G({\bf x},t,{\bf y},t') \quad {\rm and} \quad
\tilde B =G_{\rm adv}({\bf x},t,{\bf z},t'') \]
and one obtains
\begin{equation}
( G_{\rm adv}({\bf x},t,{\bf z},t''), {\cal L} G({\bf x},t,{\bf y},t'))
=( {\cal L} G_{\rm adv}({\bf x},t,{\bf z},t''),G({\bf x},t,{\bf y},t'))
\label{b4}\end{equation}
Let us indicate briefly that there are no contributions from the right hand
side of eq. (\ref{b0}). If the integration in eq. (\ref{b4}) is performed over
a large cylinder in the ${\bf x}$,$t$-space which extends over a large sphere
in ${\bf x}$-space and over all $t$ with $T_0<t<T_1$, one has surface
contributions which are to be evaluated over the large sphere at $t=T_0$ and
at $t=T_1$ and over the surface of the large sphere in ${\bf x}$-space for all
$t$ with $T_0<t<T_1$. If $T_0$ is before $t'$ and $t''$ and $T_1$ after $t'$
and $t''$ there are no contributions from the space integrals at $t=T_0$ and
at $t=T_1$, as at least one factor vanishes in $l_0$ and in the $l_i$, namely
the factors containing $G_{\rm adv}$ at $t=T_1$ and those containing $G$ at
$t=T_0$. There is also no contribution from the large surface in ${\bf
x}$-space if it is selected so large that no signal which was generated at
$t=t'$ and at ${\bf x}={\bf y}$ has reached this surface. Therefore eq.
(\ref{b4}) is true, and one can evaluate the scalar products with the
$\delta$-functions in the eqs. (\ref{b1}) and (\ref{b2}) easily, and one
obtains the equation
\begin{equation}
-G_{\rm adv}({\bf y},t',{\bf z},t'') =-G({\bf z},t'',{\bf y},t')
\label{b5}\end{equation}
which can with eq. (\ref{b3}) be rewritten as
\[
G({\bf y},-t',{\bf z},-t'';-{\bf v}({\bf y},-t)) =
G({\bf z},t'',{\bf y},t';{\bf v}({\bf z},t)).\]
This is the reciprocity principle with reversed flow.
\subsection{Variational Principle and Energy Conservation}
From the self-adjointness
on can conclude the existence of a variational principle from which eq.
(\ref{a5}) can be derived. One has
\begin{equation}
\delta L = 0 \quad {\rm mit} \quad L=\frac 12 (B,{\cal L} B) -(B,q_{tot})
\label{a8} \end{equation}
as
\begin{equation}
\delta L =\frac 12 (\delta B,{\cal L} B) + \frac 12 (B,{\cal L} \delta B )
-(\delta B,q_{tot})= (\delta B,{\cal L} B -q_{tot}).
\label{a9} \end{equation}
The Lagrangian from (\ref{a8}) can be simplified somewhat, if the second
derivatives in ${\cal L}$ are eliminated with integration by parts. One finds
then
\begin{equation}
L=\int\int \Bigl[\frac{\rho}{2a ^2} \bigl(\frac{D\, B}{D\, t}\bigr)^2
-\frac {\rho}2(\nabla B)^2- q_{tot} B \Bigr]
d^3x\,dt.
\label{a10a}\end{equation}
It seems very remarkable that this variational principle seems to be the
simplest possible extension of the well known principle for the wave equation
which is invariant with respect to Galilei-transformations. The Lagrangian in
eq. (\ref{a8}) differs from the Lagrangian of the wave equation only by the
fact, that partial derivatives with respect to time are replaced by material
derivatives formed with the velocity field ${\bf v}$.
Now it is possible to obtain from a variational principle an energy theorem,
rsp. an energy conservation law, if the Lagrangian density does not depend
explicitly from the time. If $l$ denotes the Lagrangian density from eq.
(\ref{a10a}), i.e.
\begin{equation}
l=\frac{\rho}{2a ^2} \bigl(\frac{D\, B}{D\, t}\bigr)^2
-\frac {\rho}2(\nabla B)^2 - q_{tot} B,
\label{a10}\end{equation}
one has for the energy theorem
\begin{equation}
\frac{\partial}{\partial t}\Bigl(\dot B\frac{\partial l}{\partial \dot B} -l
\Bigr)
+\frac{\partial}{\partial x^i} \dot B\frac{\partial l}{\partial B_{x^i}}
= \frac{\partial l}{\partial t}
\label{a11} \end{equation}
where the time derivative on the right hand side acts only on the explicit
time dependance in $l$, i.e. here in $\rho$, $a$, ${\bf v}$ and $q_{tot}$.
The variable $B$ and its derivatives are to be kept constant. The time
derivative on the left hand side of eq.(\ref{a11}) acts also on the implicit
dependance in $B$ and its derivatives $\dot B$ und $B_{x^i}$. Only the spatial
coordinates $x^i$ are to be kept constant. An energy conservation law is
obtained from (\ref{a11}) if the Lagrangian density does not contain the time
explicitly. In general one obtains for the energy flux
${\displaystyle U_i= \dot B\frac{\partial l}{\partial B_{x^i}}}$
and the energy density
${\displaystyle e= \dot B\frac{\partial l}{\partial \dot B} -l}$
the explicit expressions
\begin{equation}
{\bf U} = \rho \dot B \Bigl(\frac1{a ^2} \frac{D\, B}{D\, t} {\bf v}
- \nabla B \Bigr)
\label{a12} \end{equation}
and
\[
e = \frac{\rho}{a^2}\dot B \frac{D\, B}{D\, t}
-\frac{\rho}{2a ^2} \bigl(\frac{D\, B}{D\, t}\bigr)^2
+\frac {\rho}2(\nabla B)^2 = \frac{\rho}{a^2} {\dot B}^2
+\frac {\rho}2(\nabla B)^2 -
\frac{\rho}{2a^2}\Bigl({\bf v} \cdot \nabla B \Bigr)^2
\]
which shows that the energy density $e$ is positive for subsonic ${\bf v}$.
The energy theorem is then of the form
\begin{displaymath}
{\partial e \over \partial t} + \dvg {\bf U} =q_{\rm En}
\end{displaymath}
with an energy density $e$ and an energy source densitye $q_{\rm En}$.
A useful relation is obtained if one integrates this equation over the time
for finite time events or if one averages this equation for the case of steady
phenomena. If one denotes the resulting quantities by an overbar, one obtains
\begin{equation}
\dvg {\bar{\bf U}} ={\bar q_{\rm En}}.
\end{equation}
Another important conclusion can be drawn from the energy theorem if one
applies it to the solution of an initial value problem with vanishing right
hand side $q_{tot}$. If one assumes that the solution vanishes for large
$|{\bf x}|$, one obtains
\[ \abl{\;}{t}\int e\, d^3\, x \; = 0,\]
i.e. the toatal energy in the sound field remains constant. As it is a sum of
positive contributions, none of these -- e.g. $\dot{B}$ -- can grow
exponentially in time, i.e. instabilities cannot occur.
The physical meaning of this energy flux becomes clearer if one considers an
irrotational isentropic region. There on may write (\ref{a12}) with the eqs.
(\ref{a2}) and (\ref{a6}) as
\begin{equation}
{\bf U} = \dot B \Bigl(\frac 1{a ^2} \dot p {\bf v}+ \rho \nabla \Phi_t \Bigr).
\label{a13}\end{equation}
One may compare this energy flux with the flux from the Blokhintzev energy
theorem which is valid for the propagation of irrotational sound waves in an
irrotational mean flow in linear approximation. One has neglected quantities
which are quadratically in the acoustical quantities. Notice that no
linearization has been assumed in the derivation of the energy theorem
(\ref{a10},\ref{a11}). It is insofar an exact identity, only dissipative
effects have been ignored. If one denotes in the Blokhintzev energy flux with
${\bf U}_{\rm Bl}$, the density, the speed of sound and the velocity of the
irrotational mean flow with $\rho_0$, $a_0$ and ${\bf v_0}$ and with $p'$ and
${\bf \phi'}$ the acoustic pressure and the acoustic potential, one can write
\begin{equation}
{\bf U}_{\rm Bl} = (\frac 1{\rho_0} p' + {\bf v_0} \cdot \nabla \phi')
(\frac 1{a_0^2} p'{\bf v_0} + \rho_0\nabla \phi').
\label{a13a}\end{equation}
A comparison of eq. (\ref{a13a}) with eq. (\ref{a13}) shows, that both energy
fluxes are products of two factors, where the factors of eq. (\ref{a13}) are
just the time derivatives of the factors of the Blokhintzev energy flux
(\ref{a13a}). If one thinks of the sound field as a superposition of temporal
Fourier modes, one finds that the average energy flux consists of a
superposition from fluxes of the modes. In the energy flux of eq. (\ref{a12})
all contributions contain an additional factor $\omega^2$ if $\omega$ denotes
the angular frequency of the Fourier modes.
\subsection{Solutions}
As a first application one may consider the case of constant values of the
velocity ${\bf v}$, density $\rho$ and speed of sound $a$. Then vorticity and
entropy inhomogeneities are convected with the velocity ${\bf v}$. These
inhomogeneities and also the total enthalpy $B$ are then functions of
${\bf x} - {\bf v} t$ only, i.e. $B= B(${\bf x} - {\bf v} t$)$,
$s= F_s(${\bf x} - {\bf v} t$)$
and $\omega = {\bf F}_{\omega}(${\bf x} - {\bf v} t$)$.
Then eq. (\ref{a5}) leads to
\begin{displaymath}
\rho \Delta B = q_{tot}, \qquad
q_{tot} =
\left(\frac{\partial}{\partial t}(\rho_s \frac{\partial}{\partial t}+
\dvg \rho_s {\bf v} \frac{\partial}{\partial t})
+ \dvg \rho T \nabla \right)F_s + \dvg \rho {\bf v}\times F_{\omega}.
\end{displaymath}
One may then introduce $B_{\omega}$ and $B_s$ by
\begin{displaymath}
\rho \Delta B_{\omega} = F_{\omega} \quad {\rm und} \quad
\rho \Delta B_s = F_s
\end{displaymath}
and one obtains
\begin{equation}
B=
\left(\frac{\partial}{\partial t}(\rho_s \frac{\partial}{\partial t}+
\dvg \rho_s {\bf v} \frac{\partial}{\partial t})
+ \dvg \rho T \nabla \right)B_s + \dvg \rho {\bf v}\times B_{\omega}.
\label{a13b}\end{equation}
This shows -- as one might have expected -- that passively convected entropy
and vorticity inhomogeneities do not radiate sound. In the general case these
quantities will not be passively convected and one needs for its determination
extra equations. For the entropy one may use the equation of entropy
conservation
\begin{displaymath}
\frac{D\,s}{D\, t} =0
\end{displaymath}
and for the vorticity the Beltrami vortex theorem
\begin{displaymath}
\frac{D}{D\, t} {\omega \over \rho} = {\omega \over \rho} \cdot \nabla
{\bf v}.
\end{displaymath}
Here we consider especially the two-dimensional case. Then the right hand side
of the Beltrami vortex theorem, which is related to the stretching of vortex
lines, vanishes.
In addition to the differential equations one needs boundary conditions. If
one is interested in cases where the vorticity vanishes at solid walls in the
flow region, one may use the relation (\ref{a6}) between total enthalpy $B$
and potential $\Phi$ and one finds that the normal component of the velocity
vanishes at a fixed surface if the normal derivative of $B$ vanishes there.
With eq. (\ref{a12}) one notices that the normal component of the energy flux
(\ref{a12}) vanishes at rigid walls if the normal component of the velocity
${\bf v}$ vanishes there. One would expect this of course.
In addition one needs conditions of no-reflexion at the boundary of the
computation region. We will here assume the simplest quasi-onedimensional
condition and require there
\begin{displaymath}
\frac{\partial B}{\partial t} = -({\bf v} + {\bf n} a)\cdot \nabla B,
\end{displaymath}
where ${\bf n}$ denotes the outer normal of the computation region.
As a numerical example, we consider a localized vortex of radius 1 and of
vanishing total vorticity which is convected in a flow around a circular
cylinder of radius $1/2$ and situated at $x=0$ and $y=0$. The initial azimuthal
velocity $w$ around the center of the vortex, which is situated initially at
$x_0=-3$, $y_0=0.5$, is assumed to be
\[ w=(1-4r^2)(1-r^2)^2, \]
$r$ denotes the distance from the vortex center. For the velocity, we assume
an incompressible potential flow of velocity 1 in $x$-direction at $x=-\infty$.
The density is chosen as 1, the speed of sound as 2. Initial values for $B$
are obtained from eq.(\ref{a13b}) and are given by
\[B=0.5 (y-y_0)(1-r^2)^3 \quad {\rm for} \quad r<1. \]
This problem is treated with the general purpose PDE-solver PDEase/2 which
runs on a PC. A grayplot of the $B$-field at two different times is shown in
figure 1.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=5cm]{bild1.eps}
\qquad
\includegraphics[width=5cm]{bild2.eps}
\end{center}
\hskip1cm
Figure 1. The $B$-field generated by a vortex convected along a rigid
cylinder. Grayscales correspond to $|B|$, white to $B=0$.
\end{figure}
The left half shows a very early stage with the vortex to the left
of the cylinder. As the vortex is inserted in an inhomogeneous velocity field,
sound radiation begins immediately. The right half shows a later stage, where
the vortex has approached the cylinder. As the flow Mach number is not small,
one notices significant deviations from a dipole character. Hydrodynamic
instabilities are not observed, but numerical small-scale errors are
noticeable. A reliable numerical solution of eq.(\ref{a5}) requires obviously
more efforts.
|
{
"timestamp": "2010-09-21T02:02:49",
"yymm": "1009",
"arxiv_id": "1009.3766",
"language": "en",
"url": "https://arxiv.org/abs/1009.3766"
}
|
\section{Introduction}
Critical points, across which a continuous phase transition happens, are interesting and important for their universality, meaning that they can be simply classified according to very few critical exponents. In particular, a quantum Lifshitz point, where fluctuation is driven by zero point energy and characterised by anisotropic scaling of space and time, might be realized in some antiferromagnetic matters with strongly correlated electrons. As an alternative to the conventional lattice approach toward nonperturbative computation, application of AdS/CFT correspondence, originally proposed as a duality between strings in weakly curved AdS space and operators in strongly coupled super Yang-Mills\cite{Maldacena:1997re,Gubser:1998bc,Witten:1998qj}, to quantum critical points in strongly coupled systems has demonstrated some interesting results\cite{Faulkner:2009wj}. In this paper, we would like to study the quantum criticality in a more general background of Einstein-Maxwell-Dilaton gravity. From the theoretical perspective, descending from the (super)gravity in higher dimensional spacetimes, it is very common to find a gravity system in lower dimensions couple nonminimally to a number of dilatons, gauge fields, higher ranked tensor and form fields. From the practical viewpoint, there are at least two advantages along this line of generalization, which will become clear later:
\begin{enumerate}
\item The nonzero dilaton field supports the Lifshitz-like scaling as the isometry of background metric, such that quantum Lifshitz point becomes accessible.
\item The nonminimal couple between a probe scalar and the Maxwell field, as well as the direct couple between two scalars, provide tunable parameters in addition to scalar masses. Since the mass of a scalar will map to the conformal dimension of corresponding condensate\footnote{For this statement to be true, we have implied that our background is asymptotically $AdS_4$ at infinity.}, we are able to approach the quantum critical point while keeping the scaling dimension unaltered.
\end{enumerate}
This paper is organized as follows. The gravity model and its probe limit is introduced in the section II. The effect of nonminimal couple to the quantum criticality at $AdS_2$, as a special case, will be discussed in the section III. The quantum criticality at Lifshitz point for generic critical exponent is discussed in the section IV and BKT phase transition in the section V. We will discuss the solution beyond the probe limit in the section VI and discussion and comments in the last section. In the appendix we give a quantum mechanics treatment for solvable case $z=2$.
\section{The gravity model and its probe limit}
We will consider the following Lagrangian as a generalized model of Einstein-Maxwell-Dilaton gravity\cite{Aprile:2010yb}:
\begin{equation}
2\kappa_G^2 (-g)^{-1/2}{\cal L} = R + \frac{6}{L^2} - G(\psi,\chi) F^{\mu\nu}F_{\mu\nu} -|D_{\mu}\psi|^2 - |D_{\mu}\chi|^2 - V(\psi,\chi),
\end{equation}
where $\psi$ and $\chi$ are complex scalars carrying charges $q_\psi$ and $q_\chi$ under Maxwell field $F_{\mu\nu}=\partial_{[\mu} A_{\nu]}$. Since the phases of scalars are irrelevant to our discussion on uniform condensate, it is consistent to set them to zero. Similar constructions, which can be seen as special limits of this model, have been useful to simulate various condensed matter systems. To mention a few: a single charged scalar can be used to describe the superconductor\cite{Hartnoll:2008vx}, a neutral scalar probed in the charged black hole can be used to model the antiferromagnetic state\cite{Iqbal:2010eh}, and competing of two (charged) scalars was first attempted in \cite{Basu:2010fa} for mixed magnetic and superconducting states. In this paper, we adopt a generalized background where additional function $G$ is introduced to engineer possible interaction between scalars and Maxwell field, other than the minimal coupling via covariant derivative $D_\mu$. Some new features of this generalization have been observed in \cite{Aprile:2010yb} where, for instance, the critical temperature becomes tunable and phase transition other than second order can be engineered. We will choose a specific form of $G$ and $V$ for a toy model:
\begin{eqnarray}\label{model}
&&G(\chi) = 1 + \kappa |\chi|^2,\nonumber\\
&&V(\psi,\chi) = m_\psi^2 |\psi|^2 + m_\chi^2 |\chi|^2 + \eta |\psi|^2|\chi|^2.
\end{eqnarray}
We would like to study the limit similar to that in \cite{Iqbal:2010eh}, where the boundary theory is at zero temperature and finite density. To achieve this, we take a probe limit of scalar field $\chi$ such that it decouples from the rest of the fields\footnote{Notice that this limit is different from the usual probe limit for Einstein-Maxwell model where both scalar and vector fields are decoupled from the gravity sector upon sending $q\to \infty$ after scaling down both $\psi$ and $A$ by a factor of $q$.}. The above mentioned action will break down into two pieces: the background in its IR region ($u\to \infty$), supported by the Maxwell field and constant scalar $\psi$, admits a geometry respecting the Lifshitz scaling of critical exponent $z$\cite{Gubser:2009cg}:
\begin{eqnarray}
&& ds^2 = -(\frac{L_0}{u})^{2z}dt^2 + \frac{L_0^2}{u^2}(d\vec{x}^2 + du^2),\nonumber\\
&& A_t = \sqrt{2-\frac{2}{z}} (\frac{L_0}{u})^z, \qquad \psi = \psi_0
\end{eqnarray}
where the constant $\psi_0$, charge $q$, and radius of curvature at IR $L_0$ are determined by a pair $(z,m_\psi^2)$ for $m_\psi^2>0$:
\begin{equation}\label{sol_z}
\psi_0 = \frac{\sqrt{2(z-1)}}{m_\psi L_0},\qquad q_\psi^2 = \frac{zm_\psi^2}{2(z-1)}, \qquad L_0 = L \sqrt{\frac{(z+1)(z+2)}{6}}.
\end{equation}
Given such an IR geometry with $z>1$, it is unclear whether a corresponding UV solution can be exactly constructed\footnote{We remark that the charged dilatonic black hole and brane have been numerically constructed, for example, in \cite{Goldstein:2009cv,Bertoldi:2009dt}, where the near horizon geometry exhibits the Lifshitz spacetime and it becomes $AdS_4$ at asymptotical infinity.}. One well known example is given by the extremal RN black hole in $AdS_4$\footnote{Here we already rescale the horizon at $u=1$. The extremal limit is obtained for $\alpha = 1$.}:
\begin{equation}
ds_{UV}^2 = \frac{L^2}{u^2}(-f(r)dt^2+\frac{du^2}{f(r)}+d\vec{x}^2),\qquad f(r) = 1 - (1+3\alpha) u^3 + 3\alpha u^4.
\end{equation}
It flows to $AdS_2\times R^2$ in the IR region, which corresponds to a RG flow in the boundary field theory from a UV fixed point with exponent $z=1$ to an IR one with $z\to \infty$. The near horizon solution is given by
\begin{equation}\label{ads2_metric}
ds_{IR}^2 = \frac{L_0^2}{u^2}(-dt^2+du^2)+\frac{d\vec{x}^2}{L^2},
\end{equation}
and
\begin{equation}\label{ads2_at}
A_t = \frac{L_0^2\mu}{u}, \qquad \psi = 0,
\end{equation}
where the curvature radius of $AdS_2$ can be related to that of $AdS_4$ via
\begin{equation}
L_0^2=\frac{L^2}{6}.
\end{equation}
On the other hand, the probe action for scalar field $\chi$ now reads:
\begin{equation}\label{probe_L}
2\kappa_G^2(-g)^{-1/2}{\cal L} = - \frac{L^2}{4} \kappa|\chi|^2F^{\mu\nu}F_{\mu\nu}-m_\chi^2|\chi|^2-\eta|\psi|^2|\chi|^2-|\partial_\mu \chi - iq_\chi A_{\mu}\chi|^2.
\end{equation}
The condition to have instability in its IR region (for $\chi$ to condensate) would depend on not only the pair $(z,m_\psi)$, but also $(\kappa,\eta)$, representing a nontrivial interaction among $\chi$, Maxwell field $F$ and background scalar $\psi$.
\section{Quantum Criticality at $AdS_2$}
As a warm up, we will revisit the quantum criticality at $AdS_2$ before going for generic $z$. A detail treatment for a minimal coupled scalar was given in \cite{Faulkner:2009wj}, so here we only highlight the difference. Let us make a Fourier transform of the scalar field along $(t,\vec{x}$ directions:
\begin{equation}
\chi(u,t,\vec{x}) = \int{\frac{d\omega d^2k}{(2\pi)^3}}\chi(u,\omega,\vec{k})e^{-i\omega t +\vec{k}\cdot\vec{x}}, \quad |\vec{k}|\equiv k.
\end{equation}
Now consider the metric of $AdS_2\times R^2$ with a constant electric field, obtained in (\ref{ads2_metric}) and (\ref{ads2_at}). The equation of motion reads:
\begin{eqnarray}\label{eqn_inner}
&&-\partial_u^2 \chi + \left[ \frac{m_{eff}^2L_0^2}{u^2} - (\omega+\frac{\mu q_\chi L_0}{u})^2 \right]\chi = 0 ,\nonumber\\
&&m_{eff}^2 \equiv m_\chi^2 + k^2L^2 - 6\kappa \mu^2.
\end{eqnarray}
\begin{figure}[tbp]
\label{fig1} \includegraphics[width=0.45\textwidth]{ads2_BF.eps}
\caption{BF bounds set by $AdS_4$ at UV and $AdS_2$ at IR, the scalar $\chi$ will condensate if $m_\chi^2L^2$ falls inside the shadow region.}
\end{figure}
We remark that the couple between scalar and Maxwell field contributes to the last term in the definition of effective mass, and apparently it depends on chemical potential. The couple between two scalars does not enter due to a trivial $\psi_0$ in the $AdS_2$ background. Since we are interested in the positive coupling constant $\kappa$, which can be tuned to shift the effective mass to be more negative. In practice, for a neutral scalar to condensate, we ask the effective mass to satisfy $AdS_4$ Breitenlohner-Freedman (BF) bound\cite{Breitenlohner:1982bm} but violate that of $AdS_2$. Therefore we are free to tune $\kappa$ such that
\begin{equation}
-\frac{9}{4} < m_\chi^2 L^2, \qquad (m_\chi^2-6\kappa \mu^2) L^2 \le -\frac{3}{2}.
\end{equation}
We plot the admissible range of $\kappa$ and $m_\chi^2$ for condensate to happen in the Figure 1. In particular, the equality holds for a critical $\kappa_c$ given some chosen mass\footnote{In the extremal limit, we are free to replace $\mu=\sqrt{3}$.},
\begin{equation}
\kappa_c = \frac{m_\chi^2}{18} + \frac{1}{12L^2},
\end{equation}
such that one may engineer a quantum phase transition where $T_c=0$ as that in \cite{Faulkner:2009wj}. The equation (\ref{eqn_inner}) can be solved explicitly and the retarded Green function reads
\begin{eqnarray}
&&{\cal G}_k(\omega) = 2\nu_k e^{-i\pi \nu_k}\frac{\Gamma(-2\nu_k)\Gamma(\frac{1}{2}+\nu_k-iq_\chi \mu L_0)}{\Gamma(2\nu_k)\Gamma(\frac{1}{2}-\nu_k-iq_\chi \mu L_0)}(2\omega)^{2\nu_k},\nonumber\\
&&\nu_k \equiv \sqrt{m_{eff}^2 L_0^2-q_\chi^2\mu^2 L_0^2+\frac{1}{4}}.
\end{eqnarray}
Since the effect of nonminimal coupling only appears in the modification of effective mass, one expects the discussion in \cite{Faulkner:2009wj} still hold in our case. To list a few:
\begin{enumerate}
\item The low energy behavior of boundary system is uniquely determined by this IR analysis. For example, the spectral function $Im_{} G_R(\omega,k) \propto \omega^{2\nu_k}$.
\item For sufficient large $\kappa$, $\nu_k$ becomes pure imaginary and log-periodic behavior is expected and gapless excitation is responsible for this.
\item Once a nonminimal coupled Fermion can be formulated in this background, one may have a model of non-Fermi liquids characterised by coupling $\kappa$. We will leave this for future study.
\end{enumerate}
\section{Quantum criticality at Lifshitz point}
Now let us take a closer look at quantum critical point of generic Lifshitz scaling. The equation of motion reads:
\begin{eqnarray}\label{eom_z}
&&-\partial_u^2\chi + \frac{(z+1)}{u}\partial_u\chi + \left[ k^2-(\frac{u}{L_0})^{2z-2}(\omega+q_\chi\sqrt{2-\frac{2}{z}}(\frac{L_0}{u})^z)^2+\frac{m_{eff}^2L_0^2}{u^2} \right]\chi=0,\nonumber\\
&&m_{eff}^2 \equiv m_\chi^2 - \frac{12(z-1)}{L^2(z+1)(z+2)}\left[ \kappa z - \frac{\eta}{m_\psi^2} \right].
\end{eqnarray}
At small $u$, the scalar $\chi$ behaves like
\begin{equation}
\chi \sim A(\omega) u^{1+\frac{z}{2}-\nu} + B(\omega) u^{1+\frac{z}{2}+\nu},\qquad \nu \equiv \sqrt{m_{eff}^2L_0^2-q_\chi^2(2-\frac{2}{z})L_0^2+(\frac{z+2}{2})^2}.
\end{equation}
By scaling invariance, one can argue the retarded Green function should scale like
\begin{equation}
{\cal G}(\omega) \equiv \frac{B(\omega)}{A(\omega)} \propto \omega^{\frac{2\nu}{z}},\qquad
\end{equation}
It is unclear whether the remaining part of $\cal G$ is obtainable for generic $z$ because analytic solution to equation (\ref{eom_z}) may not exist. There is an exception for $z=2$. One can find that, up to a normalized factor and phase:
\begin{eqnarray}\label{Retarded_Green}
&&{\cal G} \sim \frac{\Gamma(-\nu)\Gamma(\frac{1+\nu}{2}+i\delta)}{\Gamma(\nu)\Gamma(\frac{1-\nu}{2}-i\delta)} \omega^{\nu},\nonumber\\
&&\nu \equiv \sqrt{(m_{eff}^2-q_\chi^2)L_0^2+4}, \qquad \delta \equiv \frac{k^2}{4\omega}-\frac{q_\chi}{2}
\end{eqnarray}
This solvability mainly thanks to the integrability of equation (\ref{eom_z}) in the case of $z=2$, where it can be recasted into
\begin{equation}\label{ode_z2}
-\partial_u^2 \xi + \left[\frac{\nu^2}{u^2}-\frac{\omega^2}{L_0^2}u^2\right]\xi = -4\omega \delta\xi,
\end{equation}
with a field redefinition $\chi \equiv u^{3/2}\xi$. This is nothing but a one-dimensional quantum mechanics of the Calogero particle with an inverse square potential in the harmonic trap. A detail treatment for solving this is given in the appendix.
\begin{figure}[tbp]
\label{fig2}
\includegraphics[width=0.45\textwidth]{kappa_effect.eps}
\caption{A typical $\kappa=0.05$ will widen an additional window for condensate (shaded region). We also reproduce the curves due to effect of negative $\eta$ (dotted black) and positive $\eta$ (dashed purple), below which the condensate develops.}
\end{figure}
\begin{figure}[tbp]
\label{fig3}
\includegraphics[width=0.45\textwidth]{Efimov.eps}
\caption{A typical wavefunction for $|\chi|$, where the first four Efimov states are shown to correspond to those zeros at $u=1.643,2.791,4.438,7.293$ (The curve does not appear to hit the zero at those points due to limited number of sample points in the plot). The second and higher Efimov states are higher excitation modes with one and more zero nodes.}
\end{figure}
\section{BKT phase transition at critical line}
To have condensate at IR, we ask the effective mass to violate the BF bound, that is
\begin{equation}
m_\chi^2L^2 \le \frac{12(z-1)}{(z+1)(z+2)}(\kappa z - \frac{\eta}{m_\psi^2})-\frac{3(z+2)}{2(z+1)}.
\end{equation}
In the Figure 2, we show that a shadow region is sandwiched by two curves, where the lower(upper) one associates to a BF bound with (non)zero $\kappa$. This shows that nonminimal coupling enhances the instability and raises the BF bound for $m_\chi^2$. In comparison, we also reproduce those BF bounds with positive and negative coupling $\eta$ but without $\kappa$\cite{Basu:2010fa}. The quantum Lifshitz point corresponds to where the BF bound is about to be violated.
Therefore, given masses of scalars and exponent $z$ there exists a critical line in the parameters space formed by $(\kappa, \eta)$. For $\kappa z - \frac{\eta}{m_{\psi}^2}> \delta_c$, the BF bound is violated and conformality is lost. Following similar argument in \cite{Kaplan:2009kr,Faulkner:2009wj}, an infinite tower of IR scales is generated and associated with the infinite number of Efimov states:
\begin{equation}
u_{IR}^{(n)} = u_{UV} \log(\frac{n\pi}{\sqrt{2(z-1)(\kappa z-\frac{\eta}{m_\psi^2}-\delta_c)}}), \qquad n = 1,2,\cdots.
\end{equation}
If we manage to turn on the temperature in this background, one may associate $u_{UV}$ to the scale set by $\mu$ and $u^{(1)}_{IR}$ to the scale set by a finite temperature $T_c$ as
\begin{equation}
T_c \sim (\frac{1}{u^{(1)}_{IR}})^z \sim \mu \exp{(-\frac{z\pi}{\sqrt{2(z-1)(\kappa z-\frac{\eta}{m_\psi^2}-\delta_c)}})},
\end{equation}
In the Figure 3, we show that the first few IR scales out of infinite many, while the boundary condition of vanishing wavefunction $\chi$ is imposed at a chosen UV scale.
In the Figure 4, it is also shown that the distance between UV scale and the first IR scale decreases with increasing $z$, implying that condensate is easier to form at larger $z$. We observe that critical temperature rises up with increasing $\kappa-\kappa_c$ and plot it against various $z$ in the plot to the right.
We comment on some new features as follows: the positive $\kappa$ acts like a negative coupling $\eta$ in the case of $m_\psi^2 >0$, both seem to weaken the stability by decreasing its effective mass. However, this similarity breaks down for large enough $z$, where the critical temperature is almost determined by $\kappa$ alone as follows:
\begin{equation}
T_c \sim \mu \exp(-\frac{\pi}{\sqrt{2\kappa}}),
\end{equation}
where the contribution from $\eta$ term is ignorable. We remark that the minimal coupling limit $\kappa \to 0$ drives the critical $T_c \to 0$, which can be identified as the quantum critical point observed in pure $AdS_2$ background.
\begin{figure}[tbp]
\label{fig4}
\includegraphics[width=0.45\textwidth]{UV-IR.eps}
\includegraphics[width=0.45\textwidth]{tc.eps}
\caption{To the left: IR scales against various $z$. The zero of each $\chi$ at $u=1$ is chosen for the UV cutoff (by imposing the boundary condition) and another zero at larger $u$ can be regarded as the IR cutoff. We have removed irrelevant IR scales set by higher Efimov states from the plot. From right to left, the curves correspond to that of $z=2$(black thick), $z=3$(blue thin), $z=4$(red dotted) and $z=5$(green dashed). The dynamically generated IR scales move toward the UV as $z$ increases, signaling a raise of critical temperatures. Both axes are in the log scale. To the right: The critical temperature against $\kappa-\kappa_c$ for $z=2$(blue), $z=5$(dashed red) and $z=20$(dotted black). It is expected to reach quantum Lifshitz point at $\kappa = \kappa_c$.}
\end{figure}
\begin{figure}[tbp]
\label{fig5}
\includegraphics[width=0.45\textwidth]{V2.eps}
\includegraphics[width=0.45\textwidth]{L.eps}
\caption{To the left: A typical potential for $V_2(\chi)$. At the minimum of potential, $|\chi|^2$ takes value $3/2$ at the IR cutoff. To the right: A Plot of the ratio $\rho = (\tilde{L_0}/L_0)^2$ against $z'$ (thick red). Notice for large enough $z'$, we have $\rho > 1$ (dashed blue), meaning that $\tilde{L_0}$ is larger than $L_0$. For these plots, we have fixed the variables $z=2,q_\chi^2L_0^2=2,m_\chi^2L_0^2=-1.9,\kappa L_0^2=0.05, \lambda L_0^2 = 1$}
\end{figure}
\section{Beyond the probe}
To go beyond the probe limit, one starts to consider the back reaction from the $\chi$ field. We will make the following assumptions, following similar arguments in \cite{Iqbal:2010eh}:
\begin{enumerate}
\item We assume the existence of an IR cutoff point $u=u_0$, where $\chi$ field smoothly goes to a constant $\chi_0$. This could be achieved by embedding a charged black hole and the horizon naturally introduces the cutoff\cite{Goldstein:2009cv}. For our purpose here, we simply demand $\chi(u_0)= \chi_0$ and $\chi'(u_0)=0$ at the cutoff.
\item Nonlinear potential terms of higher power are necessary in addition to those in the equation (\ref{model}), in order to arrive at some physical ground state after receiving back reaction. A simplest addition is to include a $|\chi|^4$ term.
\item After back reaction, we assume our background geometry still respects the Lifshitz scaling of some critical exponent $z'$, which, however, is not necessary to be the same as the original $z$.
\end{enumerate}
Now we are ready to discuss the consequence derived from those assumptions. Let us first investigate the trace of Einstein equation, including those parts with $\chi$ involved:
\begin{eqnarray}\label{trace}
&&R+ \frac{2(z^2+2z+3)}{L_0^2} = |D\chi|^2 + 2 V_1(\chi),\nonumber\\
&&V_1(\chi) \equiv \left[ m_\chi^2 + \eta \frac{12}{m_\psi^2 L^2}\frac{z-1}{(z+1)(z+2)}\right] |\chi|^2 + \lambda |\chi|^4.
\end{eqnarray}
Notice that we have included the quartic terms with any $\lambda >0$.
As what has been observed in \cite{Iqbal:2010eh}, it will reduce the effective $L_0$ providing that the right hand side of (\ref{trace}) is negative at IR cutoff. However, if the variation of $z$ is admissible, it could increase $L_0$ instead. To see this, we derive the effective radius of curvature for generic $z'$, denoting $\tilde{L}_0$, after receiving back reaction:
\begin{equation}
\frac{z'^2+2z'+3}{\tilde{L_0}^2} = \frac{z^2+2z+3}{L_0^2}-\left[ -q_\chi^2\frac{z-1}{z} + m_\chi^2 + \frac{2\eta}{m_\psi^2 L_0^2}(z-1) \right]|\chi_0|^2 - \lambda|\chi_0|^4.
\end{equation}
It is not difficult to see that $\tilde{L_0}$ in fact could increase if $z'$ is larger enough than $z$. Interestingly, this in turns will either increase or decrease the background condensate $\psi$ thanks to its inversely proportional to $\tilde{L_0}$ as shown in the equation (\ref{sol_z}). We plot a typical $V_2$ and the ratio $(\tilde{L_0}/L_0)^2$ in the Figure 5.
One should also investigate the equation of motion (\ref{eom_z}) around the IR cutoff:
\begin{eqnarray}
&&-u^2\partial_u^2\chi + L_0^2V_2'(\chi) = 0, \nonumber\\
&&V_2(\chi) \equiv \left[ -q_\chi^2 \frac{z-1}{z}+ m_\chi^2 - (z-1)\left( \kappa z - \frac{\eta}{m_\psi^2} \right) \right] |\chi|^2+\lambda |\chi|^4.
\end{eqnarray}
To ensure the $\chi$ field sits right on the bottom of a concave-up potential at this point, we should demand $V_2'(\chi_0) = 0$ as well as
$V_2''(\chi_0) > 0$. This pins down to the following constraint:
\begin{equation}
-q_\chi^2 \frac{z-1}{z}+ m_\chi^2 - (z-1)( \kappa z - \frac{\eta}{m_\psi^2}) < 0
\end{equation}
We remark that potential $V_2$ includes the contribution from coupling $\kappa$ but $V_1$ does not thanks to the traceless condition of Maxwell field in four dimensions.
\begin{figure}[tbp]
\label{fig6}
\includegraphics[width=0.45\textwidth]{RG_aboveBF.eps}
\includegraphics[width=0.45\textwidth]{RG_belowBF.eps}
\caption{To the left: A typical RG running for $\lambda$ for $\nu^2>0$, where nonminimal coupling $\kappa=0$(thick blue), $\kappa=0.2$(dashed red), $\kappa=0.249$(dotted black). To the right: RG running shows periodic flows for $\kappa=0.251$(thick blue), $\kappa=1$(dashed black), $\kappa=1.5$(thin red). In both plots, we have fine tuned the parameters to have critical $\kappa_c = 0.25$.}
\end{figure}
\section{Discussion}
Our model may be useful to describe a condensed matter system with two or multiple condensates, such as a two-band model in the superconductor. Since we have taken one of two scalars to be small, our scenario is suitable to the window where a second condensate just begins to develop, while the first has been strong out there. The above discussion regarding back reaction implies that the appearance of a second condensate may either enhance or suppress the first one through their direct coupling. On the other hand, this back reaction may be seen as some sort of perturbation or deformation from the critical point, as recently discussed in \cite{Faulkner:2010gj}.
We have given some detail treatment for a Lifshiz system with critical exponent $z=2$ in the appendix, thanks to its integrability. We showed that it can be translated into a one-dimensional quantum mechanics of Calogero particle in a harmonic trap potential and demonstrated that the RG running of its contact coupling shows periodic flowing once the unitarity is broken by overcritical $\kappa$. For generic $z>2$, the differential equation (\ref{eom_z}) will include trap potential terms of higher order $O(u^z)$. Since the contact potential, originally introduced for regularization, always dominates over the trap potential of any order in the region $u<u_0$, we expect that the same discussion for periodic RG running holds true for a Lifshitz system of higher $z$. In fact, this statement is confirmed in the section V by the appearance of an infinite tower of Efimov states in the bulk.
\section*{Appendix}
In this appendix, we would like to take a closer look at a special case for $z=2$. In particular, we will highlight the relation between inverse square potential and holographic RG flow, following the same treatment as found in \cite{Moroz:2009nm}. One starts with the following differential equation, obtained from (\ref{ode_z2}) after a change of variable:
\begin{eqnarray}
&&-\partial_u^2 \xi - V(u)\xi = - (k^2 - 2\omega q_\chi )\xi,\nonumber\\
&&V(u) = -\frac{\nu^2}{u^2}+\frac{\omega^2}{L_0^2}u^2.
\end{eqnarray}
In order to explore both regions of positive and negative $\nu^2$, we will analytically continue to a complex $u$ plane. The above potential is ill-defined at $u=0$ for its inverse square potential. One way to regularize it is to cut off the potential for $|u|< u_0$, and replace it by a $\delta-$function potential at $u=0$. That is,
\begin{equation}
V(u) = -\frac{\lambda}{u_0}\delta(u),\qquad |u|< u_0,
\end{equation}
where $\lambda$ is a dimensionless contact coupling and its RG flow against the cutoff $u_0$ will be studied in the following.
The wavefunction can be exactly solved; it is given by the Parabolic Cylinder function for $|u|< u_0$ and the hypergeometric function outside the cutoff. It is sufficient for us to work with the limit $u>0, y\equiv \sqrt{\omega}u \ll 1$, where $\xi(u)$ can be expanded as
\begin{equation}
\xi(y) =
\left\{\begin{matrix}
(a + b y + O(y^2)) + D (a'+ b' y + O(y^2)), \quad 0< y <|y_0|\equiv \sqrt{\omega}u_0, \\
N(c_- y^{1/2-\nu} + c_+ y^{1/2+\nu}), \quad |y| > y_0.
\end{matrix}\right.
\end{equation}
where $a,a',b,b'$ are first two coefficients of Taylor expansion of the desired Parabolic Cylinder function, and $N$ is some normalization factor of wavefunction. Their precise forms are irrelevant to our discussion here. By observing the continuity of $\xi(y)$ and discontinuity of its first derivate at $y=0$, one can determine
\begin{equation}\label{D_lambda}
\lambda = 2 y_0\frac{-b-b'D}{a+a'D}.
\end{equation}
Imposing the boundary condition of continuity of $\xi(y)$ and $\xi'(y)$ at $y=y_0$ for both expansions, one obtains the relations:
\begin{equation}\label{BC}
\left\{\begin{matrix}
a+Da' \approx N(c_-y_0^{1/2-\nu}+ c_+ y_0^{1/2+\nu}),\\
b+Db' \approx N\left[ c_-(1/2-\nu)y_0^{-1/2-\nu}+c_+(1/2+\nu)y_0^{-1/2+\nu}\right].
\end{matrix}\right.
\end{equation}
Combining (\ref{D_lambda}) and (\ref{BC}), and introducing a running variable $t=-\ln y_0$, we arrive at the RG flow for $\lambda$
\begin{equation}
\lambda (t) = -1 + 2\nu \frac{e^{\nu t}-Ce^{-\nu t}}{e^{\nu t}+ Ce^{-\nu t}},\qquad C\equiv \frac{c_+}{c_-},
\end{equation}
where $C$ is nothing but retarded Green function ${\cal G}$ given in (\ref{Retarded_Green}). The coupling $\lambda$ satisfies the general Riccati differential equation:
\begin{equation}
\partial_t \lambda = -\frac{1}{2}(\lambda + 1 - 2\nu)(\lambda + 1+ 2\nu).
\end{equation}
We plot it against various $\kappa$ in the Figure 6. We remark that the periodic flows start to appear once the unitary bound is violated by $\kappa > \kappa_c$.
\begin{acknowledgments}
The author would like to thank the hospitality of high energy theoretical group in the Caltech. The author is grateful to useful discussion with Yutin Huang, Hirosi Ooguri, and Shang-Yu Wu. This work is supported in part by the Taiwan's National Science Council and National Center for Theoretical Sciences.
\end{acknowledgments}
|
{
"timestamp": "2010-09-22T02:00:31",
"yymm": "1009",
"arxiv_id": "1009.3952",
"language": "en",
"url": "https://arxiv.org/abs/1009.3952"
}
|
\section{Introduction}
\label{sec:intro}
A problem in many applications is that of sound generated by circular
sources. These include rotors of various types such as aircraft
propellers and fans, wind turbines and cooling fans; vibrating systems
such as loudspeakers; ducts such as aircraft engines, ventilation
systems and exhausts; and distributed sources with axial symmetry such
as jets. There have been numerous studies of the noise generation and
radiation process in each of these areas extending over many
decades. These studies can be divided into those which examine the
relationship between the acoustic source and the physical processes
which give rise to it, for example the work of
Lighthill~\cite{lighthill52} and of Ffowcs Williams and
Hawkings~\cite{ffowcs-williams-hawkings69b} which relate aerodynamic
quantities to acoustic sources, and those which examine the radiated
field generated by a given source distribution, such as methods for
prediction of the field radiated by pistons and
loudspeakers~\cite{oberhettinger61a,pierce89,mellow06,mellow08} or from
a known rotating source
distribution~\cite{gutin48,wright69,chapman93,carley99}.
There are a number of areas where these issues, those of generation
and radiation, overlap. One is the general area of source
identification. There have been many attempts to develop methods which
use acoustic measurements to infer, in greater or lesser detail, the
source distribution responsible for the acoustic field. In the case of
rotating sources, some examples include cooling
fans~\cite{gerard-berry-masson05a,gerard-berry-masson05b,%
gerard-berry-masson-gervais07} and
propellers~\cite{peake-boyd93,li-zhou95,%
minniti-blake-mueller01a,minniti-blake-mueller01b}, while a number
of groups have developed methods for the inverse problem for radiation
from a duct termination~\cite{holste-neise97,lewy05,lewy08,%
castres-joseph07a,castres-joseph07b}. Such studies can have a number
of motivations. The first is to use near-field data, for example those
taken in wind-tunnel tests, to predict the far acoustic field. In this
case, the requirement is to extract information about source strength
and directivity, but there is no need to know which processes generate
the source. A second motivation, however, is the identification of the
noisiest parts of the source with a view to reduction of noise at
source, for example the identification of ``hot spots'' caused by
unsteady loading on a cooling
fan~\cite{gerard-berry-masson05a,gerard-berry-masson05b}. In this case,
the link between the aerodynamics and the source is an essential part
of the solution of the problem.
In each of the applications of source identification listed, the
authors have recognized that the problem is (very)
ill-conditioned. This can be attributed to physical causes, and is not
merely an artifact of the methods used. Recent
analysis~\cite{carley09,carley10b,carley10c} has given a framework for
the study of this ill-conditioning by quantifying the source
information which is radiated into the acoustic near and far
fields. As described below, it has been found that the source can be
decomposed into orthogonal modes based on Chebyshev polynomials, only
a limited number of which radiate a detectable acoustic field, with
the limit being fixed by the source frequency.
A second area where the issues of generation and radiation overlap is
that of jet noise. Lighthill's acoustic analogy~\cite{lighthill52} is
accepted as an exact theory for noise generation by turbulence and
there is solid evidence for the validity of his source term, as
demonstrated by high quality numerical simulation~\cite{freund01}. This
knowledge, however, is not sufficient to explain certain features of
jet noise, in particular the low radiation efficiency of subsonic jets
and the low order structure of the acoustic field. It is known that
subsonic jets radiate only a small fraction of the source energy, a
view given support by the very small changes in the flow which suffice
to give large reductions in noise, when control is
applied~\cite{freund10}. It is also known that the acoustic far field
of a jet is significantly simpler than the flow field. In a recent
study~\cite{jordan-schlegel-stalnov-noack-tinney07}, modal
decomposition of the far-field noise and of the flow field of a
Mach~0.9 jet showed that~24 modes were sufficient to capture~90\% of
the energy of the acoustic field, but~350 were required to
resolve~50\% of the flow energy. Clearly, a very large part of the
flow, however energetic it might be, simply does not radiate but it is
not obvious if this is due to the nature of the source or purely a
result of radiation effects.
The radiation effect has been explained in terms of source
cancellation~\cite{michel07,michel09} and by viewing the radiation
process as equivalent to the imposition of a spatial filter using a
wavenumber criterion. Such an approach has been used by
Freund~\cite{freund01} who found that the part of the source which
radiates is indeed that part left over after applying an appropriate
spatial filter. Similarly, Sinayoko and
Agarwal~\cite{sinayoko-agarwal10} apply a linear convolution filter to
decompose the flow into radiating and non-radiating parts.
The analysis to be presented below attempts to explain some of these
features. Previous work~\cite{carley09,carley10b,carley10c} has found
limits on the information radiated from a tonal circular source,
motivated by a desire to understand the ill-conditioning of source
identification methods. These limits have been found without recourse
to a far-field approximation, making the approach suitable for
analysis of general problems. The remainder of this paper contains an
extension of the theory to explicitly include the radial source term,
and to yield spectral quantities in the acoustic field of random
sources.
The first extension, which can be viewed as a generalization of
previous work on axisymmetric radiators~\cite{stepanishen76}, will help
explain radial cancellation effects, which have been studied in jet
noise using a far-field formulation~\cite{michel09} but not, to the
author's knowledge, in the near field. It will be found that for a
given azimuthal order, many different sources radiate identical
acoustic fields, differing only by a scaling factor. This result is
part of the explanation for the ill-conditioning of identification
methods and also opens a possible approach to the development of
control systems by identifying a class of sources which can give rise
to practically identical acoustic fields.
The second extension, to predicting the cross-spectrum between the
acoustic pressures radiated by a random source to arbitrary points in
the near and/or far field, is an extension of an earlier ring-source
model for radiation from random sources characteristic of
jets~\cite{michalke83}. In this case, it will be found that the
cross-spectrum depends on four constants, functions of observer
radial separation, which are weighted integrals of the source
cross-spectrum.
The results to be presented arise from two different exact theories
for radiation from circular
sources~\cite{carley10,carley10b,carley10c} which are combined to give
a formulation for the information in the acoustic field in terms of
radiation functions and weighted integrals of the source term. The
implications of the results are discussed in terms of the information
content of the acoustic field and with regard to some of the
measurement methods used to study noise sources.
\section{Tonal disk source}
\label{sec:analysis}
\begin{figure}
\centering
\includegraphics{jasa10b-figs-2}
\caption{Coordinate system for disk radiation calculations}
\label{fig:coordinate}
\end{figure}
The problem is initially formulated as that of calculating the
acoustic field radiated by a monopole source distributed over a
circular disk. The system for the analysis is shown in
Figure~\ref{fig:coordinate} with cylindrical coordinates
$(r,\theta,z)$ for the observer and $(a,\psi,0)$ for the source. All
lengths are non-dimensionalized on disk radius. The field from one
azimuthal mode of the acoustic source, specified as $s_{n}(a)\exp\J [n
\psi-\omega t]$, has the form $P_{n}(k,r,z)\exp\J [n \theta-\omega
t]$, with $P_{n}$ given by the Rayleigh
integral~\cite{goldstein74,carley09}:
\begin{align}
\label{equ:disk}
P_{n}(k,r,z) &=
\int_{0}^{1}
\int_{0}^{2\pi} \frac{\E^{\J(kR'+n\psi)}}{4\pi
R'}\,\D \psi s_{n}(a) a\,\D a,\\
R' &=
\left[
r^{2} + a^{2} - 2ra\cos\psi + z^{2}
\right]^{1/2},\nonumber
\end{align}
where $k$ is non-dimensional wavenumber (Helmholtz number).
\subsection{Equivalent line source expansion}
\label{sec:line}
\begin{figure}
\centering
\includegraphics{jasa10b-figs-1}
\caption{Transformation to equivalent line source}
\label{fig:sideline}
\end{figure}
The analysis of the nature of the sound field from an arbitrary disk
source is based on a transformation of the disk to an exactly
equivalent line source, an approach which has been used to study
transient radiation from pistons~\cite{oberhettinger61a,pierce89},
rotor noise~\cite{chapman93,carley99} and source identification
methods~\cite{carley09,carley10c,carley10b}.
The transformation to a line source is shown in
Figure~\ref{fig:sideline}, which shows the new coordinate system
$(r_{2},\theta_{2},z)$ centred on a sideline of constant radius
$r$. Under this transformation:
\begin{align}
\label{equ:transformed}
P_{n}(k,r,z) &= \int_{r-1}^{r+1} \frac{\E^{\J kR'}}{R'}
K(r,r_{2})r_{2}\,\D
r_{2},\\
R' &= \left(r_{2}^{2} + z^{2}\right)^{1/2},\nonumber\\
\label{equ:kfunc}
K(r,r_{2}) &= \frac{1}{4\pi}
\int_{\theta_{2}^{(0)}}^{2\pi-\theta_{2}^{(0)}} \E^{\J
n\psi}s_{n}(a)\,\D\theta_{2},
\end{align}
for observer positions with $r>1$, with the limits of integration
given by:
\begin{align}
\label{equ:theta}
\theta_{2}^{(0)} &= \cos^{-1}\frac{1-r^{2}-r_{2}^{2}}{2rr_{2}}.
\end{align}
Functions of the form of $K(r,r_{2})$ have been analyzed in previous
work~\cite{carley99} and can be written:
\begin{align}
\label{equ:kfunc:exp}
K(r,r_{2}) &= \sum_{q=0}^{\infty} u_{q}(r)U_{q}(s)(1-s^2)^{1/2},
\end{align}
where $U_{q}(s)$ is a Chebyshev polynomial of the second kind,
$s=r_{2}-r$ and the coefficients $u_{q}(r)$ are functions of $r$ but
not of $z$. Inserting Equation~\ref{equ:kfunc:exp} into
Equation~\ref{equ:transformed}:
\begin{align}
\label{equ:line:ip}
P_{n}(k,r,z) &=
\sum_{q=0}^{\infty} u_{q}(r)\lfn{q}(k,r,z),\\
\label{equ:lfunc}
\lfn{q}(k,r,z) &= \int_{-1}^{1} \frac{\E^{\J kR'}}{R'}
U_{q}(s) (r+s)(1-s^2)^{1/2}\,\D s,\\
R' &= \left[(r+s)^{2} + z^{2}\right]^{1/2}.
\end{align}
The radiation properties of the integral of Equation~\ref{equ:lfunc}
have been examined in some detail elsewhere~\cite{carley10b,carley10c},
giving an exact result for the in-plane case $z=0$:
\begin{align}
\label{equ:line:series}
\lfn{q}(k,r,0) &=
\J^{q} (q+1) \pi \E^{\J k r} \frac{\besselj{q+1}(k)}{k}.
\end{align}
For large order $q$, the Bessel function $J_{q}(k)$ is exponentially
small for $k<q$ so that the line source modes with order $q>k$
generate noise fields of exponentially small amplitude. Since the
integrals have their maximum in the plane $z=0$,
Equation~\ref{equ:line:series} says that the whole field is of
exponentially small amplitude. This gives an indication of how much of
a given source distribution radiates into the acoustic field, near or
far.
In previous analyses, two approximations to $\lfn{q}$ have been
developed. One is an asymptotic formula valid in the limit
$k\to\infty$, derived using the method of stationary
phase~\cite{carley10b,carley10c}. This will not be required here, but
we will make use of the far-field form of Equation~\ref{equ:lfunc}:
\begin{align}
\label{equ:lfunc:ff}
\lfn{q} &\approx \J^{q}\pi\frac{\E^{\J k R}}{R}\frac{q+1}{k\sin\phi}
\biggl[
\left(r+\J\frac{q+2}{k\sin\phi}\right)
J_{q+1}(k\sin\phi)
-\J J_{q}(k\sin\phi)
\biggr],
\end{align}
where $R=[r^{2}+z^{2}]^{1/2}$ and $\phi=\cos^{-1}z/R$.
Given the basic information about the form of the radiated field,
there remains to establish the relationship between the radial
structure of the source $s_{n}(a)$ and the line source coefficients
$u_{q}(r)$.
\subsection{Series expansion for spinning sound fields}
\label{sec:series}
A recently derived series~\cite{carley10} for the field radiated by a
ring source of radius $a$ can be used to find a second expression for
the sound radiated by a disk source with arbitrary radial variation:
\begin{align}
R_{n} &= \int_{0}^{2\pi} \frac{\E^{\J(kR'+n\psi)}}{4\pi R'}\,\D\psi,
\nonumber\\
&=
\J^{2n+1}\frac{\pi}{4}
\frac{1}{(aR)^{1/2}}
\sum_{m=0}^{\infty}
(-1)^m
\frac{(2n+4m+1)(2m-1)!!}{(2n+2m)!!}\nonumber\\
\label{equ:ring}
&\times
\hankel{n+2m+1/2}(kR) P_{n+2m}^{n}(\cos\phi)
\besselj{n+2m+1/2}(ka),
\end{align}
with $\hankel{\nu}(x)$ the Hankel function of the first kind of order
$\nu$, $\besselj{\nu}$ the Bessel function of the first kind and
$P_{n}^{m}$ the associated Legendre function. The observer position is
specified in the spherical polar coordinates used in
Equation~\ref{equ:lfunc:ff}.
Multiplication by the radial source term $a s_{n}(a)$ and
integration gives an expression for the field radiated by a general
source of unit radius and azimuthal order $n$:
\begin{align*}
P_{n}(k,r,z) &=
\J^{2n+1}\frac{\pi}{4}
\sum_{m=0}^{\infty}
(-1)^m
\frac{(2n+4m+1)(2m-1)!!}{(2n+2m)!!} P_{n+2m}^{n}(\cos\phi)S_{n+2m},\\
S_{n+2m}(k,r,z) &= \int_{0}^{1} s_{n}(a)
\besselj{n+2m+1/2}(ka)\hankel{n+2m+1/2}(kR)
\left(
\frac{a}{R}
\right)^{1/2}\,\D a.
\end{align*}
Setting $z=0$ ($\phi=\pi/2$, $R=r$):
\begin{align}
\label{equ:series:ip}
P_{n}(k,r,0) &=
\frac{\J\pi}{4}
\sum_{m=0}^{\infty}
A_{m}S_{n+2m},\\
A_{m} &=
\frac{1}{m!}
\frac{(2n+4m+1)(2n+2m-1)!!(2m-1)!!}{2^{m} (2n+2m)!!},\nonumber
\end{align}
where use has been made of the expression~~\cite{gradshteyn-ryzhik80}:
\begin{align}
P_{n+2m}^{n}(0) = \frac{(-1)^{m+n}}{2^{m}}\frac{(2n+2m-1)!!}{m!}.
\end{align}
\subsection{Line source coefficients}
\label{sec:coefficients}
The expressions for $P_{n}$ from section~\ref{sec:line} and
section~\ref{sec:series} are both exact and can be equated to derive a
system of equations relating the coefficients $u_{q}(r)$ to the
weighted integrals of the radial source distribution $s_{n}(a)$:
\begin{align}
\frac{\J}{4}
\sum_{m=0}^{\infty}
A_{m}
S_{n+2m}
&=
\label{equ:system}
\E^{\J k r}
\sum_{q=0}^{\infty} u_{q}(r)
\J^{q} (q+1) \frac{\besselj{q+1}(k)}{k}.
\end{align}
Under repeated differentiation, Equation~\ref{equ:system} becomes a
lower triangular system of linear equations which connects the
coefficients $u_{q}(r)$ and $S_{n+2m}$:
\begin{align}
\frac{\J}{4}
\sum_{m=0}^{\infty}
A_{m}
S_{n+2m}^{(v)} &=
\label{equ:diff}
\sum_{q=0}^{\infty} u_{q}(r)
\J^{q} (q+1)
\left[
\E^{\J k r}
\frac{\besselj{q+1}(k)}{k}
\right]^{(v)},
\end{align}
where superscript $(v)$ denotes the $v$th partial derivative with
respect to $k$, evaluated at $k=0$.
Using standard series~\cite{gradshteyn-ryzhik80}, the products of
special functions can be written:
\begin{align}
\label{equ:prod:exp:j}
\E^{\J k r}\frac{\besselj{q+1}(k)}{k} &=
\frac{1}{\J^{q}} \sum_{t=0}^{\infty}(\J k)^{t+q} E_{t,q}(r),\\
E_{t,q}(r) &= \frac{1}{2^{q+1}}
\sum_{s=0}^{[t/2]}
\frac{r^{t-2s}}{4^{s}s!(s+q+1)!(t-2s)!},\nonumber
\end{align}
where $[t/2]$ is the largest integer less than or equal to $t/2$, and
\begin{align}
\left(
\frac{a}{r}
\right)^{1/2}
\hankel{n+1/2}(kr)\besselj{n+1/2}(ka) &=
\left(
\frac{r}{2}
\right)^{2n+1}
\sum_{t=0}^{\infty}
\frac{k^{2t+2n+1}}{t!}
\left(
-\frac{r^{2}}{4}
\right)^{t}V_{n,t}(a/r) \nonumber\\
&- (-1)^n\J \sum_{t=0}^{\infty}
\frac{k^{2t}}{t!}
\left(
-\frac{r^{2}}{4}
\right)^{t}W_{n,t}(a/r),
\label{equ:prod:h:j}
\end{align}
with the polynomials $V_{n,t}$ and $W_{n,t}$ given by:
\begin{subequations}
\label{equ:vwpoly}
\begin{align}
V_{n,t}(x) &= \sum_{s=0}^{t} {t \choose s}
\frac{x^{2s+n+1}}{\Gamma(n+s+3/2)\Gamma(t-s+n+3/2)},\\
W_{n,t}(x) &= \sum_{s=0}^{t} {t \choose s}
\frac{x^{2s+n+1}}{\Gamma(n+s+3/2)\Gamma(t-s-n+1/2)}.
\end{align}
\end{subequations}
Given the power series, the derivatives at $k=0$ are readily found:
\begin{subequations}
\label{equ:derivatives}
\begin{align}
\J^{q}
\frac{\partial^{v}}{\partial k^{v}}
\left[
\E^{\J k r}\frac{\besselj{q+1}(k)}{k}
\right]_{k=0}
&=
\left\{
\begin{array}{ll}
0, & v < q;\\
\J^{v}v!E_{v-q,q}(r), & v \geq q.
\end{array}
\right.\\
\frac{\partial^{v}}{\partial k^{v}}
\left[
(a/r)^{1/2}
\hankel{n+1/2}(kr)\besselj{n+1/2}(ka)
\right]_{k=0}
&= \nonumber\\
\left\{
\begin{array}{lll}
\displaystyle
0, & v=2v'+1,& v' < n;\\
\displaystyle
\left(
\frac{r}{2}
\right)^{2n+1}
\left(
-\frac{r^{2}}{4}
\right)^{v'-n}
\frac{v!}{(v'-n)!}V_{n,v'-n}(a/r), & v=2v'+1, & v'\geq n;\\
\displaystyle
-(-1)^{n}\J \frac{(2v')!}{v'!}
\left(
-\frac{r^{2}}{4}
\right)^{v'}W_{n,v'}(a/r), &v=2v'.
\end{array}
\right.
\end{align}
\end{subequations}
Setting $v=0,1,\ldots$ yields an infinite lower triangular system of
equations for $u_{q}(r)$:
\begin{align}
\label{equ:system:1}
\mathsf{E}\mathbf{U} = \mathbf{B},
\end{align}
with $\mathbf{U}=[u_{0}\,u_{1}\,\ldots]^{T}$ and the elements of
matrix $\mathsf{E}$ and vector $\mathbf{B}$ given by:
\begin{subequations}
\label{equ:entries}
\begin{align}
E_{vq} &=
\left\{
\begin{array}{ll}
\J^{v}(q+1)v!E_{v-q,q}(r), & q\leq v;\\
0, & q > v.
\end{array}
\right.\\
B_{v} &=
\frac{\J}{4}\int_{0}^{1}T_{v}(r,a)s_{n}(a)\,\D a
\end{align}
\end{subequations}
where
\begin{align}
T_{v} &=
(-1)^{n+v'}
v!
\left(
\frac{r}{2}
\right)^{v}
\sum_{m=0}^{\infty}
A_{m}
\left\{
\begin{array}{lll}
\displaystyle
0 & v = 2v'+1, & v' < n+2m;\\
\displaystyle
\frac{V_{n+2m,v'-n-2m}(a/r)}{(v'-n-2m)!}
& v = 2v'+1, & v' \geq n+2m;\\
\displaystyle
- \frac{\J}{v'!}
W_{n+2m,v'}(a/r) & v =
2v'.
\end{array}
\right.
\label{equ:tfunc}
\end{align}
Given a radial source term $s_{n}(a)$, Equation~\ref{equ:system:1} can
be solved to find the coefficients $u_{q}(r)$ of the equivalent line
source modes. Since it is lower triangular, the first few values of
$u_{q}$ can be reliably estimated, although ill-conditioning prevents
accurate solution for arbitrary large $q$.
\subsection{Radiated field}
\label{sec:radiated}
From the relationship between the radial source term and the line
source coefficients, some general properties of the acoustic field can
be stated. The first result, already shown in previous
work~\cite{carley10b,carley10c} is that, since the line source modes
with $q+1>k$ generate exponentially small fields, the acoustic field
has no more than $k$ degrees of freedom, in the sense that the
radiated field is given by a weighted sum of the fields due to no more
than $k$ elementary sources. From Equation~\ref{equ:system:1}, this
result can be extended.
The first extension comes from the fact that $B_{2v+1}\equiv0$, for
$v'<n$, on the right hand side of Equation~\ref{equ:system:1}. This
means that $u_{q}$, $q=2v'+1$, is uniquely defined by the lower order
coefficients with $q\leq 2v'$. The result is that the acoustic field
of azimuthal order $n$, whatever might be its radial structure, has no
more than $k-n$ degrees of freedom, whether in the near or far field.
A second extension comes from examination of
Equation~\ref{equ:system:1}. The first few entries of the system of
equations are:
\begin{align}
\label{equ:system:a}
\left[
\begin{array}{rrrrr}
1/2 & 0 & 0 & 0 & \cdots \\
r/2 & 1/4 & 0 & 0 & \cdots \\
\vdots &\vdots & \vdots & 0 & \cdots
\end{array}
\right]
\left(
\begin{array}{c}
u_{0} \\ u_{1} \\ \vdots
\end{array}
\right)
=
\left(
\begin{array}{c}
B_{0} \\ 0 \\ \vdots
\end{array}
\right),
\end{align}
resulting in the solution:
\begin{align}
\label{equ:system:sol}
u_{0} = 2B_{0};\quad u_{1} = -2ru_{0} = -4rB_{0},
\end{align}
so that the ratio of $u_{0}$ and $u_{1}$ is constant, for arbitrary
$s_{n}(a)$. This means that low frequency sources of the same radius
and azimuthal order generate fields which vary only by a scaling
factor, since the higher order terms are exponentially small. Again,
this result holds in the near and in the far field.
Finally, if we attempt to isolate a source $s_{n}(a)$ associated with
a single line source mode, by setting $u_{q}\equiv1$ for some $q$,
with all other $u_{q}\equiv0$, we find that the line modes must occur
in pairs, since if $u_{2v'}\equiv1$, $u_{2v'+1}\neq0$, being fixed by
the condition $B_{2v'+1}\equiv0$, further reducing the number of
degrees of freedom or, alternatively, worsening the conditioning of
the inverse problem.
\subsection{Comparison to far-field methods}
\label{sec:far:field}
An alternative analysis which is widely used in radiation prediction
uses the far field approximations $R'\approx R - a\sin\phi\cos\psi$,
$1/R'\approx 1/R$. On this approximation:
\begin{align}
\label{equ:far:field}
P_{n} &\approx (-\J)^n\frac{\E^{\J k R}}{2R}
\int_{0}^{1}
J_{n}(ka\sin\phi) s_{n}(a) a\,\D a,
\end{align}
so that the radiated field is given by a Hankel transform of the
radial source, with a dependence on the polar angle $\phi$. In some
sense, this can also be viewed as fixing a limit on the radiated
information as in, for example, the use of ring sources to study
coherence effects on jet noise~\cite{michel09,michalke83}, or as a
spatial filter. The approach suffers, however, from its inability to
give information on the structure of the near field which might be of
use in understanding such experimental methods as near-field to
far-field correlations~\cite{laurendeau-jordan-delville-bonnet08}. The
approach presented in this paper gives the radiated field, near and
far, as the sum of products of two integrals. The first of these
integrals $\lfn{q}$ contains only radiation effects while the second
$u_{q}$ depends only on the source. The source and radiation terms are
thus `uncoupled', simplifying the problem of analysing the radiated
field, without needing to make a far-field approximation.
\section{Random disk source}
\label{sec:random}
The second problem considered is that of the noise radiated by a
random disk source. This is a general problem for broadband noise from
rotating systems and is also a model problem for jet noise, extending
the random ring source problem which has been studied previously in
order to examine the effects of source coherence on jet
noise~\cite{michalke83}. The assumptions made are that the source terms
are statistically stationary and that the statistical properties of
the source are symmetric about the source axis. It will also be
assumed that the non-dimensional wavenumber $k\lesssim2$, which is a
reasonable assumption for the frequency range of maximum noise level
for a subsonic jet. The result derived is an expression for the
cross-spectrum between the pressure at two points, which reduces to
the power spectrum when the points coincide. The expression is quite
general and, unlike previous formulae, does not require that the
points be in the acoustic far field of the source.
The starting point is an expression for the pressure radiated from a
source distributed over a unit disk:
\begin{align}
\label{equ:disk:1}
p(r,\theta,z,t) &= \int_{0}^{1}\int_{0}^{2\pi}
\frac{q(a,\psi,t-R/c)}{4\pi R}a\,\D \psi\,\D a,
\end{align}
from which the correlation between $p$ measured at two points
$(r_{1},\theta_{1},z_{1})$ and $(r_{2},\theta_{2},z_{2})$ is:
\begin{align}
\overline{p(r_{1},\theta_{1},z_{1},t)p(r_{2},\theta_{2},z_{2},t+\tau)}
&=\nonumber\\
\frac{1}{(4\pi)^{2}}
\int_{0}^{1}\int_{0}^{2\pi}
\int_{0}^{1}\int_{0}^{2\pi}
\frac{\overline{q(a_{1},\psi_{1},t-R_{1}/c)q(a_{2},\psi_{2},t-R_{2}/c+\tau)}}
{R_{1}R_{2}}a_{1}a_{2}
\,\D \psi_{1}\,\D a_{1}
\,\D \psi_{2}\,\D a_{2}.
\end{align}
Fourier transforming to find the cross spectrum between the points:
\begin{align}
\label{equ:xspec}
W_{12}(f) &= \frac{1}{(4\pi)^{2}}
\int_{0}^{1}\int_{0}^{2\pi}
\int_{0}^{1}\int_{0}^{2\pi}
\frac{\E^{\J k(R_{2}-R_{1})}}{R_{1}R_{2}}
Q_{12}(a_{1},\psi_{1};a_{2},\psi_{2}) a_{1}a_{2}
\,\D \psi_{1}\,\D a_{1}
\,\D \psi_{2}\,\D a_{2},\\
Q_{12}(a_{1},\psi_{1};a_{2},\psi_{2}) &= \int_{-\infty}^{\infty}
\overline{q(a_{1},\psi_{1},t)q(a_{2},\psi_{2},t+\tau)}\E^{\J 2\pi f
\tau}\,\D \tau,
\end{align}
where $Q_{12}$ is the correlation between the source at two points
$(a_{1},\psi_{1})$ and $(a_{2},\psi_{2})$, assumed real.
On the assumption of axial symmetry, the source correlation can depend
only on the angular separation between two points $\psi_{2}-\psi_{1}$,
so that $Q_{12}$ and $W_{12}$ can be expanded in Fourier series in
azimuth:
\begin{align*}
Q_{12}(a_{1},\psi_{1};a_{2},\psi_{2}) &=
\sum_{m=-\infty}^{\infty}Q_{12}^{(m)}(a_{1},a_{2})\E^{\J m
(\psi_{2}-\psi_{1})},\\
W_{12}(r_{1},\theta_{1},z_{1};r_{2},\theta_{2},z_{2})
&=
\sum_{m=-\infty}^{\infty}W_{12}^{(m)}(r_{1},z_{1};r_{2},z_{2})
\E^{\J m (\theta_{2}-\theta_{1})},
\end{align*}
with:
\begin{align}
\label{equ:xspec:2}
W_{12}^{(m)}(r_{1},\theta_{1},z_{1};r_{2},\theta_{2},z_{2})
&= \frac{1}{(4\pi)^{2}}
\int_{0}^{1}\int_{0}^{2\pi}
\frac{\E^{-\J (kR_{1}+m\psi_{1})}}{R_{1}}\\
&\times\left[
\int_{0}^{1}\int_{0}^{2\pi}
\frac{\E^{\J (kR_{2}+m\psi_{2})}}{R_{2}}
Q_{12}^{(m)}(a_{1},a_{2})
a_{2}
\,\D \psi_{2}\,\D a_{2}\,
\right]
a_{1}
\,\D \psi_{1}\,\D a_{1}. \nonumber
\end{align}
Transforming to the equivalent line source form, as above:
\begin{align*}
\frac{1}{4\pi}
\int_{0}^{1}\int_{0}^{2\pi}
\frac{\E^{\J (kR_{2}+m\psi_{2})}}{R_{2}}
Q_{12}^{(m)}(a_{1},a_{2})
a_{2}
\,\D \psi_{2}\,\D a_{2}
&=
\sum_{q_{2}=0}^{\infty}
u_{q_{2}}(r_{2},a_{1})\lfn{q_{2}}(k,r_{2},z_{2}),
\end{align*}
which results in:
\begin{align*}
W_{12}^{(m)} &= \sum_{q_{2}} \lfn{q_{2}}(k,r_{2},z_{2})
\frac{1}{4\pi}
\int_{0}^{1}\int_{0}^{2\pi}
\frac{\E^{-\J (kR_{1}+m\psi_{1})}}{R_{1}}
u_{q_{2}}(r_{2},a_{1})
a_{1}\,\D a_{1}\,\D\psi_{1},\\
&= \sum_{q_{1}}\sum_{q_{2}} u_{q_{1}}(r_{1},r_{2})
\lfn{q_{2}}(k,r_{2},z_{2})\lfn{q_{1}}^{*}(k,r_{1},z_{1}),
\end{align*}
where $*$ denotes complex conjugation. The coefficients $u_{q_{1}}$
are found by treating $u_{q_{2}}$ as the radial source in the
$(a_{1},\psi_{1})$ integral. Up to this point, the analysis is exact
but to simplify the development, we introduce the assumption $k<2$ so
that only modes of order~0 and~1 contribute to the acoustic field.
Solving Equation~\ref{equ:system:1} yields:
\begin{align*}
u_{0} &= 2 B_{0},\\
u_{1} &= 4(B_{1} - rB_{0}),
\end{align*}
with:
\begin{align*}
B_{0} &= \int_{0}^{1}s_{m}(a)w_{m}(a/r)\,\D a,\\
B_{1} &= \int_{0}^{1}s_{m}(a)v_{m}(a)\,\D a,
\end{align*}
where:
\begin{align*}
w_{m}(x) &= \frac{1}{2\pi}\sum_{q=0}^{\infty}
\frac{1}{q!} \frac{(2m+2q-1)!!(2q-1)!!}{2^{q}(2m+2q)!!}x^{m+2q+1},\\
v_{m}(x) &=
\left\{
\begin{array}{ll}
x/2\pi, & m = 0;\\
0, & m \neq 0.
\end{array}
\right.
\end{align*}
The result is that the $m$th azimuthal component of the cross-spectrum
between two field points for $k\lesssim2$ is given by:
\begin{align}
W_{12}^{(m)} &=
\lfn{0}(k,r_{2},z_{2})
\left[
u_{00}\lfn{0}^{*}(k,r_{1},z_{1}) +
u_{01}\lfn{1}^{*}(k,r_{1},z_{1})
\right]
\nonumber\\
\label{equ:xspec:3}
&+ \lfn{1}(k,r_{2},z_{2})
\left[
u_{10}\lfn{0}^{*}(k,r_{1},z_{1}) +
u_{11}\lfn{1}^{*}(k,r_{1},z_{1})
\right],
\end{align}
where
\begin{subequations}
\label{equ:uij}
\begin{align}
u_{00} &=
4\int_{0}^{1}\int_{0}^{1}
Q_{12}^{(m)}(a_{1},a_{2})w_{m}(a_{1}/r_{1})w_{m}(a_{2}/r_{2})\,\D
a_{1}\, \D a_{2},\\
u_{01} &=
8\int_{0}^{1}\int_{0}^{1}
Q_{12}^{(m)}(a_{1},a_{2})w_{m}(a_{2}/r_{2})
[v_{m}(a_{1}) - r_{1}w_{m}(a_{1}/r_{1})]
\,\D
a_{1}\, \D a_{2},\\
u_{10} &=
8\int_{0}^{1}\int_{0}^{1}
Q_{12}^{(m)}(a_{1},a_{2})w_{m}(a_{1}/r_{1})
[v_{m}(a_{2}) - r_{2}w_{m}(a_{2}/r_{2})]
\,\D
a_{1}\, \D a_{2},\\
u_{11} &=
16\int_{0}^{1}\int_{0}^{1}
Q_{12}^{(m)}(a_{1},a_{2})
[v_{m}(a_{1}) - r_{1}w_{m}(a_{1}/r_{1})]
[v_{m}(a_{2}) - r_{2}w_{m}(a_{2}/r_{2})]
\,\D
a_{1}\, \D a_{2}.
\end{align}
\end{subequations}
The modal coefficients of the cross-spectrum of a jet noise field, at
the wavenumbers of interest in practice, are thus fixed by four
coefficients, functions of the radial separations $r_{1}$ and $r_{2}$,
which are weighted integrals of the source cross-spectrum.
\section{Results}
\label{sec:results}
To check the analyses presented above, and to consider their
implications, some results are presented for tonal and random disk
sources.
\subsection{Line source coefficient evaluation}
\label{sec:coefficient}
\begin{figure}
\centering
\begin{tabular}{c}
\includegraphics{jasa10b-figs-3} \\
$n=2$ \\
\includegraphics{jasa10b-figs-5} \\
$n=16$
\end{tabular}
\caption{Line source mode coefficients computed using the method of
section~\ref{sec:coefficients} (solid lines) and directly from
analytical formulae (symbols) for $r=5/4$, $s=r_{1}^{\gamma}$,
$\gamma=0$ (circles), $\gamma=2$ (squares) and $\gamma=4$
(diamonds) for $n=2$ and $16$.}
\label{fig:cfft:compare}
\end{figure}
The first results are a check on the calculation of the coefficients
$u_{q}(r)$ comparing those computed using Equation~\ref{equ:system:1}
and those computed directly from exact closed-form
expressions~\cite{carley99} for $K(r,r_{2})$ in the case when the
radial source term is a monomial in radius
$s_{n}=r_{1}^{\gamma}$. Figure~\ref{fig:cfft:compare} compares the two
sets of coefficients for $\gamma=0,2,4$, with the plots terminated at
a value of $q$ where the difference between the two sets of results
becomes noticeable, $q\approx20$. This gives an indication of the
effect of the ill conditioning of Equation~\ref{equ:system:1}. For
$q\lesssim 20$, the computed values of $u_{q}$ are reliable. It is
noteworthy that for small $q$, the coefficients are practically equal
for all values of $\gamma$ so that for low frequency radiation, the
radiated fields will be practically indistinguishable.
\subsection{Tonal radiation from a disk}
\label{sec:disk}
\begin{figure*}
\centering
\begin{tabular}{cc}
\includegraphics{jasa10b-figs-18} &
\includegraphics{jasa10b-figs-19}
\end{tabular}
\caption{Acoustic field predicted by full numerical integration
(lines) and line source summation (symbols) for $n=8$,
$r=5/4$. Real part shown solid, imaginary part dashed. Left hand
plot: $k=5$; right hand plot: $k=9$.}
\label{fig:disk}
\end{figure*}
As a test of the ability to predict radiation from tonal sources, we
present data for the acoustic field of a disk source with $n=8$,
$s_{n}=J_{n}(a_{n1}a)$, where $a_{n1}$ is the first non-zero root of
$J_{n}(x)$. Full numerical integration and line source calculations
have been performed for two wavenumbers, $k=5$ and $k=9$,
respectively. The first~11 line source modes were used in each case,
with the modal coefficients being found from
Equation~\ref{equ:system:1}. Sample results are shown in
Figure~\ref{fig:disk}, with the data scaled on the value at $z=0$, and
it is clear that the line source model gives accurate results, even
when only a subset of the modes is used. From these, and other, data,
the reliability of the model for tonal sources is confirmed.
\subsection{Low frequency random source}
\label{sec:low:random}
In order to generate data to test the random disk source model, we
must assume a form for the source correlation.
Michalke~\cite{michalke83} gives a form suitable for a ring source
which meets the symmetry requirements laid out above. With the
addition of radially varying terms, Michalke's expression can be
extended:
\begin{align}
\label{equ:coherence:1}
Q_{12}(a_{1},\psi_{1};a_{2},\psi_{2}) &=
q(a_{1})q(a_{2})
\exp
\left[
-\frac{(a_{1}-a_{2})^{2}}{\beta^{2}}
\right]
\exp
\left[
-\frac{1-\cos(\psi_{1}-\psi_{2})}{\alpha^{2}}
\right]
\end{align}
with $\alpha$ being an azimuthal length scale $\beta$ controlling the
correlation in radius. Equation~\ref{equ:coherence:1} can be
interpreted as the product of the local source strengths $q(a_{1})$
and $q(a_{2})$ with a coherence function, given by the exponentials,
which is symmetric in source position and has unit value when the
source points coincide.
The azimuthal components of $Q_{12}$ can be found from mathematical
tables~\cite{gradshteyn-ryzhik80,michalke83} as:
\begin{align}
\label{equ:coherence:2}
Q_{12}^{(m)} &= q(a_{1})q(a_{2})
\exp
\left[
-\frac{(a_{1}-a_{2})^{2}}{\beta^{2}}
\right]
\exp
\left[
-\frac{1}{\alpha^{2}}
\right]
I_{m}(1/\alpha^{2})
\end{align}
where $I_{m}$ is a modified Bessel function.
\begin{figure*}
\centering
\begin{tabular}{cc}
\includegraphics{jasa10b-figs-10} &
\includegraphics{jasa10b-figs-12} \\
\textit{a} & \textit{b} \\
\includegraphics{jasa10b-figs-15} &
\includegraphics{jasa10b-figs-17} \\
\textit{c} & \textit{d}
\end{tabular}
\caption{Cross-spectrum $W_{12}^{(m)}(z_{2})$ scaled on
$W_{12}^{(m)}(0)$, $r_{1}=5/4$, $z_{1}=0$, $r_{2}=5$. Numerical
evaluation shown as solid line (real part) and dashed line
(imaginary part); Equation~\ref{equ:xspec:3} with numerical
evaluation of $L_{q}$ shown as circles; Equation~\ref{equ:xspec:3}
with far-field approximation shown as squares. Parameters:
\textit{a}: $k=1$, $m=0$, $\alpha=1$, $\beta=100$;
\textit{b}: $k=1$, $m=0$, $\alpha=3$, $\beta=0.01$;
\textit{c}: $k=2$, $m=1$, $\alpha=1$, $\beta=100$;
\textit{d}: $k=2$, $m=1$, $\alpha=3$, $\beta=0.01$.}
\label{fig:xspec}
\end{figure*}
Figure~\ref{fig:xspec} shows sample results for the predicted cross
spectrum between pressure at a point $r_{1}=5/4$, $z_{1}=0$ and
$r_{2}=5$, $0\leq z_{2}\leq8$, for a disk source of unit strength. The
reference results are the cross-spectra found by full numerical
integration of Equation~\ref{equ:xspec:2}. The first comparison is
with Equation~\ref{equ:xspec:3} where the functions $L_{q}$ have been
evaluated by numerical integration. In the second comparison, the
functions $L_{q}$ have been evaluated using the exact in-plane result,
Equation~\ref{equ:line:series}, for $z_{1}=0$, and the far-field
approximation, Equation~\ref{equ:lfunc:ff}, for $r_{2}=5$, $0\leq
z_{2}\leq 8$. All data have been scaled on the numerically evaluated
cross-spectrum at $z_{2}=0$.
The first obvious point from Figure~\ref{fig:xspec} is the similarity
of the cross-spectra, even for quite large variations in the parameter
$\beta$: changing $m$ changes the form of the radiated field, as might
be expected, but changes in the source correlation have little effect
on the radiated field. The second point is that the line source
approach gives very good results, even for $k=2$ where, in principle,
the approximation used should start to break down. Finally, although
computational efficiency is not the primary aim of the method, we note
that the line source approach converts the four dimensional integral,
Equation~\ref{equ:xspec:2}, required at each field point, into four
two-dimensional integrals which are functions of radial separation
only, Equation~\ref{equ:uij}, and four one-dimensional integrals
$\lfn{i}$, giving a large saving in calculation time.
\subsection{Noise cancellation by an equivalent source}
\label{sec:cancellation}
One implication of the results of this paper is that it is not
possible to tell different sources apart if, to within a scaling
factor, they have same line source coefficients $u_{q}$, for those
line source modes with $q<k$. Even without considering errors from
background noise or other causes, this is equivalent to a condition on
weighted integrals of the radial source $s_{n}$. Any sources which
yield the same, or nearly the same, integrals $B_{v}$ for $v<V$, with
$V$ a positive integer, in Equation~\ref{equ:entries}, will have
indistinguishable acoustic fields for $k<V$.
This conclusion can also be read as a statement about noise
cancellation, such as in active noise control. The acoustic field of a
given source can be cancelled by any source which has the same set of
line source coefficients.
\begin{figure}
\centering
\includegraphics{jasa10b-figs-20} \\
\includegraphics{jasa10b-figs-21}
\caption{Cancellation effects for radial source terms with $n=2$,
$k=1$, $r=5/4$: top figure radiated field from original
$s_{n}(a)$ (solid) and modified source $s_{n}(a)-\zeta
s_{n}'(a)$ (dashed); bottom row source terms $s_{n}(a)$ (solid)
and $\zeta s_{n}'(a)$ dashed.}
\label{fig:field:compare}
\end{figure}
An example of this cancellation is shown in
Figure~\ref{fig:field:compare}. The original field is generated using
a source term $s_{n}(a)$ and the line source coefficients $u_{q}$ of
$s_{n}$ are calculated. A secondary source term $s_{n}'(a)$ is
generated and its line source coefficients $u_{q}'$ are computed. The
secondary source $s_{n}'$ is then scaled by a factor
$\zeta=u_{0}/u_{0}'$. As a test, $s_{n}=J_{n}(a_{n2}a)$, with $a_{n2}$
the second extremum of $J_{n}(x)$, and $s_{n}'\equiv1$. The first plot
in Figure~\ref{fig:field:compare} shows the field due to $s_{n}$ and
that radiated by $s_{n}-\zeta s_{n}'$. The large reduction,
20\deci\bel, in the radiation near the source plane is obvious,
although there is a small increase in the noise field around
$z=1$. The source terms are shown in the second plot of
Figure~\ref{fig:field:compare}. The secondary source $\zeta s_{n}'$ is
of much smaller amplitude than $s_{n}$ even though it generates a
nearly-equivalent field: the effect of matching the line source
coefficients has been to produce a field which is very similar to that
of the original source, even though the source distributions are quite
different in form and in amplitude.
\section{Conclusions}
\label{sec:conclusions}
The radiation properties of disk sources of arbitrary radial variation
have been analyzed to establish the part of the source which radiates
into the acoustic field, without recourse to a far field
approximation. Limits have been established on the number of degrees
of freedom of the part of the source which radiates and the
implications of these limits have been discussed for the problems of
rotor noise and studies of source mechanisms in jets. The analysis has
been developed for tonal and for random sources, with implications for
applications in active control of noise from rotors and experimental
analysis of jet noise sources.
|
{
"timestamp": "2010-09-21T02:02:41",
"yymm": "1009",
"arxiv_id": "1009.3748",
"language": "en",
"url": "https://arxiv.org/abs/1009.3748"
}
|
\section{Introduction}
During the past decade the gravitational average action \cite{mr} has been
used both as a framework within which the asymptotic safety scenario for a
consistent microscopic quantum theory of gravity can be tested \cite{mr}-\cite{livrev} and
as a convenient tool for finding the leading quantum gravity corrections to
various classical spacetimes. The latter investigations exploited the
effective field theory properties of the average action $\Gamma _{k}$ in an
essential way. It can be regarded as a one parameter family of effective
field theories, one for each value of the built-in infrared cutoff $k$ \cite
{avact}-\cite{ymrev}. In single scale problems involving a typical covariant
momentum scale $k$ a tree-level evaluation of $\Gamma _{k}$ encapsulates the
leading quantum effects at this scale. Thanks to this property the running
couplings contained in $\Gamma _{k}$ can be used in order to
``renormalization group improve'' the classical field equations or solutions
thereof \cite{bh1}-\cite{mof}.
The possibility of interpreting $\Gamma _{k}$ as a ``running effective field
theory'' distinguishes the effective average action \cite{avact} from
alternative functionals satisfying exact renormalization group (RG)
equations. The functional evolved by Polchinski\'{}s equation, for instance,
has the interpretation of a bare action. Therefore
it cannot be used for ``improvement'' purposes in the same way \cite{avactrev}.
\bigskip
Knowing the gravitational average action with some accuracy (i.e. , in some
truncation) means that we know the scale dependence of a set of generalized
gravitational couplings; typically it includes Newton\'{}s constant, for instance.
These running couplings can be used in order to ``RG improve'' classical spacetimes.
The basic idea is as follows. One starts by picking a solution of the classical field equation.
This solution will in general depend on the classical gravitational couplings. Then one
replaces the classical ones by their $k$-dependent counterparts and tries to
express the value of $k$ by means of a ``cutoff identification'' in terms of
the relevant geometrical or dynamical scale.
\bigskip
In refs. \cite{bh2,evap} this approach has been applied to stationary and
spherically symmetric, uncharged black holes. The classical starting point
was the Schwarzschild metric which involves the classical Newton\'{}s constant
G_{0}$ in the familiar way. The improvement consisted in replacing the
classical $G_{0}$ by the running Newton\'{}s constant $G\left( k\right) $
obtained from the functional \ RG equation for the effective average action.
A subtle point is finding a suitable cutoff identification. It should be
chosen in such a way that higher values of $k$ correspond to a ``zooming''
into the details of the black hole. One can try to find a meaningful
identification in the form $k=k\left( \mathcal{P}\right) $ which associates scales to
spacetime points $\mathcal{P}$. It is plausible that this map should be such that $k$
is smaller (larger) at larger (smaller) distances from the center of the
black hole. In the analogous situation in flat space one would set
k \propto 1/r$ where $r$ is the radial distance; with this identification
one can obtain the quantum corrected Coulomb potential from the $k
-dependence of the fine structure constant, for instance. In gravity the
assignment of scales to points should be diffeomorphism invariant, i.e. upon
introducing coordinates $x^{\mu }$ the relationship $k=k\left( \mathcal{P}\right) $
should be represented by a \textit{scalar} function $x^{\mu }\mapsto k\left(
x^{\mu }\right) $. In \cite{bh2,evap} the following class of cutoff
identification was considered:
\begin{equation}
k\left( \mathcal{P}\right) =\xi /d\left( \mathcal{P}\right) \label{1.1}
\end{equation}
\begin{equation}
d\left( \mathcal{P}\right) =\int_{\mathcal{C}}\sqrt{\left| ds^{2}\right| } \label{1.1B}
\end{equation}
Here $\xi $ is a constant of order unity and $d\left( \mathcal{P}\right) $ is a
distance scale typical of the point $\mathcal{P}$. According to (\ref{1.1B}) it is
given by the length of a certain curve $\mathcal{C}$. This curve is supposed to end at
$\mathcal{P}$, and to start at some reference point $\mathcal{P}_{0}$. The line element $ds^{2}$
refers to the classical metric. While diffeomorphism invariant by
construction, the above ansatz is still very general and different choices
are possible for $\mathcal{C}$. They correspond to different ways of ``re-focusing''
the ``microscope'' with which spacetime is observed when one goes from one
point to another. In refs. \cite{bh2, evap} a straight radial line from the
center to the point $\mathcal{P}$ has been employed, and this choice has been
motivated in detail. The only running parameter considered in this analysis
was Newton\'{}s constant. Its $k$-dependence had been assumed to be given by the formula
\begin{equation}
G\left( k\right) =\frac{G_{0}}{1+wG_{0}k^{2}} \label{1.2}
\end{equation}
Here $G_{0}$ is the classical (macroscopic) Newton\'{}s constant, and $w$ is a
positive constant. This equation is a rather precise approximation to
G\left( k\right) $ as obtained from the Einstein-Hilbert truncation \cite{mr}
for all RG trajectories with a negligible cosmological constant in the
classical regime. According to (\ref{1.2}), the running Newton\'{}s constant
interpolates between $G_{0}$ for $k\rightarrow 0$ and the non-Gaussian fixed
point behavior $G\left( k\right) \propto 1/k^{2}\rightarrow 0$ for
k\rightarrow \infty $. With (1.1) inserted into (\ref{1.2}) we obtain the
position dependent Newton\'{}s constant
\begin{equation}
G\left( \mathcal{P}\right) =\frac{G_{0}d^{2}\left( \mathcal{P}\right) }{d^{2}\left( \mathcal{P}\right)
\bar{w}G_{0}} \label{1.3}
\end{equation}
with $\bar{w}=w\xi ^{2}$.
\bigskip
The RG improved Schwarzschild metric was obtained by replacing
G_{0}\rightarrow G\left( \mathcal{P}\right) $ in the classical metric. It has been
analysed in great detail in \cite{bh2}. In particular, its horizon structure
was investigated. One finds that besides the usual Schwarzschild horizon
there exists a new inner horizon which merges with the (standard) outer one
at a critical value of the mass. An ``extremal'' black hole of this kind has
vanishing Hawking temperature. In fact the improvement suggests a very
attractive scenario for the final state of black hole evaporation: In the
early stages the temperature increases with decreasing mass, as predicted by
the conventional semiclassical analysis. However, once the mass approaches
the Planck mass, the quantum gravity effects reduce the temperature, and
ultimately ``switch off'' the Hawking radiation. For further details on the
RG improved Schwarzschild black hole we refer to \cite{bh2} and to \cite{evap}
where a dynamical picture of the evaporation process by means of a quantum
corrected Vaidya metric has been developed. The generalization to higher dimensions was considered in \cite{Falls}.
\bigskip
The purpose of the present paper is to perform a similar analysis for
rotating black holes. We shall construct and analyse an RG-improved version
of the Kerr metric. In Boyer-Lindquist coordinates the classical Kerr metric
reads \cite{BoyerL}
\begin{equation}
ds_{\text{class}}^{2}=-\left( 1-\frac{2MG_{0}r}{\rho }\right) dt^{2}+\frac
\rho ^{2}}{\Delta }dr^{2}+\rho ^{2}d\theta ^{2}+\frac{\Sigma \sin ^{2}\theta
}{\rho ^{2}}d\varphi ^{2}-\frac{4MG_{0}ra\sin ^{2}\theta }{\rho ^{2}
dtd\varphi \label{1.4}
\end{equation}
Here we used the traditional abbreviations
\begin{equation}
\rho ^{2}\equiv r^{2}+a^{2}\cos ^{2}\theta \label{1.5}
\end{equation}
\begin{equation}
\Delta \equiv r^{2}+a^{2}-2MG_{0}r \label{1.6}
\end{equation}
\begin{equation}
\Sigma \equiv \left( r^{2}+a^{2}\right) ^{2}-a^{2}\Delta \sin ^{2}\theta
\label{1.7}
\end{equation}
Kerr black holes are characterized by two parameters, their mass $M$ and
angular momentum $J=aM$ \cite{Kerr63,Cohen,Carter}.
\bigskip
Applying the method outlined above we shall ``improve'' $ds_{\text{class
}^{2}\equiv g_{\mu \nu }^{\text{class}}dx^{\mu }dx^{\nu }$ by replacing
G_{0}\rightarrow G\left( k\right) $ and using a cutoff identification of the
type (\ref{1.1}). To start with, we are going to analyse various plausible curves $\mathcal{C}$, including
a straight radial line again, and discuss their physical properties.
\bigskip
The classical Kerr spacetime has two spherical horizons $H_{\pm }$ at the
radii \cite{Adler,Taylor}
\begin{equation}
r_{\pm }=m\pm \sqrt{m^{2}-a^{2}} \label{1.8}
\end{equation}
and two static limit surfaces $S_{\pm }$ at
\begin{equation}
r_{S_{\pm }}\left( \theta \right) =m\pm \sqrt{m^{2}-a^{2}\cos ^{2}\theta }
\label{1.9}
\end{equation}
(Here
\begin{equation}
m\equiv MG_{0} \label{1.10}
\end{equation}
denotes the ``geometric mass'' which actually has the dimension of a
length.) We shall discuss in detail the analogous critical surfaces
(horizons and static limit surfaces) of the improved metric. In particular
we demonstrate that, contrary to the Schwarzschild case, the improvement
does not lead to the formation of additional horizons.
\bigskip
As compared to the Schwazschild metric, the Kerr spacetime displays several
new features which are interesting from a conceptual point of view. One of
them is the existence of an ergosphere and the possibility of extracting
energy from the black hole via the Penrose process \cite{Taylor,MTW,Christ1}. We shall analyse in
detail how the quantum gravity effects influence the structure of the
ergosphere and the ``phase space'' available for the Penrose process.
Another new feature of the Kerr spacetime becomes apparent when one asks
whether the improved black holes still satisfy a set of (quantum corrected) laws of
black hole thermodynamics. In full generality this is an extremely difficult
question. Here we can only analyze whether there exists an entropy-like
state function satisfying a modified version of the first law. In the case
of Kerr black holes the space of states, labeled by $M$ and $J$, is
2-dimensional. As a result, it turns out that the mere \textit{existence} of an
entropy is a non-trivial issue. (For the Schwarzschild metric the space is
1-dimensional and so the existence of an entropy for the improved black hole
is guaranteed.) We shall see that, within the present approach, a state
function with the interpretation of an entropy can exist only if the
corresponding Hawking temperature is no longer proportional to the surface
gravity, as it is semiclassically. At least in the limit of small angular
momentum we shall find unambiguously defined relations $T=T\left( J,M\right)
$ and $S=S\left( J,M\right) $ for the temperature and entropy of the
improved rotating black holes.
\bigskip
The remaining sections of this paper are organized as follows. In section 2
we discuss the cutoff identification we are going to employ, and in section
3 we introduce the RG improved Kerr metric and analyse some of its general
properties; in particular we derive formulas for the modified static
limit and horizon surfaces, we reexpress the metric in a set of
appropiately generalized Eddington-Finkelstein coordinates, and compute the
surface gravity of the rotating quantum black holes. Then, in section 4 and
5 we analyse the detailed structure of the critical surfaces and the phase
space of the Penrose mechanism (negative energy states), respectively, usign
both analytical and numerical methods. In section 6 we reinterprete the
improved vacuum black hole as a classical one in presence of a certain kind
of fictitious matter which mimicks the quantum effects, and we investigate
the positivity properties of this matter system. In section 7 we show how
the ``bare'' mass and angular momentum of these black holes get ``dressed''
by the quantum effects in according with the antiscreening character of
Quantum Einstein Gravity. Finally, in section \ref{Seccion 8} we take a first step towards
an RG improved black hole thermodynamics; in particular we derive a modified
first law satisfied by the improved Kerr black holes. Section 9 contains a
summary of the results.
\bigskip
\section{The cutoff identification}
After replacing $G_{0}\rightarrow G\left( k\right) $ we would like to
express the scale $k$ as a scalar function on spacetime so that Newton\'{}s constant becomes position dependent:
\begin{equation}
G\left( r,\theta \right) \equiv G\Bigl( k=k\left( r,\theta \right) \Bigr)
\label{2.1}
\end{equation}
Here we have indicated that for symmetry reasons $k$ and $G$ can depend on
the Boyer-Lindquist coordinates $r$ and $\theta $ only. The classical
spacetime is stationary and invariant under rotations about the $z$-axis ;
we require that the corresponding Killing vectors \cite{Poisson,MTW}
\begin{equation}
\mathbf{t}\equiv t^{\mu }\partial _{\mu }=\frac{\partial }{\partial t},\
\boldsymbol{\varphi}\equiv \varphi ^{\mu }\partial _{\mu }=\frac{\partial }
\partial \varphi } \label{2.2}
\end{equation}
are Killing vectors of the improved metric, too. If $G\left( x^{\mu }\right) $
is annihilated by $\mathbf{t}$ and $\boldsymbol{\varphi}$ this is indeed the case. In the
Boyer-Lindquist (BL) system this means that $G=G\left( r,\theta \right) $.
\bigskip
When an explicit form of the ``RG trajectory'' $G=G\left( k\right) $ is
needed we shall use the relationship (\ref{1.2}). However, for our mostly
qualitative discussion the precise details of this function are not
important. What matters is only that it smoothly interpolates between $G= const$ in the infrared $\left( k\rightarrow 0\right) $ and $G\left( k\right)
\propto 1/k^{2}$ in the ultraviolet $\left( k\rightarrow \infty \right) $.
Furthermore, we assume, as in the previous analyses \cite{bh2, evap} that
k\left( \mathcal{P}\right) =\xi /d\left( \mathcal{P}\right) $, which is given by the integral (\ref{1.1B}). In the case at hand it reads
\begin{equation}
d\left( r,\theta \right) =\int_{\mathcal{C}\left( r,\theta \right) }\sqrt{\left|
ds^{2}\right| } \label{2.3}
\end{equation}
where $\mathcal{C}\left( r,\theta \right) $ is a path associated to the point $\mathcal{P}$ with
BL coordinates $\left( t,r,\theta ,\varphi \right) $. By stationarity and
axial symmetry, $\mathcal{C}$ and $d$ must not depend on $t$ and $\varphi$. The line
element $ds^{2}$ in (\ref{2.3}) is the one of the classical Kerr metric.
\bigskip
The choice for $\mathcal{C}$ which appears most natural is a radial path from the
origin to $\mathcal{P}$. Along this path, $dt=d\theta =d\varphi =0$ and, by (\ref{1.4}),
$ds^{2}=\left( \rho ^{2}/\Delta \right) \;dr^{2}$. Hence we have in this
case
\begin{equation}
d\left( r,\theta \right) =\int_{0}^{r}d\bar{r}\sqrt{\left| \frac{\bar{r
^{2}+a^{2}\cos ^{2}\theta }{\bar{r}^{2}+a^{2}-2m\bar{r}}\right| }
\label{2.4}
\end{equation}
This integral is easy to perform only in the equatorial plane, i.e. for
\theta =\pi /2$. One obtains
\begin{equation}
d\left( r\right) \equiv d\left( r,\pi /2\right) =
\begin{cases}
d_{1}\left( r\right) & \text{ if }r<r_{-} \\
d_{2}\left( r\right) & \text{ if\ }r_{-}<r<r_{+} \\
d_{3}\left( r\right) & \text{ if\ }r_{+}<r
\end{cases}
\label{2.5}
\end{equation}
where $r_{\pm }$ are the radii of the classical horizons given in (\ref{1.8}), and \cite{Tesis}
\begin{eqnarray}
d_{1}\left( r\right) &=&\sqrt{r^{2}+a^{2}-2mr}+m\ln \left( \frac{-r+m-\sqrt
r^{2}+a^{2}-2mr}}{\left| a-m\right| }\right) -a \label{2.6} \\
d_{2}\left( r\right) &=&\frac{m}{2}\ln \left| \frac{m+a}{m-a}\right| -a
\sqrt{2mr-r^{2}-a^{2}} \\
&&+m\arctan \left( \frac{r-m}{\sqrt{2mr-r^{2}-a^{2}}}\right) +\frac{m\pi }{2} \nonumber
\\
d_{3}\left( r\right) &=&\sqrt{r^{2}+a^{2}-2mr}+m\ln \left( r-m+\sqrt
r^{2}+a^{2}-2mr}\right) + \\
&&\pi m-a-m\ln \left| m-a\right| \nonumber
\end{eqnarray}
\centerline{}
\begin{center}
\begin{pspicture}(-1.2,2)(4.2,7.5)
\includegraphics{dr_Kerr_2.tex_gr1.eps}
\rput[l]{90}(-11.6,5.6){$d\left(r\right)$}
\rput[l](-10.25,8.6){$\theta=90^{\circ}$}
\rput[l](-8.25,8.6){$m=10$}
\rput[l](-6.85,7.6){$a \approx 10$}
\rput[l](-7.2,7.2){\tiny{(Extremal Case)}}
\rput[l](-6.95,6.9){\tiny{($r_-=r_+$)}}
\psline[linestyle=dashed,linewidth=.5pt](-8.18,3.1)(-8.18,3.5)
\psline[linestyle=dashed,linewidth=.5pt](-4.6,3.1)(-4.6,6)
\rput[l](-8.18,2.8){$r_-$ \tiny{(for $a=9$)}}
\rput[l](-4.6,2.8){$r_+$ \tiny{(for $a=9$)}}
\rput[l](-4.75,6.6){$a= 9$}
\rput[l](-4.45,5.6){$a=8$}
\rput[l](-2,5.6){$a=0$}
\rput[l](-3.3,4.1){$d\left(r\right)=r$}
\rput[l](-5.5,1.9){$r$}
\rput[l](-12.5,0.8){Fig. 1:}
\rput[l](-10.8,0.8){The radial distance $d\left(r\right)$ in the equatorial plane for $m=10$}
\rput[l](-10.8,0.3){and various values of $a$. All quantities are expressed in Planck}
\rput[l](-10.8,-.2){units. The grayscale runs from black to gray for increasing $a$.}
\end{pspicture}
\end{center}
\centerline{}
\centerline{}
\centerline{}
\centerline{}
The function $d\left( r\right) $ for the equatorial plane is displayed in
Fig. 1 for a black hole with a mass of $10m_{\rm Pl}$ and for various values of
the angular momentum parameter $a$. (In this and the following figures all
dimensionful quantities are expressed in units of the Planckian quantities
formed with the infrared value of Newton\'{}s constant, $\ell_{\rm Pl}=m_{\rm Pl}^{-1}=\sqrt{G_{0}}$. Since $a,m,r$ and $d\left(
r\right) $ have the dimension of a length they are measured in units of
\ell_{\rm Pl}$. As $m\equiv G_{0}M$ by definition, the geometric mass $m$ equals
the actual mass $M$ when Planck units are used.)
The main features of the $d\left( r\right) $ curves are as follows. For $a<m$
not too close to the extreme case $a=m$, the curves run essentially parallel
to the dashed line in Fig. 1, representing the function $d\left( r\right) =r
. At their respective values of $r_{-}$ and $r_{+}$, all curves have a
vertical tangent. Near the classical horizon radii $r_{\pm }$ the functions
d\left( r\right) $ shift away from the $d\left( r\right) =r$ - line by a
kind of smoothed-out step function. At a sufficient distance from $r_{\pm }$
they run parallel to $d\left( r\right) =r$. In particular for $r>>r_{+}$ the
exact $d\left( r\right) $ is approximately of the form $d\left( r\right)
\approx r+\Delta d$ where $\Delta d$ is a constant independent of $r$.
Obviously, for $r$ large enough so that $\Delta d/r<<1$, we can approximate
the $d\left( r\right) $ curves simply by $d\left( r\right) =r$. For smaller
r$ there is the step-like behavior near $r_{-}$ and $r_{+}$, but in most of
our qualitative investigations it will not play a role. The deviations from
d\left( r\right) =r$ become significant when $a$ approaches $m$ which
corresponds to the situation of an extremal classical black hole.
For $\theta \neq \pi /2$ it is easy to evaluate the integral (\ref{2.4})
numerically. It turns out that $d\left( r,\theta \right) $ has a similar $r
-dependence for all values of $\theta $. For $\theta <\pi /2$ the shift
\Delta d$ is somewhat larger than at the equator, but nevertheless all
curves are essentially parallel to $d\left( r\right) =r$ again.
A more precise asymptotic analysis of the integral (\ref{2.4}) reveals that
d\left( r,\theta \right) $ has the following structure for $r\rightarrow
\infty $:
\begin{equation}
d\left( r,\theta \right) =r+m\ln \left( r\right) +F\left( \theta \right)
+O\left( \frac{1}{r}\right) \label{2.7}
\end{equation}
There are three types of terms which do not vanish for $r\rightarrow \infty
: a linearly increasing one, a logarithmically increasing one, and one which
is $r$-independent. Among the three, only the $r$-independent one depends on
the angle $\theta $. Since $F\left( \theta \right) $ is subdominant we see
that, to logarithmic accuracy, $d\left( r,\theta \right) $ is actually
independent of $\theta $ at large $r$.
An alternative definition of the distance scale $d\left( r,\theta \right) $
could be as follows \cite{Taylor}. Let $\mathcal{C}\left( r,\theta \right) $ be a circular path of
coordinate radius $r$, contained in the $\theta =$const plane and centered
about the origin. In this case we define $d\left( r,\theta \right) $ to be
the \textit{reduced circumference} of this path, i.e.
its proper length divided by $2\pi $. In flat space the reduced
circumference would equal $r$; in the Kerr background there are
corrections. A detailed numerical analysis \cite{Tesis} shows that, for $r$ not too
small, and $a$ not too close to $m$, the resulting distance functions
d\left( r,\theta \right) $ have similar qualitative properties as those fom
the radial path.
For concreteness we shall use the distance function obtained from the radial
path whenever a concrete expression is needed. Since our analysis is mostly
at a qualitative or ``semi-quantitative'' level we shall be concerned with
leading order effects only. For this reason we shall neglect the subdominant
$\theta $-dependence of $d\left( r,\theta \right) $ and assume that $d\equiv
d\left( r\right) $ and, as a result $G,$ depends on $r$ only:
\begin{equation}
G\left( r\right) \equiv G\Bigl( k=\xi /d\left( r \right) \Bigr) \label{2.8}
\end{equation}
The implications of the $\theta $-dependence are presumably too weak to be
accesible by our present method.
\bigskip
\section{General properties of the improved Kerr metric}
From now on we assume that we are given a $r$-dependent Newton\'{}s constant,
G=G\left( r\right) $. It may arise by inserting the cutoff identification
k\propto 1/d\left( r\right) $ into a solution of the RG equation such as
\ref{1.2}), but for most parts of our discussion the actual
origin of the $r$-dependence is irrelevant.
\subsection{The quantum corrected metric}
Substituting $G_{0}\rightarrow G\left( r\right) $ in (\ref{1.4}) we arrive
at the improved Kerr metric in BL coordinates
\begin{equation}
ds_{I}^{2}=g_{tt}dt^{2}+2g_{t\varphi }dtd\varphi +g_{rr}dr^{2}+g_{\theta
\theta }d\theta ^{2}+g_{\varphi \varphi }d\varphi ^{2} \label{3.1.a}
\end{equation}
with the components
\begin{eqnarray}
g_{tt} &=&-\left( 1-\frac{2MG\left( r\right) r}{\rho ^{2}}\right)
\;,\;g_{rr}=\frac{\rho ^{2}}{\Delta _{I}\left( r\right) }\;,\;g_{\varphi
\varphi }=\frac{\Sigma _{I}\left( r,\theta \right) \sin ^{2}\theta }{\rho
^{2}} \label{3.1.b} \\
g_{\theta \theta } &=&\rho ^{2}\;,\;g_{t\varphi }=-\frac{2MG\left( r\right)
ra\sin ^{2}\theta }{\rho ^{2}}
\end{eqnarray}
Here $\rho ^{2}\equiv r^{2}+a^{2}\cos ^{2}\theta $ is unchanged, but $\Delta
$ and $\Sigma $ contain $G\left( r\right) $ now:
\begin{equation}
\Delta _{I}\left( r\right) \equiv r^{2}+a^{2}-2MG\left( r\right) r
\label{3.2}
\end{equation}
\begin{equation}
\Sigma _{I}\left( r,\theta \right) \equiv \left( r^{2}+a^{2}\right)
^{2}-a^{2}\Delta _{I}\left( r\right) \sin ^{2}\theta \label{3.3}
\end{equation}
For later use we also note the components of the inverse metric tensor:
\begin{eqnarray}
g^{tt} &=&-\frac{\Sigma _{I}}{\rho ^{2}\Delta _{I}}\;,\;g^{rr}=\frac{\Delta
_{I}}{\rho ^{2}}\;,\;g^{\varphi \varphi }=\frac{\Delta _{I}-a^{2}\sin
^{2}\theta }{\rho ^{2}\Delta _{I}\sin ^{2}\theta } \label{3.4} \\
g^{\theta \theta } &=&\frac{1}{\rho ^{2}}\;,\;g^{t\varphi }=-\frac{2MG\left(
r\right) ra}{\rho ^{2}\Delta _{I}}
\end{eqnarray}
In the rest of this section we shall describe various general properties of
the metric (\ref{3.1.a}), (\ref{3.1.b}). The discussion parallels the
classical case to some extent \cite{Poisson}, but the results collected here
will be needed for an analysis of the quantum effects.
\subsection{Killing vectors and conserved quantitites}
We mentioned already that the improved metric has the Killing vector $\boldsymbol{t}$ and $\boldsymbol{\varphi}$ of eq. (\ref{2.2}). If we employ BL coordinates its components
are obviously
\begin{equation}
t^{\mu }=\delta _{t}^{\mu }\;,\;\varphi ^{\mu }=\delta _{\varphi }^{\mu
}\;\;\;\;\;\;\;\text{(BL)} \label{3.5}
\end{equation}
Considering a point particle of mass $m$ which moves along the trajectory
x^{\mu }\left( \tau \right) $ with four-velocity $u^{\mu }\equiv dx^{\mu
}/d\tau $ and momentum $p^{\mu }=mu^{\mu }$ these Killing vectors imply a
conserved energy and angular momentum about the symmetry axis \cite{MTW}:
\begin{eqnarray}
E &=&-t_{\mu }p^{\mu }\equiv -mt_{\mu }u^{\mu } \label{3.6} \\
L &=&-\varphi _{\mu }p^{\mu }\equiv -m\varphi _{\mu }u^{\mu } \notag
\end{eqnarray}
\subsection{Zero angular momentum, static, and \\ stationary observers}
We consider three classes of special ``observers'' (actually point
particles) following a world line $x^{\mu }\left( \tau \right) $,
parametrized by the proper time $\tau $, with the velocity $u^{\mu }=dx^{\mu
}\left( \tau \right) /d\tau \equiv \dot{x}^{\mu }$, $u^{\mu }u_{\mu }=-1$.
(A dot will always denote the derivative with respect to $\tau $.)
\subsubsection{Zero angular momentum observers}
By definition zero angular momentum observers (or ``ZAMOs'') are particles
with vanishing $L$: $0=L=mg_{\mu \nu }\dot{x}^{\mu }\varphi ^{\nu }$. When
evaluated in BL coordinates, this condition reads $g_{t\varphi }\dot{t}+g_{\varphi \varphi }\dot{\varphi}=0$. Parametrizing the ZAMO\'{}s world
line by the coordinate time $t$ rather than the proper time $\tau $,
the condition assumes the form $g_{t\varphi }+g_{\varphi \varphi }\left(
d\varphi /dt\right) =0$. Therefore, introducing the angular velocity with
respect to the coordinate time,
\begin{equation}
\Omega \equiv \frac{d\varphi }{dt}, \label{3.7}
\end{equation}
as well as the convenient abbreviation
\begin{equation}
\omega \equiv \omega \left( r,\theta \right) \equiv -\frac{g_{t\varphi }}
g_{\varphi \varphi }}=-\frac{2G\left( r\right) Mar}{\Sigma _{I}} \label{3.8}
\end{equation}
we conclude that even though they have no angular momentum, the ZAMOs rotate
around the $z$-axis with the angular velocity
\begin{equation}
\Omega ^{\text{ZAMO}}=\omega \label{3.9}
\end{equation}
The quantity $\omega \geq 0$ is the coordinate angular velocity with which
inertial frames are dragged along \cite{Taylor,Lense,Adler,MTW}. It is affected by the $r$-dependence of
G $ on which it depends both explicitly and via $\Sigma _{I}$.
\subsubsection{Static observers}
By definition, the four-velocity of static observers is proportional to the
Killing vector $\boldsymbol{t}$, i.e. $u^{\mu }=\gamma t^{\mu }$ where $\gamma $ is chosen as $\gamma =\left[ -g_{\mu \nu }t^{\mu
}t^{\nu }\right] ^{-\frac{1}{2}}$ in order to achieve $u_{\mu }u^{\mu }=-1$.
The motion of static observers is not geodesic. To follow their world line
they will need a rocket engine say. Static observers exist only in those
portions of the improved Kerr spacetime in which $\boldsymbol{t}$ is timelike. The \textit{``static limit''}
is reached when $\boldsymbol{t}$ becomes null, i.e. when $\gamma ^{-2}=-g_{\mu \nu }t^{\mu }t^{\nu }=0$. In BL coordinates this is the
case where $g_{tt}=0$, or explicitly,
\begin{equation}
r^{2}-2G\left( r\right) Mr+a^{2}\cos ^{2}\theta =0 \label{3.10_}
\end{equation}
In the classical case the solution to this condition are two static limit
surfaces $S_{\pm }$ which can be parametrized as $r=r_{S_{\pm }}\left(
\theta \right) $ with $r_{S_{\pm }}\left( \theta \right) $ given in (\ref
{1.9}). For the improved metric the situation will be more complicated;
depending on the values of $M$ and $a$ there can be two or one, or no static
limit surface $S$ at all. Also in the improved case, since $g_{tt}=0$ on $S
, static limit surfaces are surfaces of infinite redshift.
\subsubsection{Stationary observers} \label{3.3.3}
A way of defining event horizons, different from their characterization as
one-way surfaces, is related to stationary observers. By definition a
stationary observer moves with a constant angular velocity $\Omega =d\varphi
/dt$ in the $\varphi $-direction. Its four-velocity is proportional to the
Killing vector $\boldsymbol{\xi}=\boldsymbol{t}+\Omega \boldsymbol{\varphi}$, i.e. $u^{\mu }=\gamma
\left( t^{\mu }+\Omega \varphi ^{\mu }\right) =\gamma \xi ^{\mu }$. This
class of observers is stationary in the sense that they perceive no time
variation of the gravitational field. They exist only if $\Omega $ and the
\textit{constant} parameters of their orbit, $r$ and
\varphi $, are such that $\gamma ^{-2}=-g_{\mu \nu }\xi ^{\mu }\xi ^{\nu }>0
. In BL coordinates this condition boils down to
\begin{equation}
q\left( \Omega \right) \equiv \Omega ^{2}-2\omega \Omega +g_{tt}/g_{\varphi
\varphi }<0 \label{3.11}
\end{equation}
If
\begin{equation}
\Omega _{\pm }=\omega \pm \sqrt{\omega ^{2}-g_{tt}/g_{\varphi \varphi }}
\label{3.12}
\end{equation}
is real, the function $q$ has two zeros on the real axis, and (\ref{3.11})
is satisfied if $\Omega _{-}<\Omega <\Omega _{+}$. Depending on whether
g_{tt}$, evaluated at the $\left( r,\theta \right) $-values of the orbit, is
negative, zero, or positive qualitatively different situations can
occur. The corresponding graph of $q\left( \Omega \right) $ is sketched in
Fig. 2. Let us discuss the 4 cases depicted there in turn.
\newpage
\begin{minipage}[t]{.55\linewidth}
\begin{center}
\begin{pspicture}(2.3,0.5)(5.3,6)
\includegraphics[width=\linewidth]{Polin_Om1.tex_gr1_II_09.eps}
\rput[l]{90}(-9,3.8){$q\left(\Omega\right)$}
\rput[l](-0.5,1.4){$\Omega$}
\rput[l](-8,5.55){$\Omega_{-}$}
\rput[l](-4.5,1.4){a)}
\end{pspicture}
\end{center}
\end{minipage}\hfill
\begin{minipage}[t]{.55\linewidth}
\begin{center}
\begin{pspicture}(5.3,0.5)(1.3,6)
\includegraphics[width=\linewidth]{Polin_Om3.tex_gr1.eps}
\rput[l]{90}(-9,3.8){$q\left(\Omega\right)$}
\rput[l](-0.5,1.4){$\Omega$}
\rput[l](-6.7,4.75){$\Omega_{-}$}
\rput[l](-0.8,4.75){$\Omega_{+}$}
\rput[l](-4.5,1.5){b)}
\end{pspicture}
\end{center}
\end{minipage}\hfill
\begin{minipage}[t]{.55\linewidth}
\begin{center}
\begin{pspicture}(1.2,1.5)(3.4,7)
\includegraphics[width=\linewidth]{Polin_Om2.tex_gr1.eps}
\rput[l]{90}(-9,3.8){$q\left(\Omega\right)$}
\rput[l](-0.5,1.2){$\Omega$}
\rput[l](-5.9,3.62){$\Omega_{-}$}
\rput[l](-2,3.62){$\Omega_{+}$}
\rput[l](-4.5,1.2){c)}
\end{pspicture}
\end{center}
\end{minipage}\hfill
\begin{minipage}[t]{.55\linewidth}
\begin{center}
\begin{pspicture}(.3,1.5)(3.3,7)
\includegraphics[width=\linewidth]{Polin_Om4.tex_gr1.eps}
\rput[l]{90}(-9,3.8){$q\left(\Omega\right)$}
\rput[l](-0.5,1.2){$\Omega$}
\rput[l](-4.85,4.25){$\Omega_{-}=\Omega_{+}=\Omega_{\text{H}}$}
\rput[l](-4.5,1.2){d)}
\end{pspicture}
\end{center}
\end{minipage}
\begin{center}
\rput[l](-5.4,-1){Fig. 2:}
\rput[l](-3.7,-1){The function $q\left(\Omega\right)$ in the 4 cases discussed in the text.}
\end{center}
\centerline{}
\begin{enumerate}
\item The case $g_{tt}<0$: In this case $\sqrt{\omega
^{2}-g_{tt}/g_{\varphi \varphi }}=\sqrt{\omega ^{2}+\left| g_{tt}/g_{\varphi
\varphi }\right| }>\omega $ since $g_{\varphi \varphi }>0$ for all $r>0$ and
$\theta \neq 0,\pi $. Therefore since $\omega \geq 0$, it follows that
\Omega _{-}<0$ and $\Omega _{+}<0$. Stationary observers exist for $\Omega
\in \left( \Omega _{-},\Omega _{+}\right) $. Those with $\Omega \in \left(
\Omega _{-},0\right) $ are rotating in the opposite direction as the black
hole, those with $\Omega \in \left( 0,\Omega _{+}\right) $ rotate in the
same direction. Static observers correspond to the special case $\Omega =0$.
The case $g_{tt}<0$ is depicted in Fig. 2a.
\item The case $g_{tt}=0$: In this case $\Omega _{-}=0$ and $\Omega
_{+}=2\omega >0$. There are no counter-rotating $\left( \Omega <0\right) $
observers any more; stationary observers are necesarily co-rotating with the
black hole. Counter-rotating light rays are bound to stay static with $\Omega \equiv
\Omega _{-}=0$. (see Fig.2b).
\item The case $g_{tt}>0$: Here $\sqrt{\omega ^{2}-g_{tt}/g_{\varphi
\varphi }}<\omega $ and therefore $\Omega _{-}>0$ and $\Omega _{+}>0$. All
stationary observers are co-rotating with strictly positive angular velocity
$\Omega \in \left( \Omega _{-},\Omega _{+}\right) $. There are no static
observers.
\item The case $\Delta _{I}=0$: Using the explicit form of the metric
components, the frequencies $\Omega _{\pm }$ can always be written as \cite{Poisson,Tesis}
\begin{equation}
\Omega _{\pm }=\omega \pm \frac{\Delta _{I}^{1/2}\rho ^{2}}{\Sigma _{I}\sin
\theta } \label{3.13}
\end{equation}
This implies that when $\Delta _{I}=0$ the two frequencies become equal:
\left. \Omega _{+}\right| _{\Delta _{I}=0}=\left. \Omega _{-}\right|
_{\Delta _{I}=0}=\left. \omega \right| _{\Delta _{I}=0}=0$. At a radius $r$
such that $\Delta _{I}\left( r\right) =0$ stationary observers are forced to
rotate precisely with the angular velocity $\omega $ about the black hole.
The condition $\Delta _{I}=0$ is equivalent to $g^{rr}=0$. Therefore using
the same argument as classically \cite{Poisson}, one sees that it defines an
event horizon of the improved Kerr spacetime.
\end{enumerate}
We shall find that under the condition $M\gg m_{\rm Pl}$ the improved spacetime
has two spherical horizons $H_{\pm }$ and two limit surfaces $S_{\pm }$ exactly like the classical one. The radii of the static limit
surfaces, $r_{S_{\pm }}^{I}\left( \theta \right) \equiv r_{S}^{I}\left(
\theta \right) $, satisfy
\begin{equation}
g^{rr}=0\Leftrightarrow \left( r_{S}^{I}\right) ^{2}-2G\left(
r_{S}^{I}\right) Mr_{S}^{I}+a^{2}\cos ^{2}\theta =0 \label{3.14}
\end{equation}
while the radii of the horizons, $r_{_{\pm }}^{I}\equiv r_{H}^{I}$, are such
that
\begin{equation}
\Delta _{I}\left( r_{H}^{I}\right) =0\Leftrightarrow \left( r_{H}^{I}\right)
^{2}-2G\left( r_{H}^{I}\right) Mr_{H}^{I}+a^{2}=0 \label{3.15}
\end{equation}
The 4 surfaces can be ordered by increasing radius:
\begin{equation}
r_{S_{-}}^{I}\left( \theta \right) \leq r_{_{-}}^{I}\leq r_{_{+}}^{I}\leq
r_{S_{+}}^{I}\left( \theta \right) \label{3.16}
\end{equation}
Here as always, the label ``$I$'' stands for ``improved''.
\bigskip
If one decreases $r$ at fixed $\theta $ the 4 cases occur in the above
order: For $r>r_{S_{+}}^{I}\left( \theta \right) $, outside the static
limit, case (1) is realized. At $r=r_{S_{+}}^{I}\left( \theta \right) $ we
have $g_{tt}=0$ and case (2) applies. Between $S_{+}$ and $H_{+}$, in the
ergosphere, we have (3), and all stationary observers with $r\in \left(
r_{_{+}}^{I},r_{S_{+}}^{I}\left( \theta \right) \right) $ necessarily rotate
in the direction of the black hole. When we approach $r=r_{_{+}}^{I}$ from
above the only allowed angular velocity is
\begin{equation}
\Omega _{+}=\Omega _{-}=\omega \left( r_{_{+}}^{I},\theta \right) \equiv
\Omega _{H} \label{3.17}
\end{equation}
For $r<r_{_{+}}^{I}$ , there exist no stationary observers any longer: once
it has crossed the horizon $H_{+}$, a particle necessarily falls into the
black hole. We shall refer to $\Omega _{H}$ as ``the angular velocity of the
black hole''. Noting that $\Omega _{H}=2G\left( r_{+}^{I}\right)
Mar_{+}^{I}/\Sigma _{I}\left( r_{+}^{I},\theta \right) $ with $\Sigma
_{I}\left( r_{+}^{I},\theta \right) =\left[ \left( r_{+}^{I}\right)
^{2}+a^{2}\right] ^{2}-a^{2}\Delta _{I}\left( r_{+}^{I}\right) \sin
^{2}\theta =\left[ \left( r_{+}^{I}\right) ^{2}+a^{2}\right] ^{2}$ we
observe that $\Omega _{H}$ is actually independent of the angle $\theta $
and depends only on the parameters $M$ and $a$:
\begin{equation}
\Omega _{H}\left( M,a\right) =\frac{a}{r_{+}^{I}\left(M,a\right) ^{2}+a^{2}}
\label{3.18}
\end{equation}
This formula looks like its classical counterpart \cite{Poisson}; however, the improvement
changes the $M$ and $a$ dependence of $r_{+}^{I}$.
\subsection{Generalized Eddington-Finkelstein coordinates}
The systems of Boyer-Lindquist coordinates $\left( t,r,\theta ,\varphi
\right) $ breaks down when $\Delta _{I}=0$ i.e. on a possible horizon. In
order to reexpress the improved Kerr metric in a system of coordinates which
remains regular there, we define a generalization of the familiar advanced
time (or ingoing) Eddington-Finkelstein (EF) coordinates \cite{Poisson,MTW}:
\begin{eqnarray}
v =t+r^{\ast }\left( r\right)\;,\; r =r\;,\;\theta =\theta\;,\;
\psi =\varphi +r^{\#}\left( r\right) \label{3.19}
\end{eqnarray}
Here the functions $r^{\ast }$ and $r^{\#}$ are given by
\begin{eqnarray}
r^{\ast }\left( r\right) &\equiv &\int^{r}dr^{\prime }\frac{r^{\prime
2}+a^{2}}{\Delta \left( r^{\prime }\right) }=\int^{r}dr^{\prime }\frac
r^{\prime 2}+a^{2}}{r^{\prime 2}-2Mr^{\prime }G\left( r^{\prime }\right)
+a^{2}} \label{3.20} \\
r^{\#}\left( r\right) &\equiv &\int^{r}dr^{\prime }\frac{a}{\Delta \left(
r^{\prime }\right) }=\int^{r}dr^{\prime }\frac{a}{r^{\prime 2}-2Mr^{\prime
}G\left( r^{\prime }\right) +a^{2}} \notag
\end{eqnarray}
For a constant $G\left( r\right) $ these integrals can be performed in
closed form. For the improved metric this is not possible in general.
Luckily the explicit forms of $r^{\ast }$ and $r^{\#}$ are not needed in
order to express the metric in terms of the new coordinates $x^{\mu }=\left(
v,r,\theta ,\psi \right) $. It is enough to use that by (\ref{3.20})
dt=dv-\left( r^{\prime 2}+a^{2}\right) \Delta _{I}^{-1}dr$ and $d\varphi
=d\psi -a\Delta _{I}^{-1}dr$. Inserting these differentials into (\ref{3.1.a}) we obtain the following line element for the improved Kerr metric in
ingoing EF coordinates:
\begin{eqnarray}
ds_{I}^{2} &=&-\left( 1-\frac{2G\left( r\right) Mr}{\rho ^{2}}\right)
dv^{2}+2drdv-2a\sin ^{2}\theta d\psi dr+ \label{3.21} \\
&&-\frac{4G\left( r\right) Mar\sin ^{2}\theta }{\rho ^{2}}d\psi dv+
\frac{\Sigma _{I}\sin ^{2}\theta }{\rho ^{2}}d\psi ^{2}+\rho ^{2}d\theta ^{2} \notag
\end{eqnarray}
We shall also need the Killing vector $\boldsymbol{\xi} =\boldsymbol{t}+\Omega _{H}\boldsymbol{\varphi}$ in EF
coordinates. It is trivial to see that \ $\boldsymbol{\xi} =\frac{\partial }{\partial v
+\Omega _{H}\frac{\partial }{\partial \varphi }$ , i.e.
\begin{equation}
\xi ^{v}=1\;,\;\xi ^{r}=0\;,\;\xi ^{\theta }=0\;,\;\xi ^{\psi }=\Omega _{H}
\label{3.22}
\end{equation}
Using the metric (\ref{3.21}) one obtains the following expression for the
square $\boldsymbol{\xi} ^{2}=g_{\mu \nu }\xi ^{\mu }\xi ^{\nu }$:
\begin{equation}
\boldsymbol{\xi} ^{2}=\frac{\Sigma _{I}\sin ^{2}\theta }{\rho ^{2}}\left( \omega -\Omega
_{H}\right) ^{2}-\frac{\rho ^{2}\Delta _{I}}{\Sigma _{I}} \label{3.23}
\end{equation}
This scalar function is well defined both away from and directly on $H_{+}$.
In fact, it vanishes on the horizon, $\left. \boldsymbol{\xi} ^{2}\right| _{H_{+}}=0$,
since $\Delta _{I}=0$ and $\omega =\Omega _{H}$ there. This is exactly as it
should be: In subsection \ref{3.3.3} we saw that $\gamma ^{-2}=-$ $\boldsymbol{\xi} ^{2}\propto q\left( \Omega _{H}\right) $ , and since $q\left( \Omega _{H}\right) =0$,
the Killing vector becomes null on the horizon.
\subsection{Quantum corrections to the surface gravity}
As the improved Kerr metric admits a Killing vector which is null at the
event horizon and tangent to the horizon\'{}s null generators we may define the surface gravity $\kappa $ in the usual
way \cite{Poisson}:
\begin{equation}
-D_{\mu }\boldsymbol{\xi} ^{2}\left( r_{+}^{I}\right) =2\kappa \xi _{\mu }\left(
r_{+}^{I}\right) \label{3.24_}
\end{equation}
To determine $\kappa $ we shall evaluate (\ref{3.24_}) in the generalized EF
coordinates introduced in the previous subsection. On the RHS of (\ref{3.24_
) we insert $\xi _{\mu }=g_{\mu v}+\Omega _{H}g_{\mu \psi }$ which, in EF
coordinates, evaluates to
\begin{eqnarray}
\xi _{\mu }\left( r_{+}^{I}\right) &=&\left[ 1-a\Omega _{H}\sin ^{2}\theta
\right] \partial _{\mu }r \label{3.25} \\
&=&\frac{\left( r_{+}^{I}\right) ^{2}+a^{2}\cos ^{2}\theta }{\left(
r_{+}^{I}\right) ^{2}+a^{2}}\partial _{\mu }r \notag
\end{eqnarray}
In deriving (\ref{3.25}) we made repeated use of the horizon condition (\ref
{3.15}). On the LHS of (\ref{3.24_}) we need the derivative $D_{\mu }\boldsymbol{\xi}^{2}\equiv \partial _{\mu }\boldsymbol{\xi} ^{2}$ of the function $\boldsymbol{\xi} ^{2}$ given in eq.
(\ref{3.23}), evaluated at $r=r_{+}^{I}$. Since $\Delta _{I}=0$ and $\left(
\omega -\Omega _{H}\right) =0$ there, one easily finds
\begin{equation}
-D_{\mu }\boldsymbol{\xi} ^{2}\left( r_{+}^{I}\right) =\frac{\left( r_{+}^{I}\right)
^{2}+a^{2}\cos ^{2}\theta }{\left[ \left( r_{+}^{I}\right) ^{2}+a^{2}\right]
^{2}}\Delta _{I}^{\prime }\left( r_{+}^{I}\right) \partial _{\mu }r
\label{3.26}
\end{equation}
As a result, the surface gravity is given by
\begin{equation}
\kappa =\frac{1}{2}\frac{\Delta _{I}^{\prime }\left( r_{+}^{I}\right) }
\left( r_{+}^{I}\right) ^{2}+a^{2}} \label{3.27}
\end{equation}
where the prime, as always, denotes a derivative with respect to the
argument. More explicitly,
\begin{equation}
\kappa =\frac{r_{+}^{I}-G\left( r_{+}^{I}\right) M-r_{+}^{I}G^{\prime
}\left( r_{+}^{I}\right) M}{\left( r_{+}^{I}\right) ^{2}+a^{2}} \label{3.28}
\end{equation}
Several comments are in order here.\\
\textbf{(a)} For $G\left( r\right) =$const, eq. (\ref{3.28}) coincides with
the classical result. The quantum corrections modify $\kappa $ both
explicitly, by the $G^{\prime }\left( r_{+}^{I}\right) $-term, and
implicitly, via the shift in the radius $r_{+}^{I}$.\\
\textbf{(b) }The surface gravity of the improved metric has turned out
independent of $\theta $. It is constant on $H_{+}$ therefore. This is
nontrivial since the symmetry assumptions imply only $\varphi $-, but no
\theta $-independence. Classically, $\kappa =$const constitutes the zeroth
law of black hole thermodynamics where $\kappa $ is related to the
Bekenstein-Hawking temperature via $T=\kappa /2\pi $ \cite{Hawking-Bardeen-C,Wald-Racz}. In section 8 we shall
address the question whether a similar interpretation can hold in the
improved case.\\
\textbf{(c)} As in the classical case, $\kappa $ vanishes for extremal
black holes. Their $\Delta \left( r\right) $ has a double zero at the
horizon, implying $\Delta =\Delta ^{\prime }=0$ there.\\
\textbf{(d)} Sometimes it is convenient to rewrite $\kappa $ in a way which
removes any explicit $a$-dependence. Again exploiting $\Delta _{I}\left(
r_{+}^{I}\right) =0$ yields
\begin{equation}
\kappa =\frac{1}{2G\left( r_{+}^{I}\right) M}-\frac{1}{2r_{+}^{I}}-\frac
G^{\prime }\left( r_{+}^{I}\right) }{2G\left( r_{+}^{I}\right) }
\label{3.29}
\end{equation}
Of course $\kappa $ continues to be implicitly $a$-dependent via $r_{+}^{I}$.\\
\textbf{(e) }For $a=0$ the horizon condition is $r_{+}^{I}=2G\left(
r_{+}^{I}\right) M$. Using this relation in (\ref{3.29}) we obtain the
surface gravity for the improved Schwarzschild metric:
\begin{equation}
\kappa =\frac{1}{4G\left( r_{+}^{I}\right) M}-\frac{G^{\prime }\left(
r_{+}^{I}\right) }{2G\left( r_{+}^{I}\right) } \label{3.30_}
\end{equation}
Assuming the validity of $T=\kappa /2\pi $ for the Schwarzschild black hole,
eq. (\ref{3.30_}) implies exactly the Hawking temperature which had been found in
ref. \cite{bh2} using a rather different argument.
\section{Horizons and static limit surfaces}
\subsection{Critical surfaces}
In this section we determine the horizons and the static limit surfaces of
the improved Kerr metric. We shall collectively refer to them as ``critical surfaces''. In Section 3 we saw that the radii $r_{S}^{I}\left( \theta \right)
$ and $r_{H}^{I}$ of a static limit surface $S$ and a horizon $H$ are given
by eqs. (\ref{3.14}) and (\ref{3.15}), respectively. By defining
\begin{equation}
b\equiv\left\{
\begin{tabular}{ll}
$a\cos \theta $ & for $S$ \\
$a$ & for $H
\end{tabular}
\right. \label{4.1}
\end{equation}
those two equations can be combined into one, namely
\begin{equation}
r^{2}-2G\left( r\right) Mr+b^{2}=0 \label{4.2}
\end{equation}
With a $G\left( r\right) $ of the form (\ref{1.3}), i.e.
\begin{equation}
G\left( r\right) =\frac{G_{0}d^{2}\left( r\right) }{d^{2}\left( r\right)
\bar{w}G_{0}} \label{4.3}
\end{equation}
this condition becomes
\begin{equation}
\tilde{d}^{2}\left( \tilde{r}\right) \left( \tilde{r}^{2}+\tilde{b}^{2}-
\tilde{m}\tilde{r}\right) +\bar{w}\left( \tilde{r}^{2}+\tilde{b}^{2}\right)
=0 \label{4.4}
\end{equation}
Here and in the following the tilde means that the corresponding quantity is
expressed in terms of the Planck units related to $G_{0}$. In particular,
\tilde{r}=r/\ell_{\rm Pl}$, $\tilde{m}=m/\ell_{\rm Pl}$, $\tilde{M}=M/m_{\rm Pl}$, $\tilde{a
=a/\ell_{\rm Pl}$, $\tilde{b}=b/\ell_{\rm Pl}$ and $\tilde{d}=d/\ell_{\rm Pl}$ where $G_{0}$
\equiv m_{\rm Pl}^{-2}$ $\equiv \ell_{\rm Pl}^{2}$. Thus we are led to investigate
possible zeros of the family of functions
\begin{equation}
Q_{\tilde{b}}^{\bar{w}}\left( \tilde{r}\right) \equiv \tilde{d}^{2}\left(
\tilde{r}\right) \left( \tilde{r}^{2}+\tilde{b}^{2}-2\tilde{m}\tilde{r
\right) +\bar{w}\left( \tilde{r}^{2}+\tilde{b}^{2}\right) \label{4.5}
\end{equation}
Depending on our choice for the parameters $\tilde{b}$ and $\bar{w}$ the
equation (\ref{4.5}) describes the critical surfaces of the following
metrics:
\begin{enumerate}
\item Classical Schwarzschild metric: $\bar{w}=0$, $\tilde{b}=0$
\item Classical Kerr metric: $\bar{w}=0$, $\tilde{b}\neq 0$
\item Improved Schwarzschild metric: $\bar{w}\neq 0$, $\tilde{b}=0$
\item Improved Kerr metric: $\bar{w}\neq 0$, $\tilde{b}\neq 0$
\end{enumerate}
\bigskip
We shall analyse (\ref{4.5}) for the distance function $d\left( r\right) $
obtained from the straight radial path $\mathcal{C}$ discussed in section 2. We
proceed in two steps: We first employ the simple approximation $d\left(
r\right) =r$ for an analytic discussion of the problem and then in a second
step, we use numerical methods to show that, qualitatively, the results
obtained analytically are indeed representative and provide us with a
correct picture of the new features which are due to the nonzero angular
momentum of the black hole.
\subsection{The approximation $d\left( r\right) =r$}
For $d\left( r\right) =r$ the function $Q_{\tilde{b}}^{\bar{w}}$ becomes a
quartic polynomial:
\begin{equation}
Q_{\tilde{b}}^{\bar{w}}\left( \tilde{r}\right) \equiv \tilde{r}^{4}-2\tilde{
}\tilde{r}^{3}+\left( \tilde{b}^{2}+\bar{w}\right) \tilde{r}^{2}+\bar{w
\tilde{b}^{2} \label{4.6}
\end{equation}
Before turning to the general case of the improved Kerr metric it is
instructive to see how the critical surfaces arise in the special cases (1),
(2), and (3):
\begin{enumerate}
\item \textit{The classical Schwarzschild metric}: In this
case the polynomial simplifies to
\begin{equation}
Q_{0}^{0}\left( \tilde{r}\right) \equiv \tilde{r}^{3}\left( \tilde{r}-
\tilde{m}\right) \label{4.7}
\end{equation}
It has a triple zero at $\tilde{r}=0$ and a simple zero at $\tilde{r}=
\tilde{m}$, or $r=2G_{0}M$.
\item \textit{The classical Kerr metric}: Here the
function (\ref{4.6}) becomes
\begin{equation}
Q_{\tilde{b}}^{0}\left( \tilde{r}\right) \equiv \tilde{r}^{2}\left( \tilde{r
^{2}-2\tilde{m}\tilde{r}+\tilde{b}^{2}\right) \label{4.8}
\end{equation}
It has a double zero at $\tilde{r}=0$ and two simple zeros at
\begin{equation}
\tilde{r}_{\pm }=\tilde{m}\pm \sqrt{\tilde{m}^{2}-\tilde{b}^{2}} \label{4.9}
\end{equation}
if $\tilde{m}\neq \tilde{b}$, or one double zero at $\tilde{r}=\tilde{m}$ if
\tilde{m}=\tilde{b}$. These zeros give rise to the familiar static limit
surfaces $S_{\pm }$ and horizons $H_{\pm }$ at
\begin{equation}
r_{S_{\pm }}\left( \theta \right) =G_{0}M\pm \sqrt{\left( G_{0}M\right)
^{2}-a^{2}\cos ^{2}\theta } \label{4.10}
\end{equation}
\begin{equation}
r_{\pm }\equiv r_{H_{\pm }}=G_{0}M\pm \sqrt{\left( G_{0}M\right) ^{2}-a^{2}}
\label{4.11}
\end{equation}
In the case $\tilde{m}=\tilde{a}$ the two horizons $H_{+}$ and $H_{-}$ merge
to a simple one with the ``critical'' radius $\tilde{r}=\tilde{m}$. We then
have an extremal black hole with $a=G_{0}M$, or $J\equiv aM=G_{0}M^{2}$, and
$r_{\text{crit}}=G_{0}M_{\text{crit}}=a$ \cite{Bardeen}.
\item \textit{The improved Schwarzschild metric}: In this
case (\ref{4.6}) reads
\begin{equation}
Q_{0}^{\bar{w}}\left( \tilde{r}\right) =\tilde{r}^{2}\left( \tilde{r}^{2}-
\tilde{m}\tilde{r}+\bar{w}\right) \label{4.12}
\end{equation}
This function has a double zero at $\tilde{r}=0$ and two simple zeros at
\begin{equation}
\tilde{r}_{\pm }^{\text{I}}=\tilde{m}\pm \sqrt{\tilde{m}^{2}-\bar{w}}
\label{4.13}
\end{equation}
if $\tilde{m}^{2}\neq \bar{w}$, or one double zero at $\tilde{r}^{\text{I}}
\tilde{m}$ if $\tilde{m}^{2}=\bar{w}$. As a result, the quantum-corrected
Schwarzschild spacetime hast two spherical horizons $H_{\pm }$ at
\begin{equation}
r_{\pm }^{\text{I}}=G_{0}M\pm \sqrt{\left( G_{0}M\right) ^{2}-\bar{w}G_{0}}
\label{4.14}
\end{equation}
If $\tilde{m}^{2}=\bar{w}$ the two horizons coalesce to a single one at the
critical radius $r_{\text{cr}}=\sqrt{\bar{w}}\ell{\rm Pl}=G_{0}M_{\text{cr}}$.
This new type of an extremal black hole is realized when the mass equals the
critical mass $M_{\text{cr}}=\sqrt{\bar{w}}\;m_{\rm Pl}$. Since $\bar{w}=O\left(
1\right) $ extremal black holes have a mass of the order of $m_{\rm Pl}$. For
M<M_{\text{cr}}$ the improved Schwarzschild metric has no horizon at all.
The improved Schwarzschild metric has been discussed in detail in ref. \cite
{bh2} to which the reader is refered for further details.
As for the existence of horizons it is also interesting to note that there
is a close analogy between the \textit{classical} Kerr
metric and the \textit{improved} Schwarzschild metric.
The above formulae are identical if one identifies $\tilde{a}^{2}$ with
\bar{w}$ or, for the dimensionful quantities $a^{2}$ with $\bar{w}G_{0}$.
\bigskip
Note that in going from case (1) to either case (2) or case (3) the
triple zero at $\tilde{r}=0$ turns into a double zero at $\tilde{r}=0$, plus
a simple zero at $\tilde{r}>0$.
\item \textit{The improved Kerr metric}: Finally we
discuss the zeros of $Q_{\tilde{b}}^{\bar{w}}$ with both $\bar{w}$ and
\tilde{b}$ nonzero. In principle their dependence on $\tilde{m}$, $\tilde{b}$
and $\bar{w}$ could be written down in closed form but the formulas are not
very instructive. The following indirect reasoning shows the essential
points more clearly.
The first and second derivatives of $Q_{\tilde{b}}^{\bar{w}}$ are
\begin{equation}
\frac{d}{d\tilde{r}}Q_{\tilde{b}}^{\bar{w}}\left( \tilde{r}\right) =2\tilde{
}\left[ 2\tilde{r}^{2}-3\tilde{m}\tilde{r}+\left( \tilde{b}^{2}+\bar{w
\right) \right] \label{4.15}
\end{equation}
\begin{equation}
\frac{d^{2}}{d\tilde{r}^{2}}Q_{\tilde{b}}^{\bar{w}}\left( \tilde{r}\right)
=12\tilde{r}^{2}-12\tilde{m}\tilde{r}+2\left( \tilde{b}^{2}+\bar{w}\right)
\label{4.16}
\end{equation}
The derivative (\ref{4.15}) vanishes at the $\tilde{r}$-values $\tilde{r
_{0} $, $\tilde{r}_{1}$, and $\tilde{r}_{2}$ given by
\begin{eqnarray}
\tilde{r}_{0} &=&0 \label{4.17} \\
\tilde{r}_{1} &=&\frac{3}{4}\tilde{m}\left[ 1-\sqrt{1-\frac{8}{9}\frac
\tilde{b}^{2}+\bar{w}}{\tilde{m}^{2}}}\right] \notag \\
\tilde{r}_{2} &=&\frac{3}{4}\tilde{m}\left[ 1+\sqrt{1-\frac{8}{9}\frac
\tilde{b}^{2}+\bar{w}}{\tilde{m}^{2}}}\right] \notag
\end{eqnarray}
Provided
\begin{equation}
\frac{8}{9}\left( \tilde{b}^{2}+\bar{w}\right) \leq \tilde{m}^{2}
\label{4.18}
\end{equation}
the square roots in (\ref{4.17}) are real so that $\tilde{r}_{1}$, and
\tilde{r}_{2}$ are real and positive. As a result, $Q_{\tilde{b}}^{\bar{w}}$
has 3 different extrema for $\tilde{r}\geq 0$, except when the equality sign
holds in (\ref{4.18}). Then two extrema merge to an inflection point.
Inserting (\ref{4.17}) into (\ref{4.16}) one finds that the second
derivative is negative at $\tilde{r}_{1}$ and positive at $\tilde{r}_{0}$,
and $\tilde{r}_{2}$. Therefore in the nondegenerate case, $\tilde{r}_{0}$,
and $\tilde{r}_{2}$ are minima, and $\tilde{r}_{1}$ is a maximum of $Q_
\tilde{b}}^{\bar{w}}$. If $\frac{8}{9}\left( \tilde{b}^{2}+\bar{w}\right)
\tilde{m}^{2}$ there is a minimum at $\tilde{r}_{0}=0$ and an inflection
point at $\tilde{r}_{1}=\tilde{r}_{2}=3\tilde{m}/4$, and if $\frac{8}{9
\left( \tilde{b}^{2}+\bar{w}\right) >\tilde{m}^{2}$ the only critical point
is the minimum at $\tilde{r}_{0}=0$.
\end{enumerate}
Let us come back to the zeros of $Q_{\tilde{b}}^{\bar{w}}$. Regarded a
function of the complex variable $\tilde{r}\in\mathbb{C}$, it has 4 zeros on the complex plane; only those on the positive
real axis are physically relevant though. Furthermore, regarded a function
on the full real line, $Q_{\tilde{b}}^{\bar{w}}\left( \tilde{r}\right) $ is
the sum of 4 terms all of which are positive if $\tilde{r}<0$. As a
consequence, $Q_{\tilde{b}}^{\bar{w}}$ has no zeros at strictly negative
\tilde{r}$. A priori $Q_{\tilde{b}}^{\bar{w}}$ could have 4 zeros at $\tilde
r}>0$. This case is already excluded, however, since we saw that the
function has at most one maximum and one minimum at strictly positive
\tilde{r}$. Therefore, as far as zeros at $\tilde{r}>0$ are concerned, only
the following 3 cases can occur: (a) 2 simple zeros, (b) 1 double zero, (c)
no zero at all.
In Fig. 3 we show an example of each case.
In this figure and all similar diagrams the notation $\tilde{r}_{\tilde{b}_{\pm} }^{\text{I}}$ stands for either $\tilde{r}_{H_{\pm} }^{\text{I}}\equiv \tilde{r
_{\pm }^{\text{I}}$ or $\tilde{r}_{S_{\pm} }^{\text{I}}$, depending on the
interpretation of $\tilde{b}$. The superscript ``I'' indicates that the respective
radii refer to the improved metric.
\newpage
\centerline{}
\centerline{}
\centerline{}
\begin{minipage}[t]{.55\linewidth}
\begin{center}
\begin{pspicture}(2.3,-4)(12.7,2)
\includegraphics[width=\linewidth]{Graphic2.tex_gr2_II_09.eps}
\rput[l]{90}(-9.1,2.5){$\tilde{Q}(\tilde{r})$}
\rput[l](-4.25,-0.1){$\tilde{r}$ }
\rput[l](-2.55,0.72){$\tilde{r}_2$ }
\rput[l](-5,3.25){$\tilde{r}_1$}
\rput[l](-3,4){$\tilde{b}=7.8$}
\rput[l](-3,3.5){$\tilde{m}=8$}
\rput[l](-3,1.35){$\tilde{r}^{\text{I}}_{\tilde{b}-}$}
\rput[l](-1.35,1.35){$\tilde{r}^{\text{I}}_{\tilde{b}+}$}
\rput[l](-7,-0.3){a)}
\end{pspicture}
\end{center}
\end{minipage}\hfill
\begin{minipage}[t]{.55\linewidth}
\begin{center}
\begin{pspicture}(1.5,0.07)(12,2.07)
\includegraphics[width=\linewidth]{Graphic2.tex_gr3_II_09.eps}
\rput[l]{90}(-9.1,2.5){$\tilde{Q}(\tilde{r})$}
\rput[l](-4.25,-0.1){$\tilde{r}$}
\rput[l](-3,4){$\tilde{b}=7.88$}
\rput[l](-3,3.5){$\tilde{m}=8$}
\rput[l](-1.95,1.4){$=\tilde{r}^{\text{I}}_{\tilde{b}-}$}
\rput[l](-3.2,1.4){$=\tilde{r}^{\text{I}}_{\tilde{b}+}$}
\rput[l](-3.5,1.4){$\tilde{r}_2$}
\rput[l](-5,3.25){$\tilde{r}_1$}
\rput[l](-7,-0.3){b)}
\end{pspicture}
\end{center}
\end{minipage}
\begin{minipage}[t]{.55\linewidth}
\begin{center}
\begin{pspicture}(-3.4,2.6)(2.4,-2.4)
\includegraphics[width=\linewidth]{Graphic2.tex_gr4_II_09.eps}
\rput[l]{90}(-9.1,2.5){$\tilde{Q}(\tilde{r})$}
\rput[l](-3,4){$\tilde{b}=8$}
\rput[l](-3,3.5){$\tilde{m}=8$}
\rput[l](-7,-0.3){c)}
\rput[l](-4.25,-0.1){$\tilde{r}$ }
\rput[l](-11,-1.7){Fig. 3:}
\rput[l](-9.3,-1.7){The figures show examples of the 3 possible configurations}
\rput[l](-9.3,-2.2){the function $Q^{\bar{w}}_{\tilde{b}}$ of eq. (4.6) can assume, with two, one, and}
\rput[l](-9.3,-2.7){no zero on the positive real axis.}
\end{pspicture}
\end{center}
\end{minipage}
\centerline{}
\centerline{}
\centerline{}
From the definition of $Q_{\tilde{b}}^{\bar{w}}$, eq. (\ref{4.6}), it is
obvious that the occurrence of zeros is the more likely the larger is
\tilde{m}$ and the smaller are $\tilde{b}$ and $\bar{w}$. The reason is that
for $\tilde{m}$ large and $\tilde{b}$, $\bar{w}$ small the second term on
the RHS of (\ref{4.6}) $-2\tilde{m}\tilde{r}^{3}<0$, becomes very negative
and the positive terms $\left( \tilde{b}^{2}+\bar{w}\right) \tilde{r}^{2}>0$
and $\bar{w}\tilde{b}^{2}>0$ are small which favors zeros. Therefore we
expect that, for $\tilde{a}$ (and $\bar{w}$) fixed, there are two zeros for
large $\tilde{m}$ ( case (a) ) and no zero for small $\tilde{m}$ ( case (c) ). In between there is a
critical mass at which the extremal situation of a simple double zero is
realized ( case (b) ).
In Fig. 4 we show that this is indeed the case. Here the radii of both
horizons and critical limit surfaces are displayed; this amounts to $\tilde{
}=\tilde{a}$ and $\tilde{b}=\tilde{a}\cos \theta $ in the formulas above. In
all diagrams we fixed $\tilde{a}=5$ (and $\bar{w}=1$), and plotted the
classical and improved radii as a function of $\tilde{m}$. The 4 diagrams
correspond to different values of $\theta $. Generically ( cases (b) and (c) )
we find 4 different improved radii $\tilde{r}_{\pm }^{\text{I}}$, $\tilde{r
_{S_{\pm} }^{\text{I}}\left( \theta \right) $ when $\tilde{m}$ is very large.
When we lower $\tilde{m}$ we reach a point at which the two horizons
coalesce, $\tilde{r}_{+}^{\text{I}}=\tilde{r}_{-}^{\text{I}}$, and below
which there is no horizon any longer, but there still exist two critical
limit surfaces. Lowering $\tilde{m}$ even further the two static limit
surfaces coalesce at a certain critical mass, $\tilde{r}_{S_{+}}^{\text{I
}\left( \theta \right) =\tilde{r}_{S_{-}}^{\text{I}}\left( \theta \right) $,
and for even smaller $\tilde{m}$ there exist neither a horizon nor a static
limit surface.
Fig. 4a) applies to the poles \ $\left( \theta =0,\pi \right) $ where the
event horizons and static limit surfaces touch, $\tilde{r}_{+}^{\text{I}}
\tilde{r}_{S_{+}}^{\text{I}}\left( \theta \right) $, $\tilde{r}_{-}^{\text{I}}
\tilde{r}_{S_{-}}^{\text{I}}\left( \theta \right) $. Fig. 4d) refers to the
equatorial plane ($\theta =\pi /2$) in which, classically, $r_{S_{-}}=0$,
r_{S_{+}}=2m$.
In the 2-dimensional diagrams of Fig. 5 we display the $\theta $-dependence
of the various radii. Here we picked the parameter values $\tilde{m}=6$,
\tilde{a}=5$ for which there exist two horizons $H_{\pm }$ and two static
limits $S_{\pm }$. (For the constant $\bar{w}$ we chose $\bar{w}=4$.)
Both Fig. 4 and 5 show that the quantum effects are the larger the smaller is
\tilde{m}$. For $\tilde{m}\equiv M/m_{\rm Pl}\gg 1$ the critical surfaces of the
improved black hole coincide essentially with those of the classical one.
Lowering $M$ we find that the radius of the outer horizon $H_{+}$ is always
smaller than in the classical case, while the radius of $H_{-}$ is always
larger than classically. Similarly we see that $\tilde{r}_{S_{+}}^{\text{I
}\left( \theta \right) <\tilde{r}_{S_{+}}\left( \theta \right) $ whereas $\tilde{r}_{S_{-}}^{\text{I}}\left( \theta \right)
>\tilde{r}_{S_{-}}\left( \theta \right) $. Both for horizons and static limits the extremal points where the
upper and the lower branch of the curves meet are shifted towards larger
masses by the quantum corrections.
\newpage
\begin{minipage}[t]{.55\linewidth}
\begin{center}
\begin{pspicture}(2.5,0.5)(5.5,6)
\includegraphics[width=\linewidth]{Graf_d_Eq_r_1.tex_gr1.eps}
\rput[l]{90}(-8.9,4){$r_{b\pm}$}
\rput[l](-7,6.3){$\theta=0,\pi$}
\rput[l](-0.5,1.35){$m$}
\rput[l](-7.5,1.35){a)}
\end{pspicture}
\end{center}
\end{minipage}\hfill
\begin{minipage}[t]{.55\linewidth}
\begin{center}
\begin{pspicture}(5.3,0.5)(1.3,6)
\includegraphics[width=\linewidth]{Graf_d_Eq_r_2.tex_gr1.eps}
\rput[l]{90}(-8.9,4){$r_{b\pm}$}
\rput[l](-7,6.3){$\theta=\frac{\pi}{6},\frac{5\pi}{6}$}
\rput[l](-0.5,1.35){$m$}
\rput[l](-7.5,1.35){b)}
\end{pspicture}
\end{center}
\end{minipage}\hfill
\begin{minipage}[t]{.55\linewidth}
\begin{center}
\begin{pspicture}(1.5,0.4)(4.7,5.9)
\includegraphics[width=\linewidth]{Graf_d_Eq_r_3.tex_gr3_II_09_3.eps}
\rput[l]{90}(-8.9,3){$r_{b\pm}$}
\rput[l](-7,5){$\theta=\frac{2\pi}{6},\frac{4\pi}{6}$}
\rput[l](-0.5,0.1){$m$}
\rput[l](-7.5,0.1){c)}
\end{pspicture}
\end{center}
\end{minipage}\hfill
\begin{minipage}[t]{.55\linewidth}
\begin{center}
\begin{pspicture}(0.5,1.5)(3.5,7)
\includegraphics[width=\linewidth]{Graf_d_Eq_r_4.tex_gr1.eps}
\rput[l]{90}(-8.9,4){$r_{b\pm}$}
\rput[l](-6.5,6){$\theta=\frac{\pi}{2}$}
\rput[l](-0.5,1.2){$m$}
\rput[l](-7.5,1.2){d)}
\end{pspicture}
\end{center}
\end{minipage}
\centerline{}
\centerline{}
\centerline{}
\centerline{}
\centerline{}
\begin{center}
\rput[l](-9.4,2.5){Fig. 4:}
\rput[l](-7.7,2.5){The figures show the $m$-dependence of the radii $r_{b\pm}$ (thick lines) and $r^{\text{I}}_{b\pm}$ (thin}
\rput[l](-7.7,2){lines) for $a=5$ and several values of $\theta$. The continuous lines represent $r_{\pm}$ and $r^{\text{I}}_{\pm}$.}
\rput[l](-7.7,1.5){The dashed lines represent $r_{S\pm}$ and $r^{\text{I}}_{S\pm}$.}
\end{center}
\newpage
\centerline{}
\centerline{}
\centerline{}
\centerline{}
\begin{minipage}[t]{.55\linewidth}
\begin{center}
\begin{pspicture}(2.3,-1.8)(12.7,3.8)
\includegraphics[width=\linewidth]{2DPlotBH_D_eq_r.tex_gr1.eps}
\rput[l](-8,0.5){a)}
\rput[l](-8,-0.5){Fig. 5: The figures show a cross section through the event horizons (continuous lines) and}
\rput[l](-8,-1.0){static limit surfaces (dashed lines) in the $xz$-plane
for a quantum black hole with $\tilde{m}=6$}
\rput[l](-8,-1.5){and $\tilde{a}=5$. To facilitate the comparison with the classical case, in Fig. 5b) the corresponding}
\rput[l](-8,-2.0){classical surfaces are superimposed (thick lines).}
\end{pspicture}
\end{center}
\end{minipage}\hfill
\begin{minipage}[t]{.55\linewidth}
\begin{center}
\begin{pspicture}(2,1.75)(12.5,3.75)
\includegraphics[width=\linewidth]{2DPlotBH_Compo.tex_gr1.eps}
\rput[l](-8,0.5){b)}
\end{pspicture}
\end{center}
\end{minipage}
\centerline{}
\centerline{}
\subsection{The quantum extremality condition}
Let us determine the condition on $\tilde{m}$ and $\tilde{b}$ which implies
a double zero of $Q_{\tilde{b}}^{\bar{w}}\left( \tilde{r}\right) $. If $b=a$
this is the condition for the two horizons $H_{+}$ and $H_{-}$ to coincide,
i.e. for the quantum black hole to be extremal. When $Q_{\tilde{b}}^{\bar{w
}\left( \tilde{r}\right) $ has a double zero at some value of $\tilde{r}$,
the function must have a (local) minimum there. Since $\tilde{r}=\tilde{r
_{2}$ of (\ref{4.17}) is the only minimum it has for $\tilde{r}>0$, it
follows that the extremal case is realized precisely if $Q_{\tilde{b}}^{\bar
w}}$ vanishes at $\tilde{r}_{2}$: $\left. Q_{\tilde{b}}^{\bar{w}}\left(
\tilde{r}_{2}\right) \right| _{\text{extremal}}=0$. Inserting (\ref{4.17})
into (\ref{4.6}) we obtain
\begin{equation}
Q_{\tilde{b}}^{\bar{w}}\left( \tilde{r}_{2}\right) =-\frac{27\tilde{m}^{4}}
32}\left[ \left( 1-\frac{8}{9}\frac{\tilde{b}^{2}+\bar{w}}{\tilde{m}^{2}
\right) ^{\frac{3}{2}}+\frac{8}{27}\frac{\left( \tilde{b}^{2}-\bar{w}\right)
^{2}}{\tilde{m}^{4}}-\frac{4}{3}\frac{\tilde{b}^{2}+\bar{w}}{\tilde{m}^{2}}+
\right] \label{4.19}
\end{equation}
As a result, setting $b=a$, the condition for $H_{+}=H_{-}$ reads
\begin{equation}
\left( 1-\frac{8}{9}\frac{\tilde{a}^{2}+\bar{w}}{\tilde{m}^{2}}\right) ^
\frac{3}{2}}+\frac{8}{27}\frac{\left( \tilde{a}^{2}-\bar{w}\right) ^{2}}
\tilde{m}^{4}}-\frac{4}{3}\frac{\tilde{a}^{2}+\bar{w}}{\tilde{m}^{2}}+1=0
\label{4.20}
\end{equation}
We shall refer to (\ref{4.20}) as the ``quantum extremality condition''. If
\bar{w}=0$ it reduces to $\tilde{m}=\tilde{a}$ for the classical Kerr
metric, and if $\tilde{a}=0$ to $\tilde{m}=\sqrt{\bar{w}}$ which is the
correct result for the extremal version of the improved Schwarzschild black
hole, see ref. \cite{bh2}. In the general case $\bar{w
\neq 0$, $\tilde{a}\neq 0$ the condition (\ref{4.20}) can be solved for
\tilde{m}=\tilde{m}\left( \tilde{a}\right) $ only numerically. The result is
shown in Fig. 6. We observe that $\tilde{m}\left( \tilde{a}\right) $
approaches the classical $\tilde{m}=\tilde{a}$ for large $a$, but deviates
significantly for $\tilde{a}\rightarrow 0$.
\begin{minipage}[t]{.55\linewidth}
\begin{center}
\begin{pspicture}(-2.4,7.6)(3.4,2.6)
\includegraphics[width=\linewidth]{Gr4_Einzig_math.tex_gr1.eps}
\rput[l]{90}(-9.2,4){$\tilde{m}(\tilde{a})$}
\rput[l](-1,2.4){$\tilde{a}$ }
\rput[l](-8,3.7){$\sqrt{\bar{w}}$}
\rput[l](-11.5,0.3){Fig. 6:}
\rput[l](-9.8,0.3){The solution $\tilde{m}\left(\tilde{a}\right)$ of the ``quantum extremality condition''}
\rput[l](-9.8,-0.2){for the improved Kerr black hole (with $d(r)=r$ and $\bar{w}=1$).}
\rput[l](-9.8,-0.7){The dashed line represents the $\tilde{m}\left(\tilde{a}\right)=\tilde{a}$
dependence of the}
\rput[l](-9.8,-1.2){classical Kerr spacetime.
For $\tilde{a} \to 0$, $\tilde{m}$ assumes its minimum}
\rput[l](-9.8,-1.7){value at $\sqrt{\bar{w}}$, while it approaches the classical behavior for}
\rput[l](-9.8,-2.2){$\tilde{a} \to \infty$.}
\end{pspicture}
\end{center}
\end{minipage}
\centerline{}
\centerline{}
\centerline{}
\centerline{}
\centerline{}
\centerline{}
\centerline{}
\subsection{Exact distance function}
Up to now we employed the simplified distance function $d\left( r\right) =r$
which has the virtue that all calculations can be performed analytically.
Using numerical techniques we have repeated the above analysis for the
``exact'' distance function (\ref{2.5}), (\ref{2.6}). It turns out that,
qualitatively, the results found with the ``exact'' $d\left( r\right) $ are
exactly the same as those from the $d\left( r\right) =r$ approximation. This
concerns in particular the number of horizons and critical surfaces, the
systematics of their mass and angular momentum dependence, and their
dissappearing at extremal configurations. (See Fig. 7 for an example.)
Thus one of the main results is that the classical and the improved Kerr
metric, sufficiently far away from extremality, have \textit{the same} number of horizons and static limit surfaces. This was
different for the Schwarzschild metric: the classical spacetime has 1
horizon, but the improved spacetime has 2. So, a priori one might have
expected a similar doubling in the case of the Kerr metric. Actually this is
not what happens: The quantum corrections do not generate new critical
surfaces but rather smoothly deform the classical ones.
In the language of ``catastrophe theory'' \cite{Poston,Milnor} this can be
understood from the structural stability properties of the zeros and
critical points of $Q_{\tilde{b}}^{\bar{w}}$. The corresponding function for
the classical Schwarzschild metric has a ``structurally unstable'' triple
zero at $\tilde{r}=0$; giving a nonzero value to $\bar{w}$ it dissolves
into a double zero at $\tilde{r}=0$ and a simple one at $\tilde{r}>0$. The
very same transition from a triple to a double plus a simple zero happens to
the Kerr metric already classically by a nonzero value of $\tilde{a}$. If,
in addition, the quantum parameter $\bar{w}\propto O\left( \hbar \right) $
is given a nonzero value, no further zero is generated. It is easy to
formally prove the structural stability of the classical Kerr zeros \cite{Tesis}.
\newpage
\begin{minipage}[t]{.55\linewidth}
\begin{center}
\begin{pspicture}(2,0.5)(5,6)
\includegraphics[width=\linewidth]{q0s5_II_09.eps}
\rput[l](-1,0.2){$m$}
\rput[l]{90}(-8,3.2){$r\left(m\right)$}
\rput[l](-6.5,5.5){$a=0$}
\rput[l](-7.5,0.2){a)}
\end{pspicture}
\end{center}
\end{minipage}\hfill
\begin{minipage}[t]{.55\linewidth}
\begin{center}
\begin{pspicture}(6.1,0.5)(2.1,6)
\includegraphics[width=\linewidth]{q5s5_II_09.eps}
\rput[l](-1,0.2){$m$}
\rput[l]{90}(-8,3.2){$r\left(m\right)$}
\rput[l](-6.5,5.5){$a=5$}
\rput[l](-7.5,0.2){b)}
\end{pspicture}
\end{center}
\end{minipage}\hfill
\begin{minipage}[t]{.55\linewidth}
\begin{center}
\begin{pspicture}(1,2.5)(4.5,8.3)
\includegraphics[width=\linewidth]{q10s5_II_09.eps}
\rput[l]{90}(-8,3.2){$r\left(m\right)$}
\rput[l](-6.5,5.5){$a=10$}
\rput[l](-7.5,0.2){c)}
\rput[l](-1,0.2){$m$}
\end{pspicture}
\end{center}
\end{minipage}\hfill
\begin{minipage}[t]{.55\linewidth}
\begin{center}
\begin{pspicture}(1.1,2.8)(3.6,8.3)
\includegraphics[width=\linewidth]{q15s5_II_09.eps}
\rput[l](-1,0.2){$m$}
\rput[l]{90}(-8,3.2){$r\left(m\right)$}
\rput[l](-6.5,5.5){$a=15$}
\rput[l](-7.5,0.2){d)}
\end{pspicture}
\end{center}
\end{minipage}
\centerline{}
\centerline{}
\centerline{}
\centerline{}
\centerline{}
\centerline{}
\centerline{}
\centerline{}
\centerline{}
\begin{center}
\rput[l](-9.2,3){Fig. 7:}
\rput[l](-7.5,3){The figures show the $m$-dependence of the improved radii $r^{\text{I}}_{b\pm}$ obtained from the}
\rput[l](-7.5,2.5){exact $d\left(r\right)$ given in (\ref{2.5}), (\ref{2.6})
at $\theta=\frac{\pi}{2}$, for $\bar{w}=5$ and several values of $a$. The}
\rput[l](-7.5,2){outer curves are the improved static limits $r^{\text{I}}_{S\pm}$, the inner ones the improved event}
\rput[l](-7.5,1.5){horizons $r^{\text{I}}_{\pm}$. The structure of the curves is essentially the same as in the $d\left(r\right)=r$}
\rput[l](-7.5,1){approximation.}
\end{center}
\newpage
\section{Penrose process}
One of the most remarkable features of rotating black holes is the
possibility of extracting energy from them, by means of the Penrose process
for instance \cite{Taylor}. This is possible since under certain kinematical conditions
test particles in the Kerr metric can be in a state of negative energy. In
fact, let us consider a composite system \rm{A}, consisting of two particles \rm{B}
and C, which crosses the static limit. It disintegrates into B and C near
the event horizon whereby particle B is in a state of negative energy.
Subsequently B falls through the horizon, thus making a negative
contribution to the black hole\'{}s internal energy. The other particle, C, leaves the ergosphere and reaches
its final state at infinity. The conservation of the total energy for the
black hole and the test particles implies an increased energy for the test
particle C. The energy it gains equals minus the change in the internal
energy of the black hole.
As the possibility of energy extraction is intimately limited to the
existence of negative energy states we shall now analyze this issue for the
improved Kerr metric in order get a first impression of the impact the
quantum gravity corrections have on the region of the test particle phase
space with $E<0$.
The conserved energy of a point particle is given by eq. (\ref{3.6}). If we
use BL coordinates and parametrize its trajectory by the proper time $\tau $
we have explicitly, with the angular velocity $\Omega \equiv d\varphi /dt$,
\begin{eqnarray}
E =-mt^{\mu }g_{\mu \nu }\frac{dx^{\nu }}{d\tau } \label{5.1}
=-m\left[ g_{tt}\frac{dt}{d\tau }+g_{\varphi t}\frac{d\varphi }{d\tau }
\right]=-m\left[ g_{tt}+g_{\varphi t}\Omega \right] \frac{dt}{d\tau }
\end{eqnarray}
Using the explicit form of the improved Kerr metric the negative energy
constraint $E\leq 0$ boils down to
\begin{equation}
\Omega \leq \Omega _{0}\equiv -\frac{g_{tt}}{g_{\varphi t}}=\frac{2MG\left(
r\right) r-\rho ^{2}}{2MG\left( r\right) ra\sin ^{2}\theta } \label{5.2}
\end{equation}
Following \cite{Taylor} it is convenient to reexpress the inequality (\ref{5.2})
in terms of the tangential ``bookkeeper velocity''
\begin{equation}
v_{\text{tan}} \equiv R\left( r,\theta \right) \frac{d\varphi }{dt}=R\left(
r,\theta \right) \Omega \label{5.3}
\end{equation}
with the reduced circumference
\begin{equation}
R\left( r,\theta \right) \equiv \sqrt{g_{\varphi \varphi }}=\sqrt{\frac
\Sigma _{I}\sin ^{2}\theta }{\rho ^{2}}} \label{5.4}
\end{equation}
(The reduced circumference is defined such that $ds^{2}=R^{2}d\varphi ^{2}$
if $dt=dr=d\theta =0$.) In terms of $v_{\text{tan}}$ the negative energy condition $\Omega
\leq \Omega _{0}$ becomes $v_{\text{tan}}\leq R\Omega _{0}$, or
\begin{equation}
v_{\text{tan}}\left( r\right) \leq v_{0}\equiv R\left( r,\theta \right)
\left( \frac{2MG\left( r\right) r-\rho ^{2}}{2MG\left( r\right) ra\sin
^{2}\theta }\right) \label{5.5}
\end{equation}
In the following we shall restrict our analysis to the equatorial plane,
\theta =\pi /2$. In this case the condition (\ref{5.5}) assumes the form
\begin{equation}
v_{\text{tan}}\left( r\right) \leq \frac{1}{a}\sqrt{r^{2}+a^{2}+\frac
2Ma^{2}G\left( r\right) }{r}}\left( 1-\frac{r}{2MG\left( r\right) }\right)
=v_{0}^{\rm{eq}}\left( r\right) \label{5.6}
\end{equation}
Here $v_{0}^{\rm{eq}}$ denotes the bookkeeper tangential
velocity, i.e. the velocity refering to the \textit{coordinate} time $t$ of a
particle which moves in the equatorial plane and has vanishing energy, $E=0$.
The phase space for the rotational motion of a massive test particle is
bounded by the $v\left( r\right) $-curves for co- and counter-rotating light
rays:
\begin{equation}
v_{-}^{\rm{light}}\left( r\right) <v_{\text{tan}}\left(
r\right) <v_{+}^{\rm{light}}\left( r\right) \label{5.7}
\end{equation}
The bookkeeper tangential velocities for light follow from (\ref{3.12}):
\begin{equation}
v_{\pm }^{\rm{light}}\left( r\right) =R\left( r,\theta
\right) \Omega _{\pm }=R\left( r,\theta \right) \left( \omega \pm \sqrt
\omega ^{2}-\frac{g_{tt}}{g_{\varphi \varphi }}}\right) \label{5.8}
\end{equation}
In the $\left( r,v_{\text{tan}}\right) $-plane, the part of the test
particle phase space corresponding to $E<0$ is obtained by intersecting the
regions defined by the inequalities (\ref{5.6}) and (\ref{5.7}),
respectively.
The situation is sketched qualitatively in Fig. 8. Besides $v_{\pm }^{\rm{light}}$ and $v_{0}^{\rm{eq}}$
the figure shows also the $r$-dependence of the dragging velocity $v_{\text{dragging}}=R\left(
r,\theta \right) \omega $. It is not difficult to prove that for any
function $G\left( r\right)$, the $v_{0}^{\rm{eq}}$- and $v_{-}^{\rm{light}}$-curves intersect at the static limit $\left( r=r_{S_{+}}\right) $, and that $v_{0}^{\rm{eq}}=v_{+}^{\rm{light}}=v_{-}^{\rm{light}}=v_{\text{dragging}}$ at the horizon $\left( r=r_{+}\right) $.
In Figs. 9 and 10 we show the corresponding realistic plots which were
obtained numerically. Fig. 9 corresponds to the classical, and Fig. 10 to the
improved case. All plots refer to the equator, $\theta =\pi /2$, and in the
improved case the function $G\left( r\right) $ was taken as in eq. (\ref{4.3
) with $d\left( r\right) =r$. Next to each $\left( r,v\right)$-plot we
display the $M$-dependence of the radii $r_{\pm }$, $r_{S\pm }$ and indicate
by a dashed vertical bar the $M$-value used in the corresponding plot on the
LHS. This presentation makes it obvious if, and how many critical surfaces
exist for the corresponding $M$-value.
When varying the mass $m=MG$ in the Figs. 9 and 10 we keep the ratio $a/m$
fixed. The reason is that, classically, $r_{\pm }$ and $r_{S\pm }$ are
\textit{linear} functions of $m$ if we readjust $a$ such that
$a/m=const$; see eqs. (\ref{1.8}), (\ref{1.9}). As a consequence, the
negative energy region for the classical metric changes its size with $m$, but not its shape. This
can be seen in Fig. 9. Hence changes of the shape are entirely due to the
quantum effects.
Fig. 10a) shows the region of negative energy for $M=5m_{\text{pl}}$, $a=4.5
. Since we are still sufficiently away from the Planck region the shape of
the improved negative energy region is not too different from the
classical one. In Fig. 10b) we have changed $M$ from 5 to 4 Planck masses
for which the shape of the negative energy region is almost
unchanged. Besides the $E<0$ region discussed above figures 10a) and 10b) show an internal negative
energy region bounded by $r_{S_-}^{\text{I}}$ and $r_{-}^{\text{I}}$. Since
the possibility of extraction of energy relies on the existence of
stationary states with negative energy outside $r_{+}^{\text{I}}$, the
internal region cannot be considered physically relevant.
Figures 10c) to 10f) were obtained for the regime $M\approx m_{\text{pl}}$.
Drastic changes in the shape of the negative energy regions are visible. Since
the reliability of our method is questionable in this regime, conclusions about this region have to be considered
with some care. We analyse
these cases nevertheless since they hint at the possibility of interesting
new features.
In Fig. 10c) the quantum extremal black hole with $M=M_{cr}$ and $r_{-}^
\text{I}}=r_{+}^{\text{I}}=r_{\text{extr}}^{\text{I}}$ has been reached. The
internal and external negative energy regions touch at $r_{\text{extr}}^
\text{I}}$.
Fig. 10d) shows a hypothetical configuration for $M<M_{\text{cr}}$ with two
static limits $S_{\pm }^{\text{I}}$ and no event horizon. The internal and
external negative energy regions merged into just one. This region is
bounded by the static limit surfaces at $r_{S_-}^{\text{I}}$ and $r_{S_+}^{\text{I}}$.
In this case there exists an ergosphere from where energy can be
extracted, but no horizons.
Figures 10e) and f) show configurations in which no extraction of energy is
possible. At the extremal static limit configuration shown in figure 10e)
the negative energy region is reduced to zero size.
This analysis suggests that, while it is possible to extract energy from
classical black holes with arbitrarily small masses and angular momenta,
there exists a lowest mass for the Penrose mechanism in the improved Kerr
spacetime. It is close to the Planck mass and defined by the extremal static
limit. However, since the reliability of our method is questionable in the
regime $M\approx m_{\rm{Pl}}$, it would be desirable to investigate this
possibility by independent methods.
\centerline{}
\begin{minipage}[t]{.65\linewidth}
\begin{center}
\begin{pspicture}(-1.5,0.1)(1.5,6.6)
\includegraphics[width=\linewidth]{Penrose_Scheme_II_09_Ver_4.eps}
\rput[l](0,3.3){\mbox{$r$}}
\rput[l](-10.8,6.8){\mbox{$v(r)$}}
\rput[l](-2.1,5.2){\mbox{$v_+^{\rm light}$}}
\rput[l](-2.1,2.27){\mbox{$v_-^{\rm light}$}}
\rput[l](-2.1,4.1){\mbox{$v_{\rm dragging}$}}
\rput[l](-4.5,1.68){\mbox{$v_0^{\rm eq}$}}
\rput[l](-7,3.3){\mbox{$r_{+}$}}
\rput[l](-5.8,3.3){\mbox{$r_{S+}$}}
\end{pspicture}
\end{center}
\end{minipage}
\centerline{}
\centerline{}
\centerline{}
\centerline{}
\centerline{}
\centerline{}
\begin{center}
\rput[l](-9.2,3){Fig. 8:}
\rput[l](-7.5,3){The figure shows schematically the $r$-dependence of $v_{\pm}^{\rm light}$, $v_0^{\rm eq}$ and $v_{\rm dragging}$}
\rput[l](-7.5,2.5){at the equatorial plane. The hatched region corresponds to pairs $\left(r,v\right)$ for which}
\rput[l](-7.5,2){the test particle has negative energy.}
\end{center}
\newpage
\begin{minipage}[t]{.55\linewidth}
\begin{center}
\begin{pspicture}(2.5,0.5)(5.5,6)
\includegraphics[width=\linewidth]{Penrose_Poster9.tex_gr1.eps}
\rput[l](-6,7.3){M=15\ ,\ a=13.5\ ,\ $\bar{w}$=0}
\rput[l](-.8,1.35){\mbox{$r$}}
\rput[l](-9,4.5){\mbox{$v(r)$}}
\rput[l](-6.2,4.2){\mbox{$r_{-}$}}
\rput[l](-3.3,4.2){\mbox{$r_{+}$}}
\rput[l](-1.5,4.2){\mbox{$r_{S+}$}}
\rput[l](-3.1,6.3){\tiny{Allowed}}
\rput[l](-3.1,6){\tiny{Negative Energy}}
\rput[l](-3.1,5.7){\tiny{Region}}
\psline{->}(-2,5.8)(-2.3,5)
\rput[l](-8,1.35){a)}
\end{pspicture}
\end{center}
\end{minipage}\hfill
\begin{minipage}[t]{.55\linewidth}
\begin{center}
\begin{pspicture}(5.3,0.5)(1.3,6)
\includegraphics[width=\linewidth]{Penrose_Poster8_1.tex_gr1.eps}
\rput[l](-.8,1.35){\mbox{$M$}}
\rput[l](-9.5,4.1){\mbox{$r(M)$}}
\rput[l](-2.77,5.95){\mbox{$r_{S+}$}}
\rput[l](-2.63,5){\mbox{$r_+$}}
\rput[l](-2.63,3.3){\mbox{$r_-$}}
\psline[linestyle=dashed,linewidth=0.5pt](-2.2,1.98)(-2.24,5.83)
\end{pspicture}
\end{center}
\end{minipage}\hfill
\begin{minipage}[t]{.55\linewidth}
\begin{center}
\begin{pspicture}(1.5,1.4)(4.7,6.9)
\includegraphics[width=\linewidth]{Penrose_Poster10.tex_gr1.eps}
\rput[l](-6,7.3){M=14\ ,\ a=12.6\ ,\ $\bar{w}$=0}
\rput[l](-.8,1.35){\mbox{$r$}}
\rput[l](-9,4.5){\mbox{$v(r)$}}
\rput[l](-3.1,6.3){\tiny{Allowed}}
\rput[l](-3.1,6){\tiny{Negative Energy}}
\rput[l](-3.1,5.7){\tiny{Region}}
\psline{->}(-2,5.8)(-2.4,4.9)
\rput[l](-6.2,4.2){\mbox{$r_{-}$}}
\rput[l](-3.6,4.2){\mbox{$r_{+}$}}
\rput[l](-1.92,4.2){\mbox{$r_{S+}$}}
\rput[l](-8,1.35){b)}
\end{pspicture}
\end{center}
\end{minipage}\hfill
\begin{minipage}[t]{.55\linewidth}
\begin{center}
\begin{pspicture}(0.5,1.5)(3.5,7)
\includegraphics[width=\linewidth]{Penrose_Poster8_1.tex_gr1.eps}
\rput[l](-3.20,5.77){\mbox{$r_{S+}$}}
\rput[l](-3.08,4.77){\mbox{$r_+$}}
\rput[l](-3.08,3.22){\mbox{$r_-$}}
\psline[linestyle=dashed,linewidth=0.5pt](-2.62,1.98)(-2.62,5.6)
\rput[l](-0.8,1){\mbox{$M$}}
\rput[l](-9.5,4.1){\mbox{$r(M)$}}
\end{pspicture}
\end{center}
\end{minipage}
\centerline{}
\centerline{}
\centerline{}
\centerline{}
\centerline{}
\centerline{}
\centerline{}
\begin{center}
\rput[l](-9.2,3){Fig. 9:}
\rput[l](-7.5,3){The plots on the LHS of this figure display the $r$-dependence of $v_{\pm}^{\rm light}$ and $v_0^{\rm eq}$
and}
\rput[l](-7.5,2.5){are analogous to Fig. 8. They refer to classical Kerr black holes with $M=15m_{\rm pl}$,}
\rput[l](-7.5,2){$a=13.5m_{\rm pl}$ and $M=15m_{\rm pl}$, $a=12.6m_{\rm pl}$, respectively.
They have the same ratio}
\rput[l](-7.5,1.5){$a/m=0.9$. On the RHS the radius of the critical surfaces is displayed for all ma-}
\rput[l](-7.5,1){sses up to $20m_{\rm pl}$, for the constant ratio $a/m=0.9$ as in the corresponding plots on}
\rput[l](-7.5,0.5){the LHS. The dashed vertical line in the plots on the RHS symbolizes the mass va-}
\rput[l](-7.5,0){lues used on the LHS.}
\end{center}
\newpage
\begin{minipage}[t]{.55\linewidth}
\begin{center}
\begin{pspicture}(2.5,0.5)(5.5,6)
\includegraphics[width=\linewidth]{Penrose_Poster1.tex_gr1.eps}
\rput[l](-6,7.3){M=5\ ,\ a=4.5\ ,\ $\bar{w}$=1}
\rput[l](-5.7,4.15){\mbox{$r^{\text{I}}_{-}$}}
\rput[l](-7.74,4.15){\mbox{$r^{\text{I}}_{S-}$}}
\rput[l](-3.57,4.15){\mbox{$r^{\text{I}}_{+}$}}
\rput[l](-1.75,4.15){\mbox{$r^{\text{I}}_{S+}$}}
\rput[l](-.8,1.35){\mbox{$r$}}
\rput[l](-9,4.5){\mbox{$v(r)$}}
\rput[l](-3.1,6.3){\tiny{Allowed}}
\rput[l](-3.1,6){\tiny{Negative Energy}}
\rput[l](-3.1,5.7){\tiny{Region}}
\psline{->}(-2,5.8)(-2.5,5)
\rput[l](-8,1.35){a)}
\end{pspicture}
\end{center}
\end{minipage}\hfill
\begin{minipage}[t]{.55\linewidth}
\begin{center}
\begin{pspicture}(5.3,0.5)(1.3,6)
\includegraphics[width=\linewidth]{Penrose_Poster7_1.tex_gr1.eps}
\rput[l](-.8,1.35){\mbox{$M$}}
\rput[l](-9.5,4.1){\mbox{$r(M)$}}
\rput[l](-3.64,5.37){\mbox{$r^{\text{I}}_{S+}$}}
\rput[l](-3.52,4.44){\mbox{$r^{\text{I}}_+$}}
\rput[l](-3.48,3.45){\mbox{$r^{\text{I}}_-$}}
\rput[l](-3.64,2.49){\mbox{$r^{\text{I}}_{S-}$}}
\psline[linestyle=dashed,linewidth=0.5pt](-3.08,2.1)(-3.08,5.15)
\end{pspicture}
\end{center}
\end{minipage}\hfill
\begin{minipage}[t]{.55\linewidth}
\begin{center}
\begin{pspicture}(1.5,1.4)(4.7,6.9)
\includegraphics[width=\linewidth]{Penrose_Poster2.tex_gr1.eps}
\rput[l](-6,7.3){M=4\ ,\ a=3.6\ ,\ $\bar{w}$=1}
\rput[l](-.8,1.35){\mbox{$r$}}
\rput[l](-9,4.5){\mbox{$v(r)$}}
\rput[l](-6,4.2){\mbox{$r^{\text{I}}_{-}$}}
\rput[l](-7.74,4.2){\mbox{$r^{\text{I}}_{S-}$}}
\rput[l](-4.6,4.2){\mbox{$r^{\text{I}}_{+}$}}
\rput[l](-3.2,4.2){\mbox{$r^{\text{I}}_{S+}$}}
\rput[l](-4.1,6.3){\tiny{Allowed}}
\rput[l](-4.1,6){\tiny{Negative Energy}}
\rput[l](-4.1,5.7){\tiny{Region}}
\psline{->}(-3,5.8)(-3.5,5)
\rput[l](-8,1.35){b)}
\end{pspicture}
\end{center}
\end{minipage}\hfill
\begin{minipage}[t]{.55\linewidth}
\begin{center}
\begin{pspicture}(0.5,1.5)(3.5,7)
\includegraphics[width=\linewidth]{Penrose_Poster7_1.tex_gr1.eps}
\rput[l](-.8,1.35){\mbox{$M$}}
\rput[l](-9.5,4.1){\mbox{$r(M)$}}
\rput[l](-4.58,4.8){\mbox{$r^{\text{I}}_{S+}$}}
\rput[l](-4.5,3.97){\mbox{$r^{\text{I}}_+$}}
\rput[l](-4.48,3.33){\mbox{$r^{\text{I}}_-$}}
\rput[l](-4.58,2.5){\mbox{$r^{\text{I}}_{S-}$}}
\psline[linestyle=dashed,linewidth=0.5pt](-4.03,2.1)(-4.03,4.58)
\end{pspicture}
\end{center}
\end{minipage}
\newpage
\begin{minipage}[t]{.55\linewidth}
\begin{center}
\begin{pspicture}(2.5,0.5)(5.5,6)
\includegraphics[width=\linewidth]{Penrose_Poster3.tex_gr1.eps}
\rput[l](-6,7.3){M=3\ ,\ a=2.7\ ,\ $\bar{w}$=1}
\rput[l](-5.9,4.15){\mbox{$r^{\text{I}}_{-}$}}
\rput[l](-5.9,3.55){\mbox{$r^{\text{I}}_{+}$}}
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\rput[l](-4.35,4.15){\mbox{$r^{\text{I}}_{S+}$}}
\rput[l](-5.1,6.3){\tiny{Allowed}}
\rput[l](-5.1,6){\tiny{Negative Energy}}
\rput[l](-5.1,5.7){\tiny{Region}}
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\end{pspicture}
\end{center}
\end{minipage}\hfill
\begin{minipage}[t]{.55\linewidth}
\begin{center}
\begin{pspicture}(5.3,0.5)(1.3,6)
\includegraphics[width=\linewidth]{Penrose_Poster7_1.tex_gr1.eps}
\rput[l](-.8,1.35){\mbox{$M$}}
\rput[l](-9.5,4.1){\mbox{$r(M)$}}
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\end{minipage}\hfill
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\begin{center}
\begin{pspicture}(1.5,1.4)(4.7,6.9)
\includegraphics[width=\linewidth]{Penrose_Poster4.tex_gr1.eps}
\rput[l](-6,7.3){M=2\ ,\ a=1.8\ ,\ $\bar{w}$=1}
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\end{center}
\end{minipage}\hfill
\begin{minipage}[t]{.55\linewidth}
\begin{center}
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\end{center}
\end{minipage}
\newpage
\begin{minipage}[t]{.55\linewidth}
\begin{center}
\begin{pspicture}(2.5,0.5)(5.5,6)
\includegraphics[width=\linewidth]{Penrose_Poster5.tex_gr1.eps}
\rput[l](-6,7.3){M=1\ ,\ a=0.9\ ,\ $\bar{w}$=1}
\rput[l](-7.37,4.05){\mbox{$r^{\text{I}}_{S-}$}}
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\rput[l](-5.1,6.3){\tiny{No Allowed}}
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\end{center}
\end{minipage}\hfill
\begin{minipage}[t]{.55\linewidth}
\begin{center}
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\includegraphics[width=\linewidth]{Penrose_Poster7_1.tex_gr1.eps}
\rput[l](-.8,1.35){\mbox{$M$}}
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\end{center}
\end{minipage}\hfill
\begin{minipage}[t]{.55\linewidth}
\begin{center}
\begin{pspicture}(1.5,1.4)(4.7,6.9)
\includegraphics[width=\linewidth]{Penrose_Poster6.tex_gr1.eps}
\rput[l](-6,7.3){M=0.5\ ,\ a=0.45\ ,\ $\bar{w}$=1}
\rput[l](-.8,1.35){\mbox{$r$}}
\rput[l](-9,4.5){\mbox{$v(r)$}}
\rput[l](-8,1.2){f)}
\rput[l](-5.1,6.6){\tiny{No Allowed}}
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\end{center}
\end{minipage}\hfill
\begin{minipage}[t]{.55\linewidth}
\begin{center}
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\includegraphics[width=\linewidth]{Penrose_Poster7_1.tex_gr1.eps}
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\begin{center}
\centerline{}
\centerline{}
\centerline{}
\centerline{}
\centerline{}
\centerline{}
\centerline{}
\rput[l](-9.2,3){Fig. 10:}
\rput[l](-7.5,3){The same type of plots as in Fig. 9, but now for the improved Kerr black hole, with masses}
\rput[l](-7.5,2.5){ranging from $M=5m_{\rm pl}$ down to $M=0.5m_{\rm pl}$. All examples considered have an identical}
\rput[l](-7.5,2){ratio $a/m=0.9$.}
\end{center}
\newpage
\section{Vacuum energy-momentum tensor and \\ energy conditions} \label{Sec Vac}
We may reinterpret the RG improved vacuum Kerr metric $g_{\mu \nu }^{\text
imp}}$ as a \textit{classical} spacetime in presence of
matter. Knowing $g_{\mu \nu }^{\text{imp}}$ explicitly, we can compute its
Einstein tensor and insist on the validity of the classical field equation
\begin{equation}
G_{\mu \nu }\left( g^{\text{imp}}\right) =8\pi G_{0}\;T_{\mu \nu }^{\text{Q}} \label{6.1}
\end{equation}
This equation then defines a vacuum energy momentum tensor which describes the
energy and momentum of a fictitious ``pseudo matter'' which reproduces the
quantum corrections found by the RG improvement by means of the conventional
Einstein equation. The explicit calculation yields, after a fair amount of algebra,
\begin{equation}
T_{\mu \nu }^{\text{Q}}\left( r,\theta \right) =\frac{M}{32\pi G_{0}\rho
^{6}\Delta }\left[
\begin{array}{cccc}
q_{1} & 0 & 0 & v \\
0 & q_{2} & 0 & 0 \\
0 & 0 & q_{3} & 0 \\
v & 0 & 0 & q_{4}
\end{array}
\right] \label{6.2}
\end{equation}
with the entries ($n=$1,2,3,4)
\begin{eqnarray}
q_{n}\left( r,\theta \right) &\equiv &\alpha _{n}\left( r,\theta \right)
G^{\prime }\left( r\right) +\beta _{n}\left( r,\theta \right) G^{\prime
\prime }\left( r\right) \; \label{6.3} \\
v\left( r,\theta \right) &\equiv &\alpha _{\nu }\left( r,\theta \right)
G^{\prime }\left( r\right) +\beta _{\nu }\left( r,\theta \right) G^{\prime
\prime }\left( r\right) \notag
\end{eqnarray}
Here the coefficient functions are given by
\begin{eqnarray}
\alpha_{1}\left( r,\theta \right) &\equiv &-\left( a^{2}+r^{2}\right) \left[
8r^{2}\left( a^{2}+r^{2}\right) -a^{4}\left( \sin 2\theta \right) ^{2}\right]
\nonumber \\&&-16ra^{2}MG\sin ^{2}\theta \cos ^{2}\theta \label{6.4} \\
\alpha _{2} &\equiv &8r^{2}\Delta ^{2}\;,\;\alpha _{3}\equiv 8\Delta
a^{2}\cos ^{2}\theta \\
\alpha _{4} &\equiv &\csc ^{2}\theta \alpha _{3}-8a^{2}r^{2}\;,\;\alpha
_{\nu }\equiv 8ar^{2}\left( r^{2}+a^{2}\right) -a\alpha _{3} \\
\beta _{1}\left( r,\theta \right) &\equiv &4\Delta r\rho ^{2}a^{2}\sin
^{2}\theta \;,\;\beta _{2}\left( r,\theta \right) \equiv 0
\end{eqnarray}
\begin{eqnarray}
\beta _{3}\left( r,\theta \right) &\equiv &4\Delta r\rho ^{2}\;,\;\beta
_{4}\left( r,\theta \right) \equiv 4\Delta r\rho ^{2}\csc ^{2}\theta \\
\beta _{\nu }\left( r,\theta \right) &\equiv &-4a\Delta r\rho ^{2}
\end{eqnarray}
The rows and columns of the $T_{\mu \nu }^{\text{Q}}$ matrix above are
ordered in the sequence $t-r-\theta -\varphi $. The matrix is diagonal
except for the $t\varphi $ entry. A nonzero value of $T_{t \varphi }^{\text{Q}}$ was to be expected, of course, since this corresponds precisely
to matter rotating about the $z$-axis.
It is not difficult to diagonalize $T_{\mu \nu }^{\text{Q}}.$ In its
eigenbasis it reads
\begin{equation}
T_{\mu \nu }^{\text{Q}}\left( r,\theta \right) =\frac{M}{32\pi G_{0}\rho
^{6}\Delta }\;\ \rm{diag}\left[
\begin{array}{cccc}
l_{1}, & l_{2}, & l_{3}, & l_{4} \end{array} \right] \label{6.5}
\end{equation}
with the diagonal matrix elements
\begin{eqnarray}
l_{1} &\equiv &\frac{1}{2}\left[ q_{1}+q_{4}+\sqrt{q_{1}^{2}-2q_{1}q_{4}+q_{4}^{2}+4v^{2}}\right] \label{6.6} \\
l_{2} &\equiv &q_{2}\;,\;l_{3}\equiv q_{3} \notag \\
l_{4} &\equiv &\frac{1}{2}\left[q_{1}+q_{4}-\sqrt{q_{1}^{2}-2q_{1}q_{4}+q_{4}^{2}+4v^{2} }\right] \notag
\end{eqnarray}
Despite the formal analogy it would be premature to conclude that the
vacuum quantum effects can be mimicked by the presence of matter. The reason
is that $T_{\mu \nu }^{\text{Q}}$ turns out to violate all the
positivity conditions which are usually assumed to be satisfied by
physically realizable matter \cite{Hawking-Ellis}. For a diagonalized energy momentum tensor
$T_{\mu }^{\;\nu }$ $=\rm{diag}\left[
\begin{array}{cccc}
-\rho, & p_{1}, & p_{2}, & p_{3} \end{array}
\right] $ one distiguishes the following ``energy conditions'' \cite{Poisson,Hawking-Ellis}:
\begin{eqnarray}
\text{weak energy condition} &:&\text{ }\rho \geqslant 0\;,\;\rho +p_{i}>0
\label{6.7} \\
\text{null energy condition} &:&\;\rho +p_{i}\geqslant 0 \notag \\
\text{dominant energy condition} &:&\text{ }\rho \geqslant 0\;,\;\rho
\geqslant \left| p_{i}\right| \notag \\
\text{strong energy condition} &:&\;\rho +p_{i}\geqslant 0\;,\;\rho
+\sum_{i}p_{i}\geqslant 0\; \notag
\end{eqnarray}
From (\ref{6.5}) with (\ref{6.6}) we can read off the energy density $\rho $
and the pressures $p_{i},\;i=1,2,3,$ corresponding the energy momentum
tensor $T_{\mu \nu }^{\text{Q}}$. It is then straightforward to check
numerically whether or not the energy conditions (\ref{6.7}) are satisfied.
The result is that \textit{all four energy conditions are violated, at least in a part of the improved Kerr spacetime.}
This result does not come completely unexpected; also the vacuum expectation value of
energy momentum operators (as in the case of the Casimir effect, for
instance) typically violates the energy conditions. As a consequence, the quantum gravity effects are qualitatively different
from those due to ordinary matter. From the practical point of view this
means that the analysis of the improved black hole does not reduce to
applying the many known results and theorems which are available for
classical black holes with matter. The reason is that in most cases their
derivation assumes the validity of one or the other of the conditions (\ref
{6.7}). For instance, for deriving the focusing theorem for timelike
geodesic congruences from Raychaudhuri's equation one needs the strong
energy condition \cite{Poisson}. Furthermore, the thermodynamics of the improved black
holes is \textit{not} a special case of the familiar (semi-) classical black
hole thermodynamics with matter.
\section{Dressing of mass and angular momentum}
The improved Kerr metric describes an isolated object in an asymptotically
flat spacetime. As this spacetime posesses the two Killing vectors $\boldsymbol{t}$ and $\boldsymbol{\varphi}$ we can
ascribe a mass and an angular momentum to this object by means of the Komar integrals \cite{Komar,Poisson}:
\begin{equation}
M_{\text{Komar}}=-\frac{1}{8\pi G_{0}}\int_{S}\nabla ^{\alpha }t^{\beta
}dS_{\alpha \beta } \label{7.1}
\end{equation}
\begin{equation}
J_{\text{Komar}}=\frac{1}{16\pi G_{0}}\int_{S}\nabla ^{\alpha }\varphi
^{\beta }dS_{\alpha \beta } \label{7.2_rep}
\end{equation}
Here $S$ is a two-sphere at spatial infinity. Its surface element
dS_{\alpha \beta }$ is given by $dS_{\alpha \beta }=-2n_{\left[ \alpha
\right. }r_{\left. \beta \right] }\sqrt{\sigma }d^{2}\theta $ where
n_{\alpha }$ and $r_{\alpha }$ are the timelike and spacelike normals to $S
. Here $\sigma $ is the determinant of $\sigma _{ab}$, the metric induced
from $g_{\alpha \beta }$ in the 2-d surface $S$, and $d^{2}\theta
\equiv d\theta ^{1}d\theta ^{2}$ with $\theta ^{a}$ angular coordinates on
\mathrm{S}$. The integrals for $M_{\mathrm{Komar}}$ and $J_{\mathrm{Komar}}$
probe the metric only at spatial infinity. Since the improved Kerr metric
equals the classical one far away from the black hole, the values of $M_
\mathrm{Komar}}$ and $J_{\mathrm{Komar}}$ are not changed by the RG
improvement. It is well known \cite{Poisson} that for the classical Kerr
metric they coincide with the mass and angular momentum parameters which it
contains:
\begin{equation}
M_{\mathrm{Komar}}=M\ ,\ J_{\mathrm{Komar}}=J \label{7.3}
\end{equation}
Thus, for $S$ a surface at spatial infinity, (\ref{7.3}) holds true also
in the improved case.
The mass and angular momentum of the object as measured at infinity receives
a contribution from the pseudo-matter mimicking the quantum effects. To
identify it we break up $M_{\mathrm{Komar}}$ and $J_{\mathrm{Komar}}$ into
two pieces, one which contains only the effect of the pseudo-matter within
the outer horizon $\mathrm{H}\equiv H_+$, and one which is due to the matter
distribution outside $\mathrm{H}$. The first contribution yields quantities
M_{H}$ and $J_{H}$ which we refer to as the mass and angular momentum of the
black hole, meaning here only the portion of space bounded by $\mathrm{H}$.
The second contribution describes the ``dressing'' of this intrinsic mass and angular momentum
by matter external to the black hole.
The relation between the parameters $M$ and $J$ calculated at the spatial
infinity and the quantities $M_{\text{H}}$ and $J_{\text{H}}$ calculated at
the event horizon can be derived if we consider a 3-d spacelike hypersurface $\Sigma $ extending from the event horizon to spatial infinity. Its inner
boundary is $\text{H}$, a two dimensional cross section of the event
horizon, and its outer boundary is $S$. Using Gauss' theorem and the field
equation (\ref{6.1}) we find that $M$ and $J$ can be decomposed as:
\begin{eqnarray}
M &=&M_{\text{H}}+2\int_{\Sigma }\left( T^{\text{Q}}_{\alpha \beta }-\frac{1}{2
T^{\text{Q}}g_{\alpha \beta }\right) n^{\alpha }t^{\beta }\sqrt{h}d^{3}y \label{7.4} \\
J &=&J_{\text{H}}-\int_{\Sigma }\left( T^{\text{Q}}_{\alpha \beta }-\frac{1}{2
T^{\text{Q}}g_{\alpha \beta }\right) n^{\alpha }\varphi ^{\beta }\sqrt{h}d^{3}y
\label{7.5}
\end{eqnarray}
Here $h_{ab}$ is the metric induced in $\Sigma $ and $y^{a}\ \left(
a=1,2,3\right) $ are coordinates intrisic to this hypersurface. $M_{\text{H
} $ and $J_{\text{H}}$ are the ``genuine'' black-hole mass and angular
momentum, respectively. They are given by surface integrals over $\text{H}$:
\begin{equation}
M_{\mathrm{H}}=-\frac{1}{8\pi G_{0}}\int_{\mathrm{H}}\nabla ^{\alpha
}t^{\beta }ds_{\alpha \beta } \label{7.6}
\end{equation}
\begin{equation}
J_{\mathrm{H}}=\frac{1}{16\pi G_{0}}\int_{\mathrm{H}}\nabla ^{\alpha
}\varphi ^{\beta }ds_{\alpha \beta } \label{7.7}
\end{equation}
The surface element $ds_{\alpha \beta }=2\xi _{\left[ \alpha \right.
}N_{\left. \beta \right] }\sqrt{\sigma }d^{2}\theta =\left( \xi _{\alpha
}N_{\beta }-\xi _{\beta }N_{\alpha }\right) \sqrt{\sigma }d^{2}\theta $
involves an auxiliary null vector $N_{\alpha }$ which satisfies $N_{\alpha
}\xi ^{\alpha }=-1$ and $N_{\alpha }N^{\alpha }=0$ \cite{Poisson}.
The relations (\ref{7.4}) and (\ref{7.5}) can be interpreted as follows: The
total mass $M$ (angular momentum $J$) is given by a contribution $M_{H}$ (
J_{H}$) from the black hole, plus a contribution from the matter
distribution outside. If the black hole is in vacuum, then $M=M_{H}$ and
J=J_{H}$. According to the discussion of section \ref{Sec Vac} we expect that $M_{H}\neq
M$ and $J_{H}\neq J$ when the contributions of the ``quantum fluid'' are
taken into account, i.e. that the mass and the angular momentum of the black
hole get ``renormalized'' or ``dressed'' by the matter surrounding it.
This interpretation is confirmed by an explicit evaluation of the integrals
\ref{7.6}) and (\ref{7.7}). The calculation is somewhat lengthy but similar
to the classical one. The final answer reads \cite{Tesis}
\begin{equation}
M_{H}=M\frac{G\left( r_{+}\right) }{G_{0}}\left\{ 1-\left[ \frac{\left(
r_{+}^{2}+a^{2}\right) G^{\prime }\left( r_{+}\right) }{aG\left(
r_{+}\right) }\right] \arctan \left( \frac{a}{r_{+}}\right) \right\}
\label{7.8}
\end{equation}
\begin{equation}
J_{H}=\left\{ J+\left[ 1-\frac{2MG\left( r_{+}\right) }{a}\arctan \left(
\frac{a}{r_{+}}\right) \right] \left[ \frac{M^{2}G^{\prime }\left(
r_{+}\right) r_{+}^{2}}{a}\right] \right\} \frac{G\left( r_{+}\right) }{G_{0
} \label{7.9}
\end{equation}
These results have a number of remarkable properties: \\
\textbf{(A) }One can verify that for any pair of black hole parameters,
\left( M,J\right) $, the ratio $M_{H}/M$ is always \textit{smaller }than
unity; it approaches unity only asymptotically, for $M\rightarrow \infty $,
when the quantum effects become insignificant. The interpretation is that
the black hole posseses a ``genuine'' (positive) mass $M_{H}$ to which the
quantum matter adds another \textit{positive} contribution to make up the mass
measured at infinity, $M$. Given the fact that the pseudo matter satisfies no
standard positivity condition it is by no means trivial that $M$ is larger
than $M_{H}$. However this is exactly what one would expect if quantum
gravity is \textit{antiscreening:} the metric fluctuations dress any test
mass (here the black hole interior) in such a way that the mass increases
with the distance \cite{mr}. The same is found to hold true for the angular
momentum: $J_{H}/J$ is always smaller than unity, i.e. the pseudo matter
increases the spin of the test mass. \\
\textbf{(B) }Despite their somewhat complicated structure, the results (\ref{7.8}) and (\ref{7.9}) satisfy the same Smarr formula which is valid for
classical black holes \cite{Tesis,Larry}:
\begin{equation}
M_{\text{H}}=2\Omega _{\text{H}}J_{\text{H}}+\frac{\kappa \mathcal{A}}{4\pi
G_{0}} \label{7.10}
\end{equation}
Here $\Omega _{\text{H}}$ and $\kappa $ are given by the ``improved''
equations (\ref{3.18}) and (\ref{3.28}), respectively, and $\mathcal{A}$
denotes the surface of the outer horizon H. Both in the classical and the
improved case it can be written as
\begin{equation}
\mathcal{A}=4\pi \left( r_{+}^{2}+a^{2}\right) \label{7.11}
\end{equation}
but for improved black holes the dependence of $r_{+}$ on $M$ and $J$ (or $a
) is much more complicated. \\
\textbf{(C)} The results (\ref{7.8}), (\ref{7.9}) are strikingly similar to
the corresponding formulas for the \textit{classical} Kerr-Newman spacetime \cite{Poisson} which, besides mass and angular momentum, is characterized by an electric charge $Q$. The expressions coincide \textit{exactly} if we identify
\begin{equation}
Q^{2}\;\hat{=}\;2Mr_{+}^{2}G^{\prime }\left( r_{+}\right) /G_{0} \label{7.12}
\end{equation}
This coincidence does not come completely unexpected. In \cite{bh1} where
the $a=0-$case had been analysed it turned out that the improved
Schwarzschild metric has many features in common with the classical
Reissner-Nordstr\"{o}m metric (a minimum of the lapse function $f\left(
r\right) $, causal structure, etc.). For $a\neq 0$ there is still a
corresponding similarity between the improved Kerr metric and the classical
Kerr-Newman spacetime. The exact coincidence of the Komar integrals is
somewhat surprising though. It is intriguing to speculate that it might
have a deeper meaning.
\section{A modified first law of black hole thermodynamics} \label{Seccion 8}
The first law of \textit{classical} black hole thermodynamics states that
the one-form $2\pi \left( \delta M-\Omega _{\text{H}}\delta J\right) /\kappa$ is exact, i.e. that it can be written as the differential of a state function
S=S\left( M,J\right) .$ Hence
\begin{equation}
\delta M-\Omega _{\text{H}}\delta J=T\delta S, \label{8.1}
\end{equation}
where one interpretes
\begin{equation}
T\left( M,J\right) =\frac{\kappa \left( M,J\right) }{2\pi } \label{8.2}
\end{equation}
and $S$ as the black hole temperature and entropy, respectively \cite{Bek1,Hawking-Bardeen-C}. In terms of
its surface area $\mathcal{A}$ the latter is given by $S=\mathcal{A}/4G_{0}$ \cite{Hawking Area}. For these results to hold the functions (zero forms) $\kappa $ and $\Omega_{\text{H}}$ must have a very special $M-$ and $J-$ dependence. In section 3
we found the corresponding relations for the improved case, namely
\begin{eqnarray}
\kappa \left( M,J\right) &=&\frac{r_{+}^{\text{I}}-M\left[ r_{+}^{\text{I
}G^{\prime }\left( r_{+}^{\text{I}}\right) +G\left( r_{+}^{\text{I}}\right)
\right] }{\left( r_{+}^{\text{I}}\right) ^{2}+\left( J/M\right) ^{2}}
\label{8.3} \\
\Omega _{\text{H}}\left( M,J\right) &=&\frac{\left( J/M\right) }{\left(
r_{+}^{\text{I}}\right) ^{2}+\left( J/M\right) ^{2}}
\label{8.4}
\end{eqnarray}
Here $r_{+}^{\text{I}}\equiv r_{+}^{\text{I}}\left( M,J\right) $, but this
relationship cannot be written down in closed form.
In this section we analyze whether the RG improved black holes satisfy a
quantum corrected version of the first law (\ref{8.1}), and if so, how the
temperature and entropy get modified.
\subsection{Preliminaries}
The states an improved Kerr black hole can be in are labeled by the two
parameters $M$ and $J$. We visualize the corresponding state space as (part
of) the 2-dimensional euclidean plane with cartesian coordinates $x^{1}=M$,
x^{2}=J$. Using the convenient language of differential forms, state
functions are zero forms on this space, i.e. scalars $f=f\left( x\right)
\equiv f\left( M,J\right) $. Defining the exterior derivative as\footnote
To conform with the standard notation of thermodynamics we denote the
exterior derivative by $\delta $ rather than $d$.}
\begin{equation*}
\delta =\delta M\frac{\partial }{\partial M}+\delta J\frac{\partial }
\partial J}
\end{equation*}
a differential form $\boldsymbol{\alpha}$ is \textit{closed} if $\delta
\boldsymbol{\alpha}=0$, and it is \textit{exact} if $\boldsymbol{\alpha
=\delta \boldsymbol{\beta}$ where $\boldsymbol{\beta}$ denotes a $\left(
p-1\right) $-form when $\boldsymbol{\alpha}$ is a $p$-form. The state space
being 2-dimensional, the only case of interest is $p=1$. A general 1-form
has the expansion $\boldsymbol{\alpha}=P\left( M,J\right) \delta M+N\left(
M,J\right) \delta J$. This 1-form is closed if
\begin{equation}
\frac{\partial P}{\partial J}=\frac{\partial N}{\partial M} \label{8.5}
\end{equation}
and it is exact if there exists a zero-form $S\left( M,J\right) $ such that
\boldsymbol{\alpha}=\delta S$ or, in components, $P=\partial S/\partial
M\;,\;N=\partial S/\partial J$. We assume that the states $\left( M,J\right)
$ form a simply connected subset of the euclidean plane so that $\delta
\boldsymbol{\alpha}=0$ is necessary and sufficient for the exactness of
\boldsymbol{\alpha}$.
If $\boldsymbol{\alpha}$ is not exact, one can try to find an integrating
factor $\mu \left( M,J\right) $ such that the product $\mu
\boldsymbol{\alpha}$ is exact: $\mu \left( M,J\right) \boldsymbol{\alpha
=\delta S$. Hence $\delta \left( \mu \boldsymbol{\alpha}\right) =0$, or
\partial \left( \mu P\right) /\partial J=\partial \left( \mu N\right)
/\partial M$, which implies a quasi-linear partial differential equation for
the $0-$form $\mu \left( M,J\right)$ \cite{Simmons,Courant,Caratheodory}:
\begin{equation}
P\left( \frac{\partial \mu }{\partial J}\right) -N\left( \frac{\partial \mu
}{\partial M}\right) =\mu \left[ \left( \frac{\partial N}{\partial M}\right)
-\left( \frac{\partial P}{\partial J}\right) \right] \label{8.6}
\end{equation}
\subsection{Does there exist an entropy-like state function?}
The $1-$form we are actually interested in is
\begin{equation}
\boldsymbol{\alpha}=\frac{2\pi }{\kappa \left( M,J\right) } \Bigl( \delta M-\Omega _{\text{H}}\left( M,J \right) \delta J\Bigr) \label{8.7}
\end{equation}
with
\begin{equation}
P\equiv \frac{2\pi }{\kappa }\;,\;N\equiv -\frac{2\pi \Omega _{\text{H}}}{\kappa } \label{8.8}
\end{equation}
involving the surface gravity and angular velocity of eqs. (\ref{8.3}) and (\ref{8.4}). The crucial question is whether $\boldsymbol{\alpha}$ is closed,
i.e. whether its components (\ref{8.8}) satisfy the integrability condition
\ref{8.5}). The explicit calculation reveals that for a generic $G\left(
r\right) $ this is actually \textit{not} the case: The $1-$form (\ref{8.7})
with the quantum corrected versions of $\kappa $ and $\Omega _{\text{H}}$ is
not closed and, as a consequence, not exact. (This calculation is
straightforward in principle, but rather tedious \cite{Tesis}. One has to be careful
about differentiating all the implicit $M-$ and $J-$ dependencies that enter
via $r_{+}^{\text{I}}\left( M,J\right) $. One does not need the explicit
form of this function; its partial derivatives can be expressed in terms of
r_{+}^{\text{I}}$ itself by differentiating the horizon condition $\Delta\left( r_{+}^{\text{I}}\right) =0$.)
As $\boldsymbol{\alpha}$ is not exact in the improved case we must conclude
that there does \textit{not} exist a differential relation of the type
\begin{equation}
\delta M-\Omega _{\text{H}}\delta J=\left( \frac{\kappa }{2\pi }\right)
\delta \left( \frac{\mathcal{A}}{4G_{0}}+\text{quantum corrections}\right)
\label{8.9}
\end{equation}
which could play the role of a modified first law for quantum black holes.
The interpretation of (\ref{8.9}) would have been clear: The
Bekenstein-Hawking temperature of the improved black holes is related to the
surface gravity by $T=\kappa /2\pi $, as in the classical case, and there
exists a state function $S\left( M,J\right) $ which equals the classical
\mathcal{A}/4G_{0}$ plus correction terms. Since $\boldsymbol{\alpha}$ is
actually not exact we must conclude that \textit{either} \textit{there
exists no entropy-like state function for the improved black holes or the
classical relation }$T=\kappa /2\pi $ \textit{does not hold true for them.}
We see that for quantum Kerr black holes even the very existence of an
entropy is a nontrivial issue. The situation was different for the improved
Schwarzschild black holes \cite{ bh2}. Since there the state space is
1-dimensional, $\boldsymbol{\alpha}\equiv \left( 2\pi /\kappa \right) \delta
M$ is trivially exact, $T=\kappa /2\pi $ continues to be valid and the
entropy one finds has indeed the structure $\mathcal{A}/4G_{0}+$quantum
corrections \cite{bh2,evap}.
Thus we are led to conclude that if there exists a modified, i.e. quantum
version of black hole thermodynamics which is accesible by RG improvement
then the temperature cannot be simply proportional to the surface gravity,
T\neq \kappa /2\pi $. While a priori it is perhaps not very surprising that
the semi-classical relation $T=\kappa /2\pi $ is subject to quantum gravity
correction this causes a difficulty of principle. Within the present
approach we were able to find the corrected $M-$ and $J-$ dependence of
\kappa $ and $\Omega _{\text{H}}$ and, as a result, we know the corrected
1- $form $\boldsymbol{\alpha}$. However without additional input, knowledge
of $\boldsymbol{\alpha}$ is not enough to deduce the two functions $T\left(
M,J\right) $ and $S\left( M,J\right) $. There exist infinitely many pairs
\left( T,S\right) $ such that $\boldsymbol{\alpha}=T\delta S$ for a
prescribed $\boldsymbol{\alpha}$. In a full-fledged quantum gravity version
of black hole thermodynamics it might be possible to find the ``correct'' one, presumably.
A general theory of this kind is beyond the scope of the present paper. Here
we only consider the possible structure of a modified first law. As we shall
see in the next subsection, progress can be made by restricting the
discussion to black holes of small angular momentum. To leading order in a
J^{2}$ expansion the corrections to the temperature and entropy are found to
be uniquely fixed.
\subsection{Temperature and entropy to order $J^{2}$}
By time reflection symmetry, the small$-J$ expansions of the temperature and
entropy read
\begin{equation}
T\left( M,J\right) =T_{0}\left( M\right) +T_{2}\left( M\right) J^{2}+O\left(
J^{4}\right) \label{8.10}
\end{equation}
\begin{equation}
S\left( M,J\right) =S_{0}\left( M\right) +S_{2}\left( M\right) J^{2}+O\left(
J^{4}\right) \label{8.11}
\end{equation}
The terms of lowest order, $T_{0}\left( M\right) $ and $S_{0}\left( M\right)
$, refer to the RG improved Schwarzschild spacetime \cite{bh2,evap}. They
satisfy $\delta M=T_{0}\delta S_{0}$ or $1/T_{0}\left( M\right)
=dS_{0}\left( M\right) /dM$. In \cite{bh2} this relation has been integrated
in order to find the entropy of the improved Schwarzschild black hole:
\begin{equation}
S_{0}=\int_{M_{\text{cr}}}^{M}\frac{dM^{\prime }}{T_{0}\left( M^{\prime
}\right) } \label{8.12}
\end{equation}
In the approximation $d\left( r\right) =r$ the
temperature was found to be given by
\begin{eqnarray}
T_{0}\left( M\right) &=&\frac{1}{4\pi G_{0}M_{\text{cr}}}\frac{\sqrt{Y\left(
1-Y\right) }}{1+\sqrt{1-Y}} \label{8.13} \\
&=&\frac{1}{8\pi G_{0}M}\left[ 1-\frac{1}{4}\left( \frac{M_{\text{cr}}}{M
\right) ^{2}-\frac{1}{8}\left( \frac{M_{\text{cr}}}{M}\right) ^{4}+O\left(
M^{-6}\right) \right] \nonumber
\end{eqnarray}
Here $Y\equiv M_{\text{cr}}^{2}/M^{2}$ and $M_{\text{cr}}\equiv \sqrt{\bar{w}}\;m_{\rm Pl}$.
(The ``critical'' mass $M_{\text{cr}}$ is the smallest mass
for which the improved Schwarzschild spacetime has an event horizon \cite
{bh2}.) Using (\ref{8.13}) in (\ref{8.12}) yields
\begin{eqnarray}
S_{0}\left( M\right) &=&S_{0}\left( M_{\text{cr}}\right) +2\pi \bar{w}\left[
Y^{-1}\sqrt{1-Y}\left( 1+\sqrt{1-Y}\right) +\arctan \sqrt{1-Y}\right]
\notag \\
&=&S_{0}\left( M_{\text{cr}}\right) +\frac{\mathcal{A}_{\text{Class}}^{\text
Sch}}}{4G_{0}}+\notag \\ &&2\pi \bar{w}\left[ \ln \left( \frac{2M}{M_{\text{cr}}}\right)
-\frac{3}{2}-\frac{3}{8}\left( \frac{M_{\text{cr}}}{M}\right) ^{2}-\frac{5}
32}\left( \frac{M_{\text{cr}}}{M}\right) ^{4}+O\left( M^{-6}\right) \right] \label{8.14}
\end{eqnarray}
Here $\mathcal{A}_{\text{Class}}^{\text{Sch}}\equiv 4\pi \left(
2G_{0}M\right) ^{2}$ is the classical Schwarzschild surface area. The first
few terms of the large$-M$ expansions given in (\ref{8.13}) and (\ref{8.14})
are rather reliable predictions probably since for $M\gg m_{\rm Pl}$ the
classical spacetime is only weakly distorted by quantum effects.
Next we try to determine $T_{2}$ and $S_{2}$ such that $\delta M-\Omega _
\text{H}}\delta J=T\delta S$ is satisfied to order $J^{2}$. Inserting the
ans\"{a}tze for $T$ and $S$ we have
\begin{eqnarray}
\delta M-\Omega _{\text{H}}\delta J &=&\left[ T_{0}\left( M\right)
+T_{2}\left( M\right) J^{2}\right] \delta \left[ S_{0}\left( M\right)
+S_{2}\left( M\right) J^{2}\right] \label{8.15} \\
&=&T_{0}\delta S_{0}+\delta S_{0}T_{2}J^{2}+\delta \left( S_{2}J^{2}\right)
T_{0}+O\left( J^{3}\right) \notag
\end{eqnarray}
Exploiting that $T_{0}\delta S_{0}=\delta M$ we are left with
\begin{eqnarray}
-\Omega _{\text{H}}\delta J &=&\delta S_{0}T_{2}J^{2}+\delta \left(
S_{2}J^{2}\right) T_{0}+O\left( J^{3}\right) \label{8.16} \\
&=&T_{2}J^{2}\left( \frac{dS_{0}}{dM}\right) \delta M+T_{0}\left[
J^{2}\left( \frac{dS_{2}}{dM}\right) \delta M+2JS_{2}\delta J\right]
+O\left( J^{3}\right) \notag
\end{eqnarray}
Equating the coefficients of $\delta J$ and $\delta M$ we find the following
two coupled equations which determine $S_{2}$ and $T_{2}$:
\begin{equation}
T_{2}\left( \frac{dS_{0}}{dM}\right) +T_{0}\left( \frac{dS_{2}}{dM}\right)
=0+O\left( J^{4}\right) \label{8.17}
\end{equation}
\begin{equation}
2JT_{0}S_{2}+\Omega _{\text{H}}=0+O\left( J^{3}\right) \label{8.18}
\end{equation}
In eq. (\ref{8.18}) we need $\Omega _{\text{H}}$ to linear order in $J$
only. From (\ref{8.4}) we obtain
\begin{equation}
\Omega _{\text{H}}\left( M,J\right) =\frac{J}{Mr_{\text{Sch}+}^{\text{I
}\left( M\right) ^{2}}+O\left( J^{3}\right) \label{8.19}
\end{equation}
where $r_{\text{Sch}+}^{\text{I}}\equiv r_{+}^{\text{I}}\left( J=0\right) $
refers to the improved Schwarzschild black hole. In the approximation
d\left( r\right) =r$ we are using here this radius is explicitly given by
\cite{bh2}
\begin{equation}
r_{\text{Sch}+}^{\text{I}}=G_{0}M\left[ 1+\sqrt{1-Y}\right] \label{8.20}
\end{equation}
With (\ref{8.19}) in (\ref{8.18}) we can solve for the function $S_{2}$:
\begin{equation}
S_{2}\left( M\right) =-\Bigl[ 2M\;T_{0}\left( M\right) r_{\text{Sch}+}
^{\text{I}}\left( M\right) ^{2}\Bigr]^{-1} \label{8.21}
\end{equation}
Furthermore, taking advantage of $dS_{0}/dM=1/T_{0}$ again, we can solve
\ref{8.17}) for $T_{2}$ in terms of the, by now known, function $S_{2}$:
\begin{equation}
T_{2}\left( M\right) =-T_{0}\left( M\right) ^{2}\;\frac{dS_{2}\left( M\right)
}{dM} \label{8.22}
\end{equation}
In deriving the relations (\ref{8.21}) and (\ref{8.22}) we were able to find
a well defined and \textit{unique} answer for the coefficients of the
J^{2}- $terms. Eq. (\ref{8.21}) for $S_{2}\left( M\right) $ involves only
the known Schwarzschild quantities $T_{0}$ and $r_{\text{Sch}+}^{\text{I}}$,
and once $S_{2}$ is known also $T_{2}$ is completely fixed by eq. (\ref{8.22
).
Using the results from the Schwarzschild case we obtain the following final
result for the temperature and entropy to order $J^{2}$:
\begin{eqnarray}
T\left( M,J\right) &=&\frac{1}{8\pi G_{0}M}\left[ 1-\frac{1}{4}\left( \frac
M_{\text{cr}}}{M}\right) ^{2}-\frac{1}{8}\left( \frac{M_{\text{cr}}}{M
\right) ^{4}+O\left( M^{-6}\right) \right] \label{8.23} \\
&&-\frac{J^{2}}{32\pi M^{5}G_{0}^{3}}\left[ 1+\left( \frac{M_{\text{cr}}}{M
\right) ^{2}+\frac{15}{16}\left( \frac{M_{\text{cr}}}{M}\right) ^{4}+O\left(
M^{-6}\right) \right] +O\left( J^{4}\right) \notag
\end{eqnarray}
\begin{eqnarray}
S\left( M,J\right) &=&\frac{\mathcal{A}_{\text{class}}^{\text{Sch}}}{4G_{0}
+2\pi \bar{w}\left[ \ln \left( \frac{2M_{\text{cr}}}{M}\right) -\frac{3}{2}
\frac{3}{8}\left( \frac{M_{\text{cr}}}{M}\right) ^{2}-\frac{5}{32}\left(
\frac{M_{\text{cr}}}{M}\right) ^{4}+O\left( M^{-6}\right) \right] \notag \\
&&-\left( \frac{\pi J^{2}}{M^{2}G_{0}}\right) \left[ \allowbreak 1+\frac{3}{
}\left( \frac{M_{\text{cr}}}{M}\right) ^{2}+\frac{5}{8}\left( \frac{M_{\text
cr}}}{M}\right) ^{4}+O\left( M^{-6}\right) \right] +O\left( J^{4}\right) \label{8.24}
\end{eqnarray}
In writing down the result for the entropy we fixed the undetermined
constant of integration such that $S=0$ for $M=M_{\text{cr}}$ and $J=0$.
We observe that the angular momentum dependent terms in (\ref{8.23}) and
\ref{8.24}) \textit{decrease} both the black hole's temperature and entropy
as compared to the corresponding Schwarzschild quantities. We also see that
the size of the $J^{2}$-corrections increases with $M_{\text{cr}}/M$, i.e.
these corrections grow as the mass $M$ of the black hole becomes smaller
during the evaporation process.
In summarizing the most important aspects of the modified black hole
thermodynamics discussed in this section we recall that $2\pi T$ does not
agree with the surface gravity $\kappa $ here as it is the case in the familiar
(semi-) classical situation. We demonstrated that a modified first law can
exist only when we give up the relationship $T=\kappa /2\pi $. We also
showed that, to order $J^{2}$, there is a uniquely determined modification
of this relationship which allows for the existence of a state function
S\left( M,J\right) $ with the interpretation of an entropy.
\section{Summary and conclusion}
In this paper we tried to assess the impact of the leading quantum gravity
corrections on the properties of rotating black holes within the framework
of Quantum Einstein Gravity (QEG). Using the gravitational average action as
the basic tool we developed a scale-dependent picture of the spacetime
structure. We exploited that $\Gamma _{k}$ is a family of effective field
theories labeled by $k$. More precisely, to each point $\mathcal{P}$ we associated a
coarse-graining scale $k=k\left(\mathcal{P}\right) $ and then described a
neighborhood of $\mathcal{P}$ by the specific effective action $\Gamma _{k\left(\mathcal{P}\right) }$. In principle there could be several plausible choices of the
map $\mathcal{P}\rightarrow k\left( \mathcal{P}\right) $. They lead to different ``pictures'' of
the same physical system. Using the analogy of a microscope with a variable
resolving power \cite{avactrev} we are using a microscope with a position
dependent resolving power, and clearly the ``picture'' we see depends on how
we change the resolving power from point to point. For the black hole the
choice of $k\left( \mathcal{P}\right) $ is made less ambiguous than for a generic
spacetime since we would like the ``picture'', the improved metric, to have
the same symmetries as the classical metric. We chose $k\left( \mathcal{P}\right) $ to
be monotonically decreasing in the radial direction, giving the best
``resolution'' to points near the black hole and the worst to those
asymptotically far away. The experience with similar ``RG improvements''
indicates that in this way the improved metric encodes the leading quantum
corrections at least at a qualitative level.
The results we obtained can be summarized as follows. In general the quantum
corrections are small for heavy black holes ($M\gg m_{\rm Pl}$), but become
appreciable for light ones. Heavy quantum black holes have the same number
of critical surfaces as the classical ones, namely two static limit surfaces
and two horizons. (For $J=0$ the improvement had led to the formation of a
new horizon.) As one lowers $M$ towards the Planck mass, the two horizons
coalesce and then disappear. At an even smaller mass the static limit
surfaces coalesce and then disappear as well.
Even though the reliability of the improvement method becomes questionable when the corrected metric is
very different from the classical one we believe that the disappearence of
the horizons below a certain critical mass is a fairly reliable prediction.
In fact, this phenomenon has a very simple interpretation: The existence of
a horizon means that the gravitational field is so strong that it can trap
light; if, however, the strength of the gravitational interaction is reduced
at small distances by the RG running of $G$ then it is quite plausible that
very small objects with a low mass cannot prevent light from scaping.
Whether or not these objects have a naked singularity remains an open question. The method used
here is likely to loose its validity close to the black hole's center.
Also on the basis of earlier investigations \cite{bh2}, it is
likely though that the quantum corrections soften the singularity (again
because $G$ is ``switched off'' at short distances); it is even conceivable
that it disappears altogether \cite{bh2}.
A particularly intriguing feature of the Kerr black hole is the possibility
of energy extraction. As the Penrose process is related to the existence of
negative energy states of test particles we analyzed the ``phase space'' of
such negative energy states in detail. In paticular we saw that, while it is
possible to extract energy from classical black holes of arbitrary small
mass and angular momentum, in the improved Kerr spacetime there exists a
minimum mass for energy extraction. It is defined by the extremal
configuration of the static limit surfaces.
We explained that even though the quantum black holes in the vacuum can be
reinterpreted as classical black holes in presence of a special kind of
matter mimicking the quantum fluctuations, many of their mechanical and in
particular thermodynamical properties are nevertheless nonstandard since
this ``pseudo matter'' does not satisfy any of the familiar energy
conditions.
As a first step towards an ``RG improved black hole thermodynamics'' we
analyzed the problem of identifying a state function which could possibly be
interpreted as an entropy. We saw that in the quantum case the 1-form
\left( \delta M-\Omega _{H}\delta J\right) /\kappa $ is no longer exact or,
stated differently, the surface gravity is not an integrating factor of
\delta M-\Omega _{H}\delta J$. We concluded that if an entropy is to exist
also for the improved black hole, their temperature cannot simply be
proportional to $\kappa $. We also saw that for small angular momentum, to
order $J^{2}$, there exist unambiguously defined modified relationships for
the $M$- and $J$-dependence of temperature and entropy. We hope to come back
to a more detailed discussion of these thermodynamical issues elsewhere.
\section{Acknowledgments}
E. T. Would like to thank the German Service of Academic Exchange (DAAD) and the Institute of Physics in Mainz for the financial support during the development of his \newline Ph. D. thesis.
|
{
"timestamp": "2010-09-21T02:00:33",
"yymm": "1009",
"arxiv_id": "1009.3528",
"language": "en",
"url": "https://arxiv.org/abs/1009.3528"
}
|
\section{Introduction}
\label{sec:intro}
The ultradiscrete QRT (uQRT) maps form an eight-parameter family of integrable two-dimensional piecewise linear maps, which includes reductions of the ultradiscrete KP equation \cite{QCS01,Nobe06,KIMS07} and the ultradiscretization of non-autonomous limit of the discrete Painlev\'e equations \cite{TTGOR97}.
Each map $(x,y)\mapsto(\bar x,\bar y)$ of the family has the following form \cite{QCS01}:
\begin{align}
\bar x
=
F_{1}(y)-F_{3}(y)-x
\qquad
\bar y
=
G_{1}(\bar x)-G_{3}(\bar x)-y,
\label{eq:uQRTmap}
\end{align}
where $F_{i}$ and $G_{i}$ ($i=1,3$) are tropical polynomials \cite{RGST03} containing eight parameters $\alpha_{00}$, $\alpha_{01}$, $\alpha_{02}$, $\alpha_{10}$, $\alpha_{12}$, $\alpha_{20}$, $\alpha_{21}$, $\alpha_{22}\in{\mathbb R}\cup\{-\infty\}$:
\begin{align*}
&F_{1}(y)
:=
\max\left(
\alpha_{20}+2y,
\alpha_{21}+y,
\alpha_{22}\right)
&&
F_{3}(y)
:=
\max\left(
\alpha_{00}+2y,
\alpha_{01}+y,
\alpha_{02}
\right)\\
&G_{1}(\bar x)
:=
\max\left(
\alpha_{02}+2\bar x,
\alpha_{12}+\bar x,
\alpha_{22}
\right)
&&
G_{3}(\bar x)
:=
\max\left(
\alpha_{00}+2\bar x,
\alpha_{10}+\bar x,
\alpha_{20}
\right).
\end{align*}
Let $F$ be a tropical polynomial in $x$ and $y$
\begin{align}
F(\text{\mathversion{bold}{$\alpha$}};x,y)
:=
\max
\left(
F_{1}(y),F_{2}(y)+x,F_{3}(y)+2x
\right),\label{eq:F}
\end{align}
where $F_{2}(y):=\max\left(\alpha_{10}+2y,\alpha_{11}+y,\alpha_{12}\right)$ and $\text{\mathversion{bold}{$\alpha$}}=(\alpha_{ij})_{0\leq i,j\leq2}$ is a $3\times3$ real matrix.
The map \eqref{eq:uQRTmap} has a one-parameter family of the invariant curves $\{\Sigma_{\kappa}\}_{\kappa\in{\mathbb R}}$ filling the plane, each of which is at most a convex octagon
\begin{align}
\Sigma_{\kappa}:\quad
\kappa+x+y
=
F(\text{\mathversion{bold}{$\alpha$}}_{-\infty};x,y),
\label{eq:uQRTinvariant}
\end{align}
where $\text{\mathversion{bold}{$\alpha$}}_{-\infty}$ is the matrix $\text{\mathversion{bold}{$\alpha$}}$ with ${\alpha_{11}=-\infty}$.
The uQRT maps can be derived from the QRT maps \cite{QRT89}, which form an 18-parameter family of two-dimensional paradigmatic integrable maps, through the ultradiscretization procedure \cite{TTMS96,IMNS06}.
In \cite{Nobe08}, the author showed that the uQRT map is nothing but the addition of points on a tropical elliptic curve.
This is a natural tropicalization of the geometry of the QRT maps found by Tsuda \cite{Tsuda04}.
Through this description, the uQRT map is linearized on the tropical Jacobian of the corresponding tropical elliptic curve, and the initial value problem can be solved.
Several studies on ultradiscrete systems via tropical geometry \cite{IT08,IT09,IT09-2,Iwao10} and the above fact suggest that the tropical geometric approach is effective to examine ultradiscrete integrable systems.
We consider a piecewise linear map $\Phi:(x,y)\mapsto(\bar x,\bar y)$:
\begin{align}
\bar x
=
\xi(x,y)
:=
-x
+
\left|y\right|
\qquad
\bar y
=
\xi(y,\bar{x})
=
-y
+
\left|\bar{x}\right|.
\label{eq:hky2dn}
\end{align}
Since \eqref{eq:hky2dn} is introduced by Brown \cite{Brown85}, we call it the Brown map.
The Brown map has the following remarkable property.
\begin{proposition}[see \cite{GKP89,HT03,HY02}
\label{prop:Brown}
\begin{enumerate}
\item Let $\{P_{n}\}_{n\in{\mathbb Z}_{\geq0}}$ be a sequence of points in ${\mathbb R}^{2}$ satisfying $P_{n+1}=\Phi(P_{n})$ for $n\in{\mathbb Z}_{\geq0}$.
Then $\{P_{n}\}_{n\in{\mathbb Z}_{\geq0}}$ is periodic with period nine\footnote{No member of the uQRT family has a constant period more than eight \cite{Nobe03}.}.
\item Let $\Upsilon_{\kappa}$ be the concave nonagon (see figure \ref{fig:ipname}) defined by
\begin{align}
\kappa
=
\min\left(H(x,y),H(y,x)\right)
\qquad
\mbox{for $\kappa\in{\mathbb R}_{>0}$},
\label{eq:UK}
\end{align}
where
\begin{align*}
H(x,y)
:=
\max(x,-y,-x+y,-x-y).
\end{align*}
Then $\{\Upsilon_{\kappa}\}_{\kappa\in{\mathbb R}_{\geq0}}$ is the family of the invariant curves of the Brown map.
\end{enumerate}
\end{proposition}
Although proposition \ref{prop:Brown} implies that the Brown map is not a member of the uQRT family, we can give its tropical geometric description composing the additions of points on a pair of tropical elliptic curves.
We reformulate the map on the tropical Jacobians of the corresponding tropical elliptic curves, and present the general solution to the initial value problem in terms of the ultradiscrete theta function.
\section{Geometry of uQRT maps}
\label{sec:uQRT}
Let $\text{\mathversion{bold}{$\alpha$}}$ be a $3\times3$ real matrix.
Consider a tropical polynomial in $x$ and $y$ containing a parameter $\mu\in{\mathbb R}$
\begin{align}
E_{\mu}(\text{\mathversion{bold}{$\alpha$}};x,y)
=
\max\left(
-\mu+F(\text{\mathversion{bold}{$\alpha$}};x,y),F(\text{\mathversion{bold}{$\beta$}};x,y)
\right),
\label{eq:tpoly}
\end{align}
where $F$ is the tropical polynomial \eqref{eq:F}, $\text{\mathversion{bold}{$\beta$}}=-\lambda+\text{\mathversion{bold}{$\alpha$}}_{\lambda}$, and $\text{\mathversion{bold}{$\alpha$}}_{\lambda}$ is the matrix $\text{\mathversion{bold}{$\alpha$}}$ with $\alpha_{11}=\lambda$.
Here we assume $\alpha_{11}<0,\lambda$ and $\alpha_{00}=-\infty$.
Let us denote the tropical curve given by $E_{\mu}(\text{\mathversion{bold}{$\alpha$}};x,y)$ by $\mathcal{T}\left(E_{\mu}(\text{\mathversion{bold}{$\alpha$}};x,y)\right)$\footnote{A tropical curve given by a tropical polynomial is defined as the set of points at which the polynomial is not smooth \cite{RGST03}}.
For generic choice of the parameters, $\mathcal{T}\left(E_{\mu}(\text{\mathversion{bold}{$\alpha$}};x,y)\right)$ is a tropical elliptic curve and has an additive group structure \cite{Vigeland04}.
Let us consider a pencil of the tropical elliptic curves
\begin{align}
\left\{\mathcal{T}\left(E_{\mu}(\text{\mathversion{bold}{$\alpha$}};x,y)\right)\right\}_{\mu\in{\mathbb R}}.
\label{eq:pencil}
\end{align}
Note that we have
\begin{align*}
\mathcal{T}\left(E_{\mu}(\text{\mathversion{bold}{$\alpha$}};x,y)\right)
=
\begin{cases}
\mathcal{T}\left(F(\text{\mathversion{bold}{$\beta$}};x,y)\right)
&
\mbox{for $\mu\geq\lambda$}\\
\mathcal{T}\left(F(\text{\mathversion{bold}{$\alpha$}};x,y)\right)
&\mbox{for $\mu\leq\alpha_{11}$}.\\
\end{cases}
\end{align*}
All members of \eqref{eq:pencil} have seven tentacles (half lines) in common, hence the pencil \eqref{eq:pencil} has seven base points\footnote{If $\alpha_{00}>-\infty$ the pencil \eqref{eq:pencil} has 8 base points in analogy to biquadratic curves in ${\mathbb P}^1\times{\mathbb P}^1$, counting multiplicities. Therefore, it is natural to consider that \eqref{eq:pencil} has 6 single and a double base points for $\alpha_{00}=-\infty$.} (see figure \ref{fig:basepoint}).
\begin{figure}[htbp]
\begin{center}
{\unitlength=.03in{\def1.0{1.0}
\begin{picture}(80,55)(-40,-30)
\thicklines
\dottedline(10,10)(2,2)
\dottedline(10,-10)(5,-10)
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\dottedline(-5,-15)(-5,-7)
\dottedline(-15,-5)(-7,-5)
\dottedline(-15,5)(-10,5)
\dottedline(-10,10)(-10,5)
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\dashline[10]{3}(18,-10)(18,18)
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\dashline[10]{3}(-23,5)(-23,-5)
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\dashline[10]{3}(5,-23)(18,-10)
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\dottedline(-7,-5)(-7,2)
\dottedline(-7,2)(2,2)
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\put(-15,-5){\line(-1,0){15}}
\put(-15,5){\line(-1,0){15}}
\put(-10,10){\line(0,1){15}}
\put(13,0){\makebox(0,0){$P$}}
\put(0,13){\makebox(0,0){$C$}}
\put(-13,11){\makebox(0,0){${\mathcal{O}}$}}
\put(11,-13){\makebox(0,0){${T}$}}
\put(10,10){\line(0,-1){20}}
\put(10,-10){\line(-1,-1){5}}
\put(5,-15){\line(-1,0){10}}
\put(-5,-15){\line(-1,1){10}}
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\put(-15,5){\line(1,1){5}}
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\put(10,0){\circle*{1}}
\put(-10,10){\circle*{1}}
\put(10,-10){\circle*{1}}
\end{picture}
}}
\caption{Several members of the pencil \eqref{eq:pencil}.
The dotted and broken lines represent the curve $\mathcal{T}\left(F(\text{\mathversion{bold}{$\alpha$}};x,y)\right)$ and $\mathcal{T}\left(F(\text{\mathversion{bold}{$\beta$}};x,y)\right)$ respectively.
The solid line is the unique curve $C$ of \eqref{eq:pencil} passing through $P$.
}
\label{fig:basepoint}
\end{center}
\end{figure}
Let $P=(x,y)$ be a point in ${\mathbb R}^{2}$.
Choose such $\alpha_{11}$ and $\lambda$ as to satisfy
\begin{align*}
\alpha_{11}
<
F(\text{\mathversion{bold}{$\alpha$}}_{-\infty};x,y)-x-y
<
\lambda.
\end{align*}
Note the uQRT map to be independent of the choice of $\alpha_{11}$ and $\lambda$.
Then we have a unique member of the pencil \eqref{eq:pencil} passing through $P$ (see figure \ref{fig:basepoint}).
We denote the curve by $C$.
The complement of the tentacles of $C$, denoted by $\bar C$, coincides with the invariant curve $\Sigma_{\mu}$ of the uQRT map.
Put $\mathcal{V}_{1}={\mathcal{O}}\in\bar C$, the additive identity element.
Let $\mathcal{V}_{i+1}$ be the vertex next to $\mathcal{V}_{i}$ in counterclockwise direction for $i=1,2,\ldots,n$.
Let $\mathcal{E}_{i}$ be the edge connecting $\mathcal{V}_{i}$ with $\mathcal{V}_{i+1}$ for $i=1,2,\ldots,n$ ($\mathcal{V}_{n+1}:=\mathcal{V}_{1}$).
Let $\varepsilon_{i}=1/|\mbox{\boldmath{$v_{i}$}}|$, where $\mbox{\boldmath{$v_{i}$}}$ is the primitive tangent vector along $\mathcal{E}_{i}$ and $|\mbox{\boldmath{$v_{i}$}}|$ denotes its Euclidian length.
Define the total lattice length $\mathcal{L}$ and the tropical Jacobian $J(\bar C)$ of $\bar C$ respectively \cite{Vigeland04}:
\begin{equation*}
\mathcal{L}=\sum_{i=1}^{n}\varepsilon_{i}|\mathcal{E}_{i}|
\qquad
\mbox{and}
\qquad
J(\bar C)
=
{\mathbb R}/\mathcal{L}{\mathbb Z}.
\end{equation*}
Let $\eta:\bar C{\to} J(\bar C)$ be the Abel-Jacobi map \cite{MZ06,IT08}, which is linear on each edge $\mathcal{E}_{i}$ of $\bar C$ and is inductively defined by the formula
\begin{align*}
\left\{
\begin{array}{l}
\eta(\mathcal{V}_{1})
=
0\\[3pt]
\eta(\mathcal{V}_{i+1})
=
\eta(\mathcal{V}_{i})
+
{\varepsilon_{i}|\mathcal{E}_{i}|}
\quad
(i=1,2,\ldots,n-1).
\end{array}
\right.
\end{align*}
Then the addition $\oplus$ in the group $(\bar C,\mathcal{O})$ is defined by the formula
\begin{align*}
d_{C}(\mathcal{O},P\oplus Q)
=
d_{C}(\mathcal{O},P)
+
d_{C}(\mathcal{O},Q),
\end{align*}
where $P$ and $Q$ are points on $\bar C$ and
\begin{align*}
d_{C}(P,Q)
:=
\eta(Q)
-
\eta(P).
\end{align*}
We have the following theorem.
\begin{theorem}[theorem 3 in \cite{Nobe08}
\label{thm:uQRTadd}
The uQRT map $P\mapsto\bar P$ \eqref{eq:uQRTmap} is equivalent to the addition formula of $(\bar C,\mathcal{O})$:
\begin{equation*}
P\oplus T
=
\bar P.
\end{equation*}
\end{theorem}
\begin{corollary}[corollary 1 in \cite{Nobe08}
The uQRT map $P\mapsto\bar P$ \eqref{eq:uQRTmap} is linearized on the tropical Jacobian
$J(C)$ in terms of the Abel-Jacobi map $\eta:\bar C\to J(\bar C)$
\begin{equation*}
\eta(P)
\mapsto
\eta(\bar P)
=
\eta(P)+\eta( T).
\end{equation*}
\end{corollary}
\section{Brown map}
\subsection{Geometric construction}
\label{subsec:geomconst}
Let $P=(x,y)$ be a point in ${\mathbb R}^{2}$.
Then there exists a unique nonagon $\Upsilon_{\kappa}$ passing through $P$.
Let the vertex of $\Upsilon_{\kappa}$ whose coordinate is $(\kappa,2\kappa)$ be $\mathcal{V}_{1}$ (see figure \ref{fig:ipname}).
Successively let the vertices and the edges of $\Upsilon_{\kappa}$ be $\mathcal{V}_{i}$ and $\mathcal{E}_{i}$ in counterclockwise direction as in section \ref{sec:uQRT}, respectively.
\begin{figure}[htbp]
\begin{center}
{\unitlength=.03in{\def1.0{1.0}
\begin{picture}(100,55)(-50,-20)
\thicklines
\put(-15,15){\line(1,0){15}}
\put(15,15){\line(1,0){15}}
\put(30,15){\line(-1,-1){15}}
\put(0,-15){\line(-1,1){15}}
\put(-15,0){\line(0,1){15}}
\put(15,-15){\line(0,1){15}}
\put(15,15){\line(0,1){15}}
\put(15,30){\line(-1,-1){15}}
\put(0,-15){\line(1,0){15}}
\put(-25,7.2){\color{red}\line(1,0){60}}
\put(-25,7){\color{red}\line(1,0){60}}
\put(-25,6.8){\color{red}\line(1,0){60}}
\put(22,-20){\color{blue}\line(0,1){55}}
\put(22.2,-20){\color{blue}\line(0,1){55}}
\put(21.8,-20){\color{blue}\line(0,1){55}}
\put(-15,7){\circle*{2}}
\put(22,7){\circle*{2}}
\put(22,15){\circle*{2}}
\put(15,33){\makebox(0,0){$\mathcal{V}_{1}$}}
\put(-12,4){\makebox(0,0){$P$}}
\put(26,3){\makebox(0,0){$Q$}}
\put(26,19){\makebox(0,0){$\bar P$}}
\put(-28,7){\makebox(0,0){$L_{1}$}}
\put(26,-18){\makebox(0,0){$L_{2}$}}
\end{picture}}}
\caption{The concave nonagon $\Upsilon_{\kappa}$.}
\label{fig:ipname}
\end{center}
\end{figure}
Assume $P\not\in\mathcal{E}_{2}\cup\mathcal{E}_{5}\cup\mathcal{E}_{8}$.
Consider the line $L_{1}$ parallel to the $x$-axis and passing through $P$.
Since $L_{1}$ intersects $\Upsilon_{\kappa}$ at two points, we denote the second intersection point by $Q=(\bar x,y)$ (see figure \ref{fig:ipname})\footnote{If $P=\mathcal{V}_{1}$ then $Q=\mathcal{V}_{1}$.}.
Since both $P$ and $Q$ are on $\Upsilon_{\kappa}$, we have
\begin{align*}
\kappa
=
\min\left(H(x,y),H(y,x)\right)
\qquad
\mbox{and}
\qquad
\kappa
=
\min\left(H(\bar x,y),H(y,\bar x)\right).
\end{align*}
Eliminating $\kappa$ from these equations, we obtain the $x$-component of \eqref{eq:hky2dn}:
\begin{align}
\bar x
&=
-x
+
\max(y,0)
-
\min(y,0)
=
\xi(x,y).
\label{eq:xcomp}
\end{align}
On the other hand, if $P\in \mathcal{E}_{2}\cup \mathcal{E}_{5}\cup \mathcal{E}_{8}$ then we translate $P$ to the point $P^{\prime}=(x^{\prime},y)$ on the nearest vertex to the left, and do the same procedure as above.
Then we obtain $Q^{\prime}:=(\bar x^{\prime},y)$\footnote{If $P^{\prime}=\mathcal{V}_{3}$, $\mathcal{V}_{5}$, or $\mathcal{V}_{9}$ then $Q^{\prime}=\mathcal{V}_{8}$, $\mathcal{V}_{6}$, or $\mathcal{V}_{2}$, respectively.}, where $\bar x^{\prime}=\xi(x,y)+\zeta$ and we put $\zeta:=x-x^{\prime}$.
Let $Q=(\bar x,y)$ be the point which is the translation of $Q^{\prime}$ by $\zeta$ to the left.
Then $\bar x$ is given by \eqref{eq:xcomp}.
The $y$-component of \eqref{eq:hky2dn} is obtained similarly.
Thus the correspondence between $P$ and $\bar P$ is equivalent to the Brown map.
This implies the second part of proposition \ref{prop:Brown}.
\subsection{Decomposition into uQRT maps}
\label{subsec:decomp}
Now we justify the above geometric construction of the Brown map.
Let us consider a pair $(\mathcal{P},\check{\mathcal{P}})$ of tropical elliptic pencils:
\begin{align*}
\mathcal{P}=\left\{\mathcal{T}\left(E_{\mu}(\text{\mathversion{bold}{$\alpha$}};x,y)\right)\right\}_{\mu\in{\mathbb R}}
\qquad
\mbox{and}
\qquad
\check{\mathcal{P}}=\left\{\mathcal{T}\left(E_{\mu}({}^t\text{\mathversion{bold}{$\alpha$}};x,y)\right)\right\}_{\mu\in{\mathbb R}},
\end{align*}
where $\text{\mathversion{bold}{$\alpha$}}$ is chosen as follows
\begin{align}
\text{\mathversion{bold}{$\alpha$}}
=
\left(
\begin{matrix}
-\infty&0&-\infty\\
-\infty&-\infty&0\\
0&-\infty&0\\
\end{matrix}
\right).
\label{eq:No10}
\end{align}
Let the member of $\mathcal{P}$ for $\mu=\kappa$ be $\Gamma_{\kappa}$.
Then the complement of the tentacles, denoted by $\bar{\Gamma}_{\kappa}$, is given by $\kappa=H(x,y)$.
Also let the member of $\check{\mathcal{P}}$ for $\mu=\kappa$ be $\check{\Gamma}_{\kappa}$.
Then $\bar{\check{\Gamma}}_{\kappa}$ is given by $\kappa=H(y,x)$.
For the choice \eqref{eq:No10} of $\text{\mathversion{bold}{$\alpha$}}$, the uQRT map $(x,y)\mapsto(\bar x,\bar y)$ is
\begin{align}
\bar x
=
\xi(x,y)
\qquad
\bar y
=
-y+\max(\bar x,0),
\label{eq:uQRT1}
\end{align}
and the invariant curve of \eqref{eq:uQRT1} is $\bar{\Gamma}_{\kappa}$.
The map \eqref{eq:uQRT1} is periodic with period seven for any initial value \cite{Nobe03}.
\begin{remark
By definition given by Vigeland \cite{Vigeland04}, the tropical curve $\Gamma_{\kappa}$ is not a tropical elliptic curve because the vertex $(-\kappa,0)$ has singularity.
Nevertheless, we can construct the tropical Jacobian $J(\bar{\Gamma}_{\kappa})$ and the Abel-Jacobi map $\eta:\bar{\Gamma}_{\kappa}\to J(\bar{\Gamma}_{\kappa})$ as in the previous section because $\Gamma_{\kappa}$ has genus one.
Therefore, we call $\Gamma_{\kappa}$ a tropical elliptic curve ignoring its singularity.
\end{remark}
On the other hand, for the choice of $\text{\mathversion{bold}{$\alpha$}}$ as the transpose of \eqref{eq:No10}, the uQRT map $(x,y)\mapsto(\bar x,\bar y)$ is
\begin{align}
\bar x
=
-x+\max(y,0)
\qquad
\bar y
=
\xi(y,\bar{x}).
\label{eq:uQRT2}
\end{align}
The map \eqref{eq:uQRT2} is symmetric to \eqref{eq:uQRT1} with respect to $y=x$ and is also periodic with period seven for any initial value.
The invariant curve of \eqref{eq:uQRT2} is $\bar{\check{\Gamma}}_{\kappa}$.
The Brown map can be decomposed into the uQRT maps \eqref{eq:uQRT1} and \eqref{eq:uQRT2}.
Let $P$, $Q$, $\bar P$, $L_{1}$, and $L_{2}$ as in \ref{subsec:geomconst}.
Then there exists a unique member ${\Gamma}_{\kappa^{\prime}}$ of the pencil $\mathcal{P}$ passing through $P$ \footnote{Note that the value $\kappa^{\prime}$ of ${\Gamma}_{\kappa^{\prime}}$ does not always coincide with the value $\kappa$ of $\Upsilon_{\kappa}$.} (see figure \ref{fig:ipdecomp}).
If we consider a tropical line $TL_{1}$ passing through $P$ and $\mathcal{V}_{6}=(\kappa^{\prime},-\kappa^{\prime})$ then it intersects $\bar{\Gamma}_{\kappa^{\prime}}$ at $Q^{\prime}$ again.
This intersection defines a map $P\mapsto Q^{\prime}$ on $\bar{\Gamma}_{\kappa^{\prime}}$, which is the $x$-component of \eqref{eq:uQRT1}.
\begin{figure}[htbp]
\begin{center}
{\unitlength=.03in{\def1.0{1.0}
\begin{picture}(100,75)(-20,-25)
\dashline[10]{2}(40,-25)(40,45)
\dashline[10]{2}(40,12)(90,12)
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\dottedline(20,40)(20,-20)
\dottedline(20,-20)(0,-20)
\dottedline(0,-20)(-20,0)
\put(-15,15){\line(1,0){45}}
\put(30,15){\line(-1,-1){30}}
\put(0,-15){\line(-1,1){15}}
\put(-15,0){\line(0,1){15}}
\put(-18,8){\makebox(0,0){$P$}}
\put(23,2){\makebox(0,0){$Q$}}
\put(23,18){\makebox(0,0){$\bar P$}}
\put(-29,5){\makebox(0,0){$L_{1}$}}
\put(25,-13){\makebox(0,0){$L_{2}$}}
\put(-25,5){\color{red}\line(1,0){60}}
\put(-25,5.2){\color{red}\line(1,0){60}}
\put(-25,4.8){\color{red}\line(1,0){60}}
\put(20,-15){\color{blue}\line(0,1){50}}
\put(20.2,-15){\color{blue}\line(0,1){50}}
\put(19.8,-15){\color{blue}\line(0,1){50}}
\put(-15,5){\circle*{2}}
\put(20,5){\circle*{2}}
\put(20,15){\circle*{2}}
\put(60,35){\line(1,0){30}}
\put(90,35){\line(-1,-1){20}}
\put(70,15){\line(-1,1){10}}
\put(60,25){\line(0,1){10}}
\put(70,28){\makebox(0,0){$\bar{\check{\Gamma}}_{\kappa^{\prime\prime}}$}}
\put(80,40){\makebox(0,0){$\bar P^{\prime}$}}
\put(85,25){\makebox(0,0){$Q^{\prime}$}}
\put(60,40){\makebox(0,0){$\mathcal{V}_{3}$}}
\put(49,35){\makebox(0,0){$TL_{2}$}}
\put(80,35){\color{blue}\line(-1,0){25}}
\put(80,35.2){\color{blue}\line(-1,0){25}}
\put(80,34.8){\color{blue}\line(-1,0){25}}
\put(80,35){\color{blue}\line(0,-1){20}}
\put(80.2,35){\color{blue}\line(0,-1){20}}
\put(79.8,35){\color{blue}\line(0,-1){20}}
\put(80,35){\color{blue}\line(1,1){5}}
\put(80.2,35){\color{blue}\line(1,1){5}}
\put(79.8,35){\color{blue}\line(1,1){5}}
\put(80,25){\circle*{1.5}}
\put(80,35){\circle*{1.5}}
\put(60,35){\circle*{1.5}}
\dottedline(60,-10)(80,10)
\dottedline(80,10)(80,-20)
\dottedline(80,-20)(70,-20)
\dottedline(70,-20)(60,-10)
\put(73,-13){\makebox(0,0){$\bar{\Gamma}_{\kappa^{\prime}}$}}
\put(62.5,-3){\makebox(0,0){$P$}}
\put(85,-9){\makebox(0,0){$Q^{\prime}$}}
\put(85,-20){\makebox(0,0){$\mathcal{V}_{6}$}}
\put(49,-7.5){\makebox(0,0){$TL_{1}$}}
\put(80,-7.5){\color{red}\line(-1,0){25}}
\put(80,-7.7){\color{red}\line(-1,0){25}}
\put(80,-7.3){\color{red}\line(-1,0){25}}
\put(80,-7.5){\color{red}\line(0,-1){15}}
\put(80.2,-7.5){\color{red}\line(0,-1){15}}
\put(79.8,-7.5){\color{red}\line(0,-1){15}}
\put(80,-7.5){\color{red}\line(1,1){5}}
\put(80.2,-7.5){\color{red}\line(1,1){5}}
\put(79.8,-7.5){\color{red}\line(1,1){5}}
\put(62.5,-7.5){\circle*{1.5}}
\put(80,-7.5){\circle*{1.5}}
\put(80,-20){\circle*{1.5}}
\end{picture}}}
\caption{Decomposition of the Brown map into two uQRT maps.}
\label{fig:ipdecomp}
\end{center}
\end{figure}
On the other hand, there exits a unique member ${\bar{\check\Gamma}}_{\kappa^{\prime\prime}}$ of the pencil $\check{\mathcal{P}}$ passing through $Q^{\prime}$.
A map $Q^{\prime}\mapsto\bar P^{\prime}$ on $\bar{\check\Gamma}_{\kappa^{\prime\prime}}$ can also be defined in terms of the intersection between $\bar{\check\Gamma}_{\kappa^{\prime\prime}}$ and a tropical line $TL_{2}$ passing through $Q^{\prime}$ and $\mathcal{V}_{3}=(-\kappa^{\prime\prime},\kappa^{\prime\prime})$.
This is the $y$-component of \eqref{eq:uQRT2}.
If $Q^{\prime}=Q$ and $\bar P^{\prime}=\bar P$ then the composition of the uQRT maps \eqref{eq:uQRT1} and \eqref{eq:uQRT2} is equivalent to the Brown map.
To show this we return to the uQRT map with generic choice of $\text{\mathversion{bold}{$\alpha$}}$ for a while.
\subsection{Integrability of a map arising from a pair of tropical elliptic pencils}
\label{subsec:suff}
Let us consider a pencil of tropical elliptic curves
\begin{align}
\left\{\mathcal{T}\left(E_{\mu}({}^t\text{\mathversion{bold}{$\alpha$}};x,y)\right)\right\}_{\mu\in{\mathbb R}}
=
\left\{\mathcal{T}\left(E_{\mu}(\text{\mathversion{bold}{$\alpha$}};y,x)\right)\right\}_{\mu\in{\mathbb R}}.
\label{eq:pencil2}
\end{align}
Each member of the pencil \eqref{eq:pencil2} is symmetric to that of \eqref{eq:pencil} with respect to $y=x$.
Let ${C}_{\kappa}$ be the member of \eqref{eq:pencil2} then $\bar{{C}}_{\kappa}$ is given by $\kappa+x+y=F({}^t\text{\mathversion{bold}{$\alpha$}}_{-\infty};x,y)$
and is the invariant curve of the following uQRT map
\begin{align}
\bar x
=
G_{1}(y)-G_{3}(y)-x
\qquad
\bar y
=
F_{1}(\bar x)-F_{3}(\bar x)-y.
\label{eq:uQRTmapsym}
\end{align}
Now we consider a pair of pencils \eqref{eq:pencil} and \eqref{eq:pencil2}.
Let the member of \eqref{eq:pencil} be $\check{C}_{\kappa}$.
Then, the pair $({C}_{\kappa},\check{{C}}_{\kappa})$ of the members of \eqref{eq:pencil2} and \eqref{eq:pencil} for $\mu=\kappa$ defines the set of points satisfying
\begin{align}
\kappa+x+y
=
\min\left(
F(\text{\mathversion{bold}{$\alpha$}}_{-\infty};x,y),F({}^t\text{\mathversion{bold}{$\alpha$}}_{-\infty};x,y)
\right).
\label{eq:iccomp}
\end{align}
We denote the set by $U_{\kappa}$ and consider the one-parameter family $\{U_{\kappa}\}_{\kappa\in{\mathbb R}_{\geq0}}$ which fills the plane.
The pair $({C}_{\kappa},\check{{C}}_{\kappa})$ also defines a map whose $x$-component (resp. $y$-component) is that of \eqref{eq:uQRTmapsym} (resp. \eqref{eq:uQRTmap}) \footnote{For the choice \eqref{eq:No10} of $\text{\mathversion{bold}{$\alpha$}}$, \eqref{eq:uQRTC} and \eqref{eq:iccomp} reduce to the Brown map and its invariant curve, respectively.}:
\begin{align}
\bar x
=
G_{1}(y)-G_{3}(y)-x
\qquad
\bar y
=
G_{1}(\bar x)-G_{3}(\bar x)-y.
\label{eq:uQRTC}
\end{align}
If $(\bar x,\bar y)\in U_{\kappa}$ holds for any $(x,y)\in U_{\kappa}$ then \eqref{eq:uQRTC} is an integrable map whose invariant curve is given by \eqref{eq:iccomp}.
We have the following lemma (proof is given in \ref{app:proof}).
\begin{lemma
\label{lem:cond}
If the following holds for any $(x,y)\in U_{\kappa}\setminus\bar{{C}}_{\kappa}$
\begin{align}
F_{1}(y)+G_{3}(y)
=
G_{1}(y)+F_{3}(y)
\label{eq:cond}
\end{align}
then $(\bar x,y),(\bar x,\bar y)\in U_{\kappa}$.
\end{lemma}
Let the parameters of \eqref{eq:uQRTC} be as in \eqref{eq:No10}.
Then \eqref{eq:cond} reduces to
\begin{align*}
2y
=
\max\left(2y,y\right)
\quad
\Longleftrightarrow
\quad
y\geq0.
\end{align*}
This is satisfied for any $(x,y)\in\Upsilon_{\kappa}\setminus\bar{{\Gamma}}_{\kappa}$.
Then we have $Q^{\prime}, \bar P^{\prime}\in \Upsilon_{\kappa}$, therefore we conclude $Q^{\prime}=Q$ and $\bar P^{\prime}=\bar P$ as desired.
Thus the Brown map can be regarded as an integrable map arising from the pair $(\mathcal{P},\check{\mathcal{P}})$ of the tropical elliptic pencils.
\subsection{Linearization on tropical Jacobians}
Put ${\mathcal{O}}=\mathcal{V}_{1}$, ${\check{\mathcal{O}}}=\mathcal{V}_{3}$,
$T=\mathcal{V}_{6}$, and $\check{T}=\mathcal{V}_{8}$.
We define the Abel-Jacobi maps $\eta:\bar \Gamma_{\kappa}\to J(\bar{\Gamma}_{\kappa})={\mathbb R}/7\kappa{\mathbb Z}$ and $\check{\eta}:\bar{\check{\Gamma}}_{\kappa}\to J(\bar{\check{\Gamma}}_{\kappa})={\mathbb R}/7\kappa{\mathbb Z}$.
Then the uQRT maps \eqref{eq:uQRT1} and \eqref{eq:uQRT2} is the translations by $\eta(T)=\check{\eta}(\check{T})=4\kappa$ on the tropical Jacobians $J(\bar{\Gamma}_{\kappa})$ and $J(\bar{\check{\Gamma}}_{\kappa})$, respectively.
The total lattice length of $\Upsilon_{\kappa}$ is $\mathcal{L}=\sum_{i=1}^{9}\varepsilon_{i}|\mathcal{E}_{i}|=9\kappa$.
Put $J(\Upsilon_{\kappa})={\mathbb R}/\mathcal{L}{\mathbb Z}={\mathbb R}/9\kappa{\mathbb Z}$.
Consider the bijection $\nu:\Upsilon_{\kappa}\to J(\Upsilon_{\kappa})$ linear on each $\mathcal{E}_{i}$ and defined on the vertices $\mathcal{V}_{i}$ ($i=1,2,\ldots,9$) by the formula
\begin{align}
\begin{cases}
\nu({\mathcal{O}})={\eta}({\mathcal{O}})=0\\
\nu(\mathcal{V}_{i+1})=\nu(\mathcal{V}_{i})+\eta(\mathcal{V}_{i+1})-\eta(\mathcal{V}_{i})
&\mbox{if $\mathcal{E}_{i}\in \bar \Gamma_{\kappa}$}\\
\nu(\mathcal{V}_{i+1})=\nu(\mathcal{V}_{i})+\check{\eta}(\mathcal{V}_{i+1})-\check{\eta}(\mathcal{V}_{i})
&\mbox{if $\mathcal{E}_{i}\in \bar{\check{\Gamma}}_{\kappa}$},\\
\end{cases}
\label{eq:nu}
\end{align}
where the subscripts are reduced modulo nine.
Let us consider a piecewise linear function $\phi:{\mathbb R}\to{\mathbb R}$
\begin{align*}
\phi(u)
=
\max_{n\in{\mathbb Z}}\left(
\min\left(
2u-5n,u+2n+1
\right),
\min\left(
2u-5n-3,u+2n+2
\right)
\right).
\end{align*}
Define the injections $\psi:J(\bar{\Gamma}_{\kappa^{\prime}})\to J(\Upsilon_{\kappa})$ and $\check{\psi}:J(\bar{\check{\Gamma}}_{\kappa^{\prime\prime}})\to J(\Upsilon_{\kappa})$:
\begin{align*}
{\psi}(u)
:=
\kappa\phi\left(\frac{u}{\kappa^{\prime}}-1\right)+\kappa
\qquad
\mbox{and}
\qquad
\check{\psi}(u)
:=
\kappa\phi\left(\frac{u}{\kappa^{\prime\prime}}-5\right)+8\kappa.
\end{align*}
Note the following diagram to commute for any $(x,y)\in{\mathbb R}^2$
\begin{align*}
\begin{CD}
J(\bar \Gamma_{\kappa^{\prime}})@>\psi>> J(\Upsilon_{\kappa})@<\check{\psi}<< J(\bar{\check{\Gamma}}_{\kappa^{\prime\prime}})\\
@AA\eta A@AA\nu A@AA\check{\eta}A\\
\bar \Gamma_{\kappa^{\prime}}@<\iota<< \Upsilon_{\kappa} @>\check{\iota}>> \bar{\check{\Gamma}}_{\kappa^{\prime\prime}}\\
\end{CD}
\end{align*}
where $\iota$ and $\check{\iota}$ are the natural identifications of points.
\begin{proposition
\label{prop:JBM}
The Brown map $P\mapsto\bar P$ is linearized on $J(\Upsilon_{\kappa})$ in terms of $\nu:\Upsilon_{\kappa}\to J(\Upsilon_{\kappa})$:
\begin{align}
\nu(P)
\mapsto
\nu(\bar P)
=
\nu(P)+5\kappa.
\label{eq:prop}
\end{align}
\end{proposition}
(Proof)\qquad
Since the points $P$, $Q$, and $\mathcal{V}_{6}$ on $\bar{{C}}_{\kappa^{\prime}}$ (resp. $\mathcal{V}_{3}={\check{\mathcal{O}}}$, $\bar P$, and $Q$ on $\bar{\check{\Gamma}}_{\kappa^{\prime\prime}}$) are on the tropical line $TL_{1}$ (resp. $TL_{2}$) (see figure \ref{fig:ipdecomp}), we have
\begin{align*}
P\oplus Q={\mathcal{O}}
\qquad
\mbox{and}
\qquad
\bar P\oplus Q=\mathcal{V}_{4}.
\end{align*}
It immediately follows
\begin{align}
&{\eta}(P)+{\eta}(Q)
=0
\qquad \ \
\mbox{on $J(\bar \Gamma_{\kappa^{\prime}})$}
\label{eq:PQO}
\\
&\check{\eta}(\bar P)+\check{\eta}(Q)
=\kappa^{\prime\prime}
\qquad
\mbox{on $J(\bar{\check{\Gamma}}_{\kappa^{\prime\prime}})$}.
\label{eq:PQV2}
\end{align}
Since ${\psi}$ fixes zero, ${\psi}$ reduces \eqref{eq:PQO} to
\begin{align}
\nu(P)+\nu(Q)
=0
\qquad
\mbox{on $J(\Upsilon_{\kappa})$}.
\label{eq:nuPQ}
\end{align}
On the other hand, since $\check\psi(0)=2\kappa$, $\check\psi$ reduces \eqref{eq:PQV2} to
\begin{align}
\nu(\bar P)+\nu(Q)
=5\kappa
\qquad
\mbox{on $J(\Upsilon_{\kappa})$}.
\label{eq:nubPQ}
\end{align}
The relations \eqref{eq:nuPQ} and \eqref{eq:nubPQ} imply \eqref{eq:prop}.
\hfill\hbox{\rule[-2pt]{6pt}{6pt}}
It is easy to see that the fundamental period of a point in \eqref{eq:hky2dn} \cite{Nobe08} is
\begin{align*}
\frac{\mathcal{L}}{{\rm gcd}\left(\mathcal{L},5\kappa\right)}
=
\frac{9\kappa}{{\rm gcd}\left(9\kappa,5\kappa\right)}
=
9.
\end{align*}
This implies the first part of proposition \ref{prop:Brown}.
\subsection{General solution}
Let us consider the ultradiscrete elliptic theta function\footnote{$\Theta(u;\theta)$ is an ultradiscretization of the elliptic theta function $\vartheta_{01}(z,\tau)$.} $\Theta:{\mathbb R}\to{\mathbb R}$ \cite{Nobe06}:
\begin{align*}
\Theta(u;\theta)
=
-\theta \left(u-\frac{1}{2}\right)^{2}
+
\theta\max_{n\in{\mathbb Z}}\left(
2nu-n-n^{2}
\right),
\end{align*}
where $\theta\in{\mathbb R}$.
We define the piecewise linear function $\Xi:{\mathbb R}\to{\mathbb R}$
\begin{align*}
\Xi(u;\alpha,\beta,\gamma,\theta)
:=&
\left\{
\Theta\left(\frac{u}{\alpha};\theta\right)
-
\Theta\left(\frac{u-\beta}{\alpha};\theta\right)
\right\}\\
&\qquad-
\left\{
\Theta\left(\frac{u-\gamma}{\alpha};\theta\right)
-
\Theta\left(\frac{u-\beta-\gamma}{\alpha};\theta\right)
\right\},
\end{align*}
where we assume $\alpha\geq\beta+\gamma$ and $\beta\geq\gamma>0$.
General solutions to the uQRT maps can be given by using $\Xi$ \cite{Nobe08}.
Let $P_{0}=(x_{0},y_{0})$.
Also let $\kappa=\min\left(H(x_{0},y_{0}),H(y_{0},x_{0})\right)$.
Then the general solution to the Brown map with the initial value $P_{0}$ is given as follows
\begin{align*}
&x_{n}
=
\Omega(u_{n})
:=
\max\left(
\Xi\left(u_{n};9\kappa,6\kappa,2\kappa,\frac{9}{2}\kappa\right),
\Xi\left(u_{n}-\kappa;9\kappa,3\kappa,3\kappa,\frac{9}{2}\kappa\right)
\right)\\
&y_{n}
=
\Omega(u_{n}-2\kappa),
\end{align*}
where we put $u_{n}:=\nu(P_{0})-3\kappa+5\kappa n$ (see figure \ref{fig:sol}).
\begin{figure}[htbp]
\centering
{\unitlength=.05in{\def1.0{1.0}
\begin{picture}(60,35)(-30,-20)
\put(-33,-4){\vector(1,0){70}}
\put(-28,-15){\vector(0,1){30}}
\put(41,-4){\makebox(0,0){$u_{n}$}}
\dashline[10]{1}(-7,-15)(-7,13)
\dashline[10]{1}(14,-15)(14,13)
\dashline[10]{1}(35,-15)(35,13)
\dashline[10]{1}(-30,-11)(37,-11)
\dashline[10]{1}(-30,3)(37,3)
\dashline[10]{1}(-30,10)(37,10)
\thicklines
\put(-28,3){\line(1,-1){14}}
\put(-14,-11){\line(1,0){7}}
\put(-7,-11){\line(1,1){14}}
\put(7,3){\line(1,0){7}}
\put(14,3){\line(1,1){7}}
\put(21,10){\line(1,-1){7}}
\put(28,3){\line(1,0){7}}
\dottedline(-14,3)(0,-11)
\dottedline(0,-11)(7,-11)
\dottedline(7,-11)(21,3)
\dottedline(21,3)(28,3)
\dottedline(28,3)(35,10)
\dottedline(-28,10)(-21,3)
\dottedline(-21,3)(-14,3)
\put(-7,-18){\makebox(0,0){$3\kappa$}}
\put(-28,-18){\makebox(0,0){$0$}}
\put(14,-18){\makebox(0,0){${6\kappa}$}}
\put(35,-18){\makebox(0,0){$9\kappa$}}
\put(-35,-4){\makebox(0,0){$0$}}
\put(-35,-11){\makebox(0,0){$-\kappa$}}
\put(-35,3){\makebox(0,0){$\kappa$}}
\put(-35,10){\makebox(0,0){$2\kappa$}}
\end{picture}
}}
\caption{The general solution $x_n=\Omega(u_n)$ (solid line) and $y_{n}=\Omega(u_{n}-2\kappa)$ (dotted line) to the Brown map.
}
\label{fig:sol}
\end{figure}
\section{Concluding remarks}
We show that the Brown map \eqref{eq:hky2dn} can be obtained geometrically from the one-parameter family $\{\Upsilon_{\kappa}\}_{\kappa\in{\mathbb R}}$ of the concave nonagons, each of which is the invariant curve of \eqref{eq:hky2dn}.
This is because \eqref{eq:hky2dn} is arising from the pair $(\mathcal{P},\check{\mathcal{P}})$ of tropical elliptic pencils, and therefore it decomposes into two uQRT maps which are mutually symmetric with respect to $y=x$.
Through this description, we reformulate \eqref{eq:hky2dn} as a translation on $J(\Upsilon_{\kappa})$ in terms of the addition formulae of the tropical elliptic curves.
We then give the general solution to the initial value problem by using the ultradiscrete theta function.
In \ref{subsec:suff}, we give a sufficient condition for the map \eqref{eq:uQRTC} arising from the pair of tropical elliptic pencils \eqref{eq:pencil} and \eqref{eq:pencil2} to be integrable (see lemma \ref{lem:cond}).
Using this criterion, we search all possible choices of the parameter $\text{\mathversion{bold}{$\alpha$}}$ for the Brown-type map, i.e., for the map which has the property that the invariant curve is a concave polygon and the period is constant for any initial value.
Then we find several choices of $\text{\mathversion{bold}{$\alpha$}}$ passing the criterion; however, all of them except the Brown map are contained in the uQRT family.
Thus, the Brown map is distinctive among ultradiscrete integrable maps.
The procedure we discuss here, which composes maps arising from a pair of mutually symmetric tropical elliptic pencils, is considered to be generic.
Considering a wide class of symmetry to find Brown-type maps is a further problem.
\section*{Acknowledgment}
This work was partially supported by grants-in-aid for scientific research, Japan society for the promotion of science (JSPS) 19740086 and 22740100.
|
{
"timestamp": "2010-09-22T02:01:02",
"yymm": "1009",
"arxiv_id": "1009.3997",
"language": "en",
"url": "https://arxiv.org/abs/1009.3997"
}
|
\section*{Temperature and power dependence}
Figures \ref{fig:power}a and b show the temperature dependence of
the resonance frequency and quality factor. These values are
measured at large bias currents where the backaction is
negligible. The frequency change due to temperature is small
compared to the observed backaction (see main text). The intrinsic
quality factor decreases significantly with increasing
temperature. This rules out that the observed frequency shift and
Q-factor change are caused by heating of the resonator due to
Joule heating in the junctions: We observe an \emph{increase} in
quality factor with increasing bias current and voltage setpoint
(i.e. dissipation), but a \emph{decrease} in quality factor with
increasing temperature. An increased damping at higher
temperatures is seen more often in micro- or nanomechanical
resonators \cite{yamaguchi_ASS_qfactor, huettel_NL_highQ,
lahaye_science, zolfagharkhani_PRB_qfactor}.
The observed frequency shift and change in damping do not depend
on the driving power. As shown in Figure \ref{fig:power}c, the
measured resonator response stays the same in all panels. Although
the driving power is changed by three orders of magnitude, the
only effect is that the signal-to-noise ratio becomes better when
increasing the power.
For the highest driving power ($P_d = -75 \unit{dBm}$) and highest
Q-factor ($Q \sim 5800$) the amplitude of the resonator motion is
$u_{\mathrm{max}} = 20 \unit{pm}$, as determined using the
calibrated displacement responsivity
\cite{etaki_NP_squid_position_detector}. The flux through the dc
SQUID is then modulated with an amplitude $a B \ell
u_{\mathrm{max}} = 0.02 \unit{\Phi_0}$. So even for the largest
resonator motion, the change in flux is much smaller than a flux
quantum. Exactly on resonance, the piezo motion is $Q$ times
smaller than $u_{\mathrm{max}}$, about $3 \unit{fm}$. The driving
force $F_d$ is then given by the resonator mass $m$ times the
acceleration of the piezo element, $F_d = m \omega^2 u_p \approx m
\omega_0^2 u_p$. The measurements shown in the main text are done
with a driving power of $- 80 \unit{dBm}$ ($F_d = 48 \un{fN}$).
\begin{figure}[tb]
\centering
\includegraphics[width=8.6cm]{supp_power.eps}
\caption{Temperature dependence of the intrinsic quality factor
$Q_0$ (a) and resonator frequency $f_0$ (b). (c) Colorscale plot
of the oscillator response at $B = 100 \unit{mT}$ at different
driving powers $P_d$. This measurement is done with a voltage
setpoint halfway between $V_\mathrm{min}$ and $V_\mathrm{max}$.
\label{fig:power}}
\end{figure}
\section*{Device B}
All effects that we have observed in the device that we discuss in
the main text have also been measured in a second device. The
resonator in this dc SQUID operates around $2 \unit{MHz}$. The
maximum critical current of device B is $\icmax = 2I_0 = 2.4
\unit{\mu A}$ at $B = 115 \unit{mT}$ and $\icmax = 1.0 \unit{\mu
A}$ at $B = 130 \unit{mT}$.
In the measurements on device B, the feedback loop could not
maintain the SQUID voltage for low voltage setpoints. Also, a less
sensitive, room temperature amplifier was used for the resonator
signal. Its lower gain resulted in an increased scatter in the
data and the inability to explore the region with the highest
backaction, i.e., at low bias current and low voltage setpoints.
The measurements in Fig. \ref{fig:devB} show qualitatively the
same backaction as the data presented in the main text.
\begin{figure}[tb]
\centering
\includegraphics[width=8.6cm]{supp_devB.eps}
\caption{Backaction measurements of device B. (a) Frequency shift
and (b) quality factor plotted versus the normalized bias current
at $B = 115 \un{mT}$ (circles) and $B = 130 \un{m}T$ (triangles).
The inset depicts the voltage setpoint for these measurements with
respect to $V_\mathrm{min}$ and $V_\mathrm{max}$. Bias current and
setpoint dependence of the frequency shift (c) and damping (d) at
$B = 115$ mT. The solid lines indicate the measured
$V_\mathrm{min}$ and $V_\mathrm{max}$. \label{fig:devB}}
\end{figure}
\section*{The RCSJ model for the dc SQUID}
To calculate the backaction, the dc SQUID is modelled using the
resistively- and capacitively-shunted junction (RCSJ) model. This
widely-used model is discussed in detail in Ref.
\cite{squidhandbook}. The introduction to this model presented
here is largely based on this review. A current $I_B$ is sent
through the SQUID and the circulating current $J$ redistributes
the current over the two junctions, which we assume to be
identical. In the RCSJ model, the two junctions (labelled with
$i=1,2$) are modelled as a resistor ($R$), capacitor ($C$) and an
``ideal'' Josephson junction with critical current $I_{0}$ in
parallel. The voltage over each junction is related to the time
derivative of the phase difference $\delta_i$ of the
superconducting wave function: $V_i = \Phi_0 \ \dot \delta_i/
2\pi$, where $\Phi_0 = h/2e = 2.05 \times 10^{-15} \unit{T m^2}$
is the flux quantum. Current conservation yields two second-order
differential equations, governing the time-dependence of the phase
differences $\delta_{1,2}$ of the junctions:
\begin{eqnarray}
\frac{\Phi_0}{2\pi} C \ddot{\delta}_{1} + \frac{\Phi_0}{2\pi}
\frac{1}{R}\dot{\delta}_{1} + I_{0} \sin \delta_{1} & = & \half
I_{B} + J,\\
\frac{\Phi_0}{2\pi} C \ddot{\delta}_{2} + \frac{\Phi_0}{2\pi}
\frac{1}{R}\dot{\delta}_{2} + I_{0} \sin \delta_{2} & = & \half
I_{B} - J.\end{eqnarray}
These equations are coupled to each other by the amount of flux
piercing the loop:
\begin{equation}
\delta_2 - \delta_1 = 2\pi\cdot\Phi_{\mathrm{tot}}/\Phi_0.
\end{equation}
The total flux $\Phi_{\mathrm{tot}}$ has two contributions: the
externally applied flux $\Phi$ (which also includes the flux due
to the resonator displacement) and the flux due to the circulating
current flowing through the inductance of the loop $L$, i.e.,
$\Phi_{\mathrm{tot}} = \Phi + LJ$.
The equations are scaled to simplify their numerical integration.
This yields:
\begin{eqnarray}
\bc \ddot{\delta_1} + \dot{\delta_1} + \sin \delta_1 &=& \imath_B/2+\jmath \\
\bc \ddot{\delta_2} + \dot{\delta_2} + \sin \delta_2 &=& \imath_B/2-\jmath \\
2\pi (\phi + \bl \jmath/2) & = & \delta_2 - \delta_1 .
\end{eqnarray}
The bias current and circulating current are normalized using the
critical current: $\imath_B = I_B/I_0$ and $\jmath = J/I_0$.
Furthermore, time is scaled using the characteristic frequency
$\omega_c = 2\pi RI_0/\Phi_0$, fluxes using the flux quantum,
i.e., $\phi = \Phi/\Phi_0$, and voltages using the characteristic
voltage $RI_0$ so that $v = V/RI_0 = (\dot \delta_1+\dot
\delta_2)/2$. The parameter $\bl = 2I_0L/\Phi_0$ and $\bc = 2\pi
I_0 R^2 C/\Phi_0 $ are the inductive screening parameter and the
Stewart-McCumber number respectively. The inductive screening
parameter indicates how much a change in flux is screened by the
circulating current $J$ flowing through the self-inductance of the
loop $L$, whereas $\bc$ indicates the importance of inertial terms
due to the junction capacitance $C$.
The three equations are integrated numerically for different bias
conditions, i.e., different values for the bias current $I_B$ and
for the flux $\Phi$ through the SQUID. Figure
\ref{fig:simulation}a shows typical examples of calculated
time-traces of the circulating current $\jmath$ and voltage $v$.
Both are rapidly oscillating at a frequency of $0.69~\omega_c$,
which is the Josephson frequency that equals the average value of
the voltage $\overline v$ \cite{squidhandbook}. The maximum
current that the dc SQUID can carry without generating a voltage
equals the sum of the critical currents of the two junctions:
$\icmax = 2I_{0}$.
\begin{figure}[tb]
\centering
\includegraphics{supp_simulation.eps}
\caption{(a) Calculated timetrace of the circulating current
$\jmath(t)$ and voltage $v(t)$ in the absence of noise. (b)
Circulating current for a small ($\phi_\mathrm{mod}= 0.01$)
modulation of the flux with $\omega_\mathrm{mod} = 0.02$. The
time-averaged value of the circulating current $\overline \jmath
(t)$ has a phase shift with respect to the modulation $\phi(t)$ as
indicated by the orange lines. (c) Absolute value of the Fourier
transform of the timetraces of the SQUID voltage and circulating
current shown in (b). (d) The modulation frequency dependence of
the real (dark gray) and imaginary (light gray) parts of the
transfer function. The lines are a guide to the eye. These
simulations were done for the dc SQUID parameters from the main
text ($\bl = 0.21$ and $\bc = 0.23$) at $\imath_B = 2$ and $\phi =
0.25$. \label{fig:simulation}}
\end{figure}
The critical current ($I_0 = 1.1 \unit{\mu A}$) and the
normal-state resistance ($R = 15.6 \unit{\Omega}$) of the
junctions are estimated from the IV-characteristics (Fig. 1b of
the main text). Using finite-element simulations we estimate $L =
175 \unit{pH}$ for our device. Finally, the capacitance $C = 0.6
\unit{pF}$ is obtained from the position of the LC resonance in
the dc SQUID \cite{squidhandbook}.
\section*{Calculation of the transfer functions}
When the resonator moves, the flux through the dc SQUID loop is
altered, which in turn changes the average circulating current
$J$. In principle, $J(t)$ could depend on the all the past
displacements, $u(t')$ for $t' < t$. However, the dc SQUID reacts
at a frequency ($\sim \omega_c/2\pi \sim 8 \un{GHz}$) that is much
faster than that of the resonator ($1 \un{MHz}$), so the
circulating current $J(t)$ is expected to depends on the
\emph{instantaneous} displacement $u(t)$. For small amplitudes ($u
\ll \Phi_0/aB\ell$) the response is linear and gives a
contribution $J_1(t) = c_1 u(t)$. Another contribution comes from
the velocity of the resonator, $\dot u $, which causes a
time-varying flux. This generates an electromotive force in the
SQUID loop (Faraday's induction law), which also changes the
circulating current by $J_2(t) = c_2 \dot u (t)$. Combining these
two effects gives $J(t) = J(u,\dot u ) = c_1 u(t) + c_2 \dot
u(t)$. The values of the parameters $c_1$ and $c_2$ depend on the
dynamics of the dc SQUID and are $c_1 = \pderl{J}{u}$ and $c_2 =
\pderl{J}{\dot u}$. As discussed in the main text, these
quantities are related to the intrinsic flux-to-current transfer
functions, $\jphi = \Phi_0 / (a B \ell I_0) \times \pderl{J}{u}$
and $\jphidot = \omega_c \Phi_0 / (a B \ell I_0) \times
\pderl{J}{\dot u}$.
In principle $\jmath_\phi$ can be obtained by calculating the
average circulating current at different fluxes and then
numerically differentiating this to obtain $\jmath_\phi =
\partial \jmath / \partial \phi$. However, for the
velocity-dependent transfer function this is not possible. Our
method for calculating these transfer functions $\jphi$ and
$\jphidot$ works as follows: We calculate the steady-state
response of the circulating current with a small modulation added
to the applied flux, $\phi \rightarrow \phi + \phi_\mathrm{mod}
\cos(\omega_\mathrm{mod}t)$. Figure \ref{fig:simulation}b shows
that this modulates the circulating current $\jmath(t)$. Figure
\ref{fig:simulation}c shows the Fourier transform of the
circulating current and the voltage. In the spectrum of both
$\jmath$ and $v$ a peak appears at the modulation frequency. The
real part of the peak corresponds to the derivative $\jmath_\phi =
\real[\jmath_\mathrm{mod}/\phi_\mathrm{mod}]$, while
$\imag[\jmath_\mathrm{mod}/\phi_\mathrm{mod}] = -
\omega_\mathrm{mod} \jmath_{\dot \phi}$ and similar for
$v_\mathrm{mod}$. The frequency dependence in Fig
\ref{fig:simulation}d, shows a constant
$\real[\jmath_\mathrm{mod}]$ and a linearly increasing
$\imag[\jmath_\mathrm{mod}]$ as indicated by the dotted lines. The
transfer functions do not depend on the modulation frequency,
provided that it is sufficiently low. This confirms that the
circulating current only depends on the instantaneous displacement
and velocity as was postulated at the beginning of this Section.
\end{document}
|
{
"timestamp": "2010-09-22T02:02:26",
"yymm": "1009",
"arxiv_id": "1009.4115",
"language": "en",
"url": "https://arxiv.org/abs/1009.4115"
}
|
\section{Introduction}
\label{intro}
The problem of high temperature superconductivity \cite{Ginzburg} has been opened for experimental study by the discovery of high-$T_c$ cuprates back in 1986 but among a number of models suggested in the subsequent years none has been generally accepted so far. In this situation it is reasonable to ask what are the chances for resolving the problem in the near future and what are the reasons for reading yet another review on this subject. We believe, it is the present level of experimental techniques, developed, to the large extent, in the endeavor to solve the high-$T_c$ problem, that suggests positive answers to the both questions. At the present time, it seems that both the experimental accuracy and understanding of how to decipher the key interactions from the experimental data have just reached the complexity level of the problem.
In particular, the angle resolved photoemission spectroscopy (ARPES), continuously improving \cite{ARPES}, nowadays provides a direct view of the rich spectrum of one-particle fermionic excitations, which encapsulates all the interactions of the electrons in crystal, with the accuracy better than 0.25 \% of the Brillouin zone (0.004 {\AA}$^{-1}$) in momentum \cite{KoralekRSI2007}, a few meV in energy \cite{KissRSI2008,IshizakaPRB2008}, and down to 1 K sample temperature \cite{BorisenkoLFA,KordyukLFA}. On the other hand, the progress in understanding the structure of ARPES spectra is proven by the successful bridging ARPES to other experimental techniques such as inelastic neutron scattering (INS) \cite{ChatterjeePRB2007,InosovPRB2007,DahmNP2009}, scanning tunneling spectroscopy (STS) \cite{McElroyPRL2006,ChatterjeePRL2006,KordyukJES2007}, Raman spectroscopy \cite{Raman}, as well as to the macroscopic probes, such as resistivity and Hall measurements \cite{EvtushinskyTaSe2}, $\mu$SR \cite{EvtushinskyNJP}, etc. So, the evaluation of the existing models for consistency with different experimental probes as well as their reevaluation with the improved experimental accuracy appears timely and should be important for establishing the true mechanism of high-$T_c$ superconductivity.
In this paper, we summarize a consistent view of electronic interactions in HTSC cuprates, in which, the spin-fluctuations play a decisive role in formation of the fermionic excitation spectrum in the normal state and are sufficient to explain the high transition temperatures to the superconducting state while the pseudogap phenomenon is a consequence of a Peierls-type intrinsic instability of electronic system to formation of the spin density waves. We evaluate the robustness of our conclusions and room for the electron-phonon interaction in future experiment. Ironically, a similar analysis being applied to the iron pnictides reveals especially strong electron-phonon coupling that suggests important role of phonons in high-$T_c$ superconductivity of pnictides.
\section{Role of spin-fluctuations in cuprates}
\label{sec:2}
At this point, it is reasonable to ask whether searching for a mediator or a `pairing glue' \cite{AndersonSci2007} is relevant for the superconductivity in cuprates at all? Leaving a detailed answer to this question for a future study we note that spectroscopically, i.e. from the point of view of one-particle excitation spectrum, known with present experimental accuracy, the superconducting state of the cuprates reveals nothing\footnote{We do not consider the issue of Fermi-liquidity here but the pseudo-gap phenomenon is discussed in Sec.~\ref{sec:3}.} beyond the conventional BCS model where both the spin-fluctuations and phonons have provided numerous evidences for the role of the `pairing glue'. Therefore, here we focus on spectroscopic differences that can help to choose unambiguously between those two scenarios and stay open for any inconsistencies of the one-particle spectral function with BCS theory, hoping that, if the mechanism of superconductivity is principally different, the difference can be eventually identified by ARPES.
Starting from the first proposals and up to now the main argument between the promoters of either phonons or spin-fluctuations against the opposite scenario is that the coupling strength of the relevant excitations to electrons is by far not sufficient to provide the high-$T_c$ pairing \cite{KeePRL2002,AbanovPRL2002,Giustino}. We do not discuss this kind of arguments here but approach the problem from the empirical side, trying to formulate a set of critical experimental observations which can be described by one scenario but not by the other.
The idea of this approach is to understand the constituents of the quasiparticle spectrum of cuprates in the normal state, identify all the essential interactions which form this spectrum and estimate the strength of their coupling to electrons. Naturally, one should aim at a complete understanding of the quasiparticle spectrum in the whole Brillouin zone (BZ) but even its simple mapping with a sufficient accuracy requires tremendous experimental efforts. Therefore, as the first challenge, a successful model should be able to describe the main peculiarities of this spectrum known as `nodal kink' and `antinodal dip', reasonably explaining their behavior with doping and temperature. And since the key requirement for the empirical approach is experimental accuracy, we focus on two compounds: bilayer Bi(Pb)$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ (BSCCO), the most `arpesable' high-$T_c$ compound \cite{KordyukLTP2006}, and YBa$_2$Cu$_3$O$_{7-\delta}$ (YBCO), the best compound to compare ARPES and neutron scattering experiments \cite{DahmNP2009}.
\subsection{Antinode}
\label{sec:2.1}
\begin{figure}
\centering
\resizebox{0.52\columnwidth}{!}{%
\includegraphics{Fig1left.pdf}}
\resizebox{0.37\columnwidth}{!}{%
\includegraphics{Fig1abcd.pdf} }
\caption{(left) Occupied low energy electronic band structure of the bi-layer high-$T_c$ cuprates is sketched in the first Brillouin zone. (a-d) ARPES spectra from the antinodal cut trough the saddle point illustrate strong dependence of the renormalization strength on doping and temperature. After \cite{BorisenkoPRL2003}.}
\label{fig:1}
\end{figure}
The expected conduction band structure of the bi-layer cuprates in the first BZ is sketched in Fig.~1. The areas of the primary interest, the nodal and antinodal regions, lie along the BZ diagonal and around the Fermi surface crossings by the BZ boarder, respectively. The most outstanding feature of the quasiparticle spectrum has been observed on the energy distribution curves (EDC) from the antinodal region back in 1991 \cite{DessauPRL1991} and known as the `peak-dip-hump' lineshape. Considering the `dip' as a consequence of very strong scattering of the electrons by a `bosonic mode' immediately nominates the appropriate bosons for the role of the pairing glue. So, the understanding of its mechanism is called for.
Taking into account the role of the bilayer splitting, which has appeared to be the main responsible for the peak-dip-hump lineshape in the overdoped BSCCO \cite{KordyukPRL2002}, the true doping and temperature dependence of the coupling to the mode has been revealed: it emerges below $T_c$ but its strength decreases monotonically with overdoping \cite{BorisenkoPRL2003,GromkoPRB2003}, vanishing at about 24\% of holes per unit cell \cite{KimPRL2003}. The examples of the ARPES spectra taken along the antinodal direction in Fig. 1 (a)-(d) illustrate this. The energy scale and location in momentum of this strong renormalization, as well as its abrupt emergence below $T_c$, point unambiguously to the famous magnetic resonance \cite{EschrigPRB2003,ChubukovPRB2004,EschrigAiP2006} as the main scatterer. However, the observed strong doping dependence of the renormalization strength seems to be the most crucial evidence for the spin-fluctuation scenario against phonons. Within the former scenario, the strong doping dependence can be naturally understood as a proximity to the antiferromagnet (see Fig. 2(d)), while the complete vanishing of renormalization with overdoping is hard to reconcile with phonons.
\subsection{Node}
\label{sec:2.2}
Exactly the opposite argument has been applied to the nodal direction where an ubiquity of the kink \cite{VallaSci99} in the quasiparticle dispersion for different compounds, doping levels, and temperatures has been considered as an evidence for the strong coupling to a phonon mode \cite{LanzaraNature01}. One should note, however, that since the renormalization strength along the nodal direction is much smaller than in the antinodal region, its parametrization requires a much more careful analysis. In other words, in order to conclude about the properties of the bosonic contribution to quasiparticle spectrum one should single it out first. Practically this means that the electron-boson part, $\Sigma_{b}$, should be singled out from the total quasiparticle self-energy, which, in turn, should be derived from the ARPES spectrum and disentangled from the artificial effects such as bilayer splitting \cite{KordyukPRB2004}, superstructure, photoemission matrix elements, etc. As a result, while the nodal experimental dispersion looks similar for different compounds, doping levels and temperatures (see \cite{LanzaraNature01,ZhouNature03} or Fig. 2(a)), the derived $\Sigma_{b}$ varies essentially and, when the experimental artifacts are properly accounted for, exhibits critical dependence on doping and temperature \cite{KordyukPRL2004,KordyukPRB2005,KordyukPRL2006}.
\begin{figure}
\center{
\resizebox{0.56\columnwidth}{!}{%
\includegraphics{Fig2abc.pdf} }
\resizebox{0.37\columnwidth}{!}{%
\includegraphics{Fig2d.pdf} }}
\caption{Temperature and doping dependence of renormalization along the nodal direction. (a) MDC dispersions of an optimally doped BSCCO ($T_c = 92$K) at different temperatures \cite{KordyukPRL2006}. (b) Real part of the self-energy for the overdoped (OD 75K) and underdoped (UD 77K) samples \cite{KordyukPRB2005}. (c) Dependence of the total coupling strength on doping and temperature \cite{KordyukPRL2006}. (d) A sketch to illustrate similarly strong $xT$-dependence of $T^*$, pseudogap, and coupling strength in terms of proximity to antiferromagnet: $\lambda^*$ refers to both the nodal electron-phonon coupling strength, $\lambda_b$, in the pseudogap state and antinodal renormalization strength below $T_c$; $\lambda_b(x, T<T_c)$ is also illustrated by the color gradient.}
\label{fig:2}
\end{figure}
In Ref. \cite{KordyukPRL2006} both the nodal quasiparticle self-energy and the bare band dispersion have been derived from ARPES spectra with a self-consistent procedure \cite{KordyukPRB2005}. An example of temperature dependence of the experimental dispersion is shown in Fig. 2(a) for the optimally doped BSCCO while Fig. 2(b) illustrates the doping dependence of the real part of the self-energy. The data suggests that the total self-energy can be considered a sum of three components: $\Sigma(\omega, T, x) = \Sigma_{imp}(T, x) + \Sigma_{el}(\omega) + \Sigma_{b}(\omega, T, x)$. Here $\Sigma_{imp}$ stands for elastic scattering on impurities that generally increases with $T$ and is sample- rather than doping-dependent \cite{EvtushinskyPRB2006}. $\Sigma_{el}$ has been associated with the electron-electron scattering due to Coulumb interaction \cite{KordyukPRL2004}. It is important to stress that the electron-electron scattering in HTSC cuprates makes an essential contribution to the low energy bare band renormalization with $\lambda_{el} \approx 0.5$.\footnote{Here we define the coupling strength as $\lambda = -\Sigma'(\omega)/\omega$ at $\omega \rightarrow 0$.} This component almost does not depend on $T$ and $x$ but is a smooth function of $\omega$, though, in terms of its real part, with the maximum on the scale of the band width \cite{KordyukPRB2005}. The last contribution, $\Sigma_{b}$, is left for the electron-boson coupling. Its imaginary part is a step-like function, and it is this component which is only responsible for the kink in the dispersion at 50--80 meV binding energy (depending on cuprate family, doping and temperature). What is important, $\Sigma_{b}$ depends critically on $T$ and $x$, following the same trend on the phase diagram associated with the `proximity to antiferromagnet' (see the sketch in Fig. 2(d)). Fig. 2(c) shows the total coupling strength, $\lambda$, as function of $x$ and $T$ that can be summarized as follows: $\lambda(x,T) = \lambda_{el} + \lambda_b(x,T)$. So, both the coupling strength to the bosonic mode in the antinodal region below $T_c$ and the electron-boson coupling strength along the nodal direction just above $T_c$ shows the same trend with doping, as indicated by the solid red line in Fig. 2(d). We consider such a dependence as the most robust evidence in favor of the spin-fluctuation rather than phonon origin of the nodal kink.
The revealed $xT$-dependence of the nodal and antinodal renormalizations leaves some open questions. The most straightforward one is the `kink puzzle'. Namely, why the antinodal renormalization disappears abruptly above $T_c$, while the nodal kink survives up to much higher temperatures? It has been suggested that the nodal kink is the result of coupling to rather persistent high energy part of the spin-fluctuation spectrum known as a gapped continuum \cite{ChubukovPRB2004,KordyukPRL2006}. To clarify this issue, the momentum dependence of both the quasiparticle and bosonic spectra should be taken into account.
\subsection{Bridging ARPES and INS}
\label{sec:2.3}
\begin{figure}
\center{
\resizebox{0.6\columnwidth}{!}{%
\includegraphics{Fig3ab.pdf} }
\resizebox{0.31\columnwidth}{!}{%
\includegraphics{Fig3c.pdf} }}
\caption{Bridging ARPES and INS. (a) The Fermi surface of YBCO in the 1st BZ derived from ARPES data \cite{ZabolotnyyPRB2007} represents the fermionic Green's function. (b) The intensity of spin excitations along $Q = q(2\pi,2\pi)$ resulting from numerical fits to the INS spectra measured by V. Hinkov and B. Keimer (MPI, Stuttgart) \cite{DahmNP2009}. (c) Comparison of experimental (upper row) and theoretical (lower row) fermionic spectra (see Ref. \cite{DahmNP2009} for details), by T. Dahm (University of T\"{u}bingen).}
\label{fig:3}
\end{figure}
The aim of this section is to show that the spin-fluctuation spectrum, $\chi(\Omega, \textbf{q})$, is indeed consistent with the spectrum of one-particle fermionic excitations, $A(\omega, \textbf{k})$ $\propto$ Im($G$), in the whole reciprocal space. We start from the final empirical conclusion about the structure of the fermionic Green's function of the cuprates that can be formulated by a conceptual equation
\begin{equation}\label{E1}
G^{-1} = G^{-1}_0 - \bar{U}^2 G \star G \star G,
\end{equation}
which is the Dyson equation $G^{-1} = G^{-1}_0 - \Sigma$ extended by the following definition of fermionic self-energy and electronic susceptibility
\begin{eqnarray}\label{E2}
\Sigma &=& \bar{U}^2 \chi \star G,\\
\chi &=& G \star G.
\end{eqnarray}
The sense of these formulas is that the Green's functions of bare electrons and fermionic quasiparticles, $G_0$ and $G$, are related by a single parameter, $\bar{U}$, a spin-fermion coupling constant. The `$\star$' signs denote here the operations with the meaning of cross-correlation, but the exact relations for $\Sigma$ and $\chi$ can be found in Refs. \cite{DahmNP2009} and \cite{InosovPRB2007}, respectively. The justification of Eq. \ref{E1} can be separated, consequently, into two steps, according to Eqs.~2 and 3.
The first step is a search for `fingerprints' of the bosonic spectrum in the fermionic one. It can be also formulated as $G^{-1} = G^{-1}_0 - \bar{U}^2 \chi \star G$. Deriving $G(\omega, \textbf{k})$ and $G_0(\omega, \textbf{k})$ from ARPES and $\chi(\Omega, \textbf{q})$ from INS, one can check their mutual consistency and, if it is the case, estimate $\bar{U}$. Such calculations have been performed by T. Dahm \cite{DahmNP2009} on the basis of ARPES and INS spectra (see Fig. 3) measured for YBCO crystals from the same batch. The details of ARPES on YBCO can be found elsewhere \cite{ZabolotnyyPRB2007}. The overall similarity of experimental and calculated fermionic spectral functions, shown in Fig. 3(c), demonstrates clearly that the spin fluctuations can explain all the peculiarities of the electronic scattering in cuprates including its doping, temperature and momentum dependence. In particular, they provide a natural explanation for the `kink puzzle': As illustrated in Fig. 3(a), the nodal kink is a result of the interband scattering on the spin-fluctuations from the upper, universal, weakly temperature-dependent branch of the spectrum ($Q_2$ vector), while the scattering between the antinodal regions ($Q_1$ vector) is determined by the middle of the spin-fluctuation spectrum where a large peak, known as a `resonance mode', appears just below $T_c$.
The determined value\footnote{For the exact formulas see \cite{DahmNP2009}, since the numeric factors are omitted here.} of $\bar{U}$ = 1.59 eV gives an estimate of $T_c$ exceeding 150 K \cite{DahmNP2009}. This demonstrates that the spin fluctuations have sufficient strength to mediate high-temperature superconductivity. The fact that the estimated value of $T_c$ is about two times higher than the measured one seems natural since the actual transition temperature can be reduced by a variety of effects. On the other hand, it is important to mention that, within the described scenario, $T_c$ should increase monotonously with underdoping, in clear contradiction to experiment. The resolution of this controversy is related to understanding of the pseudogap phenomenon, as discussed in Section \ref{sec:3}.
Having shown that the spin-fluctuation spectrum, measured by INS, describes perfectly the renormalization of the one-particle fermionic spectrum, measured by ARPES, and even can mediate high-$T_c$ superconductivity, one can follow a similar procedure described by Eq.~3 to clarify its origin. If the spin-fluctuation spectrum is formed by two-particle fermionic excitations, it should be proportional to the imaginary part of the dynamic spin susceptibility \cite{ChubukovReview}. One should note, however, that Eq.~3 gives the bare susceptibility, $\chi_0$. The dynamic one can be then derived within the random phase approximation (RPA). Indeed, the RPA spin susceptibility, modeled based on electronic band structure parameters, reproduces the low energy part of the INS spectrum, including the ($\pi,\pi$) resonance, as well as gives similar explanation for the nodal kink as produced by coupling to the upper branch of the spectrum \cite{EreminPRL2005}.
In Ref.~\cite{InosovPRB2007}, the dynamic susceptibility, for both odd and even channels, has been derived from the fermionic spectral function of BSCCO accurately mapped by ARPES. The detailed comparison with INS results supports the idea that the magnetic response below $T_c$ (or at least its major part) can be explained by the itinerant magnetism. Namely, the itinerant component of $\chi$, at least near optimal doping, has enough intensity to account for the experimentally observed magnetic resonance in both INS channels. Taking into account the out-of-plane exchange interaction, the energy difference between the odd and even resonances as well as their intensity ratio are perfectly described. Moreover, the calculated incommensurate resonance structure is similar to that observed in the INS experiment \cite{InosovPRB2007}.
\subsection{The role of phonons}
\label{sec:2.4}
One should note that similarly good description of both the nodal kink and antinodal dip has been obtained with an anisotropic coupling to two phonon modes \cite{DevereauxPRL2004}. Moreover, the strong temperature dependence of the antinodal renormalization, namely its abrupt increase below $T_c$, can be also explained as a result of the superconducting gap opening. The same can be said about other effects which we do not discuss here, the `odd scattering' \cite{Odd,OddYBCO} and the `magnetic isotope effect' \cite{TerashimaZn,Zn}. Both have been suggested as evidences for the spin-fluctuation origin of the renormalization in cuprates and later digested by the multiple phonons scenario \cite{LeePRB2007}.
In this situation, the critical (as shown in Fig.~2(d)) doping dependence of the strength of both the nodal kink \cite{LanzaraNature01,KordyukPRL2004,KordyukPRL2006} and antinodal dip \cite{BorisenkoPRL2003,KimPRL2003} remains the main problem for the phonon scenario. The idea of carrier density dependent screening of some phonon modes may explain some doping dependence of the energy scales (the positions of kink and dip) \cite{LeePRB2007,JohnstonACM2010} and only a moderate doping dependence of the renormalization strength, in an agreement with some experiments \cite{JohnstonACM2010}. In this case, one should stress the disagreement between those experiments and the experiments reviewed in the previous sections.
Having said that, we should admit that the question about the role of phonons in cuprates stays open. Here we argue that the overall fermionic spectrum, $A(\textbf{k},\omega)$, with its most remarkable features, the nodal kink and the antinodal dip, can be well described by coupling between electrons, but coupling to the lattice should also be present and is expected to be responsible for the fine structure in $A(\textbf{k},\omega)$ \cite{ZhouPRL2005,ZhaoX2010}. Moreover, revealing fine structure in ARPES spectra can essentially change our estimate for the renormalization at the very low binding energies. An example for this is recently discovered 10 meV kink \cite{Plumb}. It may appear that the interaction responsible for this kink plays an important role in superconducting pairing.
\section{Abnormal normal state}
\label{sec:3}
\begin{figure}
\center{
\resizebox{0.9\columnwidth}{!}{%
\includegraphics{Fig4.pdf} }}
\caption{Nonmonotonic pseudogap. (a) The temperature map which consists of a number of momentum integrated energy distribution curves (EDCs) measured at different temperatures at a `hot spot'. The gap is seen as a shift of the leading edge midpoint (LEM) which corresponds to white color close to the Fermi level. (b) The position of LEM as function of temperature for an underdoped Tb-BSCCO with $T_c$ = 77 K and $T^*$ = 170 K is remarkably similar to the pseudo-gap in a transition-metal dichalcogenide 2H-TaSe$_2$ (c) with the transitions to the commensurate and incommensurate CDW phases at $T_{ICC}$ = 90 K and $T_{NIC}$ = 122 K, respectively. After \cite{KordyukPG}.}
\label{fig:4}
\end{figure}
An evident weakness of Eq.~1 is that it does not take into account the `pseudogap' (PG) \cite{TimuskPG,NormanPG}. Recent progress in ARPES measurements has led to the `two gaps' idea \cite{Tanaka2G,Kondo2G}, according to which the superconducting gap and pseudogap have different origin and compete for the phase space. At the same time, there is no general consensus on this issue \cite{NormanPG,Tanaka2G,Kondo2G,LeePG,VallaPG,HuefnerPG}. Here we argue that careful temperature- and momentum-resolved photoemission experiments suggest that the origin of the pseudogap is a spin density ordering.
In Ref.~\cite{KordyukPG} it is shown that the depletion of the spectral weight in slightly underdoped Bi(Tb)-2212 superconductor, usually associated with the pseudogap, exhibits an unexpected nonmonotonic temperature dependence: decreases linearly approaching $T^*$ at which it reveals a sharp transition but does not vanish and starts to increase gradually again at higher temperature. Fig.~4(a) shows the temperature evolution of the pseudogap as a temperature map. The gap is seen as a shift of the leading edge midpoint (LEM) of a gapped EDC. Since the leading edge of the momentum integrated EDC of the non-gapped spectrum is expected to stay at zero binding energy for any temperature \cite{VallaPG,KordyukPRB2003}, the finite shift of the LEM is a good empirical measure for a gap of unknown origin. From the presented temperature map one can see an unusual temperature evolution of the gap (in terms of the colorscale, the LEM corresponds to the white color): first it decreases with increasing temperature up to about 170 K, then it starts to increase again. The temperature dependence of the LEM is summarized in Fig. 4(b) and compared to the similar quantity measured for TaSe$_2$ (panel c), for which it is known that the pseudogap results from the incommensurate charge density wave \cite{BorisenkoTaSe2,BorisenkoNbSe2}. The observed one-to-one correspondence between the temperature dependences of the pseudogap for Bi-2212 and TaSe$_2$, which is discussed in details in \cite{KordyukPG}, suggests that density wave ordering also appears in cuprates and, reducing the electron density of states at the Fermi level, competes with superconductivity. One may assume that the spin-fluctuations, being a dominant mediator for electronic interactions in the cuprates, play also the role of the main driving force for the electronic instability resulting in the spin density wave formation. This assumption is based on the same `proximity-to-antiferromaget' argument: the increase of the pseudogap with underdoping and vanishing with overdoping.
\section{Phonons in ferropnictides}
\label{sec:4}
\begin{figure}
\center{
\resizebox{0.7\columnwidth}{!}{%
\includegraphics{Fig5.pdf} }}
\caption{Fingerprints of phonons in LiFeAs. First and second derivatives of the MDC dispersions and MDC widths, respectively, on top of the phonon density of states (gray color scale). After \cite{KordyukLFA}.}
\label{fig:5}
\end{figure}
The iron based pnictides, a newly discovered family of high-$T_c$ superconductors, seems to provide a clear case where the phonons definitely lose vs spin-fluctuations in the nomination for the most probable pairing glue. First, the spin-fluctuations are generally expected to be strong in all the ferropnictides \cite{MazinX2009MazinNature,KorshunovPRL2009,EreminPRB2010}. Second, a number of estimates have suggested that the electron-phonon coupling there is by far not sufficient to mediate the pairing \cite{BoeriPRL2008,JishiAiCMP2010}. So, it would be interesting to apply a similar analysis of the fermionic self-energy to the pnictides.
LiFeAs seems to be a key compound for this task. While in other pnictides the self-energy analysis is complicated by essentially three-dimensional electronic band structure and magnetic ordering \cite{ZabolotnyyN2009}, LiFeAs does not show any static magnetic ordering but has rather high critical temperature ($T_c$ = 18 K) and a sizeable isotropic superconducting gap with $2\Delta/kT_c$ = 4 \cite{BorisenkoLFA,InosovLFA}. It also provides the simplest case from an experimental point of view. First, it is a stoichiometric compound that exhibits superconductivity at ambient pressure without chemical doping, thus can be easily studied by the experimental techniques which require samples without impurities. Second, although the its electronic band structure is similar to other pnictides \cite{BoeriPRL2008,JishiAiCMP2010,NekrasovJETPL2008,SinghPRB2008}, it has a perfectly two-dimensional electronic band \cite{BorisenkoLFA} well separated in momentum space from other bands. Finally, LiFeAs cleaves between the two layers of Li atoms resulting in equivalent and neutral counterparts, offering a unique opportunity to overcome the problems arising from a polar surface that can be crucial for the surface sensitive methods \cite{ZabolotnyyPRB2007}. This altogether has allowed us to derive precisely the quasiparticle self-energy from ARPES spectra and analyze its fine structure \cite{KordyukLFA}.
Fig.~5 summarizes this analysis, presenting 1st and 2nd derivatives of the MDC dispersions and MDC widths, respectively, on top of the phonon density of states calculated in \cite{JishiAiCMP2010}. The peaks on those functions are expected to coincide with peaks in the corresponding bosonic spectrum, and one can see that the correspondence is remarkable. This result, together with the estimated electron-phonon coupling strength $\lambda_{ph} = 1.38$, shows that electron-phonon coupling in pnictides is much higher than in cuprates and may be important for superconducting pairing \cite{KordyukLFA}.
\section{Conclusions}
\label{sec:5}
A careful and systematic study of fermionic spectrum of high-$T_c$ cuprates by ARPES has allowed us to derive the fermionic self-energy, analyze its structure, and identify the `fingerprints' of the spin-fluctuation spectrum. The uncovered strong dependence of the intensity of those fingerprints (of the electron-boson coupling strength) on doping and temperature unambiguously supports the magnetic origin of the key interaction. Therefore, we conclude that the spin-fluctuations play a decisive role in formation of the fermionic excitation spectrum in the normal state and are sufficient to explain the high transition temperatures to the superconducting state. The pseudogap phenomenon is consistent with this scenario and is a consequence of a Peierls-type intrinsic instability of electronic system to formation of an incommensurate spin density wave. Ironically, a similar analysis being applied to the iron pnictides reveals especially strong electron-phonon coupling that suggests important role of phonons for high-$T_c$ superconductivity in in these compounds.
\vspace{6 pt}
We acknowledge discussions with I. Eremin, A. Chubukov, I. I. Mazin, T. Dahm, D. J. Scalapino, S.-L. Drechsler, M.~L.~Kuli\'{c}, W. Hanke, B. Keimer, V. Hinkov, P. Bourges, R. Hackl, T. Valla, T. P. Devereaux, A. M. Gabovich, Yu. V. Kopaev, M. V. Sadovskii, E.~G. Maksimov, V. M. Loktev, E. A. Pashitskii, A. Semenov, J. Fink, M. Golden, M. Knupfer, A. Koitzsch, R. Schuster, and technical support from R. H\"{u}bel. The work is supported by DFG (Forschergruppe FOR538) and BMBF.
|
{
"timestamp": "2010-09-23T02:01:50",
"yymm": "1009",
"arxiv_id": "1009.4336",
"language": "en",
"url": "https://arxiv.org/abs/1009.4336"
}
|
\part{Candidates}
\author[G. Gelmini, P. Gondolo]{Graciela Gelmini$^a$, Paolo Gondolo$^b$ \\ $^a$Department of Physics and Astronomy, UCLA,
475 Portola Plaza, Los Angeles, CA 90095, USA \\ $^b$ Department of Physics, University of Utah,
115 S 1400 E \# 201, Salt Lake City, UT 84112, USA}
\chapter{DM production mechanisms}
\vspace{-2.5cm}
{\small{(Chapter 7 of the book {\it{Particle Dark Matter: Observations, Models and Searches}} edited by Gianfranco Bertone, Published by Cambridge University Press, 2010)}}
\section{Dark matter particles: relics from the pre-BBN era}
A general class of candidates for non-baryonic cold dark matter are weakly interacting massive particles (WIMPs). The interest in WIMPs as dark matter candidates stems from the fact that WIMPs in chemical equilibrium in the early universe naturally have the right abundance to be cold dark matter. Moreover, the same interactions that give the right WIMP density make the detection of WIMPs possible. The latter aspect is important as it provides a means to test the WIMP hypothesis.
The argument showing that WIMPs are good dark matter candidates is old~\cite{Hut:1977zn,Lee:1977ua,Vysotsky:1977pe,Sato:1977ye,Dicus:1977nn}. The density per comoving volume of non relativistic particles in equilibrium in the early Universe decreases exponentially with decreasing temperature, due to the Boltzmann factor, until the reactions which change the particle number become ineffective.
At this point, when the annihilation rate becomes smaller than the Hubble expansion rate,
the WIMP number per comoving volume becomes constant. This moment of chemical decoupling or freeze-out happens later, i.e.\ at smaller WIMP densities, for larger WIMP annihilation cross section $\sigma_{\rm ann}$. If there is no subsequent change of entropy in matter plus radiation, the present relic density of WIMPs is approximately
\begin{equation}
\label{eq:omegawimp}
\Omega h^2 \approx \frac{ 3 \times 10^{-27} {\rm ~cm^3/s} }{ \langle \sigma_{\rm ann} v \rangle }.
\end{equation}
For weak cross sections this gives the right order of magnitude of the DM density (and a temperature $T_{f.o.} \simeq m/20$ at freeze-out for a WIMP of mass $m$). This is a ballpark argument. A more precise derivation will be presented in Section~\ref{sec:Gelmini:2}.
It is important to realize that the determination of the WIMP relic density depends on the history of the Universe before Big Bang Nucleosynthesis (BBN), an epoch from which we have no data. BBN (200 s after the Big Bang, $T\simeq 0.8$ MeV) is the earliest episode from which we have a trace, namely the abundance of light elements
D, $^4$He and $^7$Li. The next observable in time is the Cosmic Microwave Background radiation (produced $3.8\times10^4$ yr after the Big Bang, at $T\simeq$ eV) and the next one is the Large Scale Structure of the Universe. WIMPs have their number fixed at $T_{f.o.} \simeq m/20$, thus WIMPs with $m \gsim 100$ MeV would freeze out at $T\gsim 4$ MeV and would thus be the earliest remnants. If discovered, they would for the first time give information on the pre-BBN phase of the Universe.
As things stand now, to compute the WIMP relic density we must make assumptions about the pre-BBN epoch. The standard computation of the relic density relies on the
assumptions that the entropy of matter and radiation was conserved,
that WIMPs were produced thermally, i.e.\ via interactions with the particles in the plasma, that they decoupled while the Universe expansion was dominated by radiation, and that they were in kinetic and chemical equilibrium before they decoupled. These are just assumptions, which do not hold in all cosmological models. In particular, in order for BBN and all the subsequent history of the Universe to proceed as usual, it is enough that the earliest and highest temperature during the radiation dominated period, the so called
reheating temperature $T_{RH}$, is larger than 4 MeV~\cite{Hannestad:2004px}. At temperatures higher than 4 MeV, when the WIMP freeze out is expected to occur, the content and expansion history of the Universe may differ from the standard assumptions. In non-standard cosmological models, the WIMP relic abundance may be higher or lower than the standard abundance. The density may be decreased by reducing the rate of thermal production (through a low $T_{RH} < T_{f.o.}$) or by producing radiation after freeze-out (entropy dilution). The density may also be increased by creating WIMPs from decays of particles or extended objects (non-thermal production) or by increasing the expansion rate of the Universe at the time of freeze-out.
Non-thermal production mechanisms may also be at work within standard cosmological scenarios. For example, WIMPs may be produced in the out of equilibrium decay of other particles whose density may be
fixed by thermal processes. A particular type of heavy WIMPs, WIMPZILLAS, could be formed during the reheating phase at the end of an inflationary period through gravitational interactions. Another production mechanism involves quantum-mechanical oscillations, for example a sterile neutrino may be produced in the early universe by the oscillation of active (interacting) neutrinos into sterile neutrinos (for the latter, see the chapter of M. Shaposhnikov in this book).
In the rest of the Chapter we review the standard production mechanism and some of the non-standard scenarios.
\section{Thermal production in the standard cosmology}
\label{sec:Gelmini:2}
\subsection{The standard production mechanism}
In the standard scenario, it is assumed that in the early universe WIMPs were produced in collisions between particles of the thermal plasma during the radiation dominated era. Important reactions were the production and annihilation of WIMP pairs in particle-antiparticle collisions, such as
\begin{equation}
\chi \bar{\chi} \leftrightarrow e^+e^-, \mu^+\mu^-, q\bar{q}, W^+W^-, ZZ, HH, \ldots .
\end{equation}
At temperatures much higher than the WIMP mass, $T \gg m_\chi$, the colliding particle-antiparticle pairs in the plasma had enough energy to create WIMP pairs efficiently. Also, the inverse reactions that convert WIMPs into standard model particles (annihilation) were initially in equilibrium with the WIMP-producing processes. Their common rate was given by
\begin{equation}
\Gamma_{\rm ann} = \langle \sigma_{\rm ann} v \rangle n_{\rm eq} ,
\end{equation}
where $\sigma_{\rm ann}$ is the WIMP annihilation cross section, $v$ is the relative velocity of the annihilating WIMPs, $n_{\rm eq}$ is the WIMP number density in chemical equilibrium, and the angle brackets denote an average over the WIMP thermal distribution.
As the universe expanded, the temperature of the plasma became smaller than the WIMP mass. While annihilation and production reactions remained in equilibrium, the number of WIMPs produced decreased exponentially as $e^{-m_\chi/T}$ (the Boltzmann factor), since only particle-antiparticle collisions with kinetic energy in the tail of the Boltzmann distribution had enough energy to produce WIMP pairs. At the same time, the expansion of the universe decreased the number density of particles $n$, and with it the production and annihilation rates, which are proportional to $n$. When the WIMP annihilation rate $\Gamma_{\rm ann}$ became smaller than the expansion rate of the universe $H$, or equivalently the mean free path for WIMP-producing collisions became longer than the Hubble radius, production of WIMPs ceased (chemical decoupling). After this, the number of WIMPs in a comoving volume remained approximately constant (or in other words, their number density decreased inversely with volume).
In many of the current theories, WIMPs are their own antiparticles. For this kind of WIMPs (e.g.\ neutralinos and Majorana neutrinos), the WIMP density is necessarily equal to the antiWIMP density. In the following we restrict our discussion to this case. We refer the reader interested in cosmological WIMP-antiWIMP asymmetries, as might apply for example to a Dirac neutrino, to~\cite{Griest:1986yu}.
The current density of WIMPs can be computed by means of the rate equation for the WIMP number density $n$ and the law of entropy conservation:
\begin{eqnarray}
\frac{dn}{dt}&=&-3Hn-\left<\sigma_{\rm ann} v\right>(n^2-n_{\rm eq}^2)\,,\label{eq:first}\\
\frac{ds}{dt}&=&-3Hs\,.
\label{eq:sstd}
\end{eqnarray}
Here $t$ is time, $s$ is the entropy density, $H$ is the Hubble parameter, and as before $n_{\rm eq}$ and $\left<\sigma_{\rm ann} v\right>$ are the WIMP equilibrium number density and the thermally averaged total annihilation cross section. The first and the second term on the right hand side of Eq.~\ref{eq:first} take into account the expansion of the Universe and the change in number density due to annihilations and inverse annihilations, respectively.
It is customary (see e.g.~\cite{Kolb:1985nn,Srednicki:1988ce,Gondolo:1990dk,Edsjo:1997bg}) to combine Eqs.~(\ref{eq:first}) and~(\ref{eq:sstd}) into a single one for $Y=n/s$, and to use $x=m/T$, with $T$ the photon temperature, as the independent variable instead of time. This gives:
\begin{equation}
\frac{dY}{dx}=\frac{1}{3H}\frac{ds}{dx}\left<\sigma v\right>(Y^2-Y_{eq}^2)\,.
\label{eq:std}
\end{equation}
Here and in the rest of the Chapter we will simply write $\left<\sigma v\right>$ for $\left<\sigma_{\rm ann} v\right>$ when no ambiguity can arise.
\begin{figure}[!b]
\includegraphics[width=0.7\textwidth]{yyy.eps}
\caption{Typical evolution of the WIMP number density in the early universe during the epoch of WIMP chemical decoupling (freeze-out).}
\label{fig:2}
\end{figure}
According to the Friedman equation, the Hubble parameter is determined by the mass-energy density $\rho$ as
\begin{equation}
H^2=\frac{8\pi}{3M_P^2} \rho\,,
\end{equation}
where $M_P=1.22\times10^{19}~{\rm GeV}$ is the Planck mass.
The energy and entropy densities are related to the photon temperature by the equations
\begin{equation}
\rho=\frac{\pi^2}{30}g_{\rm eff}(T) T^4\,,\quad s=\frac{2 \pi^2}{45}h_{\rm eff}(T) T^3,
\label{eq:eff}
\end{equation}
where $g_{\rm eff}(T)$ and $h_{\rm eff}(T)$ are effective degrees of freedom for the energy density and entropy density respectively. Recent computations of $g_{\rm eff}(T)$ and $h_{\rm eff}(T)$ that include QCD effects can be found in Ref.~\cite{Hindmarsh:2005ix}. If the degrees of freedom parameter $g_*^{1/2}$ is defined as
\begin{equation}
g_*^{1/2}=\frac{h_{\rm eff}}{g_{\rm eff}^{1/2}}\left(1+\frac{1}{3}\frac{T}{h_{\rm eff}}\frac{dh_{\rm eff}}{dT}\right),
\end{equation}
then Eq.~(\ref{eq:std}) can be written in the following way,
\begin{equation}
\frac{dY}{dx}=-\left(\frac{45}{\pi M_P^2}\right)^{-1/2}\frac{g_*^{1/2}m}{x^2}\left<\sigma v\right> (Y^2-Y_{eq}^2)\,.
\label{eq:stdfi}
\end{equation}
This single equation is then numerically solved with the initial condition $Y=Y_{eq}$ at $x\simeq 1$ to obtain the present WIMP abundance $Y_0$. From it, the WIMP relic density can be computed as
\begin{equation}
\Omega_{\chi}h^2=\frac{\rho_\chi^0h^2}{\rho_c^0}=\frac{m_\chi s_0 Y_0h^2}{\rho_c^0}=2.755\times 10^8\, Y_0 m_\chi/\mathrm{GeV}\,,
\label{eq:relic}
\end{equation}
where $\rho_c^0$ and $s_0$ are the present critical density and entropy density respectively. In obtaining the numerical value in Eq.~(\ref{eq:relic}) we used $T_0=2.726\,\mathrm{K}$ for the present background radiation temperature and $h_{\rm eff}(T_0)=3.91$ corresponding to photons and three species of neutrinos.
The numerical solution of Eq.~(\ref{eq:stdfi}), see Fig.~\ref{fig:2} for an illustration, shows that at high temperatures $Y$ closely tracks its equilibrium value $Y_{eq}$. In fact, the interaction rate of WIMPs is strong enough to keep them in thermal and chemical equilibrium with the plasma. But as the temperature decreases, $Y_{eq}$ becomes exponentially suppressed and $Y$ is no longer able to track its equilibrium value. At the freeze out temperature ($T_{\rm f.o.}$), when the WIMP annihilation rate becomes of the order of the Hubble expansion rate, WIMP production becomes negligible and the WIMP abundance per comoving volume reaches its final value. In the standard cosmological scenario, the WIMP freeze out temperature is about $T_{\rm f.o.} \simeq m_\chi/20$, which corresponds to a typical WIMP speed at freeze-out of $v_{\rm f.o.}= (3T_{\rm f.o.}/2m_\chi)^{1/2} \simeq 0.27c$.
An important property that Fig.~\ref{fig:2} illustrates is that smaller annihilation cross sections lead to larger relic densities (``The weakest wins.'') This can be understood from the fact that WIMPs with stronger interactions remain in chemical equilibrium for a longer time, and hence decouple when the universe is colder, wherefore their density is further suppressed by a smaller Boltzmann factor. This leads to the inverse relation between $\Omega_\chi h^2$ and $\langle \sigma_{\rm ann} v \rangle$ in Eq.~(\ref{eq:omegawimp}).
From this discussion follows that the freeze out temperature plays a prominent role in determining the WIMP relic density. In general, however, the freeze out temperature depends not only on the mass and interactions of the WIMP but also, through the Hubble parameter, on the content of the Universe. Some examples of how modifications of the Hubble parameter affect the WIMP density are discussed in Section~\ref{sec:Gelmini:4} below.
\subsection{Annihilations and coannihilations}
A pedagogical example of the dependence of the relic density on the WIMP mass is provided by a thermally-produced fourth-generation Dirac neutrino $\nu$ with Standard Model interactions and no lepton asymmetry, although it is excluded as a cold dark matter candidate by a combination of LEP and direct detection limits~\cite{Griest:1989pr,Angle:2008we,Ahmed:2008eu}.
Figure~\ref{fig:3} summarizes its relic density $\Omega_\nu h^2$ as a function of its mass $m_\nu$. The narrow band between the horizontal lines is the current cosmological measurement of the cold dark matter density $\Omega_{\rm cdm} h^2 = 0.1131\pm0.0034$~\cite{Hinshaw:2008kr}. Neutrinos with $\Omega_\chi>\Omega_{\rm cdm}$ are said to be overadundant, those with $\Omega_\chi<\Omega_{\rm cdm}$ are called underabundant. A neutrino lighter than $\sim1$ MeV freezes out while relativistic. If it is so light to be still relativistic today ($m_\nu \lsim 0.1 {\rm ~meV}$), its relic density is $\rho_\nu = 7\pi^2 T_\nu^4/120$. If it was massive enough to have become non-relativistic after freeze out, its relic density is determined by its equilibrium number density as $\rho_\nu=m_\nu 3\zeta(3)T_\nu^3/2\pi^2$. Here $T_\nu = (3/11)^{1/3} T_\gamma$, where $T_\gamma = 2.725 \pm 0.002 {\rm K}$ is the cosmic microwave background temperature. A neutrino heavier than $\sim1$ MeV freezes out while non-relativistic. Its relic density is determined by its annihilation cross section, as in Eq.~(\ref{eq:omegawimp}). The shape of the relic density curve above $\sim1$ MeV in Figure~\ref{fig:3} is a reflection of the behavior of the annihilation cross section into lepton-antilepton and quark-antiquark pairs $f\bar{f}$: the Z-boson resonance at $m_\nu \simeq m_Z/2$ gives rise to the characteristic V shape in the $\Omega h^2$ curve. Above $m_\nu \sim 100 {\rm ~GeV}$, new annihilation channels into Z- or W-boson pairs open up (thresholds at $m_nu = m_W$ and $m_\nu=m_Z$, respectively). When the new channel annihilation cross sections dominate the relic density decreases. Soon, however, the perturbative expansion of the cross section in powers of the (Yukawa) coupling constant becomes untrustworthy (the question mark in Figure~\ref{fig:3}). A general unitarity argument~\cite{Griest:1989wd} limits the relic density to the dashed curve on the right,
\begin{equation}
\Omega_\nu h^2 \simeq 3.4\times 10^{-6} \sqrt{\frac{m_\nu}{T_{\rm f.o.}}} \left( \frac{m_\nu}{1{\rm ~TeV}}\right)^2.
\end{equation}
\begin{figure}[t]
\includegraphics[width=0.9\textwidth]{omega_nu.eps}
\caption{Relic density $\Omega_\nu h^2$ of a thermal Dirac neutrino with standard-model interactions as a function of the neutrino mass $m_\nu$ (solid line). The very close horizontal dashed lines enclose the current 1$\sigma$ band for the cold dark matter density \cite{Hinshaw:2008kr}.}
\label{fig:3}
\end{figure}
The relic density of other WIMP candidates exhibits features similar to that of the Dirac neutrino just discussed. In general, because of the presence of resonances and thresholds in the annihilation cross section, one should not rely on a Taylor expansion of $\sigma_{\rm ann} v$ in powers of $v$, because it would lead to unphysical negative cross sections. Let us remark that resonant and threshold annihilation, including the coannihilation thresholds discussed below, are ubiquitous for neutralino dark matter (see the of chapter of J. Ellis in this book). In the non-relativistic limit $v\to0$, the product $\sigma_{\rm ann} v$ tends to a constant, because of the exoenergetic character of the annihilation process that makes the annihilation cross section $\sigma_{\rm ann}$ diverge as $1/v$ as $v\to0$. One can safely Taylor expand $\sigma_{\rm ann} v$ in powers of $v^2$ if $\sigma_{\rm ann} v$ varies slowly with $v$, $\sigma_{\rm ann} v = a + b v^2 + \cdots$. Then the thermal average is $\langle \sigma_{\rm ann} v \rangle = a + b \tfrac{3T}{2m} + \cdots $. Close to resonances and thresholds, however, $\sigma_{\rm ann}$ varies rapidly with $v$ and more sophisticated procedures (described in the following) should be used~\cite{Griest:1990kh,Gondolo:1990dk}.
State-of-the-art calculations of WIMP relic densities strive to achieve a precision comparable to the observational one, which is currently around a few percent. Since the speed of the WIMPs at freeze-out is about $c/3$, relativistic corrections must be included. Fully relativistic formulas for any cross section, with or without resonances, thresholds, and coannihilations, were obtained in~\cite{Gondolo:1990dk,Edsjo:1997bg}. In the simplest case without coannihilations, one has
\begin{equation}
\langle \sigma_{\rm ann} v \rangle = \frac{ \int_0^\infty dp \, p ^2 \, W_{\chi\chi}(s) \, K_1(\sqrt{s}/T) } { m_\chi^4 \, T \left[ K_2(m_\chi/T) \right]^2 } ,
\end{equation}
where $W_{\chi\chi}(s)$ is the $\chi\chi$ annihilation rate per unit volume and unit time (a relativistic invariant), $s = 4 ( m_\chi^2 + p^2) $ is the center-of-mass energy squared, and $K_1(x)$, $K_2(x)$ are modified Bessel functions. The Lorentz-invariant annihilation rate $W_{ij}(p)$ for the collision of two particles of 4-momenta $p_i$ and $p_j$ is related to the annihilation cross section $\sigma_{ij}$ through
\begin{equation}
W_{ij}(s) = \sigma_{ij} F_{ij} ,
\end{equation}
where
\begin{equation}
F_{ij} = 4 \sqrt{ (p_i \cdot p_j)^2 - m_i^2 m_j^2 }
\end{equation}
is the Lorentz-invariant flux factor.
Coannihilations are an essential ingredient in the calculation of the WIMP relic density. They are processes that deplete the number of WIMPs through a chain of reactions, and occur when another particle is close in mass to the dark matter WIMP (mass difference $\Delta m \sim $ temperature $T$). In this case, scattering of the WIMP off a particle in the thermal `soup' can convert the WIMP into the slightly heavier particle, since the energy barrier that would otherwise prevent it (i.e.\ the mass difference) is easily overcome. The particle participating in the coannihilation may then decay and/or react with other particles and eventually effect the disappearance of WIMPs. We give two examples in the context of the Minimal Supersymmetric Standard Model. Neutralino coannihilation with charginos $\tilde{\chi}^\pm$ may proceed, for instance, through
\begin{equation}
\tilde{\chi}_1^0 e^- \to \tilde{\chi}_2^- \nu_e, \qquad \tilde{\chi}_2^- \to \tilde{\chi}_2^0 d \bar{u} , \qquad \tilde{\chi}_2^0 \tilde{\chi}_1^0 \to W^+ W^- .
\end{equation}
Neutralino coannihilation with tau sleptons $\tilde{\tau}$ may instead involve the processes
\begin{equation}
\tilde{\chi}_1^0 \tau \to \tilde{\tau} \gamma, \qquad \tilde{\tau} \tilde{\chi}_1^+ \to \tau W^+ .
\end{equation}
\begin{figure}
\includegraphics[width=0.75\textwidth]{rateex.eps}
\caption{The effective invariant annihilation rate $W_{\rm eff}$
as a function of $p_{\rm eff}$ for a particular supersymmetric model examined in~\protect\cite{Edsjo:1997bg}. The final state threshold for
annihilation into $W^+ W^-$ and the coannihilation thresholds
appearing in Eq.~(\protect\ref{eq:weff}) are indicated.
The $\chi_2^0 \chi_2^0$ coannihilation threshold is too small to
be seen.}
\label{fig:effrate}
\end{figure}
Coannihilations were first included in the study of near-degenerate heavy neutrinos in \cite{Binetruy:1983jf} and were brought to general attention in \cite{Griest:1990kh}. The relativistic treatment was formulated in \cite{Edsjo:1997bg}. Under the conditions described below, which are reasonable during WIMP freeze-out, one replaces $\langle \sigma_{\rm ann} v \rangle$ in Eq.~(\ref{eq:first}) with
\begin{equation} \label{eq:sigmavefffin2}
\langle \sigma_{\rm{eff}}v \rangle = \frac{\int_0^\infty
dp_{\rm{eff}} \, p_{\rm{eff}}^2 \, W_{\rm{eff}}(s) \, K_1(\sqrt{s}/T) } { m_\chi^4 \, T \left[ \sum_{i=1}^{N} \frac{g_i}{g_\chi}
\frac{m_i^2}{m_\chi^2} K_2(m_i/T) \right]^2},
\end{equation}
where $s=4 p_{\rm eff} + 4 m_\chi^2$, $g_i$ is the number of internal degree of freedom (statistical weight factor) for the $i$-th coannihilating particle, and
\begin{equation} \label{eq:weff}
W_{\rm{eff}}(s) = \sum_{ij}\frac{F_{ij}}{F_{\chi\chi}}
\frac{g_ig_j}{g_\chi^2} W_{ij}(s) .
\end{equation}
The sums extend over all the $N$ coannihilating particles, including the $\chi$, and $m_1=m_\chi$, $g_1=g_\chi$. The assumptions underlying Eq.~(\ref{eq:sigmavefffin2}) are: (1) all coannihilating particles decay into the lightest one, which is stable, and their decay rate is much faster than the expansion rate of the
universe -- so the final WIMP abundance is
simply described by the sum of the density of all coannihilating
particles; (2) the scattering cross sections of coannihilating particles off the thermal background are of the same
order of magnitude as their annihilation cross sections -- since the relativistic background particle density is much larger than each of
the non-relativistic coannihilating particle densities, the scattering rate is much faster and the momentum distributions of the coannihilating particles remain in
thermal equilibrium; (3) all coannihilating particle are semi-relativistic, so the Fermi-Dirac and Bose-Einstein thermal distributions can be replaced by Maxwell-Boltzmann distribution $f_{i} = e^{-E_{i}/T}$.
An important aspect of the effective annihilation rate in Eq.~(\ref{eq:sigmavefffin2}) is that coannihilations appear
as thresholds at a value of $\sqrt{s}$ equal to the sum of the masses of the
coannihilating particles. As an example of this,
Fig.~\ref{fig:effrate}, taken from Ref.~\cite{Edsjo:1997bg}, shows that coannihilation thresholds and regular final state thresholds appear on the same footing in the invariant annihilation rate $W_{\rm eff}$.
Computations of WIMP relic densities can become quite involved, especially in the presence of coannihilations. There exist publicly available software~\cite{Gondolo:2004sc,Belanger:2006is} that can handle these calculations for generic WIMPs (see the chapter of F. Boudjemain in this book).
\section{Non-thermal production in the standard cosmology}
\subsection{Gravitational mechanisms}
WIMPZILLAs \citep{Chung:1998zb,Chung:1998ua,Chung:1998rq,Kuzmin:1998uv,Kuzmin:1998kk,Chung:2001cb} illustrate a fascinating idea for generating matter in the expanding universe: the gravitational creation of matter in an accelerated expansion. This mechanism is analogous to the production of Hawking radiation around a black hole, and of Unruh radiation in an accelerated reference frame.
WIMPZILLAs are very massive relics from the Big Bang: they can be the dark matter in the universe if their mass is $\approx 10^{13}$ GeV. They might be produced at the end of inflation through a variety of possible mechanisms: gravitationally, during preheating, during reheating, in bubble collisions. It is possible that their relic abundance does not depend on their interaction strength but only on their mass, giving great freedom in their phenomenology. To be the dark matter today, they are assumed to be stable or to have a lifetime of the order of the age of the universe. In the latter case, their decay products may give rise to the highest energy cosmic rays.
\begin{figure}[t]
\centering
\includegraphics[width=0.8\textwidth]{f08.eps}
\vspace{-12pt}
\caption{Relic density of gravitationally-produced WIMPZILLAs as a function of their mass $M_X$. $H_I$ is the Hubble parameter at the end of inflation, $T_{\rm rh}$ is the reheating temperature, and $M_{\rm pl} \approx 3 \times 10^{19} {\rm ~GeV}$ is the Planck mass. The dashed and solid lines correspond to inflationary models that smoothly end into a radiation or matter dominated epoch, respectively. The dotted line is a thermal distribution at the Gibbons-Hawking temperature $T=H_I/2\pi$. Outside the `thermalization region' WIMPZILLAs cannot reach thermal equilibrium.
(Figure from \citealp{Chung:1998zb}.)}
\label{fig:8}
\end{figure}
Gravitational production of particles is an important phenomenon that is worth describing here. Consider a scalar field (particle) $X$ of mass $M_X$ in the expanding universe. Let $\eta$ be the conformal time and $a(\eta)$ the time dependence of the expansion scale factor. Assume for simplicity that the universe is flat. The scalar field $X$ can be expanded in spatial Fourier modes as
\begin{equation}
X(\vec{x},\eta) = \int \frac{ d^3 k}{ (2\pi)^{3/2} a(\eta) } \left[ a_k h_k(\eta) e^{i \vec{k} \cdot \vec{x} } + a_k^\dagger h_k^*(\eta) e^{-i \vec{k} \cdot \vec{x}} \right] .
\end{equation}
Here $a_k$ and $a_k^\dagger$ are creation and annihilation operators, and $h_k(\eta)$ are mode functions that satisfy (a) the normalization condition
$
h_k h_k^{\prime *} - h'_k h_k^* = i
$
(a prime indicates a derivative with respect to conformal time),
and (b) the mode equation
\begin{equation}
\label{eq:modeequation}
h''_k(\eta) + \omega_k^2(\eta) \, h_k(\eta) = 0 ,
\end{equation}
where
\begin{equation}
\omega_k^2(\eta) = k^2 + M_X^2 a^2 + (6 \xi-1) \frac{a''}{a} .
\end{equation}
The parameter $\xi$ is $\xi=0$ for a minimally-coupled field and $\xi=\tfrac{1}{6}$ for a conformally-coupled field. The mode equation, Eq.~(\ref{eq:modeequation}), is formally the same as the equation of motion of a harmonic oscillator with time-varying frequency $\omega_k(\eta)$. For a given complete set of positive-frequency solutions $h_k(\eta)$, the vacuum $ | 0_{h} \rangle$ of the field $X$, i.e.\ the state with no $X$ particles, is defined as the state that satisfies $a_k | 0_{h} \rangle = 0$ for all $k$. Since Eq.~(\ref{eq:modeequation}) is a second order equation and the frequency depends on time, the normalization condition is in general not sufficient to specify the positive-frequency modes uniquely, contrary to the case of constant frequency $\omega_0$ for which $h_k^0(\eta) = e^{-i\omega_0 \eta}/(2\omega_0)^{1/2}$. Different boundary conditions for the solutions $h_k(\eta)$ define in general different creation and annihilation operators $a_k$ and $a_k^\dagger$, and thus in general different vacua.\footnote{The precise definition of a vacuum in a curved space-time is still subject to some ambiguities. We refer the interested reader to \protect\cite{Fulling:1979ac,Fulling:1989nb,Birrell:1982ix,Wald:1995yp} and to the discussion in \protect\cite{Chung:2003wn} and references therein.}
For example,
solutions which satisfy the condition of having only positive-frequencies in the distant past,
\begin{equation}
h(\eta) \sim e^{-i \omega_k^{-} \eta} \quad {\rm for~}\eta\to -\infty,
\end{equation}
contain both positive and negative frequencies in the distant future,
\begin{equation}
\label{eq:bogolubov}
h(\eta) \sim \alpha_k e^{-i \omega_k^{+} \eta} + \beta_k e^{+ i \omega_k^{+} \eta } \quad {\rm for~}\eta\to +\infty.
\end{equation}
Here $\omega_k^{\pm} = \lim_{\eta\to\pm\infty} \omega_k(\eta)$. As a consequence, an initial vacuum state is no longer a vacuum state at later times, i.e.\ particles are created. The number density of particles is given in terms of the Bogolubov coefficient $\beta_k$ in Eq.~(\ref{eq:bogolubov}) by
\begin{equation}
n_X = \frac{1}{(2\pi a)^3} \int d^3k |\beta_k|^2.
\end{equation}
These ideas have been applied to gravitational particle creation at the end of inflation by \cite{Chung:1998zb} and \cite{Kuzmin:1998uv}. Particles with masses $M_X$ of the order of the Hubble parameter at the end of inflation, $H_I \approx 10^{-6} M_{\rm Pl} \approx 10^{13} {\rm ~GeV}$, may have been created with a density which today may be comparable to the critical density. Figure~\ref{fig:8} shows the relic density $\Omega_X h^2$ of these WIMPZILLAs as a function of their mass $M_X$ in units of $H_I$. Curves are shown for inflation models that have a smooth transition to a radiation dominated epoch (dashed line) and a matter dominated epoch (solid line). The third curve (dotted line) shows the thermal particle density at temperature $T=H_I/2\pi$. Also shown in the figure is the region where WIMPZILLAs are thermal relics. It is clear that it is possible for dark matter to be in the form of heavy WIMPZILLAs generated gravitationally at the end of inflation.
\subsection{Decays}
Dark matter may be produced in the decay of other particles. If the DM particles are non-interacting when the decay occurs, they inherit (except for some entropy dilution factor) the density of the parent particle $P$
\begin{equation}
\Omega_{\rm DM} ~h^2 \simeq \frac{m_{\rm DM}}{m_P} \Omega_P~ h^2~.
\end{equation}
This is the case of superWIMPs (see the chapter J. Feng in this book), extremely weakly interacting particles produced in the late decays of WIMPs (e.g. axinos or gravitinos from the decay of neutralinos or sleptons) which practically only interact gravitationally and cannot be directly detected. In some models
the superWIMP may produce WIMPs through its decay. This is the case, for example, of gravitinos producing Winos (which otherwise would have a very low thermal relic density) with the right DM abundance through their decay~\cite{Frieman:1989vt, Gherghetta:1999sw}.
\section{Thermal and non-thermal production in non-standard cosmologies}
\label{sec:Gelmini:4}
The relic density (and also the velocity distribution before structure formation) of WIMPs and other DM candidates such as heavy sterile neutrinos and axions, depends on the characteristics of the Universe (expansion rate, composition, etc.) immediately before BBN, i.e.\ at temperatures $T\gsim$ 4 MeV~\cite{Hannestad:2004px}.
The standard computation of relic densities relies on the assumption that radiation domination began before the main epoch of production of the relics and that the entropy of matter and radiation has been conserved during and after this epoch. Any modification of these assumptions would lead to different relic density values. Any extra contribution to the energy density of the Universe would increase the Hubble expansion rate $H$ and lead to larger relic densities (since the decreasing interaction rate $\Gamma$ becomes smaller than $H$ earlier, when densities are larger). This can happen in the Brans-Dicke-Jordan~\cite{Kamionkowski:1990ni}
cosmological model, models with anisotropic expansion~\cite{Barrow:1982ei, Kamionkowski:1990ni, Profumo:2003hq}, scalar-tensor~\cite{Santiago:1998ae, Damour:1998ae, Catena:2004ba, Catena:2007ix} or
kination~\cite{Salati:2002md, Profumo:2003hq} models and other models~\cite{Barenboim:2006rx, Barenboim:2007tu, Arbey:2008kv} In some scalar-tensor models $H$ may be decreased, leading to smaller relic densities~\cite{Catena:2007ix}. These models alter the thermal evolution of the Universe without an extra entropy production.
Not only the value of $H$ but the dependence of the temperature $T$ on the scale factor of the Universe could be different, if entropy in matter and radiation is produced. This is the case if a scalar field $\phi$ oscillating around its true minimum while decaying is the dominant component of the Universe just before BBN. This field may be an inflaton or another late decaying field, such as a modulus in supersymmetric models. Models of this type include some with moduli fields, either the Polonyi field~\cite{Moroi:1994rs, Kawasaki:1995cy} or others~\cite{Moroi:1999zb}) or an Affleck-Dine field and Q-ball decay~\cite{Fujii:2002kr, Fujii:2003iq}, and thermal inflation~\cite{Lyth:1995ka}.
Moduli fields correspond to flat directions in the supersymmetric potential, which are lifted by the same mechanisms that give mass to the supersymmetric particles of the order of a few to 10's of TeV, and they usually have interactions of gravitational strength. The decays of the $\phi$ field finally reheat the Universe to a low reheating temperature $T_{\rm RH}$, which could be not much larger than 5 MeV. In these low temperature reheating (LTR) models there can be direct production of DM relics in the decay of $\phi$ which increase the relic density, and there is entropy generation, through the decay of $\phi$ into radiation, which suppresses the relic abundance.
Thus, in non-standard cosmological scenarios, the relic density of WIMPs
$\Omega_\chi$ may be larger or smaller than in standard cosmologies $\Omega_{\rm std}$.
The density may be decreased by reducing the rate of thermal production (through a low $T_{\rm RH} < T_{f.o.}$), by reducing the expansion rate of the Universe at freeze-out or by producing radiation after freeze-out (entropy dilution). The density may be increased by creating WIMPs from particle (or extended objects) decay (non-thermal production) or by increasing the expansion rate of the Universe at freeze-out. Usually these scenarios contain additional parameters that can be adjusted to modify
the WIMP relic density. However these are due to physics that does not manifest itself in accelerator or detection experiments.
Let us comment that not only the relic density of WIMPs but their characteristic speed before structure formation in the Universe can differ in standard and non-standard pre-BBN cosmological models.
If kinetic decoupling (the moment when the exchange of momentum between WIMPs and radiation ceases to be effective) happens during the reheating phase of LTR models, WIMPs can have much smaller characteristic speeds, i.e. be much ``colder"
~\cite{Gelmini:2008sh}, with free-streaming lengths several orders of magnitude smaller than in the standard scenario. Much smaller DM structures could thus be formed, a fraction of which may persist as minihaloes within our galaxy and be detected in indirect DM searches. The signature would be a much larger boost factor of the annihilation signal than expected in standard cosmologies for a particular WIMP candidate. WIMPs may instead be much ``hotter" than in standard cosmologies too, they may even be warm DM instead of cold, which would leave an imprint on the large scale structure spectrum ~\cite{Lin:2000qq, Hisano:2000dz, Gelmini:2006vn}.
\subsection{Low temperature reheating (LTR) models}
Let us consider a late decaying
scalar field $\phi$ of mass $m_\phi$ and decay width $\Gamma_\phi$ which dominates the energy density of the Universe while oscillating
about the minimum of its potential and decays reheating the Universe to a low reheating
temperature $T_{\rm RH}$, with 5 MeV $\lsim
T_{\rm RH} \lsim T_{\rm f.o.} $ for $T_{\rm RH}$, so BBN is not affected.
The usual choice for the parameter $T_{\rm RH}$ is the
temperature the Universe would attain under the assumption that the
$\phi$ decay and subsequent thermalization are instantaneous,
\begin{equation} \Gamma_\phi = H_{\rm decay} =
\sqrt{\left(\frac{8\pi}{3}\right) \rho_R} = \sqrt{\frac{8}{90} \pi^3g_\star}~
\frac{T_{\rm RH}^2}{M_P}.
\label{TRH-def}
\end{equation}
Here, $\Gamma_\phi$ is the decay width of the $\phi$ field, $ \Gamma_\phi \simeq m_\phi^3/ \Lambda_{\rm eff}^2$.
If $\phi$ has non-suppressed gravitational couplings, as is usually
the case for moduli fields, the effective energy scale $\Lambda_{\rm
eff} \simeq M_P$ (but $\Lambda_{\rm
eff}$ could be smaller~\cite{Khalil:2002mu}). Thus, with $g_\star \simeq
10$,
\begin{equation} T_{\rm RH} \simeq 10~{\rm MeV}\left(\frac{m_\phi}{\rm
100~TeV}\right)^{3/2} \left(\frac{M_P}{\Lambda_{\rm eff}}\right).
\label{TRH}
\end{equation}
Numerical calculations in which the approximation of instantaneous decay is not made show that the parameter $T_{\rm RH}$
provides a good estimate of the first temperature of the radiations dominated epoch (see Fig.~\ref{gg05_fig1}).
Both thermal and non thermal production mechanisms in LTR modesl have been discussed
\cite{McDonald:1989jd, Kamionkowski:1990ni, Moroi:1994rs, Kawasaki:1995cy, Chung:1998rq, Moroi:1999zb, Giudice:2000ex, Allahverdi:2002nb, Allahverdi:2002pu,
Khalil:2002mu, Fornengo:2002db, Pallis:2004yy, Gelmini:2006pw, Gelmini:2006pq, Drees:2006vh, Endo:2006ix, Drees:2007kk}, mostly in supersymmetric models where the WIMP is the neutralino. The decay of
$\phi$ into radiation increases the entropy, diluting the WIMP number density. The decay of $\phi$ into WIMPs increases the WIMP number density. In supersymmetric models $\phi$ decays into supersymmetric particles, which eventually decay into the lightest such particles (the LSP, typically a neutralino). Call $b$ the net number of WIMPs
produced on average per $\phi$ decay, which is a highly model dependent parameter~~\cite{Moroi:1999zb, Allahverdi:2002nb, Allahverdi:2002pu, Gelmini:2006pw}.
A combination of $T_{\rm RH}$ and the
ratio $b/m_\phi$ can bring the
relic WIMP density to the desired value $\Omega_{\rm cdm}$~\cite{Gelmini:2006pw}.
The equations which describe the
evolution of the Universe depend only on the
combination $b/m_\phi$ and not on $b$ and $m_\phi$ separately. They are
\begin{eqnarray}
\label{eq:evol1}
\dot\rho &=& -3H(\rho+p)+\Gamma_\phi \rho_\phi \\
\dot n_\chi &=& -3Hn_\chi-{{\langle\sigma v\rangle}}
\left(n_\chi^2-n^2_{\chi,\rm eq}\right) + \frac{b}{m_\phi} \, \Gamma_\phi \rho_\phi \\
\dot\rho_\phi &=&-3H \rho_\phi -\Gamma_\phi \rho_\phi \\
H^2 &=& \frac{8\pi}{3M_P^2}
(\rho + \rho_\phi).
\label{eq:evol4}
\end{eqnarray}
In Eqs.~(\ref{eq:evol1}-\ref{eq:evol4}), a dot indicates a time
derivative, $\rho_\phi$ is the energy density in the $\phi$ field,
which is assumed to behave like non-relativistic matter; $\rho$ and
$p$ are the total energy density and pressure of matter and radiation
at temperature $T$; $n_\chi$ is the number density of
WIMPs (which are assumed to be in kinetic but not
necessarily chemical equilibrium) and $n_{\chi,\rm eq}$ is its value in chemical equilibrium;
finally, $H=\dot a/a$ is the Hubble parameter, with $a$ the scale
factor. The first principle of thermodynamics in the form $d(\rho
a^3)+d(\rho_\phi a^3)+pda^3=Td(s a^3)$ can be used to rewrite
Eq.~(\ref{eq:evol1}) as
\begin{equation} \dot s = - 3 H s + \frac{\Gamma_\phi \rho_\phi}{T}.
\end{equation}
where $s=(\rho+p-m_\chi n_\chi)/T$ is the entropy density of the matter and radiation.
For $\rho_\phi\rightarrow 0$
these equation reduce to the standard scenario.
During the $\phi$-oscillation-dominated epoch, $H \propto T^4$~\cite{McDonald:1989jd}.
This can be seen using Eq.~(\ref{eq:evol1}) while the matter content is negligible. In Eq.~(\ref{eq:evol1}) with
$p=\rho/3$) substitute $\rho \simeq T^4$ and
$\rho_\phi \simeq M_P^2 H^2$. Then use $H\sim t^{-1}$, write $T \propto t^{\alpha}$,
where $\alpha$ is a constant, match the powers of $t$ in all terms, and determine that $\alpha=-(1/4)$.
Hence, $H \propto t^{-1} \propto T^4$
(and $\rho_{\phi} \propto H^2 \propto T^8$).
Since $H$
equals $T_{\rm RH}^2/M_P$ at $T=T_{\rm RH}$, it is $H \simeq T^4/(T_{\rm RH}^2
M_P)$.
The initial conditions are specified through the value $H_I$ of the Hubble
parameter at the beginning of the $\phi$-oscillations dominated epoch. This amounts
to giving the initial energy density $\rho_{\phi,I}$ in the $\phi$
field at the beginning of the reheating phase, or equivalently the maximum temperature of the radiation
$T_{\rm MAX}$. Indeed, one has $H_I \simeq \rho_{\phi,I}^{1/2}/M_P
\simeq T_{\rm MAX}^4/(T_{\rm RH}^2 M_P)$. The latter relation can be derived from
$\rho_\phi \simeq T^8/T_{\rm RH}^4$ and the consideration that the maximum
energy in the radiation equals the initial (maximum) energy
$\rho_{\phi,I}$. As the $\phi$ begins to decay, the temperature of the radiation bath
rises sharply to $T_{\rm MAX}$~\cite{Chung:1998rq}, decreases slowly as function of the scale factor $a$ during the $\phi$-oscillating dominated phase, as $T \sim a^{-3/8}$ until it reaches $T_{\rm RH}$, when the radiation dominated phase starts and $T\sim a^{-1}$.
\begin{figure}
\vspace{-15pt}
\includegraphics[width=0.60\textwidth]{gg05_fig1.eps}
\vspace{25pt}\\
{\includegraphics[width=0.60\textwidth]{evolt2.eps}}
\vspace{15pt}\\
{\includegraphics[width=0.60\textwidth]{evol.eps}}
\vspace{-10pt}
\caption{a. (top) WIMP density $\Omega_\chi h^2$ as function of the
reheating temperature $T_{\rm RH}$ for illustrative values of the ratio
$\eta=b (100$TeV$/m_\phi)$~\cite{Gelmini:2006pw}.
b. (middle) Evolution of the neutralino $\chi$ abundance for different values of $T_{RH}$ and $\eta=0$ in
an mSUGRA model with $M_{1/2}=m_0=600 {\rm GeV}$, $A_0=0$, $\tan\beta=10$, $\mu>0$, $m_\chi=246 {\rm GeV}$ and standard relic density $ \Omega_{\rm std} h^2\simeq 3.6$ ~\cite{Gelmini:2006pq}. The short vertical lines indicate $T_{RH}$.~\cite{Gelmini:2006pq}.
c. (bottom) Same as b. but for $T_{RH}=1\,\mathrm{GeV}$ and several values of $\eta$.
}
\label{gg05_fig1}
\vspace{-10pt}
\end{figure}
Fig.~\ref{gg05_fig1}a shows how the WIMP density $\Omega_{\chi} h^2$ depends on
$T_{\rm RH}$ for illustrative values of the
parameter $\eta = b (100{\rm TeV}/m_\phi)$, both for WIMPs which are underdense and for WIMPs that are
overdense in usual cosmologies. The behavior of the relic density as a function of $T_{\rm RH}$ is easy to
understand physically.
The usual thermal production scenario occurs for $T_{\rm RH} > T_{\rm f.o.}$.
For $T_{\rm RH}<T_{\rm f.o.}$, there are four different ways in which the density
$\Omega h^2$ depends on $T_{\rm RH}$. There are four cases~\cite{Gelmini:2006pw}: (1) Thermal production without
chemical equilibrium, for which $\Omega_\chi \sim T_{\rm RH}^7$~\cite{Chung:1998rq}.
(2) Thermal production with chemical equilibrium, in which case
the WIMP freezes out while the universe is
dominated by the $\phi$ field. Its freeze-out density is larger than
usual, but it is diluted by the entropy produced in
$\phi$ decays (Fig.~\ref{gg05_fig1}b).
In this case
$\Omega_{\chi} \propto T_{\rm RH}^4$.
(3) Non-thermal production
without chemical equilibrium, where $\Omega_{\chi} \propto \eta T_{\rm RH}$
(independently of any assumption on neutralino kinetic equilibrium) (Fig.~\ref{gg05_fig1}c).
(4) Non-thermal production with
chemical equilibrium, where $\Omega \propto T_{\rm RH}^{-1}$ (Fig.~\ref{gg05_fig1}c).
For the validity of the annihilation term in Eq.\ref{eq:evol1} one needs to assume that WIMPs
enter into kinetic equilibrium before production ceases.
In any event the solutions just presented should remain qualitatively valid because kinetic
equilibrium affects only solutions which interpolate in $T_{\rm RH}$ between two correct solutions,
namely the solution of the standard cosmology at high
$T_{\rm RH}$ for which WIMPs are initially in kinetic equilibrium, and the WIMP production purely through
the scalar field decay (case 3), for which kinetic equilibrium is irrelevant.
For all overdense ($\Omega_{\rm std} >
\Omega_{\rm cdm}$) WIMPs, given one value of $\eta\lsim 10^{-4}$ $(100{\rm
GeV}/m_\chi)$ there is only one value of $T_{\rm RH}$ for which
$\Omega_\chi = \Omega_{\rm cdm}$. The exception is a severely overabundant light WIMP with
$\Omega_{\rm std} \gsim10^{12}$ $(m_\chi/ $ $100 {\rm GeV})^4$ (if the production is
thermal with chemical equilibrium as is usual).
Underdense ($\Omega_{\rm std} <
\Omega_{\rm cdm}$) WIMPs have one or two solutions $\Omega_\chi=\Omega_{\rm cdm}$ per $\eta$, if $ \Omega_{\rm std} \gsim 10^{-5} (100 {\rm GeV}/m_\chi)$ and $\eta \gsim 10^{-7}$ $(100 {\rm GeV}/m_\chi)^2$ $(\Omega_{\rm cdm}/\Omega_{\rm std})$ (for $T_{\rm RH} > 5$ MeV)~\cite{Gelmini:2006pw}.
In particular the neutralino density can be that of cold DM in almost any supersymmetric model, provided
$10^{12} (m_\chi/ 100 {\rm GeV})^4 \gsim \Omega_{\rm std} \gsim 10^{-5} (100 {\rm GeV}/m_\chi)$ and the high energy theory
accomodates the necessary combinations of values of $b/m_\phi$ and
$T_{\rm RH}$.
Let us comment on other DM candidates. Sterile neutrinos $\nu_s$ would also be remnants of the pre-BBN era. If they are produced through oscillations with active neutrinos $\nu_a$ their production rate has a sharp peak at $T_{\rm max} \simeq 13 {\rm\,MeV} (m_s/ 1{\rm\,eV})^{1/3}$)~\cite{Barbieri:1990vx, Enqvist:1990dq, Enqvist:1990ek, Dodelson:1993je} which for $m_s > 10^{-3}$ eV is above 1 MeV. ``Visible" $\nu_s$ (i.e. those that could be found soon in neutrino experiments) must necessarily have mixings $\sin(\theta)$ with $\nu_a$ large enough to be overabundant, and thus be rejected, in standard cosmologies. In LTR with $T_{\rm RH} < T_{\rm max}$, the relic abundance of visible $\nu_s$ could be reduced enough for them to be cosmologically acceptable, both if they are lighter or heavier than 1MeV~\cite{Giudice:2000dp, Gelmini:2004ah, Yaguna:2007wi, Gelmini:2008fq}. E.g. for $\nu_s$ lighter than 1 MeV produced through oscillations, $n_s/ n_a \simeq 10 \sin^2 {2\theta}
\left({T_{\rm RH}}/{5~{\rm MeV}}\right)^3$ ~\cite{ Gelmini:2004ah,Yaguna:2007wi}
thus $n_s$ is small for low $T_{\rm RH}$, even if $\sin\theta$ is large.
Another example is that of thermally produced axions, whose abundance
can be strongly suppressed if $T_{\rm RH}$ is smaller than their freeze-out temperature $\sim$ 50 MeV in standard cosmologies~ \cite{Giudice:2000ex, Grin:2007yg}. Also superWIMPs may be produced in LTR models~\cite{Okada:2007na}.
Finally, let us remark that LTR scenarios are more complicated than the
standard cosmology and no consistent all-encompassing
scenario exists yet. In particular Baryogenesis should happen during the reheating epoch too,
possibly through the Affleck-Dine mechanism~\cite{Moroi:1994rs, Fujii:2002kr, Dolgov:2002vf, Fujii:2003iq}.
\subsection{Models that only change the pre-BBN Hubble parameter}
\begin{figure}
{\includegraphics[width=0.7\textwidth]{H_T.eps}}
\caption{The Hubble parameter $H$ as a function of the photon temperature $T$ before primordial nucleosynthesis for several cosmological models.}
\label{H-T}
\end{figure}
We will consider two of these models, in which the change in WIMP relic density is more modest than in LTR: kination and scalar-tensor gravitational models. An homogeneous field $\phi$, e.g. a candidate for quintessence, has an energy density $\rho_\phi = \dot{\phi}^2/2 + V(\phi)$. Kination is an epoch in which the kinetic term dominates over the potential $V(\phi)$ so $\rho_{\rm total} \simeq \dot{\phi}^2/2 \sim a^{-6}$. No entropy is produced in this period, so $T\sim a^{-1}$ as usual. Thus $H \sim \sqrt{\rho_{\rm total} } \sim T^3$ (see line ``K" in Fig~\ref{H-T}). This case is intermediate between LTR, for which $H\sim T^4$ (see the line ``LTR" in Fig~\ref{H-T}) and the standard radiation domination case, for which $H\sim T^3$ (see the line ``RD" in Fig~\ref{H-T}). Thus kination yields freeze-out temperatures $T_{f.o.}$ larger than the standard, somewhere in between the LTR and the standard values. The only entropy dilution of the density comes from the conversion of a larger number degrees of freedom present at the higher $T_{f.o.}$ into photon degrees of freedom at low temperatures, as particles annihilate and heat up the photon bath, and this effect is modest. The contribution of the $\phi$ kinetic energy to the total density is usually quantified through the ratio of $\phi$ -to-photon energy density, $\eta_\phi = \rho_\phi/ \rho_\gamma$ at $T\simeq 1$ MeV so that at higher temperatures $H\simeq \sqrt{\eta_\phi} (T/ 1{\rm MeV}) H_{\rm standard}$. Notice that at $T\simeq 1$ MeV, i.e. during BBN, the quintessence field cannot be dominant, thus $\eta_\phi <1$. Ref.~\cite{Salati:2002md} finds that the enhancement of the relic density of WIMPs in kination models is
\begin{equation}
\Omega_{\rm kination}/ \Omega_{\rm std} \simeq \sqrt{\eta_\phi} 10^{3} (m_\chi/100 {\rm GeV})
\end{equation}
Thus, WIMPs that are underdense in the standard cosmology could account for the whole of the dark matter.
Scalar-tensor theories of gravity~\cite{Santiago:1998ae, Damour:1998ae, Catena:2004ba, Catena:2007ix} incorporate a scalar field coupled only through the metric tensor to the matter fields. In many of these models the expansion of the Universe drives the scalar field towards a state where the theory is indistinguishable from General Relativity, but the effect of the scalar field changes the expansion rate of the Universe at earlier times, either increasing or decreasing it. Theories with a single matter sector typically predict an enhancement of $H$ before BBN. In Ref.~\cite{Catena:2004ba} the $H$ is enhanced by a factor $A$, which is $A \simeq 2.19 \times 10^{14} (T_0/T)$ ($T_0$ is the present temperature of the Universe) for large temperatures $T> T_\phi$. At $T_\phi$, $A$ drops sharply to values close to 1 before BBN sets is (see the line ``ST1" in Fig~\ref{H-T}). WIMPs freeze-out at $T > T_\phi$ while $H\sim T^{1.2}$, but at the transition temperature $T_\phi$, $H$ drops sharply to the standard value, and becomes smaller than the WIMP reaction rate. The already frozen WIMPs are still abundant enough at $T_\phi$ to start annihilating again. This is a post freeze-out ``reannihilation phase" peculiar to these models. The WIMP relic abundance is reduced in this phase, but nonetheless remains much larger than in the standard case. The amount of increase in the WIMP relic abundance was found in Ref.~\cite{Catena:2004ba} to be between 10 and 10$^3$.
With more than one matter sector, of which only one is ``visible" and the other ``hidden", scalar-tensor models may also produce a reduction of $H$ by as much as 0.05 of the standard value (see line ``ST2" in Fig~\ref{H-T}) before the transition temperature $T_\phi$ at which $H$ increases sharply to the standard value before BBN~\cite{Catena:2007ix}. The maximum reduction of the WIMP relic abundance is larger for larger WIMP masses, ranging from a factor of 0.8-0.9 for masses close to 10 GeV to 0.1-0.2 for those close to 500 GeV~~\cite{Catena:2004ba}.
\vspace{0.5cm}
{\bf{Acknowlegements}} G.G. was supported in part by the US Department of Energy Grant
DE-FG03-91ER40662, Task C and P.G. was supported in part by the NFS
grant PHY-0456825 at the University of Utah.
\bibliographystyle{plainyr}
|
{
"timestamp": "2010-09-21T02:02:08",
"yymm": "1009",
"arxiv_id": "1009.3690",
"language": "en",
"url": "https://arxiv.org/abs/1009.3690"
}
|
\section{Introduction}
\section{Introduction}
The following setting
is considered in this introductory
section.
Let
$
( H, \left\| \cdot \right\|_H,
\left< \cdot, \cdot \right>_H
)
$,
$
( U, \left\| \cdot \right\|_U,
\left< \cdot, \cdot \right>_U
)
$
and
$
( V, \left\| \cdot \right\|_V,
\left< \cdot, \cdot \right>_V
)
$
be separable real
Hilbert spaces,
let
$
\left( \Omega, \mathcal{F}, \P \right)
$
be a probability space
with a normal filtration
$
( \mathcal{F}_t )_{ t \in [0,\infty) }
$,
let
$
( W_t )_{ t \in [0,\infty) }
$
be a cylindrical
standard
$ ( \mathcal{F}_t )_{ t \in [0,\infty) }
$-Wiener process on $ U $,
let
$ A \colon D(A) \subset H \to H $
be a generator of an analytic
semigroup and let
$
\alpha \in
( - 1, 0]
$,
$
\beta \in
( - \frac{ 1 }{ 2 }, 0]
$,
$
\eta \in [0,\infty)
$
be real numbers
such that
$
\eta - A
$
is bijective and
positive.
Moreover,
to simplify the notation
define
$
\| v \|_{ H_r }
:=
\| ( \eta - A )^r v \|_H
$
for all
$ v \in H_r := D( ( \eta - A )^r ) $
and all $ r \in \R $
and
let
$
F \colon H \to H_{ \alpha }
$
and
$
B \colon H \to
HS( U, H_{ \beta } )
$
be globally Lipschitz
continuous functions
and let
$
X \colon [0,\infty)
\times \Omega \to H
$
be an adapted stochastic
process with continuous sample
paths satisfying
$
\sup_{ s \in [0,t] }
\E[
\| X_s \|_H^p
]
< \infty
$
and
\begin{equation}
\label{eq:SPDE.intro}
X_t
=
e^{ A t } X_0
+
\int_0^t
e^{ A (t - s) }
F( X_s ) \, ds
+
\int_0^t
e^{ A (t - s) }
B( X_s ) \, dW_s
\end{equation}
$ \P $-a.s.\ for
all $ t, p \in [0,\infty) $.
The stochastic process
$
X \colon [0,\infty) \times
\Omega \to H
$
is thus a {\it mild solution}
of the stochastic
partial differential equation (SPDE) \eqref{eq:SPDE.intro}
(SPDEs have been extremely
intensively studied in the last decades;
see, e.g.,
the books
\cite{r90,dz92,GreckschTudor1996,DaPratoZabczyk1996,cr99,DaPrato2004,SanzSole2005,c07,PeszatZabczyk2007,Kotelenez2008,Holdenetal2009}
and lecture notes \cite{Walsh1986,KallianpurXiong1995,KrylovRoecknerZabczyk1999,PrevotRoeckner2007,AlbeverioFlandoliSinai2008,Dalangetal2009,Hairer2009}
and the references therein).
A simple example of this framework
is the following setting:
If
$ H = U = L^2( (0,1), \R ) $
is the Hilbert space of equivalence
classes of Lebesgue square integrable
functions,
if $ A \colon D(A) \subset H \to H $
is the Laplacian with Dirichlet
boundary conditions on $ (0,1) $
and if
$
\big( F( v ) \big)(x)
= f( x, v(x) )
$
and
$
\big( B( v ) u \big)(x)
= f( x, v(x) ) \cdot u(x)
$
for all $ x \in (0,1) $,
$ u, v \in H $ where
$
f, b \colon (0,1) \times \R \to \R
$
are continuously differentiable
functions with globally bounded
derivatives, then the above
framework is fulfilled with
$ \alpha = \eta = 0 $
and
$
\beta \in
(- \frac{ 1 }{ 2 }, - \frac{ 1 }{ 4 }
)
$
and \eqref{eq:SPDE.intro}
reduces to
the SPDE
\begin{equation}
\label{eq:SPDE.intro.ex}
d X_t(x)
=
\left[
\tfrac{ \partial^2 }{
\partial x^2
}
X_t(x)
+
f(x, X_t(x) )
\right]
dt
+
b(x, X_t(x) ) \, dW_t(x)
\end{equation}
with
$
X_t(0) = X_t(1) = 0
$
for
$ x \in (0,1) $ and
$ t \in [0,\infty) $.
Further examples of the
above described framework
and existence and uniqueness
results for \eqref{eq:SPDE.intro}
can,
e.g., be found
in
Da Prato \& Zabczyk~\cite{dz92,DaPratoZabczyk1996},
Brze\'{z}niak~\cite{b97b}
(see Theorem~4.3 in \cite{b97b}),
Van Neerven, Veraar \&
Weis~\cite{vvw08}
(see Theorem~6.2 and
Section~10 in \cite{vvw08})
and in the references
therein.
Our aim is to derive an It\^{o}
type formula for the
solution process $ X $
of the SPDE~\eqref{eq:SPDE.intro}.
Let us briefly review some
related It\^{o} type formula results
from the literature.
First, note that if
$ \alpha = \beta = 0 $
and if the mild solution process
$ X $
of the SPDE~\eqref{eq:SPDE.intro}
is also a $ D(A) $-valued strong
solution of the
SPDE~\eqref{eq:SPDE.intro},
then
the standard It\^{o} formula
(see It\^{o}~\cite{Ito1951})
in infinite dimensions
can be applied to $ X $.
More precisely, in that case,
Theorem~2.4 in Brze\'{z}niak,
Van Neerven, Veraar
\citationand\
Weis~\cite{bvvw08}
implies
\begin{equation}
\label{eq:standardito.intro}
\begin{split}
\varphi( X_t )
& =
\varphi( X_{ t_0 } )
+
\int_{ t_0 }^t
\varphi'( X_s )
\left[
A X_s + F(X_s)
\right]
ds
+
\int_{ t_0 }^t
\varphi'( X_s )
B( X_s ) \,
dW_s
\\ & \quad +
\frac{ 1 }{ 2 }
\sum_{ j \in \mathcal{J} }
\int_{ t_0 }^t
\varphi''( X_s )
\big(
B( X_s ) g_j ,
B( X_s ) g_j
\big)
\,
ds
\end{split}
\end{equation}
$ \P $-a.s.\ for all
$ t_0, t \in [0,\infty) $
with $ t_0 \leq t $
and all twice continuously
Fr\'{e}chet differentiable
functions
$ \varphi \in C^2( H, V ) $
where
$
\mathcal{J}
$
is a set and
$
( g_j )_{ j \in \mathcal{J} }
\subset U
$
is an arbitrary orthonormal basis
of $ U $.
The case where
$ X $ is not $ D(A) $-valued
and thus not
a strong solution
of \eqref{eq:SPDE.intro}
is more subtle.
There are a few
results in the literature
in this direction.
First,
in the case
$ \alpha \geq - \frac{ 1 }{ 2 } $
and
$ \beta = 0 $
(i.e., $ B $ maps
from $ H $ to $ HS( U, H ) $),
in the case where
$
A \colon D(A) \subset H
\to H
$
is self-adjoined
and in the case of the special
test function
$
\varphi(v) =
\| v \|^2_H
$
for all $ v \in H $,
\eqref{eq:standardito.intro}
can be
generalized and
then reads as
\begin{equation}
\label{eq:itosquare.intro}
\begin{split}
\left\|
X_t
\right\|^2_H
& =
\left\|
X_{ t_0 }
\right\|^2_H
+
2
\int_{ t_0 }^t
\left< X_s,
A X_s
\right>_H
ds
+
2
\int_{ t_0 }^t
\left< X_s,
F(X_s)
\right>_H
ds
+
2
\int_{ t_0 }^t
\left< X_s,
B( X_s ) \,
dW_s
\right>_H
\\ & \quad +
\int_{ t_0 }^t
\left\|
B( X_s )
\right\|^2_{ HS( U, H ) }
ds
\end{split}
\end{equation}
$ \P $-a.s.\ for all
$ t_0, t \in [0,\infty) $
with $ t_0 \leq t $;
see
Pardoux's
pioneering work \cite{p72,Pardoux1975,Pardoux1975Thesis}
and see, e.g., also
\cite{kr79,gk81,gk8182,p87,op89,rrw07,PrevotRoeckner2007}
for generalizations and reviews of
this It\^{o} formula for the squared
norm (the above mentioned results
from the literature
consider slightly different frameworks
and, in particular, often allow
$ A $ to be nonlinear too).
Note that in that case
$ X $ enjoys values in
$
H_{ 1 / 2 }
=
D\big(
( \eta - A )^{ 1 / 2 }
\big)
$
(see Theorem~4.2 in
Kruse \citationand\
Larsson~\cite{KruseLarsson2011})
and therefore,
the integral
$
\int_0^t
\left< X_s,
A X_s
\right>_H
ds
:=
\eta
\int_0^t
\left\| X_s \right\|_H^2
ds
-
\int_0^t
\|
(\eta - A)^{ \frac{ 1 }{ 2 } } X_s
\|_H^2
\, ds
$
in \eqref{eq:itosquare.intro}
is well defined.
Formula~\eqref{eq:itosquare.intro}
is a crucial ingredient in
the {\it variational approach}
for SPDEs
(see the monographs
\cite{Pardoux1975Thesis,kr79,r90,PrevotRoeckner2007}).
Formula~\eqref{eq:itosquare.intro}
is an It\^{o} formula for
possibly non-strong
solutions of SPDEs
in the case of the special
test function
$
\left\| \cdot \right\|_H^2
$.
There are also a few
results in the literature
which establish It\^{o}
type formulas for possible
non-strong solutions
of SPDEs
for more general test functions
than the squared norm
$
\left\| \cdot \right\|_H^2
$;
see \cite{p87,z06,gnt05,l07,lt08,l09b}.
In Theorem~5.1
in Pardoux~\cite{p87},
formula~\eqref{eq:itosquare.intro}
is generalized to a special
class of test functions which have
similar topological properties
as the function
$
\left\| \cdot \right\|_H^2
$.
In Zambotti~\cite{z06}, the standard
It\^{o} formula is applied to
regularized versions of the solution
process of the stochastic
heat equation with
additive noise and then the
limit of these regularized
It\^{o} formulas is made sense through
a suitable renormalization term
that appears in the resulting formula.
In Gradinaru, Nourdin
\citationand\ Tindel~\cite{gnt05},
Malliavin calculus and a
Skorokhod integral is used
to prove an It\^{o} type formula
for the solution of the
stochastic heat equation with
additive noise
(see also
Leon \citationand\ Tindel~\cite{lt08}
for a related It\^{o} formula
result for the
stochastic heat equation with additive fractional
noise).
In Lanconelli~\cite{l07},
a Wick product is used to
formulate an It\^{o} type formula
for the solution process
of the stochastic heat equation
with additive noise
and the relation between the
formulas in \cite{z06,gnt05}
is analyzed
(see Section~3 in \cite{l07}
and see also
Lanconelli~\cite{l09b}
for some consequences
of this Wick produckt It\^{o}
type formula for the stochastic
heat equation with additive noise).
In general it is
not
clear how and whether
\eqref{eq:standardito.intro}
can be generalized
to the case where
$
X \colon [0,\infty) \times \Omega \to H
$
is not a $ D(A) $-valued
strong solution of
\eqref{eq:SPDE.intro}.
This article suggests a different
approach for deriving an It\^{o}
formula for solutions of
\eqref{eq:SPDE.intro}.
We
do not aim for a suitable
generalization of
\eqref{eq:standardito.intro}
to the case of
non-strong solutions but
instead we suggest
a somehow different It\^{o}
type formula for
\eqref{eq:SPDE.intro}
which naturally holds
for \eqref{eq:SPDE.intro}
in its full generality
for all smooth
test functions.
More precisely,
we establish
in Corollary~\ref{cor:ito}
in Subsection~\ref{sec:solutionSPDE}
below
the identity
\begin{equation}
\label{eq:mildito_intro}
\begin{split}
\varphi( X_t )
& =
\varphi( e^{ A( t - t_0 ) } X_{ t_0 } )
+
\int_{ t_0 }^t
\varphi'( e^{ A( t - s ) } X_s ) \,
e^{ A( t - s ) }
F( X_s ) \, ds
+
\int_{ t_0 }^t
\varphi'( e^{ A( t - s ) } X_s ) \,
e^{ A( t - s ) }
B( X_s ) \, dW_s
\\ & \quad+
\frac{1}{2}
\sum_{ j \in \mathcal{J} }
\int_{ t_0 }^t
\varphi''( e^{ A( t - s ) } X_s )
\left(
e^{ A( t - s ) }
B( X_s ) g_j,
e^{ A( t - s ) }
B( X_s ) g_j
\right) ds
\end{split}
\end{equation}
$ \mathbb{P} $-a.s.\ for
all $ t_0, t \in [0,\infty) $
with $ t_0 \leq t $
and all
$
\varphi \in
\cup_{
r <
\min( \alpha + 1, \beta + 1 / 2 )
}
C^2( H_r, V )
\supset
C^2( H, V )
$.
Corollary~\ref{cor:ito}
also ensures that all terms
in \eqref{eq:mildito_intro}
are well defined
(see
\eqref{eq:well1d}--\eqref{eq:well3d}
in Subsection~\ref{sec:solutionSPDE}).
In the case of
\eqref{eq:SPDE.intro.ex},
natural examples for the
test functions
$
\varphi \in
\cup_{
r <
\min( \alpha + 1, \beta + 1 / 2 )
}
C^2( H_r, V )
$
in \eqref{eq:mildito_intro}
are Nemytskii operators
and nonlinear
integral operators
such as
$
H_r \ni
v \mapsto
\int_0^1 \psi( x, v(x) ) \, dx
\in \R
$
for any
$
0 < r <
\min( \alpha + 1 , \beta + 1 / 2 )
$
and any smooth function
$ \psi \colon (0,1) \times \R \to \R $
with globally bounded
derivatives.
In the special case
$
\varphi = id_H \colon
H \ni v \mapsto v
\in H = V
$,
equation~\eqref{eq:mildito_intro}
reduces to
the variant of constants
formula~\eqref{eq:SPDE.intro}
and in that sense,
\eqref{eq:mildito_intro}
is somehow a {\it mild It\^{o}
formula}.
In the deterministic case
$ B \equiv 0 $,
equation~\eqref{eq:mildito_intro}
is somehow a
{\it mild chain rule};
see Example~\ref{ex:2}
in Section~\ref{sec:mildito} below
for more details.
The identity~\eqref{eq:mildito_intro}
can be generalized to a much larger
class of stochastic processes
than solution processes of
the SPDE~\eqref{eq:SPDE.intro}.
To be more precise,
in Definition~\ref{propdef}
in Subsection~\ref{sec:mildprocesses}
a class of stochastic processes
which exhibit a similar algebraic
structure as \eqref{eq:SPDE.intro}
is introduced and referred
as
{\it mild It\^{o} processes}.
Examples of mild It\^{o} processes
are solution processes of
SPDEs such as \eqref{eq:SPDE.intro}
(see Subsection~\ref{sec:solutionSPDE})
as well as their numerical
approximation processes
(see Subsection~\ref{sec:numerics}).
The identity~\eqref{eq:mildito_intro}
is then a special case of
equation~\eqref{eq:itoformel_start}
in Theorem~\ref{thm:ito}
below
in which a mild It\^{o} formula
for mild It\^{o} processes
is established.
Let us outline how
\eqref{eq:mildito_intro}
and Theorem~\ref{thm:ito}
respectively are established.
A central idea in the proof
of \eqref{eq:mildito_intro}
is to consider a suitable
transformation of the
solution process
$
X \colon [0,\infty) \times \Omega
\to H
$
of the
SPDE~\eqref{eq:SPDE.intro}.
The transformed stochastic process
is then a standard It\^{o} process
to which the standard
It\^{o} formula
(see
\eqref{eq:standardito.intro})
can be applied.
Relating then the transformed
stochastic process in an
appropriate way to the original
solution process
$
X \colon [0,\infty) \times
\Omega \rightarrow H
$
of the SPDE~\eqref{eq:SPDE.intro}
finally results
in the mild It\^{o}
formula~\eqref{eq:mildito_intro}.
Two types of transformations
are well suited for this job.
One possibility is, roughly speaking,
to multiply the solution process
$ X $ of the
SPDE~\eqref{eq:SPDE.intro}
by $ e^{ - A t } $, $ t \in [0,\infty) $,
where
$ e^{ - A t } $, $ t \in [0,\infty) $,
has to be understood in an
appropriate large Hilbert space
(see Subsection~\ref{sec:mildito}
below for details).
In that sense the transformed stochastic
process becomes rougher
than the solution process $ X $
of the SPDE~\eqref{eq:SPDE.intro}.
This transformation
has been suggested in
Teichmann~\cite{t09}
and
Filipovi{\'c}, Tappe
\citationand\
Teichmann~\cite{ftt10}
(see also
Hausenblas
\citationand\
Seidler~\cite{hs01,hs08}).
The other possible transformation
goes into the other direction
and, roughly speaking,
consists of
multiplying the solution
process $ X $ of the
SPDE~\eqref{eq:SPDE.intro}
by
$ e^{ A ( T - t ) } $,
$ t \in [0,T] $,
for some large value $ T \in (0,\infty) $.
This transformation is based on an idea
in
Conus
\citationand\
Dalang~\cite{cd08}
and
Conus~\cite{c08}
(see also
Debussche
\citationand\
Printems~\cite{dp09},
Lindner \citationand\
Schilling~\cite{ls10}
and
Kov{\'a}cs, Larsson
\citationand\
Lindgren~\cite{kll11}).
The second transformation,
which makes the transformed process
smoother than the
solution process $ X $ of the
SPDE~\eqref{eq:SPDE.intro},
turns out to be more flexible and allows
us to prove Theorem~\ref{thm:ito}
in its full generality.
For more details on the proofs
of \eqref{eq:mildito_intro} and
Theorem~\ref{thm:ito}
respectively, the reader
is referred to
Subsection~\ref{sec:mildito}
below.
In the remainder of this
introductory section, a few
consequences
of the mild Ito
formula~\eqref{eq:mildito_intro}
and its generalization in
Theorem~\ref{thm:ito}
are illustrated.
For this
let
$
X^x \colon [0,\infty)
\times \Omega \to H
$,
$ x \in H $,
be a family of
adapted stochastic processes
with continuous sample paths
satisfying
$
X_t
=
e^{ A t } x
+
\int_0^t
e^{ A (t - s) }
F( X_s^x ) \, ds
+
\int_0^t
B( X_s^x ) \, dW_s
$
$ \P $-a.s.\ for
all $ t \in [0,\infty) $
and all $ x \in H $
(see, e.g.,
Theorem~4.3 in
Brze\'{z}niak~\cite{b97b}
or
Theorem~6.2 in
Van Neerven, Veraar \&
Weis~\cite{vvw08}
for the up to indistinguishability
unqiue existence of such processes).
Then
for every
$
r \in
\big(
- \infty,
\min( \alpha + 1, \beta + 1 / 2 )
\big)
$
and every
at most polynomially
growing continuous function
$
\varphi \in C( H_r, V )
$
define
the continuous
function
$
u_{ \varphi } \colon
[0,\infty) \times H_r
\to V
$
through
$
u_{ \varphi }(t,x) :=
\E\big[
\varphi( X^x_t )
\big]
$
for all
$ (t,x) \in [0,\infty) \times H_r $.
Under the assumption that
$ \alpha = \beta = 0 $
and that
$ F $ and $ B $ are three times continuously
Fr\'{e}chet differentiable with globally
bounded derivatives,
the functions
$
u_{ \varphi } \colon [0,\infty)
\times H \to V
$,
$
\varphi \colon H \to V
$
twice continuous Fr\'{e}chet differentiable
with globally bounded derivatives,
are strict solutions of
the infinite dimensional
Kolmogorov
partial differential
equation (PDE)
$
\tfrac{ \partial }{
\partial t
}
u_{ \varphi }(t,x)
=
(L u_{ \varphi })(t,x)
$
with
$
u_{ \varphi }(0,x) = \varphi(x)
$
for
$
(t,x) \in (0,\infty)
\times H_1
$
where
$
L \colon C^2( H, V )
\to
C( H_1, V )
$
is defined through
\begin{equation}
(L \varphi)(x)
:=
\frac{ 1 }{ 2 }
\text{Tr}\Big(
\big( B(x) \big)^{ \! * }
\varphi''( x ) \,
B(x)
\Big)
+
\varphi'( x )
\big[ A x + F(x) \big]
\end{equation}
for all $ x \in H_1 = D(A) $
and all
$
\varphi
\in C^2( H, V )
$
(see
Theorem~7.5.1
in
Da Prato \citationand\
Zabczyk~\cite{dz02}).
Infinite dimensional Kolmogorov
equation have
been intensively investigated
in the last two decades
(see, e.g., the monographs
\cite{MaRoeckner1992,c01,dz02b,DaPrato2004}
and articles
\cite{Zabczyk1999,Roeckner1999,RoecknerSobol2006,DaPrato2007}
and the references mentioned therein).
We prove here
that
the functions
$
u_{ \varphi } \colon [0,\infty)
\times H \to V
$,
$ \varphi $ sufficiently smooth,
also solve
another kind of
Kolmogorov equation.
More precisely,
from
\eqref{eq:mildito_intro}
we derive
in
Subsection~\ref{sec:kolmogorov}
below
(see
\eqref{eq:mildP2})
the identity
\begin{equation}
\label{eq:mildKolmogorov.intro}
u_{ \varphi }(t,x)
=
\varphi( e^{ A t } x )
+
\int_{ 0 }^t
u_{
L_{ t - s }( \varphi )
}(s,x)
\, ds
\end{equation}
for all
$ (t,x) \in (0,\infty) \times H $
and all
$
\varphi \in
\cup_{
r <
\min( \alpha + 1, \beta + 1 / 2 )
}
C^2( H_r, V )
$
with at most polynomially growing
derivatives
where
$
L_t \colon
\cup_{
r <
\min( \alpha + 1, \beta + 1 / 2 )
}
C^2( H_r, V )
\to
C( H, V)
$,
$ t \in (0,\infty) $,
is a family of bounded linear
operators defined through
\begin{equation}
\begin{split}
\big(
L_{ t } \varphi
\big)( x )
:=
\frac{ 1 }{ 2 }
\text{Tr}\Big(
\big( e^{ A t } B(x) \big)^{ \! * }
\varphi''( e^{ A t } x ) \,
e^{ A t } B(x)
\Big)
+
\varphi'( e^{ A t } x )
\, e^{ A t } F(x)
\end{split}
\end{equation}
for all
$ x \in H $,
$
\varphi \in
\cup_{
r <
\min( \alpha + 1, \beta + 1 / 2 )
}
C^2( H_r, V )
$,
$ t \in (0,\infty) $.
Equation~\eqref{eq:mildKolmogorov.intro}
is somehow a {\it mild Kolmogorov
backward equation}.
From
\eqref{eq:mildKolmogorov.intro}
we derive new regularity
properties
of solutions of second-order
PDEs in Hilbert spaces.
More precisely,
using \eqref{eq:mildKolmogorov.intro}
we establish
in Theorem~\ref{thm:continuity}
below the existence
of real numbers
$
c_{ \delta, \rho, q, T }
\in [0,\infty)
$,
$
\delta, \rho, q, T \in [0,\infty)
$,
such that the
{\it regularity estimate}
\begin{equation}
\label{eq:Schauder}
\sup_{
x \in
H_{ \delta }
}
\!
\left(
\,
\frac{
\left\|
u_{ \varphi }(t, x)
\right\|_V
}{
\left(
1 +
\| x
\|_{
H_\delta
}
\right)^{ (q + 2) }
}
\,
\right)
\leq
c_{ \delta, \rho, q, T }
\cdot
\| \varphi \|_{ t, q }^{ \delta, \rho }
\end{equation}
holds for all
$ t \in (0,T] $,
$
\varphi \in
C^2(
H_{ \rho }
,
V
)
$
with
$
\sup_{ x \in H_{ \rho } }
\frac{
\| \varphi''(x) \|_{
L^{ (2) }( H_{ \rho }, V )
}
}{
( 1 + \| x \|_{ H_{ \rho } } )^q
}
< \infty
$,
$
q, \delta, T \in [ 0, \infty )
$,
$
\rho \in
[
0,
\min(
\alpha + 1,
\beta + \frac{ 1 }{ 2 }
)
)
$
where
\begin{equation}
\begin{split}
&
\| \varphi \|_{ t, q }^{ \delta, \rho }
:=
\\ &
\sup_{ x \in H_{ \delta } }
\left[
\frac{
\left\|
\varphi( e^{ A t } x )
\right\|_V
}{
\left(
1 + \| x \|_{ H_{ \delta } }
\right)^{ (q + 2) }
}
\right]
+
\int_0^t
\left( t - s \right)^{
\min( \delta - \rho, 0 )
}
\sup_{
x \in H_{ \rho }
}
\left[
\frac{
\left\|
(K_t \varphi)'( x )
\right\|_{
L( H_{ \alpha }, V )
}
}{
\left(
1 + \| x \|_{ H_{ \rho } }
\right)^{ ( q + 1 ) }
}
+
\frac{
\left\|
( K_t \varphi )''( x )
\right\|_{
L^{(2)}( H_{ \beta }, V )
}
}{
\left(
1 + \| x \|_{ H_{ \rho } }
\right)^{ q }
}
\right]
ds
\end{split}
\end{equation}
for all
$
\varphi \in
C^2(
H_{ \rho }
,
V
)
$
with
$
\sup_{ x \in H_{ \rho } }
\frac{
\| \varphi''(x) \|_{
L^{ (2) }( H_{ \rho }, V )
}
}{
( 1 + \| x \|_{ H_{ \rho } } )^q
}
< \infty
$,
$
t, q, \delta \in [ 0, \infty )
$,
$
\rho \in
[
0,
\min(
\alpha + 1,
\beta + \frac{ 1 }{ 2 }
)
)
$
and where
$ K_t \colon C( H_1, V ) \to C( H, V ) $,
$ t \in (0,\infty) $,
is defined through
$
(K_t \varphi)(x) = \varphi( e^{ A t } x )
$
for all
$ x \in H $,
$ t \in (0,\infty) $.
The constants
$
c_{ \delta, \rho, q, T }
$,
$
\delta, \rho, q, T \in [0,\infty)
$,
appearing
in \eqref{eq:Schauder}
are described explicitly
in Theorem~\ref{thm:continuity}
below.
Next a direct consequence
of the regularity
estimate~\eqref{eq:Schauder}
is presented.
For this let
$
(
C^2_{ Lip }( H, \R ) ,
\left\| \cdot
\right\|_{ C^2_{ Lip }( H, \R ) }
)
$
be the real Banach space of all twice
continuously differentiable
globally Lipschitz continuous
real valued functions on $ H $
with globally Lipschitz
continuous derivatives
(see \eqref{eq:defCnLip} in
Subsection~\ref{sec:setting}
for details).
Moreover,
for every $ t \in (0,\infty) $
let
$
(
\mathcal{G}_t( H, \R ) ,
\left\| \cdot
\right\|_{ \mathcal{G}_t( H, \R ) }
)
$
be the completion of
the normed real vector
space
$
(
C^2_{ Lip }( H, \R ) ,
\left\| \cdot
\right\|_{ t, 0 }^{ 0, 0 }
)
$.
Then consider the mapping
$
\mathcal{I} \colon
\{
\mu \colon \mathcal{B}(H)
\to [0,1]
\text{ probability measure}
\colon
\int \| x \|_H \, \mu( dx ) < \infty
\}
\to
(
C^2_{ Lip }( H, \R )
)'
$
given by
$
\big(
\mathcal{I}( \mu )
\big)( \varphi )
=
\int \varphi( x ) \, \mu( dx )
$
for all
$
\varphi \in
C^2_{ Lip }( H, \R )
$
and all probability measures
$
\mu \colon \mathcal{B}(H)
\to [0,1]
$
with
$
\int \| x \|_H \, \mu( dx ) < \infty
$.
Lemma~\ref{lem:embedding} below
proves that $ \mathcal{I} $
is injective, that is,
$ \mathcal{I} $ embeds the
probability measures with finite
first absolute moments into
linear forms on $ C^2_{ Lip }( H, \R ) $.
Next note that
$
\mathcal{I}( \P_{ X_t } )
\in
(
C^2_{ Lip }( H, \R )
)'
$
for all $ t \in [0,\infty) $
where
$
\P_{ X_t }[ A ]
=
\P\big[ X_t \in A \big]
$
for all
$ A \in \mathcal{B}(H) $,
$ t \in [0,\infty) $
are the probability measures
of the solution process
$ X_t $, $ t \in [0,\infty) $,
of the SPDE~\eqref{eq:SPDE.intro}.
From \eqref{eq:Schauder} we then
infer
for every $ t \in (0,\infty) $
that
$
\mathcal{I}( \P_{ X_t } )
\in
(
C^2_{ Lip }( H, \R )
)'
$
is not only in
$
(
C^2_{ Lip }( H, \R )
)'
$
but in the
smaller space
$
(
\mathcal{G}_t( H, \R )
)'
$
too
(the embedding
$
(
\mathcal{G}_t( H, \R )
)'
\subset
(
C^2_{ Lip }( H, \R )
)'
$
continuously is proved
in Lemma~\ref{lem:norm}
below).
We thus have established
more {\it regularity of the
probability measures}
$ \P_{ X_t } $,
$ t \in (0,\infty) $,
of the solution process
of the SPDE~\eqref{eq:SPDE.intro}.
Another application
of the regularity
estimate~\eqref{eq:Schauder}
and the
mild Kolmogorov backward
equation~\eqref{eq:mildKolmogorov.intro}
is the analysis
of continuity properties of solutions of
second-order PDEs in Hilbert spaces
(see, e.g., the books
\cite{MaRoeckner1992,c01,dz02b,DaPrato2004}).
More precisely,
Corollary~\ref{cor:temporal}
in Section~\ref{sec:weakregularity}
below proves that
there exist real
number
$
c_{ r, \delta, \rho, T } \in [0,\infty)
$,
$
r, \delta, \rho, T \in [0,\infty)
$,
such that
\begin{equation}
\label{eq:Hoelder.intro}
\left\|
u_{ \varphi }(t_1, x_1) -
u_{ \varphi }(t_2, x_2)
\right\|_V
\leq
\left[
\tfrac{
c_{ r, \delta, \rho, T }
\,
(
1
+
\| x_1
\|_{
H_{ \delta }
}^3
+
\| x_2
\|_{
H_{ \delta }
}^3
)
\,
\| \varphi \|_{
C^2_{ Lip }( H_{ \rho }, V )
}
}{
\left|
\min( t_1, t_2 )
\right|^{
\max( r + \rho - \delta, 0 )
}
}
\right]
\Big[
\left| t_1 - t_2 \right|^{
r
}
+
\left\| x_1 - x_2 \right\|_{
H_{ \delta }
}
\Big]
\end{equation}
for all
$
t_1, t_2 \in (0,T]
$,
$
x_1, x_2 \in D( (-A)^{ \delta } )
$,
$
\varphi \in
C^2_{ Lip }( H_{ \rho }, V)
$,
$
r \in
[0, 1+ \alpha - \rho ) \cap
[0, 1+ 2 \beta - 2 \rho )
$,
$
\delta, T \in [0, \infty)
$,
$
\rho \in
[
0,
\min(
\alpha + 1,
\beta + \frac{ 1 }{ 2 }
)
)
$.
Inequality~\eqref{eq:Hoelder.intro}
thus proves
H\"{o}lder continuity of
the solutions
$
u_{ \varphi } \colon
[0,\infty) \times H \to V
$,
$ \varphi \in C^2_{ Lip }( H, V ) $,
of second-order Kolmogorov PDEs
in infinite dimensions.
In particular,
in the case of the
example
SPDE~\eqref{eq:SPDE.intro.ex},
inequality~\eqref{eq:Hoelder.intro}
ensures that
\begin{equation}
\label{eq:Hoelder2.intro}
\sup_{
\substack{
t_1, t_2 \in [0,T]
\\
t_1 < t_2
}
}
\left(
\frac{
|t_1|^{ r } \,
\big\|
\mathbb{E}\big[
\varphi( X_{ t_1 } )
\big]
-
\mathbb{E}\big[
\varphi( X_{ t_2 } )
\big]
\big\|_V
}{
| t_1 - t_2 |^r
}
\right)
< \infty
\end{equation}
for all
$
\varphi \in C^2_{ Lip }( H, V )
$,
$ T \in (0,\infty) $
and all
$
r \in
[0, \frac{ 1 }{ 2 } )
$.
Results in the literature imply
that \eqref{eq:Hoelder2.intro}
holds for all
$ r \in [ 0, \frac{ 1 }{ 4 } ] $.
More formally,
in the case of
the SPDE~\eqref{eq:SPDE.intro.ex},
we get
from the
global Lipschitz continuity of
$ \varphi $
that
\begin{equation}
\label{eq:strong.intro}
\begin{split}
&
\sup_{
\substack{
t_1, t_2 \in [0,T]
\\
t_1 < t_2
}
}
\frac{
|t_1|^{ r } \,
\big\|
\mathbb{E}\big[
\varphi( X_{ t_1 } )
\big]
-
\mathbb{E}\big[
\varphi( X_{ t_2 } )
\big]
\big\|_V
}{
| t_1 - t_2 |^r
}
\leq
\left[
\sup_{
\substack{
x, y \in H
\\
x \neq y
}
}
\tfrac{
\| \varphi(x) - \varphi(y) \|_V
}{
\| x - y \|_H
}
\right]
\left[
\sup_{
\substack{
t_1, t_2 \in [0,T]
\\
t_1 < t_2
}
}
\frac{
|t_1|^{ r } \,
\mathbb{E}\big[
\|
X_{ t_1 }
-
X_{ t_2 }
\|_{ H }
\big]
}{
| t_1 - t_2 |^r
}
\right]
< \infty
\end{split}
\end{equation}
for all
$
\varphi \in C^2_{ Lip }( H, V )
$,
$ T \in (0,\infty) $
and all
$
r \in
[0, \frac{ 1 }{ 4 }]
$
where the second factor
on the right hand side
of \eqref{eq:strong.intro}
is finite due to
Theorem~6.3 in
Van Neerven, Veraar \&
Weiss~\cite{vvw08}
for the case $ r \in [0,\frac{1}{4}) $
and due to
Corollaries~A.16 and A.35 in
\cite{jk12}
for the case $ r = \frac{ 1 }{ 4 } $
(see also
Brze\'{z}niak~\cite{b97b},
Kruse \citationand\
Larsson~\cite{KruseLarsson2011},
Van Neerven, Veraar \&
Weiss~\cite{vvw11}
for related results).
This shows that
regularity results in the literature
ensure
that \eqref{eq:Hoelder2.intro}
holds for all
$ r \in [ 0, \frac{ 1 }{ 4 } ] $.
Up to our best knowledge,
this is the first result in
the literature which
establishes
that \eqref{eq:Hoelder2.intro}
also holds in the regime
$ r \in (\frac{1}{4}, \frac{1}{2}) $.
A further application
of the regularity
estimate~\eqref{eq:Schauder}
and the
mild Kolmogorov backward
equation~\eqref{eq:mildKolmogorov.intro}
is the weak
approximation of SPDEs.
Let us illstrate this in the case of
spectral Galerkin projections
for the example
SPDE~\eqref{eq:SPDE.intro.ex}.
More precisely,
in the case of the
SPDE~\eqref{eq:SPDE.intro.ex},
Corollary~\ref{cor:galerkin}
in Section~\ref{sec:weakregularity}
implies
that
there exist real
numbers
$ C_{ r, T } \in [0,\infty) $,
$ r, T \in [0,\infty) $,
such that
\begin{equation}
\label{eq:weaknumeric_intro}
\left\|
\E\big[
\varphi\big( X_T \big)
\big]
-
\E\big[
\varphi\big( P_N( X_T )
\big)
\big]
\right\|_V
\leq
\frac{
C_{ r, T }
\left\|
\varphi
\right\|_{ C^2_{ Lip }( H, V ) }
}{
N^r
}
\end{equation}
for all $ N \in \mathbb{N} $,
$ T \in (0,\infty) $,
$
\varphi \in
C^2_{ Lip }( H, V )
$
and all
$
r \in [0, 1)
$
where
$ P_N \in L(H) $,
$ N \in \N $,
are spectral Galerkin
projections defined by
$
( P_N v)(x)
:=
\sum_{ n = 1 }^N
2 \sin( n \pi x )
\int_0^1
v(y) \, \sin( n \pi y )
\, dy
$
for all
$ x \in (0,1) $,
$
v \in H = L^2( (0,1), \R )
$
and all $ N \in \N $.
Inequality~\eqref{eq:weaknumeric_intro}
and
Corollary~\ref{cor:galerkin}
respectively
are a straightforward consequence
of the regularity
estimate~\eqref{eq:Schauder}
(see Section~\ref{sec:weakregularity}
for details).
In the case of the
stochastic heat equation with
additive noise
$ f(x,y) = 0 $
and $ b(x,y) = 1 $
for all $ x \in (0,1) $,
$ y \in \R $
in \eqref{eq:SPDE.intro},
inequality~\eqref{eq:weaknumeric_intro}
follows
for all $ r \in [0,1) $
from the results in
\cite{dp09,ls10,kll11,KovacsLarssonLindgren2012}
at least in the case of bounded test
functions
(see also \cite{h03b,dd06,gkl09,h10c,d11,Brehier2012,Kruse2012}
for further numerical
weak approximation results for SPDEs).
In addition, in the general
setting of the
SPDE~\eqref{eq:SPDE.intro},
it is well know that
inequality~\eqref{eq:weaknumeric_intro}
holds for all $ r \in [0,\frac{1}{2}] $.
Indeed,
in that case,
we get from
the global
Lipschitz continuity of
$ \varphi $ that
\begin{equation}
\begin{split}
&
\left\|
\E\big[
\varphi\big( X_T \big)
\big]
-
\E\big[
\varphi\big( P_N( X_T )
\big)
\big]
\right\|_V
\leq
\left\|
\varphi
\right\|_{
C^2_{ Lip }( H, V )
}
\E\big[
\|
( I - P_N )
X_T
\|_H
\big]
\\ & \leq
\left\|
\varphi
\right\|_{
C^2_{ Lip }( H, V )
}
\E\big[
\|
X_T
\|_{ H_{ 1 / 4 } }
\big]
\|
( I - P_N )
( \eta - A )^{ - 1 / 4 }
\|_{ L(H) }
\leq
\frac{
\left\|
\varphi
\right\|_{
C^2_{ Lip }( H, V )
}
\E\big[
\|
X_T
\|_{ H_{ 1 / 4 } }
\big]
}{
\sqrt{ N \pi }
}
< \infty
\end{split}
\end{equation}
for all
$ N \in \mathbb{N} $,
$ T \in (0,\infty) $
and all
$
\varphi \in
C^2_{ Lip }( H, V )
$
where finiteness of
$
\E\big[
\|
X_T
\|_{ H_{ 1 / 4 } }
\big]
$
for all $ T \in (0,\infty) $
follows, e.g.,
from
Theorem~5.1 in \cite{jk12}
(see also
Kruse \citationand\
Larsson~\cite{KruseLarsson2011}
and
Van Neerven, Veraar
\citationand\
Weiss~\cite{vvw11}
and the references therein
for similar results).
This shows that regularity results
from the literature ensure that
\eqref{eq:weaknumeric_intro}
holds for all
$ r \in [0,\frac{1}{2}] $.
The present article
proves that
\eqref{eq:weaknumeric_intro}
also holds for all
$ r \in [0,1) $.
Observe that
\eqref{eq:weaknumeric_intro}
estimates the weak
approximation error
of spatial spectral
Galerkin projections
only instead of more complicated
spatial approximations
(see also
Corollary~\ref{cor:galerkin}
in Section~\ref{sec:weakregularity}
below for a generalization
of \eqref{eq:weaknumeric_intro})
and also the time interval
and the noise are not discretized
in \eqref{eq:weaknumeric_intro}.
We believe that the mild
Kolmogorov backward
equation~\eqref{eq:mildKolmogorov.intro}
can also be used to solve
these more complicated
weak numerical approximation
problems for SPDEs
and plan to develop these results
in a future publication.
Another application
of the mild It\^{o}
formula~\eqref{eq:mildito_intro}
and the mild
Kolmogorov backward
equation~\eqref{eq:mildKolmogorov.intro}
respectively
are the derivation of strong
and weak stochastic Taylor
expansions of solutions
of SPDEs.
Details can be found
in Subsection~\ref{sec:tay}
below.
These Taylor expansions can
then be used to derive
higher order numerical schemes
for SPDEs.
In Subsection~\ref{sec:mil}
below this is illustrated
in the case of Milstein
scheme for SPDEs.
\subsubsection*{Acknowledgements}
We thank Josef
Teichmann for his instructive
presentations on the method of
the moving frame at the conference
``Rough paths in interaction''
at the Institut Henri Poincar\'{e}.
In addition, we thank Daniel
Conus for pointing out his instructive
thesis to us at the conference
``Recent Advances in the Numerical
Approximation of Stochastic Partial
Differential Equations'' at
the Illinois Institute of Technology.
Moreover, we are grateful to
Raphael Kruse, Stig Larsson and
Jan van Neerven for
their very helpful comments.
Finally, we gratefully acknowledge
Meihua~Yang for fruitful discussions
on the analysis of weak temporal regularity
of solutions of SPDEs
(see Corollary~\ref{cor:temporal})
and its consequences in
numerical analysis.
This work has been
partially supported
by the Collaborative
Research Centre $701$
``Spectral Structures
and Topological Methods
in Mathematics'',
by the
International Graduate
School ``Stochastics and Real
World Models'',
by the research
project ``Numerical solutions
of stochastic differential equations
with non-globally Lipschitz continuous
coefficients'' (all funded by the German
Research Foundation)
and by the
BiBoS Research Center.
The support of Issac Newton
Institute for Mathematical
Sciences in Cambridge is
also gratefully acknowledged
where part of this was done
during the special semester
on ``Stochastic Partial Differential
Equations''.
\section{Mild stochastic
calculus}
\label{sec:mildcalc}
Throughout this section
assume that the
following setting is fulfilled.
Let $ \mathbbm{I} \subset [0,\infty) $
be a
closed and convex subset
of $ [0,\infty) $
with nonempty interior,
let
$ \left( \Omega, \mathcal{F}, \mathbb{P} \right) $
be a probability space
with a normal filtration
$ ( \mathcal{F}_t )_{ t \in \mathbbm{I} } $
and let
$
\big(
\check{H},
\left< \cdot , \cdot \right>_{ \check{H} },
\left\| \cdot \right\|_{ \check{H} }
\big)
$,
$
\big(
\tilde{H},
\left< \cdot , \cdot \right>_{ \tilde{H} },
\left\| \cdot \right\|_{ \tilde{H} }
\big)
$,
$
\big(
\hat{H},
\left< \cdot , \cdot \right>_{ \hat{H} },
\left\| \cdot \right\|_{ \hat{H} }
\big)
$
and
$
\left(
U,
\left< \cdot , \cdot \right>_U,
\left\| \cdot \right\|_U
\right)
$
be separable
$\mathbb{R}$-Hilbert spaces
with
$
\check{H}
\subset
\tilde{H}
\subset
\hat{H}
$
continuously and densely.
In addition, let
$
Q \colon U \rightarrow U
$ be
a bounded nonnegative symmetric linear
operator and
let
$ \left( W_t \right)_{ t \in \mathbbm{I} }
$
be a cylindrical $ Q $-Wiener
process with respect
to
$
( \mathcal{F}_t )_{ t \in \mathbbm{I} }
$.
Moreover, by
$
\left(
U_0,
\left< \cdot , \cdot \right>_{ U_0 },
\left\| \cdot \right\|_{ U_0 }
\right)
$
the $ \mathbb{R} $-Hilbert space
with
$ U_0 = Q^{ 1/2 }( U ) $
and
$
\| u \|_{ U_0 }
= \| Q^{ - 1/2 }( u ) \|_U
$
for all $ u \in U_0 $
is denoted
(see, for example,
Proposition~2.5.2 in
Pr\'{e}v\^{o}t
\citationand\
R\"{o}ckner~\cite{PrevotRoeckner2007}).
Here and below
$
S^{ - 1 } \colon
\text{im}(S) \subset U
\rightarrow U
$
denotes the pseudo inverse
of a bounded linear
operator
$
S \colon U \rightarrow U \in L(U)
$
(see, e.g., Appendix~C in \cite{PrevotRoeckner2007}).
In addition, let
$
i_v \colon
L( \hat{H}, \check{H} )
\rightarrow \check{H}
\in
L\big(
L( \hat{H}, \check{H} ), \check{H}
\big)
$,
$ v \in \hat{H} $,
be a family of bounded
linear operators defined
through
$ i_v A = A v $
for all
$ A \in L( \hat{H}, \check{H} ) $ and all
$ v \in \hat{H} $.
Then by
$
\mathcal{S}( \hat{H}, \check{H} )
:=
\sigma(
\,
\cup_{ v \in \hat{H} } \,
i_v^{-1}( \mathcal{B}( \check{H} ) )
\,
)
=
\sigma(
\{
i_v^{-1}( \mathcal{A} )
\subset L( \hat{H}, \check{H} )
\colon
v \in \hat{H},
\mathcal{A} \in
\mathcal{B}( \check{H} )
\}
)
$
the strong sigma algebra on
$ L( \hat{H}, \check{H} ) $
is denoted
(see, for instance,
Section~1.2 in
Da Prato
\citationand\ Zabczyk~\cite{dz92}).
Finally, let
$ \angle \subset \mathbbm{I}^2 $
be a set defined
through
$
\angle :=
\left\{
(t_1, t_2) \in \mathbbm{I}^2 \colon
t_1 < t_2
\right\}
$
and let
$ \tau \in \mathbbm{I} $
be defined throught
$ \tau := \inf( \mathbbm{I} )
$.
\subsection{Mild stochastic processes}
\label{sec:mildprocesses}
\begin{definition}[Mild
It\^{o} process]
\label{propdef}
Let
$
S \colon
\angle
\rightarrow
L( \hat{H}, \check{H} )
$
be a
$
\mathcal{B}( \angle )
$/$
\mathcal{S}( \hat{H}, \check{H} )
$-measurable
mapping satisfying
$
S_{ t_2, t_3 }
S_{ t_1, t_2 }
=
S_{ t_1, t_3 }
$
for all $ t_1, t_2, t_3
\in \mathbbm{I} $
with $ t_1 < t_2 < t_3 $.
Additionally,
let
$
Y \colon \mathbbm{I}
\times \Omega
\rightarrow \hat{H}
$
and
$
Z \colon \mathbbm{I}
\times \Omega
\rightarrow HS( U_0, \hat{H} )
$
be two predictable
stochastic processes
with
$
\int_{ \tau }^{ t }
\left\| S_{ s, t } Y_s
\right\|_{ \check{H} }
ds < \infty
$
$ \mathbb{P} $-a.s.\ and
$
\int_{ \tau }^{ t }
\left\| S_{ s, t } Z_s
\right\|_{ HS( U_0, \check{H} ) }^2
ds < \infty
$
$ \mathbb{P} $-a.s.\ for
all $ t \in \mathbbm{I} $.
Then a predictable stochastic
process
$
X \colon \mathbbm{I} \times \Omega
\rightarrow \tilde{H}
$
satisfying
\begin{equation}
\label{eq:mildito}
X_t
=
S_{ \tau, t } \,
X_{ \tau }
+
\int_{ \tau }^t
S_{ s, t } \,
Y_s
\, ds
+
\int_{ \tau }^t
S_{ s, t } \,
Z_s
\, dW_s
\end{equation}
$ \mathbb{P} $-a.s.\
for all
$
t \in \mathbbm{I}
\cap ( \tau, \infty )
$
is called
a \underline{mild It\^{o} process}
(with \underline{semigroup} $S$,
\underline{mild drift} $ Y $
and \underline{mild diffusion} $ Z $).
\end{definition}
Note that if
$
(
\check{H},
\left< \cdot , \cdot \right>_{ \check{H} },
\left\| \cdot \right\|_{ \check{H} }
)
=
(
\hat{H},
\left< \cdot , \cdot \right>_{ \hat{H} },
\left\| \cdot \right\|_{ \hat{H} }
)
$
and if the semigroup
$
S \colon \angle \rightarrow
L( \hat{H} )
$
satisfies
$ S_{ t_1, t_2 } = I $
for all $ (t_1, t_2) \in \angle $,
then the mild It\^{o}
process~\eqref{eq:mildito}
is nothing else but a standard
It\^{o} process.
(Throughout this article the terminology
``standard It\^{o} process'' instead
of simply ``It\^{o} process''
is used in order to distinguish
more clearly from the above
notion of a mild It\^{o} process.)
Every standard It\^{o} process
is thus a mild It\^{o} process too.
However, a mild It\^{o} process
is, in general,
not a standard It\^{o} process
(see Section~\ref{sec:appl}
for some examples).
The concept of a mild It\^{o} process
in Definition~\ref{propdef}
thus generalizes the concept
of a standard It\^{o} process.
In concrete examples of mild
It\^{o} processes it will be crucial
that the semigroup
$
S \colon
\angle \rightarrow
L( \hat{H}, \check{H} )
$
in Definition~\ref{propdef}
depends explicitly on
both variables
$ t_1 $ and $ t_2 $
with $ (t_1, t_2) \in \angle $
instead of on
$ t_2 - t_1 $ only
(see Subsection~\ref{sec:numerics}
for details).
The assumptions
$
\int_{ \tau }^{ t }
\left\| S_{ s, t } Y_s
\right\|_{ \check{H} }
ds < \infty
$
$ \mathbb{P} $-a.s.\ and
$
\int_{ \tau }^{ t }
\left\| S_{ s, t } Z_s
\right\|_{ HS( U_0, \check{H}) }^2
ds < \infty
$
$ \mathbb{P} $-a.s.\ for
all $ t \in \mathbbm{I} $
in Definition~\ref{propdef}
ensure that both
the deterministic and the
stochastic integral
in \eqref{eq:mildito}
are well defined.
In the next step an immediate
consequence of
Definition~\ref{propdef}
is presented.
\begin{prop}
\label{propsimple}
Let
$
X \colon \mathbbm{I} \times \Omega
\rightarrow \tilde{H}
$
be a mild It\^{o} process
with semigroup
$
S \colon \angle
\rightarrow L( \hat{H}, \check{H} )
$,
mild drift
$
Y \colon \mathbbm{I} \times \Omega
\rightarrow \hat{H}
$
and mild
diffusion
$
Z \colon \mathbbm{I} \times \Omega
\rightarrow HS( U_0, \hat{H} )
$.
Then
\begin{equation}
\label{eq:itovolterra}
X_{ t_2 }
=
S_{ t_1, t_2 } \,
X_{ t_1 }
+
\int_{t_1}^{t_2}
S_{ s, t_2 } \,
Y_s
\, ds
+
\int_{t_1}^{t_2}
S_{ s, t_2 } \,
Z_s
\, dW_s
\end{equation}
$ \mathbb{P} $-a.s.\
for all
$ t_1, t_2 \in \mathbbm{I} $
with $ t_1 < t_2 $.
\end{prop}
Proposition~\ref{propsimple}
follows directly from
Theorem~\ref{thm:ito}
below.
Obviously,
equation~\eqref{eq:itovolterra}
in Proposition~\ref{propsimple}
generalizes
equation~\eqref{eq:mildito}
in the definition
of a mild It\^{o} process.
Let us complete this subsection
on mild It\^{o} processes
with the notion of a certain subclass of
mild It\^{o} processes.
\begin{definition}[Mild
martingale]
A
mild It\^{o} process
$
X \colon \mathbbm{I} \times \Omega
\rightarrow \tilde{H}
$
with semigroup
$
S \colon \angle
\rightarrow L( \hat{H}, \check{H} )
$,
mild drift
$
Y \colon \mathbbm{I} \times \Omega
\rightarrow \hat{H}
$
and mild
diffusion
$
Z \colon \mathbbm{I} \times \Omega
\rightarrow HS( U_0, \hat{H} )
$
satisfying
$
\mathbb{E}\big[ \| X_t \|_{ \tilde{H} }
\big] < \infty
$
for all $ t \in \mathbbm{I} $
is called a
\underline{mild martingale}
(with respect to the filtration
$
\mathcal{F}_t,
t \in \mathbbm{I},
$
and with respect to
the semigroup $ S $) if
\begin{equation}
\label{eq:mildmartingale}
\mathbb{E}\!\left[
X_{ t_2 }
\big|
\mathcal{F}_{ t_1 }
\right]
=
S_{ t_1, t_2 } \,
X_{ t_1 }
\end{equation}
$ \mathbb{P} $-a.s.\ for
all $ t_1, t_2 \in \mathbbm{I} $
with $ t_1 < t_2 $.
\end{definition}
\begin{prop}[Stochastic convolutions]
\label{propmartingale}
Let
$
X \colon \mathbbm{I} \times \Omega
\rightarrow \tilde{H}
$
be a mild It\^{o} process
with semigroup
$
S \colon
\angle
\rightarrow L( \hat{H}, \check{H} )
$,
mild drift
$
Y \colon \mathbbm{I} \times \Omega
\rightarrow \hat{H}
$
and mild
diffusion
$
Z \colon \mathbbm{I} \times \Omega
\rightarrow HS( U_0, \hat{H} )
$
satisfying
$
\mathbb{E}\big[
\| X_t \|_{ \tilde{H} }^2
\big] < \infty
$
for all $ t \in \mathbbm{I} $.
If $ \mathbb{P}\big[ Y_t = 0 \big] = 1 $
for all $ t \in \mathbbm{I} $, then
$ X $ is a mild martingale
with respect to the
filtration $ \mathcal{F}_t $,
$ t \in \mathbbm{I} $, and
with respect to
the semigroup $ S $.
\end{prop}
\begin{proof}[Proof
of Propostion~\ref{propmartingale}]
Propostion~\ref{propsimple}
yields
\begin{equation}
\label{eq:convolution}
X_{ t_2 }
=
S_{ t_1, t_2 } \, X_{ t_1 }
+
\int_{ t_1 }^{ t_2 }
S_{ s, t_2 }
\, Z_s \, dW_s
\end{equation}
$ \mathbb{P} $-a.s.\ for
all $ t_1, t_2 \in \mathbbm{I} $
with $ t_1 < t_2 $.
Equation~\eqref{eq:convolution}
and the assumption
$
\mathbb{E}\big[
\| X_t \|_{ \tilde{H} }^2
\big]
< \infty
$
for all $ t \in \mathbbm{I} $
imply
equation~\eqref{eq:mildmartingale}.
The proof of Proposition~\ref{propmartingale}
is thus completed.
\end{proof}
\subsection{Mild It\^{o} formula}
\label{sec:mildito}
Let $ \mathcal{J} $ be a set and let
$ g_j \in U_0 $,
$ j \in \mathcal{J} $,
be an arbitrary orthonormal basis
of the $ \mathbb{R} $-Hilbert
space
$ \left(
U_0,
\left< \cdot, \cdot
\right>_{ U_0 },
\left\| \cdot
\right\|_{ U_0 }
\right) $.
For an
$ \mathbb{R} $-vector space
$ ( V,
\left\| \cdot \right\|_V )
$
and a function
$
\varphi \in
C^{ 1, 2 }(
\mathbbm{I} \times \check{H},
V
)
$
we denote by
$
\partial_1 \varphi
\in
C( \mathbbm{I} \times \check{H}, V )
$,
$
\partial_2 \varphi
\in
C( \mathbbm{I} \times \check{H},
L(\check{H}, V)
)
$
and
$
\partial_2^2 \varphi
\in
C( \mathbbm{I} \times \check{H},
L^{(2)}( \check{H}, V)
)
$
the partial
Fr\'{e}chet
derivatives of $ \varphi $
given by
$
( \partial_1 \varphi)(t,x)
=
\big(
\tfrac{ \partial \varphi }{ \partial t }
\big)( t, x )
$,
$
( \partial_2 \varphi)(t,x)
=
\big(
\tfrac{ \partial \varphi }{ \partial x }
\big)( t, x )
$
and
$
( \partial_2^2 \varphi)(t,x)
=
\big(
\tfrac{ \partial^2 \varphi }{ \partial x^2 }
\big)( t, x )
$
for all
$ t \in \mathbbm{I} $
and all
$ x \in \check{H} $.
\begin{theorem}[Mild
It{\^o} formula]
\label{thm:ito}
Let
$
X \colon \mathbbm{I} \times \Omega
\rightarrow \tilde{H}
$
be a mild It{\^o} process
with semigroup
$
S \colon \angle
\rightarrow L( \hat{H}, \check{H} )
$,
mild drift
$
Y \colon \mathbbm{I} \times
\Omega \rightarrow \hat{H}
$
and mild
diffusion
$
Z \colon \mathbbm{I} \times
\Omega \rightarrow
HS(U_0,\hat{H})
$
and let $ ( V,
\left< \cdot, \cdot \right>_V,
\left\| \cdot \right\|_V ) $
be a separable
$ \mathbb{R} $-Hilbert
space.
Then
\begin{equation}
\label{eq:well1}
\mathbb{P}\!\left[
\int_{ t_0 }^t
\left\|
( \partial_2 \varphi)( s, S_{ s, t } X_s )
S_{ s, t } Y_s
\right\|_V
+
\left\|
( \partial_2 \varphi)( s, S_{ s, t } X_s )
S_{ s, t } Z_s
\right\|_{ HS(U_0, V ) }^2
ds
< \infty
\right] = 1,
\end{equation}
\begin{equation}
\label{eq:well2}
\mathbb{P}\!\left[
\int_{ t_0 }^t
\left\|
( \partial_1 \varphi)( s, X_s )
\right\|_V
+
\|
(\partial^2_2 \varphi)( s, S_{ s, t } X_s )
\|_{ L^{(2)}( \check{H}, V ) } \,
\|
S_{ s, t } Z_s
\|_{ HS(U_0, \check{H} ) }^2 \,
ds
< \infty
\right] = 1
\end{equation}
and
\begin{equation}
\label{eq:itoformel_start}
\begin{split}
&
\varphi( t, X_t )
=
\varphi( t_0,
S_{ t_0, t }
X_{ t_0 }
)
+
\int_{ t_0 }^t
(\partial_1
\varphi)( s,
S_{ s, t }
X_s
) \, ds
+
\int_{ t_0 }^t
(\partial_2
\varphi)( s,
S_{ s, t }
X_s
) \,
S_{ s, t } \,
Y_s \, ds
\\&\quad+
\int_{ t_0 }^t
(\partial_2
\varphi)( s,
S_{ s, t }
X_s
) \,
S_{ s, t } \,
Z_s \, dW_s
+
\frac{1}{2}
\sum_{ j \in \mathcal{J} }
\int_{ t_0 }^t
( \partial_2^2 \varphi )( s,
S_{ s, t }
X_s
)
\left(
S_{ s, t }
Z_s g_j,
S_{ s, t }
Z_s g_j
\right) ds
\end{split}
\end{equation}
$ \mathbb{P} $-a.s.\ for
all $ t_0, t \in \mathbbm{I} $
with $ t_0 < t $
and all
$
\varphi \in C^{1,2}(
\mathbb{I} \times \check{H}, V
)
$.
\end{theorem}
Note that \eqref{eq:well1}
and \eqref{eq:well2}
ensure that the
possibly infinite sum
and all integrals in
\eqref{eq:itoformel_start}
are well defined.
Indeed,
equation~\eqref{eq:well2}
implies
\begin{equation}
\begin{split}
\lefteqn{
\sum_{ j \in \mathcal{J} }
\int_{ t_0 }^t
\left\|
(\partial^2_2 \varphi)( s,
S_{ s, t }
X_s
)
\!\left(
S_{ s, t }
Z_s g_j,
S_{ s, t }
Z_s g_j
\right)
\right\|_{ V }
ds
}
\\ & \leq
\int_{ t_0 }^t
\left\|
(\partial^2_2 \varphi)( s,
S_{ s, t }
X_s
)
\right\|_{ L^{(2)}(\check{H}, V) }
\left(
\sum\nolimits_{ j \in \mathcal{J} }
\left\|
S_{ s, t }
Z_s g_j
\right\|_{ \check{H} }^2
\right)
ds
\\ & =
\int_{ t_0 }^t
\left\|
(\partial^2_2 \varphi)( s,
S_{ s, t }
X_s
)
\right\|_{ L^{(2)}( \check{H}, V ) }
\left\|
S_{ s, t }
Z_s
\right\|_{ HS( U_0, \check{H} ) }^2
ds
< \infty
\end{split}
\end{equation}
$ \mathbb{P} $-a.s.\ for
all $ t_0, t \in \mathbbm{I} $
with $ t_0 < t $
and all
$
\varphi
\in C^{ 1, 2 }(
\mathbbm{I} \times
\check{H}, V
)
$.
Moreover, note that
the mild It\^{o}
formula~\eqref{eq:itoformel_start}
is independent of the
particular choice
of the orthonormal
basis $ g_j \in U_0 $,
$ j \in \mathcal{J} $,
of $ U_0 $.
In the next step a certain
flow property of the mild
It\^{o} formula~\eqref{eq:itoformel_start}
is observed. To be more precise,
the mild It\^{o} formula~\eqref{eq:itoformel_start}
on the time
interval $ [\hat{t}, t] $
applied to the test function
$ \varphi( s, v ) $,
$ s \in [\hat{t}, t] $,
$ v \in \check{H} $,
reads as
\begin{equation}
\label{eq:sgprop1}
\begin{split}
&
\varphi( t, X_t )
=
\varphi(
\hat{t}, S_{ \hat{t}, t }
X_{ \hat{t} }
)
+
\int_{ \hat{t} }^t
(\partial_1 \varphi)( s,
S_{ s, t }
X_s
) \,
ds
+
\int_{ \hat{t} }^t
(\partial_2 \varphi)( s,
S_{ s, t }
X_s
) \,
S_{ s, t } \,
Y_s \, ds
\\&\quad+
\int_{ \hat{t} }^t
(\partial_2 \varphi)( s,
S_{ s, t }
X_s
) \,
S_{ s, t } \,
Z_s \, dW_s
+
\frac{1}{2}
\sum_{ j \in \mathcal{J} }
\int_{ \hat{t} }^t
(\partial_2^2 \varphi)( s,
S_{ s, t }
X_s
)
\left(
S_{ s, t }
Z_s g_j,
S_{ s, t }
Z_s g_j
\right) ds
\end{split}
\end{equation}
$ \mathbb{P} $-a.s.\ for
all $ \hat{t}, t \in \mathbbm{I} $
with $ \hat{t} < t $
and all
$
\varphi \in
C^{1,2}(
\mathbbm{I} \times
\check{H},
V
)
$.
Moreover, observe that
the mild It\^{o}
formula~\eqref{eq:itoformel_start}
on the time interval
$ [t_0,\hat{t}] $
applied to the test function
$ \varphi( s, S_{ \hat{t}, t } v ) $,
$ s \in [t_0,\hat{t}] $,
$ v \in \check{H} $,
reads as
\begin{equation}
\label{eq:sgprop2}
\begin{split}
&
\varphi( \hat{t}, S_{ \hat{t}, t }
X_{ \hat{t} } )
=
\varphi(
t_0,
S_{ t_0, t }
X_{ t_0 }
)
+
\int_{ t_0 }^{ \hat{t} }
(\partial_1 \varphi)( s,
S_{ s, t }
X_s
) \,
ds
+
\int_{ t_0 }^{ \hat{t} }
(\partial_2 \varphi)( s,
S_{ s, t }
X_s
) \,
S_{ s, t } \,
Y_s \, ds
\\&\quad+
\int_{ t_0 }^{ \hat{t} }
(\partial_2 \varphi)( s,
S_{ s, t }
X_s
) \,
S_{ s, t } \,
Z_s \, dW_s
+
\frac{ 1 }{ 2 }
\sum_{ j \in \mathcal{J} }
\int_{ t_0 }^{ \hat{t} }
(\partial_2^2 \varphi)( s,
S_{ s, t }
X_s
)
\left(
S_{ s, t }
Z_s g_j,
S_{ s, t }
Z_s g_j
\right) ds
\end{split}
\end{equation}
$ \mathbb{P} $-a.s.\ for
all $ t_0, \hat{t}, t \in \mathbbm{I} $
with $ t_0 < \hat{t} < t $
and all
$
\varphi \in
C^{1,2}(
\mathbbm{I} \times
\check{H},
V
)
$.
Putting \eqref{eq:sgprop2}
into \eqref{eq:sgprop1}
then results in the mild It\^{o}
formula~\eqref{eq:itoformel_start}
on the time interval $ [t_0,t] $
for $ t_0, t \in \mathbbm{I} $
with $ t_0 < t $.
If the test function
$
( \varphi(t, x) )_{
t \in \mathbbm{I},
x \in \check{H}
}
\in
C^{1,2}(
\mathbbm{I} \times \check{H}, V
)
$
in the mild It\^{o}
formula~\eqref{eq:itoformel_start}
does not depend on
$ t \in \mathbbm{I} $, then the
mild It\^{o} formula in Theorem~\ref{thm:ito}
reads as follows.
\begin{cor}
\label{cor:itoauto}
Let
$
X \colon \mathbbm{I} \times \Omega
\rightarrow \tilde{H}
$
be a mild It{\^o} process
with semigroup
$
S \colon \angle
\rightarrow
L( \hat{H}, \check{H} )
$,
mild drift
$
Y \colon \mathbbm{I} \times
\Omega \rightarrow \hat{H}
$
and mild
diffusion
$
Z \colon \mathbbm{I} \times
\Omega \rightarrow
HS(U_0,\hat{H})
$
and let $ ( V,
\left< \cdot, \cdot \right>_V,
\left\| \cdot \right\|_V ) $
be a separable
$ \mathbb{R} $-Hilbert
space.
Then
\begin{equation}
\label{eq:well1NON}
\mathbb{P}\!\left[
\int_{ t_0 }^t
\left\|
\varphi'( S_{ s, t } X_s )
S_{ s, t } Y_s
\right\|_V
+
\left\|
\varphi'( S_{ s, t } X_s )
S_{ s, t } Z_s
\right\|_{ HS(U_0, V ) }^2
ds
< \infty
\right] = 1,
\end{equation}
\begin{equation}
\label{eq:well2NON}
\mathbb{P}\!\left[
\int_{ t_0 }^t
\|
\varphi''( S_{ s, t } X_s )
\|_{ L^{(2)}( \check{H}, V ) } \,
\|
S_{ s, t } Z_s
\|_{ HS(U_0, \check{H} ) }^2 \,
ds
< \infty
\right] = 1
\end{equation}
and
\begin{align}
\label{eq:itoformel_startNON}
\varphi( X_t )
&=
\varphi(
S_{ t_0, t }
X_{ t_0 }
)
+
\int_{ t_0 }^t
\varphi'(
S_{ s, t }
X_s
) \,
S_{ s, t } \,
Y_s \, ds
+
\int_{ t_0 }^t
\varphi'(
S_{ s, t }
X_s
) \,
S_{ s, t } \,
Z_s \, dW_s
\nonumber
\\&\quad+
\frac{1}{2}
\sum_{ j \in \mathcal{J} }
\int_{ t_0 }^t
\varphi''(
S_{ s, t }
X_s
)
\left(
S_{ s, t }
Z_s g_j,
S_{ s, t }
Z_s g_j
\right) ds
\end{align}
$ \mathbb{P} $-a.s.\ for
all
$ t_0, t \in \mathbbm{I} $
with $ t_0 < t $
and all
$
\varphi \in C^{2}(
\check{H}, V
)
$.
\end{cor}
Let us illustrate
Theorem~\ref{thm:ito}
and Corollary~\ref{cor:itoauto}
by two simple examples.
The first one is a mild
version of the stochastic
integration by parts
formula (see, e.g., Corollary~2.6
in \cite{bvvw08}).
\begin{example}[Mild stochastic
integration
by parts]
\label{ex:1}
Let
$
(
\check{\mathcal{H}},
\left< \cdot , \cdot
\right>_{ \check{\mathcal{H}} },
\left\| \cdot \right\|_{
\check{\mathcal{H}}
}
)
$,
$
(
\tilde{\mathcal{H}},
\left< \cdot , \cdot
\right>_{ \tilde{\mathcal{H}} },
\left\| \cdot
\right\|_{ \tilde{\mathcal{H}} }
)
$,
$
(
\hat{\mathcal{H}},
\left< \cdot , \cdot
\right>_{ \hat{\mathcal{H}} },
\left\| \cdot \right\|_{ \hat{\mathcal{H}} }
)
$,
$
(
\mathcal{U},
\left< \cdot , \cdot \right>_\mathcal{U},
\left\| \cdot \right\|_{ \mathcal{U} }
)
$
and
$
( V,
\left< \cdot, \cdot \right>_V,
\left\| \cdot \right\|_V )
$
be separable
$\mathbb{R}$-Hilbert spaces
with
$
\check{\mathcal{H}}
\subset
\tilde{\mathcal{H}}
\subset
\hat{\mathcal{H}}
$
continuously and densely,
let
$
X \colon \mathbbm{I} \times \Omega
\rightarrow \tilde{H}
$
be a mild It{\^o} process
with semigroup
$
S \colon \angle
\rightarrow L( \hat{H}, \check{H} )
$,
mild drift
$
Y \colon \mathbbm{I} \times
\Omega \rightarrow \hat{H}
$
and mild
diffusion
$
Z \colon \mathbbm{I} \times
\Omega \rightarrow
HS(U_0,\hat{H})
$
and
let
$
\mathcal{X} \colon
\mathbbm{I} \times \Omega
\rightarrow \tilde{\mathcal{H}}
$
be a mild It{\^o} process
with semigroup
$
\mathcal{S} \colon \angle
\rightarrow
L(
\hat{\mathcal{H}},
\check{\mathcal{H}}
)
$,
mild drift
$
\mathcal{Y}
\colon \mathbbm{I} \times
\Omega \rightarrow \hat{\mathcal{H}}
$
and mild
diffusion
$
\mathcal{Z} \colon \mathbbm{I} \times
\Omega \rightarrow
HS(U_0,\hat{\mathcal{H}})
$.
Corollary~\ref{cor:itoauto} then shows
\begin{align}
&
\varphi\big(
X_t, \mathcal{X}_t
\big)
=
\varphi\big(
S_{ t_0, t } X_t,
\mathcal{S}_{ t_0, t } \mathcal{X}_t
\big)
+
\int_{ t_0 }^t
\varphi\big(
S_{ s, t } Y_s,
\mathcal{S}_{ s, t } \mathcal{X}_s
\big)
\, ds
+
\int_{ t_0 }^t
\varphi\big(
S_{ s, t } X_s,
\mathcal{S}_{ s, t } \mathcal{Y}_s
\big)
\,
ds
\\ & +
\int_{ t_0 }^t
\varphi\big(
S_{ s, t } Z_s( \cdot ),
\mathcal{S}_{ s, t } \mathcal{X}_s
\big)
\, dW_s
+
\int_{ t_0 }^t
\varphi\big(
S_{ s, t } X_s,
\mathcal{S}_{ s, t }
\mathcal{Z}_s( \cdot )
\big)
\,
dW_s
+
\sum_{ j \in \mathcal{J} }
\int_{ t_0 }^t
\varphi\big(
S_{ s, t } Z_s g_j,
\mathcal{S}_{ s, t } \mathcal{Z}_s
g_j
\big)
\,
ds
\nonumber
\end{align}
$ \mathbb{P} $-a.s.\ for
all $ t_0, t \in \mathbbm{I} $
with $ t_0 < t $
and all bounded bilinear mappings
$
\varphi \colon
\check{H} \times \check{\mathcal{H}}
\rightarrow
V
$.
\end{example}
\begin{example}[Mild chain rule]
\label{ex:2}
Let
$
S \colon
\angle
\rightarrow
L( \hat{H}, \check{H} )
$
be an
$
\mathcal{B}( \angle )
$/$
\mathcal{S}( \hat{H}, \check{H} )
$-measurable
mapping satisfying
$
S_{ t_2, t_3 }
S_{ t_1, t_2 }
=
S_{ t_1, t_3 }
$
for all $ t_1, t_2, t_3
\in \mathbbm{I} $
with $ t_1 < t_2 < t_3 $
and
let
$
x \colon \mathbbm{I}
\rightarrow \tilde{H}
$
and
$
y \colon \mathbbm{I}
\rightarrow \hat{H}
$
be two Borel measurable
functions
with
$
\int_{ \tau }^{ t }
\left\| S_{ s, t } y_s
\right\|_{ \tilde{H} }
ds < \infty
$
and
$
x_t
=
S_{ \tau, t } \,
x_{ \tau }
+
\int_{ \tau }^t
S_{ s, t } \,
y_s
\, ds
$
for all $ t \in \mathbbm{I} $.
Corollary~\ref{cor:itoauto}
then shows
\begin{equation}
\label{eq:mildchain}
\varphi( x_t )
=
\varphi( S_{ t_0, t} x_{ t_0 } )
+
\int_{ t_0 }^t
\varphi'( S_{ s, t } x_s ) \,
S_{ s, t } \,
y_s \, ds
\end{equation}
for all $ t_0, t \in \mathbbm{I} $
with $ t_0 < t $
and all
$
\varphi \in
C^2( \check{H}, V )
$.
Equation~\eqref{eq:mildchain}
is somehow a mild
chain rule for
the mild process
$
x \colon \mathbbm{I} \rightarrow
\tilde{H}
$.
\end{example}
Let us now concentrate on proofs
of the mild It\^{o}
formula~\eqref{eq:itoformel_start}.
A central difficulty in order
to establish an It{\^o} formula
for the stochastic process
$
X \colon \mathbbm{I}
\times \Omega \rightarrow
\tilde{H}
$
is that this stochastic process
is, in general,
not a standard It\^{o} process
to which
the standard
It{\^o} formula
(see, e.g.,
Theorem~4.17 in Section~4.5 in
Da Prato \citationand\
Zabczyk~\cite{dz92}) could be
applied.
(Here and below the terminology
``standard It\^{o} formula'' instead
of simply ``It\^{o} formula'' is used
in order to distinguish more clearly
from the above mild It\^{o} formula.)
The stochastic process
$
X \colon \mathbbm{I} \times
\Omega \rightarrow \tilde{H}
$
is, in general, not a standard
It\^{o} process
since it satisfies the
It{\^o}-Volterra type
equation~\eqref{eq:mildito}
in which the integrand processes
$ S_{ s, t } \, Y_s $,
$ s \in [\tau, t] $,
and $ S_{ s, t } \, Z_s $,
$ s \in [\tau, t] $,
depend explicitly
on $ t \in \mathbbm{I} $ too
(this was the reason for introducing
the notion of a
mild It\^{o} process;
see Definition~\ref{propdef}).
Below we present two proofs
which overcome
this difficulty and which establish
the mild It\^{o} formula~\eqref{eq:itoformel_start}.
Both proofs consider
appropriate transformations
of the mild It\^{o} process
$ X \colon \mathbbm{I} \times \Omega
\rightarrow \tilde{H} $.
The transformed stochastic processes
are then standard It\^{o} processes
to which the standard
It\^{o} formula can be applied.
Relating then the transformed
stochastic processes in a
suitable way to the original
mild It\^{o} process
$
X \colon \mathbbm{I} \times
\Omega \rightarrow \tilde{H}
$
finally results
in the mild It\^{o}
formula~\eqref{eq:itoformel_start}.
The main difference of the two
proofs is the type of transformation
applied to the mild It\^{o}
process
$
X \colon \mathbbm{I} \times \Omega
\rightarrow \tilde{H}
$.
The first proof makes use of
a transformation in
Teichmann~\cite{t09}
and
Filipovi{\'c}, Tappe
\citationand\
Teichmann~\cite{ftt10}
(see equations (1.3)
and (1.4) in Teichmann~\cite{t09}
and Section~8 in
Filipovi{\'c}, Tappe
\citationand\
Teichmann~\cite{ftt10}
and see also
Hausenblas
\citationand\
Seidler~\cite{hs01,hs08}).
The first proof does not show
Theorem~\ref{thm:ito} in the
general case but in the case
in which the semigroup of the
mild It\^{o} process is
pseudo-contractive
(see below for the precise
description of
the used assumptions).
Under this additional assumption,
the semigroup
$
( S_{ t_1, t_2 } )_{
(t_1, t_2) \in \angle
}
$
on the Hilbert space $ \hat{H} $
can be dilated to a group
$
( \mathcal{U}_t )_{ t \in \mathbb{R} }
$
on
a larger Hilbert space
(see, e.g.,
Sz\"{o}kefalvi-Nagy~\cite{na53,na54}
and Theorem~I.81 in
Sz\"{o}kefalvi-Nagy
\citationand\
Foia{\lfhook{s}}~\cite{naf70}
for the so-called dilations of the unitary theorem).
On this larger Hilbert space,
the mild It\^{o} process~\eqref{eq:mildito} can
be transformed into a
standard It\^{o} process
by -- roughly speaking -- multiplying with
$ \mathcal{U}_{ - t } $
for $ t \in \mathbbm{I} $.
Next the standard It\^{o} formula
can be applied to the transformed
standard It\^{o} process.
Relating this transformed
standard It\^{o}
process then in a suitable way
to the original mild It\^{o} process
finally results
in the mild It\^{o}
formula~\eqref{eq:itoformel_start}.
The second proof
establishes
Theorem~\ref{thm:ito} in the general
case.
It makes use of an idea
in Conus
\citationand\
Dalang~\cite{cd08}
and
Conus~\cite{c08}
(see Section~6 in
Conus \citationand\
Dalang~\cite{cd08}
and
equations~(1.7)
and (7.6) in
Conus~\cite{c08}
and see also
Section~3 in
Debussche
\citationand\
Printems~\cite{dp09},
Theorem~4 in
Lindner \citationand\
Schilling~\cite{ls10}
and Theorem~3.1 in
Kov{\'a}cs, Larsson
\citationand\
Lindgren~\cite{kll11})
and exploits a more elementary
transformation.
Roughly speaking, the
mild It\^{o} process
$ X \colon \mathbbm{I} \times \Omega
\rightarrow \tilde{H} $
is transformed in the second
proof by multiplying
with $ S_{ t, T } $ for
$ t \in [\tau, T) $
with a fixed $ T \in \mathbbm{I} $
(compare that
the transformation in first proof
is based on multiplying with the group
at the negative time value $ - t $).
Since
$ T - t > 0 $, the transformed
process of the
$ \tilde{H} $-valued process
$ X \colon \mathbbm{I} \times \Omega
\rightarrow \tilde{H} $
enjoys values in
$ \tilde{H} $ too
(this is in contrast to the first
proof where the transformed process
of $ X \colon \mathbbm{I} \times
\Omega \rightarrow \tilde{H} $
takes values in a larger
Hilbert space in which $ \tilde{H} $
is continuously embedded).
Nonetheless, as in the first proof,
the transformed
stochastic process is
a standard It\^{o} process
to which the standard It\^{o} formula
can be applied.
Relating the transformed
standard It\^{o} process
in a suitable way
to the original mild It\^{o}
process then again results
in the mild It\^{o}
formula~\eqref{eq:itoformel_start}.
Both proofs thus essentially consist
of three steps: a
{\it transformation},
an {\it application of
the standard
It\^{o} formula}
and the use of a
suitable {\it relation}
of the transformed
standard It\^{o}
process and the original
mild It\^{o} process.
The second proof also uses
the following simple result.
\begin{lemma}
\label{lem:easy}
Let $ Y, Z \colon \mathbbm{I}
\times \Omega \rightarrow [0,\infty) $
be two product measurable
stochastic processes with
$
\mathbb{P}\big[
Y_t = Z_t
\big] = 1
$
for all $ t \in \mathbbm{I} $
and with
$
\mathbb{P}\big[
\int_{ \mathbbm{I} } Y_s \, ds < \infty
\big] = 1
$.
Then
$
\mathbb{P}\big[
\int_{ \mathbbm{I} } Z_s \, ds < \infty
\big] = 1
$.
\end{lemma}
The proof of Lemma~\ref{lem:easy}
is clear and therefore omitted.
Instead the first proof
of Theorem~\ref{thm:ito}
in the special case of
a pseudo-contractive semigroup
is now presented.
\begin{proof}
{\it Proof
of Theorem~\ref{thm:ito}
in the case
where
the partial Fr\'{e}chet
derivatives
$ \partial_1 \varphi $,
$ \partial_2 \varphi $
and
$ \partial_2^2 \varphi $
of $ \varphi $
are globally bounded,
where
$ Y \colon \mathbbm{I} \times \Omega
\rightarrow \hat{H}
$
and
$ Z \colon \mathbbm{I} \times \Omega
\rightarrow HS( U_0, \hat{H} )
$
have
continuous sample paths,
where
$
(
\check{H},
\left< \cdot , \cdot \right>_{ \check{H} },
\left\| \cdot \right\|_{ \check{H} }
)
=
(
\tilde{H},
\left< \cdot , \cdot \right>_{ \tilde{H} },
\left\| \cdot \right\|_{ \tilde{H} }
)
=
(
\hat{H},
\left< \cdot , \cdot \right>_{ \hat{H} },
\left\| \cdot \right\|_{ \hat{H} }
)
$,
where
$ U_t \in L( \tilde{H} ) $,
$ t \in [0, \infty) $,
is a strongly continuous pseudo-contractive semigroup
on $ \tilde{H} $
and where
$
S_{ t_1, t_2 } = U_{ ( t_2 - t_1 ) }
\in L( \tilde{H} )
$
for all $ (t_1, t_2) \in \angle $.}
First, observe that,
under these additional
assumptions,
\eqref{eq:well1}
and \eqref{eq:well2}
are obviously fulfilled.
Moreover, due to Proposition~8.7
in \cite{ftt10},
there exists
a separable $ \mathbb{R} $-Hilbert
space $ \left( \mathcal{H}, \left< \cdot, \cdot \right>_{ \mathcal{H} }, \left\| \cdot \right\|_{ \mathcal{H} } \right) $
with $ \tilde{H} \subset \mathcal{H} $ and $ \left\| v \right\|_{ \tilde{H} } = \left\| v \right\|_{ \mathcal{H} } $ for
all $ v \in \tilde{H} $ and
a strongly continuous group
$ \mathcal{U}_t \in L( \mathcal{H} ) $,
$ t \in \mathbb{R} $, such that
\begin{equation}
\label{eq:put}
U_t( v )
=
P\big(
\mathcal{U}_t( v )
\big)
\end{equation}
for all
$ v \in \tilde{H} \subset \mathcal{H} $
and all $ t \in [0,\infty) $
where
$
P \colon
\mathcal{H} \rightarrow \tilde{H}
$
is the orthogonal projection from $ \mathcal{H} $ to $ \tilde{H} $.
In this first proof
the mild It\^{o} process
$
X \colon \mathbbm{I} \times \Omega
\rightarrow \tilde{H}
$
is now transformed into
a standard It\^{o} process
by - roughly speaking - multiplying
with $ \mathcal{U}_{ - t } $
for $ t \in \mathbbm{I} $.
In a more concrete setting
this transformation
has been proposed
in Teichmann~\cite{t09} and
Filipovi{\'c}, Tappe
\citationand\
Teichmann~\cite{ftt10};
see equations~(1.3) and (1.4) in
Teichmann~\cite{t09} and
Section~8 in Filipovi{\'c},
Tappe
\citationand\
Teichmann~\cite{ftt10}
and see also
Hausenblas \citationand\
Seidler~\cite{hs01,hs08}.
Let us now go into details.
Let
$
\bar{X} \colon \mathbbm{I} \times
\Omega \rightarrow \mathcal{H}
$
be the up to
indistinguishability
unique adapted stochastic process with continuous
sample paths satisfying
\begin{equation}
\label{eq:barX}
\bar{X}_t
=
X_{ \tau }
+
\int_{ \tau }^t
\mathcal{U}_{-s} \, Y_s \, ds
+
\int_{ \tau }^t
\mathcal{U}_{-s} \, Z_s \, dW_s
\end{equation}
$ \mathbb{P}
$-a.s.\ for
all $ t \in \mathbbm{I} $
({\it Transformation};
see also equation~(1.4)
in \cite{t09}
and equation~(8.6)
in \cite{ftt10}).
Next observe that
the identity
$
X_t =
\mathcal{P}\big(
\mathcal{U}_t(
\bar{X}_t
)
\big)
$
$ \mathbb{P} $-a.s.\ (see
also
Theorem~8.8 in
\cite{ftt10})
and
the standard
It{\^o} formula
in infinite dimensions (see
Theorem~2.4 in Brze\'{z}niak,
Van Neerven, Veraar
\citationand\
Weis~\cite{bvvw08})
applied to the test function
$ \varphi\big( s,
P\big( \mathcal{U}_t( v ) \big)
\big) $,
$ s \in [t_0,t] $,
$ v \in \mathcal{H} $,
give
\begin{equation}
\label{eq:important}
\begin{split}
&
\varphi( t,
X_t
)
=
\varphi\big( t,
P\big( \mathcal{U}_t( \bar{X}_t ) \big)
\big)
=
\varphi\big( t_0,
P\big( \mathcal{U}_t( \bar{X}_{ t_0 } )
\big)
\big)
+
\int_{t_0}^t
(\partial_1 \varphi)\big( s,
P\big( \mathcal{U}_t( \bar{X}_{ s } )
\big)
\big)
\,
ds
\\ & \quad
+
\int_{t_0}^t
(\partial_2 \varphi)\big( s,
P\big( \mathcal{U}_t( \bar{X}_{ s } )
\big)
\big)
\,
P \;
\mathcal{U}_{(t-s)} \,
Y_s \, ds
+
\int_{t_0}^t
(\partial_2 \varphi)\big( s,
P\big( \mathcal{U}_t( \bar{X}_{ s } )
\big)
\big)
\,
P \;
\mathcal{U}_{(t-s)} \,
Z_s \, dW_s
\\ & \quad
+
\frac{1}{2}
\sum_{ j \in \mathcal{J} }
\int_{t_0}^t
(\partial_2^2 \varphi)\big( s,
P\big( \mathcal{U}_t( \bar{X}_{ s } )
\big)
\big) \!
\left(
P \, \mathcal{U}_{(t-s)} Z_s g_j,
P \, \mathcal{U}_{(t-s)} Z_s g_j
\right) ds
\end{split}
\end{equation}
$ \mathbb{P} $-a.s.\ for
all $ t_0, t \in \mathbbm{I} $
with $ t_0 \leq t $
and all
$
\varphi \in
C^{ 1, 2 }(
\mathbbm{I} \times \tilde{H}, V
)
$
({\it Application of the
standard It\^{o} formula}).
Next note that
equation~\eqref{eq:put} gives
\begin{equation}
\label{eq:backtransform0}
\begin{split}
P\big( \mathcal{U}_t( \bar{X}_{ s } )
\big)
& =
P\big( \mathcal{U}_t( X_{ \tau } )
\big)
+
\int_{ \tau }^s
P \;
\mathcal{U}_{(t-u)} \, Y_u \, du
+
\int_{ \tau }^s
P \;
\mathcal{U}_{(t-u)} \, Z_u \,
dW_u
\\ & =
S_{\tau, t} \, X_{ \tau }
+
\int_{ \tau }^s
S_{u,t} \, Y_u \, du
+
\int_{ \tau }^s
S_{ u, t } \, Z_u \,
dW_u
\\ & =
S_{ s, t }
\left(
S_{ \tau, s } \, X_{ \tau }
+
\int_{ \tau }^s
S_{ u, s } \,
Y_u \, du
+
\int_{ \tau }^s
S_{ u, s } \,
Z_u \, dW_u
\right)
=
S_{ s, t } \, X_s
\end{split}
\end{equation}
$ \mathbb{P} $-a.s.\ for
all $ s, t \in \mathbbm{I} $
with $ s \leq t $
({\it Relation} of the transformed
standard It\^{o} process
$
\bar{X}
\colon \mathbbm{I} \times \Omega
\rightarrow \mathcal{H}
$
and the original mild It\^{o}
process
$
X \colon \mathbbm{I} \times
\Omega \rightarrow \tilde{H}
$).
Using
\eqref{eq:put}
and
\eqref{eq:backtransform0}
in
\eqref{eq:important}
finally shows
\eqref{eq:itoformel_start}.
The proof
is thus completed.
\end{proof}
In the next step
the proof
of Theorem~\ref{thm:ito}
in the general case is
given.
Above an outline of this second proof
is given.
\begin{proof}[Proof
of Theorem~\ref{thm:ito}]
In this second proof
the time variable $ t \in \mathbbm{I} $
within the integrand processes
in \eqref{eq:mildito} is fixed
and then, the standard It\^{o} formula
is applied to the resulting
standard It\^{o} process.
In a more concrete setting
this trick has been
proposed in
Conus \citationand\
Dalang~\cite{cd08}
and
Conus~\cite{c08};
see Section~6 in
Conus \citationand\ Dalang~\cite{cd08}
and
equations~(1.7) and (7.6) in
Conus~\cite{c08} and see
also
Section~5 in
Lindner \citationand\
Schilling~\cite{ls10} and
Section~3 in
Kovacs, Larsson \citationand\
Lindgren~\cite{kll11}.
Another related result can
be found in Section~3
in Debussche \citationand\
Printemps~\cite{dp09}.
Let us now go into details.
Let
$
\bar{X}^{t} \colon
[ \tau, t ] \times \Omega
\rightarrow
\check{H}
$,
$
t \in
\mathbbm{I} \cap
( \tau, \infty )
$,
be a family of adapted
stochastic processes
with continuous sample paths
given by
\begin{equation}
\label{eq:semiX}
\bar{X}_u^t
=
S_{ \tau, t } \, X_{ \tau }
+
\int_{ \tau }^u
S_{ s, t } \, Y_s \, ds
+
\int_{ \tau }^u
S_{ s, t } \, Z_s \, dW_s
\end{equation}
$ \mathbb{P} $-a.s.~for
all $ u \in [\tau, t] $
and all
$
t \in
\mathbbm{I} \cap
( \tau, \infty )
$
({\it Transformation};
see also
Section~6 in \cite{cd08},
Section~7 in \cite{c08},
Section~5 in \cite{ls10} and
Section~3 in \cite{kll11}).
Note that
the assumptions
$
\mathbb{P}\big[
\int_{ \tau }^t
\left\| S_{ s, t } Y_s \right\|_{
\check{H}
}
ds < \infty
\big] = 1
$
and
$
\mathbb{P}\big[
\int_{ \tau }^t
\left\| S_{ s, t } Z_s \right\|_{
HS( U_0, \check{H} )
}^2
ds < \infty
\big] = 1
$
for all
$
t \in
\mathbbm{I}
$
(see Definition~\ref{propdef})
ensure that
$
\bar{X}^t \colon
[\tau,t] \times \Omega
\rightarrow \check{H}
$,
$
t \in \mathbbm{I} \cap
( \tau, \infty )
$,
in \eqref{eq:semiX}
are indeed
well defined
adapted
stochastic processes
with continuous sample paths.
In the next step
the
continuity
of the partial derivatives
of
$
\varphi \colon
\mathbbm{I} \times
\check{H}
\rightarrow V
$,
the continuity
of the sample paths
of
$
\bar{X}^t
\colon [\tau,t]
\times \Omega
\rightarrow \check{H}
$
and again
the assumptions
$
\mathbb{P}\big[
\int_{ \tau }^t
\left\| S_{ s, t } Y_s \right\|_{
\check{H}
}
ds < \infty
\big] = 1
$
and
$
\mathbb{P}\big[
\int_{ \tau }^t
\left\| S_{ s, t } Z_s \right\|_{
HS( U_0, \check{H} )
}^2
ds < \infty
\big] = 1
$
in Definition~\ref{propdef}
imply
\begin{equation}
\label{eq:well1c}
\mathbb{P}\!\left[
\int_{ t_0 }^t
\left\|
(\partial_2 \varphi)( s, \bar{X}^t_s )
S_{ s, t } Y_s
\right\|_V
+
\left\|
(\partial_2 \varphi)( s, \bar{X}^t_s )
S_{ s, t } Z_s
\right\|_{ HS(U_0, V ) }^2
ds
< \infty
\right] = 1
\end{equation}
and
\begin{equation}
\label{eq:well2c}
\mathbb{P}\!\left[
\int_{ t_0 }^t
\left\|
(\partial_1 \varphi)( s, \bar{X}^t_s )
\right\|_V
+
\left\|
(\partial_2^2 \varphi)( s, \bar{X}^t_s )
\right\|_{ L^{(2)}( \check{H}, V ) }
\left\|
S_{ s, t } Z_s
\right\|_{ HS(U_0, \check{H} ) }^2
ds
< \infty
\right] = 1
\end{equation}
for all
$
t_0 \in [ \tau, t ]
$,
$
t \in
\mathbbm{I} \cap
( \tau, \infty )
$
and all
$
\varphi \in C^{1,2}(
\mathbbm{I} \times \check{H}, V
)
$.
Moreover,
the identity
$
X_t =
\bar{X}_t^t
$
$ \mathbb{P} $-a.s.\ and
the standard
It{\^o} formula
(see
Theorem~2.4 in Brze\'{z}niak,
Van Neerven, Veraar
\citationand\
Weis~\cite{bvvw08})
give
\begin{equation}
\label{eq:itoformel2inProof}
\begin{split}
&
\varphi( t, X_{ t } )
=
\varphi( t, \bar{X}_t^{ t } )
=
\varphi( t_0, \bar{X}_{ t_0 }^{ t } )
+
\int_{t_0}^t
(\partial_1 \varphi)( s, \bar{X}_{ s }^{ t } )
\, ds
+
\int_{t_0}^t
(\partial_2 \varphi)( s, \bar{X}_{ s }^{ t } ) \,
S_{ s, t } \, Y_s \, ds
\\&\quad+
\int_{t_0}^t
(\partial_2
\varphi)( s, \bar{X}_{ s }^{ t } ) \,
S_{ s, t } \, Z_s \, dW_s
+
\frac{1}{2}
\sum_{ j \in \mathcal{J} }
\int_{t_0}^t
(\partial_2^2 \varphi)( s,
\bar{X}_{ s }^{ t }
)
\left(
S_{ s, t } \, Z_s \, g_j,
S_{ s, t } \, Z_s \, g_j
\right) ds
\end{split}
\end{equation}
$ \mathbb{P} $-a.s.\ for all
$
t_0, t \in \mathbbm{I}
$
with $ t_0 < t $
and all
$
\varphi \in C^{1,2}(
\mathbbm{I} \times \check{H}, V
)
$
({\it Application
of the
standard It\^{o} formula};
see also Section~6
in \cite{cd08},
equations~(1.7) and (7.6)
in \cite{c08},
Theorem~4 in \cite{ls10}
and Theorem~3.1 in
\cite{kll11}).
Equation~\eqref{eq:itoformel2inProof}
is an expansion formula
for the stochastic processes
$ \varphi( t, X_t ) $,
$
t \in \mathbbm{I} \cap
( \tau, \infty )
$,
for
$
\varphi \in C^{1,2}(
\mathbbm{I} \times \check{H}, V
)
$.
Nevertheless, this formula
seems to be of limited use
since the integrands
in~\eqref{eq:itoformel2inProof} contain
the transformed stochastic processes
$ \bar{X}_s^t $,
$ s \in [t_0,t] $,
$ t_0, t \in \mathbbm{I} $,
$ t_0 < t $,
instead of
the mild It\^{o} process
$ X_s $,
the mild drift
$ Y_s $
and the mild diffusion
$ Z_s $
for $ s \in [t_0,t] $,
$ t_0, t \in \mathbbm{I} $,
$ t_0 < t $,
only.
However, a key
observation
here is to
exploit the elementary
identity
\begin{equation}
\label{eq:fact}
\begin{split}
\bar{X}^{ t }_s
&=
S_{ \tau, t } \, X_{ \tau }
+
\int_{ \tau }^s
S_{ u, t } \,
Y_u \, du
+
\int_{ \tau }^s
S_{ u, t } \,
Z_u \, dW_u
\\&=
S_{ s, t }
\left(
S_{ \tau, s } \, X_{ \tau }
+
\int_{ \tau }^s
S_{ u, s } \,
Y_u \, du
+
\int_{ \tau }^s
S_{ u, s } \,
Z_u \, dW_u
\right)
=
S_{ s, t } \,
X_s
\end{split}
\end{equation}
$ \mathbb{P} $-a.s.\ for
all $ s, t \in \mathbbm{I} $
with $ s < t $
in
equation~\eqref{eq:itoformel2inProof}
({\it Relation} of the transformed
standard It\^{o} processes
$
\bar{X}^t
\colon [\tau,t] \times \Omega
\rightarrow \mathcal{H}
$,
$
t \in \mathbbm{I}
\cap ( \tau, \infty )
$,
and the original mild It\^{o}
process
$
X \colon \mathbbm{I} \times
\Omega \rightarrow \tilde{H}
$).
This enables us to obtain a
closed formula for the stochastic
processes
$ \varphi( t, X_t ) $,
$
t \in \mathbbm{I} \cap
( \tau, \infty )
$,
for
$
\varphi \in C^{1,2}(
\mathbbm{I} \times \check{H}, V
)
$.
More precisely,
\eqref{eq:fact},
\eqref{eq:well1c},
\eqref{eq:well2c}
and Lemma~\ref{lem:easy}
imply
\eqref{eq:well1} and \eqref{eq:well2}.
Putting~\eqref{eq:fact} into~\eqref{eq:itoformel2inProof}
then gives
\eqref{eq:itoformel_start}.
The proof of Theorem~\ref{thm:ito}
is thus completed.
\end{proof}
Let us close this section on mild
stochastic calculus with a remark
on possible generalizations.
\begin{remark}
\label{rem:moregeneral}
Note that here
mild It\^{o} processes,
mild drifts and mild diffusions
with values in
separable Hilbert spaces
are considered. Instead one could
develop the above notions and
the above mild It\^{o} formula for
stochastic processes with values
in an appropriate class
of possibly non-separable Banach spaces too.
Indeed, the standard It\^{o} formula
also holds for stochastic processes
with values in UMD Banach
spaces (see Theorem~2.4 in
Brze\'{z}niak, Van Neerven, Veraar
\citationand\
Weis~\cite{bvvw08}).
Details on the stochastic
integration
in UMD Banach spaces
can be found in
Van Neerven,
Veraar \citationand\
Weis~\cite{vvw07,vvw08}
and in the references therein.
Another possible
generalization
is to consider more general integrators
than the cylindrical Wiener process
$ ( W_t )_{ t \in \mathbbm{I} } $.
This leads to the concept
of a {\it mild semimartingale} instead
of a mild It\^{o} process
in Definition~\ref{propdef}.
In particular, the fourth
integral in the
mild It\^{o} formula~\eqref{eq:itoformel_start}
then needs to be replaced by
an integral involving the
quadratic
variation of the
driving noise process.
\end{remark}
\section{Stochastic
partial differential
equations (SPDEs)}
\label{sec:appl}
\subsection{Setting and
assumptions}
\label{sec:setting}
Throughout this section
suppose that the
following setting and the following assumptions are fulfilled.
Let $ T \in (0,\infty) $
be a real number,
let
$ \left( \Omega, \mathcal{F}, \mathbb{P} \right) $
be a probability space
with a normal filtration
$ ( \mathcal{F}_t )_{ t \in [0,\infty) } $
and let
$
\left(
H,
\left< \cdot , \cdot \right>_H,
\left\| \cdot \right\|_H
\right)
$,
$
\left(
U,
\left< \cdot , \cdot \right>_U,
\left\| \cdot \right\|_U
\right)
$
and
$
\left(
V,
\left< \cdot , \cdot \right>_V,
\left\| \cdot \right\|_V
\right)
$
be separable
$ \mathbb{R} $-Hilbert spaces.
In addition, let
$ Q \colon U \rightarrow U $
be
a bounded nonnegative symmetric linear
operator and
let $ ( W_t )_{ t \in [0,\infty) } $
be a cylindrical $ Q $-Wiener
process with respect
to $ ( \mathcal{F}_t )_{ t \in [0,\infty) } $.
Moreover, by
$
\left(
U_0,
\left< \cdot , \cdot \right>_{ U_0 },
\left\| \cdot \right\|_{ U_0 }
\right)
$
the separable
$\mathbb{R}$-Hilbert space
with
$ U_0 = Q^{ 1/2 }( U ) $
and
$
\| u \|_{ U_0 }
= \| Q^{ - 1/2 }( u ) \|_U
$
for all $ u \in U_0 $
is denoted.
\begin{assumption}[Linear operator A]\label{semigroup}
Let $ A \colon D(A) \subset H
\rightarrow H $ be a generator
of a strongly continuous analytic
semigroup
$ e^{ A t } \in L(H) $,
$ t \in [0,\infty) $.
\end{assumption}
Let $ \eta \in [0,\infty) $ be a nonnegative real number such that
$
\sigma(A)
\subset
\{
\lambda \in \mathbb{C}
\colon
\text{Re}( \lambda ) < \eta
\}
$ where
$ \sigma(A) \subset \mathbb{C}
$ denotes
as usual the spectrum
of the linear
operator
$ A \colon D(A) \subset
H \rightarrow H $.
Such a real number exists
since $ A $ is assumed to be a generator
of a strongly continuous semigroup
(see Assumption~\ref{semigroup}).
By
$
H_r :=
D\!\left(
\left( \eta - A \right)^r
\right)
$
equipped with
the norm
$
\left\| v \right\|_{ H_r }
:=
\left\|
\left( \eta - A \right)^r \! v
\right\|_H
$
for all $ v \in H_r $
and all $ r \in \mathbb{R} $,
the $ \mathbb{R} $-Hilbert spaces
of domains of fractional powers
of the linear operator
$ \eta - A \colon D(A) \subset H
\rightarrow H $ are denoted
(see, e.g., Subsection~11.4.2
in Renardy \citationand\
Roggers~\cite{rr93}).
\begin{assumption}[Drift term $F$]\label{drift}
Let $ \alpha, \gamma \in \mathbb{R} $
be real numbers with
$ \gamma - \alpha < 1 $
and let
$ F \colon H_{ \gamma }
\rightarrow H_{ \alpha } $
be globally
Lipschitz continuous.
\end{assumption}
\begin{assumption}[Diffusion term $B$]\label{diffusion}
Let $ \beta \in \mathbb{R} $ be a
real number with
$ \gamma - \beta < \frac{1}{2} $
and
let $ B \colon H_{ \gamma }
\rightarrow HS(U_0,H_{ \beta }) $
be globally Lipschitz continuous.
\end{assumption}
\begin{assumption}[Initial value $\xi$]\label{initial}
Let
$ p \in [2,\infty) $
be a real number and
let
$
\xi \colon \Omega
\rightarrow H_{ \gamma }
$
be an
$ \mathcal{F}_{0}
$/$ \mathcal{B}\left(
H_{ \gamma } \right)
$-measurable
mapping
with
$
\mathbb{E}\big[
\| \xi \|^p_{ H_{ \gamma } }
\big] < \infty
$.
\end{assumption}
Furthermore,
similar as
in Section~\ref{sec:mildcalc},
let
$ \angle \subset [0,T]^2 $
be defined
through
$
\angle :=
\left\{
(t_1, t_2) \in [0,T]^2 \colon
t_1 < t_2
\right\}
$.
In addition to the above assumptions,
the following notations will be used
in the remainder of this
article.
For two
$ \mathbb{R} $-Banach spaces
$ ( V_1, \left\| \cdot \right\|_{ V_1 } ) $
and
$
( V_2, \left\| \cdot \right\|_{ V_2 } )
$
and real numbers
$
n \in \{ 0, 1, 2, \ldots \}
$
and
$ q \in [0,\infty) $
define
$
\left\| v
\right\|_{ L^{ (0) }( V_1, V_2 ) }
:=
\left\| v
\right\|_{ V_2 }
$
for every
$ v \in V_1 $,
define
\begin{equation}
\| \varphi \|_{
G^n_q( V_1, V_2 )
}
:=
\sum_{ i = 0 }^{ n - 1 }
\| \varphi^{(i)}(0) \|_{
L^{(i)}( V_1, V_2 )
}
+
\sup_{ v \in V_1 }
\left(
\frac{
\| \varphi^{(n)}(v) \|_{
L^{(n)}( V_1, V_2 )
}
}{
\left(
1 + \| v \|_{ V_1 }
\right)^q
}
\right)
\in [0,\infty] ,
\end{equation}
\begin{equation}
\| \varphi \|_{
\text{Lip}^{ n + 1 }_q( V_1, V_2 )
}
:=
\sum_{ i = 0 }^{ n }
\| \varphi^{(i)}(0) \|_{
L^{(i)}( V_1, V_2 )
}
+
\sup_{
\substack{
v, w \in V_1
\\
v \neq w
}
}
\left(
\frac{
\|
\varphi^{(n)}(v) -
\varphi^{(n)}(w)
\|_{
L^{(n)}( V_1, V_2 )
}
}{
\left(
1 +
\max(
\| v \|_{ V_1 },
\| w \|_{ V_1 }
)
\right)^q
\left\|
v - w
\right\|_{ V_1 }
}
\right)
\in [0,\infty]
\end{equation}
and
\begin{equation}
\label{eq:defCnLip}
\| \varphi \|_{
C^n_{ Lip }( V_1, V_2 )
}
:=
\| \varphi(0) \|_{ V_2 }
+
\sum_{ i = 1 }^{ n }
\| \varphi^{(i)} \|_{
L^{ \infty }( V_1, L^{(i)}(V_1,V_2) )
}
+
\sup_{
\substack{
v, w \in V_1
\\
v \neq w
}
}
\left(
\frac{
\|
\varphi^{ (n) }(v) -
\varphi^{ (n) }(w)
\|_{
L^{(n)}( V_1, V_2 )
}
}{
\left\|
v - w
\right\|_{ V_1 }
}
\right)
\in [0,\infty]
\end{equation}
for every
$
\varphi \in C^n( V_1, V_2 )
$,
define
$
G^n_q( V_1, V_2 )
:=
\{
\varphi \in C^n( V_1, V_2 ) \colon
\| \varphi \|_{
G^n_{q}( V_1, V_2 )
}
< \infty
\}
$,
define
$
\text{Lip}^{ n + 1 }_q( V_1, V_2 )
$
$
:=
\{
\varphi \in C^n( V_1, V_2 ) \colon
\| \varphi \|_{
\text{Lip}^{ n + 1 }_{ q }( V_1, V_2 )
}
< \infty
\}
$
and define
$
C_{ Lip }^n( V_1, V_2 )
:=
\{
\varphi \in C^n( V_1, V_2 ) \colon
\| \varphi \|_{ C^n_{ Lip }( V_1, V_2 ) }
< \infty
\}
$
and note that
$
\left\| \varphi \right\|_{
G^{ m }_q( V_1, V_2 )
}
=
\left\| \varphi \right\|_{
\text{Lip}^{ m }_q( V_1, V_2 )
}
$
for every
$
\varphi \in C^{ m }( V_1, V_2 )
$
and every
$ m \in \N $.
Let us collect a few simple properties
of the defined objects.
More precisely, observe that
\begin{align}
\label{eq:Gnq1}
\| \varphi \|_{
G^n_q( V_1, V_2 )
}
& =
\sum\nolimits_{ i = 0 }^{ n - 1 }
\| \varphi^{(i)}(0) \|_{
L^{(i)}( V_1, V_2 )
}
+
\| \varphi^{(n)} \|_{
G^0_{ q }( V_1, V_2 )
}
\\
\label{eq:Gnq2}
\|
\varphi
\|_{
G^{ 0 }_{ q + n }( V_1, V_2 )
}
& \leq
\|
\varphi
\|_{
G^{ n - k }_{ q + k }( V_1, V_2 )
}
\leq
\|
\varphi
\|_{
G^n_{ q }( V_1, V_2 )
} ,
\\
\label{eq:growthestimateGnq}
\|
\varphi^{ ( k ) }(v)
\|_{
L^{ ( k ) }( V_1, V_2 )
}
& \leq
\|
\varphi
\|_{
G^n_q( V_1, V_2 )
}
\left( 1 + \| v \|_{ V_1 } \right)^{
( q + n - k )
}
\\
\label{eq:lipestimateGnq}
\|
\varphi^{ (k) }(v)
-
\varphi^{ (k) }(w)
\|_{
L^{ (k) }( V_1, V_2 )
}
& \leq
\|
\varphi
\|_{
\mathrm{Lip}^{ n + 1 }_{ q }( V_1, V_2 )
}
\,
\big(
1 + \max( \| v \|_{ V_1 }, \| w \|_{ V_1 }
)
\big)^{
( q + n - k )
}
\,
\| v - w \|_{ V_1 }
\end{align}
for all
$ v, w \in V_1 $,
$ \varphi \in C^n( V_1, V_2 ) $,
$
k \in \{ 0, 1, \dots, n \}
$,
$ n \in \{ 0, 1, \dots \} $,
$ q \in [0,\infty) $
and all
$ \mathbb{R} $-Banach spaces
$ ( V_1, \left\| \cdot \right\|_{ V_1 } ) $
and
$
( V_2, \left\| \cdot \right\|_{ V_2 } )
$.
Moreover,
note
for all $ n \in \{ 0, 1, 2, \dots \} $,
$ q \in [0,\infty) $
and all
$ \mathbb{R} $-Banach spaces
$ ( V_1, \left\| \cdot \right\|_{ V_1 } ) $
and
$
( V_2, \left\| \cdot \right\|_{ V_2 } )
$
that the pairs
$
(
G^n_q(V_1, V_2) ,
\left\| \cdot
\right\|_{ G^n_q( V_1, V_2 ) }
)
$,
$
(
\text{Lip}^{n+1}_q(V_1, V_2) ,
\left\| \cdot
\right\|_{
\text{Lip}^{n+1}_q( V_1, V_2 )
}
)
$
and
$
(
C^n_{ Lip }(V_1, V_2) ,
$
$
\left\| \cdot
\right\|_{
C^{ n }_{ Lip }( V_1, V_2 )
}
)
$
are
$ \mathbb{R} $-Banach
spaces
with
$
G^{ n + 1 }_q(V_1, V_2)
\subset
\text{Lip}^{n+1}_q(V_1, V_2)
\subset
G^{ n }_{ q + 1 }(V_1, V_2)
$
continuously.
More function spaces
of similar type can be found
in D\"{o}rsek \citationand\
Teichmann~\cite{dt10}.
\subsection{Solution processes
of SPDEs}
\label{sec:solutionSPDE}
The following proposition
shows that
the setting in
Section~\ref{sec:setting}
ensures that the
SPDE~\eqref{eq:SPDE} below
admits an up to modifications
unique mild solution process.
It is similar to special cases of
Theorem~4.3
in Brze\'{z}niak~\cite{b97b}
and
Theorem~6.2 in
Van Neerven, Veraar
\citationand\
Weis~\cite{vvw08}.
Its proof is clear and therefore
omitted.
\begin{prop}
\label{prop:prop}
Assume that the setting
in Section~\ref{sec:setting}
is fulfilled.
Then there exists an
up to modifications unique
predictable stochastic
process
$
X \colon [0,T] \times
\Omega
\rightarrow H_{ \gamma }
$
which fulfills
$
\sup_{ t \in [0,T] }
\mathbb{E}\big[
\| X_t \|_{ H_{ \gamma } }^p
\big]
< \infty
$
and
\begin{equation}
\label{eq:SPDE}
X_t = e^{ A t } \,
\xi
+
\int_{0}^t
e^{ A (t - s) }
F( X_s ) \,
ds
+
\int_{0}^t
e^{ A (t - s) }
B( X_s ) \,
dW_s
\end{equation}
$\mathbb{P}$-a.s.\ for
all $ t \in [0,T] $.
In addition, we have
$
X
\in
\cap_{ r \in (-\infty, \gamma] }
\,
C^{ \min( \gamma - r, 1/2 ) }\!\left(
[0,T] ,
L^p( \Omega; H_{ r } )
\right)
$.
\end{prop}
Proposition~\ref{prop:prop},
in particular, ensures that
the mild solution process
$ X \colon [0,T] \times \Omega
\rightarrow H_{ \gamma } $
of the SPDE~\eqref{eq:SPDE}
is a mild It\^{o} process
with semigroup
$
e^{ A (t_2 - t_1) }
\in
L(
H_{ \min( \alpha, \beta. \gamma ) } ,
H_{ \gamma }
)
$,
$ (t_1, t_2) \in \angle $,
with mild drift
\begin{equation}
\label{eq:milddrift}
F( X_t ) , \;
t \in [0,T] ,
\end{equation}
and with mild diffusion
\begin{equation}
\label{eq:milddiffusion}
B( X_t ) , \; t \in [0,T] .
\end{equation}
This fact now enables us to apply
the mild It\^{o} formula~\eqref{eq:itoformel_start}
to the solution process $ X $
of the
SPDE~\eqref{eq:SPDE}.
To this end let $ \mathcal{J} $ be a set
and let
$ g_j \in U_0 $,
$ j \in \mathcal{J} $,
be an arbitrary orthonormal basis
of the $ \mathbb{R} $-Hilbert
space
$
(
U_0,
\left< \cdot, \cdot
\right>_{ U_0 },
\left\| \cdot
\right\|_{ U_0 }
)
$.
A direct consequence
of Theorem~\ref{thm:ito}
and Corollary~\ref{cor:itoauto}
is the next corollary.
\begin{cor}[A new -
somehow mild -
It{\^o} formula for solutions
of SPDEs]
\label{cor:ito}
Assume that
the setting in Section~\ref{sec:setting}
is fulfilled.
Then
\begin{equation}
\label{eq:well1d}
\mathbb{P}\!\left[
\int_{ t_0 }^t
\big\|
\varphi'( e^{ A(t-s) } X_s ) \,
e^{ A(t-s) } F( X_s )
\big\|_V \,
ds
< \infty
\right] = 1,
\end{equation}
\begin{equation}
\label{eq:well2d}
\mathbb{P}\!\left[
\int_{ t_0 }^t
\big\|
\varphi'( e^{ A(t-s) } X_s ) \,
e^{ A(t-s) } B( X_s )
\big\|_{ HS(U_0, V ) }^2
\,
ds
< \infty
\right] = 1,
\end{equation}
\begin{equation}
\label{eq:well3d}
\mathbb{P}\!\left[
\int_{ t_0 }^t
\big\|
\varphi''( e^{ A( t- s) } X_s )
\big\|_{ L^{(2)}( H_{ r }, V ) }
\,
\big\|
e^{ A (t - s) } B( X_s )
\big\|_{ HS(U_0, H_{ r }) }^2
\,
ds
< \infty
\right] = 1
\end{equation}
and
\begin{align}
\label{eq:itoformel}
\varphi( X_t )
&=
\varphi( e^{ A( t - t_0 ) } X_{ t_0 } )
+
\int_{ t_0 }^t
\varphi'( e^{ A( t - s ) } X_s ) \,
e^{ A( t - s ) }
F( X_s ) \, ds
\nonumber
\\&\quad+
\int_{ t_0 }^t
\varphi'( e^{ A( t - s ) } X_s ) \,
e^{ A( t - s ) }
B( X_s ) \, dW_s
\\&\quad+
\nonumber
\frac{1}{2}
\sum_{ j \in \mathcal{J} }
\int_{ t_0 }^t
\varphi''( e^{ A( t - s ) } X_s )
\left(
e^{ A( t - s ) }
B( X_s ) g_j,
e^{ A( t - s ) }
B( X_s ) g_j
\right) ds
\end{align}
$ \mathbb{P} $-a.s.\ for
all $ t_0, t \in [0,T] $
with $ t_0 < t $,
all
$
\varphi \in
C^2( H_{ r }, V )
$
and all
$
r \in
( - \infty,
\min( \alpha + 1, \beta + \frac{ 1 }{ 2 } )
)
$.
\end{cor}
First, observe that the
possibly infinite sum
and all integrals in
\eqref{eq:itoformel} are
well defined due to
\eqref{eq:well1d}--\eqref{eq:well3d}.
Next define mappings
$
K_0
\colon
\cup_{ r \in \mathbb{R} }
C( H_{ r }, V )
\rightarrow
\cup_{ r \in \mathbb{R} }
C( H_{ r }, V )
$
and
$
K_t
\colon
\cup_{ r \in \mathbb{R} }
C( H_{ r }, V )
\rightarrow
C( H_{ \min(\alpha,\beta,\gamma) }, V )
$,
$ t \in (0,\infty) $,
through
$ K_0( \varphi ) := \varphi $
and
\begin{equation}
\label{eq:defK}
\big( K_t \varphi \big)(x)
:=
\varphi( e^{ A t } x )
\end{equation}
for all
$ x \in H_{ \min( \alpha, \beta, \gamma) } $,
$
\varphi \in
\cup_{ r \in \mathbb{R} }
C( H_{ r }, V )
$
and all
$ t \in (0,\infty) $.
Note that
$
K_{ t_1 } \circ K_{ t_2 } =
K_{ t_1 + t_2 }
$
for all $ t_1, t_2 \in [0,\infty) $.
In addition, define
linear operators
$
L^{(0)} \colon
C^2(
H_{ \min( \alpha, \beta, \gamma ) },
V
)
\rightarrow
C(
H_{ \gamma },
V
)
$
and
$
L^{(1)} \colon
C^1(
H_{ \min( \beta, \gamma ) },
V
)
\rightarrow
C(
H_{ \gamma },
V
)
$
through
\begin{equation}
\label{eq:defL0}
\begin{split}
( L^{ (0) } \varphi )(x)
& :=
\varphi'(x) F(x)
+
\frac{1}{2}
\sum_{ j \in \mathcal{J} }
\varphi''(x) \big(
B(x) g_j, B(x) g_j
\big)
\\ & =
\varphi'(x) F(x)
+
\frac{1}{2}
\text{Tr}\Big(
\big( B(x) \big)^{ * }
\varphi''(x) \,
B(x)
\Big)
\end{split}
\end{equation}
for all
$
x \in H_{ \gamma }
$,
$
\varphi \in
C^{2}(
H_{ \min( \alpha, \beta, \gamma ) },
V
)
$
and through
$
( L^{ (1) } \varphi
)(x)
:=
\varphi'(x) B(x)
$
for all
$
x \in H_{ \gamma }
$,
$
\varphi \in
C^{1}(
H_{ \min( \beta, \gamma ) },
V
)
$.
Furthermore,
define mappings
$
L_t^{(0)} \colon
\cup_{ r \in \mathbb{R} }
C^{ 2 }( H_{ r }, V )
\rightarrow
C( H_{ \gamma }, V )
$,
$ t \in (0,\infty) $,
and
$
L_t^{(1)} \colon
\cup_{ r \in \mathbb{R} }
C^{ 1 }( H_{ r }, V )
\rightarrow
C( H_{ \gamma }, HS( U_0, V) )
$,
$ t \in (0,\infty) $,
through
$
L_{ t }^{(0)}( \varphi )
:=
L^{ (0) }( K_t( \varphi ) )
$
for all
$
\varphi \in
\cup_{ r \in \mathbb{R} }
C^2( H_r , V )
$
and through
$
L_{ t }^{(1)}( \varphi )
:=
L^{ (1) }( K_t( \varphi ) )
$
for all
$
\varphi \in
\cup_{ r \in \mathbb{R} }
C^1( H_r , V )
$.
Note that these definitions imply
\begin{equation}
\label{eq:defL0t}
\begin{split}
\big(
L_{ t }^{(0)} \varphi
\big)( x )
& =
\varphi'( e^{ A t } x )
\, e^{ A t } F(x)
+
\frac{ 1 }{ 2 }
\sum_{ j \in \mathcal{J} }
\varphi''( e^{ A t } x )\big(
e^{ A t } B(x) g_j,
e^{ A t } B(x) g_j
\big)
\\ & =
\varphi'( e^{ A t } x )
\, e^{ A t } F(x)
+
\frac{ 1 }{ 2 }
\text{Tr}\Big(
\big( e^{ A t } B(x) \big)^{ \! * }
\varphi''( e^{ A t } x ) \,
e^{ A t } B(x)
\Big)
\end{split}
\end{equation}
for all
$ x \in H_{ \gamma } $,
$
\varphi \in
\cup_{ r \in \mathbb{R} }
C^2( H_{ r }, V )
$,
$ t \in (0,\infty) $
and
\begin{equation}
\label{eq:defL1}
\big(
L_{ t }^{(1)} \varphi
\big)( x )
=
\varphi'( e^{ A t } x )
\, e^{ A t } B(x)
\end{equation}
for all
$ x \in H_{ \gamma } $,
$
\varphi \in
\cup_{ r \in \mathbb{R} }
C^1( H_{ r }, V )
$,
$ t \in (0,\infty) $.
The mild It\^{o}
formula~\eqref{eq:itoformel} can
thus be written as
\begin{equation}
\label{eq:operatorIto}
\begin{split}
\varphi( X_t )
& =
\varphi( e^{ A (t - t_0) } X_{ t_0 } )
+
\int_{ t_0 }^t
\!
\big(
L_{ ( t - s ) }^{(0)} \varphi
\big) ( X_s )
\, ds
+
\int_{ t_0 }^t
\!
\big(
L_{ ( t - s ) }^{(1)} \varphi
\big) ( X_s )
\, dW_s
\\ & =
\big(
K_{ ( t - t_0 ) } \varphi
\big)( X_{ t_0 } )
+
\int_{ t_0 }^t
\!
\big(
L_{ ( t - s ) }^{(0)} \varphi
\big) ( X_s )
\, ds
+
\int_{ t_0 }^t
\!
\big(
L_{ ( t - s ) }^{(1)} \varphi
\big) ( X_s )
\, dW_s
\\ & =
\big(
K_{ ( t - t_0 ) } \varphi
\big)( X_{ t_0 } )
+
\int_{ t_0 }^t
\!
\big(
L^{(0)} K_{(t-s)} \varphi
\big) ( X_s )
\, ds
+
\int_{ t_0 }^t
\!
\big(
L^{(1)} K_{(t-s)} \varphi
\big) ( X_s )
\, dW_s
\end{split}
\end{equation}
$ \mathbb{P} $-a.s.\ for all
$ t_0, t \in [0,T] $ with $ t_0 < t $
and all
$
\varphi \in
\cup_{
r <
\min( \alpha + 1, \beta + \frac{ 1 }{ 2 } )
}
C^2( H_r, V )
$.
Moreover, taking
expectations on both
side of \eqref{eq:operatorIto}
gives
\begin{equation}
\label{eq:operatorIto2}
\begin{split}
\mathbb{E}\Big[
\varphi( X_t )
\Big]
& =
\mathbb{E}\Big[
\big(
K_{ (t - t_0) }
\varphi
\big)( X_{ t_0 } )
\Big]
+
\int_{ t_0 }^t
\mathbb{E}\Big[
\big(
L_{ ( t - s ) }^{(0)} \varphi
\big) ( X_s )
\Big]
\, ds
\\ & =
\mathbb{E}\Big[
\big(
K_{ ( t - t_0 ) } \varphi
\big)( X_{ t_0 } ) \,
\Big]
+
\int_{ t_0 }^t
\mathbb{E}\Big[
\big(
L^{(0)} K_{(t-s)} \varphi
\big) ( X_s ) \,
\Big]
\, ds
\end{split}
\end{equation}
for all $ t_0, t \in [0,T] $
with $ t_0 < t $
and all
$
\varphi \in
\cup_{
r <
\min( \alpha + 1, \beta + \frac{ 1 }{ 2 } )
}
G^2_0( H_r, V )
$.
Based on \eqref{eq:operatorIto2}
a mild Kolmogorov
backward equation is derived
in Subsection~\ref{sec:kolmogorov} below.
Other kinds of It{\^o} type formulas
for solutions of SPDEs can
be found
in \cite{c07,dz92,gnt05,gk81,gk8182,kr79,l07,l09b,lt08,op89,p72,p87,PrevotRoeckner2007,rrw07,z06}.
In the next step
Corollary~\ref{cor:ito}
is illustrated by two simple
examples.
\begin{example}[Identity]
Assume that the setting
in Section~\ref{sec:setting}
is fulfilled,
let
$ V = H_{ \gamma } $,
let
$ \left\| v \right\|_V
= \left\| v \right\|_{ H_\gamma } $
for all
$ v \in H_{ \gamma } $
and
let $ \varphi \colon H_{\gamma}
\rightarrow H_{\gamma} $
be the identity on
$ H_{\gamma} $,
i.e., $ \varphi(v) = v $
for all $ v \in H_{\gamma} $.
The mild It\^{o} formula~\eqref{eq:itoformel}
in Corollary~\ref{cor:ito}
then reduces to
\begin{equation}
X_t
=
e^{ A( t - t_0 ) } X_{ t_0 }
+
\int_{ t_0 }^t
e^{ A( t - s ) }
F( X_s ) \,
ds
+
\int_{ t_0 }^t
e^{ A( t - s ) } B( X_s ) \,
dW_s
\end{equation}
$ \mathbb{P} $-a.s.\ for
all $ t_0, t \in [0,T] $
with $ t_0 \leq t $.
This is nothing else but
the mild
formulation of
the SPDE~\eqref{eq:SPDE}.
In this sense, the
formula~\eqref{eq:itoformel}
is somehow a mild
It{\^o} formula for SPDEs.
\end{example}
\begin{example}[Squared norm]
Assume that the setting
in Section~\ref{sec:setting}
is fulfilled,
let $ V = \mathbb{R} $,
$
\left\| v \right\|_V =
\left| v \right|
$
for all
$ v \in V = \mathbb{R} $,
assume
$
\min( \alpha + 1, \beta + \frac{ 1 }{ 2 } )
> 0
$
and
let
$
\varphi \colon H
\rightarrow V
$
be given
by
$
\varphi( v ) = \left\| v \right\|_H^2
$
for all
$ v \in H
$.
The mild It\^{o} formula~\eqref{eq:itoformel}
in Corollary~\ref{cor:ito}
then reduces to
\begin{equation}
\label{eq:squarenorm}
\begin{split}
\left\| X_t \right\|^2_H
&=
\big\|
e^{ A( t - t_0 ) } X_{ t_0 }
\big\|_H^2
+
2 \int_{ t_0 }^t
\left<
e^{ A( t - s ) } X_s,
e^{ A( t - s ) } F( X_s )
\right>_{ \! H } ds
\\&\quad+
2 \int_{ t_0 }^t
\left<
e^{ A( t - s ) } X_s,
e^{ A( t - s ) } B( X_s ) \, dW_s
\right>_{ \! H }
+
\int_{ t_0 }^t
\big\|
e^{ A( t - s ) } B( X_s )
\big\|_{ HS( U_0, H ) }^2
\, ds
\end{split}
\end{equation}
$ \mathbb{P} $-a.s.\ for
all $ t_0, t \in [0,T] $
with $ t_0 < t $
(see also Example~\ref{ex:1}
above).
We refer to
\cite{gk81,gk8182,kr79,op89,p72,PrevotRoeckner2007,rrw07}
for other It{\^o} type
formulas
with the particular test
function
$
\varphi(v) =
\left\| v \right\|_H^2
$,
$ v \in H $.
If
$ X_0 = 0 $
and
$ F(v) = 0 $
for all $ v \in H $
in addition to
the above assumptions,
then \eqref{eq:squarenorm}
simplifies to
\begin{equation}
\label{eq:squarenorm2}
\left\| X_t \right\|^2_H
=
2 \int_{ 0 }^t
\left<
e^{ A( t - s ) } X_s,
e^{ A( t - s ) } B( X_s ) \, dW_s
\right>_{ \! H }
+
\int_{ 0 }^t
\big\|
e^{ A( t - s ) } B( X_s )
\big\|_{ HS( U_0, H ) }^2
\, ds
\end{equation}
$ \mathbb{P} $-a.s.\ for
all $ t \in [0,T] $ and this,
in particular, gives
\begin{equation}
\label{eq:squarenorm3}
\mathbb{E}\!\left[
\left\| X_t \right\|^2_H
\right]
=
\int_{ 0 }^t
\mathbb{E}\!\left[
\big\|
e^{ A( t - s ) } B( X_s )
\big\|_{ HS( U_0, H ) }^2
\right]
ds
\end{equation}
for all $ t \in [0,T] $.
Clearly,
equation~\eqref{eq:squarenorm3}
is nothing else but
a special case of It\^{o}'s
isometry.
\end{example}
\subsubsection{SPDEs
with time dependent
coefficients}
In addition to the setting
in Section~\ref{sec:setting}
assume in this subsection
that
$
\tilde{F} \colon
[0,\infty) \times
H_{ \gamma }
\rightarrow
H_{ \alpha }
$
and
$
\tilde{B} \colon
[0,\infty) \times H_{ \gamma }
\rightarrow
HS( U_0, H_{ \beta } )
$
are two globally Lipschitz
continuous mappings.
Then there exists an up to
modifications unique predictable
stochastic process
$
\tilde{X} \colon [0,\infty) \times
\Omega \rightarrow H_{ \gamma }
\in
C( [0,\infty),
L^p( \Omega; H_{ \gamma } )
)
$
which fulfills
\begin{equation}
\tilde{X}_t
=
e^{ A t } \xi
+
\int_0^t
e^{ A (t - s) }
\tilde{F}( s, \tilde{X}_s ) \, ds
+
\int_0^t
e^{ A (t - s) }
\tilde{B}( s, \tilde{X}_s ) \, dW_s
\end{equation}
$ \mathbb{P} $-a.s.\ for
all $ t \in [0,\infty) $.
Next define mappings
$
L^{ (0) }_{ s, t }
\colon
\cup_{ r \in \mathbb{R} }
C^2( H_r, V )
\rightarrow
C( H_{ \gamma }, V )
$,
$
s, t \in [0,\infty)
$,
$
s < t
$,
and
$
L^{ (1) }_{ s, t }
\colon
\cup_{ r \in \mathbb{R} }
C^1( H_r, V )
\rightarrow
C( H_{ \gamma }, V )
$,
$
s, t \in [0,\infty)
$,
$
s < t
$,
through
\begin{equation}
\big(
L^{(0)}_{ s, t } \varphi
\big)( x )
:=
\varphi'( e^{ A (t - s) } x )
\,
e^{ A (t - s) }
\tilde{F}(s, x)
+
\frac{ 1 }{ 2 }
\sum_{
j \in \mathcal{J}
}
\varphi''( e^{ A (t - s) } x )
\big(
e^{ A (t - s) } \tilde{B}( s, x ) g_j,
e^{ A (t - s) } \tilde{B}( s, x ) g_j
\big)
\end{equation}
for all
$ x \in H_{ \gamma }
$,
$
\varphi \in
\cup_{ r \in \mathbb{R} }
C^2( H_{ r }, V )
$
and all
$
s, t \in [0,\infty)
$
with $ s < t $
and through
\begin{equation}
\big(
L^{ (1) }_{ s, t } \varphi
\big)( x )
:=
\varphi'( e^{ A (t - s) } x ) \,
e^{ A (t - s) }
\tilde{B}(s,x)
\end{equation}
for all
$ x \in H_{ \gamma } $,
$
\varphi \in
\cup_{ r \in \mathbb{R} }
C^1( H_{ r }, V )
$
and all
$
s, t \in [0,\infty)
$
with $ s < t $.
Corollary~\ref{cor:itoauto}
then implies
\begin{equation}
\label{eq:timedependent}
\varphi( \tilde{X}_t )
=
\big(
K_{ ( t - t_0 ) } \varphi
\big)( \tilde{X}_{ t_0 } )
+
\int_{ t_0 }^t
\!
\big(
L^{ (0) }_{ s, t } \varphi
\big) ( \tilde{X}_s )
\, ds
+
\int_{ t_0 }^t
\!
\big(
L^{(1)}_{ s, t } \varphi
\big) ( \tilde{X}_s )
\, dW_s
\end{equation}
for all
$ t_0, t \in [0,\infty) $
with $ t_0 < t $
and all
$
\varphi \in
\cup_{
r <
\min( \alpha + 1, \beta + \frac{ 1 }{ 2 } )
}
C^2( H_r, V )
$.
The mild It\^{o}
formula~\eqref{eq:timedependent}
is nothing else but the
counterpart of
\eqref{eq:itoformel}
for SPDEs with
time dependent coefficients.
\subsubsection{Mild Kolmogorov
backward equation for
SPDEs}
\label{sec:kolmogorov}
Based on \eqref{eq:operatorIto2}
a mild Kolmogorov
backward equation is derived
in this subsection.
Proposition~\ref{prop:prop}
implies the existence of
predictable stochastic processes
$
X^x \colon [0,\infty) \times \Omega
\rightarrow H_{ \gamma }
\in
\cap_{ q \in [1,\infty) }
C(
[0,\infty),
L^q( \Omega; H_{ \gamma } )
)
$,
$ x \in H_{ \gamma }
$,
such that
\begin{equation}
\label{eq:SPDE00}
X_t^x = e^{ A t } x
+
\int_0^t e^{ A (t - s) }
F( X_s^x ) \, ds
+
\int_0^t e^{ A (t - s) }
B( X_s^x ) \, dW_s
\end{equation}
$ \mathbb{P} $-a.s.\ for all
$ t \in [0,\infty) $ and all
$ x \in H_{ \gamma } $.
Proposition~\ref{prop:prop}
also implies that
$
\P\big[
X^x_t \in
H_r
\big] = 1
$
for all $ t \in (0,\infty) $
and all
$
r \in
( - \infty,
\min( \alpha + 1 ,
\beta + \frac{ 1 }{ 2 } )
)
$.
Then define mappings
$
P_t \colon
\cup_{ q \in [0,\infty) }
\cup_{
r \in ( - \infty,
\min( \alpha + 1, \beta + \frac{ 1 }{ 2 }
)
)
}
G^0_q( H_{ r }, V )
\rightarrow
\cup_{ q \in [0,\infty) }
\cup_{
r \in ( - \infty,
\min( \alpha + 1, \beta + \frac{ 1 }{ 2 }
)
)
}
G^0_q( H_{ r }, V )
$,
$ t \in [0,\infty) $,
through
$
P_0( \varphi ) := \varphi
$
for all
$
\varphi \in
\cup_{ q \in [0,\infty) }
\cup_{
r \in ( - \infty,
\min( \alpha + 1, \beta + \frac{ 1 }{ 2 }
)
)
}
$
and through
$
P_t( \varphi ) \in
\cup_{ q \in [0,\infty) }
G^0_q( H_{ \gamma }, V )
$
and
\begin{equation}
\label{eq:defPt}
(
P_t \varphi
)( x )
:=
\mathbb{E}\big[
\varphi( X^{ x }_t )
\big]
\end{equation}
for all
$ x \in H_{ \gamma } $,
$
\varphi \in
\cup_{ q \in [0,\infty) }
\cup_{
r \in
( - \infty,
\min( \alpha + 1, \beta + \frac{ 1 }{ 2 }
)
)
}
$
and all
$ t \in (0,\infty) $.
Note that
$
P_t\big(
G^0_q( H_r, V )
\big)
\subset
G^0_q( H_{ \gamma }, V )
$
for all
$ t \in (0,\infty) $,
$
r \in
( - \infty,
\min( \alpha + 1,
\beta + \frac{ 1 }{ 2 }
)
)
$
and all
$
q \in [0,\infty)
$.
The next lemma collect a few
simple properties of
the linear operators
$ P_t $, $ t \in [0,\infty) $.
\begin{lemma}[Properties
of $ P_t $, $ t \in [0,\infty) $]
\label{lem:Pt}
Assume that the setting in
Section~\ref{sec:setting}
is fulfilled. Then
$
P_t \in
L( G^0_q( H_{ \gamma }, V ) )
$
and
$
\sup_{ s \in [0,t] }
\| P_s \|_{
L( G^0_q( H_{ \gamma }, V ) )
}
< \infty
$
for all $ t, q \in [0,\infty) $
and it holds for every
$ q \in [0,\infty) $
that the function
$
[0,\infty) \ni t
\mapsto
P_t \in
L(
Lip^1_{ q }( H_{ \gamma }, V),
G^0_{ q + 1 }( H_{ \gamma }, V)
)
$
is locally
$ \frac{ 1 }{ 2 } $-H\"{o}lder
continuous.
\end{lemma}
The proof of
Lemma~\ref{lem:Pt}
is a straightforward
consequence of
inequality~\eqref{eq:lipestimateGnq}
and is therefore omitted.
Next the
mild It\^{o} formula
in \eqref{eq:operatorIto}
implies
\begin{equation}
\label{eq:mildP2}
\begin{split}
\big(
P_t \varphi
\big)( x )
& =
\mathbb{E}\Big[
\big(
K_{ (t - t_0) }
\varphi
\big)( X_{ t_0 }^x )
\Big]
+
\int_{ t_0 }^t
\mathbb{E}\Big[
\big(
L_{ ( t - s ) }^{(0)} \varphi
\big) ( X_s^x )
\Big]
\, ds
\\ & =
\big(
P_{ t_0 } K_{ ( t - t_0 ) }
\varphi
\big)(x)
+
\int_{ t_0 }^t
\big(
P_s
L_{ ( t - s ) }^{(0)} \varphi
\big) ( x )
\, ds
\\ & =
\big(
P_{ t_0 } K_{ ( t - t_0 ) }
\varphi
\big)(x)
+
\int_{ t_0 }^t
\big(
P_s \,
L^{(0)} K_{ ( t - s ) } \varphi
\big) ( x )
\, ds
\end{split}
\end{equation}
for all $ t_0, t \in [0,\infty) $
with $ t_0 < t $,
$ x \in H_{ \gamma } $
and all
$
\varphi \in
\cup_{ q \in [0,\infty) }
\cup_{
r \in
( - \infty,
\min( \alpha + 1, \beta + \frac{ 1 }{ 2 }
)
)
}
$.
In the next subsection
we will use \eqref{eq:mildP2} to
study regularity properties
of solutions of SPDEs.
In this regularity analysis
we also use the following
two lemmas.
\begin{lemma}[Estimates
for $ K_t $, $ t \in (0,\infty) $]
\label{lem:Ktbound}
Assume that the setting
in Section~\ref{sec:setting}
is fulfilled. Then
the function
$
(0,\infty) \ni t
\mapsto
K_t \in
L\!\left(
Lip^{ n + 1 }_q( H_{ r_1 }, V )
,
G^n_{ q + 1 }(
H_{ r_2
},
V
)
\right)
$
is continuous
for every $ r_1, r_2 \in \R $,
$ q \in [0,\infty) $,
$ n \in \{ 0, 1, \dots \} $
and it holds that
$
K_t \in
L(
Lip^n_q( H_{ r_1 }, V )
,
Lip^n_{ q }(
H_{ r_2
},
V
)
)
$
and that
\begin{align}
\label{eq:normuse1}
\|
K_t( \varphi_0 )
\|_{
G^0_q( H_{ r_1 }, V )
}
& \leq
\max\!\big(
1,
\|
e^{ A t }
\|_{
L( H_{ r_2 }, H_{ r_1 } )
}^q
\big)
\,
\|
\varphi_0
\|_{
G^0_q( H_{ r_2 }, V )
},
\\
\label{eq:normuse2}
\|
( K_t \varphi_1 )'
\|_{
G^0_{q}( H_{ \gamma },
L( H_{ \alpha }, V)
)
}
& \leq
\max\!\big(
1, \| e^{ A t } \|_{ L(H) }^q
\big)
\,
\|
e^{ A t }
\|_{ L(H_{\alpha},H_{\gamma}) }
\,
\|
\varphi_1'
\|_{
G^0_{q}(
H_{ \gamma }, L( H_{\gamma}, V)
)
} ,
\\
\label{eq:normuse3}
\|
( K_t \varphi_2 )''
\|_{
G^0_{q}( H_{ \gamma },
L^{(2)}( H_{ \beta }, V)
)
}
& \leq
\max\!\big( 1, \| e^{ A t } \|_{ L(H) }^q
\big)
\,
\|
e^{ A t }
\|_{ L(H_{\beta},H_{\gamma}) }^2
\,
\|
\varphi_2''
\|_{
G^0_{q}(
H_{ \gamma },
L^{(2)}( H_{\gamma}, V)
)
}
\end{align}
for all
$
\varphi_0 \in
C( H_{ r_2 }, V )
$,
$
\varphi_1 \in
C^1( H_{ \gamma }, V )
$,
$
\varphi_2 \in
C^2( H_{ \gamma }, V )
$,
$ q \in [0,\infty) $,
$ n \in \N $,
$ r_1, r_2 \in \R $
and all
$
t \in (0,\infty)
$.
\end{lemma}
\begin{lemma}[Estimates
for $ L^{(0)} $]
\label{kolm:bound0}
Assume that the setting
in Section~\ref{sec:setting}
is fulfilled. Then
it holds that
$
L^{ (0) }
\in
L(
G^2_{ q }(
H_{ \min(\alpha,\beta, \gamma)
},
V
),
G^0_{ q + 2 }( H_{ \gamma }, V )
)
$,
that
$
L^{ (0) }
\in
L(
Lip^3_{ q }(
H_{ \min(\alpha,\beta, \gamma)
},
V
),
\text{Lip}^1_{ q + 2
}( H_{ \gamma }, V )
)
$
and that
\begin{equation}
\label{eq:estimateL0}
\begin{split}
&
\|
L^{(0)}( \varphi )
\|_{
G^0_{ q + 2 }( H_{ r }, V )
}
\\ & \leq
\| F \|_{
G_1^0( H_{ r }, H_{ \alpha } )
}
\,
\|
\varphi'
\|_{
G_{ q + 1 }^0( H_{ r },
L( H_{ \alpha }, V )
)
}
+
\tfrac{ 1 }{ 2 } \,
\| B \|_{
G_1^0( H_{ r },
HS( U_0, H_{ \beta } ) )
}^2
\,
\|
\varphi''
\|_{
G_{ q }^0( H_{ r },
L^{(2)}( H_{ \beta }, V )
)
} ,
\\ & \leq
\max\!\big(
\| F \|_{
G_1^0( H_{ r }, H_{ \alpha } )
},
\| B \|_{
G_1^0( H_{ r },
HS( U_0, H_{ \beta } ) )
}^2
\big)
\big(
\|
\varphi'
\|_{
G_{ q + 1 }^0( H_{ r },
L( H_{ \alpha }, V )
)
}
+
\|
\varphi''
\|_{
G_{ q }^0( H_{ r },
L^{(2)}( H_{ \beta }, V )
)
}
\big)
\end{split}
\end{equation}
for all
$ r \in [ \gamma, \infty )
$,
$
\varphi \in C^{2}(
H_{ \min( \alpha, \beta, \gamma ) },
V)
$
and all $ q \in [0,\infty) $.
\end{lemma}
The proofs of
Lemma~\ref{lem:Ktbound}
and Lemma~\ref{kolm:bound0}
are straightforward
and therefore omitted.
The proof of Lemma~\ref{lem:Ktbound}
makes use of
inequality~\eqref{eq:lipestimateGnq}
above.
The next corollary follows
from
Lemmas~\ref{lem:Pt}--\ref{kolm:bound0}.
\begin{cor}
\label{cor:continuity}
Assume that the setting in
Section~\ref{sec:setting} is fulfilled.
Then
the function
$
(t_0, t) \ni
s \mapsto
P_s \, L^{(0)} K_{ t - s }
\in
L(
Lip^3_q( H_{ \gamma }, V ),
G^0_{ q + 3 }( H_{ \gamma }, V )
)
$
is continuous and satisfies
$
\int_{ t_0 }^t
\|
P_s \, L^{ (0) } K_{ t - s }
\|_{
L(
Lip^3_q( H_{ \gamma }, V ),
G^0_{ q + 3 }( H_{ \gamma }, V )
)
}
\, ds
< \infty
$
for every $ t_0, t \in [0,\infty) $
with $ t_0 < t $
and every
$ q \in [0,\infty) $.
\end{cor}
\begin{proof}[Proof
of Corollary~\ref{cor:continuity}]
The triangle inequality implies that
\begin{equation}
\label{eq:continuity}
\begin{split}
&
\big\|
P_{ s } L^{(0)} K_{ t - s }
-
P_{ s_0 } L^{(0)} K_{ t - s_0 }
\big\|_{
L(
Lip^3_q( H_{ \gamma }, V ),
G^0_{ q + 3 }( H_{ \gamma }, V )
)
}
\\ & \leq
\big\|
P_{ s } L^{(0)} K_{ t - s }
-
P_{ s } L^{(0)} K_{ t - s_0 }
\big\|_{
L(
Lip^3_q( H_{ \gamma }, V ),
G^0_{ q + 3 }( H_{ \gamma }, V )
)
}
\\ &
+
\big\|
P_{ s } L^{(0)} K_{ t - s_0 }
-
P_{ s_0 } L^{(0)} K_{ t - s_0 }
\big\|_{
L(
Lip^3_q( H_{ \gamma }, V ),
G^0_{ q + 3 }( H_{ \gamma }, V )
)
}
\\ & \leq
\| P_{ s } \|_{
L(
G^0_{ q + 3 }( H_{ \gamma }, V )
)
}
\,
\| L^{ (0) } \|_{
L(
G^2_{ q + 1 }(
H_{
\min( \alpha, \beta, \gamma)
},
V
),
G^0_{ q + 3 }( H_{ \gamma }, V )
)
}
\\ & \quad \cdot
\|
K_{ t - s }
-
K_{ t - s_0 }
\|_{
L(
Lip^3_q( H_{ \gamma }, V ),
G^2_{ q + 1 }(
H_{
\min( \alpha, \beta, \gamma)
}, V
)
)
}
\\ &
+
\big\|
P_{ s }
-
P_{ s_0 }
\big\|_{
L(
Lip^1_{ q + 2 }( H_{ \gamma }, V ),
G^0_{ q + 3 }( H_{ \gamma }, V )
)
}
\,
\| L^{ (0) } \|_{
L(
Lip^3_q(
H_{ \min( \alpha, \beta, \gamma )
}, V
) ,
Lip^1_{ q + 2 }( H_{ \gamma }, V )
)
}
\\ & \quad \cdot
\|
K_{ t - s_0 }
\|_{
L(
Lip^3_q( H_{ \gamma }, V ) ,
Lip^3_q(
H_{ \min( \alpha, \beta, \gamma )
}, V
)
)
}
\\ & \leq
\underbrace{
\left[
\| L^{ (0) } \|_{
L(
G^2_{ q + 1 }(
H_{
\min( \alpha, \beta, \gamma)
},
V
),
G^0_{ q + 3 }( H_{ \gamma }, V )
)
}
\cdot
\sup_{ u \in [t_0,t] }
\| P_{ u } \|_{
L(
G^0_{ q + 3 }( H_{ \gamma }, V )
)
}
\right]
}_{ < \infty
\text{ due to
Lemmas~\ref{kolm:bound0}
and \ref{lem:Pt}}
}
\\ & \quad \cdot
\|
K_{ t - s }
-
K_{ t - s_0 }
\|_{
L(
Lip^3_q( H_{ \gamma }, V ),
G^2_{ q + 1 }(
H_{
\min( \alpha, \beta, \gamma)
}, V
)
)
}
\\ &
+
\underbrace{
\left[
\| L^{ (0) } \|_{
L(
Lip^3_q(
H_{ \min( \alpha, \beta, \gamma )
}, V
) ,
Lip^1_{ q + 2 }( H_{ \gamma }, V )
)
}
\cdot
\|
K_{ t - s_0 }
\|_{
L(
Lip^3_q( H_{ \gamma }, V ) ,
Lip^3_q(
H_{ \min( \alpha, \beta, \gamma )
}, V
)
)
}
\right]
}_{ < \infty
\text{ due to
Lemmas~\ref{kolm:bound0}
and \ref{lem:Ktbound}}
}
\\ & \quad \cdot
\|
P_{ s }
-
P_{ s_0 }
\|_{
L(
Lip^1_{ q + 2 }( H_{ \gamma }, V ),
G^0_{ q + 3 }( H_{ \gamma }, V )
)
}
\end{split}
\end{equation}
for all
$ s_0, s \in (t_0, t) $,
$ q \in [0,\infty) $
and
$ t_0, t \in [0,\infty) $
with $ t_0 \leq t $.
Combining \eqref{eq:continuity}
with Lemma~\ref{lem:Pt}
and Lemma~\ref{lem:Ktbound}
shows that
the function
$
(t_0, t) \ni
s \mapsto
P_s \, L^{(0)} K_{ t - s }
\in
L(
Lip^3_q( H_{ \gamma }, V ),
G^0_{ q + 3 }( H_{ \gamma }, V )
)
$
is continuous
for every $ t_0, t \in [0,\infty) $
with $ t_0 < t $
and every
$ q \in [0,\infty) $.
Combining this
with
Lemmas~\ref{lem:Pt}--\ref{kolm:bound0}
completes the proof
of Corollary~\ref{cor:continuity}.
\end{proof}
In the following we reformulate
\eqref{eq:mildP2} in a suitable
abstract way
by using Corollary~\ref{cor:continuity}.
More precisely,
combining Corollary~\ref{cor:continuity}
with equation~\eqref{eq:mildP2}
shows that
\begin{equation}
\label{eq:mildP}
\begin{split}
P_t( \varphi )
& =
P_{ t_0 }\!\left(
K_{ ( t - t_0 ) }( \varphi )
\right)
+
\int_{ t_0 }^{ t }
P_{ s }\big(
L^{ (0) }_{ ( t - s ) }( \varphi )
\big)
\,
ds
\\ & =
P_{ t_0 }\!\left(
K_{ ( t - t_0 ) }( \varphi )
\right)
+
\int_{ t_0 }^{ t }
P_{ s }\big(
L^{ (0) }\big( K_{ ( t - s ) }( \varphi ) \big)
\big)
\,
ds
\end{split}
\end{equation}
in
$
G^0_{ q + 3 }( H_{ \gamma }, V )
$
for all $ t_0, t \in [0,\infty) $
with $ t_0 \leq t $,
$
\varphi \in
Lip^3_{ q }( H_{ \gamma }, V )
$
and all
$ q \in [0,\infty) $
where the integrals in \eqref{eq:mildP}
are Bochner integrals
in $ \mathbb{R} $-Banach
space
$
G^0_{ q + 3 }( H_{ \gamma }, V )
$.
According to
Corollary~\ref{cor:continuity},
these Bochner integrals
are indeed well defined.
We would like to add
to the mild Kolmogorov
backward equation~\eqref{eq:mildP}
that
the mild Kolmogorov operators
$ L^{(0)}_t $, $ t \in (0,\infty) $,
appearing in \eqref{eq:mildP}
do, in general, not commute with
the semigroup operators
$ P_t $, $ t \in [0,\infty) $,
i.e, we do, in general, not have
that
$
( P_t L^{ (0) }_s )( \varphi )
=
( L^{ (0) }_s P_t )( \varphi )
$
for all $ s, t \in (0,\infty) $
and all
$
\varphi \in G^2_0( H_{ \gamma }, V )
$.
This is in contrast
to the standard Kolmogorov backward
equation where the semigroup and
the Kolmogorov operator do commute
(see, e.g., Section~8.1
in Oeksendal~\cite{Oeksendal2000}).
In the
next step
let
$
\mathcal{K}_t
\colon
L(
G^2_{ 1 }(
H_{ \min( \alpha, \beta, \gamma )
},
V ),
G^0_{ 3 }( H_{ \gamma }, V )
)
\rightarrow
L(
Lip^3_0(
H_{ \gamma },
V
),
G^0_3( H_{ \gamma }, V )
)
$,
$
t \in (0,\infty)
$,
let
$
\mathcal{K}_0
\colon
L(
G^0_3( H_{ \gamma }, V )
)
\rightarrow
L(
Lip^3_{ 0 }( H_{ \gamma }, V ),
G^0_{ 3 }( H_{ \gamma }, V )
)
$
and let
$
\mathcal{L}
\colon
L(
G^0_{ 3 }( H_{ \gamma }, V )
)
\rightarrow
L(
G^2_{ 1 }(
H_{ \min( \alpha, \beta, \gamma )
},
V ),
G^0_{ 3 }( H_{ \gamma }, V )
)
$
be bounded linear
operators defined through
\begin{equation}
\label{eq:defLK2}
( \mathcal{K}_t \Phi )( \varphi )
:=
\Phi\!\left( K_t( \varphi )
\right)
\end{equation}
for all
$ t \in (0,\infty) $,
$
\varphi \in
Lip^3_0(
H_{ \gamma
}, V
)
$
and all
$
\Phi \in
L(
G^2_{ 1 }(
H_{ \min( \alpha, \beta, \gamma )
},
V ),
G^0_{ 3 }( H_{ \gamma }, V )
)
$,
through
$
( \mathcal{K}_0 \Phi )( \varphi )
:= \Phi( \varphi )
$
for all
$
\varphi \in
Lip^3_{ 0 }( H_{ \gamma }, V )
$
and all
$
\Phi \in
L(
G^0_{ 3 }( H_{ \gamma }, V )
)
$
and through
\begin{equation}
\label{eq:defLK}
(
\mathcal{L}
\Phi
)( \varphi )
:=
\Phi\big( L^{ (0) }( \varphi ) \big)
\end{equation}
for all
$
\varphi \in
G^2_{ 1 }(
H_{ \min( \alpha, \beta, \gamma )
}
$
and all
$
\Phi \in
L(
G^0_{ 3 }( H_{ \gamma }, V )
)
$.
Lemmas~\ref{lem:Ktbound}
and \ref{kolm:bound0} ensure
that
$ \mathcal{K}_t $, $ t \in [0,\infty) $,
and
$ \mathcal{L} $
are indeed well defined
bounded linear operators.
The next corollary follows
from
Lemmas~\ref{lem:Pt}--\ref{kolm:bound0}.
\begin{cor}
\label{cor:welldefinedMKB}
Assume that the setting
in Section~\ref{sec:setting}
is fulfilled.
Then
the function
$
(t_0, t) \ni s
\mapsto
\mathcal{K}_{ t - s }
(
\mathcal{L}(
P_s )
)
\in
L(
Lip^3_{ 0 }( H_{ \gamma }, V ),
G^0_{ 3 }( H_{ \gamma }, V )
)
$
is continuous and satisfies
$
\int_{ t_0 }^t
\|
\mathcal{K}_{ t - s }
(
\mathcal{L}(
P_s )
)
\|_{
L(
Lip^3_{ 0 }( H_{ \gamma }, V ),
G^0_{ 3 }( H_{ \gamma }, V )
)
}
\, ds
< \infty
$
for every $ t_0, t \in [0,\infty) $
with $ t_0 < t $.
\end{cor}
\begin{proof}[Proof
of Corollary~\ref{cor:welldefinedMKB}]
Note that the triangle
inequality implies that
\begin{equation}
\label{eq:continuity2}
\begin{split}
&
\big\|
\mathcal{K}_{ t - s }
(
\mathcal{L}(
P_{ s } )
)
-
\mathcal{K}_{ t - s_0 }
(
\mathcal{L}(
P_{ s_0 } )
)
\big\|_{
L(
Lip^3_{ 0 }( H_{ \gamma }, V ),
G^0_{ 3 }( H_{ \gamma }, V )
)
}
\\ & \leq
\big\|
\mathcal{K}_{ t - s }
(
\mathcal{L}(
P_{ s } )
)
-
\mathcal{K}_{ t - s_0 }
(
\mathcal{L}(
P_{ s } )
)
\big\|_{
L(
Lip^3_{ 0 }( H_{ \gamma }, V ),
G^0_{ 3 }( H_{ \gamma }, V )
)
}
\\ & +
\big\|
\mathcal{K}_{ t - s_0 }
(
\mathcal{L}(
P_{ s } )
)
-
\mathcal{K}_{ t - s_0 }
(
\mathcal{L}(
P_{ s_0 } )
)
\big\|_{
L(
Lip^3_{ 0 }( H_{ \gamma }, V ),
G^0_{ 3 }( H_{ \gamma }, V )
)
}
\\ & \leq
\|
\mathcal{K}_{ t - s }
-
\mathcal{K}_{ t - s_0 }
\|_{
L\left(
L(
G^2_{ 1 }(
H_{ \min( \alpha, \beta, \gamma )
},
V ),
G^0_{ 3 }( H_{ \gamma }, V )
) ,
L(
Lip^3_{ 0 }( H_{ \gamma }, V ),
G^0_{ 3 }( H_{ \gamma }, V )
)
\right)
}
\\ & \cdot
\underbrace{
\|
\mathcal{L}
\|_{
L\left(
L(
G^0_{ 3 }(
H_{ \gamma
},
V )
),
L(
G^2_{ 1 }(
H_{ \min( \alpha, \beta, \gamma )
},
V ),
G^0_{ 3 }( H_{ \gamma }, V )
)
\right)
}
}_{
< \infty
\text{ due to
Lemma~\ref{kolm:bound0}}
}
\cdot
\underbrace{
\left[
\sup_{ u \in [t_0,t] }
\|
P_{ u }
\|_{
L(
G^0_{ 3 }( H_{ \gamma }, V )
)
}
\right]
}_{
< \infty
\text{ due to
Lemma~\ref{lem:Pt}}
}
\\ & +
\underbrace{
\left[
\sup_{
\substack{
\Phi \in
L(
G^0_{ 3 }( H_{ \gamma }, V )
)
\\
\| \Phi \|_{
L(
Lip^1_{ 2 }( H_{ \gamma }, V ),
G^0_{ 3 }( H_{ \gamma }, V )
)
} \leq 1
}
}
\|
\mathcal{K}_{ t - s_0 }(
\mathcal{L}( \Phi )
)
\|_{
L(
Lip^3_{ 0 }(
H_{ \gamma
},
V ),
G^0_{ 3 }( H_{ \gamma }, V )
)
}
\right]
}_{ < \infty
\text{ due to
Lemmas~\ref{lem:Ktbound}
and \ref{kolm:bound0}}
}
\cdot
\,
\|
P_{ s }
-
P_{ s_0 }
\|_{
L(
Lip^1_{ 2 }( H_{ \gamma }, V ),
G^0_{ 3 }( H_{ \gamma }, V )
)
}
\end{split}
\end{equation}
for all
$ s_0, s \in (t_0, t) $
and all
$ t_0, t \in [0,\infty) $
with $ t_0 \leq t $.
Combining \eqref{eq:continuity2}
with Lemma~\ref{lem:Ktbound}
and Lemma~\ref{lem:Pt}
completes the proof
of Corollary~\ref{cor:welldefinedMKB}.
\end{proof}
We now use
Corollary~\ref{cor:welldefinedMKB}
to reformulate
equation~\eqref{eq:mildP}.
More precisely,
combining
Corollary~\ref{cor:welldefinedMKB},
equation~\eqref{eq:mildP},
definition~\eqref{eq:defLK}
and
definition~\eqref{eq:defLK2}
shows that
\begin{equation}
\label{eq:kolm1}
P_t
=
\mathcal{K}_{ (t - t_0) }(
P_{ t_0 } )
+
\int_{ t_0 }^{ t }
\mathcal{K}_{ (t-s) }
\big(
\mathcal{L}(
P_s )
\big)
\, ds
\end{equation}
in
$
L(
Lip^3_{ 0 }( H_{ \gamma }, V ),
G^0_{ 3 }( H_{ \gamma }, V )
)
$
for all $ t_0, t \in [0,\infty) $
with $ t_0 \leq t $
and this, in particular, implies that
\begin{equation}
\label{eq:kolm2}
P_t
=
\mathcal{K}_{ t }(
P_{ 0 } )
+
\int_{ 0 }^{ t }
\mathcal{K}_{ (t - s) }\big(
\mathcal{L}(
P_s )
\big)
\, ds
\end{equation}
in
$
L(
Lip^3_{ 0 }( H_{ \gamma }, V ),
G^0_{ 3 }( H_{ \gamma }, V )
)
$
for all $ t \in [0,\infty) $
where the integrals in
\eqref{eq:kolm1} and
\eqref{eq:kolm2}
are understood to be
Bochner integrals in
$
L(
G^3_{ 0 }( H_{ \gamma }, V ),
G^0_{ 3 }( H_{ \gamma }, V )
)
$.
According to
Corollary~\ref{cor:welldefinedMKB},
these Bochner integrals are
indeed well defined.
Equation~\eqref{eq:kolm2}
and equation~\eqref{eq:kolm1}
are somehow
{\it mild Kolmogorov
backward equations}
for the $ P_t $, $ t \in [0,\infty) $,
(see \eqref{eq:defPt})
associated to the
SPDE~\eqref{eq:SPDE00}.
\subsubsection{Weak regularity
for solutions of SPDEs}
\label{sec:weakregularity}
Another consequence
of the mild It\^{o}
formula~\eqref{eq:itoformel}
is to study weak regularity of
solutions of SPDEs.
To be more
precise, in this subsection
regularity
of the
probability
measures
$
\mathbb{P}_{ X_t }
$,
$
t \in (0,T]
$,
of the solution process
$
X_t
$,
$
t \in [0,T]
$,
of the SPDE~\eqref{eq:SPDE}
are studied by using the mild
Kolmogorov backward
equation~\eqref{eq:mildP2} above.
Below (see the illustrations
below Lemma~\ref{lem:embedding})
we also describe in more detail
what we understand by regularity
of a probability measure.
While strong regularity
of solutions
of SPDEs have been
intensively analyzed
in the literature (see, e.g.,
Da Prato \citationand\
Zabczyk~\cite{dz92, DaPratoZabczyk1996},
Brze\'{z}niak~\cite{b97b},
Brze\'{z}niak,
Van Neerven, Veraar
\citationand\ Weis~\cite{bvvw08},
Van Neerven, Veraar
\citationand\
Weis~\cite{vvw07,vvw08,vvw11},
Jentzen \citationand\
R\"{o}ckner~\cite{jr12},
Kruse \citationand\
Larsson~\cite{KruseLarsson2011}
and the references therein),
weak regularity
for solutions of SPDEs
seem to be much less investigated.
Let us now go into details.
An important ingredient in our
analysis on weak regularity
of solutions of SPDEs are
the following mappings.
Let
$
\left\| \cdot \right\|_{
t, q
}^{
\delta, \rho
}
\colon
G^2_{ q }(
H_{ \rho },
V
)
\rightarrow
[0,\infty)
$,
$ t \in (0,\infty) $,
$ q \in [0,\infty) $,
$ \delta \in (\rho - 1,\infty) $,
$
\rho \in \R
$,
be a family of functions
defined through
\begin{align}
\label{eq:supernorm}
&
\| \varphi \|_{ t, q }^{ \delta, \rho }
\\ &
:=
\|
K_t( \varphi )
\|_{
G^0_{ q + 2 }( H_{ \delta }, V )
}
+
\int_0^t
\left( t - s \right)^{
\min( \delta - \rho, 0 )
}
\left(
\|
( K_s \varphi )'
\|_{
G_{ q + 1 }^0( H_{ \rho },
L( H_{ \alpha }, V )
)
}
+
\|
( K_s \varphi )''
\|_{
G_{ q }^0( H_{ \rho },
L^{(2)}( H_{ \beta }, V )
)
}
\right)
ds
\nonumber
\end{align}
for all
$ t \in (0,\infty) $,
$
\varphi \in
G^2_{ q }( H_{ \rho }, V )
$,
$ \delta \in (\rho - 1,\infty)
$,
$ q \in [0,\infty) $
and all
$
\rho \in
\R
$.
Please note that the integrand
in \eqref{eq:supernorm}
is indeed Borel measurable
in $ s \in [0,\infty) $
since $ H_{ \rho } $ is
separable
for every $ \rho \in \R $.
The next lemma collects
some properties of the
functions
$
\left\| \cdot
\right\|_{ t, q }^{
\delta, \rho
}
\colon
G^2_{ q }( H_{ \rho }, V )
\rightarrow
[0,\infty)
$,
$ t \in (0,\infty) $,
$ q \in [0,\infty) $,
$ \delta \in (\rho - 1,\infty)
$,
$
\rho \in
\R
$.
\begin{lemma}[Properties
of
$
\left\| \cdot \right\|_{ t, q
}^{ \delta, \rho }
$,
$ t \in (0,\infty) $,
$ q \in [0,\infty) $,
$ \delta \in (\rho - 1,\infty) $,
$
\rho \in
\R
$]
\label{lem:norm}
Assume that the setting
in Section~\ref{sec:setting}
is fulfilled and let
$
\left\| \cdot \right\|_{ t, q
}^{ \delta, \rho }
$,
$ t \in (0,\infty) $,
$ q \in [0,\infty) $,
$ \delta \in (\rho - 1,\infty) $,
$
\rho \in
\R
$,
be defined through
\eqref{eq:supernorm}. Then
\begin{equation}
\label{eq:norm}
\begin{split}
\left\| \varphi
\right\|_{ t, q }^{ \delta, \rho }
& \leq
\| \varphi \|_{
G^2_{ q }( H_{ \rho }, V )
}
\,
\Bigg(
\max\!\big(
1,
\|
e^{ A t }
\|_{
L( H_{ \rho }, H_{ \delta } )
}^{ ( q + 2 ) }
\big)
\\ & \quad
+
\max\!\big( 1,
\| e^{ A t } \|_{ L(H) }^{ (q + 1) }
\big)
\int_0^t
\left( t - s \right)^{
\min( \delta - \rho, 0 )
}
\left[
\|
e^{ A s }
\|_{ L(H_{ \alpha },H_{ \rho } ) }
+
\|
e^{ A s }
\|_{ L(H_{ \beta }, H_{ \rho }) }^2
\right]
ds
\Bigg)
< \infty
\end{split}
\end{equation}
for all
$
\varphi \in
G^2_{ q }( H_{ \rho }, V )
$,
$ q \in [0,\infty) $,
$
t \in (0,\infty)
$,
$
\delta \in ( \rho - 1, \infty )
$,
$
\rho \in
( - \infty,
\min(
\alpha + 1 ,
\beta + \frac{ 1 }{ 2 }
)
)
$
and it holds
for every
$ t \in (0,\infty) $,
$ q \in [0,\infty) $,
$
\rho \in
( - \infty,
\min(
\alpha + 1 ,
\beta + \frac{ 1 }{ 2 }
)
)
$,
$
\delta \in (\rho - 1,\infty)
$
that the mapping
$
\left\| \cdot \right\|_{ t, q }^{
\delta, \rho
}
\colon
G^2_{ q }( H_{ \rho }, V )
\rightarrow
[0,\infty)
$
is a norm on
$
G^2_{ q }( H_{ \rho }, V )
$.
\end{lemma}
\begin{proof}[Proof
of Lemma~\ref{lem:norm}]
Combining
\eqref{eq:normuse1}--\eqref{eq:normuse3},
\eqref{eq:Gnq1}
and \eqref{eq:Gnq2}
shows
\eqref{eq:norm}.
Next observe
for every
$ t \in (0,\infty) $,
$ q \in [0,\infty) $,
$
\rho \in
( - \infty,
\min(
\alpha + 1 ,
\beta + \frac{ 1 }{ 2 }
)
)
$
and every
$
\delta \in ( \rho - 1, \infty )
$
that
$
\left\| \cdot \right\|_{ t, q }^{
\delta, \rho
}
\colon
G^2_{ q }( H_{ \rho }, V )
\rightarrow
[0,\infty)
$
is a semi-norm on
$
G^2_{ q }( H_{ \rho }, V )
$.
In addition, note
for every
$ \rho, \delta \in \R $,
$ q \in [0,\infty) $,
$ t \in (0,\infty) $
and every
$
\varphi \in
G^0_{ q }( H_{ \rho }, V )
$
that if
$
\|
K_t( \varphi )
\|_{
G^0_q( H_{ \delta }, V )
}
= 0
$,
then
$
\sup_{
x \in e^{ A t }( H_{ \delta } )
}
\|
\varphi(x)
\|_V
$ = 0.
The fact that
for every
$ \rho, \delta \in \R $
the set
$
e^{ A t }( H_{ \delta } )
$
is dense in
$
H_{ \rho }
$
therefore shows
for every
$ t \in (0,\infty) $,
$ q \in [0,\infty) $,
$
\rho \in
( - \infty,
\min(
\alpha + 1 ,
\beta + \frac{ 1 }{ 2 }
)
)
$
and every
$
\delta \in ( \rho - 1, \infty )
$
that
$
\left\| \cdot \right\|_{ t, q }^{
\delta, \rho
}
\colon
G^2_{ q }( H_{ \rho }, V )
\rightarrow
[0,\infty)
$
is indeed a norm on
$
G^2_{ q }( H_{ \rho }, V )
$.
The proof of
Lemma~\ref{lem:norm}
is thus completed.
\end{proof}
In the next step
we denote
for every
$ t \in (0,\infty ) $,
$ q \in [0,\infty) $,
$
\rho \in
( - \infty,
\min(
\alpha + 1 ,
\beta + \frac{ 1 }{ 2 }
)
)
$,
$
\delta \in ( \rho - 1 , \infty )
$
by
$
(
\mathcal{G}^{ 2, \delta }_{ t, q }(
H_{ \rho }, V ),
\left\| \cdot
\right\|_{ t, q }^{
\delta, \rho
}
)
$
the completion of the normed
$ \mathbb{R} $-vector space
$
(
G^2_{ q }( H_{ \rho }, V ),
$
$
\left\| \cdot
\right\|_{ t, q }^{
\delta, \rho
}
)
$.
The pairs
$
(
\mathcal{G}^{ 2, \delta }_{ t, q
}( H_{ \rho }, V ),
\left\| \cdot \right\|_{ t, q }^{
\delta, \rho
}
)
$
for
$ t \in (0,\infty ) $,
$ q \in [0,\infty) $,
$ \delta \in ( \rho - 1, \infty ) $
and
$
\rho \in
( - \infty,
\min(
\alpha + 1 ,
\beta + \frac{ 1 }{ 2 }
)
)
$
are thus
$ \mathbb{R} $-Banach
spaces.
\begin{theorem}[Weak regularity
for $ P_t $, $ t \in (0,\infty) $]
\label{thm:continuity}
Assume that the setting in
Section~\ref{sec:setting}
is fulfilled.
Then
$
P_t
\in
L(
\mathcal{G}^{ 2, \delta }_{ t, q - 2
}( H_{ \rho }, V ) ,
G^0_{ q }( H_{ \delta }, V )
)
$
and
\begin{align}
&
\|
P_t
\|_{
L\left(
\mathcal{G}^{ 2, \delta
}_{ t, q - 2
}( H_{ \rho }, V ) ,
G^0_{ q }( H_{ \delta }, V )
\right)
}
\\ & \leq
\nonumber
\max\!\Big(
1,
\| F \|_{
G^0_1( H_{ \rho }, H_{ \alpha } )
}
,
\| B \|_{
G^0_1( H_{ \rho },
HS( U_0, H_{ \beta } ) )
}^2
\Big)
\max\!
\left(
1,
\sup_{ s \in (0, t) }
\left[
s^{ \max( \rho - \delta, 0 ) }
\|
P_s
\|_{
L(
G^0_{ q }( H_{ \rho }, V ) ,
G^0_{ q }( H_{ \delta }, V )
)
}
\right]
\right)
< \infty
\end{align}
for all
$ t \in (0,\infty) $,
$
\delta \in [ \gamma, \infty )
\cap
( \rho - 1 , \infty )
$,
$ q \in [2,\infty) $
and all
$
\rho \in
[
\gamma,
\min(
\alpha + 1,
\beta + \frac{ 1 }{ 2 }
)
)
$.
\end{theorem}
\begin{proof}[Proof
of Theorem~\ref{thm:continuity}]
Equation~\eqref{eq:mildP2}
implies
\begin{equation}
\label{eq:use1}
\begin{split}
&
\|
P_t( \varphi )
\|_{
G^0_{ q }( H_{ \delta }, V )
}
\\ & \leq
\|
K_t( \varphi )
\|_{
G^0_{ q }( H_{ \delta }, V )
}
\\ & \quad
+
\left(
\sup_{ s \in (0,t) }
\left[
s^{ \max( \rho - \delta, 0 ) }
\|
P_s
\|_{
L(
G^0_{ q }( H_{ \rho }, V ) ,
G^0_{ q }( H_{ \delta }, V )
)
}
\right]
\right)
\left(
\int_0^t
\left( t - s
\right)^{ - \max( \rho - \delta, 0 ) }
\|
L^{ (0) }_s( \varphi )
\|_{
G_{ q }^0( H_{ \rho }, V )
}
\,
ds
\right)
\\ & \leq
\max\!\left(
1 ,
\sup_{ s \in (0,t) }
\left[
s^{ \max( \rho - \delta, 0 ) }
\|
P_s
\|_{
L(
G^0_{ q }( H_{ \rho }, V ) ,
G^0_{ q }( H_{ \delta }, V )
)
}
\right]
\right)
\\ & \quad
\cdot
\left(
\|
K_t( \varphi )
\|_{
G^0_{ q }( H_{ \delta }, V )
}
+
\int_0^t
\left( t - s
\right)^{ \min( \delta - \rho, 0 ) }
\|
L^{ (0) }( K_s( \varphi ) )
\|_{
G_{ q }^0( H_{ \rho }, V )
}
\,
ds
\right)
\end{split}
\end{equation}
for all
$ t \in (0,\infty) $,
$
\varphi \in
G^2_{ q }( H_{ \rho }, V )
$,
$
\delta \in [\gamma,\infty)
\cap
( \rho - 1, \infty )
$,
$
q \in [2,\infty)
$
and all
$
\rho \in
[
\gamma,
\min(
\alpha + 1,
\beta + \frac{ 1 }{ 2 }
)
)
$.
Inequality~\eqref{eq:use1}
and Lemma~\ref{kolm:bound0}
then complete
the proof of
Theorem~\ref{thm:continuity}.
\end{proof}
Below we illustrate
Theorem~\ref{thm:continuity}
through some consequences.
To do so, we need the following
elementary lemma for probability
measures on separable
Hilbert spaces.
\begin{lemma}[An embedding
for probability measures]
\label{lem:embedding}
Let
$
(
H,
\left\| \cdot \right\|_H,
\left< \cdot, \cdot \right>_H
)
$
and
$
(
V,
\left\| \cdot \right\|_V,
\left< \cdot, \cdot \right>_V
)
$
be separable real
Hilbert spaces
with $ V \neq \{ 0 \} $
and let
$
\mu_1, \mu_2 \colon
\mathcal{B}(H)
\to [0,1]
$
be two probability measures
with
$
\int_{ H }
\varphi( x ) \, \mu_1( dx )
=
\int_{ H }
\varphi( x ) \, \mu_2( dx )
$
for all infinitely often
Fr\'{e}chet differentiable
functions
$
\varphi \colon H \to V
$
with compact support.
Then $ \mu_1 = \mu_2 $.
\end{lemma}
\begin{proof}[Proof
of Lemma~\ref{lem:embedding}]
First of all,
we denote throughout
this proof
for every $ x \in H $
and every $ r \in (0,\infty) $
by
$
B_r( x )
:=
\{
y \in H
\colon
\left\| x - y \right\|_H \leq r
\}
$
the ball in $ H $
on $ x $ with radius $ r $.
Next let $ v_0 \in V $
be a vector
which satisfies
$ \left\| v_0 \right\|_V = 1 $.
Such a vector does indeed
since we assumed that
$ V \neq \{ 0 \} $.
Furthermore, let
$
\psi_k \colon \R \to [0,1]
$,
$ k \in \N $,
be a sequence of infinitely
often differentiable functions
with
$
\psi_k(x) = 1
$
for all $ x \in [-1,1] $
and all $ k \in \N $
and with
$
\psi_k(x) = 0
$
for all
$
x \in
(- \infty, - 1 - \frac{ 1 }{ k } )
\cup
(1 + \frac{ 1 }{ k }, \infty)
$
and all $ k \in \N $.
Moreover, let
$ N \in \N $,
let
$ x_1, \dots, x_N \in H $
and let
$
r_1, \dots, r_N \in (0,\infty)
$.
In the next step we
define a sequence
$ \varphi_k \colon H \to V $,
$ k \in \N $,
of functions
by
\begin{equation}
\varphi_k(x)
:=
v_0
\cdot
\prod_{ n = 1 }^N
\psi_k\!\left(
\tfrac{
\left\| x - x_n \right\|_H^2
}{
\left( r_n \right)^2
}
\right)
\end{equation}
for all $ x \in H $
and all $ k \in \N $.
Note for every $ k \in \N $
that $ \varphi_k $ is infinitely
often Fr\'{e}chet differentiable
with a compact support.
Therefore, we obtain
\begin{equation}
\label{eq:mu1mu2}
\int_{ H } \varphi_k(x) \, \mu_1( dx )
=
\int_{ H } \varphi_k(x) \, \mu_2( dx )
\end{equation}
for all $ k \in \N $.
In the next step observe that
$
\varphi_k(x) = v_0
$
for all
$
x \in
\cap_{ n = 1 }^N
B_{ r_n }( v_n )
$
and all
$ k \in \N $,
that
$
\sup_{ k \in \N }
\sup_{ x \in H }
\left\| \varphi_k(x) \right\|_V
\leq 1
$
and that
\begin{equation}
\lim_{ k \to \infty }
\varphi_k(x)
=
\begin{cases}
1 &
\colon
x \in
\cap_{ n = 1 }^N
B_{ r_n }( v_n )
\\
0 &
\colon
x \in
H \backslash
\left(
\cap_{ n = 1 }^N
B_{ r_n }( v_n )
\right)
\end{cases}
\end{equation}
for all $ x \in H $.
Combining this and
\eqref{eq:mu1mu2}
with Lebesgue's theorem
on dominated convergence
then proves that
$
\mu_1\big(
\cap_{ n = 1 }^N
B_{ r_n }( x_n )
\big)
=
\mu_2\big(
\cap_{ n = 1 }^N
B_{ r_n }( x_n )
\big)
$.
Combining this, the fact that the set
\begin{equation}
\cup_{ M \in \N }
\left\{
\cap_{ m = 1 }^M
B_{ s_m }( y_m )
\subset H
\colon
s_1, \dots, s_M \in (0,\infty),
y_1, \dots, y_M \in H
\right\}
\end{equation}
is a $ \cap $-stable
generator of the Borel
sigma-algebra
$ \mathcal{B}(H) $
and the uniqueness theorem
for measures (see, e.g.,
Lemma~1.42 in Klenke~\cite{k08b})
then completes the proof
of Lemma~\ref{lem:embedding}.
\end{proof}
Let us now illustrate
Theorem~\ref{thm:continuity}
by a simple application.
First, we denote by
$
G^2_{ 0 }(
H_{ \gamma },
\R
)'
:=
L(
G^2_{ 0 }(
H_{ \gamma },
\R
),
\R
)
$
and
$
\mathcal{G}^{ 2, \gamma }_{ t, 0
}(
H_{ \gamma },
\R
)'
:=
L(
\mathcal{G}^{ 2, \gamma
}_{ t, 0 }(
H_{ \gamma },
\R
),
\mathbb{R}
)
$
for
$ t \in (0,\infty) $
the topological
dual spaces of
$
G^2_{ 0 }(
H_{ \gamma },
\R
)
$
and
$
\mathcal{G}^{ 2, \gamma }_{ t, 0
}(
H_{ \gamma },
\R
)
$
for $ t \in (0,\infty) $
respectively.
Moreover, we denote by
$
\mathcal{M}_2( H_{ \gamma } )
$
the set of all
probability measures
$
\mu \colon
\mathcal{B}( H_{ \gamma } )
\to [0,1]
$
which satisfy
$
\int_{ H_{ \gamma } }
\left\| x \right\|_{
H_{ \gamma }
}^2
\mu( dx ) < \infty
$
and we consider
the mapping
$
\mathcal{I} \colon
\mathcal{M}_2( H_{ \gamma } )
\to
G^2_{ 0 }(
H_{ \gamma },
\R
)'
$
given by
$
( \mathcal{I} \mu)( \varphi )
=
\int_{
H_{ \gamma }
}
\varphi( x ) \, \mu( dx )
$
for all
$
\varphi \in G^2_0( H_{ \gamma }, \R )
$
and all
$
\mu \in
\mathcal{M}_2( H_{ \gamma } )
$.
Lemma~\ref{lem:embedding}
then proves that
$ \mathcal{I} $ is injective
and through $ \mathcal{I} $
we can thus identify the
probability measures
$ \mathcal{M}_2( H_{ \gamma } ) $
with finite second moment
as a subset of linear forms in
$
G^2_{ 0 }(
H_{ \gamma },
\R
)'
$.
Next note
that
Proposition~\ref{prop:prop}
proves
that
the probability
measure
$
\mathbb{P}_{ X_t }
$
of the solution process
of the SPDE~\eqref{eq:SPDE}
at every time $ t \in (0,T] $
has a finite second moment and
is thus in
$ \mathcal{M}_2( H_{ \gamma } ) $.
Hence,
the linear form
$
\mathcal{I}(
\mathbb{P}_{ X_t }
)
=
\int_{ H_{ \gamma } }
( \cdot ) \,
d\mathbb{P}_{ X_t }
\in
G^2_{ 0 }(
H_{ \gamma },
\R
)'
$
corresponding to
the probability
measure
$
\mathbb{P}_{ X_t }
$
of the solution of the SPDE~\eqref{eq:SPDE}
at time
$
t \in (0,T]
$
is in
$
G^2_{ 0 }(
H_{ \gamma },
V
)'
$.
In addition, observe that
Theorem~\ref{thm:continuity},
in particular,
implies that
$
\int_{ H_{ \gamma } }
( \cdot ) \,
d\mathbb{P}_{ X_t }
\in
\mathcal{G}^{ 2, \gamma }_{ t, 0 }(
H_{ \gamma },
V
)'
$
for all $ t \in (0,T] $.
Moreover,
note that
Lemma~\ref{lem:norm} implies
that
$
\mathcal{G}^{ 2, \gamma
}_{ t, 0 }(
H_{ \gamma },
V
)'
\subset
G^2_{ 0 }(
H_{ \gamma },
V
)'
$
continuously
for all
$ t \in (0,\infty) $.
Theorem~\ref{thm:continuity}
thus proves
for every $ t \in (0,T] $
that
$
\mathcal{I}(
\mathbb{P}_{ X_t }
)
=
\int_{ H_{ \gamma } }
( \cdot ) \,
d\mathbb{P}_{ X_t }
$
does not only lie in
$
G^2_{ 0 }(
H_{ \gamma },
V
)'
$
but also
in the smaller space
$
\mathcal{G}^{ 2, \gamma
}_{ t, 0 }(
H_{ \gamma },
V
)'
$
too.
In this sense
Theorem~\ref{thm:continuity}
proves more regularity
of the probability measures
$
\mathbb{P}_{ X_t }
$,
$ t \in (0,T]
$,
of the solution of the
SPDE~\eqref{eq:SPDE}.
It thus establishes ``weak
regularity'' for the solution
of the SPDE~\eqref{eq:SPDE}.
In the remainder
of this subsection some further
consequences of
Theorem~\ref{thm:continuity}
are derived.
\begin{cor}
\label{cor:weakest}
Assume that the setting
in Section~\ref{sec:setting}
is fulfilled,
assume
$ \alpha \leq \gamma $,
assume
$ \beta \leq \gamma $,
let
$
\rho \in
[
\gamma,
\min(
\alpha + 1,
\beta + \frac{ 1 }{ 2 }
)
)
$
be a real number,
let
$
(
\tilde{H},
\left< \cdot , \cdot \right>_{ \tilde{H} },
\left\| \cdot \right\|_{ \tilde{H} }
)
$
be a separable $ \mathbb{R} $-Hilbert
space, let
$
R, \tilde{R} \in
L( H_{ \rho }, \tilde{H} )
$, let
$
\varphi \in
C^2_{ Lip }( \tilde{H}, V )
$
and let
$
\psi \colon H_{ \rho }
\rightarrow
\tilde{H}
$
be given by
$
\psi(x)
=
\varphi( R x )
-
\varphi( \tilde{R} x )
$
for all $ x \in H_{ \rho } $.
Then
\begin{equation}
\label{eq:weak_crucial}
\begin{split}
&
\|
P_t( \psi )
\|_{
G^0_{ q }( H_{ \delta }, V )
}
\\ & \leq
\max\!\left(
1,
\| F \|_{
G^0_1( H_{ \rho }, H_{ \alpha } )
}
,
\| B \|_{
G^0_1( H_{ \rho },
HS( U_0, H_{ \beta } ) )
}^2
\right)
\max\!
\left(
1,
\sup_{ s \in (0, t) }
\left[
s^{ \max( \rho - \delta, 0 ) }
\|
P_s
\|_{
L(
G^0_{ q }( H_{ \rho }, V ) ,
G^0_{ q }( H_{ \delta }, V )
)
}
\right]
\right)
\\ & \quad
\cdot
\| \varphi \|_{
C^2_{ Lip }( \tilde{H}, V )
}
\,
\frac{
\max( t, 1 )
}{
t^{ \max( r + \rho - \delta, 0 ) }
}
\,
\|
R - \tilde{R}
\|_{
L( H_{ \rho + r }, \tilde{H} )
}
\left[
\sup_{
u \in [ \rho - \delta, 1]
\cup [0,1]
}
\sup_{ s \in (0,t] }
\left(
s^{ \max( u, 0 ) }
\|
e^{ A s }
\|_{
L( H, H_{ u } )
}
\right)
\right]^3
\\ & \quad
\cdot
\big[
1 +
\| R \|_{ L( H_{ \rho }, \tilde{H} )
} +
\| \tilde{R}
\|_{ L( H_{ \rho }, \tilde{H} )
}
\big]^2
\left(
1
+
\int_0^1
\left( 1 - s \right)^{
\min( \delta - \rho, 0 )
}
s^{
\left[
\min\left(
\alpha - \rho ,
2 \beta - 2 \rho
\right) - r
\right]
}
\, ds
\right)
< \infty
\end{split}
\end{equation}
for all
$ t \in (0,\infty) $,
$ q \in [3,\infty) $,
$
\delta \in [\gamma, \infty)
$
and all
$
r \in
[
0,
\min( 1 + \alpha - \rho,
1 + 2 \beta - 2 \rho
)
)
$.
In particular,
we have
\begin{equation}
\label{eq:weak_crucialB}
\begin{split}
&
\sup_{
\Phi \in
C^2_{ Lip }( H_{ \rho }, V )
\backslash \{ 0 \}
}
\sup_{
S \in L( H_{ \rho } )
}
\sup_{
t \in (0,T]
}
\left(
\frac{
t^{ \max( r + \rho - \delta, 0 ) }
\,
\|
P_t( \Phi ) -
P_t\big( \Phi( S( \cdot ) ) \big)
\|_{
G^0_{ 3 }( H_{ \delta }, V )
}
}{
\|
I - S
\|_{
L( H_{ \rho + r }, H_{ \rho } )
}
\,
\big(
1 +
\| S \|_{ L( H_{ \rho } )
}^2
\big)
\,
\| \varphi \|_{
C^2_{ Lip }( H_{ \rho }, V )
}
}
\right)
< \infty
\end{split}
\end{equation}
for all
$
\delta \in [\gamma, \infty)
$
and all
$
r \in
[
0,
\min( 1 + \alpha - \rho,
1 + 2 \beta - 2 \rho
)
)
$.
\end{cor}
\begin{proof}[Proof
of Corollary~\ref{cor:weakest}]
Throughout this the proof
the real numbers
$ \kappa_{ r, t } \in [1,\infty) $,
$ r \in [0,\infty) $,
$ t \in (0,\infty) $,
defined by
\begin{equation}
\kappa_{ r, t }
:=
\sup_{ u \in [-r,1] }
\sup_{ s \in (0,t] }
\left(
s^{ \max( u, 0 ) }
\|
e^{ A s }
\|_{
L( H, H_{ u } )
}
\right)
< \infty
\end{equation}
for all $ r \in [0,\infty) $
are used.
The quantities
$ \kappa_{ r, t } $,
$ r \in [0,\infty) $,
$ t \in (0,\infty) $,
are indeed
finite since $ e^{ A t } $,
$ t \in [0,\infty) $,
is an analytic semigroup.
The estimate
\begin{equation}
\|
e^{ A t }
\|_{ L( H_{ a }, H_{ b } ) }
=
\|
e^{ A t }
\|_{ L( H, H_{ ( b - a ) } ) }
\leq
\kappa_{
\max( a - b, 0 ), t
}
\,
t^{
\min\left( a - b, 0\right)
}
\end{equation}
for all
$ t \in (0,\infty) $
and all $ a, b \in \R $
with $ b - a \leq 1 $ then
shows
\begin{equation}
\label{eq:weakest1}
\begin{split}
&
\|
K_t( \psi )
\|_{
G_{q}^0( H_{ \delta }, V )
}
\leq
\| \varphi' \|_{
L^{ \infty }( \tilde{H},
L( \tilde{H}, V)
)
}
\,
\| R - \tilde{R} \|_{
L( H_{ r }, \tilde{H} )
}
\,
\| e^{ A t } \|_{
L( H_{ \delta }, H_r )
}
\\ & \leq
\| \varphi \|_{
C^2_{ Lip }( \tilde{H}, V )
}
\,
\| R - \tilde{R} \|_{
L( H_{ r }, \tilde{H} )
}
\,
\kappa_{
\max( \delta - r, 0 ), t
}
\,
t^{ \min( \delta - r, 0 ) }
\end{split}
\end{equation}
and
\begin{equation}
\begin{split}
&
\|
( K_t \psi )'
\|_{
G_{ q - 1 }^0( H_{ \rho },
L( H_{ \alpha }, V )
)
}
\\ & \leq
\sup_{ x \in H_{ \rho } }
\frac{
\|
( \varphi'( R e^{ A t } x ) -
\varphi'( \tilde{R} e^{ A t } x ) )
R e^{ A t }
\|_{
L( H_{ \alpha }, V )
}
}{
\big(
1 +
\| x \|_{ H_{ \rho } }
\big)^{ (q - 1) }
}
\\ & \quad +
\sup_{ x \in H_{ \rho } }
\frac{
\|
\varphi'( \tilde{R} e^{ A t } x )
(R - \tilde{R})
e^{ A t }
\|_{
L( H_{ \alpha }, V )
}
}{
\big(
1 +
\| x \|_{ H_{ \rho } }
\big)^{ (q-1) }
}
\\ & \leq
\| \varphi'' \|_{
L^{ \infty }( \tilde{H},
L^{ (2) }( \tilde{H}, V )
)
}
\,
\| R \|_{
L( H_{ \rho }, \tilde{H} )
}
\,
\| R - \tilde{R} \|_{
L( H_{ r }, \tilde{H} )
}
\,
\underbrace{
\| e^{ A t } \|_{
L( H_{ \alpha }, H_{ \rho } )
}
\,
\| e^{ A t } \|_{
L( H_{ \rho }, H_r )
}
}_{
\leq
\left(
\kappa_{ 0, t }
\right)^2
t^{
\left(
\alpha - r
\right)
}
}
\\ & \quad +
\| \varphi' \|_{
L^{ \infty }( \tilde{H},
L( \tilde{H}, V )
)
}
\,
\| R - \tilde{R} \|_{
L( H_{ r }, \tilde{H} )
}
\,
\underbrace{
\| e^{ A t } \|_{
L( H_{ \alpha }, H_{ r } )
}
}_{
\leq
\kappa_{ 0, t } \,
t^{
\left(
\alpha - r
\right)
}
}
\end{split}
\end{equation}
and
\begin{equation}
\label{eq:weakest3}
\begin{split}
&
\|
( K_t \psi )''
\|_{
G_{q-2}^0( H_{ \rho },
L^{(2)}( H_{ \beta }, V )
)
}
\\ & \leq
\sup_{ x \in H_{ \rho } }
\sup_{
\substack{
\| v \|_{ H_{ \beta } } =
\| w \|_{ H_{ \beta } } = 1
}
}
\frac{
\big\|
\big( \varphi''( R e^{ A t } x ) -
\varphi''( \tilde{R} e^{ A t } x ) \big)(
R e^{ A t } v,
R e^{ A t } w
)
\big\|_{
V
}
}{
\big(
1 +
\| x \|_{ H_{ \rho } }
\big)^{ (q-2) }
}
\\ & \quad +
\sup_{ x \in H_{ \rho } }
\sup_{
\substack{
\| v \|_{ H_{ \beta } } =
\| w \|_{ H_{ \beta } } = 1
}
}
\frac{
\big\|
\varphi''( \tilde{R} e^{ A t } x )\big(
(R - \tilde{R})
e^{ A t } v,
R e^{ A t } w
\big)
\big\|_{
V
}
}{
\big(
1 +
\| x \|_{ H_{ \rho } }
\big)^{ (q - 2) }
}
\\ & \quad +
\sup_{ x \in H_{ \rho } }
\sup_{
\substack{
\| v \|_{ H_{ \beta } } =
\| w \|_{ H_{ \beta } } = 1
}
}
\frac{
\big\|
\varphi''( \tilde{R} e^{ A t } x )\big(
\tilde{R} e^{ A t } v ,
(R - \tilde{R})
e^{ A t } w
\big)
\big\|_{
V
}
}{
\big(
1 +
\| x \|_{ H_{ \rho } }
\big)^{ (q - 2) }
}
\\ & \leq
\left[
\sup_{
\substack{
x, y \in \tilde{H}
\\
x \neq y
}
}
\frac{
\| \varphi''(x) - \varphi''(y) \|_{
L^{ (2) }( \tilde{H}, V )
}
}{
\| x - y \|_{ \tilde{H} }
}
\right]
\| R \|_{
L( H_{ \rho }, \tilde{H} )
}^2
\,
\| R - \tilde{R} \|_{
L( H_{ r }, \tilde{H} )
}
\,
\underbrace{
\| e^{ A t } \|_{
L( H_{ \beta }, H_{ \rho } )
}^2
\,
\| e^{ A t } \|_{
L( H_{ \rho }, H_r )
}
}_{
\leq
\left(
\kappa_{
0, t
}
\right)^3
t^{
\left(
2 \beta - \rho - r
\right)
}
}
\\ & \quad +
\| \varphi'' \|_{
L^{ \infty }( \tilde{H},
L^{ (2) }( \tilde{H}, V )
)
}
\left[
\| R \|_{
L( H_{ \rho }, \tilde{H} )
}
+
\| \tilde{R} \|_{
L( H_{ \rho }, \tilde{H} )
}
\right]
\| R - \tilde{R} \|_{
L( H_{ r }, \tilde{H} )
}
\,
\underbrace{
\| e^{ A t } \|_{
L( H_{ \beta }, H_{ \rho } )
}
\,
\| e^{ A t } \|_{
L( H_{ \beta }, H_{ r } )
}
}_{
\leq
\left(
\kappa_{
0, t
}
\right)^2
t^{
\left(
2 \beta - \rho - r
\right)
}
}
\end{split}
\end{equation}
for all $ q \in [3,\infty) $,
$ t \in (0,T] $,
$
\delta \in [\gamma,\infty)
$
and all
$
r \in
[
\rho,
\min( 1 + \alpha ,
1 + 2 \beta - \rho
)
)
$.
Combining
\eqref{eq:weakest1}--\eqref{eq:weakest3}
implies
\begin{equation}
\label{eq:specialnorm_est}
\begin{split}
&
\left\|
\psi
\right\|_{
t, q - 2
}^{
\delta, \rho
}
\leq
\| \varphi \|_{
C^2_{ Lip }( \tilde{H}, V )
}
\,
\| R - \tilde{R} \|_{
L( H_{ r }, \tilde{H} )
}
\left[
\kappa_{
\max( \delta - r, 0 ), t
}
\right]^3
\big[
1 +
\| R \|_{ L( H_{ \rho }, \tilde{H} )
} +
\| \tilde{R}
\|_{ L( H_{ \rho }, \tilde{H} )
}
\big]^2
\\ & \quad
\cdot
\left(
t^{ \min( \delta - r, 0 ) }
+
\int_0^t
\left( t - s \right)^{
\min( \delta - \rho, 0 )
}
\max\!\left(
s^{
\left(
\alpha - r
\right)
}
,
s^{
\left(
2 \beta - \rho - r
\right)
}
\right)
ds
\right)
\\ & \leq
\| \varphi \|_{
C^2_{ Lip }( \tilde{H}, V )
}
\,
\| R - \tilde{R} \|_{
L( H_{ r }, \tilde{H} )
}
\left[
\kappa_{
\max( \delta - r, 0 ), t
}
\right]^3
\big[
1 +
\| R \|_{ L( H_{ \rho }, \tilde{H} )
} +
\| \tilde{R}
\|_{ L( H_{ \rho }, \tilde{H} )
}
\big]^2
\max\!\big(
1, t^{
| 2 \beta - \rho - \alpha |
}
\big)
\\ & \quad
\cdot
\left(
t^{ \min( \delta - r, 0 ) }
+
t^{
\left[
\min( \delta - \rho, 0 )
+
\min\left(
\alpha ,
2 \beta - \rho
\right)
+ 1 - r
\right]
}
\int_0^1
\left( 1 - s \right)^{
\min( \delta - \rho, 0 )
}
s^{
\min\left(
\alpha - r ,
2 \beta - \rho - r
\right)
}
\, ds
\right)
\\ & \leq
\| \varphi \|_{
C^2_{ Lip }( \tilde{H}, V )
}
\,
\| R - \tilde{R} \|_{
L( H_{ r }, \tilde{H} )
}
\left[
\kappa_{
\max( \delta - r, 0 )
}
\right]^3
\big[
1 +
\| R \|_{ L( H_{ \rho }, \tilde{H} )
} +
\| \tilde{R}
\|_{ L( H_{ \rho }, \tilde{H} )
}
\big]^2
\max( t, 1 )
\\ & \quad
\cdot
t^{ \min( \delta - r, 0 ) }
\left(
1
+
\int_0^1
\left( 1 - s \right)^{
\min( \delta - \rho, 0 )
}
s^{
\min\left(
\alpha - r ,
2 \beta - \rho - r
\right)
}
\, ds
\right)
\end{split}
\end{equation}
for all
$ t \in (0,T] $,
$ q \in [3,\infty) $,
$
\delta \in [\gamma, \infty)
$
and all
$
r \in
[
\rho,
\min( 1 + \alpha ,
1 + 2 \beta - \rho
)
)
$.
Next observe that
Theorem~\ref{thm:continuity}
implies that
\begin{equation}
\label{eq:theorem_reg_cons}
\begin{split}
\|
P_t( \psi )
\|_{
G^0_{ q }( H_{ \delta }, V )
}
& \leq
\max\!\left(
1,
\| F \|_{
G^0_1( H_{ \rho }, H_{ \alpha } )
}
,
\| B \|_{
G^0_1( H_{ \rho },
HS( U_0, H_{ \beta } ) )
}^2
\right)
\\ & \quad
\cdot
\max\!
\left(
1,
\sup_{ s \in (0, t) }
\left[
s^{ \max( \rho - \delta, 0 ) }
\|
P_s
\|_{
L(
G^0_{ q }( H_{ \rho }, V ) ,
G^0_{ q }( H_{ \delta }, V )
)
}
\right]
\right)
\cdot
\left\|
\psi
\right\|_{
t, q - 2
}^{
\delta, \rho
}
\end{split}
\end{equation}
for all
$ t \in (0,T] $,
$ q \in [3,\infty) $
and all
$
\delta \in [\gamma, \infty)
$.
Combining
\eqref{eq:specialnorm_est}
and \eqref{eq:theorem_reg_cons}
then shows \eqref{eq:weak_crucial}.
Inequality~\eqref{eq:weak_crucial}
implies \eqref{eq:weak_crucialB}.
This completes the proof
of Corollary~\ref{cor:weakest}.
\end{proof}
In the remainder of this subsection,
Corollary~\ref{cor:weakest}
is illustrated
by three simple consequences
(Corollary~\ref{cor:spatial},
Corollary~\ref{cor:temporal}
and
Corollary~\ref{cor:galerkin}).
Corollary~\ref{cor:spatial}
follows immediately
from inequality~\eqref{eq:weak_crucialB}
in Corollary~\ref{cor:weakest}
and its proof is therefore
omitted.
\begin{cor}[Spatial
weak semigroup regularity]
\label{cor:spatial}
Assume that the setting
in Section~\ref{sec:setting}
is fulfilled and
assume
$ \alpha \leq \gamma $
and
$ \beta \leq \gamma $.
Then
\begin{equation}
\label{eq:weakspatialsemigroup1}
\sup_{
\substack{
\varphi \in
C^2_{ Lip }( H_{ \rho }, V )
\backslash \{ 0 \}
}
}
\sup_{
t \in (0,T]
}
\sup_{
h \in (0,T]
}
\left(
\frac{
t^{
\max( \rho - \delta + r, 0 )
} \,
\|
P_{ t }( K_h( \varphi ) )
-
P_{ t }( \varphi )
\|_{
G^0_3( H_{ \delta }, V )
}
}{
h^{ r } \,
\| \varphi \|_{
C^2_{ Lip }( H_{ \rho }, V )
}
}
\right)
< \infty
\end{equation}
for all
$
\delta \in [\gamma, \infty)
$,
$
r \in
[0,1+ \alpha - \rho) \cap
[0,1+2 \beta - 2 \rho)
$
and all
$
\rho \in
[
\gamma,
\alpha + 1
)
\cap
[
\gamma,
\beta + \frac{ 1 }{ 2 }
)
$.
In particular, if
the real number
$ p \in [2,\infty) $
in Assumption~\ref{initial}
satisfies
$ p \geq 3 $,
then
\begin{equation}
\sup_{
\substack{
\varphi \in
C^2_{ Lip }( H_{ \gamma }, V )
\backslash \{ 0 \}
}
}
\sup_{
t \in [0,T]
}
\sup_{
h \in (0,T]
}
\left(
\frac{
t^{ r } \,
\big\|
\mathbb{E}\big[
\varphi( e^{ A h } X_{ t } )
\big]
-
\mathbb{E}\big[
\varphi( X_{ t } )
\big]
\big\|_V
}{
h^r \,
\| \varphi \|_{
C^2_{ Lip }( H_{ \gamma }, V )
}
}
\right)
< \infty
\end{equation}
for all
$
r \in
[0,1+ \alpha - \gamma) \cap
[0,1+2 \beta - 2 \gamma)
$.
\end{cor}
\begin{cor}[Temporal
weak regularity]
\label{cor:temporal}
Assume that the setting
in Section~\ref{sec:setting}
is fulfilled and
assume
$ \alpha \leq \gamma $
and
$ \beta \leq \gamma $.
Then
\begin{equation}
\sup_{
\substack{
t_1, t_2 \in (0,T]
\\
t_1 < t_2
}
}
\left(
\frac{
| t_1 |^{
\max( \rho - \delta + r, 0 )
}
\,
\|
P_{ t_2 }
-
P_{ t_1 }
\|_{
L( C^2_{ Lip }( H_{ \rho }, V ),
G^0_3( H_{ \delta }, V ) )
}
}{
| t_2 - t_1 |^r
}
\right)
< \infty
\end{equation}
for all
$
\delta \in [\gamma, \infty)
$,
$
r \in
[0,1+ \alpha - \rho) \cap
[0,1+2 \beta - 2 \rho)
$
and all
$
\rho \in
[
\gamma,
\alpha + 1
)
\cap
[
\gamma,
\beta + \frac{ 1 }{ 2 }
)
$.
In particular, if
the real number
$ p \in [2,\infty) $
in Assumption~\ref{initial}
satisfies
$ p \geq 3 $,
then
\begin{equation}
\sup_{
\substack{
\varphi \in
C^2_{ Lip }( H_{ \gamma }, V )
\backslash \{ 0 \}
}
}
\sup_{
\substack{
t_1, t_2 \in [0,T]
\\
t_1 \neq t_2
}
}
\left(
\frac{
|t_1|^{ r } \,
\big\|
\mathbb{E}\big[
\varphi( X_{ t_2 } )
\big]
-
\mathbb{E}\big[
\varphi( X_{ t_1 } )
\big]
\big\|_V
}{
| t_2 - t_1 |^r \,
\| \varphi \|_{
C^2_{ Lip }( H_{ \gamma }, V )
}
}
\right)
< \infty
\end{equation}
for all
$
r \in
[0,1+ \alpha - \gamma) \cap
[0,1+2 \beta - 2 \gamma)
$.
\end{cor}
\begin{proof}[Proof
of Corollary~\ref{cor:temporal}]
First, define real
numbers
$
c_{ \rho, \delta, r } \in [0,\infty)
$,
$
r \in
[0,1+ \alpha - \rho) \cap
[0,1+2 \beta - 2 \rho)
$,
$
\delta \in [\gamma, \infty)
$,
$
\rho \in
[
\gamma,
\alpha + 1
)
\cap
[
\gamma,
\beta + \frac{ 1 }{ 2 }
)
$,
through
\begin{equation}
c_{ \rho, \delta, r }
:=
\sup_{
\substack{
\varphi \in
C^2_{ Lip }( H_{ \rho }, V )
\backslash \{ 0 \}
}
}
\sup_{
t \in (0,T]
}
\sup_{
h \in (0,T]
}
\left(
\frac{
t^{
\max( \rho - \delta + r, 0 )
} \,
\|
P_{ t }( K_h( \varphi ) )
-
P_{ t }( \varphi )
\|_{
G^0_3( H_{ \delta }, V )
}
}{
h^{ r } \,
\| \varphi \|_{
C^2_{ Lip }( H_{ \rho }, V )
}
}
\right)
< \infty
\end{equation}
for all
$
r \in
[0,1+ \alpha - \rho) \cap
[0,1+2 \beta - 2 \rho)
$,
$
\delta \in [\gamma, \infty)
$
and all
$
\rho \in
[
\gamma,
\alpha + 1
)
\cap
[
\gamma,
\beta + \frac{ 1 }{ 2 }
)
$.
Corollary~\ref{cor:spatial}
shows that these real numbers
are indeed finite.
In the next step we combine
\eqref{eq:mildP2}
and the definition
of
$
c_{ \rho, \delta, r } \in [0,\infty)
$
to obtain that
\begin{equation}
\label{eq:esttemporal}
\begin{split}
&
\frac{
| t_1
|^{
\max\left(
\rho - \delta + r, 0
\right)
}
\left\|
(
P_{ t_2 } \varphi
)( x )
-
(
P_{ t_1 } \varphi
)( x )
\right\|_{ V }
}{
| t_2 - t_1 |^r
}
\\ & \leq
\frac{
| t_1
|^{
\max\left(
\rho - \delta + r, 0
\right)
}
\left\|
(
P_{ t_1 } K_{ ( t_2 - t_1 ) }
\varphi
)(x)
-
(
P_{ t_1 } \varphi
)( x )
\right\|_V
}{
| t_2 - t_1 |^r
}
+
\int_{ t_1 }^{ t_2 }
\left[
\frac{
s^{
\max\left(
\rho - \delta + r, 0
\right)
}
\,
\|
(
P_s \,
L^{ (0) }_{ ( t_2 - s ) } \varphi
) ( x )
\|_V
}{
| t_2 - t_1 |^r
}
\right]
ds
\\ & \leq
c_{ \rho, \delta, r }
\left[
1 + \| x \|_{ H_{ \delta } }
\right]^{ 3 }
\|
\varphi
\|_{
C^2_{ Lip }( H_{ \rho }, V )
}
+
\left( 1 + T \right)
\int_{ t_1 }^{ t_2 }
\left[
\frac{
s^{
\max\left(
\rho - \delta, 0
\right)
}
\,
\|
(
P_s \,
L^{ (0) }_{ ( t_2 - s ) } \varphi
) ( x )
\|_V
}{
| t_2 - t_1 |^{
\min(
1 + \alpha - \rho,
1 + 2 \beta - 2 \rho
)
}
}
\right]
ds
\\ & \leq
c_{ \rho, \delta, r }
\left[
1 + \| x \|_{ H_{ \delta } }
\right]^{ 3 }
\|
\varphi
\|_{
C^2_{ Lip }( H_{ \rho }, V )
}
\\ & \quad +
\left( 1 + T \right)
\left[
\sup_{
t \in (0,T]
}
\sup_{
s \in (0, t)
}
\left(
s^{
\max\left(
\rho - \delta, 0
\right)
}
\,
| t - s |^{
\max(
\rho - \alpha,
2 \rho - 2 \beta
)
}
\,
\|
(
P_s \,
L^{ (0) }_{ t - s } \varphi
) ( x )
\|_V
\right)
\right]
\end{split}
\end{equation}
for all $ t_1, t_2 \in (0,T] $
with $ t_1 < t_2 $,
$
x \in H_{ \delta }
$,
$
r \in
[0,1 + \alpha - \rho) \cap
[0,1 + 2 \beta - 2 \rho)
$,
$
\delta \in [\gamma, \infty)
$,
$
\varphi \in
C^2_{ Lip }( H_{ \rho }, V )
$
and all
$
\rho \in
[
\gamma,
\alpha + 1
)
\cap
[
\gamma,
\beta + \frac{ 1 }{ 2 }
)
$.
Furthermore, observe that
Lemma~\ref{kolm:bound0}
shows that
\begin{equation}
\label{eq:esttemporal2}
\begin{split}
&
\sup_{ s \in (0,t) }
\left[
s^{
\max( \rho - \delta , 0 )
}
\left( t - s \right)^{
\max( \rho - \alpha, 2 \rho - 2 \beta )
}
\,
\|
P_{ s } (
L^{ (0) }_{ t - s }( \varphi ) )
\|_{
G^0_3( H_{ \delta }, V )
}
\right]
\\ & \leq
\sup_{ s \in (0, t) }
\left[
s^{
\max( \rho - \delta , 0 )
}
\|
P_{ s }
\|_{
L(
G^0_3( H_{ \rho }, V ) ,
G^0_3( H_{ \delta }, V )
)
}
\left( t - s \right)^{
\max( \rho - \alpha, 2 \rho - 2 \beta )
}
\|
L^{ (0) }_{ t - s }( \varphi ) )
\|_{
G^0_3( H_{ \delta }, V )
}
\right]
\\ & \leq
\left[
\sup_{ s \in (0, t) }
s^{
\max( \rho - \delta , 0 )
}
\|
P_{ s }
\|_{
L(
G^0_3( H_{ \rho }, V ) ,
G^0_3( H_{ \delta }, V )
)
}
\right]
\max\!\big(
\| F \|_{
G^0_1( H_{ \rho }, H_{ \alpha } )
}
,
\| B \|_{
G^0_1( H_{ \rho },
HS( U_0, H_{ \beta } ) )
}^2
\big)
\\ & \quad
\cdot
\left[
\sup_{ s \in (0,t) }
s^{
\max( \rho - \alpha, 2 \rho - 2 \beta )
}
\Big(
\|
( K_s \varphi )'
\|_{
G_{ 2 }^0( H_{ \rho },
L( H_{ \alpha }, V )
)
}
+
\|
( K_s \varphi )''
\|_{
G_{ 1 }^0( H_{ \rho },
L^{(2)}( H_{ \beta }, V )
)
}
\Big)
\right]
\\ & \leq
\left[
\sup_{ s \in (0, T] }
s^{
\max( \rho - \delta , 0 )
}
\|
P_{ s }
\|_{
L(
G^0_3( H_{ \rho }, V ) ,
G^0_3( H_{ \delta }, V )
)
}
\right]
\max\!\big(
\| F \|_{
G^0_1( H_{ \rho }, H_{ \alpha } )
}
,
\| B \|_{
G^0_1( H_{ \rho },
HS( U_0, H_{ \beta } ) )
}^2
\big)
\\ & \quad
\cdot
\max\!\left(
1,
\sup_{ s \in [0,T] }
\| e^{ A s } \|_{ L(H) }^2
\right)
\\ & \quad
\cdot
\left[
\sup_{ s \in (0,T] }
s^{
\max( \rho - \alpha, 2 \rho - 2 \beta )
}
\left(
\|
e^{ A s }
\|_{ L(H_{ \alpha }, H_{ \rho }) }
\,
\|
\varphi'
\|_{
G^0_2(
H_{ \rho }, L( H_{ \rho }, V)
)
}
+
\|
e^{ A s }
\|_{ L( H_{ \beta }, H_{ \rho } )
}^2
\,
\|
\varphi''
\|_{
G^0_1(
H_{ \rho },
L^{(2)}( H_{ \rho }, V)
)
}
\right)
\right]
\\ & \leq
\left[
\sup_{ s \in (0, T] }
s^{
\max( \rho - \delta , 0 )
}
\,
\|
P_{ s }
\|_{
L(
G^0_3( H_{ \rho }, V ) ,
G^0_3( H_{ \delta }, V )
)
}
\right]
\max\!\big(
\| F \|_{
G^0_1( H_{ \rho }, H_{ \alpha } )
}
,
\| B \|_{
G^0_1( H_{ \rho },
HS( U_0, H_{ \beta } ) )
}^2
\big)
\\ & \quad
\cdot
\max\!\left(
1,
\sup_{ s \in [0,T] }
\| e^{ A s } \|_{ L(H) }^2
\right)
\left(
\|
\varphi'
\|_{
G^0_2(
H_{ \rho }, L( H_{ \rho }, V)
)
}
+
\|
\varphi''
\|_{
G^0_1(
H_{ \rho },
L^{(2)}( H_{ \rho }, V)
)
}
\right)
\\ & \quad
\cdot
\left[
\sup_{ s \in (0,T] }
s^{
\max( \rho - \alpha, 2 \rho - 2 \beta )
}
\max\!
\big(
\|
e^{ A s }
\|_{
L( H_{ \alpha }, H_{ \rho } )
}
,
\|
e^{ A s }
\|_{
L( H_{ \beta }, H_{ \rho } )
}^2
\big)
\right]
\end{split}
\end{equation}
for all
$
t \in (0,T]
$,
$
\delta \in [\gamma, \infty)
$,
$
\varphi \in
G^2_{ 1 }( H_{ \rho }, V )
$
and all
$
\rho \in
[
\gamma,
\alpha + 1
)
\cap
[
\gamma,
\beta + \frac{ 1 }{ 2 }
)
$.
Combining \eqref{eq:esttemporal}
and \eqref{eq:esttemporal2}
completes the proof of
Corollary~\ref{cor:temporal}.
\end{proof}
\begin{cor}[Galerkin
approximations]
\label{cor:galerkin}
Assume that the setting
in Section~\ref{sec:setting}
is fulfilled,
assume
$ \alpha \leq \gamma $
and
$ \beta \leq \gamma $,
let
$
\rho \in
[
\gamma,
\alpha + 1
)
\cap
[
\gamma,
\beta + \frac{ 1 }{ 2 }
)
$
and
$
\delta \in [\gamma, \infty)
$
be real numbers
and let
$ P_N \in L(H_{ \rho }) $,
$ N \in \N $,
be a sequence of bounded
linear operators
with
$
\sup_{ N \in \N }
\| P_N \|_{ L( H_{ \rho } ) }
< \infty
$.
Then
\begin{equation}
\sup_{
\substack{
\varphi \in
C^2_{ Lip }( H_{ \rho }, V )
\backslash \{ 0 \}
}
}
\sup_{
N \in \mathbb{N}
}
\sup_{
t \in (0,T]
}
\left(
\frac{
t^{
\max( \rho - \delta + r, 0 )
} \,
\|
P_{ t }( \varphi ) -
P_{ t }( \varphi( P_N( \cdot ) ) )
\|_{
G^0_3( H_{ \delta }, V )
}
}{
\| I - P_N \|_{
L( H_{ \rho + r }, H_{ \rho } )
}
\,
\| \varphi \|_{
C^2_{ Lip }( H_{ \rho }, V )
}
}
\right)
< \infty
\end{equation}
for all
$
r \in
[0,1+ \alpha - \rho) \cap
[0,1+2 \beta - 2 \rho)
$.
In particular, if
$
( \lambda_n )_{ n \in \mathbb{N} }
\subset (0,\infty)
$
is a non-decreasing
sequence
of real numbers,
if
$
( e_n )_{ n \in \mathbb{N} }
\subset H
$
is an orthonormal basis
of $ H $ with
$
D(A) = \{
v \in H \colon
\sum_{ n = 1 }^{ \infty }
| \lambda_n |^2
| \left< e_n, v \right>_H |^2
< \infty
\}
$
and
$
A v =
\sum_{ n = 1 }^{ \infty }
- \lambda_n \left< e_n, v \right>_H
e_n
$
for all $ v \in D(A) $,
if $ \rho = \gamma = 0 $
and $ p \geq 3 $
and if
$
P_N( v )
=
\sum_{ n = 1 }^N
\left< e_n, v \right>_H
e_n
$
for all $ v \in H $,
$ N \in \mathbb{N} $,
then
\begin{equation}
\sup_{
\substack{
\varphi \in
C^2_{ Lip }( H, V )
\backslash \{ 0 \}
}
}
\sup_{
N \in \mathbb{N}
}
\sup_{
t \in (0,T]
}
\left(
\frac{
t^{
\max( r - \delta, 0 )
} \,
\|
P_{ t }( \varphi ) -
P_{ t }( \varphi( P_N( \cdot ) ) )
\|_{
G^0_3( H_{ \delta }, V )
}
}{
( \lambda_N )^{ - r }
\,
\| \varphi \|_{
C^2_{ Lip }( H, V )
}
}
\right)
< \infty
\end{equation}
and
\begin{equation}
\sup_{
\substack{
\varphi \in
C^2_{ Lip }( H, V )
\backslash \{ 0 \}
}
}
\sup_{
N \in \mathbb{N}
}
\sup_{
t \in (0,T]
}
\left(
\frac{
t^r \,
\big\|
\mathbb{E}\big[
\varphi( X_t )
\big]
-
\mathbb{E}\big[
\varphi( P_N( X_t ) )
\big]
\big\|_V
}{
( \lambda_N )^{-r} \,
\| \varphi \|_{
C^2_{ Lip }( H, V )
}
}
\right)
< \infty
\end{equation}
for all
$
r \in
[0,1+ \alpha) \cap
[0,1+2 \beta)
$.
\end{cor}
Corollary~\ref{cor:galerkin}
follows directly from
Corollary~\ref{cor:weakest}
and its proof
is therefore
omitted.
Corollary~\ref{cor:galerkin}
is a certain spatial weak numerical
approximation result for SPDEs.
Further weak numerical
approximation results
for SPDEs
can be found in
\cite{h03b,dd06,dp09,gkl09,h10c,ls10,d11,kll11,KovacsLarssonLindgren2012,Brehier2012,Kruse2012}.
\subsubsection{Stochastic Taylor
expansions for solutions of SPDEs}
\label{sec:tay}
A further application of the mild
It\^{o} formula~\eqref{eq:itoformel}
is the derivation of stochastic
Taylor expansions for solutions
of stochastic partial differential
equations.
In Kloeden \citationand\
Platen~\cite{kp92}
stochastic Taylor expansions
are derived for solutions of
finite dimensional
stochastic ordinary differential
equations by an iterated application
of the standard It\^{o} formula.
Clearly, this strategy can not be
accomplished in the infinite dimensional
SPDE setting since the standard It\^{o}
formula can, in general,
not be applied to
the solution process of an SPDE.
However, by using the
mild It\^{o}
formula~\eqref{eq:itoformel}
instead of the standard It\^{o}
formula, this approach can
be generalized to solutions
of SPDEs in a
straightforward way.
The main difference
to the finite dimensional setting in
Kloeden \citationand\
Platen~\cite{kp92}
is that the linear
operators
$ L^{ (0) }_t $, $ t \in (0,T] $,
and
$ L^{ (1) }_t $, $ t \in (0,T] $,
in \eqref{eq:defL0}
and \eqref{eq:defL1}
here
depend explicitly
on the time variable
$ t \in (0,T] $ too
(compare \eqref{eq:defL0}
and \eqref{eq:defL1} here
with (1.13) and (1.14)
in Chapter~5 in \cite{kp92};
see also
Theorem~\ref{thm:strongtay}
and Theorem~\ref{thm:weaktay}
below for more details).
Similar and related
stochastic Taylor expansions for SPDEs
can be found
in Buckdahn \citationand\
Ma~\cite{bm02},
Bayer \citationand\
Teichmann~\cite{bt08},
Conus~\cite{c08},
Jentzen \citationand\
Kloeden~\cite{jk09c},
Buckdahn,
Bulla \citationand\
Ma~\cite{bbm10}
and
Jentzen~\cite{j10}.
For formulating the
stochastic
Taylor expansions below
some notations
are introduced
(see also Chapter~5
in \cite{kp92}).
By
$
\mathcal{M}
:= \{ \emptyset \}
\cup
\left(
\cup_{ n = 1 }^{ \infty }
\{ 0, 1 \}^n
\right)
$
the set of
multi-indices is denoted.
Moreover, define
two functions
$
\left| \cdot \right|
\colon \mathcal{M}
\rightarrow \{ 0, 1, 2, \dots \}
$
and
$
-( \cdot ) \colon
\mathcal{M} \backslash
\{ \emptyset \}
\rightarrow \mathcal{M}
$
by
$
| \emptyset | := 0
$,
by
$
| (\alpha_1, \alpha_2,\dots, \alpha_n) |
:= n
$
and by
$
- (\alpha_1, \alpha_2, \dots, \alpha_n)
:=
(\alpha_2, \alpha_3, \dots, \alpha_n)
$
for all
$
\alpha_1, \alpha_2, \dots, \alpha_n \in
\{ 0, 1 \}
$
and all
$
n \in \mathbb{N}
$.
Thus note that
$
| \alpha | \geq 1
$
and
$
\alpha_1, \dots ,
\alpha_{ | \alpha | }
\in \{ 0, 1 \}
$
for all
$
\alpha \in
\mathcal{A}
\backslash
\{ \emptyset \}
$.
Furthermore, a finite
nonempty
subset $ \mathcal{A}
\subset \mathcal{M} $
of $ \mathcal{M} $
is called {\it hierarchical set}
if $ - \alpha \in \mathcal{A} $
for all $ \alpha \in \mathcal{A}
\backslash \{ \emptyset \} $.
Next define a function
$
\mathbb{B} \colon
\mathcal{P}( \mathcal{M} )
\rightarrow
\mathcal{P}( \mathcal{M} )
$
by
$
\mathbb{B}(\mathcal{A})
:=
\{
\alpha \in \mathcal{M}
\backslash \mathcal{A}
\colon
-\alpha \in \mathcal{A}
\}
$ for all
$ \mathcal{A} \subset \mathcal{M} $.
Finally,
let
$
W^0 \colon [0,T]
\rightarrow \mathbb{R}
$
be a function
and let
$
( W^1_t )_{ t \in [0,T] }
$
be a cylindrical $ Q $-Wiener process
defined by
$
W^0(t) := t
$
and
$
W^1_t
:=
W_t
$
for all $ t \in [0,T] $.
Using this notation
the mild It\^{o}
formula~\eqref{eq:operatorIto}
can be written as
\begin{equation}
\label{eq:mildito3}
\varphi( X_t )
=
\varphi( e^{ A (t - t_0) } X_{ t_0 } )
+
\sum_{ i = 0 }^{ 1 }
\int_{ t_0 }^t
\!
\big(
L_{ ( t - s ) }^{(i)} \varphi
\big) ( X_s )
\, dW^i_s
\end{equation}
$ \mathbb{P} $-a.s.\ for
all $ t_0, t \in [0,T] $
with $ t_0 \leq t $
and all
$
\varphi \in
C^2( H_{ \gamma }, V )
$.
Moreover, for two normed
$ \mathbb{R} $-vector spaces
and
$ n \in \{ 0, 1, 2, \dots \} $
we define
$
C_b^n( V_1, V_2 )
:=
\left\{
\varphi \in C^n( V_1, V_2 )
\colon
\| \varphi
\|_{
L^{ \infty }(
V_1, V_2
)
}
+
\sum_{ k = 1 }^n
\| \varphi^{ (k) }
\|_{
L^{ \infty }(
V_1, L^{ (k) }(V_1, V_2)
)
}
< \infty
\right\}
$,
$
C_b( V_1, V_2 )
:=
C_b^0( V_1, V_2 )
$
and
$
C^{ \infty }_b( V_1, V_2 )
:=
\cap_{ k \in \mathbb{N} }
C^k_b( V_1, V_2 )
$.
We are now ready to present the
stochastic Taylor expansions
based on the mild It\^{o}
formula~\eqref{eq:mildito3}.
\begin{theorem}[Strong
stochastic Taylor
expansions]
\label{thm:strongtay}
Assume that the setting
in Section~\ref{sec:setting}
is fulfilled,
assume
$
F \in
C^{ \infty }_b(
H_{ \gamma },
H_{ \alpha }
)
$,
assume
$
B \in
C^{ \infty }_b(
H_{ \gamma },
HS( U_0, H_{ \beta } )
)
$
and let
$
\varphi
\in C^{ \infty }_b( H_{ \gamma }, V )
$.
Then
\begin{align}
\label{eq:strongtay}
&
\varphi( X_{ t } )
=
\varphi\big( e^{ A( t - t_0) } X_{ t_0 } \big)
\\&
+
\sum_{
\substack{
\alpha
\in \mathcal{A}
\\
\alpha \neq \emptyset
}
}
\int_{ t_0 }^{ t }
\int_{ t_0 }^{ s_{ |\alpha| } }
\dots
\int_{ t_0 }^{ s_2 }
\Big(
L_{
(
s_2 - s_1
)
}^{
( \alpha_1 )
}
\dots
L_{
(
s_{ |\alpha| } - s_{ |\alpha| - 1}
)
}^{
( \alpha_{ |\alpha| - 1 } )
} \,
L_{
(
t - s_{ |\alpha| }
)
}^{
( \alpha_{ |\alpha| } )
} \,
\varphi
\Big)\big(
e^{
A ( s_1 - t_0 )
}
X_{ t_0 }
\big) \,
dW^{ \alpha_1 }_{ s_1 } \,
dW^{ \alpha_2 }_{ s_2 }
\dots
dW^{ \alpha_{ |\alpha| } }_{
s_{ |\alpha| }
}
\nonumber
\\&+
\sum_{
\alpha
\in \mathbb{B}(\mathcal{A})
}
\int_{ t_0 }^{ t }
\int_{ t_0 }^{ s_{ |\alpha| } }
\dots
\int_{ t_0 }^{ s_2 }
\Big(
L_{
(
s_2 - s_1
)
}^{
( \alpha_1 )
}
\dots
L_{
(
s_{ |\alpha| } - s_{ |\alpha| - 1}
)
}^{
( \alpha_{ |\alpha| - 1 } )
} \,
L_{
(
t - s_{ |\alpha| }
)
}^{
( \alpha_{ |\alpha| } )
} \,
\varphi
\Big)\big(
X_{ s_1 }
\big) \,
dW^{ \alpha_1 }_{ s_1 } \,
dW^{ \alpha_2 }_{ s_2 }
\dots
dW^{ \alpha_{ |\alpha| } }_{
s_{ |\alpha| }
}
\nonumber
\end{align}
for all $ t_0, t \in [0,T] $
with $ t_0 \leq t $ and
all hierarchical sets
$ \mathcal{A} \subset \mathcal{M} $.
\end{theorem}
\begin{proof}[Proof
of Theorem~\ref{thm:strongtay}]
Theorem~\ref{thm:strongtay}
immediately follows from an
iterated application of the
mild It\^{o} formula~\eqref{eq:mildito3}.
\end{proof}
The term
$
\varphi\big(
e^{ A (t-t_0) } X_{ t_0 }
\big)
+
\sum_{
\alpha \in \mathcal{A},
\alpha \neq \emptyset
}
\dots
$,
$ t \in [t_0,T] $,
on the left hand side of
\eqref{eq:strongtay}
is referred as {\it strong stochastic
Taylor approximation}
(or truncated strong stochastic
Taylor expansion) of
$ \varphi( X_t ) $,
$ t \in [t_0,T] $,
corresponding to the
hierarchical set
$ \mathcal{A}
\subset \mathcal{M} $
for $ t_0 \in [0,T] $.
The expression
$
\sum_{
\alpha \in \mathbb{B}(\mathcal{A})
}
\dots
$,
$ t \in [t_0,T] $,
on the left hand side of
\eqref{eq:strongtay}
is called {\it remainder term}
of the strong stochastic
Taylor expansions of
$ \varphi( X_t ) $,
$ t \in [t_0,T] $,
corresponding to the
hierarchical set
$ \mathcal{A}
\subset \mathcal{M} $
for $ t_0 \in [0,T] $.
Next observe that,
in the case
$ H = \mathbb{R}^d $
with $ d \in \mathbb{N} $
and $ A = 0 $,
Theorem~\ref{thm:strongtay}
essentially reduces to
Theorem~5.5.1
in
Kloeden \citationand\
Platen~\cite{kp92}.
Let us also add the following remark
on possible generalizations
of Theorem~\ref{thm:strongtay}.
\begin{remark}
The assumption
in Theorem~\ref{thm:strongtay}
that $ F $, $ B $
and $ \varphi $ are infinitely
often Fr\'{e}chet differentiable
can be relaxed.
To be more precise,
to obtain \eqref{eq:strongtay}
for a given hierarchical
set
$
\mathcal{A} \subset \mathcal{M}
$,
it is sufficient to assume
that
$
F \in C_b( H_{ \gamma }, H_{ \alpha } )
$
is
$
\max_{
\alpha \in \mathbb{B}( \mathcal{A} ),
\alpha_1 = 0
}
\min\big\{
2 k - 2 -
\sum_{ i = 1 }^{ k - 1 }
\alpha_i
\colon
k \in \{ 1, \dots, |\alpha|\},
\alpha_{ k + 1 } =
\ldots =
\alpha_{ | \alpha | } =
1
\big\}
$--times,
that
$
B \in
C_b(
H_{ \gamma },
HS( U_0, H_{ \beta } )
)
$
is
$
\max_{
\alpha \in \mathbb{B}( \mathcal{A} )
}
(
2 | \alpha | - 2
-
\sum_{ i = 1 }^{ | \alpha | - 1 }
\alpha_i
)
$--times
and that
$
\varphi \in
C_b(
H_{ \gamma },
V
)
$
is
$
\max_{
\alpha \in \mathbb{B}( \mathcal{A} )
}
(
2 | \alpha |
-
\sum_{ i = 1 }^{ | \alpha | }
\alpha_i
)
$--times
continuously
Fr\'{e}chet
differentiable
with globally
bounded
Fr\'{e}chet derivatives.
Moreover,
the boundedness
assumptions on $ F, B $
and $ \varphi $ and its derivatives
can be reduced
if $ p \in [2,\infty) $
in Assumption~\ref{initial}
is assumed
to be sufficiently large.
\end{remark}
In the next step
Theorem~\ref{thm:strongtay}
is illustrated with two possible
examples.
First, in the case of the
hierarchical set
$ \mathcal{A} = \{ \emptyset \} $,
equation~\eqref{eq:strongtay}
reduces to
\eqref{eq:mildito3}, i.e., we have
\begin{equation}
\varphi( X_t )
=
\underbrace{
\varphi( e^{ A (t - t_0) } X_{ t_0 } )
}_{
\substack{
\text{strong stochastic Taylor}
\\
\text{approximation corresponding}
\\
\text{to the hierarchical set }
\mathcal{A} = \{ \emptyset \}
}
}
+
\underbrace{
\sum_{ i = 0 }^{ 1 }
\int_{ t_0 }^t
\!
\big(
L_{ ( t - s ) }^{(i)} \varphi
\big) ( X_s )
\, dW^i_s
}_{
\substack{
\text{remainder term corresponding}
\\
\text{to the hierarchical set }
\mathcal{A} = \{ \emptyset \}
}
}
\end{equation}
$ \mathbb{P} $-a.s.\ for
all $ t_0, t \in [0,T] $
with $ t_0 \leq t $
and all
$
\varphi \in
C^{ \infty }_b( H_{ \gamma }, V )
$.
Second, in the case of the
hierarchical set
$
\mathcal{A}
=
\{
\emptyset,
(1)
\}
$,
equation~\eqref{eq:strongtay}
simplifies to
\begin{equation}
\begin{split}
\varphi( X_t )
& =
\underbrace{
\varphi( e^{ A (t - t_0) } X_{ t_0 } )
+
\int_{ t_0 }^t
\varphi'(
e^{ A (t - t_0) } X_{ t_0 }
)
\, e^{ A (t - s) }
B(
e^{ A (s - t_0) } X_{ t_0 }
)
\, dW_s
}_{
\text{strong stochastic Taylor
approximation corresponding
to }
\mathcal{A} = \{ \emptyset, (1) \}
}
\\ & \quad
\underbrace{
\begin{split}
&
+
\int_{ t_0 }^t
\big(
L_{ ( t - s ) }^{(0)} \varphi
\big) ( X_s )
\,
ds
+
\int_{ t_0 }^t
\int_{ t_0 }^s
\big(
L_{ ( s - u ) }^{(0)}
L_{ ( t - s ) }^{(1)} \varphi
\big) ( X_u )
\,
du
\,
dW_s
\\&
+
\int_{ t_0 }^t
\int_{ t_0 }^s
\big(
L_{ ( s - u ) }^{(1)}
L_{ ( t - s ) }^{(1)} \varphi
\big) ( X_u )
\,
dW_u
\,
dW_s
\end{split}
}_{
\text{remainder term corresponding
to }
\mathcal{A} = \{ \emptyset, (1) \}
}
\end{split}
\end{equation}
$ \mathbb{P} $-a.s.\ for
all $ t_0, t \in [0,T] $
with $ t_0 \leq t $
and all
$
\varphi \in
C^{ \infty }_b( H_{ \gamma }, V )
$.
After having presented strong
stochastic Taylor expansions
in Theorem~\ref{thm:strongtay},
we now formula the corresponding
weak stochastic Taylor expansions
based on the mild It\^{o}
formula~\eqref{eq:mildito3}.
\begin{theorem}[Weak
stochastic Taylor
expansions]
\label{thm:weaktay}
Assume that the setting
in Section~\ref{sec:setting}
is fulfilled,
let
$ n \in \mathbb{N} $,
assume
$
F \in
C_b^{ (2n - 2) }( H_{ \gamma },
H_{ \alpha } )
$,
assume
$
B \in
C_b^{ (2n - 2) }( H_{ \gamma },
HS( U_0, H_{ \beta } ) )
$
and let
$
\varphi \in
C_b^{ 2n }( H_{ \gamma }, V )
$.
Then
\begin{align}
\label{eq:weaktay}
&
\nonumber
\mathbb{E}\big[ \varphi( X_{ t } ) \big]
=
\mathbb{E}\big[
\varphi\big( e^{ A (t - t_0) } X_{ t_0 }
\big)
\big]
\\ &
\nonumber
+
\sum_{
k = 1
}^{ n - 1 }
\int_{ t_0 }^{ t }
\int_{ t_0 }^{ s_{ k } }
\dots
\int_{ t_0 }^{ s_2 }
\mathbb{E}\!\left[
\big(
L_{
(
s_2 - s_1
)
}^{
( 0 )
}
\dots
L_{
(
s_{ k } - s_{ k - 1}
)
}^{
( 0 )
} \,
L_{
(
t - s_{ k }
)
}^{
( 0 )
} \,
\varphi
\big)\!\left(
e^{
A ( s_1 - t_0 )
}
X_{ t_0 }
\right)
\right]
ds_1 \,
ds_2
\dots
ds_{ k }
\\ & +
\int_{ t_0 }^{ t }
\int_{ t_0 }^{ s_{ n } }
\dots
\int_{ t_0 }^{ s_2 }
\mathbb{E}\!\left[
\big(
L_{
(
s_2 - s_1
)
}^{
( 0 )
}
\dots
L_{
(
s_{ n } - s_{ n - 1}
)
}^{
( 0 )
} \,
L_{
(
t - s_{ n }
)
}^{
( 0 )
} \,
\varphi
\big)\!\left(
X_{ s_1 }
\right)
\right]
ds_1 \,
ds_2
\dots
ds_{ n }
\end{align}
for all $ t_0, t \in [0,T] $
with $ t_0 \leq t $.
\end{theorem}
\begin{proof}[Proof of Theorem~\ref{thm:weaktay}]
Equation~\eqref{eq:weaktay}
immediately follows by
taking expectations on both
sides of
equation~\eqref{eq:strongtay}
with the hierarchical set
$
\mathcal{A} =
\{
\alpha \in \mathcal{M}
\colon
| \alpha | \leq n,
\sum_{ i = 1 }^{ |\alpha| }
\alpha_i = 0
\}
$.
\end{proof}
Using definition~\eqref{eq:defPt},
the weak stochastic Taylor expansions
in Theorem~\ref{thm:weaktay}
can also be written in the following form.
\begin{cor}
\label{cor:weaktay}
Assume that the setting
in Section~\ref{sec:setting}
is fulfilled,
let
$ n \in \mathbb{N} $,
assume
$
F \in
C_b^{ (2n - 2) }( H_{ \gamma },
H_{ \alpha } )
$,
assume
$
B \in
C_b^{ (2n - 2) }( H_{ \gamma },
HS( U_0, H_{ \beta } ) )
$
and let
$
\varphi \in
C_b^{ 2n }( H_{ \gamma }, V )
$.
Then
\begin{equation}
\label{eq:weaktay2}
\begin{split}
\big( P_t \varphi \big)( x )
&=
\big( K_t \varphi
\big)(x)
\\ & \quad
+
\sum_{
k = 1
}^{ n - 1 }
\int_{ 0 }^{ t }
\int_{ 0 }^{ s_{ k } }
\dots
\int_{ 0 }^{ s_2 }
\big(
K_{ s_1 } \,
L_{
(
s_2 - s_1
)
}^{
( 0 )
}
\dots
L_{
(
s_{ k } - s_{ k - 1}
)
}^{
( 0 )
} \,
L_{
(
t - s_{ k }
)
}^{
( 0 )
} \,
\varphi
\big)\!\left(
x
\right)
ds_1 \,
ds_2
\dots
ds_{ k }
\\ & \quad +
\int_{ 0 }^{ t }
\int_{ 0 }^{ s_{ n } }
\dots
\int_{ 0 }^{ s_2 }
\big(
P_{ s_1 } \,
L_{
(
s_2 - s_1
)
}^{
( 0 )
}
\dots
L_{
(
s_{ n } - s_{ n - 1}
)
}^{
( 0 )
} \,
L_{
(
t - s_{ n }
)
}^{
( 0 )
} \,
\varphi
\big)(
x
) \,
ds_1 \,
ds_2
\dots
ds_{ n }
\end{split}
\end{equation}
for all
$ x \in H_{ \gamma } $
and all
$ t \in [0,\infty) $.
\end{cor}
\subsubsection{Further mild
It\^{o} formulas for
solutions of SPDEs}
This subsection presents
two slightly different
variants
(Corollary~\ref{cor:anotherito1}
and Proposition~\ref{prop:secondito})
of the mild It\^{o} formula
in Corollary~\ref{cor:ito}.
Both variants assume that
the test function $ \varphi $ in
Corollary~\ref{cor:ito}
fulfills additional regularity.
The first variant
(see Corollary~\ref{cor:anotherito1}
below)
is a direct consequence of
Corollary~\ref{cor:ito}.
\begin{cor}[Another -
somehow mild -
It{\^o} type formula
for solutions of SPDEs]
\label{cor:anotherito1}
Assume that
the setting in Section~\ref{sec:setting}
is fulfilled.
Then
\begin{equation}
\label{eq:welldefinedOTHER1}
\mathbb{P}\!\left[
\int_{ t_0 }^t
\big\|
\varphi'\big(
(
I +
e^{ A (t - t_0) } -
e^{ A (t - s) }
)
X_{ t_0 }
\big)
\big\|_{
L( H_r, V )
}
\,
\big\|
A \, e^{ A( t - s ) } \,
X_{ t_0 }
\big\|_{
H_r
}
\,
ds
< \infty
\right] = 1
\end{equation}
and
\begin{equation}
\label{eq:OTHERITO1}
\begin{split}
\varphi( X_t )
&=
\varphi( X_{ t_0 } )
+
\int_{ t_0 }^t
\varphi'\big(
(
I +
e^{ A (t - t_0) } -
e^{ A (t - s) }
)
X_{ t_0 }
\big)
A \, e^{ A( t - s ) } \,
X_{ t_0 } \, ds
\\ & \quad
+
\int_{ t_0 }^t
\varphi'( e^{ A( t - s ) } X_s ) \,
e^{ A( t - s ) }
F( X_s ) \, ds
+
\int_{ t_0 }^t
\varphi'( e^{ A( t - s ) } X_s ) \,
e^{ A( t - s ) }
B( X_s ) \, dW_s
\\&\quad+
\frac{1}{2}
\sum_{ j \in \mathcal{J} }
\int_{ t_0 }^t
\varphi''( e^{ A( t - s ) } X_s )
\big(
e^{ A( t - s ) }
B( X_s ) g_j,
e^{ A( t - s ) }
B( X_s ) g_j
\big) \, ds
\end{split}
\end{equation}
$ \mathbb{P} $-a.s.\ for
all
$ t_0, t \in [ 0, T ] $
with $ t_0 \leq t $,
$
\varphi \in
C^2( H_{ r }, V )
$
and all
$
r \in (-\infty, \gamma )
$.
\end{cor}
\begin{proof}[Proof
of Corollary~\ref{cor:anotherito1}]
Let $ r \in (-\infty,\gamma) $
be a fixed real number
and define
a family
$
\bar{X}^{ t_0, t }
\colon
[t_0,t] \times \Omega
\rightarrow H_{ r }
$,
$
(t_0, t) \in \angle
$,
of adapted stochastic
processes with continuous
sample paths
by
\begin{equation}
\begin{split}
\bar{X}^{ t_0, t }_u
& :=
X_{ t_0 } +
\int_{ t_0 }^u
A \, e^{ A ( t - s ) } X_{ t_0 } \,
ds
=
X_{ t_0 } +
e^{ A (t - u) }
\left(
e^{ A (u - t_0) } - I
\right)
X_{ t_0 }
\\ & =
\left(
I +
e^{ A (t - t_0) } -
e^{ A (t - u) }
\right)
X_{ t_0 }
\end{split}
\end{equation}
for all $ u \in [t_0,t] $
and all $ t_0, t \in [0,T] $
with $ t_0 \leq t $.
The fundamental theorem of
calculus then implies
\begin{equation}
\label{eq:hauptsatz}
\begin{split}
\varphi( e^{ A( t - t_0 ) } X_{ t_0 } )
& =
\varphi\!\left(
X_{ t_0 } +
\int_{ t_0 }^t
A \, e^{ A ( t - s ) } X_{ t_0 } \,
ds
\right)
=
\varphi( \bar{X}^{t_0,t}_t )
\\ & =
\varphi( X_{ t_0 } )
+
\int_{ t_0 }^{ t }
\varphi'( \bar{X}^{ t_0, t }_s ) \, A \,
e^{ A (t - s) } \, X_{ t_0 }
\, ds
\end{split}
\end{equation}
for
all $ t_0, t \in [ 0, T ] $
with $ t_0 \leq t $
and all
$
\varphi \in
C^2( H_{ r }, V )
$.
Combining \eqref{eq:hauptsatz}
and Corollary~\ref{cor:anotherito1}
then completes the proof
of Corollary~\ref{cor:ito}.
\end{proof}
Observe
that
equations~\eqref{eq:well1d}--\eqref{eq:well3d}
and
equation~\eqref{eq:welldefinedOTHER1}
ensure that all deterministic
and stochastic integrals in
\eqref{eq:OTHERITO1}
are well defined.
\begin{prop}[A further -
somehow mild - It{\^o} type
formula for solutions of SPDEs]
\label{prop:secondito}
Assume that
the setting in Section~\ref{sec:setting}
is fulfilled. Then
\begin{equation}
\label{eq:thirdito_well01}
\mathbb{P}\!\left[
\int_{ t_0 }^t
\big\|
\varphi'\big(
X_{ t_0 }
+
e^{ A (t - s) }
(
X_s - X_{ t_0 }
)
\big)
\big\|_{ L( H_r, V ) }
\,
\big\|
A \, e^{ A (t - s) } X_{ t_0 }
\big\|_{ H_r }
\,
ds < \infty
\right] = 1
,
\end{equation}
\begin{equation}
\mathbb{P}\!\left[
\int_{ t_0 }^t
\big\|
\varphi'\big(
X_{ t_0 }
+
e^{ A (t - s) }
(
X_s - X_{ t_0 }
)
\big)
\big\|_{ L( H_r, V ) }
\,
\big\|
e^{ A( t - s ) }
F( X_s )
\big\|_{ H_r }
\,
ds < \infty
\right] = 1
,
\end{equation}
\begin{equation}
\mathbb{P}\!\left[
\int_{ t_0 }^t
\big\|
\varphi'\big(
X_{ t_0 }
+
e^{ A (t - s) }
(
X_s - X_{ t_0 }
)
\big) \,
e^{ A(t-s) } B( X_s )
\big\|_{ HS(U_0, V ) }^2
\,
ds
< \infty
\right] = 1,
\end{equation}
\begin{equation}
\label{eq:thirdito_well04}
\mathbb{P}\!\left[
\int_{ t_0 }^t
\big\|
\varphi''\big(
X_{ t_0 }
+
e^{ A (t - s) }
(
X_s - X_{ t_0 }
)
\big)
\big\|_{ L^{(2)}( H_{ r }, V ) }
\,
\big\|
e^{ A (t - s) } B( X_s )
\big\|_{ HS(U_0, H_{ r }) }^2
\,
ds
< \infty
\right] = 1
\end{equation}
and
\begin{equation}
\label{eq:anotherito}
\begin{split}
\varphi( X_t )
& =
\varphi( X_{ t_0 } )
+
\int_{ t_0 }^t
\varphi'\big(
X_{ t_0 }
+
e^{ A (t - s) }
(
X_s - X_{ t_0 }
)
\big)
\big[
A \, e^{ A (t - s) } X_{ t_0 }
+
e^{ A( t - s ) }
F( X_s )
\big] \, ds
\\ & \quad +
\int_{ t_0 }^t
\varphi'\big(
X_{ t_0 }
+
e^{ A (t - s) }
(
X_s - X_{ t_0 }
)
\big) \,
e^{ A( t - s ) }
B( X_s ) \, dW_s
\\ & \quad +
\frac{1}{2}
\sum_{ j \in \mathcal{J} }
\int_{ t_0 }^t
\varphi''\big(
X_{ t_0 }
+
e^{ A (t - s) }
(
X_s - X_{ t_0 }
)
\big)
\big(
e^{ A( t - s ) }
B( X_s ) g_j,
e^{ A( t - s ) }
B( X_s ) g_j
\big) \, ds
\end{split}
\end{equation}
$ \mathbb{P} $-a.s.\ for
all
$ t_0, t \in [ 0, T ] $
with $ t_0 \leq t $,
$
\varphi \in
C^2( H_{ r }, V )
$
and all
$
r \in (-\infty, \gamma )
$.
\end{prop}
\begin{proof}[Proof
of Proposition~\ref{prop:secondito}]
First, observe that
the well known identity
$
e^{ A t } v =
v +
\int_0^t A \, e^{ A s } v \, ds
$
for all $ v \in H_{ \gamma } $
and all $ t \in [0,\infty) $
shows
\begin{equation}
X_t
=
X_{ t_0 }
+
\int_{ t_0 }^t
\left[
A \,
e^{ A (t - s) }
X_{ t_0 }
+
e^{ A (t - s) } F( X_s )
\right] ds
+
\int_{ t_0 }^t
e^{ A (t - s) } B( X_s ) \, dW_s
\end{equation}
$ \mathbb{P} $-a.s.\ for all
$ t_0, t \in [0,T] $
with $ t_0 \leq t $.
In the next step
let $ r \in (-\infty, \gamma) $
be a fixed real number and
let
$
\bar{X}^{ t_0, t }
\colon [t_0,t] \times
\Omega \rightarrow H_{ r }
$,
$
(t_0, t) \in \angle
$,
be a family
of adapted
stochastic processes with
continuous sample paths
given by
\begin{equation}
\begin{split}
\bar{X}^{t_0, t}_u
& =
X_{ t_0 }
+
\int_{ t_0 }^u
\left[
A \,
e^{ A (t - s) }
X_{ t_0 }
+
e^{ A (t - s) } F( X_s )
\right]
ds
+
\int_{ t_0 }^u
e^{ A (t - s) } B( X_s ) \, dW_s
\\ & =
X_{ t_0 }
+
e^{ A (t - t_0) }
X_{ t_0 }
-
e^{ A (t - u) }
X_{ t_0 }
+
\int_{ t_0 }^u
e^{ A (t - s) } F( X_s ) \,
ds
+
\int_{ t_0 }^u
e^{ A (t - s) } B( X_s ) \, dW_s
\end{split}
\end{equation}
$ \mathbb{P} $-a.s.\ for
all $ u \in [t_0,t] $
and all
$ t_0, t \in [0,T] $
with $ t_0 \leq t $
(see also \eqref{eq:semiX} above).
The standard
It{\^o} formula in infinite
dimensions (see
Theorem~2.4 in Brze\'{z}niak,
Van Neerven, Veraar
\citationand\
Weis~\cite{bvvw08})
then gives
\begin{equation}
\label{eq:itoformel2}
\begin{split}
\varphi( \bar{X}_u^{ t_0, t } )
&=
\varphi( \bar{X}_{ t_0 }^{ t_0, t } )
+
\int_{t_0}^u
\varphi'( \bar{X}_{ s }^{ t_0, t } )
\left[
A \, e^{ A (t - s) } \, X_{ t_0 }
+
e^{ A (t - s) } F( X_s )
\right] ds
\\&\quad
+
\int_{t_0}^u
\varphi'( \bar{X}_{ s }^{ t_0, t } ) \,
e^{ A (t - s) } B( X_s ) \, dW_s
\\ & \quad +
\frac{1}{2}
\sum_{ j \in \mathcal{J} }
\int_{t_0}^u
\varphi''( \bar{X}_{ s }^{ t_0, t } )
\left(
e^{ A (t - s) } B( X_s ) g_j,
e^{ A (t - s) } B( X_s ) g_j
\right) ds
\end{split}
\end{equation}
$ \mathbb{P} $-a.s.\ for
all $ u \in [t_0,t] $,
$ t_0, t \in [0,T] $
with $ t_0 \leq t $
and all
$ \varphi \in C^2( H_{ r }, V ) $.
This, in particular, shows
\begin{equation}
\label{eq:OTHERITOitoformel2}
\begin{split}
\varphi( \bar{X}^{ t_0, t }_t )
=
\varphi( X_t )
&=
\varphi( X_{ t_0 } )
+
\int_{t_0}^t
\varphi'( \bar{X}_{ s }^{ t_0, t } )
\left[
A \, e^{ A (t - s) } \, X_{ t_0 }
+
e^{ A (t - s) } F( X_s )
\right] ds
\\&\quad
+
\int_{t_0}^t
\varphi'( \bar{X}_{ s }^{ t_0, t } ) \,
e^{ A (t - s) } B( X_s ) \, dW_s
\\ & \quad +
\frac{1}{2}
\sum_{ j \in \mathcal{J} }
\int_{t_0}^t
\varphi''( \bar{X}_{ s }^{ t_0, t } )
\left(
e^{ A (t - s) } B( X_s ) g_j,
e^{ A (t - s) } B( X_s ) g_j
\right) ds
\end{split}
\end{equation}
$ \mathbb{P} $-a.s.\ for
all $ t_0, t \in [0,T] $
with $ t_0 \leq t $
and all
$ \varphi \in C^2( H_{ r }, V ) $
(see also \eqref{eq:itoformel2inProof}
above).
Putting the identity
\begin{equation}
\begin{split}
\bar{X}^{ t_0, t }_s
& =
X_{ t_0 }
+
\int_{ t_0 }^s
\left[
A \,
e^{ A (t - u) }
X_{ t_0 }
+
e^{ A (t - u) } F( X_u )
\right]
ds
+
\int_{ t_0 }^s
e^{ A (t - u) } B( X_u ) \, dW_u
\\ & =
X_{ t_0 }
+
e^{ A (t - s) }
\int_{ t_0 }^s
\left[
A \,
e^{ A (s - u) }
X_{ t_0 }
+
e^{ A (s - u) } F( X_u )
\right]
ds
\\ & \quad
+
e^{ A (t - s) }
\int_{ t_0 }^s
e^{ A (s - u) } B( X_u ) \, dW_u
=
X_{ t_0 }
+
e^{ A (t - s) }
\left(
X_s - X_{ t_0 }
\right)
\end{split}
\end{equation}
$ \mathbb{P} $-a.s.\ for
all $ s \in [t_0,T] $
and all
$ t_0, t \in [0,T] $
with $ t_0 \leq t $
(see also \eqref{eq:fact} above)
into \eqref{eq:OTHERITOitoformel2}
finally shows \eqref{eq:anotherito}.
The proof of
Proposition~\ref{prop:secondito}
is thus completed.
\end{proof}
Note
that
equations~\eqref{eq:thirdito_well01}--\eqref{eq:thirdito_well04}
imply that all deterministic
and stochastic integrals in
\eqref{eq:anotherito}
are well defined.
Finally, observe that
the It\^{o} type formulas
in Corollary~\ref{cor:anotherito1}
and Proposition~\ref{prop:secondito}
can be generalized to the more general
case of mild It\^{o} processes
(or mild semimartingales, cf.\ Remark~\ref{rem:moregeneral})
if additional assumptions on the
semigroup are fulfilled.
\subsection{Numerical
approximations processes
for SPDEs}
\label{sec:numerics}
This subsection demonstrates
how different types of numerical
approximation processes for SPDEs
can be formulated
as mild It\^{o} processes.
To this end the following notation is used.
Let
$
\lfloor \cdot \rfloor_N
\colon
[0,T]
\rightarrow
[0,T]
$,
$ N \in \mathbb{N} $,
be a sequence of mappings
given by
\begin{equation}
\lfloor t \rfloor_N
:=
\max\!\Big(
s \in
\left\{
0, \tfrac{T}{N},
\tfrac{2 T}{N}, \dots,
\tfrac{ (N-1) T }{ N }, T
\right\}
\colon
s \leq t
\Big)
\end{equation}
for all $ t \in [0,T] $
and all $ N \in \mathbb{N} $.
Both Euler (see
Subsection~\ref{sec:euler})
and
Milstein (see
Subsection~\ref{sec:mil})
type approximations for SPDEs
are formulated as mild It\^{o}
processes. We begin
with Euler type approximations
for SPDEs in
Subsection~\ref{sec:euler}.
\subsubsection{Euler type
approximations for SPDEs}
\label{sec:euler}
It is illustrated here how
exponential Euler
approximations, accelerated
exponential Euler approximations,
linear implicit Euler approximations
and linear implicit Crank-Nicolson
approximations can be formulated as
mild It\^{o} processes.
\paragraph{Exponential Euler
approximations for SPDEs}
Let
$
Y^N
\colon [0,T]
\times
\Omega
\rightarrow H_{ \gamma }
$,
$
N \in \mathbb{N}
$,
be a sequence of predictable
stochastic
processes given by
\begin{equation}
\label{eq:split0}
\begin{split}
Y^N_t
& =
e^{ A t } \, \xi
+
\int_0^t
e^{
A
\left( t -
\lfloor s \rfloor_N
\right)
}
\,
F\big(
Y^N_{
\lfloor s \rfloor_N
}
\big)
\,
ds
+
\int_0^t
e^{
A
\left( t -
\lfloor s \rfloor_N
\right)
}
\,
B\big(
Y^N_{
\lfloor s \rfloor_N
}
\big)
\,
dW_s
\\ & =
e^{ A t } \, \xi
+
\int_0^t
e^{ A (t - s) }
\;
e^{
A
\left( s -
\lfloor s \rfloor_N
\right)
}
\,
F\big(
Y^N_{
\lfloor s \rfloor_N
}
\big)
\,
ds
+
\int_0^t
e^{ A (t - s) }
\;
e^{
A
\left( s -
\lfloor s \rfloor_N
\right)
}
\,
B\big(
Y^N_{
\lfloor s \rfloor_N
}
\big)
\,
dW_s
\end{split}
\end{equation}
$ \mathbb{P} $-a.s.\ for
all $ t \in [0,T] $
and all $ N \in \mathbb{N} $.
Observe that
for each $ N \in \mathbb{N} $
the stochastic
process
$
Y^N \colon
[0,T] \times \Omega
\rightarrow H_{ \gamma }
$
is a mild It\^{o} process
with semigroup
$
e^{ A (t_2 - t_1) } \in
L( H_{ \min( \alpha, \beta, \gamma ) },
H_{ \gamma }
)
$, $ (t_1,t_2) \in \angle $,
with mild drift
\begin{equation}
\label{eq:split_milddrift}
e^{
A
\left( t -
\lfloor t \rfloor_N
\right)
}
\,
F\big(
Y^N_{
\lfloor t \rfloor_N
}
\big),
\qquad
t \in [0,T],
\end{equation}
and with mild diffusion
\begin{equation}
\label{eq:split_milddiffusion}
e^{
A
\left( t -
\lfloor t \rfloor_N
\right)
}
\,
B\big(
Y^N_{
\lfloor t \rfloor_N
}
\big),
\qquad
t \in [0,T] .
\end{equation}
Proposition~\ref{propsimple}
hence shows
\begin{equation}
\label{eq:propcons}
Y^N_{ t }
=
e^{
A \left( t - t_0 \right)
}
\,
Y^N_{ t_0 }
+
\int_{ t_0 }^{
t
}
e^{
A \left( t - \lfloor s \rfloor_N \right)
}
\,
F\big(
Y^N_{ \lfloor s \rfloor_N }
\big) \,
ds
+
\int_{ t_0 }^{
t
}
e^{
A \left( t - \lfloor s \rfloor_N \right)
}
\,
B\big(
Y^N_{ \lfloor s \rfloor_N }
\big) \,
dW_s
\end{equation}
$ \mathbb{P} $-a.s.\ for
all $ t_0, t \in [0,T] $
with $ t_0 \leq t $ and
from \eqref{eq:propcons}
we conclude
\begin{equation}
\label{eq:split}
\begin{split}
Y^N_{
\frac{ (n+1) T }{ N }
}
& =
e^{
A \frac{ T }{ N }
}
\,
Y^N_{ \frac{ n T }{ N } }
+
\int_{ \frac{ n T }{ N } }^{
\frac{ (n+1) T }{ N }
}
e^{
A \frac{ T }{ N }
}
\,
F\big(
Y^N_{ \frac{ n T }{ N } }
\big) \,
ds
+
\int_{ \frac{ n T }{ N } }^{
\frac{ (n+1) T }{ N }
}
e^{
A \frac{ T }{ N }
}
\,
B\big(
Y^N_{ \frac{ n T }{ N } }
\big) \,
dW_s
\\ & =
e^{
A \frac{ T }{ N }
}
\left(
Y^N_{ \frac{ n T }{ N } }
+
\frac{ T }{ N } \cdot
F\big(
Y^N_{ \frac{ n T }{ N } }
\big)
+
\int_{ \frac{ n T }{ N } }^{
\frac{ (n+1) T }{ N }
}
B\big(
Y^N_{ \frac{ n T }{ N } }
\big) \,
dW_s
\right)
\end{split}
\end{equation}
$ \mathbb{P} $-a.s.\ for
all
$ n \in \left\{ 0, 1, \dots, N-1
\right\} $
and all
$ N \in \mathbb{N} $.
The mild It\^{o} processes
$ Y^N $, $ N \in \mathbb{N} $,
are thus nothing else but appropriate
time continuous interpolations
of exponential Euler approximations
(a.k.a.~splitting-up
approximations or
exponential integrator
approximations;
see, e.g.,
\cite{gk03a,lr04,cv10,lt11}
and the references therein).
Note that the mild
drift~\eqref{eq:split_milddrift}
and the mild
diffusion~\eqref{eq:split_milddiffusion}
of the exponential Euler
approximations~\eqref{eq:split}
contain the
{\it correction
term}
$
e^{
A
\left( t -
\lfloor t \rfloor_N
\right)
}
$, $ t \in [0,T] $,
when compared to
the mild drift~\eqref{eq:milddrift}
and the mild diffusion~\eqref{eq:milddiffusion}
of the exact solution of the
SPDE~\eqref{eq:SPDE}.
\paragraph{Accelerated
exponential Euler
approximations for SPDEs}
This paragraph demonstrates
that accelerated
exponential Euler
approximations
can be written as mild It\^{o}
processes.
Let
$
\tilde{Y}^N
\colon [0,T]
\times
\Omega
\rightarrow H_{ \gamma }
$,
$
N \in \mathbb{N}
$,
be a sequence of predictable
stochastic
processes given by
\begin{equation}
\label{eq:lin0}
\begin{split}
\tilde{Y}^N_t
& =
e^{ A t } \,
\xi
+
\int_0^t
e^{
A
( t -
s
)
}
\,
F\big(
\tilde{Y}^N_{
\lfloor s \rfloor_N
}
\big)
\,
ds
+
\int_0^t
e^{
A
( t -
s
)
}
\,
B\big(
\tilde{Y}^N_{
\lfloor s \rfloor_N
}
\big)
\,
dW_s
\end{split}
\end{equation}
$ \mathbb{P} $-a.s.\ for
all $ t \in [0,T] $
and all $ N \in \mathbb{N} $.
Note that
for each $ N \in \mathbb{N} $
the stochastic
process
$
\tilde{Y}^N \colon
[0,T] \times \Omega
\rightarrow H_{ \gamma }
$
is a mild It\^{o} process
with semigroup
$
e^{ A (t_2 - t_1) } \in
L(
H_{ \min( \alpha, \beta, \gamma ) } ,
H_{ \gamma }
)
$, $ (t_1,t_2) \in \angle $,
with mild drift
\begin{equation}
\label{eq:lin_milddrift}
F\big(
\tilde{Y}^N_{
\lfloor t \rfloor_N
}
\big),
\qquad
t \in [0,T],
\end{equation}
and with mild diffusion
\begin{equation}
\label{eq:lin_milddiffusion}
B\big(
\tilde{Y}^N_{
\lfloor t \rfloor_N
}
\big),
\qquad
t \in [0,T] .
\end{equation}
Proposition~\ref{propsimple}
therefore gives
\begin{equation}
\label{eq:propcons2}
\tilde{Y}^N_{ t }
=
e^{
A \left( t - t_0 \right)
}
\,
\tilde{Y}^N_{ t_0 }
+
\int_{ t_0 }^{
t
}
e^{
A \left( t - s \right)
}
\,
F\big(
\tilde{Y}^N_{ \lfloor s \rfloor_N }
\big) \,
ds
+
\int_{ t_0 }^{
t
}
e^{
A \left( t - s \right)
}
\,
B\big(
\tilde{Y}^N_{ \lfloor s \rfloor_N }
\big) \,
dW_s
\end{equation}
$ \mathbb{P} $-a.s.\ for
all $ t_0, t \in [0,T] $
with $ t_0 \leq t $ and
this implies
\begin{equation}
\label{eq:lin}
\begin{split}
\tilde{Y}^N_{
\frac{ (n+1) T }{ N }
}
=
e^{
A \frac{ T }{ N }
}
\,
\tilde{Y}^N_{ \frac{ n T }{ N } }
+
\left(
\int_0^{ \frac{T}{N} } e^{ A s } \, ds
\right)
F\big(
\tilde{Y}^N_{ \frac{ n T }{ N } }
\big)
+
\int_{
\frac{ n T }{ N }
}^{
\frac{ ( n + 1 ) T }{ N }
}
e^{
A \left(
\frac{ (n+1) T }{ N } - s
\right)
} \,
B\big(
\tilde{Y}^N_{ \frac{ n T }{ N } }
\big) \,
dW_s
\end{split}
\end{equation}
$ \mathbb{P} $-a.s.\ for
all
$ n \in \left\{ 0, 1, \dots, N-1
\right\} $
and all
$ N \in \mathbb{N} $.
In particular,
in the case of additive
noise, i.e., $ B(v) = B(0) $
for all $ v \in H_{ \gamma } $,
equation~\eqref{eq:lin}
reduces to
\begin{equation}
\label{eq:linB}
\begin{split}
&
\tilde{Y}^N_{
\frac{ (n+1) T }{ N }
}
=
e^{
A \frac{ T }{ N }
}
\,
\tilde{Y}^N_{ \frac{ n T }{ N } }
+
\left(
\int_0^{ \frac{T}{N} } e^{ A s } \, ds
\right)
F\big(
\tilde{Y}^N_{ \frac{ n T }{ N } }
\big)
+
\int_{
\frac{ n T }{ N }
}^{
\frac{ ( n + 1 ) T }{ N }
}
e^{
A \left(
\frac{ (n+1) T }{ N } - s
\right)
} \,
B(0)
\,
dW_s
\end{split}
\end{equation}
$ \mathbb{P} $-a.s.\ for
all
$ n \in \left\{ 0, 1, \dots, N-1
\right\} $
and all
$ N \in \mathbb{N} $.
The mild It\^{o} processes
$ \tilde{Y}^N $, $ N \in \mathbb{N} $,
are thus nothing else but appropriate
time continuous interpolations
of the numerical approximations
in \cite{jk09b}
in the case of additive noise
(see~(3.3) in \cite{jk09b})
and in \cite{j10}
in the general case
(see~(21) and (50)
in \cite{j10}).
Note that the mild
drift~\eqref{eq:lin_milddrift}
and the
diffusion~\eqref{eq:lin_milddiffusion}
of the approximation
processes~\eqref{eq:lin0}
do not contain the correction
term
$
e^{
A
\left( t -
\lfloor t \rfloor_N
\right)
}
$, $ t \in [0,T] $,
in mild
drift~\eqref{eq:split_milddrift}
and the mild
diffusion~\eqref{eq:split_milddiffusion}
of the exponential Euler
approximations~\eqref{eq:split0}.
The approximation
processes~\eqref{eq:lin0}
thus seem to be more close
to the exact solution~\eqref{eq:SPDE}
than the exponential Euler
approximations~\eqref{eq:split0}
(compare the mild drifts~\eqref{eq:lin_milddrift},
\eqref{eq:split_milddrift},
\eqref{eq:milddrift}
and the mild
diffusions~\eqref{eq:lin_milddiffusion},
\eqref{eq:split_milddiffusion},
\eqref{eq:milddiffusion}).
Indeed, under suitable assumptions,
it has been shown
(see \cite{jk09b,jkw11} for details)
that
$ \tilde{Y}^N $, $ N \in \mathbb{N} $,
converges
to $ X $ significantly
faster than
$ Y^N $,
$ N \in \mathbb{N} $.
This motivates to call approximations
of the form~\eqref{eq:lin}
and \eqref{eq:linB} as
{\it accelerated exponential
Euler approximations}.
The crucial point in
the accelerated exponential Euler
approximations is that
they contain
the stochastic integrals
\begin{equation}
\label{eq:integrals}
\int_{
\frac{ n T }{ N }
}^{
\frac{ ( n + 1 ) T }{ N }
}
e^{
A \left(
\frac{ (n+1) T }{ N } - s
\right)
} \,
B\big(
\tilde{Y}^N_{ \frac{ n T }{ N } }
\big) \,
dW_s
\end{equation}
for
$ n \in \left\{ 0, 1, \dots, N-1
\right\} $
and
$ N \in \mathbb{N} $
in the scheme
instead of simply increments
of driving noise process.
This enables them to converge,
under suitable assumptions,
significantly faster to $ X $ than
schemes using only
increments of the
driving noise process
such as \eqref{eq:split}.
In addition,
in the case of additive noise,
the
stochastic integrals~\eqref{eq:integrals}
in \eqref{eq:linB}
depends linearly on the cylindrical
Wiener process $ W_t $, $ t \in [0,T] $
and are easy to simulate.
Therefore,
the accelerated exponential
Euler approximations~\eqref{eq:linB}
can in the case of
additive noise be simulated
quite efficiently
(see Section~3 in \cite{jk09b}
and, particularly,
see Figure~2 in \cite{jkw11}
for details).
Further investigations
and related results on approximation
methods that make use
of stochastic integrals of the
form \eqref{eq:integrals}
can, e.g., be found in
\cite{jk09b,jk09d,lt10,lt10b,d10,jkw11,j11,mtac11,xd11}.
\paragraph{Linear
implicit Euler
approximations for SPDEs}
Next it is shown that
linear implicit Euler approximations
can be formulated as mild It\^{o}
processes.
For this we assume $ \eta = 0 $
in the following
in order to avoid trivial complications.
Moreover, let
$
\bar{S}^N \colon
\angle
\rightarrow
L( H_{ \gamma } )
$,
$ N \in \mathbb{N} $,
be a sequence of mappings
given by
\begin{equation}
\label{eq:lineulersg}
\bar{S}^N_{ t_1, t_2 }
:=
\Big(
I - A
\left( t_1 - \lfloor t_1 \rfloor_N \right)
\Big)
\Big(
I - A
\left( t_2 - \lfloor t_2 \rfloor_N \right)
\Big)^{ \! -1 }
\Big(
I - \tfrac{ T }{ N } A
\Big)^{
\left(
\lfloor t_1
\rfloor_N
-
\lfloor t_2 \rfloor_N
\right)
\frac{ N }{ T }
}
\end{equation}
for all
$ t_1, t_2 \in [0,T] $
with $ t_1 < t_2 $
and
all $ N \in \mathbb{N} $.
Moreover,
let
$
\bar{Y}^N
\colon [0,T]
\times
\Omega
\rightarrow H_{ \gamma }
$,
$
N \in \mathbb{N}
$,
be a sequence of predictable
stochastic
processes given by
$
\bar{Y}^N_0 = \xi
$
and
\begin{equation}
\label{eq:lineuler0}
\begin{split}
\bar{Y}^N_t
& =
\bar{S}^N_{ 0, t } \,
\xi
+
\int_0^t
\bar{S}_{ \lfloor s \rfloor_N, t }^N
\,
F\big(
\bar{Y}^N_{
\lfloor s \rfloor_N
}
\big)
\,
ds
+
\int_0^t
\bar{S}_{ \lfloor s \rfloor_N, t }^N
\,
B\big(
\bar{Y}^N_{
\lfloor s \rfloor_N
}
\big)
\,
dW_s
\\ & =
\bar{S}^N_{ 0, t } \,
\xi
+
\int_0^t
\bar{S}_{ s, t }^N
\,
\big(
I -
A
\left( s -
\lfloor s \rfloor_N
\right)
\big)^{ -1 }
\,
F\big(
\bar{Y}^N_{
\lfloor s \rfloor_N
}
\big)
\,
ds
\\ & \quad
+
\int_0^t
\bar{S}_{ s, t }^N
\,
\big(
I -
A
\left( s -
\lfloor s \rfloor_N
\right)
\big)^{ -1 }
\,
B\big(
\bar{Y}^N_{
\lfloor s \rfloor_N
}
\big)
\,
dW_s
\end{split}
\end{equation}
$ \mathbb{P} $-a.s.\ for
all $ t \in (0,T] $
and all $ N \in \mathbb{N} $.
Observe that
for each $ N \in \mathbb{N} $
the stochastic
process
$
\bar{Y}^N \colon
[0,T] \times \Omega
\rightarrow H_{ \gamma }
$
is a mild It\^{o} process
with semigroup
$ \bar{S}^N $,
with mild drift
\begin{equation}
\mathbbm{1}_{ (0,\infty) }(
t - \lfloor t \rfloor_N
) \;
\big(
I -
A
\left( t -
\lfloor t \rfloor_N
\right)
\big)^{ -1 }
\,
F\big(
\bar{Y}^N_{
\lfloor t \rfloor_N
}
\big),
\qquad
t \in [0,T],
\end{equation}
and with mild diffusion
\begin{equation}
\mathbbm{1}_{ (0,\infty) }(
t - \lfloor t \rfloor_N
) \;
\big(
I -
A
\left( t -
\lfloor t \rfloor_N
\right)
\big)^{ -1 }
\,
B\big(
\bar{Y}^N_{
\lfloor t \rfloor_N
}
\big),
\qquad
t \in [0,T] .
\end{equation}
Proposition~\ref{propsimple}
therefore implies
\begin{equation}
\label{eq:lineul}
\begin{split}
\bar{Y}^N_{
\frac{ (n+1) T }{ N }
}
& =
\Big(
I -
\tfrac{ T }{ N } A
\Big)^{ \! -1 }
\left(
\bar{Y}^N_{ \frac{ n T }{ N } }
+
\frac{ T }{ N } \cdot
F\big( \bar{Y}^N_{ \frac{ n T }{ N } } \big)
+
\int_{ \frac{ n T }{ N } }^{
\frac{ ( n + 1 ) T }{ N }
}
B\big( \bar{Y}^N_{ \frac{ n T }{ N } } \big) \,
dW_s
\right)
\end{split}
\end{equation}
$ \mathbb{P} $-a.s.\ for
all
$ n \in \left\{ 0, 1, \dots, N-1
\right\} $
and all
$ N \in \mathbb{N} $.
This shows that
the stochastic
processes
$
\bar{Y}^N
$,
$ N \in \mathbb{N} $,
are nothing else but
appropriate time
continuous interpolations
of linear
implicit Euler approximations
(see, e.g., \cite{g99,h02,h03a,w05b,mrw07,mr07b,k11,cv12}
and the references therein)
for the SPDE~\eqref{eq:SPDE}.
Note that the semigroups~\eqref{eq:lineulersg}
of the linear implicit Euler
approximations~\eqref{eq:lineuler0}
depend explicitly on both variables
$ t_1 $ and $ t_2 $
with $ 0 \leq t_1 \leq t_2 \leq T $
instead of on the difference
$ t_2 - t_1 $ only
although the semigroup
$ e^{ A t } $, $ t \in [0,T] $,
of the underlying SPDE~\eqref{eq:SPDE}
depends on one variable only.
\paragraph{Linear
implicit Crank-Nicolson
approximations for SPDEs}
Finally, in this paragraph it is
demonstrated
that linear implicit Crank-Nicolson
approximations
can be formulated as mild It\^{o}
processes too.
As in the case of the linear
implicit Euler approximations
we assume
$ \eta = 0 $
in the following in order to avoid
trivial complications.
Let
$
\hat{S}^N \colon
\angle
\rightarrow
L( H_{ \gamma } )
$,
$ N \in \mathbb{N} $,
be a sequence of mappings
given by
\begin{equation}
\label{eq:lincranksg}
\hat{S}^N_{ t_1, t_2 }
:=
\Big(
I -
A
\tfrac{
\left( t_1 - \lfloor t_1 \rfloor_N \right)
}{ 2 }
\Big)
\Big(
I -
A
\tfrac{
\left( t_2 - \lfloor t_2 \rfloor_N \right)
}{ 2 }
\Big)^{ \! -1 }
\Big(
I - \tfrac{ T }{ 2 N } A
\Big)^{
\left(
\lfloor t_1 \rfloor_N
-
\lfloor t_2 \rfloor_N
\right)
\frac{ N }{ T }
}
\end{equation}
for all
$ t_1, t_2 \in [0,T] $
with $ t_1 < t_2 $
and
all $ N \in \mathbb{N} $.
Moreover,
let
$
\hat{Y}^N
\colon [0,T]
\times
\Omega
\rightarrow H_{ \gamma }
$,
$
N \in \mathbb{N}
$,
be a sequence of predictable
stochastic
processes given by
$
\hat{Y}^N_0 = \xi
$
and
\begin{equation}
\label{eq:crank0}
\begin{split}
\hat{Y}^N_t
& =
\hat{S}^N_{ 0, t } \,
\xi
+
\int_0^t
\hat{S}_{ \lfloor s \rfloor_N, t }^N \,
\Big(
\tfrac{ 1 }{ 2 } A \,
\hat{Y}^N_{
\lfloor s \rfloor_N
}
+
F\big(
\hat{Y}^N_{
\lfloor s \rfloor_N
}
\big)
\Big) \,
ds
+
\int_0^t
\hat{S}_{ \lfloor s \rfloor_N, t }^N
\,
B\big(
\hat{Y}^N_{
\lfloor s \rfloor_N
}
\big)
\,
dW_s
\\ & =
\hat{S}^N_{ 0, t } \,
\xi
+
\int_0^t
\hat{S}_{ s, t }^N
\left(
I -
A
\tfrac{
\left( s -
\lfloor s \rfloor_N
\right)
}{ 2 }
\right)^{ \! - 1 }
\Big(
\tfrac{ 1 }{ 2 } A \,
\hat{Y}^N_{
\lfloor s \rfloor_N
}
+
F\big(
\hat{Y}^N_{
\lfloor s \rfloor_N
}
\big)
\Big)
\,
ds
\\ & \quad
+
\int_0^t
\hat{S}_{ s, t }^N
\left(
I -
A
\tfrac{
\left( s -
\lfloor s \rfloor_N
\right)
}{ 2 }
\right)^{ \! - 1 }
B\big(
\hat{Y}^N_{
\lfloor s \rfloor_N
}
\big)
\,
dW_s
\end{split}
\end{equation}
$ \mathbb{P} $-a.s.\ for
all $ t \in (0,T] $
and all $ N \in \mathbb{N} $.
Observe
for every $ N \in \mathbb{N} $
that
the stochastic
process
$
\hat{Y}^N \colon
[0,T] \times \Omega
\rightarrow H_{ \gamma }
$
is a mild It\^{o} process
with semigroup
$ \hat{S}^N $,
with mild drift
\begin{equation}
\mathbbm{1}_{ (0,\infty) }(
t - \lfloor t \rfloor_N
) \,
\left(
I -
A
\tfrac{
\left( t -
\lfloor t \rfloor_N
\right)
}{ 2 }
\right)^{ \! - 1 }
\Big(
\tfrac{ 1 }{ 2 } A \,
\hat{Y}^N_{
\lfloor t \rfloor_N
}
+
F\big(
\hat{Y}^N_{
\lfloor t \rfloor_N
}
\big)
\Big) ,
\;
t \in [0,T],
\end{equation}
and with mild diffusion
\begin{equation}
\mathbbm{1}_{ (0,\infty) }(
t - \lfloor t \rfloor_N
) \,
\left(
I -
A
\tfrac{
\left( t -
\lfloor t \rfloor_N
\right)
}{ 2 }
\right)^{ \! - 1 }
B\big(
\hat{Y}^N_{
\lfloor t \rfloor_N
}
\big),
\;
t \in [0,T] .
\end{equation}
Proposition~\ref{propsimple}
hence gives
\begin{equation}
\label{eq:lineul}
\begin{split}
\hat{Y}^N_{
\frac{ (n+1) T }{ N }
}
=
\Big(
I -
\tfrac{ T }{ 2 N } A
\Big)^{ \! -1 }
\left(
\Big(
I +
\tfrac{ T }{ 2 N } A
\Big) \,
\hat{Y}^N_{ \frac{ n T }{ N } }
+
\frac{ T }{ N } \cdot
F\big(
\hat{Y}^N_{ \frac{ n T }{ N } }
\big)
+
\int_{ \frac{ n T }{ N } }^{
\frac{ ( n + 1 ) T }{ N }
}
B\big(
\hat{Y}^N_{ \frac{ n T }{ N } }
\big) \,
dW_s
\right)
\end{split}
\end{equation}
$ \mathbb{P} $-a.s.\ for
all
$ n \in \left\{ 0, 1, \dots, N-1
\right\} $
and all
$ N \in \mathbb{N} $.
This shows that
the stochastic
processes
$
\hat{Y}^N
$,
$ N \in \mathbb{N} $,
are nothing else but
appropriate time
continuous interpolations
of linear
implicit Crank-Nicolson
approximations
(see, e.g., \cite{s99,w05b,brsv08}
and the references therein)
for the
SPDE~\eqref{eq:SPDE}.
\subsubsection{Milstein type
approximations for SPDEs}
\label{sec:mil}
The stochastic Taylor expansions
in Subsection~\ref{sec:tay}
can be used to derive higher order
numerical approximation
methods for SPDEs. In the sequel
this is illustrated for Milstein type
approximations for SPDEs
(see \cite{gk96,ks01,ms06,lcp10,jr12,bl11a,bl11b,WangGan2012,b12}).
For these derivations we
assume that the diffusion term
$
B \colon H_{ \gamma }
\rightarrow
HS( U_0, H_{ \beta } )
$
is twice continuously
Fr\'{e}chet differentiable
with globally bounded derivatives
and that $ \beta = \gamma $.
First note,
in view of the
mild It\^{o}
formula~\eqref{eq:operatorIto},
that
equation~\eqref{eq:SPDE}
can also be written as
\begin{equation}
\label{eq:SPDE2}
X_t
=
e^{ A (t - t_0) } X_{ t_0 }
+
\int_{ t_0 }^t
\big(
L^{ (0) }_{ (t - s) } \text{id}
\big)( X_s ) \,
ds
+
\int_{ t_0 }^t
\big(
L^{ (1) }_{ (t - s) } \text{id}
\big)( X_s ) \,
dW_s
\end{equation}
$ \mathbb{P} $-a.s.\ for all
$ t_0, t \in [0,T] $
with $ t_0 \leq t $
where
$
\text{id}
= \text{id}_{ H_{ \gamma } }
\colon H_{ \gamma }
\rightarrow
H_{ \gamma }
$ is the identity on $ H_{ \gamma } $.
Next the mild It\^{o} formula~\eqref{eq:operatorIto}
is applied to the
test function
$
\big(
L^{ (1) }_{ (t - s) } \text{id}
\big)( x )
\in HS( U_0, H_{ \beta } )
$,
$
x \in H_{ \gamma }
$,
to obtain
\begin{equation}
\label{eq:Bito}
\begin{split}
&
\big(
L^{ (1) }_{ (t - s) } \text{id}
\big)( X_s )
\\&
=
\big(
L^{ (1) }_{ (t - s) } \text{id}
\big)\big(
e^{ A (s - t_0) } X_{ t_0 }
\big)
+
\int_{ t_0 }^s
\big(
L^{ (0) }_{ (s - u) }
L^{ (1) }_{ (t - s) }
\text{id}
\big)( X_u )
\, du
+
\int_{ t_0 }^s
\big(
L^{ (1) }_{ (s - u) }
L^{ (1) }_{ (t - s) }
\text{id}
\big)( X_u )
\, dW_u
\end{split}
\end{equation}
$ \mathbb{P} $-a.s.\ for all
$ t_0, s, t \in [0,T] $ with
$ t_0 \leq s \leq t $.
Putting \eqref{eq:Bito}
into \eqref{eq:SPDE2}
then gives
\begin{align}
\label{eq:milsteintay}
X_t
&=
e^{ A (t - t_0) } X_{ t_0 }
+
\int_{ t_0 }^t
\big(
L^{ (0) }_{ (t - s) } \text{id}
\big)( X_s ) \,
ds
+
\int_{ t_0 }^t
\big(
L^{ (1) }_{ (t - s) } \text{id}
\big)\big( e^{ A (s - t_0) } X_{ t_0 }
\big) \,
dW_s
\nonumber
\\ &
+
\int_{ t_0 }^t
\int_{ t_0 }^s
\big(
L^{ (0) }_{ (s - u) }
L^{ (1) }_{ (t - s) }
\text{id}
\big)( X_u )
\, du
\, dW_s
+
\int_{ t_0 }^t
\int_{ t_0 }^s
\big(
L^{ (1) }_{ (s - u) }
L^{ (1) }_{ (t - s) }
\text{id}
\big)( X_u )
\, dW_u
\, dW_s
\\ & =
\nonumber
e^{ A (t - t_0) } X_{ t_0 }
+
\int_{ t_0 }^t
e^{ A (t - s) } F( X_s ) \,
ds
+
\int_{ t_0 }^t
e^{ A (t - s) }
B\big( e^{ A (s - t_0) } X_{ t_0 }
\big) \,
dW_s
\\ &
+
\int_{ t_0 }^t
\int_{ t_0 }^s
\big(
L^{ (0) }_{ (s - u) }
L^{ (1) }_{ (t - s) }
\text{id}
\big)( X_u )
\, du
\, dW_s
+
\int_{ t_0 }^t
\int_{ t_0 }^s
e^{ A (t - s) }
B'\big(
e^{ A (s - u) }
X_u
\big) \,
e^{ A (s - u) }
B( X_u )
\, dW_u
\, dW_s
\nonumber
\end{align}
$ \mathbb{P} $-a.s.\ for
all $ t_0, t \in [0,T] $
with $ t_0 \leq t $.
The identity~\eqref{eq:milsteintay}
corresponds to the strong stochastic
Taylor expansion in
Theorem~\ref{thm:strongtay}
which is
described by the hierarchical
set
$
\mathcal{A} = \{ \emptyset, (1) \}
$;
see Subsection~\ref{sec:tay}
for details.
Using the approximations
$ e^{ A h } \approx I $
and
$ X_{ t_0 + h } \approx X_{ t_0 } $
for small $ h \in [0,T] $
and omitting the
integral
$
\int_{ t_0 }^t
\int_{ t_0 }^s
\big(
L^{ (0) }_{ (s - u) }
L^{ (1) }_{ (t - s) }
\text{id}
\big)( X_u )
\, du
\, dW_s
$
in \eqref{eq:milsteintay}
then results in the approximation
\begin{align}
\label{eq:milsteinapprox}
\nonumber
X_t
& \approx
e^{ A (t - t_0) } X_{ t_0 }
+
\int_{ t_0 }^t
e^{ A (t - s) } F( X_{ t_0 } ) \,
ds
+
\int_{ t_0 }^t
e^{ A (t - s) }
B( X_{ t_0 }
) \,
dW_s
\\ & \quad
+
\int_{ t_0 }^t
\int_{ t_0 }^s
e^{ A (t - s) }
B'(
X_{ t_0 }
) \,
e^{ A (s - u) }
B( X_{ t_0 } )
\, dW_u
\, dW_s
\\ & \approx
e^{ A (t - t_0) }
\left(
X_{ t_0 }
+
F( X_{ t_0 } )
\cdot \left( t - t_0 \right)
+
\int_{ t_0 }^t
B( X_{ t_0 }
) \,
dW_s
+
\int_{ t_0 }^t
\int_{ t_0 }^s
B'(
X_{ t_0 }
) \,
B( X_{ t_0 } )
\, dW_u
\, dW_s
\right)
\nonumber
\end{align}
for $ t_0, t \in [0,T] $
with small $ t - t_0 \geq 0 $.
The
approximation~\eqref{eq:milsteinapprox}
can then be used to define exponential
Milstein approximations for SPDEs
(see equations~\eqref{eq:expmil}
and \eqref{eq:expmil2} below
for details).
In the following it is
demonstrated how different
types of Milstein
approximations for SPDEs
can be formulated as
mild It\^{o} processes.
To this end
we assume in the remainder
of this subsection
that
$ B \colon H_{ \gamma }
\rightarrow HS( U_0, H_{ \beta } ) $
in Assumption~\ref{diffusion}
is once continuously Fr\'{e}chet
differentiable
and that
$ \beta = \gamma $.
\paragraph{Exponential
Milstein approximations
for SPDEs}
This paragraph formulates
exponential Milstein
approximations as mild It\^{o}
processes.
To be more precise, let
$
Z^N
\colon [0,T]
\times
\Omega
\rightarrow H_{ \gamma }
$,
$
N \in \mathbb{N}
$,
be a sequence of predictable
stochastic
processes given by
\begin{equation}
\label{eq:expmil}
\begin{split}
Z^N_t
& =
e^{ A t } \, \xi
+
\int_0^t
e^{
A
\left( t -
\lfloor s \rfloor_N
\right)
}
\,
F\big(
Z^N_{
\lfloor s \rfloor_N
}
\big)
\,
ds
+
\int_0^t
e^{
A
\left( t -
\lfloor s \rfloor_N
\right)
}
\,
B\big(
Z^N_{
\lfloor s \rfloor_N
}
\big)
\,
dW_s
\\ & \quad +
\int_0^t
e^{
A
\left( t -
\lfloor s \rfloor_N
\right)
}
\,
B'\big(
Z^N_{
\lfloor s \rfloor_N
}
\big)
\bigg(
\int_{ \lfloor s \rfloor_N
}^{ s }
B\big(
Z^N_{
\lfloor s \rfloor_N
}
\big) \,
dW_u
\bigg)
\,
dW_s
\end{split}
\end{equation}
$ \mathbb{P} $-a.s.\ for
all $ t \in [0,T] $
and all $ N \in \mathbb{N} $.
Note
for every $ N \in \mathbb{N} $
that the stochastic
process
$
Z^N \colon
[0,T] \times \Omega
\rightarrow H_{ \gamma }
$
is a mild It\^{o} processes
with semigroup
$
e^{ A (t_2 - t_1) } \in
L( H_{ \min( \alpha, \beta, \gamma ) } ,
H_{ \gamma }
)
$, $ (t_1,t_2) \in \angle $,
with mild drift
\begin{equation}
e^{
A
\left( t -
\lfloor t \rfloor_N
\right)
}
\,
F\big(
Z^N_{
\lfloor t \rfloor_N
}
\big),
\;
t \in [0,T],
\end{equation}
and with mild diffusion
\begin{equation}
\label{eq:expmil_milddiffusion}
e^{
A
\left( t -
\lfloor t \rfloor_N
\right)
}
\left(
B\big(
Z^N_{
\lfloor t \rfloor_N
}
\big)
+
B'\big(
Z^N_{
\lfloor t \rfloor_N
}
\big)
\bigg(
\int_{ \lfloor t \rfloor_N
}^{ t }
B\big(
Z^N_{
\lfloor t \rfloor_N
}
\big) \,
dW_s
\bigg)
\right), \;
t \in [0,T] .
\end{equation}
Proposition~\ref{propsimple}
hence yields
\begin{equation}
\label{eq:expmil2}
\begin{split}
Z^N_{
\frac{ (n+1) T }{ N }
}
& =
e^{
A \frac{ T }{ N }
}
\,
\Bigg(
Z^N_{ \frac{ n T }{ N } }
+
\frac{ T }{ N } \cdot
F\big(
Z^N_{ \frac{ n T }{ N } }
\big)
+
\int_{ \frac{ n T }{ N } }^{
\frac{ ( n + 1 ) T }{ N }
}
B\big(
Z^N_{ \frac{ n T }{ N } }
\big) \,
dW_s
\\ & \quad +
\int_{ \frac{ n T }{ N } }^{
\frac{ ( n + 1 ) T }{ N }
}
B'\big(
Z^N_{ \frac{ n T }{ N } }
\big) \bigg(
\int_{ \frac{ n T }{ N } }^s
B\big(
Z^N_{ \frac{ n T }{ N } }
\big) \,
dW_u
\bigg) \,
dW_s
\Bigg)
\end{split}
\end{equation}
$ \mathbb{P} $-a.s.\ for
all
$ n \in \left\{ 0, 1, \dots, N-1
\right\} $
and all
$ N \in \mathbb{N} $.
The mild It\^{o} processes
$ Z^N $, $ N \in \mathbb{N} $,
are thus nothing else but appropriate
time continuous interpolations
of exponential Milstein
approximations
(see \cite{ms06,jr12,bl11b}).
\paragraph{Linear implicit
Euler-Milstein approximations
for SPDEs}
In this paragraph
linear implicit Euler-Milstein
approximations
are formulated as mild It\^{o}
processes.
Let
$
\bar{Z}^N
\colon [0,T]
\times
\Omega
\rightarrow H_{ \gamma }
$,
$
N \in \mathbb{N}
$,
be a sequence of predictable
stochastic
processes given by
$
\bar{Z}^N_0 = \xi
$
and
\begin{equation}
\begin{split}
\bar{Z}^N_t
& =
\bar{S}^N_{0, t} \, \xi
+
\int_0^t
\bar{S}^N_{
\lfloor s \rfloor_N, t
}
\,
F\big(
\bar{Z}^N_{
\lfloor s \rfloor_N
}
\big)
\,
ds
+
\int_0^t
\bar{S}^N_{
\lfloor s \rfloor_N, t
}
\,
B\big(
\bar{Z}^N_{
\lfloor s \rfloor_N
}
\big)
\,
dW_s
\\ & \quad +
\int_0^t
\bar{S}^N_{
\lfloor s \rfloor_N, t
}
\,
B'\big(
\bar{Z}^N_{
\lfloor s \rfloor_N
}
\big)
\bigg(
\int_{ \lfloor s \rfloor_N
}^{ s }
B\big(
\bar{Z}^N_{
\lfloor s \rfloor_N
}
\big) \,
dW_u
\bigg)
\,
dW_s
\end{split}
\end{equation}
$ \mathbb{P} $-a.s.\ for
all $ t \in (0,T] $
and all $ N \in \mathbb{N} $
(see \eqref{eq:lineulersg} for the
definition of the mappings
$ \bar{S}^N \colon \angle
\rightarrow L( H_{ \gamma } ) $,
$ N \in \mathbb{N} $).
Observe
for every $ N \in \mathbb{N} $
that the stochastic
process
$
\bar{Z}^N \colon
[0,T] \times \Omega
\rightarrow H_{ \gamma }
$
is a mild It\^{o} process
with semigroup
$ \bar{S}^N $,
with mild drift
\begin{equation}
\mathbbm{1}_{ (0,\infty) }(
t - \lfloor t \rfloor_N
) \;
\bar{S}^N_{ \lfloor t \rfloor_N, t }
\,
F\big(
\bar{Z}^N_{
\lfloor t \rfloor_N
}
\big),
\;\;
t \in [0,T],
\end{equation}
and with mild diffusion
\begin{equation}
\mathbbm{1}_{ (0,\infty) }(
t - \lfloor t \rfloor_N
) \;
\bar{S}^N_{ \lfloor t \rfloor_N, t }
\left(
B\big(
\bar{Z}^N_{
\lfloor t \rfloor_N
}
\big)
+
B'\big(
\bar{Z}^N_{
\lfloor t \rfloor_N
}
\big)
\bigg(
\int_{ \lfloor t \rfloor_N
}^{ t }
B\big(
\bar{Z}^N_{
\lfloor t \rfloor_N
}
\big) \,
dW_s
\bigg)
\right),
\;\;
t \in [0,T] .
\end{equation}
Proposition~\ref{propsimple}
hence gives
\begin{equation}
\begin{split}
\bar{Z}^N_{
\frac{ (n+1) T }{ N }
}
& =
\Big(
I -
\tfrac{ T }{ N } A
\Big)^{ \! -1 }
\,
\Bigg(
\bar{Z}^N_{ \frac{ n T }{ N } }
+
\frac{ T }{ N } \cdot
F\big(
\bar{Z}^N_{ \frac{ n T }{ N } }
\big)
+
\int_{ \frac{ n T }{ N } }^{
\frac{ ( n + 1 ) T }{ N }
}
B\big(
\bar{Z}^N_{ \frac{ n T }{ N } }
\big) \,
dW_s
\\ & \quad +
\int_{ \frac{ n T }{ N } }^{
\frac{ ( n + 1 ) T }{ N }
}
B'\big(
\bar{Z}^N_{ \frac{ n T }{ N } }
\big) \bigg(
\int_{ \frac{ n T }{ N } }^s
B\big(
\bar{Z}^N_{ \frac{ n T }{ N } }
\big) \,
dW_u
\bigg) \,
dW_s
\Bigg)
\end{split}
\end{equation}
$ \mathbb{P} $-a.s.\ for
all
$ n \in \left\{ 0, 1, \dots, N-1
\right\} $
and all
$ N \in \mathbb{N} $.
The mild It\^{o} processes
$ \bar{Z}^N $, $ N \in \mathbb{N} $,
are thus nothing else but appropriate
time continuous interpolations
of linear implicit Euler-Milstein
approximations
(see \cite{ks01,ms06,bl11b,b12}).
\paragraph{Linear implicit
Crank-Nicolson-Milstein
Milstein approximations
for SPDEs}
Finally, let
$
\hat{Z}^N
\colon [0,T]
\times
\Omega
\rightarrow H_{ \gamma }
$,
$
N \in \mathbb{N}
$,
be a sequence of predictable
stochastic
processes given by
$
\hat{Z}^N_0 = \xi
$
and
\begin{equation}
\begin{split}
\hat{Z}^N_t
& =
\hat{S}^N_{0, t} \, \xi
+
\int_0^t
\hat{S}^N_{
\lfloor s \rfloor_N, t
}
\,
\Big(
\tfrac{ 1 }{ 2 } A \,
\hat{Z}^N_{
\lfloor s \rfloor_N
}
+
F\big(
\hat{Z}^N_{
\lfloor s \rfloor_N
}
\big)
\Big) \,
ds
+
\int_0^t
\hat{S}^N_{
\lfloor s \rfloor_N, t
}
\,
B\big(
\hat{Z}^N_{
\lfloor s \rfloor_N
}
\big)
\,
dW_s
\\ & \quad +
\int_0^t
\hat{S}^N_{
\lfloor s \rfloor_N, t
}
\,
B'\big(
\hat{Z}^N_{
\lfloor s \rfloor_N
}
\big)
\bigg(
\int_{ \lfloor s \rfloor_N
}^{ s }
B\big(
\hat{Z}^N_{
\lfloor s \rfloor_N
}
\big) \,
dW_u
\bigg)
\,
dW_s
\end{split}
\end{equation}
$ \mathbb{P} $-a.s.\ for
all $ t \in (0,T] $
and all $ N \in \mathbb{N} $
(see \eqref{eq:lincranksg}
for the definition
the mappings
$ \hat{S}^N \colon
\angle \rightarrow L( H_{ \gamma } ) $,
$ N \in \mathbb{N} $)
and note
for each $ N \in \mathbb{N} $
that the stochastic
process
$
\hat{Z}^N \colon
[0,T] \times \Omega
\rightarrow H_{ \gamma }
$
is a mild It\^{o} processes
with semigroup
$ \hat{S}^N $,
with mild drift
\begin{equation}
\mathbbm{1}_{ (0,\infty) }(
t - \lfloor t \rfloor_N
) \;
\hat{S}^N_{ \lfloor t \rfloor_N, t }
\,
\Big(
\tfrac{ 1 }{ 2 } A \,
\hat{Y}^N_{
\lfloor t \rfloor_N
}
+
F\big(
\hat{Y}^N_{
\lfloor t \rfloor_N
}
\big)
\Big) ,
\quad
t \in [0,T],
\end{equation}
and with mild diffusion
\begin{equation}
\mathbbm{1}_{ (0,\infty) }(
t - \lfloor t \rfloor_N
) \;
\hat{S}^N_{ \lfloor t \rfloor_N, t }
\left(
B\big(
\hat{Z}^N_{
\lfloor t \rfloor_N
}
\big)
+
B'\big(
\hat{Z}^N_{
\lfloor t \rfloor_N
}
\big)
\bigg(
\int_{ \lfloor t \rfloor_N
}^{ t }
B\big(
\hat{Z}^N_{
\lfloor t \rfloor_N
}
\big) \,
dW_s
\bigg)
\right),
\quad
t \in [0,T] .
\end{equation}
Proposition~\ref{propsimple}
therefore shows
\begin{equation}
\begin{split}
\hat{Z}^N_{
\frac{ (n+1) T }{ N }
}
& =
\Big(
I -
\tfrac{ T }{ 2 N } A
\Big)^{ \! -1 }
\,
\Bigg(
\Big(
I +
\tfrac{ T }{ 2 N } A
\Big) \,
\hat{Z}^N_{ \frac{ n T }{ N } }
+
\frac{ T }{ N } \cdot
F\big(
\hat{Z}^N_{ \frac{ n T }{ N } }
\big)
\\ & \quad +
\int_{ \frac{ n T }{ N } }^{
\frac{ ( n + 1 ) T }{ N }
}
B\big(
\hat{Z}^N_{ \frac{ n T }{ N } }
\big) \,
dW_s
+
\int_{ \frac{ n T }{ N } }^{
\frac{ ( n + 1 ) T }{ N }
}
B'\big(
\hat{Z}^N_{ \frac{ n T }{ N } }
\big) \bigg(
\int_{ \frac{ n T }{ N } }^s
B\big(
\hat{Z}^N_{ \frac{ n T }{ N } }
\big) \,
dW_u
\bigg) \,
dW_s
\Bigg)
\end{split}
\end{equation}
$ \mathbb{P} $-a.s.\ for
all
$ n \in \left\{ 0, 1, \dots, N-1
\right\} $
and all
$ N \in \mathbb{N} $.
The mild It\^{o} processes
$ \hat{Z}^N $, $ N \in \mathbb{N} $,
are thus nothing else but appropriate
time continuous interpolations
of linear implicit
Crank-Nicolson-Milstein
approximations for SPDEs
(see \cite{ms06,bl11b}).
\bibliographystyle{acm}
|
{
"timestamp": "2012-07-26T02:07:16",
"yymm": "1009",
"arxiv_id": "1009.3526",
"language": "en",
"url": "https://arxiv.org/abs/1009.3526"
}
|
\section{Introduction}
An understanding of cold quark matter is both one of the most challenging problem in
particle physics and a prerequisite to understand the true nature
of pulsars and pulsar-like objects. However, due to both non-perturbative
nature of strong interaction at low energy and complexity presented
by the quantum many-body problem, it is almost impossible to understand
such state theoretically from first principles.
Over the decades various approaches to bypass these difficulties has been developed,
both perturbatively such as that of color super-conductivity~\cite{alford08} and non-perturbatively such as
lattice QCD and QCD-based effective models. On the other hand, it has been conjectured~\cite{xu03}
that quark matter could be in a solid state at extreme low temperature present
in pulsar interior. This possibility could combine naturally with several previous works suggesting
the possibility that deconfined quark matter might contain quark clusters of $3N$ valence
quarks~\cite{clark86,schulz87,bpzszp88} into a reasonable conjecture that quark clusters
could form crystal lattices.
Because of the difficulty to obtain detail of the interaction between quark clusters and therefore equation of state
of cold quark matter at a few times nuclear densities,
it is interesting to apply simple phenomenological models. If we can use astronomical
observations to constrain parameters in such models, we will be able to gain some insight on
properties of low-energy QCD or rule out such form of cold quark matter within pulsars.
In several previous works (\cite{xylai09Poly},\cite{xylai09LJ}), different models have been tried to investigate the possible equation of state of solid quark matter and have provided possibility to explain stiffness in equation of state required by observed massive pulsars~\cite{massiveNS08}.
In \cite{xylai09LJ} the Lennard-Jones potential which was introduced to model interaction between inert gas
molecules~\cite{LJ} is used as potential between two quark clusters. The Lennard-Jones potential shares some basic properties with nuclear forces such as short-range repulsion and longer range attraction (c.f.~\citep{kranebook88}).
In this article we adopt a more realistic parametrization that shares great similarity with various models
depicting hyperon-hyperon potential. It has been shown that
the interaction between two H-dibaryons -- cluster of 6 valence quarks may also share this general feature~\citep{sakai97}.
This article is arranged as follows. The model of inter-cluster potential is presented in Section 2.
Parameter space used for calculation is discussed in Section 3. Section 4 shows result of calculation.
Conclusion and some discussions are presented in section 5.
\section{Inter-cluster potential}
As in \cite{xylai09LJ} we consider quark clusters
with $3N$ valence quarks with $N=1,6$ and a simple cubic lattice structure is adopted for simplicity.
In the context of strange quark matter, these are particles
with the same valence quark composition as hyperons and the hypothetical `quark-alpha'~\citep{michel88}.
By extension of the Lennard-Jones potential, we adopt the following simple parametrization for effective interaction between
two quark clusters localized on crystal lattice sites,
\begin{equation}
v(r)=V_1e^{-\left(\frac{r}{r_1}\right)^2}-V_2e^{-\left(\frac{r}{r_2}\right)^2},
\end{equation}
where possible spin-dependent interactions are omitted for simplicity.
As is mentioned above, with the condition $V_1>V_2$, $r_1<r_2$, this potential qualitatively
reproduces the general feature of various successful phenomenological potentials
of nuclear interactions: soft-core repulsion at short range and attraction at longer range.
It turned out that maximum mass and mass-radius curve is
not sensitively depend on the value of $r_2$. Therefore
it is reasonable to fix $r_2$ at 2fm which is a typical range of nuclear force.
For simplicity we assume that cluster center-of-mass has a Gaussian wave packet with width $w$
as wave function
\begin{equation}
\psi_{\boldsymbol{r}_0,w}(\boldsymbol{r})=\frac{1}{\pi^{3/4}w^{3/2}}e^{-\frac{\vert \boldsymbol{r}-\boldsymbol{r}_0 \vert^2}{2w^2}},
\end{equation}
In~\cite{xylai09LJ} it is assumed that potential well created by surrounding clusters
with Lennard-Jones interaction is deep enough to trap quark clusters in the potential well.
To ensure that soft-core potential with given parameter can also achieve this we
adopt variational method, i.e. to determine the value of $w$ by minimizing total energy of a single cluster which is a sum of kinetic energy of the wave packet and potential energy
contributed by surrounding cluster lattices.
The result shows that with the range of
of parameters considered in this work, the width of wave packet is rather small
compared to inter-cluster distance and hence it makes sense to
speak of this system of clusters as quark clusters trapped in periodic lattice.
With this small width, the overlap between adjacent wave packets is negligible. Thus
it is reasonable to omit the difference
between fermionic and bosonic quark clusters.
To calculate total contribution to single cluster potential energy, a sum is taken
over a cube of $21^3$ lattices centered around the quark cluster under consideration.
The size of the cube is enough since cluster number density in this work will
not exceed $\sim 10 n_0$. Hence, twice the total contribution
to potential energy for a single cluster is
\end{multicols}
\ruleup
\begin{equation}
\label{one}
V(n)\equiv \left(\sum_{k_1=-10}^{10}\sum_{k_2=-10}^{10}\sum_{k_3=-10}^{10}\right)' \tilde{v}\left(w;\frac{\sqrt{k_1^2+k_2^2+k_3^2}}{n^{1/3}}\right),
\end{equation}
\ruledown \vspace{0.5cm}
\begin{multicols}{2}
\noindent where the prime means that the sum omits $k_1=k_2=k_3=0$, and
\begin{equation}
\tilde{v}(w;r)=\int d^3\boldsymbol{r}'\psi^*_{\textbf{0},w}(\boldsymbol{r}')v(r')\psi_{\boldsymbol{r},w}(\boldsymbol{r}'),
\end{equation}
is the expectation value of potential energy between two clusters.
The total energy density is then
\begin{equation}
\epsilon=\frac{n}{2}V(n,w)+nm+\frac{3}{4mw^2}+\frac{9}{8}(6\pi^2)^{1/3}\hbar vn^{4/3},
\end{equation}
where the third term comes from contribution of kinetic energy
and $w$ is treated as a function of number density $n$. Same as in~\cite{xylai09LJ} the fourth term
comes from zero-point energy of phonon in Debye's approximation, with
\begin{equation}
\frac{1}{v^3}=\frac{1}{3}\left(\frac{1}{v_\parallel^3}+\frac{2}{v_\perp^3}\right),
\end{equation}and $v_\parallel,v_\perp$ stands for sound velocity of longitude
and transverse modes respectively. Following the argument of~\cite{xylai09LJ}
we take $v=c/3$ because its value have only very small influence on the final result.
Ignoring effect of temperature the pressure can be derived as
\begin{align}
P=n^2\frac{d}{dn}\left(\frac{\epsilon}{n}\right)=\frac{n^2}{2}\frac{\partial V}{\partial n}+\frac{n^2}{2}\frac{\partial V}{\partial w}\frac{dw}{dn}-\frac{3}{2mw^3}\frac{dw}{dn}.
\end{align}
With energy density and pressure one can establish equation of state
and solve Tolman-Oppenheimer-Volkoff (TOV) equation with varying central density
to get the mass-radius relation.
In practice, it is more convenient to skip (numerical) determination of equation of state
and write the TOV equation in terms of $n(r)$ and $M(r)$,
\end{multicols}
\ruleup
\begin{align}
\frac{dn}{dr}&=-\frac{G}{r^2-2GMr}(P(n)+\epsilon(n))(M+4\pi r^3P(n))\left(\frac{dP}{dn}\right)^{-1}\\
\frac{dM}{dr}&=4\pi r^2\epsilon(n)\\
M(0)&=0, \quad n(0)=n_c
\end{align}
\ruledown \vspace{0.5cm}
\begin{multicols}{2}
\noindent where $n_c$ is central number density of quark cluster.
Because quark matter is usually expected to be self-bound at zero pressure (for instance
under bag model equation of state),
it is reasonable to simply adopt a truncation baryon number density $n_{\text{surf}}$ at the surface. In this
work we adopt $n_{\text{surf}}=2n_0$ where $n_0$ is baryon number density of normal nuclear matter.
\section{Parameter space}
As is stated above, $r_2$ is fixed to $2$fm. Therefore, we have 4 free parameters: quark cluster mass $m$; height of the two Gaussians in the potential $V_1,V_2$; range of repulsive core $r_1$.
It is appropriate to expect that the depth of attractive part of the potential might be of the same order of magnitude
as typical potential between two nucleons in nuclear matter.
Therefore we fix $V_2$ at $50$MeV and $100$MeV respectively. On the other hand, we fix the value of $m$ to $1$GeV and $6$GeV for 3-quark clusters and the aforementioned hypothetical `quark-alpha'. Here mass of 3-quark cluster is take from mass of $\Lambda$ hyperon ($1115$MeV). The 18-quark cluster -- the `quark-alpha' is assumed to contain 6 quarks of each flavor which may have a mass less than $6m_\Lambda$ but we omit this possibly small difference for simplicity. Thus all that are left is $V_1$ and $r_1$. We adopt a condition,
\begin{equation}
V_1r_1^3>V_2r_2^3,
\end{equation}which ensures that potential energy is always positive (i.e., repulsive) when density is very high.
\section{Results and conclusions}
With the above settings of parameter space, we drew 4 contour plots of maximum mass calculated for 4 different sets of $(m,V_2)$ which are shown in Fig.\ref{fig1}.
\begin{center}
\includegraphics[width=8cm]{f1.eps}
\figcaption{\label{fig1} Maximum mass in unit of solar mass of solid quark star. \emph{Upper left}: ~$m=1$GeV, $V_2=100$MeV; \emph{Upper right}: $m=1$GeV, $V_2=50$MeV; \emph{Lower left}: $m=6$GeV, $V_2=100$MeV; \emph{Lower right}: $m=6$GeV, $V_2=50$MeV. Boundary of contour plots are moved a little upper right than $V_1r_1^3=V_2r_2^3$ to avoid parameters that lead to large error in numerical calculations}
\end{center}
Typical mass-radius relation curves of these settings with maximum mass exceeding $1.9 M_\odot$ are also shown (Fig.\ref{fig2}). From Fig.\ref{fig1} we can see that for our simple soft-core parametrization, maximum mass can range roughly from below $1M_\odot$ to about $\sim 3M_\odot$ for solid quark stars with 3-quark cluster forming crystal lattice and from below $0.5M_\odot$ to about $2.1M_\odot$ for solid quark stars made up of `quark-alpha' particles. On the other hand typical mass-radius relations shown in Fig.\ref{fig2} are very similar to those calculated within bag model EoS (shown as gray curves in Fig.\ref{fig2}, adopted from model SS1, SS2 in~\cite{SS1SS2}) and M-R curves calculated in \cite{xylai09LJ}. This shows that at least for some region of parameter space our simple parametrization can also produce heavy maximum mass supported by observed value of $2M_\odot$. Inversely, with current observation we can already restrict parameters in this very simple model with only 4 parameters. For instance, to get maximum mass larger than $1.9M_\odot$ for $m=1$GeV and $V_1<6$GeV, we have to restrict $r_1$ to below $0.75$fm when $V_2=100$MeV and restrict $r_1\lesssim 0.6$fm and for $m=6$GeV it requires $r_1\lesssim 1$fm and $V_1 \gtrsim 8$GeV.
\begin{center}
\includegraphics[width=7.5cm]{f2.eps}
\figcaption{\label{fig2} Mass-radius relation curves for typical parameters giving large maximum masses. \emph{Solid}: $m=1$GeV, $V_1=3.2$GeV, $V_2=100$MeV, $r_1=0.68$fm; \emph{Dashed}: $m=1$GeV, $V_1=5$GeV, $V_2=50$MeV, $r_1=0.54$fm, \emph{Dotted}: $m=6$GeV, $V_1=9$GeV, $V_2=100$MeV, $r_1=0.95$fm; \emph{Gray solid and dashed}: Bag model EoS SS1 and SS2 adopted from~\citep{SS1SS2} }
\end{center}
\section{Discussion}
In cold quark matter at baryon number densities realistic for compact stars, the interaction between quarks could be strong enough that instead of condensating in momentum space to form color-superconductive phase it is possible that dressed quarks undergo condensation in position space to form quark clusters. As is stated in \cite{xylai09LJ} if the potential well formed by neighboring clusters are deep enough to trap quark cluster, cold quark matter could form crystal solid in low temperature.
In this work we discussed simple-cubic lattice structure formed by 3 and 18-quark clusters using a simple two Gaussian component parameterization of soft-core potential to simulate the interaction between quark clusters. This parametrization shares the basic properties of nucleon-nucleon interaction mediated by meson exchange -- short range repulsion, medium and long range attraction and a finite range. These properties are also shared by Lennard-Jones potential adopted in \cite{xylai09LJ}. However unlike Lennard-Jones potential with $r^{-12}$ pole at origin, soft-core potential adopted in this work can be treated by non-relativistic quantum mechanics. By minimization total energy we found that at realistic densities this soft-core potential can lead to a stable lattice structure. It is entirely possible that other unit cell structure (e.g. body-centered cubic) is more stable, but we expect the difference to be quantitative instead of qualitative.
It is also worth mentioning that despite great similarity between mass-radius relation obtained with bag model equation of state and those calculated here, the underlying picture is drastically different. In bag model, quark star is degenerate Fermi gas of free quarks sustained by vacuum energy and associated negative pressure. In this work and \cite{xylai09LJ} pressure is mainly provided by repulsive core of inter-cluster potential instead of mere degenerate pressure.
Although maximum mass cannot be easily tuned to $\sim 6M_\odot$ as in \cite{xylai09LJ} due to soft-core nature of the interaction, our parametrization can still provide a stable lattice crystal structure with maximum mass exceeding $2M_\odot$ which is in accordance with observed maximum mass\cite{massiveNS08}. Inversely, the observed maximum mass can be used to put constraints on parameters of this simple model which will possibly give some insights into the form of interaction between quark clusters if such phase exists.
\acknowledgments{We thank pulsar group at Peking University for useful discussions.}
\end{multicols}
\vspace{105mm}
\vspace{-1mm}
\centerline{\rule{80mm}{0.1pt}}
\vspace{2mm}
\begin{multicols}{2}
|
{
"timestamp": "2010-09-23T02:00:49",
"yymm": "1009",
"arxiv_id": "1009.4247",
"language": "en",
"url": "https://arxiv.org/abs/1009.4247"
}
|
\section{Introduction}
In a seminal paper \cite{Pawel} Pawe{\l} Horodecki provided an
example of a density operator living in $\Cd \ot \Cd$ which
represents entangled state positive under partial transposition
(PPT)
\begin{equation}\label{RHO}
\rho_a\, =\, N_a\left(\begin{array}{ccc|ccc|ccc}
a&\cdot&\cdot&\cdot&a&\cdot&\cdot&\cdot&a\\
\cdot&a&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&\cdot&a&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\hline
\cdot&\cdot&\cdot&a&\cdot&\cdot&\cdot&\cdot&\cdot\\
a&\cdot&\cdot&\cdot&a&\cdot&\cdot&\cdot&a\\
\cdot&\cdot&\cdot&\cdot&\cdot&a&\cdot&\cdot&\cdot\\
\hline
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&b&\cdot&c\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&a&\cdot\\
a&\cdot&\cdot&\cdot&a&\cdot&c&\cdot&b
\end{array}\right) \ ,
\end{equation}
with
\begin{equation}\label{}
N_a = \frac{1}{8a+1}\ ,\ \ \ \ b = \frac{1+a}{2}\ , \ \ \ \ c = \frac{\sqrt{1-a^2}}{2}\ ,
\end{equation}
and $a \in [0,1]$. The above matrix representation corresponds to
the standard computational basis $|ij\> = |i\> \ot |j\>$ in $\Cd\ot
\Cd$ and to make the picture more transparent we replaced all zeros
by dots. Since the partial transposition $\rho_a^\Gamma = (\oper \ot
{\rm T})\rho_a \geq 0$ the state is PPT for all $a\in [0,1]$. It is
easy to show that for $a=0$ and $a=1$ the state is separable and it
was shown \cite{Pawel} that for $a \in (0,1)$ the state is entangled
(for the recent reviews of quantum entanglement and the methods of
its detection see \cite{HHHH} and \cite{Guhne}). Actually, the
family (\ref{RHO}) provides one of the first examples of bound
entanglement. In this Letter we analyze the structure of
(\ref{RHO}). In particular we study its symmetry group.
\section{Symmetry group}
Let $G$ be a subgroup of the unitary group $U(d)$ (a group of
unitary $d \times d$ matrices). A state $\rho$ living in
$\mathbb{C}^d \ot \mathbb{C}^d$ is $G \ot \overline{G}$--invariant
if
\begin{equation}\label{iso}
U \ot \overline{U} \rho = \rho\, U \ot \overline{U} \ ,
\end{equation}
where $U \in G$, and $\overline{U}$ denotes the complex conjugation
of the matrix elements with respect to the computational basis
$|i\>$. It is clear that if $\rho$ is $G \ot
\overline{G}$--invariant then its partial transposition is $G \ot
G$--invariant, that is
\begin{equation}\label{werner}
U \ot {U} \rho = \rho\, U \ot U \ ,
\end{equation}
where $U \in G$. Recall, that if $G = U(d)$, then $G \ot
\overline{G}$--invariant states define a class of isotropic states
\cite{ISO}, whereas $G \ot G$--invariant states define a class of
Werner states \cite{Werner1}. Recently \cite{PPT-nasza} we found a
class of $G \ot \overline{G}$--invariant states, where $G$ defines a
maximal abelian subgroup of $U(d)$ defined as follows:
\begin{equation}\label{U-x}
U_\mathbf{x} = \exp\left(i \sum_{k=1}^{d} x_k |k\>\<k| \right)\ ,
\end{equation}
and $\mathbf{x}=(x_1,\ldots,x_{d}) \in \mathbb{R}^d$. It was shown
\cite{PPT-nasza} that states invariant under the maximal abelian
subgroup have the following structure
\begin{equation}\label{}
\rho = \sum_{i,j=1}^d a_{ij}\, |i\>\<j| \ot |i\>\<j| + \sum_{i\neq
j=1}^d d_{ij}\, |i\>\<i| \ot |j\>\<j|\ ,
\end{equation}
where the matrix $||a_{ij}|| \geq 0$, and the numbers $d_{ij} \geq
0$. The normalization condition gives
\[ \sum_{i=1}^d a_{ii} + \sum_{i\neq j=1}^d d_{ij} = 1 \ . \]
The corresponding matrix representation for $d=3$ reads as follows
\begin{equation}\label{RHO-abelian}
\rho\, =\, \left(\begin{array}{ccc|ccc|ccc}
a_{11}&\cdot&\cdot&\cdot&a_{12}&\cdot&\cdot&\cdot&a_{13}\\
\cdot&d_{12}&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&\cdot&d_{13}&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\hline
\cdot&\cdot&\cdot&d_{21}&\cdot&\cdot&\cdot&\cdot&\cdot\\
a_{21}&\cdot&\cdot&\cdot&a_{22}&\cdot&\cdot&\cdot&a_{23}\\
\cdot&\cdot&\cdot&\cdot&\cdot&d_{23}&\cdot&\cdot&\cdot\\
\hline
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&d_{31}&\cdot&\cdot\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&d_{32}&\cdot\\
a_{31}&\cdot&\cdot&\cdot&a_{32}&\cdot&\cdot&\cdot&a_{33}
\end{array}\right) \ .
\end{equation}
Let us observe that (\ref{RHO-abelian}) is PPT if and only if
\begin{equation}\label{}
d_{ij} d_{ji} \geq |a_{ij}|^2\ , \ \ \ \ i\neq j\ .
\end{equation}
Surprisingly many well know states considered in the literature
belong to this class (see \cite{PPT-nasza} for examples). Note,
however, that Horodecki state (\ref{RHO}) does not belong to
(\ref{RHO-abelian}) unless $a=1$. Consider now a subgroup $G_0$ of
the $G$ defined by (\ref{U-x}) with $x_1=x_3$. One finds the
following structure of invariant states
\begin{equation} \label{RHO-13}
\rho = \left( \begin{array}{ccc|ccc|ccc}
\rho_{11} & \cdot & \rho_{13} & \cdot & \rho_{15} & \cdot & \rho_{17} & \cdot & \rho_{19} \\
\cdot & \rho_{22} & \cdot & \cdot & \cdot & \cdot & \cdot &
\rho_{28} &
\cdot \\
\rho_{31} & \cdot & \rho_{33} & \cdot & \rho_{35} & \cdot &
\rho_{37} & \cdot & \rho_{39} \\ \hline \cdot & \cdot & \cdot &
\rho_{44} & \cdot & \rho_{46} & \cdot & \cdot &
\cdot \\
\rho_{51} & \cdot & \rho_{53} & \cdot & \rho_{55} & \cdot &
\rho_{57} & \cdot &
\rho_{59} \\
\cdot & \cdot & \cdot & \rho_{64} & \cdot & \rho_{66} & \cdot & \cdot & \cdot \\
\hline
\rho_{71} & \cdot & \rho_{73} & \cdot & \rho_{75} & \cdot & \rho_{77} & \cdot & \rho_{79} \\
\cdot & \rho_{82} & \cdot & \cdot & \cdot & \cdot & \cdot &
\rho_{88} &
\cdot \\
\rho_{91} & \cdot & \rho_{93} & \cdot & \rho_{95} & \cdot &
\rho_{97} & \cdot & \rho_{99}
\end{array} \right) \ ,
\end{equation}
and it evidently contains Horodecki state (\ref{RHO}).
Interestingly, invariant states (\ref{RHO-13}) have almost perfect
chessboard structure \cite{Bruss} (see also the recent paper
\cite{Djokovic}. Note, however, that only a subclass of states
considered in \cite{Bruss,Djokovic} are $G_0 \ot G_0$--invariant.
The characteristic feature of (\ref{RHO-13}) is that $\rho$ has a
direct sum structure $\rho = \rho_1 \oplus \rho_2 \oplus \rho_3$\,
where the corresponding operators $\rho_k$ are supported on
$\mathcal{H}_k$
\begin{eqnarray} \label{3H}
\mathcal{H}_1 &=& \rm{span}_{\,\mathbb{C}} \{\, |11\>,\ |13\>,\ |22\>,\ |31\>,\ |33\>\, \} \ , \nonumber\\
\mathcal{H}_2 &=& \rm{span}_{\,\mathbb{C}} \{\, |12\>,\ |32\>\, \} \ , \\
\mathcal{H}_3 &=& \rm{span}_{\,\mathbb{C}} \{\, |21\>,\ |23\>\, \} \nonumber \
,
\end{eqnarray}
giving rise to the direct sum decomposition
$ \mathbb{C}^3 \ot \mathbb{C}^3 = \mathcal{H}_1 \oplus
\mathcal{H}_2 \oplus \mathcal{H}_3$.
Similarly, the partial transposition
\begin{equation}\label{Gamma}
\rho^\Gamma = \left( \begin{array}{ccc|ccc|ccc}
\rho_{11} & \cdot & \rho_{31} & \cdot & \cdot & \cdot & \rho_{17} & \cdot & \rho_{37} \\
\cdot & \rho_{22} & \cdot & \rho_{15} & \cdot & \rho_{35} & \cdot &
\rho_{28} &
\cdot \\
\rho_{13} & \cdot & \rho_{33} & \cdot & \cdot & \cdot & \rho_{19} &
\cdot & \rho_{39} \\ \hline \cdot & \rho_{51} & \cdot & \rho_{44} &
\cdot & \rho_{64} & \cdot & \rho_{57} &
\cdot \\
\cdot & \cdot & \cdot & \cdot & \rho_{55} & \cdot & \cdot & \cdot &
\cdot \\
\cdot & \rho_{53} & \cdot & \rho_{46} & \cdot & \rho_{66} & \cdot & \rho_{59} & \cdot \\
\hline
\rho_{71} & \cdot & \rho_{91} & \cdot & \cdot & \cdot & \rho_{77} & \cdot & \rho_{97} \\
\cdot & \rho_{82} & \cdot & \rho_{75} & \cdot & \rho_{95} & \cdot &
\rho_{88} &
\cdot \\
\rho_{73} & \cdot & \rho_{93} & \cdot & \cdot & \cdot & \rho_{79} &
\cdot & \rho_{99}
\end{array} \right)
\end{equation}
has a direct sum structure $\rho^\Gamma = \widetilde{\rho}_1 \oplus
\widetilde{\rho}_2 \oplus \widetilde{\rho}_3$\, where the
corresponding operators $\widetilde{\rho}_k$ are supported on
$\widetilde{\mathcal{H}}_k$
\begin{eqnarray} \label{3Ha}
\widetilde{\mathcal{H}}_1 &=& \rm{span}_{\,\mathbb{C}} \{\, |11\>,\ |13\>,\ |31\>,\ |33\>\, \} \ , \nonumber\\
\widetilde{\mathcal{H}}_2 &=& \rm{span}_{\,\mathbb{C}} \{\, |12\>,\ |21\>,\ |23\>,\ |32\>\, \} \ , \\
\widetilde{\mathcal{H}}_3 &=& \rm{span}_{\,\mathbb{C}} \{\, |22\>\, \} \nonumber \
,
\end{eqnarray}
together with $ \mathbb{C}^3 \ot \mathbb{C}^3 =
\widetilde{\mathcal{H}}_1 \oplus \widetilde{\mathcal{H}}_2 \oplus
\widetilde{\mathcal{H}}_3$. Interestingly one has
\begin{equation}\label{HHHH}
\widetilde{\mathcal{H}}_1 \oplus \widetilde{\mathcal{H}}_3 =
\mathcal{H}_1\ , \ \ \ \mathcal{H}_2 \oplus \mathcal{H}_3 =
\widetilde{\mathcal{H}}_2\ .
\end{equation}
Hence to check for PPT one needs to check positivity of two $4\times
4$ leading submatrices of (\ref{Gamma}). Note, that decompositions
(\ref{3H}) and (\ref{3Ha}) remind the characteristic circulant
decompositions \cite{CIRCULANT}. There is however important
difference: (\ref{3H}) and (\ref{3Ha}) are governed by the symmetry
group $G_0$ whereas the circulant decompositions are not directly
related to any symmetry. For other types of decompositions which
simplify PPT conditions see also \cite{Brazylia}.
\section{Another representations of the Horodecki state}
Consider now another commutative subgroup $G_0'$ defined by
$x_1=x_2$. It is clear that
\begin{equation}\label{}
G_0' = S' G_0 S'^\dagger\ ,
\end{equation}
where $S'$ represents permutation $(1,2,3) \rightarrow (1,3,2)\,$,
that is
\begin{equation}\label{}
S' = \left( \begin{array}{ccc} 1 & \cdot & \cdot \\ \cdot & \cdot & 1 \\
\cdot & 1 & \cdot \end{array} \right) \ .
\end{equation}
Hence a class of $G_0' \ot \overline{G}_0'$--invariant states is
defined by
\begin{equation}\label{}
\rho' = S' \ot S'\, \rho\, S'^\dagger \ot S'^\dagger\ ,
\end{equation}
where $\rho$ is $G_0 \ot \overline{G}_0$--invariant. The
corresponding matrix representation of $\rho'$ has the following
form
\begin{equation}\label{1=2}
\rho' = \left( \begin{array}{ccc|ccc|ccc}
\rho_{11} & \rho_{12} & \cdot & \rho_{14} & \rho_{15} & \cdot & \cdot & \cdot & \rho_{19} \\
\rho_{21} & \rho_{22} & \cdot & \rho_{24} & \rho_{25} & \cdot &
\cdot & \cdot &
\rho_{29} \\
\cdot & \cdot & \rho_{33} & \cdot & \cdot & \rho_{36} & \cdot & \cdot & \cdot \\
\hline \rho_{41} & \rho_{42} & \cdot & \rho_{44} & \rho_{45} & \cdot
& \cdot & \cdot & \rho_{49}
\\
\rho_{51} & \rho_{52} & \cdot & \rho_{54} & \rho_{55} & \cdot &
\cdot & \cdot &
\rho_{59} \\
\cdot & \cdot & \rho_{63} & \cdot & \cdot & \rho_{66} & \cdot & \cdot & \cdot \\
\hline \cdot & \cdot
& \cdot & \cdot & \cdot & \cdot & \rho_{77} & \rho_{78} & \cdot \\
\cdot & \cdot
& \cdot & \cdot & \cdot & \cdot & \rho_{87} & \rho_{88} & \cdot \\
\rho_{91} & \rho_{92} & \cdot & \rho_{94} & \rho_{95} & \cdot &
\cdot & \cdot & \rho_{99}
\end{array} \right)\ .
\end{equation}
In particular one obtains the following representation of the
Horodecki state invariant under $G_0'$
\begin{equation}\label{}
{\rho_a}' = S' \ot S'\, \rho_a\, S'^\dagger \ot S'^\dagger\ ,
\end{equation}
or in the matrix form
\begin{equation}\label{RHO-12}
{\rho_a}'\, =\, N_a\left(\begin{array}{ccc|ccc|ccc}
b&c&\cdot&\cdot&a&\cdot&\cdot&\cdot&a\\
c&b&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&\cdot&a&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\hline
\cdot&\cdot&\cdot&a&\cdot&\cdot&\cdot&\cdot&\cdot\\
a&\cdot&\cdot&\cdot&a&\cdot&\cdot&\cdot&a\\
\cdot&\cdot&\cdot&\cdot&\cdot&a&\cdot&\cdot&\cdot\\
\hline
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&a&\cdot&\cdot\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&a&\cdot\\
a&\cdot&\cdot&\cdot&a&\cdot&\cdot&\cdot&a
\end{array}\right) \ .
\end{equation}
The characteristic feature of (\ref{1=2}) is that $\rho'$ has a
direct sum structure $\rho' = \rho_1' \oplus \rho_2' \oplus
\rho_3'$\, where the corresponding operators $\rho_k$ are supported
on $\mathcal{H}_k'$
\begin{eqnarray}
\mathcal{H}_1' &=& (S'\ot S')\mathcal{H}_1 = \rm{span}_{\,\mathbb{C}} \{\, |11\>,\ |12\>,\ |21\>,\ |22\>,\ |33\>\, \} \ , \nonumber\\
\mathcal{H}_2' &=& (S'\ot S')\mathcal{H}_2 =\rm{span}_{\,\mathbb{C}} \{\, |13\>,\ |23\>\, \} \ , \\
\mathcal{H}_3' &=& (S'\ot S')\mathcal{H}_3 =\rm{span}_{\,\mathbb{C}} \{\, |31\>,\ |32\>\, \} \nonumber
\ .
\end{eqnarray}
One easily finds for the partial transposition
\begin{equation}\label{}
{\rho'}^\Gamma = \left( \begin{array}{ccc|ccc|ccc}
\rho_{11} & \rho_{21} & \cdot & \rho_{14} & \rho_{24} & \cdot & \cdot & \cdot & \cdot \\
\rho_{12} & \rho_{22} & \cdot & \rho_{15} & \rho_{25} & \cdot &
\cdot & \cdot &
\cdot \\
\cdot & \cdot & \rho_{33} & \cdot & \cdot & \rho_{36} & \rho_{19} & \rho_{29} & \cdot \\
\hline \rho_{41} & \rho_{51} & \cdot & \rho_{44} & \rho_{54} & \cdot
& \cdot & \cdot & \cdot
\\
\rho_{42} & \rho_{52} & \cdot & \rho_{45} & \rho_{55} & \cdot &
\cdot & \cdot &
\cdot \\
\cdot & \cdot & \rho_{63} & \cdot & \cdot & \rho_{66} & \rho_{49} & \rho_{59} & \cdot \\
\hline \cdot & \cdot
& \rho_{91} & \cdot & \cdot & \rho_{94} & \rho_{77} & \rho_{87} & \cdot \\
\cdot & \cdot
& \rho_{92} & \cdot & \cdot & \rho_{95} & \rho_{78} & \rho_{88} & \cdot \\
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot &
\rho_{99}
\end{array} \right)\ .
\end{equation}
It is evident that ${\rho'}^\Gamma$ has a direct sum structure
${\rho'}^\Gamma = \widetilde{\rho}_1' \oplus \widetilde{\rho}_2'
\oplus \widetilde{\rho}_3'$\, where the corresponding operators
$\widetilde{\rho}_k'$ are supported on $\mathcal{H}_k'$
\begin{eqnarray}
\widetilde{\mathcal{H}}_1' &=& (S'\ot S')\widetilde{\mathcal{H}}_1 = \rm{span}_{\,\mathbb{C}} \{\, |11\>,\ |12\>,\ |21\>,\ |22\>,\, \} \ , \nonumber\\
\widetilde{\mathcal{H}}_2' &=& (S'\ot S')\widetilde{\mathcal{H}}_2 = \rm{span}_{\,\mathbb{C}} \{\, |13\>,\ |23\>, \, |31\>,\ |32\>\, \} \ , \\
\widetilde{\mathcal{H}}_3' &=& (S'\ot S')\widetilde{\mathcal{H}}_3 = \rm{span}_{\,\mathbb{C}} \{\, |33\>\, \} \nonumber
\ .
\end{eqnarray}
Again the analog of the formulae (\ref{HHHH}) holds, that is
\begin{equation}\label{HHHH-prime}
\widetilde{\mathcal{H}}_1' \oplus \widetilde{\mathcal{H}}_3' =
\mathcal{H}_1'\ , \ \ \ \mathcal{H}_2' \oplus \mathcal{H}_3' =
\widetilde{\mathcal{H}}_2'\ .
\end{equation}
Finally, let us consider another commutative subgroup $G_0''$ of $G$
defined by $x_2=x_3$. It is clear that
\begin{equation}\label{}
G_0'' = S'' G_0 S''^\dagger\ ,
\end{equation}
where $S''$ represents permutation $(1,2,3) \rightarrow (2,1,3)\,$,
that is
\begin{equation}\label{}
S'' = \left( \begin{array}{ccc} \cdot & 1 & \cdot \\ 1 & \cdot & \cdot \\
\cdot & \cdot & 1 \end{array} \right) \ .
\end{equation}
Hence a class of $G_0'' \ot \overline{G}_0''$--invariant states is
defined by
\begin{equation}\label{}
\rho'' = S'' \ot S''\, \rho\, S''^\dagger \ot S''^\dagger\ ,
\end{equation}
where $\rho$ is $G_0 \ot \overline{G}_0$--invariant. The
corresponding matrix representation of $\rho''$ has the following
form
\begin{equation}\label{2=3}
\rho'' = \left( \begin{array}{ccc|ccc|ccc}
\rho_{11} & \cdot & \cdot & \cdot & \rho_{15} & \rho_{16} & \cdot & \rho_{18} & \rho_{19} \\
\cdot & \rho_{22} & \rho_{23} & \cdot & \cdot & \cdot & \cdot &
\cdot & \cdot \\
\cdot & \rho_{32} & \rho_{33} & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\
\hline \cdot & \cdot & \cdot & \rho_{44} & \cdot & \cdot & \rho_{47}
& \cdot & \cdot \\
\rho_{51} & \cdot & \cdot & \cdot & \rho_{55} & \rho_{56} & \cdot &
\rho_{58} & \rho_{59} \\
\rho_{61} & \cdot & \cdot & \cdot & \rho_{65} & \rho_{66} & \cdot & \rho_{68} & \rho_{69} \\
\hline \cdot & \cdot
& \cdot & \rho_{74} & \cdot & \cdot & \rho_{77} & \cdot & \cdot \\
\rho_{81} & \cdot
& \cdot & \cdot & \rho_{85} & \rho_{86} & \cdot & \rho_{88} & \rho_{89} \\
\rho_{91} & \cdot & \cdot & \cdot & \rho_{95} & \rho_{96} & \cdot
& \rho_{98} & \rho_{99}
\end{array} \right)\ .
\end{equation}
In particular one obtains the following representation of the
Horodecki state invariant under $G''_0$
\begin{equation}\label{}
{\rho_a}'' = S'' \ot S''\, \rho_a\, S''^\dagger \ot S''^\dagger\ ,
\end{equation}
that is,
\begin{equation}\label{RHO-23}
{\rho_a}''\, =\, N_a\left(\begin{array}{ccc|ccc|ccc}
a&\cdot&\cdot&\cdot&a&\cdot&\cdot&\cdot&a\\
\cdot&a&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&\cdot&a&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\hline
\cdot&\cdot&\cdot&a&\cdot&\cdot&\cdot&\cdot&\cdot\\
a&\cdot&\cdot&\cdot&b&c&\cdot&\cdot&a\\
\cdot&\cdot&\cdot&\cdot&c&b&\cdot&\cdot&\cdot\\
\hline
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&a&\cdot&\cdot\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&a&\cdot\\
a&\cdot&\cdot&\cdot&a&\cdot&\cdot&\cdot&a
\end{array}\right) \ .
\end{equation}
Again, the characteristic feature of (\ref{2=3}) is that $\rho''$
has a direct sum structure $\rho'' = \rho_1'' \oplus \rho_2'' \oplus
\rho_3''$\, where the corresponding operators $\rho_k$ are supported
on $\mathcal{H}_k''$
\begin{eqnarray}
\mathcal{H}_1'' &=& (S''\ot S'')\mathcal{H}_1 = \rm{span}_{\,\mathbb{C}} \{\, |11\>,\ |23\>,\ |22\>,\ |32\>,\ |33\>\, \}\ , \nonumber\\
\mathcal{H}_2'' &=& (S''\ot S'')\mathcal{H}_2 = \rm{span}_{\,\mathbb{C}} \{\, |21\>,\ |31\>\, \} \ , \\
\mathcal{H}_3'' &=& (S''\ot S'')\mathcal{H}_3 = \rm{span}_{\,\mathbb{C}} \{\, |12\>,\ |13\>\, \} \nonumber
\ .
\end{eqnarray}
One easily finds for the partial transposition
\begin{equation}\label{2=3-Gamma}
{\rho''}^\Gamma = \left( \begin{array}{ccc|ccc|ccc}
\rho_{11} & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\
\cdot & \rho_{22} & \rho_{32} & \rho_{15} & \cdot & \cdot &
\rho_{18} &
\cdot & \cdot \\
\cdot & \rho_{23} & \rho_{33} & \rho_{16} & \cdot & \cdot & \rho_{19} & \cdot & \cdot \\
\hline \cdot & \rho_{51} & \rho_{61} & \rho_{44} & \cdot & \cdot &
\rho_{47}
& \cdot & \cdot \\
\cdot & \cdot & \cdot & \cdot & \rho_{55} & \rho_{65} & \cdot & \rho_{58} & \rho_{68} \\
\cdot & \cdot & \cdot & \cdot & \rho_{56} & \rho_{66} & \cdot & \rho_{59} & \rho_{69} \\
\hline \cdot & \rho_{81}
& \rho_{91} & \rho_{74} & \cdot & \cdot & \rho_{77} & \cdot & \cdot \\
\cdot & \cdot
& \cdot & \cdot & \rho_{85} & \rho_{95} & \cdot & \rho_{88} & \rho_{98} \\
\cdot & \cdot & \cdot & \cdot & \rho_{86} & \rho_{96} & \cdot &
\rho_{89} & \rho_{99}
\end{array} \right)\ ,
\end{equation}
which is supported the direct product of three subspaces
\begin{eqnarray}
\widetilde{\mathcal{H}}_1'' &=& (S''\ot S'')\widetilde{\mathcal{H}}_1 = \rm{span}_{\,\mathbb{C}} \{\, |21\>,\ |23\>,\ |32\>,\ |33\>\, \} \ , \nonumber\\
\widetilde{\mathcal{H}}_2'' &=& (S''\ot S'')\widetilde{\mathcal{H}}_2 = \rm{span}_{\,\mathbb{C}} \{\, |12\>,\ |21\>,\ |13\>,\ |31\>\, \} \ , \\
\widetilde{\mathcal{H}}_3'' &=& (S''\ot S'')\widetilde{\mathcal{H}}_3 = \rm{span}_{\,\mathbb{C}} \{\, |11\>\, \} \nonumber
\ .
\end{eqnarray}
It is evident that the analog of (\ref{HHHH}) is satisfied for
$\mathcal{H}_k''$ and $ \widetilde{\mathcal{H}}_k''$.
\section{Conlcusions}
We shown that the celebrated Horodecki state \cite{Pawel} belongs to
a class of states invariant under a commutative subgroup $G_0$ of
$U(3)$. Taking conjugate subgroups $G_0'$ and $G_0''$ we provided
another classes of invariant states. In particular we found
equivalent representations of the Horodecki state invariant under
$G_0'$ and $G_0''$, respectively (cf. formulae (\ref{RHO-12}) and
(\ref{RHO-23})). Interestingly, known entanglement witnesses
detecting PPT entangled state (\ref{RHO}) display $G_0$-invariance
(see \cite{W1,W2}). It should be clear that our discussion can be
immediately generalized from $3 \ot 3$ to $d \ot d$ ($d$ arbitrary
but finite). Now, the maximal commutative subgroup of $U(d)$ defined
by (\ref{U-x}) gives rise to a number of subgroups corresponding to
$x_{k_1} = \ldots = x_{k_l}$. In particular using a subgroup defined
by $x_1=x_d$ one may introduce the generalized Horodecki state in
$d \ot d$. We believe that our discussion opens new perspectives to
study symmetric states of composite quantum systems. It would be
interesting to generalize our analysis to multipartite case
\cite{multi1,multi2}.
\section*{Acknowledgments}
This work was partially supported by
the Polish Ministry of Science and Higher Education Grant No
3004/B/H03/2007/33.
|
{
"timestamp": "2010-09-23T02:02:20",
"yymm": "1009",
"arxiv_id": "1009.4385",
"language": "en",
"url": "https://arxiv.org/abs/1009.4385"
}
|
\section*{Acknowledgments}
\section{Introduction and Motivation}
\input{intro}
\section{Volatility Contributions under Serial Correlation}
\input{volaContrib}
\section{Scaling in ARMA and VARMA Models}
\input{tsModels}
\section*{Appendix: Proofs}
\begin{proof}[Proof of Proposition~\ref{prop:scaling.univariate} and Corollary~\ref{cor:scaling.univariate}]
Considering that $\sigma(\lambda,d)$ is the square-root of $\var{\sum_{i=1}^d X_i(\lambda)}$ and writing down this variance in matrix form using
weak stationarity we get
\begin{align*}
\var{\sum_{i=1}^d X_i(\lambda)} &=\textbf{1}^T \begin{pmatrix} \covar{X_1(\lambda)}{X_1(\lambda)} & \dots & \covar{X_1(\lambda)}{X_d(\lambda)}\\
\covar{X_2(\lambda)}{X_1(\lambda)} &\dots & \covar{X_2(\lambda)}{X_d(\lambda)}\\
\vdots&\ddots&\vdots\\
\covar{X_d(\lambda)}{X_1(\lambda)} &\dots & \covar{X_d(\lambda)}{X_d(\lambda)}\end{pmatrix} \textbf{1} \\
&=\textbf{1}^T \begin{pmatrix} \gamma(0) & \gamma(1) &\dots & \gamma(d-1)\\
\gamma(1) & \gamma(0) &\dots & \gamma(d-2)\\
\vdots&\vdots&\ddots&\vdots\\
\gamma(d-1) & \gamma(d-2) &\dots & \gamma(0)\end{pmatrix} \textbf{1},
\end{align*}
where $\gamma(\cdot)$ denotes the auto-covariance function of $(X_t(\lambda))_{t \in \za}$ and $\textbf{1}=(1,\dots,1)^T$. Summing up along the diagonals and using symmetry we get Equation~\eqref{eq:scaling.unvariate}.
For proving~\eqref{eq:scaling.unvariate.constant} note that the auto-covariances can be expressed in terms of the auto-correlation and the variance in the following sense:
\begin{align*}
\rho(k)=\frac{\gamma(k)}{\gamma(0)} \Leftrightarrow \gamma(k) = \gamma(0) \rho(k) = \sigma(\lambda)^2 \rho(k),
\end{align*}
for $k = 0,1,\ldots$
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:vola.contrib.multi} and Corollary~\ref{cor:scaling.vola.corr}]
Following the Euler allocation rule the volatility contributions are given by
\[
\sigma_i(\lambda,d) = \lambda_i \frac{\covar{\sum_{k=1}^d X^i_k}{\sum_{k=1}^d X_k(\lambda)}}{\sigma(\lambda,d)},
\]
where we can calculate the denominator $\sigma(\lambda,d)$ by~\eqref{eq:scaling.unvariate} and~\eqref{eq:acf.pf.mult}.
For the numerator we get
\begin{align*}
\covar{\sum_{k=1}^d X^i_k}{\sum_{k=1}^d X_k(\lambda)} &= \textbf{1}^T \begin{pmatrix} \covar{X_1^i}{X_1(\lambda)} &\dots & \covar{X_1^i}{X_{d}(\lambda)}\\
\covar{X_2^i}{X_1(\lambda)} &\dots & \covar{X_2^i}{X_{d}(\lambda)} \\
\vdots&\ddots&\vdots\\
\covar{X_d^i}{X_1(\lambda)} &\dots & \covar{X_d^i}{X_d(\lambda)}\end{pmatrix} \textbf{1}\\
&= \textbf{1}^T \begin{pmatrix} \gamma_i(0) & \gamma_i(1) &\dots & \gamma_i(d-1)\\
\gamma_i(1) & \gamma_i(0) &\dots & \gamma_i(d-2)\\
\vdots&\vdots&\ddots&\vdots\\
\gamma_i(d-1) & \gamma_i(d-2) &\dots & \gamma_i(0)\end{pmatrix} \textbf{1},
\end{align*}
where $\textbf{1} = (1,\ldots,1)^T$ and the $\gamma_i(k)$ are given in~\eqref{eq:multi.covar1} for $i=1,\ldots,n$ and $k = 0,1,\ldots$
Now again use the symmetries and sum up along the diagonals to get the result.
To prove the corollary, recall that $\sigma(\lambda,d) = \sigma(\lambda) \delta(d)$. Plugging this
into the denominator of~\eqref{eq:vola.contrib.multi} and recalling that the one day risk contribution is given by $\sigma_i(\lambda) = \lambda_i \frac{\gamma_i(0)}{\sigma(\lambda)}$ we get
\begin{align*}
\sigma_i(\lambda,d) = \lambda_i \frac{\gamma_i(0)}{\sigma(\lambda)} \left(d + \frac2{\gamma_i(0)} \sum_{k=1}^{d-1} (d-k) \gamma_i(k)\right) / \delta(d),
\end{align*}
which gives the form of the factor $\delta(i,d)$ for $i=1,\ldots,n$.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:VAR.AR.1} and Corollary~\ref{cor:VAR.AR.1}]
Considering Proposition~\ref{prop:VAR.AR.1} we use that by assumption $(X_t(\lambda))_{t \in \za}$ is an $\AR{p}$-process as in~\eqref{eq:VAR.AR.1.ARP} and the identity $\lambda \X_t = X_t$:
\begin{align*}
\lambda^T \X_t &= X_t(\lambda) \\
\Leftrightarrow \sum_{k=1}^p \lambda^T \Phi_k \X_{t-k} + \lambda^T\Z_t &= \sum_{k=1}^p \phi_k \lambda^T \X_{t-k} + \lambda^T\Z_t \\
\Leftrightarrow \lambda^T \sum_{k=1}^p (\Phi_k - \phi_k I_n)\X_{t-k}& = 0.
\end{align*}
As $(\X_t)_{t \in \za}$ takes values in $\re^n$ this forces
\begin{align*}
\lambda^T (\Phi_k - \phi_k I_n) = 0\quad\text{for } k= 1,\ldots,p.
\end{align*}
To prove Corollary~\ref{cor:VAR.AR.1} consider that the above condition is true for arbitrary $\lambda \in \re^n$ if and only if
the rank of the matrix $\Phi_k - \phi_k I_n$ is zero for $k= 1,\ldots,p$ which concludes the proof.
\end{proof}
\subsection{Closing time problem}
Before we consider the closing time problem it is essential to understand the interplay between multivariate time series and portfolios constructed from these. Creating a portfolio return $(X_t)_{t\in\za}$ from asset returns modelled by a multivariate time series $(\X_t)_{t\in\za}$ using the weights $\lambda$ by $X_t = \lambda^T \X_t$ means to apply a linear transformation to $\X_t$. To understand this transformation we quote the following proposition on $\VMA{q}-$processes~\cite[Proposition~ 11.1]{Luethkepohl:NewIntroMultipleTS}:
\begin{prop}[Linear transformation of a $\VMA{q}$-process]
Let $(\Z_t)_{t \in \za}$ be an n-dimensional white noise process with nonsingular covariance
matrix $\Sigma$ and let
\[
\X_t = \sum_{j=1}^q \Theta_j \Z_{t-j} + \Z_t, \quad \text{for } t \in \za,
\]
be an n-dimensional invertible $\VMA{q}$-process. Furthermore, let F be an $(M \times
n)$ matrix of rank M. Then the M-dimensional process $\Y_t = F \X_t$ has an
invertible $\VMA{\tilde{q}}$ representation,
\[
\Y_t = \sum_{j=1}^{\tilde{q}} \tilde{\Theta}_j \Ztilde_{t-j} + \Ztilde_t, \quad \text{for } t \in \za,
\]
where $(\Ztilde_t)_{t \in \za}$ is M-dimensional white noise with nonsingular covariance matrix, the $\tilde{\Theta}_j$ are coefficient matrices and $\tilde{q} \le q$.
\end{prop}
This proposition can be applied to analyze the situation if we form a portfolio. Then $F = \lambda^T$ is $1 \times n$ and we get a moving average process of order equal or less the order of the vector process. For the closing time problem $q = 1$ which leads to $\tilde{q} = 1$ and we can summarize our findings for this problem so far:
\begin{obs}\label{obs:obsMA}
The multivariate process of closing-time returns of assets traded in different time zones can be modelled as a $\VMA{1}$ process of the form
\[
\X_{t} = \Theta_1 \Z_{t-1} + \Z_{t} , \quad \text{for } t \in \za.
\]
Creating a portfolio with asset weights $\lambda$ results in a $\MA{1}$ process of the from
\[
\lambda^T \X_{t} =: X_{t} = \theta_1 Z_{t-1} + Z_{t} , \quad \text{for } t \in \za.
\]
\end{obs}
For a $\MA{1}$-process and the scaling constant of Example~\ref{ex:maq} simplifies to
\begin{align}\label{eq:dMA1}
\delta(d) = \sqrt{d + 2 (d-1) \frac{\theta_1}{1+\theta_1^2} }, \quad \text{for } d > 1.
\end{align}
This allows a top-down modelling of the returns if we think that the only auto-correlations in the portfolio come from the closing time problem. We just have to fit the parameters $\theta$ and $\sigma^2$ by~\eqref{eq:MAk.ACF} and apply~\eqref{eq:dMA1} for correct scaling of portfolio volatility. Alternatively, we can of course directly estimate the autocorrelation $\rho(1)$ of portfolio returns and plug it into~\eqref{eq:scaling.unvariate.constant}. However, estimating the full model would give as more insight.
\paragraph{The $\VMA{1}$-model}
In this paragraph we derive the details of the formulas of Section~\ref{sec:volacontrib} for the $\VMA{1}$-case.
Below in Example~\ref{ex:portfolio2} we apply these formulas to the closing time problem started in Example~\ref{ex:portfolio}.
By~\eqref{eq:cross.covar} and~\eqref{eq:autovoar.functions} we get the following for the $\VMA{1}$-model:
\begin{align}
\Gamma(0) &= \Theta_1 \Sigma \Theta_1^T + \Sigma, \quad \text{ and} \notag \\
\Gamma(1) &= \Theta_1 \Sigma, \label{eq:GammaVMA}
\end{align}
which gives
\begin{align}
\gamma(0) &= \lambda^T \left(\Theta_1 \Sigma \Theta_1^T + \Sigma\right) \lambda, \quad \text{ and}\notag \\
\gamma(1) &=\lambda^T \left( \Theta_1 \Sigma \right) \lambda, \label{eq:gammaVMA}
\end{align}
for the portfolio time series by~\eqref{eq:acf.pf.mult}.
Thus the scaling constant~\eqref{eq:scaling.unvariate.constant} is given by
\begin{align} \label{eq:VMA.deltad.pf}
\delta(d) = \sqrt{d + 2 (d-1) \frac{\lambda^T \Theta_1 \Sigma \lambda}{ \lambda^T \left(\Theta_1 \Sigma \Theta_1^T + \Sigma\right) \lambda} },
\end{align}
and
the scaling of contributions~\eqref{eq:scaling.constant.contrib} is given by
\begin{align} \label{eq:VMA.deltad.contrib}
\delta(i,d) = \left(d + 2 (d-1) \frac{ (\Theta_1 \Sigma \lambda)_i}{\left((\Theta_1 \Sigma \Theta_1^T + \Sigma) \lambda\right)_i} \right) / \delta(d),
\end{align}
where $\delta(d)$ is calculated in~\eqref{eq:VMA.deltad.pf} above. Again as in Example~\ref{ex:maq} one can estimate $\Gamma(0)$ and $\Gamma(1)$ and plug them into~\eqref{eq:scaling.constant.contrib}. But the estimator for $\Theta_1$ and $\Sigma$ or transformations of it will reveal interesting structures (see~e.g.~\cite[Chapter~8.2.1 Reduced and Structural Forms]{tsay2005analysis} or~\cite[Chapter~2.3.2 Impulse Response Analysis]{Luethkepohl:NewIntroMultipleTS}).
\paragraph{Scaling volatility of closing-time returns in $\VMA{1}$ compared to scaling volatility of contemporaneous returns}
In this paragraph we have a closer look at the Newey-West estimator~\eqref{eq:NWintro_real} and the na\"{\i}ve estimator~\eqref{eq:NWintro} in the the context of volatility scaling.
Having estimated the lag-zero covariance matrix of asset-returns $\Gamma(0)$ and the lag-one covariance matrix $\Gamma(1)$ we find the volatility of the $d$ days portfolio closing-time return by~\eqref{eq:acf.pf.mult} as
\begin{align}\label{eq:comp.eq1}
\sigma(\lambda,d) = \sqrt{d \lambda^T \Gamma(0) \lambda + 2 (d-1) \lambda^T \Gamma(1) \lambda}, \quad \text{for } d > 1.
\end{align}
Considering the contemporaneous returns together with their covariance matrix given by the na\"{\i}ve estimator $\tilde{\Sigma}$ from~\eqref{eq:NWintro} and assuming zero autocorrelations among contemporaneous returns the
$d$ days volatility of the contemporaneous portfolio return $\tilde{\sigma}(\lambda,d)$ is given by
\begin{align}\label{eq:comp.eq2}
\tilde{\sigma}(\lambda,d) &= \sqrt{d \lambda^T \tilde{\Sigma} \lambda} \notag \\
&= \sqrt{d \lambda^T (\Gamma(0) + \Gamma(1) + \Gamma(1)^T) \lambda} \notag \\
&= \sqrt{d \lambda^T \Gamma(0) \lambda + 2 d \lambda^T \Gamma(1) \lambda}, \quad \text{for } d > 1.
\end{align}
Considering the ratio of the scaled volatility of the portfolio closing-time return~\eqref{eq:comp.eq1} over the scaled volatility of the contemporaneous portfolio return~\eqref{eq:comp.eq2} we see that this quantity converges to 1:
\begin{align}\label{eq.naivelimit1}
\lim_{d \rightarrow \infty} \frac{\sigma(\lambda,d)}{\tilde{\sigma}(\lambda,d)} = \lim_{d \rightarrow \infty} \frac{\sqrt{d \lambda^T \Gamma(0) \lambda + 2 (d-1) \lambda^T \Gamma(1) \lambda}}{\sqrt{d \lambda^T \Gamma(0) \lambda + 2 d \lambda^T \Gamma(1) \lambda}} = 1.
\end{align}
Thus for large $d$ the risk figures for the two procedures coincide.
However, applying the Newey-West estimator up to lag~1~\eqref{eq:NWintro_real} the limit of the corresponding ratio is given by
\begin{align*}
\lim_{d \rightarrow \infty}&\frac{\sqrt{d \lambda^T \Gamma(0) \lambda + 2 (d-1) \lambda^T \Gamma(1) \lambda}}{\sqrt{d \lambda^T \Gamma(0) \lambda + d \lambda^T \Gamma(1) \lambda}} \\
=& \frac{\sqrt{ \lambda^T \Gamma(0) \lambda + 2 \lambda^T \Gamma(1) \lambda}}{\sqrt{\lambda^T \Gamma(0) \lambda + \lambda^T \Gamma(1) \lambda}} \neq 1.
\end{align*}
Thus the volatility estimates by the Newey-West estimator~\eqref{eq:NWintro_real} for contemporaneous returns, in this set-up, do not coincide with the result of applying the $\VMA{1}$-model for closing-time returns nor with the result of applying the na\"{\i}ve (but in this case correct and useful) estimator~\eqref{eq:NWintro}. This shows that the compatibility of long-term risk estimates in the two notions, closing-time return and contemporaneous return, depends on the estimator used for the covariance matrix of contemporaneous returns.
As already mentioned the estimator $\tilde{\Sigma}$ defined in~\eqref{eq:NWintro} can, in general, be invalid which by~\eqref{eq.naivelimit1} questions the $\VMA{1}$-model. In this case we should analyze the situation in more detail and make sure that the assumption that the closing-time problem is the only source of auto-correlation is acceptable. In the following we apply our findings to the data from Example~\ref{ex:portfolio} and all estimators considered are mathematically valid.
\begin{example}[Example~\ref{ex:portfolio} continued]\label{ex:portfolio2}
The conclusion of Example~\ref{ex:portfolio} is that we can model the closing-time returns as $\VMA{1}$-process.
We apply this to the data of Example~\ref{ex:portfolio}. Note that we can not solve~\eqref{eq:GammaVMA} for $\Theta_1$ or $\Sigma$. Thus we apply maximum likelihood estimation for this task as it is provided in the R-package DSE~\cite{Gilbertdse}. Note that positive definiteness of the covariance matrix of residuals of the $\VMA{1}$-model is assured in the estimation procedure applied \footnote{We acknowledge personal communications with the author, Paul Gilbert, on this topic.}. This fact is required for valid estimators in~\eqref{eq:GammaVMA}.
On the other hand we can directly estimate $\Gamma(0)$ and $\Gamma(1)$ from the data, knowing that $\Gamma(k) = 0$ for $k\ge2$. But looking at $\Theta_1$ and $\Sigma$ gives us some insight in the problem at hand. As risk and risk contributions is the focus of this article we do not go through the whole impulse-response analysis but refer to~\cite[Chapter~8.2.1 Reduced and Structural Forms]{tsay2005analysis} or~\cite[Chapter~2.3.2 Impulse Response Analysis]{Luethkepohl:NewIntroMultipleTS}) or the generalized impulse-response of~\cite{Pesaran:genirf}.
In Figure~\ref{fig:scalingVMA} we see the scaling constant for the portfolio time-series by the just mentioned MLE estimation of the $\VMA{1}$-model of the seven assets, a $\MA{1}$-model estimated directly on the portfolio time-series and the square-root-rule. The reason why the upper lines do not match perfectly are estimation errors but this plot gives us some confidence for the MLE estimators for $\Theta_1$ and $\Sigma$.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.8\textwidth]{scalingVMA}
\end{center}
\caption{Scaling constants $\delta(d)$ for ranging $d$ for a full $\VMA{1}$-model (solid, gray), a univariate $\MA{1}$-model (dashed, black) and the square-root-of-time (dotted, black).}
\label{fig:scalingVMA}
\end{figure}
Considering Table~\ref{tab:pf.contrib} we see the contributions to volatility p.a. on one hand by assuming zero serial correlations between the assets in the portfolio and on the other hand by modelling the asset returns in the portfolio as $\VMA{1}$-process.
Applying the square-root-of-time rule to this portfolio we get a volatility p.a. of $ 20.07\%$ while the volatility p.a. increases by approximately $17\%$ to $ 23.46\%$ in the $\VMA{1}$-model. Concerning the analysis of sources of risk note that the risk contribution by the geographically most distant market Japan of $1.58\%$ looks quite small when ignoring serial cross-correlations but it increases to $3.84\%$ taking them into account. The risk contributions of the markets leading the portfolio do not change dramatically. The increase of the portfolio volatility can be attributed to the Asian assets whose risk contribution increases significantly as serial cross correlations to the US and Europe are taken into account. This example shows that not only the accuracy of total volatility of the portfolio increases but also the attribution to single assets fits economic considerations much better!
\begin{table}[!htb]
\begin{center}
\begin{tabular}{l |c c c c c}
Asset & Currency & Exposure & Square-root rule & $\VMA{1}$ & Difference \\
\hline\hline
Portfolio & EUR & 105\% & 20.07\% & 23.46\% & 3.38\% \\
\hline
Topix & JPY &15\% & 1.58\% & 3.84\% & 2.26\% \\
H-shares & HKD & 15\% & 3.56\% & 5.40\% & 1.84\% \\
DJ Euro Stoxx 50 & EUR & 15\% & 3.28\% & 2.96\% & -0.32\% \\
Swiss Market & CHF & 15\%& 2.24\% & 2.11\% & -0.13\% \\
JSE TOP 40 & ZAR & 15\% & 2.62\% & 2.96\% & 0.34\% \\
Russell 2000 & USD &15\% & 3.94\% & 3.43\% & -0.51\% \\
NASDAQ 100 & USD &15\% & 2.85\% & 2.75\% & -0.11\%
\end{tabular}
\end{center}
\caption{Contributions to volatility p.a. in a the global portfolio of Example~\ref{ex:portfolio} applying the square-root-of-time rule and modelling a $\VMA{1}$-process.}
\label{tab:pf.contrib}
\end{table}
\end{example}
\subsection{Genuine auto-correlations}
We conclude our theoretical study of auto-correlated portfolio returns by a short detour to genuine auto-correlations. In the literature studies can be found (see e.g~\cite{Andersonetal:StockReturn} and references therein) which provide evidence that, besides the spurious effects of non-contemporaneous trading, genuine effects such as partial price adjustment and time-varying risk premia can lead to genuine auto-correlations in asset returns. We stay in the class of vector-autoregressive models but focus on $\VAR{p}$, especially, $\VAR{1}$-models in this context as opposed to the $\VMA{1}$-model for the closing time problem above.
Before we consider an example of first order genuine auto-correlations we go one step deeper into understanding the interplay between multivariate time series and portfolios constructed from these. An important result on linear transformations of $\VARMA{p}{q}$ processes is the following~\cite[Corollary~ 6.1.1]{Luethkepohl:NewIntroMultipleTS}:
\begin{theorem}[Linear transformations of $\VARMA{p}{q}$ processes]\label{th:varmaqptilde}
Let $(\X_t)_{t\in\za}$ be an $n$-dimensional, stable, invertible $\VARMA{p}{q}$ process and let $F$ be an $M \times n$ matrix of rank $M$. Then the process $(F \X_t)_{t\in\za}$ has a $\VARMA{\tilde{p}}{\tilde{q}}$ representation with
\begin{align*}
\tilde{p} \le n p \quad \text{and} \quad \tilde{q} \le (n-1)p + q.
\end{align*}
\end{theorem}
The above theorem tells us that a portfolio which is a simple linear transformation of a $\VARMA{p}{q}$ process can not be guaranteed to have an $\ARMA{p}{q}$ representation of the same order. Furthermore it is important to note that as a consequence the class of $\VAR{p}$-models is not closed with respect to linear transformation as, in general, the result of the transformation can be some $\VARMA{\tilde{p}}{\tilde{q}}$ process with $\tilde{q}>0$.
Concerning multivariate models we nevertheless focus on $\VAR{p}$-models due to known identification problems of
$\VARMA{p}{q}$-models if $q>0$ (see for example~\cite{Luetkepohl:Forecast}).
As a preparation for our key result on portfolios constructed out of $\VAR{p}$-process we state the following proposition:
\begin{prop}[AR portfolios built from VAR processes]\label{prop:VAR.AR.1}
Let $(\X_t)_{t \in \za}$ be a $\VAR{p}$-process in $\re^n$ for $n>0$ of the form
\[
\X_t = \sum_{k=1}^p \Phi_k \X_{t-k} + \Z_t
\]
and let $\lambda = (\lambda_1,\ldots,\lambda_n)$ be a vector of weights in $\re^n$. Then the portfolio process $(X_t(\lambda))_{t \in \za}$ is an $\AR{p}$-process of the form
\begin{align}\label{eq:VAR.AR.1.ARP}
X_t(\lambda) = \sum_{k=1}^p \phi_k X_{t-k}(\lambda) + \lambda^T \Z_t
\end{align}
if and only if
\begin{align}\label{prop:VAR.AR.1.item2}
\lambda^T \Phi_k = \phi_k \lambda^T\quad \text{for } k=1,\ldots,p.
\end{align}
\end{prop}
Note that as in Corollary~\ref{cor:acf.pf.mult} $(\lambda^T \Z_t)_{t \in \za}$ is clearly a white noise process. The full
proof of the proposition can be found in the appendix. Condition~\eqref{prop:VAR.AR.1.item2} means
that only portfolios with $\lambda$ being an eigenvector of all coefficient
matrices of the $\VAR{p}$-process admit the $\AR{p}$ representation~\eqref{eq:VAR.AR.1.ARP} which one could expect to hold in general, at first glance. The following corollary concludes these considerations.
\begin{corollary}[AR portfolios built from VAR processes]\label{cor:VAR.AR.1}
In the setting of Proposition~\ref{prop:VAR.AR.1} the process $(X_t(\lambda))_{t \in \za}$ is an $\AR{p}$-process for any portfolio weighting $\lambda \in \re^n$ if and only if the coefficient matrices of the VAR-process are diagonal and of the following form
\[
\Phi_k = \phi_k I_n, \quad\text{for } k=1,\ldots,p.
\]
\end{corollary}
The consequence of the above corollary is that modelling genuinely auto-correlated assets we will in general not observe portfolio returns consistent with an $\AR{1}$-model. This would only be possible if the coefficient matrix $\Phi_1$ were of the form
\[
\begin{pmatrix} a & 0 & \dots & 0 \\
0 & a & \dots & 0 \\
\vdots& 0 & \ddots & \vdots \\
0 & \dots & \dots & a \\
\end{pmatrix}
\]
for some fixed value for $a$ - all the same for each asset.
The conclusion is that the weighted $\VAR{1}$-model is richer than an $\AR{1}$-model.
\paragraph{The $\VAR{1}$-model}
After these general considerations we focus on the $\VAR{1}$-model of the form
\[
\X_t = \Phi_1 \X_{t-1} + \Z_t, \quad \text{for } t\in \za,
\]
since this model will be the natural choice to capture genuine serial correlations.
In this case we get the following expressions for the covariance matrices
\begin{align}
\Gamma(0) &= \Sigma (I - \Phi_1^2)^{-1} \quad\text{and} \nonumber \\
\Gamma(k) &= \Phi_1^k \Gamma(0), \quad \text{for } k\ge1, \label{eq.var1.2}
\end{align}
which gives
\begin{align}
\gamma(0) &= \lambda^T \Gamma(0) \lambda, \quad \text{ and}\notag \\
\gamma(k) &=\lambda^T \left( \Phi_1^k \Gamma(0) \right) \lambda, \quad \text{for } k\ge1, \label{eq:gammaVAR}
\end{align}
for the portfolio time series by~\eqref{eq:acf.pf.mult}.
Using~\eqref{eq.var1.2} and~\eqref{eq:gammaVAR} the scaling constant~\eqref{eq:scaling.unvariate.constant} for $d>1$ is given by
\begin{align} \label{eq:VAR.deltad.pf}
\delta(d) = \sqrt{d + 2 \sum_{k=1}^{d-1} (d-k) \frac{\lambda^T \Phi_1^k \Gamma(0) \lambda}{\lambda^T \Gamma(0) \lambda}},
\end{align}
and the scaling of contributions~\eqref{eq:scaling.constant.contrib} is given by
\begin{align} \label{eq:VAR.deltad.contrib}
\delta(i,d) = \frac1{\delta(d)} \left(d + 2 \sum_{k=1}^{d-1} (d-k) \frac{( \Phi_1^k \Gamma(0) \lambda)_i}{(\Gamma(0) \lambda)_i} \right),
\end{align}
where $\delta(d)$ is calculated in~\eqref{eq:VAR.deltad.pf} above. In contrast to the closing-time problem, in this case, we would have to estimate $d-1$ lagged covariance matrices $\Gamma(k), k = 1,\ldots, d-1$ if we wanted to plug them into~\eqref{eq:scaling.constant.contrib} directly. This is clearly not feasible for large values of $d$ which justifies the use of a specific time-series model in these cases.
The following concrete example illustrates the above issues. We furthermore analyse
the trade-off when approximating such a portfolio with first order genuine auto-correlations with an $\AR{1}$-model, although it is not theoretically justified.
\begin{example}\label{example:pfAB}
Consider a portfolio consisting of two contemporaneously traded assets A and B with annual volatilities of $25\%$ and $20\%$ and a correlation of $70\%$. Asset A has a negative genuine first order autocorrelation of $-5\%$ while asset B exhibits $2.5\%$ genuine first order autocorrelation. Note that these values are consistent with findings in~\cite{Andersonetal:StockReturn}. Thus we consider the following covariance matrices
\begin{align*}
\Gamma(0) &=\text{diag}\begin{pmatrix}
0.25 & 0.2 \\
\end{pmatrix} \begin{pmatrix}
1 & 0.7 \\
0.7 &1 \\
\end{pmatrix}
\text{diag}\begin{pmatrix}
0.25 & 0.2 \\
\end{pmatrix} \frac1{250} \quad \text{ and } \\
\Gamma(1) &=
\text{diag}\begin{pmatrix}
0.25 & 0.2 \\
\end{pmatrix} \begin{pmatrix}
-0.05 & 0 \\
0 & 0.025 \\
\end{pmatrix}
\text{diag}\begin{pmatrix}
0.25 & 0.2 \\
\end{pmatrix} \frac1{250}.
\end{align*}
We can model these two assets by a $\VAR{1}$-model and calculate the coefficient matrix $\Phi_1$ by~\eqref{eq.var1.2} and get
\[
\Phi_1 = \Gamma(1) \Gamma(0)^{-1} = \begin{pmatrix}
-0.0980 & 0.0858 \\
-0.0275 & 0.0490 \\
\end{pmatrix}.
\]
Modelling a portfolio with the weighting $\lambda = (\frac12,\frac12)^T$ we expect an ARMA(2,1) process by Theorem~\ref{th:varmaqptilde}. However, for a pure $\AR{1}$-model the coefficient $\phi_1$ is given by
\[
\phi_1 = \gamma(1)/\gamma(0) = \frac{2.362}{14.53} = -0.0123.
\]
Using the results for scaling in univariate models from Example~\ref{example:ARMA1} and Equation~\eqref{eq:VAR.deltad.pf} we compare the resulting scaling constants in Table~\ref{table:scaling.comparison}. We see that the $\AR{1}$-model performs well especially for shorter holding periods in approximating the result of the multivariate model. However, such a simple approximation should be used with care. Furthermore the $\VAR{1}$-model tells us more details about the risk contributions as we will see below.
\begin{table}[!htb]
\centering
\begin{tabular}{l|c c c}
d & $\VAR{1}$ & $\AR{1}$ & SRTR \\ \hline
2 & 1.405 &1.405 &1.414 \\
5 & 2.218 &2.214 & 2.236 \\
10 & 3.134 & 3.127 &3.162 \\
30 & 5.427 &5.412 & 5.477 \\
90 & 9.398 & 9.372 & 9.487 \\
20 & 15.662 &15.619 &15.811 \end{tabular}
\caption{Volatility scaling factors $\delta(d)$ for $\VAR{1}$, $\AR{1}$ and the square-root of time for various holding periods $d$.}
\label{table:scaling.comparison}
\end{table}
We conclude this detour on genuine auto-correlations by an analysis of the relative risk contributions, i.e. risk contributions in percentage of total volatility. Applying~\eqref{eq:VAR.deltad.contrib} to our example we see in Table~\ref{table:scaling.risk.contributions} that the contribution of asset A is dominant as it has the higher volatility. But with increasing holding period the relative risk contribution of asset A decreases which reflects its negative auto-correlation and the positive auto-correlation of asset $B$. This is a feature that only the multivariate approach can offer.
\begin{table}[!htb]
\centering
\begin{tabular}{l|c c }
$d$ & $ \frac{\sigma(d,A)}{\sigma(d)} $ & $ \frac{\sigma(d,B)}{\sigma(d)} $ \\ \hline
1 & 56.52 &43.48\\
2 & 55.39 &44.61\\
5 & 54.77 &45.23\\
10 & 54.56 &45.44\\
30 &54.42 &45.58\\
90 & 54.37 &45.63\\
250 & 54.36& 45.64\\
\end{tabular}
\caption{Relative risk contributions (percentage) of assets A and B for various holding periods $d$ in the $\VAR{1}$-model.}
\label{table:scaling.risk.contributions}
\end{table}
\end{example}
\section{Conclusions}
In this article we first clarify the notion of closing-time returns and contemporaneous returns in global portfolios. In Example~\ref{ex:portfolio} we illustrate these notions in a setting of time-shifted multivariate Brownian motion. Serial correlations naturally occur when analyzing portfolios of geographically diversified assets traded in distant time zones and we motivate the use of a $\VMA{1}$-model for the closing-time returns. We then address the problem of calculating portfolio volatility of closing-time returns for holding periods of more than one day.
We show that ignoring serial correlations leads on one hand to biased estimates of volatility and on the other hand
to misleading risk contributions as Example~\ref{ex:portfolio2} illustrates.
We propose formulas for calculating accurate volatility scaling modelling the portfolio closing-time return as a univariate process as well as in a multivariate setting. Moreover in the multivariate setting we also provide explicit formulas for genuine risk contributions that take the time series structure of the assets involved into account. Modelling the asset returns as a vector moving average process of order one we derive handy formulas and perform a complete analysis of risk and risk contributions and compare this approach to the Newey-West estimator of contemporaneous returns and another simple but useful estimator in the same spirit.
Finally we take a short detour to genuine auto-correlations and propose the application of a $\VAR{1}$-model to tackle this problem.
Applying the findings of this article to the calculation of the tracking error, i.e. the volatility of the additional return of the portfolio above a given benchmark, can improve the analysis of relative risk which is often an aim in asset management.
Besides the analysis of market risk our findings can be applied to portfolio optimization as well as portfolio construction techniques such as risk-parity (also known as equally-weighted risk contributions, see~\cite{Roncalli:PropEquallyWeighted}) where risk contributions by assets are the driving input.
As another direction of further research the findings of this article may also be applied to the VEC specification of
multivariate GARCH models, since they admit a VARMA representation (see~\cite{Luethkepohl:NewIntroMultipleTS}).
\subsection{The univariate model with auto-correlations}\label{subsec:volacontrib.uni}
First we make the following definition for the auto-covariance and auto-correlation function of portfolio returns $(X_t(\lambda))_{t \in \za}$:
\begin{definition}[Auto-covariances of a univariate time series]\label{defi:univariate.covar}
Let $(X_t(\lambda))_{t \in \za}$ denote a univariate weakly stationary stochastic process, then the auto-\-covariance function and the auto-correlation function is denoted by
\begin{align}
\gamma(k) &:=\covar{X_t(\lambda)}{X_s(\lambda)} \quad \text{and} \label{eq:uni.acf}\\
\rho(k)&:=\corr{X_t(\lambda)}{X_s(\lambda)} = \gamma(k)/\gamma(0), \label{eq:uni.acorf}
\end{align}
for $t,s \in \za$ where $ |t-s| = k$, respectively.
\end{definition}
In the presence of auto-correlations in the portfolio returns the scaling by the square-root of time is not accurate and the following proposition states the correct scaling in this setting.
\begin{prop}[Volatility for a holding period of $d$ days]\label{prop:scaling.univariate}
Let $(X_t(\lambda))_{t \in \za}$ be a univariate weakly stationary stochastic process with auto-covariance function $\gamma(\cdot)$ as defined in Definition~\ref{defi:univariate.covar} then the volatility of the return when holding the portfolio over $d\ge2$ days is given by
\begin{align}\label{eq:scaling.unvariate}
\sigma(\lambda,d) = \sigma(\sum_{k=1}^d X_k(\lambda)) = \sqrt{d \gamma(0) + 2 \sum_{k=1}^{d-1} (d-k) \gamma(k)}.
\end{align}
\end{prop}
The proof of Proposition~\ref{prop:scaling.univariate} is given in the appendix and the correction to scaling by the square-root of time is clearly seen in the following corollary.
\begin{corollary}[Scaling rule for the univariate model]\label{cor:scaling.univariate}
Let $(X_t(\lambda))_{t \in \za}$ as in Proposition~\ref{prop:scaling.univariate} and $\rho(\cdot)$ be its auto-correlation function then
\begin{align}\label{eq:scaling.unvariate.rule}
\sigma(\lambda,d) = \sigma(\lambda) \delta(d),
\end{align}
where $\sigma(\lambda)$ is the one-day volatility from~\eqref{eq:vola} and the factor $\delta(d)$ is given by
\begin{align}\label{eq:scaling.unvariate.constant}
\delta(d) = \sqrt{d + 2 \sum_{k=1}^{d-1} (d-k) \rho(k)}, \quad \text{for } d\ge2.
\end{align}
\end{corollary}
The short proof of this corollary can also be found in the appendix. Considering~\eqref{eq:scaling.unvariate.constant} we see that in our univariate model the scaling is given by the square-root of time corrected by an expression taking into account all relevant auto-correlations. If these are zero then~\eqref{eq:scaling.unvariate.constant} reduces to the well-known formula~\eqref{eq:sqrtrule}.
The next corollary gives a crude estimate of how much higher the true scaling factor can be compared to the square-root-of-time rule.
\begin{corollary}[Error when using the square-root-of-time rule]\label{cor:square.root.error}
Let $(X_t(\lambda))_{t \in \za}$ as in Proposition~\ref{prop:scaling.univariate} and $\delta(d)$ be the correct scaling factor, then
the proportion of correct scaling to an application of the square-root-of-time rule can be estimated as
\begin{align}\label{eq:square.root.error}
\frac{\delta(d)}{\sqrt{d}} \le \sqrt{d}, \quad\text{for } d\ge2.
\end{align}
\end{corollary}
\begin{proof}
Considering~\eqref{eq:scaling.unvariate.constant} and noting that $|\rho(k)| \le 1$ for all $k \in \za$, we get
\[
\delta(d) \le \sqrt{d + 2 \sum_{k=1}^{d-1} (d-k)} = d,
\]
where we apply that $\sum_{k=1}^{d-1} (d-k) = \frac12(d^2-d)$.
\end{proof}
The above corollary states that the proportion of the correct scaling factor and the square-root of time when the necessary assumptions are not fulfilled grows with the square-root of time. This conservative estimate does not assume anything non-trivial about the auto-correlations. In Figure~\ref{fig:scalingVMA} in Section~\ref{sec:time.series} we will see that the concrete picture is not always that bad.
\subsection{The multivariate model with auto-correlations}\label{subsec:volacontrib.multi}
Next we will analyze the situation if the constituent assets are known. In this setting a bottom-up
modelling of the portfolio structure is possible.
For the the covariance structure of the asset returns $(\X_t)_{t \in \za}$ we need the following definition:
\begin{definition}[Covariance matrix function of a multivariate time series]
Let $(\X_t)_{t \in \za}$ denote a weakly stationary process in $\re^n$. Then we denote the matrices of serial covariances of lag $k =0,1,2,\ldots$ by
\begin{align}\label{eq:covar.matrix}
\Gamma(k):=\covar{\X_{t+k}}{\X_t},
\end{align}
for $t \in \za$.
\end{definition}
Consider that the element at position $(i,j)$ in the matrix $\Gamma(k)$ is given by
\[
\Gamma(k)_{i,j} = \covar{X_{t+k}^i}{X_t^j},
\]
thus modelling multivariate time series it holds that
\[
\Gamma(k) = \Gamma(-k)^T,
\]
as matrices are not necessarily symmetric. This means that $\covar{X_{t+k}^i}{X_t^j}$ is in general not equal to $\covar{X_{t+k}^j}{X_t^i}$ for $k>0$. In the lead-lag setting the leading market's return `yesterday' is strongly correlated to the lagging one's return `today' but not vice-versa.
Analogously to~\eqref{eq:varvonstrib} the key to volatility contributions in this setting are the covariances of
assets with the portfolio return. The following proposition gives the corresponding expressions for our setting.
\begin{prop}\label{prop:multi.var.pf.stationary}
Let $(X_t(\lambda))_{t \in \za}$ denote the portfolio return when weighting the asset returns $(\X_t)_{t \in \za}$ by $\lambda$, i.e.
\[
X_t(\lambda) = \sum_{i=1}^n \lambda_i X_t^i,\quad\text{for } t \in \za,
\]
then $(X_t(\lambda))_{t \in \za}$ is weakly stationary and
it holds that
\begin{align}\label{eq:multi.covar1}
\gamma_i(k)&:=\covar{X_{t+k}^i}{X_t(\lambda)}= (\Gamma(k) \lambda)_i,
\end{align}
for $k =0,1,\ldots$ and $i=1,\ldots,n$ where $(\Gamma(k) \lambda)_i$ denotes the the $i_{th}$ element of the vector $\Gamma(k) \lambda$.
\end{prop}
\begin{proof}
The expectation of $(X_t(\lambda))_{t \in \za}$ is calculated straightforward. Furthermore it holds that
\[
\covar{X_{t+k}}{X_t(\lambda)} = \covar{\lambda^T \X_{t+k}}{\lambda^T \X_{t}} = \lambda^T \Gamma(k) \lambda,
\]
where $\Gamma(k)$ denotes the covariance matrix of $(\X_t)_{t \in \za}$ for $k=0,1,\ldots$.
The above expression depends on the absolute value of the lag $k$ only as
\[
\lambda^T \Gamma(-k) \lambda = \lambda^T \Gamma(k)^T \lambda = \lambda^T \Gamma(k)\lambda,
\]
which concludes the proof of weak stationarity.
To prove~\eqref{eq:multi.covar1} consider that by the bilinearity of covariance we get
\begin{align*
\covar{X_{t+k}^i}{X_t(\lambda)} &=\covar{X_{t+k}^i}{\lambda^T \X_t}\\
&=\sum_{j=1}^n \lambda_j \covar{X_{t+k}^i}{X_t^j} = (\Gamma(k)\lambda)_i,
\end{align*}
for $i = 1,\ldots n$ and $k = 0,1,\ldots$.
\end{proof}
Using~\eqref{eq:multi.covar1} we get useful expressions for the auto-covariance structure of the portfolio returns as well as risk contributions in this setting.
\begin{corollary}[Portfolio auto-covariance in the multivariate model]\label{cor:acf.pf.mult}
Let $(X_t(\lambda))_{t \in \za}$ be the portfolio return where the asset returns have covariance structure as given in ~\eqref{eq:covar.matrix} then the auto-covariance function~\eqref{eq:uni.acf} is given by
\begin{align}\label{eq:acf.pf.mult}
\gamma(k) = \covar{\lambda^T \X_{t+k}}{\lambda^T \X_t} = \lambda^T \Gamma(k) \lambda= \sum_{i=1}^n \lambda_i \gamma_i(k)
\end{align}
for $i=1,\ldots,n$ and $k = 0,1,\ldots$ where $\gamma_i(k)$ is given in~\eqref{eq:multi.covar1}.
\end{corollary}
Having calculated the auto-covariance function of $(X_t(\lambda))_{t \in \za}$ we find the volatility scaling for any $d\ge1$ by Proposition~\ref{prop:scaling.univariate} and Corollary~\ref{cor:scaling.univariate}.
To conclude this section we analyze how to calculate contributions to volatility and derive formulas how these contributions change over time.
\begin{prop}[Volatility contributions with serial correlations]\label{prop:vola.contrib.multi}
Let $(\X_t)_{t \in \za}$ denote a weakly stationary process in $\re^n$ and $(X_t(\lambda))_{t \in \za}$ the process of portfolio returns, then the volatility contributions by the Euler-allocation rule when holding this portfolio for $d \ge 2$ days are given by
\begin{align}\label{eq:vola.contrib.multi}
\sigma_i(\lambda,d) = \frac{\lambda_i}{\sigma(\lambda,d)} \left(d \gamma_i(0) + 2 \sum_{k=1}^{d-1} (d-k) \gamma_i(k)\right),
\end{align}
where $\gamma_i(k)$ is given in~\eqref{eq:multi.covar1} for $i=1,\ldots,n$ and $k = 0,1,\ldots$.
\end{prop}
The following corollary states how risk contributions scale in the above framework.
\begin{corollary}[Scaling of volatility contributions with serial correlations]\label{cor:scaling.vola.corr}
Let $\sigma_i(\lambda,d)$ as in Proposition~\ref{prop:vola.contrib.multi}, then for $i=1,\ldots,n$
\begin{align*}
\sigma_i(\lambda,d) = \sigma_i(\lambda) \delta(i,d),
\end{align*}
where $\sigma_i(\lambda)$ is the one-day volatility contribution as defined in~\eqref{eq:varvonstrib} and the factor $\delta(i,d)$ is given by
\begin{align}\label{eq:scaling.constant.contrib}
\delta(i,d) = \left(d + \frac2{\gamma_i(0)} \sum_{k=1}^{d-1} (d-k) \gamma_i(k)\right) / \delta(d),
\end{align}
with $\gamma_i(k)$ defined in~\eqref{eq:multi.covar1} and $\delta(d)$ is the scaling factor for the respective portfolio volatility.
\end{corollary}
The proofs of the two statements above can be found in the appendix.
\begin{remark}
Corollary~\ref{cor:scaling.vola.corr} shows that in this modelling approach the relative volatility contribution changes depending on the holding period and the whole covariance structure of the returns involved. Thus assets whose relative risk contribution increases with the holding period can be identified. This is a feature that neither the model with uncorrelated asset returns nor the univariate model has. Finally, it is easily seen that~\eqref{eq:scaling.constant.contrib} reduces to
\begin{align*}
\delta(i,d) = \frac{d}{\sqrt{d}} = \sqrt{d}, \quad\text{for } i = 1,\ldots,n,
\end{align*}
in the case of no auto-correlations.
\end{remark}
Considering the results in Proposition~\ref{prop:scaling.univariate} and~\ref{prop:vola.contrib.multi} in general we have to estimate $d-1$ auto-covariances respectively auto-covariance matrices to find a correct scaling for $d$ days. Considering volatility per annum this corresponds to $d-1=249$ or $251$, which is clearly not appealing.
In the following section we apply these findings to classical auto-regressive time series models which reduces the number of parameters tremendously.
|
{
"timestamp": "2011-11-30T02:03:41",
"yymm": "1009",
"arxiv_id": "1009.3638",
"language": "en",
"url": "https://arxiv.org/abs/1009.3638"
}
|
\section{INTRODUCTION}
The publication of comparatively large catalogs of gamma-ray
bursts based on the results of the surveys performed within the
framework of space missions of the Italo-Dutch BeppoSAX
satellite\footnote{\tt http://www.asdc.asi.it/bepposax/}
(Satellite per Astronomia X, and Beppo in the honor of Giuseppe
Occhialini) \cite{bepposax}, and NASA's Compton observatory, where
the BATSE experiment\footnote{\tt
http://www.batse.msfc.nasa.gov/batse/} (Burst and Transient Source
Experiment) \cite{batse} was performed, made it possible to study
the properties of the space distribution of gamma-ray bursts,
which, to a first approximation, are uniformly distributed in the
sky.
Among recent studies concerning this subject we distinguish the
analysis of the possible relation between the distribution of
gamma-ray bursts from the BATSE catalog and the large-scale
structure by Williams and Frey ~\cite{grb_apm}. The authors
analyze the effect of the local large-scale structure on the
apparent positions of gamma-ray bursts in the case of weak
lensing. This effect may show up for distant ($z>4$) events as an
anticorrelation of the positions of gamma-ray bursts and galaxies
in clusters. The authors studying gamma-ray bursts report such an
anticorrelation for galaxies at $z \sim 0.2-0.3$, which they found
based on the optical magnitudes and positions of galaxies measured
using the APM (Automatic Plate Measuring Facility).
\begin{figure*
\centerline{
\psfig{figure=fig1_bep_bat_map.ps,angle=-90,width=7cm}
}
\caption{Distribution of gamma-ray bursts on the sky. The gray
and white symbols (the circles and crosses correspond to short and
long bursts, respectively) show the BeppoSAX and BATSE data,
respectively. }
\label{f1}
\end{figure*}
\begin{figure*
\centerline{\vbox{
\hbox{
\psfig{figure=fig2a_bep_s.ps,angle=-90,width=7cm}
\psfig{figure=fig2b_bep_l.ps,angle=-90,width=7cm}
} \hbox{
\psfig{figure=fig2c_bat_s.ps,angle=-90,width=7cm}
\psfig{figure=fig2d_bat_l.ps,angle=-90,width=7cm}
}}}
\caption{Distribution of the gamma-ray bursts on the sky. The top left panel shows the
BeppoSAX data for bursts with $t<2$\,s. The top right panel shows
the BeppoSAX data for bursts with $t>2$\,s. The bottom left panel
shows the BATSE data for $t<2$\,s. The bottom right panel shows
the BATSE data for $t>2$\,s. }
\label{f2}
\end{figure*}
In another paper, M\'esz\'aros~et al. \cite{grb_vor} use various
methods (the Voronoi tesselation diagrams, the minimal spanning
tree, and multifractal spectra) to analyze the distribution of the
BATSE gamma-ray bursts. The above authors break the list of
objects into subsamples of sources with different signal durations
($t<2$\,s, \mbox{$2<t<10$\,s,} $t>10$\,s) and apply the above
methods to each subsample. They found the results for the first
two groups to deviate from the results obtained for uniformly
distributed simulated data. Based on the data obtained the
authors \cite{grb_vor} discuss the validity of the cosmological
principle.
We use the BeppoSAX (781 source, the energy interval covered:
0.1--200\,keV) and BATSE (2037 sources, 20\,keV--2\,MeV) catalogs
adopted from the sites of the corresponding experiments.
We subdivide each catalog into two subsamples containing short
(i.e., lasting $t<2$\,s) and long ($t>2$\,s) events. We thus
obtain four source subsamples, which we analyze separately using
the same procedure. Figure~1 shows the positions of all gamma-ray
bursts from the BeppoSAX and BATSE catalogs. Figure~2 shows the
positions of short and long BeppoSAX and BATSE gamma-ray bursts on
separate sky charts.
In this paper we analyze the statistical correlation properties of
the GRB distribution on the sky with respect to the distribution
of the cosmic microwave background (CMB). We proceed from the
assumption that gamma-ray bursts should be associated with massive
galaxies, and that the
positions of these galaxies are correlated with the large-scale
structure. In this case there is hope that we may find deviations
in the sky distribution of CMB fluctuations in the areas where the
bursts are projected. Note that the large-scale structure may show
up in CMB fluctuations via the Zel'dovich--Sunyaev effect and the
integrated Sachs--Wolfe effect~\cite{swe}. This former effect
manifests itself via the interaction of CMB photons with hot
electrons of galaxy clusters \cite{zse} and is observed on the
angular scales of about 4\arcmin\, or smaller. The latter effects
occurs when the photons move in a gravitational field with a
variable potential that arises during
the formation of the large-scale structure and expansion of the
Universe, and should be observed on the angular scales greater
than 200\arcmin.
To study the correlation properties of the positions of gamma-ray
bursts and CMB peaks, we use the CMB map based on the results of
the experiment that has been conducted for seven years within the
framework of the WMAP\footnote{{\tt http://lambda.gsfc.nasa.gov}}
(Wilkinson Microwave Anisotropy Probe) experiment aimed to
integrate the signal from the entire sky \cite{wmap7ytem}. The CMB
signal was reconstructed from multifrequency observations using
the method of internal linear combination (ILC) of the background
components~\cite{wmapresults}. This procedure yielded the CMB map,
which is also referred to as the ILC map, and which is used to
analyze low-order harmonics with multipole numbers $\ell\le150$.
The ILC map is based on observations made in five channels:
23\,GHz (the K band), 33\,GHz (the Ka band), 41\,GHz (the Q
band), 61\,GHz (the V band), and 94\,GHz (the W band).
Given that one of the central problems in statistical studies of
gamma-ray bursts is due to the large error box size, about
1\degr$\times$1\degr, of the available source positions, we
operate with pixels of about the same or even greater size in
order to avoid uncertainties in the analysis of the sky
distribution of the events.
The methods we use in this paper and our results are laid out in
the following way. First we describe the methods that we use to
pixelize the distribution of bursts and to correlate the maps. We
then analyze the statistics of CMB deviations in the gamma-ray
burst areas. At the next stage, we compare the BATSE--BeppoSAX and
BATSE--CMB correlation maps. We finally discuss the results in the
concluding section.
\section{MOSAIC CORRELATION METHOD}
To analyze the map properties on different angular scales, we
expand the signal distributed on the sphere into the spherical
harmonics (multipoles):
\begin{equation}
\Delta S(\theta,\phi)= \sum_{\ell=1}^{\infty}\sum_{m=-\ell}^{m=\ell}
a_{\ell m} Y_{\ell m} (\theta, \phi)\,.
\label{eq1}
\end{equation}
Here $\Delta S(\theta,\phi)$ are the variations of signal on the
sphere in polar coordinates; $\ell$ is the number of the
multipole, and $m$ is the number of the mode of the multipole. The
spherical harmonics are determined as
\begin{eqnarray}
Y_{\ell m}(\theta,\phi) = \sqrt{{(2\ell+1)\over 4\pi}{(\ell-m)!
\over (\ell+m)!}}P_\ell^m(x) e^{i m\phi}, \\
~x=\cos\theta\,,\nonumber \label{eq2}
\end{eqnarray}
where $P_\ell^m(x)$ are the associated Legendre polynomials. The
expansion coefficients $a_{\ell m}$ for a continuous function
$\Delta S(x,\phi)$ can be written as:
\begin{equation}
a_{\ell m}=\int^1_{-1}dx\int^{2\pi}_0 \Delta S(x,\phi)
Y^{*}_{\ell m}(x,\phi) d\phi\,,
\label{eq3}
\end{equation}
where $Y^{*}_{\ell m}$ denotes the complex conjugate of $Y_{\ell m}$.
The correlation properties of two maps of the signal distribution
on the sphere can be described, on a given angular scale, by a
correlation coefficient for the corresponding multipole $\ell$ as:
\begin{equation}
K(\ell) = \frac{1}{2} \frac{\sum\limits_{m=-\ell}^\ell t_{\ell
m}s^*_{\ell m} + t^*_{\ell m}s_{\ell m}}
{(\sum\limits_{m=-\ell}^\ell |t_{\ell m}|^2
\sum\limits_{m=-\ell}^\ell |s_{\ell m}|^2)^{1/2}}\,,
\label{eq4}
\end{equation}
where $t_{\ell m}$ and $s_{\ell m}$ are the variations of two
signals in a harmonic representation. The coefficient $K(\ell)$
can be used to assess the correlation between the harmonics on the
sphere, i.e., to compare the properties of maps on a given angular
scale. However, in the case of a search for correlated areas,
which do not repeat in other regions of the sphere, this approach
smears such single areas in the process of averaging over the
sphere within a certain harmonic. In this case it becomes
practically impossible to identify the correlated areas.
Verkhodanov et al.~\cite{cormap} proposed an approach, which was
implemented in the second release of the GLESP package
\cite{glesp2} (the {\tt difmap} utility). The proposed procedure
makes it possible to find correlations between two maps in the
areas of a certain angular size. In this method, each pixel with
number $p$ subtending the solid angle $\Xi_p$ is assigned the
cross-correlation coefficient between the data of the two maps on
the corresponding area. Thus a correlation map is constructed for
two signals $T$ and $S$, where the value of each pixel $p$
($p=1,2,...,N_0$, \mbox{and $N_0$ is the} total number of pixels
on the sphere) with the subtending angle $\Xi_p$, and computed for
the sphere maps with the initial resolution determined by
$\ell_{max}$ is \mbox{equal to}
\begin{eqnarray}
\nonumber K(\Xi_p|\ell_{max}) = \hspace{3cm} \\
\frac{\sum\sum\limits_{p_{ij}\in\Xi_p}
(T(\theta_i,\phi_j) - \overline{T(\Xi_p))}
(S(\theta_i,\phi_j) - \overline{S(\Xi_p))}}
{\sigma_{T_p}\sigma_{S_p}}\,.
\label{eq5}
\end{eqnarray}
Here $T(\theta_i,\phi_j)$ is the value of the signal $T$ in the
pixel with the coordinates $(\theta_i,\phi_j)$ for the initial
resolution of the pixelization of the sphere; $S(\theta_i,\phi_j)$
is the value of the other signal in the same area;
$\overline{T(\Xi_p)}$ and \mbox{$\overline{S(\Xi_p)}$ } are the
mean values averaged over the area $\Xi_p$ and obtained from the
data of the maps with higher resolution determined by
$\ell_{max}$, and $\sigma_{T_p}$ and $\sigma_{S_p}$ are the
corresponding standard deviations in the area considered.
\section{CMB SIGNAL STATISTICS IN THE AREAS OF GAMMA-RAY BURSTS}
\begin{figure*
\centerline{\vbox{
\hbox{
\psfig{figure=fig3a_mbep_s_lmax20.ps,angle=-90,width=7cm}
\psfig{figure=fig3b_mbep_l_lmax20.ps,angle=-90,width=7cm}
} \hbox{
\psfig{figure=fig3c_mbat_s_lmax20.ps,angle=-90,width=7cm}
\psfig{figure=fig3d_mbat_l_lmax20.ps,angle=-90,width=7cm}
}}}
\caption{Positions of gamma-ray bursts from different subsamples on the CMB maps
with a resolution of $\ell_{max}=20$. The top left and top right
panels show the BeppoSAX data for bursts with $t<2$\,s and
$t>2$\,s, respectively. The bottom left and bottom right panels
show the data for the BATSE bursts with $t<2$\,s and $t>2$\,s,
respectively.}
\label{f3}
\end{figure*}
\begin{figure*
\centerline{\vbox{
\hbox{
\psfig{figure=fig4a_mbep_s_lmax150.ps,angle=-90,width=7cm}
\psfig{figure=fig4b_mbep_l_lmax150.ps,angle=-90,width=7cm}
} \hbox{
\psfig{figure=fig4c_mbat_s_lmax150.ps,angle=-90,width=7cm}
\psfig{figure=fig4d_mbat_l_lmax150.ps,angle=-90,width=7cm}
}}}
\caption{Positions of gamma-ray bursts from different subsamples
on the CMB maps with a resolution of $\ell_{max}=150$. The top
left and top right panels show the BeppoSAX data for bursts with
$t<2$\,s and $t>2$\,s, respectively. The bottom left and bottom
right panels show the data for the BATSE bursts with $t<2$\,s
and $t>2$\,s, respectively. }
\label{f4}
\end{figure*}
In the first method the statistical properties of distributions
of GRB positions versus the CMB signal are tested by simply
analyzing the pixel value measurements on the CMB maps. We
performed pixel measurements using the {\tt mapcut} procedure of
the GLESP package. Figures~3 and 4 show the positions of gamma-ray
bursts of the BeppoSAX and BATSE catalog subsamples on the CMB
maps with the resolution of 260\arcmin\, \mbox{($\ell_{max}=20$)}
and 36\arcmin\, ($\ell_{max}=150$). We chose the resolution of
these maps to match the expected angular scale of the Sachs--Wolfe
effect and the limiting resolution of the WMAP mission.
To search for the eventual correlations, we computed the number of
GRB positions that fall onto the CMB pixels with negative signal
fluctuations (which may be due to the effects described above) in
the CMB maps of different resolution. The Table lists the
statistics of CMB pixel values in the gamma-ray burst areas for
the long- and short-event BATSE and BeppoSAX subsamples. It
includes the total number of sources in subsamples; the number of
sources located in CMB pixels with negative fluctuation values;
the expected number of pixels with negative CMB amplitudes
according to the results of 200 random Gaussian CMB signal
simulations modeled in terms of the $\Lambda$CDM cosmology, and
the 1$\sigma$-scatter of these quantities.
\begin{table}
\caption{{\bf Table.}~Statistics of the
CMB pixel values in the GRB areas for the BATSE and BeppoSAX
subsamples. The columns are: $t$, the duration (s); mission name;
resolution of the CMB map (the number of the multipole); total
number of gamma-ray sources in the subsample ($N_t$); the number
of sources ($N_e$) located in the CMB pixels with negative
fluctuation values; expected number of pixels with negative CMB
amplitudes according to the results of 200 random Gaussian CMB
signal simulations, modelled in terms of the $\Lambda$CDM
cosmology, and the 1$\sigma$-scatter of these quantities}
\begin{tabular}{clrrrc}
\hline
$t$, s & Mission &$\ell_{max}$ & $N_t$ & $N_e$ & Model \\
\hline
$<$2 & BATSE & 150 & 497 & 244 & 249$\pm$11 \\
$>$2 & BATSE & 150 & 1540 & 763 & 769$\pm$19 \\
$<$2 & BATSE & 20 & 497 & 250 & 248$\pm$13 \\
$>$2 & BATSE & 20 & 1540 & 781 & 768$\pm$32 \\
\hline
$<$2 & BeppoSAX & 150 & 87 & 43 & 44$\pm$5 \\
$>$2 & BeppoSAX & 150 & 694 & 339 & 347$\pm$15 \\
$<$2 & BeppoSAX & 20 & 87 & 50 & 44$\pm$5 \\
$>$2 & BeppoSAX & 20 & 694 & 346 & 348$\pm$18 \\
\hline
\end{tabular}
\end{table}
\begin{figure*
\centerline{\vbox{
\hbox{
\psfig{figure=fig5a_sbep_s_lmax20.eps,angle=-90,width=7cm}
\psfig{figure=fig5b_sbep_l_lmax20.eps,angle=-90,width=7cm}
} \hbox{
\psfig{figure=fig5c_sbat_s_lmax20.eps,angle=-90,width=7cm}
\psfig{figure=fig5d_sbat_l_lmax20.eps,angle=-90,width=7cm}
} }}
\caption{ Distribution of CMB fluctuations in the WMAP pixels
corresponding to the GRB positions on the maps with a resolution
of $\ell_{max}=20$. The top left and top right panels show the
BeppoSAX data for the bursts with $t<2$\,s and $t>2$\,s,
respectively. The bottom left and bottom right panels show the
data for the BATSE bursts with $t<2$\,s and $t>2$\,s,
respectively. The dashed lines show the 1$\sigma$ scatter of CMB
values in the $\Lambda$CDM cosmological model.}
\label{f5}
\end{figure*}
\begin{figure*
\centerline{\vbox{
\hbox{
\psfig{figure=fig6a_sbep_s_lmax150.eps,angle=-90,width=7cm}
\psfig{figure=fig6b_sbep_l_lmax150.eps,angle=-90,width=7cm}
} \hbox{
\psfig{figure=fig6c_sbat_s_lmax150.eps,angle=-90,width=7cm}
\psfig{figure=fig6d_sbat_l_lmax150.eps,angle=-90,width=7cm}
}}}
\caption{ Distribution of the CMB fluctuations in the WMAP
pixels corresponding to GRB positions on the maps with a
resolution of $\ell_{max}=150$. The top left and top right panels
show the BeppoSAX data for the bursts with $t<2$\,s and $t>2$\,s,
respectively. The bottom left and bottom right panels show the
data for the BATSE bursts with $t<2$\,s and $t>2$\,s,
respectively. The dashed lines show the 1$\sigma$ scatter of CMB
values in the $\Lambda$CDM cosmological model.}
\label{f6}
\end{figure*}
Figures~5 and 6 show the distributions of the CMB fluctuations for
four gamma-ray burst subsamples and two CMB maps with different
resolutions. The dashed lines show the expected 1$\sigma$ scatter
of CMB values in the $\Lambda$CDM cosmological model. The bottom
left panel in Fig.~6, which shows the distribution of fluctuations
on the CMB map with a resolution of $\ell_{max}=150$ in the areas
of short gamma-ray bursts of the BATSE catalog, stands out
conspicuously. In the positive part of the plot there is a peak,
which makes the distribution deviate from Gaussianity. Figure~7
shows the positions of these bursts superimposed onto the CMB
quadrupole. The statistical significance of this feature, which we
estimated by generating 10000 simulated CMB realizations with the
power spectrum corresponding to the $\Lambda$CDM model, is equal
to \mbox{$7\times10^{-4}$.}
\begin{figure*
\centerline{
\psfig{figure=fig7_ilc2bat1.ps,angle=-90,width=7cm,%
bblly=0pt,bbllx=0pt,bburx=500pt,bbury=840pt,clip=}
}
\caption{Positions of short BATSE bursts corresponding to the excess in the histogram
in Fig.\,6 (the bottom left panel) on the map of the quadrupole ILC component.}
\label{f7}
\end{figure*}
To analyze the distribution of GRB positions corresponding to the
excess peak, we used the GLESP software package \cite{glesp2} to
pixelize the map of burst positions. We chose the pixel size
700\arcmin$\times$700\arcmin\ in order to make the maximum pixel
value (the number of events inside the corresponding area) no less
than three, and to provide a significant dynamic range for the
harmonic analysis. Figure~8 shows the map of selected bursts
pixelized in such a way. The distribution of events on the sphere
shown in the image is clearly non-uniform: it is concentrated near
the ecliptic and/or equatorial poles, which is immediately
apparent in the smoothed map (Fig.\,9). Figure~9 also shows the
ecliptic and equatorial coordinate grids, and it is evident from
this figure that the asymmetry shows up both in the distribution
of the signal power with respect to the equatorial plane, and in
the number of spots that concentrate in the Southern Hemisphere in
both coordinate systems. Note also that the hot spot above the
Galactic center lies in the equatorial plane.
\begin{figure*
\centerline{
\psfig{figure=fig8_mct_bat1P.ps,angle=-90,width=7cm}
}
\caption{Pixelized map of positions of short gamma-ray bursts of
the BATSE catalog corresponding to the excess in the histogram in
Fig.\,6 (the bottom left panel). The pixel size
700\arcmin$\times$700\arcmin\ is chosen in a way to make the
maximum pixel value---the number of events inside the
corresponding sky area---greater than or equal to three.}
\label{f8}
\end{figure*}
\begin{figure*
\centerline{\hbox{
\psfig{figure=fig9a_mst_bat1Pe.ps,angle=-90,width=7cm}
\psfig{figure=fig9b_mst_bat1Pq.ps,angle=-90,width=7cm}
}}
\caption{Smoothed sky map corresponding to the pixelization used in Fig.\,8,
with the ecliptic (right) and equatorial (left) coordinate grids superimposed. }
\label{f9}
\end{figure*}
\begin{figure*
\centerline{\hbox{
\psfig{figure=fig10a_mst_bat1PL2e.ps,angle=-90,width=7cm,%
bblly=0pt,bbllx=0pt,bburx=500pt,bbury=840pt,clip=}
\psfig{figure=fig10b_mst_bat1PL2q.ps,angle=-90,width=7cm,%
bblly=0pt,bbllx=0pt,bburx=500pt,bbury=840pt,clip=}
}}
\caption{A quadrupole of the smoothed map (Fig.\,9)
with the ecliptic (right) and equatorial (left) coordinate grids superimposed. }
\label{f10}
\end{figure*}
To reveal the asymmetries more clearly, we demonstrate a
quadrupole of the smoothed map (Fig.\,10) with the ecliptic and
equatorial coordinate grids superimposed, as we do it in Fig.\,9.
It is evident from the positions of quadrupole spots that: (1) the
poles of both coordinate systems are located on the outskirts of
hot spots; (2) cold spots are located symmetrically with respect
to the equatorial planes, and (3) the Galactic center is located
in the saddle-shaped area between cold and hot spots.
Our analysis of gamma-ray burst positions in the area of the peak
in the distribution of short BATSE bursts with respect to the CMB
revealed their unexpected sensitivity to local (near-Earth)
coordinate systems. To analyze this problem in more detail, we
pixelize all the four subsamples of the GRB catalogs using the
method of mosaic correlation described in the previous section.
\section{GRB AND WMAP ILC DISTRIBUTION CORRELATION MAPS}
To verify and refine the correlation properties of the maps of GRB
positions and CMB fluctuations, we pixelized the maps of GRB
positions for four gamma-ray burst subsamples (Fig.\,11). Like at
the previous stage, we chose the pixel size
200\arcmin$\times$200\arcmin\, ($\ell_{max}=26$) in order to
ensure that the maximum pixel value---the number of events in the
corresponding area---would be not less than three.
\begin{figure*
\centerline{\vbox{
\hbox{
\psfig{figure=fig11a_mct_bep1p.ps,angle=-90,width=7cm}
\psfig{figure=fig11b_mct_bep2p.ps,angle=-90,width=7cm}
} \hbox{
\psfig{figure=fig11c_mct_bat1p.ps,angle=-90,width=7cm}
\psfig{figure=fig11d_mct_bat2p.ps,angle=-90,width=7cm}
}}}
\caption{Pixelized maps of event positions in gamma-ray burst subsamples. The pixel
size is 200$'\times200'$. The top left and top right panels show
the BeppoSAX data for bursts with $t<2$\,s and $t>2$\,s,
respectively. The bottom left and bottom right panels show the
data for the BATSE bursts with $t<2$\,s and $t>2$\,s,
respectively. }
\label{f11}
\end{figure*}
We then performed the mosaic correlation for the BATSE--BeppoSAX
and BATSE--CMB pairs (Fig.\,12). The pixel size for these
correlations was equal to 500\arcmin$\times$500\arcmin.
\begin{figure*
\centerline{\vbox{
\hbox{
\psfig{figure=fig12a_mapc_bbep1e.ps,angle=-90,width=7cm}
\psfig{figure=fig12b_mapc_bbep2.ps,angle=-90,width=7cm}
} \hbox{
\psfig{figure=fig12c_mapc_cbep1.ps,angle=-90,width=7cm}
\psfig{figure=fig12d_mapc_cbep2.ps,angle=-90,width=7cm}
}}}
\caption{Correlation maps for the CMB data ($\ell_{max}=26$) and gamma-ray burst positions
in the Galactic coordinate system. The pixel size for these correlations is
equal to 500$'\times500'$. The top left panel shows the correlation between BATSE and
BeppoSAX data, $t<2$\,s, with the ecliptic coordinate grid superimposed.
The top right panel shows the correlation between BATSE ($t<2$\,s) and CMB data.
The bottom left panel shows the correlation between BATSE and BeppoSAX data, $t>2$\,s.
The bottom right panel shows the correlation between BATSE ($t>2$\,s) and CMB
data. }
\label{f12}
\end{figure*}
We can now use the expansion defined by formula (\ref{eq1}) to
compute the angular power spectrum \mbox{of the map:}
\begin{equation}
C(\ell) =
\frac{1}{2\ell+1}\left[|a_{\ell 0}|^2 +2\sum_{m=1}^\ell
|a_{\ell,m}|^2\right]\,.
\end{equation}
The power spectrum allows us to identify the main harmonics
contributing to the correlation map. Figure~13 shows the power
spectra of the correlation coefficient maps (mosaic correlation)
between the BATSE and CMB data ($\ell_{max}=26$) and between the
BATSE and BeppoSAX gamma-ray positions.
\begin{figure*
\centerline{\vbox{
\hbox{
\psfig{figure=fig13a_batb1rand.eps,angle=-90,width=7cm}
\psfig{figure=fig13b_batb2rand.eps,angle=-90,width=7cm}
} \hbox{
\psfig{figure=fig13c_batc1rand.eps,angle=-90,width=7cm}
\psfig{figure=fig13d_batc2rand.eps,angle=-90,width=7cm}
} }} \caption{Power spectra of the correlation coefficient maps
between the BATSE GRB positions and BeppoSAX and CMB data
($\ell_{max}=26$) (the solid line). The pixel size of the
correlations is 500\arcmin$\times$500\arcmin. The top left panel
shows the spectrum of correlation between the BATSE and BeppoSAX
data, $t<2$\,s. The top right panel shows the spectrum of
correlation between the BATSE ($t<2$\,s) and CMB data. The bottom
left panel shows the spectrum of correlation between the BATSE and
BeppoSAX data, $t>2$\,s. The bottom right panel shows the spectrum
of correlation between the BATSE ($t>2$\,s) and CMB data. The
dashed lines show the $\pm \sigma$, $\pm 2\sigma$, and $\pm
3\sigma$ deviations from the mean, where $\sigma$ is estimated
from the spectra of 200 simulated random realizations of the zero
hypothesis, which assumes a uniform distribution of gamma-ray
bursts in the sky and the Gaussian CMB amplitude distribution
corresponding to the $\Lambda$CDM cosmology. }
\label{f13}
\end{figure*}
\begin{figure
\centerline{
\psfig{figure=fig14_prob.eps,angle=-90,width=7cm}
}
\caption{Significance levels for the deviations from the mean level for the power
spectra of the correlation maps: short gamma-ray burst
BATSE-BeppoSAX correlations (the thin dashed line); long gamma-ray
burst BATSE-BeppoSAX correlations (the thick dashed line); short
BATSE bursts and CMB fluctuations (the thin solid line), and long
BATSE bursts and CMB fluctuations (the thick solid line). The
statistical significance was estimated via direct numerical
simulations involving 10000 random realizations of the zero
hypothesis, which consists in the uniform GRB distribution in the
sky and the Gaussian CMB amplitude distribution corresponding to
the $\Lambda$CDM cosmology.}
\label{f14}
\end{figure}
Three correlation power spectra stand out among other
correlations: those between the positions of short BATSE and
BeppoSAX bursts (the top left panel in Fig.\,13) and between short
and long BATSE bursts and CMB fluctuations (top and bottom right
panels, respectively in Fig.\,13). The peaks of these spectra
exhibit features that formally lie beyond the 2$\sigma$ level,
and, in the case of the harmonics $\ell =1$, 4, 7, 9, even beyond
3$\sigma$. To estimate the statistical significance of these
deviations, we performed numerical simulations of the zero
\mbox{hypothesis} by generating: (1) 10,000 random realizations of
the set of sky positions of the BeppoSAX gamma-ray bursts assuming
that they are uniformly distributed in the sky and (2) random CMB
maps with the Gaussian amplitude distribution corresponding to the
$\Lambda$CDM cosmology, and computed the correlations and the
corresponding power spectra for these simulated data in the same
way as we did it for real data. Figure~14 shows the probabilities
for a simulated realization to reach the deviation levels of the
features observed in Fig.\,13 within the framework of the zero
\mbox{hypothesis} i.e., the significance levels of the peaks in
the power spectrum. The resulting significance levels for the
multipoles $\ell =1$, 4, 7, and 9 of the correlation maps between
the short BATSE and BeppoSAX bursts are equal to 0.0011, 0.0012,
0.0001, and less than 0.0001, respectively. At the same time, the
significance level for the octupole of the correlations between
the long BATSE and BeppoSAX bursts is equal to 0.0325, and that
for the quadrupole of the correlation map between the long BATSE
bursts and CMB fluctuations is equal to 0.0614. Figure~15 shows
the map of one of the harmonics ($\ell=7$) in the distribution of
the correlation coefficients between the positions of short BATSE
and BeppoSAX bursts. Figure~16 shows the maps of the peak
harmonics ($\ell=3$ for short and $\ell=2$ for long bursts) in the
spectrum of the maps of correlations between the BATSE and CMB.
\begin{figure
\centerline{
\psfig{figure=fig15_map_bbep1L7e.ps,angle=-90,width=7cm}
}
\caption{Map of the seventh harmonic ($\ell=7$) in the expansion of the map of correlations
between the positions of short BATSE and BeppoSAX bursts with the equatorial coordinate
grid superimposed. }
\label{f15}
\end{figure}
\begin{figure*
\centerline{\vbox{
\psfig{figure=fig16a_maps_bbep2L3q.ps,angle=-90,width=7cm}
\psfig{figure=fig16b_maps_cbat2L2q.ps,angle=-90,width=7cm}
}}
\caption{Maps of the harmonics identified in the power spectrum (Fig.\,13). Correlation maps
between the BATSE burst positions ($t>2$ s) and the BeppoSAX data
(the top panel, $\ell=3$) and between the BATSE burst positions
($t>2$ s) and the CMB (the bottom panel, $\ell=2$) with the
equatorial coordinate grid superimposed. }
\label{f16}
\end{figure*}
A distinguishing feature of the maps shown with superimposed
coordinate grids are the positions of the poles of coordinate
systems. It is evident from Fig.\,15 that the ecliptic poles are
located at special \mbox{points---in} the saddles between the
maxima and minima of the signal distribution on the map.
Similarly, the equatorial poles are located in the saddles in the
map of the correlation octupole (the left panel in Fig.\,16). We
can see in the right panel (the quadrupole) of Fig.~16 that the
poles are located in the minima of the quadrupole. We use the
method described above based on the generation of 10,000 random
realizations of the GLESP pixelization with 102 pixels at the
equator to estimate the statistical significance of such a
configuration i.e., the probability for the minima of the
quadrupole to occur in the 5-degree radius areas centered on the
equatorial poles, and find it to be equal to 0.0035.
\section{DISCUSSION}
Our analysis of correlations between the maps of GRB positions and
the CMB shows that such correlations do exist. Of great interest
is the orientation (phase properties) of correlation maps, where
we can point out two points: (1) in the case of the correlations
between the short GRB positions in different catalogs and their
correlations with the CMB, the features found (the positions of
the poles) are observed both in the equatorial and ecliptic
coordinate systems; moreover, despite a small number of short
BeppoSAX and BATSE events, the correlations between their
positions in 500\arcmin$\times$500\arcmin\ windows exhibit a chain
of events in the ecliptic plane (the top left panel in Fig.\,12),
and a predominant occurrence of the correlated pixels in the
Southern Hemisphere; (2) the correlations between the long BATSE
events and CMB fluctuations exhibit features in the equatorial
coordinate system, while the joint probability of high quadrupole
amplitude and of the quadrupole minimum occurrence at the
equatorial pole is practically equal to zero in the case of
correlation with random maps.
The correlations found between the GRB positions and the CMB, that
are sensitive to the equatorial coordinate system must be due to
the systematic effects. This way, a relatively greater number of
gamma-ray events near the equatorial poles may be due to
longer exposures of the satellite cameras due to the observing
method employed. Both satellites that observed the gamma-ray
bursts moved in rather low Earth orbits, rendering the areas
located near the celestial equator periodically unobservable. At
the same time, for the CMB data such a sensitivity to the
equatorial coordinate system is impossible to explain in terms of
such a simple model, as the microwave background data were
obtained onboard the WMAP satellite, which rotates about the L2
Lagrangian point. Note that in our previous paper~\cite{cor_ecl}
we also found some correlations to be ``aware'' of the equatorial
system.
We do not rule out the possible contribution from the
Earth's magnetic field, which has a large extent and shows up as
large-scale correlations of the microwave background. However, we
do not understand the mechanism of such correlations.
Neither do we understand the possible interrelation between the
ecliptic plane and the positions of gamma-ray bursts. Here the
situation is reversed: the ecliptic features in the CMB data have
already been discussed recently~\cite{diego,dikarev,cor_ecl},
however, further studies are needed to understand what may link
gamma-ray evens and the ecliptic plane. One of the possible
hypotheses explains this feature as a selection effect due to the
fact that the observing instruments of the satellites are turned
away from the Sun, and this may put the plane of the ecliptic in a
special position.
Note also that the correlation properties of the CMB, which show
up in the ``near-Earth'' coordinate systems when the data with
``randomly'' distributed events are used, are indicative of the
non-Gaussian nature of the data for low-order multipoles, which
may be due either to the systematic effects, or to a hitherto
unexplored effect in the near-Earth space. We consider further
studies of this correlation to be of especially great interest
given that the new high-quality data are expected to be provided
by the Fermi and Planck missions.
\noindent
{\small
{\bf Acknowledgments}.
We are grateful to Valery Larionov (St. Petersburg State
University) for useful discussions of the results of this work. We
thank the NASA for making available the NASA Legacy Archive, from
where we adopted the WMAP data. We are also grateful to the
authors of the HEALPix\footnote{\tt
http://www.eso.org/science/healpix/} \cite{healpix} package, which
we used to transform the WMAP7 maps into the coefficients $a_{\ell
m}$. This work made use of the \mbox{GLESP\footnote{\tt
http://www.glesp.nbi.dk} \cite{glesp,glesp1}} package for the
further analysis of the CMB data on the sphere. This work was
supported by the Program for the Support of Leading Scientific
Schools of Russia (the School of S.~E.~Khaikin) and the Russian
Foundation for Basic Research (grant nos.~09-02-00298 and
\mbox{08-02-00486.} O.V.V. also acknowledges partial support from
the Foundation for the Support of Domestic Science (the program
``Young Doctors of Science of the
Russian Academy of Sciences'') and the Dynasty Foundation.
\section{INTRODUCTION}
The publication of comparatively large catalogs of gamma-ray
bursts based on the results of the surveys performed within the
framework of space missions of the Italo-Dutch BeppoSAX
satellite\footnote{\tt http://www.asdc.asi.it/bepposax/}
(Satellite per Astronomia X, and Beppo in the honor of Giuseppe
Occhialini) \cite{bepposax}, and NASA's Compton observatory, where
the BATSE experiment\footnote{\tt
http://www.batse.msfc.nasa.gov/batse/} (Burst and Transient Source
Experiment) \cite{batse} was performed, made it possible to study
the properties of the space distribution of gamma-ray bursts,
which, to a first approximation, are uniformly distributed in the
sky.
Among recent studies concerning this subject we distinguish the
analysis of the possible relation between the distribution of
gamma-ray bursts from the BATSE catalog and the large-scale
structure by Williams and Frey ~\cite{grb_apm}. The authors
analyze the effect of the local large-scale structure on the
apparent positions of gamma-ray bursts in the case of weak
lensing. This effect may show up for distant ($z>4$) events as an
anticorrelation of the positions of gamma-ray bursts and galaxies
in clusters. The authors studying gamma-ray bursts report such an
anticorrelation for galaxies at $z \sim 0.2-0.3$, which they found
based on the optical magnitudes and positions of galaxies measured
using the APM (Automatic Plate Measuring Facility).
\begin{figure*
\centerline{
\psfig{figure=fig1_bep_bat_map.ps,angle=-90,width=7cm}
}
\caption{Distribution of gamma-ray bursts on the sky. The gray
and white symbols (the circles and crosses correspond to short and
long bursts, respectively) show the BeppoSAX and BATSE data,
respectively. }
\label{f1}
\end{figure*}
\begin{figure*
\centerline{\vbox{
\hbox{
\psfig{figure=fig2a_bep_s.ps,angle=-90,width=7cm}
\psfig{figure=fig2b_bep_l.ps,angle=-90,width=7cm}
} \hbox{
\psfig{figure=fig2c_bat_s.ps,angle=-90,width=7cm}
\psfig{figure=fig2d_bat_l.ps,angle=-90,width=7cm}
}}}
\caption{Distribution of the gamma-ray bursts on the sky. The top left panel shows the
BeppoSAX data for bursts with $t<2$\,s. The top right panel shows
the BeppoSAX data for bursts with $t>2$\,s. The bottom left panel
shows the BATSE data for $t<2$\,s. The bottom right panel shows
the BATSE data for $t>2$\,s. }
\label{f2}
\end{figure*}
In another paper, M\'esz\'aros~et al. \cite{grb_vor} use various
methods (the Voronoi tesselation diagrams, the minimal spanning
tree, and multifractal spectra) to analyze the distribution of the
BATSE gamma-ray bursts. The above authors break the list of
objects into subsamples of sources with different signal durations
($t<2$\,s, \mbox{$2<t<10$\,s,} $t>10$\,s) and apply the above
methods to each subsample. They found the results for the first
two groups to deviate from the results obtained for uniformly
distributed simulated data. Based on the data obtained the
authors \cite{grb_vor} discuss the validity of the cosmological
principle.
We use the BeppoSAX (781 source, the energy interval covered:
0.1--200\,keV) and BATSE (2037 sources, 20\,keV--2\,MeV) catalogs
adopted from the sites of the corresponding experiments.
We subdivide each catalog into two subsamples containing short
(i.e., lasting $t<2$\,s) and long ($t>2$\,s) events. We thus
obtain four source subsamples, which we analyze separately using
the same procedure. Figure~1 shows the positions of all gamma-ray
bursts from the BeppoSAX and BATSE catalogs. Figure~2 shows the
positions of short and long BeppoSAX and BATSE gamma-ray bursts on
separate sky charts.
In this paper we analyze the statistical correlation properties of
the GRB distribution on the sky with respect to the distribution
of the cosmic microwave background (CMB). We proceed from the
assumption that gamma-ray bursts should be associated with massive
galaxies, and that the
positions of these galaxies are correlated with the large-scale
structure. In this case there is hope that we may find deviations
in the sky distribution of CMB fluctuations in the areas where the
bursts are projected. Note that the large-scale structure may show
up in CMB fluctuations via the Zel'dovich--Sunyaev effect and the
integrated Sachs--Wolfe effect~\cite{swe}. This former effect
manifests itself via the interaction of CMB photons with hot
electrons of galaxy clusters \cite{zse} and is observed on the
angular scales of about 4\arcmin\, or smaller. The latter effects
occurs when the photons move in a gravitational field with a
variable potential that arises during
the formation of the large-scale structure and expansion of the
Universe, and should be observed on the angular scales greater
than 200\arcmin.
To study the correlation properties of the positions of gamma-ray
bursts and CMB peaks, we use the CMB map based on the results of
the experiment that has been conducted for seven years within the
framework of the WMAP\footnote{{\tt http://lambda.gsfc.nasa.gov}}
(Wilkinson Microwave Anisotropy Probe) experiment aimed to
integrate the signal from the entire sky \cite{wmap7ytem}. The CMB
signal was reconstructed from multifrequency observations using
the method of internal linear combination (ILC) of the background
components~\cite{wmapresults}. This procedure yielded the CMB map,
which is also referred to as the ILC map, and which is used to
analyze low-order harmonics with multipole numbers $\ell\le150$.
The ILC map is based on observations made in five channels:
23\,GHz (the K band), 33\,GHz (the Ka band), 41\,GHz (the Q
band), 61\,GHz (the V band), and 94\,GHz (the W band).
Given that one of the central problems in statistical studies of
gamma-ray bursts is due to the large error box size, about
1\degr$\times$1\degr, of the available source positions, we
operate with pixels of about the same or even greater size in
order to avoid uncertainties in the analysis of the sky
distribution of the events.
The methods we use in this paper and our results are laid out in
the following way. First we describe the methods that we use to
pixelize the distribution of bursts and to correlate the maps. We
then analyze the statistics of CMB deviations in the gamma-ray
burst areas. At the next stage, we compare the BATSE--BeppoSAX and
BATSE--CMB correlation maps. We finally discuss the results in the
concluding section.
\section{MOSAIC CORRELATION METHOD}
To analyze the map properties on different angular scales, we
expand the signal distributed on the sphere into the spherical
harmonics (multipoles):
\begin{equation}
\Delta S(\theta,\phi)= \sum_{\ell=1}^{\infty}\sum_{m=-\ell}^{m=\ell}
a_{\ell m} Y_{\ell m} (\theta, \phi)\,.
\label{eq1}
\end{equation}
Here $\Delta S(\theta,\phi)$ are the variations of signal on the
sphere in polar coordinates; $\ell$ is the number of the
multipole, and $m$ is the number of the mode of the multipole. The
spherical harmonics are determined as
\begin{eqnarray}
Y_{\ell m}(\theta,\phi) = \sqrt{{(2\ell+1)\over 4\pi}{(\ell-m)!
\over (\ell+m)!}}P_\ell^m(x) e^{i m\phi}, \\
~x=\cos\theta\,,\nonumber \label{eq2}
\end{eqnarray}
where $P_\ell^m(x)$ are the associated Legendre polynomials. The
expansion coefficients $a_{\ell m}$ for a continuous function
$\Delta S(x,\phi)$ can be written as:
\begin{equation}
a_{\ell m}=\int^1_{-1}dx\int^{2\pi}_0 \Delta S(x,\phi)
Y^{*}_{\ell m}(x,\phi) d\phi\,,
\label{eq3}
\end{equation}
where $Y^{*}_{\ell m}$ denotes the complex conjugate of $Y_{\ell m}$.
The correlation properties of two maps of the signal distribution
on the sphere can be described, on a given angular scale, by a
correlation coefficient for the corresponding multipole $\ell$ as:
\begin{equation}
K(\ell) = \frac{1}{2} \frac{\sum\limits_{m=-\ell}^\ell t_{\ell
m}s^*_{\ell m} + t^*_{\ell m}s_{\ell m}}
{(\sum\limits_{m=-\ell}^\ell |t_{\ell m}|^2
\sum\limits_{m=-\ell}^\ell |s_{\ell m}|^2)^{1/2}}\,,
\label{eq4}
\end{equation}
where $t_{\ell m}$ and $s_{\ell m}$ are the variations of two
signals in a harmonic representation. The coefficient $K(\ell)$
can be used to assess the correlation between the harmonics on the
sphere, i.e., to compare the properties of maps on a given angular
scale. However, in the case of a search for correlated areas,
which do not repeat in other regions of the sphere, this approach
smears such single areas in the process of averaging over the
sphere within a certain harmonic. In this case it becomes
practically impossible to identify the correlated areas.
Verkhodanov et al.~\cite{cormap} proposed an approach, which was
implemented in the second release of the GLESP package
\cite{glesp2} (the {\tt difmap} utility). The proposed procedure
makes it possible to find correlations between two maps in the
areas of a certain angular size. In this method, each pixel with
number $p$ subtending the solid angle $\Xi_p$ is assigned the
cross-correlation coefficient between the data of the two maps on
the corresponding area. Thus a correlation map is constructed for
two signals $T$ and $S$, where the value of each pixel $p$
($p=1,2,...,N_0$, \mbox{and $N_0$ is the} total number of pixels
on the sphere) with the subtending angle $\Xi_p$, and computed for
the sphere maps with the initial resolution determined by
$\ell_{max}$ is \mbox{equal to}
\begin{eqnarray}
\nonumber K(\Xi_p|\ell_{max}) = \hspace{3cm} \\
\frac{\sum\sum\limits_{p_{ij}\in\Xi_p}
(T(\theta_i,\phi_j) - \overline{T(\Xi_p))}
(S(\theta_i,\phi_j) - \overline{S(\Xi_p))}}
{\sigma_{T_p}\sigma_{S_p}}\,.
\label{eq5}
\end{eqnarray}
Here $T(\theta_i,\phi_j)$ is the value of the signal $T$ in the
pixel with the coordinates $(\theta_i,\phi_j)$ for the initial
resolution of the pixelization of the sphere; $S(\theta_i,\phi_j)$
is the value of the other signal in the same area;
$\overline{T(\Xi_p)}$ and \mbox{$\overline{S(\Xi_p)}$ } are the
mean values averaged over the area $\Xi_p$ and obtained from the
data of the maps with higher resolution determined by
$\ell_{max}$, and $\sigma_{T_p}$ and $\sigma_{S_p}$ are the
corresponding standard deviations in the area considered.
\section{CMB SIGNAL STATISTICS IN THE AREAS OF GAMMA-RAY BURSTS}
\begin{figure*
\centerline{\vbox{
\hbox{
\psfig{figure=fig3a_mbep_s_lmax20.ps,angle=-90,width=7cm}
\psfig{figure=fig3b_mbep_l_lmax20.ps,angle=-90,width=7cm}
} \hbox{
\psfig{figure=fig3c_mbat_s_lmax20.ps,angle=-90,width=7cm}
\psfig{figure=fig3d_mbat_l_lmax20.ps,angle=-90,width=7cm}
}}}
\caption{Positions of gamma-ray bursts from different subsamples on the CMB maps
with a resolution of $\ell_{max}=20$. The top left and top right
panels show the BeppoSAX data for bursts with $t<2$\,s and
$t>2$\,s, respectively. The bottom left and bottom right panels
show the data for the BATSE bursts with $t<2$\,s and $t>2$\,s,
respectively.}
\label{f3}
\end{figure*}
\begin{figure*
\centerline{\vbox{
\hbox{
\psfig{figure=fig4a_mbep_s_lmax150.ps,angle=-90,width=7cm}
\psfig{figure=fig4b_mbep_l_lmax150.ps,angle=-90,width=7cm}
} \hbox{
\psfig{figure=fig4c_mbat_s_lmax150.ps,angle=-90,width=7cm}
\psfig{figure=fig4d_mbat_l_lmax150.ps,angle=-90,width=7cm}
}}}
\caption{Positions of gamma-ray bursts from different subsamples
on the CMB maps with a resolution of $\ell_{max}=150$. The top
left and top right panels show the BeppoSAX data for bursts with
$t<2$\,s and $t>2$\,s, respectively. The bottom left and bottom
right panels show the data for the BATSE bursts with $t<2$\,s
and $t>2$\,s, respectively. }
\label{f4}
\end{figure*}
In the first method the statistical properties of distributions
of GRB positions versus the CMB signal are tested by simply
analyzing the pixel value measurements on the CMB maps. We
performed pixel measurements using the {\tt mapcut} procedure of
the GLESP package. Figures~3 and 4 show the positions of gamma-ray
bursts of the BeppoSAX and BATSE catalog subsamples on the CMB
maps with the resolution of 260\arcmin\, \mbox{($\ell_{max}=20$)}
and 36\arcmin\, ($\ell_{max}=150$). We chose the resolution of
these maps to match the expected angular scale of the Sachs--Wolfe
effect and the limiting resolution of the WMAP mission.
To search for the eventual correlations, we computed the number of
GRB positions that fall onto the CMB pixels with negative signal
fluctuations (which may be due to the effects described above) in
the CMB maps of different resolution. The Table lists the
statistics of CMB pixel values in the gamma-ray burst areas for
the long- and short-event BATSE and BeppoSAX subsamples. It
includes the total number of sources in subsamples; the number of
sources located in CMB pixels with negative fluctuation values;
the expected number of pixels with negative CMB amplitudes
according to the results of 200 random Gaussian CMB signal
simulations modeled in terms of the $\Lambda$CDM cosmology, and
the 1$\sigma$-scatter of these quantities.
\begin{table}
\caption{{\bf Table.}~Statistics of the
CMB pixel values in the GRB areas for the BATSE and BeppoSAX
subsamples. The columns are: $t$, the duration (s); mission name;
resolution of the CMB map (the number of the multipole); total
number of gamma-ray sources in the subsample ($N_t$); the number
of sources ($N_e$) located in the CMB pixels with negative
fluctuation values; expected number of pixels with negative CMB
amplitudes according to the results of 200 random Gaussian CMB
signal simulations, modelled in terms of the $\Lambda$CDM
cosmology, and the 1$\sigma$-scatter of these quantities}
\begin{tabular}{clrrrc}
\hline
$t$, s & Mission &$\ell_{max}$ & $N_t$ & $N_e$ & Model \\
\hline
$<$2 & BATSE & 150 & 497 & 244 & 249$\pm$11 \\
$>$2 & BATSE & 150 & 1540 & 763 & 769$\pm$19 \\
$<$2 & BATSE & 20 & 497 & 250 & 248$\pm$13 \\
$>$2 & BATSE & 20 & 1540 & 781 & 768$\pm$32 \\
\hline
$<$2 & BeppoSAX & 150 & 87 & 43 & 44$\pm$5 \\
$>$2 & BeppoSAX & 150 & 694 & 339 & 347$\pm$15 \\
$<$2 & BeppoSAX & 20 & 87 & 50 & 44$\pm$5 \\
$>$2 & BeppoSAX & 20 & 694 & 346 & 348$\pm$18 \\
\hline
\end{tabular}
\end{table}
\begin{figure*
\centerline{\vbox{
\hbox{
\psfig{figure=fig5a_sbep_s_lmax20.eps,angle=-90,width=7cm}
\psfig{figure=fig5b_sbep_l_lmax20.eps,angle=-90,width=7cm}
} \hbox{
\psfig{figure=fig5c_sbat_s_lmax20.eps,angle=-90,width=7cm}
\psfig{figure=fig5d_sbat_l_lmax20.eps,angle=-90,width=7cm}
} }}
\caption{ Distribution of CMB fluctuations in the WMAP pixels
corresponding to the GRB positions on the maps with a resolution
of $\ell_{max}=20$. The top left and top right panels show the
BeppoSAX data for the bursts with $t<2$\,s and $t>2$\,s,
respectively. The bottom left and bottom right panels show the
data for the BATSE bursts with $t<2$\,s and $t>2$\,s,
respectively. The dashed lines show the 1$\sigma$ scatter of CMB
values in the $\Lambda$CDM cosmological model.}
\label{f5}
\end{figure*}
\begin{figure*
\centerline{\vbox{
\hbox{
\psfig{figure=fig6a_sbep_s_lmax150.eps,angle=-90,width=7cm}
\psfig{figure=fig6b_sbep_l_lmax150.eps,angle=-90,width=7cm}
} \hbox{
\psfig{figure=fig6c_sbat_s_lmax150.eps,angle=-90,width=7cm}
\psfig{figure=fig6d_sbat_l_lmax150.eps,angle=-90,width=7cm}
}}}
\caption{ Distribution of the CMB fluctuations in the WMAP
pixels corresponding to GRB positions on the maps with a
resolution of $\ell_{max}=150$. The top left and top right panels
show the BeppoSAX data for the bursts with $t<2$\,s and $t>2$\,s,
respectively. The bottom left and bottom right panels show the
data for the BATSE bursts with $t<2$\,s and $t>2$\,s,
respectively. The dashed lines show the 1$\sigma$ scatter of CMB
values in the $\Lambda$CDM cosmological model.}
\label{f6}
\end{figure*}
Figures~5 and 6 show the distributions of the CMB fluctuations for
four gamma-ray burst subsamples and two CMB maps with different
resolutions. The dashed lines show the expected 1$\sigma$ scatter
of CMB values in the $\Lambda$CDM cosmological model. The bottom
left panel in Fig.~6, which shows the distribution of fluctuations
on the CMB map with a resolution of $\ell_{max}=150$ in the areas
of short gamma-ray bursts of the BATSE catalog, stands out
conspicuously. In the positive part of the plot there is a peak,
which makes the distribution deviate from Gaussianity. Figure~7
shows the positions of these bursts superimposed onto the CMB
quadrupole. The statistical significance of this feature, which we
estimated by generating 10000 simulated CMB realizations with the
power spectrum corresponding to the $\Lambda$CDM model, is equal
to \mbox{$7\times10^{-4}$.}
\begin{figure*
\centerline{
\psfig{figure=fig7_ilc2bat1.ps,angle=-90,width=7cm,%
bblly=0pt,bbllx=0pt,bburx=500pt,bbury=840pt,clip=}
}
\caption{Positions of short BATSE bursts corresponding to the excess in the histogram
in Fig.\,6 (the bottom left panel) on the map of the quadrupole ILC component.}
\label{f7}
\end{figure*}
To analyze the distribution of GRB positions corresponding to the
excess peak, we used the GLESP software package \cite{glesp2} to
pixelize the map of burst positions. We chose the pixel size
700\arcmin$\times$700\arcmin\ in order to make the maximum pixel
value (the number of events inside the corresponding area) no less
than three, and to provide a significant dynamic range for the
harmonic analysis. Figure~8 shows the map of selected bursts
pixelized in such a way. The distribution of events on the sphere
shown in the image is clearly non-uniform: it is concentrated near
the ecliptic and/or equatorial poles, which is immediately
apparent in the smoothed map (Fig.\,9). Figure~9 also shows the
ecliptic and equatorial coordinate grids, and it is evident from
this figure that the asymmetry shows up both in the distribution
of the signal power with respect to the equatorial plane, and in
the number of spots that concentrate in the Southern Hemisphere in
both coordinate systems. Note also that the hot spot above the
Galactic center lies in the equatorial plane.
\begin{figure*
\centerline{
\psfig{figure=fig8_mct_bat1P.ps,angle=-90,width=7cm}
}
\caption{Pixelized map of positions of short gamma-ray bursts of
the BATSE catalog corresponding to the excess in the histogram in
Fig.\,6 (the bottom left panel). The pixel size
700\arcmin$\times$700\arcmin\ is chosen in a way to make the
maximum pixel value---the number of events inside the
corresponding sky area---greater than or equal to three.}
\label{f8}
\end{figure*}
\begin{figure*
\centerline{\hbox{
\psfig{figure=fig9a_mst_bat1Pe.ps,angle=-90,width=7cm}
\psfig{figure=fig9b_mst_bat1Pq.ps,angle=-90,width=7cm}
}}
\caption{Smoothed sky map corresponding to the pixelization used in Fig.\,8,
with the ecliptic (right) and equatorial (left) coordinate grids superimposed. }
\label{f9}
\end{figure*}
\begin{figure*
\centerline{\hbox{
\psfig{figure=fig10a_mst_bat1PL2e.ps,angle=-90,width=7cm,%
bblly=0pt,bbllx=0pt,bburx=500pt,bbury=840pt,clip=}
\psfig{figure=fig10b_mst_bat1PL2q.ps,angle=-90,width=7cm,%
bblly=0pt,bbllx=0pt,bburx=500pt,bbury=840pt,clip=}
}}
\caption{A quadrupole of the smoothed map (Fig.\,9)
with the ecliptic (right) and equatorial (left) coordinate grids superimposed. }
\label{f10}
\end{figure*}
To reveal the asymmetries more clearly, we demonstrate a
quadrupole of the smoothed map (Fig.\,10) with the ecliptic and
equatorial coordinate grids superimposed, as we do it in Fig.\,9.
It is evident from the positions of quadrupole spots that: (1) the
poles of both coordinate systems are located on the outskirts of
hot spots; (2) cold spots are located symmetrically with respect
to the equatorial planes, and (3) the Galactic center is located
in the saddle-shaped area between cold and hot spots.
Our analysis of gamma-ray burst positions in the area of the peak
in the distribution of short BATSE bursts with respect to the CMB
revealed their unexpected sensitivity to local (near-Earth)
coordinate systems. To analyze this problem in more detail, we
pixelize all the four subsamples of the GRB catalogs using the
method of mosaic correlation described in the previous section.
\section{GRB AND WMAP ILC DISTRIBUTION CORRELATION MAPS}
To verify and refine the correlation properties of the maps of GRB
positions and CMB fluctuations, we pixelized the maps of GRB
positions for four gamma-ray burst subsamples (Fig.\,11). Like at
the previous stage, we chose the pixel size
200\arcmin$\times$200\arcmin\, ($\ell_{max}=26$) in order to
ensure that the maximum pixel value---the number of events in the
corresponding area---would be not less than three.
\begin{figure*
\centerline{\vbox{
\hbox{
\psfig{figure=fig11a_mct_bep1p.ps,angle=-90,width=7cm}
\psfig{figure=fig11b_mct_bep2p.ps,angle=-90,width=7cm}
} \hbox{
\psfig{figure=fig11c_mct_bat1p.ps,angle=-90,width=7cm}
\psfig{figure=fig11d_mct_bat2p.ps,angle=-90,width=7cm}
}}}
\caption{Pixelized maps of event positions in gamma-ray burst subsamples. The pixel
size is 200$'\times200'$. The top left and top right panels show
the BeppoSAX data for bursts with $t<2$\,s and $t>2$\,s,
respectively. The bottom left and bottom right panels show the
data for the BATSE bursts with $t<2$\,s and $t>2$\,s,
respectively. }
\label{f11}
\end{figure*}
We then performed the mosaic correlation for the BATSE--BeppoSAX
and BATSE--CMB pairs (Fig.\,12). The pixel size for these
correlations was equal to 500\arcmin$\times$500\arcmin.
\begin{figure*
\centerline{\vbox{
\hbox{
\psfig{figure=fig12a_mapc_bbep1e.ps,angle=-90,width=7cm}
\psfig{figure=fig12b_mapc_bbep2.ps,angle=-90,width=7cm}
} \hbox{
\psfig{figure=fig12c_mapc_cbep1.ps,angle=-90,width=7cm}
\psfig{figure=fig12d_mapc_cbep2.ps,angle=-90,width=7cm}
}}}
\caption{Correlation maps for the CMB data ($\ell_{max}=26$) and gamma-ray burst positions
in the Galactic coordinate system. The pixel size for these correlations is
equal to 500$'\times500'$. The top left panel shows the correlation between BATSE and
BeppoSAX data, $t<2$\,s, with the ecliptic coordinate grid superimposed.
The top right panel shows the correlation between BATSE ($t<2$\,s) and CMB data.
The bottom left panel shows the correlation between BATSE and BeppoSAX data, $t>2$\,s.
The bottom right panel shows the correlation between BATSE ($t>2$\,s) and CMB
data. }
\label{f12}
\end{figure*}
We can now use the expansion defined by formula (\ref{eq1}) to
compute the angular power spectrum \mbox{of the map:}
\begin{equation}
C(\ell) =
\frac{1}{2\ell+1}\left[|a_{\ell 0}|^2 +2\sum_{m=1}^\ell
|a_{\ell,m}|^2\right]\,.
\end{equation}
The power spectrum allows us to identify the main harmonics
contributing to the correlation map. Figure~13 shows the power
spectra of the correlation coefficient maps (mosaic correlation)
between the BATSE and CMB data ($\ell_{max}=26$) and between the
BATSE and BeppoSAX gamma-ray positions.
\begin{figure*
\centerline{\vbox{
\hbox{
\psfig{figure=fig13a_batb1rand.eps,angle=-90,width=7cm}
\psfig{figure=fig13b_batb2rand.eps,angle=-90,width=7cm}
} \hbox{
\psfig{figure=fig13c_batc1rand.eps,angle=-90,width=7cm}
\psfig{figure=fig13d_batc2rand.eps,angle=-90,width=7cm}
} }} \caption{Power spectra of the correlation coefficient maps
between the BATSE GRB positions and BeppoSAX and CMB data
($\ell_{max}=26$) (the solid line). The pixel size of the
correlations is 500\arcmin$\times$500\arcmin. The top left panel
shows the spectrum of correlation between the BATSE and BeppoSAX
data, $t<2$\,s. The top right panel shows the spectrum of
correlation between the BATSE ($t<2$\,s) and CMB data. The bottom
left panel shows the spectrum of correlation between the BATSE and
BeppoSAX data, $t>2$\,s. The bottom right panel shows the spectrum
of correlation between the BATSE ($t>2$\,s) and CMB data. The
dashed lines show the $\pm \sigma$, $\pm 2\sigma$, and $\pm
3\sigma$ deviations from the mean, where $\sigma$ is estimated
from the spectra of 200 simulated random realizations of the zero
hypothesis, which assumes a uniform distribution of gamma-ray
bursts in the sky and the Gaussian CMB amplitude distribution
corresponding to the $\Lambda$CDM cosmology. }
\label{f13}
\end{figure*}
\begin{figure
\centerline{
\psfig{figure=fig14_prob.eps,angle=-90,width=7cm}
}
\caption{Significance levels for the deviations from the mean level for the power
spectra of the correlation maps: short gamma-ray burst
BATSE-BeppoSAX correlations (the thin dashed line); long gamma-ray
burst BATSE-BeppoSAX correlations (the thick dashed line); short
BATSE bursts and CMB fluctuations (the thin solid line), and long
BATSE bursts and CMB fluctuations (the thick solid line). The
statistical significance was estimated via direct numerical
simulations involving 10000 random realizations of the zero
hypothesis, which consists in the uniform GRB distribution in the
sky and the Gaussian CMB amplitude distribution corresponding to
the $\Lambda$CDM cosmology.}
\label{f14}
\end{figure}
Three correlation power spectra stand out among other
correlations: those between the positions of short BATSE and
BeppoSAX bursts (the top left panel in Fig.\,13) and between short
and long BATSE bursts and CMB fluctuations (top and bottom right
panels, respectively in Fig.\,13). The peaks of these spectra
exhibit features that formally lie beyond the 2$\sigma$ level,
and, in the case of the harmonics $\ell =1$, 4, 7, 9, even beyond
3$\sigma$. To estimate the statistical significance of these
deviations, we performed numerical simulations of the zero
\mbox{hypothesis} by generating: (1) 10,000 random realizations of
the set of sky positions of the BeppoSAX gamma-ray bursts assuming
that they are uniformly distributed in the sky and (2) random CMB
maps with the Gaussian amplitude distribution corresponding to the
$\Lambda$CDM cosmology, and computed the correlations and the
corresponding power spectra for these simulated data in the same
way as we did it for real data. Figure~14 shows the probabilities
for a simulated realization to reach the deviation levels of the
features observed in Fig.\,13 within the framework of the zero
\mbox{hypothesis} i.e., the significance levels of the peaks in
the power spectrum. The resulting significance levels for the
multipoles $\ell =1$, 4, 7, and 9 of the correlation maps between
the short BATSE and BeppoSAX bursts are equal to 0.0011, 0.0012,
0.0001, and less than 0.0001, respectively. At the same time, the
significance level for the octupole of the correlations between
the long BATSE and BeppoSAX bursts is equal to 0.0325, and that
for the quadrupole of the correlation map between the long BATSE
bursts and CMB fluctuations is equal to 0.0614. Figure~15 shows
the map of one of the harmonics ($\ell=7$) in the distribution of
the correlation coefficients between the positions of short BATSE
and BeppoSAX bursts. Figure~16 shows the maps of the peak
harmonics ($\ell=3$ for short and $\ell=2$ for long bursts) in the
spectrum of the maps of correlations between the BATSE and CMB.
\begin{figure
\centerline{
\psfig{figure=fig15_map_bbep1L7e.ps,angle=-90,width=7cm}
}
\caption{Map of the seventh harmonic ($\ell=7$) in the expansion of the map of correlations
between the positions of short BATSE and BeppoSAX bursts with the equatorial coordinate
grid superimposed. }
\label{f15}
\end{figure}
\begin{figure*
\centerline{\vbox{
\psfig{figure=fig16a_maps_bbep2L3q.ps,angle=-90,width=7cm}
\psfig{figure=fig16b_maps_cbat2L2q.ps,angle=-90,width=7cm}
}}
\caption{Maps of the harmonics identified in the power spectrum (Fig.\,13). Correlation maps
between the BATSE burst positions ($t>2$ s) and the BeppoSAX data
(the top panel, $\ell=3$) and between the BATSE burst positions
($t>2$ s) and the CMB (the bottom panel, $\ell=2$) with the
equatorial coordinate grid superimposed. }
\label{f16}
\end{figure*}
A distinguishing feature of the maps shown with superimposed
coordinate grids are the positions of the poles of coordinate
systems. It is evident from Fig.\,15 that the ecliptic poles are
located at special \mbox{points---in} the saddles between the
maxima and minima of the signal distribution on the map.
Similarly, the equatorial poles are located in the saddles in the
map of the correlation octupole (the left panel in Fig.\,16). We
can see in the right panel (the quadrupole) of Fig.~16 that the
poles are located in the minima of the quadrupole. We use the
method described above based on the generation of 10,000 random
realizations of the GLESP pixelization with 102 pixels at the
equator to estimate the statistical significance of such a
configuration i.e., the probability for the minima of the
quadrupole to occur in the 5-degree radius areas centered on the
equatorial poles, and find it to be equal to 0.0035.
\section{DISCUSSION}
Our analysis of correlations between the maps of GRB positions and
the CMB shows that such correlations do exist. Of great interest
is the orientation (phase properties) of correlation maps, where
we can point out two points: (1) in the case of the correlations
between the short GRB positions in different catalogs and their
correlations with the CMB, the features found (the positions of
the poles) are observed both in the equatorial and ecliptic
coordinate systems; moreover, despite a small number of short
BeppoSAX and BATSE events, the correlations between their
positions in 500\arcmin$\times$500\arcmin\ windows exhibit a chain
of events in the ecliptic plane (the top left panel in Fig.\,12),
and a predominant occurrence of the correlated pixels in the
Southern Hemisphere; (2) the correlations between the long BATSE
events and CMB fluctuations exhibit features in the equatorial
coordinate system, while the joint probability of high quadrupole
amplitude and of the quadrupole minimum occurrence at the
equatorial pole is practically equal to zero in the case of
correlation with random maps.
The correlations found between the GRB positions and the CMB, that
are sensitive to the equatorial coordinate system must be due to
the systematic effects. This way, a relatively greater number of
gamma-ray events near the equatorial poles may be due to
longer exposures of the satellite cameras due to the observing
method employed. Both satellites that observed the gamma-ray
bursts moved in rather low Earth orbits, rendering the areas
located near the celestial equator periodically unobservable. At
the same time, for the CMB data such a sensitivity to the
equatorial coordinate system is impossible to explain in terms of
such a simple model, as the microwave background data were
obtained onboard the WMAP satellite, which rotates about the L2
Lagrangian point. Note that in our previous paper~\cite{cor_ecl}
we also found some correlations to be ``aware'' of the equatorial
system.
We do not rule out the possible contribution from the
Earth's magnetic field, which has a large extent and shows up as
large-scale correlations of the microwave background. However, we
do not understand the mechanism of such correlations.
Neither do we understand the possible interrelation between the
ecliptic plane and the positions of gamma-ray bursts. Here the
situation is reversed: the ecliptic features in the CMB data have
already been discussed recently~\cite{diego,dikarev,cor_ecl},
however, further studies are needed to understand what may link
gamma-ray evens and the ecliptic plane. One of the possible
hypotheses explains this feature as a selection effect due to the
fact that the observing instruments of the satellites are turned
away from the Sun, and this may put the plane of the ecliptic in a
special position.
Note also that the correlation properties of the CMB, which show
up in the ``near-Earth'' coordinate systems when the data with
``randomly'' distributed events are used, are indicative of the
non-Gaussian nature of the data for low-order multipoles, which
may be due either to the systematic effects, or to a hitherto
unexplored effect in the near-Earth space. We consider further
studies of this correlation to be of especially great interest
given that the new high-quality data are expected to be provided
by the Fermi and Planck missions.
\noindent
{\small
{\bf Acknowledgments}.
We are grateful to Valery Larionov (St. Petersburg State
University) for useful discussions of the results of this work. We
thank the NASA for making available the NASA Legacy Archive, from
where we adopted the WMAP data. We are also grateful to the
authors of the HEALPix\footnote{\tt
http://www.eso.org/science/healpix/} \cite{healpix} package, which
we used to transform the WMAP7 maps into the coefficients $a_{\ell
m}$. This work made use of the \mbox{GLESP\footnote{\tt
http://www.glesp.nbi.dk} \cite{glesp,glesp1}} package for the
further analysis of the CMB data on the sphere. This work was
supported by the Program for the Support of Leading Scientific
Schools of Russia (the School of S.~E.~Khaikin) and the Russian
Foundation for Basic Research (grant nos.~09-02-00298 and
\mbox{08-02-00486.} O.V.V. also acknowledges partial support from
the Foundation for the Support of Domestic Science (the program
``Young Doctors of Science of the
Russian Academy of Sciences'') and the Dynasty Foundation.
|
{
"timestamp": "2010-09-21T02:02:22",
"yymm": "1009",
"arxiv_id": "1009.3720",
"language": "en",
"url": "https://arxiv.org/abs/1009.3720"
}
|
\section{Introduction}
The extreme ISM conditions in the central $\sim$ 200 pc of the Galaxy render the region more akin to a star-bursting system \citep[e.g.,][]{Launhardt2002} than to
almost any region in the Galactic disk.
The similarities
include:
i) a high areal star-formation and (consequent) supernova rates;
ii) a flatish overall radio spectrum within the star-forming region \citep[cf.][]{Niklas1997,Thompson2006};
iii) a region surrounding the star-forming nucleus of bright but diffuse, non-thermal radio emission;
iv) the existence of diffuse $\gamma$-ray emission also apparently associated with star formation \citep[cf.][on NGC 253 and M82]{Abdo2009,Acero2009,Acciari2009}; and
v) a rather strong magnetic field \citep[$>50 \ \mu$G;][]{Crocker2010}.
Here we argue for another similarity: a strong outflow with a speed 400-1200 km/s (comparable to the escape speed) and energetically consistent with being driven by current star-formation \citep{Veilleux2005,Strickland2009}.
Massive, young stars are copious producers of UV and optical light which is reprocessed into IR emission by the dust of the stars' natal molecular envelopes \citep{Devereux1990}.
On the other hand,
cosmic ray (CR) electrons and ions --
ultimately powered by supernovae \citep[e.g.,][]{Hillas2005} -- produce their own (non-thermal) radiative signatures.
These include $\sim$GHz radio continuum (RC) synchrotron emission and
inverse Compton (IC) and bremsstrahlung emission at $\gamma$-ray wavelengths by CR electrons and
$\gamma$-rays from neutral meson decay following hadronic collisions between
CR
ions and gas.
\vspace{-0.1cm}
Given the connection of these radiative processes back to massive ($M_\star > 8 {\,M_\odot}$) star formation \citep{Voelk1989},
one might expect that they be globally correlated.
Such is observed \citep{Dickey1984,deJong1985,Helou1985}: an extremely tight (dispersion of $\sim 0.26$ dex; Yun et al. 2001) FIR-RC correlation (FRC) is found \citep[e.g.,][]{Condon1992} to hold over five orders of magnitude in RC luminosity \citep{Yun2001}, and both globally and at sub-galactic scales \citep{Hughes2006,Tabatabaei2007}.
Likewise, one might also expect \citep{Thompson2006,Thompson2007}
a global scaling between FIR and $\gamma$-ray production (`F$\gamma$S').
As we show below, however -- and in interesting contrast to star-bursting systems \citep{Thompson2006} --
the GC does not fall on these scaling relations: we detect far less non-thermal emission than expected given the region's star-formation rate.
This deficit is ultimately explained by a large-scale, powerful outflow from the region.
\section{Correlations and Scalings}
\label{sctn_Correlations}
The H.E.S.S. Imaging Air-Cherenkov $\gamma$-ray Telescope has detected hard-spectrum, diffuse $\sim$TeV $\gamma$-ray emission surrounding the GC over the region defined by $| l | < 0.8^\circ$ and $| b | < 0.3^\circ$ with an intensity of $1.4 \times 10^{-20}$ cm$^{-2}$ eV$^{-1}$ s$^{-1}$ sr$^{-1}$ at 1 TeV (with the point TeV source coincident with Sgr A$^*$ subtracted).
Only dimmer diffuse TeV emission is detected outside this (hereinafter) `HESS field'.
Of note, is that the spectral index, $\gamma$, of the GC diffuse $\sim$TeV emission, where $F_\gamma \propto E_\gamma^{-\gamma}$, is $2.3 \pm 0.07_\textrm{\tiny{stat}} \pm 0.20_\textrm{\tiny{sys}}$, significantly harder than the spectral index of the CR ion population threading the Galactic disk
and the diffuse $\gamma$-ray emission it generates.
Disk CRs experience
energy-dependent confinement and their steady-state distribution is, therefore, steepened from the
injection distribution into the softer $\sim E^{-2.75}$ spectrum observed at earth \citep[see, e.g.,][]{Aharonian2006}.
The GC TeV $\gamma$-ray spectral index (and that inferred for the parent CR ions) is close to that inferred for the {\it injection} spectrum of Galactic disk CRs, itself within the reasonable range of
$\sim$ 2.1-2.2 expected \citep{Hillas2005} for 1st-order Fermi acceleration at astrophysical shocks.
Empirically the 1.4 GHz RC (spectral) luminosity and the total IR luminosity ($L_{TIR}[8-1000] \ \mu m$; \citealt{Calzetti2000}) are connected as \citep{Yun2001,Thompson2007}
\begin{equation}
\nu L_\nu(1.4 \ \textrm{GHz}) \simeq 1.1 \times 10^{-6} \ L_{TIR}
\end{equation}
with a scatter of $\sim 0.26$ dex.
On the basis of IRAS data \citep{Launhardt2002}
the $L_{TIR}$ of the HESS field is $1.6 \times 10^{42}$ erg/s, implying \citep{Kennicutt1998} a SFR of 0.08 ${\,M_\odot}$/yr for the HESS field.
In useful units, the 1.4 GHz RC luminosity \citep[]{Reich1990} of the HESS field is $1.7 \times 10^{35}$ erg/s[\footnote{We have removed the contribution from synchrotron emission from relativistic electrons in the Galactic plane but out of the GC: see Crocker, Jones, Aharonian et al. 2010, {\it to be submitted}, henceforth Paper II.}], {\it $\sim$1.0 dex or $\sim4 \sigma$ short of the expectation from the
FRC}.
\citet{Thompson2007}
use the empirically-established connection between the SFR and the total infrared luminosity
to relate the power, injected by supernovae into CRs, to $L_{TIR}$ and hence to
{\it predict} that the TIR and $\gamma$-ray emission from luminous star-forming galaxies should scale as
\begin{equation}
\nu L_\nu(\textrm{GeV}) \simeq 2.0 \times 10^{-5} \ \eta_{0.10} \ L_{TIR}
\label{eqn_GeVFIR}
\end{equation}
where the proton spectrum is assumed $\propto E_p^{-2}$ up to $E_p^\textrm{\tiny{max}} \simeq 10^{15}$ eV and we have renormalized the equation of \citet{Thompson2007} assuming $\eta_{0.10}$ 10\% of the $10^{51}$ ergs per supernova goes into relativistic ions.
This relation assumes that the region under consideration is calorimetric to CR ions.
On the basis of the results presented by \citet{Meurer2009}, Fermi observes a luminosity of $\sim 3 \times 10^{36}$ erg/s for $E_\gamma >$ GeV for emission from the central $1^\circ \times 1^\circ$ field,
only $\sim$10\% of that expectated from the FIR emission.
The Fermi observations are, however, substantially polluted by line-of-sight and point source emission \citep[including from a source coincident with Sgr A*:][]{Chernyakova2010} so they only constitute an upper limit to the true diffuse $\gamma$-ray emission from the region.
We can consider the HESS data by scaling eq.~\ref{eqn_GeVFIR} from $L_\gamma(E_\gamma >$ GeV) to $L_\gamma(E_\gamma >$ TeV).
For the TeV spectral index of $\sim 2.3$ and assuming an hadronic origin to the TeV $\gamma$-rays, $L_\gamma(E_\gamma >$ TeV) $\simeq 0.2 \ L_\gamma(E_\gamma >$ GeV).
The TeV luminosity we infer for the HESS field of 1.2$\times 10^{35}$ erg/s (integrating to 100 TeV)
{\it is only $\sim$2\% of the prediction from the suitably-scaled version of eq.~\ref{eqn_GeVFIR}.}
Thus the FRC fails badly in the case of the HESS field: far less RC than expected is detected given its FIR output.
Likewise, the $\gamma$-ray luminosity of the region is significantly in deficit given the region's FIR output (and implied star-formation rate).
There are three potential explanations of these discrepancies:
Firstly, a RC deficit could arise if a starburst event occurred more recently ($\lower.5ex\hbox{$\; \buildrel < \over \sim \;$} 10^7$ years) than the lifetime of the massive stars whose supernova remnants accelerate the CR electrons which generate synchrotron emission.
Although we expect some stochastic variation in the GC's overall
SFR, we find, however, that the current SFR is close to the long-term ($\lower.5ex\hbox{$\; \buildrel > \over \sim \;$} 10^7$) average value \citep[cf.][]{Serabyn1996,Figer2004}.
A strong piece of evidence for this is that a number of other handles on the GC supernova rate we describe in Paper II that are sensitive to long-term average values of this quantity -- through, e.g., studies of the region's pulsar population \citep{Lazio2008} -- are consistent with the supernova rate implied by the current SFR as traced by FIR, viz. 0.04/century in the HESS field.
Secondly, it may be that GC SNRs are intrinsically low-efficiency \citep[cf.][]{Erlykin2007} CR accelerators \citep[plausible because of their -- {\it on average} -- dense environs:][]{Fatuzzo2005}.
However, the detailed numerical modelling set out in Paper II shows
that GC supernovae do, indeed, accelerate CRs with typical \citep[e.g.,][]{Hillas2005} efficiency: about 10\% of the total $10^{51}$ erg mechanical energy per supernova goes into non-thermal particles.
Given the above rate, this implies that supernovae
inject $ \sim 10^{39}$ erg/s into the GC CR population \citep[cf.][]{Crocker2010b}.
Lastly, given the half-height of the region is only $\sim 40$ pc, a reasonable reaction to the breakdown of the FRC is that it is unsurprising; many studies \citep{Murgia2005} find a break-down in the correlation at $\sim$ kpc, often proposed to be due to electron transport.
On the other hand, studies, e.g., of the Large Magellanic Cloud \citep{Hughes2006}, the Scd galaxy M33 \citep{Tabatabaei2007}, and within the Milky Way \citep{Zhang2010} reveal a tight connection between RC and FIR emission down to scales $\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}$ 50 pc.
A potential
fourth explanation of why the HESS field falls off the FRC is that power fed into non-thermal electrons is `lost' to ionization and bremsstrahlung and/or inverse Compton emission \citep{Thompson2006,Thompson2007} rather than synchrotron emission (plausible because of the GC's dense gas and radiation environment).
Given, however, the HESS field also falls short of the F$\gamma$S, this explanation is, at least, seriously incomplete.
In summary here, it seems that {\it CR transport out of the HESS field is by far the most plausible explanation for why it falls off the global scalings discussed}; below we show that the transport mechanism is a wind.
\section{Prior Evidence for an Outflow from the GC}
\label{sectn_outflow}
There is multi-wavelength
evidence in support of the existence of GC outflow.
Recent infra-red observations show that the GC's massive stellar clusters are blowing a bubble into their environment \citep[e.g.,][]{Bally2010}.
\citet{Keeney2006} and \citet{Zech2008} have found evidence for high-velocity gas consistent with a GC outflow or fountain in UV absorption features towards, respectively, two AGN and a GC globular cluster.
The region's spectacular non-thermal radio filaments \citep{Yusef-Zadeh1987}
may be due to a fast outflow \citep[e.g.,][]{Shore1999}.
RC evidence of an outflow
was found in 10 GHz radio continuum emission by \citet{Sofue1984} in the form of a $\sim1^\circ$ (or $\sim 140$ pc) tall and diameter $<130$ pc shell of emission rising north of the Galactic plane called the Galactic Centre lobe (GCL).
RC emission from the lobe's eastern part has HI absorption that clearly puts it in the GC region \citep{Lasenby1989} and its ionized gas has a high metalicity \citep{Law2009}.
Filamentary structures coincident with the radio have been discovered at mid-infrared wavelengths \citep{Bland-Hawthorn2003} and
the structure interpreted as evidence for a previous episode of either starburst \citep{Bland-Hawthorn2003} or nuclear activity \citep{Melia2001}.
\citet{Law2010} has found that the formation of the GCL is consistent with currently-observed pressures and rates of
star-formation in the central few $\times 10$ pc of the Galaxy.
Finally, \citet{Law2010} determined the $\sim$GHz spectral index of the GCL steepens with increasing distance (both north and south) of the Galactic plane.
This constitutes strong evidence for synchrotron ageing of a CR electron population transported out of the plane.
Thus, a natural interpretation is that the GCL's RC emission is due to
CR electrons advected from the inner GC (essentially the HESS region) on a wind \citep[cf.][on, e.g., NGC 253 and M82]{Zirakashvili2006,Heesen2009}.
This interpretation requires that
\begin{itemize}
\item
The spectrum of the electrons leaving the HESS region (as given by Eq.\ref {solutionSmpl}) must match the spectrum {\it at injection} required for the GCL electrons with spectral index 2.0 -- 2.4 \citep{Crocker2010}.
This will be well-satisfied if an energy-independent transport process like a wind removes CR electrons -- accelerated into an in-situ $\sim E^{-2}$ distribution -- from the inner GC.
\item
The power in electrons leaving the HESS region must be enough to support the GCL electron population, viz. $(3-10) \times 10^{37}$ erg/s \citep{Crocker2010}.
This is well satisfied given the SN rate in the HESS region.
\item The time to transport electrons over the extent of the GCL must be less than the loss time over the same scale.
This implies a wind speed of strictly $>$ 150 km/s and probably $\lower.5ex\hbox{$\; \buildrel > \over \sim \;$}$ 300 km/s: see fig.~\ref{plotRqrdWindSpeed}.
\end{itemize}
\begin{figure}
\epsfig{file=plotRqrdWindSpeed.pdf,width=\columnwidth}
\caption{Lower bound on the wind speed required for electrons advected out of HESS field to synchrotron-illuminate the entire extent of the GC lobe within their loss times given by ionization, bremsstrahlung, synchrotron, and IC emission for environmental parameters of $B$ and $n_H$ and an interstellar radiation field energy density $U_{ISRF} \simeq 20$ eV cm$^{-3}$.
We infer from \citet{Ferriere2007} that the volumetric average $n_H$ in the GCL is $\sim 10$ cm$^{-3}$.
The strict lower limit to the GCL magnetic field at 50 $\mu$G \citep[and probable value 100 $\mu$G:][]{Crocker2010} imply a conservative lower limit to the GC outflow speed of $>$ 150 km/s (and probably $\lower.5ex\hbox{$\; \buildrel > \over \sim \;$} 300$ km/s).
}
\label{plotRqrdWindSpeed}
\end{figure}
\section{Non-Thermal Hints of an Outflow from the GC}
\label{sctn_Hints}
An important consideration is why the GC CR ion population is so hard in comparison to the diffusion-steepened, local population.
There are three reasonable interpretations of this:
i) the system is out of steady state with less time having passed since the CR injection event than required for diffusion steepening \citep[cf.~the interpretation adopted by][that a single CR-injection event $\sim10^4$ year ago at the GC explains the observed diminution in the $\gamma$-ray to molecular column ratio beyond $|l| \sim 1^\circ$]{Aharonian2006}; {\it or} there is a smallest relevant timescale defined by an energy-independent ii) {\it loss} or iii) {\it escape} process.
We argue here that iii) is preferred by all the evidence.
We can dismiss ii) on the basis of our results above which show the system falls far short of being a calorimeter for protons.
A number of factors also tell against i):
firstly, as argued, other evidence indicates that the system is close to its steady state;
secondly,
the spectral index of the $\sim$TeV emission is a constant $\sim 2.3$ over the HESS region (within errors) presenting, therefore,
no evidence of diffusion hardening at the leading edge of a (putative) diffusion sphere; and
lastly, these spectral considerations apply also to the relativistic electron population: the hard radio spectrum of the region,
$\alpha \lower.5ex\hbox{$\; \buildrel < \over \sim \;$} 0.54$ (for $S_\nu \propto \nu^{-\alpha}$ and radio data 1.4--10 GHz: see Paper II),
requires that the synchrotron-emitting electron population is also very hard, $\sim E_e^{-2.1}$.
Given the rather short loss times associated with synchrotron and IC emission in the GC environment, this hard electron spectrum constitutes independent evidence for {\it rather quick and energy-independent CR transport} \citep[cf.][]{Lisenfeld2000}.
Consider then the region's non-thermal particle population which,
in steady state, approximates to:
\begin{equation}
\label{solutionSmpl}
n_x\!(E_x) \simeq \frac{\tau_\textrm{\tiny{loss}}\!(E_x) \tau_\textrm{\tiny{esc}}}{\tau_\textrm{\tiny{loss}}\!(E_x)+(\gamma-1)\tau_\textrm{\tiny{esc}}} {\dot Q}_x\!(E_x)
\end{equation}
where ${\dot Q}_e(E_e)$ denotes
the injection rate of particles of type $x \in \{e,p\}$; we account for both escape and energy loss
over $\tau_\textrm{\tiny{esc}}$ and $\tau_\textrm{\tiny{loss}}$ with the escape time assumed to be energy-independent; and
$\gamma$ is the spectral index of the (assumed) power-law (in momentum) proton or electron spectrum at injection.
Turning now to the CR ion population (henceforth protons for simplicity), we have already seen that we only detect $\sim$2\% of the TeV $\gamma$-ray flux expected in the calorimetric limit.
Given, then, that $pp$ collisions are by far the dominant energy loss process for high-energy CR protons, this deficit implies that there is significant escape of accelerated ions (with accompanying adiabatic losses) -- i.e., the system is quite far from calorimetric.
We define $R_\textrm{\tiny{TeV}} \equiv L_\textrm{\tiny{TeV}}^\textrm{\tiny{obs}}/L_\textrm{\tiny{TeV}}^\textrm{\tiny{thick}} \simeq 10^{-2}$ (uncertain by a factor $\sim$2) as the ratio of the observed flux of TeV $\gamma$-ray emission to the expected in the calorimetric limit
\citep[cf. fractions $\sim 0.01$ and $\sim 0.05$ for the Galactic disk and NGC 253;][]{Acero2009}.
From Eq.\ref{solutionSmpl} and accounting for adiabatic losses with timescale $\tau^p_\textrm{\tiny{adbtc}} = 3 \tau^p_\textrm{\tiny{esc}}$:
\begin{equation}
R_\textrm{\tiny{TeV}} \simeq 10^{-2}
\sim \frac{3 \tau_\textrm{\tiny{esc}}^{p}}{3 \tau_\textrm{\tiny{esc}}^{p} + 4 \tau_{pp}}\ .
\label{eqn_hadronic}
\end{equation}
By analogy with the hadronic case, we define $R_\textrm{\tiny{radio}} \equiv L_\textrm{\tiny{synch}}^\textrm{\tiny{obs}}/L_\textrm{\tiny{synch}}^\textrm{\tiny{thick}} \simeq 10^{-1}$ (again uncertain by a factor $\sim$2).
Given the very flat radio spectral index, this deficit is potentially explained as a result of electron energy loss into bremsstrahlung, adiabatic deceleration or advective escape.
Eq.\ref{solutionSmpl} then gives
\begin{equation}
R_\textrm{\tiny{radio}} \simeq 0.1 \sim
\frac{\tau_\textrm{\tiny{esc}}^{e} ( \tau_\textrm{\tiny{brems}} + 3 \tau^e_\textrm{\tiny{esc}})}{ \tau_\textrm{\tiny{synch}}(\tau_\textrm{\tiny{brems}} + 4\tau_\textrm{\tiny{esc}}^{e} )}
\ .
\label{eqn_leptonic}
\end{equation}
Now, given the foregoing, particle escape is both energy-independent and the same for CR electrons and protons ($\tau_\textrm{\tiny{esc}}^{e} \equiv \tau_\textrm{\tiny{esc}}^{p} =$ const) as would be expected for a wind.
This means that eq.~\ref{eqn_hadronic} and \ref{eqn_leptonic}
yield a combined constraint on the required velocity of the outflow responsible for particle removal: see fig.~\ref{plotvWind}.
Also shown in fig.~\ref{plotvWind} are minimum and maximum values for the speed of the star-formation-driven
`super-wind' expected on the basis of observations of the nuclei of external, star-forming galaxies
and the GC's high areal SFR \citep{Strickland2009}.
The asymptotic speed of such a wind
scales as
$v_{\tiny{wind}} \sim \sqrt{2 \ \eta \ \dot{E}/\dot{M}}$ where $0 < \eta < 1$ is the thermalization efficiency, typically ranging between 0.1 for relatively quiescent star formation and almost 1 for star-bursts \citep{Strickland2009}.
%
Adopting $\eta_{\tiny{min}} \equiv 0.1$, $\eta_{\tiny{max}} \equiv 1.0$, $\dot{E} = 1.4 \times 10^{40}$ erg/s and 0.025 ${\,M_\odot}$/year (see Paper II), we find $v_{\tiny{wind}}^{min} \simeq 400$ km/s and $v_{\tiny{wind}}^{max} \simeq 1200$ km/s.
Putting some of these considerations in a different form, we expect a TeV luminosity from the HESS region which satisfies
$
L_\gamma (E_\gamma > \textrm{TeV}) \sim 1/3 \ U_\textrm{\tiny{CR}}(E_p > 10 \ \textrm{TeV})/\tau_{pp} \ V \leq L_\gamma^\textrm{\tiny{obs}} (E_\gamma > \textrm{TeV}) \equiv 1.2 \times 10^{35} \ \textrm{erg/s}$
where
$U_\textrm{\tiny{CR}}(E_p > 10 \ \textrm{TeV}) \sim 1/20 \times 1.4 \times 10^{39}$
erg/s
$\times \ d/v_{\tiny{wind}}/V$
is the energy density in CR protons sufficiently energetic to generate TeV $\gamma$-rays,
$d \simeq 40$ pc and $V \simeq 10^{62}$ cm$^{-3}$ for the HESS region, and $n_H$ is the {\it effective} gas density the protons sample.
This implies $n_H \lower.5ex\hbox{$\; \buildrel < \over \sim \;$} 6$ cm$^{-3} \ (v_{\tiny{wind}}/1200$ km/s), cf. the volumetric average gas density through the HESS region $\sim$120 cm$^{-3}$ summing over all phases and $\sim$6 cm$^{-3}$ including only plasma phases.
Likewise, the total gas mass the protons sample satisfies $M_{\tiny{gas}} \lower.5ex\hbox{$\; \buildrel < \over \sim \;$} 5 \times 10^5 {\,M_\odot} \ (v_{\tiny{wind}}/1200$ km/s) which is much less than the $\sim 10^7 {\,M_\odot}$ of gas in the region.
In order that the region's protons {\it not} sample all the molecular gas in the region they should be removed in a time shorter than the convection time into the
dense regions of the molecular clouds: $t_{\tiny{wind}} \equiv d/v_{\tiny{wind}} < t_{\tiny{cloud}} \sim 10$ pc/30 km/s \citep[adopting 30 km/s as a typical internal velocity dispersion for the region's giant molecular clouds, conservatively, of radius $\sim$10 pc: e.g.][]{Morris1996} which also implies a lower limit: $v_{\tiny{wind}} \lower.5ex\hbox{$\; \buildrel > \over \sim \;$} 130$ km/s.
Typical timescales are plotted in fig.~\ref{plotTimescales}.
\begin{figure}
\epsfig{file=plotvWind.pdf,width=\columnwidth}
\caption{Outflow speed inferred given the departures from calorimetry for both protons (`$p$') and electrons (`$e$'): $R_\textrm{\tiny{TeV}} = 0.01 $ and $R_\textrm{\tiny{radio}} \simeq 0.1$ as described in the text (the width of
the bands reflects the uncertainty of $\sim$2 in both $R_\textrm{\tiny{TeV}}$ and $R_\textrm{\tiny{radio}}$).
Protons cool via their hadronic collisions with ambient gas (hence the linear dependence between wind speed and gas density, $n_H$)
and adiabatic deceleration.
In addition to bremssrahlung (and ionization), electrons also cool via synchrotron (so the magnetic field enters as a parameter) and IC emission and adiabatic deceleration.
As the wind escape time is the same for both electrons and protons, {\it the intersection of the electron and proton bands describes a valid gas density and wind velocity for the HESS environment for each magnetic field sampled}.
The horizontal dashed line shows the approximate maximum allowed wind speed ($\sim1200$ km/s) balancing the total power assumed injected into the system by supernovae and massive stars ($1.4 \times 10^{40}$ erg/s) with the kinetic power advected by the wind plasma at its asymptotic velocity (assuming 100\% thermalization efficiency).
The horizontal dot-dashed line shows the approximate minimum plausible wind speed ($\sim400$ km/s) for thermalization efficiency of 10\%.
}
\label{plotvWind}
\end{figure}
\begin{figure}
\epsfig{file=plotTimescales.pdf,width=\columnwidth}
\caption{HESS region timescales for central parameter values suggested by
our analysis, viz. $n_H = 10$ cm$^{-3}$, and $v_\textrm{\tiny{wind}} = 700$ km/s
with
i) (horizontal solid band) the inverse of the supernova rate;
ii) (dashed horizontal line) particle escape with energy-independent velocity of 700 km/s;
iii) (solid red lines) electron cooling for (thick) $B = 2 \times 10^{-4}$ G and (thin) the limiting case of vanishing magnetic field (IC cooling dominant at high energy);
iv) (blue dotted line) proton cooling.
Calorimetry generically requires $t_{loss} < t_{esc}$.
}
\label{plotTimescales}
\end{figure}
\section{Discussion and Conclusions}
\label{sctn_Discussion}
A clear picture emerges from the considerations above.
Given the morphological and spectral data on the GC lobe, we can infer that it is illuminated with CR electrons injected in the HESS region carried from the plane on an outflow with a speed 150--1000 km/s.
The spectral data on the HESS region itself imply that most CR electrons and protons accelerated in situ
are advected from the region; electrons lose only $O[10 \ \%]$ of their power to synchrotron emission in the HESS region while protons lose only $O[1 \ \%]$ of their power to $pp$ collisions on ambient gas in the same region.
Self-consistently and given our understanding of outflows from external, star-forming galaxies, the same star-formation and subsequent supernova processes that drive the thermal and non-thermal radiation from the HESS region will also drive an outflow with a speed 400--1200 km/s.
This implies that the magnetic field in the HESS field lies in the range 100-300 $\mu$G and the effective gas density encountered by the CRs is in the range 3--20 cm$^{-3}$.
The latter is much less than the
volumetric average $n_H$ over the HESS region suggesting that even super-TeV CRs do not `sample' all $H_2$ before escaping the region.
We suspect that the outflow we identify plays many important roles \citep[see Paper II and][]{Crocker2010b} including advecting positrons into the Galactic bulge \citep[thereby explaining the $\sim$kpc extension of the 511 keV annihilation radiation:][]{Weidenspointner2008}, carrying CR ions accelerated by GC supernovae out to very large heights ($\sim$ 10 kpc) thereby explaining the WMAP `haze' and Fermi 'bubbles' \citep{Finkbeiner2004,Dobler2009,Su2010,Crocker2010b}, and generally keeping the energy density of the non-thermal components of the GC ISM in check \citep{Breitschwerdt2002}.
\vspace{-0.73cm}
\section{Acknowledgements}
The authors gratefully acknowledge correspondence with Rainer Beck, Joss Bland-Hawthorn, Valenti Bosch-Ramon, Sabrina Casanova, Roger Clay, John Dickey, Ron Ekers, Katia Ferri{\` e}re, Stanislav Kel'ner, Mitya Khangulyan, Jasmina Lazendic-Galloway, Mark Morris, Giovanni Natale, Emma de O\~na Wilhelmi, Ray Protheroe, Brian Reville, Frank Rieger, Ary Rodr{\'{\i}}guez-Gonz{\'a}lez, Gavin Rowell, and Andrew Taylor. RMC particularly thanks Heinz V{\"o}lk for many enlightening discussions. The authors are indebted to the referee, Professor John Bally, for an expeditious and insightful review.
\vspace{-0.7cm}
|
{
"timestamp": "2010-11-16T02:03:07",
"yymm": "1009",
"arxiv_id": "1009.4340",
"language": "en",
"url": "https://arxiv.org/abs/1009.4340"
}
|
\section{Introduction}
The Reissner-Nordst\"om solution is the unique, asymptotically flat, static and spherically symmetric solution of the source-free Einstein-Maxwell theory in $3+1$ dimensions, and gives the second simplest class of black holes.
The existence of the electric field makes the geometry qualitatively different from that of the Schwarzschild black hole. In fact, the solution has two distinct horizons, one is outer event horizon and the other is inner Cauchy horizon.
Strings may be regarded as extremal forms of electromagnetic fields, that is, infinitely thin flux tube. Unlike the electromagnetic fields, however, strings have no transverse stresses though they have the same equation of states as that of the electromagnetic fields in the direction they extend.
Ensembles of such strings form ``string matter" that is intrinsically anisotropic.
We are interested in constructing black holes by making use of the string matter and investigate the internal structure of the solutions. Since the strings have tension and tend to shrink, the string matter should have charges in order to stabilize the configurations.
Thus the solutions inevitably provide the Reissner-Nordst\"om solution outside the matter, but the internal structure, that is our main concern, turns out to be different.
In addition to the strings and charges, we will introduce a spherical membrane that is self-gravitating.
A spherically symmetric membrane has no singularity, because its interior is flat spacetime.
Furthermore, strings can be attached to membranes, and we have more intriguing solutions if we combine them.
Thus we can make various static solutions that have the same metric as the Reissner-Nordst\"om solution outside the outer horizon but have different internal structures.
The existence of these solutions may shed some light, though not conclusive, on the problem of the origin of the black hole entropy.
The organization of this paper is as follows.
In section 2 we investigate systems consisting of strings, membranes and charges in the Einstein-Maxwell theory in $3+1$ dimensions.
We first introduce a certain charged stringy configuration which has a conical singularity at the origin. We find an event horizon if the string tension becomes larger than a critical value.
Next we study a charged spherical membrane that may approach an extremal black hole, though the interior is a flat spacetime.
Then we combine these two solutions to make configurations that have horizons but no singularity.
Section 3 is devoted to brief discussions on the entropy of the models by considering random walk model of strings.
Throughout this paper we use units in which $G=1$ and $c=1$.
\section{Stringy hedgehog configurations}
We investigate static and spherically symmetric configurations in the Einstein-Maxwell theory with sources consisting of strings, membranes and electric charges.
In the Schwarzschild coordinate system, the metric is given by
\begin{eqnarray}
ds^2=-e^{2\phi(r)}dt^2+h(r)dr^2+r^2(d\theta^2+\sin^2\theta d\varphi^2) .
\end{eqnarray}
Since we consider hedgehog like configurations and do not assume local isotropy,
the energy-momentum tensor is written in the form ,
\begin{eqnarray}
{T^\mu}_\nu =
\left(
\begin{array}{cccc}
-\rho(r) & 0 & 0 &0 \\0 & P_r(r) & 0 & 0 \\0 & 0 & P_\perp(r) &0 \\0 & 0 & 0 & P_\perp(r)
\end{array}
\right) ,
\end{eqnarray}
where $\rho, P_r, P_\perp$ are the energy density, pressures in the radial and perpendicular directions, respectively.
The Einstein-Maxwell equations are
\begin{eqnarray}
R_{\mu \nu}-{1\over2}g_{\mu\nu}R=-8\pi T_{\mu \nu} ,
\label{Einstein}
\end{eqnarray}
and
\begin{eqnarray}
{1\over \sqrt{-g}}\partial_\mu\left(\sqrt{-g} g^{\mu \nu} g^{\rho \sigma} F_{\nu\sigma} \right) = J^\rho ,
\end{eqnarray}
where $J^\rho$ is the electric current, and $T_{\mu\nu}$ is the energy-momentum tensor given by
\begin{eqnarray}
T_{\mu\nu}=T^{Maxwell}_{\mu\nu}+T^{matter}_{\mu\nu} .
\end{eqnarray}
Here $T^{Maxwell}$ is the contribution of the Maxwell field,
\begin{eqnarray}
T^{Maxwell}_{\mu\nu}=-F_{\mu\rho}{F_{\nu}}^\rho+{1\over 4} g_{\mu\nu} F_{\rho\sigma}F^{\rho\sigma} ,
\end{eqnarray}
and $T^{matter}_{\mu\nu}$ is that of the matter whose concrete form is determined by the source we choose.
Here we assume that the charges are distributed on a thin spherical shell.
Then from the symmetry and the equation of motion in vacuum, we have
the field strength outside the shell as
\begin{eqnarray}
F_{0r}={q \over 4\pi r^2} ,
\end{eqnarray}
where $q$ is the total charge of the shell, and all the other components vanish.
There is no field strength inside the shell.
Then the energy density and pressure outside the shell are given by
\begin{eqnarray}
\rho^{Maxwell}(r)= {Q^2 \over 8\pi r^4} ,
\quad
P_r^{Maxwell}(r)=- {Q^2 \over 8\pi r^4} ,
\quad
P_\perp^{Maxwell}(r)= {Q^2 \over 8\pi r^4} ,
\label{MaxwellEMT}
\end{eqnarray}
where $Q^2=q^2/(4\pi)$.
The Einstein equation \eqref{Einstein} reads
\begin{eqnarray}
8\pi \rho&=&{1\over r^{2}}{d \over dr} r(1-h^{-1}) ,
\label{rhoeq}
\\
8\pi P_r&=&{2\phi' \over r h}-{1 \over r^2}(1-h^{-1}) ,
\label{Peq}
\\
8\pi P_\perp&=&{\phi'' \over h}+\left(\phi'-{h' \over 2h} +{1\over r}\right){\phi' \over h}
-{h' \over rh^2},
\label{Qeq}
\end{eqnarray}
where the prime denotes the derivative with respect to $r$.
We rearrange these equations so that physical meanings are transparent.
We first integrate the first equation \eqref{rhoeq} to have
\begin{eqnarray}
h(r) = \left[1- {2m(r) \over r}
\right]^{-1} ,
\label{h}
\end{eqnarray}
where
\begin{eqnarray}
m(r) = \int_0^r dr' 4\pi r'^2 \rho(r')
\label{m}
\end{eqnarray}
is the mass inside the sphere of radius $r$.
The ADM mass $m_\infty $ is defined by $m(r)$ evaluated at $r=\infty$,
\begin{eqnarray}
m_{\infty} = \lim_{r\to\infty}m(r) .
\end{eqnarray}
Substituting \eqref{m} into the second equation \eqref{Peq}, we find the gradient of the gravitational potential,
\begin{eqnarray}
{d \phi \over dr} ={h \over r^{2}} \left( m+4 \pi r^3P_r \right) .
\label{dphidr}
\end{eqnarray}
Finally, we differentiate the second equation \eqref{Peq}, subtract the third equation \eqref{Qeq} multiplied by $2/r$ to eliminate the term $\phi''$, and add the second equation \eqref{Peq} multiplied by $2/r$ to get
\begin{eqnarray}
{dP_r \over dr} + {2 \over r} (P_r-P_\perp)
+{h \over r^{2}} \left( m+4 \pi r^3P_r \right)(\rho+P_r) = 0 .
\label{balance}
\end{eqnarray}
This is the equation of equilibrium for anisotropic matter \cite{Lemaitre}.
In fact, it shows that the gravitational force is balanced with the gradient of the pressure plus the difference of the pressures in the radial and perpendicular directions. Here the anisotropy of the source is responsible for the difference.\footnote{When $P_r=P_\perp$, the equation \eqref{balance} becomes the Tolman-Oppenheimer-Volkoff equation \cite{Oppenheimer:1939ne,Tolman:1939jz} for isotropic fluids.}
\subsection{String hedgehog}
\label{String hedgehog}
As is explained in the introduction, we are interested in constructing the solutions by making use of strings that may be seen as infinitely thin electric flux tubes and have the same equation of states as the electromagnetic field in the direction they extend; on the other hand there is no transverse tension, as we will see shortly. Interestingly enough, it is possible to construct a charged black hole solution that has a single horizon.
Let us consider a static configuration that is composed of $N$ open strings with tension $\mu$. We assume that one end of each string is joined together at a point $O$, while the other end can move freely.
We attach an electric charge $e$ at the free end of each string, and the total charge amounts to $q=Ne$.
The charged end points tend to spread out because of electric repulsion, but they are pulled back by the strings.
Therefore in large $N$ limit the strings extend radially and the end points distribute uniformly on a sphere of radius $r_1$ so that the whole configuration becomes spherically symmetric. We call this configuration ``charged string hedgehog".\footnote{For neutral stringy hedgehogs, see Refs. \citen{Guendelman:1991qb,Delice:2003zp}.}
The region outside is a vacuum with electric field and the geometry is described by the Reissner-Nordstr\"om solution.
Note that there is a singularity (which turns out to be conical) at the center $O$.
We will remove it later, but in this subsection we leave it as it is.
We also mention that we do not introduce any point mass or charge at the center.
\begin{figure}[h]
\begin{center}
\resizebox{5cm}{5.5cm}{\includegraphics{hedgehog.eps}}
\end{center}
\caption{The charged string hedgehog. Two dimensional slice.}
\label{fig:hedgehog}
\end{figure}
The energy-momentum tensor of the $N$ strings is given by \eqref{emtNstrings} in Appendix \ref{emtstring}. Adding the contribution from the Maxwell field \eqref{MaxwellEMT}, we have
\begin{eqnarray}
\rho(r) &=& {\sigma \over 4 \pi r^2} \theta(r_1-r) + {Q^2 \over 8 \pi r^4}\theta(r-r_1) ,
\label{rhoHH}
\\P_r(r)&=&-\rho(r) ,
\label{prHH}
\\P_\perp(r)&=&{Q^2 \over 8 \pi r^4}\theta(r-r_1) ,
\label{pperpHH}
\end{eqnarray}
where $\sigma = N \mu$.
It follows from \eqref{m} that
\begin{eqnarray}
m(r)
&=&
\left\{
\begin{array}{ll}
\sigma r &(r < r_1)
\\
m_\infty-{Q^2 \over 2 r} &(r \geq r_1)
\end{array}
\right. ,
\label{mr}
\end{eqnarray}
where
\begin{eqnarray}
m_\infty = \sigma r_1+{Q^2 \over 2 r_1}
\label{minfHH}
\end{eqnarray}
is the ADM mass of the string hedgehog.
We immediately obtain the metric components from \eqref{h} and \eqref{dphidr},
\begin{eqnarray}
h(r)= e^{-2\phi(r)}=
\left\{
\begin{array}{ll}
{1 \over 1-2\sigma} &(r < r_1)
\\
{1 \over \left(1-{r_+ \over r}\right)\left(1-{r_-\over r}\right)} &(r \geq r_1)
\end{array}
\right. ,
\label{hHH}
\end{eqnarray}
where $r_\pm = m_\infty \pm \sqrt{m_\infty^2 - Q^2}$.
Inside the hedgehog $h$ and $\phi$ are constant, and therefore we have a conical singularity at the origin.
The size $r_1$ is determined by solving the equilibrium equation \eqref{balance}.
Inserting \eqref{rhoHH}, \eqref{prHH} and \eqref{pperpHH} into it we have
\begin{eqnarray}
\left[
{\sigma \over 4\pi r^2}-{Q^2 \over 8\pi r^4}
\right]\delta(r-r_1)=0 ,
\end{eqnarray}
and we find
\begin{eqnarray}
r_1={Q \over \sqrt{2\sigma}} .
\label{r1Qsigma}
\end{eqnarray}
Here and hereafter $Q$ is understood as the absolute value of the charge.
There is an alternatively and physically more transparent way to determine $r_1$. Indeed one can obtain \eqref{r1Qsigma} by minimizing the ADM mass $m_\infty$ \eqref{minfHH} with respect to $r_1$. We will see that this situation holds for more general cases and use it in later subsections.
By substituting \eqref{r1Qsigma} into \eqref{minfHH}, the ADM mass is given by
\begin{eqnarray}
m_\inft
=\sqrt{2\sigma} \, Q .
\end{eqnarray}
Next we examine the behavior of the metric given by \eqref{mr} and \eqref{hHH}.
When $\sigma <1/2$, we find $m(r) < r/2$ and thus $h(r)$ is positive everywhere, which means there is no horizon. The metric is non-singular except for the origin.
In this case we have $m_\infty <Q$, and $r_\pm$ are not real.
The configuration is not like a black hole but like a star (see Figure \ref{fig:mVSr_stringOnly1}).
\begin{figure}[h]
\begin{center}
\resizebox{6cm}{5cm}{\includegraphics{mVSr_stringOnly1.eps}}
\resizebox{6cm}{5cm}{\includegraphics{penrose_stringOnly1.eps}}
\end{center}
\caption{The plot of $m(r)$ of the charged string hedgehog (left) and the Penrose diagram (right) in the case $\sigma<1/2$.}
\label{fig:mVSr_stringOnly1}
\end{figure}
As $\sigma$ approaches $1/2$, $m_\infty$ becomes close to $Q$, and $h$ becomes infinitely large for $r<r_1$, while the metric in the region $r\geq r_1$ approaches the extremal Reissner-Nordstr\"om solution.
If $\sigma$ is very close to $1/2$, outside observers can hardly distinguish the configuration from the extremal Reissner-Nordstr\"om black hole in the sense that it takes very long time for a particle to get close to the edge where the metric is different from the black hole.
On the other hand, when $\sigma \geq 1/2$, we have a horizon.\footnote{Rigorously speaking, we should avoid the coordinate singularity by choosing a new coordinate system like the Kruskal coordinates. However, we continue to use the Schwarzschild coordinates for the sake of simplicity.}
The metric components change their signs at $r=r_+$ as is seen from \eqref{hHH}.
The geometry in the region $r\geq r_1$ coincides with the Reissner-Nordstr\"om black hole solution.
However, $r_-$ has no physical meaning though it is real, because $r_-$ is smaller than $r_1$ and the geometry in the region $r < r_1$ is not described by the Reissner-Nordstr\"om, but a conical geometry.
Thus we conclude that the configuration is a black hole which has only the outer horizon at $r=r_+$ and a conical singularity at the origin.
It has the same causal structure as the Schwarzschild geometry.
Note that the trajectories of the end points of the strings lie on the surface $r=r_1$ which now should be regarded as spacelike, because the time and space directions are interchanged inside the horizon.
What happens is that the strings are produced from the electric field at ``time" $r=r_1$.
See the Penrose diagram in Figure \ref{fig:mVSr_stringOnly3}.
\begin{figure}[h]
\begin{center}
\resizebox{6cm}{5cm}{\includegraphics{mVSr_stringOnly3.eps}}
\resizebox{6cm}{4.3cm}{\includegraphics{penrose_stringOnly3.eps}}
\end{center}
\caption{The plot of $m(r)$ of the charged string hedgehog (left) and the Penrose diagram (right) in the case $\sigma>1/2$.
There are two string world sheets, one in the upper triangle and the other in the lower one, in the Penrose diagram, but they are not necessary the same because the two triangle regions are independent. Actually we may replace, for example, the lower triangle by the one that appears in the usual Schwarzschild geometry in that there is the ordinary singularity and no string at the base of the triangle.}
\label{fig:mVSr_stringOnly3}
\end{figure}
So far we have considered string hedgehog configurations, in which the density and the pressure diverge at the origin, and the spacetime has a conical singularity. In the following subsections, we show that such singularity can be removed by introducing a membrane that encloses the origin to make its interior flat.
\subsection{Charged membrane}
Before introducing a membrane to the charged string hedgehog, we consider a system
consisting of a membrane and charges without strings.\footnote{For studies on (un)charged spherical shells in gravitational theory, see for Refs. \citen{Israel:1966,Israel:1967,Bekenstein:1971ej,Chase:1970,Boulware:1973}, and for more recent studies see e.g. Refs. \citen{Guendelman:2008ip,Belinski:2008bn,Bicak:2010zz} and references therein.}
Let us consider a spherical static membrane with tension $\kappa$ which is charged uniformly with net charge $Q$.
We assume that the membrane is balanced at radius $r_0$ which will be determined below.
\begin{figure}[h]
\begin{center}
\resizebox{5cm}{5cm}{\includegraphics{membrane.eps}}
\end{center}
\caption{The charged spherical membrane.}
\label{fig:membrane}
\end{figure}
More precisely, we introduce charged particles that are constrained to move on the membrane, and assume that they couple to the Maxwell field. Then they spread uniformly on the membrane in order to lower the energy. Furthermore, for the sake of simplicity, we set the mass of the particles to zero, so that the energy momentum tensor simply consists of two contributions, one from the Nambu-Goto action of the membrane and the other from the Maxwell field.
The energy-momentum tensor of the membrane is given in Appendix \ref{emtmemb} and that of the Maxwell field in \eqref{MaxwellEMT}.
Therefore we have
\begin{eqnarray}
\rho(r) &=& {\kappa \over \sqrt{h(r_0)}}\delta(r-r_0)+ {Q^2 \over 8 \pi r^4}\theta(r-r_0) ,
\label{dmemb}
\\P_r(r)&=&- {Q^2 \over 8 \pi r^4}\theta(r-r_0) ,
\label{prmemb}
\\P_\perp(r)&=&-{\kappa \over \sqrt{h(r_0)}}\delta(r-r_0) + {Q^2 \over 8 \pi r^4}\theta(r-r_0).
\label{ppmemb}
\end{eqnarray}
In order for the energy-momentum tensor to be real, the condition $h(r_0) > 0$ should be satisfied.
Also, it is clear that $h(r)$ is not continuous at $r=r_0$ because it is determined through \eqref{h} and \eqref{m} with the density \eqref{dmemb} that contains a delta function.
The ADM mass is in principle obtained by evaluating $m(\infty )$ from \eqref{m} and \eqref{dmemb} as
\begin{eqnarray}
m_{\infty } = \int_0^\infty dr 4\pi r^2 \left[{\kappa \over \sqrt{h(r_0)}}\delta(r-r_0)
+ {Q^2 \over 8 \pi r^4}\theta(r-r_0)\right] .
\end{eqnarray}
While the second term is easily evaluated, the first term is rather subtle because $h(r)$ has a discontinuity at $r=r_0$.
In order to circumvent it, we start with the equation,
\begin{eqnarray}
{d \over dr} m(r) = 4\pi r^2 \rho(r) .
\label{mdiffeq}
\end{eqnarray}
Here $\rho(r)$ is given by \eqref{dmemb}, and contains $h(r)$, which is
expressed in terms of $m(r)$ as \eqref{h}.
Multiplying $\sqrt{h(r)}$ and integrating over a small region around $r_0$, from $r=r_0-\epsilon$ to $r=r_0+\epsilon$, we obtain
\begin{eqnarray}
\int_{r_0-\epsilon}^{r_0+\epsilon} dr {1 \over \sqrt{1- {2 m(r) \over r}}} {d m \over dr}
=
\int_{r_0-\epsilon}^{r_0+\epsilon} dr
\left[
4\pi \kappa r^2 \delta(r-r_0)+{Q^2 \over 2 r^2} \sqrt{h(r)}\theta(r-r_0)
\right] .
\end{eqnarray}
In the small $\epsilon$ limit, we may safely replace $2m(r)/r$ with $2m(r)/r_0$ in the left hand side, and the integration of the second term in the right hand side vanishes because it is a finite function.
Performing the integration we thus find
\begin{eqnarray}
\sqrt{1-{2m(r_0+\epsilon) \over r_0}}-\sqrt{1-{2m(r_0-\epsilon) \over r_0}}
=-4\pi \kappa r_0 .
\label{membranemasseq}
\end{eqnarray}
Because there is no matter inside the membrane, we set $m(r_0-\epsilon)=0$,
and obtain the mass of the membrane,
\begin{eqnarray}
m_0=4\pi \kappa r_0^2\left(1- 2\pi \kappa r_0 \right) ,
\label{mpCmemb}
\end{eqnarray}
where $m_0=\lim_{\epsilon \to 0} m(r_0+\epsilon)$.
This equation indicates that the mass of the membrane is reduced by the gravitational binding energy.
Adding the contribution from the electric field, we obtain the ADM mass of the whole system,
\begin{eqnarray}
m_\infty=4\pi \kappa r_0^2\left(1-2\pi \kappa r_0 \right)+{Q^2\over 2 r_0} .
\label{minfCM0}
\end{eqnarray}
Now we turn to the problem of finding $r_0$. For this purpose, it is convenient to rescale the variables as
\begin{eqnarray}
\tilde{m}_\infty := 4\pi \kappa m_\infty , \quad
\tilde{r}_0 := 4\pi \kappa r_0 , \quad
\tilde{Q} := 4\pi \kappa Q ,
\end{eqnarray}
so that \eqref{minfCM0} becomes
\begin{eqnarray}
\tilde{m}_\infty=\tilde{r}_0^2\left(1-{\tilde{r}_0 \over 2} \right)+{\tilde{Q}^2\over 2 \tilde{r}_0} .
\label{minfCM}
\end{eqnarray}
Although the radius $r_0$ can be determined by solving the Einstein equation as is done in the previous section, it is more convenient and physically transparent to employ the minimization of the ADM mass with respect to $\tilde{r}_0$.
We show the equivalence of the two procedures in Appendix \ref{eom}.
As is shown in Figure \ref{fig:minfVSr0}, the function \eqref{minfCM} has two extrema when $\tilde{Q} < 1$, and no extremum when $\tilde{Q} > 1$.
\begin{figure}[h]
\begin{center}
\resizebox{7cm}{6.7cm}{\includegraphics{minfVSr0_case1.eps}}
\resizebox{6.7cm}{6cm}{\includegraphics{minfVSr0_case2.eps}}
\end{center}
\caption{The plots of $\tilde{m}_\infty$ as functions of $\tilde{r}_0$ for $\tilde{Q} <1$ (left) and $\tilde{Q} > 1$ (right).}
\label{fig:minfVSr0}
\end{figure}
The extremal points are determined by $d\tilde{m}_\infty /d\tilde{r}_0=0$, which is expressed as
\begin{eqnarray}
\tilde{Q}^2 = \tilde{r}_0^3\left(4-3\tilde{r}_0\right) .
\label{admmvary}
\end{eqnarray}
As is depicted in Figure \ref{fig:Q2VSr0}, we have two positive roots when $\tilde{Q} < 1$, one in the region $0 < \tilde{r}_0 < 1$ and the other in $1 < \tilde{r}_0 <4/3$.
The former is a local minimum of $\tilde{m}_\infty(\tilde{r}_0)$, and the system is stable, while the latter is a local maximum, and the system is unstable.
\begin{figure}[h]
\begin{center}
\resizebox{8.5cm}{7cm}{\includegraphics{Q2VSr0.eps}}
\end{center}
\caption{The plot of $\tilde{Q}^2=\tilde{r}_0^3(4-3\tilde{r}_0)$.}
\label{fig:Q2VSr0}
\end{figure}
Since the solution of the quartic equation \eqref{admmvary} is complicated, we do not try to express various quantities as explicit functions of $\tilde{Q}^2$. Instead, we will see that the equations become simple if we express them in terms of $\tilde{r}_0$.
Indeed the ADM mass is obtained by substituting \eqref{admmvary} into \eqref{minfCM} as
\begin{eqnarray}
\tilde{m}_\infty = \tilde{r}_0^2(3-2\tilde{r}_0) .
\label{admmass}
\end{eqnarray}
The equations \eqref{admmass} and \eqref{admmvary} can be regarded as a parametric representation of the $\tilde{m}_\infty$-$\tilde{Q}$ curve, which is plotted in Figure \ref{fig:mQmemb}.
The solid and dotted curves correspond to $0 < \tilde{r}_0 < 1$ and
$1 < \tilde{r}_0 <4/3$, and represent stable and unstable configurations, respectively.
For later use we calculate
\begin{eqnarray}
{d \tilde{m}_\infty \over d\tilde{Q}} =\sqrt{4\tilde{r}_0-3\tilde{r}_0^2},
\label{dmdQ}
\end{eqnarray}
and
\begin{eqnarray}
{d^2 \tilde{m}_\infty \over d\tilde{Q}^2} ={2-3\tilde{r}_0 \over 6\tilde{r}_0(1-\tilde{r}_0)},
\end{eqnarray}
which shows that the $\tilde{m}_\infty$-$\tilde{Q}$ curve has an inflection point at $(\tilde{Q}, \tilde{m}_\infty)=(4/(3\sqrt{3}), 20/27)$ where $\tilde{r}_0=2/3$.
Note that it is also an inflection point of the mass of the membrane \eqref{mpCmemb}.
We can also show $\tilde{m}_\infty < \tilde{Q}$ for $0<\tilde{r}_0<1$, and
$\tilde{m}_\infty > \tilde{Q}$ for $1<\tilde{r}_0<4/3$.
In the following discussions, we will consider only stable configurations, $0<\tilde{r}_0\leq 1$.
\begin{figure}[h]
\begin{center}
\resizebox{9cm}{8cm}{\includegraphics{mQ_onlybrane.eps}}
\end{center}
\caption{$\tilde{m}_\infty$-$\tilde{Q}$ plot. The curve has a inflection point at $(\tilde{Q}, \tilde{m}_\infty)=(4/(3\sqrt{3}), 20/27)$ where $\tilde{r}_0=2/3$ and is tangent to the slope $\tilde{m}_\infty=\tilde{Q}$ at $\tilde{Q}=1$ where $\tilde{r}_0=1$.
The slope $\tilde{m}_\infty=\tilde{Q}$ represents the extremal relation.
There is also an unstable branch given by $1 < \tilde{r}_0 \leq 4/3.$}
\label{fig:mQmemb}
\end{figure}
The metric is also determined by a simple calculation.
Because there is nothing in the region $r < r_0$, and only the electric field in
$r > r_0$, we have
\begin{eqnarray}
m(r)=\left(m_\infty-{Q^2 \over 2 r}\right)\theta(r-r_0) .
\label{m(r)memb}
\end{eqnarray}
Then from \eqref{h} we have
\begin{eqnarray}
h(r)
=
\left\{
\begin{array}{ll}
1 &(r < r_0)
\\
{1\over 1-{2m_\infty \over r} + {Q^2 \over r^2}} &(r \geq r_0)
\end{array}
\right. ,
\label{h(r)memb}
\end{eqnarray}
and from \eqref{dphidr}, \eqref{prmemb}, \eqref{m(r)memb} and \eqref{h(r)memb} we obtain
\begin{eqnarray}
e^{2\phi(r)}
=
\left\{
\begin{array}{ll}
1-{2m_\infty \over r_0} + {Q^2 \over r_0^2} &(r < r_0)
\\
1-{2m_\infty \over r} + {Q^2 \over r^2} &(r \geq r_0)
\end{array}
\right. .
\end{eqnarray}
Note that $\phi(r)$ is continuous at $r=r_0$, while $h(r)$ is not \cite{Jacobson:2007tj}.
The function $m(r)$ for various charges (radius) are plotted in Figure \ref{fig:mVSr_braneOnly}.
\begin{figure}[h]
\begin{center}
\resizebox{8cm}{7cm}{\includegraphics{mVSr_braneOnly.eps}}
\end{center}
\caption{The plot of the functions $m(r)$ of the charged membrane for $\tilde{Q}=1~ (\tilde{r}_0=1)$, $\tilde{Q}=16/27~(\tilde{r}_0=2/3)$ and $\tilde{Q}=1/9~(\tilde{r}_0=1/3)$.
These curves are tangent to the curve of the membrane mass $m=4\pi \kappa r^2(1-2\pi \kappa r)$.}
\label{fig:mVSr_braneOnly}
\end{figure}
To summarize, if we consider stable configurations, the charged membrane has no horizons except for the extremal limit $\tilde{Q}=1$. Furthermore there is no singularity even in the extremal limit.
Before closing this subsection, it is worth mentioning that the system considered here can be regarded as a generalization of the Poincar\'e-Schwinger (Abraham-Lorentz) model of electron in which gravity is included (though spin is not included).
However, we find that the effect of gravity does not help, because the radius of the membrane is always larger than the classical electron radius, as is obvious from \eqref{minfCM},
\begin{eqnarray}
r_0={4\pi \kappa r_0^3 \over m_\infty}\left(1-2\pi \kappa r_0 \right) + {Q^2 \over 2 m_\infty}
\geq {Q^2 \over 2 m_\infty} = r_{classical} .
\end{eqnarray}
Note that the membrane mass $4\pi \kappa r_0^2 \left(1-2\pi \kappa r_0 \right)$ is always positive, because $4\pi \kappa r_0<4/3$ even if we allow unstable configurations. The gravitational interaction decreases the mass but does not make it negative.
\subsection{String-membrane hedgehog}
Now we use the membrane solution just studied in the last subsection to remove the conical singularity in the charged hedgehog configuration we found in subsection \ref{String hedgehog}.
In this way, we shall obtain a configuration that has horizons without any singularity at the origin.
We insert a neutral spherical membrane around the origin, which is connected to the strings in a charged hedgehog \cite{Guendelman:1991qb}.
The membrane tends to shrink to the origin while the strings pull it radially so that the membrane is suspended.
In addition, the strings are balanced with the Coulomb force at the other end points. See Figure \ref{fig:hedgehog-membrane}.
\begin{figure}[h]
\begin{center}
\resizebox{5.5cm}{6cm}{\includegraphics{hedgehog-membrane.eps}}
\end{center}
\caption{The charged hedgehog with membrane inside. Two dimensional slice.}
\label{fig:hedgehog-membrane}
\end{figure}
The energy density and the pressure consist of three contributions, one from the membrane, one from the strings, and one from the electric field,
\begin{eqnarray}
\rho(r) &=& {\kappa \over \sqrt{h(r_0)}}\delta(r-r_0)
+{\sigma \over 4 \pi r^2} \theta(r-r_0)\theta(r_1-r)
+{Q^2 \over 8 \pi r^4}\theta(r-r_1) ,
\label{rhoStringMembrane}
\\P_r(r)&=&-{\sigma \over 4 \pi r^2} \theta(r-r_0)\theta(r_1-r)
-{Q^2 \over 8 \pi r^4}\theta(r-r_1) ,
\label{PrStringMembrane}
\\P_\perp(r)&=&-{\kappa \over \sqrt{h(r_0)}}\delta(r-r_0)+{Q^2 \over 8 \pi r^4}\theta(r-r_1) ,
\end{eqnarray}
where $r_0$ and $r_1$ are the positions of the inner and outer end points of the strings, respectively.
The ADM mass also consists of three pieces,
\begin{eqnarray}
m_\infty=4\pi \kappa r_0^2\left(1-2\pi \kappa r_0 \right)+\sigma(r_1-r_0)+{Q^2\over 2 r_1} ,
\label{ADMorigi}
\end{eqnarray}
where the first term is the membrane mass \eqref{mpCmemb}, the second term is the mass of the strings, and the third term is the contribution from the electric field.
As in the case of the charged membrane we introduce rescaled variables,
\begin{eqnarray}
\tilde{m}_\infty = 4\pi \kappa m_\infty , \quad
\tilde{r}_0 = 4\pi \kappa r_0 , \quad
\tilde{r}_1 = 4\pi \kappa r_1 , \quad
\tilde{Q} = 4\pi \kappa Q ,
\label{redefStringMembrane}
\end{eqnarray}
in which \eqref{ADMorigi} becomes
\begin{eqnarray}
\tilde{m}_\infty=\tilde{r}_0^2\left(1-{\tilde{r}_0\over 2}\right)+\sigma(\tilde{r}_1-\tilde{r}_0)+{\tilde{Q}^2\over 2 \tilde{r}_1} .
\label{ADMredef}
\end{eqnarray}
By taking variations with respect to $\tilde{r}_0$ and $\tilde{r}_1$, we find that $\tilde{m}_\infty$ is minimized at
\begin{eqnarray}
\tilde{r}_1={\tilde{Q} \over \sqrt{2 \sigma}} ,
\label{r1}
\end{eqnarray}
and
\begin{eqnarray}
\tilde{r}_0={2 \over 3} \left(1 - \sqrt{1-{3 \over 2}\sigma}\right) .
\label{r0}
\end{eqnarray}
In \eqref{r0} we have selected the solution of
\begin{eqnarray}
\sigma={\tilde{r}_0 \over 2} \left(4-3\tilde{r}_0\right) ,
\label{sigma-r0}
\end{eqnarray}
that gives the local minimum of $\tilde{m}_\infty$, so that the system is stable.
We also have a necessary condition for the existence of a minimum,
\begin{eqnarray}
\sigma \leq {2\over 3} .
\label{sigma_lt_2/3}
\end{eqnarray}
This inequality means that if the string tension exceeds $2/3$, the membrane will be torn by the strings no matter how large the membrane tension is.
From \eqref{r0} we find that the radius of the membrane is bounded as
\begin{eqnarray}
\tilde{r}_0 \leq {2 \over 3} .
\end{eqnarray}
Furthermore, from the inequality $\tilde{r}_1 \geq \tilde{r}_0$, we have
\begin{eqnarray}
\tilde{Q} \geq \sqrt{2 \sigma}\tilde{r}_0.
\label{Qbound}
\end{eqnarray}
Substituting \eqref{r1} and \eqref{sigma-r0} back into \eqref{ADMredef}, we find that the ADM mass can be written as
\begin{eqnarray}
\tilde{m}_\infty
=-\tilde{r}_0^2(1-\tilde{r}_0)+\sqrt{2 \sigma} \tilde{Q} .
\label{minf}
\end{eqnarray}
This equation has a simple and interesting physical interpretation, if we regard it as expressing the allowed region in the $\tilde{m}_\infty$-$\tilde{Q}$ plane.
For a given $\tilde{r}_0$ (or $\sigma$ by \eqref{sigma-r0}), because of the restriction \eqref{Qbound}, the equation \eqref{minf} represents a half line that starts from the point $(\tilde{Q},\tilde{m}_\infty)=(\sqrt{2\sigma}\tilde{r}_0, 3\tilde{r}_0^2-2\tilde{r}_0^3)$.
This expression coincides with the parametric representation of the curve of the charged membrane solution in the previous section, namely \eqref{admmvary} and \eqref{admmass}.
Furthermore, the half line \eqref{minf} is tangent to the curve as is seen from \eqref{dmdQ}.
Therefore, as $\tilde{r}_0$ varies from $0$ to $2/3$, we have a set of half lines whose envelope is the membrane curve as is depicted in Figure \ref{fig:MQ}.
Recall that the point at $\tilde{r}_0=2/3$ is the inflection point of the curve.
These are naturally understood, if one recognizes that the charged membrane can be obtained from the charged hedgehog-membrane by setting the string tension to a critical value so that the length of the strings becomes zero.
However the charged membrane in the parameter region $2/3 <\tilde{r}_0 <1$ can not be obtained by such limit (the dotted line in Figure \ref{fig:MQ}).
As we will see below, the region above the line $\tilde{m}_\infty=\tilde{Q}$ corresponds to the solutions which should be regarded as black holes. We would like to mention that we can recover the string hedgehog solutions, if we take the large membrane tension limit. However, it makes sense only under the condition \eqref{sigma_lt_2/3}.
\begin{figure}[h]
\begin{center}
\resizebox{9cm}{7cm}{\includegraphics{string-brane.eps}}
\end{center}
\caption{$\tilde{m}_\infty$-$\tilde{Q}$ graph of the hedgehog-membrane. The shaded are the regions that can be realized by the hedgehog-membrane. Horizons appear in the region above the line $\tilde{m}_\infty = \tilde{Q}$.}
\label{fig:MQ}
\end{figure}
Now it is straightforward to calculate $h(r)$ and $\phi(r)$.
From \eqref{m} and \eqref{rhoStringMembrane}, we have
\begin{eqnarray}
m(r)
=
\left\{
\begin{array}{ll}
0 &(r < r_0)
\\
m_0 + \sigma (r-r_0) &(r_0 \leq r < r_1)
\\
m_\infty-{Q^2 \over 2 r} &(r \geq r_1)
\label{m(r)StringMembrane}
\end{array}
\right. ,
\end{eqnarray}
where $m_0$ and $m_\infty$ ($\tilde{m}_\infty$) are given by \eqref{mpCmemb} and \eqref{minf}, respectively.
Then from \eqref{h}, \eqref{dphidr} and \eqref{PrStringMembrane} it follows that
\begin{eqnarray}
h(r)
=
\left\{
\begin{array}{ll}
1 &(r < r_0)
\\
{1 \over 1- {2m_0 \over r}-2\sigma \left(1-{r_0\over r}\right)} &(r_0 \leq r < r_1)
\\
{1 \over 1-{2 m_\infty \over r} + {Q^2 \over r^2}} &(r \geq r_1)
\end{array}
\right. ,
\end{eqnarray}
and
\begin{eqnarray}
e^{2 \phi(r)}
=
\left\{
\begin{array}{ll}
1- {2m_0 \over r_0} &(r < r_0)
\\
1- {2m_0 \over r}-2\sigma \left(1-{r_0\over r}\right) &(r_0 \leq r < r_1)
\\
1-{2 m_\infty \over r} + {Q^2 \over r^2} &(r \geq r_1)
\end{array}
\right. .
\end{eqnarray}
If $\sigma<1/2$, it is clear from \eqref{m(r)StringMembrane} that $m(r)<r/2$ everywhere, and there is no horizon. On the other hand, if $\sigma>1/2$, the function $m(r)$ may exceed the line $r/2$ and horizons may appear depending on the values of $\kappa $ and $Q$.
As we will see below, we can make non-extremal black holes as well as extremal ones.
For a fixed $\sigma$, the solutions are characterized by the rescaled charge $\tilde{Q}$,
and in fact we have the following three cases.
Firstly, if $\tilde{Q}$ is so small that
\begin{eqnarray}
\tilde{Q} < \tilde{Q}_1,
\end{eqnarray}
where $\tilde{Q}_1={\tilde{r}_0^2(1-\tilde{r}_0) \over \sqrt{2\sigma}-1}$,
$m(r)$ does not exceed $r/2$, and we find no horizon (see Figure \ref{fig:mVSr_string-brane1}).
In this case the configuration is seen as a charged star. There is no singularity at the origin, unlike the case of $\tilde{Q} > \tilde{m}_\infty$ in the Reissner-Nordstr\"om solution.
\begin{figure}[h]
\begin{center}
\resizebox{7cm}{7cm}{\includegraphics{mVSr_string-brane1.eps}}
\resizebox{6.5cm}{6cm}{\includegraphics{penrose_noHorizon.eps}}
\end{center}
\caption{The plot of $m(r)$ of the charged string-membrane hedgehog with a fixed $\sigma$, and $\tilde{Q}<\tilde{Q}_1$ (left), and the corresponding Penrose diagram (right).}
\label{fig:mVSr_string-brane1}
\end{figure}
Secondly, if $\tilde{Q}$ is in the region,
\begin{eqnarray}
\tilde{Q}_1 < \tilde{Q} < \tilde{Q}_2,
\label{Qineq}
\end{eqnarray}
where $\tilde{Q}_2={2\tilde{r}_0^2 \sqrt{2\sigma}\over 3\tilde{r}_0-1}$,
the Coulomb potential part of $m(r)$ exceeds $r/2$,
and we have two horizons at $r_+$ and $r_-$ (see Figure \ref{fig:mVSr_string-brane3}).
In this case, $r_+$ and $r_-$ are the same as those of the Reissner-Nordstr\"om solution.
When $\tilde{Q}=\tilde{Q}_1$, $\tilde{r}_+$ and $\tilde{r}_-$ merge, and we have an extremal black hole.
When $\tilde{Q}=\tilde{Q}_2$, the outer end points of strings reach the inner horizon, $\tilde{r}_1=\tilde{r}_-$.
\begin{figure}[h]
\begin{center}
\resizebox{6.5cm}{6cm}{\includegraphics{mVSr_string-brane3.eps}}
\resizebox{6cm}{6cm}{\includegraphics{penrose_BH_case1.eps}}
\end{center}
\caption{The plot of $m(r)$ of the charged string-membrane hedgehog with a fixed $\sigma$, and $\tilde{Q}_1 < \tilde{Q} < \tilde{Q}_2$ (left), and the corresponding Penrose diagram (right).}
\label{fig:mVSr_string-brane3}
\end{figure}
Lastly, if $\tilde{Q}$ is so large that
\begin{eqnarray}
\tilde{Q}_2 < \tilde{Q} ,
\end{eqnarray}
we have a new type of solutions (see Figure \ref{fig:mVSr_string-brane5}).
Although the outer horizon is still the same as that of the Reissner-Nordstr\"om solution $r_+$,
the inner horizon is now given by the intersection $r_-'$ of the string part of $m(r)$ and $r/2$.
By a simple calculation we have
\begin{eqnarray}
r'_-={8\pi \kappa r_0^2(1-4\pi \kappa r_0) \over 2\sigma -1} < r_-.
\end{eqnarray}
\begin{figure}[h]
\begin{center}
\resizebox{6.5cm}{6cm}{\includegraphics{mVSr_string-brane5.eps}}
\resizebox{6.5cm}{6cm}{\includegraphics{penrose_BH_case2.eps}}
\end{center}
\caption{The plot of $m(r)$ of the charged string-membrane hedgehog with a fixed $\sigma$, and $\tilde{Q}_2 < \tilde{Q}$ (left), and the corresponding Penrose diagram (right).}
\label{fig:mVSr_string-brane5}
\end{figure}
A comment is in order here.
Although we have found horizons that are not accompanied with singularities, this is not a contradiction because membranes do not satisfy the strong energy condition on which the singularity theorem is based \cite{Hawking:1973uf}. Actually, the condition can be written as \cite{Wald:1984rg}
\begin{eqnarray}
\rho+P_r+2P_\perp \geq 0, \quad {\mbox{and}} \quad \rho+P_i \geq 0 ~ (i=r, \perp),
\label{strongene}
\end{eqnarray}
and as is easily seen from \eqref{dmemb}, \eqref{prmemb} and \eqref{ppmemb}, the first inequality is not satisfied.
\section{Discussion and summary}
Although the configurations we considered in section 2 are static (zero temperature), once we take into account fluctuations around the solutions a lot of degrees of freedom will emerge. Furthermore, a large redshift inside the configurations makes such the degrees of freedom quite huge which might be a source of the entropy of the corresponding black holes.
In this discussion section we consider, instead of the thermal fluctuation around the solutions, the simplest model of thermal motion to capture the essential point of the problem. We note that the thermal motion can make the strings stretched without charges and the system may have a finite size.
We consider a system consisting of $N$ open strings. This time we assume both ends of the strings are fixed to a point $O$, and we do not introduce any charge. Since the ground state of such system has zero size classically, we consider finite temperature states.
A long string with finite temperature is well approximated by a random walk. Because the both end points are fixed to the origin, the averaged mass density $\rho(r) $ of the string is given by the square of Green's function $G(r)$,
\begin{eqnarray}
\rho(r)=G(r)^2 .
\label{rhoFiniteTemp}
\end{eqnarray}
Here, $G(r)$ is determined by the diffusion equation \cite{Horowitz:1997jc},
\begin{eqnarray}
{e^{-\phi(r)} \over r^2 \sqrt{h(r)}} \partial_r \left({r^2 e^{\phi(r)} \over \sqrt{h(r)}} \partial_r G(r)\right)
=\left(e^{2\phi(r)} \beta^2 -\beta_H^2\right) G(r) ,
\label{diffusionEq}
\end{eqnarray}
where $\beta$ is the inverse temperature observed at infinity, and $\beta_H$ is the inverse of the Hagedorn temperature that is of order of the string scale.
If $h(r)$ and $\phi(r)$ are constant as in the case of the charged string hedgehog, \eqref{diffusionEq}
is easily solved as
\begin{eqnarray}
G(r) \propto {e^{-\xi \sqrt{h_0} r} \over r} .
\end{eqnarray}
Here we have assumed $h(r)=h_0$ and $\phi(r)=\phi_0$, and $\xi=e^{2\phi_0} \beta^2 -\beta_H^2$ is a constant.
If the local temperature inside the configuration is at the Hagedorn temperature namely $e^{2\phi_0} \beta^2 -\beta_H^2 =0$, the density \eqref{rhoFiniteTemp} becomes
\begin{eqnarray}
\rho \propto {1 \over r^2} .
\end{eqnarray}
If we have $N$ strings, we should multiply it by $N$, but at any rate $\rho$ is proportional to $1/r^2$.
Then the $h(r)$ obtained from \eqref{m} is a constant, which verifies the self consistency of the initial assumption.
In general, the entropy of a thermal string is proportional to its length.
Therefore, the entropy in our case is given by the proper mass, which is equal to the total intrinsic length of the strings,
\begin{eqnarray}
S \simeq
l_s \int_{0}^{r_0} dr 4\pi r^2 \sqrt{h_0} \rho(r)
= l_s\sqrt{h_0} M ,
\label{stringyEntropy}
\end{eqnarray}
where $l_s$ is the string length scale and $M$ is the ADM mass,
\begin{eqnarray}
M =
\int_{0}^{r_0} dr 4\pi r^2 \rho(r) .
\end{eqnarray}
Here we have assumed that $\rho (r)$ rapidly decreases to zero around $r\sim r_0$. Therefore, if we have a relation like
\begin{eqnarray}
\sqrt{h_0} \simeq {r_s \over l_s} ,
\label{sqrth0}
\end{eqnarray}
where $r_s$ is the Schwarzschild radius, the entropy \eqref{stringyEntropy} becomes $S \simeq r_s M $, which coincides with the entropy of the Schwarzschild black hole up to a numerical factor.
Although it would be difficult to show \eqref{sqrth0} rigorously, here we present a possible scenario to get it.
We assume that the system has no horizon, but the system chooses $r_0$ very close to $r_s$ in order to maximize the entropy,
\begin{eqnarray}
r_0 = r_s +\delta, \qquad \delta \ll r_s .
\label{R}
\end{eqnarray}
The density changes from some finite value to zero as $r$ varies from $r_s$ to $r_0$. It is natural to imagine that such change occurs in the string scale,
which means that the physical distance between $r_s$ and $r_0$ is of order of the string scale $l_s$, that is,
\begin{eqnarray}
\int_{r_s}^{r_0} dr \sqrt{h_0} \simeq \delta \sqrt{h_0} \simeq l_s.
\label{deltasqrth}
\end{eqnarray}
On the other hand, if we evaluate \eqref{h} around $r\simeq r_0$, we have
\begin{eqnarray}
h_0 \simeq \left( 1-{r_s \over r_0} \right)^{-1} \simeq {r_s \over \delta} .
\label{hatr0}
\end{eqnarray}
Then \eqref{sqrth0} follows from \eqref{deltasqrth} and \eqref{hatr0}.
To summarize, in this paper we have studied systems consisting of strings, membranes and charges in the Einstein-Maxwell theory in $3+1$ dimensions.
We constructed the string hedgehog solution, which has, though charged, the
same causal structure as the Schwarzschild black hole (and thus has a single
horizon), whereas the singularity is replaced by a conical one.
Then we studied the gravitational charged membrane solution. We gave a
simple derivation of the self-energy of the membrane. The gravitational binding
energy that reduces the mass leads to the existence of unstable solutions as
well as stable ones.
We found that for each fixed value of the membrane tension there is a
maximal charge (and mass) where the solution approaches the extremal black hole,
while the interior of the membrane is flat Minkowski spacetime.
Finally, we constructed solutions by combining these two configurations to
show there are black hole solutions that have no singularities inside the
horizons.
We studied in detail the behavior of the solutions by varying the magnitude
of the charge (times the membrane tension), which is the only parameter of the
solutions if we fix the string tension.
There are several directions to generalize the solutions we have studied in this paper. For example, it is interesting to include the effect of rotations to see the relation to the Kerr-Newman solution.
It is also interesting to allow the radial motion to study dynamical nature of the models.
To generalize the present arguments to higher dimensions might also be interesting.
We hope to report some of these issues in the near future.
\section*{Acknowledgements}
The authors would like to thank K. Murakami, M. Ninomiya, Y. Sekino and F. Sugino for valuable discussions and comments.
T.M. thanks the particle theory group of Department of Physics at Kyoto University for hospitality during the completion of this work. This work was supported by the Grant-in-Aid for the Global COE Program "The Next Generation of Physics, Spun from Universality and Emergence" from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan.
|
{
"timestamp": "2011-04-26T02:00:33",
"yymm": "1009",
"arxiv_id": "1009.4028",
"language": "en",
"url": "https://arxiv.org/abs/1009.4028"
}
|
\section{Introduction}
Let $\Omega$ be a bounded multiply connected domain in ${\Bbb R}^n,\,
n=2,3$, with Lipschitz boundary $\partial\Omega$ consisting of $N$
disjoint components $\Gamma_j$, i.e.
$\partial\Omega=\Gamma_1\cup\ldots\cup\Gamma_N$ and
$\Gamma_i\cap\Gamma_j=\emptyset,\, i\neq j$. In $\Omega$ consider
the stationary Navier--Stokes system with nonhomogeneous
boundary conditions
\begin{displaymath}
\left\{ \begin{array}{rcl}
-\nu \Delta{\bf u}+\big({\bf u}\cdot \nabla\big){\bf u} +\nabla p & = & {0}\qquad \hbox{\rm in }\;\;\Omega,\\[4pt]
\hbox{\rm div}\,\,{\bf u} & = & 0 \qquad \hbox{\rm in }\;\;\Omega,\\[4pt]
{\bf u} & = & {\bf a} \qquad \hbox{\rm on }\;\;\partial\Omega.
\end{array}\right.
\eqno(1)
\end{displaymath}
Starting from the famous J. Leray's paper \cite{Leray} published in
1933, problem (1) was a subject of investigation in many papers
\cite{Amick}, \cite{BOPI}--\cite{Galdibook},
\cite{Hopf}--\cite{Lad}, \cite{Lions}, \cite{Takashita},
\cite{Temam}, \cite{VorJud}. The continuity equation $(1_2)$ implies
the necessary compatibility condition for the solvability of problem
(1):
$$
\intop\limits_{\partial\Omega}{\bf a}\cdot{\bf
n}\,dS=\sum\limits_{j=1}^N\intop\limits_{\Gamma_j}{\bf a}\cdot{\bf
n}\,dS=0,\eqno(2)
$$
where ${\bf n}$ is a unit vector of the outward (with respect to
$\Omega$) normal to $\partial\Omega$. However, for a long time the
existence of a weak solution ${\bf u}\in W^{1,2}(\Omega)$ to
problem (1) was proved only under the condition
$$
{{\mathscr{F}}}_j=\intop\limits_{\Gamma_j}{\bf a}\cdot{\bf n}\,dS=0,\qquad
j=1,2,\ldots,N, \eqno(3)
$$
(see \cite{Leray}, \cite{Lad1}, \cite{Fu}, \cite{VorJud}
\cite{Lad}, etc.). Condition (3) requires the fluxes ${{\mathscr{F}}}_j$ of the
boundary datum ${\bf a}$ to be zero separately on all components
$\Gamma_j$ of the boundary $\partial\Omega$, while the compatibility
condition (2) means only that the total flux is zero. Thus, (3) is
stronger than (2) and (3) does not allow the presence of sinks and
sources.
Problem (1), (3) was first studied by J. Leray \cite{Leray} who
initiated two different approaches to prove its solvability. In both
approaches the problem is reduced to an operator equation with a
compact operator and the existence of a fixed--point is obtained by
using the Leray--Schauder theorem. The main difference in these
approaches is in getting an a priori estimate of the solution. The
first method uses the extension of boundary data ${\bf a}$ into
$\Omega$ as ${\bf A}(\varepsilon,
x)={curl}\,\big(\zeta(\varepsilon,x) {\bf b}(x)\big)$, where
$\zeta(\varepsilon,x)$ is Hopf's cut--off function \cite{Hopf}.
For such extension there holds an estimate (see, e.g., \cite{Lad})
$$
\Big|\intop\limits_\Omega \big({\bf v}\cdot\nabla\big){\bf A}\cdot{\bf
v}\,dx\big|\leq \varepsilon c \intop\limits_\Omega |\nabla{\bf
v}|^2\,dx\quad\forall\;\; {\bf v}\in\mathop{\mathaccent "7017 {W}^{1,2}}(\Omega),\eqno(4)
$$
with $c$ being independent of $\varepsilon$ and $\varepsilon >0$
taken sufficiently small (so that $\varepsilon c<\nu$). Obviously,
the extension of the boundary data in the form of $curl$ is possible
only if condition (3) is satisfied. A. Takashita \cite{Takashita}
has constructed a counterexample showing that estimate (4) is false
whatever the choice of the extension ${\bf A}$ can be, if the
condition (3) is not valid. Thus, the first approach may be applied
only when (3) is valid.
The second approach is to prove an a priory estimate by
contradiction. Such arguments also could be found in the book of
O.A. Ladyzhenskaya \cite{Lad}. Later, a slight modification of this
argument was proposed independently by L.V. Kapitanskii and K.
Pileckas \cite{KaPi1}, and by Ch.J. Amick \cite{Amick}. This
modification has the advantage that it allows to take any solenoidal
extension of the boundary data and requires (unlike Hopf's
construction) only the Lipschitz regularity of the boundary
$\partial\Omega$. We should mention that the method used in
\cite{Amick}, \cite{KaPi1} was already contained in the basic paper
of J.Leray \cite{Leray}. In \cite{KaPi1} the solvability of problem
(1) was proved by this method only under "stronger" condition (3),
while in \cite{Amick} was constructed a class of plane domains with
special symmetry on $\Omega$ and on ${\bf a}=\big(a_1, a_2\big)$,
where problem (1) is solvable for arbitrary fluxes ${{\mathscr{F}}}_j$,
assuming only condition (2). More precisely, it is proved in
\cite{Amick} that problem (1) has at least one solution for all
values of ${{\mathscr{F}}}_j$, if $\Omega\subset {\Bbb R}^2$ is symmetric with
respect to the $x_1$--axis and all components $\Gamma_j$ intersect
the line $\{x: \;x_2=0\}$, ${a}_1$ is an even function, while
${a}_2$ is an odd function with respect to $x_2$. Note that Amick's
result was proved by contradiction and does not contain an effective
a priori estimate for the Dirichlet integral of the solution. An
effective estimate for the solution of the Navier--Stokes problem
with the above symmetry conditions was first obtained by H. Fujita
\cite{Fu1} (see also \cite{Morimoto}). Recently V.V. Pukhnachev has
established an analogous estimate for the solution to problem (1)
in the case of three--dimensional stationary fluid motion with two
mutually perpendicular planes of symmetry (private communication).
The assumption on ${{\mathscr{F}}}_j$ to be zero (see (3)) was relaxed in
\cite{Galdi1} where it is shown that problem (1) still admits a
solution provided that $|{{\mathscr{F}}}_j|$ are sufficiently
small\footnote{As far as we are aware, the idea of requiring
smallness of $|{{\mathscr{F}}}_j| $ instead of its vanishing appears for the
first time in \cite{Finn} (see also \cite{Fu}).}. In \cite{BOPI}
estimates for $|{{\mathscr{F}}}_j|$ are expressed in terms of simple geometric
characteristics of $\Omega$ which can be easily verified for
arbitrary domains. These results have been extended to solutions
corresponding to boundary data in Lebesgue's spaces in \cite{Russo}.
As far as exterior domains are concerned, the hypothesis of zero
flux at the boundary has been replaced by the assumptions of small
flux in \cite{RussoA}.
An interesting contribution to the Navier--Stokes problem is due to
H.Fuji-ta and H. Morimoto \cite{FM} (see also
\cite{RussoAStarita}). They studied problem (1) in a domain $\Omega$
with two components of the boundary $\Gamma_1$ and $\Gamma_2$.
Assuming that ${\bf a}={{\mathscr{F}}} \nabla u_0+\boldsymbol{\alpha}$, where ${{\mathscr{F}}}\in{
{\Bbb R}}$, $u_0$ is a harmonic function, and $\boldsymbol{\alpha}$ satisfies
condition (3), they proved that there is a countable subset
${\mathscr{N}}\subset {{\Bbb R}}$ such that if ${{\mathscr{F}}}\not\in{\mathscr{N}}$ and $\boldsymbol{\alpha}$ is
small (in a suitable norm), then system (1) has a weak solution.
Moreover, if $\Omega\subset {\Bbb R}^2$ is an annulus and $u_0=\log|x|$,
then ${\mathscr{N}}=\emptyset$.
To the best of our knowledge this is the state of art of the
Navier--Stokes problem with nonhomogeneous boundary conditions in
bounded multiply connected domains. As a consequence, the
fundamental question whether problem (1) is solvable for all
values of ${{\mathscr{F}}}_j$ (Leray's problem) is still open despite of
efforts of many mathematicians.
In this paper we study problem (1) in a plane domain
$$
\Omega= \Omega_1\setminus\overline{\Omega}_2, \quad
\overline{\Omega}_2\subset\Omega_1, \eqno(5)
$$
where $\Omega_1$ and $\Omega_2$ are bounded simply connected domains
of ${\Bbb R}^2$ with Lip-schitz boundaries $\partial \Omega_1=\Gamma_1$,
$\partial \Omega_2=\Gamma_2$. Without loss of generality we may
assume that $\Omega_2\supset \{x\in{\Bbb R}^2: |x|<1\}$. Since $\Omega$
has only two components of the boundary, condition (2) may be
rewritten in the form
$$
{\mathscr{F}}=\intop\limits_{\Gamma_2}{\bf a}\cdot{\bf
n}\,dS=-\intop\limits_{\Gamma_1}{\bf a}\cdot{\bf n}\,dS\eqno(6)
$$
(${\bf n}$ is an outward normal with respect to the domain
$\Omega$). Using some suggestions from \cite{Amick}, we prove that
problem (1) is solvable without any restriction on the value of
$|{\mathscr{F}}|$ provided ${{\mathscr{F}}}\ge 0$ (outflow condition). Note that this is
the first result on Leray's problem which does not require smallness
or symmetry conditions of the data.
This results was first announced in the "International Conference on
Mathematical Fluid Mechanics: a Tribute to Giovanni Paolo Galdi",
May 21-25, 2007, Portugal
(http://cemat.ist.utl.pt/gpgaldi/abs/russo.pdf).
\section{Notation and preliminary results}
Everywhere in the paper
$\Omega=\Omega_1\setminus\overline{\Omega}_2\subset{\Bbb R}^2$ is a
bounded domain defined above by (5). We assume that the boundary
$\partial\Omega$ is Lipschitz \footnote{$\partial\Omega$ is
Lipschitz, if for every $\xi\in\partial\Omega$, there is a
neighborhood of $\xi$ in which $\partial\Omega$ is the graph of a
Lipschitz continuous function (defined on an open interval).}. We
use standard notations for function spaces: $C(\overline\Omega)$,
$C(\partial\Omega)$, $W^{k,q}(\Omega)$, $\mathop{\mathaccent "7017 {W}^{k,q}}(\Omega)$,
$W^{\alpha,q}(\partial\Omega)$, where $\alpha\in(0,1), k\in{\Bbb
N}_0, q\in[1,+\infty]$. ${\cal H}^1({\Bbb R}^2)$ denotes the Hardy space
on ${\Bbb R}^2$. In our notation we do not distinguish function
spaces for scalar and vector valued functions; it is clear from the
context whether we use scalar or vector (or tensor) valued function
spaces. $H(\Omega)$ is subspace of all divergence free vector fields
from $\mathop{\mathaccent "7017 {W}^{1,2}}(\Omega)$ with the norm
$$
\|{\bf u}\|_{H(\Omega)}=\|\nabla{\bf u}\|_{L^2(\Omega)}.
$$
Note that for function ${\bf u}\in H(\Omega)$ the norm
$\|\,\cdot\,\|_{H(\Omega)}$ is equivalent to
$\|\,\cdot\,\|_{W^{1,2}(\Omega)}$.\\
Let us collect auxiliary results that we shall
use below to prove the solva-bility of problem (1).
\\
{\bf Lemma 1.} {\it Let $\Omega $ be a bounded domain with Lipschitz
boundary. If ${\bf a}\in W^{1/2,2}(\partial\Omega)$
and
$$
\intop\limits_{\partial\Omega }{\bf a}\cdot{\bf n}\,dS=0,
$$
then there exists a divergence free extension ${\bf A}\in
W^{1,2}(\Omega)$ of ${\bf a}$ such that
$$
\|{\bf
A}\|_{W^{1,2}(\Omega)}\leq c \|{\bf
a}\|_{W^{1/2,2}(\partial\Omega)}.\eqno(7)
$$}
Lemma 1 is well known (see \cite{LadSol1}).
\\
{\bf Lemma 2.} (see \cite{SolSca}). {\it Let $\Omega $ be a bounded
domain with Lipschitz boundary and let $R(\boldsymbol{\eta})$
be a continuous linear functional defined on $\mathop{\mathaccent "7017 {W}^{1,2}}(\Omega)$. If
$$
R(\boldsymbol{\eta})=0\qquad\forall\;\;\boldsymbol{\eta}\in H(\Omega),
$$
then there exists a function $p\in L^2(\Omega)$ with
$\intop\limits_\Omega p(x)\,dx=0$ such that
$$
R(\boldsymbol{\eta})= \intop\limits_\Omega p\,{\rm
div}\,\boldsymbol{\eta}\,dx\qquad\forall\;\;\boldsymbol{\eta}\in \mathop{\mathaccent "7017 {W}^{1,2}}(\Omega).
$$
Moreover, $\|p\|_{L^2(\Omega)}$ is equivalent to
$\|R\|_{(\mathop{\mathaccent "7017 {W}^{1,2}}(\Omega))^*}$. }
\\
{\bf Lemma 3.} {\it Let $f\in {\cal H}^1({\Bbb R}^2)$ and let
$$
J(x)=\intop_{{\Bbb R}^2}\log|x-y|\,f(y)\,dy.\eqno(8)
$$
Then \\
(i) $J\in C({\Bbb R}^2)$;\\
(ii) $\nabla J\in L^{2}({\Bbb R}^2)$, $D^\alpha J\in L^{1}({\Bbb R}^2),
|\alpha|=2$. }
\\
Lemma 3 is well known; a proof of the property (i) could be found in
\cite{Tailorbook} (see Theorem 5.12 and Corollary 12.12 at p.
82--83), and the property (ii) is proved, for example, in
\cite{catafalco}
(see Theorem 5.13, p. 208).\\
{\bf Lemma 4.} {\it Let ${\bf w}\in W^{1,2}({\Bbb R}^2)$ and $\hbox{\rm div}\,{\bf
w}=0$. Then
$$
\hbox{\rm div}\,\big[\big({\bf w}\cdot\nabla\big){\bf
w}\big]=\sum\limits_{i,j=1}^2\frac{\partial w_i}{\partial
x_j}\frac{\partial w_j}{\partial x_i}\in {\cal H}^1(\mathbb{R}^2).
$$
}\\
Lemma 4 follows from div-curl lemma with two cancelations (see,
e.g., Theorem II.1 in \cite{CLMS}).\\
{\bf Lemma 5.} {\it Let $\Omega\subset{\Bbb R}^2$ be a bounded domain
with Lipschitz boundary and let $h\in C(\partial\Omega)$. If $h$
could be extended into domain $\Omega$ as a function $H\in
W^{1,2}(\Omega)$, then there exists a unique weak solution $v\in
W^{1,2}(\Omega)$ of the problem
\begin{displaymath} \left\{\begin{array}{rcl}
-\Delta{v} & = & 0\qquad \hbox{\rm in }\Omega,\\[4pt]
{v} & = & {h} \qquad \hbox{\rm on }\partial\Omega,
\end{array}\right.\eqno(9)
\end{displaymath}
such that $v\in C(\overline{\Omega})$.
}
\\
The proof of Lemma 5 could be found in \cite{Littman} (see also
Theorem 4.2 in \cite{Landis}). Note that not every continuous on
$\partial\Omega$ function $h$ could be extended into $\Omega $ as a
function $H$ from $W^{1,2}(\Omega)$. If this is the case, then
there exists a weak solution $v$ of (9) satisfying only $v\in
W_{loc}^{1,2}(\Omega)\cap C(\overline{\Omega})$ (see Chapter II in
\cite{Landis}).\\
\section{Euler equation}
In this section we collect some properties of a solution to the
Euler system
\begin{displaymath}
\left\{\begin{array}{rcl} \big({\bf
w}\cdot\nabla\big){\bf w}+\nabla p & = & 0,\\[4pt]
\hbox{\rm div}\,{\bf w} & = & 0,
\end{array}\right. \eqno(10)
\end{displaymath}
that are used below to prove the main result of the paper.
\\
Assume that ${\bf w}\in W^{1,2}(\Omega)$ and $p\in
W^{1,2}(\Omega)$ satisfy the Euler equations (10) for almost all
$x\in\Omega$ and let $\intop\limits_{\Gamma_i}{\bf w}\cdot{\bf
n}dS=0,\;i=1,2$. Then there exists a continuous stream function
$\psi\in W^{2,2}(\Omega)$ such that $\nabla\psi=(-w_2, w_1)$. Denote
by $\Phi= p+\frac{|{\bf w}|^2}{2}$ the total head pressure
corresponding to the solution $({\bf w}, p)$. Obviously, $\Phi\in
W^{1,s}(\Omega)$ for all $s\in [1,2)$. By direct calculations one
can easily get the identity
$$
\nabla\Phi\equiv \Big(\frac{\partial w_2}{\partial
x_1}-\frac{\partial w_1}{\partial x_2}\Big)\big(w_2,
-w_1\big)=(\Delta \psi)\nabla\psi.\eqno(11)
$$
If all functions are smooth, from this identity the classical
Bernoulli law follows immediately: {\it the total head pressure
$\Phi(x)$ is constant along any streamline of the flow}.
In the general case the following assertion holds.
\\
{\bf Lemma 6.}\cite{korob1}. {\it Let ${\bf w}\in W^{1,2}(\Omega)$
and $p\in W^{1,2}(\Omega)$ satisfy the Euler equations (10) for
almost all $x\in\Omega$ and let $\intop\limits_{\Gamma_i}{\bf
w}\cdot{\bf n}dS=0,\;i=1,2$. Then for any connected set $K\subset
\overline\Omega$ such that
$$
\psi(x)\big|_{K}=const,\eqno(12)
$$
the identity
$$
\Phi(x)=const \quad \mathfrak{H}^1-almost\quad everywhere\quad on
\quad K\eqno(13)
$$
holds. Here $\mathfrak{H}^1$ denotes one-dimensional Hausdorff
measure\footnote{$\mathfrak{H}^1(F)=\lim\limits_{t\to 0+}\mathfrak{H}^1_t(F)$, where
$\mathfrak{H}^1_t(F)=\inf\{\sum\limits_{i=1}^\infty {\rm diam}
F_i:\, {\rm diam} F_i\leq t, F\subset \bigcup\limits_{i=1}^\infty
F_i\}$. }.
In particular, if ${\bf w}=0$ on ${\partial \Omega}$ $($in the sense
of trace$)$, then the pressure $p(x)$ is constant on
$\partial\Omega$. Note that $p(x)$ could take different constant
values $p_j=p(x)\big|_{\Gamma_j}, j=1,2$, on different components
$\Gamma_j$ of the boundary $\partial\Omega$. }\\
Here and henceforth we understand connectedness in the sense of
general topology. Note that the proof of the above lemma is based on
classical results of \cite{Kronrod} and on recent results obtained
in \cite{korob}. The last statement of Lemma 6 was proved in
\cite{KaPi1} (see Lemma 4) and in \cite{Amick} (see Theorem 2.2).
\\
{\bf Lemma 7.} {\it Let $({\bf w}, p)$ satisfy the Euler equations
(10) for almost all $x\in\Omega$, ${\bf w}\in W^{1,2}(\Omega)$ and
${\bf w}(x)\big|_{\partial\Omega}=0$. Then
$$
p\in C(\overline{\Omega})\cap W^{1,2}(\Omega).\eqno(14) $$}
{\bf Proof.} From Euler equations (10) it follows that $p\in
W^{1,s}(\Omega)$ for any $s\in[1,2)$ and
$$
\|p\|_{W^{1,s}(\Omega)}\leq c\|{\bf w}\|_{H(\Omega)}^2.
$$
Multiply (10) by
$\boldsymbol{\varphi}=\nabla\xi$, where $\xi\in C_0^\infty(\Omega)$:
$$
\intop_\Omega\nabla p\cdot\nabla\xi\,dx=-\intop\limits_\Omega\big({\bf
w}\cdot\nabla\big){\bf w}\cdot\nabla\xi\,dx \quad\forall\xi\in
C^\infty_0(\Omega).
$$
Thus, $p\in W^{1,q}(\Omega)$ is the unique weak solution of the
boundary value problem for the Poisson equations
\begin{displaymath}
\left\{\begin{array}{rcl} -\Delta p & = & \hbox{\rm div}\,\big[\big({\bf
w}\cdot\nabla\big){\bf w}\big]\qquad \hbox{\rm in }\;\;\Omega,
\\[4pt]
p(x)& = & p_1 \qquad\qquad\qquad\quad\;\; \hbox{\rm on
}\;\,\Gamma_1,\\[4pt]
p(x) & = & p_2 \,\;\qquad\qquad\qquad\quad\; \hbox{\rm on
}\;\;\Gamma_2.
\end{array}\right.\eqno(15)
\end{displaymath}
According to Lemma 4, $\hbox{\rm div}\,\big[\big({\bf w}\cdot\nabla\big){\bf
w}\big] \in {\cal H}^1({\Bbb R}^2)$ (here we assume that ${\bf w}\in
H(\Omega)$ is extended by zero to ${\Bbb R}^2$). Define the function
$J_1(x)$ by the formula
$$
J_1(x)=-\frac{1}{2\pi}\intop_{{{\Bbb R}}^2}\log|x-y|\,{\rm
div}_y\,\big[\big({\bf w}(y)\cdot\nabla_y\big){\bf w}(y)\big]dy.
$$
In virtue of Lemma 3, $J_1\in C({\Bbb R}^2)$, $\nabla J_1\in
L^{2}({{\Bbb R}}^2)$, $D^\alpha J_1\in L^{1}({{\Bbb R}}^2), |\alpha|=2$. Since
$-\Delta J_1(x)= \hbox{\rm div}\,\big[\big({\bf w}\cdot\nabla\big){\bf w}\big]$
in ${\Bbb R}^2$, we get for $J_2(x)=p(x)-J_1(x)$ the following problem
\begin{displaymath}
\left\{\begin{array}{rcl} -\Delta J_2 & = & 0\qquad\qquad\qquad
\hbox{\rm in }\;\;\Omega,
\\[4pt]
J_2\big|_{\partial\Omega} & = & j_2-j_1\;\;\qquad\;\;\;\; \hbox{\rm
on }\;\;\partial\Omega,
\end{array}\right.\eqno(16)
\end{displaymath}
where $j_1(x)=J_1(x)\big|_{\partial\Omega}$,
$$
j_2(x)=\left\{\begin{array}{rcl} {p}_1 \qquad\; \hbox{\rm on}\;\;\Gamma_1,\\[4pt]
{p}_2\qquad\; \hbox{\rm on }\;\;\Gamma_2.
\end{array}\right.
$$
The function $j_1$ is a trace on $\partial\Omega$ of $J_1\in
W^{1,2}(\Omega)\cap C(\overline{\Omega})$, while $j_2\in
C(\partial\Omega)$ and $j_2$ obviously could be extended to $\Omega$
as a function from $W^{1,2}(\Omega)$. Thus, by Lemma 5 problem (16)
has a unique weak solution $J_2\in W^{1,2}(\Omega)$ such that
$J_2\in C(\overline{\Omega})$. By uniqueness $ p(x)=J_1(x)+J_2(x)$.
Hence, $p\in C(\overline{\Omega})\cap
W^{1,2}(\Omega)$. \\
We say that the function $f\in W^{1,s}(\Omega)$ satisfies a {\it
weak one-side maximum principle locally} in $\Omega$, if
$$
\mathop{\hbox{\rm ess}\,\hbox{\rm sup}}_{x\in\Omega^\prime}\,f(x)\leq \mathop{\hbox{\rm
ess}\,\hbox{\rm sup}}_{x\in\partial\Omega^\prime}\,f(x) \eqno(17)
$$
holds for any strictly interior subdomain $\Omega^\prime$
($\overline{\Omega^\prime}\subset\Omega)$ with the boundary
$\partial\Omega^\prime$ that does not contain singleton connected
components. (In (17) negligible sets are the sets of 2--dimensional
Lebesgue measure zero in the left ess sup, and the sets of
1--dimensional Hausdorff measure zero in the right ess sup.) If
(17) holds for any $\Omega^\prime \subset\Omega$ with the boundary
$\partial\Omega^\prime$ not containing singleton connected
components, then we say that $f\in W^{1,s}(\Omega)$
satisfies a {\it weak one-side maximum principle } in $\Omega$
(since the boundary $\partial\Omega$ is Lipschitz, we can take
$\Omega^\prime=\Omega$ in (17)).
\\
{\bf Lemma 8.} \cite{korob1}. {\it Let ${\bf w}\in W^{1,2}(\Omega)$
and $p\in W^{1,2}(\Omega)$ satisfy the Euler equations (10) for
almost all $x\in\Omega$ and let $\intop\limits_{\Gamma_i}{\bf
w}\cdot{\bf n}dS=0,\;i=1,2$. Assume that there exists a sequence of
functions $\{\Phi_\mu\}$ such that $\Phi_\mu\in
W^{1,s}_{loc}(\Omega)$ and $\Phi_\mu\rightharpoonup\Phi$ in the
space $W^{1,s}_{loc}(\Omega)$ for all $s\in (1,2)$. If all
$\Phi_\mu$ satisfy the weak one-side maximum principle locally in
$\Omega$, then $\Phi$ satisfies the weak one-side maximum principle
in $\Omega$.
In particular, if ${\bf w}\big|_{\partial\Omega}=0$, then
$$
\mathop{\hbox{\rm ess}\,\hbox{\rm sup}}_{x\in\Omega}\,\Phi(x)\leq
\mathop{\hbox{\rm ess}\,\hbox{\rm
sup}}_{x\in\partial\Omega}\,\Phi(x)= \max\{p_1, p_2\}.\eqno(18)
$$}
The proof of the above lemma is based on Lemma 6, classical results
of \cite{Kronrod}, and on recent results obtained in \cite{korob}.
Note that the weaker version of Lemma 8 was proved by Ch. Amick
\cite{Amick} (see Theorem 3.2 and Remark thereafter).
\\
\section{Existence theorem}
Let us consider Navier--Stokes problem (1) in the domain $\Omega$
defined by (5) and assume that $\partial\Omega$ is at least
Lipschitz. If the boundary datum ${\bf a}\in
W^{1/2,2}(\partial\Omega)$ and ${\bf a}$ satisfies the condition
(6), i.e.,
$$
\intop\limits_{\partial\Omega}{\bf a}\cdot{\bf
n}\,dS=\intop\limits_{\Gamma_1}{\bf a}\cdot{\bf
n}\,dS+\intop\limits_{\Gamma_2}{\bf a}\cdot{\bf n}\,dS=0,
$$
then by Lemma 1 there exists a divergence free extension ${\bf
A}\in W^{1,2}(\Omega)$ of ${\bf a}$ and there holds estimate (7).
Using this fact and standard results (see, e.g. \cite{Lad}) we can
find a weak solution ${\bf U}\in W^{1,2}(\Omega)$ of the Stokes
problem such that ${\bf U}-{\bf A}\in H(\Omega)$ and
$$
\nu\intop\limits_\Omega\nabla{\bf U}\cdot\nabla\boldsymbol{\eta}\,dx= 0
\quad\forall\;\boldsymbol{\eta}\in H(\Omega). \eqno(19)
$$
Moreover,
$$
\|{\bf U}\|_{W^{1,2}(\Omega)}\leq c\|{\bf
a}\|_{W^{1/2,2}(\partial\Omega)}.\eqno(20)
$$
By a {\it weak solution} of problem (1) we understand a function
${\bf u}$ such that ${\bf w}={\bf u}-{\bf A}\in H(\Omega)$ and
satisfies the integral identity
$$
\nu\intop\limits_\Omega\nabla{\bf
w}\cdot\nabla\boldsymbol{\eta}\,dx-\intop\limits_\Omega\big(({\bf w}+{\bf
U})\cdot\nabla\big)\boldsymbol{\eta}\cdot{\bf
w}\,dx-\intop\limits_\Omega\big({\bf w}\cdot\nabla\big)\boldsymbol{\eta}\cdot{\bf
U}\,dx
$$
$$
=\intop\limits_\Omega\big({\bf U}\cdot\nabla\big)\boldsymbol{\eta}\cdot{\bf
U}\,dx \qquad\forall\boldsymbol{\eta}\in H(\Omega).\eqno(21)
$$
We shall prove the following
\\
{\bf Theorem 1. } {\it Assume that ${\bf a}\in
W^{1/2,2}(\partial\Omega)$ and let condition $(6)$ be fulfilled. If
${\mathscr{F}}=\intop\limits_{\Gamma_2}{\bf a}\cdot{\bf n}\,dS\ge 0$, then
problem $(1)$ admits at least one weak solution.}
\\
{\bf Proof.} 1. We follow a contradiction argument of J. Leray
\cite{Leray}. Although, this argument was used also in many other
papers (e.g. \cite{Lad1}, \cite{Lad}, \cite{KaPi1}, \cite{Amick}),
we reproduce here, for the reader convenience, some details of it.
It is well known (e.g. \cite{Lad}) that integral identity (21) is
equivalent to an operator equation in the space $H(\Omega)$ with a
compact operator, and, therefore, in virtue of the Leray--Schauder
fixed--point theorem, to prove the existence of a weak solution to
Navier--Stokes problem (1) it is sufficient to show that all
possible solutions of the integral identity
$$
\nu\intop\limits_\Omega\nabla{\bf
w}\cdot\nabla\boldsymbol{\eta}\,dx-\lambda\intop\limits_\Omega\big(({\bf w}+{\bf
U})\cdot\nabla\big)\boldsymbol{\eta}\cdot{\bf
w}\,dx-\lambda\intop\limits_\Omega\big({\bf
w}\cdot\nabla\big)\boldsymbol{\eta}\cdot{\bf U}\,dx
$$
$$
=\lambda\intop\limits_\Omega\big({\bf
U}\cdot\nabla\big)\boldsymbol{\eta}\cdot{\bf U}\,dx \qquad \forall\;\boldsymbol{\eta}\in
H(\Omega)\eqno(22)
$$
are uniformly bounded (with respect to $\lambda\in[0,\nu^{-1}]$) in
$H(\Omega)$. Assume this is false. Then there exist sequences $
\{\lambda_k\}_{k\in{\Bbb N}}\subset [0, \nu^{-1}]$ and $\{{\bf
w}_k\}_{k\in{\Bbb N}}\in H(\Omega)$ such that
$$
\nu\intop\limits_\Omega\nabla{\bf
w}_k\cdot\nabla\boldsymbol{\eta}\,dx-\lambda_k\intop\limits_\Omega\big(({\bf
w}_k+{\bf U})\cdot\nabla\big)\boldsymbol{\eta}\cdot{\bf
w}_k\,dx-\lambda_k\intop\limits_\Omega\big({\bf
w}_k\cdot\nabla\big)\boldsymbol{\eta}\cdot{\bf U}\,dx
$$
$$
=\lambda_k\intop\limits_\Omega\big({\bf
U}\cdot\nabla\big)\boldsymbol{\eta}\cdot{\bf U}\,dx\qquad\forall\,\boldsymbol{\eta}\in
H(\Omega), \eqno(23)
$$
and
$$
\lim\limits_{k\to\infty}\lambda_k=\lambda_0\in[0, \nu^{-1}],\quad
\lim\limits_{k\to\infty}J_k=\lim\limits_{k\to\infty}\|{\bf
w}_k\|_{H(\Omega)}=\infty.\eqno(24)
$$
Let us take in (23) $\boldsymbol{\eta}=J_k^{-2}{\bf w}_k$ and denote
$\widehat{\bf w}_k=J_k^{-1}{\bf w}_k$. Since
$$
\intop\limits_\Omega\big(({\bf w}_k+{\bf U})\cdot\nabla\big){\bf
w}_k\cdot{\bf w}_k\,dx=0,
$$
we get
$$
\nu\intop\limits_\Omega|\nabla\widehat{\bf
w}_k|^2\,dx=\lambda_k\intop\limits_\Omega\big(\widehat{\bf
w}_k\cdot\nabla\big)\widehat{\bf w}_k\cdot{\bf U}\,dx+
J_k^{-1}\lambda_k\intop\limits_\Omega\big({\bf
U}\cdot\nabla\big)\widehat{\bf w}_k\cdot{\bf U}\,dx. \eqno(25)
$$
Since $\|\widehat{\bf w}_k\|_{H(\Omega)}=1$, there exists a
subsequence $\{\widehat{\bf w}_{k_l}\}$ converging weakly
in $H(\Omega)$ to a vector field $\widehat{\bf w}\in H(\Omega)$.
Because of the compact imbedding
$$
H(\Omega)\hookrightarrow L^r(\Omega) \quad
\forall\,r\in(1,\infty),
$$
the subsequence $\{\widehat{\bf w}_{k_l}\}$ converges strongly in
$L^r (\Omega)$. Therefore, we can pass to a limit as $k_l\to\infty$
in equality (25). As a result we obtain
$$
\nu=\lambda_0\intop\limits_\Omega\big(\widehat{\bf
w}\cdot\nabla\big)\widehat{\bf w}\cdot{\bf U}\,dx.\eqno(26)
$$
2. Let us return to integral identity (23). Consider the functional
$$
R_k(\boldsymbol{\eta})=\intop\limits_\Omega\Big(\nu\nabla{\bf
w}_k\cdot\nabla\boldsymbol{\eta}-\lambda_k\big(({\bf w}_k+{\bf
U})\cdot\nabla\big)\boldsymbol{\eta}\cdot{\bf w}_k-\lambda_k\big({\bf
w}_k\cdot\nabla\big)\boldsymbol{\eta}\cdot{\bf U}\Big)\,dx
$$
$$
-\lambda_k\intop\limits_\Omega\big({\bf
U}\cdot\nabla\big)\boldsymbol{\eta}\cdot{\bf U}\,dx \qquad \forall
\;\boldsymbol{\eta}\in\mathop{\mathaccent "7017 {W}^{1,2}}(\Omega).
$$
Obviously, $R_k(\boldsymbol{\eta})$ is a linear functional, and using (20) and
the imbedding theorem, we obtain
$$
\big|R_k(\boldsymbol{\eta})\big|\leq c\Big(\|{\bf w}_k\|_{H(\Omega)}+\|{\bf
w}_k\|_{H(\Omega)}^2
+\|{\bf a}\|_{W^{1/2,2}(\partial\Omega)}^2\Big)
\|\boldsymbol{\eta}\|_{H(\Omega)},
$$
with constant $c$ independent of $k$. It follows from (23) that
$$
R_k(\boldsymbol{\eta})=0\qquad\forall\;\boldsymbol{\eta}\in H(\Omega).
$$
Therefore, by Lemma 2, there exist functions $p_k\in \widehat
L^2(\Omega)=\{q\in L^2(\Omega):\; \intop\limits_\Omega q(x)\,dx=0\}$
such that
$$
R_k(\boldsymbol{\eta})=\intop\limits_\Omega p_k\hbox{\rm div}\,
\boldsymbol{\eta}\,dx\qquad\forall\;\boldsymbol{\eta}\in \mathop{\mathaccent "7017 {W}^{1,2}}(\Omega),
$$
and
$$
\|p_k\|_{L^2(\Omega)}\leq c\Big(\|{\bf w}_k\|_{H(\Omega)}+\|{\bf
w}_k\|_{H(\Omega)}^2
+ \|{\bf a}\|_{W^{1/2,2}(\partial\Omega)}^2\Big). \eqno(27)
$$
The pair $\big({\bf w}_k, p_k)$ satisfies the integral identity
$$
\nu\intop\limits_\Omega\nabla{\bf
w}_k\cdot\nabla\boldsymbol{\eta}\,dx-\lambda_k\intop\limits_\Omega\big(({\bf
w}_k+{\bf U})\cdot\nabla\big)\boldsymbol{\eta}\cdot{\bf
w}_k\,dx-\lambda_k\intop\limits_\Omega\big({\bf
w}_k\cdot\nabla\big)\boldsymbol{\eta}\cdot{\bf U}\,dx
$$
$$
-\lambda_k\intop\limits_\Omega\big({\bf
U}\cdot\nabla\big)\boldsymbol{\eta}\cdot{\bf U}\,dx =\intop\limits_\Omega p_k\hbox{\rm div}\,
\boldsymbol{\eta}\,dx\qquad\forall \;\boldsymbol{\eta}\in\mathop{\mathaccent "7017 {W}^{1,2}}(\Omega).\eqno(28)
$$
Let ${\bf u}_k={\bf w}_k+{\bf U}$. Then identity (28) takes the form
(see (19))
$$
\nu\intop\limits_\Omega\nabla{\bf
u}_k\cdot\nabla\boldsymbol{\eta}\,dx-\intop\limits_\Omega p_k\,{\rm
div}\,\boldsymbol{\eta}\,dx =-\lambda_k\intop\limits_\Omega({\bf
u}_k\cdot\nabla\big){\bf u}_k\cdot\boldsymbol{\eta}\,dx \;\;
\forall\,\boldsymbol{\eta}\in\mathop{\mathaccent "7017 {W}^{1,2}}(\Omega).
$$
Thus, $\big({\bf u}_k, p_k)$ might be considered as a weak
solution to the Stokes problem
\begin{displaymath}
\left\{\begin{array}{rcl}-\nu\Delta{\bf u}_k +\nabla p_k & = & {\bf
f}_k \qquad \;\hbox{\rm in }\;\;\Omega,
\\[4pt]
\hbox{\rm div}\,{\bf u}_k & = & 0 \;\qquad \;\hbox{\rm in }\;\;\Omega,
\\[4pt]
{\bf u}_k & = & {\bf a}
\qquad\;\, \hbox{\rm on }\;\;\partial\Omega,
\end{array}\right.
\end{displaymath}
with the right--hand side ${\bf f}_k=-\lambda_k\big({\bf
u}_k\cdot\nabla\big){\bf u}_k$. Obviously, ${\bf f}_k\in
L^s(\Omega)$ for $s\in(1,2)$ and
$$
\|{\bf f}_k\|_{L^s(\Omega)}\leq c\|\big({\bf
u}_k\cdot\nabla\big){\bf u}_k\|_{L^s(\Omega)} \leq c\|{\bf
u}_k\|_{L^{2s/(2-s)}(\Omega)}\|\nabla{\bf u}_k\|_{L^{2}(\Omega)}
$$
$$
\leq c \Big( \big(\|{\bf w}_k\|_{H(\Omega)}+\|{\bf
U}\|_{W^{1,2}(\Omega)}\big)^2\Big) \leq c\Big(\|{\bf
w}_k\|_{H(\Omega)}^2
+\|{\bf a}\|_{W^{1/2,2}(\partial\Omega)}^2\Big),
$$
where $c$ is independent of $k$. By well known local regularity
results for the Stokes system (see \cite{Lad}, \cite{Galdibook}) we
have ${\bf w}_k\in W^{2,s}_{loc}(\Omega)$, $p_k\in
W_{loc}^{1,s}(\Omega)$, and the estimate
$$
\|{\bf
w}_k\|_{W^{2,s}(\Omega^\prime)}+\|p_k\|_{W^{1,s}(\Omega^\prime)}\leq
c\Big( \|{\bf f}_k\|_{L^s(\Omega)}+\|{\bf
u}_k\|_{W^{1,2}(\Omega)}+\|{p}_k\|_{L^2(\Omega)}\Big)
$$
$$
\leq c\Big(\|{\bf w}_k\|_{H(\Omega)}^2+\|{\bf w}_k\|_{H(\Omega)}
+\|{\bf a}\|_{W^{1/2,2}(\partial\Omega)} +\|{\bf
a}\|_{W^{1/2,2}(\partial\Omega)}^2\Big), \eqno(29)
$$
holds, where $\Omega^\prime$ is arbitrary domain with
$\overline{\Omega}\,^\prime\subset\Omega$ and the constant $c$
depends on ${\rm dist}\,(\Omega^\prime,
\partial\Omega)$ but not on $k$.
Denote $\widehat p_k=J_k^{-2} p_k $. It follows from (27) and (29)
that
$$
\|\widehat p_k\|_{L^{2}(\Omega)}\leq const, \quad \|\widehat
p_k\|_{W^{1,s}(\Omega^\prime)}\leq const
$$
for any $\overline\Omega^\prime\subset\Omega$ and $s\in(1,2)$.
Hence, from $\{\widehat p_{k_l}\}$ can be extracted a subsequence,
still denoted by $\{\widehat p_{k_l}\}$, which converges weakly in
$\widehat L^2(\Omega)$ and $W^{1,s}_{loc}(\Omega)$ to some function
$\widehat p\in W_{loc}^{1,s}(\Omega)\cap \widehat L^2(\Omega)$. Let
$\boldsymbol{\varphi}\in C_0^\infty(\Omega)$. Taking in (28)
$\boldsymbol{\eta}=J_{k_l}^{-2}\boldsymbol{\varphi}$ and letting $k_l\to\infty$ yields
$$
-\lambda_0\intop\limits_\Omega\big(\widehat{\bf
w}\cdot\nabla\big)\boldsymbol{\varphi}\cdot\widehat{\bf
w}\,dx=\intop_\Omega\widehat p\,{\rm div}\varphi\,dx
\quad\forall\boldsymbol{\varphi}\in C^\infty_0(\Omega).
$$
Integrating by parts in the last equality, we derive
$$
\lambda_0\intop\limits_\Omega\big(\widehat{\bf
w}\cdot\nabla\big)\widehat{\bf
w}\cdot\boldsymbol{\varphi}\,dx=-\intop_\Omega\nabla\widehat p\cdot\varphi\,dx
\quad\forall\boldsymbol{\varphi}\in C^\infty_0(\Omega).\eqno(30)
$$
Hence, the pair $\big(\widehat{\bf w}, \widehat p\big)$ satisfies
for almost all $x\in\Omega$ the Euler equations
\begin{displaymath}
\left\{\begin{array}{rcl} \lambda_0\big(\widehat{\bf
w}\cdot\nabla\big)\widehat{\bf w}+\nabla\widehat p & = & 0,\\[4pt]
{\rm div }\,\widehat{\bf w} & = & 0,
\end{array}\right. \eqno(31)
\end{displaymath}
and $ \widehat{\bf w}\big|_{\partial\Omega}=0$. By Lemmas 6 and 7,
$\widehat p\in C(\overline{\Omega})\cap W^{1,2}(\Omega)$ and the
pressure $\widehat p(x)$ is constant on $\Gamma_1$ and $\Gamma_2$.
Denote by $\widehat p_1$ and
$\widehat p_2$ values of $\widehat p(x)$ on $\Gamma_1$ and
$\Gamma_2$, respectively. Multiplying equations (31) by ${\bf U}$
and integrating by parts, we derive
$$
\lambda_0\intop\limits_\Omega\big(\widehat{\bf
w}\cdot\nabla\big)\widehat{\bf w}\cdot {\bf
U}\,dx=-\intop\limits_\Omega\nabla\widehat p\cdot{\bf
U}\,dx=-\intop\limits_{\partial\Omega}\widehat p \,{\bf a}\cdot{\bf
n}\,dS
$$
$$
= -\widehat p_1\intop\limits_{\Gamma_1} {\bf a}\cdot{\bf
n}\,dS-\widehat p_2\intop\limits_{\Gamma_2}{\bf a}\cdot{\bf
n}\,dS={{\mathscr{F}}}(\widehat p_1-\widehat p_2)\eqno(32)
$$
(see formula (6)). If either ${{\mathscr{F}}}=0$ or $\widehat p_1=\widehat
p_2$, it follows from (32) that
$$
\lambda_0\intop\limits_\Omega\big(\widehat{\bf
w}\cdot\nabla\big)\widehat{\bf w}\cdot {\bf U}\,dx=0.\eqno(33)
$$
The last relation contradicts equality (26). Therefore, the norms
$\|{\bf w}\|_{H(\Omega)}$ of all possible solutions to identity (22)
are uniformly bounded with respect to $\lambda\in[0,\nu^{-1}]$ and
by Leray--Schauder fixed--point theorem problem (1) admits at least
one weak solution ${\bf u}\in
W^{1,2}(\Omega)$.\\
3. Up to this point our arguments were standard and followed those
of Leray \cite{Leray} (see also \cite{KaPi1} and \cite{Amick}).
However, by the our assumptions ${{\mathscr{F}}}>0$ and, in general, $\widehat
p_2\neq\widehat p_1$ (see a counterexample in \cite{Amick}). Thus,
(33) may be false. In order to prove that $\widehat p_1$ and
$\widehat p_2$ do coincide in the case ${{\mathscr{F}}}>0$, we use the property
of $\big(\widehat{\bf w},\widehat p\big)$ to be a limit (in some
sense) of solutions to the Navier--Stokes equations. Note that the
possibility of using this fact was already pointed up by Amick
\cite{Amick}.
Let $\Phi_{k_l}=p_{k_l}+\dfrac{\lambda_{k_l}}{2}|{\bf
u}_{k_l}|^2$, where ${\bf u}_{k_l}={\bf w}_{k_l}+{\bf U} $, be a
total head pressures corresponding to the solutions $\big({\bf
w}_{k_l}, p_{k_l}\big)$ of identities (25). Then $\Phi_{k_l}\in
W^{2,s}_{loc}(\Omega),\; s\in (1,2)$, satisfy almost everywhere in
$\Omega$ the equations
$$
\nu \Delta\Phi_{k_l}-\lambda_{k_l}\big({\bf
u}_{k_l}\cdot\nabla\big)\Phi_{k_l}=\nu \Big(\frac{\partial
{u}_{1k_l}}{\partial x_2}-\frac{\partial {u}_{2k_l}}{\partial
x_1}\Big)^2.
$$
It is well known \cite{GilbTrud1}, \cite{GilbTrud2} (see also
\cite{Mi2}) that for $\Phi_{k_l}$ one-side maximum principle holds
locally (since the boundary is only Lipschitz, $\Phi_{k_l}$ do not
have second derivatives up to the boundary). Set $\widehat
\Phi_{k_l}=J_{k_l}^{-2}\Phi_{k_l}$. It follows from (27), (29) that
the sequence
$\widehat \Phi_{k_l}$ weakly converges to $\widehat
\Phi=\widehat p+\dfrac{\lambda_0}{2}|\widehat{\bf u}|^2$ in
$L^2(\Omega)\cap W^{1,s}_{loc}(\Omega),\, s\in (1,2)$. Therefore, by
Lemma 8, $\widehat \Phi$ satisfies the weak one-sided maximum
principle and
$$
{\mathop{\hbox{\rm ess}\,\hbox{\rm
sup}}_{x\in\Omega}}\,\widehat\Phi(x)\leq {\mathop{\hbox{\rm
ess}\,\hbox{\rm
sup}}_{x\in\partial\Omega}}\,\widehat\Phi(x)=\max\{\widehat
p_1,\widehat p_2\}\eqno(34)
$$
(see (18)).
We conclude from equalities (26) and (32)
$$
(\widehat p_1-\widehat p_2){{\mathscr{F}}}=\nu >0.
$$
So, if ${{\mathscr{F}}}>0$, then
$$
\widehat p_2 < \widehat p_1.\eqno(35)
$$
Now, it follows from (34), (35) that
$$
\intop\limits_\Omega\widehat\Phi(x)\,dx \leq {\mathop{\hbox{\rm
ess}\,\hbox{\rm sup}}_{x\in\Omega}}\,\widehat\Phi(x)|{\Omega}|\leq
\widehat p_1|\Omega|,\eqno(36)
$$
where $|\Omega|$ means the measure of $\Omega$.\\
On the other hand, from equation $(31_1)$ we obtain the identity
$$
0=x\cdot\nabla \widehat p(x)+\lambda_0 x\cdot\big(\widehat {\bf
w}(x)\cdot\nabla \big)\widehat{\bf w}(x)=\hbox{\rm div}\,\big[x\,\widehat p(x)+
\lambda_0 \big(\widehat {\bf w}(x)\cdot x\big)\widehat {\bf
w}(x)\big]
$$
$$
-\widehat p(x)\,\hbox{\rm div}\, x-\lambda_0 |\widehat {\bf
w}(x)|^2=\hbox{\rm div}\,\big[x\,\widehat p(x)+ \lambda_0 \big(\widehat {\bf
w}(x)\cdot x\big)\widehat {\bf w}(x)\big]-2\widehat\Phi(x).
$$
Integrating this identity we derive
$$
2\intop\limits_\Omega\widehat\Phi(x)\,dx=\intop\limits_{\partial\Omega}\widehat
p(x) \big(x\cdot{\bf n}\big)\,dS=\widehat p_1\intop\limits_{\Gamma_1}
\big(x\cdot{\bf n}\big)\,dS+\widehat
p_2\intop\limits_{\Gamma_2}\big(x\cdot{\bf n}\big)\,dS
$$
$$
=\widehat p_1\intop\limits_{\Omega_1}{\rm div}\,x\,dx-\widehat
p_2\intop\limits_{\Omega_2}{\rm div}\,x\,dx=2\big(\widehat
p_1|\Omega_1|-\widehat p_2|\Omega_2|\big).
$$
Hence,
$$
\intop\limits_\Omega\widehat\Phi(x)\,dx=\widehat
p_1|\Omega_1|-\widehat p_2|\Omega_2|=\widehat
p_1|\Omega|+\big(\widehat p_1-\widehat p_2\big)|\Omega_2|.\eqno(37)
$$
Inequalities (36) and (37) yield
$$
\widehat p_1\leq\widehat p_2.
$$
This contradicts inequality (35). Thus, all solutions of integral
identity (22) are uniformly bounded in $H(\Omega)$ and by the
Leray--Schauder fixed--point theorem there exists at least one weak
solution of problem (1). $\qquad\qquad\qquad\quad\Box$
\\
\\
{\bf Remark 2.} Let $\Omega=\{x:1<|x|<2\}$ be the annulus and let
$(r,\theta)$ be polar coordinates in ${\Bbb R}^2$. If $f\in
C^\infty_0(1,2)$, then the pair $\widehat{\bf
w}=\big(\widehat{w}_r,\widehat{w}_\theta\big)$ and $\widehat p$ with
$$
\widehat{w}_r(r,\theta)=0,\;\; \widehat{w}_\theta(r,\theta)= f(r),
\quad \widehat
p(r,\theta)=\lambda_0\intop_1^{r}\frac{f^2(t)}{t}\,dt\eqno(38)
$$
satisfy both equations (31) and the boundary condition
$\widehat{\bf w}\big|_{\partial\Omega}=0$ ($\widehat{w}_r$ and
$\widehat{w}_\theta$ are components of the velocity field in polar
coordinate system). On the other hand,
$$
0=\widehat p(x)\big|_{r=1}\neq \widehat
p(x)\big|_{r=2}=\lambda_0\intop\limits _1^{2}\frac{f^2(t)}{t}\,dt>0.
$$
This simple example, due to Ch.J. Amick \cite{Amick} (see also
\cite{Galdi1}, v. II, p. 59), shows that, in general, the pressure
$\widehat p$ corresponding to the solution of Euler equations (31)
could have not equal constant values on different components of the
boundary.
It is interesting to observe that for the solution like (38)
necessarily holds $\widehat p_1>\widehat p_2$. Indeed, writing the
Euler equations (31) in polar coordinates and integrating over
$\Omega$ yields
$$
\lambda_0\intop_\Omega\frac{\widehat
{w}_\theta^2(r)}{r}\,drd\theta=\lambda_0\intop_\Omega\frac{f^2(r)}{
r}\,drd\theta=\intop_\Omega\frac{\partial p(r)}{\partial r}\,drd\theta
=\widehat p_1-\widehat p_2> 0.\eqno(39)
$$
The solution (38) cannot be a limit of solutions to Navier--Stokes
problem (in the sense described in the proof of Theorem 1). If it is
so, then we conclude from (26), (32) and (39) that ${\mathscr{F}}>0$. But
this, as it is proved in Theorem 1, leads to a contradiction.
\\
We emphasize that in the case when ${\mathscr{F}}<0$ (inflow condition)
problem (1) remains unsolved. However, in this case we do not know
any counterexample showing that for the solution of Euler equations
(31) the inequality $\widehat p_2>\widehat p_1$ holds.
\\
It is well known (see \cite{BOPI}, \cite{Galdi1}) that independently
of the sign of ${\mathscr{F}}$ problem (1) has a solution, if $|{\mathscr{F}}|$ is
sufficiently small. Using this result Theorem 1 can
be strengthened as follows \\
{\bf Theorem 2.} {\it Assume that ${\bf a}\in
W^{1/2,2}(\partial\Omega)$ and let condition $(6)$ be fulfilled.
Then there exists ${\mathscr{F}}_0>0$ such that for any ${\mathscr{F}}\in(-{\mathscr{F}}_0,+\infty)$
problem $(1)$ admits at least one weak solution.}
{\small
|
{
"timestamp": "2011-10-31T01:02:01",
"yymm": "1009",
"arxiv_id": "1009.4024",
"language": "en",
"url": "https://arxiv.org/abs/1009.4024"
}
|
\section{Introduction}
\label{sec:intro}
Recently, a highly significant ($\sim 4\sigma$) indication that there is a spatial gradient in values of the fine-structure constant, $\alpha$, was reported~\cite{webb10arxiv}. That is, in one direction on the sky $\alpha$ was larger in the past, while in the opposite direction it seems to have been smaller. This result has massive implications for the ``fine-tuning'' problem. It is well known that the constants of nature are finely tuned to allow life to exist. For example, the production of carbon from helium in stars (the famous triple-$\alpha$ reaction) is sensitive to the position of a low-energy resonance, which in turn is very sensitive to variations in coupling constants. If the coupling constants were slightly different, there would be no resonance and hence no carbon-based life. There are many other examples. With the detection of spatial variation of coupling constants we begin to have a natural explanation of fine-tuning: we simply we appeared in the region of the Universe where values of fundamental constants are suitable for our existence.
We can only detect the variation of dimensionless fundamental constants. In this review we will discuss systems that are sensitive to variation of the fine-structure constant, $\alpha = e^2/\hbar c$, the proton $g$-factor, $g_p$, and the dimensionless mass ratios $\mu = m_e/m_p$ and $X_q=m_q/\Lambda_\textrm{QCD}$. Here $m_e$, $m_p$, and $m_q$ are the electron, proton, and light-current quark masses, respectively, and $\Lambda_\textrm{QCD}$ is the quantum chromodynamics scale, defined as the position of the Landau pole in the logarithm of the running strong coupling constant, $\alpha_s(r) \sim 1/\ln{(\Lambda_\textrm{QCD} r/\hbar c)}$. The proton mass $m_p$ is proportional to $\Lambda_\textrm{QCD}$ (if we neglect the few percent contribution of quark masses to the proton mass), therefore the relative variation of $\mu=m_e/m_p$ is approximately equal to the relative variation of $X_e=m_e/\Lambda_\textrm{QCD}$. In the Standard Model electron and quark masses are proportional to the vacuum expectation value of the Higgs field. In the same vein, $g_p$ is not a fundamental constant, but we can express its variation in terms of variation of light quark mass using the relationship $\delta g_p/g_p \approx -0.1\,\delta X_q/X_q$~\cite{flambaum04prd} (accurate calculations for nuclear $g$-factors can be found in \cite{flambaum06prc}).
The evidence for a cosmological spatial gradient in $\alpha$ motivates us to interpret other data in terms of spatial variation. However, many of the limits on variations are best interpreted as limits on fundamental constants other than $\alpha$, since the system being examined is more sensitive to these other constants. A minimal hypothesis is to expect all fundamental constants to vary along the same direction as the $\alpha$-gradient. There are some good theoretical justifications for this postulation. If the constants vary because they are coupled to a scalar field $\Phi$ that varies over space-time, for example the quintessence field $\Phi/c^2$ or a dimensionless dilaton field, then a fundamental constant, $X$, will vary according to
\begin{equation}
\label{eq:delphi}
\frac{\delta X}{X} = k_X \delta\Phi
\end{equation}
where $k_X$ is a dimensionless coupling coefficient. Then all constants will vary in the direction of the gradient of the scalar field $\Delta\Phi$, which in turn must be directed along the axis found by the $\alpha$-variation data in~\cite{webb10arxiv}.
Equation~\eref{eq:delphi} implies that the relative variation of different constants can be related to each other using equations like
\begin{equation}
\label{eq:k_mu_alpha}
k_\mu = R^\alpha_\mu\, k_\alpha
\end{equation}
where the constants $R^{X'}_X$ can be determined from observations and compared with theories of spatial variation. For example, applying Grand Unification of the interactions of the Standard Model to the relative variation of $\alpha$ and $\alpha_s$ (see, e.g.,~\cite{flambaum09ijmpa}) we find
\[
\frac{\delta X_q}{X_q} \sim 35\, \frac{\delta\alpha}{\alpha}
\]
That is, the relative variation in $X_q$ may be much larger than the relative variation in $\alpha$. The coefficent here is model dependent, but large values are prevalent among models in which the variations come from high-energy scales. Therefore, variation of fundamental constants provides a unique probe of the predictions of Unification Theories (see, e.g.,~\cite{marciano84prl,langacker02plb,calmet02epjc,wetterich03jcap,dent03npb}).
In this review we discuss measurements of variation of fundamental constants in quasar absorption spectra and in atomic clocks. We interpret the astronomical measurements in the context of the observed spatial gradient in $\alpha$, relating other constants to this variation using expressions such as \eref{eq:k_mu_alpha}. Atomic clocks may prove to be the most reliable way to corroborate the detection of spatial $\alpha$-variation, and we discuss some of the proposed experiments where huge enhancements of the variation can be expected. There are many other systems which are not discussed here, including Big Bang Nucleosynthesis where hints of variation have been reported~\cite{dmitriev04prd,berengut10plb}. We refer the reader to recent reviews~\cite{flambaum09ijmpa,lea07rpp}.
\section{Quasar absorption spectra}
\label{sec:quasar}
We start our discussion with quasar absorption systems, since the remarkable recent detection of spatial variation in $\alpha$ from optical absorption spectra~\cite{webb10arxiv}, will provide us with a benchmark for all other studies.
\subsection{Optical atomic spectra}
\label{sec:quasaralpha}
It is natural to analyse fine-structure intervals in the search for variation of $\alpha$. Measurements of $\alpha$-variation by comparison of cosmic and laboratory optical spectra were first performed by Savedoff as early as 1956~\cite{savedoff56nat}. There were numerous works successfully implementing this ``alkali-doublet'' method (see review~\cite{uzan03rmp}).
In 1999, a different approach was developed: the many-multiplet method~\cite{dzuba99prl,dzuba99pra}. It exploits the fact that relativistic corrections to atomic transition frequencies can exceed the fine-structure interval between the excited levels by an order of magnitude (for example, an $s$-wave electron does not have the spin-orbit splitting but it has the maximal relativistic correction to energy). We can express the $\alpha$-dependence of a transition frequency $\omega$ as
\begin{equation}
\label{eq:q}
\omega = \omega_0 + q x
\end{equation}
where
\begin{equation}
x = (\alpha/\alpha_0)^2 - 1 \approx 2\,\frac{\alpha - \alpha_0}{\alpha_0} ,
\end{equation}
$\alpha_0$ is the laboratory value of $\alpha$ and $\omega_0$ is the laboratory frequency of a particular transition. The coefficients $q$ can vary strongly from atom to atom and can have opposite signs in different transitions (for example, in $s$--$p$ versus $d$--$p$ transitions). Thus, any variation of $\alpha$ may be revealed by
comparing different transitions in different atoms in cosmic and laboratory
spectra. A statistical gain is also realised because many more spectral lines in different elements can be used. This method improves the sensitivity to any variation of $\alpha$ by more than an order of magnitude compared to the alkali-doublet method.
Relativistic many-body calculations are used to reveal the dependence of atomic frequencies on $\alpha^2$ (the $q$ coefficients). We have performed accurate many-body calculations of the $q$ coefficient for all transitions of astrophysical interest~\cite{dzuba99pra,dzuba99prl,dzuba03pra0,dzuba05pra,dzuba02praA,berengut04praB, berengut05pra,berengut06pra,dzuba08pra}. These are strong E1 transtions from the ground state in Mg\,I, Mg\,II, Ti\,II, Fe\,I, Fe\,II, Cr\,II, Ni\,II, Al\,II, Al\,III, Si\,II, Zn\,II, Mn\,II, as well as many other atoms and ions which are seen in quasar absorption spectra, but have not yet been used in the quasar measurements because of the absence of accurate UV transition laboratory wavelengths. For a ``shopping list'' of needed measurements, see Ref.~\cite{berengut09mmsait}.
From the very first analyses of quasar data that utilised the new method, hints of $\alpha$-variation were reported~\cite{webb99prl,webb01prl}. The largest analysis, with three independent samples of data containing 143 absorption systems spread over redshift range \mbox{$0.2 < z < 4.2$}, suggested that $\alpha$ was smaller in the past: \mbox{$\delta \alpha/\alpha = (-0.543 \pm 0.116) \times 10^{-5}$}~\cite{murphy03mnras}. However, these studies all used spectra taken at the Keck telescope. Similar studies from another group, using our method and calculations but a much smaller sample of data taken at the VLT, at first showed a stringent null constraints~(Srianand~\etal~\cite{srianand04prl}). More careful analysis of this sample suggested that the errors were underestimated by a large factor~\cite{murphy07prl,murphy08mnras,srianand07prl}.
The latest results, combining the Keck data and a new sample of 153 measurements from the VLT, shows a spatial variation in $\alpha$. This gradient, which we will refer to as the ``Australian dipole'', has a declination of around $-60^\circ$. This explains why the Keck data, restricted mainly to the northern sky since the telescope is in Hawaii at a latitude of 20$^\circ$~N, originally suggested a time-varying $\alpha$ that was smaller in the past. The VLT is in Chile, at latitude $25^\circ$~S, giving the new study much more complete sky coverage. The new results are entirely consistent with previous ones. The ``Australian dipole'' of $\alpha$-variation found by \cite{webb10arxiv} is
\begin{equation}
\label{eq:ausdipole}
\frac{\delta\alpha}{\alpha_0} = (1.10\pm0.25)\E{-6}\, r \cos \psi
\end{equation}
where $\delta \alpha/\alpha_0 = (\alpha(\mathbf{r})-\alpha_0)/\alpha_0$ is the relative variation of $\alpha$ at a particular place $\mathbf{r}$ in the Universe (relative to Earth at $\mathbf{r}=0$). The function $r\cos\psi$ describes the geometry of the spatial variation: $\psi$ is the angle between the direction of the measurement and the axis of the Australian dipole, ($17.4\,(0.9)$~h, $-58\,(9)^\circ$) in equatorial coordinates. The distance function is the light-travel distance $r = ct$ measured in giga-lightyears. This is model dependent for large redshifts: the standard $\Lambda_\textrm{CDM}$ cosmology parametrized by WMAP5~\cite{hinshaw09apjss} is used to determine the light-travel time $t$. It is assumed here that $\delta\alpha/\alpha_0 = 0$ at zero redshift, which is supported by the data, however this assumption should be tested using the same absorption methods as are used at high redshift (e.g. by using absorbers within our own galaxy).
The new results of $\alpha$-variation are particularly striking in that if the VLT data and Keck data are considered separately, there is a high level of agreement between their dipole fits. In their paper~\cite{webb10arxiv}, the authors estimate the probability that the observed alignment between the independent Keck and VLT dipoles is due to chance at $\sim4\%$. Similarly, if one breaks the data into subsamples consisting of absorbers at $z<1.8$ and those at $z>1.8$ (approximately half the data in each subsample), then there is again agreement between the directions of the independent dipole axes.
In addition to the previously mentioned many-multiplet method results of Srianand~\etal from the small VLT sample~\cite{srianand04prl} (see also~\cite{chand04aap}), there are some high-accuracy single-absorption system $\alpha$-variation results using alkali-doublets (Si\,IV)~\cite{chand05aap} and single-ion (Fe\,II)~\cite{quast04aap,levshakov05aap,levshakov06aap,levshakov07aap} measurements. The alkali-doublet result suffers from the same problems as the small-sample many-multiplet result of~\cite{srianand04prl} (namely, sharp fluctuations in chi-squared vs. $\delta\alpha/\alpha$ graph which indicate failings in the chi-squared minimisation routine~\cite{murphy07prl}).
The Fe\,II results give stringent constraints in two absorbing systems: at $z=1.15$ towards HE0515--4414~\cite{levshakov06aap} and at $z=1.84$ towards Q1101--264~\cite{levshakov07aap}. It is interesting to compare the results of these analyses with the variation expected in these systems if the Australian dipole result is correct. Using \eref{eq:ausdipole}, we obtain an expected variation of $\delta\alpha/\alpha = (1.9\pm1.5)\E{-6}$ for the former, which was measured to be $(-0.07\pm0.84)\E{-6}$~\cite{levshakov06aap}. The absorber towards Q1101--264 has a larger expected variation of $\delta\alpha/\alpha = (3.8\pm2.0)\E{-6}$, and was measured to be $(5.4\pm2.5)\E{-6}$~\cite{levshakov07aap}. The single system results are therefore seen to be consistent with the Australian dipole.
\subsection{Molecular rotational quasar spectra}
Limits on variation of $\mu$ at high redshift can be obtained by comparison of different rotational-electronic excitations in molecular hydrogen. Taking advantage of newly available H$_2$ wavelengths, Ref.~\cite{reinhold06prl} reported a non-zero cosmological variation of $\mu$ in quasar spectra using molecular hydrogen transitions in the Ly-$\alpha$ forest. The authors obtained $\delta \mu/\mu = (2.4 \pm 0.6)\E{-5}$ at redshifts $z \approx 2.6 - 3.0$, i.e. a decrease of $\mu$ in the past 12 Gyr at the $3.5\sigma$ confidence level.
More recent determinations, using the laboratory wavelengths and sensitivity coefficients presented in \cite{reinhold06prl,ubachs07jms,salumbides08prl} and carefully controlling systematics and accounting for known calibration errors in the VLT, obtain a more stringent constraint: $\delta\mu/\mu = (3.4\pm2.7)\E{-6}$~\cite{king08prl,malec10mnras} from four quasar absorption systems.
In the context of an observed dipole in $\alpha$-variation~\cite{webb10arxiv}, it makes sense to check whether the (limited) data supports interpretation as a spatial variation. In Ref.~\cite{berengut10arxiv1} the data was shown to support a statistically significant spatial variation of $\mu$ aligned with the Australian dipole, although it was noted that the paucity of data prevents any firm conclusion.
\subsection{Comparison of hydrogen hyperfine and UV transitions}
A comparison of the 21-cm hyperfine transition in atomic hydrogen with UV metal lines was performed for 9 quasar spectra with redshifts $0.23 \le z \le 2.35$~\cite{tzanavaris05prl,tzanavaris07mnras}. The ratio of the transition frequencies is proportional to the parameter $x=\alpha^2 \mu g_p$, and this was constrained to
\begin{equation}
\label{x_var}
\delta x/x = (6.3\pm 9.9)\times 10^{-6}\ .
\end{equation}
It was found that there was much more scatter in the data than would be expected from the statistical errors alone~\cite{tzanavaris07mnras}. In principle this could mean that the model is wrong, e.g. a spatial variation should be considered rather than a time variation. However, even the best fit dipole model (which does not align with the Australian dipole) cannot explain the scatter~\cite{berengut10arxiv1}. It is probable that velocity offsets between the hydrogen and metal absorbers cause dominating systematics. Two new measurements comparing H\,I 21-cm with UV lines in neutral carbon in single absorption systems have reported slightly stronger constraints in $x$~\cite{kanekar10apjlett}; these are shown in the first two lines of \tref{tab:singles}.
\subsection{Comparisons involving hyperfine and molecular rotational transitions}
Individual measurements of fundamental constant variation in quasar absorption spectra can be compared with the observed spatial $\alpha$-variation~\cite{berengut10arxiv1}. That is, we can regard the $\alpha$-variation, according to \eref{eq:ausdipole} with $r$ and $\psi$ calculated for the object, as the ``expected'' variation and compare other measurements with it (this is presented in \tref{tab:singles}).
\begin{table}[t]
\caption{\label{tab:singles} Comparison of expected variation, given by \eref{eq:ausdipole}, and measured variation of fundamental constants in different quasar absorption systems.
$R^\alpha_{g_p}$ and $R^\alpha_\mu$ are defined by equations like \eref{eq:k_mu_alpha}. As noted in Section~\ref{sec:intro}, $R^\alpha_{g_p}\approx -0.1\,R^\alpha_{q}$ and $R^\alpha_\mu \approx R^\alpha_q$ where $R^\alpha_q$ is the variation of light quark mass $X_q$ relative to $\alpha$-variation.
Errors in the expected $\alpha$-variation (the prefactors in the third column) are of the order $\sim1.5\E{-6}$.
}
\begin{tabular}{llccc}
\hline
System & Constant & Expected variation & Measurement & Ref. \\
& &$(\times 10^{-6})$ & $(\times 10^{-6})$ \\
\hline
\mbox{H\,I} + \mbox{C\,I} & $\alpha^2 \mu g_p$
& $1.12\,(2 + R^\alpha_\mu + R^\alpha_{g_p})$
& $6.64 \pm 0.84_\textrm{stat} \pm 6.7_\textrm{sys}$ & \cite{kanekar10apjlett} \\
&
& $-5.20\,(2 + R^\alpha_\mu + R^\alpha_{g_p})$
& $7.0 \pm 1.8_\textrm{stat} \pm 6.7_\textrm{sys}$ & \cite{kanekar10apjlett} \\
\mbox{H\,I} + mol. rot. & $\alpha^2 g_p$
& $0.50\,(2 + R^\alpha_{g_p})$
& $-2.0 \pm 4.4$ & \cite{murphy01mnrasD} \\
&
& $-5.47\,(2 + R^\alpha_{g_p})$
& $-1.6 \pm 5.4$ & \cite{murphy01mnrasD} \\
\mbox{H\,I} + OH & $(\alpha^2/\mu)^{1.57} g_p$
& $-1.04\,(3.14 - 1.57 R^\alpha_\mu + R^\alpha_{g_p})$
& $4.4 \pm 3.6_\textrm{stat} \pm 10_\textrm{sys}$ & \cite{kanekar05prl} \\
OH & $(\alpha^2/\mu)^{1.85} g_p$
& $0.50\,(3.70 - 1.85 R^\alpha_\mu + R^\alpha_{g_p})$
& $-11.8 \pm 4.6$ & \cite{kanekar10apjlett0} \\
NH$_3$ & $\mu$
& $-5.47\,R^\alpha_\mu$
& $< 1.8\ (2\sigma)$ & \cite{murphy08sci} \\
&
& $1.34\,R^\alpha_\mu$
& $< 1.4\ (3\sigma)$ & \cite{henkel09aap} \\
\hline
\end{tabular}
\end{table}
The frequency of the hydrogenic hyperfine line is proportional to $\alpha^2\mu g_p$; molecular rotational frequencies are proportional to $\mu$. Comparison of the two placed limits on variation of the parameter $\alpha^2 g_p$~\cite{drinkwater98mnras} in two quasar absorption spectra. A similar analysis was repeated using more accurate data for the same objects~\cite{murphy01mnrasD}, at $z=0.247$ and at $z=0.6847$, resulting in the limits shown in lines 3 and 4 of \tref{tab:singles}, respectively. The object at $z=0.6847$ is associated with the gravitational lens toward quasar B0218+357 and corresponds to lookback time $\sim 6.2$ Gyr.
Comparison of OH 18-cm with H$\,$I 21-cm, and different conjugate-satellite OH 18-cm lines, can be used to measure the combinations of fundamental constants $(\alpha^2/\mu)^{1.57}g_p$ and $(\alpha^2/\mu)^{1.85}g_p$, respectively. Resulting measurements are shown in lines 5 and 6 of \tref{tab:singles}.
\subsection{Enhancement of variation of $\mu$ in the inversion spectrum of ammonia}
\label{sec:NH3}
In 2004, van Veldhoven \etal\ suggested using a decelerated molecular beam of ND$_3$ to search for the variation of $\mu$ in laboratory experiments\cite{van-veldhoven04epjd}. The ammonia molecule has a pyramidal shape and the inversion frequency depends on the exponentially small tunneling of three hydrogen (or deuterium) nuclei through the potential barrier. Because of that, it is very sensitive to any changes of the parameters of the system, and particularly to the reduced mass for this vibrational mode.
High precision data on the redshifts of NH$_3$ inversion lines exist for the previously mentioned object B0218+357 at $z=0.6847$~\cite{henkel05aap}. Comparing them with the redshifts of rotational lines of CO, HCO$^+$, and HCN molecules from Ref.~\cite{combes97apj} allows strong limits on variation of $\mu$, and hence $X_e$, to be obtained~\cite{flambaum07prl}. The most accurate measurements, utilising new rotational spectra, give very stringent limits on $\mu$-variation in this quasar absorption system and the object PKS1830--211 at $z=0.8858$ (last two lines of \tref{tab:singles}).
\section{Clocks}
\label{sec:clocks}
\subsection{Optical atomic clocks}
Atomic clocks can be used to measure time-variation of fundamental constants in the Earth frame. Different optical atomic clocks utilize transitions that have positive, negative or small contributions of the relativistic corrections to frequencies, so comparison of these clocks can be used to measure $\alpha$-variation. The same methods of relativistic many-body calculations used in the quasar absorption studies can be used to calculate the dependence on $\alpha$ of different clocks~\cite{dzuba99pra,dzuba00pra, dzuba03pra0,angstmann04pra,angstmann04pra0,dzuba08pra0}. A summary of results is presented in Ref.~\cite{flambaum09cjp}. The relativistic effects are proportional
to $(Z\alpha)^2$, therefore the $q$ coefficients for optical clock transitions may be substantially larger than in cosmic transitions since the clock transitions are often in heavy atoms (Hg\,II, Yb\,II, Yb\,III, etc.) while cosmic spectra contain mostly light atoms ($Z\lesssim 33$).
The temporal variation of $\alpha$ measured in the laboratory can be compared with the variation expected from the observed spatial gradient in $\alpha$~\cite{webb10arxiv} because the solar system moves along the axis of the dipole. The expected variation from the dipole was calculated to be~\cite{berengut10arxiv0}
\begin{equation}
\label{eq:clock_req}
\dot\alpha/\alpha = 1.35\E{-18} \cos\psi~\ensuremath{\textrm{yr}^{-1}}
\end{equation}
where $\psi$ is the angle between the motion of the Sun and the dipole. The best fit from~\cite{webb10arxiv} gives $\cos\psi \sim 0.07$ but this has an uncertainty of $\sim 0.15$. The $\alpha$-variation is modulated by the annual motion of the Earth around the Sun, $\delta\alpha/\alpha = 1.4\E{-20} \cos\omega t$~\cite{berengut10arxiv0} where $\omega$ refers to the angular frequency of the yearly orbit.
The current best constraint on time-variation of $\alpha$ was achieved by precisely measuring the frequency ratio of Hg\,II and Al\,II clocks several times over the course of a year~\cite{rosenband08sci}. Using our calculations, the rate of change of $\alpha$ is measured at \mbox{$\dot\alpha/\alpha = (-1.6\pm 2.3)\E{-17}~\ensuremath{\textrm{yr}^{-1}}$}. This limit will need to be improved by around two orders-of-magnitude in order to confirm or contradict the observed spatial gradient in $\alpha$. On the other hand, the quasar observations do not exclude time-variation of $\alpha$ below the rate of $\sim 10^{-16}~\ensuremath{\textrm{yr}^{-1}}$, and here the current laboratory limits are already competitive.
\subsection{Enhanced effect of $\alpha$-variation in Dy atom}
The sensitivity required by \eref{eq:clock_req} may be obtained by finding systems where $\alpha$-variation is strongly enhanced. Transitions between two almost degenerate levels in Dy atom can give a very high relative enhancement because these levels move in opposite directions if $\alpha$ varies~\cite{dzuba99prl,dzuba03pra0,dzuba08pra0}. The relative variation may be presented as $\delta \omega/\omega=K \delta \alpha /\alpha$ where the coefficient $K$ exceeds $10^8$ ($q=30\,000$ cm$^{-1}$, $\omega \sim 10^{-4}$ cm$^{-1}$). The values of $K=2 q/\omega$ are different for different hyperfine components and isotopes because $\omega$ changes. An experiment is currently underway to place limits on $\alpha$ variation using this transition~\cite{nguyen04pra,cingoz07prl}, however one of the levels has quite a large linewidth and this limits the accuracy. The current limit is $\dot{\alpha}/\alpha=(-2.7 \pm 2.6) \times 10^{-15}$~yr$^{-1}$. Several other enhanced effects of $\alpha$ variation in atoms have been calculated~\cite{dzuba05pra0,angstmann06jpb}.
\subsection{Enhanced effects of $\alpha$-variation in highly charged ions}
Sensitivity to $\alpha$-variation increases with ion charge as $(Z_i+1)^2$~\cite{berengut10prl}. The most sensitive atomic systems will maximize the contributions from three factors: high nuclear charge $Z$, high ionization degree, and significant differences in the configuration composition of the states involved. Unfortunately, the interval between different energy levels in an ion also increases as $\sim (Z_i+1)^2$, which can quickly take the transition frequency out of the range of lasers as $Z_i$ increases. The phenomena of Coulomb degeneracy and configuration crossing can be used to combat this tendency. In a neutral atom, an electron orbital with a larger angular momentum is significantly higher than one with smaller angular momentum but with the same principal quantum number $n$. On the other hand, in the hydrogen-like limit orbitals with different angular momentum but the same principal quantum number are nearly degenerate. Therefore, somewhere in between there can be a crossing point where two levels with different angular momentum and principal quantum number can come close together: in such cases the excitation energy may be within laser range.
In Ref.~\cite{berengut10prl} we showed, using the Ag isoelectronic sequence as an example, why high $q$-values can occur in highly charged ions, and how the tendency of such systems towards large transition frequencies could be overcome. A two-valence-electron ion, Sm$^{14+}$, was identified, which has optical transitions that are the most sensitive to potential variation of $\alpha$ ever found. While atomic spectroscopy in electron beam ion traps is currently not competitive with optical frequency standards (see, e.g.,~\cite{draganic03prl,crespo08cjp} and review~\cite{beiersdorfer09pscr}) the technology continues to improve, and with the enhancements in sensitivity, highly-charged ions may prove to be a good system for detecting variation of $\alpha$.
\subsection{Enhanced effect of variation in UV transition of $^{229}$Th nucleus}
\label{sec:Th}
The $^{229}$Th nucleus has the lowest known excited state, lying just $7.6\pm 0.5$ eV above the ground state~\cite{beck07prl}. The position of this level was determined from the energy differences of many high-energy $\gamma$-transitions to the ground and first-excited states. The subtraction produces the large uncertainty in the position of the 7.6 eV excited state. The width of this level is estimated to be about $10^{-4}$ Hz \cite{tkalya00prc}, which explains why it is so hard to find the direct radiation in this very weak transition. Nevertheless, the search for the direct radiation continues.
Because the $^{229}$Th transition is very narrow and can be investigated with laser spectroscopy, it is a possible reference for an optical clock of very high accuracy~\cite{peik03epl}. The near degeneracy of these isomers is a result of cancellation between very large energy contributions (order of MeV). Since these contributions would have different dependences on fundamental constants, this transition would be a very sensitive probe of possible variation of fundamental constants~\cite{flambaum06prl}. A rough estimate for the relative variation of the $^{229}$Th transition frequency is
\begin{equation}\label{deltaf}
\frac{\delta \omega}{\omega} \approx 10^5 \left(
0.1 \frac{\delta \alpha}{\alpha} + \frac{\delta X_q}{X_q }\right)\,.
\end{equation}
Therefore, the experiment would have the potential of improving the sensitivity to temporal variation of the fundamental constants by many orders of magnitude.
More accurate nuclear calculations give different values for the sensitivity of this transition to $\alpha$. Refs.~\cite{hayes07plb,hayes08prc} claim that both isomers have identical deformations and therefore there is no enhancement of $\alpha$-variation. Other calculations give enhancement factors in the range $10^2$ -- $10^5$, depending on particulars of the model used~\cite{he07jpg,he08jpg,flambaum09epl,litvinova09prc}. To resolve this, we have proposed a method of extracting sensitivity to $\alpha$-variation using direct laboratory measurements of the change in nuclear mean-square charge radius between the isomers (isomeric shift)~\cite{berengut09prl}.
From \eref{deltaf}, we obtain the following energy shift in the 7.6 eV $^{229}$Th transition:
\begin{equation}\label{delta3}
\delta \omega \approx
\frac{\delta X_q}{X_q}\ \textrm{MeV}\,.
\end{equation}
This corresponds to the frequency shift $\delta \nu \approx 3\cdot 10^{20}\,\delta X_q/X_q$ Hz. The width of this transition is $10^{-4}$ Hz so one may hope to get the sensitivity to the variation of $X_q$ about $10^{-24}$ per year. This is $10^{10}$ times better than the current atomic clock limit on the variation of $X_q$.
Note that there are other narrow low-energy levels in nuclei, for example the 76~eV level in $^{235}$U with lifetime 26.6 minutes is the second-lowest known. One may expect a similar enhancement there. Unfortunately, this level cannot be reached with usual lasers. In principle, it may be investigated using a free-electron laser or synchrotron radiation. However, the accuracy of the frequency measurements is much lower in this case.
\ack
This work is supported by the Australian Research Council, Marsden grant and ECT*.
\bibliographystyle{iopart-num}
\providecommand{\newblock}{}
|
{
"timestamp": "2010-09-21T02:02:09",
"yymm": "1009",
"arxiv_id": "1009.3693",
"language": "en",
"url": "https://arxiv.org/abs/1009.3693"
}
|
\section{Introduction}
\label{sec:intro}
For $A \in GL_n(\numbersys{R})$ and $y \in \numbersys{R}^n$, we define the dilation operator
on $L^2(\numbersys{R}^n)$ by
$\dila[A] f(x)=\abs{\ensuremath{\determinant}{A}}^{1/2}f(Ax)$ and the translation operator by
$\tran[y]f(x)=f(x-y)$. Given a $n\times n$ real, expansive
matrix $A$ and a lattice of the form $\ensuremath\lat{\Gamma}=P \numbersys{Z}^n$ for
$P \in GL_n(\numbersys{R})$, we consider wavelet systems of the form
\[ \set{\dila \tran \psi}_{j \in \numbersys{Z}, \gamma \in \ensuremath\lat{\Gamma}}, \]
where the Fourier transform of $\psi$ has compact support.
Our aim is, for any given real, expansive dilation matrix $A$, to construct
wavelet frames with good regularity properties and with a dual frame
generator of the form
\begin{equation}
\label{eq:26}
\phi = \sum_{j = a}^b c_j \dila \psi
\end{equation}
for some explicitly given coefficients $c_j \in \numbersys{C}$ and $a,b \in \numbersys{Z}$.
This will generalize and extend the one-dimensional results on
constructions of dual wavelet frames in \cite{lemvig_constr_paris,MR2311859} to
higher dimensions. The extension is non-trivial since it is unclear
how to determine the translation lattice $\ensuremath\lat{\Gamma}$ and how to
control the support of the generators in the Fourier domain. This will
be done by considering suitable norms in $\numbersys{R}^n$ and non-overlapping
packing of ellipsoids in lattice arrangements.
The construction of redundant wavelet representations in higher
dimensions is usually based on extension principles
\cite{ehler_han,id_bh_ar_zs_03,MR1968120,MR1992289,MR2000c:42037,MR98c:42035,MR2355010,MR2274841,MR1848710}.
By making use of extension principles one is restricted to considering
expansive dilations $A$ with integer coefficients. Our constructions
work for any real, expansive dilation. Moreover, in the extension
principle the number of generators often increases with the smoothness
of the generators. We will construct pairs of dual wavelet frames
generated by one smooth function with good time localization.
It is a well-known fact that a wavelet frame need not have dual frames
with wavelet structure. In \cite{MR2286929} frame wavelets with
compact support and explicit analytic form are constructed for real
dilation matrices. However, no dual frames are presented for these
wavelet frames. This can potentially be a problem because it might be
difficult or even impossible to find a dual frame with wavelet
structure. Since we exhibit \emph{pairs} of dual wavelet frames, this issue
is avoided.
The principal importance of having a dual
generator of the form (\ref{eq:26}) is that it will inherit properties
from $\psi$ preserved by dilation and linearity, \textit{e.g.\@\xspace} vanishing
moments, good time localization and regularity properties. For a more
complete account of such matters we refer to
\cite{lemvig_constr_paris}.
In the rest of this introduction we review basic definitions. A frame for a separable
Hilbert space $\mathcal{H}$ is a countable collection of vectors $\{f_j\}_{j \in
\mathbb{J}}$ for which there are
constants $0 < C_1 \leq C_2 < \infty$ such that
\[
C_1 \norm{f}^2 \leq \sum_{j \in \mathbb{J}} \abs{\innerprod{f}{f_j}}^2 \leq
C_2 \norm{f}^2 \qquad\text{for all }f\in \mathcal{H}.
\]
If the upper bound holds in the above inequality, then $\{f_j\}$ is
said to be a Bessel sequence with Bessel constant $C_2$.
For a Bessel sequence $\{f_j\}$ we define the frame operator by
\[
\Frame\colon \mathcal{H} \to \mathcal{H}, \qquad Sf = \sum_{j \in
\mathbb{J}} \innerprod{f}{f_j} f_j.
\]
This operator is bounded, invertible, and positive. A frame $\{f_j\}$
is said to be \emph{tight} if we can choose $C_1 = C_2$; this is
equivalent to $\Frame = C_1 I$ where $I$ is the identity operator.
Two Bessel sequences $\{f_j\}$ and $\{g_j\}$ are said to be
\emph{dual} frames if
\[ f = \sum_{j \in \mathbb{J}} \innerprod{f}{g_j}f_j \quad \forall f
\in \mathcal{H}. \] It can be shown that two such Bessel sequences are
indeed frames. Given a frame $\{f_j\}$, at least one dual always
exists; it is called the canonical dual and is given by $\{S^{-1} f_j
\}$. Only a frame, which is not a basis, has several duals.
For $f \in L^1(\numbersys{R}^n)$ the Fourier transform
is defined by $\hat f(\xi) = \int_{\numbersys{R}^n} f(x)\expo{-2 \pi i
\einnerprod{\xi}{x}} \mathrm{d}x$ with
the usual extension to $L^2(\numbersys{R}^n)$.
Sets in $\numbersys{R}^n$ are, in general, considered equal if they are equal
up to sets of measure zero.
The boundary of a set $E$ is denoted by $\partial E$, the interior by
$E^\circ$, and the closure by $\overline{E}$. Let $B\in GL_n(\numbersys{R})$. A
\emph{multiplicative tiling set} $E$ for $\setpropsmall{B^j}{j\in \numbersys{Z}}$
is a subset of positive measure such that
\begin{gather}
\absBig{\numbersys{R}^n \setminus \bigcup_{j \in \numbersys{Z}} B^j(E)} = 0 \quad \text{and} \quad
\abs{B^j(E) \cap B^l(E)} =0 \quad \text{for $l\neq j$.}
\end{gather}
In this case we say that $\setprop{B^j(E)}{j \in \numbersys{Z}}$ is an almost
everywhere partition of $\numbersys{R}^n$, or that it tiles $\numbersys{R}^n$. A multiplicative
tiling set $E$ is \emph{bounded} if $E$ is a bounded set and $0 \notin
\overline{E}$. By $B$-dilative periodicity of a function $f \colon \numbersys{R}^n
\to \numbersys{C}$ we understand $f(x)=f(Bx)$ for a.e.} %\@\xspace}\ $x \in \numbersys{R}^n$, and by a
$B$-dilative partition of unity we understand $\sum_{j \in \numbersys{Z}} f(B^j
x) =1$; note that the functions in the ``partition of unity'' are not
assumed to be non-negative, but can take any real or complex value.
A (full-rank) lattice $\ensuremath\lat{\Gamma}$ in $\numbersys{R}^n$ is a point set of the form $\ensuremath\lat{\Gamma}=P \numbersys{Z}^n$ for
some $P \in GL_n(\numbersys{R})$. The determinant of $\ensuremath\lat{\Gamma}$ is
$d(\ensuremath\lat{\Gamma})=\abs{\ensuremath{\determinant}{P}}$; note that the generating matrix $P$ is not
unique, and that $d(\ensuremath\lat{\Gamma})$ is independent of the particular choice of $P$.
\section{The general form of the construction procedure}
\label{sec:general-form-Rn}
Fix the dimension $n \in \numbersys{N}$. We let $A\in GL_n(\numbersys{R})$ be
expansive, \textit{i.e.,\xspace} all eigenvalues of $A$ have absolute
value greater than one, and denote the transpose matrix by $B = A^t$.
For any such
dilation $A$, we want to construct a pair of
functions that generate dual wavelet frames for some translation
lattice.
Our construction is based on the following result which is a
consequence of the characterizing equations for dual wavelet frames by
Chui, Czaja, Maggioni, and Weiss \cite[Theorem 4]{MR1891728}.
\begin{theorem} \label{thm:dual-charac-Rn}
Let $A \in GL_n(\numbersys{R})$ be expansive, let $\ensuremath\lat{\Gamma}$ be a lattice in $\numbersys{R}^n$, and
let $\Psi=\{\psi_1, \dots, \psi_L\}$, $\tilde \Psi=\{\tilde\psi_1, \dots,
\tilde\psi_L\} \subset L^2(\numbersys{R}^n)$. Suppose that the two wavelet systems
$\setpropsmall{\dila \tran \psi_l}{j\in \numbersys{Z}, \gamma \in \ensuremath\lat{\Gamma}, l = 1, \dots, L}$
and $ \setpropsmall{\dila \tran \tilde{\psi}_l }{j\in \numbersys{Z}, \gamma\in \ensuremath\lat{\Gamma}, l =
1, \dots, L}$ form Bessel families. Then $\setsmall{\dila \tran
\psi_l}$ and $\setsmall{\dila \tran \tilde\psi_l}$ will be dual
frames if the following conditions hold
\begin{align}
\label{eq:diagonalterm-Rn}
&\sum_{l=1}^L \sum_{j\in \numbersys{Z}} \hat{\tilde{\psi_l}}(B^j \xi)
\overline{\hat{\psi_l}(B^j \xi)} = d(\ensuremath\lat{\Gamma}) &\quad &\text{a.e.} %\@\xspace}\ } \xi
\in
\numbersys{R}^n, \\
\label{eq:nondiagonalterm-Rn}
&\sum_{l=1}^L \hat{\tilde{\psi_l}}(\xi) \overline{\hat
\psi_l(\xi+\gamma)} = 0 &\quad &\text{a.e.} %\@\xspace}\ } \xi \in
\numbersys{R}^n \text{ for } \gamma \in \ensuremath\lat{\Gamma}^\ast \setminus \{0\}.
\end{align}
\end{theorem}
\begin{proof}
By $\xi = B^j \omega$ for $j \in \numbersys{Z}$, condition
(\ref{eq:nondiagonalterm-Rn}) becomes
\begin{align}\label{eq:1}
\sum_{l=1}^L \hat{\tilde{\psi_l}}(B^j \omega) \overline{\hat
\psi_l(B^j \omega+\gamma)} = 0 \quad \text{a.e.} %\@\xspace}\ } \omega \in
\numbersys{R}^n \text{ for } \gamma \in \ensuremath\lat{\Gamma}^\ast \setminus \{0\}.
\end{align}
We use the notation as in \cite{MR1891728}, thus
$\Lambda (A, \ensuremath\lat{\Gamma}) = \setpropsmall{\alpha \in \numbersys{R}^n}{ \exists (j, \gamma) \in \numbersys{Z} \times
\ensuremath\lat{\Gamma}^\ast : \alpha = B^{-j}\gamma}$ and
$I_{A,\ensuremath\lat{\Gamma}}(\alpha) = \setpropsmall{(j, \gamma) \in \numbersys{Z} \times \ensuremath\lat{\Gamma}^\ast }{\alpha = B^{-j}\gamma}$.
Since $I_{A,\ensuremath\lat{\Gamma}}(\alpha) \subset \numbersys{Z} \times (\ensuremath\lat{\Gamma}^\ast \setminus \{0\})$ for any
$\alpha \in \Lambda(A,\ensuremath\lat{\Gamma}) \setminus \{0\}$, equation~(\ref{eq:1}) yields
\begin{align*}
\frac{1}{d(\ensuremath\lat{\Gamma})} \sum_{(j,\gamma)\in I_{A,\ensuremath\lat{\Gamma}}(\alpha)} \sum_{l=1}^L
\hat{\tilde{\psi_l}}(B^j \omega) \overline{\hat
\psi_l(B^j( \omega+B^{-j}\gamma))} = 0 \quad \text{a.e.} %\@\xspace}\ } \omega \in
\numbersys{R}^n
\end{align*}
for $\alpha \neq 0$. By $I_{A,\ensuremath\lat{\Gamma}}(0)= \numbersys{Z} \times \{0\}$, we can
rewrite \eqref{eq:diagonalterm-Rn} as
\begin{align*}
\frac{1}{d(\ensuremath\lat{\Gamma})} \sum_{(j,\gamma)\in I_{A,\ensuremath\lat{\Gamma}}(0)} \sum_{l=1}^L
\hat{\tilde{\psi_l}}(B^j \omega) \overline{\hat
\psi_l(B^j(\omega+B^{-j}\gamma))} = 1 \quad \text{a.e.} %\@\xspace}\ } \omega \in
\numbersys{R}^n,
\end{align*}
using that $B^{-j}\gamma=0$ for all $j\in \numbersys{Z}$.
Gathering the two equations displayed above yields
\begin{align*}
\frac{1}{d(\ensuremath\lat{\Gamma})} \sum_{(j,\gamma)\in I_{A,\ensuremath\lat{\Gamma}}(\alpha)} \sum_{l=1}^L
\hat{\tilde{\psi_l}}(B^j \omega) \overline{\hat
\psi_l(B^j(\omega+B^{-j}\gamma))} = \delta_{\alpha,0} \quad \text{a.e.} %\@\xspace}\ } \omega \in
\numbersys{R}^n,
\end{align*}
for all $\alpha \in \Lambda(A,\ensuremath\lat{\Gamma})$. The conclusion follows now from
\cite[Theorem 4]{MR1891728}.
\end{proof}
The following result, Lemma~\ref{thm:bessel2-Rn}, gives a sufficient
condition for a wavelet system to form a Bessel sequence; it is an
extension of \cite[Theorem 11.2.3]{oc_03} from $L^2(\numbersys{R})$ to
$L^2(\numbersys{R}^n)$.
\begin{lemma} \label{thm:bessel2-Rn} Let $A \in GL_n(\numbersys{R})$ be
expansive, $\ensuremath\lat{\Gamma}$ a lattice in $\numbersys{R}^n$, and $\phi \in L^2(\numbersys{R}^n)$.
Suppose that, for some set $M \subset \numbersys{R}^n$ satisfying $\cup_{l \in \numbersys{Z}} B^{l}(M) =
\numbersys{R}^n$,
\begin{align}\label{eq:14}
C_2 = \frac{1}{d(\ensuremath\lat{\Gamma})} \sup_{\xi \in M} \sum_{j \in \numbersys{Z}}
\sum_{\gamma \in \ensuremath\lat{\Gamma}^\ast}
\abs{\hat \phi(B^j \xi) \hat \phi(B^j \xi + \gamma)} < \infty{}.
\end{align}
Then the wavelet system $\{\dila \tran \phi\}_{j \in \numbersys{Z}, \gamma \in
\ensuremath\lat{\Gamma}}$ is a Bessel sequence with bound $C_2$. Further, if also
\begin{align}\label{eq:lemma-suff-lower}
C_1 = \frac{1}{d(\ensuremath\lat{\Gamma})} \inf_{\xi \in M} \left(\sum_{j \in \numbersys{Z}}
\abs{\hat \phi(B^j \xi)}^2 - \sum_{j \in \numbersys{Z}}
\sum_{\gamma \in \ensuremath\lat{\Gamma}^\ast \setminus \{0\}}
\abs{\hat \phi(B^j \xi) \hat \phi(B^j \xi + \gamma)} \right) > 0 ,
\end{align}
holds, then $\{\dila \tran \phi\}_{j \in \numbersys{Z}, \gamma \in \ensuremath\lat{\Gamma}}$ is
a frame for $L^2(\numbersys{R}^n)$ with frame bounds $C_1$ and $C_2$.
\end{lemma}
\begin{proof}
The statement follows directly by applying Theorem 3.1 in \cite{chris_rahimi} on
generalized shift invariant systems to wavelet systems.
In the general result for generalized
shift invariant systems \cite[Theorem 3.1]{chris_rahimi}, the supremum/infimum
is taken over $\numbersys{R}^n$, but because of the $B$-dilative periodicity of
the series in (\ref{eq:14}) and (\ref{eq:lemma-suff-lower}) for wavelet systems, it
suffices to take the supremum/infimum over a set $M \subset \numbersys{R}^n$ that has the
property that $\cup_{l \in \numbersys{Z}} B^{l}(M) = \numbersys{R}^n$ up to sets of measure
zero.
\end{proof}
Theorem~\ref{thm:dual-charac-Rn} and Lemma~\ref{thm:bessel2-Rn} are all we
need to prove the following result on pairs of dual wavelet frames.
\begin{theorem}\label{thm:constr-dual-wavelet-Rn-general}
Let $A \in GL_n(\numbersys{R})$ be expansive and $\psi \in L^2(\numbersys{R}^n)$.
Suppose that $\hat \psi$ is a bounded, real-valued function with $\supp \hat
\psi \subset \cup_{j=0}^d B^{-j}(E) $ for some $d \in \numbersys{N}_0$ and
some bounded multiplicative tiling set $E$ for $\setprop{B^j}{j \in
\numbersys{Z}}$, and that
\begin{equation}
\label{eq:dyadic-part2-Rn}
\sum_{j \in \mathbb{Z}} \hat \psi (B^j \xi) =1
\quad \text{for a.e.} %\@\xspace}\ }\xi \in \numbersys{R}^n.
\end{equation}
Let $b_j \in \numbersys{C}$ for $j=-d, \dots, d$ and let $\overline{m} =
\max\setprop{j}{b_j \neq 0}$ and $\underline{m} = -
\min\setprop{j}{b_j \neq 0}$. Take a lattice $\ensuremath\lat{\Gamma}$ in $\numbersys{R}^n$ such that
\begin{equation}
\label{eq:13}
\Bigl(\bigcup_{j=0}^d B^{-j}(E)+\gamma\Bigr) \cap \bigcup_{j=-\underline{m}}^{\overline{m}+d}B^{-j}(E) = \emptyset \quad
\text{for all $\gamma \in \ensuremath\lat{\Gamma}^\ast \setminus \{0\}$,}
\end{equation}
and define the function $\phi$ by
\begin{align}
\label{eq:dual-generator1-Rn}
\phi(x) = d(\ensuremath\lat{\Gamma}) \sum_{j=-\underline{m}}^{\overline{m}} b_j \abs{\ensuremath{\determinant}{A}}^{-j} \psi(A^{-j}x) \quad
\text{for } x \in \numbersys{R}^n.
\end{align}
If $b_0 = 1$ and $b_j + b_{-j} = 2$ for $j=1,2, \dots, d$, then
the functions $\psi$ and $\phi$ generate dual frames $\setsmall{\dila \tran \psi}_{j
\in \numbersys{Z}, \gamma \in \ensuremath\lat{\Gamma}}$ and $\setsmall{\dila \tran \phi}_{j \in \numbersys{Z}, \gamma
\in \ensuremath\lat{\Gamma}}$ for $L^2(\numbersys{R}^n)$.
\end{theorem}
\begin{proof}
On the Fourier side, the definition in (\ref{eq:dual-generator1-Rn}) becomes
\begin{align*}
\hat \phi(\xi) = d(\ensuremath\lat{\Gamma}) \sum_{j=-\underline{m}}^{\overline{m}} b_j \hat \psi(B^{j}\xi).
\end{align*}
Since $\hat \psi$ by assumption is compactly
supported in a ``ringlike'' structure bounded away from the origin,
this will also be the case for $\hat \phi$. This property
implies that $\psi$ and $\phi$ will generate wavelet Bessel sequences.
The details are as follows. The support of $\hat \psi$ and $\hat \phi$
is
\begin{align}\label{eq:supp-generators-general}
\supp \hat \psi \subset \bigcup_{j=0}^d B^{-j}(E),
&&\supp \hat \phi \subset
\bigcup_{j=-\underline{m}}^{\overline{m}+d}B^{-j}(E).
\end{align}
Note that $0 \le \underline{m}, \overline{m} \le
d$.
The sets $\setprop{B^j(E)}{j \in \numbersys{Z}}$ tiles $\numbersys{R}^n$, whereby we see that
\begin{align}\label{eq:18}
\abs{\,\supp \hat \psi(B^j \cdot) \cap B^{-d}(E) }&= 0 \quad \text{for $j <0$
and $j > d$,} \\
\intertext{and,}
\abs{\,\supp \hat \phi(B^j \cdot) \cap B^{-d}(E) }&= 0 \quad \text{for $j
<-\underline{m}$ and $j > \overline{m}+d$.} \label{eq:28}
\end{align}
Since $\underline{m}, \overline{m} \ge 0$, condition (\ref{eq:13})
implies that $\hat \psi(B^j \xi) \hat \psi(B^j \xi + \gamma) = 0$ for
$j \ge 0$ and $\gamma \in \ensuremath\lat{\Gamma}^\ast \setminus \{0\}$. Therefore, using
\eqref{eq:18}, we find that
\begin{align*}
\sum_{j\in\numbersys{Z}} \sum_{\gamma \in \ensuremath\lat{\Gamma}^\ast } \abs{ \hat \psi(B^j \xi)
\hat \psi(B^j \xi + \gamma)}
= \sum_{j=0}^{d} \left( \hat \psi(B^j \xi) \right)^2 < \infty
\qquad \text{for $\xi \in B^{-d}(E)$.}
\end{align*}
An application of Lemma~\ref{thm:bessel2-Rn} with $M=B^{-d}(E)$ shows that
$\psi$ generates a Bessel sequence. Similar calculations using
\eqref{eq:28} will show that $\phi$ generates a Bessel sequence; in
this case the sum over $\gamma \in \ensuremath\lat{\Gamma}^\ast$ will be finite, but it will
in general have more than one nonzero term.
To conclude that $\psi$ and $\phi$ generate dual wavelet frames we
will show that conditions~\eqref{eq:diagonalterm-Rn} and
\eqref{eq:nondiagonalterm-Rn} in Theorem~\ref{thm:dual-charac-Rn}
hold. By $B$-dilation periodicity of the sum in condition
\eqref{eq:diagonalterm-Rn}, it is sufficient to verify this condition
on $B^{-d}(E)$.
For $\xi \in B^{-d}(E) $ we have by \eqref{eq:18},
\begin{align*}
\frac{1}{d(\ensuremath\lat{\Gamma})} \sum_{j \in \numbersys{Z}} \overline{\hat \psi (B^j \xi)} \hat \phi (B^j \xi)
& = \frac{1}{d(\ensuremath\lat{\Gamma})} \sum_{j =0}^{d} \hat \psi (B^j \xi) \hat \phi (B^j \xi) \\ &=
\hat \psi (\xi) \left[ b_0 \hat \psi (\xi) + b_{1} \hat \psi(B \xi)
+ \dots + b_{d} \hat
\psi(B^{d}\xi) \right] \\
&\phantom{= }\;+ \hat \psi(B \xi) \left[ b_{-1} \hat \psi( \xi) +
b_0 \hat \psi (B \xi) +
\dots + b_{d-1}\hat \psi(B^{d}\xi) \right] + \cdots \\
&\phantom{= }\;+ \hat \psi(B^{d}\xi) \left[ b_{-d}\hat
\psi(\xi) + \dots + b_{-1} \hat \psi(B^{d-1}\xi) + b_0 \hat
\psi(B^{d}\xi) \right],
\intertext{and further, by an expansion of these terms, }
&= \sum_{ j,l = 0}^{d} b_{l-j} \hat\psi(B^j \xi) \hat\psi(B^l \xi)\\
&= b_0 \sum_{j=0}^{d} \hat\psi(B^j \xi)^2 + \sum_{ \substack{j,l = 0 \\ j >
l}}^{d} (b_{j-l}+b_{l-j}) \hat\psi(B^j \xi) \hat\psi(B^l \xi) .
\end{align*}
Using that $b_0 = 1$ and $b_{j-l}+b_{l-j}= 2$ for $j\neq l$ and $j,l
=0,\dots, d$, we arrive at
\begin{align*}
\frac{1}{d(\ensuremath\lat{\Gamma})} \sum_{j \in \numbersys{Z}} \overline{\hat \psi (B^j \xi)} \hat
\phi (B^j \xi) &= \sum_{j=0}^{d} \hat\psi(B^j \xi)^2 + \sum_{
\substack{j,l = 0 \\ j >
l}}^{d} 2\hat\psi(B^j \xi) \hat\psi(B^l \xi) \\
&= \biggl(\sum_{j =0}^{d} \hat \psi (B^j \xi) \biggr)^2 =
\biggl(\sum_{j \in \numbersys{Z}} \hat \psi (B^j \xi) \biggr)^2 = 1,
\end{align*}
exhibiting that $\psi$ and $\phi$ satisfy condition \eqref{eq:diagonalterm-Rn}.
By \eqref{eq:supp-generators-general} we see that condition
\eqref{eq:13} implies that the functions $\hat
\phi$ and $\hat \psi (\cdot + \gamma)$ will have disjoint support
for $\gamma \in \ensuremath\lat{\Gamma}^\ast \setminus \{0 \}$,
hence (\ref{eq:nondiagonalterm-Rn}) is satisfied.
\end{proof}
\begin{remark}
The use of the parameters $b_j$ in the definition of the dual
generator together with the condition $b_{-j}+b_j=2$ was first seen in
the work of Christensen and Kim~\cite{OC_RYK_dual-Gabor-poly} on
pairs of dual Gabor frames.
\end{remark}
We can restate
Theorem~\ref{thm:constr-dual-wavelet-Rn-general} for wavelet
systems with standard translation lattice $\numbersys{Z}^n$ and dilation
$\widetilde A =P^{-1}AP$, where $P \in GL_n(\numbersys{R})$ is so that $\ensuremath\lat{\Gamma} = P
\numbersys{Z}^n$. The result follows directly by an application of the relations
$\dila[\widetilde{A}^j] \dila[P] = \dila[P] \dila$ for $j \in \numbersys{Z}$
and $\dila[P]\tran[Pk]=\tran[k] \dila[P]$ for $k \in \numbersys{Z}^n$, and the
fact that $\dila[P]$ is unitary as an operator on $L^2(\numbersys{R}^n)$.
\begin{corollary}
Suppose $\psi$, $\{b_j\}$, $A$ and $\ensuremath\lat{\Gamma}$ are as in
Theorem~\ref{thm:constr-dual-wavelet-Rn-general}. Let $P \in
GL_n(\numbersys{R})$ be such that $\ensuremath\lat{\Gamma} = P \numbersys{Z}^n$, and let $\widetilde A
=P^{-1}AP$. Then the functions $\tilde \psi = \dila[P]{\psi}$ and
$\tilde \phi = \dila[P] \phi$, where $\phi$ is defined in
(\ref{eq:dual-generator1-Rn}), generate dual frames
$\setsmall{\dila[\widetilde A^j] \tran[k] \tilde \psi}_{j \in \numbersys{Z}, k
\in \numbersys{Z}^n}$ and $\setsmall{\dila[\widetilde{A}^j] \tran[k] \tilde
\phi}_{j \in \numbersys{Z}, k \in \numbersys{Z}^n}$ for $L^2(\numbersys{R}^n)$.
\end{corollary}
The following Example~\ref{ex:adhoc-construc-Rn} is an application of
Theorem~\ref{thm:constr-dual-wavelet-Rn-general} in $L^2(\numbersys{R}^2)$ for
the quincunx matrix. In particular, we construct a partition of unity
of the form (\ref{eq:dyadic-part2-Rn}) for the quincunx matrix.
\begin{example}\label{ex:adhoc-construc-Rn}
The quincunx matrix is defined as
\begin{equation*}
A =
\begin{pmatrix}
1 & -1 \\ 1 & 1
\end{pmatrix},
\end{equation*}
and its action on $\numbersys{R}^2$ corresponds to a counter clockwise rotation
of 45 degrees and a dilation by $\sqrt{2} I_{2 \times 2}$.
\begin{figure}[htbp]
\psset{xunit=5cm,yunit=5cm}%
\begin{pspicture}(-.13,-.13)(1.2,1.2
\psaxes[Dx=0.25, Dy=0.25,tickstyle=bottom]{->}(0,0)(-.13,-0.13)(1.2,1.2)
\pspolygon(1,0)(1,1)(0.5,0.5)
\pspolygon(0,1)(1,1)(0.5,0.5)
\pspolygon(0,1)(0,0.5)(0.5,0.5)
\pspolygon(0.5,0)(0,0.5)(0.5,0.5)
\pspolygon(0.5,0)(1,0)(0.5,0.5)
\rput[c](0.5, 0.75){$J_5$}
\rput[c]( 0.75, 0.5){$J_4$}
\rput[c]( 0.625, 0.175){$J_2$}
\rput[c]( 0.175, 0.625){$J_3$}
\rput[c]( 0.35, 0.35){$J_1$}
\uput[-90](1.17,-2.4pt){$x_1$}
\uput[145](-2.4pt,1.17){$x_2$}
\end{pspicture}\centering
\caption{Sketch of the triangular domains $J_i$, $i = 1,2,3,4,5$.}
\label{fig:quincunx-support-sets}
\end{figure}
Define the tent shaped, piecewise linear function $g$ by
\begin{equation*}
g(x_1,x_2)=\begin{cases}
-1+2x_1 + 2x_2, \quad &\text{for $(x_1,x_2) \in J_1,$}\\
2x_2, \quad &\text{for $(x_1,x_2) \in J_2,$}\\
2x_1, \quad &\text{for $(x_1,x_2) \in J_3,$}\\
2-2x_1, \quad &\text{for $(x_1,x_2) \in J_4,$}\\
2-2x_2, \quad &\text{for $(x_1,x_2) \in J_5,$}\\
0 & \text{otherwise,}
\end{cases}
\end{equation*}
where the sets $J_i$ are the triangular domains sketched in
Figure~\ref{fig:quincunx-support-sets}.
Note that the value at ``the top of the tent'' is $g(1/2,1/2)=1$.
Define $\hat\psi $ as a mirroring of $g$ in the $x_1$ axis and the $x_2$ axis:
\begin{equation*}
\hat \psi(\xi_1, \xi_2) =
\begin{cases}
g(\xi_1,\xi_2)\quad &\text{for $(\xi_1,\xi_2) \in \itvco{0}{\infty}
\times \itvco{0}{\infty},$}\\
g(\xi_1,-\xi_2)\quad &\text{for $(\xi_1,\xi_2) \in \itvco{0}{\infty}
\times \itvoo{-\infty}{0},$}\\
g(-\xi_1,\xi_2)\quad &\text{for $(\xi_1,\xi_2) \in \itvoo{-\infty}{0}
\times \itvco{0}{\infty},$}\\
g(-\xi_1,-\xi_2)\quad &\text{for $(\xi_1,\xi_2) \in \itvoo{-\infty}{0}
\times \itvoo{-\infty}{0}.$}
\end{cases}
\end{equation*}
Since the transpose $B$ of the quincunx matrix also corresponds to a
rotation of 45 degrees (but clockwise) and a dilation by $\sqrt{2} I_{2 \times 2}$, we
see that $ \sum_{j \in \mathbb{Z}} \hat \psi (B^j \xi) =1 $.
We are now ready to apply
Theorem~\ref{thm:constr-dual-wavelet-Rn-general} with $E =
\itvcc{-1}{1}^2 \setminus B^{-1}(\itvcc{-1}{1}^2) = \itvcc{-1}{1}^2
\setminus I_1$ and $d=2$; the set $E$ is the union of the domians
$J_4$ and $J_5$ and their mirrored versions. We
choose $b_{-2}=b_{-1}=0$ and $b_1=b_2=2 d(\ensuremath\lat{\Gamma})$, hence
$\underline{m}=0$ and $\overline{m}=2$. Therefore,
\[ \bigcup_{j=0}^d B^{-j}(E),
\bigcup_{j=-\underline{m}}^{\overline{m}+d}B^{-j}(E) \subset
\itvcc{-1}{1}^2,\]
that shows that we can take
$\ensuremath\lat{\Gamma}^\ast = 2\numbersys{Z}^2$ or $\ensuremath\lat{\Gamma} = 1/2 \numbersys{Z}^2$, since
$ (\itvcc{-1}{1}^2+
\gamma) \cap \itvcc{-1}{1}^2 = \emptyset$ whenever $0\neq \gamma \in
2\numbersys{Z}^2$.
Defining the dual generator according to \eqref{eq:dual-generator1}
yields
\begin{equation}
\label{eq:11}
\phi(x) = (1/4) \psi(x) + (1/4) \psi(A^{-1}x) + (1/8) \psi(A^{-2}x);
\end{equation}
using that $d(\ensuremath\lat{\Gamma})=1/4$, and we remark that $\hat \phi$ is a piecewise
linear function since this is the case for $\hat\psi$. The conclusion
from Theorem~\ref{thm:constr-dual-wavelet-Rn-general} is that $\psi$
and $\phi$ generate dual frames $\setsmall{\dila \tran[k/2] \psi}_{j,k
\in \numbersys{Z}}$ and $\setsmall{\dila \tran[k/2] \phi}_{j,k \in \numbersys{Z}}$ for
$L^2(\numbersys{R}^2)$.
The frame bounds can be found using Lemma~\ref{thm:bessel2-Rn} since the
series~(\ref{eq:14}) and (\ref{eq:lemma-suff-lower}) are finite sums on
$E$; for $\setsmall{\dila \tran[k/2] \psi}$ one finds $C_1=4/3$
and $C_2 = 4$.
\end{example}
When the result on constructing pairs of dual wavelet frames is
written in the generality of
Theorem~\ref{thm:constr-dual-wavelet-Rn-general}, it is not always
clear how to choose the set $E$ and the lattice $\ensuremath\lat{\Gamma}$. In
Example~\ref{ex:adhoc-construc-Rn} we showed how this can be done for
the quincunx dilation matrix and constructed a pair of dual frame
wavelets. In Section~\ref{sec:spec-case-Rn} and
Theorem~\ref{thm:constr-dual-wavelet-Rn-special} we specify how to
choose $E$ and $\ensuremath\lat{\Gamma}$ for general dilations. The issue of exhibiting
functions $\psi$ satisfying the condition (\ref{eq:dyadic-part2-Rn})
is addressed in Section~\ref{sec:examples-rn}.
In one dimension, however, it is straightforward to make good choices
of $E$ and $\ensuremath\lat{\Gamma}$ as is seen by the following corollary of
Theorem~\ref{thm:constr-dual-wavelet-Rn-general}. The corollary
unifies the construction procedures in Theorem~2 and Proposition~1
from \cite{lemvig_constr_paris} in a general procedure.
\begin{corollary}
\label{thm:constr-dual-wavelet-gen-phi}
Let $d \in \numbersys{N} _0$, $a>1$, and $\psi \in L^2(\numbersys{R})$. Suppose that $\hat
\psi$ is a bounded, real-valued function with $\supp \hat \psi \subset
\itvccs{-a^{c}}{-a^{c-d-1}} \cup \itvccs{a^{c-d-1}}{a^{c}}$ for some $c
\in \numbersys{Z}$, and that
\begin{equation}
\label{eq:dyadic-part2}
\sum_{j \in \mathbb{Z}} \hat \psi (a^j \xi) =1
\quad \text{for a.e.} %\@\xspace}\ } \xi \in \numbersys{R}.
\end{equation}
Let $b_j \in \numbersys{C}$ for $j=-d, \dots, d$, let $m = - \min\setprop{j}{\{b_j
\neq 0\}}$, and define the function $\phi$ by
\begin{align}
\label{eq:dual-generator1}
\phi(x) = \sum_{j=-m}^{d} b_j a^{-j} \psi(a^{-j}x) \quad
\text{for } x \in \numbersys{R}.
\end{align}
Let $b \in \itvoc{0}{a^{-c}(1+a^m)^{-1}}$. If $b_0 = b$ and $b_j +
b_{-j} = 2b$ for $j=1,2, \dots, d$, then $\psi$ and $\phi$
generate dual frames $\setsmall{\dila[a^j] \tranl[bk] \psi}_{j,k \in
\numbersys{Z}}$ and $\setsmall{\dila[a^j] \tranl[bk] \phi}_{j,k \in \numbersys{Z}}$ for
$L^2(\numbersys{R})$.
\end{corollary}
\begin{proof}
In Theorem~\ref{thm:constr-dual-wavelet-Rn-general} for $n=1$ and
$A=a$ we take $E=\itvccs{-a^c}{-a^{c-1}} \cup \itvccs{a^{c-1}}{a^c}$
as the multiplicative tiling set for $\setprop{a^j}{j \in \numbersys{Z}}$. The
assumption on the support of $\hat \psi$ becomes
\[ \supp \hat \psi \subset \bigcup_{j=0}^d a^{-j}(E) = \itvccs{-a^{c}}{-a^{c-d-1}}
\cup \itvccs{a^{c-d-1}}{a^{c}}. \]
Moreover, since
\[ \bigcup_{j=0}^d a^{-j}(E) \subset \itvcc{-a^c}{a^c}, \quad
\bigcup_{j=-m}^{2d} a^{-j}(E) \subset \itvcc{-a^{c+m}}{a^{c+m}}, \]
and
\[ (\itvcc{-a^c}{a^c}+\gamma ) \cap \itvcc{-a^{c+m}}{a^{c+m}} =
\emptyset \quad \text{for $\abs{\gamma} \ge a^c + a^{c+m} = a^c
(1+a^m)$,}\]
the choice $\ensuremath\lat{\Gamma}^\ast = b^{-1} \numbersys{Z}$ for $b^{-1} \ge a^c
(1+a^m)$ satisfies equation (\ref{eq:13}). This corresponds to $\ensuremath\lat{\Gamma}
=b\numbersys{Z}$ for $0 < b \le a^{-c} (1+a^m)^{-1}$.
\end{proof}
The assumptions in Corollary~\ref{thm:constr-dual-wavelet-gen-phi}
imply that $m \in \{0,1, \dots , d \}$; we note that in case $m=0$, the
corollary reduces to \cite[Theorem 2]{lemvig_constr_paris}.
\section{A special case of the construction procedure}
\label{sec:spec-case-Rn}
We aim for a more automated construction procedure than what we have
from Theorem~\ref{thm:constr-dual-wavelet-Rn-general}, in particular,
we therefore need to deal with good ways of choosing $E$ and $\ensuremath\lat{\Gamma}$.
The basic idea in this automation process will be to choose $E$ as a
dilation of the difference between $I_\ast$ and $B^{-1}(I_\ast)$,
where $I_\ast$ is the unit ball in a norm in which the matrix $B=A^t$
is expanding ``in all directions''; we will make this statement
precise in Section~\ref{sec:dilation-matrix}. This idea is
instrumental in the proof of
Theorem~\ref{thm:constr-dual-wavelet-Rn-special}.
\subsection{Some results on expansive matrices}
\label{sec:dilation-matrix}
We need the following well-known equivalent conditions for a
(non-singular) matrix being expansive.
\begin{proposition}\label{thm:expansive-matrix-equiv}
For $B \in GL_n(\numbersys{R})$ the following assertions are equivalent:
\begin{compactenum}[(i)]
\item $B$ is expansive, \textit{i.e.,\xspace} all eigenvalues $\lambda_i$ of $B$
satisfy $\abs{\lambda_i}>1$. \label{item:pairsRn1}
\item For any norm $\enorm{\,\cdot\,}$ on $\numbersys{R}^n$ there are
constants $ \lambda > 1$ and $c\ge 1$ such that \[
\enormsmall{B^jx} \ge 1/c \lambda^j \enorm{x}
\qquad \text{for all $j \in \numbersys{N}_0$}, \] for any $x \in
\numbersys{R}^n$. \label{enu:almost-expanding}
\item There is a Hermitian norm $\enorm[\ast]{\,\cdot\,}$ on $\numbersys{R}^n$ and a constant
$\lambda > 1$ such that \[
\enormsmall[\ast]{B^jx} \ge \lambda^j \enorm[\ast]{x} \qquad
\text{for all $j \in \numbersys{N}_0$}, \]
for any $x \in \numbersys{R}^n$. \label{enu:really-expanding}
\item $\mathcal{E} \subset \lambda \mathcal{E} \subset B \mathcal{E}$ for
some ellipsoid $\mathcal{E}=\setpropsmall{x \in \numbersys{R}^n}{\enorm{Px} \le
1}$, $P \in GL_n(\numbersys{R})$, and $\lambda >1$. \label{enu:expanding-ellipsoid}
\end{compactenum}
\end{proposition}
By Proposition~\ref{thm:expansive-matrix-equiv} we have that for a
given expansive matrix $B$, there exists a scalar product with the
induced norm $\enorm[\ast]{\,\cdot\,}$ so that
\[
\enorm[\ast]{Bx} \ge \lambda \enorm[\ast]{x} \quad \text{for } x\in
\numbersys{R}^n,
\]
holds for some $\lambda >1$. We say that $\enorm[\ast]{\,\cdot\,}$ is
a norm associated with the expansive matrix $B$. Note that such a norm
is not unique; we will follow the construction as in the proof of
\cite[Lemma 2.2]{MR2004e:42023}, so let $c$ and $\lambda$ be as in
\enumref{enu:almost-expanding} in
Proposition~\ref{thm:expansive-matrix-equiv} for the standard
Euclidean norm with $1<\lambda < \abs{\lambda_i}$ for $i =1, \dots,
n$, where $\lambda_i$ are the eigenvalues of $B$. For $k \in \numbersys{N}$
satisfying $k>2 \ln c / \ln \lambda$ we introduce the symmetric,
positive definite matrix $K \in GL_n(\numbersys{R})$:
\begin{equation}
\label{eq:15}
K = I + (B^{-1})^t B^{-1} + \dots + (B^{-k})^t B^{-k}.
\end{equation}
The scalar product associated with $B$ is then defined by
$\innerprod[\ast]{x}{y}= x^t K y$. It might not be effortless to
estimate $c$ and $\lambda$ for some given $B$, but it is obvious that
we just need to pick $k \in \numbersys{N}$ such that $ B^t K B - \lambda^2 K $
becomes positive semi-definite for some $\lambda >1$ since this
corresponds to $\innerprod{KBx}{Bx} \ge \lambda^2 \innerprod{Kx}{x}$,
that is, $\enorm[\ast]{Bx}^2 \ge \lambda^2 \enorm[\ast]{x}^2$ for all
$x \in \numbersys{R}^n$.
We let $I_\ast$ denote the unit ball in the Hermitian norm
$\enorm[\ast]{\,\cdot\,} = \enormsmall{K^{1/2}\cdot}$ associated
with $B$, \textit{i.e.,\xspace}
\begin{equation}
I_\ast = \setprop{x\in \numbersys{R}^n}{\enorm[\ast]{x} \le 1} = \setprop{x\in
\numbersys{R}^n}{\enormsmall{K^{1/2}x} \le 1} = \setprop{x\in
\numbersys{R}^n}{x^t Kx \le 1},\label{eq:25}
\end{equation}
and we let $O_\ast$ denote the annulus
\[ O_\ast =I_\ast \setminus B^{-1}(I_\ast).\]
The ringlike structure of $O_\ast$ is guaranteed by the fact that $B$
is expanding in all directions in the $\enorm[\ast]{\,\cdot\,}$ norm, \textit{i.e.,\xspace}
\begin{equation}
I_\ast \subset \lambda I_\ast \subset B(I_\ast), \qquad \lambda >1, \label{eq:9}
\end{equation}
which is \enumref{enu:expanding-ellipsoid} in Proposition
\ref{thm:expansive-matrix-equiv}. We note that by an orthogonal
substitution $I_\ast$ takes the form $\setpropsmall{x \in \numbersys{R}^n}{\mu_1
\tilde x_1^2+\dots+\mu_n \tilde x_n^2 \le 1}$, where $\mu_i$ are
the positive eigenvalues of $K$ and $x= Q\tilde x$ with the $i$th
column of $Q \in
O(n)$ comprising of the $i$th eigenvector of $K$. The annulus $O_\ast$ is a bounded multiplicative tiling set
for $\setpropsmall{B^j}{j\in \numbersys{Z}}$. This is a consequence of the
following result.
\begin{lemma}\label{thm:dilated-annuli}
Let $B \in GL_n(\numbersys{R})$ be an expansive matrix. For $x\neq 0$ there is a unique $j
\in \numbersys{Z}$ so that $B^j x \in O_\ast$; that is,
\begin{equation}
\label{eq:8}
\numbersys{R}^n \setminus \{0\} = \bigcup_{j \in \numbersys{Z}} B^j(O_\ast) \quad \text{ with disjoint union.}
\end{equation}
\end{lemma}
\begin{proof}
From equation~(\ref{eq:9}) we know that $\{B^l(I_\ast) \}_{l\in\numbersys{Z}}$ is a
nested sequence of subsets of $\numbersys{R}^n$, thus
\[ B^{l}(I_\ast) \setminus B^{l-1} (I_\ast) = B^l(O_\ast), \quad l
\in\numbersys{Z} ,\]
are disjoint sets. Since $\enorm[\ast]{B^{-j}x} \le
\lambda^{-j}\enorm[\ast]{x}$ and $\enorm[\ast]{B^{j}x} \ge
\lambda^{j}\enorm[\ast]{x}$ for $j \ge 0$ and $\lambda > 1$, we
also have
\begin{gather*}
\bigcup_{m=-l+1}^{l} B^m(O_\ast) = B^l(I_\ast) \setminus B^{-l}(I_\ast)
= \setprop{x\in\numbersys{R}^n}{\enormsmall[\ast]{B^{-l}x} \le 1 \text{ and }
\enormsmall[\ast]{B^{l}x} > 1}
\\ \supset \setprop{x\in\numbersys{R}^n}{\lambda^{-l}\enorm[\ast]{x} \le 1 \text{ and }
\lambda^l \enorm[\ast]{x} > 1} = \setprop{x\in\numbersys{R}^n}{\lambda^{-l} < \enorm[\ast]{x} \le \lambda^l}.
\end{gather*}
Taking the limit $l \to \infty$ we get (\ref{eq:8}).
\end{proof}
\begin{example}
\label{ex:hermitian-norm}
Let the following dilation matrix be given
\begin{equation}
\label{eq:17}
A =
\begin{pmatrix}
3 & -3 \\ 1 & 0
\end{pmatrix}.
\end{equation}
Here we are interested in the transpose matrix $B=A^t$ with
eigenvalues $\mu_{1,2}=3/2 \pm i \sqrt{3}/2$, hence $B$ is an
expansive matrix with $\abs{\mu_{1,2}}=\sqrt{3}>1$. The dilation
matrix $B$ is not expanding in the standard norm
$\enorm[2]{\,\cdot\,}$ in $\numbersys{R}^n$, \textit{i.e.,\xspace} $I_2 \not\subset B(I_2)$, as
shown by Figure~\ref{fig:ex3-euclidean-norm}.
\begin{figure}[ht]
\centering
\includegraphics[scale=.35]{./figure-ex3-euclidean-norm.eps}
\caption{Boundaries of the sets $I_2$, $B(I_2)$, $B^2(I_2)$, and
$B^3(I_2)$ marked by solid, long dashed, dashed, and dotted
lines, respectively. Note that $I_2 \setminus B(I_2)$ is
non-empty, and even $I_2 \setminus B^2(I_2)$ is non-empty.}
\label{fig:ex3-euclidean-norm}
\end{figure}
In order to have $B$ expanding the unit ball we need to use the
Hermitian norm from \enumref{enu:really-expanding} in
Proposition~\ref{thm:expansive-matrix-equiv} associated with $B$.
In \eqref{eq:15} we take $k=2$ so that the real, symmetric,
positive definite matrix $K$ is
\begin{equation*}
K = I + (B^{-1})^t B^{-1} + (B^{-2})^t B^{-2} =
\begin{pmatrix}
28/9 & 16/9 \\ 16/9 & 8/3
\end{pmatrix},
\end{equation*}
and let $\innerprod[\ast]{x}{y} := x^t Ky$. The choice $k=2$ suffices since
it makes $ B^t K B - \lambda^2 K $ semi-positive definite for
$\lambda =1.03$ and thus
\[ \enorm[\ast]{Bx} \ge \lambda \enorm[\ast]{x}, \qquad x \in \numbersys{R}^2, \]
holds for $\lambda=1.03$.
Figure~\ref{fig:ex3-adaptive-norm} and
\ref{fig:ex3-adaptive-norm-zoom} illustrate that $B$ indeed expands
the Hermitian norm unit ball $I_\ast$ in all directions.
\begin{figure}[ht]
\centering
\includegraphics[scale=.4]{./figure-ex3-adaptive-norm.eps}
\caption{The unit ball $I_\ast$ in the Hermitian norm
$\enorm[\ast]{\,\cdot\,}$ associated with $B$ and its dilations $B(I_\ast), B^2(I_\ast),
B^3(I_\ast)$. Only the boundaries are marked. }
\label{fig:ex3-adaptive-norm}
\end{figure}
We also remark that the Hermitian norm with $k=1$ will not make the dilation
matrix $B$ expanding in $\numbersys{R}^n$; in this case we have a situation
similar to Figure~\ref{fig:ex3-euclidean-norm}.
\begin{figure}[th!]
\centering
\includegraphics[scale=.4]{./figure-ex3-adaptive-norm-zoom.eps}
\caption{A zoom of Figure~\ref{fig:ex3-adaptive-norm}. Boundaries
of the sets $I_\ast$, $B(I_\ast)$,
$B^2(I_\ast)$, and $B^3(I_\ast)$ marked by solid,
long dashed, dashed, and dotted lines, respectively.}
\label{fig:ex3-adaptive-norm-zoom}
\end{figure}
\end{example}
\subsection{A crude lattice choice}
\label{sec:choos-transl-latt}
Let us consider the setup in
Theorem~\ref{thm:constr-dual-wavelet-Rn-general} with the set $E =
B^c(O_\ast)$ for some $c\in\numbersys{Z}$, where the norm
$\enorm[\ast]{\,\cdot\,}=\enormsmall{K^{1/2}\cdot}$ is associated with
$B$. Let $\mu$ be the smallest eigenvalue of $K$ such that $\ell =
\sqrt{1/\mu}$ is the largest semi-principal axis of the ellipsoid
$I_\ast$, \textit{i.e.,\xspace} $\ell = \max_{x \in I_\ast}\enorm[2]{x}$. Then we can
take any lattice $\ensuremath\lat{\Gamma} = P\numbersys{Z}^n$, where $P$ is a non-singular matrix
satisfying
\begin{equation}
\normsmall[2]{P} \le \frac{1}{ \ell \norm[2]{A^c} (1 +
\norm[2]{A^{\underline{m}}}) }, \label{eq:16}
\end{equation}
as our translation lattice in Theorem~\ref{thm:constr-dual-wavelet-Rn-general}. To see this, recall that we are looking for
a lattice $\ensuremath\lat{\Gamma}^\ast$ such that, for $\gamma \in \ensuremath\lat{\Gamma}^\ast \setminus \{0\}$,
\begin{align}
\supp \hat \phi \cap \supp \hat \psi(\cdot \pm \gamma) =
\emptyset. \label{eq:19}
\end{align}
For our choice of $E$ we
find that $\supp \hat \phi
\subset B^{c+\underline{m}}(I_\ast)$ and $\supp \hat \psi \subset
B^{c}(I_\ast)$. Since
\begin{align*}
\enorm[2]{B^{c+\underline{m}}x} \le \norm[2]{B^{c+\underline{m}}}
\enorm[2]{x} \le \norm[2]{B^{c+\underline{m}}} \ell \qquad \text{for any $x \in I_\ast$},
\end{align*}
and similar for $B^cx$, we have the situation in (\ref{eq:19}) whenever
$\enorm[2]{\gamma} \ge \ell (\norm[2]{A^c} +
\norm[2]{A^{c+\underline{m}}})$. Here we have used that
for the $2$-norm $\norm[2]{A}= \norm[2]{B}$.
For $z \in \numbersys{Z}^n$ we have
\begin{equation*}
\enorm[2]{z} \le \normsmall[2]{P^t} \,\enormsmall[2]{(P^t)^{-1}z} = \normsmall[2]{P} \,\enormsmall[2]{(P^t)^{-1}z},
\end{equation*}
therefore, by $\enorm[2]{z} \ge 1$ for $z \neq 0$, we have
\[ \enorm[2]{(P^t)^{-1}z} \ge \frac{1}{\norm[2]{P}} \qquad \text{for
$z \in \numbersys{Z} \setminus \{0\}$.}\] Now, by assuming that $P$ satisfies
(\ref{eq:16}), we have
\begin{equation*}
\enorm[2]{\gamma} = \enormsmall[2]{(P^t)^{-1}z} \ge 1/ \normsmall[2]{P} \ge
\ell \norm[2]{A^c} (1 + \norm[2]{A^{\underline{m}} }) \ge l (\norm[2]{A^c} +
\norm[2]{A^{c+\underline{m}}})
\end{equation*}
for $0 \neq \gamma = (P^t)^{-1}z \in (P^t)^{-1}\numbersys{Z}^n=\ensuremath\lat{\Gamma}^\ast$,
hence the claim follows.
A lattice choice based on (\ref{eq:16}) can be rather crude, and
produces consequently a wavelet system with unnecessarily many
translates. From equation~(\ref{eq:16}) it is obvious that any lattice
$\ensuremath\lat{\Gamma}=P \numbersys{Z}^n$ with $\norm{P}$ sufficiently small will work as
translation lattice for our pair of generators $\psi$ and $\phi$.
Hence, the challenging part is to find a sparse translation lattice
whereby we understand a lattice $\ensuremath\lat{\Gamma}$ with large determinant
$d(\ensuremath\lat{\Gamma}):=\abs{\ensuremath{\determinant}{P}}$. In the dual lattice system this corresponds
to a dense lattice $\ensuremath\lat{\Gamma}^\ast$ with small volume $d(\ensuremath\lat{\Gamma}^\ast)$ of the
fundamental parallelotope $I_{\ensuremath\lat{\Gamma}^\ast}$ since $d(\ensuremath\lat{\Gamma})d(\ensuremath\lat{\Gamma}^\ast)=1$.
In Theorem~\ref{thm:constr-dual-wavelet-Rn-special} in the next
section we make a better choice of the translation lattice compared to
what we have from (\ref{eq:16}).
Using a crude lattice approach as above, we can easily
transform the translation lattice to the integer lattice if we allow
multiple generators. We pick a matrix $P$ that satisfies condition
(\ref{eq:16}) and whose inverse is integer valued, \textit{i.e.,\xspace}
$Q:=P^{-1} \in GL_n(\numbersys{Z})$. The conclusion from
Theorem~\ref{thm:constr-dual-wavelet-Rn-general} is that
$\setsmall{\dila \tran[ Q^{-1}k] \psi}_{j
\in \numbersys{Z}, k \in \numbersys{Z}^n}$ and $\setsmall{\dila \tran[Q^{-1}k] \phi}_{j \in
\numbersys{Z}, k \in \numbersys{Z}^n}$ are dual frames.
The order of the quotient group $Q^{-1}\numbersys{Z}^n/\numbersys{Z}^n$ is $\abs{\ensuremath{\determinant}{Q}}$, so
let $\setpropsmall{d_i }{i=1, \dots, \abs{\ensuremath{\determinant}{Q}}}$ denote a
complete set of representatives of the quotient group, and define \[
\Psi = \setprop{\tran[d_i] \psi}{i =
1, \dots, \abs{\ensuremath{\determinant} Q}}, \quad \Phi = \setprop{\tran[d_i] \phi}{i =
1, \dots, \abs{\ensuremath{\determinant} Q}}.\]
Since $\setsmall{\dila \tran[ Q^{-1}k] \psi}_{j
\in \numbersys{Z}, k \in \numbersys{Z}^n} = \setsmall{\dila \tran[ k] \psi}_{j
\in \numbersys{Z}, k \in \numbersys{Z}^n, \psi \in \Psi}$ and likewise for the dual frame,
the statement follows.
\subsection{A concrete version of
Theorem~\protect{\ref{thm:constr-dual-wavelet-Rn-general}} }
\label{sec:special-case-Rn}
We list some standing assumptions and conventions for this section.
\paragraph{\bf General setup.} We assume $A \in GL_n(\numbersys{R})$ is
expansive. Let $\enorm[\ast]{\,\cdot\,} =
\innerprod[\ast]{\,\cdot\,}{\,\cdot\,}^{1/2}$ be a Hermitian norm as
in \enumref{enu:really-expanding} in
Proposition~\ref{thm:expansive-matrix-equiv} associated with $B =
A^t$, let $I_\ast$ denote the unit ball in the $\enorm[\ast]{\,\cdot\,}$-norm,
and let $K \in GL_n(\numbersys{R})$ be the symmetric, positive definite matrix
such that $\innerprod[\ast]{x}{y} = y^t Kx$. Let $\Lambda :=
\mathrm{diag}(\lambda_1, \dots, \lambda_n)$, where $\{\lambda_i\}$
are the eigenvalues of $K$, and let $Q \in O(n)$ be such that the
spectral decomposition of $K$ is $Q^t K Q = \Lambda$.
The following result is a special case of
Theorem~\ref{thm:constr-dual-wavelet-Rn-general}, where we, in
particular, specify how to choose the translation lattice $\ensuremath\lat{\Gamma}$. Since
we in Theorem~\ref{thm:constr-dual-wavelet-Rn-special} define $\ensuremath\lat{\Gamma}$,
it allows for a more automated construction procedure.
\begin{theorem}\label{thm:constr-dual-wavelet-Rn-special}
Let $A, I_\ast, K, Q, \Lambda$ be as in the general setup. Let $d \in \numbersys{N}
_0$ and $\psi \in L^2(\numbersys{R}^n)$. Suppose that $\hat \psi$ is a bounded,
real-valued function with $\supp \hat \psi \subset B^c (I_\ast)
\setminus B^{c-d-1} (I_\ast)$ for some $c \in \numbersys{Z}$, and that
\eqref{eq:dyadic-part2-Rn} holds. Take $\ensuremath\lat{\Gamma} = (1/2) A^c Q
\sqrt{\Lambda} \numbersys{Z}^n$. Then the function $\psi$ and the function
$\phi$ defined by
\begin{align}
\label{eq:dual-generator2-Rn}
\phi(x) = d(\ensuremath\lat{\Gamma}) \left [ \psi(x) + 2 \sum_{j=0}^{d}
\abs{\ensuremath{\determinant}{A}}^{-j} \psi(A^{-j}x)\right ] \quad
\text{for } x \in \numbersys{R}^n,
\end{align}
generate dual frames $\setsmall{\dila \tran \psi}_{j \in \numbersys{Z}, \gamma \in
\ensuremath\lat{\Gamma}}$ and $\setsmall{\dila \tran \phi}_{j \in \numbersys{Z}, \gamma \in \ensuremath\lat{\Gamma}}$ for
$L^2(\numbersys{R}^n)$
\end{theorem}
\begin{remark}
Note that $d(\ensuremath\lat{\Gamma}) = 2^{-n} \abs{\ensuremath{\determinant}{A}}^c (\lambda_1 \cdots
\lambda_n)^{1/2} $ and $\sqrt{\Lambda} =
\mathrm{diag}(\sqrt{\lambda_1}, \dots, \sqrt{\lambda_n})$.
\end{remark}
\begin{proof}
The annulus $O_\ast$ is a bounded multiplicative tiling set for the dilations
$\setprop{B^j}{j\in\numbersys{Z}}$ by Lemma~\ref{thm:dilated-annuli}, hence
this is also the case for $B^c(O_\ast)$ for $c\in \numbersys{Z}$. The support
of $\hat\psi$ is $\supp \hat \psi \subset B^c (I_\ast) \setminus
B^{c-d-1} (I_\ast) = \cup_{j=0}^d B^{c-j}(O_\ast)$. Therefore we
can apply Theorem~\ref{thm:constr-dual-wavelet-Rn-general} with
$E=B^c(O_\ast)$, $b_j=2$ and $b_{-j}=0$ for $j =1, \dots, d$ so
that $\underline{m}=0$ and $\overline{m}=d$. The only thing left
to justify is the choice of the translation lattice
$\ensuremath\lat{\Gamma}$.
We need to show that condition~(\ref{eq:13}) with $\underline{m}=0$
and $\overline{m}=d$ in
Theorem~\ref{thm:constr-dual-wavelet-Rn-general} is satisfied by
$\ensuremath\lat{\Gamma}^\ast = 2 B^{c} Q \Lambda^{-1/2} \numbersys{Z}^n$. By the orthogonal
substitution $x = Q \tilde x$ the quadratic form $x^t K x$ of
equation (\ref{eq:25}) reduces to
\[ \lambda_1 \tilde x_1^2 + \dots + \lambda_n \tilde x_n^2, \]
where $\lambda_i>0$, hence in the $\tilde x = Q^t x$ coordinates $I_\ast$ is
given by
\[ \tilde I_\ast = \setprop{\tilde x \in \numbersys{R}^n}{\biggl(\frac{\tilde
x_1}{1/\sqrt{\lambda_1}}\biggr)^2+ \dots + \biggl(\frac{\tilde
x_n}{1/\sqrt{\lambda_n}}\biggr)^2 < 1}\]
which is an ellipsoid with semi axes $\tfrac{1}{\sqrt{\lambda_1}},
\dots, \tfrac{1}{\sqrt{\lambda_n}}$.
Therefore, in the $\tilde x$ coordinates,
\[ (\tilde I_\ast + \gamma) \cap \tilde I_\ast = \emptyset \qquad \text{for
$0 \neq \gamma \in 2 \Lambda^{-1/2} \numbersys{Z}^n $},\]
or, in the $x$ coordinates,
\begin{equation*}
( I_\ast + \gamma) \cap I_\ast = \emptyset \qquad \text{for
$0 \neq \gamma \in 2 Q \Lambda^{-1/2} \numbersys{Z}^n $.}
\end{equation*}
By applying $B^c$ to this relation it becomes
\begin{equation} \label{eq:20}
\bigl( B^c(I_\ast) + \gamma \bigr) \cap B^c(I_\ast) = \emptyset \qquad \text{for
$0 \neq \gamma \in \ensuremath\lat{\Gamma}^\ast = 2 B^c Q \Lambda^{-1/2} \numbersys{Z}^n $,}
\end{equation}
whereby we see that condition~(\ref{eq:13}) is satisfied with
$\underline{m}=0$ and $\ensuremath\lat{\Gamma}^\ast = 2 B^{c} Q \Lambda^{-1/2} \numbersys{Z}^n$. The
dual lattice of $\ensuremath\lat{\Gamma}^\ast$ is $\ensuremath\lat{\Gamma} = 1/2 A^{-c} Q \Lambda^{1/2} \numbersys{Z}^n$.
It follows from Theorem~\ref{thm:constr-dual-wavelet-Rn-general} that
$\psi$ and $\phi$ generate dual frames for this choice of the translation
lattice.
\end{proof}
The frame bounds for the pair of dual frames $\setsmall{\dila \tran
\psi}_{j \in \numbersys{Z}, \gamma \in \ensuremath\lat{\Gamma}}$ and $\setsmall{\dila \tran
\phi}_{j \in \numbersys{Z}, \gamma \in \ensuremath\lat{\Gamma}}$ in
Theorem~\ref{thm:constr-dual-wavelet-Rn-special} can be given
explicitly as
\begin{align*}
C_1 &= \frac{1}{d(\ensuremath\lat{\Gamma})}\inf_{\xi \in B^{c-d}(O_\ast)} \sum_{j=0}^{d}
\left(\hat \psi (B^{j} \xi)\right)^2, & &C_2 =
\frac{1}{d(\ensuremath\lat{\Gamma})}\sup_{\xi \in B^{c-d}(O_\ast)} \sum_{j=0}^{d} \left(\hat
\psi (B^{j} \xi)\right)^2,
\intertext{and}
C_1 &= \frac{1}{d(\ensuremath\lat{\Gamma})}\inf_{\xi \in B^{c-d}(O_\ast)}
\sum_{j=-d}^{d} \left(\hat \phi (B^{j}
\xi)\right)^2, & &C_2 = \frac{1}{d(\ensuremath\lat{\Gamma})}\sup_{\xi \in B^{c-d}(O_\ast)}
\sum_{j=-d}^{d} \left(\hat \phi (B^{j}
\xi)\right)^2 ,
\end{align*}
respectively. The frame bounds do not depend on the specific structure
of $\ensuremath\lat{\Gamma}$, but only on the determinant of $\ensuremath\lat{\Gamma}$; in particular, the
condition number $C_2/C_1$ is independent of $\ensuremath\lat{\Gamma}$.
To verify these frame bounds, we note that equation (\ref{eq:20})
together with the fact $\supp \hat \psi$, $\supp \hat \phi \subset
B^c(I_\ast)$ imply that
\begin{align*}
\hat \psi(\xi) \hat \psi(\xi + \gamma) = \hat \phi(\xi) \hat
\phi(\xi + \gamma) = 0 \qquad \text{for a.e.} %\@\xspace}\ $\xi \in \numbersys{R}^n$
and $\gamma \in \ensuremath\lat{\Gamma}^\ast \setminus \{0\}$}.
\end{align*}
Therefore, by equations~(\ref{eq:18}) and (\ref{eq:28}) with
$E=B^c(O_\ast)$, $\underline{m}=0$ and $\overline{m}=d$, we have
\begin{align*}
\sum_{j\in\numbersys{Z}} \sum_{\gamma \in \ensuremath\lat{\Gamma}^\ast } \abs{ \hat \psi(B^j \xi)
\hat \psi(B^j \xi + \gamma)} = \sum_{j\in\numbersys{Z}} \abs{ \hat \psi(B^j \xi)}^2
= \sum_{j=0}^{d} \left( \hat \psi(B^j \xi) \right)^2,
\end{align*}
and
\begin{align*}
\sum_{j\in\numbersys{Z}} \sum_{\gamma \in \ensuremath\lat{\Gamma}^\ast } \abs{ \hat \phi(B^j \xi)
\hat \phi(B^j \xi + \gamma)} = \sum_{j\in\numbersys{Z}} \abs{ \hat \phi(B^j \xi)}^2
= \sum_{j=-d}^{d} \left( \hat \phi(B^j \xi) \right)^2,
\end{align*}
for $\xi \in B^{c-d}(O_\ast)$. The stated frame bounds follow from
Lemma~\ref{thm:bessel2-Rn}.
\begin{example}
Let $A$ and $K$ be as in Example~\ref{ex:hermitian-norm}.
The eigenvalues of $K$ are $\lambda_1 = (26 + 2\sqrt{65})/9 \approx
4.7$ and $\lambda_2 = (26 - 2\sqrt{65})/9 \approx 1.1$.
Let the normalized (in the standard norm) eigenvectors of $K$ be columns of
$Q \in O(2)$ and $\Lambda = \mathrm{diag}(\lambda_1, \lambda_2)$,
hence $Q^t K Q = \Lambda$.
By the orthogonal transformation
$x = Q \tilde x$ the Hermitian norm unit ball $I_\ast$ becomes
\[ \tilde I_\ast = \setprop{\tilde x \in \numbersys{R}^2}{\biggl(\frac{\tilde
x_1}{1/\sqrt{\lambda_1}}\biggr)^2+ \biggl(\frac{\tilde
x_2}{1/\sqrt{\lambda_2}}\biggr)^2 < 1} \subset I_2 \]
which is an ellipse with semimajor axis $1/ \sqrt{\lambda_2} \approx 0.95$
and semiminor axis $1/ \sqrt{\lambda_1} \approx 0.46$.
Since
$ \Lambda^{-1/2} = \mathrm{diag}(1/\sqrt{\lambda_1},
1/\sqrt{\lambda_2})$, we have
\[ \abs{(\tilde I_\ast + \gamma) \cap \tilde I_\ast} = 0 \qquad \text{for
$0 \neq \gamma \in 2 \Lambda^{-1/2} \numbersys{Z}^2 $}.\]
By the orthogonal substitution back to $x$ coordinates, we get
\[ \abs{( I_\ast + \gamma) \cap I_\ast} = 0 \qquad \text{for
$0 \neq \gamma \in 2 Q \Lambda^{-1/2} \numbersys{Z}^2 $}.\]
Suppose that $\hat \psi$ is a bounded, real-valued
function with $\supp \hat \psi \subset B^c (I_\ast) \setminus
B^{c-d-1} (I_\ast)$ for $c = 1$ that satisfies the $B$-dilative partition~\eqref{eq:dyadic-part2-Rn}.
Since $c=1$ we need to take $\ensuremath\lat{\Gamma}^\ast = 2 B^{1} Q
\Lambda^{-1/2} \numbersys{Z}^2$ and $\ensuremath\lat{\Gamma} = 1/2 A^{-1} Q \Lambda^{1/2} \numbersys{Z}^2$, see
Figure~\ref{fig:ex3-gamma-star-c1} and \ref{fig:ex3-gamma-c1}.
\begin{figure}[ht]
\centering
\includegraphics[scale=.35]{./figure-ex3-gamma-star-c1.eps}
\caption{The dual lattice $\ensuremath\lat{\Gamma}^\ast = 2B^{c}Q
{\ensuremath\lat{\Lambda}}^{-1/2} \numbersys{Z}^2$ for $c=1$ is shown by dots, and the boundary of the
set $B^c(I_\ast)$ by a solid line. Boundaries of the set
$B^c(I_\ast)$ translated to several different $\gamma \in
\ensuremath\lat{\Gamma}^\ast \setminus \{0\}$ are shown with dashed lines. Recall that
$\supp \hat \psi, \supp \hat \phi \subset B^c(I_\ast)$, hence
$\supp \hat \phi \cap \supp \hat \psi(\cdot + \gamma) =
\emptyset$ for $\gamma \in
\ensuremath\lat{\Gamma}^\ast \setminus \{0\}$. }
\label{fig:ex3-gamma-star-c1}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[scale=.35]{./figure-ex3-gamma-c1.eps}
\caption{The translation lattice $\ensuremath\lat{\Gamma} = (1/2) A^{c}Q {\ensuremath\lat{\Lambda}}^{1/2}
\numbersys{Z}^2$ for $c=1$.}
\label{fig:ex3-gamma-c1}
\end{figure}
\end{example}
\subsection{An alternative lattice choice}
\label{sec:optim-latt-choice}
Let the setup up and assumptions be as in
Theorem~\ref{thm:constr-dual-wavelet-Rn-special}, except for the
lattice $\ensuremath\lat{\Gamma}$ which we want to choose differently. As in
Section~\ref{sec:choos-transl-latt} the dual lattice $\ensuremath\lat{\Gamma}^\ast$ needs
to satisfy (\ref{eq:19}) for $\gamma \in \ensuremath\lat{\Gamma}^\ast \setminus
\{0\}$. We want to choose $\ensuremath\lat{\Gamma}^\ast$ as dense as possible since this will
make the translation lattice $\ensuremath\lat{\Gamma}$ as sparse as possible and the
wavelet system with as few translates as possible. Since $\supp \hat
\psi, \supp \hat \phi \subset B^c(I_\ast)$, we are looking for
lattices $\ensuremath\lat{\Gamma}^\ast$ that packs the ellipsoids $B^c(I_\ast) +
\gamma$, $\gamma \in \ensuremath\lat{\Gamma}^\ast$, in a non-overlapping,
optimal way. By the coordinate transformation $\hat x =
\Lambda^{-1/2}Q^t B^{-c}x$, the ellipsoid $B^c(I_\ast)$ turns into the
standard unit ball $I_2$ in $\numbersys{R}^n$. This calculations are as follows.
\begin{align*}
B^c(I_\ast) &= \setprop{B^cx}{\enorm[\ast]{x}^2 \le 1} =
\setprop{x}{\enormsmall[2]{K^{1/2}B^{-c}x}^2 \le 1} \\
&= \setprop{x}{\enorm[2]{K^{1/2}B^{-c}B^c Q \Lambda^{-1/2}\hat x}^2
\le 1} \\
&= \setprop{x}{\einnerprod[2]{\hat x}{\Lambda^{-1/2} Q^t K
Q\Lambda^{-1/2} \hat x} \le 1} =\setprop{x}{\enorm[2]{\hat x}^2
\le 1},
\end{align*}
and we arrive at a standard sphere packing problem with lattice
arrangement of non-overlapping unit $n$-balls. The proportion of the
Euclidean space $\numbersys{R}^n$ filled by the balls is called the density of the
arrangement, and it is this density we want as high as possible.
Taking $\ensuremath\lat{\Gamma}$ as in Theorem~\ref{thm:constr-dual-wavelet-Rn-special}
corresponds to a square packing of the unit $n$-balls $I_2 + k$ by the
lattice $2 \numbersys{Z}^n$, \textit{i.e.,\xspace} $k \in 2\numbersys{Z}^n$. The density of this packing is $V_n
2^{-n}$, where $V_n$ is the volume of the $n$-ball: $V_{2n}=\pi^n/(n!)$ and
$V_{2n+1}=(2^{2n+1}n!\pi^n)/(2n+1)!$.
This is not the densest packing of balls in $\numbersys{R}^n$ since there exists
a lattice with density bigger than $1.68 n2^{-n}$ for each $n\neq
1$ \cite{0030.34602}; a slight improvement of this lower bound was
obtained in \cite{0776.52006} for $n>5$. Moreover, the densest lattice
packing of hyperspheres is known up to dimension $8$, see
\cite{MR1648172}; it is precisely this dense lattice we want to use in
place of $2 \numbersys{Z}^n$ (at least whenever $n \le 8$).
In $\numbersys{R}^2$ Lagrange proved that the hexagonal packing, where each ball
touches $6$ other balls in a hexagonal lattice, has the highest
density $\pi / \sqrt{12}$. Hence using $P \numbersys{Z}^2$ with
\[ P =
\begin{pmatrix}
2 & 0 \\ 1 & \sqrt{3}
\end{pmatrix}
\]
instead of $2 \numbersys{Z}^2$ improves the packing by a factor of
\[ \frac{\pi/\sqrt{12}}{\pi/2^2} = 4/\sqrt{12} = 2/\sqrt{3}.\] It is
easily seen that this factor equals the relation between the area of
the fundamental parallelogram of the two lattices $\abs{\ensuremath{\determinant}{2I_{2
\times 2}}}/\abs{\ensuremath{\determinant}{P}}$. In
Figure~\ref{fig:ex3-gamma-star-c1} we see that each ellipse only
touches $4$ other ellipses corresponding to the square packing
$2\numbersys{Z}^n$; in the optimal packing each ellipse touch $6$ others. In
$\numbersys{R}^3$ Gauss proved that the highest density is $\pi / \sqrt{18}$
obtained by the hexagonal close and face-centered cubic packing; here
each ball touches $12$ other balls.
\section{Dilative partition of unity}
\label{sec:examples-rn}
With Theorem~\ref{thm:constr-dual-wavelet-Rn-special} at hand the only
issue left is to specify how to construct functions satisfying the
partition of unity (\ref{eq:dyadic-part2-Rn}) for any given expansive
matrix. In the two examples of this section we outline possible ways
of achieving this.
\subsection{Constructing a partition of unity}
\label{sec:constr-part-unity}
As usual we fix the dimension $n \in \numbersys{N}$ and the expansive matrix $B \in
GL_n(\numbersys{R})$.
In the examples in this section we construct functions satisfying the assumptions
in Theorem~\ref{thm:constr-dual-wavelet-Rn-special}, that is, a real-valued
function $g \in L^2(\numbersys{R}^n)$ with $\supp g \subset B^c(I_\ast)
\setminus B^{c-d-1}(I_\ast)$ for some $c \in \numbersys{Z}$ and $d \in \numbersys{N}_0$ so that
the $B$-dilative partition
\begin{equation}
\sum_{j \in \mathbb{Z}} g (B^j \xi) =1
\quad \text{for a.e.} %\@\xspace}\ }\xi \in \numbersys{R}^n,
\label{eq:21}
\end{equation}
holds.
In the construction we will use that the radial coordinate of the
surface of the ellipsoid $\partial B^j(I_\ast)$, $j \in \numbersys{Z}$, can be
parametrized by the $n-1$ angular coordinates $\theta_1, \dots,
\theta_{n-1}$.
The radial coordinate expression will be of the form
$h(\theta_1, \dots, \theta_{n-1})^{-1/2}$ for some positive,
trigonometric function $h$, where $h$ is bounded away from zero and
infinity with the specific form of $h$ depending on the dimension
$n$ and the length and orientation of the ellipsoid axes.
We illustrate this with the following example in $\numbersys{R}^4$. We want to
find the radial coordinate $r$ of the ellipsoid \[\setprop{x \in
\numbersys{R}^4}{(x_1/\ell_1)^2 + (x_2/\ell_2)^2 +(x_3/\ell_3)^2 + (x_4/\ell_4)^2 =1},
\qquad \ell_i > 0,\, i=1,2,3,4, \] as a function the angular coordinates
$\theta_1, \theta_2$ and $\theta_3$. We express $x=(x_1, x_2, x_3,
x_4) \in \numbersys{R}^4$ in the hyperspherical coordinates $(r,\theta_1,
\theta_2, \theta_3) \in \{0\} \cup \numbersys{R}_+ \times [0,\pi] \times
[0,\pi] \times [0, 2\pi)$ as follows:
\begin{align*}
x_1 &= r \cos \theta_1, && x_2 = r \sin \theta_1 \cos \theta_2, \\
x_3 &= r \sin \theta_1 \sin \theta_2 \cos \theta_3, && x_4 = r \sin
\theta_1 \sin \theta_2 \sin \theta_3.
\end{align*}
Then we substitute $x_i$, $i=1,\dots,4$, in the expression above and factor
out $r^2$ to obtain $r^2 f(\theta_1, \theta_2, \theta_{3}) =1$, where
\begin{align}\label{eq:27}
f(\theta_1,\theta_2, \theta_3) &= \ell_1^{-2}\cos^2 \theta_1 + \ell_2^{-2}\sin^2 \theta_1
\cos^2 \theta_2 \\ &\phantom{=}\,+ \ell_3^{-2}\sin^2 \theta_1 \sin^2 \theta_2 \cos^2 \theta_3 + \ell_4^{-2} \sin^2
\theta_1 \sin^2 \theta_2 \sin^2 \theta_3. \nonumber
\end{align}
The conclusion is that $r=r(\theta_1,\theta_2, \theta_3) = f(\theta_1, \theta_2,
\theta_{3})^{-1/2}$.
\begin{example}\label{exa:d-one}
For $d=1$ in Theorem~\ref{thm:constr-dual-wavelet-Rn-special} we
want $g \in C^s_0(\numbersys{R}^n)$ for any given $s \in \numbersys{N} \cup \{0\}$. The
choice $d=1$ will fix the ``size'' of the support of $g$ so that $\supp
g \subset B^c(I_\ast) \setminus B^{c-2}(I_\ast)$ for some $c\in \numbersys{Z}$.
Now let $r_1=r_1(\theta_1, \dots, \theta_{n-1})$ and
$r_2=r_2(\theta_1, \dots, \theta_{n-1})$ denote the radial
coordinates of the surface of the ellipsoids $\partial
B^{c-1}(I_\ast)$ and $\partial B^c(I_\ast)$ parametrized by $n-1$
angular coordinates $\theta_1, \dots, \theta_{n-1}$, respectively.
Let $f$ be a continuous function on the annulus
$S=\overline{B^c(O_\ast)}$ satisfying $f \vert _{\partial
B^{c-1}(I_\ast)}=1$ and $f \vert _{\partial
B^{c}(I_\ast)}=0$.
Using the parametrizations $r_1, r_2$ of the surfaces of the two
ellipsoids and fixing the $n-1$ angular coordinates we realize
that we only have to find a continuous function $f:\itvcc{r_1}{r_2}
\to \numbersys{R}$ of one variable (the radial coordinate) satisfying
$f(r_1)=1$ and $f(r_2)=0$. For example the general function $f \in
C^0(S)$ of $d$ variables can be any of the functions below:
\begin{subequations}
\label{eq:def-of-f-radius}
\begin{align}
f(x)&= f(r,\theta_1, \dots, \theta_{n-1}) =
\frac{r_2-r}{r_2-r_1}, \label{eq:22}\\
f(x)&= f(r,\theta_1, \dots, \theta_{n-1}) = \frac{(r_2-r)^2}{(r_2-r_1)^3} (2 (r -r_1)+r_2-r_1),\label{eq:23} \\
f(x)&= f(r,\theta_1, \dots, \theta_{n-1}) = \tfrac{1}{2}+\tfrac{1}{2}\cos{\pi (\tfrac{r-r_1}{r_2-r_1})} \label{eq:24},
\end{align}
\end{subequations}
where $r = \enorm{x} \in \itvcc{r_1}{r_2}$, $\theta_1, \dots,
\theta_{n-2} \in \itvcc{0}{\pi}$, and $\theta_{n-1} \in
\itvco{0}{2\pi}$; recall that $r_1=r_1(\theta_1, \dots, \theta_{n-1})$
and $r_2=r_2(\theta_1, \dots, \theta_{n-1})$. In definitions
\eqref{eq:23} and \eqref{eq:24} the function $f$ even
belongs to $C^1(S)$.
Define $g \in L^2(\numbersys{R})$ by:
\begin{equation}
\label{eq:def-C1-example-Rn}
g(x) =
\begin{cases}
1-f(Bx) \quad &\text{for } x \in B^{c-1}(I_\ast)
\setminus B^{c-2}(I_\ast), \\
f(x) \quad &\text{for } x \in B^{c}(I_\ast)
\setminus B^{c-1}(I_\ast), \\
0 \quad & \text{otherwise.}
\end{cases}
\end{equation}
This way $g$ becomes a $B$-dilative partition of unity with $\supp
g \subset B^{c}(I_\ast)
\setminus B^{c-2}(I_\ast) $, so we can
apply Theorem~\ref{thm:constr-dual-wavelet-Rn-special} with $\hat
\psi=g$ and $d=2$.
\end{example}
We can simplify the expressions for the radial coordinates $r_1, r_2$
of the surface of the ellipsoids $\partial B^{c-1}(I_\ast)$ and
$\partial B^c(I_\ast)$ from the previous example by a suitable
coordinate change. The idea is to transform the ellipsoid $
B^{c-1}(I_\ast)$ to the standard unit ball $I_2$ by a first coordinate
change $\tilde x = \Lambda^{1/2} Q^t B^{-c+1} x$. This will transform
the outer ellipsoid $B^c(I_\ast)$ to another ellipsoid. A second and
orthogonal coordinate transform $\hat x = Q_\prime^t \tilde x$ will
make the semiaxes of this new ellipsoid parallel to the coordinate
axes, leaving the standard unit ball $I_2$ unchanged. Here $Q_\prime$
comes from the spectral decomposition of $A^{-1}B^{-1}$, \textit{i.e.,\xspace}
$A^{-1}B^{-1} = Q_\prime^t \Lambda_\prime Q_\prime$. In the $\hat x$
coordinates $r_1=1$ is a constant and $r_2=f^{-1/2}$ with $f$ of the
form (\ref{eq:27}) for $n=4$ and likewise for $n \neq
4$.
In the construction in Example~\ref{exa:d-one} we assumed that $d=1$. The next
example works for all $d \in \numbersys{N}$; moreover, the constructed function
will belong to $C^\infty_0(\numbersys{R}^n)$.
\begin{example}\label{exa:all-d}
For sufficiently small $\delta >0$ define $\Delta_1, \Delta_2 \subset \numbersys{R}^n$ by
\begin{align*}
\Delta_1 &= B^{c-d-1}(I_\ast) + \unitball{0}{\delta}, \\
\Delta_2 &+ \unitball{0}{\delta} = B^{c}(I_\ast).
\end{align*}
This makes $\Delta_2 \setminus \Delta_1$ a subset of the
annulus $B^{c}(I_\ast) \setminus B^{c-d-1}(I_\ast)$; it is exactly the
subset, where points less than $\delta$ in
distance from the boundary have been removed, or in other words
\[ \Delta_2 \setminus \Delta_1 + \unitball{0}{\delta} =
B^{c}(I_\ast) \setminus B^{c-d-1}(I_\ast).
\]
For this to hold, we of course need to take $\delta >0$ sufficiently
small, \textit{e.g.\@\xspace} such that $\Delta_1 \subset r
\Delta_1 \subset \Delta_2$ holds for some $r>1$.
Let $h\in C^\infty _0(\numbersys{R}^n)$ satisfy $\supp h = \unitball{0}{1}$, $h \ge
0$, and $\int h \,\mathrm{d}\mu =1$, and define $h_\delta = \delta^{-d}
h(\delta^{-1}\cdot)$. By convoluting the characteristic function on
$\Delta_2 \setminus \Delta_1$ with $h_\delta$ we obtain a smooth
function living on the annulus $ B^{c}(I_\ast) \setminus
B^{c-d-1}(I_\ast)$. So let $p \in C^\infty _0(\numbersys{R}^n)$ be defined by
\[ p = h_\delta \ast \chi_{\Delta_2 \setminus \Delta_1},
\]
and note that $\supp p = B^{c}(I_\ast) \setminus B^{c-d-1}(I_\ast)$ since
$\supp h_\delta = \unitball{0}{\delta}$. Normalizing the function $p$
in a proper way will give us the function $g$ we are looking for. We
will normalize $p$ by the function $w$:
\begin{equation*}
w(x) = \sum_{j \in \mathbb{Z}} p (B^j x).
\end{equation*}
For a fixed $x\in \numbersys{R}^n \setminus \{0\}$ this sum has either $d$ or $d+1$ nonzero
terms, and $w$ is therefore bounded away from
$0$ and $\infty$:
\begin{equation*}
\exists c, C > 0 \, \colon c < w(x) < C \quad \text{for all } x\in
\numbersys{R}^n \setminus \{0\},
\end{equation*}
hence we can define a function $g\in C^\infty_0(\numbersys{R}^n)$ by
\begin{equation}
g(x) = \frac{p(x)}{w(x)} \quad \text{for } x\in \numbersys{R}^n \setminus \{0\},
\quad \text{and,} \quad g(0)=0.
\label{eq:definition-g-Rn}
\end{equation}
The function $g$ will be an almost everywhere $B$-dilative partition of unity as is seen by
using the $B$-dilative periodicity of $w$:
\begin{align*}
\sum_{j \in \mathbb{Z}} g (B^j x) = \sum_{j \in \mathbb{Z}}
\frac{p (B^j x)}{w(B^j x)}
= \sum_{j \in \mathbb{Z}}
\frac{p(B^j x)}{w(x)}
= \frac{1}{w(x)} \sum_{j \in \mathbb{Z}}p (B^j x) = 1.
\end{align*}
Since $p$ is supported on the annulus $B^{c}(I_\ast) \setminus
B^{c-d-1}(I_\ast)$, we can simplify the definition in
(\ref{eq:definition-g-Rn}) to get rid of the infinite sum in the
denominator; this gives us the following expression
\[ g(x) = p(x) / \sum_{j=-d}^{d}p(B^j x) \qquad \text{for
} x \in\numbersys{R}^n \setminus \{0\}. \]
We can obtain a more explicit expression for $p$ by the following
approach. Let $r_1=r_1(\theta_1, \dots, \theta_{n-1})$ and
$r_2=r_2(\theta_1, \dots, \theta_{n-1})$ denote the radial coordinates
of the surface of the ellipsoids $\partial B^{c-d-1}(I_\ast)$ and
$\partial B^c(I_\ast)$ parametrized by $n-1$ angular coordinates
$\theta_1, \dots, \theta_{n-1}$, respectively.
Finally, let $p \in C^\infty_0(\numbersys{R}^n)$ be defined by
\[ p(x) = \eta(\enorm{x}-r_1)\,\eta(r_2-\enorm{x}), \quad \text{with }
r_1= r_1(\theta_1, \dots, \theta_{n-1}) \text{ and } r_2=
r_2(\theta_1, \dots, \theta_{n-1})\]
where $\theta_1, \dots,
\theta_{n-1}$ can be found from $x$, and
\begin{equation*}
\eta(x)= \begin{cases} \mathrm{e}^{-1/x} \quad &x>0, \\ 0 \quad &x\le 0.
\end{cases}
\end{equation*}
\end{example}
|
{
"timestamp": "2010-09-23T02:02:01",
"yymm": "1009",
"arxiv_id": "1009.4351",
"language": "en",
"url": "https://arxiv.org/abs/1009.4351"
}
|
\section{Introduction}
In this paper we study nonnegative, measure-valued solutions of
the Cauchy problem for a one-dimensional drift-diffusion
equation
\begin{equation}\label{eq:rho}
\de_t{\rho}-\de_x\big(\de_x(\beta({\rho}))+V'{\rho}\big)=0 \quad {\rm in }\ (0,+\infty)\times \mathbb{R},
\qquad
{\rho}(0,\cdot)={\rho}_0 \quad \text{in } \mathbb{R}.
\tag{1.DDE}
\end{equation}
Here we assume that
\begin{equation}\label{hp:beta}
\beta\in C^1([0,+\infty)) \text{ is increasing,}\quad
\beta(0)=0,\quad
{\beta^{\infty}}:=\lim_{r\to +\infty} \beta(r)<+\infty,
\tag{1.$\beta$}
\end{equation}
and $V:\mathbb{R}\to \mathbb{R}$ is a $C^2$ driving potential,
satisfying the conditions
\begin{equation} \label{crescita-quadratica}
V''(x)\ge \lambda\quad \quad {\rm for\ all}\ x\in \mathbb{R};\qquad
{\liminf_{|x|\to+\infty}\frac{V(x)}{|x|^2}\ge0.}
\tag{1.$V$}
\end{equation}
We will look for solutions $t\mapsto \rho_t$
in the space $\PlusMeasuresTwo{\mathbb{R},\frak m}$ of nonnegative Borel measures with finite mass $\frak m={\rho}(\mathbb{R})$ and finite
quadratic momentum
\begin{equation}
\label{eq:7}
\Mom2{\rho}:=\int_\mathbb{R} |x|^2\,\d{\rho}(x)<+\infty.
\end{equation}
Conditions \eqref{hp:beta} describe the physical situation in which
the diffusion operator is very weak and {possibly} unable to smooth
out the solution if initially point masses are present.
This fact is reflected by the natural entropy functional $\mathcal F$ which
generates equations like \eqref{eq:rho} as gradient flow in
$\PlusMeasuresTwo{\mathbb{R},\frak m}$ and in particular
decays along the solutions of \eqref{eq:rho},
\begin{equation}
\label{eq:29}
\mathcal F(\rho):=\mathcal E(\rho)+\mathcal V(\rho),\quad
\mathcal E(\rho):=\int_\mathbb{R} E(u(x))\,\d x\quad\text{if }\rho=u\Leb
1+\rho^\perp,\quad
\mathcal V(\rho)=\int_\mathbb{R} V(x)\,\d\rho(x),
\end{equation}
where
{the convex energy density function $E:[0,+\infty)\to\mathbb{R}$ is defined as}
\begin{equation}
\label{eq:26}
E(r):=-\beta(r)-r\int_r^{+\infty} \frac{\beta'(s)}s\,\d s\quad
\text{so that}\quad \beta'(r) = rE''(r),\quad
E(0)=0,
\tag{1.$E$}
\end{equation}
and satisfies
\begin{equation}
\lim_{r\to+\infty}E(r)=-\beta_\infty\quad\text{and therefore}\quad
\lim_{r \to+\infty} \frac{E(r)}r=0,\label{eq:32}
\end{equation}
so that the (lower semicontinuous) integral functional $\mathcal E$ defined
by \eqref{eq:29}
{depends only on}
the regular part of a Borel measure (see for instance
\cite{DT84}).
Even worst, the energy density $E$ does not satisfy the
regularizing condition \cite[Thm. 10.4.8]{ags}
$\lim_{r\to+\infty}E(r)=-\infty$, which prevents a singular part
for measures with finite energy dissipation along \eqref{eq:rho},
thus in particular for any solution $\rho_t$ at positive time $t>0$.
\paragraph{Sub-linear diffusions and Bose-Einstein distribution.}
In order to better clarify the physical meaning of condition
\eqref{hp:beta}, let us briefly describe a situation in ${\R^d}$
in which the steady state of the
drift-diffusion equation is explicitly computable.
To this aim, for $x \in \mathbb{R}^d$, $d \ge 1$, let us fix $V(x) = |x|^2/2$, while, for a fixed
constant $\alpha >0$, the diffusion function $\beta(r)$ is defined by
\begin{equation}\label{alpha}
\beta(0)=0,\qquad \beta'(r) = \frac 1{1+ r^\alpha}.
\end{equation}
Then, since in this case the drift-diffusion equation
\[
\de_t{\rho}-\nabla_x \cdot\left( \nabla_x\beta ({\rho}) + x {\rho}\right)=0, \qquad x \in \mathbb{R}^d
\]
can be rewritten as
\begin{equation}\label{gen}
\de_t{\rho}-\nabla_x \cdot \left( \rho \nabla_x\left(\frac 1\alpha \ln \frac{\rho^\alpha}{1+ \rho^\alpha} + \frac{|x|^2}2\right)\right)=0,
\end{equation}
the steady states of \eqref{gen} are given by
\begin{equation}\label{steady-gen}
\rho_\infty (x) = \left[ e^{\alpha |x|^2/2 +\eta} - 1 \right]^{-1/\alpha}, \qquad \eta \ge 0.
\end{equation}
The (nonnegative) constant $\eta$ in \eqref{steady-gen} identifies the mass of the steady solution
\[
{\mathfrak m}_\eta = \int_{\mathbb{R}^d} \left[ e^{\alpha |x|^2/2 +\eta} - 1
\right]^{-1/\alpha} \, \d x.
\]
Since the mass ${\mathfrak m}_\eta$ is decreasing as soon as $\eta$ increases, the maximum value of ${\mathfrak m}_\eta$ is attained at $\eta =0$.
Note that, if $B_d$ denotes the measure of the unit sphere in $\mathbb{R}^d$, the value
\[
{\mathfrak m}_0 = \int_{\mathbb{R}^d} \left[ e^{\alpha |x|^2/2} - 1 \right]^{-1/\alpha} \, \d x = B_d \int_0^{+\infty} r^{d-1} \left[ e^{\alpha r^2/2} - 1
\right]^{-1/\alpha} \, \d r
\]
is bounded as soon as $\alpha > 2/d$. Whenever the constant $\alpha$ is chosen in this range, the value
\begin{equation}\label{crit}
{\mathfrak m}_{\rm c} = {\mathfrak m}_0 = B_d \int_0^{+\infty} r^{d-1} \left[ e^{\alpha r^2/2} - 1 \right]^{-1/\alpha} \, \d r < +\infty
\end{equation}
defines the so-called \emph{critical mass} of the problem, namely the maximal mass that can be achieved by a regular steady state. It is
interesting to remark that, in view of the lower bound on $\alpha$ which implies the existence of a critical mass, since in dimension one $\alpha
>2$, the function $\beta$ in \eqref{alpha} satisfies conditions \eqref{hp:beta}, in particular
\[
\lim_{r\to +\infty} \beta(r)<+\infty.
\]
This condition clearly can fail in higher dimensions.
The most relevant physical example of such type of steady states is furnished by the three-dimensional Bose-Einstein distribution
\cite{CC70}
\begin{equation}\label{BEsteady}
u_\infty (x) = \left[ e^{|x|^2/2 +\eta} - 1 \right]^{-1}
\end{equation}
that is the steady state of equation \eqref{gen} corresponding to $\alpha = 1$. In this case the function $\beta$ is explicitly computable to give
$\beta({\rho})= \ln (1+{\rho})$. Since $\alpha =1$, if the dimension $d \ge 3$, the Bose-Einstein distribution exhibits a {critical mass}. We remark
that in this case the energy functional $E(u)$ is the Bose-Einstein entropy
\[
E(u) = u\ln u - (1+u) \ln(1+u).
\]
One of the fundamental problems related to evolution equations that
relax towards a stationary state characterized by the existence of a
critical mass, is to show how, starting from an initial distribution
with a supercritical mass ${\mathfrak m}> {\mathfrak m}_{\rm c}$, the solution eventually develops
a singular part (the condensate), and, as soon as the singular part
is present, to be able to follow its evolution. We remark that in
general the condensation phenomenon is heavily dependent of the
dimension of the physical space. In dimension $d \le 2$, in fact,
the maximal mass ${\mathfrak m}_0$ of the Bose-Einstein distribution
\eqref{BEsteady} is unbounded, and the eventual formation of a
condensate is lost.
In order to simplify the mathematical difficulties, while maintaining the physical picture in which the steady state has a critical mass, in
\cite{BGT09} the one-dimensional case corresponding to a steady state of the form \eqref{steady-gen}, with $\alpha >2$ has been considered. Note
that the analysis of \cite{BGT09} refers to a linear diffusion with a super-linear drift
\[
\de_t \rho = \de_x\left( \de_x \rho + x \rho(1+ \rho^\alpha) \right),
\]
that is reminiscent of the Kaniadakis-Quarati model of Bose-Einstein particles \cite{KQ94}
\begin{equation}
\label{FPB} \de_t \rho = \nabla\cdot \left( \nabla \rho + x \rho(1+ \rho)\right).
\end{equation}
\paragraph{A measure-theoretic setting for diffusion equations.}
\
In the present paper we deal with an almost complete description of the time-evolution of the solution of problem \eqref{eq:rho} with a Borel
measure as initial datum. While the mathematical study of drift-diffusion kinetic equations with the Bose-Einstein density as steady state has
been considered before (cfr. \cite{EHV98, EMV03} and the references therein), to our knowledge, drift-diffusion equations of type \eqref{eq:rho} at
present have never been studied systematically.
Motivated by the previous remarks and by the degeneracy of the entropy functional $\mathcal F$ introduced in
\eqref{eq:29}, whose minimizers could exhibit concentration effect,
we address the study of \eqref{eq:rho} by the measure-theoretic
point of view recently developed in the framework of optimal
transport \cite{ags}. This approach, started by the pioneering papers
of \textsc{Jordan-Kinderlehrer-Otto} \cite{Jordan-Kinderlehrer-Otto98} and \textsc{Otto} \cite{Otto01},
provides a sufficiently general setting for measure-valued
solutions to \eqref{eq:rho}.
$\PlusMeasuresTwo{\mathbb{R},\frak m}$ endowed
with the so called $L^2$-Wasserstein distance is the natural ambient
space
for carrying on our analysis.
A first important fact is that
the entropy functional $\mathcal F$ \eqref{eq:29} turns out to be
displacement $\lambda$-convex, a crucial property
which holds only in the one-dimensional case, since the possibility
of entropies satisfying \eqref{eq:32} is prevented by
\textsc{McCann}'s condition
\cite{McCann97}
in higher dimensions.
Moreover, we are able to extend the results of \cite{ags} (which
for sublinear entropies covers the case when $\lim_{r\to+\infty}E(r)=-\infty$)
providing an explicit characterization of the
dissipation of $\mathcal F$, which is strictly related to
the ``Wasserstein differential'' of $\mathcal F$.
As a crucial byproduct of this analysis, we will find
the right condition that measure-valued solutions have to satisfy
in order to enjoy nice uniqueness and stability results. It
is worth mentioning here that the distributional formulation of \eqref{eq:rho}
does not provide enough information to characterize the solutions,
when a concentration on a Lebesgue negligible set occurs.
Applying the general theory
of gradient flows of displacement $\lambda$-convex functionals in
Wasserstein spaces, we can thus obtain a precise
characterization
of measure valued solutions to \eqref{eq:rho} and we can prove their
existence, uniqueness, and stability.
Further justifications showing that the notion of Wasserstein
solutions is well adapted to \eqref{eq:rho} come from natural
regularization/approximation results: we will show that
our solutions are both the limit of the simplest vanishing viscosity
approximation
of \eqref{eq:rho} and of smooth solutions generated by
regularization of the initial data.
We complete our analysis by studying the propagation of the
singularities, the structure of minimizers of $\mathcal F$ and of
stationary solutions,
and the asymptotic behavior of the solutions, showing general
convergence results to the minimizer of $\mathcal F$.
\paragraph{Plan of the paper.}
In the next section we will make precise our definition
of measure-valued solutions to \eqref{eq:rho} (\S
\ref{subsec:def})
and we
will present our main results concerning existence, uniqueness,
stability, and approximation of Wasserstein
solutions (\S \ref{subsec:main}).
The equation governing the propagation of their singularities is
considered
in \S \ref{subsec:propagation}.
\S \ref{subsec:minimizers} is devoted to a precise characterization of
minimizers
of $\mathcal F$ and of the critical mass; steady states are studied in \S
\ref{subsec:stationary} and \S \ref{subsec:asymptotic}
collects some results concerning the
asymptotic behaviour of Wasserstein solutions.
Section \ref{sec:Wass} briefly recalls some definitions and tools
of (one-dimensional) optimal transport, Wasserstein distance, and
the related (sub)differentiability properties of displacement
$\lambda$-convex functionals. Theorems \ref{thm:lsc-dissipation}
and \ref{th:charsubdiff} lie at the core
of our further developments. A last paragraph
devoted to a simple regularization of $\mathcal F$ by
$\Gamma$-convergence concludes the section.
The last section contains the proofs of all our
main results: the connection with
the general theory is discussed in \S \ref{sec:gf} and
\S \ref{sec:estimates} is devoted to the propagation
of the singularities;
the study of the minimizers of $\mathcal F$ and of the
related asymptotic behavior
of the solutions to \eqref{eq:rho} is
performed in the last part.
\section{Definitions and main results}
In this section we collect the main
definitions and results we shall prove in the rest of the paper.
\subsection{Wasserstein solutions to \eqref{eq:rho}}
\label{subsec:def}
\paragraph{The case of bounded initial densities and Lipschitz drifts.}
When (the Lebesgue density of) ${\rho}_0\in L^\infty(\mathbb{R})$ and the potential $V$ is such that
\begin{equation}
\label{eq:8}
V''(x)\le \mathsf{c}\quad\text{for every }x\in \mathbb{R},
\end{equation}
it is not difficult {(see \cite{Vazquez07} and next Corollary \ref{cor:linfty_bound})}
to show that {a smooth} solution ${\rho}_t$
{of \eqref{eq:rho}}
satisfies the \emph{a priori} estimate
\begin{equation}
\label{eq:9}
\sup_{t\in [0,T]}\|{\rho}_t\|_{L^\infty(\mathbb{R})}\le R_T:=\|{\rho}_0\|_{L^\infty(\mathbb{R})}{\mathrm e}^{\mathsf{c}\, T}\quad\text{for every }T>0,
\end{equation}
so that it is uniformly bounded in every bounded time interval $[0,T]$. We can infer from \eqref{eq:9} that the behavior of $\beta(r)$ as
$r\uparrow+\infty$ does not play any role, and a solution in $[0,T]$ could be easily obtained by solving \eqref{eq:rho} with respect to a
nonlinearity $\tilde\beta$ defined for instance by
\begin{equation}
\label{eq:10}
\tilde\beta(r):=
\begin{cases}
\beta(r)&\text{if }r\le 2 R_T,\\
\beta(2R_T)+\beta'(2R_T)(r-2 R_T)&\text{if }r>2R_T.
\end{cases}
\end{equation}
Denoting by $\mathsf{S}_t({\rho}_0)$ the solution ${\rho}_t$ generated by a bounded initial datum ${\rho}_0$, it is possible to check that $\mathsf{S}_t$ satisfies
the $L^1$ contraction property
\begin{equation}
\label{eq:11}
\big\|\mathsf{S}_t({\rho})-\mathsf{S}_t(\eta)\big\|_{L^1(\mathbb{R})}\le \|{\rho}-\eta\|_{L^1(\mathbb{R})}\quad\text{for every }{\rho},\eta\in L^1(\mathbb{R})\cap L^\infty(\mathbb{R}).
\end{equation}
Consequently $\mathsf{S}_t$ can be extended in a canonical way to a contraction semigroup in the cone $L^1_+(\mathbb{R})$ of nonnegative integrable densities.
\paragraph{Measure-valued solutions.}
In case the Lebesgue density of ${\rho}_0$ is not bounded
{or $V$ does
not satisfy \eqref{eq:8}}, the presence of a singular part in the
solution ${\rho}$ of \eqref{eq:rho} has to be taken into account, since
the boundedness
of $\beta$ is responsible of the (possible) presence of a critical
mass. {We shall see an example of a solution $\rho_t$ exhibiting a
singular part for every $t\ge0$ in the next Remark \ref{rem:singularex}.}
{In the following we will denote by $\PlusMeasures\mathbb{R}$ (resp.\
$\PlusMeasures{\mathbb{R},{\mathfrak m}}$) the space of nonnegative Borel
measures in $\mathbb{R}$ with finite mass (resp.\ with prescribed mass
${\mathfrak m}>0$)
and by $\PlusMeasuresTwo\mathbb{R}$ (resp.\ $\PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$) the
collection of measures in $\PlusMeasures\mathbb{R}$ (resp.\ in
$\PlusMeasures{\mathbb{R},{\mathfrak m}}$) with finite quadratic
momentum.}
In order to enucleate a precise notion of measure-valued solution, for every ${\rho}\in \PlusMeasures\mathbb{R}$ we
consider the classical Lebesgue decomposition
\begin{equation}
\label{eq:1}
{\rho}={\rho}^a+{\rho}^\perp,\quad
{\rho}^a=u\,\Leb1,
\end{equation}
where $u\in L^1(\mathbb{R})$ is the Lebesgue density of the absolutely continuous part ${\rho}^a$ of ${\rho}$ and ${\rho}^\perp$ is the singular part of
${\rho}$, concentrated on a set of Lebesgue measure $0$.
It is then natural to substitute the term $\beta({\rho})$ in \eqref{eq:rho} by $\beta(u)$ and then interpret \eqref{eq:rho} in the sense of
distributions. If we want
to obtain a good notion of solution,
we should add some further requirements to the
density $u$. The first one is of qualitative type,
and relies in considering $u$ as a continuous function on $\mathbb{R}$ with values in the extended set
$[0,+\infty]$, endowed with the usual topology.
\begin{definition}[Measures with continuous densities]
\label{def:right_measures}
We say that a measure ${\rho}={\rho}^a+{\rho}^\perp\in \PlusMeasures\mathbb{R}$ has
a generalized continuous density $u\in C^0(\mathbb{R};[0,+\infty])$ with
proper
domain $\Dom u:=\big\{x\in \mathbb{R}:u(x)<+\infty\big\}$ if
\begin{align}
\label{eq:2}
{\rho}^\perp\big(\Dom u\big)=0,\quad
\Leb 1\big(\mathbb{R}\setminus \Dom u\big)=0,\quad\text{and}\quad
{\rho}^a=u\,\Leb 1\restr{\Dom u}.
\end{align}
{We set $\Domp u:=\big\{x\in \mathsf D(u):u(x)>0\big\}$.}
We denote by $\CPlusMeasures\mathbb{R}$ the collection of all measures with
generalized continuous density and we set
$\CPlusMeasuresTwo\mathbb{R}:=\CPlusMeasures\mathbb{R}\cap \PlusMeasuresTwo\mathbb{R}$,
$\CPlusMeasuresTwo{\mathbb{R},{\mathfrak m}}:=\CPlusMeasures\mathbb{R}\cap \PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$.
\end{definition}
Notice that $\Dom u$ is a \emph{dense open} subset of $\mathbb{R}$, ${\rho}^\perp={\rho}\restr{\mathbb{R}\setminus \Dom u}$, and
\begin{equation}
\label{eq:4}
\lim_{x\to \bar x}u(x)=+\infty\quad\text{for every }\bar
x\in \partial \Dom u=\mathbb{R}\setminus
\Dom u.
\end{equation}
In particular, $\CPlusMeasures\mathbb{R}$ does not contain any purely singular measure: if ${\rho}^a=0$ then also ${\rho}^\perp$ vanishes.
If ${\rho}\in \CPlusMeasures\mathbb{R}$ then we will always identify its Lebesgue density $\d{\rho}/\d\Leb1$ with the (unique) continuous precise
representative $u\in C^0(\mathbb{R};[0,+\infty])$ given by Definition \ref{def:right_measures}. By \eqref{hp:beta} we can consider $\beta$ as a
continuous function defined on the extended set $[0,+\infty]$ and therefore the composition $\beta\circ u$ is a well defined real continuous
function on $\mathbb{R}$.
The second requirement is a quantitative estimate concerning
the ``generalized Fisher'' dissipation functional.
\begin{definition}[Generalized Fisher dissipation]
If ${\rho}$ belongs to $\CPlusMeasures\mathbb{R}$ with continuous density $u$
we set
\begin{equation}
\label{eq:5}
{\mathcal I}({\rho}):=\int_{{\Domp u}} \Big|\frac{\partial_x \beta(u)}{u}+V'\Big|^2 u\,\d
x+\int_{\mathbb{R}} |V'|^2\,\d{\rho}^\perp\quad \text{if}\quad\beta\circ
u\in W^{1,1}_{\rm loc}(\mathbb{R}).
\end{equation}
When $\beta\circ u\not\in W^{1,1}_{\rm loc}(\mathbb{R})$ or ${\rho}\not\in
\CPlusMeasures\mathbb{R}$, we simply set $\mathcal I({\rho}):=+\infty.$
\end{definition}
It turns out that $\mathcal I$ is a lower semicontinuous functional with respect to weak
convergence of measures in $\PlusMeasures\mathbb{R}$
(see Theorem \ref{thm:lsc-dissipation2}).
\begin{definition}[Wasserstein solutions to \eqref{eq:rho}]
\label{def:measuresolution}
We say that ${\rho}\in C^0([0,+\infty);\PlusMeasuresTwo{\mathbb{R},{\mathfrak m}})$
is a Wasserstein solution of problem \eqref{eq:rho} if,
denoting by ${\rho}_t$ the measure ${\rho}$ at the time $t$,
\begin{subequations}
\begin{equation}
\label{eq:3}
{\rho}_t\in \CPlusMeasures\mathbb{R} \text{ for $\Leb 1$-a.e.\ $t>0$,}
\end{equation}
\begin{equation}\label{eq:3bis}
\int_{T_0}^{T_1} \mathcal I({\rho}_t)\,\d t<+\infty\quad\text{for every } 0< T_0<T_1<+\infty,
\end{equation}
and
\begin{equation}\label{eq-debole}
\int_0^{+\infty} \int_\mathbb{R}\Big(-\de_t\varphi+ \de_x\varphi V'\Big)
\,\d{\rho}_t\, \d t+
\int_0^{+\infty}\int_\mathbb{R}\de_x\varphi\, \de_x\beta(u_t)\,\d
x\,\d t
=0 \quad \forall \varphi\in C_{\rm c}^\infty((0,+\infty)\times \mathbb{R}),
\end{equation}
\end{subequations}
where $u_t$ is the generalized continuous density of ${\rho}_t$ for
$\Leb1$-a.e. $t>0$.
\end{definition}
\begin{remark}[Convergence in $\PlusMeasuresTwo{\mathbb{R},\frak m}$]
{$\PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$ is a complete metric space endowed with}
the so called $L^2$-Wasserstein
distance $W_2(\cdot,\cdot)$.
More details on this distance will be given in the next section;
let us just recall that a sequence ${\rho}_n$
converges to ${\rho}$ in $\PlusMeasuresTwo{\mathbb{R},\frak m}$ as $n\uparrow+\infty$ iff
\begin{equation}
\label{eq:14}
\lim_{n\uparrow+\infty}\int_\mathbb{R}\varphi(x)\,\d{\rho}_n(x)=
\int_\mathbb{R}\varphi(x)\,\d{\rho}(x)\quad\text{for every }\varphi\in
C^0(\mathbb{R})\text{ with }
\sup_x\frac{|\varphi(x)|}{1+x^2}<+\infty.
\end{equation}
\end{remark}
\begin{remark}[The role of the generalized continuous density]
By neglecting condition \eqref{eq:3} one can easily construct
evolutions of purely
singular measures which solve \eqref{eq-debole} and are not
influenced at all by the diffusion term. We
take a finite number of $C^1$ curves $x_j:[0,+\infty)\to
\mathbb{R}$, $i=1,\cdots,N$,
which solve the differential equation $\dot x_j(t)=-V'(x_j(t))$ in
$[0,+\infty)$, and we set
\begin{equation}
\label{eq:6}
{\rho}_t:=\sum_{j=1}^N \alpha_j \delta_{x_j(t)}, \qquad \alpha_j\ge0.
\end{equation}
In this case ${\rho}^a_t\equiv0$ for every $t\ge0$, which implies
$\beta(u_t)\equiv0$ and \eqref{eq-debole} contains just the pure
transport contribution given by the first integral.
On the other hand, by taking a smooth approximating family
${\rho}^{\varepsilon} \to {\rho}_0$ in $\PlusMeasuresTwo{\mathbb{R},\frak m}$, we can see that \eqref{eq:6} is not the
limit of the corresponding solution ${\rho}^{\varepsilon}_t$ as ${\varepsilon}\downarrow0$ (see Theorem \ref{thm:main1}).
\end{remark}
\paragraph{Energy functional and Fisher dissipation.}
In order to understand both the role of the generalized Fisher
dissipation and the consequences of \eqref{eq:3bis},
let us recall the definition \eqref{eq:26} of the so-called
internal energy density $E:[0,+\infty)\to \mathbb{R}$ by the relation
\begin{equation}\label{def:E}
E(r):=-{\beta^{\infty}}+
\int_r^{+\infty} \big(1-\frac{r}s\big)\beta'(s)\,\d s
=
-\beta(r)-
r\int_r^{+\infty}\frac{\beta'(s)}s\,\d s.
\end{equation}
It is simple to check that $E$ is a convex nonpositive function satisfying
\begin{equation}\label{eq:12}
E\in C^2(0,+\infty),\quad
E(0)=0, \quad
\lim_{r\to 0^+}\frac{E(r)}{r\log r} = \beta'(0) \in [0,+\infty), \quad
{E^{\infty}}=\lim_{r\uparrow+\infty}E(r)=-{\beta^{\infty}},
\end{equation}
and
\begin{equation}\label{eq:121}
\beta'(r)=rE''(r), \quad
E'(r)=-\int_r^{+\infty}\frac{\beta'(s)}s\,\d s < 0, \qquad \forall r \in (0,+\infty).
\end{equation}
We associate the integral functional
\begin{equation}
\label{eq:54}
\mathcal E({\rho}):=\int_\mathbb{R} E(u(x))\,\d x \quad \text{whenever}\quad
{\rho}=u\Leb1 +{\rho}^\perp\in \PlusMeasures\mathbb{R},
\end{equation}
to the energy density $E$, the potential energy
\begin{equation}
\label{eq:83}
\mathcal V(\rho):=\int_\mathbb{R} V(x)\,\d{\rho}(x)
\end{equation}
to the potential $V$,
and the energy functional $\mathcal F:\PlusMeasures{\mathbb{R}}\to (-\infty,+\infty]$
\begin{equation}\label{def:EE}
\mathcal F({\rho}):=\mathcal E({\rho})+\mathcal V(\rho).
\end{equation}
Formal computations show that $\mathcal F$ and $\mathcal I$
satisfy the energy dissipation identity along solutions to \eqref{eq:rho}
\begin{equation}
\label{eq:13}
\mathcal F({\rho}_{t_1})+\int_{t_0}^{t_1} \mathcal I({\rho}_t)\,\d t=\mathcal F({\rho}_{t_0}) \qquad 0\leq t_0<t_1<+\infty.
\end{equation}
\subsection{Existence, stability, and approximation results.}
\label{subsec:main}
Recall that $\lambda\in \mathbb{R}$ is a lower bound for the second
derivative of $V$, see \eqref{crescita-quadratica}. Let us set
\begin{equation}
\label{eq:27}
\mathsf E_\lambda(t):=\int_0^t \mathrm e^{\lambda s}\,\d s=
\begin{cases}
\frac{\mathrm e^{\lambda t}-1}\lambda&\text{if }\lambda\neq 0,\\
t&\text{if }\lambda=0.
\end{cases}
\end{equation}
\begin{theorem}[Existence, uniqueness, stability, and comparison]
\label{thm:main1}
For every ${\rho}_0\in \PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$ there exists a
unique Wasserstein solution ${\rho}_t$ to \eqref{eq:rho} according to Definition
\ref{def:measuresolution}. This solution satisfies the regularization
estimate
\begin{equation}
\label{eq:18}
\mathcal F({\rho}_t)+\frac{\mathsf E_\lambda(t)}2\mathcal I({\rho}_t)\le \frak
m V(0)+\frac 1{2\mathsf E_\lambda(t)}\Mom 2{{\rho}_0}\quad\text{for every }t>0,
\end{equation}
the energy dissipation identity \eqref{eq:13}, and the dissipation inequality
\begin{equation}
\label{eq:19}
\mathcal I({\rho}_t)\le \mathcal I({\rho}_{t_0}) \mathrm e^{-2\lambda (t-t_0)}, \qquad \forall\, t\geq t_0\ge0.
\end{equation}
The map $\mathsf{S}_t:\PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}\to\PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$
defined by $\mathsf{S}_t({\rho}_0)={\rho}_t$ is a semigroup of
continuous maps in $\PlusMeasuresTwo{\mathbb{R},\frak m}$ satisfying the
stability property
\begin{equation}
\label{eq:15}
W_2(\mathsf{S}_t({\rho}_0),\mathsf{S}_t(\eta_0))\le \mathrm e^{-\lambda t}W_2({\rho}_0,\eta_0).
\end{equation}
If moreover $\rho_0\le \eta_0$ then $\mathsf{S}_t(\rho_0)\le
\mathsf{S}_t(\eta_0)$ for every $t\ge0$.
\end{theorem}
\begin{remark}[Singularities]
\label{rem:singular}
Recalling the definition \eqref{eq:5} of $\mathcal I$, the regularization
estimate \eqref{eq:18} shows that the solution given by Theorem
\ref{thm:main1} satisfies ${\rho}_t\in\CPlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$
for every $t>0$.
\end{remark}
{In the case when $V'$ is Lipschitz, the stability property
\eqref{eq:15} and a simple regularization of the initial datum
show that Wasserstein solutions are the limit of locally bounded
solutions satisfying \eqref{eq:9}.}
Another way to see that Definition \ref{def:measuresolution} provides the right notion of solution involves a classical viscous regularization of
\eqref{eq:rho} {combined with a suitable regularization of the potential $V$}.
Given a small parameter ${\varepsilon}>0$ let us consider the perturbed nonlinear functions
\begin{equation}
\label{eq:16}
\beta^{\varepsilon}(r):=\beta(r)+{\varepsilon} r,\quad r\in [0,+\infty),
\end{equation}
{and a family $V^{\varepsilon}$ of smooth and Lipschitz potentials such that
\begin{subequations}
\begin{gather}
\label{eq:82}
V^{\varepsilon}(x)\le V(x)+A|x|^2\quad
\lambda\le (V^{\varepsilon})''(x)\le \sup_\mathbb{R} V''\qquad
\text{for every }x\in\mathbb{R},\\
\label{eq:85}(V^{\varepsilon})^{(h)} \to V^{(h)}
\quad\text{as }{\varepsilon}\downarrow0
\quad\text{uniformly on compact sets of }\mathbb{R},
\quad h=0,1,2,\\
\label{eq:86}
\liminf_{|x|\to\infty}\frac{V^{\varepsilon}(x)}{|x|^2}\ge0\quad\text{uniformly
with respect to ${\varepsilon}$.}
\end{gather}
\end{subequations}
}
For every ${\rho}^{\varepsilon}_{0}\in \PlusMeasuresTwo{\mathbb{R},\frak m}$ we consider the problem
\begin{equation}\label{eq:17}
\partial_t {\rho}^{\varepsilon}-\partial_x\big(\partial_x\beta^{\varepsilon}(u^{\varepsilon})+(V^{\varepsilon})' {\rho}^{\varepsilon}\big)=0,
\quad {\rm in }\ (0,+\infty)\times \mathbb{R};
\quad
{\rho}^{\varepsilon}(0,\cdot)={\rho}^{\varepsilon}_0,\quad \text{in }\mathbb{R},
\end{equation}
the associated energy functional
\begin{equation}
\mathcal E^{\varepsilon}({\rho})=
\begin{cases}
\mathcal E({\rho}) + {\varepsilon}\displaystyle \int_\mathbb{R} u\log u \,\d x & \text{if } {\rho}=u\Leb{1} \ll \Leb{1}\\
+\infty & \text{if } {\rho}^\perp \not= 0
\end{cases}
\quad
{\mathcal V^{\varepsilon}(\rho):=\int_\mathbb{R} V^{\varepsilon}(x)\,\d\rho,\quad
\mathcal F^{\varepsilon}=\mathcal E^{\varepsilon}+\mathcal V^{\varepsilon},}
\end{equation}
and the corresponding Fisher-dissipation
\begin{equation}
\label{eq:46bis}
\mathcal I^{\varepsilon}({\rho}):=\int_\mathbb{R}\Big|\frac{\partial_x\beta^{\varepsilon}(u)}u+(V^{\varepsilon})'\Big|^2u\,\d x \qquad\text{if }{\rho}=u\Leb1,\
\beta^{\varepsilon}(u)\in W^{1,1}_{\rm loc}(\mathbb{R}).
\end{equation}
As usual $\mathcal I^{\varepsilon}({\rho})=+\infty$ if $u\not\in W^{1,1}_{\rm loc}(\mathbb{R})$
or
${\rho}\not\ll\Leb 1$.
\begin{theorem}[Convergence of viscous regularizations]
\label{thm:main2}
For every {${\rho}^{\varepsilon}_{0}=u^{\varepsilon}_0\Leb1\in \PlusMeasuresTwo{\mathbb{R},\frak
m}$
with $u^{\varepsilon}_0\in C^1_c(\mathbb{R})$},
there exists a unique {smooth} solution ${\rho}^{\varepsilon}=u^{\varepsilon}\Leb 1\in
C^0([0,+\infty); \PlusMeasuresTwo{\mathbb{R},\frak m})$
of problem \eqref{eq:17} {satisfying $\mathcal I^{\varepsilon}(\rho^{\varepsilon})\in L^1_{loc}(0,+\infty)$.}
Moreover \eqref{eq:18}, \eqref{eq:19}, and \eqref{eq:13} hold with $\mathcal F, \mathcal I$ replaced by $\mathcal F^{\varepsilon}, \mathcal I^{\varepsilon}$, respectively.
If ${\rho}^{\varepsilon}_0\to {\rho}_0$ in $\PlusMeasuresTwo{\mathbb{R},\frak m}$
and
$\sup_{\varepsilon} \mathcal F^{\varepsilon}({\rho}^{\varepsilon})<+\infty$, then ${\rho}^{\varepsilon}_t$ converges in
$\PlusMeasuresTwo{\mathbb{R},\frak m}$ to the unique Wasserstein solution ${\rho}_t$ of
problem \eqref{eq:rho}
as ${\varepsilon}\downarrow0$ for every $t>0$.
Moreover
$u^{\varepsilon}_t\to u_t$ uniformly on compact sets
of $\Dom{u_t}$ for every $t>0$.
\end{theorem}
The proofs of Theorems \ref{thm:main1} and \ref{thm:main2} {take
advantage of}
the theory of gradient flows of convex functionals with respect to the
Wasserstein distance \cite{ags} and will be given in Section \ref{sec:gf}.
\begin{remark}[Non smooth potentials]
\label{rem:nonsmooth}
Theorems \ref{thm:main1} and \ref{thm:main2} are still true in the case when $V$ is a
general $\lambda$-convex function, i.e.\ the condition
\eqref{crescita-quadratica} on the lower bound on $V''$ (which we
assumed
for the sake of simplicity) is
replaced by
\begin{equation}
\label{eq:79}
x\mapsto V(x)-\frac\lambda 2x^2\quad\text{is convex in }\mathbb{R}.
\end{equation}
\eqref{eq:79} implies that $V$ is differentiable $\Leb 1$-almost
everywhere, so that the first occurence of $V'$ in the definition
\eqref{eq:5} of $\mathcal I$
still makes sense as it is integrated with respect to $\Leb
1$. The second integral term in \eqref{eq:5} should be replaced by
\begin{equation}
\label{eq:81}
\int_\mathbb{R} |\partial^\circ V(x)|^2\,\d\rho^\perp(x)
\end{equation}
where $\partial^\circ V(x)$ denotes the element of minimal norm in
the (non empty) Frechet subdifferential $\partial V$ of $V$.
\end{remark}
\subsection{Propagation of singularities.}
\label{subsec:propagation}
In this section we want to study the evolution of the singular
part $\rho_t^\perp$ of the Wasserstein solution $\rho_t$ to
\eqref{eq:rho}.
By Remark \ref{rem:singular} we know that $\rho_t=u_t\Leb1+\rho_t^\perp\in
\CPlusMeasuresTwo\mathbb{R}$ for every $t>0$, so that the support of
$\rho_t^\perp$ coincides with
the set where the (continuous representative of the) density $u_t$
takes the value $+\infty$. We thus call
$\CDom{u_t}:=\mathbb{R}\setminus\Dom{u_t}=\big\{x\in
\mathbb{R}:u_t(x)=+\infty\big\} $
{and we will show that the evolution of $\CDom{u_t}$ follows the
flow generated by $-V'$.}
Let us first introduce the evolution semigroup $\mathsf{X}$ on $\mathbb{R}$ generated by $-V'$, thus satisfying
\begin{equation}
\label{eq:20}
\frac \d{\d t}\mathsf{X}_t(x)=-V'(\mathsf{X}_t(x)),\quad
\mathsf{X}_0(x)=x\quad\text{for every }x\in \mathbb{R}.
\end{equation}
Since $V'$ is of class $C^1$ and, by \eqref{crescita-quadratica},
$$\big(V'(x)-V'(y)\big)(x-y)\ge\lambda|x-y|^2\quad \text{for every }x,y\in \mathbb{R},$$
$\mathsf{X}_t$ is a family of diffeomorphisms {mapping $\mathbb{R}$ onto the open
set $\mathsf R_t:=\mathsf{X}_t(\mathbb{R})$.}
We set
\begin{equation}
\label{eq:21}
{\mathsf J}_t:=\mathsf{X}_t\big(\CDom{u_0}\big),\quad
{\mathsf D}_t:=\mathsf{X}_t\big(\Dom{u_0}\big),\quad t\ge0,
\end{equation}
and we notice that $\mathsf J_t={\mathsf R_t}\setminus \mathsf D_t$ is a closed
subset of {$\mathsf R_t$}, since $\mathsf D_t$ is open.
{If $\sigma\in \PlusMeasures\mathbb{R}$, the push-forward $(\mathsf{X}_t)_\#\sigma$
through
$\mathsf{X}_t$ is the Borel measure defined by}
\[
(\mathsf{X}_t)_\# \sigma(A) := \sigma\big(\mathsf{X}_t^{-1}(A)\big)\quad\text{for each Borel set $A \subset \mathbb{R}$}.
\]
\begin{theorem}[Propagation of singularities]
\label{thm:main3}
If ${\rho}_0\in\CPlusMeasuresTwo{\mathbb{R}}$ and
${\rho}_t=u_t\Leb 1+{\rho}_t^\perp\in \CPlusMeasuresTwo\mathbb{R}$ is the
unique {Wasserstein} solution of \eqref{eq:rho},
then
\begin{equation}
\label{eq:68}
\partial_t {\rho}_t^\perp-\partial_x\big({\rho}_t^\perp\,V'\big)\le
0\quad\text{in the sense of distributions,}
\quad
{\lim_{t\downarrow0}\rho_t^\perp\le \rho_0^\perp
\quad\text{weakly in }\PlusMeasures\mathbb{R}.}
\end{equation}
In particular
\begin{equation}
\label{eq:22}
\CDom{u_t}\subset \mathsf J_t, \quad
{\rho}_t^\perp\le (\mathsf{X}_t)_\#{\rho}_0^\perp,\quad
\text{for every }t\ge0,
\end{equation}
so that for every Borel set $A\subset \mathbb{R}$
\begin{equation}\label{eq:23}
{\rho}_t^\perp(A)\le {\rho}_0^\perp \big(\mathsf{X}_t^{-1}(A)\big).
\end{equation}
In particular ${\rho}_t^\perp$ is concentrated in $\mathsf{X}_t(\CDom{u_0})$
and $u_t$ is finite in $\mathsf{X}_t(\Dom{u_0})$.
\end{theorem}
The proof of Theorem \ref{thm:main3} will be carried out in Section \ref{sec:estimates}.
The case when ${\rho}_0^\perp=\sum_{j=1}^N\alpha_j\delta_{x_j}$ with $x_1<x_2<\cdots<x_N$ and $\alpha_j>0$ is of particular interest. In this case,
from Theorem \ref{thm:main3} we deduce that ${\rho}_t=u_t\Leb1+{\rho}_t^\perp$ with
\begin{equation}
\label{eq:24}
{\rho}_t^\perp=\sum_{j=1}^N \alpha_j(t)\delta_{x_j(t)},\quad
x_j(t)=\mathsf{X}_t(x_j),
\end{equation}
where $\alpha_j:[0,+\infty)\to [0,+\infty)$ { is nonincreasing.}
Theorem \ref{thm:main3} can be equivalently formulated in terms of
the density $u_t$ of the regular part of $\rho_t$:
\begin{corollary}[The regular part is a supersolution]
If $\rho_t=u_t\Leb1+\rho_t^\perp\in \CPlusMeasuresTwo\mathbb{R}$ is a
Wasserstein solution to \eqref{eq:rho} then $u_t$ is a
supersolution of \eqref{eq:rho}, i.e.
\begin{equation}
\label{eq:71}
\partial_t
u-\partial_x\big(\partial_x\beta(u)+V'\,u\big)\ge0\quad\text{in
the sense of distributions in }(0,+\infty)\times\mathbb{R}.
\end{equation}
\end{corollary}
\subsection{Minimizers of the energy functional and critical mass.}
\label{subsec:minimizers}
In this section we will assume that the potential $V$ satisfies the
coercivity condition
\begin{equation}
\label{eq:95}
\lim_{|x|\to+\infty}V(x)=+\infty
\quad \text{and we set}\quad
V_{\rm min}:= \min_\mathbb{R} V,\quad
Q:=\big\{x\in \mathbb{R}:V(x)=V_{\rm min}\big\},
\tag{2.coer}
\end{equation}
and we study the minimizers of the functional $\mathcal F$,
which are particular steady states of equation \eqref{eq:rho}.
The structure of the minimizers of $\mathcal F$ is governed by two critical constants
and two functions, with their inverses.
The first function is $r\mapsto -E'(r)$: it is a decreasing
homeomorphism between $(0,+\infty)$ and the interval $(0,\frak d)$, which
can be characterized by
the constant
\begin{displaymath}
\frak d:=-\lim_{x\to 0^+}E'(x)=\int_0^{+\infty}\frac{\beta'(s)}s\,\d
s\in (0,+\infty].
\end{displaymath}
Notice that $\frak d$ is finite if and only if $s\mapsto \beta'(s)/s$ is integrable
in a right neighborhood of $0$.
We can thus consider
the pseudo-inverse function
$H:(0,+\infty)\to [0,+\infty)$ defined by
$$
H(v)=\begin{cases}(E')^{-1}(-v)& \text{if }v\in(0,\frak d)\\
0& \text{if $\frak d<+\infty$ and }v\in [\frak d,+\infty)
\end{cases}
$$
which is decreasing in the interval $(0,\frak d)$.
The second function is
\begin{displaymath}
M_\mathbb{R}(v):=\int_\mathbb{R} H(V(x)-v)\,\d x,\quad
v\le V_{\rm min}.
\end{displaymath}
In order to avoid a degenerate situation, we will assume that $V$
satisfies the integrability condition
\begin{equation}
\label{eq:96}
\int_{\mathbb{R}\setminus \tilde Q}H(V(x)-V_{\rm min})\,\d
x<+\infty,\quad\text{for some bounded open neighborhood $\tilde Q$
of $Q$}.
\tag{2.int}
\end{equation}
\eqref{eq:96} yields $M_\mathbb{R}(v)<+\infty$ for every $v<V_{\rm min}$
so that
$M_\mathbb{R}$ is an increasing homeomorphism between $(V_{\rm min}-\frak d,V_{\rm
min})$ and the interval $(0,{\mathfrak m}_{\rm c})$,
where the critical mass ${\mathfrak m}_{\rm c}$ is defined by
\begin{equation}
\label{eq:28}
{\mathfrak m}_{\mathrm c}:=\lim_{v\uparrow V_{\rm min}}M_\mathbb{R}(v)=
\int_\mathbb{R} H(V(x)-V_{\rm min})\,\d x\in (0,+\infty].
\end{equation}
If $M^{-1}:(0,\frak m_\mathrm c)\to (V_{\rm min}-\frak d,V_{\rm min})$ denotes the
inverse map of $M$, we eventually set
\begin{equation}\label{eq:31}
\frak v:=\begin{cases}
M_\mathbb{R}^{-1}({\mathfrak m}) & \text{if } {\mathfrak m} < {\mathfrak m}_\mathrm c \\
V_{\rm min} & \text{if } {\mathfrak m}\geq {\mathfrak m}_\mathrm c.
\end{cases}
\end{equation}
\begin{theorem}[Characterization of minimizers]
\label{thm:main4}
If $V$ satisfies \eqref{eq:95} then
$\mathcal F$ attains its minimum on $\PlusMeasures{\mathbb{R},{\mathfrak m}}$.
If $V$ also satisfies \eqref{eq:96} then
a measure $\rho\in \PlusMeasures{\mathbb{R},{\mathfrak m}}$ is a minimizer of
$\mathcal F$ if and only if it belongs to $\CPlusMeasures{\mathbb{R},{\mathfrak m}}$ and
its decomposition ${\rho}_{\rm min}=u_{\rm min}\Leb 1+{\rho}_{\rm
min}^\perp$ satisfies
\begin{equation}\label{eq:30}
\begin{aligned}
u_{\rm min}(x)=H(V(x)-\frak v),\quad
{\rho}_{\rm min}^\perp(\mathbb{R}\setminus Q)=0,\quad
{\rho}_{\rm min}^\perp (Q)=(\frak m - \frak m_\mathrm c)^+.
\end{aligned}
\end{equation}
$\rho_{\rm min}$ belongs to $\PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$ if $V$
satisfies
the condition (stronger than \eqref{eq:96})
\begin{equation}
\label{eq:97}
\int_{\mathbb{R}\setminus \tilde Q}|x|^2\,H(V(x)-V_{\rm min})\,\d
x<+\infty,\quad\text{for some bounded open neighborhood $\tilde Q$ of $Q$}.
\end{equation}
\end{theorem}
\begin{remark}
\label{rem:examples}
\
\begin{itemize}
\item
In the case when $\frak m \le \frak m_{\mathrm c},$
the minimizer $\rho_{\rm min}=u_{\rm min}\Leb 1$
is unique and $\rho_{\rm min}^\perp=0$.
If ${\mathfrak m}<{\mathfrak m}_{\rm c}$, $u_{\rm min}$ is bounded, whereas
if ${\mathfrak m}={\mathfrak m}_{\rm c}$, $u_{\rm min}(x)=+\infty$ for every $x\in Q$.
Last, if $\frak m > \frak m_{\mathrm c}$
the minimizer has a nontrivial singular part and
it is unique only when $Q$ is a singleton.
\item
As already pointed out, the existence of the critical mass ${\mathfrak m}_{\rm
c}<+\infty$
depends on the behavior of the $\beta(r)$ for large values of $r$
{and
on the local behaviour of $V$ near $Q$}.
\item
{If $\frak d<+\infty$
then
the support of ${\rho}_{\rm min}$ is compact and it is contained in
the sublevel of $V$
$\{ x\in\mathbb{R}:V(x)\le \frak v+\frak d\}$.}
\item {If $Q$ is an interval (in particular if $V$ is convex) then the minimizer of $\mathcal F$ is unique.
This property is always true when ${\mathfrak m}_\mathrm c=+\infty$;
when ${\mathfrak m}_\mathrm c<+\infty$, the fact that $Q$ is a closed interval
and \eqref{eq:28} show that $Q$ is a singleton.}
\end{itemize}
\end{remark}
\subsection{Stationary solutions}
\label{subsec:stationary}
In this section we will study the stationary Wasserstein solutions
of \eqref{eq:rho},
i.e.\ constant measures $\rho\in \PlusMeasuresTwo{\mathbb{R}}$ which
solve \eqref{eq:rho}.
As a starting point, we observe that steady states can be
characterized as measures
with vanishing Fisher dissipation.
\begin{theorem}
\label{thm:Fisher=0}
A measure $\rho\in \PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$ is a stationary
Wasserstein
solution of \eqref{eq:rho} iff $\rho\in
\CPlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$ and
$\mathcal I(\rho)=0$.
\end{theorem}
Of course, any minimizer $\rho$ of $\mathcal F$ in
$\PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$ satisfies $\mathcal I(\rho)=0$ and it is a
stationary solution,
but in general one can expect that other stationary solutions exist.
Their structure depends in a crucial way on
$\frak d$; the simplest case is when $\frak d=+\infty$.
\begin{theorem}[Characterization of stationary measures I]
\label{thm:statI}
Let us suppose that $V$ satisfies
\eqref{eq:95} and \eqref{eq:96}.
If $\frak d=+\infty$ then for every $\rho\in \PlusMeasures{\mathbb{R},{\mathfrak m}}$
\begin{equation}\label{nulldissipation}
\text{$\mathcal I({\rho})=0 \quad
\Leftrightarrow\quad {\rho}$ is a minimizer for
$\mathcal F$ in $\PlusMeasures{\mathbb{R},{\mathfrak m}}$.}
\end{equation}
In particular, a measure $\rho\in
\PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$ is
a stationary solution if and only if it is a minimizer of $\mathcal F$.
\end{theorem}
The case when $\frak d<+\infty$ is more complicated and requires some
preliminary definition.
%
\begin{definition}
\label{def:adm_interval}
Let us suppose that $\frak d<+\infty$.
We say that a bounded open interval $I=(a,b)\subset\mathbb{R}$ is
an admissible local sublevel of $V$ if
\begin{equation}
\label{eq:98}
V(a)=V(b),\quad
\frak v_I:=V(a)-\frak d\le V(x)< V(a)\quad\text{for every $x\in
(a,b)$},
\end{equation}
and
\begin{equation}
\label{eq:99}
M_I:=\int_a^b H\big(V(x)-\frak v_I\big)\,\d x<+\infty.
\end{equation}
We set $Q_I:=\big\{x\in I
:V(x)=\min_I V\big\}$.
\end{definition}
Notice that $Q_I$ is not empty iff
\begin{gather}
\label{eq:98bis}
\frak v_I=V(a)-\frak d=\min_{I}V.
\end{gather}
If $Q_I$ is empty, i.e.\ $\frak v_I<\min_I V$,
then condition \eqref{eq:99} is always satisfied.
If $u:\mathbb{R}\to [0,+\infty]$ is a continuous map, we set
\begin{equation}
\label{eq:104}
\begin{aligned}
\Pos u:=&\big\{x\in \mathbb{R}:u(x)>0\big\},\\
\Conn u:=&\text{the
collection of all the connected components of $\Pos u$.}
\end{aligned}
\end{equation}
\begin{theorem}[Characterization of stationary measures II]
\label{thm:main_stationary}
Let us suppose that $V$ satisfies
\eqref{eq:95} and \eqref{eq:96}.
If $\frak d<+\infty$ a measure
${\rho}=u\Leb 1+\rho^\perp\in \CPlusMeasures{\mathbb{R},{\mathfrak m}}$ satisfies
$\mathcal I(\rho)=0$
if and only if it
satisfies the following
three conditions:
\begin{enumerate}
\item
All the connected components in $\Conn u$ of the open set
$\Pos u$ are admissible local
sublevels of $V$
according to Definition \ref{def:adm_interval}.
\item
\begin{equation}
\label{eq:103}
u\restr{I}=H(V(x)-\frak v_I)\quad\text{for every }I\in \Conn u.
\end{equation}
\item
If $Q(u):=\bigcup_{I\in
\Conn u}Q_I$
\begin{equation}
\label{eq:84}
\text{$\rho^\perp$ is concentrated on $Q(u)$,
and}
\quad\frak m=\sum_{I\in \Conn u}M_I
+\rho^\perp(\mathbb{R}).
\end{equation}
\end{enumerate}
\end{theorem}
\begin{corollary}
\label{cor:obvious}
If $V$ satisfies
\eqref{eq:95}, \eqref{eq:96},
and
\begin{equation}
\text{the set $Q$ of \eqref{eq:95} is an interval $[q_-,q_+]$, $V'\ge0$ in
$(q_+,+\infty)$, and $V'\le 0$ in $(-\infty,q_-),$}
\label{eq:110}
\end{equation}
(\eqref{eq:110} is always satisfied if $V$ is convex),
then
\eqref{nulldissipation} holds
and there exists a unique stationary measure in
$\PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$ which coincides with the unique
minimizer of $\mathcal F$ in $\PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$.
\end{corollary}
\begin{remark}
\label{rem:converse_obvious}
It is possible to prove a converse form of Corollary
\ref{cor:obvious}:
if $\frak d<+\infty$ and
for every value of ${\mathfrak m}>0$ there exists a unique steady state
in $\PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$ then $V$ satisfies \eqref{eq:110}.
\end{remark}
\begin{example}\label{ex:1}
Let us choose $\beta(r)=\arctan r$, so that $E(r)=\displaystyle r\log\left(\frac{ r}{\sqrt {1+r^2}}\right)-\arctan r$ and $E'(r)=\displaystyle
\log\left(\frac{ r}{\sqrt {1+r^2}}\right)$. Notice that $\frak d=+\infty$. One can compute explicitly $\displaystyle
H(v)=\frac{\mathrm e^{-v}}{\sqrt{1-\mathrm e^{-2v}}}$, for $v>0$. If the potential is $V(x)=|x|^\alpha$, with $\alpha>1$, the critical mass is defined by
$\displaystyle {\mathfrak m}_{\rm c}=\int_\mathbb{R} \frac{\mathrm e^{-|x|^\alpha}}{\sqrt{1-\mathrm e^{-2|x|^\alpha}}}\,\d x.$ It follows that ${\mathfrak m}_{\mathrm c}<+\infty$ if and only if $\alpha<2$.
We find that
\begin{equation}
u_{\rm min}(x)=\frac{\mathrm e^{-|x|^\alpha+\frak
v}}{\sqrt{1-\mathrm e^{-2|x|^\alpha+2\frak v}}}.
\end{equation}
If $\alpha \ge 2$, for every value of the mass ${\mathfrak m}$, the unique minimum point, which is also the unique stationary solution, can not have a
singular part, and it is bounded and positive. The same situation occurs when $\alpha<2$ and ${\mathfrak m} < {\mathfrak m}_\mathrm c$. If $\alpha<2$ and ${\mathfrak m} = {\mathfrak m}_\mathrm c$, then the unique stationary state is infinite
at $x=0$ but without a singular part, whereas for ${\mathfrak m} >{\mathfrak m}_\mathrm c$ the singular part is ${\rho}^\perp_{\rm min}=({\mathfrak m}-{\mathfrak m}_\mathrm c)\delta_0$.
\end{example}
\begin{example}\label{ex:2}
Let us choose $\beta(r)=\frac{r^2}{1+r^2}$. Then $E(r)=-r\arctan (1/r)$ and $E'(r)=\frac{r}{1+r^2}-\arctan (1/r)$. In this case
$\frak d=\frac \pi 2$.
Let us observe that $E'(r)$ has the same behavior of $r\mapsto-1/r^3$ as $r\to +\infty$. Therefore $H(v)$ has the same behavior of $v\mapsto
v^{-1/3}$ for $v\to 0^+$. Considering again the potential $V(x)=|x|^\alpha$, with $\alpha>1$, it follows that ${\mathfrak m}_{\mathrm c}<+\infty$ if and only
if $\alpha<3$.
The support of the unique stationary state
is $\{x\in\mathbb{R}: |x|\le (\frak v+\pi/2)^{1/\alpha}\}$
and it is compact for every value of ${\mathfrak m}$ and $\alpha$.
Finally we show
a measure ${\rho}\in\CPlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$ satisfying
$\mathcal I({\rho})=0$ which is not of the form \eqref{eq:30}.
To this aim we consider the double well potential $V(x)=\pi(x-1)^2(x+1)^2$. Let ${\mathfrak m}>{\mathfrak m}_\mathrm c$. Defining $u(x)=H(V(x))$ for $x>0$ and $u(x)=0$
for $x\leq 0$, we observe that $u$ is continuous on $\mathbb{R}$ with values in $[0,+\infty]$ and $\int_\mathbb{R} u(x)\,\d x = {\mathfrak m}_\mathrm c/2$. Consequently, the
measure ${\rho}= u\Leb{1}+({\mathfrak m}-{\mathfrak m}_\mathrm c/2)\delta_1$ belongs to $\CPlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$, satisfies $\mathcal I({\rho})=0$ but is not of
the form \eqref{eq:30}.
We can construct a similar example when $V$ has a local minimum greater than $V_{\rm min}$. For instance we can consider a potential $V$ defined
by $V(x)=2\pi(x+1)^2+1$ for $x<-1/2$, $V(x)=2\pi(x-1)^2$ for $x>1/2$ and suitably defined in $[-1/2,1/2]$ in order to satisfy the condition $V(x)>\pi/2$ and the
$\lambda$-convexity assumption. Then the support of ${\rho}_{\rm min}$ is contained in $[-3/2,-1/2]\cup[1/2,3/2]$. Let us define
$u(x)=H(V(x)-1)$ for $x<0$ and $u(x)=0$ for $x\geq 0$, and ${\rho}= u\Leb{1}+({\mathfrak m}-\tilde{\mathfrak m}_\mathrm c)\delta_{-1}$, where $\tilde{\mathfrak m}_\mathrm c:=
{\mathfrak m}_\mathrm c - \int_{-\infty}^0H(V(x))\,\d x$. Then ${\rho}\in\CPlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$, $\mathcal I({\rho})=0$ but ${\rho}$ is not of the form
\eqref{eq:30} and it is not a minimizer of $\mathcal F$.
\end{example}
\begin{remark}
We point out that the case $\frak d=+\infty$ reveals some analogies with a diffusion which is linear near to $0$. In this case we
have the immediate strict positivity of the solution also starting from compactly supported initial data.
On the contrary, the case $\frak d<+\infty$ corresponds to a slow diffusion near to $0$. In this case, starting from compactly supported
initial data the solution could remain compactly supported for all
time and it may happen that
as $t\to+\infty$ the solution converges to a steady state which is not
a global minimum of $\mathcal F$.
\end{remark}
\begin{remark}[Examples of singular solutions]
\label{rem:singularex}
Let $\rho:=u\Leb 1+\rho^\perp\in \CPlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$ be a
steady state
of \eqref{eq:rho} with
$\rho^\perp\neq 0$: e.g., one can consider the case when
${\mathfrak m}_\mathrm c<+\infty$ and take a minimizer of $\mathcal F$ with
${\mathfrak m}>{\mathfrak m}_\mathrm c$.
If $\tilde\rho_0=\tilde u\Leb 1+\rho^\perp\in \CPlusMeasuresTwo{R,{\mathfrak m}}$ with
$\tilde u\ge u$ then the comparison principle shows that the
Wasserstein
solution $\tilde\rho_t$ of \eqref{eq:rho} with initial datum $\tilde
\rho_0$
is singular and its singular part is $\rho^\perp$ for every $t\ge0$.
\end{remark}
\subsection{Asymptotic behaviour}
\label{subsec:asymptotic}
{Let us first considering the case of a convex potential $V$.}
Here we can
apply the general results about the asymptotic behavior
for displacement convex functionals (see \cite{ags}).
Moreover, as we observed in Remark \ref{rem:examples}, the specific
form of the functional $\mathcal F$ yields that it has only one minimizer
${\rho}_{\rm min}$ in each class $\CPlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$ which
is also the unique steady state by Theorem \ref{thm:statI} and
Corollary \ref{cor:obvious}: the study of the asymptotic behaviour
is therefore greatly simplified.
\begin{theorem}[Asymptotic behavior I: the convex case]\label{thm:main6}
Let us assume that the potential $V$ is convex (i.e.\
\eqref{crescita-quadratica} is satisfied with $\lambda=0$)
and
satisfies \eqref{eq:95} and \eqref{eq:97},
and
let $\rho_{\rm min}$ be the unique minimizer of $\mathcal F$ in
$\PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$.
If $\rho$ is a Wasserstein solution to \eqref{eq:rho} in
$\PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$
then $\rho_t$ weakly converges to $\rho_{\rm min}$ as $t\to+\infty$
in the duality
with continuous and bounded functions. Moreover, for every $t\in (0,+\infty)$
\begin{equation}
\label{eq:105}
\displaystyle \mathcal F({\rho}_t)-\mathcal F({\rho}_{\rm min}) \leq \frac{W^2_2({\rho}_0,{\rho}_{\rm min})}{2t} , \quad
\displaystyle \mathcal I({\rho}_t) \leq \frac{W^2_2({\rho}_0,{\rho}_{\rm
min})}{t^2}.
\end{equation}
If the potential $V$ also satisfies \eqref{crescita-quadratica} with
$\lambda>0$, then
for every $t>0$ we have the exponential estimates
\begin{gather}
W_2({\rho}_t,{\rho}_{\rm min}) \leq \mathrm e^{-\lambda t} W_2({\rho}_0,{\rho}_{\rm min}),\\
\mathcal F({\rho}_t)-\mathcal F({\rho}_{\rm min}) \leq \mathrm e^{-2\lambda t} \big(\mathcal F({\rho}_0)-\mathcal F({\rho}_{\rm min})\big),\quad
\mathcal I({\rho}_t) \leq \mathrm e^{-\lambda t}\frac{W^2_2({\rho}_0,{\rho}_{\rm min})}{t^2}.
\end{gather}
\end{theorem}
The last result concerns more general potentials $V$: a simple
characterization
of the asymptotic behaviour of a Wasserstein solution is possible
only when there exists a unique steady state for \eqref{eq:rho}
(which therefore coincides with the minimizer of $\mathcal F$):
this is the case when $\frak d=+\infty$ and
$V$ satisfies \eqref{eq:95} and \eqref{eq:97}, or when $\frak d<+\infty$ and $V$
satisfies
the conditions of Corollary \ref{cor:obvious}.
\begin{theorem}[Asymptotic behavior II]
\label{thm:main5}
Let us suppose that $V$ satisfies \eqref{eq:95} and \eqref{eq:96}
and
let us assume that there exists a unique steady state $\bar\rho\in
\PlusMeasures{\mathbb{R},{\mathfrak m}}$
with $\mathcal I(\bar\rho)=0$ ($\bar\rho$ is also the unique minimizer of
$\mathcal F$ in $\PlusMeasures{\mathbb{R},{\mathfrak m}}$).
If $\rho$ is a Wasserstein solution to \eqref{eq:rho} in
$\PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$ then
\begin{equation}
\label{eq:109}
\rho_t\rightharpoonup\bar\rho\quad\text{weakly as }t\to+\infty,\quad
\lim_{t\uparrow+\infty}\mathcal I(\rho_t)=0.
\end{equation}
In particular the continuous density $u_t$ converges to $\bar u$
uniformly on the compact sets of $\Dom {\bar u}$;
if moreover the support of $\rho_0^\perp$ is compact and
${\mathfrak m}<{\mathfrak m}_\mathrm c$,
then there exists a finite time $T>0$
such that $\rho_t\ll\Leb 1$ for every $t\ge T$.
\end{theorem}
\section{Wasserstein distance and differential calculus}
\label{sec:Wass}
In this Section we recall the definition and the main properties of the Wasserstein distance and differential calculus in Wasserstein spaces (we refer the interested reader to \cite{V03}, \cite{V09}, \cite{ags} for
more details). Also, the subdifferential of the energy functional $\mathcal F$ will
be characterized and discussed.
\subsection{Transport of measures, Wasserstein distance, and
differential calculus.}
If ${\rho}\in \PlusMeasures{\mathbb{R}^d,{\mathfrak m}}$ and ${\mbox{\boldmath$r$}}:\mathbb{R}^d\to \mathbb{R}^h$ is a Borel map, the push-forward of ${\rho}$ through ${\mbox{\boldmath$r$}}$ is the measure
$\mu={\mbox{\boldmath$r$}}_\#{\rho}\in \PlusMeasures{\mathbb{R}^h,{\mathfrak m}}$ defined by
\begin{equation}
\label{eq:33}
\mu(A):={\rho}({\mbox{\boldmath$r$}}^{-1}(A))\quad\text{for every Borel subset }A\subset \mathbb{R}^h.
\end{equation}
It can also be characterized by the change-of-variable formula
\begin{equation}
\label{eq:34}
\int_{\mathbb{R}^h}\varphi(y)\,\d\mu(y)=
\int_{\mathbb{R}^d}\varphi({\mbox{\boldmath$r$}}(x))\,\d{\rho}(x),
\end{equation}
{for every bounded or
nonnegative Borel function }$\varphi:\mathbb{R}^h\to\mathbb{R}.$\\
According to this definition, the marginals ${\rho}^i\in \PlusMeasures{\mathbb{R},{\mathfrak m}}$, $i=1,2$, of ${\mbox{\boldmath$\rho$}}\in \PlusMeasures{\mathbb{R}\times \mathbb{R},{\mathfrak m}}$ can be
defined by ${\rho}^i=(\pi^i)_\#{\mbox{\boldmath$\rho$}}$, where $\pi^i(x^1,x^2)=x^i$ is the projection on the $i$-th component in $\mathbb{R}\times \mathbb{R}$. In this case we say
that ${\mbox{\boldmath$\rho$}}$ is a coupling between ${\rho}^1,{\rho}^2$ and we denote by $\Gamma({\rho}^1,{\rho}^2)$ the (weakly) closed convex subset of
$\PlusMeasures{\mathbb{R}\times\mathbb{R},{\mathfrak m}}$ consisting of such couplings. We recall that a sequence of measures ${\rho}_n\in\PlusMeasures{\mathbb{R}^d,{\mathfrak m}}$
weakly converges to ${\rho}\in\PlusMeasures{\mathbb{R}^d,{\mathfrak m}}$ if
$\lim_{n\to+\infty}\int_{\mathbb{R}^d}\varphi(y)\,\d{\rho}_n(y)=\int_{\mathbb{R}^d}\varphi(y)\,\d{\rho}(y)$ for every continuous, bounded function $\varphi\in
C_{\rm b}(\mathbb{R}^d).$
For every couple of measures ${\rho}^1,{\rho}^2\in \PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$ the $L^2$-Wasserstein distance is defined by
\begin{equation}
\label{eq:35}
W_2^2({\rho}^1,{\rho}^2):=\min\Big\{\int_{\mathbb{R}\times\mathbb{R}}
|x^1-x^2|^2\,\d{\mbox{\boldmath$\rho$}}(x^1,x^2):
{\mbox{\boldmath$\rho$}}\in \Gamma({\rho}^1,{\rho}^2)\Big\}.
\end{equation}
The space $\PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$ endowed with the distance $W_2$ is a complete separable metric
space and the topology induced by the Wasserstein distance is stronger than the narrow topology:
in fact a sequence $\rho_n$ converges to $\rho$ in
$\PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$ iff
\eqref{eq:14} holds (see e.g. \cite{V03}).
{There exists a unique optimal coupling ${\mbox{\boldmath$\rho$}}_{\rm opt}$ realizing
the minimum in \eqref{eq:35}: it} admits a nice representation in terms of the cumulative distribution
functions $M_{{\rho}^i}$ of ${\rho}^1,{\rho}^2$ and of their
pseudo-inverses $Y_{{\rho}^i}$.
Let us first recall their definitions in the case of $\sigma\in
\PlusMeasures{\mathbb{R},{\mathfrak m}}$
\begin{equation}
\label{eq:36}
M_{\sigma}(x):=\sigma\big((-\infty,x]\big)\quad x\in \mathbb{R};\qquad
Y_{\sigma}(w):=\inf\Big\{x\in \mathbb{R}: M_{\sigma}(x)\ge w\Big\},\quad
w\in (0,{\mathfrak m}).
\end{equation}
Notice that $M_{\sigma}$ is a right-continuous and nondecreasing map
from $\mathbb{R}$ to $[0,{\mathfrak m}]$; if we denote by $\lambda_{\mathfrak m}=\Leb 1\restr{(0,{\mathfrak m})}$ the restriction
of the Lebesgue measure to the interval $(0,{\mathfrak m})$, it is possible to show that
\begin{equation}
\label{eq:37}
\big(Y_{{\rho}^i}\big)_\#\lambda_{\mathfrak m}={\rho}^i,\quad
\big(Y_{{\rho}^1},Y_{{\rho}^2}\big)_\#\lambda_{\mathfrak m}={\mbox{\boldmath$\rho$}}_{\rm opt}
\end{equation}
so that
\begin{equation}
\label{eq:72}
W_2^2(\rho^1,\rho^2)=\int_0^{\mathfrak m}
\big|Y_{\rho^1}(w)=Y_{\rho^2}(w)\big|^2\,\d w=
\|Y_{\rho^1}-Y_{\rho^2}\|_{L^2(0,{\mathfrak m})}^2.
\end{equation}
The map $\rho\mapsto Y_\rho$ provides an isometry between
$\PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$ and
the cone of nondecreasing function in $L^2(0,{\mathfrak m})$.
\paragraph{Displacement interpolation and displacement convexity.}
Let ${\rho}^0,{\rho}^1\in \PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$.
Their \emph{displacement interpolation} is the path ${\rho}^\vartheta\in
\PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$ with $\vartheta\in [0,1]$, defined by
\begin{equation}
\label{eq:38}
{\rho}^\vartheta:=\big((1-\vartheta)Y_{{\rho}^0}+\vartheta
Y_{{\rho}^1}\big)_\#\lambda_{\mathfrak m} = \big((1-\vartheta)\pi^1+\vartheta
\pi^2\big)_\# {\mbox{\boldmath$\rho$}}_{\rm opt}.
\end{equation}
The curve $\vartheta\mapsto \rho^\vartheta$ is the unique (minimal, constant speed)
geodesic connecting $\rho^0$ to $\rho^1$ in
$\PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$ and it corresponds to
the segment connecting $Y_{\rho^0}$ to $Y_{\rho^1}$ in
$L^2(0,{\mathfrak m})$.
We say that a functional $\mathcal G:\PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}\to
(-\infty,+\infty]$
is displacement $\lambda$-convex if for every ${\rho}^0,{\rho}^1$ in its proper domain we have
\begin{equation}
\label{eq:39}
\mathcal G({\rho}^\vartheta)\le
(1-\vartheta)\mathcal G({\rho}^0)+\vartheta\mathcal G({\rho}^1)-\frac\lambda2\vartheta(1-\vartheta)
W_2^2({\rho}^0,{\rho}^1).
\end{equation}
In the one-dimensional case, the displacement convexity of the
internal functional $\mathcal E$
is equivalent to the convexity of the energy density $E$
and it coincides with convexity along generalized geodesics (see
\cite[Definition 9.2.4]{ags}).
\begin{proposition}[Displacement $\lambda$-convexity and lower semicontinuity of $\mathcal F$]
\label{prop:lscF}
$\mathcal F$ is lower semicontinuous with respect to the Wasserstein
distance in $\PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$
and displacement $\lambda$-convex.
Moreover $\mathcal F$ satisfies the following coercivity property
\begin{equation}
\inf\Big\{\mathcal F(\rho):\rho\in \PlusMeasuresTwo{\mathbb{R},{\mathfrak m}},\quad
\int_\mathbb{R}|x|^2\,\d\rho(x) \leq C\Big\} >-\infty\quad\text{for every }C>0.
\end{equation}
\end{proposition}
\begin{proof}
Since $E$ is convex and sublinear, by \cite{DT84} it follows that $\mathcal E$ is lower semicontinuous with respect to the narrow convergence. In the
one-dimensional case the convexity of $E$ is equivalent to the
displacement convexity.
The functional ${\rho}\mapsto\int_\mathbb{R} V(x)\,\d{\rho}(x)$ is
displacement $\lambda$-convex if and only if $V$ is $\lambda$-convex;
it is also lower semicontinuous with respect to convergence in
$\PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$ since $V$ is continuous and
quadratically bounded from below.
\end{proof}
\begin{definition}[Subdifferential and slope]
\label{def:subdifferential}
Let $\mathcal G:\PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}\to (-\infty,+\infty]$ be
a displacement $\lambda$-convex and lower semcontinuous functional,
let ${\rho}^0\in \PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$ with
$\mathcal G({\rho}^0)<+\infty$ and ${\mbox{\boldmath$ \xi$}}\in L^2({\rho}^0)$.
We say that ${\mbox{\boldmath$ \xi$}}$ belongs to the $W_2$-subdifferential
of $\mathcal G$ at the point ${\rho}^0$, and we write ${\mbox{\boldmath$ \xi$}}\in \partial\mathcal G({\rho}^0)$, if for every ${\rho}^1\in
\PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$ {the optimal coupling
${\mbox{\boldmath$\rho$}}_{\rm opt}$ between ${\rho}^0$ and ${\rho}^1$ satisfies}
\begin{equation}
\label{eq:40}
\mathcal G({\rho}^1)-\mathcal G({\rho}^0)\ge
\int_{\mathbb{R}\times\mathbb{R}}\Big({\mbox{\boldmath$ \xi$}}(x)(y-x) + \frac\lambda2|y-x|^2\Big)\,\d{\mbox{\boldmath$\rho$}}_{\rm opt}(x,y).
\end{equation}
$\partial\mathcal G({\rho}^0)$ is a closed convex (and possibly empty)
subset of $L^2({\rho}^0)$.
When $\partial\mathcal G({\rho}^0)$ is not empty
we denote by
$\partial^\circ\mathcal G({\rho}^0)\in L^2(\rho^0)$ its (unique) element
of minimal $L^2({\rho}^0)$-norm.
The (metric) slope of $\mathcal G$ is defined as
\begin{equation}
\label{eq:42}
|\partial\mathcal G|({\rho}^0)=\limsup_{W_2({\rho},{\rho}^0)\to0}
\frac{\big(\mathcal G({\rho}^0)-\mathcal G({\rho})\big)^+}{W_2({\rho},{\rho}^0)}
=\sup_{\rho\neq\rho^0}
\Big(
\frac{\big(\mathcal G({\rho}^0)-\mathcal G({\rho})\big)^+}{W_2({\rho},{\rho}^0)}+
\frac \lambda2 W_2(\rho,\rho^0)\Big)^+.
\end{equation}
\end{definition}
For general displacement $\lambda$-convex functionals, one has
\begin{equation}
\label{eq:76}
|\partial\mathcal G|({\rho})\le \|\partial^\circ\mathcal G(\rho)\|_{L^2(\rho)}.
\end{equation}
When the functional $\mathcal G$ satisfies the regularity condition
\begin{equation}
\label{eq:73}
|\partial\mathcal G|(\rho^0)<+\infty\quad \Rightarrow\quad
\rho^0\ll\Leb1,
\end{equation}
then the metric slope \eqref{eq:42} can be equivalently characterized by
\begin{equation}
\label{eq:41}
|\partial\mathcal G|^2({\rho}^0):=\min\Big\{\int_\mathbb{R} |{\mbox{\boldmath$ \xi$}}|^2\,\d{\rho}^0:{\mbox{\boldmath$ \xi$}}\in \partial\mathcal G({\rho}^0)\Big\},
\end{equation}
where $|\partial\mathcal G|({\rho}^0)=+\infty$ iff
$\partial\mathcal G({\rho}^0)$ is empty.
In this case $|\partial\mathcal G|({\rho}^0)=\|\partial^\circ\mathcal G({\rho}^0)\|_{L^2({\rho}^0)}$.
\subsection{Slope and Fisher dissipation in the super-linear case.}
Let us consider the perturbed family of energy densities
$E^{\varepsilon}(r):=E(r)+{\varepsilon} r\log r$
associated to the energy functionals
\begin{equation}
\label{eq:400}
\mathcal F^{\varepsilon}({\rho}):=\int_\mathbb{R} E^{\varepsilon}(u(x))\,\d x+\int_\mathbb{R}
V^{\varepsilon}(x)\,\d{\rho}(x)\quad\text{if } {\rho}=u\,\Leb 1;\quad
\mathcal F^{\varepsilon}({\rho})=+\infty\quad\text{if }{\rho}\not\ll\Leb 1.
\end{equation}
Notice that $(rE^{\varepsilon})''(r)=\beta'(r)+{\varepsilon}=(\beta^{\varepsilon})'(r)$, where $\beta^{\varepsilon}$ is defined in \eqref{eq:16}.
Since $E^{\varepsilon}$ has a super-linear growth, the slope $|\partial
\mathcal F^{\varepsilon}|$ can be explicitly characterized
\cite[Theorem 10.4.13]{ags} and it
coincides with the square root of the associated Fisher-dissipation
\begin{equation}
\label{eq:46}
\mathcal I^{\varepsilon}({\rho}):=\int_\mathbb{R}\Big|\frac{\partial_x\beta^{\varepsilon}(u)}u+(V^{\varepsilon})'\Big|^2u\,\d x \qquad\text{if }{\rho}=u\Leb1,\quad
u\in W^{1,1}_{\rm loc}(\mathbb{R}).
\end{equation}
As usual $\mathcal I^{\varepsilon}({\rho})=+\infty$ if $u\not\in W^{1,1}_{\rm loc}(\mathbb{R})$ or even ${\rho}\not\ll\Leb 1$. Thus we have
\begin{equation}
\label{eq:47}
|\partial \mathcal F^{\varepsilon}|^2({\rho})=\mathcal I^{\varepsilon}({\rho})
\end{equation}
and the minimal subdifferential ${\mbox{\boldmath$ \xi$}}^{\varepsilon}=\partial^\circ\mathcal F^{\varepsilon}({\rho})\in L^2({\rho})$ is characterized as
\begin{equation}
\label{eq:48}
{\mbox{\boldmath$ \xi$}}^{\varepsilon}{\rho}=\partial_x\beta^{\varepsilon}(u)\Leb1+{\rho}\,(V^{\varepsilon})'\quad\text{if }{\rho}=u\Leb1\in D(\mathcal I^{\varepsilon}).
\end{equation}
The following compactness and lower semicontinuity property will play a crucial role in the sequel.
\begin{theorem}
\label{thm:lsc-dissipation}
If ${\rho}^{\varepsilon}=u^{\varepsilon}\,\Leb 1\in D(\mathcal I_{{\varepsilon}})$, ${\varepsilon}>0$, with $u^{\varepsilon}(x)>0$ for all $ x\in\mathbb{R}$,
is a family of measures satisfying
\begin{equation}\label{wconv}
{{\rho}^{\varepsilon}\rightharpoonup {\rho}\quad\text{weakly in
$\PlusMeasures{\mathbb{R},{\mathfrak m}}$ as ${\varepsilon}\downarrow0,$}}\quad
\limsup_{{\varepsilon}\downarrow0} \mathcal I^{{\varepsilon}}({\rho}^{\varepsilon})<+\infty,
\end{equation}
then we have
\begin{equation*}
{\rho}=u\,\Leb 1+{\rho}^\perp\in D(\mathcal I)\subset \CPlusMeasuresTwo{\mathbb{R},{\mathfrak m}},
\end{equation*}
\begin{equation}
\label{eq:490}
\mathcal I({\rho})\le \liminf_{{\varepsilon}\downarrow0}\mathcal I^{{\varepsilon}}({\rho}^{\varepsilon}),
\end{equation}
\begin{equation}\label{eq:219}
u^{\varepsilon} \text{ converges to } u \text{ uniformly on compact sets of } \Dom{u}.
\end{equation}
Moreover, if ${\mbox{\boldmath$ \xi$}}^{\varepsilon} = \partial^o\mathcal F^{\varepsilon}({\rho}^{\varepsilon})$
as in \eqref{eq:48}, we have
\begin{equation}
\label{eq:49}
{\mbox{\boldmath$ \xi$}}^{\varepsilon}{\rho}^{\varepsilon} \rightharpoonup {\mbox{\boldmath$ \xi$}}{\rho}=\partial_x\beta(u)\Leb1+V'{\rho},\quad\text{in
the duality with }C^0_{\rm b}(\mathbb{R}).
\end{equation}
Finally, if $f:[0,+\infty)\to \mathbb{R}$ is a continuous function
such that $\displaystyle\lim_{r\uparrow+\infty}\frac{f(r)}{r}=f_\infty \in\mathbb{R}$, then
\begin{equation}
\label{eq:65}
f(u^{\varepsilon})\Leb 1\rightharpoonup f(u)\Leb1+f_\infty{\rho}^\perp\quad\text{in
the duality with }C^0_{\rm c}(\mathbb{R}).
\end{equation}
\end{theorem}
\begin{proof}
Since $\displaystyle \mathcal I^{\varepsilon} ({\rho}^{\varepsilon}) = \int_\mathbb{R} |{\mbox{\boldmath$ \xi$}}^{\varepsilon}|^2 \d {\rho}^{\varepsilon}$, by \eqref{wconv} (see \cite[Theorem 5.4.4]{ags}) there exists ${\mbox{\boldmath$ \xi$}}\in
L^2({\rho})$ such that
\begin{equation}\label{eq:401}
{\mbox{\boldmath$ \xi$}}^{\varepsilon}{\rho}^{\varepsilon} \rightharpoonup {\mbox{\boldmath$ \xi$}}{\rho},\quad\text{in
the duality with }C^0_{\rm b}(\mathbb{R}),
\end{equation}
and
\begin{equation*}
\int_\mathbb{R} |{\mbox{\boldmath$ \xi$}}|^2 \d {\rho}\le \liminf_{{\varepsilon} \downarrow 0} \int_\mathbb{R} |{\mbox{\boldmath$ \xi$}}^{\varepsilon}|^2 \d {\rho}^{\varepsilon}.
\end{equation*}
From \eqref{wconv} and \eqref{eq:85} it follows that
\begin{equation}\label{eq:402}
(V^{\varepsilon})'u^{\varepsilon} \Leb1 \rightharpoonup V'{\rho} \quad \text{ in the duality with }C^0_{\rm c}(\mathbb{R}).
\end{equation}
Since by \eqref{eq:48}
\begin{equation}\label{eq:80}
\de_x\beta^{\varepsilon}(u^{\varepsilon})\Leb1 = {\mbox{\boldmath$ \xi$}}^{\varepsilon} u^{\varepsilon}\Leb1 - (V^{\varepsilon})'u^{\varepsilon}\Leb1,
\end{equation}
\eqref{eq:401} and \eqref{eq:402} imply that
$$\de_x\beta^{\varepsilon}(u^{\varepsilon})\Leb1 \rightharpoonup {\mbox{\boldmath$ \xi$}} {\rho} - V'{\rho} \quad\text{ in the duality with }C^0_{\rm c}(\mathbb{R}).$$
Let us now prove that ${\rho}\in \CPlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$, $\beta(u)\in W^{1,1}_{\mathrm{loc}}(\mathbb{R})$ and $\de_x\beta(u)= {\mbox{\boldmath$ \xi$}} {\rho} - V'{\rho}$.
\\
We introduce the functions
$$
G(r)=\int_0^r \frac{\beta'(s)}{\sqrt s}\,\d s, \qquad G^{\varepsilon}(r)=G(r)+2{\varepsilon}\sqrt r.
$$
Since $u^{\varepsilon} \in W^{1,1}_{\mathrm{loc}}(\mathbb{R})$, $u^{\varepsilon}(x)>0$ and $G$ is locally Lipschitz in $(0,+\infty)$, we have
\begin{equation}\label{sss}
\de_xG^{\varepsilon}(u^{\varepsilon}) = \frac{\de_x(\beta^{\varepsilon}(u^{{\varepsilon}}))}{\sqrt{u^{\varepsilon}}}.
\end{equation}
Let $I=(a,b)$ be an arbitrary bounded interval of $\mathbb{R}$. Since $\beta'(0^+) < +\infty$ we have that $G^{\varepsilon} (r)\le M\sqrt r$, for some $M>0$.
Therefore
\begin{equation}\label{bound-11}
\sup_{\varepsilon} \int_I\left|G^{\varepsilon}(u^{\varepsilon})\right|^2 \,\d x < +\infty.
\end{equation}
By \eqref{eq:80} and \eqref{wconv} we have
\begin{equation} \label{bound-l2}
\int_I\left|\frac{\de_x(\beta^{\varepsilon}(u^{{\varepsilon}}))}{\sqrt{u^{\varepsilon}}}\right|^2 \d x =
\int_I |{\mbox{\boldmath$ \xi$}}^{\varepsilon} - (V^{\varepsilon})'|^2u^{\varepsilon}\d x \le
2
\int_I |{\mbox{\boldmath$ \xi$}}^{\varepsilon}|^2u^{\varepsilon}\d x + 2
\int_I |(V^{\varepsilon})'|^2u^{\varepsilon}\d x
\end{equation}
so that
\begin{equation}
\label{eq:84bis}
\sup_{{\varepsilon}>0}
\int_I\left|\frac{\de_x(\beta^{\varepsilon}(u^{{\varepsilon}}))}{\sqrt{u^{\varepsilon}}}\right|^2 \d x<+\infty.
\end{equation}
By \eqref{bound-11}, \eqref{eq:84bis} and \eqref{sss}, we infer that the family $\{G^{\varepsilon}(u^{\varepsilon})\}_{{\varepsilon}>0}$ is bounded in $H^1_{\mathrm{loc}}(\mathbb{R})$. Thus,
for every sequence ${\varepsilon}_j\to 0$ we can extract a sub-sequence, still denoted by $\{{\varepsilon}_j\}$, such that $G_{{\varepsilon}_j}(u_{{\varepsilon}_j})$ converges weakly in $H^1_{\mathrm{loc}}(\mathbb{R})$, and uniformly on the compact sets of $\mathbb{R}$,
to some continuous function $g \in H^1_{\mathrm{loc}}(\mathbb{R})$.
Since
\begin{equation}\label{bvest}
\sup_{\varepsilon}\int_I |\de_x(\beta^{\varepsilon}(u^{{\varepsilon}}))|\,\d x=\sup_{\varepsilon}\int_I | {\mbox{\boldmath$ \xi$}}^{\varepsilon}-(V^{\varepsilon})'|u^{{\varepsilon}}\,\d x\le \sup_{\varepsilon}\sqrt{{\mathfrak m}}\bigg(\int_I | {\mbox{\boldmath$ \xi$}}^{\varepsilon}-(V^{\varepsilon})'|^2u^{{\varepsilon}}\,\d x\bigg)^\frac 1 2 <+\infty,
\end{equation}
and $\{\beta^{\varepsilon}(u^{\varepsilon})\}_{{\varepsilon}>0}$ is bounded in $L^1(\mathbb{R})$,
the family $\{\beta^{\varepsilon}(u^{\varepsilon})\}_{{\varepsilon}>0}$ is bounded in $L^\infty(I)$.
Therefore the family $\{{\varepsilon} u^{\varepsilon} = \beta^{\varepsilon}(u^{\varepsilon})-\beta(u^{\varepsilon})\}_{{\varepsilon}>0}$ is bounded in $L^\infty(I)$. Since $0\leq
G^{\varepsilon}(u^{\varepsilon})-G(u^{\varepsilon})=2\sqrt{{\varepsilon}} \sqrt{{\varepsilon} u^{\varepsilon}}$, we conclude that $G(u_{{\varepsilon}_j})$ converges uniformly on the
compact sets of $\mathbb{R}$ to $g$, as $j\to +\infty$. The inequality
$$
0\leq G \leq G_\infty=\int_0^{+\infty} \frac{\beta'(s)}{\sqrt s}\d s,
$$
together with the previous observations, gives $0\le g \le G_\infty$. Since $G$ is increasing and $G_\infty < +\infty$, we can define
the function
$$u(x):=\begin{cases}G^{-1}(g(x)) & \text{if } g(x)<G_\infty,\\
+\infty & \text{if } g(x)=G_\infty\end{cases}$$
which turns out to be continuous on the open set $\Dom u:=\{ x\in\mathbb{R}: g(x)<G_\infty\}$. Since $G(u_{{\varepsilon}_j}) \to g$ uniformly on the compact
sets of $\mathbb{R}$, we have that $u_{{\varepsilon}_j} = G^{-1}(G(u_{{\varepsilon}_j}))\to u$ on the compact sets of $\Dom u$ and $u_{{\varepsilon}_j}(x)\to+\infty$ for every $x\in\mathbb{R}\setminus
\Dom u$. By Fatou's Lemma we obtain that $u\in L^1(\mathbb{R})$ and $\Leb{1}(\mathbb{R}\setminus \Dom u)=0$.
For every $\psi\in C^0_{\rm c}(\Dom u)$, using (\ref{wconv}) we have
$$
\lim_{j\to+\infty} \int_\mathbb{R}\psi(x)\,\d{\rho}_{{\varepsilon}_j}= \int_\mathbb{R}\psi(x)\,\d{\rho} =\int_\mathbb{R}\psi(x)u(x)\,\d x.
$$
Thus
\begin{equation} \label{eq:decomp}
{\rho}_{|\Dom u}=u\Leb1 \quad \text{ and }\quad {\rho}_{|\mathbb{R}\setminus \Dom u}= {\rho}^\perp.
\end{equation}
This shows that ${\rho}\in \CPlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$. Moreover, we deduce that the whole family $u^{\varepsilon}$ converges to $u$ uniformly on compact sets of $\Dom u$, as ${\varepsilon} \downarrow 0$.
For any bounded interval $I=(a,b)$, we have proved that $\{\beta^{\varepsilon}(u^{\varepsilon})\}_{{\varepsilon}>0}$ is bounded in $W^{1,1}(I)$.
Then, by ${\rm BV}$ compactness (see e.g. \cite{afp}) there exists $h\in {\rm BV}_{\rm loc}(\mathbb{R})$ such that, up to subsequences as before, $\beta^{\varepsilon}(u^{\varepsilon})\to h$ in $L^1_{\rm loc}(\mathbb{R})$ and $\Leb{1}$-a.e. and $\partial_x\beta^{\varepsilon}(u^{\varepsilon})\Leb{1} \rightharpoonup \partial_x h$ in duality with $C_{\mathrm c}^0(\mathbb{R})$.
Since $ 0 \le \beta^{\varepsilon} (u^{\varepsilon}) - \beta(u^{\varepsilon}) = {\varepsilon} u^{\varepsilon}$ and ${\varepsilon} u^{\varepsilon}(x) \to 0$ pointwise in $\Dom u$, we have that $\beta(u^{\varepsilon}) \to h$, $\Leb{1}$-a.e.
On the other hand, by the continuity of $\beta$, $\beta(u^{\varepsilon}) \to \beta(u)$ $\Leb{1}$-a.e. Hence $h=\beta(u)$.
Moreover, by using \eqref{eq:80}, it is easy to see that $\partial_x\beta(u)\Leb1= {\mbox{\boldmath$ \xi$}} {\rho} - V'{\rho}$.
The last identity and \eqref{eq:decomp} yield $\beta(u)\in {\rm BV}_{\rm loc}(\mathbb{R}) \cap W^{1,1}_{\mathrm{loc}}(\Dom u)$.
Finally, we prove that $\beta(u)\in W^{1,1}_{\mathrm{loc}}(\mathbb{R})$ and $\de_x(\beta(u))=\de_x(\beta(u))_{|\Dom u}$.
Since $\Dom u$ is open, we can write
$$
\Dom u=\bigcup_{n\in\mathbb{N}} (a_n,b_n)
$$
where the intervals are pairwise disjoint;
recalling that $\beta(u(a_n))=\beta(u(b_n))={\beta^{\infty}}$, we have for every $\zeta\in C_{\rm c}^\infty(\mathbb{R})$
\begin{align*}
\int_\mathbb{R} \zeta'\beta(u)\,\d x&=\int_{\Dom u}
\zeta'\,\beta(u)\,\d x+{\int_{\mathbb{R}\setminus \Dom
u}\zeta'\,\beta(u)\,\d x}=\sum_n\int_{a_n}^{b_n}
\zeta'\beta(u)\,\d x
{+{\beta^{\infty}}\int_{\mathbb{R}\setminus\Dom u}\zeta'\,\d x}\\
&=\sum_n\bigg(-\int_{a_n}^{b_n}
\zeta\,\de_x(\beta(u))\,\d x+\big(\zeta({b_n})-\zeta({a_n})\big){\beta^{\infty}}\bigg)
{+{\beta^{\infty}}\int_{\mathbb{R}\setminus\Dom u}\zeta'\,\d x}\\
&=-\int_{\Dom u} \zeta\,\de_x(\beta(u))\,\d x+\sum_n{\beta^{\infty}}
\int_{a_n}^{b_n} \zeta'\,\d x
{+{\beta^{\infty}}\int_{\mathbb{R}\setminus\Dom u}\zeta'\,\d x}\\
&=-\int_{\Dom u} \zeta\,\de_x(\beta(u))\,\d x+{\beta^{\infty}} \int_\mathbb{R} \zeta'\,\d x
=-\int_{\Dom u} \zeta\,\de_x(\beta(u))\,\d x .
\end{align*}
We eventually prove \eqref{eq:65}.
By possibly substituting $f(r)$ with $f(r)-f_\infty r$ it is not
restrictive to assume $f_\infty=0$, i.e.
\begin{equation}
\label{eq:74}
\lim_{r\to+\infty}\frac{f(r)}r=0\quad\text{or, equivalently,}\quad
\forall\,\eta>0\ \exists\, M_\eta:\quad
|f(r)|\le M_\eta+\eta r\quad\text{for every }r\ge0.
\end{equation}
Property \eqref{eq:74} easily shows that the family
$\{f(u^{\varepsilon})\}_{{\varepsilon}>0}$ is equi-integrable in $\mathbb{R}$:
for every $\delta>0$ and choosing $\eta:= \delta/2{\mathfrak m} $,
every Borel set $A$ with
measure $\Leb1(A)\le \delta/2M_\eta$ satisfies
\begin{equation}
\label{eq:75}
\int_A |f(u^{\varepsilon}(x))|\,\d x\le \int_A \Big(M_\eta +\eta
u_{\varepsilon}(x)\Big)\,\d x\le M_\eta\, \Leb 1(A)+\eta\, {\mathfrak m}\le
\delta\quad
\text{for every }{\varepsilon}>0.
\end{equation}
The previous equi-integrability estimate and the tightness of
$\rho^{\varepsilon}$ given by \eqref{wconv} show that the family
$f(u^{\varepsilon})$ is weakly
compact in $L^1(\mathbb{R})$. On the
other hand, $f(u^{\varepsilon})\to f(u)$ locally uniformly in $\Dom u$. Since
$\Leb1(\mathbb{R}\setminus\Dom u)=0$ it follows that $f(u)$ is also the weak
limit of $f(u^{\varepsilon})$ in $L^1(\mathbb{R})$.
\end{proof}
By a similar and even simpler argument it is possible to prove the following lower semi
continuity result for the Fisher dissipation
$\mathcal I$ with respect to weak convergence. Lower semicontinuity with
respect to Wasserstein convergence will follow
by \eqref{th:charsubdiff} and the representation
\eqref{eq:42} of the metric slope for a displacement
$\lambda$-convex functional \cite[Corollary 2.4.10]{ags}.
\begin{theorem}[Lower semi continuity of $\mathcal I$]
\label{thm:lsc-dissipation2}
If ${\rho}_n=u_n\,\Leb 1 +{\rho}^\perp_n \in D(\mathcal I)$
is a sequence of measures weakly convergent to a measure ${\rho}$ and satisfying
\begin{equation}\label{upbound2}
\limsup_{n\to+\infty} \mathcal I({\rho}_n)<+\infty,
\end{equation}
then we have
\begin{equation}\label{eq:4902}
{\rho}=u\,\Leb 1+{\rho}^\perp\in D(\mathcal I)\subset \CPlusMeasuresTwo{\mathbb{R},{\mathfrak m}},
\qquad
\mathcal I({\rho})\le \liminf_{n\to+\infty}\mathcal I({\rho}_n).
\end{equation}
Moreover
\begin{equation}\label{eq:2190}
u_{n} \text{ converges to } u \text{ uniformly on compact sets of } \Dom{u}.
\end{equation}
\end{theorem}
\subsection{Characterization of the Wasserstein subdifferential of $\mathcal F$}
\begin{theorem}[Characterization of $\de\mathcal F$]\label{th:charsubdiff}
Let ${\rho} = u\Leb{1}+{\rho}^\perp\in \PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$ with $\mathcal F({\rho})<+\infty$ and
${\mbox{\boldmath$ \xi$}}\in L^2({\rho})$.\\
${\mbox{\boldmath$ \xi$}}=\de^o\mathcal F({\rho})$ (and, in particular, $\partial\mathcal F({\rho})$ is not empty) if and only if
\begin{equation}
\label{eq:50}
{\rho}\in \CPlusMeasuresTwo{\mathbb{R},{\mathfrak m}}, \quad \mathcal I({\rho})<+\infty,
\quad
{\mbox{\boldmath$ \xi$}}{\rho}=\de_x\beta(u)\Leb{1}+V'{\rho}.
\end{equation}
In this case
\begin{equation}
\label{eq:44}
|\partial\mathcal F|^2({\rho})=\int_\mathbb{R} |{\mbox{\boldmath$ \xi$}}|^2\,\d{\rho}=\mathcal I({\rho}).
\end{equation}
\end{theorem}
\begin{proof}
Let us first suppose that ${\rho},{\mbox{\boldmath$ \xi$}}$ satisfy \eqref{eq:50} and let
us prove that ${\mbox{\boldmath$ \xi$}}\in \partial\mathcal F({\rho})$, i.e.\ \eqref{eq:40} holds with
${\rho}^0:={\rho}$;
in particular, recalling \eqref{eq:76}, this also shows that
\begin{equation}
\label{eq:77}
|\partial\mathcal F|^2(\rho)\le \int_\mathbb{R} |{\mbox{\boldmath$ \xi$}}|^2\,\d\rho=\mathcal I(\rho)<+\infty.
\end{equation}
It is not restrictive to assume $\lambda=0$. By a standard regularization and stability of the optimal couplings with respect to
weak convergence, we can also suppose that ${\rho}^1=u^1\,\Leb1$ with $u^1\in C^1(\mathbb{R})$ supported in the bounded interval $[a,b]$ with $u^1(x)>0$
for every $x\in (a,b)$. In this case $M_{{\rho}^1}\in C^2(\mathbb{R})$, the monotone rearrangement map $Y_{{\rho}^1}\in C^0([0,{\mathfrak m}])$ satisfies
$Y_{{\rho}^1}(0)=a,\ Y_{{\rho}^1}({\mathfrak m})=b$ and its restriction to $(0,{\mathfrak m})$ is of class $C^2$. We set
\begin{equation}
\label{eq:51}
\left\{
\begin{aligned}
{\mbox{\boldmath$r$}}(x):=&Y_{{\rho}^1}(M_{\rho}(x)),\ {\mbox{\boldmath$r$}}^\vartheta(x):=(1-\vartheta)x+\vartheta{\mbox{\boldmath$r$}}(x)\\
\ss(y):=&Y_{\rho}(M_{{\rho}^1}(y)),\ \ss^{\vartheta}(y):=\vartheta y+(1-\vartheta)\ss(y)
\end{aligned}
\right.
\quad\text{for
every }x,y\in \mathbb{R},\ \vartheta\in [0,1],
\end{equation}
and we observe that ${\mbox{\boldmath$r$}}^\vartheta\restr{\Dom u}\text{ is $C^1$.}$
We introduce the sets
\begin{equation}
\label{eq:56}
\mathsf D:=\Dom u,\quad \mathsf D_>:=\{x\in \Dom u:u(x)>0\},\quad
\tilde\sfD:=\mathbb{R}\setminus \mathsf D,\quad
\mathsf G:={\mbox{\boldmath$r$}}(\mathsf D)={\mbox{\boldmath$r$}}(\mathsf D_>),\quad
\tilde\mathsf G:=(a,b)\setminus \mathsf G,
\end{equation}
and we have
\begin{gather}
\label{eq:52}
({\mbox{\boldmath$\rho$}}_{\rm opt})\restr{\mathsf D\times\mathbb{R}}=({\mbox{\boldmath$i$}}\times{\mbox{\boldmath$r$}}^1)_\# (u\Leb
1)=
(\ss^0\times {\mbox{\boldmath$i$}})_\#(u^1\Leb1\restr\mathsf G)
,\quad
({\mbox{\boldmath$\rho$}}_{\rm opt})\restr{\tilde\sfD\times\mathbb{R}}=
(\ss^0\times {\mbox{\boldmath$i$}})_\#(u^1\Leb1\restr{\tilde\mathsf G})
\\
{\rho}^\vartheta\restr{{\mbox{\scriptsize\boldmath$r$}}^\vartheta(\mathsf D)}={\mbox{\boldmath$r$}}^\vartheta_\# (u\Leb 1),\quad
{\rho}^\vartheta\restr{\mathbb{R}\setminus{\mbox{\scriptsize\boldmath$r$}}^\vartheta(\mathsf D)}=\ss^{\vartheta}_\# (u^1\Leb 1\restr{\tilde\mathsf G}),\quad
\\
\label{eq:53}
u^\vartheta({\mbox{\boldmath$r$}}^\vartheta(x))({\mbox{\boldmath$r$}}^\vartheta)'(x)=u(x),\quad
u^\vartheta(\ss^\vartheta(y))(\ss^\vartheta)'(y)=u^1(y)\quad\text{for every }x\in \mathsf D,
\ y\in (a,b).
\end{gather}
Since $(\ss^0)'(y)=0$ for every $y\in \tilde G$
\begin{equation}
\label{eq:55}
\mathcal E({\rho}^\vartheta)=\int_{\mathsf D_>}E\Big(\frac{u(x)}{(1-\vartheta)+\vartheta{\mbox{\boldmath$r$}}'(x)}\Big)(1-\vartheta+\vartheta{\mbox{\boldmath$r$}}'(x))\,\d
x+\int_{\tilde \mathsf G}E\Big(\frac{u^1(y)}{\vartheta}\Big)\vartheta\,\d y.
\end{equation}
Therefore, owing to the convexity of the maps $\vartheta\mapsto
\mathcal E({\rho}^\vartheta)$
{and $s\mapsto sE(\alpha/s)$ for every $\alpha\ge0$,}
\begin{align*}
+\infty>\mathcal E({\rho}^1)-\mathcal E({\rho})\ge\lim_{\vartheta\downarrow0}\vartheta^{-1}\Big(\mathcal E({\rho}^\vartheta)-\mathcal E({\rho})\Big)=
-\int_{\mathsf D}\beta(u)({\mbox{\boldmath$r$}}'-1)\,\d x-{\beta^{\infty}}\Leb 1(\tilde \mathsf G).
\end{align*}
Let us now choose two sequences $z^-_{k}\to-\infty$, $z^+_k\to +\infty$ in $\mathsf D$, let $(a_k^-,b_k^-) $. Let $(a_k^+,b_k^+)$ be the connected
component of $\mathsf D$ containing $z_k^-$ and $z_k^+$ respectively, and let $I_k^n:=(a_k^n,b_k^n)$, $n\in \Lambda_k\subset \mathbb{N}$ be the (at most
countable) connected components of $\mathsf D\cap (b_k^-,a_k^+)$. We consider a continuous function $\psi_k:\mathbb{R}\to [0,1]$ satisfying
\begin{equation}
\label{eq:57}
\psi_k(x)=0\text{ in }\mathbb{R}\setminus [z_k^-,z_k^+],
\quad \psi_k(x)\equiv 1\text{ if }x\in [\tfrac
12(z_k^-+b_k^-),\tfrac 12 (z_k^++a_k^+)],\quad
\psi_k\restr{[z_k^-,z_k^+]}\text{ is concave.}
\end{equation}
For sufficiently big $k$ we have $\psi_k\equiv 1$ on $(a,b)$. Then
\begin{equation}
\label{eq:60}
\begin{aligned}
\beta(u(x))({\mbox{\boldmath$r$}}(x)-x)\psi_k'(x) &\ge
\beta(u(x))\big(\psi_k({\mbox{\boldmath$r$}}(x))-\psi_k(x)) \\ &\ge
\beta(u(x))(1-\psi_k(x))\ge0\quad\text{for every }x\in [z_k^-,z_k^+];
\end{aligned}
\end{equation}
\begin{equation}
\label{eq:61}
-{\beta^{\infty}}\Leb 1(\tilde \mathsf G)=
\lim_{k\to\infty}\Leb 1(\tilde\mathsf G\cap({\mbox{\boldmath$r$}}(b_k^-),{\mbox{\boldmath$r$}}(a_k^+))).
\end{equation}
Moreover
\begin{equation}
\label{eq:58}
-\int_{\mathsf D}\beta(u)({\mbox{\boldmath$r$}}'-1)\,\d x=\lim_{k\uparrow+\infty}
-\int_{\mathsf D}\beta(u)({\mbox{\boldmath$r$}}'-1)\,\psi_k(x)\,\d x
\end{equation}
and
\begin{align*}
-\int_{\mathsf D}&\beta(u)({\mbox{\boldmath$r$}}'-1)\,\psi_k(x)\,\d x\ge
\int_{a_k^+}^{z_k^+}\partial_x \beta(u)\,({\mbox{\boldmath$r$}}(x)-x)\psi_k(x)\,\d x
+
\int_{z_k^-}^{b_k^-}\partial_x \beta(u)\,({\mbox{\boldmath$r$}}(x)-x)\psi_k(x)\,\d x
\\&+
\sum_{n\in \Lambda_k} \int_{a_k^n}^{b_k^n}\partial_x
\beta(u)\,({\mbox{\boldmath$r$}}(x)-x)\,\d x
\\&+{\beta^{\infty}}\Big[({\mbox{\boldmath$r$}}(a_k^+)-a_k^+)-({\mbox{\boldmath$r$}}(b_k^-)-b_k^-)-
\sum_{n\in \Lambda_k}
\big({\mbox{\boldmath$r$}}(b_k^n)-{\mbox{\boldmath$r$}}(a_k^n)-(b_k^n-a_k^n)\big)\Big]
\\&=
\int_\mathbb{R} \partial_x\beta(u)({\mbox{\boldmath$r$}}(x)-x)\psi_k(x)\,\d x+
{\beta^{\infty}}\Leb 1(\tilde\mathsf G\cap({\mbox{\boldmath$r$}}(b_k^-),{\mbox{\boldmath$r$}}(a_k^+))),
\end{align*}
where we used the fact that $\Leb1\big((b_k^-,a_k^+)\setminus \mathsf D\big)=0$.
Combining all these estimates we get
\begin{equation}
\label{eq:62}
+\infty>\mathcal E({\rho}^1)-\mathcal E({\rho})\ge\limsup_{k\uparrow+\infty}\int_\mathbb{R} \partial_x\beta(u)({\mbox{\boldmath$r$}}(x)-x)\psi_k(x)\,\d x.
\end{equation}
On the other hand
\begin{align*}
+\infty&>\int_\mathbb{R} V(y)\,\d{\rho}^1(y)-\int_\mathbb{R} V(x)\,\d{\rho}(x)=
\int_{\mathbb{R}\times\mathbb{R}}\Big(V(y)-V(x)\Big)\,\d{\mbox{\boldmath$\rho$}}_{\rm opt}(x,y)\\&\ge
\int_{\mathbb{R}\times\mathbb{R}} V'(x)(y-x)\,\d{\mbox{\boldmath$\rho$}}_{\rm opt}(x,y)\ge
\limsup_{k\uparrow+\infty} \int_{\mathbb{R}\times\mathbb{R}} V'(x)(y-x)\psi_k(x)\,\d{\mbox{\boldmath$\rho$}}_{\rm opt}(x,y)
\end{align*}
Summing up the two contributions we have
\begin{displaymath}
\mathcal F({\rho}^1)-\mathcal F({\rho})\ge \limsup_{k\uparrow+\infty}
\int_{\mathbb{R}\times\mathbb{R}} {\mbox{\boldmath$ \xi$}}(x)(y-x)\psi_k(x)\,\d{\mbox{\boldmath$\rho$}}_{\rm opt}(x,y)=
\int_{\mathbb{R}\times\mathbb{R}} {\mbox{\boldmath$ \xi$}}(x)(y-x)\,\d{\mbox{\boldmath$\rho$}}_{\rm opt}(x,y).
\end{displaymath}
Let us now show that if $|\partial\mathcal F|(\rho)<+\infty$ then there
exists ${\mbox{\boldmath$ \xi$}}\in L^2(\rho)$ satisfying \eqref{eq:50} (thus in
particular ${\mbox{\boldmath$ \xi$}}\in \partial\mathcal F(\rho)$) with
\begin{equation}
\label{eq:78}
\mathcal I(\rho)=\int_\mathbb{R} |{\mbox{\boldmath$ \xi$}}|^2\,\d\rho\le |\partial\mathcal F|^2(\rho);
\end{equation}
recalling \eqref{eq:77}, this shows that ${\mbox{\boldmath$ \xi$}}=\partial^\circ\mathcal F(\rho)$.
We apply the forthcoming Lemma \ref{le:Gamma-convergence} and the general
approximation result \cite[Lemma 10.3.16]{ags} to find a family ${\rho}_{{\varepsilon}}$ converging to ${\rho}$ in $\PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$ and
${\mbox{\boldmath$ \xi$}}^{\varepsilon}\in\de \mathcal F^{{\varepsilon}}({\rho}_{{\varepsilon}})$
such that
\begin{equation}
\lim_{{\varepsilon}\downarrow0} |\partial\mathcal F^{\varepsilon}|^2(\rho^{\varepsilon})=
\lim_{{\varepsilon}\downarrow0} \mathcal I^{\varepsilon}(\rho^{\varepsilon})
= \lim_{{\varepsilon}\downarrow0} \int_\mathbb{R} |{\mbox{\boldmath$ \xi$}}^{\varepsilon}|^2\,\d\rho^{\varepsilon}=
|\partial\mathcal F|^2(\rho).
\end{equation}
Theorem
\ref{thm:lsc-dissipation} then yields \eqref{eq:78} and \eqref{eq:50}.
\end{proof}
\subsection{$\Gamma$-convergence of $\mathcal F^{\varepsilon}$ to $\mathcal F$}
The following lemma shows that the family of functionals
$\mathcal F^{\varepsilon}$ converges to $\mathcal F$ in a kind of $\Gamma$ convergence way
(with different convergence in the $\liminf$ and the $\limsup$ inequalities).
\begin{lemma}
\label{le:Gamma-convergence}
As ${\varepsilon}\downarrow0$ the family of functionals $\mathcal F^{\varepsilon}$ converge to $\mathcal F$
according to the following two properties:
\begin{itemize}
\item[(i)] For every family $\{{\rho}^{\varepsilon}\}\subset\PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$
such that ${\rho}^{\varepsilon}\rightharpoonup {\rho}$, as ${\varepsilon}\downarrow0$, in duality with $C^0_{\rm b}(\mathbb{R})$, and
\begin{equation}\label{boundsecmom}
M_2:=\limsup\limits_{{\varepsilon}\downarrow0}\Mom{2}{{\rho}^{\varepsilon}} <+\infty,
\end{equation}
one has
$$
\liminf_{{\varepsilon}\downarrow 0}\mathcal F^{\varepsilon}({\rho}^{\varepsilon})\ge \mathcal F({\rho}).
$$
\item[(ii)] For every ${\rho} \in \PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$ there exists a family of measures
$\{{\rho}^{\varepsilon}\}\subset \PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$ such that $W_2({\rho}^{\varepsilon},{\rho})\to 0$ as ${\varepsilon}\downarrow0$ and
$$
\limsup_{{\varepsilon}\downarrow 0}\mathcal F^{\varepsilon}({\rho}^{\varepsilon})\le \mathcal F({\rho}).
$$
\end{itemize}
\end{lemma}
\begin{proof}
(i)
The ``liminf'' inequality for the potential energy $\displaystyle\mathcal V^{\varepsilon}(\rho):=
\int_\mathbb{R} V^{\varepsilon}\,\d\rho$ under weak convergence and \eqref{boundsecmom}
follows from \eqref{eq:86} and \eqref{eq:85}, since for every
$\delta>0$ there exist $R>\delta^{-1}$ and
${\varepsilon}_0>0$ such that
\begin{displaymath}
V^{\varepsilon}(x)\ge -\delta |x|^2\quad\text{for every }x\in \mathbb{R}\setminus
[-R,R],\quad
V^{\varepsilon}(x)\ge V(x)-\delta \quad\text{for every }x\in [-2R,2R],\ 0<{\varepsilon}<{\varepsilon}_0;
\end{displaymath}
for every $0<{\varepsilon}<{\varepsilon}_0$ and every smooth function
\begin{equation}
\text{${\raise.3ex\hbox{$\chi$}}:\mathbb{R}\to[0,1]$ with ${\raise.3ex\hbox{$\chi$}}(x)=1$
if $|x|\le 1$ and ${\raise.3ex\hbox{$\chi$}}(x)=0$ if $|x|\ge2$}
\label{eq:88}
\end{equation}
we have
\begin{displaymath}
\mathcal V^{\varepsilon}(\rho^{\varepsilon})= \int_\mathbb{R}
V^{\varepsilon}(x){\raise.3ex\hbox{$\chi$}}(x/R)\,\d\rho^{\varepsilon}+
\int_\mathbb{R}
V^{\varepsilon}(x)(1-{\raise.3ex\hbox{$\chi$}}(x/R))\,\d\rho^{\varepsilon}\ge
\int_\mathbb{R}
\Big(V(x)-\delta\Big){\raise.3ex\hbox{$\chi$}}(x/R)\,\d\rho^{\varepsilon}
-\delta\int_\mathbb{R} |x|^2\,\d\rho^{\varepsilon}
\end{displaymath}
so that
\begin{displaymath}
\liminf_{{\varepsilon}\to0}\mathcal V^{\varepsilon}(\rho^{\varepsilon})\ge
\int_\mathbb{R} {\raise.3ex\hbox{$\chi$}}(x/R)V(x)\,\d\rho(x)-\delta({\mathfrak m}+M_2).
\end{displaymath}
Since $R\ge\delta^{-1}$ and the previous inequality is valid for
arbitrary $\delta>0$, passing to the limit as $\delta\to0$ we obtain
\begin{equation}
\label{eq:87}
\liminf_{{\varepsilon}\to0}\mathcal V^{\varepsilon}(\rho^{\varepsilon})\ge \mathcal V(\rho).
\end{equation}
Let us now prove the ``liminf'' inequality for $\mathcal E^{\varepsilon}$: recalling
the usual decomposition
${\rho}^{\varepsilon}=u^{\varepsilon}\Leb{1} + ({\rho}^{\varepsilon})^\perp$, thanks to the definition of $\mathcal E^{\varepsilon}$ we get
\begin{equation*}
\mathcal E^{\varepsilon}({\rho}^{\varepsilon})= \mathcal E ({\rho}^{\varepsilon}) + {\varepsilon}\int_\mathbb{R} u^{\varepsilon} \log u^{\varepsilon} \d x \ge \mathcal E({\rho}^{\varepsilon})+{\varepsilon}\int_{\{0< u^{\varepsilon}< 1\}} u^{\varepsilon} \log u^{\varepsilon} \d x.
\end{equation*}
By Cauchy-Schwarz inequality and \eqref{boundsecmom} we obtain
\begin{align} \label{stima-ulogu}
\limsup_{{\varepsilon}\downarrow0}\bigg| \int_{\{0< u^{\varepsilon}< 1\}} u^{\varepsilon} \log u^{\varepsilon} \,\d x\bigg|
& \le \limsup_{{\varepsilon}\downarrow0}\bigg(\int_\mathbb{R} (1+|x|)^2 u^{\varepsilon} \,\d x\bigg)^\frac 1 2
\bigg(\int_{\{0< u^{\varepsilon}< 1\}} \frac{u^{\varepsilon} \log^2 u^{\varepsilon} }{(1+|x|)^2} \,\d x\bigg)^\frac 1 2 <+\infty.
\end{align}
Hence
$$
\liminf_{{\varepsilon}\downarrow0}\mathcal E^{\varepsilon}({\rho}^{\varepsilon}) \ge \liminf_{{\varepsilon}\downarrow0}\mathcal E({\rho}^{\varepsilon}),
$$
and (i) follows by the lower semicontinuity of $\mathcal E$ with respect to the weak convergence.
(ii) Let ${\rho}=u\Leb1+{\rho}^\perp\in\PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$
{with
$\mathcal F(\rho)<+\infty$ (the case $\mathcal F(\rho)=+\infty$ is trivial)}. Defining $c^{\varepsilon}:={\mathfrak m}/{\rho}([-1/{\varepsilon},1/{\varepsilon}])$, and $h^{\varepsilon}:=c^{\varepsilon}
\chi_{[-1/{\varepsilon},1/{\varepsilon}]}$, we set $${\rho}^{\varepsilon}:=h^{\varepsilon}{\rho}= h^{\varepsilon} u\Leb1+h^{\varepsilon}{\rho}^\perp.$$
Since $\lim_{{\varepsilon}\downarrow0}h^{\varepsilon}(x)=1$ pointwise,
for every function $W:\mathbb{R}\to \mathbb{R}$ such that $\displaystyle \int_\mathbb{R} |W(x)|\,\d{\rho}(x) <+\infty$, the dominated convergence theorem shows that
\begin{equation}\label{contW}
\lim_{{\varepsilon}\downarrow0} \int_\mathbb{R} W(x) \d {\rho}^{\varepsilon}(x) =
\lim_{{\varepsilon}\downarrow0}\Big(\int_\mathbb{R} W(x) h^{\varepsilon}(x)u(x) \,\d x + \int_\mathbb{R} W(x) h^{\varepsilon}(x)\,\d{\rho}^\perp(x)\Big)
= \int_\mathbb{R} W(x) \,\d{\rho}(x).
\end{equation}
In particular, choosing {$W=\varphi$ as in \eqref{eq:14} we obtain that
$W_2({\rho}^{\varepsilon},{\rho})\to 0$ so that for every $\delta>0$ there exists
$R>0$ such that
\begin{equation}
\label{eq:89}
\lim_{{\varepsilon}\downarrow0}\int_\mathbb{R} |x|^2(1-{\raise.3ex\hbox{$\chi$}}(x/R))\,\d\rho^{\varepsilon}=
\int_\mathbb{R} |x|^2(1-{\raise.3ex\hbox{$\chi$}}(x/R))\,\d\rho\le \delta
\end{equation}
for every function ${\raise.3ex\hbox{$\chi$}}$ as in \eqref{eq:88}.
On the other hand, \eqref{eq:85} yields ${\varepsilon}_0>0$ such that
$V^{\varepsilon}(x)\le V(x)+\delta$ if $|x|\le 2R$ and therefore
\begin{align}
\notag\mathcal V^{\varepsilon}(\rho^{\varepsilon})&\le
\int_\mathbb{R} V^{\varepsilon}(x){\raise.3ex\hbox{$\chi$}}(x/R)\,\d\rho^{\varepsilon}+
\int_\mathbb{R} \Big(V(x)+A|x|^2\Big)(1-{\raise.3ex\hbox{$\chi$}}(x/R))\,\d\rho^{\varepsilon}\\
\label{eq:91}&\le
\int_\mathbb{R} V(x)\,\d\rho^{\varepsilon}+\delta{\mathfrak m}+A\int_\mathbb{R} |x|^2(1-{\raise.3ex\hbox{$\chi$}}(x/R))\,\d\rho^{\varepsilon}.
\end{align}
Using \eqref{contW} with $W=V$, passing to the limit as ${\varepsilon}\downarrow0$ in
\eqref{eq:91}, we obtain
\begin{equation}
\label{eq:92}
\limsup_{{\varepsilon}\downarrow0}\mathcal V^{\varepsilon}(\rho^{\varepsilon})\le V(\rho)+\delta({\mathfrak m}+A).
\end{equation}
Since $\delta>0$ is arbitrary we conclude
}
\begin{equation}\label{contV}
\limsup_{{\varepsilon}\downarrow0} \mathcal V^{\varepsilon}({\rho}^{\varepsilon})\le \mathcal V(\rho).
\end{equation}
On the other hand, since $E\leq 0$ is continuous, by Fatou's Lemma we have
\begin{equation}\label{limsupinE}
\limsup_{{\varepsilon}\downarrow0}\int_\mathbb{R} E(u^{\varepsilon}(x))dx \le \int_\mathbb{R} E(u(x))dx.
\end{equation}
Denoting by $\PlusMeasuresComp{\mathbb{R},{\mathfrak m}}$ the set of nonnegative measures with compact support and total mass ${\mathfrak m}$, we have just proved that
\begin{equation}\label{Gammacont}
\forall\,{\rho}\in\PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}\cap D(\mathcal F)\ \ \exists \{{\rho}^{\varepsilon}\}\subset \PlusMeasuresComp{\mathbb{R},{\mathfrak m}}
:\quad
W_2({\rho}^{\varepsilon},{\rho})\to 0,\quad \lim_{{\varepsilon}\downarrow0}\mathcal F({\rho}^{\varepsilon})=\mathcal F({\rho}).
\end{equation}
{A standard diagonal argument for $\Gamma$-convergence shows that
(ii) can be reduced to prove }
\begin{equation}\label{GammaLimsupComp}
\forall\,{\rho}\in\PlusMeasuresComp{\mathbb{R},{\mathfrak m}}, \quad \exists \{{\rho}^{\varepsilon}\}\subset \PlusMeasuresComp{\mathbb{R},{\mathfrak m}}
: W_2({\rho}^{\varepsilon},{\rho})\to 0,\quad \limsup_{{\varepsilon}\downarrow0}\mathcal F^{\varepsilon}({\rho}^{\varepsilon})\leq\mathcal F({\rho}).
\end{equation}
Let ${\rho}=u\Leb1+{\rho}^\perp\in\PlusMeasuresComp{\mathbb{R},{\mathfrak m}}$; denoting
by $k^{\varepsilon}={\varepsilon}^{-1}k(\cdot/{\varepsilon})$
a standard family
of {symmetric and nonnegative} mollifiers with support
$[-{\varepsilon},{\varepsilon}]$, we set $u^{\varepsilon}(x) = (k^{\varepsilon} * {\rho}) (x) = \int_\mathbb{R} k^{\varepsilon}(x-y)\,\d{\rho}(y)$ and ${\rho}^{\varepsilon}=u^{\varepsilon}\Leb{1}$. By definition of
convolution and Fubini's theorem we have
$$\int_\mathbb{R} V(x)\,\d{\rho}^{\varepsilon}(x)= \int_\mathbb{R} V(x)\int_\mathbb{R} k^{\varepsilon}(x-y)\,\d{\rho}(y)\,\d x =
\int_{\supp ({\rho})}\int_{[-1,1]}V(y+{\varepsilon} z)k(z)\,\d z\, \d{\rho}(y).$$
By the continuity of $V$,
and the dominated convergence theorem
\begin{equation}\label{contV2}
\lim_{{\varepsilon}\downarrow0} \int_\mathbb{R} V(x) \d {\rho}^{\varepsilon}(x) = \int_\mathbb{R} V(x) \d {\rho}(x).
\end{equation}
Recalling that $E$ is decreasing and applying Jensen's inequality to
the probability measure $k^{\varepsilon} (x-y)\Leb{1}(y)$ and the convex function $E$ we get
$$ E(u^{\varepsilon}(x))=E\Big(\int_\mathbb{R} k^{\varepsilon}(x-y)\,\d {\rho}(y)\Big)\le E\Big(\int_\mathbb{R} u(y)k^{\varepsilon}(x-y)\,\d y\Big) \leq \int_\mathbb{R} E(u(y)) k^{\varepsilon}(x-y)\,\d y.$$
Integrating with respect to $x$ and using Fubini's theorem we obtain
\begin{equation}\label{supE}
\int_\mathbb{R} E(u^{\varepsilon}(x))\,\d x \leq \int_\mathbb{R} E(u(x))\,\d x.
\end{equation}
Finally, since $k^{\varepsilon}\le 1/{\varepsilon}$ and $u^{\varepsilon}(x)\le {\varepsilon}^{-1}{\mathfrak m}$, we have
\begin{equation}\label{estEntr}
{\varepsilon}\int_\mathbb{R} u^{\varepsilon}\log u^{\varepsilon} \,\d x\le {\mathfrak m}\, {\varepsilon} \log \frac {\mathfrak m} {\varepsilon}.
\end{equation}
Since $W_2({\rho}^{\varepsilon},{\rho})\to 0$,
\eqref{contV2}, \eqref{supE} and
\eqref{estEntr} yield \eqref{GammaLimsupComp}.
\end{proof}
\section{Proofs of the main Theorems}
\subsection{Subdifferential characterization of the gradient flow of $\mathcal F$ and existence result.}\label{sec:gf}
\begin{proof}[Proof of Theorem \ref{thm:main1}.]
The proof of Theorem \ref{thm:main1} is based on the general results
about the generation of gradient flows for
displacement $\lambda$-convex functionals in
$\PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$ established in \cite{ags} (notice that all the theory in \cite{ags} can be applied to the space
$\PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$ and not only to the space $\PlusMeasuresTwo{\mathbb{R},1}$ considered in \cite{ags}).
By Proposition \ref{prop:lscF} the functional $\mathcal F$ is displacement
$\lambda$-convex
(in dimension $1$ generalized geodesics \cite[Definition 9.2.2]{ags}
coincide
with the displacement interpolations \eqref{eq:38})
and we can apply the general theory summarized in Theorem 11.2.1
\cite{ags}.
Since $D(\mathcal F)=\{{\rho}\in \PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}:
\mathcal F({\rho})<+\infty \}$ is dense in $\PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$,
the evolution is well defined starting from
an arbitrary element of $\PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$. Therefore, by \cite[Theorem 11.2.1]{ags},
for every ${\rho}_0\in \PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$ there exists a unique curve ${\rho}$ belonging to $
C^0([0,+\infty);\PlusMeasuresTwo{\mathbb{R},{\mathfrak m}})$ such that
$\rho_t\in D(\mathcal I)\subset D(\mathcal F)$ for every $t>0$ and
\begin{align}\label{ce}
\de_t{\rho}_t + \de_x({\rho}_t\,{\mbox{\boldmath$v$}}_t)&=0, \qquad &\text{ in } \mathscr{D}'(\mathbb{R}\times (0,+\infty)),\\
\label{nlrel}
{\mbox{\boldmath$v$}}_t&=-\de^\circ\mathcal F({\rho}_t), \qquad &\text{ for $\Leb{1}$-a.e. } t \in (0,+\infty),\\
\label{eq:43}
\mathcal F({\rho}_{t_0})-\mathcal F({\rho}_{t_1}) &= \int_{t_0}^{t_1}\int_\mathbb{R}|{\mbox{\boldmath$v$}}_t|^2\,{\rho}_t(x)\,\d t
\qquad &0\leq t_0 < t_1,
\end{align}
Moreover the map ${\rho}_0\mapsto S_t({\rho}_0):={\rho}_t$ defines a continuous semigroup satisfying the $\lambda$-contraction property
\eqref{eq:15}.
From \cite[Theorem 2.4.15]{ags} the map $t\mapsto e^{\lambda t}|\de \mathcal F|^2({\rho}_t)$ is non-increasing, and then \eqref{eq:19}
holds. The regularization estimate \eqref{eq:18}
(which implies \eqref{eq:3bis})
still follows by
Theorem 11.2.1 and by
\cite{S10} in the case $\lambda\neq 0$.
From \eqref{nlrel} and Theorem \ref{th:charsubdiff} we have
\eqref{eq:3}.
\eqref{ce}, \eqref{nlrel}, and \eqref{eq:50} yields
\eqref{eq-debole}.
{The comparison result follows from Theorem
\ref{thm:main2} and the corresponding property for solution of the
viscous regularization.}
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:main2}]
The part concerning existence of solutions to problem \eqref{eq:17} for a measure initial datum, is similar to the part concerning existence for
problem \eqref{eq:rho}, taking into account the characterization of the subdifferential of $\mathcal F^{\varepsilon}$ \eqref{eq:48}.
The stability with respect to the convergence in $\PlusMeasuresTwo{\mathbb{R},{\mathfrak m}}$ follows from Lemma \ref{le:Gamma-convergence} and Theorem 11.2.1 of
\cite{ags}. The uniform convergence follows from Theorem \ref{thm:lsc-dissipation} \eqref{eq:219}.
\end{proof}
\subsection{Localized entropy estimates and propagation of
singularities}\label{sec:estimates}
Let us consider
\begin{subequations}
\begin{gather}
\label{eq:93}
\text{a smooth convex function $\psi:[0,+\infty)\to \mathbb{R}$ with
$\psi(0)=0$,}
\intertext{and let us set (recall that $\beta^{\varepsilon}(r)=\beta(r)+{\varepsilon} r$)}
\label{eq:63}
\eta(r):=r\psi'(r)-\psi(r),\quad
\gamma(r):=\int_0^r\beta'(s)\psi'(s)\,\d s,\quad
\gamma^{\varepsilon}(r):= \gamma(r)+{\varepsilon}\psi(r)=
\int_0^r
(\beta^{\varepsilon})'(s)\psi'(s)\,\d s.
\end{gather}
\end{subequations}
\begin{theorem} \label{thm:propagsing}
If $u^{\varepsilon}$ is a smooth {bounded} solution to \eqref{eq:17}
{and $\psi,\eta,\gamma^{\varepsilon}$ satisfy \eqref{eq:93} and
\eqref{eq:63},}
then
$\psi(u^{\varepsilon})$ is a classical solution to
\begin{equation} \label{eq:59bis}
\partial_t\psi(u^{\varepsilon})-\partial_x\big(\partial_x\gamma^{\varepsilon}(u^{\varepsilon})+\psi(u^{\varepsilon})(V^{\varepsilon})'\big)\le \eta(u^{\varepsilon})(V^{\varepsilon})''.
\end{equation}
In particular, for every nonnegative $\phi\in C^2_{\rm c}(\mathbb{R}\times [0,T])$
it holds
\begin{equation}
\label{eq:59}
\begin{aligned}
\int_\mathbb{R} \psi(u^{\varepsilon}(x,T))& \phi(x,T)\,\d x +
\int_0^T\int_\mathbb{R}
\psi(u^{\varepsilon})\big(-\partial_t\phi+\partial_x \phi (V^{\varepsilon})'\big)\,\d x\,\d t\\&
- \int_0^T\int_\mathbb{R} \big(\gamma^{\varepsilon}(u^{\varepsilon})\partial^2_x\phi +\eta(u^{\varepsilon})\phi (V^{\varepsilon})''\big)\,\d x\,\d t
\le \int_\mathbb{R} \psi(u^{\varepsilon}(x,0))\phi(x,0)\,\d x.
\end{aligned}
\end{equation}
\end{theorem}
\begin{proof}
By straightforward computations we obtain that
$$
\partial_t\psi(u^{\varepsilon})-\partial_x\big(\partial_x\gamma^{\varepsilon}(u^{\varepsilon})+\psi(u^{\varepsilon})(V^{\varepsilon})'\big)
=\eta(u^{\varepsilon})(V^{\varepsilon})''-(\beta^{\varepsilon})'(u^{\varepsilon})\psi''(u^{\varepsilon})(\partial_x u^{\varepsilon})^2.
$$
Since $\psi$ is convex and $\beta^{\varepsilon}$ is increasing, $(\beta^{\varepsilon})'(u^{\varepsilon})\psi''(u^{\varepsilon})(\partial_x u^{\varepsilon})^2\ge 0$. This implies
\eqref{eq:59bis}.
\end{proof}
\noindent We will now prove the {\em a priori} estimate \eqref{eq:9}.
\begin{corollary}
\label{cor:linfty_bound}
Let us assume that \eqref{eq:8} holds and that ${\rho}_0=u_0\Leb1$ has a bounded density. Then \eqref{eq:9} holds.
\end{corollary}
\begin{proof}
{By Theorem
\ref{thm:main2} it is sufficient to show \eqref{eq:9} for the
(bounded and integrable)} solutions $\rho^{\varepsilon}=u^{\varepsilon}\Leb{1}$ of \eqref{eq:17} with initial
datum ${\rho}_0$.
Let us apply \eqref{eq:59} with $\psi(r)=r^p$, $p\ge 2$, and
$\phi(x)={\raise.3ex\hbox{$\chi$}}(x/n)$, where ${\raise.3ex\hbox{$\chi$}}$ satisfies \eqref{eq:88}.
{Since $(V^{\varepsilon})'$ is bounded and $(V^{\varepsilon})''\le \mathsf{c}$,}
it is not difficult to pass to the limit as $n \to+\infty$, getting
\begin{equation*}
\int_\mathbb{R} u^{\varepsilon}(x,T)^p \,\d x
\le \int_\mathbb{R} u_0^p(x)\,\d x+\mathsf{c} (p-1) \int_0^T\int_\mathbb{R} u^{\varepsilon}(x,t)^p \,\d x\,\d t.
\end{equation*}
From Gronwall's Lemma it follows that
\begin{equation*}
\int_\mathbb{R} u^{\varepsilon}(x,T)^p \,\d x \le \mathrm e^{\mathsf{c} (p-1) T}\int_\mathbb{R} u_0^p(x)\,\d x, \qquad \text{for all}\ T>0.
\end{equation*}
Letting $p\uparrow +\infty$ we get estimate \eqref{eq:9} for ${\rho}^{\varepsilon}$.
\end{proof}
\noindent
The following corollary of Theorem \ref{thm:propagsing}
is a preliminary step for the proof of Theorem \ref{thm:main3}
on the propagation of the singularities.
\begin{corollary}\label{cor3}
Let $\psi,\eta,\gamma$ be as in \eqref{eq:93} and \eqref{eq:63},
with
$\lim_{r\uparrow+\infty}\psi'(r)=\psi_\infty'\in (0,+\infty)$.
If ${\rho}=u\Leb 1+{\rho}^\perp$ is the measure-valued solution to
\eqref{eq:rho}
and $\psi({\rho}):=\psi(u)\Leb 1+\psi_\infty'\,{\rho}^\perp$, we have
\begin{equation}
\label{eq:66}
\partial_t
\psi({\rho})-\partial_x\big(\psi({\rho})V'\big)\le \partial^2_x(\gamma(u))+\eta(u)V''\quad
\text{in the sense of distributions}.
\end{equation}
\end{corollary}
\begin{proof}
It is sufficient to pass to the limit in \eqref{eq:59}, recalling \eqref{eq:65}
and applying the dominated convergence theorem with the estimate
$|\psi(r)|\le \|\psi'\|_{L^\infty((0,+\infty))} r$.
Notice that
\begin{displaymath}
\lim_{r\to+\infty}\frac {\eta(r)}r=
\lim_{r\to+\infty} \Big(\psi'(r)-\frac {\psi(r)}r\Big)=0
\end{displaymath}
and
\begin{displaymath}
\lim_{r\to+\infty}\frac {\gamma(r)}r=
\lim_{r\to+\infty}\frac
1r\Big(\beta(r)\psi'(r)-\beta(0)\psi'(0)-\int_0^r
\beta(s)\psi''(s)\,\d s\Big)
=0,
\end{displaymath}
since $\lim_{r\uparrow+\infty}\beta(r)=\beta_\infty<+\infty$ and we estimate the integral as follows
\begin{displaymath}
0\le \frac {1}{r} \int_0^r
\beta(s)\psi''(s)\,\d s\le
\frac{\beta_\infty}{r}\Big(\psi'(r)-\psi'(0)\Big).
\end{displaymath}
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:main3}]
Let us fix a nonnegative function $\zeta\in C^\infty_{\rm c}(\mathbb{R})$
with compact support in $[0,1]$ and integral equal to $1$. We set
$\zeta_k(r):=\zeta(r-k)$, $Z_k(r):=\int_0^r \zeta_k(s)\,\d s$,
$\psi_k(r)=\int_0^r Z_k(s)\,\d s$. It is immediate to check that $\psi_k$
satisfies the assumptions of Corollary \ref{cor3}. Moreover, the
corresponding functions $\gamma_k(r)$ and $\eta_k(r)$ are uniformly
bounded by $Cr$ and converge to $0$ pointwise as $k\to +\infty$.
Passing to the limit in
\begin{equation}
\label{eq:66bis}
\partial_t
\psi_k({\rho})-\partial_x\big(\psi_k({\rho})V'\big)\le \partial^2_x(\gamma_k(u))+\eta_k(u)V''\quad
\text{in the sense of distributions}
\end{equation}
as $k\uparrow+\infty$ we obtain \eqref{eq:68}. Now, set $\mu_t=
(\mathsf{X}_t)_\#{\rho}_0^\perp$. It is well known that $\mu_t$ solves
$\partial_t\mu_t-\partial_x(\mu_tV')=0$. Then the family of measures
$\sigma_t={\rho}_t^\perp-\mu_t$ satisfies
$\partial_t\sigma_t-\partial_x(\sigma_tV')\le 0$ with {$\sigma_0\le0$}.
By {a simple variant of} Proposition 8.1.7 of \cite{ags} we deduce that $\sigma_t\le 0$
for every $t\ge 0$.
Therefore for
every Borel set $A\subset \mathbb{R}$, $ {\rho}_t^\perp(A)\le {\rho}_0^\perp
\big(\mathsf{X}_t^{-1}(A)\big)$. Choosing $A=D_t$, the inclusion
$\CDom{u_t}\subset \mathsf J_t$ follows.
\end{proof}
\subsection{Minimizers, stationary solutions, and asymptotic properties}
\label{sec:min}
\begin{proof}[Proof of Theorem \ref{thm:main4}.]
Let us first show that
every measure
${\rho}_{\rm min}=u_{\rm min}\Leb1 +{\rho}^\perp_{\rm min}$ satisfying
\eqref{eq:30} is a minimizer for $\mathcal F$.
Notice that by construction $\rho_{\rm min}\in \CPlusMeasures{\mathbb{R},{\mathfrak m}}$.
Let {${\rho}=u\Leb 1+{\rho}^\perp$ be an arbitrary measure in
$\PlusMeasures{\mathbb{R},{\mathfrak m}}$.}
If $A=\{x\in\mathbb{R}:V(x)-\frak v<\frak d \}$
and $B=\mathbb{R}\setminus A$ denotes its complement,
$$ u_{\rm min}(x)=\begin{cases}H(V(x)-\frak v) & \text{if }x\in A,
\\ 0 & \text{if }x\in B. \end{cases}$$
Since $$ E'(H(v))=\begin{cases}-v & \text{if }v\in(0,\frak d), \\
-\frak d
& \text{if }v\in[\frak d,+\infty), \end{cases}$$
and $E$ is convex, we
get
\begin{align*}
\mathcal E({\rho})-\mathcal E({\rho}_{\rm min})& =
\int_\mathbb{R} \big(E(u(x))-E(u_{\rm min}(x))\big)\,\d x\ge
\int_\mathbb{R} E'(u_{\rm min}(x))\big(u(x)-u_{\rm min}(x)\big)\,\d x\\&=
\int_A (\frak v-V(x)) \big(u(x)-u_{\rm min}(x)\big)\,\d x
- \frak d\int_B u(x)\,\d x.
\end{align*}
Moreover, since $V(x)-\frak v\ge \frak d$
for every $x\in B$,
\begin{align*}
\mathcal F({\rho})-\mathcal F({\rho}_{\rm min})&=\mathcal E({\rho})-\mathcal E({\rho}_{\rm min})+\int_\mathbb{R}
V\,\d{\rho}-\int_\mathbb{R} V\,\d{\rho}_{\rm min}\\
&\ge \int_A (\frak v-V(x)) \big(u(x)-u_{\rm min}(x)\big)\,\d x
+ \int_B (V(x)-\frak d) u(x)\,\d x \\
&\quad + \int_A V(x) \big(u(x)-u_{\rm min}(x)\big)\,\d x
+\int_\mathbb{R} V\,\d{\rho}^\perp-\int_\mathbb{R} V\,\d{\rho}^\perp_{\rm min} \\
&\ge \int_\mathbb{R} \frak v \big(u(x)-u_{\rm min}(x)\big)\,\d x
+\int_\mathbb{R} V\,\d{\rho}^\perp-\int_\mathbb{R} V\,\d{\rho}^\perp_{\rm min}.
\end{align*}
Hence, owing to the identity
\[\displaystyle
{\rho(\mathbb{R})=\rho_{\rm min}(\mathbb{R}),\quad
\text{so that}\quad
\int_\mathbb{R} u\,\d x-
\int_\mathbb{R} u_{\rm
min}\,\d x=\int_\mathbb{R} \d {\rho}_{\rm min}^\perp-\int_\mathbb{R} \d {\rho}^\perp } ,
\]
and recalling that ${\rho}_{\rm
min}^\perp$ is concentrated in $Q$, we obtain
\begin{align*}
\mathcal F({\rho})-\mathcal F({\rho}_{\rm min})&\ge\int_\mathbb{R} \frak v \big(u(x)-u_{\rm min}(x)\big)\,\d x
+\int_\mathbb{R} V\,\d{\rho}^\perp-\int_\mathbb{R} V\,\d{\rho}^\perp_{\rm min}\\
&=\int_\mathbb{R} (V-\frak v)\,\d{\rho}^\perp-\int_\mathbb{R}(V-\frak v)\,\d{\rho}^\perp_{\rm min}
\ge -\int_\mathbb{R}(V-\frak v)\,\d{\rho}^\perp_{\rm min} =0.
\end{align*}
This shows that $\mathcal F({\rho})\ge\mathcal F({\rho}_{\rm min})$ for every
{${\rho}\in\PlusMeasures{\mathbb{R},{\mathfrak m}}$.}
We prove now that every minimizer ${\rho}=u\Leb 1+{\rho}^\perp \in
\PlusMeasures{\mathbb{R},{\mathfrak m}}$ of $\mathcal F$ in $\PlusMeasures{\mathbb{R},{\mathfrak m}}$
satisfies \eqref{eq:30}. We
consider another minimizer $\rho_{\rm min}$ given by \eqref{eq:30}
so that
equalities hold in all the previous inequalities and in particular we have
$$
0=\mathcal F({\rho})-\mathcal F({\rho}_{\rm min})= \int_\mathbb{R} (V-\frak v)\,\d{\rho}^\perp .
$$
It follows that ${\rho}^\perp$ is concentrated on $Q$ and ${\rho}^\perp =0$ when ${\mathfrak m} < {\mathfrak m}_{\rm c}$
(recall that $V(x)-\frak v \ge 0$ and equality holds if and only if $\frak v = V_{\rm min}$ and $x\in Q$).
If $u\not = u_{\rm min}$, then, by the strict convexity of $E$,
$\mathcal F((1-\theta){\rho} + \theta {\rho}_{\rm min})< \mathcal F({\rho}_{\rm min})$ for every $\theta \in (0,1)$.
Taking the continuity of $u$ into account, it follows that $u(x)=u_{\rm min}(x)$ for every $x\in \mathbb{R}$.
Consequently
${\rho}^\perp (\mathbb{R}) = {\rho}_{\rm min}^\perp (\mathbb{R})$ and we conclude.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:Fisher=0}]
It follows easily by \cite[Theorem 11.1.3]{ags}, which shows
in particular that $\rho$ is a stationary solution of the Wasserstein gradient
flow of a displacement $\lambda$-convex functional $\mathcal F$ iff
$|\partial \mathcal F|(\rho)=0$.
We can then invoke Theorem \ref{th:charsubdiff}.
\end{proof}
The proof of Theorems \ref{thm:statI} and \ref{thm:main_stationary} is based on the following lemma:
\begin{lemma}
\label{le:char}
Let $\rho=u\Leb 1+\rho^\perp\in \CPlusMeasures\mathbb{R}$ be a measure
satisfying $\mathcal I(\rho)=0$, and let us consider the open set $\Pos u:=\big\{x\in
\mathbb{R}:u(x)>0\big\}$.
If $I$ is a connected component of $\Pos u$ then
\begin{equation}
\label{eq:64}
E'(u(x))+V(x)=c_I\quad \text{for every }x\in I.
\end{equation}
\end{lemma}
\begin{proof}
Let us first show that
the function
$E'\circ u$ belongs to $W^{1,1}_{\rm loc}(\Pos u)$ with
\begin{equation}
\label{eq:100}
\partial_x \big(E'\circ u\big)=\frac {\partial_x \big(\beta\circ
u\big)}u\quad \text{in }\Pos u.
\end{equation}
We can simply write $E'\circ u=L\circ (\beta\circ u)$ where
$L:=E'\circ \beta^{-1}$ and $\beta\circ u\in W^{1,1}_{\rm loc}(\mathbb{R})$.
The function $L$ belongs
to $C^1(0,{\beta^{\infty}})$ and can be extended to ${\beta^{\infty}}$ by continuity
setting $L({\beta^{\infty}})=0$; it is easy to check that this
extension belongs to $C^1(0,{\beta^{\infty}}]$, since
\begin{displaymath}
L'(r)=\frac{E''\circ \beta^{-1}}{\beta'\circ \beta^{-1}}=
\frac{1}{\beta^{-1}},
\quad
\lim_{r\uparrow{\beta^{\infty}}}L'(r)=0.
\end{displaymath}
\eqref{eq:100} then follows by the chain rule for the composition of
a $C^1$ with a Sobolev function.
If $I$
is a connected component of $\Pos u$, we have
\begin{equation}
\label{eq:70}
0 = \frac{\partial_x\beta(u(x))}{u(x)}+V'(x) = \partial_x(E'(u(x))+V(x)) \quad\text{in }I,
\end{equation}
so that there exists a constant $c_I$ such that \eqref{eq:64} holds.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:statI}]
We have to prove only the ``right'' implication $\Rightarrow$.
A simple argument by contradictions shows that $\Pos u=\mathbb{R}$:
otherwise, if the interval
$I=(a,b)$ is a connected component of $\Pos u$ and one of its
extremes, say $a$, is finite,
we should have
\begin{displaymath}
\lim_{x\downarrow a}u(x)=0,\quad
-\frak d=\lim_{x\downarrow a}E'(u(x))=c_I-V(a)>-\infty.
\end{displaymath}
Since $\Pos u=\mathbb{R}$ Lemma \ref{le:char} yields $V(x)\ge c_I$ for every
$x\in \mathbb{R}$ and $u(x)=H(V(x)-c_I)$. Since $\rho\in
\CPlusMeasures{\mathbb{R},{\mathfrak m}}$
we conclude that \eqref{eq:30} holds and $\rho$ is a minimizer of
$\mathcal F$
by Theorem \ref{thm:main4}.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:main_stationary}]
Let $\rho=u\Leb 1+\rho^\perp\in \CPlusMeasures{\mathbb{R},{\mathfrak m}}$ with
$\mathcal I(\rho)=0$ and let $I=(a,b)$ be
a connected component of the open set $\Pos u$.
Since the range of the function $r\mapsto -E'(r)$ for $r\in
(0,+\infty]$ is the bounded interval $(0,\frak d]$ and
$\lim_{|x|\to\infty}V(x)=+\infty$
we deduce from Lemma \ref{le:char} that $I$ is bounded.
It follows that $u(a)=u(b)=0$ and therefore
$\lim_{x\downarrow a}E'(u(x))=
\lim_{x\uparrow b}E'(u(x))=-\frak d$, $c_I=V(a)-\frak d=V(b)-\frak d$.
We thus obtain \eqref{eq:98} and the representation \eqref{eq:103},
which also yields \eqref{eq:99} since $u$ is integrable in $\mathbb{R}$.
Since for every $x\in I$ $u(x)=+\infty$ iff $V(x)=V(a)-\frak d$, i.e.
$x\in Q_I$, we obtain \eqref{eq:84}.
Conversely, if $\rho=u\Leb 1+\rho^\perp\in
\CPlusMeasures{\mathbb{R},{\mathfrak m}}$ satisfies the
three conditions of Theorem \ref{thm:main_stationary},
we immediately have that $\mathcal I(\rho)=0$.
In fact, the first integral of the definition of $\mathcal I$ in
\eqref{eq:5}
vanishes by \eqref{eq:64} and \eqref{eq:100}; the second
integral, corresponding to the singular part of $\rho$ vanishes
since $\rho^\perp$ is concentrated on $Q(u)$ and
$V'$ vanishes in each point of $Q_I$, which is a local minimizer of $V$.
\end{proof}
\begin{proof}[Proof of Corollary \ref{cor:obvious}]
Remark \ref{rem:examples} shows that the minimizer of $\mathcal F$ is
unique.
We have just to check the case when $\frak d<+\infty$.
By the assumption on the first derivative of $V$ is immediate to
check that the set $\Pos u$ contains just one connected component $I=(a,b)$
with $a<q_-<q_+<b$. Theorem \ref{thm:main4}
shows that $\rho$ is a minimizer of $\mathcal F$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:main5}.]
We use the dissipation identity \eqref{eq:13} to obtain the
inequality
\begin{displaymath}
{\int_{t_0}^{t_1}\mathcal I({\rho}_t)\,\d t=
\mathcal F({\rho}_{t_0})-\mathcal F({\rho}_{t_1})\le \mathcal F({\rho}_{t_0})-\mathcal F(\bar
\rho)<+\infty\quad
\text{for every }0<t_0<t_1<+\infty.}
\end{displaymath}
Passing to the limit as $t_1\uparrow+\infty$ we get $\mathcal I(\rho_t)\in
L^1(t_0,+\infty)$, so that
\begin{equation}
\label{eq:106}
\sum_{n=2}^{+\infty}\int_{n-1}^{n}\mathcal I(\rho_t)\,\d t<+\infty.
\end{equation}
Since by \eqref{eq:19} $\mathcal I(\rho_t)\ge \mathrm e^{-2\lambda^-}\mathcal I(\rho_n)$
if $t\in (n-1,n)$ we obtain $\sum_{n=2}^{+\infty}\mathcal I(\rho_n)<+\infty$;
in particular
\begin{equation}
\label{eq:108}
\lim_{n\uparrow+\infty}\mathcal I(\rho_n)=0\quad\text{and a further
application of \eqref{eq:19} yields}\quad
\lim_{t\uparrow+\infty}\mathcal I(\rho_t)=0.
\end{equation}
Since
$\mathcal F({\rho}_{t})\leq \mathcal F({\rho}_{t_0})$ for every $t\geq t_0$, by \eqref{eq:95}
we infer that {$\{{\rho}_{t}\}_{t\ge t_0}$ is tight;}
by Theorem \ref{thm:lsc-dissipation2}
any weak limit point $\rho_\infty$ of $\rho_t$ as $t\uparrow+\infty$
satisfies
$\mathcal I({\rho}_\infty)=0$ and therefore $\rho_\infty=\bar\rho$. It follows
that $\rho_t\rightharpoonup \bar\rho$ weakly as $t\uparrow+\infty$.
Theorem \ref{thm:lsc-dissipation2} yields the uniform convergence of
$u_t$ to $\bar u$
on compact sets of $\Dom{\bar u}$ as $t\to +\infty$.
When ${\mathfrak m}<{\mathfrak m}_\mathrm c$, $\bar{\rho}$ has a
bounded density and therefore for every compact subset $K\subset \mathbb{R}$
there exists a time $T>0$ such that ${\rho}_{t}$ is bounded on $K$
for every $t\ge T$. Choosing as $K:=\big\{x\in \mathbb{R}:V(x)\le c\big\}$
for a constant $c$ sufficiently big so that $K$ contains the support
of $\rho_0^\perp$, Theorem \ref{thm:main3}
shows that the support of $\rho_t^\perp$ is contained in $K$
for every $t>0$ and therefore $\rho_t^\perp=0$ for $t\ge T$.
\end{proof}
\bibliographystyle{siam}
\def$'${$'$}
|
{
"timestamp": "2010-09-23T02:01:25",
"yymm": "1009",
"arxiv_id": "1009.4305",
"language": "en",
"url": "https://arxiv.org/abs/1009.4305"
}
|
\section{Introduction}
Given a family $\mathcal{F}$ of sets, a graph $G=(V,E)$ is called an
\emph{intersection graph} of sets from $\mathcal{F}$, if there exists a
map $f~:~V(G) \rightarrow \mathcal{F}$ such that
$(u,v) \in E(G) \Leftrightarrow f(u) \cap f(v) \neq \emptyset$. If the sets
in $\mathcal{F}$ are intervals on a real line, then we call $G$ an \emph{interval
graph}. In other words, interval graphs are intersection graphs of
intervals on the real line. In $\mathbb{R}^k$, an axis parallel $k$-dimensional
box or a \emph{$k$-box} is a cartesian product $R_1 \times R_2 \times \cdots \times R_k$, where
each $R_i$ is a closed interval $[a_i,b_i]$ on the real line.
A graph $G$ is said to have a $k$-box representation if there exists a mapping from the
vertices of $G$ to $k$-boxes in the $k$-dimensional eucledian space
such that two vertices in $G$ are adjacent if and only if their corresponding $k$-boxes
have a non-empty intersection.
\emph{Boxicity} of $G$, denoted
by $\ensuremath{box}(G)$, is
the minimum positive integer $k$ such that $G$ has a
$k$-box representation. As each interval can also be viewed as an
axis parallel $1$-dimensional box, interval graphs are precisely the
class of graphs with boxicity 1.
We take the boxicity of a complete graph to be 1.
\subsection{Background}
The concept of boxicity was introduced by F.S. Roberts in 1969 \cite{Roberts}.
Cozzens \cite{Coz} showed that computing the boxicity of a graph is
NP-hard. Yannakakis in \cite{Yan1} improved this result. Finally,
Kratochvil \cite{Kratochvil} showed that deciding whether the boxicity
of a graph is at most 2 itself is NP-complete.
Box representation of graphs finds application in niche overlap (competition) in
ecology and to problems of fleet maintenance in operations research
(see \cite{CozRob}). Given a low dimensional box representation,
some well known NP-hard problems become polynomial time solvable.
For instance, the max-clique problem is polynomial time solvable for graphs with
boxicity $k$ because the number of maximal cliques in such graphs is only
$O((2n)^k)$.
Roberts proved that for every graph $G$
on $n$ vertices, $\ensuremath{box}(G) \leq \lfloor \frac{n}{2} \rfloor$. He gave a tight example to this
by showing that a complete $\frac{n}{2}$-partite graph with 2 vertices in each part
has its boxicity equal to $\frac{n}{2}$. In \cite{chintan}, it was shown that if
$t$ denotes the size of a
minimum vertex cover of $G$, then $\ensuremath{box}(G) \leq \lfloor \frac{t}{2}\rfloor + 1$.
Chandran, Francis and Sivadasan showed in \cite{tech-rep} that,
for any graph $G$ on $n$ vertices having maximum degree $\Delta$,
$\ensuremath{box}(G) \leq (\Delta + 2)\ln n $. An upper bound solely in terms of the maximum
degree $\Delta$, which says $\ensuremath{box}(G) \leq 2\Delta^2$, is proved in \cite{CFNMaxdeg}.
Esperet \cite{Esperet} improved this bound to $\Delta^2 + 2$. Recently
Adiga, Bhowmick and Chandran \cite{DiptAdiga} showed that $\ensuremath{box}(G)=O(\Delta \log^2 \Delta)$.
Chandran and Sivadasan in \cite{CN05} found a relation between treewidth and boxicity
which says $\ensuremath{box}(G) \leq \mathrm{tw}(G)+ 2$,
where $\mathrm{tw}(G)$ denotes the treewidth of graph $G$.
Attempts on finding better bounds for boxictiy of special graph
classes can also be seen in the literature. Scheinerman \cite{Scheiner}
showed that outerplanar graphs have boxicity at most 2. Thomassen \cite{Thoma1}
proved that the boxicity of planar graphs is not greater than 3. Cozzens
and Roberts \cite{CozRob} have done a study on the boxicity of split graphs.
Results on the boxicity of Chordal graphs, AT-free graphs, permutation graphs
etc. can be seen in \cite{CN05}. Better bounds for
the boxicity of Circular Arc graphs and AT-free graphs can be seen in \cite{Dipt,Bhowmick20101536}.
In \cite{SunMatRog} it was shown that, there exist chordal bipartite graphs with
arbitrarily high boxicity.
\subsection{An Equivalent Definition for Boxicity}
Let $G,G_1,G_2,\ldots,G_b$ be a collection of graphs with $V(G) = V(G_i)$,
for any $i \leq b$. We say $G= \bigcap_{i=1}^{b}G_i$
when $E(G) = \bigcap_{i=1}^{b}E(G_i)$.
The following lemma gives the relationship between interval graphs
and intersection graphs of $k$-boxes.
\begin{lemma}[Roberts\cite{Roberts}]
\label{Robertslemma}
For any graph $G$, $\ensuremath{box}(G) \leq k$ if and only if
there exist $k$ interval graphs $I_1, I_2, \ldots, I_k$
such that $G = \bigcap_{i=1}^{k}I_i$.
\end{lemma}
From the above lemma, we can say that boxicity of a graph $G$ is the minimum positive integer $k$ for which
there exist $k$ interval graphs $I_1,I_2 \ldots, I_k$ such that $G= \bigcap_{i=1}^kI_i$.
We have seen that intervals graphs are intersection graphs of
intervals on the real line. Hence for any interval graph
$I$, there exists a map $f~:~V(I)\rightarrow \{X \subseteq \mathbb{R}~|~
X\mbox{ is a closed interval}\}$ such that, for any $u,v \in V(I)$,
$(u,v) \in E(I)$ if and only if $f(u) \cap f(v) \neq \emptyset$.
Such a map $f$ is called an \emph{interval representation} of $I$.
An interval graph can have more than one interval representation.
It is known that given an interval graph $I$, we can find an interval
representation for $I$ in which no two intervals share any endpoints.
\subsection{Preliminaries}
Except in Theorem \ref{linegraphtheorem}, Section \ref{linegraphsection}, we consider only finite, undirected,
and simple graphs. In Theorem \ref{linegraphtheorem}, we consider finite, undirected
multigraphs. For any finite positive integer $n$, let $[n]$ denote the
set $\{1,2,\ldots n\}$.
For a graph $G$, we use $V(G)$ and $E(G)$ to denote the set of
its vertices and edges respectively. For any $v \in V(G)$, $N_G(v) := \{u~|~(v,u) \in E(G)\}$
and $d_G(v) := |N_G(v)|$. The maximum degree of $G$ is denoted by $\Delta(G)$.
$\chi(G)$ represents the chromatic number of $G$.
We say that an edge $e_i$ is a neighbour of
another edge $e_j$ in $G$, if they share an endpoint.
Given two graphs $G$ and $H$, we say $G=H$ when $G$ is isomorphic to $H$.
We say that a graph $G$ is obtained by \emph{fully subdividing} $H$, if $G$ is
obtained as a result of subdividing every edge of $H$ exactly once.
Given a multigraph $H$, we define a graph $L(H)$ in the
following way: $V(L(H)) = E(H)$ and $E(L(H)) =
\{(e_1,e_2)~|~e_1,e_2 \in E(H),~e_1 \mbox{ and } e_2 \mbox{ share an
endpoint in }$ $ H \}$. A graph $G$ is a \emph{line graph} if and
only if there exists a multigraph $H$ such that $G$ is isomorphic to $L(H)$.
Let $I$ be an interval graph and $f$ an interval representation of $I$.
Then, $\forall x\in V(I)$, we use $l(f(x))$ and $r(f(x))$ to denote
the left and right endpoint respectively of the interval $f(x)$.
\subsection{Our Results}
In this paper, we show that for a line graph $G$ with maximum degree
$\Delta$,
$$\ensuremath{box}(G) \leq 2\Delta(\lceil \log_2 \log_2 \Delta \rceil + 3) + 1.$$
From the above result, we also infer that if
chromatic number of $G$ is $\chi$, then $\ensuremath{box}(G) = O(\chi \log_2 \log_2 (\chi))$.
Recall that, in \cite{DiptAdiga} it was shown that for any graph $G$,
$\ensuremath{box}(G)\leq c\cdot \Delta \log^2 \Delta$, where $c$ is a large constant.
Hence, for the class of line graphs,
our result is an improvement over the best bound known for general graphs.
Moreover, in contrast with the result in \cite{DiptAdiga}, the proof here
is constructive and easily gives an efficient algorithm to get a
box representation for the given line graph. We leave the tightness of our result open.
The main
supporting result that we have used to prove the above result is the
following (this itself may be independently interesting): For a graph $G$
obtained by fully subdividing another graph $H$,
$\ensuremath{box}(G) \leq \lceil \log_2 \log_2 (\Delta) \rceil + 3$, where
$\Delta$ is the maximum degree of $G$. At the end of the paper, we point
out another consequence of this supporting result.
For the $d$-dimensional hypercube $H_d$,
$$ \ensuremath{box}(H_d) \geq \frac{\lceil \log_2 \log_2 d \rceil + 1}{2}.$$
It was shown by Chandran and Sivadasan in \cite{CN99} that
$\ensuremath{box}(H_d) \leq \frac{cd}{\log d}$, where $c$ is a constant. They had raised
the question of finding a non-trivial lower bound for $\ensuremath{box}(H_d)$.
\section{Boxicity of a Fully Subdivided Complete Graph}
Let $S=\{\sigma_1, \sigma_2, \ldots , \sigma_p\}$ be a set of permutations of
$[n]$, where $n$ is any finite positive integer.
$S$ is called \emph{$k$-suitable} for $[n]$ if for any $k$-element subset
$X \subseteq [n]$ and for any $x \in X$, there exists a permutation $\sigma \in S$ with the
following property:
\[\sigma^{-1}(x) \geq \sigma^{-1}(y), \forall y \in X.\]
The minimum cardinality of a $k$-suitable set for $[n]$ is
denoted by $N'(n,k)$. Spencer \cite{scramble} proved that
\[N'(n,3) < \log_2 \log_2 n + \frac{1}{2} \log_2 \log_2 \log_2 n + \log_2 (\sqrt{2} \pi) + o(1).\]
In this paper, we are interested in a slightly relaxed version of the notion
of $3$-suitability. Given a permutation $\sigma$ of $[n]$ and
$s,t \in [n]$, let
\begin{eqnarray}
\label{betaequation}
\beta(s,t,\sigma) & = & \{ x~|~\sigma^{-1}(s) < \sigma^{-1}(x) < \sigma^{-1}(t) \nonumber \\
& & \mbox{ or } \sigma^{-1}(t) < \sigma^{-1}(x) < \sigma^{-1}(s) \}.
\end{eqnarray}
A set $S=\{\sigma_1, \sigma_2, \ldots , \sigma_p\}$ is called \emph{simply $3$-suitable}
for $[n]$, if for each pair $s,t \in [n]$, $\bigcap_{i=1}^p \beta(s,t,\sigma_i) = \emptyset$.
In other words, for every triple $x,s,t \in [n]$ there exists a permutation $\sigma \in S$
such that either $\sigma^{-1}(x) < \min \left(\sigma^{-1}(s), \sigma^{-1}(t)\right)$ or
$\sigma^{-1}(x) > \max \left(\sigma^{-1}(s), \sigma^{-1}(t)\right)$. It
is easy to see that any $3$-suitable set is also a simply $3$-suitable set while the
converse is clearly not true. Let $N(n)$ be the minimum possible cardinality of a
simply $3$-suitable set for $[n]$. From Spencer's bound on
$N'(n,3)$, we have $N(n) \leq N'(n,3) <
\log_2 \log_2 n + \frac{1}{2} \log_2 \log_2 \log_2 n + \log_2 (\sqrt{2} \pi) + o(1)$.
But since simply $3$-suitability is a more relaxed notion than $3$-suitability,
we can get the following exact formula for $N(n)$:
\begin{lemma}
\label{scramblelemma}
$N(n) = \lceil \log_2 \log_2 n \rceil + 1$.
\end{lemma}
\begin{proof}
Erd\H{o}s and Szekeres \cite{ErdosSzekeres} proved that if $\sigma_1$ and
$\sigma_2$ are two permutations of $[n^2 + 1]$, then there exists
some $X \subset [n^2 + 1]$ with $|X|=n+1$ such that the permutation
of $X$ obtained by restricting $\sigma_1$ to $X$ is the same as
the permutation obtained by restricting $\sigma_2$ to $X$. By an
easy inductive argument (as Spencer points out in \cite{scramble})
we can show that if $\sigma_1, \sigma_2, \ldots
\sigma_{s+1}$ are permutations of $[2^{2^s} + 1]$, then there exists some
triple $\{x,y,z\}$ such that the order of these 3 elements with respect
to each permutation $\sigma_1, \sigma_2, \ldots \sigma_{s+1}$ is the same.
This implies that $N(n) \geq\lceil \log_2 \log_2 n \rceil + 1$.
We need to show that when $n\leq 2^{2^i}$, $N(n) \leq i+1$.
Note that when the permutations in a simply 3-suitable set $S$ for $[n]$ are restricted to $[n_1]$
(where $n_1 < n$), $S$ becomes a simply 3-suitable set for $[n_1]$. Hence it is
enough to prove that, when $n=2^{2^i}$, $N(n) \leq i+1$.
We prove this by induction on $i$. The base case, when $i=0$ and $n=2$, is
trivially true. For any $i<i_1$, assume $N(n) \leq i+1$.
Let $i=i_1$, $n=2^{2^{i_1}}$ and $n_1 = 2^{2^{i_1-1}}$. Then
$n=n_1 \cdot n_1$. So set $[n]$ can be partitioned into $n_1$ sets
$A_1, A_2, \ldots A_{n_1}$, where for any $p \in [n_1]$,
$A_p = \{(p-1)n_1 + 1, (p-1)n_1 + 2, \ldots , (p-1)n_1 + n_1\}$.
Clearly for any $a \in [n]$, there exist $k,p \in [n_1]$
such that $a=(p-1)n_1 + k$.
By induction hypothesis, there exists a simply 3-suitable set
$S'= \{\eta_1, \eta_2, \ldots \eta_{i_1}\}$ of $[n_1]$.
Then we define $i_1 +1$ permutations
$S=\{\sigma_1, \ldots, \sigma_{i_1+1}\}$ for $[n]$ as follows:
\begin{eqnarray*}
\label{generatescramble1}
\sigma_j^{-1}(a) & = & (\eta_j^{-1}(p) - 1)n_1 + \eta_j^{-1}(k) \mbox{, where } 1 \leq j \leq i_1. \\
\label{generatescramble2}
\sigma_{i_1+1}^{-1}(a) & = & (n_1 - \eta_{i_1}^{-1}(p))n_1 + \eta_{i_1}^{-1}(k).
\end{eqnarray*}
We claim that $S$ is a simply 3-suitable set for $[n]$ i.e., for any
$s,t \in [n]$, $\bigcap_{i=1}^{i_1+1}\beta(s,t,\sigma_i)=\emptyset$.
Let $s \in A_p$ and $t \in A_q$. Consider the 2 cases below: \\
\textbf{case 1}: If $p=q$, then there exist $k_1,k_2 \in [n_1]$
with $k_1 \neq k_2$ such that, $s=(p-1)n_1+ k_1$ and $t=(p-1)n_1 + k_2$.
Consider a permutation $\sigma_j$, where $j \in [i_1]$.
\begin{eqnarray*}
\beta(s,t,\sigma_j) & = & \{ x~|~\sigma_j^{-1}(s) < \sigma_j^{-1}(x) < \sigma_j^{-1}(t) \\
& & \mbox{ or } \sigma_j^{-1}(t) < \sigma_j^{-1}(x) < \sigma_j^{-1}(s) \} \\
& = & \{ x~|~(\eta_j^{-1}(p)-1)n_1 + \eta_j^{-1}(k_1) < \sigma_j^{-1}(x) <
(\eta_j^{-1}(p)-1)n_1 + \eta_j^{-1}(k_2) \\
& & \mbox{ or } (\eta_j^{-1}(p)-1)n_1 + \eta_j^{-1}(k_2) < \sigma_j^{-1}(x) <
(\eta_j^{-1}(p)-1)n_1 + \eta_j^{-1}(k_1) \}.
\end{eqnarray*}
If $\beta(s,t,\sigma_j) \neq \emptyset$, then consider any $x \in \beta(s,t,\sigma_j)$.
Clearly $x \in A_p$. Let $x = (p-1)n_1 + k_3$.
From the above, it is clear that either $\eta_j^{-1}(k_1) < \eta_j^{-1}(k_3) < \eta_j^{-1}(k_2)$
or $\eta_j^{-1}(k_2) < \eta_j^{-1}(k_3) < \eta_j^{-1}(k_1)$. This means that
$x \in \beta(s,t,\sigma_j) \implies k_3 \in \beta(k_1,k_2,\eta_j)$. Therefore,
$\bigcap_{j=1}^{i_1}\beta(s,t,\sigma_j) \neq \emptyset \implies
\bigcap_{j=1}^{i_1}\beta(k_1,k_2,\eta_j) \neq \emptyset$.
By induction hypothesis, we know that $\bigcap_{j=1}^{i_1}\beta(k_1,k_2,\eta_j) =
\emptyset$. Hence $\bigcap_{j=1}^{i_1}\beta(s,t,\sigma_j) = \emptyset$. \\
\textbf{case 2}: If $p \neq q$, then $\exists k_1,k_2 \in [n_1]$ such that
$s=(p-1)n_1 + k_1$ and $t=(q-1)n_1 + k_2$. Let $x = (r-1)n_1 + k_3$.
Now $x \in \bigcap_{j=1}^{i_1}\beta(s,t,\sigma_j)$ implies, for any
$j \in [n_1]$,
$
(\eta_j^{-1}(p)-1)n_1 + \eta_j^{-1}(k_1) < (\eta_j^{-1}(r)-1)n_1 + \eta_j^{-1}(k_3) <
(\eta_j^{-1}(q)-1)n_1 + \eta_j^{-1}(k_2)
\mbox{ or }
(\eta_j^{-1}(q)-1)n_1 + \eta_j^{-1}(k_2) < (\eta_j^{-1}(r)-1)n_1 + \eta_j^{-1}(k_3) <
(\eta_j^{-1}(p)-1)n_1 + \eta_j^{-1}(k_1)
$. It follows that $\eta_j^{-1}(p) \leq \eta_j^{-1}(r) \leq \eta_j^{-1}(q)$ or
$\eta_j^{-1}(q) \leq \eta_j^{-1}(r) \leq \eta_j^{-1}(p)$. If $r \notin \{p,q\}$,
then $\eta_j^{-1}(p) < \eta_j^{-1}(r) < \eta_j^{-1}(q)$ or
$\eta_j^{-1}(q) < \eta_j^{-1}(r) < \eta_j^{-1}(p)$ i.e.,
$ r \in \bigcap_{j=1}^{i_1}\beta(p,q,\eta_j)$ which contradicts the induction
hypothesis that $\bigcap_{j=1}^{i_1}\beta(p,q,\eta_j)= \emptyset$.
Therefore we infer that $r=p$ or $r=q$. Let
$r=p$ (proof is similar when $r=q$). If $x \in \bigcap_{j=1}^{i_1+1}\beta(s,t,\sigma_j)$
then we have $x \in \beta(s,t,\sigma_{i_1})$ and therefore
$\sigma_{i_1}^{-1}(s) < \sigma_{i_1}^{-1}(x) < \sigma_{i_1}^{-1}(t)$ or
$\sigma_{i_1}^{-1}(t) < \sigma_{i_1}^{-1}(x) < \sigma_{i_1}^{-1}(s)$.
Without loss of generality, let
$\sigma_{i_1}^{-1}(s) < \sigma_{i_1}^{-1}(x) < \sigma_{i_1}^{-1}(t)$. Then
$(\eta_{i_1}^{-1}(p)-1)n_1 + \eta_{i_1}^{-1}(k_1) < (\eta_{i_1}^{-1}(r)-1)n_1 + \eta_{i_1}^{-1}(k_3) <
(\eta_{i_1}^{-1}(q)-1)n_1 + \eta_{i_1}^{-1}(k_2)$. Since $p=r$, we have
$\eta_{i_1}^{-1}(p) = \eta_{i_1}^{-1}(r)$ and therefore $\eta_{i_1}^{-1}(k_1) <
\eta_{i_1}^{-1}(k_3)$. This also allows us to infer that
$(n_1 - \eta_{i_1}^{-1}(p))n_1 + \eta_{i_1}^{-1}(k_1) <
(n_1 - \eta_{i_1}^{-1}(r))n_1 + \eta_{i_1}^{-1}(k_3)$. That is
$\sigma_{i_1+1}^{-1}(s) < \sigma_{i_1+1}^{-1}(x)$. On the other hand,
$(n_1 - \eta_{i_1}^{-1}(q))n_1 + \eta_{i_1}^{-1}(k_2) <
(n_1 - \eta_{i_1}^{-1}(p))n_1 + \eta_{i_1}^{-1}(k_1)$ (since
$\eta_{i_1}^{-1}(p) < \eta_{i_1}^{-1}(q)$). Therefore,
$\sigma_{i_1+1}^{-1}(t) < \sigma_{i_1+1}^{-1}(s)$. So we have,
$\sigma_{i_1+1}^{-1}(t) < \sigma_{i_1+1}^{-1}(s) < \sigma_{i_1+1}^{-1}(x)$. Hence
$x \notin \beta(s,t,\sigma_{i_1+1})$ contradicting our assumption that
$x \in \bigcap_{j=1}^{i_1+1}\beta(s,t,\sigma_j)$.
\qed
\end{proof}
\begin{theorem}
\label{completegraphtheorem}
Let $G$ be the graph obtained by fully subdividing the complete graph $K_n$. Then
$\frac{\lceil \log_2 \log_2 n \rceil + 1}{2} \leq \ensuremath{box}(G) \leq \lceil \log_2 \log_2 n \rceil + 2$.
\end{theorem}
\begin{proof}
Let $v_1, v_2, \ldots v_n$ be the vertices of $K_n$ and $
e_1, e_2, \ldots e_m$ its edges, where $m =$ $n\choose2$. Let $u_{p\cdot q}$
denote the vertex introduced when subdividing the edge $(v_p,v_q) \in E(K_n)$, where
$p<q$. Thus the graph $G$ obtained by fully subdividing $K_n$ has the vertex set
$V(G) = \{v_1,v_2,\ldots v_n\} \cup \{u_{p\cdot q}~|~1 \leq p < q \leq n\}$
and $E(G) = \{(v_p,u_{p\cdot q})~|~1 \leq p < q \leq n\} \cup \{(v_q,u_{p\cdot q}~|~
1 \leq p<q\leq n)\}$.
We first show that $\ensuremath{box}(G) \leq \lceil \log_2 \log_2 n \rceil + 2$.
Let $k=\lceil \log_2 \log_2 n \rceil + 1$. By Lemma \ref{scramblelemma},
there exists a simply 3-suitable set $S=\{\sigma_1, \ldots, \sigma_k\}$ for
$[n]$. Using $S$, we construct a $(k+1)$-dimensional
box representation for $G$. Corresponding to each permutation $\sigma_i$ of
$[n]$ in $S$, we construct an interval graph $I_i$ as follows. Let $f_i$ denote the interval
representation of $I_i$.
\begin{eqnarray*}
\mbox{for every }v_p \in V(G), & f_i(v_p) & = [\sigma_i^{-1}(p),\sigma_i^{-1}(p)]. \\
\mbox{for every }u_{p\cdot q} \in V(G), & f_i(u_{p\cdot q}) & =
[\sigma_i^{-1}(p),\sigma_i^{-1}(q)] \mbox{, if } \sigma_i^{-1}(p)] < \sigma_i^{-1}(q). \\
\mbox{for every }u_{p\cdot q} \in V(G), & f_i(u_{p\cdot q}) & =
[\sigma_i^{-1}(q),\sigma_i^{-1}(p)] \mbox{, if } \sigma_i^{-1}(q) < \sigma_i^{-1}(p)].
\end{eqnarray*}
The interval representation $f_{k+1}$ of the $(k+1)$th interval graph $I_{k+1}$ is as follows:
\begin{eqnarray*}
\mbox{for every }v_p \in V(G), & f_{k+1}(v_p) & = [1,m]. \\
\mbox{for every }u_{p\cdot q} \in V(G), & f_{k+1}(u_{p\cdot q}) & = [j,j] \mbox{, where } u_{p\cdot q} \mbox {
was obtained by } \\
& & \mbox{subdividing edge } e_j = (v_p,v_q) \mbox{ of } K_n.
\end{eqnarray*}
By Lemma \ref{Robertslemma}, in order to prove that $\ensuremath{box}(G) \leq k+1$ it is sufficient to
show that $\bigcap_{i=1}^{k+1}I_i = G$, i.e.,\\
(i) each $I_j$ is a supergraph of $G$. \\
(ii) for any $(x,y) \notin E(G)$, there exists some interval graph $I_i$
such that $(x,y) \notin E(I_i)$.
Recall that any edge of $G$ is of the form $(v_p,u_{pq})$ or
$(v_q,u_{pq})$, where $v_p, v_q \in V(K_n)$. It is easy to verify
that, for any $i \in [k+1]$, $f_i(u_{pq}) \cap f_i(v_p) \neq \emptyset$
and $f_i(u_{pq}) \cap f_i(v_q) \neq \emptyset$. Therefore (i) is true.
Let $(x,y) \notin E(G)$. In order to prove (ii), we consider the following cases:
\textbf{case 1}: $x=v_p,y= v_q$, for some $1\leq p< q \leq n$. \\
It is easy to see that $f_1(v_p) \cap f_1(v_q) = \emptyset$ and therefore
$(v_p,v_q) \notin E(I_1)$. \\
\textbf{case 2}: $x=u_{p\cdot q}, y = u_{r\cdot s}$ and $u_{p\cdot q} \neq u_{r\cdot s}$. \\
Clearly, $f_{k+1}(u_{p\cdot q}) \cap f_{k+1}(u_{r\cdot s}) = \emptyset$ and therefore
$(u_{p\cdot q}, u_{r\cdot s}) \notin E(I_{k+1})$. \\
\textbf{case 3}: $x=v_p, y=u_{r\cdot s}$, for any $p,r,s \in [n]$,
$p \notin \{r,s\}$ and $r < s$. \\
Since $S$ is a simply 3-suitable set for $[n]$ there exists
a permutation $\sigma_j$ such that $p \notin \beta(r,s,\sigma_j)$ i.e.,
either $\sigma_j^{-1}(p) < \min(\sigma_j^{-1}(r),\sigma_j^{-1}(s))$ or
$\sigma_j^{-1}(p) > \max(\sigma_j^{-1}(r),\sigma_j^{-1}(s))$.
Now it is easy to see that, $f_j(v_p) \cap f_j(u_{r\cdot s}) =
\emptyset$ and therefore $(v_p, u_{r\cdot s}) \notin E(I_j)$.
We thus prove (ii) and thereby
prove that $\ensuremath{box}(G) \leq \lceil \log_2\log_2n \rceil + 2$.
We now show that $\ensuremath{box}(G) \geq \frac{\lceil \log_2 \log_2 n \rceil + 1}{2}$. Let
$\ensuremath{box}(G) = b$. By Lemma \ref{Robertslemma} there
exist $b$ interval graphs, say $I_1, I_2, \ldots , I_b$, such that
$G = \bigcap_{i=1}^{b}I_i$. For any $i \in [b]$, let $f_i$ be an
interval representation of $I_i$ such that no two intervals share any endpoints.
From each $f_i$, generate two permutations $L_i$ and $R_i$ of $[n]$ in the following way.
For $p,q \in [n]$, $p\neq q$, $L_i^{-1}(p) < L_i^{-1}(q) \Leftrightarrow l(f_i(v_p)) <
l(f_i(v_q))$. Similarly, $R_i^{-1}(p) < R_i^{-1}(q) \Leftrightarrow r(f_i(v_p)) <
r(f_i(v_q))$
Consider the set $S=\{L_1,R_1,L_2,R_2,\ldots L_b,R_b\}$ of permutations
of $[n]$. We claim that $S$ is a simply 3-suitable set for $[n]$.
Let $s,t \in [n]$. Then for any $i \in [b]$,
\begin{eqnarray}
\label{implications1}
x \in \beta(s,t,L_i) & \implies &
\left(L_i^{-1}(s) < L_i^{-1}(x) < L_i^{-1}(t)\right) \mbox{ or} \\
& & \left(L_i^{-1}(t) < L_i^{-1}(x) < L_i^{-1}(s)\right) \nonumber \\
& \implies & \left(l(f_i(v_s)) < l(f_i(v_x)) < l(f_i(v_t))\right) \mbox{ or } \nonumber \\
& & \left(l(f_i(v_t)) < l(f_i(v_x)) < l(f_i(v_s))\right). \nonumber \\
\label{implications2}
x \in \beta(s,t,R_{i}) & \implies &
\left(R_{i}^{-1}(s) < R_{i}^{-1}(x) < R_{i}^{-1}(t)\right) \mbox{ or} \\
& & \left(R_{i}^{-1}(t) < R_{i}^{-1}(x) < R_{i}^{-1}(s)\right) \nonumber \\
& \implies & \left(r(f_i(v_s)) < r(f_i(v_x)) < r(f_i(v_t))\right) \mbox{ or } \nonumber \\
& & \left(r(f_i(v_t)) < r(f_i(v_x)) < r(f_i(v_s))\right). \nonumber
\end{eqnarray}
Suppose, for contradiction, $x \in \bigcap_{j=1}^{b}\left(\beta(s,t,L_j)
\cap \beta(s,t,R_j)\right)$. Consider any $i \in [b]$.
Let $y=\max(l(f_i(v_s)),l(f_i(v_t)))$ and
$z=\min(r(f_i(v_s)),r(f_i(v_t)))$.
Consider the two cases below: \\
\textbf{case 1}: $y<z$.
Then by implications (\ref{implications1}) and (\ref{implications2}) it is clear that $l(f_i(v_x)) < y =
\max(l(f_i(v_s)),l(f_i(v_t)))$ and $r(f_i(v_x)) >
z=\min(r(f_i(v_s)),r(f_i(v_t)))$. Therefore, $[y,z] \subseteq f_i(v_x)$.
Now we will show that $f_i(u_{s\cdot t}) \cap [y,z] \neq \emptyset$ which will
immediately imply that $f_i(u_{s\cdot t}) \cap f_i(v_x) \neq \emptyset$.
If $f_i(u_{s\cdot t}) \cap [y,z] = \emptyset$, then either
$r(f_i(u_{s\cdot t})) < y$ or $l(f_i(u_{s\cdot t})) > z$. In both these cases,
it is easy to see that either $(u_{s\cdot t}, v_s) \notin E(I_i)$ or
$(u_{s\cdot t}, v_t) \notin E(I_i)$. This contradicts the fact that $I_i$ is a
supergraph of $G$. Hence $f_i(u_{s\cdot t}) \cap [y,z] \neq \emptyset$ and therefore
$(u_{s\cdot t}, v_x) \in E(I_i)$.\\
\textbf{case 2}: $y>z$. Since $(u_{s\cdot t}, v_s) \in E(I_i)$ and
$(u_{s\cdot t}, v_t) \in E(I_i)$, we have $r(f_i(u_{s\cdot t})) > y$ and
$l(f_i(u_{s\cdot t})) < z$. Therefore, $[z,y] \subseteq f_i(u_{s\cdot t})$.
Now we will show that $f_i(v_x) \cap [z,y] \neq \emptyset$ which will
immediately imply that $f_i(u_{s\cdot t}) \cap f_i(v_x) \neq \emptyset$.
If $f_i(v_x) \cap [z,y] = \emptyset$, then either
$r(f_i(v_x)) < z$ or $l(f_i(v_x))) > y$. In both these cases,
we contradict implications (\ref{implications1}) and (\ref{implications2}) which state that
$r(f_i(v_x))$ is sandwiched between $r(f_i(v_s))$ and
$r(f_i(v_t))$, and $l(f_i(v_x))$ is sandwiched between $l(f_i(v_s))$ and
$l(f_i(v_t))$. Hence $f_i(v_x) \cap [z,y] \neq \emptyset$ and therefore
$(u_{s\cdot t}, v_x) \in E(I_i)$.
Thus we conclude that if there exists an $x \notin \{s,t\}$ such that
$x \in \bigcap_{j=1}^{2b}\beta(s,t,$ $\sigma_j)$, then
$(u_{s\cdot t},v_x) \in E(\bigcap_{i=1}^b I_i)$ which implies that $(u_{s\cdot t},v_x) \in E(G)$.
But this contradicts the fact that $(u_{s\cdot t},v_x) \notin E(G)$ and hence
$\bigcap_{j=1}^{2b}\beta(s,t,\sigma_j) = \emptyset$ i.e., $S$ is a simply 3-suitable set.
Then by Lemma \ref{scramblelemma}, $|S| = 2b \geq \lceil \log_2 \log_2 n \rceil + 1$ or
$\ensuremath{box}(G) \geq \frac{\lceil \log_2 \log_2 n \rceil + 1}{2}$.
\qed
\end{proof}
\remark{Louis Esperet informed us that he had independently observed Theorem
\ref{completegraphtheorem}. But he has not published it.
We thank him for personal communication. In \cite{Esperet}, he also conjectures that
for any graph $G$, (i) $\ensuremath{box}(G) \leq a(G) + \kappa$, (ii) $\ensuremath{box}(G) \leq \lambda \cdot a(G)$,
where $\kappa$, $\lambda$ are constants and $a(G)$ refers to the arboricity of $G$. As
arboricity of any graph is upper bounded by its degeneracy and since fully subdivided
complete graphs are 2-degenerate, Theorem \ref{completegraphtheorem} disproves
Esperet's both conjectures.
}
\section{Boxicity of a Fully Subdivided Graph of Chromatic Number $\chi$}
\begin{theorem}
\label{sparsegraphtheorem}
Let $H$ be a graph with chromatic number $\chi$ and
let $G$ be the graph obtained by fully subdividing $H$. Then,
$\ensuremath{box}(G) \leq \lceil \log_2 \log_2 \chi \rceil + 3$.
\end{theorem}
\begin{proof}
Given a colouring of $H$ using $\chi$ colours, let $C_1, C_2 \ldots C_\chi$ represent the
$\chi$ colour classes. Let $|C_i| = c_i$ and $c_{max}=\max_i(c_i)$.
Give an arbitrary order to the vertices in each colour class. Let $v_{ij}$ denote the
$j$-th vertex in the $i$-th colour class, where $i \in [\chi]$ and $j \in [c_i]$.
Let $E(H)=\{e_1, e_2, \ldots ,e_m\}$ be the edge set of $H$. Let
$u_{pq\cdot rs}$ denote the vertex introduced while subdividing the edge $(v_{pq}, v_{rs})$,
where $p < r$.
Let $k = \lceil \log_2 \log_2 \chi \rceil + 1$. By Lemma \ref{scramblelemma}, there exists
a simply 3-suitable set $S=\{\sigma_1, \ldots \sigma_k\}$ for $[\chi]$.
We use $S$ to construct a $(k+2)$-dimensional box representation for $G$.
Corresponding to each
permutation $\sigma_i \in S$, we construct an interval graph $I_i$ as follows.
Let $f_i$ denote the interval representation of $I_i$. \\ When $i \leq k$,
\begin{eqnarray*}
\mbox{for every } v_{pq} \in E(G), & f_i(v_{pq}) & = [g_i(p,q),g_i(p,q)], \\
& & \mbox{where } g_i(p,q)=\sigma_i^{-1}(p) + \frac{q-1}{c_{max}}. \\
\mbox{for every } u_{pq\cdot rs} \in E(G), & f_i(u_{pq\cdot rs}) & = [g_i(p,q),g_i(r,s)] \mbox{, if } g_i(p,q) < g_i(r,s). \\
\mbox{for every } u_{pq\cdot rs} \in E(G), & f_i(u_{pq\cdot rs}) & = [g_i(r,s),g_i(p,q)] \mbox{, if } g_i(r,s) < g_i(p,q), \\
& & \mbox{where }g_i(p,q) = \sigma_i^{-1}(p) + \frac{q-1}{c_{max}} \\
& & \mbox{ and } g_i(r,s) = \sigma_i^{-1}(r) + \frac{s-1}{c_{max}}.
\end{eqnarray*}
The interval representations of the remaining 2 interval graphs namely $I_{k+1}$ and $I_{k+2}$
are as follows:-
\begin{eqnarray*}
\mbox{for every } v_{pq} \in E(G), & f_{k+1}(v_{pq}) & = [1,m]. \\
\mbox{for every } u_{pq\cdot rs} \in E(G), & f_{k+1}(u_{pq\cdot rs}) & = [j,j], \\
& & \mbox{where } u_{pq\cdot rs} \mbox { was obtained by } \\
& & \mbox{subdividing edge } e_j = (v_{pq},v_{rs}) \mbox{ of } H. \\
\mbox{for every } v_{pq} \in E(G), & f_{k+2}(v_{pq}) & = [h_k(p,q),h_k(p,q)], \\
& & \mbox{where } h_k(p,q)= (\chi+1) - \sigma_k^{-1}(p) + \frac{q-1}{c_{max}}. \\
\mbox{for every } u_{pq\cdot rs} \in E(G), & f_{k+2}(u_{pq\cdot rs}) & = [h_k(p,q),h_k(r,s)] \mbox{, if } h_k(p,q) < h_k(r,s). \\
\mbox{for every } u_{pq\cdot rs} \in E(G), & f_{k+2}(u_{pq\cdot rs}) & = [h_k(r,s),h_k(p,q)] \mbox{, if } h_k(r,s) < h_k(p,q), \\
& & \mbox{where }h_k(p,q) = (\chi+1) - \sigma_k^{-1}(p) + \frac{q-1}{c_{max}} \\
& & \mbox{ and } h_k(r,s) = (\chi+1) - \sigma_k^{-1}(r) + \frac{s-1}{c_{max}}.
\end{eqnarray*}
Observe that every edge in $G$ is of the form $(u_{pq \cdot rs}, v_{pq})$ or
$(u_{pq \cdot rs}, v_{rs})$ where $v_{pq}$ and $v_{rs}$ are vertices of $H$
and $u_{pq\cdot rs}$ is the vertex introduced while subdividing edge $(v_{pq},v_{rs})$.
Any interval graph $I_i$, where $1\leq i \leq k$, is clearly a supergraph of $G$ because in
$f_i$ the interval corresponding to $u_{pq\cdot rs}$ has its endpoints
on the point intervals assigned to $v_{pq}$ and $v_{rs}$. The same is true
with interval graph $I_{k+2}$. In the interval representation $f_{k+1}$ of
$I_{k+1}$ , any vertex $v_{pq}$ is assigned an interval $[1,m]$ which overlaps
with the interval of every other vertex. Hence all interval graphs $I_1, I_2,
\ldots , I_{k+2}$ are supergraphs of $G$.
In order to show that for every $(x,y) \notin E(G)$ there exists
some interval graph $I_i$ in our collection such that $(x,y) \notin E(I_i)$, we
consider the following cases:\\
\textbf{case 1}: $x= v_{pq}$, $y=v_{rs}$, where $v_{pq} \neq v_{rs}$. \\
As $f_1(v_{pq}) \cap f_1(v_{rs}) = \emptyset$,
$(v_{pq},v_{rs}) \notin E(I_1)$. \\
\textbf{case 2}: $x=u_{pq\cdot rs}, y=u_{wx\cdot yz}$, where $u_{pq\cdot rs} \neq u_{wx\cdot yz}$. \\
It is easy to verify that $f_{k+1}(u_{pq\cdot rs}) \cap f_{k+1}(u_{wx\cdot yz}) = \emptyset$ and
hence $(u_{pq\cdot rs}, u_{wx\cdot yz}) \notin E(I_{k+1})$. \\
\textbf{case 3}: $x=u_{pq \cdot rs}, y=v_{ab}$ and $a \notin \{p,r\}$. \\
Note that $p,r,a \in [\chi]$ and since $S$ is a simply 3-suitable set for $[\chi]$,
there exists a $\sigma_i \in S$ such that $a \notin \beta(p,r,\sigma_i)$ i.e.,
$\sigma_i^{-1}(a) < \min(\sigma_i^{-1}(p),\sigma_i^{-1}(r))$ or
$\sigma_i^{-1}(a) > \max(\sigma_i^{-1}(p),\sigma_i^{-1}(r))$.
$f_i(v_{ab})=[g_i(a,b),g_i(a,b)]$ and $f_i(u_{pq \cdot rs})=[g_i(p,q),g_i(r,s)]$.
Recalling that, for any $x_1 \in [\chi]$ and $x_2 \in [c_i]$,
$g_i(x_1,x_2) = \sigma_i^{-1}(x_1) + \frac{x_2-1}{c_{max}}$
it is easy to verify that $f_i(v_{ab}) \cap f_i(u_{pq \cdot rs})= \emptyset$. \\
\textbf{case 4}: $x=u_{pq \cdot rs}, y=v_{ab}$ and $a \in \{p,r\}$. \\
Assume $a=p$ (proof is similar when $a=r$).
Assume $(v_{pb},u_{pq \cdot rs}) \in E(I_i), \forall i \in \{1, 2, \ldots, k+2\}$.
It means $(v_{pb},u_{pq \cdot rs}) \in E(I_k) \implies
\sigma_k^{-1}(p) + \frac{q-1}{c_{max}} < \sigma_k^{-1}(p) + \frac{b-1}{c_{max}}
< \sigma_k^{-1}(r) + \frac{s-1}{c_{max}}
\implies q<b$ (here we assume that $\sigma_i^{-1}(p)
< \sigma_i^{-1}(r)$. Proof is similar when $\sigma_i^{-1}(p)
> \sigma_i^{-1}(r)$). In $f_{k+2}$, note that $u_{pq \cdot rs}$ is assigned the interval
$[(\chi+1) - \sigma_k^{-1}(r) + \frac{s-1}{c_{max}}, (\chi+1) - \sigma_k^{-1}(p) + \frac{q-1}{c_{max}}]$
and $v_{ab}$ ($= v_{pb}$) is assigned the interval
$[(\chi+1) - \sigma_k^{-1}(p) + \frac{b-1}{c_{max}}, (\chi+1) - \sigma_k^{-1}(p) + \frac{b-1}{c_{max}}]$.
Therefore, $(v_{pb},u_{pq \cdot rs}) \in E(I_{k+2}) \implies b<q$. But this contradicts
our earlier inference that $q<b$. Therefore, either $(v_{ab},u_{pq \cdot rs}) \notin E(I_k)$
or $(v_{ab},u_{pq \cdot rs}) \notin E(I_{k+2})$.
We have thus shown that for any $(x,y) \notin E(G)$, $\exists i \in [k+2]$
such that $(x,y) \notin E(I_i)$. As each $I_i$ is a supergraph of $G$, we
have $G=\bigcap_{i=1}^{k+2}I_i$. Applying Lemma \ref{Robertslemma}, we get
$\ensuremath{box}(G) \leq \lceil \log_2 \log_2 \chi \rceil + 3$.
\qed
\end{proof}
\begin{corollary}
\label{sparsegraphcorollary}
Given a graph $H$, let $G$ be the graph obtained by fully subdividing $H$. Then,
$\ensuremath{box}(G) \leq \lceil \log_2 \log_2 (\Delta(H)) \rceil + 3
\leq \lceil \log_2 \log_2 (\Delta(G)) \rceil + 3$
\end{corollary}
\begin{proof}
By Brooks' theorem (see chapter 5 in \cite{Diest}),
$\chi \leq \Delta(H)$ unless the graph $H$ is isomorphic to
a complete graph $K_{\Delta(H) + 1}$ or to an odd cycle.
If $H$ is isomorphic to $K_{\Delta(H) +1}$, then by Theorem \ref{completegraphtheorem},
$\ensuremath{box}(G) \leq \lceil \log_2 \log_2 (\Delta(H) + 1) \rceil + 2 \leq \lceil \log_2 \log_2 (\Delta(H))
\rceil + 3$. If $H$ is an odd cycle, then $G$ will be a cycle and hence
$\ensuremath{box}(G) \leq 2 < \lceil \log_2 \log_2 (\Delta(H)) \rceil + 3$. Therefore applying
Theorem \ref{sparsegraphtheorem}, we have $\ensuremath{box}(G) \leq
\lceil \log_2 \log_2 (\Delta(H)) \rceil + 3$. As $\Delta(H) \leq \Delta(G)$, the corollary
follows.
\qed
\end{proof}
\section{Line Graphs}
\label{linegraphsection}
For any bipartite graph $G$ with bipartition $\{A,B\}$, we use $C_A(G)$
to denote the graph with $V(C_A(G)) = V(G)$ and
$E(C_A(G))=E(G) \cup \{(x,y)~|~x,y \in A\}$. Thus $C_A(G)$ is
the graph obtained from $G$ by making $A$ a clique. Similarly one can
define $C_B(G)$.
\begin{lemma}
\label{bipartitelemma}
For any bipartite graph $G$ with bipartition $\{A,B\}$,
$\ensuremath{box}(C_A(G)) \leq 2\cdot \ensuremath{box}(G)$.
\end{lemma}
\begin{proof}
Proof of this lemma is similar to the proof of Lemma 7 in \cite{SunMatRog}. In
\cite{SunMatRog}
it is proved that $\ensuremath{box}(C_{AB}(G)) \leq 2\cdot \ensuremath{box}(G)$, where
$C_{AB}(G)$ refers to the graph obtained by making both $A$ and $B$ cliques. For
the sake of completeness, we give a proof to our lemma below.
Let $\ensuremath{box}(G) = b$. Then by Lemma \ref{Robertslemma},
there exist $b$ interval graphs, say $I_1, I_2,$ $\ldots ,I_b$, such that
$G = \bigcap_{i=1}^bI_i$. Let $f_i$ denote an interval representation of $I_i$, where
$i \in [b]$. Let $s_i=\min_{x\in A}(l(f_i(x)))$ and $t_i =
\max_{x\in A}(r(f_i(x)))$. From these $b$ interval graphs we construct $2b$ interval graphs namely
$I_1', I_2', \ldots I_b',I_1'', I_2'', \ldots I_b''$ as follows. Let $f_i'$, $f_i''$
denote interval representations of $I_i'$ and $I_i''$ respectively, where $i\in [b]$.
\begin{eqnarray*}
\mbox{Construction of } f_i': \\
\forall x\in A,~f_i'(x)& = & [s_i,r(f_i(x))]. \\
\forall x\in B,~f_i'(x)& = & f_i(x). \\
\mbox{Construction of } f_i'': \\
\forall x\in A,~f_i''(x)& = & [l(f_i(x)), t_i]. \\
\forall x\in B,~f_i''(x)& = & f_i(x). \\
\end{eqnarray*}
We claim that $C_A(G) = \bigcap_{i=1}^b (I_i' \cap I_i'')$.
Consider any $(x,y)\in E(C_A(G))$. To show that
$(x,y)\in E(I_i')$ and $(x,y)\in E(I_i'')$, $\forall i \in [b]$, we consider the
following 2 cases. If $(x,y)\in E(G)$, clearly $(x,y)\in E(I_i)$. From the
construction of $f_i'$ and $f_i''$,
it is easy to see that $I_i'$ and $I_i''$ are supergraphs of $I_i$.
Otherwise if $(x,y)\notin E(G)$, then $x,y \in A$ and therefore
$[s_i,s_i] \subseteq f_i'(x) \cap f_i'(y)$ and
$[t_i,t_i] \subseteq f_i''(x) \cap f_i''(y)$.
Now, consider any $(x,y) \notin E(C_A(G)$.
We know that $(x,y) \notin E(C_A(G))
\implies (x,y) \notin E(G) \implies
(x,y) \notin E(I_i), \mbox{ for some } i\in [b]$.
It is then easy to verify that,
(a) if $x \in A$, $y \in B$, then $\left(f_i'(x) \cap f_i'(y) = \emptyset\right)$ or
$\left(f_i''(x) \cap f_i''(y) = \emptyset \right)$.
(b) if $x,y \in B$, then $\left(f_i'(x) \cap f_i'(y) = \emptyset \right)$ and
$\left(f_i''(x) \cap f_i''(y) = \emptyset \right)$. \\
Thus we prove the claim that $C_A(G) = \bigcap_{i=1}^b(I_i' \cap I_i'')$.Therefore
by Lemma \ref{Robertslemma}, $\ensuremath{box}(C_A(G)) \leq 2 \cdot \ensuremath{box}(G)$.
\qed
\end{proof}
\begin{lemma}
\label{linegraphlemma}
Let $G$ be a bipartite graph with bipartition $\{X,Y\}$ having the following two properties:
(i) for any $y \in Y$, $d_G(y) \leq 2$ and
(ii) for any $y_1,y_2 \in Y$, if $y_1 \neq y_2$ then $N_G(y_1) \neq N_G(y_2)$.
Then, $\ensuremath{box}(G) \leq \lceil \log_2 \log_2 (\Delta(G))\rceil + 3$.
\end{lemma}
\begin{proof}
If $\Delta(G) = 1$, then $G$ is a collection of isolated edges and therefore
$\ensuremath{box}(G) = 1 \leq \lceil \log_2 \log_2 (\Delta(G))\rceil + 3$.
Let $\Delta(G) \geq 2$.
From $G$, we construct a bipartite graph
$G'$ with bipartition $\{X',Y'\}$
in the following way: To start with, let $G' = G$.
For each vertex $u \in Y'$ with $d_{G'}(u) = 1$,
we add a new vertex $n_u$ to $X'$
such that $u$ is the only neighbour of $n_u$. For each $v \in
Y'$ with $d_{G'}(v) = 0$,
delete $v$ from $Y'$.
So $X' = X \cup \{n_u~|~u \in Y \mbox{ and } d_{G}(u)=1\}$
and $Y' = Y \setminus
\{v \in Y~|~v \mbox{ is an isolated vertex}\}$. We claim that $\ensuremath{box}(G) \leq \ensuremath{box}(G')$.
This is because the graph obtained by removing isolated vertices from $G$ is an
induced subgraph of $G'$ and therefore its boxicity is at most that of $G'$. As adding
isolated vertices to any graph does not increase its boxicity, our claim follows.
From the construction of $G'$ we can say that, for every $y \in Y'$,
$d_{G'}(y) = 2$.
Let $G''$ be the subgraph induced on vertices of $X'$ in ${G'}^2$,
where ${G'}^2$ denotes the square of graph $G'$.
It is easy to see that $G'$ can be obtained by fully subdividing
$G''$ (Here note that if $G$ and thereby $G'$ had not satisfied property (ii), then
the graph obtained by fully subdividing $G''$ would have just been a subgraph of $G'$).
Therefore by our above claim and applying Corollary \ref{sparsegraphcorollary}, we get
\[\ensuremath{box}(G) \leq \ensuremath{box}(G') \leq \lceil \log_2 \log_2 (\Delta(G'))\rceil + 3.\]
From the construction of $G'$ and recalling that $\Delta(G) \geq 2$,
we infer that $\Delta(G') \leq \Delta(G)$. Therefore,
\[\ensuremath{box}(G) \leq \lceil \log_2 \log_2 (\Delta(G))\rceil + 3.\]
\qed
\end{proof}
A \emph{critical clique} of a graph $G$ is a clique $K$ where the
vertices of $K$ all have the same set of neighbours in $G\setminus K$, and
$K$ is maximal under this property. Let $\mathcal{K}$ denote the collection of
critical cliques in $G$. The \emph{critical clique graph} of a
graph $G$, denoted by $CC(G)$, has $V(CC(G)) = \mathcal{K}$ and
$E(CC(G)) = \{(K_1, K_2)~|~K_1,K_2 \in \mathcal{K} \mbox{ and }
V(K_1) \cup V(K_2) \mbox{ induces a clique in } G\}$.
Notice that
$CC(G)$ is isomorphic to some induced subgraph of $G$. For example, we
can take a representative vertex from each critical clique and the induced
subgraph on this set of vertices is isomorphic to $CC(G)$. The following
lemma is due to Chandran, Francis and Mathew \cite{SunMatRog2} :
\begin{lemma}
\label{criticalcliquelemma}
For any graph $G$, $\ensuremath{box}(G)=\ensuremath{box}(CC(G))$.
\end{lemma}
We now prove the main result of the paper. Recall that, given a
multigraph $H$, we define its line graph $L(H)$ in the
following way: $V(L(H)) := E(H)$ and $E(L(H)) :=
\{(e_1,e_2)~|~e_1,e_2 \in E(H),~e_1 \mbox{ and } e_2 \mbox{ share an
endpoint in } H \}$. A graph $G$ is a line graph if and
only if there exists a multigraph $H$ such that $G$ is isomorphic to $L(H)$.
\begin{theorem}
\label{linegraphtheorem}
Given a multigraph $H$, let $G$ be a graph
isomorphic to $L(H)$. Let $\Delta$ denote $\Delta(G)$ and $\chi$ represent $\chi(G)$.
Then,
$\ensuremath{box}(G) \leq 2\Delta(\lceil \log_2 \log_2 \Delta \rceil + 3) + 1$.
\end{theorem}
\begin{proof}
Given a vertex colouring of $G$ using $\chi$ colours,
let $D_1,D_2,\ldots ,D_{\chi}$ be the colour classes.
For any $1\leq i \leq (\chi -1)$, let $G_i$, with
$V(G_i)=V(G)$ and $E(G_i)=E(G) \cup \{(x,y)~|~x,y
\in \overline{D_i}\}$, be the split graph where $D_i$ is an
independent set and $\overline{D_i}$ a clique
(here $\overline{D_i}=\{x\in V(G)~|~x \notin D_i\}$).
Let $G_{\chi}^+$ be the graph
having $V(G_{\chi}^+) = V(G)$ and
$E(G_{\chi}^+) = \{(x,y)~|~x \in \overline{D_{\chi}}, y \in V(G)\}$.
It is easy to see that
\[G = G_1 \cap G_2 \cap \cdots \cap
G_{(\chi -1)} \cap G_{\chi}^+.\]
Therefore by Lemma \ref{Robertslemma},
\begin{eqnarray*}
\ensuremath{box}(G) & \leq & \Sigma_{i=1}^{(\chi -1)}\ensuremath{box}(G_i) + \ensuremath{box}(G_{\chi}^+).
\end{eqnarray*}
By Lemma \ref{criticalcliquelemma}, we know that $\ensuremath{box}(G_i) =
\ensuremath{box}(CC(G_i))$. Also, observe that $G_{\chi}^+$
is an interval graph and hence its boxicity is 1. Therefore,
\begin{eqnarray}
\label{criticalcliquebound}
\ensuremath{box}(G) & \leq & \Sigma_{i=1}^{(\chi -1)}\ensuremath{box}(CC(G_i)) + 1.
\end{eqnarray}
We know that, $\forall i \in [(\chi -1)]$, $G_i$ is a split graph, where $D_i$ is an independent set
and $\overline{D_i}$ a clique.
As $CC(G_i)$ is isomorphic to some subgraph of
$G_i$, it is also a split graph with $V(CC(G_i)) = X_i \uplus Y_i$,
where $X_i \subseteq D_i$ is an
independent set and $Y_i \subseteq \overline{D_i}$ a clique.
Let $H_i$ be the bipartite graph
obtained from $CC(G_i)$ by making $Y_i$ an
independent set. By Lemma \ref{bipartitelemma}, we have
$\ensuremath{box}(CC(G_i)) \leq 2 \cdot \ensuremath{box}(H_i)$.
Applying this to inequality (\ref{criticalcliquebound}), we get
\begin{eqnarray}
\label{boxicitysuminequality}
\ensuremath{box}(G) \leq 2\Sigma_{i=1}^{(\chi -1)}\ensuremath{box}(H_i) + 1.
\end{eqnarray}
\begin{claim}
\label{linegraphclaim}
For any $i \in [(\chi -1)]$ and $y \in Y_i$, $d_{H_i}(y) \leq 2$.
\end{claim}
\begin{proof}
Recall that $G=L(H)$ and therefore a proper vertex colouring of $G$
is equivalent to a proper edge colouring of $H$.
Since in any edge colouring of $H$ a given edge $e$ cannot have
more than 2 monochromatic neighbours, for any $y \in \overline{D_i}$,
$|N_{G}(y) \cap D_i| \leq 2$.
Observe that the bipartite graph $H_i$ is a subgraph of $G$. Therefore,
for any $y \in Y_i \subseteq \overline{D_i}$,
we get $|N_{H_i}(y) \cap X_i| = |N_{H_i}(y)| = d_{H_i}(y) \leq 2$.
\end{proof}
For any $i \in [(\chi -1)]$, $H_i$ is a bipartite graph with bipartition $\{X_i,Y_i\}$ satisfying the following
two properties: \\
(i) by Claim \ref{linegraphclaim}, for any $ y \in Y_i$, $d_{H_i}(y) \leq 2$. \\
(ii) for any $y_1,y_2 \in Y_i$, if $y_1 \neq y_2$ then $N_{H_i}(y_1) \neq N_{H_i}(y_2)$.
Assume for contradiction that there exist some $y_1,y_2 \in Y_i$ with $y_1 \neq y_2$
and $N_{H_i}(y_1) = N_{H_i}(y_2)$. Then we have $N_{CC(G_i)}(y_1) = N_{CC(G_i)}(y_2)$
which contradicts the fact that $CC(G_i)$ is the critical clique graph of $G_i$.
Therefore by Lemma \ref{linegraphlemma}, we get
$\ensuremath{box}(H_i) \leq \lceil \log_2 \log_2 (\Delta(H_i))\rceil + 3$.
Since $H_i$ is a subgraph of $G$, $\Delta(H_i) \leq \Delta$. Hence,
$$\ensuremath{box}(H_i) \leq \lceil \log_2 \log_2 \Delta\rceil + 3.$$
We thus rewrite inequality (\ref{boxicitysuminequality}) as,
\begin{eqnarray*}
\ensuremath{box}(G) \leq 2(\chi -1)(\lceil \log_2 \log_2 \Delta \rceil + 3) + 1 \leq
2\Delta(\lceil \log_2 \log_2 \Delta \rceil + 3) + 1.
\end{eqnarray*}
As $G=L(H)$, $\Delta \leq 2(\Delta(H)-1) \leq 2(\chi -1)$.
Therefore,
\begin{eqnarray*}
\ensuremath{box}(G) & \leq & 2(\chi -1)(\lceil \log_2 \log_2 (2(\chi -1)) \rceil + 3) + 1.
\end{eqnarray*}
\qed
\end{proof}
\section{Lower Bound for Boxicity of a Hypercube}
For any non-negative integer $d$, a $d$-dimensional
hypercube $H_d$ has its vertices corresponding to
the $2^d$ binary strings each of length $d$.
Two vertices are adjacent if and only if their binary strings
differ from each other in exactly one bit position.
\begin{theorem}
$\ensuremath{box}(H_d) \geq \frac{\lceil \log_2 \log_2 d \rceil + 1}{2}$
\end{theorem}
\begin{proof}
For any vertex $v \in V(H_d)$,
let $g(v)$ denote the number of ones in the bit
string associated with $v$. Let $X=\{v \in V(H)~|~g(v)=1 \mbox{ or }
g(v)=2\}$. Let $H'$ be the subgraph of $H$ induced on the vertex set
$X$.
We can see that $H'$ is a bipartite graph with bipartition
$\{A, B\}$, where $A = \{v\in V(H')~|~g(v)=1\}$ and
$B=\{v\in V(H')~|~g(v)=2\}$.
It is easy to observe that $H'$ is a graph obtained by fully subdividing
$K_{|A|}$, where $K_{|A|}$ refers to a complete graph on $|A|=d$ vertices.
Then by Theorem \ref{completegraphtheorem}, we can say that
\[\ensuremath{box}(H') \geq \frac{\lceil \log_2 \log_2 d \rceil + 1}{2}.\]
As $H'$ is an induced subgraph of $H$,
\[\ensuremath{box}(H) \geq \ensuremath{box}(H') \geq \frac{\lceil \log_2 \log_2 d \rceil + 1}{2}.\]
\qed
\end{proof}
|
{
"timestamp": "2010-09-24T02:00:20",
"yymm": "1009",
"arxiv_id": "1009.4471",
"language": "en",
"url": "https://arxiv.org/abs/1009.4471"
}
|
\section{Introduction}
In solid state physics, one first studies crystallized matter with a
perfectly regular atomic structure where the atoms are located on a
periodic lattice. However, most crystals are not perfectly periodic;
in fact,
the regular pattern of atoms may be disturbed by various defects which
fall into two main classes: there are defects which leave the
lattice unchanged (like impurities or vacancies), and there are
more serious ``geometric'' defects of the lattice itself, cf.\ [AM],
which may involve translations and rotation of portions of the
lattice. Such lattice dislocations occur, in particular, at grain boundaries in alloys.
These models are deterministic but may be generalized to include randomness.
Many of the geometric defects mentioned above are accessible to
mathematical analysis only after some idealization which leads to
the following type of problem, cf.\ [DS]: there is a periodic potential $V \colon
\R^d \to \R$ with period lattice $\Z^d$ and a Euclidean transformation
$T \colon \R^d \to \R^d$ such that the potential coincides with $V$
in the half-space $\{ x \in \R^d \mid x_1 \ge 0\}$ and with $V \circ T$ in $\{x_1 < 0\}$.
In the simplest cases $T$ is translation in the direction
of one of the coordinate axes, with again two main subcases:
translation orthogonal to the hyperplane $\{ x_1 = 0 \}$ or translations that
keep the $x_1$-coordinate fixed.
In the present paper, we discuss the case $d=2$ (where the coordinates are denoted by
$x$ and $y$) and we will mainly focus on translation in the $x$-direction.
In a forthcoming companion paper [HK] we will then study
some aspects of the {\it rotation problem} where we take the given periodic potential
$V$ in the right half-plane and a rotated version $V \circ M_\theta$
in the left half-plane with $M_\theta$ denoting rotation by the angle
$\theta$; some results from the present paper will be essential for [HK].
The one-dimensional dislocation problem is particularly simple:
Let $V:\R\to\R$ be a periodic potential with period $1$ and let
$$
W_t(x):=\left\
\begin{array}{lll}
V(x), && x\geq 0,\\
V(x+t), && x<0,\\
\end{array
\right.\eqno{(1.1)}
$$
for $t\in[0,1]$. The (self-adjoint) operator $H_t:=-\frac{\text{d}^2}
{\text{d}x^2} + W_t$ is called the {\it dislocation operator},
$t$ the {\it dislocation parameter}. There is quite a number of results
available on this problem:
it is well known and easy to see that the essential spectrum of $H_t$
does not depend on $t$ for $0 \le t \le 1$; also $H_t$
cannot have any embedded eigenvalues.
Furthermore, there is no singular continuous spectrum, cf.\ [DS].
For $0 < t < 1$, the operators $H_t$ may have bound states (discrete eigenvalues)
located in the gaps of the essential spectrum. These eigenvalues and
the corresponding resonances have been studied by Korotyaev
[K1, K2] in great detail, using powerful results from analytic function
theory which are specific to the one-dimensional, periodic case.
While, predictably, our results for the one-dimensional periodic case are weaker
than Korotyaev's, our method of proof is very elementary and can
be generalized in several directions; most importantly, we can
apply our techniques to dislocation problems in dimensions greater than 1.
In one dimension, we also give a more systematic treatment of
regularity properties of the eigenvalue ``branches''; in particular,
it is shown that the eigenvalue branches are Lipschitz-continuous if $V$
is (locally) of bounded variation.
The one-dimensional dislocation problem is mainly included to
introduce and test our variational approach which is inspired by [DH, ADH]:
we use approximations by problems on intervals $(-n-t,n)$ with
periodic boundary conditions where it is easy to control the spectral flow, and let $n$ tend to
$\infty$. This idea can be adapted to the study of the translation problem
for the strip $\Sigma := \R \times (0,1)$ in $\R^2$ with periodic
boundary conditions in the $y$-variable, say. In $\R^2$, we consider
dislocation potentials $W_t$ defined by
$$
W_t(x,y):=\left\
\begin{array}{lll}
V(x,y), && x\geq 0,\\
V(x+t,y), && x<0,\\
\end{array
\right.\eqno{(1.2)}
$$
for $t \in [0,1]$. On the strip $\Sigma$ we obtain
existence results for eigenvalues of $S_t := -\Delta + W_t$ in the spectral
gaps of $S_0$. From that we easily derive that $D_t := -\Delta + W_t$, acting in $L_2(\R^2)$,
will have {\it surface states} with a non-zero
density on an appropriate scale, for suitable $t \in (0,1)$.
To distinguish the bulk from the surface density of states for this problem,
we consider the operators $-\Delta + W_t$ on squares
$Q_n = (-n,n)^2$ with Dirichlet boundary conditions, for
$n$ large, count the number of eigenvalues inside a compact
subset of a non-degenerate spectral gap of $D_0$ and scale with $n^{-2}$
for the bulk and with $n^{-1}$ for the surface states. Taking the limits $n\to\infty$
(which exist as explained in [DS, EKSchrS]), we obtain
the integrated density of states measures
$\rho_{\text{bulk}}(D_t,I)$ for the bulk and
$\rho_{\text{surf}}(D_t,J)$ for the surface states of this model;
here $I \subset \R$ and $J \subset \R\backslash\sigma(D_0)$ are open intervals and
$\overline J \subset \R\backslash\sigma(D_0)$.
Our main result can be described as follows:
If $(a,b)$ is a (non-trivial) spectral gap of
the periodic operator $-\Delta + V$, acting in $L_2(\R^2)$, then
for any compact interval $[\alpha,\beta] \subset (a,b)$ with $\alpha <
\beta$ there is a $t \in (0,1)$ such that
$\rho_{\text{surf}}(D_t,(\alpha,\beta))>0$. Upper bounds for the surface density of states are discussed
in [HK].
Our paper is organized as follows. Section 2 deals with
dislocation on the real line. Here it is shown that the $k$-th gap
in the essential spectrum of $H_t$ (if it is open) is crossed by
effectively $k$ eigenvalues of $H_t$ as $t$ increases from $0$ to
$1$. As an example, we discuss a periodic step potential in
Section 3 where one can compute the eigenvalues of the dislocation
operator numerically. Note that our calculations yield numbers
which are exact up to finding the zeros of some transcendental
functions. Related pictures can be found in [DPR] where a
different numerical approach has been used.
In Section 4 we adapt the method of Section 2 to the dislocation
problem on the strip $\Sigma$. The results obtained for the strip then
easily yield spectral information for the dislocation problem in
the plane. Section 5 presents examples from the class of
{\it muffin tin} potentials where one can ``see'' the motion of
the eigenvalues rather directly for either translation in the
$x$-direction or in the $y$-direction. Finally, we include an
Appendix on regularity properties of the functions describing the
eigenvalues of the dislocation operator $H_t$ in one
dimension.
For basic notation and definitions concerning self-adjoint operators
in Hilbert space, we refer to [K, RS-I].\\[.25cm]\indent
{\it Acknowledgements.} The authors would like to thank Andreas
Ruschhaupt (Hannover) and Evgeni Korotyaev (St.\ Petersburg) for
helpful discussions. The authors are particularly indebted to
J\"urgen Voigt, Dresden, who kindly contributed Lemma A.6 in the
Appendix.
\section{Dislocation on the real line}
In this section, we study perturbations of periodic Schr\"odinger
operators on the real line where the potential is obtained from a
periodic potential by a coordinate shift on the left half-axis.
Let $h_0$ denote the (unique) self-adjoint extension of
$-\frac{\d^2}{\dx^2}$ defined on $\Cci{\R}$. Our basic class of
potentials is given by
$$
\PP : = \left\{ V \in
L_{1,\text{loc}}(\R,\R);
\forall x \in \R :
\, V(x+1) = V(x) \right\}.
\eqno{(2.1)}
$$
Potentials $V \in \PP$ belong to the class
$L_{1,\text{loc,unif}}(\R)$ which coincides with the Kato-class on
the real line; in particular, any $V \in \P$ has relative
form-bound zero with respect to $h_0$ and thus the form-sum $H$ of
$h_0$ and $V \in \P$ is well defined (cf. [CFrKS]). For $V \in \P$
given, we define the dislocation potentials $W_t$ as in
eqn.~(1.1), for $0 \le t \le 1$; as before, the form-sum $H_t$ of
$h_0$ and $W_t$ is well defined.
We begin with some well-known results pertaining to the spectrum
of $H = H_0$. As explained in [E, RS-IV], we have
$$
\sigma(H) = \sigmaess(H) = \cup_{k=1}^\infty [\gamma_k,
\gamma'_k],
\eqno{(2.2)}
$$
where the $\gamma_k$ and $\gamma'_k$ satisfy $\gamma_k < \gamma_k'
\le \gamma_{k+1}$, for all $k \in \N$, and $\gamma_k \to \infty$
as $k\to\infty$. Moreover, the spectrum of $H$ is purely
absolutely continuous. The intervals $[\gamma_k,\gamma_k']$ are
called the {\it spectral bands} of $H$. The open intervals
$\Gamma_k := (\gamma_k',\gamma_{k+1})$ are the {\it spectral gaps}
of $H$; we say the $k$-th gap is {\it open} or {\it
non-degenerate} if $\gamma_{k+1} > \gamma_k'$.
In order to determine the essential spectrum of $H_t$ for $0<t<1$,
we introduce Dirichlet boundary conditions at $x=0$ for the
operator $H_0$ and at $x=0$ and $x = -t$ for $H_t$ to obtain the
operators
$$
H_D=H^-\oplus H^+,\quad H_{t,D}=H_t^-\oplus H_{(-t,0)}\oplus H^+,
\eqno{(2.3)}
$$
where $H^{\pm}$ acts in $\R^{\pm}$ with a Dirichlet boundary
condition at $0$, $H_t^-$ in $(-\infty,-t)$ with Dirichlet
boundary condition at $-t$ and $H_{(-t,0)}$ in $(-t,0)$ with
Dirichlet boundary conditions at $-t$ and $0$. Since $H_{(-t,0)}$
has purely discrete spectrum and since the operators $H_t^-$ and
$H^-$ are unitarily equivalent, we conclude that
$\sigmaess(H_D)=\sigmaess(H_{t,D})$. It is well known that
decoupling by (a finite number of) Dirichlet boundary conditions
leads to compact perturbations of the corresponding resolvents (in
fact, perturbations of finite rank) and thus Weyl's essential
spectrum theorem yields $\sigmaess(H_D)=\sigmaess(H)$ and
$\sigmaess(H_{t,D})=\sigmaess(H_t)$.
In addition to the essential spectrum, the operators $H_t$ may
have discrete eigenvalues below the infimum of the essential
spectrum and inside any (non-degenerate) gap, for $t \in (0,1)$;
these eigenvalues are simple. The eigenvalues of $H_t$ in the gaps
of $H$ depend continuously on $t$; cf.\ the Appendix for a brief
exposition of the relevant perturbational arguments, which are
fairly standard. A more complete and precise picture is
established in the following lemma which says that the discrete
eigenvalues of $H_t$ inside a given gap $\Gamma_k$ of $H$ can be
described by an (at most) countable, locally finite family of
continuous functions, defined on suitable subintervals of $[0,1]$.
\lem Let $k \in \N$ and suppose that the gap $\Gamma_k$ of $H$ is
open, i.e., $\gamma_k' < \gamma_{k+1}$. Then there is a (finite or
countable) family of continuous functions $f_j \colon
(\alpha_j,\beta_j) \to \Gamma_k$, where $0 \le \alpha_j < \beta_j
\le 1$, with the following properties:
\begin{enumerate}
\item[$(i)$] $f_j(t)$ is an eigenvalue of $H_t$, for all $\alpha_j
< t < \beta_j$ and for all $j$. Conversely, for any $t \in (0,1)$
and any eigenvalue $E \in \Gamma_k$ of $H_t$ there is a unique
index $j$ such that $f_j(t) = E$.
\item[$(ii)$] As $t \downarrow \alpha_j$ (or $t \uparrow
\beta_j$), the limit of $f_j(t)$ exists and belongs to the set
$\{\gamma_k',\gamma_{k+1}\}$.
\item[$(iii)$] For all but a finite number of indices $j$ the
range of $f_j$ does not intersect a given compact subinterval
$[a',b'] \subset \Gamma_k$.
\end{enumerate}
\endlem\rm
For the convenience of the reader, we include a proof in the
Appendix. Under stronger assumptions on $V$ one can show that the
eigenvalue branches are H\"older- or Lipschitz-continuous, or even
analytic; cf.\ the Appendix. Additional information on the
eigenvalue functions $f_j$ can be found in [K1, K2].
It is our aim in this section to show that at least $k$ eigenvalues move
from the upper to the lower edge of the $k$-th gap
as the dislocation parameter ranges from $0$ to $1$.
Using the notation of Lemma 2.1
and writing $f_i(\alpha_i) := \lim_{t\downarrow\alpha_i}f_i(t)$,
$f_i(\beta_i) := \lim_{t\uparrow\beta_i}f_i(t)$, we now define
$$
\NN_k := \#\{ i \mid f_i(\alpha_i) = \gamma_{k+1}, \,\,\, f_i(\beta_i) = \gamma_k' \}
- \#\{ i \mid f_i(\alpha_i) = \gamma_k', \,\,\, f_i(\beta_i) = \gamma_{k+1} \}.
\eqno{(2.4)}
$$
Thus $\NN_k$ is precisely the number of eigenvalue branches of
$H_t$ that cross the $k$-th gap moving from the upper to the lower
edge minus the number crossing from the lower to the upper edge.
Put differently, $\NN_k$ is the spectral multiplicity which {\it
effectively} crosses the gap $\Gamma_k$ in downwards direction as
$t$ increases from $0$ to $1$.
Our main result in this section says that $\NN_k = k$, provided
the $k$-th gap is open:
\thm Let $V \in \PP$ and suppose that the $k$-th spectral gap of
$H$ is open, i.e., $\gamma_k' < \gamma_{k+1}$. Then $\NN_k = k$.
\endthm\rm
Again, the results obtained by Korotyaev in [K1, K2] are more detailed; e.g.,
it is shown that, for any $t \in (0,1)$, the dislocation operator $H_t$ has
two unique states (an eigenvalue and a resonance) in any given gap of the
periodic problem.
On the other hand, our variational arguments are more flexible
and allow an extension to higher dimensions, as will be seen in the sequel.
In this sense, the importance of this section lies in testing our
approach in the simplest possible case. For further reading concerning the spectral
flow through the gaps of perturbed Schr\"odinger operators, we
recommend [P, Saf].
The main idea of our proof---somewhat reminiscent of [DH,
ADH]---goes as follows: consider a sequence of approximations on
intervals $(-n-t,n)$ with associated operators $H_{n,t} =
-\frac{\text{d}^2}{\text{d}x^2}+ W_t$ with periodic boundary
conditions. We first observe that the gap $\Gamma_k$ is free of
eigenvalues of $H_{n,0}$ and $H_{n,1}$ since both operators are
obtained by restricting a periodic operator on the real line to
some interval of length equal to an entire multiple of the period,
with periodic boundary conditions. Second, the operators $H_{n,t}$
have purely discrete spectrum and it follows from Floquet theory
(cf. [E, RS-IV]) that $H_{n,0}$ has precisely $2n$ eigenvalues in
each band while $H_{n,1}$ has precisely $2n+1$ eigenvalues in each
band. As a consequence, effectively $k$ eigenvalues of $H_{n,t}$
must cross any fixed $E\in\Gamma_k$ as $t$ goes from $0$ to $1$.
To obtain the result of Theorem 2.2 we only have to take the limit
$n \to \infty$. Here we employ several technical lemmas. In the
first one, we show that the eigenvalues of the family $H_{n,t}$
depend continuously on the dislocation parameter.
\lem The eigenvalues of $H_{n,t}$ depend continuously on $t \in
[0,1]$.\endlem\rm
\proof We may assume that the eigenvalues of $H_{n,t}$ are
numbered according to min-max. Since the Hilbert space
$L_2(-n-t,n)$ depends on $t$, we use the unitary mappings
$$
U_{n,t} \colon L_2(-n-t,n) \to L_2(-n,n),
\qquad
(U_{n,t} f)(x) := \sqrt{\sigma_{n,t}} f(\sigma_{n,t} x),
\eqno{(2.5)}
$$
where $\sigma_{n,t} := \frac{2n + t}{2n}$. Let ${\tilde H}_{n,t} := U_{n,t} H_{n,t}
U_{n,t}^{-1}$ and ${\tilde W}_t(x) := W_t(\sigma_{n,t} x)$ so that
(writing $\sigma = \sigma_{n,t}$)
$$
{\tilde H}_{n,t} = \sigma^{-2} h_0 + {\tilde W}_t(x)
= \sigma^{-2}(h_0 + \sigma^2 {\tilde W}_t(x)).
\eqno{(2.6)}
$$
It is easy to see that the mapping $[0,1] \ni t \mapsto \sigma^2
{\tilde W}_t \in L_1(-n,n)$ is continuous. Now the usual
perturbational and variational arguments for quadratic forms ([K]
and the Appendix) imply that the eigenvalues of $h_0 + \sigma^2
{\tilde W}_t$ depend continuously on $t$, and then the same is
true for the eigenvalues of $H_{n,t}$.
\endproof
The next lemma is to establish a connection between the spectra of
$H_t$ and $H_{n,t}$ for $0 \le t \le 1$ and $n$ large. In the
proof and henceforth, we will make use of the following cut-off
functions: We pick some $\phi\in\Cci{-2,2}$ with $0\leq\phi\leq 1$
and $\phi(x)=1$ for $|x| \le 1$. For $k \in (0,\infty)$ we then
define $\phi_k(x):=\phi(x/k)$ so that
$\supp\phi_k\subset(-2k,2k)$, $\phi_k(x)=1$ for $|x|\leq k$,
$|\phi_k'(x)|\leq Ck^{-1}$ and $|\phi_k''(x)|\leq Ck^{-2}$.
Finally, we let $\psi_k:=1-\phi_k$. For any self-adjoint operator
$T$ we denote the spectral projection associated with an interval
$I \subset \R$ by $P_I(T)$ and we write $\dim\, P_I(T)$ to denote
the dimension of the range of the projection $P_I(T)$.
\lem Let $k \in \N$ with $\Gamma_k\neq\emptyset$.
Let $t \in (0,1)$ and suppose that $\eta_1 < \eta_2 \in \Gamma_k$
are such that $\eta_1, \eta_2 \notin \sigma(H_t)$. Then there is
an $n_0 \in \N$ such that $\eta_1, \eta_2 \notin \sigma(H_{n,t})$
for $n \ge n_0$, and\rm
$$
\dim\, P_{(\eta_1, \eta_2)}(H_t) = \dim\, P_{(\eta_1, \eta_2)}(H_{n,t}),
\qquad n \ge n_0.
\eqno{(2.7)}
$$
\endlem\rm
\proof In the subsequent calculations, we always take $k := n/4$,
for $n\in\N$.
(1) Let $E \in (\eta_1,\eta_2) \cap \sigma(H_t)$ with associated
normalized eigenfunction $u$. Then $u_k:=\phi_ku\in D(H_{n,t})$,
$H_{n,t}u_k=H_tu_k$ and $\norm{u_k}\to 1$ as $n \to \infty$. Since
$$
\norm{H_{n,t}u_k-E u_k}\leq 2\cdot\infnorm{\phi_k'}\norm{u'}
+\infnorm{\phi_k''}\norm{u},\eqno{(2.8)}
$$
it is now easy to conclude that $ \dim\, P_{(\eta_1,
\eta_2)}(H_{n,t}) \ge \dim\, P_{(\eta_1, \eta_2)}(H_t)$ for $n$
large.
(2) We next assume for a contradiction that $\eta \in \Gamma_k$
satisfies
$\eta \in \sigma(H_{n,t})$ for infinitely many $n \in \N$.
Then there is a subsequence $(n_j)_{j\in\N} \subset \N$ s.th.\
$\eta \in \sigma(H_{n_j,t})$; we let $u_{n_j,t} \in D(H_{n_j,t})$
denote a normalized eigenfunction and set
$$
v_{1,n_j} := \phi_{k_j} u_{n_j,t},
\qquad v_{2,n_j} := \psi_{k_j} u_{n_j,t},\eqno{(2.9)}
$$
so that $v_{1,n_j} \in D(H_t)$ and $\norm{(H_t - \eta)v_{1,n_j}}
\to 0$ as $j \to \infty$ by a similar estimate as in part (1) (and
using a simple bound for $\norm{u_{n,t}'}$ which follows from the
fact that $V$ has relative form-bound zero w.r.t.\ $h_0$.) Let us
now show that $v_{2,n_j} \to 0$ (and hence $\norm{v_{1,n_j}} \to
1$) as $j \to \infty$:
The function
$$
\tilde{v}_{2,n_j}:= \left\
\begin{array}{lll}
v_{2,n_j}(x), && x \geq 0, \\
v_{2,n_j}(x-t), && x < 0, \\
\end{array
\right.
\eqno{(2.10)}
$$
belongs to the domain of $H_{n_j,0}$ and
$H_{n_j,0}\tilde{v}_{2,n_j}=[H_{n_j,t}v_{2,n_j}]^\sim$\,, where
$[\cdot]^\sim$ is defined in analogy with eqn.~(2.10). Since we also have
$(H_{n_j,t} - \eta)v_{2,n_j} \to 0$, as $j \to \infty$,
we see that $(H_{n_j,0}-\eta)\tilde v_{2,n_j} \to 0$. But
$\dist(\eta,\sigma(H_{n,0})) \ge \delta_0 > 0$ for all $n$ and the
Spectral
Theorem implies that $\norm{\tilde{v}_{2,n_j}}\to 0$ as
$j\to\infty$. We have thus shown that $\norm{v_{1,n_j}} \to 1$ and
$\norm{(H_t - \eta)v_{1,n_j}} \to 0$ which implies that $\eta \in
\sigma(H_t)$.
(3) It remains to show that $ \dim\, P_{(\eta_1, \eta_2)}(H_{n,t})
\le \dim\, P_{(\eta_1, \eta_2)}(H_t)$, for $n$ large. The proof
by contradiction follows the lines of part (2); instead of a
sequence of functions $u_{n_j}$ we work with an orthonormal system
$u_{n_j}^{(1)}, \ldots, u_{n_j}^{(\ell)}$ of eigenfunctions where
$\ell = \dim\, P_{(\eta_1,\eta_2)}(H_t + 1)$. We leave the details
to the reader.
\endproof
\rem In fact, using standard exponential decay estimates for
resolvents of Schr\"odinger operators, cf.~[S], it can be shown
that the eigenvalues of $H_t$ and $H_{n,t}$ in the gap $\Gamma_k$
are exponentially close, for $n$ large; e.g., if $E \in
\sigma(H_t) \cap \Gamma_k$ for some $t \in (0,1)$, then there are
constants $c \ge 0$ and $\alpha > 0$ s.th.\ the operators
$H_{n,t}$ have an eigenvalue in $(E- c \ee^{-\alpha n}, E + c
\ee^{-\alpha n})$, for $n$ large. There is a similar converse
statement with the roles of $H_t$ and $H_{n,t}$ exchanged; cf.\
also Remark 4.2 for further discussion.\endrem\rm
The desired connection between the spectral flow for $(H_{n,t})_{0
\le t \le 1}$ and $(H_t)_{0 \le t \le 1}$ is obtained by applying
Lemma 2.4 at suitable $t_i \in [0,1]$ and $\eta_{1,i} < \eta_{2,i}
\in \Gamma_k$. We now construct an appropriate partition of the
parameter interval $[0,1]$.
\lem Let $k \in \N$ with $\Gamma_k\neq\emptyset$.
Then there exists a partition $0 = t_0
< t_1 < \ldots < t_{K-1} < t_K = 1$ and there exist $E_j \in
\Gamma_k$ and $n_0 \in \N$ such that
$$
E_j \notin \sigma(H_t) \cup \sigma(H_{n,t}), \qquad \forall t \in [t_{j-1},
t_j], \quad j=1, \ldots, K, \quad n \ge n_0.\eqno{(2.11)}
$$
\endlem\rm
\proof For any $t \in [0,1]$ there exists $\eta_t \in \Gamma_k$
such that $\eta_t \notin \sigma(H_t)$. Since the spectrum of $H_t$
depends continuously on the parameter $t$ there also exists $\eps
= \eps_t > 0$ such that $\eta_t \notin \sigma(H_\tau)$ for all
$\tau \in (t -\eps_t, t + \eps_t)$. By compactness, we can find a
partition $(\tau_j)_{0 \le j \le K}$ (with $\tau_0 = 0$, $\tau_K =
1$) such that the intervals $(\tau_j - \eps_j, \tau_j + \eps_j)$
cover $[0,1]$. Set $E_j := \eta_{\tau_j}$. We next pick arbitrary
points $t_j \in (\tau_j , \tau_j + \eps_j) \cap (\tau_{j+1}
-\eps_{j+1}, \tau_{j+1})$, for $j=1, \ldots, K-1$, set $t_0=0$,
$t_K=1$ and see that $E_j \notin \sigma(H_t)$ for $t_{j-1} \le t
\le t_j$, $j = 1, \ldots, K$.
By Lemma 2.4, using Lemma 2.3 combined with a simple compactness argument,
we then find that we also have $E_j \notin \sigma(H_{n,t})$
for $t \in [t_{j-1},t_j]$ and $n$ large.
\endproof
We are now ready for the proof of Theorem 2.2.\\[.2cm]
\emph{Proof of Theorem 2.2.} Let $E_j$ be as in Lemma 2.6 and $\NN_k$
as in
eqn.~(2.4). We fix some ${\tilde E} \in \Gamma_k$
such that ${\tilde E} > E_j$ for $j = 0,\ldots, K$ and ${\tilde E}
\notin \sigma(H_{t_j}) \cup \sigma(H_{n,t_j})$ for $j = 0, \ldots,
K$ and for all $n$ large. It is then easy to see that
$$
\NN_k = \sum_{j=1}^K \left( \dim\, P_{(E_j, {\tilde E})}(H_{t_j})
- \dim\, P_{(E_j, {\tilde E})}(H_{t_{j-1}}) \right)\eqno{(2.12)}
$$
and that
\begin{align}
\dim\, P_{(-\infty, {\tilde E})}(H_{n,1}) & - \dim\, P_{(-\infty, {\tilde
E})}(H_{n,0}) \nonumber\\
& = \sum_{j=1}^K \bigg( \dim\, P_{(E_j, {\tilde E})}(H_{n,t_j})
- \dim\, P_{(E_j, {\tilde E})}(H_{n,t_{j-1}})
\bigg) .\nonumber
\end{align}
\vspace{-.8cm}
$$\eqno{(2.13)}$$
The LHS of (2.13) equation equals $k$. Furthermore, by Lemma
2.4, we have
$$
\dim\, P_{(E_j, {\tilde E})}(H_{t_j}) = \dim\, P_{(E_j, {\tilde E})}(H_{n,t_j})\eqno{(2.14)}
$$
for all $j$ and all $n$ large, and the desired result follows.
\hfill$\square$
\section{A one-dimensional periodic step
potential}
In this section, we study the one-dimensional $2\pi$-periodic
potential
$$
V(x):=\left\
\begin{array}{lll}
-1, && x\in[0,\pi], \\
1, && x\in(\pi,2\pi). \\
\end{array
\right.\eqno{(3.1)}
$$
(While the other sections of this paper deal with $1$-periodic potentials,
we have preferred to work here with period $2\pi$ in order to keep
the explicit calculations done by hand as simple as possible.)
To obtain the band-gap structure of $H=-\frac{\d^2}{\dx^2}+V$, we
compute the discriminant function
$$
D(E):=\phi_1(2\pi;E)+\phi_2'(2\pi;E)=\hbox{tr}
\left
\begin{array}{cc}
\phi_1(2\pi;E) & \phi_1'(2\pi;E) \\
\phi_2(2\pi;E) & \phi_2'(2\pi;E) \\
\end{array
\right)\eqno{(3.2)}
$$
where $\phi_1(\,\cdot\,;E)$ and $\phi_2(\,\cdot\,;E)$ solve the
equation
$$-u''+(V-E)u=0\eqno{(3.3)}$$
and satisfy the boundary conditions
$$\phi_1(0;E)=\phi_2'(0;E)=1\quad\hbox{and}\quad\phi_1'(0;E)=\phi_2(0;E)=0.\eqno{(3.4)}$$
The matrix $M(E)$ on the RHS of (3.2) is called the {\it monodromy
matrix}. A simple computation shows that $[-1/2,1/2]\subset\Gamma_1$,
where $\Gamma_1$ is the first spectral gap of $H$ (with numbering
according to Floquet theory). Note that the gap edges of
$\Gamma_1$ also equal the first eigenvalue in the (semi-)periodic
eigenvalue problem for $-\frac{\text{d}^2}{\text{d}x^2}+V$ in
$L_2(0,2\pi)$, cf., e.g., [E, CL].
As explained in [E, RS-IV], for any $E\notin\sigma(H)$, there are
two solutions $\phi_\pm(x;E)\in C^1(\R)$, square integrable at
$\pm\infty$, of (3.3); in fact, the functions $\phi_\pm(x;E)$ are
exponentially decaying at $\pm\infty$ and exponentially increasing
at $\mp\infty$. In our example, the dislocation potential $W_t$
for $t\in(0,1)$ will produce a bound state at $E$ if and only if
the boundary conditions coming from $\phi_+(0;E)$ and
$\phi_-(t;E)$ match up, i.e.,
$$\phi_-(t;E)=\phi_+(0;E)\quad\text{and}\quad\phi_-'(t;E)=\phi_+'(0;E).\eqno{(3.5)}$$
An equivalent condition for (3.5) is the equality of the ratio
functions $\frac{\phi_-(t;E)}{\phi_-'(t;E)}$ and
$\frac{\phi_+(0;E)}{\phi_+'(0;E)}$, cf. [DPR].
We compute the Floquet solutions $\varphi_{\pm}$ by solving the
equation $-u''+(V-E)u=0$ for $x<0$ and $x>0$ and for $E$ varying
in $[-1/2,1/2]$, assuming that $(u(0),u'(0))$ equals an
appropriate eigenvector of $M(E)$. Note that, since $D(E)<-2$,
both eigenvalues of $M(E)$ are negative and not equal to $-1$.
Finally, we divide $[-1/2,1/2]$ into 100 subintervals of equal
length and compute numerical values for $t$ such that
$$\left|\frac{\phi_-(t;E)}{\phi_-'(t;E)}-\frac{\phi_+(0;E)}{\phi_+'(0;E)}\right|<\eps,\eqno{(3.6)}$$
where the error $\eps>0$ is small. This leads to the following
plot of $t\mapsto E(t)$, see Fig.~1.
\begin{figure}[H]
\begin{center}
\includegraphics[width=8cm]{trajectory}
\caption{An eigenvalue branch of $H_t$ in the first spectral gap.}
\end{center}
\end{figure}
\section{Periodic potentials on the strip
and the plane}
Let $V \colon \R^2 \to \R$ be $\Z^2$-periodic and Lipschitz-continuous and let $\Sigma = \R \times (0,1)$ denote the infinite
strip of width $1$. We denote by $S_t$ the (self-adjoint) operator
$-\Delta + W_t$, acting in $L_2(\Sigma)$, with periodic boundary
conditions in the $y$-variable and with $W_t$ defined as in
eqn.~(1.2);
again, the parameter
$t$ ranges between $0$ and $1$. Since $S_0$ is periodic in the
$x$-variable, its spectrum has a band-gap structure.
We first observe that the essential spectrum of the family $S_t$
does not depend on the parameter $t$, i.e.,
$\sigmaess(S_t)=\sigmaess(S_0)$ for all $t\in[0,1]$. As in Section
2, this follows from the compactness of
$(S_t-c)^{-1}-(S_{t,D}-c)^{-1}$, where $S_{t,D}$ is $S_t$ with an
additional Dirichlet boundary condition at $x=0$, say. (While, in
one dimension, adding in a Dirichlet boundary condition at a
single point causes a rank-one perturbation of the resolvent, the
resolvent difference is now Hilbert-Schmidt, which can be seen
from the following well-known line of argument: If
$-\Delta_\Sigma$ denotes the (negative) Laplacian in $L_2(\Sigma)$
and $-\Delta_{\Sigma;D}$ is the (negative) Laplacian in
$L_2(\Sigma)$ with an additional Dirichlet boundary condition at
$x=0$, then $(-\Delta_\Sigma + 1)^{-1}
- (-\Delta_{\Sigma;D} + 1)^{-1}$ has an integral kernel which can be
written down explicitly using the Green's function for
$-\Delta_\Sigma$ and the reflection principle.)
While the essential spectrum of the family $S_t$ does not change
as $t$ ranges through $[0,1]$, $S_t$ will have discrete
eigenvalues in the spectral gaps of $S_0$ for appropriate values
of $t$.
We have the following result.
\thm Let $(a,b)$, $a<b$, denote a spectral gap of $S_t$ and let $E
\in (a,b)$. Then there exists $t = t_E \in (0,1)$ such that $E$ is
a discrete eigenvalue of $S_t$.
\endthm\rm
\proof (1) As on the real line, we work with approximating
problems on finite size sections of the infinite strip $\Sigma$.
Let
$$
\Sigma_{n,t}:=(-n-t,n)\times(0,1),\quad n\in\N,\eqno{(4.1)}
$$
and consider $S_{n,t}:=-\Delta+W_t$ acting in $L_2(\Sigma_{n,t})$
with periodic boundary conditions in both coordinates. The
operator $S_{n,t}$ has compact resolvent and purely discrete
spectrum accumulating only at $+\infty$. The rectangles
$\Sigma_{n,0}$ (respectively, $\Sigma_{n,1}$) consist of $2n$
(respectively, $2n+1$) period cells. By routine arguments (see,
e.g., [RS-IV, E]), the number of eigenvalues below the gap $(a,b)$
is an integer multiple of the number of cells in these rectangles;
we conclude, that eigenvalues of $S_{n,t}$ must cross the gap as
$t$ increases from $0$ to $1$.\\
(2) Let $E\in(a,b)$. According to (1), for any $n\in\N$ we can
find $t_n\in(0,1)$ such that $E\in\sigmadisc(S_{n,t_n})$; then
there are eigenfunctions $u_n\in D(S_{n,t_n})$ with $S_{n,t_n}u_n
= Eu_n$, $\norm{u_n}=1$, and $\norm{\nabla u_n} \le C$ for some
constant $C \ge 0$. We now choose cut-off functions $\phi_n$ as in
Section 2 and denote the natural extension to $\R^2$ again by
$\phi_n$. We also let $\psi_n = 1 - \phi_n$. Clearly,
$$
\norm{(S_{t_n} - E)(\phi_{n/4} u_n)}, \hskip1ex
\norm{(S_{n,t_n} - E)(\psi_{n/4} u_n)} \le c/n,
\eqno{(4.2)}
$$
for some $c \ge 0$. There is a subsequence $(t_{n_j})_{j\in\N}
\subset (t_n)_{n\in\N}$ and $\cl t \in [0,1]$ s.th.\ $t_{n_j} \to
\cl t$ as $j\to\infty$. Since $V$ is Lipschitz, we may infer from
$(4.2)$ that
$$
\norm{(S_{\cl t} - E)(\phi_{n_j/4} u_{n_j})} \to 0, \qquad j \to \infty,
\eqno{(4.3)}
$$
and it remains to show that $\norm{\psi_{n/4} u_n} \to 0$ so that
$\norm{\phi_{n/4} u_n} \to 1$. We associate with functions $v
\colon \Sigma_{n,t} \to \C$ functions $\tilde{v} \colon
\Sigma_{n,0} \to \C$ by
$$
\tilde{v}(x,y) :=
\left\
\begin{array}{lll}
v(x,y), && x>0, \\
v(x-t,y), && x < 0,\\
\end{array
\right.
\eqno{(4.4)}
$$
in analogy with eqn.~(2.10). Then $[\psi_{n/4} u_n]^{\sim} \in
D(S_{n,0})$ and
$$
\norm{(S_{n,0} - E)[\psi_{n/4} u_n]^{\sim}}
= \norm{(S_{n,t_n} - E)(\psi_{n/4} u_n)} \le c/n.
\eqno{(4.5)}
$$
Since $(a,b) \cap \sigma(S_{n,0}) = \emptyset$ for all $n\in\N$,
and since $E \in (a,b)$, the Spectral Theorem implies that
$[\psi_{n/4} u_n]^{\sim} \to 0$ (and therefore also $\psi_{n/4}u_n
\to 0$) as $n \to \infty$.
We therefore have shown that the functions $v_{n_j} :=
\phi_{n_j/4} u_{n_j}$ for $j\in\N$ satisfy $\norm{(S_{\cl t} - E)
v_{n_j}} \to 0$ and $\norm{v_{n_j}} \to 1$ as $j \to \infty$ which
implies $E \in \sigma(S_{\cl t})$.
\endproof
\rem By a well-known line of argument, one can obtain {\it
exponential localization} of the eigenfunctions of $S_t$ near the
interface $\{(x,y) \mid x = 0 \}$. Since we will use exponential
localization in a more systematic way in the forthcoming paper
[HK] we only give a brief sketch here: Suppose that $E \in (a,b)$
and $t \in (0,1)$ satisfy $E \in \sigma(S_t)$. Let $u \in D(S_0) =
D(S_t)$ denote a normalized eigenfunction and let $\phi_n$, $n \in
\N$, be as in the proof of Theorem 4.1. As above, we have
$$
(S_t - E) (\phi_n u) = -2 \nabla \phi_n \cdot \nabla u - (\Delta \phi_n) u
= : r_n,
\eqno{(4.6)}
$$
where $\norm{r_n} \le c/n$, for $n \in \N$. Since $r_n$ has
support in the interval $(-2n-1, 2n)$ we now see that there exist
constants $C \ge 0$ and $\alpha > 0$ such that
$$
\norm{\chi_{|x| \ge 4n} u}
\le \norm{\chi_{|x| \ge 4n} (S_t - E)^{-1} r_n}
\le C \ee^{-\alpha n},
\eqno{(4.7)}
$$
by standard exponential decay estimates for the resolvent kernel
of Schr\"odinger operators (cf., e.g., [S], [HK]).
\endrem\rm
We now turn to the dislocation problem on the plane $\R^2$ where
we study the operators
$$
D_t=-\Delta+W_t,\quad 0\leq t\leq 1.
\eqno{(4.8)}
$$
Denote by $S_t(\theta)$ the operator $S_t$ with $\theta$-periodic
boundary conditions in the $y$-variable. Since $W_t$ is periodic
with respect to $y$, we have
$$
D_t\simeq\int_{[0,2\pi]}^{\oplus}S_t(\theta)\frac{\d\theta}{2\pi},
\eqno{(4.9)}
$$
and hence the spectrum of $D_t$ has a band-gap structure;
furthermore, $D_t$ has no singular continuous part, cf.\ [DS, FS].
As for the spectrum of $S_t$ inside the gaps of $S_0$, Theorem 4.1
leads to the following result.
\thm Let $(a,b)$ denote a spectral gap of $D_0$, $a > \inf
\sigmaess(D_0)$, and let $E \in (a,b)$. Then there exists $t = t_E
\in (0,1)$ with $E \in \sigma(D_t)$.
\endthm\rm
\proof Let $\phi_n u_n\in D(S_t)$ as in part (2) of the proof of
Theorem 4.1 denote an approximate solution of the eigenvalue
problem for $S_t$ and $E$. We extend $u_n$ to a function ${\tilde
u}_n(x,y)$ on $\R^2$ which is periodic in $y$. Writing $\Phi_n =
\Phi_n(x,y) := \phi_n(x) \phi_n(y)$ we compute
\begin{align}
(D_t - E) (\Phi_n\tilde u_n)
& = \left(- \partial_x^2 - \partial_y^2 + W_t - E \right) (\phi_n(x)
\phi_n(y) \tilde u_n(x,y)) \nonumber\\
& = \phi_n(y) {[(S_t - E) (\phi_n(x) u_n)]^{\sim} }
- \phi_n(x) \left(2 \phi_n'(y) \partial_y {\tilde u}_n + \phi_n''(y) {\tilde u}_n
\right). \nonumber
\end{align}
\vspace{-.65cm}
$$\eqno{(4.10)}$$
The norms of the three terms on the RHS can be estimated (up to a
constant which is independent of $n$) by $\eps n$, ${1 \over n} n$
and ${1 \over n^2} n$, respectively, and we see that
$$
\norm{(D_t - E) (\Phi_n\tilde u_n)} \le c_0 (1 + n\eps),
\eqno{(4.11)}
$$
while $\norm{\Phi_n\tilde u_n} \ge c_0 n$ with a constant $c_0 >
0$. This implies the desired result.
\endproof
\rem We learn from the above proof that there are functions
$$
v_n = v_n(x,y) := {1 \over \norm{\Phi_n\tilde u_n}} \Phi_n\tilde u_n
\eqno{(4.12)}
$$
that satisfy $\norm{v_n} =1$, $\supp v_n \subset [-n,n]^2$ and
$$
(D_t - E) v_n \to 0, \qquad n \to \infty.
\eqno{(4.13)}
$$
These functions play a key role in our analysis of the rotation
problem at small angle in [HK].
\endrem\rm
We finally turn to a brief discussion of the i.d.s.\ (the
integrated density of states [V]) for the dislocation operators $D_t$.
We adopt the natural distinction of [DS, EKSchrS, KS] between {\it
bulk} and {\it surface} states. Roughly speaking, the bulk states
correspond to states away from the interface with energies in the
spectral bands while the surface states for $0< t < 1$ are
produced by the interface and are (exponentially) localized near
the interface.
The (integrated) density of states measures for the bulk and surface states
use a different scaling factor in the following
definition: restricting $D_t$ to large squares $Q_n = (-n,n)^2$
and taking Dirichlet boundary conditions, we obtain the operators
$D^{(n)}_t$. For $I \subset \R$ an open interval, let
$N(I,D^{(n)}_t)$ denote the number of eigenvalues of $D^{(n)}_t$
in $I$, counting multiplicities. We then define for open intervals
$I \subset \R$ and $J \subset \R \setminus \sigma (D_0)$ with
$\overline{J} \subset \R \setminus \sigma (D_0)$
$$
\rho_{\text{bulk}}(I, D_t) = \lim_{n \to \infty} \frac{1}{4n^2} N(I,D^{(n)}_t),
\quad
\rho_{\text{surf}}(J, D_t) = \lim_{n \to \infty} \frac{1}{2n}
N(J,D^{(n)}_t).
\eqno{(4.14)}
$$
The existence of the limits in (4.14) has been established in
[EKSchrS, KS] for ergodic Schr\"odinger operators. Note that the
surface density of states measure is defined (and possibly
non-zero) for subintervals of the spectral bands, but then
eqn.~(4.14) is not suited to capture the surface states (cf.\
[EKSchrS, KS]).
The fact that the surface density of states exists does not mean
it is non-zero and there are only rare examples where we know
$\rho_{\text{surf}}$ to be non-trivial. It is one of the main
results of the present paper to show that dislocation moves enough
states through the gap to have a non-trivial surface density of
states, for suitable parameters $t$. Indeed, it is now
easy to derive the following result:
\cor Let $(a,b)$ be a spectral gap of $D_0$ with $a > \inf
\sigmaess(D_0)$, and let $\emptyset \ne J \subset (a,b)$ be an
open interval. Then there is a $t \in (0,1)$ such that
$\rho_{\text{\rm surf}}(J,D_t) > 0$.
\endcor\rm
\proof
Let $[\alpha,\beta] \subset J$ with $\alpha < \beta$, fix
$E \in (\alpha,\beta)$, and let $0 < \eps < \min \{ E-\alpha,
\beta - E\}$. By Theorem~4.3 and Remark~4.4 there exist $t = t_E
\in (0,1)$ and a function $u_0$ in the domain of $D_t$ satisfying
$\norm{u_0} = 1$, $\supp u_0$ compact, and $\norm{(D_t - E) u_0} < \eps$.
Let $\nu \in \N$ be such that $\supp u_0 \subset (-\nu,\nu)^2$; note that,
in the present proof, $\nu$ corresponds to the $n$ of Remark 4.4.
We then see that the functions $\varphi_k$, defined by
$\varphi_k(x,y) := u_0(x, y - 2 k \nu)$ for $k \in \N$, have pairwise
disjoint supports, are of norm $1$, and satisfy $\norm{(D_t - E) \phi_k} <
\eps$. Furthermore, we have $\supp \varphi_k \subset (-n,n)^2$ provided
$(2k + 1) \nu < n$. Denoting
$\MM_n := \hbox{\rm span} \{ \phi_k \mid k \in \N, \> k \le {1 \over
2} ({n \over \nu} - 1) \}$,
it is clear that $\dim \MM_n \ge n/(3\nu)$, for all $n$ large.
Let $\NN_n$ denote the range of the spectral projection
$P_{(\alpha,\beta)}(D_t^{(n)})$ of $D_t^{(n)}$ associated with the interval
$(\alpha,\beta)$; we will show that $\dim \NN_n \ge \dim \MM_n$ which
implies the desired result. If we assume for a contradiction
that $\dim \NN_n < \dim \MM_n$ for some $n \in \N$,
we can find a function $v \in \MM_n \cap \NN_n^\perp$ of norm $1$. By
the Spectral Theorem, $|\!|(D_t^{(n)} - E) v |\!| \ge \eps$. On the
other hand, $v$ is a finite linear combination of the $\phi_k$,
which implies $|\!|(D_t^{(n)} - E) v|\!| < \eps$.
\endproof
We will continue the discussion of bulk versus surface states in
the companion paper [HK] where a corresponding upper bound of the
form $N(J, D_t^{(n)}) \le c n \log n$ is provided.
\section{Muffin tin potentials}
Here we present some simple examples where one can see the
behavior of surface states directly. We will deal with
$\Z^2$-periodic muffin tin potentials of infinite height (or
depth) on the plane $\R^2$ which can be specified by fixing a
radius $0 < r < 1/2$ for the discs where the potential vanishes,
and the center $P_0 = (x_0, y_0) \in [0,1)^2$ for the generic
disc. In other words, we consider the periodic sets
$$
\Omega_{r,P_0} := \cup_{(i,j) \in \Z^2} B_r(P_0 + (i,j)),
$$
and we let $V = V_{r,P_0}$ be zero on $\Omega_{r,P_0}$ while we
assume that $V$ is infinite on $\R^2 \setminus \Omega_{r,P_0}$. If
$H_{ij}$ is the Dirichlet Laplacian of the disc $B_r(P_0 +
(i,j))$, then the form-sum of $-\Delta$ and $V_{r,P_0}$ is
$\oplus_{(i,j)\in \Z^2} H_{ij}$. Without loss of generality, we
may assume $y_0 = 0$ henceforth.\\[.25cm]
{\bf (1) Dislocation in the $x$-direction.} Here muffin tin
potentials yield an illustration for some of the phenomena
encountered in Section 4. In the simplest case we would take $x_0
= 1/2$ so that the disks $B_r(1/2 + i, j)$, for $i \in \N_0$ and
$j \in \Z$, will not intersect or touch the interface $\{(x,y)
\mid x=0\}$. Defining the dislocation potential $W_t$ as in
Section 4, we see that there are bulk states given by the
Dirichlet eigenvalues of all the discs that do not meet the
interface, and there may be surface states given as the Dirichlet
eigenvalues of the sets $B_r(1/2 - t, j) \cap \{ x < 0\}$ for $j
\in \Z$ and $1/2 - r < t < 1/2 + r $.
More precisely, let $\mu_k = \mu_k(r)$ denote the Dirichlet
eigenvalues of the Laplacian on the disc of radius $r$, ordered by
min-max and repeated according to their respective multiplicities.
The Dirichlet eigenvalues of the domains $B_r(1/2 - t, 0) \cap \{x
< 0\}$, $1/2 - r < t < 1/2 + r$, are denoted as $\lambda_k(t) =
\lambda_k(t,r)$; they are continuous, monotonically decreasing
functions of $t$ and converge to $\mu_k$ as $t \uparrow 1/2 + r$
and to $+\infty$ as $t \downarrow 1/2 - r$. In this simple model,
the eigenvalues $\mu_k$ correspond to the bands of a periodic
operator. We see that the gaps are crossed by surface states as
$t$ increases from $0$ to $1$, in accordance with the results of
Section 4 (Corollary 4.5).
Along the same lines, one can easily analyze examples where $x_0$
is different from $1/2$; here more complicated geometric shapes
may come into play. In [HK] we will again use muffin tin
potentials as examples for the rotation problem. In that paper, we
will also discuss approximations by muffin tin potentials of
height $n$ and their limit as $n \to \infty$.\\[.25cm]
{\bf (2) Dislocation in the $y$-direction.} This problem has not
been considered so far. We include a brief discussion of this case
for two reasons: on the one side, we observe a new phenomenon
which did not appear so far; on the other hand, one can see from
our example that, presumably, there is no general theorem for
translation of the left half-plane in the $y$-direction.
Let $V = V_r$ denote the muffin tin potential defined above, with $x_0 = y_0 = 0$.
We then let ${\tilde W}_t$ coincide with
$V$ in the right half-plane, while we take ${\tilde W}_t(x,y)
= V(x, y - t)$ in the left half-plane. At the interface $\{x=0\}$ we see
half-discs on the left and on the right with the half-discs on
the right being fixed while the half-discs on the left are shifted
by $t$ in the $y$-direction. The surface states correspond to the
states of the Dirichlet Laplacian on the union
$\Omega_{t,r;\text{surf}}$
of these half-discs. There are two cases: either $\Omega_{t,r;\text{surf}}$
is connected and we have a scattering channel along the interface,
or $\Omega_{t,r;\text{surf}}$ is the disjoint union of a sequence of
bounded domains; cf.\ Figure 2. In the second case, the eigenvalues on such
domains start at the Dirichlet eigenvalues of the disc of radius $r$,
increase up to the corresponding eigenvalues of a half-disc, and then
move down again to where they started. For $1/4 < r < 1/2$, the picture
is more complicated: If we let $\tau_0 = 1 - 2r$, $\tau_1 = 2r$, we find
that the sets $\Omega_{t,r;\text{surf}}$ are disconnected for
$0\le t \le \tau_0$ and for $\tau_1 \le t \le 1$; for $\tau_0 < t < \tau_1$,
however, $\Omega_{t,r;\text{surf}}$ is connected and forms a
periodic wave guide with purely a.c.\ spectrum [SW]; cf.\ also [DS].
We therefore observe a dramatic change in the spectrum
of the dislocation operators: for $t \in [0,\tau_0] \cup [\tau_1,1]$ the surface states
in the gap are given by eigenvalues of infinite multiplicity
while for $t \in (\tau_0, \tau_1)$ the surface states form bands of a.c.\ spectrum in the gaps.
\begin{figure}[H]
\begin{center}
\includegraphics[width=5.2cm]{muffintin}
\includegraphics[width=5.2cm]{muffintintwo}
\caption{Muffin tins: two cases for dislocation in the
$y$-direction.}
\end{center}
\end{figure}
Note that, if we had chosen $x_0 = 1/2$, then nothing at all
would have happened for translation in the $y$-direction.
(The authors thank A.~Ruschhaupt, Hannover, for asking about translation
in the $y$-direction.)
\section{Appendix: continuity and regularity of eigenvalues}
In this appendix, we discuss several basic facts concerning
continuity and regularity of the eigenvalue branches for the
one-dimensional dislocation problem. We first consider potentials
$V$ from the class $\P \subset L_{1,\text{loc,unif}}(\R)$ as in
(2.1) where the eigenvalues are continuous functions of the
dislocation parameter $t$. In the subsequent estimates we will use
$$
\norm{V}_{1,\text{loc,unif}} := \sup_{y \in \R} \int_y^{y+1}
|V(x)| \d x
\eqno{(A.1)}
$$
as a natural norm on $L_{1,\text{loc,unif}}(\R)$. As is well known
(cf., e.g., [CFrKS]), any potential $V \in
L_{1,\text{loc,unif}}(\R)$ is relatively form-bounded with respect
to $h_0$ with relative form-bound zero. More precisely, we have
the following lemma.\\[.25cm]
{\bf A.1.~Lemma.} {\it For any $\eps > 0$ there exists a constant
$C_\eps \ge 0$ such that for any $V \in L_{1,\text{\rm
loc,unif}}(\R)$ we have
$$
\int_\R |V| \, |\phi|^2 \d x
\le \norm{V}_{1,\text{\rm loc,unif}}\,\left(\eps \norm{\phi'}^2 +
C_\eps \norm{\phi}^2 \right), \qquad \phi \in \mathcal H^1(\R).
\eqno{(A.2)}
$$
}
\vspace{-.5cm} \proof For $f\in\Ccinf(\R)$ with support contained
in $(0,\eps)$ we have $\infnorm{f}\leq\sqrt\eps\norm{f'}$. Let
$(\zeta_n)_{n\in\N}$ denote a (locally finite) partition of unity
on the real line with the properties:
$\supp\zeta_1\subset(0,\eps)$, each $\zeta_n$ is a translate of
$\zeta_1$, $M:=\sup_{x\in\R}\sum_{n\in\N}|\zeta_{n}'(x)|^2$ is
finite and $\sum_{n\in\N}\zeta_{n}^2(x)=1$ for all $x\in\R$. By
the IMS localization formula (see [CFrKS]), we have for any
$\varphi\in\Ccinf(\R)$,
$$\norm{\varphi'}^2=\ska{-\varphi''}{\varphi}=\sum_{n=1}^{\infty}\norm{(\zeta_n\varphi)'}^2
-\sum_{n=1}^{\infty}\norm{\zeta_n'\varphi}^2\geq\sum_{n=1}^{\infty}\norm{(\zeta_n\varphi)'}^2-M\norm{\varphi}^2,$$
so that
\begin{align}
\int|V(x)||\varphi(x)|^2\dx
& \le\sum_{n=1}^{\infty}\infnorm{\zeta_n\varphi}^2\int_{\supp\zeta_n}|V(x)|\dx \nonumber\\
& \leq \eps\,\left(\norm{\varphi'}^2+M\norm{\varphi}^2\right)
\,\norm{V}_{1,\text{\rm loc,unif}}. \nonumber
\end{align}
The general case follows by approximation and Fatou's lemma.
\endproof
For $V \in \P$, the function
$$
\vartheta_V(s) := \int_0^1 |V(x + s) - V(x)| \d x, \qquad 0\le s \le 1,
\eqno{(A.3)}
$$
is continuous and $\theta_V(s) \to 0$, as $s \to 0$. Furthermore,
for $W_t$ is as in eqn.~(1.1), we have $\norm{W_t - W_{t'}}_{1,
\text{loc,unif}} = \theta_V(t - t')$. This leads to the following
lemma.
\vskip1em\noindent {\bf A.2.~Lemma.} {\it Let $V \in \P$, $E_0 \in
\R \setminus \sigma(H_{t_0})$, and write $\eps_0 := \text{\rm
dist}(E_0, \sigma(H_{t_0}))$. Then there is $\tau_0 > 0$ such that
$H_t$ has no spectrum in $(E_0-\eps_0/2, E_0 + \eps_0/2)$ for $|t
- t_0| < \tau_0$. Furthermore, there exists a constant $C \ge 0$
such that for some $\tau_1 \in
(0,\tau_0)$
$$
\norm{(H_t - E_0)^{-1} - (H_{t_0} - E_0)^{-1}} \le C \theta_V(t - t_0),
\qquad |t - t_0| < \tau_1.
\eqno{(A.4)}
$$
}
\vspace{-.5cm}\proof Without loss of generality we may assume that
$V \ge 1$. Let ${\bold h}_t$ denote the quadratic form associated
with $H_t$. Applying Lemma A.1 (with $\eps := 1$) we see that
$$
\left|{\bold h}_{t_0}[u] - {\bold h}_t[u] \right|
\le \int_\R |W_t - W_{t_0}| \, |u|^2 \d x
\le C_1 \theta_V(t - t_0) {\bold h}_{t_0} [u], \qquad
u \in {\mathcal H}^1(\R),
$$
with some constant $C_1$. The desired result now follows by [K;
Thm.~VI-3.9].
\endproof
\vskip.25em We therefore see that $H_{t_n} \to H_{t_0}$ in the
sense of norm resolvent convergence if $t_0 \in [0,1]$,
$(t_n)_{n\in\N} \subset [0,1]$ and $t_n \to t_0$. By standard
arguments, this implies that the discrete eigenvalues of $H_t$
depend continuously on $t$. We are now prepared for the proof of
Lemma 2.1.
\vskip1em
{\it Proof of Lemma 2.1.} We consider $t \in \T$, the flat
one-dimensional torus, and we denote the spectral gap by $(a,b)$.
Let $[a',b']\subset(a,b)$.
{(1)} Let $(\eta,\tau) \in (a,b) \times \T$. Since $\sigma(H_\tau)
\cap (a,b)$ is a discrete set, and since $\sigma(H_t)$ depends
continuously on $t$, there is a neighborhood $U_{\eta,\tau}
\subset (a,b) \times \T$ of $(\eta,\tau)$ of the form
$U_{\eta,\tau} = (\eta_1, \eta_2) \times (\tau_1, \tau_2) $
belonging to either of the two following types:
\vskip.5ex {\bf Type (1):} For $\tau_1 < t < \tau_2$ we have
$\sigma(H_t) \cap (\eta_1, \eta_2) = \emptyset$, \vskip.5ex
or
\vskip.5ex {\bf Type (2):} $\eta$ is an eigenvalue of $H_\tau$ and
there is a continuous function $f \colon (\tau_1, \tau_2) \to
(\eta_1, \eta_2)$ such that $f(t)$ is an eigenvalue of $H_t$;
$H_t$ has no further eigenvalues in $(\eta_1,\eta_2)$, for $\tau_1
< t < \tau_2$.
\vskip.5ex
Now the family $\{ U_{\eta,\tau} \mid (\eta,\tau) \in (a,b) \times
\T\}$ is an open cover of $(a,b) \times \T$ and there exists a
finite selection $\{ U_{\eta_i,\tau_i} \}_{i=1,\ldots,N}$ such
that
$$
[a',b'] \times \T \subset \cup_{i=1}^N U_{\eta_i,\tau_i}.
$$
As a first consequence, we see that there is at most a finite
number of functions that describe the spectrum of $H_t$ in the
open set $ \cup_{i=1}^N U_{\eta_i,\tau_i} \supset [a',b'] \times
\T$.
{(2)} Suppose that $(\eta,\tau) \in (a,b) \times \T$ is such that
$\eta \in \sigma(H_\tau)$ and let $f \colon (\tau_1,\tau_2) \to
(\eta_1,\eta_2)$ as above. Consider a sequence $(t_j)_{j\in\N}
\subset (\tau_1,\tau_2)$ with $t_j \to \tau_1$. We can find a
subsequence $(t_{j_k})_{k\in\N}$ such that $f(t_{j_k}) \to {\bar
\eta}$ for some ${\bar\eta} \in [\eta_1,\eta_2]$. It is easy to
see that ${\bar\eta} \in \sigma(H_{\tau_1})$.
If ${\bar\eta} \in (a,b)$
the point $({\bar\eta},\tau_1)$ has a neighborhood
$U_{{\bar\eta},\tau_1}$ of type $(2)$ and we can extend the
domain of definition of $f$ beyond $\tau_1$. It follows that there
exist a maximal open interval $(\alpha,\beta) \subset (0,1)$ and
a (unique) continuous extension ${\tilde f} \colon (\alpha,\beta)
\to (a,b)$ of $f$ such that ${\tilde f}(t)$ is an eigenvalue of
$H_t$ for all $t \in (\alpha,\beta)$.
{(3)} It remains to show that $\tilde f(t)$ converges to a band
edge as $t \downarrow \alpha$ and as $t \uparrow \beta$. By the
same argument as above, we find that any sequence $(t_j)_{j\in\N}
\subset (\alpha,\beta)$ satisfying $t_j \to \alpha$ has a
subsequence $(t_{j_k})_{k\in\N}$ such that $\tilde f(t_{j_k}) \to
{\bar\eta}$ for some ${\bar\eta} \in [a,b]$. Here ${\bar\eta}
\notin (a,b)$ because otherwise we could again extend the domain
of definition of ${\tilde f}$ beyond $\alpha$, contradicting the
maximality property of the interval
$(\alpha,\beta)$.
Suppose there are sequences $(t_j)_{j\in\N}, (s_j)_{j\in\N}
\subset (\alpha,\beta)$ such that $t_j \to \alpha$ and $s_j \to
\alpha$ and ${\tilde f}(t_j) \to a$ while ${\tilde f}(s_j) \to b$
as $j \to \infty$. Then for any $\eta' \in (a,b)$ there is a
sequence $(r_j)_{j\in\N} \subset (\alpha,\beta)$ such that $r_j
\to \alpha$ and ${\tilde f}(r_j) \to \eta'$, whence $\eta' \in
\sigma(H_\alpha)$. This would imply that $(a,b) \subset
\sigma(H_\alpha)$, which is impossible. \hfill$\square$
\vskip1.0em
We next turn our attention to the question of Lipschitz-continuity
of the functions $f_j$ in Lemma 2.1. With $\theta_V \colon [0,1]
\to [0,\infty)$ as in (A.3), we study potentials from the classes
$$
\PP_\alpha : = \{ V \in \PP \mid \exists C \ge 0 \colon
\theta_V(s) \le C
s^\alpha, \forall 0< s \le 1\} ,
\eqno{(A.5)}
$$
where $0 < \alpha \le 1$. The class $\PP_\alpha$ consists of all
periodic functions $V \in \P$ which are (locally)
$\alpha$-H\"older-continuous in the $L_1$-mean; for $\alpha = 1$
this is a Lipschitz-condition in the $L_1$-mean. The class $\PP_1$
is of particular practical importance since it contains the
periodic step functions. It will be shown below that $\P_1$
coincides with the class of periodic functions on the real line
which are locally of bounded variation. We first prove
Lipschitz-continuity of the eigenvalues of $H_t$ for $V \in \P_1$.
\vskip1em\noindent {\bf A.3.~Proposition.} {\it For $V \in \P_1$,
let $(a,b)$ denote any of the gaps $\Gamma_k$ of $H$ and let $f_j
\colon (\alpha_j, \beta_j) \to (a,b)$ be as in Lemma 2.1. Then the
functions $f_j$ are uniformly Lipschitz-continuous. More
precisely, for each gap $\Gamma_k$ there exists a constant $C_k\ge
0$ such that for all $j$
$$
|f_j(t) - f_j(t')| \le C_k |t - t'|, \qquad \alpha_j \le t, t' \le \beta_j.
$$
}
\vspace{-.5cm} \proof As in the proof of Lemma 2.6 we can find a
finite number of levels $E_1, \ldots, E_\ell \in (a,b)$ and a
partition $0=\tau_0 < \tau_1 < \ldots < \tau_{\ell-1} < \tau_\ell
= 1$ such that $E_j \notin \sigma(H_t)$ for all $t \in I_j :=
[\tau_{j-1}, \tau_j]$ and for $j=1, \ldots, \ell$. Now $V \in
\P_1$ implies $\norm{W_t - W_{t'}}_{1,\text{loc,unif}} =
\theta_V(t-t') \le C |t-t'|$ and we conclude with the aid of Lemma
A.2 that there are constants $c_1, \ldots, c_\ell \ge 0$ such that
$$
\norm{(H_t - E_j)^{-1} - (H_{t'} - E_j)^{-1}} \le c_j |t-t'|,
\qquad t, t' \in I_j.
$$
This implies that the min-max-values $\mu_k(s)$ of $(H_s -
E_j)^{-1}$ satisfy
$$
|\mu_k(t) - \mu_k(t')| \le c_j |t - t'|,
\qquad t, t' \in I_j.
$$
By the spectral mapping theorem, the eigenvalues of $H_t$ in
$(E_j, b)$ are in bijection with the eigenvalues of $(H_t -
E_j)^{-1}$ in $(\frac{1}{b-E_j},\infty)$. We now let $C :=
\max\{c_1, \ldots, c_\ell\}$ to finish our proof.
\endproof
\vskip.5em\noindent
\noindent{\bf A.4.~Remarks.}
(a) By the same argument, we obtain the following result on
H\"older-continuity: If $0<\alpha <1$ and $V \in \P_\alpha$, then
each of the functions $f_j \colon (\alpha_j,\beta_j) \to (a,b)$ is
locally uniformly H\"older-continuous (as defined in [GT]), i.e.,
for any compact subset $[\alpha_j', \beta_j'] \subset (\alpha_j,
\beta_j)$ there is a constant $C = C(j, \alpha_j', \beta_j')$ such
that $|f_j(t) - f_j(t')| \le C |t - t'|^\alpha$, for all $t, t'
\in [\alpha_j', \beta_j']$. Note that our method does not
necessarily yield a uniform constant for the whole interval
$(\alpha_j, \beta_j)$, much less a constant that would be uniform
for all $j$.
\vskip.5ex
(b) For analytic potentials $V$, it is shown in [K1] that the
eigenvalue branches $f_j$ in Lemma 2.1 depend analytically on $t$.
This is a simple consequence of the fact that, for real analytic
$V$, the $H_t$ form a holomorphic family of self-adjoint operators
in the sense of Kato. In [K2], the author proves that the $f_j$
are squares of $W^1_2$-functions near the gap edges if the
potential is in $L_2(\T)$.
\vskip1em
We finally give a characterization of the class $\P_1$.
\vskip1em\noindent {\bf A.5.~Proposition.} {\it Let $BV_{\text{\rm
loc}}(\R)$ denote the space of real-valued functions which are of
bounded variation over any compact subset of the real line.
Then $\P_1 = \P \cap BV_{\text{\rm loc}}(\R)$.
}
\vskip1em
It is easy to see that any $V \in \P \cap BV_{\text{\rm loc}}(\R)$
belongs to $\P_1$: certainly, any $V \in \P$ which is monotonic
over $[0,1]$ is an element of $\P_1$ and any function of bounded
variation can be written as the difference of two monotonic
functions.
The converse direction is established by the following result due
to J.\ Voigt, Dresden; cf.\ also [EG, Chapter 5] for related
material on $BV$-functions of several variables.
\vskip1em\noindent {\bf A.6.~Lemma.} {\it Let $f \in
L_{1,\text{\rm loc}}\,(\R,\R)$ be periodic with period $1$ and
suppose that there are $c \ge 0$, $\eps > 0$ such that
$$
\int_0^1 |f(x+t) - f(x)| \d x \le c t, \qquad \forall 0 < t < \eps.
\eqno{(A.6)}
$$
Consider $f$ as a function in $L_1(\T)$, with $\T$ denoting the
one-dimensional torus.
We then have: the distributional derivative $\partial f$ is a
(signed) Borel-measure $\mu$ on $\T$ and there is a number $a \in
\R$ such that $f(x) = a + \mu([0,x))$, a.e. in $[0,1) \simeq \T$.
In particular, $f$ has a left-continuous representative of bounded
variation. } \vskip1em
\noindent{\it Proof.} Defining $\eta \colon C^1(\T) \to \R$ by
$$\eta(\phi) := - \int_0^1 \phi' f \d x,$$ we may compute
\begin{align}
- \int_0^1 \phi' f \d x
& = \lim_{t \to 0} \int_0^1 \frac{1}{t} (\phi(x-t) - \phi(x)) f(x) \d x
\nonumber\\
& = \lim_{t \to 0} \int_0^1 \phi(x) \frac{1}{t} (f(x+t) -
f(x)) \d x,
\nonumber
\end{align}
and the assumption (A.6) yields the estimate $|\eta(\phi)| \le c
\norm{\phi}_\infty$. Since $C^1(\T)$ is dense in $C(\T)$, the
functional $\eta$ has a unique continuous extension to all of
$C(\T)$; we denote the extension by the same symbol $\eta$. By the
Riesz representation theorem there is a measure $\mu$ such that
$\eta(\phi) = \int \phi \d \mu$ for all $\phi \in C(\T)$.
Furthermore, for $\phi \in C^1(\T)$ we have $ - \int_0^1 \phi' f
\d x= \int_0^1 \phi \d \mu$, and we see that $\mu = \partial f$ on
$\T$ in the distributional sense.
The choice $\phi := 1$ yields $\int_\T \d \mu = - \int_0^1 \phi' f
\d x = 0$ and the function ${\tilde f}(x) := \mu([0,x))$ satisfies
$\partial {\tilde f} = \mu$. This is easy to check: for $\phi \in
C^1(\T)$ we have
\begin{align}
\int {\tilde f} \phi' \d x
& = \int_0^1 \int_{0 \le y < x} \d \mu(y) \phi'(x) \d x
\nonumber\\
& = \int_{0 \le y < 1} \int_y^1 \phi'(x) \d x \d \mu(y)
= - \int_{[0,1)} \phi(y) \d \mu(y).
\nonumber
\end{align}
We therefore see that $\partial (f - {\tilde f}) = 0$; hence there
is some $a$ such that $f - {\tilde f} = a$. \hfill$\square$
|
{
"timestamp": "2011-05-04T02:01:31",
"yymm": "1009",
"arxiv_id": "1009.3581",
"language": "en",
"url": "https://arxiv.org/abs/1009.3581"
}
|
\section{Introduction}\label{intro}
Amongst approaches for scattered data approximation on the sphere,
the hybrid interpolation scheme of von Golitschek \& Light \cite{GL}
and Sloan \& Sommariva \cite{SS06},
which employs both radial basis functions and spherical polynomials,
seems an attractive
method, especially when the data is concentrated in some regions (such as over mountain ranges and trenches on the Earth's surface), yet relatively sparse in other regions.
The underlying idea is that radial basis functions can give good approximation for rapidly
varying data over short distances, whereas the polynomial component can more effectively
represent smooth variations on a global scale. The radial basis functions are centered
at data points which are supposed given, and the linear combination of radial basis
functions is constrained to be orthogonal, in a natural sense, to the finite dimensional
space of polynomials.
However, the hybrid scheme poses difficulties in implementation,
compared with a pure radial basis function approximation, when the
number of centers is large. In the case of a pure radial basis function
approximation with a (strictly) positive definite kernel, the resulting
linear system has a matrix that is positive definite, allowing an
iterative solution by the conjugate gradient method, and
preconditioning by, for example, the additive Schwarz method,
see \cite{GST06}. For the hybrid scheme, in contrast, the
linear equations
for the relevant expansion
coefficients have the saddle-point form, see \cite{SS06},
\begin{equation}\label{saddle_general}
\left[\begin{array}{cc}A &Q\\Q^T & 0\end{array}\right]
\left[\begin{array}{c}{{\mathbb \alpha}}\\ {{\mathbb \beta}}\end{array}\right] =
\left[\begin{array}{c}{{\bf f}}_X\\ {\bf{0}} \end{array}\right],\end{equation}
where $A\in\mathbb{R}^{N\times N}$ is a positive definite matrix arising from the radial basis function part of the function approximation, and $Q\in \mathbb{R}^{N\times M}$ is a matrix of
spherical harmonics evaluated at the data centers, with $M\le N$. (The matrices $A$ and $Q$ are defined properly in Section 2).
The saddle-point structure means that the overall matrix is not positive
definite, and that the conjugate gradient method is no longer a suitable
iterative solver. More fundamentally, because the matrix has both
positive and negative eigenvalues, the problem of constructing a good
preconditioner becomes more delicate. For a thorough review of strategies
and challenges for the numerical solution of saddle-point problems,
see \cite{BenziGolubLiesen,ESW05}.
In this manuscript we concentrate on the stability of the saddle-point
formulation
of this hybrid scheme,
and devise and validate a rapid preconditioned iterative solution method for
the solution of the equations from the approximation. We make use of the Brezzi stability and convergence theorem well known
in the context of mixed finite elements, along with the new inf-sup
condition of \cite{SW08} to establish convergence of the approximation
scheme; and then use the inf-sup condition to obtain an optimal
preconditioner.
A leading contender amongst the solution methods for equations with the
structure \eqref{saddle_general} (see \cite{BenziGolubLiesen}) is the block
preconditioning method of \cite{MurGolWat00} employing an approximation
to the Schur complement
\[
S:=Q^TA^{-1}Q.
\]
The use of
Schur complement approximations in preconditioners is by now well established (see for
example \cite {ESW05}) in the setting of mixed finite element methods. In particular, Verf\"urth \cite{Verf} showed that for the mixed finite element approximation of the Stokes flow problem the Schur complement is spectrally equivalent to the identity operator
(or to the mass matrix or $L_2$ projection matrix in the finite element setting),
making this a suitable approximation to the Schur complement. Verf\"urth's proof makes
essential use of the Babuska-Brezzi or inf-sup condition (Assumption 2.1 in \cite{Verf}),
see \cite{Brezzi74,Brezzi}.
In the present setting, the Schur complement turns out to be spectrally equivalent not to the identity operator/matrix, but rather to a specific diagonal but non-constant matrix. This spectral equivalence, a main result of the paper, is stated in Theorem \ref{spectral_equiv}.
The key ingredient here, in analogy with the known inf-sup condition for the Stokes flow problem, is the inf-sup stability condition recently established by
Sloan and Wendland \cite{SW08} for the hybrid approximation problem.
An approximate solver for the primal operator
(the radial basis function interpolation matrix in this case) is
also required. This could be provided, for example, by the domain decomposition
method of Le Gia, Sloan and Tran \cite{GST06}, or by any other preconditioner
for the pure radial basis function problem. The resulting block
diagonal preconditioner is symmetric and positive definite, hence
the preconditioned MINRES method (\cite{PS75},\cite{ESW05}) is
applicable to the full problem (which is symmetric but not positive
definite).
In Section 2 we formulate the hybrid approximation scheme and establish
notation. Then in Section 3 we describe the inf-sup condition of \cite{SW08},
and use the Brezzi theorem to establish stability and convergence of the
scheme. In Section 4 we turn to preconditioning, establishing there the
main spectral
equivalence result. In Section 5 numerical results are presented (using the
primal preconditioner of \cite{GST06}) and we conclude in Section 6.
\section{Problem formulation}\label{form}
Let $X=X_N=\{{\bf x}_1,{\bf x}_2,\ldots,{\bf x}_N\}$ be a set of $N$ distinct
points on the sphere ${\mathbb S}^d$ in $ {\mathbb R}^{d+1}$. Using these as centers we define
a radial basis function approximation space
$${\mathcal X}_{X_N}={\mathcal X}_N :=\{\sum_{i=1}^N \alpha_i \phi(\cdot,{\bf x}_i) : \alpha_1,\ldots,\alpha_N \in {\mathbb R}\}$$
with a suitable kernel function $\phi$.
The kernel is assumed to be (strictly) positive definite, that is
$$\sum_{i=1}^N\sum_{j=1}^N\alpha_i\phi(\ensuremath{\mathbf{x}}_i,\ensuremath{\mathbf{x}}_j)\alpha_j\ge 0$$
for every set of points $X_N=\{\ensuremath{\mathbf{x}}_1,\ldots,\ensuremath{\mathbf{x}}_N\}\in {\mathbb S}^d$ and for all $N\in {\mathbb N}$, with equality for distinct points $\ensuremath{\mathbf{x}}_j$ only if $\alpha_1=\alpha_2= \ldots =\alpha_N=0.$
The native space $\mathcal{N}_{\phi}$ is defined as the completion under the inner product
\begin{equation}\label{IP}\langle \sum_{i} \alpha_i \phi(\cdot,{\bf x}_i),
\sum_{j} \alpha_j^{\prime} \phi(\cdot,{\bf x}_j)\rangle_{\phi} =
\sum_{i}\sum_{j}\alpha_i \alpha_j^{\prime}\phi({\bf x}_i,{\bf x}_j)\end{equation}
of the linear space
$$F_{\phi} := \{\sum_{j=1}^N \alpha_j\phi(\cdot,{\bf x}_j),
\alpha_j\in {\mathbb R}, {\bf x}_j\in{\mathbb S}^d, j=1,\ldots,N,N\in {\mathbb N}\},$$
where we insist that the points ${\bf x}_j$ are distinct.
The norm is as usual defined by
$$\|\cdot\|_{\phi} = \langle \cdot,\cdot\rangle_{\phi}^{1/2} .$$
It is well-known that $\mathcal{N}_{\phi}$ is a reproducing kernel Hilbert space
(see \cite{Ar}) with the reproducing kernel $\phi(\cdot,\cdot)$. That is
\begin{eqnarray*}
\phi({\bf x},{\bf y}) &=& \phi({\bf y},{\bf x}),\quad {\bf x},{\bf y}\in{\mathbb S}^d\\
\phi(\cdot,{\bf y}) &\in& \mathcal{N}_{\phi}, \quad {\bf y}\in{\mathbb S}^d\end{eqnarray*}
and for $f\in \mathcal{N}_{\phi}$
\begin{equation}\label{rep}\langle f,\phi(\cdot,{\bf y})\rangle_{\phi} = f({\bf y}), \quad {\bf y}\in{\mathbb S}^d .\end{equation}
The kernel function $\phi$ needs to be positive definite in
order that the inner product $\langle\cdot,\cdot\rangle_{\phi}$ satisfy the
positivity axiom for an inner product. Equivalently,
the matrices $A_{X}$ defined by
\begin{equation}\label{AXphi}
\left( A_{X}\right)_{i,j} := \phi({\bf x}_i,{\bf x}_j) ,\quad i,j = 1,\ldots,N
\end{equation}
are positive definite as well as symmetric for every $X$ and every $N\in {\mathbb N}$.
Taking now a fixed $N\in {\mathbb N}$ and a fixed set $X_N \subset {\mathbb S}^d$, we may define the usual radial basis function interpolant to a continuous function $f$ on ${\mathbb S}^d$ by
$$f_N({\bf x}) = \sum_{j=1}^N \alpha_j \phi({\bf x},{\bf x}_j), $$
where $\alpha_1,\ldots,\alpha_N$ are such that
$$f_N({\bf x}_i) = f({\bf x}_i),\quad i= 1,\ldots,N ,$$
which we may write as
$$\sum_{j=1}^N\phi(\ensuremath{\mathbf{x}}_i,\ensuremath{\mathbf{x}}_j)\alpha_j=f(\ensuremath{\mathbf{x}}_i) \textrm{ for }i= 1,\ldots,N. $$
That is, the vector
${\mathbb \alpha} = (\alpha_1,\ldots,\alpha_N)^T$ of coefficients satisfies
\begin{equation}\label{radb}
A_{X}{\mathbb \alpha} = {\bf f}_X,\end{equation}
where
\begin{equation}\label{fdef}{\bf f}_X :=(f({\bf x}_1),\ldots,f({\bf x}_N))^T.\end{equation}
The hybrid approximation scheme of von Golitschek \& Light \cite{GL} and
Sloan \& Sommariva, see \cite{SW08}, employs not only the radial basis
functions, but also
spherical polynomials of total degree up to some conveniently chosen $L\ge 0.$ We define
$${\mathcal P}_L = \hbox{span}\{Y_{\ell,k}:\, k=1,\ldots,M(d,\ell),\,\ell=0,\ldots,L\},$$
where $Y_{\ell,k}$ is a spherical harmonic of degree $\ell$, that is, the restriction to ${\mathbb S}^d$ of a homogeneous harmonic
polynomial in $ {\mathbb R}^{d+1}$ of degree $\ell$, and $M(d,\ell)$ is the dimension of the space spanned by the spherical harmonics of degree $\ell$. Then ${\mathcal P}_L$ is the set of spherical polynomials of degree $\le L$.
We shall assume that $\{Y_{\ell,k}:\, k=1,\ldots,M(d,\ell),\,\ell=0,1,\ldots\}$ is an orthonormal set with respect to the usual $L_2$ inner product, that is
\[ \int_{{\mathbb S}^d}Y_{\ell,k}({\bf x})Y_{\ell',k'}({\bf x})d\omega({\bf x})=\delta_{\ell,\ell'}\delta_{k,k'},\]
where $d\omega({\bf x})$ denotes surface measure on ${\mathbb S}^d$.
Then it is well known that
$\{Y_{\ell,k}:\, k=1,\ldots,M(d,\ell),\,\ell=0,1,\ldots\}$ is a complete orthonormal basis for
$L_2({\mathbb S}^d)$.
For a given function $f$, the hybrid approximation scheme
is then to find
\begin{equation}\label{unl}
u_{N,L}({\bf x})= \sum_{j=1}^N \alpha_j \phi({\bf x},{\bf x}_j) \in{\mathcal X}_N
\end{equation}
and
\begin{equation}\label{pnl}
p_{N ,L}({\bf x})= \sum_{\ell=0}^L \sum_{k=1}^{M(d,\ell)} \beta_{l,k} Y_{\ell,k}({\bf x})\in{\mathcal P}_L
\end{equation}
such that
\begin{equation}\label{interp}
u_{N,L}(\ensuremath{\mathbf{x}}_i)+p_{N,L}(\ensuremath{\mathbf{x}}_i)=f(\ensuremath{\mathbf{x}}_i),
\end{equation}
or equivalently,
\begin{equation}\label{1st}
\sum_{j=1}^N \alpha_j \phi({\bf x}_i,{\bf x}_j) +
\sum_{\ell=0}^L \sum_{k=1}^{M(d,\ell)} \beta_{\ell,k} Y_{\ell,k}({\bf x}_i) = f({\bf x}_i),
\quad i= 1,\ldots, N ,\end{equation}
which is to be solved subject to the side condition
\begin{equation}\label{2nd}
\sum_{j=1}^N \alpha_j q({\bf x}_j) = 0\quad \forall q\in{\mathcal P}_L.\end{equation}
The condition (\ref{2nd}) is equivalent, via \eqref{rep}, to $\langle q, u_{N,L}\rangle_\phi=0$ for all $q\in{\mathcal P}_L$, forcing the radial basis function component to be $\mathcal{N}_\phi$-orthogonal to ${\mathcal P}_L$. It also ensures that the defining linear system is square and symmetric.
The conditions (\ref{1st}),(\ref{2nd}) can also be seen to be those which
derive from the solution of the constrained optimization problem
$$
\min_{u_{N,L}\in {\mathcal X}_N}\;\,\frac{1}{2}\| u_{N,L} - f\|_\phi^2\;\; \mbox{subject to} \;\;
\langle Y_{\ell,k},u_{N,L}\rangle_\phi=0$$
for all $\ell=0,\ldots,L , k=1,\ldots,{M(d,\ell)}$, the coefficients $\beta_{\ell,k}$ being the
Lagrange multipliers in the Lagrangian
\begin{eqnarray*}
{\mathcal L} &=&\frac{1}{2}\| u_{N,L} - f\|_\phi^2 +
\sum_{\ell=0}^L \sum_{k=1}^{M(d,\ell)} \beta_{\ell,k}\langle Y_{\ell,k},u_{N,L}\rangle_\phi\\
&=& \frac{1}{2}\| u_{N,L} - f\|_\phi^2 + \langle p_{N ,L},u_{N,L}\rangle_\phi .
\end{eqnarray*}
This is therefore another way of expressing the hybrid approximation problem.
By choosing the spherical harmonic functions as the basis for ${\mathcal P}_L$ in (\ref{2nd}),
we can write
(\ref{1st}),(\ref{2nd}) as a so-called `saddle-point' linear system of equations
\begin{equation}\label{saddle}
\left[\begin{array}{cc}A_{X} &Q_{X,L}\\Q_{X,L}^T & 0\end{array}\right]
\left[\begin{array}{c}{{\mathbb \alpha}}\\ {{\mathbb \beta}}\end{array}\right] =
\left[\begin{array}{c}{{\bf f}}_X\\ {\bf{0}} \end{array}\right],\end{equation}
where $A_{X}$ is defined by (\ref{AXphi}),
${\mathbb \alpha}$
is the vector of coefficients $\alpha_j$ as defined above,
${\mathbb \beta}$ is a vector containing the
coefficients $\beta_{\ell,k}$ for $k=1,\ldots,M(d,\ell),\,\ell=0,\ldots,L$,
and $Q_{X,L}$ is the $N \times M$ matrix defined by
\begin{equation}\label{Qdef}(Q_{X,L})_{i,\ell k} := Y_{\ell,k}({\bf x}_i) ,
\quad i = 1,\ldots,N, \; k=1,\ldots,M(d,\ell), \; \ell=0,\ldots,L,
\end{equation}
and
$$M := \sum_{\ell=0} M(d,\ell) = \mbox{dim }({\mathcal P}_L).$$
In the present application we need to prescribe more precisely the nature of the kernel $\phi(\ensuremath{\mathbf{x}},\ensuremath{\mathbf{y}})$. In the first place we shall assume that it is zonal, meaning that
$$ \phi(\ensuremath{\mathbf{x}},\ensuremath{\mathbf{y}})=\Phi(\ensuremath{\mathbf{x}}\cdot\ensuremath{\mathbf{y}})$$
for some function $\Phi\in C[-1,1]$, where $\ensuremath{\mathbf{x}}\cdot\ensuremath{\mathbf{y}}$ denotes the Euclidean inner product in $\ensuremath{\mathbb{R}}^{d+1}$. More precisely, we shall assume that $\phi(\ensuremath{\mathbf{x}},\ensuremath{\mathbf{y}})$ has an expansion of the form
\begin{equation}\label{ass2} \phi(\ensuremath{\mathbf{x}},\ensuremath{\mathbf{y}})=\sum_{\ell=0}^\infty\sum_{k=1}^{M(d,\ell)}a_\ell Y_{\ell,k}(\ensuremath{\mathbf{x}})Y_{\ell,k}(\ensuremath{\mathbf{y}}),\end{equation}
with $a_\ell>0$ for all $\ell\ge0$.
That the expansion is zonal follows from the addition theorem for spherical harmonics,
\[\sum_{k=1}^{M(d,\ell)}Y_{\ell,k}({\bf x})Y_{\ell,k}({\bf y})=\frac{M(d,\ell)}{\omega_d}P_\ell(d+1,{\bf x}\cdot{\bf y}),\]
where $P_\ell(d+1,z)$ is the Legendre polynomial of degree $\ell$ in dimension $d+1$ normalized to $P_\ell(d+1,1)=1$, and $\omega_d$ is the total surface measure of ${\mathbb S}^d$,
\[\omega_d=\int_{{\mathbb S}^d}d\omega({\bf x}).\]
In this situation it is well known that the inner product in $\mathcal{N}_\phi$ can be written as
\begin{equation}\label{fiprod}\langle u,v\rangle_\phi = \sum_{\ell=0}^\infty\sum_{k=1}^{M(d,\ell)}\dfrac{\widehat{u}_{\ell,k}\widehat{v}_{\ell,k}}{a_\ell},\end{equation}
where
$$\widehat{u}_{\ell,k}= \int_{{\mathbb S}^d}u(\ensuremath{\mathbf{x}})Y_{\ell,k}(\ensuremath{\mathbf{x}})d\omega(\ensuremath{\mathbf{x}}).$$
Indeed, as a special case of (\ref{fiprod}) we find
$$\langle f,\phi(\cdot,\ensuremath{\mathbf{y}})\rangle_\phi = \sum_{\ell=0}^\infty\sum_{k=1}^{M(d,\ell)}\dfrac{\widehat{f}_{\ell,k}a_\ell Y_{\ell,k}}{a_\ell}=\sum_{\ell=0}^\infty\sum_{k=1}^{M(d,\ell)}\widehat{f}_{\ell,k} Y_{\ell,k}(\ensuremath{\mathbf{y}})=f({\bf y}),$$
thus verifying the reproducing kernel property (\ref{rep}).
If we further assume that for large $\ell$
\begin{equation}\label{ass3}
a_\ell\sim(\ell+1)^{-2s},
\end{equation}
then it follows from (\ref{fiprod}) and (\ref{ass3}) that the native space $\mathcal{N}_\phi$ is equivalent to the Sobolev space $H^s({\mathbb S}^d)$ with inner product
\begin{equation}\label{Hsnorm}\langle u,v\rangle_{H^s}=\sum_{\ell=0}^\infty\sum_{k=1}^{M(d,\ell)}(\ell+1)^{2s}\widehat{u}_{\ell,k}\widehat{v}_{\ell,k}.\end{equation}
Technicalities aside, we remark that the essential difficulty in analysing the hybrid approximation is that ${\mathcal X}_N$ and ${\mathcal P}_L$ are both subsets of $\mathcal{N}_\phi$, and that in an appropriate sense both can approximate $\mathcal{N}_\phi$ as $N$ or $L$ tend to $\infty$. We need the inf-sup condition, now to be introduced, to allow both subspaces to coexist comfortably within the one approximation.
\section{Inf-sup condition, and the Brezzi theorem}\label{sec_infsup}
Typically, uniqueness of the solution and optimal error estimates for saddle-point problems
follow from so-called inf-sup conditions together with appropriate coercivity of the primal operator. For us an essential tool will be the following inf-sup theorem proved in \cite{SW08}.
In this theorem $h_X$, for a given point set $X=X_N\subset{\mathbb S}^d$, is the mesh norm, defined by
$$h_X:=\sup_{\ensuremath{\mathbf{x}}\in{\mathbb S}^d}\inf_{\ensuremath{\mathbf{x}}_j\in X}\rm{cos}^{-1}(\ensuremath{\mathbf{x}}\cdot\ensuremath{\mathbf{x}}_j).$$
In words, $h_X$ is the maximum geodesic distance from a point on ${\mathbb S}^d$ to the nearest point of $X$.
\begin{theorem}\label{SW_theorem}
Let $\phi$ be a kernel satisfying (\ref{ass2}) and (\ref{ass3}) for
some $s>d/2$. There exist constants $\gamma>0$ and $\tau>0$ depending only on $d$ and $s$ such that for all $L\ge 1$ and all $X_N = \{\ensuremath{\mathbf{x}}_1,\ldots,\ensuremath{\mathbf{x}}_N\}\subset {\mathbb S}^d$ satisfying $h_X\le\tau/L$ the following inequality holds:
$$\inf_{p\in {\mathcal P}_L\setminus\{0\}}\sup_{v\in{\mathcal X}_N\setminus\{0\}}\dfrac{\langle p,v\rangle_\phi}{\|v\|_\phi\|p\|_\phi}\ge\gamma.$$
\end{theorem}
To use this to prove stability we start with the following well-known theorem from Brezzi \cite{Brezzi}, which is at the heart of most analyses of mixed finite elements.
\begin{theorem}\label{brezzi} Let $H$ and $J$ be real Hilbert spaces, $a(\xi_1,\xi_2)$
be a continuous bilinear form on $H\times H$ and $b(\psi,\xi)$ be a continuous bilinear form
on $J\times H$. Let $\{H_N: N\in {\mathbb N}\}$ and $\{J_L : L\in {\mathbb N}\}$ be
sequences of subspaces of $H$ and $J$ respectively. Set
$$K=\{\eta \in H : b(\theta,\eta)=0 \;\;\forall \theta\in J \},\;\; K_{N,L}=\{\eta \in H_N : b(\theta,\eta)=0 \;\;\forall \theta\in J_L \} .$$
If
\begin{equation}\exists \gamma_0 >0\; \text{ such that }\quad a(\eta,\eta) \geq \gamma_0\|\eta\|_H^2
\quad \forall \eta\in K \cup K_{N,L},\label{coerc}\end{equation}
and
$$\exists \gamma_1 >0\; \text{ such that }\quad \sup_{\eta\in H \setminus\{0\}}
\frac{b(\theta,\eta)}{\|\eta\|_H} \geq \gamma_1 \|\theta\|_J\quad\ \forall \theta\in J
$$
\begin{equation}\text{ and }\sup_{\eta\in H_N \setminus\{0\}}
\frac{b(\theta,\eta)}{\|\eta\|_H} \geq \gamma_1 \|\theta\|_J\quad\ \forall \theta\in J_L,
\label{infsup}\end{equation}
then for every $\ell_1\in H^{\prime}$ and $\ell_2\in J^{\prime}$ and every $N,L>0$
the discrete problem of finding $\xi_{N,L}\in H_N$ and $\psi_{N,L}\in J_L$ such that
\begin{eqnarray*}
a(\xi_{N,L},\eta)+b(\psi_{N,L},\eta) &=& \langle \ell_1,\eta\rangle\quad\forall \eta\in H_N\\
b(\theta,\xi_{N,L})&=&\langle\ell_2,\theta\rangle\quad\forall\theta\in J_L
\end{eqnarray*}
has a unique solution, and there exists a constant $C=C(\gamma_0,\gamma_1)>0$
such that
$$\|\xi-\xi_{N,L}\|_H + \|\psi - \psi_{N,L}\|_J \leq C
\left(\inf_{\widehat{\xi}_N\in H_N} \|\xi-\widehat{\xi}_N\|_H +
\inf_{\widehat{\psi}_L\in J_L} \|\psi-\widehat{\psi}_L\|_J \right)$$
where $\xi\in H$ and $ \psi\in J$ are defined by
\begin{eqnarray*}
a(\xi,\eta)+b(\psi,\eta) &=& \langle \ell_1,\xi\rangle\quad\forall \eta\in H,\\
b(\theta,\xi)&=&\langle\ell_2,\theta\rangle\quad\forall\theta\in J.
\end{eqnarray*}
\end{theorem}
To apply this theorem, we first observe that the hybrid approximation with its defining equations (\ref{1st}) and (\ref{2nd}) can be written, using the reproducing kernel property (\ref{rep}), as the problem of finding $u_{N,L}\in {\mathcal X}_N$ and $p_{N,L} \in {\mathcal P}_L$ such that
\begin{equation}\label{firsts}\langle u_{N,L}, \eta\rangle_\phi+\langle p_{N,L},\eta\rangle_\phi=\langle f,\eta\rangle_\phi \quad\forall\eta\in{\mathcal X}_N,\end{equation}
\begin{equation}\label{seconds} \langle q,u_{N,L}\rangle_\phi=0\qquad \forall q\in{\mathcal P}_L.\end{equation}
To use Theorem \ref{brezzi} we take
$H=\mathcal{N}_{\phi}$, $J={\mathcal P}_L$, $H_N={\mathcal X}_N$ and $J_L={\mathcal P}_L$, with
the inner product on $\mathcal{N}_{\phi}$ being defined by (\ref{IP}), and the bilinear forms $a(\cdot,\cdot)$ and $b(\cdot,\cdot)$
both equal to the $\mathcal{N}_{\phi}$ inner product.
The coercivity condition
(\ref{coerc}) is trivially satisfied on the whole space $H=\mathcal{N}_\phi$ with $\gamma_0=1$ since
$$a(u,u)=\langle u,u\rangle_{\phi} = \|u\|_{\phi}^2 .$$
The existence of a constant $\gamma_1$ independent of $N$ and $L$
satisfying (\ref{infsup}) is ensured by Theorem \ref{SW_theorem}, provided $h_X\le\tau/L$.
The first part of Theorem \ref{brezzi} then confirms that the solution of the system (\ref{firsts}) and (\ref{seconds}) exists and is unique provided $h_X\le \tau/L$. The last part of that theorem defines the comparison quantities: it defines $u_L\in \mathcal{N}_\phi$ and $p_L\in {\mathcal P}_L$ such that
$$\langle u_L,\eta\rangle_\phi+\langle p_L,\eta\rangle_\phi=\langle f,\eta\rangle_\phi \quad\forall\eta\in \mathcal{N}_\phi,$$
$$ \langle q,u_L\rangle_\phi=0\qquad \forall q\in{\mathcal P}_L.$$
The second equation says that $u_L$ is orthogonal to the space ${\mathcal P}_L$. In principle the orthogonality is with respect to the $\mathcal{N}_\phi$ inner product, but because of the zonal property of the kernel it is easy to see that this is the same as the $L_2$ orthogonal projection.
Indeed, from \eqref{fiprod} we have
\[\langle q,u_L\rangle_\phi=0\,
\forall q\in{\mathcal P}_L\implies(\widehat u_L)_{\ell,k}=0\mbox{ for }\ell\in[0,L]
\implies\langle q,u_L\rangle_{L_2}=0\,\forall q\in{\mathcal P}_L.\]
The first of the latter set of equations then becomes, on specialising the choice of $\eta$ to $q\in{\mathcal P}_L$,
$$\langle p_L,q\rangle_\phi=\langle f,q\rangle_\phi \quad\forall q\in{\mathcal P}_L,$$
thus $p_L$ is the orthogonal projection of $f$ on the subspace ${\mathcal P}_L$ with respect to either the $L_2$ or the $\mathcal{N}_\phi$ inner products. We write this orthogonal projection as $P_L f$. Now we can write $p_L=P_L f$, and $u_L=f-p_L=f-P_L f$. Theorem \ref{brezzi} with
$\xi = u_L$ and $\psi = p_L$
now gives the following convergence result, recovering a result obtained by a direct argument in \cite{SW08}. Note that even though we have taken $J={\mathcal P}_L$ in the theorem, the constants $\gamma_0$ and $\gamma_1$ do not depend on $L$, and hence neither does $C$.
\begin{theorem}
Let $\phi$ be a kernel satisfying (\ref{ass2}) and (\ref{ass3}) for some $s>d/2$. There exist constants $C>0$ and $\tau>0$ depending only on $d$ and $s$ such that for all $L\ge 1$ and all $X = X_N = \{\ensuremath{\mathbf{x}}_1,\ldots,\ensuremath{\mathbf{x}}_N\}\subset {\mathbb S}^d$ satisfying $h_X\le\tau/L$ the solutions of \eqref{unl},\eqref{pnl} and \eqref{interp} satisfy
\begin{equation}\label{stability}
\|f-u_{N,L}-p_{N,L}\|_\phi
\leq C
\inf_{\widehat{\xi}_N\in {\mathcal X}_N}
\|(f-P_Lf)-\widehat{\xi}_N\|_\phi.
\end{equation}
\end{theorem}
Explicit error bounds in the $L_2$ norm for $f\in H^s$ and $f\in H^{2s}$ can then be obtained as in \cite{SW08}.
\section{Preconditioning}\label{linear}
Now we turn our attention to the linear algebra aspects of the hybrid approximation described in Section 2.
We have noted already that the hybrid approximation can be written as the linear system (\ref{saddle}),
with $A_X, Q_{X,L}$ and ${\bf f}$ defined by (\ref{AXphi}),(\ref{Qdef}), and (\ref{fdef}).
The solution of saddle-point linear systems such as (\ref{saddle_general})
has received much attention in recent years - see
\cite{BenziGolubLiesen} for an overview of possible approaches. In
particular, it was shown in \cite{MurGolWat00} that
a suitable preconditioner for the saddle point system
\begin{equation}\label{saddle_mat}
\left[\begin{array}{cc}A &Q\\Q^T & 0\end{array}\right]
\end{equation}
with positive definite $A$ is
\begin{equation}\label{mgw}
\left[\begin{array}{cc}A &0\\0 & S\end{array}\right],
\end{equation}
where
$$S = Q^T A^{-1} Q$$
is the Schur complement. This is because of the remarkable fact that the product of (\ref{saddle_mat}) by the inverse of (\ref{mgw}) is a diagonalisable matrix with just three distinct eigenvalues, namely $1, (1\pm\sqrt{5})/2$. Thus an appropriate Krylov subspace
iteration such as MINRES (see \cite{PS75}) will converge in just
three iterations. While this is generally not a practical preconditioner, an approximate preconditioner of the form
\begin{equation}
\left[\begin{array}{cc}\widehat{A} &0\\0 & \widehat{S}\end{array}\right],
\end{equation}
where $\widehat{A}$ is a preconditioner for the
problem (\ref{radb}) involving only $A$, and $\widehat{S}$
is a suitable Schur complement approximation, will lead to rapid convergence.
In the present work we shall assume that an approximate preconditioner $\widehat{A}$ for $A=A_X$ is already available; one example would be the domain decomposition preconditioner from \cite{GST06} - this is the one we employ in the numerical results presented in the next section. Our interest here is in finding an appropriate approximation to the Schur complement $S_X=Q^T_{X,L}A_X^{-1}Q_{X,L}$. We shall see that this is handed to us by the inf-sup result in Theorem \ref{SW_theorem}.
That inf-sup condition can be stated as
$$\sup_{v\in {\mathcal X}_N \setminus\{0\}}
\frac{\langle p,v \rangle_{\phi}}{\|v\|_{\phi}} \geq
\gamma_1 \|p\|_\phi\quad\ \forall p\in {\mathcal P}_L,$$
provided $h_X\le\tau/L$. With the help of the Cauchy-Schwarz inequality, this can be strengthened to a two-sided inequality,
\begin{equation}
\|p\|_\phi =
\sup_{v\in\mathcal{N}_\phi\setminus\{0\}}
\frac{\langle p,v \rangle_{\phi}}{\|v\|_{\phi}}
\ge\sup_{v\in {\mathcal X}_N \setminus\{0\}}
\frac{\langle p,v \rangle_{\phi}}{\|v\|_{\phi}} \geq
\gamma_1 \|p\|_\phi
\quad\ \forall p\in {\mathcal P}_L, \label{sup}
\end{equation}
provided $h_X\le\tau/L$.
To find an equivalent matrix expression, we write $p\in{\mathcal P}_L$ and $v\in{\mathcal X}_N$ as
$$p= \sum_{\ell=0}^{L} \sum_{k=1}^{M(d,\ell)}\beta_{\ell,k}Y_{\ell,k}, \qquad v=\sum_{i=1}^N\alpha_i\phi(\cdot,\ensuremath{\mathbf{x}}_i).$$
With the help of the reproducing property (\ref{rep}), we find
\begin{align*}\langle p,v\rangle_\phi & =\sum_{\ell=0}^{L} \sum_{k=1}^{M(d,\ell)}\sum_{i=1}^N\beta_{\ell,k}\alpha_i \langle Y_{\ell,k},\phi(\cdot,\ensuremath{\mathbf{x}}_i)\rangle_\phi\\
& =\sum_{\ell=0}^{L} \sum_{k=1}^{M(d,\ell)}\sum_{i=1}^N\beta_{\ell,k}\alpha_i Y_{\ell,k}(\ensuremath{\mathbf{x}}_i)={\mathbb \beta}^TQ^T_{X,L}{\mathbb \alpha},
\end{align*}
and
$$\|v\|_\phi=\langle v,v\rangle_\phi^{1/2}=
(\sum_{i=1}^N\sum_{j=1}^N\alpha_i\alpha_j\phi(\ensuremath{\mathbf{x}}_i,\ensuremath{\mathbf{x}}_j))^{1/2}
=({\mathbb \alpha}^TA_X{\mathbb \alpha})^{1/2}.$$
Also, with the aid of
(\ref{fiprod})
we obtain
$$\|p\|_\phi=\left(\sum_{\ell=0}^L\sum_{k=1}^{M(d,\ell)}\dfrac{ \beta_{\ell,k}^2}{a_\ell}\right)^{{\frac{1}{2}}}
=\left({\mathbb \beta}^T\Lambda_L {\mathbb \beta}\right)^{\frac{1}{2}},$$
where $\Lambda_L$ is the $M\times M$ diagonal matrix given by
\begin{equation}\label{Lambda}
(\Lambda_L)_{\ell k,\ell^\prime k^\prime}=\delta_{\ell\ell^\prime}\delta_{kk^\prime}/a_{\ell}.
\end{equation}
Thus in matrix terms (\ref{sup}) can be written as
\begin{equation}
({\mathbb \beta}^T \Lambda_L {\mathbb \beta})^{{\frac{1}{2}}} \geq
\sup_{{\mathbb \alpha}\in {\mathbb R}^N\setminus\{0\}}
\frac{{\mathbb \beta}^T Q^T_{X,L}{\mathbb \alpha}}{({\mathbb \alpha}^T A_{X}{\mathbb \alpha})^{{\frac{1}{2}}}}
\geq \gamma_1 ({\mathbb \beta}^T \Lambda_L {\mathbb \beta})^{{\frac{1}{2}}}\quad \forall {\mathbb \beta}\in {\mathbb R}^M.
\end{equation}
Because $A_{X}$ is symmetric and positive definite, the central term can be simplified by the substitution ${\mathbb \alpha}=A_X^{-{\frac{1}{2}}}{\bf a}$, making it
$$
\sup_{{\bf a}\in {\mathbb R}^N\setminus\{0\}}
\frac{{\mathbb \beta}^T Q_{X,L}^T A_{X}^{-{\frac{1}{2}}}{\bf a}}{({\bf a}^T {\bf a})^{{\frac{1}{2}}}}\,=\,({\mathbb \beta}^T Q_{X,L}^TA_X^{-1}Q_{X,L}{\mathbb \beta})^{\frac{1}{2}},
$$
with the last step following because the supremum over ${\bf a}$ is clearly achieved by
${\bf a}=({\mathbb \beta}^T Q^T_{X,L} A_X^{-{\frac{1}{2}}})^T$.
Thus in matrix terms (\ref{sup}) can be expressed as
\begin{equation}\label{schur}
{\mathbb \beta}^T \Lambda_L {\mathbb \beta} \geq {\mathbb \beta}^T Q^T_{X,L} A^{-1}_XQ_{X,L}{\mathbb \beta}\geq \gamma_1^2{\mathbb \beta}^T\Lambda_L{\mathbb \beta}\quad\forall{\mathbb \beta}\in {\mathbb R}^M.
\end{equation}
Through the above arguments we have established the following theorem.
\begin{theorem}\label{spectral_equiv}
Let $\phi$ be a kernel satisfying (\ref{ass2}) and (\ref{ass3}) for some $s>d/2$, and let $A_X$ and $Q_{X,L}$ be given by (\ref{AXphi}) and (\ref{Qdef}). For all $L\ge 1$ and $h_X\le\tau/L$, where $\tau$ is as in Theorem \ref{SW_theorem}, the Schur complement $S_X=Q_{X,L}^T A_X^{-1}Q_{X,L}$ is spectrally equivalent to the diagonal matrix $\Lambda_L$ given by (\ref{Lambda}).
\end{theorem}
It follows from the theorem that our practical recommendation for the hybrid problem is a preconditioner of the form
\begin{equation}
\left[\begin{array}{cc}\widehat{A} &0\\0 & \Lambda_L\end{array}\right],
\label{prec}\end{equation}
where $\widehat{A}$ is an approximation to $A$, and $\Lambda_L$ is defined by (\ref{Lambda}).
\section{Numerical examples}\label{numer}
We will use the following kernel
\[
\phi(\ensuremath{\mathbf{x}},\ensuremath{\mathbf{y}}) = \psi(|\ensuremath{\mathbf{x}}-\ensuremath{\mathbf{y}}|) = \psi(\sqrt{2-2\ensuremath{\mathbf{x}} \cdot \ensuremath{\mathbf{y}}}),
\quad \ensuremath{\mathbf{x}},\ensuremath{\mathbf{y}} \in {\mathbb S}^2,
\]
where the radial basis function $\psi(r)$ is one of the three choices
\[
\psi_0(r) = (1-r)^2_{+} , \quad \psi_1(r) = (1-r)^4_{+} (4r+1) , \quad
\psi_2(r) = (1-r)^6_{+} (35 r^2+18r+3)
\]
with $(x)_{+} = x$ for $x\ge 0 $ and $0$ otherwise. Note that
$\psi_0\in C^0( {\mathbb R}^3), \psi_1\in C^2( {\mathbb R}^3)$ and $\psi_2\in C^4( {\mathbb R}^3)$ and each
is positive definite (see \cite{Wend}).
We comment that we have not here employed any scaling of the compactly supported
RBFs.
Using functions with smaller support would improve matrix
conditioning, but because it would also reduce approximation accuracy we have chosen
not to use any scaling.
It is shown theoretically in \cite{NarWar02} and verified
numerically in Figure \ref{fig:aell}
that the coefficients $a_\ell$ in the
expansion of the kernel $\phi$ defined from $\psi_1$ are of order $(1+\ell)^{-5}$. This
is consistent with (\ref{ass3}) but because the constants in the equivalence are large we
have chosen to work directly with the Fourier-Legendre coefficients $a_{\ell}$,
which being $1$-dimensional integrals are easily evaluated numerically.
For the sphere ${\mathbb S}^2$, we have
\[
a_\ell = 2\pi \int_{-1}^1 \Phi(t) P_\ell(3,t) dt,
\quad \mbox{ where } \Phi(t) = \psi(\sqrt{2-2t}).
\]
\begin{figure}[ht]
\begin{center}
\scalebox{0.65}{\includegraphics{a_ell100.eps}}
\caption{Numerical values of $(\ell+1)^5a_\ell$}\label{fig:aell}
\end{center}
\end{figure}
For the preconditioner, as described in \cite{GST06}, the matrix
$A$ is preconditioned using a domain decomposition technique.
First, given a fixed parameter $0<\nu<\pi$, an appropriate set
of centers $\{\ensuremath{\mathbf{p}}_1,\ldots,\ensuremath{\mathbf{p}}_J\} \subset X$ is chosen over the
whole sphere so that
\[
\min_{i\ne j} \cos^{-1} (\ensuremath{\mathbf{p}}_i \cdot \ensuremath{\mathbf{p}}_j) \ge \nu.
\]
Second, with another fixed parameter $0 < \mu< \pi/3$, we
decompose the point set $X$ into a collection of smaller sets
$X_j$, for $j=1,\ldots,J$, defined by
\[
X_j := \{ \ensuremath{\mathbf{x}} \in X : \cos^{-1}(\ensuremath{\mathbf{x}} \cdot \ensuremath{\mathbf{p}}_j) \le \mu \}.
\]
The sets $X_j$ with cardinality $m_j$, for $j=1,\ldots,J$, may overlap
each other and must satisfy $\cup_{j=1}^J X_j = X$. The restriction operator from $ {\mathbb R}^N$
to $ {\mathbb R}^{m_j}$ is denoted by $R_j$ and the extension operator from $ {\mathbb R}^{m_j}$
to $ {\mathbb R}^N$ is $R^T_j$. Given a vector $\ensuremath{\mathbf{r}} \in {\mathbb R}^N$, the
action of the preconditioner $\widehat{A}$ is given by
\[
\widehat{A}^{-1} \ensuremath{\mathbf{r}} = \sum_{j=1}^J R^T_j (A_j)^{-1} R_j \ensuremath{\mathbf{r}}
\]
where the matrix $A_j$ is the restriction of the full matrix
$A$ on the subdomain $X_j$ (see \cite{GST06} for more details).
By Theorem \ref{spectral_equiv} the diagonal matrix $\Lambda_L$ defined by (\ref{Lambda})
is spectrally equivalent to $S_X = Q^{T}_{X,L} A^{-1}_X Q_{X,L}$.
The block diagonal preconditioner for (\ref{saddle_mat}) is therefore the matrix (\ref{prec}).
The results here are for interpolation of the function
\[
f(x,y,z) = \exp(x+y+z)+ [0.01-x^2-y^2-(z-1)^2]^2_{+},
\]
consisting of a smooth first term and a second term whose support is a cap
of Euclidean radius $0.1$.
Using each of the kernel functions $\phi$ obtained from the radial basis functions
$\psi_m, m=0,1,2$, we employ $N = 2000$, $4000$, $8000$, $16000$ and $32000$
points, and maximum polynomial degree $L = 0,5,10,15,20,25$. In each case, a thousand of the
points were generated in a cap about the $z$ axis subtending an angle of $0.1$ radians at the
origin and the remaining points distributed outside this cap. The Saff-Kuijlars equal area algorithm described in \cite{SaffK} was used to generate these points in the following manner.
Firstly the Saff-Kuijlars points are generated on the whole sphere and those in the cap region are discarded. Then a similar equal area construction only for the cap is used to generate $1000$ points in this region.
The number of MINRES iterations and the CPU time in seconds required for convergence
to a residual norm tolerance of $10^{-9}$ are tabulated for the unpreconditioned case
in Table~\ref{tab:unprec} and for the preconditioner introduced here in Table~\ref{tab:prec}. The computer code is written in Fortran 90, compiled with the Intel compiler and run
on a single core of an SGI Altix XE320 with two Intel Xeon X5472 CPUs.
\begin{table}
\begin{center}
\begin{tabular}{|c|l|r|r|r|r|r|}
\hline
$m$ & $N$ & 2000 & 4000 & 8000 & 16000 & 32000 \\
\hline
0 & $L= 0$ & 207 ( 14)& 228 ( 62)& 273 ( 297)& 294 ( 1229) & 367 ( 6277)\\
& $L= 5$ & 813 ( 57)& 1116 (304)& 1531 (1662)&1996 ( 8297) & 2535 (43513)\\
& $L=10$ & 1053 ( 77)& 1488 (413)& 2240 (2463)&3016 (12573) & 4001 (67005)\\
& $L=15$ & 1080 ( 85)& 1492 (498)& 2208 (2475)&3037 (12788) & 4270 (71728)\\
& $L=20$ & 1126 ( 99)& 1466 (464)& 2032 (2334)&2777 (17227) & 3832 (66025)\\
& $L=25$ & 1262 (123)& 1467 (499)& 1976 (2382)&2562 (11069) & 3461 (58451)\\
\hline
1 & $L= 0$ & 1041 ( 75)& 790 ( 218)& 695 ( 755)& 912 ( 3954)& 995 (17436)\\
& $L= 5$ & 2654 (193)& 4293 (1194)& 3564 (3889)& 3172 (13736)& 2321 (40754)\\
& $L=10$ & 3647 (280)& 4370 (1237)& 4047 (4488)& 5335 (42445)& 3381 (59228)\\
& $L=15$ & 3267 (265)& 4308 (1281)& 4021 (4594)& 3792 (16908)& 2808 (48999)\\
& $L=20$ & 2883 (264)& 2923 ( 935)& 3329 (6906)& 3977 (17987)& $>$24 hours \\
& $L=25$ & 2659 (265)& 2837 (1006)& 3295 (4048)& 2756 (12735)& 3461 (61280)\\
\hline
2& $L= 0$ & 1079 ( 82) & 1578 (457) & 1903 ( 2164) & 2743 (12233) & 2671 (48177)\\
& $L= 5$ & 1978 (152) & 2594 (757) & 14424 (16476) & $>$24 hours & $>$40 hours \\
& $L=10$ & 2205 (176) & 3130 (929) & 13569 (15603) & $>$24 hours & $>$40 hours \\
& $L=15$ & 2402 (205) & 2803 (873) & 11970 (14053) & $>$24 hours & $>$40 hours \\
& $L=20$ & 1796 (169) & 1869 (624) & 8710 (10687) & 13551 (63016) &$>$40 hours\\
& $L=25$ & 1615 (168) & 1758 (629) & 6417 ( 8239) & 10299 (48420) &$>$40 hours\\
\hline
\end{tabular}
\caption{MINRES iteration count (CPU time) without preconditioning}
\label{tab:unprec}
\end{center}
\end{table}
\begin{table}
\begin{center}
\begin{tabular}{|c|l|r|r|r|r|r|}
\hline
$m$ & $N$ & 2000 & 4000 & 8000 & 16000 & 32000 \\
\hline
0 & $L=0$ & 31 ( 6)& 39 (18) & 30 ( 51)& 29 (225) & 39 ( 959)\\
& $L=5$ & 59 (11)& 71 (31) & 58 ( 96)& 62 (465) & 75 (1802)\\
& $L=10$ & 70 (13)& 88 (39) & 70 (116)& 71 (532) & 89 (2120)\\
& $L=15$ & 76 (15)& 93 (43) & 76 (128)& 76 (572) & 96 (2279)\\
& $L=20$ & 83 (17)& 98 (48) & 80 (138)& 82 (780) & 99 (2393)\\
& $L=25$ & 95 (20)& 97 (50) & 80 (142)& 84 (649) & 104 (2505)\\
\hline
1 & $L=0$ & 43 ( 8)& 75 (34)& 35 ( 66)& 29 (227)& 47 (1161) \\
& $L=5$ & 76 (15)& 128 (57)& 83 (138)& 74 (557)& 105 (2559)\\
& $L=10$ & 91 (18)& 148 (67)& 98 (163)& 94 (705)& 136 (3296)\\
& $L=15$ & 98 (19)& 168 (78)& 96 (163)& 100 (758)& 148 (3615)\\
& $L=20$ & 107 (22)& 170 (83)& 103 (180)& 103 (788)& 153 (3754)\\
& $L=25$ & 106 (23)& 174 (89)& 103 (185)& 113 (870)& 161 (3972)\\
\hline
2& $L= 0$ & 64 (13) & 149 ( 61) & 46 ( 81) & 30 ( 243) & 61 (1419) \\
& $L= 5$ & 95 (19) & 157 ( 70) & 88 (151) & 103 ( 797) & 140 (3380) \\
& $L=10$ & 95 (19) & 171 ( 71) & 102 (176) & 111 ( 861) & 146 (3425) \\
& $L=15$ & 112 (23) & 187 (123) & 113 (199) & 118 ( 923) & 165 (3614) \\
& $L=20$ & 119 (25) & 197 ( 86) & 115 (207) & 133 (1052) & 196 (4301) \\
& $L=25$ & 125 (28) & 201 ( 91) & 119 (220) & 131 (1046) & 203 (4433) \\
\hline
\end{tabular}
\caption{MINRES iteration count(CPU time) with preconditioning
$\widehat{A}$ and $\widehat{S}$}
\label{tab:prec}
\end{center}
\end{table}
The preconditioning is seen to be effective: as anticipated from the theory above,
the number of iterations remains approximately constant over all choices of
$N$ for each degree $L$. Indeed, for each $N$ aside from the simple case $L=0$
(in which the radial basis function approximation matrix is only
supplemented by one row and one column),
the iteration counts grow only slowly
with increasing $L$.
In order to determine the descriptiveness of the bound \eqref{schur}
we have also computed the generalised eigenvalues $\lambda_i$ of the pencil
$Q_{X,L}^T A_X^{-1} Q_{X,L} - \lambda\Lambda_L$ for the case $N=4000$ points, for $m=0,1$
and for $L=5,10,15,20,25$. Note that the generalised eigenvalues are exactly
the eigenvalues of $\Lambda^{-1}_L (Q_{X,L}^T A_X^{-1} Q_{X,L})$.
The minimum and maximum computed eigenvalues
are given in Table~\ref{tab:eigvals}. It is noticeable how close the largest
eigenvalue is to the analytical upper bound of $1$ and that, although the lowest
eigenvalue does decrease for larger $L$, it remains reasonably
close to $1$. (Note that for fixed $X$ and increasing $L$ the inf-sup condition
must eventually break down.)
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
$m$ & $L$ & 5 & 10 & 15 & 20 & 25 \\
\hline
0 &$\lambda_{\min}$&0.9987434&0.9899326&0.9623012&0.9068357&0.8348191\\
&$\lambda_{\max}$&0.9997653&0.9997658&0.9997674&0.9997753&0.9998099\\
\hline
1 &$\lambda_{\min}$&0.9999955&0.9999125&0.9993989&0.9973949&0.9908182\\
&$\lambda_{\max}$&0.9999986&0.9999986&0.9999986&0.9999986&0.9999989\\
\hline
\end{tabular}
\caption{Extreme eigenvalues of $Q_{X,L}^T A_X^{-1} Q_{X,L} - \lambda\Lambda_L$ for $N=4000$} \label{tab:eigvals}
\end{center}
\end{table}
\section{Conclusions}\label{concl}
By employing a recent inf-sup stability result of Sloan and Wendland,
we have derived an effective preconditioned iterative solver for the
hybrid radial basis function and spherical polynomial approximation
scheme of Sloan and Sommariva. The preconditioner requires only a
good approximation for the radial basis function interpolation
problem and a simple diagonal scaling matrix for the Schur complement
based on the Fourier-Legendre coefficients of the kernel function used
in the radial basis function interpolant. We have established theoretically
that the preconditioner for the Schur complement is optimal.
\vspace*{0.3in}
{\bf Acknowledgements.}
The support of the Australian Research Council
is gratefully acknowledged.
\bibliographystyle{amsplain}
|
{
"timestamp": "2010-09-23T02:01:03",
"yymm": "1009",
"arxiv_id": "1009.4275",
"language": "en",
"url": "https://arxiv.org/abs/1009.4275"
}
|
\section{Introduction}
Alice has a perfectly fair penny---one that lands heads exactly 50\% of the time. Unfortunately, the penny is mixed in with a jar of ordinary, imperfect pennies. The truly fair penny can never be distinguished from the other pennies, since no amount of experimentation can identify it with certainty. Still, Alice has discovered a workable solution. Whenever she needs a fair coin flip, she flips all the pennies and counts the Lincolns; an even number means heads, and an odd number means tails.
Alice's technique is an example of ``robust coin flipping.'' She samples many random sources, some specified number of which are unreliable, and still manages to simulate a desired coin flip. Indeed, Alice's technique works even if the unreliable coin flips somehow fail to be independent.
Bob faces a sort of converse problem. He's marooned on an island, and the nearest coin is over three hundred miles away. Whenever \emph{he} needs a fair coin flip, he calls up two trustworthy friends who don't know each other, asking for random equivalence classes modulo two. Since the sum of the classes is completely mysterious to either of the friends, Bob may safely use the sum to make private decisions.
Bob's technique seems similar to Alice's, and indeed we shall see that the two predicaments are essentially the same. We shall also see that the story for biased coin flips is much more complex.
\subsection{Preliminaries and Definitions}
Informally, we think of a random source as a (possibly remote) machine capable of sampling from certain probability spaces. Formally, a \textbf{random source} is a collection $\mathcal{C}$ of probability spaces that is closed under quotients. That is, if $X \in \mathcal{C}$ and there is a measure-preserving map\footnote{A measure-preserving map (morphism in the category of probability spaces) is a function for which the inverse image of every measurable set is measurable and has the same measure. Any measure-preserving map may be thought of as a quotient ``up to measure zero.''} $X \to Y$, then $Y \in \mathcal{C}$. Random sources are partially ordered by inclusion: We say that $\mathcal{C}$ is \textbf{stronger than} $\mathcal{D}$ iff $\mathcal{C} \supset \mathcal{D}$.
The quotients of a probability space $X$ are precisely the spaces a person can model with $X$. For example, one can model a fair coin with a fair die: Label three of the die's faces ``heads" and the other three ``tails." Similarly, one can model the uniform rectangle $[0,1]^2$ with the uniform interval $[0,1]$: Take a decimal expansion of each point in $[0,1]$, and build two new decimals, one from the odd-numbered digits and one from the even-numbered digits.\footnote{In fact, this defines an isomorphism of probability spaces between the rectangle and the interval.} Thus, forcing $\mathcal{C}$ to be closed under quotients is not a real restriction; it allows us to capture the notion that ``a fair die is more powerful that a fair coin.''\footnote{It would also be natural (albeit unnecessary) to require that $\mathcal{C}$ is closed under finite products.}
We define an \textbf{infinite random source} to be one that contains an infinite space.\footnote{An infinite space is one that is not isomorphic to any finite space. A space with exactly 2012 measurable sets will always be isomorphic to a finite space, no matter how large it is as a set.} A \textbf{finite random source}, on the other hand, contains only finite probability spaces. Further, for any set of numbers $\mathbb{S}$, we define an \textbf{$\mathbb{S}$-random source} to be one which is forced to take probabilities in $\mathbb{S}$. That is, all the measurable sets in its probability spaces have measures in $\mathbb{S}$.
Sometimes we will find it useful to talk about the strongest random source in some collection of sources. We call such a random source \textbf{full-strength} for that collection. For instance, a full-strength finite random source can model any finite probability space, and a full-strength $\mathbb{S}$-random source can model any $\mathbb{S}$-random source.
In practice, when $p$ people simulate a private random source for someone else, they may want to make sure that privacy is preserved even if a few people blab about the data from their random sources or try to game the system. Define an \textbf{$r$-robust} function of $p$ independent random variables to be one whose distribution does not change when the joint distribution of any $r$ of the random variables is altered. Saying that $p$ people simulate a random source $r$-robustly is equivalent to asserting that the privacy of that source is preserved unless someone learns the data of more than $r$ participants. Similarly, to simulate a random source using $p$ sources, at least $q$ of which are working properly, Alice must run a $(p-q)$-robust simulation.
By a \textbf{robust} function or simulation, we mean a $1$-robust one.
We use $J$ to denote the all-ones tensor of appropriate dimensions. When we apply $J$ to a vector or hypermatrix, we always mean ``add up the entries.''
\subsection{Results}\label{results}
This paper answers the question ``When can a function sampling from $p$ independent random sources be protected against miscalibration or dependency among $p-q$ of them?'' (Alice's predicament), or equivalently, ``When can $p$ people with random sources simulate a \emph{private} random source for someone else\footnote{Later, we give an application to secure multiparty computation in which the output of the simulated random source has no single recipient, but is utilized by the group without any individual gaining access; see Section \ref{application}.} in a way that protects against gossip among any $p-q$ of them?'' (Bob's predicament). In the first question, we assume that at least $q$ of the sources are still functioning correctly, but we don't know which. In the second question, we assume that at least $q$ of the people keep their mouths shut, but we don't know who. In the terminology just introduced, we seek a $(p-q)$-robust simulation.
Consider the case of $p$ full-strength finite random sources. We prove: If $1 \leq q \leq p/2$, the people may simulate any finite $\mathbb{Q}$-random source and nothing better; if $p/2 < q < p$, they may simulate any finite $\overline{\mathbb{Q}}$-random source and nothing better. The proof uses projective varieties, convex geometry, and the probabilistic method. We also deal briefly with the case of infinite random sources, in which full-strength simulation is possible, indeed easy (see Appendix \ref{infapp}).
\subsection{Yao's robust coin flipping}
Our work fits in the context of secure multiparty computation, a field with roots in A. C. Yao's influential paper \cite{yao}.
In the last section of his paper, entitled ``What cannot be done?'', Yao presents (a claim equivalent to) the following theorem:
\begin{thm}[A. C. Yao]
Alice has several finite random sources, and she wants to generate a random bit with bias $\alpha$. Unfortunately, she knows that one of them may be miscalibrated, and she doesn't know which one. This annoyance actually makes her task impossible if $\alpha$ is a transcendental number.
\end{thm}
\noindent It does not not suffice for Alice to just program the distribution $(\alpha \hspace*{8pt} 1-\alpha)$ into one of the random sources and record the result; this fails because she might use the miscalibrated one! We require---as in our jar of pennies example---that Alice's algorithm be robust enough to handle unpredictable results from any single source
Unfortunately, Yao provides no proof of the theorem, and we are not aware of any in the literature. Yao's theorem is a special case of the results we described in the previous section.
\section{Simulating finite random sources}
The following result is classical.
\begin{prop}\label{diecon}
If $p$ players are equipped with private $d$-sided dice, they may $(p-1)$-robustly simulate a $d$-sided die.
\end{prop}
\begin{proof}
We provide a direct construction. Fix a group $G$ of order $d$ (such as the cyclic group $\mathbb{Z}/d\mathbb{Z}$). The $i^{\rm th}$ player uses the uniform measure to pick $g_i \in G$ at random. The roll of the simulated die will be the product $g_1g_2 \cdots g_p$.
It follows from the $G$-invariance of the uniform measure that any $p$-subset of \begin{equation} \{g_1, g_2,...,g_p, g_1g_2 \cdots g_p\} \end{equation} is independent! Thus, this is a $(p-1)$-robust simulation.
\end{proof}
\noindent For an example of this construction, consider how Alice and Bob may robustly flip a coin with bias $2/5$. Alice picks an element $a \in \mathbb{Z}/5\mathbb{Z}$, and Bob picks an element $b \in \mathbb{Z}/5\mathbb{Z}$; both do so using the uniform distribution. Then, $a,b,$ and $a+b$ are pairwise independent! We say that the coin came up heads if $a+b \in \{0,1\}$ and tails if $a+b \in \{2,3,4\}$.
This construction exploits the fact that several random variables may be pairwise (or $(p-1)$-setwise) independent but still dependent overall. In cryptology, this approach goes back to the one-time pad. Shamir \cite{shamir} uses it to develop secret-sharing protocols, and these are exploited in multiparty computation to such ends as playing poker without cards \cite{poker, game}.
\begin{cor}\label{qcon}
If $p$ players are equipped with private, full-strength finite $\mathbb{Q}$-random sources, they may $(p-1)$-robustly simulate a private, full-strength finite $\mathbb{Q}$-random source for some other player.
\end{cor}
\begin{proof}
Follows from Proposition \ref{diecon} because any finite rational probability space is a quotient of some finite uniform distribution.
\end{proof}
\subsection{Cooperative numbers}
We define a useful class of numbers.
\begin{defn}
If $p$ people with private full-strength finite random sources can robustly simulate a coin flip with bias $\alpha$, we say $\alpha$ is \textbf{p-cooperative}. We denote the set of $p$-cooperative numbers by $\coop{p}$.
\end{defn}
\noindent The ability to robustly simulate coin flips of certain bias is enough to robustly simulate any finite spaces with points having those biases, assuming some hypotheses about $\coop{p}$ which we will later see to be true.
\begin{lem}
Suppose that, if $\alpha, \alpha^\prime \in \coop{p}$ and $\alpha < \alpha^\prime$, then $\alpha/\alpha^\prime \in \coop{p}$. If $p$ people have full-strength finite random sources, they can robustly simulate precisely finite $\coop{p}$-random sources.
\end{lem}
\begin{proof}
Clearly, any random source they simulate must take $p$-cooperative probabilities, because any space with a subset of mass $\alpha$ has the space $(\alpha \hspace*{8pt} 1-\alpha)$ as a quotient.
In the other direction, consider a finite probability space with point masses \begin{equation} (\begin{array}{cccc} \alpha_1 & \alpha_2 & \cdots & \alpha_n \end{array}) \end{equation} in $\coop{p}$. Robustly flip a coin of bias $\alpha_1$. In the heads case, we pick the first point. In the tails case, we apply induction to robustly simulate \begin{equation} (\begin{array}{ccc} \alpha_2/(1-\alpha_1) & \cdots & \alpha_n/(1-\alpha_1) \end{array}). \end{equation} This is possible because $1-\alpha_1 \in \coop{p}$ by symmetry, and so the ratios $\alpha_i/(1-\alpha_1) \in \coop{p}$ by assumption.
\end{proof}
\subsection{Restatement using multilinear algebra}
Consider a $\{$heads, tails$\}$-valued function of several independent finite probability spaces that produces an $\alpha$-biased coin flip when random sources sample the spaces. If we model each probability space as a stochastic vector---that is, a nonnegative vector whose coordinates sum to one---we may view the product probability space as the Kronecker product of these vectors. Each entry in the resulting tensor represents the probability of a certain combination of outputs from the random sources. Since the sources together determine the flip, some of these entries should be marked ``heads,'' and the rest ``tails.''
For instance, if we have a fair die and a fair coin at our disposal, we may cook up some rule to assign ``heads'' or ``tails'' to each combination of results:
\begin{equation}
\left(
\begin{array}{c}
\frac16 \\
\vspace*{-9pt} \\
\frac16 \\
\vspace*{-9pt} \\
\frac16 \\
\vspace*{-9pt} \\
\frac16 \\
\vspace*{-9pt} \\
\frac16 \\
\vspace*{-9pt} \\
\frac16
\end{array}
\right)
\otimes
\left(
\begin{array}{cc}
\frac12 & \frac12
\end{array}
\right)
=\left(
\begin{array}{cc}
\frac1{12} & \frac1{12} \\
\vspace*{-9pt} \\
\frac1{12} & \frac1{12} \\
\vspace*{-9pt} \\
\frac1{12} & \frac1{12} \\
\vspace*{-9pt} \\
\frac1{12} & \frac1{12} \\
\vspace*{-9pt} \\
\frac1{12} & \frac1{12} \\
\vspace*{-9pt} \\
\frac1{12} & \frac1{12}
\end{array}
\right)
\longrightarrow
\left(
\begin{array}{cccccc}
H & T \\
\vspace*{-9pt} \\
H & T \\
\vspace*{-9pt} \\
T & H \\
\vspace*{-9pt} \\
H & T \\
\vspace*{-9pt} \\
T & H \\
\vspace*{-9pt} \\
T & H
\end{array}
\right)
\end{equation}
If we want to calculate the probability of heads, we can substitute $1$ for $H$ and $0$ for $T$ in the last matrix and evaluate
\begin{equation}
\left(
\begin{array}{cccccc}
\frac16 & \frac16 & \frac16 & \frac16 & \frac16 & \frac16
\end{array}
\right)
\left(
\begin{array}{cc}
1 & 0 \\
1 & 0 \\
0 & 1 \\
1 & 0 \\
0 & 1 \\
0 & 1
\end{array}
\right)
\left(
\begin{array}{c}
\frac12 \\
\vspace*{-9pt} \\
\frac12
\end{array}
\right)
= \frac12.
\end{equation}
This framework gives an easy way to check if the algorithm is robust in the sense of Yao. If one of the random sources is miscalibrated (maybe the die is a little uneven), we may see what happens to the probability of heads:
\begin{equation}
\left(
\begin{array}{cccccc}
\frac1{12} & \frac1{10} & \frac16 & \frac14 & \frac1{15} & \frac13
\end{array}
\right)
\left(
\begin{array}{cc}
1 & 0 \\
1 & 0 \\
0 & 1 \\
1 & 0 \\
0 & 1 \\
0 & 1
\end{array}
\right)
\left(
\begin{array}{c}
\frac12 \\
\vspace*{-9pt} \\
\frac12
\end{array}
\right)
= \frac12.
\end{equation}
It's unaffected! In fact, defining
\begin{equation}
A\left(\ex{1}, \ex{2}\right) = \ex{1} \left(
\begin{array}{cc}
1 & 0 \\
1 & 0 \\
0 & 1 \\
1 & 0 \\
0 & 1 \\
0 & 1
\end{array}
\right)
\ex{2}^\top,
\end{equation}
we see that letting
$\beeta{1} = \left(
\begin{array}{cccccc}
\frac16 & \frac16 & \frac16 & \frac16 & \frac16 & \frac16
\end{array}
\right)$
and
$\beeta{2} = \left(
\begin{array}{cc}
\frac12 & \frac12
\end{array}
\right)$
gives us
\begin{eqnarray}
A\left(\ex{1}, \beeta{2}\right) & = & \frac12 \nonumber \\
A\left(\beeta{1}, \ex{2}\right) & = & \frac12
\end{eqnarray}
\noindent for all $\ex{1}$ and $\ex{2}$ of mass one. These relations express Yao's notion of robustness; indeed, changing at most one of the distributions to some other distribution leaves the result unaltered. As long as no two of the sources are miscalibrated, the bit is generated with probability $1/2$.
If $\alpha$ denotes the bias of the bit, we may write the robustness condition as
\begin{eqnarray}
A\left(\ex{1}, \beeta{2}\right) & = & \alpha J\left(\ex{1}, \beeta{2}\right) \nonumber \\
A\left(\beeta{1}, \ex{2}\right) & = & \alpha J\left(\beeta{1}, \ex{2}\right)
\end{eqnarray}
since the $\beeta{i}$ both have mass one. (Here as always, $J$ stands for the all-ones tensor of appropriate dimensions.) These new equations hold for all $\ex{i}$, by linearity. Subtracting, we obtain
\begin{eqnarray}
0 & = & (\alpha J - A)\left(\ex{1}, \beeta{2}\right) \nonumber \\
0 & = & (\alpha J - A)\left(\beeta{1}, \ex{2}\right)
\end{eqnarray}
which says exactly that the bilinear form $\left( \alpha J - A \right)$ is degenerate, i.e., that
\begin{equation}
\Det(\alpha J - A) = 0.\footnote{If the matrix $(\alpha J - A)$ is not square, this equality should assert that all determinants of maximal square submatrices vanish. }
\end{equation}
These conditions seem familiar: Changing the all-ones matrix $J$ to the identity matrix $I$ would make $\alpha$ an eigenvalue for the left and right eigenvectors $\beeta{i}$. By analogy, we call $\alpha$ a \emph{mystery-value} of the matrix $A$ and the vectors $\beeta{i}$ \emph{mystery-vectors}. Here's the full definition:
\begin{defn}
A $p$-linear form $A$ is said to have \textbf{mystery-value} $\alpha$ and corresponding \textbf{mystery-vectors} $\beeta{i}$ when, for any $1 \leq j \leq p$,
\begin{equation}
0 = (\alpha J - A) \left( \beeta{1}, \ldots , \beeta{j-1} , \ex{j} ,\beeta{j+1}, \ldots , \beeta{p} \right) \mbox{ for all vectors $\ex{j}$.}
\end{equation}
We further require that $J(\beeta{i}) \not = 0$.
\end{defn}
\noindent We will see later that these conditions on $\left( \alpha J - A \right)$ extend the notion of degeneracy to multilinear forms in general. This extension is captured by a generalization of the determinant---the hyperdeterminant.\footnote{Hyperdeterminants were first introduced in the $2 \times 2 \times 2$ case by Cayley \cite{cayley}, and were defined in full generality and studied by Gelfand, Kapranov, and Zelevinsky \cite[Chapter 14]{GKZ}.} Hyperdeterminants will give meaning to the statement $\Det (\alpha J - A) = 0 $, even when $A$ is not bilinear.
This organizational theorem summarizes our efforts to restate the problem using multilinear algebra.
\begin{thm}
A function from the product of several finite probability spaces to the set $\{H, T\}$ generates an $\alpha$-biased bit robustly iff the corresponding multilinear form has mystery-value $\alpha$ with the probability spaces as the accompanying mystery-vectors.
\end{thm}
\noindent We may now show the equivalence of robustness and privacy more formally. Privacy requires that $(\alpha J - A)\left(\otimes \beeta{i} \right)$ remains zero, even if one of the distributions in the tensor product collapses to some point mass, that is, to some basis vector.\footnote{That is, the simulated bit remains a ``mystery'' to each player, even though she can see the output of her own random source.} This condition must hold for all basis vectors, so it extends by linearity to Yao's robustness.
\subsection{Two players}
The case $p=2$ leaves us in the familiar setting of bilinear forms.
\begin{prop}[Uniqueness]
Every bilinear form has at most one mystery-value.
\end{prop}
\begin{proof}
Suppose $\alpha$ and $\alpha^\prime$ are both mystery-values for the matrix $A$ with mystery-vectors $\beeta{i}$ and $\beeta{i}^\prime$, respectively. We have four equations at our disposal, but we will only use two:
\begin{eqnarray}
A\left(\ex{1} \hspace{1.73pt}, \rule[0pt]{0pt}{12pt} \hspace{1.73pt} \beeta{2} \right) & = & \alpha \nonumber \\
A\left(\beeta{1}^\prime, \rule[0pt]{0pt}{12pt} \ex{2} \right) & = & \alpha^\prime
\end{eqnarray}
We observe that a compromise simplifies both ways:
\begin{equation}
\alpha = A\left(\beeta{1}^\prime, \beeta{2} \right) = \alpha^\prime,
\end{equation}
so any two mystery-values are equal.
\end{proof}
\begin{cor}
Two players may not simulate an irrationally-biased coin.
\end{cor}
\begin{proof}
Say the $\{0,1\}$-matrix $A$ has mystery-value $\alpha$. Any field automorphism $\sigma \in \Gal(\mathbb{C}/\mathbb{Q})$ respects all operations of linear algebra, so $\sigma(\alpha)$ is a mystery-value of the matrix $\sigma(A)$. But the entries of $A$ are all rational, so $\sigma(A)=A$. Indeed, $\sigma(\alpha)$ must also be a mystery-value of $A$ itself. By the uniqueness proposition, $\sigma(\alpha) = \alpha$. Thus, $\alpha$ is in the fixed field of every automorphism over $\mathbb{Q}$ and cannot be irrational.
\end{proof}
\begin{thm}\label{2rat}
$\coop{2} = \mathbb{Q} \cap [0,1]$. Two people with finite random sources can robustly simulate only $\mathbb{Q}$-random sources; indeed, they can already simulate a full-strength finite $\mathbb{Q}$-random source if they have full-strength finite $\mathbb{Q}$-random sources.
\end{thm}
\begin{proof}
The previous corollary shows that no probability generated by the source can be irrational, since it could be used to simulate an irrationally-biased coin. The other direction has already been shown in Corollary \ref{qcon}.
\end{proof}
\begin{prop}\label{ratcon}
If $p$ people have full-strength finite $\mathbb{Q}$-random sources, they may $(p-1)$-robustly simulate any finite $\mathbb{Q}$-random source.
\end{prop}
\begin{proof}
Follows from Proposition \ref{diecon} just as the constructive direction of Theorem \ref{2rat} does.
\end{proof}
\subsection{Three or more players: what can't be done}
Even if three or more players have private finite random sources, it remains impossible to robustly simulate a transcendentally-biased coin. The proof makes use of algebraic geometry, especially the concept of the dual of a complex projective variety. We describe these ideas briefly in Appendix \ref{geoapp}. For a more thorough introduction, see \cite[Lec. 14, 15, 16]{harris} or \cite[Ch. 1]{GKZ}.
Let $A$ be a rational multilinear functional of format $n_1 \times \cdots \times n_p$ (see Section \ref{format}), and let $X$ be the Segre variety of the same format. Set $n := n_1 \cdots n_p - 1$, the dimension of the ambient projective space where $X$ lives. In what follows, we prove that $A$ has algebraic mystery-values. This is trivial when $A$ is a multiple of $J$, and for convenience we exclude that case.
\begin{prop}\label{tangency}
Let $A$ have mystery-value $\alpha$ with corresponding mystery-vectors $\beeta{i}$. Define $\beta = \otimes \beeta{i}$, and let $\mathfrak{B}$ denote the hyperplane of elements of $(\mathbb{P}^n)^\ast$ that yield zero when applied to $\beta$. Now $(\mathfrak{B}, \left( \alpha J - A \right))$ is in the incidence variety $W_{X^\vee}$ (see Section \ref{geoapp1}).
\end{prop}
\begin{proof}
By the biduality theorem \ref{biduality}, the result would follow from the statement,
\begin{equation} \mbox{``The hyperplane } \left\{\hspace{.02in} x \hspace{.015in} : \hspace{.025in} (\alpha J - A)(x) = 0 \right\} \mbox{ is tangent to $X$ at $\beta$.''} \end{equation}
But this statement is true by the partial derivatives formulation (Definition \ref{pderiv}) of the degeneracy of $\left( \alpha J - A \right)$.
\end{proof}
\noindent It is a standard fact (see \textit{e.g.} \cite[p. 6]{mumford}) that any variety has a stratification into locally closed smooth sets. The first stratum of $X^{\vee}$ is the Zariski-open set of smooth points of the variety. This leaves a subvariety of strictly smaller dimension, and the procedure continues inductively. Equations for the next stratum may be found by taking derivatives and determinants.
Since $X^{\vee}$ itself is defined over $\mathbb{Q}$, it follows that each of its strata is as well. We conclude that there must be some subvariety $S \subseteq X^{\vee}$, defined over $\mathbb{Q}$, that contains $\left( \alpha J - A \right)$ as a smooth point.
\begin{thm}
Any mystery-value of $A$ must be an algebraic number.
\end{thm}
\begin{proof}
Let $A' = \alpha J - A$, and let $\ell$ be the unique projective line through $A$ and $J$. Let $\mathbb{A}$ be some open affine in $(\mathbb{P}^n)^\ast$ containing $A'$ and $J$. The hyperplane $\frak{B} \cap \mathbb{A}$ is the zero locus of some degree one regular function $f$ on $\mathbb{A}$. On $\ell \cap \mathbb{A}$, this function will be nonzero at $J$ (since $J(\beta) \neq 0$), so $f$ is linear and not identically zero. It follows that $f(A) = 0$ is the unique zero of $f$ on $\ell$, occurring with multiplicity one. Thus, the restriction of $f$ to the local ring of $\ell$ at $A'$ is in the maximal ideal but not its square:
\begin{equation}
f \neq 0 \in \frak{m}_{\ell}/\frak{m}_{\ell}^2 = T^*_{A'}(\ell) \mbox{\hspace{15pt} where $\frak{m}_{\ell}$ denotes the maximal ideal in $ \mathcal{O}_{\ell,A'} $}.
\end{equation}
On the other hand, Proposition \ref{tangency} shows that $(\frak{B}, A') \in W_{X^\vee}$. Consequently, $\frak{B}$ must be tangent to $S$, that is, $f$ restricted to $S$ \emph{is} in the square of the maximal ideal of the local ring of $S$ at $A'$:
\begin{equation}
f = 0 \in \frak{m}_{S}/\frak{m}_{S}^2 = T^*_{A'}(S) \mbox{\hspace{15pt} where $\frak{m}_{S}$ denotes the maximal ideal in $ \mathcal{O}_{S,A'} $}.
\end{equation}
The function $f$ must be zero in the cotangent space of the intersection $S \cap \ell$ since the inclusion $S \cap \ell \hookrightarrow S$ induces a surjection
\begin{equation} T^*_{A'}(S) \twoheadrightarrow T^*_{A'}(S \cap \ell), \end{equation}
so the corresponding surjection
\begin{equation} T^*_{A'}(\ell) \twoheadrightarrow T^*_{A'}(S \cap \ell) \end{equation}
must kill $f$. This first space is the cotangent space of a line, hence one dimensional. But $f$ is nonzero in the first space, so the second space must be zero. It follows that $S \cap \ell$ is a zero dimensional variety.
Of course, $[\alpha : 1]$ lies in $S \cap \ell$, which is defined over $\mathbb{Q}$! The number $\alpha$ must be algebraic.
\end{proof}
\noindent Therefore, the set of $p$-cooperative numbers is contained in $\overline\mathbb{Q} \cap [0,1]$, and we have established the following proposition:
\begin{prop}
If several people with finite random sources simulate a private random source for someone else, that source must take probabilities in $\overline\mathbb{Q}$.
\end{prop}
\subsection{Three players: what can be done}
We prove that three players with private full-strength finite random sources are enough to simulate any private finite $\overline\mathbb{Q}$-random source. First, we give a construction for a hypermatrix with stochastic mystery-vectors for a given algebraic number $\alpha$, but whose entries may be negative. Next, we use it to find a nonnegative hypermatrix with mystery-value $(\alpha+r)/s$ for some suitable natural numbers $r$ and $s$. Then, after a bit of convex geometry to ``even out'' this hypermatrix, we scale and shift it back, completing the construction.
\begin{rmk}
Our construction may easily be made algorithmic, but in practice it gives hypermatrices that are far larger than optimal. An optimal algorithm would need to be radically different to take full advantage of the third person. The heart of our construction (see Proposition \ref{construction}) utilizes $2 \times (n+1) \times (n+1)$ hypermatrices, but the degree of the hyperdeterminant polynomial grows much more quickly for (near-)diagonal formats \cite[Ch. 14]{GKZ}. We would be excited to see a method of producing (say) small cubic hypermatrices with particular mystery-values.
\end{rmk}
\subsubsection{Hypermatrices with cooperative entries}
Recall that a $\{$heads, tails$\}$-function of several finite probability spaces may be represented by a $\{1,0\}$-hypermatrix. The condition that the entries of the matrix are either $1$ or $0$ is inconvenient when we want to build simulations for a given algebraic bias. Fortunately, constructing a matrix with cooperative entries will suffice.
\begin{lem}\label{rationalmat}
Suppose that $A$ is a $p$-dimensional hypermatrix with p-cooperative entries in $[0,1]$ and stochastic mystery-vectors $\beeta{1}, \ldots, \beeta{p}$ for the mystery-value $\alpha$. Then, $\alpha$ is $p$-cooperative.
\end{lem}
\begin{proof}
Let the hypermatrix $A$ have entries $w_1, w_2, \ldots , w_n $. Each entry $w_k$ is $p$-cooperative, so it is the mystery-value of some $p$-dimensional $\{0,1\}$-hypermatrix $A_k$ with associated stochastic mystery-vectors $\beeeta{k}{1}, \beeeta{k}{2}, \ldots, \beeeta{k}{p}$. We now build a $\{0,1\}$-hypermatrix $A'$ with $\alpha$ as a mystery-value. The hypermatrix $A'$ has blocks corresponding to the entries of $A$. We replace each entry $w_i$ of $A$ with a Kronecker product:
\begin{equation}
w_i \mbox{ becomes } J_1 \otimes J_2 \otimes \cdots \otimes J_{i-1} \otimes A_i \otimes J_{i+1} \otimes \cdots \otimes J_n.
\end{equation}
It is easy to check that the resulting tensor $A'$ has $\alpha$ as a mystery-value with corresponding mystery-vectors $\beeta{i} \otimes \beeeta{1}{i} \otimes \beeeta{2}{i} \otimes \cdots \otimes \beeeta{n}{i}$.
\end{proof}
\noindent Because rational numbers are $2$-cooperative, this lemma applies in particular to rational $p$-dimensional hypermatrices, for $p \geq 2$. In this case and in others, the construction can be modified to give an $A'$ of smaller format.
Readers who have been following the analogy between mystery-values and eigenvalues will see that Lemma \ref{rationalmat} corresponds to an analogous result for eigenvalues of matrices. Nonetheless, there are striking differences between the theories of mystery-values and eigenvalues. For instance, we are in the midst of showing that it is always possible to construct a nonnegative rational hypermatrix with a given nonnegative algebraic mystery-value and stochastic mystery-vectors. The analogous statement for matrix eigenvalues is false, by the Perron-Frobenius theorem: any such algebraic number must be greater than or equal to all of its Galois conjugates (which will also occur as eigenvalues). Encouragingly, the inverse problem for eigenvalues has been solved: Every ``Perron number'' may be realized as a ``Perron eigenvalue'' \cite{lind}. Our solution to the corresponding inverse problem for mystery-values uses different techniques. It would be nice to see if either proof sheds light on the other.
\subsubsection{Constructing hypermatrices from matrices}
\begin{prop}\label{companion}
If $\lambda$ is a real algebraic number of degree $n$, then there is some $M \in \M_n(\mathbb{Q})$ having $\lambda$ as an eigenvalue with non-perpendicular positive left and right eigenvectors.
\end{prop}
\begin{proof}
Let $f \in \mathbb{Q}[x]$ be the minimal polynomial for $\lambda$ over $\mathbb{Q}$, and let $L$ be the companion matrix for $f$. That is, if
\begin{equation}
f(x) = x^n + \sum_{k=0}^{n-1} a_k x^k \mbox{ for } a_k \in \mathbb{Q},
\end{equation}
then
\begin{equation}
L=\left(
\begin{array}{ccccc}
0 & 0 & \cdots & 0 & -a_0 \\
1 & 0 & \cdots & 0 & -a_1 \\
0 & 1 & \cdots & 0 & -a_2 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & \cdots & 1 & -a_{n-1}
\end{array}
\right).
\end{equation}
The polynomial $f$ is irreducible over $\mathbb{Q}$, so it has no repeated roots in $\mathbb{C}$. The matrix $L$ is therefore diagonalizable, with diagonal entries the roots of $f$. Fix a basis for which $L$ is diagonal, with $\lambda$ in the upper-left entry. In this basis, the right and left eigenvectors, $v_0$ and $w_0$, corresponding to $\lambda$ are zero except in the first coordinate. It follows that $v_0(w_0) \neq 0$.
The right and left eigenvectors may now be visualized as two geometric objects: a real hyperplane and a real vector not contained in it. It's clear that $\GL_n(\mathbb{R})$ acts transitively on the space $\mathcal{S} := \{(v,w) \in (\mathbb{R}^n)^* \times \mathbb{R}^n : v(w) = v_0(w_0) \}$. Moreover, $\GL_n(\mathbb{Q})$ is dense in $\GL_n(\mathbb{R})$, so the orbit of $(v_0,w_0)$ under the action of $\GL_n(\mathbb{Q})$ is dense in $\mathcal{S}$. The set of positive pairs in $\mathcal{S}$ is non-empty and open, so we may rationally conjugate $L$ to a basis which makes $v_0$ and $w_0$ positive.
\end{proof}
\begin{prop}\label{construction}
If $\lambda$ is real algebraic, then there exist integers $r \geq 0$, $s > 0$ such that $(\lambda+r)/s \in \coop{3}$.
\end{prop}
\begin{proof}
By Proposition \ref{companion}, there is a rational $n \times n$ matrix $M$ with non-perpendicular positive right and left eigenvectors $v,w$ for the eigenvalue $\lambda$. Rescale $w$ so that $v(w) = 1$, and choose an integer $q \geq \max{\{J(v), J(w)\}}$. Define the block $2 \times (n+1) \times (n+1)$ hypermatrix
\begin{equation}
A := \left(
\begin{array}{ccc}
\begin{array}{c|cccr}
0 \ & \ 0 & \hspace{6pt} \cdots & & 0 \\ \hline
0 \ & & & & \\
\vdots \ & & \hspace{8pt} q^2 M & & \\
0 \ & & & &
\end{array}
& \rule[-34pt]{1.1pt}{74pt} &
\begin{array}{l|cccr}
1 \ & \ 1 & & \ \hspace{8pt} \cdots & 1 \\ \hline
1 & & & & \\
\vdots & \multicolumn{4}{c}{q^2 (M - I) + J} \\
1 & & & &
\end{array}
\end{array} \right),
\end{equation}
where $I$ and $J$ are the $n \times n$ identity and all-ones matrices, respectively. Consider $A$ as a trilinear form, where the metacolumns correspond to the coordinates of the first vector, the rows the second, and the columns the third. Define the block vectors
\begin{equation}
\begin{array}{rclccccccrl}
\beeta{1} &=& \left( \hspace*{2.5pt} 1-\lambda \right. & \lambda \left. \right), & & & & & & &\\
\beeta{2} &=&
\multicolumn{2}{l}{\left( \right. 1 - J(v)/q } & |
& v_1 / q & v_2 / q & & \cdots & v_n / q \left. \right)& \hspace{-0pt} \mbox{, and} \\
\beeta{3} &=& \multicolumn{2}{l}{\left( \right. 1 - J(w)/q } & | & w_1 / q & w_2 / q & & \cdots & w_n / q \left. \right)& \hspace{-0pt}.
\end{array}
\end{equation}
Clearly, these are all probability vectors. It's easy to verify that
\begin{eqnarray}
A\left(\ex{1}, \beeta{2}, \beeta{3} \right) &=& \lambda J\left(\ex{1}\right),\nonumber\\
A\left(\beeta{1}, \ex{2}, \beeta{3} \right) &=& \lambda J\left(\ex{2}\right), \mbox{ and}\nonumber\\
A\left(\beeta{1}, \beeta{2}, \ex{3} \right) &=& \lambda J\left(\ex{3}\right).
\end{eqnarray}
Choose a nonnegative integer $r$ large enough so that all the entries of $A + r J$ are positive, and then a positive integer $s$ so that all the entries of $A' := (A + r J)/s$ are between $0$ and $1$.
\begin{eqnarray}
A'\left(x^{(1)}, \beeta{2}, \beeta{3}\right) &=& \frac{\lambda + r}{s} J\left(\ex{1}\right),\nonumber\\
A'\left(\beeta{1}, x^{(2)}, \beeta{3}\right) &=& \frac{\lambda + r}{s} J\left(\ex{2}\right), \mbox{ and}\nonumber\\
A'\left(\beeta{1}, \beeta{2}, x^{(3)}\right) &=& \frac{\lambda + r}{s} J\left(\ex{3}\right).
\end{eqnarray}
By Lemma \ref{rationalmat}, it follows that $(\lambda + r)/s$ is $3$-cooperative.
\end{proof}
\subsubsection{Finishing the Proof}
The following lemma, which we we prove later, enables us to complete the goal of this section: to classify which private random sources three or more people can simulate.
\begin{lem}[Approximation lemma]\label{approximation}
Let $\alpha$ be a $p$-cooperative number. Now for any $\varepsilon > 0$ there exists a $p$-dimensional rational hypermatrix whose entries are all within $\varepsilon$ of $\alpha$, having $\alpha$ as a mystery-value with stochastic mystery-vectors.
\end{lem}
\begin{thm}\label{constructive}
$\coop{p} = \overline{\mathbb{Q}} \cap [0,1]$ for each $p \geq 3$.
\end{thm}
\begin{proof}
Certainly $0$ and $1$ are $3$-cooperative. Let $\alpha$ be an algebraic number in $(0,1)$. By Proposition \ref{construction}, there are integers $r \geq 0$, $s>0$ so that $(\alpha + r)/s$ is $3$-cooperative. Let $\varepsilon := \left( \min \{\alpha, 1-\alpha\} \right) / s$.
By Proposition \ref{approximation}, there is some three-dimensional rational hypermatrix $A$ whose entries are all within $\varepsilon$ of $(\alpha + r)/s$, having $(\alpha + r)/s$ as a mystery-value with stochastic mystery-vectors. Then, $s A - r J$ is a three-dimensional rational hypermatrix with entries between $0$ and $1$, having $\alpha$ as a mystery-value with stochastic mystery-vectors. By Lemma \ref{rationalmat}, $\alpha$ is $3$-cooperative.
We already showed that all cooperative numbers are algebraic. Thus, for $p \geq 3$,
\begin{equation} \overline{\mathbb{Q}} \cap [0,1] \subseteq \coop{3} \subseteq \coop{p} \subseteq \overline{\mathbb{Q}} \cap [0,1], \end{equation}
so $\coop{p} = \overline{\mathbb{Q}} \cap [0,1]$.
\end{proof}
In conclusion, we have the following theorem.
\begin{thm}\label{main}
Three or more people with finite random sources can robustly simulate only $\overline\mathbb{Q}$-random sources. Indeed, if they have full-strength finite $\overline\mathbb{Q}$-random sources, they can already robustly simulate a full-strength finite $\overline\mathbb{Q}$-random source.
\end{thm}
\subsubsection{Proof of the approximation lemma}\label{aapp}
The proof that follows is a somewhat lengthy ``delta-epsilon'' argument broken down into several smaller steps. As we believe our construction of a hypermatrix with mystery-value $\alpha$ to be far from optimal, we strive for ease of exposition rather than focusing on achieving tight bounds at each step along the way.
Recall that a finite probability space may be usefully modeled by a positive\footnote{We may leave out points of mass zero.} vector of mass one. Let $\beta$ be such a vector. We denote by $\#\beta$ the number of coordinates of $\beta$ . We say $\beta^\prime$ is a \emph{refinement} of $\beta$ when $\beta$ is the image of a measure-preserving map from $\beta^\prime$; that is, when the coordinates of $\beta^\prime$ may be obtained by splitting up the coordinates of $\beta$.
The following easy lemma states that any positive vector of unit mass can be refined in such a way that all the coordinates are about the same size.
\begin{lem}[Refinement lemma]
Let $\beta$ be a positive vector of total mass $1$. For any $\delta > 0$ there exists a refinement $\beta^\prime$ of $\beta$ with the property that
\begin{equation} \min_j \beta_j^\prime \geq \frac{1-\delta}{\#\beta^\prime}. \end{equation}
\end{lem}
\begin{proof}
Without loss of generality, assume that $\beta_1$ is the smallest coordinate of $\beta$. Let $\gamma = \beta_1 \delta$, and let $k = \#\beta$. The vector $\beta$ is in the standard open $k$-simplex
\begin{equation}
\Delta^k = \{\mbox{positive vectors of mass 1 and dimension $k$}\}.
\end{equation}
The rational points in $\Delta^k$ are dense (as in any rational polytope), and
\begin{equation}
U := \{x \in \Delta^k : (\forall i) \left|\beta_i -x_i\right| < \gamma \mbox{ and } \beta_1 < x_1\}
\end{equation}
is an open subset of the simplex. So $U$ contain a rational point $\left(\frac{n_1}{n}, \ldots, \frac{n_k}{n}\right)$, with $n = \sum n_i$. Thus, $\left|\beta_i - \frac{n_i}{n}\right| < \gamma$ and $\beta_1 < \frac{n_1}{n}$, so
\begin{equation}
\left|\frac{\beta_i}{n_i} - \frac{1}{n}\right| < \frac{\gamma}{n_i} \leq \frac{\gamma}{n_1} < \frac{\gamma}{\beta_j n} = \frac{\delta}{n}.
\end{equation}
Let $\beta^\prime$ be the refinement of $\beta$ obtained by splitting up $\beta_i$ into $n_i$ equal-sized pieces. We have $\#\beta^\prime = n$, and the claim follows from this last inequality.
\end{proof}
\begin{rmk}
The best general bounds on the smallest possible $\#\beta^\prime$ given $\beta$ and $\delta$ are not generally known, but fairly good bounds may be obtained from the multidimensional version of Dirichlet's theorem on rational approximation, which is classical and elementary \cite{davenport}. Actually calculating good simultaneous rational approximations is a difficult problem, and one wishing to make an algorithmic version of our construction should consult the literature on multidimensional continued fractions and Farey partitions, for example, \cite{lagarias, farey}.
\end{rmk}
\noindent The next proposition is rather geometrical. It concerns the $n \times n$ matrix $S_{\delta} := (1-\delta)(J/n ) + \delta I$, which is a convex combination of two maps on the standard simplex: the averaging map and the identity map. Each vertex gets mapped almost to the center, so the action of $S_{\delta}$ can be visualized as shrinking the standard simplex around its center point. The proposition picks up where the refinement lemma left off:
\begin{prop}\label{second}
If a stochastic vector $\beta$ satisfies
\begin{equation} \min_i \beta_i \geq \frac{1-\delta}{\#\beta} \end{equation} then its image under the map $S^{-1}_{\delta} $ is still stochastic.
\end{prop}
\begin{proof}
First note that
$\left[ (1-\delta) \left(J / \# \beta \right) + \delta I \right] \left[ (1-1/\delta)\left(J / \# \beta \right) + (1/\delta) I \right] = I$,
so we have an explicit form for $S^{-1}_{\delta}$.
We know that $ \min_i \beta_i \geq (1-\delta)/\#\beta$, so the vector \begin{equation} E=\frac{1}{\delta} \left[ \beta - \left( \frac{1-\delta}{\#\beta} \right) J \right] \end{equation}
is still positive. Now
$\beta = (1-\delta) \left(J / \#\beta \right) + \delta E$,
a convex combination of two positive vectors. The vector $\beta$ has mass $1$, and $ \left(J / \#\beta \right)$ as well, so $E$ also has mass $1$.
Now compute:
\begin{eqnarray}
S^{-1}_{\delta} \beta & = & \biggl[ (1-1/\delta) \left(J / \#\beta \right) + (1/\delta) I \biggr] \biggl[ (1-\delta) \left(J / \#\beta \right) + \delta E \biggr] \nonumber \\
& = & \biggl[ (1 - 1/\delta)(1 - \delta) + (1/\delta)(1 - \delta) + (1 - 1/\delta)\delta \biggr] \left(J / \#\beta \right) + E \nonumber \\
& = & E.
\end{eqnarray}
This completes the proof.
\end{proof}
\noindent The following proposition shows that applying the matrix $S_{\delta}$ in all arguments of some multilinear functional forces the outputs to be close to each other.
\begin{prop}\label{third}
Let $A$ be a hypermatrix of format $n_1 \times n_2 \times \cdots \times n_p$ with entries in $[0,1]$, and take $\delta := \varepsilon / (2p)$. Now the matrix $A'$ defined by
\begin{equation} A^\prime\left(\otimes x^{(i)}\right) := A\left(\otimes S_{\delta} x^{(i)}\right) \end{equation}
satisfies $| A'(x) - A'(x') | \leq \varepsilon$
for any two stochastic tensors $x$ and $x'$.
\end{prop}
\begin{proof}
Let $m := A\left(\otimes (J / n_i)\right)$, the mean of the entries of $A$. We show that for any stochastic vectors $x^{(i)}$,
\begin{equation} \left|A'\left(\otimes x^{(i)}\right) - m\right| \leq \varepsilon/2. \end{equation}
Since any other stochastic tensor is a convex combination of stochastic pure tensors, it will follow that
$|A'(x) - m| \leq \varepsilon/2$.
Then the triangle inequality will yield the result.
It remains to show that $A'$ applied to a stochastic pure tensor gives a value within $\varepsilon /2 $ of $m$.
\begin{eqnarray}
A'\left(\otimes x^{(i)}\right) & = & A\left(\otimes S_{\delta} x^{(i)}\right) \nonumber \\
& = & A\left( \otimes \left[ (1-\delta)(J /n_i) + \delta I \right] x^{(i)} \right) \nonumber \\
& = & A\left( \otimes \left[(1-\delta)(J /n_i) + \delta x^{(i)} \right] \right). \label{bad}
\end{eqnarray}
Each argument of $A$---that is, factor in the tensor product---is a convex combination of two stochastic vectors. Expanding out by multilinearity, we get convex combination with $2^p$ points. Each point---let's call the $k^{\mbox{\tiny{th}}}$ one $y_k$---is an element of $[0,1]$ since it is some weighted average of the entries of $A$. This convex combination has positive $\mu_k$ such that $\sum \mu_k = 1$ and
\begin{equation} A'\left(\otimes x^{(i)}\right) = \sum_{k=1}^{2^p} \mu_k y_k. \end{equation}
Taking the first vector in each argument of $A$ in \eqref{bad}, we see that $y_1 = A\left(\otimes (J / n_i)\right)=m$, the average entry of $A$. Thus, the first term in the convex combination is
$\mu_1 y_1 = (1-\delta)^p m$.
The inequality $(1-\varepsilon/2) \leq (1-\delta)^p$ allows us to split up the first term. Let $\mu_0 := 1-\varepsilon/2$ and $\mu_1' := \mu_1 - \mu_0 \geq 0$. We have
$\mu_1 y_1 = (\mu_0 + \mu_1') y_1 = (1-\varepsilon/2) m + \mu_1' m$.
After splitting this term, the original convex combination becomes
\begin{equation}
A'\left(\otimes x^{(i)}\right) = (1-\varepsilon/2) m + \mu_1' m + \sum_{k=2}^{2^p} \mu_k y_k.
\end{equation}
Let $e$ denote the weighted average of the terms after the first. We may rewrite the convex combination
\begin{equation} A'\left(\otimes x^{(i)}\right) = (1-\varepsilon/2) m + (\varepsilon/2) e. \end{equation}
Since $m, e \in [0,1]$,
\begin{equation} m-\varepsilon / 2 \leq (1-\varepsilon /2) m \leq A'\left(\otimes x^{(i)}\right) \leq (1-\varepsilon /2) m + \varepsilon / 2 \leq m + \varepsilon /2, \end{equation}
and
\begin{equation} \left| A'\left(\otimes x^{(i)}\right) - m \right| \leq \varepsilon / 2,\end{equation}
so we are done.
\end{proof}
These results are now strong enough to prove the approximation lemma \ref{approximation}.
\begin{proof}
The number $\alpha$ is $p$-cooperative, so it comes with some $p$-dimensional nonnegative rational hypermatrix $A$ and positive vectors $\beeta{1}, \beeta{2}, \ldots, \beeta{p}$ of mass one, satisfying (in particular) $A\left( \otimes \beeta{i} \right) = \alpha$. The refinement lemma allows us to assume that each $\beeta{i}$ satisfies
\begin{equation} \min_j \beta^{(i)}_j \geq \frac{1-\delta}{\#\beta^{(i)}}. \end{equation}
If one of the $\beeta{i}$ fails to satisfy this hypothesis, we may replace it with the refinement given by the lemma, and duplicate the corresponding slices in $A$ to match.
Now, by Proposition \ref{second}, each $S^{-1}_{\delta} \beeta{i}$ is a stochastic vector.
Let $A'$ be as in Proposition \ref{third}. It will still be a rational hypermatrix if we pick $\varepsilon$ to be rational. We know
\begin{equation} A'\left( \otimes S^{-1}_{\delta} \beeta{i} \right) = \alpha. \end{equation}
On the other hand, any entry of the matrix $A'$ is given by evaluation at a tensor product of basis vectors. Both $\alpha$ and any entry of $A'$ can be found by evaluating $A'$ at a stochastic tensor. Thus, by Proposition \ref{third}, each entry of $A'$ is within $\varepsilon$ of $\alpha$.
\end{proof}
\subsection{Higher-order robustness}
We complete the proof of our main theorem.
\begin{prop}\label{onlyrat}
If $r \geq p/2$, then $p$ people with finite random sources may $r$-robustly simulate only finite $\mathbb{Q}$-random sources.
\end{prop}
\begin{proof}
Consider an $r$-robust simulation. Imagine that Alice has access to half of the random sources (say, rounded up), and Bob has access to the remaining sources. Because Alice and Bob have access to no more than $r$ random sources, neither knows anything about the source being simulated. But this is precisely the two-player case of ordinary $1$-robustness, so the source being simulated is restricted to rational probabilities.
\end{proof}
\noindent In the constructive direction, we show the following:
\begin{prop}\label{bigcon}
If $r < p/2$, then $p$ people with full-strength finite $\overline \mathbb{Q}$-random sources may $r$-robustly simulate a full-strength finite $\overline \mathbb{Q}$-random source.
\end{prop}
\noindent The proof is to simulate simulations (and simulate simulations of simulations, etc.). We treat the $p=3$ case of our $1$-robust simulation protocol as a black box. If a majority of the random sources put into it are reliable, the one that comes out (the simulated random source) will also be reliable. This viewpoint leads us into a discussion of majority gates.
\begin{defn}
A \textbf{p-ary majority gate} is a logic gate that computes a boolean function returning $1$ if a majority of its inputs are $1$ and $0$ if a majority of its inputs are $0$. (The output doesn't matter when there are ties.)
\end{defn}
\begin{lem}[Bureaucracy]
A $p$-ary majority gate may be built by wiring together ternary majority gates.
\end{lem}
\noindent The proof of the bureaucracy lemma is a straightforward application of the probabilistic method, and is covered in detail in Appendix \ref{bapp}. Now, by iterating simulations of simulations according to the wiring provided by the bureaucracy lemma, we can overcome any minority of malfunctioning sources. So the bureaucracy lemma, together with the ``black box'' of our three-player construction, implies Proposition \ref{bigcon}.
Now we're finally ready to prove our main result. The statement here is equivalent to the ones in the abstract and in Section \ref{results} but uses the language of robustness.
\begin{thm}\label{main}
Say $p$ people have full-strength finite random sources. If $p/2 \leq r < p$, the people may $r$-robustly simulate any finite $\mathbb{Q}$-random source and nothing better; if $1 \leq r < p/2$, they may $r$-robustly simulate any finite $\overline{\mathbb{Q}}$-random source and nothing better.
\end{thm}
\begin{proof}
The claim simply combines Proposition \ref{ratcon}, Proposition \ref{onlyrat}, Theorem \ref{main}, and Proposition \ref{bigcon}.
\end{proof}
\section{Application to Secure Multiparty Computation and Mental Poker}\label{application}
We begin with the classical case: Three gentlemen wish to play poker, but they live far away from each other, so playing with actual cards is out of the question. They could play online poker, in which another party (the remotely hosted poker program) acts as a dealer and moderator, keeping track of the cards in each player's hand, in the deck, etc., and giving each player exactly the information he would receive in a physical game. But this solution require our gentlemen to trust the moderator! If they fear the moderator may favor one of them, or if they wish to keep their game and its outcome private, they need another system.
A better solution is to use secure multiparty computation. Our gentlemen work to \emph{simulate} a moderator in a way that keeps the outcomes of the moderator's computations completely hidden from each of them. An unconditionally-secure method of playing poker (and running other games/computations) ``over the phone'' has been described in \cite{poker}.
In the classical case, the players may perform finite computations, communicate along private channels, and query full-strength finitary private random sources. The simulated moderator has the almost same abilities as the players, except that its private random source is limited to rational probabilities. The work of this paper expands this to all algebraic probabilities, and shows that one can do no better.
To see how this may be useful, think back to our poker players. They may be preparing for a poker tournament, and they may want to simulate opponents who employ certain betting strategies. But poker is a complicated multiplayer game (in the sense of economic game theory), and Nash equilibria will occur at mixed strategies with algebraic coefficients.\footnote{The appearance of algebraic (but not transcendental) coefficients in mixed strategies is explained by R. J. Lipton and E. Markakis here \cite{nash}.}
|
{
"timestamp": "2012-01-20T02:00:48",
"yymm": "1009",
"arxiv_id": "1009.4188",
"language": "en",
"url": "https://arxiv.org/abs/1009.4188"
}
|
\section{Introduction}
One of the most striking aspects of the physics of Bose-Einstein condensed
systems is their response to rotation. The rotation plays the role of a
uniform magnetic field, which frustrates the uniform condensate, forcing it into a state containing quantized vortices and carrying non-vanishing
currents.\cite{blochdz,fetterreview,advances} Theory shows that at
sufficiently high vortex density this frustration can lead to the breakdown of
Bose-Einstein condensation, and the formation of a series of strongly
correlated quantum phases which can be viewed as bosonic analogues of the
fractional quantum Hall states.\cite{advances}
In typical magnetically trapped Bose gases\cite{SchweikhardCEMC92} practical
limitations on the rotation rate (vortex density) are such that strongly
correlated phases are expected only at a very low particle density where the interaction energy scale is very small.\cite{advances} As a
result, it has proved difficult to reach this strongly correlated
regime. (However, see Ref.~\onlinecite{gemelke} for interesting recent results
for systems with small particle numbers.)
It has been proposed that one can exploit the strong interactions that are
available in systems of bosonic atoms confined to optical
lattices\cite{blochdz} to enhance the possibility of achieving these
correlated phases. In this context, the natural model to consider is the
Bose-Hubbard model with uniform effective magnetic flux
[Eq.~(\ref{eq:hamiltonian})]. This ``frustrated'' Bose-Hubbard model can show
very interesting physics, far beyond the physics of the usual Bose-Hubbard
model.\cite{fisherbh} Atomic systems well-described by this frustrated
Bose-Hubbard have been studied experimentally by using rotating optical
lattices,\cite{tung,footlattice} albeit so far limited to situations of large
lattice constants and large numbers of particles per lattice site which are
outside the strongly correlated regime. However, a series of
theoretical
proposals\cite{JakschZoller,mueller,sorensen:086803,palmer,palmer:013609,hafezi-2007,gerbier}
indicate that it should be possible to imprint strong gauge fields on an optical lattice, and thereby realize a regime where interactions are
strong, and with both the particle number per site, $n$, and vortex number per
plaquette, $n_\phi$, of order one. In this regime, theory shows that there are
strongly correlated phases representative of the continuum quantum Hall states
limit\cite{sorensen:086803,hafezi-2007,hafezi-epl} as well as related
interesting strongly correlated phases that are stabilized by the lattice
itself.\cite{mollercooper-cf} Other candidates are related to Mott physics.\cite{Umucalilar:2010p395,Powell:2010p303}
Our confidence in the existence of strongly correlated phases of the
frustrated Bose-Hubbard model relies on the results of large-scale numerical
exact diagonalization
studies.\cite{sorensen:086803,hafezi-2007,hafezi-epl,mollercooper-cf}
However, these studies have found evidence for strongly correlated phases only
in a relatively small region of parameter space (spanned by the particle density
per site $n$, flux per plaquette $n_\phi$, and interaction strength
$U/J$). There are surely competing {\it condensed} phases, which can be viewed
as vortex lattices that are pinned by the lattice.\cite{duriclee} An important question emerges
from the point of view of these numerical approaches: How does one determine
condensation
in exact diagonalization studies? In conventional condensed systems, one
looks for the maximum eigenvalue of the single particle density matrix of the
groundstate.\cite{yang} However, here the condensed states are (pinned)
vortex lattices, and therefore break translational symmetry. As a result, one
expects a degeneracy of the spectrum in the thermodynamic
limit.\cite{cwg,advances} How does one quantify the degree of condensation?
In this paper we propose a powerful general numerical method that can be used
to identify and characterize condensed groundstates which break a symmetry of
the Hamiltonian. We use this to study several cases of interest in the context
of optically induced gauge potentials.\cite{dalibardreview} Optically induced
gauge potentials have recently been implemented experimentally without an
optical lattice.\cite{lin:130401,spielmanfield} These successes encourage a
high degree of optimism that the related schemes on optical
lattices\cite{JakschZoller,gerbier} will also be successful.
Motivated by the proposals of Jaksch and Zoller\cite{JakschZoller} and Gerbier
and Dalibard,\cite{gerbier} in this paper we focus not on the case of a
uniform magnetic field, but on the case of a two-dimensional square lattice
with a {\it staggered} magnetic field, with a flux per plaquette of magnitude
$\alpha$ that alternates in sign along one of the principal axes. This flux
configuration involves much a simpler experimental implementation than the
case of uniform flux. As described below, for the special case of $\alpha=1/2$
this is equivalent to the uniform flux. In this case, the model simulates a
quantum version of the ``fully frustrated'' XY model. For other values of
$\alpha$, it represents a class of frustrated quantum spin models. Related
but different staggered flux Hamiltonians can be generated by time-dependent
lattice potentials as discussed in Ref.~\onlinecite{morais}.
Based on numerical exact diagonalizations, we provide evidence
showing that the groundstate breaks translational invariance, and is condensed
for all flux densities and (repulsive) interaction strengths. We evaluate the
condensate fraction and condensate wavefunctions from the exact
diagonalization results. We describe how evidence for translational symmetry
breaking can be found in measurements of the real-space and momentum space
(expansion) profiles, and how these can be used to determine the condensate
depletion. As part of this work, we explain some important general aspects of
expansion imaging of systems involving optically induced gauge potentials.
\section{Model}
We shall study the properties of the two-dimensional Bose-Hubbard model
subject to an abelian gauge potential, as described by the Hamiltonian
\begin{eqnarray}
\nonumber
\hat{H} & = & -J \sum_{\langle i,j\rangle } \left[\hat{a}_i^\dag\hat{a}_j e^{i A_{ij}} +
\hat{a}_j^\dag\hat{a}_i e^{i A_{ji}}\right] + \frac{U}{2}\sum_i \hat{n}_i(\hat{n}_i-1)
\label{eq:hamiltonian}
\end{eqnarray}
The operator $\hat{a}_i^{(\dag)}$ destroys (creates) a boson on the lattice
site $i$, which we choose to form a square lattice; $U$ describes the onsite
repulsion ($U\geq 0$ is assumed throughout); $J$ is the nearest-neighbour
tunneling energy. The Hamiltonian conserves the total number of bosons,
$\hat{N} = \sum_i \hat{n}_i = \sum_i \hat{a}^\dag_i\hat{a}_i$. Throughout this
work, we shall consider the system to be uniform, with $N$ chosen such that
the mean particle density per lattice site is $n$. The results of these
studies can be used within the local density approximation to model
experimental systems which have an additional trapping potential.
The fields $A_{ij}$ (which satisfy $A_{ij} = -A_{ji}$) describe the imposed
gauge potential. All of the physics of the system defined by the Hamiltonian
(\ref{eq:hamiltonian}) (energy spectrum, response functions, etc.) is
gauge-invariant. Therefore, its properties depend only on the fluxes through plaquettes
\begin{equation} n^{a}_\phi \equiv \frac{1}{2\pi} \sum_{i,j\in a}
A_{ij}
\label{eq:flux}
\end{equation}
where $a$ labels the plaquette, and the sum represents the
directed sum of the gauge fields around that plaquette (the discrete
version of the line integral), as illustrated in
Fig.~\ref{fig:plaquette}(a).
\begin{figure}
\includegraphics[width=0.8\columnwidth]{fig1}
\caption{(a) The gauge-invariant flux through the plaquette, $n_\phi$, is
defined by $n_\phi \equiv \frac{1}{2\pi} \sum_{i,j} A_{ij} = \frac{1}{2\pi}\left(A_{12}+A_{23}+A_{34}+A_{41}\right)$
(b) The simplest optically induced gauge potential to imprint on the square lattice\protect\cite{JakschZoller,gerbier} has an alternating pattern of fluxes of magnitude $\alpha$, Eq.~(\ref{eq:staggered}).}
\label{fig:plaquette}
\end{figure}
Since each phase $A_{ij}$ is defined modulo $2\pi$, the gauge
invariant fluxes (\ref{eq:flux}) are defined modulo $1$ (i.e. are invariant under
$n_\phi^a \to n_\phi^a +1$), so they can be restricted to
the interval $-1/2< n^a_\phi \leq 1/2$.
The gauge-invariant fluxes through the plaquettes lead to an intrinsic
``frustration'' of condensed (superfluid) phases on the Bose-Hubbard
system. This is best understood in the case of strong interactions, $U
\gg J$, where double occupancy is excluded. In this hard-core limit,
the Bose-Hubbard model is equivalent to a spin-1/2 quantum magnet,
using the standard mapping $\hat{s}_i^z = \hat{n}_i-\frac{1}{2}$,
$\hat{s}_i^+ = \hat{a}_i^\dag$, $\hat{s}_i^- = \hat{a}_i$, with
Hamiltonian (up to a constant shift in energy)
\begin{eqnarray}
\hat{H}_{\rm h-c} & = & -J \sum_{\langle i,j\rangle} \left[\hat{s}_i^+\hat{s}^-_j e^{i A_{ij}} +
\hat{s}_j^+\hat{s}^-_i e^{-i A_{ij}} \right]
\label{eq:spham}
\end{eqnarray}
(The
conservation of particle number becomes conservation of $\hat{S}^z =
\sum_i \hat{s}^z_i = \hat{N} -1/2$.)
This Hamiltonian describes a quantum spin-1/2 magnet, experiencing XY
nearest neighbour spin exchange interactions. These exchange
interactions are {\it ``frustrated"} by the gauge fields.
The ``frustration'' can be seen by considering the natural mean-field limit of
the spin Hamiltonian (\ref{eq:spham}), generalizing from spin-$1/2$ to
spin-$S$ and taking the $S\to \infty$ limit.\cite{duriclee} Then, the (vector
of) spin operators $\hat{s}_i$ can be replaced by the classical vector
$\vec{s}$ of fixed length $S$. It is convenient to parameterize this vector as
\begin{equation} \vec{s} = S(\sin\theta \cos\phi,\sin\theta
\cos\phi,\cos\theta)
\label{eq:means}
\end{equation}
which represents the spin by the polar and azimuthal angles $\theta,\phi$.
The Hamiltonian becomes the
(classical) energy functional
\begin{equation}
H_{\rm mft} = -2JS^2 \sum_{\langle i,j\rangle} \sin\theta_i\sin\theta_j
\cos(\phi_i-\phi_j+ A_{ij})
\label{eq:sphammft}
\end{equation}
and it is natural define the fractional occupation of the lattice
sites by $n_i = (1/2) (1+ \cos\theta_i)$.
The restricted space of configurations with $\theta_i=\pi/2$ is
important for groundstate configurations with (uniform) density $n =
1/2$. In this case, $s_z =0$, so all spins lie in the $x$-$y$-plane and
the Hamiltonian becomes that of the frustrated XY
model.\cite{teitel} Here, ``frustration'' refers to the fact that,
with $n_\phi \neq 0$ for any plaquette, the angles $\phi_i$ around
this plaquette cannot be chosen in order to maximally satisfy the XY
exchange couplings. This is illustrated in Fig.~\ref{fig:choices}.
\begin{figure}
\includegraphics[width=0.8\columnwidth]{frustrated}$\qquad\;$
\caption{Illustration of the frustrated coupling of the XY spin
model around a plaquette with non-zero flux $n_\phi$. For a given
phase $\phi_1$ at site 1, one can choose $\phi_2$, $\phi_3$, and
$\phi_4$ to maximally reduce the energy on three of the bonds
(with a contribution of $-JS^2$ to the energy for each). However, the exchange energy on the remaining
bond is $ -JS^2\cos(A_{12} + A_{23} + A_{34} + A_{41}) =
-JS^2\cos(2\pi n_\phi)$. For $n_\phi\neq 0$, this bond cannot also
be fully satisfied, indicating the magnetic frustration. Maximal
frustration occurs for $n_\phi =1/2$.}
\label{fig:choices}
\end{figure}
Maximal frustration occurs for $n_\phi = 1/2$. This situation is referred to
as the ``fully frustrated'' case. The ``fully frustrated'' classical XY
model has been much studied as an interesting frustrated classical magnet with a
non-standard thermal phase transition.\cite{villain} The frustrated
Bose-Hubbard model that we study here is a class of frustrated spin-$1/2$
quantum systems that are analogous to this frustrated classical model. We
shall focus on the nature of the groundstates of the system.
\section{Staggered Fluxes}
Motivated by proposals for optically induced gauge
potentials,\cite{JakschZoller,gerbier} we consider a situation in
which the flux is staggered in the $x$ direction. Specifically, we
choose the gauge (and geometry) described in
Ref.~\onlinecite{gerbier}, for which
\begin{equation}
A_{ij} = 2\pi \alpha \, y_i
(x_i-x_j)\times(-1)^{x_i}
\label{eq:gauge}
\end{equation}
where $(x_i,y_i)$ is the pair of integers that define the position of
lattice site $i$ ({\it i.e.} the cartesian co-ordinates position in
units of the lattice constants, $a_x$, $a_y$, in the $x$- and
$y$-directions, as shown in Fig.\ref{fig:plaquette}(b)). Since the lattice is square with nearest-neighbour
hopping, this has the effect that hopping in the $y$ direction has no
gauge field ($A_{ij}=0$) while hopping in the $x$ direction involves a
phase $A_{ij} = 2\pi \alpha y_i (-1)^{x_i}$ that alternates from one
row to the next. The magnitude of the flux per plaquette is
\begin{equation}
n^{a}_\phi = (-1)^{x_a} \alpha \,,
\label{eq:staggered}
\end{equation}
where the position $x_a$ is defined by the $x$-position of the
``bottom-left'' corner of the plaquette ({\it i.e.} the minimum $x_i$ for
all sites $i$ surrounding the plaquette $a$). The flux is staggered
in the $x$-direction, as shown in Fig.~\ref{fig:plaquette}(b).
A special situation arises for $\alpha=1/2$. Then, the case of alternating
fluxes of $n^a_\phi = (-1)^{x_a}\frac{1}{2}$ is gauge-equivalent to that of
uniform flux $n^a_\phi = \frac{1}{2}$. The (gauge-invariant)
properties of this case have higher translational symmetry than the
case of $\alpha\neq 0,1/2$. Furthermore, a gauge can be chosen in
which $e^{i A_{ij}}$ is real, meaning that the Hamiltonian is
time-reversal symmetric. Indeed the gauge (\ref{eq:gauge}) has this
property.
While gauge invariance allows the physics of (\ref{eq:hamiltonian}) to
be studied in any gauge, as we shall describe below, the {\it
expansion images} of the atomic gas are gauge-dependent. We shall
therefore be clear to specify the gauge considered.
Under the conditions that we study, significant insight into the
physics of the frustrated Bose-Hubbard model can be obtained by studying
its properties within mean-field theory. Indeed, one important goal of
this work is to show how the mean-field groundstates emerge from the
results of exact diagonalization studies.
\subsection{Single Particle Spectrum}
We first ignore interactions, and study the single particle energy
eigenstates. For the gauge field we consider (\ref{eq:gauge}), the unit cell
can be chosen to have size $2\times 1$ (in the $x$- and $y$-directions), and
the energy eigenstates are then
\begin{equation}
\psi_i = e^{i (k_x x_i + k_y y_i)} \times \left\{ \begin{array}{l}
\psi_e \quad\quad\quad\quad x_i \;\mbox{even}\\
\psi_o e^{i 2\pi\alpha y_i} \quad x_i \;\mbox{odd}
\end{array}
\right.
\end{equation}
with the momenta in the ranges $-\pi/2\leq k_x < \pi/2$ and $-\pi\leq k_y <
\pi$. [We express $k_x$ and $k_y$ in units of $1/a_x$ and $1/a_y$
respectively.]
The energy eigenvalues $E$ and eigenfunctions
within the unit cell, $(\psi_e,\psi_o)$,
follow from
\begin{equation}
-2J \left(\begin{array}{cc}
\cos(k_y) & \cos(k_x) \\
\cos(k_x) & \cos(k_y+2\pi\alpha)
\end{array}\right)
\left(\begin{array}{c}
\psi_e\\
\psi_o
\end{array}\right)
= E
\left(\begin{array}{c}
\psi_e\\
\psi_o
\end{array}\right)
\label{eq:matrix}
\end{equation}
The lowest energy state has $k_x=0$. For $\alpha <
\frac{1}{\pi}\arccos\left((\sqrt{5}-1)/2\right) \simeq 0.288$, there
is a single minimum at $k_y=\pi\alpha$. However, for $\alpha > 0.288$
this minimum splits in two, and there are two {\it degenerate} minima
defining single particle states $\psi^A_i$ and $\psi^B_i$. The
wavevectors of these states, $k_y^{A,B}$ are shown in
Fig.~\ref{fig:minima}. They are related by $k^A_y+k_y^B = -2\pi
\alpha$.\footnote{Note that the positions $k^{A/B}$ of these minima are gauge dependent.}
Hence $\cos(k^{A/B}_y) = \cos(k^{B/A}_y+2\pi\alpha)$, so the
Hamiltonian (\ref{eq:matrix}) is the same for both states provided the
two sites of the unit cell are swapped. Therefore, $\psi^A$ and $\psi^B$
are exactly degenerate, and their wavefunctions related by
$\psi^{A}_{e/o} = \psi^{B}_{o/e}$.
\begin{figure}
\includegraphics[width=0.9\columnwidth]{kpoints}
\caption{The single-particle groundstate is non-degenerate for $\alpha< 0.289$, but is two-fold degenerate for $\alpha > 0.289$. The two minima are at wavevectors $k^A_y, k^B_y$ shown above (both have $k_x=0$).
}
\label{fig:minima}
\end{figure}
The appearance of two degenerate minima in the single-particle spectrum leads
to the question of whether the groundstate is a ``simple'' condensate, or
``fragmented''.\cite{mueller:033612} Following the general
result,\cite{nozieresfragment} one expects that weak repulsive interactions
will lead to a simple condensate in which all particles condense the same
single particle state. The nature of this condensate can depend on the
properties of the two states, and the nature of the interactions. However, if
the condensate wavefunction contains non-zero weights of both states, with
wavevectors $k_y^A$ and $k_y^B$, then the condensate breaks
translational symmetry in the $y$ direction.
Similar physics -- of a two-component Bose gas -- has been discussed recently
in the context of optically induced gauge potentials in the
continuum.\cite{spielmannonab,hononabelian,Wang:arXiv1006.514} For the case we consider,
one principle difference is
that there is an underlying lattice periodicity. Phases with broken
translational symmetry can therefore ``lock'' to the lattice periodicity,
leading to condensed groundstates with a broken {\it discrete} translational
symmetry. Furthermore, the unit cells of the phases that we describe contain
particle currents, which can be viewed as arrays of vortices and antivortices.
In the following we shall explore the nature of this symmetry
breaking. A principle result will be to show that the results of exact
diagonalization studies are consistent with simple condensation in a
symmetry broken state. We shall focus on two cases:
(i) $\alpha =1/2$. Here, the model is equivalent to the case of
uniform fluxes $n_\phi^a = 1/2$. The (gauge-invariant) system
therefore enjoys the full translational symmetry of the lattice. The
minima are spaced by $\Delta k_y = \pi$, suggesting that the broken
symmetry state will have translational period $2$ in the
$y$-direction. The symmetry broken state will have two
degenerate configurations.
(ii) $\alpha =
\frac{1}{\pi}\arccos\left(\frac{1}{2}\sqrt{\frac{1}{2}(9-\sqrt{65})}\right)
\simeq 0.389$. This value is chosen such that $\Delta k_y = 2\pi/3$,
such that the broken translational symmetry state has period $3$ in
the $y$-direction. The symmetry broken state will have three
degenerate configurations.
Throughout this work, we concentrate on cases where the particle density $n$
is non-integer. Thus, we do not discuss the properties of the Mott insulator
states. While frustration by the gauge fields can affect the stability of the
Mott insulating states,\cite{oktel,goldbaummueller} the qualitative form of these
insulating states is unchanged. We focus on the nature of the superfluid
states, the properties of which are much changed in the presence of the
frustrating gauge fields.
\subsection{Gross-Pitaevskii Theory}
One can understand the effects of weak interactions, {\it i.e.} $U\ll nJ$, within
Gross-Pitaevskii mean-field theory. One assumes complete condensation,
forming a many-body state
\begin{equation} |\Psi_c\rangle \equiv \left( \sum_i \psi^c_i
\hat{a}^\dag_i\right)^N |0\rangle
\label{eq:condensedstate}
\end{equation}
where $\psi^c_i$ is the (normalized) condensate wavefunction. The
condensate wavefunction is
determined by viewing (\ref{eq:condensedstate})
as a variational state, and
minimizing the average energy per particle
\begin{eqnarray}
\label{eq:ke}
\frac{\langle \hat{H}\rangle}{N} & = & -J \sum_{\langle i,j\rangle } \left[\psi^{c*}_i\psi^c_j e^{i A_{ij}} +\psi^{c*}_j\psi^c_i e^{i A_{ji}}\right] \\
& & + \frac{U}{2}(N-1)\sum_i |\psi^c_i|^4 - \frac{U}{2N}
\label{eq:int}
\end{eqnarray}
For large systems $N\gg 1$ the typical interaction energy
(\ref{eq:int}) is of the order of $Un$, while the kinetic energy
(\ref{eq:ke}) is of order $J$.
In the weak coupling limit, $nU \ll J$, the groundstates of the
Gross-Pitaevskii equation can be obtained by assuming that the
condensate consists only of those states that minimize the
single-particle kinetic energy. For $\alpha < \alpha_c $, there is a
single minimum, so the condensate will be formed from this state
alone. For $\alpha > \alpha_c$, there are two degenerate minima, and
we should write
\begin{equation}
\psi^c = A \psi_{k_A} + B \psi_{k_B}
\label{eq:linear}
\end{equation}
which (schematically) denotes a linear superposition of the states in these
two minima. Then one chooses $A$ and $B$ to minimize the interaction energy
$\sum_i |\psi^c_i|^4$.
\subsubsection{The case $\alpha = 1/2$}
\label{sec:alpha_0.5}
This is the case of the fully frustrated Bose-Hubbard model, $\alpha=1/2$,
which is gauge equivalent to uniform plaquette fluxes of $n_\phi = 1/2$.
Minimizing the interaction energy within states of the form (\ref{eq:linear})
leads to two solutions which we can denote schematically by
\begin{equation}
\psi^c_\pm = \frac{1}{\sqrt{2}}\left[ \psi_{k_A} \pm i \psi_{k_B}\right]
\label{eq:staggeredstates}
\end{equation}
In detail, the wavefunctions are
\begin{equation}
\psi^c_{\pm,i} = \frac{1}{2\sqrt{2-\sqrt{2}}} \times
\left\{ \begin{array}{l}
1\pm i(\sqrt{2}-1)(-1)^{y_i} \quad\quad x_i \;\mbox{even}\\
(\sqrt{2}-1)(-1)^{y_i} \pm i \quad\quad\quad x_i \;\mbox{odd}
\end{array}
\right.
\label{eq:detailed}
\end{equation}
where $i$ refers to the labelling in Fig.~\ref{fig:choices}. The wavefunctions
are illustrated in Fig.~\ref{fig:villain}.
The pattern of phases is equivalent to that of the ordered groundstate of the fully-frustrated classical XY model\cite{villain}.
Since the condensed state is a superposition of states with different
$k_y$, it is a state with broken translational invariance. The
difference in wavevectors is $\Delta k_y = k_y^A - k_y^B = \pi$, so
the new unit cell in the $y$-direction has size $\Delta y = 2$ (in
units of the lattice constant $a_y$).
It is useful to recall that the Hamiltonian at $\alpha=1/2$ is
time-reversal invariant, so all of its eigenvectors can be chosen
real. The fact that the condensed wavefunctions
(\ref{eq:detailed}) are imaginary shows that these break
time-reversal symmetry. The two states are time-reversed partners,
$\psi^c_\pm = \left(\psi^c_\mp\right)^*$.
The two states are the two (time-reversed) partners of a so-called
``staggered flux'' phase, a related version of which has been
discussed in the context of the cuprate superconductors.\cite{marston} This phase is characterised by circulating currents
around the plaquettes, which are arranged in a staggered
``checkerboard'' pattern. It is straightforward to show that the
states $\psi^c_\pm$ do carry these staggered gauge-invariant
currents.
\begin{figure}
\includegraphics[width=0.99\columnwidth]{staggered_flux_diagram}
\caption{Illustration of the
two (degenerate)
mean-field groundstates of the fully frustrated model, $n_\phi =
1/2$. The arrows on each lattice site represent the phase of the condensate
wavefunction (the amplitude is constant). The two phases are characterized by
staggered circulating currents, in directions illustrated within the plaquettes.
Double lines mark links where a `phase' $e^{i\pi}$ is imprinted.
}
\label{fig:villain}
\end{figure}
We emphasize that the underlying Hamiltonian is both translationally
invariant and time-reversal invariant, so the emergence of these
staggered flux states is a result of the breaking of both of these
symmetries.
The same condensed states have been shown to describe the groundstates
of a related Bose-Hubbard model.\cite{morais} In constrast to the
model that we study, in the model of Ref.~\onlinecite{morais}, the
staggered flux $\alpha$ is applied directly in a checkerboard pattern.
However, for staggered flux of $\alpha=1/2$ the two models coincide
(up to a choice of gauge, as described below).
The derivation of the states (\ref{eq:staggeredstates}) described
above was presented in the weak-coupling limit, $nU \ll J$.
However, note that the state has the special property that the
density is uniform $|\psi_i^c|^2= \mbox{constant}$.
Therefore, these
condensates already minimize the interaction energy $\sum_i
|\psi_i^c|^4$ at fixed average density. With increasing
interaction strength $nU \sim J$, the condensate does not change.
(We have confirmed this with extensive numerical simulations of the
mean-field theory.\footnote{This was also found in the studies of Ref.~\protect\onlinecite{duriclee}. (D.K.K. Lee, private communication)})
Even in the regime $U\gg J$ where
Gross-Pitaevskii theory becomes unreliable owing to the suppression
of number fluctuations and a Gutzwiller mean-field theory is
required, the condensate wavefunction remains the same.
\subsubsection{The case $\alpha =0.389$}
\label{seq:alpha_0.389}
\label{sec:3fold}
Choosing $\alpha =
\frac{1}{\pi}\arccos\left(\frac{1}{2}\sqrt{\frac{1}{2}(9-\sqrt{65})}\right)
\simeq 0.389$ leads to the situation in which $\Delta k_y = 2\pi/3$.
Then, a state of the form (\ref{eq:linear}) with non-zero $A$ and $B$
breaks translational symmetry with a period of $\Delta y = 3$ in the
$y$-direction.
Minimizing the interaction energy over states of the form
(\ref{eq:linear}) leads to the conclusion that the optimum condensate
wavefunction is the superposition
\begin{equation}
\psi^c_\pm = \frac{1}{\sqrt{2}}\left[ \psi_{k_A} + e^{i\chi}
\psi_{k_B}\right].
\label{eq:d3}
\end{equation}
Thus, the condensate does break translational symmetry, with the expected period
of $\Delta y = 3$. Unlike the (special) case of $\alpha=1/2$, for these
functions the density is not uniform, so spatial patterns of the
density show this periodicity.
One interesting feature of this mean-field result is that the
interaction energy is independent of the phase $\chi$ appearing in
(\ref{eq:d3}). Thus, there is an infinite set of condensed groundstates, which
are related by the different choices of the phase $\chi$. In a continuum
model, this infinite degeneracy would correspond to the broken continuous
translational symmetry, the different choices of $\chi$ denoting translations
of the same state. Here the model is explicitly defined on a lattice, so there
is no continuous translational symmetry. Since the
wavefunction is defined only on lattice sites, the states with different
values of $\chi$ are not just related by simple translations.
Nevertheless, for weak interactions
$nU \ll J$, we find that, surprisingly, there emerges a continuous degeneracy, beyond the expected threefold degeneracy from translations.
This gives an additional ``Goldstone'' mode, associated with slow
spatial variations of $\chi$ as a function of position.
(The same feature arises for values of $\alpha$ for which $\Delta k_y = 2\pi/p$ with $p\neq 2$.)
Since this additional emergent degeneracy is not protected by a symmetry of
the Hamiltonian, one does not expect it to survive in general (e.g. to
stronger interactions or to quantum fluctuations). Indeed, we find that by
performing mean-field theory to stronger interaction strengths, the continuous
degeneracy is lost. Including energy corrections to order $(nU)^2/J$ we find
that the three cases $\chi = 0, \pm 2\pi/3$ are selected as the energy minima.
This gives rise to a set of three degenerate groundstates, with a unit cell of
size $\Delta y =3$ and all related by translations in the $y$ direction. The
Goldstone mode described above develops a gap.
\subsubsection{The case of general $\alpha$}
For typical cases of $\alpha$ in the range $\alpha_c < \alpha < 1/2$,
there are two minima in the single particle energy spectrum, but the
spacing $\Delta k_y$ is {\it incommensurate} with the underlying
lattice. In the weak coupling limit $nU \ll J$, one expects the
groundstate to remain incommensurate with the lattice, having a
continuous degeneracy (along the lines of that described by the phase
$\chi$ above). However, for sufficiently strong interactions, the
density wave will ``lock'' to the lattice, via an
incommensurate-commensurate phase
transition.\cite{pokrovskytalapov}
Additionally, for staggered flux $\alpha \gtrsim \alpha_c$, near the bifurcation
in Fig.~\ref{fig:minima}, the single-particle dispersion becomes very flat, and
we expect a regime of large fluctuations where potential condensate groundstates
will be strongly depleted.
\section{Numerical Methods}
In order to investigate the quantum groundstate of the frustrated
Bose-Hubbard model, we have performed large scale exact
diagonalization studies of the model (\ref{eq:hamiltonian}).
We
define the system on a square lattice of $N_s = L_x\times L_y$ sites, and
impose periodic boundary conditions to minimize finite size effects.
For a total number of bosons, $N$, we therefore study a system with
mean density $n= N/(L_xL_y)$.
The possible geometries and system sizes are constrained by the form
of the gauge potential that is applied. Furthermore, the plaquette
fluxes determine the translational symmetries of the Hamiltonian, and
hence the conserved momenta. For staggered flux (\ref{eq:staggered}),
in the general case ($\alpha\neq 0,1/2$) the translational symmetries are
those of the unit cell with size $2\times 1$ (in $x$ and $y$ directions).
Higher symmetry arises for the cases $\alpha=0$ and $\alpha=1/2$, for
which the flux is uniform; in the latter case, the (magnetic)
translational symmetries of many-particle systems follow from
Ref.~\onlinecite{kolread}.
By definition, the Hamiltonian commutes with the unitary
transformations that effect these translational symmetries. The energy
eigenstates are therefore also eigenstates of these translational
symmetry operators ({\it i.e.} eigenstates of conserved lattice
momentum). In any exact numerical calculation, the energy eigenvectors
will also be eigenstates of the conserved lattice momentum. However,
as described above, in general the mean-field states break the
translational symmetry. Thus the mean-field states are not eigenstates
of momentum. In order to make comparisons with the mean-field states,
and the possibility of condensation, one must allow for this breaking
of translational symmetry.
\subsection{Condensate Fraction}
Often the groundstates of (repulsive) interacting bosons can be
understood in terms of Bose-Einstein condensation. Interactions
between the particles lead to depletion of the condensate. If the
effects of interactions are very strong, these may even drive a phase
transition from the condensed phase into non-condensed phases. It is
therefore very important to know if the groundstate remains condensed.
(If not, then the system may be described by a novel, uncondensed,
and possibly strongly correlated phase of matter.)
The condensate fraction is quantified by the general definition
introduced by Yang.\cite{yang}
From the many-particle groundstate $|\Psi_0\rangle$, one forms the single particle density operator
\begin{equation}
\label{eq:sp}
\rho_{ij} = \langle \Psi_0 | \hat{a}^\dag_i\hat{a}_j|\Psi_0\rangle \,,
\end{equation}
a Hermitian operator, the trace of which $\sum_i \rho_{ii}$ is the
mean (total) number of particles, $N$. Then, one finds the
eigenvalues of $\rho_{ij}$. For ``simple'' BECs,\cite{Leggett01} the
spectrum has one eigenvalue which is of order {$N$},\cite{yang} and
which is therefore much larger than all others for large $N$ (the
thermodynamic limit). Denoting this largest eigenvalue $\lambda_0$,
the condensate density $n_c$ and condensate fraction $x_c$ for
average density $n$ are defined by\cite{yang}
\begin{equation}
\label{eq:xc}
x_c \equiv \frac{n_c}{n} \equiv \frac{\lambda_0}{N}\,.
\end{equation} The eigenvector of $\rho_{ij}$ corresponding to the largest eigenvalue
is the condensate wavefunction,
$\psi_i^{0}$.
While this method is applicable in the simplest of situations, it gives
misleading results in cases where the groundstate of the system breaks a
symmetry of the Hamiltonian in the thermodynamic limit. In that case it is
well known that, for a finite sized system (in which the ground state is an
eigenstate of all symmetry operators), an analysis of the density matrix
states shows a ``fragmented'' condensate in which there is more than one
eigenvalue of order $N$. (See Ref.~\onlinecite{mueller:033612} for a discussion of
fragmentation in the context of cold atomic gases.) The origin of this
fragmentation, and its relationship to symmetry breaking, is well-understood
for simple model systems with no condensate depletion, for example for
condensation of $N$ bosons in two
orbitals.\cite{mueller:033612,dalibardparity} From the practical point of
view of exact numerical calculations, it is important to have a prescription
for how to quantify the degree of condensation in general -- {\it i.e.} in cases
where there are many degrees of freedom and condensate depletion can be
significant.
\subsection{Condensate Fraction with Symmetry Breaking}
\label{subsec:symmetrybreaking}
We propose a method to determine, on the basis of numerical exact
diagonalization studies, the condensate fraction in cases where the
condensed state breaks a symmetry in the thermodynamic limit. Given
the context of this paper, we focus on the case of translational
symmetry. The method, however, is very general: no specific knowledge
is required of the condensed state, or indeed of the symmetry that is
broken. These emerge directly from the numerical calculations in an
unbiased way.
The starting point is to determine the energy spectrum, in order to identify
if the groundstate may have a broken symmetry in the thermodynamic limit. As
ever with numerical studies, one must study the spectrum with varying systems
size (up to as large a system size as can be achieved), in order to glean
information about the properties of the spectrum in the thermodynamic limit.
It is well-known that the signature of (translational) symmetry breaking in a
finite size calculation is the appearance of a set of quasi-degenerate energy
levels in the spectrum, with different eigenvalues of the conserved quantity
associated with the symmetry ({\it i.e.} momentum in the case of translational
symmetry breaking). In the context of cold atomic gases, this has been
illustrated for spin-rotational invariance,\cite{LawPB98,hospins}
translational and rotational symmetry breaking,\cite{cwg,advances,zhaimtm} and
parity.\cite{parkeparity,dalibardparity} In the thermodynamic limit, these
states become degenerate, and it becomes valid to superpose the states
(e.g. as selected by an arbitrary weak symmetry breaking perturbation). Any
superposition of all these states forms a groundstate which has broken
symmetry. Before superposition, each of the states (eigenstates of the
symmetry operators) can be viewed as fragmented
condensates.\cite{mueller:033612}
Based on the results of these calculations of the energy spectrum, one
can look to see if, in the thermodynamic limit, several states are
tending to become degenerate. The emergence of this degeneracy
appears when the system size is sufficiently large; if the degeneracy
is not well resolved, then this suggests that the numerical
calculations are not on a sufficiently large system size to be
conclusive. In many cases, an emergent quasi-degeneracy can be very
convincingly
established.\cite{cwg,advances,parkeparity,dalibardparity} In many
practical cases of interest where the degeneracy is not fully
established, it can still be of value to make the hypothesis that a
small number of low-energy states will be degenerate in the
thermodynamic limit, and to test if this hypothesis is borne out by a
high condensate fraction.
Suppose that an analysis of the energy spectrum suggests that there
are $D$ such states, $|\Psi_0^{\mu}\rangle$, with $\mu=1,2,\ldots,D$,
which tend towards degeneracy in the thermodynamic limit. We assume
that these $D$ states can be distinguished by eigenvalues of symmetry
operators (e.g. no two have the same momenta).
In order to investigate the possibility of simple BEC in a
broken symmetry state, we propose that one forms the superposition
state
\begin{equation}
|\Psi^{c}_0\rangle \equiv \sum_{\mu=1}^D c_\mu |\Psi_0^\mu\rangle
\label{eq:psic}
\end{equation}
which depends on the $D$ complex amplitudes $c_\mu$. Then, for this
superposition state -- which is not an eigenstate of the
(translational) symmetry -- one should determine the single particle
density matrix (\ref{eq:sp}) and find the condensate fraction
(\ref{eq:xc}), each of which are functions of the parameters $c_\mu$.
We define the condensate fraction of the broken symmetry state by in
terms of the optimal choice
\begin{equation}
X_c \equiv \max_{c_\mu}\left[
x_c(c_\mu) \right]\,.
\end{equation}
The corresponding optimizing coefficients $c_\mu$ define the associated
condensate wavefunction (\ref{eq:psic}). Since this is a broken symmetry
state, in general there are $D$ sets of coefficients $c_\mu$ which give the
(same) maximum condensate fraction, and hence $D$ such condensed states.
These correspond to the $D$ broken symmetry states, and are related by
applications of the symmetry operations.
Below, we illustrate the application of this approach for the cases of
the Bose-Hubbard model with staggered flux, at $\alpha=1/2$ (fully
frustrated) where $D=2$, and at $\alpha=0.389$ where $D=3$.
\subsection{Unfrustrated Bose-Hubbard Model}
As a warm-up, and to test the quantitative validity of exact
diagonalization in determining the condensate fraction, we study the
Bose-Hubbard model in the absence of gauge fields (all plaquette
fluxes vanish, $n^a_\phi =0$). In this case, for non-integer particle
density, it is known that the groundstate is condensed,\cite{fisherbh}
and the condensate fraction has been established by detailed numerical
studies including quantum Monte Carlo,\cite{Hebert:2001p35}
as well as in spin-wave theory.\cite{Bernardet:2002p34} For $n=1/2$, the
condensate fraction is $x_c = 1$ for $U/J=0$, and falls to $x_c \simeq
0.4$ in the hard-core limit $U/J\to \infty$.\cite{Bernardet:2002p34}
We have used ED results on systems of up to $L_x \times L_y =
5 \times 6$ to determine the condensate fraction. Consistent with the lack
of symmetry breaking, in all cases the spectrum shows a clear
groundstate. (The broken gauge invariance of this state will appear in
a emergent quasi-degeneracy at different particle numbers. We work at
fixed particle number.) By forming the single particle density
operator, and finding its maximal eigenvalue, we find that for
hard-core bosons at $n=1/2$ the condensate fraction is $0.433(6)$. The
favourable comparison with the Monte Carlo result illustrates that,
for this case, the system sizes amenable to ED are sufficiently large
to allow accurate quantitative determination of $x_c$.
\subsection{Fully Frustrated Bose-Hubbard Model, $\alpha =1/2$}
We now turn to the case of the fully frustrated Bose-Hubbard model,
$\alpha=1/2$, which is gauge equivalent to uniform plaquette fluxes
$n_\phi = 1/2$. To analyse this case, we study system sizes for which
the translational symmetry\cite{kolread} is the largest, implying the
largest possible Brillouin zone for the conserved (many-particle)
momentum, and no degeneracy associated merely with the magnetic
translations. (This is the ``preferred'' case of $d=1$ in the
terminology and notation of Ref.~\onlinecite{kolread}.)
In these cases, where no \emph{many-body} degeneracy is expected,
the groundstate in the non-interacting system still remains degenerate
due to the degeneracy of the \emph{single-particle} groundstate. This degeneracy
is split due to interactions in the system. We find an emerging
two-fold quasi-degeneracy in the spectrum for sufficiently large
system sizes even in the case of hardcore interactions,
as shown in the inset of Fig.~\ref{fig:fffraction} (left frame). Following the
procedure of \S\ref{subsec:symmetrybreaking}, we recognize this as a sign
of possible symmetry-breaking with $D=2$. We apply the prescription
in \S\ref{subsec:symmetrybreaking} to determine the maximal condensate
fraction.
\begin{figure}
\includegraphics[width=0.95\columnwidth]{condensate_fraction_dependencies}
\caption{Exact diagonalisation results for
the condensate fraction $x_c$ for the maximally condensed state at
$n_\phi=1/2$, {\it i.e.} incorporating translational symmetry breaking: as
a function of the interaction strength $U$ (left); and as a function
of particle density $n$ in the hard-core limit (right). In addition, the inset in
the left frame shows the low-lying spectrum of the system for the case of hardcore
interactions at half filling. These spectra clearly confirm the emergence of a twofold
degenerate groundstate in large systems. At $N=12$, the splitting is barely discernable
at the scale of the figure.
}
\label{fig:fffraction}
\end{figure}
In Fig.~\ref{fig:fffraction} we present the results of these
calculations of the (maximal) condensate fraction for the fully
frustrated Bose-Hubbard model, both as a function of $U/J$ for
$n=1/2$, and as a function of $n$ in the hard-core limit $U/J\to
\infty$. In all cases, we find that the groundstate appears to be fully condensed.
The smallest condensate fraction is for $n=1/2$ and $U/J\to \infty$.
Here, an extrapolation in $1/N$ to the thermodynamic limit yields
a condensate fraction of about $x_c \simeq 0.39$.
In the graphs of
Fig.~\ref{fig:fffraction}(b) there appears to be a reduction of
condensate fraction at about $n\simeq 0.25$. We associate this with
the fact that, on the lattice, the Laughlin state of bosons would
appear for full frustration $n_\phi =1/2$ at density $n=1/2$.\cite{mollercooper-cf}
Extrapolation of the condensed fraction in this case yields $x_c \sim
0.48(2)$. Although the possibility of Laughlin correlations may act to
destabilize the condensed state, the groundstate at this density is a
condensed (superfluid) phase.
In all cases, the nature of the groundstate that we find closely matches the
predictions of mean-field theory. First, we can check that symmetry
breaking is of the same form. Furthermore, the condensate wavefunction that
we obtain is {\it exactly} that described above in Eq.~(\ref{eq:staggeredstates}).
As discussed previously, these states have uniform density, which makes them
highly robust to details of the interaction potential.
The results provide the first evidence from exact diagonalizations
that, under all conditions, the groundstate of the fully frustrated
Bose-Hubbard model is condensed. Our model includes the
possibility of particle-number fluctuations, and thus goes beyond
previous studies using the picture of Josephson-junction arrays
that is based on phase fluctuations only.\cite{Polini:2005p392}
Furthermore, our results provide a quantitative measure of the condensed fraction.
It is interesting to compare the results to those that would be
obtained from a Gutwiller ansatz. In the hard-core limit, the
Gutwiller mean-field state has $x_c = 1-n$.
Thus, at $n=1/2$ the Gutzwiller theory predicts a condensate fraction
of $0.5$. This is close to the value we obtain from exact
diagonalisation results, $\simeq 0.39$, indicating that Gutzwiller theory is
quantitatively fairly accurate in this case.
At $n=1/4$ the Gutzwiller theory predicts a condensate fraction of
$0.75$. The exact diagonalization result of $\simeq 0.48$, shows a large
quantitative discrepancy. As described above, we attribute this to the
competition introduced by Laughlin-like correlations which act to
destabilise the condensate.
Another interesting observation can be made by comparing the condensate
fractions for the frustrated Bose Hubbard model at $n_\phi=1/2$ with the
unfrustrated zero-field case. For half filling, the gauge-field reduces the
condensate fraction seen in our exact diagonalizations from about $x_c(n_\phi=0)=0.43$
to $x_c(n_\phi=1/2)=0.39$. For $n=1/4$ on the other hand, the reduction is more significant,
with $x_c(n_\phi=0)=0.66(1)$ being reduced to $x_c(n_\phi=1/2)=0.48$ in the presence
of the field.
\subsection{Staggered Flux Bose-Hubbard Model, $\alpha=0.389$}
This is the staggered flux value at which $\Delta k_y = 2\pi/3$, so we
expect a broken symmetry state with unit cell size $\Delta y =3$, and thus a
groundstate degeneracy of $D=3$ in the thermodynamic limit. As described
above, this is borne out in mean-field theory, at least for sufficiently
strong interactions.
The numerical results are consistent with these expectations. For large
interactions, a clear three-fold degeneracy appears in the groundstate.
Assuming $D=3$ for all interaction strengths leads to the condensate fraction
shown in Fig.~\ref{fig:3fold} for density $n=1/2$.
For small $U/J$ and/or density $n$, the results of our analysis show that the
groundstate is condensed in the manner predicted by mean-field theory.
For the strongest interactions (hard-core interactions and $n=1/2$) finite-size
effects remain significant, and it is difficult to be sure that extrapolation to the
thermodynamic limit will leave a non-zero condensate fraction (see inset of Fig.~\ref{fig:3fold}).
In finite size systems, the condensed wavefunction obtained from the numerical
procedure is in good qualitative agreement with the results of mean-field theory
described above. At very weak interactions, the presence of the Goldstone mode
discussed in \S\ref{seq:alpha_0.389} is also visible. For $U/J\lesssim 0.1$, the
results in Fig.~\ref{fig:3fold} show a discontinuous drop in $x_c$. This occurs
when the splitting of the three-fold groundstate degeneracy (due to finite
size effects) is larger than the energy scale which mean-field
theory shows is required to ``lock'' the density wave to the underlying
lattice. Thus, this reduction in $x_c$ at small $U/J$ is a finite-size
effect. In this regime, we can recover a large condensate fraction, $x_c \simeq 1$, by including additional levels ($D>3$) to account for the higher degree of symmetry breaking.
\begin{figure}
\includegraphics[width=0.9\columnwidth]{condensate_fraction_3xY_n_1_2_u}
\caption{ Exact diagonalisation results for
the condensate fraction $x_c$ as a function of interaction strength $U/J$,
for the staggered flux with $\alpha=0.389$ where the groundstate shows
three-fold translational symmetry breaking. Results are at half filling,
$n=1/2$, calculated for $N=6$ particles on a system of dimensions $L_x\times
L_y=4\times 3$. Small symbols show optimisations over the lowest three
eigenstates. For $U/J\lesssim 0.1$ a maximally condensed state requires to
also include the low-lying Goldstone modes (large symbols; see main text).
The condensate fraction calculated in the hardcore limit is
shown as a dashed line. Inset: Scaling of the condensate fraction with
system size for different $U$. For the largest system with $N=12$ particles
in $N_s=24$ sites, the two available lattice geometries yield significantly
different $x_c$.}
\label{fig:3fold}
\end{figure}
\section{Experimental Consequences}
The phases described above break translational symmetry of the
underlying lattice (with 2-fold and 3-fold symmetry breaking in the
cases discussed in detail in sections \ref{sec:alpha_0.5} and \ref{seq:alpha_0.389}).
Following the usual expectations for
symmetry breaking states, in an experiment on a system with a large
number of atoms, one expects that very small perturbations (perhaps in
the state preparation) which break the perfect symmetry of the
underlying model will cause the system to select one of the broken
symmetry groundstates. (It is also possible that domains will form,
separated by domain walls that can be long-lived and survive as
metastable configurations.)
In general, one expects that this translational symmetry breaking will
appear in the real-space images of the system ({\it i.e.} if in situ imaging
on the scale of the lattice constant is possible).
This is the case for the $3$-fold symmetry breaking described in
\S \ref{sec:3fold}. There, the real-space image of the particle density
will have spatial structure with a unit cell that has size $3\times
2$, and is therefore 3 times larger than the unit cell of the
underlying microscopic model. The three broken symmetry phases can be
distinguished by the three possible positions of this unit cell.
However, for $n_\phi=1/2$, where there is a 2-fold degeneracy of the
groundstate, the two broken symmetry phases {\it cannot} be
distinguished by the real-space image. Here, the groundstate is the
staggered flux phase. This has both broken translational symmetry and
broken time reversal symmetry. However, it is invariant under the
combined action of translation and time reversal. Since density is
time-reversal invariant, the state has uniform density. Thus, one
cannot distinguish the symmetry breaking in real-space images. (In
non-equilibrium situations where there can be domain walls separating
different phases, there may appear density inhomogeneities associated
with the domain walls.) As we now discuss, there are ways to detect
these symmetry broken states in the expansion images.
\subsection{Expansion Images}
The nature of the groundstate can be probed by expansion imaging. We
assume that all fields are released rapidly (compared to the subband
splitting of the lattice), and that the particles (of mass $M$) expand
ballistically -- without any potentials, or gauge fields and neglecting
further interactions -- according to the free-particle Hamiltonian,
denoted $\hat{H}_{\rm free}$. Then the expansion image after a time
$t$ is given by\cite{blochdz}
\begin{equation}
n({\bm x}) = \left(M/\hbar t\right)^3|\tilde{w}({\bm k})|^2 G({\bm k})
\label{eq:exp}
\end{equation}
where ${\bm k} = M{\bm x}/\hbar t$, $\tilde{w}({\bm k})$ is the Fourier transform of the Wannier state of the lowest Bloch band, and
\begin{equation}
\label{eq:ft}
G({\bm k}) = \frac{1}{N_s}\sum_{i,j} e^{i{\bm k}\cdot\left({\bm r}_i -{\bm
r}_j\right)}\langle \hat{a}^\dag_i \hat{a}_j\rangle
\end{equation}
is the Fourier transform of the single particle density matrix.
For states that are well-described as condensates ({\it i.e.} with small condensate
depletion), the density matrix is $\langle \hat{a}^\dag_i \hat{a}_j\rangle
\simeq (\psi^c_i)^*\psi^c_j$, where $\psi^c_i$ is the condensate wavefunction.
Thus, the expansion image directly provides (the Fourier transform of) this
condensate wavefunction.
As described in detail below in \S\ref{seq:MeasureCondensateFraction}, depletion leads to a reduction of the
amplitudes of the ``condensate'' peaks in the expansion image and to the
appearance of an additional incoherent background.
Compared to the usual Bose-Hubbard model, in the situation that we
consider here -- of an optically induced gauge potential on the
lattice -- there are three important new considerations concerning
expansion images.
$\bullet$ The first aspect relates to {\it gauge invariance}. Consider making a change
of the vector potential in the Bose-Hubbard Hamiltonian
(\ref{eq:hamiltonian}), from $A_{ij}$ to
\begin{equation}
A'_{ij} = A_{ij} + S_i - S_j
\label{eq:gauget}
\end{equation}
where $S_i$ is any set of real numbers. This ``gauge transformation''
leaves the fluxes (\ref{eq:flux}) unchanged, which are therefore
said to be ``gauge-invariant''. The new Hamiltonian (with $A'_{ij}$ in
place of $A_{ij}$) can be brought back to its original form
by introducing the operators
\begin{equation}
\hat{a}'_i \equiv \hat{a}_i e^{-iS_i} \quad
\hat{a}'^{\dag}_i \equiv \hat{a}^\dag_i e^{iS_i} \,.
\label{eq:redef}
\end{equation}
In this way, the Bose-Hubbard Hamiltonian adopts the same form as at the start
(again with gauge fields $A_{ij}$), but now with $\hat{a}'$ replacing
$\hat{a}$. Any property that is insensitive to the distinction between
$\hat{a}'$ and $\hat{a}$ in (\ref{eq:redef}) remains the same, and is
therefore gauge invariant. Such quantities include the energy
spectrum, density response functions, etc.; indeed any observable of
the closed system described by (\ref{eq:hamiltonian}) is
gauge-invariant. All of these gauge-invariant properties depend only
on the gauge-invariant fluxes (\ref{eq:flux}).
An important point is that
the expansion image is {\it not} gauge-invariant.
Under the transformation (\ref{eq:redef}) the single particle density
operator becomes
\begin{equation}
\langle \hat{a}'^{\dag}_i \hat{a}'_j \rangle =
e^{i (S_i - S_j)}\langle \hat{a}^{\dag}_i \hat{a}_j \rangle
\end{equation}
The gauge transformation affects the Fourier transform of the density
operator (\ref{eq:ft}), and therefore the expansion image
(\ref{eq:exp}). There is no inconsistency with general principles of
gauge invariance. As described above, prior to expansion, all
physical properties of the Bose-Hubbard Hamiltonian
(\ref{eq:hamiltonian}) are completely unchanged. The essential point
is that the expansion image involves the evolution of the system under
the free-space Hamiltonian, $\hat{H}_{\rm free}$, and this Hamiltonian
is unchanged ({\it i.e.} the gauge transformation was not applied to it).
Indeed, if for the expansion imaging all optical dressing is switched
off, then this Hamiltonian is always in a fixed gauge with vanishing
vector potential. Provided the expansion image is taken under the
evolution of a Hamiltonian $\hat{H}_{\rm free}$ with vanishing gauge
potential, the expansion image measures the {\it canonical} momentum
distribution of the particles, and therefore depends on the gauge
used in the Bose-Hubbard Hamiltonian $\hat{H}$ before expansion.
$\bullet$ The second consideration relates to the fact that, owing to the
optical dressing, {\it the atoms are in
more than one internal state}. In particular, for the schemes of
Refs.~\onlinecite{gerbier} and \onlinecite{JakschZoller}, the atoms on alternating sites along
the $x$ axis are, in turn, in the ground state, $|g\rangle$, and an excited
state, $|e\rangle$, of the atom. Therefore, upon release of the cloud,
one has the possibility to study expansion images of several different
types. The results described above (\ref{eq:exp},\ref{eq:ft}) apply
only if the image at time $t$ is formed in such a way that the
measurement does not distinguish between the different internal states
of the atom. However, since the electronic states, $|g\rangle$ and
$|e\rangle$, are very different, it is also possible to perform
measurements which are state-specific: one can form the expansion
image of the $|g\rangle$ atoms, or of the $|e\rangle$ atoms. In these
cases, one should replace $G({\bm k})$ in (\ref{eq:exp}) by
\begin{equation}
G_{g/e}({\bm k}) = \frac{1}{N_s}\sum_{i,j\in g/e} e^{i{\bm k}\cdot\left({\bm r}_i -{\bm
r}_j\right)}\langle \hat{a}^\dag_i \hat{a}_j\rangle
\end{equation}
where the change is that the sums should be over those sites $i,j$ on
which the atoms are of type $g$ or $e$.
$\bullet$ Finally, the third consideration -- although not restricted to systems of this
type -- arises naturally from the ability to address sites of type $g$ and $e$
separately by spectroscopic methods. This allows for much more interesting and
useful possibilities in the expansion imaging, including the possibility to
{\it imprint phase patterns prior to expansion}. Specifically, immediately
prior to expansion, one can choose to drive (coherent) transfer of the atoms
from one state to an other. For example, using coherent laser fields of the
same type as used to provide the laser-assisted
tunneling,\cite{JakschZoller,gerbier} one can choose to transfer all atoms
(initially of both $e$ and $g$ type) into a given ``target'' state (a
superposition of $e$ and $g$) while adding a spatially dependent phase,
$S^{\rm exp}_i$. Alternatively, the phases can be imprinted by site-selective
potentials $V_i$ applied for a short time $t$, leading to $S^{\rm exp}_i =
V_it/\hbar$. In the language of the earlier discussion, immediately prior to expansion one
has effectively applied a gauge transformation to the initial wavefunction. The expansion
image follows from (\ref{eq:exp}) but now with
\begin{equation}
\langle \hat{a}'^{\dag}_i \hat{a}'_j \rangle =
e^{i (S^{\rm exp}_i - S^{\rm exp}_j)}\langle \hat{a}^{\dag}_i \hat{a}_j \rangle
\end{equation}
replacing the density operator in (\ref{eq:ft}). (Since we chose the
same target state for all atoms, they are indistinguishable in the
final image so all sites contibute.)
This additional freedom is not restricted to dressed atomic systems of
the type we have described. Indeed, the application of a
spatially-varying potential $V({\bm r}$) over a time $t$ to a
one-component condensate will cause the local phase to wind by $V({\bm
r})t/\hbar$. If the time $t$ is short compared to microscopic
timescales, this can be viewed as an instantaneous phase-imprinting
prior to expansion. Techniques of this kind could be used to tune out
the ``shearing'' in the expansion images of Ref.~\onlinecite{spielmanfield}.
\begin{figure*}
\includegraphics[width=1.8\columnwidth]{expansion_alpha_0_5_epsilon_collate2}
\caption{The expansion image for the staggered flux phase, as determined by the structure factor as a function of $(k_x,k_y)$.
(a) For the gauge field considered ($\epsilon=0$)
the two broken symmetry states have the same expansion image.
(b, c) With a phase imprinting before expansion, denoted by $\epsilon$, the two states have different expansion images.}
\label{fig:staggeredspectrum}
\end{figure*}
The additional freedom to imprint phases greatly enhances the
information that can be extracted from expansion images. As we
illustrate below, it allows important information regarding the nature
of the phases to be extracted.
\subsubsection{Expansion Images for Full Frustration, $\alpha=1/2$.}
\label{sec:expansion_1_2}
As explained above, the groundstate at $n_\phi =1/2$ is the ``staggered flux''
phase, which is two-fold degenerate.
In the gauge (\ref{eq:gauge}) we have
been considering, the condensate wavefunctions
are given by Eq.~(\ref{eq:detailed}), and
therefore have a unit cell of size $2\times 2$.
A straightforward calculation of $G^{\pm}({\bm k})$ shows that
the two broken symmetry states have the same
Fourier transform, $G^{+}({\bm k})=G^{-}({\bm k})$, illustrated in
Fig.~\ref{fig:staggeredspectrum} (a). Therefore, the expansion
image cannot discriminate between whether the groundstate is in state
$\psi^c_+$
or in state $\psi^c_-$.
On the other hand,
the expansion image can distinguish these states from the
expansion images of condensates formed from all particles condensed in
either one or the other of the two single-particle states. These
individual single-particle states have the full translational symmetry
of the underlying system -- namely under $x\to x+2a_x$ and $y\to
y+a_y$ -- so the Fourier components of a condensate formed from either
one has peaks spaced by the reciprocal lattice vectors ${\bm
K}_x=(\pi/a_x,0)$ and ${\bm K}_y=(0,2\pi/a_y)$. On the other hand,
the ``staggered flux phase'' has broken translational invariance in
the $y$ direction, being invariant only under the translations $y\to
y+2a_y$, so the Fourier components are spaced by the $(\pi/a_x,0)$ and
$(0,\pi/a_y)$. The appearance of this smaller periodicity in the
$k_y$ direction is indicative of the broken spatial periodicity.
\begin{figure}
\includegraphics[width=0.65\columnwidth]{phase_imprint}
\caption{A pattern of phases to imprint before expansion. For $\epsilon\neq 0$, this causes the
expansion images of the two staggered flux states to differ.}
\label{fig:phaseimprint}
\end{figure}
For the gauge used in this work, the expansion image does not
distinguish between the two different staggered flux states. However,
in the gauge used in Ref.~\onlinecite{morais} these two states have very
different expansion images. Indeed, in the gauge of Ref.~\onlinecite{morais}
one state has a condensate wavefunction with uniform density and phase (and
therefore enjoys the full symmetry of the lattice), while the other has a phase
pattern with a $2\times 2$ unit cell; these give rise to very different
Fourier transforms and therefore expansion images.
As described above, the ability to locally address sites of the
lattice leads to the possibility to apply a spatial phase pattern
$e^{iS_i}$ to the system prior to expansion. This can be used to
discriminate between the two staggered flux states. Specifically, we
consider the case in which potentials or optical dressing is used to
imprint the phase pattern
\begin{equation}
S_i = \epsilon\times{\rm mod}(x_i,2){\rm mod}(y_i,2)\,.
\end{equation}
Thus, for the atoms on sites with $x_i= \mbox{odd}$
({\it i.e.} which are all of ``excited'' or all ``ground'' states in the
proposal of Ref.~\onlinecite{gerbier}), the phase of every other one
along the $y$-direction is advanced by $\epsilon$, as illustrated in
Fig.~\ref{fig:phaseimprint}. This can be achieved, for example, by
applying a state-selective potential with period $\Delta y = 2$.
A calculation of the resulting expansion images shows that this phase pattern
causes the two staggered flux phases to have different expansion images,
$G_+({\bm k},\epsilon)\neq G_-({\bm k},\epsilon)$. Owing to the relation
$\psi^c_+ = (\psi^c_-)^*$, it is straightforward to show that $G_-({\bm
k},\epsilon)= G_+({\bm k},-\epsilon)$. The results are illustrated for
$\epsilon = \pi/2$ in Fig.~\ref{fig:staggeredspectrum}(b) and (c). There is a
clear distinction between the expanstion images of the two states: e.g. the signal
at $(k_x,k_y)=(1,1)$ is absent for $\psi_-$, while being strong for $\psi_+$. It is
not, however, necessary to impose large phase difference $S_i$ to obtain large
effects. The change in the spectrum increases linearly for non-zero
$\epsilon$. As shown in Fig.~\ref{fig:staggeredspectrum2} even very small
changes of phase $\epsilon$ can give rise to notable changes in the expansion
images.
\begin{figure}
\includegraphics[width=0.95\columnwidth]{expansion_alpha_0_5_epsilon=piby10}
\caption{The expansion image of the staggered flux state is very sensitive
to $\epsilon\neq 0$, here shown for $\psi^c_+$. Even a phase of $\epsilon = \pi/10$ allows for notable
change. Since $G_-({\bm k},\epsilon)=
G_+({\bm k},-\epsilon)$, this change allows for a discrimination between $\psi^c_+$ and $\psi^c_-$.}
\label{fig:staggeredspectrum2}
\end{figure}
\subsection{Measurement of the Condensate Fraction}
\label{seq:MeasureCondensateFraction}
The condensate fraction can be measured experimentally by analysing the expansion images of the lattice gas.
For a perfect condensate only few a coherent peaks are visible within the Brillouin zone.
Condensate depletion from strong interactions in the atomic gas results in the appearance of an additional
background, i.e. some of the density of particles is spread out in the Brillouin zone.
Generally, we may write the density matrix of the depleted condensate as
\begin{equation}
\label{eq:depletedCondensateRho}
\rho_{ij} \equiv n_c (\psi^c_i)^* \psi^c_j + \delta\rho_{ij},
\end{equation}
where $n_c$ is the condensate density and $\psi^c_i$ is its wavefunction; this equation thus defines $\delta\rho_{ij}$.
Similarly, the amplitudes of the expansion image can be written as
\begin{equation}
\label{eq:correctedGk}
G({\bf{k}}) \equiv n_c G_c({\bf{k}}) + \Delta G({\bf{k}}),
\end{equation}
where the coherent part $G_c(k)$ derives from the non-interacting condensate
\begin{equation}
G_c({\bf{k}}) = (1/N_s) \sum_{i,j} (\psi^c_i)^* \psi^c_j \,e^{i{\bf{k}}({\bf{r}}_i-{\bf{r}}_j)}.
\end{equation}
This defines $\Delta G(k)$. Experimentally, one can only measure $G({\bf{k}})$.
In order to extract $n_c=n x_c$, one needs to make some assumption about $\Delta G({\bf{k}})$.
Numerically, we find that this background of the expansion image has some internal structure (and this data could be used to
build a more accurate model), but to a first approximation we may assume that it is homogeneous.
This translates into making the simple assumption that $\Delta G({\bf{k}})$ is independent of ${\bf{k}}$. One then obtains
\begin{equation}
\label{eq:depletedCondensate}
G({\bf{k}}) \equiv n_c G_c({\bf{k}}) + (n-n_c),
\end{equation}
which satisfies the proper normalisation of the Fourier amplitudes $\sum_{\bf{k}} G({\bf{k}}) = \sum_i \rho_{ii} = N$.
\begin{figure}
\includegraphics[width=0.8 \columnwidth]{expansion_4x4_n_8_u_corr2}
\caption{a) Expansion images $G({\bf{k}})$ of interacting Bose-Einstein condensates, showing the effect of condensate depletion. The
data show the evolution of the condensate at $n=n_\phi=1/2$ on a square lattice of size $L_x=L_y=4$ for weak to hardcore interactions. The depletion
of the coherent peaks is given approximately by the condensate fraction. b) The bottom panel shows the magnitude of the corrective
term $\Delta G({\bf{k}})$ in Eq.~(\ref{eq:correctedGk}). This data shows some structure, in particular, the background contribution is smaller at
the $k$-points where coherent peaks were present then it is elsewhere.}
\label{fig:depletion}
\end{figure}
Let us test the accuracy of this assumption for the example of a half filled lattice at $\alpha=1/2$ that was discussed in \S\ref{sec:expansion_1_2}.
To visualize the effect of condensate depletion, Fig.~\ref{fig:depletion} displays the evolution of the expansion
image for the case already displayed in Fig.~\ref{fig:staggeredspectrum}(a) for a weakly interacting gas.
Unlike in the single particle picture, calculations of the actual many-body wavefunction for the interacting system are limited
to finite size. Data in Fig.~\ref{fig:depletion} were obtained for a lattice of size $4\times 4$, so the `incoherent' background occurs at
peaks spaced by $\Delta k_x = 2\pi/L_x = \pi/(2a_x)$ and $\Delta k_y = 2\pi/L_y = \pi/(2a_y)$.
The main features of the expansion image remain those of the pure condensate even with strong interactions, except
for the partial suppression of the coherent peaks that proceeds according to (\ref{eq:correctedGk}) with $\Delta G({\bf{k}})$ given by small amplitudes.
The Brillouin zone can thus be partitioned into areas with peaks $\mathcal{A}_\text{P}$ and the remaining background area $\mathcal{A}_\text{BG}$.
The magnitude of this corrective term $\Delta G({\bf{k}})$ is shown in Fig.~\ref{fig:staggeredspectrum}(b). Its spatial dependency is weak,
excepting the notably smaller correction within $\mathcal{A}_\text{P}$ as compared to the overall background $\mathcal{A}_\text{BG}$.
Therefore, the average amplitude in the background
\begin{equation}
\label{eq:DepletionProxy}
n-n_c \simeq \langle I(\mathcal{A}_\text{BG}) \rangle = \mathcal{A}_\text{BG}^{-1} \sum_{{\bf{k}}\in\mathcal{A}_\text{BG}} G({\bf{k}})
\end{equation}
is a proxy for the level of condensate depletion $(n-n_c)$.
Applied to the case with hardcore interactions in Fig.~\ref{fig:depletion}, we deduce a condensate fraction of $n_c=0.214$ from the intensity of the background, which
should be compared to the exact value of $n_c=0.256$. This estimate is rather crude, as the background also contributes some signal within $\mathcal{A}_\text{P}$.
A more accurate value is obtained allowing for such a contribution $\langle\delta I(\mathcal{A}_\text{P}) \rangle \equiv \kappa \langle I(\mathcal{A}_\text{BG})\rangle$,
where $\kappa$ is a `coherence' factor for the addition between the coherent condensate wavefunction and the incoherent background. The corrected estimate becomes
\begin{equation}
n-n_c = \frac{\mathcal{A}_\text{BG} + \kappa \mathcal{A}_\text{P}}{\mathcal{A}_\text{BG}+\mathcal{A}_\text{P}} \langle I(\mathcal{A}_\text{BG}) \rangle,
\end{equation}
which reduces to (\ref{eq:DepletionProxy}) in the limit of sharp coherence peaks $\mathcal{A}_\text{P}\to 0$. For the data in Fig.~\ref{fig:depletion}, we find exact values
of $\kappa$ in the range of 0.4 to 0.5. Assuming $\kappa = 0.45$, the exact condensate depletion is reproduced to within 1\% accuracy.
\section{Summary}
We have studied the groundstates of two-dimensional frustrated Bose-Hubbard
models. We have focused on the situation in which the imposed gauge fields
give rise to a pattern of staggered fluxes, of magnitude $\alpha$ and
of alternating sign along one of the principal axes. For $\alpha=1/2$ this
model is equivalent to the case of uniform flux $n_\phi=1/2$, which is the
``fully frustrated'' XY model with time-reversal symmetry. We have shown
that, for $\alpha_c < \alpha < 1/2$, with $\alpha_c \approx 0.389$,
the mean-field groundstate breaks
translational invariance, giving rise to a density wave pattern. For $\alpha
=1/2$ the mean-field groundstate breaks both translational symmetry and
time-reversal symmetry, forming the staggered flux phase which has uniform
density but circulating gauge-invariant currents.
We have introduced a general numerical technique to detect broken
symmetry condensates in exact diagonalization studies. Using this technique
we have shown that, for all cases studied, the Bose-Hubbard model with
staggered flux $\alpha$ is condensed. We have obtained quantitative
determinations of the condensate fractions. In
particular, our results establish that the fully frustrated quantum XY model
is condensed at zero temperature, with a condensate fraction of $x_c\simeq
0.4$.
The low-temperature condensed phases that appear in this system are of significant interest in
connection with their thermal phase transitions into the high-temperature
normal phase. The groundstate of the fully frustrated system breaks both
$U(1)$ and $Z_2$ symmetries (owing to both Bose-Einstein condensation, and the
combination of translational symmetry breaking and time-reversal symmetry
breaking). The transition to the high-temperature phase is interesting, combining the physics of
the Kosterlitz-Thouless transition [$U(1)$] with an Ising transition ($Z_2$),
and its properties have stimulated much theoretical debate.\cite{martinoli,Fazio2001235} The cold
atom system described here will allow this transition to be studied also in a highly quantum regime, with of order one particle per lattice site,
that is inaccessible in frustrated Josephson junction arrays.\cite{Polini:2005p392} Our results show that
the staggered flux model leads also to other cases that break $U(1)$ and $Z_3$
(or higher $Z_p$ symmetry depending on the flux $\alpha$). There is also the
possibility to study commensurate/incommensurate transitions driven by locking
of the density wave order to the underlying lattice.
We discussed in detail the experimental consequences of our results. In our
discussion, we have explained the meaning of gauge-invariance in ultracold
atom systems subject to optically induced gauge potentials: Expansion images
are gauge-dependent, and (provided gauge fields are absent during expansion)
measure the canonical momentum distribution. Furthermore, we have shown how
the ability to imprint phase patterns prior to expansion (analogous to an
instantaneous change of gauge) can allow very useful additional information to
be extracted from expansion images.
\acknowledgments
GM gratefully acknowledges support from Trinity Hall Cambridge, and would like
to thank Nordita for their hospitality. NRC has been supported by
EPSRC Grant EP/F032773/1.
{\it Note added:} After completing this work, we noticed a related preprint by Powell et al.,\cite{Powell:2010p391} which has
some overlap with our discussion of the case $\alpha=1/2$.
|
{
"timestamp": "2010-09-23T02:02:35",
"yymm": "1009",
"arxiv_id": "1009.4420",
"language": "en",
"url": "https://arxiv.org/abs/1009.4420"
}
|
\section{Introduction}
Mystery is the origin of tiny neutrino masses that are
indicated from the neutrino oscillation data.
How can we understand the smallness of neutrino masses
as compared to the electroweak scale?
A simple way of the explanation may be based on the seesaw
mechanism~\cite{Minkowski:1977sc,Yanagida:1979,GellMann:1979,Mohapatra:1979ia},
introducing
right-handed neutrinos with large Majorana masses
at the scale such as that of grand unification.
Although this is an attractive scenario,
introduction of such large masses causes
another hierarchy among mass scales.
In addition, such a large mass scale is beyond the experimental
reach and the theory would be untestable directly.
In the Standard Model (SM),
the Higgs sector, on the other hand, is the last uncharted
part.
Although the SM Higgs sector is the simplest scenario
with a scalar isospin doublet,
the true Higgs sector may take a non-minimal form.
Such an extended Higgs sector
may be closely related to the mechanism to induce tiny neutrino masses
at the TeV scale.
Such a possibility is interesting because the model
is in principle testable directly at on-going and
future collider experiments,
such as the Fermilab Tevatron, the CERN Large Hadron Collider (LHC)
and the International Linear Collider (ILC).
If neutrino masses are of the Majorana type, they are generated
through the lepton number violating effective operators.
In the usual seesaw scenarios, the neutrino masses are derived
from the dimension-five operator $\nu\nu\phi\phi/\Lambda$,
where $\nu$ represents left-handed neutrinos,
$\phi$ does the Higgs boson, and
$\Lambda$ is a scale of the new physics.
In a class of models where neutrino masses are radiatively
generated, such a dimension-five operator is induced at
the loop level by the TeV scale dynamics.
For example, in the model proposed by A.~Zee~\cite{Zee:1980ai,Zee:1985rj},
the dimension-five operator is generated at the one-loop level
via the lepton number violating interaction and dynamics of
the extended Higgs sector.
In the model proposed by E.~Ma~\cite{Ma:2006km},
the dimension-five operator is also
generated at the one-loop level via the physics of the extra scalar
doublet and the TeV scale right-handed neutrinos, where the both of
new fields are assigned odd quantum number under the discrete
$Z_2$ symmetry.
Such a one-loop generation of neutrino masses from the TeV
scale dynamics, however, still requires unnaturally small
coupling constants for reproducing the tiny neutrino masses.
There are several models in which neutrino masses are generated at
the two-loop level~\cite{Zee:1985id,Babu:1988ki,Aoki:2010ib,Babu:2010vp}
and also the three-loop level~\cite{Krauss:2002px,Cheung:2004xm,Aoki:2008av,Aoki:2009vf},
where such fine tuning is not necessary because of the sufficient
suppression by additional loop factors.
In all these models, dimension-five operators are
induced at the loop level.
Recently, a new idea has been proposed where
tiny neutrino masses are generated via the operators whose
dimension is higher than
five~\cite{Gogoladze:2008wz,Babu:2009aq,Bonnet:2009ej,Picek:2009is,Liao:2010rx}.
In Ref.~\cite{Bonnet:2009ej},
some concrete examples are examined,
in which neutrino masses are generated
via the dimension-seven operator $\nu\nu\phi\phi\phi\phi/\Lambda^3$
which are induced at the tree level with the extend scalar dynamics.
In this case, there is an additional suppression factor of
$(v/\Lambda)^2$ as compared to neutrino masses
via the dimension-five operators, where $v$ ($\simeq 246$ GeV)
is the vacuum expectation value (vev) of the Higgs boson.
Although these models are interesting,
a sort of fine tuning is still required especially
to reproduce the scale of neutrino masses,
when $\Lambda$ is assumed to be of TeV scale.
\begin{figure*}[htb]
\unitlength=1cm
\begin{picture}(16,3)
\put(0,0){\includegraphics[width=7.5cm]{seesaw.ps}}
\put(8,0){\includegraphics[width=7.5cm]{Ma.ps}}
\put(0,0){$L$}
\put(7.2,0){$L$}
\put(1.9,2.5){$H_{2}$}
\put(5.4,2.5){$H_{2}$}
\put(3.5,-0.2){$N_{R}$}
\put(8,0){$L$}
\put(15.2,0){$L$}
\put(9.9,2.8){$H_{2}$}
\put(13.3,2.8){$H_{2}$}
\put(10,1.5){$\eta$}
\put(13.3,1.5){$\eta$}
\put(11.5,-0.2){$N_{R}$}
\thicklines
\put(7.2,1){\vector(1,0){1}}
\end{picture}
\caption{Schematic explanation of the method to make
a loop digram from the tree diagram for neutrino masses.
The Higgs doublets $H_{2}$ in the tree seesaw diagram (left)
are substituted by the inert doublets $\eta$ with odd parity,
and the loop of the inert doublet is closed
by the quartic coupling of $(\eta^{\dagger} H_{2})^{2}$.
This loop dimension-five model (right)
was proposed in Ref.~\cite{Ma:2006km}.
Here $N_{R}$ represents right-handed neutrinos.}
\label{Fig:seesaw-to-Ma}
\end{figure*}
In this paper, we propose a scenario in which neutrino masses are
generated via higher-dimensional operators
$\nu\nu (\phi \phi)^{(d-3)/2}/\Lambda^{d-4}$
($d = 7,9,11 \cdots$)
which are induced by quantum effect.
In general, the size of neutrino masses
from the operator with the mass dimension $d$,
which arises from a $n$-loop diagram,
can be estimated as
\begin{align}
m_{\nu}
\sim
v
\times
\left(
\frac{1}{16\pi^{2}}
\right)^{n}
\times
\left(\frac{v}{\Lambda}\right)^{d-4}.
\label{eq:mNu-symbolic}
\end{align}
In the models with $d=7$ and $n=1$,
neutrino masses are further suppressed by
the one-loop factor $1/(16 \pi^{2})$
and the factor $(v/\Lambda)^2$
as compared to the tree-induced dimension-five operator case
(=the ordinary seesaw model).
In such models,
the new physics scale $\Lambda$ may be set on the TeV scale
without assuming any unnatural small coupling constant.
In order to realize this scenario, we impose
an exact $Z_2$ parity~\cite{Krauss:2002px,Ma:2006km}
and an approximate discrete
symmetry~\cite{Bonnet:2009ej,Giudice:2008uua}
to forbid the appearance of the dimension-five operator
as well as
the dimension-seven operators induced at the tree level.
In such models, the lightest $Z_{2}$ odd particle can be a
Dark Matter (DM) candidate as long as it is electrically
neutral.
We show two concrete examples of the models along this line.
It is demonstrated that the models can reproduce the neutrino
data for the masses and mixings without fine tuning among
coupling constants due to the TeV scale dynamics of the models.
We discuss the constraint on parameters of the models from
the data of lepton flavor violation~\cite{Amsler:2008zzb,Adam:2009ci}.
In these models, extended scaler sectors appear
with the exact $Z_2$ symmetry,
which provide rich phenomenological predictions.
We mention the test of the models at current and future collider
experiments at the LHC and the ILC.
This paper is organized as follows:
In Sec.~\ref{Sec:method}, we briefly recapitulate the method to realize
the higher dimensional neutrino mass generation with
an approximate discrete symmetry.
We also review the way to make a tree diagram for neutrino masses
become the loop diagram by introduction of $Z_{2}$ parity.
Combining with these two methods, we construct two concrete models
in Sec.~\ref{Sec:models},
in which neutrino masses arise from the effective
dimension-seven operator which is induced at the one-loop level.
In Sec.~\ref{Sec:Summary},
we discuss some phenomenological aspects of the models.
\section{Method}
\label{Sec:method}
Before we come on to descriptions of the concrete models,
let us look briefly at the essentials for
the tree-level dimension-seven neutrino mass
generation~\cite{Bonnet:2009ej}.
There are two key components to produce the effective
dimension-seven operator for neutrino masses
at the electroweak scale:
\begin{itemize}
\item An additional symmetry to forbid the dimension-five
$\nu\nu \phi\phi/\Lambda$ operator.
The simplest choice for non-supersymmetric models is $Z_{5}$.
\item The extended Higgs sector with two Higgs doublets
so that the combination $(H_{1} H_{2})$
can carry a charge under the additional symmetry.
Here the hypercharge of $H_{1}$ is given to be $-1/2$ and
that of $H_{2}$ is $+1/2$.
\end{itemize}
Taking the setups and assigning appropriate charges
to the standard model particles,
we can forbid the dimension-five operator and make
\begin{align}
\mathcal{L}_{\text{eff}}
=
\frac{\mathcal{C}}{\Lambda^{3}}
LLH_{2} H_{2} H_{2} H_{1}
\label{eq:effop-dim7}
\end{align}
to be the leading contribution to neutrino masses,
where $\mathcal{C}$ is a mass dimensionless coefficient\footnote{%
The choice of dimension-seven operators which contribute to
neutrino masses is not unique~\cite{Bonnet:2009ej}.
In this paper, we concentrate on the operator shown
in Eq.~\eqref{eq:effop-dim7}.
}.
The possible models for this tree-level dimension-seven
neutrino mass generation mechanism
are listed in Ref.~\cite{Bonnet:2009ej}.
In the models including the SM singlet fermions
at the high energy scale,
one can see that the $Z_{5}$ symmetry forbids the fermions
(=right-handed neutrinos) to have the Majorana mass term.
Because of the absence of the Majorana mass term,
the dimension-five operator cannot arise at the electroweak
scale and the dimension-seven operator dominates
the contribution to neutrino masses.
Extending these models,
we consider the models,
in which neutrino masses are generated via the
dimension-seven operator
but the effective operator is induced through a one-loop
diagram.
To construct such loop-induced models, we follow the method with
the exact $Z_{2}$ symmetry, which was developed
in Refs.~\cite{Krauss:2002px,Ma:2006km}.
The essential is introduction of the inert doublet
with the odd parity under the $Z_{2}$ symmetry.
Due to the exact $Z_{2}$ parity, the inert doublet
cannot take a vacuum expectation value (vev).
Let us describe the maneuver, taking the ordinary type-I
seesaw model as an example.
The procedure is schematically illustrated
in Fig.~\ref{Fig:seesaw-to-Ma}.
Assigning the odd parity to the right-handed neutrinos $N_{R}$
and substituting the inert doublet $\eta$ for
the Higgs doublet $H_{2}$ in the neutrino Yukawa interaction,
one can forbid the
tree-level contribution to neutrino masses.
The inert doublets in the diagram Fig.~\ref{Fig:seesaw-to-Ma}
are converted to the Higgs doublets
through a quartic interaction,
\begin{align}
\mathcal{L} = \frac{\lambda}{2}
(\eta^{\dagger} H_{2}) (\eta^{\dagger} H_{2})
+{\rm H.c.}.
\end{align}
In other words, the inert Higgs legs are closed by the quartic
interaction and make a loop.
This leads to the one-loop diagram for neutrino masses,
which was proposed in Ref.~\cite{Ma:2006km}.
In the following sections, we will apply this procedure
to the models in which neutrino masses are
generated through the dimension-seven operator which
is induced via tree diagrams,
and build the loop-induced dimension-seven models.
\section{Models}
\label{Sec:models}
We here consider two examples
to illustrate the method to build the models in which
neutrino masses are generated through the effective dimension-seven
operator induced from a one-loop diagram.
\subsection{Model A}
The renormalizable models to induce the effective
interaction Eq.~\eqref{eq:effop-dim7}
from tree diagrams at the electroweak scale are
listed in Ref.~\cite{Bonnet:2009ej}.
In this subsection, we employ the model described
as Decomposition~\#1 among them, in which the SM
gauge singlet Dirac fermion $\psi$
and the singlet scalar $\varphi$ are introduced.
The particle contents and the charge assignments
are summarized in Table~\ref{Tab:Decom1-charge}.
Here, the charges for the quarks and leptons
are assigned so as to reproduce the Yukawa interactions
of type-II two-Higgs-doublet model (THDM)\footnote{%
\label{footnote:THDM}
In general, there are four possibilities
for the Yukawa interaction in THDM
under the (softly-broken) discrete $Z_{2}$
symmetry~\cite{Barger:1989fj,Grossman:1994jb,Akeroyd:1994ga,Akeroyd:1996di,Akeroyd:1998ui,Aoki:2009ha,Su:2009fz,Logan:2009uf}.
All the possibilities can also be realized with appropriate
charge assignments in the case of the $Z_{5}$ symmetry.
}.
For detailed arguments for this model and the
(softly broken) $Z_{5}$ symmetry, see Sec.~3.1
in Ref.~\cite{Bonnet:2009ej}.
\begin{table}[tb]
\begin{tabular}{ccccccccccc}
\hline \hline
& $L$ & $e^{c}$ & $Q$ & $u^{c}$ & $d^{c}$ & $H_{2}$ & $H_{1}$
& $\psi ({\bf 1}^{D}_{0})$ & $\eta ({\bf 2}^{s}_{1/2})$
& $\varphi({\bf 1}^{s}_{0})$
\\
\hline
softly broken $Z_{5}$
&
1 &1 &0 & 0 & 2 & 0 & 3 & 1 & 0 & 3 \\
exact $Z_{2}$
&
+ & + & + & + & + & + & +
&
$-$ & $-$ & +
\\
\hline \hline
\end{tabular}
\caption{Particle contents and charge assignments for
Model A.
The symbol ${\bf X}_{Y}^{\mathcal{L}}$ indicates
the representations of the fields;
${\bf X}$ for $SU(2)_{L}$, $Y$ for $U(1)_{Y}$,
and $\mathcal{L}$ for Lorenz group; i.e.,
Dirac spinor ($D$) and scalar ($s$).
}
\label{Tab:Decom1-charge}
\end{table}
In this letter, we are interested in the neutrino masses
induced from the dimension-seven operator Eq.~\eqref{eq:effop-dim7}
but the effective interaction is realized by a loop diagram.
In order to forbid arising the dimension-seven operator from
a tree diagram, we introduce the exact $Z_{2}$ parity
and an inert doublet $\eta$, and assign the odd charge
for the inert doublet and the
singlet Dirac fermion $\psi$.
The Lagrangian of the fundamental interactions for the neutrino mass
generation is given as
\begin{align}
\mathcal{L}
=&
\mathcal{L}_{\rm SM}
\nonumber
\\
&+
\biggl[
{(Y_{\nu})_{a}}^{\alpha}
\overline{\psi}^{a}
{\rm P}_{L}
(\eta {\rm i} \tau^{2} L_{\alpha})
+
(\kappa_{L})^{ab}
\varphi
\overline{\psi^{c}}_{a}
{\rm P}_{L}
\psi_{b}
\nonumber
\\
&\hspace{0.5cm}+
(\kappa_{R})^{ab}
\varphi
\overline{\psi^{c}}_{a}
{\rm P}_{R}
\psi_{b}
+
\mu \varphi^{*}
(H_{1} {\rm i} \tau^{2} H_{2})
+
{\rm H.c.}
\biggr]
\nonumber
\\
&
+
{M_{a}}^{b}
\overline{\psi}^{a}
\psi_{b}
+
m_{\varphi}^{2}
\varphi^{*} \varphi
+
m_{\eta}^{2} (\eta^{\dagger} \eta)
\nonumber
\\
&+
\left[
\frac{\lambda}{2}
(\eta^{\dagger} H_{2})
(\eta^{\dagger} H_{2})
+
{\rm H.c.}
\right]
-
\mathcal{V}_{\text{scalar}},
\label{eq:L-Decom1}
\end{align}
where $a$, $b$ and $\alpha$ represent
the flavour indices.
Let us first focus on neutrino masses which are
our main concern, and we will take up some phenomenological
consequences of this model and the part
$\mathcal{V}_{\text{scalar}}$ of the scalar potential later.
With the interactions shown in Eq.~\eqref{eq:L-Decom1},
the dimension-seven operator for neutrino masses
is induced by the one-loop diagram described
in Fig.~\ref{Fig:mNu-Decom1}, which is evaluated as
\begin{align}
\mathcal{L}_{\text{eff}}
=&
\frac{1}{(4\pi)^{2}}
\frac{\lambda \mu}{m_{\varphi}^{2} m_{\eta}^{2}}
{(Y_{\nu}^{\sf T})^{\alpha}}_{a}
{(Y_{\nu})_{b}}^{\beta}
\nonumber
\\
&\times
\Bigl[
(\kappa_{L})^{a b}
\frac{M_{a} M_{b}}{m_{\eta}^{2}}
\mathcal{I}(x_{a},x_{b})
+
(\kappa_{R})^{a b}
\mathcal{J}(x_{a},x_{b})
\Bigr]
\nonumber
\\
&\times
(
\overline{L^{c}}_{\alpha}
{\rm i} \tau^{2} H_{2}
)
(H_{2}
{\rm i} \tau^{2} L_{\beta})
(H_{1} {\rm i} \tau^{2} H_{2}),
\end{align}
where the functions $\mathcal{I}$
and $\mathcal{J}$ are defined as
\begin{align}
\mathcal{I} (x_{a},x_{b})
=&
\frac{1}{(1-x_{a})(1-x_{b})}
\nonumber
\\
&\hspace{-1.5cm}\times
\Biggl[
1
+
\frac{
(1-x_{b}) x_{a} \ln x_{a}
}{
(x_{a} - x_{b})(1-x_{a})}
-
\frac{(1-x_{a}) x_{b} \ln x_{b}}{
(x_{a} - x_{b})
(1-x_{b})}
\Biggr],
\\
\mathcal{J}
(x_{a},x_{b})
=&
\frac{1}{(1-x_{a})(1-x_{b})}
\nonumber
\\
&\hspace{-1.5cm}\times
\Biggl[
1
+
\frac{
(1-x_{b}) x_{a}^{2} \ln x_{a}
}{
(x_{a} - x_{b})(1-x_{a})}
-
\frac{(1-x_{a}) x_{b}^{2} \ln x_{b}}{
(x_{a} - x_{b})
(1-x_{b})}
\Biggr],
\end{align}
with $x_{a} \equiv M_{a}^{2}/m_{\eta}^{2}$.
\begin{figure}[tb]
\unitlength=1cm
\begin{picture}(8,5)
\put(0,0){\includegraphics[width=8cm]{Decom1.ps}}
\put(0,2){$L$}
\put(7.8,2){$L$}
\put(2.6,1.5){$\psi$}
\put(5,1.5){$\psi$}
\put(4.2,1.3){$\varphi$}
\put(2,3){$\eta$}
\put(5.8,3){$\eta$}
\put(2.05,4.6){$H_{2}$}
\put(5.6,4.6){$H_{2}$}
\put(2.05,0){$H_{2}$}
\put(5.6,0){$H_{1}$}
\end{picture}
\caption{Diagram for neutrino masses in Model A.
The lepton number is violated at
the interaction of $\psi$-$\psi$-$\varphi$,
which is shown with a fat blob.}
\label{Fig:mNu-Decom1}
\end{figure}
We obtain neutrino masses
\begin{align}
(m_{\nu})^{\alpha \beta}
=
-\frac{v^{2}}{2} \sin^{2} \beta
{(Y^{\sf T})^{\alpha}}_{a}
(M_{\text{eff}}^{-1})^{ab}
{(Y_{\nu})_{b}}^{\beta},
\label{eq:mNu-A}
\end{align}
which are the same form as those derived from
the ordinary type-I seesaw scenario.
{\it The effective mass} $M_{\text{eff}}$
{\it for the right-handed neutrinos} is given to be
\begin{align}
(M_{\text{eff}}^{-1})^{ab}
=&
\frac{1}{(4\pi)^{2}}
\frac{v^{2}}{2}
\sin 2\beta
\frac{\lambda \mu}{m_{\varphi}^{2} m_{\eta}^{2}}
\nonumber
\\
&\hspace{-1.2cm}\times
\left[
(\kappa_{L})^{a b}
\frac{M_{a} M_{b}}{m_{\eta}^{2}}
\mathcal{I}(x_{a},x_{b})
+
(\kappa_{R})^{a b}
\mathcal{J}(x_{a},x_{b})
\right].
\label{eq:Meff-A}
\end{align}
Therefore, it is guaranteed that
this model can reproduce all the features of the
neutrino flavour in the canonical type-I seesaw model.
From the expressions Eqs.~\eqref{eq:mNu-A}
and~\eqref{eq:Meff-A},
it turns out that
neutrino masses of the order of one eV
is compatible with TeV scale masses for new fields
without assuming extremely tiny couplings in the model.
One can expect that the collider experiments
are accessible to those fields.
Let us discuss the scalar potential
and the softly-broken discrete symmetry in this model.
With the exact $Z_{5}$ symmetry,
the part of the scalar potential, which is only
including $H_{1}$ and $H_{2}$ is described as
\begin{align}
\mathcal{V}_{\text{THDM}}
=&
m_{1}^{2} |H_{1}|^{2}
+
m_{2}^{2} |H_{2}|^{2}
+
\frac{\lambda_{1}}{2}
|H_{1}|^{4}
+
\frac{\lambda_{2}}{2}
|H_{2}|^{4}
\nonumber
\\
&+
\lambda_{3}
|H_{1}|^{2} |H_{2}|^{2}
+
\lambda_{4}
|H_{1} {\rm i} \tau^{2} H_{2}|^{2},
\end{align}
and it actually respects the global $U(1)$ symmetry
including $Z_{5}$~\cite{Bonnet:2009ej}.
If the $U(1)$ symmetry is spontaneously broken
with the vevs of the Higgs doublets,
it leads a Nambu-Goldstone boson.
To dodge this problem, here we assume $Z_{5}$ is an approximate
symmetry at the new physics scale $\Lambda$ and
introduce an explicit and soft-breaking term of $Z_{5}$
\begin{align}
\mathcal{V}_{\text{soft}}
=
m_{3}^{2} H_{1} {\rm i} \tau^{2} H_{2} + {\rm H.c.},
\label{eq:Z5soft}
\end{align}
by setting the scale $m_{3}$ at the electroweak scale.
This term does not invoke the dimension-five operator at the tree level,
but at the loop level. The dimension-five contribution
is only constructed through the dimension-seven operator
of Fig.~\ref{Fig:mNu-Decom1} by connecting outer
legs of $H_{2}$ and $H_{1}$.
Therefore, setting $m_{3}$ is smaller than $\Lambda$
(but large enough to avoid the bound to the pseudo
Nambu-Goldstone boson), we can keep the contribution
sub-dominant against that arises from the original
dimension-seven diagram.
Notice that by appropriate assignment of $Z_5$ charges
for right-handed quarks and charged leptons we can
have a Yukawa interaction without flavor changing neutral
current at the tree level (See also footnote \ref{footnote:THDM}).
Phenomenological constraints and implications to
the scalar sector are mentioned in Sec.~\ref{Sec:Summary}
together with the other example which will
be illustrated in the next subsection.
Before we turn to another example,
let us briefly discuss constraints from lepton flavour
violation in this model.
Since the Higgs fields do not mediate
flavour changing neutral currents at the tree level,
the leading contribution to the lepton flavour
violating processes arises from a diagram
with a loop between two Yukawa interactions.
The contribution is exactly the same as that in the original
dark doublet model~\cite{Ma:2006km}, which was calculated
in Ref.~\cite{Ma:2001mr,Kubo:2006yx}:
\begin{align}
{\rm Br}(\mu \rightarrow e \gamma)
=
\frac{3 \alpha_{\rm em}}{64 \pi (G_{F} m_{\eta}^{2})^{2}}
\left|({\mathcal{C}_{\text{A}})_{e}}^{\mu}\right|^{2},
\label{eq:Brmueg-Decom1}
\end{align}
where the mass-dimensionless coefficient
$\mathcal{C}_{\text{A}}$
for Model A is given as
\begin{align}
{(\mathcal{C}_{\text{A}})_{e}}^{\mu}
=&
{(Y_{\nu}^{\dagger})_{e}}^{a}
\mathcal{F}(x_{a})
{(Y_{\nu})_{a}}^{\mu},
\end{align}
and the function $\mathcal{F}$ is
\begin{align}
\mathcal{F}(x_{a})
\equiv
\frac{1- 6 x_{a} + 3 x_{a}^{2} + 2 x_{a}^{3} - 6 x_{a}^{2} \ln x_{a}}
{6 (1-x_{a})^{4}}.
\end{align}
We can see that
it might be essential to assume a large
value for $m_{\eta}$
enough to avoid a sizable
$\mu \rightarrow e \gamma$ effect.
An alternative way to circumvent the large LFV process
is discussed in Ref.~\cite{Kubo:2006yx}.
\subsection{Model B}
\begin{table}[tb]
\begin{tabular}{ccccccccccc}
\hline \hline
& $L$ & $e^{c}$ & $Q$ & $u^{c}$ & $d^{c}$ & $H_{2}$ & $H_{1}$
& $\psi ({\bf 1}^{D}_{0})$ & $\eta ({\bf 2}^{s}_{1/2})$
& $\eta'({\bf 2}^{s}_{-1/2})$
\\
\hline
soft br. $Z_{5}$
&
1 &1 &0 & 0 & 2 & 0 & 3 & 1 & 0 & 2 \\
exact $Z_{2}$
&
+ & + & + & + & + & + & +
&
$-$ & $-$ & $-$
\\
\hline \hline
\end{tabular}
\caption{Particle contents and charge assignments
for the softly broken $Z_{5}$ and the exact $Z_{2}$
in Model B.}
\label{Tab:Decom13-charge}
\end{table}
Let us show the second example with the different
type of Decomposition (\#~13).
We introduce two inert doublets.
This allows to have two types of Yukawa interactions for neutrinos:
one is the ordinary one with right-handed neutrinos $\psi_{R}$,
and the other appears with left-handed component $\psi_{L}$
of the SM singlet fermion and violates the
lepton number.
The particle contents and their charge assignments are summarized
in Tab.~\ref{Tab:Decom13-charge}.
The interaction is given by
\begin{align}
\mathcal{L} =& \mathcal{L}_{\rm SM}
\nonumber
\\
&+
\biggl[
{(Y_{\nu})_{a}}^{\alpha}
\overline{\psi}^{a} {\rm P}_{L}
\eta {\rm i} \tau^{2} L_{\alpha}
+
{(Y_{\nu}')^{a \alpha}}
\overline{\psi^{c}}_{a}
{\rm P}_{L} \eta'^{\dagger} L_{\alpha}
\nonumber
\\
&\hspace{0.5cm}+
\zeta (H_{1} {\rm i} \tau^{2} H_{2})
(\eta' {\rm i} \tau^{2} \eta)
+
\frac{\lambda}{2}
(\eta^{\dagger} H_{2}) (\eta^{\dagger} H_{2})
\nonumber
\\
&
\hspace{0.5cm}
+
{\rm H.c.}
\biggr]
\nonumber
\\
&
+
{M_{a}}^{b} \overline{\psi}^{a} \psi_{b}
+
m_{\eta'}^{2} \eta'^{\dagger} \eta'
+
m_{\eta}^{2} \eta^{\dagger} \eta
-
\mathcal{V}_{\text{scalar}}.
\label{eq:L-Decom13}
\end{align}
The scalar potential is obviously different from
that of Model A.
However we assume also
that it includes the soft violation term of the
$Z_{5}$ symmetry, which was shown in Eq.~\eqref{eq:Z5soft},
to avoid the problem of the Nambu-Goldstone boson.
With the Lagrangian in Eq.~\eqref{eq:L-Decom13},
neutrino masses are constructed as shown in Fig.~\ref{Fig:mNu-Decom13},
and they are calculated to be
\begin{align}
(m_{\nu})^{\alpha \beta}
=&
-
\lambda \zeta
\frac{v^{4}}{8}\sin^{2} \beta \sin 2 \beta
\nonumber
\\
&\times
\biggl[
(Y_{\nu}'^{{\sf T}})^{\alpha a}
M_{a} \mathcal{I} (x_{a},y)
{(Y_{\nu})_{a}}^{\beta}
\nonumber
\\
&\hspace{0.5cm}+
{(Y_{\nu}^{\sf T})^{\alpha}}_{a}
M_{a}
\mathcal{I} (x_{a},y)
(Y_{\nu}')^{a \beta}
\biggr],
\label{eq:mNu-ModelB}
\end{align}
where $y \equiv m_{\eta'}^{2} /m_{\eta}^{2}$.
The flavour structure of this model is rather involved,
and it cannot be understood with the ordinary seesaw formula
because of two independent
Yukawa matrices $Y_{\nu}$ and $Y_{\nu}'$.
With the assumption that
$Y_{\nu}'$ takes the same flavour structure as
$Y_{\nu}$, Eq.~\eqref{eq:mNu-ModelB}
is reduced to the ordinary type-I seesaw formula.
Therefore, this model can obviously reproduce
the mass matrices which are consistent with
the observed mass squared differences and the mixings.
The lepton number violating Yukawa interaction
gives an additional contribution to the LFV process
$\ell_{\alpha} \rightarrow \ell_{\beta} \gamma$.
The decay branching ratio in Model B can be obtained
by substituting
\begin{align}
{(\mathcal{C}_{\text{B}})_{e}}^{\mu}
=
{(Y_{\nu}^{\dagger})_{e}}^{a}
\mathcal{F}(x_{a})
{(Y_{\nu})_{a}}^{\mu}
+
(Y_{\nu}'^{\dagger})_{e a}
\mathcal{F}(x_{a})
(Y_{\nu}')^{a \mu},
\label{eq:LFVcoeff-B}
\end{align}
for $\mathcal{C}_{\text{A}}$ in Eq.~\eqref{eq:Brmueg-Decom1}.
\begin{figure}[t]
\unitlength=1cm
\begin{picture}(8,3.5)
\put(0,0){\includegraphics[width=8cm]{Decom13.ps}}
\put(0,0){$L$}
\put(7.8,0){$L$}
\put(3.9,0.6){$\psi$}
\put(5.95,0.8){$\eta'$}
\put(1.9,0.8){$\eta$}
\put(4,2.5){$\eta$}
\put(1.5,2.7){$H_{2}$}
\put(1,1.2){$H_{2}$}
\put(6.2,2.7){$H_{2}$}
\put(6.65,1.2){$H_{1}$}
\end{picture}
\caption{Diagram for neutrino masses in
Model B.
The lepton number is violated at
the interaction of $\psi$-$L$-$\eta'$,
which is shown with a fat blob.}
\label{Fig:mNu-Decom13}
\end{figure}
\section{Summary and Discussion}
\label{Sec:Summary}
We have proposed the new scenario in which tiny neutrino masses
are generated via loop-induced $d>$ 5 operators.
In such a scenario, the scale of tiny neutrino masses can be
reproduced from the TeV scale physics
in a natural way without extreme fine tuning because
the combination of the loop factor $1/(16\pi^2)^{n}$
and the additional coefficient of $(v/\Lambda)^{d-5}$
provides the sufficient suppression factor.
We have in particular discussed as examples two concrete models where
neutrino masses are generated via one-loop induced
dimension-seven operators due to
the dynamics of extended Higgs sector and a vector-like Dirac
neutrino whose mass is assumed to be
at the TeV scale under the imposed exact discrete $Z_2$ symmetry.
We have shown that in these models neutrino masses can be reproduced
and that the neutrino mixing data are also satisfied without contradicting
the constraint from the LFV data~\cite{Amsler:2008zzb,Adam:2009ci}.
We here give a comment on phenomenological implications
in these models. However, the detailed discussion is
beyond the scope of this paper, and it is given
elsewhere~\cite{in-preparation}.
First of all, a common feature of these models is the
extended Higgs sectors, in which there are two $Z_2$-even
Higgs doublets and one or two $Z_2$-odd doublets.
Phenomenology of the THDM has been
discussed in literature.
The Higgs potential is constrained by the perturbative
unitarity~\cite{Lee:1977eg,Kanemura:1993hm,Akeroyd:2000wc,Ginzburg:2003fe}, the vacuum stability~\cite{Deshpande:1977rw,Nie:1998yn,Kanemura:1999xf}, and
also electroweak precision data~\cite{Lim:1983re,Haber:1992py,Pomarol:1993mu}.
When the type-II THDM is assumed,
the bounds from $b\rightarrow s \gamma$~\cite{Ciuchini:1997xe},
$B \rightarrow \tau \nu$~\cite{Hou:1992sy,Isidori:2006pk,Isidori:2007jw}
and
the leptonic tau decay~\cite{Krawczyk:2004na} have also to be taken
care.
The discovery of extra Higgs bosons in addition to the lightest
(SM-like) Higgs boson and the measurement of their properties
are important to test these models.
In these models, the induced neutrino masses are multiplied by the
factor of $\sin^{2} \beta \sin 2 \beta$, so that a large value of
$\tan \beta$ gives a further suppression factor.
This may bring an interesting correlation between
neutrino masses and the physics of the Higgs sector.
The experimental confirmation of the $Z_{2}$ odd sector
is essentially important too.
Especially, the lightest $Z_2$ odd particle can be a
candidate of dark matter if it is electrically (and colour) neutral.
In these models, there are two possibilities for the DM candidate;
i.e., 1) the lightest $\eta^0$ boson is the DM or
2) the Dirac neutrino $\psi$ is the DM.
In Case 1), the phenomenology of such $Z_2$ odd sector has been
studied with the physics of the DM candidate in the context of
the dark doublet model~\cite{Barbieri:2006dq}
and the radiative seesaw
models~\cite{Krauss:2002px,Ma:2006km,Aoki:2010tf}.
An interesting signature of DM may be the
invisible decay of the (SM-like) Higgs boson when DM is
lighter than a half of the Higgs boson mass.
It is expected that the branching ratio of the Higgs boson
invisible decay of greater than
50 \% (1\%) can be detected at the LHC (at the ILC).
The direct DM search is also important for the case of 1).
The multi Higgs portal dark matter has been discussed in
Ref.~\cite{Aoki:2009pf}.
The detailed comprehensive study for models of the Higgs portal dark matter
has been done in Ref.~\cite{Kanemura:2010sh} in a specific scenario where
only the Higgs boson and the DM candidate are electroweak scale and the
other new particles are supposed to be decoupled.
The collider phenomenology of the Higgs sectors with dark doublet fields
has been studied in \cite{Lundstrom:2008ai} at the LEP,
in Ref.~\cite{Cao:2007rm,Dolle:2009fn,Dolle:2009ft,Miao:2010rg}
at the LHC and in Ref.~\cite{Aoki:2010tf} at the ILC.
For the test of our model, many parts
of these previous studies can be applied.
If $\psi$ is dark matter corresponding to
the case of 2), the situation may be similar to
the case in the model by Ma where the right-handed neutrino is
the DM candidate which
has been studied in detail in Ref.~\cite{Kubo:2006yx}.
However, $\psi$ is a Dirac neutrino, not a Majorana neutrino,
so that the DM number can be assigned. The DM number may
be dynamically generated in the context of asymmetric DM.
Details of these issues are discussed in Ref.~\cite{in-preparation}.
\begin{center}
{\bf Acknowledgments}
\end{center}
This work was supported, in part, by Grant-in-Aid for Scientific Research,
Japan Society for the Promotion of Science, Nos.~18034004 (C)
and 22244031 (A).
|
{
"timestamp": "2010-09-21T02:03:37",
"yymm": "1009",
"arxiv_id": "1009.3845",
"language": "en",
"url": "https://arxiv.org/abs/1009.3845"
}
|
\section{Introduction}
The complete automation of the 1-loop calculations is nowadays a
feasible task~\cite{nlopapers}.
The advent of the OPP reduction method~\cite{opp},
together with the concept of multiple cuts~\cite{britto},
allowed to revitalize Unitarity~\cite{unitarity} based Techniques,
such as Generalized Unitarity (GU)~\cite{genunit}, by reducing
the computation of 1-loop amplitudes to a problem with the same conceptual
complexity of a tree level calculation, resulting in achievements that
were inconceivable only a few years ago~\cite{proof}.
As a matter of principle, any program capable of producing tree level results
can be transformed nowadays into a NLO calculator by either {\em cutting}
the 1-loop diagrams, in the OPP method, or by {\em gluing} tree level
structures in the GU approach.
Both OPP and GU, when applied in 4 dimensions, allow the extraction of
the Cut Constructible (CC) part of the amplitude, while a
left over piece, the rational part ${\rm R}$, needs to be derived separately.
In the Generalized Unitarity approaches, this is achieved by computing
the amplitude in different numbers of space-time dimensions, or
via bootstrapping techniques~\cite{boot}, while,
in the OPP approach, ${\rm R}$ is split in 2 pieces ${\rm R= R_1+R_2}$.
The first piece, ${\rm R_1}$, is derivable in the same framework used
to reconstruct the CC part of the amplitude, while ${\rm R_2}$ is computable
through a special set of Feynman rules for the theory at hand~\cite{rational},
to be used in a tree level-like computation.
The OPP treatment of ${\rm R}$ in its present formulation
has one advantage and one drawback.
The advantage is that no calculation in dimensions other than four
is needed, avoiding the use of 6 and 8 dimensional
explicit representations of the external particle wave functions.
The drawback is that, for each theory that needs to be studied,
a different special set
of Feynman Rules has to be explicitly computed once for all.
On the other hand, in the OPP framework, the speed for computing
the Rational part is very high, so that we prefer it.
The full set of ${\rm R_2}$ Feynman rules has been already derived for QED
in~\cite{rational}, for QCD in~\cite{qcdrational}, and, for the
Standard Model (SM) of the Electroweak (EW) interactions in the
't Hooft-Feynman gauge in~\cite{ewrational}.
It is the main aim of the present paper to present the ${\rm R_2}$ Feynman
rules for the Electroweak Standard Model in a general renormalizable
$R_{\xi}$ gauge and in the Unitary gauge. On the one hand,
this completes the theoretical picture on $R_2$ and,
on the other hand, it allows tree level packages based on gauges
other that the 't Hooft-Feynman one to be transformed
into 1-loop calculators with the help of the mentioned
OPP or GU techniques.
In addition, the use of a general renormalizable $R_{\xi}$ gauge, can be used
to verify the correctness and the numerical stability
of the 1-loop predictions by studying the invariance
of the results under a change in the numerical value of $\xi$.
The outline of the paper is as follows. In section~\ref{sec:2} we recall the origin of ${\rm R_2}$.
In section~\ref{sec:3} we fix our notation and our calculational framework.
Section~\ref{sec:4} contains the complete
list of all possible special ${\rm R_2}$ EW SM vertices in the $R_{\xi}$ gauge and in the Unitary gauge.
Finally, section~\ref{checks} describes the tests we performed on our formulae.
\section{Theory of ${\rm R_2}$ \label{sec:2}}
The presence of a rational part ${\rm R}$ in a generic 1-loop amplitude is due to the regularization
procedure needed before carrying out the calculation.
In dimensional regularization, one computes the 1-loop integrals in
$n=~4+\epsilon$ dimensions, so that a generic $m$-point one-loop (sub-) amplitude reads
\begin{eqnarray}
\label{eq:1}
{\cal A}= \frac{1}{(2 \pi)^4} \int d^n \bar q \frac{\bar N(\bar q)}{\db{0}\db{1}\cdots \db{m-1}}\,,~~~
\db{i} = ({\bar q} + p_i)^2-m_i^2\,,~~~\bar q = q + \tld{q}\,,
\end{eqnarray}
where ${\bar q}$ is the integration momentum, a bar denotes objects living
in $n$ dimensions and a tilde represents $\epsilon$-dimensional quantities.
Notice that the external momenta $p_i$ are always kept in 4 dimensions.
The numerator function $\bar{N}(\bar q)$ can be
split into a $4$-dimensional plus an $\epsilon$-dimensional part
\begin{eqnarray}
\label{eq:split}
\bar{N}(\bar q) = N(q) + \tld{N}(\tld{q}^2,q,\epsilon)\,.
\end{eqnarray}
$N(q)$ brings information on the CC part of the amplitude (and within the OPP
framework can also be used to compute a part of the rational piece called ${\rm R_1}$), while
$\tld{N}(\tld{q}^2,q,\epsilon)$ gives rise to a second piece of the rational part called ${\rm R_2}$, defined as
\begin{eqnarray}
\label{eqr2}
{\rm R_2} \equiv \frac{1}{(2 \pi)^4}\int d^n\,\bar q
\frac{\tld{N}(\tld{q}^2,q,\epsilon)}{\db{0}\db{1}\cdots \db{m-1}} \,.
\end{eqnarray}
Due to possible ambiguities when passing from $N(q)$ to $\bar{N}(\bar q)$, the actual form of
$\tld{N}(\tld{q}^2,q,\epsilon)$ can only be read, to the best of our knowledge, starting from the original theory in
$n$ dimensions. In the OPP framework, that is achieved by computing
analytically tree-level like Feynman rules, by splitting the Feynman diagrams according to the following
three rules
\begin{eqnarray}
\label{qandg}
\bar q_{\bar \mu} &=& q_\mu + \tld{q}_{\tld{\mu}}\,, \nonumber \\
\bar \gamma_{\bar \mu} &=& \gamma_{\mu}+ \tld{\gamma}_{\tld{\mu}}\,,\nonumber \\
\bar g^{\bar \mu \bar \nu} &=& g^{\mu \nu}+ \tld{g}^{\tld{\mu} \tld{\nu}}\,.
\end{eqnarray}
Effective vertices are then generated by calculating the ${\rm R_2}$ parts coming from
all possible one-particle irreducible Green functions of the theory at hand,
up to four external legs.
The fact that four external legs are enough to account for ${\rm R_2}$
is guaranteed by the ultraviolet nature of the rational terms,
proved in~\cite{directcomp1}
\footnote{In GU approaches, the entire calculation is instead performed in $n$ dimensions, at the price of
introducing, as already mentioned in the Introduction, explicit 6 and 8-dimensional polarization vectors for the particles glued together
to form the loop amplitude.}.
Some freedom is however left in the choice of the regularization procedure, so that, instead of
Eq.~\ref{eqr2}, one could also use the definition
\begin{eqnarray}
\label{eqr2fdh}
{\rm R_2} \Bigl |_{FDH} = \frac{1}{(2 \pi)^4}\int d^n\,\bar q
\frac{\tld{N}(\tld{q}^2,q,\epsilon= 0)}{\db{0}\db{1}\cdots \db{m-1}} \,,
\end{eqnarray}
provided the same prescription is used in all parts of the calculation.
The choice in Eq.~\ref{eqr2fdh} corresponds to the so called Four Dimensional Helicity scheme~\cite{fdhqcd} (FDH).
In such a scheme, when using dimensional regularization, the only object
to be continued in $n$ dimensions is
\begin{eqnarray}
\label{eq:repl}
q^2 \to q^2 + \tld{q}^2\,,
\end{eqnarray}
and it would be nice if one could be able to use this information
to have access to ${\rm R_2}$ starting uniquely from the theory in 4 dimensions.
Unfortunately, the replacement in Eq.~\ref{eq:repl} is still too ambiguous, in the sense that different ways
of writing $N(q)$ may lead to different $n$-dimensional continuations, as already observed in~\cite{ewrational}, so that no better solution can be found, at present, than relying on the original
$n$-dimensional theory.
It is worth mentioning that only the combination ${\rm R}= {\rm R_1}+{\rm R_2}$ is gauge invariant, not, in general,
${\rm R_1}$ or ${\rm R_2}$ separately. In this respect, the {\em right} analytical continuation from
$N(q)$ to $\bar{N}(\bar q)$ by means of Eq.~\ref{eq:repl}, is that one that preserves all the Ward Identities of the theory.
In the following sections, we present the result of the explicit calculation we performed of all possible 2, 3 and 4-point
effective vertices in the Electroweak Standard Model in a general $R_{\xi}$ gauge
and in the Unitary gauge.
\section{Notations and Feynman rules\label{sec:3}}
The vector boson fields (generically symbolized by $V$)
are denoted by $A$, $Z$, $W^\pm$. The physical scalar Higgs field
is written as $H$ while $\chi$ and $\phi^\pm$ denote the neutral and the charged
scalar goldstone bosons, respectively. All scalar fields are
generically symbolized by $S$.
We work in the 1-fermion-family approximation, with lepton and quark doublets
given by
\begin{eqnarray}
\label{doublets}
\left(
\begin{tabular}{l}
$\nu_l$ \\
$l$
\end{tabular}
\right)
\hspace{1cm}{\rm and}\hspace{1cm}
\left(
\begin{tabular}{l}
$u$ \\
$d$
\end{tabular}
\right)\,.
\end{eqnarray}
Fermions are generically symbolized by $f$, and the charge,
the third isospin component and the mass of a fermion
by $Q_f$, $I_{3f}$ and $m_f$, respectively.
The sine and cosine of the Weinberg angle, the $W$ and the $Z$ mass
are denoted by $c_w$, $s_w$, $M_W$ and $M_Z$, respectively.
Following reference~\cite{denner}, we introduce the two quantities
$V_{ud}$ and $V^{\dagger}_{du}$ in the coupling of the $W$ boson with
the quark doublet of Eq.~\ref{doublets}.
This allows one to keep track of the CKM matrix and to easily
generalize the results to the 3-families case.
Finally, we use projector operators denoted by
$\Omega^\pm = \frac{1 \pm \gamma_5}{2}$.
The set of Feynman rules we use for our calculation
is that one given in~\cite{denner}, with some modifications due to the fact that
the expressions in that paper refer to the 't Hooft-Feynman gauge, while
we want to work in the $R_\xi$ gauge.
In the computation of ${\rm R_2}$, the ghost fields never enter, so that,
in order to pass from the the 't Hooft-Feynman gauge to the $R_\xi$ one,
we just need to modify the propagators of the scalar goldstone bosons and
of the vector bosons as follows
\begin{center}
\begin{picture}(300,100)
\SetScale{0.5}
\SetWidth{0.5}
\SetColor{Black}
\SetOffset(0,105)
\Text(64,-17)[]{{\Black{$p$}}}
\LongArrow(105,-48)(140,-48)
\DashLine(56,-62)(188,-62){8}
\Text(104,-31)[]{$S$}
\Text(18,-31)[]{$S$}
\Text(120,-31)[l]{$\displaystyle = \frac{i}{p^2-\xi M_S^2} $}
\SetOffset(0,50)
\Text(64,-17)[]{{\Black{$p$}}}
\LongArrow(105,-48)(140,-48)
\Photon(56,-62)(188,-62){5.5}{8}
\Text(104,-31)[]{$V_\beta$}
\Text(18,-31)[]{$V_\alpha$}
\Text(120,-31)[l]{$\displaystyle =
\frac{-i}{p^2-M_V^2} \left(g_{\alpha\beta}
-(1-\xi)\frac{p_\alpha p_\beta}{p^2-\xi M_V^2}\right)\,.
$}
\end{picture}
\end{center}
To compute our results in the Unitary gauge, we simply take the
limit $\xi \to \infty$ in the above propagators {\em before integrating}
over the loop momentum
\footnote{See section~\ref{checks} for more discussions on this issue.}.
Then the unphysical scalar particles decouple and the massive
gauge boson propagators become
\begin{eqnarray}
\label{eqmasspro}
\frac{-i}{p^2-M_V^2} \left(g_{\alpha\beta}-\frac{p_\alpha p_\beta}{M_V^2}\right)\,,
\end{eqnarray}
while for the photon we use
\begin{eqnarray}
\frac{-i}{p^2} g_{\alpha\beta}.
\end{eqnarray}
Notice that the choice in Eq.~\ref{eqmasspro} is mandatory in the framework
of the OPP method.
In fact, taking the limit $\xi \to \infty$ {\em after} integration over
the loop momentum would imply a nonviable numerical
cancellation between ${\rm R_1}$ and ${\rm R_2}$, since the two parts are treated separately.
A last comment is in order with respect to our
treatment of $\gamma_5$ in vertices
containing fermionic lines. When computing all contributing Feynman diagrams,
we pick up a ``special'' vertex in the loop and anticommute all $\gamma_5$'s
to reach it before performing the $n$-dimensional algebra, and, when a trace
is present, we start reading it from this vertex. This treatment
produces, in general, a term proportional to the totally antisymmetric $\epsilon$
tensor, whose coefficient may be different
depending on the choice of the ``special''
vertex. However, when summing over all quantum numbers of each fermionic
family, we checked that all contributions proportional to $\epsilon$ cancel.
\section{Results \label{sec:4}}
In this section, we present our results.
We omit, in this paper, the gauge invariant contributions
coming from fermion loops, because they can be recovered with the help
of the formulae we already worked out in the case of the 't Hooft-Feynman
gauge in~\cite{ewrational}.
In fact, the fermion loop part can be easily separated from the rest since it
always involves a sum $\sum_{i}$ over fermions or fermion families.
A parameter $\lambda_{HV}$ is introduced in our formulae such that
$\lambda_{HV}= 1$ corresponds to the 't Hooft-Veltman scheme
and $\lambda_{HV}= 0$ to the FDH scheme of eq.~\ref{eqr2fdh}.
We explicitly write down, in this publication, all the formulae
in the 2-point case, while,
for the 3 the and 4-point vertices, we just classify the non vanishing ones.
In fact the expression we obtained are rather lengthy, and there is no point
in writing them down on paper. We rather provide the full set of results as
FORM~\cite{form} files~\cite{formfiles}. The notation used in
those files closely follows that one introduced in the previous section.
In Fig.~\ref{fig:1}-\ref{fig:3} we present the generic non vanishing
2-point, 3-point and 4-point vertices that appear in our calculation,
that also serve to further fix our notations.
\begin{figure}
\begin{center}
\begin{tabular}{l}
\begin{picture}(200,50)
\SetScale{0.5}
\SetWidth{0.5}
\SetColor{Black}
\SetOffset(0,55)
\Vertex(142,-62){6.0}
\Text(50,-17)[]{{\Black{$p_1$}}}
\LongArrow(75,-48)(110,-48)
\DashLine(56,-62)(142,-62){8}
\DashLine(142,-62)(228,-62){8}
\Text(124,-31)[]{$S_2$}
\Text(18,-31)[]{$S_1$}
\Text(138,-31)[l]{$= ~{\rm Vert}(S_1,S_2)$}
\Text(-90,-31)[]{$(a)$}
\end{picture} \\
\begin{picture}(200,50)
\SetOffset(0,42)
\SetScale{0.5}
\SetWidth{0.5}
\SetColor{Black}
\Photon(56,-34)(133,-34){5.5}{3.6}
\DashLine(135,-34)(228,-34){8}
\Vertex(142,-34){6.0}
\Text(50,-3)[]{{\Black{$p_1$}}}
\LongArrow(75,-20)(110,-20)
\Text(18,-17)[]{$V_{\alpha}$}
\Text(124,-17)[]{$S$}
\SetOffset(0,55)
\Text(138,-31)[l]{$= ~{\rm Vert}(V,S)$}
\Text(-90,-31)[]{$(b)$}
\end{picture} \\
\begin{picture}(200,50)
\SetOffset(0,42)
\SetScale{0.5}
\SetWidth{0.5}
\SetColor{Black}
\Photon(56,-34)(133,-34){5.5}{3.6}
\Vertex(142,-34){6.0}
\Photon(135,-34)(228,-34){5.5}{4}
\Text(50,-3)[]{{\Black{$p_1$}}}
\LongArrow(75,-20)(110,-20)
\Text(18,-17)[]{$V_{1\alpha}$}
\Text(124,-17)[]{$V_{2\beta}$}
\SetOffset(0,55)
\Text(138,-31)[l]{$= ~{\rm Vert}(V_1,V_2)$}
\Text(-90,-31)[]{$(c)$}
\end{picture} \\
\begin{picture}(200,50)
\SetOffset(0,55)
\SetWidth{0.5}
\SetScale{0.5}
\SetColor{Black}
\ArrowLine(56,-62)(138,-62)
\ArrowLine(141,-62)(228,-62)
\Vertex(142,-63){6.0}
\Text(50,-17)[]{{\Black{$p_1$}}}
\LongArrow(75,-48)(110,-48)
\DashLine(56,-62)(142,-62){8}
\Text(18,-31)[]{$f_1$}
\Text(124,-31)[]{$\bar f_2$}
\Text(138,-31)[l]{$= ~{\rm Vert}(f_1,f_2)$}
\Text(-90,-31)[]{$(d)$}
\end{picture}
\end{tabular}
\end{center}
\caption{\label{fig:1} All possible 2-point vertices.}
\end{figure}
\begin{figure}
\begin{center}
\begin{tabular}{l}
\begin{picture}(210,70)
\SetOffset(-10,25)
\SetWidth{0.5}
\SetScale{0.5}
\SetColor{Black}
\Vertex(146,15){6.0}
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\DashLine(200,-27)(146,15){8}
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\Text(52,18)[b]{{\Black{$p_1$}}}
\LongArrow(180,62.3)(156,43)
\LongArrow(180,-31)(156,-12)
\Text(75,29)[lb]{{\Black{$p_2$}}}
\Text(75,-13)[lt]{{\Black{$p_3$}}}
\Text(33,7.5)[r]{\Black{$S_1$}}
\Text(107, 30)[l]{$S_2$}
\Text(107,-15)[l]{$S_3$}
\Text(125,7.5)[l]{$= ~{\rm Vert}(S_1,S_2,S_3)$}
\end{picture} \\
\begin{picture}(210,70)
\SetOffset(-10,25)
\SetWidth{0.5}
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\LongArrow(180,-31)(156,-12)
\Text(75,29)[lb]{{\Black{$p_2$}}}
\Text(75,-13)[lt]{{\Black{$p_3$}}}
\Text(33,7.5)[r]{\Black{$V_\alpha$}}
\Text(107, 30)[l]{$S_1$}
\Text(107,-15)[l]{$S_2$}
\Text(125,7.5)[l]{$= ~{\rm Vert}(V,S_1,S_2)$}
\end{picture} \\
\begin{picture}(210,70)
\SetOffset(-10,25)
\SetWidth{0.5}
\SetScale{0.5}
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\Photon(200,-27)(146,15){5.5}{4}
\LongArrow(85,28)(120,28)
\Text(52,18)[b]{{\Black{$p_1$}}}
\LongArrow(180,62.3)(156,43)
\LongArrow(180,-31)(156,-12)
\Text(75,29)[lb]{{\Black{$p_2$}}}
\Text(75,-13)[lt]{{\Black{$p_3$}}}
\Text(33,7.5)[r]{\Black{$S$}}
\Text(107, 30)[l]{$V_{1\beta}$}
\Text(107,-15)[l]{$V_{2\gamma}$}
\Text(125,7.5)[l]{$= ~{\rm Vert}(S,V_1,V_2)$}
\end{picture} \\
\begin{picture}(210,70)
\SetOffset(-10,25)
\SetWidth{0.5}
\SetScale{0.5}
\SetColor{Black}
\Vertex(146,15){6.0}
\Photon(72,15)(146,15){5.5}{4}
\Photon(146,15)(200,57){5.5}{4}
\Photon(200,-27)(146,15){5.5}{4}
\LongArrow(85,28)(120,28)
\Text(52,18)[b]{{\Black{$p_1$}}}
\LongArrow(180,62.3)(156,43)
\LongArrow(180,-31)(156,-12)
\Text(75,29)[lb]{{\Black{$p_2$}}}
\Text(75,-13)[lt]{{\Black{$p_3$}}}
\Text(33,7.5)[r]{\Black{$V_{1\alpha}$}}
\Text(107, 30)[l]{$V_{2\beta}$}
\Text(107,-15)[l]{$V_{3\gamma}$}
\Text(125,7.5)[l]{$= ~{\rm Vert}(V_1,V_2,V_3)$}
\end{picture} \\
\begin{picture}(210,70)
\SetOffset(-10,25)
\SetWidth{0.5}
\SetScale{0.5}
\SetColor{Black}
\Vertex(146,15){6.0}
\DashLine(72,15)(144,15){8}
\ArrowLine(146,15)(200,57)
\ArrowLine(200,-27)(146,15)
\LongArrow(85,28)(120,28)
\Text(52,18)[b]{{\Black{$p_1$}}}
\LongArrow(180,62.3)(156,43)
\LongArrow(180,-31)(156,-12)
\Text(75,29)[lb]{{\Black{$p_2$}}}
\Text(75,-13)[lt]{{\Black{$p_3$}}}
\Text(33, 7.5)[r]{$S$}
\Text(102, 30)[l]{$\bar f_1$}
\Text(102,-15)[l]{$f_2$}
\Text(125,7.5)[l]{$= ~{\rm Vert}(S,f_1,f_2)$}
\end{picture} \\
\begin{picture}(210,70)
\SetOffset(-10,25)
\SetWidth{0.5}
\SetScale{0.5}
\SetColor{Black}
\Vertex(146,15){6.}
\ArrowLine(146,15)(200,57)
\ArrowLine(200,-27)(146,15)
\Photon(72,15)(146,15){5.5}{4}
\LongArrow(85,28)(120,28)
\Text(52,18)[b]{{\Black{$p_1$}}}
\LongArrow(180,62.3)(156,43)
\LongArrow(180,-31)(156,-12)
\Text(75,29)[lb]{{\Black{$p_2$}}}
\Text(75,-13)[lt]{{\Black{$p_3$}}}
\Text(30,7.5)[r]{\Black{$V_\alpha$}}
\Text(102, 30)[l]{$\bar f_1$}
\Text(102,-15)[l]{$f_2$}
\Text(125,7.5)[l]{$= ~{\rm Vert}(V,f_1,f_2)$}
\end{picture}
\end{tabular}
\end{center}
\caption{\label{fig:2} All possible non vanishing 3-point vertices.}
\end{figure}
\begin{figure}
\begin{center}
\begin{tabular}{l}
\begin{picture}(200,100)
\SetOffset(-10,45)
\SetWidth{0.5}
\SetScale{0.5}
\SetColor{Black}
\DashLine(57,-36)(161,54){8}
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\Vertex(110,9){6.0}
\LongArrow(60,-17)(85,3)
\Text(27.5,-7.5)[tr]{{\Black{$p_1$}}}
\LongArrow(60,33)(85,13)
\Text(26,20)[tr]{{\Black{$p_2$}}}
\LongArrow(145,55)(120,33)
\Text(55,25)[lb]{{\Black{$p_3$}}}
\LongArrow(145,-37)(120,-14)
\Text(55,-15)[lt]{{\Black{$p_4$}}}
\Text(22,-18)[tr]{$S_1$}
\Text(22, 27)[br]{$S_2$}
\Text(90, 27)[bl]{$S_3$}
\Text(90,-18)[tl]{$S_4$}
\Text(115,5)[l]{$= ~{\rm Vert}(S_1,S_2,S_3,S_4)$}
\end{picture} \\
\begin{picture}(200,100)
\SetOffset(-10,45)
\SetWidth{0.5}
\SetScale{0.5}
\SetColor{Black}
\DashLine(57,-36)(110,9){8}
\Photon(110,9)(161,54){5.5}{4}
\DashLine(57,54)(110,9){8}
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\Vertex(110,9){6.0}
\LongArrow(60,-17)(85,3)
\Text(27.5,-7.5)[tr]{{\Black{$p_1$}}}
\LongArrow(60,33)(85,13)
\Text(26,20)[tr]{{\Black{$p_2$}}}
\LongArrow(145,55)(120,33)
\Text(55,25)[lb]{{\Black{$p_3$}}}
\LongArrow(145,-37)(120,-14)
\Text(55,-15)[lt]{{\Black{$p_4$}}}
\Text(22,-18)[tr]{$S_1$}
\Text(22, 27)[br]{$S_2$}
\Text(90, 27)[bl]{$V_{1\gamma}$}
\Text(90,-18)[tl]{$V_{2\delta}$}
\Text(115,5)[l]{$= ~{\rm Vert}(S_1,S_2,V_1,V_2)$}
\end{picture}\\
\begin{picture}(200,100)
\SetOffset(-10,45)
\SetWidth{0.5}
\SetScale{0.5}
\SetColor{Black}
\Photon(57,-36)(161,54){5.5}{8}
\Photon(57,54)(161,-36){5.5}{8}
\Vertex(110,9){6.0}
\LongArrow(60,-17)(85,3)
\Text(27.5,-7.5)[tr]{{\Black{$p_1$}}}
\LongArrow(60,33)(85,13)
\Text(26,20)[tr]{{\Black{$p_2$}}}
\LongArrow(145,55)(120,33)
\Text(55,25)[lb]{{\Black{$p_3$}}}
\LongArrow(145,-37)(120,-14)
\Text(55,-15)[lt]{{\Black{$p_4$}}}
\Text(22,-18)[tr]{$V_{1\alpha}$}
\Text(22, 27)[br]{$V_{2\beta}$}
\Text(90, 27)[bl]{$V_{3\gamma}$}
\Text(90,-18)[tl]{$V_{4\delta}$}
\Text(115,5)[l]{$= ~{\rm Vert}(V_1,V_2,V_3,V_4)$}
\end{picture}
\end{tabular}
\end{center}
\caption{\label{fig:3} All possible non vanishing 4-point vertices.}
\end{figure}
\subsection{The $R_\xi$ gauge}
\subsubsection{Bosonic contribution to the vertices with 2 legs}
\vspace{0.3cm}
\leftline{{\bf Scalar-Scalar effective vertices}}
\vspace{0.3cm}
\noindent The generic effective vertex is
\begin{eqnarray}
{\rm Vert}(S_1,S_2) = \frac{ie^2}{16 \pi^2 s_w^2} C
\end{eqnarray}
with ${\rm Vert}(S_1,S_2)$ given in fig.~\ref{fig:1} $(a)$ and
with the actual values of $S_1$, $S_2$ and $C$
\begin{eqnarray}
\label{eq:vert1}
HH~~:~~C & = &
\frac{m_W^2}{4} \left(1+2\xi-\xi^2-12\lambda_{HV}\right)
\left(1+\frac{1}{2 c_w^4} \right)
+ p_1^2\,\frac{9-11\xi}{24} \left(1+\frac{1}{2 c_w^2} \right)
\nonumber \\
\nonumber \\
\chi\chi~~:~~C & = &
\frac{m_W^2}{24 c_w^4} \left(1+2\xi^2-12\lambda_{HV}\right)
+\frac{m_W^2}{12} \left(1-2\xi+7\xi^2-12\lambda_{HV}\right)
\nonumber \\
&&- \frac{m_H^2}{12 c_w^2} \left(1-\frac{5}{2}\xi \right)
+ p_1^2\,\frac{9-11\xi}{24} \left(1+\frac{1}{2 c_w^2} \right)
\nonumber \\
\nonumber \\
\phi^-\phi^+ ~~:~~C & = &
\frac{m_W^2}{24 c_w^4} \left(1+2\xi^2-12\lambda_{HV}\right)
+\frac{m_W^2}{2 c_w^2} \left(\xi-\frac{3}{2}\xi^2 \right)
\nonumber \\
&& +\frac{m_W^2}{12} \left(1-8\xi+16\xi^2-12\lambda_{HV}\right)
\nonumber \\
&& - \frac{m_H^2}{12} \left(1-\frac{5}{2}\xi \right)
+ p_1^2\,\frac{9-11\xi}{24} \left(1+\frac{1}{2 c_w^2} \right)
\end{eqnarray}
\vspace{0.3cm}
\leftline{{\bf Vector-Scalar effective vertices}}
\vspace{0.3cm}
\noindent The generic effective vertex is
\begin{eqnarray}
{\rm Vert}(V,S) = \frac{ie^2}{\pi^2}\, C\, p_{1\alpha}
\end{eqnarray}
with ${\rm Vert}(V,S)$ given in fig.~\ref{fig:1} $(b)$ and
with the actual values of $V$, $S$ and $C$
\begin{eqnarray}
\label{eq:vert2}
W^- \phi^+~~:~~C & = &
-(1-\xi) \frac{M_W}{192 c_w^2 s_w^2}
\nonumber \\
\nonumber \\
W^+\phi^-~~:~~C & = &
(1-\xi) \frac{M_W}{c_w^2 s_w^2}
\nonumber \\
\nonumber \\
Z \chi ~~:~~C & = &
(1-\xi) \frac{i\,M_Z}{192 c_w^2 s_w^2}\left(1+2c_w^2 s_w^2 \right)
\nonumber \\
\nonumber \\
A \chi ~~:~~C & = &
(1-\xi) \frac{i\,M_Z}{96}\frac{c_w}{s_w}
\end{eqnarray}
Notice that all these vertices vanish in the 't Hooft-Feynman gauge
($\xi = 1$).
\vspace{0.3cm}
\leftline{{\bf Vector-Vector effective vertices}}
\vspace{0.3cm}
\noindent The generic effective vertex is
\begin{eqnarray}
{\rm Vert}(V_1,V_2) = \frac{ie^2}{8 \pi^2}
\left(C_1 \,p_{1\alpha} p_{1\beta} + C_2 \,g_{\alpha \beta}\right)
\end{eqnarray}
with ${\rm Vert}(V_1,V_2)$ given in fig.~\ref{fig:1} $(c)$ and
with the actual values of $V_1$, $V_2$, $C_1$ and $C_2$
\begin{eqnarray}
\label{eq:vert3}
AA~~:~~C_1 & = & K_1\nonumber \\
C_2 & = & K_2
\nonumber \\
\nonumber \\
AZ~~:~~C_1 & = & - \frac{c_w}{s_w} K_1 \nonumber \\
C_2 & = & - \frac{c_w}{s_w} K_2
\nonumber \\
\nonumber \\
ZZ~~:~~C_1 & = & \frac{c_w^2}{s_w^2} K_1\nonumber \\
C_2 & = & \frac{c_w^2}{s_w^2} K_2
\nonumber \\
\nonumber \\
W^-W^+~~:~~C_1 & = & \frac{1}{s_w^2} K_1\nonumber \\
C_2 & = & \frac{1}{s_w^2} K_2
\end{eqnarray}
where
\begin{eqnarray}
K_1 &=& - \frac{1}{3} \lambda_{HV}+\frac{3}{4}(1-\xi)
\nonumber \\
\nonumber \\
K_2 &=& p^2\left(\frac{21\xi-17}{24}+\frac{\lambda_{HV}}{3}\right)
- \xi \frac{\xi+3}{4} m_W^2
\end{eqnarray}
\vspace{0.3cm}
\leftline{{\bf Fermion-Fermion effective vertices}}
\vspace{0.3cm}
\noindent The generic effective vertex is
\begin{eqnarray}
{\rm Vert}(f_1,f_2) = \frac{ie^2}{\pi^2}
\left[\left(C_- \Omega^- + C_+\Omega^+\right)\rlap/p_1 + C_0
\right]
\end{eqnarray}
with ${\rm Vert}(f_1,f_2)$ given in fig.~\ref{fig:1} $(d)$ and
with the actual values of $f_1$, $f_2$, $C_-$, $C_+$ and $C_0$
\begin{eqnarray}
\label{eq:vert4}
u u ~~:~~C_- & = & \frac{Q_u^2}{c_w^2}
\left(\frac{\lambda_{HV}}{16}-\frac{1-\xi}{24} \right)\nonumber\\
C_+ & = & \left(\frac{I_{3u}^2}{s_w^2 c_w^2}
- \frac{ 2 Q_{u} I_{3u}}{c_w^2} + \frac{Q_u^2}{c_w^2}
+ \frac{1}{2 s_w^2}
\left(V_{u d} V_{d u}^\dagger\right)
\right)
\left(\frac{\lambda_{HV}}{16}-\frac{1-\xi}{24} \right)
\nonumber \\
C_0 & = & \frac{m_{u} Q_{u} }{8 c_w^2} \left( Q_{u} - I_{3u} \right)
\left( \lambda_{HV}-\frac{1-\xi}{4} \right)
\nonumber \\
\nonumber \\
d d ~~:~~C_- & = & \frac{Q_{d}^2}{c_w^2}
\left(\frac{\lambda_{HV}}{16}-\frac{1-\xi}{24} \right) \nonumber \\
C_+ & = & \left(
\frac{I_{3{d}}^2}{s_w^2 c_w^2}
- \frac{ 2 Q_{d} I_{3d}}{c_w^2} + \frac{Q_{d}^2}{c_w^2}
+ \frac{1}{2 s_w^2}
\left(V_{u d} V_{d u}^\dagger\right)
\right)
\left(\frac{\lambda_{HV}}{16}-\frac{1-\xi}{24} \right)
\nonumber \\
C_0 & = & \frac{m_{d} Q_{d}}{8 c_w^2} \left( Q_{d} - I_{3d} \right)
\left( \lambda_{HV}-\frac{1-\xi}{4} \right)
\nonumber \\
\nonumber \\
l l ~~:~~C_- & = & \frac{Q_{l}^2}{c_w^2}
\left(\frac{\lambda_{HV}}{16}-\frac{1-\xi}{24} \right) \nonumber \\
C_+ & = & \left(\frac{I_{3l}^2}{s_w^2 c_w^2}
- \frac{ 2 Q_{l} I_{3l}}{c_w^2} + \frac{Q_{l}^2}{c_w^2}
+ \frac{1}{2 s_w^2} \right)
\left(\frac{\lambda_{HV}}{16}-\frac{1-\xi}{24} \right)
\nonumber \\
C_0 & = & \frac{m_{l}Q_{l}}{8 c_w^2} \left( Q_{l} - I_{3l} \right)
\left( \lambda_{HV}-\frac{1-\xi}{4} \right)
\nonumber \\
\nonumber \\
\nu_l \nu_l ~~:~~C_- & = & 0 \nonumber \\
C_+ & = & \frac{1}{s_w^2} \left(\frac{I^2_{3\nu_l}}{c_w^2}
+ \frac{1}{2} \right) \left(\frac{\lambda_{HV}}{16}-\frac{1-\xi}{24} \right)
\nonumber \\
C_0 & = & 0
\end{eqnarray}
\subsubsection{Bosonic contribution to the vertices with 3 legs}
The generic 3-point vertices appearing in our calculation are drawn in Fig.~\ref{fig:2}. As already pointed out,
we limit ourselves to list the non vanishing cases, while
the full set of results is available in~\cite{formfiles}. We found 43 non zero ${\rm R_2}$
vertices in the $R_\xi$ gauge, classified in Table~\ref{tab:tab1}.
\begin{table}
\begin{center}
{\small Scalar-Scalar-Scalar vertices:}
\vspace{0.4cm}
\begin{tabular}{lll}
${\rm Vert}(H,H,H)$, & ${\rm Vert}(H,\chi,\chi)$, & ${\rm Vert}(H,\phi^+,\phi^-)$.
\end{tabular}
\end{center}
\begin{center}
{\small Vector-Scalar-Scalar vertices:}
\vspace{0.4cm}
\begin{tabular}{llll}
${\rm Vert}(A,H,\chi)$, & ${\rm Vert}(A,\phi^+,\phi^-)$, & ${\rm Vert}(Z,H,\chi)$, & ${\rm Vert}(Z,\phi^+,\phi^-)$, \\
${\rm Vert}(W^-,H,\phi^+)$, & ${\rm Vert}(W^-,\chi,\phi^+)$, &
${\rm Vert}(W^+,H,\phi^-)$, & ${\rm Vert}(W^+,\chi,\phi^-)$.
\end{tabular}
\end{center}
\begin{center}
{\small Scalar-Vector-Vector vertices:}
\vspace{0.4cm}
\begin{tabular}{llll}
${\rm Vert}(H,A,A)$, & ${\rm Vert}(H,A,Z)$, & ${\rm Vert}(H,Z,Z)$,
& ${\rm Vert}(H,W^+,W^-)$, \\
${\rm Vert}(\phi^-,A,W^+)$, & ${\rm Vert}(\phi^+,A,W^-)$ & ${\rm Vert}(\phi^-,Z,W^+)$, & ${\rm Vert}(\phi^+,Z,W^-)$.
\end{tabular}
\end{center}
\begin{center}
{\small Vector-Vector-Vector vertices:}
\vspace{0.4cm}
\begin{tabular}{ll}
${\rm Vert}(A,W^+,W^-)$, & ${\rm Vert}(Z,W^+,W^-)$.
\end{tabular}
\end{center}
\begin{center}
{\small Scalar-Fermion-Fermion vertices:}
\vspace{0.4cm}
\begin{tabular}{llll}
${\rm Vert}(H,u,u)$, & ${\rm Vert}(H,d,d)$, & ${\rm Vert}(H,l,l)$,
& \\
${\rm Vert}(\chi,u,u)$, & ${\rm Vert}(\chi,d,d)$, & ${\rm Vert}(\chi,l,l)$,
& \\
${\rm Vert}(\phi^-,d,u)$, & ${\rm Vert}(\phi^-,l,\nu_l)$, & ${\rm Vert}(\phi^+,u,d)$, & ${\rm Vert}(\phi^+,\nu_l,l)$.
\end{tabular}
\end{center}
\begin{center}
{\small Vector-Fermion-Fermion vertices:}
\vspace{0.4cm}
\begin{tabular}{llll}
${\rm Vert}(A,u,u)$, & ${\rm Vert}(A,d,d)$, & ${\rm Vert}(A,\nu_l,\nu_l)$,
& ${\rm Vert}(A,l,l)$, \\
${\rm Vert}(Z,u,u)$, & ${\rm Vert}(Z,d,d)$, & ${\rm Vert}(Z,\nu_l,\nu_l)$,
& ${\rm Vert}(Z,l,l)$, \\
${\rm Vert}(W^-,d,u)$, & ${\rm Vert}(W^-,l,\nu_l)$, & ${\rm Vert}(W^+,u,d)$,
& ${\rm Vert}(W^+,\nu_l,l)$.
\end{tabular}
\end{center}
\begin{center}
\caption{\label{tab:tab1} {The 43 non zero 3-point effective vertices
in the $R_\xi$ gauge.
In the Unitary gauge there are 23 non vanishing vertices, namely the 22
listed here that do not contain $\chi$ or $\phi^\pm$ fields plus
${\rm Vert}(H,\nu_l,\nu_l)$.}}
\end{center}
\end{table}
\subsubsection{Bosonic contribution to the vertices with 4 legs}
All non vanishing generic 4-point vertices that appear in our calculation
are drawn in Fig.~\ref{fig:3}. The full set of results can be found in~\cite{formfiles}.
The 35 non zero ${\rm R_2}$ vertices in the $R_\xi$ gauge are classified in Table~\ref{tab:tab2}.
\begin{table}
\begin{center}
{\small Scalar-Scalar-Scalar-Scalar vertices:}
\vspace{0.4cm}
\begin{tabular}{lll}
${\rm Vert}(H,H,H,H)$, & ${\rm Vert}(H,H,\chi,\chi)$, & ${\rm Vert}(H,H,\phi^-,\phi^+)$, \\
${\rm Vert}(\chi,\chi,\chi,\chi)$, & ${\rm Vert}(\chi,\chi,\phi^-,\phi^+)$,
& ${\rm Vert}(\phi^-,\phi^+,\phi^-,\phi^+)$.
\end{tabular}
\end{center}
\begin{center}
{\small Scalar-Scalar-Vector-Vector effective vertices:}
\vspace{0.4cm}
\begin{tabular}{llll}
${\rm Vert}(H,H,A,A)$,$\!\!$ & ${\rm Vert}(H,H,A,Z)$,$\!\!$ &
${\rm Vert}(H,H,Z,Z)$,$\!\!$ & ${\rm Vert}(H,H,W^-,W^+)$, $\!$\\
${\rm Vert}(H,\phi^+,W^-,A)$,$\!\!$ & ${\rm Vert}(H,\phi^+,W^-,Z)$,$\!\!$ &
${\rm Vert}(\chi,\chi,A,A)$,$\!\!$ & ${\rm Vert}(\chi,\chi,A,Z)$, $\!$\\
${\rm Vert}(\chi,\chi,Z,Z)$,$\!\!$ & ${\rm Vert}(\chi,\chi,W^-,W^+)$,$\!\!$ &
${\rm Vert}(\chi,\phi^+,W^-,A)$,$\!\!$ & ${\rm Vert}(\chi,\phi^+,W^-,Z)$, $\!$\\
${\rm Vert}(\phi^-,H,A,W^+)$,$\!\!$ & ${\rm Vert}(\phi^-,H,Z,W^+)$,$\!\!$ &
${\rm Vert}(\phi^-,\chi,A,W^+)$,$\!\!$ & ${\rm Vert}(\phi^-,\chi,Z,W^+)$, $\!$\\
${\rm Vert}(\phi^-,\phi^+,A,A)$,$\!\!$ & ${\rm Vert}(\phi^-,\phi^+,A,Z)$, $\!\!$&
${\rm Vert}(\phi^-,\phi^+,Z,Z)$,$\!\!$ & ${\rm Vert}(\phi^-,\phi^+,W^-,W^+)$. $\!$
\end{tabular}
\end{center}
\begin{center}
{\small Vector-Vector-Vector-Vector effective vertices:}
\vspace{0.4cm}
\begin{tabular}{lll}
${\rm Vert}(A,A,A,A)$, & ${\rm Vert}(A,A,A,Z)$, & ${\rm Vert}(A,A,Z,Z)$, \\
${\rm Vert}(A,Z,Z,Z)$, & ${\rm Vert}(Z,Z,Z,Z)$, & ${\rm Vert}(A,A,W^-,W^+)$, \\
${\rm Vert}(A,Z,W^-,W^+)$, & ${\rm Vert}(Z,Z,W^-,W^+)$, & ${\rm Vert}(W^-,W^+,W^-,W^+)$.
\end{tabular}
\end{center}
\begin{center}
\caption{\label{tab:tab2} {The 35 non zero 4-point effective vertices
in the $R_\xi$ gauge. In the Unitary gauge there are 14
non vanishing vertices, namely all those ones that do not contain
$\chi$ or $\phi^\pm$ fields.}}
\end{center}
\end{table}
\subsection{The Unitary gauge}
We follow again the notations of Fig.~\ref{fig:1}.
\subsubsection{Bosonic contribution to the vertices with 2 legs}
\vspace{0.3cm}
\leftline{{\bf Scalar-Scalar effective vertices}}
\vspace{0.3cm}
\noindent The generic effective vertex is
\begin{eqnarray}
{\rm Vert}(S_1,S_2) = \frac{ie^2}{16 \pi^2 s_w^2} C
\end{eqnarray}
with ${\rm Vert}(S_1,S_2)$ given in fig.~\ref{fig:1} $(a)$ and
with the actual values of $S_1$, $S_2$ and $C$
\begin{eqnarray}
HH~~:~~C & = &
\frac{5}{6}\, p_1^2 \left(1+\frac{1}{2 c_w^2} \right)
-\frac{9}{40}\frac{p_1^4}{m_W^2}
- m_W^2 \left(1+\frac{1}{2 c_w^4} \right)
\left(\frac{1}{4}+ 3 \lambda_{HV} \right)
\end{eqnarray}
\vspace{0.3cm}
\leftline{{\bf Vector-Scalar effective vertices}}
\vspace{0.3cm}
\noindent No contribution is found in the Unitary gauge.
\vspace{0.3cm}
\leftline{{\bf Vector-Vector effective vertices}}
\vspace{0.3cm}
\noindent The generic effective vertex is
\begin{eqnarray}
{\rm Vert}(V_1,V_2) = \frac{ie^2}{8 \pi^2}
\left(C_1 \,p_{1\alpha} p_{1\beta} + C_2 \,g_{\alpha \beta}\right)
\end{eqnarray}
with ${\rm Vert}(V_1,V_2)$ given in fig.~\ref{fig:1} $(c)$ and
with the actual values of $V_1$, $V_2$, $C_1$ and $C_2$
\begin{eqnarray}
AA~~:~~C_1 & = & K_1\nonumber \\
C_2 & = & K_2
\nonumber \\
\nonumber \\
AZ~~:~~C_1 & = & - \frac{c_w}{s_w} K_1 \nonumber \\
C_2 & = & - \frac{c_w}{s_w} K_2
\nonumber \\
\nonumber \\
ZZ~~:~~C_1 & = & \frac{c_w^2}{s_w^2} K_1\nonumber \\
C_2 & = & \frac{c_w^2}{s_w^2} K_2
\nonumber \\
\nonumber \\
W^-W^+~~:~~C_1 & = & \frac{1}{s_w^2} K_3\nonumber \\
C_2 & = & \frac{1}{s_w^2} K_4
\end{eqnarray}
where
\begin{eqnarray}
K_1 &=& -\frac{1}{3} \left(\lambda_{HV}-5\right)-\frac{17}{60} \,\frac{p_1^2}{m_W^2}
\nonumber \\
\nonumber \\
K_2 &=& \frac{3}{4} m_W^2+\frac{1}{3}\,p_1^2\,\left(\lambda_{HV}-\frac{23}{4}
\right)+\frac{37}{120}\,\,\frac{p_1^4}{m_W^2}
\nonumber \\
\nonumber \\
K_3 &=& -\frac{1}{3} \left(\lambda_{HV}-\frac{5}{2}
-\frac{9}{8} c_w^2 \right)
+\frac{11}{24} c_w^4-\frac{17}{120}\,\frac{p_1^2}{m_W^2}
\left(1+ c_w^4 \right)
\nonumber \\
\nonumber \\
K_4 &=& \frac{3}{8} \frac{m_W^2}{c_w^2}\left(s_w^2+c_w^4+c_w^6 \right)
+ p_1^2\left[\frac{\lambda_{HV}}{3}-\frac{7}{8}
-\frac{7}{16} c_w^2\left(1+\frac{29}{21}c_w^2 \right) \right]
\nonumber \\
&&+\frac{37}{240}\frac{p_1^4}{m_W^2} \left(1+c_w^4 \right)
\end{eqnarray}
\vspace{0.3cm}
\leftline{{\bf Fermion-Fermion effective vertices}}
\vspace{0.3cm}
\noindent The generic effective vertex is
\begin{eqnarray}
{\rm Vert}(f_1,f_2) = \frac{ie^2}{\pi^2}
\left[\left(C_- \Omega^- + C_+\Omega^+\right)\rlap/p_1 + C_0
\right]
\end{eqnarray}
with ${\rm Vert}(f_1,f_2)$ given in fig.~\ref{fig:1} $(d)$ and
with the actual values of $f_1$, $f_2$, $C_-$, $C_+$ and $C_0$
\begin{eqnarray}
u u ~~:~~C_- & = &
\frac{Q_u^2}{16 c_w^2}
\left[
\lambda_{HV}
+\frac{s_w^2}{m_Z^2} \left(\frac{p_1^2}{4}-\frac{2}{3} m_Z^2
-\frac{5}{6} m_u^2 \right)
\right]
\nonumber\\
C_+ & = & \frac{\lambda_{HV}}{16} \left[\frac{I_{3u}^2}{s_w^2 c_w^2}
- \frac{ 2 Q_{u} I_{3u}}{c_w^2} + \frac{Q_u^2}{c_w^2}
+ \frac{1}{2 s_w^2}
\left(V_{u d} V_{d u}^\dagger\right)
\right]
\nonumber \\
&&+ \frac{s_w^2}{16 m_Z^2 c_w^2}
\left(\frac{p_1^2}{4}-\frac{2}{3} m_Z^2
-\frac{5}{6} m_u^2 \right)
\left(Q_u-\frac{I_{3u}}{s_w^2} \right)^2
\nonumber \\
&& + \frac{V_{u d} V_{d u}^\dagger}{32 m_W^2 s_w^2}
\left(\frac{p_1^2}{4}-\frac{2}{3} m_W^2
-\frac{5}{6} m_d^2 \right)
\nonumber \\
C_0 & = & \frac{Q_u m_u}{8 c_w^2} \left[
\lambda_{HV} \left(Q_u-I_{3u} \right)
+\frac{s_W^2}{4 m_Z^2}
\left(Q_u-\frac{I_{3u}}{s_w^2} \right)
\left(\frac{p_1^2}{3}-m_Z^2-m_u^2 \right) \right]
\nonumber \\
\nonumber \\
d d ~~:~~C_- & = &
\frac{Q_d^2}{16 c_w^2}
\left[
\lambda_{HV}
+\frac{s_w^2}{m_Z^2} \left(\frac{p_1^2}{4}-\frac{2}{3} m_Z^2
-\frac{5}{6} m_d^2 \right)
\right]
\nonumber \\
C_+ & = & \frac{\lambda_{HV}}{16} \left[\frac{I_{3d}^2}{s_w^2 c_w^2}
- \frac{ 2 Q_{d} I_{3d}}{c_w^2} + \frac{Q_d^2}{c_w^2}
+ \frac{1}{2 s_w^2}
\left(V_{u d} V_{d u}^\dagger\right)
\right]
\nonumber \\
&&+ \frac{s_w^2}{16 m_Z^2 c_w^2}
\left(\frac{p_1^2}{4}-\frac{2}{3} m_Z^2
-\frac{5}{6} m_d^2 \right)
\left(Q_d-\frac{I_{3d}}{s_w^2} \right)^2
\nonumber \\
&& + \frac{V_{u d} V_{d u}^\dagger}{32 m_W^2 s_w^2}
\left(\frac{p_1^2}{4}-\frac{2}{3} m_W^2
-\frac{5}{6} m_u^2 \right)
\nonumber \\
C_0 & = & \frac{Q_d m_d}{8 c_w^2} \left[
\lambda_{HV} \left(Q_d-I_{3d} \right)
+\frac{s_W^2}{4 m_Z^2}
\left(Q_d-\frac{I_{3d}}{s_w^2} \right)
\left(\frac{p_1^2}{3}-m_Z^2-m_d^2 \right) \right]
\nonumber \\
\nonumber \\
l l ~~:~~C_- & = &
\frac{Q_l^2}{16 c_w^2}
\left[
\lambda_{HV}
+\frac{s_w^2}{m_Z^2} \left(\frac{p_1^2}{4}-\frac{2}{3} m_Z^2
-\frac{5}{6} m_l^2 \right)
\right]
\nonumber \\
C_+ & = & \frac{\lambda_{HV}}{16} \left[\frac{I_{3l}^2}{s_w^2 c_w^2}
- \frac{ 2 Q_{l} I_{3l}}{c_w^2} + \frac{Q_l^2}{c_w^2}
+ \frac{1}{2 s_w^2} \right]
\nonumber \\
&&+ \frac{s_w^2}{16 m_Z^2 c_w^2}
\left(\frac{p_1^2}{4}-\frac{2}{3} m_Z^2
-\frac{5}{6} m_l^2 \right)
\left(Q_l-\frac{I_{3l}}{s_w^2} \right)^2
\nonumber \\
&& + \frac{1}{32 m_W^2 s_w^2}
\left(\frac{p_1^2}{4}-\frac{2}{3} m_W^2\right)
\nonumber \\
C_0 & = & \frac{Q_l m_l}{8 c_w^2} \left[
\lambda_{HV} \left(Q_l-I_{3l} \right)
+\frac{s_W^2}{4 m_Z^2}
\left(Q_l-\frac{I_{3l}}{s_w^2} \right)
\left(\frac{p_1^2}{3}-m_Z^2-m_l^2 \right) \right]
\nonumber \\
\nonumber \\
\nu_l \nu_l ~~:~~C_- & = & 0 \nonumber \\
C_+ & = &
\frac{\lambda_{HV}}{16 s_w^2}
\left(\frac{1}{2}+\frac{I^2_{3\nu_l}}{c_w^2} \right)
+ \frac{I^2_{3\nu_l}}{16 m_Z^2 c_w^2 s_w^2}
\left(\frac{p_1^2}{4}-\frac{2}{3} m_Z^2\right)
\nonumber \\
&& + \frac{1}{32 m_W^2 s_w^2}
\left(\frac{p_1^2}{4}-\frac{2}{3} m_W^2
-\frac{5}{6} m_l^2 \right)
\nonumber \\
C_0 & = & 0
\end{eqnarray}
\subsubsection{Bosonic contribution to the vertices with 3 legs}
The generic 3-point vertices appearing in our calculation are drawn in Fig.~\ref{fig:2}. As before
the full set of results is available in~\cite{formfiles}.
We found 23 non zero ${\rm R_2}$ vertices in the Unitary gauge, classified in Table~\ref{tab:tab1}.
\subsubsection{Bosonic contribution to the vertices with 4 legs}
All non vanishing generic 4-point vertices that appear in our calculation
are drawn in Fig.~\ref{fig:3}. The full set of results is presented in~\cite{formfiles}.
The 14 non zero ${\rm R_2}$ vertices in the Unitary gauge are classified in Table~\ref{tab:tab2}.
\section{\label{checks}Checks}
All our formulae have been obtained cross-checking two independent calculations.
To further check our results, we used the fact that the ${\rm R}= {\rm R_1}+{\rm R_2}$ contribution to physical quantities
should be independent of the chosen gauge.
In particular, parametrizing the gauge boson self-energies as follows
\begin{eqnarray}
\Sigma_V^{\mu\nu}(p) &=& g^{\mu\nu}\, \Sigma_{V0}(p^2) +p^\mu p^\nu\,\Sigma_{V1}(p^2)~~~~{\rm with}~~~{V= Z,W,\gamma}\,,
\end{eqnarray}
we verified that the ${\rm R}$ contribution to
$\Sigma_{W0}(M_W^2)$, $\Sigma_{Z0}(M_Z^2)$ and $\Sigma_{\gamma 0}(0)$ is the same in both the $R_\xi$ and the Unitary gauge.
In addition, in the case of both gauges, we checked all of
the 2-point like Ward Identities presented in~\cite{ewrational} involving
${\rm Vert}(S_1,S_2)$, ${\rm Vert}(V,S)$ and ${\rm Vert}(V_1,V_2)$.
To test the 3-point sector, we computed the ${\rm R}= {\rm R_1}+{\rm R_2}$ contribution to the process
$
H \to \gamma \gamma\,.
$
Again, we found the same answer working in both gauges, obtaining an expression
for ${\rm R}$ in full agreement with that one presented in~\cite{Bardin:1999ak}.
As for the 4-point sector, we checked that, in the limit $\xi \to 1$, we fully reproduce the effective vertices
presented in~\cite{ewrational}.
Finally, in the case of the $R_\xi$ gauge, we computed ${\rm R_2}$ using
both the following two equivalent representations for the massive
gauge boson propagators
\begin{eqnarray}
&& {-i} \left(\frac{g_{\alpha\beta}}{p^2-M_V^2}
-(1-\xi)\frac{p_\alpha p_\beta}{(p^2-M_V^2)(p^2-\xi M_V^2)}\right)
\,~~{\rm and}~~\nonumber \\
&& {-i} \left(\frac{g_{\alpha\beta}}{p^2-M_V^2}
-\frac{p_\alpha p_\beta} {M_V^2(p^2-M_V^2)}
+\frac{p_\alpha p_\beta} {M_V^2(p^2-\xi M_V^2)} \right)\,,
\end{eqnarray}
always finding the same results. Since the two expressions lead
to different integrals in the intermediate stages of the calculation,
this provides a strong consistency check of our procedure.
As a last remark notice that, when working in the Unitary gauge, we take the
limit $\xi \to \infty$ {\em before} integrating over the loop momentum. The fact that this gives the same result for ${\rm R}$ as in a generic $R_\xi$ gauge
in the above mentioned cases
{\em provided the same prescription is used in the calculation of ${\rm R_1}$} is an explicit check of the equivalence
of the limits $\xi \to \infty$ after or before the loop momentum integration in the definition of the Unitary gauge at 1-loop.
\section{Conclusions}
We presented the full set of Feynman rules producing the rational terms of kind ${\rm R_2}$ needed to perform
any 1-loop calculation in the Electroweak Standard Model in the $R_{\xi}$ gauge and in the Unitary gauge.
In a few physical cases we also checked the independence of the full rational piece
${\rm R}= {\rm R_1}+{\rm R_2}$ of the chosen gauge and, in the case
of the Unitary gauge, of the order between the limit $\xi \to \infty$
and the integration over the loop momentum.
Our results can be used to transform tree level packages based on gauges
other that the 't Hooft-Feynman one into 1-loop calculators with the help of the
OPP or Generalized Unitarity techniques.
\section*{Acknowledgments}
R.P.'s and I.M.'s research was partially supported by the RTN
European Programme MRTN-CT-2006-035505 (HEPTOOLS, Tools and Precision
Calculations for Physics Discoveries at Colliders).
M.V.G.'s research was supported by INFN.
The research of R.P. and M.V.G. was also supported
by the MEC project FPA2008-02984.
R.P. also acknowledges the financial support of the bilateral
INFN/MICINN program ACI2009-1045 (Aspects of Higgs physics at the LHC).
|
{
"timestamp": "2010-09-23T02:01:24",
"yymm": "1009",
"arxiv_id": "1009.4302",
"language": "en",
"url": "https://arxiv.org/abs/1009.4302"
}
|
\section{Introduction}
The positivity of the Hilbert space scalar product gives place to an infinite series of inequalities involving the correlation functions of any number of variables in a quantum field theory (QFT). In the real-time formulation we have, for any finite sequence of test functions
$f_0\in \mathbb{C}$, $f_1(x_1)$, ..., $f_k(x_1,...,x_k)$, the inequality
\begin{equation}
\sum_{i,j=0}^k \int \,dx_1 ...dx_i \, dy_1... dy_j \, (f_i(x_1,...,x_i))^* {\cal W}_{i+j}(x_i,...,x_1,y_1,...,y_j) f_j(y_1,...,y_j)\ge 0\,,\label{wig}
\end{equation}
where ${\cal W}_{n}(x_1,...,x_n)$ are the Wightman distributions (for a real scalar field) \cite{w1}. This property allows for the reconstruction of the Hilbert space from the correlation functions and plays a central role in the Wightman axiomatic framework \cite{sw}. In (\ref{wig}) the integrations are over all the $d$-dimensional Minkowski space, $d\ge 2$. These inequalities always involve the singularities of the Wightman functions at coinciding points, i.e. when $x_i=y_j$. Therefore, the content of (\ref{wig}) for a finite number of functions at a finite number of points is far from being transparent, and its consequences are sometimes more easily seen in momentum space.
A different situation holds in the euclidean framework \cite{os,os1}. The relation corresponding to (\ref{wig}) in this case is called reflection positivity, and writes
\begin{equation}
\sum_{i,j=0}^k \int \,dx_1...dx_i \, dy_1... dy_j \, (f_i(x_1,...,x_i))^* {\cal E}_{i+j}
(\hat{x}_i,...,\hat{x}_1,y_1,...,y_j) f_j(y_1,...,y_j)\ge 0\,. \label{rp}
\end{equation}
Here ${\cal E}_{n}
(x_1,...,x_n)$ are the Schwinger functions (euclidean correlators), the integration is over the euclidean space and $\hat{x}=(-x^0,x^1,...,x^{d-1})$ is the euclidean time-reflected point corresponding to $x=(x^0,x^1,...,x^{d-1})$. The test functions $f_j(x_1,...,x_j)$ have support only for the time ordered points on the positive-time half-space $0<x_1^0<...<x_j^0$. Thus, the inequalities (\ref{rp}), in contrast to the ones in real time (\ref{wig}), only involve non coinciding points for the correlators. The reflection positivity property provides the connection between euclidean statistical interpretation of the Schwinger functions and the quantum interpretation in terms of a relativistic QFT.
The purpose of this paper is to show the Wightman functions satisfy a positivity relation resembling reflection positivity. This seems to have escaped previous attention. In order to introduce these inequalities let us define the wedge in Minkowski space as the open set $\mathbb{W}=\{x\in \mathbb{R}^d;\, x^1> \vert x^0 \vert\}$. This is bounded by the two null planes intersecting on the $(d-2)$-dimensional spatial plane $\{x\in \mathbb{R}^d;\, x=(0,0,x^2,...,x^{d-1}) \}$.
We define an order relation on the points of $\mathbb{R}^d$ as
$x\triangleleft y$ iff $y-x \in \mathbb{W}$. In particular, if $x\triangleleft y$ holds, $x$ and $y$ are space-like separated. The
wedge reflection positivity (WRP) relations for a hermitian scalar field read
\begin{equation}
\sum_{i,j=0}^k \int \,dx_1 ...dx_i \, dy_1... dy_j \, (f_i(x_1,...,x_i))^* {\cal W}_{i+j}
(\bar{x}_i,...,\bar{x}_1,y_1,...,y_j) f_j(y_1,...,y_j)\ge 0\,. \label{wedgereflection}
\end{equation}
Here the wedge reflection is $\bar{x}=(-x^0,-x^1,x^2,...,x^{d-1})$, and the inequalities hold for any finite sequence of test functions $f_0$, $f_1(x_1)$, ..., $f_k(x_1,...,x_k)$,
where $f_j(x_1,...,x_j)$ can be non zero only if the points $x_1$,...,$x_j$ are wedge ordered, $0\triangleleft x_1\triangleleft...\triangleleft x_j$.
The WRP is naturally understood as a consequence of the Tomita-Takesaki theory for the algebra of operators on the wedge, revealing the TCP theorem has an associated positivity property. However, the inequalities (\ref{wedgereflection}) are valid in greater generality. We prove them from the positivity, covariance and spectral properties of Wightman functions, without using the TCP theorem, or, equivalently, weak local commutativity \cite{ot}.
The WRP does not involve the correlators at coinciding points. Then, specific inequalities for a finite number of Wightman functions evaluated at definite points can be derived. Given any collection of wedge ordered sets of points, $A_i=\{x_1^{(i)},...,x_{n_i}^{(i)}\}$, $0\triangleleft x_1^{(i)}\triangleleft...\triangleleft x_{n_i}^{(i)}$, for $i=1,...,m$, we have from (\ref{wedgereflection}), taking the limit of localized test functions,
\begin{equation}
\textrm{det}\left(\{{\cal W}(\bar{A}_{i}A_j)\}_{i,j=1...m}\right)\ge 0\,,\label{cuatro}
\end{equation}
where we write $\bar{A}_i=\{\bar{x}_{n_i}^{(i)},...,\bar{x}_{1}^{(i)}\}$.
Recently, we have shown that the exponentials $e^{(n-1)I_n(\bar{A}_i , A_j)}$ of the Renyi mutual information $I_n(\bar{A}_i,A_j)$ of integer index $n$, between disjoint regions bounded by the sets of points $\bar{A}_i$ and $A_j$ in two-dimensional QFT, obeys the inequalities (\ref{cuatro}) \cite{ca}. The Renyi entropies measure essentially the entanglement of the vacuum state. The inequalities indicate they are given by the vacuum expectation values of some local operators. This is also indicated by the path integral representations of the Renyi entropies in the euclidean framework \cite{cc}. An interesting application of the WRP inequalities would be to show the validity of this mapping between vacuum entanglement and field operators with full rigor. We postpone the study of a reconstruction theorem in this sense, based on the WRP inequalities, to a future work.
\section{Proof of wedge reflection positivity}
In order to make the exposition more clear, let us consider the case of a hermitian scalar field first, and then show the necessary changes for the case of fields with spin. In order to prove WRP we use analyticity of the Wightman functions. The proof is very similar in form to the one of reflection positivity for the Schwinger functions.
Let us first introduce some notation, which closely follows the one in \cite{os}. The open future light cone is $V^+=\{x:\, x.x> 0,\, x^0>0\}$, where $x.x=(x^{0})^2-\sum_{i=1}^{d-1} (x^i)^2$ is the Minkowski scalar product, and we call the closed cone $V^+_c$. We write the Minkowski metric $G=\textrm{diag}(1,-1,...,-1)$. Let ${\cal S}(\mathbb{R}^{d n})$ be the Schwartz space of infinite differentiable complex test functions of fast decrease on $\mathbb{R}^{d n}$, with the usual
topology. Using the notation for the partial derivatives
$D^\alpha=\partial^{\vert\alpha\vert}/((\partial x_1^0)^{\alpha_1} ...(\partial x_n^{d-1})^{\alpha_{(d-1)n}})$, where $\alpha=(\alpha_1,...,\alpha_{(d-1)n})$ we also define the following closed subspaces of
${\cal S}(\mathbb{R}^{d n})$
\begin{eqnarray}
{\cal S}_+(\mathbb{R}^{dn})&=&\{ f\in {\cal S}(\mathbb{R}^{d n}); D^\alpha f (x_1,...,x_n)=0\,\,\, \forall \alpha \,\,\, \textrm{unless} \,\,\,\, 0\triangleleft x_1\triangleleft ... \triangleleft x_n \} \,,\\
{\cal S}_\triangleleft(\mathbb{R}^{dn})&=&\{ f\in {\cal S}(\mathbb{R}^{d n}); D^\alpha f (x_1,...,x_n)=0\,\,\, \forall \alpha \,\,\, \textrm{unless} \,\,\,\, x_1\triangleleft...\triangleleft x_n \}\,.
\end{eqnarray}
For each test function space ${\cal S}$ we call ${\cal S}^\prime$ the corresponding dual space of distributions.
Let us start the proof by recalling some well-known facts. Because of translation invariance we have
\begin{equation}
{\cal W}_n(x_1,x_2,...,x_n)=W_{n-1}(\xi_1,..., \xi_{n-1})
\end{equation}
for a distribution $W_{n-1}\in {\cal S}^\prime(\mathbb{R}^{d(n-1)})$, with $\xi_1=x_1-x_2$,..., $\xi_{n-1}=x_{n-1}-x_n$. Because of their spectral properties, the functions $W_{n-1}(\xi_1,..., \xi_{n-1})$ can be continued analytically to the forward tube ${\cal T}_{n-1}$. This is formed by the arrays $(\zeta_1,\zeta_2,...,\zeta_{n-1})$ of $(n-1)$ complex vectors $\zeta_j=\xi_j-i \eta_j$, where $\xi_j$ and $\eta_j$ are real vectors in $\mathbb{R}^d$ and $\eta_j\in V^+$ for $j=1,...,n-1$. The analytic continuation is done by the Laplace transform
\begin{equation}
W_{n-1}(\zeta_1,...,\zeta_{n-1})
= (2\pi)^{-d(n-1)}\int dp_1...dp_{n-1}\, e^{-i \sum_{j=1}^{n-1} p_j (\xi_j-i \eta_j)} \tilde{W}_{n-1}(p_1,...,p_{n-1}).
\end{equation}
Here $\tilde{W}$ is the Fourier transform of $W$, defined as
\begin{equation}
\tilde{W}_{n-1}(p_1,...,p_{n-1})=\int d\xi_1...d\xi_{n-1}\,e^{i \sum_{j=1}^{n-1} p_j \xi_j}
W_{n-1}(\xi_1,...,\xi_{n-1})\,.
\end{equation}
The domain of analyticity can be augmented further to the extended tube ${\cal T}^{\prime (n-1)}$ by prolonging Lorentz covariance to proper complex Lorentz transformations, that is, to the group of complex matrices $\Lambda$ of unit determinant satisfying $\Lambda^T G \Lambda=G$ \cite{sw}. For a scalar field we have
\begin{equation}
W_{n-1}(\zeta_1,..., \zeta_{n-1})=W_{n-1}(\Lambda\zeta_1,..., \Lambda\zeta_{n-1})\,.\label{lorr}
\end{equation}
The extended tube includes in particular all the Jost points, which are the arrays $(\xi_1,...,\xi_{n-1})$ of real vectors which are all included in some wedge $\mathbb{W}^\prime$, a Lorentz transform of $\mathbb{W}$, $\mathbb{W}^\prime= \Lambda \mathbb{W}$ with $\Lambda$ any proper or unproper Lorentz transformation \cite{sw,jos}.
In particular the case $x_1\triangleleft x_2...\triangleleft x_n$ gives $\xi_j=x_j-x_{j+1}\in -\mathbb{W}$, and $(\xi_1,...,\xi_{n-1})$ is a Jost point. Consequently ${\cal W}_n(x_1,x_2,...,x_n)=W_{n-1}(\xi_1,..., \xi_{n-1})$ is an analytic function for $x_1\triangleleft...\triangleleft x_n$.
Consider the complex Lorentz transformation $x^\prime=\Lambda_0\, x$ which leaves the coordinates $x^{2 \prime }=x^2$,..., $x^{(d-1)\prime }=x^{(d-1)}$ invariant and transforms the first two coordinates as $x^{0 \prime }=i x^1$, $x^{1 \prime }= i x^0$. That is,
\begin{equation}
\Lambda_0=
\left(\begin{array}{cccc}
0 & i & \ldots & 0 \\
i & 0 & \ldots & 0 \\
\vdots & \vdots &\ddots & 0\\
0 & 0& 0& 1
\end{array}\right)\,.
\end{equation}
This transforms a vector $\xi \in -\mathbb{W}$, $\xi^1 <-\vert\xi^0\vert$, into a vector in ${\cal T}^1$, since
\begin{equation}
\Lambda_0 \xi= \Lambda_0 (\xi^0 , \xi^1, \xi^2, ... , \xi^{d-1})=(i\xi^1 , i \xi^0, \xi^2, ... , \xi^{d-1}) \,.\label{termo}
\end{equation}
This is, $\Lambda_0 \xi=\xi^\prime -i \xi^{\prime\prime}$, with $\xi^\prime=(0,0,\xi^2,...,\xi^n)$ and $\xi^{\prime\prime}=(-\xi^1 , - \xi^0, 0, ... , 0) \in V^+$.
Then we can write for a Wightman
function on $x_1\triangleleft x_2...\triangleleft x_n$,
\begin{eqnarray}
{\cal W}_n(x_1,x_2,...,x_n)&=&W_{n-1}(\Lambda_0\xi_1,..., \Lambda_0\xi_{n-1}) = W_{n-1}(\xi_1^\prime-i \xi^{\prime\prime}_1,...,\xi_n^\prime-i \xi^{\prime\prime}_n)\label{parenq}\\
&&= (2\pi)^{-d(n-1)}\int dp_1...dp_{n-1}\, e^{-i \sum_{j=1}^{n-1} p_j (\xi_j^\prime-i\xi^{\prime\prime}_j)} \tilde{W}_{n-1}(p_1,...,p_{n-1})\,.\nonumber
\end{eqnarray}
The Wightman function of the left hand side can be understood as a distribution in ${\cal S}^\prime_\triangleleft(\mathbb{R}^{dn})$, which result from a restriction of the Wightman distribution in ${\cal S}^\prime(\mathbb{R}^{dn})$ to the test functions in ${\cal S}_\triangleleft(\mathbb{R}^{dn})$.
For a sequence of test functions $f_0$, $f_1(x_1)$,..., $f_k(x_1,...,x_k)$, with $f_j\in {\cal S}_+(\mathbb{R}^{dj})$, the left hand side of the WRP relation (\ref{wedgereflection}), can be written according to (\ref{parenq})
\begin{eqnarray}
&& \sum_{i,j=0}^k \int \,dx_1 ...dx_i \, dy_1... dy_j \, (f_i(\bar{x}_i,...,\bar{x}_1))^* {\cal W}_{i+j}
(x_1,...,x_i,y_1,...,y_j) f_j(y_1,...,y_j) \nonumber \\
&&=(2\pi)^{-d(i+j-1)}\sum_{i,j=0}^k \int \,d\xi_1 ...d\xi_{i+j-1} dx_i \,(f^{-}_i(-\bar{x}_i,-\bar{\xi}_{i-1},..., -\bar{\xi}_{1}))^* f^-_j(\xi_i-x_i,\xi_{i+1},..., \xi_{i+j-1}) \nonumber \\
&&\hspace{4cm}\times \int dp_1...dp_{i+j-1}\,e^{-i \sum_{q=1}^{i+j-1} p_q (\xi_q^\prime-i\xi^{\prime\prime}_q)}\tilde{W}_{i+j-1}(p_1,...,p_{i+j-1})\,,\label{meni}
\end{eqnarray}
where we have defined $
f^-_n(\chi_1,..., \chi_{n})=f_n(x_1,...,x_n)$, $\chi_1=-x_1$, $\chi_k=x_{k-1}-x_{k}$ for $k=2,...,n$. The points $\chi_1$,...,$\chi_n\in -\mathbb{W}$. We have that for any vector $\xi$ it is $\bar{\xi^\prime}=\xi^\prime$, and $\bar{\xi^{\prime\prime}}=-\xi^{\prime\prime}$. Using this, and interchanging the order of the coordinate and momentum integrals, we can check that (\ref{meni}) becomes
\begin{equation}
\sum_{i,j=0}^k \int \,dp_1 ... dp_{i+j-1} \, (\hat{f}_i(p_i,...,p_1))^* \tilde{W}_{i+j-1}
(p_1,...,p_{i+j-1}) \hat{f}_j(p_i,...,p_{i+j-1})\,,\label{doce}
\end{equation}
where
\begin{equation}
\hat{f}_n(p_1,...,p_n)=(2 \pi)^{-d (n-1/2)}\int d\chi_1 ... d\chi_n f^-_n(\chi_1,..., \chi_n) e^{-\sum_{q=1}^n(p_q \chi_q^{\prime \prime}+i p_q \chi^\prime_q )}\,.
\label{trece}
\end{equation}
The change of the order of the integrals in (\ref{meni}) for the variables $p_l^2,...,p_l^d$ and $\xi_l^2,...,\xi_l^d$ is just the definition of the Fourier transform of a distribution. For the components $p_l^0, p_l^1$ and $\xi_l^0, \xi_l^1 $ the justification comes from the same arguments as in the Lemma 8.4 in \cite{os}.
Note that in eq. (\ref{doce}) the distribution $\tilde{W}_{n-1}(p_1,...,p_{n-1})$ has support on $p_l\in V_c^+$ because of the spectral condition. In this domain the functions $\hat{f}_n(p_1,...,p_n)$ of (\ref{trece}) are infinitely differentiable and of fast decrease (see for example Lemma 8.2 in \cite{os}). Thus, they can be thought as restrictions of functions $\underline{\hat{f}}(p_1,...,p_n)$ in ${\cal S}(\mathbb{R}^{d n})$ to $V_c^+$ in (\ref{doce}) (see Lemma 2.1 in \cite{os1}).
We then write the right hand side of eq. (\ref{doce}) as
\begin{equation}
\sum_{i,j=0}^k \int \,dx_1 ...dx_i \, dy_1 ... dy_j \, (\check{f}_i(x_1,...,x_i))^* {\cal W}_{i+j}
(x_i,...,x_1,y_1,...,y_j) \check{f}_j(y_1,...,y_j)\,,\label{malv}
\end{equation}
where $\check{f}_j(x_1,...,x_j)=(2 \pi)^{-d/2}\int dp_1 ... dp_j\, \underline{\hat{f}}(p_1,...,p_j)\, e^{-i p_1 x_1 +i \sum_{q=2}^{j-1} p_q(x_{q-1}-x_{q}) }$. The quantity (\ref{malv}) is positive by the standard positivity property (\ref{wig}) for the Wightman distributions. We have then finished the proof of (\ref{wedgereflection}).
\subsection{Fields with spin}
It is not difficult to find the changes to (\ref{wedgereflection}) which are necessary in order to allow for fields with charge or spin. Let ${\cal W}^{(\nu, \kappa)}_n(x_1,...,x_n)=W^{(\nu, \kappa)}_{n-1}(\xi_1,...,\xi_{n-1})=\langle 0 \vert \psi^{\nu_1 \kappa_1}(x_1)...\psi^{\nu_n \kappa_n}(x_n)\vert 0 \rangle $. We use $(\nu \kappa)$ as an abbreviation of $(\nu_1...\nu_n,\kappa_1...\kappa_n)$. The $\nu_i$ represent the index corresponding to the finite dimensional representation of the covering group of the Lorentz group for the field $\psi^{\nu_i\kappa_i}$, labeled by $\kappa_i$. The transformation law for the field reads
\begin{equation}
U(\Lambda) \psi^{\nu \kappa}(x)U(\Lambda)^{-1}=S^\kappa(\Lambda^{-1})^{\nu}_{\nu^\prime}\psi^{\nu^\prime \kappa}(\Lambda x)\,.
\end{equation}
The adjoint field is represented as $(\psi^{\nu \kappa}(x))^\dagger=\psi^{\nu^* \kappa^*}(x)$. It is labeled by $\kappa^*$, and the corresponding representation of the (real) Lorentz group is the complex conjugate representation to the one corresponding to $\kappa$. We also write $(\bar{\nu})= (\nu_n^*...\nu_1^*)$ and $(\bar{\kappa})=(\kappa_n^*...\kappa_1^*)$, both, taking the adjoint fields and inverting the ordering of the indices.
The covariant transformation law for the Wightman distributions now reads
\begin{equation}
W^{(\nu, \kappa)}_{n-1}(\xi_1,..., \xi_{n-1})=\sum_{\mu} S^{(\kappa)}(\Lambda^{-1})_{(\mu)}^{(\nu)} W^{(\mu, \kappa)}_{n-1}(\Lambda\xi_1,..., \Lambda\xi_{n-1})\,,\label{piro}
\end{equation}
where we have introduced the notation $S^{(\kappa)}(\Lambda^{-1})_{(\mu)}^{(\nu)}=S^{\kappa_1}(\Lambda^{-1})_{\mu_1}^{\nu_1}...S^{\kappa_n}(\Lambda^{-1})_{\mu_n}^{\nu_n}$.
In the extended tube, this equation holds for the complex Lorentz transformations, and in particular it extends to $\Lambda_0$, where $S^{\kappa}(\Lambda_0^{-1})^{\mu}_{\nu}$ is the matrix corresponding to the representation of the field $\kappa$ evaluated for $\Lambda_0$ in the complex Lorentz group.
In order to cancel these matrix factors coming from the Lorentz transformation $\Lambda_0$ (see eq. (\ref{parenq})) we have to include extra factors to (\ref{wedgereflection}). This leads to a WRP for general fields, which writes
\begin{eqnarray}
\sum_{\stackrel{i,j}{\stackrel{(\nu_i,\kappa_i)}{(\nu_j,\kappa_j)}}} \int \,dx_1 ...dx_i \, dy_1... dy_j \, (f_i^{(\nu_i, \kappa_i)}(x_1,...,x_i))^*S^{(\bar{\kappa}_i)}(\Lambda_0)_{(\bar{\mu}_{i})}^{(\bar{\nu}_{i})}
S^{(\kappa_{j})}(\Lambda_0)_{(\mu_j)}^{(\nu_j)}&& \nonumber\\
\hspace{3.cm} {\cal W}_{i+j}^{(\bar{\mu}_i\mu_j, \bar{\kappa}_i \kappa_j)}
(\bar{x}_i,...,\bar{x}_1,y_1,...,y_j) f^{(\nu_j, \kappa_j)}_j(y_1,...,y_j)&\ge& 0\,, \label{dicete}
\end{eqnarray}
where $f^{(\nu_l \kappa_l)}_l(x_1,...,x_l)\in {\cal S}_+(\mathbb{R}^{dl})$.
In order to find the extra matrix factors more explicitly, we can write the field representations of the one dimensional subgroup of boosts in the $x_1$ direction
\begin{equation}
\Lambda(\phi)=
\left(\begin{array}{cccc}
\cosh (\phi) & \sinh (\phi) & \ldots & 0 \\
\sinh (\phi) & \cosh (\phi) & \ldots & 0 \\
\vdots & \vdots &\ddots & 0\\
0 & 0& 0& 1
\end{array}\right)\,,
\end{equation}
as $S^{\kappa}(\Lambda(\phi))=e^{\phi K}$, with $\phi$ the boost parameter. Then we have $S^{\kappa}(\Lambda_0)=e^{i\frac{\pi}{2}K}$ and $S^{\kappa^*}(\Lambda_0)=e^{i\frac{\pi}{2}K^*}$, with $K^*$ the complex conjugate of the matrix $K$.
The $S^{(\kappa_{j})}(\Lambda_0)_{(\mu_j)}^{(\nu_j)}$ can be absorbed in the test functions in (\ref{dicete}), leaving no factors for the unbarred indices, at the expense of changing the matrix factors $S^{\kappa^*}(\Lambda_0)$ for the barred indices by their squares $(S^{\kappa^*}(\Lambda_0))^2=e^{i \pi K^*}$.
This gives, for each barred index, a factor of the wedge parity $P_{\mathbb{W}}=\Lambda_0^2=\textrm{diag}(-1,-1,1...1)$ on the vector indices, a $e^{i \frac{\pi}{2} (\alpha^{1})^*}=i(\alpha^{1})^*$ for each Dirac spinor one, and $i \alpha^{1}$ for the adjoint spinors, where $\alpha^1=\gamma^0\gamma^1$ is the Dirac matrix. In two dimensions, for a field of spin $s$, transforming as $
U(\Lambda(\phi)) \psi(x)U(\Lambda(\phi))^{-1}=e^{-s \, \phi} \psi(\Lambda(\phi) x)$, we have a factor $e^{i \pi s^*}$ on the barred indices.
\section{WRP, TCP and the Bisognano-Wichmann theorem}
A generalized form of the WRP inequalities can be derived in a general quantum mechanical setting using the Tomita-Takesaki modular theory \cite{tt}. Given a cyclic and separating vector state $\vert 0 \rangle$ in a von Neumann algebra ${\cal A}$, we can define the antilinear operator $S$ by
\begin{equation}
S {\cal O}\vert 0 \rangle={\cal O}^\dagger \vert 0 \rangle\,,
\end{equation}
for any ${\cal O}\in {\cal A}$. $S$ can be decomposed as $S=J \Delta^{\frac{1}{2}}$, with $J$ antiunitary and $\Delta$ self-adjoint and positive definite. The crucial point of the Tomita Takesaki theory is that $J$ maps the algebra ${\cal A}$ into its commutant algebra ${\cal A}^\prime$.
One also has $\Delta \vert 0 \rangle=\vert 0 \rangle$, $J \vert 0 \rangle=\vert 0 \rangle$, $J=J^\dagger=J^{-1}$ and $J \Delta = \Delta^{-1} J$. Then it follows, for any ${\cal O}\in {\cal A}$, and writing $\bar{{\cal O}}=J{\cal O}J$ for the "reflected" operator, $\bar{{\cal O}}\in {\cal A}^\prime$,
\begin{equation}
\langle 0\vert \bar{{\cal O}} {\cal O} \vert 0 \rangle=\langle 0\vert {\cal O} J {\cal O} \vert 0 \rangle^*=\langle 0\vert {\cal O} \Delta^{\frac{1}{2}} S {\cal O} \vert 0 \rangle^*=\langle 0\vert {\cal O} \Delta^{\frac{1}{2}} {\cal O}^\dagger \vert 0 \rangle\geq 0\,.\label{rere}
\end{equation}
This is a general quantum mechanical reflection positivity property.
The connection with QFT is given by the Bisognano-Wichmann theorem \cite{aa}. This gives the modular reflection $J$ corresponding to the vacuum state and the algebra ${\cal A}_{\mathbb{W}}$ generated by the operators localized in the wedge $\mathbb{W}$. Consider the theory of a hermitian scalar field in four space-time dimensions obeying the Wightman axioms (including local commutativity), which are the hypothesis of the Bisognano-Wichmann theorem. Then $J=U(R(e_1,\pi))\Theta$, where $U(R(e_1,\pi))$ is the unitary operator corresponding to a rotation of angle $\pi$ around the $(0,1,0,0)$ axis, and $\Theta$ is the TCP operator. The modular reflection $J$ acts geometrically on the field operators as a wedge reflection $J \phi(x) J =\phi(\bar{x})$.
Let $K_1$ be the boost generator in the direction of the first spatial coordinate.
Specifically, Bisognano and Wichmann prove that for an element ${\cal O}$ of the polynomial algebra of the field in the
wedge
\begin{equation}
{\cal O}=\sum_{i=0}^p \int dx_1 ... dx_i \, f_i(x_1,...,x_i) \phi(x_1)... \phi(x_i) \,,\label{hylo}
\end{equation}
where $f_i(x_1,...,x_i)$ is a test function with support on $\mathbb{W}$, we have
\begin{equation}
e^{ \pi K_1 } \, {\cal O}^\dagger \vert 0\rangle=J {\cal O} \, \vert 0 \rangle\,.
\end{equation}
From this relation it follows
\begin{equation}
\langle 0 \vert J {\cal O} J {\cal O} \, \vert 0\rangle = \langle 0\vert {\cal O} J {\cal O} \vert 0 \rangle^*= \langle 0\vert {\cal O} e^{ \pi K_1 } \, {\cal O}^\dagger
\vert 0\rangle \ge 0\,.\label{quete}
\end{equation}
The last inequality follows from positivity of the operator $e^{ \pi K_1 }$. This also identifies $\Delta=e^{2 \pi K_1}$. The reflected operator $\bar{{\cal O}}=J {\cal O} J$ is
\begin{equation}
J {\cal O} J=\sum_{i=0}^p \int dx_1 ... dx_i (f_i(x_1,...,x_i))^* \phi(\bar{x}_1)... \phi(\bar{x}_i) \,.
\end{equation}
Thus, from (\ref{quete}) we have
\begin{equation}
\sum_{i,j=1}^k \int \,dx_1...dx_i \, dy_1... dy_j \, (f_i(x_1,...,x_i))^* {\cal W}_{i+j}
(\bar{x}_1,...,\bar{x}_i,y_1,...,y_j) f_j(y_1,...,y_j)\ge 0\,. \label{culpa}
\end{equation}
A remark about the relation of this inequality with (\ref{wedgereflection}) is in order.
First, in (\ref{culpa}) the test functions have support inside the wedge, but there is no restriction to the ordering of the points, nor they have to be spatially separated to each other. The WRP, eq. (\ref{wedgereflection}), follows from (\ref{culpa}) for the specific case of $f_i\in {\cal S}_+(\mathbb{R}^{dn})$, and using local commutativity in order to obtain the correct ordering for the points inside the Wightman functions (note the difference in ordering between (\ref{wedgereflection}) and (\ref{culpa})). Thus, in this sense, the relation (\ref{culpa}) is stronger than (\ref{wedgereflection}). However, (\ref{wedgereflection}) follows without the need of local commutativity (LC), or weak local commutativity, which is the condition for the validity of the TCP theorem \cite{jos}.
It is also possible to express the WRP for any spin representation and for operators with even fermion number, in terms of a relation involving a TCP operator, if the local commutativity holds. In order to write this relation in any dimension we use a version of the TCP theorem which does not involve a reflection for all space-time coordinates, but a wedge reflection \cite{cpt}. This is equivalent to the standard TCP theorem in even dimensional space-times, because both operations are related by a rotation. However, the "wedge TCP theorem" also holds in odd dimensions, where the inversion of coordinates $x\rightarrow -x$ has determinant $(-1)$ and cannot be reached continuously from the identity by the complex Lorentz transformations.
This wedge TCP theorem follows from the same arguments as the standard one: The matrix $P_{\mathbb{W}}$ belongs to the complex Lorentz group, and the analyticity of the Wightman functions on the Jost points implies
\begin{equation}
{\cal W}^{(\nu, \kappa)}_{n}(x_1,...x_n)=\sum_{\mu} \left(S^{(\kappa)}(P_{\mathbb{W}})^{-1}\right)^{ (\nu)}_{(\mu)} {\cal W}^{(\mu, \kappa)}_{n}(\bar{x}_1,..., \bar{x}_{n})\label{cristina}
\end{equation}
if $x_1\triangleleft x_2...\triangleleft x_n$. Then, if LC holds, with either commuting or anticommuting fields at space-like separated points,
eq. (\ref{cristina}) can be interpreted as the expression of the existence of a symmetry. This is
\begin{equation}
\langle 0\vert \psi^{\nu_1 \kappa_1}(x_1)... \psi^{\nu_n \kappa_n}(x_n) \vert0 \rangle=\langle 0\vert J\psi^{\nu_1 \kappa_1}(x_1)J^{-1}...\, J\psi^{\nu_n \kappa_n}(x_n)J^{-1} \vert 0\rangle^*\,,
\end{equation}
where the antiunitary operator $J$ keeps the vacuum invariant $J\vert0\rangle=\vert0\rangle$, and transform the fields as
\begin{equation}
J \psi^{\nu \kappa}(x) J^{-1}=i^F (S^{\kappa^*}(P_{\mathbb{W}}))^{\nu}_{\mu}\psi^{\mu^* \kappa^*}(\bar{x})\,.\label{compa}
\end{equation}
Here $F=0$ for a bosonic field and $F=1$ for a fermionic one.
We have used $(S^\kappa(P_{\mathbb{W}})^{-1})^*=S^{\kappa^*}(P_{\mathbb{W}})$.
Compatibility of (\ref{compa}) with anticommutation relations for fermion fields implies $J^{-1}\neq J$ on the fermion sector. This is unlike the Tomita-Takesaki theory, where $J=J^{-1}$.
When ${\cal O}$ is formed by polynomials with even number of fermion fields,
\begin{equation}
{\cal O}=\sum \int dx_1...dx_l f^{(\nu_l \kappa_l)}_l(x_1,...,x_l) \psi^{\nu_l^1 \kappa_l^1}(x_1)...\psi^{\nu_l^l \kappa_l^l}(x_l)\,,
\end{equation}
and the components of $f_l^{(\nu_l \kappa_l)}$ belong to ${\cal S}_+(\mathbb{R}^{dl})$, we can rewrite (\ref{dicete}), using eq. (\ref{compa}) and LC, as $\langle 0 \vert J {\cal O} J^{-1} {\cal O} \, \vert 0\rangle \ge 0$.
\section{Final remarks}
The WRP is a positivity property of the Wightman functions at the Jost points. We think it might be possible to prove the Wightman axioms from the properties of analyticity, covariance and WRP for a series of functions defined exclusively at the Jost points. A proof of a reconstruction theorem in a similar fashion to the one for the euclidean axiomatic system \cite{os} is under construction. In this new "mixed" axiomatic system one would retain Lorentz covariance, but have some of the features of the euclidean system. For example, the nature of the distributions at coinciding points is not relevant, and the spectrum condition would follow from the other axioms.
We note this scheme has resemblances to some investigations in algebraic QFT, where it was found that parting from the adequately positioned wedge regions it is possible to reconstruct the whole theory \cite{reco}. Also, as mentioned in the introduction, it is a natural system in order to study whether the Renyi entanglement entropies for the vacuum state actually define field operators \cite{cc}. These Renyi entropies are only defined for spatially separated regions, giving place to correlators only for the Jost points. They may provide standard Wightman fields for a class of QFT defined algebraically.
\section*{Acknowledgments}
This work was partially supported by CONICET and Universidad Nacional de Cuyo, Argentina.
|
{
"timestamp": "2010-09-21T02:03:30",
"yymm": "1009",
"arxiv_id": "1009.3832",
"language": "en",
"url": "https://arxiv.org/abs/1009.3832"
}
|
\section{}
The nucleonic ground state as given in Eq.(1) is
\begin{equation}
|56,0^+\rangle\equiv|N_S\rangle
\end{equation}
where
\begin{equation}
|N_S\rangle=\frac{1}{\sqrt{2}}\left(\chi^\rho\phi^\rho+\chi^\lambda\phi^\lambda\right)\psi^0_{00S}
\end{equation}
where for example
\begin{equation}
\phi^\lambda_p=-\frac{1}{\sqrt{6}}\left(udu+duu-2uud\right)
\end{equation}
\begin{equation}
\phi^\rho_p=\frac{1}{\sqrt{2}}\left(udu-duu\right)
\end{equation}
and
\begin{equation}
\chi^\lambda_{\frac{1}{2}}=-\frac{1}{\sqrt{6}}\left(\uparrow\downarrow\uparrow+
\downarrow\uparrow\uparrow-2\uparrow\uparrow\downarrow\right)
\end{equation}
\begin{equation}
\chi^\rho_{\frac{1}{2}}=\frac{1}{\sqrt{2}}\left(\uparrow\downarrow\uparrow
-\downarrow\uparrow\uparrow\right)
\end{equation}
And
\begin{equation}
|70,2^+\rangle\equiv|N_D\rangle
\end{equation}
where
\begin{equation}
|N_D\rangle=\frac{1}{\sqrt{2}}\left\{\phi^\rho\left[\chi^S\psi^2_{2m\rho}\right]^{J=\frac{1}{2}}+
\phi^\lambda\left[\chi^S\psi^2_{2m\lambda}\right]^{J=\frac{1}{2}}\right\}
\end{equation}
where for example $(X=\rho \mbox{or} \lambda)$
\begin{equation}
\left[\chi^S\psi^2_{2mX}\right]^{J=\frac{1}{2}}_{J_Z=\frac{1}{2}}=
-\frac{1}{\sqrt{10}}\chi^S_{\frac{3}{2}}\psi^2_{2-1X}
+\frac{1}{\sqrt{5}}\chi^S_{\frac{1}{2}}\psi^2_{20X}
-\frac{3}{\sqrt{10}}\chi^S_{-\frac{1}{2}}\psi^2_{21X}
+\frac{2}{\sqrt{5}}\chi^S_{-\frac{3}{2}}\psi^2_{22X}
\end{equation}
For full details see Ref.\cite{c} and Ref.\cite{d}-\cite{i}.
|
{
"timestamp": "2012-01-24T02:02:10",
"yymm": "1009",
"arxiv_id": "1009.4276",
"language": "en",
"url": "https://arxiv.org/abs/1009.4276"
}
|
\section{Introduction}
\label{section_introduction}
\setcounter{equation}{0}
Stanley introduced
the notion of exponential structures,
that is,
a family of posets that have
the partition lattice $\Pi_n$ as the
archetype~\cite{Stanley_e_s, Stanley_EC_II}.
His original motivation was
to explain certain permutation phenomena.
His theory ended up inspiring
many mathematicians
to study the partition lattice and other exponential structures
from enumerative, representation theoretic and homological perspectives.
For example, Stanley studied the
$r$-divisible partition lattice $\Pi_{n}^{r}$
and computed its
M\"obius numbers~\cite{Stanley_e_s}.
Calderbank, Hanlon and Robinson~\cite{Calderbank_Hanlon_Robinson}
derived plethystic formulas in order
to determine
the character of the representation of the symmetric group on its top
homology, while
Wachs determined the homotopy type,
gave explicit bases for the homology and cohomology
and
studied the $\hbox{\german S}_{n}$ action on the top homology~\cite{Wachs_1}.
For the poset of partitions
with block sizes divisible by~$r$ and having
cardinality at least $rk$,
a similar array of questions have been considered
by
Bj\"orner and Wachs, Browdy,
Linusson, Sundaram and
Wachs~\cite{Bjorner_Wachs_nonpure, Browdy,
Linusson, Sundaram_applications_Hopf,
Wachs_2}.
Other related work can be found
in~\cite{Bjorner_Lovasz, Bjorner_Welker,
Gottlieb_Wachs, Sundaram_homology, Sundaram_Wachs, Welker},
as well as work of
Sagan~\cite{Sagan}, who showed certain examples of exponential structures
are $CL$-shellable.
In this paper
we extend Stanley's notion of exponential structures
to that
of {\em exponential Dowling
structures}.
The prototypical example is
the Dowling lattice~\cite{Dowling}.
It can most easily be viewed
as the intersection lattice of the
complex hyperplane arrangement
in~(\ref{equation_complex_hyperplane_arrangement}).
See Section~\ref{section_Dowling_lattice} for a review
of the Dowling
lattice.
In Section~\ref{section_Dowling_exponential_structures}
we introduce exponential Dowling structures.
We derive the compositional formula
for exponential Dowling structures
analogous
to Stanley's theorem
on the compositional formula
for exponential structures~\cite{Stanley_e_s}.
As an application, we give the generating function for the
M\"obius numbers of an exponential Dowling structure.
An important method to generate new exponential Dowling structures
from old ones is given in
Example~\ref{example_r_k}.
Loosely speaking,
in this new structure
an
$r$-divisibility condition holds for the ``non-zero blocks''
and the cardinality of the ``zero block'' satisfies
the more general condition of being greater
than or equal to $k$ and congruent to $k$ modulo $r$.
We will return to many important special cases of this example
in later sections.
In Section~\ref{section_Mobius_function}
we consider restricted forms
of both exponential and
exponential Dowling structures.
In the case the exponential Dowling structure
is restricted to elements whose
type satisfies a semigroup condition,
the generating function for the M\"obius function of this poset
is particularly elegant.
See Corollary~\ref{corollary_semigroup}
and Proposition~\ref{proposition_M}.
When the blocks have even size, the
generating function is nicely expressed
in terms of the hyperbolic functions.
See Corollary~\ref{corollary_hyperbolic}.
In Section~\ref{section_permutations}
we continue to develop the connection
between permutations and structures
first studied by Stanley
in the case of exponential structures.
In particular we consider
the lattice
$\Pi_{m}^{r,j}$,
an extension
of the
$r$-divisible partition lattice $\Pi_{m}^{r}$.
In Section~\ref{section_EL} we verify that
Wachs' $EL$-labeling of the $r$-divisible
partition lattice $\Pi_{m}^{r}$ naturally extends to
the new lattice $\Pi_{m}^{r,j}$.
We end with remarks and open questions regarding
further exponential Dowling structures and their connections
with permutation statistics.
\section{The Dowling lattice}
\label{section_Dowling_lattice}
\setcounter{equation}{0}
Let $G$ be a finite group of order $s$.
The {\em Dowling lattice} $L_{n}(G) = L_{n}$
has the following combinatorial description.
For the original formulation,
see Dowling's paper~\cite{Dowling}.
Define an {\em enriched block} $\widetilde{B} = (B,f)$ to be
a non-empty subset $B$ of $\{1,\ldots,n\}$ and
a function $f : B \longrightarrow G$.
Two enriched blocks $\widetilde{B} = (B,f)$ and
$\widetilde{C} = (C,g)$ are said to be equivalent if
$B = C$ and the functions $f$ and $g$ differ only by a multiplicative
scalar, that is,
there exists $\alpha \in G$ such that
$f(b) = g(b) \cdot \alpha$ for all $b$ in $B$.
Hence there are only $s^{|B| - 1}$ possible ways to enrich a
non-empty set~$B$, up to equivalence.
Let $\widetilde{B} = (B,f)$ and $\widetilde{C} = (C,g)$
be two disjoint enriched blocks and
let~$\alpha$ be an element in~$G$.
We can define a function~$h$ on the block $B \cup C$
by
$$ h(b) = \left\{ \begin{array}{c c l}
f(b) & \mbox{ if } & b \in B, \\
\alpha \cdot g(b) & \mbox{ if } & b \in C.
\end{array} \right. $$
Since the group element
$\alpha$ can be chosen in $s$ possible ways,
there are $s$ possible
ways to merge two enriched blocks.
For $E$ a subset of $\{1, \ldots, n\}$,
an {\em enriched partition}
$\widetilde{\pi} = \{\widetilde{B}_{1}, \ldots, \widetilde{B}_{m}\}$
on the set $E$
is a partition $\pi = \{B_{1}, \ldots, B_{m}\}$ of $E$,
where each block $B_i$
is enriched with a function $f_{i}$.
The elements of the Dowling lattice~$L_{n}$
are the collection
$$ L_{n}
=
\left\{ (\widetilde{\pi}, Z)
\:\: : \:\: Z \subseteq \{1, \ldots, n\}
\mbox{ and }
\widetilde{\pi}
\mbox{ is an enriched partition of }
\overline{Z} = \{1, \ldots, n\} - Z \right\} . $$
The set $Z$ is called the {\em zero block.}
Define the cover relation on
$L_{n}$ by the following two relations:
$$ \begin{array}{r c l}
(\{\widetilde{B}_1, \widetilde{B}_2, \ldots, \widetilde{B}_m\}, Z)
& \prec &
(\{\widetilde{B}_2, \ldots, \widetilde{B}_m\}, Z \cup B_1) , \\
(\{\widetilde{B}_1, \widetilde{B}_2, \ldots, \widetilde{B}_m\}, Z)
& \prec &
(\{\widetilde{B}_1 \cup \widetilde{B}_2, \ldots, \widetilde{B}_m\}, Z) .
\end{array} $$
The first relation says that a block is allowed
to merge with the zero set.
The second relation says that two blocks are allowed
to be merged together.
The minimal element $\hat{0}$ corresponds to the partition
having all singleton blocks and empty zero block,
while the maximal element $\hat{1}$ corresponds to the partition
where all the elements lie in the zero block.
Observe that the Dowling lattice $L_{n}$ is
graded of rank $n$.
When the group $G$ is the cyclic group of order $s$,
that is, $\hbox{\Cp Z}_{s}$,
the Dowling lattice has the following geometric description.
Let $\zeta$ be a
primitive $s$th root of unity.
The {\em Dowling lattice}
$L_{n}(\hbox{\Cp Z}_{s})$ is the intersection lattice of
the complex hyperplane arrangement
\begin{equation}
\left\{
\begin{array}{c c c l}
z_i & = & \zeta^{h} \cdot z_j &
\mbox{ for } 1 \leq i < j \leq n
\mbox{ and } 0 \leq h \leq s-1, \\
z_i & = & 0 & \mbox{ for } 1 \leq i \leq n ,
\end{array}
\right.
\label{equation_complex_hyperplane_arrangement}
\end{equation}
that is, the collection of all possible intersections
of these hyperplanes ordered by reverse inclusion.
In the notation we will suppress
the Dowling lattice's
dependency on the group $G$.
Only the order~$s$ of the group will
matter in this paper.
In Section~\ref{section_permutations}
the order $s$ will be specialized to the value $1$.
For an element $x = (\widetilde{\pi}, Z)$
in the Dowling lattice $L_{n}$,
define the {\em type} of $x$ to be
$(b; a_1, a_2, \ldots, a_n)$,
where
$a_i$ is the number of blocks in
$\widetilde{\pi}$
of size $i$ in $x$ and
$b$ is the size of the zero block $Z$.
Observe
that
the interval $[x,\hat{1}]$ in the Dowling lattice
is isomorphic to $L_{n-\rho(x)}$
where $\rho$ denotes the rank function.
Moreover,
the interval
$[\hat{0},x]$
is isomorphic to
$L_b \times \Pi_1^{a_1} \times \cdots \times \Pi_n^{a_n}$,
where
$(b; a_1, a_2, \ldots, a_n)$ is the type of $x$
and
$\Pi_j^{a_j}$ denotes the Cartesian product of
$a_j$ copies of the partition lattice on $j$ elements.
\begin{lemma}
In the Dowling lattice
$L_n$ there are
$$\frac{s^n \cdot n!}
{s^b \cdot b! \cdot (s \cdot 1!)^{a_1} \cdot a_1!
\cdot (s \cdot 2!)^{a_2} \cdot a_2!
\cdots (s \cdot n!)^{a_n} \cdot a_n!}
$$
elements of type
$(b; a_1, a_2, \ldots, a_n)$.
\label{lemma_count_type}
\end{lemma}
\begin{proof}
For an element
of type $(b;a_1, a_2, \ldots, a_n)$
in the Dowling lattice $L_n$
we can
choose the $b$ elements in the zero-set in ${n \choose b}$ ways.
The underlying partition on the remaining $n-b$ elements can be
chosen in
$$ \displaystyle
{n-b \choose
\underbrace{1, \ldots, 1}_{a_1},
\underbrace{2, \ldots, 2}_{a_2},
\ldots,
\underbrace{n, \ldots, n}_{a_n}}
\cdot
\frac{1}{a_1 ! \cdot a_2! \cdots a_n!} $$
ways.
For a block of size $k$
there are $s^{k-1}$ signings,
so the result follows.
\end{proof}
\section{Dowling exponential structures}
\label{section_Dowling_exponential_structures}
\setcounter{equation}{0}
Stanley introduced the notion of an exponential
structure. See~\cite{Stanley_e_s} and~\cite[Section 5.5]{Stanley_EC_II}.
\begin{definition}
An {\em exponential structure}
${\bf Q} = (Q_{1}, Q_{2}, \ldots)$
is a sequence of posets
such that
\begin{itemize}
\item[(E1)]
The poset $Q_{n}$ has a unique maximal element $\hat{1}$
and every maximal chain in $Q_{n}$ contains
$n$ elements.
\item[(E2)]
For an element $x$ in $Q_n$ of rank $k$,
the interval $[x,\hat{1}]$ is isomorphic to the
partition lattice on $n-k$ elements, $\Pi_{n-k}$.
\item[(E3)]
The lower order ideal generated by $x \in Q_n$
is isomorphic to
$Q_1^{a_1} \times \cdots \times Q_n^{a_n}$.
We call $(a_1, \ldots, a_n)$ the {\em type}
of $x$.
\item[(E4)]
The poset $Q_n$ has $M(n)$ minimal elements.
The sequence $(M(1), M(2), \ldots)$ is called the
{\em denominator sequence}.
\end{itemize}
\end{definition}
Analogous to the definition of an exponential structure,
we introduce
the notion of an exponential Dowling structure.
\begin{definition}
An {\em exponential Dowling structure}
${\bf R} = (R_{0}, R_{1}, \ldots)$
associated to an exponential structure
${\bf Q} = (Q_1, Q_2, \ldots)$
is a sequence of posets
such that
\begin{itemize}
\item[(D1)]
The poset $R_{n}$ has a unique maximal element $\hat{1}$
and every maximal chain in $R_{n}$ contains
$n+1$ elements.
\item[(D2)]
For an element $x \in R_n$,
$[x,\hat{1}] \cong L_{n - \rho(x)}$.
\item[(D3)]
Each element $x$ in $R_{n}$
has a {\em type} $(b; a_1, \ldots, a_n)$ assigned
such that
the lower order ideal generated by $x$ in $R_{n}$
is isomorphic to
$R_b \times Q_1^{a_1} \times \cdots \times Q_n^{a_n}$.
\item[(D4)]
The poset $R_n$ has $N(n)$ minimal elements.
The sequence $(N(0), N(1), \ldots)$ is called the
{\em denominator sequence}.
\end{itemize}
\end{definition}
Observe that $R_{0}$ is the one element poset and thus $N(0) = 1$.
Also note if $x$ has type
$(b; a_1, \ldots, a_n)$
then
$a_{n-b+1} = \cdots = a_{n} = 0$.
Condition {\it (D3)} has a different formulation than
condition {\it (E3)}. The reason is that there could be
cases where the lower order ideal generated by an element
does not factor uniquely into the form
$R_b \times Q_1^{a_1} \times \cdots \times Q_n^{a_n}$.
However,
in the examples we consider the type of an element will be clear.
\begin{proposition}
Let
${\bf R} = (R_{0}, R_{1}, \ldots)$
be an exponential Dowling structure
with associated exponential structure
${\bf Q} = (Q_1, Q_2, \ldots)$.
The number of elements in $R_{n}$ of
type $(b;a_1, \ldots, a_n)$
is given by
\begin{equation}
\label{equation_pairs}
\frac{N(n) \cdot s^n \cdot n!}
{N(b) \cdot s^b \cdot b! \cdot
(M(1) \cdot s \cdot 1!)^{a_1} \cdot a_1!
\cdots
(M(n) \cdot s \cdot n!)^{a_n} \cdot a_n!
}
\end{equation}
\end{proposition}
\begin{proof}
Consider pairs of elements $(x,y)$
satisfying
$y \leq x$, where
the element $x$ has type $(b;a_1, \ldots, a_n)$
and $y$ is a minimal element of~$R_n$.
We count such pairs in two ways.
The number of minimal elements $y \in R_n$ is given
by~$N(n)$. Given such a minimal element $y$, the number
of $x$'s is given in Lemma~\ref{lemma_count_type}.
Alternatively,
we wish to count the number of $x$'s.
The number of $y$'s given an element $x$ equals the number
of minimal elements occurring in the lower order ideal generated by
$x$.
This equals the number of minimal elements in
$R_b \times Q_1^{a_1} \times \cdots \times Q_n^{a_n}$,
that is,
$N(b) \cdot M(1)^{a_1} \cdots M(n)^{a_n}$.
Thus the answer is as in~(\ref{equation_pairs}).
\end{proof}
Let ${\bf Q}$ be an exponential structure
and $r$ a positive integer.
Stanley defines the exponential structure ${\bf Q}^{(r)}$
by letting $Q^{(r)}_{n}$ be the subposet $Q_{r n}$ of all
elements $x$ of type $(a_{1}, a_{2}, \ldots)$ where
$a_{i} = 0$ unless $r$ divides $i$.
The denominator sequence of ${\bf Q}^{(r)}$ is
given by
$$ M^{(r)}(n) = \frac{M(r n) \cdot (r n)!}
{M(r)^{n} \cdot n! \cdot r!^{n}} . $$
\begin{example}
{\rm
Let ${\bf R}$ be an exponential Dowling structure associated with the
exponential structure~${\bf Q}$.
Let $r$ be a positive integer and $k$ a non-negative integer.
Let $R^{(r,k)}_{n}$ be the subposet of $R_{r n + k}$ consisting of all
elements $x$ of type $(b; a_{1}, a_{2}, \ldots)$
such that
$b \geq k$, $b \equiv k \bmod r$
and
$a_{i} = 0$ unless $r$ divides $i$.
Then ${\bf R}^{(r,k)} = (R^{(r,k)}_{0}, R^{(r,k)}_{1}, \ldots)$
is an exponential Dowling structure associated with the
exponential structure ${\bf Q}^{(r)}$.
The minimal elements of $R^{(r,k)}_{n}$
are the elements of $R_{r n + k}$ having
types given by $b = k$, $a_{r} = n$
and $a_{i} = 0$ for $i \neq n$.
The denominator sequence of ${\bf R}^{(r,k)}$
is given by
$$ N^{(r,k)}(n) =
\frac{N(r n + k) \cdot (r n + k)! \cdot s^{(r-1) \cdot n}}
{N(k) \cdot k!
\cdot
M(r)^{n} \cdot r!^{n} \cdot n! } . $$
}
\label{example_r_k}
\end{example}
Stanley~\cite{Stanley_e_s} proved the following structure theorem.
\begin{theorem}
(The Compositional Formula for Exponential Structures)
Let ${\bf Q} = (Q_{1}, Q_{1}, \ldots)$ be an exponential structure
with denominator sequence
$(M(1), M(2), \ldots)$.
Let
$f: \hbox{\Cp P} \rightarrow \hbox{\Cp C}$
and
$g: \hbox{\Cp N} \rightarrow \hbox{\Cp C}$
be given functions such that $g(0) = 1$.
Define the function
$h : \hbox{\Cp N} \rightarrow \hbox{\Cp C}$ by
\begin{equation}
h(n) = \sum_{x \in Q_{n}} f(1)^{a_1} \cdot f(2)^{a_2} \cdots
f(n)^{a_n} \cdot g(a_1 + \cdots + a_n),
\label{equation_f_g_h}
\end{equation}
for $n \geq 1$, where
$\hbox{\rm type}(x) = (a_1, \ldots, a_n)$,
and $h(0) = 1$.
Define the formal power series $F, G, K \in \hbox{\Cp C}[[x]]$ by
\begin{eqnarray*}
F(x) & = & \sum_{n \geq 1} f(n) \cdot \frac{x^n}{M(n) \cdot n!} \\
G(x) & = & \sum_{n \geq 0} g(n) \cdot \frac{x^n}{n!} \\
H(x) & = & \sum_{n \geq 0} h(n) \cdot \frac{x^n}{M(n) \cdot n!} .
\end{eqnarray*}
Then
$H(x) = G(F(x))$.
\label{theorem_Stanley_compositional_formula}
\end{theorem}
For Dowling structures we have an analogous theorem.
\begin{theorem}
(The Compositional Formula for Exponential Dowling Structures)
Let ${\bf R} = (R_0, R_1, \ldots)$ be an exponential Dowling structure
with denominator sequence
$(N(0), N(1), \ldots)$ and
associated exponential structure
${\bf Q} = (Q_1, Q_2, \ldots)$ with
denominator sequence
$(M(1), M(2), \ldots)$.
Let
$f: \hbox{\Cp P} \rightarrow \hbox{\Cp C}$,
$g: \hbox{\Cp N} \rightarrow \hbox{\Cp C}$
and
$k: \hbox{\Cp N} \rightarrow \hbox{\Cp C}$
be given functions.
Define the function
$h : \hbox{\Cp N} \rightarrow \hbox{\Cp C}$ by
\begin{equation}
h(n) = \sum_{x \in R_n} k(b) \cdot f(1)^{a_1} \cdot f(2)^{a_2} \cdots
f(n)^{a_n} \cdot g(a_1 + \cdots + a_n),
\label{equation_f_g_h_k}
\end{equation}
for $n \geq 0$, where
$\hbox{\rm type}(x) = (b;a_1, \ldots, a_n)$.
Define the formal power series $F, G, K, H \in \hbox{\Cp C}[[x]]$ by
\begin{eqnarray*}
F(x) & = & \sum_{n \geq 1} f(n) \cdot \frac{x^n}{M(n) \cdot n!}\\
G(x) & = & \sum_{n \geq 0} g(n) \cdot \frac{x^n}{n!} \\
K(x) & = & \sum_{n \geq 0} k(n) \cdot \frac{x^n}{N(n) \cdot n!} \\
H(x) & = & \sum_{n \geq 0} h(n) \cdot \frac{x^n}{N(n) \cdot n!}
\end{eqnarray*}
Then
$H(x) = K(x) \cdot G(1/s \cdot F(s \cdot x))$.
\label{theorem_compositional_formula}
\end{theorem}
\begin{proof_special}
By applying the compositional formula of generating functions
to the (exponential) generating functions
$1/s \cdot F(s x) = \sum_{n\geq 1} f(n)/(M(n) \cdot s) \cdot (s x)^{n}/n!$
and $G(x)$,
we obtain
\begin{eqnarray*}
G(1/s \cdot F(s x))
& = &
\sum_{n \geq 0}
\sum_{\pi \in \Pi_{n}}
\prod_{B \in \pi}
\frac{f(|B|)}{M(|B|) \cdot s}
\cdot
g(|\pi|)
\cdot
\frac{(s x)^{n}}{n!}\\
& = &
\sum_{n \geq 0}
\sum_{1 \cdot a_1 + \cdots + n \cdot a_n = n}
\frac{s^{n} \cdot n!}
{(M(1) \cdot s \cdot 1!)^{a_1} \cdot a_1!
\cdots
(M(n) \cdot s \cdot n!)^{a_n} \cdot a_n!}
\\
& & \hspace{50 mm}
\cdot
f(1)^{a_1}
\cdots
f(n)^{a_n}
\cdot
g(a_1 + \cdots + a_n)
\cdot
\frac{x^{n}}{n!} .
\end{eqnarray*}
Multiply this identity with
the (exponential) generating function
$K(x) = \sum_{n \geq 0} k(n)/N(n) \cdot x^{n}/n!$
to obtain
\begin{eqnarray*}
& &
K(x) \cdot G(1/s \cdot F(s x)) \\
& = &
\sum_{n \geq 0}
\sum_{b = 0}^{n}
\sum_{1 \cdot a_1 + \cdots + (n-b) \cdot a_{n-b} = n-b}
{n \choose b}
\cdot
\frac{k(b)}{N(b)}
\\
& & \hspace{35 mm}
\cdot
\frac{s^{n-b} \cdot (n-b)!}
{(M(1) \cdot s \cdot 1!)^{a_1} \cdot a_1!
\cdots
(M(n-b) \cdot s \cdot (n-b)!)^{a_{n-b}} \cdot a_{n-b}!}
\\
& & \hspace{35 mm}
\cdot
f(1)^{a_1}
\cdots
f(n)^{a_n}
\cdot
g(a_{1} + \cdots + a_{n-b})
\cdot
\frac{x^{n}}{n!} \\
& = &
\sum_{n \geq 0}
\sum_{b = 0}^{n}
\sum_{1 \cdot a_1 + \cdots + n \cdot a_n = n-b}
\frac{s^{n} \cdot n!}
{ N(b) \cdot s^{b} \cdot b!
\cdot
(M(1) \cdot s \cdot 1!)^{a_1} \cdot a_1!
\cdots
(M(n) \cdot s \cdot n!)^{a_n} \cdot a_n!}
\\
& & \hspace{50 mm}
\cdot
k(b)
\cdot
f(1)^{a_1}
\cdots
f(n)^{a_n}
\cdot
g(a_1 + \cdots + a_n)
\cdot
\frac{x^{n}}{n!} \\
\hspace{10 mm}
& = &
\sum_{n \geq 0}
\sum_{x \in R_{n}}
k(b)
\cdot
{f(1)}^{a_1}
\cdots
{f(n)}^{a_n}
\cdot
g(a_1 + \cdots + a_n)
\cdot
\frac{x^{n}}{N(n) \cdot n!}
= H(x) .
\hspace{30 mm} \mbox{$\Box$}\vspace{\baselineskip}
\end{eqnarray*}
\end{proof_special}
\begin{example}
{\rm
Let ${\bf R} = (R_0, R_1, \ldots)$ be an exponential Dowling structure
with denominator sequence
$(N(0), N(1), \ldots)$ and
associated exponential structure
${\bf Q} = (Q_1, Q_2, \ldots)$ with
denominator sequence
$(M(1), M(2), \ldots)$.
Let $V_{n}(t)$ be the polynomial
$$ V_{n}(t)
=
\sum_{x \in Q_{n}} t^{\rho(x,\hat{1})} . $$
In Example~5.5.6 in~\cite{Stanley_EC_II}
Stanley obtains the generating function
$$ \sum_{n \geq 0} V_{n}(t) \cdot \frac{x^n}{M(n) \cdot n!}
=
\exp\left(
\sum_{n \geq 1} \frac{x^n}{M(n) \cdot n!}
\right)^{t} , $$
by setting $f(n) = 1$ and $g(n) = t^{n}$
in Theorem~\ref{theorem_Stanley_compositional_formula}.
Similarly, defining $W_{n}(t)$ by
$$ W_{n}(t)
=
\sum_{x \in R_{n}} t^{\rho(x,\hat{1})} , $$
we obtain
$$ \sum_{n \geq 0} W_{n}(t) \cdot \frac{x^n}{N(n) \cdot n!}
=
\left(
\sum_{n \geq 0} \frac{x^n}{N(n) \cdot n!}
\right)
\cdot
{\exp\left(
\sum_{n \geq 1} \frac{(s \cdot x)^n}{M(n) \cdot n!}
\right)}^{\frac{t}{s}} , $$
by setting $f(n) = 1$,
$g(n) = t^{n}$ and $k(n) = 1$
in Theorem~\ref{theorem_compositional_formula}.
}
\end{example}
\begin{corollary}
Let ${\bf R} = (R_0, R_1, \ldots)$ be an exponential Dowling structure
with denominator sequence
$(N(0), N(1), \ldots)$ and
associated exponential structure
${\bf Q} = (Q_1, Q_2, \ldots)$ with
denominator sequence
$(M(1), M(2), \ldots)$.
Then the M\"obius function of
the posets
$Q_{n} \cup \{\hat{0}\}$,
respectively
$R_{n} \cup \{\hat{0}\}$,
has the generating function:
\begin{eqnarray}
\sum_{n \geq 1}
\mu(Q_{n} \cup \{\hat{0}\})
\cdot
\frac{x^{n}}{M(n) \cdot n!}
& = &
-
\ln\left(
\sum_{n \geq 0}
\frac{x^{n}}{M(n) \cdot n!}
\right) ,
\label{equation_one_Mobius} \\
\sum_{n \geq 0}
\mu(R_{n} \cup \{\hat{0}\})
\cdot
\frac{x^{n}}{N(n) \cdot n!}
& = &
-
\left(
\sum_{n \geq 0}
\frac{x^{n}}{N(n) \cdot n!}
\right)
\cdot
\left(
\sum_{n \geq 0}
\frac{(s \cdot x)^{n}}{M(n) \cdot n!}
\right)^{-1/s} .
\label{equation_two_Mobius}
\end{eqnarray}
\label{corollary_Mobius}
\end{corollary}
\begin{proof}
Setting $f(n) = 1$ and $g(n) = (-1)^{n-1} \cdot (n-1)!$ and using
that
$$
\mu(Q_{n} \cup \{\hat{0}\})
=
-
\sum_{x \in Q_{n}}
\mu(x,\hat{1})
=
-
\sum_{x \in Q_{n}}
g(a_1 + \cdots + a_n) ,
$$
equation~(\ref{equation_one_Mobius}) follows
by Theorem~\ref{theorem_Stanley_compositional_formula}.
Similarly, to prove the second
identity~(\ref{equation_two_Mobius}),
redefine $g(n)$ to
be the M\"obius function of the Dowling lattice
$L_{n}$ of rank $n$, that is,
$$ g(n) = (-1)^{n} \cdot 1 \cdot (s+1) \cdot (2 \cdot s + 1)
\cdots ((n-1) \cdot s + 1) . $$
By the binomial theorem we have
$\sum_{n \geq 0} g(n) \frac{x^{n}}{n!}
=
(1 + s \cdot x)^{-1/s}$.
Moreover, let $k(n) = 1$.
Using the recurrence
$$
\mu(R_{n} \cup \{\hat{0}\})
=
-
\sum_{x \in R_{n}}
\mu(x,\hat{1})
=
\sum_{x \in R_{n}}
g(a_1 + \cdots + a_n) ,
$$
and Theorem~\ref{theorem_compositional_formula},
the result follows.
\end{proof}
\section{The M\"obius function of restricted structures}
\label{section_Mobius_function}
\setcounter{equation}{0}
Let $I$ be a subset of the positive integers $\hbox{\Cp P}$.
For an exponential structure
${\bf Q} = (Q_{1}, Q_{2}, \ldots)$
define the {\em restricted poset} $Q_{n}^{I}$ to be all elements $x$
in $Q_{n}$ whose type
$(a_{1}, \ldots, a_{n})$
satisfies $a_{i} > 0$ implies $i \in I$.
For $n \in I$ let $\mu_{I}(n)$ denote the
M\"obius function of the poset $Q_{n}^{I}$ with a $\hat{0}$ adjoined,
that is, the poset
$Q_{n}^{I} \cup \{\hat{0}\}$.
For $n \not\in I$ let $\mu_{I}(n) = 0$.
For any positive integer $n$ define
$$ m_{n} = \sum_{x \in Q_{n}^{I} \cup \{\hat{0}\}} \mu_{I}(\hat{0},x)
= 1 + \sum_{x \in Q_{n}^{I}} \mu_{I}(\hat{0},x) . $$
Observe that for $n \in I$ we have
that $Q_{n}^{I}$ has a maximal element and
hence $m_{n} = 0$.
Especially for $n \not\in I$ we have
the expansion
$$ m_{n} = 1
-
\sum_{\sum_{i \in I} i \cdot a_{i} = n}
(-1)^{\sum_{i \in I} a_{i}}
\frac{M(n) \cdot n!}
{(M(1) \cdot 1!)^{a_1} \cdot a_1!
\cdots
(M(n) \cdot n!)^{a_n} \cdot a_n!}
\cdot
\mu_{I}(1)^{a_1} \cdots \mu_{I}(n)^{a_n} . $$
The following theorem
was inspired by
work of Linusson~\cite{Linusson}.
\begin{theorem}
$$
\sum_{i \in I} \mu_{I}(i) \frac{x^{i}}{M(i) \cdot i!}
=
-
\ln \left(
\sum_{n \geq 0} \frac{x^{n}}{M(n) \cdot n!}
-
\sum_{n \not\in I} m_{n} \cdot \frac{x^{n}}{M(n) \cdot n!}
\right) .
$$
\label{theorem_I}
\end{theorem}
\begin{proof}
Expand the product
\begin{eqnarray*}
- 1 + \prod_{i \in I} \exp\left(-\mu_{I}(i) \frac{x^{i}}{M(i) \cdot i!}
\right)
& = &
- 1 + \prod_{i \in I}
\left(1 - \mu_{I}(i) \frac{x^{i}}{M(i) \cdot i!}
+ \frac{1}{2} \cdot
\left(\mu_{I}(i) \frac{x^{i}}{M(i) \cdot i!}\right)^{2}
- \cdots \right) \\
& = &
\sum_{n \geq 1}
\sum_{\sum_{i \in I} i \cdot a_{i} = n}
\prod_{i \in I}
\frac{1}{a_{i}!} \cdot
\left(- \mu_{I}(i) \frac{x^{i}}{M(i) \cdot i!} \right)^{a_i} \\
& = &
\sum_{n \geq 1}
\sum_{\sum_{i \in I} i \cdot a_{i} = n}
(-1)^{\sum_{i \in I} a_{i}}
\cdot
\left( \mu_{I}(1)^{a_1} \cdots \mu_{I}(n)^{a_n} \right) \\
& \cdot &
\frac{M(n) \cdot n!}
{(M(1) \cdot 1!)^{a_1} \cdot a_{1}! \cdots
(M(n) \cdot n!)^{a_n} \cdot a_{n}!}
\frac{x^{n}}{M(n) \cdot n!} \\
& = &
\sum_{n \geq 1} (1 - m_{n}) \cdot \frac{x^{n}}{M(n) \cdot n!} \\
& = &
\sum_{n \geq 1} \frac{x^{n}}{M(n) \cdot n!}
-
\sum_{n \not\in I} m_{n} \cdot \frac{x^{n}}{M(n) \cdot n!} .
\end{eqnarray*}
The result now follows.
\end{proof}
Let $I$ be a subset of the positive integers $\hbox{\Cp P}$
and $J$ be a subset of the natural numbers $\hbox{\Cp N}$.
For an exponential Dowling structure
${\bf R} = (R_{0}, R_{1}, \ldots)$,
define the {\em restricted poset} $R_{n}^{I,J}$ to be all elements $x$
in $R_{n}$ whose type
$(b; a_{1}, \ldots, a_{n})$
satisfies
$b \in J$
and
$a_{i} > 0$ implies $i \in I$.
For $n \in J$ define $\mu_{I,J}(n)$ to be the
M\"obius function of the poset
$R_{n}^{I,J} \cup \{\hat{0}\}$, that is,
$R_{n}^{I,J}$ with a minimal element $\hat{0}$ adjoined.
For $n \not\in J$ let $\mu_{I,J}(n) = 0$.
Define for any non-negative integer $n$
$$ p_{n} = \sum_{x \in R_{n}^{I,J} \cup \{\hat{0}\}} \mu_{I,J}(\hat{0},x)
= 1 + \sum_{x \in R_{n}^{I,J}} \mu_{I,J}(\hat{0},x) . $$
Observe that for $n \in J$ we have $p_{n} = 0$
since the poset $R_{n}^{I,J} \cup \{\hat{0}\}$ has a maximal element.
For $n \not\in J$ we have
\begin{eqnarray*}
p_{n}
& = &
1
+
\sum_{(b;a_{1}, \ldots, a_{n})}
(-1)^{a_1 + \cdots + a_{n}}
\cdot
\frac{N(n) \cdot s^{n} \cdot n!}
{N(b) \cdot s^{b} \cdot b!
\cdot
(M(1) \cdot s \cdot 1!)^{a_1} \cdot a_1!
\cdots
(M(n) \cdot s \cdot n!)^{a_n} \cdot a_n!} \\
& &
\hspace*{50 mm}
\cdot
\mu_{I,J}(b)
\cdot
\mu_{I}(1)^{a_1} \cdots \mu_{I}(n)^{a_n} ,
\end{eqnarray*}
where the sum is over all types
$(b; a_{1}, \ldots, a_{n})$ where
$b \in J$,
$a_{i} > 0$ implies $i \in I$,
and
$b + \sum_{i \in I} i \cdot a_{i} = n$.
\begin{theorem}
\begin{eqnarray*}
\sum_{b \in J} \mu_{I,J}(b) \cdot
\frac{x^{b}}{N(b) \cdot b!}
& = &
\frac{\displaystyle
-
\sum_{n \geq 0}
\frac{x^{n}}{N(n) \cdot n!}
+
\sum_{n \not\in J}
p_{n} \cdot
\frac{x^{n}}{N(n) \cdot n!}
}
{\displaystyle
\left(
\sum_{n \geq 0} \frac{(s \cdot x)^{n}}{M(n) \cdot n!}
-
\sum_{n \not\in I} m_{n} \cdot \frac{(s \cdot x)^{n}}{M(n) \cdot n!}
\right)^{1/s}}
\end{eqnarray*}
\label{theorem_I_J}
\end{theorem}
\begin{proof}
By similar reasoning as in the proof
of Theorem~\ref{theorem_I}, we have
$$
\exp\left( -
\sum_{i \in I} \mu_{I}(i) \cdot
\frac{x^{i}}{M(i) \cdot s \cdot i!}
\right)
=
\sum_{(a_{1}, \ldots, a_{n})}
\prod_{i \in I}
\frac{1}{a_{i}!} \cdot
\left(
- \frac{\mu_{I}(i) \cdot x^{i}}{M(i) \cdot s \cdot i!}
\right)^{a_{i}} .
$$
Multiplying with
$\sum_{b \in J} \mu_{I,J}(b) \cdot
\frac{x^{b}}{N(b) \cdot s^{b} \cdot b!}$
and expanding, we obtain
\begin{eqnarray*}
& &
\left(\sum_{b \in J} \mu_{I,J}(b) \cdot
\frac{x^{b}}{N(b) \cdot s^{b} \cdot b!}
\right)
\cdot
\exp\left( -
\sum_{i \in I} \mu_{I}(i) \cdot
\frac{x^{i}}{M(i) \cdot s \cdot i!}
\right) \\
& = &
\sum_{n \geq 0}
\sum_{b \in J}
\sum_{(a_1, \ldots, a_{n-b})}
\left( \frac{ \mu_{I,J}(b) \cdot x^{b}}{N(b) \cdot s^{b} \cdot b!}
\right)
\cdot
\prod_{i \in I}
\frac{1}{a_{i}!} \cdot
\left(
- \frac{\mu_{I}(i) \cdot x^{i}}{M(i) \cdot s \cdot i!}
\right)^{a_{i}} \\
& = &
\sum_{n \geq 0}
(p_{n} - 1) \cdot
\frac{x^{n}}{N(n) \cdot s^{n} \cdot n!}
\end{eqnarray*}
Substituting $x \longmapsto s x$, we can rewrite this
equation as
\begin{eqnarray*}
& &
\sum_{b \in J} \mu_{I,J}(b) \cdot
\frac{x^{b}}{N(b) \cdot b!} \\
& = &
\left(
\sum_{n \geq 0}
(p_{n} - 1) \cdot
\frac{x^{n}}{N(n) \cdot n!}
\right)
\cdot
\exp\left( \frac{1}{s} \cdot
\sum_{i \in I} \mu_{I}(i) \cdot
\frac{(s \cdot x)^{i}}{M(i) \cdot i!}
\right) \\
& = &
\left(
\sum_{n \geq 0}
(p_{n} - 1) \cdot
\frac{x^{n}}{N(n) \cdot n!}
\right)
\cdot
\exp\left(
\sum_{i \in I} \mu_{I}(i) \cdot
\frac{(s \cdot x)^{i}}{M(i) \cdot i!}
\right)^{1/s}
\end{eqnarray*}
By applying
Theorem~\ref{theorem_I}
to the last term, the result follows.
\end{proof}
As a corollary to
Theorems~\ref{theorem_I}
and~\ref{theorem_I_J},
we have
\begin{corollary}
Let $I \subseteq \hbox{\Cp P}$ be a semigroup
and $J \subseteq \hbox{\Cp N}$ such that
$I + J \subseteq J$.
Then the M\"obius function of
the restricted poset
$Q_{n}^{I} \cup \{\hat{0}\}$
and
$R_{n}^{I,J} \cup \{\hat{0}\}$
respectively
has the generating function:
\begin{eqnarray}
\sum_{n \in I}
\mu_{I}(n)
\cdot
\frac{x^{n}}{M(n) \cdot n!}
& = &
-
\ln\left(
\sum_{n \in I \cup \{0\}}
\frac{x^{n}}{M(n) \cdot n!}
\right) ,
\label{equation_one_corollary_semigroup} \\
\sum_{n \in J}
\mu_{I,J}(n)
\cdot
\frac{x^{n}}{N(n) \cdot n!}
& = &
-
\left(
\sum_{n \in J}
\frac{x^{n}}{N(n) \cdot n!}
\right)
\cdot
\left(
\sum_{n \in I \cup \{0\}}
\frac{(s \cdot x)^{n}}{M(n) \cdot n!}
\right)^{-1/s} .
\label{equation_two_corollary_semigroup}
\end{eqnarray}
\label{corollary_semigroup}
\end{corollary}
\begin{proof}
The semigroup condition implies that
the poset $Q_{n}^{I}$ is empty when $n \not\in I$
and hence $m_{n} = 1$. Similarly,
the other condition implies that
the poset $R_{n}^{I}$ is empty when $n \not\in J$,
so $p_{n} = 1$.
\end{proof}
Let ${\bf D}$ be the Dowling structure consisting of
the Dowling lattices, that is,
${\bf D} = (L_0, L_1, \ldots)$.
\begin{proposition}
For the exponential Dowling structure
${\bf D}^{(r,k)}$ we have
$$ \sum_{n \geq 0}
\mu\left(D_{n}^{(r,k)} \cup \{\hat{0}\}\right)
\cdot
\frac{x^{r n + k}}{(r n + k)!}
=
\left(
\sum_{n \geq 0} \frac{x^{r n + k}}{(r n + k)!}
\right)
\cdot
\left(
\sum_{n \geq 0} \frac{(s \cdot x)^{r n}}{(r n)!}
\right)^{-1/s} . $$
\label{proposition_M}
\end{proposition}
This can be proven from
Corollary~\ref{corollary_semigroup}
using $I = r \cdot \hbox{\Cp P}$ and $J = k + r \cdot \hbox{\Cp N}$.
This also follows from
Corollary~\ref{corollary_Mobius}
by using the Dowling structure ${\bf D}^{(r,k)}$.
When $r=1$ we have the following corollary.
\begin{corollary}
Let $k \geq 1$.
Then the M\"obius function
of the poset $D_{n}^{(1,k)} \cup \{\hat{0}\}$
is given by
$$ \mu\left(D_{n}^{(1,k)} \cup \{\hat{0}\}\right)
=
(-1)^{n} \cdot {{n+k-1} \choose {k-1}} . $$
Furthermore, the M\"obius function does not depend on the order $s$.
\end{corollary}
\begin{proof_special}
We have
$$ \sum_{n \geq 0}
\mu\left(D_{n}^{(1,k)} \cup \{\hat{0}\}\right)
\cdot
\frac{x^{n+k}}{(n+k)!}
=
\left(
\sum_{n \geq 0} \frac{x^{n+k}}{(n+k)!}
\right)
\cdot
\exp(-x) . $$
Differentiate with respect to $x$ gives
\begin{eqnarray*}
\sum_{n \geq 0}
\mu\left(D_{n}^{(1,k)} \cup \{\hat{0}\}\right)
\cdot
\frac{x^{n+k-1}}{(n+k-1)!}
& = &
\left(
\sum_{n \geq 0} \frac{x^{n+k-1}}{(n+k-1)!}
\right)
\cdot
\exp(-x)
-
\left(
\sum_{n \geq 0} \frac{x^{n+k}}{(n+k)!}
\right)
\cdot
\exp(-x) \\
& = &
\frac{x^{k-1}}{(k-1)!}
\cdot
\exp(-x) \\
& = &
\sum_{n \geq 0} (-1)^{n} \cdot
{{n+k-1} \choose {k-1}}
\cdot \frac{x^{n+k-1}}{(n+k-1)!} .
\hspace{30 mm} \mbox{$\Box$}\vspace{\baselineskip}
\end{eqnarray*}
\end{proof_special}
When $r=2$ we can express the generating function
for the M\"obius function
in terms of hyperbolic functions. We have two cases,
depending on whether $k$ is even or odd.
\begin{corollary}
The M\"obius function
of the poset $D_{n}^{(2,k)} \cup \{\hat{0}\}$
is given by
\begin{eqnarray}
\sum_{n \geq 0}
\mu\left(D_{n}^{(2,2j)} \cup \{\hat{0}\}\right)
\cdot
\frac{x^{2n+2j}}{(2n+2j)!}
& = &
\left(
\cosh(x)
-
\sum_{i = 0}^{j-1} \frac{x^{2i}}{(2i)!}
\right)
\cdot
\mbox{\rm sech}\left( s \cdot x \right)^{1/s}
, \\
\sum_{n \geq 0}
\mu\left(D_{n}^{(2,2j+1)} \cup \{\hat{0}\}\right)
\cdot
\frac{x^{2n+2j+1}}{(2n+2j+1)!}
& = &
\left(
\sinh(x)
-
\sum_{i = 0}^{j-1} \frac{x^{2i+1}}{(2i+1)!}
\right)
\cdot
\mbox{\rm sech}\left( s \cdot x \right)^{1/s}.
\end{eqnarray}
\label{corollary_hyperbolic}
\end{corollary}
\section{Permutations and partitions with restricted block sizes}
\label{section_permutations}
\setcounter{equation}{0}
For a permutation $\sigma = \sigma_{1}\sigma_{2}\cdots\sigma_{n}$
in the symmetric group
$\hbox{\german S}_{n}$ define the descent set of $\sigma$ to be
the set
$\{ i \: : \: \sigma_{i} < \sigma_{i+1}\}$.
An equivalent notion is
the {\em descent word} of $\sigma$,
which is the ${\bf a}{\bf b}$-word
$u = u_{1} u_{2} \cdots u_{n-1}$ of degree $n-1$ where
$u_{i} = {\bf a}$ if $\sigma_{i} < \sigma_{i+1}$
and
$u_{i} = {\bf b}$ otherwise.
For an ${\bf a}{\bf b}$-word $u$ of length $n-1$
let $\Des{u}$ be the number of permutations $\sigma$
in $\hbox{\german S}_{n}$ with descent word $u$.
Similarly, define the $q$-analogue $\Desq{u}$
to be the sum
$$ \Desq{u}
=
\sum_{\sigma} q^{\mbox{\scriptsize\rm inv}(\sigma)} , $$
where the sum ranges over all permutations $\sigma$
in $\hbox{\german S}_{n}$ with descent word $u$
and $\mbox{\rm inv}(\sigma)$ is the number of
inversions of $\sigma$.
Let $[n]$ denote $1 + q + \cdots + q^{n-1}$
and $[n]! = [1] \cdot [2] \cdots [n]$.
Finally, let $\qb{n}{k}$ denote the Gaussian coefficient
$[n]!/([k]! \cdot [n-k]!)$.
\begin{lemma}
For two ${\bf a}{\bf b}$-words $u$ and $v$ of degree $n-1$, respectively $m-1$,
the following identity holds:
$$ \qb{n+m}{n}
\cdot
\Desq{u} \cdot \Desq{v}
=
\Desq{u \cdot {\bf a} \cdot v}
+
\Desq{u \cdot {\bf b} \cdot v} . $$
\end{lemma}
This is ``the Multiplication Theorem'' due to
MacMahon~\cite[Article~159]{MacMahon}.
Using this identity, we obtain
the following lemma for Eulerian generating functions.
\begin{lemma}
Let $\left( u_{n} \right)_{n \geq 1}$
and $\left( v_{n} \right)_{n \geq 1}$
be two sequences of ${\bf a}{\bf b}$-words such that
the $n$th word has degree $n-1$.
Then the following Eulerian generating function identity holds:
\begin{eqnarray*}
& &
\left(
\sum_{n \geq 1} c_{n} \cdot \Desq{u_{n}} \cdot \frac{x^{n}}{[n]!}
\right)
\cdot
\left(
\sum_{n \geq 1} d_{n} \cdot \Desq{v_{n}} \cdot \frac{x^{n}}{[n]!}
\right) \\
& = &
\sum_{n \geq 2} \sum_{{i+j=n} \atop {i,j \geq 1}}
c_{i} \cdot d_{j} \cdot
\left(\Desq{u_{i} \cdot {\bf a} \cdot v_{j}}
+
\Desq{u_{i} \cdot {\bf b} \cdot v_{j}}\right)
\cdot \frac{x^{n}}{[n]!} .
\end{eqnarray*}
\end{lemma}
Now we obtain the following proposition.
In the special case when $w={\bf a}^{i}$, where
$0 \leq i \leq r-1$, the result is due to
Stanley~\cite{Stanley_binomial}. See
also~\cite[Section~3.16]{Stanley_EC_I}.
\begin{proposition}
Let $w$ be an ${\bf a}{\bf b}$-word of degree $k-1$.
Then Eulerian generating function for
the descent statistic $\Desq{({\bf a}^{r-1} {\bf b})^{n} \cdot w}$
is given by
$$ \sum_{n \geq 0}
(-1)^{n} \cdot
\Desq{({\bf a}^{r-1} {\bf b})^{n} \cdot w} \cdot
\frac{x^{r n + k}}{[r n + k]!}
=
\frac{{\displaystyle
\sum_{n \geq 0} \Desq{{\bf a}^{r n} \cdot w} \cdot
\frac{x^{r n + k}}{[r n + k]!}
}}{{\displaystyle
\sum_{n \geq 0} \frac{x^{r n}}{[r n]!}
}} . $$
\label{proposition_permutations}
\end{proposition}
\begin{proof}
Consider the following product of generating functions:
\begin{eqnarray*}
& &
\left(
\sum_{n \geq 1}
\Desq{{\bf a}^{r n-1}} \cdot
\frac{x^{r n}}{[r n]!}
\right)
\cdot
\left(
\sum_{n \geq 0}
(-1)^{n} \cdot
\Desq{({\bf a}^{r-1} {\bf b})^{n} \cdot w} \cdot
\frac{x^{r n + k}}{[r n + k]!}
\right) \\
& = &
\sum_{n \geq 0}
\left(
\sum_{{i+j = n} \atop {i \geq 1}}
(-1)^{j} \cdot
\left(
\Desq{{\bf a}^{r i} ({\bf a}^{r-1} {\bf b})^{j} \cdot w}
+
\Desq{{\bf a}^{r (i-1)} ({\bf a}^{r-1} {\bf b})^{j+1} \cdot w}
\right)
\right)
\cdot
\frac{x^{r n + k}}{[r n + k]!} \\
& = &
\sum_{n \geq 0}
\left(
\Desq{{\bf a}^{r n} \cdot w}
+
(-1)^{n-1} \cdot
\Desq{({\bf a}^{r-1} {\bf b})^{n} \cdot w}
\right)
\cdot
\frac{x^{r n + k}}{[r n + k]!} .
\end{eqnarray*}
Now add
$\sum_{n \geq 0}
(-1)^{n} \cdot
\Desq{({\bf a}^{r-1} {\bf b})^{n} \cdot w} \cdot
x^{r n + k}/[r n + k]!$
to both sides and the desired identity is established.
\end{proof}
For $r$ a positive integer and $n$ a non-negative
integer let $m = r n$. Define
the poset $\Pi_{m}^{r}$ to be
the collection of all partitions $\pi$
of the set
$\{1, \ldots, m\}$
such that each block size is divisible by $r$
together with a minimal element $\hat{0}$ adjoined.
This is the well-known and well-studied
{\em $r$-divisible partition lattice}.
See~\cite{Calderbank_Hanlon_Robinson, Sagan, Stanley_e_s, Wachs_1}.
Other restrictions of the partition lattice and the Dowling lattice
can be found in~\cite{Bjorner_Sagan, Gottlieb_I, Gottlieb_II}.
A natural extension of the $r$-divisible partition lattice
is the following.
For $r$ a positive integer, and $n$ and $j$ non-negative integers,
let $m = r n + j$.
Define the poset $\Pi_{m}^{r,j}$ to be
the collection of all partitions $\pi$
of the set
$\{1, \ldots, m\}$
such that
\vspace*{-2 mm}
\begin{itemize}
\item[(i)] a block $B$ of $\pi$ containing the element $m$
must have cardinality at least $j$,
\vspace*{-2 mm}
\item[(ii)] a block $B$ of $\pi$ not containing the element $m$
must have cardinality divisible by $r$,
\end{itemize}
\vspace*{-2 mm}
together with a minimal element $\hat{0}$ adjoined to the poset.
We order all such partitions in the usual way by refinement.
For instance, $\Pi_{m}^{1,1}$ is the classical
partition lattice $\Pi_{m}$ with $\hat{0}$ adjoined.
Observe that the poset $\Pi_{m}^{r,j} -\{\hat{0}\}$
is a filter (upper order ideal) of the partition lattice~$\Pi_{m}$.
Hence~$\Pi_{m}^{r,j}$ is a finite semi-join lattice
and we can conclude that it is a lattice.
The same argument holds for~$\Pi_{m}^{r}$.
By combining Propositions~\ref{proposition_M}
and~\ref{proposition_permutations},
we obtain the next result.
\begin{theorem}
Let $r$ and $k$ be positive integers and $n$ a non-negative
integer and let $m = r n + k + 1$.
Then the M\"obius function of
the lattice $\Pi_{m}^{r,k+1}$
is given by
the sign $(-1)^{n}$ times
the number of permutations
on $m-1$ elements
with the descent set
$\{r, 2 r, \ldots, n r\}$,
that is,
$$ \mu(\Pi_{m}^{r,k+1})
=
(-1)^{n}
\cdot
\Des{({\bf a}^{r-1} {\bf b})^{n} \cdot {\bf a}^{k-1}} . $$
\label{theorem_m_plus_1}
\end{theorem}
\begin{proof}
Begin to observe that $\Pi_{m}^{r,k+1}$ is
isomorphic to the poset $D^{(r,k)}_{n}$ when $s=1$.
Namely, remove the element $m$ from the block $B$ that contains
this element and rename this block to be the zero block.
The result follows now by observing
that setting $w = {\bf a}^{k-1}$ and $q=1$ in
Proposition~\ref{proposition_permutations}
gives the same generating function as
setting $s=1$ in
Proposition~\ref{proposition_M}.
\end{proof}
For completeness, we also consider the case $j=1$.
\begin{theorem}
Let $r$ and $n$ be positive integers and let $m = r n + 1$.
Then the M\"obius function of
the lattice $\Pi_{m}^{r,1}$
is $0$.
\label{theorem_j_1}
\end{theorem}
\begin{proof}
This follows directly from
Proposition~\ref{proposition_M}
by setting $k=0$ and $s=1$.
A direct combinatorial argument is the following.
Each of the atoms of the lattice $\Pi_{m}^{r,1}$
has the element $m$ in a singleton block.
The same holds for the join of all
the atoms and hence the join of all
the atoms is not the maximal element $\hat{1}$ of the lattice.
Thus by Corollary~3.9.5 in~\cite{Stanley_EC_I} the result is obtained.
\end{proof}
Setting $k = r-1$ in Theorem~\ref{theorem_m_plus_1},
we obtain the following corollary due
to Stanley~\cite{Stanley_e_s}.
\begin{corollary}
For $r \geq 2$ and $m = r n$
the M\"obius function of
the $r$-divisible partition lattice $\Pi_{m}^{r}$
is given by
the sign $(-1)^{n-1}$
times the number of
permutations of $r n - 1$ elements
with the descent set
$\{r, 2 r, \ldots, (n-1) r\}$,
that is,
$$ \mu(\Pi_{m}^{r})
=
(-1)^{n-1}
\cdot
\Des{({\bf a}^{r-1} {\bf b})^{n-1} \cdot {\bf a}^{r-2}} . $$
\end{corollary}
When $r=2$ this corollary reduces to
$(-1)^{n-1} \cdot E_{2n-1}$, where $E_{i}$ denotes
the $i$th Euler number.
This result is originally due to
G.\ S.\ Sylvester~\cite{Sylvester}.
The odd indexed Euler numbers are known as
the {\em tangent numbers}
and the even indexed ones as
the {\em secant numbers}.
Setting $r=2$ and $k=2$
in Theorem~\ref{theorem_m_plus_1}
we obtain
that the M\"obius function of the
partitions where all blocks have even size
except the block containing the largest element,
which has an odd size greater than or equal to three,
is given by
the secant numbers,
that is, $(-1)^{n-1} \cdot E_{2n}$.
\section{EL-labeling}
\label{section_EL}
\setcounter{equation}{0}
It is a natural question
to ask if the poset $\Pi_{m}^{r,j}$ occurring
in Theorems~\ref{theorem_m_plus_1}
and~\ref{theorem_j_1} is
$EL$-shellable. The answer is positive.
An $EL$-labeling that works is the one using Wachs'
$EL$-labeling~\cite{Wachs_1} for the $r$-divisible
partition lattice $\Pi_{m}^{r}$,
which we state here for the extended
partition lattice $\Pi_{m}^{r,j}$.
Let $r$ and $j$ be positive integers and $n$ a non-negative
integer and let $m = r n + j$.
Define the labeling $\lambda$ as follows.
First consider the edges in the Hasse diagram not adjacent to
the minimal element $\hat{0}$.
Let $x$ and $y$ be two elements in $\Pi_{m}^{r,k+1} - \{\hat{0}\}$ such
that $x$ is covered by $y$ and $B_{1}$ and $B_{2}$ are the blocks of
$x$ that are merged to form the partition $y$.
Assume that $\max(B_{1}) < \max(B_{2})$.
Set
\begin{equation}
\lambda(x,y)
=
\left\{
\begin{array}{c l}
-\max(B_{1}) & \mbox{ if } \max(B_{1}) > \min(B_{2}), \\
\max(B_{2}) & \mbox{ otherwise.}
\end{array}
\right.
\label{equation_lambda_1}
\end{equation}
Now consider the edges between the minimal element
$\hat{0}$ and the atoms. There are
$M = (m-1)!/(n! \cdot r!^{n} \cdot (j-1)!)$
number of atoms.
For each atom $a = \{B_{1}, B_{2}, \ldots, B_{n+1}\}$
order the blocks such that $\min(B_{1}) < \min(B_{2}) <
\cdots < \min(B_{n+1})$.
Let $\widetilde{a}$ be the permutation in
$\hbox{\german S}_{m}$ that is obtained by going through the blocks
in order and writing down the elements
in each block in increasing order.
For instance, for the atom $a = 16|23|459|78$ we obtain
the permutation $\widetilde{a} = 162345978$.
It is straightforward to see that different atoms give rise
to different permutations by considering where the largest
element $m$ is.
Finally, order the atoms $a_{1} < a_{2} < \cdots < a_{M}$
such that the permutations
$\widetilde{a_{1}} < \widetilde{a_{2}} < \cdots < \widetilde{a_{M}}$
are ordered in lexicographic order.
Define the label of the edge from the minimal element to an atom
by
\begin{equation}
\lambda(\hat{0},a_{i})
=
0_{i}
\label{equation_lambda_2}
\end{equation}
Order the labels by
$$ \{-m < -(m-1) < \cdots < -1
< 0_{1} < 0_{2} < \cdots < 0_{M} <
1 < \cdots < m \} . $$
Let $A_{m}^{r,j}$ be the collection
of all permutations $\sigma \in \hbox{\german S}_{m}$ such
that the descent set of $\sigma$ is
$\{r, 2r, \ldots, n r\}$ and $\sigma(m) = m$.
Note that when $j=1$ there are no such permutations
since the condition $\sigma(m) = m$ forces
$n r$ to be an ascent. Given
a permutation $\sigma \in A_{m}^{r,j}$,
let $t_{1}, \ldots, t_{n}$ be
the permutation of $1, \ldots, n$ such that
$$ \sigma(r t_{1}) >
\sigma(r t_{2}) >
\cdots >
\sigma(r t_{n}) . $$
Define the maximal chain $f_{\sigma}$
in $\Pi_{m}^{r,j}$ whose $i$-block
partition is obtained by splitting $\sigma$
at $r t_{1}$, $r t_{2}$, $\ldots$, $r t_{i-1}$.
As an example, for $\sigma = 562418379$
where $r = 2$, $n=3$, $j=3$ and $m=9$, we have
the maximal chain
$$ f_{562418379}
=
\{\hat{0} <
56|24|18|379 <
56|2418|379 <
562418|379 <
562418379 = \hat{1}\} . $$
Observe that different permutations
in $A_{m}^{r,j}$ give different maximal chains.
\begin{theorem}
The labeling
$(\lambda(x,y), -\rho(x))$
where $\lambda$ is defined in equations~(\ref{equation_lambda_1})
and~(\ref{equation_lambda_2}),
$\rho$ denotes the rank function and
the ordering is lexicographic on the pairs,
is an $EL$-labeling for the poset
$\Pi_{m}^{r,j}$.
The falling maximal chains are given by
$\{ f_{\sigma} \: : \: \sigma \in A_{m}^{r,j}\}$.
\end{theorem}
The proof that this labeling is an $EL$-labeling mimics the
proof of Theorem~5.2 in Wachs' paper~\cite{Wachs_1} and hence is omitted.
We distinguish between the cases $j=1$ and $j \geq 2$
in the following two corollaries.
\begin{corollary}
The chain complex of $\Pi_{m}^{r,1}$
is contractible.
\end{corollary}
\begin{corollary}
The chain complex of $\Pi_{m}^{r,j}$
is homotopy equivalent to a wedge of
$\Des{({\bf a}^{r-1} {\bf b})^{n} \cdot {\bf a}^{j-2}}$ number of
$(n-1)$-dimensional spheres.
Hence all the poset homology of the
poset
$\Pi_{m}^{r,j}$
is concentrated in the top homology
which has rank
$\Des{({\bf a}^{r-1} {\bf b})^{n} \cdot {\bf a}^{j-2}}$
\end{corollary}
\section{Concluding remarks}
\label{section_concluding_remarks}
\setcounter{equation}{0}
Can more examples of exponential Dowling structures be given? For
instance, find the Dowling extension of counting matrices with
non-negative integer entries having a fixed row and column sum.
See~\cite[Chapter~5]{Stanley_EC_II}.
Theorem~\ref{theorem_m_plus_1}
has been generalized
in~\cite{Ehrenborg_Readdy_restricted}.
As we have seen in
this theorem
the generating function
for the M\"obius function
of $D_{n}^{(r,k)} \cup \{\hat{0}\}$
in
Proposition~\ref{proposition_M}
in the case when the order $s$ is equal to $1$
has a permutation enumeration analogue.
It would be interesting to find a permutation
interpretation for this generating function
for general values of the order $s$.
Similar generating functions
have appeared when enumerating classes
of $r$-signed permutations.
A few examples are
$(\sin(px) + \cos((r-p)x)/\cos(rx)$
counting
$p$-augmented $r$-signed permutations
in~\cite{Ehrenborg_Readdy_Sheffer},
$\sqrt[r]{1/(1 - \sin(rx))}$
counting
augmented Andr\'e $r$-signed permutations
in~\cite{Ehrenborg_Readdy_r-cubical},
and
$\sqrt[r]{1/(1 - rx)}$
counting
$r$-multipermutations
in~\cite{Park}.
There are several other questions to raise.
Is there a $q$-analogue of the partition lattice
such that a natural $q$-analogue of
Theorem~\ref{theorem_m_plus_1} also holds?
We only use the case $w = {\bf a}^{k-1}$ in
Proposition~\ref{proposition_permutations}.
Are there other poset statistics that correspond to
other ${\bf a}{\bf b}$-words $w$?
The symmetric group $\hbox{\german S}_{m-1}$ acts naturally on
the lattice $\Pi_{m}^{r,j}$. Hence it also acts
on the top homology group of
$\Pi_{m}^{r,j}$.
In a forthcoming paper we study
the representation of this $\hbox{\german S}_{m-1}$ action.
Similar questions arise concerning the poset
$D_{n}^{(r,k)} \cup \{\hat{0}\}$;
see Proposition~\ref{proposition_M}.
Is this poset shellable?
Is the homology of
this poset
concentrated in the top homology?
Note that the wreath product
$G \wr \hbox{\german S}_{n}$ acts on
the Dowling lattice $L_{n}(G) = L_{n}$.
Hence $G \wr \hbox{\german S}_{n}$ acts on
the exponential Dowling structure
$D_{n}^{(r,k)} \cup \{\hat{0}\}$.
What can be said about the action of
the wreath product
$G \wr \hbox{\german S}_{n}$
on
the homology group(s) of
$D_{n}^{(r,k)} \cup \{\hat{0}\}$?
\section*{Acknowledgements}
The first author was partially supported by National Science
Foundation grant 0200624. Both authors thank the
Mittag-Leffler Institute where a portion of this research was completed
during the Spring 2005 program in Algebraic Combinatorics.
The authors also thank the referee for suggesting
additional references.
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{\small
|
{
"timestamp": "2010-09-23T02:00:17",
"yymm": "1009",
"arxiv_id": "1009.4202",
"language": "en",
"url": "https://arxiv.org/abs/1009.4202"
}
|
\section{Introduction}
The notion of subgroup distortion was first formulated by Gromov in \cite{gromov}. For a group $G$ with finite generating set $T$ and a subgroup $H$ of $G$ finitely generated by $S$, the distortion function of $H$ in $G$ is $$\Delta_{H}^{G}(l) = \max \{ |w|_S : w \in H, |w|_T \leq l \},$$ where $|w|_S$ represents the word length with respect to the given generating set $S$, and similarly for $|w|_T$. This function measures the difference in the word metrics on $G$ and on $H$.
As usual, we only study distortion up to a natural equivalence relation. For non-decreasing functions $f$ and $g$ on $\mathbb{N}$, we say that $f \preceq g$ if there exists an integer $C>0$ such that $f(l) \leq Cg(Cl)$ for all $l \geq 0$. We say two functions are equivalent, written $f \approx g$, if $f \preceq g$ and $g \preceq f$.
When considered up to this equivalence, the distortion function becomes independent of the choice of finite generating sets. If the subgroup $H$ is infinite then the growth of the
distortion function is at least linear, and therefore one does not extend the
equivalence classes using the equivalence defined by the inequality $f(l) \leq Cg(Cl) +Cn.$ A subgroup $H$ of $G$ is said to be undistorted if $\Delta^{G}_{H}(l) \approx l$. If a subgroup $H$ is not undistorted, then it is said to be distorted, and its distortion refers to the equivalence class of $\Delta_H^G(l)$.
\begin{rem}\label{poa}
Suppose there exists a subsequence of $\mathbb{N}$ given by $\{l_i\}_{i \in \mathbb{N}}$ where $l_i < l_{i+1}$ for $i \geq 1$. If there exists $c>0$ such that $\frac{l_{i+1}}{l_i} \leq c$, for all $i \geq 1$, and $f(l_i) \geq g(l_i)$, then $f \succeq g$.
\end{rem}
Here we study the effects of distortion in various subgroups of the wreath products ${\mathbb Z}^k \textrm{ wr } {\mathbb Z}$, for $0<k \in {\mathbb Z}$, and more generally, in $A \textrm{ wr } {\mathbb Z}$ where $A$ is finitely generated abelian.
Note that wreath products $A \textrm{ wr } B$ where $A$ is abelian play a very important role in group theory for many reasons. Given any semidirect product $G=C\lambda D$ with abelian normal subgroup $C,$ then any two homomorphisms from $A \to C$ and $B \to D$ (uniquely) extend to a homomorphism from $A \textrm{ wr } B$ to $G.$ Also, if $B$ is presented as a factor-group $F/N$ of a $k$-generated free group $F,$ then the maximal extension $F/[N,N]$ of $B$ with abelian kernel is canonically embedded in ${\mathbb Z}^k \textrm{ wr } B$ (see \cite{magnus}.) Wreath products of abelian groups give an inexhaustible source of examples and counter-examples in group theory.
For instance, the group ${\mathbb Z}\textrm{ wr } {\mathbb Z}$ is the simplest example of a finitely generated (though not finitely presented) group containing a free abelian group of infinite rank. In \cite{sapir} the group ${\mathbb Z} \textrm{ wr } {\mathbb Z}$ is studied in connection with diagram groups and in particular with Thompson's group. In the same paper, it is shown that for $H_d=( \cdots ({\mathbb Z} \textrm{ wr } {\mathbb Z}) \textrm{ wr } {\mathbb Z}) \cdots \textrm{ wr } {\mathbb Z})$, where the group ${\mathbb Z}$ appears $d$ times, there is a subgroup $K \leq H_d \times H_d$ having distortion function $\Delta_K^{H_d \times H_d}(l) \succeq l^d$. In contrast to the study of these iterated wreath products, here we obtain polynomial distortion of arbitrary degree in the group ${\mathbb Z} \textrm{ wr } {\mathbb Z}$ itself. In \cite{cleary} the distortion of ${\mathbb Z} \textrm{ wr } {\mathbb Z}$ in Baumslag's metabelian group is shown to be at least exponential, and an undistorted embedding of ${\mathbb Z} \textrm{ wr } {\mathbb Z}$ in Thompson's group is constructed.
In this note, rather than embedding the group ${\mathbb Z} \textrm{ wr } {\mathbb Z}$ into larger groups, or studying multiple wreath products, we will study distorted and undistorted subgroups in the wreath products $A \textrm{ wr } {\mathbb Z}$ with $A$ finitely generated abelian. The main results are as follows.
\begin{theorem}\label{x}
Let $A$ be a finitely generated abelian group.
\begin{enumerate}
\item For any finitely generated infinite subgroup $H \leq A \textrm{ wr } {\mathbb Z}$ there exists $m \in \mathbb{N}$ such that the distortion of $H$ in $A \textrm{ wr } {\mathbb Z}$ is $$\Delta_H^{A \textrm{ wr } {\mathbb Z}}(l) \approx l^{m}.$$
\item If $A$ is finite, then $m=1$; that is, all subgroups are undistorted.
\item If $A$ is infinite, then for every $m \in \mathbb{N}$, there is a $2$-generated subnormal subgroup $H$ of $A \textrm{ wr } {\mathbb Z}$ having distortion function $$\Delta_H^{A \textrm{ wr } {\mathbb Z}}(l) \approx l^{m}.$$
\end{enumerate}
\end{theorem}
The following will be explained in Subsection \ref{fs}.
\begin{cor}\label{ot}
For every $m \in \mathbb{N}$, there is a $2$-generated subgroup $H$ of the free $n$-generated metabelian group $S_{n,2}$ having distortion function $$\Delta_H^{S_{n,2}}(l) \succeq l^{m}.$$
\end{cor}
\begin{cor}\label{xx}
If we let the standard generating set for ${\mathbb Z} \textrm{ wr } {\mathbb Z}$ be $\{ a, b \}$, then the subgroup $H_m= \langle b, [ \cdots [a,b],b], \cdots, b] \rangle$, where the commutator is $(m-1)$-fold, is $m-1$ subnormal, isomorphic to the whole group ${\mathbb Z} \textrm{ wr } {\mathbb Z}$, with distortion $l^m$. In particular the normal subgroup $\langle b, [a,b] \rangle$ has quadratic distortion.
\end{cor}
Corollary \ref{xx} is proved at the end of this paper. Because the subgroup $\langle [a,b], b \rangle$ of ${\mathbb Z} \textrm{ wr } {\mathbb Z}$ is normal, it follows by induction that the distorted subgroup $H_m$ is subnormal.
\begin{rem}\label{mmm}
There are distorted embeddings from the group ${\mathbb Z} \textrm{ wr } {\mathbb Z}$ into itself as a normal subgroup. For example, the map defined on generators by $b \mapsto b, a \mapsto [a,b]$ extends to an embedding, and the image is a quadratically distorted subgroup by Corollary \ref{xx}. By Lemma \ref{t2}, ${\mathbb Z} \textrm{ wr } {\mathbb Z}$ is the smallest example of a metabelian group embeddable to itself as a normal subgroup with distortion.
\end{rem}
\begin{cor}\label{sc}
There is a distorted embedding of ${\mathbb Z} \textrm{ wr } {\mathbb Z}$ into Thompson's group $F$.
\end{cor}
Under the embedding of Remark \ref{mmm}, ${\mathbb Z} \textrm{ wr } {\mathbb Z}$ embeds into itself as a distorted subgroup. It is proved in \cite{sapir} that ${\mathbb Z} \textrm{ wr } {\mathbb Z}$ embeds to $F$. Therefore, Corollary \ref{sc} is true.
It is interesting to contrast Theorem \ref{x} part $(2)$ with the following, which will be discussed in Section \ref{ppq}. Throughout this paper, we use the convention that ${\mathbb Z}_n$ represents the finite group ${\mathbb Z}/n{\mathbb Z}$.
\begin{prop}\label{pqp}
The group $G={\mathbb Z}_n \textrm{ wr } {\mathbb Z}^k$ for $n\ge 1$, has a finitely generated subgroup $H$ with distortion at least $l^{k}$.
\end{prop}
Some of the techniques to be introduced in this paper include some computations with polynomials. We will use the theory of modules over principal ideal domains in Section \ref{mt} to reduce the problem of subgroup distortion in ${\mathbb Z}^k \textrm{ wr } {\mathbb Z}$ to the consideration of certain $2$-generated subgroups in ${\mathbb Z} \textrm{ wr } {\mathbb Z}$. Every such subgroup is associated with a polynomial, and therefore we need to define and compute the distortion of arbitrary polynomial, as in Theorem \ref{tgs}. All of these techniques are used in conjunction with Theorem \ref{t6}, which provides a formula for computing the word length in arbitrary wreath product and makes computing subgroup distortion more tangible in the examples we consider.
\section{Background and Preliminaries}
\subsection{Subgroup Distortion}
Here we provide some examples of distortion as well as some basic facts to be used later on.
\begin{ex}
\item 1. Consider the three-dimensional Heisenberg group $\mathcal{H}^3 = \langle a, b, c | c=[a,b], [a,c]=[b,c] = 1 \rangle.$ It has cyclic subgroup $\langle c \rangle_{\infty}$ with quadratic distortion, which follows from the equation $c^{l^2} = [a^l,b^l]$.
\item 2. The Baumslag-Solitar Group $BS(1,2) = \langle a, b | b a b^{-1} = a^2 \rangle$ has cyclic subgroup $\langle a \rangle_{\infty}$ with at least exponential distortion, because $a^{2^l} =b^lab^{-l}.$
\end{ex}
However, there are no similar mechanisms distorting subgroups in ${\mathbb Z} \textrm{ wr } {\mathbb Z}$. Therefore, a natural conjecture would be that free metabelian groups or the group ${\mathbb Z} \textrm{ wr } {\mathbb Z}$ do not contain distorted subgroups. This conjecture was brought to the attention of the authors by Denis Osin. The result of Theorem \ref{x} shows that the conjecture is not true.
The following facts are well-known and easily verified. When we discuss distortion functions, it is assumed that the groups under consideration are finitely generated.
\begin{lemma}\label{wkf}
\item 1. If $H \leq G$ and $[G:H] < \infty$ then $\Delta_H^G(l) \approx l$.
\item 2. If $H \leq K \leq G$ then $\Delta_{H}^K(l) \preceq \Delta_H^G(l).$
\item 3. If $H \leq K \leq G$ then $\Delta_{H}^G(l) \preceq \Delta_K^G((\Delta_H^K(l)).$
\item 4. If $H$ is a retract of $G$ then $\Delta_H^G(l) \approx l$.
\item 5. If $G$ is a finitely generated abelian group, and $H \leq G$, then $\Delta_H^G(l) \approx l$.
\end{lemma}
\subsection{Wreath Products}
We consider the wreath products $A \textrm{ wr } B$ of finitely generated groups $A=\textrm{gp} \langle S \rangle = \langle \{y_1, \dots, y_s\} \rangle$ and $B = \textrm{gp} \langle T \rangle = \langle \{ x_1, \dots, x_t \} \rangle$. We introduce the notation that $A \textrm{ wr } B$ is the semidirect product $W\lambda B$, where $W$ is the direct product $\displaystyle\times_{g \in B}A_g,$ of isomorphic copies $A_g$ of the group $A.$ We view elements of $W$ as functions from $B$ to $A$ with finite support, where for any $f \in W$, the support of $f$ is $\supp(f)=\{g \in B: f(g) \ne 1\}$. The (left) action $\circ$ of $B$ on $W$ by automorphisms is given by the following formula: for any $f \in W, g \in B$ and $x \in B$ we have that $(g\circ f)(x)=f(xg)$.
Any element of the group $A \textrm{ wr } B$ may be written uniquely as $wg$ where $g \in B, w \in W$. The formula for multiplication in the group $A \textrm{ wr } B$ is given as follows. For $g_1, g_2 \in B, w_1, w_2 \in W$ we have that $(w_1g_1)(w_2g_2)=(w_1(g_1\circ w_2))(g_1g_2).$ In particular,
$B$ acts by conjugation on $W$ in the wreath product: $gwg^{-1}=g\circ w.$
Therefore the wreath product is generated by the subgroups $B$ and $A_1\le W,$ where non-trivial
functions from $A_1$ have support $\{1\}.$ In what follows,
the subgroup $A_1$ is identified with $A,$ and so $A_g=gAg^{-1},$ and $S\cup T$ is a
finite set of generators in $A \textrm{ wr } B.$ In particular, ${\mathbb Z} \textrm{ wr } {\mathbb Z}$ is generated by
$a$ and $b$ where $a$ generates the left (passive) infinite cyclic group and $b$ generates
the right (active) one.
Here we observe that a finitely generated abelian subgroup of $G=A \textrm{ wr } B $ with finitely generated abelian $A$ and $B$ is undistorted. It should be remarked that the authors are aware that the proof of the fact that abelian subgroups of ${\mathbb Z}^k \textrm{ wr } {\mathbb Z}$ are undistorted is available in \cite{sapir}. In that paper it is shown that ${\mathbb Z}^k \textrm{ wr } {\mathbb Z}$ is a subgroup of the Thompson group $F$, and that every finitely generated abelian subgroup of $F$ is undistorted. However, our observation is elementary and so we include it.
\begin{lemma}\label{abel} Let $A$ and $B$ be finitely generated abelian groups. Then
every finitely generated abelian subgroup $H$ of $ A \textrm{ wr } B$ is
undistorted.
\end{lemma}
\begin{proof}
It follows from the classification of finitely generated abelian groups $G$ that every subgroup $S$ is a retract of a subgroup of finite index in $G$, and so we are done if $H$ is a subgroup of $A$ or $B$, or if $H \cap W = \{1\}$, by Lemma \ref{wkf}. Therefore we assume that
$H \cap W \ne \{1\}.$ Since $H$ is abelian, this implies that the the factor-group
$HW/W$ is finite. Then it suffices to prove the lemma for $H_1=H\cap W$ since $[H:H_1]\le \infty.$
Because $H_1$ is finitely generated, it is contained in a finite product of conjugate copies of $A$. That is to say, $H_1 \subset A'$ for a wreath product $A' \textrm{ wr } B' = W\lambda B'$ where $B'$ has finite index in $B$. We are now reduced to our earlier argument, thus completing the proof.
\end{proof}
\begin{rem} In fact, under the assumptions of Lemma \ref{abel}, $H$ is a retract
of a subgroup having finite index in $A \textrm{ wr } B.$
\end{rem}
We now return to one of the motivating ideas of this paper, and complete the explanation of Remark \ref{mmm}.
\begin{lemma}\label{t2}
The group ${\mathbb Z} \textrm{ wr } {\mathbb Z}$ is the smallest metabelian group which embeds to itself as a normal distorted subgroup in the following sense. For any metabelian group $G$, if there is an embedding $\phi: G \rightarrow G$ such that $\phi(G) \unlhd G$ and $\phi(G)$ is a distorted subgroup in $G$, then there exists some subgroup $H$ of $G$ for which $H \cong {\mathbb Z} \textrm{ wr } {\mathbb Z}.$
\end{lemma}
\begin{proof}
By Lemma \ref{wkf}, we have that the group $G/\phi(G)$ is infinite, else $\phi(G)$ would be undistorted. Being a finitely generated solvable group, $G/\phi(G)$ must have a subnormal factor isomorphic to ${\mathbb Z}$. Because $\phi(G) \cong G$, one may repeat this argument to obtain a subnormal series in $G$ with arbitrarily many infinite cyclic factors. Therefore, the derived subgroup $G'$ has infinite (rational) rank.
Since the group $B=G/G'$ is finitely presented, the action of $B$ by conjugation makes $G'$ a finitely generated left $B$ module. Hence, $G'=\langle B \circ C\rangle$ for some finitely generated $C \leq G'$. Because it is a finitely generated abelian group, $B=\langle b_k \rangle \cdots \langle b_1 \rangle$ is a product of cyclic groups. Therefore for some $i$ we have a subgroup $A=\langle \langle b_{i-1} \rangle \cdots \langle b_1 \rangle \circ C \rangle$ of finite rank in $G'$ but $\langle \langle b_i \rangle \circ A \rangle$ has infinite rank. Then $A$ has an element $a$ such that the $\langle b_i \rangle$-submodule generated by $a$ has infinite rank, and so it is a free $\langle b_i \rangle$-module. It follows that $a$ and $b$, where $b_i=bG'$, generate a subgroup of the form ${\mathbb Z} \textrm{ wr } {\mathbb Z}$.
\end{proof}
\subsection{Connections with Free Solvable Groups}\label{fs}
In \cite{magnus}, Magnus shows that if $F=F_k$ is an absolutely free group of rank $k$ with normal subgroup $N$, then the group $F/[N,N]$ embeds into ${\mathbb Z}^k \textrm{ wr } F/N= {\mathbb Z}^k \textrm{ wr } G$.
This wreath product is a semidirect product $W\lambda G$ where the action of $G$ by
conjugation turns $W$ into a free left ${\mathbb Z}[G]$-module with $k$ generators.
For more information in an easy to read exposition, refer to \cite{rem}.
\begin{rem}
The monomorphism $\alpha : F/[N,N] \rightarrow {\mathbb Z}^k \textrm{ wr } G$ is called the Magnus embedding.
\end{rem}
We let $S_{k,l}$ denote the $k$-generated derived length $l$ free solvable group.
\begin{lemma}\label{y}
If $k,l \geq 2$, then the group $S_{k,l}$ contains a subgroup isomorphic to ${\mathbb Z} \textrm{ wr } {\mathbb Z}$.
\end{lemma}
\proof It is well known (and follows from the Magnus embedding) that any nontrivial $a \in S_{k,l}^{(l-1)}$ and $b \notin S_{k,l}^{(l-1)}$ generate ${\mathbb Z} \textrm{ wr } {\mathbb Z}$.\endproof
\medskip
It should be noted that by results of \cite{shmelkin}, the group ${\mathbb Z} \textrm{ wr } {\mathbb Z}^2$ can not be embedded into any free metabelian or free solvable groups.
\medskip
Subgroup distortion has connections with the membership problem.
It was observed in \cite{gromov} and proved in \cite{farb} that for a finitely generated subgroup $H$ of a finitely generated group $G$ with solvable word problem, the membership problem is solvable in $H$ if and only if the distortion function $\Delta_{H}^{G}(l)$ is bounded by a recursive function.
By Theorem 2 of \cite{umirbaev}, the membership problem for free solvable groups of length greater than two is undecidable. Therefore, because of the connections between subgroup distortion and the membership problem just mentioned, we restrict our primary attention to the case of free metabelian groups. It is worthwhile to note that the membership problem for free metabelian groups is solvable (see \cite{romanovskii}).
Lemma \ref{y} motivates us to study distortion in ${\mathbb Z} \textrm{ wr } {\mathbb Z}$ in order to better understand distortion in free metabelian groups. Distortion in free metabelian groups is similar to distortion in wreath products of free abelian groups, by Lemma \ref{y} and the Magnus embedding. In particular, if $k \geq 2$ then $${\mathbb Z} \textrm{ wr } {\mathbb Z} \leq S_{k,2} \leq {\mathbb Z}^k \textrm{ wr } {\mathbb Z}^k.$$ Thus by Lemma \ref{wkf}, given $H \leq {\mathbb Z} \textrm{ wr } {\mathbb Z}$ we have $$\Delta_{H}^{{\mathbb Z} \textrm{ wr } {\mathbb Z}}(l) \preceq \Delta_{H}^{S_{k,2}}(l).$$ This explains Corollary \ref{ot}. On the other hand, given $L \leq S_{k,2}$ then we have $$\Delta_{L}^{S_{k,2}}(l) \preceq \Delta_{L}^{{\mathbb Z}^k \textrm{ wr } {\mathbb Z}^k}(l).$$ Based on this discussion, we ask the following. An answer would be helpful in order to more fully understand subgroup distortion in free metabelian groups.
\begin{question}
\item What effects of subgroup distortion are possible in ${\mathbb Z}^k \textrm{ wr } {\mathbb Z}^k$ for $k>1$?
\end{question}
\section{Canonical Forms and Word Metric}\label{s2}
Here we aim to further understand how the length of an element of a wreath product
$A\textrm{ wr } B$ depends on the canonical form of this element.
Let us start with $G={\mathbb Z}^k \textrm{ wr } {\mathbb Z}=W\lambda\langle b \rangle,$ where ${\mathbb Z}^k=\textrm{gp}\{a_1,\dots,a_k\}.$
Because the subgroup $W$ of ${\mathbb Z}^k \textrm{ wr } {\mathbb Z} = W \lambda {\mathbb Z}$ is abelian, we also use additive notation to represent elements of $W$.
\begin{rem}\label{yyy}
In the case of ${\mathbb Z} \textrm{ wr } {\mathbb Z}=\langle a \rangle \textrm{ wr } \langle b \rangle$, we use module language to write any element as $$w={\sum_{i=-\infty}^{\infty}} m_i (b^i \circ a)=f(b)a=f(x)a \textrm{, where } f(x)={\sum_{i=-\infty}^{\infty}} m_ix^i$$ is a Laurent polynomial in $x$, and the sums are finite.
Similarly, if $A$ is a finitely generated abelian group, then by the definition of $A \textrm{ wr } {\mathbb Z},$
arbitrary element in $A \textrm{ wr } {\mathbb Z}$ is
of the form $$wb^t=\left(\sum_{i=1}^kf_i(x)a_i \right) b^t,$$ where $f_i(x)$ are Laurent polynomials and $a_i$-s generate $A$. This form is unique if $A={\mathbb Z}^k$ is a free abelian
with basis $a_1,\dots,a_k$.
\end{rem}
We will use the notation that $(w)_i$ equals the conjugate $bwb^{-i}$ for
$i \in {\mathbb Z}$ and $w \in W.$
The normal form described in Remark \ref{yyy} for elements of $A\textrm{ wr } {\mathbb Z}$
is necessary to obtain a general formula for computing the word length.
\begin{rem}\label{yy}
Arbitrary element of $A \textrm{ wr } {\mathbb Z}$ may be written in a normal form, following \cite{ct}, as
$$\left( (u_1)_{\iota_1}+ \cdots +(u_N)_{\iota_N}+(v_1)_{-\epsilon_1}+ \cdots +(v_M)_{-\epsilon_M}
\right)b^t$$
where $0 \leq \iota_1 < \cdots <\iota_N, 0 < \epsilon_1 < \cdots < \epsilon_M$, and $u_1, \dots, u_N, v_1, \dots, v_M$ are elements in $A\backslash\{1\}.$
\end{rem}
The following formula for the word length in $A \textrm{ wr } {\mathbb Z}$ is given in \cite{ct}.
\begin{lemma}\label{t5}
Given an element in $A \textrm{ wr } {\mathbb Z}$ having normal form as in Remark \ref{yy}, its length is given by the formula $$\sum_{i=1}^N |u_i|_{A} + \sum_{i=1}^M |v_i|_{A} + \min \{ 2 \epsilon_M +\iota_N +|t-\iota_N|, 2\iota_N+\epsilon_M+|t+\epsilon_M|\}.$$
where $|*|_A$ is the length in the group $A.$
\end{lemma}
The formula from Lemma \ref{t5} becomes more intelligible if one extends it to
wreath products $A \textrm{ wr } B$ of arbitrary finitely generated groups. We want to
obtain such a generalization in this section since we consider non-cyclic active
groups in Section \ref{ppq}. We fix the notation that, with respect to the symmetric generating set $T=T^{-1}$, the Cayley graph $\cay(B)$ is defined as follows. The set of vertices is all elements of $G$. For any $g \in G, t \in T$, $g$ and $gt$ are joined by an edge pointing from $g$ to $gt$ whose label is $t$.
Any $u \in A \textrm{ wr } B$ can be expressed as follows:
\begin{equation}\label{r8}
(b_1\circ a_1)\dots (b_r\circ a_r) g
\end{equation}
where $g \in B, w=(b_1\circ a_1)\dots (b_r\circ a_r)\in W, 1 \ne a_j \in A, b_j \in B$ and for $i \ne j$ we have $b_i \ne b_j$. The expression $(\ref{r8})$ is unique, up to a rearrangement of the (commuting) factors $b_j\circ a_j$.
For any $u=wg \in A \textrm{ wr } B$ with canonical form as in Equation $(\ref{r8})$ we consider the set $P$ of paths in the Cayley graph $\cay(B)$ which start at $1$, go through every vertex $b_1, \dots, b_r$ and end at $g$. We introduce the notation that $$\trace(u)=\min\{||p||: p \in P\},$$ $$\spann(u)=\textrm{ the particular } p \in P \textrm{ realizing } \trace(u)=||p||.$$ We also define the norm of any such representative $w$ of $W$ by $$||w||_A= \sum_{j=1}^r|a_j|_S.$$
We have the following formula for word length, which generalizes that given for the case where $B= {\mathbb Z}$ in the paper \cite{ct}. ({\bf Caution}: The {\it right}-action definition of wreath product
would be incompatible with the standard definition of Cayley graph in the proof of Theorem \ref{t6}.)
\begin{theorem}\label{t6}
For any element $u=wg \in A \textrm{ wr } B$, we have that $$|wg|_{S,T} = ||w||_A+\trace(u)$$ where $u=(b_1\circ a_1)\dots (b_r\circ a_r) g$ is the canonical form of Equation $(\ref{r8})$.
\end{theorem}
\proof We will use the following pseudo-canonical (non-unique) form in the proof. This is just the expression of Equation $(\ref{r8})$ but without the assumption that all $b_j$ are distinct
or $a_j$-s are non-trivial.
For any element $u \in A \textrm{ wr } B$ which is expressed in pseudo-canonical form we may define a quantity depending on the given factorization by $$\Psi((b_1\circ a_1)\dots (b_r\circ a_r) g)=\sum_{j=1}^r|a_j|_S+|b_1|_T +|b_1^{-1}b_2|_T+\dots + |b_{r-1}^{-1}b_r|_T+|b_r^{-1}g|_T.$$
First we show that for $u$ in canonical form $(\ref{r8}),$ it holds that $|u|_{S,T} \geq ||w||_A+\trace(u)$.
By the choice of generating set $\{S,T\}$ of $A \textrm{ wr } B$, we have that any element $u \in A \textrm{ wr } B$ may be written as
\begin{equation}\label{t7}
u=g_0h_1g_1 \cdots h_mg_m
\end{equation}
where $m\ge 0, g_i\in B, h_j\in A, g_0$ and $g_m$ can be trivial, but all other factors are non-trivial. We may choose the expression \eqref{t7} so that $|u|_{S,T}=\sum_{j=1}^m|h_j|_S+\sum_{i=0}^m|g_i|_T .$ Observe that we may use the expression from Equation $(\ref{t7})$ to write
\begin{equation}\label{r7}
u=(x_1\circ h_1) \dots (x_{m}\circ h_m)g
\end{equation}
where $g=g_0 \dots g_m$ and $x_j=g_0\dots g_{j-1}$, for $j=1, \dots, m$.
Then we have by definition that for the pseudo-canonical form $(\ref{r7})$,
\begin{equation}\label{ppd}
\Psi((x_1\circ h_1) \dots (x_{m}\circ h_m)g)=\sum_{j=1}^m|h_j|_S+|x_1|_T+ |x_1^{-1}x_2|_T +\dots + |x_{m-1}^{-1}x_m|_T +|x_m^{-1}g|_T$$
$$=\sum_{j=1}^m|h_j|_S+\sum_{i=0}^m|g_i|_T = |u|_{S,T}.
\end{equation}
It is possible that in the form of Equation $(\ref{r7})$, some $x_i = x_j$ for $1 \leq i \ne j \leq m$.
When taking $u$ to the canonical form $wg=(b_1\circ a_1) \dots (b_{r}\circ a_r)g$ of Equation $(\ref{r8})$, we claim that
\begin{equation}\label{one}
||w||_A \leq \displaystyle\sum_{j=1}^m|h_j|_S
\end{equation} and that
\begin{equation}\label{two}
\trace(u) \leq |x_1|_T+ |x_1^{-1}x_2|_T+\dots + |x_{m-1}^{-1}x_m|_T +|x_m^{-1}g|_T.
\end{equation}
Obtaining the canonical form requires a finite number of steps of the following nature. We take an expression such as $$(x_1\circ h_1) \dots (x_i\circ h_i)\dots (x_i\circ h_j)\dots (x_{m}\circ h_m)$$ and replace it with
$$(x_1\circ h_1) \dots (x_i\circ h_ih_j)\dots (x_{j-1}\circ h_{j-1})(x_{j+1}\circ h_{j+1})\dots (x_{m}\circ h_m).$$ The assertion of Equation $(\ref{one})$ follows because $$|h_ih_j|_S \leq |h_i|_S+|h_j|_S.$$ Equation $(\ref{two})$ is true because $$|x_{j-1}^{-1}x_{j+1}|_T \leq |x_{j-1}^{-1}x_j|_T+|x_j^{-1}x_{j+1}|_T,$$ which implies that
$$|b_1|_T +|b_1^{-1}b_2|_T+\dots + |b_{r-1}^{-1}b_r|_T+|b_r^{-1}g|_T$$ $$\le |x_1|_T +|x_1^{-1}x_2|_T+\dots + |x_{m-1}^{-1}x_m|_T+|x_m^{-1}g|_T.$$
Finally, we have that $$\trace(u) \leq |b_1|_T +|b_1^{-1}b_2|_T+\dots + |b_{r-1}^{-1}b_r|_T+|b_r^{-1}g|_T,$$ because the right hand side is the length of a particular path in $P$: the path which travels from $1$ to $b_1$ to $b_2, \dots,$ to $b_r$ to $g$. It follows that the length of this path is at least as large as the length of $\spann(u)$.
Thus for a canonical form $u=(b_1\circ a_1)\dots (b_r\circ a_r) g$ we see by Equations $(\ref{ppd})$, $(\ref{one})$ and $(\ref{two})$ that $$||w||_A + \trace(u) \leq \Psi((x_1\circ h_1) \dots (x_{m}\circ h_m)g)=|u|_{S,T}.$$
To obtain the reverse inequality, take $u=(b_1\circ a_1) \dots (b_{r}\circ a_r)g$ in $A \textrm{ wr } B$ in canonical form. By the definition, $\spann(u)$ will be a path that starts at $1$, goes in some order directly through all of $b_1, \dots, b_r$, and ends at $g.$
We may rephrase this to say that for some $\sigma \in \textrm{Sym}(r),$ there is a path $p=\spann(u)$ in $P$ such that $|p|_T=|b_{\sigma(1)}|_T+ |b_{\sigma(1)}^{-1}b_{\sigma(2)}|_T+... + |b_{\sigma(r-1)}^{-1}b_{\sigma(r)}|_T +|b_{\sigma(r)}^{-1}g|_T$. In other words,
$$\trace(u)=|b_{\sigma(1)}|_T+ |b_{\sigma(1)}^{-1}b_{\sigma(2)}|_T+... + |b_{\sigma(r-1)}^{-1}b_{\sigma(r)}|_T +|b_{\sigma(r)}^{-1}g|_T.$$
Moreover, in the wreath product we have that $$u=(b_{\sigma(1)}\circ a_{\sigma(1)}) \cdots (b_{\sigma(r)}\circ a_{\sigma(r)})g$$ $$=b_{\sigma(1)}a_{\sigma(1)}b_{\sigma(1)}^{-1}b_{\sigma(2)}a_{\sigma(2)} \cdots b_{\sigma(r-1)}^{-1}b_{\sigma(r)}a_{\sigma(r)}b_{\sigma(r)}^{-1}g.$$ This implies that $$|u|_{S,T} \leq |b_{\sigma(1)}|_T+|a_{\sigma(1)}|_S+|b_{\sigma(1)}^{-1}b_{\sigma(2)}|_T+ \cdots +|a_{\sigma(r)}|_S+|b_{\sigma(r)}^{-1}g|_T$$ $$= \sum_{j=1}^r|a_j|_S+\trace(u)=||w||_A+\trace(u).$$
\endproof
\section{Distortion in ${\mathbb Z}_p \textrm{ wr } {\mathbb Z}^k$}\label{ppq}
We begin with the following result, the proof of which exploits the formula of Theorem \ref{t6}.
\begin{prop}\label{t3}
The group ${\mathbb Z}_2 \textrm{ wr } {\mathbb Z}^2$ contains distorted subgroups.
\end{prop}
This is interesting in contrast to the case of ${\mathbb Z}_2 \textrm{ wr } {\mathbb Z}$ which has no effects of subgroup distortion. The essence in the difference comes from the fact that the Cayley graph of ${\mathbb Z}$ is one-dimensional, and that of ${\mathbb Z}^2$ is asymptotically two-dimensional, which gives us more room to create distortion using Theorem \ref{t6}.
We will use the following notation in the case of $G={\mathbb Z}_2 \textrm{ wr } {\mathbb Z}^2$: $a$ generates the passive group of order $2$ while $b$ and $c$ generate the active group ${\mathbb Z}^2.$
The canonical form of Equation $(\ref{r8})$ will be denoted by $$((g_1+ \cdots +g_k) a)g$$ for $g_1, \dots, g_k$ distinct elements of ${\mathbb Z}^2$ and $g \in {\mathbb Z}^2$. We may do this because any nontrivial element of ${\mathbb Z}_2$ is just equal to the generator $a$.
\begin{lemma}\label{ss}
Let $H \leq G$ be generated by a nontrivial element $w \in W$ as well as the generators $b,c$ of ${\mathbb Z}^2$. Then $H \cong G$.
\end{lemma}
We know that $W=\displaystyle\bigoplus_{g \in {\mathbb Z}^2}\langle g\circ a\rangle$ is a free module over ${\mathbb Z}_2[{\mathbb Z}^2]$. Therefore, we may think of $W$ as being the Laurent polynomial ring in two variables, say, $x$ for $b$ and $y$ for $c$. We can use the module language to express any element as $w=f(x,y)a=(x^{i_1}y^{j_1}+ \cdots+x^{i_k}y^{j_k})a$, where for $p \ne q$ we have that $x^{i_p}y^{j_p} \ne x^{i_q}y^{j_q}$. This corresponds to the canonical form $w=(g_1+ \cdots +g_k)a$ where $g_p=b^{i_p}c^{j_p}$ for $p=1, \dots, k$.
We now have all the required facts to prove Proposition \ref{t3}.
\begin{proof} of Proposition \ref{t3}:
Let $G={\mathbb Z}_2 \textrm{ wr } {\mathbb Z}^2 = \textrm{gp} \langle a, b, c \rangle$ and $H = \textrm{gp} \langle b, c, w \rangle$ where $w=[a,b]=(1+x)a$. By Lemma \ref{ss} we have that $H \cong G$. Let $$f_l(x)=\displaystyle\sum_{i=0}^{l-1}x^i \textrm{ and } g_l(x)=(1+x)f_l(x).$$
The element $f_l(x)f_l(y)w \in H$ is in canonical form, when written in the additive group notation as $\sum_{i,j=0}^{l-1} b^ic^j\circ w.$
By Theorem \ref{t6}, we have that its length in $H$ is at least $l^2+l^2$ since
the support of it has cardinality $l^2$, and the length of arbitrary loop going through $l^2$ different vertices is at least $l^2$.
\begin{center}
\small{Figure $1$: The $l^2$ vertices (left) and the rectangle with perimeter $2l+2(l-1)$ (right)}
\includegraphics[scale=.58]{sidebyside.pdf}\\
\end{center}
Now we compute the length of $f_l(x)f_l(y)w$ in $G$. We have that $$f_l(x)f_l(y)w=(1+x)f_l(x)f_l(y)a=g_l(x)f_l(y)a=\bigg[\sum_{i=0}^{l-1} (y^i+y^ix^{l})\bigg]a.$$ Theorem \ref{t6} shows that $|f_l(x)f_l(y)w|_G=2l+2(l-1)+2l$. This is because the shortest path in $\cay({\mathbb Z}^2)$ starting at $1$, passing through $1, c, \dots, c^{l-1}$ and $b^l, cb^l, \dots, c^{l-1}b^l$ and ending at $1$ is given by traversing the perimeter of the rectangle, and so gives the length of $2(l-1)+2l$.
Therefore the subgroup $H$ is at least quadratically distorted.
\end{proof}
\begin{rem}
The subgroup $H$ is not normal in $G$ because the element $aca^{-1}$ is not in $H$.
\end{rem}
The proof of Proposition \ref{t3} can be generalized as follows. Consider the group $G={\mathbb Z}_n \textrm{ wr } {\mathbb Z}^k=\textrm{gp} \langle a, b_1, \dots, b_k\rangle$ for $n\ge 2$ and $k>1$. Then the subgroup $H = \textrm{gp} \langle w, b_1, \dots, b_k\rangle$ where $w=(1-x_1) \cdots (1-x_{k-1})a=[...[a,b_1],b_2],...b_{k-1}]$ has distortion at least $l^{k}$. This is a restatement of Proposition \ref{pqp}.
By (the analogue of) Lemma \ref{ss} we have that $H \cong G$ and so we can compute lengths using Theorem \ref{t6}. Consider the element $f_l(x_1) \cdots f_l(x_k)w$ in $H$. Then it has length in $H$ at least equal to $l^k+l^k$ because the path in $\cay({\mathbb Z}^k)$ arising from Theorem \ref{t6} would need to pass through at least $l^k$ vertices: ${b_1}^{\alpha_1} \cdots {b_k}^{\alpha_{k}}$ for $\alpha_i \in \{0, \dots, l-1\}, i=1, \dots, k$. In the group $G$, $$f_l(x_1) \cdots f_l(x_k)w=g_l(x_1) \cdots g_l(x_{k-1})f_l(x_k)a.$$ This has linear length, which follows because the vertices of the support are placed along the edges of a
$k$-dimensional parallelotope, such that the length of any edge of the parallelotope is at most $l$.
\section{Estimating Word Length}
Although the notion of equivalence has only been defined for functions from $\mathbb{N}$ to $\mathbb{N}$, we would like to define a notion of equivalence for functions on a finitely generated group. We say that two functions $f, g: G \rightarrow \mathbb{N}$ are equivalent if there exists $C>0$ such that for any $x \in G$ we have $$\frac{1}{C}f(x)-C \leq g(x) \leq Cf(x)+C.$$ If there is a function $f: G \rightarrow \mathbb{N}$ such that $f \approx | \cdot |_G$, then for any subgroup $H$ of $G$, $\Delta_H^G(l) \approx \max \{ |x|_H : x \in H, f(x) \leq l\}$.
We need to establish a looser way of estimating lengths in ${\mathbb Z} \textrm{ wr } {\mathbb Z}$, than the formula introduced in Lemma \ref{t5}.
Recall that this group has standard generators $a\in W$ (passive)
and $b$ (active).
Here we call {\it exemplary} any sugroup $H=\langle b,w \rangle \leq {\mathbb Z} \textrm{ wr } {\mathbb Z}$ where
$w \in W \backslash 1.$ We have $w=h(x)a,$ where $h(x)= \sum_{j=0}^{t}d_{j}x^j,$ and $d_0\ne 0.$
This follows without loss of generality by conjugating of $w$ by a power of $b$. Thus
we associate a polynomial $h(x)\in{\mathbb Z}[x]$ with any exemplary subgroup $H.$
\begin{lemma}\label{xxx} The mapping
$a\mapsto w, b\mapsto b$ extends to a monomorphism of the wreath product ${\mathbb Z} \textrm{ wr } {\mathbb Z}$ onto the exemplary subgroup
\end{lemma}
\proof This follows because in this case $W$ is a free module with one generator $a$ over the
domain ${\mathbb Z}[\langle b\rangle],$ $w=h(x)a,$ and the mapping
$u\to h(x)u$ ($u\in W$) is an injective module homomorphism. \endproof
Then for any element $g \in H$, we may write $g= f(x)w=(f(x)h(x)a)b^n,$ where $f(x)=\sum_{q=s}^{s+p}z_qx^q$ is
a Laurent polynomial. Denote by $S(f)$ the sum $\sum_{q=s}^{s+p}|z_q|.$
For this element, consider the norms
$$e(g) = S(fh) \textrm{ and } e_H(g) = S(f)$$
Letting $\iota(g)=\max\{t+s+p,0\}, \varepsilon(g)=\min\{s,0\}, \iota_H(g)=\max\{s+p,0\},$
$\varepsilon_H(g)=\min\{s,0\},$ we define $u_H(g)=\iota_H(g)-\varepsilon_H(g) \textrm{ and } u(g)=\iota(g)-\varepsilon(g).$
Consider the function $$\delta(l)= \max \{e_H(g): g \in H \cap W, e(g) \leq l \textrm{ and } u(g) \leq l\}.$$ The following Lemma shows that we may simplify computations of word length in exemplary subgroups.
\begin{lemma}\label{r2}
Let $H=\langle b, w \rangle \leq {\mathbb Z} \textrm{ wr } {\mathbb Z}$ be an exemplary subgroup. Then we have that $$\Delta_H^{{\mathbb Z} \textrm{ wr } {\mathbb Z}}(l) \approx \delta(l).$$
\end{lemma}
\begin{proof}
Recall that by Lemma \ref{t5}, we have the following formulas. For $g \in H$ with the notation established above, we have that:
$$|g|_H=e_H(g)+\min\{-2\varepsilon_H(g)+\iota_H(g)+|n-\iota_H(g)|, 2\iota_H(g)-\varepsilon_H(g)+|n-\varepsilon_H(g)|\}$$ and
$$|g|_{{\mathbb Z} \textrm{ wr } {\mathbb Z}}=e(g)+\min\{-2\varepsilon(g)+\iota(g)+|n-\iota(g)|, 2\iota(g)-\varepsilon(g)+|n-\varepsilon(g)|\}.$$
The following inequality follows from the definitions:
\begin{equation}\label{bb}
\max \{e(g), u(g), |n| \} \leq |g|_{{\mathbb Z}^r \textrm{ wr } {\mathbb Z}}.
\end{equation}
Similarly, we have that
\begin{equation}\label{bbb}
|g|_H \leq e_H(g)+2u_H(g)+|n| \textrm{ and } |g|_{{\mathbb Z} \textrm{ wr } {\mathbb Z}} \leq e(g)+2u(g)+|n|.
\end{equation}
Observe that for $g \in H \cap W$ we have that
\begin{equation}\label{c}
|g|_H \geq \max\{e_H(g), u_H(g)\}.
\end{equation}
Observe that
\begin{equation}\label{cc}
\max \{u_H(g) : g \in H, u(g) \leq l\} \leq l.
\end{equation}
Thus, $$\Delta_H^{{\mathbb Z}\textrm{ wr } {\mathbb Z}}(l) \leq \max \{e_H(g):g \in H, e(g) \leq l, u(g) \leq l\}$$ $$+ \max \{2u_H(g) : g \in H, u(g) \leq l\}+ \max \{|n| :g \in H, |n| \leq l\} \leq \delta(l)+3l.$$
The first inequality follows from Equation (\ref{bb}), the second from Equation (\ref{bbb}).
On the other hand, we have that $$\Delta_H^{{\mathbb Z} \textrm{ wr } {\mathbb Z}}(l) \geq \max\{e_H(g): g \in H \cap W, e(g) \leq l/4, u(g) \leq l/4\}$$ $$-\max\{u_H(g): g \in H \cap W, e(g) \leq l/4, u(g) \leq l/4\} \geq \delta(l/4)-l/4.$$
The first inequality follows from Equation (\ref{bbb}), the second from Equation (\ref{c}), and the third from Equation (\ref{cc}).
Thus $\Delta_H^{{\mathbb Z} \textrm{ wr } {\mathbb Z}}(l)$ and $\delta(l)$ are equivalent.
\end{proof}
\section{Distortion of Polynomials}
In order to understand distortion in exemplary subgroups of ${\mathbb Z} \textrm{ wr } {\mathbb Z}$, we will introduce the notion of distortion of a polynomial.
\begin{defn}\label{pd}
Let $R$ be a subring of a field with a real valuation, and consider the polynomial ring $R[x]$. We will define the norm function $S: R[x] \rightarrow \mathbb{R}^+$ which takes any $f(x)=\sum_{i=0}^na_ix^i \in R[x]$ to $S(f)=\sum_{i=0}^n|a_i|.$ For any $h \in R[x]$ and $c>0$, we define the distortion of the polynomial $h$ from $\mathbb{N}$ to $\mathbb{N}$ by
\begin{equation}\label{3811}
\Delta_{h,c}(l) = \sup\{ S(f) : \deg(f) \leq cl, \textrm{ and } S(hf) \leq cl \}.
\end{equation}
\end{defn}
\begin{rem}
Taking into account the inequality $S(hf) \le cl$, one can easily find some explicit upper boundes $C_i=C_i(h,c,l)$ for the modules of the coefficient at $x^i$ of $f(x)$ in Formula \eqref{pd}, starting with the lowest coefficients. Therefore the supremum in Equation \eqref{3811} is finite. Furthermore, if $R = {\mathbb Z}, \mathbb{R}$ or $\mathbb{C}$ then the supremum is taken over a compact set of polynomials of bounded degree with bounded coefficients, and since $S$ is a continuous function, one may replace $\sup$ by $\max$ in Definition \ref{pd}.
\end{rem}
Note that the distortion does not depend on the constant $c$, up to equivalence, and so we will consider $\Delta_h(l)$.
The following fact makes concrete our motivation for studying distortion of polynomials.
\begin{lemma}\label{hpu}
Let $H$ be an exemplary subgroup
$\langle b, w \rangle \leq {\mathbb Z} \textrm{ wr } {\mathbb Z},$ and $w=h(x)a$ for $h=d_0+\cdots+d_tx^t \in {\mathbb Z}[x]$. Then $$\Delta_h(l) \approx \Delta_H^{{\mathbb Z} \textrm{ wr } {\mathbb Z}}(l).$$
\end{lemma}
\begin{proof}
By Lemma \ref{r2}, we have that $\Delta_H^{{\mathbb Z} \textrm{ wr } {\mathbb Z}}(l) \approx \delta(l)=\max\{e_H(g):g \in H \cap W, e(g) \leq l, u(g) \leq l\}.$ Let $g_l=f_l(x)w \in H \cap W$ be so that $\delta(l)=e_H(g_l)$.
There exists $n \in {\mathbb Z}$ so that $\bar g_l=b^ng_lb^{-n} \in H$ and $\bar g_l=\bar f_l(x)w$ where $\bar f_l(x)$ is a regular polynomial. It is easy to check that $e_H(g_l)=e_H(\bar g_l), e(g_l)=e(\bar g_l)$ and $u(\bar g_l) \leq u(g_l).$ Now observe that $\deg(\bar f_l) \leq u(\bar g_l) \leq u(g_l) \leq l$ and $S(h\bar f_l) =e(\bar g_l)=e(g_l)\leq l$. Therefore, $\Delta_h(l) \succeq S(\bar f_l) = e_H(\bar g_l) = e_H(g_l) \approx \Delta_H^{{\mathbb Z} \textrm{ wr } {\mathbb Z}}(l).$
On the other hand, let us choose any polynomials $f_l(x)$ such that $\deg f_l\le l,$
$S(hf_l)\le l,$ and $\Delta_h(l)=\Delta_{h,1}(l)= S(f_l).$ Then by Lemma \ref{t5},
$|f_l(x)w|_H \ge S(f) = \Delta_h(l)$ while $$|f_l(x)w|_{{\mathbb Z}\textrm{ wr }{\mathbb Z}}= |f_l(x)h(x)a|_{{\mathbb Z}\textrm{ wr }{\mathbb Z}}
\le S(hf) + 2l \le 3l.$$ It follows that $\Delta_H^{{\mathbb Z} \textrm{ wr } {\mathbb Z}}(l) \succeq \Delta_h(l),$
and the lemma is proved.
\end{proof}
\section{Lower Bounds on Polynomial Distortion}
Given any polynomial $h=\sum_{j=0}^td_jx^j \in {\mathbb Z}[x], d_0,d_t \ne 0$ with complex, real or integer coefficients, we are able to compute the equivalence class of its distortion function.
\begin{lemma}\label{hnc}
The distortion $\Delta_h(l)$ of $h$ with respect to the ring of polynomials over ${\mathbb Z}, \mathbb{R},$ or $\mathbb{C}$ is bounded from below by $l^{\kappa+1}$, up to equivalence, where $c$ is a complex root of $h$ of multiplicity $\kappa$ and modulus one.
\end{lemma}
\begin{proof}
Let $c$ be a complex root of $h$ of multiplicity $\kappa$ and modulus $1$. That is, $$h(x)=(x-c)^{\kappa}\tilde{h}(x)$$ over $\mathbb{C}$. Let $$v_l(x)=x^{l-1}+cx^{l-2}+\cdots+c^{l-2}x+c^{l-1}.$$ Then the product $$h(x)v_l^{\kappa+1}(x)=(x^l-c^l)^{\kappa}\tilde{h}(x)v_l(x)$$ satisfies $S(hv_l^{\kappa+1})=O(l)$, because $S(v_l)=O(l)$. On the other hand, because $|c|=1$, we have that $S(v_l^{\kappa+1}) \geq |v_l(c)^{\kappa+1}|=l^{\kappa+1}.$ This implies that if $c \in \mathbb{R}$; i.e. $c = \pm 1$, then $\Delta_h(l) \succeq l^{\kappa+1}$, where the distortion is considered over $\mathbb{C}$, $\mathbb{R}$ or over $\mathbb{Z}$.
We will show that a similar computation holds over $\mathbb{R}$ and over ${\mathbb Z}$ even in the case when $c \in \mathbb{C} - \mathbb{R}$. Let $\bar c$ be the complex conjugate of $c$. By hypothesis that $c \notin \mathbb{R}$ we know that $\bar c \ne c.$ Then $\bar c=c^{-1}$ is a root of $h(x)$ of multiplicity $\kappa$ as well, and $$h(x) = (x-c)^{\kappa}(x-\bar c)^{\kappa} H(x)$$ where $H(x)$ has real coefficients. Consider the product $v_l(x)\bar v_l(x),$ where $$\bar v_l(x) = x^{l-1}+ \bar c x^{l-2}+\cdots +\bar c ^{l-1}.$$ A simple calculation shows that each of the coefficients of this product is a sum of the form $$\sum_{i+j=k} c^i\bar{c}^j= \sum_{i+j=k} c^{i-j}=c^{k} + c^{k-2}+\cdots+ c^{-k}.$$ This is a geometric progression with common ratio $c^2 \ne 1$. Therefore, the modulus of every such coefficient is at most $\frac{2}{|1-c^2|}$ and so $S(v_l\bar v_l)$ is $O(l)$. This computation implies that the products $$h(x)v_l^{\kappa+1}(x)\bar v_l^{\kappa+1}(x) = (x^l - c^l)^{\kappa} (x^l - \bar c^l)^{\kappa}H(x)v_l(x)\bar v_l(x)$$ have the sum of the modules of their coefficients which are $O(l)$.
The polynomial $v_l^{\kappa+1}(x)\bar v_l^{\kappa+1}(x)$ has real coefficients. There is a polynomial $F_l(x)$ with integer coefficients such that each coefficient of $F_l(x)- v_l^{\kappa+1}(x)\bar v_l^{\kappa+1}(x)$ has modulus at most one. Thus $S(hF_l)$ is also $O(l)$.
We will show that the sums of modules of coefficients of $F_l(x)$ grow at least as $l^{\kappa+1}$ on a subsequence from Remark \ref{poa}. It suffices to obtain the same property for $ v_l^{\kappa+1}(x)\bar v_l^{\kappa+1}(x).$ Since $|c|=1$, we have that the sum of the modules of the coefficients of $ v_l^{\kappa+1}(x)\bar v_l^{\kappa+1}(x)$ is at least $$|v_l^{\kappa+1}(c)\bar{v_l}^{\kappa+1}(c)| = l^{\kappa+1}|\bar{v_l}^{\kappa+1}(c)|.$$ We will show that there exists a subsequence $\{l_i\}$ so that on this sequence, $$|\bar v_{l_i}^{\kappa+1}(c)| \geq \frac{1}{2}.$$ We have that $$\bar v_l(c) = c^{l-1}+c^{l-2}\bar c+\cdots+\bar c^{l-1}=c^{l-1}+ c^{l-3} +\cdots+ c^{1-l}$$ because $\bar c=c^{-1}$. Therefore $|\bar v_l(c)| = |1+c^2+\cdots+c^{2l-2}|$ and similarly, $|\bar f_{l+1}(c)| = |1+c^2+\cdots+c^{2l}|.$ One of these two numbers must be at least one half because $|\bar v_l(c) - \bar v_{l+1}(c)| = |c^{2l}| =1$. Thus either $l$ or $l+1$ can be included in the sequence $\{l_i\}$, and all required properties are shown.
\end{proof}
\section{Upper Bounds on Distortion of Polynomials}
In order to obtain upper bounds on distortion of polynomials we require some facts from linear algebra.
Fix an integer $k \geq 1$ and let $n>0$ be arbitrary.
\begin{lemma}\label{ob}
Let $Y_1, \dots, Y_{n}, C_2, \dots, C_{n}$ be $k \times 1$ column vectors. Suppose that the sum of the modules of all coordinates of $C_2, \dots, C_{n}$ is bounded by some constant $c$, and that the modulus of each coordinate of $Y_1$ and $Y_{n}$ is also bounded by $c$. Suppose further we have the relationship
\begin{equation}\label{mnt}
Y_i=AY_{i-1}+C_i, i=2, \dots, n
\end{equation}
where $A$ is a $k \times k$ matrix, in Jordan normal form, having only one Jordan block. Then the modulus of each coordinate of arbitrary $Y_i, 2 \leq i \leq n-1$ is bounded by $dcn^{k-1}$ where $d$ depends on $A$ only. In the case where the eigenvalue of $A$ does not have modulus one, the modulus of each coordinate of arbitrary $Y_i, 2 \leq i \leq n-1$ is bounded by $cd$, where $d$ depends on $A$ only. All matrix entries are assumed to be complex.
\end{lemma}
\begin{proof}
Let $\lambda$ be the eigenvalue of $A$, so that $A=\begin{pmatrix}
\lambda & 0 & 0 \hdots & 0 \\
1 & \lambda & 0 \hdots & 0\\
\vdots & \ddots & \ddots & \vdots \\
0 & 0 \hdots & 1 & \lambda \\
\end{pmatrix}.$
We will consider cases.
\begin{itemize}
\item First suppose that $|\lambda| < 1$.
\end{itemize}
From Formula \eqref{mnt} we derive:
\begin{equation}\label{zx}
Y_i=A(AY_{i-2}+C_{i-1})+C_i=(A)^2Y_{i-2}+AC_{i-1}+C_{i} = \cdots $$
$$= (A)^{i-1}Y_1+(A)^{i-2}C_2+ \cdots + AC_{i-1}+C_i.
\end{equation}
The following formula for $A^r$ is well-known because $A$ is assumed to be a Jordan block; it may also be checked easily using induction. We have that
$$A^r=
\begin{pmatrix}
\lambda^r & 0 & 0 \hdots & 0 \\
r\lambda^{r-1} & \lambda^r& 0 \hdots & 0\\
\frac{r(r-1)}{2!}\lambda^{r-2} & r\lambda^{r-1} & \lambda^r \hdots & 0\\
\vdots & \ddots & \ddots & \vdots \\
\frac{r!}{(r-(k-1))!(k-1)!}\lambda^{r-(k-1)} \hdots & \frac{r(r-1)}{2!}\lambda^{r-2} & r\lambda^{r-1} & \lambda^r \\
\end{pmatrix},
$$ with the understanding that if $r < k-1$, any terms of the form $\binom{r}{j}\lambda^{r-j}$ where $r<j$ are $0$.
Arbitrary nonzero element of the matrix $A^r$ is of the form $\binom{r}{j}\lambda^{r-j}$ for some $j \leq k-1$.
Let $a_{s,t}(r)$ denote the $s,t$ entry of $A^r$. Then $a_{s,t}(r)$ is either zero or of the form $\binom{r}{j}\lambda^{r-j}$ for some $0\leq j\leq k-1$ depending on $s$ and $t$. Then $$\sum_{r=1}^i|a_{s,t}(r)| \leq \sum_{r=1}^{\infty}|a_{s,t}(r)|=\sum_{r=1}^{\infty}|\binom{r}{j}\lambda^{r-j}|$$ which is a constant depending on $A$ and not on $i$, because the series on the right is convergent when $|\lambda|<1$. Let $$c_1=\max_{1 \leq s,t \leq k}\{\sum_{r=1}^{\infty}|a_{s,t}(r)|\}.$$ Let $\bar{A}$ be the $k \times k$ matrix whose $s,t$ entry is $\sum_{r=1}^{\infty}|a_{s,t}(r)|$, and the column $\bar{C}$ be obtained by placing in the $j^{th}$ row the sum of the modules of the entries of the $j^{th}$ row of all $C_i$ and $Y_1$. Then every entry of $\bar{C}$ is bounded by $2c$. Observe that the modulus of every entry in the right side of \eqref{zx} is bounded by an entry of $\bar{A}\bar{C}$, which is in turn bounded by $2cc_1$, which does not include any power of $n$ at all.
\begin{itemize}
\item Let $|\lambda| > 1$.
\end{itemize}
Because $\lambda^{-1}$ is an eigenvalue of $A^{-1}$, there exists a decomposition $A^{-1}=SJS^{-1}$ where $$J=\begin{pmatrix}
\frac{1}{\lambda} & 0 & 0 \hdots & 0 \\
1 & \frac{1}{\lambda} & 0 \hdots & 0\\
\vdots & \ddots & \ddots & \vdots \\
0 & 0 \hdots & 1 & \frac{1}{\lambda} \\
\end{pmatrix}.$$ Letting $Y_i'=S^{-1}Y_i$ and $C_i'=S^{-1}C_i$ we see by Equation \eqref{mnt} that $$Y_{n-r}'=J^{r}Y_n'+J^{r}C_n'+ J^{r-1}C_{n-1}'+\cdots + JC_{n-r+1}',$$ for $r=1,\dots,n-2$. Observe that the sum of modules of coordinates of $Y_{n-r}'$ is less than or equal to $ksc$, where $s$ depends on $S$ (and hence on $A$) only. Similarly, the sum of all modules of all coordinates of $C_2',\dots,C_n'$ is bounded above by $ksc$. This case now follows just as the previous one to obtain constant upper bounds on the modules of the entries in $Y_2',\dots,Y_{n-1}'$. Finally, the modulus of any coordinate of $Y_{n-r}$ is bounded by $ks$ times the modulus of a coordinate of $Y_{n-r}'$.
\begin{itemize}
\item Let $|\lambda| = 1$.
\end{itemize}
In this case, we have that $$|\binom{r}{j}\lambda^{r-j}| = \binom{r}{j} = \frac{r(r-1) \cdots (r-(j-1))}{j!}$$ $$\leq r(r-1) \cdots (r-(j-1)) \leq r^j \leq n^{k-1}.$$
It follows from Equation $(\ref{zx})$ that every entry of $Y_i$ is bounded above by $2cn^{k-1}$.
\end{proof}
\begin{lemma}\label{yx}
Let $Y_1, \dots, Y_{n}, C_2, \dots, C_{n}$ be $k \times 1$ column vectors. Suppose that the sum of the modules of all coordinates of $C_2, \dots, C_{n}$ is bounded by some constant $c$, and that the modulus of each coordinate of $Y_1$ and $Y_{n}$ is also bounded by $c$. Suppose further we have the relationship $$Y_i=AY_{i-1}+C_i, i=2, \dots, n $$where $A$ is a $k \times k$ matrix. Then the modulus of each coordinate of arbitrary $Y_i, 2 \leq i \leq n-1$ is bounded by $dcn^{\kappa-1}$ where $d$ depends on $A$ only, and $\kappa \leq k$ is the maximal size of any Jordan block of the Jordan form of $A$ having eigenvalue with modulus one.
\end{lemma}
\begin{proof}
There exists a Jordan decomposition, $A=SA'S^{-1}$.
Let $S^{-1}=(s_{i,j})_{1 \leq i,j \leq k}$ and let $s=\max |s_{i,j}|$. Then for $C_i'=S^{-1}C_i$ and $Y_i'=S^{-1}Y_i$ we have that
\begin{equation}\label{s}
Y_i'=A'Y_{i-1}'+C_i'.
\end{equation}
By hypothesis, the sum of the modules of all coordinates of $C_2',\dots,C_n'$ is bounded by $ksc=c'$ and the coordinates of $Y_1'$ and $Y_{n}'$ are bounded by $c'$ as well. As we will explain, our problem can be reduced to the similar problem for $Y_i'$ in (\ref{s}). Suppose that the modules of coordinates of every $Y_i'$ are bounded by $dc'n^{\kappa-1}$ where $d$ depends on $A$ only. Then, letting $S=(\tilde{s_{i,j}})_{1 \leq i,j \leq k}$ and $\tilde{s}=\max|\tilde{s_{i,j}}|$ we have by definition of $Y_i'$ that arbitrary element of $Y_i$ has modulus bounded above by $k\tilde{s}dc'n^{\kappa-1}=d'cn^{\kappa-1}$ where $d'=k^2s\tilde{s}d$ only depends on $A'$, as required.
Lemma \ref{ob} says that if $A'$ has only one Jordan block, then the bound is constant if the eigenvalue does not have modulus one. Otherwise, we have in this case that $k=\kappa$ and the claim is true. If there is more than one Jordan block present in $A'$, the problem is decomposed into at most $k$ subproblems, each with only one Jordan block of size smaller than $k$. Again, we are done by Lemma \ref{ob}.
\end{proof}
We will use Lemma \ref{yx} to prove the following fact, which requires establishing some notation prior to being introduced. Let $d_0, \dots, d_t \in {\mathbb Z}$ where $d_0,d_k \ne 0$. Let the $(n+k) \times n$ matrix $M$ have $j^{th}$ column, for $j=1, \dots, n$, given by $[0, \dots, 0, d_0, d_1, \dots, d_k, 0, \dots, 0]^{T}$, where $d_0$ first appears as the $j^{th}$ entry in this $j^{th}$ column. Given the matrix $M$, we may also construct the matrix
\begin{equation}\label{8u}
A=
\begin{pmatrix}
0 & 1 & 0 \hdots & 0 \\
0 & 0 & 1 \hdots & 0\\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 \hdots & 0 & 1 \\
a_1 & a_2 \hdots & a_{k-1} & a_{k}, \\
\end{pmatrix}
\end{equation}
where $a_j=-\frac{d_{k-j+1}}{d_0}$, for $j=1,\dots,k$. Let $\kappa$ be the maximal size of a Jordan block of the Jordan form of $A$ having eigenvalue with modulus one.
\begin{lemma}\label{r9}
Suppose that $X=[x_1, x_2, \dots, x_n]^T$ is a solution to the system of equations $MX=B$, where $B=[b_1, b_2, \dots, b_{n+k}]^T$. Then it is possible to bound the modules of all coordinates $x_1, \dots, x_n$ of the vector $X$ such that $|x_i| \leq cbn^{\kappa-1}$ where $b=\sum_j\{|b_j|\}$ for $1 \leq j \leq n+k$ and the constant $c$ depends upon $d_0, \dots, d_k$ only.
\end{lemma}
Prior to proving Lemma \ref{r9} we prove an easier special case.
\begin{lemma}\label{a}
It is possible to bound the coordinates $x_1, \dots, x_{k}$ of the vector $X$ from Lemma \ref{r9} from above by $b\tilde{\gamma}$ where $b=\sum_j\{|b_j|\}$ and $\tilde{\gamma}=\tilde{\gamma}(d_0, \dots, d_{k-1})$.
\end{lemma}
\begin{proof}
By Cramer's Rule, we have the explicit formula that $$|x_i|=\bigg|\frac{\det(L_i)}{\det(L)}\bigg|$$ where $L$ is the $k \times k$ upper left submatrix of $M$ corresponding to the first $k$ equations, and $L_i$ is obtained by replacing column $i$ in $L$ with $[b_1, \dots, b_{k}]^T$. Because $\det(L)=d_0^{k}$, it suffices to show that the desired bounds exist for $\det(L_i);$ that is, we must show that there exists a constant $\tilde{\gamma}$ depending on $d_0, \dots, d_{k-1}$ only such that $|\det(L_i)| \leq b\tilde{\gamma}$ for $i=1, \dots, k$. By expanding along the $i^{th}$ column in $L_i$, we find that $$\det(L_i)=\pm b_1f_1(d_0, \dots, d_{k-1})\pm b_2f_2(d_0, \dots, d_{k-1}) \pm \cdots \pm b_{k}f_{k}(d_0, \dots, d_{k-1}),$$ where for each $i=1, \dots, k$, $f_i$ is a function of $d_0, \dots, d_{k-1}$ only obtained as the determinant of a submatrix containing none of $b_1, \dots, b_{k}$. The proof is complete by the triangle inequality.
\end{proof}
Note that the $|x_j|$ for $j=n-k+1, \dots, n$ are similarly bounded by $b\overline{\gamma}$ for the same $b$ and some $\overline{\gamma}=\overline{\gamma}(d_{0}, \dots, d_{k-1})$ as in Lemma \ref{a}. It is clear according to Lemma \ref{a} that we may assume that $|x_i| \leq b\gamma$ for the same $\gamma=\gamma(d_0, \dots, d_{k-1})$ for all $i=1, \dots, k, n-k+1, \dots, n$.
We proceed with the proof of Lemma \ref{r9}.
\begin{proof}
It suffices to obtain upper bounds for $|x_i|$ when $n-k \geq i \geq k+1$.
For such indices, we have that $$d_{k}x_{i-k}+d_{k-1}x_{i+1-k}+ \cdots + d_0x_i=b_i.$$ In other words, $$x_i=\xi_i+a_1x_{i-k}+a_2x_{i+1-k}+\cdots+a_{k}x_{i-1},$$ where $\xi_i=\frac{b_i}{d_0}$ and $a_j=-\frac{d_{k-j+1}}{d_0}$. Let $X_i=[x_{i-k+1}, \dots, x_i]^{T}$ and let $\Xi_i=[0, \dots, 0, \xi_i]^T$. Then for the matrix $A$ of Equation \eqref{8u} we have the recursive relationship $$X_i=AX_{i-1}+\Xi_i$$ for $i=k, \dots, n$. Observe that the matrix $A$ depends on $d_0, \dots, d_k$ only, and that the sum of modules of the entries in all $\Xi_i$ are bounded by $\frac{b}{|d_0|}$.
We see by Lemma \ref{a} that Lemma \ref{yx} applies to our situation. Therefore, the modules of coordinates of arbitrary $X_i$, $k+1 \leq i \leq n-k$ are bounded by $dc(n-k+1)^{\kappa-1}\leq dcn^{\kappa-1}$, where $d$ depends only on $d_0, \dots, d_k$, $c=\max\{\frac{b}{|d_0|},\gamma b\}$.
\end{proof}
\begin{lemma}\label{cz}
Let $h(x)=d_0+\dots+d_tx^t$, where $d_0,d_t \ne 0$. Then the distortion of $h$ is at most $l^{\kappa+1}$ where $\kappa$ is the maximal size of a Jordan block of the Jordan form of $$A=
\begin{pmatrix}
0 & 1 & 0 \hdots & 0 \\
0 & 0 & 1 \hdots & 0\\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 \hdots & 0 & 1 \\
-\frac{d_t}{d_0} & -\frac{d_{t-1}}{d_0} \hdots & -\frac{d_2}{d_0} & -\frac{d_1}{d_0} \\
\end{pmatrix}
$$ of Equation \eqref{8u} with eigenvalue having modulus one.
\end{lemma}
\begin{proof}
Consider any $f=\sum_{q=s}^{s+p}z_qx^q$ as in Definition \ref{pd}. Then consider $hf=\sum_{j=s}^{s+p+t}y_jx^j$. The coefficients $y_j$ are given by the matrix equation $MZ=Y,$ where $Z=[z_{s}, \dots, z_{s+p}]^T, Y=[y_{s}, \dots, y_{s+p+t}]^T$ and
$$M=
\begin{pmatrix}
d_{0} & 0 & 0 & \hdots & 0 \\
d_{1} & d_{0} & 0 & \hdots & 0 \\
d_{2} & d_{1} & d_{0}& \hdots & 0\\
\vdots & \hdots & \ddots & \ddots & \vdots \\
d_{t} & d_{t-1} & \hdots & d_{1} \hdots & 0 \\
0 & d_{t} & \hdots & d_{2} \hdots & 0 \\
\vdots & & \ddots & & \vdots \\
0 & \hdots & 0 & d_{t} & d_{t-1} \\
0 & \hdots & 0 & 0 & d_{t} \\
\end{pmatrix}
$$
is an $(p+t+1) \times (p+1)$ matrix.
By Lemma \ref{r9} we have that for each $q=s, \dots, s+p$ that $|z_q| \leq cy(p+1)^{\kappa-1}$ where $c=c(d_0,\dots,d_t), y=\sum_j |y_{j}| \leq l.$ Therefore, $$\Delta_h(l) \leq S(f) = \sum_{q=s}^{s+p}|z_q| \leq c(l+1)^{\kappa+1}.$$
\end{proof}
The following theorem shows that the upper and lower bounds are the same, and so we can compute exactly the distortion of a polynomial.
\begin{theorem}\label{tgs}
Let $h(x)=d_0+\dots+d_tx^t$ be a polynomial in ${\mathbb Z}[x]$. Then the distortion of $h$ is equivalent to a polynomial. Further, the degree of this polynomial is exactly one plus the maximal multiplicity of a (complex) root of $h(x)$ having modulus one.
\end{theorem}
\begin{proof}
On the one hand, Lemma \ref{hnc} shows that the distortion is bounded from below by the
polynomial of degree one plus the maximal multiplicity $\kappa$ of a root of $h(x)$ having modulus one.
On the other hand, the characteristic polynomial $\chi(x)$ of the matrix $A$ in Lemma \ref{cz} equals $x^t+\frac{d_1}{d_0}x^{t-1}+\cdots+\frac{d_{t-1}}{d_0}x+\frac{d_t}{d_0}=x^th(x^{-1})/d_0$.
And so the real polynomials $\chi(x)$ and $h(x)$ have the same roots with modulus $1$ (and with the
same multiplicities). Since the size of a Jordan block does not exceed
the multiplicity of the root of the characteristic polynomial, we have $\Delta_h(l) \preceq l^{\kappa +1}$ by Lemma \ref{cz}. The theorem is proved.
\end{proof}
\begin{rem} Theorem \ref{tgs} will be used here for polynomials with integer coefficients,
but it is valid (with the same proof) for polynomials with complex or real coefficients.
\end{rem}
Theorem \ref{tgs} and Lemma \ref{cz} imply the following.
\begin{cor} \label{exempl} The distortion of any exemplary subgroup $H$ of ${\mathbb Z}\textrm{ wr }{\mathbb Z}$ is equivalent to a polynomial. The degree of this polynomial is exactly one plus the maximal multiplicity of a (complex) root having modulus one of the polynomial $h(x)$ associated with $H$.
\end{cor}
\section{Tame Subgroups}
For every $k\ge 1,$ the wreath product ${\mathbb Z}\textrm{ wr }{\mathbb Z}$ has subgroups $W \lambda \langle b^k\rangle$ isomorphic to ${\mathbb Z}^k \textrm{ wr } {\mathbb Z},$ and so we are forced to study distortion in the groups ${\mathbb Z}^k \textrm{ wr } {\mathbb Z}$
even we are interested in ${\mathbb Z} \textrm{ wr } {\mathbb Z}$ only. Let $a_1,\dots,a_k; b$ be canonical generators of ${\mathbb Z}^k \textrm{ wr } {\mathbb Z}.$ If a subgroup $H$ of $G={\mathbb Z}^k \textrm{ wr } {\mathbb Z}$ is generated by $b, w_1,\dots, w_k,$ where every
$w_i$ belongs to the normal closure $W_i$ of $a_i$ ($W_i$ = the submodule ${\mathbb Z}[\langle b\rangle]a_i$
of $W$) then we say that $H$ is a {\it tame} subgroup of ${\mathbb Z}^k \textrm{ wr } {\mathbb Z}.$
If $w_i\ne 1,$ then the subgroup $H_i$ is an exemplary subgroup of the wreath product
$G_i = W_i\lambda \langle b\rangle \cong {\mathbb Z} \textrm{ wr } {\mathbb Z}.$
\begin{lemma}\label{lcc}
For the tame subgroup $H$, we have that $$\Delta_H^G(l) \approx \sum_{i=1}^k \Delta_{H_i}^{G_i}(l).$$
\end{lemma}
\begin{proof}
Observe that $H_i \hookrightarrow H$ is an undistorted embedding, due to that fact that $H_i$ is a retract of $H$ (and similarly for $G_i \hookrightarrow G$). Therefore, by Lemma \ref{wkf} we have that $$\Delta_{H_i}^{G_i}(l) \preceq \Delta_{H_i}^G(l) \preceq \Delta_H^G(l),$$ for every $i$, and therefore $ \Delta_H^G(l)\succeq \sum_{i=1}^k \Delta_{H_i}^{G_i}(l).$
To prove the other inequality, we
consider an element $u=vb^t\in H$ with $|u|_G\le l.$ Then there is a unique decomposition
$v=v_1+\dots +v_k,$ where
$v_i\in H_i,$ and for $u_i=v_ib^t,$ we have $u_i\in H_i$ since $H$ is tame. Then we have $|u_i|_{G_i}
\le |u|_G\le l$ since $G_i$ is a retract of $G.$ Therefore the required inequality
will follow from the inequality $|u|_H\le \sum_i |u_i|_{H_i}.$
This inequality is true indeed by Theorem \ref{t6} because $\trace_H(u)\le \sum_i \trace _{H_i}(u_i)$ since $\supp_H(u)\subset \cup_i \supp_{H_i} (u_i),$ and $||v||_H \le
\sum_i ||v_i||_{H_i}$ since $H$ is a tame subgroup of $G.$
\end{proof}
\begin{cor} \label{tame} Every tame subgroup of ${\mathbb Z}^k\textrm{ wr }{\mathbb Z}$ has a polynomial
distortion.
\proof The statement follows from Corollary \ref{exempl} and Lemma \ref{lcc}. \endproof
\end{cor}
\section{Some Modules}\label{mt}
To get rid of the word `tame' in the formulation of Lemma \ref{tame}, we will need few remarks about modules.
The following is well known (see also \cite{fs}).
\begin{lemma}\label{r1}
The ring $F[\langle b \rangle]$ is a principal ideal domain if $F$ is a field.
\end{lemma}
\begin{lemma}\label{ccc}
Suppose that $\overline{W}$ is a submodule of a free module $\overline{V}$ of rank $k$ over a (commutative) principal ideal ring $R.$ Then $\overline V$ is a free module of rank $l \leq k$, and modules $\overline{V}$ and $\overline{W}$ have bases $e_1', \dots, e_l'$ and $f_1', \dots, f_k'$ respectively such that for some $u_i' \in R,$ $$e_i'=u_i'f_i', i=1, \dots, l$$
\end{lemma}
At first we apply this statement to the following special case of Theorem \ref{x} Part $(2)$.
\begin{lemma}\label{ta}
If $p$ is a prime, then any finitely generated subgroup $H$ of $G={\mathbb Z}_p^k \textrm{ wr } {\mathbb Z}$ containing
the generator $b$ is undistorted.
\end{lemma}
\begin{proof}
By Lemma \ref{wkf}, it suffices to show that $H$ has finite index in a retract $K$ of $G.$
Since $p$ is a prime, ${\mathbb Z}_p$ is a field. This implies by Lemma \ref{r1}, that the ring $R={\mathbb Z}_p[\langle b \rangle]$ is a principal ideal ring.
Let $V=H\cap W.$ Then $V$ is a free $R$-module by Lemma \ref{ccc}, and we have that $V$ and $W$ have bases $e_1, \dots, e_m$ and $f_1, \dots, f_k$ respectively, for $m \leq k$ such that
\begin{equation}\label{xc}
e_i=g_if_i, i=1, \dots, m
\end{equation}
for some polynomials $g_i \in R\backslash 0$. Thus we can choose the generators for $G$ and $H$ to be $\{b, f_1, \dots, f_l \}$ and $\{b, e_1, \dots, e_m\}$, respectively, and $H$ is a subgroup
of the retract $K$ of $G$, where $K$ is isomorphic to ${\mathbb Z}_p^m \textrm{ wr } {\mathbb Z}$ and is generated by
$\{b, f_1, \dots, f_m \}.$ Now $V$ is a submodule of the ${\mathbb Z}_p[\langle b\rangle]$-module
$W'$ generated by $\{f_1, \dots, f_m \},$ and the factor-module $W'/V$ is a direct sum
of cyclic modules $\langle f_i \rangle/\langle g_if_i\rangle.$ Hence $W'/V$
is finite since it is easy to see that each $\langle f_i \rangle/\langle g_if_i\rangle$
has finite order at most $p^{\deg g_i}.$ Since the subgroup $H$ contains $b,$ the index
of $H$ in $K$ is also finite.
\end{proof}
We return to our discussion of module theory. Let $H \leq {\mathbb Z}^k \textrm{ wr } {\mathbb Z}$ be generated by $b$, as well as any elements $w_1, \dots, w_k \in W$. Let $V$ be the normal closure of $w_1, \dots, w_k$ in ${\mathbb Z}^r \textrm{ wr } {\mathbb Z};$ i.e., the ${\mathbb Z}[\langle b\rangle]$-submodule of $W$ generated by $w_1,\dots, w_k.$ Let $\overline{V}= V \otimes_{{\mathbb Z}} {\mathbb Q}$ and $\overline{W} = W \otimes_{{\mathbb Z}} {\mathbb Q}$. Observe $\overline{W}$ and $\overline{V}$ are free modules over $\q[\langle b \rangle]$ of respective ranks $k$ and $l \leq k$.
\begin{rem}\label{cb}
It follows from Lemma \ref{ccc} that there exist $0<m,n \in {\mathbb Z}$ with $(me_i')=u_i(nf_i')$ where $e_i=me_i' \in V, f_i=nf_i' \in W, u_i \in \z[\langle b \rangle]$. Moreover, the modules generated by $\{e_1, \dots, e_l\}$ and $\{f_1, \dots, f_k\}$ are free.
\end{rem}
\begin{rem}\label{ca}
There is a bijective correspondence between the set of finitely generated $\z[\langle b \rangle]$ submodules $N$ of $\z[\langle b \rangle]^k$ and the set of subgroups $K=N \lambda\langle b \rangle$ of ${\mathbb Z}^k \textrm{ wr } {\mathbb Z}$ such that the finite set of generators of $K$ is of the form $b, w_1, \dots, w_k$, $w_i \in W$.
\end{rem}
\begin{rem}\label{bfz}
Let $V_1$ and $W_1$ be generated as submodules over $\z[\langle b \rangle]$ by the elements from Remark \ref{cb}: $e_1, \dots, e_l$ and $f_1, \dots, f_k$ respectively. Let $H_1$ and $G_1$ be subgroups of ${\mathbb Z}^r \textrm{ wr } {\mathbb Z}$ generated by $\{b, V_1\}$ and $\{b, W_1\}$ respectively. It follows by Remark \ref{cb} that that $G_1 \cong {\mathbb Z}^k \textrm{ wr } {\mathbb Z}$ and $H_1 \cong {\mathbb Z}^l \textrm{ wr } {\mathbb Z}$.
\end{rem}
\begin{rem}\label{bc}
Observe that under the correspondence of Remark \ref{ca} each generator $e_i$ of the group $H_1$ is in the normal closure of only one generator $f_i$ of $G_1$, i.e., $H_1$ is
a tame subgroup of $G_1.$
\end{rem}
\begin{lemma}\label{ba}
There exists $0<n', m' \in \mathbb{N}$ so that $n'W \subset W_1 \subset W,$ and $m'V \subset V_1 \subset V.$
\end{lemma}
\begin{proof}
By Remark \ref{ca} we have that $V$ is a finitely generated $\z[\langle b \rangle]$ module with generators $w_1, \dots, w_k$. For each $w_i$, we have that the element $w_i \otimes 1 \in \overline{V}$. Therefore, by Lemma \ref{ccc}, there are $\lambda_{i,j} \in \q[\langle b \rangle]$ so that $w_i=\sum_{j=1}^{l}\lambda_{i,j}e_j'.$ First observe that $mw_i=\sum_{j=1}^{l}\lambda_{i,j}e_j,$ because $e_i=me_i' \in V$.
Next, there exists $M_i \in \mathbb{N}$ so that $M_imw_i = \sum_{j=1}^{l} \mu_{i,j}e_j \in V_1$ where $\mu_{i,j} \in \z[\langle b \rangle]$. Let $m'=M_1 \dots M_k m.$ Then for any $v \in V$, we have that $v=\sum_{i=1}^kv_iw_i$ where $v_i \in \z[\langle b \rangle],$ and therefore, $m'v \in V_1$ as required. A similar argument works for obtaining $n'$.
\end{proof}
\begin{lemma}\label{ab}
Let ${\mathbb Z}^k \textrm{ wr } {\mathbb Z}=G = W \lambda \langle b \rangle$ and let $K= \langle \langle w_1, \dots, w_s \rangle \rangle^{G} \leq G$ be the normal closure of elements $w_i \in W$. Suppose that there exists $n \in \mathbb{N}$ and a finitely generated subgroup $K' \leq K$ so that $nK \leq K'.$ Then $$\Delta_{\langle b, K' \rangle}^G(l) \approx \Delta_{\langle b, K \rangle}^G(l).$$
\end{lemma}
\begin{proof}
We will use the notation that $K_1=\textrm{gp} \langle K, b \rangle, K_1' = \textrm{gp} \langle K', b \rangle, K_1'' = \textrm{gp} \langle nK, b \rangle.$
Observe that the mapping $\phi: G \rightarrow G: b \rightarrow b, w \rightarrow nw \textrm{ for } w \in W$ is an injective homomorphism which restricts to an isomorphism $K_1 \rightarrow K_1''$. An easy computation which uses Lemma \ref{t5} and the definition of $\phi$ shows that for any $g \in K_1$, we have that
\begin{equation}\label{e1}
|g|_{G} \leq |\phi(g)|_{G} \leq n|g|_{G}
\end{equation} where the lengths are computed in $G$ with respect to the usual generating set $\{a_1, \dots, a_k,b\}$.
Observe that under the map $\phi$ we have that
\begin{equation}\label{e2}
\textrm{for } x \in K_1, |x|_{K_1}= |\phi(x)|_{K_1''},
\end{equation}
where the lengths in $K_1''$ are computed with respect to the images under $\phi$ of a fixed generating set of $K_1$.
By their definitions, we have the embeddings
\begin{equation}\label{ac}
K_1'' \leq K_1' \leq K_1 \overset{\phi}{\hookrightarrow} K_1''.
\end{equation}
By Equation (\ref{ac}) there exists $k'>0$ depending only on the chosen generating sets of $K_1$ and $K_1'$ so that
\begin{equation}\label{e3}
\textrm{for any } x \in K_1', |x|_{K_1} \leq k'|x|_{K_1'}.
\end{equation}
It also follows by Equation (\ref{ac}) that there exists a constant $k>0$ depending only on the chosen generating sets of $K_1''$ and $K_1'$ so that
\begin{equation}\label{xa}
\textrm{for any } x \in K_1'', |x|_{K_1'} \leq k|x|_{K_1''}.
\end{equation}
First we show that $\Delta_{K_1''}^{G}(l) \preceq \Delta_{K_1}^{G}(l).$
Let $g \in K_1''$ be such that $|g|_{G} \leq l$ and $|g|_{K_1''}=\Delta_{K_1''}^{G}(l)$. Then there exists $g' \in K_1$ such that $\phi(g')=g$. Therefore, it follows that $\Delta_{K_1''}^{G}(l) = |g|_{K_1''} =|\phi(g')|_{K_1''}=|g'|_{K_1} \leq \Delta_{K_1}^{G}(l).$ The first and second equalities follow by definition, the third by Equation (\ref{e2}), and the inequality is true because by Equation (\ref{e1}) we have that $|g'|_G \leq |\phi(g)|_G=|g|_G \leq l.$
We claim that $\Delta_{K_1}^G(l) \preceq \Delta_{K_1'}^G(l).$
Let $g \in K_1$ be such that $|g|_{K_1}=\Delta_{K_1}^G(l)$. Then $|g|_{K_1} \leq |\phi(g)|_{K_1} \leq k'|\phi(g)|_{K_1'} \leq k'\Delta_{K_1'}^G(nl),$ which follows from Equations (\ref{e1}), (\ref{e3}) and by definition.
On the other hand, we will show that $\Delta_{K_1'}^G(l) \preceq \Delta_{K_1''}^G(l).$
Let $g \in K_1'$ be such that $|g|_{K_1'}=\Delta_{K_1'}^G(l)$. Then $|g|_{K_1'} \leq |\phi(g)|_{K_1'} \leq k|\phi(g)|_{K_1''} \leq k\Delta_{K_1''}^G(nl),$ which follows from Equations (\ref{e1}), (\ref{xa}) and by definition.
Therefore, we have that $\Delta_{K_1}^G(l) \preceq \Delta_{K_1'}^G(l) \preceq \Delta_{K_1''}^G(l) \preceq \Delta_{K_1}^G(l).$
\end{proof}
We say that $H$ is a {\it subgroup with $b$} in a wreath product $A \textrm{ wr } \langle b\rangle$ if
$H=\langle b,w_1,\dots,w_s\rangle$ where $w_1,\dots, w_s\in W$.
\begin{lemma}\label{addm}
Let $H$ be a subgroup with $b$ in $G={\mathbb Z}^k \textrm{ wr } {\mathbb Z}$. Then the distortion of $H$ in ${\mathbb Z}^k \textrm{ wr } {\mathbb Z}$ is equivalent to the distortion of a tame subgroup $H_1$ of a wreath product $G_1={\mathbb Z}^l \textrm{ wr }{\mathbb Z},$ $l\le k.$
\end{lemma}
This follows from the results of Section \ref{mt}. Recall that the tame subgroup $H_1$ of the group $G_1$ was defined in Lemma \ref{bfz}, and these groups were associated to the given $H \leq G$. It follows from Lemmas \ref{ba} and \ref{ab} that $$\Delta_{G_1}^{G}(l) \approx \Delta_{G}^{G}(l) \approx l \textrm{ and } \Delta_{H_1}^{G}(l) \approx \Delta_{H}^{G}(l),$$
and therefore $\Delta_{H_1}^{G_1}(l) \approx \Delta_{H}^{G}(l).$
\begin{cor}\label{withb} The distortion of every subgroup with $b$ in ${\mathbb Z}^k \textrm{ wr }{\mathbb Z}$ is polynomial.
\end{cor}
\proof This follows from Corollary \ref{tame} and Lemma \ref{addm}. \endproof
\section{Distortion in $A \textrm{ wr } {\mathbb Z}$}\label{vvv}
In this section, we will reduce distortion in subgroups of $A \textrm{ wr } {\mathbb Z}$ where $A$ is finitely generated abelian to that in subgroups of ${\mathbb Z}^k \textrm{ wr } {\mathbb Z}$ only.
\begin{lemma}\label{grga}
Let $A$ be a finitely generated abelian group and consider $G=A \textrm{ wr } {\mathbb Z}=A \textrm{ wr } \langle b \rangle.$ Assume that $k$ is the torsion-free rank of $a.$ If $H$ is a subgroup with $b$ in $G$ then the distortion of $H$ in $G$ is equivalent to that of a subgroup with $b$ in ${\mathbb Z}^k \textrm{ wr } {\mathbb Z}$.
\end{lemma}
\begin{proof}
There exists a series of subgroups $$A=A_0 > A_1 > \cdots > A_m \cong {\mathbb Z}^k$$ for $k \geq 0$ where $A_{i-1}/A_i$ has prime order for $i=1, \dots, m$.
We induct on $m$. If $m=0$, then $A \cong {\mathbb Z}^k$ and the claim holds.
Now let $m>0$. Observe that $A_1$ is a finitely generated abelian group with a series $A_1 > \cdots > A_m \cong {\mathbb Z}^k$ of length $m-1$. Therefore, by induction, any subgroup with $b$ in $G_2=A_1 \textrm{ wr } {\mathbb Z}$ has distortion equivalent to that of a subgroup with $b$ in ${\mathbb Z}^k \textrm{ wr } {\mathbb Z}$, for some $k$.
By Lemma \ref{ta}, all subgroups with $b$ of $G_1=(A/A_1) \textrm{ wr } {\mathbb Z}$ are undistorted.
Denote the natural homomorphism by $\phi: G \rightarrow G_1.$ Let $$U=\displaystyle\bigoplus_{\langle b \rangle}A_1=\ker(\phi).$$ Observe that $U \cdot \langle b \rangle \cong G_2.$ The product is semidirect because $U$ is a normal subgroup which meets $\langle b \rangle$ trivially, and it is isomorphic to the wreath product by definition: the action of $b$ on the module $\displaystyle\bigoplus_{\langle b \rangle} A_1$ is the same.
Let $R={\mathbb Z}[\langle b \rangle]$. Observe that $R$ is a Noetherian ring. This follows from basic algebra because ${\mathbb Z}$ is a commutative Noetherian ring. Therefore, $W$ is a finitely generated module over the Noetherian ring $R$, hence is Noetherian itself. Thus, the $R$-submodule $H \cap U$ is finitely generated.
Let $\{w_1', \dots, w_r'\}$ generate $H \cap U$ as a $R$-module. Let $\{b, w_1, \dots, w_s\}$ be a set of generators of $H$ modulo $U$; that is, the canonical images of these elements generate the subgroup $H_1=HU/U \cong H/H \cap U$ of $G_1$. Then the set $\{b, w_1, \dots, w_s, w_1', \dots, w_r'\}$ generates $H$. Furthermore, the collection $\{b, w_1', \dots, w_r'\}$ generates the subgroup $H_2= (H\cap U)\cdot \langle b \rangle$ of $G_2$.
Let $g \in H$ have $|g|_G \leq l$. Then the image $g_1=\phi(g)$ in $G_1$ belongs to $H_1$, because $g \in H$, and has length $|g_1|_{G_1} \leq l.$
It follows by Lemma \ref{ta} that $H_1$ is undistorted in $G_1.$ Therefore, there exists a linear function $f: \mathbb{N} \rightarrow \mathbb{N}$ (which does not depend on the choice of $g$) such that $|g_1|_{H_1} \le f(l).$ That is to say, there exists a product $P$ of at most $f(l)$ of the chosen generators $\{b, w_1, \dots, w_s\}$ of $H_1$ such that $P=g_1^{-1}$ in $H_1$. Taking preimages, we obtain that $gP \in U$.
Because $H$ is a subgroup of $G$, there exists a constant $c$ depending only on the choice of finite generating set of $H$ such that for any $x \in H$ we have that
\begin{equation}\label{bx}
|x|_G \leq c|x|_H.
\end{equation}
It follows by Equation (\ref{bx}) that
\begin{equation}\label{xxc}
|gP|_G \leq |g|_G + |P|_G \leq |g|_G+c|P|_H \leq l+cf(l).
\end{equation}
Observe that $gP \in H_2.$ This follows because $gP \in U$ by construction, and $g \in H$ by choice. Further, $P \in H$ because it is a product of some of the generators of $H$. Since $H_2=(H \cap U)\cdot \langle b \rangle$ we see that $gP \in H_2$. Using the fact that $G$ and $G_2$ are wreath products together with the length formula in Lemma \ref{t5}, we have that for any $x \in G_2,$
\begin{equation}\label{cx}
|x|_{G_2} \leq |x|_G.
\end{equation}
By induction, the finitely generated subgroup $H_2$ of $G_2$ has distortion function $F(l)$ equivalent to that of a subgroup $\tilde{H}_2$ with $b$ in ${\mathbb Z}^k \textrm{ wr } {\mathbb Z}$ for some $k$. That is, $F(l)=\Delta_{H_2}^{G_2}(l) \approx \Delta_{\tilde{H_2}}^{{\mathbb Z}^k \textrm{ wr } {\mathbb Z}}(l)$. In particular, for any $x \in H_2$,
\begin{equation}\label{ax}
|x|_{H_2} \leq F(|x|_{G_2}).
\end{equation}
Since $gP \in H_2$, we have that $$|gP|_{H_2} \leq F(|gP|_{G_2}) \leq F(|gP|_G) \leq F(l+cf(l)).$$ The first inequality follows from Equation (\ref{ax}), the second from Equation (\ref{cx}), and the last from Equation (\ref{xc}).
Because $H_2 \leq H$ there is a constant $k$ such that for any $x \in H_2, |x|_H \leq k |x|_{H_2}.$
Combining all previous estimates, we compute that $$|g|_H \le |gP|_H+|P|_H \le k|gP|_{H_2}+f(l)\le kF(l+cf(l))+f(l).$$
Thus, at this point we have shown that $\Delta_H^G(l) \preceq F(l) = \Delta_{H_2}^{G_2}(l),$ since $f$ is linear. On the other hand, $\Delta_{H_2}^G(l) = \Delta_H^G(l)$ by Lemma \ref{ab}. By Lemma \ref{wkf} we have that $\Delta_{H_2}^{G_2}(l) \preceq \Delta_{H_2}^{G}(l)$ and so $\Delta_H^G(l) \approx \Delta_{H_2}^{G_2}(l) \approx \Delta_{\tilde{H_2}}^{{\mathbb Z}^k \textrm{ wr } {\mathbb Z}}(l)$.
\end{proof}
\begin{cor}\label{AwrZ} For any finitely generated abelian group $A$, the distortion of every subgroup $H$ with $b$ in $A \textrm{ wr } {\mathbb Z}$ is polynomial. $H$ is undistorted if $A$ is finite.
\end{cor}
\proof This follows from Lemma \ref{grga} and Corollary \ref{withb}.
\section{Completion of the Proof of Theorem \ref{x}}
\begin{lemma}\label{r4}
Let $G$ be a group having normal subgroup $W$ and cyclic $G/W = \langle bW \rangle$. Then any finitely generated subgroup $H$ of $G$ may be generated by elements of the form $w_1b^t, w_2, \dots, w_s$ where $w_i \in W$.
\end{lemma}
The proof is elementary and follows from the assumption that $G/W$ is cyclic.
\begin{rem}\label{z}
It follows that any finitely generated subgroup in $A \textrm{ wr } {\mathbb Z}=W\lambda \langle b\rangle$ can be generated by elements \\$w_1b^t, w_2, \dots, w_s$ where $w_i \in W$.
\end{rem}
\begin{defn}
For a fixed finitely generated abelian group $A$ and any $t>0$, the group $L_t$ is the subgroup of $A \textrm{ wr } {\mathbb Z}$ generated by the subgroup $W$ and by the element $b^t$.
\end{defn}
\begin{lemma}\label{zz}
If $A$ is a fixed $r$ generated abelian group then $L_t \cong A^{t} \textrm{ wr } {\mathbb Z}$,
where $A^t = A\bigoplus\dots\bigoplus A$ ($t$ times).
\end{lemma}
\proof The statement follows from Remark \ref{yyy} with \\$A^t = A_1\bigoplus A_b\bigoplus\dots\bigoplus A_{b^{t-1}}$.\endproof
\begin{lemma}\label{zzz}
For any $w \in W$ there is an automorphism $L_t \rightarrow L_t$ identical on $W$ such that $wb^t \rightarrow b^t$, provided $t \ne 0$.
\end{lemma}
\proof This follows because the actions by conjugation of $b^t$ and $wb^t$ on $W$ coincide.
\endproof
\begin{lemma}\label{r5}
Let $H$ be a finitely generated subgroup of $A \textrm{ wr } {\mathbb Z}$ not contained in $W$, where $A$ is finitely generated abelian. Then the distortion of $H$ in $A \textrm{ wr } {\mathbb Z}$ is equivalent to the distortion of a subgroup $H'$ with $b$ in $A' \textrm{ wr } {\mathbb Z}$ where
$A' \cong A^t$ is also finitely generated abelian.
\end{lemma}
\begin{proof}
By Lemma \ref{r4} the generators of $H$ may be chosen to have the form $w_0b^t, w_1, \dots, w_s$ where $w_i \in W$. Therefore, for this value of $t$ we have that $H$ is a subgroup of $L_t$. Using the isomorphisms of Lemmas \ref{zz} and \ref{zzz} we have that $H$ is a subgroup of $A^{t} \textrm{ wr } {\mathbb Z}=A' \textrm{ wr } {\mathbb Z}$ generated by the image of $b^tw_0, w_1, \dots, w_s$ under the two isomorphisms: elements $b, x_1, \dots, x_s$. Finally, because $[A \textrm{ wr } {\mathbb Z}: L_t ]<\infty$ we have by Lemma \ref{wkf} that the distortion of $H$ in $A \textrm{ wr } {\mathbb Z}$ is equivalent to the distortion of its image in $A^{t} \textrm{ wr } {\mathbb Z}$.
\end{proof}
{\bf Proof of Theorem \ref{x}}\label{final}
Theorem \ref{x} Parts $(1)$ and $(2)$ follow from Lemma \ref{abel} if the subgroup $H$ is abelian. Otherwise they follow from Corollary \ref{AwrZ} and Lemma \ref{r5}.
Now we complete the proof of Theorem \ref{x}, Part $(3)$. Let $A$ be a finitely generated abelian group of rank $k$. Consider the $2$-generated subgroup $H \leq {\mathbb Z} \textrm{ wr } {\mathbb Z}$ constructed as follows. Let $m \in \mathbb{N}$. Consider $h(x)=(1-x)^{m-1}$. Then the distortion of the polynomial $h$ is seen to be equivalent to $l^m$, by Lemma \ref{tgs}. By Lemma \ref{hpu}, this means that the $2$-generated subgroup $\langle b,(1-x)^{m-1}a \rangle = H_m \leq {\mathbb Z} \textrm{ wr } {\mathbb Z}$ has distortion $\Delta_{H_m}^{{\mathbb Z} \textrm{ wr } {\mathbb Z}}(l) \approx l^m$. The subgroup ${\mathbb Z} \textrm{ wr } {\mathbb Z}$ is a retract of $A \textrm{ wr } {\mathbb Z}$ if $A$ is infinite. Therefore, the distortion of $H_m$ in ${\mathbb Z} \textrm{ wr } {\mathbb Z}$ and in $A \textrm{ wr } {\mathbb Z}$ are equivalent by Lemma \ref{wkf}.
\begin{rem}
If we adopt the notation that the commutator $[a,b]=aba^{-1}b^{-1}$, then we see that in ${\mathbb Z} \textrm{ wr } {\mathbb Z}$, the element of $W$ corresponding to the polynomial $(1-x)^{m-1}a$ is $[ \cdots [a,b],b], \cdots, b]$ where the commutator is $(m-1)$-fold. This explains Corollary \ref{xx}.
\end{rem}
\medskip
\noindent {\bf Acknowledgement.}
The authors are grateful to Nikolay Romanovskiy for his many valuable comments.
\addtolength{\textwidth}{.7in}
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{
"timestamp": "2011-04-11T02:01:55",
"yymm": "1009",
"arxiv_id": "1009.4364",
"language": "en",
"url": "https://arxiv.org/abs/1009.4364"
}
|
\section{Conclusion} \label{sec:Conclusion}
In this paper, a reduced complexity decoding scheme for BICMB-CP is presented. SD with initial radius calculated by ZF-DFE is used to acquire precoded bit metrics needed for the Viterbi decoder. SD can achieve the same performance as exhaustive search, and more importantly, achieves a substantial complexity reduction. Two techniques are applied to reduce both the number of executions and operations for SD substantially. Therefore, BICMB-CP can be considered as a practical application for MIMO systems requiring high throughput with the full diversity order. The reduced complexity decoding in this paper can be applied to any convolutional coded MIMO system.
\section{Reduced Complexity Decoding for BICMB-CP} \label{sec:Decoding}
Recall that $l$ is the symbol position in $\tilde{\mathbf{x}}_k^\prime$. If $P+1 \leq l \leq S$ for (\ref{eq:ML_bit_metrics_PPMB}), the complex-valued scalar symbol carrying the coded bit $c_{k^\prime}$ is non-precoded. The non-precoded bit metric is the same as BICMB and can be decoded with low complexity using the technique presented in \cite{Akay_BICM_LCD}.
If $1 \leq l \leq P$ for (\ref{eq:ML_bit_metrics_PPMB}), the complex-valued scalar symbol carrying the coded bit $c_{k^\prime}$ is precoded. The computational complexity for the precoded bit metric is much higher than the non-precoded bit metric. Exhaustive search requires exponential complexity according to the modulation alphabet size and the dimension of the constellation precoder. The total number of lattice points needed to be searched is $|\tilde{\chi}|^{P-1}|\tilde{\chi}_{c_{k'}}^i|=\frac{|\tilde{\chi}|^P}{2}$. In this section, techniques are focus on reducing the complexity of precoded bit metrics calculation.
\subsection{Calculating Precoded Bit Metrics By SD}
SD is used to reduce the complexity of exhaustive search by only searching lattice points inside a sphere with radius $\delta$ \cite{Jalden_SD}. Let $\tilde{\mathbf{G}}=\boldsymbol{\Gamma}_p \boldsymbol{\Theta}_p$, then SD is employed to solve
\begin{equation}
\gamma^{l,i}(\tilde{\mathbf{r}}_k, c_{k^\prime}) = \min\limits_{\tilde{\mathbf{x}} \in \tilde{\Omega}} \| \tilde{\mathbf{r}}_{k}^p - \tilde{\mathbf{G}} \tilde{\mathbf{x}} \|^2
\label{eq:SD_precoded_bit_metrics}
\end{equation}
where $\tilde{\Omega} \subset \tilde{\psi}_{c_{k'}}^{l,i}$, and $\Vert \tilde{\mathbf{r}}_{k}^p - \tilde{\mathbf{G}} \tilde{\mathbf{x}} \Vert^2<\delta^{2}$.
The $P$-dimensional complex-valued input-output relation of the precoded part in (\ref{eq:deteced_symbol_decomposed}) can be transformed into a 2P-dimensional real-valued problem \cite{Jalden_SD}:
\begin{equation}
\mathbf{r}_k^p = \mathbf{G} \mathbf{x}_k^p + \mathbf{n}_k^p,
\label{eq:real_preocded_input_output}
\end{equation}
where $\mathbf{r}_k^p$, $\mathbf{G}$, $\mathbf{x}_k^p$, and $\mathbf{n}_k^p$ are corresponding real-valued representations of $\tilde{\mathbf{r}}_k^p$, $\tilde{\mathbf{G}}$, $\tilde{\mathbf{x}}_k^p$, and $\tilde{\mathbf{n}}_k^p$, respectively. For square QAM where $M$ is an even integer, the first and the remaining $\frac{M}{2}$ bits of labels for the $2^M$-QAM are generally Gray coded separately as two $2^{\frac{M}{2}}$-PAM constellations, and represent the real and the imaginary axes respectively. Assume that the same Gray coded mapping scheme is used for the the real and the imaginary axes. As a result, each element of $\mathbf{x}_k^p$ belongs to a real-valued signal set $\chi$, and one bit in the label of $\mathbf{x}_k^p $ corresponds to $c_{k^\prime}$. The new position of $c_{k^\prime}$ in the label of $\mathbf{x}_k^p $ needs to be acquired as $k^\prime\rightarrow(k,\hat{l},\hat{i})$, which means $c_{k^\prime}$ lies in the $\hat{i}^{th}$ bit position of the label for the $\hat{l}^{th}$ element of real-valued vector symbol $\mathbf{x}_k^p $. Let $\chi_b^{\hat{i}}$ denote a subset of $\chi$ whose labels have $b \in \{0, 1\}$ in the $\hat{i}^{th}$ bit position. Define $\psi_{c_{k'}}^{\hat{l},\hat{i}} \subset \chi^{2P}$ as
\begin{align*}
\psi_{b}^{\hat{l},\hat{i}} = \{ \mathbf{x} = [x_1 \, \cdots \, x_{2P} ]^T : x_{v|v=\hat{l}} \in \chi_{b}^{\hat{i}}, \textrm{ and } x_{v|v \neq \hat{l}} \in \chi\}.
\end{align*}
Then (\ref{eq:SD_precoded_bit_metrics}) is rewritten as
\begin{equation}
\gamma^{l,i}(\tilde{\mathbf{r}}_k, c_{k^\prime}) = \min\limits_{\mathbf{x} \in \Omega} \| \mathbf{r}_k^p - \mathbf{G} \mathbf{x} \|^2
\label{eq:SD_real_precoded_bit_metrics}
\end{equation}
where $\Omega \subset \psi_{c_{k'}}^{\hat{l},\hat{i}}$, and $\Vert \mathbf{r}_{k}^p - \mathbf{G} \mathbf{x} \Vert^2<\delta^{2}$.
By using the QR decomposition of $\mathbf{G}=\mathbf{QR}$, where $\mathbf{R}$ is an upper triangular matrix, and the matrix $\mathbf{Q}$ is unitary, (\ref{eq:SD_real_precoded_bit_metrics}) is rewritten as
\begin{equation}
\gamma^{l,i}(\tilde{\mathbf{r}}_k, c_{k^\prime}) = \min\limits_{\mathbf{x} \in \Omega} \| \breve{\mathbf{r}}_k^p - \mathbf{R} \mathbf{x} \|^2 \label{eq:SD_real_QR_precoded_bit_metrics}
\end{equation}
where $\breve{\mathbf{r}}_k^p=\mathbf{Q}^H \mathbf{r}_k^p$.
SD can now be viewed as a pruning algorithm on a tree of depth $2P$, whose branches correspond to elements drawn from the set $\chi$, except for branches of the layer $u=\hat{l}$, which correspond to elements drawn from the set $\chi^{\hat{i}}_{c_{k'}}$. SD starts the search process from the root of the tree, and then searches down along branches until the total weight of a node exceeds the square of the sphere radius, $\delta^{2}$. At this point, the corresponding branch is pruned, and any path passing through that node is declared as improbable for a candidate solution. Then the algorithm backtracks, and proceeds down a different branch. Once a valid lattice point at the bottom level of the tree is found within the sphere, $\delta^{2}$ is set to the newly-found point weight, thus reducing the search space for finding other candidate solutions. In the end, the candidate solution corresponding to the path from the root to the leaf which is inside the sphere with the lowest weight is picked, and the corresponding weight is set to be the bit metric value. If no candidate solution is found, the tree will be searched again with a larger initial radius. SD can achieve the same performance as exhaustive search.
The node weight is calculated as \cite{Azzam_SD_NLR}, \cite{Azzam_SD_RLR}
\begin{equation}
w(\mathbf{x}^{(u)})=w(\mathbf{x}^{(u+1)})+w_{pw}(\mathbf{x}^{(u)}) \label{eq:node_weight}
\end{equation}
with $w(\mathbf{x}^{(2P+1)})=0$, $w_{pw}(\mathbf{x}^{(2P+1)})=0$, and $u=2P,2P-1,\cdots,1$, where $\mathbf{x}^{(u)}$ denotes the partial vector symbol at layer $u$. The partial weight $w_{pw}(\mathbf{x}^{(u)})$ is written as
\begin{equation}
w_{pw}(\mathbf{x}^{(u)})=|\breve{r}_{k,u}^p-\sum^{2P}_{v=u}{R_{u,v}x_v}|^{2} \label{eq:partial_weight}
\end{equation}
where $\breve{r}_{k,u}^p$ is the $u^{th}$ element of $\breve{\mathbf{r}}_k^p$, $R_{u,v}$ is the $(u,v)^{th}$ element of $\bf R$, and $x_v$ is the $v^{th}$ element of $\mathbf{x} \in \psi_{b}^{\hat{j},\hat{i}}$.
\subsection{Acquiring Initial Radius By ZF-DFE}
The initial radius $\delta$ should be chosen properly, so that it is not too small or too large. Too small an initial radius results in too many unsuccessful searches and thus increases complexity, while too large an initial radius results in too many lattice points to be searched.
In this work, for $c_{k'}=b$ where $b \in \{0,1\}$, ZF-DFE is used to acquire a estimated real-valued vector symbol $\breve{\mathbf{x}}_k^b$, which is also the Baiba point \cite{Agrell_CPS}. Then the square of initial radius $\delta_b^2$, which guarantees no unsuccessful searches is calculated by
\begin{equation}
\delta_b^2=\Vert \breve{\mathbf{r}}_k^p-\mathbf{R}\breve{\mathbf{x}}_k^b \Vert^2.
\label{eq:IR_ZFDFE}
\end{equation}
The estimated real-valued vector symbol $\breve{\mathbf{x}}_k^b$ is detected successively starting from $\breve{x}_{k, 2P}^b$ until $\breve{x}_{k, 1}^b$, where $\breve{x}_{k, u}^b$ denotes the $u^{th}$ element of $\breve{\mathbf{x}}_k^b$. The decision rule on $\breve{x}_{k, u}^b$ is
\begin{equation}
\breve{x}_{k, u}^b=\left\{
\begin{array}{ll}
\arg\min\limits_{x\in\chi} {| \breve{r}_{k,u}^p-\sum^{2P}_{v=u+1}{R_{u,v}\breve{x}_{k, v}^b}-R_{u,u}x |},&u\neq \hat{l},\\ \arg\min\limits_{x\in\chi^{\hat{i}}_b} {| \breve{r}_{k,u}^p-\sum^{2P}_{v=u+1}{R_{u,v}\breve{x}_{k, v}^b}-R_{u,u}x |},&u=\hat{l}.\\
\end{array}
\right.
\label{eq:ZFDFE}
\end{equation}
The estimation of the symbols (\ref{eq:ZFDFE}) can be carried out recursively by rounding (or quantizing) to the nearest constellation element in $\chi$ or $\chi^{\hat{i}}_b$.
\subsection{Reducing Number of Executions in SD}
For the $k^{th}$ time instant, the precoded real-valued vector symbol $\mathbf{x}_k^p$ carries $MP$ bits. Since each bit generates two bit metrics for $c_{k'}=0$ and $c_{k'}=1$, then $2MP$ precoded bit metrics in total need to be acquired. However, some precoded bit metrics have the same value, hence SD can be modified to be executed less than $2MP$ times, as mentioned in \cite{Studer_SOSD}.
Define $\hat{\mathbf{x}}_k$, $\hat{\mathbf{x}}_k^{c_{k'}}$, and $\gamma_{k}$ as
\begin{equation}
\hat{\mathbf{x}}_k = \arg\min\limits_{\mathbf{x} \in \chi^{2P}} \| \breve{\mathbf{r}}_k^p - \mathbf{R} \mathbf{x} \|^2,
\label{eq:ML_symbol}
\end{equation}
\begin{equation}
\hat{\mathbf{x}}_k^{c_{k'}} = \arg\min\limits_{\mathbf{x} \in \psi_{c_{k'}}^{\hat{l},\hat{i}}} \| \breve{\mathbf{r}}_k^p - \mathbf{R} \mathbf{x} \|^2,
\label{eq:bit_metric_symbol_estimated}
\end{equation}
and
\begin{equation}
\gamma_{k} = \| \breve{\mathbf{r}}_k^p - \mathbf{R} \hat{\mathbf{x}}_k \|^2,
\label{eq:ML_symbol_weight}
\end{equation}
respectively.
Note that $\psi_0^{\hat{l},\hat{i}} \cup \psi_1^{\hat{l},\hat{i}} = \chi^{2P}$ and $\psi_0^{\hat{l},\hat{i}} \cap \psi_1^{\hat{l},\hat{i}} = \emptyset$. Then
\begin{equation}
\gamma_{k} = \min{\{\gamma^{l,i}(\tilde{\mathbf{r}}_k, c_{k^\prime}=0), \gamma^{l,i}(\tilde{\mathbf{r}}_k, c_{k^\prime}=1) \} },
\label{eq:symbol_weight_relation}
\end{equation}
which means that, for the $MP$ bits corresponding to $\mathbf{x}_k^p$, the smaller precoded bit metric for each bit of $c_{k'}=0$ and $c_{k'}=1$ have the same value $\gamma_{k}$.
Let $\hat{b}_{\hat{i}}^{\hat{l}}\in \{0,1\}$ denotes the value of the $\hat{i}^{th}$ bit in the label of $\hat{x}_{k,\hat{l}}$, which is the $\hat{l}^{th}$ element of $\hat{\mathbf{x}}_k$. Then
\begin{equation}
\gamma^{l,i}(\tilde{\mathbf{r}}_k, c_{k^\prime}=\hat{b}_{\hat{i}}^{\hat{l}})= \gamma^k.
\label{eq:bit_metric_smaller}
\end{equation}
First, two bit metrics $\gamma^{l,i}(\tilde{\mathbf{r}}_k, c_{k^\prime}=0)$ and $\gamma^{l,i}(\tilde{\mathbf{r}}_k, c_{k^\prime}=1)$ for one of the $MP$ bits corresponding to $\mathbf{x}_k^p$ and their related $\hat{\mathbf{x}}_k^{c_{k'}}$ are derived by SD. Then the $\hat{\mathbf{x}}_k^{c_{k'}}$ corresponding to the smaller bit metric is chosen to be $\hat{\mathbf{x}}_k$, and $\gamma_{k}$ is acquired by (\ref{eq:symbol_weight_relation}). For each of the other $MP-1$ bits, $\gamma^{l,i}(\tilde{\mathbf{r}}_k, c_{k^\prime}=\hat{b}_{\hat{i}}^{\hat{l}})$ is acquired by (\ref{eq:bit_metric_smaller}), and $\gamma^{l,i}(\tilde{\mathbf{r}}_k, c_{k^\prime}=\bar{\hat{b}}_{\hat{i}}^{\hat{l}})$ is calculated by SD. Consequently, the execution number of SD for one time instant is reduced from $2MP$ to $MP+1$.
\subsection{Reducing Number of Operations in SD}
In our previous work \cite{Li_RCSD_arXiv}, a technique was introduced to implement SD with low computational complexity, which achieves the same performance as exhaustive search. The technique in this paper can be employed to achieve substantial further complexity reduction for BICMB-CP. In this subsection, a brief description of the technique is presented for reducing the number of real multiplications.
Note that for one channel realization, both $\mathbf{R}$ and $\chi$ are independent of time. In other words, to decode different received symbols for one channel realization, the only term in (\ref{eq:partial_weight}) which depends on time is $r_{k,u}^p$. Consequently, a check-table $\mathbb{T}$ is constructed to store all terms of $R_{u,v}x$, where $R_{u,v}\neq0$ and $x \in \chi$, before starting the tree search procedure. Equations (\ref{eq:node_weight}) and (\ref{eq:partial_weight}) imply that only one real multiplication is needed by using $\mathbb{T}$ instead of $2P-u+2$ for each node to calculate the node weight. As a result, the number of real multiplications can be significantly reduced.
Note that $\chi$ can be divided into two smaller sets $\chi_{1}$ with negative elements and $\chi_{2}$ with positive elements. Any negative element in $\chi_{1}$ has a positive element with the same absolute value in $\chi_{2}$. Consequently, in order to build $\mathbb{T}$, only terms of $R_{u,v}x$, where $R_{u,v}\neq0$ and $x\in\chi_{1}$, need to be calculated and stored. Since the channel is assumed to be flat fading, only one $\mathbb{T}$ needs to be built in one burst. If the burst length is very long, its complexity can be neglected.
In our previous work \cite{Azzam_SD_NLR}, \cite{Azzam_SD_RLR}, a new lattice representation was introduced. In this work, the same lattice representation is employed to (\ref{eq:real_preocded_input_output}) but with a new application. The structure of the lattice representation becomes advantageous after applying the QR decomposition to $\mathbf{G}$. By doing so, and due to the special form of orthogonality between each pair of columns, all elements $R_{u,u+1}$ for $u=1,3,\ldots,2P-1$ in the upper triangular matrix $\mathbf{R}$ become zero. The locations of these zeros introduce orthogonality between the real and the imaginary parts of every detected symbol, which can be taken advantage of to reduce the computational complexity of SD.
Based on this feature, SD is modified in the following way: once the tree is searched in layer $u$, where $u$ is an odd number, partial weights of this node and all of its brother nodes are computed, temporally stored, and recycled when calculating partial node weights with the same grandparent node of layer $u+2$ but with different parent nodes of layer $u+1$. By implementing the modification, further complexity reduction is achieved.
\section{Introduction} \label{sec:Introduction}
Beamforming is employed in a Multi-Input Multi-Output (MIMO) system to achieve spatial multiplexing\footnotemark \footnotetext{In this paper, the term ``spatial multiplexing" is used to describe the number of spatial subchannels, as in \cite{Paulraj_ST}. Note that the term is different from ``spatial multiplexing gain" defined in \cite{Zheng_DM}.} and thereby increase the data rate, or to enhance the performance, when channel state information is available at the transmitter \cite{Jafarkhani_STC}. A set of beamforming vectors is obtained by Singular Value Decomposition (SVD) which is optimal in terms of minimizing the average Bit Error Rate (BER) \cite{Palomar_JTRBD}.
\IEEEpubidadjcol
It is known that an SVD subchannel with larger singular value provides greater diversity gain. Spatial multiplexing without channel coding results in the loss of the full diversity order \cite{Sengul_DA_SMB}. To overcome the diversity order degradation of multiple beamforming, Bit-Interleaved Coded Multiple Beamforming (BICMB) was proposed \cite{Akay_BICMB}, \cite{Akay_On_BICMB}. BICMB can achieve the full diversity order offered by the channel as long as the code rate $R_c$ and the number of subchannels used $S$ satisfy the condition $R_c S \leq 1$ \cite{Park_DA_BICMB}.
Bit-Interleaved Coded Multiple Beamforming with Constellation Precoding (BICMB-CP) converts a symbol into a precoded symbol and distributes it over subchannels \cite{Park_BICMB_CP}. The addition of the constellation precoder to BICMB, whose code rate $R_c$ is greater than $1/S$, provides the full diversity when the subchannels for transmitting the precoded symbols are properly chosen. However, BICMB-CP causes increased decoding complexity compared to BICMB.
In this paper, Sphere Decoding (SD) with initial radius acquired by Zero-Forcing Decision Feedback Equalization (ZF-DFE) is used to calculate bit metrics of precoded symbols. The initial radius calculated by ZF-DFE \cite{Han_SD_IR}, which is also the metric weight of the Baiba point \cite{Agrell_CPS}, ensures no empty spheres. Based on SD, two techniques are applied to reduce the number of executions carried out by SD and the computational complexity of each SD execution, respectively. Conventional SD substantially reduces the complexity, in terms of the average number of real multiplications needed to acquire one precoded bit metric, compared with exhaustive search. With the techniques proposed in this paper, further reductions of orders of magnitude are achieved. The reduction becomes larger as the constellation precoder dimension and the constellation size increase.
The remainder of this paper is organized as follows: In Section \ref{sec:System_model}, the description of BICMB-CP is given. In Section \ref{sec:Decoding}, a reduced complexity decoding technique for BICMB-CP is proposed. In Section \ref{sec:Results}, complexity comparisons for different constellation precoder dimensions or modulation schemes are presented. Finally, a conclusion is provided in Section \ref{sec:Conclusion}.
\textbf{Notation:} Let $\textrm{diag}[\mathbf{B}_1, \cdots, \mathbf{B}_P]$ stand for a block diagonal matrix with matrices $\mathbf{B}_1, \cdots, \mathbf{B}_P$, and let $\textrm{diag}[b_1, \cdots, b_P]$ be a diagonal matrix with diagonal entries $b_1, \cdots, b_P$. The superscripts $(\cdot)^H$, $(\cdot)^T$, and $\bar{(\cdot)}$ stand for conjugate transpose, transpose and binary complement, respectively. Let $\mathbb{R}^+$ and $\mathbb{C}$ stand for the set of positive real numbers and complex numbers, respectively. Finally, let $N_t$ and $N_r$ stand for the number of transmit and receive antennas, respectively.
\section{Simulation Results} \label{sec:Results}
Since the $P$-dimensional complex-valued input-output relation of the precoded part in (\ref{eq:deteced_symbol_decomposed}) can be viewed as a $P$-dimensional BICMB-FP, BICMB-FP is considered to verify the proposed technique. Exhaustive Search (EXH), Conventional SD (CSD), and Proposed Smart Implementation (PSI) which combines Section \ref{sec:Decoding}.C and Section \ref{sec:Decoding}.D, are applied. The average number of real multiplications, the most expensive operations in terms of machine cycles, for acquiring one bit metric is calculated at different SNR.
Fig. \ref{fig:2x2} shows comparisons for $2\times2$ $S=2$ $R_c=\frac{2}{3}$ BICMB-FP. For $4$-QAM, the complexity of EXH is reduced by $0.4$ and $0.5$ orders of magnitude at low and high SNR respectively, by CSD. PSI yields larger reductions by $1.1$ and $1.2$ orders of magnitude at low and high SNR respectively. In the case of $64$-QAM, reductions between CSD and EXH are $1.5$ and $2.1$ orders of magnitude at low and high SNR respectively, while larger reductions of $2.6$ and $3.0$ are achieved by PSI.
Similarly, Fig. \ref{fig:4x4} shows complexity comparisons for $4\times4$ $S=4$ $R_c=\frac{4}{5}$ BICMB-FP. For $4$-QAM, the complexity of EXH decreases by $1.3$ and $1.5$ orders of magnitude at low and high SNR respectively. PSI gives larger reductions by $2.3$ orders of magnitude at low SNR, and $2.4$ orders of magnitude at high SNR. For the $64$-QAM case, reductions between EXH and CSD by $3.2$ and $4.4$ orders of magnitude are observed at low and high SNR respectively, while larger reductions by $4.4$ and $5.4$ are achieved by PSI.
Simulation results show that CSD reduces the complexity substantially compared to EXH, and the complexity can be further reduced significantly by PSI. The reductions become larger as the constellation precoder dimension and the modulation alphabet size increase. One important property of our decoding technique needs to be emphasized is that the substantial complexity reduction achieved causes no performance degradation.
\ifCLASSOPTIONonecolumn
\begin{figure}[!m]
\centering
\scalebox{.7}{\includegraphics{numofmul_bicm-fpmb_2x2.eps}}
\caption{Average number of real multiplications vs. SNR for $2\times2$ $S=2$ BICMB-FP.}
\label{fig:2x2}
\end{figure}
\begin{figure}[!m]
\centering
\scalebox{.7}{\includegraphics{numofmul_bicm-fpmb_4x4.eps}}
\caption{Average number of real multiplications vs. SNR for $4\times4$ $S=2$ BICMB-FP.}
\label{fig:4x4}
\end{figure}
\else
\begin{figure}[!t]
\centering
\scalebox{.5}{\includegraphics{numofmul_bicm-fpmb_2x2.eps}}
\caption{Average number of real multiplications vs. SNR for $2\times2$ $S=2$ BICMB-FP.}
\label{fig:2x2}
\end{figure}
\begin{figure}[!t]
\centering
\scalebox{.5}{\includegraphics{numofmul_bicm-fpmb_4x4.eps}}
\caption{Average number of real multiplications vs. SNR for $4\times4$ $S=2$ BICMB-FP.}
\label{fig:4x4}
\end{figure}
\fi
\section{BICMB-CP Overview} \label{sec:System_model}
Fig. \ref{fig:system_model} represents the structure of BICMB-CP. First, the convolutional encoder with code rate $R_c = k_c/n_c$, possibly combined with a perforation matrix for a high rate punctured code \cite{Haccoun_PCC}, generates the codeword $\mathbf{c}$ from the information bits. Then, the spatial interleaver distributes the coded bits into $S\leq\min(N_t, N_r)$ streams, each of which is interleaved by an independent bit-wise interleaver $\pi$. The interleaved bits are modulated by Gray mapped square QAM onto the complex-valued symbol sequence $\tilde{\mathbf{X}} = [\tilde{\mathbf{x}}_1 \, \cdots \, \tilde{\mathbf{x}}_K]$, where $\tilde{\mathbf{x}}_k$ is an $S \times 1$ complex-valued symbol vector at the $k^{th}$ time instant. It is assumed that each stream employs the same $2^M$-QAM constellation, where $M$ is the number of bits labeling a complex-valued scalar symbol. Let $\tilde{\chi} \subset \mathbb{C}$ of size $|\tilde{\chi}| = 2^M$ denote the complex-valued signal set of the square QAM.
\ifCLASSOPTIONonecolumn
\begin{figure}[!m]
\centering \includegraphics[width = 0.6\linewidth]{system_model.eps}
\caption{Structure of BICMB-CP.} \label{fig:system_model}
\end{figure}
\else
\begin{figure}[!t]
\centering \includegraphics[width = 1.0\linewidth]{system_model.eps}
\caption{Structure of BICMB-CP.} \label{fig:system_model}
\end{figure}
\fi
The complex-valued symbol vector $\tilde{\mathbf{x}}_k$ is multiplied by the $S \times S$ precoder $\boldsymbol{\Theta}$, which is defined as
\begin{align}
\mathbf{\Theta} = \left[ \begin{array}{cc}
\mathbf{\Theta}_p & \mathbf{0} \\
\mathbf{0} & \mathbf{I}_{S-P}
\end{array} \right]
\label{eq:precoder_def}
\end{align}
where $\mathbf{\Theta}_p$ is the $P \times P$ unitary constellation precoding matrix that precodes the first $P$ modulated
entries of $\tilde{\mathbf{x}}_k$. The system is called Bit-Interleaved Coded Multiple Beamforming with Full Precoding (BICMB-FP) when all of the $S$ modulated entries are precoded, otherwise it is called Bit-Interleaved Coded Multiple Beamforming with Partial Precoding (BICMB-PP). The symbol generated by $\boldsymbol{\Theta}$ is multiplied by $\mathbf{T}$, which is an $S \times S$ permutation matrix, to map the precoded and non-precoded symbols onto the predetermined subchannels. Let us define $\mathbf{b}_p = \left[ b_p(1) \, \cdots \, b_p(P) \right]$ as a vector whose element $b_p(u)$ is the subchannel on which the precoded symbols are transmitted, and ordered increasingly such that $b_p(u) < b_p(v)$ for $u < v$. In the same way, $\mathbf{b}_n = \left[ b_n(1) \, \cdots \, b_n(S-P) \right]$ is defined as an increasingly ordered vector whose element $b_n(u)$ is the subchannel which carries the non-precoded symbols.
The MIMO channel $\mathbf{H} \in \mathbb{C}^{N_r \times N_t}$ is assumed to be quasi-static, Rayleigh, and flat fading, and perfectly known to both the transmitter and the receiver. Assume that the channel coefficients remain constant for a block of $K$ symbols. The beamforming vectors are determined by the SVD of the MIMO channel, i.e., $\mathbf{H} = \mathbf{U \Lambda V}^H$ where $\mathbf{U}$ and $\mathbf{V}$ are unitary matrices, and $\mathbf{\Lambda}$ is a diagonal matrix whose $s^{th}$ diagonal element, $\lambda_s \in \mathbb{R}^+$, is a singular value of $\mathbf{H}$ in decreasing order. When $S$ scalar symbols are transmitted at the same time, then the first $S$ vectors of $\mathbf{U}$ and $\mathbf{V}$ are chosen to be used as beamforming matrices at the receiver and the transmitter, respectively. In Fig. \ref{fig:system_model}, $\mathbf{U}_S$ and $\mathbf{V}_S$ denote the first $S$ column vectors of $\mathbf{U}$ and $\mathbf{V}$ respectively.
The spatial interleaver arranges the complex-valued symbol vector as $\tilde{\mathbf{x}}_k^\prime = [(\tilde{\mathbf{x}}_k^p)^T \, \vdots \, (\tilde{\mathbf{x}}_k^n)^T]^T = [\tilde{x}_{k,b_p(1)} \, \cdots \,
\tilde{x}_{k,b_p(P)} \, \vdots$ $\, \tilde{x}_{k,b_n(1)} \, \cdots \, \tilde{x}_{k, b_n(S-P)}]^T$, where $\tilde{\mathbf{x}}_k^p$ and $\tilde{\mathbf{x}}_k^n$ are the modulated entries to be transmitted on the subchannels specified in $\mathbf{b}_p$ and $\mathbf{b}_n$, respectively. Then, the $S \times 1$ received complex-valued symbol vector at the $k^{th}$ time instant $\tilde{\mathbf{r}}_k = [ (\tilde{\mathbf{r}}_{k}^p)^T \, \vdots \, (\tilde{\mathbf{r}}_{k}^n)^T]^T = [\tilde{r}_{k,1} \, \cdots \, \tilde{r}_{k, P} \, \vdots\, \tilde{r}_{k, P+1} \, \cdots \, \tilde{r}_{k,S}]^T$ is
\begin{align}
\tilde{\mathbf{r}}_k = \boldsymbol{\Gamma} \mathbf{\Theta} \tilde{\mathbf{x}}_k^\prime + \tilde{\mathbf{n}}_k, \label{eq:detected_symbol}
\end{align}
where $\boldsymbol{\Gamma} = \textrm{diag}[\boldsymbol{\Gamma}_p ,\boldsymbol{\Gamma}_n]$ is a block diagonal matrix, with diagonal matrices $\boldsymbol{\Gamma}_p = \textrm{diag}[\lambda_{b_p(1)}, $ $\, \cdots, \, \lambda_{b_p(P)}]$ and $\boldsymbol{\Gamma}_n = \textrm{diag}[\lambda_{b_n(1)}, \, \cdots, \, \lambda_{b_n(S-P)}]$, and $\tilde{\mathbf{n}}_k = [ (\tilde{\mathbf{n}}_{k}^p)^T \, \vdots \, (\tilde{\mathbf{n}}_{k}^n)^T]^T = [\tilde{n}_{k,1} \, \cdots \, \tilde{n}_{k, P} \, \vdots \, \tilde{n}_{k, P+1} \, \cdots \, \tilde{n}_{k,S}]^T$ is a complex-valued additive white Gaussian noise vector with zero mean and variance $N_0 = S / SNR$. The channel matrix $\mathbf{H}$ is complex Gaussian with zero mean and unit variance, and to make the received Signal-to-Noise Ratio (SNR) $SNR$, the total transmitted power is scaled as $S$. The input-output relation in
(\ref{eq:detected_symbol}) is decomposed into two equations as
\begin{equation}
\begin{split}
\tilde{\mathbf{r}}_{k}^p = \boldsymbol{\Gamma}_p \boldsymbol{\Theta}_p \tilde{\mathbf{x}}_k^p + \tilde{\mathbf{n}}_k^p, \\
\tilde{\mathbf{r}}_{k}^n = \boldsymbol{\Gamma}_n \tilde{\mathbf{x}}_k^n + \tilde{\mathbf{n}}_k^n. \label{eq:deteced_symbol_decomposed}
\end{split}
\end{equation}
The location of the coded bit $c_{k'}$ within the complex-valued symbol sequence $\tilde{\mathbf{X}}$ is known as $k' \rightarrow (k, l, i)$, where $k$, $l$, and $i$ are the time instant in $\tilde{\mathbf{X}}$, the symbol position in $\tilde{\mathbf{x}}_k^\prime$, and the bit position on the label of the scalar symbol $\tilde{x}_{k,l}^\prime$, respectively. Let $\tilde{\chi}_{b}^{i}$ denote a subset of $\tilde{\chi}$ whose labels have $b \in \{0, 1\}$ in the $i^{th}$ bit position. By using the location information and the input-output relation in (\ref{eq:detected_symbol}), the receiver calculates the Maximum Likelihood (ML) bit metrics for $c_{k'}$ as
\begin{align}
\gamma^{l,i}(\tilde{\mathbf{r}}_{k}, c_{k'}) = \min_{\tilde{\mathbf{x}}^ \in \tilde{\xi}_{c_{k'}}^{l,i}} \| \tilde{\mathbf{r}}_{k} - \boldsymbol{\Gamma} \boldsymbol{\Theta} \tilde{\mathbf{x}} \|^2, \label{eq:ML_bit_metrics}
\end{align}
where $\tilde{\xi}_{c_{k'}}^{l,i}$ is a subset of $\tilde{\chi}^S$, defined as
\begin{align*}
\tilde{\xi}_{b}^{l,i} = \{ \tilde{\mathbf{x}} = [\tilde{x}_1 \, \cdots \, \tilde{x}_S ]^T : \tilde{x}_{s|s=l} \in \tilde{\chi}_{b}^{i}, \textrm{ and } \tilde{x}_{s|s \neq l} \in \tilde{\chi}\}.
\end{align*}
In particular, the bit metrics, equivalent to
(\ref{eq:ML_bit_metrics}) for partial precoding, are
\begin{align}
\gamma^{l,i}(\tilde{\mathbf{r}}_{k}, c_{k'}) = \left\{
\begin{array}{ll}
\min\limits_{\tilde{\mathbf{x}} \in \tilde{\psi}_{c_{k'}}^{l,i}} \| \tilde{\mathbf{r}}_{k}^p - \boldsymbol{\Gamma}_p \boldsymbol{\Theta}_p \tilde{\mathbf{x}} \|^2, & \textrm{ if $1 \leq l \leq P$}, \\
\min\limits_{\tilde{x} \in \tilde{\chi}_{c_{k'}}^{i}} |r_{k,l} - \lambda_{l'} \tilde{x} |^2, & \textrm{ if $P+1 \leq l \leq S$},
\end{array} \right.
\label{eq:ML_bit_metrics_PPMB}
\end{align}
where $\tilde{\psi}_{b}^{l,i}$ is a subset of $\tilde{\chi}^P$, defined as
\begin{align*}
\tilde{\psi}_{b}^{l,i} = \{ \tilde{\mathbf{x}} = [\tilde{x}_1 \, \cdots \, \tilde{x}_P ]^T : \tilde{x}_{v|v=l} \in \tilde{\chi}_{b}^{i}, \textrm{ and } \tilde{x}_{v|v \neq l} \in \tilde{\chi}\},
\end{align*}
and $l'$ is an entry in $\mathbf{b}_n$, corresponding to the subchannel mapped by $\mathbf{T}$. Finally, the ML decoder, which uses Viterbi decoding, makes decisions according to the rule
\begin{align}
\mathbf{\hat{c}} = \arg\min_{\mathbf{c}} \sum_{k'} \gamma^{l,i}(\tilde{\mathbf{r}}_{k}, c_{k'}).
\label{eq:Decision_Rule}
\end{align}
|
{
"timestamp": "2011-09-16T02:00:31",
"yymm": "1009",
"arxiv_id": "1009.3514",
"language": "en",
"url": "https://arxiv.org/abs/1009.3514"
}
|
\section{Introduction}
\label{sect-1}
The spatial and temporal variability of dimensionless physical
constants has become a topic of considerable interest in laboratory
and astrophysical studies as a test of the Einstein equivalence
principle of local position invariance (LPI), which states that
outcomes of nongravitational experiments should be independent of
their position in space-time (e.g., Dent 2008).
The violation of LPI is anticipated in some extensions of Standard Model
and, in particular, in those dealing with dark energy
(e.g., Hui {et al.}\ 2009; Damour \& Donoghue 2010).
A concept of dark energy with negative pressure ($p = -\rho$)
appeared in physics long before the discovery of the accelerating
universe through observations of nearby and distant (at redshif $z
\sim 1$) supernovae type Ia \citep{Per98, Rie98}.
Examples of dark energy in a form of a scalar field with a self-interaction
potential can be found in reviews by \cite{PR03}, \cite{Cop06}, and by \cite{Uz10}.
Since that time many sophisticated models have been suggested to explain the nature of
dark energy and among them the scalar fields which are ultra-light
in cosmic vacuum but possess a large mass locally when they are
coupled to ordinary matter by the so-called chameleon mechanism
\citep{KW04, Br04, Ave08, Br10a, Br10b}.
A subclass of these models
considered by \cite{OP08} predicts that fundamental physical
quantities such as elementary particle masses and low-energy
coupling constants may also depend on the local matter density.
Since the mass of the electron $m_{\rm e}$ is proportional to the
Higgs vacuum expectation value (VEV $\sim 200$ GeV), and the mass of
the proton $m_{\rm p}$ is proportional to the quantum chromodynamics
(QCD) scale $\Lambda_{\rm QCD} \sim 220$ MeV, we may probe the ratio
of the electroweak scale to the strong scale through the
measurements of the dimensionless mass ratio $\mu = m_{\rm e}/m_{\rm p}$
in high density laboratory (terrestrial) environment, $\mu_{\rm lab}$,
and in low density interstellar clouds, $\mu_{\rm space}$
($\rho_{\rm lab}/\rho_{\rm space} > 10^{10}$). In this way we are
testing whether the scalar field models have chameleon-type
interaction with ordinary matter.
Several possibilities to detect chameleons from astronomical observations
were discussed in \cite{BDS09}, \cite{DSS09}, \cite{BZ10}, and \cite{Av10}.
First experiments constraining these models were recently carried out
in Fermilab \citep{USW10} and in Lawrence Livermore National Laboratory \citep{Ryb10}.
At the moment, the most accurate relative changes in the mass ratio
$\Delta \mu/\mu = (\mu_{\rm space} - \mu_{\rm lab})/\mu_{\rm lab}$
can be obtained with the ammonia method \citep{VKB04,FK07}.
{NH$_3$}\ is a molecule whose inversion frequencies are very sensitive to any
changes in $\mu$ because of the quantum mechanical tunneling of the
N atom through the plane of the H atoms. The sensitivity coefficient
to $\mu$-variation of the {NH$_3$}\ $(J,K) = (1,1)$ inversion
transition at 24 GHz is $Q_{\rm inv}=4.46$. This means that
the inversion frequency scales as $\Delta \omega/\omega = 4.46(\Delta
\mu/\mu)$. In other words, sensitivity to $\mu$-variation is 4.46
times higher than that of molecular rotational
transitions, where $Q_{\rm rot}=1$. Thus, by comparing the observed
radial velocity of the inversion transition of {NH$_3$}, $V_{\rm inv}$,
with a suitable rotational transition, $V_{\rm rot}$, of another
molecule arising co-spatially with ammonia, a limit on the spatial
variation of $\mu$ can be determined:
\begin{equation}\label{eq1}
\frac{\Delta \mu}{\mu}
= \frac{V_{\rm rot} - V_{\rm inv}}{c(Q_{\rm inv} - Q_{\rm rot})}
\approx 0.3 \frac{\Delta V}{c},
\end{equation}
where $c$ is the speed of light and $\Delta V = V_{\rm rot} - V_{\rm inv}$.
Surprisingly, recent observations of a sample of nearby (distance $R
\sim$ 140 pc) cold molecular cores ($T_{\rm kin} \sim 10$K, $n =
10^4-10^5$ cm$^{-3}$, $B < 10$ $\mu$G) in lines of {NH$_3$}\ $(J,K) =
(1,1)$ at 24 GHz, HC$_3$N $J = 2-1$ at 18 GHz, and N$_2$H$^+$ $J =
1-0$ at 93 GHz reveal a statistically significant positive velocity
offset between the low-$J$ rotational and inversion transitions:
$\Delta V = V_{\rm rot} - V_{\rm inv} = 27 \pm 4_{\rm stat} \pm
3_{\rm sys}$ m~s$^{-1}$, which gives $\Delta \mu/\mu = (26 \pm
4_{\rm stat} \pm 3_{\rm sys})\times 10^{-9}$ \citep{LML10}\footnote{Presented are the
corrected values of $\Delta V$ and $\Delta \mu/\mu$ discussed in \cite{LLH10}.}.
A few molecular cores from this sample were mapped in the {NH$_3$}\ (1,1) and
HC$_3$N (2--1) lines and it was found that in two of them (L1498 and
L1512) these lines trace the same material and show the offset
$\Delta V = 26.9 \pm 1.2_{\rm stat} \pm 3.0_{\rm sys}$ m~s$^{-1}$
throughout the entire clouds \citep{LLH10}. It was also demonstrated
that for these clouds the frequency shifts caused by external
electric and magnetic fields and by the cosmic black body
radiation-induced Stark effect are less than 1 m~s$^{-1}$.
Optical depth effects in these clouds were studied from the
analysis of unsaturated ($\tau < 1$) and slightly saturated ($\tau \approx 1-2$)
hyperfine components of the corresponding molecular transitions and
it was found that both groups of lines have similar radial
velocities within the $1\sigma$ uncertainty intervals.
The nonzero $\Delta\mu$ implies that at deep interstellar vacuum the
electron-to-proton mass ratio increases by $\sim 3\times10^{-8}$ as
compared with its terrestrial value and, hence, LPI is broken. In
view of the potentially important application of this discrepancy to
the fundamental physics, one has to be sure that the nonzero $\Delta \mu$
is not caused by some overlooked systematic errors. An obvious
way to tackle this problem is to use other molecular transitions
which have different sensitivity coefficients $Q_\mu$. It has
already been suggested to measure $\Lambda$-doublet lines of light
diatomic molecules OH and CH \citep{Koz09}, and microwave
inversion-rotational transitions in partly deuterated ammonia
NH$_2$D and ND$_2$H \citep{KLL10}.
In the present paper, we propose to use tunneling and rotation
transitions in the hydronium ion {H$_3$O$^+$}. Like ammonia, it also has a
double minimum vibrational potential. The inversion transitions
occurs when the oxygen atom tunnels through the plane of the
hydrogen atoms. This leads to an inversion splitting of the
rotational levels. The splitting of {H$_3$O$^+$}\ is very large,
$55.3462\pm0.0055$ cm$^{-1}$ \citep{LO85}
as compared to $1.3$ cm$^{-1}$ splitting in {NH$_3$}. Consequently, the ground-state
inversion-rotational spectrum of {H$_3$O$^+$}\ is observed in the
submillimeter-wave region \citep{PHL85,BDD85,VTM88}, whereas pure
inversion transitions in the far-infrared region
\citep{VVT89,YDPP09}.
\begin{figure}[t!]
\epsscale{1.1}
\plotone{fig1.eps}
\caption{The level scheme for {H$_3$O$^+$}. The depicted frequencies are in GHz. \label{fig1}}
\end{figure}
{H$_3$O$^+$}\ has both ortho- and para-modifications (see \fref{fig1}).
In the submillimeter,~-- the range accessible from high altitude
ground-based telescopes,~-- there are three low-lying transitions at
307, 364, and 388 GHz which belong to para-{H$_3$O$^+$}, and one
ortho-{H$_3$O$^+$}\ transition at 396 GHz. The 388 GHz line is, however,
blocked by water vapor in the atmosphere. The other lines were observed
in the interstellar molecular clouds
\citep{HCH86,WBB86,WMT91,vdT06,PvD92}.
The 364 GHz line was also
observed in two galaxies: \object{M~82} and \object{Arp~220} \citep{vdT08}.
In far-IR, {H$_3$O$^+$}\ lines were detected from aboard the space
observatories at $\omega = 4.31$ THz \citep{TNP96},
$\omega = 1.66$, 2.97, and 2.98 THz \citep{GC01,LBS06,PBS07}, $\omega = 984$ GHz
\citep{GdL10}, and $\omega = 1.03, 1.07$, and 1.63 THz \citep{BBvD10}.
The observed transitions of {H$_3$O$^+$}\ arise in the warm ($T_{\rm kin}
\sim 100$ K) and dense ($n \approx 10^5-10^6$ cm$^{-3}$)
star-forming regions surrounding protostars where hydronium appears
to be one of the most abundant species with the abundance as high as
$X$({H$_3$O$^+$}) $\approx 5\times10^{-9}$ \citep{WMT91,BBvD10}.
\section{Sensitivity coefficient of inversion transition}
\label{inv}
Sensitivity of the {H$_3$O$^+$}\ inversion transition to $\mu$-variation can be
estimated from the analytical Wentzel-Kramers-Brillouin (WKB)
approximation.
Following \cite{LL77}, we write for the inversion frequency
(used units are $\hbar=|e|=m_e=1$):
\begin{equation}\label{inv1}
\omega_\mathrm{inv}\approx \frac{2E_0}{\pi}\,\mathrm{e}^{-S},
\end{equation}
where $S$ is the action over classically forbidden region and $E_0$
is the ground state vibrational energy. Expression (\ref{inv1})
gives the following sensitivity to $\mu$-variation:
\begin{equation}\label{inv2}
Q_\mathrm{inv}\approx \frac{S+1}{2}
+\frac{S\,E_0}{2(U_\mathrm{max}-E_0)}\,,
\end{equation}
where $U_\mathrm{max}$ is the barrier hight and we are not using an
additional approximation $E_0=\omega_v/2$.
According to \cite{RNVH04} and \cite{DN06}, we can take
$U_\mathrm{max}=651$~cm$^{-1}$\ and $E_0\approx 400$~cm$^{-1}$. The inversion
frequency for {H$_3$O$^+$}\ is 55.3~cm$^{-1}$. Thus, Eqs. (\ref{inv1},\ref{inv2})
give:
\begin{equation}\label{inv3}
S\approx 1.5\,,\qquad Q_\mathrm{inv}\approx 2.5\,.
\end{equation}
\cite{DN06} report the inversion frequencies for {H$_3$O$^+$},
H$_2$DO$^+$, HD$_2$O$^+$, and D$_3$O$^+$ to be 55.3 cm$^{-1}$, 40.5 cm$^{-1}$,
27.0 cm$^{-1}$, and 15.4 cm$^{-1}$, respectively.
Neglecting the weak dependence of
the reduced mass, $m_r$, on the inversion coordinate, we take $m_r$ to be 0.7,
0.8, 1.0, and 1.25, respectively \citep{DN06}.
Figure~\ref{fig2} shows the inversion frequency as a function of $m_r$.
From this plot we can estimate the sensitivity coefficient for {H$_3$O$^+$}\ to be:
\begin{equation}\label{inv4}
Q_\mathrm{inv}\approx 2.46\,,
\end{equation}
which is in a perfect agreement with \Eref{inv3}. We can conclude that the inversion
transition in {H$_3$O$^+$}\ is almost two times less sensitive to $\mu$-variation,
than similar transition in {NH$_3$}, where $Q_\mathrm{inv}=4.5$ \citep{FK07}.
\section{Sensitivity coefficients of mixed transitions}
\label{mix}
The spectrum of the rotational and inversion transitions of {H$_3$O$^+$}\ is
studied in \cite{YDPP09}. For the lowest vibrational state we can
write the simplified inversion-rotational Hamiltonian as:
\begin{eqnarray} \label{mix1}
H\!\!\!\! & = &\!\!\!\! BJ(J+1) + (C-B)K^2 -D_J[J(J+1)]^2\quad \nonumber \\
\!\!\!\! & &\!\!\!\! -D_{JK}J(J+1)K^2 -D_K K^4 + \dots \\
\!\!\!\! & &\!\!\!\! +\frac{s}{2} \left\{W_0 + W_J J(J+1) + W_K K^2 +\dots \right\}\, . \nonumber
\end{eqnarray}
Here we neglected higher terms of expansion in $J$ and $K$; $s=\pm 1$ for
symmetric and antisymmetric inversion state; total parity $p=(-1)^K s$.
Numerical values are given in \cite{YDPP09} (MHz):
\begin{equation}
\begin{array}{cccc}
B & C-B & D_J & D_{JK} \nonumber\\
334406 & -148804 & 35 & -70 \nonumber\\
D_K & W_0 & W_J & W_K \nonumber\\
41 & -1659350 & 5988 & -8458 \nonumber
\end{array}
\end{equation}
Note that we write Hamiltonian (\ref{mix1}) in such a way that terms which
determine inversion splitting are collected in the last line. Therefore, we
have the following relation with parameters used in \cite{YDPP09}:
\begin{eqnarray}\label{mix1a}
B & = & \left[B(0^+)+B(0^-)\right]/2\,, \nonumber \\
W_J & = & B(0^+)-B(0^-)\,,
\end{eqnarray}
and similarly for $C-B$ and $W_K$. Parameters $D_J$, $D_{JK}$, and $D_K$ are
averaged over inversion states $s=\pm 1$.
\begin{figure}[t!]
\epsscale{1.10}
\plotone{fig2c.eps}
\caption{Inversion frequency as a function of the reduced mass for
hydronium ion isotopologues. \label{fig2}}
\end{figure}
To estimate sensitivities of the mixed transitions it is sufficient to
account for $\mu$-dependence of the dominant parameters $B$, $C$,
and $W_0$. It is clear that $B,\,C \sim\mu$ and $W_0$ scales as
$\mu^{Q_\mathrm{inv}}$. It follows, that for rotational part of the
energy we have $Q_\mathrm{rot}=1$ and for inversion part
$Q_\mathrm{inv}$ is given by \Eref{inv3} or (\ref{inv4}).
This leads to the expressions, used earlier for NH$_2$D
\citep{KLL10}:
\begin{equation}\label{mix2}
\omega_\mathrm{mix} =\omega_\mathrm{rot}\pm\omega_\mathrm{inv}\, ,
\end{equation}
and
\begin{equation}\label{mix3}
Q_\mathrm{mix}
=\frac{\omega_\mathrm{rot}}{\omega_\mathrm{mix}}Q_\mathrm{rot}
\pm\frac{\omega_\mathrm{inv}}{\omega_\mathrm{mix}}Q_\mathrm{inv}\, .
\end{equation}
We use Hamiltonian (\ref{mix1}) and expression (\ref{mix3}) to calculate the
frequencies and sensitivities of the mixed transitions.
The obtained results are presented in \tref{tab_mix}.
Final results are very sensitive to the parameter $Q_\mathrm{inv}$.
A good agreement between two different estimates of $Q_\mathrm{inv}$
from \Eref{inv3} and from \fref{fig1} shows that this parameter is
known with 10\% accuracy, or better. In the next approximation, we
need to weight independently all terms of the Hamiltonian
(\ref{mix1}) with different scalings. However, this does not lead to
any significant changes in sensitivities $Q_\mu$ of the low-$J$
transitions from \tref{tab_mix}.
\begin{table*}
\begin{center}
\caption{Frequencies and sensitivities to $\mu$-variation of the
inversion-rotation transitions in {H$_3$O$^+$}. Experimental frequencies
are taken from \cite{JPL_Catalog,YDPP09}. \label{tab_mix}}
\begin{tabular}{cccccc r@{.}l c r c}
\tableline\tableline
\multicolumn{6}{c}{Transition} &\multicolumn{4}{c}{Frequency (MHz)} &\multicolumn{1}{c}{$Q_\mu$}\\
$J$&$K$&$s$&$J'$&$K'$&$s'$&\multicolumn{2}{c}{Exper.}
& \multicolumn{1}{c}{error} &\multicolumn{1}{c}{\Eref{mix1}} \\
\tableline
1 & 1 &$-1$& 2 & 1 &$+1$ & 307192&410& 0.05 & 307072&$+9.0$ \\
3 & 2 &$+1$& 2 & 2 &$-1$ & 364797&427& 0.10 & 365046&$-5.7$ \\
3 & 1 &$+1$& 2 & 1 &$-1$ & 388458&641& 0.08 & 389160&$-5.2$ \\
3 & 0 &$+1$& 2 & 0 &$-1$ & 396272&412& 0.06 & 397198&$-5.1$ \\
0 & 0 &$-1$& 1 & 0 &$+1$ & 984711&907& 0.30 & 984690&$+3.5$ \\
4 & 3 &$+1$& 3 & 3 &$-1$ & 1031293&738& 0.30 & 1031664&$-1.4$ \\
4 & 2 &$+1$& 3 & 2 &$-1$ & 1069826&632& 0.30 & 1071154&$-1.2$ \\
3 & 2 &$-1$& 3 & 2 &$+1$ & 1621738&993& 2.00 & 1621326&$+2.5$ \\
2 & 1 &$-1$& 2 & 1 &$+1$ & 1632090&98 & & 1631880&$+2.5$ \\
1 & 1 &$-1$& 1 & 1 &$+1$ & 1655833&910& 1.50 & 1655832&$+2.5$ \\
\tableline
\end{tabular}
\end{center}
\end{table*}
\section{Discussion and conclusions}
\label{dis}
We have shown above that the rest-frame frequencies of the
inversion-rotational transitions of {H$_3$O$^+$}\ are very sensitive to the
value of $\mu$. For a given transition from \tref{tab_mix},
$\omega_i$, with the sensitivity coefficient $Q_i$, the expected
frequency shift, $\Delta \omega_i/\omega_i$, due to a change in
$\mu$ is given by \citep{LML10}:
\begin{equation}\label{disEq1}
\frac{\Delta \omega_i}{\omega_i}
\equiv \frac{\tilde{\omega}_i - \omega_i}{\omega_i} =
Q_i\frac{\Delta \mu}{\mu}\, ,
\end{equation}
where $\omega_i$ and $\tilde{\omega_i}$ are the frequencies
corresponding to the laboratory value of $\mu$ and to an altered
$\mu$ in a low-density environment, respectively.
By analogy with \Eref{eq1}, we can estimate the value of $\Delta \mu/\mu$
from two transitions with different sensitivity coefficients $Q_i$ and $Q_j$:
\begin{equation}\label{disEq2}
\frac{\Delta \mu}{\mu} = \frac{V_j - V_i}{c(Q_i - Q_j)}\, ,
\end{equation}
where $V_j$ and $V_i$ are the apparent radial velocities of the
corresponding {H$_3$O$^+$}\ transitions.
Consider two lowest frequency transitions from \tref{tab_mix}:
$1_1^- \rightarrow 2_1^+$ and $3_2^+ \rightarrow 2_2^-$ of
para-{H$_3$O$^+$}\ at, respectively, 307 and 364 GHz. Here $\Delta Q =
Q_{307} - Q_{364} = 14.7$, which is 4 times larger then the $\Delta Q$
value from the ammonia method. This means that the offset $\Delta V \sim 27$ m~s$^{-1}$,
detected in the ammonia method, should correspond to
the relative velocity shift between these transitions, $\Delta V =
V_{364} - V_{307}$, of about $100$ m~s$^{-1}$.
Published results on interstellar {H$_3$O$^+$}\ allow us to put an upper
limit on $\Delta \mu/\mu$. The observations of the 307, 364, and 396
GHz lines carried out at the 10.4-m telescope of the Caltech
Submillimeter Observatory (CSO) by \cite{PvD92} and the observations of the 307 and 364
GHz lines at the 12-m APEX telescope (Atacama Pathfinder Experiment) by
\cite{vdT06} have accuracy of about 1 km~s$^{-1}$\ that provides a limit
on $\Delta \mu/\mu < 2\times10^{-7}$, which is consistent with the
signal $\sim 3\times10^{-8}$ revealed by the ammonia method
\citep{LML10,LLH10}.
In order to check ammonia results we need to improve the accuracy of
{H$_3$O$^+$}\ observations by more than one order of magnitude. According to
\tref{tab_mix}, the uncertainties of the laboratory frequencies of
the transitions at 307, 364, and 396 GHz are, respectively, 50, 80,
and 45 m~s$^{-1}$. Therefore, we also need a factor of few improvement of
the laboratory accuracy to be able to detect reliably an expected signal
$\Delta V \sim 100$ m~s$^{-1}$\ and to
check the non-zero ammonia results.
An important advantage of the hydronium method is that it is based
on only one molecule. In the ammonia method there is
unavoidable Doppler noise caused by relative velocity shifts due to
spatial segregation of {NH$_3$}\ and other molecules.
When using hydronium, the only
source of the Doppler noise may arise from possible kinetic
temperature fluctuations within the molecular cloud since the
submillimeter {H$_3$O$^+$}\ transitions have different upper level
energies: $E_u = 80, 139$, and 169 K for the 307, 364, and 396 GHz
transitions, respectively. It is also important that two {H$_3$O$^+$}\ transitions
have similar $Q$ values: $Q_{364} \approx Q_{396}$. This allows us to
control the Doppler noise and to measure accurately the relative position
of the 307 GHz line.
The analysis of other possible sytematic effects for hydronium is mostly
similar to what was done in detail for ammonia in Levshakov {et al.}\ (2010b).
The systematic shifts caused by pressure effects are about a few m~s$^{-1}$, or lower.
As mentioned above in Sect.~\ref{sect-1}, the frequency shifts caused by external
electric and magnetic fields and by the cosmic black body
radiation-induced Stark effect are less or about 1 m~s$^{-1}$\ for {NH$_3$}.
An additional source of
systematic for {H$_3$O$^+$}\ can come from the unresolved hyperfine structure (HFS)
in combination with possible non-thermal HFS populations in the ISM.
As noted by \cite{KR10}, HFS lines reduce the effective optical depth
of the molecular rotational transition by spreading the emission out over
a wider bandwidth.
To our knowledge, the HFS has not been resolved yet for H$_3{}^{16}$O$^+$
either in laboratory, or astronomical measurements.
An expected size of the hyperfine splittings can be estimated using analogy with ammonia.
For the latter the main hyperfine
splitting is associated with the spin of nitrogen $\bm{I}_1$.
The maximum hyperfine
splitting caused by the hydrogenic spin $\bm{I}$ is about 40 kHz
\citep{HT83}. This splitting includes interaction with the nitrogen spin
$\sim (\bm{I}_1\cdot\bm{I})$ and with molecular rotation
$\sim (\bm{J}\cdot\bm{I})$. In hydronium, the oxygen nucleus is spinless and there is
only interaction with rotation $\sim (\bm{J}\cdot\bm{I})$. Hydronium
has similar electronic structure and close rotational constants to ammonia,
so its spin-rotational interaction should be $\la 40$ kHz.
Thus, we can expect less than 40 m~s$^{-1}$\ of the hyperfine bandwidth for 300 GHz lines and
smaller splittings at higher frequencies.
At $T_{\rm kin} \sim 100$~K,~-- a typical kinetic temperature
of warm and dense gas in the star-forming regions,~-- the
thermal width of the {H$_3$O$^+$}\ lines is comparable with this HFS splitting.
For para-{H$_3$O$^+$}, transitions at 307, 388 and 364 GHz have 3 HFS components each
(two transitions with $\Delta F = \Delta J$ are strong, and the remaining one is weak).
For ortho-{H$_3$O$^+$}, the 396 GHz transition has 9 HFS components
(4 strong, 3 weaker, and 2 weakest).
The difference in the magnitude of the energies of
the hyperfine and rotational transitions in molecular spectra in the
sub-millimiter range is considerable:
milli-Kelvin for the hyperfine levels and tens of Kelvin
between rotational levels.
This means that the HFS levels may be populated approximately in statistical equilibrium
even if the rotational levels are not \citep{KR10}.
Therefore, we may suggest that the line centers of the sub-millimiter {H$_3$O$^+$}\
transitions are not affected significantly by relative populations of the hyperfine
levels and that the exepected velocity shifts are of a few m~s$^{-1}$.
To sum up, the systematic shifts of the line centers
caused by possible non-thermal HFS populations and pressure effects are likely
not larger than 10 m~s$^{-1}$, which is about 10\% of the expected relative shift
between the para-{H$_3$O$^+$}\ $J_K =
1_1^- \rightarrow 2_1^+$ and $3_2^+ \rightarrow 2_2^-$ transitions
due to $\mu$-variation.
A more accurate analysis of the HFS-induced systematics will be possible
after the HFS is either measured, or calculated theoretically.
Finally, we would like to note that other isotopologues of the hydronium ion
also must have large sensitivity coefficients $Q_\mu$. To use them
we need laboratory studies of the low-frequency mixed transitions in
the spectra of the partly deuterated hydronium ions H$_2$DO$^+$,
HD$_2$O$^+$, and in D$_3$O$^+$.
In the near future, high-precision measurements in the submillimeter and FIR ranges
with greatly improved sensitivity
will be available with the
Atacama Large Millimeter/submillimeter Array (ALMA), the Stratospheric Observatory For
Infrared Astronomy (SOFIA), the Cornell Caltech Atacama Telescope (CCAT), and others.
Thus, any further advances in exploring $\Delta \mu/\mu$ depend crucially on
accurate laboratory measurements ($\Delta \omega/\omega \la 10^{-8}$)
of relevant molecular transitions in the submillimeter and FIR ranges
where reliable spectroscopic data are still relatively poor.
\acknowledgments
We are grateful to V. Kokoouline for bringing inversion spectra of
{H$_3$O$^+$}\ to our attention. The work has been supported in part by the
RFBR grants 08-02-00460, 09-02-12223, and 09-02-00352, by the
Federal Agency for Science and Innovations grant NSh-3769.2010.2,
and by the Chinese Academy of Sciences,
grant No. 2009J2-6.
|
{
"timestamp": "2010-11-03T01:00:41",
"yymm": "1009",
"arxiv_id": "1009.3672",
"language": "en",
"url": "https://arxiv.org/abs/1009.3672"
}
|
\section{Introduction}
Let $Q=\mathbb P^1\times \mathbb P^1$ and let
$X\subset Q$ be a $0$-dimensional scheme. Let $R$ and $C$ be,
respectively, a $(1,0)$ and a $(0,1)$-line not containing any point of $X$ and let
$Z$ be a $0$-dimensional scheme given by $X$ and some points on $R$ or
$C$. In this paper we deal with the problem of finding the Hilbert matrix
(function) of $Z$ with respect to the Hilbert matrix of $X$. A first approach
was given in a very particular case in 1992 in \cite{GMR}, with the only
perspective of comprehending the Hilbert functions of ACM
$0$-dimensional schemes in $Q$. This paper is the first real step towards the characterization of
the Hilbert functions of $0$-dimensional schemes in $Q$ that are not ACM.
In Theorem \ref{T:1} and Theorem \ref{T:2} we improve
the result in \cite{GMR}, under some geometric and algebraic conditions that, as we see in
Example \ref{ex}, can not be suppressed without any further
assumption. In Theorem
\ref{T:5} and Theorem \ref{T:6} we see
that the result holds for all ACM schemes $X$ without any additional
condition. As an application in Section \ref{Esempio} we compute the
Hilbert matrix of any non ACM reduced set of points in $Q$ having a certain position in a grid of $(1,0)$ and $(0,1)$-lines. This
previously could be done and was known just for ACM $0$-dimensional schemes.
A good reference for a general discussion on $0$-dimensional schemes on $\mathbb P^1\times
\mathbb P^1$ is \cite{GMR}, in which there are the most important
results about the Hilbert function. Further results on the Hilbert
function has been obtained just in the particular case of fat
points (see for example \cite{GVT} and \cite{VT2}).
\section{Notation and preliminary results}
Let $k$ be an algebraically closed field,let $\mathbb P^1=\mathbb
P^1_k$, let $Q=\mathbb P^1\times \mathbb P^1$ and let
$\mathscr O_Q$ be its structure sheaf. For any sheaf $\mathscr F$ we
denote:
\[
\mathscr F(a,b)=\mathscr F\otimes \mathscr O_Q(a,b).
\]
Let us consider the bi-graded ring:
\[
S=H^0_*(\mathscr O_Q)=\bigoplus_{a,b\ge 0}H^0(\mathscr O_Q(a,b)).
\]
For any bi-graded $S$-module $N$ let $N_{(i,j)}$ the component of
degree $(i,j)$.
Given $X\subset Q$ $0$-dimensional scheme, let $I(X)\subset S$ be the
associated saturated sheaf, $S(X)=S/I(X)$ and $\mathscr I_X\subset
\mathscr O_Q$ its ideal sheaf.
\begin{defin}
The function:
\[
M_X\colon \mathbb Z\times \mathbb Z\rightarrow \mathbb N
\]
defined by:
\[
M_X(i,j)=\dim_k {S(X)}_{(i,j)}=\dim_k S_{(i,j)}-\dim_k {I(X)}_{(i,j)}
\]
is called \emph{Hilbert function} of $X$. The function $M_X$ can be
represented as a matrix with infinite integers entries:
\[
M_X=(M_X(i,j))=(m_{ij})
\]
called \emph{Hilbert matrix} of $X$.
\end{defin}
Note that $M_X(i,j)=0$ for either $i<0$ or $j<0$ and so we restrict
ourselves to the range $i\ge 0$ and $j\ge 0$. Moreover, for $i\gg 0$
and $j\gg 0$ $M_X(i,j)=\deg X$.
\begin{defin}
Given the Hilbert matrix $M_X$ of a $0$-dimensional scheme $X\subset Q$, the \emph{first difference of the Hilbert function} of $X$ is the
matrix:
\[
\Delta M_X=(c_{ij}),
\]
where:
\[
c_{ij}=m_{ij}-m_{i-1j}-m_{ij-1}+m_{i-1j-1}.
\]
\end{defin}
We consider the following matrices:
\[
\Delta^R M_X=(a_{ij})\mbox{ and }\Delta^C M_X=(b_{ij}),
\]
with $a_{ij}=m_{ij}-m_{ij-1}$ and $b_{ij}=m_{ij}-m_{i-1j}$. Note that:
\[
c_{ij}=a_{ij}-a_{i-1j}=b_{ij}-b_{ij-1}
\]
and
\[
m_{ij}=\sum_{\substack{h\le i\\ k\le j}}c_{hk}.
\]
\begin{thm}[{\cite[Theorem 2.11]{GMR}}] \label{T0}
Given a $0$-dimensional scheme $X\subset Q$ and given its Hilbert matrix
$M_X$, the first difference $\Delta M_X=(c_{ij})$ satisfies the
following conditions:
\begin{enumerate}
\item $c_{ij}\le 1$ and $c_{ij}=0$ for $i\gg 0$ or $j\gg 0$;
\item if $c_{ij}\le 0$, then $c_{rs}\le 0$ for any $(r,s)\ge (i,j)$;
\item for every $(i,j)$ $0\le \sum_{t=0}^j c_{it}\le
\sum_{t=0}^jc_{i-1t}$ and $0\le \sum_{t=0}^i c_{tj}\le \sum_{t=0}^i c_{tj-1}$.
\end{enumerate}
\end{thm}
\begin{remark} \label{rm}
If $X\subset Q$ is a $0$-dimensional scheme, let us consider $a=\min\{i\in \mathbb
N\mid I(X)_{(i,0)}\ne 0\}-1$ and $b=\min\{j\in \mathbb
N\mid I(X)_{(0,j)}\ne 0\}-1$. Then by Theorem \ref{T0} $\Delta M_X$ is zero out of the
rectangle with opposite vertices $(0,0)$ and $(a,b)$. In this case we
say that $\Delta M_X$ is of syze $(a,b)$.
\end{remark}
Let $M_X=(m_{ij})$ be the Hilbert matrix of a $0$-dimensional scheme
$X\subset Q$. Using the notation in \cite{GMR}, for every $j\ge 0$ we set:
\[
i(j)=\min\{t\in \mathbb N\mid m_{tj}=m_{t+1j}\}=\min\{t\in \mathbb
N\mid b_{t+1j}=0\}.
\]
and for every $i\ge 0$ we set:
\[
j(i)=\min\{t\in \mathbb N\mid m_{it}=m_{it+1}\}=\min\{t\in \mathbb
N\mid a_{it+1}=0\}
\]
In particular, we see that $i(0)=a$ and $j(0)=b$.
Let $X\subset Q$ be a $0$-dimensional scheme and let $L$ be a line
defined by a form $l$. Let $J=(I(X),l)$ and let $d=\deg(\operatorname{sat} J)$. Then we call $d$ the number of points
of $X$ on the line $L$ and, by abuse of notation, we make the position
$d=\#(X\cap L)$. We say that $L$ is disjoint from $X$ if $d=0$.
The key result used in Section \ref{main} is the following:
\begin{thm}[{\cite[Theorem 2.12]{GMR}}] \label{T}
Let $X\subset Q$ be a $0$-dimensional scheme and let $M_X=(m_{ij})$
be its Hilbert matrix. Then for every $j\ge 0$ there are just
$a_{i(0)j}-a_{i(0)j+1}$ lines of type $(1,0)$ each containing just
$j+1$ points of $X$ and, similarly, for every $i\ge 0$ there are
just $b_{ij(0)}-b_{i+1j(0)}$ lines of type $(0,1)$ each containing
just $i+1$ points of $X$.
\end{thm}
The result that in this paper we improve is given by the following:
\begin{thm}[{\cite[Lemma 2.15]{GMR}}] \label{T2}
Let $X\subset Q$ be a 0-dimensional scheme and let $M_X$ be its Hilbert
matrix. Let $R_0$,\dots, $R_a$
and $C_0$,\dots,$C_b$ be, respectively, the
$(1,0)$ and $(0,1)$-lines containing $X$ and at least one point of $X$.
Let $R$ be a $(1,0)$-line disjoint from $X$ and let $Y=X\cup
R\cap(C_0\cup\dots\cup C_n)$, with $n\ge b$ and
$C_{b+1}$,\dots,$C_n$ arbitrary $(0,1)$-lines. Then:
\[
\Delta M_Y^{(i,j)}=
\begin{cases}
1 & \text{for } i=0,\, j\le n\\
0 & \text{for }i=0,\, j\ge n+1\\
\Delta M_X^{(i-1,j)} & \text{for } i\ge 1.
\end{cases}
\]
\end{thm}
Of course a similar result can be proved by adding $m+1$ points on a
$(0,1)$-line $C$ disjoint from $X$. So, with the previous notation, it
is possible to prove the following result.
\begin{thm}
Let $C$ be a $(0,1)$-line disjoint from $X$. Let $Y=X\cup
C\cap(R_0\cup\dots\cup R_m)$, $m\ge a$, and
$R_{a+1}$,\dots,$R_m$ arbitrary $(1,0)$-lines. Then:
\[
\Delta M_Y^{(i,j)}=
\begin{cases}
1 & \text{for } i\le m,\,j=0\\
0 & \text{for } i\ge m+1,\, j=0\\
\Delta M_X^{(i,j-1)} & \text{for } j\ge 1.
\end{cases}
\]
\end{thm}
\section{The first difference of the Hilbert function} \label{main}
Let $X\subset Q$ be a 0-dimensional scheme and let $M_X$ be its Hilbert
matrix. In all this paper we suppose that $\Delta M_X$ is of size
$(a,b)$ and we denote by $R_0$,\dots, $R_a$
and $C_0$,\dots,$C_b$, respectively, the
$(1,0)$ and $(0,1)$-lines containing $X$ and at least one point of $X$.
\begin{thm} \label{T:1}
Let $R$ be a $(1,0)$-line disjoint from $X$. Let $C_{b+1}$,\dots,$C_n$,
$n\ge b$, be arbitrary $(0,1)$-lines and $i_1$,\dots,$i_r\in
\{0,\dots,b\}$. Let $\mathcal P=\{R\cap C_i\mid i\in\{0,\dots,n\},\,
i\ne i_1,\dots,i_r\}$ and let $Z=X\cup \mathcal P$. Suppose also that on the $(0,1)$-line $C_{i_k}$
there are $q_k$ points of $X$ for $k=1,\dots,r$ and that $q_1\le
q_2\le \dots \le q_r$. Then, given
$T=\{(q_1,n),(q_2,n-1),\dots,(q_r,n-r+1)\}$, we have:
\[
\Delta M_Z^{(i,j)}=
\begin{cases}
1 & \text{if } i=0,\, j\le n\\
0 & \text{if }i=0,\, j\ge n+1\\
\Delta M_X^{(i-1,j)} & \text{if } i\ge 1\text{ and }(i,j)\notin T\\
\Delta M_X^{(i-1,j)}-1 & \text{if } i\ge 1\text{ and }(i,j)\in T
\end{cases}
\]
if one of the following conditions holds:
\begin{enumerate}
\item $r=1$;
\item $r\ge 2$, $q_{r-1}<q_r$ and for any $k\in \{1,\dots,r-1\}$ and $i\ge q_k$\, $\Delta
M_X^{(i,n-k+1)}=0$;
\item $r\ge 2$, $q_{r-1}=q_r$ and for any $k\in \{1,\dots,r\}$ and $i\ge q_k$\, $\Delta
M_X^{(i,n-k+1)}=0$.
\end{enumerate}
\end{thm}
\begin{proof}
Let $Y=X\cup (R\cap (\bigcup_{i=0}^n C_i))$. By Theorem \ref{T2} it is
sufficient to prove that:
\[
\Delta M_Z(i,j)=
\begin{cases}
\Delta M_Y(i,j) & \text{if }(i,j)\notin T\\
\Delta M_Y(i,j)-1& \text{if }(i,j)\in T.
\end{cases}
\]
We divide the proof in different steps.
\begin{step} \label{s:1}
$\Delta M_Z^{(0,j)}=\Delta M_Y^{(0,j)}=1$ for $j\le n$, $\Delta
M_Z^{(0,j)}=\Delta M_Y^{(0,j)}=0$ for $j\ge n+1$ and $\Delta M_Z^{(i,j)}=\Delta
M_Y^{(i,j)}=\Delta M_X^{(i-1,j)}$ for any $(i,j)$ with $j<n-r+1$ and $i\ge 1$.
\end{step}
It is easy to see that $\Delta M_Z^{(0,j)}=1$ for $j\le n$, because
for such values of $j$ $h^0(\mathscr I_Z(0,j))=0$. Moreover, $\Delta
M_Z^{(0,j)}=0$ for $j\ge n+1$ by Remark \ref{rm}.
Taken $(i,j)$, with $j< n-r+1$ and $i\ge 1$, any $(i,j)$-curve containing $Z$ must
contain $R$ and so $h^0(\mathscr
I_Z(i,j))=h^0(\mathscr I_X(i-1,j))$ and $\Delta M_Z^{(i,j)}=\Delta
M_X^{(i-1,j)}$.
\\
Let $r_1$,\dots,$r_{t+1}$ be a sequence of positive integers such
that
$q_{1}=\dots=q_{r_1}<q_{r_1+1}=\dots=q_{r_2}<\dots<q_{r_t+1}=\dots=q_{r_{t+1}}=q_r$
and let $r_0=0$.
\begin{step} \label{s:2}
If $h\in \{1,\dots,t+1\}$, then $\Delta M_Z^{(i,j)}=\Delta
M_Y^{(i,j)}$ for any $(i,j)\le (q_{r_{h-1}+1}-1,n-r_{h-1})$ and for
$(i,j)=(q_{r_{h-1}+1},j)$ with $j<n-r_h+1$.
\end{step}
Taken $(i,j)\le (q_{r_{h-1}+1}-1,n-r_{h-1})$, then any $(i,j)$-curve
containing $Z$ must contain $C_{i_{r_{h-1}+1}}$,\dots,$C_{i_r}$ and so it must contain
$R$. This means that $h^0(\mathscr I_Z(i,j))=h^0(\mathscr I_Y(i,j))$
and so that $\Delta M_Z^{(i,j)}=\Delta M_Y^{(i-1,j)}$.
Taken $(i,j)=(q_{r_{h-1}+1},j)$ with $j<n-r_h+1$, then any
$(i,j)$-curve containing $Z$ must contain
$C_{i_{r_h+1}}$,\dots,$C_{i_r}$ and so it must contain $R$. Again this
implies $h^0(\mathscr
I_Z(i,j))=h^0(\mathscr I_Y(i,j))$ and $\Delta M_Z^{(i,j)}=\Delta
M_Y^{(i,j)}$.
\begin{step} \label{s:3}
For any $1\le h\le t+1$ one of the following conditions holds:
\begin{enumerate}
\item there exists $\overline j$ with $n-r_h+1\le \overline j\le n-r_{h-1}$ such that $\Delta
M_Z^{(q_{r_{h-1}+1},j)}=\Delta M_Y^{(q_{r_{h-1}+1},j)}$ for any $j< \overline j$ and
$\Delta M_Z^{(q_{r_{h-1}+1},\overline j)}<\Delta M_Y^{(q_{r_{h-1}+1},\overline j)}$;
\item $\Delta
M_Z^{(q_{r_{h-1}+1},j)}=\Delta M_Y^{(q_{r_{h-1}+1},j)}$ for any $n-r_h+1\le j\le n-r_{h-1}$.
\end{enumerate}
\end{step}
Since $Z\subset Y$ we see that $M_Z(q_{r_{h-1}+1},n-r_h+1)\le
M_Y(q_{r_{h-1}+1},n-r_h+1)$. Moreover, by Step \ref{s:2} we
see that $M_Z(i,j)=M_Y(i,j)$ for any $(i,j)<(q_{r_{h-1}+1},n-r_h+1)$. This
implies that:
\[
\Delta M_Z^{(q_{r_{h-1}+1},n-r_h+1)}\le \Delta M_Y^{(q_{r_{h-1}+1},n-r_h+1)}.
\]
If $\Delta M_Z^{(q_{r_{h-1}+1},n-r_h+1)}=\Delta M_Y^{(q_{r_{h-1}+1},n-r_h+1)}$, then we
can repeat the previous procedure to show that $\Delta
M_Z^{(q_{r_{h-1}+1},n-r_h+2)}\le \Delta M_Y^{(q_{r_{h-1}+1},n-r_h+2)}$. By iterating this
procedure we get the conclusion of Step \ref{s:3}.
\begin{step} \label{s:4}
\
\begin{enumerate}
\item If $r\ge 2$ and $q_{r-1}=q_r$, given $h\in \{1,\dots,t+1\}$ and $j\in\{n-r_h+1,\dots,n-r_{h-1}\}$,
we have:
\[
\sum_{i=q_{r_{h-1}+1}}^{a+1}\Delta M_Z^{(i,j)}=\Delta M_Y^{(q_{r_{h-1}+1},j)}-1;
\]
\item If $r\ge 2$ and $q_{r-1}<q_r$, given $h\in \{1,\dots,t\}$ and $j\in\{n-r_h+1,\dots,n-r_{h-1}\}$,
we have:
\[
\sum_{i=q_{r_{h-1}+1}}^{a+1}\Delta M_Z^{(i,j)}=\Delta M_Y^{(q_{r_{h-1}+1},j)}-1
\]
and
\[
\sum_{i=q_r}^{a+1}\Delta M_Z^{(i,n-r+1)}=\sum_{i=q_r}^{a+1}\Delta M_Y^{(i,n-r+1)}-1.
\]
\end{enumerate}
\end{step}
Let us first note that by Theorem \ref{T}:
\[
a_{i(0)n-r}(Z)-a_{i(0)n-r+1}(Z)=
\sum_{i\le a+1}
\Delta M_Z^{(i,n-r)}-\sum_{i\le a+1} \Delta M_Z^{(i,n-r+1)}
\]
is equal to the number of $(1,0)$-lines containing precisely $n-r+1$ points of
$Z$, while:
\[
a_{i(0)n-r}(Y)-a_{i(0)n-r+1}(Y)=
\sum_{i\le a+1}
\Delta M_Y^{(i,n-r)}-\sum_{i\le a+1} \Delta M_Y^{(i,n-r+1)}
\]
is equal to the number of $(1,0)$-lines containing precisely $n-r+1$ points of
$Y$. By hypothesis it must be:
\begin{multline*}
\sum_{i\le a+1}
\Delta M_Z^{(i,n-r)}-\sum_{i\le a+1} \Delta M_Z^{(i,n-r+1)}=\\
=\sum_{i\le a+1}
\Delta M_Y^{(i,n-r)}-\sum_{i\le a+1} \Delta M_Y^{(i,n-r+1)}+1.
\end{multline*}
By Step \ref{s:1} this implies that:
\[
\sum_{i\le a+1} \Delta M_Z^{(i,n-r+1)}=\sum_{i\le a+1} \Delta M_Y^{(i,n-r+1)}-1.
\]
Let us now suppose that for some $j\ge n-r+1$, with $j<n$, we have:
\begin{equation}
\label{eq:4}
\sum_{i\le a+1} \Delta M_Z^{(i,j)}=\sum_{i\le a+1} \Delta M_Y^{(i,j)}-1.
\end{equation}
We will show that:
\begin{equation}
\label{eq:5}
\sum_{i\le a+1} \Delta M_Z^{(i,j+1)}=\sum_{i\le a+1} \Delta M_Y^{(i,j+1)}-1.
\end{equation}
Again, by Theorem \ref{T} $\sum_{i\le a+1}
\Delta M_Z^{(i,j)}-\sum_{i\le a+1} \Delta M_Z^{(i,j+1)}$ is equal
to the number of $(1,0)$-lines containing precisely $j+1$ points of
$Z$, while $\sum_{i\le a+1}
\Delta M_Y^{(i,j)}-\sum_{i\le a+1} \Delta M_Y^{(i,j+1)}$ is equal
to the number of $(1,0)$-lines containing precisely $j+1$ points of
$Y$. By hypothesis it must be:
\[
\sum_{i\le a+1}
\Delta M_Z^{(i,j)}-\sum_{i\le a+1} \Delta M_Z^{(i,j+1)}=
\sum_{i\le a+1}
\Delta M_Y^{(i,j)}-\sum_{i\le a+1} \Delta M_Y^{(i,j+1)}.
\]
By \eqref{eq:4} it means that \eqref{eq:5} holds, so that:
\[
\sum_{i\le a+1} \Delta M_Z^{(i,j)}=\sum_{i\le a+1} \Delta M_Y^{(i,j)}-1
\]
for any $j$ with $n-r+1\le j\le n$. Now the hypotheses on $X$, Step \ref{s:1} and Step \ref{s:2} give us the conclusion of Step \ref{s:4}.
\begin{step} \label{s:5}
If $r\ge 2$, $\Delta M_Z^{(i,j)}=\Delta
M_Y^{(i,j)}=0$ for any $(i,j)\ge (q_1+1,n-r_1+1)$ and $\Delta M_Z^{(i,j)}=\Delta
M_Y^{(i,j)}-1$
for any $(i,j)\in \{(q_1,n),(q_1,n-1),\dots,(q_1,n-r_1+1)\}$.
\end{step}
By Theorem \ref{T} we know that:
\[
b_{q_1-1j(0)}(Z)-b_{q_1j(0)}(Z)=\sum_{j\le n} \Delta
M_Z^{(q_1-1,j)}-\sum_{j\le n} \Delta M_Z^{(q_1,j)}
\]
is equal to the number
of $(0,1)$-lines containing exactly $q_1$ points of $Z$ and, in the
same way that:
\[
b_{q_1-1j(0)}(Y)-b_{q_1j(0)}(Y)=\sum_{j\le n} \Delta
M_Y^{(q_1-1,j)}-\sum_{j\le n} \Delta M_Y^{(q_1,j)}
\]
is equal to the number of $(0,1)$-lines containing exactly $q_1$ points of
$Y$. So by construction we have:
\begin{equation} \label{eq:1}
\sum_{j\le n} \Delta
M_Z^{(q_1-1,j)}-\sum_{j\le n} \Delta M_Z^{(q_1,j)}=\sum_{j\le n} \Delta
M_Y^{(q_1-1,j)}-\sum_{j\le n} \Delta M_Y^{(q_1,j)}+r_1.
\end{equation}
By what we proved in Step \ref{s:2} we see that:
\[
\sum_{j\le n} \Delta
M_Z^{(q_1-1,j)}=\sum_{j\le n} \Delta
M_Y^{(q_1-1,j)}
\]
so that by \eqref{eq:1} and again by Step \ref{s:2}
we get:
\begin{equation}
\label{eq:2}
\sum_{n-r_1+1}^{n} \Delta M_Z^{(q_1,j)}=\sum_{n-r_1+1}^{n} \Delta M_Y^{(q_1,j)}-r_1.
\end{equation}
By this equality and by Step \ref{s:3} we see that there exists $n-r_1+1\le \overline j\le
n$ such that $\Delta
M_Z^{(q_1,j)}=\Delta M_Y^{(q_1,j)}$ for any $j<\overline j$ and
$\Delta M_Z^{(q_1,\overline j)}<\Delta M_Y^{(q_1,\overline j)}$. In
particular, $\Delta M_Z^{(q_1,\overline j)}\le 0$ and so by Theorem \ref{T0} we have:
\[
\Delta M_Z^{(i,\overline j)}\le 0
\]
for any $i\ge q_1$. By Step \ref{s:4} we have:
\[
\sum_{i=q_1}^{a+1} \Delta M_Z^{(i,\overline j)}=\Delta
M_Y^{(q_1,\overline j)}-1
\]
\[
\Rightarrow 0\ge \sum_{i=q_1+1}^{a+1} \Delta M_Z^{(i,\overline
j)}=\Delta M_Y^{(q_1,\overline j)}-\Delta M_Z^{(q_1,\overline
j)}-1\ge 0.
\]
This means that $\Delta M_Y^{(q_1,\overline j)}-\Delta M_Z^{(q_1,\overline
j)}-1=0$ and that $\Delta M_Z^{(i,\overline j)}=0$ for any $i\ge
q_1+1$.
Now take any $j>\overline j$. By the fact that $\Delta M_Z^{(i,\overline j)}=0$ for any $i\ge
q_1+1$ and by Theorem \ref{T0} we can say that $\Delta
M_Z^{(i,j)}\le 0$ for any $i\ge q_1+1$ and any $j>\overline j$. By
Step \ref{s:4} we get:
\begin{equation}
\label{eq:10}
0\ge \sum_{i=q_1+1}^{a+1} \Delta M_Z^{(i,j)}=\Delta M_Y^{(q_1,j)}-\Delta M_Z^{(q_1,j)}-1
\end{equation}
so that:
\[
\Delta M_Z^{(q_1,j)}\ge \Delta M_Y^{(q_1,j)}-1
\]
for any $j>\overline j$. So we can say that:
\begin{equation}
\label{eq:6}
\sum_{j=n-r_1+1}^n \Delta M_Z^{(q_1,j)}\ge \sum_{j=n-r_1+1}^n \Delta
M_Y^{(q_1,j)}-n+\overline j-1.
\end{equation}
This fact compared to \eqref{eq:2} gives us that $\overline j\le
n-r_1+1$, but by hypothesis $\overline j\ge n-r_1+1$ and so it must be
$\overline j=n-r_1+1$. This implies that the inequality in
\eqref{eq:6} is an equality, which means that:
\[
\Delta M_Z^{(q_1,j)}=\Delta M_Y^{(q_1,j)}-1
\]
for any $j\ge \overline j=n-r_1+1$, with $j\le n$, and by \eqref{eq:10} $\Delta M_Z^{(i,j)}=\Delta
M_Y^{(i,j)}=0$ for any $(i,j)\ge (q_1+1,n-r_1+1)$.
\begin{step} \label{s:6}
If $r\ge 2$, $q_{r-1}=q_r$ and for any $k\in \{1,\dots,r\}$ and $i\ge q_k$\,
$\Delta M_X^{(i,n-k+1)}=0$, then $\Delta M_Z^{(i,j)}=\Delta
M_Y^{(i,j)}=0$ for any $(i,j)\ge (q_k+1,n-k+1)$ and $k\in \{1,\dots,r\}$
and $\Delta M_Z^{(i,j)}=\Delta
M_Y^{(i,j)}-1$ for any $(i,j)\in \{(q_1,n),(q_2,n-1),\dots,(q_{r},n-r+1)\}$.
\end{step}
We proceed iterating the procedure given in Step \ref{s:5}. So let us suppose that for some $h\in \{2,\dots,t+1\}$
the equalities in the claim hold for any $i<q_{r_{h-1}+1}$ and for
any $j\ge n-r_{h-1}+1$. We will show that they hold also for
$q_{r_{h-1}+1}\le i<q_{r_h+1}$ and for any $j\ge n-r_h+1$.
To this end, we repeat what we did in Step \ref{s:5}. So, as done
before, we see that $\sum_{j\le n} \Delta
M_Z^{(q_{r_{h-1}+1}-1,j)}-\sum_{j\le n} \Delta M_Z^{(q_{r_{h-1}+1},j)}$ is the number of
$(0,1)$-lines containing precisely $q_{r_{h-1}+1}$ points of $Z$, while $\sum_{j\le n} \Delta
M_Y^{(q_{r_{h-1}+1}-1,j)}-\sum_{j\le n} \Delta M_Y^{(q_{r_{h-1}+1},j)}$ is the number of
$(0,1)$-lines containing precisely $q_{r_{h-1}+1}$ points of $Y$. By hypothesis it must
be:
\begin{itemize}
\item[\addtocounter{num}{1}\thenum)] if $q_{r_{h-1}+1}-1>q_{r_{h-1}}$:
\begin{multline*}
\sum_{j\le n} \Delta
M_Z^{(q_{r_{h-1}+1}-1,j)}-\sum_{j\le n} \Delta
M_Z^{(q_{r_{h-1}+1},j)}=\\
=\sum_{j\le n} \Delta
M_Y^{(q_{r_{h-1}+1}-1,j)}-\sum_{j\le n} \Delta M_Y^{(q_{r_{h-1}+1},j)}+r_h-r_{h-1};
\end{multline*}
\item[\addtocounter{num}{1}\thenum)] if $q_{r_{h-1}+1}-1=q_{r_{h-1}}$:
\begin{multline*}
\sum_{j\le n} \Delta
M_Z^{(q_{r_{h-1}+1}-1,j)}-\sum_{j\le n} \Delta
M_Z^{(q_{r_{h-1}+1},j)}=\\
=\sum_{j\le n} \Delta
M_Y^{(q_{r_{h-1}+1}-1,j)}-\sum_{j\le n} \Delta M_Y^{(q_{r_{h-1}+1},j)}+r_h-r_{h-1}-(r_{h-1}-r_{h-2}).
\end{multline*}
\end{itemize}
By what we proved in Step \ref{s:1} and Step \ref{s:2} and by inductive
hypothesis we see that these equalities are both equivalent to the following:
\begin{equation}
\label{eq:7}
\sum_{n-r_h+1}^{n-r_{h-1}} \Delta
M_Z^{(q_{r_{h-1}+1},j)}=\sum_{n-r_h+1}^{n-r_{h-1}} \Delta M_Y^{(q_{r_{h-1}+1},j)}-r_h+r_{h-1}.
\end{equation}
By this equality and by Step \ref{s:3} we see that there exists $n-r_h+1\le \overline j\le
n-r_{h-1}$ such that $\Delta
M_Z^{(q_{r_{h-1}+1},j)}=\Delta M_Y^{(q_{r_{h-1}+1},j)}$ for any $j<\overline j$ and
$\Delta M_Z^{(q_{r_{h-1}+1},\overline j)}<\Delta
M_Y^{(q_{r_{h-1}+1},\overline j)}$. In
particular, $\Delta M_Z^{(q_{r_{h-1}+1},\overline j)}\le 0$ and so by
Theorem \ref{T0} we have:
\[
\Delta M_Z^{(i,\overline j)}\le 0
\]
for any $i\ge q_{r_{h-1}+1}$. By Step \ref{s:4} we have:
\[
\sum_{i=q_{r_{h-1}+1}}^{a+1} \Delta M_Z^{(i,\overline j)}=\Delta
M_Y^{(q_{r_{h-1}+1},\overline j)}-1
\]
\[
\Rightarrow 0\ge \sum_{i=q_{r_{h-1}+1}+1}^{a+1} \Delta M_Z^{(i,\overline
j)}=\Delta M_Y^{(q_{r_{h-1}+1},\overline j)}-\Delta M_Z^{(q_{r_{h-1}+1},\overline
j)}-1\ge 0.
\]
This means that $\Delta M_Y^{(q_{r_{h-1}+1},\overline j)}-\Delta M_Z^{(q_{r_{h-1}+1},\overline
j)}-1=0$ and that $\Delta M_Z^{(i,\overline j)}=0$ for any $i\ge
q_{r_{h-1}+1}+1$.
Now take any $j>\overline j$. By the fact that $\Delta M_Z^{(i,\overline j)}=0$ for any $i\ge
q_{r_{h-1}+1}+1$ and by Theorem \ref{T0} we can say that $\Delta
M_Z^{(i,j)}\le 0$ for any $i\ge q_{r_{h-1}+1}+1$ and any $j>\overline j$. By
Step \ref{s:4} we get:
\begin{equation}
\label{eq:11}
0\ge \sum_{i=q_{r_{h-1}+1}+1}^{a+1} \Delta M_Z^{(i,j)}=\Delta M_Y^{(q_{r_{h-1}+1},j)}-\Delta M_Z^{(q_{r_{h-1}+1},j)}-1
\end{equation}
so that:
\[
\Delta M_Z^{(q_{r_{h-1}+1},j)}\ge \Delta M_Y^{(q_{r_{h-1}+1},j)}-1
\]
for any $j>\overline j$. So we can say that:
\begin{equation}
\label{eq:9}
\sum_{j=n-r_h+1}^{n-r_{h-1}} \Delta M_Z^{(q_{r_{h-1}+1},j)}\ge \sum_{j=n-r_h+1}^{n-r_{h-1}} \Delta
M_Y^{(q_{r_{h-1}+1},j)}-n+r_{h-1}+\overline j-1.
\end{equation}
This fact compared to \eqref{eq:7} gives us that $\overline j\le
n-r_h+1$, but by hypothesis $\overline j\ge n-r_h+1$ and so it must be
$\overline j=n-r_h+1$. This implies that the inequality in
\eqref{eq:9} is an equality, which means that:
\[
\Delta M_Z^{(q_{r_{h-1}+1},j)}=\Delta M_Y^{(q_{r_{h-1}+1},j)}-1
\]
for any $j\ge \overline j=n-r_h+1$, with $j\le n-r_{h-1}$, and by
\eqref{eq:11} and by inductive hypothesis
$\Delta M_Z^{(i,j)}=\Delta M_Y^{(i,j)}=0$ for any $i\ge
q_{r_{h-1}+1}$ and any $j\ge n-r_h+1$.
In this way we have proved the conclusion holds for any $(i,j)$, with
$i<q_{r_h+1}$ and the proof works by iteration.
\begin{step} \label{s:7}
If either $r=1$ or $r\ge 2$, $q_{r-1}<q_r$ and for any $k\in \{1,\dots,r-1\}$ and $i\ge q_k$ $\Delta
M_X^{(i,n-k+1)}=0$, then $\Delta M_Z^{(i,j)}=\Delta
M_Y^{(i,j)}=0$ for any $(i,j)>(q_k,n-k+1)$ and $k\in \{1,\dots,r\}$ and $\Delta M_Z^{(i,j)}=\Delta
M_Y^{(i,j)}-1$ for any $(i,j)\in \{(q_1,n),(q_2,n-1),\dots,(q_{r},n-r+1)\}$.
\end{step}
Let us first suppose that $r\ge 2$ and that $q_{r-1}<q_r$. In this
case the procedure given in Step \ref{s:6} can be repeated for any $h\in
\{2,\dots,t\}$. This means that the equalities in the conclusion of Step
\ref{s:7} hold for
any $i<q_r$, for any $j\ge n-r+2$ and also for any $j\le n-r$ by Step
\ref{s:1}, i.e. for any $j\ne n-r+1$.
If $r=1$, then by Step \ref{s:1} and Remark \ref{rm} we see that the
conclusion holds for any $(i,j)$ with $j\ne n-r+1$ and for $(i,n)$ with
$i<q_1$. So in both cases we will show that $\Delta M_Z^{(q_r,n-r+1)}=\Delta
M_Y^{(q_r,n-r+1)}-1$ and that $\Delta M_Z^{(i,n-r+1)}=\Delta
M_Y^{(i,n-r+1)}=0$ for $i>q_r$.
As done before, we see that $\sum_{j\le n} \Delta
M_Z^{(q_r-1,j)}-\sum_{j\le n} \Delta M_Z^{(q_r,j)}$ is the number of
$(0,1)$-lines containing precisely $q_r$ points of $Z$, while $\sum_{j\le n} \Delta
M_Y^{(q_r-1,j)}-\sum_{j\le n} \Delta M_Y^{(q_r,j)}$ is the number of
$(0,1)$-lines containing precisely $q_r$ points of $Y$. By hypothesis it must
be:
\[
\sum_{j\le n} \Delta
M_Z^{(q_r-1,j)}-\sum_{j\le n} \Delta M_Z^{(q_r,j)}=\sum_{j\le n} \Delta
M_Y^{(q_r-1,j)}-\sum_{j\le n} \Delta M_Y^{(q_r,j)}+1.
\]
Since the equalities in the claim hold for any $i<q_r$ and for any $j\ne n-r+1$, we see that this equality is equivalent to the following:
\begin{equation}
\label{eq:8}
\Delta M_Z^{(q_r,n-r+1)}=\Delta M_Y^{(q_r,n-r+1)}-1.
\end{equation}
Since the $(0,1)$-lines containing exactly $q_r+1$ points of $Z$ are
one less than those containing exactly $q_r+1$ points of $Y$, we see
that:
\[
\sum_{j\le n} \Delta
M_Z^{(q_r,j)}-\sum_{j\le n} \Delta M_Z^{(q_r+1,j)}=\sum_{j\le n} \Delta
M_Y^{(q_r,j)}-\sum_{j\le n} \Delta M_Y^{(q_r+1,j)}-1,
\]
which, by our hypotheses, implies:
\[
\Delta M_Z^{(q_r+1,n-r+1)}=\Delta M_Y^{(q_r+1,n-r+1)}.
\]
By iterating the procedure, taken any $i\ge q_r+2$, the $(0,1)$-lines containing exactly $i$ points of $Z$ are
also those containing exactly $i$ points of $Y$, so that:
\[
\sum_{j\le n} \Delta
M_Z^{(i-1,j)}-\sum_{j\le n} \Delta M_Z^{(i,j)}=\sum_{j\le n} \Delta
M_Y^{(i,j)}-\sum_{j\le n} \Delta M_Y^{(i,j)},
\]
which, by our hypotheses, implies:
\[
\Delta M_Z^{(i,n-r+1)}=\Delta M_Y^{(i,n-r+1)}.
\]
\end{proof}
In the same way, with the above notation, we can prove the following theorem:
\begin{thm} \label{T:2}
Let $C$ be a $(0,1)$-line disjoint from $X$. Let $R_{a+1}$,\dots,$R_m$,
$m\ge a$, be arbitrary $(1,0)$-lines and $j_1$,\dots,$j_r\in
\{0,\dots,a\}$. Let $\mathcal P=\{C\cap R_j\mid j\in\{0,\dots,m\},\,
j\ne j_1,\dots,j_r\}$ and let $Z=X\cup \mathcal P$. Suppose also that on the $(1,0)$-line $R_{j_k}$
there are $p_k$ points of $X$ for $k=1,\dots,r$ and that $p_1\le
p_2\le \dots \le p_r$. Then, given
$T=\{(m,p_1),(m-1,p_2),\dots,(m-r+1,p_r)\}$, we have:
\[
\Delta M_Z^{(i,j)}=
\begin{cases}
1 & \text{if } i\le m,\, j=0\\
0 & \text{if }i\ge m+1,\, j=0\\
\Delta M_X^{(i,j-1)} & \text{if } j\ge 1\text{ and }(i,j)\notin T\\
\Delta M_X^{(i,j-1)}-1 & \text{if } j\ge 1\text{ and }(i,j)\in T
\end{cases}
\]
if one of the following conditions holds:
\begin{enumerate}
\item $r=1$;
\item $r\ge 2$, $p_{r-1}<p_r$ and for any $k\in \{1,\dots,r-1\}$ and $j\ge p_k$\, $\Delta M_X^{(m-k+1,j)}=0$;
\item $r\ge 2$, $p_{r-1}=p_r$ and for any $k\in \{1,\dots,r\}$ and $j\ge p_k$\, $\Delta M_X^{(m-k+1,j)}=0$.
\end{enumerate}
\end{thm}
\begin{proof}
The proof works as in Theorem \ref{T:1}.
\end{proof}
Under the notation of Theorem \ref{T:1} we prove the following:
\begin{thm} \label{T:3}
If one the following conditions holds:
\begin{enumerate}
\item $q_{r-1}<q_r$ and $n\ge b+r-1$,
\item $q_{r-1}=q_r$ and $n\ge b+r$,
\end{enumerate}
then:
\[
\Delta M_Z^{(i,j)}=
\begin{cases}
1 & \text{if } i=0,\, j\le n\\
0 & \text{if }i=0,\, j\ge n+1\\
\Delta M_X^{(i-1,j)} & \text{if } i\ge 1\text{ and }(i,j)\notin T\\
\Delta M_X^{(i-1,j)}-1 & \text{if } i\ge 1\text{ and }(i,j)\in T.
\end{cases}
\]
\end{thm}
\begin{proof}
\
\begin{enumerate}
\item By Remark \ref{rm} our hypothesis imply that $\Delta M_X^{(i,j)}=0$ for any
$i\ge 0$ and for any $j\ge n-r+2$, so that the hypothesis of Theorem
\ref{T:1} holds;
\item in this case by hypothesis we have that $\Delta M_X^{(i,j)}=0$ for any
$i\ge 0$ and for any $j\ge n-r+1$, so that the hypothesis of Theorem
\ref{T:1} holds.
\end{enumerate}
\end{proof}
In the same way under the notation of Theorem \ref{T:2} we prove the
following result:
\begin{thm} \label{T:4}
If one the following
conditions holds:
\begin{enumerate}
\item $p_{r-1}<p_r$ and $m\ge a+r-1$,
\item $p_{r-1}=p_r$ and $m\ge a+r$,
\end{enumerate}
then:
\[
\Delta M_Z^{(i,j)}=
\begin{cases}
1 & \text{if } i\le m,\, j=0\\
0 & \text{if }i\ge m+1,\, j=0\\
\Delta M_X^{(i,j-1)} & \text{if } j\ge 1\text{ and }(i,j)\notin T\\
\Delta M_X^{(i,j-1)}-1 & \text{if } j\ge 1\text{ and }(i,j)\in T.
\end{cases}
\]
\end{thm}
\begin{proof}
The works as in Theorem \ref{T:3}.
\end{proof}
\begin{ex} \label{ex}
In these examples we will show that if the hypothesis of Theorem \ref{T:1}
does not hold, then the conclusion is not necessarily true. As a notation,
we represent the $(1,0)$-lines as horizontal lines and the
$(0,1)$-lines as vertical lines.
\begin{enumerate}
\item Let us consider a scheme $X$ union of three generic points
and its first difference $\Delta M_X$.
\begin{figure}[H]
\begin{preview}
\begin{center}
\subfloat[$X$]{
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.5cm,y=0.5cm]
\clip(-1,-1) rectangle (4,6);
\fill [color=black] (1,1) circle (2pt);
\fill [color=black] (2,2) circle (2pt);
\fill [color=black] (3,3) circle (2pt);
\draw (1,0.5) -- (1,3.5);
\draw (2,1.5) -- (2,3.5);
\draw (3,2.5) -- (3,3.5);
\draw (0.5,1) -- (1.5,1);
\draw (0.5,2) -- (2.5,2);
\draw (0.5,3) -- (3.5,3);
\draw[color=black] (1,4.5) node {\footnotesize $C_0$};
\draw[color=black] (2,4.5) node {\footnotesize $C_1$};
\draw[color=black] (3,4.5) node {\footnotesize $C_2$};
\draw[color=black] (-0.5,1) node {\footnotesize $R_2$};
\draw[color=black] (-0.5,2) node {\footnotesize $R_1$};
\draw[color=black] (-0.5,3) node {\footnotesize $R_0$};
\end{tikzpicture}
}
\hspace{1cm}
\subfloat[$\Delta M_X$]{
\begin{tikzpicture}[x=0.7cm,y=0.7cm]
\clip(0,0) rectangle (7,7);
\draw[style=help lines,xstep=1,ystep=1] (1,1) grid (6,6);
\foreach \x in {1,...,5} \draw (\x,1) +(.5,.5) node {\dots};
\foreach \y in {2,...,5} \draw (5,\y) +(.5,.5) node {\dots};
\foreach \x in {1,...,4} \draw (\x,2.5) +(.5,0) node {$0$};
\foreach \y in {3,...,5} \draw (4.5,\y) +(0,.5) node {$0$};
\draw (1.5,3.5) node {$1$};
\draw (2.5,3.5) node {$-1$};
\draw (3.5,3.5) node {$0$};
\draw (1.5,4.5) node {$1$};
\draw (2.5,4.5) node {$0$};
\draw (3.5,4.5) node {$-1$};
\foreach \x in {1,...,3} \draw (\x,5.5) +(.5,0) node {$1$};
\foreach \x in {0,...,4} \draw (\x,6.5) +(1.5,0) node {$\x$};
\foreach \y in {0,...,4} \draw (0.5,5.5-\y) node {$\y$};
\end{tikzpicture}
}
\end{center}
\end{preview}
\end{figure}
Let $R$ be a $(1,0)$-line disjoint from $X$ and let $Z$ be the following scheme.
\begin{figure}[H]
\begin{preview}
\begin{center}
\subfloat[$Z$]{
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.5cm,y=0.5cm]
\clip(-1,-1) rectangle (4,6);
\fill [color=black] (1,1) circle (2pt);
\fill [color=black] (2,2) circle (2pt);
\fill [color=black] (3,3) circle (2pt);
\fill [color=black] (3,4) circle (2pt);
\draw (1,0.5) -- (1,4.5);
\draw (2,1.5) -- (2,4.5);
\draw (3,2.5) -- (3,4.5);
\draw (0.5,1) -- (1.5,1);
\draw (0.5,2) -- (2.5,2);
\draw (0.5,3) -- (3.5,3);
\draw (0.5,4) -- (4.5,4);
\draw[color=black] (1,5.5) node {\footnotesize $C_0$};
\draw[color=black] (2,5.5) node {\footnotesize $C_1$};
\draw[color=black] (3,5.5) node {\footnotesize $C_2$};
\draw[color=black] (-0.5,1) node {\footnotesize $R_2$};
\draw[color=black] (-0.5,2) node {\footnotesize $R_1$};
\draw[color=black] (-0.5,3) node {\footnotesize $R_0$};
\draw[color=black] (-0.5,4) node {\footnotesize $R$};
\end{tikzpicture}
}
\hspace{1cm}
\subfloat[$\Delta M_Z$]{
\begin{tikzpicture}[x=0.7cm,y=0.7cm]
\clip(0,0) rectangle (7,8);
\draw[style=help lines,xstep=1,ystep=1] (1,1) grid (6,7);
\foreach \x in {1,...,5} \draw (\x,1) +(.5,.5) node {\dots};
\foreach \y in {2,...,6} \draw (5,\y) +(.5,.5) node {\dots};
\foreach \x in {1,...,4} \draw (\x,2.5) +(.5,0) node {$0$};
\foreach \y in {3,...,6} \draw (4.5,\y) +(0,.5) node {$0$};
\draw (1.5,3.5) node {$1$};
\draw (2.5,3.5) node {$-1$};
\draw (3.5,3.5) node {$0$};
\draw (1.5,4.5) node {$1$};
\draw (2.5,4.5) node {$-1$};
\draw (3.5,4.5) node {$0$};
\foreach \x in {1,...,2} \draw (\x,5.5) +(.5,0) node {$1$};
\draw (3.5,5.5) node {$-1$};
\foreach \x in {1,...,3} \draw (\x,6.5) +(.5,0) node {$1$};
\foreach \x in {0,...,4} \draw (\x,7.5) +(1.5,0) node {$\x$};
\foreach \y in {0,...,5} \draw (0.5,6.5-\y) node {$\y$};
\end{tikzpicture}
}
\end{center}
\end{preview}
\end{figure}
In this case, under the notation of
Theorem \ref{T:1} we have $r=2$, $n=2$, $q_1=q_2=1$ and $\Delta
M_X(q_1,n)=\Delta M_X(1,2)\ne 0$. In this case, we see that $\Delta
M_Z^{(1,2)}=-1\ne \Delta M_X^{(0,2)}-1=0$.
\item Let us consider a scheme $X$ of degree $4$ with $2$ points on a
$(1,0)$-line $R_0$ and other $2$ points on a $(0,1)$-line $C_0$ and its first
difference $\Delta M_X$.
\begin{figure}[H]
\begin{preview}
\begin{center}
\subfloat[$X$]{
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.5cm,y=0.5cm]
\clip(-1,-1) rectangle (4,5);
\fill [color=black] (1,1) circle (2pt);
\fill [color=black] (1,2) circle (2pt);
\fill [color=black] (2,3) circle (2pt);
\fill [color=black] (3,3) circle (2pt);
\draw (1,0.5) -- (1,3.5);
\draw (2,2.5) -- (2,3.5);
\draw (3,2.5) -- (3,3.5);
\draw (0.5,1) -- (1.5,1);
\draw (0.5,2) -- (1.5,2);
\draw (0.5,3) -- (3.5,3);
\draw[color=black] (1,4.5) node {\footnotesize $C_0$};
\draw[color=black] (2,4.5) node {\footnotesize $C_1$};
\draw[color=black] (3,4.5) node {\footnotesize $C_2$};
\draw[color=black] (-0.5,1) node {\footnotesize $R_2$};
\draw[color=black] (-0.5,2) node {\footnotesize $R_1$};
\draw[color=black] (-0.5,3) node {\footnotesize $R_0$};
\end{tikzpicture}
}
\hspace{1cm}
\subfloat[$\Delta M_X$]{
\begin{tikzpicture}[x=0.7cm,y=0.7cm]
\clip(0,0) rectangle (7,7);
\draw[style=help lines,xstep=1,ystep=1] (1,1) grid (6,6);
\foreach \x in {1,...,5} \draw (\x,1) +(.5,.5) node {\dots};
\foreach \y in {2,...,5} \draw (5,\y) +(.5,.5) node {\dots};
\foreach \x in {1,...,4} \draw (\x,2.5) +(.5,0) node {$0$};
\foreach \y in {3,...,5} \draw (4.5,\y) +(0,.5) node {$0$};
\draw (1.5,3.5) node {$1$};
\draw (2.5,3.5) node {$0$};
\draw (3.5,3.5) node {$-1$};
\draw (1.5,4.5) node {$1$};
\draw (2.5,4.5) node {$0$};
\draw (3.5,4.5) node {$0$};
\foreach \x in {1,...,3} \draw (\x,5.5) +(.5,0) node {$1$};
\foreach \x in {0,...,4} \draw (\x,6.5) +(1.5,0) node {$\x$};
\foreach \y in {0,...,4} \draw (0.5,5.5-\y) node {$\y$};
\end{tikzpicture}
}
\end{center}
\end{preview}
\end{figure}
If $R$ is a $(1,0)$-line disjoint from $X$ and $Z=X\cup (R\cap C_2)$, then, under the notation of Theorem
\ref{T:1}, we have $r=2$, $n=2$, $q_1=1$, $q_2=2$ and $\Delta
M_X^{(2,2)}\ne 0$. However, $\Delta M_Z^{(2,1)}=0\ne \Delta M_X^{(1,1)}-1=-1$.
\begin{figure}[H]
\begin{preview}
\begin{center}
\subfloat[$Z$]{
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.5cm,y=0.5cm]
\clip(-1,-1) rectangle (4,6);
\fill [color=black] (1,1) circle (2pt);
\fill [color=black] (1,2) circle (2pt);
\fill [color=black] (2,3) circle (2pt);
\fill [color=black] (3,3) circle (2pt);
\fill [color=black] (3,4) circle (2pt);
\draw (1,0.5) -- (1,4.5);
\draw (2,2.5) -- (2,4.5);
\draw (3,2.5) -- (3,4.5);
\draw (0.5,1) -- (1.5,1);
\draw (0.5,2) -- (1.5,2);
\draw (0.5,3) -- (3.5,3);
\draw (0.5,4) -- (3.5,4);
\draw[color=black] (1,5.5) node {\footnotesize $C_0$};
\draw[color=black] (2,5.5) node {\footnotesize $C_1$};
\draw[color=black] (3,5.5) node {\footnotesize $C_2$};
\draw[color=black] (-0.5,1) node {\footnotesize $R_2$};
\draw[color=black] (-0.5,2) node {\footnotesize $R_1$};
\draw[color=black] (-0.5,3) node {\footnotesize $R_0$};
\draw[color=black] (-0.5,4) node {\footnotesize $R$};
\end{tikzpicture}
}
\hspace{1cm}
\subfloat[$\Delta M_Z$]{
\begin{tikzpicture}[x=0.7cm,y=0.7cm]
\clip(0,0) rectangle (7,8);
\draw[style=help lines,xstep=1,ystep=1] (1,1) grid (6,7);
\foreach \x in {1,...,5} \draw (\x,1) +(.5,.5) node {\dots};
\foreach \y in {2,...,6} \draw (5,\y) +(.5,.5) node {\dots};
\foreach \x in {1,...,4} \draw (\x,2.5) +(.5,0) node {$0$};
\foreach \y in {3,...,6} \draw (4.5,\y) +(0,.5) node {$0$};
\draw (1.5,3.5) node {$1$};
\draw (2.5,3.5) node {$-1$};
\draw (3.5,3.5) node {$0$};
\draw (1.5,4.5) node {$1$};
\draw (2.5,4.5) node {$0$};
\draw (3.5,4.5) node {$-1$};
\foreach \x in {1,...,2} \draw (\x,5.5) +(.5,0) node {$1$};
\draw (3.5,5.5) node {$0$};
\foreach \x in {1,...,3} \draw (\x,6.5) +(.5,0) node {$1$};
\foreach \x in {0,...,4} \draw (\x,7.5) +(1.5,0) node {$\x$};
\foreach \y in {0,...,5} \draw (0.5,6.5-\y) node {$\y$};
\end{tikzpicture}
}
\end{center}
\end{preview}
\end{figure}
\end{enumerate}
\end{ex}
\section{ACM case}
In this section we show that, if $X$ is an ACM scheme, then Theorem \ref{T:1} and Theorem \ref{T:2}
hold without any further assumption on $X$. The following result is
well known, but it is difficult to find a good reference and so we
give a short proof here.
\begin{prop} \label{P}
Let $X$ be an ACM $0$-dimensional scheme. Let $p_i=\#(X\cap R_i)$, for
$i=0$,\dots,$a$ and let $q_j=\#(X\cap C_j)$, for
$j=0$,\dots,$b$. Then:
\[
\Delta M_X^{(i,j)}=
\begin{cases}
1 & \text{if } i\le q_j-1\text{ and }0\le j\le b\\
0 & \text{otherwise}
\end{cases}
\]
or equivalently:
\[
\Delta M_X^{(i,j)}=
\begin{cases}
1 & \text{if } j\le p_i-1\text{ and }0\le i\le a\\
0 & \text{otherwise.}
\end{cases}
\]
\end{prop}
\begin{proof}
We show that:
\[
\Delta M_X^{(i,j)}=
\begin{cases}
1 & \text{if } i\le q_j-1\text{ and }0\le j\le b\\
0 & \text{otherwise}
\end{cases}
\]
The proof that also:
\[
\Delta M_X^{(i,j)}=
\begin{cases}
1 & \text{if } j\le p_i-1\text{ and }0\le i\le a\\
0 & \text{otherwise}
\end{cases}
\]
is similar.
It is well known (see, for example, \cite{GM}, \cite{Gu} and
\cite{VT}) that $X$ can be described after a suitable permutation of
lines in such a way that the following conditions holds:
\begin{enumerate}
\item for every $i\in \{0,\dots,a\}$ there exists $j(i)\in
\{0,\dots,b\}$ such that $R_i\cap C_j\in X$ for $j\in
\{0,\dots,j(i)\}$ and $R_i\cap C_j\notin X$ for $j\in
\{j(i)+1,\dots,b\}$;
\item $j(0)\ge j(1)\ge \dots \ge j(a)$.
\end{enumerate}
Moreover, if any scheme $X$ satisfies these conditions, then $X$ is an
ACM scheme. Using this fact we can easily compute $\Delta M_X$ by induction on $a$. If
$a=0$, then the equality follows by the fact that $h^0(\mathscr
I_X(0,b))=0$ and $h^0(\mathscr I_X(0,b+1))=1$.
If the equality holds for $a-1$, then we apply Theorem \ref{T2} and we
get the equality.
\end{proof}
\begin{thm} \label{T:5}
Let $X$ be an ACM scheme and let $R$ be a $(1,0)$-line disjoint from $X$. Let $C_{b+1}$,\dots,$C_n$,
$n\ge b$, be arbitrary $(0,1)$-lines and $i_1$,\dots,$i_r\in
\{0,\dots,b\}$. Let $\mathcal P=\{R\cap C_i\mid i\in\{0,\dots,n\},\,
i\ne i_1,\dots,i_r\}$ and let $Z=X\cup \mathcal P$. Suppose also that on the $(0,1)$-line $C_{i_k}$
there are $q_k$ points of $X$ for $k=1,\dots,r$ and that $q_1\le
q_2\le \dots \le q_r$. Then, given
$T=\{(q_1,n),(q_2,n-1),\dots,(q_r,n-r+1)\}$, we have:
\[
\Delta M_Z^{(i,j)}=
\begin{cases}
1 & \text{if } i=0,\, j\le n\\
0 & \text{if }i=0,\, j\ge n+1\\
\Delta M_X^{(i-1,j)} & \text{if } i\ge 1\text{ and }(i,j)\notin T\\
\Delta M_X^{(i-1,j)}-1 & \text{if } i\ge 1\text{ and }(i,j)\in T.
\end{cases}
\]
\end{thm}
\begin{proof}
The conclusion follows by Theorem \ref{T:1}, by Proposition \ref{P} and by
the fact that $\#(C_{b-k+1}\cap X)\le q_k$.
\end{proof}
In the same way:
\begin{thm} \label{T:6}
Let $X$ be an ACM scheme and let $C$ be a $(0,1)$-line disjoint from $X$. Let $R_{a+1}$,\dots,$R_m$,
$m\ge a$, be arbitrary $(1,0)$-lines and $j_1$,\dots,$j_r\in
\{0,\dots,a\}$. Let $\mathcal P=\{C\cap R_j\mid j\in\{0,\dots,m\},\,
j\ne j_1,\dots,j_r\}$ and let $Z=X\cup \mathcal P$. Suppose also that on the $(1,0)$-line $R_{j_k}$
there are $p_k$ points of $X$ for $k=1,\dots,r$ and that $p_1\le
p_2\le \dots \le p_r$. Then, given
$T=\{(m,p_1),(m-1,p_2),\dots,(m-r+1,p_r)\}$, we have:
\[
\Delta M_Z^{(i,j)}=
\begin{cases}
1 & \text{if } i\le m,\, j=0\\
0 & \text{if }i\ge m+1,\, j=0\\
\Delta M_X^{(i,j-1)} & \text{if } j\ge 1\text{ and }(i,j)\notin T\\
\Delta M_X^{(i,j-1)}-1 & \text{if } j\ge 1\text{ and }(i,j)\in T.
\end{cases}
\]
\end{thm}
\begin{proof}
The works as in Theorem \ref{T:3}.
\end{proof}
\section{Example} \label{Esempio}
Now we show how it is possible to apply Theorem \ref{T:1}, Theorem
\ref{T:3} and Theorem \ref{T:5} to compute the first difference of the
Hilbert matrix of a scheme $X$ whose points can be distributed on a
grid of $(1,0)$ and $(0,1)$-lines in the following way:
\begin{figure}[H]
\begin{preview}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.5cm,y=0.5cm]
\foreach \x in {-1,1,2,3,4,5,6,7,8,9}
\foreach \y in {-1,1,2,3,4,5,6,7,8}
\clip(-1,0) rectangle (10,10);
\fill [color=black] (1,1) circle (2pt);
\fill [color=black] (2,1) circle (2pt);
\fill [color=black] (3,1) circle (2pt);
\fill [color=black] (1,2) circle (2pt);
\fill [color=black] (2,2) circle (2pt);
\fill [color=black] (3,2) circle (2pt);
\fill [color=black] (4,2) circle (2pt);
\fill [color=black] (1,3) circle (2pt);
\fill [color=black] (2,3) circle (2pt);
\fill [color=black] (3,3) circle (2pt);
\fill [color=black] (4,3) circle (2pt);
\fill [color=black] (5,3) circle (2pt);
\fill [color=black] (1,4) circle (2pt);
\fill [color=black] (2,4) circle (2pt);
\fill [color=black] (3,4) circle (2pt);
\fill [color=black] (4,4) circle (2pt);
\fill [color=black] (5,4) circle (2pt);
\fill [color=black] (6,5) circle (2pt);
\fill [color=black] (1,6) circle (2pt);
\fill [color=black] (2,6) circle (2pt);
\fill [color=black] (5,6) circle (2pt);
\fill [color=black] (6,6) circle (2pt);
\fill [color=black] (1,7) circle (2pt);
\fill [color=black] (2,7) circle (2pt);
\fill [color=black] (4,7) circle (2pt);
\fill [color=black] (5,7) circle (2pt);
\fill [color=black] (6,7) circle (2pt);
\fill [color=black] (3,8) circle (2pt);
\fill [color=black] (7,8) circle (2pt);
\fill [color=black] (8,8) circle (2pt);
\fill [color=black] (9,8) circle (2pt);
\draw (1,0.5) -- (1,8.5);
\draw[color=black] (1,9.5) node {\footnotesize $C_0$};
\draw (2,0.5) -- (2,8.5);
\draw[color=black] (2,9.5) node {\footnotesize $C_1$};
\draw (3,0.5) -- (3,8.5);
\draw[color=black] (3,9.5) node {\footnotesize $C_2$};
\draw (4,1.5) -- (4,8.5);
\draw[color=black] (4,9.5) node {\footnotesize $C_3$};
\draw (5,2.5) -- (5,8.5);
\draw[color=black] (5,9.5) node {\footnotesize $C_4$};
\draw (6,4.5) -- (6,8.5);
\draw[color=black] (6,9.5) node {\footnotesize $C_5$};
\draw (7,7.5) -- (7,8.5);
\draw[color=black] (7,9.5) node {\footnotesize $C_6$};
\draw (8,7.5) -- (8,8.5);
\draw[color=black] (8,9.5) node {\footnotesize $C_7$};
\draw (9,7.5) -- (9,8.5);
\draw[color=black] (9,9.5) node {\footnotesize $C_8$};
\draw (0.5,1) -- (3.5,1);
\draw[color=black] (-0.5,1) node {\footnotesize $R_7$};
\draw (0.5,2) -- (4.5,2);
\draw[color=black] (-0.5,2) node {\footnotesize $R_6$};
\draw (0.5,3) -- (5.5,3);
\draw[color=black] (-0.5,3) node {\footnotesize $R_5$};
\draw (0.5,4) -- (5.5,4);
\draw[color=black] (-0.5,4) node {\footnotesize $R_4$};
\draw (0.5,5) -- (6.5,5);
\draw[color=black] (-0.5,5) node {\footnotesize $R_3$};
\draw (0.5,6) -- (6.5,6);
\draw[color=black] (-0.5,6) node {\footnotesize $R_2$};
\draw (0.5,7) -- (6.5,7);
\draw[color=black] (-0.5,7) node {\footnotesize $R_1$};
\draw (0.5,8) -- (9.5,8);
\draw[color=black] (-0.5,8) node {\footnotesize $R_0$};
\end{tikzpicture}
\caption{The scheme $X$}
\end{preview}
\end{figure}
We compute $\Delta M_X$ by adding the points of the $(1,0)$-lines. The points on $R_4$,
$R_5$, $R_6$ and $R_7$ are an aCM scheme,
so that, by using Proposition \ref{P}, we get its first difference:
\begin{figure}[H]
\begin{preview}
\begin{center}
\subfloat
{
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.5cm,y=0.5cm]
\clip(-1,-1) rectangle (6,5);
\foreach \x in {1,...,3} \fill [color=black] (\x,1) circle (2pt);
\foreach \x in {1,...,4} \fill [color=black] (\x,2) circle (2pt);
\foreach \x in {1,...,5} \fill [color=black] (\x,3) circle (2pt);
\foreach \x in {1,...,5} \fill [color=black] (\x,4) circle (2pt);
\draw (1,0.5) -- (1,4.5);
\draw (2,0.5) -- (2,4.5);
\draw (3,0.5) -- (3,4.5);
\draw (4,1.5) -- (4,4.5);
\draw (5,2.5) -- (5,4.5);
\draw (0.5,1) -- (3.5,1);
\draw (0.5,2) -- (4.5,2);
\draw (0.5,3) -- (5.5,3);
\draw (0.5,4) -- (5.5,4);
\end{tikzpicture}
}
\hspace{1cm}
\subfloat{
\begin{tikzpicture}[x=0.7cm,y=0.7cm]
\clip(0,0) rectangle (9,8);
\draw[style=help lines,xstep=1,ystep=1] (1,1) grid (8,7);
\foreach \x in {1,...,7} \draw (\x,1) +(.5,.5) node {\dots};
\foreach \y in {2,...,6} \draw (7,\y) +(.5,.5) node {\dots};
\foreach \x in {1,...,6} \draw (\x,2.5) +(.5,0) node {$0$};
\foreach \y in {3,...,6} \draw (6.5,\y) +(0,.5) node {$0$};
\foreach \x in {1,...,3} \draw (\x,3.5) +(.5,0) node {$1$};
\foreach \x in {1,...,4} \draw (\x,4.5) +(.5,0) node {$1$};
\foreach \x in {1,...,5} \draw (\x,5.5) +(.5,0) node {$1$};
\foreach \x in {1,...,5} \draw (\x,6.5) +(.5,0) node {$1$};
\draw (4.5,3.5) node {$0$};
\draw (5.5,3.5) node {$0$};
\draw (5.5,4.5) node {$0$};
\foreach \x in {0,...,6} \draw (\x,7.5) +(1.5,0) node {$\x$};
\foreach \y in {0,...,5} \draw (0.5,6.5-\y) node {$\y$};
\end{tikzpicture}
}
\end{center}
\end{preview}
\end{figure}
Now we add the point on the line $R_3$ and by Theorem \ref{T:5} we
compute its first difference:
\begin{figure}[H]
\begin{preview}
\begin{center}
\subfloat
{
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.5cm,y=0.5cm]
\clip(-1,-1) rectangle (7,6);
\foreach \x in {1,...,3} \fill [color=black] (\x,1) circle (2pt);
\foreach \x in {1,...,4} \fill [color=black] (\x,2) circle (2pt);
\foreach \x in {1,...,5} \fill [color=black] (\x,3) circle (2pt);
\foreach \x in {1,...,5} \fill [color=black] (\x,4) circle (2pt);
\fill [color=black] (6,5) circle (2pt);
\draw (1,0.5) -- (1,5.5);
\draw (2,0.5) -- (2,5.5);
\draw (3,0.5) -- (3,5.5);
\draw (4,1.5) -- (4,5.5);
\draw (5,2.5) -- (5,5.5);
\draw (6,4.5) -- (6,5.5);
\draw (0.5,1) -- (3.5,1);
\draw (0.5,2) -- (4.5,2);
\draw (0.5,3) -- (5.5,3);
\draw (0.5,4) -- (5.5,4);
\draw (0.5,5) -- (6.5,5);
\end{tikzpicture}
}
\hspace{1cm}
\subfloat
{
\begin{tikzpicture}[x=0.7cm,y=0.7cm]
\clip(0,0) rectangle (10,9);
\draw[style=help lines,xstep=1,ystep=1] (1,1) grid (9,8);
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\foreach \y in {2,...,7} \draw (8,\y) +(.5,.5) node {\dots};
\foreach \x in {1,...,7} \draw (\x,2.5) +(.5,0) node {$0$};
\foreach \y in {3,...,7} \draw (7.5,\y) +(0,.5) node {$0$};
\draw (1.5,3.5) node {$1$};
\draw (2.5,3.5) node {$0$};
\draw (3.5,3.5) node {$0$};
\draw (4.5,3.5) node {$-1$};
\draw (5.5,3.5) node {$0$};
\draw (6.5,3.5) node {$0$};
\foreach \x in {1,...,4} \draw (\x,4.5) +(.5,0) node {$1$};
\draw (5.5,4.5) node {$-1$};
\draw (6.5,4.5) node {$0$};
\foreach \x in {1,...,5} \draw (\x,5.5) +(.5,0) node {$1$};
\draw (6.5,5.5) node {$-1$};
\foreach \x in {1,...,5} \draw (\x,6.5) +(.5,0) node {$1$};
\draw (6.5,6.5) node {$0$};
\foreach \x in {1,...,6} \draw (\x,7.5) +(.5,0) node {$1$};
\foreach \x in {0,...,7} \draw (\x,8.5) +(1.5,0) node {$\x$};
\foreach \y in {0,...,6} \draw (0.5,7.5-\y) node {$\y$};
\end{tikzpicture}
}
\end{center}
\end{preview}
\end{figure}
In the same way we add the points on $R_2$ and by Theorem \ref{T:1} we
compute the first difference:
\begin{figure}[H]
\begin{preview}
\begin{center}
\subfloat
{
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.5cm,y=0.5cm]
\clip(-1,-1) rectangle (7,7);
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\foreach \x in {1,...,5} \fill [color=black] (\x,3) circle (2pt);
\foreach \x in {1,...,5} \fill [color=black] (\x,4) circle (2pt);
\fill [color=black] (6,5) circle (2pt);
\foreach \x in {5,6} \fill [color=black] (\x,6) circle (2pt);
\foreach \x in {1,2} \fill [color=black] (\x,6) circle (2pt);
\draw (1,0.5) -- (1,6.5);
\draw (2,0.5) -- (2,6.5);
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\draw (4,1.5) -- (4,6.5);
\draw (5,2.5) -- (5,6.5);
\draw (6,4.5) -- (6,6.5);
\draw (0.5,1) -- (3.5,1);
\draw (0.5,2) -- (4.5,2);
\draw (0.5,3) -- (5.5,3);
\draw (0.5,4) -- (5.5,4);
\draw (0.5,5) -- (6.5,5);
\draw (0.5,6) -- (6.5,6);
\end{tikzpicture}
}
\hspace{1cm}
\subfloat
{
\begin{tikzpicture}[x=0.7cm,y=0.7cm]
\clip(0,0) rectangle (10,10);
\draw[style=help lines,xstep=1,ystep=1] (1,1) grid (9,9);
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\draw (1.5,3.5) node {$1$};
\draw (2.5,3.5) node {$0$};
\draw (3.5,3.5) node {$0$};
\draw (4.5,3.5) node {$-1$};
\draw (5.5,3.5) node {$0$};
\draw (6.5,3.5) node {$0$};
\foreach \x in {1,2} \draw (\x,4.5) +(.5,0) node {$1$};
\foreach \x in {3,4} \draw (\x,4.5) +(.5,0) node {$1$};
\draw (5.5,4.5) node {$-2$};
\draw (6.5,4.5) node {$0$};
\foreach \x in {1,...,5} \draw (\x,5.5) +(.5,0) node {$1$};
\draw (6.5,5.5) node {$-2$};
\foreach \x in {1,...,5} \draw (\x,6.5) +(.5,0) node {$1$};
\draw (6.5,6.5) node {$0$};
\foreach \x in {1,...,6} \draw (\x,7.5) +(.5,0) node {$1$};
\foreach \x in {1,...,6} \draw (\x,8.5) +(.5,0) node {$1$};
\foreach \x in {0,...,7} \draw (\x,9.5) +(1.5,0) node {$\x$};
\foreach \y in {0,...,7} \draw (0.5,8.5-\y) node {$\y$};
\end{tikzpicture}
}
\end{center}
\end{preview}
\end{figure}
Now we add the points on $R_1$ and again by Theorem \ref{T:1} we
compute its first difference:
\begin{figure}[H]
\begin{preview}
\begin{center}
\subfloat
{
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.5cm,y=0.5cm]
\clip(-1,-1) rectangle (7,8);
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\foreach \x in {1,...,5} \fill [color=black] (\x,3) circle (2pt);
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\fill [color=black] (6,5) circle (2pt);
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\fill [color=black] (2,7) circle (2pt);
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\draw (1,0.5) -- (1,7.5);
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\draw (3,0.5) -- (3,7.5);
\draw (4,1.5) -- (4,7.5);
\draw (5,2.5) -- (5,7.5);
\draw (6,4.5) -- (6,7.5);
\draw (0.5,1) -- (3.5,1);
\draw (0.5,2) -- (4.5,2);
\draw (0.5,3) -- (5.5,3);
\draw (0.5,4) -- (5.5,4);
\draw (0.5,5) -- (6.5,5);
\draw (0.5,6) -- (6.5,6);
\draw (0.5,7) -- (6.5,7);
\end{tikzpicture}
}
\hspace{1cm}
\subfloat
{
\begin{tikzpicture}[x=0.7cm,y=0.7cm]
\clip(0,0) rectangle (10,11);
\draw[style=help lines,xstep=1,ystep=1] (1,1) grid (9,10);
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\foreach \y in {2,...,9} \draw (8,\y) +(.5,.5) node {\dots};
\foreach \x in {1,...,7} \draw (\x,2.5) +(.5,0) node {$0$};
\foreach \y in {3,...,9} \draw (7.5,\y) +(0,.5) node {$0$};
\draw (1.5,3.5) node {$1$};
\draw (2.5,3.5) node {$0$};
\draw (3.5,3.5) node {$0$};
\draw (4.5,3.5) node {$-1$};
\draw (5.5,3.5) node {$0$};
\draw (6.5,3.5) node {$0$};
\foreach \x in {1,2} \draw (\x,4.5) +(.5,0) node {$1$};
\foreach \x in {3,4} \draw (\x,4.5) +(.5,0) node {$1$};
\draw (5.5,4.5) node {$-2$};
\draw (6.5,4.5) node {$0$};
\foreach \x in {1,...,5} \draw (\x,5.5) +(.5,0) node {$1$};
\draw (6.5,5.5) node {$-3$};
\foreach \x in {1,...,5} \draw (\x,6.5) +(.5,0) node {$1$};
\draw (6.5,6.5) node {$0$};
\foreach \x in {1,...,6} \draw (\x,7.5) +(.5,0) node {$1$};
\foreach \x in {1,...,6} \draw (\x,8.5) +(.5,0) node {$1$};
\foreach \x in {1,...,6} \draw (\x,9.5) +(.5,0) node {$1$};
\foreach \x in {0,...,7} \draw (\x,10.5) +(1.5,0) node {$\x$};
\foreach \y in {0,...,8} \draw (0.5,9.5-\y) node {$\y$};
\end{tikzpicture}
}
\end{center}
\end{preview}
\end{figure}
Finally, by applying again Theorem \ref{T:1} we get the first
difference $\Delta M_X$ of $X$.
\begin{figure}[H]
\begin{preview}
\begin{center}
\subfloat{
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.5cm,y=0.5cm]
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\fill [color=black] (2,2) circle (2pt);
\fill [color=black] (3,2) circle (2pt);
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\fill [color=black] (1,3) circle (2pt);
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\fill [color=black] (3,3) circle (2pt);
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\fill [color=black] (5,3) circle (2pt);
\fill [color=black] (1,4) circle (2pt);
\fill [color=black] (2,4) circle (2pt);
\fill [color=black] (3,4) circle (2pt);
\fill [color=black] (4,4) circle (2pt);
\fill [color=black] (5,4) circle (2pt);
\fill [color=black] (6,5) circle (2pt);
\fill [color=black] (1,6) circle (2pt);
\fill [color=black] (2,6) circle (2pt);
\fill [color=black] (5,6) circle (2pt);
\fill [color=black] (6,6) circle (2pt);
\fill [color=black] (1,7) circle (2pt);
\fill [color=black] (2,7) circle (2pt);
\fill [color=black] (4,7) circle (2pt);
\fill [color=black] (5,7) circle (2pt);
\fill [color=black] (6,7) circle (2pt);
\fill [color=black] (3,8) circle (2pt);
\fill [color=black] (7,8) circle (2pt);
\fill [color=black] (8,8) circle (2pt);
\fill [color=black] (9,8) circle (2pt);
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\draw (3,0.5) -- (3,8.5);
\draw (4,1.5) -- (4,8.5);
\draw (5,2.5) -- (5,8.5);
\draw (6,4.5) -- (6,8.5);
\draw (7,7.5) -- (7,8.5);
\draw (8,7.5) -- (8,8.5);
\draw (9,7.5) -- (9,8.5);
\draw (0.5,1) -- (3.5,1);
\draw (0.5,2) -- (4.5,2);
\draw (0.5,3) -- (5.5,3);
\draw (0.5,4) -- (5.5,4);
\draw (0.5,5) -- (6.5,5);
\draw (0.5,6) -- (6.5,6);
\draw (0.5,7) -- (6.5,7);
\draw (0.5,8) -- (9.5,8);
\end{tikzpicture}
}
\hspace{1cm}
\subfloat
{
\begin{tikzpicture}[x=0.7cm,y=0.7cm]
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\draw (6.5,4.5) node {$-1$};
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\draw (6.5,5.5) node {$-3$};
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\draw (8.5,5.5) node {$0$};
\draw (9.5,5.5) node {$0$};
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\draw (6.5,6.5) node {$0$};
\draw (7.5,6.5) node {$-1$};
\draw (8.5,6.5) node {$-1$};
\draw (9.5,6.5) node {$0$};
\foreach \x in {1,...,6} \draw (\x,7.5) +(.5,0) node {$1$};
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\draw (9.5,7.5) node {$-1$};
\foreach \x in {1,...,6} \draw (\x,8.5) +(.5,0) node {$1$};
\foreach \x in {7,...,9} \draw (\x,8.5) +(.5,0) node {$0$};
\foreach \x in {1,...,6} \draw (\x,9.5) +(.5,0) node {$1$};
\foreach \x in {7,...,9} \draw (\x,9.5) +(.5,0) node {$0$};
\foreach \x in {1,...,9} \draw (\x,10.5) +(.5,0) node {$1$};
\foreach \x in {0,...,10} \draw (\x,11.5) +(1.5,0) node {$\x$};
\foreach \y in {0,...,9} \draw (0.5,10.5-\y) node {$\y$};
\end{tikzpicture}
}
\end{center}
\end{preview}
\end{figure}
|
{
"timestamp": "2010-09-22T02:02:13",
"yymm": "1009",
"arxiv_id": "1009.4095",
"language": "en",
"url": "https://arxiv.org/abs/1009.4095"
}
|
\section{Introduction}
\label{sec:intro}
There is a significant body of experimental and theoretical work
on muon captures in light nuclei, motivated by the fact that the
theoretical framework used to study these reactions is the same as
that used for weak capture reactions
of astrophysical interest, not accessible experimentally.
Muon captures, whose rates can be measured, can therefore provide
a valuable test of this theoretical framework~\cite{SFII}.
Very recently~\cite{Mar10},
the muon capture reactions $^2$H($\mu^-,\nu_\mu$)$nn$ and
$^3$He($\mu^-,\nu_\mu$)$^3$H have been studied
simultaneously in a consistent framework.
In particular, the initial and final $A=2$ and 3
nuclear wave functions have been obtained from the
Argonne $v_{18}$ (AV18)~\cite{Wir95}
or the chiral N$^3$LO (N3LO)~\cite{Ent03} two-nucleon potential,
in combination with, respectively, the Urbana IX (UIX)~\cite{Pud95}
or chiral N$^2$LO (N2LO)~\cite{Nav07}
three-nucleon potentials in the case of $A=3$.
The weak current consists of polar- and axial-vector components. The
former are related to the isovector piece of the electromagnetic
current via the conserved-vector-current (CVC) hypothesis.
These and the
axial current have been derived within two
different frameworks, the standard nuclear
physics approach (SNPA), and chiral
effective field theory.
The first one goes beyond the impulse approximation, by including
meson-exchange current contributions and terms arising
from the excitation of $\Delta$-isobar degrees of freedom.
The second approach includes two-body contributions
derived in
heavy-baryon chiral perturbation theory
within a systematic expansion, up to
N$^3$LO~\cite{Par03,Son09}.
To be noticed that, since the transition operator matrix elements
are calculated using phenomenological wave functions, it should be
viewed as a hybrid chiral effective field theory approach (EFT*).
Both SNPA and EFT* frameworks have been used in studies
of weak $pp$ and $hep$ capture reactions in the energy regime relevant to
astrophysics~\cite{Par03,Sch98,Mar00}.
The only parameter in the SNPA
nuclear weak current model is present in the axial current
(the $N$-to-$\Delta$ axial coupling constant)
and is determined by fitting the experimental
Gamow-Teller matrix element in tritium $\beta$-decay (GT$^{\rm EXP}$).
The SNPA weak vector current, related
to the isovector electromagnetic current via CVC,
reproduces the trinucleon magnetic moments to better than
1 \%~\cite{Mar10}.
In the case of EFT*, three
low-energy constants (LECs) appear: one in the axial-vector
component, and two in the electromagnetic current. Of these, only
one is relevant to the weak vector current, since the other
appears in front of an isoscalar operator.
The corresponding coupling constants are parameters
fixed to reproduce, respectively,
GT$^{\rm EXP}$ and $A=3$ magnetic moments.
To be noticed that the EFT* currents are obtained performing the
Fourier transform from momentum- to coordinate-space with
a Gaussian regulator characterized by a cutoff $\Lambda$, varied
between 500 and 800 MeV.
The total capture rates have been found to be
392.0(2.3) s$^{-1}$ for $A=2$ and
1484(13) s$^{-1}$ for $A=3$.
The spread accounts for the model dependence, i.e.,
the dependence on the input Hamiltonian model,
the model for the nuclear transition operator, and, in the EFT*
calculation, the cutoff sensitivity.
This weak model dependence is a consequence of the procedure
adopted to constrain the weak current. These results
are in very good agreement with the experimental data,
in particular with the very
accurate measurement
of Ref.~\cite{Ack98} for the total rate in muon capture on $^3$He.
The muon capture on deuteron has been studied
also using the SNPA and the EFT* framework in Ref.~\cite{Ric10}.
The SNPA retains two-body meson-exchange currents
derived from the hard pion chiral Lagrangians of the
$N\Delta\pi\rho\omega a_1$ system and are significantly
different from those of Ref.~\cite{Mar10}. On the other hand,
the EFT* currents are similar to those of Ref.~\cite{Mar10},
but two differences need to be remarked: (i) the LEC appearing in the
axial-vector component ($d_R$) is not fixed
to reproduce GT$^{\rm EXP}$, rather the doublet capture rate
calculated in SNPA; (ii) a term is added to the leading axial two-body
currents, in order to satisfy the partially-conserved-axial-current (PCAC)
hypothesis, as constructed in Ref.~\cite{Mos05} (called
there, and from now on, potential current). The calculated
SNPA values for the total capture rate are
in the range of 416--430 s$^{-1}$,
depending on the potential model used, resulting in a
model dependence much larger than in Ref.~\cite{Mar10}.
It is also argued that ``omitting the potential current causes
an enhancement of the doublet transition rate $\Lambda_{1/2}$
by $\simeq$ 1\%''~\cite{Ric10}.
In the present work we repeat the calculation of Ref.~\cite{Mar10},
in the EFT* approach, adding the potential currents as
in Ref.~\cite{Ric10}. We restrict our calculation to
the AV18 and AV18/UIX potential models, and to
a cutoff value of 600 MeV,
since, as shown in Ref.~\cite{Mar10}, the dependence on these
inputs is less than 1 \%. We fit the $d_R$ coefficient to GT$^{\rm EXP}$, and
consistently calculate the total rates
for muon capture on deuteron and $^3$He. The comparison with the results
of Ref.~\cite{Mar10} will give and indication of how significant are
the potential current contributions for these muon captures.
The paper is organized as follows: in Sec.~\ref{sec:thform} the theoretical
formalism used in the calculation is briefly reviewed. In Sec.~\ref{sec:wcur}
the EFT* model for the weak current is described, with the addition of
the potential currents. In Sec.~\ref{sec:res},
the results are presented and discussed, and some concluding remarks are
given.
\section{Theoretical formalism}
\label{sec:thform}
We briefly review the formalism used in the calculation for the muon
capture processes, discussed at length in Refs.~\cite{Mar10,Mar02}.
The muon capture on deuteron
and $^3$He is induced by the weak interaction
Hamiltonian~\cite{Wal95},
$H_{W}={G_{V}\over{\sqrt{2}}} \int {\rm d}{\bf x}
\, l_{\sigma}({\bf x}) j^{\sigma}({\bf x})$,
where $G_{V}$ is the Fermi coupling constant,
$G_{V}$=1.14939 $\times 10^{-5}$ GeV$^{-2}$~\cite{Har90},
and $l_\sigma$ and $j^\sigma$ are the leptonic and
hadronic current densities, respectively.
The transition amplitude can be written as
\begin{eqnarray}
&&T_W (f,f_z;s_1,s_2,h_\nu) \equiv
\langle nn, s_1, s_2; \nu, h_\nu \,|\, H_W \,|\,
(\mu,d);f,f_z \rangle
\nonumber \\
&&\>\>\>\>\simeq {G_V \over \sqrt{2}} \psi_{1s}^{\rm av}
\sum_{s_\mu s_d}
\langle {1 \over 2}s_{\mu}, 1 s_d | f f_z \rangle\,
l_\sigma(h_\nu,\,s_\mu)\,
\langle \Psi_{{\bf p}, s_1 s_2}(nn) | j^{\sigma}({\bf q}) |
\Psi_d(s_d)\rangle \ , \label{eq:h2ffz}
\end{eqnarray}
for muon capture on deuteron, ${\bf p}$ being the $nn$ relative
momentum, and~\cite{Mar02}
\begin{eqnarray}
&&T_W (f,f_z;s^\prime_{3},h_\nu) \equiv
\langle ^3{\rm H}, s^\prime_{3}; \nu, h_\nu \,|\, H_W \,|\,
(\mu,^3\!{\rm He});f,f_z \rangle
\nonumber \\
&&\>\>\>\>\>\>\simeq {G_V \over \sqrt{2}} \psi_{1s}^{\rm av}
\sum_{s_\mu s_3}
\langle {1 \over 2}s_{\mu}, {1 \over 2} s_3 | f f_z \rangle\,
l_\sigma(h_\nu,\,s_\mu)\,
\langle \Psi_{^3{\rm H}}(s^\prime_{3}) | j^{\sigma}({\bf q}) |
\Psi_{^3{\rm He}} (s_3)\rangle \ , \label{eq:h3ffz}
\end{eqnarray}
for muon capture on $^3$He. In order to account for the hyperfine structure
in the initial system, the muon and deuteron or $^3$He
spins are coupled to states with total spin $f=1/2$ or 3/2 in the
deuteron case, and $f=0$ or 1 in the $^3$He case.
In Eqs.~(\ref{eq:h2ffz}) and~(\ref{eq:h3ffz}) we have defined
with $s_\mu$ ($h_\nu$) the muon spin (muon neutrino helicity).
The Fourier transform of the nuclear weak current
has been introduced as
\begin{equation}
j^\sigma({\bf q})=\int {\rm d}{\bf x}\,
{\rm e}^{ {\rm i}{\bf q} \cdot {\bf x} }\,j^\sigma({\bf x})
\equiv (\rho({\bf q}),{\bf j}({\bf q}))
\label{eq:jvq} \>\>\>,
\end{equation}
with the leptonic momentum transfer ${\bf q}$ defined
as ${\bf q} = {\bf k}_\mu-{\bf k}_\nu \simeq -{\bf k}_\nu$,
${\bf k}_\mu$ and ${\bf k}_\nu$ being the muon and muon neutrino momenta.
The function $\psi_{1s}^{\rm av}$ has been introduced to take into account
the initial bound state of the muon in the atom and the charge
distribution of the nucleus. It is typically approximated
as~\cite{Wal95} $|\psi_{1s}^{\rm av}|^2 \,=\,
{(\alpha\, \mu_{\mu d})^3\over \pi}$ for muon capture on deuteron,
and~\cite{Mar02} $|\psi_{1s}^{\rm av}|^2 \,=\,
{\cal {R}}\,{(2\,\alpha\, \mu_{\mu ^3{\rm He}})^3\over \pi}$
for muon capture on $^3$He, where
$\alpha$ is the fine structure constant ($\alpha=1/137$),
$\mu_{\mu d}$ and $\mu_{\mu ^3{\rm He}}$
are the reduced masses of the $(\mu,d)$ and ($\mu,^3$He) systems,
and the factor ${\cal {R}}$ approximately accounts
for the finite extent of the nuclear charge
distribution~\cite{Wal95} and is taken to be 0.98, as in Ref.~\cite{Mar10}.
In the case of muon capture on deuteron, the final state wave
function $\Psi_{{\bf p}, s_1 s_2}(nn)$ is expanded in partial waves,
and the calculation is restricted
to total angular momentum $J\leq 2$ and orbital angular momentum
$L\leq 3$, i.e.,
in a spectroscopic notation, to $^1S_0$, $^3P_0$, $^3P_1$, $^3P_2$--$^3F_2$
and $^1D_2$.
Standard techniques~\cite{Mar00,Wal95} are
now used to carry out the multipole expansion
of the weak charge, $\rho({\bf q})$, and current,
${\bf j}({\bf q})$, operators. Details
of the calculation can be found in Ref.~\cite{Mar10}.
Here we only note that all the contributing multipole operators
selected by parity and angular momentum selection rules
are included, as explained in Ref.~\cite{Mar10}.
The total capture rate for the two reactions under consideration
is then defined as
\begin{equation}
d\Gamma = 2\pi\delta(\Delta E) \overline{|T_W|^2} \times({\rm phase \, space})
\ ,
\label{eq:dgamma}
\end{equation}
where $\delta(\Delta E)$ is the energy-conserving $\delta$-function,
and the phase space is
$d{\bf p}\,d{\bf k}_\nu/(2\pi)^6$
for muon capture on deuteron and just
$d{\bf k}_\nu/(2\pi)^3$ for muon capture on $^3$He.
The following notation has been introduced: (i) for muon capture
on deuteron
\begin{equation}
\overline{|T_W|^2} = \frac{1}{2f+1}\sum_{s_1 s_2 h_\nu}\sum_{f_z}
|T_W(f,f_z;s_1,s_2, h_\nu)|^2 \ ,
\label{eq:hw2}
\end{equation}
and the initial hyperfine state has been fixed to be $f=1/2$;
(ii) for muon capture on $^3$He
\begin{equation}
\overline{|T_W|^2} = \frac{1}{4}\, \sum_{s_3^\prime h_\nu}\sum_{f f_z}
|T_W(f,f_z;s^\prime_3, h_\nu)|^2 \ ,
\label{eq:hw3}
\end{equation}
and the factor 1/4 follows from assigning the same probability
to all different hyperfine states.
After carrying out the spin sums,
the differential rate for muon capture on deuteron
($d\Gamma^D/dp$)
and the total rate for muon capture on $^3$He ($\Gamma_0$) are easily
obtained, and their expressions can be found in Ref.~\cite{Mar10}.
In order to obtain the total rate $\Gamma^D$ for muon capture on deuteron,
$d\Gamma^D/dp$ is plotted versus $p$ and numerically integrated.
Bound and continuum wave functions for both two- and three-nucleon
systems entering in Eqs.~(\ref{eq:h2ffz}) and~(\ref{eq:h3ffz})
are obtained with the hyperspherical-harmonics (HH)
expansion method. This method, as implemented in the case of
$A=3$ systems, has been reviewed in considerable detail in a series
of recent publications~\cite{Kie08,Viv06,Mar09}. We have used the
same method
in the context of $A=2$ systems, for which of course
wave functions could have been obtained by direct solution
of the Schr\"odinger equation. A detailed discussion for the
$A=2$ wave functions is given in Ref.~\cite{Mar10}.
\section{The nuclear weak current operator}
\label{sec:wcur}
The chiral effective field theory weak current transition operator
is taken from Refs.~\cite{Par03} and~\cite{Son09}, as reviewed
in Ref.~\cite{Mar10}. It is
derived in covariant perturbation theory based on the heavy-baryon
formulation of chiral Lagrangians by retaining corrections
up to N$^3$LO.
The one-body operators are those listed in
Eqs.~(17) of Ref.~\cite{Par03} and~(4.13)--(4.14) of Ref.~\cite{Mar10}.
The vector charge and axial current operators retain terms up to $1/m^2$,
while the axial charge and vector current operators retain terms
up to $1/m^3$, $m$ being the nucleon mass. Both $1/m^2$ and $1/m^3$
contributions arise when the non-relativistic reduction of the
single-nucleon covariant current is pushed to next-to-leading order.
The two-body vector currents are obtained from the two-body
electromagnetic currents via CVC. These are
decomposed into four terms~\cite{Son09}:
the soft one-pion-exchange ($1\pi$) term, vertex corrections to the one-pion
exchange ($1\pi C$), the two-pion exchange ($2\pi$), and a contact-term
contribution. Their explicit expressions can be found in
Ref.~\cite{Son09}. All the
$1\pi$, $1\pi C$ and $2\pi$ contributions contain low-energy constants
estimated using resonance saturation arguments, and
Yukawa functions obtained by performing the Fourier transform from
momentum- to coordinate-space with a Gaussian regulator characterized
by a cutoff $\Lambda$. Here, as discussed above,
we fixed the value of $\Lambda=600$ MeV.
The contact-term electromagnetic contribution
is given as sum of two terms, isoscalar and isovector, each one
with a coefficient in front ($g_{4S}$ and $g_{4V}$) fixed to
reproduce the experimental values of triton and $^3$He magnetic
moments. For the AV18/UIX Hamiltonian model, with $\Lambda=600$ MeV,
$g_{4S}=0.55(1)$ and $g_{4V}=0.793(6)$, the error being due to
numerics~\cite{Mar10}. Note that only the isovector contribution
is of interest here, but anyway it turns out to be negligible.
The two-body axial current operator consists of two contributions:
a one-pion exchange term and a (non-derivative) two-nucleon
contact-term. The explicit expression for the contact term
can be found in Ref.~\cite{Par03}. Here we review the one-pion
exchange term, since we add, in accordance with Ref.~\cite{Mos05},
the potential current contributions. Therefore, in momentum-space,
the one-pion exchange term reads:
\begin{eqnarray}
{\bf j}_{ij}^{\pi}({\bf q}; A)&=&
\frac{g_A}{2 m f_\pi^2}\Bigl[({\bm \tau}_i\times{\bm \tau}_j)^a
\Bigl[\frac{\rm i}{2}\,(1-g_A^2)\,\frac{{\bf p}_i+{\bf p}_i^\prime}{2}
+\big(\frac{1}{4}+{\hat c}_4\big)\,{\bm\sigma}_i\times{\bf k}_j
\nonumber \\
&&\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>
+(\frac{1+c_6+g_A^2}{4})\,{\bm\sigma}_i\times{\bf q}\Bigr]
+2{\hat c}_3{\bm \tau}_j^a{\bf k}_j
\nonumber \\
&&\>\>\>\>\>\>\>\>\>\>\>
-\frac{g_A^2}{4}{\bm\tau}_j^a\,\Bigl({\bf q}+{\rm i}{\bm \sigma}_i\times
({\bf p}_i+{\bf p}_i^\prime)\Bigr)\Bigr]
\frac{{\bm\sigma}_j\cdot{\bf k}_j}{m_\pi^2+{\bf k}_j^2} + i\leftrightarrow j
\ .
\label{eq:jwa}
\end{eqnarray}
where ${\bf k}_{i,j}={\bf p}_{i,j}^\prime-{\bf p}_{i,j}$,
with ${\bf p}_{i,j}$ and ${\bf p}_{i,j}^\prime$ being the initial
and final single nucleon momenta, ${\bf q}={\bf k}_i+{\bf k}_j$,
$g_A=1.2654$ is the axial-vector coupling constant, and $f_\pi=93$ MeV
is the pion decay constant.
The values used for the coupling constants ${\hat c}_3$, ${\hat c}_4$,
and $c_6$, as obtained from $\pi N$ data,
are ${\hat c}_3=-3.66$, ${\hat c}_4=2.11$ and $c_6=5.83$~\cite{Par03}.
The terms proportional to $g_A^2$ in Eq.~(\ref{eq:jwa})
are the potential currents. They
are the same as in Eq.~(21) of Ref.~\cite{Mos05} or
Eq.~(A.17) of Ref.~\cite{Ric10}.
The low-energy constant $d_R$,
determining the strength of the contact-term two-body axial
contribution, has been fixed
by reproducing GT$^{\rm EXP}$, finding $d_R=1.54(8)$. This value should
be compared with the corresponding one given in Ref.~\cite{Mar10}
(see Table V), $d_R=1.75(8)$. The difference between these two values
of $\simeq$ 13 \% is due to the presence of the potential currents
and is comparable with that of Ref.~\cite{Ric10}.
\section{Results}
\label{sec:res}
We present in Table~\ref{tab:res} the results for the total rates
of muon capture on deuteron, in the doublet hyperfine state ($\Gamma^D$),
and on $^3$He ($\Gamma_0$).
The deuteron, $nn$, $^3$He and $^3$H wave functions have been
calculated with the
AV18~\cite{Wir95} two- and, when necessary, UIX~\cite{Pud95}
three-nucleon interactions.
The model for the nuclear weak transition operator has been
presented in Sec.~\ref{sec:wcur}.
We compare our results with those of Ref.~\cite{Mar10},
obtained with the same Hamiltonian model and cutoff $\Lambda$,
but without the two-body potential currents elaborated
in Ref.~\cite{Mos05} and discussed in Sec.~\ref{sec:wcur}.
Note that here $d_R=1.54(8)$, while in
Ref.~\cite{Mar10} $d_R=1.75(8)$.
From inspection of the table, we conclude that the two calculations
are in remarkable agreement with each other: the differences in $\Gamma_0$
and $\Gamma^D$ are $\leq 0.1$ \%, well below the theoretical
uncertainties. The largest difference, of the
order of 2 \%, is in the $^3P_0$ partial wave contribution
to $\Gamma^D$. However, when all the partial
wave contributions are summed up, the difference in $\Gamma^D$
returns well below the 1 \% level.
In conclusion, we have studied the potential currents
dictated by PCAC, as elaborated in Refs.~\cite{Ric10,Mos05},
and we have found that their contributions to the total rates
of muon capture on deuteron and $^3$He are tiny.
This result is a consequence of the procedure
adopted to constrain the weak current.
Finally, we expect that the potential currents
will give tiny contributions also in
weak capture reactions of astrophysical interest and in those
processes whose momentum transfer is small.
\begin{table}
\caption{Total rate for muon capture on deuteron and $^3$He, in
s$^{-1}$. In the $A=2$ case,
the different partial wave contributions are indicated. The
numbers among parentheses indicate the theoretical uncertainties
arising from the adopted fitting procedures.
Such uncertainty is not indicated when
less than 0.1 s$^{-1}$.
The AV18 and AV18/UIX interactions have been used to calculate the
$A=2$ and $A=3$ wave functions.
The corresponding results of Ref.~\protect\cite{Mar10} are
also listed.}
\label{tab:res}
\begin{tabular}{llllllll|l}
\hline\noalign{\smallskip}
& $^1S_0$ & $^3P_0$ & $^3P_1$
& $^3P_2$ & $^1D_2$ & $^3F_2$
& $\Gamma^D$ & $\Gamma_0$ \\
\noalign{\smallskip}\hline\noalign{\smallskip}
Present work & 250.1(8) & 20.2 & 46.1 & 71.3 & 4.5 & 0.9 & 393.1(8) & 1488(9) \\
Ref.~\protect\cite{Mar10} &
250.0(8) & 19.8 & 46.3 & 71.1 & 4.5 & 0.9 & 392.6(8) & 1488(9) \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}
|
{
"timestamp": "2010-09-22T02:01:21",
"yymm": "1009",
"arxiv_id": "1009.4016",
"language": "en",
"url": "https://arxiv.org/abs/1009.4016"
}
|
\section{Introduction}
In many applications, one considers a process that is modeled as a stochastic differential equation (SDE), while one is ultimately interested in the time evolution of the expectation of a certain function of the state, i.\,e., in weak approximation. Consider, for instance, the micro/macro simulation of dilute solutions of polymers \cite{Laso:1993p10000}, which will be the motivating example in this paper. Here, an SDE models the evolution of the configuration of an individual polymer driven by the flow field, and the function of interest is a non-Newtonian stress tensor (the expectation of a function of the polymer configuration). For this type of problem, one often resorts to Monte Carlo simulation \cite{caflisch98mca}, i.\,e., the simulation of a large ensemble of realizations of the SDE, combined with ensemble averaging to obtain an approximation of the quantity of interest at the desired moments in time. For concreteness, we introduce the SDE
\begin{equation}\label{eq:SDE}
d\bs{X}(t)=\bs{a}\big(t,\bs{X}(t)\big)~dt+\bs{b}\big(t,\bs{X}(t)\big)\star d\bs{W}(t),\quad t \inI:=[t^0,T],\quad \bs{X}(t^0)=\bs{X}_0,
\end{equation}
in which $\bs{a}:I\times\mathbb{R}^d\to\mathbb{R}^d$ is the drift, $\bs{b}:I\times\mathbb{R}^d\to\mathbb{R}^{d\times m}$ is the diffusion,
and $\bs{W}(t)$ is an $m$-dimensional Wiener process. The initial value $\bs{X}_0$ is independent of $\bs{W}$ and follows some known distribution with density $\varphi_0(\bs{x})$. As usual, \eqref{eq:SDE} is an abbreviation of the integral form
\[
\bs{X}(t)=\bs{X}_0+\int_{t^0}^t\bs{a}\big(s,\bs{X}(s)\big)~ds+\int_{t^0}^t\bs{b}\big(s,\bs{X}(s)\big)\star d\bs{W}(s),\quad t \inI.
\]
The integral with respect to $\bs{W}$ can be interpreted, e.\,g., as an It\^{o} integral with $\star\; d\bs{W}(s) = d\bs{W}(s)$ or as a Stratonovich integral with $\star\; d\bs{W}(s) = \circ \;d\bs{W}(s)$.
The function of interest for the Monte Carlo simulation is defined as the expectation $\operatorname{E}$ of a function $\bs{f}\big(\bs{X}(t)\big)$,
\[
\bar{\bs{f}}(t)=\operatorname{E}\bs{f}\big(\bs{X}(t)\big).
\]
The numerical properties of Monte Carlo simulations have been analyzed extensively in the literature. We mention studies on the order of weak convergence of explicit \cite{milstein95nio,kloeden99nso,komori07wso,roessler07sor,debrabant09foe,debrabant10rkm} and implicit \cite{kloeden99nso,komori08wfo,debrabant09ddi,debrabant08bao,debrabant11bao} time discretizations of SDE \eqref{eq:SDE}, the investigation of stability \cite{hernandez92aso,hernandez93cas,saito96sao,higham00msa,tocino05mss}, and techniques for variance reduction \cite{newton94vrf,giles08mmc,giles08imm}. For more references, we refer to \cite{kloeden99nso}.
Also for strong approximation, there has been a growing interest in the study of numerical methods for stiff SDEs \cite{kloeden99nso,milstein98bim,burrage01sar,tian01itm,burrage02pcm,tian02tss,milstein03nmf,burrage04isr,abdulle08src,abdulle08srm}.
In this paper, we present and analyze a \emph{micro/macro} acceleration technique for the Monte Carlo simulation of SDEs of the type \eqref{eq:SDE} in which there exists a time-scale separation between the (fast) time-scale on which individual trajectories of the SDE need to be simulated and the (slow) time-scale on which the function $\bar{\bs{f}}(t)$ evolves. The proposed method is motivated by the development of recent generic multiscale techniques, such as \emph{equation-free} \cite{Kevrekidis:2009p7484,KevrGearHymKevrRunTheo03} and \emph{heterogeneous multiscale} methods \cite{EEng03,E:2007p3747}. We use the simulation of a dilute polymer solution as an illustrative example. The \emph{microscopic} level is defined via an ensemble $\bs{\mcX}\equiv (\bs{X}_j)_{j=1}^J$ of $J$ realizations evolving according to \cref{eq:SDE}; the \emph{macroscopic} level will be defined by a set of $L$ \emph{macroscopic state variables} $\bs{U}\equiv (U_l)_{l=1}^L$, with $U_l(t)=\operatorname{E} u_l(\bs{X}(t))$, for some appropriately chosen functions $u_l$.
The method exploits a separation in time scales by combining short bursts of \emph{microscopic} simulation with the SDE \eqref{eq:SDE} with a \emph{macroscopic} extrapolation step, in which only the macroscopic state $\bs{U}$ is extrapolated forward in time. One time step of the algorithm can be written as follows: (1) microscopic \emph{simulation} of the ensemble using the SDE \eqref{eq:SDE}; (2) \emph{restriction}, i.\,e., extraction of (an estimate of) the macroscopic state (or macroscopic time derivative); (3) forward in time \emph{extrapolation} of the macroscopic state; and (4) \emph{matching} of the ensemble that was available at the end of the microscopic simulation with the extrapolated macroscopic state.
Remark that the resulting method is fully explicit as soon as the microscopic simulation is explicit, and that the method can readily be implemented as a higher order method by an appropriate choice of the extrapolation.
The main contributions of the present paper are the following:
\begin{itemize}
\item From a numerical analysis viewpoint, we study convergence
of the proposed micro/macro acceleration method in the absence of statistical error. Specifically, we discuss how the error that is introduced during the matching depends on the number of macroscopic state variables, and how the deterministic error depends on the extrapolation time step. Additionally, we also comment on the effects of the extrapolation on the statistical error. Finally, we give a general convergence result.
\item From a practical viewpoint, we provide numerical results for a nontrivial test case, showing the interplay between the different sources of numerical error. We illustrate the effects of the choice of macroscopic state variables, as well as the dependence of the numerical error on the chosen extrapolation strategy.
\end{itemize}
A stability analysis of the proposed method will be given in a separate publication
The remainder of the paper is organized as follows. In
\cref{sec:preliminaries}, we first introduce the mathematical setting of the
paper. We discuss the necessary assumptions on the SDE \eqref{eq:SDE} for our
Monte Carlo setting, and introduce the illustrative example that will be used
for the numerical experiments.
In \cref{subsec:new-algo}, we
propose the algorithm that will be the focus of this paper.
\Cref{sec:proj} provides some results on the matching operator, whereas the
extrapolation operator is discussed in \cref{subsec:accel-extrap}. We provide
a general convergence result in \cref{sec:conv}. \Cref{sec:numerical} provides
numerical illustrations, which are chosen to illuminate the properties of the
proposed method. We conclude in \cref{sec:concl}, where we also outline some
directions for future research.
\section{Mathematical setting\label{sec:preliminaries}}
In this section, we introduce in detail the notations that we will use (\cref{sec:math-set}), as well as the illustrative example that will be considered throughout the paper (\cref{sec:model}).
\subsection{Notations\label{sec:math-set}}
We first define the appropriate function spaces. Let $C_P^r(\mathbb{R}^d, \mathbb{R})$ denote the space of all $g \in C^r(\mathbb{R}^d,\mathbb{R})$ fulfilling that there are constants $\tilde{C}>0$ and $\kappa>0$ such that $|\partial^i_{\bs{x}} g(\bs{x})|\leq\tilde{C}(1+\|\bs{x}\|^\kappa)$ for any partial derivative of order $i \leq r$ and all $\bs{x}\in\mathbb{R}^d$. Further, let $g \in C_P^{q,r}(I\times\mathbb{R}^d, \mathbb{R})$ if $g(\cdot,\bs{x}) \in C^{q}(I,\mathbb{R})$, $g(t,\cdot) \in C^r(\mathbb{R}^d, \mathbb{R})$ for all $t \inI$ and $\bs{x} \in \mathbb{R}^d$, and $|\partial^i_{\bs{x}}g(s,\bs{x})|\leq\tilde{C}(1+\|\bs{x}\|^\kappa)$ holds for $0\leq i\leq r$ uniformly with respect to $s\in[t^0,t]$ and for all $\bs{x}\in\mathbb{R}^d$ \cite{kloeden99nso,milstein95nio}.
We consider the SDE \eqref{eq:SDE}.
Besides the exact solution $\bs{X}(t)$ of SDE \eqref{eq:SDE} starting from $\bs{X}(t^0)=\bs{X}_0$, we also introduce the exact solution of an auxiliary initial value problem for the SDE: the solution of the SDE starting from an initial value $\bs{Z}$ at time $t$ will be denoted as $\bs{X}^{t,\bs{Z}}$. With this notation, we state the following definition:
\begin{defi}[Uniform weak continuity of the SDE]\label{def:weakcontinuitySDE}
Consider a class of random variables. The
SDE \eqref{eq:SDE} is called uniformly weakly continuous for this class if, for all $g \in C_P^{0,2}(I\times\mathbb{R}^d, \mathbb{R})$, there exist constants $\Delta t_0>0$ and $C$ such that
\begin{equation}
|\operatorname{E} g\big(s,\bs{X}^{t,\bs{Z}}(t+\Delta t)\big)-\operatorname{E} g(s,\bs{Z})|\leq C\Delta t
\end{equation}
holds for all initial values $\bs{Z}$ in the considered class, $\Delta t\in[0,\Delta t_0]$, and all $t\in[t^0,T-\Delta t]$, $s\inI$.
\end{defi}
Next, we discretize \cref{eq:SDE} in time with step size $\delta t$ and denote the numerical approximation $\bs{Y}^{k}=\bs{Y}(t^k)\approx \bs{X}(t^k)$ with $t^k=t^0+k\delta t$. Leaving the concrete choice of discretization undecided for now, we introduce a short-hand notation for an abstract one step discretization scheme,
\begin{equation}\label{eq:sde_discr}
\bs{Y}^{k+1}=\phi(t^k,\bs{Y}^k;\delta t), \qquad \bs{Y}^0=\bs{X}_0, \qquad k\ge 0.
\end{equation}
In the Monte Carlo setting, we are interested in weak approximation of the SDE \eqref{eq:SDE}. We define the weak order of consistency of the discretization $\phi$ as follows (compare \cite{milstein95nio}):
\begin{defi}[Weak order of consistency of SDE discretization]
Assume that for all $g \in C_P^{0,2(p_{\os}+1)}(I\times\mathbb{R}^d, \mathbb{R})$ there exists a $C_g\in C_P^0(\mathbb{R}^d,\mathbb{R})$ such that
\begin{equation*}
|\operatorname{E} g\big(s,\phi(t,\bs{Z};\delta t)\big)-\operatorname{E} g\big(s,\bs{X}^{t,\bs{Z}}(t+\delta t)\big)| \leq C_g(\bs{Z}) \,\delta t^{p_{\os}+1}
\end{equation*}
is valid for $\bs{Z} \in \mathbb{R}^d$, $s\inI$, and $t$, \mbox{$t+\delta t \in [t^0,T]$}. Then the one step method $\phi$ is called weakly consistent of order $p_{\os}$.
\end{defi}
Since we can only simulate a finite number of realizations of approximations of \eqref{eq:SDE} (via its discretization \eqref{eq:sde_discr}), we also need to approximate the expectation $\operatorname{E}$ by an empirical mean $\widehat{\E}$. Using the ensemble $\bs{\mcY}\equiv (\bs{Y}_j)_{j=1}^J$ of realizations of \eqref{eq:sde_discr}, the empirical mean $\widehat{\E}$ is defined as
\[
\widehat{\E} \bs{f}(\bs{\mcY})=\frac{1}{J}\sum_{j=1}^J \bs{f}(\bs{Y}_j).
\]
The numerical integration scheme for the ensemble $\bs{\mcY}$ will be denoted as
\begin{equation}\label{eq:sde_discr_ens}
\bs{\mcY}^{k+1}=\phi_{\bs{\mcY}}(t^k,\bs{\mcY}^k;\delta t)=\left(\phi(t^k,\bs{\mcY}^k_j;\delta t)\right)_{j=1}^J.
\end{equation}
The total error of a Monte Carlo simulation of \eqref{eq:SDE} consists of a \emph{deterministic} error due to the time discretization, and a \emph{statistical} error due to the finite number of realizations. In general, for a given tolerance, a Monte Carlo simulation will be most efficient if the statistical and deterministic error are balanced. However, as for stiff systems of ODEs, one might encounter situations in which the time step $\delta t$ cannot be increased because of stability problems. The acceleration method that we will propose in \cref{subsec:new-algo} will prove to be particularly useful to accelerate Monte Carlo simulations in such situations.
\begin{remark}[Probability density functions]\label{rem:FokkerPlanck}
One can equivalently describe the process \eqref{eq:SDE} via an advection-diffusion equation, also known as Fokker--Planck equation (see, e.\,g., \cite{risken1996fokker}), which in the It\^{o} case takes the form
\begin{equation}
\partial_t\varphi=-\nabla \left(\bs{a}\;\varphi\right)
+ \frac{1}{2}\nabla\cdot\left[\nabla\cdot\left(\bs{b}^T\bs{b}\;\varphi\right)\right],
\end{equation}
and which describes the evolution of the probability density function $\varphi(t,\bs{x})$ of $\bs{X}(t)$, starting from the initial density $\varphi(t^0,\bs{x})=\varphi_0(\bs{x})$. The functional of interest then becomes
\begin{equation}\label{eq:intexp}
\bar{\bs{f}}(t)=\int \bs{f}(\bs{x}) \varphi(t,\bs{x})d\bs{x}.
\end{equation}
For simulation purposes, however, the Monte Carlo algorithm is generally preferred, due to the possibly high number of dimensions of the Fokker--Planck equation.
\end{remark}
\begin{remark}[Spatial dimension] While the model \eqref{eq:SDE} can have arbitrary dimension, we will consider a one-dimensional version in the numerical illustrations for ease of visualization. In that case the state vector reduces to a scalar, $X(t)$. Whenever we consider the one-dimensional case, all bold typesetting in \cref{eq:SDE} will be removed.
\end{remark}
\subsection{A motivating model problem: FENE dumbbells\label{sec:model}}
To illustrate the behavior of the proposed numerical methods, we will consider the micro/macro simulation of the evolution of immersed polymers in a solvent. Here, one models the evolution of the configuration of a polymer ensemble via an SDE of the type \eqref{eq:SDE}, driven by the flow field, for each of the individual polymers. This results in a polymer configuration distribution at each spatial point. This microscopic model is coupled to a Navier--Stokes equation for the solvent, in which the effect of the immersed polymers is taken into account via a non-Newtonian stress tensor. We refer to~\cite{Hulsen:1997p7027,Laso:1993p10000,le-bris-lelievre-09} for an introduction to the literature on this subject.
In this paper, we consider only the Monte Carlo simulation of the microscopic model, leaving the coupling with the Navier--Stokes equations for future work. We eliminate the spatial dependence by considering the microscopic equations along the characteristics of the flow field, i.\,e., in a Lagrangian frame. In general, a microscopic model describes an individual polymer as a series of beads, connected by nonlinear springs, resulting in a coupled system of SDEs for the position of each of the beads. In the simplest case, that we will also use as an illustrative example here, one represents the polymers as non-interacting dumbbells, connecting two beads by a spring that models intramolecular interaction. The state of the polymer chain is described by the end-to-end vector $\bs{X}(t)$ that connects both beads, and whose evolution is modelled using the non-dimensionalised SDE
\begin{equation}
d\bs{X}(t)=\left[\boldsymbol{\kappa}(t) \, \bs{X}(t)-\dfrac{1}{2\textrm{We}}\bs{F}\big(\bs{X}(t)\big)\right]dt + \dfrac{1}{\sqrt{\textrm{We}}}d\bs{W}(t), \label{eq:fene-3d}
\end{equation}
where $\boldsymbol{\kappa}(t)$ is the velocity gradient of the solvent, $\textrm{We}$ is the Weissenberg number, and $\bs{F}$ is an entropic force, here considered to be finitely extensible nonlinearly elastic (FENE),
\begin{equation}\label{eq:springs}
\bs{F}(\bs{X})=\dfrac{\bs{X}}{1-\|\bs{X}\|^2/\gamma},
\end{equation}
with $\gamma$ a non-dimensional parameter that is related to the maximal polymer length. The resulting non-Newtonian stress tensor is given by the Kramers' expression,
\begin{equation}
\bs{\tau}_p(t) = \dfrac{\epsilon}{\textrm{We}} \bigg( \operatorname{E}\Big( \bs{X}(t)\otimes \bs{F}\big(\bs{X}(t)\big)\Big)-\textrm{\bf{Id}} \bigg), \label{eq:kramers}
\end{equation}
in which $\epsilon$ represents the ratio of polymer and total viscosity; see \cite{le-bris-lelievre-09} for details and further references.
This model, which is of the type \eqref{eq:SDE}, takes into account Stokes drag (due to the solvent velocity field), intramolecular elastic forces, and Brownian motion (due to collisions with solvent molecules). The functional of interest in the Monte Carlo simulation is $\bar{\bs{f}}(t)=\bs{\tau}_p(t)$.
\Cref{eq:fene-3d,eq:springs} ensure that the length of the end-to-end vector, $\|\bs{X}\|$, cannot exceed the maximal value $\sqrt{\gamma}$~\cite{jourdain-lelievre-02}. However, a naive explicit discretization scheme might yield polymer lengths beyond this maximal value. This can be avoided via an accept-reject strategy, see, e.\,g., \cite[Section 4.3.2]{Ott96}. Here, for each polymer, the state after each time step is rejected if the calculated polymer length exceeds $\sqrt{(1-\sqrt{\delta t})\gamma}$, and a new random number is tried until acceptance. To prevent the distribution of the approximation process to be heavily influenced, the microscopic time-step has to be chosen small enough. Alternatively, one can use an implicit method \cite{Ott96}, which alleviates the time-step restriction. However, even for implicit SDE discretizations, the maximal time step is limited when coupling the Monte Carlo simulation with a discretization of the Navier--Stokes equations for the solvent. This is due to the fact that the coupling between the Monte Carlo and Navier--Stokes parts is, in most existing work, done explicitly in time, creating an additional stability constraint due to the coupling. (Some notable exceptions are given in \cite{Laso:2004p12587,Somasi:2000p12659}.) With the algorithm that we propose, one would extrapolate both the Monte Carlo and the Navier--Stokes part of this coupled simulation simultaneously. This will be done in future work.
Here, we simply conclude that, for this model problem, the required time step for a stable SDE (or coupled) simulation may indeed be small compared to the time scale of the evolution of the stress.
As in, e.\,g., \cite{Keunings:1997p9982}, we will consider a one-dimensional version in the numerical illustrations; the stress tensor then reduces to a scalar $\tau_p(t)$. As time discretization we will use the explicit Euler--Maruyama scheme, combined with an accept-reject strategy.
\section{Micro/macro acceleration method\label{sec:accel}\label{subsec:new-algo}}
In this section, we describe the micro/macro acceleration algorithm that is the focus of the present paper. As said above, the goal of the method is to be faster than a full microscopic simulation, while converging to a full microscopic simulation when the extrapolation time step vanishes (see \cref{sec:conv}). The algorithm combines short bursts of microscopic simulation of an ensemble $\bs{\mcX}$ of $J$ realizations of the SDE \eqref{eq:SDE} with a macroscopic extrapolation step. During this macroscopic extrapolation, only a set of $L$ macroscopic state variables $\bs{U}$ are extrapolated forward in time, and the microscopic ensemble then needs to be \emph{matched} onto the extrapolated macroscopic state.
\subsection{Description of algorithm}
Before giving a detailed description of the algorithm, let us first elaborate slightly on the macroscopic state variables, $\bs{U}=\left( U_l\right)_{l=1}^L$, which are defined as expectations of scalar functions $u_l$ of the state $\bs{X}$ and time $t$,
\begin{equation}\label{eq:state-vars}
U_l(t) = \operatorname{E} u_l\big(t,\bs{X}(t)\big).
\end{equation}
\begin{remark}[Choice of macroscopic state variables]
The choice of the functions $u_l$ is problem-dependent and will be specified with the numerical illustrations for the examples considered in this text. For the exposition in this section, it may be helpful to think about the standard moments of the distribution in a one-dimensional setting, i.\,e., $u_l(t,x)=x^l$. Note that by allowing $u_l$ to depend directly on time $t$, centralized moments $u_1(t,x)=x$, $u_l(t,x)=(x-U_1(t))^l$ for $l\geq2$, for instance, can also be considered.
\end{remark}
We introduce two abstract operators to connect both levels of description: a
\emph{restriction} operator
\begin{equation}
\label{eq:intro_restriction}
\mathcal{R}: \bs{\mcY} \mapsto \bs{U}=\mathcal{R}(\bs{\mcY}),
\end{equation}
that maps a microscopic ensemble onto a macroscopic state, and a \emph{matching} operator
\begin{equation}
\mathcal{P}: \bs{U},\bs{\mcY}^* \mapsto \bs{\mcY}=\mathcal{P}(\bs{U},\bs{\mcY}^*),
\end{equation}
which matches a given microscopic ensemble $\bs{\mcY}^*$ with an imposed macroscopic state $\bs{U}$. Both operators will be discussed in detail in \cref{sec:match-restrict}.
We further introduce some notation. Let ${\I^{\Dt}} = \{t^0, t^1, \ldots, t^{N}\}$ be a (macroscopic) time discretization of the time interval $I$, with $t^0 < t^1 < \ldots < t^{N} =T$ and step
sizes $\Delta t_{n} = t^{{n}+1}-t^{n}$ for ${n}=0,1, \ldots, N-1$.
We also introduce the discrete time instances $t^{{n},k}=t^{n}+k\delta t$ that are defined on a microscopic grid, and, correspondingly, the discrete approximations $\bs{U}^{{n},k}\approx\bs{U}(t^{{n},k})$ and $\bs{U}^{{n}}\approx\bs{U}(t^{n})$, and analogously for $\bs{\mcY}$ and $\bs{Y}$. Clearly, $(\cdot)^{n}=(\cdot)^{{n},0}$.
One step of the micro/macro acceleration method then reads:
\begin{algorithm}[Micro/macro acceleration]\label{algo:accel}Given a microscopic state $\bs{\mcY}^{{n}}$ at time $t^n$, advance to a microscopic state $\bs{\mcY}^{{n}+1}$ at time $t^{n+1}$ via a three-step procedure:
\begin{itemize}
\item[(i)] \emph{Simulate} the microscopic system over $K$ time steps of size $\delta t$, \[\bs{\mcY}^{{n},k}=\phi_{\bs{\mcY}}(t^{{n},k-1},\bs{\mcY}^{{n},k-1};\delta t), \qquad 1\le k \le K,\]
and record the restrictions $\bs{U}^{{n},k}=\mathcal{R}({\bs{\mcY}^{{n},k}})$, as well as an approximation of the function of interest $\hat{\bs{f}}^{{n},k}=\widehat{\E}\bs{f}(\bs{\mcY}^{{n},k})$.
\item[(ii)] \emph{Extrapolate} the macroscopic state $\bs{U}$ from (some of) the time points $t^{i,k}$,$i=0,\ldots,n$, $k=1,\dots,K$, to a new macroscopic state $\bs{U}^{{n}+1}$ at time $t^{{n}+1}$,
\begin{equation}\label{eq:extrap}
\bs{U}^{{n}+1} = {\mathcal E}\left(\left(\bs{U}^{i,k}\right)_{i,k=0,0}^{n,K};(\Delta t_i)_{i=0}^n,\delta t\right).
\end{equation}
\item[(iii)] \emph{Match} the microscopic state $\bs{\mcY}^{{n},K}$ with the extrapolated macroscopic state $\bs{\mcY}^{{n}+1}=\mathcal{P}(\bs{\mcY}^{{n},K},\bs{U}^{{n}+1})$.
\end{itemize}
\end{algorithm}
Some basic requirements for the matching and restriction operators are given in \Cref{sec:match-restrict}. Specific choices and convergence properties for the algorithmic components are given in \Cref{sec:proj} (matching) and \Cref{subsec:accel-extrap} (extrapolation).
\begin{remark}[Closure approximation]
Due to the possibly high computational cost of Monte Carlo simulation, another route has been followed in the literature, in which one derives an approximate macroscopic model to describe the system; see, e.\,g., \cite{Herrchen:1997p8915,Hyon:2008p9897,Keunings:1997p9982,Lielens:1998p6790,Lielens:1999p9945,Sizaire:1999p9912} for derivations of macroscopic closures for FENE dumbbell models. The goal then is to obtain a closed system of $L$ evolution equations for the macroscopic state variables
$\bs{U}$, complemented with a constitutive equation, for the observable $\bar{\bs{f}}$ of interest as a function of these macroscopic state variables.
In~\cite{Ilg:2002p10825}, a \emph{quasi-equilibrium} approach is proposed, based on thermodynamical considerations; although the method has been formulated for the FENE dumbbell case, it is applicable to general SDEs of the type \eqref{eq:SDE}. Several algorithms have been presented to simulate the evolution of the quasi-equilibrium model numerically \cite{SamLelLeg10,Wang:2008}. Note that, in contrast with the method presented in the present paper, numerical closures introduce the modeling assumption that a closed model in terms of the macroscopic state variables exists. The micro/macro acceleration method presented here only uses these macroscopic state variables for computational purposes, and maintains convergence to the full microscopic dynamics (see \cref{sec:conv}).
\end{remark}
\subsection{Matching and restriction operators}\label{sec:match-restrict}
Next, we give some detail on the \emph{restriction} and \emph{matching} operators that connect the microscopic and macroscopic levels of description. To obtain the macroscopic state from the microscopic ensemble, the restriction operator is defined, which can readily be obtained by replacing the expectation $\operatorname{E}$ by the empirical mean~$\widehat{\E}$,
\begin{equation}
U_l(t) =\mathcal{R}_l\big(\bs{\mcY}(t)\big) = \widehat{\E} u_l\big(t,\bs{\mcY}(t)\big).
\end{equation}
Due to the constraint $\mathcal{R}(\bs{\mcY}^{{n}+1})=\bs{U}^{{n}+1}$, during the matching step, the elements of $\bs{\mcY}^{{n}+1}$ are in general not independent, and Monte Carlo error estimates are not straightforward to obtain. For example, the Central Limit Theorem is not directly applicable \cite{caflisch98mca}.
Therefore, to discuss the matching operator
$\mathcal{P}$, we first turn to the idealized operators $\overline{\mathcal{P}}$ and $\overline{\mathcal{R}}$ obtained in the limit $J \to \infty$, eliminating statistical error. Rather than acting on ensembles of configurations, these operators are defined directly on the random variables. More precisely, the restriction operator $\overline{\mathcal{R}}$ reduces a random variable $\bs{Z}$ to macroscopic state variables,
\begin{equation}\label{eq:Rbar}
\overline{\mathcal{R}}(\bs{Z})=\big(\overline{\mathcal{R}}_l(\bs{Z})\big)_{l=1}^L \text{ with } \overline{\mathcal{R}}_l(\bs{Z}) =U_l= \operatorname{E} u_l(\bs{Z}) \text{ for $l=1,\ldots,L$}.
\end{equation}
The matching operator $\overline{\mathcal{P}}$ maps a random variable $\bs{Z}$ onto a new random variable $\bs{Z}^*$ that corresponds to a macroscopic state $\bs{U}^*$,
\begin{equation}
\bs{Z}^*=\overline{\mathcal{P}}(\bs{Z},\bs{U}^*)\text{ with }\overline{\mathcal{R}}(\bs{Z}^*)=\bs{U}^*.
\end{equation}
When analyzing the matching, we will consider the idealized operators $\overline{\mathcal{P}}$ and $\overline{\mathcal{R}}$.
A priori, we allow for considerable freedom in the definition of the matching operator, requiring only a few properties to be satisfied for any reasonable matching. First, we impose a projection property:
\begin{defi}[Projection property]\label{def:selfconsistency}
A matching operator $\bar{\mathcal{P}}$ associated to a restriction operator $\bar{\mathcal{R}}$ satisfies the projection property if
\[\bs{U} =\bar{\mathcal{R}}(\bs{Z}) \Rightarrow \bs{Z}=\bar{\mathcal{P}}\big(\bs{Z},\bs{U}\big)=\bar{\mathcal{P}}_{\bs{U}}(\bs{Z})\]
for any suitable random variable $\bs{Z}$, i.\,e., if $\bar{\mathcal{P}}_{\bs{U}}$ is a projection operator for every $\bs{U}$.
\end{defi}
The projection property states that a random variable remains unaffected by projection if its macroscopic state is equal to the macroscopic state on which one wants to project. We remark also that \cref{def:selfconsistency} implies $\bar{\mathcal{P}}_{\bs{U}}^2=\bar{\mathcal{P}}_{\bs{U}}$, which justifies the use of the term \emph{projection}.
Next, we consider the number of macroscopic state variables $L$ to vary, and define a sequence of vectors of macroscopic state variables $\left(\bs{U}_{[L]}\right)_{L=1,2,\dots}$, such that $\bs{U}_{[L]}=(U_l)_{l=1}^{L}$, i.\,e., for increasing $L$, additional macroscopic state variables are added. The corresponding sequences of matching and restriction operators are denoted as $\left(\overline{\mathcal{P}}_{[L]}\right)_{L=1,2,\ldots}$ and $\left(\overline{\mathcal{R}}_{[L]}\right)_{L=1,2,\ldots}$, respectively.
Using this notation, we are ready to formulate the definitions of continuity and consistency of the matching step:
\begin{defi}[Continuity of matching]
Consider a set of random variables and a set of sequences of macroscopic states. A sequence of matching operators $\left(\overline{\mathcal{P}}_{[L]}\right)_{L=1,2,\dots}$ is called continuous for these sets if, for all $g \in C_P^{0,2}(I\times\mathbb{R}^d, \mathbb{R})$, there exists a constant $C$, depending only on $g$, such that
\begin{equation}\label{eq:continuity}
|\operatorname{E} g\big(s,\overline{\mathcal{P}}_{[L]}(\bs{Z},\bs{U}_{[L]}^*)\big)-\operatorname{E} g\big(s,\overline{\mathcal{P}}_{[L]}(\bs{Z},\bs{U}^+_{[L]})\big)|\leq C\|\bs{U}^*_{[L]}-\bs{U}^+_{[L]}\|
\end{equation}
holds for all $L\geq1$, all sequences $\left(\bs{U}^*_{[L]}\right)_{L=1,2,\dots}$, $\left(\bs{U}^+_{[L]}\right)_{L=1,2,\dots}$ of macroscopic states and all $\bs{Z}$ in the considered sets, and all $s\inI$.
\end{defi}
Here and in the following, the norm $\|\cdot\|$ should be chosen such that it remains bounded for $L\to\infty$ for component-wise bounded sequences.
\begin{defi}[Consistency of matching]\label{def:consistencyprojection}
Consider a sequence of matching operators $\left(\overline{\mathcal{P}}_{[L]}\right)_{L=1,2,\dots}$. This sequence is called consistent for a class of sequences of triples $\left(\bs{Z}^*_{[L]},\bs{Z}^+_{[L]},\bs{U}_{[L]}\right)_{L=1,2,\dots}$ of random variables $\bs{Z}^*_{[L]}$, $\bs{Z}^+_{[L]}$, and macroscopic states $(\bs{U}_{[L]})$, if for all $g\in C_P^{0,2}(I\times\mathbb{R}^d, \mathbb{R})$ there exist constants $C_L$, with $C_L\to0$ for $L\to\infty$, and $L_0$ such that it holds
\begin{equation}\label{eq:consistency}
|\operatorname{E} g\big(s,\overline{\mathcal{P}}_{[L]}(\bs{Z}^*_{[L]},\bs{U}_{[L]})\big)-\operatorname{E} g\big(s,\overline{\mathcal{P}}_{[L]}(\bs{Z}^+_{[L]},\bs{U}_{[L]})\big)|
\leq C_L
|\operatorname{E} g(s,\bs{Z}^*_{[L]})-\operatorname{E} g(s,\bs{Z}^+_{[L]})|
\end{equation}
for all $L\geq L_0$ and all sequences of triples $\left(\bs{Z}^*_{[L]},\bs{Z}^+_{[L]},\bs{U}_{[L]}\right)_{L=1,2,\dots}$ in the considered class.
\end{defi}
The possible dependence of the random variables on $L$ is present since the random variables will be considered to have been generated using the micro/macro acceleration algorithm, and therefore depend on $L$.
Whereas continuity measures (in a weak sense) the difference between the matching of a random variable with two different macroscopic states, the consistency measures the difference between the matching of two different random variables with the same macroscopic state.
\Cref{def:selfconsistency,def:consistencyprojection,def:weakcontinuitySDE} immediately imply the following corollary:
\begin{cor}\label{cor:consensemblegeneration}
Consider a sequence of matching operators $\left(\overline{\mathcal{P}}_{[L]}\right)_{L=1,2,\dots}$, and assume that this sequence is consistent for a set of sequences of triples
\[\left(\bs{Z}_{[L]},\bs{X}^{t^-,\bs{Z}_{[L]}}(t^*),\overline{\mathcal{R}}_{[L]}\big(\bs{X}^{t^-,\bs{Z}_{[L]}}(t^*)\big)\right)_{L=1,2,\dots},\]
where $Z_{[L]}$ are random variables, $t^-=t^*-\Delta t$, $\Delta t\in[0,t^*-t^0]$, $t^*\inI$. Suppose further that SDE \eqref{eq:SDE} is uniformly weakly continuous for the set of all $\bs{Z}_{[L]}$.
Then for all $g \in C_P^{0,2}(I\times\mathbb{R}^d, \mathbb{R})$ there exist constants $\Delta t_0>0$ and $C_L$, with $C_L\to0$ for $L\to\infty$, and $L_0$, such that it holds
\begin{equation}
|\operatorname{E} g\big(s,\bs{X}^{t^-,\bs{Z}_{[L]}}(t^*)\big)-\operatorname{E} g\big(s,\overline{\mathcal{P}}_{[L]}(\bs{Z}_{[L]},\bs{U}_{[L]}^*)\big)|\leq C_L\Delta t
\end{equation}
for all $L\geq L_0$, all $\left(\bs{Z}_{[L]},\bs{X}^{t^-,\bs{Z}_{[L]}}(t^*),\bs{U}_{[L]}^*=\overline{\mathcal{R}}_{[L]}\big(\bs{X}^{t^-,\bs{Z}_{[L]}}(t^*)\big)\right)_{L=1,2,\dots}$ in the considered set, $\Delta t\in[0,\Delta t_0]$, and all $t^*\in[t^0+\Delta t,T]$, $s\inI$.
\end{cor}
This corollary states that the difference, measured in a weak sense, between the exact distribution at some time instance $t^*$ and a matching from a previous time $t^-=t^*-\Delta t$ with the exact macroscopic state at $t^*$ vanishes when matching with more macroscopic state variables or letting $\Delta t\to0$.
In the remainder of the text, we will only use the properties stated above. Therefore, one may use any matching operator that would come to mind for a particular problem, as long as the above properties are satisfied.
\subsection{Matching failure and adaptive time-stepping\label{sec:proj-failure}}
In the numerical experiments, one might encounter situations in which the distributions evolve on time-scales that are not significantly slower than those of the macroscopic functions of interest. In that case, when taking a large extrapolation time step, the extrapolated macroscopic state differs significantly from the state corresponding to the last available polymer ensemble, and it is possible that matching the ensemble with the desired macroscopic state fails. Consequently, a failure in the matching can be used as an indication to decrease the extrapolation time step. Based on this observation, we propose the following criterion to adaptively determine the macroscopic step size $\Delta t$: If the matching fails, we reject the step and try again with a time step
\begin{equation}\label{eq:ad_1}
\Delta t_{\text{new}}=\max(\underline{\alpha} \Delta t, K\delta t), \qquad\underline{\alpha} < 1,
\end{equation}
whereas, when the matching succeeds, we accept the step and propose
\begin{equation}\label{eq:ad_2}
\Delta t_{\text{new}}=\min(\overline{\alpha} \Delta t,\Delta t_{\text{max}}),\qquad \overline{\alpha}>1
\end{equation}
for the next step. If the macroscopic step size $\Delta t_{\text{new}} = K\delta t$, matching becomes trivial (the identity operator), since there is no extrapolation. Note that, when this happens, the criterion will ensure that the larger time steps are tried after the next burst of microscopic simulation.
\section{Matching operator\label{sec:proj}}
In this section, we analyze the matching operator in more detail. We first define the specific matching operator that will be considered in this paper (\cref{sec:PropertiesProjectionstep}). We then prove a consistency result in a very special case (\cref{sec:prop-proj-result}) and give and illustrate then a conjecture for the general case (\cref{sec:prop-proj-conj}).
\subsection{Definition of specific matching operator\label{sec:PropertiesProjectionstep}}
The matching operator $\mathcal{P}$ can be defined in several ways. One option is to minimize the difference between the given ensemble and the result of the matching, for instance in the $2-norm$, i.\,e.,
\begin{equation}\label{eq:AnsatzVertBerech}
\bs{\mcY}^{{n}+1}=\argmin_{\bs{\mcZ}:~\mathcal{R}(\bs{\mcZ})=\bs{U}^{{n}+1}}\frac{1}{2}\|\bs{\mcZ} -\bs{\mcY}^{{n},K}\|^2.
\end{equation}
An approximation of the resulting ensemble can be obtained using a Lagrange multiplier technique:
\begin{equation}\label{eq:proj1}
\left\{ \begin{aligned} &\bs{\mcY}^{{n}+1}=\bs{\mcY}^{{n},K}+\sum_{l=1}^L
\lambda_l \nabla_{\bs{\mcY}} \mathcal{R}_l (\bs{\mcY}^{{n},K}),\\ &\text{ with
$\Lambda=\{\lambda_l\}_{l=1}^L$ such that $\mathcal{R}_l (\bs{\mcY}^{{n}+1}) =
U_l^{{n}+1}$ for $l=1,\ldots,L$.} \end{aligned} \right. \end{equation}
Then, $\bs{\mcY}^{{n}+1}=\mathcal{P}(\bs{\mcY}^{{n},K},\bs{U}^{{n}+1})$ results after
obtaining $\Lambda$ from a Newton procedure that solves the $L$-dimensional nonlinear system that defines the constraints.
(Note that one can show that the resulting ensemble exactly satisfies \eqref{eq:AnsatzVertBerech} if we write an implicitly defined
gradient $\nabla_{\bs{\mcY}} \mathcal{R}_l (\bs{\mcY}^{{n}+1})$ in the first line of
\eqref{eq:proj1}.)\\
By applying the implicit function theorem, we obtain the following lemma:
\begin{lem}
The problem \eqref{eq:proj1} has a locally unique solution if
\begin{itemize}
\item in the case of standard empirical moments $U_l^n$,
\begin{equation}\label{eq:conduniquesolstempmom}
\det\left(U_{i+k-2}^n\right)_{i,k=1,\dots,L}\neq0,
\end{equation}
\item in the case of empirical centralized moments $U_l^n$,
\[
\det\left(U_{i+k-2}^n-U_{i-1}^nU_{k-1}^n\right)_{i,k=2,\dots,L}\neq0.
\]
\end{itemize}
\end{lem}%
\begin{remark}The determinant
\[
\det\left(\operatorname{E} X^{i+k-2}(t^n)\right)_{i,k=2,\dots,L},
\]
obtained by replacing the empirical moments in \eqref{eq:conduniquesolstempmom} by the moments themselves, is, from the theory of the Hamburger moment problem, known to be positive for distributions with finite support.
\end{remark}
Alternative matching operators can be defined by measuring the difference between random variables differently, for instance using the Kullback-Leibler divergence (relative entropy) \cite{KullbackLeibler51}. For some choices of the macroscopic state variables $\bs{U}$ one
could also consider classical moment matching, as described in
\cite{caflisch98mca}.
\begin{remark}[Matching for the FENE dumbbells] For the FENE dumbbells, an accept-reject strategy is applied during the combined evolution and matching, i.\,e., if the state of a polymer would become unphysical during the matching, we reject the trial move for that specific polymer in the evolution step and repeat the time step for this polymer, after which the matching of the ensemble is tried again.
\end{remark}
\subsection{Particular results for normal distributions\label{sec:prop-proj-result}}
For simplicity of the argument, and without loss of generality, we restrict ourselves to a one-dimensional notation for the rest of this section.
In the case of scalar normally distributed random variables, the following result can be proved.
\begin{lem}\label{lem:normproject}
Consider a scalar random variable $Z$ and suppose that the macroscopic state variables are given by $u_1(z)=z$ and $u_2(z)=(z-U_1)^2$. Suppose further that in analogy to \eqref{eq:AnsatzVertBerech}, $\overline{\mathcal{P}}$ is given by
\begin{equation}\label{eq:AnsatzVertBerechideal}
\overline{\mathcal{P}}(Z,\bs{U}^*)=\argmin_{\overline{\mathcal{R}}(Z^*)=\bs{U}^*}\frac{1}{2}\operatorname{E}\left((Z^*-Z)^2\right).
\end{equation}
Then, the random variable $\overline{\mathcal{P}}(Z,\bs{U}^*)$ is also normally distributed with mean and variance given by $\bs{U}^*$.
\end{lem}
\begin{proof}Denote by
\[
\begin{pmatrix}
\mu\\\sigma^2
\end{pmatrix}
=
\overline{\mathcal{R}}(Z),\quad
\begin{pmatrix}
\mu^*\\(\sigma^*)^2
\end{pmatrix}
=
\bs{U}^*,\quad\sigma,\sigma^*>0.
\]
Then, \eqref{eq:AnsatzVertBerechideal} yields
\begin{equation}\label{eq:Ynpsend}
\overline{\mathcal{P}}(Z,\bs{U}^*)=\pm\sqrt{\frac{(\sigma^*)^2}{\sigma^2}}\left(Z-\mu\pm \mu^*\sqrt{\frac{\sigma^2}{(\sigma^*)^2}}\right).
\end{equation}
Thus, if $Z$ is normally distributed, so is $\overline{\mathcal{P}}(Z,\bs{U}^*)$.
\end{proof}
With the help of this lemma, we can easily show the following two corollaries.
\begin{cor}
Suppose that the hierarchy of macroscopic state variables is defined using
$u_1(z)=z$ and $u_l(z)=(z-U_1)^l$ for $l\geq2$. Suppose further that $\overline{\mathcal{P}}$ is given by
\eqref{eq:AnsatzVertBerechideal}.
Then, the sequence of matching operators is consistent with $C_L=0$ for $L\geq2=L_0$ for all sequences of triples $\left(Z_{[L]}^+,Z_{[L]}^-,\bs{U}_{[L]}^*\right)_{L=1,2,\dots}$, where $Z_{[L]}^+$ and $Z_{[L]}^-$ are normally distributed and $\left(\bs{U}_{[L]}^*\right)_{L=1,2,\dots}$ are sequences of centralized moment values consistent with normal distributions.
\end{cor}
\begin{proof}
We first consider the case $L=2$.
As the normal distribution is uniquely determined by its first two (centralized) moments, \cref{lem:normproject} implies that for two normally distributed random variables $Z_1$ and $Z_2$, $\overline{\mathcal{P}}_{[2]}(Z_1,\bs{U}_{[2]}^*)$ and $\overline{\mathcal{P}}_{[2]}(Z_2,\bs{U}_{[2]}^*)$ are identically distributed, and thus $C_2=0$. For the same reason, \eqref{eq:Ynpsend} holds also for $L>2$, and also in this case $C_L=0$, and the matching is consistent.
\end{proof}
\begin{cor}
Suppose that the hierarchy of macroscopic state variables is defined using
$u_1(z)=z$ and $u_l(z)=(z-U_1)^l$ for $l\geq2$. Suppose further that $\overline{\mathcal{P}}$ is given by
\eqref{eq:AnsatzVertBerechideal}.
Then, the sequence of matching operators is continuous for all normally distributed random variables, and sequences of centralized moment values consistent with normal distributions.
\end{cor}
\begin{proof}
For $L\geq2$, \cref{lem:normproject} implies again that $Z^*=\overline{\mathcal{P}}_{[L]}(Z,\bs{U}^*_{[L]})$ and $Z^+=\overline{\mathcal{P}}_{[L]}(Z,\bs{U}^+_{[L]})$ are normally distributed if $Z$ is normally distributed and $\bs{U}^*_{[L]}$, $\bs{U}^+_{[L]}$ are sequences of centralized moment values consistent with normal distributions. Denoting the corresponding expectations by $\mu^*$, resp.~$\mu^+$, and variances by
$(\sigma^*)^2$, resp.~$(\sigma^+)^2$,
it holds for all $f \in C_P^{1}(\mathbb{R}, \mathbb{R})$
\begin{align*}
|\operatorname{E} f(s,Z^*)-\operatorname{E} f(s,Z^+)|
=&\left|\int_{\mathbb{R}}\big(f(s,\sigma^* z+\mu^*)-f(s,\sigma^+ z+\mu^+)\big)\frac1{\sqrt{2\pi}}e^{-z^2/2}~dz\right|\\
=&\left|\int_{\mathbb{R}}f'(s,\xi_z)[(\sigma^*-\sigma^+)z+\mu^*-\mu^+]\frac{1}{\sqrt{2\pi}}e^{-z^2/2}~dz\right|,
\end{align*}
where $\xi_z\in[\sigma^* z+\mu^*,\sigma^+ z+\mu^+]$. As there exist constants $\tilde{C}$ and $r \in \mathbb{N}$ such that $|f'(s,\xi_z)|\leq\tilde{C}(1+|\max\{\sigma^*,\sigma^+\}z+\max\{\mu^*,\mu^+\}|^{2r})$, this implies also that the sequence of matching operators is continuous (the corresponding \cref{eq:continuity} can be verified similarly in the case $L=1$).
\end{proof}
Several observations can be made. First, one can obtain a similar result whenever the distributions are defined by a finite number of moments (for instance, a lognormal distribution) by taking this knowledge into account when defining the matching operator.
Second, we remark that, if the random variables are normally distributed at all moments in time, this implies that they represent solutions of a linear SDE
in the narrow sense,
\begin{equation}\label{eq:linSDE}
dX(t)=\big(a_1(t)X(t)+a_2(t)\big)~dt+b(t)\star dW(t),\quad t\inI,
\end{equation}
with normally distributed initial values. For this equation, it is clear that the complete time evolution of the distributions can be completely described by a system of two ODEs for $U_1=\mu=\operatorname{E} X$ and $U_2=\sigma^2=\operatorname{E}\left(\left(X-\mu\right)^2\right)$, namely
\begin{equation}\label{eq:linear-closed}
\begin{cases}
dU_1/dt &= a_1(t)\; U_1 + a_2(t),\\
dU_2/dt &= 2a_1(t)\; U_2 + b(t)^2.
\end{cases}
\end{equation}
As a consequence, the matching operator \eqref{eq:AnsatzVertBerechideal} corresponds to the reconstruction of the normal distribution corresponding to the given macroscopic state.
\subsection{Conjecture for general distributions\label{sec:prop-proj-conj}}
We now turn to more general distributions. We assume that the distribution of the random variable is uniquely determined by its moments; this is the case if, for instance, the moment generating function $\sum_{i=0}^\infty\frac{\operatorname{E}(Z^i)t^i}{i!}$ is bounded in an interval around $0$.
In this setting, we propose the following conjecture:
\begin{conjecture}
Consider a sequence of restriction operators $\left({\overline \mathcal{R}_{[L]}}\right)_{L=1,2,\dots}$ in which the macroscopic state variables corresponding to $\overline \mathcal{R}_{[L]}$ are defined as the first $L$ centralized moments of the distribution, i.\,e., $u_1(z)=z$, $u_l(z)=(z-U_1)^l$, for $l=2,\ldots, L$, and define the corresponding sequence of matching operators $\left(\overline{\mathcal{P}}_{[L]}\right)_{L=1,2,\dots}$ via \eqref{eq:AnsatzVertBerechideal}.
Consider further a set $S$ of random variables for which all (centralized) moments of its members exist and uniquely determine the corresponding distribution function, and each moment can be uniformly bounded. Then, the sequence of matching operators is continuous for $S$ and all sequences of macroscopic states $\bs{U}_{[L]}$, and consistent for all sequences of triples $\left(Z_{[L]}^*,Z_{[L]}^+,\bs{U}_{[L]}\right)_{L=1,2,\dots}$ where $Z_{[L]}^*,Z_{[L]}^+\in S$.
\end{conjecture}
We illustrate the main properties of the matching operator for general distributions by means of numerical experiments. Below, we consider \cref{eq:fene-3d} in one space dimension, with $\kappa(t)\equiv 2$, $F(X)$ the FENE force \eqref{eq:springs} with $\gamma=49$, $\textrm{We}=1$. We discretize in time with the classical Euler-Maruyama scheme with time step $\delta t=2\cdot 10^{-4}$, and simulate $J=1\cdot 10^5$ realizations, whose initial state at time $t^0=0$ is taken from the invariant distribution of \cref{eq:fene-3d} for $\kappa(t)\equiv 0$.
As the macroscopic state $\bs{U}_{[L]}$, we consider the first $L$ \emph{even} centralized moments,
since for the exact solution, the odd (centralized) moments vanish due to symmetry.
\subsubsection{Error dependence on the number of moments}
We first simulate up to time $t^*=1.15$ and record the microscopic state $\bs{\mcY}^*$ and corresponding macroscopic states $\bs{U}^*_{[L]}$ for $L=1,\ldots,10$, at time $t^*$, as well as the microscopic state $\bs{\mcY}^{-}$ at time $t^-=1$. We then project the ensemble $\bs{\mcY}^{-}$ onto the macroscopic state $\bs{U}^*_{[L]}$ and compare the density of $\mathcal{P}_{[L]}(\bs{\mcY}^{-},\bs{U}^*_{[L]})$ with that of $\bs{\mcY}^*$. Since the absolute value of the moments increases quickly with the order of the moment, the residuals in the Newton procedure for the matching are scaled relative to the requested value of the corresponding moment; the Newton iterations are stopped if the norm of the residual is smaller than $1\cdot 10^{-9} $.
We perform three tests to examine the convergence (in empirical density) of $\mathcal{P}_{[L]}(\bs{\mcY}^{-},\bs{U}^*_{[L]})$ to $\bs{\mcY}^*$ for $L\to\infty$. First, we visually inspect the corresponding empirical probability density functions, see \cref{fig:conjecture-distr-1left}.
\newlength{\figwidth}
\setlength{\figwidth}{0.485 \textwidth}
\begin{figure}[tbp]\hspace*{\fill}
\subfigure[Empirical probability density functions\label{fig:conjecture-distr-1left}]{\includegraphics[scale=0.9]{fene-distr-paper}}
\subfigure[Error of the $l$-th moment as a function of $l$\label{fig:conjecture-distr-1right}]{\includegraphics[scale=0.9]{fene-error-moments}}
\hspace*{\fill}
\caption{\label{fig:conjecture-distr-1}Results after projecting a prior ensemble of FENE dumbbells onto the first $L$ even centralized moments of a reference ensemble for several values of $L$. Simulation details are given in the text.}
\hspace*{\fill}
\end{figure}
Shown are histogram approximations of the empirical density $\hat{\varphi}^{-}$ of $|\bs{\mcY}^-|$ (the initial condition for the matching), the reference empirical density $\hat{\varphi}^*$ of $|\bs{\mcY}^*|$, and approximations $\hat{\varphi}_{[L]}$ of $\mathcal{P}_{[L]}(\bs{\mcY}^-,\bs{U}_{[L]})$ for several values of $L$. The figure visually suggests that, when increasing the number of macroscopic state variables, the reference empirical density gets approximated more accurately. We now take a closer look to the projected ensembles by computing the relative difference between the $l$-th even empirical moment of the projected ensemble, $U_l$, and the corresponding empirical moment of the reference ensemble $U^*_l$ as $(U_l-U^*_l)/U^*_l$. \Cref{fig:conjecture-distr-1right} shows this error as a function of $l$ for different values of the number of macroscopic state variables $L$.
\begin{table}
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|}
\hline
$L$ &2 &3 &4 & 5 & 6 & 7 & 8 & 9 & 10 \\\hline
$p$ & $0.000$ & $0.000$ & $1.197\cdot 10^{-3}$ & $0.840$ & $0.862$ & $0.999$ & $0.999$ & $0.999$ & $0.999$ \\
\hline
\end{tabular}
\caption{\label{fig:conjecture-distr-2}The p-values of a two-sample Kolmogorov--Smirnov test that compares the reference and projected empirical distributions. Simulation details are in the text.}
\end{table}
We make two key observations. First, for $l<L$, the relative difference in the corresponding moment is of the order of the tolerance of the Newton procedure. This is expected, since these are the macroscopic state variables onto which the distribution is projected. We see that this very small error increases nevertheless with $l$; this can be explained by pointing out that the value of the moments increases very quickly with $l$, and that the equations in the Newton procedure have been rescaled accordingly.
Second, the error in the higher moments ($l>L$) also decreases with increasing $L$. This indicates that the convergence of $\hat{\varphi}_{[L]}$ to $\hat{\varphi}^{*}$ for $L\to \infty$ is not only due to the fact that we project onto more moments, but also because the approximation of the higher order moments improves. So far, we have no complete theoretical justification for this observation.
Finally, we compare the reference and projected empirical distributions using a two-sample Kolmogorov--Smirnov test \cite{Lehn85eid}; this classical hypothesis test results in a high p-value ($\leq1$) if the two samples are likely to have been drawn from the same probability distribution. The results are shown in \cref{fig:conjecture-distr-2}. We clearly see a $p$-value that approaches $1$ for increasing $L$.
\subsubsection{Error dependence on the time step}
In a next experiment, we simulate up to time $t^*=1.54$, and record the microscopic state $\bs{\mcY}^{-}$ at time $t^-=1.5$, as well as the macroscopic states $\bs{U}_{[L]}(t^-+\Delta t)$ for $L=3,4,5$, and the function of interest $\tilde{\tau}_p(t^-+\Delta t)$ for $\Delta t \in [0,t^*-t^-]$. We then project the ensemble $\bs{\mcY}^{-}$ onto the macroscopic state $\bs{U}_{[L]}(t^-+\Delta t)$, obtaining $\mathcal{P}_{[L]}\big(\bs{\mcY}^{-},\bs{U}_{[L]}(t^-+\Delta t)\big)$, and denote the corresponding value of the stress as $\hat{\tau}_p(t^-+\Delta t)$. We record the relative error $|\hat{\tau}_p(t)-\tilde{\tau}_p(t)|/\tilde{\tau}_p(t)$ as a function of $\Delta t$.
To reduce the statistical error, we report the averaged results of $100$ realizations of this experiment. The results are shown in \cref{fig:projection-Dt}, where we used $\epsilon=1$ in \eqref{eq:kramers}.
\begin{figure}
\begin{center}
\includegraphics[scale=0.8]{fig-projection-Dt}
\end{center}
\caption{\label{fig:projection-Dt}Error of the stress after projecting a prior ensemble of FENE dumbbells onto the first $L$ even centralized moments of a reference ensemble, as a function of $\Delta t$ for several values of $L$. Simulation details are given in the text.}
\end{figure}
We indeed see the linear increase of the matching error as a function of $\Delta t$; notice also, as was shown in the previous experiment, that the matching error decreases with increasing $L$. From the figure, we conclude that the remaining statistical error is at least lower than $10^{-5}$.
\section{Extrapolation operator\label{subsec:accel-extrap}}
Next, we need to specify how the extrapolation is performed. Let the order of consistency be defined as follows:
\begin{defi}[Consistency of extrapolation]
Consider a certain class of sufficiently smooth functions.
An extrapolation operator ${\mathcal E}$ is called consistent of order $p_e>0$ for this class if there exist $\Delta t_0>0$ and $C$ such that for all $\Delta t\in[0,\Delta t_0]$, all $n\leqN$ and all functions $U$ in the considered class it holds
\begin{equation}\label{eq:ConsistencyExtrap}
\left\| \tilde{U}^{n+1}-U(t^{n+1})\right\|\le C \Delta t^{p_e+1},
\end{equation}
with
\[
\tilde{U}^{n+1} = {\mathcal E}\left(\left(U(t^{i,k}) \right)_{i,k=0,0}^{n,K};(\Delta t_i)_{i=0}^n, \delta t \right).
\]
\end{defi}
Further, we will also use the following definition of continuity:
\begin{defi}[Continuity of extrapolation]\label{def:continuityextrapolation}
Consider a certain class of sufficiently smooth functions.
An extrapolation operator ${\mathcal E}$ consistent of order $p_e>0$ for this class is called continuous for this class, if there exist $\Delta t_0>0$ and $C$ such that for all $\Delta t\in[0,\Delta t_0]$, all $n\leqN$, and all functions $U_1,U_2$ in the considered class it holds
\begin{equation}\label{eq:ContinuityExtrap}
\begin{split}
\|{\mathcal E}\left(\left(U_1(t^{i,k}) \right)_{i,k=0,0}^{n,K};(\Delta t_i)_{i=0}^n, \delta t \right)
-{\mathcal E}\left(\left(U_2(t^{i,k}) \right)_{i,k=0,0}^{n,K};(\Delta t_i)_{i=0}^n, \delta t \right)\|\\
\leq C\left(\frac{\Delta t}{\delta t}\right)^{p_e}\sum_{i,k=0,0}^{n,K}\|U_1(t^{i,k})-U_2(t^{i,k})\|.
\end{split}
\end{equation}
\end{defi}
In the remainder of this section, we consider two extrapolation strategies: projective extrapolation (\Cref{sec:proj-extrap}) and multistep state extrapolation (\Cref{sec:msem-extrap}). Numerical illustrations are given in \Cref{sec:num-extrap}.
\subsection{Projective extrapolation\label{sec:proj-extrap}}
The first approach that we consider, proposed in \cite{Gear:2003p2171}, is to extrapolate $\bs{U}$ to time $t^{n+1}$ using only (some of) the time points $t^{n,k}$, $k=0,\dots,K$, i.\,e., by using the sequence of points obtained in the last burst of microscopic simulation. If this \emph{coarse projective extrapolation} is based on the interpolating polynomial of degree $p_e$ (with $p_e\leq K$) through the parameter values at times $t^{{n},k}$, $k=K-p_e,\dots,K$, we obtain
\begin{align}\label{eq:PIextrap}
\bs{U}^{{n}+1}=\sum_{s=0}^{p_e} l_s(\alpha_{n})\bs{U}^{{n},K-s},
\end{align}
with \begin{equation}\label{eq:alpha}
\alpha_{n}=\frac{\Delta t_{n}}{\delta t}-K,
\end{equation} and the Lagrange polynomials
\begin{equation}\label{eq:lsm}
l_s(\alpha)=\frac{\alpha(\alpha+1)\cdots(\alpha+{p_e})}{s!({p_e}-s)!(-1)^s(\alpha+s)}.
\end{equation}
Clearly, \eqref{eq:PIextrap} is continuous in the sense of \cref{def:continuityextrapolation}.
\begin{example}[Coarse projective forward Euler]\label{ex:coarseprojectiveforwardEuler}
The simplest, first order version of the above method is called coarse projective forward Euler. In this case, the procedure can be rewritten as
\begin{equation}\label{eq:cpfe}
\bs{U}^{{n}+1} = \bs{U}^{{n},K}+(\Delta t_{n} - K\delta t)\bs{\overline{\mathcal{H}}}^{n}, \qquad\bs{\overline{\mathcal{H}}}^{n} = \frac{\bs{U}^{{n},K}-\bs{U}^{{n},K-1}}{\delta t}.
\end{equation}
\end{example}
The procedure described above is reminiscent of a Taylor method \cite{Gear:2003p2171}.
Consequently, time integration based on this extrapolation will resemble a Taylor method when repeatedly extrapolating forward in time, and the global deterministic error will be dominated by a term of the form $C\Delta t^{p_e}$ (assuming $\delta t \ll \Delta t$), as results from an accuracy analysis of coarse projective integration for deterministic microscopic models \cite{Vandekerckhove:2008p891}.
To assess qualitatively the influence of coarse projective extrapolation on the statistical error, we apply it to the linear test equation
\begin{equation}\label{eq:staterrorlinSDE}
dX(t)=a X(t)~dt+b dW(t).
\end{equation}
Application of the one step method $\phi$ to \eqref{eq:staterrorlinSDE} yields
\begin{align*}
Y^{{n},k}=&\tilde{R}_{\os}(a,\delta t,\eta^{{n},k-1})Y^{{n},k-1}+\tilde{S}_{\os}(a,b,\delta t,\eta^{{n},k-1}),
\end{align*}
where $\tilde{R}_{\os}$ and $\tilde{S}_{\os}$ are functions depending on $\phi$ and $\eta^{{n}+1,k-1}$ are (vectors of) the i.\,i.\,d.\ random variables used by $\phi$.
In the following, we assume that $\tilde{R}_{\os}(a,\delta t,\eta^{{n},i})$ is independent of $\eta^{{n},i}$ and can be written as $\tilde{R}_{\os}(a,\delta t,\eta^{{n},i})={R}_{\os}(a\delta t)$; this holds, e.\,g., for typical Runge-Kutta methods.
The above assumptions imply
\begin{align*}
Y^{{n},k}=&{R}_{\os}(a\delta t)^kY^{{n},0}+\sum_{i=0}^{k-1}\tilde{S}_{\os}(a,b,\delta t,\eta^{{n},i}){R}_{\os}(a\delta t)^{k-i-1},\\
\operatorname{E} Y^{{n},k}=&{R}_{\os}(a\delta t)^k\operatorname{E} Y^{{n},0}+\operatorname{E}\tilde{S}_{\os}(a,b,\delta t,\eta^{{n},0})\sum_{i=0}^{k-1}{R}_{\os}(a\delta t)^{i}.
\end{align*}
If we now apply the extrapolation step \eqref{eq:PIextrap}, we obtain
\begin{align}\nonumber
&\operatorname{E} Y^{{n}+1,0}=\sum_{s=0}^{p_e}l_s(\alpha)\operatorname{E} Y^{{n},K-s}\\\label{eq:staterrorlinsdeonesteppiassump}
=&\underbrace{\left(\sum_{s=0}^{p_e}l_s(\alpha){R}_{\os}(a\delta t)^{K-s}\right)}_{=:R
_{\operatorname{E}}(a\delta t)}\operatorname{E} Y^{{n},0}+\operatorname{E}\tilde{S}_{\os}(a,b,\delta t,\eta^{{n},0})\sum_{s=0}^{p_e}l_s(\alpha)\sum_{i=0}^{K-s-1}{R}_{\os}(a\delta t)^{i},
\end{align}
with $\alpha$ and $l_s(\alpha)$ given by \eqref{eq:alpha} and \eqref{eq:lsm}.
In analogy to \eqref{eq:staterrorlinsdeonesteppiassump} we obtain also
\begin{align*}
\widehat{\E} Y^{{n}+1,0}=R_{\operatorname{E}}(a\delta t)\widehat{\E} Y^{{n},0}+\sum_{s=0}^{p_e}l_s(\alpha)\sum_{i=0}^{K-s-1}{R}_{\os}(a\delta t)^{i}\widehat{\E}\tilde{S}_{\os}(a,b,\delta t,\eta^{{n},i}).
\end{align*}
Thus $\operatorname{E}\widehat{\E} Y^{{n}+1,0}=\operatorname{E} Y^{{n}+1,0}$, but
\begin{align}\nonumber
\operatorname{Var}\widehat{\E} Y^{{n}+1,0}=&\frac1{J}R_{\operatorname{E}}(a\delta t)^2\operatorname{Var} Y^{{n},0}\\\label{eq:VarhECPI}
&+\frac1{J}\operatorname{Var}\tilde{S}_{\os}(a,b,\delta t,\eta^{{n},0})\sum_{i=0}^{K-1}{R}_{\os}(a\delta t)^{2i}\left(\sum_{s=0}^{\min\{p_e,K-1-i\}}l_s(\alpha)\right)^2.
\end{align}
The second summand in \eqref{eq:VarhECPI} behaves as $\delta t \alpha^{2p_e}/J$ for large $\alpha=\Delta t/\delta t - K$. Assuming $\Delta t \gg \delta t$, this results in an amplification of the statistical error with a factor
$\Delta t^{p_e}/\delta t^{p_e}$ during extrapolation. A natural question is then: how many realizations $\tilde{J}$ are needed to obtain the same variance using a fully microscopic simulation? In that case, we have
\begin{align}\nonumber
\operatorname{Var}\widehat{\E} Y^{{n},\alpha+K}=&\frac1{\tildeJ}{R}_{\os}(a\delta t)^{2(\alpha+K)}\operatorname{Var} Y^{{n},0}\\\label{eq:VarhECPIfull}
&+\frac1{\tildeJ}\operatorname{Var}\tilde{S}_{\os}(a,b,\delta t,\eta^{{n},0})\sum_{i=0}^{\alpha+K-1}{R}_{\os}(a\delta t)^{2i}.
\end{align}
Thus, for large $\alpha$, the required number of realizations for a full microscopic simulation is smaller by a factor $1/\alpha^{2p_e-1}$, i.e, $\tilde{J}\sim\frac{J}{\alpha^{2p_e-1}}$, whereas the computational costs per realization increases by a factor $\alpha$. This means that for large $\alpha$ and $p_e=1$, the computational cost of the micro/macro acceleration technique with coarse projective extrapolation is similarly to that of a full microscopic simulation for a given variance. For large $\alpha$ and $p_e>1$, coarse projective extrapolation is even more expensive than a full microscopic simulation.
To reduce statistical error, it has been proposed to use a chord based approximation, for instance, using $\bs{U}^{{n},K-K_1}$ for the time derivative estimate instead of $\bs{U}^{{n},K-1}$ in equation~\eqref{eq:cpfe} \cite{RicoGearKevr04}. Instead of taking Lagrange polynomials in equation~\eqref{eq:PIextrap}, we then have
\begin{equation}\bs{U}^{n+1}=\sum_{s=0}^{K}l_s(\alpha_n)\bs{U}^{n,K-s},
\end{equation}
in which $l_{K}(\alpha)=1+\frac{\alpha}{K-K_1}$, $l_{K_1}(\alpha)=-\frac{\alpha}{K-K_1}$, and $l_s(\alpha)=0$ otherwise (see \cref{ex:coarseprojectiveforwardEuler}) reduces the variance by a factor $1/(K-K_1)$. However, the conclusion on the computational cost remains the same.
\subsection{Multistep state extrapolation\label{sec:msem-extrap}}
Because of the amplification of statistical error, we look into alternative extrapolation strategies. One approach, proposed in \cite{SOMMEIJER:1990p2657,vandekerckhove07nsa}, is to extrapolate $\bs{U}$ to time $t^{{n}+1}$ using only (some of) the time points $t^{i,K}$, $i=1,\dots,{n}$, i.\,e., by using \emph{the last point of each sequence of microscopic simulations}, instead of a sequence of points from the last microscopic simulation. If this \emph{multistep state extrapolation method} is based on the interpolating polynomial of degree ${p_e}$ through the parameter values at times $t^{i,K}$, $i={n}-{p_e},\dots,{n}$, and we assume equidistant coarse time steps $\Delta t$, we obtain
\begin{align}\label{eq:MSextrap}
\bs{U}^{{n}+1}=\sum_{s=0}^{p_e}l_s(\beta)\bs{U}^{{n}-s,K},
\end{align}
where
\begin{equation}\label{eq:beta}
\beta=\frac{\alpha}{\alpha+K}
\end{equation} is the fraction of the interval $\Delta t$ over which we extrapolate, and $\alpha$ and $l_s$ are defined as in \eqref{eq:alpha} and \eqref{eq:lsm}. Note that such an extrapolation strategy requires a separate starting procedure.
For a detailed comparison of the accuracy and stability properties of acceleration of the numerical integration of ODEs using projective extrapolation and multistep state extrapolation, we refer to \cite{Vandekerckhove:2008p891,vandekerckhove07nsa}.
Here, we only remark that, while the local error using multistep state extrapolation only differs by a factor two with respect to the local error of projective extrapolation, the global error is affected quite significantly. This is due to the fact that, when increasing $\Delta t$, also the time derivative estimate itself is taken over a larger time interval. It has been shown in \cite{SOMMEIJER:1990p2657,Vandekerckhove:2008p891} that this results in an error constant (see, e.\,g., \cite[Section III.2]{HairerWanner}) of the form $C\alpha\Delta t^{p_e}$. This amplification effect will be illustrated in \cref{sec:num-extrap}.
Let us now look into the statistical error, again using the linear test equation \eqref{eq:staterrorlinSDE}. One extrapolation step of coarse multistep state extrapolation yields
\begin{align*}
\widehat{\E} Y^{{n}+1,0}=&\sum_{s=0}^{p_e}l_s(\beta)\widehat{\E} Y^{{n}-s,K}.
\end{align*}
Thus again $\operatorname{E}\widehat{\E} Y^{{n}+1,0}=\operatorname{E} Y^{{n}+1,0}$, but now
\begin{align*}
\operatorname{Var}\widehat{\E} Y^{{n}+1,0}\leq\frac1{J}\max_{s=0}^{p_e}\operatorname{Var} Y^{{n}-s,K}\left(\sum_{s=0}^{p_e}|l_s(\beta)|\right)^2.
\end{align*}
As $\beta<1$, the last factor can be bounded (independently of $\alpha$). Consequently, the amplification of statistical error during the extrapolation does not depend on $\alpha$, whereas the corresponding computational costs per simulation path are reduced by a factor $\alpha$ compared to a full microscopic simulation.
\subsection{Numerical illustration\label{sec:num-extrap}}
We now provide a numerical result to illustrate the effects of extrapolation on the deterministic and statistical error. To avoid effects of the matching step, we consider the linear equation \eqref{eq:linSDE} with $a_2(t)=-a_1(t)=b(t)\equiv 1$, for which we know that macroscopic evolution closes in terms of the first two moments of the distribution. This microscopic SDE is discretized using an Euler-Maruyama scheme with $\delta t=2\cdot 10^{-4}$. We consider $500$ realizations of a computational experiment with $J=1000$ SDE realizations. As an initial condition, we sample from a standard normal distribution. We compare the sample mean behavior and sample standard deviation of a full microscopic simulation (which we will call the reference simulation) with the micro/macro acceleration algorithm using $\Delta t=1\cdot 10^{-3}$, $2\cdot 10^{-3}$, $4\cdot 10^{-3}$, and $8\cdot 10^{-3}$. The function of interest is chosen to be $\bar{f}(t)=\operatorname{E}(X(t)^2)$. We denote by $\tilde{f}(t)$ the approximation to the function of interest calculated from one realization of the reference simulation using $J$ SDE realizations, and by $\hat{f}(t)$ the function of interest obtained via one realization of the micro/macro acceleration technique. As extrapolation techniques, we use first order projective extrapolation and first and second order multistep state extrapolation.
\Cref{fig:stat-lin-proj} shows the results for first order projective extrapolation.
\begin{figure}
\begin{center}
\includegraphics[width=\linewidth]{statistical-linear-proj}
\end{center}
\caption{\label{fig:stat-lin-proj}Results of micro/macro acceleration of the linear equation \eqref{eq:linSDE} using $L=2$ moments and projective extrapolation for different values of the time step $\Delta t$, as well as a full microscopic (reference) simulation. Top left: evolution of the sample means of the function of interest $\tilde{f}$ and $\hat{f}$. Bottom left: deterministic error on $\hat{f}$. Right: evolution of the sample standard deviation of $\hat{f}$. Simulation details are given in the text.}
\end{figure}
The left figure clearly shows, as expected, that the deterministic error grows with increasing $\Delta t$. However, from the right figure follows that, for an individual realization of the experiment, the error is dominated by the statistical error. The zoom shows that, for small $t$, the sample standard deviation grows linearly as a function of time, with a slope that is larger for larger $\Delta t$. This is in agreement with the theoretical result on the local propagation of statistical error.
Next, we look at first order multistep state extrapolation, for which the results are shown in \cref{fig:stat-lin-msem-1}.
\begin{figure}
\begin{center}
\includegraphics[width=\linewidth]{statistical-linear-msem-order1}
\end{center}
\caption{\label{fig:stat-lin-msem-1}Results of micro/macro acceleration of the linear equation \eqref{eq:linSDE} using $L=2$ moments and first order multistep state extrapolation for different values of the time step $\Delta t$, as well as a full microscopic (reference) simulation. Top left: evolution of the sample means of the stresses $\tilde{f}$ and $\hat{f}$. Bottom left: deterministic error on $\hat{f}$. Right: evolution of the sample standard deviation of $\hat{f}$. Simulation details are given in the text.}
\end{figure}
The left figure indicates that, when comparing with projective extrapolation, the deterministic error grows much more rapidly with increasing $\Delta t$. On the right, we see that, while the sample standard deviation is larger than for the reference simulation, the sample standard deviation does not depend crucially on $\Delta t$, as is also expected from the analysis of the local propagation of statistical error. Note that the lower sample standard deviation for $\Delta t=8\cdot 10^{-3}$ is related to the fact that $\hat{f}(t)$ itself is much lower as a consequence of the large deterministic error (see left figure). The zoom shows that, for small $t$, the statistical error grows linearly as a function of time, with a slope that is independent of $\Delta t$, and is identical to the slope for the reference simulation. These results are in agreement with the theoretical result on the local propagation of statistical error.
Finally, we consider second order multistep state extrapolation. The results are shown in \cref{fig:stat-lin-msem-2}.
\begin{figure}
\begin{center}
\includegraphics[width=\linewidth]{statistical-linear-msem-order2}
\end{center}
\caption{\label{fig:stat-lin-msem-2}Results of micro/macro acceleration of the linear equation \eqref{eq:linSDE} using $L=2$ moments and second order multistep state extrapolation for different values of the time step $\Delta t$, as well as a reference fully microscopic simulation. Top left: evolution of the sample means of the function of interest $\tilde{f}$ and $\hat{f}$. Bottom left: deterministic error on $\hat{f}$. Right: evolution of the sample standard deviation of $\hat{f}$. Simulation details are given in the text.}
\end{figure}
The left figure shows that the deterministic error is much better than for the first order version. However, the behavior of the statistical error is more intriguing. When zooming in to the behavior for small $t$, we observe that, over a short time interval, the sample standard deviation using the micro/macro acceleration technique increases at the same rate as the sample standard deviation in the reference simulation. This corresponds to the theoretical result on the local propagation of statistical error. However, on longer time scales, the sample standard deviation for large $t$ grows rapidly, and seems to be larger for larger $\Delta t$. As such, second order multistep state extrapolation behaves similarly to first order projective integration on long time scales. We suspect that the loss of this favorable error propagation is due to accumulation effects. This is indicated by the fact that the length of the time interval on which the local theoretical results is observed is longer for larger values of $\Delta t$. Indeed, the figure indicates that the effects of accumulated statistical errors start to appear \emph{after a given number of extrapolations}, independently of the size of the extrapolation step. This behavior requires additional analysis.
\section{Convergence results\label{sec:conv}}
Using the above, we are now ready to give a definition of convergence for the proposed algorithm:
\begin{defi}
Consider a sequence of restriction operators $\left(\overline{\mathcal{R}}_{[L]}\right)_{L=1,2,\dots}$ and a sequence of matching operators $\left(\overline{\mathcal{P}}_{[L]}\right)_{L=1,2,\dots}$, and denote the corresponding numerical approximation process obtained by using $\overline{\mathcal{R}}_{[L]}$ and $\overline{\mathcal{P}}_{[L]}$ in \Cref{algo:accel} by $\bs{Y}_{[L]}$, and the maximum step size by $\Delta t$, $\Delta t=\max_{{n}=1}^{N}\Delta t_{n}$. The accelerated micro/macro Monte Carlo simulation is then called weakly convergent to the solution $\bs{X}$ of SDE \eqref{eq:SDE} as $\Delta t \rightarrow 0$ and $L\to\infty$ at any time $t \in {\I^{\Dt}}$ with time order $p$ if for each $f \in
C_P^{2(p+1)}(\mathbb{R}^d, \mathbb{R})$ there exist constants $\Delta t_0>0$, $L_0$, $C_L$, and $\tilde{C}_L$, with $C_L\to0$ for $L\to\infty$, such that
\begin{equation}
| \operatorname{E} f\big(\bs{Y}_{[L]}(t)\big) - \operatorname{E} f\big(\bs{X}(t)\big) |\leq C_L+\tilde{C}_L(\Delta t)^{p}
\end{equation}
holds for all $t\in\I^{\Dt}$, all $L\geq L_0$, and all $\Delta t\in[0,\Delta t_0]$.
\end{defi}
We first discuss convergence when extrapolation is performed as in coarse projective integration (\cref{sec:conv-cpi}). Due to the multistep nature of the extrapolation, proving convergence for the multistep state extrapolation method is more involved; \cref{sec:conv-msem} contains a result for a linear SDE.
\subsection{Convergence using projective extrapolation\label{sec:conv-cpi}}
The following theorem generalizes the theorem for the convergence of one step methods due to Milstein (see \cite{milstein95nio,milstein04snf} or also \cite{debrabant08cwa}).
\begin{theorem}
Suppose the following conditions hold:
\begin{enumerate}[(i)]
\item \label{St-lg-cond1} The coefficient functions $\bs{a}(\bs{x})$ and $\bs{b}^i(\bs{x})$ (where $\bs{b}^i$ denotes the $i$-th column of $\bs{b}$) are continuous, satisfy a Lipschitz condition with respect to $\bs{x}$, and belong to $C_P^{p+1,2(p+1)}(I\times\mathbb{R}^d, \mathbb{R})$, $i=1,\dots,m$.
For non It\^{o} SDEs, we require in addition that $b^i$ is differentiable and that also ${b^i}'b^i$ satisfies a Lipschitz condition and belongs to $C_P^{p,2(p+1)}(I\times\mathbb{R}^d,\mathbb{R})$, $i=1,\dots,m$.
\item \label{St-lg-cond2} For sufficiently large $r$ the moments $\operatorname{E}(\|\bs{Y}_{[L]}^{n,k}\|^{2r})$ exist for $k=0,\dots,K$ and ${n}=0,1,\ldots,N$ and are uniformly bounded with respect to $L$, ${N}$.
\item SDE \eqref{eq:SDE} is uniformly weakly continuous for the set of all $\bs{Y}_{[L]}^{n,K}$.
\item \label{St-lg-cond3} The one step method $\phi$ is weakly consistent of order $p_{\os}$.
\item \sloppy The sequence of matching operators is continuous for the numerical approximation process and all sequences of macroscopic states, and consistent for all sequences of triples $\left(\bs{Y}_{[L]}^{n,K},\bs{X}^{t^{n,K},\bs{Y}_{[L]}^{n,K}}(t^{n+1,0}),\overline{\mathcal{R}}_{[L]}(\bs{Y}_{[L]}^{n+1,0})\right)_{L=1,2,\dots}$.
\item The extrapolation is consistent of order $p_e\geq1$ and continuous for the class of all functions $U(t)=\overline{\mathcal{R}}_{[L]}\big(\bs{X}^{t^{i,k},\bs{Y}_{[L]}^{i,k}}(t)\big)$.
\end{enumerate}
Then, the micro/macro acceleration algorithm with projective extrapolation is weakly convergent with time order $p=\min\{p_e,p_{\os}\}$.
\end{theorem}
\begin{proof}
Let $g(s,\bs{x}):=\operatorname{E}\Big(f\big(\bs{X}(t^{{n}+1})\big)|\bs{X}(s)=\bs{x}\Big)$ for $s\inI$, $\bs{x}\in\mathbb{R}^d$, and $t^{{n}+1}\in\I^{\Dt}$ with $s \leq t^{{n}+1}$. Due to condition~(\ref{St-lg-cond1}) $g\in C_P^{0,2(p+1)}$ \cite{milstein95nio}. Therefore, the consistency of $\phi$ implies that $g$ satisfies
\begin{equation}\label{eq:Konsuuniform}
|\operatorname{E} g\big(s ,\bs{X}^{t,\bs{x}}(t+\delta t)\big) -\operatorname{E} g\big(s ,\phi(t,\bs{x};\delta t)\big)|\leq C_g(\bs{x}) \,\delta t^{p_{\os}+1}
\end{equation}
uniformly w.\,r.\,t.\ $s \in [t^0, t^{{n}+1}]$ for some $C_g\in C_P^{0}(\mathbb{R}^d,\mathbb{R})$.
Analogously, \cref{cor:consensemblegeneration} and the continuity of the matching imply that $g$ satisfies
\begin{align*}
&|\operatorname{E} g\big(s,\bs{X}^{t^{i,K},\bs{Y}_{[L]}^{i,K}}(t^{i+1,0})\big)-\operatorname{E} g\big(s,\bs{Y}_{[L]}^{i+1,0}\big)|
\\&
\leq|\operatorname{E} g\big(s,\bs{X}^{t^{i,K},\bs{Y}_{[L]}^{i,K}}(t^{i+1,0})\big)
-\operatorname{E} g\bigg(s,\overline{\mathcal{P}}_{[L]}\Big(\bs{Y}_{[L]}^{i,K},\overline{\mathcal{R}}_{[L]}\big(\bs{X}^{t^{i,K},\bs{Y}_{[L]}^{i,K}}(t^{i+1,0})\big)\Big)\bigg)|
\\&
+|\operatorname{E} g\bigg(s,\overline{\mathcal{P}}_{[L]}\Big(\bs{Y}_{[L]}^{i,K},\overline{\mathcal{R}}_{[L]}\big(\bs{X}^{t^{i,K},\bs{Y}_{[L]}^{i,K}}(t^{i+1,0})\big)\Big)\bigg)
-\operatorname{E} g\Big(s,\overline{\mathcal{P}}_{[L]}\big(\bs{Y}_{[L]}^{i,K},\overline{\mathcal{R}}_{[L]}(\bs{Y}_{[L]}^{i+1,0})\big)\Big)|
\\&
\leq C_L\Delta t+C_1\|\overline{\mathcal{R}}_{[L]}\big(\bs{X}^{t^{i,K},\bs{Y}_{[L]}^{i,K}}(t^{i+1,0})\big)-\overline{\mathcal{R}}_{[L]}(\bs{Y}_{[L]}^{i+1,0})\|.
\end{align*}
The last summand can be expanded as follows:
\begin{align*}
&{\|\overline{\mathcal{R}}_{[L]}\big(\bs{X}^{t^{i,K},\bs{Y}_{[L]}^{i,K}}(t^{i+1,0})\big)-\overline{\mathcal{R}}_{[L]}(\bs{Y}_{[L]}^{i+1,0})}\|
\\
\leq&
\|\overline{\mathcal{R}}_{[L]}\big(\bs{X}^{t^{i,K},\bs{Y}_{[L]}^{i,K}}(t^{i+1,0})\big)-\overline{\mathcal{R}}_{[L]}\big(\bs{X}^{t^{i,0},\bs{Y}_{[L]}^{i,0}}(t^{i+1,0})\big)\|
\\&+
\|\overline{\mathcal{R}}_{[L]}\big(\bs{X}^{t^{i,0},\bs{Y}_{[L]}^{i,0}}(t^{i+1,0})\big)-{\mathcal E}\left(\left( \overline{\mathcal{R}}_{[L]}\big(\bs{X}^{t^{i,0},\bs{Y}_{[L]}^{i,0}}(t^{i,k})\big) \right)_{k=0}^{K};\Delta t_i, \delta t \right)\|
\\&+
\|{\mathcal E}\left(\left( \overline{\mathcal{R}}_{[L]}\big(\bs{X}^{t^{i,0},\bs{Y}_{[L]}^{i,0}}(t^{i,k})\big) \right)_{k=0}^{K};\Delta t_i, \delta t \right)-{\mathcal E}\left(\left(\overline{\mathcal{R}}_{[L]}\big( \bs{Y}_{[L]}^{i,k}\big)\right)_{k=0}^{K};\Delta t_i, \delta t \right)
\|.
\end{align*}
Let $\tilde{g}_{[L]}(s,\bs{x};t):=\bigg(\operatorname{E}\Big(g_l\big(\bs{X}(t)\big)|\bs{X}(s)=\bs{x}\Big)\bigg)_{l=1}^L$ for $s\inI$, $\bs{x}\in\mathbb{R}^d$, and $t\in\I^{\Dt}$ with $s \leq t$. With this definition, the continuity and consistency of the extrapolation imply
\begin{align*}
&{\|\overline{\mathcal{R}}_{[L]}\big(\bs{X}^{t^{i,K},\bs{Y}_{[L]}^{i,K}}(t^{i+1,0})\big)-\overline{\mathcal{R}}_{[L]}(\bs{Y}_{[L]}^{i+1,0})}\|
\\
\leq&
\|\overline{\mathcal{R}}_{[L]}\big(\bs{X}^{t^{i,K},\bs{Y}_{[L]}^{i,K}}(t^{i+1,0})\big)-\overline{\mathcal{R}}_{[L]}\big(\bs{X}^{t^{i,0},\bs{Y}_{[L]}^{i,0}}(t^{i+1,0})\big)\|
+C_2(\Delta t)^{p_e+1}
\\&+C_3\left(\frac{\Delta t}\delta t\right)^{p_e}\sum_{k=1}^K\|\overline{\mathcal{R}}_{[L]}\big(\bs{X}^{t^{i,0},\bs{Y}_{[L]}^{i,0}}(t^{i,k})\big)-\overline{\mathcal{R}}_{[L]}\big( \bs{Y}_{[L]}^{i,k}\big)\|
\\
=&
\|\operatorname{E}\tilde{g}_{[L]}\big(t^{i,K},\bs{Y}_{[L]}^{i,K};t^{i+1,0}\big)-\operatorname{E}\tilde{g}_{[L]}\big(t^{i,K},\bs{X}^{t^{i,0},\bs{Y}_{[L]}^{i,0}}(t^{i,K});t^{i+1,0}\big)\|
+C_2(\Delta t)^{p_e+1}
\\&+C_3\left(\frac{\Delta t}\delta t\right)^{p_e}\sum_{k=1}^K\|\left(\operatorname{E} g_l\big(\bs{X}^{t^{i,0},\bs{Y}_{[L]}^{i,0}}(t^{i,k})\big)-\operatorname{E} g_l\big( \bs{Y}_{[L]}^{i,k}\big)\right)_{l=1}^L\|,
\end{align*}
where we made also use of $\bs{X}^{t^{i,0},\bs{Y}_{[L]}^{i,0}}(t^{i+1,0})=\bs{X}^{t^{i,K},\bs{X}^{t^{i,0},\bs{Y}_{[L]}^{i,0}}(t^{i,K})}(t^{i+1,0})$.
Due to the consistency of $\phi$, altogether we obtain
\begin{align}
|\operatorname{E} g\big(s,\bs{X}^{t^{i,K},\bs{Y}_{[L]}^{i,K}}(t^{i+1,0})\big)-\operatorname{E} g\big(s,\bs{Y}_{[L]}^{i+1,0}\big)|
\leq C_L\Delta t+\tilde{C}_L(\Delta t)^{p_e}\delta t^{p_{\os}}+\tilde{C}_L\delta t^{p_{\os}+1}\label{eq:KonsuEnsembleGen}
\end{align}
uniformly w.\,r.\,t.\ $s \in [t^0, t^{{n}+1}]$ for some constants $C_L$ and $\tilde{C}_L$ with $C_L\to0$ for $L\to\infty$.
For ease of notation, in the following we will neglect the $L$-dependency of $Y$.
Then
\begin{align*}
&{\operatorname{E} f\big(\bs{X}^{t^0,X_0}(t^{{n}+1})\big)-\operatorname{E} f\big(\bs{Y}^{{n}+1,0}\big)}\\
=&\sum_{i=0}^{{n}}\sum_{k=0}^{K-1}
\left(\operatorname{E} f\big(\bs{X}^{t^{i,k},\bs{Y}^{i,k}}(t^{{n}+1})\big)-\operatorname{E} f\big(\bs{X}^{t^{i,k+1},\bs{Y}^{i,k+1}}(t^{{n}+1})\big)\right)\\
&+\sum_{i=0}^{{n}-1}\left(\operatorname{E} f\big(\bs{X}^{t^{i,K},\bs{Y}^{i,K}}(t^{{n}+1})\big)-\operatorname{E} f\big(\bs{X}^{t^{i+1,0},\bs{Y}^{i+1,0}}(t^{{n}+1})\big)\right)
\\
&+\operatorname{E} f\big(\bs{X}^{t^{{n},K},\bs{Y}^{{n},K}}(t^{{n}+1})\big)-\operatorname{E} f\big(\bs{Y}^{{n}+1,0}\big).
\end{align*}
Making use of $\bs{X}^{t^{i,k},\bs{Y}^{i,k}}(t^{{n}+1})=\bs{X}^{t^{i,k+1},\bs{X}^{t^{i,k},\bs{Y}^{i,k}}(t^{i,k+1})}(t^{{n}+1})$ and combining the last two summands we obtain
\begin{align*}
&{\operatorname{E} f\big(\bs{X}^{t^0,X_0}(t^{{n}+1})\big)-\operatorname{E} f\big(\bs{Y}^{{n}+1,0}\big)}\\
=&\sum_{i=0}^{{n}}\sum_{k=0}^{K-1}
\left(\operatorname{E} f\big(\bs{X}^{t^{i,k+1},\bs{X}^{t^{i,k},\bs{Y}^{i,k}}(t^{i,k+1})}(t^{{n}+1})\big)-\operatorname{E} f\big(\bs{X}^{t^{i,k+1},\bs{Y}^{i,k+1}}(t^{{n}+1})\big)\right)\\
&+\sum_{i=0}^{{n}}\left(\operatorname{E} f\big(\bs{X}^{t^{i,K},\bs{Y}^{i,K}}(t^{{n}+1})\big)-\operatorname{E} f\big(\bs{X}^{t^{i+1,0},\bs{Y}^{i+1,0}}(t^{{n}+1})\big)\right).
\end{align*}
Using $\bs{X}^{t^{i,K},\bs{Y}^{i,K}}(t^{{n}+1})=\bs{X}^{t^{i+1,0},\bs{X}^{t^{i,K},\bs{Y}^{i,K}}(t^{i+1,0})}(t^{{n}+1})$ and the definition of $g$ this implies
\begin{align*}
&{\operatorname{E} f\big(\bs{X}^{t^0,X_0}(t^{{n}+1})\big)-\operatorname{E} f\big(\bs{Y}^{{n}+1,0}\big)}\\
=&\sum_{i=0}^{{n}}\sum_{k=0}^{K-1}
\left(\operatorname{E} g\big(t^{i,k+1},\bs{X}^{t^{i,k},\bs{Y}^{i,k}}(t^{i,k+1})\big)-\operatorname{E} g\big({t^{i,k+1},\bs{Y}^{i,k+1}}\big)\right)\\
&+\sum_{i=0}^{{n}}\left(\operatorname{E} f\big(\bs{X}^{t^{i+1,0},\bs{X}^{t^{i,K},\bs{Y}^{i,K}}(t^{i+1,0})}(t^{{n}+1})\big)-\operatorname{E} f\big(\bs{X}^{t^{i+1,0},\bs{Y}^{i+1,0}}(t^{{n}+1})\big)\right)
\\
=&\sum_{i=0}^{{n}}\sum_{k=0}^{K-1}
\left(\operatorname{E} g\big(t^{i,k+1},\bs{X}^{t^{i,k},\bs{Y}^{i,k}}(t^{i,k+1})\big)-\operatorname{E} g\big({t^{i,k+1},\bs{Y}^{i,k+1}})\big)\right)\\
&+\sum_{i=0}^{{n}}\left(\operatorname{E} g\big(t^{i+1,0},\bs{X}^{t^{i,K},\bs{Y}^{i,K}}(t^{i+1,0})\big)-\operatorname{E} g\big({t^{i+1,0},\bs{Y}^{i+1,0}}\big)\right).
\end{align*}
Thus, \eqref{eq:Konsuuniform} and \eqref{eq:KonsuEnsembleGen} imply
\begin{align*}
&|\operatorname{E} f\big(\bs{X}^{t^0,X_0}(t^{{n}+1})\big)-\operatorname{E} f\big(\bs{Y}_{[L]}(t^{{n}+1})\big)|
\\
\leq&\sum_{i=0}^{{n}}\sum_{k=0}^{K-1}\operatorname{E} C_g(\bs{Y}_{[L]}^{i,k})\delta t^{p_{\os}+1}
+\sum_{i=0}^{{n}}\left(
C_L\Delta t+\tilde{C}_L(\Delta t)^{p_e}\delta t^{p_{\os}}+\tilde{C}_L\delta t^{p_{\os}+1}
\right),
\end{align*}
which yields together with condition (\ref{St-lg-cond2}) the desired convergence.
\end{proof}
\subsection{A result using multistep state extrapolation\label{sec:conv-msem}}
For multistep state extrapolation, the analysis is complicated by the multistep nature of the method.
In this subsection we therefore restrict ourselves to consider linear SDEs \eqref{eq:linSDE}
with normally distributed initial values and restriction operator
\begin{equation}\label{eq:restrictideal}
\overline{\mathcal{R}}(Y)=\begin{pmatrix}
\operatorname{E} Y\\
\operatorname{Var} Y
\end{pmatrix}.
\end{equation}
If we assume now equidistant coarse time steps and that the extrapolation step is given by \eqref{eq:MSextrap}, then we obtain
\begin{align*}
\operatorname{E} \bs{Y}^{{n}+1,0}=\sum_{s=0}^{p_e}l_s(\beta)\operatorname{E} \bs{Y}^{{n}-s,K},\quad \operatorname{Var} \bs{Y}^{{n}+1,0}=\sum_{s=0}^{p_e}l_s(\beta)\operatorname{Var} \bs{Y}^{{n}-s,K}
\end{align*}
with $l_s$ and $\beta$ given in \eqref{eq:lsm} and \eqref{eq:beta}. Application of the one step method $\phi$ to \eqref{eq:linSDE} yields
\begin{align}\label{eq:osmssexlinsde}
\bs{Y}^{{n}+1,K}=&\hat{R}_{\os}(a_1,t^{{n}+1,K-1},\delta t,\eta^{{n}+1,K-1})\bs{Y}^{{n}+1,K-1}
\nonumber\\
&+\hat{S}_{\os}(a_1,a_2,b,t^{{n}+1,K-1},\delta t,\eta^{{n}+1,K-1})\\\nonumber
=&\prod_{k=0}^{K-1}\hat{R}_{\os}(a_1,t^{{n}+1,k},\delta t,\eta^{{n}+1,k})\bs{Y}^{{n}+1,0}
\\&+\sum_{k=0}^{K-1}\hat{S}_{\os}(a_1,a_2,b,t^{{n}+1,k},\delta t,\eta^{{n}+1,k})\prod_{i=k}^{K-1}\hat{R}_{\os}(a_1,t^{{n}+1,i},\delta t,\eta^{{n}+1,i}),\nonumber
\end{align}
where $\hat{R}_{\os}$ and $\hat{S}_{\os}$ are functions depending on $\phi$, similar to the stability function in the deterministic case, and $\eta^{{n}+1,k}$ are (vectors of) i.\,i.\,d.\ random variables used by $\phi$. Assuming that $\hat{R}_{\os}(a_1,t^{{n}+1,k},\delta t,\eta^{{n},k})$ is independent of $\eta^{{n},k}$, which holds, e.\,g., for typical Runge-Kutta methods, we obtain the multistep formulas
\begin{align}\nonumber
\operatorname{E} \bs{Y}^{{n}+1,K}=&\prod_{k=0}^{K-1}\hat{R}_{\os}(a_1,t^{{n}+1,k},\delta t)\sum_{s=0}^{p_e}l_s(\beta)\operatorname{E} \bs{Y}^{{n}-s,K}
\\&+\sum_{k=0}^{K-1}\operatorname{E}\hat{S}_{\os}(a_1,a_2,b,t^{{n}+1,k},\delta t,\eta^{{n}+1,k})\prod_{i=k}^{K-1}\hat{R}_{\os}(a_1,t^{{n}+1,i},\delta t),\label{eq:msemlinsdeexp}\\
\operatorname{Var} \bs{Y}^{{n}+1,K}=&\prod_{k=0}^{K-1}\hat{R}_{\os}(a_1,t^{{n}+1,k},\delta t)^2\sum_{s=0}^{p_e}l_s(\beta)\operatorname{Var} \bs{Y}^{{n}-s,K}\nonumber
\\&+\sum_{k=0}^{K-1}\operatorname{Var}\hat{S}_{\os}(a_1,a_2,b,t^{{n}+1,k},\delta t,\eta^{{n}+1,k})\prod_{i=k}^{K-1}\hat{R}_{\os}(a_1,t^{{n}+1,i},\delta t)^2.\label{eq:msemlinsdevar}
\end{align}
As due to the consistency of $\phi$ we have $\hat{R}_{\os}(a_1,t^{{n}+1,k},0)=1$ and \[\operatorname{E}\hat{S}_{\os}(a_1,a_2,b,t^{{n}+1,k},0,\eta^{{n}+1,k})=\operatorname{Var}\hat{S}_{\os}(a_1,a_2,b,t^{{n}+1,k},0,\eta^{{n}+1,k})=0,\]
the corresponding characteristic polynomial is given for both equations by
\begin{equation*}
P(\xi;\beta,{p_e})=\xi^{{p_e}+1}-\sum_{s=0}^{p_e}l_s(\beta)\xi^s.
\end{equation*}
As in the deterministic case (theory of linear multistep methods) we then obtain the following theorem.
\begin{theorem}
Assume that all roots $\xi$ of $P(\xi;\beta,{p_e})=0$ lie within the unit circle and that all roots with absolute value one are simple. If then the one-step method is weakly consistent of order $p_{\os}$ and given by \eqref{eq:osmssexlinsde} with ${R}_{\os}$ independent of $\eta$, then coarse multistep state extrapolation with restriction operator \eqref{eq:restrictideal} and extrapolation given by \eqref{eq:MSextrap} is convergent of order $p=\min\{p_e,p_{\os}\}$ for linear SDEs \eqref{eq:linSDE} with normally distributed initial values.
\end{theorem}
\section{Numerical results\label{sec:numerical}}
In this section, we provide further numerical results. We first illustrate the dependence of the local error in one accelerated time step on the step size (\cref{sec:num-extrapol}). Subsequently, we perform a number of long-term simulations (\cref{sec:num-long}).
Below, we consider \cref{eq:fene-3d} in one space dimension, with $F(X)$ the FENE force \eqref{eq:springs} with $\gamma=49$, $\textrm{We}=1$. As the velocity field, we choose $\kappa(t)= 2\cdot(1.1+\sin\left(\pi t)\right)$, and we again sample the initial states from the invariant distribution of \cref{eq:fene-3d} for $\kappa(t)\equiv 0$, and use $\epsilon=1$ in \eqref{eq:kramers}. We discretize in time with the classical Euler-Maruyama scheme with time step $\delta t=2\cdot 10^{-4}$. As before, the macroscopic state $\bs{U}_{[L]}$ consists of the first $L$ \emph{even} centralized moments. We study the micro/macro acceleration algorithm with first order projective extrapolation, and with first and second order multistep state extrapolation. In all cases, we perform $K=1$ microscopic steps before extrapolation.
\subsection{Local error\label{sec:num-extrapol}}
We simulate $J=1\cdot 10^5$ realizations up to time $t^*=1.6$, and record the microscopic state $\bs{\mcY}^{-}$ at time $t^-=1.4$, as well as the macroscopic states $\bs{U}_{[L]}(t^-+\Delta t)$ for $L=3,4,5$, and the approximated function of interest $\tilde{\tau}_p(t^-+\Delta t)$ for $\Delta t \in [0,t^*-t^-]$. We then extrapolate from time $t^-$ to time $t=t^-+\Delta t$, and project the ensemble $\bs{\mcY}^{-}$ onto $\bs{U}_{[L]}(t^-+\Delta t)$. Subsequently, we compute the corresponding value of the stress as $\hat{\tau}_p(t^-+\Delta t)$. We record the relative error with respect to the reference solution, $|\hat{\tau}_p(t)-\tilde{\tau}_p(t)|/\tilde{\tau}_p(t)$, as a function of $\Delta t$.
To reduce the statistical error, we report the averaged results of $50$ realizations of this experiment. Then, the statistical error has an order of magnitude of about $10^{-5}$. The results are shown in \cref{fig:extrap-many}.
\begin{figure}
\begin{center}
\includegraphics[width=0.7\linewidth]{fig-extrapolation-many}
\end{center}
\caption{\label{fig:extrap-many}Error of the stress after extrapolating and projecting a prior ensemble of $J=1\cdot 10^5$ FENE dumbbells onto the first $L$ even centralized moments of a reference ensemble, as a function of $\Delta t$ for several values of $L$. Displayed is the result averaged over $50$ realizations of the experiment. Left: First order projective extrapolation. Right: First order multistep state extrapolation. Simulation details are given in the text.}
\end{figure}
For projective integration, we clearly see a first order behavior as a function of $\Delta t$; this is a consequence of the amplification of the statistical error during projective extrapolation. Note that, due to the presence of three competing sources of errors (extrapolation, matching, and statistical error), which may be of opposite signs, the effect of extrapolating more macroscopic state variables is not so clearly visible as in \cref{fig:projection-Dt}. Note that for $L=4$ the statistical and matching errors seem to be a bit lower, such that the second order behavior of the extrapolation error is already apparent for the largest displayed time steps.
For multistep state extrapolation, the situation is slightly different. Here, we see that, for modest gains (small $\Delta t$), the statistical error remains more or less unaffected. For $\Delta t>2\cdot 10^{-3}$ (gain factor $10$), however, the error increases as $\Delta t^2$, as a consequence of the large extrapolation error. Note that, as soon as the extrapolation error dominates, the error appears to be independent of the number of moments used.
To emphasize the effect of the statistical error, we repeat the experiment using $J=1000$ realizations (averaged over $20$ realizations of the experiment). The results are shown in \cref{fig:extrap-less}.
\begin{figure}
\begin{center}
\includegraphics[width=0.7\linewidth]{fig-extrapolation-less}
\end{center}
\caption{\label{fig:extrap-less} Error of the stress after extrapolating and projecting a prior ensemble of $J=1\cdot 10^3$ FENE dumbbells onto the first $L$ even centralized moments of a reference ensemble, as a function of $\Delta t$ for several values of $L$. Displayed is the result averaged over $20$ realizations of the experiment. Left: First order projective extrapolation. Right: First order multistep state extrapolation. Simulation details are given in the text.}
\end{figure}
Compared to \cref{fig:extrap-many}, we see qualitatively the same behavior. Again, the error of projective extrapolation increases linearly (but it is now an order of magnitude larger), while, for multistep state extrapolation, larger gains appear to be possible since the statistical error now dominates for a wider range of extrapolation step sizes $\Delta t$.
\subsection{Long-term simulation\label{sec:num-long}}
We now turn to a long-term simulation, and compare the behavior of the sample mean and sample standard deviation of a full microscopic simulation (which we will call the reference simulation) with the micro/macro acceleration algorithm.
We denote by $\tilde{\tau}_p(t)$ the approximation to the function of interest calculated from one realization of the reference simulation using $J$ SDE realizations, and by $\hat{\tau}_p(t)$ the function of interest obtained via one realization of the micro/macro acceleration technique.
As extrapolation techniques, we use first order projective extrapolation and second order multistep state extrapolation. (For the set-up in this example, the deterministic error of first order multistep state extrapolation is too high to be considered further.)
\Cref{fig:stat-fene-proj-mom} (top left) shows the evolution of the stress as a function of time. We see that the simulation exhibits a periodic behavior, with a fast increase of the stress followed by a relaxation. Note that, in this problem, due to the fast variations in $\kappa(t)$, there is not a very strong time-scale separation between the evolution of the stress tensor and the evolution of individual polymers. Hence, using the time-step adaptation strategy (see \Cref{sec:proj-failure}) will prove crucial to obtain an efficient algorithm. During the fast increase of the stress, we observed that the matching operator fails for time steps $\Delta t>1\cdot 10^{-3}$, whereas larger accelerations are possible during the relaxation. Therefore, in this experiment we will use adaptive macroscopic time steps, as outlined in \cref{sec:proj-failure}, choosing $\underline{\alpha}=0.2$ and $\overline{\alpha}=1.2$. On average, we obtained a speed-up factor of $4$, meaning that microscopic simulation has been performed over $1/4$ of the time domain.
In a first experiment, we use $\Delta t=1\cdot 10^{-3}$ and vary the number $L$ of macroscopic state variables. The results for 500 realizations of this experiment, each with an ensemble size of $J=5000$, are shown in \cref{fig:stat-fene-proj-mom}.
\begin{figure}
\begin{center}
\includegraphics[width=\linewidth]{statistical-fene-proj-mom}
\end{center}
\caption{\label{fig:stat-fene-proj-mom}Results of micro/macro acceleration of the FENE model \eqref{eq:fene-3d} using $\Delta t=1\cdot 10^{-3}$ and projective extrapolation for different numbers $L$ of macroscopic state variables, as well as a full microscopic (reference) simulation. Top left: evolution of the sample means of the stresses $\tilde{\tau}_p$ and $\hat{\tau}_p$. Bottom left: deterministic error on $\hat{\tau}_p$. Right: evolution of the sample standard deviation of $\hat{\tau}_p$. Simulation details are given in the text.}
\end{figure}
We make two main observations. First, the deterministic error decreases with increasing $L$, whereas the sample standard deviation is independent of $L$. (The different lines in the plot are nearly indistinguishable.) Note also that the variance on the sample standard deviation is quite large in this example. Second, from \cref{fig:stat-fene-proj-mom}, we see that the error of the micro/macro acceleration algorithm with respect to the reference simulation also decreases as a function of time, until it reaches a level of the order of the statistical error. This behavior can be attributed to the fact that, in this example, the macroscopic behavior of the system on long time scales is determined by only a few macroscopic state variables. The results for multistep state extrapolation (not shown) in terms of $L$ are similar.
In a second experiment, we fix $L=3$ and consider varying $\Delta t$. (This experiment and its conclusions closely resemble the one in \cref{sec:num-extrap}.) The results for $500$ realizations, each with an ensemble size of $J=1000$, are shown in \cref{fig:stat-fene-proj}.
\begin{figure}
\begin{center}
\includegraphics[width=\linewidth]{statistical-fene-proj}
\end{center}
\caption{\label{fig:stat-fene-proj}Results of micro/macro acceleration of the FENE model \eqref{eq:fene-3d} using $L=3$ moments and projective extrapolation for different values of $\Delta t$, as well as a full microscopic (reference) simulation. Top left: evolution of the sample means of the stresses $\tilde{\tau}_p$ and $\hat{\tau}_p$. Bottom left: deterministic error on $\hat{\tau}_p$. Right: evolution of the sample standard deviation of $\hat{\tau}_p$. Simulation details are given in the text.}
\end{figure}
\Cref{fig:stat-fene-proj} (bottom left) shows again that the deterministic error grows with increasing $\Delta t$, whereas the right figure illustrates that the sample standard deviation is larger for larger $\Delta t$. For second order multistep state extrapolation, we obtain \cref{fig:stat-fene-msem-2}.
\begin{figure}
\begin{center}
\includegraphics[width=\linewidth]{statistical-fene-msem-order2}
\end{center}
\caption{\label{fig:stat-fene-msem-2}Results of micro/macro acceleration of the FENE model \eqref{eq:fene-3d} using $L=3$ moments and second order multistep state extrapolation for different numbers $L$ of macroscopic state variables, as well as a full microscopic (reference) simulation. Top left: evolution of the sample means of the stresses $\tilde{\tau}_p$ and $\hat{\tau}_p$. Bottom left: deterministic error on $\hat{\tau}_p$. Right: evolution of the sample standard deviation of $\hat{\tau}_p$. Simulation details are given in the text.}
\end{figure}
The behavior of the sample standard deviation is similar to the linear case. When zooming in to the behavior for small $t$, we observe that, over a short time interval, the sample standard deviation using the micro/macro acceleration technique increases at the same rate as the sample standard deviation in the reference simulation, which corresponds to the theoretical result on the local propagation of statistical error. However, on longer time scales, we again see that the sample standard deviation for large $t$ grows rapidly, and seems to be larger for larger $\Delta t$, due to accumulation effects. Note that, in this case, second order multistep state extrapolation even behaves worse than first order projective integration on long time scales for sufficiently large $\Delta t$.
\section{Conclusions and outlook\label{sec:concl}}
We presented and analyzed a micro/macro acceleration technique for the Monte Carlo simulation of stochastic differential equations (SDEs) in which short bursts of simulation using an ensemble of microscopic SDE realizations are combined with an extrapolation of an estimated macroscopic state forward in time. The method is designed for problems in which the required time step for each realization of the SDE is small compared to the time scales on which the function of interest evolves. For such systems, one often needs to take a very small microscopic time step, which results in a deterministic error that is much smaller than the statistical error.
We showed that the proposed procedure converges in the absence of statistical error, provided the matching operator satisfies a number of natural conditions, and we introduced a matching operator that satisfies these conditions for Gaussian random variables. We also conjectured that this matching operator is suitable for general distributions, and provided numerical evidence to support this conjecture.
Concerning the statistical error, a local analysis of projective extrapolation shows that the amplification of statistical error depends on the ratio $\alpha$ of macroscopic (extrapolation) and microscopic (simulation) time steps, while this is not the case for multistep state extrapolation. Numerical evidence, however, suggests that, when using higher order multistep state extrapolation, accumulation of statistical error over macroscopic time scales may nevertheless induce an $\alpha$-dependent statistical error.
This paper has not focused on quantifying the computational gains that can be expected from this method. It is clear from the description that the method will be more efficient when there is a bigger separation in time scales between the microscopic and macroscopic levels. The numerical examples in this text do not exhibit such a strong time-scale separation; they were mainly chosen for their ability to clearly illustrate the effects of the different sources of numerical error. However, some conclusions on efficiency can nevertheless be drawn. For a given required variance on the solution, the computational cost using first order projective extrapolation is comparable to that of a full simulation, since the former requires more SDE realizations due to the $\alpha$-dependent amplification of statistical error. For first order multistep state extrapolation, extrapolation without such a drastic amplification of statistical error is possible, at the cost of an amplified deterministic error. This can be acceptable if the macroscopic function of interest changes slowly compared to the time step of the SDE.
We note that, for the model problem of coupled micro/macro simulation of dilute polymer solutions, the amplification of statistical error using projective extrapolation need not be dramatic: while the computational cost of a micro/macro accelerated simulation and a full microscopic simulation are comparable for given variance, this is no longer true when coupling this Monte Carlo simulation to a PDE for the solvent. Indeed, when extrapolating the complete coupled system forward in time, a computational gain is obtained since the PDE for the solvent also does not need to be simulated on the whole time domain.
In future work, we will study stability and propagation of statistical error on long time scales. The numerical experiments indicate that these issues can be studied in a linear setting. Another open question is for which distributions the conjecture can be proved. From an algorithmic point of view, this work raises questions on the adaptive/automatic selection of all method parameters (number of moments to extrapolate, macroscopic time step, number of SDE realizations) to ensure a reliable computation with minimal computational cost. Also, a numerical comparison with other approaches, such as implicit approximations, could be envisaged.
|
{
"timestamp": "2011-11-08T02:01:06",
"yymm": "1009",
"arxiv_id": "1009.3767",
"language": "en",
"url": "https://arxiv.org/abs/1009.3767"
}
|
\section{Introduction}
Hadronic correlation functions which involve quark disconnected
contributions are notoriously hard to compute in lattice
QCD~\cite{Neff:2001zr,Foley:2005ac}.
Such contributions however appear in several
quantities related to properties of flavour singlet particles
(e.g. the $\eta$ and $\eta'$ mesons) or to matrix elements of
flavour singlet operators (as for the strangeness content of the nucleon),
or to electro-magnetic interactions (e.g. the hadronic contribution to
the anomalous magnetic moment of the muon or to the nucleon electric
dipole moment).
There is no conceptual difficulty in treating disconnected quark diagrams
but the computational effort is immense
compared to quark-connected contributions. The disconnected part is
therefore often neglected, not always providing solid arguments that the
systematic uncertainty introduced in this way is under control.
Here we present a method that allows to predict the quark disconnected
contribution to correlation functions in (partially quenched)
chiral perturbation theory \cite{Gasser:1983yg,Gasser:1984gg,
Bernard:1992mk,Bernard:1993sv,Sharpe:2000bc}.
We first give the general argument and then present an explicit example
by deriving predictions for the magnitude of the quark disconnected
contribution to the hadronic vacuum polarisation of the photon for
$N_f=2$ dynamical flavours, for $N_f=2$ dynamical flavours
with a quenched strange quark and also for
the case of $N_f=2+1$ active flavours.
The vacuum polarisation is the main ingredient in
computations of the leading hadronic contribution to the anomalous
magnetic moment of the muon and hence of relevance to precision tests of
the Standard Model (see Ref.~\cite{Jegerlehner:2009ry} for a review of the
subject).
We will stay in the Euclidean continuum and infinite volume
for most of the discussion. While having applications to lattice QCD in mind,
reference to this regularisation is only made where we think that
it helps the better understanding of our arguments. We will
discuss however the finite volume case in order to
show how this technique allows for using
{\it partial twisting} \cite{Sachrajda:2004mi,Bedaque:2004ax,Bedaque:2004kc,deDivitiis:2004kq,Tiburzi:2005hg,Flynn:2005in} also for form-factors and polarisations
involving flavour-diagonal operators.
\section{General argument}
We consider a $n$-point fermionic correlation function in QCD with
$N_f$
dynamical flavours (not necessarily degenerate). In general several
Wick contractions contribute
to the correlator. By introducing valence quarks which are degenerate
with the dynamical flavours, each Wick contraction can be rewritten in terms
of a single fermionic correlation function defined in an un-physical
theory. The
physical result is recovered by summing over the correlation functions
in the un-physical, partially quenched, theory
\cite{Bernard:1992mk,Bernard:1993sv,Sharpe:2000bc}.
In particular, the
approach can be used to separate the contributions from quark
disconnected diagrams from those coming from quark connected diagrams.
These contributions taken on their own are un-physical, it is therefore
natural that in order to define them as correlation functions one has
to resort to un-physical theories.
Investigations of
quark-disconnected
diagrams along these lines can also be found in
Ref.~\cite{Sharpe:2000bc}.
The number of valence quarks $N_v(i)$ degenerate with the $i$th
dynamical flavour, which has to be introduced depends on the particular
correlation function. Given a $n$-point fermionic correlation function,
$N_v(i)$ is related in an obvious way to the largest number
$N_{\rm D}^{\rm max}(i)$ of disconnected quark loops involving
the $i$th flavour, which can appear as all possible Wick contractions
are considered. In particular $N_v(i)=N_{\rm D}^{\rm max}(i)-1$ at most,
in such a way that a different {\it flavour} can be {\it
attached} to each disconnected quark loop.
In this approach the partially quenched theory, in which
each Wick contraction of the original correlator can be
written as an independent correlation function,
is not unique but rather depends on the specific $n$-point
function we started from.
Partially quenched chiral perturbation theory (PQ$\chi$PT)
\cite{Bernard:1992mk,Bernard:1993sv,Sharpe:2000bc}, which is an extension
of chiral perturbation theory \cite{Gasser:1984gg,Gasser:1983yg},
provides
an asymptotic low energy description of partially quenched QCD (PQQCD)
and can therefore be used to obtain
predictions for, and algebraic relations amongst,
the disconnected and the connected part of a
fermionic correlation function. There is a rather large body of literature on
PQ$\chi$PT and PQQCD, we found the reviews in Ref.~\cite{Sharpe:2006pu}
and Ref.~\cite{Golterman:2009kw} very clear and useful.
In short, following Ref.~\cite{Sharpe:2006pu}, PQQCD can be formulated
in terms of a local theory by introducing a commuting spin-1/2 field,
a ghost \cite{Morel:1987xk}, labelled by $\tilde{q}$ for each valence quark $q$ and by
extending the fermionic QCD action by including terms of the form
$\bar{q}(\slash{{\rm D}}+m_q)q+\tilde{q}^{\dagger}
(\slash{{\rm D}}+m_q)\tilde{q}$, which violate the spin statistics theorem. As long as all masses are positive, the
determinant produced by the integral over the valence quark fields is
cancelled by the corresponding integral over the ghost fields and the
QCD partition function is reproduced. The advantage is that this
formulation provides field-theoretic expressions for partially quenched
correlation functions. As discussed in
Refs.~\cite{Sharpe:2006pu,Golterman:2009kw},
a low energy description of the theory (PQ$\chi$PT) is
obtained by constructing a chiral Lagrangian encoding the apparent
chiral symmetry group of the extended action, which is a
$SU\hspace{-.5mm}\left(N_f+\sum_{i=1}^{N_f}N_v(i)\;|\;
\sum_{i=1}^{N_f}N_v(i)\right)$
graded group.
For the application discussed here the strategy can be exemplified
for the case of a meson 2-point function
constructed of flavour-diagonal quark-bilinears,
\begin{eqnarray}\label{eq:example}
C_{\rm QCD}(y,x)&=&\sum_{q=u,d,s,\dots}\big< \bar q(y)\Gamma^\prime q(y) \bar q(x)\Gamma q(x)\big>_{\rm QCD}
\nonumber\\[-2mm]
&&\\[-2mm]
&=&-\sum_{q=u,d,s,\dots}\big<
\Tr\left\{ S_q(x,y)\Gamma^\prime S_q(y,x)\Gamma \right\}\big>_{\rm QCD} \nonumber \\
&&+ \sum_{q=u,d,s,\dots}\big< \Tr\left\{S_q(y,y)\Gamma^\prime\right\}
\Tr\left\{S_q(x,x)\Gamma\right\}\big>_{\rm QCD}\,,\nonumber\
\end{eqnarray}
where $S_q$ is the propagator of the quark field $q$
and $\Gamma^{(\prime)}$ may contain Dirac- as well as colour-structures.
The trace is over Dirac- and colour-indices.
We will rewrite the correlation function $C_{\rm QCD}(y,x)$ as the sum of two new
correlation functions: one being equivalent to the quark-connected
contribution and the other being equivalent to the quark-disconnected
contribution.
To this end for each quark $q$ we add a valence quark $q_v$ which is
mass degenerate to it, together with the corresponding ghost field.
The correlation $C_{\rm QCD}(y,x)$ can be rewritten as
\begin{eqnarray}\label{eqn:master}
C_{\rm QCD}(y,x)&=&\;\;\;\sum_{q=u,d,s,\dots}\big< \bar q(y)\Gamma^\prime q_v(y)
\bar q_v(x)\Gamma q(x)\big>_{\rm PQQCD}
\\&&+\sum_{q=u,d,s,\dots}
\big< \bar q(y)\Gamma^\prime q(y)
\bar q_v^\prime(x)\Gamma q_v(x)
\big>_{\rm PQQCD}\nonumber\\[0mm]
&\equiv&C_{\rm PQQCD}^{\rm Conn}(y,x)+C_{\rm PQQCD}^{\rm Disc}(y,x)\nonumber\,.
\end{eqnarray}
The Wick contractions of the first correlator on the r.h.s. of the first
equation lead to quark connected diagrams
only, while those for the second correlator produce quark-disconnected
diagrams only (see illustrations in figure \ref{fig:Wicks}).
If we imagine for a moment having regularised the theory on the lattice it
is clear that the above equality holds non-perturbatively, since on each gauge
configuration the quark propagators and the fermionic determinant (and
therefore the weight of the configuration in the path integral) are the same in
the two theories.
In the following
\begin{figure}
\centering
\subfigure[]{
\begin{picture}(60,15)(-30,-15)
\Oval(0,0)(12,25)(0)
\GCirc(-25,0){3}{0.5}\GCirc(25,0){3}{0.5}
\end{picture}
}
\hspace{15mm}\subfigure[]{
\begin{picture}(60,15)(-30,-15)
\Oval(-15,0)(12,12)(0)
\Oval(15,0)(12,12)(0)
\GCirc(-27,0){3}{0.5}\GCirc(27,0){3}{0.5}
\end{picture}
}
\caption{Wick contractions: (a) connected and (b) disconnected
diagram.}
\label{fig:Wicks}
\end{figure}
we will show how $C_{\rm PQQCD}^{\rm Disc}(y,x)$ and $C_{\rm PQQCD}^{\rm Conn}(y,x)$ can be
expressed in a suitable chiral effective theory, depending on the
flavours entering the sum in Eq.~\ref{eq:example}.
\section{Hadronic vacuum polarisation}\label{sec:3}
The Euclidean hadronic vacuum polarisation (VP) tensor is defined as
\begin{equation}\label{eqn:pol_tensor}
\Pi_{\mu\nu}^{(N_f)}(q)=
i\int d^4xe^{iqx}\langle
J_\mu^{(N_f)}(x)J_\nu^{(N_f)}(0)
\rangle = i C_{\mu\nu}(q)\,,
\end{equation}
where $J_\mu^{(N_f)}(x)=\sum\limits_{q=1}^{N_f}Q_q\bar q(x)\gamma_\mu q(x)$. For
$N_f=2$, $q=(u,d)$ and $Q_q=(2/3,-1/3)$ and for $N_f=2+1$,
$q=(u,d,s)$ and $Q_q=(2/3,-1/3,-1/3)$.
Euclidean invariance and current conservation imply
\begin{equation}
\Pi_{\mu\nu}^{(N_f)}(q)=(q_\mu q_\nu-g_{\mu\nu}q^2)\Pi^{(N_f)}(q^2)\,.
\end{equation}
For space-like momenta, the relation between $\Pi_{\mu\nu}^{(N_f)}(q^2)$ and
the lowest order hadronic contribution $a_\mu^{\rm HLO}$
to the anomalous magnetic moment of the
muon has been derived in Ref.~\cite{Blum:2002ii,Gockeler:2003cw} and reads (suppressing the index $N_f$)
\begin{equation}
a_\mu^{\rm HLO}= \left( {{\alpha}\over{\pi}} \right)^2 \int_0^\infty dq^2 \, f(q^2)
\hat{\Pi}(q^2)
\;,
\label{eq:amu}
\end{equation}
with
\be
f(q^2)={{m_\mu^2q^2Z^3(1-q^2Z)}\over{1+m_\mu^2q^2Z^2}}\;, \quad \quad
Z=-{{q^2-\sqrt{q^4+4m_\mu^2q^2 }}\over{2m_\mu^2q^2}}\,,
\ee
and $\hat{\Pi}(q^2)=4\pi^2\left[\Pi(q^2)-\Pi(0)\right]$.
We consider the iso-scalar meson two-point function $C_{\mu\nu}(q)$ in
Eq.~(\ref{eqn:pol_tensor}).
The Wick contractions for that correlator
lead to quark diagrams of type
(a) and (b) as illustrated in figure \ref{fig:Wicks}.
By considering in some detail the two flavour
case we will first separate the disconnected contributions from the
connected ones in the way discussed in the previous section and then
set-up a computation in the resulting PQ$\chi$PT framework.
According to the discussion above we would need two valence quarks, one
degenerate with the $u$ quark and one degenerate with the $d$ quark.
However,
as we will always assume iso-spin to be an exact symmetry, it is enough to
introduce one valence quark, which we call $r$, with the corresponding
ghost $r_{\rm g}$. We therefore have an $SU(3|1)$ chiral group.
The correlation in Eq.~(\ref{eqn:pol_tensor}) can then be decomposed as
\begin{eqnarray}
C^{}_{\mu\nu}(q)&\hspace{-2.5mm}=\hspace{-2.5mm}& \int d^4xe^{iqx}\bigg(
{{4}\over{9}}\Big< j^{ur}_\mu(x) j^{ru}_\nu(0)\Big> +
{{1}\over{9}}\Big< j^{dr}_\mu(x) j^{rd}_\nu(0)\Big> +\nonumber\\[-1mm]
\\[-1mm]
&&\qquad\qquad\;\;\;\;
{{4}\over{9}}\Big< j^{uu}_\mu(x) j^{rr}_\nu(0)\Big> +
{{1}\over{9}}\Big< j^{dd}_\mu(x) j^{rr}_\nu(0)\Big> -
{{4}\over{9}}\Big< j^{uu}_\mu(x) j^{dd}_\nu(0)\Big> \bigg),\nonumber
\label{eq:su31splitting}
\end{eqnarray}
where $j^{q_1 q_2}_\mu(x)=\bar q_1(x) \gamma_\mu q_2(x)$.
The first two correlators on the r.h.s. are connected, whereas the last
three represent the disconnected contributions to the hadronic VP tensor.
It is convenient to cast the quark-fields into a four-component vector
$\psi^T=(u,d,r,r_{\rm g})$ and introduce the generators $T^a$,
$a=1,\dots,15$ of the graded group $SU(3|1)$.
We use the conventions also employed in \cite{Giusti:2008vb},
\be
T^a=(T^a)^\dagger\;, \quad \quad {\rm Str}\left\{T^a\right\}=0\;, \quad
{\rm Str}\left\{T^aT^b\right\}={{1}\over{2}}g^{ab}\;, \quad a=1,\dots,15\,,
\ee
with the \textit{super-trace}
${\rm Str}\left\{A\right\}=A_{11}+A_{22}+A_{33}-A_{44}$.
The matrix $g^{ab}$ reads
\begin{equation}
g=\begin{pmatrix}1\cr
&\ddots\cr
& &1\cr
& & &-\sigma_2\cr
& & & &-\sigma_2\cr
& & & & &-\sigma_2\cr
& & & & & &-1\cr\end{pmatrix}
\begin{matrix}\left.\vphantom{\begin{matrix}1\cr
&\ddots\cr
& &1\cr \end{matrix}}\right\}&
\kern-1.5ex1-8\hfill\cr
\left.\vphantom{\begin{matrix}-\tau^2\cr
&\ddots\cr
& &-\tau^2\cr\end{matrix}}\right\}&
\kern-1.5ex9-14\hfill\cr
\left.\vphantom{\begin{matrix} 1\cr \end{matrix} }\right\}&
\kern-1.5ex15\hfill\cr
\end{matrix}
\end{equation}
where $\sigma_2$ is the second Pauli matrix. $T^1,\dots, T^8$ are the
generators of the $SU(3)$ subgroup that acts on the sea and valence
components, $T^9,\dots, T^{14}$ mix the quark with the ghost components,
$T^{15}$ is the diagonal matrix ${\rm diag}(1,1,1,3)/(2\sqrt{3})$.
We also add $T^0$ with ${\rm Str}\{T^0\}=1/\sqrt{2}$
which is proportional to the unit matrix
in order to describe iso-scalar interactions.
For our choice of the generators $T^0,\,\dots,\,T^{15}$,
the currents in Eq.~(\ref{eq:su31splitting}) can then be rewritten as
{\large\begin{equation}
\begin{array}{rcl}
j^{ur}_\mu(x)&=&\bar\psi(x)\gamma_\mu\left(T^4+i T^5\right)\psi(x)\,,\\[1mm]
j^{ru}_\mu(x)&=&\bar\psi(x)\gamma_\mu\left(T^4-i T^5\right)\psi(x)\,,\\[1mm]
j^{dr}_\mu(x)&=&\bar\psi(x)\gamma_\mu\left(T^6+i T^7\right)\psi(x)\,,\\[1mm]
j^{rd}_\mu(x)&=&\bar\psi(x)\gamma_\mu\left(T^6-i T^7\right)\psi(x)\,,\\[1mm]
j^{uu}_\mu(x)&=&\bar\psi(x)\gamma_\mu\left(
\sqrt{2} T^0 - \frac{1}{\sqrt{3}} T^{15} +
\frac{1}{\sqrt{3}} T^8 + T^3\right)\psi(x)\,,\\[2mm]
j^{dd}_\mu(x)&=&\bar\psi(x)\gamma_\mu\left(
\sqrt{2} T^0 - \frac{1}{\sqrt{3}} T^{15} +
\frac{1}{\sqrt{3}} T^8 - T^3\right)\psi(x)\,,\\[2mm]
j^{rr}_\mu(x)&=&\bar\psi(x)\gamma_\mu\left(
\sqrt{2} T^0 - \frac{1}{\sqrt{3}} T^{15} -
\frac{2}{\sqrt{3}} T^8\right)\psi(x)\,.
\end{array}
\end{equation}}
\noindent This form is more suited for the PQ$\chi$PT computation, as it will
become clear in the following.
Before concluding this section we note that by using again iso-spin
symmetry we could have separated the disconnected and connected parts
in the correlator above without introducing any additional valence quark.
We have used this completely equivalent approach in Ref.~\cite{Juttner:2009yb}.
Here however, we preferred to introduce a graded flavour group
to provide an example
for how one has to proceed for the the more
general but also more complicated case of $2+1$ flavours.
\section{PQ$\chi$PT for the connected and disconnected parts of the hadronic VP}
In this section we briefly introduce those parts of the (partially quenched)
chiral Lagrangian up to $O(p^4)$ which contribute to the connected as well
as to the disconnected piece of the VP \cite{Gasser:1984gg,Gasser:1983yg}. We took care that
the discussion applies to any choice for the (graded) symmetry group. In the
next section we will then present results for the cases of
$SU(3|1)$, $SU(4|2)$, and $SU(4|1)$ flavour groups. These are the relevant
symmetry groups for the description of the contributions to the VP in the $N_f=2$-theory without and with a quenched
strange quark and for the $N_f=2+1$-theory, respectively.
\subsection{$O(p^2)$-Lagrangian}
For a generic graded flavour group, the leading order chiral Lagrangian is
\cite{Gasser:1984gg,Gasser:1983yg,Bernard:1992mk,Bernard:1993sv,Sharpe:2000bc}
\begin{equation}
\mathcal{L}^{(2)}=\frac{F^2}{4}{\rm Str}\left\{D_\mu U D_\mu U^\dagger\right\}
-\frac 12 B F^2 {\rm Str}\left\{M U^\dagger + M^\dagger U\right\}\,,
\end{equation}
where $D_\mu U=\partial_\mu U+i v_\mu U -i U v_\mu$ provides the
coupling of the meson field to an external vector source $v_\mu$
and where
\begin{equation}
M={\rm diag}\left(m_1,\dots,m_{N_f},
m_1^{\rm valence},\dots,m_{N}^{\rm valence},
m_1^{\rm ghost},\dots,m_{N}^{\rm ghost} \right)\,,
\end{equation}
contains the dynamical, valence-, and ghost-quark masses.
By chirally expanding\linebreak
$U=\exp\left(2i\frac {\phi^a T^a} F\right)$, where $T^a$ are the
$\tilde N=(N_f+2N)^2-1$ generators of the corresponding
$SU(N_f+N|N)$ graded symmetry group
(with the generators normalised as in the previous section)
and where the $\phi^a$ are the Goldstone-boson/fermion fields,
we arrive at the $O(p^2)$-expression
\begin{equation}\label{eqn:O2Lag}
\mathcal{L}^{(2)}=
\frac 12 \,g^{ab} \partial_\mu \phi^a\partial_\mu \phi^b
+ 2 B \tilde M^{ab}\phi^a \phi^b+\mathcal{L}^{(2)}_{\rm int}\,,
\end{equation}
where $\tilde M^{ab}={\rm Str}\left\{MT^a T^b\right\}$. With
$v_\mu=v_\mu^{a}{T^a}$, the part of the interaction Lagrangian that
is of relevance here is
\begin{equation}
\mathcal{L}^{(2)}_{\rm int}=
\underbrace{
-\sum\limits_{k=1}^{\tilde N}
C_k^{\;bc}g^{ak}\partial_\mu \phi^a v_{\mu}^{b}\phi^c
}_{a)}
\underbrace{
+\frac 12
\sum\limits_{k=1}^{\tilde N}\sum\limits_{l=1}^{\tilde N}
C_k^{\;ab}C_l^{\;cd}g^{kl}v_{\mu}^a \phi^b\phi^c v_{\mu}^{d}}_{b)}\,,
\end{equation}
where
\begin{equation}
C_a^{\;bc}= -2 i \sum\limits_{k=1}^{\tilde N}
{\rm Str} \left\{[T^b,T^c]\,T^k \right\}g^{k a}\,,
\end{equation}
are the structure constants of the underlying (graded)
symmetry group with the\linebreak (anti-)commutator
\begin{equation}
[T^a,T^b]\equiv T^a T^b-(-)^{\eta_a\,\eta_b}\,T^b T^a\,.
\end{equation}
The $\eta_a$ (and correspondingly for $\eta_b$)
are 1 if $T^a$ mixes valence or sea
quarks with a ghost and 0 otherwise (cf.~\cite{Weinberg3}).
The Feynman-rules for the vertices illustrated in figure
\ref{fig:vertices} (a) and (b) are then determined as
\begin{equation}
\begin{array}{cl}
a)& \qquad\qquad\;\;
-\Big(
\tilde C_a^{\;bc}\,p_\mu+
\tilde C_c^{\;ba}\,p^{\,\prime}_\mu
\Big)\,,\\
b)& i\eta_{\mu\nu}\sum\limits_{k,l=1}^{\tilde N}g^{kl}
\Big( C_{k}^{\;ab}C_{l}^{\;cd}+ C_{k}^{\;ac}C_{l}^{\;bd}\Big)\,,
\end{array}
\end{equation}
with $\tilde C_a^{\;\;bc}= C_i^{\;\,bc}g^{ai}$.
\begin{figure}
\centering
\subfigure[]{
\begin{picture}(120,60)(-60,-30)
\Photon(-30,0)(00,0){4}{4}
\Text(-42, 0)[c]{$\mu, b$}
\Text(+38, 20)[c]{$p,\,a$}
\Text(+38,-20)[c]{$p^{\,\prime},\,c$}
\ArrowLine(30,30)(0,0)
\ArrowLine(30,-30)(0,0)
\end{picture}
}
\subfigure[]{
\begin{picture}(120,60)(-60,-30)
\Photon(-30,-30)(00,0){4}{4}
\Photon(0,0)(30,-30){4}{4}
\ArrowLine(-30,30)(0,0)
\ArrowLine(30,30)(0,0)
\Text(-44,-30)[c]{$\mu,a$}
\Text(+44,-30)[c]{$\nu,d$}
\Text(-40,+30)[c]{$b$}
\Text(+40,+30)[c]{$c$}
\end{picture}
}
\subfigure[]{
\label{fig:feyn_c}
\begin{picture}(80,40)(-40,-30)
\Photon(-40,0)(-10,0){4}{3}
\Photon(10,0)(40,0){4}{3}
\GCirc(0,0){10}{1}
\Text(0,0)[c]{4}
\end{picture}
}
\caption{Contributing vertices in the effective theory.}\label{fig:vertices}
\end{figure}
\subsection{$O(p^4)$-Lagrangian}
The relevant terms of the $O(p^4)$ Lagrangian \cite{Gasser:1984gg,Gasser:1983yg,Kaiser:2000ck,Kaiser:2000gs}, i.e. those parts that
have a non-zero matrix element between single external vector-sources, are
\begin{equation}\label{eqn:Lcounter}
\begin{array}{rcl}
\mathcal{L}^{(4)}&=&
X_1\,{\rm Str}\{ \hat v_{\mu\nu} \hat v_{\mu\nu}\}
+ X_2\,{\rm Str}\{ v_{\mu\nu} \} {\rm Str} \{v_{\mu\nu} \}\,,
\end{array}
\end{equation}
with
\begin{equation}
v_{\mu\nu}=\partial_\mu v_\nu -\partial_\nu v_\mu\,,
\end{equation}
and where $\hat v_{\mu\nu}$ is the trace-less part
$v_{\mu\nu}-\frac 1{N_f}\,{\rm Str}\{{v_{\mu\nu}}\}$.
As summarised in table
\ref{tab:GLcoeffs} the coefficient $X_{1}$ is the shorthand
notation for the Gasser-Leutwyler
low energy constants of the underlying symmetry group (see for example
\cite{Gasser:1983yg,Gasser:1984gg,Bijnens:1999hw}) and $X_2$
parameterises the effective dynamics of flavour-diagonal
contributions.
\begin{table}
\begin{center}
\begin{tabular}{lcclllll}
\hline\hline\\[-4mm]
&$X_1$ &$X_2$ \\
\hline&&&&\\[-4mm]
$\SUtwo$, $SU(3|1)$, $SU(4|2)$ &$-4h_2$ &$h_s$ \\
$\SUthree$, $SU(4|1)$ &$L_{10}+2H_1$ &$H_s$ \\[1mm]
\hline\hline
\end{tabular}\caption{Gasser-Leutwyler coefficients.}\label{tab:GLcoeffs}
\end{center}
\end{table}
The vertices corresponding to these Lagrangians have the following form
(cf. figure \ref{fig:vertices} (c)):
\begin{equation}\label{eqn:counter:SU2}
\begin{array}{cll}
c)& i\,2\,X_1\,g^{ab}
\left(\eta_{\mu\nu}p^2-p_\mu p_\nu \right)\, &{\rm for\,}a,b>0\,,\\
c)& i\,2\, X_2 \,\;\;\;\;\;\left(\eta_{\mu\nu}p^2-p_\mu p_\nu \right)\,& {\rm for\,}a=b=0\,.
\end{array}
\end{equation}
\subsection{Contributing diagrams}
At NLO only the diagrams in figure
\ref{fig:FMdiags}~(a) and \ref{fig:FMdiags}~(b)
contribute dynamically to the VP and the diagram
in figure~\ref{fig:FMdiags}~(c) provides counter terms.
\begin{figure}
\centering
\subfigure[]{
\begin{picture}(80,40)(-40,-11)
\Photon(-40,0)(-20,0){4}{2}
\Photon(20,0)(40,0){4}{2}
\GOval(0,0)(15,20)(0){1}
\GCirc(-20,0){4}{.3}
\GCirc(20,0){4}{.3}
\Text(-45,0)[c]{a}
\Text(+45,0)[c]{b}
\end{picture}
\label{fig:feyn_a}
}\hspace{8mm}
\subfigure[]{
\label{fig:feyn_b}
\begin{picture}(80,40)(-40,-11)
\Photon(-40,0)(40,0){4}{8}
\GOval(0,20)(20,10)(00){1}
\GCirc(0,0){4}{.3}
\Text(-45,0)[c]{a}
\Text(+45,0)[c]{b}
\end{picture}
}
\hspace{8mm}
\subfigure[]{
\label{fig:feyn_c2}
\begin{picture}(80,40)(-40,-11)
\Photon(-40,0)(-10,0){4}{3}
\Photon(10,0)(40,0){4}{3}
\GCirc(0,0){10}{1}
\Text(0,0)[c]{4}
\Text(-45,0)[c]{a}
\Text(+45,0)[c]{b}
\end{picture}
}
\caption{Feynman diagrams contributing to the hadronic VP at
1-loop level: (a) unitary, (b)
tadpole, (c) $O(p^4)$-insertion}
\label{fig:FMdiags}
\end{figure}
Let $\psi$ be a flavour vector of $N_f$ dynamical quarks, $N$ partially
quenched valence quarks and $N$ corresponding ghost-quarks.
The expression in the effective theory for
the Fourier transform of the correlator
$i\langle \bar\psi(x) \gamma_\mu T^a\psi(x)\, \bar\psi(0) \gamma_\nu T^b\psi(0)\rangle$
at NLO in chiral perturbation theory is of the form
\begin{equation}
\Pi_{\mu\nu}^{(a,b)}(q,\mu)=\Pi_{\mu\nu}^{(a,b),\rm unit.}(q)
+\Pi_{\mu\nu}^{(a,b),\rm tadp.}(q)
+\Pi_{\mu\nu}^{(a,b),\rm count.}(\mu)\,,
\end{equation}
where $\mu$ is the renormalisation scale.
The first term on the r.h.s. corresponds to the unitary diagram,
\begin{eqnarray}\label{eq:unitary}
\displaystyle
\Pi_{\mu\nu}^{(a,b),\rm unit.}(q)&=&-\frac i2\,
\tilde C_i^{\;aj}\,\tilde C_k^{\;bl}\\
&&\hspace{-1cm}\times \int\frac{d^d k}{(2\pi)^d}
\, (q+2k)_\mu(q+2k)_\nu\,
G^{ik}\big(k^2\big)\,G^{jl}\big((q-k)^2\big)\nonumber\,,
\end{eqnarray}
where the overall
factor $\frac 12$ is a
symmetry factor.
For the second term, the tadpole-diagram (b), one derives
\begin{eqnarray}\label{eq:tadpole}
\Pi_{\mu\nu}^{(a,b)\,\rm tadp.}
&=&i\,\eta_{\mu\nu}\,C_i^{\;aj}\,\tilde C_{i}^{\;kb}
\int\frac{d^d k}{(2\pi)^d}\,
G^{jk}(k^2)\,.
\end{eqnarray}
For the $SU(3|1)$ theory the
meson propagators $G^{ab}(k^2)$ can be found
e.g. in Ref.~\cite{Giusti:2008vb} and we provide the corresponding
expressions for $SU(4|1)$ and $SU(4|2)$ in appendix \ref{app:props}.
The last term, $\Pi_{\mu\nu}^{(a,b),\rm count.}(\mu)$,
contains the counter terms. We compute $\Pi_{\mu\nu}^{(a,b)}(q,\mu)$
in dimensional regularisation as
explained in detail in Ref.~\cite{Golowich:1995kd} where also the solutions
for the loop integrals have been derived and reduced to the
integral $\bar B_{21}(q^2,M^2)$ defined in appendix \ref{app:B21}.
\section{Results and applications to Lattice QCD}
Here we present the results for the VP at NLO in chiral
perturbation theory for various underlying graded flavour-symmetry groups. In
particular, we consider the theories with $N_f=2$ flavours without or with
an additional quenched strange quark and also the $N_f=2+1$ theory.
In each case we present the expressions in the effective theory for the full
VP and also for the contributions from quark-connected and
quark-disconnected diagrams.
\subsection{$N_f=2$ in $SU(3|1)$ PQ$\chi$PT}
For the theory with two dynamical and degenerate light quarks only we found the following expressions for the VP:
\begin{equation}
\begin{array}{l@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{0mm}}l@{\hspace{0cm}}l@{\hspace{1mm}}c@{\hspace{1mm}}lll}
\Pi_{\rm Full}^{(3|1)}(q^2)&=-&&\Big(\Lambda^{(3|1)}(\mu)
&+\,\frac{2}{9}h_s&+&i4\bar B_{21}(\mu^2,q^2,M^2_\pi)\Big)\,,\\[3mm]
\Pi_{\rm Conn}^{(3|1)}(q^2)&=-&\frac{10}{9}&\Big(\Lambda^{(3|1)}(\mu)&&+&i 4\bar B_{21}(\mu^2,q^2,M_\pi^2)\Big)\,,\\[3mm]
\Pi_{\rm Disc}^{(3|1)}(q^2)&=\phantom{-}&\frac{1}{9}&\Big(\Lambda^{(3|1)}(\mu)&-\,{2}h_s&+&i4\bar B_{21}(\mu^2,q^2,M_\pi^2)\Big)\,,
\end{array}
\end{equation}
where
\begin{equation}
\begin{array}{l@{\hspace{1mm}}c@{\hspace{1mm}}l@{\hspace{1mm}}c@{\hspace{1mm}}r@{\hspace{1mm}}c@{\hspace{1mm}}r@{\hspace{1mm}}ccc}
\Lambda^{(3|1)}(\mu)&=&-8 h_2(\mu)\,.\\[3mm]
\end{array}
\end{equation}
The integral $\bar B_{21}(\mu^2,q^2,M^2)$ is defined in appendix
\ref{app:B21}.
All contributions are parameterised in terms of the low-energy constant
$h_2$ and the pion mass $M_\pi$.
The full expression as well as the disconnected piece also depend
on the parameter $h_s$. This is
a peculiarity of the two-flavour theory and we will see in the next
sub-sections that the corresponding expressions in the presence of a
strange valence quark do not depend on this additional parameter.
Hence, a prediction of the quark-disconnected
diagram is not possible without the knowledge of $h_2$
and $h_s$. These parameters however,
can in principle be determined from simulations of lattice QCD and $h_s$ can
also be obtained by matching to the expressions in the next section.
The application we have in mind when discussing the VP is the
leading hadronic contribution to the muon anomalous magnetic moment defined
in Eq.~(\ref{eq:amu}). As discussed in section \ref{sec:3},
$a_\mu^{\rm HLO}$ depends on
$\hat{\Pi}(q^2)=4\pi^2\left[\Pi(q^2)-\Pi(0)\right]$. We observe that
all reference to the low-energy constants disappears in this difference
at NLO in the effective theory and we find \cite{Juttner:2009yb}
\begin{equation}
{{\hat{\Pi}_{\rm Disc}(q^2)}\over{\hat{\Pi}_{\rm Conn}(q^2)}}=-{{1}\over{10}}\,.
\end{equation}
At this order in the effective theory it is therefore sufficient to
compute the quark-connected piece contributing to $a_\mu^{\rm HLO}$ and
then correct for the quark-disconnected piece using the above relation which
predicts a 10\% negative shift for all values of the momentum (note in this
context the discussion in section \ref{subsec:disclaimer}).
The disconnected contribution at NLO turns out to have the same
momentum and quark-mass dependence as the connected piece.
\subsection{$N_f=2$ plus a quenched strange quark in $SU(4|2)$ PQ$\chi$PT}\label{subsec:4bar2}
For the theory with two dynamical and degenerate light quarks and one
quenched strange quark one needs to consider $SU(4|2)$ PQ$\chi$PT,
where we found the following expressions for the vacuum
polarisation:
\begin{equation}
\begin{array}{l@{\hspace{0mm}}c@{\hspace{0mm}}c@{\hspace{0mm}}c@{\hspace{0mm}}r@{\hspace{1mm}}c@{\hspace{1mm}}r@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{0mm}}c}
\Pi_{\rm Full}^{(4|2)}(q^2)&=&\Lambda^{(4|2)}(\mu)&-4i\Big(&
\bar B_{21}(\mu^2,q^2,M_\pi^2)
&-&\frac 19 \bar B_{21}(\mu^2,q^2,M_{ss}^2)
&+&\frac {4}9 \bar B_{21}(\mu^2,q^2,M_{K}^2)&\Big)\,,\\[5mm]
\Pi^{(4|2)}_{\rm Conn}(q^2)&=&\Lambda^{(4|2)}(\mu) &-4i\Big(&
\frac{10}{9}\bar B_{21}(\mu^2,q^2,M_\pi^2)
&&&+&\frac{2 }{9}\bar B_{21}(\mu^2,q^2,M_{K}^2)&\Big)\,,\\[5mm]
\Pi^{(4|2)}_{\rm Disc}(q^2)&=&&-4i\Big(&
-\frac{1}{9}\bar B_{21}(\mu^2,q^2,M_\pi^2)
&-&\frac{1 }{9}\bar B_{21}(\mu^2,q^2,M_{ss}^2)
&+&\frac{2 }{9}\bar B_{21}(\mu^2,q^2,M_{K}^2)&\Big)\,,
\end{array}
\end{equation}
where
\begin{equation}
\begin{array}{l@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{1mm}}r@{\hspace{1mm}}c@{\hspace{1mm}}r@{\hspace{1mm}}ccc}
\Lambda^{(4|2)}(\mu)&=&8h_2(\mu)\,.
\end{array}
\end{equation}
The full expression and the connected piece depend on the low energy constant
$h_2$ and the pion and kaon mass. An additional dependence on the
mass of the strange-quark enters in terms of the mass
$M^2_{ss}=2B m_s$ of an
un-physical meson made by the two quenched
strange quarks.
It is well known that in the $N_f=2+1$-theory
the quark-disconnected contribution to the
hadronic VP vanishes in the limit of equal quark-masses.
This $SU(3)$
symmetry prohibits the presence of low-energy constants in the expression
for the quark-disconnected diagrams at NLO and this is indeed what we found
when computing $\Pi^{(4|2)}_{\rm Disc}(q^2)$. At NLO in the effective theory
we therefore
provide an entirely parameter-free prediction of the quark-disconnected
diagram.
Moreover, as in the case of the $N_f=2$-theory, also $\hat{\Pi}(q^2)$ is free of
low-energy constants and a
parameter-free prediction for the ratio of the quark-disconnected
contribution to the quark-connected contribution can be made.
As can be seen in figure \ref{fig:results} (a),
\begin{figure}
\centering
\subfigure[]{
\begin{minipage}{.45\linewidth}
\psfrag{xlabel}[t][t][1][0]{$q^2/{\rm GeV^2}$}
\psfrag{ylabel}[c][t][1][0]{
$\hat\Pi_{\rm Disc}(q^2)/\hat\Pi_{\rm Conn}(q^2)$}
\hspace{-15mm}\epsfig{scale=.9,file=plots/Nf2.eps}\\[2mm]
\end{minipage}
}\\
\subfigure[]{
\begin{minipage}{.45\linewidth}
\psfrag{xlabel}[t][t][1][0]{$q^2/{\rm GeV^2}$}
\psfrag{ylabel}[c][t][1][0]{
$\hat\Pi_{\rm Disc}(q^2)/\hat\Pi_{\rm Conn}(q^2)$}
\hspace{-15mm}\epsfig{scale=.9,file=plots/Nf3.eps}\\[2mm]
\end{minipage}
}
\caption{The plots illustrate the ratio of the contributions
from quark-disconnected and quark-connected diagrams
to the VP as a function of the momentum $q^2$
at NLO in chiral perturbation theory for the cases (a)
$N_f=2$ with a quenched strange quark,
(b) $N_f=2+1$. In both cases the lines are from top to bottom:
the limit $M_\pi =M_K$, then fixed $M_K=495$MeV and
$M_\pi=400,\,300,\,200,\,139$MeV. The bottom most-line
at -1/10 is the result for the $N_f=2$ theory.
}
\label{fig:results}
\end{figure}
in the limits $M_K\to\infty$ and $M_K=M_\pi$, the above formulae reproduce the
results for the
$N_f=2$ theory and the vanishing of the disconnected piece in the
$M_K=M_\pi$-limit, respectively.
\subsection{$N_f=2+1$ in $SU(4|1)$ PQ$\chi$PT}\label{subsec:4bar1}
For the theory with two dynamical and degenerate light quarks and one dynamical
strange quark we found the following expressions for the vacuum
polarisation:
\begin{equation}
\begin{array}{l@{\hspace{0mm}}c@{\hspace{0mm}}c@{\hspace{0mm}}c@{\hspace{0mm}}r@{\hspace{1mm}}c@{\hspace{1mm}}r@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{0mm}}c}
\Pi^{(4|1)}_{\rm Full}(q^2)&=&\Lambda^{(4|1)}(\mu)&-4i\Big(&
\bar B_{21}(\mu^2,q^2,M_\pi^2)
&&&+& \bar B_{21}(\mu^2,q^2,M_{K}^2)&\Big)\,,\\[5mm]
\Pi^{(4|1)}_{\rm Conn}(q^2)&=&\Lambda^{(4|1)}(\mu) &-4i\Big(&
\frac{10}{9}\bar B_{21}(\mu^2,q^2,M_\pi^2)
&+&\frac{1 }{9}\bar B_{21}(\mu^2,q^2,M_{ss}^2)
&+&\frac{7}{9}\bar B_{21}(\mu^2,q^2,M_{K}^2)&\Big)\,,\\[5mm]
\Pi^{(4|1)}_{\rm Disc}(q^2)&=&&-4i\Big(&
-\frac{1}{9}\bar B_{21}(\mu^2,q^2,M_\pi^2)
&-&\frac{1 }{9}\bar B_{21}(\mu^2,q^2,M_{ss}^2)
&+&\frac{2 }{9}\bar B_{21}(\mu^2,q^2,M_{K}^2)&\Big)\,,
\end{array}
\end{equation}
where
\begin{equation}
\begin{array}{l@{\hspace{1mm}}c@{\hspace{1mm}}l@{\hspace{1mm}}c@{\hspace{1mm}}r@{\hspace{1mm}}c@{\hspace{1mm}}r@{\hspace{1mm}}ccc}
\Lambda^{(4|1)}&=&-2\left( L_{10}(\mu)+2 H_1(\mu)\right)\,.\\[3mm]
\end{array}
\end{equation}
As in the previous sub-section,
the full expression and the connected piece depend on low-energy constants,
this time
$L_{10}$ and $H_1$, and the pion and kaon mass. Also here
a dependence on the
mass of the strange-quark enters in terms of the mass
$M^2_{ss}=2B m_s$ of an
un-physical meson made of a dynamical and a quenched
strange quark.
In the same way as in the previous section $SU(3)$
symmetry prohibits the presence of low-energy constants in the expression
for the quark-disconnected diagram at NLO.
At this order in the effective theory we are therefore able to
provide an entirely parameter-free prediction of the quark-disconnected
diagram.
Again, $\hat{\Pi}(q^2)$ is free of
low-energy constants and a
parameter-free prediction for the ratio of the quark-disconnected
contribution to the quark-connected contribution can be made.
We illustrate this result in figure \ref{fig:results} (b).
In the same way as for the $N_f=2$ theory with a quenched strange quark,
in the limits limits $M_K\to\infty$ and $M_K=M_\pi$, the above formulae reproduce
the results for the $N_f=2$ theory and the vanishing of the disconnected piece in the
$SU(3)$-limit, respectively. As a further check, our formulae reproduce
the result in Ref.~\cite{Golowich:1995kd}.
From the comparison
of the plots (a) and (b) we see little influence of the dynamical strange quark.
\subsection{Discussion}\label{subsec:disclaimer}
Before we suggest applications to simulations of lattice QCD we would like
to point out obvious
limitations of the above formulae. While the underlying idea of the approach,
i.e. Eq.~(\ref{eqn:master}), is exact in QCD and in the chiral effective theory,
the results presented
here are only valid up to NLO chiral perturbation theory.
If at all, this approximation only holds for small quark-masses and only
for small values of the momentum. There are
in fact doubts that the
physical strange quark mass can be described reliably within chiral perturbation
theory \cite{Allton:2008pn}. One might hope that cancellations in the
ratio $\hat\Pi_{\rm Disc}(q^2)/\hat\Pi_{\rm Conn}(q^2)$ will improve the
convergence
of the expansion in the meson masses and the momentum. However,
only expensive numerical simulations in lattice QCD and higher order
computations in the effective theory will be able to tell.
At least as important is the fact that instead of
coupling to two pions, the photon
will also couple to vector resonances. It is \textit{a priori} not clear
if at all or in which kinematical regime chiral perturbation theory
parameterises
these dynamics correctly
in terms of the low-energy constants of the chiral effective
Lagrangian. Aubin and Blum \cite{Aubin:2006xv} for example found
indications for vector-dominance in the analysis of their lattice QCD
data for the hadronic VP.
We expect sizeable corrections to the expressions found here.
There do exist models, generally known as ``resonace $\chi$PT `` models,
where the vector resonances are
dynamical degrees of freedom (see e.g. \cite{Ecker:1988te}).
Conceptually these models are not as solid as $\chi$PT, to the extent that
no systematic power counting can be formulated.
While clearly desirable,
a study of how the vector degrees of freedom will modify the expressions
is beyond the scope of the current work but in principle
technically straight forward within the framework of resonace $\chi$PT,
which is however subject to the theoretical issues mentioned above.
Given these remarks, the results presented here can only provide an
order-of-magnitude estimate and should be used with great care.
In the absence of
any other quantitative knowledge of the quark-disconnected diagrams, the
results can be used in order to correct the lattice
data for the connected contributions.
\subsection{Applications to lattice QCD}\label{subsec:lat-app}
{\bf Partially twisted boundary conditions for the VP:} Partially twisted
boundary conditions \cite{Sachrajda:2004mi,Bedaque:2004ax,Bedaque:2004kc,deDivitiis:2004kq,Tiburzi:2005hg,Flynn:2005in} by now have become a standard
tool in lattice hadron phenomenology. For the boundary condition
$q_{i}(x_k+L)=e^{i\theta_{k,i}}q_{i}(x)$
for the valence quark-flavours $q_1$ and $q_2$
of a pseudo-scalar meson of mass $m$ in a finite lattice
of spatial extent $L$, the
dispersion relation takes the form
$E({\vec\theta}_1,{\vec\theta}_2)=\sqrt{m^2+({\vec \theta}_1/L-{\vec \theta}_2/L)^2}$ \cite{deDivitiis:2004kq,Sachrajda:2004mi,Flynn:2005in},
which shows how the periodicity of the fermionic fields can be modified
in order to induce spacial momentum to hadrons.
Besides an exponentially suppressed and computable finite size effect
the choice of quark-field
boundary conditions for processes with only one initial and/or
final hadron state introduces no further systematics
\cite{Sachrajda:2004mi}.
The effect of partial twisting cancels however
in flavour-neutral mesons, as is clear from the form of the above
dispersion relation. Due to the flavour-diagonal
structure of the electro-magnetic current,
partial twisting would therefore naively have
no effect on the hadronic VP. The
prescription introduced here however,
allows for isolating a flavour-off-diagonal connected contribution
and thus for inducing arbitrary values of the momentum
for the VP. In other words we can assign different twisting angles
to the quarks $q$ and $q_v$ in Eq.~\ref{eqn:master}, which has the net
effect of inducing momentum in the connected part.
The effectiveness of this method in the numerical computation of the connected
part of the hadronic VP will be illustrated in a forthcoming
publication~\cite{Mainz:g-2}.
The relations in the chiral effective theory between the quark-connected
and the quark-disconnected piece then allow to predict also the
momentum-dependence of the disconnected piece.
A very similar argument was already used
in Refs.~\cite{Boyle:2007wg,Jiang:2008te} in order to justify the use of
partially twisted boundary conditions for the pion's vector form factor.
The exponential suppression in the volume of the isospin breaking
introduced by partially twisting only one of the light quarks is also
discussed in Refs.~\cite{Boyle:2007wg,Jiang:2008te}.
\section{Propagators in chiral perturbation theory}\label{app:props}
Here we provide the definition of propagators in PQ$\chi$PT
with the graded symmetry groups $SU(4|1)$ and $SU(4|2)$.
For simplicity all results are quoted in the
limit where the Goldstone bosons and the corresponding ghosts are
mass-degenerate. This limit
is sufficient for the discussion in the main body of this paper.
For completeness
we note that the case $SU(3|1)$ has been worked out nicely
in \cite{Giusti:2008vb}
and for most of the discussion we adopt the conventions laid out there.
\subsection{Propagators in $SU(4|1)$ chiral perturbation theory}
We consider the case of the effective theory with $N_f=2+1$ dynamical
flavours with two degenerate
light quarks of mass $m$, one strange quark
of mass $m_s$ and one quenched $r$ quark with
$m_s=m_r= m_{r_g}$.
The dynamical (not necessarily physical)
degrees of freedom in the effective theory have masses
\begin{equation}
M^2_\pi=2 B m\,,\qquad M^2_K=B(m+m_s)\,,\qquad
M^2_{ss}=2 B m_s\,.
\end{equation}
The propagator $G^{ij}(k^2)$ is defined as follows:
The Lagrangian in Eq.~(\ref{eqn:O2Lag}) with
$\mathcal{L}_{\rm int}^{(2)}=0$ can be rewritten as
\begin{equation}
\mathcal{L}^{(2)}=
\frac 12\, g^{ab}\left(
\partial_\mu \phi^a \partial_\mu \phi^b+M_a^2\phi^a\phi^b\right)+
\frac{1}{6}h^{ab}(M_\pi^2-M_K^2) \phi^a \phi^b\,,
\end{equation}
where
\begin{equation}
M_a^2=\left\{\begin{array}{lll}
M_\pi^2&{\rm for}&a=1,2,3\,,\\
\frac 12\left(M_\pi^2+M_K^2\right)&{\rm for}&a=4,5,6,7,9,10,11,12,16,17,18,19\,,\\
M_{ss}^2&{\rm for}&a=8,13,14,15,20,21,22,23,24\,,\\
\end{array}\right.
\end{equation}
and where for our choice of the $SU(4|1)$-generators
\begin{equation}
\begin{array}{lcl}
h^{ab}&=&\;\;\;(\sqrt{2}\,g^{a\,8}+g^{a\,15}-g^{a\,24})(\sqrt{2}\,g^{b\,8}+g^{b\,15}-g^{b\,24})\,,\\[1mm]
l^{ab}&=&-(\sqrt{2}\,g^{a\,8}+g^{a\,15}+g^{a\,24})(\sqrt{2}\,g^{b\,8}+g^{b\,15}+g^{b\,24})\,.
\end{array}
\end{equation}
In terms of these matrices the propagator is given by
\begin{eqnarray}
G^{ab}(k^2)&=&
g^{ab}G_1(k^2,M_a^2)
+\frac{1}{3}\,l^{ab}\, (M_\pi^2-M_K^2)
G_2(k^2,M_\eta^2,M_{ss}^2)\,,
\end{eqnarray}
where we assume $M_\eta^2=2B(2m_s+m)/3$
and where
\begin{eqnarray}
G_1(k^2,M^2)&=&
\frac{1}{(k^2+M^2)}\,,\nonumber\\[-3mm]
\label{eq:finalprops}\\[-3mm]
G_2(k^2,M_1^2,M_2^2)&=&
\frac {1}{(k^2+M_1^2)(k^2+M_2^2)}\,.\nonumber
\end{eqnarray}
\subsection{Propagators in $SU(4|2)$ chiral perturbation theory}
We consider the case of the effective theory with $N_f=2$ dynamical
flavours with two degenerate light quarks of mass $m$ and two
quenched quarks $s$ and $r$ with the mass of the strange quark
$m_s=m_{s_g}=m_r= m_{r_g}$.
The dynamical degrees of freedom in the effective theory have masses
\begin{equation}
M^2_\pi=2 B m\,,\qquad M^2_K=B(m+m_s)\,,\qquad
M^2_{ss}=2 B m_s\,.
\end{equation}
The propagator $G^{ab}(k^2)$ is defined as follows:
The Lagrangian in Eq.~(\ref{eqn:O2Lag}) restricted to lowest order and with
$\mathcal{L}_{\rm int}^{(2)}=0$ can be rewritten as
\begin{eqnarray}
\mathcal{L}^{(2)}&=&
\frac 12g^{ab}\left(
\partial_\mu \phi^a \partial_\mu \phi^b+M_a^2\phi^a\phi^b\right)+
\frac{1}{6}h^{ab}(M_\pi^2-M_K^2) \phi^a \phi^b\,,
\end{eqnarray}
where
\begin{equation}
M_a^2=\left\{\begin{array}{lll}
M_\pi^2&{\rm for}&a=1,2,3\,,\\
\frac 12\left(M_\pi^2+M_K^2\right)&{\rm for}&a=4,5,6,7,9,10,11,12,16,17,18,19,25,26,27,28\,,\\
M_{ss}^2&{\rm for}&a=8,13,14,15,20,21,22,23,24,29,30,31,32,33,34,35\,,\\
\end{array}\right.
\end{equation}
and where for our choice of the $SU(4|2)$-generators
\begin{equation}
\begin{array}{l@{\hspace{1mm}}c@{\hspace{1mm}}l}
h^{ab}&=&\;\;\;(\sqrt{2}\,g^{a\,8}+ g^{a\, 15}-g^{a\,24}-\sqrt{2}\,g^{a\,35})
(\sqrt{2}\,g^{b\,8}+ g^{b\, 15}-g^{b\,24}-\sqrt{2}\,g^{b\,35})\,,\\
l^{ab}&=&-(\sqrt{2}\,g^{a\,8}+ g^{a\, 15}+g^{a\,24}+\sqrt{2}\,g^{a\,35})
(\sqrt{2}\,g^{b\,8}+ g^{b\, 15}+g^{b\,24}+\sqrt{2}\,g^{b\,35})\;.
\end{array}
\end{equation}
In terms of these matrices the propagator is given by
\begin{eqnarray}
\hspace{-1cm}G^{ab}(k^2)&=&
g^{ab}G_1(k^2,M_a^2)\nonumber
+ \frac{1}{3}l^{ab} (M_\pi^2-M_K^2)G_2(k^2,M_{ss}^2,M_{ss}^2) \,,
\end{eqnarray}
where $G_1$ and $G_2$ are defined as in Eq.~(\ref{eq:finalprops}).
\section{The integral $\bar B_{21}(q^2,M^2)$}\label{app:B21}
Here we just quote the integral itself. Its derivation from the sum of the
unitary contribution in Eq.~(\ref{eq:unitary}) and the tadpole in
Eq.~(\ref{eq:tadpole}) can be found for example in the
appendix of Ref.~\cite{Golowich:1995kd}:
\begin{equation}
\bar B_{21}(q^2,M^2)=\frac{1}{12}\left[
\left(
1-\frac{4 M^2}{q^2}\right)\bar B(q^2,M^2)-\frac i{48\pi^2}
\right]\,,
\end{equation}
where for $q^2<4M^2$
\begin{eqnarray}
\bar B(q^2,M^2)&=&-\frac{i}{16\pi^2}\int\limits_0^1
dx \log\left(
1-x(1-x)\frac{q^2}{M^2}
\right)\nonumber\\
&\stackrel{x=\frac{4M^2}{q^2}}{=}&-\frac{i}{16\pi^2}\left[
-2+\sqrt{1-x}
\log\left(
\frac{\sqrt{1-x}+1}{\sqrt{1-x}-1}
\right)
\right]\nonumber\\
&=&\frac{i}{96\pi^2}\frac{q^2}{M^2}+\frac{i}{960\pi^2}\frac{q^4}{M^4}+
\dots\,.
\end{eqnarray}
In the body of this paper we found it convenient to absorb a logarithmic term
into the expression for $\bar B_{21}$:
\begin{equation}
\bar B_{21}(\mu^2,p^2,M^2)=\bar B_{21}(p^2,M^2)-\frac i4 \frac{1}{48\pi^2}
\log\left(\frac{M^2}{\mu^2}\right)\,.
\end{equation}
\section{Conclusions and outlook}
In this paper we show how quark-disconnected
contributions to hadronic $n$-point functions can be estimated
within the framework of
partially quenched chiral perturbation theory. As an example we derive predictions
for the quark-disconnected contribution to the photon's
vacuum polarisation for QCD with two degenerate light quarks,
with and without a quenched strange quark, and also for the case of
the three-flavour theory. In the presence of a quenched strange quark
the prediction for the quark-disconnected contribution turns out to be
parameter-free.
The vacuum polarisation is currently computed by
various lattice collaborations with the aim to make first-principles predictions
for the leading hadronic contribution to the muon anomalous moment
\cite{Aubin:2006xv,Renner:2009by,Juttner:2009yb}. Since
precise computation of quark-disconnected correlators from first
principles
still remain numerically extremely expensive they are often neglected in
such computations. The
approach presented here will allow to improve estimates of the systematic
error introduced in this way.
The method can be applied to other physical observables of interest that
receive contributions from quark-disconnected diagrams, like
the strange-quark
content in the nucleon, the nucleon electric dipole moment
or the pion scalar form-factor.
The discussion presented here is limited to NLO chiral perturbation theory.
As a next step we plan to include the vector degrees of
freedom and it would also be important to
extend the computation for the case of the vacuum polarisation to higher
orders in the chiral expansion
in order to assess how robust the predictions reported here are.
The inclusion of the leading lattice artifacts in the effective theory
is also straight forward within the framework set up by
Sharpe and Singleton in Ref.~\cite{Sharpe:1998xm}.
A spin-off of the ideas presented here is a method that allows for projecting
(the connected part of) correlators containing flavour-diagonal currents
onto any desired momentum using partially twisted boundary
conditions.\\[2mm]
|
{
"timestamp": "2010-11-19T02:02:03",
"yymm": "1009",
"arxiv_id": "1009.3783",
"language": "en",
"url": "https://arxiv.org/abs/1009.3783"
}
|
\subsection[#1]{\sc #1}}
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\begin{document}
\title{An inverse theorem for the Gowers $U^{s+1}[N]$-norm}
\author{Ben Green}
\address{Centre for Mathematical Sciences\\
Wilberforce Road\\
Cambridge CB3 0WA\\
England }
\email{b.j.green@dpmms.cam.ac.uk}
\author{Terence Tao}
\address{Department of Mathematics\\
UCLA\\
Los Angeles, CA 90095\\
USA}
\email{tao@math.ucla.edu}
\author{Tamar Ziegler}
\address{Department of Mathematics \\
Technion - Israel Institute of Technology\\
Haifa, Israel 32000}
\email{tamarzr@tx.technion.ac.il}
\subjclass{11B30}
\begin{abstract} We prove the \emph{inverse conjecture for the Gowers $U^{s+1}[N]$-norm} for all $s \geq 1$; this is new for $s \geq 4$. More precisely, we establish that if $f : [N] \rightarrow [-1,1]$ is a function with $\Vert f \Vert_{U^{s+1}[N]} \geq \delta$ then there is a bounded-complexity $s$-step nilsequence $F(g(n)\Gamma)$ which correlates with $f$, where the bounds on the complexity and correlation depend only on $s$ and $\delta$. From previous results, this conjecture implies the Hardy-Littlewood prime tuples conjecture for any linear system of finite complexity.
\end{abstract}
\maketitle
\setcounter{tocdepth}{1}
\tableofcontents
\section{Introduction}
The purpose of this paper is to establish the general case of a conjecture named the \emph{Inverse Conjecture for the Gowers norms} by the first two authors in \cite[Conjecture 8.3]{green-tao-linearprimes}. If $N$ is a (typically large) positive integer then we write $[N] := \{1,\dots,N\}$. For each integer $s \geq 1$ the inverse conjecture $\GI(s)$, whose statement we recall shortly, describes the structure of $1$-bounded functions $f : [N] \rightarrow \C$ whose $(s+1)^{\operatorname{st}}$ Gowers norm $\Vert f \Vert_{U^{s+1}[N]}$ is large. These conjectures together with a good deal of motivation and background to them are discussed in \cite{green-icm,green-tao-u3inverse,green-tao-linearprimes}. The conjectures $\GI(1)$ and $\GI(2)$ have been known for some time, the former being a straightforward application of Fourier analysis, and the latter being the main result of \cite{green-tao-u3inverse} (see also \cite{sam} for the characteristic $2$ analogue). The case $\GI(3)$ was also recently established by the authors in \cite{u4-inverse}. The aim of the present paper is to establish the remaining cases $\GI(s)$ for $s \geq 3$, in particular reestablishing the results in \cite{u4-inverse}.
We begin by recalling the definition of the Gowers norms. If $G$ is a finite abelian group, $d \geq 1$ is an integer, and $f : G \rightarrow \C$ is a function then we define
\begin{equation}\label{ukdef}
\Vert f \Vert_{U^{d}(G)} := \left( \E_{x,h_1,\dots,h_d \in G} \Delta_{h_1} \ldots \Delta_{h_d} f(x)\right)^{1/2^d},
\end{equation}
where $\Delta_h f$ is the multiplicative derivative
$$ \Delta_h f(x) := f(x+h) \overline{f(x)}$$
and $\E_{x \in X} f(x) := \frac{1}{|X|} \sum_{x \in X} f(x)$ denotes the average of a function $f: X \to \C$ on a finite set $X$. Thus for instance we have
\[ \Vert f \Vert_{U^2(G)} := \left( \E_{x,h_1,h_2 \in G} f(x) \overline{f(x+h_1) f(x+h_2)} f(x+h_1 + h_2)\right)^{1/4}.\]
One can show that $U^d(G)$ is indeed a norm on the functions $f: G \to \C$ for any $d \geq 2$, though we will not need this fact here.
In this paper we will be concerned with functions on $[N]$, which is not quite a group. To define the Gowers norms of a function $f : [N] \rightarrow \C$, set $G := \Z/\tilde N\Z$ for some integer $\tilde N \geq 2^d N$, define a function $\tilde f : G \rightarrow \C$ by $\tilde f(x) = f(x)$ for $x = 1,\dots,N$ and $\tilde f(x) = 0$ otherwise, and set
\[ \Vert f \Vert_{U^d[N]} := \Vert \tilde f \Vert_{U^d(G)} / \Vert 1_{[N]} \Vert_{U^d(G)},\] where $1_{[N]}$ is the indicator function of $[N]$. It is easy to see that this definition is independent of the choice of $\tilde N$. One could take $\tilde N := 2^d N$ for definiteness if desired.
The \emph{Inverse conjecture for the Gowers $U^{s+1}[N]$-norm}, abbreviated as $\GI(s)$, posits an answer to the following question.
\begin{question}
Suppose that $f : [N] \rightarrow \C$ is a function bounded in magnitude by $1$, and let $\delta > 0$ be a positive real number. What can be said if $\Vert f \Vert_{U^{s+1}[N]} \geq \delta$?
\end{question}
Note that in the extreme case $\delta = 1$ one can easily show that $f$ is a phase polynomial, namely $f(n)=e(P(n))$ for some polynomial $P$ of degree at most $s$. Furthermore, if $f$ correlates with a phase polynomial, that is to say if $|\E_{n \in [N]} f(n) \overline{e( P(n))}| \geq \delta$, then it is easy to show that
$\Vert f \Vert_{U^{s+1}[N]} \geq c(\delta)$. It is natural to ask whether the converse is also true - does a large Gowers norm imply correlation with a polynomial phase function?
Surprisingly, the answer is no, as was observed by Gowers \cite{gowers-4aps} and, in the related context of \emph{multiple recurrence}, somewhat earlier by Furstenberg and Weiss \cite{furst, fw-char}. The work of Furstenberg-Weiss and Conze-Lesigne \cite{conze} draws attention to the role of homogeneous spaces $G/\Gamma$ of nilpotent Lie groups, and subsequent work of Host and Kra \cite{host-kra} provides a link, in an ergodic-theoretic context, between these spaces and certain seminorms with a formal similarity to the Gowers norms under discussion here. Later work of Bergelson, Host and Kra \cite{bhk} highlights the role of a class of functions arising from these spaces $G/\Gamma$ called \emph{nilsequences}. The inverse conjecture for the Gowers norms, first formulated precisely in \cite[\S 8]{green-tao-linearprimes}, postulates that this class of functions (which contains the polynomial phases) represents the full set of obstructions to having large Gowers norm.
We now recall that precise formulation. Recall that an \emph{$s$-step nilmanifold} is a manifold of the form $G/\Gamma$, where $G$ is a connected, simply-connected nilpotent Lie group of step at most $s$ (i.e. all $s+1$-fold commutators of $G$ are trivial), and $\Gamma$ is a discrete, cocompact\footnote{A subgroup $\Gamma$ of a topological group $G$ is \emph{cocompact} if the quotient space $G/\Gamma$ is compact.} subgroup of $G$.
\begin{conjecture}[$\GI(s)$]\label{gis-conj} Let $s \geq 0$ be an integer, and let $0 < \delta \leq 1$. Then there exists a finite collection ${\mathcal M}_{s,\delta}$ of $s$-step nilmanifolds $G/\Gamma$, each equipped with some smooth Riemannian metric $d_{G/\Gamma}$ as well as constants $C(s,\delta), c(s,\delta) > 0$ with the following property. Whenever $N \geq 1$ and $f : [N] \rightarrow \C$ is a function bounded in magnitude by $1$ such that $\Vert f \Vert_{U^{s+1}[N]} \geq \delta$, there exists a nilmanifold $G/\Gamma \in {\mathcal M}_{s,\delta}$, some $g \in G$ and a function $F: G/\Gamma \to \C$ bounded in magnitude by $1$ and with Lipschitz constant at most $C(s,\delta)$ with respect to the metric $d_{G/\Gamma}$ such that
$$ |\E_{n \in [N]} f(n) \overline{F(g^n x)}| \geq c(s,\delta).$$
\end{conjecture}
We remark that there are many equivalent ways to reformulate this conjecture. For instance, instead of working with a finite family ${\mathcal M}_{s,\delta}$ of nilmanifolds, one could work with a single nilmanifold $G/\Gamma = G_{s,\delta}/\Gamma_{s,\delta}$, by taking the Cartesian product of all the nilmanifolds in the family. Other reformulations include an equivalent formulation using polynomial nilsequences rather than linear ones (see Conjecture \ref{gis-poly}) and an ultralimit formulation (see Conjecture \ref{gis-conj-nonst}). One can also formulate the conjecture using bracket polynomials, or local polynomials; see \cite{green-tao-u3inverse} for a discussion of these equivalences in the $s=2$ case.
Let us briefly review the known partial results on this conjecture:
\begin{enumerate}
\item $\GI(0)$ is trivial.
\item $\GI(1)$ follows from a short Fourier-analytic computation.
\item $\GI(2)$ was established about five years ago in \cite{green-tao-u3inverse}, building on work of Gowers \cite{gowers-4aps}.
\item $\GI(3)$ was established, quite recently, in \cite{u4-inverse}.
\item In the extreme case $\delta = 1$ one can easily show that $f(n)=e(P(n))$ for some polynomial $P$ of degree at most $s$, and every such function \emph{is} an $s$-step nilsequence by a direct construction. See, for example, \cite{green-tao-u3inverse} for the case $s = 2$.
\item In the almost extremal case $\delta \geq 1- \eps_s$, for some $\eps_s > 0$, one may see that $f$ correlates with a phase $e(P(n))$ by adapting arguments first used in the theoretical computer-science literature \cite{akklr}.
\item The analogue of $\GI(s)$ in ergodic theory (which, roughly speaking, corresponds to the asymptotic limit $N \to \infty$ of the theory here; see \cite{host-kra-uniformity} for further discussion) was formulated and established in \cite{host-kra}, work done independently of the work of Gowers (see also the earlier paper \cite{hk1}). This work was the first place in the literature to link objects of Gowers-norm type (associated to functions on a measure-preserving system $(X, T,\mu)$) with flows on nilmanifolds, and the subsequent paper \cite{bhk} was the first work to underline the importance of \emph{nilsequences}. The formulation of $\GI(s)$ by the first two authors in \cite{green-tao-linearprimes} was very strongly influenced by these works. For the closely related problem of analysing multiple ergodic averages, the relevance of flows on nilmanifolds was earlier pointed out in \cite{furst, fw-char,lesigne-nil}, building upon earlier work in \cite{conze}. See also \cite{hk0,ziegler} for related work on multiple averages and nilmanifolds in ergodic theory.
\item The analogue of $\GI(s)$ in finite fields of large characteristic was established by ergodic-theoretic methods in \cite{bergelson-tao-ziegler,tao-ziegler}.
\item A weaker ``local'' version of the inverse theorem (in which correlation takes place on a subprogression of $[N]$ of size $\sim N^{c_s}$) was established by Gowers \cite{gowers-longaps}. This paper provided a good deal of inspiration for our work here.
\item The converse statement to $\GI(s)$, namely that correlation with a function of the form $n \mapsto F(g^n x)$ implies that $f$ has large $U^{s+1}[N]$-norm, is also known. This was first established in \cite[Proposition 12.6]{green-tao-u3inverse}, following arguments of Host and Kra \cite{host-kra} rather closely. A rather simple proof of this result is given in \cite[Appendix G]{u4-inverse}.
\end{enumerate}
The main result of this paper is a proof of Conjecture \ref{gis-conj}:
\begin{theorem}\label{mainthm} For any $s \geq 3$,
the inverse conjecture for the $U^{s+1}[N]$-norm, $\GI(s)$, is true.
\end{theorem}
By combining this result with the previous results in \cite{green-tao-linearprimes,green-tao-mobiusnilsequences} we obtain a quantitative Hardy-Littlewood prime tuples conjecture for all linear systems of finite complexity; in particular, we now have the expected asymptotic for the number of primes $p_1 < \ldots < p_k \leq X$ in arithmetic progression, for every fixed positive integer $k$. We refer to \cite{green-tao-linearprimes} for further discussion, as we have nothing new to add here regarding these applications. Several further applications of the $\GI(s)$ conjectures are given in \cite{fhk,green-tao-arithmetic-regularity}. \vspace{11pt}
\section{Strategy of the proof}\label{strategy-sec}
The proof of Theorem \ref{mainthm} is long and complicated, but broadly speaking it follows the strategy laid out in previous works \cite{gowers-4aps,gowers-longaps,green-tao-u3inverse,u4-inverse,sam}. We induct on $s$, assuming that $\GI(s-1)$ has already been established and using this to prove $\GI(s)$. To explain the argument, let us first summarise the main steps taken in \cite{u4-inverse} in order to deduce $\GI(3)$, the inverse theorem for the $U^4$-norm, from $\GI(2)$, the inverse theorem for the $U^3$ norm (established in \cite{green-tao-u3inverse}). Once this is done we will explain some of the extra difficulties involved in handling the general case. For a more extensive (but informal) discussion of the proof strategy, see \cite{gtz-announce}. Once we set up some technical machinery, we will also be able to give a more detailed description of the strategy in \S \ref{overview-sec}.
Here, then, is an overview of the argument in \cite{u4-inverse}.
\begin{enumerate}
\item (Apply induction) If $\Vert f \Vert_{U^4[N]} \gg 1$ then, for many $h$, $\Vert \Delta_h f \Vert_{U^3[N]} \gg 1$ and so $\Delta_h f$ correlates with a $2$-step nilsequence $\chi_h$.
\item (Nilcharacter decomposition) $\chi_h$ may be decomposed as a sum of a special type of nilsequence called a \emph{nilcharacter}, essentially by a Fourier decomposition. For the sake of illustration, these $2$-step nilcharacters may be supposed to have the form
\[ \chi_h(n) = e(\{\alpha_h n\} \beta_h n),\]
although these are not quite nilcharacters due to the discontinuous nature of the fractional part function $x \mapsto \{x\}$, and in any event a general $2$-step nilcharacter will be modeled by a linear combination of such ``bracket quadratic monomials'', rather than by a single such monomial (see \cite{green-tao-u3inverse} for further discussion).
\item (Rough linearity) The fact that $\Delta_h f$ correlates with $\chi_h$ forces $\chi_h$ to behave weakly linearly in $h$. To get a feel for why this is so, suppose that $|f| \equiv 1$; then we have the cocycle identity
\[ \Delta_{h+k} f(n) = \Delta_h f(n+k) \Delta_k f(n).\] To capture something like the same behaviour in the much weaker setting where $\Delta_h f$ correlates with $\chi_h$, we use an extraordinary argument of Gowers \cite{gowers-4aps} relying on the Cauchy-Schwarz inequality. Roughly speaking, the information obtained is of the form
\begin{equation}\label{linear-eq} \chi_{h_1} \chi_{h_2} \sim \chi_{h_3} \chi_{h_4} \quad \mbox{modulo lower order terms} \end{equation} for many $h_1, h_2, h_3, h_4$ with $h_1 + h_2 = h_3 + h_4$.
\item (Furstenberg-Weiss) An argument of Furstenberg and Weiss \cite{fw-char} is adapted in order to study \eqref{linear-eq}. The quantitative distribution theory of nilsequences developed in \cite{green-tao-nilratner} is a major input here. It is concluded that we may assume that the frequency $\beta_h$ does not actually depend on $h$. Note that this step appeared for the first time in the proof of $\GI(3)$; it did not feature in the proof of $\GI(2)$ in \cite{green-tao-u3inverse}.
\item (Linearisation) A similar argument allows one to then assert that
\begin{equation}\label{additive-eq} \alpha_{h_1} + \alpha_{h_2} \approx \alpha_{h_3} + \alpha_{h_4} \pmod{1}
\end{equation}
for many $h_1,h_2,h_3,h_4$ with $h_1 + h_2 = h_3 + h_4$.
\item (Additive Combinatorics) By arguments from additive combinatorics related to the Balog-Szemer\'edi-Gowers theorem \cite{balog,gowers-4aps} and Freiman's theorem, as well as some geometry of numbers, we may then assume that $\alpha_h$ varies ``bracket-linearly'' in $h$, thus
\begin{equation}\label{bracket-lin} \alpha_h = \gamma_1 \{ \eta_1 h\} + \dots + \gamma_d \{\eta_d h\}. \end{equation}
Up to top order, then, the nilcharacter $\chi_h(n)$ can now be assumed to take the form $e(\psi(h,n,n))$, where $\psi$ is ``bracket-multilinear''; it is a sum of terms such as $\{\gamma \{\eta h\} n\} \beta n$.
\item (Symmetry argument) The bracket multilinear form $\psi$ obeys an additional symmetry property. This is a reflection of the identity $\Delta_h \Delta_k f = \Delta_k \Delta_h f$, but transferring this to the much weaker setting in which we merely have correlation of $\Delta_h f$ with $\chi_h$ requires another appeal to Gowers' Cauchy-Schwarz argument from (iii). In fact, the key point is to look at the second order terms in \eqref{linear-eq}.
\item (Integration) Assuming this symmetry, one is able to express
\[ \chi_h(n) \sim \Theta(n+h) \overline{\Theta'(n)}\]
for some bracket cubic functions $\Theta, \Theta'$, which morally take the form \[ \Theta(n), \Theta'(n) \sim e(\psi(n,n,n)/3)\] (for much the same reason that $x^3/3$ is an antiderivative of $x^{2}$). Thus we morally have
\[ \Delta_h f(n) \sim \Theta(n+h) \overline{\Theta'(n)}\]
\item (Construction of a nilsequence) Any bracket cubic form like $e(\psi(n,n,n))$ ``comes from'' a 3-step nilmanifold; this construction is accomplished in \cite{u4-inverse} in a rather \emph{ad hoc} manner.
\item From here, one can analyse lower order terms by the induction hypothesis $\GI(2)$. This is a relatively easy matter.
\end{enumerate}
Let us now discuss the argument of this paper in the light of each point of this outline.
A more detailed outline is given in \S \ref{overview-sec}.
Assume that $\GI(s-1)$ has been established.
\begin{enumerate}
\item (Apply induction) If $\Vert f \Vert_{U^{s+1}[N]} \gg 1$ then, for many $h$, $\Vert \Delta_h f \Vert_{U^s[N]} \gg 1$ and so $\Delta_h f$ correlates with an $(s-1)$-step nilsequence $\chi_h$. This is straightforward (see \S \ref{overview-sec}).
\item (Nilcharacter decomposition) $\chi_h$ may be decomposed into nilcharacters; this is fairly straightforward as well. It is somewhat reassuring to think of $\chi_h(n)$ as having the form $e(\psi_h(n))$, where $\psi_h(n)$ is a bracket polynomial ``of degree $s-1$'', but we will not be working explicitly with bracket polynomials much in this paper, except as motivation and as a source of examples. One of the main challenges one is faced with during an attempt to prove $\GI(4)$ by a direct generalisation of our arguments from \cite{u4-inverse} is the fact that already bracket cubic polynomials are rather complicated to deal with and can take different forms such as $\{\alpha n\}\{\beta n\}\gamma n$ and $\{ \{\alpha n\} \beta n\} \gamma n$.
Instead of objects such as $e(\alpha n\{\beta n\})$, then, we will work with the rather more abstract notion of a \emph{symbol}. This notion, which is fairly central to our paper, is defined and discussed in \S \ref{nilcharacters}. One additional technical point is worth mentioning here. This is the fact that $e(\alpha n\{\beta n\})$ (say) cannot be realised as a nilsequence $F(g^n \Gamma)$ with $F$ \emph{continuous}, and therefore the distributional results of \cite{green-tao-nilratner} do not directly apply. In \cite{u4-inverse} these discontinuities could be understood quite explicitly, but here we take a different approach: we decompose $G/\Gamma$ into $D$ pieces using a smooth partition of unity for some $D=O(1)$, and then work instead with the (smooth) $\C^D$-valued nilsequence consisting of these pieces.
We discuss this device more fully in \S \ref{nilcharacters}, but we emphasise that this is a technical device and the reader is advised not to give this particular aspect of the proof too much attention.
\item (Rough linearity) $\chi_h$ varies roughly linearly in $h$; this is another fairly straightforward modification of the arguments of Gowers, already employed in \cite{u4-inverse}, which is performed in \S \ref{cs-sec}.
\item (Furstenberg-Weiss) This proceeds along similar lines to the corresponding argument in \cite{u4-inverse} but is, in a sense, rather easier once one has developed the device of $\C^D$-valued nilsequences, which allow one to remain in the smooth category; this is accomplised in \S \ref{linear-sec}, after a substantial amount of preparatory material in \S \ref{freq-sec}, \S \ref{reg-sec} and Appendix \ref{equiapp}.
\item (Linearisation) This is also quite similar to the corresponding argument in \cite{u4-inverse}, and is performed in \S \ref{linear-sec}. In both of parts (iv) and (v), the ``bracket calculus'' from \cite{u4-inverse} is replaced by the more conceptual ``symbol calculus'' developed in Appendix \ref{basic-sec}.
\item (Additive Combinatorics) The additive combinatorial input is much the same as in \cite{u4-inverse}. For the convenience of the reader we sketch it in Appendix \ref{app-f}.
\item (Construction of a nilsequence) Our argument differs quite substantially from that in \cite{u4-inverse} at this point. The $s$-step nilobject, which is now a two-variable object $\chi(h,n)$, is constructed \emph{before} the symmetry argument and in a more conceptual manner. This may be compared with the rather \emph{ad hoc} approach taken in \cite{green-tao-u3inverse, u4-inverse}, where various bracket polynomials were merely exhibited as arising from nilsequences. We perform this construction in \S \ref{multi-sec}.
\item (Symmetry argument) We replace $\chi(h,n)$ with an equivalent nilcharacter $\tilde \chi(h,n,\ldots,n)$ where $\tilde \chi$ is a nilcharacter in $s$ variables, that is symmetric in the last $s-1$ variables. The symmetry argument given in \S \ref{symsec} shows that $\tilde \chi(h,n,\ldots,n)$ is equivalent to $\tilde \chi(n,h,\ldots,n)$. Again the key idea in the analysis is to look at the second order terms in \eqref{linear-eq}.
\item (Integration) With the symmetry in hand, we can use the calculus of multilinear nilcharacters essentially express $\tilde \chi(h,n,\ldots,n)$ as the derivative of an expression which is roughly of the form $\tilde \chi(n,\ldots,n)/s$; see \S \ref{symsec} for details.
\item The final step of the argument is relatively straightforward, as before; see \S \ref{overview-sec}.
\end{enumerate}
In our previous paper \cite{u4-inverse} it was already rather painful to keep proper track of such notions as ``many'' and ``correlates with''. Here matters are even worse, and so to organise the above tasks it turns out to be quite convenient to first take an ultralimit of all objects being studied, effectively placing one in the setting of \emph{nonstandard analysis}. This allows one to easily import results from infinitary mathematics, notably the theory of Lie groups and basic linear algebra, into the finitary setting of functions on $[N]$. In \S \ref{nsa-sec} and Appendix \ref{nsa-app} we review the basic machinery of ultralimits that we will need here; we will not be exploiting any particularly advanced aspects of this framework. The reader does not really need to understand the ultrafilter language in order to comprehend the basic structure of the paper, provided that he/she is happy to deal with concepts like ``dense'' and ``correlates with'' in a somewhat informal way, resembling the way in which analysts actually talk about ideas with one another (and, in fact, analogous to the way we wrote this paper). It is possible to go through the paper and properly quantify all of these notions using appropriate parameters $\delta$ and (many) growth functions $\mathcal{F}$. This would have the advantage of making the paper on some level comprehensible to the reader with an absolute distrust of ultrafilters, and it would also remove the dependence on the axiom of choice and in principle provide explicit but very poor bounds. However it would cause the argument to be significantly longer, and the notation would be much bulkier.
Our exposition will be as follows. We will begin by spending some time introducing the ultrafilter language and then, motivated by examples, the notions of nilsequence, nilcharacter and symbol. Once that is done we will, in \S \ref{overview-sec}, give the high-level argument for Theorem \ref{mainthm}; this consist of detailing points (i), (ii) and (x) of the outline above and giving proper statements of the other main points.
The discussion above concerning points (iii), (iv), (v) and (vi) has been simplified for the sake of exposition. In actual fact, these points are dealt with together by a kind of iterative loop, in which more and more bracket-linear structure is placed on the nilcharacters $\chi_h(n)$ by cycling from (iii) to (vi) repeatedly.
We remark that a quite different approach using ultrafilters to the structural theory of the Gowers norms is in the process of being carried out in \cite{szeg-1,szeg-2,szeg-3}; this seems related to the work of Host and Kra, whereas our work ultimately derives from the work of Gowers.
We also make the minor remark that our proof of $\GI(s)$ is restricted to the case $s \geq 3$ case for minor technical reasons. In particular, we take advantage of the non-trivial nature of the degree $s-2$ ``lower order terms'' in the Gowers Cauchy-Schwarz argument (Proposition \ref{gcs-prop}) in the symmetry argument step; and we will also observe that the various ``smooth'' and ``periodic'' error terms arising from the equidistribution theory in Appendix \ref{equiapp} are of degree $1$ and thus negligible compared with the main terms in the analysis, which are of degree $s-1$. The arguments can be modified to give a proof of $\GI(2)$, although this proof would basically be a notationally intensive repackaging of the arguments in \cite{green-tao-u3inverse}.
\emph{Acknowledgements.} BG was, for some of the period during which this work was carried out, a fellow of the Radcliffe Institute at Harvard. He is very grateful to the Radcliffe Institute for providing excellent working conditions. TT is supported by NSF Research Award DMS-0649473, the NSF Waterman award and a grant from the MacArthur Foundation. TZ is supported by ISF grant 557/08, an Alon fellowship and a Landau fellowship of the Taub foundation. All three authors are very grateful to the University of Verona for allowing them to use classrooms at Canazei during a week in July 2009. This work was largely completed during that week.
\section{Basic notation}\label{notation-sec}
We write $\N := \{0,1,2,\ldots\}$ for the natural numbers, and $\N^+ := \{1,2,\ldots\}$ for the positive natural numbers. Given two integers $N,M$, we write $[N,M]$ for the discrete interval $[N,M] := \{ n: N \leq n \leq M\}$. We also make the abbreviations $[N] :=[1,N]$, and , and $[[N]]:=[-N,N]$. If $x$ is a real number, we write $x \mod 1$ for the associated residue class in the unit circle $\T := \R/\Z$, and write $x=y \mod 1$ if $x$ and $y$ differ by an integer.
We will rely frequently on the following two elementary functions: the \emph{fundamental character} $e: \R \to \C$ (or $e: \T \to \C$) defined by
$$ e(x) := e^{2\pi i x},$$
and the \emph{signed fractional part function}\footnote{The signed fractional part will be slightly more convenient to work with than the unsigned fractional part, as it is equal to the identity near the origin.} $\{\}: \R \to I_0$, where $I_0$ is the \emph{fundamental domain}
$$ I_0 := \{ x \in \R: -1/2 < x \leq 1/2\}$$
and $\{x\}$ is the unique real number in $I_0$ such that $x = \{x\} \mod 1$. We will often rely on the identity
$$ e(x) = e(\{x\}) = e( x \mod 1 )$$
without further comment.
For technical reasons, we will need to manipulate vector-valued complex quantities in a manner analogous to scalar complex quantities.
If $v = (v_i)_{i=1}^D$ and $w = (w_i)_{i = 1}^{D'}$ are vectors in $\C^D$ and $\C^{D'}$ respectively then we form the \emph{tensor product} $v \otimes w \in \C^{DD'}$ by the formula
\[ v \otimes w := (v_1 w_1,\dots, v_{D} w_{D'})\]
and the \emph{complex conjugate} $\overline{v}\in \C^D$ by the formula
\[ \overline{v} := (\overline{v_1},\dots,\overline{v_D}).\]
Similarly, if $X$ is some set and $f : X \rightarrow \C^D$ and $g : X \rightarrow \C^{D'}$ are functions then we write $f \otimes g: X \rightarrow \C^{DD'}$ for the function defined by $(f\otimes g)(x) := f(x) \otimes g(x)$, and similarly define $\overline{f}: X \to \C^D$.
If $G = (G,+)$ is an additive group, $k \in \N$, $\vec g = (g_1,\ldots,g_k) \in G^k$, and $\vec a = (a_1,\ldots,a_k) \in \Z^k$, we define the dot product
$$ \vec a \cdot \vec g := a_1 g_1 + \ldots + a_k g_k.$$
Given a set $H$ in an additive group, define an \emph{additive quadruple} in $H$ to be a quadruple $(h_1,h_2,h_3,h_4) \in H$ with $h_1+h_2=h_3+h_4$. The number of additive quadruples in $H$ is known as the \emph{additive energy} of $H$ and is denoted $E(H)$.
A map $\phi: H \to G$ from $H$ to another additive group $G$ is said to be a \emph{Freiman homomorphism} if it preserves additive quadruples, i.e. if $\phi(h_1)+\phi(h_2)=\phi(h_3)+\phi(h_4)$ for all additive quadruples $(h_1,h_2,h_3,h_4)$ in $H$.
Given a multi-index $\vec d = (d_1,\ldots,d_k) \in \N^k$, we write $|\vec d| := d_1+\ldots+d_k$.
We now briefly review and clarify some standard notation from group theory.
When we do not assume a group $G$ to be abelian, we will always write $G$ multiplicatively: $G = (G,\cdot)$. However, when dealing with abelian groups, we reserve the right to use additive notation instead.
We view an $n$-tuple $(a_1,\ldots,a_n)$ of labels as a finite ordered set with the ordering $a_1 < \ldots < a_n$. If $A = (a_1,\ldots,a_n)$ is a finite ordered set and $(g_a)_{a \in A}$ are a collection of group elements in a multiplicative group $G$, we define the ordered products
$$ \prod_{a \in A} g_a := g_{a_1} \ldots g_{a_n}, \; \; \prod_{i=1}^n g_i := g_1 \ldots g_n \; \; \mbox{and} \; \; \prod_{i=n}^1 g_i := g_n \ldots g_1$$
for any $n \geq 0$, with the convention that the empty product is the identity. We extend this notation to infinite products under the assumption that all but finitely many of the factors are equal to the identity.
Given a subset $A$ of a group $G$, we let $\langle A \rangle$ denote the subgroup of $G$ generated by $A$. Given a family $(H_i)_{i \in I}$ of subgroups of $G$, we write $\bigvee_{i \in I} H_i$ for the smallest subgroup of $G$ that contains all of the $H_i$.
Given two elements $g, h$ of a multiplicative group $G$, we define the \emph{commutator}
$$ [g,h] := g^{-1}h^{-1}gh.$$
We write $H \leq G$ to denote the statement that $H$ is a subgroup of $G$. If $H, K \leq G$, we let $[H,K]$ be the subgroup generated by the commutators $[h,k]$ with $h \in H$ and $k \in K$, thus $[H,K] = \left\langle \{ [h,k]: h \in H, k \in K \} \right\rangle$.
If $r \geq 1$ is an integer and $g_1,\ldots,g_r \in G$, we define an $(r-1)$-\emph{fold iterated commutator} of $g_1,\ldots,g_r$ inductively by declaring $g_1$ to be the only $0$-fold iterated commutator of $g_1$, and for $r>1$ defining an $(r-1)$-fold iterated commutator to be any expression of the form $[w,w']$, where $w$ and $w'$ are $(s-1)$-fold and $(s'-1)$-fold commutators of $g_{i_1},\ldots,g_{i_s}$ and $g_{i'_1},\ldots,g_{i'_{s'}}$ respectively, where $s, s' \geq 1$ are such that $s+s'=r$, and $\{i_1,\ldots,i_s\} \cup \{ i'_1,\ldots,i'_{s'} \} = \{1,\ldots,r\}$ is a partition of $\{1,\ldots,r\}$ into two classes. Thus for instance $[[g_3,g_1],[g_2,g_4]]$ and $[g_2,[g_1,[g_3,g_4]]]$ are $3$-fold iterated commutators of $g_1,\ldots,g_4$.
The following lemma will be useful for computing commutator groups.
\begin{lemma}\label{normal} Let $H = \langle A \rangle, K = \langle B \rangle$ be normal subgroups of a nilpotent group $G$ that are generated by sets $A \subset H$, $B \subset K$ respectively. Then $[H,K]$ is normal, and is also the subgroup generated by the $i+j-1$-fold iterated commutators of $a_1,\ldots,a_i,b_1,\ldots,b_j$ with $a_1,\ldots,a_i \in A$, $b_1,\ldots,b_j \in B$ and $i,j \geq 1$.
\end{lemma}
\begin{proof} The normality of $[H,K]$ is follows from the identity
\[
g[H,K]g^{-1} = [gHg^{-1},gKg^{-1}].
\]
It is then clear that $[H,K]$ contains the group generated by the iterated commutators of elements in $A,B$ that involve at least one element from each. The converse follows inductively using the identities
\begin{equation}\label{com-ident}
[x,y]=[y,x]^{-1}, \; \; [xy,z]=[x,z][[x,z],y][y,z] \; \; \mbox{and} \; \; [x,y^{-1}]=[y,x][[y,x],y^{-1}].
\end{equation}
This concludes the proof.
\end{proof}
As a corollary of the above lemma, we have the distributive law
$$ \left[ \bigvee_{i \in I} H_i, \bigvee_{j \in J} K_j \right] = \bigvee_{i \in I, j \in J} [H_i, K_j]$$
whenever $(H_i)_{i \in I}, (K_j)_{j \in J}$ are families of normal subgroups of a nilpotent group $G$.
If $H \lhd G$ is a normal subgroup of $G$, and $g \in G$, we use $g \mod H$ to denote the coset representative $gH$ of $g$ in $G/H$. For instance, $g = g' \mod H$ if $gH = g' H$.\vspace{11pt}
At various stages in the paper we will need the (discrete) \emph{Baker-Campbell-Hausdorff formula} in the following weak form:
\begin{equation}\label{bch}
g_1^{n_1} g_2^{n_2} = g_2^{n_2} g_1^{n_1} \prod_a g_a^{P_a(n_1,n_2)}
\end{equation}
for all $g_1,g_2$ in a nilpotent group $G$ and all integers $n_1,n_2$, where $g_a$ ranges over all iterated commutators of $g_1, g_2$ that involve at least one copy of each (note from nilpotency that there are only finitely many non-trivial $g_a$), with the $a$ ordered in some arbitrary fashion, and $P_a: \Z \times \Z \to \Z$ are polynomials. Furthermore, if $g_a$ involves $d_1$ copies of $g_1$ and $d_2$ copies of $g_2$, then $P_a$ has degree at most $d_1$ in the $n_1$ variable and $d_2$ in the $n_2$ variable.
Let $G$ be a connected, simply connected, nilpotent Lie group (or \emph{nilpotent Lie group} for short). Then we denote the Lie algebra of $G$ as $\log G$. As is well known (see e.g. \cite{bourbaki}), the exponential map $\exp: \log G \to G$ is a homeomorphism, inverted by the logarithm map $\log: G \to \log G$, and we can then define the exponentiation operation $g^t$ for any $g \in G$ and $t \in \R$ by the formula
$$ g^t := \exp( t \log g ).$$
There is a continuous version of the Baker-Campbell-Hausdorff formula:
\begin{equation}\label{bch-cont}
g_1^{t_1} g_2^{t_2} = g_2^{t_2} g_1^{t_1} \prod_a g_a^{P_a(t_1,t_2)}
\end{equation}
for all $t_1,t_2 \in \R$ and $g_1, g_2 \in G$, where $P_a$ are the polynomials occurring in \eqref{bch}. We also observe the variant formulae
$$
(g_1 g_2)^{t} = g_1^t g_2^t \prod_a g_a^{Q_a(t)}$$
for some polynomials $Q_a$ and all $t \in \R$, $g_1, g_2 \in G$, and
$$
\exp( t_1 \log g_1 + t_2 \log g_2 ) = g_1^{t_1} g_2^{t_2} \prod_a g_a^{R_a(t_1,t_2)}$$
for some further polynomials $R_a$ and all $t_1, t_2 \in \R$, $g_1, g_2 \in G$. We refer to all of these formul{\ae} collectively as \emph{the Baker-Campbell-Hausdorff formula}.
If $A$ is a subset of a nilpotent Lie group $G$, we let $\langle A \rangle_\R$ be the smallest connected Lie subgroup of $G$ containing $A$, or more explicitly
$$ \langle A \rangle_\R := \langle \{ a^t: a \in A; t \in \R \} \rangle.$$
Equivalently, $\log \langle A \rangle_\R$ is the Lie algebra generated by $\log A$.
A \emph{lattice} of a nilpotent Lie group $G$ is a discrete cocompact subgroup $\Gamma$ of $G$. Thus for instance, we see from \eqref{bch} that for any finite set $A$ in $G$, $\langle A \rangle$ will be a cocompact subgroup of $\langle A \rangle_\R$, and will thus be a lattice if $\langle A \rangle$ is discrete.
A connected Lie subgroup $H$ of $G$ is said to be \emph{rational} with respect to $\Gamma$ if $\Gamma \cap H$ is cocompact in $H$. For instance, if $G = \R^2$, $\Gamma$ is the standard lattice $\Z^2$, and $\alpha \in \R$, then the connected Lie subgroup $H := \{ (x,\alpha x): x \in \R \}$ is rational if and only if $\alpha$ is rational.\vspace{11pt}
\textsc{Further notation.} Here is a list of further notation used in the paper for reference, together with the place in the paper where each piece is defined and discussed.
\noindent\begin{tabular}{lll}
$\poly(H_\N \to G_\N)$ & polynomial maps from one filtered group $H_\N$ to $G_\N$ & \ref{poly-map-def}\\
$\poly(\Z_\N \to G_\N)$ & polynomial maps with the degree filtration & \ref{poly-map-def} \\
$\poly(\Z^k_{\N^k} \to G_{\N^k})$ & polynomial maps with the multidegree filtration & \ref{poly-map-def} \\
$\poly(\Z_{\DR} \to G_{\DR})$ & polynomial maps with the degree-rank filtration & \ref{poly-map-def} \\
$L^\infty(\Omega \to \overline{\C}^D)$ & bounded limit functions to $\ultra \C^d$& \eqref{sigma-bounded}\\
$L^\infty(\Omega \to \overline{\C}^w)$ & bounded limit functions (also $L^{\infty}(\Omega)$) & \eqref{sigma-bounded}\\
$\Lip(\ultra(G/\Gamma) \to \overline{\C}^D)$ & bd'd limit functions with bounded Lipschitz constant & \ref{lip-def} \\
$\Nil^{d}([N])$ & nilsequences of degree $\le d$ on $[N]$ & \ref{nilseq} \\
$\Nil^{\subset J}(\Omega)$ & nilsequences of degree $\subset J$ & \ref{nilch-def-gen} \\
$\Xi^d([N])$ & space of degree $d$ nilcharacters on $[N]$ & \ref{nilch-def} \\
$\Xi^{(d_1,\ldots,d_k)}_\MD(\Omega)$ & multidegree nilcharacters & \ref{nilch-def-gen} \\
$\Xi^{(d,r)}_\DR(\Omega)$ & degree-rank nilcharacters & \ref{nilch-def-gen} \\
$\Symb^d([N])$ & equiv. classes of degree $d$ nicharacters in $\Xi^d([N])$ & \ref{symbol-def} \\
$\Symb^{(d_1,\ldots,d_k)}_{\MD}(\Omega)$ & equiv. classes of multidegree nicharacters & \ref{equiv-def} \\
$\Symb^{(d,r)}_\DR(\Omega)$ & equiv. classes of degree-rank nicharacters & \ref{equiv-def} \\
$G^{\vec D},G^{\vec D, \leq (s-1,r_*)}$ & universal nilpotent Lie group of degree-rank $(s-1,r_*)$ & \ref{universal-nil}\\
$\Horiz_i(G)$ & $i$'th horizontal space of $G$ & \ref{horton} \\
$\Taylor_i(g)$ & $i'$th horizontal Taylor coefficient of a polynomial map & \ref{horton} \\
$(\vec D, \eta, \F)$ & total frequency representation of a nilcharacter & \ref{representation-def}
\end{tabular}
\section{The polynomial formulation of $\GI(s)$}\label{polysec}
The inverse conjecture $\GI(s)$, Conjecture \ref{gis-conj}, has been formulated using \emph{linear} nilsequences $F(g^n x\Gamma)$. This is largely for compatibility with the earlier paper \cite{green-tao-linearprimes} of the first two authors on linear equations in primes, where this form of the conjecture was stated in precisely this form as Conjecture 8.3. Subsequently, however, it was discovered that it is more natural to deal with a somewhat more general class of object called a \emph{polynomial nilsequence} $F(g(n)\Gamma)$. This is particularly so when it comes to discussing the distributional properties of nilsequences, as was done in \cite{green-tao-nilratner}. Thus, we shall now recast the inverse conjecture in terms of polynomial nilsequences, which is the formulation we will work with throughout the rest of the paper.
Let us first recall the definition of a polynomial nilsequence of degree $d$.
\begin{definition}[Polynomial nilsequence]
Let $G$ be a (connected, simply-connected) nilpotent Lie group. By a \emph{filtration} $G_\N = (G_i)_{i \in \N}$ of degree $\leq d$ we mean a nested sequence $G \supseteq G_{0} \supseteq G_{1} \supseteq G_{2} \supseteq \dots \supseteq G_{d+1} = \{\id\}$ with the property that $[G_{i}, G_{j}] \subseteq G_{i+j}$ for all $i, j \geq 0$, adopting the convention that $G_{i}=\{\id\}$ for all $i>d$. By a \emph{polynomial sequence} adapted to $G_\N$ we mean a map $g : \Z \rightarrow G$ such that $\partial_{h_i} \dots \partial_{h_1} g \in G_i$ for all $i \geq 0$ and $h_1,\dots, h_i \in \Z$, where $\partial_h g(n) := g(n+h) g(n)^{-1}$. Write $\poly(\Z_\N \to G_{\N})$ for the collection of all such polynomial sequences.
Let $\Gamma \leq G$ be a lattice in $G$ (i.e. a discrete and cocompact subgroup), so that the quotient $G/\Gamma$ is a nilmanifold, and assume that each of the $G_i$ are \emph{rational} subgroups (i.e. $\Gamma_i := \Gamma \cap G_i$ is a cocompact subgroup of $G_i$). We refer to the pair $G/\Gamma = (G/\Gamma,G_\N)$ as a \emph{filtered nilmanifold}. A \emph{polynomial orbit} $\orbit: \Z \to G/\Gamma$ is a sequence of the form $\orbit(n) := g(n) \Gamma$, where $g \in \poly(\Z_\N \to G_\N)$; we let $\poly(\Z_\N \to (G/\Gamma)_\N)$ denote the space of all such polynomial orbits.
If $F : G/\Gamma \rightarrow \C$ is a $1$-bounded, Lipschitz function then the sequence $F \circ \orbit = (F(g(n)\Gamma))_{n \in \Z}$ is called a \emph{polynomial nilsequence} of degree $d$.
\end{definition}
The subscripts $\N$ will become more relevant later in this paper, when we start filtering nilpotent groups and nilmanifolds by other index sets $I$ than the natural numbers $\N$. Note that we do not require $G_0$ or $G_1$ to equal $G$; this freedom will be convenient for some minor technical reasons, although ultimately it will not enlarge the space of polynomial nilsequences.
Let us give the basic examples of nilsequences and polynomials:
\begin{example}[Linear nilsequences are polynomial nilsequences]\label{polylin} Let $G$ be a $d$-step nilpotent Lie group, and let $\Gamma$ be a lattice of $G$. Then, as is well known (see e.g. \cite{bourbaki}), the \emph{lower central series filtration} defined by $G_{0} = G_1 := G$, $G_{2} := [G, G_{1}]$, $G_{3} := [G, G_{2}], \dots, G_{d+1} := [G, G_{d}] = \{\id\}$ is a filtration on $G$. Using the Baker-Campbell-Hausdorff formula \eqref{bch-cont} it is not difficult to show that the lower central series filtration is rational with respect to $\Gamma$, so the nilmanifold $G/\Gamma$ becomes a filtered nilmanifold. If $g(n) := g_1^n g_0$ for some $g_0, g_1 \in G$, then $\partial_{h_1} g(n) = g_1^{h_1}$ and $\partial_{h_i} \dots \partial_{h_1} g(n) = \id$ for $i \geq 2$: therefore $g$ is a polynomial sequence, and so every linear orbit $n \mapsto g^n x$ with $g \in G$ and $x \in G/\Gamma$ is a polynomial orbit also. As a consequence we see that every $d$-step linear nilsequence $n \mapsto F(g^n x)$ is automatically a polynomial nilsequence of degree $\leq d$.
\end{example}
\begin{example}[Polynomial phases are polynomial nilsequences]\label{polyphase} Let $d \geq 0$ be an integer. Then we can give the unit circle $\T$ the structure of a degree $\leq d$ filtered nilmanifold by setting $G := \R$ and $\Gamma := \Z$, with $G_i := \R$ for $i \leq d$ and $G_i := \{0\}$ for $i>d$. This is clearly a filtered nilmanifold. If $\alpha_0,\ldots,\alpha_d$ are real numbers, then the polynomial $P(n) := \alpha_0 + \ldots + \alpha_d n^d$ is then
polynomial with respect to this filtration, with $n \mapsto P(n) \mod 1$ being a polynomial orbit in $\T$. Thus, for any Lipschitz function $F: \T \to \C$, the sequence $n \mapsto F(P(n))$ is a polynomial nilsequence of degree $\leq d$; in particular, the polynomial phase $n \mapsto e(P(n))$ is a polynomial nilsequence.
\end{example}
\begin{example}[Combinations of monomials are polynomials]\label{lazard-ex} By Corollary \ref{laz}, we see that if $G = (G,(G_i)_{i \in\N})$ is a filtered group of degree $\leq d$, then any sequence of the form
$$ n \mapsto \prod_{j=1}^k g_j^{P_j(n)},$$
in which $g_j \in G_{d_j}$ for some $d_j \in \N$, and $P_j: \Z \to \R$ is a polynomial of degree $\leq d_j$, will be a polynomial map. Thus for instance
$$ n \mapsto g_d^{\binom{n}{d}} \ldots g_2^{\binom{n}{2}} g_1^n g_0$$
is a polynomial map whenever $g_j \in G_j$ for $j=0,\ldots,d$. In fact, all polynomial maps can be expressed in such a fashion via a \emph{Taylor expansion}; see Lemma \ref{taylo}.
\end{example}
We will give several further examples and properties of polynomial maps and polynomial nilsequences in \S \ref{nilcharacters}.
As a consequence of Example \ref{polylin}, the following variant of the inverse conjecture $\GI(s)$ is ostensibly weaker than that stated in the introduction.
\begin{conjecture}[$\GI(s)$, polynomial formulation]\label{gis-poly} Let $s \geq 0$ be an integer, and let $0 < \delta \leq 1$. Then there exists a finite collection ${\mathcal M}_{s,\delta}$ of filtered nilmanifolds $G/\Gamma = (G/\Gamma,G_\N)$, each equipped with some smooth Riemannian metric $d_{G/\Gamma}$ as well as constants $C(s,\delta), c(s,\delta) > 0$ with the following property. Whenever $N \geq 1$ and $f : [N] \rightarrow \C$ is a function bounded in magnitude by $1$ such that $\Vert f \Vert_{U^{s+1}[N]} \geq \delta$, there exists a filtered nilmanifold $G/\Gamma \in {\mathcal M}_{s,\delta}$, some $g \in \poly(\Z_\N \to G_{\N})$ and a function $F: G/\Gamma \to \C$ bounded in magnitude by $1$ and with Lipschitz constant at most $C(s,\delta)$ with respect to the metric $d_{G/\Gamma}$ such that
$$ |\E_{n \in [N]} f(n) \overline{F(g(n)\Gamma)}| \geq c(s,\delta).$$
\end{conjecture}
It turns out that this conjecture is actually \emph{equivalent} to Conjecture \ref{gis-conj}; we shall prove this equivalence in Appendix \ref{lift-app}. We remark that, though it might seem odd to put a non-trivial part of the proof of our main theorem in an appendix, we would rather encourage the reader to regard the proof of Conjecture \ref{gis-poly} as our main theorem. The rationale behind this is that everything that is done with linear nilsequences $F(g^nx \Gamma)$ in \cite{green-tao-linearprimes} could have been done equally well, and perhaps more naturally, with polynomial nilsequences $F(g(n)\Gamma)$. Further remarks along these lines were made in the introduction to our earlier paper \cite{u4-inverse}, where the polynomial formulation was emphasised from the outset. Here, however, we have felt a sense of duty to formally complete the programme outlined in \cite{green-tao-linearprimes}.
Henceforth we shall refer simply to a \emph{nilsequence}, rather than a polynomial nilsequence.
In \S \ref{nilcharacters} we will need to generalise the notion of a (polynomial) nilsequence by allowing more exotic filtrations $G_I$ on the group $G$, indexed by more complicated index sets $I$ than the natural numbers $\N$. In particular, we shall introduce the \emph{multidegree filtration}, which allows us to define nilsequences of several variables, as well as the \emph{degree-rank} filtration which provides a finer classification of polynomial sequences than merely the degree. We will discuss these using examples, and then develop a more unified theory that contains all three.
\section{Taking ultralimits}\label{nsa-sec}
The inverse conjecture, Conjecture \ref{gis-poly}, is a purely finitary statement, involving functions on a finite set $[N] = \{1,\ldots,N\}$ of integers. As such, it is natural to look for proofs of this conjecture which are also purely finitary, and much of the previous literature on these types of problems is indeed of this nature.
However there is a very notable exception, namely the portion of the literature that exploits the \emph{Furstenberg correspondence principle} between combinatorial problems and ergodic theory. See \cite{furstenberg} for the original application to Szemer\'edi's theorem, or \cite{tao-ziegler} for a more recent application to Gowers norms over finite fields. Here we use a somewhat different type of limit object, namely an \emph{ultralimit}. We are certainly not the first to employ ultralimits (a.k.a. \emph{nonstandard analysis}) in additive number theory; see for example \cite{jin}.
The ultralimit formalism allows us to convert a ``finitary'' or ``standard'' statement such as Conjecture \ref{gis-poly} into an equivalent statement concerning \emph{limit objects}, constructed as ultralimits of standard objects. This procedure is closely related to the use of the \emph{transfer principle} in nonstandard analysis, but we have elected to eschew the language of nonstandard analysis in order to reduce confusion, instead focusing on the machinery of ultralimits.
Here is a brief and somewhat vague list of the advantages of using the ultralimit approach.
\begin{itemize}
\item Pigeonholing arguments are straightforward (due to the fact that a limit function taking finitely many values is constant);
\item Book-keeping of constants: one can talk rigorously about such concepts as ``bounded'' functions without a need to quantify the bounds;
\item One may make rigorous sense of such statements as ``the function $f: [N] \to \C$ and the function $g: [N] \to \C$ are equivalent modulo degree $s$ nilsequences''.
\item In the infinitary context one may easily perform \emph{rank reduction} arguments in which one seeks to find the ``minimal bounded-complexity'' representation of a given system.
\end{itemize}
There are also some drawbacks of the approach:
\begin{itemize}
\item It becomes quite difficult to extract any quantitative bounds from our results, in particular we do not give explicit bounds on the constant $c(s,\delta)$ or on the complexity of the nilsequence in Conjecture \ref{gis-conj} or Conjecture \ref{gis-poly}. It is in principle possible to expand the ultralimit proof into a standard proof, but the bounds are quite poor (of Ackermann type) due to the repeated use of ``rank reduction arguments'' and other highly iterative schemes that arise in the conversion of ultralimit arguments to standard ones. For further discussion of the relation of ultralimit analysis to finitary analysis see \cite[\S 1.3, \S 1.5]{structure}.
\item The language of ultrafilters adds one more layer of notational complexity to an already notationally-intensive paper; however, there are gains to be made elsewhere, most notably in eliminating many quantitative constants (e.g. $\eps$, $N$) and growth functions (e.g. ${\mathcal F}$).
\end{itemize}
\textsc{Limit formulation of $\GI(s)$.} The basic notation and theory of ultralimits are reviewed in Appendix \ref{nsa-app}. We now use this formalism to convert the inverse conjecture, $\GI(s)$, into an equivalent statement formulated in the framework of ultralimits. We first consider a limit version of the concept of a Lipschitz function on a nilmanifold. For technical reasons we will need to consider vector-valued functions, taking values in $\C^D$ or $\overline \C^D$ rather than $\C$ or $\overline\C$.
\begin{definition}[Lipschitz functions]\label{lip-def} Let $G/\Gamma$ be a standard nilmanifold, and let $D \in \N^+$ be standard.
\begin{itemize}
\item We let $\Lip(G/\Gamma \to \C^D)$ be the space of standard Lipschitz functions $F: G/\Gamma \to \C^D$. (Here we endow the compact manifold $G/\Gamma$ with a smooth metric in an arbitrary fashion; the exact choice of metric is not relevant.)
\item We let $\Lip(\ultra(G/\Gamma) \to \overline{\C}^D)$ be the space of bounded limit functions $F: \ultra(G/\Gamma) \to \overline{\C}^D$ whose Lipschitz constant is bounded (or equivalently, $F$ is an ultralimit of uniformly bounded functions $F_\n: G/\Gamma \to \C^D$ with uniformly bounded Lipschitz constant).
\item We let $\Lip(\ultra(G/\Gamma) \to \overline{S^{2D-1}})$ be the functions in $\Lip(\ultra(G/\Gamma) \to \overline{\C}^D)$ that take values in the (limit) complex sphere
$$ \overline{S^{2D-1}} := \{ z \in \overline{\C}^D: |z| = 1\}.$$
\item We write \[ \Lip(\ultra(G/\Gamma) \to \overline{\C}^\omega) := \bigcup_{D \in \N^+} \Lip(\ultra(G/\Gamma) \to \overline{\C}^D)\] and \[\Lip(\ultra(G/\Gamma) \to \overline{S^\omega}) := \bigcup_{D \in \N^+} \Lip(\ultra(G/\Gamma) \to \overline{S^{2D-1}}).\]
\end{itemize}
We will often abbreviate these spaces as $\Lip(G/\Gamma)$ or $\Lip(\ultra(G/\Gamma))$ when the range of the functions involved is not relevant to the discussion.
\end{definition}
\emph{Remark.} As $G/\Gamma$ is compact, we see from the Arzel\`a-Ascoli theorem that $\Lip(G/\Gamma \to \C^D)$ is locally compact in the $L^\infty(G/\Gamma \to \C^D)$ topology. As a consequence, if we embed $\Lip(G/\Gamma \to \C^D)$ into $\Lip(\ultra(G/\Gamma) \to \overline{\C}^D)$ in the obvious manner, then the former is a dense subspace of the latter in the (standard) uniform topology, in the sense that for every $F \in \Lip(\ultra(G/\Gamma) \to \overline{\C}^D)$ and every standard $\eps > 0$ there exists $F' \in \Lip(G/\Gamma \to \C^D)$ such that $|F(x)-F'(x)| \leq \eps$ for all $x \in \ultra(G/\Gamma)$.
\emph{Remark.} Observe that the spaces $\Lip(\ultra(G/\Gamma) \to \overline{\C}^D)$ and $\Lip(\ultra(G/\Gamma) \to \overline{\C}^\omega)$ are vector spaces over $\overline{\C}$. The spaces $\Lip(\ultra(G/\Gamma) \to \overline{\C}^\omega)$ and $\Lip(\ultra(G/\Gamma) \to \overline{S^\omega})$ are also closed under tensor product (as defined in \S \ref{notation-sec}). All the spaces defined in Definition \ref{lip-def} are closed under complex conjugation.
Using the above notion, we can define the limit version of a (polynomial) nilsequence.
\begin{definition}[Nilsequence]\label{nilseq} Let $s \geq 0$ be standard. A \emph{nilsequence} of degree $\leq s$ is any limit function $\psi: \ultra \Z \to \ultra \C$ of the form $\psi(n) := F(g(n) \Gamma)$, where $G/\Gamma = (G/\Gamma,G_\N)$ is a standard filtered nilmanifold of degree $\leq s$, $g: \ultra \Z \to \ultra G$ is a limit polynomial sequence (i.e. an ultralimit of polynomial sequences $g_\n: \Z \to G$), and $F \in \Lip(\ultra(G/\Gamma) \to \overline{\C})$.
\end{definition}
Given any limit subset $\Omega$ of $\ultra \Z$, we denote the space of degree $d$ nilsequences, restricted to $\Omega$, as $\Nil^{d}(\Omega) = \Nil^{d}(\Omega \to \overline{\C}^\omega)$; this is a subset of $L^\infty(\Omega \to \overline{\C}^\omega)$. We write $\Nil^{d}(\Omega \to \overline{\C}^D)$ for the nilsequences that take values in $\overline{\C}^D$; this is a subspace (over $\overline{\C}$) of $L^\infty(\Omega \to \overline{\C}^D)$. We make the technical remark that $\Nil^{d}(\Omega)$ is a $\sigma$-limit set, since one can express this space as the union, over all standard $M$ and dimensions $D$, of the nilsequences taking values in $\overline{\C}^D$ arising from a nilmanifold of ``complexity'' $M$ and a Lipschitz function of constant at most $M$, where one defines the complexity of a nilmanifold in some suitable fashion. In particular, the limit selection lemma in Corollary \ref{mes-select} can be applied to this set.
We also define the Gowers uniformity norm $\Vert f\Vert_{U^{s+1}[N]}$ of an ultralimit $f= \lim_{\n \to p} f_\n$ of standard functions $f_\n: [N_\n] \to \C$ in the usual limit fashion
$$ \|f\|_{U^{s+1}[N]} := \lim_{\n \to p} \|f_\n\|_{U^{s+1}[N_\n]}.$$
If $f$ is vector-valued instead of scalar valued, say $f = (f_1,\ldots,f_d)$, then we define the uniformity norm by the formula
$$ \|f\|_{U^{s+1}[N]} := (\sum_{i=1}^d \|f_i\|_{U^{s+1}[N]}^{2^{s+1}})^{1/2^{s+1}}.$$
(The exponent $2^{s+1}$ is not important here, but has some very slight aesthetic advantages over other equivalent formulations of the vector-valued norm.)
The ultralimit formulation of $\GI(s)$ can then be given as follows:
\begin{conjecture}[Ultralimit formulation of $\GI(s)$]\label{gis-conj-nonst} Let $s \geq 0$ be standard and $N \geq 1$ be a limit natural number. Suppose that $f \in L^\infty([N] \to \overline{\C})$ is such that $\Vert f \Vert_{U^{s+1}[N]}$ $\gg 1$. Then $f$ correlates with a degree $\leq s$ nilsequence on $[N]$.
\end{conjecture}
See Definition \ref{linfty} for the definition of \emph{correlation} in this context.
We now show why, for any fixed standard $s$, Conjecture \ref{gis-conj-nonst} is equivalent to its more traditional counterpart, Conjecture \ref{gis-poly}.
\begin{proof}[Proof of Conjecture \ref{gis-conj-nonst} assuming Conjecture \ref{gis-poly}] Let $f$ be as in Conjecture \ref{gis-conj-nonst}. We may normalise the bounded function $f$ to be bounded by $1$ in magnitude throughout. By hypothesis, there exists a standard $\delta > 0$ such that $\Vert f \Vert_{U^{s+1}[N]} \geq \delta$. Writing $N$ and $f$ as the ultralimits of $N_\n$, $f_\n$ respectively for some $f_\n: [N_\n] \to \C$ bounded in magnitude by $1$, and applying Conjecture \ref{gis-poly}, we conclude that for $\n$ sufficiently close to $p$, we have the correlation bound
$$ |\E_{n_\n \in [N_\n]} f_\n(n_\n) \overline{F_\n(g_\n(n_\n) \Gamma_\n)}| \geq c(s,\delta)> 0$$
where $G_\n/\Gamma_\n, g_\n, x_\n, F_\n$ are as in Conjecture \ref{gis-conj}. Writing $G/\Gamma, g, x, F$ for the ultralimits of $G_\n/\Gamma_\n, g_\n, x_\n, F_\n$ respectively, we thus have
$$ |\E_{n \in [N]} f(n) \overline{F(g(n) \ultra \Gamma)}| \gg 1.$$
By the pigeonhole principle (cf. Appendix \ref{nsa-app}), we see that $G/\Gamma$ is a standard degree $\leq s$ nilmanifold, while $g: \ultra \Z \to \ultra G$ and $x \in G/\Gamma$ remain limit objects. The limit function $F$ lies in $\Lip(\ultra(G/\Gamma) \to \overline{\C})$ by construction, and the claim follows.
\end{proof}
\emph{Proof of Conjecture \ref{gis-poly} assuming Conjecture \ref{gis-conj-nonst}.} Observe (from the theory of Mal'cev bases \cite{malcev}) that there are only countably many degree $\leq s$ nilmanifolds $G/\Gamma$ up to isomorphism, which we may enumerate as $G_\n/\Gamma_\n$. We endow each of these nilmanifolds arbitrarily with some smooth Riemannian metric $d_{G_\n/\Gamma_\n}$.
Suppose for contradiction that Conjecture \ref{gis-poly} failed.
Carefully negating all the quantifiers, we may thus find a $\delta > 0$, a sequence $N_\n$ of standard integers, and a function $f_\n: [N_\n] \to \C$ bounded in magnitude by $1$ with
$\|f_\n\|_{U^{s+1}[N]} \geq \delta$, such that
\begin{equation}\label{george}
|\E_{n_\n \in [N_\n]} f_\n(n_\n) \overline{F(g(n_\n) \Gamma_{\n'}))}| \leq 1/\n
\end{equation}
whenever $\n' \leq \n$, $g \in \poly(\Z_\N \to (G_{\n'})_\N)$, and $F: G_{\n'}/\Gamma_{\n'} \to \C$ is bounded in magnitude by $1$ and has a Lipschitz constant of at most $\n$ with respect to $d_{G_\n/\Gamma_\n}$.
On the other hand, viewing $f$ as a bounded limit function, we can apply Conjecture \ref{gis-conj-nonst} and conclude that there exists a standard filtered nilmanifold $G/\Gamma$ with some smooth Riemannian metric $d_{G/\Gamma}$, a limit polynomial $g: \ultra \Z \to \ultra G$, and some ultralimit $F \in \Lip(\ultra(G/\Gamma) \to \overline{\C})$ of functions $F_\n: G/\Gamma \to \C$ with uniformly bounded Lipschitz norm, such that
$$ |\E_{n \in [N]} f(n) \overline{F(g(n) \ultra \Gamma)}| \geq \eps$$
for some standard $\eps > 0$.
By construction, $G/\Gamma$ is isomorphic to $G_{\n_0}/\Gamma_{\n_0}$ for some $\n_0$, so we may assume without loss of generality that $G/\Gamma = G_{\n_0}/\Gamma_{\n_0}$; since all smooth Riemannian metrics on a compact manifold are equivalent, we can also assume that $d_{G/\Gamma} = d_{G_{\n_0}/\Gamma_{\n_0}}$. We may also normalise $F$ to be bounded in magnitude by $1$. But this contradicts \eqref{george} for $\n$ sufficiently large, and the claim follows.\endproof
Thus, to establish Theorem \ref{mainthm}, it will suffice to establish Conjecture \ref{gis-conj-nonst} for $s \geq 3$. This is the objective of the remainder of the paper.
\emph{Remark.} We transformed the finitary linear inverse conjecture, Conjecture \ref{gis-conj}, into a nonstandard polynomial formulation, Conjecture \ref{gis-conj-nonst}, via the finitary polynomial inverse conjecture, Conjecture \ref{gis-poly}. One can also swap the order of these equivalences, transforming the finitary linear inverse conjecture into a nonstandard linear formulation by arguing as above, and then transforming the latter into a nonstandard polynomial formulation by using Proposition \ref{lift}. Of course the two arguments are essentially equivalent.
Conjecture \ref{gis-conj-nonst} is trivial when $N$ is bounded, since every function in $L^\infty[N]$ is then a nilsequence of degree at most $s$. For the remainder of the paper we shall thus adopt the convention that $N$ denotes a fixed \emph{unbounded} limit integer.
To conclude this section we reformulate Conjecture \ref{gis-poly} by introducing the important notion of \emph{bias}.
\begin{definition}[Bias and correlation] Let $\Omega$ be a limit finite subset of $\Z$, and let $d \in \N$.
We say that $f, g \in L^\infty(\Omega \to \overline{\C}^\omega)$ \emph{$d$-correlate} if we have
$$|\E_{n \in \Omega} f(n) \otimes \overline{g(n)} \otimes \psi(n)| \gg 1$$
for some degree $d$ nilsequence $\psi \in \Nil^{d}(\Omega \to \overline{\C}^\omega)$. We say that $f$ is \emph{$d$-biased} if $f$ $d$-correlates with the constant function $1$, and \emph{$d$-unbiased} otherwise.
\end{definition}
With this definition, Conjecture \ref{gis-conj-nonst} can be reformulated in the following manner.
\begin{conjecture}[Limit formulation of $\GI(s)$, II]\label{gis-conj-nonst-2} Let $s \geq 0$ be standard. Suppose that $f \in L^\infty([N] \to \overline{\C})$ is such that $\Vert f \Vert_{U^{s+1}[N]} \gg 1$. Then $f$ is $s$-biased.
\end{conjecture}
From previous literature, we see that Conjecture \ref{gis-conj-nonst-2} has already been proven for $s \leq 2$; we need to establish it for all $s \geq 3$. We also make the basic remark that while the conjecture is only phrased for scalar-valued functions $f \in L^\infty([N] \to \overline \C)$, it automatically generalises to vector-valued functions $f \in L^\infty([N] \to \overline \C^\omega)$, since if a vector-valued function $f$ has large $U^{s+1}[N]$ norm, then so does one of its components.
Finally we remark that the converse implication is known.
\begin{proposition}[Converse $\GI(s)$, ultralimit formulation]\label{inv-nec-nonst} Let $s \geq 0$ be standard. Suppose that $f \in L^\infty([N] \to \overline{\C})$ is $\leq s$-biased. Then $\Vert f \Vert_{U^{s+1}[N]} \gg 1$.
\end{proposition}
\begin{proof} This follows from \cite[Proposition 12.6]{green-tao-u3inverse}, \cite[\S 11]{green-tao-linearprimes}, or \cite[Proposition 1.4]{u4-inverse}, transferred to the ultralimit setting in the usual fashion.
\end{proof}
\section{Nilcharacters and symbols in one and several variables}\label{nilcharacters}
Conjecture \ref{gis-conj-nonst} asserts that a function in $L^\infty([N] \to \overline{\C})$ on an unbounded interval $[N]$ correlates with a degree $\leq s$ nilsequence. For inductive reasons, it is useful to observe that this conclusion implies a strengthened version if itself, in which $f$ correlates with a special type of degree $\leq s$ nilsequence, namely a degree $s$ \emph{nilcharacter}. A nilcharacter is a special type of nilsequence and should be thought of, very roughly speaking, as a generalisation of characters $e(\alpha n)$ in the degree $1$ setting, or objects such as $e(\alpha n \{\beta n\})$ in the degree $2$ setting; these were crucial in our paper on $\GI(3)$ \cite{u4-inverse}, although the notation there was slightly different in some minor ways. See \cite{gtz-announce} for further informal discussion of nilcharacters.
In the $s=1$ case, a nilcharacter is essentially (ignoring constants) the same thing as a linear phase function $n \mapsto e(\xi n)$, and the frequency $\xi$ can be viewed as living in the Pontryagin dual of $\ultra \Z$ (or, in some sense, of $[N]$, even though the latter set is not quite a locally compact abelian group). It will turn out that more generally, a degree $s$ nilcharacter will have a ``symbol'' (analogous to the frequency $\xi$) that takes values in a ``higher order Pontryagin dual'' $\Symb^s([N])$ of $[N]$; this symbol can be interpreted as the ``top order term'' of a nilcharacter, for instance the symbol of the degree $3$ nilcharacter $n \mapsto e(\alpha n^3 + \beta n^2 + \gamma n + \delta)$ is basically\footnote{This is an oversimplification; it would be more accurate to say that the symbol is given by $\alpha$ modulo $\ultra \Z + \Q + O(N^{-3})$.} $\alpha$. This higher order dual obeys a number of pleasant algebraic properties, and the primary purpose of this section is to develop those properties.
There are various additional complications to be taken into account:
\begin{itemize}
\item We will require multidimensional generalisations of these concepts (think of the two-dimensional sequence $(n_1,n_2) \mapsto e(\alpha n_1 \{\beta n_2\})$) together with appropriate notions of \emph{multidegree} in order to make sense of ``top-order'' and ``lower-order terms'';
\item We will be dealing with $\C^D$-valued (or, rather, $S^{2D-1}$-valued) nilsequences rather than merely scalar ones. This is so that we may continue to work in the smooth category, as discussed in the introduction;
\item The language of ultrafilters will be used.
\end{itemize}
Our main focus here will be on the first of these points. The second is largely a technicality, whilst the third is actually helpful in that the notion of symbol (for example) is rather clean and does not require discussion of complexity bounds. \vspace{11pt}
\textsc{Motivation and one-dimensional definitions.} We now give the definitions of a (one-dimensional) nilcharacter and its symbol, and give a few examples. However, we will hold off for now on actually proving too much about these concepts, because we will shortly need to generalise these notions to a more abstract setting in which one also allows multidimensional nilcharacters, and nilcharacters that are atuned not just to a specific degree, but also to a specific ``rank'' inside that degree.
\begin{definition}[Nilcharacter]\label{nilch-def} Let $d \geq 0$ be a standard integer. A \emph{nilcharacter} $\chi$ of degree $d$ on $[N]$ is a nilsequence $\chi(n) = F(\orbit(n)) = F(g(n) \ultra \Gamma)$ on $[N]$ of degree $\leq d$, where the function $F \in \Lip(\ultra(G/\Gamma) \to \overline{\C}^\omega)$ obeys two additional properties:
\begin{itemize}
\item $F \in \Lip(\ultra(G/\Gamma) \to \overline{S^{\omega}})$ (thus $|F|=1$ pointwise, and hence $|\chi|=1$ pointwise also); and
\item $F( g_d x ) = e( \eta(g_d) ) F(x)$ for all $x \in G/\Gamma$ and $g_d \in G_{d}$, where $\eta: G_{d} \to \R$ is a continuous standard homomorphism which maps $\Gamma_{d}$ to the integers (or equivalently, $\eta$ is an element of the Pontryagin dual of the torus $G_d/\Gamma_d$). We call $\eta$ the \emph{vertical frequency} of $F$.
\end{itemize}
The space of all nilcharacters of degree $d$ on $[N]$ is denoted $\Xi^d([N])$.
\end{definition}
\begin{example} When $d=1$, the only examples of nilcharacters are the linear phases $n \mapsto e( \alpha n + \beta )$ for $\alpha, \beta \in \ultra \R$.
\end{example}
\begin{example} For any $\alpha_0,\ldots,\alpha_d \in \ultra \R$, the function $n \mapsto e(\alpha_0 + \ldots + \alpha_d n^d)$ is a nilcharacter of degree $\leq d$. To see this, we set $G/\Gamma$ to be the unit circle $\T$ with the filtration $G_i := \R$ for $i \leq d$ and $G_i := \{0\}$ for $i>d$ (thus $G/\Gamma$ is of degree $d$), let $g(n) := \alpha_0 + \ldots + \alpha_d n^d$, and let $F(x) := e(x)$. The vertical frequency $\eta: \R \to \R$ is then just the identity function.
\end{example}
Now we give an instructive \emph{near}-example of a nilcharacter. Let $G$ be the free $2$-step nilpotent Lie group on two generators $e_1,e_2$, thus
\begin{equation}\label{heisen}
G := \langle e_1,e_2\rangle_\R = \{ e_1^{t_1} e_2^{t_2} [e_1,e_2]^{t_{12}}: t_1,t_2,t_{12} \in \R\}
\end{equation}
with the element $[e_1,e_2]$ being central, but with no other relations between $e_1, e_2$ and $[e_1,e_2]$. This is a degree $\leq 2$ nilpotent group if we set $G_0, G_1 := G$ and
$$G_2 := \langle [e_1,e_2] \rangle_\R = \{ [e_1,e_2]^{t_{12}}: t_{12} \in \R \}.$$
We let
$$\Gamma := \langle e_1,e_2 \rangle = \{ e_1^{n_1} e_2^{n_2} [e_1,e_2]^{n_{12}}: n_1,n_2,n_{12} \in \Z\}$$
be the discrete subgroup of $G$ generated by $e_1,e_2$, then $G/\Gamma$ is a degree $\leq 2$ filtered nilmanifold, known as the \emph{Heisenberg nilmanifold}, and elements of $G/\Gamma$ can be uniquely expressed using the fundamental domain
$$ G/\Gamma = \{ e_1^{t_1} e_2^{t_2} [e_1,e_2]^{t_{12}} \Gamma: t_1,t_2,t_{12} \in I_0 := (-1/2,1/2]\}.$$
If we then set $g: \ultra \Z \to \ultra G$ to be the limit polynomial sequence $g(n) := e_2^{\beta n} e_1^{\alpha n}$ for some fixed $\alpha,\beta \in \ultra \R$, and let $F: G/\Gamma \to \C$ be the function defined on the fundamental domain by the formula
\begin{equation}\label{fdef}
F( e_1^{t_1} e_2^{t_2} [e_1,e_2]^{t_{12}} \Gamma ) := e( -t_{12} )
\end{equation}
for $t_1,t_2,t_{12} \in I_0$, then one easily computes that
$$ F( g(n) \ultra \Gamma ) = e( \{\alpha n\} \beta n )$$
where $\{\}: \R \to I_0$ is the signed fractional part function. The function $n \mapsto e( \{\alpha n\} \beta n )$ is then \emph{almost} a nilcharacter of degree $2$, with vertical frequency given by the function $\eta: [e_1,e_2]^{t_{12}} \mapsto -t_{12}$. All the properties required to give a nilcharacter in Definition \ref{nilch-def} are satisfied, save for one: the function $F$ is not Lipschitz on all of $G/\Gamma$, but is instead merely \emph{piecewise} Lipschitz, being discontinuous at some portions of the boundary of the fundamental domain. To put it another way, one can view $n \mapsto e(\{ \alpha n \} \beta n)$ as a \emph{piecewise} nilcharacter of degree $2$.
Indeed, a topological obstruction prevents one from constructing \emph{any} scalar function $F \in \Lip(\ultra(G/\Gamma) \to \overline{S^1})$ of unit magnitude on the Heisenberg nilmanifold with the above vertical frequency. By taking standard parts, we may assume that $F$ comes from a standard Lipschitz function $F: G/\Gamma \to S^1$ with the same vertical frequency. For any standard $t \in [-1/2,1/2]$, consider the loop $\gamma_t := \{ e_1^t e_2^s \Gamma: s \in I_0\}$. The image $F(\gamma_t)$ of this loop lives on the unit circle and thus has a well-defined winding number (or degree). As this degree must vary continuously in $t$ while remaining an integer, it is constant in $t$; in particular, $F(\gamma_{-1/2})$ and $F(\gamma_{1/2})$ must have the same winding number. On the other hand, from the Baker-Campbell-Hausdorff formula \eqref{bch} we see that
$$F( e_1^{1/2} e_2^s \Gamma ) = F( e_1^{-1/2} e_2^s e_1 [e_1,e_2]^s \Gamma ) = e(s) F( e_1^{-1/2} e_2^s \Gamma )$$
and so the winding number of $F(\gamma_{1/2})$ is one larger than the winding number of $F(\gamma_{-1/2})$, a contradiction.
If however we allow ourselves to work with higher dimensions $D$, then this topological obstruction disappears. Indeed, let us take a smooth partition of unity $1 = \sum_{k=1}^D \varphi_k^2(t,s)$ on $\T^2$, where $D \in \N^+$ and each $\varphi_k$ is supported in $B_k \mod \Z^2$, where $B_k$ is a ball of radius $1/100$ (say) in $\R^2$. Then if we define $F := (F_1,F_2,\ldots,F_D)$, where
\begin{equation}\label{fkts}
F_k( e_1^t e_2^s [e_1,e_2]^u \ultra \Gamma) := \varphi_k(t,s) e(u)
\end{equation}
whenever $(t,s) \in \ultra B_k$ and $u \in \ultra \R$, with $F_k = 0$ if no such representation of the above form exists, then one easily verifies that $F$ lies in $\Lip(\ultra(G/\Gamma) \to \overline{S^{2D-1}})$ with the vertical frequency $\eta$, and so the vector-valued sequence $\chi: n \mapsto F( g(n) \ultra \Gamma)$ is a nilcharacter of degree $2$. A computation shows that each component $\chi_k$ of this nilcharacter $\chi = (\chi_1,\ldots,\chi_D)$ takes the form
$$ \chi_k(n) = e( \{ \alpha n - \theta_k \} \beta n ) \psi_k(n)$$
for some offset $\theta_k \in \ultra \R$ and some degree $1$ nilsequence $\psi_k$. Thus we see that $\chi$ is in some sense ``equivalent modulo lower order terms'' with the bracket polynomial phase $n \mapsto e( \{ \alpha n \} \beta n)$. We refer to the vector-valued nilsequence $\chi$ as a \emph{vector-valued smoothing} of the piecewise nilsequence $n \mapsto e(\{\alpha n \} \beta n)$; we will informally refer to this smoothing operation several times in the sequel when discussing further examples of nilsequences that are associated with bracket polynomials.
Similar computations can be made in higher degree. For instance, bracket cubic phases such as $n \mapsto e( \{ \{ \alpha n \} \beta n \} \gamma n )$ or $n \mapsto e( \{ \alpha n^2 \} \beta n )$ with $\alpha,\beta,\gamma \in \ultra \R$ can be viewed as near-examples of degree $3$ nilcharacters (with the problem again being that $F$ is discontinuous on the boundary of the fundamental domain), but there exist vector-valued smoothings of these phases which are genuine degree $3$ nilcharacters. We will not detail these computations here, but they can essentially be found in \cite[Appendix E]{u4-inverse}. More generally, one can view bracket polynomial phases of degree $d$ as near-examples of nilcharacters of degree $d$ that can be converted to genuine examples using vector-valued smoothings; this fact can be made precise using the machinery from \cite{leibman}, but we will not need this machinery here.
\emph{Remark.} The above topological obstruction is quite annoying; it is the sole reason that we are forced to work with vector-valued functions. There are two other approaches to avoid this topological obstruction that we know of. One is to work with \emph{piecewise} Lipschitz functions rather than Lipschitz functions. This allows one in particular to build (piecewise) nilcharacters out of \emph{bracket polynomials}. This is the approach taken in \cite{u4-inverse}; however, it requires one to develop a certain amount of ``bracket calculus'' to manipulate these polynomials, and some additional arguments are also needed to deal with the discontinuities at the edges of the piecewise components of the nilmanifold. Another approach is to work with randomly selected fundamental domains of the nilmanifold (cf. \cite{green-tao-longaps}) which eliminates topological obstructions, with the randomness being used to ``average out'' the effects of the boundary of the domain. While all three methods will eventually work for the purposes of establishing the inverse conjecture, we believe that the vector-valued approach introduces the least amount of artificial technicality.
By definition, every nilcharacter of degree $d$ is a nilsequence of degree $\leq d$. The converse is far from being true; however, one can approximate nilsequences of degree $\leq d$ as bounded linear combinations of nilcharacters of degree $d$. More precisely, we have the following lemma.
\begin{lemma} Let $\psi \in \Nil^d([N] \to \overline{\C})$ be a scalar nilsequence of degree $d$, and let $\eps > 0$ be standard. Then one can approximate $\psi$ uniformly to error $\eps$ by a bounded linear combination \textup{(}over $\overline{\C}$\textup{)} of the components of nilcharacters in $\Xi^d([N])$.
\end{lemma}
\begin{proof} Unpacking the definitions, it suffices to show that for every degree $d$ filtered nilmanifold $G/\Gamma$, every $F \in \Lip(\ultra(G/\Gamma) \to \overline{\C})$, and every standard $\eps>0$, one can approximate $F$ uniformly to error $\eps$ by a bounded linear combination of functions in the class ${\mathcal F}(G/\Gamma)$ of components of standard Lipschitz functions $F' \in \Lip( G/\Gamma \to S^\omega )$ that have a vertical frequency in the sense of Definition \ref{nilch-def}.
By taking standard parts, we may assume that $F$ is a standard Lipschitz function. Observe that ${\mathcal F}(G/\Gamma)$ is closed under multiplication and complex conjugation. By the Stone-Weierstrass theorem, it thus suffices to show that ${\mathcal F}(G/\Gamma)$ separates any two distinct points $x, y \in G/\Gamma$. If $x, y$ do not lie in the same orbit of the $G_d$, then this is clear from a partition of unity (taking $\eta = 0$). If instead $x = g_d y$ for some $g_d \in G_d$, then the distinctness of $x,y$ forces $g_d \not \in \Gamma_d$, and hence by Pontryagin duality there exists a vertical frequency $\eta$ with $\eta(g_d) \neq 0$. If one then builds a nilcharacter with this frequency (by adapting the vector-valued smoothing construction \eqref{fkts}) we obtain the claim.
\end{proof}
We remark that this lemma can also be proven, with better quantitative bounds, by Fourier-analytic methods: see \cite[Lemma 3.7]{green-tao-nilratner}. As a corollary of the lemma, we have the following.
\begin{corollary}\label{nilch-cor} Suppose that $f \in L^\infty([N] \to \overline{\C}^\omega)$. Then $f$ is $d$-biased if and only if $f$ correlates with a nilcharacter $\chi \in \Xi^d([N])$.
\end{corollary}
It is easy to see that if $\chi, \chi'$ are two nilcharacters of degree $d$, then the tensor product $\chi \otimes \chi'$ and complex conjugate $\overline{\chi}$ are also nilcharacters. If all nilcharacters were scalar, this would mean that the space $\Xi^d([N])$ of degree $d$ nilcharacters form a multiplicative abelian group. Unfortunately, nilcharacters can be vector-valued, and so this statement is not quite true. However, it becomes true if one only focuses on the ``top order'' behaviour of a nilcharacter. To isolate this behaviour, we adopt the following key definition.
\begin{definition}[Symbol]\label{symbol-def} Let $d \geq 0$. Two nilcharacters $\chi, \chi' \in \Xi^d([N])$ of degree $d$ are \emph{equivalent} if $\chi \otimes \overline{\chi'}$ is equal on $[N]$ to a nilsequence of degree $\leq d-1$. This can be shown to be an equivalence relation (see Lemma \ref{equiv-lemma}); the equivalence class of a nilcharacter $\chi$ will be called the \emph{symbol} of $\chi$ and is denoted $[\chi]_{\Symb^d([N])}$. The space of all such symbols will be denoted $\Symb^d([N])$; we will show later (see Lemma \ref{symbolic}) that this is an abelian multiplicative group.
\end{definition}
When $d=1$, two nilcharacters $n \mapsto e(\alpha n + \beta)$ and $n \mapsto e( \alpha' n + \beta')$ are equivalent if and only if $\alpha-\alpha'$ is a limit integer, and $\Symb^1([N])$ is just $\ultra\T$ in this case. However, the situation is more complicated in higher degree.
To get some feel for this, consider two polynomial phases
$$ \chi: n \mapsto e(\alpha_0 + \ldots + \alpha_d n^d)$$
and
$$ \chi': n \mapsto e(\alpha'_0 + \ldots + \alpha'_d n^d)$$
with $\alpha_0,\ldots,\alpha_d,\alpha'_0,\alpha'_d \in \ultra \R$, and consider the problem of determining when $\chi$ and $\chi'$ are equivalent nilcharacters of degree $d$. Certainly this is the case if $\alpha_d$ and $\alpha'_d$ are equal, or differ by a limit integer. When $d \geq 2$, there are two further important cases in which equivalence occurs. The first is when $\alpha'_d = \alpha_d + O(N^{-d})$, because in this case the top degree component $e( (\alpha_d - \alpha'_d) n^d)$ of $\chi \overline{\chi'}$ can be viewed as a Lipschitz function of $n/2N \mod 1$ (say) on $[N]$ and is thus a $1$-step nilsequence. The second is when $\alpha'_d = \alpha_d + a/q$ for some standard rational $q$, since in this case the top degree component
$e( (\alpha_d - \alpha'_d) n^d)$ of $\chi \overline{\chi'}$ is periodic with period $q$ and can thus be viewed as a Lipschitz function of $n/q \mod 1$ and is therefore again a $1$-step nilsequence. We can combine all these cases together, and observe that $\chi$ and $\chi'$ are equivalent when $\alpha'_d = \alpha_d + a/q + O(N^{-d}) \mod 1$ for some standard rational $a/q$. It is possible to use the quantitative equidistribution theory of nilmanifolds (see \cite{green-tao-nilratner}) to show that these are in fact the \emph{only} cases in which $\chi$ and $\chi'$ are equivalent; this is a variant of the classical theorem of Weyl that a polynomial sequence is (totally) equidistributed modulo $1$ if and only if at least one non-constant coefficients is irrational. In view of this, we see that $\Symb^d([N])$ contains $\ultra \R / (\ultra \Z + \Q + N^{-d} \overline{\R})$ as a subgroup, and the symbol of $n \mapsto e(\alpha_0 + \ldots + \alpha_d n^d)$ can be identified with
$$\alpha_d \hbox{ mod } 1, \Q, O(N^{-d}) := \alpha + \ultra \Z + \Q + N^{-d} \overline{\R}.$$
However, the presence of bracket polynomials (suitably modified to avoid the topological obstruction mentioned earlier) means that when $d \geq 2$, that $\Symb^d([N])$ is somewhat larger than the above mentioned subgroup. We illustrate this with the following (non-rigorous) discussion. Take $d=2$ and consider two degree $2$ nilcharacters $\chi, \chi'$ of the form
$$ \chi(n) \approx e( \{ \alpha n \} \beta n + \gamma n^2 )$$
and
$$ \chi'(n) \approx e( \{ \alpha' n \} \beta' n + \gamma' n^2 )$$
for some $\alpha, \beta, \gamma, \alpha', \beta', \gamma' \in \ultra \R$, where we interpet the symbol $\approx$ loosely to mean that $\chi, \chi'$ are suitable vector-valued smoothings of the indicated bracket phases, of the type discussed earlier in this section. These may also involve some lower order nilsequences of degree $1$.
As before, we consider the question of determining those values of $\alpha,\beta,\gamma,\alpha',\beta',\gamma'$ for which $\chi$ and $\chi'$ are equivalent. There are a number of fairly obvious ways in which equivalence can occur. For instance, by modifying the previous arguments, one can show that equivalence holds when $\alpha=\alpha', \beta=\beta'$, and $\gamma-\gamma'$ is equal to a limit integer, a standard rational, or is equal to $O(N^{-2})$. Similarly, equivalence occurs when $\beta=\beta'$, $\gamma=\gamma'$, and $\alpha-\alpha'$ is equal to a limit integer, a standard rational, or is equal to $O(N^{-1})$.
However, there are also some slightly less obvious ways in which equivalence can occur. Observe that the expression $e( \{ \alpha n\} \{\beta n\} )$ is a Lipschitz function of the fractional parts of $\alpha n$ and $\beta n$ and is thus a (piecewise) nilsequence of degree $1$ (and will become a genuine nilsequence after one performs an appropriate vector-valued smoothing). On the other hand, we have the obvious identity
$$ e( (\alpha n - \{ \alpha n \}) (\beta n - \{\beta n\}) ) = 1$$
since the exponent is the product of two (limit) integers. Expanding this out and rearranging, we obtain the (slightly imprecise) relation
\begin{equation}\label{brackalg}
e( \{ \alpha n \} \beta n ) \approx e( - \{ \beta n \} \alpha n + \alpha \beta n^2 )
\end{equation}
where we again interpret $\approx$ loosely to mean ``after a suitable vector-valued smoothing, and ignoring lower order factors''. This gives an additional route for $\chi$ and $\chi'$ to be equivalent. A similar argument also gives the variant
$$ e( \{ \alpha n \} \beta n ) \approx e( \frac{1}{2} \alpha \beta n^2 )$$
whenever $\alpha,\beta$ are \emph{commensurate} in the sense that $\alpha/\beta$ is a standard rational. We thus see that the notion of equivalence is in fact already somewhat complicated in degree $2$, and the situation only becomes worse in higher degree. One can describe equivalence of bracket polynomials explicitly using \emph{bracket calculus}, as developed in \cite{leibman} (see also the earlier works \cite{bl,ha1,ha2,ha3}), but this requires a fair amount of notation and machinery. Fortunately, in this paper we will be able to treat the notion of a symbol \emph{abstractly}, without requiring an explicit description of the space $\Symb^d([N])$.\vspace{11pt}
\textsc{More general types of filtration.}
The notion of a one-dimensional polynomial $n \mapsto \alpha_0 + \ldots + \alpha_d n^d$ of degree $\leq d$ can of course be generalised to higher dimensions. For instance, we have the notion of a multidimensional polynomial
$$ (n_1,\ldots,n_k) \mapsto \sum_{i_1,\ldots,i_k \geq 0: i_1+\ldots+i_k \leq d} \alpha_{i_1,\ldots,i_k} n_1^{i_1} \ldots n_k^{i_d}$$
of degree $\leq d$. We also have the slightly different notion of a multidimensional polynomial
$$ (n_1,\ldots,n_k) \mapsto \sum_{i_1,\ldots,i_k \geq 0: i_j \leq d_j \hbox{ for } 1 \leq j \leq k} \alpha_{i_1,\ldots,i_k} n_1^{i_1} \ldots n_k^{i_d}$$
of \emph{multidegree} $\leq (d_1,\ldots,d_k)$ for some integers $d_1,\ldots,d_k\geq 0$. We can unify these two concepts into the notion of a multi-dimensional polynomial
\begin{equation}\label{multipoly}
(n_1,\ldots,n_k) \mapsto \sum_{(i_1,\ldots,i_k) \in J} \alpha_{i_1,\ldots,i_k} n_1^{i_1} \ldots n_k^{i_d}
\end{equation}
of \emph{multidegree} $\subset J$ for some finite \emph{downset} $J \subset \N^k$, i.e. a finite set of tuples with the property that $(i_1,\ldots,i_k) \in J$ whenever $(i_1,\ldots,i_k) \in \N^k$ and $i_j \leq i'_j$ for all $j=1,\ldots,k$ for some $(i'_1,\ldots,i'_k) \in J$. Thus for instance the two-dimensional polynomial
$$ (h,n) \mapsto \alpha h n + \beta h n^2 + \gamma n^3$$
for $\alpha,\beta,\gamma \in \ultra \R$ is of multidegree $\subset J$ for
\[
J := \{ (0,0), (0,1), (0,2), (0,3), (1,0), (1,1), (1,2) \},\]
and is also of multidegree $\leq (1,3)$ and of degree $\leq 3$. (One can view the downset $J$ as a variant of the \emph{Newton polytope} of the polynomial.)
In our subsequent arguments, we will need to similarly generalise the notion of a one-dimensional nilcharacter $n \mapsto \chi(n)$ of degree $\leq d$ to a multidimensional nilcharacter $(n_1,\ldots,n_k) \mapsto \chi(n_1,\ldots,n_k)$ of degree $\leq d$, of multidegree $\leq (d_1,\ldots,d_k)$, or of multidegree $\subset J$. We will define these concepts precisely in a short while, but we mention for now that the polynomial phase
$$ (h,n) \mapsto e( \alpha h n + \beta h n^2 + \gamma n^3 )$$
will be a two-dimensional nilcharacter of multidegree $\subset J$, multi-degree $\leq (1,3)$, and degree $\leq 3$ where $J$ is as above. Moreover, variants of this phase, such as (a suitable vector-valued smoothing of)
$$ (h,n) \mapsto e( \{ \alpha_1 h\} \alpha_2 n + \{ \{ \beta_1 n \} \beta_2 h \} \beta_3 n + \{ \gamma_1 n^2 \} \gamma_2 n ),$$
will also have the same multidegree and degree as the preceding example.
The multidegree of a nilcharacter $\chi(n_1,\ldots,n_k)$ is a more precise measurement of the complexity of $\chi$ than the degree, because it separates the behaviour of the different variables $n_1,\ldots,n_k$. We will also need a different refinement of the notion of degree, this time for a one-dimensional nilcharacter $n \mapsto \chi(n)$, which now separates the behaviour of different top degree components of $\chi$, according to their ``rank''. Heuristically, the rank of such a component is the number of fractional part operations $x \mapsto \{ x \}$ that are needed to construct that component, plus one; thus for instance
$$ n \mapsto e( \alpha n^3 ) $$
has degree $3$ and rank $1$,
$$ n \mapsto e( \{ \alpha n^2 \} \beta n ) $$
has degree $3$ and rank $2$ (after vector-valued smoothing),
$$ n \mapsto e( \{ \{ \alpha n \} \beta n \} \gamma n) $$
has degree $3$ and rank $3$ (after vector-valued smoothing), and so forth. We will then need a notion of a nilcharacter $\chi$ \emph{of degree-rank $\leq (d,r)$}, which roughly speaking means that all the components used to build $\chi$ either are of degree $<d$, or else are of degree exactly $d$ but rank at most $r$. Thus for instance,
$$ n \mapsto e( \{ \alpha n \} \beta n + \gamma n^3 )$$
has degree-rank $\leq (3,1)$ (after vector-valued smoothing), while
$$ n \mapsto e( \{ \alpha n \} \beta n + \gamma n^3 + \{ \delta n^2 \} \epsilon n )$$
has degree-rank $\leq (3,2)$ (after vector-valued smoothing), and
$$ n \mapsto e( \{ \alpha n \} \beta n + \gamma n^3 + \{ \delta n^2 \} \epsilon n+ \{ \{ \mu n \} \nu n \} \rho n)$$
has degree-rank $\leq (3,3)$ (after vector-valued smoothing).
In order to make precise the notions of multidegree and degree-rank for nilcharacters, it is convenient to adopt an abstract formalism that unifies degree, multidegree, and degree-rank into a single theory. We need the following abstract definition.
\begin{definition}[Ordering]\label{order-def} An \emph{ordering} $I = (I, \prec, +, 0)$ is a set $I$ equipped with a partial ordering $\prec$, a binary operation $+: I \times I \to I$, and a distinguished element $0 \in I$ with the following properties:
\begin{enumerate}
\item The operation $+$ is commutative and associative, and has $0$ as the identity element.
\item The partial ordering $\prec$ has $0$ as the minimal element.
\item If $i, j \in I$ are such that $i \prec j$, then $i + k \prec j+k$ for all $k \in I$.
\item For every $d \in I$, the initial segment $\{ i \in I: i \prec d \}$ is finite.
\end{enumerate}
A \emph{finite downset} in $I$ is a finite subset $J$ of $I$ with the property that $j \in J$ whenever $j \in I$ and $j \prec i$ for some $i \in J$.
\end{definition}
In this paper, we will only need the following three specific orderings (with $k$ a standard positive integer):
\begin{enumerate}
\item The \emph{degree ordering}, in which $I = \N$ with the usual ordering, addition, and zero element.
\item The \emph{multidegree ordering}, in which $I = \N^k$ with the usual addition and zero element, and with the product ordering, thus $(i'_1,\ldots,i'_k) \preceq (i_1,\ldots,i_k)$ if $i'_j \leq i_j$ for all $1 \leq j \leq k$.
\item The \emph{degree-rank ordering}, in which $I$ is the sector $\DR := \{ (d,r) \in \N^2: 0 \leq r \leq d \}$ with the usual addition and zero element, and the lexicographical ordering, that is to say $(d',r') \prec (d,r)$ if $d' < d$ or if $d'=d$ and $r'<r$.
\end{enumerate}
It is easy to verify that each of these three explicit orderings obeys the abstract axioms in Definition \ref{order-def}. In the case of the degree or degree-rank orderings, $I$ is totally ordered (for instance, the first few degree-ranks are $(0,0), (1,0), (1,1), (2,0)$, $(2,1), (2,2), (3,0), \ldots$), and so the only finite downsets are the initial segments. For the multidegree ordering, however, the initial segments are not the only finite downsets that can occur.
The one-dimensional notions of a filtration, nilsequence, nilcharacter, and symbol can be easily generalised to arbitrary orderings. We give the bare definitions here, and defer the more thorough treatment of these concepts to Appendix \ref{poly-app} and Appendix \ref{basic-sec}. We will however remark that when $I$ is the degree ordering, then all of the notions defined below simplify to the one-dimensional counterparts defined earlier.
\begin{definition}[Filtered group]\label{filtered-group} Let $I$ be an ordering and let $G$ be a group. By an \emph{$I$-filtration} on $G$ we mean a collection $G_{I} = (G_{ i})_{i \in I}$ of subgroups indexed by $I$, with the following properties:
\begin{enumerate}
\item (Nesting) If $i,j \in I$ are such that $i \prec j$, then $G_i \supseteq G_j$.
\item (Commutators) For every $i,j \in I$, we have $[G_{i}, G_{ j}] \subseteq G_{i+j}$.
\end{enumerate}
If $d \in I$, we say that $G$ has \emph{degree} $\leq d$ if $G_i$ is trivial whenever $i \not \preceq d$. More generally, if $J$ is a downset in $I$, we say that $G$ has \emph{degree} $\subseteq J$ if $G_i$ is trivial whenever $i \not \in J$.
\end{definition}
Let us explicitly adapt the above abstract definitions to the three specific orderings mentioned earlier.
\begin{definition} If $(d_1,\ldots,d_k) \in \N^k$, we define a \emph{nilpotent Lie group of multi-degree $\leq (d_1,\ldots,d_k)$} to be a nilpotent $I$-filtered Lie group of degree $\leq (d_1,\ldots,d_k)$, where $I = \N^k$ is the multidegree ordering. Similarly, if $J$ is a downset, define the notion of a nilpotent Lie group of multidegree $\subseteq J$.
If $(d,r) \in \DR$, define a \emph{nilpotent Lie group of degree-rank $\leq (d,r)$} to be a nilpotent $\DR$-filtered Lie group $G$ of degree $\leq (d,r)$, with the additional axioms $G_{(0,0)}=G$ and $G_{(d,0)} = G_{(d,1)}$ for all $d \geq 1$.
We define the notion of a filtered nilmanifold of multidegree $\leq (d_1,\ldots,d_k)$, multidegree $\subseteq J$, or degree-rank $\leq (d,r)$ similarly.
\end{definition}
Note that the degree-rank filtration needs to obey some additional axioms, which are needed in order for the rank $r$ to play a non-trivial role. As such, the unification here of degree, multidegree, and degree-rank, is not quite perfect; however this wrinkle is only of minor technical importance and should be largely ignored on a first reading.
\begin{example} If $G$ is a filtered nilpotent group of multidegree $\leq (1,1)$, then the groups $G_{(1,0)}$ and $G_{(0,1)}$ must be abelian normal subgroups of $G_{(0,0)}$, and their commutator $[G_{(1,0)}, G_{(0,1)}]$ must lie inside the group $G_{(1,1)}$, which is a central subgroup of $G_{(0,0)}$.
If $G$ is a filtered nilpotent group of degree-rank $\leq (d,d)$, then $(G_{(i,0)})_{i \geq 0}$ is a $\N$-filtration of degree $\leq d$. But if we reduce the rank $r$ to be strictly less than $d$, then we obtain some additional relations between the $G_{(i,0)}$ that do not come from the filtration property. For instance, if $G$ has degree-rank $\leq (3,2)$, then the group $[G_{(1,0)},[G_{(1,0)},G_{(1,0)}]]$ must now be trivial; if $G$ has degree-rank $\leq (3,1)$, then the group $[G_{(1,0)}, G_{(2,0)}]$ must also be trivial. More generally, if $G$ has degree-rank $\leq (d,r)$, then any iterated commutator of $g_{i_1},\ldots,g_{i_m}$ with $g_j \in G_{(i_j,0)}$ for $j=1,\ldots,m$ will be trivial whenever $i_1+\ldots+i_m > d$, or if $i_1+\ldots+i_m=d$ and $m>r$.
\end{example}
\begin{example}\label{inclusions} If $(G_i)_{i \in \N}$ is an $\N$-filtration of $G$ of degree $\leq d$, then $(G_{|\vec i|})_{\vec i \in \N^k}$ is an $\N^k$-filtration of $G$ of multidegree $\subset \{\vec i \in \N^k: |\vec i| \leq d \}$, where we recall the notational convention $|(i_1,\ldots,i_k)| = i_1 + \ldots + i_k$. Conversely, if $J$ is a finite downset of $\N^k$ and $(G_{\vec i})_{\vec i \in \N^k}$ is a $\N^k$-filtration of $G$ of multidegree $\subset J$, then
$$ \left( \bigvee_{\vec i: |\vec i| \leq i} G_{\vec i} \right)_{i \in \N}$$
is easily verified (using Lemma \ref{normal}) to be an $\N$-filtration of degree $\leq \max_{\vec i \in J} |\vec i|$, where $\bigvee_{a \in A} G_a$ is the group generated by $\bigcup_{a \in A} G_a$. In particular, any multidegree $\leq (d_1,\ldots,d_k)$ filtration induces a degree $\leq d_1+\ldots+d_k$ filtration.
In a similar spirit, every degree-rank $\leq (d,r)$ filtration $(G_{(d',r')})_{(d',r') \in \DR}$ of a group $G$ induces a degree $\leq d$ filtration $(G_{(i,0)})_{i \in \N}$. In the converse direction, if $(G_i)_{i \in \N}$ is a degree $\leq d$ filtration of $G$ with $G=G_0$, then we can create a degree-rank $\leq (d,d)$ filtration $(G_{(d',r')})_{(d',r') \in \DR}$ by setting $G_{(d',r')}$ to be the space generated by all the iterated commutators of $g_{i_1},\ldots,g_{i_m}$ with $g_j \in G_{(i_j,0)}$ for $j=1,\ldots,m$ for which either $i_1+\ldots+i_m > d'$, or $i_1+\ldots+i_m=d$ and $m \geq \max(r',1)$; this can easily be verified to indeed be a filtration, thanks to Lemma \ref{normal}.
\end{example}
\begin{example}\label{dr-f} Let $d \geq 1$ be a standard integer. We can give the unit circle $\T$ the structure of a degree-rank filtered nilmanifold of degree-rank $\leq (d,1)$ by setting $G=\R$ and $\Gamma=\Z$ with $G_{(d',r')} := \R$ for $(d',r') \leq (d,1)$ and $G_{(d',r')} := \{0\}$ otherwise. This is also the filtration obtained from the degree $\leq d$ filtration (see Example \ref{polyphase}) using the construction in Example \ref{inclusions}.
\end{example}
\begin{example}[Products]\label{prodeq} If $G_{I}$ and $G'_I$ are $I$-filtrations on groups $G, G'$ then we can give the product $G \times G'$ an $I$-filtration in an obvious way by setting $(G \times G')_i := G_i \times G'_i$. The degree of $G \times G'$ is the union of the degrees of $G$ and $G'$. Similarly the product $G_1/\Gamma_1 \times G_2/\Gamma_2$ of two $I$-filtered nilmanifolds is an $I$-filtered nilmanifold.
\end{example}
\begin{example}[Pushforward and pullback]\label{pushpull} Let $\phi: G \to H$ be a homomorphism of groups. Then any any $I$-filtration $H_I = (H_{ i})_{i \in I}$ of $H$ induces a \emph{pullback $I$-filtration} $\phi^* H_I := (\phi^{-1}(H_{i}))_{i \in I}$. Similarly, any $I$-filtration $G_{I} = (G_{i})_{i \in I}$ on $G$ induces a \emph{pushforward $I$-filtration} $\phi_* G_{I} := (\phi(G_{i}))_{i \in I}$ on $H$. In particular, if $\Gamma$ is a subgroup of $G$, then we can pullback a filtration $G_{I} = (G_{ i})_{i \in I}$ of $G$ by the inclusion map $\iota : \Gamma \hookrightarrow G$ to create the \emph{restriction} $\Gamma_{I} := (\Gamma_{i})_{i \in I}$ of that filtration. It is a trivial matter to check that the subgroups of this filtration are given by $\Gamma_{ i} := \Gamma \cap G_{i}$.
\end{example}
\begin{definition}[Filtered quotient space]\label{quot} A \emph{$I$-filtered quotient space} is a quotient $G/\Gamma$, where $G$ is an $I$-filtered group and $\Gamma$ is a subgroup of $G$ (with the induced filtration, see Example \ref{pushpull}).
A \emph{$I$-filtered homomorphism} $\phi: G/\Gamma \to G'/\Gamma'$ between $I$-filtered quotient spaces is a group homomorphism $\phi: G \to G'$ which maps $\Gamma$ to $\Gamma'$, and also maps $G_i$ to $G'_i$ for all $i \in I$. Note that such a homomorphism descends to a map from $G/\Gamma$ to $G'/\Gamma'$.
If $G$ is a nilpotent $I$-filtered Lie group, and $\Gamma$ is a discrete cocompact subgroup of $G$ which is rational with respect to $G_I$ (thus $\Gamma_i := \Gamma \cap G_i$ is cocompact in $G_i$ for each $i \in I$), we call $G/\Gamma = (G/\Gamma, G_I)$ an \emph{$I$-filtered nilmanifold}.
We say that $G/\Gamma$ has degree $\leq d$ or $\subseteq J$ of $G$ has degree $\leq d$ or $\subseteq J$.
\end{definition}
\begin{example}[Subnilmanifolds] Let $G/\Gamma$ be an $I$-filtered nilmanifold of degree $\subset J$. If $H$ is a rational subgroup of $G$, then $H/(H \cap \Gamma)$ is also a filtered nilmanifold degree $\subset J$ (using Example \ref{pushpull}), with an inclusion homomorphism from $H/(H \cap \Gamma)$ to $G/\Gamma$; we refer to $H/(H \cap \Gamma)$ as a \emph{subnilmanifold} of $G/\Gamma$.
\end{example}
We isolate three important examples of a filtered group, in which $G$ is the additive group $\Z$ or $\Z^k$.
\begin{definition}[Basic filtrations]\label{basic-filter} We define the following filtrations:
\begin{itemize}
\item The \emph{degree filtration} $\Z^k_\N$ on $G = \Z^k$, in which $I = \N$ is the degree ordering and $G_i = G$ for $i \leq 1$ and $G_i = \{0\}$ otherwise. In many cases $k$ will equal $1$ or $2$.
\item The \emph{multidegree filtration} $\Z^k_{\N^k}$ on $G = \Z^k$, in which $I=\N^k$ is the multidegree ordering and $G_{\vec{0}} = \Z^k$, $G_{\vec{e}_i} = \langle \vec{e}_i\rangle$, $i = 1,\dots,k$, and $G_{\vec{v}} = \{ 0\}$ otherwise, with $e_1,\ldots,e_k$ being the standard basis for $\Z^k$;
\item The \emph{degree-rank filtration} $\Z_\DR$ on $G = \Z$, in which $I=\DR$ is the degree-rank ordering and $G_{(0,0)} = G_{(1,0)} = \Z$ and $G_{(d,r)} = \{0\}$ otherwise.
\end{itemize}
\end{definition}
\begin{definition}[Polynomial map]\label{poly-map-def}
Suppose that $H$ and $G$ are $I$-filtered groups with $H = (H,+)$ abelian\footnote{This is not actually a necessary assumption; see Appendix \ref{poly-app}. However, in the main body of the paper we will only be concerned with polynomial maps on additive domains.}. Then for any map $g : H \rightarrow G$ we define the derivative
\begin{equation}\label{partial-def} \partial_h g(n) := g(n+h) g(n)^{-1}.\end{equation}
We say that $g : H \rightarrow G$ is \emph{polynomial} if
\begin{equation}\label{polynomial-sequence-def} \partial_{h_1} \dots \partial_{h_m} g (n) \in G_{i_1 + \dots + i_m}\end{equation}
for all $m \geq 0$, all $i_1,\dots, i_m \in I$ and all $h_j \in H_{i_j}$ for $j = 1,\dots, m$, and for all $n \in H_0$.
We denote by $\poly(H_I \to G_I)$ the space of all polynomial maps from $H_I$ to $G_I$. As usual, we use $\ultra \poly(H_I \to G_I)$ to denote the space of all limit polynomial maps from $\ultra H_I$ to $\ultra G_I$ (i.e. ultralimits of polynomial maps in $\poly(H_I \to G_I)$).
\end{definition}
Many facts about these spaces (in some generality) are established in Appendix \ref{poly-app} where, in particular, a remarkable result essentially due to Lazard and Leibman \cite{lazard,leibman-group-1,leibman-group-2} is established: $\poly(H_I \to G_I)$ is a group. The material in Appendix \ref{poly-app} is formulated in the general setting of abstract orderings $I$ and for arbitrary (and possibly non-abelian) groups $H_I$, but for our applications we are only interested in the special case when $H_I$ is $\Z$ or $\Z^k$ with the degree, multidegree, or degree-rank filtration as defined above.
Before moving on let us be quite explicit about what the notion of a polynomial map is in each of the three cases, since the definitions take a certain amount of unravelling.
\begin{itemize}
\item (Degree filtration) If $H = \Z^k$ with the degree filtration $\Z^k_\N$, then $\poly(\Z^k_\N \to G_\N)$ consists of maps $g : \Z^k \rightarrow G$ with the property that
\[ \partial_{h_1} \dots \partial_{h_m} g(n) \in G_m\] for all $m \geq 0$, $h_1,\dots,h_m \in \Z$ and all $n \in G_0$. This space is precisely the same space as the one considered in \cite[\S 6]{green-tao-nilratner}. The space $\ultra \poly(\Z^k \to G_\N)$ is defined similarly, except that $g: \ultra \Z^k \to \ultra G$ is now a limit map, and all spaces such as $\Z$ and $G_m$ need to be replaced by their ultrapowers. (Similarly for the other two examples in this list.)
\item (Multidegree filtration) If $H = \Z^k$ with the multidegree filtration $\Z^k_{\N^k}$, then $\poly(\Z^k_{\N^k} \to G_{\N^k})$ consists of maps $g : \Z^k \rightarrow G$ with the property that
\[ \partial_{\vec{e}_{i_1}} \dots \partial_{\vec{e}_{i_m}} g(\vec{n}) \in G_{\vec{e}_{i_1} + \dots + \vec{e}_{i_m}}\]
for all $k \ge 0$, all $i_1,\dots, i_m$ and all $\vec{n} \in \Z^k$. To relate this space to the analogous spaces for the degree ordering, observe (using Example \ref{inclusions}) that
$$ \poly(\Z^k_\N \to (G_i)_{i \in \N} ) = \poly(\Z^k_{\N^k} \to (G_{|\vec i|})_{\vec i \in \N^k} )$$
for any $\N$-filtration $(G_i)_{i \in \N}$, and conversely one has
$$ \poly(\Z^k_{\N^k} \to (G_{\vec i})_{\vec i \in \N^k} ) \subset \poly\left(\Z^k_\N \to ( \bigvee_{|\vec i| = i} G_{\vec i} )_{i \in \N} \right)$$
for any $\N^k$-filtration $(G_{\vec i})_{\vec i \in \N^k}$. This is of course related to the obvious fact that a polynomial of multidegree $\leq (d_1,\ldots,d_k)$ is automatically of degree $\leq d_1+\ldots+d_k$.
\item (Degree-rank filtration) If $H = \Z$ with the degree-rank filtration $\Z_\DR$, $\poly(\Z_{\DR} \to G_{\DR})$ consists of maps $g : \Z \rightarrow G$ with the property that
\[ \partial_{h_1} \dots \partial_{h_m} g(n) \in G_{(m,0)}\]
whenever $m \geq 0$, $h_1,\dots,h_m \in \Z$ and $n \in G_0$. We observe (using Example \ref{inclusions}) the obvious equality
\begin{equation}\label{dreq}
\poly(\Z_\DR \to (G_{(d,r)})_{(d,r) \in \DR} ) = \poly(\Z_\N \to (G_{(i,0)})_{i \in \N} )
\end{equation}
for any $\DR$-filtration $(G_{(d,r)})_{(d,r) \in \DR}$. Thus, a degree-rank filtration $G_\DR$ on $G$ does not change the notion of a polynomial sequence, but instead gives some finer information on the group $G$ (and in particular, it indicates that certain iterated commutators of the $G_{(d,r)}$ vanish, which is information that cannot be discerned just from the knowledge that $(G_{(i,0)})_{i \in \N}$ is a $\N$-filtration).
\end{itemize}
\begin{definition}[Nilsequences and nilcharacters]\label{nilch-def-gen} Let $I$ be an ordering, and let $J$ be a finite downset in $I$. Let $H$ be an abelian $I$-filtered group. A (polynomial) nilsequence of degree $\subset J$ is any function of the form
\[\chi(n) = F(g(n) \ultra \Gamma),\] where
\begin{itemize}
\item $G/\Gamma = (G/\Gamma,G_I)$ is an $I$-filtered nilpotent manifold of degree $\subset J$;
\item $g \in \ultra\poly(H_{I} \to G_{I})$ is a limit polynomial map from $\ultra H_I$ to $\ultra G_I$; and
\item $F \in \Lip(\ultra (G/\Gamma) \rightarrow \overline{C}^{\omega})$.
\end{itemize}
The space of all such nilsequences will be denoted $\Nil^{\subset J}(\ultra H)$. We define the notion of a nilsequence of degree $\leq d$ for some $d \in I$, and the space $\Nil^{\leq d}(\ultra H)$, similarly. If $\Omega$ is a limit subset of $\ultra H$, the restriction of the nilsequences in $\Nil^{\subset J}(\ultra H)$ to $\Omega$ will be denoted $\Nil^{\subset J}(\Omega)$, and we define $\Nil^{\leq d}(\Omega)$ similarly.
We refer to the map $n \mapsto g(n) \ultra \Gamma$ as a \emph{limit polynomial orbit} in $G/\Gamma$, and denote the space of such orbits as $\ultra \poly(H_I \to (G/\Gamma)_I)$.
Suppose that $d \in I$. Then $\chi$ is said to be a \emph{degree $d$ nilcharacter} if $\chi$ is a degree $\leq d$ nilsequence with the following additional properties:
\begin{itemize}
\item $F \in \Lip(\ultra(G/\Gamma) \to \overline{S^{\omega}})$ (thus $|F|=1$) and
\item $F( g_d x ) = e( \eta(g_d) ) F(x)$ for all $x \in G/\Gamma$ and $g_d \in G_{d}$, where $\eta: G_{d} \to \R$ is a continuous standard homomorphism which maps $\Gamma_{d}$ to the integers. We call $\eta$ the \emph{vertical frequency} of $F$.
\end{itemize}
The space of all degree $d$ nilcharacters on $\ultra H$ will be denoted $\Xi^d(\ultra H)$. If $\Omega$ is a limit subset of $\ultra H$, the restriction of the nilcharacters in $\Xi^d(\ultra H)$ to $\Omega$ will be denoted $\Xi^d(\Omega)$.
With the multidegree ordering, a degree $(d_1,\ldots,d_k)$ nilcharacter will be referred to as a multidegree $(d_1,\ldots,d_k)$ nilcharacter, and the space of such characters on $\Omega$ denoted $\Xi^{(d_1,\ldots,d_k)}_\MD(\Omega)$; we similarly write $\Nil^{\subset J}(\Omega)$ or $\Nil^{\leq (d_1,\ldots,d_k)}(\Omega)$ as $\Nil^{\subset J}_\MD(\Omega)$ or $\Nil^{\leq (d_1,\ldots,d_k)}(\Omega)$ for emphasis.
Similarly, with the degree-rank ordering, and assuming $G/\Gamma$ is a filtered nilmanifold of degree-rank $\leq (d,r)$ (so in particular, we enforce the axioms $G_{(0,0)} = G$ and $G_{(d,0)}=G_{(d,1)}$), a degree $(d,r)$ nilcharacter will be referred to as a degree-rank $(d,r)$ nilcharacter. The space of nilcharacters on $\Omega$ of degree-rank $(d,r)$ will be denoted $\Xi^{(d,r)}_\DR(\Omega)$ (note that this is distinct from the space $\Xi^{(d_1,d_2)}_\MD(\Omega)$ of two-dimensional nilcharacters of multidegree $(d_1,d_2)$), and the nilsequences on $\Omega$ of degree-rank $\leq (d,r)$ will similarly be denoted $\Nil^{\leq (d,r)}_\DR(\Omega)$.
\end{definition}
\begin{example} Let $J \subset \N^k$ be a finite downset. Then any sequence of the form
$$ (n_1,\ldots,n_k) \mapsto F\left( \sum_{(i_1,\ldots,i_k) \in J} \alpha_{i_1,\ldots,i_k} n_1^{i_1} \ldots n_k^{i_k} \mod 1\right),$$
where $\alpha_{i_1,\ldots,i_k} \in \ultra \R$ and $F \in \Lip( \ultra \T \to \overline{\C}^\omega )$, is a nilsequence on $\Z^k$ of multidegree $\subseteq J$, as can easily be seen by giving $G := \R$ the $\Z^k$-filtration $G_i := \R$ for $i \in J$ and $G_i := \{0\}$ otherwise, and setting $\Gamma := \Z$ and $g \in \ultra \poly( \Z^k \to \R )$ to be the limit polynomial $n \mapsto \sum_{(i_1,\ldots,i_k) \in J} \alpha_{i_1,\ldots,i_k} n_1^{i_1} \ldots n_k^{i_k}$.
For similar reasons, any sequence of the form
$$ (n_1,\ldots,n_k) \mapsto e\left( \sum_{(i_1,\ldots,i_k) \in \N^k: i_1+\ldots+i_k \leq d} \alpha_{i_1,\ldots,i_k} n_1^{i_1} \ldots n_k^{i_k} \mod 1\right),$$
is a degree $d$ nilcharacter on $\Z^k$ of degree $d$, and any sequence of the form
$$ (n_1,\ldots,n_k) \mapsto e\left( \sum_{(i_1,\ldots,i_k) \in \N^k: i_j \leq d_j \hbox{ for } j=1,\ldots,k} \alpha_{i_1,\ldots,i_k} n_1^{i_1} \ldots n_k^{i_k} \mod 1\right),$$
is a multidegree $(d_1,\ldots,d_k)$ nilcharacter on $\Z^k$.
\end{example}
\begin{example}\label{abn} Any degree $2$ nilsequence of magnitude $1$ is automatically a degree-rank $\leq (3,0)$ nilcharacter, since every degree $\leq 2$ nilmanifold is automatically a degree-rank $\leq (2,2)$ nilmanifold, which can then converted trivially to a degree-rank $\leq (3,0)$ nilmanifold (with a trivial group $G_{(3,0)}$). Thus for instance for $\alpha,\beta \in \R$,
$$ n \mapsto e( \{ \alpha n \} \beta n )$$
is nearly a degree-rank $(3,0)$ nilcharacter, and becomes a genuine degree-rank $(3,0)$ nilcharacter after vector-valued smoothing.
If $\alpha \in \ultra \R$, then the sequence
$$ n \mapsto e( \alpha n^3 )$$
is a degree-rank $(3,1)$ nilcharacter. Indeed, we can give $G=\R$ a degree-rank $\leq (3,1)$ filtration $G_\DR$ by setting $G_{(d,r)} := \R$ for $(d,r) \leq (3,1)$, and $G_{(d,r)} := \{0\}$ otherwise.
Next, if $\alpha, \beta \in \ultra \R$, then the sequence
\begin{equation}\label{abn-eq}
n \mapsto e( \{ \alpha n^2 \} \beta n )
\end{equation}
is \emph{nearly} a degree-rank $(3,2)$ nilcharacter (and becomes a genuinely so after vector-valued smoothing). To see this, let $G$ be the Heisenberg nilpotent group \eqref{heisen}, which we give the following degree-rank filtration:
\begin{align*}
G_{(0,0)} = G_{(1,0)} = G_{(1,1)} &:= G \\
G_{(2,0)} = G_{(2,1)} &:= \langle e_1, [e_1,e_2] \rangle_\R = \{ e_1^{t_1} [e_1,e_2]^{t_{12}}: t_1,t_{12} \in \R \} \\
G_{(2,2)} = G_{(3,0)} = G_{(3,1)} = G_{(3,2)} &:= \langle [e_1,e_2] \rangle_\R = \{ [e_1,e_2]^{t_{12}}: t_{12} \in \R \} \\
G_{(d,r)} &:= \{\id\} \hbox{ for all other } (d,r) \in \DR.
\end{align*}
One easily verifies that this is a degree-rank $\leq (3,2)$ filtration. If we then set $g: \ultra \Z \to \ultra G$ to be the limit sequence $g(n) := e_2^{\beta n} e_1^{\alpha n^2}$, one easily verifies that $g$ is a limit polynomial with respect to this degree-rank filtration. If one then lets $F$ be the piecewise Lipschitz function \eqref{fdef}, then we see that
$$ F( g(n) \ultra \Gamma ) = e( \{ \alpha n^2 \} \beta n )$$
and so we see that $n \mapsto e( \{ \alpha n^2 \} \beta n )$ is a indeed piecewise degree-rank $(3,2)$ nilcharacter.
A similar argument (using the free $3$-step nilpotent manifold on three generators, which has degree $\leq 3$ and hence degree-rank $\leq (3,3)$) shows that
$$ n \mapsto e( \{ \{ \alpha n \} \beta n \} \gamma n )$$
is nearly a degree-rank $(3,3)$ nilcharacter, and becomes a genuine degree-rank $(3,3)$ nilcharacter after applying vector-valued smoothing; see \cite[Appendix E]{u4-inverse} for the relevant calculations.
These examples should help illustrate the heuristic that a degree-rank $(d,r)$ nilcharacter is built up using (suitable vector-valued smoothings of) bracket monomials which either have degree less than $d$, or have degree exactly $d$ and involve at most $r-1$ applications of the fractional part operation.
\end{example}
We observe (using Example \ref{inclusions}) the following obvious inclusions:
\begin{enumerate}
\item A multidegree $\leq (d_1,\ldots,d_k)$ nilsequence on $\Z^k$ is automatically a degree $\leq d_1+\ldots+d_k$ nilsequence.
\item A multidegree $(d_1,\ldots,d_k)$ nilcharacter on $\Z^k$ is automatically a degree $d_1+\ldots+d_k$ nilcharacter.
\item A multidegree $(d_1,\ldots,d_{k-1},0)$ nilsequence on $\Z^k$ is constant in the $n_k$ variable, and descends to a multidegree $(d_1,\ldots,d_{k-1})$ nilsequence on $\Z^{k-1}$.
\item A degree-rank $\leq (d,r)$ nilsequence on $\Z$ is automatically a degree $\leq d$ nilsequence.
\item A degree $\leq d$ nilsequence on $\Z$ is automatically a degree-rank $\leq (d,d)$ nilsequence.
\item A degree $d$ nilcharacter on $\Z$ is automatically a degree-rank $\leq (d,d)$ nilcharacter.
\end{enumerate}
It is not quite true, though, that a degree-rank $(d,r)$ nilcharacter is a degree $d$ nilcharacter if $r>1$, because the former need not exhibit vertical frequency behaviour for degree-ranks $(d,r')$ with $r'<r$.
\begin{definition}[Equivalence and symbols]\label{equiv-def}
Let $H$ be an $I$-filtered group, let $d \in I$, and let $\Omega$ be a limit subset of $\ultra H$. Two nilcharacters $\chi, \chi' \in \Xi^d(\Omega)$ are said to be \emph{equivalent} if $\chi\otimes\overline{\chi'}$ is a nilsequence of degree strictly less than $d$.
Write $[\chi]_{\Symb^d(\Omega)}$ for the equivalence class of $\chi$ with respect to this relation; this we shall refer to as the \emph{symbol} of $\chi$. Write $\Symb^d(\Omega)$ for the space of all such equivalence classes.
\end{definition}
We write $\Symb^{(d_1,\ldots,d_k)}_{\MD}(\Omega)$ for the symbols of nilcharacters $\chi \in \Xi^{(d_1,\ldots,d_k)}_\MD(\Omega)$ of multidegree $(d_1,\ldots,d_k)$, and $\Symb^{(d,r)}_\DR(\Omega)$ for the symbols of nilcharacters $\chi \in \Xi^{(d,r)}_\DR(\Omega)$ of degree-rank $(d,r)$. The basic properties of such symbols are set out in Appendix \ref{basic-sec}.
\section{A more detailed outline of the argument}\label{overview-sec}
Now that we have set up the notation to describe nilcharacters and their symbols, we are ready to give a high-level proof of Conjecture \ref{gis-conj-nonst-2} (and hence Theorem \ref{mainthm}), contingent on some key sub-theorems which will be proven in later sections. This corresponds to the realisation of points (i), (ii) and (ix) from the overview in \S \ref{strategy-sec}.
As the cases $s=1,2$ of this conjecture are already known, we assume that $s \geq 3$. We also assume inductively that the claim has already been proven for smaller values of $s$. Henceforth $s$ is fixed.
Let $f \in L^\infty[N]$ be such that
\begin{equation}\label{fus}
\|f\|_{U^{s+1}[N]} \gg 1.
\end{equation}
Define $f$ to be zero outside of $[N]$. Raising \eqref{fus} to the power $2^{s+1}$, we see that
$$ \E_{h \in [[N]]} \| \Delta_h f \|_{U^{s}[N]}^{2^s} \gg 1$$
and thus
$$ \| \Delta_h f \|_{U^{s}[N]} \gg 1$$
for all $h$ in a dense subset $H$ of $[[N]]$. Applying the inductive hypothesis, we thus see that $\Delta_h f$ is $(s-1)$-biased for all $h \in H$.
By definition, we now know that $\Delta_h f$ correlates with a nilsequence of degree $(s-1)$. By Lemma \ref{nilch-cor}, we see that for each $h \in H$, $\Delta_h f$ correlates with a nilcharacter $\chi_h \in \Xi^{s-1}([N])$. It is not hard to see that the space of such nilcharacters is a $\sigma$-limit set (see Definition \ref{separ}), so by Lemma \ref{mes-select} we can ensure that $\chi_h$ depends in a limit fashion on $h$.
The aim at this point is to obtain, in several stages, information about the dependence of $\chi_h$ on $h$. A key milestone in this analysis is a \emph{linearisation} of $\chi_h$ on $h$. In the case $s = 2$, treated in \cite{gowers-4aps,green-tao-u3inverse}, the $\chi_h(n)$ were essentially just linear phases $e(\xi_h n)$, and the outcome of the linearisation analysis was that the frequencies $\xi_h$ may be assumed to vary in a bracket-linear fashion with $h$. In the case $s = 3$ (treated in \cite{u4-inverse} but also dealt with in our present work), a model special case occurs when $\chi_h(n) \approx e(\{\alpha_h n\} \beta_h n)$ (interpreting $\approx$ loosely). The outcome of the linearisation analysis in that case was that at most one of $\alpha_h, \beta_h$ really depends on $h$, and furthermore that this dependence on $h$ is bracket-linear in nature.
Now we formally set out the general case of this linearisation process.
\begin{theorem}[Linearisation]\label{linear-thm} Let $f \in L^\infty[N]$, let $H$ be a dense subset of $[[N]]$, and let $(\chi_h)_{h \in H}$ be a family of nilcharacters in $\Xi^{s-1}([N])$ depending in a limit fashion on $h$, such that $\Delta_h f$ correlates with $\chi_h$ for all $h \in H$. Then there exists a multidegree $(1, s-1)$-nilcharacter $\chi \in \Xi^{(1,s-1)}_\MD(\ultra \Z^2)$ such that $\Delta_h f$ $(s-2)$-correlates with $\chi(h,\cdot)$ for many $h \in H$.
\end{theorem}
This statement represents the outcome of points (iii) to (vii) of the outline in \S \ref{strategy-sec} and must therefore address the following points:
\begin{itemize}
\item For some suitable notion of ``frequency'', the symbol of $\chi_h(n)$ contains only one frequency that genuinely depends on $h$;
\item That frequency depends on $h$ in a bracket-linear manner;
\item Once this is known, it follows that, for many $h$, $\Delta_h f$ $(s-2)$-correlates with $\chi(h, n)$, where $\chi$ is a certain $2$-variable nilsequence.
\end{itemize}
These three tasks are, in fact, established together and in an incremental fashion. The nilcharacter $\chi_h(n)$ is gradually replaced by objects of the form $\chi'(h,n)\otimes \chi'_h(n)$ where $\chi'(h,n)$ is a $2$-dimensional nilcharacter of multidegree $(1, s-1)$ and, at each stage, the nilcharacter $\chi'_h(n)$ (which has so far not been shown to vary in any nice way with $h$) is ``simpler'' than $\chi_h(n)$. The notion of \emph{simpler} in this context is measured by the degree-rank filtration, a concept that was introduced in the previous section. Thus the result of a single pass over the three points listed above is the following subclaim.
\begin{theorem}[Linearisation, inductive step]\label{linear-induct} Let $1 \leq r_* \leq s-1$, let $f \in L^\infty[N]$, let $H$ be a dense subset of $[[N]]$, let $\chi \in \Xi^{(1,s-1)}_\MD(\ultra \Z^2)$, let $(\chi_h)_{h \in H}$ be a family of nilcharacters of degree-rank $(s-1,r_*)$ depending in a limit fashion on $h$, such that $\Delta_h f$ $(s-2)$-correlates with $\chi(h,\cdot) \otimes \chi_h$ for all $h \in H$. Then there exists a dense subset $H'$ of $H$, a multidegree $(1, s-1)$-nilcharacter $\chi' \in \Xi^{(1,s-1)}_\MD(\ultra \Z^2)$ and a family $(\chi'_h)_{h \in H}$ of nilcharacters of degree-rank $(s-1,r_*-1)$ depending in a limit fashion on $h$, such that $\Delta_h f$ $(s-2)$-correlates with $\chi'(h,\cdot) \otimes \chi'_h$ for all $h \in H'$.
\end{theorem}
Theorem \ref{linear-thm} follows easily by inductive use of this statement, starting with $r_*$ equal to $s-1$ and using Theorem \ref{linear-induct} iteratively to decrease $r_*$ all the way to zero.
To prove Theorem \ref{linear-induct}, we follow steps (iii) to (vii) in the outline quite closely. The first step, which is the realisation of (iii), is a Gowers-style Cauchy-Schwarz inequality to eliminate the function $f$ as well as the $2$-dimensional nilcharacter $\chi(h,n)$ and therefore obtain a statement concerning only the (so far) unstructured-in-$h$ object $\chi_h(n)$. Here is a precise statement of the outcome of this procedure; the proof of this proposition is the main business of \S \ref{cs-sec}.
\begin{proposition}[Gowers Cauchy-Schwarz argument]\label{gcs-prop} Let $f,H,\chi,(\chi_h)_{h \in H}$ be as in Theorem \ref{linear-induct}. Then the sequence
\begin{equation}\label{gowers-cs-arg}
n \mapsto \chi_{h_1}(n) \otimes \chi_{h_2} (n + h_1 - h_4) \otimes \overline{\chi_{h_3}(n)} \otimes \overline{\chi_{h_4}(n + h_1 - h_4)}
\end{equation}
is $(s-2)$-biased for many additive quadruples $(h_1,h_2,h_3,h_4)$ in $H$.
\end{proposition}
With this in hand, we reach the most complicated part of the argument. This is the use of Proposition \ref{gcs-prop} to study the ``frequencies'' of the nilcharacters $\chi_h$ and the way they depend on $h$. Roughly speaking, the aim is to interpret the tensor product \eqref{gowers-cs-arg}
as a nilsequence itself (depending on $h_1, h_2, h_3, h_4$) and use results from \cite{green-tao-nilratner} to analyse its equidistribution and bias properties.
To make proper sense of this one must first find a suitable ``representation'' of the $\chi_h(n)$ in which the frequencies are either independent of $h$, depend in a bracket-linear fashion on $h$, or are appropriately \emph{dissociated} in $h$, in the sense that the frequencies associated to \eqref{gowers-cs-arg} are ``linearly independent'' for most additive quadruples $h_1+h_2=h_3+h_4$. This task is one of the more technical part of the papers and is performed in in \S \ref{reg-sec}; it incorporates the additive combinatorial step (vi) of the outline from \S \ref{strategy-sec}. The precise statement of what we prove is Lemma \ref{sunflower}, the ``sunflower decomposition''.
The representation of the $\chi_h$ (and hence of \eqref{gowers-cs-arg}) involves constructing a suitable polynomial orbit on something resembling a free nilpotent Lie group $\tilde G$; this device also featured in \cite[\S 5]{u4-inverse}. Once this is done, one applies the results from \cite{green-tao-nilratner} to examine the orbit of this polynomial sequence on the corresponding nilmanifold $\tilde G/\tilde \Gamma$. The results of \cite{green-tao-nilratner} assert (roughly speaking) that this orbit is close to the uniform measure on a subnilmanifold $H\tilde\Gamma/\tilde\Gamma$, where $H \leq \tilde G$ is some closed subgroup. In \S \ref{linear-sec}, we then crucially apply a commutator argument of Furstenberg and Weiss that exploits some equidistribution information on projections of $H$ to say something about this group $H$. The upshot of this critical phase of the argument is that the $h$-dependence of the frequencies of $\chi_h$ cannot be dissociated in nature, and must instead be completely bracket-linear; the precise statement here is Theorem \ref{slang-petal}.
At this point in the argument, we have basically shown that the top-order behaviour (in the degree-rank order) of the nilcharacters $\chi_h(n)$ is bracket-linear in $h$. To complete the proof of Theorem \ref{linear-induct} (and hence of Theorem \ref{linear-thm}) it remains to carry out part (vii) of the outline, that is to say to interpret this bracket-linear part of $\chi_h(n)$ as a multidegree $(1,s-1)$ nilcharacter $\chi'(h,n)$. This is the first part of the argument where some sort of ``degree $s$ nil-object'' is actually constructed, and is thus a key milestone in the inductive derivation of $\GI(s)$ from $\GI(s-1)$. As remarked previously, our construction here is a little more conceptual (and abstractly algebraic) than in previous works, which have been somewhat \emph{ad hoc}. The construction is given in \S \ref{multi-sec}. At the end of that section we wrap up the proof of Theorem \ref{linear-thm}: by this point, all the hard work has been done.
With Theorem \ref{linear-thm} in hand, we have completed the first seven steps of the outline. The only remaining substantial step is step (viii), the symmetry argument. Here is a formal statement of it:
\begin{theorem}[Symmetrisation]\label{aderiv} Let $f \in L^\infty[N]$, let $H$ be a dense subset of $[[N]]$, and let $\chi \in \Xi^{(1,s-1)}_\MD(\ultra \Z^2)$ be such that $\Delta_h f$ $<s-2$-correlates with $\chi(h,\cdot)$ for all $h \in H$. Then there exists a nilcharacter $\Theta \in \Xi^{s}(\ultra \Z)$ \textup{(}with the degree filtration\textup{)} and a nilsequence $\Psi \in \Nil^{\subset J}_\MD(\ultra \Z^2)$, with $J \subset \N^2$ given by the downset
\begin{equation}\label{lower}
J := \{ (i,j) \in \N^2: i+j \leq s-1 \} \cup \{ (i,s-i): 2 \leq i \leq s \},
\end{equation}
such that $\chi(h,n)$ is a bounded linear combination of $\Theta(n+h) \otimes \overline{\Theta(n)} \otimes \Psi(h,n)$.
\end{theorem}
The proof is given in \S \ref{symsec}. Informally, this theorem asserts that the multidimensional degree $(1,s-1)$ nilcharacter $\chi(h,n)$ can be expressed as a derivative $\Theta(n+h) \otimes \overline{\Theta(n)}$ of a degree $s$ nilcharacter $\Theta$, modulo ``lower order terms'', which in this context means multidimensional nilsequences $\Psi(h,n)$ that either have total degree $\leq s-1$, or are of degree at most $s-2$ in the $n$ variable.
The remaining task for this section is to show how to complete the proof of Conjecture \ref{gis-conj-nonst} (and Theorem \ref{mainthm}) from this point.
From the discussion at the beginning of this section, we have already arrived at a situation in which the given function $f \in L^\infty[N]$ has the property that $\Delta_h f$ correlates with $\chi_h$ for all $h$ in a dense subset $H$ of $[[N]]$, where $(\chi_h)_{h \in H}$ be a family of nilcharacters in $\Xi^{s-1}([N])$ depending in a limit fashion on $h$.
From Theorem \ref{linear-thm} and Theorem \ref{aderiv} we see that for many $h \in [[N]]$, $\Delta_h f$ $\leq s-2$-correlates with the sequence
$$ n \mapsto \Theta(n+h) \otimes \overline{\Theta(n)} \otimes \Psi(h,n).$$
The next step is to break up $J$ and $\Psi$ into simpler components, and our tool for this purpose shall be Lemma \ref{approx}.
Applying this lemma for $\eps$ sufficiently small, followed by the pigeonhole principle, one can thus find scalar-valued nilsequences $\psi, \psi'$ on $\ultra \Z^2$ (with the multidegree filtration) of multidegree
$$ \subset \{ (i,0) \in \N^2: i \leq s-1 \}$$
and
$$ \subset \{ (i,j) \in \N^2: i \leq s-2; i+j \leq s \}$$
respectively, such that for many $h \in [[N]]$, $\Delta_h f$ $\leq (s-2)$-correlates with
$$ n \mapsto \Theta(n+h) \otimes \overline{\Theta(n)} \psi(h,n) \psi'(h,n).$$
For fixed $h$, the nilsequence $\psi'(h,n)$ has degree $\leq s-2$ and can thus be ignored. Also, $\psi(h,n) = \psi(n)$ is of multidegree $\leq (s-1,0)$ and is thus independent of $h$, with $n \mapsto \psi(n)$ being a degree $\leq s-1$ nilsequence. Thus, for many $h \in [[N]]$, $\Delta_h f$ $\leq s-2$-correlates with
$$ n \mapsto \Theta(n+h) \otimes \overline{\Theta(n)} \psi(n).$$
Applying the pigeonhole principle again, we can thus find scalar nilsequences $\theta, \theta' \in \Nil^{\leq s}(\ultra \Z)$ such that for many $h \in [[N]]$, $\Delta_h f$ $\leq (s-2)$-correlates with
$$ n \mapsto \theta(n+h) \theta'(n)$$
(indeed one takes $\theta, \theta'$ to be coefficients of $\Theta$ and $\overline{\Theta} \psi$ respectively).
Applying the converse to $\GI(s)$ (Proposition \ref{inv-nec-nonst}), we conclude
$$ \| f\overline{\theta}(\cdot+h) \overline{f\theta'}(\cdot) \|_{U^{s-1}[N]} \gg 1$$
for many $h \in H$. Averaging over $h$ (using Corollary \ref{auton-2} to obtain the required uniformity), we conclude that
$$ \E_{h \in [[N]]} \| f\overline{\theta}(\cdot+h) \overline{f\theta'}(\cdot) \|_{U^{s-1}[N]}^{2^{s-1}} \gg 1.$$
Applying the Cauchy-Schwarz-Gowers inequality (see e.g. \cite[Equation (11.6)]{tao-vu}) we conclude that
$$ \| f\overline{\theta} \|_{U^s[N]} \gg 1$$
and hence by the inductive hypothesis (Conjecture \ref{gis-conj-nonst-2} for $s-1$), $f\overline{\theta}$ is $\leq (s-1)$-biased. Since $\theta$ is a degree $\leq s$ nilsequence, we conclude that $f$ is $\leq s$-biased, as required.
This concludes the proof of Conjecture \ref{gis-conj-nonst-2}, Conjecture \ref{gis-conj-nonst}, and hence Theorem \ref{mainthm}, contingent on Theorem \ref{linear-thm} and Theorem \ref{aderiv}.
\section{A variant of Gowers's Cauchy-Schwarz argument}\label{cs-sec}
The aim of this section is prove Proposition \ref{gcs-prop}. Thus, we have standard integers $1 \leq r_* \leq s-1$, a function $f \in L^\infty[N]$, a dense subset $H$ of $[[N]]$, a two-dimensional nilcharacter $\chi \in \Xi^{(1,s-1)}_\MD(\ultra \Z^2)$ of multidegree $(1,s-1)$, and a family $(\chi_h)_{h \in H}$ of nilcharacters of degree-rank $(s-1,r_*)$ depending in a limit fashion on $h$. We are given that $\Delta_h f$ $(s-2)$-correlates with $\chi(h,\cdot) \otimes \chi_h$ for all $h \in H$. Our objective is to show that, for many additive quadruples $(h_1,h_2,h_3,h_4)$ in $H$, the expression
\begin{equation}\label{biasing}
n \mapsto \chi_{h_1}(n) \otimes \chi_{h_2} (n + h_1 - h_4) \otimes \overline{\chi_{h_3}(n)} \otimes \overline{\chi_{h_4}(n + h_1 - h_4)}
\end{equation}
(where we extend the $\chi_h$ by zero outside of $[N]$) is $(s-2)$-biased.
The strategy, following the work of Gowers \cite{gowers-4aps}, is to start with the $\leq s-2$-correlation between $\Delta_h f$ and $\chi(h,\cdot) \chi_h$ and then apply the Cauchy-Schwarz inequality repeatedly to eliminate all terms involving $f$, $\chi(h,\cdot)$, finally arriving at a correlation statement that only involves $\chi_h$ (and lower order terms).
Unfortunately, there is a technical issue that prevents one from doing this directly, namely that the behaviour of $\chi(h,\cdot)$ in $h$ is not quite linear enough to ensure that these terms are completely eliminated by a Cauchy-Schwarz procedure. In order to overcome this issue, one must first prepare $\chi$ into a better form, as follows. We need the following technical notion (which will not be used outside of this section):
\begin{definition}\label{lindef} A \emph{linearised $(1,s-1)$-function} is a limit function $\chi: (h,n) \to \overline{\C}^\omega$ which has a factorisation
\begin{equation}\label{chan}
\chi(h,n) = c(n)^h \psi(n)
\end{equation}
where $\psi \in L^\infty(\Z \to \overline{\C}^\omega)$ and $c \in L^\infty(\Z \to S^1)$ are such that, for every $h,l \in \Z$, the sequence
$$ n \mapsto c(n-l)^h \overline{c(n)}^h$$
is a degree $\leq s-2$ nilsequence.
\end{definition}
\begin{remark} Heuristically, one should think of a linearised $(1,s-1)$-function as (a vector-valued smoothing of) a function of the form
$$ (h,n) \mapsto e( P(n) + h Q(n) )$$
where $P, Q$ are bracket polynomials of degree $s-1$; for instance,
$$ (h,n) \mapsto e( \{ \alpha n \} \beta n + \{ \gamma n \} \delta n h )$$
is morally a linearised $(1,2)$ function. This should be compared with more general multidegree $(1,2)$ nilcharacters, such as
$$ (h,n) \mapsto e( \{ \{ \alpha h \} \beta n \} \gamma n )$$
which are not quite linear in $h$ because the dependence on $h$ is buried inside one or more fractional part operations. Intuitively, the point is that one can use the laws of bracket algebra (such as \eqref{brackalg}) to move the $h$ outside of all the fractional part expressions (modulo lower order terms). While one can indeed develop enough of the machinery of bracket calculus to realise this intuition concretely, we will instead proceed by the more abstract machinery of nilmanifolds in order to avoid having to set up the bracket calculus.
\end{remark}
The key preparation for this is the following.
\begin{proposition}\label{prepare} Let $\chi \in \Xi^{(1,s-1)}_\MD(\ultra \Z^2)$ be a two-dimensional nilcharacter of multidegree $(1,s-1)$, and let $\eps > 0$ be standard. Then one can approximate $\chi$ to within $\eps$ in the uniform norm by a bounded linear combination of linearised $(1,s-1)$-functions.
\end{proposition}
\begin{proof}
From Definition \ref{nilch-def}, we can express
$$ \chi(h,n) = F(g(h,n) \ultra \Gamma)$$
where $G/\Gamma$ is a $\N^2$-filtered nilmanifold of multidegree $\leq (1,s-1)$, $g \in \ultra \poly(\Z^2_{\N^2} \to G_{\N^2})$ (with $\Z^2$ being given the multidegree filtration $\Z^2_{\N^2}$), and $F \in \Lip(\ultra(G/\Gamma) \to \overline{S^\omega})$ has a vertical frequency $\eta: G_{(1,s-1)} \to \R$.
We consider the quotient map $\pi: G/\Gamma \to G/(G_{(1,0)}\Gamma)$ from $G/\Gamma$ onto the nilmanifold $G/(G_{(1,0)}\Gamma)$, which can be viewed as an $\N$-filtered nilmanifold of degree $\leq s-1$ (where we $\N$-filter $G/G_{(1,0)}$ using the subgroups $G_{(0,i)} G_{(1,0)} / G_{(1,0)}$). The fibers of this map are isomorphic to $T := G_{(1,0)} / \Gamma_{(1,0)}$. Observe that $G_{(1,0)}$ is abelian, and so $T$ is a torus; thus $G/\Gamma$ is a torus bundle over $G/(G_{(1,0)}\Gamma)$ with structure group $T$. The idea is to perform Fourier analysis on this large torus $T$, as opposed to the smaller torus $G_{(1,s-1)}/\Gamma_{(1,s-1)}$, to improve the behaviour of the nilcharacter $\chi$.
We pick a metric on the base nilmanifold $G/(G_{(1,0)}\Gamma)$ and a small standard radius $\delta>0$, and form a smooth partition of unity $1 = \sum_{k=1}^K \varphi_k$ on $G/(G_{(1,0)}\Gamma)$, where each $\varphi_k \in \Lip(G/(G_{(1,0)}\Gamma) \to \C)$ is supported on an open ball $B_k$ of radius $r$. This induces a partition $\chi = \sum_{k=1}^K \tilde \chi_k$, where
$$ \tilde \chi_k(h,n) = F(g(h,n) \ultra \Gamma) \varphi_k(\pi(g(h,n) \ultra \Gamma)).$$
Now fix one of the $k$. Then we have
$$ \tilde \chi_k(h,n) = \tilde F_k(g(h,n) \ultra \Gamma)$$
where $\tilde F_k$ is compactly supported in the cylinder $\pi^{-1}(B_k)$.
If $r$ is small enough, we have a smooth section $\iota: B_k \to G$ that partially inverts the projection from $G$ to $G/(G_{(1,0)}\Gamma)$, and so we can parameterise any element $x$ of $\pi^{-1}(B_k)$ uniquely as $\iota(x_0) t \Gamma$ for some $x_0 \in B_k$ and $t \in T$ (noting that $t\Gamma$ is well-defined as an element of $G/\Gamma$). Similarly, we can parameterise any element of $\ultra \pi^{-1}(B_k)$ uniquely as $\iota(x_0) t \Gamma$ for $x_0 \in \ultra B_k$ and $t\in \ultra T$.
We can now view the Lipschitz function $F_k \in \Lip(\ultra(G/\Gamma))$ as a compactly supported Lipschitz function in $\Lip(\ultra(B_k \times T))$. Applying a Fourier (or Stone-Weierstrass) decomposition in the $T$ directions (cf. Lemma \ref{limone}), we thus see that for any standard $\eps > 0$ we can approximate $\tilde F_k$ uniformly to error $\eps/K$ by a sum $\sum_{k'=1}^{K'} \tilde F_{k,k'}$, where $K'$ is standard and each $F_{k,k'} \in \Lip(\ultra(B_k \times T))$ is compactly supported and has a character $\xi_{k'}: T \to \T$ such that
\begin{equation}\label{fan}
\tilde F_{k,k'}(\iota(x_0) t\Gamma) = e(\xi_{k'}(t)) \tilde F_{k,k'}(\iota(x_0) \Gamma)
\end{equation}
for all $x_0 \in \ultra(2B_k)$ and $t \in \ultra T$. It thus suffices to show that for each $k, k'$, the sequence
$$ \tilde \chi_{k,k'}: (h,n) \mapsto \tilde F_{k,k'}( g(h,n) \ultra \Gamma )$$
is a linearised $(1,s-1)$-function.
Fix $k,k'$. Performing a Taylor expansion (Lemma \ref{taylo}) of the polynomial sequence $g \in \ultra\poly(\Z^2_{\N^2} \to G_{\N^2})$, we may write
$$ g(h,n) = g_0(n) g_1(n)^h$$
where $g_0 \in \ultra \poly(\Z_\N \to G_\N)$ is a one-dimensional polynomial map (giving $G$ the $\N$-filtration $G_\N := (G_{(i,0)})_{i \in \N}$), and $g_1 \in \ultra \poly(\Z \to (G_{(1,0)})_\N)$ is another one-dimensional polynomial map (giving the abelian group $G_{(1,0)}$ the $\N$-filtration $(G_{(1,0)})_\N := (G_{(1,i)})_{i \in \N}$). In particular, we see that $\tilde \chi_{k,k'}(h,n)$ is only non-vanishing when $\pi( g_0(n) \ultra \Gamma ) \in B$. Furthermore, in that case we see from \eqref{fan} that
\begin{equation}\label{chimn}
\tilde \chi_{k,k'}(h,n) =
e( h \xi( g_1(n) \mod \Gamma_{(1,0)} ) ) \tilde F_{k,k'}(g_0(n) \ultra \Gamma),
\end{equation}
which gives the required factorisation \eqref{chan} with $c(n) := e( \xi( g_1(n) \mod \Gamma_{(1,0)} ) )$ and $\psi(n) := \tilde F_{k,k'}(g_0(n) \ultra \Gamma)$.
The only remaining task is to establish that for any given $h, l$, the sequence $n \mapsto c(n-l)^h \overline{c(n)}^h$ is a degree $\leq s-2$ nilsequence. We expand this sequence as
$$n \mapsto e( h ( \xi( g_1(n-l) \mod \Gamma_{(1,0)} ) - \xi( g_1(n) \mod \Gamma_{(1,0)} ) ) )$$
But from the abelian nature of $G_{(1,0)}$, the map $n \mapsto \xi(g_1(n) \mod \Gamma_{(1,0)})$ is a polynomial map from $\ultra \Z$ to $\ultra \T$ of degree at most $s-1$, and the claim follows.
\end{proof}
We now return to the proof of Theorem \ref{gcs-prop}.
With this multiplicative structure, we can now begin the Cauchy-Schwarz argument. By hypothesis, for each $h \in H$ we can find a scalar nilsequence $\psi_h$ of degree $\leq s-2$ such that
$$ |\E_{n \in [N]} \Delta_h f(n) \overline{\chi(h,n)} \otimes \overline{\chi_h(n)} \overline{\psi_h(n)}| \gg 1.$$
By Corollary \ref{mes-select}, we may ensure that $\psi_h$ varies in a limit fashion on $h$. Applying Corollary \ref{auton-2}, this lower bound is uniform in $h$.
Applying Proposition \ref{prepare} (with a sufficiently small $\eps$) and using the pigeonhole principle, we may then find a linearised $(1,s-1)$-function $(h,n) \mapsto c(n)^h \psi(n)$ such that
$$ |\E_{n \in [N]} \Delta_h f(n) c(n)^{-h} \overline{\psi(n)} \otimes \overline{\chi_h(n)} \overline{\psi_h(n)}| \gg 1.$$
By Corollary \ref{auton-2} again, the lower bound is still uniform in $h$. We may then average in $h$ (extending $\psi_h, \chi_h$ by zero for $h$ outside of $H$) and conclude that
$$ \E_{h \in [[N]]} |\E_{n \in [N]} \Delta_h f(n) c(n)^{-h} \overline{\psi(n)} \otimes \overline{\chi_h(n)} \overline{\psi_h(n)}| \gg 1,$$
thus there exists a scalar function $b \in L^\infty[[N]]$ such that
$$ |\E_{h \in [[N]]} \E_{n \in [N]} b(h) f(n+h) \overline{f}(n) c(n)^{-h} \overline{\psi(n)} \otimes \overline{\chi_h(n)} \overline{\psi_h(n)}| \gg 1.$$
By absorbing $b(h)$ into the $\psi_h$ factor, we may now drop the $b(h)$ factor. We write $n+h = m$ and obtain
$$|\E_{m \in [N]} f(m) \E_{h \in [[N]]} c(m-h)^{-h} f'(m-h) \otimes \overline{\chi_h(m-h)} \overline{\psi_h(m-h)}| \gg 1$$
where $f' := \overline{f} \overline{\psi}$ (recall that $f$ is extended by zero outside of $[N]$), which by Cauchy-Schwarz implies that
\begin{align*}
|\E_{m \in [N]} \E_{h,h' \in [[N]]} c(m-h)^{-h} c(m-h')^{h'} &f'(m-h) \otimes \overline{f(m-h')} \\
\otimes \overline{\chi_h(m-h)} \otimes \chi_{h'}(m-h') &\overline{\psi_h(m-h)} \psi_{h'}(m-h')| \gg 1.
\end{align*}
Making the change of variables $h' = h+l$, $n = m-h$, we obtain
\begin{align*}
|\E_{h,l \in [[2N]]; n \in [N]} c(n)^{-h} c(n-l)^{h+l} &f'(n) \otimes \overline{f'}(n-l) \\
\otimes \overline{\chi_h(n)}\otimes \chi_{h+l}(n-l) &\overline{\psi_h(n)} \psi_{h+l}(n-l)| \gg 1.
\end{align*}
We then simplify this as
\begin{equation}\label{hank}
|\E_{h,l \in [[2N]];n \in [N]} c_2(l,n) \otimes \overline{\chi_h(n)} \otimes \chi_{h+l}(n-l) \psi_{h,l}(n)| \gg 1
\end{equation}
where
\begin{align*}
c_2(l,n) &:= c(n-l)^l f'(n) \otimes \overline{f'(n-l)} \\
\psi_{h,l}(n) &= c(n-l)^h c(n)^{-h} \overline{\psi_h(n)} \psi_{h+l}(n-l)
\end{align*}
Clearly $c_2$ is bounded. As for $\psi_{h,l}$, we see from Definition \ref{lindef} and Corollary \ref{alg} that $\psi_{h,l}$ is a nilsequence of degree $\leq s-2$ for each $h,l$.
Returning to \eqref{hank}, we use the pigeonhole principle to conclude that for many $k\in [[2N]]$, we have
$$ |\E_{h \in [[2N]]; n \in [N]} c_2(k,n) \otimes \overline{\chi_h(n)} \otimes \chi_{h+k}(n-k) \psi_{h,k}(n)| \gg 1.$$
Let $k$ be such that the above estimate holds. Applying Cauchy-Schwarz in the $n$ variable to eliminate the $c_2(k,n)$ term, we have
$$
|\E_{h,h' \in [[2N]]; n \in [N]} \overline{\chi_h(n)} \otimes \chi_{h+k}(n-k) \otimes
\overline{\chi_{h'}(n)} \otimes \chi_{h'+k}(n-k) \psi_{h,k}(n)| \gg 1$$
and thus for many $k,h,h' \in [[2N]]$, we have
$$
|\E_{n \in [N]} \overline{\chi_h(n)} \otimes \chi_{h+k}(n-k) \otimes
\overline{\chi_{h'}(n)} \otimes \chi_{h'+k}(n-k) \psi_{h,k}(n)| \gg 1,$$
which implies that
$$ n \mapsto \overline{\chi_h(n)} \otimes \chi_{h+k}(n-k) \otimes
\overline{\chi_{h'}(n)} \otimes \chi_{h'+k}(n-k)$$
is $(s-2)$-biased on $[N]$. Note that this forces $h,h+k, h',h'+k$ to be an additive quadruple in $H$, as otherwise the expression vanishes. Applying a change of variables, we obtain Proposition \ref{gcs-prop}.
For future reference we observe that a simpler version of the same argument (in which the $\chi$ and $\psi_h$ factors are not present) gives
\begin{proposition}[Cauchy-Schwarz]\label{cs}
Let $f \in L^\infty[N]$, let $H$ be a dense subset of $[[N]]$, and suppose that one has a family of functions $\chi_h \in L^\infty(\ultra \Z)$ depending in a limit fashion on $h$, such that $\Delta_h f$ correlates with $\chi_h$ on $[N]$ for all $h \in H$. Then for many \textup{(}i.e. for $\gg N^3$\textup{)} additive quadruples $(h_1,h_2,h_3,h_4)$ in $H$, the sequence
\begin{equation}\label{slam}
n \mapsto \chi_{h_1}(n) \otimes \chi_{h_2} (n + h_1 - h_4) \otimes \overline{\chi_{h_3}(n)} \otimes \overline{\chi_{h_4}(n + h_1 - h_4)}
\end{equation}
is biased.
\end{proposition}
This proposition in fact has quite a simple proof; see \cite{gtz-announce}. Note how we can conclude \eqref{slam} to be biased and not merely $(s-2)$-biased. As such, Proposition \ref{cs} saves some ``lower order'' information that was not present in Proposition \ref{gcs-prop}; this lower order information will be crucial later in the argument, when we establish the symmetry property in Theorem \ref{aderiv}.
\section{Frequencies and representations}\label{freq-sec}
We will use Proposition \ref{gcs-prop} to analyse the ``frequency'' of the nilcharacters $(\chi_h)_{h \in H}$ appearing in Theorem \ref{linear-induct}. To motivate the discussion, let us first suppose that we are in the (significantly simpler) $s=2$ case, rather than the actual case $s \geq 3$ of interest. When $s=2$, we can represent $\chi_h$ as a linear phase $\chi_h(n) = e(\xi_h n + \theta_h)$ for some $\xi_h, \theta_h \in \ultra\T$; one can then interpret $\xi_h$ as the \emph{frequency} of $h$.
In order to describe how this frequency $\xi_h$ behaves in $h$, it will be convenient to \emph{represent} $\xi_h$ as a linear combination
\begin{equation}\label{xih}
\xi_h = a_{1,h} \xi_{1,h} + \ldots + a_{D,h} \xi_{D,h}
\end{equation}
of other frequencies $\xi_{1,h},\ldots,\xi_{D,h} \in \ultra\T$, where the $a_{i,h} \in \Z$ are (standard) integer coefficients, and the $(\xi_{i,h})_{h \in H}$ are families of frequencies which have better properties with regards to their dependence on $h$; for instance, they might be ``core frequencies'' $\xi_{i,h} = \xi_{*,i}$ that are independent of $h$, or they might be ``bracket-linear petal'' frequencies that depend in a bracket-linear fashion on $h$, or they might be ``regular petal'' frequencies which behave in a suitably ``dissociated'' manner in $h$.
We can schematically depict the relationship \eqref{xih} as
$$
[\chi_h] \approx \eta_h(\F_h)
$$
where $[\chi_h]$ is some sort of ``symbol'' of $\chi_h$ (which, in the linear case $s=2$, is just $\xi_h \mod 1$), $\F_h \in \ultra \T^D$ is the \emph{frequency vector} $\F_h = (\xi_{1,h},\ldots,\xi_{D,h})$, and $\eta_h: \ultra \T^D \to \ultra \T$ is the \emph{vertical frequency}
\begin{equation}\label{etaxd}
\eta_h(x_1,\ldots,x_D) := a_{1,h} x_1 + \ldots + a_{D,h} x_D.
\end{equation}
We will need to find analogues of the above type of representation in higher degree $s \geq 3$. Heuristically, we will wish to represent the symbol $[\chi]_{\Xi^{(s-1,r_*)}_\DR([N])}$ of a nilcharacter $\chi$ on $[N]$ of degree-rank $(s-1,r_*)$ (which will ultimately depend on a parameter $h$, though we will not need this parameter in the current discussion) heuristically as
\begin{equation}\label{chih-abstract}
[\chi]_{\Xi^{(s-1,r_*)}_\DR([N])} \approx \eta(\F)
\end{equation}
where $\F = (\xi_{i,j})_{1 \leq i \leq s-1; 1 \leq j \leq D_i}$ is a \emph{horizontal frequency vector} of frequencies $\xi_{i,j} \in \ultra \T$ associated to a \emph{dimension vector} $\vec D = (D_1,\ldots,D_{s-1})$, and $\eta$ is a \emph{vertical frequency} that generalises \eqref{etaxd}, but whose precise form we are not yet ready to describe precisely. We then say that the triple $(\vec D, \eta, \F)$ forms a \emph{total frequency representation} of $\chi$.
In the previous paper \cite{u4-inverse} that treated the $s=3$ case, such a representation was implicitly used via the description of degree-rank $(2,2)$ nilcharacters $\chi_h$ as essentially being bracket quadratic phases $e(\sum_{j=1}^J \{ \alpha_{h,j} n \} \beta_{h,j} n)$ modulo lower order terms (and ignoring the issue of vector-valued smoothing for now). In our current language, this would correspond to a dimension vector $\vec D = (2J,0)$ and a horizontal frequency vector of the form $(\alpha_{h,1},\ldots,\alpha_{h,J},\beta_{h,1},\ldots,\beta_{h,J})$, and a certain vertical frequency $\eta$ depending only on $J$ that we are not yet ready to describe explicitly here. Bracket-calculus identities such as \eqref{brackalg} could then be used to manipulate such a universal frequency representation into a suitably ``regularised'' form.
In principle, one could also use bracket calculus to extract the symbol of $\chi_h$ in terms of frequencies such as $\alpha_{h,j}$ and $\beta_{h,j}$ for higher values of $s$. However, as we are avoiding the use of bracket calculus machinery here, we will proceed instead using the language of nilmanifolds, and in particular by lifting the nilmanifold $G_h/\Gamma_h$ up to a \emph{universal nilmanifold} in order to obtain a suitable space (independent of $h$) in which to detect relationships between frequencies such as $\alpha_{h,j}, \beta_{h,j}$. In some sense, this universal nilmanifold will play the role that the unit circle $\T$ plays in Fourier analysis.
We first define the notion of universal nilmanifold that we need.
\begin{definition}[Universal nilmanifold]\label{universal-nil} A \emph{dimension vector} is a tuple \[ \vec D = (D_1,\ldots,D_{s-1}) \in \N^{s-1} \] of standard natural numbers. Given a dimension vector, we define the \emph{universal nilpotent group} $G^{\vec D} = G^{\vec D, \leq (s-1,r_*)}$ of degree-rank $(s-1,r_*)$ to be the Lie group generated by formal generators $e_{i,j}$ for $1 \leq i \leq s-1$ and $1 \leq j \leq D_i$, subject to the following constraints:
\begin{itemize}
\item Any $(m-1)$-fold iterated commutator of $e_{i_1,j_1},\ldots,e_{i_m,j_m}$ with $i_1+\ldots+i_m \geq s$ is trivial.
\item Any $(m-1)$-fold iterated commutator of $e_{i_1,j_1},\ldots,e_{i_m,j_m}$ with $i_1+\ldots+i_m = s-1$ and $m \geq r+1$ is trivial.
\end{itemize}
We give this group a degree-rank filtration $(G^{\vec D}_{(d,r)})_{(d,r) \in \DR}$ by defining $G^{\vec D}_{(d,r)}$ to be the Lie group generated by $(m-1)$-fold iterated commutators of $e_{i_1,j_1},\ldots,e_{i_m,j_m}$ with $1 \leq i_l \leq s-1$ and $1 \leq j_l \leq D_{i_l}$ for all $1 \leq l \leq n$ for which either $i_1+\ldots+i_m > d$, or $i_1+\ldots+i_m=d$ and $m \geq r$. It is not hard to verify that this is indeed a filtration of degree-rank $\leq (s-1,r_*)$. We then let $\Gamma^{\vec D}$ be the discrete group generated by the $e_{i,j}$ with $1 \leq i \leq s-1$ and $1 \leq j \leq D_i$, and refer to $G^{\vec D}/\Gamma^{\vec D}$ as the \emph{universal nilmanifold} with dimension vector $\vec D$.
A \emph{universal vertical frequency} at dimension vector $\vec D$ is a continuous homomorphism $\eta: G^{\vec D}_{(s-1,r_*)} \to \R$ which sends $\Gamma^{\vec D}_{(s-1,r_*)}$ to the integers (i.e. a filtered homomorphism from $G^{\vec D}_{(s-1,r_*)} / \Gamma^{\vec D}_{(s-1,r_*)}$ to $\T$).
\end{definition}
\emph{Remark.} One can give an explicit basis for this nilmanifold in terms of certain iterated commutators of the $e_{i,j}$, following \cite{leibman,mks}. This can then be used to relate nilcharacters to bracket polynomials, as in \cite{leibman}, and it is then possible to develop enough of a ``bracket calculus'' to substitute for some of the nilpotent algebra performed in this paper. However, we will not proceed by such a route here (as it would make the paper even longer than it currently is), and in fact will not need an explicit basis for universal nilmanifolds at all.
\begin{example} The unit circle with the degree $\leq d$ filtration (see Example \ref{polyphase}) is isomorphic to the universal nilmanifold $G^{(0,\ldots,0,1),\leq (d,1)}$, thus for instance the unit circle with the lower central series filtration is isomorphic to $G^{(1),\leq (1,1)}$. A universal vertical frequency for any of these nilmanifolds is essentially just a map of the form $\eta: x \mapsto nx$ for some integer $n$.
\end{example}
\begin{example} The Heisenberg group \eqref{heisen} (with the lower central series filtration) is the universal nilpotent group $G^{(2,0)} = G^{(2,0), \leq (2,2)}$ of degree-rank $(2,2)$ (after identifying $e_1,e_2$ with $e_{1,1}$ and $e_{1,2}$ respectively), and the Heisenberg nilmanifold $G/\Gamma$ is the corresponding universal nilmanifold $G^{(2,0)}/\Gamma^{(2,0)}$. If we reduce the degree-rank from $(2,2)$ to $(2,1)$, then the commutator $[e_1,e_2]$ now trivialises, and $G^{(2,0), \leq (2,1)}$ collapses to the abelian Lie group $\R^2 \equiv G^{2, \leq (1,1)}$, with universal nilmanifold $\T^2$.
If, instead of the lower central series filtration, one gives the Heisenberg group \eqref{heisen} the filtration used in Example \ref{abn} to model the sequence \eqref{abn-eq}, then this group is isomorphic to the universal nilpotent group $G^{(1,1), \leq (3,2)}$, with the two generators $e_1, e_2$ of the Heisenberg group now being interpreted as $e_{1,1}$ and $e_{2,1}$ respectively.
\end{example}
\begin{example}
Consider the universal nilpotent group $G^{(D_1,D_2,D_3),\leq (3,3)}$. This group is generated by ``degree $1$'' generators $e_{1,1},\ldots,e_{1,D_1}$, ``degree $2$'' generators $e_{2,1},\ldots,e_{2,D_2}$, and ``degree $3$'' generators $e_{3,1},\ldots,e_{3,D_3}$, with any iterated commutator of total degree exceeding three vanishing (thus for instance the degree $3$ generators are central, and the degree $2$ generators commute with each other). If one drops the degree-rank from $(3,3)$ to $(3,2)$, then all triple commutators of degree $1$-generators, such as $[[e_{1,i}, e_{1,j}],e_{1,k}]$ now vanish, reducing the dimension of the nilpotent group. Dropping the degree-rank further to $(3,1)$ also eliminates the commutators of degree $1$ and degree $2$ generators (thus making the degree $2$ generators central). Finally, dropping the degree-rank to $(3,0)$ eliminates the degree $3$ generators completely, and indeed $G^{(D_1,D_2,D_3), \leq (3,0)}$ is isomorphic to $G^{(D_1,D_2), \leq (2,2)}$.
\end{example}
\begin{example}
The free $s$-step nilpotent group on $D$ generators, in our notation, becomes $G^{(D,0,\ldots,0), \leq (s,s)}$. We may thus view the universal nilpotent groups $G^{\vec D, \leq (d,r)}$ as generalisations of the free nilpotent groups, in which some of the generators are allowed to be weighted to have degrees greater than $1$, and there is an additional rank parameter to cut down some of the top-order behaviour.
\end{example}
It will be an easy matter to lift a nilcharacter $\chi$ from a general degree-rank $\leq (s-1,r_*)$ nilmanifold $G/\Gamma$ to a universal nilmanifold $G^{\vec D}/\Gamma^{\vec D}$ for some sufficiently large dimension vector $\vec D$ (see Lemma \ref{existence} below). Once one does so, we will need to extract the various ``top order frequencies'' present in that nilcharacter. For instance, if $s=4$ and $\chi$ is (some vector-valued smoothing of) the degree $3$ phase
$$ n \mapsto e( \{ \alpha n \} \beta n^2 + \gamma n^3 + \delta n^2 + \{ \epsilon n \} \mu n + \nu n + \theta )$$
then we will need to extract out the ``degree $3$'' frequency $\gamma$, the ``degree $2$'' frequency $\beta$, and the ``degree $1$'' frequency $\alpha$. (The remaining parameters $\delta,\epsilon,\mu,\nu,\theta$ only contribute to terms of degree strictly less than $3$, and will not need to be extracted.)
As it turns out, the degree $i$ frequencies will most naturally live in the \emph{$i^\th$ horizontal torus} of the relevant universal nilmanifold; we now pause to define these torii precisely. (These torii also implicitly appeared in \cite[Appendix A]{green-tao-arithmetic-regularity}.)
\begin{definition}[Horizontal Taylor coefficients]\label{horton} Let $G = (G, (G_{(d,r)})_{(d,r) \in \DR})$ be a degree-rank-filtered nilpotent group. For every $i \geq 0$, define the \emph{$i^{\th}$ horizontal space} $\Horiz_i(G)$ to be the abelian group
$$ \Horiz_i(G) := G_{(i,1)} / G_{(i,2)},$$
with the convention that $G_{(d,r)} := G_{(d+1,0)}$ if $r>d$ (so in particular, $G_{(1,2)} = G_{(2,0)}$).
For any polynomial map $g \in \poly(\Z_\N \to G_\N)$, we define the \emph{$i^{th}$ horizontal Taylor coefficient} $\Taylor_i(g) \in \Horiz_i(G)$ to be the quantity
$$ \Taylor_i(g) := \partial_{1} \ldots \partial_{1} g(n) \mod G_{(i,2)}$$
for any $n \in \Z$. Note that this map is well-defined since $\partial_{1} \ldots \partial_{1} g$ takes values in $G_{(i,1)}$ and has first derivatives in $G_{(i+1,1)}$ and hence in $G_{(i,2)}$.
If $\Gamma$ is a subgroup of $G$, we define
$$ \Horiz_i(G/\Gamma) := \Horiz_i(G) / \Horiz_i(\Gamma)$$
and for a polynomial orbit $\orbit \in \poly(\Z_\N \to (G/\Gamma)_\N) := \poly(\Z_\N \to G_\N) / \poly(\Z_\N \to \Gamma_\N)$, we define the \emph{$i^{\th}$ horizontal Taylor coefficient} $\Taylor_i(\orbit) \in \Horiz_i(G/\Gamma)$ to be the quantity defined by
$$ \Taylor_i( g \Gamma ) := \Taylor_i(g) \mod \Horiz_i(\Gamma)$$
for any $g \in \poly(\Z_\N \to G_\N)$; it is easy to see that this quantity is well-defined.
These concepts extend to the ultralimit setting in the obvious manner; thus for instance, if $\orbit \in \ultra \poly(H_\N \to (G/\Gamma)_\N)$, then $\Taylor_i(\orbit)$ is an element to $\ultra \Horiz_i(G/\Gamma)$.
\end{definition}
If $G/\Gamma$ is a degree-rank filtered nilmanifold, it is easy to see that the horizontal spaces $\Horiz_i(G)$ are abelian Lie groups, and that $\Horiz_i(\Gamma)$ is a sublattice of $\Horiz_i(G)$, so $\Horiz_i(G/\Gamma)$ is a torus, which we call the \emph{$i^{\th}$ horizontal torus} of $G/\Gamma$.\vspace{11pt}
\emph{Remark.} The above definition can be generalised by replacing the domain $\Z$ with an arbitrary additive group $H = (H,+)$. In that case, the Taylor coefficient $\Taylor_i(g)$ is not a single element of $\Horiz_i(G)$, but is instead a map $\Taylor_i(g): H^i \to \Horiz_i(G)$
defined by the formula
$$ \Taylor_i(g)(h_1,\ldots,h_k) := \partial_{h_1} \ldots \partial_{h_k} g(n) \mod G_{(i,2)}$$
for $h_1,\ldots,h_k \in H$. Using Corollary \ref{collox} we easily see that this map is symmetric and multilinear; thus for instance when $H=\Z$ we have
$$ \Taylor_i(g)(h_1,\ldots,h_k) = h_1 \ldots h_k \Taylor_i(g).$$
However, we will not need this generalisation here.
A further application of Corollary \ref{collox} shows that the map $g \mapsto \Taylor_i(g)$ is a homomorphism. As a corollary, we see that any translate $g(\cdot+h) = (\partial_h g) g$ of $g$ will have the same Taylor coefficients as $g$: $\Taylor_i(g(\cdot+h)) = \Taylor_i(g)$.
\begin{example} Consider the unit circle $G/\Gamma = \T$ with the degree $\leq d$ filtration (see Example \ref{polyphase}). Then the $d^\th$ horizontal torus is $\T$, and all other horizontal tori are trivial. If $\alpha_0,\ldots,\alpha_d \in \ultra \R$, then the map $\orbit: n \mapsto \alpha_0 + \ldots + \alpha_d n^d \mod 1$ is a polynomial orbit in $\ultra \poly(\Z_\N \to \T_\N)$, and the $d^{th}$ horizontal Taylor coefficient is the quantity $d! \alpha_d \mod 1$ from $\ultra \Z^d$ to $\ultra\T$. (All other horizontal Taylor coefficients are of course trivial.) Thus we see that the horizontal coefficient captures most of the top order coefficient $\alpha_d$, but totally ignores all lower order terms.
\end{example}
\begin{example}
Let $G=G^{(2,1)}=G^{(2,1),\leq (2,2)}$ be the universal nilpotent group of degree-rank $(2,2)$. Thus $G$ is generated
by $e_{1,1},e_{1,2},e_{2,1}$, with relations
\[
[[e_{1,1},e_{1,2}], e_{1,i}]=[e_{1,i},e_{2,1}]=1 \quad \text{ for $i=1,2$}.
\]
and with the degree-rank filtration
\begin{align*}
G_{(0,0)}=G_{(1,0)}=G_{(1,1)}&=G \\
G_{(2,0)}=G_{(2,1)}&= \langle [e_{1,1},e_{1,2}], e_{2,1}\rangle_\R \\
G_{(2,2)}&=\langle [e_{1,1},e_{1,2}] \rangle_\R
\end{align*}
and the lattice
$$ \Gamma = \Gamma^{(2,2)} = \Gamma^{(2,2), \leq (2,1)} := \langle e_{1,1},e_{1,2},e_{2,1} \rangle.$$
Let $\alpha,\beta,\gamma \in \ultra \R$, and consider the orbit $\orbit \in \ultra\poly(\Z_\N \to (G/\Gamma)_\N)$ defined by the formula
\[
\orbit(n):=e^{n \alpha}_{1,1} e_{1,2}^{n \beta} e_{2,1}^{n^2 \gamma};
\]
this is polynomial by Example \ref{lazard-ex}.
Then
\[
\Taylor_1(g) = \partial_{1} g(n) \mod \ultra G_{(2,0)} = e^{\alpha}_{1,1}e_{1,2}^{\beta} \mod \ultra G_{(2,0)},
\]
and
\[
\Taylor_2(g) = e_{2,1}^{2\gamma} \mod \ultra G_{(2,2)}.
\]
Then $\Taylor_0(g(n)\ultra \Gamma) = g(n)\ultra \Gamma$,
\[
\Taylor_1(g\ultra \Gamma) = e^{\alpha}_{1,1}e_{1,2}^{\beta} \mod G_{(2,0)}\ultra \Gamma
\]
and
\[
\Taylor_2(g\ultra \Gamma) = e_{2,1}^{2\gamma} \mod \ultra G_{(2,2)}\Gamma_{(2,0)}.
\]
\end{example}
\begin{example}\label{heist} Let $G/\Gamma$ be the Heisenberg nilmanifold \eqref{heisen} with the lower central series filtration. Thus $G/\Gamma$ is a degree $\leq 2$ nilmanifold, which can then be viewed as a degree-rank $\leq (2,2)$ nilmanifold by Example \ref{inclusions}. The first horizontal torus $\Horiz_1(G/\Gamma)$ is isomorphic to the $2$-torus $\T^2$, with generators given by $e_1, e_2 \mod G_2 \Gamma$. The second horizontal torus $\Horiz_2(G/\Gamma)$ is trivial, since $G_{(2,1)} = [G,G]$ is equal to $G_{(2,0)} = G_2$. If $\orbit \in \ultra \poly(\Z_\N \to (G/\Gamma)_\N)$ is the polynomial orbit $\orbit: n \mapsto e_2^{\beta n} e_1^{\alpha n} \ultra \Gamma$, then the first Taylor coefficient is the quantity $(\alpha, \beta)$. Note also that if one modified the polynomial orbit by a further factor of $[e_1,e_2]^{\gamma n^2 + \delta n + \epsilon}$, this would not impact the Taylor coefficients at all. Thus we see that the Taylor coefficients only capture the frequencies associated to raw generators such as $e_1$ and $e_2$, and not to commutators such as $[e_1,e_2]$.
\end{example}
\begin{example} Now consider the Heisenberg group \eqref{heisen} with the filtration used in Example \ref{abn} to model the sequence \eqref{abn-eq}. This is now a degree $\leq 3$ nilmanifold, whose first horizontal torus $\Horiz_1(G/\Gamma)$ is isomorphic to the one-torus $\T$ with generator $e_2 \mod G_{(2,0)} \Gamma$, whose second horizontal torus $\Horiz_2(G/\Gamma)$ is isomorphic to the one-torus $\T$ with generator $e_1 \mod G_{(2,2)} \Gamma_{(2,1)}$, and whose third horizontal torus $\Horiz_3(G/\Gamma)$ is trivial. If $\orbit \in \ultra \poly(\Z_\N \to (G/\Gamma)_\N)$ is the polynomial orbit $\orbit: n \mapsto e_2^{\beta n} e_1^{\alpha n^2} \ultra \Gamma$, then the first Taylor coefficient is the linear limit map $n \mapsto \beta n \mod 1$, and the second Taylor coefficient is the quantity $2! \alpha \mod 1$.
\end{example}
We now have enough notation to be able to formally assign frequencies to a nilcharacter, by means of a package of data which we shall call a \emph{representation}.
\begin{definition}[Representation]\label{representation-def} Let $\chi \in L^\infty[N]$ be a nilcharacter of degree-rank $\leq (s-1,r_*)$. A \emph{representation} of $\chi$ is a collection of the following data:
\begin{enumerate}
\item A filtered nilmanifold $G/\Gamma$ of degree-rank $\leq (s-1,r_*)$;
\item A filtered nilmanifold $G_0/\Gamma_0$ of degree-rank $\leq (s-1,r_*-1)$;
\item A function $F \in \Lip(\ultra(G/\Gamma \times G_0/\Gamma_0) \to \overline{S^\omega})$;
\item Polynomial orbits $\orbit \in \ultra \poly(\Z_\N \to (G/\Gamma)_\N)$ and $\orbit_0 \in \ultra \poly(\Z_\N \to (G_0/\Gamma_0)_\N)$;
\item A dimension vector $\vec D = (D_1,\ldots,D_{s-1}) \in \N^{s-1}$;
\item A universal vertical frequency $\eta: G^{\vec D}_{(s-1,r_*)} \to \R$ at dimension $\vec D$ on the universal nilmanifold $G^{\vec D}/\Gamma^{\vec D}$ of degree-rank $(s-1,r_*)$;
\item A filtered homomorphism $\phi: G^{\vec D}/\Gamma^{\vec D} \to G/\Gamma$ (see Definition \ref{quot});
\item A \emph{horizontal frequency vector} $\F = (\xi_{i,j})_{1 \leq i \leq s-1; 1 \leq j \leq D_i}$ of frequencies $\xi_{i,j} \in \ultra\T$.
\end{enumerate}
which obeys the following properties:
\begin{enumerate}
\item For all $n \in [N]$, one has
\begin{equation}\label{chin}
\chi(n) = F( \orbit(n), \orbit_0(n)).
\end{equation}
\item For every $t \in G^{\vec D}_{(s-1,r_*)}$, all $x \in G/\Gamma$, and $x_0 \in G_0/\Gamma_0$, one has
\begin{equation}\label{vert}
F( \phi(t) x, x_0 ) = e( \eta(t) ) F(x,x_0).
\end{equation}
\item For every $1 \leq i \leq s-1$, one has
\begin{equation}\label{taylor}
\Taylor_i(\orbit) = \pi_{\Horiz_i(G/\Gamma)}\left(\phi( \prod_{j=1}^{D_i} e_{i,j}^{\xi_{i,j}} )\right),
\end{equation}
where $\pi_{\Horiz_i(G/\Gamma)}: G_{i} \to \Horiz_i(G/\Gamma)$ is the projection map; observe that the right-hand side is well-defined even though $\xi_{i,j}$ is only defined modulo $1$.
\end{enumerate}
We call the triplet $(\vec D, \F, \eta)$ a \emph{total frequency representation} of the nilcharacter $\chi$.
\end{definition}
This is a rather complicated definition, and we now illustrate it with a number of examples. We begin with the $s=2$, $r_*=1$ case, taking $\chi$ to be the degree-rank $(1,1)$ nilcharacter
$$ \chi(n) := e( \xi n + \theta )$$
for some $\xi, \theta \in \ultra \R$. Let $D_1 \geq 1$ be an integer, let $\F = (\xi_{1,1},\ldots,\xi_{1,D_1}) \in \ultra \T^{D_1}$ be a collection of frequencies, and let $\eta: \R^{D_1} \to \R$ be the universal vertical frequency $\eta(x_1,\ldots,x_{D_1}) := a_1 x_1 + \ldots + a_{D_1} x_{D_1}$ for some integers $a_1,\ldots,a_{D_1} \in \Z$. Then $((D_1),\F,\eta)$ will be a total frequency representation of $\chi$ if $\xi = a_1 \xi_{1,1} + \ldots + a_{D_1} \xi_{1,D_1}$. Indeed, in that case, one can take
$G/\Gamma = \T$ (with the degree-rank $\leq (1,1)$ filtration, see Example \ref{dr-f}), $G_0/\Gamma_0$ to be trivial, $F$ equal to the exponential function $(x,()) \mapsto e(x)$, $\phi: \T^{D_1} \to \T$ to be the filtered homomorphism
$$ \phi(x_1,\ldots,x_{D_1}) := a_1 x_1 + \ldots + a_{D_1} x_{D_1},$$
and $\orbit \in \ultra \poly(\Z_\N \to \T_\N)$ to be the orbit $n \mapsto \xi n + \theta \mod 1$. This should be compared with \eqref{chih-abstract} and the discussion at the start of the section.
For a slightly more complicated example, we take $s=3, r_* = 1$, and let $\chi$ be the degree-rank $(2,1)$ nilcharacter
$$ \chi(n) := e( \alpha n^2 + \beta n + \gamma ).$$
We let $D_2 \geq 1$ be an integer, set $D_1 := 0$, let $\F = ((),(\xi_{2,1},\ldots,\xi_{2,D_2})) \in \ultra \T^{0} \times \ultra \T^{D_2}$ be a collection of frequencies, and let $\eta: \R^{D_2} \to \R$ be the universal vertical frequency $\eta(x_1,\ldots,x_{D_2}) := a_1 x_1 + \ldots + a_{D_2} x_{D_2}$ for some integers $a_1,\ldots,a_{D_2} \in \Z$. Then $((0,D_2), \F, \eta)$ will be a total frequency representation of $\chi$ if $\xi = a_1 \xi_{2,1} + \ldots + a_{D_2} \xi_{2,D_2}$ (cf. \eqref{chih-abstract}). Indeed, we can take $G/\Gamma = \T$ with the degree-rank $\leq (2,1)$ filtration (see Example \ref{dr-f}), $G_0/\Gamma_0 = \T$ with the degree-rank $\leq (1,1)$ filtration, the orbit
$$ \orbit(n) := ( \alpha n^2 \mod 1, \beta n + \gamma \mod 1 )$$
and $F: G/\Gamma \times G_0/\Gamma_0 \to S^1$ to be the function
$$ F(x, y) := e(x) e(y),$$
and $\phi: \T^{D_2} \to \T$ to be the filtered homomorphism
$$ \phi(x_1,\ldots,x_{D_1}) := a_1 x_1 + \ldots + a_{D_1} x_{D_1}.$$
Note how the lower order terms $\beta_n + \gamma$ in the phase of $\chi$ are shunted off to the lower degree-rank nilmanifold $G_0/\Gamma_0$ and thus do not interact at all with the data $\F, \eta$. In this particular case, this shunting off was unnecessary, and one could have easily folded these lower order terms into the dynamics of the primary nilmanifold $G/\Gamma$; but in the next example we give, the lower order behaviour does genuinely need to be separated from the top order behaviour by placing it in a separate nilmanifold.
We now turn to a genuinely non-abelian example of a universal representation. For this, we take $s=3$, $r_*=2$, and let $\chi$ be a degree-rank $(2,2)$ nilcharacter that is a suitable vector-valued smoothing of the bracket polynomial phase
$$ n \mapsto e( \{ \alpha n \} \beta n + \gamma n^2 ).$$
We can express this nilcharacter as
$$ \chi(n) = F( \orbit(n), \orbit_0(n) ),$$
where $\orbit \in \ultra \poly(\Z_\N \to (G/\Gamma)_\N)$ is the orbit
$$ \orbit(n) := e_2^{\beta n} e_1^{\alpha n} \Gamma$$
into the Heisenberg nilmanifold \eqref{heisen} (which we give the degree-rank $\leq (2,2)$ filtration), $\orbit_0 \in \ultra \poly(\Z_\N \to (G/\Gamma)_\N)$ is the orbit
$$ \orbit_0(n) := \gamma n^2 \mod 1$$
into the unit circle $G_0/\Gamma_0 = \T$ (which we give the degree-rank $\leq (2,1)$ filtration, see Example \ref{dr-f}), and $F$ is a suitable vector-valued smoothing of the map
$$ ( e_1^{t_1} e_2^{t_2} [e_1,e_2]^{t_{12}} \Gamma, y ) \mapsto e( t_{12} ) e(y)
$$
for $t_1, t_2, t_{12} \in I_0$. By Example \ref{heist}, we have $\Taylor_1(\orbit) = (\alpha \mod 1,\beta \mod 1)$ and $\Taylor_2(\orbit)$ is trivial.
Now let $D_1 \geq 1$ be an integer, set $D_2 := 0$, let $\F = ((\xi_{1,1},\ldots,\xi_{1,D_1}),()) \in \ultra \T^{D_1} \times \ultra \T^{0}$ be a collection of frequencies. The subgroup $G^{(D_1,0)}_{(2,2)}$ of the universal nilmanifold $G^{(D_1,0)} = G^{(D_1,0),\leq (2,2)}$ is then the abelian Lie group generated by the commutators $[e_{1,i},e_{1,j}]$ for $1 \leq i < j \leq D_1$. We let $a_1,\ldots,a_{D_1},b_1,\ldots,b_{D_1} \in \Z$ be integers, and let $\phi: G^{(D_1,0)}/\Gamma^{(D_1,0)} \to G/\Gamma$ be the filtered homomorphism that maps $e_{1,i}$ to $e_1^{a_i} e_2^{b_i}$ for $i=1,\ldots,D_1$, thus
\begin{align*}
\phi( &\prod_{i=1}^{D_1} e_{1,i}^{t_i} \prod_{1 \leq i < j \leq D_1} [e_{1,i},e_{1,j}]^{t_{i,j}} \Gamma^{(D_1,0)} ) \\
&= \prod_{i=1}^{D_1} (e_1^{a_1} e_2^{b_i})^{t_i} \prod_{1 \leq i < j \leq D_1} [e_1^{a_i} e_2^{b_i}, e_1^{a_j} e_2^{b_j}]^{t_{i,j}} \Gamma \\
&= e_1^{\sum_{i=1}^{D_1} a_i t_i} e_2^{\sum_{i=1}^{D_1} b_i t_i} [e_1,e_2]^{-\sum_{i=1}^{D_1} a_i b_i \binom{t_i}{2} - \sum_{1 \leq i < j \leq d} b_i a_j t_i t_j + \sum_{1 \leq i < j \leq d} (a_i b_j - a_j b_i)t_{i,j}} \Gamma.
\end{align*}
Let us now see what conditions are required for $((D_1,0),\eta,\F)$ to be a total frequency representation of $\chi$. The condition \eqref{taylor} becomes the constraints
\begin{align*}
\alpha &= \sum_{i=1}^{D_1} a_i \xi_{1,i} \\
\beta &= \sum_{i=1}^{D_1} b_i \xi_{1,i},
\end{align*}
while the condition \eqref{vert} becomes
\begin{equation}\label{etaij}
\eta( [e_{1,i}, e_{1,j}] ) = a_i b_j - a_j b_i
\end{equation}
for all $1 \leq i < j \leq D_1$, or equivalently
$$ \eta( \prod_{1 \leq i < j \leq D_1} [e_{1,i}, e_{1,j}]^{t_{i,j}} ) = \sum_{1 \leq i < j \leq D_1} (a_i b_j - a_j b_i) t_{i,j}$$
Conversely, with these constraints we obtain a total frequency representation of $\chi$ by $((D_1,0),\eta,\F)$. This should be compared with the heuristic \eqref{chih-abstract}. (Note from \eqref{brackalg} that the top order component $\{\alpha n \} \beta n$ of $\chi$ is morally anti-symmetric in $\alpha,\beta$ modulo lower order terms, which is consistent with the anti-symmetry observed in \eqref{etaij}.) Note also that the term $\gamma n^2$, which has lesser degree-rank than the top order term $\{ \alpha n \} \beta n$, plays no role, due to it being shunted off to the lower degree-rank nilmanifold $G_0/\Gamma_0$. If instead we placed this term as part of the principal nilmanifold, then this would create a non-trivial second Taylor coefficient $\Taylor_2(\orbit)$ which would then require a non-zero value of $D_2$ in order to recover a total frequency representation. Thus we see that in order to neglect terms of lesser degree-rank (but equal degree) it is necessary to create the secondary nilmanifold $G_0/\Gamma_0$ as a sort of ``junk nilmanifold'' to hold all such terms.
We make the easy remark that every nilcharacter $\chi$ of degree-rank $\leq (s-1,r_*)$ has at least one representation.
\begin{lemma}[Existence of representation]\label{existence} Let $\chi$ be a nilcharacter of degree-rank $(s-1,r_*)$ on $[N]$. Then there exists at least one total frequency representation $(\vec D, \F, \eta)$ of $\chi$.
\end{lemma}
\begin{proof} By definition, $\chi = F \circ \orbit$ for some degree-rank $\leq (s-1,r_*)$ nilmanifold $G/\Gamma$, some $\orbit \in \ultra \poly(\Z_\N \to (G/\Gamma)_\N)$, and some $F \in \Lip(\ultra(G/\Gamma))$ with a vertical frequency. For each $1 \leq i \leq s-1$, let $f_{i,1},\ldots,f_{i,D_i}$ be a basis of generators for $\Gamma_{i}$, and let $\vec D := (D_1,\ldots,D_{s-1})$ be the associated dimension vector. Then we have a filtered homomorphism $\phi: G^{\vec D} \to G$ which maps $e_{i,j}$ to $f_{i,j}$ for all $1 \leq i \leq s-1$ and $1 \leq j \leq D_i$. It is easy to see that $\phi$ is surjective from $G^{\vec D}_{i}$ to $G_{i}$ for each $i$, and so the map $\pi_{\Horiz_i(G/\Gamma)} \circ \phi$ is surjective from $G^{\vec D}_{i}$ to $\Horiz_i(G/\Gamma)$. It is now an easy matter to locate frequencies $\xi_{i,j}$ obeying \eqref{taylor}, and the vertical frequency property of $F$ can be pulled back via $\phi$ to give \eqref{vert}. Setting $G_0/\Gamma_0$ to be trivial, we obtain the claim.
\end{proof}
To conclude this section, we now give some basic facts about total frequency representations. These facts will not actually be used in this paper, but may serve to consolidate one's intuition about the nature of these representations. We first observe some linearity in the vertical frequency $\eta$.
\begin{lemma}[Linearity] Suppose that $\chi, \chi'$ are two nilcharacters of degree-rank $(s-1,r_*)$ on $[N]$ that have total frequency representations $(\vec D, \F, \eta)$ and $(\vec D, \F, \eta')$ respectively. Then $\overline{\chi}$ has a total frequency representation $(\vec D, \F, -\eta)$, and $\chi \otimes \chi'$ has a total frequency representation $(\vec D, \F, \eta+\eta')$.
\end{lemma}
\begin{proof} This is a routine matter of chasing down the definitions, and noting that nilmanifolds, polynomial orbits, etc. behave well with respect to direct sums.
\end{proof}
\begin{lemma}[Triviality] Suppose that $\chi$ is a nilcharacter of degree-rank $(s-1,r_*)$ on $[N]$ that has a total frequency representation $(\vec D, \F, 0)$. Then $\chi$ is a nilsequence of degree-rank $\leq (s-1,r_*-1)$ \textup{(}i.e. $[\chi]_{\Symb^{(s-1,r_*)}_\DR([N])} = 0$\textup{)}.
\end{lemma}
\begin{proof} By construction, we have
$$ \chi(n) = F( \orbit(n), \orbit_0(n) )$$
for some limit polynomial orbits $\orbit \in \ultra \poly(\Z_\N \to (G/\Gamma)_\N)$, $\orbit_0 \in \ultra \poly(\Z_\N \to (G_0/\Gamma_0)_\N)$ into filtered nilmanifolds $G/\Gamma, G_0/\Gamma_0$ of degree-rank $\leq (s-1,r_*)$ and $\leq (s-1,r_*-1)$ respectively, where $F \in \Lip(\ultra(G/\Gamma \times G_0/\Gamma_0) \to \overline{S^\omega})$. Furthermore, there exists a filtered homomorphism $\phi: G^{\vec D}/\Gamma^{\vec D} \to G/\Gamma$ such that
\eqref{taylor} holds, and such that
\begin{equation}\label{flat}
F( \phi(t) x, x_0 ) = F(x,x_0).
\end{equation}
for all $t \in G^{\vec D}_{(s-1,r_*)}$.
Let $T$ be the closure of the set $\{ \phi(t) \mod \Gamma_{(s-1,r_*)}: t \in G^{\vec D}_{(s-1,r_*)}\}$; this is a subtorus of the torus $G_{(s-1,r_*)}/\Gamma_{(s-1,r_*)}$, and thus acts on $G/\Gamma$. As $F$ is continuous and obeys the invariance \eqref{flat}, we see that $F$ is $T$-invariant; we may thus quotient out by $T$ and assume that $T$ is trivial. In particular, $\phi$ now annihilates $G^{\vec D}_{(s-1,r_*)}$.
We give $G$ a new degree-rank filtration $(G'_{(d,r)})_{(d,r) \in \DR}$ (smaller than the existing filtration $(G_{(d,r)})_{(d,r) \in \DR}$), by defining $G'_{(d,r)}$ to be the connected subgroup of $G$ generated by $G_{(d,r+1)}$ (recalling the convention $G_{(d,r)} := G_{(d+1,0)}$ when $r > d$) together with the image $\phi( G^{\vec D}_{(d,r)} )$ of $G^{\vec D}_{(d,r)}$. It is easy to see that this is still a filtration, and that $G/\Gamma$ remains a filtered nilmanifold with this filtration, but now the degree-rank is $\leq (s-1,r_*-1)$ rather than $\leq (s-1,r_*)$. Furthermore, from \eqref{taylor} we see that $\orbit$ is still a polynomial orbit with respect to this new filtration. As such, $\chi$ is a nilsequence of degree-rank $\leq (s-1,r_*-1)$ as required.
\end{proof}
Combining the above two lemmas we obtain the following corollary.
\begin{corollary}[Representation determines symbol] Suppose that $\chi, \chi'$ are two nilcharacters of degree-rank $(s-1,r_*)$ on $[N]$ that have a common total frequency representation $(\vec D, \F, \eta)$. Then $\chi, \chi'$ are equivalent. In other words, the symbol $[\chi]_{\Xi^{(s-1,r_*)([N])}}$ depends only on $(\vec D, \F, \eta)$.
\end{corollary}
Note that the above results are consistent with the heuristic \eqref{chih-abstract}.
\section{Linear independence and the sunflower lemma}\label{reg-sec}
A basic fact of linear algebra is that every finitely generated vector space is finite-dimensional. In particular, if $v_1,\ldots,v_l$ are a finite collection of vectors in a vector space $V$ over a field $k$, then there exists a finite linearly independent set of vectors $v'_1,\ldots,v'_{l'}$ in $V$ such that each of the vectors $v_1,\ldots,v_l$ is a linear combination (over $k$) of the $v'_1,\ldots, v'_{l'}$. Indeed, one can take $v'_1,\ldots,v'_{l'}$ to be a set of vectors generating $v_1,\ldots,v_l$ for which $l'$ is minimal, since any linear relation amongst the $v'_1,\ldots,v'_{l'}$ can be used to decrease\footnote{Indeed, one can recast this argument as a rank reduction argument instead of a minimal rank argument, for the same reason that the principle of infinite descent is logically equivalent to the well-ordering principle. In this infinitary (ultralimit) setting, there is very little distinction between the two approaches, although the minimality approach allows for slightly more compact notation and proofs. But in the finitary setting, it becomes significantly more difficult to implement the minimality approach, and the rank reduction approach becomes preferable. See \cite{u4-inverse} for finitary ``rank reduction'' style arguments analogous to those given here.} the ``rank'' $l'$, contradicting minimality (cf. the proof of classical Steinitz exchange lemma in linear algebra).
We will need analogues of this type of fact for frequencies $\xi_1,\ldots,\xi_l$ in the limit unit circle $\ultra \T$. However, this space is not a vector space over a field, but is merely a module over a commutative ring $\Z$. As such, the direct analogue of the above statement fails; indeed, any standard rational in $\ultra \T$, such as $\frac{1}{2} \mod 1$, clearly cannot be represented as a linear combination (over $\Z$) of a finite collection of frequencies in $\ultra \T$ that are linearly independent over $\Z$.
However, the standard rationals are the \emph{only} obstruction to the above statement being true. More precisely, we have
\begin{lemma}[Baby regularity lemma]\label{baby} Let $l \in \N$, and let $\xi_1,\ldots,\xi_l \in \ultra \T$. Then there exists $l',l'' \in \N$ and $\xi'_1,\ldots,\xi'_{l'}, \xi''_1,\ldots,\xi''_{l''} \in \ultra \T$ such that $\xi'_1,\ldots,\xi'_{l'}$ are linearly independent over $\Z$ \textup{(}i.e. there exist no standard integers $a_1,\ldots,a_{l'}$, not all zero, such that $a_1 \xi'_1+\ldots+a_{l'} \xi'_{l'} = 0$\textup{)}, each of the $\xi''_i$ are rational \textup{(}i.e. they live in $\Q \mod 1$\textup{)}, and each of the $\xi_1,\ldots,\xi_l$ are linear combinations \textup{(}over $\Z$\textup{)} of the $\xi'_1,\ldots,\xi'_{l'}, \xi''_1,\ldots,\xi''_{l''}$.
\end{lemma}
\begin{proof} Fix $l,\xi_1,\ldots,\xi_l$. Define a \emph{partial solution} to be a collection of objects $l', l''$, $\xi'_1, \ldots,\xi'_{l'}$, $\xi''_1,\ldots,\xi''_{l''}$ satisfying all of the required properties, except possibly for the linear independence of the $\xi'_1,\ldots,\xi'_{l'}$. Clearly at least one partial solution exists, since one can take $l' := l$, $l'' := 0$, and $\xi'_i := \xi_i$ for all $1 \leq i \leq l$. Now let $l',l'',\xi'_1,\ldots,\xi'_{l'}, \xi''_1,\ldots,\xi''_{l''}$ be a partial solution for which $l'$ is minimal. We claim that $\xi'_1,\ldots,\xi'_{l'}$ is linearly independent over $\Z$, which will give the lemma. To see this, suppose for contradiction that there existed $a_1,\ldots,a_{l'} \in \Z$, not all zero, such that $a_1 \xi'_1 + \ldots + a_{l'} \xi'_{l'} = 0$. Without loss of generality we may assume that $a_1$ is non-zero. For each $2 \leq j \leq l'$, let $\tilde \xi'_j \in \ultra \T$ be such that $a_1 \tilde \xi'_j = \xi'_j$. We then have
$$ \xi'_1 = - \sum_{j=2}^{l'} \frac{a_j}{a_1} \xi'_j + q \mod 1$$
for some standard rational $q \in \Q$. If we then replace $\xi'_1,\ldots,\xi'_{l'}$ by $\tilde \xi'_2,\ldots,\tilde \xi'_{l'}$ (decrementing $l'$ to $l'-1$) and append $q$ to $\xi''_1,\ldots,\xi''_{l''}$, then we obtain a new partial solution with a smaller value of $l'$, contradicting minimality. The claim follows.
\end{proof}
This lemma is too simplistic for our applications, and we will need to modify it in a number of ways. The first is to introduce an error term.
\begin{definition}[Linear independence] Let $\eps > 0$ be a limit real, and let $l \in \N$. A set of frequencies $\xi_1,\ldots,\xi_l \in \ultra \T$ is said to be \emph{independent modulo $O(\eps)$} if there do not exist any collection $a_1,\ldots,a_l \in \Z$ of standard integers, not all zero, for which
$$ a_1 \xi_1 + \ldots + a_l \xi_l = O(\eps) \mod 1$$
(Thus, for instance, the empty set (with $k=0$) is trivially independent modulo $O(\eps)$.) Equivalently, $\xi_1,\ldots,\xi_l$ are linearly independent over $\Z$ after quotienting out by the subgroup $\eps \overline{\R} \mod 1$.
\end{definition}
This definition is only non-trivial when $\eps$ is an infinitesimal (i.e. $\eps=o(1)$). In practice, $\eps$ will be a negative power of the unbounded integer $N$.
We have the following variant of Lemma \ref{baby}.
\begin{lemma}[Regularising one collection of frequencies]\label{toddler} Let $l \in \N$, let $\xi_1,\ldots,\xi_l \in \ultra \T$, and let $\eps > 0$ be a limit real. Then there exist $l',l'',l''' \in \N$ and
\[
\xi'_1,\ldots,\xi'_{l'}, \xi''_1,\ldots,\xi''_{l''},\xi'''_1,\ldots,\xi'''_{l'''} \in \ultra \T
\]
such that $\xi'_1,\ldots,\xi'_{l'}$ are linearly independent modulo $O(\eps)$, each of the $\xi''_i$ are rational, each of the $\xi'''_i$ are $O(\eps)$, and each of the $\xi_1,\ldots,\xi_l$ are linear combinations \textup{(}over $\Z$\textup{)} of the $\xi'_1,\ldots,\xi'_{l'}, \xi''_1,\ldots,\xi''_{l''}, \xi'''_1,\ldots,\xi'''_{l'''}$.
\end{lemma}
One can view Lemma \ref{baby} as the degenerate case $\eps=0$ of the above lemma.
\begin{proof} We repeat the proof of Lemma \ref{baby}. Define a \emph{partial solution} to be a collection of objects $l',l'',l''$, $\xi'_1,\ldots,\xi'_{l'}, \xi''_1,\ldots,\xi''_{l''},\xi'''_1,\ldots,\xi'''_{l'''}$ obeying all the required properties except possibly for the linear independence property. Again it is clear that at least one partial solution exists, so we may find a partial solution for which $l'$ is minimal. We claim that this is a complete solution. For if this is not the case, we have
$$ a_1 \xi'_1 + \ldots + a_{l'} \xi'_{l'} = O(\eps) \mod 1$$
for some $a_1,\ldots,a_{l'} \in \Z$, not all zero. Again, we may assume that $a_1 \neq 0$. We again select $\tilde \xi'_2,\ldots,\tilde \xi'_{l'} \in \ultra \T$ with $a_1 \tilde \xi'_j = \xi'_j$ for all $2 \leq j \leq l'$, and observe that
$$ \xi'_1 = - \sum_{j=2}^{l'} \frac{a_j}{a_1} \xi'_j + q + s\mod 1$$
for some standard rational $q \in \Q$ and some $s = O(\eps)$. If we then replace $\xi'_1,\ldots,\xi'_{l'}$ by $\tilde \xi'_2,\ldots,\tilde \xi'_{l'}$, and append $q$ and $s$ to $\xi''_1,\ldots,\xi''_{l''}$ and $\xi'''_1,\ldots,\xi'''_{l'''}$ respectively, we contradict minimality, and the claim follows.
\end{proof}
This lemma is still far too simplistic for our needs, because we will not be needing to regularise just one collection $\xi_1,\ldots,\xi_l$ of frequencies, but a whole \emph{family} $\xi_{h,1},\ldots,\xi_{h,l}$ of frequencies, where $h$ ranges over a parameter set $H$. Such frequencies can exhibit a range of behaviour in $h$; at one extreme, they might be completely independent of $h$, while at the other extreme, the frequencies may vary substantially as $h$ does. It turns out that in some sense, the general case is a combination of these extreme cases.
In this direction we have the following stronger version of Lemma \ref{toddler}.
\begin{lemma}[Regularising many collections of frequencies]\label{kid} Let $l \in \N$, let $\eps > 0$ be a limit real, let $H$ be a limit finite set, and for each $h \in H$, let $\xi_{h,1},\ldots,\xi_{h,l}$ be frequencies in $\ultra \T$ that depend in a limit fashion on $h$. Then there exists a dense subset $H'$ of $H$, standard natural numbers, $l_*, l',l''_*,l''' \in \N$, ``core'' frequencies $\xi_{*,1},\ldots,\xi_{*,l_*}, \xi''_{*,1},\ldots,\xi''_{l''_*} \in \ultra \T$, and ``petal'' frequencies \[ \xi'_{h,1},\ldots,\xi'_{h,l'}, \xi'''_{h,1},\ldots,\xi'''_{h,l'''} \in \ultra \T\] for each $h \in H'$ depending in a limit fashion on $h$, and obeying the following properties:
\begin{itemize}
\item[(i)] \textup{(Independence)} For almost all triples $(h_1,h_2,h_3) \in (H')^3$ \textup{(}i.e. for all but $o(|H'|^3)$ such triples\textup{)}, the frequencies
\[
\xi_{*,1},\ldots,\xi_{*,l_*}, \xi'_{h_1,1},\ldots,\xi'_{h_1,l'}, \xi'_{h_2,1},\ldots,\xi'_{h_2,l'}, \xi'_{h_3,1},\ldots,\xi'_{h_3,l'}
\]
are linearly independent modulo $O(\eps)$.
\item[(ii)] \textup{(Rationality)} For each $1 \leq j \leq l''$, $\xi''_{*,j}$ is a standard rational.
\item[(iii)] \textup{(Smallness)} For each $h \in H'$ and $1 \leq j \leq l'''$, $\xi'''_{h,j} = O(\eps)$.
\item[(iv)] \textup{(Representation)} For each $h \in H'$, the $\xi_{h,1},\ldots,\xi_{h,l}$ are linear combinations over $\Z$ of the frequencies \[ \xi_{*,1},\ldots,\xi_{*,l_*}, \xi'_{h,1},\ldots,\xi'_{h,l'}, \xi''_{*,1},\ldots,\xi''_{*,l''}, \xi'''_{h,1},\ldots,\xi'''_{h,l'''}.\]
\end{itemize}
\end{lemma}
Note that Lemma \ref{kid} collapses to Lemma \ref{toddler} if $H$ is a singleton set.
\begin{proof} We again use the usual argument. Define a \emph{partial solution} to be a collection of objects $H', l_*, l', l''_*, l''', \xi_{*,j}, \xi'_{h,j}, \xi''_{*,j}, \xi'''_{h,j}$ obeying all the required properties except possibly for the independence property. Again, at least one partial solution exists, since we may take $H' := H$, $l_* := l'' := l''' := 0$, $l' := l$, and $\xi'_{j,h} := \xi_{j,h}$ for all $h \in H$ and $1 \leq j \leq l$. We may thus select a partial solution for which $l'$ is minimal; and among all such partial solutions with $l'$ minimal, we choose a solution with $l_*$ minimal for fixed $l'$ (i.e. we minimise with respect to the lexicographical ordering on $l'$ and $l_*$). We claim that this doubly minimal solution obeys the independence property, which would give the claim.
Suppose the independence property fails. Carefully negating the quantifiers and using Lemma \ref{dense-dich}, we conclude that there exist standard integers $a_{*,j}$ for $1 \leq j \leq l_*$ and $a'_{i,j}$ for $i=1,2,3$ and $1 \leq j \leq l'$, not all zero, such that one has the relation
$$ a_{*,1} \xi_{*,1} + \ldots + a_{*,l_*} \xi_{*,l_*} + \sum_{i=1}^3 \sum_{j=1}^{l'} a'_{i,j} \xi'_{h_i,j} = O(\eps) \mod 1$$
for many triples $(h_1,h_2,h_3) \in (H')^3$.
Suppose first that all of the $a'_{i,j}$ vanish, so that we have a linear relation
$$ a_{*,1} \xi_{*,1} + \ldots + a_{*,l_*} \xi_{*,l_*} = O(\eps) \mod 1$$
that only involves core frequencies. Then the situation is basically the same as that of Lemma \ref{toddler}; without loss of generality we may take $a_{*,1} \neq 0$, and if we then choose $\tilde \xi_{*,2},\ldots,\tilde \xi_{*,l_*}$ so that $a_{*,1} \tilde \xi_{*,j} = \xi_{*,j}$, then we can rewrite
$$ \xi_{*,1} = -\sum_{j=2}^{l'} a_{*,j} \tilde \xi_{*,j} + q + s \mod 1$$
for some $q \in \Q$ and $s = O(\eps)$, and one can then replace the $\xi_{*,1},\ldots,\xi_{*,l_*}$ with $\tilde \xi_{*,2},\ldots,\tilde \xi_{*,l_*}$ (decrementing $l_*$ by $1$) and append $q$ and $s$ to each of the collections $\xi''_{h,1},\ldots,\xi''_{h,l''}$ and $\xi'''_{h,1},\ldots,\xi'''_{h,l'''}$ respectively for each $h \in H$, contradicting minimality.
Now suppose that not all of the $a'_{i,j}$ vanish; without loss of generality we may assume that $a'_{1,1}$ is non-zero. By the pigeonhole principple, we can find $h_2, h_3 \in H'$ such that
$$ a_{*,1} \xi_{*,1} + \ldots + a_{*,l_*} \xi_{*,l_*} + \sum_{i=1}^3 \sum_{j=1}^{l'} a'_{i,j} \xi'_{h_1,j} = O(\eps) \mod 1$$
for all $h_1$ in a dense subset $H''$ of $H'$. Now let $\tilde \xi_{*,j} \in \ultra \T$ for $1 \leq j \leq l_*$ and $\tilde \xi'_{h,j} \in \ultra \T$ for $h_1 \in H'$ and $1 \leq j \leq l'$ be such that $a'_{1,1} \tilde \xi_{*,j} = \xi_{*,j}$ and $a'_{1,1} \tilde \xi'_{h,j} = \xi'_{h,j}$, then we have
$$ \xi'_{h_1,1} = - \sum_{j=2}^{l'} a'_{1,j} \tilde \xi'_{h_1,j} - \sum_{j=1}^{l_*} a_{*,j} \tilde \xi_{*,j} - \sum_{i=2}^3 \sum_{j=1}^{l'} a'_{i,j} \tilde \xi'_{i,j} + q_{h_1} + s_{h_1} \mod O(1)$$
for some standard rational $q_{h_1}$ and some $s_{h_1} = O(\eps)$. Furthermore one can easily ensure that $q_{h_1}, s_{h_1}$ depend in a limit fashion on $h_1$. By Lemma \ref{dense-dich} (and refining $H'$) we may assume that $q_{h_1} = q_*$ is independent of $h_1$.
We may thus replace $H'$ by $H''$ and replace $\xi'_{h,1},\ldots,\xi'_{h,l'}$ by $\tilde \xi'_{h,2},\ldots,\tilde \xi'_{h,l'}$ (decrementing $l'$ by $1$), while appending $q_*$ and $s_h$ to $\xi''_{*,1},\ldots,\xi''_{*,l''}$ and $\xi'''_{h,1},\ldots,\xi'''_{h,l'''}$ respectively, and replacing $\xi_{*,1},\ldots,\xi_{*,l_*}$ by $\tilde \xi_{*,1},\ldots,\tilde \xi_{*,l_*}, \tilde \xi'_{h_2,1},\ldots,\tilde \xi_{h_2,l'}, \tilde \xi'_{h_3,1},\ldots,\tilde \xi_{h_3,l'}$ (incrementing $l_*$ as necessary). This contradicts the minimality of the partial solution, and the claim follows.
\end{proof}
This is still too simplistic for our applications, as the independence hypothesis on triples $(h_1,h_2,h_3)$ will not quite be strong enough to give everything we need. Ideally, (in view of Proposition \ref{gcs-prop}) we would like to have independence of the
$\xi_{*,1},\ldots,\xi_{*,l_*}, \xi'_{h_1,1},\ldots,\xi'_{h_4,l'}$
for almost all additive quadruples $h_1+h_2=h_3+h_4$ in $H'$. Unfortunately, this need not be the case; indeed, if the original $\xi_{h,i}$ are linear in $h$, say $\xi_{h,i} = \alpha_i h$ for some $\alpha_i \in \ultra \T$ and all $1 \leq i \leq l'$, then we have $\xi_{h_1,i} + \xi_{h_2,i} = \xi_{h_3,i} + \xi_{h_4,i}$ for all additive quadruples $h_1+h_2=h_3+h_4$ in $H'$ and all $1 \leq i \leq l'$, and as a consequence it is not possible to obtain a decomposition as in Lemma \ref{kid} with the stronger independence property mentioned above. A similar obstruction occurs if the $\xi_{h,i}$ are \emph{bracket}-linear in $h$, for instance if $\xi_{h,i} = \{ \alpha_i h \} \beta_i \mod 1$ for some $\alpha_i \in \ultra \T$ and $\beta_i \in \ultra \R$.
By using tools from additive combinatorics, we can show that bracket-linear frequencies are the \emph{only} obstructions to independence on additive quadruples. More precisely, we have
\begin{lemma}\label{teenager} Let $l \in \N$, let $\eps > 0$ be a limit real, let $H$ be a dense limit subset of $[[N]]$, and for each $h \in H$, let $\xi_{h,1},\ldots,\xi_{h,l}$ be frequencies in $\ultra \T$ that depend in a limit fashion on $h$. Then there exists a dense subset $H'$ of $H$, standard natural numbers, $l_*, l',l''_*,l''',l'''' \in \N$, ``core'' frequencies $\xi_{*,1},\ldots,\xi_{*,l_*}, \xi''_{*,1},\ldots,\xi''_{*,l''_*} \in \ultra \T$, and ``petal'' frequencies $\xi'_{h,1},\ldots,\xi'_{h,l'},\xi'''_{h,1},\ldots,\xi'''_{h,l'''} \xi''''_{h,1},\ldots,\xi''''_{h,l''''} \in \ultra \T$ for each $h \in H'$ depending in a limit fashion on $h$, obeying the following properties:
\begin{itemize}
\item[(i)] \textup{(Independence)} For almost all additive quadruples $h_1+h_2=h_3+h_4$ in $H'$ (i.e. for all but $o(|H'|^3)$ such quadruples), the frequencies $\xi_{*,j}$ for $1 \leq j \leq l_*$, $\xi'_{h_i,j}$ for $i=1,2,3,4$ and $1 \leq j \leq l'$, and $\xi''''_{h_i,j}$ for $i=1,2,3$ and $1 \leq j \leq l''''$ are jointly linearly independent modulo $O(\eps)$.
\item[(ii)] \textup{(Rationality)} For each $1 \leq j \leq l''_*$, $\xi''_{*,j}$ is a standard rational.
\item[(iii)] \textup{(Smallness)} For each $h \in H'$ and $1 \leq j \leq l'''$, $\xi'''_{h,j} = O(\eps)$.
\item[(iv)] \textup{(Bracket-linearity)} For each $1 \leq j \leq l''''$, there exist $\alpha_j \in \ultra \T$ and $\beta_j \in \ultra \R$ such that $\xi''''_{h,j} = \{ \alpha_j h \} \beta_j \mod 1$ for all $h \in H'$. Furthermore, the map $h \mapsto \xi''''_{h,j}$ is a Freiman homomorphism on $H'$ \textup{(}see \S \ref{notation-sec} for the definition of a Freiman homomorphism\textup{)}.
\item[(v)] \textup{(Representation)} For each $h \in H'$, the $\xi_{h,1},\ldots,\xi_{h,l}$ are linear combinations over $\Z$ of the frequencies \[ \xi_{*,1},\ldots,\xi_{*,l_*}, \xi'_{h,1},\ldots,\xi'_{h,l'}, \xi''_{*,1},\ldots,\xi''_{*,l''}, \xi'''_{h,1},\ldots,\xi'''_{h,l'''}, \xi''''_{h,1},\ldots,\xi''''_{h,l''''}.\]
\end{itemize}
\end{lemma}
\begin{proof} As usual, we define a \emph{partial solution} to be a collection of objects $H'$, $l_*, l',l''_*,l''',l''''$, $\xi_{*,1},\ldots,\xi''''_{h,l''''}$, obeying all of the required properties except possibly for the independence property. Again, there is clearly at least one partial solution, so we select a partial solution with a minimal value of $l'$, and then (for fixed $l'$) a minimal value of $l''''$, and then (for fixed $l',l''''$) a minimal value of $l_*$. We claim that this partial solution obeys the independence property, which will give the lemma.
Suppose for contradiction that this were not the case; then by Lemma \ref{dense-dich}, there exist standard integers $a_{*,j}$ for $1 \leq j \leq l_*$, $a'_{i,j}$ for $1 \leq i \leq 4$ and $1 \leq j \leq l'$, and $a''_{i,j}$ for $1 \leq i \leq 3$ and $1 \leq j \leq l''''$, not all zero, such that
$$ \sum_{j=1}^{l_*} a_{*,j} \xi_{*,j} + \sum_{i=1}^4 \sum_{j=1}^{l'} a'_{i,j} \xi'_{h_i,j} + \sum_{i=1}^3 \sum_{j=1}^{l'''} a''''_{i,j} \xi''''_{h_i,j} = O(\eps) \mod 1$$
for many additive quadruples $h_1+h_2=h_3+h_4$ in $H'$.
Suppose first that all the $a'_{i,j}$ and $a''''_{i,j}$ vanished. Then we have a relation
$$ \sum_{j=1}^{l_*} a_{*,j} \xi_{*,j} = O(\eps) \mod 1$$
that only involves core frequencies; arguing as in Lemma \ref{kid} we can thus find another partial solution with a smaller value of $l_*$ (and the same value of $l'$, $l''''$), contradicting minimality.
Next, suppose that the $a'_{i,j}$ all vanished, but the $a''''_{i,j}$ did not all vanish. Then we have a relation
\begin{equation}\label{triplicate}
\sum_{j=1}^{l_*} a_{*,j} \xi_{*,j} + \sum_{i=1}^3 \sum_{j=1}^{l''''} a''''_{i,j} \xi''''_{h_i,j} = O(\eps) \mod 1
\end{equation}
for many triples $h_1,h_2,h_3$ in $H'$.
Without loss of generality let us suppose that $a''''_{1,1}$ is non-zero. By the pigeonhole principle, we may find $h_2,h_3 \in H'$ such that \eqref{triplicate} holds for all $h_1$ in a dense subset $H''$ of $H'$. As in previous arguments, we then find $\tilde \xi_{*,j} \in \ultra \T$ such that $a''''_{1,1} \tilde \xi_{*,j} = \xi_{*,j}$ for each $1 \leq j \leq l_*$, and also find $\tilde \beta_j \in \ultra \R$ such that $a''''_{1,1} \tilde \beta_j = \beta_j$ for all $1 \leq j \leq l''''$. If we then set $\tilde \xi''''_{h,j} := \{ \alpha_j h \} \tilde \beta_j$ for each $h \in H'$ and $1 \leq j \leq l''''$, then $a''''_{1,1} \tilde \xi''''_{h,j} = \xi''''_{h,j}$, and so for any $h_1 \in H'$ we have
$$ \xi''''_{h_1,1} = - \sum_{j=1}^{l_*} a_{*,j} \tilde \xi_{*,j} - \sum_{j=2}^{l''''} a''''_{1,j} \tilde \xi''''_{h_1,j} - \sum_{i=2}^3 \sum_{j=1}^{l''''} a''''_{i,j} \tilde \xi''''_{h_i,j} + q_{h_1} + s_{h_1} \mod 1$$
for some standard rational $q_{h_1}$ and some $s_{h_1} = O(\eps)$, both depending on a limit fashion on $h_1$. By refining $H'$ if necessary (and using the bracket-linear nature of the $\tilde \xi''''_{h,j}$) we may assume that the map $h \mapsto \tilde \xi''''_{h,j}$ is a Freiman homomorphism on $H'$, and by Lemma \ref{dense-dich} we may make $q_{h_1} = q_*$ independent of $h_1$. If we then argue as in the proof of Lemma \ref{kid}, we may find a new partial solution with a smaller value of $l''''$ and the same value of $l'$, contradicting minimality.
Finally, suppose that the $a'_{i,j}$ did not all vanish. Using the Freiman homomorphism property to permute the $i$ indices if necessary, we may assume that $a'_{4,1}$ does not vanish. We then have
$$ \Xi_1(h_1) + \Xi_2(h_2) + \Xi_3(h_3) + \Xi_4(h_4) = O(\eps)$$
for many additive quadruples $h_1+h_2=h_3+h_4$ in $H'$, where the limit functions $\Xi_i: H \to \ultra \T$ are defined by
$$ \Xi_i(h) := \sum_{j=1}^{l'} a'_{i,j} \xi'_{h,j} + \sum_{j=1}^{l''''} a''''_{i,j} \xi''''_{h,j} \mod 1$$
for $i=1,2,3$ and $h \in H$, and
$$ \Xi_4(h) := \sum_{j=1}^{l_*} a_{*,j} \xi_{*,j} +
\sum_{j=1}^{l'} a'_{4,j} \xi'_{h,j} \mod 1.$$
We can use this additive structure to ``solve'' for $\Xi_4$, using a result from additive combinatorics which we present here as Lemma \ref{lin}. Applying this lemma, we can then find a dense limit subset $H'$ of $H$, a standard integer $K$, and frequencies $\alpha'_1,\ldots,\alpha'_K, \delta \in \ultra \T$ and $\beta'_1,\ldots,\beta'_K \in \ultra \R$ such that
$$ \Xi_4(h) = \sum_{k=1}^K \{ \alpha'_k h \} \beta'_k + \delta + O(\eps) \mod 1$$
and thus
$$ a'_{4,1} \xi'_{h,1} = \sum_{k=1}^K \{ \alpha'_k h \} \beta'_k + \delta - \sum_{j=1}^{l_*} a_{*,j} \xi_{*,j} +
\sum_{j=2}^{l'} a'_{4,j} \xi'_{h,j} + O(\eps) \mod 1$$
for all $h \in H'$.
As usual, we now find $\tilde \beta_k \in \ultra \R$ for $1 \leq k \leq K$, $\tilde \beta_j \in \ultra \R$ for $1 \leq j \leq l''''$, $\tilde \delta \in \T$ and $\tilde \xi_{*,j}$ for $1 \leq j \leq l_*$ such that $a'_{4,1} \tilde \beta_k = \beta_k$, $a'_{4,1} \tilde \beta_j = \beta_j$, $a'_{4,1} \tilde \delta = \delta$, and $a'_{4,1} \tilde \xi_{*,j} = \xi_{*,j}$. We then set $\tilde \xi'_{h,j} := \{ \alpha_j h \} \tilde \beta_j \mod 1$, and we conclude that
$$ \xi'_{h,1} = \sum_{k=1}^K \{ \alpha'_k h \} \tilde \beta'_k + \tilde \delta - \sum_{j=1}^{l_*} a_{*,j} \tilde \xi_{*,j} +
\sum_{j=2}^{l'} a'_{4,j} \tilde \xi'_{h,j} + q_h + s_h \mod 1$$
for all $h \in H'$, where $q_h \in \Q$ and $s_h = O(\eps)$ depend in a limit fashion on $h$. By refining $H'$ we may take $q_h = q_*$ independent of $h$.
We can then use relation to build a new partial solution that decreases $l'$ by $1$, at the expense of enlarging the other dimensions $l_*, l'', l''', l''''$ (and also refining $H$ to $H'$), again contradicting minimality, and the claim follows.
\end{proof}
We now apply the above lemma to the language of horizontal frequency vectors introduced in the previous section. We need some definitions:
\begin{definition}[Properties of horizontal frequency vectors] Let
\[ \F = (\xi_{i,j})_{1 \leq i \leq s-1; 1 \leq j \leq D_i}\; \; \mbox{and} \; \; \F' = (\xi'_{i,j})_{1 \leq i \leq s-1; 1 \leq j \leq D'_i}\]
be horizontal frequency vectors.
\begin{itemize}
\item We say that $\F$ is \emph{independent} if, for each $1 \leq i \leq d$, the tuple $(\xi_{i,j})_{1 \leq j \leq D_i}$ is independent modulo $O(N^{-i})$.
\item We say that $\F$ is \emph{rational} if all the $\xi_{i,j}$ are standard rationals.
\item We say that $\F$ is \emph{small} if one has $\xi_{i,j} = O(N^{-i})$ for all $1 \leq i \leq s-1$ and $1 \leq j \leq D_i$.
\item We define the \emph{disjoint union} $\F \uplus \F' = (\xi''_{i,j})_{1 \leq i \leq s-1; 1 \leq j \leq D_i+D'_i}$ by declaring $\xi''_{i,j}$ to equal $\xi_{i,j}$ if $j \leq D_i$ and $\xi'_{i,j-D_i}$ if $D_i < j \leq D_i+D'_i$. This is clearly a horizontal frequency vector with dimensions $(D_1+D'_1,\ldots,D_{s-1}+D'_{s-1})$.
\item We say that $\F$ is \emph{represented} by $\F'$ if for every $1 \leq i \leq s-1$ and $1 \leq j \leq D_i$, $\xi_{i,j}$ is a standard integer linear combination of the $\xi'_{i,j'}$ for $1 \leq j' \leq D'_i$.
\end{itemize}
\end{definition}
\begin{lemma}[Sunflower lemma]\label{sunflower-basic} Let $H$ be a dense subset of $[[N]]$, and let $(\F_h)_{h \in H}$ be a family of horizontal frequency vectors depending in a limit fashion on $h$, whose dimension vector $\vec D = \vec D_h$ is independent of $h$. Then we can find the following objects:
\begin{itemize}
\item A dense subset $H'$ of $H$;
\item Dimension vectors $\vec D_* = \vec D_{*,\ind} + \vec D_{*,\rat}$ and $\vec D' = \vec D'_\lin + \vec D'_\ind + \vec D'_\sml$, which we write as $\vec D_* = (D_{*,i})_{i=1}^{s-1}$, $\vec D_{*,\ind} = (D_{*,\ind,i})_{i=1}^{s-1}$, etc.;
\item A \emph{core horizontal frequency vector} $\F_* = (\xi_{*,i,j})_{1 \leq i \leq s-1; 1 \leq j \leq D_{*,i}}$, which is partitioned as $\F_* = \F_{*,\ind} \uplus \F_{*,\rat}$, with the indicated dimension vectors $\vec D'_\ind, \vec D'_\rat$;
\item A \emph{petal horizontal frequency vector} $\F'_h = (\xi'_{h,i,j})_{1 \leq i \leq s-1; 1 \leq j \leq D'_i}$, which is partitioned as $\F'_h = \F'_{h,\lin} \uplus \F'_{h,\ind} \uplus \F'_{h,\sml}$, which is a limit function of $h$ and with the indicated dimension vectors $\vec D'_\lin, \vec D'_\ind, \vec D'_\sml$
\end{itemize}
which obey the following properties:
\begin{itemize}
\item For all $h \in H'$, $\F'_{h,\sml}$ are small.
\item $\F_{*,\rat}$ is rational.
\item For every $1 \leq i \leq d$ and $1 \leq j \leq D'_{i,\lin}$, there exists $\alpha_{i,j} \in \ultra\T$ and $\beta_{i,j} \in \ultra \R$ such that \eqref{xih-def} holds for all $h \in H'$, and furthermore that the map $h \mapsto \xi'_{h,i,j}$ is a Freiman homomorphism on $H'$.
\item For all $h \in H$, $\F_h$ is represented by $\F_* \cup \F'_h$
\item \textup{(Independence property)} For almost all additive quadruples $(h_1,h_2,h_3,h_4)$ in $H$,
$$\F_{*,\ind} \uplus \biguplus_{i=1}^4 \F'_{h_i,\ind} \uplus \biguplus_{i=1}^3 \F'_{h_i,\lin}$$
is independent.
\end{itemize}
\end{lemma}
\begin{proof} Write $\F_h = (\xi_{h,i,j})_{1 \leq i \leq s-1; 1 \leq j \leq D_i}$. For each $1 \leq i \leq s-1$ in turn, apply Lemma \ref{teenager} to the collections $(\xi_{h,i,1},\ldots,\xi_{h,i,D_i})_{h \in H}$ and $\eps = O(N^{-i})$, refining $H$ once for each $i$. The claim then follows by relabeling.
\end{proof}
To apply this lemma to families of nilcharacters, we will need two additional lemmas.
\begin{lemma}[Change of basis]\label{basis-change} Suppose that $\chi \in \Xi^{(s-1,r_*)}_\DR([N])$ is a degree-rank $(s-1,r_*)$ nilcharacter with a total frequency representation $(\vec D, \F, \eta)$, and suppose that $\F$ is represented by another horizontal frequency vector $\F'$ with a dimension vector $\vec D'$. Then there exists a vertical frequency $\eta': G^{\vec D'}_{s-1} \to \R$ such that $\chi$ has a total frequency representation $(\vec D', \F', \eta')$.
\end{lemma}
\begin{proof} By hypothesis, each element $\xi_{i,j}$ of $\F$ can be expressed as a standard linear combination $\xi_{i,j} = \sum_{j'=1}^{D'_i} c_{i,j,j'} \xi'_{i,j'}$ of elements $\xi'_{i,j'}$ of $\F'$ of the same degree, where $c_{i,j,j'} \in \Z$.
Now let $\psi: G^{\vec D'} \to G^{\vec D}$ be the unique filtered homomorphism that maps $e'_{i,j'}$ to $\prod_{j=1}^{D_i} e_{i,j}^{c_{i,j,j'}}$ (this can be viewed as an ``adjoint'' of the representation of $\F$ by $\F'$). By hypothesis, $\chi$ has a representation $\chi(n) = F( \orbit(n), \orbit_0(n))$ of $\chi$ with
$$
\Taylor_i(\orbit) = \pi_{\Horiz_i(G/\Gamma)}\left(\phi( \prod_{j=1}^{D_i} e_{i,j}^{\xi_{i,j}} )\right)
$$
for some filtered homomorphism $\phi: G^{\vec D} \to G$. A brief calculation shows that the right-hand side can also be expressed as
$$ \pi_{\Horiz_i(G/\Gamma)}\left(\phi \circ \psi( \prod_{j=1}^{D'_i} (e'_{i,j})^{\xi'_{i,j}} )\right).$$
As $\phi \circ \psi: G^{\vec D'} \to G$ is a filtered homomorphism, and $\eta \circ \psi: G^{\vec D'}_{(s-1,r_*)} \to \R$ is a vertical frequency, we obtain the claim.
\end{proof}
\begin{lemma}\label{discard} Let $\F$ be a horizontal frequency vector of dimension $\vec D$ of the form
$$ \F = \F_{\rat} \uplus \F_{\sml} \uplus \F'$$
where $\F_\rat$ is rational and $\F_\sml$ is small, and $\F'$ has dimension $\vec D'$. Suppose that $\chi \in \Xi^{(s-1,r_*)}_\DR([N])$ is a nilcharacter with a total frequency representation $(\vec D, \F, \eta)$. Then there exists a vertical frequency $\eta': G^{\vec D'}_{s-1} \to \R$ such that $\chi$ has total frequency $(\vec D', \F'/M, \eta')$ for some standard integer $M \geq 1$.
\end{lemma}
\emph{Remark.} This lemma crucially relies on the hypothesis $s \geq 3$, as it makes the (degree $1$) contributions of rational and small frequencies to be of lower order. Because of this, the inverse conjecture for $s > 2$ is in a very slight way a little bit simpler than the $s \leq 2$ theory, though it is of course more complicated in many other ways.
\begin{proof} By induction we may assume that $\F$ is formed from $\F'$ by adding a single frequency $\xi_{i_0,D_{i_0}}$, which is either rational or small.
Let us first suppose that we are adding a single frequency which is not just rational, but is in fact an integer. Then if $\chi(n) = F(g(n)\ultra \Gamma, g_0(n) \ultra \Gamma_0)$ is a nilcharacter with a total frequency representation $(\vec D,\F,\eta)$, then we have a filtered homomorphism $\phi: G^{\vec D}/\Gamma^{\vec D} \to G/\Gamma$ such that
$$
g_i = \prod_{j=1}^{D_i} \phi(e_{i,j})^{\xi_{i,j}} \hbox{ mod } G_{(i,1)}
$$
for all $1 \leq i \leq s_*-1$, where $g_i$ are the Taylor coefficients of $g$. Specialising to the degree $i_0$ and using the integer nature of $\xi_{i_0,D_{i_0}}$, we have
$$ g_{i_0} = g'_{i_0} \gamma_{i_0}$$
where $\gamma_{i_0}$ is an element of $\Gamma_{i_0}$, and
$$g'_i = \prod_{j=1}^{D_i-1} \phi(e_{i,j})^{\xi_{i,j}} \hbox{ mod } G_{(i,1)}.$$
From this and the Baker-Campbell-Hausdorff formula \eqref{bch}, we can write $g(n) = g'(n) \gamma_{i_0}^{\binom{n}{i_0}}$, where $g'$ is a polynomial sequence with a horizontal frequency representation $(\vec D', \phi', \F')$, where $\vec D'$ is $\vec D$ with $D_{i_0}$ decremented by one, and $\phi'$ is the restriction of $\phi$ to the subnilmanifold $G^{\vec D'}/\Gamma^{\vec D'}$. Since $g(n) \ultra \Gamma = g'(n) \ultra \Gamma$, we see that $\chi$ has a total frequency representation $(\vec D', \F', \eta')$, where $\eta'$ is the restriction of $\eta: G^{\vec D}_{(s-1,r_*)} \to \R$ to $G^{\vec D'}_{(s-1,r_*)}$. This gives the claim in this case (with $M=1$).
Now suppose that $\xi_{i_0,D_{i_0}}$ is merely rational rather than integer. Then we can argue as before, except that now $\gamma_{i_0}$ is a rational element of $G_{i_0}$, so that $\gamma_{i_0}^m \in \Gamma_{i_0}$ for some standard positive integer $m$. As such, there exists a standard positive integer $q$ such that $\gamma_{i_0}^{\binom{n}{i_0}} \mod \ultra \Gamma$ is periodic with period $q$. As a consequence, there exists a bounded index subgroup $\Gamma'$ of $\Gamma$ such that the point
$$ g'(n) \gamma_{i_0}^{\binom{n}{i_0}} \mod \ultra \Gamma$$
in $G/\Gamma$ can be expressed as a Lipschitz function of
$$ g'(n) \mod \ultra \Gamma'$$
and of the quantity $n/q \mod 1$. Repeating the previous arguments, we thus obtain a total frequency representation $(\vec D', \tilde \F', \eta')$ for some $\eta'$, and some $\tilde \F'$ whose coefficients are rational combinations of those of $\F'$; note that the $n/q$ dependence can be easily absorbed into the lower order term $G_0/\Gamma_0$ since $s \geq 3$. The claim then follows from Lemma \ref{basis-change}.
Finally, suppose that $\xi_{i_0,D_{i_0}}$ is small rather than rational. Then we can write
$$ g_{i_0} = c_{i_0} g'_{i_0}$$
where $g'_{i_0}$ is as before, and $c_{i_0} \in G_{i_0}$ is at a distance $O(N^{-i_0})$ from the origin. We can thus write
$$ g(n) = c_{i_0}^{\binom{n}{i_0}} g'(n)$$
where $g'$ is a polynomial sequence with horizontal frequency representation \[ (\vec D', \phi',\F').\] On $[N]$, the sequence $c_{i_0}^{\binom{n}{i_0}}$ is can be expressed as a bounded Lipschitz function of $n/2N \hbox{ mod } 1$. As a consequence, we can thus write $\chi$ in the form
$$ \chi(n) = F'( g'(n) \ultra \Gamma, g_0(n) \ultra \Gamma_0, n/2N \hbox{ mod } 1 )$$
for some $F' \in \Lip(\ultra( G/\Gamma \times G_0/\Gamma_0 \times \T ))$. As $s \geq 3$, the final term $\T$ can be absorbed into the degree-rank $\leq (s-1,r_*-1)$ nilmanifold $G_0/\Gamma_0$, and the claim follows (with $M=1$).
\end{proof}
Finally, we can state the main result of this section.
\begin{lemma}[Sunflower lemma]\label{sunflower} Let $H$ be a dense subset of $[[N]]$, and let $(\chi_h)_{h \in H}$ be a family of nilcharacters $\chi_h \in \Xi^{(s-1,r_*)}_\DR([N])$ depending in a limit fashion on $H$. Then we can find
\begin{enumerate}
\item A dense subset $H'$ of $H$;
\item Dimension vectors $\vec D_*$ and $\vec D' = \vec D'_\lin + \vec D'_\ind$, which we write as $\vec D_* = (D_{*,i})_{i=1}^{s-1}$, $\vec D' = (D'_{i})_{i=1}^{s-1}$, $\vec D'_\lin = (D'_{\lin,i})_{i=1}^{s-1}$, $\vec D'_\ind = (D'_{\ind,i})_{i=1}^{s-1}$;
\item A \emph{core horizontal frequency vector} $\F_* = (\xi_{*,i,j})_{1 \leq i \leq d; 1 \leq j \leq D_{*,i}}$;
\item A \emph{petal horizontal frequency vector} $\F'_h = (\xi'_{h,i,j})_{1 \leq i \leq d; 1 \leq j \leq D'_i}$, which is partitioned as $\F'_h = \F'_{h,\lin} \uplus \F'_{h,\ind}$, which is a limit function of $h$, where $\F'_{h,\lin}$, $\F'_{h,\ind}$ have dimensions $\vec D'_\lin$, $\vec D'_\ind$ respectively;
\item A vertical frequency $\eta: G^{\vec D_* + \vec D'}_{(s-1,r_*)} \to \R$ with dimension vector $\vec D_* + \vec D'$
\end{enumerate}
which obey the following properties:
\begin{enumerate}
\item \textup{($\F'_{h,\lin}$ is bracket-linear)} For every $1 \leq i \leq d$ and $1 \leq j \leq D'_{i,\lin}$, there exists $\alpha_{i,j} \in \ultra\T$ and $\beta_{i,j} \in \ultra \R$ such that
\begin{equation}\label{xih-def}
\xi'_{h,i,j} = \{ \alpha_{i,j} h \} \beta_{i,j} \mod 1
\end{equation}
for all $h \in H'$, and furthermore that the map $h \mapsto \xi'_{h,i,j}$ is a Freiman homomorphism on $H'$.
\item \textup{(Independence)} For almost all additive quadruples $(h_1,h_2,h_3,h_4)$ in $H$,
$$\F_{*,\ind} \uplus \biguplus_{i=1}^4 \F'_{h_i,\ind} \uplus \biguplus_{i=1}^3 \F'_{h_i,\lin}$$
is independent.
\item \textup{(Representation)} For all $h \in H'$, $\chi_h$ has a total frequency representation $( \vec D_* + \vec D', \F_* \cup \F'_h, \eta )$.
\end{enumerate}
\end{lemma}
\begin{proof} Each $\chi_h$ thus has a total frequency representation $(\vec D_h, \F_h, \eta_h)$. The space of representations is a $\sigma$-limit set, so by Lemma \ref{int-select} we may assume that $(\vec D_h, \F_h, \eta_h)$ depends in a limit fashion on $h$.
The number of possible dimension vectors is countable. Applying Lemma \ref{dense-dich}, and passing from $H$ to a dense subset, we may assume that $\vec D = \vec D_h$ is independent of $h$.
We then apply Lemma \ref{sunflower-basic} to the $(\F_h)_{h \in H}$, obtaining a dense subset $H'$ of $H$, dimension vectors $\vec D_* = \vec D_{*,\ind} + \vec D_{*,\rat}$ and $\vec D' = \vec D'_\lin + \vec D'_\ind + \vec D'_\sml$, a core horizontal frequency vector $\F_* = \F_{*,\ind} \uplus \F_{*,\rat}$, and petal horizontal frequency vectors $\F'_h = \F'_{h,\lin} \uplus \F'_{h,\ind} \uplus \F'_{h,\sml}$ for each $h \in H'$ with the stated properties.
Applying Lemma \ref{basis-change}, we see that for each $h \in H'$, $\chi_h$ has a total frequency representation
$$ (\vec D_* + \vec D', \F_* \uplus \F'_h, \eta'_h )$$
for some vertical frequency $\eta'_h$. Applying Lemma \ref{discard}, we conclude that $\chi_h$ has a total frequency representation
$$ (\vec D_{*,\ind} + \vec D'_\lin + \vec D'_\ind, \F_{*,\ind} \uplus \F'_{h,\lin} \uplus \F'_{h,\ind}, \eta''_h )$$
for some vertical frequency $\eta'_h$. The number of vertical frequencies $\eta''_h$ is countable, so by Lemma \ref{dense-dich} we may assume that $\eta = \eta''_h$ is also independent of $h$. The claim then follows.
\end{proof}
\section{Obtaining bracket-linear behaviour}\label{linear-sec}
We return now to the task of proving Theorem \ref{linear-induct}. To recall the situation thus far, we are given a two-dimensional nilcharacter $\chi \in \Xi^{(1,s-1)}_\MD(\ultra \Z^2)$ and a family of degree-rank $(s-1,r_*)$ nilcharacters $(\chi_h)_{h \in H}$ depending in a limit fashion on a parameter $h$ in a dense subset $H$ of $[[N]]$, with the property that there is a function $f \in L^\infty[N]$ such that $\chi(h,\cdot) \otimes \chi_h$ $(s-2)$-correlates with $f$ for all $h \in H$. Using Proposition \ref{gcs-prop} to eliminate $f$ and $\chi$, and refining $H$ to a dense subset if necessary, we conclude that the nilcharacter \eqref{gowers-cs-arg} is $(s-2)$-biased for many additive quadruples $h_1+h_2=h_3+h_4$ in $H$. We make the simple but important remark that this conclusion is ``hereditary'' in the sense that it continues to hold if we replace $H$ with an arbitrary dense subset $H'$ of $H$, since the hypothesis of Proposition \ref{gcs-prop} clearly restricts from $H$ to $H'$ in this fashion.
Next, we apply Lemma \ref{sunflower} to obtain a dense refinement $H'$ on $H$ for which the $\chi_h$ have a frequency representation involving various types of frequencies: a core set of frequencies $\F_*$, a bracket-linear family $(\F'_{h,\lin})_{h \in H'}$ of petal frequencies and an independent family $(\F'_{h,\ind})_{h \in H'}$ of petal frequencies.
<<<<<<< .mine
The main result of this section uses the bias of \eqref{gowers-cs-arg}, combined with the quantitative equidistribution theory on nilmanifolds (as reviewed in Appendix \ref{equiapp}) to obtain an important milestone towards establishing Theorem \ref{linear-induct}, namely that the independent petal frequencies $\F'_{h,\ind}$ do not actually have any influence on the top-order behaviour of the nilcharacters $\chi_h$, and that the bracket-linear frequencies only influence this top-order behaviour in a linear fashion. For this, we use an argument of Furstenberg and Weiss \cite{fw-char}, also used in the predecessor \cite{u4-inverse} to this paper. See also \cite{gtz-announce} for another exposition of this argument.
=======
The main result of this section uses the bias of \eqref{gowers-cs-arg}, combined with the quantitative equidistribution theory on nilmanifolds (as reviewed in Appendix \ref{equiapp}) to obtain an important milestone towards establishing Theorem \ref{linear-induct}, namely that the independent petal frequencies $\F'_{h,\ind}$ do not actually have any influence on the top-order behaviour of the nilcharacters $\chi_h$, and that the bracket-linear frequencies only influence this top-order behaviour in a linear fashion. For this, we use an argument of Furstenberg and Weiss \cite{fw-char} that was also used in the predecessor \cite{u4-inverse} to this paper. See \cite{gtz-announce} for another, somewhat simplified, exposition of this argument.
>>>>>>> .r207
We begin by formally stating the result we will prove in this section.
\begin{theorem}[No petal-petal or regular terms]\label{slang-petal} Let $f,H,\chi,(\chi_h)_{h \in H}$ be as in Theorem \ref{linear-induct} and let $H', \vec D_*, \vec D', \vec D'_\lin, \vec D'_\ind, \F_*, \F'_h, \F'_{h,\lin}, \F'_{h,\ind}, \eta$ be as in Lemma \ref{sunflower}. Let $w \in G^{\vec D_* + \vec D'}$ be an $r_*-1$-fold commutator of $e_{i_1,j_1},\ldots,e_{i_{r_*},j_{r_*}}$, where $1 \leq i_1,\ldots,i_{r_*} \leq s-1$, $i_1+\ldots+i_{r_*}=s-1$, and $1 \leq j_l \leq D_{*,i_l} + D'_{i_l}$ for all $l$ with $1 \leq l \leq r_*$.
\begin{enumerate}
\item \textup{(No petal-petal terms)} If $j_l > D_{*,i_l}$ for at least two values of $l$, then $\eta(w)=0$.
\item \textup{(No regular terms)} If $j_l > D_{*,i_l} + D'_{\lin,i_l}$ for at least one value of $l$, then $\eta(w)=0$.
\item \textup{(No petal-petal terms)} If $j_l > D_{*,i_l}$ for at least two values of $l$ then $\eta(w)=0$.
\item \textup{(No regular terms)} If $j_l > D_{*,i_l} + D'_{\lin,i_l}$ for at least one value of $l$ then $\eta(w)=0$.
\end{enumerate}
\end{theorem}
The remainder of this section is devoted to the proof of Theorem \ref{slang-petal}.
Let the notation and assumptions be as in the above theorem.
From Proposition \ref{gcs-prop} we know that, for many additive quadruples $(h_1,h_2,h_3,h_4)$ in $H'$, the sequence \eqref{gowers-cs-arg} is $(s-2)$-biased. Also, from Lemma \ref{sunflower}, we see that for almost all of these quadruples, the horizontal frequency vectors
\begin{equation}\label{jinnai-1}
\F_{*,\ind} \uplus \biguplus_{i=1}^4 \F_{h_i,\ind} \uplus \biguplus_{i=a,b,c} \F_{h_i,\lin}
\end{equation}
are independent for all distinct $a,b,c \in \{1,2,3,4\}$. We may therefore find an additive quadruple $(h_1,h_2,h_3,h_4)$ for which \eqref{gowers-cs-arg} is $(s-2)$-biased, and for which \eqref{jinnai-1} is independent for all choices of distinct $a,b,c \in \{1,2,3,4\}$.
Fix $(h_1,h_2,h_3,h_4)$ with these properties. We convert the above information to a non-equidistribution result concerning a polynomial orbit.
For each $i=1,2,3,4$, we see from Lemma \ref{sunflower} that $\chi_{h_i}$ has a total frequency representation
$$ ( \vec D_* + \vec D', \F_* \uplus \F'_{h_i}, \eta ).$$
We write
$$ \F_* \uplus \F'_{h_i} = ( \xi_{h_i,j,k} )_{1 \leq j \leq s-1; 1 \leq k \leq D_j},$$
where
$$ D_j = D_{*,j} + D'_j;$$
thus the frequencies associated to $\F_{*}$, $\F'_{h_i,\ind}$, $\F'_{h_i,\lin}$ correspond to the ranges $1 \leq k \leq D_{*,j}$, $D_{*,j} < k \leq D_{*,j}+D'_{\ind,j}$, and $D_{*,j} + D'_{\ind,j} < k \leq D_j$ respectively.
As \eqref{gowers-cs-arg} is $(s-2)$-biased, we conclude that
\begin{equation}\label{expect}
|\E_{n \in [N]} \chi_{h_1}(n) \otimes \chi_{h_2}(n+h_1-h_4) \otimes \overline{\chi_{h_3}}(n) \otimes \overline{\chi_{h_4}}(n+h_1-h_4) \psi_{h_1,h_2,h_3,h_4}(n)| \gg 1
\end{equation}
for some degree $\leq (s-2)$ nilsequence $\psi_{h_1,h_2,h_3,h_4}$, where $\chi_h$ is defined to be zero outside of $[N]$. As any cutoff to an interval can be approximated to arbitrary standard accuracy by a degree $1$ nilsequence, and $s \geq 3$, we see that the same claim holds if $\chi_h$ is instead extended to be a nilsequence on all of $\ultra \Z$.
From Definition \ref{nilch-def} and the total frequency representation of the $\chi_{h_i}$, we can rewrite the sequence inside the expectation of \eqref{expect} as a degree-rank $\leq (s-1,r_*)$ nilsequence $n \mapsto F(\orbit(n))$. Here $G/\Gamma$ is the product nilmanifold\footnote{Unfortunately, there will be several types of subscripts on nilpotent Lie groups $G$ in this argument. Firstly one has the factor groups $G_{(i)}$. Then one also has the degree filtration groups $G_d$ and the degree-rank filtration groups $G_{(d,r)}$ of $G$ (and also the analogous subgroups $(G_{(i)})_d$, $(G_{(i)})_{(d,r)}$ of the factor groups $G_{(i)}$), as well as the free nilpotent groups $G^{\vec D} = G^{\vec D}_{(s-1,r_*)}$. Finally, a Ratner subgroup $G_P$ of $G$ will also make an appearance later. We hope that these notations can be kept separate from each other.}
$$ G/\Gamma := \left(\prod_{i=1}^4 G_{(i)}/\Gamma_{(i)}\right) \times G_{(0)}/\Gamma_{(0)}$$
for some filtered nilmanifold $G_{(0)}/\Gamma_{(0)}$ of degree-rank $<(s-1,r_*-1)$ and filtered nilmanifolds $G_{(i)}/\Gamma_{(i)}$ of degree-rank $\leq(s-1,r_*)$ for $i=1,2,3,4$. The orbit $\orbit$ is defined by
$$\orbit = (\orbit_1,\orbit_2,\orbit_3,\orbit_4,\orbit_0) \in \ultra \poly(\Z_\N \to (G/\Gamma)_\N)$$
where, for each $i,j$ with $1 \leq i \leq 4$ and $1 \leq j \leq s-1$ we have
\begin{equation}\label{gij-spin}
\Taylor_j(\orbit_{(i)}) = \pi_{\Horiz_j(G_{(i)}/\Gamma_{(i)})}\left(\phi_{(i)}(\prod_{1 \leq k \leq D_j} e_{j,k}^{\xi_{h_i,j,k}})\right)
\end{equation}
where $\vec D := (D_1,\ldots,D_{s-1})$, $\phi_{(i)}: G^{\vec D}/\Gamma^{\vec D} \to G_{(i)}/\Gamma_{(i)}$ is a filtered homomorphism and $\pi_{\Horiz_j(G_{(i)}/\Gamma_{(i)})}: (G_{(i)})_j \to \Horiz_j(G_{(i)}/\Gamma_{(i)})$ is the projection to the $j^{\operatorname{th}}$ horizontal torus. Finally $F \in \Lip(\ultra(G/\Gamma))$ is defined by
\begin{align}\nonumber
F( \phi_{(1)} & (t_{(1)}) x_{(1)}, \ldots, \phi_{(4)}(t_{(4)}) x_{(4)}, y ) = \\
& e( (\eta(t_{(1)})+\eta(t_{(2)})-\eta(t_{(3)})-\eta(t_{(4)})) ) F(x_{(1)},\ldots,x_{(4)},y)\label{fallow}
\end{align}
for all $(x_{(1)},\ldots,x_{(4)},y) \in G/\Gamma$ and $t_{(1)},\ldots,t_{(4)} \in G^{\vec D}_{(s-1,r_*)}$. (Note that the shifts by $h_1-h_4$ in \eqref{expect} do not affect the Taylor coefficients of $\orbit_{(i)}$, thanks to the remarks following Definition \ref{horton}.)
By hypothesis, we have
$$ |\E_{n \in [N]} F( \orbit(n) )| \gg 1.$$
Applying Theorem \ref{ratt}, we conclude that
\begin{equation}\label{gapp}
|\int_{G_P / \Gamma_P} F(\eps x)\ d\mu(x)| \gg 1
\end{equation}
for some bounded $\eps \in G$ and some rational subgroup $G_P$ of $G$ with the property that
\begin{equation}\label{soo}
\pi_{\Horiz_j(G)}(G_P \cap G_{(i)}) \geq \Xi_j^\perp
\end{equation}
for all $1 \leq j \leq s-1$, where
$$ \Xi_j^\perp := \{ x \in \Horiz_j(G) : \xi_j(x) = 0 \hbox{ for all } \xi_j \in \Xi_j \}$$
and $\Xi_j \leq \widehat{\Horiz_j(G/\Gamma)}$ is the group of all (standard) continuous homomorphisms $\xi_j: \Horiz_j(G/\Gamma) \to \T$ such that
$$ \xi_j( \Taylor_j(\orbit) ) = O( N^{-j} ).$$
From \eqref{fallow} and \eqref{gapp} we conclude the following lemma.
\begin{lemma}\label{gapp-vanish} The group $G_P \cap ((G_{(1)})_{(s-1,r_*)} \times \{\id\} \times \{\id\} \times \{\id\} \times \{\id\})$ is annihilated by $\eta$.
\end{lemma}
\begin{proof} Let $g = (g_{(1)},\id,\id,\id,\id)$ lie in the indicated group. Then $g$ is central, and so from the invariance of Haar measure we have
$$
\int_{G_P / \Gamma_P} F(\eps x)\ d\mu(x) = \int_{G_P / \Gamma_P} F(g \eps x)\ d\mu(x).$$
On the other hand, from \eqref{fallow} we have
$$ \int_{G_P / \Gamma_P} F(g \eps x)\ d\mu(x) = e(\eta(g)) \int_{G_P / \Gamma_P} F(\eps x)\ d\mu(x).$$
Comparing these relationships with \eqref{gapp} we obtain the claim.
\end{proof}
We now analyse the group $G_P$ further. For each $1 \leq j \leq s-1$, let $V_{123,j}$ denote the subgroup of $\Horiz_j(G_{(1)}) \times \Horiz_j(G_{(2)}) \times \Horiz_j(G_{(3)})$ generated by the diagonal elements
$$ (\phi_{(1)}(e_{j,k}), \phi_{(2)}(e_{j,k}), \phi_{(3)}(e_{j,k}))$$
for $1 \leq k \leq D_{*,j}$, and by the elements
$$ (\phi_{(1)}(e_{j,k}), \id, \id), (\id, \phi_{(2)}(e_{j,k}), \id), (\id, \id, \phi_{(3)}(e_{j,k}))$$
for $D_{*,j} < k \leq D_j$. We define the subgroup $V_{124,j}$ of $\Horiz_j(G_{(1)}) \times \Horiz_j(G_{(2)}) \times \Horiz_j(G_{(4)})$ similarly by replacing $(3)$ with $(4)$ throughout.
\begin{lemma}[Components of $G_P$]\label{gp-comp} Let $1 \leq j \leq s-1$. Then the projection of $G_P \cap G_{j}$ to $\Horiz_j(G_{(1)}) \times \Horiz_j(G_{(2)}) \times \Horiz_j(G_{(3)})$ contains $V_{123,j}$. Similarly, the projection to $\Horiz_j(G_{(1)}) \times \Horiz_j(G_{(2)}) \times \Horiz_j(G_{(4)})$ contains $V_{124,j}$.
\end{lemma}
\begin{proof} We shall just prove the first claim; the second claim is similar (but uses $\{a,b,c\} = \{1,2,4\}$ instead of $\{a,b,c\}=\{1,2,3\}$).
Suppose the claim failed for some $j$. Using \eqref{soo} and duality, we conclude that there exists a $\xi_j \in \Xi_j$ which annihilates the kernel of the projection to $\Horiz_j(G_{(1)}) \times \Horiz_j(G_{(2)}) \times \Horiz_j(G_{(3)})$, and which is non-trivial on $V_{123,j}$. As $\xi_j$ annihilates the kernel of the projection to $\Horiz_j(G_{(1)}) \times \Horiz_j(G_{(2)}) \times \Horiz_j(G_{(3)})$, we have a decomposition of the form
$$ \xi_j(x_{(1)},x_{(2)},x_{(3)},x_{(4)},x_{(0)}) = \xi_{(1),j}(x_{(1)}) + \xi_{(2),j}(x_{(2)}) + \xi_{(3),j}(x_{(3)})$$
for $x_{(i)} \in \Horiz_j(G_{(i)})$ for $i=1,2,3,4,0$, where $\xi_{(i),j}: \Horiz_j(G_{(i)}) \to \R$ for $i=1,2,3$ are characters.
By definition of $\Xi_j$, we conclude that
$$ \xi_{(1),j}( \Taylor_j(\orbit_{(1)}) ) + \xi_{(2),j}( \Taylor_j(\orbit_{(2)}) ) + \xi_{(3),j}( \Taylor_j(\orbit_{(3)}) ) = O(N^{-j}).$$
However, from \eqref{gij-spin} we have
\begin{equation}\label{star}
\xi_{(i),j}(\Taylor_j(\orbit_{(i)})) = \sum_{k=1}^{D_j} c_{(i),j,k} \xi_{h_i,j,k}
\end{equation}
where the $c_{(i),j,k}$ are standard integers, defined by the formula
\begin{equation}\label{cdef}
c_{(i),j,k} := \xi_{(i),j}(\phi_{(i)}(e_{j,k})).
\end{equation}
From the independence of \eqref{jinnai-1} with $\{a,b,c\}=\{1,2,3\}$, we conclude that the $c_{(i),j,k}$ all vanish for $i=1,2,3$ and $D_{*,j} < k \leq D_j$, and that the sum $c_{(1),j,k}+c_{(2),j,k}+c_{(3),j,k}$ vanishes for $1 \leq k \leq D_{*,j}$. But this forces $\xi_j$ to vanish on $V_{123,j}$, contradiction.
\end{proof}
We now take commutators in the spirit of an argument of Furstenberg and Weiss \cite{fw-char} (see also \cite{hrush,ribet} for similar arguments in completely different settings) to conclude the following result which roughly speaking asserts that all ``petal-petal interactions'' are trivial.
\begin{corollary}[Furstenberg-Weiss commutator argument]\label{fw} Let $w$ be an $r_*-1$-fold iterated commutator of generators $e_{j_1,k_1},\ldots,e_{j_{r_*},k_{r_*}}$ with $1 \leq j_l \leq s-1$, $1 \leq k_l \leq D_l$ for $l=1,\ldots,r_*$ and $j_1+\ldots+j_{r_*} = s-1$ \textup{(}thus $w$ has ``degree-rank $(s-1,r_*)$'' in some sense\textup{)}. Suppose that at least two of the generators, say $e_{j_1,k_1}, e_{j_2,k_2}$, are ``petal'' generators in the sense that $k_1 > D_{*,j_1}$ and $k_2 > D_{*,j_2}$. Then $(\phi_{(1)}(w),\id,\id,\id,\id) \in G_P$.
\end{corollary}
\begin{proof}
For $e_{j_1,k_1}$, we may invoke Lemma \ref{gp-comp} and find an element $g_{j_1,k_1}$ of $G_P \cap G_{j_1}$ for which the coordinates $1,2,3$ are equal (modulo projection to \[ \Horiz_{j_1}(G_{(1)}) \times \Horiz_{j_1}(G_{(2)}) \times \Horiz_{j_1}(G_{(3)}))\] to $(\phi_1(e_{j_1,k_1}),\id,\id)$. Similarly, we may find an element $g'_{j_2,k_2}$ of $G_P \cap G_{j_2}$ for which the coordinates $1,2,4$ are equal (modulo projection to \[ \Horiz_{j_2}(G_{(1)}) \times \Horiz_{j_2}(G_{(2)}) \times \Horiz_{j_2}(G_{(4)}))\] to $(\phi_1(e_{j_2,k_2}),\id,\id)$. Finally, for all of the other $e_{j,k}$, we can find elements $g''_{j,k}$ of $G_P \cap G_{j}$ for which the first coordinate is equal (modulo projection to $\Horiz_j(G_{(1)})$) to $\phi_{(1)}(e_{j,k})$. If one then takes iterated commutators of the $g_{j_1,k_1}, g'_{j_2,k_2}, g''_{j,k}$ in the order indicated by $w$, we see (using the filtration property, the homomorphism property of $\phi_{(1)}$, and the fact that the $G_i/\Gamma_i$ have degree $\leq (s-1,r_*)$ for $i=1,2,3,4$ and degree $<(s-1,r_*-1)$ for $i=0$) that we obtain the element $(\phi_{(1)}(w),\id,\id,\id,\id)$. Since the iterated commutator of elements in $G_P$ stays in $G_P$, the claim follows.
\end{proof}
From Lemma \ref{gapp-vanish} and Corollary \ref{fw} we immediately obtain the first part (i) of Theorem \ref{slang-petal}. We now turn to the second part of the theorem. For this, we need two further variants of Lemma \ref{gp-comp}. For any $1 \leq j \leq s-1$, let $V_{\ind,j}$ be the subspace of $\Horiz_j(G_{(1)}) \times \Horiz_j(G_{(2)}) \times \Horiz_j(G_{(3)}) \times \Horiz_j(G_{(4)})$ generated by the elements
$$ (\phi_{(1)}(e_{j,k}), \phi_{(2)}(e_{j,k}), \phi_{(3)}(e_{j,k}),\phi_{(4)}(e_{j,k}))$$
for $1 \leq k \leq D_{*,j}$ and the elements
$$ (\phi_{(1)}(e_{j,k}), \id, \id,\id), (\id, \phi_{(2)}(e_{j,k}), \id,\id), (\id, \id, \phi_{(3)}(e_{j,k}),\id),
(\id, \id,\id, \phi_{(4)}(e_{j,k}))
$$
for $D_{*,j} < k \leq D_{*,j}+D'_{\ind,j}$.
\begin{lemma}[Components of $G_P$, II]\label{gp-comp2} Let $1 \leq j \leq s-1$. Then the projection of $G_P \cap G_{j}$ to $\Horiz_j(G_{(1)}) \times \Horiz_j(G_{(2)}) \times \Horiz_j(G_{(3)}) \times \Horiz_j(G_{(4)})$ contains $V_{\ind,j}$.
\end{lemma}
\begin{proof}
Suppose the claim failed for some $j$. Using \eqref{soo} and duality, we conclude that there exists a $\xi_j \in \Xi_j$ which annihilates the kernel of the projection to $\Horiz_j(G_{(1)}) \times \Horiz_j(G_{(2)}) \times \Horiz_j(G_{(3)}) \times \Horiz_j(G_{(4)})$, and which is non-trivial on $V_{\ind,j}$. In particular, we have a decomposition of the form
\begin{equation}\label{xij}
\xi_j(x_{(1)},x_{(2)},x_{(3)},x_{(4)},x_{(0)}) = \sum_{i=1}^4 \xi_{(i),j}(x_{(i)})
\end{equation}
for $x_{(i)} \in \Horiz_j(G_{(i)})$ for $i=1,2,3,4,0$, where $\xi_{(i),j}: \Horiz_j(G_{(i)}) \to \R$ for $i=1,2,3,4$ are characters.
By definition of $\Xi_j$, we conclude that
$$\sum_{i=1}^4 \xi_{(i),j}( \Taylor_j(\orbit_{(i)}) )
= O(N^{-j}).$$
Inserting \eqref{star}, we conclude that
\begin{equation}\label{star2}
\sum_{k=1}^{D_j} \sum_{i=1}^4 c_{(i),j,k} \xi_{h_i,j,k} = O(N^{-j}).
\end{equation}
The left-hand side is an integer linear combination of the degree $j$ frequencies in
$$ \F_{*,\ind} \uplus \biguplus_{i=1}^4 \F_{h_i,\ind} \uplus \biguplus_{i=1}^4 \F_{h_i,\lin}.$$
Using the Freiman homomorphism property from Lemma \ref{sunflower} we can eliminate the role of $\F_{h_4,\lin}$, leaving only
$$ \F_{*,\ind} \uplus \biguplus_{i=1}^4 \F_{h_i,\ind} \uplus \biguplus_{i=1}^3 \F_{h_i,\lin}.$$
But this is just \eqref{jinnai-1} for $\{a,b,c\}=\{1,2,3\}$. We conclude that the coefficients of the left-hand side of
\eqref{star2} in this basis vanish, which in terms of the original coefficients $c_{(i),j,k}$ means that
$$ \sum_{i=1}^4 c_{(i),j,k}=0$$
for $1 \leq k \leq D_{*,j}$, and
$$ c_{(i),j,k} = 0$$
for $D_{*,j} < k \leq D_{*,j} + D'_{\ind,j}$. But this forces $\xi_j$ to vanish on $V_{\ind,j}$, a contradiction.
\end{proof}
We now apply the commutator argument to show that ``independent'' frequencies also ultimately have a trivial effect.
\begin{corollary}[Furstenberg-Weiss commutator argument, II]\label{fw2} Let $w$ be an $(r_*-1)$-fold iterated commutator of generators $e_{j_1,k_1},\ldots,e_{j_{r_*},k_{r_*}}$ with $1 \leq j_l \leq s-1$, $1 \leq k_l \leq D_l$ for $l=1,\ldots,r_*$ and $j_1+\ldots+j_{r_*} = s-1$. Suppose that at least one of the generators, say $e_{j_1,k_1}$, is an ``independent'' generator in the sense that $D_{*,j_1} < k_1 \leq D_{*,j_1} + D'_{\ind,j_1}$. Then $(\phi_{(1)}(w),\id,\id,\id,\id) \in G_P$.
\end{corollary}
\begin{proof} We may assume that $k_l \leq D_{*,j_l}$ for all $2 \leq l \leq r_*$, as the claim would follow from Corollary \ref{fw} otherwise.
For $e_{j_1,k_1}$, we may invoke Lemma \ref{gp-comp2} and find an element $g_{j_1,k_1}$ of $G_P \cap G_{j_1}$ for which the first $4$ coordinates are equal (modulo projection to $\Horiz_{j_1}(G_{(1)}) \times \Horiz_{j_1}(G_{(2)}) \times \Horiz_{j_1}(G_{(3)}) \times \Horiz_{j_1}(G_{(4)})$) is equal to $(\phi_{(1)}(e_{j_1,k_1}),\id,\id,\id)$. For the other $e_{j,k}$, we can find elements $g'_{j,k}$ of $G_P \cap G_{j}$ for which the first coordinate is equal (modulo projection to $\Horiz_{j}(G_{(1)})$) to $\phi_{(1)}(e_{j,k})$. Taking commutators of $g_{j_1,k_1}$ and $g'_{j,k}$ in the order indicated by $w$, we obtain the claim.
\end{proof}
Combining Corollary \ref{fw2} with Lemma \ref{gapp-vanish} we obtain the second part of Theorem \ref{slang-petal}.
\section{Building a nilobject}\label{multi-sec}
The aim of this section is to at last build an object coming from an $s$-step nilmanifold. Recall from the discussion in \S \ref{overview-sec} that this object will be a multidegree $(1,s-1)$-nilcharacter $\chi'(h,n)$, and that this completes the proof of Theorem \ref{linear-induct}. This in turn was used iteratively to prove Theorem \ref{linear-thm}, the heart of our whole paper. It will then remain to supply the \emph{symmetry argument}, which will take us from a 2-dimensional nilsequence to a 1-dimensional one; this will be accomplished in the next section.
Let $f,H,\chi,(\chi_h)_{h \in H}$ be as in Theorem \ref{linear-induct}. If we apply Lemma \ref{sunflower}, we obtain the following objects:
\begin{itemize}
\item A dense subset $H'$ of $H$;
\item Dimension vectors $\vec D_* = \vec D_{*,\ind} + \vec D_{*,\rat}$ and $\vec D' = \vec D'_\lin + \vec D'_\ind + \vec D'_\sml$, which we write as $\vec D_* = (D_{*,i})_{i=1}^{s-1}$, $\vec D_{*,\ind} = (D_{*,\ind,i})_{i=1}^{s-1}$, etc.;
\item A core horizontal frequency vector $\F_* = (\xi_{*,i,j})_{1 \leq i \leq s-1; 1 \leq j \leq D_{*,i}}$, which is partitioned as $\F_* = \F_{*,\ind} \uplus \F_{*,\rat}$, with the indicated dimension vectors $\vec D'_\ind, \vec D'_\rat$;
\item A petal horizontal frequency vector $\F'_h = (\xi'_{h,i,j})_{1 \leq i \leq s-1; 1 \leq j \leq D'_i}$, which is partitioned as $\F'_h = \F'_{h,\lin} \uplus \F'_{h,\ind} \uplus \F'_{h,\sml}$, which is a limit function of $h$ and with the indicated dimension vectors $\vec D'_\lin, \vec D'_\ind, \vec D'_\sml$;
\item Nilmanifolds $G_h/\Gamma_h$ and $G_{0,h}/\Gamma_{0,h}$ of degree-rank $\leq (s-1,r_*)$ and $\leq (s-1,r_*-1)$ respectively for each $h \in H'$, depending in a limit fashion on $h$;
\item Polynomial sequences $g_h, g_{0,h} \in \ultra \poly(\Z_\N \to (G_h)_\N)$ for each $h \in H'$, depending in a limit fashion on $h$;
\item Lipschitz functions $F_h \in \Lip(\ultra(G_h/\Gamma_h \times G_{0,h}/\Gamma_{0,h})\to \overline{S^{\omega}})$ for each $h \in H'$, depending in a limit fashion on $h$;
\item a filtered $\phi_h: G^{\vec D_* + \vec D'} \to G_h$ for each $h \in H'$, depending in a limit fashion on $h$; and
\item a character $\eta_h: G^{\vec D_* + \vec D'}_{(s-1,r_*)} \to \R$ for each $h \in H'$, depending in a limit fashion on $h$
\end{itemize}
that obey the following properties:
\begin{itemize}
\item For every $1 \leq i \leq d$ and $1 \leq j \leq D'_{i,\lin}$, there exists $\alpha_{i,j} \in \ultra\T$ and $\beta_{i,j} \in \ultra \R$ such that \eqref{xih-def} holds, and furthermore that the map $h \mapsto \xi'_{h,i,j}$ is a Freiman homomorphism on $H'$.
\item For almost all additive quadruples $(h_1,h_2,h_3,h_4)$ in $H$,
$$\F_{*,\ind} \uplus \biguplus_{i=1}^4 \F'_{h_i,\ind} \uplus \biguplus_{i=1}^3 \F'_{h_i,\lin}$$
is independent.
\item We have the representation
$$ \chi_h(n) = F_h( g_h(n) \ultra \Gamma_h, g_{0,h}(n) \ultra \Gamma_{0,h} )$$
for every $h \in H'$.
\item $\phi_h: G^{\vec D_* + \vec D'} \to G_h$ is a filtered homomorphism such that
\begin{equation}\label{phil}
F_h( \phi_h(t) x, x_0 ) = e( \eta_h(t) ) F_h(x,x_0)
\end{equation}
for all $t \in G^{\vec D_* + \vec D'}_{(s-1,r_*)}$, $x \in G_h/\Gamma_h$, and $x_0 \in G_{0,h}/\Gamma_{0,h}$;
\item One has the Taylor coefficients
\begin{equation}\label{thune}
\Taylor_i(g_h\Gamma_h) = \pi_{\Horiz_i(G_h/\Gamma_h)}(\phi_h( \prod_{j=1}^{D_{*,i}+D'_i} e_{i,j}^{\xi_{h,i,j}} ))
\end{equation}
for all $1 \leq i \leq s-1$.
\end{itemize}
There are only countably many nilmanifolds $G/\Gamma$ up to isomorphism, so by passing from $H'$ to a dense subset using Lemma \ref{dense-dich} we may assume that
$$ G_h/\Gamma_h = G/\Gamma \quad \mbox{and} \quad G_{0,h}/\Gamma_{0,h} = G_0/\Gamma_0$$
are independent of $h$. Similarly we may take $\eta_h = \eta$ and $\phi_h = \phi$ to be independent of $h$. From the Arzel\`a-Ascoli theorem, the space of possible $F_h$ is totally bounded, and so (shrinking $\eps$ slightly if necessary) we may also assume that $F_h = F$ is independent of $h$.
For $j$ with $1 \leq j \leq D_{*,i}$, since $\xi_{h,i,j}$ is independent of $h$, we can ensure that $\xi_{h,i,j} =\gamma_{i,j}$ is also independent of $h$. Meanwhile, for $D_{*,i} < j \leq D_{*,i}+D'_{i,\lin}$, from \eqref{xih-def} we may assume that $\xi_{h,i,j}$ takes the form
$$ \xi_{h,i,j} = \{ \alpha_{i,j} h \} \beta_{i,j} \mod 1$$
for some $\alpha_{i,j} \in \ultra\T$ and $\beta_{i,j} \in \ultra \R$. By passing to a dense subset of $H'$ using the pigeonhole principle, we may assume for each $i,j$, that $\{ \alpha_{i,j} h \}$ is contained in a subinterval $\ultra I_{i,j}$ around $\ultra 0$ of length at most $1/10$ (say).
We now wish to apply Theorem \ref{slang-petal} to obtain more convenient equivalent representatives (in $\Xi_{\DR}^{(s-1,r_*)}([N])$ ) $\tilde \chi_h$ for the nilcharacters $\chi_h$. Let $\tilde G$ be the free Lie group generated by the generators $\tilde e_{i,j}$ for $1 \leq i \leq s-1$ and $1 \leq j \leq D_{*_i} + D'_{\lin,i}$ subject to the following relations:
\begin{itemize}
\item Any $(r-1)$-fold iterated commutator of $\tilde e_{i_1,j_1},\ldots,\tilde e_{i_r,j_r}$ with $i_1+\ldots+i_r > s-1$ vanishes;
\item Any $(r-1)$-fold iterated commutator of $\tilde e_{i_1,j_1},\ldots,\tilde e_{i_r,j_r}$ with $i_1+\ldots+i_r = s-1$ and $r > r_*$ vanishes;
\item Any $(r-1)$-fold iterated commutator of $\tilde e_{i_1,j_1},\ldots,\tilde e_{i_r,j_r}$ in which $j_l > D_{*,i_l}$ for at least two values of $l$ vanishes.
\end{itemize}
We give this group a $\DR$-filtration $\tilde G_\DR$ by defining $\tilde G_{(d,r)}$ to be the group generated by the $(r'-1)$-fold iterated commutators of $\tilde e_{i_1,j_1},\ldots,\tilde e_{i_{r'},j_{r'}}$ with $i_1+\ldots+i_{r'} \geq d$ and $r' \geq r$. We then let $\tilde \Gamma$ be the discrete group generated by the $\tilde e_{i,j}$; $\tilde G/\tilde \Gamma$ is then a nilmanifold of degree-rank $\leq (s-1,r_*)$.
Let $G^*$ be the subgroup of $G^{\vec D_* + \vec D'}$ generated by $(r-1)$-fold iterated commutators $\tilde e_{i_1,j_1},\ldots,\tilde e_{i_r,j_r}$ with $i_1+\ldots+i_r = s-1$ in which $j_l > D_{*,i_l}$ for at least two values of $l$, or $j_l > D_{*,i_l} + D'_{\lin,i_l}$ for at least one value of $l$. Then $G^*$ is a subgroup of the central group $G^{\vec D_* + \vec D'}_{(s-1,r_*)}$ of $G^{\vec D_* + \vec D'}$, and $\tilde G$ is isomorphic to the quotient of $G^{\vec D_* + \vec D'}$ by $G^*$. We let $\tilde \phi: G^{\vec D_* + \vec D'} \to \tilde G$ denote the quotient map. From Theorem \ref{slang-petal}, the character $\eta: G^{\vec D_* + \vec D'}_{(s-1,r_*)} \to \R$ annihilates $G^*$, and thus descends to a vertical character $\tilde \eta: \tilde G_{(s-1,r_*)} \to \R$.
We select a function $\tilde F \in \Lip( \tilde G/\tilde \Gamma \to S^\omega)$ with vertical frequency $\tilde \eta$; such a function can be built using the construction \eqref{fkts}.
We then define the polynomial sequences $ g_0, \tilde g_h \in \ultra \poly(\Z_\N \to \tilde G_\N)$ by the formulae
\begin{align}
g_0(n) &:= \prod_{i=1}^{s-1} \prod_{j=1}^{D_{*,i}} \tilde e_{i,j}^{\gamma_{i,j} \binom{n}{i}}\label{g0-def}\\
\tilde g_h(n) &:= \prod_{i=1}^{s-1} \prod_{j=D_{*,i}+1}^{D_{*,i}+D'_{\lin,i}} \tilde e_{i,j}^{\{\alpha_{i,j} h\} \beta_{i,j} \binom{n}{i}}\label{gh-def}
\end{align}
and consider the nilcharacter
\begin{equation}\label{chok}
\tilde \chi_h(n) := \tilde F( g_0(n) \tilde g_h(n) \ultra \tilde \Gamma ).
\end{equation}
These nilcharacters are equivalent to $\chi_h$ in $\Symb_{\DR}^{(s-1,r_*)}([N])$, as the following lemma shows.
\begin{lemma} For each $h \in H'$, $\chi_h$ and $\tilde \chi_h$ are equivalent \textup{(}as nilcharacters of degree-rank $(s-1,r_*)$\textup{)} on $[N]$.
\end{lemma}
\begin{proof} Fix $h$. It suffices to show that $\chi_h \otimes \overline{\tilde \chi_h}$ is a nilsequence of degree $<s-1$. We can write this sequence as
\begin{equation}\label{fhg}
n \mapsto F'_h( g'_h(n) \ultra \Gamma'),
\end{equation}
where $G' := G \times G_0 \times \tilde G$, $\Gamma := \Gamma \times \Gamma_0 \times \tilde \Gamma$,
$g'_h \in \ultra \poly( \Z_\N \to G'_\N )$ is the sequence
\[
g'_h(n) := ( g_h(n), g_{0,h}(n), g_0(n) \tilde g_h(n) )
\]
and $F'_h \in \Lip(\ultra(G'/\Gamma'))$ is the function
\[
F'_h(x,x_0,y) := F_h(x,x_0) \otimes \overline{\tilde F(y)}.
\]
We define a $\DR$-filtration $G'_\DR$ on $G'$ by defining $G'_{(d,r)}$ for $(d,r) \in \DR$ with $r \geq 1$ to be the Lie group generated by the following sets:
\begin{enumerate}
\item $G_{(d,r+1)} \times (G_0)_{(d,r)} \times \tilde G_{(d,r+1)}$;
\item $\{ (\phi(g), \id, \tilde \phi (g) ): g \in G^{\vec D_*+\vec D'}_{(d,r)} \}$,
\end{enumerate}
with the convention that $(d,d+1) = (d+1,0)$.
We also set $G'_{(d,0)} := G'_{(d,1)}$ for $d \geq 1$. One easily verifies that this is a filtration.
We claim that $g'$ is polynomial with respect to this filtration. Indeed, the sequence $n \mapsto (\id,g_{0,h}(n),\id)$ is already polynomial in this filtration, so by Corollary \ref{laz} it suffices to verify that the sequence
\begin{equation}\label{gig}
n \mapsto (g_h(n), \id, g_0(n) \tilde g_h(n))
\end{equation}
is polynomial. We use Lemma \ref{taylo} to Taylor expand $g_h(n) = \prod_{i=0}^{s-1} g_{h,i}^{\binom{n}{i}}$
where $g_{h,i} \in G_{(i,0)}$. From \eqref{thune}, one has
\[
g_{h,i} = \phi\big( \prod_{j=1}^{D_{*,i}+D'_i} e_{i,j}^{\xi_{h,i,j}} \big) \mod G_{(i,2)}.
\]
By construction of the filtration of $G'$, this implies that
\[
\big( g_{h,i}, \id, \prod_{j=1}^{D_{*,i}+D_i'} e_{i,j}^{ \xi_{h,i,j}} \mod G^* \big) \in G'_{(i,1)}.
\]
Applying Corollary \ref{laz}, we conclude that the sequence
\[
n \mapsto \big(g_h(n), \id, \prod_{i=0}^{s-1}
( \prod_{j=1}^{D_{*,i}+D_i'} e_{i,j}^{ \xi_{h,i,j}})^{\binom{n}{i}} \mod G^* \big)
\]
is polynomial with respect to the $G'$ filtration.
Applying the Baker-Campbell-Hausdorff formula repeatedly, and using \eqref{g0-def}, \eqref{gh-def}, we see that
\[
n \mapsto
\prod_{i=0}^{s-1} (\prod_{j=1}^{D_{*,i}+D_i'} e_{i,j}^{ \xi_{h,i,j}})^{\binom{n}{i}} \mod G^*
\]
differs from the sequence $n \mapsto g_0(n) \tilde g_h(n)$ by a sequence which is polynomial in the shifted filtration $(\tilde G_{(d,r+1)})_{(d,r) \in \DR}$. We conclude that \eqref{gig} is polynomial as required.
Next, we claim that $F'_h$ is invariant with respect to the action of the central group
$$
G'_{(s-1,r_*)} = \{ (\phi(g), \id, \tilde \phi (g) ): g \in G^{\vec D}_{(s-1,r_*)} \}.
$$
It suffices to check this for generators $(\phi(w),\id,w \mod G^*)$, where $w$ is an $(r_*-1)$-fold commutator of
$e_{i_1,j_1},\ldots,e_{i_{r_*},j_{r_*}}$ in $G^{\vec D}$ with $i_1+\ldots+i_r = s-1$. There are two cases. If one has $j_l > D_{*,i_l} + D'_{\lin,i_l}$ for some $l$, then $w$ lies in $G^*$ and is also annihilated by $\eta$, and the claim follows from \eqref{phil}. If instead one has $j_l \leq D_{*,i_l} + D'_{\lin,i_l}$ for all $l$, then the claim again follows from \eqref{phil} together with the construction of $\tilde \eta$ and $\tilde F$.
We may now quotient out $G'_{(0,0)}$ by $G'_{(s-1,r_*)}$ and obtain a representation of
\eqref{fhg} as a nilsequence of degree-rank $<(s-1,r_*)$, as desired.
\end{proof}
From this lemma and Lemma \ref{symbolic}(ii) we can express $\chi_h$ as a bounded linear combination of $\tilde \chi_h \otimes \psi_h$ for some nilsequence $\psi_h$ of degree-rank $\leq (s-1,r_*-1)$. Thus, to prove Theorem \ref{linear-induct} it suffices to show that there is a nilcharacter $\tilde \chi \in \Xi^{(1,s-1)}(\ultra \Z^2)$, such that $\tilde \chi_h(n) = \tilde \chi(h,n)$ for many $h \in H'$ and all $n \in [N]$.
We illustrate the construction with an example. Let
$$G:= G^{(2,0)} = \{ e_1^{t_1} e_2^{t_2} [e_1,e_2]^{t_{12}}: t_1,t_2,t_{12} \in \R \}$$
be the universal degree $2$ nilpotent group \eqref{heisen} generated by $e_1,e_2$. Let $F$ be the Lipschitz function in equation (\ref{fkts}). Suppose
\[ \chi_h(n) := F(g_h(n)\ultra \Gamma)
\]
with $g_h(n) := e_2^{\beta n} e_1^{\alpha_h n} $, where $\alpha_h:=\{\delta h\} \gamma$, and $\alpha,\beta,\gamma \in \ultra \R$. As computed in \S \ref{nilcharacters}, we have
\[F_k(g_h(n)\ultra \Gamma)= \phi_k(\alpha_hn \mod 1,\beta n \mod 1)e(\{\alpha_h n \} \beta n)\]
for some Lipschitz function $\phi_k: \T^2 \to \C$.
We would like to interpret the function $(h,n) \mapsto \chi_h(n)$ as a nilcharacter in $ \Xi_{\MD}^{(1,2)}(\ultra \Z^2)$.
The first task is to identify a subgroup $G_{\petal}$ of the group $G$ representing that part of $G$ that is ``influenced by'' the petal frequency $\alpha_h$; more specifically, we take $G_{\petal}$
to be the subgroup of $G$ generated by $e_1$ and $[e_1, e_2]$, that is to say
$$ G_\petal = \langle e_1, [e_1,e_2] \rangle_\R = \{ e_1^{t_1} [e_1,e_2]^{t_{12}}: t_1,t_{12} \in \R \}.$$
Note that $G_{\petal}$ is abelian and normal in $G$. In particular $G$ acts on $G_{\petal}$ by conjugation, and we may form the semidirect product
$$G \ltimes G_{\petal} := \{ (g,g_1): g \in G, g_1 \in G_\petal \},$$
defining multiplication by
\[
(g, g_1)\cdot (g', g'_1) = (gg', g_1^{g'} g'_1),
\]
where $a^b := b^{-1} a b$ denotes conjugation.
Now consider the action $\rho$ of $\R$ on $G \ltimes G_{\petal}$ defined by
\[
\rho(t)(g, g_1) := (g g_1^t, g_1).
\]
We may form a further semidirect product
\[ G' := \R \ltimes_{\rho} (G \ltimes G_{\petal}),\] in which the product operation is defined by
\[
(t, (g, g_1)) \cdot (t', (g', g'_1)) = (t + t', \rho(t')(g, g_1) \cdot (g', g'_1)).
\]
$G'$ is a Lie group; indeed, one easily verifies that it is $3$-step nilpotent. We give $G'$ a $\N^2$-filtration:
\begin{align*}
G'_{(0,0)}&:= G' \\
G'_{(1,0)}&:=\{(t,(g,\id)): t \in \R, g \in G_\petal \} \\
G'_{(1,1)}&:=\{(0,(g,\id)): g \in G_\petal\},\\
G'_{(1,2)}&:=\{(0,(g,\id)): g \in [G,G]\}, \\
G'_{(0,1)}&:=\{(0,(g,g_1)): g \in G_\petal; g_1 \in G_{\petal}\},\\
G'_{(0,2)}&:=\{(0,(g,g_1)): g, g_1\in [G,G]\},
\end{align*}
with $G'_{i,j}:=\{\id\}$ for all other $(i,j) \in \N^2$. One easily verifies that this is a filtration.
Inside $G'$ we take the lattice
\[
\Gamma' := \Z \ltimes_{\rho} (\Gamma \ltimes \Gamma_{\petal}),
\]
where $\Gamma_{\petal} := \Gamma \cap G_{\petal}$. Now consider the polynomial $g':\Z^2 \to G'$ defined by
\[
g'(h, n) := (0, (e_2^{\beta n}, e_1^{\gamma n})) \cdot (\delta h, (\id, \id))
\]
and observe that
\begin{align*} g'(h,n)\Gamma' & = (0, (e_2^{\beta n}, e_1^{\gamma n})) \cdot (\{\delta h\} , (\id, \id)) \Gamma' \\ & = (\{\delta h\}, (e_2^{\beta n} e_1^{\{\delta h\}\gamma n}, e_1^{\gamma n}))\Gamma'.
\end{align*}
For a dense subset $H''$, $\{\delta h\}$ is in a small interval $I$, and let $\psi$ be a smooth cutoff function supported on $2I$.
Take $ F' : G'/ \Gamma' \rightarrow \C^D$ to be the function defined by
\[ F'((t, (g, g'))\Gamma') := \psi(t) F(g\Gamma)\] whenever $ t \in I$ and $0$ otherwise. Then we have for $h \in H''$
\[ F'(g'(h,n)\tilde\Gamma) = F(e_2^{\beta n} e_1^{\{\delta h\}\gamma n}\Gamma)=\chi_h(n),\]
giving the desired representation of $(h,n) \mapsto \chi_h(n)$ as an (almost) degree $(1,2)$ nilcharacter.\vspace{11pt}
We now turn to the general case. Our construction shall proceed by an abstract algebraic construction.
Let $\tilde G_{\petal}$ be the subgroup of $\tilde G$ generated by $(r-1)$-fold ($r \ge 1$) iterated commutators of $\tilde e_{i_1,j_1},\ldots,\tilde e_{i_r,j_r}$ in which $j_l > D_{*,i_l}$ for exactly one value of $l$. Then $\tilde G_{\petal}$ is a rational abelian normal subgroup of $\tilde G$. To see that $\tilde G_{\petal}$ is normal, ones uses the equalities
\[
\tilde e^{-1}_{i,j}[g,h] \tilde e_{i,j}=[\tilde e^{-1}_{i,j}g\tilde e_{i,j},\tilde e^{-1}_{i,j}h \tilde e_{i,j}] \quad \mbox{and} \quad \tilde e^{-1}_{i,j}g \tilde e_{i,j}= g[g,\tilde e_{i,j}],
\]
the commutator identities in equation (\ref{com-ident}), and the fact that any iterated commutators of $\tilde e_{i_1,j_1},\ldots,\tilde e_{i_r,j_r}$ in which $j_l > D_{*,i_l}$ for more than one value of $l$ is trivial in $\tilde G$.
In particular, $\tilde G$ acts on $\tilde G_{\petal}$ by conjugation, leading to the semidirect product $\tilde G \ltimes \tilde G_{\petal}$ of pairs $(g,g_1)$ with the product
$$ (g,g_1) (g',g'_1) := (gg', g_1^{g'} g'_1).$$
Next, let $R$ be the commutative ring of tuples $t = (t_{i,j})_{1 \leq i \leq s-1; D_{*,i} < j \leq D_{*,i}+D'_{\lin,i}}$ with $t_{i,j} \in \R$, which we endow with the pointwise product. For each $t \in R$, we can define an homomorphism $g \mapsto g^t$ on $\tilde G$, which we define on generators by mapping $\tilde e_{i,j}$ to $\tilde e_{i,j}^t$ for $D_{*,i} < j \leq D_{*,i}+D'_{\lin,i}$, but preserving $\tilde e_{i,j}$ for $j \leq D_{*,i}$. Such a homomorphism is well-defined as it preserves the defining relations of $\tilde G$. We observe the composition law
$$ (g^t)^{t'} = g^{tt'}$$
for $g \in \tilde G$ and $t,t' \in R$. Also, on the abelian subgroup $\tilde G_{\petal}$ on $\tilde G$, we see that
\begin{equation}\label{g0g}
g^t g^{t'} = g^{t+t'}
\end{equation}
as can be seen from the Baker-Campbell-Hausdorff formula \eqref{bch}. We can thus express
\begin{equation}\label{chok2}
\tilde g_h(n) = g_1(n)^{\{ \alpha h \}}
\end{equation}
where $g_1 \in \ultra \poly(\Z_\N \to (\tilde G_{\petal})_\N)$ is the polynomial sequence
\[
g_1(n) := \prod_{i=1}^{s-1} \prod_{j=D_{*,i}+1}^{D_{*,i}+D'_{\lin,i}} \tilde e_{i,j}^{ \beta_{i,j} \binom{n}{i}}
\]
and $\{ \alpha h \} \in R$ is the element
$$ \{ \alpha h \} := ( \{ \alpha_{i,j} h \} )_{1 \leq i \leq s-1; D_{*,i} < j \leq D_{*,i}+D'_{\lin,i}}.$$
The homomorphism $g \mapsto g^t$ preserves $\tilde G_{\petal}$, and is the identity once $\tilde G_{\petal}$ is quotiented out. As a consequence we see that
\begin{equation}\label{g1g}
(g g_1 g^{-1})^t = g g_1^t g^{-1}
\end{equation}
for any $g \in \tilde G$ and $g_1 \in \tilde G_{\petal}$.
We can now define an action $\rho$ of $R$ (viewed now as an additive group) on $\tilde G \ltimes \tilde G_{\petal}$ by defining
$$ \rho(t)( g, g_1 ) := (g g_1^t, g_1);$$
the properties \eqref{g0g}, \eqref{g1g} ensure that this is indeed an action. We can then define the semi-direct product $G' := R \ltimes_\rho (\tilde G \ltimes \tilde G_{\petal})$ to be the set of pairs $(t, (g,g_1) )$ with the product
$$ (t, (g,g_1)) (t', (g',g'_1)) = (t+t', \rho(t')(g,g_1) (g',g'_1)).$$
This is a Lie group. We can give it a $\N^2$-filtration $(G'_{(d_1,d_2)})_{(d_1,d_2) \in \N^2}$ as follows:
\begin{enumerate}
\item If $d_1 > 1$, then $G'_{(d_1,d_2)} := \{\id\}$.
\item If $d_1=1$ and $d_2 > 0$, then $G'_{(1,d_2)}$ consists of the elements $(0,(g,\id))$ with $g \in \tilde G_{d_2} \cap \tilde G_\petal$.
\item If $d_1=1$ and $d_2 = 0$, then $G'_{(1,0)}$ consists of the elements $(t,(g,\id))$ with $t \in R$ and $g \in \tilde G_\petal$.
\item If $d_1=0$ and $d_2 > 0$, then $G'_{(0,d_2)}$ consists of the elements $(0,(g,g_1))$ with $g \in \tilde G_{d_2}$ and $g_1 \in \tilde G_{\petal} \cap \tilde G_{d_2}$.
\item $G'_{(0,0)} = G'$.
\end{enumerate}
One easily verifies that this is a filtration of degree $\leq (1,s-1)$ with $G'_{(0,0)} = G'$.
We let $\Gamma'$ be the subgroup of $\tilde G$ consisting of pairs $(t,(g,g_1))$ with $g \in \tilde \Gamma$, $g_1 \in \tilde \Gamma_{\petal}$, and with all coefficients of $t$ integers. One easily verifies that $\Gamma'$ is a cocompact subgroup of $G'$, and that the above $\N^2$-filtration of $G'$ is rational with respect to $\Gamma'$, so that $G'/\Gamma'$ has the structure of a filtered nilmanifold.
We consider the orbit $\orbit' \in \ultra \poly(\Z^2_{\N^2} \to (G'/\Gamma')_{\N^2})$ defined by
$$ \orbit'(h,n) := (0,(g_0(n),g_1(n))) (\alpha h, (\id,\id)) \ultra \Gamma',$$
where
$$ \alpha h := ( \alpha_{i,j} h )_{1 \leq i \leq s-1; D_{*,i} < j \leq D_{*,i}+D'_{\lin,i}}.$$
As $g_0$, $g_1$ were already known to be polynomial maps, and the linear map $h \mapsto\alpha h$ is clearly polynomial also, we see from Corollary \ref{laz} and the choice of filtration on $G'$ that $\orbit'$ is a polynomial orbit.
Now we simplify the orbit. Working on the abelian group $R$, we see that
$$ (\alpha h, (\id,\id)) \ultra \Gamma' = (\{\alpha h\}, (\id,\id)) \ultra \Gamma',$$
and then commuting this with $(0,(g_0(n),g_1(n)))$, we obtain
\begin{equation}\label{orb}
\orbit'(h,n) = (\{\alpha h\}, (g_0(n) g_1(n)^{\{\alpha h\}}, g_1(n) ) ) \ultra \Gamma'.
\end{equation}
Recall that for many $h \in H$ that each component $\{ \alpha_{i,j} h\}$ of $\{\alpha h \}$ lies in an interval $I_{i,j}$ of length at most $1/10$. Let $2I_{i,j}$ be the interval of twice the length and with the same centre as $I_{i,j}$, and let $\varphi_{i,j}: \R \to \R$ be a smooth cutoff function supported on $I_{i,j}$. We then define a function $F': G'/\Gamma' \to \C^\omega$ by setting
$$ F'( ((t_{i,j})_{1 \leq i \leq s-1; D_{*,i} < j \leq D_{*,i}+D'_{\lin,i}}, (g, g_1)) \ultra \Gamma' ) := \big(\prod_{i=1}^{s-1} \prod_{j=D_{*,i}+1}^{D_{*,i}+D'_{\lin,i}} \varphi_{i,j}(t_{i,j})\big) \tilde F(g \ultra \tilde \Gamma)$$
whenever $(g,g_1) \in \tilde G \ltimes \tilde G_{\petal}$ and $t_{i,j} \in 2I_{i,j}$ for all $1 \leq i \leq s-1$ and $D_{*,i} < j \leq D_{*,i}+D'_{\lin,i}$, with $F'$ set equal to zero whenever no representation of the above form exists. One can easily verify that $F'$ is well-defined and Lipschitz. Since $\tilde F$ has vertical frequency $\tilde \eta$, $F'$ has vertical frequency $\eta': G'_{(1,s-1)} \to \R$, defined by the formula
$$ \eta'( (0, (g, \id) ) := \tilde \eta(g)$$
for all $g \in \tilde G_{s-1}$. From \eqref{chok}, \eqref{chok2} and \eqref{orb}, we see that for many $h \in H'$ we have
$$ \tilde \chi_h(n) = F' \circ \orbit'(h,n)$$
for all $n \in [N]$. By construction, $F' \circ \orbit' \in \Xi_{\MD}^{(1,s-1)}(\ultra \Z^2)$, and Theorem \ref{linear-thm} follows.\\
\section{The symmetry argument}\label{symsec}
In this, the last section of the main part of the paper, we supply the symmetry argument, Theorem \ref{aderiv}; we recall that statement now.
\begin{theorem74-repeat} Let $f \in L^\infty[N]$, let $H$ be a dense subset of $[[N]]$, and let $\chi \in \Xi^{(1,s-1)}(\ultra \Z^2)$ be such that $\Delta_h f$ $<(s-2)$-correlates with $\chi(h,\cdot)$ for all $h \in H$. Then there exists a nilcharacter $\Theta \in \Xi^{s}(\ultra \Z)$ \textup{(}with the degree filtration\textup{)} and a nilsequence $\Psi \in \Nil^{\subset J}(\ultra \Z^2)$ \textup{(}with the multidegree filtration\textup{)}, with $J$ given by the downset
\begin{equation}\label{lower-again}
J := \{ (i,j) \in \N^2: i+j \leq s-1 \} \cup \{ (i,s-i): 2 \leq i \leq s \},
\end{equation}
such that $\chi(h,n)$ is a bounded linear combination of $\Theta(n+h) \otimes \overline{\Theta(n)} \otimes \Psi(h,n)$.
\end{theorem74-repeat}
\begin{example} Suppose that $s=2$, $\chi(h,n) = e(P(h,n))$, and $P(h,n): \ultra \Z^2 \to \ultra \R$ is a symmetric bilinear form in $n,h$. Then observe that
\begin{equation}\label{chan-sym}
\chi(h,n) = \Theta(n+h) \overline{\Theta(n)} \Psi(h,n)
\end{equation}
where $\Theta(n) := e( \frac{1}{2} P(n,n) )$ and $\Psi(h,n) := e( - \frac{1}{2} P(h,h) )$, which illustrates a special case of Theorem \ref{aderiv}. More generally, if $s \geq 2$ and $\chi(h,n) = e(P(h,n,\ldots,n))$ with $P(h,n_1,\ldots,n_{s-1}): \ultra \Z^s \to \ultra \R$ a symmetric multilinear form, then we have \eqref{chan-sym} with $\Theta(n) := e( \frac{1}{s} P(n,\ldots,n) )$, and $\Psi(h,n)$ a polynomial phase involving terms of multidegree $(i,s-i)$ in $h,n$ with $2 \leq i \leq s$. Thus we again obtain a special case of Theorem \ref{aderiv}. Note how the symmetry of $P$ is crucial in order to make these examples work, which explains why we refer to Theorem \ref{aderiv} as a symmetrisation result. Morally speaking, this type of symmetry property ultimately stems from the identity $\Delta_h \Delta_k f = \Delta_k \Delta_h f$. We remark that an analogous symmetrisation result was crucial to the analogous proof of $\GI(2)$ in \cite{green-tao-u3inverse} (see also \cite{sam}), although our arguments here are slightly different.
\end{example}
From the inclusions at the end of \S \ref{nilcharacters}, $\chi(h,n)$ is a nilcharacter on $\Z^2$ (with the degree filtration) of degree $\leq s$. For similar reasons, any nilsequence $\Psi(h,n)$ of degree $\leq s-1$ (using the degree filtration on $\Z^2$) will automatically be of the form required for Theorem \ref{aderiv}. In view of this and Lemma \ref{symbolic}, we see that it will suffice to obtain a factorisation of the form
$$ [\chi]_{\Xi^s([[N]] \times [N])} = [\Theta(n+h)]_{\Xi^s([[N]] \times [N])} - [\Theta(n)]_{\Xi^s([[N]] \times [N])}
+ [\Psi(h,n)]_{\Xi^s([[N]] \times [N])}$$
where $\Theta \in \Xi^s(\ultra \N)$ is a one-dimensional nilcharacter of degree $\leq s$ (which automatically makes $(h,n) \mapsto \Theta(n)$ and $(h,n) \mapsto \Theta(n+h)$ two-dimensional nilcharacters of degree $\leq s$, by Lemma \ref{symbolic}(vi)), and $\Psi \in \Xi^s(\ultra \N^2)$ is a two-dimensional nilcharacter of multidegree
\begin{equation}\label{slosh}
\subset \{ (i,j) \in \N^2: i+j \leq s; j \leq s-2 \}.
\end{equation}
The set of classes $[\Psi(h,n)]_{\Xi^s([[N]] \times [N])}$, with $\Psi$ of the above form, is a subgroup of the space $\Symb^s([[N]] \times [N])$ of all symbols of degree $s$ nilcharacters on $[[N]] \times [N]$. Denoting the equivalence relation induced by these classes as $\equiv$, our task is thus to show that
$$ [\chi]_{\Xi^s([[N]] \times [N])} \equiv [\Theta(n+h)]_{\Xi^s([[N]] \times [N])} - [\Theta(n)]_{\Xi^s([[N]] \times [N])}.$$
In view of Theorem \ref{multilinearisation} and Lemma \ref{symbolic} (vii), there is a nilcharacter $\tilde \chi$ on $\ultra \Z^s$ of degree $(1,\ldots,1)$ which is symmetric in the last $s-1$ variables, and such that
\begin{equation}\label{change}
[ \chi(h,n) ]_{\Xi^s(\ultra \Z^2)} = s [ \tilde \chi(h,n,\ldots,n) ]_{\Xi^s(\ultra \Z^2)}.
\end{equation}
Inspired by the polynomial identity
$$ s h n^{s-1} = (n+h)^s - n^s - \ldots$$
where the terms in $\ldots$ are of degree $s$ in $h,n$ but of degree at most $s-2$ in $n$, we now choose
$$ \Theta(n) := \tilde \chi(n,\ldots,n).$$
From Lemma \ref{symbolic} (vi) we see that $\Theta$ is a nilcharacter of degree $\leq s$. Our task is now to show that
\begin{align}\nonumber
[\tilde \chi(n+h,\ldots,n+h)]_{\Xi^s([[N]] \times [N])} -&
[\tilde \chi(n,\ldots,n)]_{\Xi^s([[N]] \times [N])} - \\ & -
s[\tilde \chi(h,n\ldots,n)]_{\Xi^s([[N]] \times [N])} \equiv 0.\label{tilch}
\end{align}
To manipulate this, we use the following lemma.
\begin{lemma}[Multilinearity]\label{multil} Let $\tilde \chi$ be a nilcharacter on $\Z^s$ \textup{(}with the multidegree filtration\textup{)} of degree $(1,\ldots,1)$. Let $m \geq 1$ be standard, and let $L_1,\ldots,L_s: \Z^m \to \Z$ and $L'_1: \Z^m \to \Z$ be homomorphisms. Then we have linearity in the first variable, in the sense that
\begin{align*}
[\tilde \chi(L_1(\vec n)+L'_1(\vec n),L_2(\vec n),\ldots,L_s(\vec n))]_{\Xi^s(\ultra \Z^m)}
&= [\tilde \chi(L_1(\vec n),L_2(\vec n),\ldots,L_s(\vec n))]_{\Xi^s(\ultra \Z^m)}\\
&\quad + [\tilde \chi(L'_1(\vec n),L_2(\vec n),\ldots,L_s(\vec n)]_{\Xi^s(\ultra \Z^m)},
\end{align*}
where $\vec n = (n_1,\ldots,n_m)$ are the $m$ independent variables of $\ultra \Z^m$, and $\Z^m$ is given the degree filtration. We similarly have linearity in the other $s-1$ variables.
\end{lemma}
\begin{proof} We prove the claim for the first variable, as the other cases follow from symmetry. From Lemma \ref{baby-calculus} and Lemma \ref{symbolic}(vi), it will suffice to show that the expression
\begin{equation}\label{touch}
\tilde \chi(h_1+h'_1,h_2,\ldots,h_s) \otimes \overline{\tilde \chi}(h_1,h_2,\ldots,h_s) \otimes \overline{\tilde \chi}(h'_1,h_2,\ldots,h_s)
\end{equation}
is a degree $<s$ nilsequence in $h_1,h'_1,h_2,\ldots,h_s$ (using the degree filtration).
Write $\tilde \chi(h_1,\ldots,h_s) = F( g(h_1,\ldots,h_s) \ultra \Gamma)$, where $G/\Gamma$ is a $\N^s$-filtered nilmanifold of degree $\leq (1,\ldots,1)$, $F \in \Lip(\ultra(G/\Gamma))$ has a vertical frequency, and $g \in \ultra \poly(\Z^s_{\N^s} \to G_{\N^s})$. Then the expression \eqref{touch} takes the form
$$ \tilde F( \tilde g(h_1,h'_1,h_2,\ldots,h_s) \ultra \Gamma^3 )$$
where $\tilde g: \ultra \Z^{s+1} \to G^3$ is the map
$$ \tilde g(h_1,h'_1,h_2,\ldots,h_s) := ( g( h_1+h'_1,h_2,\ldots,h_s), g( h_1,h_2,\ldots,h_s), g(h'_1,h_2,\ldots,h_s) )$$
and $\tilde F \in \Lip(\ultra(G/\Gamma)^3)$ is the map
$$ \tilde F( x_1,x_2,x_3) = F(x_1) \otimes \overline{F(x_2)} \otimes \overline{F(x_3)}.$$
By Lemma \ref{taylo}, we can expand
$$ g(h_1,\ldots,h_s) = \prod_{i_1,\ldots,i_s = \{0,1\}} g_{i_1,\ldots,i_s}^{\binom{h_1}{i_1} \ldots \binom{h_s}{i_s}}$$
for some $g_{i_1,\ldots,i_s} \in G_{(i_1,\ldots,i_s)}$, where we order $\{0,1\}^s$ lexicographically (say).
We now $\N$-filter $G^3$ by defining $(G^3)_{i}$ to be the group generated by $(G_{(i_1,\ldots,i_s)})^3$ for all $i_1,\ldots,i_s \in \N$ with $i_1+\ldots+i_s>i$, together with the groups $\{ (g_1g_2,g_1,g_2): g_1,g_2 \in G_{(i_1,\ldots,i_s)} \}$ for $i_1+\ldots+i_s = i$. From the Baker-Campbell-Hausdorff formula \eqref{bch} one verifies that this is a rational filtration of $G^3$. From the Taylor expansion we also see that $\tilde g$ is polynomial with respect to this filtration (giving $\Z^{s+1}$ the degree filtration). Finally, as $F$ has a vertical character, we see that $\tilde F$ is invariant with respect to the action of $(G^3)_{s} = \{ (g_1g_2,g_1,g_2): g_1,g_2 \in G_{(1,\ldots,1)}\}$. Restricting $G^3$ to $(G^3)_{0}$ and quotienting out by $(G^3)_{s}$ we obtain the claim.
\end{proof}
Using this lemma repeatedly, together with the symmetry of $\tilde \chi$ in the final $s-1$ variables, we see that we can expand
\[\begin{split}
& [\tilde \chi(n+h,\ldots,n+h)]_{\Xi^s(\ultra \Z^2)} =\\
& \sum_{j=0}^{s-1} \binom{s-1}{j}
\left( [\tilde \chi(n,h,\ldots,h,n,\ldots,n)]_{\Xi^s(\ultra \Z^2)} + [\tilde \chi(h,h,\ldots,h,n,\ldots,n)]_{\Xi^s(\ultra \Z^2)} \right),
\end{split}\]
where in the terms on the right-hand side, the final $j$ coefficients are equal to $n$, the first coefficient is either $n$ or $h$, and the remaining coefficients are $h$. Note that a term with $j$ $h$ factors and $(s-j)$ $n$ factors will have degree \eqref{slosh} and thus be negligible as long as $j \geq 2$. Neglecting these terms, we obtain the simpler expression
\[ \begin{split}
[\tilde \chi(n+h,\ldots,n+h)]_{\Xi^s(\ultra \Z^2)}
\equiv &
[\tilde \chi(n,\ldots,n)]_{\Xi^s(\ultra \Z^2)}
+
[\tilde \chi(h,n,\ldots,n)]_{\Xi^s(\ultra \Z^2)} \\
& + (s-1) [\tilde \chi(n,h,n,\ldots,n)]_{\Xi^s(\ultra \Z^2)}.
\end{split}
\]
Comparing this with \eqref{slosh}, we will be done as soon as we can show the symmetry property
\begin{equation}\label{total-sym}
(s-1) [\tilde \chi(h,n,\ldots,n)]_{\Xi^s([[N]] \times [N])} = (s-1) [\tilde \chi(n,h,n,\ldots,n)]_{\Xi^s([[N]] \times [N])}.
\end{equation}
This property does not automatically follow from the construction of $\tilde \chi$. Instead, we must use the correlation properties of $\chi$, as follows.
By hypothesis and Lemma \ref{limone}, we have that for all $h$ in
a dense subset $H$ of $[[N]]$, we can find a degree $\leq s-2$ nilcharacter $\varphi_h$ such that $f_1(\cdot+h)f_2(\cdot)$ correlates with $\chi(h,\cdot,\ldots,\cdot) \otimes \varphi_h$. By Corollary \ref{mes-select}, we may assume that the map $h \mapsto \varphi_h$ is a limit map. We set $\varphi_h=0$ for $h \not \in H$.
To use this information, we return\footnote{Here is a key place where we use the hypothesis $s \geq 3$ (the other is Lemma \ref{discard}). For $s=2$ the lower order terms in Proposition \ref{cs} are useless; however a variant of the argument below still works, see \cite{green-tao-u3inverse}.} to Proposition \ref{cs}. Invoking that proposition, we see that for many additive quadruples $(h_1,h_2,h_3,h_4)$ in $[[N]]$, the sequence
\begin{align*}
n &\mapsto \chi(h_1,n) \otimes \chi(h_2,n+h_1-h_4) \otimes \overline{\chi(h_3,n)} \otimes \overline{\chi(h_4,n+h_1-h_4)}\\
&\quad \otimes \varphi_{h_1}(n) \otimes \varphi_{h_2}(n + h_1 - h_4) \otimes \overline{\varphi_{h_3}(n)} \otimes \overline{\varphi_{h_4}(n + h_1 - h_4)}
\end{align*}
is biased.
We make the change of variables $(h_1,h_2,h_3,h_4) = (h+a,h+b,h+a+b,h)$ and then pigeonhole in $h$, to conclude the existence of an $h_0$ for which
$$ n \mapsto \tau(a,b,n) \otimes \varphi_{h_0+a}(n) \otimes \varphi_{h_0+b}(n+a) \otimes \overline{\varphi}_{h_0+a+b}(n) \otimes \overline{\varphi_{h_0}(n+a)}$$
is biased for many pairs $a,b \in [[2N]]$, where $\tau = \tau_{h_0}$ is the expression
\begin{equation}\label{tabn}
\tau(a,b,n) := \chi(h_0+a,n) \otimes \chi(h_0+b,n+a) \otimes \overline{\chi(h_0+a+b,n)} \otimes \overline{\chi(h_0,n+a)}.
\end{equation}
Henceforth $h_0$ is fixed, and we will suppress the dependence of various functions on this parameter. From Lemma \ref{baby-calculus}, $\tau$ is a degree $\leq 3$ nilcharacter on $\ultra \Z^3$ (with the degree filtration). We record its top order symbol:
\begin{lemma}\label{calc-1} We have
$$ [\tau(a,b,n)]_{\Xi^s(\ultra \Z^3)} \equiv s(s-1) [\tilde \chi(b,a,n,\ldots,n)]_{\Xi^s(\ultra \Z^3)}$$
where by $\equiv$ we are quotienting by all symbols of degree $\leq s-3$ in $n$.
\end{lemma}
\begin{proof} From \eqref{change}, \eqref{tabn}, Lemma \ref{baby-calculus} and Lemma \ref{symbolic} one has
\begin{align*} [\tau(a,b,n)]_{\Xi^s(\ultra \Z^3)} = & s( [\tilde \chi(a,n,\ldots,n)]_{\Xi^s(\ultra \Z^3)} + [\tilde \chi(b,n+a,\ldots,n+a)]_{\Xi^s(\ultra \Z^3)} - \\ & - [\tilde \chi(a+b,n,\ldots,n)]_{\Xi^s(\ultra \Z^3)}).
\end{align*}
Applying Lemma \ref{multil} in the first variable we simplify this as
$$ s
( [\tilde \chi(b,n+a,\ldots,n+a)]_{\Xi^s(\ultra \Z^3)} - [\tilde \chi(a,n,\ldots,n)]_{\Xi^s(\ultra \Z^3)}).$$
Applying Lemma \ref{multil} in all the other variables and gathering terms using the symmetry of $\tilde \chi$ in those variables, we arrive at
$$ \sum_{j=0}^{s-2} s \binom{s-1}{j} [\tilde \chi(b,a,\ldots,a,n,\ldots,n)]_{\Xi^s(\ultra \Z^3)},$$
where there are $j$ occurrences of $n$ and $s-1-j$ occurrences of $a$. All the terms with $j<s-2$ are of degree $\leq s-2$ in $n$, and the claim follows.
\end{proof}
From Lemma \ref{symbolic}, we know that $\varphi_{h_0+b}(n+a)$ is a bounded linear combination of $\varphi_{h_0+b}(n) \otimes \psi_{a,b}(n)$ for some degree $\leq s-3$ nilsequence $\psi_{a,b}$. Similarly for $\varphi_{h_0}(n+a)$. We conclude that
$$ n \mapsto \tau(a,b,n) \otimes \varphi_{h_0+a}(n) \otimes \varphi_{h_0+b}(n) \otimes \overline{\varphi}_{h_0+a+b}(n) \otimes \overline{\varphi_{h_0}(n)}$$
is $\leq (s-3)$-biased for many $a,b \in [[2N]]$.
We will now eliminate the $\varphi_h$ terms in order to focus attention on $\tau$. Applying Corollary \ref{mes-select}, we may thus find a scalar degree $\leq s-3$ nilsequence $\psi_{a,b}$ depending in a limit fashion on $a, b \in [[2N]]$, such that
\begin{align*} |\E_{a,b \in [[2N]]; n \in [N]}
\tau(a,b,n) \otimes \varphi_{h_0+a}(n) \otimes \varphi_{h_0+b}(n) \otimes & \overline{\varphi_{h_0+a+b}(n)} \otimes \\ & \otimes \overline{\varphi_{h_0,k'}(n+a)} \psi_{a,b}(n)| \gg 1.\end{align*}
We pull out the $b$-independent factors $\varphi_{h_0+a}(n) \otimes \overline{\varphi}_{h_0}(n)$ and Cauchy-Schwarz in $a,n$ to conclude that
\begin{align*}
|\E_{a,b,b' \in [[2N]]; n \in [N]}
\tau(a,b,n) &\otimes \overline{\tau(a,b',n)} \otimes \varphi_{h_0+b}(n) \otimes \overline{\varphi_{h_0+b'}(n)} \\
&\otimes \overline{\varphi_{h_0+a+b}(n)} \otimes \varphi_{h_0+a+b'}(n) \psi_{a,b,b'}(n)| \gg 1,
\end{align*}
where $(a,b,b') \mapsto \psi_{a,b,b'}$ is a limit map assigning a scalar degree $\leq s-3$ nilsequence to each $a,b,b'$.
Next, we make the substitution $c := a+b+b'$ and conclude that
\begin{align*}
|\E_{c,b,b' \in [[3N]]; n \in [N]}&
\tau(c-b-b',b,n) \otimes \overline{\tau(c-b-b',b',n)} \\
&\otimes \varphi_{h_0+b}(n) \otimes \overline{\varphi_{h_0+b'}}(n) \otimes \overline{\varphi_{h_0+c-b'}(n)} \varphi_{h_0+c-b}(n) \psi'_{c,b,b'}(n)| \gg 1
\end{align*}
where $(c,b,b') \mapsto \psi'_{c,b,b'}$ is a limit map assigning a scalar degree $\leq s-3$ nilsequence to each $c,b,b'$.
By the pigeonhole principle, we can thus find a $c_0$ such that
\begin{equation}\label{retour}
|\E_{b,b' \in [[3N]]; n \in [N]}
\alpha(b,b',n) \otimes \varphi'_b(n) \otimes \overline{\varphi'_{b'}(n)} \psi'_{c_0,b,b'}(n)| \gg 1
\end{equation}
where $\alpha = \alpha_{c_0}$ is the form
\begin{equation}\label{abab}
\alpha(b,b',n) := \tau(c_0-b-b',b,n) \otimes \overline{\tau(c_0-b-b',b',n)}
\end{equation}
and $\varphi'_b = \varphi'_{b,c_0}$ is the quantity
$$ \varphi'_b(n) := \varphi_{h_0+b,k}(n) \otimes \overline{\varphi_{h_0+c_0-b}(n)}.$$
We fix this $c_0$. Again by Lemma \ref{baby-calculus}, $\alpha$ is a degree $\leq s$ nilcharacter on $\ultra \Z^3$, and we pause to record its symbol in the following lemma.
\begin{lemma}\label{calc-2} We have
$$ [\alpha(b,b',n)]_{\Xi^s(\ultra \Z^3)} \equiv -s(s-1) [\tilde \chi(b+b',b-b',n,\ldots,n)]_{\Xi^s(\ultra \Z^3)}$$
where by $\equiv$ we are quotienting by all symbols of degree $\leq s-3$ in $n$.
\end{lemma}
\begin{proof} From \eqref{abab} and Lemma \ref{symbolic} we can write the left-hand side as
$$
[\tau(-b-b',b,n)]_{\Xi^s(\ultra \Z^3)} - [\tau(-b-b',b',n)]_{\Xi^s(\ultra \Z^3)}.$$
Applying \eqref{calc-1}, we can write this as
$$
s(s-1)
( [\tilde \chi(-b-b',b,n,\ldots,n)]_{\Xi^s(\ultra \Z^3)} - [\tilde \chi(-b-b',b',n,\ldots,n)]_{\Xi^s(\ultra \Z^3)} ).$$
The claim then follows from some applications of Lemma \ref{multil}.
\end{proof}
We return now to \eqref{retour}, and Cauchy-Schwarz in $b',n$ to eliminate the $\varphi'_{b'}(n)$ factor, yielding
$$ |\E_{b_1,b_2,b' \in [[3N]]; n \in [N]}
\alpha(b_1,b',n) \otimes \overline{\alpha(b_2,b',n)} \otimes \varphi'_{b_1}(n) \otimes \overline{\varphi'_{b_2}(n)} \psi''_{b_1,b_2,b'}(n)| \gg 1$$
where $(b_1,b_2,b') \mapsto \psi''_{b_1,b_2,b'}$ is a limit map assigning a scalar degree $\leq s-3$ nilsequence to each $b_1,b_2,b'$. Finally, we Cauchy-Schwarz in $b_1,b_2,n$ to eliminate the $\varphi'_{b_1}(n) \overline{\varphi'_{b_2}(n)}$ factor, yielding
\begin{align*} |\E_{b_1,b_2,b'_1,b'_2 \in [[3N]]; n \in [N]}
\alpha(b_1,b'_1,n) \otimes & \overline{\alpha(b_2,b'_1,n)} \otimes \overline{\alpha(b_1,b'_2,n)} \otimes \\ & \otimes \alpha(b_2,b'_2,n) \psi''_{b_1,b_2,b'_1,b'_2}(n)| \gg 1.\end{align*}
Note how the $\varphi$ terms have now been completely eliminated. To eliminate the $\psi''$ terms, we first use the pigeonhole principle to find $b_0,b'_0$ such that
\begin{equation}\label{ebony}
|\E_{b,b' \in [[3N]]; n \in [N]}
\alpha'(b,b',n) \psi''_{b,b_0,b',b'_0}(n)| \gg 1
\end{equation}
where $\alpha' = \alpha'_{b_0,b'_0}$ is the expression
\begin{equation}\label{abab2}
\alpha'(b,b',n) := \alpha(b,b',n) \otimes \overline{\alpha(b_0,b',n)} \otimes \overline{\alpha(b,b'_0,n)} \otimes \alpha(b_0,b'_0,n).
\end{equation}
We fix this $b_0,b'_0$. Again, $\alpha'$ is a degree $\leq s$ nilcharacter on $\ultra \Z^3$. From Lemma \ref{calc-2} and Lemma \ref{multil} (and using Lemma \ref{symbolic} to eliminate shifts by $b_0$) we conclude
\begin{equation}\label{calc-3}
[\alpha'(b,b',n)]_{\Xi^s(\ultra \Z^3)} \equiv s(s-1)
([\tilde \chi(b,b',n,\ldots,n)]_{\Xi^s(\ultra \Z^3)} - [\tilde \chi(b',b,n,\ldots,n)]_{\Xi^s(\ultra \Z^3)}).
\end{equation}
Note the similarity here with \eqref{total-sym}.
From \eqref{ebony}, we conclude that the sequence $n \mapsto \alpha'(b,b',n)$ is $\leq s-3$-biased for many $b,b' \in [[3N]]$. Applying Proposition \ref{inv-nec-nonst}, we conclude that
$$ \| \alpha'(b,b',n) \|_{U^{s-2}[N]} \gg 1$$
for many $b,b' \in [[3N]]$. We conclude (using Corollary \ref{auton-2} to obtain the needed uniformity) that
$$ \E_{b,b' \in [[3N]]} \| \alpha'(b,b',n) \|_{U^{s-2}[N]}^{2^{s-2}} \gg 1.$$
By definition of the Gowers norm, this implies that
\begin{equation}\label{sorba}
|\E_{b,b',h_1,\ldots,h_{s-2} \in [[3N]]; n \in [N]}
\sigma( b, b', h_1, \ldots, h_{s-2}, n )
1_\Omega(h_1,\ldots,h_{s-2},n) | \gg 1,
\end{equation}
where $\Omega$ is the polytope
$$ \Omega := \{ (h_1,\ldots,h_{s-2},n): n+\sum_{j=1}^{s-2} \omega_j h_{s-2} \in [N] \hbox{ for all } \omega \in \{0,1\}^{s-2} \}$$
and $\sigma$ is the expression
\begin{equation}\label{sdef}
\sigma( b, b', h_1, \ldots, h_{s-2}, n) :=
\bigotimes_{\omega \in \{0,1\}^{s-2}} {\mathcal C}^{|\omega|} \alpha'(b,b',n+\sum_{j=1}^{s-2} \omega_j h_{s-2}),
\end{equation}
with ${\mathcal C}$ being the conjugation map.
From Lemma \ref{baby-calculus}, $\sigma$ is a nilcharacter of degree $s$ on $\ultra \Z^{s+1}$. In the following lemma we compute its symbol.
\begin{lemma} We have
\begin{equation}\label{sorba-2}
\begin{split}
[\sigma(b,b',h_1,\ldots,h_{s-2},n)]_{\Xi^s(\ultra \Z^{s+1})} = &s!
([\tilde \chi(b,b',h_1,\ldots,h_{s-2})]_{\Xi^s(\ultra \Z^{s+1})} \\
&\quad - [\tilde \chi(b',b,h_1,\ldots,h_{s-2})]_{\Xi^s(\ultra \Z^{s+1})}).
\end{split}
\end{equation}
\end{lemma}
\begin{proof} From \eqref{sdef} and Lemma \ref{symbolic} we can write the left-hand side as
\begin{equation}\label{flip}
\sum_{\omega \in \{0,1\}^{s-2}} (-1)^{|\omega|} [\alpha'(b,b',n+\sum_{j=1}^{s-2} \omega_j h_{s-2})]_{\Xi^s(\ultra \Z^{s+1})};
\end{equation}
one should think of this as an $s-2$-fold ``derivative'' of $[\alpha'(b,b',n)]_{\Xi^s(\ultra \Z^3)}$ in the $n$ variable.
From \eqref{calc-3} we can write
\begin{align*}
[\alpha'(b,b',n)]_{\Xi^s(\ultra \Z^3)} &= s(s-1)
([\tilde \chi(b,b',n,\ldots,n)]_{\Xi^s(\ultra \Z^3)} - [\tilde \chi(b',b,n,\ldots,n)]_{\Xi^s(\ultra \Z^3)}) \\
&\quad + [\beta(b,b',n)]_{\Xi^s(\ultra \Z^3)}
\end{align*}
where $\beta$ is of degree at most $s-3$ in $n$. In fact, by inspection of the derivation of $\beta$, and heavy use of Lemma \ref{multil}, one can express
$[\beta(b,b',n)]_{\Xi^s(\ultra \Z^3)}$ as a linear combination of classes of the form
$$ [\tilde \chi(n_1,\ldots,n_s)]_{\Xi^s(\ultra \Z^3)}$$
where each of $n_1,\ldots,n_s$ is equal to either $b$, $b'$, or $n$, with at most $s-3$ copies of $n$ occurring. If one then substitutes this expansion into \eqref{flip} and applies Lemma \ref{multil} repeatedly, one obtains the claim.
\end{proof}
On the other hand, from \eqref{sorba} and Lemma \ref{bias}, we see that on $[[3N]]^{s+1}$, $\sigma$ is equal to a nilsequence of degree $\leq s-1$, and thus by Lemma \ref{symbolic}
$$
[\sigma(b,b',h_1,\ldots,h_{s-2},n)]_{\Xi^s([[3N]]^{s+1})} = 0$$
and thus by Lemma \eqref{sorba-2}
$$
s!
([\tilde \chi(b,b',h_1,\ldots,h_{s-2})]_{\Xi^s([[3N]]^{s+1})} - [\tilde \chi(b',b,h_1,\ldots,h_{s-2})]_{\Xi^s([[3N]]^{s+1})}) = 0.$$
Applying Lemma \ref{baby-calculus} we conclude that
$$
s!
([\tilde \chi(h,n,\ldots,n)]_{\Xi^s([[N]] \times [N])} - [\tilde \chi(n,h,n,\ldots,n)]_{\Xi^s([[N]] \times [N])}) = 0.$$
The claim \eqref{total-sym} now follows from Lemma \ref{torsion}. The proof of Theorem \ref{aderiv} is now complete.
|
{
"timestamp": "2011-03-29T02:01:31",
"yymm": "1009",
"arxiv_id": "1009.3998",
"language": "en",
"url": "https://arxiv.org/abs/1009.3998"
}
|
\section{Introduction}
Several papers have recently appeared concerning the topology of
random simplicial complexes \cite{clique, bhk, neighborhood, Linial,
Meshulam, triangulated, geometric}. The results so far identify
thresholds for vanishing of homology, or compute the
expectation of the Betti numbers $\mathbb{E}[\beta_k]$ (i.e. the expected rank
of these groups). In this article we prove Poisson and normal
approximation theorems for $\beta_k$ for three models of random
simplicial complex. The complexes themselves are defined precisely
and given further motivation in the following sections but we first outline our
results.
The first model considered is that of the Erd\H{o}s-R\'enyi random clique complex $X(n,p)$,
a higher dimensional analogue of the Erd\H{o}s-R\'enyi random graph
$G(n,p)$. It was shown in \cite{clique} that for each $k$ and a
certain range of $p=p(n)$, $\beta_k \neq 0$ asymptotically almost surely (a.a.s\.), and in this regime, a formula for the
asymptotic size of $\mathbb{E}[\beta_k]$in terms of $p$ is given.
(Outside of this regime it is conjectured that $\beta_k =0 $ a.a.s.\, and some evidence for the conjecture is given in \cite{clique}.)
Here we prove
a Central Limit Theorem for $\beta_k$. That is, we show that $$ \frac{ \beta_k -
\mathbb{E}[\beta_k] }{ \sqrt{\mathrm{Var} [ \beta_k]}} \Rightarrow
\mathcal{N}(0,1),$$ as $n \to \infty$, where $\mathcal{N}(0,1)$ is
the normal distribution with mean $0$ and variance $1$.
\begin{figure}\label{ER-fig}
\begin{centering}
\includegraphics{gnp.png}
\end{centering}
\caption{The Betti numbers of $X(n,p)$ plotted vertically against edge
probability $p$; in this example $n=100$. \emph{Computation and
graphic courtesy of Afra Zomorodian.}}
\label{fig:gnp}
\end{figure}
The second model considered is the random \v{C}ech complex.
This model is a higher-dimensional analog of the random geometric graph;
the underlying graph is
a random geometric graph and the presence of
$(k-1)$-dimensional faces is determined by
$k$-fold intersections of balls centered about the vertices. \v{C}ech complexes are homotopy
equivalent to Edelsbrunner and M\"{u}cke's {\it alpha shapes}, widely
applied in computational geometry and topology \cite{alpha}. The
analysis needed to obtain limit theorems for the Betti numbers
of random \v{C}ech complexes
is more subtle that what is needed for
the Erd\"os-R\'enyi model; to prove the
normal and Poisson approximation theorems we must first establish limit
theorems for certain hypergraph counts, extending some of Mathew
Penrose's results for subgraph counts for geometric random graphs
\cite{penrose}.
The final type of complex considered
is the random Vietoris-Rips complex, denoted $VR(n,r)$. This is similar to the random \v{C}ech complex;
the construction is to take the clique complex of a random geometric
graph. (A useful
reference for geometric random graphs is
\cite{penrose}.) The topology is very different than for the clique complex of the Erd\H{o}s-R\'enyi random graph;
for the contrast between $X(n,p)$ and $VR(n,r)$ see Figures
\ref{fig:gnp} and \ref{fig:geom}. The analysis needed to obtain limit theorems
for the Betti numbers of $VR(n,r)$ is nevertheless
essentially identical to that needed
for the random \v{C}ech complex. A minor example of this fact is that
in both cases, since $\beta_0$ counts the number of connected components
for the \v{C}ech and Rips complexes, $\beta_0$ is actually the same in
each of these cases and
is equal to the number of components of the random geometric graph.
This has already been treated in
detail by Penrose \cite{penrose}, and so when convenient we will restrict
attention to $\beta_k$ for $k\ge 1$.
The techniques throughout the paper are a combination of inequalities derived from combinatorial and topological considerations with Stein's method. (For an introduction to topological combinatorics see \cite{Bjorner}; for a survey of
Stein's method in proving Poisson approximation theorems see \cite{CDM}, and
for an introduction to Stein's method for normal approximation, see
\cite{RR}.)
\subsection{Notation and conventions}
Throughout this article, we use Bachmann-Landau big-$O$, little-$O$, and related notations. In particular, for non-negative functions $g$ and $h$, we write the following.
\begin{itemize}
\item $g(n) = O(h(n))$ means that there exists $n_0$ and $k$ such that for $n > n_0$, we have that $g(n) \le k \cdot h(n)$. (i.e.\ $g$ is asymptotically bounded above by $h$, up to a constant factor.)
\item $g(n) = \Omega(h(n))$ means that there exists $n_0$ and $k$ such that for $n > n_0$, we have that $g(n) \ge k \cdot h(n)$. (i.e.\ $g$ is asymptotically bounded below by $h$, up to a constant factor.)
\item $g(n) = \Theta(h(n))$ means that $g(n) = O(h(n))$ and $g(n) = \Omega(h(n))$. (i.e.\ $g$ is asymptotically bounded above and below by $h$, up to constant factors.)
\item $g(n) = o(h(n))$ means that for every $\epsilon > 0$, there exists $n_0$ such that for $n > n_0$, we have that $g(n) \le \epsilon \cdot h(n)$. (i.e.\ $g$ is dominated by $h$ asymptotically.)
\item $g(n) = \omega(h(n))$ means that for every $k >0$, there exists $n_0$ such that for $n > n_0$, we have that $g(n) \ge k \cdot h(n)$. (i.e.\ $g$ dominates $h$ asymptotically.)
\end{itemize}
We may also write $A_n\simeq B_n$ if $\lim_{n\to\infty}\frac{A_n}{B_n}=1$, and
$A_n\lesssim B_n$ if there is a constant $c$ such
that $A_n\le c B_n$ for all $n$.
A sequence $\{X_n\}_{n=1}^\infty$ of random variables is said to {\it converge
weakly} to a limiting random variable $X$ (written $X_n\Rightarrow X$)
if $\lim_{n\to\infty}\mathbb{E}[f(X_n)]=\mathbb{E}[f(X)]$ for all bounded continuous
functions $f$ (there are several other equivalent definitions).
The {\it total variation distance} between random variables $X$ and $Y$
is defined by
$$d_{TV}(X,Y):=\sup_f\big|\mathbb{E}[f(X)]-\mathbb{E}[f(Y)]\big|,$$
with the supremum taken over all continuous functions bounded by one. Clearly,
if $d_{TV}(X_n,X)\to0$ as $n\to\infty$, then $X_n\Rightarrow X$; however, the
topology induced by the total variation distance is stronger than the
topology of weak convergence.
The {\it $L_1$-Wasserstein distance} or {\it Kantorovich-Rubenstein distance}
between $X$ and $Y$ is defined by
$$d_1(X,Y):=\sup_f\big|\mathbb{E}[f(X)]-\mathbb{E}[f(Y)]\big|,$$
where the supremum is over all functions $f$ with $\sup_{x\neq y}\frac{|f(x)-f(y)|}{
|x-y|}\le 1.$ This distance also induces a topology stronger than the topology
of weak convergence.
Finally, the normal distribution with mean $\mu$ and variance $\sigma^2$ is
denoted $\mathcal{N}(\mu,\sigma^2)$, and the distribution function of the
standard normal distribution is denoted $\Phi(t)$.
\section{Erd\H{o}s-R\'enyi random clique complexes}
Perhaps the first type of random simplicial complex studied was the
$1$-dimensional version studied by Erd\H{o}s and R\'enyi \cite{Erd1}.
\begin{definition}
The {\it Erd\H{o}s-R\'enyi random graph} $G(n,p)$ is the probability
space of all graphs on vertex set $[n] = \{1, 2, \dots, n \}$ with
each edge included independently with probability
$p$.
\end{definition}
The ``clique complex'' is used to generalize $G(n,p)$ from graphs to
higher dimensional simplicial complexes.
\begin{definition}
The {\it clique complex} $X(H)$ of a graph $H$ is a the simplicial
complex with vertex set $V(H)$ and a face for
each set of vertices spanning a
complete subgraph of $H$.
\end{definition}
In other words, the clique complex $X(H)$ of a graph $H$ is the
maximal simplicial complex with $1$-skeleton $H$.
This section concerns the clique complex of the Erd\H{o}s-R\'enyi random
graph, i.e.\ $X(G(n,p))$. For simplicity in notation, \
this is denoted $X(n,p)$.
There are several motivations for using
$X(n,p)$ as a model of a
random simplicial complex. One motivation is that $X(n,p)$ provides a
natural higher-dimensional generalization of $G(n,p)$, which has
proved extremely useful in graph theory as well as in applications. (Other
higher-dimensional generalizations are studied in \cite{bhk, Linial, Meshulam}.) Another
motivation comes from the fact that every simplicial complex is
homeomorphic to the clique complex of some graph (e.g. by barycentric
subdivision) \cite{Hatcher}.
One interesting feature of $X(n,p)$ is that it provides homological
analogues of the Erd\H{o}s-R\'enyi theorem, but in a {\it
non-monotone} setting: If edges are
added at random to an empty graph, the Erd\H{o}s-R\'enyi theorem
characterizes the number of edges needed
before the graph becomes connected. Connectivity is a monotone
graph property -- if one adds edges to a connected graph, it is still
connected.
Topologically, connectivity is equivalent to a statement about zeroth
homology $H_0(G(n,p))$ but if one asks about $H_k(X(n,p))$, $k>0$,
there is a problem -- adding edges generates higher $k$-dimensional
faces and $(k+1)$-dimensional faces at the same time. Since
generators and relations are both being added, there is no reason that
things have to behave in a monotone way. In fact, it
is not just that things might not be monotone; they are non-monotone in an essential way. In
particular, there seem to be two thresholds for higher homology -- one
where $H_k$ passes from vanishing to non-vanishing, and another where
it passes back to vanishing.
The following theorem was proved in
\cite{clique}. For any fixed
$k>0$, let $\beta_k$ denote the dimension of $k$th homology,
i.e.\ $\beta_k = \dim [ H_k (\Delta, \mathbb{Q}) ].$
\begin{theorem}\label{exp_er}
If $p = \omega(n^{-1/k})$ and $p=o( n^{-1/(k+1)})$ then $$ \lim_{n \to
\infty}{\mathbb{E}[\beta_k (X(n,p))] \over n^k p^{k+1 \choose 2} } = {1 \over
( k+1)!}.$$
\end{theorem}
(In \cite{clique} explicit nontrivial homology classes are exhibited, and several
partial converses of Theorem \ref{exp_er} are proved; in particular it is shown that if $p =
O(n^{-1/k - \epsilon})$ or $p = \Omega( n^{-1/(2k+1) + \epsilon})$ for
some constant $\epsilon > 0$, then a.a.s.\ $\beta_k = 0$.)
The remainder of this section is
devoted to showing that in the
same regime, $\beta_k$ obeys a central limit theorem.
\begin{theorem} \label{clt_er}
If $p = \omega(n^{-1/k})$ and $p=o( n^{-1/(k+1)})$ then $$ \frac{
\beta_k (X(n,p)) - \mathbb{E}[\beta_k(X(n,p))] }{ \sqrt{\mathrm{Var} [ \beta_k]}} \Rightarrow
\mathcal{N}(0,1).$$
\end{theorem}
\begin{proof}
For a finite simplicial complex $\Delta$, let $f_i(\Delta)$ (or simply
$f_i$ if context is clear) denote the number of
$i$-dimensional faces of $\Delta$. A useful fact when proving
Theorems \ref{exp_er} and \ref{clt_er} is that
$\beta_k$ satisfies the following ``Morse''
inequalities:
\begin{equation}\label{morse}
-f_{k-1}+f_k-f_{k+1} \le \beta_k \le f_k,
\end{equation}
for all $k$. These
inequalities follow from the definition of simplicial homology and the
rank-nullity law \cite{Hatcher}.
The next observation to make is that $X(n,p)$ is a clique complex,
so $f_k$ counts the number of $(k+1)$-cliques.
Since there are $\binom{n}{k+1}$ possible $(k+1)$-cliques and each appears with
probability $p^{k+1 \choose 2}$,
$$ \lim_{n \to \infty} {\mathbb{E}[f_k] \over n^{k+1} p^{k+1 \choose 2}} = \frac{1}{
(k+1)!}.$$
If $p = \omega(n^{-1/k})$ then
$${\mathbb{E}[f_{k-1}] \over \mathbb{E}[f_k]} ={ n^k p^{k \choose 2} \over n^{k+1}
p^{k+1 \choose 2}}= {1 \over n p^k}= o(1),$$
and the same argument shows that if $p =
o(n^{-1/(k+1)})$ then $${\mathbb{E}[f_{k +1 }] \over \mathbb{E}[
f_{k} ]} = o(1).$$
That is, in the regime of Theorems \ref{exp_er} and \ref{clt_er},
$$\lim_{n\to\infty}\frac{\mathbb{E}[f_k]}{\mathbb{E}[-f_{k-1}+f_k-f_{k+1}]}=1,$$
which, in light of \eqref{morse}, reproves Theorem \ref{exp_er}.
Let $\tilde{f}_k:=-f_{k-1}+f_k-f_{k+1}.$
The following claim together with \eqref{morse} is used
to show that $\beta_k$ satisfies a central limit
theorem.
\begin{claim}
\
\begin{enumerate}
\item\label{var_equiv} \begin{equation*}\lim_{n\to\infty}\frac{\mathrm{Var}(f_k)}{
\mathrm{Var}(\tilde{f}_k)}=1.\end{equation*}
\item\label{clt_above} \begin{equation*} \frac{f_k-\mathbb{E}[f_k]}{\sqrt{\mathrm{Var}(f_k)}}
\Rightarrow\mathcal{N}(0,1)\quad {\rm as\, } n\to\infty.
\end{equation*}
\item \label{clt_below}\begin{equation*} \frac{\tilde{f}_k-\mathbb{E}[\tilde{f}_k]}{
\sqrt{\mathrm{Var}(\tilde{f}_k)}}\Rightarrow\mathcal{N}(0,1)\quad
{\rm as\, } n\to\infty.\end{equation*}
\end{enumerate}
\end{claim}
For $t\in\mathbb{R}$, it follows from \eqref{morse} that
$$\P\left[\frac{f_k-\mathbb{E}[f_k]}{\sqrt{\mathrm{Var}(f_k)}}\le
t\right]\le\P\left[\frac{\beta_k-\mathbb{E}[f_k]}{\sqrt{\mathrm{Var}(f_k)}}\le
t\right]\le\P\left[\frac{\tilde{f}_k-\mathbb{E}[f_k]}{\sqrt{\mathrm{Var}(f_k)}}\le
t\right].$$
The left-hand side tends to $\Phi(t)$ as $n\to\infty$ by part \ref{clt_above}
of the claim. For the right-hand side, let $\epsilon>0$ and observe that
\begin{equation}\begin{split}\label{fiddling}
\P\left[\frac{\tilde{f}_k-\mathbb{E}[f_k]}{\sqrt{\mathrm{Var}(f_k)}}\le
t\right]&\le\P\left[\frac{\tilde{f}_k-\mathbb{E}[\tilde{f}_k]}{\sqrt{\mathrm{Var}(\tilde{f}_k)
}}\le t-\epsilon\right]+\P\left[\left|\frac{\tilde{f}_k-\mathbb{E}[\tilde{f}_k]}{
\sqrt{\mathrm{Var}(\tilde{f}_k)}}-\frac{\tilde{f}_k-\mathbb{E}[f_k]}{\sqrt{\mathrm{Var}(f_k)
}}\right|>\epsilon\right]\\&
+\P\left[\frac{\tilde{f}_k-\mathbb{E}[f_k]}{\sqrt{\mathrm{Var}(f_k)}}\le
t, \left|\frac{\tilde{f}_k-\mathbb{E}[\tilde{f}_k]}{\sqrt{\mathrm{Var}(\tilde{f}_k)
}}-t\right|\le \epsilon\right].
\end{split}\end{equation}
Now, it follows from part \ref{clt_below} of the claim that the first
term of the right-hand side of \eqref{fiddling} tends to
$\Phi(t-\epsilon)$ and that the last is asymptotically bounded above by
$\Phi(t+\epsilon)-\Phi(t-\epsilon)$.
For the second term, first require $n$ to be large enough that
$$\left|\frac{\mathbb{E}[f_k]}{\sqrt{\mathrm{Var}(f_k)}}-\frac{\mathbb{E}[\tilde{f}_k]}{\sqrt{\mathrm{Var}(
\tilde{f}_k)}}\right|<\frac{\epsilon}{2}.$$
This condition together with Chebychev's inequality implies that
\begin{equation*}\begin{split}
\P\left[\left|\frac{\tilde{f}_k-\mathbb{E}[\tilde{f}_k]}{
\sqrt{\mathrm{Var}(\tilde{f}_k)}}-\frac{\tilde{f}_k-\mathbb{E}[f_k]}{\sqrt{\mathrm{Var}(f_k)
}}\right|>\epsilon\right]&\le\P\left[\tilde{f_k}\left|\frac{1}{\sqrt{
\mathrm{Var}(f_k)}}-\frac{1}{\sqrt{\mathrm{Var}(\tilde{f}_k)}}\right|>\frac{\epsilon}{2}
\right]\\&\le4\epsilon^{-2}\left(\frac{\sqrt{\mathrm{Var}(\tilde{f}_k)}}{\sqrt{
\mathrm{Var}(f_k)}}-1\right)^2,
\end{split}\end{equation*}
which tends to zero for fixed $\epsilon>0$ by part \ref{var_equiv}
of the claim. It thus follows that the right-hand side of \eqref{fiddling}
is asymptotically bounded above by $\Phi(t+\epsilon)$ as $n\to\infty$;
as $\epsilon$ is arbitrary, this completes the proof of the central limit
theorem for $\beta_k$, modulo proof of the claim.
\medskip
To prove part \ref{var_equiv} of the claim, first write
$$f_k=\sum_{\substack{A\subseteq\{1,\ldots,n\}\\|A|=k+1}}\xi_A,$$
where $\xi_A$ is the indicator that $A$ spans a face in $X(n,p)$; that
is, that $A$ spans a complete graph in $G(n,p)$. Then, enumerating
pairs of subsets of size $k+1$ of $\{1,\ldots,n\}$ by the size $r$ of
their interesection,
\begin{equation*}\begin{split}
\mathrm{Var}(f_k)&=\sum_{A,B}\mathbb{E}[\xi_A\xi_B]-\left[\binom{n}{k+1}p^{\binom{k+1}{2}}
\right]^2\\&=\binom{n}{k+1}\sum_{r=0}^{k+1}\binom{k+1}{r}\binom{n-k-1}{k+1-r}
p^{2\binom{k+1}{2}-\binom{r}{2}}-\left[\binom{n}{k+1}p^{\binom{k+1}{2}}
\right]^2.
\end{split}\end{equation*}
Now, it is not
hard to see that in the range of $p$ considered here, only the $r=0,1,2$
terms contribute in the limit; there is cancellation of the terms
of order $n^{k+1}$ and $n^k$, so that the main contribution is
in fact from the $r=2$ term and
\begin{equation}\label{f_k_var}\lim_{n\to\infty}n^{-2k}p^{(-2\binom{k+1}{2}+1)}\mathrm{Var}(f_k)=c_k,\end{equation}
for some constant $c$ depending only on $k$. From this it follows immediately
that
$$\frac{\mathrm{Var}(f_{k-1})}{\mathrm{Var}(f_{k})}=o(1)\quad{\rm and}\quad
\frac{\mathrm{Var}(f_{k+1})}{\mathrm{Var}(f_{k})}=o(1),$$
for $p$ in the range specified in the statement of the theorem.
Expanding the same way as above, it is clear that
$$\mathrm{Cov}(f_k,f_{k+1})=\binom{n}{k+1}p^{\binom{k+1}{2}+ \binom{k+2}{2}}
\left[\sum_{r=0}^{k+1}\binom{k+1}{r}\binom{n-k-1}{k+2-r}p^{-\binom{r}{2}}-
\binom{n}{k+2}\right];$$
again there is cancellation of the terms of order $n^{k+2}$
and $n^{k+1}$ so that the leading contribution is from the $r=2$ term
and
$$\lim_{n\to\infty}n^{-2k-1}p^{-\left(\binom{k+1}{2}+ \binom{k+2}{2}-1\right)}
\mathrm{Cov}(f_k,f_{k+1})=c_k$$
for a (different) constant $c_k$ depending only on $k$. Thus in the
range of $p$ being considered,
$$\frac{\mathrm{Cov}(f_k,f_{k+1})}{\mathrm{Var}(f_k)}=o(1).$$
In exactly the same way, one can show that
$$\frac{\mathrm{Cov}(f_k,f_{k-1})}{\mathrm{Var}(f_k)}=o(1)\quad{\rm and}\quad
\frac{\mathrm{Cov}(f_{k-1},f_{k+1})}{\mathrm{Var}(f_k)}=o(1),$$
completing the proof of part \ref{var_equiv} of the claim.
\smallskip
The proofs of the second and third parts both
follow from an abstract normal approximation theorem for
dissociated random variables proved (via Stein's method) in \cite{BKR}.
Part \ref{clt_above} is in fact proved there; the following is a
a straightforward modification of their proof which obtains a central
limit theorem for the lower bound $\tilde{f}_k$. One
can also recover the proof of part \ref{clt_above} from what is
given below, simply by ignoring the extra terms present in $\tilde{f}_k$
beyond those coming from $f_k$.
A set $\{X_{\bf j}:{\bf j}=(j_1,\ldots,j_r)\in J\}$ for $J$ a set of $r$-tuples
is {\it dissociated} if two subcollections of the random variables
$\{X_{\bf j}:{\bf j}\in K\}$ and $\{X_{\bf j}:{\bf j}\in L\}$ are
independent whenever $\left(\cup_{{\bf j}\in K}\{j_1,\ldots,j_r\}\right)
\cap \left(\cup_{{\bf j}\in L}\{j_1,\ldots,j_r\}\right)=\emptyset.$
Let $W:=\sum_{{\bf j}\in J}X_{\bf j},$ and for each ${\bf j}\in J$, let
$L_{\bf j}:=\{{\bf k}\in J:\{k_1,\ldots,k_r\}\cap\{j_1,\ldots,j_r\}\neq
\emptyset\}.$ That is, $L_{\bf j}$ is a dependency neighborhood for
${\bf j}$. If $\mathbb{E} X_{\bf j}=0$ and $\mathbb{E} W^2=1$, then it is shown in
\cite{BKR} that
\begin{equation}\label{BKR-thm}
d_1(W,Z)\le K\sum_{{\bf j}\in J}\sum_{{\bf k},
{\bf l}\in L_{\bf j}}\Big[\mathbb{E}|X_{\bf j}X_{\bf k}X_{\bf l}|+\mathbb{E}|X_{\bf j}
X_{\bf k}|\mathbb{E}|X_{\bf l}|\Big],
\end{equation}
where $Z$ is a standard normal random variable.
To show that $\tilde{f}_k$ satisfies a central limit theorem, let the index
set $J$ be the potential edge sets for complete graphs on $k+e$
($e\in\{0,1,2\}$) vertices
in $G(n,p)$; that is, an element of $J$ is a $\binom{k+e}{2}$-tuple of
edges spanning a given set of $k+e$ vertices. Each ${\bf j}\in J$
can thus be
associated with its spanning set $A_{\bf j}$ of vertices. If the random
variables $X_{\bf j}$ are defined by
$$X_{\bf j}:=\sigma^{-1}(\xi_{A_{\bf j}}-\mathbb{E}[\xi_{A_{\bf j}}]),$$ where
$\sigma^2=\mathrm{Var}(f_k)$, then $\{X_{\bf j}\}$
are evidently dissociated.
The second half of the sum from \eqref{BKR-thm}
is fairly straightforward to bound in this context. For each ${\bf j}$,
partition $L_{\bf j}$ into the sets $L_{\bf j}^e$ of indices whose spanning
sets have size $k+e$.
Observe that for each ${\bf j}$, if $e_j=|L_{\bf j}|-k,$ then
$$|L_{\bf j}^e|=\binom{n}{k+e}-\binom{n-k-e_j}{k+e}-(k+e_j)
\binom{n-k-e_j}{k+e-1}=
O(n^{k+e-2}).$$
Decomposing as
in the variance estimate by the size $r$ of the intersection of
$A_{\bf j}$ and $A_{\bf k}$ and using the bound
above for $|L_{\bf j}^f|$ yields
\begin{equation*}\begin{split}
\sum_{{\bf j}\in J}&\sum_{{\bf k}\in L_{\bf j}^{e}}\sum_{
{\bf l}\in L_{\bf j}^f}\mathbb{E}|X_{\bf j}X_{\bf k}|\mathbb{E}|X_{\bf l}|\\&\le
\sigma^{-3}
c_kn^{k+f-2}p^{\binom{k+f}{2}}\binom{n}{k+e_j}\sum_{r=2}^{k+(e_j\wedge e)}
\binom{k+e_j}{r}\binom{
n-k-e_j}{k+e-r}p^{\binom{k+e}{2}+\binom{k+e_j}{2}-\binom{r}{2}}\\&\le
\sigma^{-3}c_kn^{3k+e_j+e+f-4}p^{\binom{k+e_j}{2}+\binom{k+e}{2}+\binom{k+f}{2}-1},
\end{split}\end{equation*}
since the $r=2$ term yields the top-order contribution in the
range of $p$ considered here. Moreover, it is easy to check that this
expression is maximized for $e_j=e=f=1$.
Combining this estimate with \eqref{f_k_var}
shows that
the contribution to the error from the second
sum is bounded above by
$$\sigma^{-3}c_kn^{3k-1}p^{3\binom{k+1}{2}-1}\le \frac{c_k\sqrt{p}}{n},$$
which tends to zero as $n$ tends to infinity.
The first half of the sum is bounded similarly, although it requires
that the intersections
of three spanning sets of vertices be considered. Let
$r$ denote the number of points common to $A_{\bf j}$ and
$A_{\bf k}$. Let $p_1:=|A_{\bf j}\cap A_{\bf l}\cap A_{\bf k}^c|$,
$p_2:=|A_{\bf j}\cap A_{\bf l}\cap A_{\bf k}|$ and $p_3:=|A_{\bf j}^c
\cap A_{\bf l}\cap A_{\bf k}|$. Then
$$\mathbb{E}|X_{\bf j}X_{\bf k}X_{\bf l}|\le
c\sigma^{-3}p^{\binom{k+e_j}{2}+\binom{k+e_k}{2} +\binom{k+e_l}{2}-
\binom{p_1+p_2}{2}-\binom{p_2+p_3}{2}-\binom{r}{2}+\binom{p_2}{2}},$$
where the constant $c$ simply accounts for the fact that the $X_{\bf
j}$ have been centered. The number of ways to choose ${\bf j}$,
${\bf k}$ and ${\bf l}$ is
\begin{align*}
\binom{n}{k+e_j}\binom{k+e_j}{r}\binom{n-k-e_j}{k+e_k-r}\binom{k+e_j-r}{p_1}\\
\times \binom{r}{p_2}\binom{k+e_k-r}{p_3}\binom{n-2k-e_j-e_k+r}{k+e_l-p_1-p_2-p_3}.
\end{align*}
Combining these two facts, it is perhaps slightly unpleasant but not too
hard to see that the main contribution to the error arises from the case that
$r=2$, $p_1+p_2=2$ (in fact only when $p_1\neq 0$), and $e_j=e_k=e_l=1$.
It follows that
\begin{equation*}\begin{split}
\sum_{{\bf j}\in J}\sum_{{\bf k},
{\bf l}\in L_{\bf j}}&\mathbb{E}|X_{\bf j}X_{\bf k}X_{\bf l}|\le
\sigma^{-3}c_kn^{3k-1}p^{3\binom{k+1}{2}-2}\le \frac{c_k}{n\sqrt{p}},
\end{split}\end{equation*}
which also tends to zero as $n$ tends to infinity. This completes the
proof of part
\ref{clt_below} of the claim, finishing the proof of Theorem \ref{clt_er}.
\end{proof}
\section{Random \v{C}ech complexes}
The second model of random simplicial complex considered is
the random \v{C}ech complex. This is a higher-dimensional analog of
a geometric random graph, constructed explicitly below. In order to
analyze this model, we use the same techniques used by Penrose \cite{penrose}
in his study of subgraph counts of random geometric graph. The additional
spacial dependence that is inherent in the random variables we consider
presents an additional technical challenge, and means that Penrose's
results cannot be applied directly to the problem.
Suppose that $\{X_i\}_{i=1}^\infty$ is an i.i.d.\ sequence of random
vectors in $\mathbb{R}^d$, with bounded density $f$. Let
$\{r_n\}_{n=1}^\infty\subseteq\mathbb{R}_+$, such that $nr_n^d\xrightarrow{n
\to\infty}0$ (the so-called ``sparse'' regime of geometric random
graphs), and construct a
random \v{C}ech complex $\mathcal{C}(X_1,\ldots,X_n)$ on
$\{X_i\}_{i=1}^n$ as follows. If $|X_i-X_j|\le2r_n$, put an edge
between $X_i$ and $X_j$; that is,
the 1-skeleton of the complex is a
random geometric graph. More generally, make the convex hull of $\{X_{i_1}\ldots,X_{i_k}\}$
a face of the complex
if the balls of
radius $r_n$ about the points $\{X_{i_1}\ldots,X_{i_k}\}$ have non-trivial
intersection.
\begin{definition}
The points $\{x_1,\ldots,x_k\}\subseteq\mathbb{R}^d$ form an
{\it empty $( k-1)$-simplex} with respect to $r$ if for each
$j_o\in\{1,\ldots,k\}$, the intersection $\displaystyle\bigcap_{\substack{1\le
j\le k\\j\neq j_o}}B_{r}(x_j)$ is non-empty, but the intersection
$\displaystyle\bigcap_{1\le j\le k}B_{r}(x_j)=\emptyset.$
\end{definition}
Let
$h_r(x_1,\ldots,x_k)$ be the indicator that $\{x_1,\ldots,x_k\}$ form
an empty $( k-1)$-simplex with respect to $r$, and for a multiindex ${\bf
i}=(i_1,\ldots,i_k)$ with $1\le i_1<\cdots<i_k\le n$, let $\xi_{\bf
i}=h_{r_n}(X_{i_1},\ldots,X_{i_k})$.
Let
$$S_{n,k}:=\sum_{\substack{{\bf i}=(i_1,\ldots,i_k)\\1\le i_1<\cdots<i_k\le n}}
\xi_{\bf i};$$
that is, $S_{n,k}$ is the number of empty $(k-1)$-simplices in $
\mathcal{C}(X_1,\ldots,X_n).$
Another object of equal importance in what follows is $\widetilde{S}_{n,k}$,
the number of {\it isolated}
empty $k$-simples. That is, if $\zeta_{(i_1,\ldots,i_k)}$ is the indicator
that $\{X_{i_1},\ldots,X_{i_k}\}$ form an empty $( k-1)$-simplex with respect to $r_n$
and that there are no edges between $\{X_j\}_{j\in\{i_1,\ldots,i_k\}}$ and
$\{X_j\}_{j\notin\{i_1,\ldots,i_k\}}$, then
$$\widetilde{S}_{n,k}=\sum_{\substack{{\bf i}=(i_1,\ldots,i_k)\\1\le i_1<\cdots<i_k\le n}}
\zeta_{\bf i}.$$
The random variables $S_{n,k}$ and $\widetilde{S}_{n,k}$ are related to $\beta_{k-1}$
as follows. Firstly,
$\beta_{k-1}$ is bounded below by the number of isolated empty
$k$-simplices; that is, $\beta_{k-1}(\mathcal{C}(X_1,\ldots,X_n))\ge
\widetilde{S}_{n,k}.$ Furthermore, any contribution to $\beta_{k-1}$ not coming
from an isolated empty $( k-1)$-simplex comes from a component in
$\mathcal{C}(X_1,\ldots,X_n)$ on at least $k+1$ vertices. In order
for such a component to contribute to $\beta_{k-1}$, $(k-2)$-dimensional faces. Such faces are
necessarily triangulated (by the construction of
$\mathcal{C}(X_1,\ldots,X_n)$), and so any further contribution to
$\beta_{k-1}$ contains at least one simplex on $k-1$ vertices, with
either an extra edge attached to each of two different vertices
(terminating in different places), or else an extra path of length two
attached to one vertex.
Let $Y_{n,k}$ denote the number of
simplices in $\mathcal{C}(X_1,\ldots,X_n)$ on $k-1$ vertices with two extra
edges attached, counted once for each simplex on $k-1$ vertices which occurs
and for each distinct pair of simplex vertices with an extra edge.
Similarly, let $Z_{n,k}$ denote the number of
simplices in $\mathcal{C}(X_1,\ldots,X_n)$ on $k-1$ vertices with at
least one extra path of length 2 attached, counted once for each simplex which
occurs and for each vertex with a path of length two attached.
The argument above shows that
\begin{equation}\label{bounds}
\widetilde{S}_{n,k}\le\beta_{k-2}(\mathcal{C}(X_1,\ldots,X_n))\le S_{n,k}+Y_{n,k}+Z_{n,k},
\end{equation}
where the trivial bound $\widetilde{S}_{n,k}\le S_{n,k}$ has also been used.
The limiting distribution of $\beta_{k-1}$ will follow as in the previous
section by proving the same
limit theorems for the upper and lower bounds of \eqref{bounds}. The theorem
is the following.
\begin{thm}\label{CC_clt}
\
\begin{enumerate}
\item \label{CC_clt_zero}If $n^kr_n^{d(k-1)}\to0$ as $n\to\infty$, then
$$\beta_k(\mathcal{C}(X_1,\ldots,X_n))\rightarrow0\quad a.a.s.\ as\ n\to\infty.$$
\item \label{CC_clt_poisson}If $n^kr_n^{d(k-1)}\to\alpha\in(0,\infty)$ as $n\to\infty$, then
$$d_{TV}(\beta_k(\mathcal{C}(X_1,\ldots,X_n)),Y)\le cnr_n^d,,$$
where $Y$ is a Poisson random variable with $\mathbb{E}[Y]=\mathbb{E}[\beta_k]$ and $c$
is a constant depending only on $d$, $k$, and $f$.
\item \label{CC_clt_normal}If $n^kr_n^{d(k-1)}\to\infty$ as $n\to\infty$ and $nr_n^d\to0$ as $n\to\infty$,
then
$$\frac{\beta(\mathcal{C}(X_1,\ldots,X_n))-\mathbb{E}[\beta(\mathcal{C}(X_1,\ldots,X_n))]}{
\sqrt{\mathrm{Var}(\beta(\mathcal{C}(X_1,\ldots,X_n)))}}\Rightarrow\mathcal{N}(0,1).$$
\end{enumerate}
\end{thm}
The first step in proving Theorem
\ref{CC_clt} is to determine the order in $n$ and $r_n$ of
$\mathbb{E}[\widetilde{S}_{n,k}]$ and $\mathbb{E}[S_{n,k}+Y_{n,k}+Z_{n,k}]$. In fact, slightly more is needed. Let $A$ be an open subset of $\mathbb{R}^d$
such that $vol(\partial A)=0$. Let $\mathcal{X}$ be a finite subset of $\mathbb{R}^d$,
and call $x\in\mathcal{X}$ the ``left-most'' point of $\mathcal{X}$ (denoted $LMP(\mathcal{X})$)
if $x$ is the first element of $\mathcal{X}$ when $\mathcal{X}$ is ordered
lexicographically. Now, define $S_{n,k,A}$ to be the number of empty
$( k-1)$-simplices formed from $X_1,\ldots,X_n$, such that
the left-most point of the $k$-simplex is in $A$. Define $\widetilde{S}_{n,k,A}$
in the analogous way.
\begin{lemma}\label{exp-order}
For $k>1$, let $$\mu_A:=\left(\int_{A}f(x)^kdx\right)\int_{(\mathbb{R}^d)^{k-1}}h_1(0,y_2,\ldots,y_k)
d(y_2,\ldots,y_k).$$
Then
$$\lim_{n\to\infty}n^{-k}r_n^{-d(k-1)}\mathbb{E}\left[S_{n,k,A}\right]=
\lim_{n\to\infty}n^{-k}r_n^{-d(k-1)}\mathbb{E}[\widetilde{S}_{n,k,A}]=\frac{\mu_A}{k!}.$$
\end{lemma}
Observe that $\mu_A$ depends only on $f$ and $A$
and can be trivially bounded by
$\|f\|_\infty^{k-1}(2^d\theta_d)^{k-1},$ where $\theta_d$ is the volume of the
unit ball in $\mathbb{R}^d$.
\begin{comment}
\begin{proof}
In what follows, $A_n\simeq B_n$ is used to mean $\lim_{n\to\infty}\frac{A_n}{
B_n}=1.$ Let $h_{r_n,A}(x_1,\ldots,x_k)$ be the indicator that
$\{x_1,\ldots,x_k\}$ form an empty $k$-simplex with left-most point in
$A$. Then by definition of $S_{n,k,A}$,
\begin{equation*}\begin{split}
\mathbb{E}\left[S_{n,k,A}\right]&=\binom{n}{k}\mathbb{E}\left[\xi_{(1,\ldots,k)}\1_{\{LMP(X_{
1},\ldots,X_{k})\in A\}}\right]\\&
\simeq\frac{n^k}{k!}\int_{(\mathbb{R}^d)^k}h_{r_n,A}(x_1,\ldots,x_k)\prod_{i=1}^kf(x_i)
d(x_1,\ldots,x_k)\\&=\frac{n^k}{k!}\int_{(\mathbb{R}^d)^k}h_{r_n,A}(x_1,\ldots,x_k)
f(x_1)^kd(x_1,\ldots,x_k)\\&\qquad\qquad+\frac{n^k}{k!}\int_{(\mathbb{R}^d)^k}
h_{r_n,A}(x_1,\ldots,x_k)\left[\prod_{i=1}^kf(x_i)-f(x_1)^k\right]
d(x_1,\ldots,x_k).
\end{split}\end{equation*}
Let $I_1$ denote the first expression and $I_2$ the second.
To analyze $I_1$, make the change of variables $y_i=r_n^{-1}(x_i-x_1)$ for
$i\ge 2$. Then
$$I_1=\frac{n^kr_n^{d(k-1)}}{k!}\int_{(\mathbb{R}^d)^k}h_{r_n,A}(x_1,x_1+r_ny_2,\ldots,
x_k+r_ny_k)f(x_1)^kd(x_1,y_2,\ldots,y_k).$$
Recall that $h_{r_n,A}(x_1,\ldots,x_k)$ is the indicator that $\{x_1,\ldots,
x_k\}$ form an empty $k$-simplex, with respect to $r_n$, whose left-most
point is in $A$. Note that, since $A$ is open, if $x_1\in A$ and $r_n$
is sufficiently small, then if
$\{x_1,\ldots,x_k\}$ form an empty $k$-simplex with respect to $r_n$,
its left-most point is in $A$. On the other hand, if $x_1\notin A
\cup\partial A$, then for $r_n$ small enough, $h_{r_n,A}(x_1,\ldots,x_k)=0$.
It follows then, by translation
and rescaling, that for $n$ sufficiently large (depending on $x_1$),
$$h_{r_n}(x_1,x_1+r_ny_2\ldots,x_1+r_ny_k)=h_1(0,y_2,\ldots,y_k)\1_A(x_1).$$
It follows by the dominated convergence theorem that
$\lim_{n\to\infty}n^{-k}r_n^{-d(k-1)}I_1=(k!)^{-1}\mu_A,$
and it remains to show that $\lim_{n\to\infty}n^{-k}r_n^{-d(k-1)}I_2=0.$
To do this, observe first taking $A=\mathbb{R}^d$ only makes $I_2$ larger.
Furthermore, for $h_{r_n}(x_1,\ldots,x_k)$ to be nonzero,
it must be the case that $x_i\in B_{2r_n}(x_1)$ for each $i\ge 2$. One can
thus write
\begin{equation*}\begin{split}
n&^{-k}r_n^{-d(k-1)}\big|I_2\big|\\&=\frac{1}{(k!)r_n^{d(k-1)}}\left|\int_{(\mathbb{R}^d)^k}
h_{r_n}(x_1,
\ldots,x_k)f(x_1)\left(\prod_{i=2}^kf(x_i)-f(x_1)^{k-1}\right)d(x_1,\ldots,
x_k)\right|\\&\le C \int_{\mathbb{R}^d}\left[\frac{1}{
\left[(2r_n)^{d}\theta_d\right]^{k-1}}
\int_{B_{2r_n}(x_1)^{k-1}}\left|\prod_{i=2}^kf(x_i)-f(x_1)^{k-1}\right|
d(x_2,\ldots,x_k)\right]f(x_1)dx_1,
\end{split}\end{equation*}
where $$C=\frac{2^{d(k-1)}\theta_d^{k-1}}{k!}.$$
From this last expression, it follows from the dominated convergence theorem
that if $f$ is continuous at $x_1$, then the inner integral over
$(x_2,\ldots,x_k)$ tends to zero as $n$ tends to infinity, for $d$ and $k$
fixed. In fact, the following inductive argument shows that $x_1$ only
needs to be a Lebesgue point of $f$; that is, it suffices that
$$\lim_{r\downarrow 0}\left(r^{-d}\int_{B_r(x_1)}|f(y)-f(x)|dy\right)=0.$$
If $k=2$, that it suffices for $x_1$ to be a Lebesgue point of $f$
is obvious. For $k>2$, bound the integrand above by
\begin{equation*}\begin{split}|f(x_k)-f(x_1)|\prod_{i=2}^{k-1}f(x_i)&+
f(x_1)\left|\prod_{i=2}^{k-1}f(x_i)-f(x_1)^{k-2}\right|\\&\le
|f(x_k)-f(x_1)|\|f\|_\infty^{k-2}+f(x_1)\left|\prod_{i=2}^{k-1}
f(x_i)-f(x_1)^{k-2}\right|.\end{split}\end{equation*}
Now, $\frac{1}{(2r_n)^d}\int_{B_{2r_n}(x_1)}|f(x_k)-f(x_1)|dx_k\xrightarrow{n
\to\infty}0$ by definition of a Lebesgue point, and
$$\frac{1}{(2r_n)^{d(k-2)}}\int_{(B_{2r_n}(x_1))^{k-2}}\left|\prod_{i=2}^{k-1}
f(x_i)-f(x_1)^{k-2}\right|d(x_2,\ldots,x_{k-1})\xrightarrow{n
\to\infty}0$$ by induction hypothesis.
Since almost every point of $\mathbb{R}^d$ is a Lebesgue point of $f$, it now follows
by the dominated convergence theorem that $n^{-k}r_n^{-d(k-1)}I_2\to0$ as
$n\to\infty$.
\end{proof}
\end{comment}
\begin{lemma}\label{exp-upper}
Let $$\mu':=\left(\int_{\mathbb{R}^d}f(x)^{k+1}dx\right)\int_{(\mathbb{R}^d)^k}
g_1^{1,2}(0,y_1,\ldots,y_k)dy_1\cdots d y_k,$$
where $g_1^{1,2}(x_0,\ldots,x_k)$ is the indicator that
$\{x_0,\ldots,x_{k-2}\}$ form a simplex (where a
complex is built as described on $x_0,\ldots,x_{k}$ with
threshhold radius $1$) and that $\{x_0,x_{k-1}\}$ and
$\{x_1,x_{k}\}$ are edges.
Let $$\mu'':=\left(\int_{\mathbb{R}^d}f(x)^{k+1}dx\right)\int_{(\mathbb{R}^d)^k}
k_1^{1}(0,y_1,\ldots,y_k)dy_1\cdots d y_k.$$
Let $k^{1}_{1}(x_0,\ldots,x_{k})$
be the indicator that $\{x_0,\ldots,x_{k-2}\}$ form a simplex and
that $\{x_0,x_{k-1}\}$ and $\{x_{k-1},x_{k}\}$ are edges.
Then
$$\lim_{n\to\infty}n^{-(k+1)}r_n^{-dk}\mathbb{E}[Y_{n,k}]=\frac{\mu'}{2(k-3)!},$$
and
$$\lim_{n\to\infty}n^{-(k+1)}r_n^{-dk}\mathbb{E}[Z_{n,k}]=\frac{\mu''}{(k-2)!}.$$
\end{lemma}
\begin{cor}
For $S_{n,k},Y_{n,k},Z_{n,k}$ as above,
$$\mathbb{E}[S_{n,k}+Y_{n,k}+Z_{n,k}]\simeq\mathbb{E}[\widetilde{S}_{n,k}].$$
\end{cor}
The proofs of these facts are identical to the proofs of the corresponsing
facts for subgraph counts of random geometric graphs given in Chapter 3 of
\cite{penrose}.
\begin{comment}
\begin{proof}
Let $B_{n,A}$ be the event that $\{X_1,\ldots,X_k\}$ form an isolated
empty $k$-simplex with respect to $r_n$, with left-most point in $A$.
Given that $\{X_1,\ldots,X_k\}$ form an empty $k$-simplex with respect to
$r_n$ with left-most point in $A$,
the conditional probability of $B_{n,A}$ is simply the probability that no
edges connect $\{X_i\}_{1\le i\le k}$ to $\{X_i\}_{k< i\le n}$. Since
$\{X_1,\ldots,X_k\}$ form an empty $k$-simplex, they must all lie inside
a ball of radius $2r_n$, and so the conditional probability in question
is bounded below by $(1-\|f\|_\infty\theta_d(4r_n)^d)^{n-k}$. It follows
that
\begin{align*}
(1-\|f\|_\infty\theta_d(4r_n)^d)^{n-k}\mathbb{E}[\xi_{(1,\ldots,k)}\1_{\{LMP(X_1, \ldots,X_k)\in A\}}] & \le \P[B_n] \\
& \le \mathbb{E}[\xi_{(1,\ldots,k)}\1_{\{LMP(X_1,\ldots,X_k)\in A\}}],
\end{align*}
and thus
$$(1-\|f\|_\infty\theta_d(4r_n)^d)^{n-k}\mathbb{E}[S_{n,k,A}]\le\mathbb{E}[\widetilde{S}_{n,k,A}]
\le\mathbb{E}[S_{n,k,A}].$$
Since $r_n=o\left(n^{-1/d}\right)$, $(1-\|f\|_\infty\theta_d(4r_n)^d)^{n-k}$
tends to one as $n$ tends to infinity.
\end{proof}
\end{comment}
\medskip
This last corollary is already enough to prove part \ref{CC_clt_zero} of Theorem
\ref{CC_clt}: if $n^kr_n^{d(k-1)}\to0$ as $n\to\infty$, then
$$\P\big[\beta_k(\mathcal{C}(X_1,\ldots,X_n)\ge 1\big]\le\mathbb{E}\big[
\beta_k(\mathcal{C}(X_1,\ldots,X_n)\big]\le\mathbb{E}\big[S_{n,k}+Y_{n,k}+
Z_{n,k}\big]\xrightarrow{n\to\infty}0.$$
\medskip
In order to prove part \ref{CC_clt_poisson}, the following
abstract approximation theorem of Arratia, Goldstein, and Gordon is needed.
\begin{thm}[\cite{agg}]\label{Poi-approx}
Let $(\xi_i,i\in I)$ be a finite collection of Bernoulli random variables
with dependency graph $(I,\sim)$. Let $p_i:=\mathbb{E}[\xi_i]$ and $p_{ij}:=\mathbb{E}[\xi_i
\xi_j].$ Let $\lambda:=\sum_{i\in I}p_i,$ and let $W:=\sum_{i\in I}\xi_i$.
Then $$d_{TV}(W,Poi(\lambda))\le\min(3,\lambda^{-1})\left(\sum_{i\in I}\sum_{
\substack{j\sim i\\j\neq i}}p_{ij}+\sum_{i\in I}\sum_{j\sim i}p_ip_j\right). $$
\end{thm}
Penrose \cite{penrose} used this theorem to prove Poisson approximation
results for subgraph counts of random geometric graphs; one can follow
this approach essentially without change to prove the following
result, which holds in the entire sparse regime.
\begin{thm}\label{Poisson-bd}
With definitions as above,
$$d_{TV}\big(S_{n,k},Poi(\mathbb{E}[S_{n,k}])\big)\le c_{k,d,f}\big[nr_n^d\big],$$
for a constant $c_{d,k,f}$ depending only on $d$, $k$, and $\|f\|_\infty$.
\end{thm}
\begin{comment}
\begin{proof}[Proof of Theorem \ref{Poisson-bd}]
Recall that $S_{n,k}=\sum_{{\bf i}\in I}\xi_{\bf i},$ with
$$I=\{{\bf i}=(i_1, \ldots,i_k):1\le i_1<i_2< \cdots<i_k\le
n\}$$ and $\xi_{\bf i}$ the indicator that the i.i.d. points
$X_{i_1},\ldots,X_{i_k}$ form an empty $( k-1)$-simplex
with respect to $r_n$. Observe that if
$\{i_1,\ldots,i_k\}\cap\{j_1,\ldots,j_k\}=\emptyset$, then the
corresponding random variables $\xi_{(i_1,\ldots,i_k)}$ and
$\xi_{(i_1, \ldots,i_k)}$ are independent, thus the dependency
relation $\sim$ can be taken to be
$(i_1,\ldots,i_k)\sim(j_1,\ldots,j_k)$ if and only if
$\{i_1,\ldots,i_k\}\cap\{j_1,\ldots,j_k\}\neq\emptyset.$
Now, note also that for $x_1,\ldots,x_k$ to form an empty $k$-simplex
with respect to $r_n$, it must be that each of the points $x_2,\ldots,x_k$
is within a distance $2r_n$ of $x_1$. It follows that
\begin{equation}\label{pi}
p_{(i_1,\ldots,i_k)}=\mathbb{E}\left[\xi_{(i_1,\ldots,i_k)}\right]\le
\left[(2r_n)^d\theta_d\|f\|_\infty\right]^{k-1}.
\end{equation}
Similarly, if $\big|\{i_1,\ldots,i_k\}\cap\{j_1,\ldots,j_k\}\big|=\ell,$
then
\begin{equation}\label{pij}
p_{(i_1,\ldots,i_k),(j_1,\ldots,j_k)}=\mathbb{E}\left[\xi_{(i_1,\ldots,i_k)}
\xi_{(j_1,\ldots,j_k)}\right]\le
\left[(2r_n)^d\theta_d\|f\|_\infty\right]^{2k-\ell-1}.
\end{equation}
Now, given ${\bf i}\in I$, the number of ${\bf j}\in I$ with ${\bf i}\sim
{\bf j}$ (including ${\bf i}$ itself) is
\begin{equation}\label{Ni}
\binom{n}{k}-\binom{n-k}{k}=\frac{k^2n^{k-1}}{k!}+O\left(n^{k-2}\right);
\end{equation}
for $(i_1,\ldots,i_k)$ given, the number of $(j_1,\ldots,j_k)$ with
$\big|\{i_1,\ldots,i_k\}\cap\{j_1,\ldots,j_k\}\big|=\ell$ is
\begin{equation}\label{Nil}
\binom{k}{\ell}\binom{n-k}{k-\ell}=\binom{k}{\ell}\frac{1}{(k-\ell)!}n^{k-\ell}+
O\left(n^{k-\ell-1}\right).
\end{equation}
It then follows from Theorem \ref{Poi-approx} and Lemma \ref{exp-order} that
\begin{align*}
d_{TV}&\big(S_{n,k},Poi(\mathbb{E}[S_{n,k}])\big)\\ & \lesssim \min\left(3,\frac{1}{\mathbb{E}[S_{n,k}]}
\right)\left[\binom{n}{k}\sum_{\ell=1}^{k-1}\binom{k}{\ell}\frac{1}{(k-\ell)!}
n^{k-\ell}\left[(2r_n)^d\theta_d\|f\|_\infty\right]^{2k-\ell-1}\right.\\
& \left.\phantom{\sum_{\ell=1}^{k-1}bbbbbbbbbbbbbbbbbb}+\binom{n}{k}
\frac{k^2n^{k-1}}{k!}\left[(2r_n)^d\theta_d\|f\|_\infty\right]^{2k-2}\right]\\
&\lesssim \frac{1}{\mu(k!)^2}\sum_{\ell=1}^{k-1}\binom{k}{\ell}\frac{1}{
(k-\ell)!}\left[2^d\theta_d\|f\|_\infty\right]^{2k-\ell-1}\left[nr_n^d\right]^{k-
\ell}\\
&\phantom{\sum_{\ell=1}^{k-1}bbbbbbbbbbbbbbbbbb}+\frac{k^2}{\mu(k!)^3}\left[2^d\theta_d\|f\|_\infty\right]^{2k-2}
\left[nr_n^d\right]^{k-1}\\&\lesssim c_{k,d,f}\big[nr_n^d\big],
\end{align*}
for a constant $c_{k,d,f}$ depending on $k$, $d$, and $\|f\|_\infty$.
\end{proof}
\end{comment}
\begin{cor}\label{component-Poisson}
If $n^kr_n^{d(k-1)}\to\alpha\in(0,\infty)$ as $n\to\infty$, then
$$d_{TV}\big(\widetilde{S}_{n,k},Poi(\mathbb{E}[\widetilde{S}_{n,k}])\big)\le \tilde{c}_{d,k,f}\alpha(nr_n^d).$$
\end{cor}
That is, in the regime of part \ref{CC_clt_poisson} of the theorem,
the lower bound for $\beta_k$ given in \eqref{bounds} is approximately Poisson.
\begin{proof}
Note that $S_{n,k}-\widetilde{S}_{n,k}$ is the number of empty $( k-1)$-simplices among
$\{X_,\ldots,X_n\}$ which are not isolated, and is thus bounded above
by the number of connected subsets of $\{X_,\ldots,X_n\}$ with $k+1$
points, $k$ of which form an empty $k$-simplex. The expected
number of such sets is
bounded by $$\binom{n}{k+1}k\|f\|_\infty^{k+1}\theta_d^{k+1}(2r_n)^{d(k-1)}(4r_n)^d
\simeq\left(\frac{k\|f\|_\infty^{k+1}\theta_d^{k+1}2^{d(k+1)}}{(k+1)!}\right)
n^{k+1}r_n^{dk},$$ so that
\begin{equation*}\begin{split}
d_{TV}(S_{n,k},\widetilde{S}_{n,k})&=\big|\P[S_{n,k}\in A]-\P[\widetilde{S}_{n,k}\in A]\big|\\&=
\big|\P[S_{n,k}\in A,S_{n,k}\neq\widetilde{S}_{n,k}]-\P[\widetilde{S}_{n,k}\in A,S_{n,k}\neq\widetilde{S}_{n,k}]
\big|\\&\le c_{d,k,f}n^{k+1}r_n^{dk}\\&\le\tilde{c}_{d,k,f}\alpha nr_n^d.
\end{split}\end{equation*}
Moreover, it is easy to see in general that if $Y_\alpha$ and $Y_\beta$ have Poisson
distributions with means $\alpha$ and $\beta$, respectively, then
$d_{TV}(Y_\alpha,Y_\beta)\le |\alpha-\beta|$, and so
$$d_{TV}(Poi(\mathbb{E}[S_{n,k}]),Poi(\mathbb{E}[\widetilde{S}_{n,k}]))\le c_{d,k,f}\alpha nr_n^d$$
as well.
\end{proof}
\medskip
The
following result, proved below using Theorem \ref{Poi-approx},
holds throughout the sparse regime.
\begin{thm}\label{upper-Poisson}
There is a constant $c_{d,k,f}$ depending on $d$, $k$, and $f$ only,
so that with $S_{n,k},Y_{n,k},Z_{n,k}$ as above,
$$d_{TV}(S_{n,k}+Y_{n,k}+Z_{n,k}, Poi(\mathbb{E}[\widetilde{S}_{n,k}]))\le c_{d,k,f}nr_n^d.$$
\end{thm}
The inequalities in \eqref{bounds} together with Corollary
\ref{component-Poisson} and Theorem \ref{upper-Poisson} yield part
\ref{CC_clt_poisson} almost immediately.
\begin{proof}[Proof of part \ref{CC_clt_poisson} of Theorem \ref{CC_clt}]
By the left-hand inequality in \eqref{bounds} and Corollary
\ref{component-Poisson},
$$\P[\beta_{k-1}\le m]\le\P[\widetilde{S}_{n,k}\le m]\le \P[Y\le m]+c_{d,k,f}nr_n^d,$$
where $Y$ is a Poisson random variable with mean $\mathbb{E}[\widetilde{S}_{n,k}]$.
By the right-hand inequality in \eqref{bounds} and Theorem
\ref{upper-Poisson},
$$\P[\beta_{k-1}\le m]\ge\P[S_{n,k}+Y_{n,k}+Z_{n,k}\le m]\ge \P[Y\le m]-
c_{d,k,f}nr_n^d.$$
As in the previous proof, $Y$ can be replaced by a Poisson random variable
with mean $\mathbb{E}[\beta_k(\mathcal{C}(X_1,\ldots,X_n))]$ with only a change of
constant in the error term.
\end{proof}
\begin{proof}[Proof of Theorem \ref{upper-Poisson}]
For notational convenience, let $W_{n,k}:=S_{n,k}+Y_{n,k}+Z_{n,k}$.
For $1\le
p<q\le k-1$, let $g^{p,q}_{r_n}(x_1,\ldots,x_{k+1})$ be the
indicator that $\{x_1,\ldots,x_{k-1}\}$ form a simplex (where a
complex is built as described on $x_1,\ldots,x_{k+1}$ with
threshhold radius $r_n$) and that $\{x_p,x_k\}$ and
$\{x_q,x_{k+1}\}$ are edges. Let $k^{p}_{r_n}(x_1,\ldots,x_{k+1})$
be the indicator that $\{x_1,\ldots,x_{k-1}\}$ form a simplex and
that $\{x_p,x_k\}$ and $\{x_k,x_{k+1}\}$ are edges. For ${\bf j}=
(j_1,\ldots,j_{k+1})$, let $\gamma^{p,q}_{\bf
j}=g^{p,q}_{r_n}(X_{j_1}, \ldots,X_{j_{k+1}})$ and let
$\eta^p_{\bf j}=k^p_{r_n}(X_{j_1},\ldots,X_{j_{k+1}}).$ Then
\begin{align*}
W_{n,k}&=\sum_{1\le i_1<\cdots<i_k\le n}\xi_{\bf i}
+\sum_{\substack{
1\le j_1<\cdots<j_{k-1}\le
n\\j_k,j_{k+1}\notin\{j_1,\ldots,j_{k-1}\}\\j_k\neq j_{k+1}}}\sum_{1\le p<q\le
k-1}\gamma^{p,q}_{\bf j}\\
& \qquad\qquad+\sum_{\substack{
1\le j_1<\cdots<j_{k-1}\le
n\\j_k,j_{k+1}\notin\{j_1,\ldots,j_{k-1}\}\\j_k\neq j_{k+1}}}\sum_{1\le p\le
k-1}\eta^{p}_{\bf j}.
\end{align*}
The proof that $W_{n,k}$ has an approximate Poisson distribution
proceeds along the same lines as the proof given by Penrose for subgraph
counts. For the
Bernoulli random variables in the sum above, one can take a dependency
graph to be ${\bf i}\sim{\bf j}$ if ${\bf i}\cap{\bf j}\neq\emptyset$.
(Abusing notation, ${\bf i}$ is also used here to denote the set of
indices from the multiindex ${\bf i}$.) Note that it is not important
that ${\bf i}$ and ${\bf j}$ be the same size.
Now, $\mathbb{E}[\xi_{\bf i}]\le [(2r_n)^d\theta_d\|f\|_\infty]^{k-1}$
and if $|{\bf i}\cap{\bf i'}|=\ell$,
then $$\mathbb{E}[\xi_{\bf i}\xi_{\bf i'}]\le [(2r_n)^d\theta_d\|f\|_\infty]^{2k-\ell-1},$$
since if set of $k$ points forms a simplex, they must
all be in the ball of radius $2r_n$ about the first point.
Given ${\bf i}=(i_1,\ldots,i_k)$, the number of ${\bf i'}=(i_1',\ldots,
i_k')$ with ${\bf i}\sim
{\bf i'}$ (including ${\bf i}$ itself) is
\begin{equation*}
\binom{n}{k}-\binom{n-k}{k}=\frac{k^2n^{k-1}}{k!}+O\left(n^{k-2}\right);
\end{equation*}
for ${\bf i}$ as above, the number of ${\bf i}=(i_1',\ldots,i_k')$ with
$\big|{\bf i}\cap{\bf i'}\big|=\ell$ is
\begin{equation*}
\binom{k}{\ell}\binom{n-k}{k-\ell}=\binom{k}{\ell}\frac{1}{(k-\ell)!}n^{k-\ell}+
O\left(n^{k-\ell-1}\right).
\end{equation*}
This means that the contribution to the error term (without the $\min(3,
\lambda^{-1})$ factor in front) from Theorem \ref{Poi-approx} of the form
$p_{\bf i}p_{\bf i'}$ for ${\bf i}\sim{\bf i'}$ is, to top-order in $n$,
$$\frac{kn^{2k-1}}{k!
(k-1)!}\left[(2r_n)^d\theta_d\|f\|_\infty\right]^{2k-2},$$
and the contribution from terms of the form $p_{\bf ii'}$ is (to top order)
$$\binom{n}{k}\sum_{\ell=1}^{k-1}\binom{k}{\ell}\frac{1}{(k-\ell)!}n^{k-\ell}
[(2r_n)^d\theta_d\|f\|_\infty]^{2k-\ell-1}\lesssim n^{k+1}r_n^{dk}.$$
Similar to above, $\mathbb{E}[\gamma^{p,q}_{\bf j}]\le
2^d\left[(2r_n)^d\theta_d\|f\|_\infty\right]^k$ and if $|{\bf j}\cap{\bf
j'}|=\ell$, then $$\mathbb{E}[\gamma_{\bf j}^{p,q}\gamma_{\bf
j'}^{p',q'}]\le
2^{3d}\left[(2r_n)^d\theta_d\|f\|_\infty\right]^{2k+1-\ell}.$$
Given ${\bf j}=(j_1,\ldots,j_{k+1})$, the number of ${\bf j'}=
(j_1',\ldots,j_{k+1}')$ with ${\bf j}\sim{\bf j'}$ is
$$\frac{(k+1)^2n^k}{(k+1)!}+O(n^{k-1})$$
and the number of ${\bf j'}$ with $|{\bf j}\cap{\bf j'}|=\ell$ is
$$\binom{k+1}{\ell}\frac{n^{k+1-\ell}}{(k+1-\ell)!}+O(n^{k-\ell}).$$
This yields a top-order contribution to the error from Theorem
\ref{Poi-approx} from the $\mathbb{E}[\gamma_{\bf j}]\mathbb{E}[\gamma_{\bf j'}]$ and
$\mathbb{E}[\gamma_{\bf j}\gamma_{\bf j'}]$ terms of order
\begin{equation*}\begin{split}
\frac{(k+1)^2n^{2k+1}}{[(k+1)!]^2}&\binom{k-1}{2}^22^{2d}\left[
(2r_n)^d\theta_d\|f\|_\infty\right]^{2k}\\&+\binom{n}{k+1}\sum_{\ell=1}^{k+1}
\binom{k-1}{2}^2\binom{k+1}{\ell}\frac{n^{k+1-\ell}}{(k+1-\ell)!}
2^{3d}\left[(2r_n)^d\theta_d\|f\|_\infty\right]^{2k+1-\ell}\\
&\lesssim n^{k+1}r_n^{dk}.
\end{split}\end{equation*}
In the same way,
$\mathbb{E}[\eta_{\bf j}^p]\le
2^d\left[(2r_n)^d\theta_d\|f\|_\infty\right]^k,$
and if $|{\bf j}\cap{\bf
j'}|=\ell$, then $$\mathbb{E}[\eta_{\bf j}^{p}\eta_{\bf
j'}^{p'}]\le
2^{3d}\left[(2r_n)^d\theta_d\|f\|_\infty\right]^{2k+1-\ell},$$
thus the contribution from the terms of the form $\mathbb{E}[\eta_{\bf j}]\mathbb{E}[\eta_{
\bf j'}]$ and of the form $\mathbb{E}[\eta_{\bf j}\eta_{\bf j'}]$ is of the same
order as the contribution above from the corresponding $\gamma$ terms.
The cross terms are essentially the same: if $|{\bf i} \cap{\bf j}|=\ell$,
then
\begin{equation*}\begin{split}
\mathbb{E}[\xi_{\bf i}\gamma_{\bf j}^{p,q}]\le2^{2d}\left[(2r_n)^d\theta_d\|f\|_\infty
\right]^{2k-\ell}\qquad&\qquad\mathbb{E}[\xi_{\bf i}\eta_{\bf j}^{p}]\le2^{3d}
\left[(2r_n)^d\theta_d\|f\|_\infty\right]^{2k-\ell}\\
\mathbb{E}[\gamma^{p,q}_{\bf i}\eta_{\bf j}^{r}]\le2^{4d}&\left[(2r_n)^d\theta_d
\|f\|_\infty\right]^{2k+1-\ell}.
\end{split}\end{equation*}
The number of ${\bf j}=(j_1,\ldots,j_{k+1})$ with ${\bf i}\sim{\bf j}$
is
$$\binom{n}{k+1}-\binom{n-k}{k+1}=\frac{n^k}{(k-1)!}+O(n^{k-1}).$$
and the number of such ${\bf j}$ with $|{\bf i}\cap{\bf j}|=\ell$ is
$$\binom{k}{\ell}\binom{n-k}{k+1-\ell}=\binom{k}{\ell}\frac{n^{k+1-\ell}}{
(k+1-\ell)!}+O(n^{k-\ell}).$$
This yields a contribution from the $\xi$-$\gamma$ cross-terms of
\begin{equation*}\begin{split}
\frac{n^{2k}}{k!(k-1)!}\binom{k-1}{2}&2^d\left[(2r_n)^d\theta_d\|f\|_\infty
\right]^{2k-1}\\&+\binom{n}{k}\sum_{\ell=0}^k\binom{k-1}{2}\binom{k}{\ell}
\frac{n^{k+1-\ell}}{(k+1-\ell)!}2^{2d}\left[(2r_n)^d\theta_d\|f\|_\infty
\right]^{2k-\ell}\\&\lesssim n^{k+1}r_n^{dk}.
\end{split}\end{equation*}
The contribution from the $\xi$-$\eta$ cross terms is the same up to
constants depending only on $k$ and $d$, and the contribution from the
$\gamma$-$\eta$ cross terms is
\begin{eqnarray*}
& &\frac{(k+1)^2n^{2k+1}}{[(k+1)!]^2}(k-1)\binom{k-1}{2}2^{2d}\left[(2r_n)^d
\theta_d\|f\|_\infty\right]^{2k}\\
&+&\binom{n}{k+1}\sum_{\ell=0}^{k+1}
(k-1)\binom{k-1}{2}\binom{k+1}{\ell}\frac{n^{k+1-\ell}}{(k+1-\ell)!}
2^{4d}\left[(2r_n)^d\theta_d
\|f\|_\infty\right]^{2k+1-\ell}\\
&\lesssim & n^{k+1}r_n^{dk}.
\end{eqnarray*}
Collecting terms and using that $\lambda=
\mathbb{E}[W_{n,k}]\simeq n^kr_n^{d(k-1)}
\left(\frac{\mu}{k!}\right)$, Theorem \ref{Poi-approx} yields
$$d_{TV}(W,Poi(\lambda))\le c_{d,k,f}nr_n^d.$$
Again, one can replace $\lambda$ with $\mathbb{E}[\widetilde{S}_{n,k}]$ with only a loss in the
value of the constant $c_{d,k,f}$.
\end{proof}
\bigskip
The remainder of the section is devoted to the proof of part \ref{CC_clt_normal}
of Theorem \ref{CC_clt}. A central limit theorem for the
recentered, renormalized upper bound of $\beta_k$ given in
\eqref{bounds} follows
immediately from Theorem \ref{upper-Poisson} in this
range of $r_n$, by the classical
result that a Poisson random variable with mean
tending to infinity tends to a Gaussian random variable when recentered
and renormalized.
\begin{thm}\label{upper-normal}
If $nr_n^d\xrightarrow{n\to\infty}0$ and $n^kr_n^{d(k-1)}\xrightarrow\infty,$
then
$$\frac{S_{n,k}+Y_{n,k}+Z_{n,k}-\mathbb{E}[\widetilde{S}_{n,k}]}{\sqrt{\mathbb{E}[\widetilde{S}_{n,k}]}}\Longrightarrow
\mathcal{N}(0,1)$$
as $n$ tends to infinity.
\end{thm}
Clearly the approach to the lower bound of \eqref{bounds}
taken in the regime in which $n^kr_n^{d(k-1)}\to\alpha\in(0,\infty)$
also works in the case that
$n^kr_n^{d(k-1)}$ tends to infinity but $n^{k+1}r_n^{dk}$ tends to
zero to show that $\widetilde{S}_{n,k}$ is approximately Gaussian in that regime as
well. However, to deal with the regime
in which $r_n=o(n^{-1/d})$ but $n^{k+1}r_n^{dk}$ is bounded away from zero,
a different argument is needed for the lower bound of \eqref{bounds}.
Following Penrose, the approach taken
here is to consider the Poissonized version of the problem (the vertices
distributed as a Poisson process of intensity $nf(\cdot)$ instead of i.i.d.\ with
density $f$), and then to recover the i.i.d.\ case.
Let $N_n$ be a Poisson random variable with mean $n$, and let $\mathcal{P}_n=
\{X_1,\ldots,X_{N_n}\},$ where $\{X_i\}_{i=1}^\infty$ is an i.i.d.\ sequence
of random points in $\mathbb{R}^d$ with density $f$. Then $\mathcal{P}_n$ is a Poisson
process with intensity $nf(\cdot)$, and one can define $S^P_{n,k}$ and
$\widetilde{S}^P_{n,k}$ for the random points $\mathcal{P}_n$ analogously to the earlier
definitions. In what follows, assume that $k\ge 3$; that is,
the empty $( k-1)$-simplices are at least empty triangles. Empty
1-simplices
are simply pairs of vertices which are not connected, and
different arguments are needed in that case.
In order to compute expectations for the expressions which arise in
the Poissonized case, the following results are
useful.
\begin{thm}[See \cite{penrose}]\label{one}
Let $\lambda>0$ and let $\mathcal{P}_\lambda$ be a Poisson process with intensity
$\lambda f(\cdot)$. Let
$j\in\mathbb{N}$, and suppose that $h(\mathcal{Y},\mathcal{X})$ is a bounded measurable
function on pairs $(\mathcal{Y},\mathcal{X})$ with $\mathcal{X}$ a finite subset of $\mathbb{R}^d$ and
$\mathcal{Y}\subseteq\mathcal{X}$, such that $h(\mathcal{Y},\mathcal{X})=0$ unless $|\mathcal{Y}|=j$. Then
$$\mathbb{E}\left[\sum_{\mathcal{Y}\subseteq\mathcal{P}_\lambda}h(\mathcal{Y},\mathcal{P}_\lambda)\right]=\frac{\lambda^j}{j!}
\mathbb{E} h(\mathcal{X}_j',\mathcal{X}_j'\cup\mathcal{P}_\lambda),$$
where $\mathcal{X}_j'$ is a set of $j$ i.i.d. points in $\mathbb{R}^d$ with density $f$,
independent of $\mathcal{P}_\lambda$.
\end{thm}
From this, one can prove the following.
\begin{thm}\label{product}
Let $\lambda>0$ and $k,j_1,\ldots,j_k\in\mathbb{N}$; define $j:=\sum_{i=1}^kj_i$.
For $1\le i\le k$, suppose
$h_i(\mathcal{Y},\mathcal{X})$ is a bounded measurable function of pairs $(\mathcal{Y},\mathcal{X})$ of
finite subsets of $\mathbb{R}^d$ with $\mathcal{Y}\subseteq\mathcal{X}$, such that $h_i(\mathcal{Y},\mathcal{X})=0$
if $|\mathcal{Y}|\neq j_i$. Then
$$\mathbb{E}\left[\sum_{\mathcal{Y}_1,\subseteq\mathcal{P}_\lambda}\cdots\sum_{\mathcal{Y}_k\subseteq\mathcal{P}_\lambda}\left(
\prod_{i=1}^kh_i(\mathcal{Y}_i)\right)\1_{\{\mathcal{Y}_i\cap\mathcal{Y}_j=\emptyset\,{\rm for }\,
i\neq j\}}\right]=\mathbb{E}\left[\prod_{i=1}^k\left(\frac{\lambda^{j_i}}{j_i!}\right)
h_i(\mathcal{X}_{j_i}',\mathcal{X}_j'\cup\mathcal{P}_n)\right],$$
\end{thm}
where $\mathcal{X}_j'$ are $j$ i.i.d points in $\mathbb{R}^d$ with density $f$, $\mathcal{P}_\lambda$
is a Poisson process with intensity $\lambda f(\cdot)$, and $\mathcal{X}_j'$ and
$\mathcal{P}_\lambda$ are independent.
\begin{proof}
Consider the case $k=2$ for simplicity (the case of larger $k$ is the same
with more notation). Define $h(\mathcal{Y},\mathcal{X})$ on subsets $\mathcal{Y}$ of $\mathcal{X}$
of size $j_1+j_2$ by
$$h(\mathcal{Y},\mathcal{X}):=\sum_{\substack{\mathcal{Y}_1\subseteq\mathcal{Y}\\|\mathcal{Y}_1|=j_1}}
h_1(\mathcal{Y}_1,\mathcal{X})h_2(\mathcal{Y}\setminus\mathcal{Y}_1,\mathcal{X}).$$
Then by Theorem \ref{one},
\begin{equation*}\begin{split}
\mathbb{E}&\left[\sum_{\mathcal{Y}_1,\subseteq\mathcal{P}_\lambda}\sum_{\mathcal{Y}_2,\subseteq\mathcal{P}_\lambda}h_1(\mathcal{Y}_1,\mathcal{P}_n)
h_2(\mathcal{Y}_2,\mathcal{P}_n)\1_{\{\mathcal{Y}_1\cap\mathcal{Y}_2=\emptyset\}}\right]\\
&\qquad=\mathbb{E}\left[\sum_{\mathcal{Y}\subseteq\mathcal{P}_n}
h(\mathcal{Y},\mathcal{P}_n)\right]\\&\qquad=\frac{\lambda^{j_1+j_2}}{(j_1+j_2)!}\mathbb{E} h(\mathcal{X}_j',\mathcal{X}_j'\cup
\mathcal{P}_n)\\&\qquad=\frac{\lambda^{j_1+j_2}}{j_1!j_2!}\mathbb{E}\left[h_1(\mathcal{X}_{j_1}',\mathcal{X}_j'\cup\mathcal{P}_n)h_2
(\mathcal{X}_j'\setminus\mathcal{X}_{j_1}',\mathcal{X}_j'\cup\mathcal{P}_n)\right].
\end{split}\end{equation*}
\end{proof}
\medskip
One can apply these results to compute the mean and variance of $\widetilde{S}_{n,k,A}^P$, the number of isolated
empty $k$-simplices in $\mathcal{P}_n$ whose left-most vertex is in the set $A$.
Recall that
$A$ is assumed to be open
with $\mathop{\mathrm{vol}}(\partial A)=0$.
\begin{lemma}\label{CC_Poisson_means}
For $\mu_A$ as in Lemma \ref{exp-order},
$$\lim_{n\to\infty}n^{-k}r_n^{-d(k-1)}\mathbb{E}\left[\widetilde{S}_{n,k}^P\right]=
\lim_{n\to\infty}n^{-k}r_n^{-d(k-1)}\mathrm{Var}\left[\widetilde{S}_{n,k}^P\right]=\frac{\mu_A}{k!}.$$
\end{lemma}
\begin{proof}
Let $\tilde{h}_{r_n,A}(\{x_1,\ldots, x_k\},\mathcal{X})$ be the indicator that
$\{x_1,\ldots,x_k\}\subseteq\mathcal{X}$ form an isolated empty $(k-1)$-simplex in
$\mathcal{X}$, whose left-most point is in $A$. Then
\begin{equation}\begin{split}\label{mean-unPoisson}
\mathbb{E}[\widetilde{S}_{n,k,A}^P]&=\mathbb{E}\left[\sum_{\mathcal{Y}\subseteq\mathcal{P}_\lambda}\tilde{h}_{r_n,A}(\mathcal{Y},\mathcal{P}_n)\right]
=\frac{n^k}{k!}\mathbb{E}\left[\tilde{h}_{r_n,A}(\mathcal{X}_k',\mathcal{X}_k'\cup\mathcal{P}_n)\right].
\end{split}\end{equation}
Now, $\mathbb{E}\left[\tilde{h}_{r_n,A}(\mathcal{X}_k',\mathcal{X}_k'\cup\mathcal{P}_n)\right]\le\mathbb{E}\left[h_{r_n,A}(\mathcal{X}_k')
\right]\simeq r_n^{d(k-1)}\mu_A$. Note that the conditional probability that
$\mathcal{X}_k'$ is isolated from $\mathcal{P}_n$ given that $\mathcal{X}_k'$ forms an empty
$(k-1)$-simplex with left-most vertex in $A$
is bounded below by the probability that there are no points of $\mathcal{P}_n$ in
the ball of radius $4r_n$ about $X_1$, which is given by $e^{-n\mathop{\mathrm{vol}}_f(B_{4r_n}(X_1)
)}\ge e^{-n\|f\|_\infty\theta_d(4r_n)^d},$ since $\mathcal{P}_n$ is a Poisson
process with intensity $nf(\cdot)$. It thus follows that
$$ \mathbb{E}\left[\tilde{h}_{r_n,A}(\mathcal{X}_k',\mathcal{X}_k'\cup\mathcal{P}_n)\right]\ge e^{-n\|f\|_\infty\theta_d(4r_n)^d}
\mathbb{E}[h_{r_n,A}(\mathcal{X}_k')]\simeq e^{-n\|f\|_\infty\theta_d(4r_n)^d}r_n^{d(k-1)}\mu_A.$$
Since $nr_n^d\to0$, this shows that
\begin{equation*}
\mathbb{E}[\widetilde{S}_{n,k}^P]\simeq \frac{n^kr_n^{d(k-1)}\mu_A}{k!}.\end{equation*}
A similar approach is taken to compute the variance:
\begin{equation*}\begin{split}
\mathbb{E}\left[(\widetilde{S}_{n,k,A}^P)^2\right]&
=\mathbb{E}\left[\sum_{\mathcal{Y}\subseteq\mathcal{P}_n}\tilde{h}_{r_n,A}(\mathcal{Y},\mathcal{P}_n)
\right]\\&\qquad\qquad
+\mathbb{E}\left[\sum_{j=0}^{k-1}\sum_{\mathcal{Y},\mathcal{Y}'\subseteq\mathcal{P}_n}\tilde{h}_{r_n,A}(\mathcal{Y},\mathcal{P}_n)
\tilde{h}_{r_n,A}(\mathcal{Y}',\mathcal{P}_n)\1_{\{|\mathcal{Y}\cap\mathcal{Y}'|=j\}}\right].
\end{split}\end{equation*}
The first summand has already been analyzed: $\mathbb{E}\left[\widetilde{S}_{n,k,A}^P\right]
\simeq\frac{n^kr_n^{d(k-1)}\mu_A}{k!}$.
For the second, observe first that the terms corresponding to $j\neq 0$
vanish:\\ $\tilde{h}_{r_n,A}(\mathcal{Y},\mathcal{P}_n)\tilde{h}_{r_n,A}(\mathcal{Y}',\mathcal{P}_n)\equiv 0$ if $|\mathcal{Y}\cap\mathcal{Y}'|=j$,
because if $\mathcal{Y}$ and $\mathcal{Y}'$ both form empty $k$-simplices, then neither is
isolated.
When $j=0$, applying Theorem \ref{product} yields
\begin{equation*}\begin{split}
\mathbb{E}&\left[\sum_{\mathcal{Y},\mathcal{Y}'\subseteq\mathcal{P}_n}\tilde{h}_{r_n,A}(\mathcal{Y},\mathcal{P}_n)\tilde{h}_{r_n,A}
(\mathcal{Y}',\mathcal{P}_n)\1_{\{\mathcal{Y}\cap\mathcal{Y}'=\emptyset\}}
\right]\\&\qquad\qquad
=\frac{n^{2k}}{(k!)^2}\mathbb{E}\left[\tilde{h}_{r_n,A}(\mathcal{X}_k',\mathcal{X}_{2k}'\cup\mathcal{P}_n)
\tilde{h}_{r_n,A}(\mathcal{X}_{2k}'\setminus\mathcal{X}_k',\mathcal{X}_{2k}'\cup\mathcal{P}_n)\right],
\end{split}\end{equation*}
and thus (making use of \eqref{mean-unPoisson}),
\begin{equation*}\begin{split}
\mathrm{Var}\left[\widetilde{S}_{n,k,A}^P\right]=\mathbb{E}\left[\widetilde{S}_{n,k,A}^P\right]+\frac{n^{2k}}{(k!)^2}
\Big(\mathbb{E}&\left[\tilde{h}_{r_n,A}(\mathcal{X}_k',\mathcal{X}_{2k}'\cup\mathcal{P}_n)
\tilde{h}_{r_n,A}(\mathcal{X}_{2k}'\setminus\mathcal{X}_k',\mathcal{X}_{2k}'\cup\mathcal{P}_n)\right]\\&-\left(\mathbb{E}\left[
\tilde{h}_{r_n,A}(\mathcal{X}_k',\mathcal{X}_k'\cup\mathcal{P}_n)\right]\right)^2\Big),
\end{split}\end{equation*}
Now, let $\mathcal{P}_n'$ be an independent copy of $\mathcal{P}_n$. For notational
convenience,
denote $\mathcal{X}_{2k}'\setminus\mathcal{X}_k'$ by $\mathcal{Y}_k'$ and abbreviate
$\tilde{h}_{r_n,A}$ by $\tilde{h}$. Then
\begin{equation*}\begin{split}
\mathbb{E}&\left[\tilde{h}(\mathcal{X}_k',\mathcal{X}_{2k}'\cup\mathcal{P}_n)
\tilde{h}(\mathcal{Y}_{k}',\mathcal{X}_{2k}'\cup\mathcal{P}_n)\right]-\left(\mathbb{E}\left[
\tilde{h}(\mathcal{X}_k',\mathcal{X}_k'\cup\mathcal{P}_n)\right]\right)^2\\&=
\mathbb{E}\left[\tilde{h}(\mathcal{X}_k',\mathcal{X}_{2k}'\cup\mathcal{P}_n)
\tilde{h}(\mathcal{Y}_{k}',\mathcal{X}_{2k}'\cup\mathcal{P}_n)-\tilde{h}(\mathcal{X}_k',\mathcal{X}_k'\cup\mathcal{P}_n)
\tilde{h}(\mathcal{Y}_{k}',\mathcal{Y}_{k}'\cup\mathcal{P}'_n)\right]\\&=
\mathbb{E}\left[\left(\tilde{h}(\mathcal{X}_k',\mathcal{X}_{2k}'\cup\mathcal{P}_n)-\tilde{h}(\mathcal{X}_k',\mathcal{X}_{k}'\cup\mathcal{P}_n)\right)
\tilde{h}(\mathcal{Y}_{k}',\mathcal{X}_{2k}'\cup\mathcal{P}_n)\right]\\&\quad+
\mathbb{E}\left[\tilde{h}(\mathcal{X}_k',\mathcal{X}_{k}'\cup\mathcal{P}_n)
\left(\tilde{h}(\mathcal{Y}_{k}',\mathcal{X}_{2k}'\cup\mathcal{P}_n)-\tilde{h}(\mathcal{Y}_k',\mathcal{Y}_k'\cup\mathcal{P}_n)
\right)\right]\\&\quad+\mathbb{E}\left[
\tilde{h}(\mathcal{X}_k',\mathcal{X}_k'\cup\mathcal{P}_n)\left(\tilde{h}(\mathcal{Y}_{k}',
\mathcal{Y}_{k}'\cup\mathcal{P}_n)-
\tilde{h}(\mathcal{Y}_{k}',\mathcal{Y}_{k}'\cup\mathcal{P}'_n)\right)\right]\\&=E_1+E_2+E_3.
\end{split}\end{equation*}
Now, observe that in fact $E_1=0$: the difference is non-zero if and
only if $\mathcal{X}_k'$ and $\mathcal{Y}_k'$ are connected by an edge, in
which case the second factor is zero.
Observe that the difference in $E_2$ is non-positive. Furthermore,
it is non-zero if and only if $\mathcal{X}_k'$ and $\mathcal{Y}_k'$ are
connected by an edge, and both $\mathcal{X}_k'$ and $\mathcal{Y}_k'$
form empty $k$-simplices. This probability is bounded above by
$\|f\|_\infty^{2k-1}\theta_d^{2k-1}(2r_n)^{2d(k-1)}(8r_n)^d.$
Finally, if $\left[\cup_{i=1}^kB_{2r_n}(X'_i)\right]\cap
\left[\cup_{i=k+1}^{2k}B_{2r_n}(X'_i)\right]=\emptyset$, then the
two terms of $E_3$ have the same distribution by the spacial
independence property of the Poisson process. A contribution from
$E_3$ therefore only arises if in particular $|X_1-X_j|\le 2r_n$ for
each
$2\le j\le k$, if $|X_{k+1}-X_j|\le 2r_n$ for $k+2\le j\le 2k$, and
$|X_1-X_{k+1}|\le 8r_n$. The probability of this event is bounded
above by $\|f\|_\infty^{2k-1}\theta_d^{2k-1}(2r_n)^{2d(k-1)}(8r_n)^d.$
It follows that
\[\mathrm{Var}\left[\widetilde{S}_{n,k,A}^P\right]= \mathbb{E}\left[\widetilde{S}_{n,k,A}^P\right]+E,\]
and
\[
|E|\le\frac{n^{2k}(2r_n)^{2dk-d}}{(k!)^2}2\|f\|_\infty^{2k-1}\theta_d^{2k-1}4^d=C(f,d,k)
(nr_n^d)^k(n^kr_n^{d(k-1)}),\]
where $C(f,k,d)$ is a constant depending on $f$, $d$, and $k$. This
completes the proof.
\begin{comment}
Let $\xi_1$ be the indicator that $\mathcal{X}_k'$ forms an empty
$(k-1)$-simplex,
and $\xi_2$ the indicator that $\mathcal{X}_{2k}'\setminus\mathcal{X}_k'$
forms an empty $(k-1)$-simplex.
Let $\eta_1$ be the indicator
that there are no edges between $\mathcal{X}_k'$ and $(\mathcal{X}_{2k}'\setminus\mathcal{X}_k')\cup\mathcal{P}_n$,
and $\eta_2$ the indicator
that there are no edges between $\mathcal{X}_{2k}'\setminus\mathcal{X}_k'$ and $\mathcal{X}_k'\cup\mathcal{P}_n$.
Then
\begin{equation*}\begin{split}
\mathbb{E}&\left[\tilde{h}_{r_n,A}(\mathcal{X}_k',\mathcal{X}_{2k}'\cup\mathcal{P}_n)
\tilde{h}_{r_n,A}(\mathcal{X}_{2k}'\setminus\mathcal{X}_k',\mathcal{X}_{2k}'\cup\mathcal{P}_n)\right]=\mathbb{E}[
\xi_1\eta_1\xi_2\eta_2].
\end{split}\end{equation*}
Now, if $diam(A)\le 2r_n$, then $\mathbb{E}[\xi_1\eta_1\xi_2\eta_2]=0$,
because if the left-most points of $\mathcal{X}_k'$ and $\mathcal{X}_{2k}'\setminus\mathcal{X}_k'$
are both in $A$, then they are connected and so $\eta_1=\eta_2=0$.
Once $A$ is sufficiently large,
\begin{equation*}\begin{split}
\P\big[\eta_1=1\big|\xi_1=\xi_2=\eta_2=1\big]&=\P\left[\left.\mathcal{P}_n\cap\left(
\cup_{i=1}^kB_{2r_n}(X_i)\right)=\emptyset\right|\xi_1=\xi_2=\eta_2=1\right],
\end{split}\end{equation*}
and furthermore, since $\xi_1=\xi_2=\eta_2=1$ implies that
$\mathcal{X}_k'$ and $\mathcal{X}_{2k}'\setminus\mathcal{X}_k'$ both form empty $k$-simplices and
are not connected to each other, the probability that $\mathcal{X}_k'$ is isolated
in $\mathcal{X}_{2k}'\cup\mathcal{P}_n$ given $\xi_1=\xi_2=\eta_2=1$
is simply the conditional probability that no points
of $\mathcal{P}_n$ are connected to $\mathcal{X}_k'$, given that none are connected to
$\mathcal{X}_{2k}'\setminus\mathcal{X}_k'$. Since $\mathcal{P}_n$ is a Poisson process, these events
are almost independent. They are independent if the distance from $\mathcal{X}_k'$ to
$\mathcal{X}_{2k}'\setminus\mathcal{X}_k'$ is at least $4r_n$, in which case the conditional
probability that $\eta_1=1$ is
bounded below by $e^{-n\|f\|_\infty\theta_d(4r_n)^d}$, which tends to one
as $n$ tends to infinity. The probability that the distance from $\mathcal{X}_k'$ to
$\mathcal{X}_{2k}'\setminus\mathcal{X}_k'$ is at least $4r_n$, given that both $\mathcal{X}_k'$
and $\mathcal{X}_{2k}'\setminus\mathcal{X}_k'$ form empty $k$-simplices with left-most points
in $A$, is bounded below by
the probability that $X_1$ and $X_{k+1}$ are more than $8r_n$ apart (given
the same information), which
is bounded below by
\begin{equation*}\begin{split}
1&-\frac{\P\left[|X_1-X_{k+1}|\le 8r_n, \xi_2=1\big|\xi_1=1\right]}{\P[\xi_2=1]}
\\&\qquad\qquad\qquad\ge
1-\frac{\|f\|_\infty^k\theta_d^k(8r_n)^d(2r_n)^{d(k-1)}}{\P[\xi_2=1]}
\simeq 1-\frac{\|f\|_\infty^k\theta_d^k(8r_n)^d(2r_n)^{d(k-1)}}{\mu_Ar_n^{d(k-1)}}
.\end{split}\end{equation*}
For $A$ fixed with
$\mu_A\neq 0$, this quantity tends to one as $n$ tends to infinity.
Next, let $\eta_{12}$ be the indicator that no edges connect
$\mathcal{X}_k'$ to $\mathcal{X}_{2k}'\setminus\mathcal{X}_k'$. Then
\begin{equation*}\begin{split}
\P\left[\eta_2=1\big|\xi_1=\xi_2=1\right]&=\P\left[\eta_2=1\big|\xi_1=\xi_2=1,
\eta_{12}=1\right]\P\left[\eta_{12}=1\big|\xi_1=\xi_2=1\right].
\end{split}\end{equation*}
Similarly to the argument above,
$$\P\left[\eta_2=1\big|\xi_1=\xi_2=1,
\eta_{12}=1\right]\ge e^{-n\|f\|_\infty\theta_d(4r_n)^d},$$
and
$$\P\left[\eta_{12}=1\big|\xi_1=\xi_2=1\right]\ge\P\left[|X_1-X_k|\ge 8r_n
\big|\xi_1=\xi_2=1\right]\ge
1-\frac{\|f\|_\infty^k\theta_d^k(2^{k+2}r_n)^d}{\mu_A}.$$
Finally, $\P[\xi_1=\xi_2=1]=\left(\P[\xi_1=1]\right)^2\simeq
r_n^{2d(k-1)}\mu_A^2,$ by independence of the points of $\mathcal{X}_{2k}'$.
All together, this shows that
$$\mathbb{E}[\xi_1\xi_2\eta_1\eta_2]\simeq r_n^{2d(k-1)}\mu_A^2,$$
and so
\begin{equation*}
\mathrm{Var}\left[\widetilde{S}_{n,k,A}^P\right]
\simeq\frac{n^kr_n^{
d(k-1)}\mu_A}{k!}.
\end{equation*}
\end{comment}
\end{proof}
The following abstract normal approximation theorem is another version
of the dependency graph approach to Stein's method. It is used in what
follows to prove a central limit theorem for $\widetilde{S}_{n,k}^P$.
\begin{thm}[Penrose]\label{normal}
Suppose $\{\xi_i\}_{i\in I}$ is a finite collection of random variables with
dependency graph $(I,\sim)$ with maximum degree $D-1$, with $\mathbb{E}[\xi_i]=0$
for each $i$. Set $W:=\sum_{i\in I}\xi_i$; suppose $\mathbb{E}[W^2]=1$. Let $Z$
be a standard normal random variable. Then for all $t\in\mathbb{R}$,
$$\big|\P[W\le t]-\P[Z\le t]\big|\le\frac{2}{\sqrt[4]{2\pi}}\sqrt{D^2\sum_{
i\in I}\mathbb{E}|\xi_i|^3}+6\sqrt{D^3\sum_{i\in I}\mathbb{E}|\xi_i|^4}.$$
\end{thm}
Making use of this result, we prove the following.
\begin{thm}\label{Poissonized_normal}
With notation as above, and for $n^kr_n^{d(k-1)}\to\infty$ and $nr_n^d\to0$,
$$\frac{\widetilde{S}_{n,k}^P-\mathbb{E}\left[\widetilde{S}_{n,k}^P\right]}{\sqrt{\mathrm{Var}\left[\widetilde{S}_{n,k}^P\right]}}
\Rightarrow\mathcal{N}(0,1).$$
\end{thm}
\begin{proof}
To define a dependency graph
for the summands of $\widetilde{S}_{n,k}^P$, the independence
properties of the Poisson process are exploited. Let $\{Q_{i,n}\}_{i\in \mathbb{N}}$
be a partition of $\mathbb{R}^d$ into cubes of side length $r_n$. For the moment, assume that $A$ is a bounded set, and let $I_A$ be the set of
indices $i$ such that $diam(A\cap Q_{i,n})>2r_n$. Write
\begin{equation}\label{grid}
\widetilde{S}_{n,k,A}^P=\sum_{i\in I_A} \sum_{\mathcal{Y}\subseteq\mathcal{P}_n}\tilde{h}_{r_n,A\cap Q_{i,n}}(\mathcal{Y},\mathcal{P}_n).
\end{equation}
Observe that if one defines a relation $\sim$ on $I_A$ by $i\sim j$ if and
only if the Euclidean distance from $Q_{i,n}$ to $Q_{j,n}$ is
less than
$8r_n$, then $(I_A,\sim)$ is a dependency graph for the summands in
\eqref{grid}. The degree of vertices in this dependency graph is then
bounded by $17^d$.
Let $\xi_i:=\sum_{\mathcal{Y}\subseteq\mathcal{P}_n}\tilde{h}_{r_n,A\cap Q_{i,n}}(\mathcal{Y},\mathcal{P}_n);$ to apply Theorem
\ref{normal}, bounds are needed for $\mathbb{E}|\xi_i-\mathbb{E}\xi_i|^p$ for $p=3,4$,
for which it suffices to have bounds on
$\mathbb{E}|\xi|^p$ for $p=3,4$. Observe that if $Z_{i}$ is the number of points
within $2r_n$ of $Q_{i,n}$, then $Z_{i,n}$ is distributed as a Poisson
random variable with mean $n\mathop{\mathrm{vol}}_f((Q_{i,n})_{2r_n})$, and
$$|\xi_i|\le (Z_i)(Z_i-1)\cdots(Z_i-k+1)=:(Z_i)_k.$$
It follows that there is a constant $c$ depending only on $d$ and $f$, such
that for $\rho_n:=nr_n^d$,
$$\mathbb{E}|\xi_i|^p\le\mathbb{E} (Z_i)_k^p\le\sum_{m=k}^\infty(m)_k^p\frac{e^{-c\rho_n}(c
\rho_n)^m}{m!}\le c'\rho_n^k$$
for some new constant $c'$ depending only on $d$, $f$, and $k$.
Note that since $A$ is bounded,
$|I_A|$ is at worst of the order $r_n^{-d}$, with coefficient
depending on
$A$. Applying Theorem \ref{normal} to $\frac{\xi_i-\mathbb{E}\xi_i}{\sqrt{\mathrm{Var}(\widetilde{S}_{n,
k,A})}}$ gives
$$\left|\P\left[\frac{\widetilde{S}_{n,k,A}^P-\mathbb{E} \widetilde{S}_{n,k,A}^P}{\sqrt{\mathrm{Var}(\widetilde{S}_{n,k,A}^P)}}
\le t\right]-\P[Z\le t]\right|\le c''[n^kr_n^{d(k-1)}]^{-1/4},$$
which tends to zero as $n$ tends to infinity.
To move to $A=\mathbb{R}_d$, let $\zeta_{n,k}(A):=\frac{\widetilde{S}_{n,k,A}^P
-\mathbb{E}[\widetilde{S}_{n,k,A}^P]}{\sqrt{n^kr_n^{d(k-1)}}}$ and consider $A_K:=(-K,K)^d$
and $A^K:=\mathbb{R}^d\setminus[-K,K]^d.$ Given $t\in\mathbb{R}$ and $\epsilon>0$,
\begin{equation*}\begin{split}
\P[\zeta_{n,k}(\mathbb{R}^d)\le t]=\P[\zeta_{n,k}&(A_K)\le t-\epsilon]-\P[\{\zeta_{n,k}(A_K)
\le t-\epsilon\}\cap\{\zeta_{n,k}(\mathbb{R}^d)>t\}]\\&
+\P[\{|\zeta_{n,k}(A_K)-t|<\epsilon
\}\cap\{\zeta_{n,k}(\mathbb{R}^d)\le t\}]\\&+\P[\{\zeta_{n,k}(A_K)\ge t+\epsilon\}\cap\{
\zeta_{n,k}(\mathbb{R}^d)\le t\}].
\end{split}\end{equation*}
Now, $\zeta_{n,k}(\mathbb{R}^d)=\zeta_{n,k}(A_K)+\zeta_{n,k}(A^K)$ almost surely since
$vol(A_K^c\cup (A^K)^c)=0$, so
$$\big|\P[\zeta_{n,k}(\mathbb{R}^d)\le t]-\P[\zeta_{n,k}(A_K)\le t-\epsilon]\big|\le
\P[|\zeta_{n,k}(A^K)|\ge\epsilon]+\P[|\zeta_{n,k}(A_K)-t|<\epsilon].$$
By Chebychev's inequality and the central limit theorem already
established for bounded sets, this last expression is bounded above by
\begin{equation*}\begin{split}
\frac{1}{\epsilon^2}\mathrm{Var}(\zeta_{n,k}(A^K))&+
\P\left[\left|\sqrt{\frac{\mathrm{Var}(\widetilde{S}_{n,k,A_K}^P)}{n^kr_n^{d(k-1)}}}Z-t\right|<
\epsilon\right]+c_K\left[(n^kr_n^{d(k-1)})^{-1/4}\right]\\&\le
\frac{1}{\epsilon^2}\mathrm{Var}(\zeta_{n,k}(A^K))+\frac{2\epsilon\sqrt{
n^kr_n^{d(k-1)}}}{\sqrt{2\pi\mathrm{Var}(\widetilde{S}_{n,k,A_K}^P)}}+c_K\left[(n^kr_n^{d(k
-1)})^{-1/4}\right]\\&\simeq \frac{1}{\epsilon^2}\frac{\mu_{A^K}}{k!}
+\frac{2\epsilon\sqrt{k!}}{\sqrt{2\pi\mu_{A_K}}}+c_K\left[(n^kr_n^{d(k
-1)})^{-1/4}\right],
\end{split}\end{equation*}
for a constant $c_K$ depending on $K$.
Taking $n$ to infinity for $K$ and $\epsilon$ fixed yields
$$\limsup_{n\to\infty}\big|\P[\zeta_{n,k}(\mathbb{R}^d)\le t]-\P[\zeta_{n,k}(A_K)\le
t-\epsilon]\big|\le \frac{1}{\epsilon^2}\frac{\mu_{A^K}}{k!}
+\frac{2\epsilon\sqrt{k!}}{\sqrt{2\pi\mu_{A_K}}},$$
which, together with
the central limit theorem for $\zeta_{n,k}(A_K)$, implies that
$$\limsup_{n\to\infty}\left|\P[\zeta_{n,k}(\mathbb{R}^d)\le t]-\P\left[
\sqrt{\frac{\mathrm{Var}(\widetilde{S}_{n,k,A_K}^P)}{n^kr_n^{d(k-1)}}}Z\le
t-\epsilon\right]\right|\le \frac{1}{\epsilon^2}\frac{\mu_{A^K}}{k!}
+\frac{2\epsilon\sqrt{k!}}{\sqrt{2\pi\mu_{A_K}}}.$$
Now,
\begin{eqnarray*}
&&\P\left[\sqrt{\frac{\mathrm{Var}(\widetilde{S}_{n,k,A_K}^P)}{n^kr_n^{d(k-1)}}}Z\le
t-\epsilon\right]\\
&=&\Phi\left(\sqrt{\frac{n^kr_n^{d(k-1)}}{\mathrm{Var}(\widetilde{S}_{n,k,A_K}^P)}}(t-
\epsilon)\right)\xrightarrow{n\to\infty}\Phi\left(\sqrt{\frac{k!}{\mu_{A_K}}}
(t-\epsilon)\right);
\end{eqnarray*}
that is,
$$\limsup_{n\to\infty}\left|\P[\zeta_{n,k}(\mathbb{R}^d)\le t]-\Phi\left(\sqrt{
\frac{k!}{\mu_{A_K}}}(t-\epsilon)\right)\right|\le \frac{1}{\epsilon^2}
\frac{\mu_{A^K}}{\mu}+\frac{2\epsilon\sqrt{k!}}{\sqrt{2\pi\mu_{A_K}}}.$$
Recall that $\lim_{K\to\infty}\mu_{A_K}=\mu$ and $\lim_{K\to\infty}\mu_{A^K}=
0$. Thus for $n$ and $K$ large enough,
$$\left|\P[\zeta_{n,k}(\mathbb{R}^d)\le t]-\Phi\left(\sqrt{
\frac{k!}{\mu}}(t-\epsilon)\right)\right|\le \frac{2\epsilon\sqrt{k!}}{\sqrt{2\pi\mu}}+\epsilon.$$
Since $\Phi\left(\sqrt{
\frac{k!}{\mu}}(t-\epsilon)\right)\xrightarrow{\epsilon\to0}\Phi\left(
\sqrt{
\frac{k!}{\mu}}t\right)$ and $\epsilon$ was arbitrary, this finally
shows that
$$\lim_{n\to\infty}\left|\P[\widetilde{S}_{n,k}^P\le t]-\Phi\left(\sqrt{
\frac{k!}{\mu}}t\right)\right|=0.$$
\end{proof}
\medskip
The remaining work is to use this result to obtain the same
result for $\widetilde{S}_{n,k}$ itself.
To do so, the following ``de-Poissonization result'' is used.
\begin{thm}[See \cite{penrose}]\label{de-Poisson}
Suppose that for each $n\in\mathbb{N}$, $H_n(\mathcal{X})$ is a real-valued functional on finite
sets $\mathcal{X}\subseteq\mathbb{R}^d$. Suppose that for some $\sigma^2\ge 0$,
\begin{enumerate}
\item $\displaystyle\frac{1}{n}\mathrm{Var}(H_n(\mathcal{P}_n))\longrightarrow\sigma^2,$
and
\item $\displaystyle\frac{1}{\sqrt{n}}\big[H_n(\mathcal{P}_n)-\mathbb{E} H_n(\mathcal{P}_n)\big]
\Longrightarrow\sigma^2Z,$
for $Z$ a standard normal random variable.
\end{enumerate}
Suppose that there are constants $\alpha\in\mathbb{R}$ and $\gamma>\frac{1}{2}$ such
that the increments $R_{m,n}=H_n(\mathcal{X}_{m+1})-H_n(\mathcal{X}_m)$ satisfy
\begin{equation}\label{means}
\lim_{n\to\infty}\left(\sup_{n-n^\gamma\le m\le n+n^\gamma}|\mathbb{E}[R_{m,n}]-\alpha|
\right)=0,
\end{equation}
\begin{equation}\label{covs}
\lim_{n\to\infty}\left(\sup_{n-n^\gamma\le m<m'\le n+n^\gamma}|\mathbb{E}[R_{m,n}R_{m',n}]
-\alpha^2|\right)=0,
\end{equation}
and
\begin{equation}\label{vars}
\lim_{n\to\infty}\left(\frac{1}{\sqrt{n}}\sup_{n-n^\gamma\le m\le n+n^\gamma}
\mathbb{E}[R_{m,n}^2]\right)=0.
\end{equation}
Finally, assume that there is a constant $\beta>0$ such that, with probability
one,
$$|H_n(\mathcal{X}_m)|\le\beta(n+m)^\beta.$$
Then $\alpha^2\le\sigma^2$ and as $n\to\infty$, $\frac{1}{n}\mathrm{Var}(H_n(\mathcal{X}_n))
\to \sigma^2-\alpha^2$ and
$$\frac{1}{\sqrt{n}}\big[H_n(\mathcal{X}_n)-\mathbb{E} H_n(\mathcal{X}_n)\big]\Longrightarrow
\sqrt{\sigma^2-\alpha^2}Z.$$
\end{thm}
In conjunction with Theorem \ref{Poissonized_normal}, this yields the following.
\begin{thm}
With notation as above, and for $n^kr_n^{d(k-1)}\to\infty$ and $nr_n^d\to0$,
$$\frac{\widetilde{S}_{n,k}-\mathbb{E}\left[\widetilde{S}_{n,k}\right]}{\sqrt{\mathrm{Var}\left[\widetilde{S}_{n,k}\right]}}
\Rightarrow\mathcal{N}(0,1).$$
\end{thm}
\begin{proof}
Theorem \ref{de-Poisson} is applied to the functional
$$H_n(\mathcal{X}):=\frac{1}{\sqrt{(nr_n^d)^{k-1}}}\sum_{\mathcal{Y}\subseteq\mathcal{X}}\tilde{h}_{r_n}
(\mathcal{Y},\mathcal{X});$$
$\sigma^2=\frac{\mu}{k!}$ and the central limit theorem
holds for $H_n(\mathcal{P}_n)$ by Theorem \ref{Poissonized_normal}.
Let $D_{m,n}:=\sum_{\mathcal{Y}\subseteq\mathcal{X}_{m+1}}\tilde{h}_{r_n}(\mathcal{Y},\mathcal{X}_{m+1})-\sum_{\mathcal{Y}\subseteq\mathcal{X}_m}
\tilde{h}_{r_n}(\mathcal{Y},\mathcal{X}_m)$, and observe that $D_{m,n}$
is the number of isolated empty $(k-1)$-simplices in
$\mathcal{X}_{m+1}$ with $X_{m+1}$ as a vertex, minus the number of empty $(k-1)$-simplices
in $\mathcal{X}_m$ which are isolated in $\mathcal{X}_m$ but connected to $X_{m+1}$.
Thus
\begin{eqnarray}\label{inc_exp}
\mathbb{E}[D_{m,n}]&=&\binom{m}{k-1}\mathbb{E}[\tilde{h}_{r_n}(\mathcal{X}_k,\mathcal{X}_{m+1})]\\
&&- \binom{m}{k}\mathbb{E}[\tilde{h}_{r_n}(\mathcal{X}_k,\mathcal{X}_m)]\P\left[X_{m+1}\in\cup_{i=1}^kB_{2r_n}(X_i)
\right]. \nonumber
\end{eqnarray}
It is clear that
$$(1-\|f\|_\infty\theta_d(4r_n)^d)^{m+1-k}r_n^{d(k-1)}\mu\le
\mathbb{E}[\tilde{h}_{r_n}(\mathcal{X}_k,\mathcal{X}_{m+1})]\le r_n^{d(k-1)}\mu,$$
with the upper bound arising from removing the condition that $\mathcal{X}_k$ be a
component in $\mathcal{C}(\mathcal{X}_{m+1})$ and the lower bound arising
by bounding below the conditional probability that $\mathcal{X}_k$ is a component,
given that it forms an empty $(k-1)$-simplex. If $\gamma<1$, then
$\lim_{n\to\infty}(1-\|f\|_\infty\theta_d(4r_n)^d)^{m+1-k}=1$, uniformly
in $m\in[n-n^\gamma,n+n^\gamma],$
thus $\mathbb{E}[\tilde{h}_{r_n}(\mathcal{X}_k,\mathcal{X}_{m+1})]\simeq r_n^{d(k-1)}\mu$ uniformly in
$m\in[n-n^\gamma,n+n^\gamma]$, and the same is
true for $\mathbb{E}[\tilde{h}_{r_n}(\mathcal{X}_k,\mathcal{X}_{m})]$.
For the second term of \eqref{inc_exp}, observe that
$$\frac{\binom{m}{k}}{\binom{m}{k-1}}\P\left[X_{m+1}\in\cup_{i=1}^kB_{2r_n}(X_i)
\right]\lesssim\frac{m}{k}\|f\|_\infty\theta_d(4r_n)^d,$$
and $\lim_{n\to\infty}mr_n^d=0$, uniformly in $m\in[n-n^\gamma,n+n^\gamma]$.
That is, the second term is of strictly
smaller order than the first. Thus
$$\lim_{n\to\infty}\sup_{n-n^\gamma\le m\le n+n^\gamma}\left|(nr_n^d)^{1-k}\mathbb{E}[D_{m,n}]
-\frac{1}{(k-1)!}\mu\right|=0.$$
This implies that
$$\lim_{n\to\infty}\sup_{n-n^\gamma\le m\le n+n^\gamma}\left|(nr_n^d)^{(1-k)/2}\mathbb{E}[
D_{m,n}]\right|=0,$$
since $nr_n^d\to0$ as $n\to\infty$, and so the first increment condition
of the theorem is satisfied with $\alpha=0$ and any choice of $\gamma\in
(\frac{1}{2},1)$.
Next, consider the quantity $\mathbb{E}[D_{m,n}D_{m',n}]$ for $m\le m'$.
Recall that
$$D_{m,n}=\sum_{\substack{\mathcal{Y}\subseteq\mathcal{X}_m\\|\mathcal{Y}|=k-1}}\tilde{h}_{r_n}(\mathcal{Y}\cup\{X_{m+1}\},
\mathcal{X}_{m+1})-\sum_{\substack{\mathcal{Y}\subseteq\mathcal{X}_{m}\\|\mathcal{Y}|=k}}\tilde{h}_{r_n}(\mathcal{Y},
\mathcal{X}_{m})\1_{\left\{X_{m+1}\in\bigcup_{y\in\mathcal{Y}}B_{2r_n}(y)\right\}}.$$
First consider the contribution to $\mathbb{E}[D_{m,n}D_{m',n}]$ from terms of the form
$$\mathbb{E}\big[\tilde{h}_{r_n}(\mathcal{Y}\cup \{X_{m+1}\},\mathcal{X}_{m+1})\tilde{h}_{r_n}(\mathcal{Y}'\cup \{X_{m'+1}\},
\mathcal{X}_{m'+1})\big]$$ for $\mathcal{Y},\mathcal{Y}'$ such that $\big(\mathcal{Y}\cup \{X_{m+1}\}\big)\cap
\mathcal{Y}'=\emptyset.$
By conditioning on the event $
\tilde{h}_{r_n}(\mathcal{Y}\cup\{X_{m+1}\},\mathcal{X}_{m+1})=1$, it follows that
\begin{equation*}\begin{split}
\mathbb{E}[\tilde{h}_{r_n}(&\mathcal{Y}\cup \{X_{m+1}\},\mathcal{X}_{m+1})\tilde{h}_{r_n}(\mathcal{Y}'\cup\{X_{m'+1}\},\mathcal{X}_{m'+1})]
\simeq r_n^{2d(k-1)}\mu^2\zeta,
\end{split}\end{equation*}
where $\zeta$ is the conditional probability that $\mathcal{Y}'\cup X_{m'+1}$
is a component in $\mathcal{X}_{m'+1}$, given that it forms an empty $(k-1)$-simplex,
and that $\mathcal{Y}\cup X_{m+1}$ forms an empty $(k-1)$-simplex which is
not connected to any other points of $\mathcal{X}_{m+1}$. Note that
if $m=m'$ then $\zeta=0$. Otherwise, simply bound $\zeta\le 1$, so that
these terms have asymptotic order bounded above by
$r_n^{2d(k-1)}\mu^2$, uniformly
in $m$. The number of such terms is bounded by $\frac{(n+n^\gamma)^{2k-2}}{
[(k-1)!]^2}.$
Note that if
$\big(\mathcal{Y}\cup \{X_{m+1}\}\big)\cap \mathcal{Y}'\neq\emptyset,$ and $m\neq m'$,
then $\tilde{h}_{r_n}(\mathcal{Y}\cup\{
X_{m+1}\},\mathcal{X}_{m+1})\tilde{h}_{r_n}(\mathcal{Y}'\cup\{X_{m'+1}\},\mathcal{X}_{m'+1})\equiv 0.$ If
$m=m'$ and then it must be that $\mathcal{Y}=\mathcal{Y}'$ to get a non-zero contribution.
In this case, one gains a contribution to $\mathbb{E}[D_{m,n}^2]$ of
$$\binom{m}{k-1}r_n^{d(k-1)}\mu\le\frac{(n+n^\gamma)^{k-1}r_n^{d(k-1)}\mu}{
(k-1)!}.$$
Moving on to the cross terms, if $m'=m$ then
$$\tilde{h}_{r_n}(\mathcal{Y}\cup\{X_{m+1}\},\mathcal{X}_{m+1})\tilde{h}_{r_n}(\mathcal{Y}',\mathcal{X}_{m})\1_{\left\{X_{m+1}
\in\bigcup_{y\in\mathcal{Y}'}B_{2r_n}(y)\right\}}\equiv 0.$$
If $m<m'$ (or $m>m'$), then
\begin{equation*}\begin{split}
\mathbb{E}&\left[\tilde{h}_{r_n}(\mathcal{Y}\cup\{X_{m+1}\},\mathcal{X}_{m+1})\tilde{h}_{r_n}(\mathcal{Y}',\mathcal{X}_{m'})\1_{\left\{X_{m'+1}
\in\bigcup_{y\in\mathcal{Y}'}B_{2r_n}(y)\right\}}\right]\\&\qquad\qquad
\le\mathbb{E}\left[\tilde{h}_{r_n}(\mathcal{Y}\cup\{X_{
m+1}\},\mathcal{X}_{m+1})\tilde{h}_{r_n}(\mathcal{Y}',\mathcal{X}_{m'})\right]\|f\|_\infty\theta_d(4r_n)^d.
\end{split}\end{equation*}
Again, to get a non-zero contribution, it must be that $(\mathcal{Y}\cup\{X_{m+1}\})
\cap\mathcal{Y}'=\emptyset.$ In this case, the expression above is bounded above
by
$$(r_n^{d(k-1)}\mu)^2\|f\|_\infty\theta_d(4r_n)^d.$$
The number of such terms is bounded by $\binom{m}{k-1}\binom{m}{k}\le
\frac{(n+n^\gamma)^{2k-1}}{k!(k-1)!}.$
For the product of the second sums from $D_{m,n}$ and $D_{m',n}$,
we have already seen
that the conditional probability that $X_{m+1}\in\bigcup_{y\in\mathcal{Y}}B_{2r_n}(y)$
given $\mathcal{Y}$ is bounded above by $\|f\|_\infty\theta_d(4r_n)^d$, and so if
$m=m'$,
\begin{eqnarray*}
\lefteqn{\mathbb{E}\left[\sum_{\mathcal{Y}\subseteq\mathcal{X}_{m'}}\left(\tilde{h}_{r_n}(\mathcal{Y},\mathcal{X}_{m'})\1_{\left\{X_{m'+1}\in
\bigcup_{y\in\mathcal{Y}}B_{2r_n}(y)\right\}}\right)^2\right] \le} \\
& & \frac{(n+n^\gamma)^k}{k!}r_n^{d(k-1)}\mu\|f\|_\infty\theta_d(4r_n)^d,
\end{eqnarray*}
while if $\mathcal{Y}\neq\mathcal{Y}'$,
$$\left(\tilde{h}_{r_n}(\mathcal{Y},\mathcal{X}_{m'})\1_{\left\{X_{m'+1}\in
\bigcup_{y\in\mathcal{Y}}B_{2r_n}(y)\right\}}\right)\left(\tilde{h}_{r_n}(\mathcal{Y}',\mathcal{X}_{m'})\1_{\left\{X_{m'+1}\in
\bigcup_{y\in\mathcal{Y}'}B_{2r_n}(y)\right\}}\right)\equiv0.$$
For $m\neq m'$,
$\mathcal{Y}\subseteq\mathcal{X}_{m}$ and $\mathcal{Y}\subseteq\mathcal{X}_{m'}$,
let $\xi$ be the indicator that $\mathcal{Y}$
forms an empty $(k-1)$-simplex and $\eta$ the indicator that it is a
component in $\mathcal{X}_m$. Let $\xi'$ and $\eta'$ be the corresponding indicators
that $\mathcal{Y}'$ is an empty $(k-1)$-simplex and that it is a component in $\mathcal{X}_{m'}$.
Let $\zeta$ and $\zeta'$ be the indicators that $X_{m+1}$ is connected to
$\mathcal{Y}$ and that $X_{m'+1}$ is connected to $\mathcal{Y}'$, respectively. Then what
is needed is
$$\mathbb{E}[\xi\eta\zeta\xi'\eta'\zeta'].$$
Note that for the product to be non-zero, it must be that
$(\mathcal{Y} \cup\{X_{m+1}\})\cap\mathcal{Y}'=\emptyset.$
Now,
$$\P\left[\zeta\zeta'=1\big|\xi\eta\xi'\eta'=1\right]\le\frac{\|f\|^2_\infty
\theta^2_d(4r_n)^{2d}}{\mathop{\mathrm{vol}}_f(\cap_{y\in\mathcal{Y}'}B_{2r_n}(y)^c)}\le
\frac{\|f\|^2_\infty
\theta^2_d(4r_n)^{2d}}{1-\|f\|_\infty\theta_d(4r_n)^d},$$
since if $\xi\eta\xi'\eta'=1$, then $\mathcal{Y}$ and $\mathcal{Y}'$ make up empty
$(k-1)$-simplices; and morover, while nothing at all is known about $X_{m'+1}$,
it is known that $X_{m+1}$ is not connected to $\mathcal{Y}'$.
Trivially,
$\P\big[\eta\eta'=1\big|\xi\xi'=1\big]\le 1$,
and $\P[\xi\xi'=1]=\P[\xi=1]\P[\xi'=1]\simeq r_n^{2d(k-1)}\mu^2,$
since $\mathcal{Y}\cap\mathcal{Y}'=\emptyset.$
Thus
\begin{equation*}\begin{split}
\mathbb{E}&\left[\sum_{\mathcal{Y}\subseteq\mathcal{X}_m}\sum_{\substack{\mathcal{Y}\subseteq\mathcal{X}_{m'}\\\mathcal{Y}'\neq\mathcal{Y}}}
\tilde{h}_{r_n}(\mathcal{Y},\mathcal{X}_m)\1_{\left\{X_{m+1}\in\bigcup_{y\in\mathcal{Y}}B_{2r_n}(y)\right\}}
\tilde{h}_{r_n}(\mathcal{Y},\mathcal{X}_{m'})\1_{\left\{X_{m'+1}\in\bigcup_{y\in\mathcal{Y}'}B_{2r_n}(y)\right\}}\right]
\\&\qquad\lesssim\frac{c_{d,f}(nr_n^d)^{2k}\mu^2}{(k!)^2}.
\end{split}\end{equation*}
It now follows that $\mathbb{E}[D_{m,n}D_{m',n}]\lesssim c_{d,f,k}(nr_n^d)^k$ for
all $m,m'\in[n-n^\gamma,n+n^\gamma]$ with $m\neq m'$, and
so
$$\lim_{n\to\infty}\sup_{n-n^\gamma\le m<m'\le n+n^\gamma}(nr_n^d)^{1-k}\mathbb{E}[D_{m,n}D_{m',n}]
=0.$$ If $m=m'$, then $\mathbb{E}[D_{m,n}^2]\lesssim c_{d,f}(nr_n^d)^{k-1},$ and so
$$\lim_{n\to\infty}\sup_{n-n^\gamma\le m\le n+n^\gamma}\frac{1}{\sqrt{n}}
(nr_n^d)^{1-k}\mathbb{E}[D_{m,n}^2]=0.$$
Thus the increment conditions of the theorem are satisfied with $\alpha=0$.
Finally, observe that
$$H_n(\mathcal{X}_m)\le\frac{\sqrt{n}m}{n^kr_n^{d(k-1)}k}\le\frac{(\sqrt{n}+m)^2}{n^kr_n^{d(k-1)}k};$$
since $n^kr_n^{d(k-1)}$ is assumed to go to infinity as $n\to\infty$,
the polynomial boundedness condition of Theorem \ref{de-Poisson}
is satisfied
and the central limit theorem for
$\widetilde{S}_{n,k}$ is proved.
\end{proof}
\bigskip
As was previously noted, that the same central limit theorem holds for
upper and lower bounds for $\beta_k$ given in \eqref{bounds}
immediately yields part \ref{CC_clt_normal}
of Theorem \ref{CC_clt}.
\begin{thm}\label{betti-normal}
$$\frac{\beta_{k-1}-\mathbb{E}[\widetilde{S}_{n,k}]}{\sqrt{\mathbb{E}[\widetilde{S}_{n,k}]}}\Longrightarrow
\mathfrak{N}(0,1).$$
\end{thm}
\section{Vietoris-Rips complexes}
Vietoris-Rips complexes were introduced by Leopold Vietoris in the
context of algebraic topology, and independently by Eliyahu Rips in
the context of geometric group theory. These complexes continue to be
a useful construction in both fields, and are also useful in
computational topology -- although they do not carry the same homotopy
information that the \v{C}ech complex does, the fact that they are
determined by their underlying graph makes them much smaller in memory
and more amenable to certain kinds of calculation.
Let $f: \mathbb{R}^d \to \mathbb{R}^{\ge 0}$ be a bounded measurable
density function and et $\mathcal{X}_n$ denote a
set of $n$ points drawn independently from this distribution. For any
$r>0$ define a (random geometric) graph $G(n,r)$
on $\mathcal{X}_n$ by inserting an edge $\{x,y\}$
whenever $d(x,y) < 2r$. Usually
$r=r(n)$ and we consider the limit
as $n$ tends to infinity.
The {\it random Vietoris-Rips complex} $VR (n,r)$ is the clique
complex of this random geometric graph; that is, the maximal
simplicial complex with 1-skeleton $G(n,r)$.
To see the contrast with $X(n,p)$, Figure \ref{VR-fig}
has a picture of the Betti numbers of a random Rips
complex $VR(n,r)$ on $100$ uniform points in a $6$-dimensional cube,
with $n=100$ and $0 \le r \le 1$; compare with Figure \ref{ER-fig}.
\begin{figure}\label{VR-fig}
\begin{centering}
\includegraphics{geom.png}
\end{centering}
\caption{The Betti numbers of $VR(n,r)$ plotted vertically against $r$
horizontally; $n=100$. \emph{Computation and graphic courtesy of
Afra Zomorodian.}}
\label{fig:geom}
\end{figure}
In the sparse range of parameter, $r = o(n^{-1/d})$, a formula for the
asymptotic expectation of $\beta_k$ was given in \cite{geometric}.
\begin{theorem} \label{exp_rips}
For $d \ge 2$, $k \ge 1$, $\epsilon>0$, and $r _n = O(n^{-1/d -
\epsilon})$, the expectation of the $k$th Betti number $\mathbb{E}[\beta_k]$
of the random Vietoris-Rips complex $VR(X_n;r_n)$ satisfies $$
\frac{\mathbb{E}[\beta_k]}{n^{2k+2} r_n^{d(2k+1)}} \to C_k,$$ as $n \to
\infty$, where $C_k$ is a constant that depends only on $k$ and the
underlying density function $f$.
\end{theorem}
In the same regime we prove limit theorems for
$\beta_k$.
\begin{theorem} \label{clt_rips}
With the same hypothesis as in Theorem \ref{exp_rips},
\begin{enumerate}
\item if $n^{2k+2} r_n^{d(2k+1)}\to0$ as $n\to\infty$, then
$$\beta_k(VR(X_n;r_n))\to0\qquad\qquad a.a.s.;$$
\item if $n^{2k+2} r_n^{d(2k+1)}\to\alpha\in(0,\infty)$ as $n\to\infty$,
then
$$d_{TV}(\beta_k(VR(X_n;r_n)),Y)\le c\alpha nr_n^d,$$
where $Y$ is a Poisson random variable with $\mathbb{E}[Y]=\mathbb{E}[\beta_k]$
and $c$ is a constant depending only on $d$, $k$, and $f$;
\item if $n^{2k+2} r_n^{d(2k+1)}\to\infty$, then
$$ \frac{ \beta_k - \mathbb{E}[\beta_k] }{
\sqrt{\mathrm{Var} [ \beta_k]}} \to \mathcal{N}(0,1).$$
\end{enumerate}
\end{theorem}
(The case $k=0$ is handled in detail by Penrose \cite{penrose}.)
The main idea of the proof of Theorem
\ref{clt_rips} is again to bound $\beta_k$
between two random variables which
satisfy the same central limit theorem. The intuition behind the bounds is
that almost all of the homology of $VR(n,r)$ is contributed
from a single source: the octahedral components.This is essentially because they
are the smallest possible support of homology (smallest in the sense
of vertex support), in the same way that empty $(k-1)$-simplices
were the smallest possible support of homology in the previous section.
\begin{definition}
The $(k+1)$-dimensional {\it cross-polytope} is
defined to be the convex hull of the $2k+2$ points $\{ \pm e_i
\}$, where $e_1, e_2, \ldots, e_{k+1}$ are the standard basis vectors of
$\mathbb{R}^{k+1}$. The boundary of this polytope is a $k$-dimensional simplicial
complex, denoted $O_{k}$.
\end{definition}
Simplicial complexes which arise as clique complexes of graphs are
sometimes called {\it flag complexes}. A useful fact in
combinatorial topology is the following; for a proof see
\cite{clique}.
\begin{lemma} \label{octa}
If $\Delta$ is a flag complex, then any nontrivial element of $k$-dimensional
homology $H_k(\Delta)$ is supported on a subcomplex $S$ with at least
$2k+2$ vertices. Moreover, if $S$ has exactly $2k+2$ vertices, then
$S$ is isomorphic to $O_k$.
\end{lemma}
\begin{definition}
Let $o_k(\Delta)$ (or $o_k$ if context is clear) denote the number of
induced subgraphs of $\Delta$
combinatorially isomorphic to the
$1$-skeleton of the cross-polytope $O_k$, and let $\tilde{o}_k(
\Delta)$ denote the
number of components of $\Delta$ combinatorially isomorphic to the
$1$-skeleton of the cross-polytope $O_k$.
\end{definition}
\begin{definition}
Let $f_k^{= i}(\Delta)$ denote the number of $k$-dimensional faces on
connected components containing with exactly $i$ vertices.
Similarly,
let $f_k^{\ge i}(\Delta)$ denote the number of $k$-dimensional faces
on connected components containing at least $i$ vertices.
\end{definition}
In \cite{penrose}, Penrose proved the following
limit theorems for subgraph counts of random geometric graphs.
\begin{thm}[Penrose]\label{subgraph-clt}
Let $\Gamma_1,\ldots,\Gamma_m$ be graphs on $v\ge2$ vertices, such that
$\P[G(v,r)\cong \Gamma_j]>0$ for each $j$. Let $G_n(\Gamma)$ denote
the number of induced subgraphs of $G(n,r_n)$ isomorphic to $\Gamma$.
Then with $r_n$ as in the
statement of Theorem \ref{clt_rips},
\begin{enumerate}
\item \label{subgraph-means}
There is a constant $\mu_j$ depending only on $\Gamma_j$ and $v$
such that $$\lim_{n\to\infty}r_n^{-d(v-1)}n^{-v}\mathbb{E}[G_n(\Gamma_j)]=\mu_j.$$
\item Let $Z_1,\ldots,Z_m$ be indpendent Poisson random variables with
$\mathbb{E} Z_j=\mathbb{E}[G_n(\Gamma_j)]$. There is a constant $c$ depending only on $m$
such that
$$d_{TV}\big[(G_n(\Gamma_1),\ldots,G_n(\Gamma_m)),(Z_1,\ldots,Z_m)\big]\le
cn^{v+1}r_n^{dv}.$$
\item Suppose that $n^vr_n^{d(v-1)}\to\infty$ as $n\to\infty$. Let
$\tau=\sqrt{n^vr_n^{d(v-1)}}$. Then the joint distribution of the random
variables $\{G_n(\Gamma_j)\}_{j=1}^m$ converges to a centered Gaussian
distribution with covariance matrix $\Sigma=diag(\mu_1,\ldots,\mu_m)$,
for $\mu_j$ as in part \ref{subgraph-means}
\end{enumerate}
\end{thm}
A dimension bound paired with Lemma \ref{octa} yields
\begin{equation}\label{octo-morse}
\tilde{o}_k \le \beta_k \le \tilde{o}_k + f_k^{\ge 2k+3},
\end{equation}
in analogy to the Morse
inequalities used in the first section.
One could work with $f_k^{\ge 2k+3}$ directly, but it turns out to be sufficient
to overestimate $f_k^{\ge 2k+3}$ as follows. For each
$k$-dimensional face, consider the underlying
$(k+1)$-clique; if it is in a component
with at least $2k+3$ vertices, extend the clique to a connected
subgraph with exactly $2k+3$ vertices and ${k+1 \choose 2} + k+2$
edges, by the following algorithm.
\begin{enumerate}
\item Set $G$ to be the $1$-skeleton of
the complex, and initialize $H$ to be the $(k+1)$-clique.
\item Find some edge connecting $V(H)$ to $V(G) - V(H)$. Add this
edge (and its endpoint) to $H$. This is always possible since by
assumption $H$ is contained in a component with at least $2k+3$
vertices.
\item Repeat step $2$ until $H$ has exactly $2k+3$ vertices.
\end{enumerate}
For example, let $k=2$; then
$$\tilde{o}_2 \le \beta_2 \le \tilde{o}_2 + f_2^{\ge
7}.$$ Up to isomorphism, the seventeen graphs that arise when
extending a $2$-dimensional face (i.e.\ a $3$-clique) to a minimal
connected graph on $7$ vertices are exhibited in Figure
\ref{fig:betti2}.
In particular,$f_2^{\ge 7} \le \sum_{i=1}^{17} s_i,$ where
$s_i$ counts the number of subgraphs isomorphic to graph $i$ for some
indexing of the seventeen graphs in Figure \ref{fig:betti2}.
\begin{figure}
\begin{centering}
\includegraphics[width=5in]{B2VR.png}
\end{centering}
\caption{The case $k=2$: the seventeen isomorphism types of subgraphs
which arise when extending a $3$-clique to a connected graph on $7$
vertices with $7$ edges. Each
subgraph isomorphic to one of these can contribute at most $1$
to the sum bounding the error term $f_2^{\ge 7}$.}
\label{fig:betti2}
\end{figure}
In general,
one can express the number of graphs on $2k+3$ vertices that can
arise from the algorithm above as a function of $k$. Moreover,
as is noted in \cite{penrose},
the number of occurances of a given graph $\Gamma$ on $v$ vertices (that is,
the subgraph count corresponding to $\Gamma$) can be written
as a linear combination of the induced subgraph counts for those
graphs on $v$ vertices which have $\Gamma$ as a subgraph. That is,
\begin{equation}\label{VR_morse}\tilde{o}_k\le\beta_k\le o_k+g_{2k+3},
\end{equation}
where $g_{2k+3}$ is a linear combination of the induced subgraph counts
of graphs on $2k+3$ vertices, the number of which depends only on $k$,
and the trivial bound $\tilde{o}_k\le o_k$ has been used on the right-hand side.
The
induced subgraph counts appearing on the right-hand side of \eqref{VR_morse}
are among the components of a random vector whose
joint distribution is identified in Theorem \ref{subgraph-clt} (for two different
values of $v$), and thus limiting distributions for
$o_k$ and $g_{2k+3}$ are known in those regimes.
Moreover, it is easy to modify Penrose's proofs (just as in the previous
section) to show that
$$d_{TV}(o_k+g_{2k+3},Y)\le c\alpha nr_n^d,$$
where $Y$ is a Poisson random variable with $\mathbb{E}[Y]=\mathbb{E}[o_k+g_{2k+3}]$,
which in particular yields a central limit theorem if $n^{2k+2}r_n^{d(2k+1)}\to
\infty$ as $n\to\infty$.
To obtain the limiting distribution for the lower bound of \eqref{VR_morse}
is also just as in the previous section; all the proofs go through
in exactly the same way, and will therefore not be repeated.
For $k=1$ there are several ways of extending a $2$-clique (i.e.\ an
edge) to a connected graph on $5$ vertices and $4$ edges. In this
case the graph must be a tree, and
it is no longer possible
to recover the clique from the connected graph.
However, there
are only three isomorphism types of trees on five vertices, shown in
Figure \ref{fig:betti1}. Counting these types of subgraphs
may therefore result in an underestimate
for $f_1^{\ge 5}$ because some edges might
get extended to the same tree. However, each tree has only four edges, and so one can obtain the bound
$$f_1^{\ge 5} \le 4 ( t_1 + t_2 + t_3),$$ where
$t_1, t_2, t_3$ count the number of subgraphs isomorphic to the three
trees in Figure \ref{fig:betti1}. The proof is then the same as in the
case $k\ge 2$.
\begin{figure}
\begin{centering}
\includegraphics[width=3.5in]{B1VR.png}
\end{centering}
\caption{The case $k=1$: the three isomorphism types of trees on five vertices. Each
subgraph isomorphic to one of these can contribute at most $4$
to the sum bounding the error term $f_1^{\ge 5}$.}
\label{fig:betti1}
\end{figure}
\section{Comments} \label{section:open}
We studied here three different kinds of random simplicial complex in order to work as generally as possible; however there are various ways in which we believe it may be possible to extend our results.\\
\paragraph 1 The random Vietoris-Rips and \v{C}ech complexes studied here are on Euclidean space, but this is mostly a matter of convenience. It would seem that the same proofs work, mutatis mutandis, for arbitrary Riemannian manifolds. This may be of interest in topological data analysis, as in earlier work of Niyogi, Smale, and Weinberger \cite{Smale}.
\medskip
\paragraph 2 It may be possible to extend the central limit theorems for the random Vietoris-Rips and \v{C}ech complexes into denser regimes, at least into the thermodynamic limit. We expect, for example, that there exists some $c>0$ such that CLT's
hold for all Betti numbers $\beta_k$ simultaneously, whenever $r \ge c n^{-1/d}$.
\medskip
\paragraph 3 An easier argument than those
presented here should yield central limit theorems
for Euler characteristic $\chi$ of geometric random complexes, in the sparse range. Again it would be nice to know this
this in denser regimes, and we would guess that it holds at
least partway into the thermodynamic limit.
\bigskip
\noindent {\bf Acknowledgements:} The authors met and began discussing
this project at the Workshop on Topological Complexity of Random Sets held
at the American Institute of Mathematics in August, 2009; many thanks to
AIM and to the organizers of the workshop. The authors also thank
Omer Bobrowski for pointing out a mistake in the original version
of the paper.
\bibliographystyle{plain}
|
{
"timestamp": "2011-01-19T02:00:31",
"yymm": "1009",
"arxiv_id": "1009.4130",
"language": "en",
"url": "https://arxiv.org/abs/1009.4130"
}
|
\section*{Introduction}
A toric arrangement is a finite set of hypersurfaces in a complex torus $T=(\C^*)^n$, in which every hypersurface is the kernel of a character $\chi \in X \subset Hom(T,\C^*) $ of $T$.
Let $\mathcal{R}_X$ be the complement of the arrangement: its geometry and topology have been studied by many authors, see for instance \cite{L2}, \cite{li}, \cite{DP}, \cite{Mo2}.
In particular, in \cite{Lo} and \cite{DP} the De Rham cohomology of $\mathcal{R}_X$ has been computed, and recently in \cite{Mo3} a \emph{wonderful model} has been built.
In the present paper we build a topological model $\Cal{S}$ for $\mathcal{R}_X$. This
model is a regular CW-complex, similar to the one introduced by Salvetti (\cite{Sa1}) for the complement of hyperplane arrangements.
Indeed for a wide class of arrangements, which we call \textit{thick},
the cells of $\Cal{S}$ are in bijection with pairs $[C \prec F]$, where
$C$ is a \textit{chamber} of the \textit{real} toric arrangement and
$F$ is a \textit{facet} adjacent to it (according to the definitions
given in Section \ref{DefTo}).
\bigskip
The model $\Cal S$ is well suited for homology and homotopy computations, which will be developed in future papers (see for instance \cite{simo2}).
Furthermore,
the jumping loci in the local system cohomology of a CW-complex are affine algebraic varieties. In the theory of hyperplane arrangements such objects, called \emph{characteristic varieties}, proved to be of fundamental importance. It is then a remarkable fact that the characteristic varieties can be defined also in the toric case.
In Section 2 we focus on a toric arrangement associated to an affine
Weyl group $\widetilde{W}$.
In this case the chambers are in bijection with the elements
of the corresponding finite Weyl group W, and the cells of $\Cal S$ are
given by the pairs $(w, \Gamma)$, where $w \in W$ and $\Gamma$ is a
proper subset of the set $S$ of generators of $\widetilde{W}$.
This generalizes a construction introduced in \cite{Sa} and \cite{boss3}.
In the last Section we give a description of the facets of the real toric arrangement defined by the Weyl group $\widetilde A_n$ in the torus corresponding to the root lattice. This description in terms of Young tableaux turns out to be interesting since it coincides with the complex
describing the space of all periodic legged gaits of a robot body (see \cite{KGCohen}).
\bigskip
\paragraph{Acknowledgements}
We are grateful to the organizers of the research program "Configuration Spaces: Geometry, Combinatorics and Topology" at Centro De Giorgi (Pisa), which provided us a significant occasion to work together. In particular we wish to thank Fred Cohen and Mario Salvetti for several valuable suggestions. We also thank Priyavrat Deshpande for many stimulating conversations we had while we were completing the present paper.
\section{The CW-complex}
\subsection{Main definitions}
Let $T=(\C^*)^n$ be a complex torus and $X \subset Hom(T,\C^*) $ be a finite set of
characters of $T$. The kernel of every $\chi \in X$ is a hypersurface
of $T$:
\begin{equation*}
H_{\chi}:=\{t \in T \, \mid \, \chi(t)=1\}.
\end{equation*}
Then $X$ defines on T the \textit{toric arrangement}:
\begin{equation*}
\mathcal{T}_X:=\{ H_{\chi} , \chi \in X\}.
\end{equation*}
Let $\mathcal{R}_X$ be the \textit{complement } of the arrangement:
\begin{equation*}
\mathcal{R}_X:= T \setminus \bigcup_{\chi \in X} H_{\chi}.
\end{equation*}
Let $\pi: V \longrightarrow T$ be the universal covering of $T$. Then
$V$ is a complex vector space of rank $n$, and $\pi$ is the quotient
map $\pi: V \longrightarrow V/ \Lambda$, where $\Lambda$ is a lattice
in $V$. Then the preimage $\pi^{-1}(H_{\chi})$ of a hypersurface
$H_{\chi} \in \mathcal{T}_X$ is the union of an infinite family of parallel hyperplanes. Thus
$$ \Cal A_X:=\{ H \mbox{ hyperplane of } V \mid \exists \chi \in X
\mbox{ s.t. } \pi(H)=H_{\chi}\}$$
is a periodic affine hyperplane arrangement in V. Let $\mathcal{M}_X$ be its
complement:
\begin{equation*}
\mathcal{M}_X:= V \setminus \bigcup_{\chi \in X}\pi^{-1}(H_{\chi}).
\end{equation*}
By definition, $\pi$ maps $\mathcal{M}_X$ on $\mathcal{R}_X$. Moreover the equations
defining the hyperplanes in $\Cal A_X$ can always be assumed to have
integral (hence real) coefficients since they are given by elements of
$\Lambda$. Thus by \cite{Sa1} there is an (infinite) CW-complex
$\widetilde{\Cal S} \subset \mathcal{M}_X$ and a map $\varphi: \mathcal{M}_X
\longrightarrow \widetilde{\Cal S}$ giving a homotopy equivalence. \\
Furthermore, we can build $\widetilde{\Cal S}$ in such a way that it is
invariant under the action of translation in $\Lambda$: for instance
by building the cells relative to a fundamental domain and then inductively,
defining for each cell above the other cells of its $\Lambda$-orbit by translation.
Thus $\pi(\widetilde{\Cal S})$ is a finite CW-complex, which will be
denoted by $\mathcal{S}$, and the image of every cell of $\widetilde{\Cal S}$
is a cell of $\mathcal{S}$. Moreover, since $\varphi$ is $\Lambda-$equivariant,
it is well defined the map
\begin{equation*}
\varphi_{\pi}(t):= (\pi \circ \varphi)(\pi^{-1}(t))
\end{equation*}
which makes the following diagram commutative:
\begin{equation} \begin{array}{ccc}\label{complesso}
\mathcal{M}_X &\xrightarrow{\varphi} &\widetilde{\Cal S} \\
\pi \downarrow & &\pi \downarrow \\
\mathcal{R}_X & \xrightarrow{\varphi_{\pi}} & \Cal S
\end{array}
\end{equation}
\begin{lem} The map $\varphi_{\pi}$ is a homotopy equivalence between
$\mathcal{R}_X$ and $\Cal S$.
\end{lem}
\textbf{Proof.} The map $\varphi$ is a homotopy equivalence hence, by
definition, there is a continuous map $\psi: \widetilde{\Cal S}
\rightarrow \mathcal{M}_X$
such that $\psi \varphi$ is homotopic to the
identity map $id_{\mathcal{M}_X}$ and $\varphi \psi$ is homotopic to
$id_{\widetilde{\Cal S}}$. Namely, since $\widetilde{\Cal S}$ is a deformation retract, the homotopy inverse $\psi$ is simply the inclusion map, which is clearly $\Lambda-$equivariant. Hence the map
$$\psi_\pi(t):= (\pi \circ \psi)(\pi^{-1}(t))$$
is well defined and makes the following diagram commutative:
\begin{equation} \begin{array}{ccc}
\widetilde{\Cal S} &\xrightarrow{\psi} & \mathcal{M}_X\\
\pi \downarrow & &\pi \downarrow \\
\Cal S & \xrightarrow{\psi_{\pi}} & \mathcal{R}_X .
\end{array}
\end{equation}
Let $I=[0,1]$ be the unit interval and $F: \mathcal{M}_X \times I \rightarrow
\mathcal{M}_X$ be the continuous map such that $F(x,0) = \psi(\varphi(x))$ and
$F(x,1)=id_{M_{X}}(x)$. Again, since $F$ is $\Lambda-$equivariant, we can define the map:
$$F_\pi(t):= (\pi \circ F)(\pi^{-1}(t))$$
In this way we get the commutative diagram:
\begin{equation} \begin{array}{ccc}
\mathcal{M}_X \times I&\xrightarrow{F} & \mathcal{M}_X \\
\pi \downarrow & &\pi \downarrow \\
\mathcal{R}_X \times I& \xrightarrow{F_{\pi}} & \mathcal{R}_X.
\end{array}
\end{equation}
By construction map $F_{\pi}$ is a continuous map such that $F_{\pi}(x,1)=id_{\mathcal{R}_X}$ and
$$F_{\pi}(x,0)= (\psi\varphi)_{\pi}(x)=\pi\psi \varphi\pi^{-1}(x)=\pi\psi \pi^{-1}\pi\varphi\pi^{-1}(x)=\psi_\pi\circ\varphi_\pi(x).$$
Hence $F_{\pi}$ gives the required homotopy equivalence. $\qquad \square$
\subsection{Salvetti complex for affine arrangements}
In order to describe the structure of $\Cal S$, we now have to focus on the real counterparts of the complex arrangements above.\\
Let $V_{\R}$ be the real part of $V$. In other words, let $V_{\R}\doteq \R^n$ be a real vector space, and let $V\doteq V_{\R}\otimes_\R \C$ be its complexification. Then we identify $V_{\R}$ with a subspace of $V$ via the map $v\mapsto v\otimes 1$.\\
Let $\Cal{A}_{X,\R}$ be the
corresponding hyperplane arrangement on $V_{\R}$ and $\mathcal{M}_{X,\R}=\mathcal{M}_{X}\cap V_{\R}$ its
complement.
Since the image of $\R$ under the map $\C \longrightarrow \C / \Z
\xrightarrow{\sim} \C^*$ is the circle
$$S^1:=\{z \in \C \, \mid \, \mid z \mid =1\}$$
we have that the image of $V_{\R}$ under the map $\pi:V \rightarrow V/\Lambda
\xrightarrow{\sim} T$ is a compact torus $T_{\R} \subset T$. A \emph{real toric
arrangement} $\Cal T_{X,\R}$ is naturally defined on $T_{\R}$ with
hypersurfaces $H_{\chi,\R}:=H_{\chi} \cap T_{\R}$ and complement
$\mathcal{R}_{X,\R} = \mathcal{R}_X \cap T_{\R}$. Furthermore $\pi$ restricts to universal
covering map $\pi: V_{\R} \longrightarrow T_{\R}$ and
$\pi(\mathcal{M}_{X,\R})=\mathcal{R}_{X,\R}$.
\bigskip
We recall the following definitions:
\begin{enumerate}
\item a \textit{chamber} of $\Cal A_{X,\R}$ is a connected component
of $\mathcal{M}_{X,\R}$;
\item a \textit{space} of $\Cal A_{X,\R}$ is an intersection of
elements in $\Cal A_{X,\R}$;
\item a \textit{facet} of $\Cal A_{X,\R}$ is the intersection of a space
and the closure of a chamber.
\end{enumerate}
Let $\mathbf{S}:=\{\widetilde{F}^k\}$ be the stratification of $V_{\mathbb{R}}$ into facets $\widetilde{F}^k$
induced by the arrangement $\Cal{A}_{X,\R}$ (see \cite{Bou}), where superscript $k$ stands
for codimension.
Then the $k$-cells of the complex $\widetilde{\Cal S}$ described in
\cite{Sa1} bijectively correspond to pairs
$$[\widetilde{C} \prec \widetilde{F}^k]$$
where $\widetilde{C}=\widetilde{F}^0$ is a chamber of $\mathbf S$ and
$\widetilde{F}^i \ \prec \widetilde{F}^j \Leftrightarrow
clos(\widetilde{F}^i)\supset \widetilde{F}^j$ is the standard partial
ordering in $\mathbf S$ (see also \cite{OT}).
Let $|\widetilde{F}|$ be the affine subspace spanned by $\widetilde{F},$ and let us consider
the subarrangement
$$\Cal A_{\widetilde{F}}\ =\ \{ H \in \Cal{A}_{X,\R} \ :\ \widetilde{F}\subset H\}.$$
A cell $[\widetilde{C}\prec \widetilde{F}^k]$ is in the boundary of $[\widetilde{D}\prec \widetilde{G}^j]$ ($k<
j$) if and only if
\begin{equation}\label{bordoaffine}
\begin{split}
& \mbox{i) } \widetilde{F}^k\prec \widetilde{G}^j \\
& \mbox{ii) } \mbox{ the chambers } \widetilde{C} \mbox{ and } \widetilde{D} \mbox{ are contained in the same chamber of } \Cal A_{\widetilde{F}^k}.
\end{split}
\end{equation}
Previous conditions are equivalent to say that
$\widetilde{C}$ is the chamber of $\Cal{A}_{X,\R}$ which is the "closest" to $\widetilde{D}$
among those which contain $\widetilde{F}^k$ in their closure. The standard notation
$[\widetilde{C}\prec \widetilde{F}^k] \in \partial_{\widetilde{\Cal S}}[\widetilde{D}\prec \widetilde{G}^j]$ will be used.
\bigskip
{It is a simple remark that the above description of the Salvetti complex
$\widetilde{\Cal S}$ is $\Lambda$-invariant. Indeed each translation $t \in \Lambda$ acts on the stratification $\mathbf{S}:=\{\widetilde{F}^k\}$ sending a $k$-facet $F^k$ into the $k$-facet
$t.F^k$. Then the translation $t$ acts on $\widetilde{\Cal S}$ sanding a $k$-cell $[C \prec F^k]$
in the $k$-cell $[t.C \prec t.F^k]$.}
\subsection{Salvetti Complex for toric arrangements}\label{DefTo}
In order to give a similar description for $\Cal S$, we introduce the
following definitions:
\begin{enumerate}
\item a \textit{chamber} of $\Cal T_{X,\R}$ is a connected component
of $R_{X,\R}$;
\item a \textit{layer} of $\Cal T_{X,\R}$ is a connected component of
an intersection of elements of $\Cal T_{X,\R}$;
\item a \textit{facet} of $\Cal T_{X,\R}$ is an intersection of a layer
and the closure of a chamber.
\end{enumerate}
\begin{lem}\label{camereefacce}
~
\begin{enumerate}
\item If $\widetilde{C}$ is a chamber of $\Cal A_{X,\R}$,
$\pi(\widetilde{C})$ is a chamber of $\Cal T_{X,\R}$;
\item If $\widetilde{L}$ is a space of $\Cal A_{X,\R}$,
$\pi(\widetilde{L})$ is a layer of $\Cal T_{X,\R}$;
\item If $\widetilde{F}$ is a facet of $\Cal A_{X,\R}$,
$\pi(\widetilde{F})$ is a facet of $\Cal T_{X,\R}$;
\end{enumerate}
\end{lem}
\textbf{Proof.} The first statement is clear, as well as the second
one since $\pi(\widetilde{L})$ must be connected. The third claim is a
direct consequence of the previous two.~$~\square~$
\bigskip
Now, let us consider pairs $$[C \prec F^k]$$
where $C=F^0$ is a chamber of $\Cal T_{X,\R}$, $F^k$ a
$k$-codimensional facet of
$\Cal T_{X,\R}$ and $F^i \prec F^j \Leftrightarrow
clos(F^i)\supset F^j$.
By Lemma \ref{camereefacce} the quotient map $\pi(\widetilde F)$ of a facet
is still a facet in the real torus and, because of the surjectivity of $\pi$, we get that any facet $F$ in $\Cal{T}_{X, \R}$ is the image
$F=\pi(\widetilde F)$ of an affine one.
We notice that
$$\pi([\widetilde C \prec \widetilde F])=\pi([\widetilde D \prec \widetilde G]) \Longrightarrow [\pi(\widetilde C) \prec \pi(\widetilde F)] =[\pi(\widetilde D) \prec \pi(\widetilde G)]. $$
Indeed if $\pi([\widetilde C \prec \widetilde F])=\pi([\widetilde D \prec \widetilde G])$ there is a translation
$t \in \Lambda$ which sends $[\widetilde C \prec \widetilde F]$ in $[\widetilde D \prec \widetilde G]$. As a simple
consequence $\widetilde D=t. \widetilde C$ and $\widetilde F =t.\widetilde G$, i.e. $\pi(\widetilde C)=\pi(\widetilde D)$ and
$\pi(\widetilde F)=\pi(\widetilde D)$.
Then there is a natural surjective map from the cells of $\Cal S$ to the set of pairs $[C \prec F]$, but this map in general is not injective.
Let us consider the simple example defined by $\Cal A=\{x \in \R \mid x \in \Z \}$.
\begin{equation*}
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\end{equation*}
The chambers $\widetilde{C}_i$ for $i \in \Z$ are the open intervals $(i,i+1)$ and the
$1$-codimensional facets are the points. The toric arrangement depends on the chosen lattice. For example we can quotient in two different way as in the above figure.
Namely, the picture on the left corresponds to the choice $\Lambda=\Z$, i.e. $\pi:x\mapsto e^{2\pi i x}$, whereas the picture on the right is given by $\Lambda=2\Z$ and $\pi:x\mapsto e^{\pi i x}$.
{As shown in the pictures the complex in the former example cannot be described by the two pairs
$[C_{-1} \prec C_{-1}]$, $[C_{-1} \prec e_0]$ since it has $3$ cells.
Furthermore, this CW-complex is not regular (the closure of its cells is not contractible). On the other hand, in the latter example we have a regular CW-complex with two 0-dimensional cells and four 1-dimensional cells.
}
\bigskip
Now we will focus on the case in wich $\Cal S$ maps bijectively on the set of pairs
$[C \prec F]$, since then the description of the complex $\mathcal{S}$ is particularly striking.
Since $\Cal S=\pi(\widetilde{\Cal S})$ is a complex homotopic to the complement $\mathcal{R}_X$,
$\Cal S$ is described by pairs of the form $[C \prec F]$ if and
only if the map
\begin{equation}\label{cond0}
\pi([\widetilde C \prec \widetilde F]) \longrightarrow [\pi(\widetilde C) \prec
\pi(\widetilde F)]
\end{equation}
is injective.
\bigskip
Moreover, if the definition (\ref{cond0}) holds then we can define the boundary of a pair
$[C \prec F]$. We need first to introduce new notations.
\bigskip
\textbf{Notations.} Let $P_0\subset V$ be a fundamental parallelogram for $\pi: V\rightarrow T$ containing the origin of $V$.
Let $\Cal{A}_{0,X}$ be the subarrangement of $\Cal{A}_{X}$ made by all the hyperplanes that intersect $P_0$ (see, for istance, figure (\ref{figura1}) in the next Section).
We will say that a maximal dimensional cell $[\widetilde C \prec \widetilde F^n]$ is in $\Cal{A}_{0,X}$ if its support $\mid \widetilde F^n \mid$ is the intersection
of some of the hyperplanes in $\Cal{A}_{0,X}$. While a $k$-cell $[\widetilde
C \prec \widetilde F^k]$ is in $\Cal{A}_{0,X}$
if it is in the boundary of a $n$-cell in $\Cal{A}_{0,X}$. Let $\widetilde {\Cal S}_{0}$ be the set of all
such cells.
\bigskip
With previous notations if (\ref{cond0}) is injective (i.e. it is a bijection) we define the boundary
as follow:
\bigskip
$[C \prec F^k]$ is in the boundary of $[D \prec G^j]$ $(k < j)$ if and only if there are cells
$[\widetilde C \prec \widetilde F^k] \in \pi^{-1}([C \prec F^k]) \cap \widetilde {\Cal S}_{0}$
and $[\widetilde D \prec \widetilde G^j] \in \pi^{-1}([D \prec G^j]) \cap \widetilde {\Cal S}_{0}$
such that $[\widetilde C \prec \widetilde F^k] \in \partial_{\widetilde{\Cal S}} [\widetilde D \prec \widetilde G^j]$.
\bigskip
Obviously this boundary map commutes with the one in $\widetilde{\Cal S}$ and
we get that the map in (\ref{cond0}) is a bijection of CW-complexes.
\bigskip
Toric arrangement for which $\Cal S$ is in bijection with pairs $[C \prec F]$ are easily characterized as follows.
\begin{df} A toric arrangement $\Cal{T}_X$ is \emph{thick} if the quotient map
$$\pi : V \longrightarrow T$$ is injective on the closure $clos(\widetilde{C})$ of every chamber
$\widetilde{C}$ of the associated affine arrangement $\Cal{A}_{X, \R}$.
\end{df}
We notice that every toric arrangement is covered by a thick one and the fiber of the covering map is finite;
hence our assumption is not very restrictive.
We have the following
\begin{lem}\label{isom} A toric arrangement $\Cal{T}_X$ is thick if and only if
$$
[\pi(\widetilde C) \prec \pi(\widetilde F)]=[\pi(\widetilde D) \prec \pi(\widetilde G)] \Longleftrightarrow \pi([\widetilde C \prec \widetilde F])=\pi([\widetilde D \prec \widetilde G])
$$
for any two cells $[\widetilde C \prec \widetilde F], [\widetilde D \prec \widetilde G] \in \widetilde{\Cal S}$
\end{lem}
\textbf{Proof.} By previous considerations, it is enough to prove that the thick condition is equivalent to
$$[\pi(\widetilde C) \prec \pi(\widetilde F)]=[\pi(\widetilde D) \prec \pi(\widetilde G)] \Longrightarrow \pi([\widetilde C \prec \widetilde F])=\pi([\widetilde D \prec \widetilde G])$$
$\Rightarrow :$ Let $\Cal{T}_X$ be thick and $[\pi(\widetilde C) \prec \pi(\widetilde F)]=[\pi(\widetilde D) \prec \pi(\widetilde G)]$ for two given $k$-cells in $\widetilde{\Cal S}$. This implies that $\pi(\widetilde C) = \pi(\widetilde D)$ and
$\pi(\widetilde F)=\pi(\widetilde G)$, i.e. there are translations $t,t^{\prime} \in \Lambda$ such that
$\widetilde D=t.\widetilde C$ and $\widetilde G=t^{\prime}.\widetilde F$.
By construction $t.\widetilde F$ is a facet in the closure
$clos(D)$. We get two facets $t. \widetilde F$ and $\widetilde G$ both in $clos(D)$ and with the same image
$\pi(t.\widetilde F)=\pi(\widetilde F)=\pi(\widetilde G)$. By hypothesis $\pi$ is injective on $clos(D)$ then
$t.\widetilde F=G$, i.e. $t=t^{\prime}$ which implies that $\pi([\widetilde C \prec \widetilde F])=\pi([\widetilde D \prec \widetilde G])$.
$\Leftarrow$ Let $\widetilde F$ and $\widetilde G$ two facets in $clos(\widetilde C)$ such that
$\pi(\widetilde F)=\pi(\widetilde G)$ then
$$\pi([\widetilde C \prec \widetilde F])=[\pi(\widetilde C) \prec \pi(\widetilde F)]=[\pi(\widetilde C) \prec \pi(\widetilde G)]=
\pi([\widetilde C \prec \widetilde G]).$$
As a consequence if $t \in \Lambda$ is the translation such that $\widetilde F=t.\widetilde G$ then
$t. \widetilde C=\widetilde C$. It follows that $t$ is the identity and we get $\widetilde F=\widetilde G$, i.e.
$\pi$ is injective on $clos(\widetilde C)$ $\qquad$ $\square$
\bigskip
By Lemma \ref{isom} the map defined in (\ref{cond0}) is a bijection if and only if
$\mathcal{T}_X$ is a thick toric arrangement. Hence the set of pairs $[C \prec F]$ is a CW-complex $\overline{\Cal S}$ and we get the following theorem
\begin{teo}
Let $\mathcal{T}_X$ be a thick toric arrangement.
Then its complement $\Cal{R}_X$ has the same
homotopy type of the CW-complex $\overline{\Cal S}$.
\end{teo}
Then in this case the complex $\mathcal{S}$ has a nice combinatorial description, totally analogue to that of the classical Salvetti complex \cite{Sa1}.
Moreover if a toric arrangement is thick then the maximal dimensional cells
$[\widetilde{C} \prec \widetilde{F}^n]$ in $\Cal{A}_{0,X}$ are in one to one correspondence with the $n$-dimensional facets of $\overline{\Cal{S}}$.
Then the boundary in a thick toric arrangement $\Cal T_X$ can be completely described knowing the boundary in the associated finite complex $\Cal A_{0,X}$.
This allows to better understand the fundamental group of the complement
and to perform computations on integer cohomology.
Furthermore, in this case $\mathcal{S}$ is a \emph{regular} CW-complex.
\begin{rmk} The number of chambers of $\Cal T_{X,\R}$ can be computed
by formulae given in \cite{ERS} and \cite{Mo2}. However, the
combinatorics of the layers in $\Cal T_{X,\R}$ is more complicated
than the one of spaces of $\Cal A_{X,\R}$. Hence, an enumeration of
the facets is not easy to provide in the general case. Thus from now on
we focus on the arrangements defined by roots systems. In this case
the chambers are parametrized by the elements of the Weyl group, and
the poset of layers has been described in \cite{Mo1}.
\end{rmk}
\section{Weyl toric arrangements}
In this section we give a simpler description of the above complex for the case of toric arrangements associated to affine Weyl groups, by taking as $\Lambda$ the coroot lattice (for the theory of Weyl groups see, for instance, \cite{Bou}).
Indeed in this case the toric arrangement is thick.
Using this description, we give an example of how the integer cohomology of these arrangements can be computed.
\subsection{Notations and Recalls.}
\paragraph{Toric arrangement associated to a Weyl group.}
Let $\Phi$ be a root system, $\Lambda=\langle\Phi^\vee\rangle$ be the
lattice spanned by the coroots, and $\Delta$ be its dual
lattice (which is called the \emph{cocharacters} lattice).
Then we define a torus $T=T_\Delta$ having $\Delta$
as group of characters. Namely, if $\mathfrak{g}$
is the semisimple complex Lie algebra associated to $\Phi$ and
$\mathfrak{h}$ is a Cartan subalgebra, $T$ is defined as the
quotient $T\doteq {\mathfrak{h}}/\Lambda$.
Each root $\alpha$ takes integer values on
$\Delta$, so it induces a map
$$e^{\alpha}: T\rightarrow {\mathbb{C}}/{\mathbb{Z}}\simeq \mathbb{C^*}$$
which is a character of the torus.
Let $X$ be the set of these
characters; more precisely, since $\alpha$ and $-\alpha$ define
the same hypersurface, we set
$$X\doteq \left\{e^{\alpha},\:\alpha\in \Phi^+\right\}.$$
In this way to every affine Weyl group $\widetilde{W}$ we associate a toric
arrangement $\mathcal{T}_{\widetilde{W}}$, with complement $\mathcal{R}_{\widetilde{W}}$.
We will call these arrangements Weyl toric arrangements. They have been studied in \cite{Mcm} and \cite{Mo1}.
\begin{rmk}\label{remsec}
~
\begin{enumerate}
\item Let $G$ be the semisimple, simply connected linear
algebraic group associated to $\mathfrak{g}$. Then
$T$ is the maximal torus of $G$ corresponding to
$\mathfrak{h}$, and $\mathcal{R}_X$ is known as the set of
\emph{regular points} of $T$.
\item One may take as $\Delta$ the root lattice (or equivalently, take as $\Lambda$ the character lattice).
But in this way one obtains as $T$ a maximal
torus of the semisimple \emph{adjoint} group $G^a$,
which is the quotient of $G$ by its center.
\end{enumerate}
\end{rmk}
Let $(\widetilde{W},S)$ be the Coxeter system associated to $\widetilde{W}$ and
$$\mathcal{A}_{\widetilde{W}}=\{H_{\widetilde{w} s_i \widetilde{w}^{-1}} \mid \widetilde{w}
\in \widetilde{W} \mbox{ and } s_i \in S\}$$
the arrangement in $\C^n$ obtained by complexifying the reflection hyperplanes of
$\widetilde{W}$, where, in a standard way, the hyperplane
$H_{\widetilde{w} s_i \widetilde{w}^{-1}}$ is the hyperplane fixed by the reflection
$\widetilde{w} s_i \widetilde{w}^{-1}$.\\
We can view $\Lambda$ as a subgroup of $\widetilde{W}$, acting by translations. Then it is well known that $\widetilde{W}/\Lambda \simeq W$, where $W$ is the finite reflection group associated to $\widetilde{W}$ (see for instance \cite{Ra}). As a consequence, the toric arrangement
can be described as:
$$
T_{\widetilde{W}}=\{H_{[w] s_i [w^{-1}]} \mid w \in W \mbox{ and } s_i \in S \}
$$
where two hypersurfaces $H_{[w] s_i [w^{-1}]}$ and $H_{[\overline{w}] s_i [\overline{w}^{-1}]}$ are equal if and only if
there is a translation $t \in \Lambda$ such that $tw s_i (tw)^{-1}=\overline{w} s_i \overline{w}^{-1}$, i.e. $\overline{w}=tw$.
By \cite{Mo1}, these hypersurfaces
intersect in
$$
\frac{\mid W \mid}{\mid W_{S \setminus \{s_i \} } \mid}
$$
local copies of the finite hyperplane arrangement $A_{W_{S \setminus \{s_i\}}}$ associated to the group
generated by $S \setminus \{s_i\}$, $s_i \in S$.
For example in the affine Weyl group $\widetilde{A}_n$ generated by $\{s_0,\ldots, s_n\}$
for any generator $s_i$ the finite reflection group associated to $S \setminus \{s_i\}$ is a copy of the finite Coxeter group $A_n$.
Then we have
{\begin{prop}The toric arrangement $\mathcal{T}_{\widetilde{W}}$ is thick. \end{prop} }
{\textbf{Proof.} Since $\Lambda$ is the coroot lattice, if $t \in \Lambda$ is a translation such that there is a $n$-codimensional facet $\widetilde F^n \in clos(\widetilde C) \cap clos(t.\widetilde C)$ for an affine chamber $\widetilde C$, then $t$ is the identity (see \cite{Bou}). }
{If $T_{\widetilde{W}}$ is not thick then there are two facets $\widetilde F_1$ and $\widetilde F_2$ in
the closure $clos(\widetilde C)$ of a chamber $\widetilde C$ such that $\pi(\widetilde F_1)=\pi(\widetilde F_2)$, i.e.
there is a translation $t \in \Lambda$ such that $\widetilde F_2= t.\widetilde F_1$. Hence $\widetilde F_2$ is a facet
in $clos(C) \cap clos(t.C)$. In particular all the $n$-codimensional facets $\widetilde F^n$ in the closure
of $\widetilde F_2$ are in the closure of both $C$ and $t.C$. This is a contradiction and it concludes the proof. $\qquad \square$}
\bigskip
Then we can construct
the Salvetti complex for these arrangements in a way which is very similar to the one known for affine Coxeter arrangements.
\paragraph{Salvetti Complex for affine Artin groups}
It is well known (see, for instance, \cite{boss3}, \cite{Sa} ) that the cells of Salvetti complex
$\widetilde {\Cal S}_W$ for arrangements $\Cal{A}_{\widetilde{W}}$ are of the form
$E(\widetilde{w},\Gamma)$ with
$\Gamma \subset S$ and $\widetilde{w} \in \widetilde{W}$. Indeed if $\widetilde{\alpha} \in \{\widetilde{w}s\widetilde{w}^{-1} | s \in S, \widetilde{w} \in \widetilde{W}\}$ is a reflection,
the chambers are in one to one correspondence with the elements of the group $\widetilde{W}$ as follows.
Fixed a base chamber $C_0$, it corresponds to $1 \in \widetilde{W}$. Now if $C$ corresponds to $\widetilde{w}$, the chamber $D$ separated from $C$ by the reflection hyperplane $H_{\widetilde{\alpha}}$ corresponds to the element
$\widetilde{\alpha}\widetilde{w} \in \widetilde{W}$. The notation
$D \simeq \widetilde{\alpha}\widetilde{w}$ will be used.
If $\widetilde F^k$ is a $k$-codimensional facet then the $k$-cell $[\widetilde C \prec \widetilde F^k]$
corresponds to the pair $E(\widetilde{w},\Gamma)$ where $\widetilde{w} \simeq \widetilde C$ and $\Gamma=\{s_{i_1},\ldots , s_{i_k}\}$ is the unique subset of cardinality $k$ in $S$ such that
$$\mid F^k \mid = \bigcap_{j=1}^k H_{\widetilde{w}s_{i_j}\widetilde{w}^{-1}}.$$
If $\widetilde{W}_{\Gamma}$ is the finite subgroup generated by $s \in \Gamma$, by \cite{boss3} the integer boundary map can be expressed as follows:
\begin{equation}
\begin{split}
\partial_k(E(\widetilde{w},\Gamma)) =
&\sum_{s_j \in \Gamma}
\sum_{\beta\in \widetilde{W}^{\Gamma\setminus\{ s_j \}}_{\Gamma}}(-1)^{l(\beta)+\mu
(\Gamma,s_j)} E(\widetilde{w}\beta,\Gamma\setminus\{ s_j\}).
\end{split}
\end{equation}
where $\widetilde{W}^{\Gamma\setminus\{\sigma\}}_{\Gamma}=\{w \in \widetilde W_{\Gamma} : l(ws) > l(w) \forall s \in \Gamma \setminus\{\sigma\} \}$ and $\mu(\Gamma, s_j)=\sharp\{s_i \in \Gamma | i \leq j \}$.
\begin{rmk}Instead of the co-boundary operator we prefer to describe its dual,
i.e. we define the boundary of a $k$-cell $E(\widetilde{w},\Gamma )$ as a linear combination of
the $(k-1)$-cells which have $E(\widetilde{w},\Gamma )$ in their co-boundary, with the
same coefficient of the co-boundary operator.
We make this choice since the boundary operator has a nicer description than
co-boundary operator in terms of the elements of $\widetilde W$.
\end{rmk}
\subsection{Description of the complex}
Let $\Cal{S}_W$ be the CW-complex associated to $\mathcal{T}_{\widetilde{W}}$. By the previous considerations, $\Cal{S}_W$ admits a description similar to that of
$\widetilde{\Cal S}_W$.
Indeed each chamber $C$ is in one to one correspondence with an equivalence class
$[w] \in \widetilde{W} / \Lambda$ and then with an element
$w \in W \simeq \widetilde{W} / \Lambda$ of the finite reflection group $W$. We will write $C \simeq [w]$.
In the same way, the pair $[C \prec F^k]$
corresponds to the cell $E([w],\Gamma) \in \Cal{S}_W$ where $C \simeq [w]$ and
$\Gamma=\{s_{i_1},\ldots , s_{i_k}\}$ is the unique subset of cardinality $k$ in $S$ such that
$$\mid F^k \mid = \bigcap_{j=1}^k H_{[w]s_{i_j}[w^{-1}]}.$$
We now want to describe the boundary of each cell: this is done in a
standard way by characterizing the cells that are in the boundary of a
given cell, and by assigning an orientation to all cells (see, for
instance, \cite{Sa}).
By construction the toric CW-complex is locally isomorphic to the affine one and it can inherit its affine orientation. Then the integer boundary operator for Weyl toric arrangements can be written as the affine one:
\begin{equation}\label{bordo}
\begin{split}
\partial_k(E([w],\Gamma)) =
&\sum_{\sigma\in \Gamma}
\sum_{\beta\in W^{\Gamma\setminus\{\sigma\}}_{\Gamma}}(-1)^{l(\beta)+\mu
(\Gamma,\sigma)} E([w\beta],\Gamma\setminus\{ \sigma \})
\end{split}
\end{equation}
where, instead of elements of the affine group $\widetilde{W}$, we have equivalence classes with representatives in the finite group $W$.
By the formula above, the complex $\Cal{S}_W$ can be effectively used for computing homotopy invariants of $\mathcal{R}_{\widetilde{W}}$. For instance we have
{\begin{prop} $$H^{\bullet}(\mathcal{R}_{\widetilde{W}},\Z) \simeq H^{\bullet}(\Cal{S}_W,\Z) $$
where the coboundary map is the dual of the map defined in (\ref{bordo}).
\end{prop}}
\bigskip
\textbf{Example.} Let us consider the affine Weyl group $\widetilde{B}_2$ (see \cite{Bou}) with Coxeter-Dynkin diagram
\begin{equation*}
\begin{array}{ccccc}
\circ & \stackrel{4}{-} &\circ & \stackrel{4}{-} &\circ \\
s_0& & s_1& &s_2 \\
\end{array}
\end{equation*}
and associated finite group $B_2$
\begin{equation*}
\begin{array}{ccc}
\circ & \stackrel{4}{->} &\circ \\
s_1& &s_2 \\
\end{array}
\end{equation*}
In this case we get translations $t_1=s_0s_1s_2s_1$ and $t_2=s_2s_1s_0s_1$ and the affine arrangement is represented as:
\begin{equation*}
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\put{$H_{\alpha_2,2}$}[r] at -3.3 -1.3
\put{$H_{\alpha_2,3}$}[r] at -3.3 -0.3
\put{$H_{\alpha_2,4}$}[r] at -3.3 0.7
\put{$H_{\alpha_2,5}$}[r] at -3.3 1.7
\put{$H_\varphi = H_{\alpha_1+\alpha_2}$}[br] at -3.3 3.3
\put{$H_{\alpha_1+\alpha_2,1}=H_{\varphi,1}=H_{\alpha_0}$}[tl] at 3.3 -2.3
\put{$H_{\alpha_1+\alpha_2,-1}$}[r] at -3.3 2.3
\put{$H_{\alpha_1+\alpha_2,-2}$}[r] at -3.3 1.3
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\put{$H_{\alpha_1+\alpha_2,-5}$}[r] at -3.3 -1.7
\put{$H_{\alpha_1+2\alpha_2,0}=H_{\alpha_1+2\alpha_2}$}[l] at 3.3 0
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\put{$\scriptstyle{A}$} at 0.2 0.5
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\put{$\scriptstyle{\varepsilon_1}$}[t] at 1 -0.1
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\put{$\bullet$} at 1 -1
\put{$\bullet$} at 0 -2
\plot -3.2 -3.2 3.2 3.2 /
\plot 3.2 -3.2 -3.2 3.2 /
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\plot 3.2 0 -3.2 0 /
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\plot 3 3.2 3 -3.2 /
\plot 3.2 -3 -3.2 -3 /
\plot 3.2 -2 -3.2 -2 /
\plot 3.2 -1 -3.2 -1 /
\plot 3.2 1 -3.2 1 /
\plot 3.2 2 -3.2 2 /
\plot 3.2 3 -3.2 3 /
\plot 3.2 -1.8 1.8 -3.2 /
\plot 3.2 -0.8 0.8 -3.2 /
\plot 3.2 0.2 -0.2 -3.2 /
\plot 3.2 1.2 -1.2 -3.2 /
\plot 3.2 2.2 -2.2 -3.2 /
\plot 2.2 3.2 -3.2 -2.2 /
\plot 1.2 3.2 -3.2 -1.2 /
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\plot -1.2 3.2 3.2 -1.2 /
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\plot -3.2 0.2 0.2 -3.2 /
\plot -3.2 1.2 1.2 -3.2 /
\plot -3.2 2.2 2.2 -3.2 /
\endpicture
\end{equation*}
If $\Cal{A}_0$ is the finite subarrangement defined in Section 2.3, then the real toric arrangement is obtained quotienting it as shown in the
following figure, where arrows indicate identified edges:
\begin{equation} \label{figura1}
\beginpicture
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\setplotarea x from -4.5 to 4.5, y from -3.5 to 4
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\put{$H_{s_1s_2s_1}$}[bl] at -1.9 3.2
\put{$H_{s_0s_1s_0}$}[bl] at 0 3.2
\put{$H_{s_1s_0s_1}$}[bl] at 3.3 1.5
\put{$H_{s_2s_1s_2}$}[bl] at -0.2 -3.5
\put{$H_{s_2}$}[bl] at 3.3 -1.9
\put{$\bullet$} at 2.3 0
\put{$\scriptstyle{s_0}$}[t] at 2.3 -0.2
\put{$\bullet$} at 0 2.3
\put{$\scriptstyle{s_1s_0}$}[r] at -0.1 2.3
\put{$\bullet$} at 2.7 2.3
\put{$\scriptstyle{s_0s_1s_0}$}[r] at 3 2.1
\put{$\bullet$} at 2.2 2.6
\put{$\scriptstyle{s_1s_0s_1s_0}$}[r] at 2.7 2.8
\put{$\bullet$} at 1.4 2.6
\put{$\scriptstyle{s_1s_0s_1}$}[r] at 1.4 2.8
\put{$\bullet$} at 2.7 1.3
\put{$\scriptstyle{s_0s_1}$}[r] at 2.9 1.1
\put{$\bullet$} at -2.4 2.3
\put{$\scriptstyle{s_1s_2s_0}$}[r] at -2.2 2.5
\put{$\bullet$} at -2.4 1
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\put{$\bullet$} at -2.4 -1.4
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\put{$\bullet$} at -2.4 -2
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\put{$\bullet$} at 1 -1
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\put{$\blacktriangleright$} at 0 -3
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\put{$\blacktriangledown$} at 3 0
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\plot 3 3.2 3 -3.2 /
\plot 3.2 -3 -3.2 -3 /
\plot 3.2 3 -3.2 3 /
\endpicture
\end{equation}
Here, for brevity, the vertices $E(w, \emptyset)$ are labelled by the element
$w \in \widetilde W$.
\bigskip
We get, for example, that the cell $E([1],\emptyset)$ is the vertex in the chamber containg
$1 \in \widetilde W$, while the vertices $E([s_0],\emptyset)$ and
$E([s_1s_2s_1],\emptyset)$ correspond to the same chamber in the toric arrangement; indeed
$s_0=t_1s_1s_2s_1$, then $[s_0]=[s_1s_2s_1]$.
Notice that the number of chambers in the real torus is $8$ in one to one correspondence with the finite Weyl group $B_2$ with cardinality $8$.
Then we get exactly:
\bigskip
$8 \qquad 0$-cells of the form $E([w],\emptyset)$ for $w \in B_2$,
$24 \qquad 1$-cells of the form $E([w],\{s_i\})$ for $w \in B_2$ and $i=0,1,2$,
$24 \qquad 2$-cells of the form $E([w],\{s_i,s_j\})$ for $w \in B_2$ and $0 \leq i<j \leq 2$.
\bigskip
These cells locally correspond to four finite Coxeter arrangements, two of type $B_2$ and two of type $A_1 \times A_1$
appearing in the figure above.
In particular the $2$-cells can be written as:
\bigskip
$E([w],\{s_i,s_{i+1}\})$ with a representative $w$ chosen in the Coxeter group $B_2$ generated by $\{s_i,s_{i+1}\})$, i=0,1;
$E([w],\{s_0,s_2\})$ and $E([s_1w],\{s_0,s_2\})$ with a representative $w$ chosen in the group
$\{1,s_0,s_2,s_0s_2\}$ generated by $\{s_0,s_2\}$.
\bigskip
The representatives can be chosen in the more suitable way for computations. The boundary
map (\ref{bordo}) for the $1$-cells is:
\begin{equation*}
\partial_1 E([w],\{s_i\})= E([w],\emptyset) - E([ws_i],\emptyset)
\end{equation*}
and it gives rise to a matrix of $24$ columns and $8$ rows with entries $0$, $1$ and $-1$.\\
On the other hand, the second boundary map is given by
\begin{equation*}
\begin{split}
\partial_2 E([w],\{s_i,s_{i+1}\})= E([w],\{s_i\}) - E([ws_{i+1}],\{s_i\}) + E([ws_{i}s_{i+1}],\{s_i\})- \\
- E([w],\{s_{i+1}\}) + E([ws_{i}],\{s_{i+1}\}) - E([ws_{i+1}s_{i}],\{s_{i+1}\})\\
\end{split}
\end{equation*}
\begin{equation*}
\partial_2 E([w],\{s_0,s_2\})= E([w],\{s_0\}) - E([ws_2],\{s_0\}) - E([w],\{s_{2}\}) + E([ws_{0}],\{s_2\}).
\end{equation*}
In this way we get that the homology, and hence the cohomology, is torsion free and $H_0(R_{B_2}, \Z) = \Z$, $H_1(R_{B_2}, \Z) = \Z^8$ and $H_2(R_{B_2}, \Z) = \Z^{15}$, which agrees with the Betti numbers computed in \cite[Ex. 5.14]{Mo1}.
In general we have the following
\begin{conj}
Let $\widetilde{W}$ be an affine Weyl group and $\mathcal{T}_{\widetilde{W}}$ be the corresponding toric arrangement. Then the integer cohomology of the complement is torsion free (and hence it coincides with the De Rham cohomology computed in \cite{DP}).
\end{conj}
This conjecture will be proved in a future paper \cite{simo2}.
\section{An example from robotics}
In this section we give an example of non-thick arrangement: the one obtained from the
affine Weyl arrangement $\Cal A_{\widetilde A_n}$, by quotienting by the coroot lattice, which we will denoted by
$\Lambda_{\widetilde A_n}$ (see the second part of Remark \ref{remsec}).
\bigskip
Indeed in this case the underlying real toric arrangement has a very nice description in terms of Young tableaux. More precisely the facets of $\Cal T_{\widetilde A_n, \R}$ are in one to one correspondence with
a family of Young tableaux which turn out to be the same tableaux describing the space of all periodic legged gaits of a robot body (see \cite{KGCohen}).
\bigskip
It is clear that, in this case, the finite arrangement $\Cal A_{0,\widetilde A_n}$ is exactly the braid
arrangement $\Cal A_{A_n}$.
\subsection{Tableaux description for the complex $\widetilde{\Cal S}_{A_n}$}
We indicate by $A_n$ the symmetric group on $n+1$ elements,
acting by permutations of the coordinates. Then $\Cal{A} =
\Cal{A}_{A_n}$ is the braid arrangement and $\widetilde{\Cal S}_{A_n}$ is the
associated CW-complex (even if the arrangement is finite we continue to use the same notation
used above for the affine case to distinguish it from the toric one).
Given a system of coordinates in $\R^{n+1}$, we describe
$\widetilde{\Cal S}_{A_n}$ through certain tableaux as follow.
Every $k$-cell $[\widetilde C \prec \widetilde F]$ is represented by a tableau with $n+1$
boxes and $n+1-k$ rows (aligned on the left), filled with all the
integers in $\{1,...,n+1\}.$ There is no monotony condition on the
lengths of the rows. One has:
\medskip
\ni - $(x_1,\ldots, x_{n+1})$ is a point in $\widetilde F$ if and only if:
\bigskip
$1.$ $i$ and $j$ belong to the same row if and only if $x_i=x_j$,
$2.$ $i$ belongs to a row preceding the one containing $j$ if and only if
$x_i < x_j$;
\bigskip
\ni - the chamber $\widetilde C$ belongs to the half-space $x_i < x_j$ if and only if:
\bigskip
$1.$ either the row which contains $i$ is preceding the one
containing $j$ or
$2.$ $i$ and $j$ belong to the same row and the column which
contains $i$ is preceding the one containing $j$.
\medskip
Notice that the facets of the real stratification are represented
by standard Young tableaux, since the order of the entries in each
row does not matter, and hence we can assume it to be strictly increasing.\\
Notice also that the geometrical action of $A_n$ on the stratification
induces a natural action on the complex $\widetilde{\Cal S}_{A_n}$ which, in terms of
tableaux, is given by a left action of $A_n$: $\sigma. \ T$ is the
tableau with the same shape as $T,$ and with entries permuted
by $\sigma.$
\subsection{Tableaux description for the facets of $\Cal T_{\widetilde{A}_n, \R}$}
Let $\Cal A_{0,\widetilde A_n} \subset \Cal A_{\widetilde A_n}$ be the braid arrangement passing through the origin and $\pi: \R^{n+1} \longrightarrow \R^{n+1}/\Lambda_{\widetilde A_n}=
T_{\R}$ the projection map.
\bigskip
If $\mathbf F_{\widetilde A_n}$ is the stratification of $\R^{n+1}$ into facets induced by the arrangement
$\Cal A_{\widetilde A_n}$, we define the set:
$$
\mathbf F_{0, \widetilde A_n}=\{\widetilde F^k \in \mathbf F_{\widetilde A_n} \mid clos(F^k) \supset
\bigcap_{H \in \Cal A_{0,\widetilde A_n} } H\}.
$$
Obviously $\mathbf F_{0,\widetilde A_n}$ is in one to one correspondence with the stratification
$\mathbf F_{A_n}$ induced by the braid arrangement $\Cal A_{A_n}$ and the restriction
$\pi_{\mathbf F_{0,\widetilde A_n}}$ is surjective on $T_{\R}$.
\bigskip
It follows that in order to understand how $\Lambda_{\widetilde A_n}$ acts on
$\mathbf F_{\widetilde A_n}$ it is enough to study how it acts on $\mathbf F_{0,\widetilde A_n}$.
Moreover it is enough to consider facets in the closure of the base chamber $\widetilde C_0$
corresponding to $1 \in \widetilde A_n$; the action on the others will be obtained by symmetry.
\bigskip
Let us remark that a facet $\widetilde F^k$ is in $\mathbf F_{0,\widetilde A_n}$ if and only if it intersects any ball $B_0$
around the origin. Let $B_0$ be a ball of sufficiently small radius and
$$x=(x_1, \ldots, x_{n+1}) \in clos(\widetilde C_0) \cap B_0$$ be a given point in a facet
$\widetilde F^k \in \mathbf F_{0,\widetilde A_n}$.
Then the $x_i$'s satisfy $x_1 \leq x_2 \leq \ldots \leq x_{n+1}$ and the standard Young
tableaux $Tb_{\widetilde F^k}$ associated to $\widetilde F^k$ will have entries increasing along both,
rows and columns.
\bigskip
Let $t_1, \ldots ,t_n \in \Lambda_{\widetilde A_n}$ be a base such that $t_i$ translates the
reflection hyperplane $H_{i,i+1}=Ker(x_i - x_{i+1})$ fixing all hyperplanes
$H_{j,j+1}=Ker(x_j-x_{j+1})$ for $j \neq i$ (i.e. each point in $H_{j,j+1}$ is sent in a point still in $H_{j,j+1}$).\\
Then we can assume that translation $t_i$ acts on the entry $x_i$ as $t_i.x_i= x_i + t$ with $x_i +t > x_{i+1}$ and, as $H_{j,j+1}$, for $j \neq i$, are invariant under the action of $t_i$, it follows that
$t_i.x_{i-1}=x_{i-1}+t$ and, by induction, $t_i.x_j=x_j+t$ for all $j < i$, while
$t_i.x_j=x_j$ for all $j >i$.
\bigskip
Recall that, by construction, given a standard Young tableaux, a point $(x_1,\ldots, x_{n+1})$ is a point in $\widetilde F$ if and only if:
\bigskip
$1.$ $i$ and $j$ belong to the same row if and only if $x_i=x_j$,
$2.$ $i$ belongs to a row preceding the one containing $j$ if and only if $x_i < x_j$;
\bigskip
It follows that if $Tb$ is a tableau such that $i \in r_k$ and
$i+1 \in r_{k+1}$ are in two different rows, then $t_i$ acts on $Tb$ sending it in a tableau
$Tb^{\prime}$ with rows $r_1^{\prime}=r_{k+1}, \ldots, r_{h-k}^{\prime}=r_h,r_{h-k+1}^{\prime}=r_1, \ldots, r_h^{\prime}=r_k$. While if $i, i+1 \in r_k$ are in the same row, then $t_i$ acts sending the corresponding facet in a facet which is not anymore in $\Cal A_{0, \widetilde A_n}$.
\bigskip
Then $\Lambda_{\widetilde A_n}$ acts on the $h$ rows of a tableau $Tb_{\widetilde F}$ as a power of the cyclic permutation $(1, \ldots , h)$.
\bigskip
Equivalently let $Y(n+1,k+1)$ be the set of standard Young tableaux with $k+1$ rows and $n+1$ entries and $Tb \in Y(n+1,k+1)$ be a tableau of rows $(r_1, \ldots , r_{k+1})$.
Then {we have the following proposition.}
{\begin{prop} The set of facets $F^k$ of the toric arrangement $\Cal T_{\widetilde A_n,\R}$ is in one to one correspondence with the set
$$Y(n+1,k+1) / \sim$$
where a tableau $Tb^{\prime} \sim Tb$ if and only if the rows of $Tb^{\prime}$ are
$(r_{\sigma^s(1)}, \ldots , r_{\sigma^s(k+1)})$ for a power $\sigma^s$ of the cyclic permutation
$\sigma=(1, \ldots , k+1)$.
\end{prop}}
In this way we get exactly the tableaux described in \cite{KGCohen}.
\bigskip
Finally let us recall that the relation $\widetilde F^k \prec \widetilde F^{k+1}$
holds if and only if the tableau $Tb_{\widetilde F^{k+1}}$
corresponding to $\widetilde F^{k+1}$ is obtained by attaching two consecutive rows of
$Tb_{\widetilde F^k}$.\\
As a consequence if $F^k$ and $F^{k+1}$ are facets in the toric arrangement
$\Cal T_{\widetilde A_n, \R}$, $ F^k \prec F^{k+1}$ if and only if the tableau $Tb_{F^{k+1}}$
corresponding to $F^{k+1}$ is obtained by attaching two consecutive rows of
$Tb_{F^k}$ or attaching the first one to the last one.
|
{
"timestamp": "2010-10-29T02:00:22",
"yymm": "1009",
"arxiv_id": "1009.3622",
"language": "en",
"url": "https://arxiv.org/abs/1009.3622"
}
|
\section{Introduction}
Frames are nowadays a standard methodology in applied mathematics, computer science, and engineering
when redundant, yet stable expansions are required. Examples include sampling theory \cite{E03}, data quantization
\cite{BP07,BLPY10}, quantum measurements \cite{EF02}, coding \cite{BDV00,SH03}, image processing \cite{CD02,KL10},
wireless communication \cite{HBP01,HP02,S01}, time-frequency analysis \cite{DHRS03,WES05}, speech recognition \cite{BCE06},
and bioimaging \cite{CK08}; see also \cite{KC07a,KC07b} for a beautiful survey and further
references. The typical application exploits the decomposition of a signal $x \in \mathbb{R}^n$ into its frame components, which requires
computation of the frame measurements, i.e., the inner products between the signal $x$ and the frame vectors $(\varphi_i)_{i=1}^N$, say.
However, if the dimension $n$ of the ambient space is large and the frame vectors have `many' non-zero entries, the computational
complexity of the computation of the frame measurements might be high; in fact, for applications with constraints
on the available computing power and bandwidth for data processing, computing the frame measurements and hence the frame decomposition might be intractable.
In this paper, \ah{we} \gk{focus on frames in finite-dimensional Hilbert spaces and} tackle this problem by constructing
frames which have very few non-zero entries, thereby reducing
the number of required additions and multiplications when computing frame measurements significantly. This viewpoint
can be also slightly generalized by assuming that there exists a unitary transformation mapping the frame into
one having this `sparsity property'. Sparsity of fusion frames, which were introduced in \cite{CKL08} as a mathematical
framework for distributed processing thereby going beyond frame theory, was already defined in \cite{CCHKP10} as a concept.
However, the paradigm we aim for in this paper differs from the one introduced in \cite{CCHKP10} for fusion frames when restricting
to the case of frames, since we here aim for an overall sparsity of the frame.
Frame constructions have a long history; browsing through the literature, however, it becomes evident that all
constructions for unit norm tight frames -- those frames most advantageous for applications -- only produce
such frames for very special cases such as harmonic frames, see also \cite{Cas04,CL06}. Very recently, a \fk{significant}
\gk{advance in} the construction of unit norm tight frames was achieved through the introduction of the so-called
{\em Spectral Tetris} algorithm in \cite{CFMWZ09}. For most combinations of the number of frame vectors and the dimension
of the ambient space, this procedure indeed generates a unit norm tight frame. An extension of Spectral Tetris to construct
unit norm frames with prescribed frame operator \ahh{if its eigenvalues are greater or equal to two}
was introduced in \cite{CCHKP10} to allow additional flexibility in the design process.
In this paper we show that \ahh{the} unit norm frames which this extended Spectral Tetris algorithm
generates are {\em optimally sparse} in the sense of the total number of non-zero entries in the frame vectors,
\ahh{provided that Spectral Tetris is performed after ordering the prescribed eigenvalues in an appropriate way}. We
also explicitly determine the exact minimum value of the non-zero entries.
Along the way, we
introduce {\em block decompositions} as a novel structural property of unit norm frames, which we anticipate to be
useful also in other settings.
\subsection{Main Contribution}
Our main contribution is hence two-fold: Firstly, we introduce sparsity of a frame as a novel paradigm in frame theory.
More precisely, we introduce the notion of a sparse frame as well as a sparsity measure for such frames, thereby
allowing for optimality results. Secondly, we analyze an extended version of Spectral Tetris and
prove that this algorithm indeed constructs optimally sparse frames \ahh{if performed after ordering the prescribed eigenvalues
blockwise}. Thus, Spectral Tetris can serve as an
algorithm for computing frames with this desirable property, and our results show that it is not possible to derive sparser frames
through a different procedure.
\subsection{Impact on Applications}
Frames are nowadays considered a fundamental tool in electrical engineering, and we wish to refer to the survey paper \cite{CK08}
as also to the introductory papers \cite{KC07a,KC07b}.
However, the application of frames for the analysis of high-dimensional data such a webpages labeled by over
a million parameters or databases of images, each image being one data point, typically suffers from the fact that frame measurements
are computationally not feasible due to constraints such as computing power and bandwidth, or even limited space to store
the synthesis matrix. With the results presented in this paper, we introduce {\em sparsity of frames} as a novel paradigm for
frame constructions, resulting in computationally highly efficient frames. Our results do not only provide a lower bound for the
maximally achievable sparsity, but with the Spectral Tetris algorithm explicit constructions of efficient frames
for high-dimensional data analysis are now possible. Certainly, the desire to construct {\em tight} frames is evident
due to the favorable reconstruction properties of such frames. But our results go beyond this case, and also enable constructions of
optimally sparse {\em non-tight} frames with prescribed eigenvalues of the frame operator.
Let us provide two \ahh{additional} exemplary
applications illustrating why such frame properties are a natural constraint and which areas our results are
anticipated to impact.
{\em Analysis of Streaming Signals.}
The structure of the frame operator plays a key role in the noise
rejection ability of the frame. When the frame coefficients are corrupted by additive white Gaussian noise, the mean-squared
error (MSE) in reconstructing the signal is minimized by choosing the frame to be tight. In the presence of colored noise,
however, a tight frame is no longer optimal and the frame operator needs to be matched to the noise covariance matrix.
In such cases, the frame needs to be designed with respect to the eigen-basis of the inverse noise covariance matrix and
its eigenvalues. This is similar in spirit to the water-filling principle for precoder design in wireless communication,
where transmit power is distributed across the eigen directions of an inverse channel-noise covariance matrix to equalize
signal-to-noise-ratio across eigen directions, see \cite{SSBGS02,PCL03}. Additional structure on the frame is typically required depending on the
application. When the signal to be decomposed is a time or space series that cannot be observed or processed over long
blocks---due to limited memory, aperture size, or computational power---then it is needed to have a frame that not only
has a prescribed operator but also requires access to the signal samples only over a small temporal or spatial window.
This motivates construction of frames with sparse elements and desired spectra. The constructions we develop in this
paper yield to 2-sparse frames in any dimension, where the frame coefficients for a signal in an $N$-dimensional space
can be computed by observing the data stream through a window of only two samples. The frame sparsity can be tailored
to any arbitrary basis. In particular, when the noise covariance is known, the frame can be made sparse with respect to
the eigen-basis of the inverse noise covariance matrix.
{\em Face Recognition.} In face recognition, one main objective is to classify faces according to some given criterion, for instance, to
distinguish male from female faces. The application of PCA (or similar algorithms) delivers a basis of eigenfaces.
Learning algorithms on some training set of faces can then, for each basis element, determine the degree of the
significance of its coefficients for determining the gender. Customarily, measurements taken to classify faces are
assumed to be affected by noise; hence frame expansions are desirable. The frame should ideally be designed to
match the degree of significance ($=$ eigenvalues) of the given basis in the sense that it should be more redundant
for the computation of the significant coefficients and less for the insignificant coefficients. This is precisely
the setting we consider in this paper, and for which we analyze and construct optimally sparse
frames.
These are just brief samples of applications which will benefit from the results and constructions developed in this
paper. We anticipate that also various other applications
are impacted.
\subsection{Outline}
This paper is organized as follows. In Section \ref{sec:frameconstruction}, we first fix the terminology we require from frame
theory and then review the extended version of the Spectral Tetris algorithm. A novel sparsity measure for a frame will then
be introduced in Section \ref{sec:sparsity} together with a notion of optimality. In Section \ref{sec:optimality result}, a structural
property of frames suitable for our analysis is first introduced, and finally we state and prove our main result Theorem \ref{theo:main}.
We finish with some conclusions and discussions in Section \ref{sec:conclusions}.
\section{Frame Construction}
\label{sec:frameconstruction}
We first review the \gk{initial} as well as the extended version of the Spectral Tetris algorithm from \cite{CCHKP10}. To stand on common ground,
we start by fixing our terminology while briefly reviewing the basic definitions and notations related to frames.
\subsection{Frames}
A sequence
$\Phi = (\varphi_i)_{i=1}^N$ in $\mathbb{R}^n$ is called a {\em frame} for $\mathbb{R}^n$, if it is a -- typically, but not necessarily
linearly dependent -- spanning set. This definition is equivalent to asking for the existence of constants $0 < A \le B < \infty$
such that
\[
A\norm{x}^2 \leq \sum_{i=1}^N |\langle x, \varphi_i \rangle |^2 \leq B\norm{x}^2
\quad \mbox{for all } x \in \mathbb{R}^n.
\]
When $A$ is chosen as the largest possible value and $B$ as the smallest for these inequalities to hold, then we call them the {\em (optimal)
frame bounds}. If $A$ and $B$ can be chosen as $A=B$, then the frame $\Phi$ is called {\em $A$-tight}, and if $A=B=1$ is possible, $\Phi$ is a
{\em Parseval frame}. $\Phi$ is called {\em equal-norm}, if there exists some $c>0$ such that $\|\varphi_i\|=c$ for all $i=1,\ldots,N$, and
it is {\em unit-norm} if $c=1$.
Frames allow the analysis of data by studying the associated {\em frame coefficients} $(\langle x, \varphi_i \rangle)_{i=1}^N$, where
the operator $T$ defined by $T: \mathbb{R}^n \to \ell_2(\{1, 2, \dots, N\})$, $x \mapsto (\langle x,\varphi_i\rangle)_{i=1}^N$
is called the \emph{analysis operator}. The adjoint $T^*$ of the analysis operator is typically
referred to as the {\em synthesis operator} and satisfies $T^*((c_i)_{i=1}^N) = \sum_{i=1}^N c_i\varphi_i$. Later, the
synthesis operator will play an essential role, and we will write it in the matrix form $[\varphi_1| \ldots | \varphi_N]$ with
the frame vectors as columns. In the sequel we refer to this matrix as the {\em synthesis matrix}.
The main operator associated with a frame, which provides a stable reconstruction process, is the {\em frame operator}
\[
S =T^* T : \mathbb{R}^n \to \mathbb{R}^n, \quad x \mapsto \sum_{i=1}^N \langle x,\varphi_i\rangle \varphi_i,
\]
a positive, self-adjoint, invertible operator on $\mathbb{R}^n$. In the case of an $A$-tight frame, we have $S= A \cdot \mbox{\rm Id}_{\mathbb{R}^n}$, and in case of
a Parseval frame, $S=\mbox{\rm Id}_{\mathbb{R}^n}$. In general, $S$ allows for the reconstruction of a signal $x \in \mathbb{R}^n$ through the
reconstruction formula
\begin{equation} \label{eq:expansion}
x = \sum_{i=1}^N \langle x,S^{-1} \varphi_i\rangle \varphi_i.
\end{equation}
Redundancy is obviously the crucial property of a frame ensuring resilience to noise and erasures while simultaneously enabling us
to choose the expansion coefficients appropriately. The particular choice of coefficients displayed in \eqref{eq:expansion} is the
smallest in $\ell_2$ norm \cite{Chr03}, hence it contains the least energy. Recently, a different view point has received rapidly
increasing attention, namely to choose the coefficient sequence to be sparse in the sense of having only few non-zero entries,
thereby allowing data compression while preserving perfect recoverability (see, e.g., \cite{BDE09}, and the references therein).
In this context, for later use, we will denote the support of a vector $x \in \mathbb{R}^n$, i.e., the \gk{set of indices of the} non-zero entries, by ${\text{\rm supp}} \, x$.
Finally, we should mention that, customarily, redundancy of a frame $(\varphi_i)_{i=1}^N $ for $\mathbb{R}^n$ was measured by $\frac{N}{n}$,
i.e., the number of frame vectors divided by the dimension of the ambient space. Since this measure is exceptionally crude and not
sensitive to local behavior of the frame vectors, the notions of {\em upper} and {\em lower redundancy} have been suggested in \cite{BCK10}
as a finer redundancy measure.
\subsection{The Spectral Tetris Algorithm}
Spectral Tetris was first introduced in \cite{CFMWZ09} as an algorithm to generate unit norm tight frames
for any number of frame vectors $N$, say, and for any ambient dimension $n$ provided that $\frac{N}{n} \ge 2$. This
algorithm \gk{is indeed significant for} frame constructions, since it is the first systematic construction
of unit norm tight frames. Before, only a number of very special classes of unit norm tight frames such as harmonic frames
have been known.
An extension to the construction of unit norm frames having a desired frame operator associated with eigenvalues
$\lambda_1, \ldots, \lambda_n \ge 2$ satisfying $\sum_{j=1}^n \lambda_j = N$ was then introduced and analyzed in
\cite{CCHKP10} -- in fact, an even more general algorithm for the construction of fusion frames was stated therein. The
frame-version of this algorithm is what we intend to analyze in this paper. Figure \ref{fig:ST} states the steps of
this version of the algorithm, which we coin {\em Spectral Tetris for Frames}; in short, STF. We wish to remark that
the original form of the algorithm in \cite{CFMWZ09} requires the sequence of eigenvalues to be in decreasing order,
i.e. $\lambda_1 \ge \ldots \ge \lambda_n$. This assumption,
however, was made only for classification reasons, and it is easily seen that it can be dropped. Since in the sequel,
we will consider carefully chosen, presumably non-decreasing, sequences of eigenvalues, the gained freedom is essential
for our analysis.
\begin{figure}[h]
\centering
\framebox{
\begin{minipage}[h]{5.0in}
\vspace*{0.3cm}
{\sc \underline{STF: Spectral Tetris for Frames}}
\vspace*{0.4cm}
{\bf Parameters:}
\begin{itemize}
\item Dimension $n \in \mathbb{N}$.
\item Number of frame elements $N \in \mathbb{N}$.
\item Sequence of eigenvalues $\lambda_1, \ldots, \lambda_n \ge 2$ satisfying $\sum_{j=1}^n \lambda_j = N$.
\end{itemize}
\vspace*{0.2cm}
{\bf Algorithm:}
\begin{itemize}
\item[1)] Set $i := 1$.
\item[2)] For $j=1,\ldots,n$ do
\item[3)] \hspace*{1cm} Repeat
\item[4)] \hspace*{2cm} If $\lambda_j < 1$ then
\item[5)] \hspace*{3cm} $\varphi_i := \sqrt{\frac{\lambda_j}{2}} \cdot e_j + \sqrt{1-\frac{\lambda_j}{2}} \cdot e_{j+1}$.
\item[6)] \hspace*{3cm} $\varphi_{i+1} := \sqrt{\frac{\lambda_j}{2}}\cdot e_j - \sqrt{1-\frac{\lambda_j}{2}} \cdot e_{j+1}$.
\item[7)] \hspace*{3cm} $i := i+2$.
\item[8)] \hspace*{3cm} $\lambda_{j+1} := \lambda_{j+1} - (2-\lambda_j)$.
\item[9)] \hspace*{3cm} $\lambda_j := 0$.
\item[10)] \hspace*{2cm} else
\item[11)] \hspace*{3cm} $\varphi_i := e_j$.
\item[12)] \hspace*{3cm} $i := i+1$.
\item[13)] \hspace*{3cm} $\lambda_j := \lambda_j - 1$.
\item[14)] \hspace*{2cm} end.
\item[15)] \hspace*{1cm} until $\lambda_j = 0$.
\item[16)] end.
\end{itemize}
\vspace*{0.2cm}
{\bf Output:}
\begin{itemize}
\item Frame STF$(N; \lambda_1, \ldots, \lambda_n) :=\{\varphi_i\}_{i=1}^N$.
\end{itemize}
\vspace*{0.01cm}
\end{minipage}
}
\caption{The Spectral Tetris algorithm for constructing an $N$-element unit norm frame STF$(N; \lambda_1, \ldots, \lambda_n)$ for $\mathbb{R}^n$
with an associated frame operator having eigenvalues $\lambda_1, \ldots, \lambda_n$.}
\label{fig:ST}
\end{figure}
Before we continue, let us give an example to provide a more intuitive feeling of how STF works and to introduce a cursor notation which
will be utilized in later proofs.
\begin{example} \label{exa:cursor}
We aim to construct a $\ahh{10}$ element unit norm frame in $\mathbb{R}^4$ having the eigenvalues $\lambda_1=\lambda_2=\lambda_3=\frac{8}{3}$ and
$\lambda_4=2$. STF will provide such a frame by generating a $\ahh{4 \times 10}$ synthesis matrix with the following properties: The columns of
the matrix have norm $1$ (guaranteeing that the frame has unit norm vectors) and the rows of the matrix are orthogonal and square sum to
the desired eigenvalues (guaranteeing that the frame operator is diagonal with the desired eigenvalues on its diagonal). In the example
we consider each of the first $3$ rows of the to-be-generated synthesis matrix have to square sum to $\frac{8}{3}$ and the last row to
square sum to $2$. The algorithm now starts with a $\ahh{4 \times 10}$ matrix of unknown entries and lets a {\em cursor} move forward along columns
and rows assigning values to certain entries. The remaining entries are set to zero in the end. When the cursor is in position $(i,j)$, we
update the variable $\lambda_i$ to be the difference between the eigenvalue assigned to row $i$ and the square sum of the entries already
assigned to row $i$, in order to keep track of how much weight still has to be assigned to row $i$ to make it square sum to the desired
eigenvalue. In general, one of the three cases occurs in each step:
{\em Case $1$}: If $\lambda_i>1$, then the current entry $(i,j)$ is set to one, we update $\lambda_i := \lambda_i-1$, and the cursor $(i,j)$
is moved to the right, i.e., $(i,j) := (i+1,j)$. This is, for example, the case when the cursor is in position $(1,1)$.
\ahh{At this point we have $\lambda_1=\frac{8}{3}$ and we update to $\lambda_1=\frac{8}{3}-1=\frac{5}{3}$.
The matrix changes as follows, where we denote the unknown matrix entries by $\cdot$ and the position of the cursor by $\odot$:}
\[
\begin{bmatrix}
\ahh{\odot}&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\end{bmatrix}
\quad \longrightarrow \quad
\begin{bmatrix}
1&\ahh{\odot}&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\end{bmatrix}
\]
{\em Case $2$}: If $0< \lambda_i<1$, then the entries $(i,j)$, $(i+1,j)$, $(i,j+1)$, and $(i+1,j+1)$ are set according to lines $5)$ and $6)$ of
STF, we update $\lambda_{i+1} := \lambda_{i+1} + \lambda_i - 2$, and the cursor is moved to $(i,j) := (i+2,j+1)$.
This is, for example, the case when the cursor is in position $(1,3)$.
\ahh{At this point we have $\lambda_1 = \frac{2}{3}$ and update $\lambda_2 = \frac{8}{3}$ to $\lambda_2 = \frac{8}{3}- (2-\frac{2}{3})=\frac{4}{3}$. The matrix changes as follows:}
\[
\begin{bmatrix}
1&1&\ahh{\odot}&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\end{bmatrix}
\quad \longrightarrow \quad
\begin{bmatrix}
1&1&\sqrt{\frac{1}{3}}&\sqrt{\frac{1}{3}}&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&\cdot&\sqrt{\frac{2}{3}}&-\sqrt{\frac{2}{3}}&\ahh{\odot}&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\end{bmatrix}
\]
This case is the crucial step of the algorithm. Note that the $2\times2$ block we inserted has the properties that its rows are orthogonal and
its columns square sum to $1$, which are properties desired for the synthesis operator.
{\em Case $3$}: If $\lambda_i = 1$, then the entry $(i,j)$ is set to one, and the cursor is moved to the right below $(i,j) := (i+1,j+1)$.
This is, for example, the case when the cursor is $(3,8)$. \ahh{At this point we have $\lambda_3=1$ and the matrix changes as follows:}
\[
\begin{bmatrix}
1&1&\sqrt{\frac{1}{3}}&\sqrt{\frac{1}{3}}&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&\cdot&\sqrt{\frac{2}{3}}&-\sqrt{\frac{2}{3}}&1&\sqrt{\frac{1}{6}}&\sqrt{\frac{1}{6}}&\cdot&\cdot&\cdot\\
\cdot&\cdot&\cdot&\cdot&\cdot&\sqrt{\frac{5}{6}}&-\sqrt{\frac{5}{6}}&\ahh{\odot}&\cdot&\cdot\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\end{bmatrix}
\quad \longrightarrow \quad
\begin{bmatrix}
1&1&\sqrt{\frac{1}{3}}&\sqrt{\frac{1}{3}}&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&\cdot&\sqrt{\frac{2}{3}}&-\sqrt{\frac{2}{3}}&1&\sqrt{\frac{1}{6}}&\sqrt{\frac{1}{6}}&\cdot&\cdot&\cdot\\
\cdot&\cdot&\cdot&\cdot&\cdot&\sqrt{\frac{5}{6}}&-\sqrt{\frac{5}{6}}&1&\cdot&\cdot\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\ahh{\odot}&\cdot
\end{bmatrix}.
\]
After performing all steps of the algorithm, the final synthesis matrix constructed by STF has the form
\[
\begin{bmatrix}
1&1&\sqrt{\frac{1}{3}}&\sqrt{\frac{1}{3}}&0&0&0&0&0&0\\
0&0&\sqrt{\frac{2}{3}}&-\sqrt{\frac{2}{3}}&1&\sqrt{\frac{1}{6}}&\sqrt{\frac{1}{6}}&0&0&0\\
0&0&0&0&0&\sqrt{\frac{5}{6}}&-\sqrt{\frac{5}{6}}&1&0&0\\
0&0&0&0&0&0&0&0&1&1
\end{bmatrix} .
\]
\end{example}
\section{New Paradigm for Frame Constructions: Sparsity}
\label{sec:sparsity}
\subsection{Classical Sparsity}
Over the past few years, sparsity has become a key concept in various areas of applied mathematics, computer science, and
electrical engineering. Sparse signal processing methodologies explore the fundamental fact that many types of signals can
be represented by only a few non-zero coefficients when choosing a suitable basis or, more generally, a frame. A signal
representable by only $k$, say, basis or frame elements is called {\em $k$-sparse}. If signals possess such a
sparse representation, they can in general be recovered from few measurements using $\ell_1$ minimization techniques (see, e.g.,
\cite{BDE09,CRT06,Don06} and the references therein).
\subsection{Sparse Frames}
In this paper, however, we pose a different question concerning sparsity, viewing sparsity from a very different standpoint.
Typically, data processing applications face low on-board computing power and/or a small bandwidth budget. When the signal
dimension is large, the decomposition of the signal into its frame measurements requires a large number of additions and
multiplications, which may be infeasible for on-board data processing. Also the space required for storing the synthesis matrix of
the frame might be huge.
It would hence be a significant improvement, if each frame vector would
contain very few non-zero entries, hence -- phrasing it differently -- be sparse in the standard unit vector basis, which ensures low-complexity
processing. Since we are interested in the performance of the whole frame, the total number of non-zero entries in the frame
vectors seems to be a suitable sparsity measure. This viewpoint can also be slightly generalized by assuming that there exists
a unitary transformation mapping the frame into one having this `sparsity' property.
\subsection{Sparseness Measure}
Taking these considerations into account, we are led to proclaim the following definition for a sparse frame:
\begin{definition} \label{def:k_sparse}
Let $(e_j)_{j=1}^n$ be an orthonormal basis for $\mathbb{R}^n$. Then a frame $(\varphi_i)_{i=1}^N$ for $\mathbb{R}^n$ is called
{\em $k$-sparse} with respect to $(e_j)_{j=1}^n$, if, for each $i \in \{1,\ldots,N\}$, there exists $J_i \subseteq \{1,\ldots,n\}$
such that
\[
\varphi_i \in \mbox{\rm span}\{e_j : j \in J_i\}
\]
and
\begin{equation} \label{eq:k_sparse_1}
\sum_{i=1}^n |J_i| = k.
\end{equation}
\end{definition}
The attentive reader will have realized that this definition differs from the definition stated in \cite{CCHKP10} for fusion
frames (see \cite{CKL08}) when restricting to the special case of frames. The exact relation is the following:
Let $(e_j)_{j=1}^n$ be an orthonormal basis for $\mathbb{R}^n$, let $(\varphi_i)_{i=1}^N$ be a frame for $\mathbb{R}^n$, and, for each
$i \in \{1,\ldots,N\}$, let $J_i \subseteq \{1,\ldots,n\}$ such that $\varphi_i \in \mbox{\rm span}\{e_j : j \in J_i\}$. Then,
in the sense of \cite{CCHKP10}, the frame is $\max\{|J_i| : i = 1, \ldots, n\}$-sparse, whereas in our Definition
\ref{def:k_sparse}, the frame is $\sum_{i=1}^n |J_i|$ sparse. Thus our definition encodes the true overall sparsity which is
the sparsity required for frame processing in contrast to the more local version of \cite{CCHKP10}.
One can certainly imagine other sparsity measures dependent on the requirements and constraints of the application at hand.
Instead of \eqref{eq:k_sparse_1}, a weighted version could be considered with the weights chosen depending on the computational
constraints of the application. Also, \eqref{eq:k_sparse_1} could be regarded as the $\ell_1$ norm of the sequence $\{|J_i| :
i = 1, \ldots, n\}$, and a different viewpoint might lead us to considering a different norm instead -- as it was done in
\cite{CCHKP10} for the $\ell_\infty$ norm.
\subsection{Notion of Optimality}
We now have the necessary machinery at hand to introduce a notion of an {\em optimally} sparse frame. Optimality will typically
-- as also in this paper - be considered within a particular class of frames, for instance, in the class of unit norm tight frames.
\begin{definition}
Let ${\mathcal{F}}$ be a class of frames for $\mathbb{R}^n$, let $(\varphi_i)_{i=1}^N \in {\mathcal{F}}$, and let $(e_j)_{j=1}^n$ be an orthonormal basis for $\mathbb{R}^n$.
Then $(\varphi_i)_{i=1}^N$ is called {\em optimally sparse in ${\mathcal{F}}$ with respect to $(e_j)_{j=1}^n$}, if $(\varphi_i)_{i=1}^N$ is
$k_1$-sparse with respect to $(e_j)_{j=1}^n$ and there does not exist a frame $(\psi_i)_{i=1}^N \in {\mathcal{F}}$ which is $k_2$-sparse
with respect to $(e_j)_{j=1}^n$ with $k_2 < k_1$.
\end{definition}
We wish to emphasize the strong dependence of sparsity on the chosen basis. Also an optimally sparse frame is in general not uniquely determined;
we present an example for this observation in Subsection \ref{subsec:maxsparsity}.
\section{An Optimality Result for Sparse Frames}
\label{sec:optimality result}
We now seek a construction for an optimally sparse unit norm frame with prescribed properties. As already elaborated upon before, the
condition we impose is having a given frame operator, which, in particular, also includes operators with equal eigenvalues corresponding
to tight frames. This frame operator will in the following be always determined by its eigenvalues. Hence we are interested in optimal
sparsity within the following class:
Let $n, N > 0$ and let the real values $\lambda_1,\ldots,\lambda_n\geq 2$ satisfy $\sum_{j=1}^n \lambda_j = N$. Then
the class of unit norm frames $(\varphi_i)_{i=1}^N$ in $\mathbb{R}^n$ whose frame operator has eigenvalues $\lambda_1, \ldots, \lambda_n$ will
be denoted by
\[
{\mathcal{F}}(N,\{\lambda_i\}_{i=1}^{n}).
\]
It is important to mention that by writing $\{\lambda_i\}_{i=1}^{n}$, we wish to indicate that the ordering does not play a role
here, however, multiplicities are counted. The just defined class ${\mathcal{F}}(N,\{\lambda_i\}_{i=1}^{n})$
\ahh{is non-empty by application of the STF. In fact, it can be shown using methods introduced in \cite{CK03} and \cite{Cas04},
that it is an infinite set for any $n, N > 0$ and real values $\lambda_1,\ldots,\lambda_n\geq 2$.}
It might be beneficial for the reader to mention at this point that we will discuss the analysis presented in Subsections
\ref{subsec:structure} to \ref{subsec:main} in the important special case of tight frames in Subsection \ref{subsec:tight} for illustrative
purposes.
\subsection{Novel Structural Property of Synthesis Matrices}
\label{subsec:structure}
Aiming for determining the \fk{maximally} achievable sparsity for a class ${\mathcal{F}}(N,\{\lambda_i\}_{i=1}^{n})$, we first need
to introduce a particular measure associated with the set of eigenvalues $\{\lambda_i\}_{i=1}^{n}$. This measure indicates the
maximal number of partial sums which are an integer; here one maximizes over all reorderings of the eigenvalues. Before
stating the precise definition, let us provide some intuition why the maximally achievable sparsity is dependent on partial
integer valued sums. Whenever the partial sum of the first $l$, say, eigenvalues
$\lambda_1,\ldots,\lambda_l$ is an integer, the cursor -- recall that the concept of a cursor was introduced in Example \ref{exa:cursor}
-- in row $l$ will be in Case $3$ of the cases discussed in Example \ref{exa:cursor}. Opposed to the setting of 4 new non-zeros entries
in Case 2, in this case only {\em one} new non-zero entry will be defined. Roughly speaking, this will allow us to reduce the analysis
to the blocks between two such integer partial sums.
The precise definition of the measure on a set of eigenvalues we require is now as follows.
\begin{definition}
A finite sequence of real values $\lambda_1,\ldots,\lambda_n$ is {\em ordered blockwise}, if for any permutation $\pi$ of $\{1,\ldots,n\}$
the set of partial sums $\{\sum_{j=1}^s\lambda_j\colon s=1,\ldots,n\}$ contains at least as many integers as the set
$\{\sum_{j=1}^s\lambda_{\pi(j)}\colon s=1,\ldots,n\}$.
The {\em maximal block number} of a finite sequence of real values $\lambda_1, \ldots, \lambda_n$, denoted by
$\mu(\lambda_1, \ldots, \lambda_n)$, is the number of integers in $\{\sum_{j=1}^s\lambda_{\sigma(j)}\colon s=1,\ldots,n\}$, where
$\sigma$ is a permutation of $\{1,\ldots,n\}$ such that $\lambda_{\sigma(1)},\ldots,\lambda_{\sigma(n)}$ is ordered blockwise.
\end{definition}
Surprisingly, the notion of maximal block number can illuminatingly be transferred to a particular decomposition
property of the synthesis matrix of a frame. Let us first define the decomposition property we are interested in:
\begin{definition}
Let $n, N > 0$, and let $(\varphi_i)_{i=1}^N$ be a frame for $\mathbb{R}^n$. Then we say that the synthesis matrix of $(\varphi_i)_{i=1}^N$
has {\em block decomposition of order $m$}, if there exists a partition $\{1, \ldots, N\} = I_1 \cup \ldots \cup I_m$ such that,
for any $k_1 \in I_{i_1}$ and $k_2 \in I_{i_2}$ with $i_1 \neq i_2$, we have ${\text{\rm supp}} \, \varphi_{k_1} \cap {\text{\rm supp}} \, \varphi_{k_2} = \emptyset$
and $m$ is maximal.
\end{definition}
The following result now connects the maximal block number of the sequence of eigenvalues of a frame operator
with the block decomposition order of an associated frame.
\begin{proposition}\label{prop:block}
Let $n, N > 0$ and let the real values $\lambda_1,\ldots,\lambda_n \ge 2$ satisfy $\sum_{j=1}^n \lambda_j = N$.
Then the synthesis matrix of any frame in the class ${\mathcal{F}}(N,\{\lambda_i\}_{i=1}^{n})$ has block decomposition
of order at most $\mu(\lambda_1, \ldots, \lambda_n)$.
\end{proposition}
\begin{proof}
Suppose $(\varphi_i)_{i=1}^N\in{\mathcal{F}}(N,\{\lambda_i\}_{i=1}^{n})$ has block decomposition of order
$\nu$ and let
$\{1, \ldots, N\} = I_1 \cup \ldots \cup I_\nu$ be a corresponding partition.
For $j=1,\ldots,\nu$, let $S_j$ be the common support set of the vectors $(\varphi_i)_{i\in I_j}$, i.e., $k\in S_j$ if and only if
$k\in{\text{\rm supp}} \,\varphi_i$ for some $i\in I_j$. Now let $r_k$ denote the $k$-th row of the synthesis matrix of $(\varphi_i)_{i=1}^N$.
Then $S_1,\ldots,S_{\nu}$ is a partition of $\{1,\ldots,n\}$
and, for every $j=1,\ldots,\nu$ \ah{we have by the fact that $(\varphi_i)_{i=1}^N$ consists of unit norm vectors and by our choice of
$I_j$ and $S_j$ that}
\begin{equation}{\label{reviewer3}}
|I_j|= \sum_{k\in I_j} \|\varphi_k\|^2 =\sum_{k\in S_j} \|r_k\|^2 = \sum_{k\in S_j} \lambda_k.
\end{equation}
\ah{The last equality holds since we, after permutation of the columns, can write the synthesis matrix of $(\varphi_i)_{i=1}^N$ as
$T^* = [T^*_1,\ldots,T^*_{\nu}]$, where $T^*_j$ has zero entries except on the rows indexed by $I_j$ and the columns indexed by
$S_j$, for $j=1,\ldots,\nu$. The frame operator $T^*T = \sum_{j=1}^{\nu} T^*_j T_j$ is block diagonal with blocks $T^*_j T_j$, hence
its eigenvalues are exactly the union of those of each matrix $T^*_j T_j$. But $\sum_{k\in I_j} \|\varphi_k\|^2 =\sum_{k\in S_j} \|r_k\|^2 $
equals the} \fk{square of the} \ah{Hilbert-Schmidt norm of $T^*_j$ and therefore the sum of the eigenvalues of $T^*_jT_j$. This shows the
last equality of (\ref{reviewer3}). Since (\ref{reviewer3}) holds for all $j=1,\ldots,\nu$, we conclude that} the maximal block number of
$\lambda_1, \ldots, \lambda_n$ is at least $\nu$.
Thus the synthesis matrix of the arbitrarily chosen frame $(\varphi_i)_{i=1}^N$ in the class ${\mathcal{F}}(N,\{\lambda_i\}_{i=1}^{n})$ has
block decomposition of order at most $\mu(\lambda_1, \ldots, \lambda_n)$.
\end{proof}
\subsection{\fk{Maximally} Achievable Sparsity}
\label{subsec:maxsparsity}
Having introduced the required new notions, we are now in a position to state the exact value for the maximally achievable
sparsity for a class ${\mathcal{F}}(N,\{\lambda_i\}_{i=1}^{n})$. It is not initially clear that this optimal sparsity can always be attained.
With Theorem \ref{theo:main} we will prove that this is indeed the case; in fact, Theorem \ref{theo:main} also provides an explicit
construction of those frames.
\begin{theorem} \label{theo:maxsparsity}
Let $n, N > 0$, and let the real values $\lambda_1,\ldots,\lambda_n \ge 2$ satisfy $\sum_{j=1}^n \lambda_j = N$. Then any frame
in ${\mathcal{F}}(N,\{\lambda_i\}_{i=1}^{n})$ has sparsity at least
\[
N+2(n-\mu(\lambda_1, \ldots, \lambda_n))
\]
with respect to any orthonormal basis.
\end{theorem}
\begin{proof}
We first study the case that $\mu(\lambda_1, \ldots, \lambda_n)=1$. For this, let $T^*$ denote the synthesis matrix of a frame in
${\mathcal{F}}(N,\{\lambda_i\}_{i=1}^{n})$ with respect to a fixed orthonormal basis. For the sake of brevity, in the sequel we will use
the phrase that two rows of $T^*$ {\em have overlap of size} $k$, if the intersection of their supports is a set of size $k$. Note that,
since the rows of $T^*$ are orthogonal, it is not possible that two rows of $T^*$ have overlap $1$.
Fix now an arbitrary row $r_1$ of $T^*$. Since, by Proposition \ref{prop:block}, $T^*$ has block decomposition of order $1$, there
exists a row $r_2$ whose overlap with $r_1$ is of size $\geq 2$. Similarly, there has to exist a row different from $r_1$ and $r_2$
which has overlap of size $\geq 2$ with $r_1$ or $ r_2$. Iterating this procedure will provide an order $r_1, r_2,\dots r_n$ such
that, for each row $r_j$, there exists some $k<j$ such that $r_j$ has overlap of size $\geq 2$ with $r_k$. Since all columns in $T^*$
are unit norm, for each column $c$, there exists a minimal $j$ for which the entry $c_{r_j}$ is non-zero. This yields $N$
non-zero entries in $T^*$. In addition, each row $r_2$ through $r_n$ has at least 2 non-zero entries coming from the overlap,
which are different from the just accounted for $N$ entries, \ah{since these entries \fk{cannot} be the
non-zero entries of minimal index of a column due to the overlap with a previous row.}
This sums up to a total of at least $2(n-1)$ non-zero coefficients.
Consequently, the synthesis matrix has at least $N+2(n-1)$ non-zero entries, as desired.
Finally, suppose $\mu:=\mu(\lambda_1, \ldots, \lambda_n)>1$. By Proposition~\ref{prop:block}, $T^*$ has block decomposition of order
at most $\mu$. Performing the same construction as above, there exist at most $\mu$ rows $r_j$ (including the first one) which do not
have overlap with a row $r_k$, $k<j$. Thus the synthesis matrix $T^*$ must at least contain $N+2(n-\mu)$ non-zero entries.
\end{proof}
It should be mentioned that an optimally sparse frame from ${\mathcal{F}}(N,\{\lambda_i\}_{i=1}^{n})$ is in general not uniquely determined.
Various examples can be constructed along the following line: For simplicity, we choose $n=4$ and $N=9$ and construct a tight frame,
i.e., $\lambda_1=\ldots=\lambda_4=\frac{9}{4}$.
Then, by Theorem \ref{theo:maxsparsity}, the maximally achievable sparsity is $9+2(4-1)=15$. The following matrices are synthesis matrices
with respect to the standard unit vector basis of two different frames in ${\mathcal{F}}(9, \{\frac{9}{4}\}_{i=1}^9)$, the first in fact being generated by
Spectral Tetris:
\[
\begin{bmatrix}
1&1&\sqrt{\frac{1}{8}}&\sqrt{\frac{1}{8}}&0&0&0&0&0\\
0&0&\sqrt{\frac{7}{8}}&-\sqrt{\frac{7}{8}}&\sqrt{\frac{1}{4}}&\sqrt{\frac{1}{4}}&0&0&0\\
0&0&0&0&\sqrt{\frac{3}{4}}&-\sqrt{\frac{3}{4}}&\sqrt{\frac{3}{8}}&\sqrt{\frac{3}{8}}&0\\
0&0&0&0&0&0&\sqrt{\frac{5}{8}}&-\sqrt{\frac{5}{8}}&1\\
\end{bmatrix}
\]
and
\[
\begin{bmatrix}
1&\sqrt{\frac{5}{8}}&\sqrt{\frac{5}{8}}&0&0&0&0&0&0\\
0&\sqrt{\frac{3}{8}}&-\sqrt{\frac{3}{8}}&\sqrt{\frac{3}{8}}&\sqrt{\frac{3}{8}}&\sqrt{\frac{3}{8}}&\sqrt{\frac{3}{8}}&0&0\\
0&0&0&\sqrt{\frac{5}{8}}&-\sqrt{\frac{5}{8}}&0&0&1&0\\
0&0&0&0&0&\sqrt{\frac{5}{8}}&-\sqrt{\frac{5}{8}}&0&1\\
\end{bmatrix}.
\]
\subsection{Main Result}
\label{subsec:main}
Having set the benchmark, we now prove that frames constructed by Spectral Tetris in fact achieve the optimal
sparsity rate. For this, we would like to remind the reader that the frame constructed by Spectral Tetris
(see Figure \ref{fig:ST}) was denoted by STF$(N; \lambda_1, \ldots, \lambda_n)$.
\begin{theorem} \label{theo:main}
Let $n, N > 0$, and let the real values $\lambda_1,\ldots,\lambda_n \ge 2$ be ordered blockwise and satisfy $\sum_{j=1}^n \lambda_j = N$.
Then the frame STF$(N; \lambda_1, \ldots, \lambda_n)$ is optimally sparse in ${\mathcal{F}}(N,\{\lambda_i\}_{i=1}^{n})$ with respect to the standard unit vector basis.
That is, this frame is $N+2(n-\mu(\lambda_1, \ldots, \lambda_n))$-sparse with respect to the standard unit vector basis.
\end{theorem}
\begin{proof}
Let $(\varphi_i)_{i=1}^N$ be the frame STF$(N; \lambda_1,\ldots,\lambda_n)$. We will first show that its synthesis matrix
has block decomposition of order $\mu:=\mu(\lambda_1, \ldots, \lambda_n)$. For this, let $k_0=0$, and let $k_1,\ldots,k_{\mu}\in\mathbb{N}$
be chosen such that $\ah{m_i:=\sum_{j=1}^{k_i}\lambda_j}$ is an integer for every $i=1,\ldots,\mu$. Moreover, let $m_0=0$.
Further, note that $k_{\mu}=n$ and $m_{\mu}=N$, since $\sum_{j=1}^n \lambda_j$ is an integer by hypothesis. The steps of Spectral Tetris (STF)
for computing STF$(m_1; \lambda_1, \ldots, \lambda_{k_1})$ and STF$(N; \lambda_1, \ldots, \lambda_n)$ \ah{coincide until the cursor }\fk{index}\ah{ in
computing STF$(N; \lambda_1, \ldots, \lambda_n)$ reach}\fk{es}\ah{ $(k_1,m_1)$}. Therefore, the first $k_1$ entries of the first $m_1$
vectors of both constructions coincide. Continuing the computation of STF$(N; \lambda_1, \ldots, \lambda_n)$ will set the remaining
entries of the first $m_1$ vectors and also the first $k_1$ entries of the remaining vectors to zero. Thus, any of the first
$k_1$ vectors has disjoint support from any of the vectors constructed later on. Repeating this argument for $k_2$ until $k_{\mu}$,
we obtain that the synthesis matrix has a block decomposition of order $\mu$; the corresponding partition of the frame vectors being
\[
\bigcup_{i=1}^{\mu}\{\varphi_{m_{i-1}+1},\ldots,\ah{\varphi_{m_{i}}}\}.
\]
To compute the number of non-zero entries in the synthesis matrix generated by Spectral Tetris, we let $i\in\{1,\ldots,\mu\}$ be
arbitrarily fixed and compute the number of non-zero entries of the vectors $\varphi_{m_{i-1}+1},\ldots,\ah{\varphi_{m_{i}}}$.
Spectral Tetris ensures that each of the rows $k_{i-1}+1$ up to $k_{i}-1$ intersects the support of the subsequent row on a set
of size $2$, \ah{since in these rows }\fk{STF will always proceed as }\ah{in case $2$ of the three cases} \fk{in}\ah{ the spectral tetris example above.}
Thus, there exist $2(k_{i}-k_{i-1}-1)$ frame vectors with two non-zero entries. The remaining
$\ah{(m_i-m_{i-1})}-2(k_i-k_{i-1}-1)$
frame vectors will have only one entry, yielding a total number of $\ah{(m_i-m_{i-1})}+2(k_i-k_{i-1}-1)$ non-zero entries in the vectors
$\varphi_{m_{i-1}+1},\ldots,\ah{\varphi_{m_{i}}}$.
Summarizing, the total number of non-zero entries in the frame vectors of $(\varphi_i)_{i=1}^N$ is
\[
\sum_{i=1}^{\mu}\ah{(m_i-m_{i-1})}+2(k_i-k_{i-1}-1)
=\left(\sum_{i=1}^{\mu}\ah{(m_i-m_{i-1})}\right)+
2\left(k_{\mu}-\left(\sum_{i=1}^{\mu}1\right)\right)
=N+2(n-\mu),
\]
which is by Theorem \ref{theo:maxsparsity} the maximally achievable sparsity.
\end{proof}
The reader will have realized that Spectral Tetris generates frames which are `only' optimally sparse with
respect to the standard unit vector basis. This seems at first sight like a drawback. However, if sparsity with respect
to a different orthonormal basis is required, Spectral Tetris can easily be modified to accommodate this request by using
vectors of this orthonormal basis instead of the standard unit vector basis when filling in the frame vectors in Steps 5,
6 and 11 in STF (cf. Figure \ref{fig:ST}). It is a straightforward exercise to show that this modified Spectral Tetris algorithm
then generates a frame which is optimally sparse with respect to this new orthonormal basis.
\subsection{Special Case: Constructing Optimally Sparse Tight Frames}
\label{subsec:tight}
In the special case of equal eigenvalues, i.e., of tight frames, with $N$ elements in $\mathbb{R}^n$, all eigenvalues need to
equal $\frac{N}{n}$ for the equality $\sum_{j=1}^n \lambda_j = N$ to be satisfied. The maximal block number
can be easily computed to be $\gcd(N,n)$. Theorem \ref{theo:main} then takes the following form:
\begin{corollary}
For $n, N > 0$, the frame STF$(N; \frac{N}{n}, \ldots, \frac{N}{n})$ is optimally sparse in
${\mathcal{F}}(N,\{\frac{N}{n}\}_{i=1}^N)$ with respect to
the standard unit vector basis.
That is, this frame is $N+2(n-\gcd(N,n))$-sparse with respect to the standard unit vector basis.
\end{corollary}
\section{Conclusions and Discussion}
\label{sec:conclusions}
In this paper we considered the design of frames which enable efficient computations of the associated frame measurements.
This led to the introduction of the notion of a sparse frame as well as a sparsity measure for such frames, thereby
introducing optimal sparsity as a new paradigm into the construction of frames. We then analyzed an extended version of
Spectral Tetris for frames and proved that the frames constructed by this algorithm are indeed optimally sparse.
This shows that Spectral Tetris can serve as an algorithm for computing frames with this desirable property, and
our results prove that it is not possible to derive sparser frames through a different procedure.
We would finally like to point out that the analysis in this paper leads to several intriguing open problems for
future research; a few examples are stated in the sequel.
\begin{itemize}
\item {\it Eigenvalues also smaller than $2$.} It is still an open problem whether and how Spectral Tetris extends
to sets of eigenvalues, if some eigenvalues are smaller than $2$. \gk{The extension of the} Spectral Tetris
\gk{algorithm} by inserting larger DFT matrices than the previously exploited $2 \times 2$-matrices, \gk{allowed }\fk{ for }\gk{
some partial results (see \cite{CFH10}). }
However, from the results in \cite{CFH10} it can be deduced that this procedure does not always lead to optimally
sparse frames even in the case when all eigenvalues are equal, i.e., the tight frame case. Hence, extensive research
will be necessary to introduce an appropriate -- in the sense of optimal sparsity -- extension of Spectral Tetris.
\item {\it Extension to other classes of frames.} Depending on the application, other desiderata might be requested
from a frame such as, for instance, equi-angularity. For such a class of frames, the question of an optimally
sparse frame can and should similarly be posed.
\item {\it Relative sparsity/Compressibility.} Taking numerical considerations and perturbations into account, it
will be necessary to extend the notion of sparsity to relative sparsity/compressibility for frames and analyze
optimality for such.
\end{itemize}
\section*{Acknowledgement}
The authors would like to thank Ali Pezeshki for enlightening discussions on this topic. We are also grateful to
the anonymous referees for valuable comments and suggestions.
The first and second author were supported by the grant AFOSR F1ATA00183G003, NSF 1008183, and DTRA/ NSF 1042701.
Part of this work was completed while the second author visited the Institute of Mathematics at University
of Osnabr\"uck. This author would like to thank this institute for its hospitality and support during his visit.
The second and third author gratefully acknowledge the support of the Institute of Advanced Study through the Park
City Math Institute, where part of this work was completed. The third author also acknowledges the support of the
Hausdorff Center for Mathematics.
The fourth author acknowledges support by DFG Grant SPP-1324, KU 1446/13 and DFG Grant, KU 1446/14. She would like
to thank the Department of Mathematics at University of Missouri for its hospitality and support during her visit,
which enabled completion of this work.
\bibliographystyle{plain}
|
{
"timestamp": "2011-06-30T02:00:24",
"yymm": "1009",
"arxiv_id": "1009.3663",
"language": "en",
"url": "https://arxiv.org/abs/1009.3663"
}
|
\section{Introduction}
Asymptotic giant branch (AGB) stars suffer mass loss episodes during the late
stages of their evolution. The combination of high densities with relatively low
temperatures makes the atmospheres of these stars favourable sites for grain
formation, and in fact they are considered one of the most efficient sources of dust
in the Galaxy \citep[see, e.g.,][]{Whittet_03}.
Planetary nebulae (PNe) are created when AGB stars reach temperatures high enough to
ionize the material previously ejected.
Several authors have studied the dust present in PNe, but it is not clear how much
dust they have and whether it is destroyed or modified during their lifetimes
\citep{Pottasch_84, Lenzuni_89, Stasinska_99}.
We study the dust in PNe through the analysis of the iron depletion factor,
the ratio between the expected abundance of iron and the one measured in the
gas phase.
The abundances of refractory elements like iron have been calculated from ultraviolet
absorption lines in several interstellar clouds toward different sight lines, and the
low values obtained, compared to
the solar ones, are generally interpreted as due to depletion into dust grains \citep[see, e.g.,][]{Morton_74}.
Iron is mostly condensed into grains and has a
relatively high cosmic abundance. Those two facts together make iron an important
contributor to the mass of refractory grains \citep{Sofia_94}, and hence the iron
depletion factor is likely to reflect the abundance of refractory elements in dust
grains. Besides, in ionized nebulae iron is the refractory element with the strongest
lines in the optical range of the spectrum.
In a previous work \citep{Delgado-Inglada_09}, we performed a
homogeneous analysis of the iron abundance in a sample of 28 Galactic
disk PNe and 8 Galactic H II regions. We obtained very low iron
abundances in all the objects, implying that more than 90\% of their
total iron abundance is condensed into dust grains. This suggests that
iron depletes very efficiently in AGB stars and molecular clouds,
whereas refractory dust is barely destroyed in the ionized gas of PNe
and H II regions. Here, we extend our analysis to include 20 Galactic
bulge PNe (observed by \citealt{Wang_07}), and study the dependence of
the results on PN morphology.
\section{The analysis}
In H II regions and low ionization PNe, Fe$^{++}$ and Fe$^{+3}$ are the dominant
ionization stages of iron. Due to the faintness of [Fe IV] lines, the gaseous iron
abundance is usually calculated from the Fe$^{++}$ abundance and an ionization correction
factor (ICF) derived from photoionization models. However, for the handful of objects
with measurements of [Fe III] and [Fe IV] lines, a discrepancy has been found between
the abundance calculated with the aforementioned method and the one found by adding
the abundances of Fe$^{++}$ and Fe$^{+3}$.
\citet{Rodriguez_05} studied this discrepancy and derived three correction schemes
that take into account what changes in all the atomic data involved in the calculations
would solve the discrepancy: 1) a decrease in the collision strengths for Fe$^{+3}$
by factors of $\sim$2--3, 2) an increase in the collision strengths for Fe$^{++}$ by
factors of $\sim$2--3, or 3) an increase in the total recombination coefficient
or the rate of the charge-exchange reaction with H$^0$ for Fe$^{+3}$ by a factor of
$\sim$10.
Since these three possibilities are equally plausible, and since the discrepancy
could be due to some combination of them, the real value of the iron abundance will
be intermediate between the extreme values derived with the three correction schemes.
As in our previous work, we select PNe with a relatively low degree of ionization
($I(\mbox{He II}\,\lambda4686)/I(\mbox{H}\beta)$ $\lesssim0.3$,
see \citealt{Delgado-Inglada_09}), and with moderate electron density (below
25\,000 cm$^{-3}$), since high densities can be associated with large density
gradients that introduce large uncertainties in the calculations. Besides, the objects
have spectra with all the lines we need to calculate the physical conditions and the
abundances of Fe$^{++}$, O$^{+}$, and O$^{++}$.
We calculate a mean electron density and two electron temperatures (using the usual
[N II] and [O III] diagnostic line ratios) for the low and high ionization regions;
and with them we derive the O$^{+}$, O$^{++}$, and Fe$^{++}$ abundances.
To derive the total oxygen abundances we use the ICF from
\citet{KB_94} and for iron the three ionization schemes of \citet{Rodriguez_05}.
See \citet{Delgado-Inglada_09} for more details on the procedure and on the atomic
data we use. The uncertainties in all the quantities were estimated via Montecarlo
simulations (Delgado-Inglada \& Rodr\'iguez, in prep.).
\section{Results}
Figure \ref{fig1} shows the values of the Fe/O abundance ratio (left axis), and
the depletion factors (right axis) for Fe/O:
$[\mbox{Fe/O}]= \log(\mbox{Fe/O}) - \log(\mbox{Fe/O})_{\odot}$. We use the solar
abundance from \citealt{Lodders_03}, $\log(\mbox{Fe/O})_{\odot} = -1.22\pm0.06$,
as the expected total abundance.
Filled symbols in this figure indicate the values derived using the correction scheme
defined by point (3) above, while empty symbols represent the values obtained with
the other two correction schemes. The real values are expected to be in the range
defined by these three values. We find that both PNe and H II regions have
consistently high depletions factors, with more than 80\% of their iron atoms
condensed into dust grains.
The differences between the Fe/O values derived with the three correction schemes
depend on the degree of ionization of the objects: the higher the degree of ionization,
the greater the difference in Fe/O values. Hence, the iron abundance is better
constrained for the objects with $\log$(O$^{+}$/O$^{++}$) $\gtrsim-1.0$, where
the differences between the correction schemes are $\leq$0.5 dex. No matter which
correction scheme we consider, the range of depletions is wide in this region, with a
difference in the iron abundance of a factor of $\sim$100 between the objects with
the lowest and the highest value.
Although these variations should be reflecting real differences between the objects,
we do not find any clear correlation between the iron abundances and parameters related
to the nebular age (such as the electron density or the surface brightness of the PN)
or with the dust chemistry (Delgado-Inglada \& Rodr\'iguez, in prep.).
The first result suggests that no significant destruction of dust grains is taking
place in these objects, and the second one argues for a similar efficiency of iron
depletion in C-rich and O-rich environments.
\begin{figure}
\epsscale{1.0}
\plotone{fig1.ps}
\caption{{\small Values of Fe/O (left axis) and the depletion factors
for Fe/O ($[\mbox{Fe/O}]= \log(\mbox{Fe/O}) - \log(\mbox{Fe/O})_{\odot}$, right
axis) as a function of the degree of ionization. Three values of Fe/O are shown for
each object, derived using the three correction schemes of \citet{Rodriguez_05}.
The PNe are classified according to their origin (disk or bulge) and their morphology.
See the text for more explanations.
\label{fig1}}}
\end{figure}
Here, we explore the relation between the iron abundances of the sample PNe and their
morphological types. There is some observational evidence that the morphological type
of a PN is related to the progenitor mass. Symmetric PNe (round or elliptical nebulae)
might descend from low mass progenitors, and asymmetric PNe (bipolar or more complex
objects) might have the most massive progenitor stars
\citep[see, e.g.,][]{Corradi_95, Stanghellini_06}. Asymmetric PNe have also been
associated with binary systems \citep[see, e.g.,][]{Corradi_95, Soker_98, deMarco_09},
and with stellar rotation and/or magnetic fields \citep[see, e.g.,][]{GarciaSegura_99}.
Although the reasons behind the shaping of PNe are still matter of debate, we study
here if there is a correlation between the morphologies and the iron depletion factors.
Figure \ref{fig1} shows the PNe morphologies given by the Planetary Nebula Image Catalog
of Bruce Balick (PNIC\footnote{http://www.astro.washington.edu/users/balick/PNIC/})
and by \citet{Stanghellini_02} for the 34 objects with available images.
Two of them are classified as round, 26 as elliptical, four as bipolar, one as
irregular, and one more as complex. This figure shows that there is no obvious trend
relating the morphological type and the amount of iron condensed in dust grains.
For example, elliptical and bipolar PNe are distributed in the whole
range of depletions. The three correction schemes used in the abundance determination
give consistent results on this issue.
\section{Conclusions}
We derive the iron abundance in a sample of 20 Galactic bulge PNe,
and compare the results with the ones previously obtained for 28
Galactic disk PNe and 8 Galactic H II regions. We find high depletion
factors in all the objects: less than 20\% of their expected total
number of iron atoms are measured in the gas phase. This result
suggests that iron depletion into dust grains is an efficient process
in AGB stars, whereas dust destruction is not very efficient in the
subsequent PN phase. Although the range of depletions in the sample
PNe is wide, covering around two orders of magnitude, we do not find
any correlation between the iron depletion factors and the morphology
of the PNe. Further details of the analysis will we presented in
Delgado-Inglada \& Rodr\'iguez (in prep.).
\acknowledgements We acknowledge support from Mexican CONACYT project 50359-F.
|
{
"timestamp": "2010-09-22T02:00:27",
"yymm": "1009",
"arxiv_id": "1009.3945",
"language": "en",
"url": "https://arxiv.org/abs/1009.3945"
}
|
\section{Introduction}
{\it Bessel processes} play a prominent role both in the theory of Brownian motion (see \cite{McKean:1960} and \cite{ItoMcKean:1974}) as well as in various theoretical and practical applications. The n-dimensional Bessel process appears quite naturally as the n-dimensional Euclidean norm of Brownian motion; the more intriguing applications are related to the celebrated {Ray-Knight} theorems describing the behaviour of the local time of Brownian motion in terms of two-dimensional (quadratic) Bessel process (see \cite{RevuzYor:2005}, \cite{Yor:1992} or \cite{BorodinSalminen:2002}). There is also an intimate relation between Bessel processes and
{\it the geometric Brownian motion} \cite{Lamperti:1972} ({the so-called Lamperti's theorem} - see Preliminaries). Bessel processes also appear when representing
important {\it jump L\'evy processes} by means of {\it traces} of some multidimensional diffusions ({\it Bessel-Brownian diffusions}); see \cite{Molcanov:1969}, \cite{BMR2:2010}. Another important application consists of the fact that {\it the hyperbolic Brownian motion}, i.e. the canonical diffusion in the real hyperbolic space, can be represented as subordination of the standard Brownian motion
via an exponential functional of geometric Brownian motion. Thus, from Lamperti's theorem, {\it the Poisson kernel of half-spaces} for hyperbolic Brownian motion can be represented via subordination of the standard Brownian motion by the hitting time of a Bessel process (see \cite{BR:2006}). We exploit this relationship for obtaining the precise bounds of
the Poisson kernel of half-spaces for hyperbolic Brownian motion with drift.
Our main goal is to study estimates of the density of the distribution of the first hitting time $T^{(\mu)}_{a}$ of a level $a>0$ by a Bessel process starting from $x>a$. Our approach is based on an integral formula for the density of $T^{(\mu)}_{a}$, given in the paper
\cite{BR:2006} {(see also \cite{BGS:2007})}. This formula, although quite complex, proves to be a very effective one.
{We deal with processes with non-positive indices}, however our conclusions are valid also for positive ones, since the processes are equivalent on the sigma-algebra generated by the process before hitting $0$.
To formulate our results denote by
$R^{(\mu)}=\{R_t^{(\mu)},t\geq 0\}$ a Bessel process with index $\mu\in\R$. We note by $\textbf{P}_x^{(\mu)}$ the probability law of a Bessel process $R^{(\mu)}$ with index $\mu$ on the canonical paths space with starting point $R^{(\mu)}_0 =x$, where $x>0$.
Let us denote the first hitting time of the level $a> 0$ by a Bessel process with index $\mu$
\begin{eqnarray*}
T^{(\mu)}_a = \inf\{t>0; R^{(\mu)}_t=a\}\/.
\end{eqnarray*}
Our main result is a sharp estimate of the density of the hitting distribution if $x>a$. By the scaling property of Bessel processes it is enough to consider $x>a=1$. The distribution of $T^{(\mu)}_1$ is well known for $\mu=\pm1/2$ when both distributions have the $1/2$-stable positive distribution with scale parameter $\lambda=x-1$, however incomplete for $\mu=1/2$.
Our upper and lower estimates are comparable in both time and space domains. We have the following:
\begin{center}
{\bf Uniform estimate for a density function of $T_1^{(\mu)}$.}
\end{center}
For every $x>1$ and $t>0$ we have
\begin{eqnarray*}
\frac{\textbf{P}_x^{(\mu)}(T_1^{(\mu)}\in dt)}{dt} \approx (x-1)\left(\frac{1}{1+x^{2\mu}}\right)\frac{ {e^{-(x-1)^2/2t}}}{t^{3/2}} \frac{ x^{2|\mu|-1} }{t^{|\mu|-1/2}+ x^{|\mu|-1/2}}\/,\quad \mu\neq 0\/.
\end{eqnarray*}
Moreover, we have
$$
\frac{\textbf{P}_x^{(0)}(T_1^{(0)}\in dt)}{dt} \approx
(x-1) e^{-(x-1)^2/2t} \frac{(x+t)^{1/2}}{x t^{3/2}}
\frac {1+\log x}{(1+\log (1+\frac tx))(1+\log (t+x))}\/.
$$
Here $ f\approx g$ means that there exist strictly positive constants $c_1$ and $c_2$ depending only on $\mu$ such that $c_1\le f/g\le c_2$.
Our setup includes the case of the hitting distribution of a unit ball by a Brownian motion starting from the exterior of the ball.
To state this result,
let $\sigma^{(n)}$ be the first hitting time of a unit ball by $n$-dimensional Brownian motion $W^{(n)}=\{W_t^{(n)},t\geq 0\}$, i.e.
\begin{eqnarray*}
\sigma^{(n)} = \inf\{t>0; |W^{(n)}_t|=1\}\/.
\end{eqnarray*}
\begin{center}
{\bf Uniform estimate for hitting the unit ball by Brownian motion.}
\end{center}
For $W^{(n)}_0 = x\in\R^n$ such that $|x|>1$ we have
\begin{eqnarray*}
\frac{ P^{x}(\sigma^{(n)}\in dt)}{dt} \approx \frac{|x|-1}{|x|}\frac{ {e^{-(|x|-1)^2/2t}}}{t^{3/2}} \frac{ 1}{t^{(n-3)/2}+ |x|^{(n-3)/2}}\/, \quad n>2,
\end{eqnarray*}
for every $t>0$. Moreover, we have
\begin{eqnarray*}
\frac{P^{x}(\sigma^{(2)}\in dt)}{dt} \approx \frac{|x|-1}{|x|} e^{-(|x|-1)^2/2t} \frac{(|x|+t)^{1/2}}{ t^{3/2}}
\frac {1+\log |x|}{(1+\log (1+\frac t{|x|}))(1+\log (t+|x|))}\/.
\end{eqnarray*}
To the best of our knowledge even for the planar Brownian motion our results are new and considerably complement existing results obtained in \cite{Grigoryan:2002} (see also \cite{Collete:2000}), where the estimates are {only} sharp in the region $t \ge |x|^2$ with sufficiently large starting point $x$. If $t<|x|^2$ the bound obtained in \cite{Grigoryan:2002} has an exponential term of the form $\exp\{-c_i\frac{|x|^2}{t}\}$ with different constants $c_1, c_2$ for the lower and the upper estimate, respectively. We remove this obstacle and provide sharp estimates which are of the same order in the full range of $t$ and $x$. We also provide sharp estimates for the survival probability $
\textbf{P}_x^{(\mu)}(t<T_1^{(\mu)}<\infty)$. The asymptotic result if $t\to \infty$ is due to Hunt \cite{Hunt:1956} in the case of the planar Brownian motion and Port \cite{Port:1969b} in the context of Brownian motion in higher dimensions. { Recently, a result about the asymtotic behaviour of the hitting density for the planar Brownian, when $t\to \infty$, was established by Uchiyama \cite{Uchiyama:2010}. His result gives a very accurate expansion of the hitting density provided $ |x|^2$ is small relative to $t$, that is when the exponential term is negligable. When this is not true the error term of the expansion in \cite{Uchiyama:2010} may be much bigger than the leading term. Therefore our estimate in the case of the planar Brownian motion is much more accurate, when we assume that $t$ is not too large with respect to $|x|^2$, since the impact of of the exponential term may be significant and it is reflected in our estimates. Moreover, we give a very exact estimate of the density in the situation when $t$ is small relative to $|x|$ (see Lemma \ref{qt:estimate:zero} and Remark \ref{remark1}). }
The organization of the paper is as follows. After Preliminaries, in Section 3, we provide uniform estimates of the density function of the first hitting time $T^{(\mu)}_{1}$. This section is basic for further applications, which are collected in the next section.
We first provide the estimates of the {\it survival times} of a killed Bessel process and, finally, compute the precise bounds of the Poisson kernel of a halfspace for hyperbolic Brownian motion with drift.
Appendix contains various estimates of quantities involved in the basic formula for the density function of $T_1^{(\mu)}$, which, although quite laborious, but at the same time, are indispensable ingredients of the proof of the main result.
Throughout the whole paper $ f\approx g$ means that there exists a strictly positive constant $c$ depending only on $\mu$ such that $c^{-1}g\le f \le c g$. If the comparability constant will also depend on some other parameters $\gamma_1, \gamma_2,\dots$ we will write $ f\stackrel c{\approx} g$, $c=c(\gamma_1, \gamma_2,\dots)$. Also in a string of inequalities a constant may change from line to line which might not be reflected in notation. Moreover we consistently do not exhibit dependence of $c$ on $\mu$ in inequalities of type $f\le c g $.
\section{Preliminaries}
\subsection{Modified Bessel functions}
Various formulas appearing throughout the paper are expressed in terms of {\it modified Bessel functions} $I_{\vartheta}$ and $K_{\vartheta}$. For convenience we collect here basic information about these functions.
The \textit{modified Bessel functions of the first} and \textit{second kind} are independent solutions to the \textit{modified Bessel differential equation}
\begin{eqnarray*}
z^2y'' + zy' - (\vartheta^2+z^2)y =0\/,
\end{eqnarray*}
where $\vartheta \in \R$. The Wronskian of the pair $\left\{K_\vartheta(z),I_\vartheta(z)\right\}$ is equal to
\begin{eqnarray}
\label{Wronskian}
W\left\{K_\vartheta(z),I_\vartheta(z)\right\} = I_\vartheta(z) K_{\vartheta+1}(z)+I_{\vartheta+1}(z)K_\vartheta(z) = \frac{1}{z}\/.
\end{eqnarray}
In the sequel we will use the asymptotic behavior of $I_\vartheta$ and $K_\vartheta$ at zero as well as at infinity. For every $\vartheta \geq 0$ we have (see \cite{AbramowitzStegun:1972} 9.6.7 and 9.6.12)
\begin{eqnarray}
\label{I_atzero}
I_\vartheta(r) &=& \frac{r^\vartheta}{2^\vartheta\Gamma(\vartheta+1)} +O(r^{\vartheta+2})\,,\quad r\to 0^{+}\/, \vartheta >0\/.
\end{eqnarray}
For $\vartheta>0$, we have (\cite{AbramowitzStegun:1972} 9.6.9 and 9.6.13)
\begin{eqnarray}
\label{K_atzero}
K_\vartheta(r)&\cong& {\frac{2^{\vartheta-1}\Gamma(\vartheta)}{r^{\vartheta}}}\,,
\quad r\to 0^+, \label{asympt_K_0}
\end{eqnarray}
where $g(r) \cong f(r) $ means that the ratio of $g$ and $f$ tends to $1$. Moreover, in the case $\vartheta = 0$, we have (see \cite{AbramowitzStegun:1972} 9.6.13)
\begin{eqnarray}
\label{K0_atzero}
K_0(r) = -\log\frac{r}{2}I_0(r) + O(1)\/,\quad r\to 0^{+}\/.
\end{eqnarray}
The behavior of $I_\vartheta$ and $K_\vartheta$ at infinity is described as follows (see \cite{AbramowitzStegun:1972} 9.7.1, 9.7.2)
\begin{eqnarray}
\label{asymp_I_infty}
I_\vartheta(r)= {\frac{e^r}{\sqrt{2\pi r}}} (1+ O(1/r))\,,\quad r\to \infty\/,
\end{eqnarray}
\begin{eqnarray}
\label{asymp_K_infty}
K_\vartheta(z) = \sqrt{\frac{\pi}{2z}}\,e^{-z}(1+O(1/z))\/,\quad |z|\to \infty\/,
\end{eqnarray}
where the last equality is true for every complex $z$ such that $|arg z|<\frac{3}{2}\pi$.
\subsection{Bessel process and exponential functionals of Brownian motion}
In the following section we introduce notation and basic facts about Bessel processes. We follow the exposition given in \cite{MatsumotoYor:2005a} and \cite{MatsumotoYor:2005b}, where we refer the Reader for more details and deeper insight into the subject (see also \cite{RevuzYor:2005}).
We denote by $\textbf{P}_x^{(\mu)}$ the probability law of a Bessel process $R^{(\mu)}$ with index $\mu$ on the canonical paths space with starting point $R^{(\mu)}_0 =x$, where $x>0$. Let $\mathcal{F}^{(\mu)}_t=\sigma\{R^{(\mu)}_s,s\leq t\}$ be the filtration of the coordinate process $R^{(\mu)}_t$. The state space of $R^{(\mu)}$ depends on the value of $\mu$ and the boundary condition at zero. For simplicity, in the case $-1<\mu<0$ (then the point $0$ is non-singular), we impose killing condition on $0$. However, the exact boundary condition at $0$ is irrelevant from our point of view, because we will only consider the process $R^{(\mu)}$ up to the first hitting time of the strictly positive level.
Let us denote the first hitting time of the level $a> 0$ by a Bessel process with index $\mu$
\begin{eqnarray*}
T^{(\mu)}_a = \inf\{t>0; R^{(\mu)}_t=a\}\/.
\end{eqnarray*}
Observe that for $\mu\leq 0$ we have $T^{(\mu)}_a<\infty$ a.s. and $\textbf{P}^{(\mu)}_x(T^{(\mu)}_a=\infty)>0$ whenever $\mu>0$. Using the scaling property of Bessel processes, which is exactly the same as the scaling property of one-dimensional Brownian motion, we get for every $b>0$ and $t>0$
\begin{eqnarray*}
{\textbf{P}_{bx}^{(\mu)}(T^{(\mu)}_{ba}<t) = \textbf{P}_{x}^{(\mu)}(b^2T^{(\mu)}_{a}<t)}\/, \quad x>a>0\/.
\end{eqnarray*}
Therefore, from now on we do assume that $a=1$ and $x>1$. We denote the density function of $T_1^{(\mu)}$ with respect to Lebesgue measure by $q_x^{(\mu)}$, i.e.
\begin{eqnarray*}
q_x^{(\mu)}(t) = \frac{\textbf{P}_{x}^{(\mu)}(T^{(\mu)}_{1}\in dt)}{dt}\/,\quad t>0\/,x>1\/.
\end{eqnarray*}
We have the absolute continuity property for the laws of the Bessel processes with different indices
\begin{eqnarray}
\label{Bessel:AC}
\left. \frac{d\textbf{P}^{(\mu)}_x}{d\textbf{P}^{(\nu)}_x}\right|_{\mathcal{F}^{(\nu)}_t} = \left(\frac{{R_t}}{x}\right)^{\mu-\nu}\exp\left(-\frac{\mu^2-\nu^2}{2}\int_0^t\frac{ds}{{(R_s)}^2}\right)\/,\quad {\textbf{P}^{(\nu)}_x}\textrm{ - a.s. on }\{T^{(\nu)}_0>t\}\/.
\end{eqnarray}
Here $T_0^{(\mu)}$ denotes the first hitting time of $0$ by $R^{(\mu)}$. If $\nu\geq 0$ then the condition $\{T^{(\nu)}_0>t\}$ can be omitted. In particular, for $\nu=-\mu$, where $\mu\geq 0$ we have
\begin{eqnarray*}
\left. \frac{d\textbf{P}^{(\mu)}_x}{d\textbf{P}^{(-\mu)}_x}\right|_{\mathcal{F}^{(-\mu)}_t} = \left(\frac{{R_t}}{x}\right)^{2\mu}\/,\quad {\textbf{P}^{(-\mu)}_x}\textrm{ - a.s. on }\{T^{(-\mu)}_0>t\}\/.
\end{eqnarray*}
Consequently, for $\mu\geq 0$ and $x>1$ we get that
\begin{eqnarray}
\label{BesselTime:indices}
q_x^{(\mu)}(t) = \left(\frac1x\right)^{2\mu}\qmu\/.
\end{eqnarray}
We denote by $B=\{B_t,t\geq 0\}$ the one-dimensional Brownian motion starting from $0$ and by $B^{(\mu)}=\{B_t^{(\mu)}=B_t+\mu t,t\geq 0\}$ the Brownian motion with constant drift $\mu\in \R$.
The process $X^{(\mu)} = \{x\exp(B^{(\mu)}_t),t\geq 0\}$ is called a \textit{geometric Brownian motion} or \textit{exponential Brownian motion} with drift $\mu\in\R$ starting from $x>0$.
For $x>1$ let $\tau$ be the first exit time of the geometric Brownian motion with drift $\mu$ from the set $(1,\infty)$
\begin{eqnarray*}
\tau = \inf\{s>0; x\exp(B_s+\mu s)=1\}\/.
\end{eqnarray*}
We have $\tau<\infty$ a.s. whenever $\mu\leq 0$ since then $\inf_{t\geq 0}B^{(\mu)}(t) = -\infty$.
For $x>0$ we consider the integral functional
\begin{eqnarray*}
\Amuplus(t)= \int_0^t (X_s^{(\mu)})^2ds = x^2\int_0^t\exp(2B_s+2\mu s)ds\/.
\end{eqnarray*}
The crucial fact which establishes the relation between Bessel processes, the integral functional $\Amuplus$ and the geometric Brownian motion is the Lamperti relation saying that there exists a Bessel process $R^{(\mu)}$ such that
\begin{eqnarray*}
x\exp(B_t^{(\mu)}) = R^{(\mu)}_{A^{(\mu)}_x(t)}\/,\quad t\geq 0\/.
\end{eqnarray*}
Consequently, we get
\begin{eqnarray}
\label{Amu_Tmu}
A^{(\mu)}_x(\tau) \stackrel{d}{=} T_1^{(\mu)}\/.
\end{eqnarray}
\subsection{Representation of hitting time density function}
We recall the result of \cite{BR:2006}, where the integral formula for the density of $\Amu(\tau)$ was given. According to (\ref{Amu_Tmu}), as immediate consequence, we obtain the formula for $\qmu$. Note also that in the paper \cite{BR:2006} different normalizations of Brownian motion and different definition of geometric Brownian motion were used and consequently we have $\qmu = {q_\mu}(t/2)/2$, where ${q_\mu}(t)$ is the density function considered in \cite{BR:2006}.
\begin{thm}[{[Byczkowski, Ryznar 2006]}]\label{rep}
For $\mu \geq 0$ there is a function $w_\lambda$ such that
\begin{eqnarray}\label{rep1}
\qmu = \lambda \frac{e^{-\lambda^2/2t}}{ \sqrt{2\pi t} }
\(\frac{x^{\mu-1/2}}{t} +
\int_0^\infty \(e^{-\kappa/2t} - 1\) w_\lambda(v)dv \) \/,
\end{eqnarray}
where $\kappa = \kappa (v)=(\lambda+v)^2 - \lambda^2 = v(2\lambda+v)$,
and $\lambda=x-1$.
\end{thm}
The function $w_\lambda$ appearing in the formulas is described in terms of the \textit{modified Bessel functions} $K_\mu$ and $I_\mu$. The function $K_\mu(z)$ extends to an entire function when $\mu-1/2$ is an integer and has a holomorphic extension to $\C\setminus (-\infty,0]$ when $\mu-1/2$ is not an integer.
Denote the set of zeros of the function $K_\mu(z)$ by $Z=\{z_1,...,z_{k_\mu}\}$ (cf. \cite{Erdelyi:1954}, p. 62). Recall that $k_\mu=\mu-1/2$
when $\mu-1/2 \in \N$. For $\mu-1/2 \notin \N$, $k_\mu$ is the even number closest to $\mu-1/2$. The functions $K_\mu$ and $K_{\mu-1}$ have no common zeros.
The function $w_\lambda$ is defined as a sum of two functions
\begin{eqnarray}
\label{wlambda:sum}
w_\lambda(v)=w_{1,\/\lambda}(v)+w_{2,\/\lambda}(v)\/,
\end{eqnarray}
where
$$w_{1,\/\lambda}(v)= -\frac{x^\mu }{ \lambda} \sum_{i=1}^{k_\mu}
\frac{z_i e^{\lambda z_i} K_\mu(xz_i) }{ K_{\mu-1}(z_i)} \, e^{z_i v}$$
and
\begin{eqnarray*}
w_{2,\/\lambda}(v)= -\cos(\pi\mu) \frac{x^\mu }{ \lambda}
\int_0^\infty \frac{
I_\mu\(xu\)K_\mu(u)-I_\mu(u)K_\mu\(xu\)}{
\cos^2(\pi\mu) K_\mu^2(u)+(\pi I_\mu(u)+\sin(\pi \mu) K_\mu(u))^2} \,
e^{-\lambda u} e^{-vu} \/u du\/.
\end{eqnarray*}
{Moreover the moments of $\kappa$ with respect to $w_\lambda(v)dv$ can be computed in the following way}
\begin{eqnarray}\label{laplace01}
x^{\mu-1/2} (\mu^2-1/4)/2x=\int_0^\infty w_\lambda(v) dv\/,
\end{eqnarray}
and, for $\mu>1/2$, we have
\begin{eqnarray}\label{laplace02}
2 x^{\mu-1/2} =\int_0^\infty \kappa w_\lambda(v) dv\/.
\end{eqnarray}
We use also the following representation of $\qmu$ for $\mu\geq 1/2$ (see \cite{BR:2006} (24))
\begin{eqnarray}
\label{rep3}
\qmu = \lambda\,\frac{e^{-\lambda^2/2t}}{\sqrt{2\pi t}}\int_0^\infty \left(e^{-\kappa/2t}-\sum_{0\leq j \leq l}(-1)^j\frac{1}{j!}\left(\frac{\kappa}{2t}\right)^j\right)w_\lambda(v)dv\/,
\end{eqnarray}
where $l=[\mu+1/2]$ if $\mu \notin \N$, and $l=\mu-1/2$ otherwise.
\section{Uniform estimates of hitting time density function}
Throughout of the rest of the paper we denote $\lambda=x-1$. The main result of the paper is the following uniform estimate for a density function of $T_1^{(\mu)}$.
\begin{thm}
For every $x>1$ and $t>0$ we have
\begin{eqnarray}
\label{Bessel:hittingtime:estimates}
q^{(\mu)}_x(t) \approx \lambda \left(\frac{1}{1+x^{2\mu}}\right)\frac{ {e^{-\lambda^2/2t}}}{t^{3/2}} \frac{ x^{2|\mu|-1} }{t^{|\mu|-1/2}+ x^{|\mu|-1/2}}\/,\quad \mu\neq 0\/.
\end{eqnarray}
Moreover, we have
$$
q^{(0)}_x(t) \approx
\left\{\begin{array}{lc} \lambda \dfrac{e^{-\lambda^2/2t} }{x t }
\dfrac {1+\log x}{(1+\log \frac tx)(1+\log t)}\/,& t>2x\/,\\
{\lambda} \dfrac{e^{-\lambda^2/2t} }{x^{1/2} t^{3/2} }\/, & t\le 2x
\end{array}\right.
$$
or equivalently
$$
q^{(0)}_x(t) \approx
\lambda e^{-\lambda^2/2t} \frac{(x+t)^{1/2}}{x t^{3/2}}
\frac {1+\log x}{(1+\log (1+\frac tx))(1+\log (t+x))}\/.
$$
\end{thm}
As corollary, putting $\mu=n/2-1$, we get the corresponding result for $n$-dimensional Brownian motion.
\begin{thm}
Let $\sigma^{(n)}$ be the first hitting time of a unit ball by $n$-dimensional Brownian motion $W^{(n)}=\{W_t^{(n)},t\geq 0\}$, i.e.
\begin{eqnarray*}
\sigma^{(n)} = \inf\{t>0; |W^{(n)}_t|=1\}\/.
\end{eqnarray*}
Then, for $W^{(n)}_0 = x\in\R^n$ such that $|x|>1$ we have
\begin{eqnarray*}
\frac{P^{x}(\sigma^{(n)}\in dt)}{dt} \approx \frac{|x|-1}{|x|}\frac{ {e^{-(|x|-1)^2/2t}}}{t^{3/2}} \frac{ 1}{t^{(n-3)/2}+ |x|^{(n-3)/2}}\/, \quad n>2,
\end{eqnarray*}
for every $t>0$. Moreover, we have
\begin{eqnarray*}
\frac{P^{x}(\sigma^{(2)}\in dt)}{dt} \approx \frac{|x|-1}{|x|} e^{-(|x|-1)^2/2t} \frac{(|x|+t)^{1/2}}{ t^{3/2}}
\frac {1+\log |x|}{(1+\log (1+\frac t{|x|}))(1+\log (t+|x|))}
\end{eqnarray*}
\end{thm}
The proof of the main theorem follows from Lemmas \ref{qt:estimate:inside}, \ref{qt:estimate:infty1}, \ref{qt:estimate:infty2}, \ref{qt:zero:estimate:infty} given below. We use the crucial estimates of the function $w_\lambda(v)$ and its components $w_{1,\,\lambda}(v)$ and $w_{2,\,\lambda}(v)$ given in Appendix (see Lemmas \ref{w1:estimate:lemma}, \ref{w2:estimate:lemma} and \ref{w:muzero:estimate:lemma}).
The following lemma provides satisfactory estimates of $\qmu$ in the case when $t$ is small relative to $x$.
{\begin{lem} \label{qt:estimate:zero}
We have the following expansion
\begin{eqnarray*}
\qmu= \lambda\frac{e^{-\lambda^2/4t}}{(2\pi)^{1/2}t^{3/2}}x^{\mu-1/2} \left(1+\frac{1-4\mu^2}8\frac t{x} + E(t,x)\right),
\end{eqnarray*}
where the error term satisfies the following estimate
$$ |E(t,x)|\le C \frac tx (\sqrt{t}\wedge \frac {t}\lambda).$$
Moreover, for $0\le \mu<1/2$ we have
\begin{eqnarray*}
\lambda\frac{e^{-\lambda^2/2t}}{(2\pi)^{1/2}t^{3/2}}x^{\mu-1/2}\le \qmu\le \lambda\frac{e^{-\lambda^2/4t}}{(2\pi)^{1/2}t^{3/2}}x^{\mu-1/2} \left(1+\frac{1-4\mu^2}8\frac t{x} \right)
\end{eqnarray*}
for every $x>1, t>0$.
\end{lem} }
\begin{proof}
{ By the basic formula (\ref{rep1}), with application of (\ref{laplace01}), we obtain
\begin{eqnarray*}
\frac{(2\pi)^{1/2}t^{3/2}}{\lambda x^{\mu-1/2}}\,e^{\lambda^2/2t}\qmu &=& 1+\frac{t}{x^{\mu-1/2}}\int_0^\infty (e^{-\kappa/2t}-1)w_\lambda(v)dv\\
& =& 1+\frac{1-4\mu^2}8\frac t{x} + \frac{t}{x^{\mu-1/2}}\int_0^\infty e^{-\kappa/2t}w_\lambda(v)dv\\
& =& 1+\frac{1-4\mu^2}8\frac t{x} + E(x,t)
\/.
\end{eqnarray*} }
{ Observe that, by Lemmas \ref{w1:estimate:lemma}, \ref{w2:estimate:lemma}
and \ref{w:muzero:estimate:lemma}, $|w_\lambda(v)|\le C x^{\mu-3/2}$, which gives the following estimate
\begin{eqnarray*}
|E(x,t)|= \left|\frac{t}{x^{\mu-1/2}}\int_0^\infty e^{-(\lambda v) /t}e^{- v^2 /2t}w_\lambda(v)dv\right|\leq
C \frac tx\int_0^\infty e^{-(\lambda v) /t}e^{- v^2 /2t}dv\le C \frac tx (\sqrt{t}\wedge \frac {t}\lambda)
\/.
\end{eqnarray*} }
This ends the proof of the first claim. For $\mu<1/2$ observe that
\begin{eqnarray*}
0\le \int_0^\infty (e^{-\kappa/2t}-1)w_\lambda(v)dv\le -\int_0^\infty w_\lambda(v)dv = x^{\mu-1/2} (1/4-\mu^2)/2x.
\end{eqnarray*}
Consequently, we get
\begin{eqnarray*}
1\leq\frac{(2\pi)^{1/2}t^{3/2}x^{1/2}}{\lambda}\,e^{\lambda^2/4t}\qmu\leq\left(1+\frac{1-4\mu^2}8\frac t{x} \right)
\end{eqnarray*}
and this completes the proof of the second claim in the case $\mu<1/2$.
\end{proof}
{\begin{rem}\label{remark1} If $ 1<x<2$ then the absolute value of the error term is bounded by $Ct^{3/2}$, while for $x>2$ it is bounded by $C(\frac tx)^2$. Observe also that the density $\qmu$ up to a multiplicative constant is close to the density of the hitting distribution of $1$ by the one-dimensional Brownian motion starting from $x$, when the fraction $\frac tx$ is small with the error precisely estimated by the above lemma. With some additional effort one can show that the error term estimate can not be improved.
\end{rem}}
{
\begin{prop}
\label{Prop:asymp:inside}
For every $\mu\neq 0$ and $c>0$ we have
\begin{eqnarray*}
\lim_{x/t\to c,\,x\to \infty} \dfrac{1+x^{2\mu}}{x^{|\mu|-1/2}}\frac{q_x^{(\mu)}(t)\sqrt{2\pi t}}{ e^{-\lambda^2/2t}} &=& \sqrt{\frac{\pi c}{2}} \frac{e^{-c}}{K_{|\mu|}(c)}\/.
\end{eqnarray*}
\end{prop}
\begin{proof}
It is enough to show the above-given convergence for strictly negative indices. The general statement follows from (\ref{BesselTime:indices}). Now we assume that $\mu>0$ and consider $\qmu$. We define
\begin{eqnarray*}
w_\infty (v) = -\sum_{i=1}^{k_\mu} \frac{\sqrt{z_i}e^{-z_i}}{K_{\mu-1}(z_i)}e^{-vz_i}-\frac{\cos(\pi\mu)}{\sqrt{2\pi}}\int_0^\infty
\frac{e^{u}K_\mu(u)e^{-vu}\sqrt{u}du}{\cos^2(\pi\mu) K_\mu^2(u)+(\pi I_\mu(u)+\sin(\pi \mu) K_\mu(u))^2}
\end{eqnarray*}
for every $v>0$. If $k_\mu=0$ then the first sum is equal to zero. Using the asymptotic expansion (\ref{asymp_K_infty}) we easily see that
\begin{eqnarray*}
\lim_{x\to\infty} \frac{\lambda}{x^{\mu-1/2}}w_{1,\,\lambda}(v) = -\sum_{i=1}^{k_\mu} \frac{\sqrt{z_i}e^{-z_i}}{K_{\mu-1}(z_i)}\,e^{-vz_i}\/.
\end{eqnarray*}
The relation (\ref{asymp_I_infty}) implies that $|I_\mu(u)|\leq c_2 \dfrac{e^u}{\sqrt{u}}$ and consequently
\begin{eqnarray*}
\sqrt{x}|I\mu(xu)K_\mu(u)-I_\mu(u)K_\mu(xu)|e^{-xu}\leq c_2 K_\mu(u) \frac{1}{\sqrt{u}}
\end{eqnarray*}
for every $u>0$. Moreover, using the estimates of $K_\vartheta$ and $I_\vartheta$ given in Preliminaries, we observe that the function
\begin{eqnarray*}
f(u,v) = \frac{e^{u}K_\mu(u)e^{-vu}\sqrt{u}du}{\cos^2(\pi\mu) K_\mu^2(u)+(\pi I_\mu(u)+\sin(\pi \mu) K_\mu(u))^2}
\end{eqnarray*}
is bounded, as a function of $u$, by $c_3 e^{-(v+2)u}u^{3/2}$ on $[1,\infty)$ and by $c_3 u^{\mu+1/2}$ on $(0,1)$ and consequently is integrable on $(0,\infty)$. Using the dominated convergence theorem we get
\begin{eqnarray*}
\lim_{x\to \infty} \frac{\lambda}{x^{\mu-1/2}}w_{2,\,\lambda}(v) = -\frac{\cos(\pi\mu)}{\sqrt{2\pi}}\int_0^\infty
\frac{e^{u}K_\mu(u)e^{-vu}\sqrt{u}du}{\cos^2(\pi\mu) K_\mu^2(u)+(\pi I_\mu(u)+\sin(\pi \mu) K_\mu(u))^2},
\end{eqnarray*}
which implies
\begin{eqnarray*}
\lim_{x\to \infty} \frac{\lambda}{x^{\mu-1/2}}w_{\lambda}(v) = w_\infty(v)\/,\quad v>0\/.
\end{eqnarray*}
Using (\ref{w1:estimate:abs}) and (\ref{w2:estimates}) from Appendix we get that
\begin{eqnarray}
\label{wlambda:estimate}
\frac{\lambda}{x^{\mu-1/2}}|w_{\lambda}(v)|\leq \frac{\lambda}{x^{\mu-1/2}}|w_{1,\,\lambda}(v)|+\frac{\lambda}{x^{\mu-1/2}}|w_{2,\,\lambda}(v)|\leq c_4 e^{-v\theta_\mu}+c_4 \frac{1}{(v+1)^{\mu+3/2}}
\end{eqnarray}
for some positive $\theta_\mu$. Next, we take advantage of the following formula of the Laplace transform of $ w_\lambda(v)$ (see Lemma 3.1 in \cite{BR:2006})
\begin{eqnarray*}
\frac{\lambda}{x^{\mu-1/2}} \int_0^\infty e^{-r v}w_\lambda(v)dv = \frac{re^{\lambda r}x^{1/2} K_\mu(xr)}{ K_\mu(r)}-(r-(\mu^2-1/4)\frac{\lambda}{2x})\/,\quad r> 0.
\end{eqnarray*}
Thus, taking the limit as $x$ tends to $\infty$ in the above relation
and applying the dominated convergence theorem we get
\begin{eqnarray}
\label{winfty:laplace}
\int_0^\infty e^{-rv}w_\infty(v)dv = \sqrt{\frac{\pi r}{2}}\frac{e^{-r}}{K_\mu(r)}-r+\frac{\mu^2-1/4}{2}
\end{eqnarray}
for every $r> 0$.
Now let $c>0$. Observe that $\lim_{x/t\to c,x\to \infty}\frac\kappa{2t} = cv$. Using (\ref{wlambda:estimate}) and the dominated convergence theorem we get
\begin{eqnarray*}
\lim_{x/t\to c,\, x\to \infty} \frac{\lambda}{x^{\mu-1/2}}\int_0^\infty e^{-\kappa/2t}w_\lambda(v)dv &=& \int_0^\infty e^{-cv}w_\infty(v)dv\/.
\end{eqnarray*}
We have
\begin{eqnarray*}
\frac{\qmu\sqrt{2\pi t}}{\lambda e^{-\lambda^2/2t}} = \frac{x^{\mu-1/2}}{t}-(\mu^2-1/4)\frac{x^{\mu-1/2}}{2x}+\int_0^\infty e^{-\kappa/2t}w_\lambda(v)dv\/.
\end{eqnarray*}
Multiplying both sides by $\dfrac{\lambda}{x^{\mu-1/2}}$, taking limit as $x/t\rightarrow c$ and $x\rightarrow \infty$ and using (\ref{winfty:laplace}) we get
\begin{eqnarray*}
\lim_{x/t\to c,\,x\to\infty} \dfrac{1}{x^{\mu-1/2}}\frac{\qmu\sqrt{2\pi t}}{ e^{-\lambda^2/2t}} &=& c-\frac{\mu^2-1/4}{2}+\int_0^\infty e^{-cv}w_\infty(v)dv = \sqrt{\frac{\pi c}{2}} \frac{e^{-c}}{K_\mu(c)}>0\/.
\end{eqnarray*}
\end{proof}
}
\begin{lem}
\label{qt:estimate:inside}
For every $C>0$ there is a constant $c_1>0$ depending on $C$ and {$\mu>0$} such that
\begin{eqnarray*}
\frac1{c_1} \lambda \frac{ e^{-\lambda^2/2t }}{t^{3/2}}\,x^{\mu-1/2}\le \qmu\le c_1 \lambda \frac{ e^{-\lambda^2/2t }}{t^{3/2}}\,x^{\mu-1/2}\/,
\end{eqnarray*}
whenever $x<Ct$, $x>1$.
\end{lem}
\begin{proof}
By Lemma~\ref{qt:estimate:zero} it is enough to consider $\mu>1/2$.
Let $0<C^\prime<C$. The fact that the limit given in Proposition~\ref{Prop:asymp:inside} exists and is strictly positive implies that for every $c\in[C^\prime,C]$ there exist $\varepsilon_c>0$, $D_c>1$ and $x_c>2$ such that
\begin{eqnarray*}
\frac{1}{D_c}\leq\dfrac{1}{x^{\mu-1/2}}\frac{\qmu\sqrt{2\pi t}}{ e^{-\lambda^2/2t}}\leq D_c
\end{eqnarray*}
for every $(x/t,x)\in(c-\varepsilon_c,c+\varepsilon_c)\times (x_c,\infty)$. The family
\begin{eqnarray*}
\left\{(c-\varepsilon_c,c+\varepsilon_c)\right\}_{c\in[C^\prime,C]}
\end{eqnarray*}
is an open cover of the compact set $[C^\prime,C]$. Consequently, there exists a finite subcover $\{(c_k-\varepsilon_{c_k},c_k+\varepsilon_{c_k})\}_{k=1,\ldots,m}$. Setting $C^*=\max\{x_{c_k}:k=1,\ldots,m\}$ and $D = \max\{D_{c_k},k=1,\ldots,m\}$ we get
\begin{eqnarray}\label{middle_estimate}
\frac{1}{D}\leq\dfrac{1}{x^{\mu-1/2}}\frac{\qmu\sqrt{2\pi t}}{ e^{-\lambda^2/2t}}\leq D
\end{eqnarray}
for every $C^\prime\leq x/t\leq C$ and $x>C^*$, which proves the lemma for this range of $x$ and $t$. Choosing small enough the constant $C^\prime$ depending on $\mu$ we infer that using Lemma \ref{qt:estimate:zero} we complete the proof in the case $ x/t\leq C$ and $x>C^*$.
The estimates for $x\leq C^*$ and $C^\prime\leq x/t\leq C$ can be deduced from the absolute continuity property for Bessel processes with different indices. Indeed, from (\ref{Bessel:AC}) we have
\begin{eqnarray*}
x^{\mu-1/2} e^{-c_\mu t}q_x^{(-1/2)}(t)\leq \qmu\leq x^{\mu-1/2}q_x^{(-1/2)}(t)\/,
\end{eqnarray*}
where $c_\mu=\frac{\mu^2-1/4}2>0$.
Moreover, we have
\begin{eqnarray*}
q_x^{(-1/2)}(t) = \lambda \frac{e^{-\lambda^2/2t}}{\sqrt{2\pi t^3}}
\end{eqnarray*}
and $\exp(-c_\mu t) \geq \exp(-c_\mu C^*/C^\prime)$. This ends the proof.
\end{proof}
In the next lemma we show a result which provides a satisfactory estimate when $t$ is large relative to $x$. This is done under some additional assumption on $\mu$.
\begin{lem}\label{qt:estimate:infty1}
{Suppose that $\mu-1/2\in \N$. We have the following expansion
$$ \qmu = \frac{(x^{2\mu}-1)}{{\Gamma(\mu)2^{\mu}}} e^{-\lambda^2/2t} \frac{1}{t^{\mu+1}}(1+ E(x, t)).$$
There is a constant $c>0$ such that for $t>0$,
$$|E(x, t)|\le c \frac xt.$$}
\end{lem}
\begin{proof} We use the following result proved in Lemma 4.4 of \cite{BR:2006}.
Let $l=\mu-1/2$.
Then
\begin{eqnarray*}
\lim_{t\to\infty} t^{l+1} \int_0^\infty w_\lambda(v)
\left(e^{-\kappa/2t}-\sum_{0\leq j\leq l} (-1)^j\frac{1}{j!}
\left(\frac\kappa {2t}\right)^j\right)dv
\end{eqnarray*}
\begin{eqnarray} \label{cm}
= \frac{(-1)^{l+1}}{2^{l+1} (l+1)!}\int_0^\infty\kappa^{l+1} w_\lambda(v)\ dv=C_{l+1}(x)>0\/.
\end{eqnarray}
Let
\begin{eqnarray*}
H(\lambda,t)&=&\int_0^\infty w_\lambda(v)
\left(e^{-\kappa/2t}-\sum_{0\leq j\leq l+1} (-1)^j\frac{1}{j!}
\left(\frac\kappa {2t}\right)^j\right)dv.
\end{eqnarray*}
Using (\ref{cm}) we may write
\begin{eqnarray*}
\lambda \frac{e^{-\lambda^2/2t} }{ \sqrt{2\pi t} } H(\lambda,t)&=& \lambda \frac{e^{-\lambda^2/2t} }{ \sqrt{2\pi t} }\int_0^\infty w_\lambda(v)
\left(e^{-\kappa/2t}-\sum_{0\leq j\leq l+1} (-1)^j\frac{1}{ j!}
\left(\frac\kappa {2t}\right)^j\right)dv\\&=&\lambda \frac{e^{-\lambda^2/2t} }{ \sqrt{2\pi t} }\int_0^\infty w_\lambda(v)
\left(e^{-\kappa/2t}-\sum_{0\leq j\leq l} (-1)^j\frac{1}{j!}
\left(\frac\kappa {2t}\right)^j\right)dv \\
&&- \lambda \frac{e^{-\lambda^2/2t} }{ \sqrt{2\pi t} }C_{l+1}t^{-l-1}\\
&=& \qmu- \lambda \frac{e^{-\lambda^2/2t} }{ \sqrt{2\pi t} }C_{l+1}t^{-l-1},
\end{eqnarray*}
where we applied (\ref{rep3}) in the last step.
Observe that $$ \kappa^{l+2}\le c( \lambda^{l+2}v^{l+2}+v^{2l+4}),$$
for some constant $c$. Next,
$$\left|e^{-\kappa/2t} - \sum_{0\leq j\leq l+1} (-1)^j\frac{1}{ j!}
(\frac\kappa {2t})^j \right| \le \left(\frac\kappa {2t}\right)^{l+2}\le c\left( \frac{\lambda^{l+2}v^{l+2}+v^{2l+4}}{t^{l+2}}\right),$$
which together with the estimate (see Lemma \ref{w1:estimate:lemma} in Appendix)
$$ |w_{\lambda}(v)|=|w_{1,\lambda}(v)|\le c x^{\mu-3/2} e^{-\theta_\mu v},$$
leads to the following bound for
$H(\lambda,t)$:
\begin{eqnarray}\label{H:estimate}
| H(\lambda,t)| \le
\int_0^\infty \left(\frac\kappa {2t}\right)^{l+2}|w_\lambda(v)|dv\le c x^{\mu-3/2} \frac{(\lambda^{l+2}+1)}{t^{l+2}}\approx
\frac{x^{2\mu}}{t^{l+2}}.
\end{eqnarray}
To complete the proof we need
to find the constant $C_{l+1}$. Let $T_0^{(-\mu)}$ denote the hitting time of $0$ if we start the process from $x$. Due to the strong Markov property and the scaling property we obtain the following equality of the distributions:
$$T_0^{(-\mu)}{\stackrel{d}{=}}\frac1 {x^2}T_0^{(-\mu)} + T_1^{(-\mu)},$$
where $T_0^{(-\mu)}$ and $ T_1^{(-\mu)}$ are independent. It follows that
$${P_x^{(-\mu)}}(T_0^{(-\mu)}>t)\cong {P_x^{(-\mu)}}(T_0^{(-\mu)}>x^2t) +{P_x^{(-\mu)}}(T_1^{(-\mu)}>t),\ t\to \infty.$$
{Note that by the result of Getoor and Sharpe \cite{GetoorSharpe:1979}} we know that $t^{\mu}{P_x^{(-\mu)}}(T_0^{(-\mu)}>t)\cong\frac {x^{2\mu}}{\Gamma(\mu+1)2^{\mu}}$, which implies that
$$t^{\mu}{P_x^{(-\mu)}}(T_1^{(-\mu)}>t)\cong\frac {x^{2\mu}-1}{\Gamma(\mu+1)2^{\mu}}.$$
Fom (\ref{H:estimate}) and
\begin{eqnarray*}
\qmu &=&
\lambda \frac{e^{-\lambda^2/2t} }{ \sqrt{2\pi t} }C_{l+1}t^{-l-1}+\lambda \frac{e^{-\lambda^2/2t} }{ \sqrt{2\pi t} } H(\lambda,t)
\end{eqnarray*}
we infer
$$t^{\mu}{P_x^{(-\mu)}}(T_1^{(-\mu)}>t)\cong \frac{\lambda }{\mu \sqrt{2\pi } }C_{l+1}.$$
This in turn shows that
$C_{l+1}(x)=\frac {\sqrt{2\pi}\mu(x^{2\mu}-1)}{\lambda\Gamma(\mu+1)2^{\mu}}=\frac {\sqrt{2\pi}(x^{2\mu}-1)}{\lambda\Gamma(\mu)2^{\mu}}$
and completes the proof.
\end{proof}
\begin{rem} Let $l= \mu-1/2\in\N$ and $k\in \N$. Since all moments of $\kappa$ with respect to $w_\lambda(v)dv$ exist we can write
$$ | {\qmu}- \lambda \frac{e^{-\lambda^2/2t} }{ \sqrt{2\pi t} }\sum_{i=1}^k \frac{C_{l+i}}{t^{l+i}}|\le c\lambda \frac{e^{-\lambda^2/2t} }{ \sqrt{2\pi t} }\frac{x^{2\mu+k-1}}{t^{l+k+1}}, $$
for some constant $c$, { depending only on $\mu$ and $k$}, where the constants $C_{l+i}= C_{l+i}(x)$ can be found by similar considerations as above.
\end{rem}
\begin{lem} Let $\mu-1/2\notin \N$ and let $l=[\mu+1/2]$. There are constants $c_1, c_2, c_3$ depending only on $\mu$ such that
\label{qt:estimate:infty2}
$$c_2\frac{\lambda x^{2\mu-1}} { t^{\mu+1}} e^{-\lambda^2/2t}\left( 1-c_3\left(\frac{x} { t}\right)^{l-\mu+1/2}\right) \le q^{(-\mu)}_x(t)\le c_1 \frac{\lambda x^{2\mu-1}} { t^{\mu+1}} e^{-\lambda^2/2t},$$
for $t>x>1$.
Note that $l-\mu+1/2>0$.
\end{lem}
\begin{proof}
Applying (\ref{rep3}) we have
\begin{eqnarray*
\qmu& =& \lambda \frac{e^{-\lambda^2/2t} }{ \sqrt{2\pi t} }
\int_0^\infty \left(e^{-\kappa/2t} - \sum_{0\leq j\leq l} (-1)^j\frac{1}{ j!}
\(\frac\kappa {2t}\right)^j \)w_\lambda(v)dv\\
& =& \lambda \frac{e^{-\lambda^2/2t} }{ \sqrt{2\pi t} }
\int_0^\infty \left(e^{-\kappa/2t} - \sum_{0\leq j\leq l} (-1)^j\frac{1}{ j!}
\(\frac\kappa {2t}\right)^j \)w_{\lambda,1}(v)dv\\
&+& \lambda \frac{e^{-\lambda^2/2t} }{ \sqrt{2\pi t} }
\int_0^\infty \left(e^{-\kappa/2t} - \sum_{0\leq j\leq l} (-1)^j\frac{1}{ j!}
\(\frac\kappa {2t}\right)^j \)w_{\lambda,2}(v)dv \\
&=& q^{(-\mu)}_{x,1}(t)+ q^{(-\mu)}_{x,2}(t)
.
\end{eqnarray*}
The upper estimate of $q^{(-\mu)}_{x,1}(t)$ is obtained using almost the same arguments as in the proof of (\ref{H:estimate}) in Lemma \ref{qt:estimate:infty1}. The resulting bound is of the following form:
\begin{eqnarray}
|q^{(-\mu)}_{x,1}(t)|\le c x^{2\mu-1}\lambda \frac{e^{-\lambda^2/2t} }{ t^{\mu+1} }\left(\frac{x}{t}\right)^{l-\mu+1/2}.\label{q1:estimate}
\end{eqnarray}
Next, we deal with $q^{(-\mu)}_{x,2}(t)$.
Observing that
$$ e^{-\kappa/2t} - \sum_{0\leq j\leq l} (-1)^j\frac{1}{ j!}
\(\frac\kappa {2t}\right)^j \approx(-1)^{l+1} \frac {\kappa^{l+1}}{t^l(\kappa +t)}$$
and using Lemma \ref{w2:estimate:lemma} from Appendix, where the estimate of $w_{\lambda,2}(v)$ is provided, we have
\begin{eqnarray*
q^{(-\mu)}_{x,2}(t)
&\approx&\lambda \frac{e^{-\lambda^2/2t} }{ t^{l+1/2} }
\int_0^\infty \frac {x^{2\mu-1}}{(v+1)^{\mu+3/2}(v+x)^{\mu+1/2}}\frac {\kappa^{l+1}}{(\kappa +t)}dv.
\end{eqnarray*}
We need to effectively estimate the integral
$$ J(t,x)=\int_0^\infty \frac {x^{2\mu-1}}{(v+1)^{\mu+3/2}(v+x)^{\mu+1/2}}\frac {\kappa^{l+1}}{(\kappa +t)}dv.$$
Using the folowing change of variables
$\frac {\kappa}t= s$ we obtain
$\frac {\kappa+v^2}{v t} dv= ds,$ which yields
\begin{eqnarray}\label{change}\frac { dv} v\le \frac { ds} s\le 2\frac { dv} v. \end{eqnarray}
Assume that $x>2$.
Thus, $\kappa=2\lambda v+v^2\approx v(v+x)$ and
$$ J(t,x)= \int_0^\infty \frac {x^{2\mu-1}(v+x)^{l-\mu+1/2}v^{l+1}}{(v+1)^{\mu+3/2}}\frac {1}{(\kappa +t)}dv=J_1(t,x)+J_2(t,x),$$
where
$$ J_1(t,x)= \int_0^1\frac {x^{2\mu-1}(v+x)^{l-\mu+1/2}v^{l+1}}{(v+1)^{\mu+3/2}}\frac {1}{(\kappa +t)}dv\approx
x^{\mu+l-1/2}\int_0^1\frac {v^{l+1}}{(\kappa +t)}dv$$
and
$$ J_2(t,x)= \int_1^\infty\frac {x^{2\mu-1}(v+x)^{l-\mu+1/2}v^{l+1}}{(v+1)^{\mu+3/2}}\frac {1}{(\kappa +t)}dv\approx
x^{2\mu-1}\int_1^\infty\frac {\kappa^{l-\mu+1/2}}{v(\kappa +t)}dv.$$
Applying the above change of variables and (\ref{change}) we obtain
$$J_2(t,x)\approx x^{2\mu-1} \int_1^\infty\frac {\kappa^{l-\mu+1/2}}{v(\kappa +t)}dv\approx x^{2\mu-1}\int_{(1+x)/t}^\infty\frac {(st)^{l-\mu+1/2}}{s(s+1)t}ds=x^{2\mu-1}\frac {t^l} {t^{\mu+1/2}}\int_{(1+x)/t}^\infty \frac{s^{l-\mu-1/2}}{s+1}ds.$$
Observing that $\int_{0}^\infty \frac{s^{l-\mu-1/2}}{s+1}ds<\infty$ we arrive at
$$J_2(t,x)\approx x^{2\mu-1} \frac {t^l} {t^{\mu+1/2}},\quad t>x.$$
The first integral $ J_1(t,x)$ we trivially estimate
$$ J_1(t,x) \le c
x^{2\mu-l}\left(\frac {x}{t}\right)^{l+1/2-\mu}t^{l-1/2-\mu}\le c J_2(t,x).$$
This yields the following estimate
\begin{eqnarray}\label{integral1}
J(t,x)\approx x^{2\mu-1} \frac {t^l} {t^{\mu+1/2}},\quad t>x.\end{eqnarray}
Next, assume that $x\le 2$.
Thus, $\kappa\approx v^2,\ v\ge 1$, and
$$ J(t,x)\approx \int_0^\infty \frac {1}{(v+1)^{2\mu+2}}\frac {(\lambda v)^{l+1}+ v^{2(l+1)} }{(\kappa +t)}dv= J_3(t,x)+ J_4(t,x),$$
where $$ J_4(t,x)= \int_1^\infty \frac {1}{(v+1)^{2\mu+2}}\frac {(\lambda v)^{l+1}+ v^{2(l+1)} }{(\kappa +t)}dv\approx \int_1^\infty \frac {v^{2(l-\mu)} }{(v^2 +t)}dv \approx \frac {t^l} {t^{\mu+1/2}}.$$
Clearly
$$ J_3(t,x)= \int_0^1 \frac {1}{(v+1)^{2\mu+2}}\frac {(\lambda v)^{l+1}+ v^{2(l+1)} }{(\kappa +t)}dv\le c\frac 1t\le c J_4(t,x)$$
if $t>1$.
Obviously this implies that (\ref{integral1}) holds in the case $x\le2$.
Using this estimate we finally obtain that
\begin{eqnarray
q^{(-\mu)}_{x,2}(t)
&\approx&\lambda \frac{e^{-\lambda^2/2t} }{ t^{l+1/2} }
J(t,x)\nonumber\\
&\approx& \lambda x^{2\mu-1} \frac{e^{-\lambda^2/2t}} { t^{\mu+1}}, \quad t>x>1.\label{q2:estimate} \end{eqnarray}
A combination of (\ref{q1:estimate}) and (\ref{q2:estimate}) ends the proof.
\end{proof}
\begin{lem} Let $\mu=0$.
\label{qt:zero:estimate:infty}
For $t>2x$,
\label{qt:estimate:large}
$$ q_x^{(0)}(t) \approx \frac{\lambda} x\frac{e^{-\lambda^2/2t} }{ t }
\frac {1+\log x}{(1+\log \frac tx)(1+\log t)}. $$
\end{lem}
\begin{proof}
Recalling the representation (\ref{rep1}) for $\qmu$ and observing that $e^{-\kappa/2t} - 1\approx \frac {-\kappa}{(\kappa +t)}$ we have (note that $w_\lambda(v)\le 0$)
\begin{eqnarray*}q_x^{(0)}(t) &=& \lambda \frac{e^{-\lambda^2/2t} }{ \sqrt{2\pi t} }
\(x^{-1/2}/2t +
\int_0^\infty \(e^{-\kappa/2t} - 1\) w_\lambda(v)dv \)\\
&\approx& \lambda \frac{e^{-\lambda^2/2t} }{ \sqrt{2\pi t} }
\(x^{-1/2}/2t +
\int_0^\infty \frac {\kappa}{(\kappa +t)}(-w_\lambda(v))dv \)
.\end{eqnarray*}
Hence, it is enough to estimate the integral
$$J(t,x)=\int_0^\infty \frac {\kappa}{(\kappa +t)}(-w_\lambda(v))dv.$$
We start with the case $x>2$. Due to Lemma \ref{w:muzero:estimate:lemma} (see Appendix) we have
$$ -w_\lambda(v)
\approx\left\{
\begin{array}{ll}
\frac{1}{x^{3/2}}, & \hbox{$v<3/2$,} \\
\frac{1}{x^{3/2}v^{3/2}\log v}, & \hbox{$3/2 \leq v<x$,}\\
\frac{\log x}{xv^2\log^2v}, &\hbox{$v\ge x>2$.}\end{array}
\right.$$
We write
\begin{eqnarray*
J(t,x)&=& \(\int_0^2 + \int_2^x + \int_x^\infty\) \frac {\kappa}{(\kappa +t)}(-w_{\lambda}(v))dv
\\
&=& J_1(t,x)+J_2(t,x)+ J_3(t,x).
\end{eqnarray*}
Note that
$$ \frac {\kappa}{(\kappa +t)}
\approx\left\{
\begin{array}{ll}
\frac {vx}{(vx +t)}, & \hbox{$v<x$,} \\
\frac {v^2}{(v^2 +t)}, & \hbox{$ v\ge x$,}\end{array}
\right.$$
which implies
$$ J_1(t,x)\approx \int_0^{3/2} \frac{1}{x^{3/2}}\frac {\kappa}{(\kappa +t)}dv\approx \int_0^{3/2} \frac{1}{x^{3/2}}\frac {vx}{(vx +t)}dv\le \frac{3}{2x^{1/2}t}, $$
$$ J_2(t,x)\approx \int_{3/2}^x \frac{1}{x^{3/2}v^{3/2}\log v}\frac {\kappa}{(\kappa +t)}dv\approx \int_{3/2}^x \frac{1}{x^{3/2}v^{3/2}\log v}\frac {vx}{(vx +t)}dv,$$
$$ J_3(t,x)(t,x)\approx \int_x^\infty \frac{\log x}{xv^2\log^2v}\frac {\kappa}{(\kappa +t)}dv\approx \int_x^\infty \frac{\log x}{xv^2\log^2v}\frac {v^2}{(v^2 +t)}dv.$$
Assume that $2x<t<x^2$. First, we deal with
\begin{eqnarray*} J_2(t,x)&\approx& \frac{1}{x^{1/2}}\int_{3/2}^{t/x}\frac{1} {v^{1/2}\log v}\frac {1}{(vx +t)}dv+\frac{1}{x^{1/2}}\int_{t/x}^x\frac{1} {v^{1/2}\log v}\frac {1}{(vx +t)}dv\\
&\approx& \frac{1}{x^{1/2}t}\int_{3/2}^{t/x}\frac{1} {v^{1/2}\log v}dv+\frac{1}{x^{3/2}}\int_{t/x}^x\frac{1} {v^{3/2}\log v}dv\\&\approx& \frac{1}{x^{1/2}t}\int_{3/2}^{t/x}\frac{1} {v^{1/2}\log v}dv+\frac{1}{x^{3/2}}\int_{t/x}^x\frac{1} {v^{3/2}\log v}dv.\end{eqnarray*}
We have
$$
\int_{3/2}^{t/x}\frac{1} {v^{1/2}\log v}dv\approx \sqrt{t/x}\frac 1{\log \frac tx}
$$
and
$$
\int_{t/x}^x\frac{1} {v^{3/2}\log v}dv\le 2 \sqrt{x/t}\frac 1{\log \frac tx},
$$
which shows that
$$ J_2(t,x)\approx \frac 1{x\sqrt{t}}\frac 1{\log \frac tx}.$$
Next,
$$ J_3(t,x)(t,x)\approx \int_x^\infty \frac{\log x}{x\log^2v}\frac {1}{(v^2 +t)}dv\approx \frac{\log x}x \int_x^\infty \frac{1}{v^2\log^2v}dv\approx \frac 1{ x^2\log x}\/. $$
Combining all the estimates we see that for $2x<t<x^2$ we have
$$ J(t,x)\approx J_2(t,x)\approx \frac 1{x\sqrt{t}}\frac 1{\log \frac tx}.$$
Next, we assume that $t>x^2$.
\begin{eqnarray*} J_3(t,x)(t,x)&\approx& \int_x^{\sqrt{t}} \frac{\log x}{x\log^2v}\frac {1}{(v^2 +t)}dv+ \int_{\sqrt{t}}^\infty \frac{\log x}{x\log^2v}\frac {1}{(v^2 +t)}dv\\
&\approx& \frac{\log x}{xt}\int_x^{\sqrt{t}} \frac{1}{\log^2v}dv+ \frac{\log x}{x}\int_{\sqrt{t}}^\infty \frac{1}{v^2\log^2v}dv. \end{eqnarray*}
Observe that
$$ \int_x^{\sqrt{t}} \frac{1}{\log^2v}dv\le \int_2^{\sqrt{t}} \frac{1}{\log^2v}dv\approx \frac{\sqrt{t}}{\log^2t}$$ and
$$\int_{\sqrt{t}}^\infty \frac{1}{v^2\log^2v}dv \approx \frac{1}{\sqrt{t}\log^2t}.$$
As a consequence we obtain
$$ J_3(t,x)(t,x)\approx \frac 1{x\sqrt{t}}\frac {\log x}{\log^2 t}.$$
Next,
\begin{eqnarray*} J_2(t,x)&\approx& \frac{1}{x^{1/2}}\int_{3/2}^x\frac{1} {v^{1/2}\log v}\frac {1}{(vx +t)}dv\\
&\approx& \frac{1}{x^{1/2}t}\int_{3/2}^{x}\frac{1} {v^{1/2}\log v}dv\\&\approx& \frac{1}{t\log x}\le C\frac 1{x\sqrt{t}}\frac {\log x}{\log^2 t},\ x^2\le t.\end{eqnarray*}
Recall that
$$ J_1(t,x)\le \frac{3}{2x^{1/2}t}. $$
Hence, in this case, it is easily seen that the integral $ J_3(t,x)$ dominates and
$$ J(t,x)\approx J_3(t,x)(t,x)\approx \frac 1{x\sqrt{t}}\frac {\log x}{\log^2 t},\ x^2\le t.$$
Summarizing all the estimates obtained for $ J(t,x)$ we have that for $4<2x<t<x^2$,
\begin{eqnarray*}
q_x^{(0)}(t) &=& \lambda \frac{e^{-\lambda^2/2t} }{ \sqrt{2\pi t} }
\(x^{-1/2}/2t +
\int_0^\infty \(e^{-\kappa/2t} - 1\) w_\lambda(v)dv \)\\
&\approx& \lambda \frac{e^{-\lambda^2/2t} }{ \sqrt{2\pi t} }
\(x^{-1/2}/2t + \frac 1{x\sqrt{t}}\frac 1{\log \frac tx}\)\\
&\approx& \frac{e^{-\lambda^2/2t} }{ t }
\frac 1{\log \frac tx}\\
&\approx&
\frac{\lambda} x\frac{e^{-\lambda^2/2t} }{ t }
\frac {1+\log x}{(1+\log \frac tx)(1+\log t)},
\end{eqnarray*}
while for $t>x^2$,
\begin{eqnarray*}
q_x^{(0)}(t) &=& \lambda \frac{e^{-\lambda^2/2t} }{ \sqrt{2\pi t} }
\(x^{-1/2}/2t +
\int_0^\infty \(e^{-\kappa/2t} - 1\) w_\lambda(v)dv \)\\
&\approx& \lambda \frac{e^{-\lambda^2/2t} }{ \sqrt{2\pi t} }
\(x^{-1/2}/2t + \frac 1{x\sqrt{t}}\frac {\log x}{\log^2 t}\)\\
&\approx& \frac{e^{-\lambda^2/2t} }{ t }
\frac {\log x}{\log^2 t} \\
&\approx& \frac{\lambda} x\frac{e^{-\lambda^2/2t} }{ t }
\frac {1+\log x}{(1+\log \frac tx)(1+\log t)}.
\end{eqnarray*}
This completes the proof in the case $x>2$.
Finally, assume that $1<x\le 2\le t$. By Lemma \ref{w:muzero:estimate:lemma} (see Appendix) we have
$$ -w_\lambda(v)
\approx\left\{
\begin{array}{ll}
1, & \hbox{$v\le 2$,} \\
\frac{1}{v^2(\log^2 v+1)}, & \hbox{$v>2 $.}\\
\end{array}
\right.$$
Thus, using the fact that for such range of $x$ we have
\begin{eqnarray*}
1-e^{-\kappa/2t} \approx \frac{\kappa}{\kappa+t}\approx \frac{v^2}{v^2+t}
\end{eqnarray*}
we obtain
\begin{eqnarray*}
q_x^{(0)}(t) &\approx& \lambda\frac{e^{-\lambda^2/2t}}{\sqrt{t}}\left(\frac{1}{x^{1/2}t}+\int_0^2\frac{v^2}{v^2+t}dv+\int_2^\infty \frac{dv}{(v^2+t)(\log^2v+1)} \right)\/.
\end{eqnarray*}
Obviously, we have
\begin{eqnarray*}
\int_0^2\frac{v^2}{v^2+t}dv \approx \frac{1}{t}\
\end{eqnarray*}
and
\begin{eqnarray*}
\int_2^\infty \frac{dv}{(v^2+t)(\log^2v+1)} = \frac{1}{\sqrt{t}}\int_{\frac{2}{\sqrt{t}}}^\infty \frac{1}{(s^2+1)}\frac{ds}{(\log(s)+\log{\sqrt{t}})^2+1}\approx \frac{1}{\sqrt{t}\log^2 t}\/.
\end{eqnarray*}
Hence, for $1<x\le 2\le t$, we have
\begin{eqnarray*}
q_x^{(0)}(t)&\approx& \lambda\frac{e^{-\lambda/4t}}{t\log^2 t} \approx \lambda\frac{e^{-\lambda/4t}}{t}\frac{1+\log x}{\log^2 t} \\ &\approx&\frac{\lambda} x\frac{e^{-\lambda^2/2t} }{ t }
\frac {1+\log x}{(1+\log \frac tx)(1+\log t)}.
\end{eqnarray*}
The proof is completed.
\end{proof}
\section{Applications}
\subsection{Survival probabilities of killed Bessel process}
In this subsection we introduce uniform estimates for the survival probabilities of Bessel process killed when exiting the half-line $(1,\infty)$.
The theorem below is formulated for processes with non-negative indices, however due to (\ref{BesselTime:indices}) we can easily derive the corresponding result for strictly negative indices.
\begin{thm}
Let $\mu>0$. Then, for every $t\geq 0$ and $x>1$, we have
\begin{eqnarray*}
\textbf{P}_x^{(\mu)}(t<T^{(\mu)}_1<\infty) \approx \frac{x-1}{\sqrt{x\wedge t}+x-1}{\frac{1}{t^{\mu}+x^{2\mu}}}\/.
\end{eqnarray*}
Moreover, for every $t\geq 0$ and $x>1$, we have
\begin{eqnarray*}
\textbf{P}_x^{(0)}(T^{(0)}_1>t) &\approx& 1\wedge\frac{\log x}{\log(1+ t^{1/2})}\/.\end{eqnarray*}
\end{thm}
\begin{proof}
Using (\ref{Bessel:hittingtime:estimates}) we get
\begin{eqnarray*}
\textbf{P}_x^{(\mu)}(t<T^{(\mu)}_1\lefteqn{<\infty) = \int_t^\infty q_x^{(\mu)}(s) ds}\\
&\approx& \frac{(x-1)}{x^{2\mu}}\left(x^{\mu-1/2}\int_{t}^{t\vee x} s^{-3/2}e^{-(x-1)^2/2s}ds + x^{2\mu-1}\int_{t\vee x}^\infty s^{-\mu-1}e^{-(x-1)^2/2s}ds \right)\\
&=& J_1(t,x)+J_2(t,x)\/.
\end{eqnarray*}
The integral $J_2(t,x)$ can be estimated as follows
\begin{eqnarray*}
J_2(t,x) &=& \frac{x-1}{x}\int_{t\vee x}^\infty s^{-\mu-1}e^{-(x-1)^2/2s}ds = \frac{(x-1)^{1-2\mu}}{x}\int_0^{\frac{(x-1)^2}{2(t\vee x)}}u^{\mu-1}e^{-u}du\\
&\approx& \frac{(x-1)^{1-2\mu}}{x}\left(\frac{(x-1)^{2\mu}}{(t\vee x)^{\mu}}\wedge 1\right) = \frac{x-1}{x}\left(\frac{1}{(t\vee x)^{\mu}}\wedge\frac{1}{(x-1)^{2\mu}}\right)\\
&\approx& \frac{x-1}{x}\frac{1}{(t\vee x)^{\mu}+(x-1)^{2\mu}}\/.
\end{eqnarray*}
Observe also that for $t\geq x$ we have
\begin{eqnarray*}
\frac{1}{(t\vee x)^{\mu}+(x-1)^{2\mu}} &=& \frac{1}{t^{\mu}+(x-1)^{2\mu}} \approx \frac{1}{t^\mu+x^{2\mu}}\/,\\
\frac{x-1}{x} &\approx& \frac{x-1}{\sqrt{x}-1+x} = \frac{x-1}{\sqrt{x\wedge t}+x-1}\/.
\end{eqnarray*}
The fact that for $t\geq x$ the integral $J_1(t,x)$ vanishes together with the above-given estimates of $J_2(t,x)$ end the proof in that case. If $t<x$, then
\begin{eqnarray*}
J_2(t,x) &\approx& \frac{x-1}{x}\frac{1}{x^{\mu}+(x-1)^{2\mu}} \approx \frac{x-1}{x}\frac{1}{t^{\mu}+x^{2\mu}}\/.
\end{eqnarray*}
Substituting $u=(x-1)^2/(2t)$ we can rewrite $J_1(t,x)$ in the following way
\begin{eqnarray*}
J_1(t,x) &=& \frac{x-1}{x^{\mu+1/2}}\int_{t}^{x} s^{-3/2}e^{-(x-1)^2/2s}ds
= \frac{\sqrt{2}}{x^{\mu+1/2}}\int_{\frac{(x-1)^2}{2x}}^{\frac{(x-1)^2}{2t}}u^{-1/2}e^{-u}du\/,\quad t<x\/.
\end{eqnarray*}
For $x\geq 2$ we have
\begin{eqnarray*}
J_1(t,x) &\leq& \frac{1}{2^{\mu}}\int_{\frac{(x-1)^2}{2x}}^\infty u^{-1/2}e^{-u}du\approx \left(\frac{(x-1)^2}{2x}\right)^{1/2}\exp\left({-\frac{(x-1)^2}{2x}}\right)
\end{eqnarray*}
and it means that $J_1(t,x)$ is dominated by $J_2(t,x)$ in that region. Consequently, we obtain
\begin{eqnarray*}
\textbf{P}_x^{(\mu)}(t<T^{(\mu)}_1<\infty) &\approx& J_2(t,x)\approx \frac{x-1}{x}\frac{1}{t^{\mu}+x^{2\mu}}\approx \frac{x-1}{\sqrt{t}+x-1}\frac{1}{t^{\mu}+x^{2\mu}
\end{eqnarray*}
whenever $t<x$ and $x\geq 2$. Moreover, for $1<x<2$, $t<x$ and $(x-1)>\sqrt{t}$ we get
\begin{eqnarray*}
\frac{1}{2^{\mu}}\int_{1/4}^{1/2} u^{-1/2}e^{-u}du\leq J_1(t,x)\leq \int_{0}^\infty u^{-1/2}e^{-u}du\/.
\end{eqnarray*}
Thus, the integral $J_1(t,x)$ dominates $J_2(t,x)$ and we have
\begin{eqnarray*}
\textbf{P}_x^{(\mu)}(t<T^{(\mu)}_1<\infty) &\approx& J_1(t,x) \approx 1\approx \frac{x-1}{\sqrt{t}+x-1}\frac{1}{t^\mu+u^{2\mu}}\/
\end{eqnarray*}
Finally, for $1<x<2$, $t<x$ and $(x-1)\leq\sqrt{t}$ we get
\begin{eqnarray*}
J_1(t,x)\approx \int_{\frac{(x-1)^2}{2x}}^{\frac{(x-1)^2}{2t}}u^{-1/2}du =\sqrt{2}(x-1)\left(\frac{1}{\sqrt{t}}-\frac{1}{\sqrt{x}}\right)\/,\quad J_2(t,x)\approx x-1\/.
\end{eqnarray*}
Thus we get $1-1/\sqrt{x}\leq (\sqrt{x}-1)/\sqrt{t}\leq \sqrt{2}/\sqrt{t}$ and consequently
\begin{eqnarray*}
\textbf{P}_x^{(\mu)}(t<T^{(\mu)}_1<\infty) &\approx& (x-1)\left(\frac{1}{\sqrt{t}}-\frac{1}{\sqrt{x}}+1\right) \approx \frac{x-1}{\sqrt{t}} \approx \frac{x-1}{\sqrt{t}+x-1}\frac{1}{t^\mu+x^{2\mu}}\/.
\end{eqnarray*}
Now we deal with the case $\mu=0$. We begin with the case of large time $t\geq 2$. We have to consider three cases. For $s\geq t\geq x^2$ we have
\begin{eqnarray*}
q_x^{(0)}(s) \approx \frac \lambda {x s}\frac {1+\log x}{\log^2 s}\/.
\end{eqnarray*}
Consequently
\begin{eqnarray}
\label{DuzeT}
\textbf{P}_x^{(0)}(T^{(0)}_1>t) &\approx& \frac{x-1}x \int_t^\infty \frac {1+\log x}{s\log^2 s}ds\approx
\frac{(x-1)(1+\log x)}{x\log t}
\approx \frac{\log x}{\log(1+ t^{1/2})}\/.
\end{eqnarray}
If $2\leq t\leq x^2$ and additionally $x\geq 2$, using the above estimate, we have
\begin{eqnarray*}
1 \geq \textbf{P}_x^{(0)}(T^{(0)}_1>t) \ge \textbf{P}_x^{(0)}(T^{(0)}_1>x^2) \approx \frac{\log x}{\log(1+ x)}\approx 1\/.
\end{eqnarray*}
Finally, for $2\leq t\leq x^2$ with $x<2$ we get $2<t<4$ and we can write
\begin{eqnarray*}
\textbf{P}_x^{(0)}(T^{(0)}_1>t) &=& \textbf{P}_x^{(0)}(t<T^{(0)}_1\le 10) + \textbf{P}_x^{(0)}(T^{(0)}_1\ge 10)\approx \lambda\/.
\end{eqnarray*}
To justify the last approximation observe that, using Lemma \ref{qt:estimate:zero}, we get
\begin{eqnarray*}
q_x^{(0)}(s) \approx\lambda,\quad 2\le t\le s\le 10\/,
\end{eqnarray*}
and it gives
\begin{eqnarray*}
\textbf{P}_x^{(0)}(t<T^{(0)}_1\le 10) = \int_t^{10} q_x^{(0)}(s)ds \approx \lambda\/.
\end{eqnarray*}
Moreover, using (\ref{DuzeT}) we also get
\begin{eqnarray*}
\textbf{P}_x^{(0)}(T^{(0)}_1\ge 10) \approx \lambda\/.
\end{eqnarray*}
Combining all cases we obtain that
\begin{eqnarray*}
\textbf{P}_x^{(0)}(T^{(0)}_1>t)
&\approx& 1\wedge\frac{\log x}{\log(1+ t^{1/2})},\quad t\ge 2 \/.
\end{eqnarray*}
In the case of small times $t\le 2$ and $1<x<2$, by Lemma \ref{qt:estimate:zero}, we have
\begin{eqnarray*}
q_x^{(0)}(s) \approx{\lambda} \dfrac{e^{-\lambda^2/2s} }{ s^{3/2} }, \quad t\le s\le 10\/.
\end{eqnarray*}
and thus
\begin{eqnarray*}
\textbf{P}_x^{(0)}(t<T^{(0)}_1\le 10)\approx \int_t^{10} {\lambda} \dfrac{e^{-\lambda^2/2s} }{ s^{3/2} }ds\approx 1\wedge \frac\lambda {t^{1/2}}
\end{eqnarray*}
Observe also that $\textbf{P}_x^{(0)}(T^{(0)}_1\ge 10)\approx \lambda$ by our previous estimates in the case of large times. Hence
\begin{eqnarray*}
\textbf{P}_x^{(0)}(T^{(0)}_1>t) &\approx& 1\wedge \frac\lambda {t^{1/2}}.
\end{eqnarray*}
Finally, for $t\le 2$ and $x>2$, using the Markov property, one can easily obtain that
\begin{eqnarray*}
\textbf{P}_x^{(0)}(T^{(0)}_1>t)\ge \textbf{P}_2^{(0)}(T^{(0)}_1>2)\approx 1\/.
\end{eqnarray*}
Again combining all the cases we easily obtain
\begin{eqnarray*}
\textbf{P}_x^{(0)}(T^{(0)}_1>t) &\approx& 1\wedge\frac \lambda{t^{1/2}}\approx 1\wedge\frac{\log x}{\log(1+ t^{1/2})}, \quad t\le 2 .
\end{eqnarray*}
This ends the proof. \end{proof}
\subsection{Poisson kernel for hyperbolic Brownian motion with drift}
Let us consider a half-space model of $n$-dimensional real hyperbolic space
\begin{eqnarray*}
\H^n = \{(y_1,\ldots,y_{n-1},y_n)\in\R^n: y_n>0\}
\end{eqnarray*}
with {Riemannian} metric
\begin{eqnarray*}
ds^2 = \frac{dy_1^2+\ldots+dy_{n-1}^2+dy_n^2}{y_n^2}\/.
\end{eqnarray*}
The hyperbolic distance $d_{\H^n}(y,z)$ is given by
\begin{eqnarray*}
\cosh d_{\H^n}(y,z) = 1+\frac{|y-z|^2}{{2y_nz_n}}\/,\quad y,z\in\H^n\/.
\end{eqnarray*}
The Laplace-Beltrami operator associated with the metric is given by
\begin{eqnarray*}
\Delta_{\H^n} = y_n^2\sum_{i=1}^n \dfrac{\partial^2}{\partial y_i^2}-(n-2)y_n\dfrac{\partial}{\partial y_n}\/.
\end{eqnarray*}
For every $\mu> 0$, we also define the following operator
\begin{eqnarray*}
\Delta_\mu = \Delta_{\H^n}-(2\mu-n+1)y_n\frac{\partial}{\partial y_n} = y_n^2\sum_{i=1}^n \dfrac{\partial^2}{\partial y_i^2}-(2\mu-1)y_n\dfrac{\partial}{\partial y_n}\/.
\end{eqnarray*}
The hyperbolic Brownian motion (HBM) with drift is a diffusion $Y^{(\mu)}=\{Y_t^{(\mu)},t\geq 0\}$ on $\H^n$ with a generator $\frac12\,\Delta_{\mu}$. For $\mu=\frac{n-1}{2}$ we obtain the standard HBM on $\H^n$ (with $\frac12\, \Delta_{\H^n}$ as a generator).
The structure of the process $Y^{(\mu)}$ starting from $(\tilde{y},y_n)\in\H^n$ can be described in terms of the geometric Brownian motion and integral functional $\Amu$ as follows. Let $\tilde{B}=\{\tilde{B}_t,t\geq 0\}$ be $(n-1)$-dimensional Brownian motion starting from $\tilde{y}\in\R^{n-1}$ independent from a geometric Brownian motion $X^{(-\mu)}$ starting from $y_n>0$. Then we have
\begin{eqnarray}
\label{HBM:structure}
Y^{(\mu)}_t \stackrel{d}{=} (\tilde{B}(A_{y_n}^{(-\mu)}(t)),X^{(-\mu)}_t)\/.
\end{eqnarray}
We consider $D=\{(y_1,\ldots,y_{n-1},y_n)\in\H^n: y_n>1\}$ and the first exit time of $Y^{(\mu)}$ from $D$
\begin{eqnarray*}
\tau_D = \inf\{t\geq 0: Y^{(\mu)}_t\notin D \} = \inf\{t\geq 0: X^{(-\mu)}_t\notin (1,\infty) \} = \tau\/,
\end{eqnarray*}
where $\tau=$ is the first exit time from the set $(1,\infty)$ of a geometric Brownian motion $X^{(-\mu)}$ defined in Preliminaries. We denote by $P^{(\mu)}(y,z)$, $y\in D$ and $z\in\partial D$, the Poisson kernel of $D$, i.e. the density of the distribution of $Y^{{\mu}}_{\tau_D}$, with $Y^{(\mu)}_0 = y_n>1$.
\begin{thm}
\label{Poisson:estimate:thm}
For every $\mu>0$ we have
\begin{eqnarray}
\label{Poisson:estimate}
P^{(\mu)}(y,z) \approx \frac{y_n-1}{|z-y|^{n}}\left(\frac{y_n} {{\cosh} d_{\H^n}(y,z)}\right)^{\mu-1/2}\/,
\end{eqnarray}
where $y=(\tilde{y},y_n)$, $y_n>1$ and $z=(\tilde{z},1)$, $\tilde{z}\in\R^{n-1}$.
\end{thm}
\begin{proof}
Let us denote by
\begin{eqnarray*}
g_t(w) = \frac{\exp(-|w|^2/2t)}{(2\pi t)^{(n-1)/2}}\/,\quad w\in\R^{n-1}
\end{eqnarray*}
the Brownian motion transition density in $\R^{n-1}$, $n=2,3,\ldots$. Using the fact that $\tilde{B}$ and $A^{(-\mu)}_{y_n}(\tau)$ are independent we obtain
\begin{eqnarray*}
P^{(\mu)}(y,z) = \int_0^\infty g_t(\tilde{z}-\tilde{y})q_{y_n}^{(-\mu)}(t) dt\/,\quad y_n>1\/, z\in\R^{n-1}\/.
\end{eqnarray*}
Using the estimates given in (\ref{Bessel:hittingtime:estimates}) we obtain
\begin{eqnarray*}
P^{(\mu)}(y,z) &\approx& \lambda \int_0^\infty e^{-(|\tilde{z}-\tilde{y}|^2+\lambda^2)/2t}\frac{y_n^{2\mu-1}}{t^{\mu-1/2}+y_n^{\mu-1/2}}\frac{dt}{t^{(n+2)/2}}\\
&\approx& \lambda \left(y_n^{\mu-1/2}\int_0^{y_n} e^{-(|\tilde{z}-\tilde{y}|^2+\lambda^2)/2t}\frac{dt}{t^{(n+2)/2}} + y_n^{2\mu-1}\int_{y_n}^\infty e^{-(|\tilde{z}-\tilde{y}|^2+\lambda^2)/2t}\frac{dt}{t^{(n+1)/2+\mu}}\right)\\
&=& \lambda\, y_n^{\mu-n/2-1/2}\left[\rho^{-n/2}\int_\rho^\infty u^{n/2-1}e^{-u}du+\rho^{1/2-\mu-n/2}\int_0^\rho u^{n/2-3/2+\mu}e^{-u}du\right]\/,
\end{eqnarray*}
where $\rho=\dfrac{|\tilde{z}-\tilde{y}|^2+\lambda^2}{2y_n}$, $\lambda=y_n-1$.
Using (\ref{large1}) (see Appendix) we can see that
\begin{eqnarray*}
P^{(\mu)}(y,z) \approx \lambda \frac{y_n^{\mu-1/2}}{(|\tilde{z}-\tilde{y}|^2+\lambda^2)^{n/2}}\/,\quad \frac{2y_n}{|\tilde{z}-\tilde{y}|^2+\lambda^2}\geq 1
\end{eqnarray*}
and
\begin{eqnarray*}
P^{(\mu)}(y,z) \approx \lambda \frac{y_n^{\mu-1/2}}{(|\tilde{z}-\tilde{y}|^2+\lambda^2)^{n/2}}\frac{(2y_n)^{\mu-1/2}}{(|\tilde{z}-\tilde{y}|^2+\lambda^2)^{\mu-1/2}}\/,\quad \frac{2y_n}{|\tilde{z}-\tilde{y}|^2+\lambda^2}< 1\/.
\end{eqnarray*}
Combining both estimates and using the formula for the hyperbolic distance we obtain
\begin{eqnarray*}
P^{(\mu)}(y,z) &\approx& \lambda\frac{y_n^{\mu-1/2}}{(|\tilde{z}-\tilde{y}|^2+\lambda^2)^{n/2}}\left(\frac{1} {1+\frac{|\tilde{z}-\tilde{y}|^2+\lambda^2}{2y_n}}\right)^{\mu-1/2}
= \frac{\lambda}{|z-y|^{n}}\left(\frac{y_n} {\cosh d_{\H^n}(y,z)}\right)^{\mu-1/2}\/.
\end{eqnarray*}
\end{proof}
\begin{rem}
The operator $\Delta_\mu$ is strongly elliptic operator on every bounded (in hyperbolic metric) subset of $\H^n$. Consequently, the hyperbolic Poisson kernels of such set are comparable with Euclidean ones. However, considered set $D$ is unbounded in $\H^n$ and the general comparison results can not be applied. {Besides}, the function $P^{(\mu)}(y,z)$ for $\mu\neq 1/2$ is no longer comparable with Euclidean Poisson kernel of upper half-space and the difference in behavior of those two functions is described by the factor
\begin{eqnarray*}
\left(\frac{y_n} {{\cosh} d_{\H^n}(y,z)}\right)^{\mu-1/2}\/.
\end{eqnarray*}
\end{rem}
\section{APPENDIX}
\subsection{Uniform estimates for some class of integrals}
\begin{lem}
\label{gamma}
For $\nu\ge0$, $0\le a<b$ and $d>0$ we have
\begin{eqnarray}
\label{large1} \int_{a}^b u^{\nu}e^{-du}du \stackrel c \approx
b^\nu\left(\frac{a+\frac 1d}{b+\frac 1d}\right)^\nu e^{-ad}\frac {b-a} {d(b-a)+1},
\end{eqnarray}
where $c=c(\nu)$.
\end{lem}
\begin{proof} Let
$F(\nu, a,b,d)=\int_{a}^b u^{\nu}e^{-u}du$. Since $F(\nu, a,b,d)=d^{-\nu-1} F(\nu, ad,bd,1)$ it is enough to prove the lemma for $d=1$. Assume that $ b\ge 1$. Then $$\int_{a}^b u^{\nu}e^{-u}du= e^{-a}\int_{0}^{b-a} (a+u)^{\nu}e^{-u}du \stackrel c \approx e^{-a}\int_{0}^{b-a} (a^{\nu}+u^{\nu})e^{-u}du \stackrel c \approx e^{-a} (a^{\nu}+1)((b-a)\wedge1),$$ which is an equivalent form of (\ref{large1}) in the case $b\ge d=1$. If $b<1$ then
$$\int_{a}^b u^{\nu}e^{-u}du \stackrel c \approx b^{\nu}(b-a),$$
which is exactly (\ref{large1}) in the case $b<d=1$. Note that in all comparisons above the constant $c$ is dependent only on $\nu$.
\end{proof}
\begin{lem}
\label{logestimate:lemma}
Let $0\le a\le 1$. Then for every $v>0$ we have
\begin{eqnarray}\int_0^a \frac{e^{-vu}u du}{\log^2 u+1}&\approx& \frac{1}{(v+1/a)^2(\log^2 (v+1/a)+1)}.\label{I}\end{eqnarray}
If additionally $av\le 1$ then
\begin{eqnarray}\int_a^1 \frac{e^{-vu}u du}{1-\log u}\approx \frac{1-a}{(v+1)^{3/2}(1+\log (v+1))}.\label{J}\end{eqnarray}
\end{lem}
\begin{proof}
Let $$J(a,v)=\int_0^a \frac{e^{-vu}u du}{\log^2 u+1}.$$
First, assume that $av<2$ then
$$J(a,v)\approx \int_0^a \frac{ u du}{\log^2 u+1}\le \frac{ a^2}{2(\log^2 a+1)}.$$
If additionally $a>1/2$ then $$J(a,v)\approx 1.$$ If $a\le 1/2$ then
$$J(a,v)\ge e^{-2} \int_{a/2}^a \frac{u du}{\log^2 u+1}\ge c \frac{ a^2}{\log^2 a+1},$$
which ends the proof (\ref{I}) in the case $av<2$.
We assume now that $av\ge 2$.
Observe that for every $q ,r \in\R$, from the fact that \\$(qr/\sqrt{r^2+1}-\sqrt{r^2+1})^2\geq 0$ we get
\begin{eqnarray*}
\label{inequality:01}
(q-r)^2+1\geq \frac{q^2}{r^2+1}\/.
\end{eqnarray*}
This implies that $$ \frac{1}{(\log v-\log s)^2+1}\leq \frac{\log^2 s+1}{\log^2 v} $$
Consequently, we obtain
\begin{eqnarray*}
J(a,v) = \frac{1}{v^2}\int_0^{av} \frac{e^{-s}s\,ds}{(\log v-\log s)^2+1}\leq \frac{1}{v^2\log^2 v}\int_0^\infty e^{-s}s(\log^2 s+1)ds\/.
\end{eqnarray*}
On the other hand we get
\begin{eqnarray*}
J(a,v) = \frac{1}{v^2}\int_0^{av} \frac{e^{-s}s\,ds}{(\log v-\log s)^2+1}\ge
\frac{1}{v^2(\log^2v+1)}\int_1^2 e^{-s}s\,ds,
\end{eqnarray*}
which ends the proof (\ref{I}).
Let \begin{eqnarray*}
J(a,v)&=&\int_{a}^1 \frac{e^{-vu}u^{1/2}}{1-\log u}\,du \\
&=& \frac{1}{v^{3/2}}\int_{va}^v \frac{e^{-s}s^{1/2}}{1-\log s+\log v}\,ds\/.
\end{eqnarray*}
If $v<2$ then
\begin{eqnarray}
J(a,v)&\approx&\int_{a}^1 \frac{u^{1/2}}{1-\log u}\,du \approx 1-a,\label{J1}
\end{eqnarray}
which is (\ref{J}) in this case.
Next, we assume that $2\le v\le 1/a$.
Using the fact that for every $r>q>0$ we have $r-q+1\geq r/(q+1)$ we get
\begin{eqnarray*}
\int_{va}^v \frac{e^{-s}s^{1/2}}{1-\log s+\log v}\,ds&\leq& \left(\int_0^1+\int_1^v\right)\frac{e^{-s}s^{1/2}}{1-\log s+\log v}\,ds\\
&\leq& \frac{c}{\log v}\int_0^1 e^{-s}s^{1/2}ds + \frac{1}{\log v}\int_1^v e^{-s}s^{1/2}(1+\log s)ds\leq \frac{c_2}{\log v}\/.
\end{eqnarray*}
Moreover, for $2<v<1/a$ we get
\begin{eqnarray*}
\int_{va}^v \frac{e^{-s}s^{1/2}}{1-\log s+\log v}\,ds&\geq& \int_1^2 \frac{e^{-s}s^{1/2}}{1-\log s+\log v}\,ds\geq \frac{c_1}{\log v}\/.
\end{eqnarray*}
We have just proved that
\begin{eqnarray}
J(a,v)\approx \frac{1}{v^{3/2}\log v},\ 2<v<1/a.\label{J2}
\end{eqnarray} Combining (\ref{J1}) and (\ref{J2}) we complete the proof of (\ref{J}).
\end{proof}
\subsection{Estimates of $w_{1,\,\lambda}$ and $w_{2,\,\lambda}$}
\begin{lem}
\label{w1:estimate:lemma}
There exist constants $c=c(\mu)>0$ and $\theta_\mu>0$ such that
\begin{eqnarray}
\label{w1:estimate:abs}
|w_{1,\,\lambda}(v)|\leq c x^{\mu-3/2}e^{-v\theta_\mu}\/,\quad v>0\/.
\end{eqnarray}
\end{lem}
\begin{proof}
Recall that the set of zeros of the function $K_\mu(z)$ is denoted by $Z=\{z_1,...,z_{k_\mu}\}$.
For every $z_i\in Z$ we have $\Re z_i<0$ and the set $Z$ is finite. Consequently, there exists constant $c_1>0$ such that for every $i=1,\ldots,k_\mu$ we have
$$\left|\frac{K_\mu(xz_i) }{ K_{\mu-1}(z_i)}\right|= \left|\frac{{K_\mu(xz_i)-K_\mu(z_i)} }{ K_{\mu-1}(z_i)}\right|\le c_1(x-1).$$
Moreover, using (\ref{asymp_K_infty}), the constant $c_1$ can be chosen to ensure that for every $x\ge 2$ we have
$$\left|\frac{e^{\lambda z_i}K_\mu(xz_i) }{ K_{\mu-1}(z_i)}\right|\le c_1 \frac1{\sqrt{x}}.$$
Hence there are $c_2>0$ and $\theta_\mu= -\max_i\{\Re z_i\}>0$ such that
$$\left|\frac{z_i e^{\lambda z_i} K_\mu(xz_i)}{ K_{\mu-1}(z_i)} \/ e^{z_i v}\right|\le c_2\frac \lambda{x^{3/2}}e^{-\theta_\mu v}$$
and it gives $$ \left|w_{1,\/\lambda}(v)\right|\le c_3 x^{\mu-3/2} e^{-\theta_\mu v}\/.$$
\end{proof}
Let us define for $u>0$ and $x>1$:
\begin{eqnarray*}
S_\mu(x,u)=I_\mu\(xu\)K_\mu(u)-I_\mu(u)K_\mu\(xu\)\/.
\end{eqnarray*}
The function $S_\mu$ appears in the formula for $w_{2,\lambda}$ and consequently, the uniform estimates of the function $S_\mu$, which are given in the next Lemma, are crucial to get the estimates of the function $w_{2,\lambda}$ given in Lemma \ref{w2:estimate:lemma} for $\mu>0$ and in Lemma \ref{w:muzero:estimate:lemma} in the case $\mu=0$.
\begin{lem}
\label{S1:estimate:lemma}
For $\mu\geq 0$ we have
\begin{eqnarray}
\label{S1:estimate:1}
\frac{\lambda}{x}\frac {K_{\mu}(xu)}{K_{\mu}(u)} \le S_\mu(x,u)\le \lambda\frac{K_{\mu}(u)}{K_{\mu}(xu)}.
\end{eqnarray}
There are constants $c_1$ and $c_2$ such that for $\mu\ge 0$, $1<x<2$ and $u>0$ we thus obtain
\begin{eqnarray}
\label{S1:estimate:2}
c_1 \lambda \, e^{-\lambda\,u} \leq S_\mu(x,u)\leq c_2 {\lambda}\, e^{\lambda\,u}\,.
\end{eqnarray}
There is a constant $c_1$ such that for $\mu>0$, $x>2$ and $u>0$ we have
\begin{eqnarray}
\label{S1:estimate:3}
c_1I_\mu\(xu\)K_\mu(u) \leq S_\mu(x,u)\leq I_\mu\(xu\)K_\mu(u)\,.
\end{eqnarray}
There is a constant $c_1$ such that for $\mu=0$, $x>2$ and $xu>1$ we have
\begin{eqnarray}
\label{S1:estimate:4}
c_1I_0\(xu\)K_0(u) \leq S_0(x,u)\leq I_0\(xu\)K_0(u)\,.
\end{eqnarray}
For $\mu=0$, $x>2$ and $xu<1$ we have
\begin{eqnarray}
\label{S1:estimate:4b}
S_0(x,u)\approx \log x.
\end{eqnarray}
\end{lem}
\begin{proof}
Write for $u>0$
\begin{eqnarray*}
\psi(u) = \frac{I_{\mu}(u)}{K_{\mu}(u)}\,.
\end{eqnarray*}
Then by (\ref{Wronskian}) we have
\begin{eqnarray*}
\psi'(u) = \frac{1}{u}\, \frac{1}{K_{\mu}^2(u)}\,.
\end{eqnarray*}
Writing
\begin{eqnarray*}
S_{\mu}(x,u) = \[\psi(xu)-\psi(u)\]\,K_{\mu}(xu)\,K_{\mu}(u) \,.
\end{eqnarray*}
we obtain from the Lagrange theorem
\begin{eqnarray*}
S_{\mu}(x,u) = \frac{xu-u}{\theta xu}\, \frac{K_{\mu}(xu)\,K_{\mu}(u)}{K_{\mu}(\theta xu)\,K_{\mu}(\theta xu)} \,.
\end{eqnarray*}
The quantity $\theta$ here has the property $1\leq \theta x \leq x$.
This and the monotonicity of the function $K_{\mu}$, give the estimate (\ref{S1:estimate:1}). The estimate (\ref{S1:estimate:2}) is a direct consequence of the limiting behaviour of the function $K_{\mu}$.
To prove (\ref{S1:estimate:3}) and (\ref{S1:estimate:4}) note that
the function $g(x,u)=\frac{\psi(u)}{ \psi(xu)}= \frac{ I_\mu\(u\)K_\mu(xu)}{K_\mu\(u\)I_\mu\(xu\)}$ as a function of $x$ is decreasing. Hence for $ x>2$, $$g(x,u)\le g(2,u)<1, u>0. $$
If $\mu>0$, then the limits at $0$ and $\infty$ of $g(2,u)$ are strictly less then $1$. By continuity
$$\sup_{u>0}g(2,u)= a<1.$$
If $\mu=0$ by the same argument for $ x>2$, $$g(x,u)\le \sup_{v\ge 1/2}g(2,v)<1, u\ge 1/2. $$
If $xu>1$, $u<1/2$ and $x>2$ then
$$g(x,u)= \frac{ I_0\(u\)K_0(xu)}{K_0\(u\)I_0\(xu\)}\le \frac{ I_0\(1/2\)K_0(1)}{K_0\(1/2\)I_0\(1\)}= g(2,1/2) <1.$$
These estimates imply that
$$g(x,u)\le \sup_{v\ge 1/2}g(2,v)=a<1, u>1/x. $$
Hence, in both cases ($\mu=0$ or $\mu>0$) we have for $x>2$ and $xu>1$
$$
S_{\mu}(x,u) = \[1-\frac {\psi(u)}{\psi(xu)}\]\,\psi(xu) K_{\mu}(xu)\,K_{\mu}(u)\ge (1-a)I_{\mu}(xu)\,K_{\mu}(u)\,,
$$
which ends the proof of (\ref{S1:estimate:3}) and (\ref{S1:estimate:4}).
It remains to consider $\mu=0, x>2$ and
$xu<1$. We apply the asymptotics of $K_0$ at $0$. Namely, by (\ref{K0_atzero}) we can write
$$K_0(z)= -\log \frac z2 \, I_0(z)+ A(z), $$where $A(z)\to c>0, z\to 0$. This yields
$$S_{0}(x,u)= I_0(ux)I_0(u)\log x + I_0(ux)A(u)- I_0(u)A(ux)\approx \log x.$$
\end{proof}
\begin{lem}
\label{w2:estimate:lemma}
For $\mu>0$ and $x>1$ we have
\begin{eqnarray}
\label{w2:estimates}
w_{2,\/\lambda}(v)\approx {(-\cos(\pi\mu))}\frac {x^{2\mu-1}}{(v+1)^{\mu+3/2}(v+x)^{\mu+1/2}}\/,\ v>0 \/.
\end{eqnarray}
\end{lem}
\begin{proof}
Let us denote
\begin{eqnarray*}
h(x,u,v)&=& \frac{ S_\mu\(x,u\)}{
\cos^2(\pi\mu) K_\mu^2(u)+(\pi I_\mu(u)+\sin(\pi \mu) K_\mu(u))^2} \/
e^{-\lambda u} e^{-vu} \/u\\
&\approx& \frac{ S_\mu\(x,u\)}{
K_\mu^2(u)+I_\mu^2(u)} \/
e^{-\lambda u} e^{-vu} \/u.
\end{eqnarray*}
Since
$w_{2,\/\lambda}(v)=
-\cos(\pi\mu) \frac{x^\mu }{ \lambda}
\int_0^\infty h(x,u,v) du$, it is enough to estimate $\int_0^\infty h(x,u,v) du$.
which is done below for two cases.
A) Case $1<x<2$.\newline
Suppose that $ 0<u<1$. By Lemma \ref{S1:estimate:lemma},
$S_\mu\(x,u\)\approx \lambda$, thus
\begin{eqnarray*}h(x,u,v)
&\approx& \frac{\lambda }{
K_\mu^2(u)+I_\mu^2(u)} \/
e^{-\lambda u} e^{-vu} \/u\\ &\approx& \lambda u^{2\mu+1}e^{-(v+1)u},\ 0<u<1. \end{eqnarray*}
For $u>1$ we have, by Lemma \ref{S1:estimate:lemma}, $S_\mu\(x,u\)\le c\lambda e^u$, which yields the following upper bound.
\begin{eqnarray*}h(x,u,v)
&\le& c \frac{\lambda e^u }{
K_\mu^2(u)+I_\mu^2(u)} \/
e^{-\lambda u} e^{-vu} \/u\\ &\le& c\lambda u\/e^{-(v+1)u}. \end{eqnarray*}
Applying Lemma \ref{gamma} to the above estimates we arrive at
$$\int_0^1 h(x,u,v)du\approx \frac {\lambda} {(v+1)^{2\mu+2}}$$
and
$$\int_1^\infty h(x,u,v)du\le c \frac {\lambda}{(v+1)^2}e^{-(v+1)}.$$
Combining both integrals we obtain
$$\int_0^\infty h(x,u,v)du\approx \frac {\lambda} {(v+1)^{2\mu+2}},$$
which proves the lemma in the case $1<x<2$.
B) Case $x>2$. \newline
By Lemma \ref{S1:estimate:lemma}, $ S_\mu(x,u)\approx I_\mu(xu)K_\mu\(u\)$, which implies
\begin{eqnarray*} h(x,u,v)
&\approx& \frac{ I_\mu(xu)K_\mu\(u\)}{K_\mu^2(u)+I^2_\mu(u)} \/
e^{-\lambda u} e^{-vu} \/u.
\end{eqnarray*}
Next, using the asymptotics of the Bessel functions, we arrive at
$$h(x,u,v)
\approx\left\{
\begin{array}{ll}
x^\mu u^{2\mu+1}e^{-(v+\lambda)u}, & \hbox{$xu<1$,} \\
x^{-1/2}u^{\mu+1/2}e^{-vu}, & \hbox{$1/x<u<1$,}\\
x^{-1/2}ue^{-(v+2)u}, &\hbox{$u>1$.}\end{array}
\right.$$
To estimate $H(x,v)=\int_0^\infty h(x,u,v)du$ we split the integral into three parts:
\begin{eqnarray*}
H(x,v)&=&\int_0^{1/x} h(x,u,v)du+\int_{1/x}^1 h(x,u,v)du+\int_{1}^\infty h(x,u,v)\\
&=&J_1(x,v)+J_2(x,v)+J_3(x,v).
\end{eqnarray*}
Applying Lemma \ref{gamma} with $a=0, b=1/x$ and $d= v+\lambda$, the first integral can be estimated in the following way:
\begin{eqnarray*}J_1(x,v)&\approx& x^\mu \int_0^{1/x}u^{2\mu+1}e^{-(v+\lambda)u}du\\&\approx& \frac{x^\mu}{(v+\lambda)^{2\mu+1}} \frac {1/x}{1+(v+\lambda)/x} \\
&\approx& \frac{ x^{\mu}}{(v+x)^{2\mu + 2}}.\end{eqnarray*}
Next, we deal with the second integral. Again, by Lemma \ref{gamma} with $a=1/x, b=1$ and $d= v$, we obtain
\begin{eqnarray*} J_2(x,v) &\approx& x^{-1/2} \int_{1/x}^1 u^{\mu+1/2}e^{-vu}du\\
&\approx& x^{-1/2}\left(\frac{\frac1 x +\frac1 v}{1+\frac1 v}\right)^{\mu+1/2} e^{-v/x}\frac {1-1/x}{1+v-v/x}\\ &\approx& x^{-1/2}\left(\frac{\frac1 x +\frac1 v}{1+\frac1 v}\right)^{\mu+1/2} e^{-v/x}\frac {\lambda }{x+\lambda v}\\ &\approx& x^{-1/2}\left(\frac{1+\frac v x }{1+ v}\right)^{\mu+1/2} e^{-v/x}\frac {1 }{ v+1}.
\end{eqnarray*}
The third integral can be estimated for $v>0$ as follows.
$$ J_3(x,v) \approx x^{-1/2} \int_{1}^\infty ue^{-(v+2)u}du \approx x^{-1/2} \frac 1{(v+2)} e^{-(v+2)}. $$
It is clear that $$ J_3(x,v) \le c \frac {x^{\mu}}{(v+1)^{\mu+3/2}(v+x)^{\mu+1/2}}.$$
Next,
$$\frac {(v+1)^{\mu+3/2}(v+x)^{\mu+1/2}} {x^{\mu}} I_1\approx \frac {(v+1)^{\mu+3/2}(v+x)^{\mu+1/2}} {x^{\mu}} \frac{ x^{\mu}}{(v+x)^{2\mu + 2}} = \frac {(v+1)^{\mu+3/2}}{(v+x)^{\mu+3/2}}$$
and
\begin{eqnarray*}\frac {(v+1)^{\mu+3/2}(v+x)^{\mu+1/2}} {x^{\mu}} I_2&\approx& \frac {(v+1)^{\mu+3/2}(v+x)^{\mu+1/2}} {x^{\mu}} x^{-1/2}\left(\frac{1+\frac v x }{1+ v}\right)^{\mu+1/2} e^{-v/x}\frac {1 }{ v+1} \\
&=& (1+ v/x)^{2\mu+1} e^{-v/x}.\end{eqnarray*}
The observation $\frac {(v+1)^{\mu+3/2}}{(v+x)^{\mu+3/2}}+ (1+ v/x)^{2\mu+1} e^{-v/x}\approx 1$ completes the proof.
\end{proof}
\begin{lem}
\label{w:muzero:estimate:lemma}
For $\mu=0$ and $x>1$ we have
\begin{eqnarray*}
-w_\lambda(v) &\approx& \frac{1}{x(v+1)^{3/2}(v+x)^{1/2}}\frac{\log (x+1)}{\log(v+2)(\log(x+1)+\log(v+2))}\/,\ v>0\/.
\end{eqnarray*}
\end{lem}
\begin{proof}
Let us denote
\begin{eqnarray*}
f(x,v) = \frac{1}{x(v+1)^{3/2}(v+x)^{1/2}}\frac{\log (x+1)}{\log(v+2)(\log(x+1)+\log(v+2))}\/.
\end{eqnarray*}
We write
\begin{eqnarray*}
\int_0^\infty \frac{S_0(x,u)\,e^{-\lambda u}}{K_0^2(u)+\pi^2I_0^2(u)}\,e^{-vu} udu &=& \left(\int_0^{1/x}+\int_{1/x}^1+\int_1^\infty\right) \frac{S_0(x,u)\,e^{-\lambda u}}{K_0^2(u)+\pi^2I_0^2(u)}\,e^{-vu} udu\\
&=& J_1(x,v)+J_2(x,v)+J_3(x,v).
\end{eqnarray*}
The estimates of $J_3(x,v)$ are exactly the same as the corresponding estimates proved in Lemma \ref{w2:estimate:lemma}.
Hence, for $x\geq 2$,
\begin{eqnarray}
J_3(x,v) &\approx&
\frac{1}{\sqrt{x}}\frac{e^{-(v+2)}}{(v+2)^2}\/,\quad v>0\/,\label{I31}
\end{eqnarray}
and for $1<x<2,$
\begin{eqnarray}
J_3(x,v) &\le & c\frac{\lambda}{(v+1)^2}e^{-(v+1)}.\label{I32}
\end{eqnarray}
To estimate $J_2(x,v)$ for $x>2$ observe that for $xu\geq 1$ and $u<1$, $S_0(x,u)\approx \dfrac{e^{xu}}{\sqrt{xu}}K_0(u)\approx \dfrac{e^{xu}}{\sqrt{xu}}(1-\log u)$. This follows from the asymptotic expansions for $I_0$ and $K_0$ (see (\ref{I_atzero}) and (\ref{K_atzero})), and Lemma \ref{S1:estimate:lemma}. Thus
\begin{eqnarray}
J_2(x,v) &\approx& \frac{1}{x^{1/2}}\int_{1/x}^1 \frac{K_0(u)e^{u}e^{-vu}u^{1/2}}{K_0^2(u)+\pi^2I_0^2(u)}\,du \approx \frac{1}{x^{1/2}}\int_{1/x}^1 \frac{e^{-vu}u^{1/2}}{1-\log u}\,du \nonumber \\
&\approx& \frac{1}{x^{1/2}}\frac{1-1/x}{(v+1)^{3/2}(1+\log (v+1))},\ v\le x, \label{I21}
\end{eqnarray}
where the last step is a consequence of (\ref{J}) with $a=1/x$. Next,
\begin{eqnarray}
J_2(x,v) &\approx& \frac{1}{x^{1/2}}\int_{1/x}^1 \frac{K_0(u)e^{u}e^{-vu}u^{1/2}}{K_0^2(u)+\pi^2I_0^2(u)}\,du \le c\frac{1}{x^{1/2}}\int_{0}^1 \frac{e^{-vu}u^{1/2}}{1-\log u}\,du \nonumber \\
&\le& c\frac{1}{x^{1/2}}\frac{1}{(v+1)^{3/2}(1+\log (v+1))},\ v>0, \label{I22}
\end{eqnarray}
where, again, the last step is a consequence of (\ref{J}) with $a=0$.
For $x<2$ and $1/x\le u<1$, by Lemma \ref{S1:estimate:lemma}, we have $S_0(x,u)\approx \lambda$ hence
\begin{eqnarray}
J_2(x,v) &\approx& \lambda \int_{1/x}^1 \frac{e^{-vu}u}{K_0^2(u)+\pi^2I_0^2(u)}\,du \approx \lambda \int_{1/x}^1 \frac{e^{-vu}u}{1+\log^2 u}\,du. \label{I24}
\end{eqnarray}
Finally, for $0<u<1/x$ we have, by Lemma \ref{S1:estimate:lemma}, $S_0(x,u)\approx \log x,\ x>0$ and consequently using (\ref{I}), with $a=1/x$, we obtain
\begin{eqnarray}
J_1(x,v) &\approx& \log x\int_0^{1/x} \frac{e^{-\lambda u}e^{-vu}u}{K_0^2(u)+\pi^2I_0^2(u)}\,du \approx \log x\int_0^{1/x} \frac{e^{-vu}u}{1+\log^2 u}\,du \label{I11}.\\
&\approx& \frac{\log x}{(v+x)^2(\log^2 (v+x)+1)}.\label{I12}
\end{eqnarray}
We are now ready to estimate the function $-w_\lambda(v)$. At first we consider $1<x<2$. Taking into account that $\log x\approx \lambda$, using (\ref{I24}) and (\ref{I11}) w arrive at
\begin{eqnarray}
J_1(x,v)+ J_2(x,v) &\approx& \lambda \int_{0}^1 \frac{e^{-vu}u}{1+\log^2 u}\,du \\
&\approx& \frac{\lambda}{(v+1)^2(\log^2 (v+1)+1)}.
\end{eqnarray}
Combining this with (\ref{I32}) we have for $1<x<2$,
\begin{eqnarray*}
-w_\lambda(v) &=& \frac{1}{\lambda}(J_1(x,v)+ J_2(x,v)+ J_3(x,v)) \approx \frac{1}{\lambda}(J_1(x,v)+ J_2(x,v))\\
&\approx& \frac{1}{(v+1)^2(\log^2 (v+1)+1)} \approx f(x,v)\/.
\end{eqnarray*}
Taking into account (\ref{I31}), (\ref{I21}) and (\ref{I12}), we infer that for $0<v<2$ and $x>2$, $J_1(x,v)+ J_2(x,v)\le c J_3(x,v),$ which yields
\begin{eqnarray*}
-w_\lambda(v) = \frac{1}{\lambda}(J_1(x,v)+ J_2(x,v)+ J_3(x,v)) \approx \frac{1}{\lambda}J_3(x,v) \approx \frac{1}{x^{3/2}}\approx f(x,v)\/.
\end{eqnarray*}
For $2 \leq v<x$, by (\ref{I12}), (\ref{I31}) and (\ref{I21}), $J_1(x,v)+ J_3(x,v)\le c J_2(x,v)$. Hence,
\begin{eqnarray*}
-w_\lambda(v) = \frac{1}{\lambda}(J_1(x,v)+ J_2(x,v)+ J_3(x,v)) \approx \frac{1}{\lambda} J_2(x,v) \approx \frac{1}{x^{3/2}v^{3/2}\log v}\approx f(x,v)\/.
\end{eqnarray*}
Finally for $v\ge x>2$, by (\ref{I22}), (\ref{I12}) and (\ref{I31}) we have $J_2(x,v)+J_3(x,v)\leq c J_1(x,v)$ for some constant $c>1$ and consequently
\begin{eqnarray*}
-w_\lambda(v) = \frac{1}{\lambda}(J_1(x,v)+ J_2(x,v)+ J_3(x,v)) \approx \frac{1}{\lambda} J_1(x,v) \approx \frac{\log x}{xv^2\log^2v}\approx f(x,v).
\end{eqnarray*}
The proof is completed.
\end{proof}
|
{
"timestamp": "2011-06-08T02:00:35",
"yymm": "1009",
"arxiv_id": "1009.3513",
"language": "en",
"url": "https://arxiv.org/abs/1009.3513"
}
|
\section{Historique \label{sec:hist}}
It is a tradition of the \textit{Rencontres de Blois} to blend scientific discussions at the highest level with culture in an atmosphere that makes us aware of our heritage, broadly conceived. This splendid chateau and the art and history revealed during the excursion to Chenonceau and Clos Luc\'{e} have enhanced our experience this week. But, in this region so rich in patrimony, there is more: two of the most celebrated historical figures \textit{d'origine Bl\'{e}soise} had connections with physics, and so may be counted among our scientific ancestors!
If you have ventured to the far end of the esplanade, you will have noticed the \textit{Mus\'{e}e de la Magie}~\cite{museemagie}, a monument to the life and work
of Jean-Eug\`{e}ne Robert-Houdin (1805--1871), the most famous magician in all of France and
the father of modern conjuring. Robert-Houdin was the
first magician to appear in formal wear, and the first to use electricity in his act. Part of his
legacy is a posthumous work~\cite{roberth} devoted to magic and ``amusing physics.''
The second famous son of Blois is Denis Papin (1647-1714), the master of steam
power who worked in London with Robert Boyle. In 1679, he invented the \textit{marmite de Papin}~\cite{denisp}, \textit{cocotte minute,} or pressure cooker, a kind of cooking-pot in which arbitrarily tough meat can be rendered soft. Papin thereby established what we recognize today as the Standard Model of English Cuisine.
\section{Arrival of the LHC \label{sec:lhc}}
The signal event of this year in particle physics is the arrival of the Large Hadron Collider at CERN as a research instrument. We celebrate both the performance of the collider itself, described by Lucio Rossi~\cite{rossi}, and the impressive early analyses carried out by the experimental teams. One measure of the machine development is that, at the time we met in Blois in July, the LHC had delivered more than $350\hbox{ nb}^{-1}$ to the ATLAS and CMS detectors, at $3.5\hbox{ TeV}$ per beam. By September 6, the integrated luminosity was an order of magnitude higher~\cite{lhclum}, and the peak luminosity had surpassed $10^{31}\hbox{ cm}^{-2}\hbox{ s}^{-1}$. The goal for the 2010-2011 run is to accumulate $\approx 1\hbox{ fb}^{-1}$ at $\sqrt{s} = 7\hbox{ TeV}$. The current expectation for post-shutdown operation is to reach $\sqrt{s} = 13\hbox{ - }14\hbox{ TeV}$ during 2013, and to progress toward the design luminosity of $10^{34}\hbox{ cm}^{-2}\hbox{ s}^{-1}$. CERN's conception of the future, described by Rolf Heuer~\cite{rolf}, includes options for a high-luminosity LHC, an electron-positron collider, and more. The European strategy for particle physics should be revisited in 2012.
Members of the experimental collaborations have reported here on their experience in commissioning and calibrating the detectors~\cite{LHCcom}, and on some early analyses~\cite{LHCearly}. To note just a few of the interesting results, I cite ALICE's studies of particle production, including measurements of the charged multiplicity~\cite{yves}, CMS measurements of two-particle correlations~\cite{roeck}, ATLAS investigations of high-transverse-momentum jets~\cite{margie}, and the observation of sequential decays of heavy quarks by LHC$b$~\cite{andreas}. It is exciting to hear that LHC$b$ will begin to confront D0's surprising dimuon charge asymmetry~\cite{Abazov:2010hv} at an integrated luminosity of $\approx 100\hbox{ pb}^{-1}$. Many more results were available one week after Blois, at ICHEP 2010 in Paris~\cite{ichepparis}. All of this testifies to the skillful planning and execution by accelerator physicists and experimenters, and also to their dedication and stamina!
For all of this impressive progress, we cannot ignore the fact that the LHC is, for now, operating at only half its design energy. This has implications for the rate at which we might anticipate discoveries, and also for relative advantages of the experimental campaigns at the LHC and Tevatron. Parton luminosities, supplemented by what we know from measurements at the Tevatron and from simulations for the 14-TeV LHC, are a reliable tool for assessing what we can expect~\cite{Quigg:2009gg}. Although every measurement or search is a special case for which we must consider both signals and backgrounds, as a general rule the LHC experiments begin to open terrain not explored by the Tevatron when their integrated luminosity exceeds a few hundred$\hbox{ pb}^{-1}$. Guido Altarelli reviewed the TeV-scale physics setting~\cite{guido} and the prospects for new insights early in the life of the LHC. Among many possibilities, I regard the discovery of a diquark resonance~\cite{Bauer:2009cc} (for which the $pp$ collisions of the LHC offer higher sensitivity than the $\bar{p}p$ collisions of the Tevatron) as not so plausible, but the early observation of a fourth-generation quark~\cite{4thgen,soni} as not so implausible.
\section{The Tevatron in Its Prime\label{sec:teva}}
Meanwhile, across the Atlantic, the Tevatron has never been in better form: the peak luminosity has reached $4 \times 10^{32}\hbox{ cm}^{-2}\hbox{ s}^{-1}$ and more than $9\hbox{ fb}^{-1}$ has been delivered to CDF and D0~\cite{tevlum}, which are reporting new results on many topics. To indicate the breadth of scientific interests, I note a D0 study of $\bar{p}p$ elastic scattering~\cite{d0el}, which manifests the expected shrinkage of the diffraction peak. This is just one among many Tevatron results on the strong interactions~\cite{tevqcd}. More generally on strong interactions, we heard a comprehensive account of QCD by Varelas~\cite{varelas}; an update on the HERA parton distribution functions~\cite{glazov}; and progress reports on investigations of the quark-gluon-plasma~\cite{qgp} and of hot and dense baryonic matter. Holographic techniques informed by gauge-gravity duality open the investigation in the strong-coupling regime of theories that are not QCD, but might have properties in common with QCD~\cite{hashi}.
Among searches, we may mention the $t^\prime$ search from D0~\cite{d0tpr} and the $Z^\prime$ search from CDF~\cite{cdfzp}, which limits $M(Z^\prime_{\mathrm{SM}}) > 1\,071\hbox{ GeV}$ using $4.6\hbox{ fb}^{-1}$. For the evolution of $Z^\prime$ searches from the Tevatron to the LHC, see~\cite{zptevlhc}, and for other new physics searches, see~\cite{tevnp}.
The Tevatron experiments have established an enviable record for precise measurements. The combined Tevatron top-quark mass is now known so precisely, $m_t = (173.3 \pm 0.6\hbox{ (stat)} \pm 0.9\hbox{ (syst)})\hbox{ GeV} = (173.3 \pm 1.1) \hbox{ GeV}$~\cite{shaba}, that it is urgent to confront the question of exactly what quantity is measured~\cite{Smith:1996xz}. We have also heard about new determinations of top-quark properties~\cite{topprop} and electroweak observables~\cite{ewprog}.
Encouraged by the increasing incisiveness of their Higgs-boson searches~\cite{daniela,tevhpar}, the Tevatron collaborations have proposed to extend running beyond the planned October 2011 cutoff, and to continue for three more years, to accumulate $\sim 16\hbox{ fb}^{-1}$ for analysis. At ICHEP, CDF and D0 excluded a standard-model Higgs boson in the range $158\hbox{ GeV} \ltap M_H \ltap 175\hbox{ GeV}$ (and also $100\hbox{ GeV} \ltap M_H \ltap 109\hbox{ GeV}$) at 95\% CL~\cite{ichepH}. The reduced energy of the current LHC run and the LHC shutdown for retrofitting in 2012-2013 also enter into a reassessment of the Tevatron's potential to contribute to the investigation of electroweak symmetry breaking. The centerpiece of the proposal is that at $16\hbox{ fb}^{-1}$ the combined-experiments / combined-channels sensitivity for the standard-model Higgs boson would exceed 3-$\sigma$ ``evidence'' for $100\hbox{ GeV} \ltap M_H \ltap 185\hbox{ GeV}$~\cite{fnalpac}. Although a decision to continue running would be nontrivial because of budgetary constraints and the interaction with Fermilab's future program, the Physics Advisory Committee has responded with considerable enthusiasm~\cite{fnalpacaug}. The verdict rests with the laboratory management and the funding agencies.
\section{Learning to See at the LHC \label{sec:L2C}}
The LHC is beginning to advance the experimental frontier of particle physics to the heart of the TeV scale, where we are confident that we will find new insights into the nature of electroweak symmetry breaking. \textit{We do not know what the new wave of exploration will find.} Precisely because we do not know the answers, it is imperative to look broadly, and this has been the thrust of many plenary talks~\cite{theorists} and contributions in the parallel sessions~\cite{LHCpar}. Along with our conviction that exploration of the 1-TeV scale will give definitive answers about electroweak symmetry breaking, we have good reason to hope that we might also find candidates for the dark matter of the universe and resolve the hierarchy problem.
I believe that we should also take advantage of the opportunity to learn more about the richness of the strong interactions, especially in the realm of ``soft'' particle production that theorists cannot describe by controlled calculations in perturbation theory. I would like to emphasize that the object of initial studies is not merely to tune PYTHIA parameters (which may not have direct physical significance)~\cite{pytune}, as useful as that exercise may be. The first conclusion of the LHC experiments is that the pre-LHC event generators did not perfectly anticipate what was observed.
Experimental studies in the 1970s established the essential features of multiple production at energies up to $\sqrt{s} = 63\hbox{ GeV}$: Feynman scaling, with distinct diffractive and ``multiperipheral'' components, the latter characterized by short-range order in rapidity.
This doesn't mean that we can regard ``soft'' particle production as settled knowledge. Tevatron studies have been informative but not exhaustive, so we can't be sure that what was learned in the 1970s accounts for all the important features at Tevatron energies and beyond. At the highest energies, well into the ($\propto \ln^2{s}$?) growth of the $pp$ total cross section, long-range correlations might show themselves in new ways. {The high density of partons carrying $p_z = 5\hbox{ to }10\hbox{ GeV}$ may give rise to hot spots in the spacetime evolution of the collision aftermath, and thus to thermalization or other phenomena not easy to anticipate from the QCD Lagrangian.} We might anticipate a growing rate of multiple-parton interactions, perhaps involving correlations among partons: the quark-diquark component of the proton might manifest itself in elementary collisions involving diquarks. The $\ln{s}$ expansion of the rapidity plateau softens kinematical constraints in the central region, and the sensitivity to high-multiplicity events (or otherwise rare occurrences) of modern experiments vastly exceeds what could be seen with bubble-chamber statistics.
For all these reasons, I suspect that a few percent of minimum-bias events collected at $\sqrt{s} \gtap 2\hbox{ TeV}$ might display unusual event structures.
\textit{Looking at events} can play an important role~\cite{Quigg:2010nn}, not only to refine our intuition, but also to discover candidate new physics that might become the object of dedicated study in the future. Because I expect the event structure to evolve with increasing collision energy, I would like to see during 2010-2011 \textit{a set of modest dedicated runs at steps in energy, e.g., at} $\sqrt{s} = 0.9, 2, 3.5, 5, 7\hbox{ TeV}$, lightly triggered, to survey the nature of particle production. Now that the essential performance of the detectors has been validated, such a survey would be well worth the disruption it would cause to routine operations and the accumulation of integrated luminosity at $7\hbox{ TeV}$. We should use this first LHC run to learn what we will want to study in depth beginning in 2013.
\section{Neutrinos}
The investigation of neutrino properties and interactions, and the search for extraterrestrial sources of neutrinos, was also well-represented in Blois~\cite{nuprog}. Among new initiatives, which include the start-up of the T2K program and fresh data from ANTARES, we welcome the first $\nu_{\mu} \to \nu_{\tau}$ candidate from OPERA~\cite{Agafonova:2010dc}. This specimen puts a face on the inference that $\nu_{\mu} \leftrightarrow \nu_{\tau}$ mixing is the dominant phenomenon in atmospheric neutrino oscillations~\cite{Ashie:2005ik}. Review talks by Kayser~\cite{boris} and Wojcicki~\cite{stanw} summarized the current status and recent experimental progress.
The MINOS collaboration has reported disappearance results from their first antineutrino run~\cite{minos}; the antineutrino mixing angle and mass-squared difference are in some tension with the corresponding neutrino values. While this may well be a transitory effect of modest statistics, it is worth stretching our minds on possible implications---especially those less radical than CPT violation. For example, it is worth asking whether nonstandard interactions that survive other experimental sieves could give rise to an apparent difference in $(\sin^22\theta,\Delta m^2)$~\cite{belen,Kopp:2010qt}.
A decade of progress in neutrino mixing leaves us with a great many other unanswered questions, including: Do neutrino masses display a normal or inverted spectrum? What is the value of the small mixing angle, $\theta_{13}$?
Are neutrinos their own antiparticles?
How many mass eigenstates are there?
Are sterile neutrinos required to understand neutrino mixing?
Do neutrinos have any peculiar lectromagnetic properties?
Is CP symmetry respected in neutrino interactions?
Do some or all of the light neutrinos experience nonstandard neutrino interactions?
What is the origin of neutrino mass?
What new surprises might nature have in store for us? All that we need to know has stimulated many promising new initiatives in neutrino physics~\cite{y2k}.
\section{Quark Flavor Physics \label{sec:qflav}}
The rich field of quark flavor physics was also strongly represented in the results presented at Blois~\cite{qflav}, along with new determinations of tau-lepton properties~\cite{tau}, and studies of hadron physics~\cite{hadron}. Christian Kiesling summarized the state of our knowledge on CP violation~\cite{kiesling}. Having established the outlines of flavor physics---charged-current interactions that exhibit, to good approximation, a three-generation $V-A$ form; the suppression of flavor-changing neutral currents by the GIM mechanism; and CP-violating phenomena prescribed by the CKM quark-mixing matrix---we now must take a deeper look, to see just how accurately our standard-model idealization conforms to reality.
On closer examination, experiments indicate a number of anomalies at a provocative level of statistical significance~\cite{soni}. Among these, the inclusive-exclusive tension in determinations of the quark-mixing-matrix element $\abs{V_{ub}}$ is notable for its persistence ~\cite{roney}. It is amusing (at least) that the discordant determinations of $\abs{V_{ub}}$ from inclusive decays ($B \to X_u \ell \nu$), the exclusive decays $B \to \pi \ell \nu$ (mediated by the vector current), and the annihilation decay $B \to \tau \nu_\tau$ (mediated by the axial current) could be reproduced if a small right-handed charged-current interaction were present~\cite{Buras:2010pz}.
Determinations of the CP-violating phase $\beta_s^{J\!/\!\psi\phi}$, measured in $B_s \to J\!/\!\psi\phi$ decays, have previously differed (at $2.1\sigma$) from standard-model expectations~\cite{cdfd0bph}. The disagreement is somewhat mitigated by a new CDF measurement that benefits from increased statistics and improved $B$ tagging~\cite{cdfbphase}. Forthcoming data, together with measurements of the lifetime difference between the two $(B_s,\overline{B}_s)$ mass eigenstates, will probe for new physics. Indeed, $B$ physics promises much new information from BaBar and Belle, CDF and D0, and the LHC experimentsl~\cite{lhcflav}. New initiatives are in the works as well~\cite{comingatt}; in particular, funding in the amount
of $10^8$\yen\ $\approx$ \EUR{87M} over the next three years
has been secured for the KEKB upgrade. We may look forward to a new round of $e^+e^-$ collisions beginning April 1, 2014!
\section{Cosmic Issues \label{sec:cosmic}}
We begin with two reports on ``conventional'' astrophysics. We were treated to a remarkable example of modern observational capabilities in the reconstruction of stellar orbits around the massive black hole at the center of our galaxy~\cite{liebundgut}. Steady progress toward the detection of gravitational waves raises the fascinating prospect of a future in which the (non)observation of gravitational waves may serve as diagnostics for astrophysical phenomena~\cite{bauer}.
Astro/Cosmo/Particle physics is rich in new results and the prospect of others, thanks to many new instruments and their planned successors~\cite{astrocont}. Over the past three decades, the hot big-bang cosmology established in the 1960s has been revised several times to incorporate new ideas and new observations. As provisional as it must be, the concordance cosmology, including an inflationary epoch and the introduction of dark matter and some form of dark energy, has a pleasing economy and consistency~\cite{malik}. A central question is whether the dark energy is a manifestation of a cosmological constant, or has a dynamical origin~\cite{khoury}. An audacious proposal to study dark energy over a range in redshifts by measuring $dz/dt$ is an element of the E-ELT Project~\cite{eelt}.
A decade after the discovery that the universe is expanding at an accelerating rate, it is worth restating how remarkable are the implications of the cosmological-constant interpretation. Not only are we living during an epoch in which the matter and dark-energy densities are comparable, we are at the threshold of a new inflationary age. It is worth bearing in mind that the inflationary $\Lambda$CDM cosmology is less a coherent theoretical framework than an assembly of modules---ideas and inventions---added in response to observations~\cite{Steinhardt:2004gk}. It is not yet grounded in general principles, so it is important that we remain skeptical, probe the idealizations, look for deviations, and seek a more holistic foundation.
A wealth of observational information from the cosmic microwave background, baryon acoustic oscillations, and the Union08 supernova data set points to a universe that is to excellent approximation flat, but dominated by something other than matter~\cite{Kowalski:2008ez,slosar}. Indeed, the latest W-MAP analysis, within the concordance cosmological model, points to a mass-energy budget of the present universe consisting of about $73\%$ dark energy, $23\%$ dark matter, and $4.5\%$ normal atomic matter~\cite{hinshaw}. It was not ever thus: according to the standard thermal history of the universe, at the surface of last scattering (age ca. $380\,000$ years), the universe was made up of roughly $63\%$ dark matter, $15\%$ photons, $12\%$ ordinary baryonic matter, and perhaps $10\%$ neutrinos. Dark energy was a trace component. The Planck satellite has completed its initial observing campaign in excellent form; we eagerly await the first wave of analyses, as well as the results of projects now in progress to measure polarizations of the cosmic microwave background radiation~\cite{paolod}.
Although we must always be alert to the possibility that we have misread the evidence, it is virtually certain that collisionless cold dark matter is a significant component of the present universe. Crafty observational strategies, such as the analysis of gravitational lensing of distant light sources that produced the COSMOS 3-dimensional map~\cite{cosmos}, begin to tell us where the dark matter resides. We know what it is not: neither baryonic nor, for the most part, neutrinos. We don't know how many species, and we should resist jumping to the conclusion that there is only one dark matter.
The quest for dark matter as weakly interacting massive particles has three main elements: the direct detection of thermal relics from the early universe by passive experiments ~\cite{dmcont}; indirect searches that aim to observe (co-)annihilation products of dark-matter particles with cosmological lifetimes~\cite{dmind}; and collider searches that could characterize the properties of dark-matter candidates in great detail, but of course cannot establish cosmological lifetimes. It remains a possibility that some or all of the dark matter is in the form of axions~\cite{axion}, which could guide us to a solution of the strong CP problem.
On the collider front, it is tantalizing that many proposals for physics beyond the standard model lead to dark-matter candidates, and that a relic density of the right magnitude naturally arises for WIMP masses that lie between $100\hbox{ GeV}$ and $1\hbox{ TeV}$, the range accessible to the Tevatron and LHC. With respect to the detection of thermal relics or their annihilation products, many new techniques and instruments are reporting results over an interesting range of masses and interaction cross sections~\cite{bernard}. For all the approaches, the experimenter must confront these questions: Is there a signal? If yes, is it background? If no, prove you are sensitive. The next five years promise a lot of excitement!
\section{The Quy Nhon International Center for Interdisciplinary Science Education}
On Monday afternoon the founding father of the \textit{Rencntres de Blois}, Tr\^{a}n Thanh V\^{a}n, presented his vision of a new international science and education center, to be built in Vietnam~\cite{van}. It is an ambitious plan---some might say, extravagant---and a natural response is, ``This time, he's gone too far.'' I venture to speculate that many of Van's past achievements, all the way back to the \textit{Rencontres de Moriond,} might have elicited that same response, when first exposed. In his \textit{Elegy} to Robert Lowell, the Irish master Seamus Heaney writes, ``The way we are living, timorous or bold, will have been our life.'' Let us follow our friend Van's example and, in science and in life, be bold!
\section*{Acknowledgments}
Fermilab is operated by the Fermi Research Alliance under contract number DE-AC02-07CH11359 with the U.S. Department of Energy. I acknowledge with pleasure the generous support of the Alexander von Humboldt Foundation, and thank Andrzej Buras for a warm welcome at Technische Universit\"{a}t M\"{u}nchen, where this report was completed. I owe special thanks to Liz Simmons and Boris Kayser for assisting in the preparation of my talk.
Pour terminer, je souhaite un tr\`{e}s grand merci \`{a} tous les participants, aux gentils organisateurs des Rencontres de Blois, \`{a} nos amies sauvetrices du secretariat, au personnel du Ch\^{a}teau de Blois, \`{a} Kim et Van.
\section*{References}
\frenchspacing
|
{
"timestamp": "2010-09-21T02:02:37",
"yymm": "1009",
"arxiv_id": "1009.3742",
"language": "en",
"url": "https://arxiv.org/abs/1009.3742"
}
|
\section{Introduction}
\label{sec:intro}
To interpret the large-scale alignments of quasar optical polarization
vectors observed at redshifts $z\sim$~1 (Hutsem\'ekers \cite{HUT98};
Hutsem\'ekers and Lamy \cite{HUT01}; Hutsem\'ekers et
al. \cite{HUT05}) polarization induced by photon-pseudoscalar mixing
along the line of sight has been invoked (Hutsem\'ekers \cite{HUT98};
Jain et al. \cite{JAI02}). Photon-pseudoscalar mixing generates
dichroism and birefringence, the latter transforming linear
polarization into circular polarization and vice-versa along the line
of sight. If photon-pseudoscalar mixing produces the linear
polarization needed to explain the observed alignments, a comparable
amount of circular polarization would be expected (Raffelt and
Stodolsky \cite{RAF88}; Jain et al. \cite{JAI02}; Das et
al. \cite{DAS04}; Gnedin et al. \cite{GNE07}; Hutsem\'ekers et
al. \cite{HUT08}; Payez et al. \cite{PAY08}). Hence, we present
accurate circular polarization measurements for a sample of quasars
whose polarization vectors are coherently oriented.
The optical circular polarization of quasars has rarely been measured.
Our new observations, data reduction, and a compilation of published
measurements are presented in Sect.~\ref{sec:data}. Implications for
the photon-pseudoscalar mixing mechanism are discussed in
Sect.~\ref{sec:discuss1}. The detection of significant circular
polarization in two objects and its consequence for quasar physics are
presented in Sect.~\ref{sec:discuss2}.
\section{Observations and data reduction}
\label{sec:data}
The observations were carried out on April 18-20, 2007 at the European
Southern Observatory (ESO, La Silla) using the 3.6m telescope equipped
with the ESO Faint Object Spectrograph and Camera EFOSC2. Circular
polarization was measured using a super-achromatic quarter-wave
($\lambda$/4) retarder plate (QWP), which transforms the circular
polarization into linear polarization, and a Wollaston prism, which
splits the linearly polarized beam into two orthogonally polarized
images of the object (Saviane et al. \cite{SAV07}). The CCD was used
in unbinned mode, which corresponds to a scale of 0.157$\arcsec$/pixel
on the sky. All measurements were performed through a Bessel V filter
(V$\#$641; central wavelength: 5476 \AA; FWHM: 1132 \AA).
At least one pair of exposures with the QWP rotated to the angles $-45
\degr$ and $+45 \degr$ was secured for each target. Frames were
dark-subtracted and flat-fielded. The circular polarization $p_{\rm
circ}$, i.e., the normalized Stokes $V/I$ parameter, was extracted
from each pair of frames using a procedure used to measure the
normalized Stokes $Q/I$ and $U/I$ parameters and described in Lamy and
Hutsem\'ekers (\cite{LAM99}) and Sluse et al. (\cite{SLU05}). Errors
were estimated from the photon noise. Seeing was typically
around~1$\arcsec$. Owing to the variable atmospheric extinction (thin
to thick cirrus), some exposures had to be repeated to reach a
sufficient signal-to-noise ratio.
The performances of the instrument were checked during our run and
during the setup night (April 17) using an unpolarized standard star
and a star with high and slowly variable circular polarization,
LP~790$-$20 (West \cite{WES89}; Jordan and Friedrich
\cite{JOR02}). The results, discussed in Saviane et
al. (\cite{SAV07}), demonstrated the quality of the instrumental
setup. LP~790$-$20 was also used to fix the sign of the circular
polarization, i.e., $p_{\rm circ} > 0$ when the electric vector
rotates counter-clockwise as seen by an observer facing the object.
To evaluate the cross-talk between linear and circular polarization,
we measured the circular polarization of linearly polarized
stars. These observations were repeated several times during our
observing run. Hilt~652 was observed during the setup night. The
results are given in Table~\ref{tab:datastd} together with the
published linear polarization (i.e. the polarization degree $p_{\rm
lin}$ and the polarization position angle $\theta_{\rm
lin}$). Uncertainties are smaller than in Saviane et
al.~(\cite{SAV07}) because of the availability of repeated
observations. Although the objects are highly linearly polarized, we
measure a null circular polarization. Combining the data of Hilt~652
and Ve~6$-$23, which have similar polarization angles, we derive the
3$\sigma$ upper limit to the circular polarization due to cross-talk
in the V filter $|p_{\rm circ} / p_{\rm lin}| \lesssim$
0.0075.
Our new measurements of quasar circular polarization are reported in
Table~\ref{tab:dataqso} with 1$\sigma$ photon-noise errors. The
targets are extracted from the sample of 355 polarized quasars defined
in Hutsem\'ekers et al.~(\cite{HUT05}), as well as their B1950
names/coordinates, their redshift $z$, and their linear polarization
degree and angle, $p_{\rm lin}$ and $\theta_{\rm lin}$.
A compilation of other measurements of quasar optical circular
polarization is given in Table~\ref{tab:prevqso}. Unless indicated
otherwise, these measurements were obtained in white light, i.e., in
the 3200--8800~\AA\ or 4000--8800~\AA\ spectral ranges, which roughly
correspond to an effective wavelength of 6000~\AA. When several
estimates of either linear or circular polarization are available,
only the value with the smallest uncertainty is considered. BL~Lac
objects, similar in many respects to highly polarized quasars (HPQs)
(e.g., Scarpa and Falomo \cite{SCA97}; Fan et al. \cite{FAN08}), are
included. Both BL Lac and HPQs belong to the blazar sub-group of
active galactic nuclei (AGN). For BL~Lac objects, the polarization is
often strongly variable. We then adopt the circular polarization with
the smallest uncertainty and, when quasi-simultaneous observations are
available, the value of the linear polarization obtained as close as
possible in time. Otherwise we select a representative value of the
linear polarization from the survey of Impey and Tapia~(\cite{IMP90}).
\begin{table}[t]
\caption{The circular polarization of linearly polarized standard stars}
\label{tab:datastd}
\begin{tabular}{lccr}\hline\hline \\[-0.10in]
Object & $p_{\rm lin}$ (\%) & $\theta_{\rm lin}$ ($\degr$) & $p_{\rm circ}$ (\%) \\
\hline \\[-0.10in]
Hilt 652 & 6.25 $\pm$ 0.03 & 179.2 $\pm$ 0.2$^{a}$ & 0.003 $\pm$ 0.021 \\
Ve~6$-$23 & 8.26 $\pm$ 0.05 & 171.6 $\pm$ 0.2$^{a}$ & $-$0.050 $\pm$ 0.035 \\
HD155197 & 4.38 $\pm$ 0.03 & 103.2 $\pm$ 0.2$^{b}$ & 0.033 $\pm$ 0.025 \\
\hline\\[-0.2cm]
\end{tabular}\\
\tiny{Note: All these polarization measurements were obtained
in the V filter. References for linear polarization: (a)~Fossati et
al. \cite{FOS07}; (b)~Turnshek et al. \cite{TUR90}. }
\end{table}
\begin{table}[t]
\caption{New circular polarization measurements of quasars}
\label{tab:dataqso}
\begin{tabular}{lcccr}\hline\hline\\[-0.10in]
Object & $z$ & $p_{\rm lin}$ (\%) & $\theta_{\rm lin}$ ($\degr$) & $p_{\rm circ}$ (\%)\\
\hline\\[-0.10in]
\hspace*{-1mm}1120$+$019 & \hspace*{-1mm}1.465 & 1.95 $\pm$ 0.27 & 9 $\pm$ 4$^{c}$ & $-$0.02 $\pm$ 0.05 \\
\hspace*{-1mm}1124$-$186 & \hspace*{-1mm}1.048 & 11.68 $\pm$ 0.36 & 37 $\pm$ 1$^{g}$ & $-$0.04 $\pm$ 0.08 \\
\hspace*{-1mm}1127$-$145 & \hspace*{-1mm}1.187 & 1.30 $\pm$ 0.40~[w] & 23 $\pm$ 10$^{a}$ & $-$0.05 $\pm$ 0.05 \\
\hspace*{-1mm}1157$+$014 & \hspace*{-1mm}1.990 & 0.76 $\pm$ 0.18 & 39 $\pm$ 7$^{f}$ & $-$0.10 $\pm$ 0.08 \\
\hspace*{-1mm}1205$+$146 & \hspace*{-1mm}1.640 & 0.83 $\pm$ 0.18 & 161 $\pm$ 6$^{f}$ & $-$0.10 $\pm$ 0.09 \\
\hspace*{-1mm}1212$+$147 & \hspace*{-1mm}1.621 & 1.45 $\pm$ 0.30 & 24 $\pm$ 6$^{c}$ & 0.15 $\pm$ 0.09 \\
\hspace*{-1mm}1215$-$002$^{\star}$ & \hspace*{-1mm}0.420 & 23.94 $\pm$ 0.70 & 91 $\pm$ 1$^{g}$ & $-$0.42 $\pm$ 0.40 \\
\hspace*{-1mm}1216$-$010 & \hspace*{-1mm}0.415 & 11.20 $\pm$ 0.17 & 100 $\pm$ 1$^{g}$ & $-$0.01 $\pm$ 0.07 \\
\hspace*{-1mm}1222$+$228 & \hspace*{-1mm}2.058 & 0.92 $\pm$ 0.14 & 169 $\pm$ 4$^{g}$ & 0.01 $\pm$ 0.10 \\
\hspace*{-1mm}1244$-$255 & \hspace*{-1mm}0.633 & 8.40 $\pm$ 0.20~[w] & 110 $\pm$ 1$^{a}$ & $-$0.23 $\pm$ 0.20 \\
\hspace*{-1mm}1246$-$057 & \hspace*{-1mm}2.236 & 1.96 $\pm$ 0.18~[w] & 149 $\pm$ 3$^{e}$ & 0.01 $\pm$ 0.03 \\
\hspace*{-1mm}1254$+$047 & \hspace*{-1mm}1.024 & 1.22 $\pm$ 0.15~[w] & 165 $\pm$ 3$^{b}$ & $-$0.02 $\pm$ 0.04 \\
\hspace*{-1mm}1256$-$229$^{\star}$ & \hspace*{-1mm}0.481 & 22.32 $\pm$ 0.15 & 157 $\pm$ 1$^{g}$ & 0.18 $\pm$ 0.04 \\
\hspace*{-1mm}1309$-$056 & \hspace*{-1mm}2.212 & 0.78 $\pm$ 0.28 & 179 $\pm$ 11$^{c}$ & $-$0.08 $\pm$ 0.06 \\
\hspace*{-1mm}1331$-$011 & \hspace*{-1mm}1.867 & 1.88 $\pm$ 0.31 & 29 $\pm$ 5$^{c}$ & $-$0.04 $\pm$ 0.06 \\
\hspace*{-1mm}1339$-$180 & \hspace*{-1mm}2.210 & 0.83 $\pm$ 0.15 & 20 $\pm$ 5$^{g}$ & $-$0.01 $\pm$ 0.07 \\
\hspace*{-1mm}1416$-$129 & \hspace*{-1mm}0.129 & 1.63 $\pm$ 0.15~[w] & 44 $\pm$ 3$^{b}$ & 0.05 $\pm$ 0.06 \\
\hspace*{-1mm}1429$-$008 & \hspace*{-1mm}2.084 & 1.00 $\pm$ 0.29 & 9 $\pm$ 9$^{c}$ & 0.02 $\pm$ 0.08 \\
\hspace*{-1mm}2121$+$050 & \hspace*{-1mm}1.878 & 10.70 $\pm$ 2.90~[w] & 68 $\pm$ 6$^{a}$ & 0.02 $\pm$ 0.15 \\
\hspace*{-1mm}2128$-$123 & \hspace*{-1mm}0.501 & 1.90 $\pm$ 0.40~[w] & 64 $\pm$ 6$^{d}$ & $-$0.04 $\pm$ 0.03 \\
\hspace*{-1mm}2155$-$152 & \hspace*{-1mm}0.672 & 22.60 $\pm$ 1.10~[w] & 7 $\pm$ 2$^{a}$ & $-$0.35 $\pm$ 0.10\\
\hline\\[-0.2cm]
\end{tabular}\\
\tiny{Notes: Linear and circular polarizations were measured in
the V filter except a series of linear polarization data from the
literature measured in white light and noted~[w]; (${\star}$)
1215$-$002 is classified as a BL~Lac by Collinge~et~al.~\cite{COL05};
Sbarufatti et al. \cite{SBA05} re-determined the redshift of
1256$-$229 ($z$=0.481) and considered this object as a BL Lac.
References for linear polarization: (a)~Impey \& Tapia \cite{IMP90};
(b)~Berriman et al. \cite{BER90}; (c)~Hutsem\'ekers et
al. \cite{HUT98b}; (d)~Visvanathan \& Wills \cite{VIS98}; (e)~Schmidt
\& Hines \cite{SCH99}; (f)~Lamy \& Hutsem\'ekers \cite{LAM00};
(g)~Sluse et al. \cite{SLU05}. }
\end{table}
\begin{table}[t]
\caption{Previous circular polarization measurements of quasars and BL~Lac objects}
\label{tab:prevqso}
\begin{tabular}{lcccr}\hline\hline \\[-0.10in]
Object & $z$ & $p_{\rm lin}$ (\%) & $\theta_{\rm lin}$ ($\degr$) & $p_{\rm circ}$ (\%) \\
\hline \\[-0.10in]
\hspace*{-1mm}0237$-$233 & 2.223 & 0.25 $\pm$ 0.29 & $-$ \ $^{d}$ & $-$0.06 $\pm$ 0.08$^{a}$ \\
\hspace*{-1mm}0955$+$326 & 0.533 & 0.18 $\pm$ 0.24 & $-$ \ $^{c}$ & 0.06 $\pm$ 0.08$^{a}$ \\
\hspace*{-1mm}1127$-$145 & 1.187 & 1.30 $\pm$ 0.40 & 23 $\pm$ 10$^{e}$ & 0.32 $\pm$ 0.20$^{a}$ \\
\hspace*{-1mm}1156$+$295 & 0.729 & 2.68 $\pm$ 0.41 & 114 $\pm$ 4$^{f}$ & 0.12 $\pm$ 0.14$^{b}$ \\
\hspace*{-1mm}1222$+$228 & 2.058 & 1.09 $\pm$ 0.16~[u] & 167 $\pm$ 4$^{i}$ & 0.23 $\pm$ 1.80$^{i}$ \\
\hspace*{-1mm}1226$+$023 & 0.158 & 0.25 $\pm$ 0.04 & 58 $\pm$ 4$^{c}$ & $-$0.01 $\pm$ 0.02$^{g}$ \\
\hspace*{-1mm}1253$-$055 & 0.536 & 9.00 $\pm$ 0.40 & 67 $\pm$ 1$^{e}$ & 0.09 $\pm$ 0.07$^{a}$ \\
\hspace*{-1mm}1308$+$326 & 0.997 & 12.10 $\pm$ 1.50 & 68 $\pm$ 3$^{e}$ & $-$0.08 $\pm$ 0.17$^{b}$ \\
\hspace*{-1mm}1634$+$706 & 1.334 & 0.24 $\pm$ 0.07~[u] & 4 $\pm$ 8$^{i}$ & $-$0.05 $\pm$ 0.09$^{i}$ \\
\hspace*{-1mm}1641$+$399 & 0.594 & 4.00 $\pm$ 0.30 & 103 $\pm$ 2$^{e}$ & $-$0.05 $\pm$ 0.23$^{b}$ \\
\hspace*{-1mm}2230$+$114 & 1.037 & 7.30 $\pm$ 0.30 & 118 $\pm$ 1$^{e}$ & $-$0.05 $\pm$ 0.17$^{a}$ \\
\hspace*{-1mm}2302$+$029 & 1.044 & 0.66 $\pm$ 0.12~[u] & 136 $\pm$ 5$^{i}$ & $-$0.39 $\pm$ 0.16$^{i}$ \\
\hspace*{-1mm}{\it 0138$-$097} & 0.733 & 3.60 $\pm$ 1.50 & 168 $\pm$ 11$^{e}$ & 0.25 $\pm$ 0.35$^{k}$ \\
\hspace*{-1mm}{\it 0219$+$428} & 0.444 & 26.11 $\pm$ 0.19 & 3 $\pm$ 1$^{k}$ & 0.16 $\pm$ 0.05$^{k}$ \\
\hspace*{-1mm}{\it 0422$+$004} & 0.310 & 10.29 $\pm$ 0.23 & 179 $\pm$ 1$^{g}$ & 0.14 $\pm$ 0.07$^{g}$ \\
\hspace*{-1mm}{\it 0735$+$178} & 0.424 & 11.69 $\pm$ 0.22 & 123 $\pm$ 1$^{k}$ & 0.03 $\pm$ 0.05$^{k}$ \\
\hspace*{-1mm}{\it 0823$-$223} & 0.910 & 14.39 $\pm$ 0.16 & 11 $\pm$ 1$^{k}$ & 0.16 $\pm$ 0.08$^{k}$ \\
\hspace*{-1mm}{\it 0851$+$202} & 0.306 & 10.80 $\pm$ 0.30 & 156 $\pm$ 1$^{e}$ & $-$0.01 $\pm$ 0.02$^{a}$ \\
\hspace*{-1mm}{\it 1101$+$384} & 0.031 & 2.59 $\pm$ 0.11 & 10 $\pm$ 1$^{h}$ & 0.02 $\pm$ 0.03$^{h}$ \\
\hspace*{-1mm}{\it 2155$-$304} & 0.116 & 4.12 $\pm$ 0.25 & 93 $\pm$ 2$^{j}$ & $-$0.02 $\pm$ 0.02$^{j}$ \\
\hspace*{-1mm}{\it 2200$+$420} & 0.068 & 4.90 $\pm$ 0.40 & 147 $\pm$ 2$^{e}$ & $-$0.07 $\pm$ 0.19$^{a}$ \\
\hline\\[-0.2cm]
\end{tabular}\\
\tiny{Notes: All but a few polarization measurements were
obtained in white light; the multi-color measurements in references
(h), (j) and (k) were averaged; [u] refers to linear and circular
polarization measurements averaged over the 2200--3200 \AA\
ultraviolet wavelength band; the circular polarization of objects
2200$+$420 and 2230$+$114 was obtained in the 4000--6000~\AA\ and
3500--5200~\AA\ bands respectively; italicized names indicate objects
classified as BL~Lac in V\'eron-Cetty \& V\'eron
\cite{VER06}. References for linear and circular polarization:
(a)~Landstreet \& Angel \cite{LAN72}; (b)~Moore \& Stockman
\cite{MOO81}; (c)~Stockman et al. \cite{STO84}; (d)~Moore \& Stockman
\cite{MOO84}; (e)~Impey \& Tapia \cite{IMP90}; (f)~Wills et
al. \cite{WIL92}; (g)~Valtaoja et al. \cite{VAL93}; (h)~Takalo \&
Sillanp\"a\"a \cite{TAK93}; (i)~Impey et al. \cite{IMP95}; (j)~Tommasi
et al. \cite{TOM01a}; (k)~Tommasi et al. \cite{TOM01b}. }
\end{table}
\section{Discussion}
\label{sec:discuss}
The measurements reported in Tables~\ref{tab:dataqso}
and~\ref{tab:prevqso} show that all quasars and BL Lac objects have
null circular polarization ($<$~3~$\sigma$) except two HPQs,
1256$-$229 and 2155$-$152, and one highly polarized BL Lac object,
0219$+$428.
We first discuss the constraints provided by the majority of null
detections on the photon-pseudoscalar mixing mechanism, and then the
consequences of the three detections for blazar physics.
\subsection{Constraints on photon-pseudoscalar mixing}
\label{sec:discuss1}
Quasars with right ascension between 11$^{\rm h}$20$^{\rm m}$ and
14$^{\rm h}$30$^{\rm m}$ belong to the region of alignment A1 defined
in Hutsem\'ekers~(\cite{HUT98}). In this region of the sky, quasars
with $1 < z < 2.3$ have their polarization angle preferentially in the
range [146$\degr$--226$\degr$] (modulo 180$\degr$), while quasars with
$0 < z < 1$ have their polarization angle preferentially in the range
[30$\degr$--120$\degr$]. Assuming that the quasar intrinsic
polarization vectors are randomly oriented, the addition of a small
systematic linear polarization $\Delta p_{\rm lin} \simeq 0.5 \%$ at a
fixed position angle can account for the observed alignments
(Hutsem\'ekers et al. \cite{HUT08}; Appendix~\ref{sec:apa}). If
photon-pseudoscalar mixing is responsible for this extra linear
polarization, one expects, on average, that $| p_{\rm circ}| \simeq
\Delta p_{\rm lin} \simeq 0.5 \%$ (Appendix~\ref{sec:apb}). Because
the light from most quasars is intrinsically linearly polarized to
some extent and not circularly polarized, limits on any additional
polarization from interactions along the line of sight cannot be
derived from the measurement of the linear polarization degree, while,
on the other hand, useful constraints can be derived from the
measurement of circular polarization.
Most of the thirteen quasars with $z > 1$ located in region A1 were
found to have $|p_{\rm circ}| \lesssim 0.25 \% $ (3$\sigma$ upper
limit), which is definitely smaller than the expected value. Averaging
over the thirteen objects, we infer that $\langle |p_{\rm circ}|
\rangle = 0.035 \pm 0.016 \%$ after neglecting the sign, from which a
stringent 3$\sigma$ upper limit on the circular polarization of
$\langle |p_{\rm circ}| \rangle \leq 0.05\%$ can be derived. This
limit is one order of magnitude smaller than the expected value $|
p_{\rm circ}| \simeq 0.5 \%$. A similar result is obtained for the
nine objects at $z < 1$ in that region.
This result rules out the interpretation of the observed alignments in
terms of photon-pseudoscalar mixing, at least in its simplest
formulation. A more complex treatment of the photon-pseudoscalar
interaction is thus required to account for the observations (Payez et
al. \cite{PAY10a,PAY10b}).
\subsection{Detection of optical circular polarization and
implication for blazar physics}
\label{sec:discuss2}
Circular polarization is detected at the 3$\sigma$ level in two HPQs:
1256$-$229 and 2155$-$152 (Table~\ref{tab:dataqso}). On April 21, we
had the opportunity to re-measure the linear polarization of these
objects in the V filter, after replacing the quarter-wave plate by the
half-wave plate (HWP) (cf. Saviane et al. \cite{SAV07}). Four
exposures with the HWP rotated to 0$\degr$, 22.5$\degr$, 45$\degr$,
and 67.5$\degr$ were secured and reduced in the standard way
(e.g. Sluse et al. \cite{SLU05}). The results are reported in
Table~\ref{tab:circqso}, together with the circular polarization
measurements from Table~\ref{tab:dataqso}. Although the optical linear
polarization of these quasars is high, the circular polarization we
measured at the same epoch is above the 3$\sigma$ upper limit on the
circular polarization generated by the instrumental cross-talk
(Sect.~\ref{sec:data}).
In Table~\ref{tab:circqso}, we also summarize the main polarization
properties of these objects, including measurements at radio
wavelengths. For completeness, we include the BL Lac object for which
circular polarization was found to be significant after averaging the
UBVRI measurements (Table~\ref{tab:prevqso}). As far as we know,
these are the only 3$\sigma$ detections of optical circular
polarization in quasars, in addition to those reported by Wagner and
Mannheim (\cite{WAG01}) for 3C279\footnote{Wagner et
al. (\cite{WAG00}) reported that $p_{\rm circ} = 0.25 \pm 0.03 \%$,
while Wagner and Mannheim (\cite{WAG01}) reported $p_{\rm circ} =
0.45 \pm 0.03 \%$. These detections were considered tentative by the
authors in view of significant instrumental effects and thus not
included in Tables~\ref{tab:prevqso} and~\ref{tab:circqso}.} (=
1253$-$055). Although variable (as commonly seen in HPQs), the
optical linear polarization is high in all three objects suggesting
that a relation exists between linear and circular polarization.
\begin{table}[t]
\caption{Radio to optical polarization characteristics of objects
with detected optical circular polarization}
\label{tab:circqso}
\begin{tabular}{lcccr}\hline\hline\\[-0.10in]
\multicolumn{5}{l}{1256$-$229~(PKS) \ \ $z$=0.481} \ \ \ HPQ, BL Lac? \ \ \\
\hline\\[-0.10in]
Date & $\nu$ (Ghz) & $p_{\rm lin}$ (\%) & $\theta_{\rm lin}$ ($\degr$) & $p_{\rm circ}$ (\%)\\
\hline\\[-0.10in]
03/2002 & 5.4 10$^5$ & 22.32 $\pm$ 0.15 & 157 $\pm$ 1$^{d}$ & $-$ \ \ \\
04/2007 & 5.4 10$^5$ & 15.42 $\pm$ 0.16 & 163 $\pm$ 1$^{j}$ & 0.18 $\pm$ 0.04$^{j}$ \\
\hline\hline\\[-0.10in]
\multicolumn{5}{l}{2155$-$152~(PKS) \ \ $z$=0.672} \ \ \ HPQ \ \ \\
\hline\\[-0.10in]
Date & $\nu$ (Ghz) & $p_{\rm lin}$ (\%) & $\theta_{\rm lin}$ ($\degr$) & $p_{\rm circ}$ (\%)\\
\hline\\[-0.10in]
08/1984 & 5.4 10$^5$ & 32.70 $\pm$ 1.30 & 6 $\pm$ 1$^{a}$ & $-$ \ \ \\
04/2007 & 5.4 10$^5$ & 17.67 $\pm$ 0.49 & 51 $\pm$ 1$^{j}$ & $-$0.35 $\pm$ 0.10$^{j}$\\
09/2007 & 85.6 & 11.04 $\pm$ 0.53 & 52 $\pm$ 1$^{i}$ & $<$ 0.93$^{i}$ \\
03/2003 & 15.4 & 3.66 & $-$ \ $^{f}$ & $<$ 0.34$^{f}$ \\
1979-1999 & 8.0 & 4.81 (mean) & $-$ \ $^{h}$ & $-$ \ \ \\
& & 15.3 (max) & $-$ \ $^{h}$ & $-$ \ \ \\
\hline\hline\\[-0.10in]
\multicolumn{5}{l}{0219$+$428~(3C66A) \ \ $z$=0.444} \ \ \ BL Lac \ \ \\
\hline\\[-0.10in]
Date & $\nu$ (Ghz) & $p_{\rm lin}$ (\%) & $\theta_{\rm lin}$ ($\degr$) & $p_{\rm circ}$ (\%)\\
\hline\\[-0.10in]
01/1992 & 5 10$^5$~[w] & 31.07 $\pm$ 0.31 & 39 $\pm$ 1$^{b}$ & 0.76 $\pm$ 0.10$^{b}$\\
12/1999 & 5 10$^5$~[w] & 26.11 $\pm$ 0.19 & 3 $\pm$ 1$^{c}$ & 0.16 $\pm$ 0.05$^{c}$\\
11/2008 & 85.6 & 4.36 $\pm$ 0.54 & 3 $\pm$ 4$^{i}$ & $<$ 0.73$^{i}$ \\
12/1996 & 5.0 & 2.9 & $-$ \ $^{e}$ & $<$ 0.20$^{e}$ \\
1974-1999 & 8.0 & 2.96 (mean) & $-$ \ $^{g}$ & $-$ \ \ \\
\hline\\[-0.2in]
\end{tabular}
$ $\\[0.1in]
\tiny{Notes: Multi-color measurements in references (b) and (c)
were averaged; upper limits are given at 3$\sigma$. References for
linear or circular polarization: (a)~Brindle et al. \cite{BRI86};
(b)~Takalo \& Sillanp\"a\"a \cite{TAK93}; (c)~Tommasi et
al. \cite{TOM01b}; (d)~Sluse et al. \cite{SLU05}; (e)~Homan et
al. \cite{HOM01}; (f)~Homan and Lister \cite{HOM06}; (g)~Fan et
al. \cite{FAN06}; (h)~Fan et al. \cite{FAN08}; (i)~Agudo et
al. \cite{AGU10}; (j)~this work. }
\end{table}
Radio circular polarization has been detected in a small number of
blazars with typical values of a few tenths of a percent (Weiler and
de Pater \cite{WEI83}; Rayner et al. \cite{RAY00}; Homan et
al. \cite{HOM01}; Homan and Lister \cite{HOM06}; Vistrishchak et
al. \cite{VIS08}). Although the origin of the radio circular
polarization is not yet understood, two main mechanisms of production
have been proposed: intrinsic circular polarization of the
relativistically beamed synchrotron radiation (which also produces the
radio linear polarization) and Faraday conversion of linear to
circular polarization (e.g. Wardle and Homan \cite{WAR03}). Since
beamed synchrotron radiation can also explain the high optical linear
polarization observed in HPQs and contribute significantly to the
optical continuum (Impey and Tapia~\cite{IMP90}; Wills et
al. \cite{WIL92}), a similar origin to both the optical and the radio
circular polarizations appears likely although they most probably
arise from different regions. Since Faraday conversion is inefficient
at visible wavelengths, the detected optical circular polarization
should be caused by synchrotron emission. Since intrinsic circular
polarization is not produced in a positron-electron plasma, this
mechanism requires the predominance of a proton-electron plasma, as
already suggested by circular polarization measurements obtained at
millimeter wavelengths (Agudo et al. \cite{AGU10}). Furthermore, if
circular polarization is intrinsic, a correlation between the linear
and the circular polarization degrees is expected, the high recorded
values indicating a rather homogeneous magnetic field whose strength
should be of the order of 1~kG (e.g. Valtaoja et
al. \cite{VAL93}). This is much higher than usually assumed in quasar
jets, and can only occur in small regions close to the quasar core
(Wardle and Homan \cite{WAR03}; Silant'ev et al. \cite{SIL09};
Piotrovich et al. \cite{PIO10}). On the other hand, the optical
continuum could predominantly arise from inverse Compton scattering of
radio synchrotron radiation, a mechanism that preserves the circular
polarization (Sciama and Rees \cite{SCI67}). This would require a
significant circular polarization at radio wavelengths, which is
apparently not observed (Table~\ref{tab:circqso}). Given the
uncertainties and the non-simultaneous observations, no firm
conclusion can be derived. Unveiling the origin of the optical
circular polarization --even a few tenths of a percent-- thus appears
challenging (see also Rieger and Mannheim \cite{RIE05}). A clearer
understanding would require simultaneous observations at radio and
optical wavelengths.
\section{Conclusions}
\label{sec:conclu}
We have reported new accurate measurements of optical circular
polarization in the V filter for a sample of 21 quasars. For most
objects, the uncertainties are smaller than 0.1\%, and smaller than
0.05\% for six of them.
All objects have null polarization within the uncertainties except two
highly linearly polarized blazars. This has allowed us to constrain
the polarization caused by photon-pseudoscalar mixing along the line
of sight, ruling out the interpretation of the observed alignments of
quasar polarization vectors in terms of photon-pseudoscalar mixing, at
least in the framework of a simple formulation.
We also found small but significant optical circular polarization in
two blazars, providing clues about the strength of the magnetic
fields, the nature of the jets and/or the dominant emission
mechanism. Our observations demonstrate that optical circular
polarization is routinely measurable with present day high-accuracy
polarimeters.
\begin{acknowledgements}
D.H. thanks Alexandre Payez and Jean-Ren\'e Cudell for useful
discussions. A fellowship from the Alexander von Humboldt Foundation
to D.S. is gratefully acknowledged. This research has made use
of data originally from the University of Michigan Radio Astronomy
Observatory, which has been supported by the University of Michigan
and the National Science Foundation.
\end{acknowledgements}
\vspace*{-5mm}
|
{
"timestamp": "2010-09-22T02:01:40",
"yymm": "1009",
"arxiv_id": "1009.4049",
"language": "en",
"url": "https://arxiv.org/abs/1009.4049"
}
|
\section{Introduction}
\label{sec1} Ultracold molecules offer new opportunities for scientific
exploration, including studies of molecular Bose-Einstein condensates,
novel quantum phases, and ultracold chemistry. For molecular
interactions that take place at microKelvin temperatures, even the
smallest activation energy exceeds the available thermal energy. This
opens up new possibilities for controlling the pathways of chemical
reactions (see, e.g., Ref.\ \cite{Krems:08}).
A major objective of current experiments on cold molecules is to
achieve quantum degeneracy, particularly for polar molecules. Two
approaches are being pursued: indirect methods, in which molecules are
formed from pre-cooled atomic gases, and direct methods, in which
molecules are cooled from room temperature. There have been very
substantial recent advances, particularly in indirect methods. In
particular, the JILA \cite{Ni:KRb:2008} and Innsbruck
\cite{Danzl:ground:2010} groups have formed deeply bound ground-state
molecules at temperatures below 1 $\mu$K, by magnetoassociation of
pairs of ultracold atoms followed by coherent state transfer with
lasers. Methods that form ultracold molecules from ultracold atoms are
however restricted at present to species formed from heavy alkali-metal
atoms.
Direct methods, such as buffer-gas cooling \cite{Doyle:98}, Stark
deceleration \cite{Meijer:99}, crossed-beam collisional cooling
\cite{Elioff} and Maxwell extraction \cite{Buuren}, are applicable to a
much larger variety of chemically interesting molecules. However, these
methods cannot yet reach temperatures below 10 to 100~mK. Finding a way
to cool these molecules further, below 1~mK, is one of the biggest
challenges facing the field. The most promising possibility is
so-called sympathetic cooling, in which cold molecules are introduced
into an ultracold atomic gas and thermalize with it. Sympathetic
cooling has been successfully used to achieve Fermi degeneracy in
$^6$Li \cite{DeMarco:1999} and Bose-Einstein condensation in $^{41}$K
\cite{Modugno:2001}, and for producing ultracold ions
\cite{Zipkes2010,Zipkes2010a,Schmid}. However, it has not yet been
achieved for molecular systems, although there are theoretical
proposals for experiments in which ultracold NH or ND$_3$ molecules are
obtained by collisions with a bath of colder atoms such as Rb, Mg or N
\cite{Zuchowski:NH3:2009,Soldan:MgNH:2009,Wallis:MgNH:2009}.
The group at Imperial College London recently succeeded in producing
samples of cold LiH molecules in the first rotationally excited state
\cite{Tokunaga:2007,Tokunaga:2009} using Stark deceleration. LiH is an
attractive molecule for cooling, since it has large dipole moment and
light mass, so that it can be controlled easily with fields. It has a
relatively large rotational constant (7.5 cm$^{-1}$), which opens up
the possibility of producing cold molecules in a single excited
rotational state. There is proposal to produce ultracold LiH molecules
by sympathetic cooling with Li \cite{Tarbutt:privatecomm}. However,
sympathetic cooling can be successful only if the rate of elastic
(thermalization) collisions is large compared to the rate of inelastic
(deexcitation) collisions, which cause trap loss. The main objects of
the present paper are to explore the interaction between Li atoms and
LiH molecules, to understand the nature of the interaction between
these two species, and to obtain a detailed and accurate potential
energy surface for the Li--LiH system.
The results of scattering calculations at ultralow temperature are very
sensitive to the details of the interaction potential
\cite{Zuchowski:NH3:2009,Wallis:MgNH:2009}. For systems containing
heavy atoms, the methods of quantum chemistry currently available
cannot generate interaction potentials with accuracy better than a few
percent. This limitation is caused by approximate treatments of
correlation effects and relativistic contributions. With potential
energy surfaces of moderate precision, it is usually possible to
extract only qualitative information from low-energy collision
calculations. By contrast, Li--LiH is a light system containing only 7
electrons and state-of-the-art {\em ab initio} electronic structure
calculations can be performed with no significant approximations. It
therefore offers a unique possibility to produce a very precise
interaction potential, which will allow a quantitative description of
Li--LiH collision dynamics, even in the ultralow temperature regime.
In electronic structure calculations one aims at approaching
the exact solution of the Schr{\"o}dinger equation, as closely
as possible within the algebraic approximation. In practice,
this is accomplished by combining hierarchies of one-electron
and $N$-electron expansions. The accuracy increases across the
hierarchies in a systematic manner, allowing the errors in the
calculations to be controlled and a systematic approach to the
the exact solution to be achieved. The standard $N$-electron
hierarchy employed in electronic structure calculations
consists of the Hartree-Fock (HF), second-order M\o
ller-Plesset perturbation theory (MP2), coupled-cluster with
single and double excitations (CCSD), and coupled-cluster with
single, double, and approximate noniterative triple excitations
[CCSD(T)] models, with the latter recovering most of the
correlation energy. Thus, CCSD(T) constitutes a robust and
accurate computational tool nowadays. All these models are
size-consistent, which means that the interaction potential
shows the correct dissociation behaviour at large
intermolecular distances. In contrast, methods based on the
configuration interaction approach with a restricted excitation
space like multireference configuration interaction limited to
single and double excitations (MRCISD) are not size-consistent
and therefore they are not well suited for calculations of the
interaction energy.
The most popular example of a one-electron hierarchy is the family of
Dunning correlation-consistent polarized valence basis sets, cc-pV$X$Z
\cite{Dunning:89} with the cardinal number $X$ going from D (double-zeta), through
T indicating triple-zeta, and so on.
These have successfully been combined with the HF,
MP2, CCSD, CCSD(T) hierarchy of wave function models for the
calculation of various molecular properties \cite{electronic_energies,basis,static_dipole_moments}.
The basis-set limit, corresponding to $X\to\infty$, may be approached either by
extrapolating the results obtained with finite cardinal numbers towards
infinite $X$ \cite{extrap1,extrap2}, or by replacing the standard
one-electron hierarchy by explicitly-correlated methods, such as
CCSD--F12 and CCSD(T)--F12
\cite{skhv08a,skhv08b,Koehn:2008,tknh07,bokhan2008,tkh2008}, in which
the interelectron distance $r_{12}$ is explicitly introduced into the
wave function\cite{Kutzelnigg:1985,Klopper:1987,Ten-no:2004}.
The F12 methods have recently been implemented
efficiently \cite{efficient_ccsd_f12,ccf12,F12Rev,F12Rev_Tew} and shown
to accelerate the convergence towards the basis-set limit for a number
of properties
\cite{response_properties_from_f12,nh07,geometries_and_frequencies_from_f12}.
In the present paper, we combine all-electron spin-unrestricted CCSD(T)--F12 calculations
with frozen-core FCI calculations to yield a highly accurate best
estimate of the Li--LiH interaction potential. We also compare the F12 interaction energies with
results obtained from standard (not explicitly correlated) CCSD(T) calculations.
We then characterize the
ground-state potential, analyze possible interactions with excited
states, and investigate channels for reactive collisions.
\section{Computational Details}
\label{sec2} We have calculated the interaction energies between the
lithium atom and the lithium hydride molecule in Jacobi coordinates
($R,r,\theta$), defined for the isotopic combination
$^7$Li--$^7$Li$^1$H. Calculations were performed for states of
$^2A^\prime$ symmetry in the $C_{s}$ point group. The LiH bond
distance, $r$, was initially kept frozen at the LiH monomer equilibrium
distance of 3.014 bohr \cite{lihre}. The distance $R$ between Li and
the center of mass of LiH ranged from 3.0 to 10.0 bohr with an interval
of 0.5 bohr, and then from 11.0 to 20.0 bohr with an interval of 1.0
bohr. Additional distances of 30.0, 40.0, and 50.0 bohr were also used.
The angle $\theta$, between the vector pointing from Li to H in the LiH
molecule and the vector pointing from the center of mass of the
molecule to the Li atom, was varied from $0^{\circ}$ to $180^{\circ}$
with an interval of $15^{\circ}$; $\theta=0^{\circ}$ corresponds to
Li--H---Li configurations. We thus used a total of 28 intermonomer
distances, $R$, which combined with the 13 values of $\theta$ yielded
364 grid points on the two-dimensional interaction energy surface.
Calculations with uncorrelated basis functions were carried out using the unrestricted version of the
coupled-cluster model CCSD(T)
with Dunning's cc-pV$X$Z(-mid) basis sets with $X=$ D, T, Q, 5, where
mid indicates the inclusion of an additional set of basis functions,
the so-called midbond-95 set \cite{mid_partridge},
placed at the middle of the Li--LiH distance $R$. All electrons were
correlated in these calculations. Additionally, for the purpose of
comparison with the FCI results (see below), the frozen-core
approximation ($1\sigma_{\mathrm{LiH}}$ and $1s_{\mathrm{Li}}$ orbitals
kept frozen) was used for the cc-pVQZ basis. All these calculations
were carried out using the {\sc molpro} package \cite{MOLPRO2008}.
The full basis set of the dimer
was used in the supermolecular calculations and the Boys and Bernardi
scheme \cite{Boys:70} was used to correct for basis-set superposition
error.
The explicitly correlated spin-unrestricted CCSD--F12 and CCSD(T)--F12
\cite{ccf12,kw08,efficient_ccsd_f12,kaw2009}
calculations were carried out with the {\sc molpro} code
\cite{MOLPRO2008} to establish the CCSD and CCSD(T) basis-set limits
for the LiH--Li interaction.
We chose to use the F12b variant \cite{ccf12,kaw2009} of the explicitly
correlated spin-unrestricted energy
implemented in the {\sc molpro} code.
Employing the fixed-amplitude ansatz for
the F12 wave function ensured the orbital invariance and
size-consistency of the CCSD-F12 and CCSD(T)-F12 results. The QZVPP
basis set \cite{def2} was employed as the orbital basis in the F12
calculations. The corresponding QZVPP-jk basis set \cite{jk} was used
as the auxiliary basis for the density-fitting approximation
\cite{df1,kw08} for many-electron integrals, while the uncontracted
version of the QZVPP-jk basis was used to approximate the
Resolution-of-Identity in the F12 integrals \cite{ks2002,v2004}. In
addition, the valence correlation in the dimer was described with the
full configuration interaction method (FCI). The FCI and standard CCSD(T) calculations
in the frozen-core approximation were carried out using the cc-pVQZ basis.
The {\sc dalton} package \cite{dalton20} and the {\sc
lucia} program \cite{lucia} were combined to yield the FCI results.
To calculate potential energy surface $V(R,\theta)$ with the LiH bond length kept fixed
at its equilibrium value
we used computational scheme which was previously applied in theoretical studies of
the ground and excited states of the calcium dimer \cite{buss0,buss1,buss2,buss3,Koch:08}.
The potential $V(R,\theta)$ was constructed according to the following expression:
\begin{equation}
V(R,\theta) =
V^{\rm CCSD(T)-F12}(R,\theta) + \delta V_{\rm v-v}^{\rm FCI}(R,\theta)
\label{Pig}
\end{equation}
where $V^{\rm CCSD(T)-F12}(R,\theta)$ contribution was
obtained from all-electron CCSD(T)-F12 calculations, while the
correction for the valence-valance correlation beyond the
CCSD(T)-F12 level, $\delta V_{\rm v-v}^{\rm FCI}(R,\theta)$,
was calculated in an orbital cc-pVQZ basis set. Both terms,
$V^{\rm CCSD(T)-F12}(R,\theta)$ and $\delta V_{\rm v-v}^{\rm
FCI}(R,\theta)$, were obtained from the standard expressions
for the supermolecule interaction energy, as given in Ref.
\cite{buss3}.
The long-range asymptotic form of the potentials is of primary
importance for cold collisions. We have therefore computed the leading
long-range coefficients that describe the induction and dispersion
interactions up to and including $R^{-10}$ and $l=4$ terms,
\begin{equation} V(R,\theta)= -\sum_{n=6}^{10}\sum_{l=0}^{n-4}
\frac{C_n^l}{R^n}P_l(\cos\theta),
\label{lr1}
\end{equation} where $l$ is even/odd for $n$ even/odd, and
$C_n^l=C_n^l({\rm ind})+C_n^l({\rm disp})$. The long-range coefficients
$C_n^l({\rm ind})$ and $C_n^l({\rm disp})$ are given by the standard
expressions (see, e.g., Refs.\ \cite{Jeziorski:94,Moszynski:08}). The
multipole moments and polarizabilities of LiH were computed with the
recently introduced explicitly connected representation of the
expectation value and polarization propagator within the
coupled-cluster method \cite{Jeziorski:93,Moszynski:05,Korona:06a},
while the Li polarizabilities (both static and at imaginary
frequencies) were taken from highly accurate relativistic calculations
from Derevianko and coworkers \cite{Derevianko:10}.
The interaction potentials were interpolated between calculated points
using the reproducing kernel Hilbert space method (RKHS) \cite{rkhs}
with the asymptotics fixed using the {\em ab initio} long-range Van der
Waals coefficients. The switching function of Ref.\ \cite{Janssen} was
used to join the RKHS interpolation smoothly with the Van der Waals
part in the interval between $R_a=18$ and $R_b=26$ bohr.
\section{Convergence of the Li--LiH interaction potential
towards the exact solution}
In sec.~\ref{sub1} we analyze the convergence of the Li--LiH
interaction potential with respect to the one-electron and $N$-electron
hierarchies. Based on the analysis, we give in sec.~\ref{sub2} our best
estimate for the ground-state interaction potential with the Li--H bond
length fixed at its monomer equilibrium value. The features of the
potential are presented in sec.~\ref{sub3}.
\subsection{Convergence of the one-electron and $N$-electron hierarchies}
\label{sub1}
In order to investigate the saturation of the Li--LiH interaction
energy in the one-electron space, we have analyzed three characteristic
points of the Li--LiH potential (the global minimum, the saddle point,
the local minimum, and one point very close to
the avoided crossing: $R=5.5$ bohr and $\theta=0.0^\circ$).
The characteristic points were obtained from
the potentials
calculated at the CCSD(T) / cc-pV$X$Z-mid level of theory, for $X =$ D,
T, Q, and 5. The interaction energies were then compared to the
corresponding energies of the spin-unrestricted CCSD(T)-F12 / QZVPP potential
(approximation F12b), which
serves as the basis-set limit. To evaluate the accuracy of the pure
one-electron basis ({\em not}\/ explicitly correlated), the relative
percentage errors, $\Delta_{\rm F12b} = (V^{{\rm cc-pV}X{\rm
Z}} - V^{\rm F12b}) \, / \, |V^{\rm F12b}| \, \cdot
\, 100\%$, were determined for each $X$ at every characteristic point.
The results are given in Table \ref{tab0}.
We have also evaluated the characteristic points from the extrapolated
interaction energy surfaces, which were generated as follows: at each
grid point, the extrapolated {\em total}\/ energies for Li, LiH, and
Li--LiH were obtained by adding the Hartree-Fock energy calculated with
cardinal number $X$ to the extrapolated correlation energy, $E^{\rm
corr}_{(X-1)X}$, obtained from the two-point extrapolation formula
\cite{extrap1,extrap2},
\begin{equation}
\label{extrapol}
E^{\rm corr}_{(X-1)X} = E^{\rm corr}_X + \frac{E^{\rm corr}_X
- E^{\rm corr}_{(X-1)}}{[1-(X)^{-1}]^{-\alpha}-1},
\end{equation}
where $E^{\rm corr}_{(X-1)}$ and $E^{\rm corr}_X$ are the correlation
energies obtained for two consecutive cardinal numbers, $(X-1)$ and
$X$, respectively. The final extrapolated interaction energy at a
single grid point is obtained by subtracting the Li and LiH
extrapolated total energies from the Li--LiH extrapolated total energy.
We used the values $\alpha = 2$ and $\alpha = 3$, which were
recommended by Jeziorska {\em et al.}\/ in their helium dimer study
\cite{mj1,mj2} as the ones most suited for extrapolating all the
components of the interaction energy. The energies of the
characteristic points obtained in this way were compared with the
CCSD(T)-F12 / QZVPP results and the corresponding values of
$\Delta_{\rm F12b}$ are included in Table \ref{tab0}.
The relative percentage errors, $\Delta_{\rm F12b}$, are plotted in
Fig.~\ref{fig1} for both plain (non-extrapolated) and extrapolated
characteristic points. For the global minimum, the plain cc-pV$X$Z
results approach the basis-set limit from above and the convergence is
smooth and fast: the error is reduced by a factor of 2 to 3 for each
increment in $X$. The extrapolation accelerates the convergence: the
$(X-1)X$ extrapolated interaction energies have a quality at least that
of the plain cc-pV$(X+1)$Z results. Though the extrapolation with
$\alpha = 2$ seems to be more efficient than that with $\alpha = 3$ for
the DT and TQ cases, it actually overshoots the basis-set limit when
the Q and 5 cardinal numbers are used. More importantly, using
$\alpha=2$ leads to irregular behaviour: the Q5 extrapolation results
in a lower quality than the TQ extrapolation. In contrast,
extrapolation with $\alpha=3$, though slightly less efficient for low
cardinal numbers, exhibits highly systematic behaviour and leads to an
error as small as 0.01\% for the Q5 extrapolation.
Similar behaviour of the extrapolation schemes is observed for
the point near the avoided crossing. Both extrapolations, with $\alpha=2$
and $\alpha=3$, converge smoothly towards the basis-set limit,
but the convergence is not as fast as in the case of the global minimum.
In contrast to the global minimum, there is no problem here with
overshooting the basis-set limit. For each pair of cardinal numbers $(X-1)X$
the extrapolation with $\alpha=2$ gives results slightly more favourable
than using $\alpha=3$, with the smallest error of $0.19\%$ for the
Q5 extrapolation.
For the saddle point and local minimum, the convergence of the
relative errors is not as smooth as for the global minimum: the
relative error for $X$=D is surprisingly small. This is obviously
accidental and does not reflect particularly high quality of the
cc-pVDZ basis set. Indeed, when the cc-pVDZ results are employed in
Eq.~(\ref{extrapol}), the extrapolation worsens the accuracy: the
errors for the DT extrapolation are much larger than the errors for
both the $X$=D and $X$=T plain results, independent of the value of the
$\alpha$ extrapolation parameter. Starting from $X$=T, the plain
results smoothly approach the basis-set limit, though the convergence
is clearly slower than in the case of the global minimum. The
extrapolation with $\alpha = 2$ is unsystematic and unpredictable, as
in the case of the global minimum, while that with $\alpha = 3$
smoothly approaches the basis-set limit. The errors of the Q5
extrapolation with $\alpha=3$ are $-0.49\%$ for the saddle point and
$-0.13\%$ for the local minimum.
Patkowski and Szalewicz \cite{Patkowski:2010} recently investigated Ar$_2$ with the CCSD(T)-F12
method. They found that the F12a and F12b variants \cite{ccf12} gave significantly different
results. They also concluded that, for Ar$_2$, calculations
with explicitly correlated functions
cannot yet compete with calculations employing extrapolation based on conventional orbital basis sets.
Indeed, while their orbital results converged smoothly towards the extrapolated results,
the CCSD(T)-F12a and CCSD(T)-F12b
results behaved erratically with respect to both the orbital and the extrapolated results. Table
\ref{tab0} shows that this is not the case for the
Li--LiH system. In our case the CCSD(T)-F12a and CCSD(T)-F12b results are quite similar and are
fully consistent with the plain and extrapolated results with conventional basis sets.
It should be stressed, however, that Ar$_2$ is bound mostly by dispersion forces,
while the main source of the bonding in Li--LiH is the induction energy, which is less sensitive to
the basis-set quality. This may at least partly explain the success of the CCSD(T)-F12 calculations
for Li--LiH.
Finally, it is important to note here that, while the interaction energy at the
characteristic points varies considerably with the basis set and
extrapolation method, the positions of the points (i.e., the distance
$R$ and angle $\theta$ at which the characteristic points occur) remain
practically unaffected by the choice of the basis set and extrapolation
scheme.
To analyze the convergence of the CCSD and CCSD(T) models in the
$N$-electron space, Fig.~\ref{fig2} compares the characteristic points
(global minimum, saddle point, local minimum, and near the avoided crossing)
of the Li--LiH
potential calculated at the CCSD / cc-pVQZ and CCSD(T) / cc-pVQZ levels
of theory with the characteristic points obtained at the FCI / cc-pVQZ
level. The $1\sigma_{\mathrm{LiH}}$ and $1s_{\mathrm{Li}}$ orbitals
were kept frozen in the calculations. As expected, the $N$-electron
error is reduced by a factor of 3 to 4 when the approximate triples
correction is included in the calculations. It can also be seen from
the figure that the global minimum is the most sensitive and the local
minimum is the least sensitive to the description of the electron
correlation.
\subsection{The best estimate of the ground-state Li--LiH potential energy surface}
\label{sub2} Because of the negligible one-electron error in the
CCSD(T)--F12 calculations and to the rather large basis set used in the
FCI / cc-pVQZ calculations, and assuming that the one-electron and
$N$-electron errors are approximately independent, the best estimate of
the ground-state interaction energy surface for the LiH-Li is
\begin{equation}
\label{best}
V^{\rm best} = V^{\rm CCSD(T)-F12} + \delta V_{\rm v-v}^{\rm FCI} + \delta V^{\rm FCI},
\end{equation}
where $V^{\rm CCSD(T)-F12}$ is the CCSD(T) basis-set limit energy
(i.e., the CCSD(T)-F12 result) and the FCI correction, $\delta V_{\rm v-v}^{\rm
FCI}$, is obtained by subtracting the CCSD(T) / cc-pVQZ
energy from the FCI / cc-pVQZ energy, both calculated in the
frozen-core approximation. The quantity $\delta V^{\rm FCI}$ accounts
for the last remaining correction (in the non-relativistic limit),
namely the effects of core-core and core-valence correlation in the FCI
/ cc-pVQZ calculations,
\begin{equation}
\delta V^{\rm FCI} = \delta V^{\rm FCI}_{\rm all-all} - \delta V^{\rm FCI}_{\rm v-v},
\end{equation}
where the subscript ``all'' refers to all electrons correlated.
The quantity $\delta V^{\rm FCI}$ is a measure of the uncertainty in
our best estimate $V^{\rm best}$. To estimate this, we may safely
assume that $\delta V^{\rm FCI}$ is at most as large as the
corresponding $\delta V^{\rm (T)}$,
\begin{equation}
\label{assumptionT}
\delta V^{\rm FCI} \le \delta V^{\rm (T)} = \delta V^{\rm (T)}_{\rm all-all} -
\delta V^{\rm (T)}_{\rm v-v},
\end{equation}
where
\begin{equation}
\delta V^{\rm (T)}_{\rm all-all} = V^{\rm CCSD(T)}_{\rm all-all} - V^{\rm CCSD}_{\rm all-all}
\end{equation}
\begin{equation}
\delta V^{\rm (T)}_{\rm v-v} = V^{\rm CCSD(T)}_{\rm v-v} - V^{\rm CCSD}_{\rm v-v},
\end{equation}
with $V^{\rm CCSD(T)}_{\rm all-all}$, $V^{\rm CCSD}_{\rm all-all}$, $V^{\rm
CCSD(T)}_{\rm v-v}$, and $V^{\rm CCSD}_{\rm v-v}$ denoting
interaction energies calculated at the CCSD(T) / cc-pVQZ or CCSD /
cc-pVQZ level, correlating all electrons or using the frozen-core
approximation, as appropriate. As can be seen from Fig.~\ref{fig2}, the
differences between CCSD(T) and CCSD are, for the characteristic points
of the potential, 2 to 3 times larger (and for the rest of the
potential at least 1.5 times larger) than the differences between FCI
and CCSD(T). Eq.~(\ref{assumptionT}) is therefore actually a
conservative estimate for $\delta V^{\rm FCI}$. The root mean square
error for $\delta V^{\rm (T)} / \delta V^{\rm (T)}_{\rm all-all}$, over the whole
potential is $4.1\%$. We thus consider that our best estimate of the
ground-state interaction energy for LiH--Li, Eq.~(\ref{best}), has a
(conservative) total uncertainty of $5\%$ of the FCI correction
($\delta V^{\rm FCI}_{\rm v-v}$). The analysis of the Li--LiH potential in
the remainder of this paper is based on the interaction energies
obtained using Eq.~(\ref{best}), unless otherwise stated.
To justify our error estimation we have performed calculations with
all electron correlated at the FCI level for the set of characteristic points
of the potential. Due to the immense memory requirements of the FCI
calculations with seven electrons we were able to apply the cc-pVDZ
basis set only. The FCI/cc-pVDZ results together with the CCSD(T)/cc-pVDZ,
both with and without the frozen-core approximation, are presented in Table \ref{tabF}.
The error in the FCI correction calculated with frozen core is
as small as 0.76 \% for the examined points.
We may see that the approximation with the FCI
valence correction added to CCSD(T), Eq. (\ref{Pig}), reproduces the exact FCI results
with accuracy better than 1\% of the FCI correction
($\delta V^{\rm FCI}_{\rm v-v}$).
This confirms our estimate of
$5\%$ uncertainty in the FCI correction $\delta V^{\rm FCI}_{\rm v-v}$.
\subsection{Features of the ground-state potential energy surface}
\label{sub3} In Table \ref{tab1} we have listed the characteristic
points of the potential energy surfaces of the ground state, which
correlates at long range with Li($^2$S) + LiH (${\rm X}\, ^1\Sigma^+$),
and the first excited state, which correlates at the long range with
Li($^2$P) + LiH (${\rm X}\, ^1\Sigma^+$). Both these states are of
$^2A^\prime$ symmetry in the $C_{s}$ point group. The latter is
included in Table \ref{tab1} since, as will be discussed in the next
section, it shows an avoided crossing with the ground-state potential
for the linear LiH--Li geometry. Table \ref{tab1} shows that the
interaction potential for the ground state of Li--LiH is deeply bound,
with a binding energy of 8743 cm$^{-1}$ at the global minimum. The
global minimum is located at a skew geometry with $R_e$=4.40 bohr and
$\theta_e$=46.5$^\circ$, and is separated by a barrier around $R$=6.3
bohr and $\theta$=136.0$^\circ$ from a shallow local minimum at the
linear Li--LiH geometry. The local minimum is at $R$=6.56 bohr, with a
well depth of only 1623 cm$^{-1}$. The excited-state potential shows
only one minimum, at $R$=5.66 bohr, with a binding energy of 4743
cm$^{-1}$.
A contour plot of the ground-state potential is shown in the left-hand
panel of Fig.\ \ref{fig3}, while the full-CI correction to the CCSD(T)
potential, $\delta V_{\rm v-v}^{\rm
FCI}$, is shown in the right-hand panel. The correction is very
small compared to the best potential. It amounts to 0.4\% around the
global minimum, and approximately 1\% at the local minimum. Thus, our
estimated error of the calculation, 5\% of the full-CI correction,
translates into 0.05\% error in the potential itself. We would like to
reiterate here that such a small error was achieved not only because
the interelectron distance was included explicitly in the {\em ab
initio} CCSD(T)--F12 calculations, but also because of the very small
valence-valence correlation beyond the CCSD(T) level. The smallness of
the valence-valence correlation beyond the CCSD(T) level is not so
surprising, since Li--LiH has only three valence electrons, and the
exact model for a three-electron system would be CCSDT, coupled-cluster
with single, double, and exact triple excitations \cite{ccsdt}. Our
results show that the triples contribution to the correlation energy
beyond the CCSD(T) model for the valence electrons is very small.
The potential for the ground state of Li--LiH is very strongly
anisotropic. This is easily seen in the left-hand panel of Fig.\
\ref{fig3}, and in Fig.\ \ref{fig4}, which shows the expansion
coefficients of the potential in terms of Legendre polynomials
$P_l(\cos\theta)$,
\begin{equation}
\label{anisotropy}
V(R,\theta) = \sum_{l=0}^{\infty} V_l(R) \, P_l(\cos\theta).
\end{equation}
Here, $V_0(R)$ is the isotropic part of the potential and
$\left\{V_l(R)\right\}_{l=1}^{\infty}$ is the set of anisotropic
coefficients. Fig.\ \ref{fig4} shows that, around the radial position
of the global minimum, $R$=4.36 bohr, the first anisotropic
contribution to the potential, $V_1(R)$, is far larger than the
isotropic term, $V_0(R)$. The higher anisotropic components, with $l=2,
3$, etc., contribute much less to the potential.
As mentioned above, calculations of collision dynamics at ultralow
temperatures require accurate values of the long-range potential
coefficients, Eq.\ (\ref{lr1}). Some important scattering properties,
such as the mean scattering length and the heights of centrifugal
barriers, are determined purely by the Van der Waals coefficients. The
calculated coefficients for Li--LiH are presented in Table \ref{tab2}.
Because of the large dipole moment of lithium hydride and the
relatively high polarizability of the lithium atom, the lowest-order,
most important, coefficients are dominated by the induction
contribution. For example, the induction part of $C_6^0$ and $C_6^2$ is
887 a.u., which accounts for 71\% of $C_6^0$ and 98\% of $C_6^2$.
\section{Interaction between the ground and excited states}
\label{sec3}
\subsection{Low-lying excited state potential, nonadiabatic coupling
matrix elements, and diabatic potentials} \label{sub4}
We encountered convergence problems with CCSD(T) calculations
at the linear LiH--Li geometry around $R$=5.6 bohr, due to the
presence of a low-lying excited state.
The excited state correlates with the
Li($^2$P)+LiH(X$^1\Sigma^+$) dissociation limit, but closer
investigation revealed that, at linear Li--HLi geometries near
the crossing with the ground state, it has ion-pair character,
Li$^+$($^1$S) + LiH$^-$($^2\Sigma$). The ion-pair state itself
has a crossing near $R=$ 9 bohr with the lowest $^2A^\prime$
state correlating with Li($^2$P) + LiH(X$^1\Sigma$). This is
shown schematically in
Fig.\ \ref{fig5}. Away from
linear Li--HLi geometries, the excited state has covalent character and
remains below the ion-pair state all the way to dissociation. The
avoided crossing between the ground state and the first excited state
is at $R$=5.66 bohr, which is near the minimum of the ground-state
potential at the linear geometry, and the energetic distance between
the two states at the avoided crossing is only 94 cm$^{-1}$.
In order to investigate how far the excited state may affect the scattering dynamics,
we computed the full potential energy
surface for the excited state in question by means of
equation-of-motion coupled-cluster method with single and double
excitations (EOM-CCSD) \cite{Monkhorst:77,Sekino:84,Stanton:93}
implemented in the {\sc qchem} code \cite{qchem}, using the orbital cc-pVQZ basis set.
Cuts through the
ground-state and excited-state potential energy surfaces at selected
values of the angle $\theta$ are shown in Fig.\ \ref{fig6}. It may be
seen that it is only near the linear LiH--Li geometry that the two
states come very close together. If we distort the system from the
linear geometry, the excited state goes up in energy very rapidly, and
around the global minimum energy, $\theta\approx 45^\circ$, it is
almost 6000 cm$^{-1}$ above the ground state. The importance of the
possible interaction between the ground and excited states can be
measured by analyzing the (vectorial) nonadiabatic coupling matrix
elements $\boldsymbol\tau_{12}$, defined as $\boldsymbol\tau_{12}= \langle\Psi_1|\nabla \Psi_2\rangle$,
where $\nabla$ is the gradient operator of the position vector
$\bbox{R}$ and $\Psi_1$ and $\Psi_2$ are the wave functions of the two
lowest states. On the two-dimensional surface, we may define radial
$\tau_{12,R}=\langle\Psi_1|\partial \Psi_2/\partial R\rangle$ and
angular $\tau_{12,\theta}=\langle\Psi_1|\partial \Psi_2/\partial
\theta\rangle$ components of the vector $\boldsymbol\tau_{12}$.
We evaluated $\boldsymbol\tau_{12}$ for all $(R,\theta)$ geometries by
means of the multireference configuration interaction method limited to
single and double excitations (MRCI) \cite{WK88,Knowles:88}, using the
{\sc molpro} code \cite{MOLPRO2008}.
The nonadiabatic coupling is largest
when the two states are very close in energy, as it can be seen in Fig.\ \ref{fig6}.
While at $\theta=0^\circ$ the radial component of the
nonadiabatic coupling approaches the Dirac delta form near the crossing
point $R_{\rm ac}$,
with increasing angle it becomes a broad function of approximately
Lorentzian shape. [Note the different scales on the vertical axes of
the different panels.]
The transformation from the adiabatic representation to a diabatic
representation may be expressed in terms of a mixing angle $\gamma$,
\begin{equation}
H_{1}=V_{2}\sin^2\gamma+ V_{1}\cos^2\gamma,
\; \; \; \;
H_{2}=V_{1}\sin^2\gamma+ V_{2}\cos^2\gamma,
\; \; \; \;
H_{12}=(V_{2}-V_{1})\sin\gamma \cos\gamma,
\label{adiab}
\end{equation}
where $V_{1}$ and $V_2$ are the ground-state and excited-state
adiabatic potentials, $H_1$ and $H_2$ are the diabatic potentials, and
$H_{12}$ is the diabatic coupling potential.
In principle, the mixing angle $\gamma$ may be
obtained by performing line integration of the nonadiabatic coupling
$\boldsymbol\tau_{12}$,
\begin{equation}
\gamma(\boldsymbol R)= \gamma(\boldsymbol R_0)+\int_{\boldsymbol R_0}^{\boldsymbol R}
\boldsymbol\tau_{12} \cdot d{\boldsymbol l},
\label{gamma}
\end{equation}
where $\boldsymbol R_0$ is the starting point of the integration. For
polyatomic molecules, however, the mixing angle $\gamma$ obtained by
integrating this equation is non-unique due to the contributions from higher states.
To circumvent the problem of path dependence, one may
assume that we deal with an ideal two-state model.
In our case, however, the ion-pair surface
Li$^+$($^1$S)+LiH$^-$($^2\Sigma$) shows another crossing at small
angles and large distances, $\theta \le 15^\circ$ and $R\approx 9$
bohr, with another excited-state potential that correlates with the
Li$(^2{\rm P})$+LiH($^1\Sigma$) dissociation limit. Thus a third state
$\Psi_3$ comes into play and a two-state model is not strictly valid.
The energy of the first excited state goes up very rapidly
with the angle $\theta$, and at the same time the contribution of the
ion-pair configuration to the wave function of the first excited state,
$\Psi_2$, diminishes rapidly. Fortunately, the nonadiabatic coupling
matrix elements between the two lower states $\boldsymbol\tau_{12}$ and
between the two higher states $\boldsymbol\tau_{23}$ are well isolated.
The maximum of $\boldsymbol\tau_{12}$ is separated from the maximum of
$\boldsymbol\tau_{23}$ by more than 4 bohr; the locations of the
crossing points between the surfaces for $\theta=0$ are shown in
Fig.\ \ref{fig5}. Moreover, the coupling
$\boldsymbol\tau_{13}$ between the ground state and the third state is
negligible over the whole configurational space. Thus, following the
discussion of Baer {\it et al.} \cite{Baer:2002} on the application of
the two-state model, we conclude that the necessary conditions are
fulfilled for the Li--LiH system. Due to the spatial separation of the
nonadiabatic couplings $\boldsymbol\tau_{12}$ and
$\boldsymbol\tau_{23}$, using the diabatization procedure based on the
two-state model is justified.
It is worth noting
that in our particular case we could not use the so-called
quasi-diabatization procedure \cite{Simah}, since it is not possible to
assign a single-reference wave function. This is due to the fact that
the excited state shows admixture from the ion-pair state.
As the starting point of the integration in Eq.\ (\ref{gamma}),
we chose $R=20$ bohr and $\theta=0^\circ$ and followed a radial
path along $\theta=0^\circ$ and subsequently angular paths at
constant $R$. The diabatic potentials were then generated
according to Eqs.\ (\ref{adiab}). Contour plots of the
adiabatic, diabatic, and coupling potentials, and of the mixing
angle $\gamma$, are presented as functions of $R$ and $\theta$
in Fig.\ \ref{fig7}. We consider first the mixing angle
$\gamma$, which is plotted in the bottom right-hand panel of
Fig.\ \ref{fig7}. As expected, the mixing angle shows an
accumulation point at $\theta=0^\circ$ at a distance $R$
corresponding to the closely avoided crossing between the
ground and excited states.
For $\theta=180^\circ$, the mixing angle is non-negligible,
even at large distances. The coupling potential $H_{12}$
vanishes quite slowly with distance $R$, as $R^{-3}$. For the
coupling between the ground and ion-pair states, this
long-range decay is exponential, because of the different
dissociation limits of the two surfaces. As expected, at large
distances the two diabatic surfaces approach the respective
adiabatic surfaces. The diabatic surface that correlates
asymptotically with the excited-state Li($^2$P)+LiH surface has
an important contribution from the ground-state adiabatic
potential only inside the avoided crossing and at small angles
$\theta$. The diabatic surface that correlates asymptotically
with the ground state resembles the ground-state adiabatic
surface rather less closely, especially at large values of
$\theta$.
The coupling between the diabatic states is small over a
significant region of $\theta$ and LiH bond length $r$ in the
vicinity of the crossing. Physically, this means that the
dynamics will be strongly nonadiabatic in this region, and to
take this rigorously into account would require a full
two-state treatment of the dynamics. However, there are no open
channels that involve the second surface, and any collisions
that cross onto it must eventually return to the original
surface. Its effect in collision calculations will therefore be
at most to cause a phase change in the outgoing wavefunction.
\subsection{Conical intersection}
\label{sub5} It is well known that potential energy surfaces for
homonuclear triatomic systems composed of hydrogen \cite{h3} or lithium
atoms \cite{li3} show conical intersections at equilateral triangular
geometries. Analogous behaviour may be expected for Li$_2$H, at
geometries where the two lithium atoms are equivalent, i.e., $C_{\rm
2v}$ geometries. Thus far, our discussion of the potential for Li--LiH
has been restricted to two dimensions with the bond length of the LiH
molecule fixed at its equilibrium value, and no conical intersection
was observed. However, if we start to vary the bond length of the LiH
molecule, conical intersections show up immediately.
At $C_{\rm 2v}$ geometries, with the two LiH bond lengths equal, there
are two low-lying electronic states, of $^2$A$_1$ and $^2$B$_2$
symmetries, that cross each other as a function of the internuclear
coordinates. Fig.\ \ref{fig8} shows contour plots of the two potential
energy surfaces and of the difference between them, and the top panel
of Fig.\ \ref{fig9} summarizes some key features of the surfaces.
The $^2$A$_1$ state has a minimum energy of $-8825$ cm$^{-1}$ at
$r$(LiH) = 3.22 bohr and an Li-H-Li angle of $95^\circ$. MRCI
calculations with all coordinates free to vary confirm that this is
indeed the absolute minimum geometry.
There is also a saddle point
on the $^2$A$_1$ surface at a linear H-Li-H geometry with $r$(LiH) =
3.04 bohr and an energy of $-4992$ cm$^{-1}$, which is a minimum in $D_{\infty{\rm h}}$ symmetry.
The $^2$B$_2$ state has a
minimum energy of $-5136$ cm$^{-1}$ at $r({\rm LiH})=$ 3.17 bohr at a
linear Li-H-Li geometry. The $^2$A$_1$ saddle point and $^2$B$_2$ linear minimum have
symmetries $^2\Sigma_g^+$ and $^2\Sigma_u^+$ respectively in
$D_{\infty{\rm h}}$ symmetry, but mix and distort if the constraint on
the LiH bond lengths is relaxed, to form a $^2\Sigma^+$ state in
$C_{\infty{\rm v}}$ symmetry with a minimum at a linear geometry with
$r$(LiH) distances of 3.00 and 3.33 bohr and an energy of $-5323$
cm$^{-1}$. Even this is a saddle point with respect to bending on the
full potential surface in $C_s$ symmetry.
The $^2$A$_1$ and $^2$B$_2$ states are of different symmetries at
$C_{\rm 2v}$ geometries, but both are of $^2$A$^\prime$ symmetry when
the geometry is distorted from $C_{\rm 2v}$ to $C_{\rm s}$ symmetry.
The two states therefore mix and repel one another at geometries where
the two LiH bond lengths are different, but a seam of conical
intersections runs along the line where the energy difference is zero
at $C_{\rm 2v}$ geometries.
The fixed LiH distance used in previous sections ($r=3.014$ bohr, shown
as a dashed line on the figure) keeps the $^2$A$_1$ surface just below
the $^2$B$_2$ surface. However, if we allow for the vibrations of LiH,
the seam of conical intersections becomes accessible at near-linear
LiH--Li geometries, where the zero of the energy difference appears for
an Li--H distance only slightly larger than 3.014 bohr. At non-linear
geometries the seam quickly moves to Li--H distances far outside the
classical turning points of the ground vibrational level of free LiH,
which are 2.72 and 3.35 bohr.
It is interesting to compare the features of the conical
intersections in Li$_2$H with those in other triatomic
molecules formed from Li and H atoms: LiH$_2$, Li$_3$ and
H$_3$. In the case of LiH$_2$, the seam of intersections occurs
at highly bent $C_{2v}$ geometries with an angle between the
two Li-H bonds of approximately $30^\circ$ and arises from
degeneracy between the surfaces of A$_1$ and B$_2$ symmetry.
The global minimum of B$_2$ symmetry is located at $r({\rm
LiH})=3.23$ bohr and a bond angle H--Li--H of $28^\circ$
\cite{Yarkony:Li:1998}. This contrasts with Li$_2$H, where the
minimum of B$_2$ symmetry is at a linear Li--H--Li
configuration. The energy of the lowest point on the seam of
intersections is about 9000 cm$^{-1}$ above the
Li($^2$S)+H$_2(X^1\Sigma_g)$ threshold, so that it is
irrelevant for low and medium-energy collisions between H$_2$
and Li in their ground states, though it is important for
quenching of Li($^2$P) by H$_2$
\cite{Yarkony:Li:1998,Martinez:1997}.
The conical intersections for the doublet states of Li$_3$ and
H$_3$ occur at equilateral triangular geometries, where the
ground state is doubly degenerate and has symmetry
$^2$E$^\prime$ in the $D_{3h}$ point group. In the case of
H$_3$, the lowest-energy point on the seam is located at an
energy more than 20000 cm$^{-1}$ above the
H($^2$S)+H$_2(X^1\Sigma_g)$ threshold, so that nonadiabatic
effects are negligible in H+H$_2$ collisions \cite{Chu:2009},
although the conical intersection also produces geometric phase
effects \cite{Juanes-Marcos:2005}. For Li$_3$, the energetics
are essentially different. The lowest-energy point on the seam
is around 4000 cm$^{-1}$ below the
Li($^2$S)+Li$_2(X^1\Sigma_g)$ threshold and only 500 cm$^{-1}$
above the $C_{2v}$ global minimum \cite{Varandas:1998}. This is
likely to produce considerable nonadiabacity in collisions of
Li$_2$ with Li.
To conclude, in all the triatomic molecules formed from H and
Li there are seams of crossings that occur at configurations of
the highest possible symmetry, either $C_{2v}$ or $D_{3h}$. For
Li$_3$ and Li$_2$H the conical intersections are accessible
during atom-molecule collisions, while for H$_3$ and LiH$_2$
nonadiabatic processes are unimportant if the colliding
partners are in their ground states and have relatively low
kinetic energy.
\section{Reaction channels}
\label{sec5} Several reaction channels exist that might affect
sympathetic cooling \cite{pz3} in Li+LiH. These are the exchange
reaction,
\begin{equation} {\rm LiH} + {\rm Li} \rightarrow {\rm Li} + {\rm HLi}
\label{exchange}
\end{equation}
and two insertion reactions,
\begin{equation} {\rm LiH} + {\rm Li} \rightarrow {\rm Li_2} + {\rm H},
\label{ins}
\end{equation}
producing Li$_2$(X$^1\Sigma_g^+$) and Li$_2$($a^3\Sigma_u^+$) plus a
ground-state H atom. The energetic location of the entrance and exit
channels of these reactions, as well as those of the potential minima
for linear and $C_{\rm 2v}$ geometries, are shown in the upper panel of
Fig.\ \ref{fig9}. The insertion reactions are highly endothermic, with
an energy difference between the entrance and exit channels of the
order of 12000 cm$^{-1}$ and 22500 cm$^{-1}$ for
Li$_2$(X$^1\Sigma_g^+$)+H and Li$_2$($a^3\Sigma_u^+$)+H, respectively.
To make the discussion more quantitative, Fig.\ \ref{fig9} also shows
two-dimensional plots of the energy as functions of the internal
coordinates. For the exchange reaction, we held Li--H--Li at linear
geometries and varied the distances from the two lithium atoms to the
hydrogen atom. For the insertion reaction, Li--Li--H was kept bent,
with the angle $\angle{\rm (HLi1Li2)}$ held constant at the $C_{\rm
2v}$ equilibrium value 42.5$^\circ$, while the Li--Li and Li--H
distances were varied. To make the plots consistent with the
correlation diagram shown on the upper panel, the zero of energy was
fixed at that of Li--LiH separated to infinite distance with the Li--H
bond length fixed at the monomer equilibrium value.
Let us consider the exchange reaction first. The two-dimensional cut
through the potential energy surface is presented in the left-hand
panel of Fig.\ \ref{fig9}. The potential energy surface of linear
Li$_2$H has two equivalent minima with an energy of $-5323$ cm$^{-1}$,
separated by a small barrier 187 cm$^{-1}$ high. The linear minima are
in any case substantially above the absolute minimum (8825 cm$^{-1}$),
so this small barrier will have no important effect on the collision
dynamics. The exchange reaction produces products that are
indistinguishable from the reactants, so reactive collisions cannot be
distinguished from inelastic collisions experimentally (unless the two
Li atoms are different isotopes).
An analogous two-dimensional cut through the potential energy surface
corresponding to Li$_2$(X$^1\Sigma_g$)+H products is presented in the
right-hand panel of Fig.\ \ref{fig9}. The plot illustrating the
reaction to form Li$_2$($a^3\Sigma_u$)+H products is not reported, as
the reaction is even more endothermic. The surface includes the
absolute minimum at an energy of $-8825$ cm$^{-1}$. The entrance
channel for this reaction corresponds to an Li--H distance of 3.014
bohr at large Li--Li distance, while in the exit channel the Li--Li
distance is approximately 5.05 bohr when the Li--H distance is very
large. However, this reaction cannot occur at low collision energies.
\section{Summary and Conclusions}
\label{sec6} In the present paper, state-of-the-art {\em ab initio}
techniques have been applied to compute the ground-state potential
energy surface for Li--LiH in the Born-Oppenheimer approximation. The
interaction potential was obtained using a combination of the
explicitly-correlated unrestricted coupled-cluster method with single, double, and
approximate noniterative triple excitations [UCCSD(T)--F12] for the
core-core and core-valence correlation, with full configuration
interaction for the valence-valence correlation. The main results of
this paper can be summarized as follows:
\begin{enumerate}
\item The Li--LiH system is strongly bound: if the LiH bondlength
is held fixed at the monomer equilibrium distance of 3.014
bohr, the potential energy surface has a global minimum 8743
cm$^{-1}$ deep at a distance $R$=4.40 bohr from the lithium
atom to the center of mass of LiH, and a Jacobi angle
$\theta=46.5^\circ$. It also shows a weak local minimum 1623
cm$^{-1}$ deep at the linear Li--LiH geometry for $R$=6.56
bohr, separated from the global minimum by a barrier at
$R$=6.28 bohr and $\theta=136^\circ$. If the LiH bond length is
allowed to vary, the potential minimum is at a depth of 8825
cm$^{-1}$, at a $C_{\rm 2v}$ geometry with LiH bond length of
3.22 bohr and an Li-H-Li angle of $95^\circ$.
\item The full-CI correction for the valence-valence correlation to
the explicitly correlated CCSD(T)--F12 potential is very small.
The remaining error in our calculations is due to the neglect
of the core-core and core-valence contributions, and is
estimated to be of the order of 0.05\% of the total potential.
\item To evaluate the performance of the conventional
orbital electron-correlated methods, CCSD and CCSD(T),
calculations were carried out using
correlation-consistent polarized valence $X$-tuple zeta
basis sets, with $X$ ranging from D to 5, and a very
large set of mid-bond functions. Simple two-point
extrapolations based on the single-power laws $X^{-2}$
and $X^{-3}$ for the basis-set truncation error
reproduce the CCSD(T)--F12 results for the
characteristic points of the potential with an error of
0.49\% at worst.
\item The potential for the ground state of Li--LiH is strongly
anisotropic. Around the distance of the global minimum, the
isotropic potential $V_0(R)$ is almost two times smaller than
the first anisotropic contribution $V_1(R)$. Higher anisotropic
components, with $l=2, 3$, etc., do not contribute much to the
potential.
\item At the linear LiH--Li geometry, the ground-state
potential shows a close avoided crossing with the first
excited-state potential, which has ion-pair character
around the avoided crossing point. The full potential
energy surface for the excited state was obtained with
the equation-of-motion method within the framework of
coupled-cluster theory with single and double
excitations. The excited-state potential has a single
minimum 4743 cm$^{-1}$ deep for the linear LiH--Li
geometry at $R$=5.66 bohr. The energy difference
between the ground and excited states at the avoided
crossing is only 94 cm$^{-1}$. An analysis of the
nonadiabatic coupling matrix elements suggests that
dynamics in the vicinity of the avoided crossing will
have nonadiabatic character.
\item When stretching the LiH bond in the Li--LiH system, a seam of
conical intersections appears for $C_{\rm 2v}$ geometries,
between the ground state of $^2$A$_1$ symmetry and an excited
state of $^2$B$_2$ symmetry. At the linear LiH--Li geometry,
the conical intersection occurs for an Li--H distance which is
only slightly larger than the equilibrium distance of the LiH
monomer, but for significantly non-linear geometries it moves
to Li--H distances far outside the classical turning points of
LiH.
\item The Li--LiH system has several possible reaction channels: an
exchange reaction to form products identical to the reactants,
and two insertion reactions that produce
Li$_2$($a^3\Sigma_u^+$) and Li$_2$(X$^1\Sigma_g^+$) plus a
ground-state hydrogen atom. The insertion reactions are highly
endothermic, with the energy difference between the entrance
and exit channels of the order of 12000 cm$^{-1}$ and 22500
cm$^{-1}$ for Li$_2$(X$^1\Sigma_g^+$)+H and
Li$_2$($a^3\Sigma_u^+$)+H, respectively.
\end{enumerate}
In a subsequent paper \cite{subs} we will analyze the dynamics of
Li--LiH collisions at ultralow temperatures, based on our best {\em ab
initio} potential. We will analyze the impact of the present
inaccuracies in the {\em ab initio} electronic
structure calculations, and discuss the prospects of sympathetic
cooling of lithium hydride by collisions with ultracold lithium atoms.
\acknowledgments{ We would like to thank Dr.~Micha{\l} Przybytek for
his invaluable technical help with the FCI calculations. We acknowledge
the financial support from the Polish Ministry of Science and Higher
Education (grant 1165/ESF/2007/03) and from the Foundation for Polish
Science (FNP) via Homing program (grant HOM/2008/10B) within EEA
Financial Mechanism. We also thank EPSRC for support under
collaborative project CoPoMol of the ESF EUROCORES Programme EuroQUAM.}
\newpage
|
{
"timestamp": "2011-02-22T02:04:17",
"yymm": "1009",
"arxiv_id": "1009.4312",
"language": "en",
"url": "https://arxiv.org/abs/1009.4312"
}
|
\section{The CUT method\label{struct:CUT}}
\subsection{Homogeneous flow equation}
Probably, unitary transformations are one of the most widely used techniques
in studies on Hamiltonians.
They render a description of a Hamiltonian possible
in a more appropriate basis in which
the physical properties can be studied more easily.
Most desirably, every Hamiltonian
can be diagonalized by a certain unitary transformation. Unfortunately, this
transformation is usually unknown.
The basic idea of the CUT-method is not to search for such a transformation in
one step, but to bring the Hamiltonian successively closer to a simpler shape
by a series of infinitesimal transformation. Therefore, a continuous flow
parameter $\ell$ is introduced that parametrizes the continuous unitary
transformation $U(\ell)$. The Hamiltonian is considered to become a function
$H(\ell)=U(\ell)H^{(b)}U^\dagger(\ell)$ of this parameter. In this way,
the initial (bare) Hamiltonian $H^{(b)}$ is linked continuously by a unitary
transformation to the renormalized Hamiltonian showing the intended structure
$H^{(r)}=H(\infty)$ in the limit of infinite $\ell$. By derivation with respect
to $\ell$, one obtains the flow equation
\begin{subequations}
\begin{align}
\partial_\ell H(\ell)=&\frac{\partial U(\ell)}{\partial \ell}U^\dagger(\ell)
H(\ell)+H(\ell)U(\ell)\frac{\partial U^\dagger(\ell)}{\partial\ell}
\\
=&\left[\eta(\ell),H(\ell)\right].
\end{align}
\label{eq:flowequation}
\end{subequations}
The antihermitian generator $\eta$ of the transformation reads
\begin{align}
\eta(\ell)=\frac{\partial U(\ell)}{\partial \ell}U^\dagger(\ell)=
-\eta^\dagger(\ell).
\label{eq:generator}
\end{align}
Equation \eqref{eq:flowequation} is linear differential equation for
the Hamiltonian. We
emphasize that also all intermediate Hamiltonians $H(\ell)$ conserve the full
information of the system because they are only written in a different basis.
Since the basis has changed during the flow, observables may not be calculated
directly using their bare operator $\mathcal{O}^{(b)}$ but have also to be
transformed by a similar flow equation
\begin{align}
\partial_\ell \mathcal{O}(\ell)=\left[\eta(\ell),\mathcal{O}(\ell)\right]
\label{eq:obsequation}.
\end{align}
The transformation of observables was used first by Kehrein and Mielke to
determine correlation functions for dissipative bosonic systems
\cite{Kehrein1997,Kehrein1998}.
\subsection{Generator schemes}
Up to here, the problem of diagonalization has only been
recast in the form of determining an appropriate generator $\eta(\ell)$.
The key ingredient of
the CUT-method is to choose the generator as manifestly antihermitian
operator depending on the flowing Hamiltonian.
We denote the superoperator $\hat \eta:
H(\ell)\to \eta(\ell)=\hat\eta[H(\ell)]$ as \emph{generator scheme} to
distinguish between the mapping $\hat\eta$ and the function $\eta(\ell)$.
In this way, the flow equation for the Hamiltonian \eqref{eq:flowequation}
becomes non-linear, while the transformation of observables
\eqref{eq:obsequation} stays linear.
The generator scheme has to be designed in a way that the flow equation has
attractive fixed points where the Hamiltonian has the desired structure.
In this manner, (block-)diagonality can be obtained by merely integrating
the flow equation \cite{Wegner1994,Mielke1998,Knetter2000,Dusuel2004}.
For the first generator scheme introduced by Weg\-ner \cite{Wegner1994},
the Hamiltonian
$H(\ell)=H_\text{d}(\ell)+H_{\text{nd}}(\ell)$ has to be decomposed into a
diagonal $H_\text{d}$ and a non-diagonal part $H_{\text{nd}}$. The generator
is defined as a commutator
\begin{align}
\eta(\ell)=\widehat \eta_\text{W}[H(\ell)]
=\left[H(\ell),H_{\text{nd}}(\ell)\right]=
\left[H_\text{d}(\ell),H_{\text{nd}}(\ell)\right]
\end{align}
of the diagonal and non-diagonal-part of the Hamiltonian.
One directly realizes that a vanishing non-diagonality yields a fixed point of
the flow. The proof of convergence for unapproximated systems was given by
Wegner \cite{Wegner1994} for finite
matrices and extended to infinite systems by Dusuel and
Uhrig \cite{Dusuel2004}. The generator decouples eigen-subspaces of different
energy eigen-values, but it is not able to treat degeneracies. In his original
work
concerning the $n$-orbital model \cite{Wegner1994}, Wegner noticed
divergences. He could avoid them via taking only terms violating the number of
quasiparticles into account in the definition of $H_{\text{nd}}$ aiming at
block-diagonality instead of diagonality. In this manner, the complexity of
the a problem can still be reduced significantly because
different quasiparticle spaces can be studied separately.
\label{pos:cut-mku}
To overcome the problem of residual off-diagonality due to degeneracies,
Mielke \cite{Mielke1998} introduced a generator scheme on the
matrix level based on a sign function of index differences
$\eta_{ij}=\text{sign}(i-j)h_{ij}$ that always yields a diagonal
Hamiltonian. Independently, Knetter and Uhrig \cite{uhrig98c,Knetter2000}
developed a similar scheme which concentrates more generally on a quasiparticle
picture. In their approach, the Hamiltonian $H(\ell)=\sum_{ij}H^i_j(\ell)$ is
decomposed into different blocks $H^i_j$ of terms with
respect to the number $i$ of
quasiparticles created and the number $j$ of quasiparticles annihilated by the
term. In this notation, the generator scheme acts as
\begin{align}
\widehat \eta_{\text{pc}}[H(\ell)]=\sum\limits_{i,j} \text{sgn}\left(i-j\right)
H^i_j(\ell).
\end{align}
In the limit of infinite $\ell$, the Hamiltonian converges to a
block-diagonal, quasiparticle conserving structure if the spectrum is bounded
from below \cite{Mielke1998,Knetter2000,Dusuel2004}. Blocks with $i\neq j$
decay exponentially with rising $\ell$. During the flow, the quasiparticle
spaces are ordered ascending to their energy
eigen-value\cite{Mielke1998,Heidbrink2002}. This implies that the vacuum state,
i.e., the state with $j=0$, is mapped to the ground state of the
Hamiltonian if it is not degenerated.
A special feature of this generator scheme is that it strictly conserves the
block-band structure of the Hamiltonian during the flow.
A similar generator was used by Stein \cite{Stein1997,Stein1998} in a case
where the sign function was not necessary. In contrast to Wegner's generator
scheme, the right-hand side of the flow equation for the Hamiltonian
\eqref{eq:flowequation} is
only quadratic in the Hamiltonian's coefficients instead of cubic.
A recent development in the field of generator schemes is the ground state
generator \cite{Fischer2010}
\begin{align}
\widehat \eta_{\text{gs}}[H(\ell)]=H^i_0(\ell)-H^0_j(\ell).\label{eq:cut-gs}
\end{align}
The definition resembles the particle conserving scheme, but it is designed
to decouple only the zero quasiparticle subspace of a
system, i.e., the ground state if not degenerated.
It was introduced by Fischer, Duffe, and Uhrig to describe quasiparticles
decays, since the picture of conserved
renormalized quasiparticles becomes very cumbersome.
Compared to the
particle conserving scheme, $\widehat\eta_{\text{gs}}$ exhibits an enhanced
numerical stability and saves computational ressources due to its fast
convergence. As a drawback, it does not conserve the block-band structure as
$\widehat\eta_\text{pc}$ does. In this work, we will make use of
both generator schemes, $\widehat\eta_\text{pc}$ and $\widehat\eta_\text{gs}$.
\subsection{Truncation scheme \label{struct:cut-truncscheme}}
\begin{figure*}
\includegraphics[width=\textwidth]{fig1}
\caption{\label{img:feynman}
Decomposition of the diagonal element for the
three-particle state $\left|\alpha\beta\gamma\right\rangle$ into
irreducible interaction
processes. By truncation of three-particle processes, only the
three-particle irreducible term $a^\dagger_\alpha a^\dagger_\beta
a^\dagger_\gamma a_\alpha a_\beta a_\gamma$ is neglected.}
\end{figure*}
One way to solve the flow equation \eqref{eq:flowequation} is to
parameterize the Hamiltonian in second quantization by an
adequate operator basis.
In this way, the differential equations for the coefficients of the operators
are found by computing the commutator on the right hand side
in \eqref{eq:flowequation} and re-expressing the result
again in the operator basis chosen. The evaluation of the commutator generates
new many-body interaction processes that are not present in the initial
Hamiltonian. They have to be incorporated in the Hamiltonian which results in
an iterative calculation of the commutator.
This proliferation of terms generically yields an infinite number of
differential equations which is intractible in practical applications. Four
different strategies have evolved to obtain a closed the set of
differential equations:
On restricting to finite systems, the flow equation can be solved exactly both
on the level of second quantization or on the level of matrix elements.
As a drawback, this
approach is subjected to the same limitation as other finite-size methods.
Furthermore, the differential equation system may still be large enough to
require further approximations for practical applications.
In some special cases, it is possible to obtain a closed system of
equations by identifying a small expansion parameter. In his original work
\cite{Wegner1994}, Wegner investigated
the $n$-orbital model. He was able to close the system of differential
equations in the limit of infinite $n$. However, these
advantageous cases do not
show up in every system or are simply out of interest. In general,
approximations have to
be applied to the system to overcome the problem of proliferation of terms.
In the method of \emph{perturbative continuous unitary transformations}
(P-CUT) introduced by Knetter and Uhrig
\cite{uhrig98c,Knetter2000,Knetter2003},
the non-diagonality $H_\text{nd}$ is considered as small
perturbation to the diagonal part $H_\text{d}$ of the bare Hamiltonian.
In this way, the flow equation can be expanded and therefore used to
apply perturbation theory up to very high orders.
The non-perturbative approach, which we will use in
this work, is dubbed
\emph{self-similar continuous unitary transformations}
(S-CUT). Here a truncation scheme is defined
that incorporates all terms considered to be important
to the problem while other terms are neglected. In view of the numerous
degrees of freedom, the choice of an adequate truncation scheme is a
non-trivial task. It has to respect the system's physical properties. Since
the structure of the Hamiltonian does not change
during the flow, it is named self-similar. As a rule of thumb, a
calculation is considered to be the more reliable the more terms are included
in the truncation scheme.
Due to truncation, the modified flow equation reads
\begin{align}
\partial_\ell H(\ell) = \widehat T \left[ \eta(\ell) , H(\ell) \right].
\label{eq:cut-truncflow}
\end{align}
Here we introduced the superoperator $\widehat T$ which denotes the
application of the truncation scheme.
A natural description for many-body problems is provided by second
quantization. Since most of the low-energy physics can be preserved by low
numbers of suitable quasiparticles, it is useful to define a truncation scheme
neglecting all terms that create or annihilate more than a given number of
quasiparticles. We emphasise that this truncation expressed second
quantization does not imply any restriction of the Hilbert space
which is to be considered a major advantage.
We stress that the action of a Hamiltonian on a state containing $n$
quasiparticles can be split into a sum of irreducible terms affecting
at most $m\le n$ quasiparticles each (see Fig.\ \ref{img:feynman}).
Thus, the truncation of high-particle irreducible processes does not imply the
complete neglect of matrix elements between states of high quasiparticle
numbers, but rather an extrapolation based on lower quasiparticle
irreducible processes.
In extended systems, often additional truncation criteria have to be applied
in order to close the set of differential equations. An obvious
choice for gapped systems with finite correlation lengt is to truncate
according to the real-space range of the physical process generated
by the term under study. We make use of
this real-space truncation in Sect.\ \ref{struct:extend}.
Truncations cause quantitative errors in the calculated
physical quantities and may possibly lead to divergences of the truncated flow,
even though convergence has been proven for the untruncated flow.
In their analysis of the flow equations for the Anderson
model \cite{Kehrein1994}, Kehrein and Mielke used a strict truncation
scheme that includes only contributions of types that are already present
in the bare Hamiltonian.
In a second step, they included a new contribution to the scheme
and analyzed the new equations in order to assess its relevance. In their
classification, terms that affect other matrix elements only by quantitative
deviations and vanish in the limit
$\ell\to\infty$ are denoted as irrelevant, or as marginal if they
converge to a finite value.
Only terms that are able to change the behavior of the flow equations
qualitatively, for instance that cause divergences if they are not treated
properly, were considered to be relevant.
This approach is suited to ensure the correct qualitative
behavior of the effective model derived from the CUT. But
it does not provide a quantitative measure of the truncation errors.
\subsection{Symmetries\label{struct:cut-sym}}
For practical computations, exploiting symmetries of the Hamiltonian
is generally very useful. To study Hamiltonians of systems with an infinite
number of sites, the use of translation symmetry is inevitable.
Let us suppose that we have chosen a basis of operators to
express the Hamiltonian. Each term in the Hamiltonian is given
by one of these basis operators multiplied by a prefactor, its coefficient.
Although the Hamiltonian is symmetric under a group of specific symmetry
transformations, the symmetry of each individual basis operator may be lower.
In this case, the coefficients of several basis operators
fulfil linear conditions to ensure that their combination is
invariant under the symmetry group. By parametrizing the Hamiltonian
in terms of linearly independent symmetric combinations of basis operators,
the number of coefficients to be tracked is significantly reduced
saving both computation time and memory.
Technically this can be done by selecting one operator as a unique
representative from which the complete symmetric combination can be obtained
by taking the sum $\sum_G$ over a specific subgroup of the symmetry group of
the Hamiltonian.
We emphasize that in general one has to clearly distinguish between
the symmetry group of the Hamiltonian and the superoperator $\sum_G$.
Even if the Hamiltonian displays a continuous symmetry such as
the $\text{SU}(n)$ spin rotation symmetry, the number of constraints
that can be derived for the coefficients of the Hamiltonian's terms is
limited. Excluding translation
symmetries, this means that only a finite number of terms
are represented by one representative operator. Therefore the number
of symmetry operations to build the symmetric combination from a single
representative is often
finite although the exploited symmetry is continuous.
In summary, we take the superoperation $\sum_G$ as a technical tool to benefit
from the Hamiltonian's underlying full symmetry group.
Moreover, the precise meaning of $\sum_G$ depends on the
representative under study.
As an example, a representative that shares the whole symmetry of the
Hamiltonian, e.g., unity, is already identical to its corresponding symmetric
combination. To this operator, $\sum_G$ acts as identity. The other limit
is a representative operator
that does not share any of the Hamiltonian's symmetries.
The superoperator $\sum_G$ applied to such a representative has to generate
the fully symmetric combination of basis operators.
Since the sums $\sum_G$ occur on both sides of Eq.\ \eqref{eq:flowequation},
the modified flow equation can be reduced to a representative expression
requiring a sum for only one argument of the commutator.
Correction factors have to be introduced since different representative basis
operators appearing in the commutator give rise to different
numbers of basis operators in the fully symmetric combination
which they represent. Details on the implementation of symmetries in S-CUT
can be found in Ref.\ \onlinecite{Reischl2006}.
\footnote{In Ref.\ \onlinecite{Reischl2006},
$\sum_G$ denotes a sum over the maximal symmetry group for
all representatives. In contrast, the superoperator
$\sum_G$ stands in our notation
only for the specific symmetry operations needed to
generate the fully symmetric combination represented
by the representative basis operator to which $\sum_G$ is applied.
This leads to correction factors for
the symmetrized flow equation instead of correction factors for coefficients
of representatives as used by Reischl.}
\section{Mathematical analysis of truncation errors\label{struct:math}}
\subsection{Effects of truncation\label{struct:math-effects}}
On truncating the flow equation, all information about the truncated terms is
lost. It is common practice to neglect terms which are considered to be
unimportant and to justify this \emph{a posteriori}.
But even if these terms are not subject of the intended analysis of the
effective Hamiltonian, their omission leads to quantitative deviations
for the coefficients of \emph{all} terms in the Hamiltonian because
they are linked by the differential equations.
We stress that the commutator of the generator and a truncated term may
result in terms that comply with the truncation scheme, i.e.,
that we want to compute quantitatively.
Thus the loss of information cannot be limited to certain terms only.
Generically, truncation introduces errors in \emph{all} coefficients
of the Hamiltonian.
Because of truncation, the transformation of a Hamiltonian $H(\ell)$ described
by the truncated flow equation \eqref{eq:cut-truncflow} does not need to be
unitary anymore. Therefore, the spectrum of $H(\ell)$ will be distorted during
the flow. Physical quantities calculated based on the effective Hamiltonian
are affected by finite inaccuracies.
In the following sections \ref{struct:math-split}-E, we present a formalism
to bound these errors rigorously.
An additional physical consequence of truncation can be derived for real-space
truncation schemes which neglect interactions beyond a certain range
$d>d_\text{max}$
(see Sect.\ \ref{struct:cut-truncscheme}). Due to the
formulation in second quantization, the S-CUT method is capable to handle
infinite systems. Nevertheless, the truncation by range affects correlations
on larger length scales.
We observed that the coefficients of representatives in a truncated infinite
system and a truncated periodic system with a certain size
$l\geq L_\text{fin}$ share
the same set of differential equations, if the algebra is local. Any
possibility to observe that the system size is actually finite is masked by
the truncation scheme if the system size is at least
$L_\text{fin}=3 d_\text{max}+1$.
Therefore also the intensive physical properties of an infinite system
determined using S-CUT with truncation range $d_\text{max}$ are
\emph{identical} to those of a finite system with a certain size
$l\geq L_\text{fin}$ and periodic
boundary conditions. The quantity $L_\text{fin}$ can be understood as an
effective size introduced by the real-space truncation scheme.
The mathematical derivation including a numerical verification is given in
Appendix \ref{struct:app-effsize}.
\subsection{Splitting the flow equation\label{struct:math-split}}
To isolate the truncation error of S-CUT, we start from
the full flow equation
\begin{align}
\partial_\ell H(\ell) = [\eta^\prime(\ell),H(\ell)]
\label{eq:fullflow}
\end{align}
for the unitarily transformed Hamiltonian $H(\ell)$ with
$H(\ell=0)=H^{(\text{b})}$. The
generator $\eta^\prime$ appears instead of $\eta$ because in practice the
generator is determined from the truncated Hamiltonian $H^{\prime}$
and not from $H$, see Eq.\ \eqref{eq:gendef}.
We decompose $H$ into two parts: the solution $H^{\prime}$ from a truncated
flow equation and the difference $H^{\prime\prime}=H-H^{\prime}$ between the
truncated and the non-truncated calculation. Next, the flow equation
\eqref{eq:fullflow} can be split into the system
\begin{subequations}
\label{eq:flows}
\begin{align}
\partial_\ell H^{\prime}(\ell) &=&
\widehat T &[\eta^\prime(\ell),H^{\prime}(\ell)]
\label{eq:tflow}\\
\partial_\ell H^{\prime\prime}(\ell) &=& (\mathbbm{1}-\widehat T)
&[\eta^\prime(\ell),H^{\prime}(\ell)]+
[\eta^\prime(\ell),H^{\prime\prime}(\ell)]
\label{eq:iflow}
\end{align}
\end{subequations}
of differential equations for $H^{\prime}$ and $H^{\prime\prime}$.
As initial conditions, we choose
\begin{subequations}
\label{eq:initial}
\begin{align}
H^{\prime}(0) \ =&\ H^{(\text{b})}
\label{eq:tinitial}\\
H^{\prime\prime}(0)\ =&\ 0\label{eq:iinitial}.
\end{align}
\end{subequations}
Obviously, the sum of the equations \eqref{eq:flows}
with the initial condition \eqref{eq:initial} reproduce
the flow equation \eqref{eq:fullflow} with its initial condition.
Up to now, the generator $\eta^\prime$ is not specified.
Using the generator scheme $\widehat\eta$, we define the generator
\begin{align}
\label{eq:gendef}
\eta^\prime(\ell) = \widehat\eta[H^{\prime}(\ell)]
\end{align}
as a function of the \emph{truncated} Hamiltonian.
Note that this choice does not violate the unitarity of the
transformation because the generator $\eta^\prime$
continues to be manifestly antihermitian.
Equation \eqref{eq:tflow} provides a \emph{closed} set of differential
equations for the
coefficients of the truncated Hamiltonian which can be treated by numerical
integration. This leads to an effective Hamiltonian
$H^{\prime}(\infty)=H^{\prime (\text{r})}$ with a structure
determined by the chosen generator scheme.
In contrast, the full Hamiltonian $H$ is transformed by a true unitary
transformation, but does not need to have any special structure in the limit
of infinite $\ell$, since it is transformed like an observable by
$\eta^\prime$. However, it is to be expected that it is close to
$H^{\prime}$ if truncation errors are small.
The difference $H^{\prime\prime}$ stores the complete
\grq non-unitarity\grq\ of the transformation of $H^{\prime}(\ell)$.
Mathematically, Eq.\ \eqref{eq:iflow} describes a transformation of
$H^{\prime\prime}$ via a flow equation with an additional \emph{inhomogeneity}
\begin{align}
\kappa(\ell) = (\mathbbm{1}-\widehat T)
&[\eta^\prime(\ell),H^{\prime}(\ell)]\label{eq:kappa}
\end{align}
depending on $H^{\prime}(\ell)$. This natural emergence of an
\emph{inhomogeneous flow equation} is quite remarkable and has not been
observed before to our knowledge. We emphasise that the number of equations
defining $\kappa(\ell)$ remains finite if $H^{\prime}$ and thus $\eta^\prime$
are restricted by the truncation scheme to a finite number of terms.
Hence the computation of $\kappa(\ell)$
is indeed feasible. Of course, this is not true for $H^{\prime\prime}$.
\subsection{Inhomogeneous flow equation}
To solve the inhomogeneous flow equation \eqref{eq:iflow}, we use the ansatz
\begin{align}
H^{\prime\prime}(\ell) = U(\ell)A(\ell)U^\dagger(\ell)
\end{align}
with $A(0)=H^{\prime\prime}(0)=0$. The unitary transformation $U(\ell)$ is
linked to the generator $\eta^\prime(\ell)$ of the transformation by
Eq.\ \eqref{eq:generator}.
The formal solution for $U(\ell)$ using the $\ell$-ordering operator
$\mathcal{L}$ reads
\begin{align}
U(\ell)=\mathcal{L}\exp \left(\int\limits_0^\ell \eta^\prime(\ell^\prime)
\mathsf{d} \ell^\prime \right).
\end{align}
Using variation of parameters
\begin{subequations}
\begin{align}
\partial_\ell H^{\prime\prime}(\ell) &=
\left[ \eta^\prime(\ell), H^{\prime\prime}(\ell)\right]+
U(\ell)\partial_\ell A(\ell)U^\dagger(\ell)
\\
&\overset{!}{=} \left[\eta^\prime(\ell),H^{\prime\prime}(\ell)\right]+
\kappa(\ell),
\end{align}
\end{subequations}
leads to the equation
\begin{align}
A(\ell) = A(0)+\int\limits_0^\ell U^\dagger(\ell^\prime)\kappa(\ell^\prime)
U(\ell^\prime)\mathsf{d} \ell^\prime.
\end{align}
Therefore, the formal solution of the inhomogeneous flow equation
\eqref{eq:iflow} is given by
\begin{align}
H^{\prime\prime}(\ell)= U(\ell)\left(H^{\prime\prime}(0)+\int\limits_0^\ell
U^\dagger(\ell^\prime)\kappa(\ell^\prime)U(\ell^\prime)
\mathsf{d} \ell^\prime\right)U^\dagger(\ell).
\label{eq:isolved}
\end{align}
This expression \eqref{eq:isolved} has a very direct interpretation:
All contributions
of the inhomogeneity up to the given value of $\ell$ are re-transformed to
$\ell=0$ and summed. This sum is evaluated after a unitary transformation to
the considered flow parameter.
\subsection{Truncation error}
The formal solution \eqref{eq:isolved} enables the calculation of the
distortion of unitarity by the truncated calculation. All effects of the
truncation are stored in $H^{\prime\prime}(\ell)$. Certainly, a direct
calculation is neither practical
nor desirable, because it is equivalent to an untruncated calculation.
But for the derviation of a bound of the truncation error only a
small part of the information is essential.
To assess the quality of the truncation, we are interested in the norm of
$H^{\prime\prime}(\ell)$. In particular, we want to focus on norms
that are unitarily invariant, i.e.,
\emph{invariant under unitary transformations}\footnote{It turned
out that the most appropriate choice is the spectral norm because it
implies rigorous bound on eigen-values, see Sect.\ \ref{struct:math-bound}.} .
We apply the norm to Eq.\ \eqref{eq:isolved}. Due to its unitary invariance,
we obtain
\begin{align}
\left|\left|H^{\prime\prime}(\ell)\right|\right|=\left|
\left|\int\limits_0^\ell
U^\dagger(\ell^\prime)\kappa(\ell^\prime)U(\ell^\prime)\mathsf{d}
\ell^\prime\right|\right|.
\end{align}
In addition, we used $H^{\prime\prime}(0)=0$ from Eq.\ \eqref{eq:iinitial}.
To avoid the complicated integration of an operator-valued function,
we apply the triangle
inequality to the Riemann integral arriving at the upper bound
\begin{subequations}
\begin{align}
\left|\left|H^{\prime\prime}(\ell)\right|\right|&\leq \int\limits_0^\ell
\left|\left|U^\dagger(\ell^\prime)\kappa(\ell^\prime)U(\ell^\prime)\right|
\right|\mathsf{d} \ell^\prime
\\
&=\int\limits_0^\ell \left|\left|\kappa(\ell^\prime)\right|\right|\mathsf{d}
\ell^\prime =: \Lambda_H(\ell),
\label{eq:def_Lambda_H}
\end{align}
\end{subequations}
where again the unitary invariance was used. We define
the derived quantity $\Lambda_H(\ell)$ as \emph{truncation error} of the
transformation. By construction, it is an upper bound for the distance between
$H^{\prime}$ and $H$ measured by the selected norm.
We emphasize that $\Lambda_H(\ell)$ is a scalar function which
depends only on the norm of the truncated terms as defined in Eq.\
\eqref{eq:kappa}.
It starts at zero and increases monotonically with the flow parameter $\ell$.
Because of the finite and constant number of terms
complying with the truncation scheme, the number of contributions to the
inhomogeneity $\kappa(\ell)$ stays also constant
during the flow. The coefficients
of the terms in $\kappa(\ell)$ can be calculated
as functions of the coefficients of
$H^{\prime}(\ell)$ already known by numerical integration.
This is the key simplification compared to the practically impossible
direct calculation of $H^{\prime\prime}(\ell)$ or $H(\ell)$.
In the above analysis, the necessary ingredients are the
flow equation and the truncation scheme. Hence
all considerations can also be carried over to the transformation of
observables. The truncation error of an observable $\mathcal{O}$ can be
estimated analogously by
\begin{align}
\Lambda_\mathcal{O}(\ell) := \int\limits_0^\ell \left|\left|(\mathbbm{1}-
\widehat T)
[\eta^\prime(\ell^\prime),\mathcal{O}^{\prime}(\ell^\prime)]
\right|\right|\mathsf{d} \ell^\prime \geq \left|
\left|\mathcal{O}^{\prime\prime}(\ell)\right|\right|
\end{align}
where $\mathcal{O}$ is decomposed in $\mathcal{O}^{\prime}$ and
$\mathcal{O}^{\prime\prime}$ in analogy to \eqref{eq:initial}.
The generator is defined by the numerically accessible
truncated Hamiltonian $H^{\prime}$.
\subsection{Rigorous bounds for observables\label{struct:math-bound}}
The truncation error $\Lambda_\mathcal{O}$ is a property of the
entire transformation of $\mathcal{O}^{\prime}$ that quantifies the
loss of accuracy by truncation. It is desirable to have rigorous bounds for
the accuracy of physical quantities calculated by truncated CUTs.
Indeed, it is possible to obtain
such bounds by calculating the truncation error defined by the spectral norm
\begin{align}
\left|\left|A\right|\right|_S := \sqrt{\max\ \operatorname{EV}
\left(A^\dagger A\right)}.
\end{align}
For hermitian operators, the spectral norm is identical to the maximum
absolute eigen-value.
We denote the lowest eigen-value of $\mathcal{O}^{\prime}$ by
$\Omega^\prime_\text{min}$ and
the associate eigen-state by $\left|\psi^\prime\right\rangle$.
For the untruncated observable $\mathcal{O}$ we use $\Omega_\text{min}$ and
$\left|\psi\right\rangle$. Since $\mathcal{O}$ is transformed by a unitary
transformation,
$\Omega_\text{min}$ does not change during the flow whereas
$\Omega^\prime_\text{min}$ is changed due to truncation, for illustration
see Fig.\ \ref{plot:toy-groundstate}.
The spectral norm of $\mathcal{O}^{\prime\prime}$ fullfills
\begin{subequations}
\begin{align}
\left|\left|\mathcal{O}^{\prime\prime}(\ell)\right|\right| &\geq \left< \
\mathcal{O}^{\prime\prime}(\ell) \ \right> {}_{{\psi^\prime(\ell)}}
\\
&= \underbrace{\left< \ \mathcal{O}(\ell) \ \right>
{}_{\psi^\prime(\ell)}}_{\geq\Omega_\text{min}}
- \underbrace{\left< \ \mathcal{O}^{\prime}(\ell) \ \right> {}_{
}{\psi^\prime(\ell)}}_{\Omega_\text{min}^\prime(\ell)}.
\end{align}
\end{subequations}
By condition, $\Omega_\text{min}$ is a lower bound for
$\left< \ \mathcal{O}(\ell) \ \right> {}_{{\psi^\prime(\ell)}}$. It follows
\begin{align}
\left|\left|\mathcal{O}^{\prime\prime}(\ell)\right|\right| \geq
\Omega_\text{min}- \Omega_\text{min}^\prime(\ell) =:
\Delta \Omega_\text{min}(\ell).
\end{align}
Analogously, one obtains the inequality
\begin{subequations}
\begin{align}
\left|\left|\mathcal{O}^{\prime\prime}(\ell)\right|\right| &\geq -\left<
\ \mathcal{O}^{\prime\prime}(\ell) \ \right> {}_{{\psi(\ell)}}
\\
&= -\underbrace{\left< \ \mathcal{O}(\ell) \ \right>
{}_{\psi(\ell)}}_{\Omega_\text{min}} +
\underbrace{\left< \ \mathcal{O}^{\prime}(\ell) \ \right>
{}_{\psi(\ell)}}_{\geq
\Omega_\text{min}^\prime(\ell)}.
\end{align}
\end{subequations}
Since $\Omega_\text{min}^\prime(\ell)$ is a lower bound for
$\left< \ \mathcal{O}^\prime(\ell) \ \right> {}_{{\psi(\ell)}}$, we obtain
\begin{align}
\left|\left|\mathcal{O}^{\prime\prime}(\ell)\right|\right| \geq -
\Omega_\text{min}+ \Omega_\text{min}^\prime(\ell)
= -\Delta \Omega_\text{min}(\ell).
\end{align}
In summary, the truncation error
\begin{align}
\Lambda_\mathcal{O}(\ell) \geq
\left|\left|\mathcal{O}^{\prime\prime}(\ell)\right|\right|\geq
\left|\Delta \Omega_\text{min}(\ell)\right|
\end{align}
is an upper bound of the deviation of the minimal eigen-value of the
effective operator due to the truncation. Analogously, one can prove that
$\Lambda_\mathcal{O}$ defines an upper bound for the deviation of the maximal
eigen-value $\Delta\Omega_\text{max}$.
A very useful result ensues by considering the special case of the truncation
error of the Hamiltonian itself
because the ground state energy can directly be
read off from the renormalized Hamiltonian $H^{(\text{r})}$.
Therefore, the exact ground state energy has to be within an interval of
$\Lambda_H(\infty)$ around the ground state energy calculated by the
truncated S-CUT
\begin{align}
\left|E_0-E_0^\prime\right|\leq \Lambda_H(\infty).
\end{align}
In this way, the truncation error defined by the spectral norm is no longer
an abstract expression, but gives a practical error bound for a physical
property of the system.
\section{Illustrative Model\label{struct:toy}}
\subsection{Double-Hard-Core-Boson}
As illustration of the formalism described above,
we investigate the truncation error of a
model of two sites which can be occupied by at most one particle each. To
describe the system in second quantization, we use the hard-core boson
language. The commutator of the associated annihilation and creation
operators on site $i$ and $j$ is given by
\begin{align}
\left[a_i,a^\dagger_j\right]=\delta_{ij} \left(\mathbbm{1} - a^\dagger_j
a_i\right) - a^\dagger_j a_i.
\end{align}
The Hamiltonian under study reads
\begin{subequations}
\label{eq:toy-dhard-core}
\begin{align}
H =
\ \epsilon\mathbbm{1} &+ \mu \left(a_1^\dagger a_1 + a_2^\dagger a_2\right)+
t \left(a_1^\dagger a_2 + a_2^\dagger a_1\right)
\label{eq:toy-dhard-core-a} \\
&+ \Gamma^{10}\left(a_1^\dagger + a_1 +a_2^\dagger +a_2\right)
\label{eq:toy-dhard-core-b} \\
& + \Gamma^{21}\left(a_1^\dagger a_2^\dagger a_2
+a_1^\dagger a_1a_2^\dagger +h.c.\right)
\label{eq:toy-dhard-core-c}\\
&
+ \Gamma^{20}\left(a_1^\dagger a_2^\dagger + a_1 a_2\right) +
V a_1^\dagger a_1a_2^\dagger a_2.
\label{eq:toy-dhard-core-d}
\end{align}
\end{subequations}
Terms in the lines \eqref{eq:toy-dhard-core-c} and \eqref{eq:toy-dhard-core-d}
are not present in the bare Hamiltonian but may emerge during the flow.
The quantity $\epsilon$ defines the vacuum energy, $\mu$ stands for the
chemical potential. The particle-particle interaction is denoted with $V$ and
$t$ is the prefactor of the hopping term. The quantities
$\Gamma^{10}$, $\Gamma^{20}$
and $\Gamma^{21}$ violate the number of quasiparticles. They represent the
non-diagonality of the Hamiltonian.
\subsection{Flow equations}
For our study, we use the particle conserving generator scheme
$\eta^\prime(\ell)=\widehat \eta_{\text{pc}}[H^{\prime}(\ell)]$,
for details see Refs.\ \onlinecite{uhrig98c,Knetter2000,Fischer2010}. The
differential equations for the coefficients of $H^{\prime}$ read
\begin{subequations}\begin{align}
\partial_\ell \epsilon^ \prime
&=\! &\! -4&\Gamma^{10\prime}\Gamma^{10\prime}
\! &\!-2&\Gamma^{20\prime}\Gamma^{20\prime}
\\
\partial_\ell \mu^\prime
&=\! &\! 4 &\Gamma^{10\prime}\Gamma^{10\prime}
\! &\!+2&\Gamma^{20\prime}\Gamma^{20\prime}
\nonumber\\
& \! &\! -2 &\Gamma^{21\prime}\Gamma^{21\prime}
\! &\!-4&\Gamma^{10\prime}\Gamma^{21\prime}
\\
\partial_l t^\prime &=\! &\! -4 &\Gamma^{10\prime}\Gamma^{21\prime}
\! &\!-2&\Gamma^{21\prime}\Gamma^{21\prime}
\\
\partial_\ell \Gamma^{10\prime}
&=\! &\! -&\Gamma^{10\prime}\mu^\prime
\! &\!- &\Gamma^{10\prime}t^\prime
\nonumber\\
& \! &\! -&\Gamma^{20\prime}\Gamma^{10\prime}
\! &\!-3&\Gamma^{20\prime}\Gamma^{21\prime}
\\
\partial_\ell \Gamma^{21\prime} &=\! &\! -&\Gamma^{21\prime}\mu^\prime
\! &\!+ &\Gamma^{21\prime}t^\prime&+2&\Gamma^{21\prime}
\Gamma^{20\prime}
\nonumber\\
& \! &\! +2&\Gamma^{10\prime}t^\prime \! &\!
+4&\Gamma^{10\prime}\Gamma^{20\prime}
\\
& \! &\! -& \Gamma^{21\prime}V^\prime \! &\!
-& \Gamma^{10\prime}V^\prime
\nonumber\\
\partial_\ell \Gamma^{20\prime}
&=\! &\! -2&\Gamma^{20\prime}\mu^\prime \! &\! -&
\Gamma^{20\prime}V^\prime
\\
\partial_\ell V^ \prime &=\! &\! 16&
\Gamma^{10\prime}\Gamma^{21\prime}\! &\! +8&
\Gamma^{21\prime}\Gamma^{21\prime}.
\end{align}
\end{subequations}
Since the block-band structure is conserved by
$\widehat\eta_{\text{pc}}$, see Ref.\
\onlinecite{Mielke1998,Knetter2000}, $\Gamma^{20\prime}$
stays zero during the flow unless it is already
present in the initial Hamiltonian. As a mi\-ni\-mal truncation scheme, we
neglect the particle-particle interaction given by
$V^\prime$ in the following
and thus the corresponding contributions to
$\partial_\ell \Gamma^{21\prime}$ and $\partial_\ell \Gamma^{20\prime}$.
Therefore the only contribution to the inhomogeneity
\begin{align}
\kappa(\ell)= \left(\Gamma^{10\prime}(\ell)\Gamma^{21\prime}(\ell) +
8\Gamma^{21\prime}(\ell)\Gamma^{21\prime}(\ell)\right) a_1^\dagger
a_1a_2^\dagger a_2
\end{align}
is given by the former derivative of the particle-particle interaction.
To make use of the possibility of calculating a rigorous bound for the
accuracy of the ground state energy and
the maximal energy eigen-value, we choose the spectral norm. Since we use
hard-core bosons, $a_1^\dagger a_1 a_2^\dagger a_2$ has the maximal
eigen-value of unity. Equation \eqref{eq:def_Lambda_H} immediately yields the
truncation error
\begin{align}
\Lambda_H(\ell) = \int\limits_0^\ell
\left|\Gamma^{10\prime}(\ell^\prime)\Gamma^{21\prime}(\ell^\prime) +
8\left(\Gamma^{21\prime}\right)^2(\ell^\prime)\right|\mathsf{d}\ell^\prime.
\end{align}
Due to the small number of couplings, we are able to calculate the quantities
$H^{\prime}$ and $H^{\prime\prime}$ directly.
To this end, we need to calculate $H(\ell)$ by
transforming it like an observable
under the flow of $\eta^\prime(\ell)$ following Eq.\ \eqref{eq:fullflow}. We
stress that $H$ is transformed in this way by an exact unitary transformation
without any truncations. We obtain the set of differential equations
\begin{subequations}
\begin{align}
\partial_\ell \epsilon &=\! &\!
-4&\Gamma^{10\prime}\Gamma^{10}-2\Gamma^{20\prime}\Gamma^{20}
\\
\partial_\ell \mu &=\! &\!
&\Gamma^{10\prime}(4 \Gamma^{10}-2\Gamma^{21})+2\Gamma^{20\prime}
\Gamma^{20}
\nonumber\\
& \! &\! -2&\Gamma^{21\prime}(\Gamma^{10}+\Gamma^{21})
\\
\partial_\ell t &=\! &\!
-2&\Gamma^{10\prime}\Gamma^{21}-2\Gamma^{21\prime}(\Gamma^{10}+\Gamma^{21})
\\
\partial_\ell \Gamma^{10} &=\! &\!
-&\Gamma^{10\prime}(\mu+t+\Gamma^{20})-\Gamma^{21\prime}\Gamma^{20}
\nonumber\\
& \! &\! -&\Gamma^{20\prime}(\Gamma^{10}+\Gamma^{21})
\\
\partial_\ell \Gamma^{21} &=\! &\!
&\Gamma^{10\prime}(2\Gamma^{20}+2t-V)+
\Gamma^{20\prime}(2\Gamma^{10}+\Gamma^{21\prime})
\nonumber\\
& \! &\! +&\Gamma^{21\prime}(-\mu+t-V+\Gamma^{20})
\\
\partial_\ell \Gamma^{20} &=\! &\!
-2&\Gamma^{10\prime}\Gamma^{21}+\Gamma^{20\prime}(-2\mu-V)+
2\Gamma^{21\prime}\Gamma^{10}
\\
\partial_\ell V &=\! &\!
8&\Gamma^{10\prime}\Gamma^{21}+8\Gamma^{21\prime}(\Gamma^{10}+\Gamma^{21}).
\end{align}
\end{subequations}
Thereby we are able to calculate $\left|\left|H^{\prime\prime}\right|\right|$
exactly as reference to estimate the quality of the truncation error
$\Lambda_H$ which yields an upper bound to
$\left|\left|H^{\prime\prime}\right|\right|$.
One should notice that the conservation of the bandstructure does not hold
for $H$ since the generator depends on $H^{\prime}$.
\subsection{Results\label{struct:toy-res}}
\begin{figure}
\includegraphics[width=\columnwidth]{fig2}
\caption{\label{plot:toy-groundstate}
The vacuum energy $\epsilon^\prime(\ell)$ of $H^{\prime}(\ell)$
converges to the ground state energy $E^\prime_0(\infty)$ of
$H^{\prime}(\infty)$. Due to truncation errors, the latter
starts to deviate from the true ground state energy $E_0$ when the flow sets
in at $\ell=0$. The calculation is carried out for
$\mu^{(\text{b})}=2, t^{(\text{b})}=1$ and $\Gamma^{10(\text{b})}=1$.
}
\end{figure}
\begin{figure*}
\begin{minipage}{0.49\textwidth}\includegraphics[width=\textwidth]{fig3a}
\end{minipage}
\begin{minipage}{0.49\textwidth}\includegraphics[width=\textwidth]{fig3b}
\end{minipage}
\begin{minipage}{0.49\textwidth}\includegraphics[width=\textwidth]{fig3c}
\end{minipage}
\begin{minipage}{0.49\textwidth}\flushleft
\caption{\label{plot:toy-errors}\hspace{23cm}}
Truncation error $\Lambda_H$, spectral distance
$\left|\left|H^{\prime\prime}\right|\right|$, deviation
$\Delta E_\text{max}$ of the highest and lowest energy eigen-value
$\Delta E_0$ of the truncated Hamiltonian for $\mu^{(\text{b})}=2$ and
$t^{(\text{b})}=1$.
Panels a and b: dependence of the flow parameter $\ell$ for a moderate
non-diagonality $\Gamma^{10{(\text{b})}}=1$ (a) and for large non-diagonality
$\Gamma^{10{(\text{b})}}=10$ (b).
Panel c: dependence of the renormalized quantities on the initial
non-diagonality $\Gamma^{10{(\text{b})}}$.
\end{minipage}
\end{figure*}
For our calculations, we used $\mu^{(\text{b})}=2$ and $t^{(\text{b})}=1$ as
initial conditions for $H^{(\text{b})}$, while $\epsilon^{(\text{b})}$,
$\Gamma^{20{(\text{b})}}$ and
$\Gamma^{21{(\text{b})}}$ are chosen to be zero. The initial non-diagonality
$\Gamma^{10{(\text{b})}}$ is used to control the degree of necessary
transformation.
For small values of $\Gamma^{10}$, the Hamiltonian is close to diagonality and
therefore only slightly changed by the CUT. A large initial non-diagonality on
the other hand requires intensive re-ordering processes in which truncation
errors are important.
Figure \ref{plot:toy-groundstate} shows the generic
behavior of the ground state
energy $E^{\prime}_0$ of ${H^{\prime}}^{(\text{r})}$ and the vacuum energy
$\epsilon$ under the truncated flow. Truncation errors have a noticable impact
due to the (large) non-diagonality $\Gamma^{10}$.
In an early stage of the flow ($\ell \lesssim 0.5$), $E^{\prime}_0(\ell)$
starts to depart from $E_0$ and remains constant for the rest of the flow.
The vacuum energy $\epsilon^\prime$ converges rapidly to the ground state
energy $E^\prime_0(\infty)$ of $H^{\prime}(\infty)$.
The truncation error $\Lambda_H$ and the spectral distance
$\left|\left|H^{\prime\prime}\right|\right|$ both
saturate in the course of the truncated flow, see
Fig.~\ref{plot:toy-errors}a and b, and
they show a strong monotonic dependence of the non-diagonality.
The spectral distance $\left|\left|H^{\prime\prime}\right|\right|$ is bounded
by the truncation error as
it has to be. Furthermore, the truncation error turns out to be also a good
approximation for the spectral norm. This can be seen in
Fig.~\ref{plot:toy-errors} (panel c). Their difference is insignificant for
small non-diagonalities and even for very large ones
($\Gamma^{10{(\text{b})}}=10$) it takes only $6\%$.
The influence of truncation on the spectrum of $H^\prime$ is studied by the
difference of the ground state energy $\Delta E_0=E_0-E^\prime_0$ and by the
difference of the energy of the highest excited level $\Delta E_\text{max}=
E_\text{max}-E^\prime_\text{max}$ compared to the values of the initial
Hamiltonian $H_0$. Both quantities stay clearly below the spectral distance,
but differ in magnitude. For small values of
$\Gamma^{10{(\text{b})}}$, $\Delta E_0$ is negligible and rises only up to
$3.5\%$ of
$\left|\left|H^{\prime\prime{(\text{r})}}\right|\right|$ for
$\Gamma^{10{(\text{b})}}=10$.
By contrast, $\Delta E_\text{max}$ is nearly identical to
$\left|\left|H^{\prime\prime{(\text{r})}}\right|\right|$ for
low non-diagonality and remains close to
$\left|\left|H^{\prime\prime{(\text{r})}}\right|\right|$
even for high values of $\Gamma^{10{(\text{b})}}$.
Thus the major impact of truncation occurs for the highest excited level.
This can be understood on recalling that the minimal truncation
of the density-density interaction expressed by $V$ means neglecting the
energy correction for the doubly occupied state and to approximate it by
$E_0+2\mu$. Because this is precisly the highest eigen-state for
vanishing non-diagonality $\Gamma^{10{(\text{b})}}$, it is directly affected
by the truncation. In contrast, the ground state is only influenced indirectly
by inaccuracies for the higher levels. Therefore, the low energy properties
can be characterized by S-CUT very accurately despite of
truncation, whereas larger inaccuracies occur at high energies.
In summary, the truncation error $\Lambda_H$ defined in Eq.\
\eqref{eq:def_Lambda_H} is illustrated as an upper bound for the spectral
distance $\left|\left|H^{\prime\prime}\right|\right|$ and for the errors of
$E_0$ and $E_\text{max}$. We stress the good agreement of
$\Lambda_H$, $\left|\left|H^{\prime\prime}\right|\right|$ and
$\Delta E_\text{max}$. The error of ground state energy, however, is
significantly lower than the bound given by
$\left|\left|H^{\prime\prime}\right|\right|$. This can be explained as a
particularity of the minimal truncation scheme which affects mainly high
energies. We highlight that the truncation error measures the non-unitarity
of the \emph{complete} transformation acting on the whole Hilbert space.
\section{Extended system\label{struct:extend}}
\subsection{Dimerized spin-$\nicefrac{1}{2}$-chain\label{struct:ext-model}}
\begin{figure}
\includegraphics[width=\columnwidth]{fig4}
\caption{\label{img:chain}Schematic representation of the dimerized spin
chain. Dark bonds stand for the coupling $J$ between two
$S=\nicefrac{1}{2}$ spins forming a dimer, light bonds denote the variable
inter-dimer coupling $\lambda J$.
The dashed line indicates an axis of reflection symmetry.}
\end{figure}
In the previous section, we studied the truncation error for a zero-dimensional
illustrative model. To illustrate the applicability of the
truncation error for more relevant models, we extend our analysis to
a one-dimensional system. Furthermore, this allows us to calculate and to
compare the truncation errors of more complex real-space truncation schemes.
As we will see, we have to use the triangle inequality again
to arrive at error bounds in extended systems.
The model studied is the one-dimensional dimerized antiferromagnetic spin
$S=\nicefrac{1}{2}$ Heisenberg chain with the Hamiltonian
\begin{align}
H=J\sum\limits_r \left(\textbf{S}_{r}^L\cdot\textbf{S}_{r}^R+
\lambda\textbf{S}_{r}^R\cdot \textbf{S}_{r+1}^L\right), \quad J>0
\label{eq:spinhammi}.
\end{align}
In this notation, $\textbf{S}_{r}^L$ stands for the operator of the left spin
in the
dimer on position $r$ and $\textbf{S}_{r}^R$ for the right spin,
see Fig.\ \ref{img:chain}. The parameter $0\leq\lambda\leq 1$ denotes the
relative strength of interdimer coupling. It is used as a control parameter
similar to $\Gamma^{10}$ in Sect.\ \ref{struct:toy}.
In the limit of $\lambda=0$, the system consists of isolated dimers. The
ground state is given by a product state of singlets with a ground state
energy of $\frac{E_0}{N}=-\frac{3}{4}J$ per dimer. The local $S=1$
excitations form equidistant spectrum with an increment of $J$.
For rising interdimer coupling, the excitations can be described by
gapped spin $S=1$ quasiparticles called triplons \cite{Schmidt2003}.
They can be seen as triplets with magnetic polarization cloud.
In the limit $\lambda=1$, the gap closes and the correlations decay
algebraically \cite{cloiz66,yang66a,yang66b,fadde81,Klumper1998,Sachdev1999}.
For $\lambda=1$, the system is the well-known homogeneous spin chain
exactly solved 1931 by Bethe \cite{Bethe1931} by what was henceforth
called Bethe ansatz. The ground state energy for the
infinite chain was calculated by Hulth\'{e}n \cite{Hulthen1938} and takes the
value $\nicefrac{1}{2}-2\ln 2$ per dimer and $J$.
With respect to the limit of isolated dimers, we choose for dimer $r$ a local
basis of singlet/triplet states
\begin{subequations}
\begin{alignat}{3}
\left|s\right\rangle_r &= \frac{1}{\sqrt{2}}
\left(\left|\uparrow\downarrow\right\rangle-
\left|\downarrow\uparrow\right\rangle\right) \ & &\ & &
\\
\left|x\right\rangle_r &= \frac{-1}{\sqrt{2}}
\left(\left|\uparrow\uparrow\right\rangle -
\left|\downarrow\downarrow\right\rangle\right) \ &=&\
t^\dagger_{x,r}&\left|s\right\rangle_r &
\\
\left|y\right\rangle_r &= \frac{i}{\sqrt{2}}
\left(\left|\uparrow\uparrow\right\rangle +
\left|\downarrow\downarrow\right\rangle\right) \
&=&\ t^\dagger_{y,r}&\left|s\right\rangle_r &
\\
\left|z\right\rangle_r &= \frac{1}{\sqrt{2}}
\left(\left|\uparrow\downarrow\right\rangle+
\left|\downarrow\uparrow\right\rangle\right) \ &=&\
t^\dagger_{z,r}&\left|s\right\rangle_r &
\end{alignat}
\end{subequations}
as in the bond operator representation
\cite{Chubukov1989-original,Chubukov1989,Sachdev1990}.
We define the reference state as the product state of singlets on each site
\begin{align}
\left|0\right\rangle:=\bigotimes\limits_r \left|s\right\rangle_r.
\end{align}
The triplet operators are defined by
\begin{subequations}
\begin{align}
t^\dagger_{\alpha,r}&:=\left|\alpha\right\rangle_r\left\langle s\right|_r
\\
t^{\phantom{\dagger}}_{\alpha,r}&:=
\left|s\right\rangle_r\left\langle\alpha\right|_r.
\end{align}
\end{subequations}
Non-physical artifacts (e.g. states with two triplets on one dimer) are
excluded. Therefore, the triplet operators obey the hard-core algebra
\begin{align}
\left[t^{\phantom{\dagger}}_{\alpha,r},t^\dagger_{\beta,s}\right]=
\delta_{rs}\delta_{\alpha\beta}\left(\mathbbm{1} -
\sum\limits_\gamma t^\dagger_{\gamma,r}
t^{\phantom{\dagger}}_{\gamma,r}\right)-
\delta_{rs}t^\dagger_{\beta,r}t^{\phantom{\dagger}}_{\alpha,r}.
\end{align}
The normal-ordered products of triplet operators (monomials) together with the
identity $\mathbbm{1}$ form the basis $\{A_i\}$ for all operators on the
lattice. In this notation, the Hamiltonian reads
\begin{align}
\label{eq:Whemmi}
H = \sum\limits_r
&\ -\frac{3}{4}\mathbbm{1} + t^\dagger_{\alpha,r}
t^{\phantom{\dagger}}_{\alpha,r}
\nonumber\\
&\ + \frac{1}{4}\lambda\left(t^\dagger_{\alpha,r} +
t^{\phantom{\dagger}}_{\alpha,r}\right)\left(t^\dagger_{\alpha,r+1} +
t^{\phantom{\dagger}}_{\alpha,r+1}\right)
\nonumber\\
&\ + \frac{i}{4}\lambda\epsilon_{\alpha\beta\gamma}
\left(\left(t^\dagger_{\alpha,r} +
t^{\phantom{\dagger}}_{\alpha,r}\right)t^\dagger_{\beta,r+1}
t^{\phantom{\dagger}}_{\gamma,r+1}\right.
\\
&\ \left. - t^\dagger_{\beta,r}t^{\phantom{\dagger}}_{\gamma,r}
\left(t^\dagger_{\alpha,r+1} +
t^{\phantom{\dagger}}_{\alpha,r+1}\right)\right)
\nonumber\\
+ \frac{1}{4}\lambda&\left(t^\dagger_{\beta,r}
t^{\phantom{\dagger}}_{\gamma,r}t^\dagger_{\gamma,r+1}
t^{\phantom{\dagger}}_{\beta,r+1}-t^\dagger_{\beta,r}
t^{\phantom{\dagger}}_{\gamma,r}t^\dagger_{\beta,r+1}
t^{\phantom{\dagger}}_{\gamma,r+1}\right).&\nonumber
\end{align}
By the CUT the triplet states are mapped to re-normalized $S=1$
excitations (triplons).
The Hamiltonian \eqref{eq:Whemmi} has three different
symmetries that can be used for simplification of the
calculation as mentioned in Sect.\ \ref{struct:cut-sym}:
(i) The Hamiltonian \eqref{eq:Whemmi} is self-adjoint. Although this is not a
symmetry in the strict sense of the word, it implies an additional constraint
for the coefficients of $H$. Since quasiparticle creating and annihilating
terms have to occur in pairs in any Hamiltonian, one of them can be chosen as
representative for the pair. Thereby the number of coefficients to be tracked
is reduced.
(ii) The Hamiltonian \eqref{eq:Whemmi} shares the reflection symmetry of the chain
$r\to -r$, see Fig.\ \ref{img:chain}.
In addition, all left-spin and right-spin
operators have to be swapped implying
$t^{\phantom{\dagger}}_{\alpha,r}\leftrightarrow-t^{\phantom{\dagger}}_{\alpha,r}$ in the triplon notation
\footnote{The choice of the reflection symmetry axis is arbitrary because
all reflection axes are equivalent due to translation symmetry.}.
(iii) The Hamiltonian \eqref{eq:spinhammi} is invariant under SU(2) rotations
in spin space. Due to this invariance, the Hamiltonian written in the
triplet algebra \eqref{eq:Whemmi} can be decomposed
into symmetric combinations of terms. The terms in a symmetric combination
differ only by permutations of triplet polarizations up to a sign factor.
The superoperator $\sum_G$ to build the symmetric combination of a maximally
asymmetric representative reads
\begin{align}
\sum_{xyz}=\sum_{xy}\sum_{\text{cyc}} = \left(\mathbbm{1}+
\widehat S_{xy} \right)
\left(\mathbbm{1}+\widehat S_\text{cyc}+\widehat S_\text{cyc}^2 \right),
\end{align}
where the cyclic permutations of triplet polarizations is denoted
by $\widehat S_\text{cyc}$ and the exchange of the polarizations
$x$ and $y$ with a negative
sign factor for each triplet operator reads $\widehat S_{xy}$.
\subsection{Real-space truncation\label{struct:ext-trunc}}
For an extended system, the omission of processes that create or
annihilate more than $N$ triplons as mentioned in Sect.\
\ref{struct:cut-truncscheme} can be insufficient because the number of
remaining terms is still infinite.
For example, even by restricting to processes of at most two
triplons, an infinite number of independent terms varying by range can emerge
in the course of the flow. Since an energy gap
implies a finite correlation length,
it is an adequate choice to neglect all processes that exceed a given range
$d_\text{max}$.
In a one dimensional system, the range can easily be defined as the
distance of the rightmost and leftmost triplet operator in a term.
In particular, it has turned out to be advantageous to use a combination
of both the quasiparticle and the range criterion as truncation scheme
\cite{Reischl2006,Fischer2010}.
For the most important processes of low quasiparticle number, e.g., the
hopping of triplons, a long range is allowed for to preserve most of the
relevant physics. For more complex processes of more quasiparticles
only a shorter range can be considered
because their number increases much more steeply with range.
But since more quasiparticles are required for such terms to
become active the reduced range does not need to imply a reduced
accuracy.
In view of the above considerations
we classify terms by the sum $n$ of created and annihilated
quasiparticles. For each value of $n$ a specific maximal range $d_n$ is
defined. The complete truncation scheme can be written as
$\textbf{d}=\left(d_2,d_3 \dots d_{2N}\right)$ where at most $N$ quasiparticles
may be created or annihilated. No maximal range needs to
be specified for $n=1$ because those terms always have
range zero and their number
is, due to translation symmetry, restricted to the six local creation
and annihilation operators. Furthermore, without magnetic field
no single annihilation or creation of triplons takes place
due to the conservation of the total spin.
\subsection{Triangle inequality\label{struct:ext-tri}}
To calculate the truncation error according to Eq.\ \eqref{eq:def_Lambda_H},
the norm of the inhomogeneity
\begin{align}
\kappa(\ell)=
\sum\limits_{A_i}\kappa_{A_i}(\ell)A_i
\end{align}
has to be calculated.
Its coefficients $\kappa_{A_i}$ are obtained using Eq.\ \eqref{eq:kappa} from
the coefficients of $H^{\prime}$ by evaluating the terms of the commutator
that are discarded in the truncation scheme.
The precise calculation of $\left|\left|\kappa\right|\right|$ is not feasible
for large systems because the effort to calculate the maximal eigen-value is
too large. Since we are only interested in an upper bound, we apply the
triangle inequality again to reach
\begin{align}
\left|\left|\kappa(\ell)\right|\right|=\left|\left|
\sum\limits_{A_i}\kappa_{A_i}(\ell){A_i}\right|\right|\leq
\sum\limits_{{A_i}}\left|\kappa_{A_i}(\ell)\right|\left|\left|A_i\right|
\right|.
\end{align}
In this way, we define an upper bound $\widetilde \Lambda_H$
or the truncation error
\begin{align}
\widetilde\Lambda_H(\ell)=\sum\limits_{{A_i}}\int\limits_0^\ell
\left|\kappa_{A_i}(\ell^\prime)\right|\mathsf{d} \ell^\prime
\left|\left|A_i\right|\right|\geq \Lambda_H(\ell).
\end{align}
Therefore, $\widetilde\Lambda_H$ is also an upper bound for $\Delta E_0$ and
$\Delta E_\text{max}$, although it is less strict than $\Lambda_H$.
Recall that our basis operators $A_i$ are normal-orderd products
of hard-core-boson creation and annihilation operators. It turns out that in
this case the spectral norm $\left|\left|A_i\right|\right|$ can be calculated
easily since the product ${A_i}^\dagger {A_i}$ is already diagonal with respect
to the dimer eigen-states. It is a product of local triplon density terms.
Thus ${A_i}^\dagger {A_i}$ has the eigen-value unity for all configurations
having triplons with the polarization of the local triplon density operators
on each site of the cluster of ${A_i}$
\footnote{The cluster of a term is the set of sites on which its
action differs from identity.}.
For different occupations, ${A_i}^\dagger {A_i}$ has the eigen-value zero.
Therefore the spectral norm for all elements of the operator basis
$\{A_i\}$ is unity.
\subsection{Optimization using symmetries\label{struct:ext-sym}}
As pointed out in Sect.\ \ref{struct:cut-sym}, exploiting symmetries
reduces the computational effort for the S-CUT method significantly.
In addition, symmetries can be used to reduce the bound
$\widetilde \Lambda$. To see this, we write $\Lambda$ using the basis of
representatives $\{C_i\}$ of operator monomials
\begin{subequations}
\begin{align}
\Lambda_H(\ell)
&= \sum\limits_{C_i} \int\limits_0^\ell
\left|\left|\kappa_{C_i}(\ell^\prime)\sum_G {C_i}\right|\right|\mathsf{d}
\ell^\prime
\\
&\leq \sum\limits_{C_i} \int\limits_0^\ell
\left|\kappa_{C_i}(\ell^\prime)\right|
\mathsf{d} \ell^\prime \cdot \left|\left|\sum_G {C_i}\right|\right| =:
\widetilde\Lambda_H(\ell).
\end{align}
\end{subequations}
We use the triangle inequality again to decompose
$\left|\left|\sum_G {C_i}\right|\right|$ into the weight $w_{C_i}$ which
denote the number of terms generated by $\sum_G$
multiplied by the norm of the representative. This yields
\begin{align}
\widetilde \Lambda_H(\ell) = \left(\int\limits_0^\ell
\left|\kappa_{C_i}(\ell^\prime)\right|\mathsf{d}\ell^\prime\right)
w_{C_i}\left|\left|{C_i}\right|\right|.
\label{eq:def_Lambda_T_H}
\end{align}
If, however, we avoid the triangle inequality for the fully
symmetric combination of basis operators we obtain a better,
stricter upper bound. This enters Eq.\ \eqref{eq:def_Lambda_T_H}
by replacing $w_{C_i}$ by a reduced effective weight $w^G_{C_i}$.
In order to determine this reduction, we have to calculate the relation
between the norm of the fully symmetric combination and the norm of a
single representative analytically
\begin{align}
\left|\left|\sum_G {C_i}\right|\right|=:w^G_{C_i}
\left|\left|{C_i}\right|\right|.
\end{align}
The effective weight factor $w^G_{C_i}$
allows us to modify Eq.\ \eqref{eq:def_Lambda_T_H} to calculate the
improved truncation bound $\widetilde\Lambda^G$.
In presence of multiple symmetries, it is possible to
improve the weight with respect to only some selected symmetries. The other
symmetries can be treated by using the triangle inequality as done in Eq.\
\eqref{eq:def_Lambda_T_H}. They continue to enter the combined weight by an
additional factor.
As an example, we consider the self-adjointness to reach
the improved truncation bound $\widetilde\Lambda^\dagger_H$.
To determine its weight, it is useful to decompose the action of a
representative
\begin{align}
C=\left|c_1\right\rangle\left\langle c_2\right|
\bigotimes\limits_{r \notin \text{cluster}}\mathbbm{1}
\end{align}
into the non-trivial action on its cluster and into
the identity on the rest of the system.
Since $C$ is a normal-ordered product of hard-core-boson operators,
the action on its cluster is given by only one non-vanishing matrix element.
To build the corresponding symmetric combination $\sum_G C$, we have to
distinguish two cases:
(i) \emph{$\left|c_1\right\rangle=\left|c_2\right\rangle$}\quad
If $C$ is self-adjoint, it is already a symmetric combination
and thushas a weight factor of unity in both cases,
i.e., using and not using the self-adjointness.
(ii) \emph{$\left|c_1\right\rangle\neq\left|c_2\right\rangle$}\quad
In this case, the symmetric combination for $C$ reads
\begin{align}
\sum_G C = \left(\mathbbm{1} + \widehat A\right) C =
\left(\left|c_1\right\rangle\left\langle c_2\right|+\left|c_2\right\rangle
\left\langle c_1\right|\right) \bigotimes\limits_{r \notin \text{cluster}}
\mathbbm{1}.
\end{align}
Due to our choice of basis, both $\left|c_1\right\rangle$ and
$\left|c_2\right\rangle$ are eigen-states
of local triplet density operators and therefore orthogonal, which implies
the maximal eigen-value of 1 for $\left|c_1\right\rangle
\left\langle c_2\right|+\left|c_2\right\rangle\left\langle c_1\right|$.
Hence $\Vert\left|c_1\right\rangle\left\langle c_2\right|\Vert=1$ and
$\Vert\left|c_1\right\rangle\left\langle c_2\right|+
\left|c_2\right\rangle\left\langle c_1\right|\Vert=1$ which implies
a gain of a factor of 2 if the triangle inequality is avoided which
would have led to $\Vert\left|c_1\right\rangle\left\langle c_2\right|+
\left|c_2\right\rangle\left\langle c_1\right|\Vert
\le \Vert\left|c_1\right\rangle\left\langle c_2\right|\Vert +
\Vert\left|c_2\right\rangle\left\langle c_1\right|\Vert =2$.
As an example, we consider the representative
$t^\dagger_{x,1}t^{\phantom{\dagger}}_{x,1}t^\dagger_{y,2}$.
On its cluster $\{1,2\}$ it
stands for a transition from $\left|x_1s_2\right\rangle$ to
$\left|x_1y_2\right\rangle$.
The matrix representation for the action of its symmetric combination
\begin{align}
\sum_G t^\dagger_{x,1}t^{\phantom{\dagger}}_{x,1}t^\dagger_{y,2} =
\left(\left|x_1y_2\right\rangle\left\langle x_1s_2\right|+
\left|x_1s_2\right\rangle\left\langle x_1y_2\right|\right)
\bigotimes\limits_{r \notin \{1,2\}}\mathbbm{1}
\end{align}
has zero matrix elements except for a $2\times2$ block
with the eigen-values -1 and +1.
In conclusion, using self-adjointness to gain an effective weight
$w^\dagger$ saves a factor of 2 for non-symmetric terms.
This effective weight can be improved further
by considering spin symmetry. The complete symmetry group
with respect to all permutations of triplet polarizations
can be decomposed into the subgroup of cyclic permutations and the subgroup
of the transposition of $x$ and $y$ triplets $\widehat S_{xy}$ plus the
identity. For simplicity, we concentrate on the latter one only and calculate
the truncation bound $\widetilde\Lambda_H^{\dagger,xy}$.
The matrix associated with the $\sum_G C$ action on its cluster can have up to
four non-vanishing matrix elements , i.e., four states of the cluster
have to be taken into account.
The representatives can be classified by the specific action
of $\sum_G$ needed to obtain the corresponding fully symmetric combination:
\emph{$\sum_G=\mathbbm{1}$:} These highly symmetric representatives
(e.g.\ $t^\dagger_{z,1}t^{\phantom{\dagger}}_{z,1}$) are invariant under both
$\widehat A$ and $\widehat S_{xy}$. They have weight $w^{\dagger,xy}$ unity.
\emph{$\sum_G=\mathbbm{1}+\widehat A$:}
These representatives are not self-adjoint, but
invariant under either transposition of $x$ and $y$ (e.g.\
$t^{\phantom{\dagger}}_{z,1}$) or
under the combination of transposing and adjunction
(e.g.\ $t^\dagger_{x,1}t^{\phantom{\dagger}}_{y,1}$). As discussed previously,
the weight $w^{\dagger,xy}$ takes the value of 1 instead of 2.
\emph{$\sum_G=\mathbbm{1}+\widehat S_{xy}$:}
For representatives that are self-adjoint, but not
invariant under transposition (e.g.\
$t^\dagger_{x,1}t^{\phantom{\dagger}}_{x,1}$), the weight
$w^{\dagger,xy}$ is reduced to the value 1 instead of 2 as well.
\emph{$\sum_G=\left(\mathbbm{1}+\widehat A\right)\left(\mathbbm{1}+
\widehat S_{xy}\right)$:}
In this asymmetric case, the norm of the symmetric combination can be
either one (e.g. $t^\dagger_{x,1}t^{\phantom{\dagger}}_{y,2}$) or $\sqrt{2}$
(e.g.\ $t^\dagger_{x,1}$). Therefore, $w^{\dagger,xy}=\sqrt{2}$ can be used
instead of $w=4$ .
The weights can be improved by exploiting further symmetries, e.g.,
cyclic spin permutations or reflection symmetry. But the complexity of the
necessary case-by-case analysis rises considerably.
Especially the use of point group symmetries of the lattice is complicated
because the cluster of monomials linked by a point group do not need to be
identical. Thus the inclusion of point group
symmetries in the calculation of the
bounds is beyond the scope of this article.
\subsection{Results for an infinite system\label{struct:inf-res}}
\begin{figure}
\includegraphics[width=\columnwidth]{fig5}
\caption{\label{plot:ext-863detail}
Renormalized truncation bound per dimer $\widetilde\Lambda_H$ and reduced
bounds $\widetilde\Lambda_H^\dagger$ (exploiting self-adjointness) and
$\widetilde\Lambda_H^{\dagger,xy}$ (exploiting self-adjointness and $xy$
symmetry) vs.\ the interdimer coupling $\lambda$ for the ground state
generator using the truncation scheme $\textbf{d}=\left(8,6,6,3,3\right)$.
}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{fig6}
\caption{\label{plot:ext-infinite}
Renormalized reduced truncation bounds per dimer
$\widetilde\Lambda_H^{\dagger,xy}$ vs.\ interdimer coupling
$\lambda$ for ground state and particle conserving generator using
various truncation schemes. }
\end{figure}
\begin{table}
\begin{tabular}{cl rrrr }
\hline
\multicolumn{1}{c}{$\widehat\eta$} &
\multicolumn{1}{c}{$\textbf{d}$} &
\multicolumn{1}{c}{$\left|\Delta E_0\right| $} &
\multicolumn{1}{c}{$\widetilde\Lambda_H$} &
\multicolumn{1}{c}{$\widetilde\Lambda_H^\dagger$} &
\multicolumn{1}{c}{$\widetilde\Lambda_H^{\dagger, xy}$}
\\
\hline
$\widehat\eta_\text{pc}$ &$ (6,3,3) $& 0.01739 &
439.37 & 219.93 & 156.74
\\
$\widehat\eta_\text{pc}$ &$ (8,6,6,3,3) $&
0.00032 & 49926.1\phantom{0} & 24974.8\phantom{0}
& 17660.2\phantom{0}
\\
$\widehat\eta_\text{gs}$ &$ (6,3,3) $& 0.02675 &
23.51 & 11.75 & 8.41
\\
$\widehat\eta_\text{gs}$ &$ (8,6,6,3,3) $&
0.00915 & 149.31 & 74.68 & 52.98
\\
$\widehat\eta_\text{gs}$ &$ (9,7,7,4,4) $&
0.00948 & 169.32 & 84.67 & 60.04
\\
\hline
\end{tabular}
\caption{\label{tab:ext-infinite} Numerical values for the renormalized
truncation bound per dimer $\widetilde\Lambda_H$ and reduced bounds
$\widetilde\Lambda_H^\dagger$, $\widetilde\Lambda_H^{\dagger,xy}$ for
$\lambda=1$ for the generators and the truncation schemes used in Fig.\
\ref{plot:ext-infinite}. The inaccuracies of the ground state energy
$\left|\Delta E_0\right|$ are calculated with respect to the analytical
result \cite{Hulthen1938}.}
\end{table}
Figure \ref{plot:ext-863detail} shows the bound $\widetilde\Lambda_H$
defined in Eq.\ \eqref{eq:def_Lambda_T_H} for the infinite
dimerized Heisenberg chain using the ground state generator scheme
$\widehat\eta_\text{gs}$ \cite{Fischer2010} defined in \eqref{eq:cut-gs} and
the truncation
scheme $\textbf{d}=(8,6,6,3,3)$ as function of the interdimer coupling
$\lambda$. For rising values of $\lambda$, the truncation bound increases
drastically similar to the behavior observed for the illustrative model.
But it attains a significantly higher absolute value finally.
For weak interdimer coupling $\lambda \lesssim 0.1$, the error bound for the
ground state energy given by $\widetilde\Lambda$
is useful as a rigorous bound.
But this estimate becomes inappropriate for medium and strong coupling since
the $\widetilde\Lambda$ grows rapidly to the same magnitude as $E_0$ and
beyond so that it does no longer represent a
meaningful bound.
For $\lambda=1$ the analytical result can be used as reference to determine
the error of the ground state energy per dimer $\left|\Delta E_0\right|$
determined as the renormalized vacuum energy of the CUT, see
Tab.\ \ref{tab:ext-infinite}. This relative error is only $1.04\%$.
In contrast, the truncation bound $\widetilde\Lambda_H$
exceeds the ground state energy by several orders of magnitude.
This discrepancy stems from the fact that the truncation error $\Lambda_H$
measures truncation effects of the entire transformation,
not only of the inaccuracies of ground state energy in particular.
The extensive use of the triangle inequality to
calculate the truncation bound $\widetilde\Lambda$ enhances this difference
additionally, although it can be reduced exploiting symmetry.
The interesting question which effect dominates
is postponed to the study of a finite system below
where all quantities are numerically accessible.
The use of the adjunction symmetry reduces the bound by a factor of about two,
and the use of $xy$ symmetry reduces it by an additional factor of about
$\sqrt{2}$. In both cases, exploiting the symmetry pays in decreasing the
bound close to the optimum which can be achieved for terms with the lowest
symmetry. Since terms with low symmetry occur much more frequently than
symmetric ones, the error bound is reduced efficiently by exploiting
symmetries. However, the gain achieved in this way
can not overcome the tremendous factor (orders of magnitude) between the
upper bound and to the error of the ground state energy.
In the extended system, various truncation schemes of different quality and
computational effort can be used. Figure \ref{plot:ext-infinite} shows the
truncation bound $\widetilde\Lambda_H^{\dagger,xy}$ for different truncations
and generator schemes. Besides the particle conserving generator
$\widehat\eta_\text{pc}$ the ground state decoupling generator
$\widehat\eta_\text{gs}$ is applied, for details see Ref.\
\onlinecite{Fischer2010}.
The numerical values for $\lambda=1$ including the
difference to the exact ground state energy are given in Tab.\
\ref{tab:ext-infinite}.
In general, the particle conserving generator scheme $\widehat\eta_\text{pc}$
yields significantly higher truncation bounds than $\widehat\eta_\text{gs}$,
although the inaccuracies of the calculated ground state energies are lower
(using equal truncation schemes $\textbf{d}$).
This is due to the fact that $\widehat\eta_\text{pc}$ performs a more
comprehensive reordering of the quasiparticle subspaces and includes much
more terms than $\widehat\eta_\text{gs}$. This implies a higher impact of
truncation errors.
Much more terms emerge in the evaluation of the commutator and have to be
incorporated in $\kappa$ resulting in a larger differential equation system
with much more coefficients.
The dependence on the truncation scheme is more complex. For weak coupling,
looser truncation schemes imply lower truncation error bounds than stricter
schemes. This is what one expects naively since the inclusion of more
and more terms, hence a less strict truncation, should describe
the system better and better. Thus the truncation error should decrease.
But this relation can be inverted for strong coupling
\emph{although} the looser schemes
reproduce the analytical result for the ground state energy with much higher
accuracy for both generator schemes as seen for $(6,3,3)$ and $(8,6,6,3,3)$.
We call this phenomenon the \emph{truncation paradoxon} because
it seems to be paradoxical at first glance. We stress that
it does not represent a logical contradiction but only a counterintuitive
behaviour.
On second thought, one realizes that
the use of a looser truncation scheme increases the number of terms in
$\kappa$ drastically. For instance, an increase from 30.972 for $(6,3,3)$ to
16.777.215 representatives for $(8,6,6,3,3)$ is found in the
$\widehat\eta_\text{pc}$
scheme. Irrespective of the consequences to the spectral norm of $\kappa$,
this massive increase of terms has a big impact on the bound
$\widetilde\Lambda$ which relies on the triangle inequality
to bound each of these terms separately. Hence the bound
is so large simply because it is very loose.
\begin{figure}
\includegraphics[width=\columnwidth]{fig7}
\caption{\label{plot:ext-finite} (Color online)
Renormalized truncation bound per dimer
$\widetilde\Lambda_H^{\dagger,xy}$,
exact renormalized truncation error per dimer $\Lambda_H$ and error of
ground state energy per dimer $\left|\Delta E_0\right|$ vs.\ interdimer
coupling $\lambda$ for the ground state generator considering at most
$N$ triplon operators per term.}
\end{figure}
\begin{table}
\begin{tabular}{cl rrrr }
\hline
$N$ & $\left|\Delta E_0\right| $ & $\Lambda_H$
& $\widetilde\Lambda_H$ & $\widetilde\Lambda_H^\dagger$
& $\widetilde\Lambda_H^{\dagger, xy}$
\\
\hline
2 & 0.04109 & 0.1556 & 20.34 & 10.17 &
7.28
\\
3 & 0.01709 & 0.0964 & 41.87 & 20.93 &
14.86
\\
4 & 0.00465 & 0.0768 & 26.78 & 13.39 &
9.48
\\
\hline
\end{tabular}
\caption{\label{tab:ext-finite}
Error of ground state energy per dimer $\left|\Delta E_0\right|$, exact
renormalized truncation error per dimer $\Lambda_H$ and renormalized
truncation bounds per dimer $\widetilde\Lambda_H^{}$,
$\widetilde\Lambda_H^{\dagger}$ and $\widetilde\Lambda_H^{\dagger,xy}$ for
$\lambda=1$ using the ground state generator scheme
$\widehat\eta_\text{gs}$
considering at most $N$ triplon operators per term.}
\end{table}
\subsection{Results for a finite extended system\label{struct:fin-res}}
The question arises whether the truncation paradox is caused by the extensive
use of triangle inequality to determine the bound $\widetilde\Lambda$,
or whether it is an intrinsic characteristics
of the truncation error $\Lambda$ itself. To investigate this issue,
it is necessary to determine the exact truncation error
$\Lambda$ without use of the triangle inequality. Thus an exact
diagonalization of $\kappa$ is required.
This restricts us to the investigation of a finite chain segment.
In the following, we study the periodic, dimerized Heisenberg chain
consisting of five dimers.
In view of the small size of the system, we use the maximal number $N$ of
interacting triplons as only truncation criterion. No extensions are
considered.
Figure \ref{plot:ext-finite} shows the truncation bound
$\widetilde\Lambda_H^{\dagger,xy}$, the exact truncation error $\Lambda$ and
the deviation of the ground state energy $\left|\Delta E_0\right|$ vs.\
$\lambda$. The values for $\lambda=1$ are given in Tab.\ \ref{tab:ext-finite}.
It turns out that \emph{both} the calculation of the exact truncation error
$\Lambda_H$ and the extensive use of the triangle inequality contribute to the
very large values of $\widetilde \Lambda_H^{\dagger,xy}$. For small values of
$\lambda$, the large value of $\Lambda_H$ dominates the truncation bound
$\widetilde\Lambda_H^{\dagger,xy}$ while for large values of $\lambda$, the
approximation using the triangle inequality to bound the very many
arising terms contributes most.
In a direct comparision, the exact truncation error $\Lambda_H$ is
overestimated by the bound $\widetilde\Lambda_H^{\dagger,xy}$ using the
triangle inequality by two orders of magnitudes, even though symmetries are
used. Nevertheless, $\Lambda_H$ is still considerably higher than the
deviation of ground state energy. In particular, it exceeds
$\left|\Delta E_0\right|$ by several orders of magnitudes for small $\lambda$.
Here we have to keep in mind that the truncation error does not only measure
the inaccuracies of ground state energy, but the effect of truncation to the
entire transformation of the Hamiltonian.
In contrast to the double hard-core boson model, no deviations in the maximal
energy eigen-value were observed (not shown). This is explained
by the fact that the fully polarized state is still an exact eigen-state
of the Hamiltonian \eqref{eq:spinhammi}.
For high values of $\lambda$, the truncation paradox occurs
again because the truncation error bound $\widetilde\Lambda_H^{\dagger,xy}$
for the three-triplon
and the four-triplon truncation exceeds the truncation error bound of the
two-triplon truncation. The exact truncation error $\Lambda_H$
does not display any paradoxical behaviour. Hence we conclude
that the extensive use of the triangle inequality is at the basis
of the truncation paradox.
\section{Summary\label{struct:facit}}
In this work, we presented a mathematically rigorous framework to bound
effects of truncation in self-similar continuous unitary transformations
\emph{a priori}. The difference ${H^{\prime\prime}}=H-H^{\prime}$ between
a unitarily transformed Hamiltonian $H$ and the Hamiltonian
$H^{\prime}$ obtained from the truncated calculation is
captured by an inhomogeneous flow equation
depending only on the truncated terms. We defined the scalar truncation error
$\Lambda$ by the norm of the truncated terms
$\left|\left|\kappa\right|\right|$. It provides an
upper bound for the norm of the difference
$\left|\left|H^{\prime\prime}\right|\right|$.
A completely analogous bound is derived for observables as well.
Using the spectral norm, the truncation error implies an upper
bound for the deviation of the minimal and maximal eigen-value of the
observable under study, which is caused by truncation.
In particular, we derived
a rigorous a priori bound for the error of ground state energy.
The analysis of the double hard-core boson model showed that the norm of the
difference ${H^{\prime\prime}}$ is bounded and approximated very well by the
truncation error. Despite the large difference to $\left|\Delta E_0\right|$,
the truncation error $\Lambda_H$ provided a good measure for the deviation in
the highest excited level $\left|\Delta E_\text{max}\right|$.
This could be understood by the special feature of the truncation scheme that
primarily affected the highest excited level.
For practical use in extended systems, an upper bound $\widetilde\Lambda$
for the truncation error is calculated using the triangle inequality.
The direct calculation of $\left|\left|\kappa\right|\right|$ is not feasible
-- even impossible for infinite systems -- because it would require an exact
diagonalization in the Hilbert space of the entire system.
In both systems studied, the double hard-core boson model and
the extended dimerized spin chain , the bound provided by
truncation error $\Lambda_H$ turned out to overestimate the actual
inaccuracies of the ground state energy significantly.
This is an inevitable consequence of the fact that the truncation error is a
measure for the non-unitarity of the \emph{whole} transformation. Therefore
it is sensitive to distortions of all eigen-values and eigen-states.
A comparison of various bounds for finite dimerized spin chain segments showed
that the truncation bound $\widetilde \Lambda_H$ overestimates the real
truncation error $\Lambda_H$ by orders of magnitudes.
Furthermore, a truncation paradoxon was observed: Looser, i.e., better,
truncation schemes implied higher truncation bounds. In the
finite system studied using the ground state generator,
this paradoxon did not occur for the exact truncation errors.
The truncation bound can be improved efficiently exploiting symmetries of the
Hamiltonian and its hermitecity. By using the transposition of the triplon
polarizations $x$ and $y$ and the hermitecity, we were able to reduce the
bound by a factor $\approx 2\sqrt{2}$. To overcome the high difference to
the truncation error $\Lambda$, much more sophisticated approximations
would be needed. Additional symmetries available are the cyclic spin
permutation, the reflection symmetry, and the translation symmetry.
We do not see a way to exploit the powerful translation symmetry
completely for the improvement of the bounds because this
requires finding bounds for operators in very large or infinite systems.
But larger subsets of terms of restricted range could indeed be analyzed
on larger clusters. The calculation of $H^{\prime\prime}$ in an extended
system is left for further investigation.
The analysis presented here provides a better understanding of
truncation errors in continuous unitary transformations.
We are confident that it
serves as seed for stricter a priori error bounds in the future.
\begin{acknowledgments}
We are grateful for fruitful discussions with K.P.\ Schmidt in particular
for suggesting the illustrative model. We acknwoledge technical support by
C.\ Raas. We want to thank S. Duffe for providing his CUT code
at the initial stage of this work and for many helpful discussions.
\end{acknowledgments}
|
{
"timestamp": "2010-09-21T02:02:39",
"yymm": "1009",
"arxiv_id": "1009.3744",
"language": "en",
"url": "https://arxiv.org/abs/1009.3744"
}
|
\section{introduction}
A well known phenomenon in the theory of mixing times\footnote{We do
not need the notion of mixing time in this paper, it is only used
for comparison. The reader unfamiliar with it may peruse the
survey \cite{MT06} or the book \cite{LPW09}.} is that occasionally certain aspects of a system mix much
faster than the system as a whole. Pemantle \cite{P94} constructed an
example of a random walk on the symmetric group $S_n$ which mixes in
time $n^{1+o(1)}$ while every $k$ elements mix in $\le C(k)\sqrt{n}$
time. Schramm showed that for the interchange process on
the complete graph --- this is another random walk on $S_n$, see below
for details --- the structure of the large cycles mixes in time
$\approx n$, and it was known before \cite{DS81} that the mixing time
of this graph is $\approx n\log n$. See \cite{S05} and also \cite{B11}. Schramm's result is related to ---
in physics' parlance, it is the \emph{mean-field} case of --- a
conjecture of B\'alint T\'oth \cite{T93} that the cycle structure of
the interchange process on the graph $\Z^d$, $d\ge 3$, exhibits a
\emph{phase-transition}. In this paper we investigate the probability of long cycles, and obtain precise formulae for any graph, using the representation theory of $S_n$. As an application, we analyse certain
variations on T\'oth's conjecture.
Let us define the interchange process. Let $G$ be a finite graph with vertex set $\{1,\dotsc,n\}$, and equip each edge $\{i,j\}$ with an alarm clock
that rings with exponential rate $a_{i,j}$. Put a marble on every
vertex of $G$, all different, and whenever the clock of $\{i,j\}$ rings,
exchange the two marbles. Each marble therefore does a standard
continuous-time random walk on the graph but the different walks are
dependent. The positions of the marbles at time $t$ is a permutation
of their original positions, and viewed this way the process is a
random walk on the symmetric group. Note that we have changed the
timing from the previous paragraph. For example, if our graph is the
complete graph and $a_{i,j}=\nicefrac{1}{n}$ for all $i$ and $j$, then the process
mixes in time $\approx\log n$ and the large cycle structure
mixes in time $\approx 1$. However, the added convenience of
having each marble do the natural continuous time random walk
outweighs the difference in notations from some of the literature.
The stronger results of this paper require representation
theory to state, but let us start with two corollaries that can be
stated elementarily. Let $s_k(t)$ be the number of cycles of length
$k$ in our permutation at time $t$. Let
$0=\lambda_0\le\lambda_1\le\dotsb\le\lambda_{n-1}$ be the eigenvalues of
the continuous time Laplacian of the random walk on the graph $G$. Then
\begin{theorem}\label{thm:n}
We have $$ \PP(s_n(t)=1)=\frac1n\prod_{i=1}^{n-1}(1-e^{-\lambda_i t}) $$
\end{theorem}
Let us demonstrate the utility of this formula on the graph
$G=\{0,1\}^d$ with weights equal to 1. There is nothing particular about
this graph, but existing literature allows for easy comparison. For
example, Wilson \cite[\S 9]{W04} showed that the mixing time of the
interchange process on $G$ is $\ge cd$ (see also \cite{M06,O10}). The eigenvalues of $G$ may be
calculated explicitly: the eigenvectors are the Walsh functions, indexed by $y\in\{0,1\}^d$ and given by $f_y(x)=(-1)^{\sum_{i=1}^d x_i y_i}$.
We get that $2k$ is an eigenvalue with multiplicity ${d \choose k}$ for
$k=0,\dotsc,d$. Inserting into the formula at times
$\frac{1\pm\epsilon}{2}\log d$ gives
\begin{align*}
\PP\Big(s_n\Big(\frac{1-\epsilon}{2}\log d\Big)=1\Big)&=
2^{-d}\prod_{k=1}^d\left(1-e^{-(1-\epsilon)k\log d}\right)^{d \choose k}\le
\intertext{and looking only at $k=K:=\lfloor d^\epsilon/2\rfloor$,}
& \le \exp\left(-d^{(\epsilon-1)K}{d\choose K}\right)\stackrel{(*)}{\le}
\exp\left(-\Big(\frac{d^{\epsilon}}{K}\Big)^K\right)\le
\exp\left(-\exp\left(cd^\epsilon\right)\right)
\end{align*}
where $(*)$ comes from
\[
{d \choose
K}=\Big(\frac{d}{K}\Big)^K\cdot\Big(\frac{1-\nicefrac{1}{d}}{1-\nicefrac{1}{K}}\cdot\frac{1-\nicefrac{2}{d}}{1-\nicefrac{2}{K}}\cdot\dotsb\Big)\ge\Big(\frac{d}{K}\Big)^K.
\]
On the other hand,
\begin{align*}
\PP\Big(s_n\Big(\frac{1+\epsilon}{2}\log d\Big)=1\Big) &=
2^{-d}\exp\bigg(\sum_{k=1}^dO(d^{-(1+\epsilon)k}){d\choose k}\bigg) =
2^{-d}(1+O(d^{-\epsilon})).
\end{align*}
We see that the probability equilibrates at $\frac{1}{2}\log d$,
before the mixing time of the whole chain. Further, the equilibration
happens sharply --- this is reminiscent of the cutoff phenomenon for
mixing times. See \cite{DS81}, \cite{LS} or \cite[\S 18]{LPW09} for
the cutoff phenomenon.
We remark that taking $t\to 0$ in Theorem \ref{thm:n} one can get a
new proof of Kirchoff's matrix-tree theorem. We fill the details in the appendix.
Another general, elementarily stated result is:
\begin{theorem}\label{thm:chuk} We have, for any graph $G$ and any $1\leq k \leq n$,
$$ \left|\EE(s_k(t))-\frac 1k\right| \leq \frac{3^n}{k}e^{-t\lambda_1} $$
\end{theorem}
The point about this result is its generality --- it holds for any
graph. In particular examples that we tried the estimate was worse
than the known or conjectured mixing time. But for general graphs it
seems to be the best known.
To proceed, let us recall a few basic facts about the representations
of $S_n$. For a full treatment see the books \cite{FH91,JK81,S00}. A representation of $S_n$ is a group homomorphism
$\tau:S_n\to \mathrm{GL}_k(\C)$ for some $k$, typically denoted by $\dim \tau$. Its character, denoted by $\chi_{\tau}$, is an
element of $L^2(S_n)$ defined by $\chi_{\tau}(g)=\tr(\tau(g))$.
Now, the irreducible representations of $S_n$ are indexed
by partitions of $n$, namely, by sequences
$\lambda=[\lambda_1,\lambda_2,\dotsc ,\lambda_k]$ with $\lambda_1\ge
\lambda_2\ge \dotsb\ge \lambda_k>0$ and $\sum_{i=1}^k \lambda_i=n$ (we
denote this by $n \vdash \lambda$). A nice graphical representation of
partitions is using \emph{Young diagrams}, i.e.\ drawing each $\lambda_i$ as a line of boxes from top
to bottom, e.g.
\[
[5,1]={\tiny\yng(5,1)}\qquad [3,2,1]={\tiny\yng(3,2,1)} \qquad [2,1^3]={\tiny\yng(2,1,1,1)}.
\]
To each partition $n \vdash \lambda$ (and hence, for each young diagram with $n$ boxes) corresponds an irreducible representation, which we shall denote by
$U_\lambda$. For brevity, we denote the character of $U_\lambda$ by $\chi_\lambda$.
Fix now some $1 \leq k \leq n$ and define
\begin{equation}\label{eq:defalpha}
\alpha_k(g)=\#\{\textrm{cycles of length $k$ in $g$}\}.
\end{equation}
Now, $\alpha_k(g)$ depends only on the cycle structure of $g$, i.e.\ is
a class function, and hence it is a linear combinations of characters
of irreducible representations. Our main result is the precise
decomposition.
\begin{theorem}\label{thm:k} For any $n$ and $k$,
\[
\alpha_k=\frac{1}{k}\sum_{n\vdash \rho}a_\rho \chi_\rho,
\]
where
\begin{equation}\label{eq:defak}
a_\rho=\begin{cases}
1&\rho=[n]\\
(-1)^{i+1} & \rho=[k-i-1,n-k+1,1^i]\mbox{ for some
}i\in\{0,\dotsc,2k-n-2\}\\
(-1)^i & \rho=[n-k,k-i,1^i]\mbox{ for some
}i\in\{\max\{2k-n,0\},\dotsc,k-1\}\\
0&\mbox{otherwise}
\end{cases}
\end{equation}
\end{theorem}
Let us describe this verbally (ignoring the diagram $[n]$ which has a somewhat special role). If $k>(n+1)/2$, start with $[k-1, n-k+1]$, with a minus sign. Now
drop boxes from the first row into the leftmost column until the first and second row are equal. Then drop in a single step two boxes, one from each of the first two rows to the leftmost column. Then start dropping
boxes from the second row until you reached a hook-shaped diagram. The sign keeps changing in each step.
If $k\le n/2$ start with the diagram $[n-k,k]$ with a plus sign, and drop boxes from the second row to the leftmost column until reaching a hook-shaped diagram, again switching sign at
each step. The case $k=(n+1)/2$ is similar except you start from
$[n-k,k-1,1]$ with a minus sign.
It is now clear what is special in the case $k=n$. In this case only
hook-shaped diagrams appear in the sum. For the hook-shaped diagrams
there is an explicit formula for the relevant eigenvalues discovered
by Bacher \cite{B94} (see also the appendix of \cite{AK09}). Let us remark that for $k<n$ the probability $\PP(s_k(t)=1)$ is not a function of the eigenvalues of the graph. In other words, one may find two \emph{isospectral} graphs for which these probabilities differ. We will
explain both facts (i.e.\ the conclusion of Theorem \ref{thm:n} from
Theorem \ref{thm:k} and the isospectral examples) in section
\ref{sec:cor} below. T\'oth's conjecture will be stated and discussed
in section \ref{sec:Toth}.
We remark that Theorem \ref{thm:k} strengthens results by Eriksen and
Hultman \cite[\S 5]{EH04} who found the decomposition of $\sum\alpha_k$,
i.e.\ of the number of cycles of a permutations. The formulas of
\cite{EH04} are quite short and reveal some patterns in the numbers
$a_\rho$. For example, for every $\rho$ of the form $[a,b,1^c]$, $a_\rho\ne 0$ for exactly two
values of $k$, with opposite signs.
\section{Notations and preliminaries}\label{sec:notations}
Let $A=\{a_{i,j}\}_{1\le
i<j\le n}$ be a collection of non-negative numbers which we consider
as a weighted graph. The random walk on $S_n$ associated with the weighted
graph $A$ is a process in continuous time starting from the identity
permutation $\mathbf{1}$ on $S_n$ and going from $g$ to $(ij)g$ with
rate $a_{i,j}$. Formally, consider $L^2(S_n)$, both as a Hilbert space with the standard inner product, and as an $\mathbb R$-algebra, via the \emph{group ring} structure.
Define the Laplacian as the element of
$L^2(S_n)$ given by
\[
\Delta=\Delta_A=\sum_{i<j}a_{i,j}(\mathbf{1}-(ij))
\]
where $\mathbf{1}$ is the element of $L^2(S_n)$ equal to 1 in the
identity permutation, and 0 everywhere else; and $(ij)$ is similarly a
singleton at the transposition $(ij)$. The distribution of the
location of our process at time $t$ is
\[
e^{-t\Delta}=\sum_{k=0}^\infty\frac{(-t\Delta)^k}{k!}
\]
In particular for $\alpha_k$
defined by (\ref{eq:defalpha}),
\[
\EE(s_k(t))=
\sum_{g \in S_n}\left(e^{-t\Delta}\right)(g)\alpha_k(g)=n!\langle e^{-t\Delta},\alpha_k\rangle
\]
where here and below $\langle\cdot,\cdot\rangle$ stands for the
standard inner product in $L^2(S_n)$, i.e.\ $\langle a,b\rangle =
\nicefrac{1}{n!}\sum_{g\in S_n} \linebreak[4]a(g)\overline{b(g)}$.
For the proof of Theorem \ref{thm:k} we will need a second set of representations of $S_n$, this time
\emph{reducible} representations. For $n \vdash \rho$, let $T_\rho<S_n$ be the
subgroup of all permutations fixing the sets $\{1,\dotsc,\rho_1\}$,
$\{\rho_1+1,\dotsc,\rho_1+\rho_2\}$, etc\@. As a group
$T_\rho\cong S_{\rho_1}\times \dotsb\times S_{\rho_r}$. Now,
$S_n$ acts on the left cosets of $T_\rho$, i.e.\ $\{hT_\rho\}_{h\in S_n}$, and using these cosets as a basis
we obtain a representation of $S_n$, which we will denote by $V_\rho$.
Readers familiar with exclusion
processes might find it convenient to think about $V_\rho$ as $\mathbb R^X$ where $X$ is the space of configurations of
the exclusion process with $\rho_1$ particles of colour 1, $\rho_2$
particles of colour 2 etc.\ --- considering $\Delta$ as an operator on
$V_\rho$ it is easy to verify that one gets an identical
process. We denote
\begin{equation}\label{eq:defVk}
\psi_\rho=\chi_{V_\rho}.
\end{equation}
It is well known that the representations $V_\rho$ are generally
reducible and their irreducible components, consist of all $U_\sigma$ for
$\sigma\trianglerighteq\rho$, where $\trianglerighteq$ is the
\emph{domination} order \cite[Corollary 4.39]{FH91} --- we say that $\sigma\trianglerighteq\rho$
when you can reach $\rho$ from $\sigma$ by a series of ``toppling'' of
a box of the Young diagram to a lower row which keep the structure
of a Young diagram. Alternatively, $\sigma\trianglerighteq\rho$ is equivalent to
\[
\sum_{i=1}^j\sigma_i \ge \sum_{i=1}^j\rho_i\qquad\forall j.
\]
\section{Character decomposition}
In this section we prove Theorem \ref{thm:k}. We go about it by describing a more general method for expressing a class function on $S_n$ as a linear combination of characters, and then applying it to our case.
Given a function $f:S_n\rightarrow \R$ that is a class function (i.e.\ satisfies $f(hgh^{-1})=f(g)$ for all $g,h\in S_n$), it can be expressed as a linear combination of the characters of $S_n$ (see, e.g., \cite[Proposition 2.30]{FH91}).
By the character orthogonality relations (ibid.), we have
\begin{equation}
f = \sum_{n \vdash \rho} \langle f, \chi_\rho \rangle \chi_\rho
\label{eq:orthog}
\end{equation}
As it is often hard to calculate the inner products $\langle f, \chi_\rho \rangle$ directly, we start by calculating $\langle f, \psi_\lambda \rangle$, where $\psi_\lambda=\chi_{V_\lambda}$ and $V_\lambda$ are the
``exclusion-like'' reducible representations defined just before
(\ref{eq:defVk}).
\begin{lemma} \label{lemma:average}
We have $\langle f, \psi_\lambda \rangle=\frac1{\#(T_\lambda)} \sum_{q\in T_\lambda} f(q) $.
\end{lemma}
\begin{proof} We have $$ \langle f, \psi_\lambda \rangle = \frac1{n!}\sum_{g \in S_n} \psi_\lambda(g) f(g) $$
Recall from \S \ref{sec:notations} that $V_\lambda$ is obtained from the action of
$S_n$ on the cosets of a $T_\lambda<S_n$. By the definition of trace, $\psi_{\lambda}(g)$ equals the number of cosets of $T_{\lambda}$ fixed by $g$. A coset $h T_\lambda$ is fixed by $g$ iff $h^{-1} g h \in T_\lambda$. Hence,
$$ \langle f, \psi_\lambda \rangle = \frac1{n!} \sum_{hT_\lambda \in S_n/T_\lambda}\; \sum_{g: h^-1gh \in T_\lambda} f(g). \label{eq:youngrule} $$
Let us make a change of variables, $q = h^{-1} g h $. Since $f$ is a class function, we have
\[ \langle f, \psi_\lambda \rangle = \frac1{n!\#(T_\lambda)}
\sum_{h\in G} \sum_{q\in T_\lambda} f(hqh^{-1}) =
\frac1{n!\#(T_\lambda)} \sum_{h\in G} \sum_{q\in T_\lambda} f(q) =
\frac1{\#(T_\lambda)} \sum_{q\in T_\lambda} f(q).
\qedhere
\]
\end{proof}
Now, by Young's rule \cite[Corollary 4.39]{FH91}, the characters $\psi_\lambda$ and the characters $\chi_\lambda$ are related by the linear equations
$$ \psi_\lambda = \sum_{n \vdash \mu} K_{\mu \lambda}\chi_\mu $$
Where the numbers $K_{\mu \lambda}$, called the Kostka numbers, are defined as follows:
Let $\lambda=[\lambda_1,\dotsc,\lambda_r]$, then $K_{\mu \lambda}$ is the number of ways the Young diagram $\mu$ can be filled with $\lambda_1$ $1$'s, $\lambda_2$ $2$'s, etc., such that each row is nondecreasing, and each column is strictly increasing. The numbers $K_{\mu \lambda}$
satisfy $K_{\mu \lambda}=0$ whenever $\mu < \lambda$ (with respect to the lexicographic order), and $K_{\mu \mu}=1$. (See \cite{FH91}, appendix A). Hence,
$$ \langle f,\psi_\lambda \rangle = \sum_{n \vdash \mu} K_{\mu \lambda}\langle f,\chi_\mu \rangle $$
In other words the numbers $\langle f,\chi_\mu\rangle$ satisfy
a system of linear equations, whose coefficient matrix $(K_{\mu \lambda})$
is triangular with $1$'s on the diagonal, hence invertible.
The resulting system of equations has a more elegant form when expressed in terms of \emph{symmetric polynomials}.
Fix an integer $m \geq n$ (whose value is not important), and consider
the ring of symmetric polynomials in $m$ variables $x_1,\dotsc ,x_m$
over $\C$. Consider the following homogeneous symmetric polynomials of degree $n$ (see \cite{FH91}, ibid. for more details):
\begin{itemize}
\item For $n \vdash \lambda=[\lambda_1,\dotsc,\lambda_r]$, $M_\lambda = \sum_\alpha x^\alpha$, where $\alpha=(\alpha_1,\dotsc,\alpha_n)$ goes over all the possible permutations of $(\lambda_1,\dotsc \lambda_r,0,\dotsc,0)$.
\item The Schur polynomials $S_\mu= \sum_{\lambda}K_{\mu \lambda}M_{\lambda}$
\item The full homogeneous polynomial $H_n$, defined as the sum of all monomials of degree $n$. It is easy to see that for all $n \vdash \lambda$, $K_{[n]\lambda}=1$. Hence, $H_n = \sum_{n \vdash \lambda} M_\lambda = \sum_{n \vdash \lambda} K_{[n]\lambda} M_\lambda = S_{[n]}$.
\end{itemize}
Recall also the Frobenius characteristic map $\ch$, defined on the class functions of $S_n$, which sends an irreducible character $\chi_{\mu}$ to its corresponding Schur polynomial $S_{\mu}$, and is extended by linearity. Clearly, decomposing a class function into irreducible characters, $f=\sum_{\mu}a_\mu \chi_{\mu}$ is equivalent to decomposing its image $\ch(f)$ into Schur polynomials, $\ch(f)=\sum_{\mu} a_{\mu} S_{\mu}$. By (\ref{eq:youngrule}), we have
$$ \ch(f)=\sum_{\mu} \langle f, \chi_{\mu} \rangle S_{\mu} =
\sum_{\mu} \langle f, \chi_{\mu} \rangle \sum_{\lambda} K_{\mu\lambda} M_{\lambda} =
\sum_{\lambda} \langle f, \psi_{\lambda} \rangle M_\lambda $$
We conclude:
\begin{lemma} \label{lemma:ch} Let $f$ be a class function on $S_n$. Then $$\ch(f) = \sum_{\lambda} \left ( \frac{1}{\#(T_{\lambda})} \sum_{g\in T_{\lambda}} f(g) \right ) M_{\lambda}. $$
\end{lemma}
We now apply this to the class functions $\alpha_k$ (Recall the definition of $\alpha_k$, (\ref{eq:defalpha})).
\begin{lemma} \label{lemma:ch_formula}
We have for all $1\leq k \leq n$, $\ch(\alpha_k)=\frac1k(\sum_{i=1}^m x_i^k)H_{n-k}(x_1,\dotsc,x_m)$.
\end{lemma}
\begin{proof} Let us define a function $\beta_k$ on the set of partitions of $n$ by
$$\beta_k([\lambda_1,\dotsc,\lambda_r])= \#\{i: \lambda_i \geq k \}. $$
By lemma \ref{lemma:average},
$$ \langle \alpha_k, \psi_\lambda \rangle = \frac1{\#(T_\lambda)} \sum_{q\in T_\lambda} \alpha_k(q). $$
The sum $\sum_{q\in T_\lambda} \alpha_k(q) $ can be evaluated by summing over all possible $k$-cycles $c\in T_\lambda$, the number of elements of $T_\lambda$ such that $c$ is one of their cycles. For any $i$ such that $\lambda_i\geq k$, there are $\binom{\lambda_i}{k} \cdot (k-1)!$ choices for a cycle $c$ in the $S_{\lambda_i}$-factor of $T_\lambda$, and $\lambda_1! \lambda_2! \dotsb (\lambda_i - k)! \dotsb \lambda_r!$ choices for an element $g\in T_\lambda$ with $c$ as a cycle. Hence each such $i$ contributes to the sum
$$ \binom{\lambda_i}{k} \cdot (k-1)! \cdot \lambda_1! \lambda_2! \dotsb (\lambda_i - k)! \dotsb \lambda_r! = \frac {\#(T_\lambda)} k$$ Obviously, if $\lambda_i < k$ then there are no $k$-cycles in the $S_{\lambda_i}$-factor, and the contribution is $0$. Hence,
\[
\langle \alpha_k, \psi_\lambda \rangle= \frac1{\#(T_\lambda)} \sum_{i:\lambda_i
\geq k} \frac {\#(T_\lambda)} k = \frac 1 k \beta_k(\lambda).
\]
By lemma \ref{lemma:ch},
$$ \ch(\alpha_k)=\frac1k\sum_{\lambda} \beta_k(\lambda)M_{\lambda}. $$
A moment's reflection shows that $ \sum_{\lambda} \beta_k(\lambda)M_{\lambda} =
(\sum_{i=1}^m x_i^k)H_{n-k}$. Indeed, each monomial
$x_1^{\alpha_1}\dotsb \discretionary{}{\mbox{$\cdot\,$}}{}
x_m^{\alpha_m}$ of degree $n$ appears on the left-hand side
with coefficient $\#\{i:\alpha_i\ge k\}$ (by the definition of
$\beta$), and the same is on the right-hand side. This finishes the lemma.
\end{proof}
Our goal is to express $\ch(\alpha_k)$ as a linear combination of Schur
polynomials. Let us start with the case of $k=n$.
\begin{lemma} \label{lemma:alpha_n} $\ch(\alpha_n) = \frac1n\sum_{i=0}^{n-1} (-1)^i S_{[n-i,1^i]} $.
\end{lemma}
\begin{proof} By lemma \ref{lemma:ch_formula}, $\ch(\alpha_n)=\frac 1n \sum_i x_i^n = \frac1n M_{[n]}$. On the other hand, for all $0\leq i \leq n-1$ we have
$$ S_{[n-i,1^i]} = \sum_{\lambda} K_{[n-i,1^i]\lambda} M_{\lambda}. $$
Let $\lambda$ have $r$ rows. By definition of the Kostka numbers, we have $K_{[n-i,1^i]\lambda}=\binom{r-1}{i}$, and $K_{[n-i,1^i]\lambda}=0$ for $i\geq r$, since the top left box of $[n-i,1^i]$ has to be numbered $1$, and the whole configuration is determined by the choice of distinct $i$ numbers out of $2,\dotsc,r$ to be placed in the leftmost column in ascending order.
Denoting by $r(\lambda)$ the number of rows in $\lambda$, we get
$$ \sum_{i=0}^{n-1}(-1)^iS_{[n-i,1^i]} = \sum_{\lambda} M_{\lambda} \sum_{i=0}^{n-1} (-1)^i \binom{r(\lambda)-1}{i} $$
By the binomial identity, the inner sum is $0$ unless $r(\lambda)=1$,
i.e.\ $\lambda=[n]$, in which case the inner sum is $1$. We get $ \sum_{i=0}^{n-1}(-1)^iS_{[n-i,1^i]}= M_{[n]}$, as desired.
\end{proof}
\begin{rem*}Lemma \ref{lemma:alpha_n} can be proved more directly by
using the Murnaghan-Nakayama rule \cite[Theorem 4.10.2]{S00} to
express $\alpha_n$ as a linear combination of characters: for any
$n\!\vdash\!\lambda$, the
scalar product $\langle\chi_\lambda,\alpha_n\rangle$ is,
up to a constant, the value of $\chi_\lambda$ at one specific permutation, namely a
cycle of length $n$. The Murnaghan-Nakayama rule, when applied to
such a cycle, takes a simple form.
\end{rem*}
We immediately conclude:
\begin{corollary} We have
$$ \alpha_n = \frac1n \sum_{i=0}^{n-1} (-1)^i \chi_{[n-i, 1^i]} $$
which is Theorem \ref{thm:k} for $k=n$.
\end{corollary}
Let us now treat the general case, using the case we already
proved. By lemma \ref{lemma:alpha_n}, applied to $k$,
$$ \frac 1k \sum_{i=1}^m x_i^k = \frac 1k \sum_{i=0}^{k-1}(-1)^i S_{[k-i,1^i]} $$
Hence, by lemma \ref{lemma:ch_formula},
$$ \ch(\alpha_k) = \frac 1k \left( \sum_{i=0}^{k-1}(-1)^i S_{[k-i,1^i]} \right ) H_{n-k}. $$
We now apply Pieri's formula (see \cite{FH91}), according to which,
$S_{[k-i,1^i]} H_{n-k}$ is the sum of all polynomials of the form
$S_{\lambda'}$, where $\lambda'$ is obtained by adding $n-k$ boxes to
$[k-i,1^i]$, without adding two boxes in the same column. Since we
have a hook-shaped diagram, our possibilities are rather limited:
we may add a box at the leftmost column or not, and the rest of the
boxes go in the first two rows. Denote therefore
$$ S_{[k-i,1^i]}H_{n-k}=A_i+B_i$$
where $A_i$ is the sum when one does not add a square at the leftmost
column, and $B_i$ is when one does. Denote also
$x(i,j)=S_{[n-i-j,1+j,1^{i-1}]}$ (the contribution coming from adding $j$ boxes to the second row of $[k-i,1^i]$, and the remaining $n-k-j$ boxes to the first row). Then
\begin{gather*}
A_0=S_{[n]}\qquad A_i = \sum_{j=0}^{\min(n-k,k-i-1)}x(i,j)\\
B_i = \sum_{j=0}^{\min(n-k-1,k-i-1)}x(i+1,j).
\end{gather*}
We now sum over $i$ and get,
$$ \left(\sum_i x_i^k \right ) H_{n-k} = \sum_{i=0}^{k-1} (-1)^i (A_i+B_i)$$
Our next goal is to find the alternating sum $\sum_{i=0}^{k-1} (-1)^i
(A_i+B_i)$. There are further cancellations here because $B_i$ and
$A_{i+1}$ are quite similar --- $B_i$ corresponds to adding a box to
the first column of $[k-i,1^i]$ while $A_{i+1}$ corresponds to not
adding a box to the first column of $[k-i-1,1^{i+1}]$. Hence most of
the terms cancel out. We get
\begin{align*}
A_{i+1}&= \left \{ \begin{array}{ll}
\sum_{j=0}^{n-k}x(i+1,j) & 0 \leq i \leq 2k-n-2 \\
\sum_{j=0}^{k-i-2}x(i+1,j) & 2k-n-1 \leq i \leq k-2 \\
\end{array}
\right.\\
B_i&= \left \{ \begin{array}{ll}
\sum_{j=0}^{n-k-1}x(i+1,j) & 0 \leq i \leq 2k-n-1 \\
\sum_{j=0}^{k-i-1}x(i+1,j) & 2k-n \leq i \leq k-1 \\
\end{array}
\right.
\end{align*}
Hence (putting $A_k=0$),
$$ B_i-A_{i+1}= \left \{ \begin{array}{ll}
\sum_{j=0}^{n-k-1}x(i+1,j)-\sum_{j=0}^{n-k}x(i+1,j)=-x(i+1,n-k) & 0 \leq i \leq 2k-n-2 \\
\sum_{j=0}^{n-k-1}x(i+1,j)-\sum_{j=0}^{n-k-1}x(i+1,j)=0& i=2k-n-1 \\
\sum_{j=0}^{k-i-1}x(i+1,j)-\sum_{j=0}^{k-i-2}x(i+1,j)=x(i+1,k-i-1)& 2k-n \le i \leq k-1 \\
\end{array}
\right.
$$
and
\begin{align*}
\sum_{i=0}^{k-1} (-1)^i (A_i+B_i)&=A_0+\sum_{i=0}^{k-1}(-1)^i(B_i-A_{i+1})=\\
&=S_{[n]}-\sum_{i=0}^{2k-n-2}(-1)^ix(i+1,n-k)+\sum_{i=2k-n}^{k-1}(-1)^ix(i+1,k-i-1)=\\
& = S_{[n]}-\sum_{i=0}^{2k-n-2}(-1)^iS_{[k-i-1,n-k+1,1^i]}+\sum_{i=2k-n}^{k-1}(-1)^i S_{[n-k,k-i,1^i]} = \sum_{\rho} a_{\rho} S_{\rho}
\end{align*}
where the numbers $a_{\rho}$ were defined in the statement of Theorem \ref{thm:k}.
Hence, by lemma \ref{lemma:ch_formula},
$$ k \cdot \ch(\alpha_k) = \left(\sum_i x_i^k \right ) H_{n-k} = \sum_{\rho} a_{\rho} S_{\rho}.$$
This ends the proof of Theorem \ref{thm:k}.\qed
\section{The probability of long cycles}\label{sec:cor}
Let $\rho$ be a partition of $n$, and
let $U_\rho:S_n\to\GL(\C^{\dim U_\rho})$ be the corresponding irreducible
representation. Let $D=\sum d_gg$ be any element of the group
ring. Then $U_\rho(D)$ is the element of $\GL(\C^{\dim U\rho})$ given by
\[
\sum_g d_gU_\rho(g).
\]
(it might be useful to think about $U_\rho(D)$ as a non-commutative Fourier transform
of $D$, with the fact that $U_\rho(D_1D_2)=U_\rho(D_1)U_\rho(D_2)$
being the non-commutative analog of
$\widehat{f*g}=\widehat{f}\widehat{\vphantom{f}g}$).
In the case that $D=\Delta_A$ we will denote the eigenvalues of this matrix by
$0\leq\lambda_1(A,\rho)\leq \dotsc \leq \lambda_{\dim(\rho)}(A,\rho)$
(it is well-known that $U_\rho(\Delta_A)$ is positive semidefinite and
in particular diagonalizable, see e.g.\ \cite{AK09}).
\begin{lemma} \label{lemma:formula}
For any $n$ and $k$ we have
\[
\EE(s_k(t))=\frac
1k\sum_{n\vdash\rho}a_\rho\sum_{j=1}^{\dim U_\rho}e^{-t\lambda_j(A,\rho)}
\]
where $a_\rho$ are as in Theorem \ref{thm:k}.
\end{lemma}
\begin{proof}
As discussed in \S\ref{sec:notations},
$$\EE(s_k(t))=n!\langle \alpha_k, e^{-t\Delta_A} \rangle = \frac1k \sum_\rho a_\rho n!\langle e^{-t\Delta_A}, \chi_\rho\rangle $$
By definition, $\chi_\rho$ attaches to each $g\in S_n$ the trace of
$g$ acting on the representation $U_\rho$. By the linearity of the trace,
\[
n!\langle e^{-t\Delta_A},\chi_\rho\rangle =
\tr\left(U_\rho\left(e^{-t\Delta_A}\right)\right)
\]
where $U_\rho(\cdot)$ is the action of a representation on an
element of the group ring as above. Further, for every representation
$U$ and any element $D$ of the group ring,
\[
U(e^D)=e^{U(D)}
\]
where the exponentiation on the left-hand side is in the group ring
while on the right-hand side we have exponentiation of matrices. Since $U_\rho(-t \Delta)$ is diagonalizable,
\[
\tr\left(U_\rho\left(e^{-t\Delta}\right)\right)=\sum_j
e^{-t\lambda_j(A,\rho)}.
\]
The proof now follows from Theorem \ref{thm:k}.
\qedhere
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:n}]
$s_n(t)$ can take only the values 0 and 1. Hence, using lemma \ref{lemma:formula} for $k=n$, we get
$$ \PP(s_n(t)=1)=\EE(s_n(t))=\frac1n\sum_{i=0}^{n-1} (-1)^i \sum_j e^{-t\lambda_j(A, [n-i,1^i])} $$
Since $[n-i,1^i]$ is a hook-shaped diagram, the eigenvalues
$\lambda_j(A,[n-i,1^i])$ are simply all the sums of $i$-tuples of the
eigenvalues $\lambda_1(A),\dotsc,\lambda_{n-1}(A)$. (See \cite{B94}
and also the appendix of \cite{AK09}). Hence,
\[
\PP(s_n(t)=1)=\frac1n\left(1+\sum_{i=1}^{n-1}(-1)^i\sum_{1\leq j_1 <
j_2 < \dotsc <j_i \leq
n-1}e^{-t(\lambda_{j_1}+\dotsc+\lambda_{j_i})}\right) =
\frac1n\prod_{i=1}^{n-1}(1-e^{-\lambda_i t}).\qedhere
\]
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:chuk}]
The partitions that appear in lemma \ref{lemma:formula} are of the
form $[a,b,1^c]$, where $a+b+c=n$, $a\geq b> 0$, and $c \geq 0$. For
such a partition a simple calculation with the hook formula \cite[\S 4.12]{FH91} gives
$$ \dim U_{\lambda} = \frac {b(a-b+1)}{(b+c)(a+c+1)} \frac {n!}{a!b!c!} \leq \binom{n}{a,b,c} $$
Hence, the total number of summands in lemma \ref{lemma:formula} is bounded by $\sum_{a+b+c=n} \binom{n}{a,b,c}=3^n$.
Also, by the celebrated Caputo-Liggett-Richthammer theorem \cite{CLR09}, we have for all $j$,
$$\lambda_j(A, [a,b,1^c]) \geq \lambda_1(A)$$
The result now follows from lemma \ref{lemma:formula}.
\end{proof}
\begin{rem*}
For a non-hook-shaped partition $\rho$, the eigenvalues
$\lambda(A,\rho)$ are, in general, not a function of the eigenvalues
of the graph $A$. Such examples exist for $n$ as low as 4. In other
words, one can find two \emph{isospectral} (weighted) graphs $A_1$,
$A_2$ with 4 vertices for which $\lambda(A_i,[2,2])$ differ. By lemma
\ref{lemma:formula}, these two isospectral graphs also have different
values for $\PP(s_{3}(t)=1)$ for general $t$. Such examples can be
found by constructing $A_2$ as a conjugation of $A_1$ (for a generic $A_1$)
by an orthogonal perturbation of the identity which preserves the vector $(1,\dotsc,1)$.
\end{rem*}
\section{T\'oth's conjecture}\label{sec:Toth}
Let us start by describing T\'oth's work on the quantum Heisenberg ferromagnet \cite{T93}. Building on earlier work by Conlon and Solovej, he found what physicists term a \emph{graphical representation} of the model, i.e.\ a rigorous translation to an (interacting) random walk question. Most relevant for us is T\'oth's formula for the \emph{spontaneous magnetization} $m(\beta)$ of the quantum Heisenberg ferromagnet at inverse temperature $\beta$. Let $c_\beta(0)$ be the size of the cycle of $0$ at time $\beta$ for the interchange process on $[-r,r]^3$. Then \cite[(5.2)]{T93}
\[
m(\beta)=\frac 12 \lim_{n\to\infty}\lim_{r\to\infty}
\frac{\mathbb E\Big(\mathbf 1\{c_\beta(0)>n\}2^{\sum_{k\ge 1} s_k(\beta)}\Big)}
{\mathbb E\big(2^{\sum_{k\ge 1} s_K(\beta)}\big)}
\]
(recall that $s_k(\beta)$ is the number of cycles of length $k$ at time $\beta$, so their sum is just the total number of cycles, again for the interchange process on $[-r,r]^3$). Notice the somewhat counterintuitive fact that the inverse temperature becomes the time in this representation. With this formula (which some readers might feel more convenient to simply take as the definition of $m(\beta)$), T\'oth's conjecture is
\begin{conjecture}\label{conj:QHF}
$m(\beta)$ admits a phase transition, i.e.\ there exists some $\beta_c$ such that $m(\beta)=0$ for $\beta<\beta_c$ and $m(\beta)>0$ for $\beta>\beta_c$.
\end{conjecture}
It is natural to try first to remove the weights and investigate only $\mathbb P(c_\beta(0)>n)$ (T\'oth himself hints that this might be an interesting toy model). One then gets the following:
\begin{conjecture}\label{conj:interchange}
The function
\[
\lim_{n\to\infty}\lim_{r\to\infty}\mathbb P(c_\beta(0)>n)
\]
Undergoes a phase transition in $\beta$: it is zero for $\beta<\beta_c$ (not necessarily the same $\beta_c$ as in the previous conjecture) and positive for $\beta>\beta_c$.
\end{conjecture}
For both conjectures, it is not difficult to show that for $\beta$
sufficiently small the corresponding limits are zero. What is wide
open, for both conjectures, is that for $\beta$ sufficiently large,
the limits are non-zero. In other words, the big open problem at this
point is not sharpness or uniqueness of the phase transition, but the
actual existence of the high $\beta$ phase (the so-called ordered
phase).
Conjecture \ref{conj:interchange} was investigated when $[-r,r]^3$ is
replaced by the complete graph, the so-called \emph{mean-field}
case. The mean-field case was solved first by Berestycki \& Durrett
\cite{BD06} (who arrived at this problem from a different angle) and
then by Schramm \cite{S05}, who gave much more information on the
structure of the large cycles. In the mean-field case, $\beta_c$ is
explicitly known. An analog of conjecture \ref{conj:interchange} for
infinite graphs was investigated for trees
\cite{A03,H12a,H12b}. Notably, for trees of sufficiently high degree,
\cite{H12b} shows that there is a phase transition without calculating
the value of $\beta_c$.
We consider our Theorem \ref{thm:k} as a stepping stone for a
representation-theoretic attack on both conjectures. For conjecture
\ref{conj:interchange}, it reduces the problem to a calculation or
estimate of the eigenvalues of only some representations. In the
mean-field case, these eigenvalues are explicitly known \cite{DS81}
which leads to a simple analysis of this problem, see \cite{BK}. For conjecture \ref{conj:QHF}, this requires an extra ingredient even in the mean-field case: the interaction between the function $c_\beta(0)$ and the function $2^{\sum s_k(\beta)}$. We hope to tackle this problem in the future.
To gain some more insight on the non-mean-field case in conjecture \ref{conj:interchange}, let us examine the case $k=n$, i.e.\ apply Theorem \ref{thm:n} to the graph $[-r,r]^3$. For this graph the eigenfunctions and eigenvalues are explicitly known. Every vector $\xi\in\{0,\dotsc,2r-1\}^3$ the function $f(v)=\exp(2\pi i\langle\xi,v\rangle/(2r-1))$ is an eigenvector with the eigenvalue being $\sum(1-\cos(2\pi\xi_j/(2r-1)))$. Plugging these values into Theorem \ref{thm:n} with a little calculation shows, for example,
\[
\min\Big\{t:\PP(s_n(t)=1)\ge \frac 1{2n}\Big\}\approx r^2\approx n^{2/3}.
\]
In other words, the probability starts approaching the limit value $\frac 1n$ only when t is of the order of $n^{2/3}$ (for general dimensions, i.e.\ the graphs $[-r,r]^d$, the value would be $n^{2/d}$).
Thus we see that, unlike what one would expect from a naive extrapolation of conjecture \ref{conj:interchange}, the probability of a cycle of length $n$ does \emph{not} equilibrate at constant time but after much longer time.
The culprit
for this slow equilibration lies in the representation $[n-1,1]$
appearing in the sum when $k=n$. It is therefore reassuring to notice
that this representation appears \emph{only} when $k=n$. Again, at
this point our estimates for the eigenvalues
$\lambda_j([-r,r]^3,\rho)$ are too weak to give good information on
T\'oth's conjecture. See \cite{T10} for more information on these eigenvalues.
\subsection*{Acknowledgements}
We wish to thank Nati Linial for asking what happens when $t\to
0$; Richard Stanley for referring us to \cite{EH04}; and Yuval Roichman
and Ron Adin for interesting discussions. GK's research partially supported by the Israel Science
Foundation.
|
{
"timestamp": "2012-05-28T02:02:54",
"yymm": "1009",
"arxiv_id": "1009.3723",
"language": "en",
"url": "https://arxiv.org/abs/1009.3723"
}
|
\section*{Preliminary notation}
This paper uses the following notation throughout. Given two real-valued functions $f,g$ with domain $D$, we write
\begin{itemize}
\item $f\ll g$, $f=O(g)$ or $g = \Omega(f)$ if there is a constant $\gamma$ such that $f(x) \leq \gamma g(x)$ for all $x \in D$. The implicit constant $\gamma$ may be different each time this notation is used.
\item $f \approx g$ if $f \ll g$ and $g \ll f$
\end{itemize}
Given two sets $A,B \subseteq \mathbb{F}_q$, we define:
\begin{itemize}
\item the \textbf{sumset} $A+B=\left\{a+b:a \in A, b \in B\right\}$
\item the \textbf{product set} $A \cdot B=\left\{ab:a \in A, b \in B\right\}$
\item the \textbf{ratio set} $\frac{A}{B}=\left\{a b^{-1}:a \in A, b \in B, b \neq 0\right\}$
\end{itemize}
\section{Introduction}
\subsection{Incidences}
This paper is about incidences between points and lines in a plane. A point is \textbf{incident} to a line if it lies on that line, and a single point can be incident to more than one line if they cross at that point. An established problem is to find upper bounds for the number of incidences between finite sets of points and lines of given cardinality.
Specifically, fix a field $F$ and an integer $n$, and let $P$ and $L$ be finite sets of points and lines respectively in the plane $F \times F$ with $\left|P\right|$=$\left|L\right|=n$. Define $$I(P,L)=\left|\left\{(p,l)\in P \times L:p \in l\right\}\right|$$ to be the cardinality of the set of incidences between $P$ and $L$. The problem is to establish upper bounds on $I(P,L)$. A straightforward exercise in combinatorics \cite{TV} shows that one always has $I(P,L)\ll n^{\frac{3}{2}}$. So non-trivial incidence bounds are those of the form $I(P,L)\ll n^{\frac{3}{2}-\epsilon}$ for positive $\epsilon$.
\subsection{Known bounds}
Different bounds are known for different choices of the field $F$. Things are largely settled in the settings $F=\mathbb{R}$ and $F=\mathbb{C}$. The result $\epsilon = 1/6$ was obtained in these settings, by Szem\'eredi and Trotter \cite{ST} and T\'oth \cite{toth} respectively. In both cases, the bound holds unconditionally and is sharp up to multiplicative constants.
Much less is known in the finite field setting $F=\mathbb{F}_q$. It is certainly not possible to have a non-trivial bound that holds in all cases, as the trivial bound $I(P,L)\approx n^{\frac{3}{2}}$ is achieved when $P=F \times F$ and $L$ is the set of lines determined by pairs of points in $P$. So one must impose some extra condition on $P$.
When $F=\mathbb{F}_p$ is a finite field of prime order this can be simply a cardinality condition. The best-known result in this setting, due to Helfgott and Rudnev \cite{HR}, requires simply that $n$ is strictly less than $p$, and guarantees that $\epsilon\geq 1/10678$ when this condition is satisfied. This result is unlikely to be best-possible, and followed work of Bourgain, Katz and Tao \cite{BKT} which established the existence of a non-trivial $\epsilon>0$ so long as $n<p^{2-\delta(\epsilon)}$, but did not quantify it.
\subsection{Bounds over general finite fields}
The Helfgott-Rudnev bound is known only in $\mathbb{F}_p$, and so one would like to extend it to general (i.e. not necessarily prime) finite fields $\mathbb{F}_q$. In particular, it would be good to extend to $\mathbb{F}_{p^2}$, as this is the finite analogue of $\mathbb{C}$. However, general finite fields can have subfields, and so stronger conditions than just cardinality are required on $P$. This is because, as with the example above, if $K$ is a subfield of $F$ then the trivial bound $I(P,L)\approx n^{\frac{3}{2}}$ can be achieved when $P$ is the subplane $K \times K$.
It is therefore an interesting problem to find conditions on $P \subseteq \mathbb{F}_q \times \mathbb{F}_q$ for which an explicit Helfgott-Rudnev-type bound holds for any $L$ with $|L|=|P|$. Progress on this problem sheds light on the relationship between the algebraic structure of fields and the geometric structure of incidences. Ultimately one would like to find an algebraic condition for $P$ that is both necessary and sufficient for an explicit incidence bound.
The natural condition to try imposing on $P$ would be to insist that it is `not too close' to being a copy of a subplane, for example by ensuring that its projection onto one of either the $x$- or $y$-axis is `not too close' to a copy of a subfield. However, the currently-known approaches for proving Helfgott-Rudnev-type bounds rely on first applying a projective transformation to $P$, which could disrupt such a condition. So any condition must, additionally, be preserved by projective transformation.
\subsection{Results}
We present an incidence result in $\mathbb{F}_q$, which holds so long as $P$ satisfies certain conditions. Informally, these are that the projection $A(P)$ of $P$ onto some co-ordinate axis has no more than `half-dimensional interaction' with `large' subfields $G$ of $\mathbb{F}_q$, where `large' will be defined relative to the cardinality $n=|P|$.
By no more than `half dimensional interaction', we mean that $A(P)$ does not intersect an affine copy of $G$ in more than $|G|^{1/2}$ places, and intersects no more than $|G|^{1/2}$ distinct translates of $G$. Since the motivation is that such sets are a long way from being fields, we shall call them `antifields' and `strong antifields'.
\begin{definition}[Antifields]
Let $F$ be a field and $\lambda>0$.
\begin{enumerate}
\item
Let $A \subseteq F$. Then
\begin{enumerate}
\item
$A$ is a $\mathbf{(1,\lambda)}$-\textbf{antifield} if $\left|A \cap (aG+b)\right|\leq \max\left\{\lambda,|G|^{\frac{1}{2}}\right\}$ for all subfields $G$ of $F$ and all $a,b \in F$.
\item
$A$ is a $\mathbf{(1,\lambda)}$-\textbf{strong-antifield} if it is a $(1,\lambda)$-antifield and, for every subfield $G$ with $|G|\geq \lambda$, it intersects strictly fewer than $\max\left\{\lambda,|G|^{\frac{1}{2}}\right\}/2$ distinct translates $G+b$ of $G$.
\end{enumerate}
\item Let $P \subset F \times F$. Then
\begin{enumerate}
\item $P$ is a $\mathbf{(2,\lambda)}$\textbf{-antifield} if the set $\left\{x:(x,y) \in P\right\}$ is a $(1,\lambda)$-antifield
\item $P$ is a $\mathbf{(2,\lambda)}$\textbf{-strong-antifield} if the set $\left\{x:(x,y) \in P\right\}$ is a $(1,\lambda)$-strong-antifield
\end{enumerate}
\end{enumerate}
\end{definition}
Note that since one can always apply a change of basis, the projection can in fact be onto any vector multiple of $\mathbb{F}_q$.
Parts $1.(a)$ and $2.(a)$ of the definition are motivated by work of Katz and Shen \cite{KS} generalising sum-product bounds in $\mathbb{F}_p$ to $\mathbb{F}_q$. Parts $1.(b)$ and $2.(b)$ are motivated by the need to avoid disruption by projective transformations. A key idea, which shall be seen later, is that certain projective images of a strong antifield will always be antifields.
We are now able to state the result:
\begin{theorem}\label{theorem:result}
There is an absolute constant $\gamma$ such that if $F$ is a finite field, $P$ and $L$ are sets of points and lines respectively in $F \times F$ with $|P|=|L|=n$, and $P$ is additionally a $\left(2,\gamma n^{\frac{2560}{6419}}\right)$-strong-antifield, then $I(P,L) \ll n^{\frac{3}{2}-\frac{1}{12838}}$.
\end{theorem}
The majority of this paper is concerned with the proof of Theorem \ref{theorem:result}. But since it is not necessarily obvious that many point sets should satisfy the conditions of the theorem, we shall first show that it is easy to construct examples in the important cases $q=p^2$ and $q=p^4$. This is demonstrated by the following two corollaries; the first corollary demonstrates the requirement for limited interaction with subfields, and the second corollary demonstrates how one can ignore `small' subfields.
\begin{corollary}[Construction when $q=p^2$]\label{theorem:p^2} Let $P \subseteq \mathbb{F}_{p^2} \times \mathbb{F}_{p^2}$ with $|P|=n$, and define $A=A(P)=\left\{x:(x,y) \in P\right\}$. Let $t$ be a defining element of $\mathbb{F}_{p^2}$ over $\mathbb{F}_p$, so that $\mathbb{F}_{p^2}=\mathbb{F}_p + t \mathbb{F}_p$. Suppose that $|A|\ll p$ and that $A=\bigcup_{j \in J}A_j$
where $J \subseteq \mathbb{F}_p$ with $|J|\ll \max\left\{p^{\frac{1}{2}},n^{\frac{2560}{6419}}\right\}$, and $A_j \subseteq \mathbb{F}_p+jt$ with $|A_j|\ll \max\left\{p^{\frac{1}{2}},n^{\frac{2560}{6419}}\right\}$ for each $j \in J$. Then we have
$I(P,L)\ll n^{\frac{3}{2}-\frac{1}{12838}} $ for all sets of lines $L$ in $\mathbb{F}_{p^2}\times \mathbb{F}_{p^2}$ with $|L|=n$.
\end{corollary}
\begin{proof}
We need to show that the hypotheses imply that $P$ is a $\left(2,\gamma n^{\frac{2560}{6419}}\right)$-strong-antifield. To do this, we first need to show that $P$ is simply a $\left(2,\gamma n^{\frac{2560}{6419}}\right)$-antifield. Note that the only sets of the form $a\mathbb{F}_p+b$ with $a, b \in \mathbb{F}_{p^2}$ are given by $\mathbb{F}_p +jt$ and $t\mathbb{F}_p +k$, where $j,k$ range over $\mathbb{F}_p$. Note further that $\left(\mathbb{F}_p +jt \right)\cap\left( t\mathbb{F}_p+k\right)=\left\{jt+k\right\}$. We know by assumption that $$|A \cap \left(\mathbb{F}_p+jt\right)|\ll \max\left\{p^{\frac{1}{2}},n^{\frac{2560}{6419}}\right\}$$ for each $j \in \mathbb{F}_p$. Observe that
\begin{align*}
|A \cap \left(t\mathbb{F}_p+k\right)|= \sum_{j \in \mathbb{F}_p}\left|A \cap \left(t \mathbb{F}_p +k \right)\cap\left(\mathbb{F}_p+jt\right) \right|= \# \left\{j \in \mathbb{F}_p:\left|A \cap \left(\mathbb{F}_p+jt\right)\right|\right\}\leq |J| \ll \max\left\{p^{\frac{1}{2}},n^{\frac{2560}{6419}}\right\}.
\end{align*}
So we conclude that $P$ is a $\left(2,\gamma n^{\frac{2560}{6419}}\right)$-antifield. Since $|J| \ll \max\left\{p^{\frac{1}{2}},n^{\frac{2560}{6419}}\right\}$ it is also a $\left(2,\gamma n^{\frac{2560}{6419}}\right)$-strong-antifield, as required.
\end{proof}
\begin{corollary}[Construction when $q=p^4$]\label{theorem:p^4} Let $P \subseteq \mathbb{F}_{p^4} \times \mathbb{F}_{p^4}$ with $|P|=n\gg p^{\frac{6419}{2560}}$, and define $A=A(P)=\left\{x:(x,y) \in P\right\}$. Let $t$ be a defining element of $\mathbb{F}_{p^4}$ over $\mathbb{F}_{p^2}$, so that $\mathbb{F}_{p^4}=\mathbb{F}_{p^2} + t \mathbb{F}_{p^2}$. Suppose that $|A|\ll p^2$ and that $A=\bigcup_{j \in J}A_j$
where $J \subseteq \mathbb{F}_{p^2}$ with $|J|\ll \max\left\{p,n^{\frac{2560}{6419}}\right\}$, and $A_j \subseteq \mathbb{F}_p+jt$ with $|A_j|\ll \max\left\{p,n^{\frac{2560}{6419}}\right\}$ for each $j \in J$. Then we have
$I(P,L)\ll n^{\frac{3}{2}-\frac{1}{12838}} $ for all sets of lines $L$ in $\mathbb{F}_{p^4}\times \mathbb{F}_{p^4}$ with $|L|=n$.
\end{corollary}
\begin{proof}
We need to show that the hypotheses imply that $P$ is a $\left(2,\gamma n^{\frac{2560}{6419}}\right)$-strong-antifield. Note that since $n\gg p^{\frac{6419}{2560}}$, we can ignore the subfield $\mathbb{F}_p$ and need check this only with respect to the subfields $\mathbb{F}_{p^2}$ and $\mathbb{F}_{p^4}$. This checking follows Corollary \ref{theorem:p^2}.
\end{proof}
\section{Structure for proving Theorem \ref{theorem:result}}
The rest of the paper is concered with proving Theorem \ref{theorem:result}. This section outlines the structure of the proof. It states results, which will be proved later, and shows how they fit together to give the overall proof. There are two components to this. The first component is a key lemma that relates the algebraic and geometric structure of antifields. The second component uses this key lemma, and a method of Katz and Shen \cite{KS}, as part of an otherwise technical generalisation of the Helfgott-Rudnev proof.
\subsection{The first component: Relating the algebraic and geometric stucture of antifields}
Recall that we defined both \textbf{antifields} and \textbf{strong-antifields}, that both are defined algebraically, and that Theorem \ref{theorem:result} is a statement about strong-antifields. The first component of the proof of Theorem \ref{theorem:result} is to relate the algebraic and geometric structure of these objects by showing that under certain projective transformations the image of a strong-antifield is an antifield.
The formal statement is expressed in terms of \textbf{cross ratios}. These are projective invariants, which means that they are preserved by projective transformations of a line and so are important in projective geometry.
\begin{definition}
Let $F$ be a field and let $a,b,c,d \in F$ with $a \neq d$ and $b \neq c$. Then define the \textbf{cross ratio} $X(a,b,c,d)$ by
$$X(a,b,c,d)=\frac{(a-b)(c-d)}{(a-d)(c-b)}$$
\end{definition}
We can now state the key lemma:
\begin{lemma}\label{theorem:key}
Let $A \subseteq F$ be a $(1,\lambda)$-strong-antifield and let $B \subseteq F$. Suppose there is a cross-ratio-preserving injection $\tau:B \to A$ (i.e. an injection $\tau$ for which $X(\tau(b_1),\tau(b_2),\tau(b_3),\tau(b_4))=X(b_1,b_2,b_3,b_4)$ whenever $b_1,b_2,b_3,b_4 \in B$). Then $B$ is a $(1,\lambda)$-antifield.
\end{lemma}
\subsection{The second component: Applying the first component in a technical modification of the Helfgott-Rudnev proof}
The structure of the second component broadly follows \cite{HR}. It begins by applying Lemma \ref{theorem:key} in an adaptation of an argument of Bourgain, Katz and Tao \cite{BKT} to replace $L$ and $P$ with a construction of lines and points of a certain form, at the expense of some incidences and of passing from a strong-antifield to an antifield.
\begin{proposition}\label{theorem:propbelow}
Let $F$ be a field, and let $P$ and $L$ be a set of lines and points respectively in $F \times F$ with $|P|=|L|=n$ such that $I(P,L)=n^{\frac{3}{2}-\epsilon}$ for some $\epsilon>0$. Let $\lambda \geq 0$. Then, if $P$ is a $(2,\lambda)$-strong-antifield there exist:
\begin{enumerate}
\item Sets $A,B \subseteq F$ with $|A|,|B| \ll n^{\frac{1}{2}+\epsilon}$ and $0 \notin B$
\item A set $L_A$ of lines through the origin with gradients in $A$.
\item A set $L_B$ of horizontal (i.e. gradient $0$) lines with $y$-intercepts in $B$
\item A $(2,\lambda)$-antifield $P^*$ with $|P^*|\leq n$, the points of which each lie on the intersection of a line in $L_A$ with a line in $L_B$.
\end{enumerate}
such that $I\left(P^*,L(P^*)\right)\gg n^{\frac{3}{2}-5\epsilon}$
where $L(P^*)$ is the set of lines determined by pairs of points in $P^*$.
\end{proposition}
Following \cite{HR} we then generalise the definition of incidences to colinear $k$-tuples for any integer $k$:
\begin{definition}[Colinear $k$-tuples]\label{theorem:colinear}
Let $F$ be a field. Let $P$ be a finite set of points in $F \times F$ and let $L$ be a finite set of lines in $F \times F$. We define the number of \textbf{colinear $k$-tuples} between $P$ and $L$, denoted $I_k(P,L)$ by
$$I_k(P,L)=\left|\left\{\left(p_1,\ldots,p_k,l\right)\in P^k \times L:p_1,\ldots,p_k \in l\right\}\right|$$
\end{definition}
This generalises the definition of incidences because $I(P,L)=I_1(P,L)$. Moreover, the following lemma shows that H\"older's inequality relates incidences to colinear $k$-tuples:
\begin{lemma}\label{theorem:relation}
Let $F$ be a field and $k \in \mathbb{N}$. Let $P,L$ be sets of points and lines in $F \times F$. Then we have $I_k\left(P,L\right) \geq \frac{I\left(P,L\right)^k}{|L|^{k-1}}$.
\end{lemma}
\begin{proof}
Define $f:L \to \mathbb{N}$ by $f(l)=\sum_{p \in P}\delta_{lp}$ where $\delta_{lp}=1$ if $p \in L$ and $0$ otherwise, i.e. $f(l)$ is the number of points in $P$ that are incident to $l$. Note that $\left\|f \right\|_k=I_k(P,L)^{\frac{1}{k}}$. H\"older's inequality implies that $\left\|f \right\|_1 \leq \left\|f \right\|_k \left\|1 \right\|_{\frac{k}{k-1}}$, which is the same as $I(P,L) \leq I_k(P,L)^{\frac{1}{k}} |L|^{\frac{k-1}{k}} $.
\end{proof}
Applying Lemma \ref{theorem:relation} with $k=3$ reinterprets Proposition \ref{theorem:propbelow} as a lower bound on colinear triples:
\begin{corollary}\label{theorem:below}
With the notation in Proposition \ref{theorem:propbelow} and Definition \ref{theorem:colinear}, we also have
$I_3\left(P^*,L(P^*)\right) \gg n^{\frac{5}{2}-15\epsilon}$
\end{corollary}
So we have a lower bound on colinear triples in $P^*$. Separately, the next proposition gives an upper bound on this quantity, which is obtained by combinatorial methods. Its proof uses the method in \cite{KS} to adapt the approach in \cite{HR}.
\begin{proposition}\label{theorem:above}
There is an absolute constant $\gamma_1$ such that if:
\begin{itemize}
\item $F$ is a field and $A$, $B$ are finite subsets of $F$ with $0 \notin B$.
\item $L_A$ is the set of lines through the origin with gradients lying in $A$.
\item $L_B$ is the set of horizontal lines crossing the $y$-axis at some $b \in B$.
\item $P$ is a set of points, each lying on the intersection of some line in $L_A$ with some line in $L_B$.
\item $T:=I_3\left(P,L(P)\right)$.
\item $P$ is, additionally, a $\left(2,\frac{\gamma_1 T^{65}}{|A|^{130}|B|^{194}}\right)$-antifield.
\end{itemize}
Then:
\begin{equation*}
T\ll \max{\left\{\left|A\right|^{\frac{643}{321}}\left|B\right|^{\frac{961}{321}},\left|A\right|^{\frac{535}{267}}\left|B\right|^{\frac{799}{267}},\left|A\right|^{\frac{499}{249}}\left|B\right|^{\frac{743}{249}} \right\}}
\end{equation*}
\end{proposition}
The results collected above then allow us to prove Theorem \ref{theorem:result}:
\paragraph{Proving Theorem \ref{theorem:result} from the propositions}
Let $|P|=|L|=n$ with $I(P,L)=n^{\frac{3}{2}-\epsilon}$. If $\epsilon > 1/12838$ then we are already done, so assume that $\epsilon \leq 1/12838$. We shall find a constant $\gamma$ such that $\epsilon \geq 1/12838$ so long as $P$ is a $\left(2,\gamma n^{\frac{1}{2}-\frac{1299}{12838}}\right)$-strong-antifield.
So let us suppose that $P$ is a $\left(2,\gamma n^{\frac{1}{2}-\frac{1299}{12838}}\right)$-strong-antifield, where $\gamma$ is a constant to be specified. Apply Proposition \ref{theorem:propbelow} and Corollary \ref{theorem:below} to obtain a particular $\left(2,\gamma n^{\frac{1}{2}-\frac{1299}{12838}}\right)$-antifield $P^*$ for which
\begin{equation}\label{eq:below}
T:=I_3\left(P^*,L(P^*)\right)\gg n^{\frac{5}{2}-15 \epsilon}
\end{equation}
and for which Proposition \ref{theorem:above} is applicable so long as
\begin{equation}\label{eq:req}
\gamma n^{\frac{1}{2}-\frac{1299}{12838}} \leq \frac{\gamma_1T^{65}}{|A|^{130}|B|^{194}}
\end{equation}
where $\gamma_1$ is an absolute constant. Note also that
\begin{equation}\label{eq:abound}
|A|,|B|\ll n^{\frac{1}{2}+\epsilon}
\end{equation}
Now, since $\epsilon \leq 1/12838$ and combining \eqref{eq:below} and \eqref{eq:abound}, we see that there is an absolute constant $\gamma_2$ such that $$n^{\frac{1}{2}-\frac{1299}{12838}}\leq n^{\frac{1}{2}-1299\epsilon} \leq \gamma_2 \frac{T^{65}}{|A|^{130}|B|^{194}}$$ So we can ensure that \eqref{eq:req} holds by taking $\gamma=\frac{\gamma_1}{\gamma_2}$. We therefore have by Proposition \ref{theorem:above} that
\begin{equation}\label{eq:above}
T\ll \max{\left\{\left|A\right|^{\frac{643}{321}}\left|B\right|^{\frac{961}{321}},\left|A\right|^{\frac{535}{267}}\left|B\right|^{\frac{799}{267}},\left|A\right|^{\frac{499}{249}}\left|B\right|^{\frac{743}{249}} \right\}}
\end{equation}
Comparing \eqref{eq:below} and \eqref{eq:above}, plugging in \eqref{eq:abound}, and taking logs then yields $\epsilon\geq 1/12838$ as required.
\subsection{The rest of this paper}
The proof of Theorem \ref{theorem:result} will be complete once Propositions \ref{theorem:propbelow} and \ref{theorem:above} have been established. Lemma Lemma \ref{theorem:key} is used for proving Propositions \ref{theorem:propbelow}. The proofs of these three results are the subject of the rest of the paper:
\begin{itemize}
\item Section \ref{section:aflemma} presents the proof of Lemma \ref{theorem:key}
\item Section \ref {section:BKTproof} presents the proof of Proposition \ref{theorem:propbelow}.
\item Section \ref{section:lemmata} collects some technical lemmata that will be useful when proving Proposition \ref{theorem:above}, some with proof and some without.
\item Finally, Section \ref{section:above} presents the proof of Proposition \ref{theorem:above}.
\end{itemize}
\section{Proving Lemma \ref{theorem:key}}\label{section:aflemma}
This section is concerned the proof of Lemma \ref{theorem:key}. Recall the statement of the lemma:
\begin{tabular}{|p{15cm}|}
\hline
\paragraph{Lemma \ref{theorem:key}}
Let $A \subseteq F$ be a $(1,\lambda)$-strong-antifield and let $B \subseteq F$. Suppose there is a cross-ratio-preserving injection $\tau:B \to A$ (i.e. an injection $\tau$ for which $X(\tau(b_1),\tau(b_2),\tau(b_3),\tau(b_4))=X(b_1,b_2,b_3,b_4)$ whenever $b_1,b_2,b_3,b_4 \in B$). Then $B$ is a $(1,\lambda)$-antifield.
\\
\space
\\
\hline
\end{tabular}
For a set $A$, define $X(A)=\left\{X(a,b,c,d):a,b,c,d \in A, a \neq d, b \neq c\right\}$. To prove Lemma \ref{theorem:key} we will need the following intermediate result:
\begin{lemma}\label{theorem:cr}
Let $F$ be a field. Suppose $A \subseteq F$ and there is a subfield $G$ of $F$ for which $X(A)\subseteq G $. Then either $|A \cap (xG+y)| \leq 2$ for all $x,y \in F$, or there exist $x,y \in F$ such that $A \subseteq xG+y$.
\end{lemma}
\begin{proof}
We show that if $|A \cap (xG+y)| \geq 3$ then $A \subseteq xG+y$. Let $a,b,c$ be three distinct elements of $A \cap \left(xG+y\right)$ and suppose for a contradiction that $A \nsubseteq xG+y$. Then we can find $d \in A$ with $d \notin xG+y$. So we have
\begin{align*}
a=g_1x+y\\
b=g_2x+y\\
c=g_3x+y\\
d=g_4x+z
\end{align*}
where $g_1,g_2,g_3,g_4 \in G$ and $\frac{z-y}{x} \notin G$. Moreover, since $a,b,c$ are distinct, we know that $g_1,g_2,g_3$ are distinct. Finally, we know that $a,b,c \neq d$. We then know by assumption that
$$\frac{(a-b)(c-d)}{(a-d)(c-b)}\in G$$
But we also have
\begin{align*}
\frac{(a-b)(c-d)}{(a-d)(c-b)}&=\frac{x(g_1-g_2)(x(g_3-g_4)+(y-z))}{(x(g_1-g_4)+(y-z))(x(g_3-g_2))}= \left(\frac{g_1-g_2}{g_3-g_2}\right) \frac{g_3-g_4 + \frac{y-z}{x}}{g_1-g_4 + \frac{y-z}{x}}
\end{align*}
Since $g_1,g_2$ and $g_3$ are distinct, this means that $$\frac{g_3-g_4 + \frac{y-z}{x}}{g_1-g_4 + \frac{y-z}{x}} \in G$$ and so there exists $g_5 \in G$ with $$\frac{g_3-g_4 + \frac{y-z}{x}}{g_1-g_4 + \frac{y-z}{x}} = g_5$$
We now split into two cases, according to whether or not $g_5=1$. If $g_5=1$ then we obtain $g_3=g_1$, which contradicts the fact that these two elements are distinct. If $g_5 \neq 1$ then we obtain $$\frac{y-z}{x}=\frac{g_5(g_1-g_4)-g_3+g_4}{1-g_5} \in G $$ which contradicts the fact that $\frac{y-z}{x}\notin G$. Either way, we are done.
\end{proof}
\begin{corollary}\label{theorem:coroll}
Let $F$ be a field, $G$ be a subfield of $F$, $A \subseteq F$ be a $(1,\lambda)$-strong-antifield, and $A' \subseteq A$ be such that $|A'|\geq \max\left\{\lambda,|G|^{\frac{1}{2}}\right\}$. Then $X(A')\nsubseteq G$.
\end{corollary}
\begin{proof}
Suppose that there exists $A' \subseteq A$ with $|A'| \geq \max \left\{\lambda,|G|^{\frac{1}{2}}\right\}$ and $X(A') \subseteq G$. Then by Lemma \ref{theorem:cr}, either $A' \subseteq aG+b$ for some $a,b \in F$, or $\left|A' \cap (aG+b)\right|\leq 2$ for all $a,b \in F$.
In the former case, we have $A' \subseteq A \cap \left(aG+b\right)$ and so $\left|A \cap \left(aG+b\right)\right|\geq \max \left\{\lambda,|G|^{\frac{1}{2}}\right\}$. In the latter case we have $|A' \cap (G+b)| \leq 2$ for all distinct translates $G+b$ of $G$, which means that $A'$ and therefore $A$ intersects at least $\max\left\{\lambda,|G|^{\frac{1}{2}}\right\}/2$ such translates.
Either way, we contradict the fact that $A$ is a $(1,\lambda)$-strong-antifield and are therefore done.
\end{proof}
We are now in a position to prove Lemma \ref{theorem:key}.
\paragraph{Proof of Lemma \ref{theorem:key}}
Suppose for a contradiction that there is a subfield $G$ of $F$ and elements $a,b \in F$ such that $$\left|B \cap (aG+b)\right|\geq \max\left\{\lambda,|G|^{\frac{1}{2}}\right\}$$
Let $B'=B \cap (aG+b)$. Then we have $\tau(B')\subseteq A$ and $|\tau(B')|=|B'|\geq \max\left\{\lambda,|G|^{\frac{1}{2}}\right\}$, but also $X(\tau(B'))=X(B')\subseteq G$. This contradicts Corollary \ref{theorem:coroll} and so we are done. This completes the proof of Lemma \ref{theorem:key}.
\section{Proof of Proposition \ref{theorem:propbelow}}\label{section:BKTproof}
We will now use Lemma \ref{theorem:key} to prove Proposition \ref{theorem:propbelow}.
Recall the statement of Proposition \ref{theorem:propbelow}:
\begin{tabular}{|p{15cm}|}
\hline
\paragraph{Proposition \ref{theorem:propbelow}}
Let $F$ be a field, and let $P$ and $L$ be a set of lines and points respectively in $F \times F$ with $|P|=|L|=n$ such that $I(P,L)=n^{\frac{3}{2}-\epsilon}$ for some $\epsilon>0$. Let $\lambda \geq 0$. Then, if $P$ is a $(2,\lambda)$-strong-antifield there exist:
\begin{enumerate}
\item Sets $A,B \subseteq F$ with $|A|,|B| \ll n^{\frac{1}{2}+\epsilon}$ and $0 \notin B$
\item A set $L_A$ of lines through the origin with gradients in $A$.
\item A set $L_B$ of horizontal (i.e. gradient $0$) lines with $y$-intercepts in $B$
\item A $(2,\lambda)$-antifield $P^*$ with $|P^*|\leq n$, the points of which each lie on the intersection of a line in $L_A$ with a line in $L_B$.
\end{enumerate}
such that $$I\left(P^*,L(P^*)\right)\gg n^{\frac{3}{2}-5\epsilon}$$
where $L(P^*)$ is the set of lines determined by pairs of points in $P^*$.
\\
\space
\\
\hline
\end{tabular}
Recall that for a point $p$ and a line $l$ we define $\delta_{pl}$ to be $1$ if $p \in l$ and $0$ otherwise. We initially follow \cite{BKT} and \cite{HR}.
The first step is to show that we may assume every point in $P$ is incident to $\gg n^{\frac{1}{2}-\epsilon}$ and $\ll n^{\frac{1}{2}+\epsilon}$ lines in $L$. Indeed, let $P_+=\left\{p \in P: p \text{ is incident to} \geq 4n^{\frac{1}{2}+\epsilon} \text{ lines } l \in L\right\}$. Then:
\begin{align*}
I\left(P_+,L\right)&=\sum_{p \in P_+}\sum_{l \in L}\delta_{pl}
\leq \frac{1}{4 n^{\frac{1}{2}+\epsilon}}\sum_{p \in P_+} \left(\sum_{l \in L}\delta_{pl}\right)^2
=\frac{1}{4 n^{\frac{1}{2}+\epsilon}}\sum_{l,l' \in L}\sum_{p \in P_+}\delta_{pl}\delta_{pl'}
\leq\frac{n^{\frac{3}{2}-\epsilon}}{2}
\end{align*}
Similarly, let $P_-=\left\{p \in P: p \text{ is incident to} \leq \frac{n^{\frac{1}{2}-\epsilon}}{3} \text{ lines } l \in L \right\}$. Then:
\begin{align*}
I\left(P_-,L\right)=\sum_{p \in P_-}\sum_{l \in L}\delta_{pl}
\leq \sum_{p \in P_-}\frac{n^{\frac{1}{2}-\epsilon}}{3}
\leq \frac{n^{\frac{3}{2}-\epsilon}}{3}
\end{align*}
So between them $P_+$ and $P_-$ contribute only five sixths of the $n^{\frac{3}{2}-\epsilon}$ incidences. Without loss of generality we shall discard them and assume from now on that $|P| \leq n$, and that every point $p \in P$ is incident to $\gg n^{\frac{1}{2}-\epsilon}$ and $\ll n^{\frac{1}{2}+\epsilon}$ lines in $L$.
Let $L_1$ be the set of ``rich'' lines in $L$ defined by
$$L_1=\left\{l \in L:l \text{ is incident to} \geq \frac{n^{\frac{1}{2}-\epsilon}}{20}\text{ points } p \in P\right\}$$
Let $P_1$ be the set of points in $P$ that are ``bushy'' relative to $L_1$, defined by
$$P_1=\left\{p \in P: p \text{ is incident to } \geq \frac{n^{\frac{1}{2}-\epsilon}}{20}\text{ lines in } L_1\right\}$$
We need to check that $P_1$ is non-empty. Note firstly that
$$I(P,L\backslash L_1)=\sum_{p \in P}\sum_{l \in L \backslash L_1}\delta_{pl}\leq \sum_{l \in L \backslash L_1}\frac{n^{\frac{1}{2}-\epsilon}}{20} \leq \frac{n^{\frac{3}{2}-\epsilon}}{20} $$
and therefore $I(P,L_1)\gg I(P,L)$. Now note that
$$
I(P \backslash P_1,L_1)= \sum_{p \in P \backslash P_1} \sum_{l \in L_1}\delta_{pl}
< \sum_{p \in P \backslash P_1}\frac{n^{\frac{1}{2}-\epsilon}}{20}
\leq \frac{n^{\frac{3}{2}-\epsilon}}{20}
$$
This means that $I(P_1,L_1)\gg I(P,L_1)\gg I(P,L)$ and so $P_1$ is certainly non-empty. Now for each $p \in P_1$ let $P_p$ be the set of points in $P$ that are joined to $p$ by a line in $L_1$. We have:
\begin{align*}
\left|P_p \right|&=\sum_{q \in P}\sum_{l \in L_1}\delta_{pl}\delta_{ql}
=\sum_{l \in L_1}\delta_{pl}\sum_{q \in P}\delta_{ql}
\gg n^{\frac{1}{2}-\epsilon}\sum_{l \in L_1}\delta_{pl}
\gg n^{1-2\epsilon}
\end{align*}
This means that:
\begin{align*}
\left|P_1\right|n^{1-2\epsilon}\ll \sum_{p \in P_1}\left|P_p\right|
\leq \sqrt{\left|P_1\right|}\sqrt{\sum_{p,q \in P_1}\left|P_p \cap P_q\right|}
\end{align*}
where the second inequality follows by Cauchy-Schwartz. So we have:
\begin{equation}\label{equat:cspigeon}
\left|P_1\right|n^{2-4\epsilon}\ll \sum_{p,q \in P_1}\left|P_p \cap P_q\right|
\end{equation}
For each $p \in P$ define $x_p$ to be the $x$-co-ordinate of $p$. And for each $x \in F$ define $P^x=\left\{p \in P:x_p=x\right\}$. It is easy to see that $|P^x|n^{\frac{1}{2}-\epsilon} \ll I(P^x,L) \leq 2n$ and so we deduce that $|P^x|\ll n^{\frac{1}{2}+\epsilon}$ for every $x \in F$. Plugging this into \eqref{equat:cspigeon} yields
$$|P_1|n^{2-4 \epsilon} \ll \sum_{p,q \in P_1:x_p \neq x_q}\left|P_p \cap P_y\right|+ \sum_{p \in P_1}\sum_{q \in P^{x_p}}\left|P_p \cap P_q\right| \ll \sum_{p,q \in P_1:x_p \neq x_q}\left|P_p \cap P_y\right|+ |P_1|n^{\frac{3}{2}+\epsilon}$$
We can therefore fix two distinct points $p,q \in P_1$ with $x_p\neq x_q$ such that
\begin{align*}
\left|P_{p} \cap P_{q}\right|&\gg \frac{n^{2-4\epsilon}}{\left|P\right|}
\gg n^{1-4\epsilon}
\end{align*}
Now let $P'=P_{p} \cap P_{q}$ and note that
\begin{align*}
I(P',L)=\sum_{p \in P'}\sum_{l \in L}\delta_{pl} \geq \left|P'\right|n^{\frac{1}{2}-\epsilon}\gg n^{\frac{3}{2}-5 \epsilon}
\end{align*}
Since $I(P^{x_p},L)\leq n$ we can discard all points in $P^{x_p}$ other than $p$ , and thereby assume $P^{x_p}=\left\{p\right\}$.
At this point we diverge from \cite{BKT} and \cite{HR}. All we shall carry forward are the facts that:
\begin{enumerate}
\item $I(P',L)\gg n^{\frac{3}{2}-5 \epsilon}$.
\item $P'$ is a $(2,\lambda)$-strong-antifield.
\item There are two points $p,q$, lying on distinct vertical lines, such that $P'=P_p \cap P_q$ where $P_p$ is a set of points lying on $O(n^{\frac{1}{2}+\epsilon})$ lines through $p$, and $P_q$ is a set of points lying on $O(n^{\frac{1}{2}+\epsilon})$ lines through $q$
\item No point in $P'$ lies on the vertical line through $p$.
\end{enumerate}
These facts are unaffected by translation of $P'$ and so without loss of generality we shall assume that $p$ is in fact the origin.
Recall that the \textbf{projective plane} $\mathbb{P}^2(F)$ is defined to be $F^3 \backslash \left(0,0,0\right)$, modulo dilations. We embed $F \times F$ in $\mathbb{P}^2(F)$ by identifying $(x,y) \in F \times F$ with $(x,y,1) \in \mathbb{P}^2(F)$. This accounts for all elements of $\mathbb{P}^2(F)$ apart from those of the form $(x,y,0)$; these are said to lie on the \textbf{line at infinity}. For our purposes, the only such point we need consider is the point $(1,0,0)$. Every line incident to this point has gradient 0, and is therefore horizontal. A \textbf{projective transformation} is an invertible linear map from $\mathbb{P}^2(F)$ to itself, i.e. a $3 \times 3$ non-singular matrix, and has the important property that it maps points to points and lines to lines.
Returning to the proof, we apply the projective transformation $\tau$ given by
\begin{equation*}
\tau=\left(
\begin{array}{ccc}
0&0&1\\
0&1&0\\
1&0&0
\end{array}\right)
\end{equation*}
Note that:
\begin{enumerate}
\item $I(\tau(P'), L(\tau(P')))\geq I(\tau(P'),\tau(L))=I(P',L)\gg n^{\frac{3}{2}-5 \epsilon}$
\item $\tau$ maps the $y$-axis to the line at infinity. In particular, it maps the origin (which we have assumed to be $p$) to the point at infinity with gradient $0$, and so the points in $\tau(P_p)$ lie on $O(n^{\frac{1}{2}+\epsilon})$ horizontal lines.
\item Since $P'$ has no points on the $y$-axis, the image $\tau(P')$ is contained in $F \times F$.
\item Since $q$ does not lie on the $y$-axis, the point $\tau(q)$ lies in $F \times F$ and not the line at infinity. Every point in $\tau(P_q)$ lies on one of $O(n^{\frac{1}{2}+\epsilon})$ lines through $\tau(q)$.
\item $\tau(x,y)=\left(\frac{1}{x},\frac{y}{x}\right)$ for each point $(x,y)$ with $x \neq 0$. So the map $x \mapsto x^{-1}$ is a cross-ratio-preserving injection from $\left\{x:(x,y) \in \tau(P') \right\}$ to $\left\{x:(x,y) \in P' \right\}$. Since $P'$ is a $(2,\lambda)$-strong-antifield, Lemma \ref{theorem:key} implies that $\tau(P)$ is a $(2,\lambda)$-antifield.
\end{enumerate}
From the above we see that we have a $(2,\lambda)$-antifield $P^*=\tau(P')$ such that:
\begin{enumerate}
\item $I(P^{*},L(P^*))\gg n^{\frac{3}{2}-5 \epsilon}$
\item Each point in $P^{*}$ lies on
\begin{enumerate}
\item one of $O(n^{\frac{1}{2}+\epsilon})$ lines that pass through a single point $s$ in $F \times F$.
\item one of $O(n^{\frac{1}{2}+\epsilon})$ horizontal lines.
\end{enumerate}
\end{enumerate}
The properties above are again invariant under translation and so without loss of generality we may assume that $s$ is the origin. And since each horizontal line in $P^*$ contributes at most $n$ incidences we can discard points to assume that $0 \notin B$. We then take $A$ to be the set of gradients of the $O(n^{\frac{1}{2}+\epsilon})$ lines through the origin, and $B$ to be the $y$-intercepts of the $O(n^{\frac{1}{2}+\epsilon})$ horizontal lines. This completes the proof of the proposition.
\section{Lemmata for proving Proposition \ref{theorem:above}}\label{section:lemmata}
This section collects the technical lemmata that will be used to prove Proposition \ref{theorem:above}.
\subsection{Pivoting results}
We will make use of some `pivoting' results. The first, Lemma \ref{theorem:pivot2}, was applied in the Helfgott-Rudnev proof \cite{HR}, and before that in e.g. \cite{GK}, \cite{garaev}, \cite{KSprime}, \cite{shen} and \cite{li}. It is stated here without proof.
\begin{lemma}[Pivoting lemma 1]\label{theorem:pivot2}
Let $F$ be a field, let $Z \subseteq F$ and let $R(Z)=\frac{Z-Z}{Z-Z}$. Let $a,b \in F$. Then if $\left|R(Z)\right|\geq \left|Z\right|^2$ there exist $z_1,z_2,z_3,z_4 \in aZ+b$ such that for all $Z' \subseteq Z$ with $\left|Z'\right|\gg \left|Z\right|$ we have $\left|Z\right|^2 \approx \left|\left(z_1-z_2\right)Z'+\left(z_3-z_4\right)Z'\right|$
\end{lemma}
The next lemma is a quick and well-known result that is a necessary tool for the lemma that follows it:
\begin{lemma}
Let $F$ be a field, let $Z \subseteq F$ and let $R(Z)=\frac{Z-Z}{Z-Z}$. If $x \notin R(Z)$ then $\left|Z+xZ\right|\approx |Z|^2$.
\end{lemma}
\begin{proof}
Clearly $\left|Z+xZ\right|\ll |Z|^2$, so we seek $\left|Z+xZ\right|\gg |Z|^2$. If there exist $z_1,z_2,z_3,z_4 \in Z$ with $z_2 \neq z_4$ and $z_1+xz_2=z_3+xz_4$, then we can write $x=\frac{z_1-z_3}{z_2-z_4}$, which contradicts the fact that $x \notin R(Z)$. So there is only one way of writing each elemnent $v\in Z+ xZ$ in the form $v=z_1+xz_2$ with $z_1,z_2 \in Z$. We therefore have $|Z+xZ|=\frac{|Z|\left(|Z|-1\right)}{2}\gg|Z|^2$, as required.
\end{proof}
Lemma \ref{theorem:pivot1}, due to Katz and Shen \cite{KS}, generalises an approach that is traditionally used in conjunction with Lemma \ref{theorem:pivot2}. The generalistation means that the result allows for the possibility of nontrivial additive subgroups.
\begin{lemma}[Pivoting lemma 2]\label{theorem:pivot1}
Let $F$ be a field and let $Z \subseteq F$ be finite such that $R(Z)=\frac{Z-Z}{Z-Z}$ is not a subfield of $F$. Let $a,b \in F$. Then either
\begin{enumerate}
\item \textbf{$\mathbf{R(aZ+b)}$ is not closed under multiplication}, in which case there exist $x_1,x_2,z_1,z_2,z_3,z_4 \in Z$ such that $\left|Z'\right|^2\leq\left|x_1\left(z_1-z_2\right)Z'-x_2\left(z_1-z_2\right)Z'+x_1\left(z_3-z_4\right)Z'\right|$ for all $Z' \subseteq Z$.
\item \textbf{$\mathbf{R(aZ+b)}$ is closed under multiplication but is not closed under addition}, in which case there exist $y_1,y_2,y_3,y_4 \in Z$ such that $\left|Z'\right|^2 \leq \left|\left(y_1-y_2\right)Z'+\left(y_3-y_4\right)Z'+\left(y_3-y_4\right)Z' \right|$ for all $Z' \subseteq Z$.
\end{enumerate}
\end{lemma}
\begin{proof}
Note that $R(aZ+b)=R(Z)$ so without loss of generality we may assume $a=1$ and $b=0$.
\paragraph{Case 1}
Since $R(Z) \cdot R(Z) \neq R(Z)$ there are $x_1,x_2,x_3,x_4,y_1,y_2,y_3,y_4 \in Z$ with
$$\frac{x_1-x_2}{x_3-x_4}\frac{y_1-y_2}{y_3-y_4} \notin R(Z) $$
This can be written as
$$\frac{x_1-x_2}{x_1}\frac{x_1}{x_1-x_3}\frac{x_1-x_3}{x_4}\frac{x_4}{x_3-x_4}\frac{y_1-y_2}{y_3-y_4} \notin R(Z)$$
and so there are $a_1,a_2,b_1,b_2,b_3,b_4 \in Z$ with $\frac{a_1-a_2}{a_1}\frac{b_1-b_2}{b_3-b_4} \notin R(Z)$. We therefore have that for any $Z' \subseteq Z$
\begin{align*}
\left|Z'\right|^2&\approx \left|Z'+\frac{a_1-a_2}{a_1}\frac{b_1-b_2}{b_3-b_4}Z'\right|\leq \left|a_1(b_1-b_2)Z' -a_2(b_1-b_2)Z' +a_1(b_3-b_4)Z'\right|
\end{align*}
This completes the proof of Case 1.
\paragraph{Case 2}
We seek $z_1,z_2,z_3,z_4 \in Z$ such that $\frac{z_1-z_2}{z_3-z_4}+1 \notin R(Z)$. We will then be done, as for any $Z' \subseteq Z$ we will have
\begin{align*}
\left|Z'\right|^2&\approx\left|Z'+\left(\frac{x_1-x_2}{x_3-x_4}+1\right)Z'\right|
\leq\left|(x_1-x_2)Z' + (x_3-x_4)Z' + (x_3-x_4)Z' \right|
\end{align*}
Since $R(Z)+R(Z) \neq R(Z)$ there are $x_1,x_2,x_3,x_4,y_1,y_2,y_3,y_4 \in Z$ with
$$\frac{x_1-x_2}{x_3-x_4}+\frac{y_1-y_2}{y_3-y_4} \notin R(Z) $$
On the other hand, since $R(Z) \cdot R(Z)=R(Z)$ there are $z_1,z_2,z_3,z_4 \in Z$ with
$$\frac{x_1-x_2}{x_3-x_4}\frac{y_3-y_4}{y_1-y_2}=\frac{z_1-z_2}{z_3-z_4}$$
Combining these two facts gives:
\begin{align*}
\frac{z_1-z_2}{z_3-z_4}+1&=\frac{x_1-x_2}{x_3-x_4}\frac{y_3-y_4}{y_1-y_2}+1
=\frac{y_3-y_4}{y_1-y_2}\left(\frac{x_1-x_2}{x_3-x_4}+\frac{y_1-y_2}{y_3-y_4}\right)
\notin R(Z)
\end{align*}
This completes the proof of Case 2 and therefore of the lemma.
\end{proof}
We will also use the following lemma, due to Katz and Shen. A proof can be found in \cite{KS}.
\begin{lemma}\label{theorem:ratiofield}
If $R(Z) \subseteq G$ for some subfield $G$ of $F$, then $Z \subseteq aG+b$ for some $a,b \in F$
\end{lemma}
\subsection{A lemma about sumsets}
The following lemma was used in the Helfgott-Rudnev paper \cite{HR}, and is originally due to Bourgain \cite{bourgain}:
\begin{lemma}\label{theorem:intersection}
Let $F$ be a field. Let $X$ and $Y$ be finite subsets of $F$ and let $K=\max_{y \in Y} \left|X+yX \right|$
Then there exist elements $x_1, x_2, x_3 \in X$ such that $\left|\left(X-x_1\right) \cap \left(x_2-x_3\right)Y \right|\gg\frac{\left|Y \right|\left|X \right|}{K}$.
\end{lemma}
\begin{proof}
Let $E$ be the number of solutions to the equation $x_1+y x_2 = x_3+yx_4 $ with $x_1,x_2,x_3,x_4 \in X$ and $y \in Y$. Then
\begin{align*}
E=\sum_{y \in Y}\sum_{k \in X+yX}\left|X \cap \left( \frac{X-k}{y}\right) \right|^2
\geq \sum_{y \in Y}\frac{\left(\sum_{k \in X+yX}\left|X \cap \left(\frac{X-k}{y}\right) \right|\right)^2}{\left|X+yX\right|}
\geq \frac{\left|X\right|^4\left|Y\right|}{K}
\end{align*}
So there exist $z_1, z_2 \in X$ such that the equation $x_1 +yz_1=z_2+yx_2$ has $\gg\frac{\left|X \right|^2 \left|Y\right|}{K}$ solutions $\left(x_1,x_2,y\right) \in X \times X \times Y$. In other words, if $X_1=X-z_1$ and $X_2=X-z_2$ then there are $\gg\frac{\left|X \right|^2 \left|Y\right|}{K}$ solutions $(u,v,y)\in X_1 \times X_2 \times Y$ to the equation $v=yu$. By averaging, there is an element $u_* =x_*-z_1\in X_1$ with $x_* \in X$ such that $v=y u_*$ has $\gg \frac{\left|Y\right|\left|X\right|}{K}$ solutions. Thus:
$$\left|\left(X-z_2\right)\cap\left(x_*-z_1\right)Y\right|=\left|X_2 \cap u_* Y\right|\gg \frac{\left|Y\right|\left|X\right|}{K} $$
\end{proof}
\subsection{Standard results from additive combinatorics}
We record some standard results from additive combinatorics. The first, below, formalises a common technique.
\begin{lemma}[Popularity pigeonholing]\label{theorem:popularity}
Let $X$ be a finite set and let $f:X \to \left[1,N \right]$ be a function. Then there is a subset $Y \subseteq X$ with $\left|Y \right|\gg \frac{\sum_{x \in X} f(x)}{N} $ such that for any $y \in Y$ we have $f(y) \gg \frac{\sum_{x \in X}f(x)}{\left|X \right|}$
\end{lemma}
\begin{proof}
Let $Y=\left\{x \in X: f(x) \geq \alpha \right\}$ where $\alpha=\frac{\sum_{x \in X}f(x)}{2\left|X \right|}$. We seek to show that $\left|Y \right|\gg \frac{\sum_{x \in X}f(x)}{N}$. We see this as follows:
\begin{align*}
\sum_{x \in X}f(x)= \sum_{x:f(x) \geq \alpha}f(x) + \sum_{x: f(x) < \alpha}f(x) \leq N\left|Y \right|+ \alpha \left|X \right|
\end{align*}
So we have
\begin{align*}
\left|Y \right|&\geq \frac{\sum_{x \in X}f(x) - \alpha \left|X \right|}{N}=\frac{\sum_{x \in X}f(x)}{2N}\gg \frac{\sum_{x \in X}f(x)}{N}
\end{align*}
\end{proof}
We will use the following form of the Pl\"unnecke-Ruzsa inequality, due to Ruzsa \cite{ruzsa}:
\begin{lemma}[Pl\"unnecke-Ruzsa inequality]\label{theorem:plunnecke}
Let $X,B_1, \ldots, B_k \subseteq \mathbb{F}_p$. Then $\left|\sum_{j=1}^k B_j \right|\ll\frac{\prod_{j=1}^k \left|X + B_j\right|}{\left|X \right|^{k-1}}$
\end{lemma}
The following lemma is a version of the Balog-Szemer\'edi-Gowers theorem. A proof can be found in \cite{TV}, but this appears to have a typographical error which leads to an exponent of $-4$, rather than the correct exponent of -5 below. See \cite{FS} for a proof yielding the exponent of $-5$.
\begin{lemma}[Balog-Szemeredi-Gowers]\label{theorem:BSG}
Let $X,Y$ be additive sets with $\left|X\right|=\left|Y\right|=n$. Suppose that there is a subset $G \subseteq X \times Y$ such that $\left|X +^{G} Y\right|<n$ and that $\left|G\right|=\alpha n^2$ for some $\alpha \in (0,1)$. Then there exist subset $X' \subseteq X$ and $Y' \subseteq Y$ with $\left|X'\right|,\left|Y'\right|\gg \alpha n$ such that
$\left|X'+Y'\right| \ll \alpha^{-5}n $
\end{lemma}
A proof of the following `covering' result can be found in \cite{shen}.
\begin{lemma}[Covering lemma]
\label{theorem:covering}
Let G be a group and $B,C \subseteq G$ be finite. Let $\epsilon \in (0,1)$. Then the number of translates of $C$ required to cover $(1- \epsilon)\left|B \right|$ elements of $B$ is $O_{\epsilon}\left(\frac{\left|B+C\right|}{\left|C\right|}\right)$.
\end{lemma}
\section{Proof of Proposition \ref{theorem:above}}
\label{section:above}
Recall the statement of Proposition \ref{theorem:above}:
\begin{tabular}{|p{15cm}|}
\hline
\paragraph{Proposition \ref{theorem:above}}
There is an absolute constant $\gamma_1$ such that if:
\begin{itemize}
\item $F$ is a field and $A$, $B$ are finite subsets of $F$ with $0 \notin B$.
\item $L_A$ is the set of lines through the origin with gradients lying in $A$.
\item $L_B$ is the set of horizontal lines crossing the $y$-axis at some $b \in B$.
\item $P$ is a set of points, each lying on the intersection of some line in $L_A$ with some line in $L_B$.
\item $T:=I_3\left(P,L(P)\right)$.
\item $P$ is, additionally, a $\left(2,\frac{\gamma_1 T^{65}}{|A|^{130}|B|^{194}}\right)$-antifield.
\end{itemize}
Then:
\begin{equation*}
T\ll \max{\left\{\left|A\right|^{\frac{643}{321}}\left|B\right|^{\frac{961}{321}},\left|A\right|^{\frac{535}{267}}\left|B\right|^{\frac{799}{267}},\left|A\right|^{\frac{499}{249}}\left|B\right|^{\frac{743}{249}} \right\}}
\end{equation*}
\\
\space
\\
\hline
\end{tabular}
This section uses the results of Section \ref{section:lemmata} to prove Proposition \ref{theorem:above}.
\subsection{Structure of the proof}
We shall assume that $P$ is a $(2,\lambda)$ antifield for some $\lambda$, and then show that the conclusion of the Proposition follows when $\lambda \approx \frac{T^{65}}{|A|^{130}|B|^{194}}$.
The proof of Proposition \ref{theorem:above} uses the following three claims, whose proofs are deferred. Instead, we shall first see how they are applied to prove the proposition. The proofs of the claims then follow.
\begin{claim}\label{theorem:bsgclaim}
There is a subset $C \subseteq \mathbb{F}_q$ with $|C| \gg \frac{T^5}{|A|^{10}|B|^{14}}$ such that for each $c \in C$ there is a pair of $(1,\lambda)$-antifields $A_c^1,A_c^2 \subseteq F$ with
\begin{equation}\label{size}
|A_c^1|,|A_c^2| \gg \frac{T}{|A||B|^3}
\end{equation}
\begin{equation}\label{sumset}
\left|A_c^1+cA_c^2\right|\ll \frac{\left|A\right|^{11}\left|B\right|^{15}}{T^5}
\end{equation}
Moreover, there exists a particular element $c_* \in C$ such that, writing $A_*=A_{c_*}$, we have
\begin{equation}\label{intersection}
\left|A_c^1 \cap A_*^1\right|, \left|A_c^2 \cap A_*^2\right|\gg \frac{T^4}{|A|^7|B|^{12}}
\end{equation}
for all $c \in C$.
\end{claim}
\begin{claim}\label{theorem:sumsetclaim}
The following bounds hold for each $c \in C$
\begin{equation}\label{claim1}
\left|A_c^1 + A_c^1\right|,\left|A_c^2 + A_c^2\right| \ll \frac{\left|A\right|^{23}\left|B\right|^{33}}{T^{11}}
\end{equation}
\begin{equation}\label{claim2}
\left|c_*A_c^2 + cA_c^2\right| \ll \frac{\left|A\right|^{59}\left|B\right|^{87}}{T^{29}}
\end{equation}
\begin{equation}\label{claim3}
\left|c_*A_*^2 + cA_c^2\right| \ll \frac{\left|A\right|^{83}\left|B\right|^{132}}{T^{44}}
\end{equation}
\begin{equation}\label{claim4}
\left|c_*A_*^2 + cA_*^2\right| \ll \frac{\left|A\right|^{119}\left|B\right|^{177}}{T^{59}}
\end{equation}
\end{claim}
\begin{claim}\label{theorem:coveringclaim}
There exists an integer $\Gamma$ with
\begin{equation}\Gamma \ll \frac{\left|A\right|^{48}\left|B\right|^{72}}{T^{24}} \label{gammabound}\end{equation}
such that given any $c \in \pm C$, $x \in \mathbb{F}_q$, and $D \subseteq A_*^2$, a constant proportion of $cD+x$ can be covered with $\Gamma$ translates of $A_*^1$
\end{claim}
\subsection{Proof of Proposition \ref{theorem:above}, assuming claims}\label{section:mainproof}
Apply Lemma \ref{theorem:intersection} with $X=A_*^2$, $Y = \frac{1}{c_*}C$ and, by inequality \eqref{claim4}, $K\ll\frac{\left|A\right|^{119}\left|B\right|^{177}}{T^{59}}$. This provides $a_1,a_2,a_3 \in A_*^2$ such that
\begin{align*}
\left|\left(A_*^2-a_1\right) \cap \left(\frac{a_2-a_3}{c_*}\right)C\right|\gg \frac{\left|A_*^2\right|\left|B_2'\right|}{K}
\gg\frac{T^{65}}{\left|A\right|^{130}\left|B\right|^{194}}
\end{align*}
For convenience, define $Z=\left(A_*^2-a_1\right) \cap \left(\frac{a_2-a_3}{c_*}\right)C$, to give the lower bound
\begin{equation}\label{zbound}
\left|Z\right|\gg \frac{T^{65}}{\left|A\right|^{130}\left|B\right|^{194}}
\end{equation}
We seek an upper bound for $\left|Z\right|$ with which to compare \eqref{zbound}. There are three possible cases:
\begin{enumerate}
\item $\mathbf{R(Z)}$\textbf{ is not closed under multiplication}. By Lemma \ref{theorem:pivot1} there are then elements $c_1,c_2,d_1,d_2,d_3,d_4 \in C$ such that for every $Z' \subseteq Z$ with $\left|Z'\right|\gg\left|Z\right|$ we have $$\left|Z\right|^2 \ll \left|c_1(d_1-d_2)Z'-c_2(d_1-d_2)Z'+c_1(d_3-d_4)Z'\right|$$
\item $\mathbf{R(Z)}$ \textbf{is closed under multiplication but is not closed under addition}. By Lemma \ref{theorem:pivot1} there are then elements $c_1,c_2,c_3,z_4 \in C$ such that for every $Z' \subseteq Z$ with $\left|Z'\right|\gg\left|Z\right|$ we have $$\left|Z\right|^2 \ll \left|(c_1-c_2)Z'+(c_1-c_2)Z'+(c_3-c_4)Z'\right|$$
\item \textbf{$\mathbf{R(Z)}$ is a field}, $G$ say. Lemma \ref{theorem:ratiofield} implies that in this case we have $Z \subseteq aG+b$ for some $a,b \in F$. So, collecting together various facts, we have
\begin{itemize}
\item $Z \subseteq A_*^2-a_1$.
\item $A_*^2$ is a $\left(1,\lambda\right)$-antifield, and therefore so is $A_*^2-a_1$.
\item $Z \subseteq aG+b$ for some $a,b \in F$.
\item $\left|Z\right|\gg \frac{T^{65}}{\left|A\right|^{130}\left|B\right|^{194}}$.
\end{itemize}
So for some $\lambda \approx \frac{T^{65}}{\left|A\right|^{130}\left|B\right|^{194}}$, the definition of a $(2,\lambda)$-antifield implies that $\left|Z\right|\leq \left|G\right|^{\frac{1}{2}}=\left|R(Z)\right|^{\frac{1}{2}}$. Lemma \ref{theorem:pivot2} then implies that there are elements $c_1,c_2,c_3,c_4 \in C$ such that for every $Z' \subseteq Z$ with $\left|Z'\right|\gg\left|Z\right|$ we have
$$\left|Z\right|^2 \ll \left|(c_1-c_2)Z'+(c_3-c_4)Z'\right| $$
\end{enumerate}
\subsubsection{Dealing with Case 1}
Given any $Z' \subseteq Z$ with $\left|Z'\right|\gg\left|Z\right|$ and any $E \subseteq A_*^2$ with $\left|E\right|\gg \left|A_*^2\right|$, apply Lemma \ref{theorem:plunnecke} with $X=c_1(d_1-d_2)E$ and $k=3$ to get
\begin{align*}
\left|Z\right|^2 &\ll \left|c_1(d_1-d_2)Z'-c_2(d_1-d_2)Z'+c_1(d_3-d_4)Z'\right|\\& \ll \frac{\left|E+Z'\right|\left|c_1E-c_2Z'\right|\left|d_1E-d_2E+d_3Z'-d_4Z'\right|}{\left|A_*^2\right|^2}
\end{align*}
By definition of $\Gamma$ from Claim \ref{theorem:coveringclaim}, there is a subset $S_1 \subseteq A_*^2$ with $\left|S_1\right|\gg \left|A_*^2\right|$ such that $d_1 S_1$ can be covered with $\Gamma$ copies of $A_*^1$. Further, there is a subset $S_2 \subseteq S_1$ with $\left|S_2\right|\gg\left|S_1\right|\gg \left|A_*^2\right|$ such that $-d_2 S_2$ can be covered with $\Gamma$ copies of $A_*^1$. And there is a subset $S_3 \subseteq S_2$ with $\left|S_3\right|\gg\left|A_*^2\right|$ such that $c_1 S_3$ can be covered with $\Gamma$ copies of $A_*^1$. Set $E=S_3$, so that $d_1E$, $-d_2E$ and $c_1E$ can be covered with $\Gamma$ copies of $A_*^1$ each.
Similarly, recall that $Z \subseteq A_*^2 - a_1$, and pick $Z' \subseteq Z$ with $\left|Z'\right|\gg\left|Z\right|$ such that $d_3Z'$,$-d_4Z'$ and $-c_2Z'$ can each be covered with $\Gamma$ copies of $A_*^1$ each. Altogether, this means that:
\begin{align*}
\left|Z\right|^2 &\ll \frac{\Gamma^6 \left|E+Z'\right|\left|A_*^1+A_*^1\right| \left|A_*^1+A_*^1+A_*^1+A_*^1\right|}{\left|A_*^2\right|^2}\\
&\leq \frac{\Gamma^6 \left|A_*^2+A_*^2\right|\left|A_*^1+A_*^1\right| \left|A_*^1+A_*^1+A_*^1+A_*^1\right|}{\left|A_*^2\right|^2}
\end{align*}
Lemma \ref{theorem:plunnecke} and the bound in Claim \ref{theorem:coveringclaim} then give
\begin{align*}
\left|Z\right|^2 &\ll \frac{\Gamma^6 \left|A_*^2+A_*^2\right|\left|A_*^1+A_*^1\right| \left|A_*^1+c_* A_*^2\right|^4}{ \left|A_*^2\right|^5}\ll\frac{\left|A\right|^{383}\left|B\right|^{573}}{T^{191}}
\end{align*}
Comparing with \eqref{zbound} gives $T \ll \left|A\right|^{\frac{643}{321}}\left|B\right|^{\frac{961}{321}}$, which satisfies the bound in the statement of the proposition.
\subsubsection{Dealing with Case 2}
Given any any $Z' \subseteq Z$ with $\left|Z'\right|\gg \left|Z\right|$ and any $E \subseteq A_*^2$ with $\left|E\right|\gg\left|A_*^2\right|$ we can apply Lemma \ref{theorem:plunnecke} with $X=\left(c_1-c_2\right)E$ and $k=2$ to get
\begin{align*}
\left|Z\right|^2 & \ll \left|(c_1-c_2)Z'+(c_1-c_2)Z'+(c_3-c_4)Z'\right|\\
&\ll \frac{\left|E+Z'+Z'\right|\left|c_1E-c_2E+c_3Z'-c_4Z'\right|}{\left|A_*^2\right|}\\
&\leq \frac{\left|A_*^2+A_*^2+A_*^2\right|\left|c_1 E-c_2E+c_3Z'-c_4Z'\right|}{\left|A_*^2\right|}
\end{align*}
As in Case 1, pick $Z'$ and $E$ so that:
$$\left|Z\right|^2\ll \frac{\Gamma^4 \left|A_*^2+A_*^2+A_*^2\right|\left|A_*^1+A_*^1+A_*^1+A_*^1\right|}{\left|A_*^2\right|} $$
Lemma \ref{theorem:plunnecke} then gives:
\begin{align*}
\left|Z\right|^2&\ll \frac{\Gamma^4 \left|A_*^1+c_*A_*^2\right|^7}{\left|A_*^1\right|^2 \left|A_*^2\right|^4}\ll\frac{\left|A\right|^{275}\left|B\right|^{411}}{T^{137}}
\end{align*}
\par
Comparing with \eqref{zbound} gives $T \ll \left|A\right|^{\frac{535}{267}}\left|B\right|^{\frac{799}{267}}$,which satisfies the bound in the statement of the proposition.
\subsubsection{Dealing with Case 3}
As with Cases 1 and 2, pick $Z'$ so that
$$\left|Z\right|^2 \ll \left|(c_1-c_2)Z'+(c_3-c_4)Z'\right| \leq \Gamma^4 \left|A_*^1+A_*^1+A_*^1+A_*^1\right|$$
Then Lemma \ref{theorem:plunnecke} gives
\begin{align*}
\left|Z\right|^2&\ll \frac{\Gamma^4\left|A_*^1+c_*A_*^2\right|^4}{\left|A_*^2\right|^3}\ll\frac{\left|A\right|^{239}\left|B\right|^{357}}{T^{119}}
\end{align*}
Comparing with \eqref{zbound} gives $T \ll \left|A\right|^{\frac{499}{249}}\left|B\right|^{\frac{743}{249}}$, which satisfies the bound in the statement of the proposition.
The proof of the proposition is therefore complete, subject to the proofs of Claims \ref{theorem:bsgclaim}, \ref{theorem:sumsetclaim} and \ref{theorem:coveringclaim}, which are given below.
\subsection{Proof of Claim \ref{theorem:bsgclaim}}
Every point in $P$ is the intersection of a horizontal line in $L_B$ (with $y$-co-ordinate lying in $B$) and a line through the origin in $L_A$ (with gradient lying in $A$). Denote the lines in $L_B$ by $h_b$ for each $b \in B$ and the lines in $L_A$ by $d_a$ for each $a \in A$. Furthermore, for each $b \in B$ define the set $X_b \subseteq F$ by
$$X_b= \left\{x:(x,b) \in h_b\cap P \right\}$$
Note that $X_b$ is a $(1,\lambda)$-antifield for each $b \in B$ as it is contained in the $(1,\lambda)$-antifield $\left\{x:(x,y) \in P\right\}$
Now, the set of lines $L(P)$ and the set of points $P$ generate $T$ colinear triples. So, by averaging, there are two distinct elements $b_1,b_2 \in B$ such that there are $\frac{T}{|B|^2}$ colinear triples $(p_1,p_2,p_3) \in P \times P \times P$ with $p_1 \in h_{b_1}$ and $p_2 \in h_{b_2}$.
By Lemma \ref{theorem:popularity} there is then a set $B' \subseteq B$ with $\left|B'\right|\gg \frac{T}{\left|A\right|^2 \left|B\right|^2}$ such that, for each $b \in B'$, there are $\gg \frac{T}{\left|B\right|^3}$ colinear triples $(p_1,p_2,p_3) \in P \times P \times P$ with $p_1 \in h_{b_1}$, $p_2 \in h_{b_2}$ and $p_3 \in h_b$.
This is the same as saying that for each $b \in B'$ there are $\gg \frac{T}{\left|B\right|^3}$ elements $x_1 \in X_{b_1}$ and $x_2 \in X_{b_2}$ for which $$x_1 \left(1-\frac{b-b_1}{b_2-b_1}\right)+x_2 \left(\frac{b-b_1}{b_2-b_1}\right)\in X_b $$
So for each $b \in B'$, we can apply the Balog-Szemeredi-Gowers theorem (Lemma $\ref{theorem:BSG}$) with $X=\left(1-\frac{b-b_1}{b_2-b_1}\right)X_{b_1}$, $Y=\frac{b-b_1}{b_2-b_1} X_{b_2}$, $n=|A|$, $G=\left\{(x_1,x_2)\in X_{b_1} \times X_{b_2}: x_1 \left(1-\frac{b-b_1}{b_2-b_1}\right)+x_2 \left(\frac{b-b_1}{b_2-b_1}\right)\in X_b \right\}$ and $\alpha=\frac{T}{\left|A\right|^2\left|B\right|^3}$ to find subsets $A_b^1 \subseteq X_{b_1}$ and $A_b^2 \subseteq X_{b_2}$ with
\begin{itemize}
\item $\left|A_b^1 + \left(\frac{b_1-b_2}{b_2-b}-1\right)A_b^2\right|=\left|(1-\frac{b-b_1}{b_2-b_1})A_b^1 + \frac{b-b_1}{b_2-b_1}A_b^2\right|\ll \frac{\left|A\right|^{11} \left|B\right|^{15} }{T^5}$
\item $\left|A_b^1\right|,\left|A_b^2\right| \gg \frac{T}{\left|A\right|\left|B\right|^3}$
\end{itemize}
Moreover, note that $A_b^1$ and $A_b^2$ are both $(1,\lambda)$-antifields for each $b \in B'$ as they are contained in the $(1,\lambda)$-antifields $X_{b_1}$ and $X_{b_2}$ respectively.
By dropping at most one element we may assume that $b_2 \notin B'$. Now let $C'=\left\{\frac{b_1-b_2}{b_2-b}-1:b \in B'\right\}$
and note that the map $b \mapsto \frac{b_1-b_2}{b_2-b}-1$ is a bijection. Define sets $A_c^1, A_c^2$ by $A_c^i=A_{b(c)}^i$ for each $c \in C'$. Then we have
\begin{itemize}
\item $\left|C'\right|=\left|B'\right|\gg \frac{T}{\left|A\right|^2 \left|B\right|^2}$
\item $\left|A_c^1 + c A_c^2\right| \ll \frac{\left|A\right|^{11} \left|B\right|^{15} }{T^5}$ for each $c \in C'$
\item $\left|A_c^1\right|,\left|A_c^2\right| \gg \frac{T}{\left|A\right|\left|B\right|^3}$ for each $c \in C'$
\end{itemize}
Let $P_c=A_c^1 \times A_c^2$, so that $\left|P_c\right|\gg\frac{T^2}{\left|A\right|^2 \left|B\right|^6}$ for each $c \in C'$. Cauchy-Schwartz implies that:
\begin{align*}
\left|C'\right|\frac{T^2}{\left|A\right|^2 \left|B\right|^6}\ll \sum_{c \in C'}\left|P_c\right|
\leq \left|A\right|\sqrt{\sum_{c,c' \in C'}\left|P_c \cap P_{c'}\right|}
\end{align*}
So there is a particular element $c^*\in C'$ such that
$$\sum_{c \in C'}\left|P_c \cap P_{c^*} \right|\gg \left|C'\right|\frac{T^4}{\left|A\right|^6\left|B\right|^{12}}\gg \frac{T^5}{\left|A\right|^8\left|B\right|^{14}}$$
Lemma \ref{theorem:popularity} then yields a subset $C \subseteq C'$ such that
\begin{itemize}
\item $\left|P_c \cap P_{c^*}\right|\gg \frac{T^4}{\left|A\right|^{6}\left|B\right|^{12}}$ for all $c \in C$
\item $\left|C\right|\gg \frac{T^5}{\left|A\right|^{10}\left|B\right|^{14}}$
\end{itemize}
Note that $\left|P_c \cap P_{c^*}\right|=\left|A_c^1 \cap A_{c^*}^1\right|\left|A_c^2 \cap A_{c^*}^2\right|$ to see that
$$\left|A_c^1 \cap A_{c^*}^1\right|,\left|A_c^2 \cap A_{c^*}^2\right|\gg \frac{T^4}{\left|A\right|^7\left|B\right|^{12}} $$
for each $c \in C$. This completes the proof of the claim.
\subsection{Proof of Claim \ref{theorem:sumsetclaim}}
The claim is proved by repeated application of Lemma \ref{theorem:plunnecke} and inequalities \eqref{size}, \eqref{sumset} and \eqref{intersection}:
\subsubsection{Proof of \eqref{claim1}}
Lemma \ref{theorem:plunnecke} and the inequalities \eqref{size} and \eqref{sumset} imply that
\begin{align*}
\left|A_c^1+A_c^1\right|\leq \frac{\left|A_c^1 + c A_c^2 \right|^2}{\left|A_c^2\right|}
\ll \frac{\left|A\right|^{23}\left|B\right|^{33}}{T^{11}}
\end{align*}
Similarly for $\left|A_c^2+A_c^2\right|$, which completes the proof of \eqref{claim1}.
\subsubsection{Proof of \eqref{claim2}}
Lemma \ref{theorem:plunnecke}, and inequalities \eqref{intersection} and \eqref{claim1}, imply that
\begin{align*}
\left|c_*A_c^2 + cA_c^2\right|&\leq \frac{\left|c_*A_c^2+c_*\left(A_c^2 \cap A_*^2\right)\right|\left|cA_c^2+c_*\left(A_c^2 \cap A_*^2\right)\right|}{\left|A_c^2 \cap A_*^2\right|}\\
&\ll \frac{\left|A_c^2+A_c^2\right|}{\left|A_c^2\cap A_*^2\right|}\left|cA_c^2+c_*\left(A_c^2 \cap A_*^2\right)\right|\\
&\ll \frac{\left|A\right|^{30}\left|B\right|^{45}}{T^{15}}\left|cA_c^2+c_*\left(A_c^2 \cap A_*^2\right)\right|
\end{align*}
Now apply Lemma \ref{theorem:plunnecke} again, with \eqref{sumset} and \eqref{intersection}, to see that
\begin{align*}
\left|cA_c^2+c_*\left(A_c^2 \cap A_*^2\right)\right|&\ll \frac{\left|\left(A_c^1\cap A_*^1\right)+cA_c^2 \right|\left|c_*\left(A_c^2 \cap A_*^2 \right)+\left(A_c^1 \cap A_*^1 \right) \right|}{\left|A_c^1 \cap A_*^1\right|}\\
&\leq \frac{\left|A_c^1+cA_c^2\right|\left|A_*^1+c_*A_*^2\right|}{\left|A_c^1 \cap A_*^1\right|}\\
& \ll \frac{\left|A\right|^{29}\left|B\right|^{42}}{T^{14}}
\end{align*}
which completes the proof of \eqref{claim2}
\subsubsection{Proof of \eqref{claim3}}
Lemma \ref{theorem:plunnecke}, and inequalities \eqref{sumset}, \eqref{intersection}, \eqref{claim1} and \eqref{claim2}, imply that:
\begin{align*}
\left|c_*A_*^2+cA_c^2\right|&\leq \frac{\left|c_*A_*^2+c_* \left(A_c^2 \cap A_*^2\right)\right|\left|cA_c^2+c_* \left(A_c^2 \cap A_*^2\right)\right|}{\left|A_c^2 \cap A_*^2\right|}\\
&\leq\frac{\left|A_*^2 + A_*^2\right|\left|c_*A_c^2+cA_c^2\right|}{\left|A_c^2 \cap A_*^2\right|}\\
& \ll \frac{\left|A\right|^{89}\left|B\right|^{132}}{T^{44}}
\end{align*}
which completes the proof of \eqref{claim3}
\subsubsection{Proof of \eqref{claim4}}
Lemma \ref{theorem:plunnecke}, and inequalities \eqref{sumset}, \eqref{intersection}, \eqref{claim1} and \eqref{claim3}, imply that
\begin{align*}
\left|c_*A_*^2+cA_*^2\right|&\ll \frac{\left|c_*A_*^2+c\left(A_c^2 \cap A_*^2\right)\right|\left|cA_*^2+c\left(A_c^2 \cap A_*^2\right)\right|}{\left|A_c^2 \cap A_*^2 \right|}\\
&\leq \frac{\left|c_*A_*^2 + cA_c^2\right|\left|A_*^2+A_*^2\right|}{\left|A_c^2 \cap A_*^2 \right|}\\
&\ll \frac{\left|A\right|^{119}\left|B\right|^{177}}{T^{59}}
\end{align*}
This completes the proof of \eqref{claim4}, and therefore of the whole claim.
\subsection{Proof of Claim \ref{theorem:coveringclaim}}
Given $D \subseteq A_*^2$, $x \in \mathbb{F}_q$ and $c \in C$, use the covering lemma (Lemma \ref{theorem:covering}) to cover a constant proportion of $cD+x$ with
$$\frac{\left|cD+\left(A_c^1 \cap A_*^1\right)\right|}{\left|A_c^1 \cap A_*^1\right|}\leq \frac{\left|cA_*^2+\left(A_c^1 \cap A_*^1\right)\right|}{\left|A_c^1 \cap A_*^1\right|}$$
translates of $A_c^1 \cap A_*^1$, and hence with the same number of translates of $A_*^1$. Lemma \ref{theorem:plunnecke} and the inequalities \eqref{sumset},\eqref{intersection} and \eqref{claim1} then give:
\begin{align*}
\frac{\left|cA_*^2+\left(A_c^1 \cap A_*^1\right)\right|}{\left|A_c^1 \cap A_*^1\right|}&\ll \frac{\left|cA_*^2+c\left(A_c^2 \cap A_*^2\right)\right|\left|\left(A_c^1 \cap A_*^1\right)+c\left(A_c^2 \cap A_*^2\right)\right|}{\left|A_c^1 \cap A_*^1\right|\left|A_c^2 \cap A_*^2\right|}\\
&\leq
\frac{\left|A_*^2+ A_*^2\right|\left|A_c^1 +cA_c^2 \right|}{\left|A_c^1 \cap A_*^1\right|\left|A_c^2 \cap A_*^2\right|}\\
&\ll \frac{\left|A\right|^{48}\left|B\right|^{72}}{T^{24}}
\end{align*}
The proof is similar when $c \in -C$. This completes the proof of the claim.
\section*{Acknowledgements}
The author is grateful to Oliver Roche-Newton and Misha Rudnev for useful discussions and for pointing out various typographical errors in earlier drafts, and to Nick Gill for asking some awkward questions.
\bibliographystyle{plain}
|
{
"timestamp": "2011-01-20T02:01:14",
"yymm": "1009",
"arxiv_id": "1009.3899",
"language": "en",
"url": "https://arxiv.org/abs/1009.3899"
}
|
\section{}
|
{
"timestamp": "2010-09-21T02:03:32",
"yymm": "1009",
"arxiv_id": "1009.3836",
"language": "en",
"url": "https://arxiv.org/abs/1009.3836"
}
|
\section{Introduction}
The presence of an important extra line-broadening mechanism (in addition to the rotational
broadening and usually called macroturbulence) affecting the spectra of O and B Sgs is
well established observationally (see Sim\'on-D\'iaz et al. 2010, and references therein).
\cite[Lucy (1976)]{Luc76} postulated that this extra\,broadening may be identified
with surface motions generated by the superposition of numerous non-radial oscillations.
More recently, \cite[Aerts et al. (2009)]{Aer09} computed time\,series of line profiles for
evolved massive stars broadened by rotation and hundreds of low amplitude non-radial gravity
mode oscillations and showed that the resulting profiles could mimic the observed ones.
Stellar oscillations are a plausible explanation for the extra\,broadening in O and B Sgs, but
this hyphotesis needs to be observationally confirmed.
\section{The macroturbulence\,--\,LPV connection}
As a first step, in \cite[Sim\'on-D\'iaz et al. (2010)]{Sim10}, we investigated the possible connection
between the macroturbulent broadening and the presence and temporal behaviour of
line-profile variations (LPVs) in a sample of 11 late-O and early-B Sgs, 2 late B-Sgs,
and 2 late-O, early-B dwarfs. To this aim, we obtained and analyzed time\,series of
high resolution (R\,$\sim$\,46000), high S/N spectra obtained with FIES@NOT in two
observing runs. We applied the Fourier transform (\cite[Gray 1976]{Gra76}) and the
goodness-of-fit techniques to disentangle and measure the contributions from
rotational ($v$\,sin$i$) and macroturbulent ($\Theta_{\rm RT}$) broadening
to the Si\,{\sc iii}\,4567 and/or the O\,{\sc iii}\,5592 line profiles.
We quantified the LPVs in these lines by means of the first,
$\langle v \rangle$, and third, $\langle v^3 \rangle$, normalized velocity
moments of the line. These moments
are related to the centroid velocity and the skewness of the line profile,
respectively, and are well suited to investigate whether an observed line profile is subject to
time-dependent line asymmetry, as expected in the case of a pulsating star.
We found a clear positive correlation between the average size of the macroturbulent
broadening, $\langle \Theta_{\rm RT} \rangle$, and the peak-to-peak amplitude of
$\langle v \rangle$ and $\langle v^3 \rangle$ variations (see Fig. \ref{f2}). To our
knowledge, this is the {\em first clear observational evidence for a connection between
extra broadening and LPVs in early B and late O Sgs.}
\begin{figure}[t!]
\begin{minipage}[l]{0.50\textwidth}
\begin{center}
\includegraphics[width=7.5cm,angle=90]{s5-28_simondiaz_fig1.eps}
\end{center}
\end{minipage}
\begin{minipage}[l]{0.07\textwidth}
\
\end{minipage}
\begin{minipage}[r]{0.40\textwidth}
\begin{center}
\caption{(Top) Empirical relations between the average size of the macroturbulent broadening
($\langle\Theta_{\rm RT}\rangle$) and the peak-to-peak amplitude of the first and third moments of the
line profile. Solid lines connect results from four stars observed in both campaigns.
(Bottom) similar plots with data
from Table 1 in \cite[Aerts et al. (2009)]{Aer09}, based on simulations of line profiles broadened
by rotation and by hundreds of low amplitude non-radial gravity mode
pulsations. The simulations lead to clear trends which are compatible with spectroscopic
observations.}
\label{f2}
\end{center}
\end{minipage}
\end{figure}
\section{Is macroturbulent broadening in OB-Sgs caused by pulsations?}
Non-radial oscillations have been often suggested as the origin of LPVs and
photosperic lines in OB Sgs; however, a firm confirmation (by means of a
rigorous seismic analysis) has not been achieved yet. From a theoretical
point of view, \cite[Saio et al. (2006)]{Sai06} showed that g-modes can
be excited in massive post-main sequence stars, as the g-modes are
reflected at the convective zone associated with the H-burning shell.
\cite{Lef07} presented observational evidence of g-mode instabilities
in a sample of photometrically variable B\,Sgs from the location of the
stars in the (log\,T$_{\rm eff}$, log\,$g$)-diagram.
These results, along with our observational confirmation of a tight
connection between macroturbulent broadening and parameters describing observed
LPVs render stellar oscillations the
most probable physical origin of macroturbulent broadening in B\,Sgs; however,
it is too premature to consider them as the only physical phenomenon to explain
the unknown broadening.
|
{
"timestamp": "2010-09-21T02:02:44",
"yymm": "1009",
"arxiv_id": "1009.3752",
"language": "en",
"url": "https://arxiv.org/abs/1009.3752"
}
|
\section{Introduction}
Given $Q=\mathbb P^1\times \mathbb P^1$, Giuffrida, Maggioni and
Ragusa in \cite{GMR} have investigated zero-dimensional schemes in
$Q$, studying in particular their Hilbert functions, which turn out to
be matrices of integers with infinite entries and with particular
numerical properties. These numerical conditions are
sufficient to characterize the Hilbert functions of arithmetically
Cohen-Macaulay zero-dimensional schemes in $Q$ (see \cite{GMR}) and
by the Hilbert function of an arithmetically
Cohen-Macaulay zero-dimensional scheme it is possible to determine a geometrical
description of the scheme. Other results about the Hilbert functions
of zero-dimensional schemes in $Q$ have been obtained
for fat points (see \cite{G}, \cite{GVT}, \cite{GVT3}, \cite{GVT2} and
\cite{VT}). In this paper in Theorem \ref{T:4} we give
numerical conditions to determine Hilbert functions of some
set of points in $Q$. In particular we describe these
schemes and we show
that any zero-dimensional scheme having in a grid of $(1,0)$ and
$(0,1)$-lines the same configuration of points has the same Hilbert function.
Given a zero-dimensional scheme $X\subset Q$ and a point $P\in X$, in Section \ref{sec:sep}
we look for the Hilbert function of $X\setminus\{P\}$ in relation to the
Hilbert function of $X$, giving a sufficient condition in Corollary
\ref{C:1}. In particular, we show that under this condition there
exists just one separator for $P\in X$ and it has minimal degree (see
\cite{M} and \cite{O}). As a consequence we can partially improve some results given in \cite{BM} on the Hilbert function of
the union of a zero-dimensional scheme $X$ with a particular set of
points of $Q$.
In Section \ref{main} we prove Theorem \ref{T:4}, in which we give
sufficient conditions to determine some Hilbert functions of set of points in $Q$. The conditions
in Theorem \ref{T:4} are quite technical, but they show a way to
new conditions for a characterization of
Hilbert functions of zero-dimensional schemes in $Q$.
In Example \ref{Ex:2} we give a matrix satisfying some of the
conditions Theorem \ref{T:4} and an application of Theorem \ref{T:4} is given in Example \ref{Ex:1},
while in Example \ref{Ex:0} we show that the conditions of Theorem
\ref{T:4} are not necessary.
\section{Notation}
Let $k$ be an algebraically closed field, let $\mathbb P^1=\mathbb
P^1_k$, let $Q=\mathbb P^1\times \mathbb P^1$ and let
$\mathscr O_Q$ be its structure sheaf. Let us consider the bi-graded
ring $S=H^0_*\mathscr O_Q=\bigoplus_{a,b\ge 0}H^0\mathscr
O_Q(a,b)$. For any sheaf $\mathscr F$ and any $a,b\in \mathbb Z$ we
define $\mathscr F(a,b)=\mathscr F\otimes_{\mathscr O_Q} \mathscr O_Q(a,b)$.
For any bi-graded $S$-module $N$ let $N_{i,j}$ be the component of
degree $(i,j)$. For any $(i_1,j_1)$, $(i_2,j_2)\in \mathbb N^2$ we write $(i_1,j_1)\ge
(i_2,j_2)$ if $i_1\ge i_2$ and $j_1\ge j_2$. Given a 0-dimensional scheme $X\subset Q$, let $I(X)\subset S$ be the
associated saturated ideal and $S(X)=S/I(X)$ the associated graded ring.
\begin{Def}
The function $M_X\colon \mathbb Z\times \mathbb Z\rightarrow \mathbb
N$ defined by:
\[
M_X(i,j)=\dim_k {S(X)}_{i,j}=(i+1)(j+1)-\dim_k
{I(X)}_{i,j}
\]
is called the \emph{Hilbert function} of $X$. The function $M_X$ can be
represented as an infinite matrix with integer entries
$M_X=(M_X(i,j))=(m_{ij})$ called \emph{Hilbert matrix} of $X$.
\end{Def}
In this paper we denote $M_X(i,j)$ also by $M_X^{(i,j)}$ to simplify
the notation. Note that $M_X(i,j)=0$ for either $i<0$ or $j<0$, so we restrict
ourselves to the range $i\ge 0$ and $j\ge 0$. Moreover, for $i\gg 0$
and $j\gg 0$ $M_X(i,j)=\deg X$.
\begin{Def}
Given the Hilbert matrix $M_X$ of a zero-dimensional scheme $X\subset Q$, the \emph{first difference of the Hilbert function} of $X$ is the
matrix $\Delta M_X=(c_{ij})$, where $c_{ij}=m_{ij}-m_{i-1j}-m_{ij-1}+m_{i-1j-1}$.
\end{Def}
We consider the matrices $\Delta^R M_X=(a_{ij})$ and $\Delta^C
M_X=(b_{ij})$, with $a_{ij}=m_{ij}-m_{ij-1}$ and
$b_{ij}=m_{ij}-m_{i-1j}$. Note that for any $i,j\ge 0$:
\begin{equation} \label{eq:7}
a_{ij}=\sum_{t=0}^ic_{tj}
\text{\quad and \quad}
b_{ij}=\sum_{t=0}^jc_{it}.
\end{equation}
For any matrix $M$ with infinite entries it is possible
to define in a similar way $\Delta M$, $\Delta^RM$ and $\Delta^CM$.
\begin{Def}[{\cite[Definition 2.2]{GMR}}]
Let $M=(m_{ij})$ be a matrix such that $m_{ij}=0$ for $i<0$ and
$j<0$. We say that $M$ is admissible if $\Delta M=(c_{ij})$
satisfies the following conditions:
\begin{enumerate}
\item $c_{ij}\le 1$ and $c_{ij}=0$ for $i\gg 0$ or $j\gg 0$;
\item if $c_{ij}\le 0$, then $c_{rs}\le 0$ for any $(r,s)\ge (i,j)$;
\item for every $(i,j)$ $0\le \sum_{t=0}^j c_{it}\le
\sum_{t=0}^jc_{i-1t}$ and $0\le \sum_{t=0}^i c_{tj}\le \sum_{t=0}^i c_{tj-1}$.
\end{enumerate}
\end{Def}
\begin{Thm}[{\cite[Theorem 2.11]{GMR}}] \label{T0}
If $X\subset Q$ is a $0$-dimensional scheme, then $M_X$ is an
admissible matrix.
\end{Thm}
If $X\subset Q$ is a zero-dimensional scheme, then $2\le
\operatorname{depth}S(X)\le 3$.
\begin{Def}
A zero-dimensional scheme $X\subset Q$ is called arithmetically
Cohen-Macaulay (ACM) if $\operatorname{depth}S(X)=2$.
\end{Def}
\begin{Thm}[{\cite[Theorem 4.1]{GMR}}]
A zero-dimensional scheme $X\subset Q$ is ACM if and only if
$c_{ij}\ge 0$ for any $(i,j)$.
\end{Thm}
Given an admissible matrix $M$, we define:
\begin{equation}
\label{eq:6}
T=\{(i,j)\in \mathbb N\times \mathbb N\mid c_{ij}<0\}.
\end{equation}
Then for any $(i,j)\in T$ we set:
\begin{equation}
\label{eq:9}
I_{ij}=\{0,\dots,-c_{ij}-1\}.
\end{equation}
\begin{Rem} \label{rm}
If $X\subset Q$ is a $0$-dimensional scheme, let us consider $a=\min\{i\in \mathbb
N\mid I(X)_{i,0}\ne 0\}-1$ and $b=\min\{j\in \mathbb
N\mid I(X)_{0,j}\ne 0\}-1$. Then by Theorem \ref{T0} $\Delta M_X$ is zero out of the
rectangle with opposite vertices $(0,0)$ and $(a,b)$, because $c_{a+10}=c_{0b+1}=0$. In this case we
say that $\Delta M_X$ is of size $(a,b)$.
\end{Rem}
Let $X\subset Q$ be a zero-dimensional scheme and let $L$ be a line
defined by a form $l$. Let $J=(I(X),l)$ and let $d=\deg(\operatorname{sat} J)$. Then we call $d$ the number of points
of $X$ on the line $L$ and, by abuse of notation, we define
$d=\#(X\cap L)$. We say that $L$ is disjoint from $X$ if $d=0$.
For any $i\ge 0$ we set $j(i)=\min\{t\in \mathbb N\mid
m_{it}=m_{it+1}\}$ and similarly for any $j\ge 0$ we set $i(j)=\min\{t\in \mathbb N\mid m_{tj}=m_{t+1j}\}$.
\begin{Thm}[{\cite[Theorem 2.12]{GMR}}] \label{T}
Let $X\subset Q$ be a zero-dimensional scheme and let $M_X=(m_{ij})$
be its Hilbert matrix. Then for every $j\ge 0$ there are just
$a_{i(0)j}-a_{i(0)j+1}$ lines of type $(1,0)$ each containing just
$j+1$ points of $X$ and, similarly, for every $i\ge 0$ there are
just $b_{ij(0)}-b_{i+1j(0)}$ lines of type $(0,1)$ each containing
just $i+1$ points of $X$.
\end{Thm}
Now we recall the following definition:
\begin{Def}
Let $X\subset Q$ be a zero-dimensional scheme and let $P\in X$. The
multiplicity of $X$ in $P$, denoted by $m_X(P)$, is the length of
$\mathscr O_{X,P}$.
\end{Def}
Given $P\in Q$, we denote by $I_P$ the maximal ideal of $S$ associated
to $P$. If $X\subset Q$ is a $0$-dimensional scheme, then
$I(X)=\cap_{P'\in X} J_{P'}$ for some ideal $J_{P'}$ such that $\sqrt{J_{P'}}=I_{P'}$.
\begin{Def}
Given a zero-dimensional scheme $X\subset Q$ and $P\in X$ such that
$m_X(P)=1$, we say that $f\in S$ is a \emph{separator} for $P\in X$ if $f(P)\ne 0$ and
$f\in \cap_{P'\in X\setminus \{P\}}J_{P'}$.
\end{Def}
This definition generalizes the definition of a separator for a point
in a reduced zero-dimensional scheme in a multiprojective space given
by \cite{GVT2}.
\section{Separators and Hilbert functions} \label{sec:sep}
Let $X\subset Q$ be a zero-dimensional scheme and let $M_X$ be its Hilbert
matrix. In all this paper we suppose that $\Delta M_X$ is of size
$(a,b)$ and we denote by $R_0$,\dots, $R_a$
and $C_0$,\dots,$C_b$, respectively, the
$(1,0)$ and $(0,1)$-lines containing $X$ and each one at least one point of
$X$.
\begin{Thm} \label{T:0}
Let $P=R_h\cap C_k\in X$ for some $h\in \{0,\dots,a\}
$ and $k\in \{0,\dots,b\}
$ and suppose that $m_X(P)=1$. Let $Z=X\setminus\{P\}$, $p=\#(Z\cap R_h)$ and $q=\#(Z\cap
C_k)$. If there exists a separator in degree $(q,p)$ for $P\in X$,
then:
\[
\Delta M_Z^{(i,j)}=
\begin{cases}
\Delta M_X^{(i,j)} & \text{if } (i,j)\ne (q,p)\\
\Delta M_X^{(i,j)}-1 & \text{if } (i,j)=(q,p).
\end{cases}
\]
\end{Thm}
\begin{proof}
It is easy to see that $\Delta M_Z^{(i,j)}=\Delta M_X^{(i,j)}$ for any $(i,j)$ with either
$i<q$ or $j<p$. Indeed, taken $(i,j)$ with $i< q$ any $(i,j)$-curve containing $Z$ must
contain $C_k$ and so $h^0\mathscr
I_Z(i,j)=h^0\mathscr I_X(i,j)$ and $\Delta M_Z^{(i,j)}=\Delta
M_X^{(i,j)}$. The proof works in a similar way if $j<p$.
By the exact sequence:
\begin{equation} \label{eq:14}
0\rightarrow \mathscr I_X\rightarrow \mathscr I_Z\rightarrow \mathscr
O_P\rightarrow 0
\end{equation}
we see that $h^0\mathscr I_Z(q,p)>h^0\mathscr I_X(q,p)$ if and
only if $h^0\mathscr I_Z(q,p)=h^0\mathscr I_X(q,p)+1$. This means
that it must be:
\[
\Delta M_Z^{(q,p)}=\Delta M_X^{(q,p)}-1.
\]
Now we only need to prove that $\Delta M_Z^{(i,j)}=\Delta M_X^{(i,j)}$ for any $(i,j)>(q,p)$.
By \eqref{eq:14} we see that for any $(i,j)$:
\begin{equation} \label{eq:15}
h^0\mathscr I_X(i,j)\le h^0\mathscr I_Z(i,j)\le h^0\mathscr I_X(i,j)+1
\end{equation}
which is equivalent to:
\[
M_X^{(i,j)}-1\le M_Z^{(i,j)}\le M_X^{(i,j)}.
\]
Since $h^0\mathscr I_Z(q,p)=h^0\mathscr I_X(q,p)+1$,
by \eqref{eq:15} we see that it must be $h^0\mathscr I_Z(i,j)=h^0\mathscr I_X(i,j)+1$
for any $(i,j)\ge (q,p)$. In particular this means that
$M_Z^{(i,j)}=M_X^{(i,j)}-1$ for any $(i,j)\ge (q,p)$. Now the conclusion follows easily.
\end{proof}
\begin{Thm} \label{T:1}
Let $P=R_h\cap C_k\in X$ for some $h\in \{0,\dots,a\}
$ and $k\in \{0,\dots,b\}
$ such that $m_X(P)=1$ and let $p+1=\#(X\cap R_h)$ and $q+1=\#(X\cap
C_k)$. Suppose that one of the following conditions holds:
\begin{enumerate}
\item $p=b$;
\item $q=a$;
\item $p<b$, $q<a$ and $\Delta
M_X^{(i,j)}=0$ for any $(i,j)\ge (q+1,p+1)$.
\end{enumerate}
Then there exists a separator for $P\in X$ in degree $(q,p)$.
\end{Thm}
\begin{proof}
We divide the proof in different steps. Let $Z=X\setminus\{P\}$.
\begin{step} \label{s:2}
There exists $\overline j$ with $p\le \overline j\le b$ such that one
the following conditions holds:
\begin{enumerate}
\item $\Delta M_Z^{(q,j)}=\Delta M_X^{(q,j)}$ for any $j< \overline j$ and
$\Delta M_Z^{(q,\overline j)}<\Delta M_X^{(q,\overline j)}$;
\item $\Delta M_Z^{(q,j)}=\Delta M_X^{(q,j)}$ for any $p\le j\le b$.
\end{enumerate}
\end{step}
Since $Z\subset X$ we see that $M_Z^{(q,p)}\le
M_X^{(q,p)}$. Moreover, as we have seen in the proof of Theorem
\ref{T:0} $M_Z^{(i,j)}=M_X^{(i,j)}$ for any $i<q$ or $j<p$. This
implies that $\Delta M_Z^{(q,p)}\le \Delta M_X^{(q,p)}$.
If $\Delta M_Z^{(q,p)}=\Delta M_X^{(q,p)}$, then we
can repeat the previous procedure to show that $\Delta
M_Z^{(q,p+1)}\le \Delta M_X^{(q,p+1)}$. By iterating this
procedure we get the conclusion of Step \ref{s:2}.
\begin{step} \label{s:3}
The following equalities hold:
\begin{enumerate}
\item $\sum_{j=p}^b\Delta M_Z^{(q,j)}=\sum_{j=p}^b\Delta M_X^{(q,j)}-1$;
\item for any $i\in \{q+1,\dots,a\}$ $\sum_{j=p}^b\Delta
M_Z^{(i,j)}=\sum_{j=p}^b\Delta M_X^{(i,j)}$.
\end{enumerate}
\end{step}
Let us first note that by Theorem \ref{T}:
\[
b_{q-1j(0)}(Z)-b_{qj(0)}(Z)=
\sum_{j\le b}
\Delta M_Z^{(q-1,j)}-\sum_{j\le b} \Delta M_Z^{(q,j)}
\]
is equal to the number of $(0,1)$-lines containing precisely $q$ points of
$Z$, while:
\[
b_{q-1j(0)}(X)-b_{qj(0)}(X)=
\sum_{j\le b}
\Delta M_X^{(q-1,j)}-\sum_{j\le b} \Delta M_X^{(q,j)}
\]
is equal to the number of $(0,1)$-lines containing precisely $q$ points of
$X$. By hypothesis it must be:
\[
b_{q-1j(0)}(Z)-b_{qj(0)}(Z)=
\sum_{j\le b}
\Delta M_Z^{(q-1,j)}-\sum_{j\le b} \Delta M_Z^{(q,j)}=\sum_{j\le b}
\Delta M_X^{(q-1,j)}-\sum_{j\le b} \Delta M_X^{(q,j)}+1
\]
Since $h^0\mathscr I_Z(i,j)=h^0\mathscr I_X(i,j)$ for any $i<q$ or
$j<p$, this implies that:
\begin{equation}
\label{eq:12}
\sum_{j\le b} \Delta M_Z^{(q,j)}=\sum_{j\le b} \Delta M_X^{(q,j)}-1.
\end{equation}
In a similar way we see that:
\[
b_{qj(0)}(Z)-b_{q+1j(0)}(Z)=
\sum_{j\le b}
\Delta M_Z^{(q,j)}-\sum_{j\le b} \Delta M_Z^{(q+1,j)}=\sum_{j\le b}
\Delta M_X^{(q,j)}-\sum_{j\le b} \Delta M_X^{(q+1,j)}-1
\]
which implies by \eqref{eq:12} that $\sum_{j\le b} \Delta M_Z^{(q+1,j)}=\sum_{j\le b} \Delta M_X^{(q+1,j)}$.
Let us now suppose that for some $i\ge q+1$, with $i<a$, we have:
\begin{equation}
\label{eq:4}
\sum_{j\le b} \Delta M_Z^{(i,j)}=\sum_{i\le b} \Delta M_X^{(i,j)}.
\end{equation}
We will show that:
\begin{equation}
\label{eq:5}
\sum_{j\le b} \Delta M_Z^{(i+1,j)}=\sum_{j\le b} \Delta M_X^{(i+1,j)}.
\end{equation}
Again, by Theorem \ref{T} $\sum_{j\le b}
\Delta M_Z^{(i,j)}-\sum_{j\le b} \Delta M_Z^{(i+1,j)}$ is equal
to the number of $(0,1)$-lines containing precisely $i+1$ points of
$Z$, while $\sum_{j\le b}
\Delta M_X^{(i,j)}-\sum_{j\le b} \Delta M_X^{(i+1,j)}$ is equal
to the number of $(0,1)$-lines containing precisely $i+1$ points of
$X$. By hypothesis it must be:
\[
\sum_{j\le b}
\Delta M_Z^{(i,j)}-\sum_{j\le b} \Delta M_Z^{(i+1,j)}=
\sum_{j\le b}
\Delta M_X^{(i,j)}-\sum_{j\le b} \Delta M_X^{(i+1,j)}.
\]
By \eqref{eq:4} it means that \eqref{eq:5} holds, so that $\sum_{j\le b} \Delta M_Z^{(i,j)}=\sum_{j\le b} \Delta M_X^{(i,j)}$
for any $i$ with $q+1\le i\le a$.
\\
The statement of the theorem is proved if we show the following:
\begin{step} \label{s:4}
$h^0\mathscr I_Z(q,p)=h^0\mathscr I_X(q,p)+1$.
\end{step}
In the cases $p=b$ and $q=a$ by Step \ref{s:3} we easily get Step
\ref{s:4}. So from now on we suppose that $p<b$ and $q<a$.
By Step \ref{s:2} and Step \ref{s:3} we see that there exists
$\overline j$ with $p\le \overline j\le b$ such that $\Delta M_Z^{(q,j)}=\Delta M_X^{(q,j)}$
for any $j< \overline j$ and $\Delta M_Z^{(q,\overline j)}<\Delta M_X^{(q,\overline j)}$.
Let us suppose that $\overline j\ge p+1$. Then $\Delta
M_Z^{(q,\overline j)}\le 0$ and by Theorem \ref{T0} we see that $\Delta M_Z^{(i,\overline j)}\le 0$ for any
$i\ge q$. By Step \ref{s:3} and by hypothesis we see that:
\[
\sum_{i=q+1}^a \Delta M_Z^{(i,\overline j)}=\Delta M_X^{(q,\overline j)}-\Delta M_Z^{(q,\overline j)}>0.
\]
So $\sum_{i=q+1}^a \Delta M_Z^{(i,\overline j)}>0$, but this
contradicts that fact that $\Delta M_Z^{(i,\overline j)}\le 0$ for any
$i\ge q$. This means that $\overline j=p$, i.e. $\Delta M_Z^{(q,p)}<\Delta M_X^{(q,p)}$.
Since, as we have seen, $h^0\mathscr I_Z(i,j)=h^0\mathscr
I_X(i,j)$ for any $(i,j)<(q,p)$, it gives us the inequality
$M_Z^{(q,p)}<M_X^{(q,p)}$, which means that $h^0\mathscr I_Z(q,p)>h^0\mathscr I_X(q,p)$. But by the exact sequence:
\[
0\rightarrow \mathscr I_X\rightarrow \mathscr I_Z\rightarrow \mathscr
O_P\rightarrow 0
\]
we see that $h^0\mathscr I_Z(q,p)>h^0\mathscr I_X(q,p)$ if and
only if $h^0\mathscr I_Z(q,p)=h^0\mathscr I_X(q,p)+1$ and the
statement is proved.
\end{proof}
\begin{Cor} \label{C:1}
Let $P=R_h\cap C_k\in X$ for some $h\in \{0,\dots,a\}
$ and $k\in \{0,\dots,b\}
$ such that $m_X(P)=1$. Given $Z=X\setminus\{P\}$, $p=\#(Z\cap R_h)$ and $q=\#(Z\cap
C_k)$, suppose that one of the following conditions holds:
\begin{enumerate}
\item $p=b$;
\item $q=a$;
\item $p<b$, $q<a$ and $\Delta
M_X^{(i,j)}=0$ for any $(i,j)\ge (q+1,p+1)$.
\end{enumerate}
Then:
\[
\Delta M_Z^{(i,j)}=
\begin{cases}
\Delta M_X^{(i,j)} & \text{if } (i,j)\ne (q,p)\\
\Delta M_X^{(i-1,j)}-1 & \text{if } (i,j)=(q,p).
\end{cases}
\]
\end{Cor}
\begin{proof}
The proof follows by Theorem \ref{T:0} and Theorem \ref{T:1}.
\end{proof}
\begin{Cor} \label{C:2}
Let $X$ be an ACM zero-dimensional scheme and let $P=R_h\cap C_k\in X$ for some $h\in \{0,\dots,a\}
$ and $k\in \{0,\dots,b\}
$ such that $m_X(P)=1$. Given $Z=X\setminus\{P\}$, $p=\#(Z\cap R_h)$ and $q=\#(Z\cap
C_k)$, we have:
\[
\Delta M_Z^{(i,j)}=
\begin{cases}
\Delta M_X^{(i,j)} & \text{if } (i,j)\ne (q,p)\\
\Delta M_X^{(i-1,j)}-1 & \text{if } (i,j)=(q,p).
\end{cases}
\]
\end{Cor}
\begin{proof}
By \cite[Proposition 4.1]{BM} we see that $\Delta M_X^{(i,j)}=0$ for
any $(i,j)\ge (q+1,p+1)$. Then the conclusion follows by Corollary \ref{C:1}.
\end{proof}
In the following we slightly improve the result given in
\cite[Theorem 3.1]{BM}.
\begin{Cor}
Let $R$ be a $(1,0)$-line disjoint from $X$. Let $C_{b+1}$,\dots,$C_n$,
$n\ge b$, be arbitrary $(0,1)$-lines and $i_1$,\dots,$i_r\in
\{0,\dots,b\}$. Let $\mathcal P=\{R\cap C_i\mid i\in\{0,\dots,n\},\,
i\ne i_1,\dots,i_r\}$ and let $W=X\cup \mathcal P$. Suppose also that on the $(0,1)$-line $C_{i_k}$
there are $q_k$ points of $X$ for $k=1,\dots,r$ and that $q_1\le
q_2\le \dots \le q_r$. Then, given
$T=\{(q_1,n),(q_2,n-1),\dots,(q_r,n-r+1)\}$, we have:
\[
\Delta M_W^{(i,j)}=
\begin{cases}
1 & \text{if } i=0,\, j\le n\\
0 & \text{if }i=0,\, j\ge n+1\\
\Delta M_X^{(i-1,j)} & \text{if } i\ge 1\text{ and }(i,j)\notin T\\
\Delta M_X^{(i-1,j)}-1 & \text{if } i\ge 1\text{ and }(i,j)\in T
\end{cases}
\]
if one of the following conditions holds:
\begin{enumerate}
\item $r=1$;
\item $r\ge 2$ and for any $k\in \{2,\dots,r\}$ and $i\ge q_k$\, $\Delta
M_X^{(i,n-k+2)}=0$.
\end{enumerate}
\end{Cor}
\begin{proof}
Let $Y=X\cup (R\cap (C_0\cup \dots \cup C_n))$. Then the statement
follows by \cite[Lemma 2.15]{GMR} and by Corollary \ref{C:1}.
\end{proof}
\section{Technical results}
In this section we prove some technical results that will be useful in
the proof of Theorem \ref{T:4}. In all this section we denote by $M$
an admissible matrix and we keep the notation given previously.
\begin{Prop
\label{P:1}
Let us suppose that for some $(i_1,j_1)$ and $(i_2,j_2)$ with
$j_1>j_2$ the
following conditions hold:
\begin{enumerate}
\item $c_{i_1j_1}<0$ and $c_{i_2j_2}\le 0$;
\item $a_{i_1j_1}+r\ge a_{i_2j_2}$, for some $r\in I_{i_1j_1}$.
\end{enumerate}
Then $i_1\le i_2$.
\end{Prop}
\begin{proof}
Let us suppose that $i_1>i_2$. Then by hypothesis we have $\sum_{t=0}^{i_1}c_{tj_1}+r\ge \sum_{t=0}^{i_2}c_{tj_2}$
and so by Theorem \ref{T0}:
\[
0\ge \sum_{t=0}^{i_2}c_{tj_1}-\sum_{t=0}^{i_2}c_{tj_2}\ge -r-\sum_{t=i_2+1}^{i_1}c_{tj_1}.
\]
This implies that:
\[
0\le r+\sum_{t=i_2+1}^{i_1}c_{tj_1}\le r+c_{i_1j_1}<0,
\]
by hypothesis and by the fact that by Theorem \ref{T0}
$c_{tj_1}\le 0$ for any $t\ge i_2$.
\end{proof}
In a similar way it is possible to prove the following:
\begin{Prop}
Let us suppose that for some $(i_1,j_1)$ and $(i_2,j_2)$ with
$i_1>i_2$ the
following conditions hold:
\begin{enumerate}
\item $c_{i_1j_1}<0$ and $c_{i_2j_2}\le 0$;
\item $b_{i_1j_1}+r\ge b_{i_2j_2}$, for some $r\in I_{i_1j_1}$.
\end{enumerate}
Then $j_1\le j_2$.
\end{Prop}
Another technical result is:
\begin{Prop} \label{P:6}
Let us suppose that for some $(i_1,j_1)$
and $(i_2,j_2)$, with $j_2<j_1-1$, the
following conditions hold:
\begin{enumerate}
\item $c_{i_1j_1}<0$ and $c_{i_2j_2}\le 0$;
\item $a_{i_1j_1}+r\ge a_{i_2j_2}$, for some $r\in I_{i_1j_1}$.
\end{enumerate}
Then there exists $(i,j)$ with $j_2<j<j_1$ and $i\le i_2$ such that
$c_{ij}<0$ and $a_{i_1j_1}+r+c_{ij}+1\le a_{ij}\le a_{i_1j_1}+r$.
\end{Prop}
\begin{proof}
First note that by Proposition \ref{P:1} it must be $i_1\le i_2$. Suppose
that for every $(i,j)$ with $j_2<j<j_1$ and $i\le i_2$ we have
$c_{ij}\ge 0$. Then this implies that $a_{i_2j_2+1}\ge
a_{i_1-1j_2+1}$, by which we get:
\[
a_{i_2j_2}\ge a_{i_2j_2+1}\ge a_{i_1-1j_2+1}\ge a_{i_1-1j_1}.
\]
However:
\[
a_{i_2j_2}\le a_{i_1j_1}+r=a_{i_1-1j_1}+c_{i_1j_1}+r<a_{i_1-1j_1},
\]
which gives us a contradiction.
Take $j$ with $j_2<j<j_1$ such that $c_{ij}<0$ for some $i\le
i_2$. Then we can choose $i$ in such a way that $a_{ij}=a_{i_2j}$. Then by Theorem \ref{T0} we see that
$a_{ij}\le a_{i_2j_2}\le a_{i_1j_1}+r$. If $a_{i_1j_1}+r+c_{ij}+1\le a_{ij}$,
then we get the conclusion. So we can suppose that:
\begin{equation}
\label{eq:3}
a_{ij}<a_{i_1j_1}+r+c_{ij}+1.
\end{equation}
Take $i'<i$ such that $c_{i'j}<0$ and $c_{kj}=0$ for
$k=i'+1,\dots,i-1$. Then $a_{ij}=c_{ij}+a_{i'j}$ and \eqref{eq:3} is
equivalent to:
\[
a_{i'j}\le a_{i_1j_1}+r.
\]
Again, if $a_{i_1j_1}+r+c_{i'j}+1\le a_{i'j}$, then the conclusion
follows. Otherwise we proceed as before. Iterating this procedure we
see that either we get the conclusion or $a_{kj}\le a_{i_1j_1}+r$ for
$k$ such that $c_{kj}=1$ and $c_{k+1j}\le 0$. So we can suppose that
such a $k$ exists. Then we see that $a_{kj}=\max\{a_{ij}\mid i\ge
0\}\ge a_{i_1-1j}$, so that $a_{i_1-1j}\le a_{i_1j_1}+r<a_{i_1-1j_1}$. But
by Theorem \ref{T0} this is not possible.
\end{proof}
In a similar way it is possible to prove the following:
\begin{Prop} \label{P:8}
Let us suppose that for some $(i_1,j_1)$
and $(i_2,j_2)$, with $i_2<i_1-1$, the
following conditions hold:
\begin{enumerate}
\item $c_{i_1j_1}<0$ and $c_{i_2j_2}\le 0$;
\item $b_{i_1j_1}+r\ge b_{i_2j_2}$, for some $r\in I_{i_1j_1}$.
\end{enumerate}
Then there exists $(i,j)$ with $i_2<i<i_1$ and $j\le j_2$ such that
$c_{ij}<0$ and $b_{i_1j_1}+r+c_{ij}+1\le b_{ij}\le b_{i_1j_1}+r$.
\end{Prop}
\begin{Rem} \label{r:1}
By Proposition \ref{P:6} it follows that, given $(i_1,j_1)$,
$(i_2,j_2)\in T$, $r_1\in I_{i_1j_1}$ and $r_2\in I_{i_2j_2}$ such that $a_{i_1j_1}+r_1=a_{i_2j_2}+r_2$, for
any $j$ with $j_1\le j\le j_2$ there exists $i$ such that $(i,j)\in
T$ and $a_{ij}+r=a_{i_1j_1}+r_1=a_{i_2j_2}+r_2$ for some $r\in I_{ij}$.
Of course, a similar result follows by Proposition \ref{P:8}.
\end{Rem}
Now we prove a result on $\Delta^R M$.
\begin{Prop} \label{P:7}
Let $(i_1,j_1)$, $(i_2,j_1)\in T$ with $i_2<i_1$. Then
$a_{i_2j_1}+s>a_{i_1j_1}+r$, for any $r\in I_{i_1j_1}$
and $s\in I_{i_2j_1}$.
\end{Prop}
\begin{proof}
Let us suppose that $a_{i_2j_1}+s\le a_{i_1j_1}+r$. Note that
$a_{i_1j_1}=a_{i_2j_1}+\sum_{i=i_2+1}^{i_1}c_{ij_1}$. Then we have:
\begin{equation} \label{eq:1}
s\le \sum_{i=i_2+1}^{i_1}c_{ij_1}+r.
\end{equation}
However $c_{i_1j_1}+r<0$ and by Theorem \ref{T0} $c_{ij_1}\le
0$ for any $i>i_2$. Then by \eqref{eq:1} we get $s<0$, which gives us
a contradiction.
\end{proof}
In a similar way it is possible to prove the following:
\begin{Prop} \label{P:10}
Let $(i_1,j_1)$, $(i_1,j_2)\in T$ with $j_2<j_1$. Then
$b_{i_1j_2}+s>b_{i_1j_1}+r$, for any $r\in I_{i_1j_1}$
and $s\in I_{i_1j_2}$.
\end{Prop}
Given the admissible matrix $M$ of size $(a,b)$, let us
consider $R_0$,\dots,$R_a$ and $C_0$,\dots,$C_b$ pairwise distinct arbitrary $(1,0)$ and
$(0,1)$-lines. Let $P_{ij}=R_i\cap C_j$ and let us consider the
following reduced ACM zero-dimensional
scheme:
\[
X=\{P_{ij}\mid c_{ij}=1\}.
\]
Under this notation we prove the following:
\begin{Prop} \label{P:3}
Let $p\in \mathbb N$ sucht that:
\[
\{(i,j)\in T\mid p+c_{ij}+1\le a_{ij}\le p\}\ne \emptyset
\]
and let:
\[
k=\max\{j\mid \exists\, (i,j)\in T,\, p+c_{ij}+1\le a_{ij}\le
p\}.
\]
Then $0\le p\le a$ and $\#(X\cap R_{p})=k+1$.
\end{Prop}
\begin{proof}
Let $(h,k)\in T$ such that $p+c_{hk}+1\le a_{hk}\le p$. Then there exists $s\in I_{hk}$ such that
$a_{hk}+s=p$. This implies that $0\le p\le a$.
Now we prove that $\#(X\cap R_{p})=k+1$. We will show that $c_{pk}=1$
and $c_{pk+1}\le 0$. Let us first note that:
\[
p=a_{hk}+s=a_{h-1k}+c_{hk}+s\le h-1<h.
\]
Let us suppose now that $c_{pk}\le 0$. In this case by \eqref{eq:7} we see that:
\[
a_{hk}+s=a_{h-1k}+c_{hk}+s<a_{hk-1}\le a_{p-1k}\le p,
\]
which contradicts the fact that $a_{hk}+s=p$. So we can say that $c_{pk}=1$.
Let us suppose now that $c_{pk+1}=1$. Then by \eqref{eq:7} we get:
\[
a_{pk+1}=p+1=a_{hk}+s+1.
\]
By Theorem \ref{T0} we see that $a_{hk}\ge a_{hk+1}$ and we
also have $a_{hk}<a_{hk}+s+1=a_{pk+1}$. This implies that $a_{hk+1}< a_{pk+1}$, but $p<h$ and so there exists $i$ with $p<i\le h$ such that
$c_{ik+1}<0$. Let $i\le h$ such that $c_{ik+1}<0$ and
$a_{ik+1}=a_{hk+1}\le a_{hk}$. By hypothesis on $k$ it must be
$a_{ik+1}<p+c_{ik+1}+1$. So, taken $i'$ such that
$c_{i'k+1}<0$ and
$c_{i'+1k+1}=\dots=c_{h-1k+1}=0$, we see that
$a_{i'k+1}\le p$. Again, by hypothesis it must be
$a_{i'k+1}<p+c_{i'k+1}+1$. Iterating the procedure we see that,
taken $m$ such that $c_{mk+1}=1$ and $c_{m+1k+1}\le 0$, it must be
$a_{mk+1}\le p$, where by \eqref{eq:7} $a_{mk+1}=m+1$. However,
$c_{pk+1}=1$ and so $m\ge p$ and so
this gives us a contradiction.
\end{proof}
In a similar way it is possible to prove the following:
\begin{Prop} \label{P:9}
Let $q\in \mathbb N$ such that:
\[
\{(i,j)\in T\mid q+c_{ij}+1\le a_{ij}\le q\}\ne \emptyset
\]
and let:
\[
h=\max\{i\mid \exists\, (i,j)\in T,\, q+c_{ij}+1\le b_{ij}\le
q\}.
\]
Then $0\le q\le b$ and $\#(X\cap C_{q})=h+1$.
\end{Prop}
\section{Main Theorem} \label{main}
In this section we give some conditions for an admissible
matrix to be the Hilbert matrix of some reduced zero-dimensional schemes. If $M$
is an admissible matrix of size $(a,b)$, it is always possible to
associate to $M$ a reduced zero-dimensional scheme $Z$ in the following way. Let $R_0$,\dots,$R_a$ and $C_0$,\dots,$C_b$ be pairwise distinct arbitrary $(1,0)$ and
$(0,1)$-lines. Let $P_{ij}=R_i\cap C_j$ and let us consider the scheme:
\[
X=\{P_{ij}\mid c_{ij}=1\}.
\]
By proceeding as in \cite[Proposition 4.1]{BM} we see that $X$ is an ACM
zero-dimensional scheme and
that:
\begin{equation}
\label{eq:11}
\Delta M_X^{(i,j)}=
\begin{cases}
1 & \text{if }(i,j)\in X\\
0 & \text{if }(i,j)\notin X.
\end{cases}
\end{equation}
Note that $(a_{ij}+r,b_{ij}+r)\in X$ for any $(i,j)\in T$ and $r\in
I_{ij}$ (see \eqref{eq:6} and \eqref{eq:9}). Then it is easy to see that:
\[
\mathcal P=\{P_{a_{ij}+r,b_{ij}+r}\mid (i,j)\in T,\, r\in I_{ij}\}\subsetneq X.
\]
\begin{Def} \label{d}
The scheme $Z=X\setminus \mathcal P$ is called \emph{zero-dimensional
scheme associated to $M$}.
\end{Def}
We call $Z$ the We want to show under which conditions
the Hilbert matrix of $Z$ is $M$. For this purpose we give the
following definitions:
\begin{Def}
Let $M$ be an admissible matrix. We say that $M$ is a
\emph{$\Delta$-regular matrix} if for any $(i_1,j_1)$,
\dots, $(i_n,j_n)\in T$ and $r_1\in I_{i_1j_1}$,
\dots, $r_n\in I_{i_nj_n}$ the following conditions hold:
\begin{enumerate}
\item if $a_{i_1j_1}+r_1=\dots=a_{i_nj_n}+r_n$, $i_1\ne
\dots \ne i_n$ and $j_1<\dots <j_n$, then
$b_{i_1j_1}+r_1\le \dots \le b_{i_nj_n}+r_n$;
\item if $b_{i_1j_1}+r_1=\dots=b_{i_nj_n}+r_n$, $j_1\ne \dots
\ne j_n$ and $i_1<\dots <i_n$, then
$a_{i_1j_1}+r_1\le \dots \le a_{i_nj_n}+r_n$.
\end{enumerate}
\end{Def}
\begin{Rem}
Given an admissible matrix $M$ and any $(i_1,j_1)$,
\dots, $(i_n,j_n)\in T$ and $r_1\in I_{i_1j_1}$,
\dots, $r_n\in I_{i_nj_n}$ such that $a_{i_1j_1}+r_1=\dots=a_{i_nj_n}+r_n$, $i_1=
\dots =i_n$ and $j_1<\dots <j_n$, then by Proposition \ref{P:10}
it must be
$b_{i_1j_1}+r_1>\dots >b_{i_nj_n}+r_n$.
Similarly, if $b_{i_1j_1}+r_1=\dots=b_{i_nj_n}+r_n$, $j_1=\dots
=j_n$ and $i_1<\dots <i_n$, then by Proposition \ref{P:7}
$a_{i_1j_1}+r_1> \dots >a_{i_nj_n}+r_n$.
\end{Rem}
\begin{Def}
An admissible matrix $M$ is called \emph{plain
matrix} if for any $(i_1,j_1)$, $(i_2,j_2)\in T$, $r_1\in
I_{i_1j_1}$, $r_2\in I_{i_2j_2}$ we have $(a_{i_1j_1}+r_1,b_{i_1j_1}+r_1)\ne (a_{i_2j_2}+r_2,b_{i_2j_2}+r_2)$.
\end{Def}
Note that, if $M$ is plain, then for
any $(i_1,j_1),(i_2,j_2)\in T$, $r_1\in I_{i_1j_1}$ and $r_2\in
I_{i_2j_2}$ we have $P_{a_{i_1j_1}+r_1,b_{i_1j_1}+r_1}\ne
P_{a_{i_2j_2}+r_2,b_{i_2j_2}+r_2}$.
\begin{Ex} \label{Ex:2}
Let us consider the following admissible matrix $M$ and its first difference $\Delta M$.
\begin{figure}[H]
\begin{preview}
\begin{center}
\subfloat[$M$]{
\begin{tikzpicture}[x=0.45cm,y=0.45cm,font=\tiny]
\clip(0,0.5) rectangle (11,8);
\draw[style=help lines,xstep=1,ystep=1] (1,1) grid (10,7);
\foreach \x in {1,...,9} \draw (\x,1) +(.5,.5) node {\dots};
\foreach \y in {2,...,6} \draw (9,\y) +(.5,.5) node {\dots};
\draw (1.5,2.5) node{$4$};
\draw (2.5,2.5) node{$8$};
\foreach \x in {3,...,8} \draw (\x,2.5) +(.5,0) node {$12$};
\draw (1.5,3.5) node{$4$};
\draw (2.5,3.5) node{$8$};
\foreach \x in {3,...,8} \draw (\x,3.5) +(.5,0) node {$12$};
\draw (1.5,4.5) node{$3$};
\draw (2.5,4.5) node{$6$};
\draw (3.5,4.5) node{$9$};
\foreach \x in {4,...,8} \draw (\x,4.5) +(.5,0) node {$12$};
\draw (1.5,5.5) node{$2$};
\draw (2.5,5.5) node{$4$};
\draw (3.5,5.5) node{$6$};
\draw (4.5,5.5) node{$8$};
\draw (5.5,5.5) node{$10$};
\foreach \x in {6,...,8} \draw (\x,5.5) +(.5,0) node {$11$};
\foreach \x in {1,...,7} \draw (\x,6.5) +(.5,0) node {$\x$};
\draw (8.5,6.5) node {$7$};
\foreach \x in {0,...,8} \draw (\x,7.5) +(1.5,0) node {$\x$};
\foreach \y in {0,...,5} \draw (0.5,6.5-\y) node {$\y$};
\end{tikzpicture}
}
\hspace{1cm}
\subfloat[$\Delta M$]{
\begin{tikzpicture}[x=0.45cm,y=0.45cm,font=\tiny]
\clip(0,0.5) rectangle (11,8);
\draw[style=help lines,xstep=1,ystep=1] (1,1) grid (10,7);
\foreach \x in {1,...,9} \draw (\x,1) +(.5,.5) node {\dots};
\foreach \y in {2,...,6} \draw (9,\y) +(.5,.5) node {\dots};
\foreach \x in {1,...,8} \draw (\x,2) +(.5,.5) node {$0$};
\foreach \y in {2,...,6} \draw (8,\y) +(.5,.5) node {$0$};
\foreach \x in {1,2,3} \draw (\x,3.5) +(.5,0) node{$1$};
\draw (4.5,3.5) node {$-3$};
\foreach \x in {5,...,7} \draw (\x,3.5) +(.5,0) node{$0$};
\foreach \x in {1,2,3,4} \draw (\x,4.5) +(.5,0) node{$1$};
\draw (5.5,4.5) node {$-2$};
\draw (6.6,4.5) node {$-1$};
\draw (7.5,4.5) node {$0$};
\foreach \x in {1,...,5} \draw (\x,5.5) +(.5,0) node{$1$};
\draw (6.5,5.5) node {$0$};
\draw (7.5,5.5) node {$-1$};
\foreach \x in {1,...,7} \draw (\x,6.5) +(.5,0) node{$1$};
\foreach \x in {0,...,8} \draw (\x,7.5) +(1.5,0) node {$\x$};
\foreach \y in {0,...,5} \draw (0.5,6.5-\y) node {$\y$};
\end{tikzpicture}
}
\end{center}
\end{preview}
\end{figure}
It is possible to see that $M$ is $\Delta$-regular and plain. Indeed,
note that $T=\{(1,6),(2,5),(2,4),(3,3)\}$ and that:
\begin{itemize}
\item $c_{16}=-1$, so that $r=0$ and $(a_{16},b_{16})=(0,4)$;
\item $c_{25}=-1$, so that $r=0$ and $(a_{25},b_{25})=(0,1)$;
\item $c_{24}=-2$, so that $r=0,1$, $(a_{24},b_{24})=(0,2)$ and
$(a_{24}+1,b_{24}+1)=(1,3)$;
\item $c_{33}=-3$, so that $r=0,1,2$, $(a_{33},b_{33})=(0,0)$ and
$(a_{33}+1,b_{33}+1)=(1,1)$ and $(a_{33}+2,b_{33}+2)=(2,2)$.
\end{itemize}
Since all these pairs are distinct, $M$ is plain. It is
$\Delta$-regular because:
\begin{itemize}
\item taken $(3,3),(2,5),(1,6)\in T$, we get $a_{33}=a_{25}=a_{16}=0$
and $b_{33}=0<b_{25}=1<b_{16}=4$;
\item taken $(3,3),(2,4),(1,6)\in T$, we get $a_{33}=a_{24}=a_{16}=0$
and $b_{33}=0<b_{24}=2<b_{16}=4$;
\item taken $(3,3),(2,4)\in T$, we get $a_{33}+1=a_{24}+1=1$ and
$b_{33}+1=1<b_{24}+1=3$;
\item taken $(2,5),(3,3)\in T$, we get $b_{25}=b_{33}+1=1$ and
$a_{25}=0<a_{33}+1=1$;
\item taken $(2,4),(3,3)\in T$, we get $b_{24}=b_{33}+2=2$ and
$a_{24}=0<a_{33}+2=2$.
\end{itemize}
\end{Ex}
Recalling Definition \ref{d}, we prove the following:
\begin{Thm} \label{T:4}
Let $M$ be a plain and $\Delta$-regular matrix such that one of the following conditions holds:
\begin{enumerate}
\item $a_{ij}\ge a_{i-1j+1}$ for any $i,j\ge 0$;
\item $b_{ij}\ge b_{i+1j-1}$ for any $i,j\ge 0$.
\end{enumerate}
Then $M^{(i,j)}=M_Z^{(i,j)}$ for any $(i,j)$.
\end{Thm}
\begin{proof}
Let us suppose that $b_{ij}\ge b_{i+1j-1}$ for any $i,j\ge 0$. Under
this hypothesis we have that for any $(i_1,j),(i_2,j)\in T$ with
$i_1>i_2$ and for any $r_1\in I_{i_1j}$ and $r_2\in I_{i_2j}$ it is
$b_{i_2j}+r_2>b_{i_1j}+r_1$. Indeed, it is sufficient to show that
$b_{i_2j}>b_{i_1j}-c_{i_1j}-1=b_{i_1j-1}-1$. By hypothesis and by the
fact that $M$ is admissible we have:
\[
b_{i_2j}\ge b_{i_2+1j-1}\ge b_{i_1j-1}>b_{i_1j-1}-1.
\]
Now we will prove that $\Delta M_Z^{(i,j)}=\Delta M^{(i,j)}$ for any
$(i,j)$. We apply Corollary \ref{C:1} by deleting one by one the points
$(a_{ij}+r,b_{ij}+r)$, that are all distinct since $M$ is
plain. We proceed in the following way: given
$(a_{i_1j_1}+r_1,b_{i_1j_1}+r_1)$ and
$(a_{i_2j_2}+r_2,b_{i_2j_2}+r_2)$, we delete first
$(a_{i_2j_2}+r_2,b_{i_2j_2}+r_2)$ if either
$a_{i_1j_1}+r_1<a_{i_2j_2}+r_2$ or $a_{i_1j_1}+r_1=a_{i_2j_2}+r_2$
and $b_{i_1j_1}+r_1<b_{i_2j_2}+r_2$.
Let us first show that it is possible to compute $M_Z$ by applying
recursively Corollary \ref{C:1}. Given the point $(a_{ij}+r,b_{ij}+r)$,
with $c_{ij}<0$ and $r\in I_{ij}$, by what we have just proved and by
the fact that $M$ is $\Delta$-regular we see that:
\[
\{(h,j)\mid b_{hj}+s=b_{ij}+r,\, h>i,\, s\in I_{hj}\}=\emptyset
\]
and
\begin{equation}
\label{eq:2}
\min\{h\mid (h,k)\in T,\,
b_{hk}+s=b_{ij}+r,\, a_{hk}+s\ge
a_{ij}+r,\, s\in I_{hk}\}=i.
\end{equation}
So, keeping the notation of Corollary \ref{C:1} and \eqref{eq:2} together with Remark
\ref{r:1} and Proposition \ref{P:9} imply:
\begin{equation}
\label{eq:8}
q=\#(X\cap C_{b_{ij}+r})-\#\{(h,k)\in T\mid
b_{hk}+s=b_{ij}+r,\, a_{hk}+s\ge a_{ij}+r,\, s\in I_{hk}\}=i.
\end{equation}
Let:
\begin{align*}
m_{ij}^{(r)}=&\min\{k\mid \exists\, (i,k)\in T,\, k\le j,\,
a_{ik}+s=a_{ij}+r,\,s\in I_{ik}\},\\
n_{ij}^{(r)}=&\max\{k\mid \exists\, (i,k)\in T,\, k\ge j\, ,
a_{ik}+s=a_{ij}+r,\, s\in I_{ik}\},\\
p_{ij}^{(r)}=&\, m_{ij}^{(r)}
+\#\{(i,k)\mid a_{ik}+s=a_{ij}+r,\, k>j,\, s\in I_{ik}\}.
\end{align*}
Note that by Remark \ref{r:1} and by Proposition \ref{P:10}:
\begin{equation} \label{eq:10}
p_{ij}^{(r)}=m_{ij}^{(r)}+n_{ij}^{(r)}-j.
\end{equation}
By by
the fact that $M$ is $\Delta$-regular, by Remark \ref{r:1} and by Propositions \ref{P:10} and \ref{P:3}:
\[
\#(X\cap R_{a_{ij}+r})
-\#\{(h,k)\in T\mid
a_{hk}+s=a_{ij}+r,\, b_{hk}+s\ge b_{ij}+r,\, s\in I_{hk}\}=p_{ij}^{(r)}
\]
that, in the notation of Corollary \ref{C:1} and together with
\eqref{eq:8}, gives that:
\begin{equation}
\label{eq:16}
(q,p)=(i,p_{ij}^{(r)}).
\end{equation}
Suppose that the first point to be deleted is $(a_{i_1j_1}+r_1,b_{i_1j_1}+r_1)$ and let $X'=X\setminus
\{(a_{i_1j_1}+r_1,b_{i_1j_1}+r_1)\}$. Then by the fact that $X$ is ACM
we can apply Corollary \ref{C:2}:
\[
\Delta M_{X'}^{(i,j)}=
\begin{cases}
\Delta M_X^{(i,j)} & \text{for } (i,j)\ne (i_1,p_{i_1j_1}^{(r_1)})\\
\Delta M_X^{(i,j)}-1 & \text{for } (i,j)=(i_1,p_{i_1j_1}^{(r_1)}).
\end{cases}
\]
Iterating the procedure, taken a point
$(a_{i_1j_1}+r_1,b_{i_1j_1}+r_1)$, let us consider:
\begin{multline*}
G=\{(a_{ij}+r,b_{ij}+r)\in \mathcal P\mid
a_{ij}+r>a_{i_1j_1}+r_1\}\cup\\
\cup \{(a_{ij}+r,b_{ij}+r)\in \mathcal P\mid
a_{ij}+r=a_{i_1j_1}+r_1,\, b_{ij}+r>b_{i_1j_1}+r_1\}
\end{multline*}
and the correspondent set:
\[
H=\{(i,p_{ij}^{(r)})\mid (a_{ij}+r,b_{ij}+r)\in G\}.
\]
If $X''=X\setminus G$, suppose that we can apply Corollary \ref{C:1}
to the scheme $X''$ by deleting one by one all the points
$(a_{ij}+r,b_{ij}+r)\in G$. In this way we see that:
\begin{equation}
\label{eq:17}
\Delta M_{X''}^{(i,j)}=
\begin{cases}
\Delta M_X^{(i,j)}& \text{if }(i,j)\notin H\\
\Delta M_X^{(i,j)}-1& \text{if }(i,j)\in H.
\end{cases}
\end{equation}
We will show that we can apply
Corollary \ref{C:1} to scheme $X'''=X''\setminus
\{(a_{i_1j_1}+r_1,b_{i_1j_1}+r_1)\}$.
By \eqref{eq:11}, \eqref{eq:16} and by \eqref{eq:17} we know $\Delta M_{X''}^{(i,j)}<0$ if and only if
$(i,j)\in H$. By \eqref{eq:16} we cannot apply
Corollary \ref{C:1} to $X'''$ if $(i_1+1,p_{i_1j_1}^{(r_1)}+1)\le
(i_2,p_{i_2j_2}^{(r_2)})$ for some $(i_2,p_{i_2j_2}^{(r_2)})\in
H$. Since $m_{ij}^{(r)}\le p_{ij}^{(r)}\le n_{ij}^{(r)}$ for every
$(i,j)\in T$, by Remark \ref{r:1} we have that
$(i_1,p_{i_1j_1}^{(r_1)}),(i_2,p_{i_2j_2}^{(r_2)})\in T$ and that:
\[
a_{i_1p_{i_1j_1}^{(r_1)}}+s_1=a_{i_1j_1}+r_1\text{\quad and \quad} a_{i_2p_{i_2j_2}^{(r_2)}}+s_2=a_{i_2j_2}+r_2,
\]
for some $s_1\in I_{i_1p_{i_1j_1}^{(r_1)}}$ and $s_2\in I_{i_2p_{i_2j_2}^{(r_2)}}$.
This means that:
\[
a_{i_1p_{i_1j_1}^{(r_1)}}+s_1=a_{i_1j_1}+r_1\le a_{i_2j_2}+r_2=a_{i_2p_{i_2j_2}^{(r_2)}}+s_2
\]
where $i_1<i_2$ and $p_{i_1j_1}^{(r_1)}<p_{i_2j_2}^{(r_2)}$, which
contradicts Proposition \ref{P:1}. So we can apply Corollary \ref{C:1} and we
see that:
\[
\Delta M_{X'''}^{(i,j)}=
\begin{cases}
\Delta M_{X''}^{(i,j)} & \text{for } (i,j)\ne (i_1,p_{i_1j_1}^{(r_1)})\\
\Delta M_{X''}^{(i,j)}-1 & \text{for } (i,j)=(i_1,p_{i_1j_1}^{(r_1)}).
\end{cases}
\]
By iterating the procedure we are able to compute $M_Z$.
Now, note that, taken $(i_1,p_{i_1j_1}^{(r_1)})$ and taken $s_1\in
I_{i_1p_{i_1j_1}^{(r_1)}}$ such that
$a_{i_1p_{i_1j_1}^{(r_1)}}+s_1=a_{i_1j_1}+r_1$, it is easy to see that:
\[
m_{i_1p_{i_1j_1}^{(r_1)}}^{(s_1)}=m_{i_1j_1}^{(r_1)}
\text{\quad and \quad}
n_{i_1p_{i_1j_1}^{(r_1)}}^{(s_1)}=n_{i_1j_1}^{(r_1)}.
\]
This implies together with \eqref{eq:10} that
$p_{p_{i_1j_1}^{(r_1)}}^{(s_1)}=m_{i_1p_{i_1j_1}^{(r_1)}}^{(s_1)}+n_{i_1p_{i_1j_1}^{(r_1)}}^{(s_1)}-p_{i_1j_1}^{(r_1)}=j$. This
means that $\Delta
M_Z^{(i,j)}=\Delta M^{(i,j)}$ for any $(i,j)$.
The proof works in a similar way if $a_{ij}\ge a_{i-1j+1}$ for any
$i,j\ge 0$.
\end{proof}
\begin{Cor} \label{C:6}
Let $M$ be an admissible matrix such that:
\[
a_{i_1j_1}-b_{i_1j_1}<a_{i_2j_2}-b_{i_2j_2}
\]
for any $(i_1,j_1)$, $(i_2,j_2)\in T$, with $i_1<i_2$ and
$j_1>j_2$. Suppose that one of the following conditions holds:
\begin{enumerate}
\item $a_{ij}\ge a_{i-1j+1}$ for any $i,j\ge 0$;
\item $b_{ij}\ge b_{i+1j-1}$ for any $i,j\ge 0$.
\end{enumerate}
Then $M^{(i,j)}=M_Z^{(i,j)}$ for any $(i,j)$.
\end{Cor}
\begin{proof}
If $a_{i_1j_1}-b_{i_1j_1}<a_{i_2j_2}-b_{i_2j_2}$
for any $(i_1,j_1)$, $(i_2,j_2)\in T$, with $i_1<i_2$ and
$j_1>j_2$, then $M$ is plain and $\Delta$-regular. Then the statement
follows by Theorem \ref{T:4}.
\end{proof}
\begin{Cor} \label{C:7}
Let $M$ be a plain matrix and let $T=\{(i_1,j_1),\dots,(i_n,j_n)\}$.
If $i_1+j_1=\dots=i_n+j_n$, then $M^{(i,j)}=M_Z^{(i,j)}$ for any $(i,j)$.
\end{Cor}
\begin{proof}
We want to prove that $M$ satisfies the hypothesis of Theorem \ref{T:4}.
By Proposition \ref{P:6}, Proposition \ref{P:8} and by hypothesis for
any $i$ there exists at most one $j\in \mathbb N$ such that $c_{ij}<0$
and, similarly, for any $j$ there exists at most one $i\in \mathbb N$
such that $c_{ij}<0$. Moreover, if $(i,j),(i+k,j-k)\in T$ for some
$i,j,k\in\mathbb N$, then $(i+1,j-1)$, \dots, $(i+k-1,j-k+1)\in T$.
Now we show that $a_{ij}\ge a_{i-1j+1}$ for any $i,j$. If $c_{ij}=1$,
then this is true because $M$ is an admissible matrix and
$a_{ij}=\sum_{k\le i}c_{kj}$. If $c_{ij}=0$, by the fact that $M$ is admissible:
\[
a_{ij}=a_{i-1j}\ge a_{i-1j+1}.
\]
If $c_{ij}<0$, then $c_{ij+1}=0$ and by the fact that $M$ is
admissible:
\[
a_{ij}\ge a_{ij+1}=a_{i-1j+1}+c_{ij+1}=a_{i-1j+1}.
\]
In a similar way it is possible to see that $b_{ij}\ge b_{i+1j-1}$.
Now we need to prove that $M$ is $\Delta$-regular. It is sufficient
to show that for any $(i,j)$, $(i+1,j-1)\in T$:
\[
b_{i+1j-1}-a_{i+1j-1}\le b_{ij}-a_{ij}.
\]
This holds because we have just proved that $b_{i+1j-1}\le b_{ij}$ and
$a_{i+1j-1}\ge a_{ij}$.
\end{proof}
In the following example we give an application of Theorem \ref{T:4}.
\begin{Ex} \label{Ex:1}
Given the following matrix $M$, it is easy to see that it satisfies the hypotheses of Theorem \ref{T:4}
and that its first difference $\Delta M$ is the following:
\begin{figure}[H]
\begin{preview}
\begin{center}
\subfloat[$M$]{
\begin{tikzpicture}[x=0.45cm,y=0.45cm,font=\tiny]
\clip(0,0.5) rectangle (12,13);
\draw[style=help lines,xstep=1,ystep=1] (1,1) grid (11,12);
\foreach \x in {1,...,10} \draw (\x,1) +(.5,.5) node {\dots};
\foreach \y in {2,...,11} \draw (10,\y) +(.5,.5) node {\dots};
\draw (1.5,2.5) node{$9$};
\draw (2.5,2.5) node{$18$};
\draw (3.5,2.5) node{$21$};
\draw (4.5,2.5) node{$23$};
\foreach \x in {5,...,9} \draw (\x,2.5) +(.5,0) node {$24$};
\draw (1.5,3.5) node{$9$};
\draw (2.5,3.5) node{$18$};
\draw (3.5,3.5) node{$21$};
\draw (4.5,3.5) node{$23$};
\foreach \x in {5,...,9} \draw (\x,3.5) +(.5,0) node {$24$};
\draw (1.5,4.5) node{$8$};
\draw (2.5,4.5) node{$16$};
\draw (3.5,4.5) node{$21$};
\draw (4.5,4.5) node{$23$};
\foreach \x in {5,...,9} \draw (\x,4.5) +(.5,0) node {$24$};
\draw (1.5,5.5) node{$7$};
\draw (2.5,5.5) node{$14$};
\draw (3.5,5.5) node{$21$};
\draw (4.5,5.5) node{$23$};
\foreach \x in {5,...,9} \draw (\x,5.5) +(.5,0) node {$24$};
\draw (1.5,6.5) node{$6$};
\draw (2.5,6.5) node{$12$};
\draw (3.5,6.5) node{$18$};
\draw (4.5,6.5) node{$23$};
\foreach \x in {5,...,9} \draw (\x,6.5) +(.5,0) node {$24$};
\draw (1.5,7.5) node{$5$};
\draw (2.5,7.5) node{$10$};
\draw (3.5,7.5) node{$15$};
\draw (4.5,7.5) node{$20$};
\draw (5.5,7.5) node{$23$};
\foreach \x in {6,...,9} \draw (\x,7.5) +(.5,0) node {$24$};
\draw (1.5,8.5) node{$4$};
\draw (2.5,8.5) node{$8$};
\draw (3.5,8.5) node{$12$};
\draw (4.5,8.5) node{$16$};
\draw (5.5,8.5) node{$19$};
\foreach \x in {6,...,9} \draw (\x,8.5) +(.5,0) node {$22$};
\draw (1.5,9.5) node{$3$};
\draw (2.5,9.5) node{$6$};
\draw (3.5,9.5) node{$9$};
\draw (4.5,9.5) node{$12$};
\draw (5.5,9.5) node{$15$};
\draw (6.5,9.5) node{$18$};
\foreach \x in {7,...,9} \draw (\x,9.5) +(.5,0) node {$19$};
\draw (1.5,10.5) node{$2$};
\draw (2.5,10.5) node{$4$};
\draw (3.5,10.5) node{$6$};
\draw (4.5,10.5) node{$8$};
\draw (5.5,10.5) node{$10$};
\draw (6.5,10.5) node{$12$};
\draw (7.5,10.5) node{$13$};
\foreach \x in {8,9} \draw (\x,10.5) +(.5,0) node {$14$};
\draw (1.5,11.5) node{$1$};
\draw (2.5,11.5) node{$2$};
\draw (3.5,11.5) node{$3$};
\draw (4.5,11.5) node{$4$};
\draw (5.5,11.5) node{$5$};
\draw (6.5,11.5) node{$6$};
\draw (7.5,11.5) node{$7$};
\foreach \x in {8,9} \draw (\x,11.5) +(.5,0) node {$8$};
\foreach \x in {0,...,9} \draw (\x,12.5) +(1.5,0) node {$\x$};
\foreach \y in {0,...,10} \draw (0.5,11.5-\y) node {$\y$};
\end{tikzpicture}}
\hspace{1cm}
\subfloat[$\Delta M$]{
\begin{tikzpicture}[x=0.45cm,y=0.45cm,font=\tiny]
\clip(0,0.5) rectangle (12,13);
\draw[style=help lines,xstep=1,ystep=1] (1,1) grid (11,12);
\foreach \x in {1,...,10} \draw (\x,1) +(.5,.5) node {\dots};
\foreach \y in {2,...,11} \draw (10,\y) +(.5,.5) node {\dots};
\foreach \x in {1,...,8} \draw (\x,2) +(.5,.5) node {$0$};
\foreach \y in {2,...,11} \draw (9,\y) +(.5,.5) node {$0$};
\foreach \x in {1,2} \draw (\x,3.5) +(.5,0) node{$1$};
\draw (3.5,3.5) node {$-2$};
\foreach \x in {4,...,8} \draw (\x,3.5) +(.5,0) node{$0$};
\foreach \x in {1,2} \draw (\x,4.5) +(.5,0) node{$1$};
\draw (3.5,4.5) node {$-2$};
\foreach \x in {4,...,8} \draw (\x,4.5) +(.5,0) node{$0$};
\foreach \x in {1,...,3} \draw (\x,5.5) +(.5,0) node{$1$};
\draw (4.5,5.5) node {$-3$};
\foreach \x in {5,...,8} \draw (\x,5.5) +(.5,0) node{$0$};
\foreach \x in {1,...,3} \draw (\x,6.5) +(.5,0) node{$1$};
\draw (4.5,6.5) node {$0$};
\draw (5.5,6.5) node {$-2$};
\draw (6.5,6.5) node {$-1$};
\foreach \x in {7,...,8} \draw (\x,6.5) +(.5,0) node{$0$};
\foreach \x in {1,...,4} \draw (\x,7.5) +(.5,0) node{$1$};
\draw (5.5,7.5) node {$0$};
\draw (6.5,7.5) node {$-2$};
\foreach \x in {7,8} \draw (\x,7.5) +(.5,0) node{$0$};
\foreach \x in {1,...,4} \draw (\x,8.5) +(.5,0) node{$1$};
\draw (5.5,8.5) node {$0$};
\draw (6.5,8.5) node {$0$};
\foreach \x in {7} \draw (\x,8.5) +(.5,0) node{$-1$};
\draw (8.5,8.5) node {$0$};
\foreach \x in {1,...,6} \draw (\x,9.5) +(.5,0) node{$1$};
\draw (7.5,9.5) node {$0$};
\draw (8.5,9.5) node{$-1$};
\foreach \x in {1,...,6} \draw (\x,10.5) +(.5,0) node{$1$};
\draw (7.5,10.5) node {$0$};
\draw (8.5,10.5) node {$0$};
\foreach \x in {1,...,8} \draw (\x,11.5) +(.5,0) node{$1$};
\foreach \x in {0,...,9} \draw (\x,12.5) +(1.5,0) node {$\x$};
\foreach \y in {0,...,10} \draw (0.5,11.5-\y) node {$\y$};
\end{tikzpicture}
}
\end{center}
\end{preview}
\end{figure}
We see that:
\begin{itemize}
\item $c_{27}=-1$, $a_{27}=0$ and $b_{27}=5$ and we get the point
$P_{05}$;
\item $c_{36}=-1$, $a_{36}=0$, and $b_{36}=3$ and we get the point
$P_{03}$;
\item $c_{45}=-2$, $a_{45}=1$ and $b_{45}=2$ and we get the points
$P_{12}$ and $P_{23}$;
\item $c_{55}=-1$, $a_{55}=0$ and $b_{55}=0$ and we get the point
$P_{00}$;
\item $c_{54}=-2$, $a_{54}=1$ and $b_{54}=1$ and we get the points
$P_{11}$ and $P_{22}$;
\item $c_{63}=-3$, $a_{63}=2$ and $b_{63}=0$ and we get the points
$P_{20}, P_{31}, P_{42}$;
\item $c_{72}=-2$, $a_{72}=5$ and $b_{72}=0$ and we get the points
$P_{50}$ and $P_{61}$;
\item $c_{82}=-2$, $a_{82}=3$ and $b_{82}=0$ and we get the points
$P_{30}$ and $P_{41}$.
\end{itemize}
By Theorem \ref{T:4} we have that $M$ is the Hilbert matrix of a
scheme $Z$ whose points can be represented in a grid of $(1,0)$ and
$(0,1)$-lines in the following way:
\begin{figure}[H]
\begin{preview}
\begin{center}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.3cm,y=0.3cm]
\clip(-1,0) rectangle (11,12);
\fill [color=black] (3,1) circle (1.5pt);
\fill [color=black] (4,1) circle (1.5pt);
\fill [color=black] (3,2) circle (1.5pt);
\fill [color=black] (4,2) circle (1.5pt);
\fill [color=black] (3,3) circle (1.5pt);
\fill [color=black] (5,3) circle (1.5pt);
\fill [color=black] (4,4) circle (1.5pt);
\fill [color=black] (5,4) circle (1.5pt);
\fill [color=black] (3,5) circle (1.5pt);
\fill [color=black] (6,5) circle (1.5pt);
\fill [color=black] (5,6) circle (1.5pt);
\fill [color=black] (6,6) circle (1.5pt);
\fill [color=black] (4,7) circle (1.5pt);
\fill [color=black] (7,7) circle (1.5pt);
\fill [color=black] (8,7) circle (1.5pt);
\fill [color=black] (3,8) circle (1.5pt);
\fill [color=black] (6,8) circle (1.5pt);
\fill [color=black] (7,8) circle (1.5pt);
\fill [color=black] (8,8) circle (1.5pt);
\fill [color=black] (4,9) circle (1.5pt);
\fill [color=black] (5,9) circle (1.5pt);
\fill [color=black] (7,9) circle (1.5pt);
\fill [color=black] (9,9) circle (1.5pt);
\fill [color=black] (10,9) circle (1.5pt);
\draw (3,0.5) -- (3,9.5);
\draw[color=black] (3,10.5) node {\tiny $C_0$};
\draw (4,0.5) -- (4,9.5);
\draw[color=black] (4,10.5) node {\tiny $C_1$};
\draw (5,2.5) -- (5,9.5);
\draw[color=black] (5,10.5) node {\tiny $C_2$};
\draw (6,4.5) -- (6,9.5);
\draw[color=black] (6,10.5) node {\tiny $C_3$};
\draw (7,6.5) -- (7,9.5);
\draw[color=black] (7,10.5) node {\tiny $C_4$};
\draw (8,6.5) -- (8,9.5);
\draw[color=black] (8,10.5) node {\tiny $C_5$};
\draw (9,8.5) -- (9,9.5);
\draw[color=black] (9,10.5) node {\tiny $C_6$};
\draw (10,8.5) -- (10,9.5);
\draw[color=black] (10,10.5) node {\tiny $C_7$};
\draw (2.5,1) -- (4.5,1);
\draw[color=black] (1.5,1) node {\tiny $R_8$};
\draw (2.5,2) -- (4.5,2);
\draw[color=black] (1.5,2) node {\tiny $R_7$};
\draw (2.5,3) -- (5.5,3);
\draw[color=black] (1.5,3) node {\tiny $R_6$};
\draw (2.5,4) -- (5.5,4);
\draw[color=black] (1.5,4) node {\tiny $R_5$};
\draw (2.5,5) -- (6.5,5);
\draw[color=black] (1.5,5) node {\tiny $R_4$};
\draw (2.5,6) -- (6.5,6);
\draw[color=black] (1.5,6) node {\tiny $R_3$};
\draw (2.5,7) -- (8.5,7);
\draw[color=black] (1.5,7) node {\tiny $R_2$};
\draw (2.5,8) -- (8.5,8);
\draw[color=black] (1.5,8) node {\tiny $R_1$};
\draw (2.5,9) -- (10.5,9);
\draw[color=black] (1.5,9) node {\tiny $R_0$};
\end{tikzpicture}
\caption{The scheme $Z$}
\end{center}
\end{preview}
\end{figure}
\end{Ex}
\begin{Ex} \label{Ex:0}
In this example we make some remarks on the hypotheses of Theorem \ref{T:4}.
\begin{enumerate}
\item Let $M$ be a plain matrix such that either condition 1 or condition 2
of Theorem \ref{T:4} holds and suppose that it is not
$\Delta$-regular. Then it might be
$M_Z\ne M$. As an example let us consider a scheme $Y$ whose points can be represented in a grid of $(1,0)$ and $(0,1)$-lines in the following way and the associated Hilbert matrix $M=M_Y$ which satisfies the previous conditions:
\begin{figure}[H]
\begin{preview}
\begin{center}
\subfloat[$Y$]{
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.3cm,y=0.3cm]
\clip(0,0) rectangle (6,6);
\fill [color=black] (1,1) circle (1.5pt);
\fill [color=black] (2,1) circle (1.5pt);
\fill [color=black] (1,2) circle (1.5pt);
\fill [color=black] (2,2) circle (1.5pt);
\fill [color=black] (3,3) circle (1.5pt);
\fill [color=black] (4,3) circle (1.5pt);
\fill [color=black] (5,3) circle (1.5pt);
\fill [color=black] (3,4) circle (1.5pt);
\fill [color=black] (4,4) circle (1.5pt);
\fill [color=black] (5,4) circle (1.5pt);
\fill [color=black] (2,5) circle (1.5pt);
\fill [color=black] (3,5) circle (1.5pt);
\fill [color=black] (4,5) circle (1.5pt);
\fill [color=black] (5,5) circle (1.5pt);
\draw (1,0.5) -- (1,5.5);
\draw (2,0.5) -- (2,5.5);
\draw (3,2.5) -- (3,5.5);
\draw (4,2.5) -- (4,5.5);
\draw (5,2.5) -- (5,5.5);
\draw (0.5,1) -- (2.5,1);
\draw (0.5,2) -- (2.5,2);
\draw (0.5,3) -- (5.5,3);
\draw (0.5,4) -- (5.5,4);
\draw (0.5,5) -- (5.5,5);
\end{tikzpicture}
}
\hspace{1cm}
\subfloat[$M=M_Y$]{
\begin{tikzpicture}[x=0.45cm,y=0.45cm,font=\tiny]
\clip(0,0) rectangle (9,9);
\draw[style=help lines,xstep=1,ystep=1] (1,1) grid (8,8);
\foreach \x in {1,...,7} \draw (\x,1) +(.5,.5) node {\dots};
\foreach \y in {2,...,7} \draw (7,\y) +(.5,.5) node {\dots};
\draw (1.5,2.5) node{$5$};
\draw (2.5,2.5) node{$10$};
\draw (3.5,2.5) node{$13$};
\foreach \x in {4,...,6} \draw (\x,2) +(.5,.5) node {$14$};
\draw (1.5,3.5) node{$5$};
\draw (2.5,3.5) node{$10$};
\draw (3.5,3.5) node{$13$};
\foreach \x in {4,...,6} \draw (\x,3) +(.5,.5) node {$14$};
\draw (1.5,4.5) node{$4$};
\draw (2.5,4.5) node{$8$};
\draw (3.5,4.5) node{$11$};
\draw (4.5,4.5) node{$13$};
\foreach \x in {5,6} \draw (\x,4) +(.5,.5) node {$14$};
\draw (1.5,5.5) node{$3$};
\draw (2.5,5.5) node{$6$};
\draw (3.5,5.5) node{$9$};
\draw (4.5,5.5) node{$12$};
\foreach \x in {5,6} \draw (\x,5) +(.5,.5) node {$14$};
\draw (1.5,6.5) node{$2$};
\draw (2.5,6.5) node{$4$};
\draw (3.5,6.5) node{$6$};
\draw (4.5,6.5) node{$8$};
\foreach \x in {5,6} \draw (\x,6) +(.5,.5) node {$10$};
\draw (1.5,7.5) node{$1$};
\draw (2.5,7.5) node{$2$};
\draw (3.5,7.5) node{$3$};
\draw (4.5,7.5) node{$4$};
\foreach \x in {5,6} \draw (\x,7) +(.5,.5) node {$5$};
\foreach \x in {0,...,6} \draw (\x,8.5) +(1.5,0) node {$\x$};
\foreach \y in {0,...,6} \draw (0.5,7.5-\y) node {$\y$};
\end{tikzpicture}
}
\end{center}
\end{preview}
\end{figure}
Indeed, we can take, as in the definition of $\Delta$-regular matrix,
$(i_1,j_1)=(4,3)$ and $i_2,j_2
=(3,4)$. So $c_{43}=c_{34}=-1$, $a_{43}=a_{34}=1$, while $b_{43}=1$
and $b_{34}=0$. This means that $b_{43}-b_{34}=1>0=a_{43}-a_{34}$.
Then, by adding the points on the $(1,0)$-lines and using
\cite[Theorem 3.1]{BM}, it is possible to see that $M_Z\ne M$:
\begin{figure}[H]
\begin{preview}
\begin{center}
\subfloat[$Z$]{
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.3cm,y=0.3cm]
\clip(0,0) rectangle (6,6);
\fill [color=black] (1,1) circle (1.5pt);
\fill [color=black] (2,1) circle (1.5pt);
\fill [color=black] (1,2) circle (1.5pt);
\fill [color=black] (2,2) circle (1.5pt);
\fill [color=black] (1,3) circle (1.5pt);
\fill [color=black] (3,3) circle (1.5pt);
\fill [color=black] (4,3) circle (1.5pt);
\fill [color=black] (3,4) circle (1.5pt);
\fill [color=black] (4,4) circle (1.5pt);
\fill [color=black] (5,4) circle (1.5pt);
\fill [color=black] (2,5) circle (1.5pt);
\fill [color=black] (3,5) circle (1.5pt);
\fill [color=black] (4,5) circle (1.5pt);
\fill [color=black] (5,5) circle (1.5pt);
\draw (1,0.5) -- (1,5.5);
\draw (2,0.5) -- (2,5.5);
\draw (3,2.5) -- (3,5.5);
\draw (4,2.5) -- (4,5.5);
\draw (5,3.5) -- (5,5.5);
\draw (0.5,1) -- (2.5,1);
\draw (0.5,2) -- (2.5,2);
\draw (0.5,3) -- (4.5,3);
\draw (0.5,4) -- (5.5,4);
\draw (0.5,5) -- (5.5,5);
\end{tikzpicture}
}
\hspace{1cm}
\subfloat[$M_Z$]{
\begin{tikzpicture}[x=0.45cm,y=0.45cm,font=\tiny]
\clip(0,0) rectangle (9,9);
\draw[style=help lines,xstep=1,ystep=1] (1,1) grid (8,8);
\foreach \x in {1,...,7} \draw (\x,1) +(.5,.5) node {\dots};
\foreach \y in {2,...,7} \draw (7,\y) +(.5,.5) node {\dots};
\draw (1.5,2.5) node{$5$};
\draw (2.5,2.5) node{$10$};
\draw (3.5,2.5) node{$13$};
\foreach \x in {4,...,6} \draw (\x,2) +(.5,.5) node {$14$};
\draw (1.5,3.5) node{$5$};
\draw (2.5,3.5) node{$10$};
\draw (3.5,3.5) node{$13$};
\foreach \x in {4,...,6} \draw (\x,3) +(.5,.5) node {$14$};
\draw (1.5,4.5) node{$4$};
\draw (2.5,4.5) node{$8$};
\draw (3.5,4.5) node{$11$};
\draw (4.5,4.5) node{$14$};
\foreach \x in {5,6} \draw (\x,4) +(.5,.5) node {$14$};
\draw (1.5,5.5) node{$3$};
\draw (2.5,5.5) node{$6$};
\draw (3.5,5.5) node{$9$};
\draw (4.5,5.5) node{$12$};
\foreach \x in {5,6} \draw (\x,5) +(.5,.5) node {$14$};
\draw (1.5,6.5) node{$2$};
\draw (2.5,6.5) node{$4$};
\draw (3.5,6.5) node{$6$};
\draw (4.5,6.5) node{$8$};
\foreach \x in {5,6} \draw (\x,6) +(.5,.5) node {$10$};
\draw (1.5,7.5) node{$1$};
\draw (2.5,7.5) node{$2$};
\draw (3.5,7.5) node{$3$};
\draw (4.5,7.5) node{$4$};
\foreach \x in {5,6} \draw (\x,7) +(.5,.5) node {$5$};
\foreach \x in {0,...,6} \draw (\x,8.5) +(1.5,0) node {$\x$};
\foreach \y in {0,...,6} \draw (0.5,7.5-\y) node {$\y$};
\end{tikzpicture}
}
\end{center}
\end{preview}
\end{figure}
\item It is easy to see that the Hilbert matrix of $3$ generic points of $\mathbb P^1\times
\mathbb P^1$ is such that $a_{ij}\ge a_{i-1j+1}$ and $b_{ij}\ge
b_{i+1j-1}$ for any $i,j\ge 0$ and that it is $\Delta$-regular, but it
is not plain:
\begin{figure}[H]
\begin{preview}
\begin{center}
\begin{tikzpicture}[x=0.45cm,y=0.45cm,font=\tiny]
\clip(0,1) rectangle (5,6);
\draw[style=help lines,xstep=1,ystep=1] (1,1) grid (5,5);
\foreach \x in {1,...,4} \draw (\x,1) +(.5,.5) node {\dots};
\foreach \y in {2,...,4} \draw (4,\y) +(.5,.5) node {\dots};
\draw (1.5,2.5) node{$3$};
\draw (2.5,2.5) node{$3$};
\draw (3.5,2.5) node{$3$};
\draw (1.5,3.5) node{$2$};
\draw (2.5,3.5) node{$3$};
\draw (3.5,3.5) node{$3$};
\draw (1.5,4.5) node{$1$};
\draw (2.5,4.5) node{$2$};
\draw (3.5,4.5) node{$3$};
\foreach \x in {0,...,3} \draw (\x,5.5) +(1.5,0) node {$\x$};
\foreach \y in {0,...,3} \draw (0.5,4.5-\y) node {$\y$};
\end{tikzpicture}
\end{center}
\end{preview}
\end{figure}
Indeed, $a_{12}=a_{21}=0$ and $b_{12}=b_{21}=0$. In this case it is
clear that $M_Z\ne M$, because $\deg Z=4\ne 3$.
\item Let $X\subset \mathbb P^1\times \mathbb P^1$ be a reduced
zero-dimensional scheme whose points can be represented on a grid
of $(1,0)$ and $(0,1)$-lines in the following way:
\begin{figure}[H]
\begin{center}
\begin{preview}
\subfloat[$X$]{
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.3cm,y=0.3cm]
\clip(0,0) rectangle (5,5);
\fill [color=black] (1,1) circle (1.5pt);
\fill [color=black] (2,1) circle (1.5pt);
\fill [color=black] (1,2) circle (1.5pt);
\fill [color=black] (2,2) circle (1.5pt);
\fill [color=black] (3,3) circle (1.5pt);
\fill [color=black] (4,3) circle (1.5pt);
\fill [color=black] (3,4) circle (1.5pt);
\fill [color=black] (4,4) circle (1.5pt);
\draw (1,0.5) -- (1,4.5);
\draw (2,0.5) -- (2,4.5);
\draw (3,2.5) -- (3,4.5);
\draw (4,2.5) -- (4,4.5);
\draw (0.5,1) -- (2.5,1);
\draw (0.5,2) -- (2.5,2);
\draw (0.5,3) -- (4.5,3);
\draw (0.5,4) -- (4.5,4);
\end{tikzpicture}
}
\hspace{1cm}
\subfloat[$M_X$]{
\begin{tikzpicture}[x=0.45cm,y=0.45cm,font=\tiny]
\clip(0,0) rectangle (8,8);
\draw[style=help lines,xstep=1,ystep=1] (1,1) grid (7,7);
\foreach \x in {1,...,6} \draw (\x,1) +(.5,.5) node {\dots};
\foreach \y in {2,...,6} \draw (6,\y) +(.5,.5) node {\dots};
\draw (1.5,2.5) node{$4$};
\foreach \x in {2,...,5} \draw (\x,2) +(.5,.5) node {$8$};
\draw (1.5,3.5) node{$4$};
\foreach \x in {2,...,5} \draw (\x,3) +(.5,.5) node {$8$};
\draw (1.5,4.5) node{$3$};
\draw (2.5,4.5) node{$6$};
\draw (3.5,4.5) node{$7$};
\draw (4.5,4.5) node{$8$};
\draw (5.5,4.5) node{$8$};
\draw (1.5,5.5) node{$2$};
\draw (2.5,5.5) node{$4$};
\draw (3.5,5.5) node{$6$};
\draw (4.5,5.5) node{$8$};
\draw (5.5,5.5) node{$8$};
\draw (1.5,6.5) node{$1$};
\draw (2.5,6.5) node{$2$};
\draw (3.5,6.5) node{$3$};
\draw (4.5,6.5) node{$4$};
\draw (5.5,6.5) node{$4$};
\foreach \x in {0,...,5} \draw (\x,7.5) +(1.5,0) node {$\x$};
\foreach \y in {0,...,5} \draw (0.5,6.5-\y) node {$\y$};
\end{tikzpicture}
}
\end{preview}
\end{center}
\end{figure}
Then it is easy to see that the Hilbert matrix $M_X$ of $X$ is plain
and $\Delta$-regular, but it does not satisfies either condition 1
or condition 2 of Theorem \ref{T:4}. Indeed, $a_{22}=1<2=a_{13}$ and
$b_{22}=1<2=b_{31}$. However, in this case $Z=X$.
\end{enumerate}
\end{Ex}
\begin{op}
Given an admissible matrix $M$, which is plain and $\Delta$-regular,
but which does not satisfy either condition 1 or condition 2, is $M$
the Hilbert function of some zero-dimensional schemes? In particular, given the associated scheme $Z$, $M_Z=M$?
\end{op}
|
{
"timestamp": "2011-09-07T02:01:21",
"yymm": "1009",
"arxiv_id": "1009.4059",
"language": "en",
"url": "https://arxiv.org/abs/1009.4059"
}
|
\section{Introduction}
\label{sec:introduction}
\input{introduction}
\section{Scenario}
\label{sec:scenario}
\input{scenario}
\section{Approach}
\label{sec:approach}
\input{approach}
\section{Case Study}
\label{sec:casestudy}
\input{casestudy}
\section{Discussion}
\label{sec:discussion}
\input{discussion}
\section{Related Work}
\label{sec:related}
\input{relatedwork}
\section{Conclusion and Future Work}
\label{sec:conclusion}
\input{conclusion}
\bibliographystyle{EPTCS/eptcs}
|
{
"timestamp": "2010-09-21T02:02:19",
"yymm": "1009",
"arxiv_id": "1009.3714",
"language": "en",
"url": "https://arxiv.org/abs/1009.3714"
}
|
\section*{Introduction}
\label{sec:introduction}
Modern lattice QCD simulations are mostly based on direct evaluation of the path integral of the theory. Such approach, while being very general and efficient for many applications, suffers from a number of problems, most notable of which are the sign problem at finite chemical potential, critical slowing down at small quark masses and large finite-volume effects as well as small signal-to-noise ratio in the analysis of excited states. These problems are inherent to standard Monte-Carlo simulations and cannot be efficiently solved by simply increasing the computation power, since the required computing time quickly increases (in the worst cases, exponentially) with the required precision. Such situation makes it tempting to devise alternative simulation algorithms for non-Abelian lattice gauge theories.
One of the efficient alternative numerical methods is the so-called Diagrammatic Monte-Carlo, a method based on stochastic summation of all the terms in the strong- or weak-coupling expansion of the observable of interest \cite{Prokofev:98:1, Wolff:09:1}. Such a method in some cases allows one to reduce or avoid completely the sign problem in the original path integral, and does not suffer from finite-volume effects. Furthermore, one can construct algorithms which yield particular correlation functions in terms of probability distributions of some random variables, which greatly facilitates the analysis of excited states \cite{Prokofev:98:1, Wolff:09:1}. This is the idea of the ``worm'' algorithm by Prokof'ev and Svistunov \cite{Prokofev:98:1}, in which the probability distribution of the positions $x$, $y$ of ``head'' and the ``tail'' of the worm yields the two-point Green function $G\lr{x, y}$. Diagrammatic Monte-Carlo and the ``worm'' algorithm have been successfully applied to a number of statistical models with discrete symmetry groups such as the Ising model, the XY model and unitary Fermi gas and showed practically no critical slowing down near quantum phase transitions.
However, application of such methods to lattice field theories with continuous field variables (such as two-dimensional $O\lr{N}$ and $CP\lr{N}$ sigma-models, Abelian gauge theories and the $\phi^4$ theory) resulted so far in quite complicated and model-dependent algorithms \cite{Wolff:09:1}. A generalization of such algorithms to $SU\lr{N}$ sigma-models or to non-Abelian gauge theories is still not found. These algorithms are in essence based on the strong-coupling expansion, and while their applicability is not limited by the strong-coupling regime, one can expect that algorithms based on the weak-coupling expansion might show better performance near the continuum limit.
Typically, the weak-coupling expansion in such lattice theories is either quite complicated or non-convergent. Up to now, divergent behavior of the weak-coupling perturbative expansions strongly limits the applicability of Diagrammatic Monte-Carlo to field theories with continuous field variables. In a recent paper \cite{Prokofev:10:1} a method was proposed to construct convergent series which approximate the non-analytic path integrals with desired precision. This method, however, is difficult to generalize to physically interesting field theories such as non-Abelian lattice gauge theories.
Another way to obtain convergent series while preserving important physical properties of the theory is to sum over diagrams with certain topology only. This corresponds to the large-$N$ limit in quantum field theories and matrix models, that is, the limit of infinite dimensionality of an internal symmetry group, such as $O\lr{N}$ or $SU\lr{N}$. For such theories, each Feynman diagram acquires a factor $N^{\chi}$, where $\chi$ is the Euler character of this diagram \cite{tHooft:75:1}. In the limit $N \rightarrow \infty$, the contribution of planar diagrams with $\chi = 2$ dominates, and the sum over all planar diagrams typically has a finite convergence radius \cite{Koplik:77:1, Brezin:78:1}.
In this paper we describe a stochastic method for summing over all planar diagrams in large-$N$ quantum field theories. The method is based on stochastic solution of Schwinger-Dyson equations, so that the correlators of field variables are obtained as stationary probability distributions of certain random variables. In this way we implement the idea of importance sampling, so that numerically small observables correspond to unlikely events. These probability distributions are sampled by the so-called nonlinear random processes. In contrast to conventional Markov chains, stationary probability distributions of such random processes satisfy nonlinear equations, and hence they can be called ``nonlinear random processes'' or ``nonlinear Markov chains'' in the terminology of \cite{Etessami:05:1, Frank:04:1}. Factorization of single-trace operators in the large-$N$ limit of quantum field theories corresponds to the phenomena of ``chaos propagation'' in random processes \cite{KacProbability}.
While in the diagrammatic Monte-Carlo and in the ``worm'' algorithm the diagrams are stored in computer memory as a whole and are updated in such a way that the detailed balance condition is satisfied at each step, the method described in this paper works only with external lines. In contrast to the standard Metropolis algorithm, one should not know explicitly the weight of each diagram, and the transition probabilities do not satisfy any detailed balance condition. Unlike the quite popular ``numerical functional methods'' in continuum gauge theories (see \cite{Pawlowski:07:1} for a review), the proposed method does not require any truncation of the hierarchy of Schwinger-Dyson equations, and work only with singlet operators w.r.t. the internal symmetry group. Another distinct feature is that the computational complexity of the method does not depend on $N$, while the standard Monte-Carlo, the functional methods and the ``worm'' algorithm all require infinite computational resources in the limit $N \rightarrow \infty$. This feature might be advantageous for numerical checks of the predictions of the holographic models which are dual to large-$N$ quantum field theories \cite{Polyakov:99:1}.
In Section \ref{sec:SDeq_general} we analyze the general structure of Schwinger-Dyson equations in large-$N$ quantum field theories on the example of a scalar matrix-valued field theory. When large-$N$ factorization is taken into account, Schwinger-Dyson equations become nonlinear equations with infinitely many unknowns. In Section \ref{sec:recursive_process} we describe nonlinear random processes of recursive type \cite{Etessami:05:1} which can be used to stochastically solve such equations. In Section \ref{sec:SDs_stochastic_solution} we apply such random processes to solve Schwinger-Dyson equations in several large-$N$ theories. In Subsection \ref{subsec:phi4_general} we consider the scalar matrix-valued field theory, for which the perturbative expansion yields the conventional Feynman diagrams in momentum space. In Subsection \ref{subsec:matrix_model} this solution is compared with the exact solution of the simplest quantum field theory in zero dimensions, that is, the Hermitian matrix model \cite{Brezin:78:1}. The convergence of such solution and the strength of the sign problem is discussed. In Subsection \ref{subsec:weingarten} we consider the Weingarten model \cite{Weingarten:80:1, Eguchi:82:3} and demonstrate how the proposed method can be used to simulate random surfaces on the hypercubic lattice. In this case, our method reproduces an ensemble of open, rather than closed, random surfaces, with critical behavior which is quite different from those of the closed planar random surfaces. Since the structure of Schwinger-Dyson equations in the Weingarten model is similar to the loop equations in large-$N$ non-Abelian lattice gauge theories \cite{Migdal:81:1}, studying this model might be helpful for further extensions of the present approach to non-Abelian gauge theories.
While the method described in Section \ref{sec:recursive_process} works well for non-compact field variables, for field theories with compact field variables, such as nonlinear sigma-models or non-Abelian lattice gauge theories, a straightforward stochastic interpretation of Schwinger-Dyson equations is only possible in the strong coupling limit. In the weak-coupling limit one expects the field correlators to contain both the perturbative part in the coupling constant $g$ as well as nonperturbative corrections of the form $\expa{-c/g^2}$ with some constant $c$. Moreover, perturbative expansion in powers of $g$ typically results in asymptotic series, and nonperturbative corrections appear as a result of resummation of such series \cite{Parisi:77:1}.
In Section \ref{sec:rps_with_mem} we show how such nonperturbative corrections can be taken into account by a further relaxation of the Markov property of the random process. The basic idea is to absorb the divergent part of the series into a self-consistent redefinition of the expansion parameter. These redefined parameters play the role of nonperturbative ``condensates'' \cite{Parisi:77:1, Kazakov:94:1}. It turns out that the redefined expansion parameters can be estimated with increasing precision from the previous history of the random process which solves the Schwinger-Dyson equations, thus leading to the emergence of the ``memory'' of the random process. The approach of the redefined parameters to their self-consistent values is reminiscent somehow of the renormalization-group flow \cite{Pawlowski:07:1}. Such dependence on the previous history makes the random process essentially non-Markovian, so that the stationary probability distribution also satisfies some nonlinear equation.
We illustrate this idea on the example of $O\lr{N}$ sigma-model in two dimensions, which is equivalent to a bosonic random walk with a self-consistent mass. Random process which simulates this model has the ``memory'' but no ``recursive'' structure. Presumably, in order to sum up both perturbative and non-perturbative corrections which arise in non-Abelian lattice gauge theories or $U\lr{N}$ sigma-models, one should devise the ``recursive'' nonlinear random process (which would sum up perturbative corrections) with memory (which would generate nonperturbative quantities in a renormalization-group-like way).
Finally, in the concluding Section we summarize the present work and discuss its extension to non-Abelian lattice gauge theories in the limit of large $N$.
\section{General structure of Schwinger-Dyson equations for large-$N$ quantum field theories}
\label{sec:SDeq_general}
In order to analyze the general structure of Schwinger-Dyson equations for large-$N$ quantum field theories, let us first consider the theory of a hermitian $N\times N$ matrix-valued field $\phi\lr{x}$ with the following Lagrangian:
\begin{eqnarray}
\label{phi4_field_action}
\mathcal{L}\lrs{\phi\lr{x}} = N \tr \phi\lr{x} \, \lr{m^2 - \Delta} \, \phi\lr{x} + \frac{N \lambda}{4} \tr \phi^{4}\lr{x} .
\end{eqnarray}
This theory is most convenient to illustrate the method described in this paper, since its perturbative expansion leads to conventional Feynman diagrams in the momentum space. Since this theory should be somehow regularized, let us assume from the very beginning that the action (\ref{phi4_field_action}) is defined on the Euclidean hypercubic $D$-dimensional lattice with total volume $V$ in lattice units. Thus, the coordinates $x$ take integer values and $\Delta$ is the lattice Laplacian (for definiteness, with periodic boundary conditions).
Schwinger-Dyson equations for a theory with the action (\ref{phi4_field_action}) read \cite{Arefeva:81:1}:
\begin{widetext}
\begin{eqnarray}
\label{phi4_SDs_original_n2}
\lr{m^2 - \Delta_1} \, G\lr{x_1, x_2} = \delta\lr{x_1, x_2} + \lambda \, G\lr{x_1, x_1, x_1, x_2} , \\
\label{phi4_SDs_original}
\lr{m^2 - \Delta_1} \, G\lr{x_1, \ldots, x_n} =
\delta\lr{x_1, x_2} \, G\lr{x_3, \ldots, x_n}
+ \nonumber \\ +
\delta\lr{x_1, x_n} \, G\lr{x_2, \ldots, x_{n-1}}
+
\sum \limits_{m=3}^{n-1} \delta\lr{x_1, x_m} \, G\lr{x_2, \ldots, x_{m-1}} \, G\lr{x_{m+1}, \ldots, x_n}
+ \nonumber \\ +
\lambda \, G\lr{x_1, x_1, x_1, x_2, \ldots, x_n} ,
\quad \quad n > 2,
\end{eqnarray}
where the single-trace correlators are $G\lr{x_1, \ldots, x_n} = \vev{ \frac{1}{N} \, \tr\lr{ \phi\lr{x_1} \ldots \phi\lr{x_n} } }$, $\Delta_1$ is the Laplacian acting on $x_1$ and we have already taken into account the factorization property in the limit $N \rightarrow \infty$ \cite{tHooft:75:1}:
\begin{eqnarray}
\label{largeNfactorization}
\vev{\frac{1}{N} \tr\lr{\phi\lr{x_1} \ldots \phi\lr{x_n} } \, \frac{1}{N} \tr\lr{\phi\lr{y_1} \ldots \phi\lr{y_m} } }
= \nonumber \\ =
\vev{\frac{1}{N} \tr\lr{\phi\lr{x_1} \ldots \phi\lr{x_n} }} \, \vev{\frac{1}{N} \tr\lr{\phi\lr{y_1} \ldots \phi\lr{y_m} } } + O\lr{\frac{1}{N^2}} .
\end{eqnarray}
These equations hold for any argument of the correlators, but the resulting system is redundant, and it is sufficient to consider only those Schwinger-Dyson equations which were obtained by the variation of the fields at $x_1$.
It is convenient now to go to the momentum representation, introducing the Green functions in momentum space $G\lr{k_1, \ldots, k_n} = \sum \limits_{x_1} \ldots \sum \limits_{x_n} \, \expa{i \sum \limits_{m} k_m \cdot x_m} \, G\lr{x_1, \ldots, x_n}$. In order to keep all expressions as symmetric as possible, we do not separate the factor $\delta\lr{\sum \limits_{m} k_m}$ in $G\lr{k_1, \ldots, k_n}$ explicitly. This condition will be automatically satisfied by the nonlinear random process which we describe in Subsection \ref{subsec:phi4_general}. The equations (\ref{phi4_SDs_original_n2}), (\ref{phi4_SDs_original}) in the momentum representation are:
\begin{eqnarray}
\label{phi4_SDs_momentum_n2}
G\lr{k_1, k_2} = G_0\lr{k_1} \, V \, \delta\lr{k_1 + k_2}
+ \nonumber \\ +
G_0\lr{k_1} \, \frac{\lambda}{V^2} \, \sum \limits_{q_1, q_2, q_3} \delta\lr{k_1 - q_1 - q_2 - q_3} \, G\lr{q_1, q_2, q_3, k_2} , \\
\label{phi4_SDs_momentum}
G\lr{k_1, \ldots, k_n}
=
G_0\lr{k_1} \, \sum \limits_{m=3}^{n-1} \delta\lr{k_1 + k_m} \, V \, G\lr{k_2, \ldots, k_{m-1}} \, G\lr{k_{m+1}, \ldots, k_n}
+ \nonumber\\ +
G_0\lr{k_1} \, \delta\lr{k_1 + k_2}\, V\, G\lr{k_3, \ldots, k_n}
+
G_0\lr{k_1} \, \delta\lr{k_1 + k_n}\, V\, G\lr{k_2, \ldots, k_{n-1}}
+ \nonumber\\ +
G_0\lr{k_1} \, \frac{\lambda}{V^2} \, \sum \limits_{q_1, q_2, q_3} \delta\lr{k_1 - q_1 - q_2 - q_3} \, G\lr{q_1, q_2, q_3, k_2, \ldots, k_n} ,
\end{eqnarray}
where $G_0\lr{k} = \lr{m^2 + \sum \limits_{\mu} 4 \sin^2\lr{k_{\mu}/2}}^{-1}$ is the free scalar propagator on the hypercubic lattice. All momenta are assumed to lie in the first Brillouin zone $-\pi \le k_{\mu} \le \pi$ and are added modulo $2 \pi$. The structure of these equations is schematically illustrated on Fig. \ref{fig:phi4_SDs_general}, where dashed blobs denote the Green functions $G\lr{k_1, \ldots, k_n}$ and empty blobs denote $G_0\lr{k}$.
\end{widetext}
\begin{figure}
\includegraphics[width=6cm]{phi4_SDs_general.eps}\\
\caption{Schematic illustration of the structure of the Schwinger-Dyson equations (\ref{phi4_SDs_momentum}). Dashed blobs denote the Green functions $G\lr{k_1, \ldots, k_n}$ and empty blobs denote the free propagator $G_0\lr{k}$.}
\label{fig:phi4_SDs_general}
\end{figure}
Thus we have obtained an infinite system of quadratic functional equations for the set of functions $G\lr{k_1, \ldots, k_n}$ with $n = 2, 4, \ldots$. Such structure is common for large-$N$ quantum field theories: Schwinger-Dyson equations are quadratic equations for infinite set of unknown variables. In the case of scalar matrix field theory considered here, the unknown variables are the functions of the sequences of momenta $\lrc{k_1, \ldots, k_n}$ for any even $n \ge 2$. In the case of lattice gauge theories or string theories Schwinger-Dyson equations are most naturally formulated in terms of the Wilson loops, which are the functions defined on the discrete space of closed loops on the lattice \cite{Migdal:81:1, Weingarten:80:1, Eguchi:82:3}. In this case, the equations are also quadratic w.r.t. the Wilson loops. For $O\lr{N}$ sigma-model, Schwinger-Dyson equations are also quadratic equations which involve only the two-point function (see Section \ref{sec:rps_with_mem}).
Typically, systems of equations with infinitely many unknowns can be efficiently solved by stochastic methods. It is advantageous to estimate the value of each unknown variable as a probability of observing some state of a random process. In this case the unknowns with numerically small values correspond to unlikely events, and the set of infinitely many unknown variables is automatically truncated to a set of unknowns with sufficiently large values. Such methods are well-known mainly in the context of kinetic equations \cite{KacProbability}. Recently they were also discussed in the context of probabilistic programming \cite{Etessami:05:1}. In the next Section we describe a discrete-time, discrete-space method of such type which is in our opinion most suitable for solving the Schwinger-Dyson equations in the large-$N$ limit.
\section{Stochastic solution of nonlinear equations by random processes of recursive type}
\label{sec:recursive_process}
We consider nonlinear equations of the following form:
\begin{eqnarray}
\label{random_process_eq}
w\lr{x} = p_{c}\lr{x} + \sum \limits_{y} p_{e}\lr{x | y} w\lr{y}
+ \nonumber \\ +
\sum \limits_{y_1, y_2} p_{j}\lr{x | y_1, y_2} w\lr{y_1} w\lr{y_2} ,
\end{eqnarray}
where $x$, $y$, $y_1$, $y_2$ are the elements of some space $X$ and $\sum \limits_{x}$ with $x \in X$ denotes summation or integration over all the elements of this space. We also assume that the functions $p_{c}\lr{x}$, $p_{e}\lr{x|y}$ and $p_{j}\lr{x | y_1, y_2}$ satisfy the inequalities
\begin{eqnarray}
\label{probability_ineq}
\sum \limits_{x} |p_{c}\lr{x}| + |p_{e}\lr{x|y_1}| + |p_{j}\lr{x | y_1, y_2}| < 1
\end{eqnarray}
for any $y_1$, $y_2$.
We would like to find a stochastic process for which $w\lr{x}$ is proportional to the probability of the occurrence of the element $x$ in some configuration space. Obviously, ordinary Markov process with configuration space $X$ cannot solve such a problem, since stationary distributions of Markov processes obey linear equations. In order to solve the nonlinear equation (\ref{random_process_eq}), one can, for example, extend somehow the configuration space. Extensions of Markov processes with stationary probability distributions which obey nonlinear equations have been considered recently in \cite{Etessami:05:1, Frank:04:1}. In this Section we concentrate on random processes similar to recursive Markov chains of \cite{Etessami:05:1}. The basic idea is that at any time one can leave the current chain and start a new one, then returning back to the old chain at some time. The initial state of a newly created chain depends on the states of older chains. Thus one has not a single Markov chain, but rather an infinite stack of chains. The random process which we describe below will be similar to these recursive Markov chains, but instead of referring to ``recursion'' we will explicitly introduce the underlying stack structure. Here we first consider the equations (\ref{random_process_eq}) with the coefficients $p_{c}\lr{x}$, $p_{e}\lr{x|y}$ and $p_{j}\lr{x | y_1, y_2}$ being all positive, and in Appendix \ref{sec:arbitrary_coefficients} we generalize to coefficients with arbitrary signs or complex phases.
Consider an extended configuration space which consists of ordered sequences $\lrc{x_1, \ldots, x_n}$ for arbitrary $n \ge 1$, with $x_1, \ldots, x_n \in X$. It is illustrative to interpret such configuration space as a stack of elements of the space $X$, so that $x_n$ is at the top of the stack. The desired random process can be specified by the following prescriptions. At each discrete time step do one of the following:
\begin{description}
\item[Create:] With probability $p_{c}\lr{x}$ create new element $x \in X$ and push it to the stack.
\item[Evolve:] With probability $p_{e}\lr{x | y}$ pop the element $y$ from the stack and push the element $x$ to the stack.
\item[Join:] With probability $p_{j}\lr{x | y_1, y_2}$ consecutively pop two elements $y_1$, $y_2$ from the stack and push a single element $x$ to the stack.
\item[Restart:] With probability $1-\sum\limits_{x}\lr{ p_{c}\lr{x} + p_{e}\lr{x|y_1} + p_{j}\lr{x | y_1, y_2}}$, where $y_1$, $y_2$ are the two topmost elements in the stack, empty the stack and push a single element $x \in X$ into it, with probability distribution proportional to $p_{c}\lr{x}$.
\end{description}
The last action is also the procedure used to initialize the random process. The ``Evolve'' action is just the evolution of a single Markov chain at the top of the stack, with transition probabilities proportional to $p_{e}\lr{x | y}$. The condition (\ref{probability_ineq}) and the positivity requirement ensures that $p_{c}\lr{x}$, $p_{e}\lr{x | y}$ and $p_{j}\lr{x | y_1, y_2}$ can be interpreted as probabilities.
Consider now an equation for the stationary probability distribution of such a Markov chain. It has a general form $p\lr{A} = \sum \limits_{B} P\lr{B \rightarrow A} \, p\lr{B}$, where $p\lr{A}$ is a stationary probability of the occurrence of a state $A$ and $P\lr{B \rightarrow A}$ is the transition probability between the states $B$ and $A$. Let $W\lr{x_1, \ldots, x_n}$ be the stationary probability to find the elements $x_1, \ldots, x_n$ in the stack. This probability distribution function is obviously normalized to unity:
$$\sum \limits_{n=1}^{\infty} \sum \limits_{x_1} \ldots \sum \limits_{x_n} W\lr{x_1, \ldots x_n} = 1 . $$
The equation for the stationary probability distribution in our case reads:
\begin{eqnarray}
\label{random_process_large_stationary_eq_n2}
W\lr{x_1}
=
\mathcal{N}_c^{-1} \, p_c\lr{x_1} \, \xi_R
+
\sum \limits_{y} p_e\lr{x_1 | y} \, W\lr{y}
+ \nonumber \\ +
\sum \limits_{y_1, y_2} p_j\lr{x_1 | y_1, y_2} \, W\lr{y_1, y_2}
\\ \label{random_process_large_stationary_eq}
W\lr{x_1, \ldots, x_n}
=
p_c\lr{x_n} \, W\lr{x_1, \ldots, x_{n-1}}
+ \nonumber \\ +
\sum \limits_{y} p_e\lr{x_n | y} \, W\lr{x_1, \ldots, x_{n-1}, y}
+ \nonumber \\ +
\sum \limits_{y_1, y_2} p_j\lr{x_n | y_1, y_2} \, W\lr{x_1, \ldots, x_{n-1}, y_1, y_2}
, n > 1 ,
\end{eqnarray}
where
\begin{eqnarray}
\label{xi_R_def}
\xi_R = \sum \limits_{n} \sum \limits_{x_1} \ldots \sum \limits_{x_n}
\left(1 - \right.
p_c\lr{x_n} \, W\lr{x_1, \ldots, x_{n-1}}
- \nonumber \\ -
\sum \limits_{y} p_e\lr{x_n | y} \, W\lr{x_1, \ldots, x_{n-1}, y}
- \nonumber \\ -
\left.
\sum \limits_{y_1, y_2} p_j\lr{x_n | y_1, y_2} \, W\lr{x_1, \ldots, x_{n-1}, y_1, y_2}
\right)
\end{eqnarray}
and $\mathcal{N}_c = \sum \limits_x p_c\lr{x}$. By a direct substitution one can check that there is a factorized solution for $W\lr{x_1, \ldots, x_n}$:
\begin{eqnarray}
\label{random_process_factorized_pd}
W\lr{x_1, \ldots, x_n} = w_{0}\lr{x_1} \, w\lr{x_2} \, \ldots w\lr{x_n} ,
\end{eqnarray}
where $w\lr{x}$ obeys exactly the equation (\ref{random_process_eq}) and $w_0\lr{x}$ obeys the following inhomogeneous linear equation:
\begin{eqnarray}
\label{random_process_eq_w0}
w_0\lr{x}
=
\mathcal{N}_c^{-1} \, p_c\lr{x} \, \xi_R
+
\sum \limits_{y} p_e\lr{x | y} \, w_0\lr{y}
+ \nonumber \\ +
\sum \limits_{y_1, y_2} p_j\lr{x | y_1, y_2} \, w_0\lr{y_1} w\lr{y_2} .
\end{eqnarray}
Thus, for any equation of the form (\ref{random_process_eq}) with positive coefficients which satisfy (\ref{probability_ineq}), there is a random process whose stationary distribution encodes the solution of this equation as in (\ref{random_process_factorized_pd}). The factorization of the stationary probability distribution of random processes with such an infinite configuration space is known as the ``propagation of chaos'' in random processes and was discovered for classical kinetic equations by McKean, Vlasov and Kac \cite{KacProbability}. Comparing the equation (\ref{random_process_eq}) with Schwinger-Dyson equations (\ref{phi4_SDs_original_n2}), (\ref{phi4_SDs_original}), (\ref{phi4_SDs_momentum_n2}) and (\ref{phi4_SDs_momentum}), we conclude that this property corresponds to the factorization of single-trace operators in large-$N$ quantum field theories. It is interesting that time reversal of the random process described above leads to the so-called branching random process \cite{KacProbability}, which has quite different properties. This is due to the fact that for such random processes there is no detailed balance condition, and hence no time reversal symmetry. We do not consider here a subtle mathematical question of the existence of solutions to equation (\ref{random_process_eq}), since in our case it is ensured by the physical applications of this equation.
Finally, let us describe a practical procedure for finding $w\lr{x}$ by simulating the random process described above. By standard statistical methods, one should sample the probability distribution $p\lr{x_n}$ of the topmost element in the stack (provided there is more than one element in it, otherwise we estimate $w_0\lr{x}$ rather than $w\lr{x}$, see (\ref{random_process_factorized_pd})). From (\ref{random_process_factorized_pd}), we get $p\lr{x_n} = s^{-1} w\lr{x_n}$, with $s = \sum \limits_{x} w\lr{x}$. It should be stressed that $w\lr{x}$ is not normalized to unity, but rather satisfies the inequality $\sum \limits_{x} w\lr{x} = s < 1$. The value of the normalization constant $s$ can be also easily found numerically, since the probability to find $n$ elements in the stack decreases as $s^n$ for $n > 1$.
\section{Stochastic solution of Schwinger-Dyson equations by recursive random processes}
\label{sec:SDs_stochastic_solution}
\subsection{Scalar matrix field theory}
\label{subsec:phi4_general}
After presenting the general method in Section \ref{sec:recursive_process}, we are ready to describe a stochastic numerical solution of Schwinger-Dyson equations (\ref{phi4_SDs_momentum_n2}), (\ref{phi4_SDs_momentum}). For simplicity, let us assume that the coupling constant $\lambda$ in (\ref{phi4_field_action}) is negative. This allows us to apply directly the results of Section \ref{sec:recursive_process}, where all the coefficients in (\ref{random_process_eq}) are assumed to be positive. In the case of positive $\lambda$, additional sign variables for each sequence of momenta can be easily introduced following Appendix \ref{sec:arbitrary_coefficients}. This will be done in the next Subsection for the Hermitian matrix model. Note that while at finite $N$ the theory with negative coupling constant is not defined and the correlators are non-analytic in $\lambda$ \cite{Prokofev:10:1}, in the leading order in $N$ perturbative series converge even when the coupling is negative, but not exceeding some critical value \cite{Brezin:78:1}. Correspondingly, in the planar approximation the correlators are analytic in $\lambda$.
The space $X$ in (\ref{random_process_eq}) should be the space of ordered sequences (of any size) of momenta $\lrc{k_1, \ldots, k_n}$ , correspondingly, the extended configuration space is a stack which contains such sequences. It is convenient also to introduce two normalization constants $\mathcal{N}$ and $c$, so that the functions $w\lr{k_1, \ldots k_n}$ which will be estimated stochastically are defined as
\begin{eqnarray}
\label{phi4_SDs_momentum_PDF}
G\lr{k_1, \ldots, k_n} = \mathcal{N} \, V^{n} \, c^{n-2} \, w\lr{k_1, \ldots, k_n} ,
\end{eqnarray}
where $V$ is again the total volume of space. The constant $c$ can be thought of as the renormalization constant for the one-particle wave functions, and $\mathcal{N}$ - as the overall wavefunction normalization.
In terms of the functions $w\lr{k_1, \ldots, k_n}$ the Schwinger-Dyson equations (\ref{phi4_SDs_momentum_n2}) and (\ref{phi4_SDs_momentum}) read:
\begin{widetext}
\begin{eqnarray}
\label{phi4_SDs_momentum_n2_rescaled}
w\lr{k_1, k_2}
=
G_0\lr{k_1} \, \mathcal{N}^{-1} \frac{\delta\lr{k_1 + k_2}}{V}
+ \nonumber \\ +
G_0\lr{k_1} \, \lambda c^2 \, \sum \limits_{q_1, q_2, q_3} \delta\lr{k_1 - q_1 - q_2 - q_3} \, w\lr{q_1, q_2, q_3, k_2} ,
\\
\label{phi4_SDs_momentum_rescaled}
w\lr{k_1, \ldots, k_n}
\, = \,
G_0\lr{k_1} \, c^{-2} \frac{\delta\lr{k_1 + k_2}}{V} \, w\lr{k_3, \ldots, k_n}
+ \nonumber \\ +
G_0\lr{k_1} \, c^{-2} \frac{\delta\lr{k_1 + k_n}}{V} \, w\lr{k_2, \ldots, k_{n-1}}
+ \nonumber \\ +
G_0\lr{k_1} \mathcal{N} \, c^{-4} \sum \limits_{m=3}^{n-1} \frac{\delta\lr{k_1 + k_m}}{V} \, w\lr{k_2, \ldots, k_{m-1}} \, w\lr{k_{m+1}, \ldots, k_n}
- \nonumber\\ -
G_0\lr{k_1} \, \lambda c^2 \, \sum \limits_{q_1, q_2, q_3} \delta\lr{k_1 - q_1 - q_2 - q_3} \, w\lr{q_1, q_2, q_3, k_2, \ldots, k_n} .
\end{eqnarray}
\end{widetext}
Comparing the Schwinger-Dyson equations (\ref{phi4_SDs_momentum_n2_rescaled}), (\ref{phi4_SDs_momentum_rescaled}) with the general equation (\ref{random_process_eq}), we arrive at the random process which stochastically solves these equations. This random process is specified by the following probabilistic choice of actions at each discrete time step:
\begin{description}
\item[Create:] With probability $G_0\lr{k} \lr{\mathcal{N} V}^{-1}$ push a new sequence of momenta $\lrc{k, -k}$ to the stack.
\item[Add:] With probability $G_0\lr{k} c^{-2}/V$ modify the topmost sequence of momenta $\lrc{k_1, \ldots, k_n}$ in the stack by adding a pair of momenta $\lrc{k, -k}$ either as $\lrc{k, k_1, \ldots, k_n, -k}$ or $\lrc{k, -k, k_1, \ldots, k_n}$.
\item[Create vertex:] With probability $ |\lambda| \, G_0\lr{q_1 + q_2 + q_3} c^2$ replace the topmost sequence $\lrc{q_1, q_2, q_3, k_2, \ldots, k_n}$ in the stack by $\lrc{q_1 + q_2 + q_3, k_2, \ldots, k_n}$. This action can only be performed if the topmost sequence contains more than two elements.
\item[Join:] With probability $G_0\lr{k} \mathcal{N}\, c^{-4} / V$ pop the two sequences $\lrc{k_1, \ldots, k_n}$, $\lrc{q_1, \ldots, q_m}$ from the stack (provided there are more than two elements in it) and join them into a single sequence as $\lrc{k_1, \ldots, k_n, k, q_1, \ldots, q_n, -k}$. Push the result to the stack.
\item[Restart:] Otherwise restart with a stack containing a sequence $\lrc{k,-k}$, $k$ being distributed with the probability proportional to $G_0\lr{k}$
\end{description}
Since the momenta are always added to the stack in pairs which sum up to zero, for all sequences in the stack the total sum of all momenta in the sequence is always zero. The $V^{-1}$ factors in (\ref{phi4_SDs_momentum_n2_rescaled}), (\ref{phi4_SDs_momentum_rescaled}) ensure that the probability distributions of the newly created momenta can be normalized to unity.
Let us check whether the inequalities (\ref{probability_ineq}) are satisfied for such a process, that is, whether the total probability of all possible actions does not exceed unity. For the free propagator $G_0\lr{k}$ one has the inequalities $G_{0}\lr{k} < 1/m^2$ and $\sum \limits_{k} G_{0}\lr{k} < V/m^2$. The total probability of all possible actions can be then estimated as $\lr{\mathcal{N}^{-1} + c^{-2} + |\lambda| c^2 + \mathcal{N}c^{-4}}/m^2$. Clearly, for sufficiently small $|\lambda|$ this estimate can be always made smaller than unity by increasing $c$ and $\mathcal{N}$. In Subsections \ref{subsec:matrix_model} and \ref{subsec:weingarten} we will analyze such bounds on coupling constants in more details for Hermitian matrix model and for the Weingarten model.
Since the constructed process involves no permutations, one can trace the history of each momenta in the stack - from creation to joining into a vertex or a ``Restart operation''. By drawing all the momenta in stack as points on the vertical lines of some two-dimensional grid and connecting the corresponding points along the horizontal lines, all planar diagrams of the theory (with an arbitrary number of external lines) can be obtained. Note also that the number of vertices in planar diagrams drawn by this random process cannot exceed the number of time steps from the previous ``Restart'' action. Thus in order to maximize the mean order of diagrams which are summed up in some fixed number of time steps, it is advantageous to maximally reduce the rate of ``Restart'' events, that is, to saturate the inequalities (\ref{probability_ineq}).
One could also try to devise a random process which would solve the Schwinger-Dyson equations (\ref{phi4_SDs_original_n2}), (\ref{phi4_SDs_original}) directly in physical space-time, rather than in the momentum space. The configuration space of such a process would be the stack of sequences of points $\lrc{x_1, \ldots, x_n}$. As compared to the algorithm in the momentum space, there would be an additional choice of moving the last point $x_n$ in the topmost sequence to adjacent lattice sites, with the probability proportional to the hopping parameter $\kappa = \lr{2 D + m^2}^{-1}$. This would correspond to drawing the worldlines of virtual and real particles by bosonic random walks. Interestingly, such worldlines can be mapped onto the string worldsheets in simplicial string theory \cite{Akhmedov:04:1}. As well, the creation of a new interaction vertex would only be possible if three such random walks would intersect in one point. However, this is an unlikely event, with probability going to zero in the continuum limit. Thus, solving the Schwinger-Dyson equations directly in the coordinate representation would lead to a less efficient numerical algorithm.
Note that for the theory (\ref{phi4_field_action}) at finite $N$ the Schwinger-Dyson equations are linear equations, which are, however, defined on much larger functional space: the set of unknown functions includes also expectation values of multi-trace operators, such as $\vev{\tr\lr{\phi\lr{x_1} \ldots \phi\lr{x_n}} \, \tr\lr{\phi\lr{y_1} \ldots \phi\lr{y_m}}}$. One can try to solve these linear equations by interpreting them as the equations for the stationary probability distribution of a Markov process. The configuration space of such a process should be a space of sequences of the form $\lrc{\lrc{x_1, \ldots, x_n}, \ldots, \lrc{y_1, \ldots, y_m}}$, thus encoding the expectation values of all multi-trace operators. However, such a straightforward procedure leads to non-normalizable transition probabilities, indicating that the series which one tries to sum up are divergent. Only when the terms subleading in $1/N$ are omitted from the Schwinger-Dyson equations, they can be interpreted as stochastic equations. At the same time, we obtain the Markov process on the extended configuration space described in Section \ref{sec:recursive_process}, which we interpret as the stack of sequences. The property of the ``propagation of chaos'' \cite{KacProbability} ensures large-$N$ factorization of single-trace operators (see equation (\ref{random_process_factorized_pd})). We are thus led to the random process of recursive type \cite{Etessami:05:1}.
\subsection{Hermitian matrix model}
\label{subsec:matrix_model}
To check the considerations of the previous Subsection, let us consider the theory (\ref{phi4_field_action}) in zero dimensions, that is, the hermitian matrix model with the following partition function:
\begin{eqnarray}
\label{matrix_model_def}
\mathcal{Z}\lr{\lambda} = \int \prod \limits_{i,j} d \phi_{ij} \,
\expa{ -N/2 \, \tr\phi^2 + \frac{\lambda N}{4}\, \tr\phi^4 } .
\end{eqnarray}
The Green functions now depend only on one integer $n$: $G_n \equiv G\lr{n} = \vev{\frac{1}{N} \, \tr \phi^{2 n}}$. The Schwinger-Dyson equations (\ref{phi4_SDs_original_n2}), (\ref{phi4_SDs_original}) also take a very simple form:
\begin{eqnarray}
\label{matrix_model_SDs}
G_1 = 1 + \lambda G_2,
\nonumber \\
G_n = 2 G_{n-1} + \sum \limits_{m = 1}^{n-2} G_m \, G_{n-m-1} + \lambda G_{n+1}, \quad n > 1.
\end{eqnarray}
Here we will assume that the coupling constant $\lambda$ can be both positive and negative, in order to illustrate the method described in Appendix \ref{sec:arbitrary_coefficients}. Let us again define the ``renormalized'' Green functions $w_n$ as $G_n = \mathcal{N} c^{n-1} \, w_n$. In the case of arbitrary sign of $\lambda$, the configuration space of the random process should be the stack which contains integer positive numbers and additional sign variables. Following Appendix \ref{sec:arbitrary_coefficients}, we introduce the variables $w_n^{\lr{+}}$ and $w_n^{\lr{-}}$ which are proportional to the probabilities to find the elements $\lrc{n, +}$ or $\lrc{n, -}$ at the top of the stack (provided the stack contains more than one element). Then $w_n = w_n^{\lr{+}} - w_n^{\lr{-}}$. We thus arrive at the following random process for stochastic evaluation of $w_n^{\lr{\pm}}$. At each discrete time step one performs at random one of the following actions:
\begin{itemize}
\item With probability $\mathcal{N}^{-1}$ add new element $\lrc{1, +}$ to the stack.
\item With probability $2 \, c^{-1}$ increase the topmost element in the stack by $1$ and do not change its sign.
\item With probability $|\lambda| \, c$ decrease the topmost element in the stack by $1$ (if it is greater than one) and multiply its sign by the sign of $\lambda$.
\item With probability $\mathcal{N}\, c^{-2}$ pop the two elements $\lrc{n, s_1}$ and $\lrc{m, s_2}$ from the stack (provided there are more than two elements) and push the element $\lrc{n + m + 1, s_1 s_2}$ to the stack.
\item Otherwise empty the stack and push into it a single element $\lrc{1, +}$.
\end{itemize}
Note that for positive $\lambda$ elements with the minus sign are not generated, so that $w_n^{\lr{-}} \equiv 0$ and the random process automatically reduces to the one described in Section \ref{sec:recursive_process}.
The inequalities (\ref{probability_ineq}) read now:
\begin{eqnarray}
\label{brezin_process_inequalities}
\mathcal{N}^{-1} + 2 \, c^{-1} + |\lambda| c + \mathcal{N}\, c^{-2} \le 1,
\quad \mathcal{N} > 0, \quad c > 0
\end{eqnarray}
As discussed in Subsection \ref{subsec:phi4_general}, in order to increase the efficiency of the algorithm it is advantageous to saturate this inequality. It is easy to see that at the same time we saturate the upper bound on the absolute value of the coupling constant $\lambda$. Maximizing this upper bound with respect to $\mathcal{N}$ and $c$, we see that $|\lambda|$ cannot exceed the value $\bar{\lambda} = 1/16 = 3/4 \, \lambda_c$, where $\lambda_c = 1/12$ is the convergence radius of the planar perturbative expansion which can be found from the exact solution of the matrix model (\ref{matrix_model_def}) \cite{Brezin:78:1}. Thus the described random process covers only some finite subrange of coupling constants for which the model (\ref{matrix_model_def}) is defined. It is easy to understand the origin of this limitation: in fact the random process described above simulates an ensemble of diagrams with an arbitrary number of external legs, with the weight of each diagram being proportional to $\lambda^{N_v}$, where $N_v$ is the number of vertices. The number of open diagrams with a given number of vertices is obviously larger than the number of closed diagrams, hence the sums over open diagrams have smaller convergence radius.
\begin{figure}[h]
\includegraphics[width=6cm, angle=-90]{moments.eps}\\
\caption{Green functions $G_n\lr{\lambda}$ in (\ref{matrix_model_SDs}) versus the coupling constant $\lambda$ for $n = 1, \ldots, 5$, obtained after $N = 10^6$ discrete time steps of the described algorithm at fixed $c = 8$ and with $\mathcal{N}$ given by ``Branch 1'' of (\ref{nn_vs_lambda}). The error bars are smaller than the symbols on the plot. Exact results of \cite{Brezin:78:1} are plotted with solid lines. }
\label{fig:brezin_moments}
\end{figure}
For $|\lambda| < \bar{\lambda}$, there is a continuous set of $\mathcal{N}$, $c$ which saturate the inequality (\ref{brezin_process_inequalities}). One can, for example, fix $c$ and express $\mathcal{N}$ as a function of $\lambda$:
\begin{eqnarray}
\label{nn_vs_lambda}
\mathcal{N} = \frac{c^2 - 2 c - |\lambda| c^3 \pm \sqrt{c^3\lr{c |\lambda| - 1}\lr{4 - c + c^2 |\lambda|}}}{2} .
\end{eqnarray}
We call the solution with the minus sign in front of the square root ``Branch~1'' and the other solution ``Branch~2''.
\begin{figure}[h]
\includegraphics[width=6cm, angle=-90]{autocorrelation_time.eps}\\
\caption{Autocorrelation time of the random process described in Subsection \ref{subsec:matrix_model} as the function of the coupling constant $\lambda$ at fixed $c = 8$ and for different choices of $\mathcal{N}$ in (\ref{nn_vs_lambda}).}
\label{fig:autocorrelation_time}
\end{figure}
On Fig. \ref{fig:brezin_moments} we plot the Green functions $G_n\lr{\lambda}$ evaluated using the described random process as functions of $\lambda$ up to $n=5$. These results were obtained after $N = 10^6$ discrete time steps at fixed $c = 8$ and with $\mathcal{N}$ given by the ``Branch 1'' of (\ref{nn_vs_lambda}). The error bars are smaller than the symbols on the plot. Solid lines are the exact results for $G_n\lr{\lambda}$ in the planar approximation, obtained using the saddle point method \cite{Brezin:78:1}.
\begin{figure}[h]
\includegraphics[width=6cm, angle=-90]{mean_stack_size.eps}\\
\caption{Mean stack size of the random process described in Subsection \ref{subsec:matrix_model} as the function of the coupling constant $\lambda$ at fixed $c = 8$ and for different choices of $\mathcal{N}$ in (\ref{nn_vs_lambda}).}
\label{fig:mean_stack_size}
\end{figure}
Autocorrelation time and mean stack size for the described random process are plotted on Figs. \ref{fig:autocorrelation_time} and \ref{fig:mean_stack_size}, respectively, as the functions of the coupling constant $\lambda$. The observable used to define the autocorrelation time was the sum of all numbers in the stack. First, we note that ``Branch 1'' is more advantageous for simulations, since with larger mean stack size one can gain more statistics. However, in this case the autocorrelation time is also larger. Interestingly, for this branch both the autocorrelation time and the mean stack size have maximum near $\lambda = 0$ rather than near the ``critical point'' of the random process $\bar{\lambda}$. For ``Branch 2'', these quantities increase slowly towards $\bar{\lambda}$.
\begin{figure}[h]
\includegraphics[width=6cm, angle=-90]{sign_problem_severity.eps}\\
\caption{The quantity $\xi^s_n$ in (\ref{matrix_model_xisn_def}) for $n = 1, 3, 5$, as the function of the coupling constant $\lambda$ at fixed $c = 8$ and for different choices of $\mathcal{N}$ in (\ref{nn_vs_lambda}). ``Br. 1,2'' is for ``Branch 1,2''. }
\label{fig:sign_problem_severity}
\end{figure}
In order to characterize the strength of the sign problem, we consider the quantity
\begin{eqnarray}
\label{matrix_model_xisn_def}
\xi^{s}_n = \lr{w_n^{\lr{+}} - w_n^{\lr{-}}}/\lr{w_n^{\lr{+}} + w_n^{\lr{-}}} .
\end{eqnarray}
$\xi^{s}_n = 1$ if the random process generates only elements with the plus sign and $\xi^s_n = 0$ if the numbers of pluses and minuses exactly cancel. In practice, it is advantageous to have as large $\xi^{s}_n$ as possible, so that the difference $w_n^{\lr{+}} - w_n^{\lr{-}}$ can be estimated with maximal precision. $\xi^{s}_n$ are plotted on Fig. \ref{fig:sign_problem_severity} as the function of $\lambda$ for $n = 1, 3, 5$. For $\lambda < 0$, $\xi_s\lr{\lambda}$ decreases with $\lambda$ and $n$. The sign cancelation is thus moderate for $n=1$ ($\xi^s_1\lr{-\bar{\lambda}} \approx 0.75$) and becomes more and more important for higher-order correlators - $\xi^s_5$ at $\lambda = - \bar{\lambda}$ is close to zero. It is interesting that $\xi^{s}_n$ are almost equal for two different choices of $\mathcal{N}$ in (\ref{nn_vs_lambda}).
Thus there are no indications of severe critical slowing down in the whole range of possible coupling constants $-\bar{\lambda} < \lambda < \bar{\lambda}$. The sign problem is also moderate for low-order correlators, but becomes more severe for higher-order correlators. It could be extremely interesting to extend the applicability of the described random process up to $\lambda = \lambda_c$ while preserving these attractive features of the algorithm.
\subsection{Random planar surfaces: the Weingarten model}
\label{subsec:weingarten}
Weingarten model \cite{Weingarten:80:1, Eguchi:82:3} is a lattice field theory which in the large-$N$ limit reproduces the sum over all closed surfaces with genus one on the hypercubic lattice. The action for each surface is proportional to its area, thus the model can be considered as a lattice regularization of bosonic strings with Nambu-Goto action. Although this model does not have a nontrivial continuum limit for any space dimensionality \cite{Durhuus:84:1}, the structure of the functional integral and of the Schwinger-Dyson equations in this model are similar to those in large-$N$ non-Abelian lattice gauge theory, and the analysis of this model might be helpful for the extension of the approach described here to non-Abelian gauge theories. In order to derive the Schwinger-Dyson equations, it is convenient to consider the reduced Weingarten model \cite{Eguchi:82:3}, which in the large-N limit is equivalent to the original model, similarly to the Eguchi-Kawai model for non-Abelian lattice gauge theory. It can be shown that in contrast to reduced lattice gauge theories, for the reduced Weingarten model additional twisting is not necessary \cite{Kawai:83:1}.
\begin{figure}
\includegraphics[width=6cm]{weingarten_loop_equations_fine.eps}\\
\caption{Schematic illustration of the structure of the loop equations (\ref{weingarten_loop_equations}) in the Weingarten model (\ref{weingarten_model_def}). These equations should hold for any link (marked by a thick line) which belongs to the loop.}
\label{fig:weingarten_loop_equations}
\end{figure}
The reduced model is defined by an integral over complex $N \times N$ matrices $U_{\mu} \equiv U^{\dag}_{-\mu}$ with $\mu = 1, \ldots, D$:
\begin{widetext}
\begin{eqnarray}
\label{weingarten_model_def}
\mathcal{Z}\lr{\beta} = \int \mathcal{D} U_{\mu} \,
\expa{ -N \sum \limits_{\mu = 1}^{D} \tr\lr{U_{\mu} U^{\dag}_{\mu}}
+ N \beta \sum \limits_{\mu \neq \nu = 1}^{D} \tr\lr{U_{\mu} U_{\nu} U_{\mu}^{\dag} U_{\nu}^{\dag}} } .
\end{eqnarray}
\end{widetext}
If one treats the second term in the exponent in (\ref{weingarten_model_def}) as a perturbation and expands $Z\lr{\beta}$ in powers of $\beta$, the resulting sum over planar diagrams is equivalent to the sum over all possible closed surfaces of genus one on the lattice with the weight $\beta^{|S|}$, where $|S|$ is the area of each surface.
A basic observable in this model is the sum over all planar surfaces which are bounded by some closed loop $C$. The loop $C$ can be uniquely specified by a sequence $\lrc{\mu_1, \ldots, \mu_n}$, where $\mu$'s take the values $\pm 1, \ldots, \pm D$. In order to reconstruct the loop $C$ from the sequence, one should start from an arbitrary point on a hypercubical lattice and move along one link in the direction $\mu_1$, forward if $\mu_1$ is positive and backward if $\mu_1$ is negative. From this new position one should similarly move in the direction $\mu_2$, and so on. From the diagrammatic expansion one can see that such a sum over surfaces is given by the following correlator:
\begin{eqnarray}
\label{weingarten_wilson_loop}
W\lr{C} = W\lr{\mu_1, \ldots, \mu_n} =
\vev{ \frac{1}{N}\, \tr\lr{U_{\mu_1} \ldots U_{\mu_n}} },
\end{eqnarray}
where one takes the conjugate variable $U_{|\mu_A|}^{\dag}$ if $\mu_A$ is negative. This observable is similar to Wilson loop in lattice gauge theory, but, unlike the Wilson loop, it does not have a ``zigzag symmetry'' \cite{Polyakov:99:1}: passing a link forward and immediately backward changes the value of the Wilson loop. In the large-$N$ limit the single-loop observables factorize, which allows us to obtain a closed set of Schwinger-Dyson equations for $W\lr{\mu_1, \ldots, \mu_n}$ \cite{Weingarten:80:1, Eguchi:82:3}:
\begin{widetext}
\begin{eqnarray}
\label{weingarten_loop_equations}
W\lr{\mu_1, \mu_2} = \delta\lr{\mu_1, -\mu_2} + \beta \sum \limits_{|\mu| \neq |\mu_1|} W\lr{\mu, \mu_1, -\mu, \mu_2}
\nonumber \\
W\lr{\mu_1, \ldots, \mu_n}
=
\delta\lr{\mu_1, -\mu_2} W\lr{\mu_3, \ldots, \mu_n}
+
\delta\lr{\mu_1, -\mu_n} W\lr{\mu_2, \ldots, \mu_{n-1}}
+ \nonumber \\ +
\sum \limits_{A = 3}^{n-1} W\lr{\mu_2, \ldots, \mu_{A-1}} \, W\lr{\mu_{A+1}, \ldots, \mu_n} \delta\lr{\mu_1, -\mu_A}
+ \nonumber \\ +
\beta \sum \limits_{|\mu| \neq |\mu_1|} W\lr{\mu, \mu_1, -\mu, \mu_2, \ldots, \mu_n}, \quad n > 2.
\end{eqnarray}
These equations should hold for any lattice link $\mu_k$ belonging to the loop $C$, but the resulting system of equations is redundant, and it is sufficient to consider only one link $\mu_1$ on the loop. The equations (\ref{weingarten_loop_equations}) are schematically illustrated on Fig. \ref{fig:weingarten_loop_equations}, where the link $\mu_1$ is marked by a thick line.
We see that the equations (\ref{weingarten_loop_equations}) again take the form similar to (\ref{random_process_eq}). Let us now define the ``renormalized'' observable $w\lr{\mu_1, \ldots, \mu_n}$ by rescaling $W\lr{\mu_1, \ldots, \mu_n}$ by the factors $\mathcal{N}$ and $q$ as $W\lr{\mu_1, \ldots, \mu_n} = \mathcal{N} q^{n} w\lr{\mu_1, \ldots, \mu_n}$. One can interpret the factor $q^{n}$ as the mass attached to the boundaries of random surfaces, somewhat like the bare quark mass in QCD. The equations (\ref{weingarten_loop_equations}) then take the following form:
\begin{eqnarray}
\label{weingarten_loop_equations_renorm}
w\lr{\mu_1, \mu_2} = \lr{\mathcal{N} q^2}^{-1} \delta\lr{\mu_1, -\mu_2} + \beta q^2 \sum \limits_{|\mu| \neq |\mu_1|} w\lr{\mu, \mu_1, -\mu, \mu_2}
\nonumber \\
w\lr{\mu_1, \ldots, \mu_n}
=
q^{-2} \delta\lr{\mu_1, -\mu_2} w\lr{\mu_3, \ldots, \mu_n}
+
q^{-2} \delta\lr{\mu_1, -\mu_n} w\lr{\mu_2, \ldots, \mu_{n-1}}
+ \nonumber \\ +
\mathcal{N} q^{-2} \, \sum \limits_{A = 3}^{n-1} w\lr{\mu_2, \ldots, \mu_{A-1}} \, w\lr{\mu_{A+1}, \ldots, \mu_n} \delta\lr{\mu_1, -\mu_A}
+ \nonumber \\ +
\beta q^2 \, \sum \limits_{|\mu| \neq |\mu_1|} w\lr{\mu, \mu_1, -\mu, \mu_2, \ldots, \mu_n}, \quad n > 2.
\end{eqnarray}
\end{widetext}
Let us now devise a random process of the type described in Section \ref{sec:recursive_process}, which solves stochastically these equations. The configuration space is now a stack which contains closed loops, that is, sequences of indices $\mu = \pm 1, \ldots, \pm D$. The desired random process is defined by the following possible actions at each discrete time step:
\begin{description}
\item[Create a new loop:] With probability $2 D \mathcal{N}^{-1} q^{-2}$ create a new elementary loop $C = \lrc{\mu, -\mu}$, where $\mu = \pm 1, \ldots, \pm D$ is random (either positive or negative).
\item[Join loops:] With probability $2 D \mathcal{N} q^{-2}$ pop the two loops $C_1 = \lrc{\mu_1, \ldots, \mu_n}$, $C_2 = \lrc{\nu_1, \ldots, \nu_m}$ from the stack and form a new loop $C$ by joining the loops $C_1$, $C_2$ with a link in the random direction $\mu$ (either positive or negative): $C = \lrc{\mu_1, \ldots, \mu_n, \mu, \nu_1, \ldots, \nu_m, -\mu}$. This action can only be performed if there are more than two loops in the stack.
\item[Flatten loop:] If the three links in the end of the sequence on the top of the stack form a boundary of the plaquette, that is, if the topmost loop has the form $C = \lrc{\mu_1, \ldots, \mu_n, \mu, \nu, -\mu}$ for some $\mu$ and $\nu$, replace these three links by a single link in the direction $\nu$ with probability $\beta q^2$: $C' = \lrc{\mu_1, \ldots, \mu_n, \nu}$.
\item[Append to loop:] With probability $4 D q^{-2}$ append a pair $\lrc{\mu, -\mu}$, where $\mu$ is random (either positive or negative), to the topmost sequence $\lrc{\mu_1, \ldots, \mu_n}$ in the stack as $\lrc{\mu, -\mu, \mu_1, \ldots, \mu_n}$ or $\lrc{\mu, \mu_1, \ldots, \mu_n, -\mu}$. The probabilities of these two choices are equal.
\item[Restart:] Otherwise start with a stack containing an elementary random loop $C = \lrc{\mu, -\mu}$, where $\mu = \pm 1, \ldots, \pm D$ is chosen randomly.
\end{description}
Again assuming that the sum of the probabilities of all possible actions is equal to one and the probability of ``Restart'' events is minimized, we obtain an equation relating $\beta$, $\mathcal{N}$ and $q$:
\begin{eqnarray}
\label{weingarten_param_relation}
\beta q^2 + 2 D q^{-2} \lr{\mathcal{N} + \mathcal{N}^{-1} + 2} = 1
\end{eqnarray}
Maximization with respect to $\mathcal{N}$ yields the relation between $q$ and $\beta$:
\begin{eqnarray}
\label{weingarten_q_vs_beta}
q = \sqrt{\frac{1 \pm \sqrt{1 - 32 D \beta}}{2 \beta}} .
\end{eqnarray}
We call the solution with the minus sign in front of the square root ``Branch 1'' and the other solution ``Branch 2''.
\begin{figure}
\includegraphics[width=6cm, angle=-90]{wgtn_w1x1_w1x0.eps}\\
\caption{The observables $W_{1\times0} \equiv W\lr{\mu, -\mu}$ ($1 \times 0$ loop) and $W_{1 \times 1} \equiv W\lr{\mu, \nu, -\mu, -\nu}$ ($1 \times 1$ loop) in the Weingarten model as a functions of the coupling constant $\beta$ for different dimensions $D$. Solid line corresponds to the first two terms in the perturbative expansion: $W\lr{\mu, -\mu} = 1 + 2 \lr{D - 1} \beta^2 + O\lr{\beta^4}$, $W\lr{\mu, \nu, -\mu, -\nu} = \beta + 8 \lr{D - 1} \beta^3 + O\lr{\beta^5}$. The data were obtained after $10^7$ iterations of the algorithm described above.}
\label{fig:wgtn_w1x1_w1x0}
\end{figure}
The value of $\beta$ in (\ref{weingarten_q_vs_beta}) cannot exceed the critical value $\bar{\beta}\lr{D} = 1/\lr{32 D}$. As we have already seen on the example of the matrix model, this critical value does not necessarily coincide with the true critical point $\beta_c\lr{D}$ at which the sum over planar surfaces diverges. Indeed, $\bar{\beta}\lr{D}$ does not exceed the lower bound $\beta_c\lr{D} > \lr{24 \lr{D - 1}}^{-1}$ obtained in \cite{Eguchi:82:3}, and is significantly lower than the critical values obtained numerically in \cite{Kawai:83:1}. In fact, for $\beta = \bar{\beta}_{D}$ all the observables are still dominated by the lowest-order perturbative contributions. Expectation values of the observables $W_{1\times0} \equiv W\lr{\mu, -\mu}$ ($1 \times 0$ loop) and $W_{1 \times 1} \equiv W\lr{\mu, \nu, -\mu, -\nu}$ ($1 \times 1$ loop), which were obtained after $10^7$ iterations of the random process described above (with $q$ given by ``Branch 1'' of (\ref{weingarten_q_vs_beta})), are plotted on Fig. \ref{fig:wgtn_w1x1_w1x0} as the functions of the coupling constant $\beta$. Solid line corresponds to the first two terms in the perturbative expansion: $W_{1 \times 0} = 1 + 2 \lr{D - 1} \beta^2 + O\lr{\beta^4}$, $W_{1 \times 1} = \beta + 8 \lr{D - 1} \beta^3 + O\lr{\beta^5}$. Within statistical errors, one sees only the lowest-order perturbative contributions. It should be stressed that the proposed random process implements stochastic summation of diagrams of \emph{all} orders, but due to the smallness of $\beta < \bar{\beta}\lr{D}$, a very large computational time is required to see the contributions of higher-order terms.
\begin{figure}
\includegraphics[width=4cm]{branched_polymers_illustrated.eps}\\
\vspace{-0.5cm}
\caption{Two simple configurations of branched polymers on the lattice. The configurations on the left and on the right count as different configurations.}
\label{fig:branched_polymers_illustrated}
\end{figure}
Note that when the coupling constant $\beta$ tends to zero (that is, the ``bare string tension'' of the random surfaces tends to infinity) and $q$ lies between the two solutions of (\ref{weingarten_q_vs_beta}), the above random process describes just the growth of ``branched polymers'', whose branches are bosonic random walks and hence correspond to particles rather than ``strings''. These branches consist of loops in which every lattice link is passed twice and which hence sweep out zero area.
Taking the limit $\beta \rightarrow 0$ in (\ref{weingarten_q_vs_beta}), we find that the minimal value of $q$ is $\bar{q} = \sqrt{8 D}$. In order to understand this critical value we first note that in the limit $\beta \rightarrow 0$ the observables $W\lr{\mu_1, \ldots, \mu_n}$ are all equal to one if the links $\mu_1, \ldots, \mu_n$ form a loop which sweeps out zero area and zero otherwise. The probabilities to encounter such loops in the described random process is hence proportional to $w\lr{\mu_1, \ldots, \mu_n} \sim q^{-n}$. Simple examples of such loops, which can be also thought of as the random tree-like graphs on the lattice, are shown on Fig. \ref{fig:branched_polymers_illustrated}. Now imagine adding to some loop $k$ links stemming from some lattice site. Since the loop includes each link twice, the probability decreases by $q^{-2 k}$. The number of possible configurations of $k$ links is $\lr{2 \times 2 D}^{k}$ since each of $k$ links can point along any of $D$ directions both forward and backward. An additional factor of $2$ appears since the zero-area loops pass twice through each point and the new links can be inserted between the links pointing either forward or backward (for example, compare the configurations on the left and on the right of Fig. \ref{fig:branched_polymers_illustrated}). Finally, one can add any number $k = 1, 2, \ldots, \infty$ of branches to any point belonging to the branched polymer. At the criticality, adding any number of random links to some configuration should not change its overall weight. Therefore, the change of the weight due to the added links times the number of ways to add them should be equal to unity. We are thus led to the following equation for $\bar{q}$:
\begin{eqnarray}
\label{branched_polymers_weight}
\sum \limits_{k = 1}^{+\infty} \lr{4 D q^{-2}}^k = \frac{4 D q^{-2}}{1 - 4 D q^{-2}} = 1 ,
\end{eqnarray}
or $8 D \bar{q}^{- 2} = 1$. Thus, in the limit $\beta \rightarrow 0$ we indeed reproduce branched polymers with the correct critical behavior.
At nonzero $\beta$, deviation from trivial branched polymer configurations can be characterized by the rate of the ``Flatten loop'' events. Indeed, since the probability of $n$ such events is proportional to $\beta^n$, such sequence of events corresponds to a random surface (which is in general open) consisting of $n$ lattice plaquettes, plus some number of random trees. One can therefore think of the described random process as of the process of drawing random loops which sweep out random planar surfaces. The average rate of ``flattening'' events is plotted on Fig. \ref{fig:wgtn_mfr} on the right as a function of the coupling constant $\beta$ for different dimensions $D$ and for different choices of $q$ in (\ref{weingarten_q_vs_beta}). One can see that in the whole range of coupling constants $0 < \beta < \bar{\beta}\lr{D}$ the rate of flattening events is numerically very small. On the other hand, the number of links in the loops, as well as the number of loops stored in the stack, are quite large. Mean stack size and mean length of the topmost loop in the stack are plotted on Fig. \ref{fig:wgtn_mss_mll} as a function of the coupling constant $\beta$ for different dimensions $D$ and for different choices of $q$ in (\ref{weingarten_q_vs_beta}). One can conclude therefore that the ``branched polymers'' actually dominate in the properties of the described random process. Critical behavior of these random trees is universal for any dimension $D$, that is why such observables as the mean stack size or the mean loop length, which are mainly sensitive to the length of loops rather than to the area of random surfaces, practically do not depend on space dimensionality.
While the closed planar surfaces in the vicinity of the true critical point $\beta_c\lr{D}$ of the Weingarten model are also dominated by ``branched polymers'' \cite{Durhuus:84:1}, in our ensemble of open random surfaces this dominance can be thought of as the manifestation of the tachyonic instability of open, rather than closed, strings. The fact that the critical coupling $\bar{\beta}\lr{D}$ in our case is smaller than the true critical point $\beta_c\lr{D}$ can be explained by the fact that the number of open surfaces with a given area is obviously larger than the number of closed surfaces with the same area. The true critical coupling constant $\beta_c\lr{D}$ can be quite easily found by a very simple re-weighting procedure, which will be described in details in a separate publication.
\begin{figure}
\includegraphics[width=6cm, angle=-90]{wgtn_mean_flattening_rate.eps}\\
\caption{Mean rate of ``flattening'' events for the random process solving the loop equations in the Weingarten model as a function of the coupling constant $\beta$ at different dimensions $D$ and for different choices of $q$ in (\ref{weingarten_q_vs_beta}). ``Br. 1,2'' is for ``Branch 1, 2''.}
\label{fig:wgtn_mfr}
\end{figure}
\begin{figure*}
\includegraphics[width=6cm, angle=-90]{wgtn_mean_stack_size.eps}
\includegraphics[width=6cm, angle=-90]{wgtn_mean_loop_length.eps}\\
\caption{Mean stack size (on the left) and mean length of the topmost loop in the stack (on the right) for the random process solving the loop equations in the Weingarten model as a function of the coupling constant $\beta$ at different dimensions $D$ and for different choices of $q$ in (\ref{weingarten_q_vs_beta}).}
\label{fig:wgtn_mss_mll}
\end{figure*}
\section{Resummation of divergent series and random processes with memory}
\label{sec:rps_with_mem}
In Section \ref{sec:SDs_stochastic_solution} we have described a stochastic method for the solution of the Schwinger-Dyson equations for theories with noncompact variables. This method works only at small coupling constants and implements stochastic summation of perturbative series. For theories with compact variables, such as nonlinear $\sigma$-models or lattice gauge theories, the structure of the Schwinger-Dyson equations is such that the described method can be straightforwardly applied only in the strong coupling regime, where one can stochastically sum all terms in the strong-coupling expansion. However, the continuum limit of such theories typically corresponds to the weak-coupling limit. An additional complication is that for physically interesting theories the observables cannot be expressed as convergent power series in the small coupling constant $g$, but rather contain non-analytic part which is typically of the form $\expa{-c/g^2}$ with some constant $c$ \cite{Parisi:77:1}.
In this Section we point out one possible way to deal with this problem. The basic idea is to absorb non-perturbative corrections into some self-consistent redefinition of the expansion parameter \cite{Parisi:77:1, Kazakov:94:1}. Recently, a similar resummation method was also considered in \cite{Prokofev:10:2}. Solving the self-consistency condition leads to the concept of a nonlinear random process with memory \cite{Frank:04:1}, in which all previous history of the process is used to estimate the value of the self-consistent expansion parameter.
Let us illustrate this idea on the simplest example of $O\lr{N}$ sigma model in the limit of large $N$. The model is defined by the following path integral over unit $N$-component vectors $n\lr{x}$ living on the sites of the $D$-dimensional hypercubic lattice:
\begin{eqnarray}
\label{sigma_model_pf}
\mathcal{Z} = \int\limits_{|n\lr{x}|=1} \mathcal{D}n(x) \expa{ \frac{N}{\lambda} \, \sum \limits_{<xy>} n\lr{x} \cdot n\lr{y} },
\end{eqnarray}
where summation goes over all neighboring lattice sites. Despite its simplicity, this model in $D = 2$ dimensions is asymptotically free and has a mass gap which depends nonperturbatively on the coupling constant $\lambda$. Schwinger-Dyson equations in this theory can be written in terms of the two-point function $\xi\lr{x, y} = \vev{n\lr{x}~\cdot~n\lr{y}}$, $\xi\lr{x} \equiv \xi\lr{x, 0}$ as:
\begin{eqnarray}
\label{sigma_model_sd}
\xi\lr{x} = \frac{1}{\lambda} \sum \limits_{\mu} \lr{\xi\lr{x \pm e_{\mu}} - \xi\lr{x} \xi\lr{\pm e_{\mu}}} + \delta\lr{x, 0} .
\end{eqnarray}
Clearly, these equations have the structure similar to (\ref{random_process_eq}), but the inequalities (\ref{probability_ineq}) are satisfied only for sufficiently large $\lambda$, that is, in the strong-coupling regime. Therefore, the continuum limit at $\lambda \rightarrow 0$ cannot be reached by the method described in Section \ref{sec:SDs_stochastic_solution}.
Let us, however, rewrite the equation (\ref{sigma_model_sd}) as
\begin{eqnarray}
\label{sigma_model_sd_as_rw}
\xi\lr{x} = \frac{1}{\lambda + \sum \limits_{\mu} \xi\lr{\pm e_\mu}} \, \lr{ \sum \limits_{\mu} \xi\lr{x \pm e_{\mu}} + \lambda \delta\lr{x} } ,
\end{eqnarray}
and introduce the ``hopping parameter''
\begin{eqnarray}
\label{sigma_model_kappa_def}
\kappa = \frac{1}{\lambda + \sum \limits_{\mu} \xi\lr{\pm e_\mu}} .
\end{eqnarray}
Now the equation (\ref{sigma_model_sd}) in the form (\ref{sigma_model_sd_as_rw}) looks like the equation for the free massive scalar propagator on the lattice with the mass $m^2 = \kappa^{-1} - 2 D$ in lattice units. Note that in the weak-coupling limit $\lambda \rightarrow 0$ $\xi\lr{\pm e_\mu} \rightarrow 1$, $m^2 \rightarrow 0$ and we approach the continuum limit.
Let us now solve the equation (\ref{sigma_model_sd_as_rw}) stochastically, assuming that $\xi\lr{x}$ is proportional to the stationary probability distribution $w\lr{x}$ of some random process: $\xi\lr{x} = c \, w\lr{x}$, $\sum \limits_{x} w\lr{x} = 1$. From (\ref{sigma_model_sd_as_rw}) we get $c = \frac{\lambda \kappa}{1 - 2 D \kappa}$. The equation (\ref{sigma_model_sd_as_rw}) now looks as
\begin{eqnarray}
\label{sigma_model_rw_eq}
w\lr{x} = \kappa \sum \limits_{\mu} w\lr{x \pm e_\mu} + \lr{1 - 2 D \kappa} \, \delta\lr{x} .
\end{eqnarray}
Combining this equation with the definition (\ref{sigma_model_kappa_def}), it is easy to show that $\kappa$ obeys the following self-consistency condition:
\begin{eqnarray}
\label{sigma_model_kappa_return_prob}
\kappa = \frac{1}{2 D + \lambda w\lr{0}} .
\end{eqnarray}
The equation (\ref{sigma_model_rw_eq}) has the form (\ref{random_process_eq}) without the nonlinear term and thus can be interpreted as the equation for the stationary probability distribution of the position of an ordinary bosonic random walk, defined by the following possible actions at each discrete time step:
\begin{description}
\item[Move:] With probability $2 D \kappa$ move along the random unit lattice vector $\pm e_{\mu}$.
\item[Restart:] With probability $\lr{1 - 2 D \kappa}$ start again at the origin $x = 0$.
\end{description}
This ensures that $w\lr{0} > 0$ and hence $\kappa$ never exceeds its critical value $\kappa_c = \lr{2 D}^{-1}$. Therefore $\xi\lr{x}$ and $w\lr{x}$ can be expanded in powers of $\kappa$.
Thus we have defined a new expansion parameter $\kappa$, which should obey the self-consistency equation (\ref{sigma_model_kappa_return_prob}), and obtained a well-defined convergent expansion, namely, the sum over all paths on the lattice with the weight $\kappa^L$, where $L$ is the length of the path. Note that the quantity $\xi\lr{\pm e_\mu}$ in fact plays the role similar to the gluon condensate (which is expressed in terms of the mean plaquette in lattice theory) in non-Abelian gauge theory: one can absorb all the divergences into the self-consistent definition of condensates \cite{Parisi:77:1, Kazakov:94:1}.
The final step in the construction of the nonlinear random process which solves the equations (\ref{sigma_model_sd}) is the solution of the self-consistency equation (\ref{sigma_model_kappa_return_prob}). One possible solution is to use the iterations
\begin{eqnarray}
\label{sigma_model_iterations}
\kappa_{i+1} = \frac{1}{2 D + \lambda w\lr{0; \kappa_i}} .
\end{eqnarray}
Here $w\lr{0; \kappa}$ is the return probability of a bosonic random walk with hopping parameter $\kappa$. In practice, one should simulate the bosonic random walk at fixed $\kappa = \kappa_i$ for some number $T$ of discrete time steps, and then estimate $w\lr{0; \kappa_i}$ as $w\lr{0; \kappa_i} \approx t\lr{0}/T$, where $t\lr{0}$ is the number of discrete time steps spent at $x = 0$. From (\ref{sigma_model_iterations}) one then gets $\kappa_{i+1}$, and the process is repeated until the value of $\kappa$ stabilizes with sufficient numerical precision. We call such algorithm ``Algorithm A''. One can also consider an ultimate case, for which the return probability is updated and estimated as $t\lr{0}/t$ every time the point $x = 0$ is reached. Now $t$ is the time from the start of the random process and $t\lr{0}$ is the number of time steps spent at $x = 0$. Such algorithm will be called ``Algorithm B''.
Mathematically, such random processes are not Markov processes, since the transition probabilities at each next step depend (via $\kappa_i$) on the behavior of the process at all previous time steps. Stationary probability distributions of such processes obey nonlinear equations (such as (\ref{sigma_model_sd})) \cite{Frank:04:1}, and and hence they are also called nonlinear random processes.
As an interesting side remark, let us discuss such a theory at finite temperature, which is described by a bosonic random walk on the cylinder. Clearly, an ordinary bosonic random walk does not feel this compactification of space, and its stationary probability distribution is just a periodic linear combination of the corresponding distribution in infinite space. Such behavior cannot lead to any nonlinear finite-temperature effects such as phase transitions. On the other hand, if the parameters of the random walk depend on the return probability, as in (\ref{sigma_model_kappa_return_prob}), there is a nonlinear feedback mechanism since in the compactified space the returns are more likely. Thus finite temperature indeed affects the local behavior of the random walker with memory and might lead to interesting critical phenomena.
\begin{figure}
\includegraphics[width=6cm, angle=-90]{approach_history.eps}\\
\caption{The process of convergence of the random process with memory which solves the Schwinger-Dyson equations (\ref{sigma_model_sd}) for $D = 2$. The quantity plotted is the estimate of the lattice spacing $a\lr{\lambda, t}$ after $t$ discrete time steps, with the exact result $a\lr{\lambda, t \rightarrow \infty}$ subtracted.}
\label{fig:sigma_model_convergence}
\end{figure}
In order to illustrate such a stochastic solution of the equations (\ref{sigma_model_sd}), we consider the case $D = 2$. In two dimensions the model (\ref{sigma_model_pf}) is asymptotically free, and one can introduce the lattice spacing by fixing the value of mass in physical units (we set $m_{phys} = 1$): $m_{phys} \, a\lr{\lambda} = m_{latt}\lr{\lambda} = \sqrt{\kappa^{-1}\lr{\lambda} - 2 D}$. The process of convergence of the lattice spacing to its exact value is illustrated on Fig. \ref{fig:sigma_model_convergence} for both the algorithms ``A'' and ``B''. For algorithm ``A'' we have used $T = 5 \cdot 10^5$. The algorithm ``A'' converges much faster than the algorithm ``B''. The values of lattice spacing obtained using both algorithms are compared with the exact solution on Fig. \ref{fig:spacing_vs_lambda}. In agreement with asymptotic freedom, lattice spacing quickly decreases with $\lambda$. Again, algorithm ``A'' yields more precise results in the same number of time steps.
\begin{figure}
\includegraphics[width=6cm, angle=-90]{spacing_vs_lambda.eps}\\
\caption{Numerical estimates of the lattice spacing as a function of the coupling constant for the two-dimensional large-$N$ $O\lr{N}$ sigma model, compared with the exact result. The estimates were obtained using both algorithms ``A'' and ``B'' with different number of time steps.}
\label{fig:spacing_vs_lambda}
\end{figure}
\section*{Discussion and conclusions}
\label{sec:conclusions}
In this paper we have presented numerical strategies for the stochastic summation and re-summation of perturbative expansions in large-$N$ quantum field theories. Our basic approach was to interpret the Schwinger-Dyson equations as the equations for the stationary probability distribution of some random process. Since Schwinger-Dyson equations in such theories are nonlinear equations, we had to use so-called nonlinear random processes, rather than ordinary Markov processes whose stationary probability distributions always obey linear equations.
It is interesting to note that since the configuration spaces of random processes described in this paper are discrete, their numerical implementation require floating-point operations only for the random choice of actions. Thus such algorithms can be potentially much faster than the standard Monte-Carlo simulations based on floating-point arithmetic, and can be advantageous for machines based on GPUs.
Our final goal is to extend the presented approach to non-Abelian lattice gauge theories. However, in this case direct stochastic interpretation of Schwinger-Dyson equations is only possible at strong coupling, while the continuum limit of such theories corresponds to the weak-coupling limit. In Section \ref{sec:rps_with_mem}, we have discussed a way to access the weak-coupling limit, which, however, was implemented numerically only for $O\lr{N}$ sigma-model at large $N$. The basic idea is to absorb the divergences into a self-consistent redefinition of the expansion parameter and solve the self-consistency conditions using random processes with memory. In some sense, $O\lr{N}$ sigma-model can be thought of as the bosonic random walk in its own condensate, and the approach to the self-consistent value of mass gap (see Fig. \ref{fig:sigma_model_convergence}) - as a renormalization-group flow.
For non-Abelian gauge theories the redefined expansion parameters can emerge as the lagrange multipliers for the ``zigzag symmetry'' of the QCD string and should also satisfy some self-consistency conditions \cite{Kazakov:94:1}. Zigzag symmetry means that when one adds a line which is passed forward and backward to the boundary of the fluctuating string, the amplitudes should not change. In lattice gauge theory, this condition is equivalent to the unitarity of the link variables $U\lr{x, \mu}$, which is similar to the condition $|n\lr{x}| = 1$ in $O\lr{N}$ sigma-model. These redefined parameters can be also related to the gluon condensate \cite{Kazakov:94:1}. By analogy with the sigma-model, one can think that non-Abelian gauge theories are similar to strings moving in some self-consistent condensates. Such a picture is also close to the idea of holographic AdS/CFT duality for non-Abelian gauge theories, where the dual string lives in some self-consistent gravitational background, and the parameters of this background can be related to gluon condensates in gauge theory \cite{Polyakov:99:1}. In fact, the requirement that the metric of the holographic background approaches that of the AdS space-time ensures the zigzag symmetry of the strings which end on the AdS boundary \cite{Polyakov:99:1}.
In view of these qualitative considerations, our hope is that the loop equations in non-Abelian gauge theories can be solved stochastically by a random process similar to the one which was devised for the Weingarten model of random surfaces (see Subsection \ref{subsec:weingarten}), but with some self-consistent choice of parameters, which might be implemented as the ``memory'' in the random process.
Among other possible applications of the presented method one can think of the solution of Schwinger-Dyson equations in continuum gauge theories, combined with the Renormalization Group methods \cite{Pawlowski:07:1}, numerical analysis of quantum gravity models described by various matrix models, and numerical solution of hydrodynamical equations \cite{Migdal:94:2}.
It should be noted here that several attempts at the stochastic solution of the loop equations in large-$N$ gauge theories have been already described in the literature quite a long time ago \cite{Migdal:86:1}. These algorithms were, in essence, based on the so-called branching random processes, so that the Wilson loop $W\lr{C}$ is proportional to the probability of transition from the initial loop configuration $C$ to the empty configuration with no loops. In particular, in contrast to the algorithm described in Subsection \ref{subsec:weingarten}, where one of the basic steps is to join loops, in the algorithms described in \cite{Migdal:86:1} the basic step was to split a self-intersecting loop into two loops. As a result, these algorithms did not implement the importance sampling and were not able to produce any sensible results for the four-dimensional gauge theory. Generally, branching random processes similar to those considered in \cite{Migdal:86:1} can be obtained from the ``recursive'' nonlinear random process described in this paper by time reversal. However, since such processes do not satisfy any detailed balance condition, they are not invariant under this operation, and lead to very different numerical algorithms.
\begin{acknowledgments}
I am grateful to Drs. M. I. Polikarpov, Yu. M. Makeenko, A. S. Gorsky, N. V. Prokof'ev and I. Ya. Aref'eva for interesting and stimulating discussions. I'd like also to thank Drs. F. Bruckmann and A. Schaefer for their kind hospitality at the University of Regensburg, where a part of this work was written. This work was partly supported by grants RFBR 09-02-00338-a, RFBR 08-02-00661-a, a grant for the leading scientific schools NSh-6260.2010.2, by the Federal Special-Purpose Programme ``Personnel'' of the Russian Ministry of Science and Education, and by personal grants from the ``Dynasty'' foundation and from the FAIR-Russia Research Center (FRRC).
\end{acknowledgments}
|
{
"timestamp": "2011-02-25T02:01:46",
"yymm": "1009",
"arxiv_id": "1009.4033",
"language": "en",
"url": "https://arxiv.org/abs/1009.4033"
}
|
\section{Background}
Transition metal oxides have drawn the attention of the scientific community for the last 50 years. Particularly, cobalt oxides are becoming increasingly important because of their interesting properties such as superconductivity,\cite{superconductivity} colossal magnetoresistance \cite{colossal} or phase separation.\cite{phase}
To elaborate models for strongly correlated electron materials, one-dimensional (1D) systems are key since they are the easiest to study because all the interesting phenomena take place along one direction. In this regard, there has been much interest in analyzing the homologous series A$_{n+2}$B$'$B$_n$O$_{3n+3}$\cite{sugiyama,est_1,est_3} (A alkaline or alkaline earth cations, B$'$ and B commonly corresponding to Co cations in a trigonal prismatic and octahedral position, respectively, and n$\in$[1,$\infty$)) where the one-dimensional chain is represented by B$'$B$_n$O$_{3n+3}$. Boulahya \textit{et al.} \cite{est_2}, reported the integer terms of the series which can be stabilized with Co occupying both octahedral and prismatic sites, by varying the nature and the proportion of alkaline-earth cations.
In particular, both the end members of the series (Ca$_{3}$Co$_{2}$O$_{6}$ (n=1) and BaCoO$_{3}$ (n=$\infty$)) have focused much attention for the past years. The n=1 compound (Ca$_{3}$Co$_{2}$O$_{6}$) has been analyzed in several previous works.\cite{sugiyama,cacoomaignan,cacoo_aasland,cacoo_aasland_2,cacoo_kageyama,cacoo_fontcuberta,cacoo_khomskii,cacoo_goodenough} The valence state of Co ions in this case is assigned to be 3+, with low spin state for Co ions in the CoO$_6$ octahedron (S=0) and high spin state within the trigonal prism (S=2). This compound shows a paramagnetic behavior at high temperature. In-chain ferromagnetic (FM) interactions arise below 80~K, reaching 1D FM order at 30~K. Below this temperature, interchain two-dimensional (2D) antiferromagnetic (AFM) interactions appear, evolving into a ferrimagnetic order below 24~K (FM order within the chains that are partly antiferromagnetically coupled). According to Wu \textit{et al.},\cite{cacoo_khomskii} the FM intrachain interactions obey an Ising type model due to the strong spin-orbit coupling effects on the Co ions within a trigonal prismatic environment. However, Cheng \textit{et al.}\cite{cacoo_goodenough} affirm that the FM intrachain-AFM interchain competition invalidates the application of an Ising model. This material has attracted much attention due to the magnetization plateaus observed in the magnetization versus field curves.\cite{cacoomaignan}
The n=$\infty$ member (BaCoO$_{3}$) has also drawn considerable interest.\cite{bacoo_est,bacoo_yamaura,bacoo_struct,bacoo_abinit,bacoo_clusters,sugiyama,bacoo_ps,bacoo_2DAF,bacoo_vpardo} It crystallizes in a 2H hexagonal pseudo-perovskite structure, in which there are just face-sharing CoO$_6$ octahedra forming the 1D CoO$_3$ chain. In this case, the valence state of Co atoms is 4+. \textit{Ab initio} calculations predicted a FM ground state along the chains.\cite{bacoo_abinit} The c-axis is assigned to be an easy direction for the magnetization, leading to a large value of the orbital angular momentum. Also a large, Ising-type, magnetocrystalline anisotropy is estimated.\cite{bacoo_vpardo}
The introduction of prisms in the structure of 2H-BaCoO$_3$ is accompanied by a decreasing of the c$_{2H}$ parameters. Keeping B= Co, to increase the P/O ratio, (P= prism, O= octahedra) control of both temperature and annealing time besides the adequate selection of A cation is required to prepare these materials. The distance between A cations is one of the main factors governing the structural type which can be stabilized in the A$_{n+2}$B$'$B$_n$O$_{3n+3}$ series.
For this reason, there are fewer works on the magnetic or electronic structure properties for the compounds with 2$\leq$n$\leq$$\infty$. The work by Sugiyama \textit{et al.} \cite{sugiyama} studied the electronic structure and magnetic properties of the members n= 1,~2,~3,~5, and $\infty$. The study reported an n-dependence of the charge and spin distribution of the Co chains. They proposed a charge distribution within the chains based on Co$^{4+}$ cations (located in octahedra) and Co$^{3+}$ ones (located in both trigonal prisms and octahedra). The existence of a magnetic transition was shown for all the compounds. Above the temperature atributed to this transition (T$_C^{on}$), a relatively strong 1D FM order appears. They suggested that, for the compounds with n=1,~2,~3, and 5, T$_C^{on}$ is induced by an interchain 2D AFM interaction. Another magnetic study reported by Sugiyama \textit{et al.}\cite{sugiyama2} confirmed the role of this 2D AFM interaction in the series. The structural work carried out by Harrison \textit{et al.}\cite{harrison} for the Sr$_6$Co$_5$O$_{15}$ phase (n=4) reported that this compound is a 2H-hexagonal perovskite related oxide, isostructural with Ba$_6$Ni$_5$O$_{15}$, phase described by Camp\'a \textit{et al}.\cite{campa} Structure, magnetic properties and electronic structure of single crystals of the oxygen-deficient compound Sr$_6$Co$_5$O{$_{14.7}$} have been studied by Sun \textit{et al.}\cite{sunand} They showed this compound to have unique polyhedral chains, consisting of a random composite of octahedra+trigonal prisms and octahedra+intermediate polyhedra. The magnetic properties of the compound can be understood according to that structural picture being the Co$^{4+}$ ions located in the octahedra and the Co$^{2+}$ ones in either the trigonal prisms or the intermediate polyhedra.
Whangbo \textit{et al.} \cite{srcoo_magn} proposed an interpretation of the electronic structure of Sr$_6$Co$_5$O{$_1$$_5$} using a H\"uckel tight binding calculation.\cite{huckel} According to their model, the polyhedral chains are composed by Co$^{4+}$ ions in the octahedral sites and Co$^{2+}$ ions in the trigonal prismatic ones.
The electrical resistivity and Seebeck coefficient dependence with the temperature of Sr$_6$Co$_5$O{$_1$$_5$} have been measured in some previous works\cite{iwasaki,takami_2} and also for the closely related compound (Sr$_{0.75}$Ba$_{0.25}$)$_6$Co$_5$O{$_1$$_5$}.\cite{takami}
The purpose of this paper is to analyze the electronic structure and special magnetic properties of the n=4 member of the series, Sr$_6$Co$_5$O{$_1$$_5$}. We have synthesized polycrystalline samples of the compound, characterized and analyzed its magnetic properties experimentally. Moreover, we have studied its electronic structure by ab initio methods, analyzing the plausible magnetic configurations and obtaining the magnetic ground state of the system. Also, we have calculated the thermopower using the standard Boltzmann transport theory based on the electronic structure obtained by first principles.
\section{Experimental and computational details}
The SrCoO$_{3-\delta}$ , ``H'' polymorph, Ba$_6$Ni$_5$O$_{15}$-like, was obtained in polycrystalline form by a citrate technique. Stoichiometric amounts of analytical grade Sr(NO$_3$)$_2$ and Co(NO$_3$)$_2.6$H$_2$O were dissolved in citric acid. The solution was slowly evaporated, leading to an organic resin which was dried at 140 $^\circ$C and slowly decomposed at 600 $^\circ$C for 12 h. The sample was then heated at 900 $^\circ$C in air. The hexagonal phase was obtained by slowly cooling in the furnace. The reaction product was characterized by X-ray diffraction (XRD) for phase identification and to asses phase purity. The characterization was performed using a Bruker-axs D8 diffractometer (40 kV, 30 mA) in Bragg-Brentano reflection geometry with Cu K$\alpha$ radiation.
Neutron powder diffraction (NPD) diagrams were collected at the Institut Laue-Langevin, Grenoble (France). The diffraction patterns were acquired at the high-resolution D2B diffractometer with $\lambda$ = 1.594 \AA, at 295 K and at 5 K in the angular range 10$^\circ$ $<$ 2$\theta$ $<$ 156$^\circ$ with a 0.05$^\circ$ step. NPD diffraction patterns were analyzed by the Rietveld method,\cite{rietveld} using the FULLPROF refinement program.\cite{carvajal} A pseudo-Voigt function was chosen to generate the line shape of the diffraction peaks. The coherent scattering lengths for Sr, Co, and O were: 7.020, 2.490, and 5.803 fm respectively. The following parameters were refined in the final run: scale factor, background coefficients, zero-point error, pseudo-Voigt corrected for asymmetry parameters, positional coordinates, and isotropic thermal factors.
The magnetization (M) between 5 and 320 K was measured in a superconducting quantum interference device magnetometer (Quantum Design) under a dc magnetic field H= 100 Oe. Data were taken upon heating in both zero-field-cooling (ZFC) and field-cooling (FC) regimes.
The electronic structure calculations were performed with the WIEN2k code,\cite{wien2k} based on density functional theory (DFT) utilizing the augmented plane wave plus local orbitals method (APW+lo). For the calculations of the transport
properties we used the BoltzTraP code \cite{boltztrap}, that
takes the energy bands obtained using the WIEN2k
software.
For this moderately correlated transition metal oxide, we used the LDA+U \cite{sic} approach including self-interaction corrections in the so-called ``fully localized limit'' with U= 4.8 eV and J= 0.7 eV. This method has proven reliable for transition metal oxides, since it improves over the generalized gradient approximation (GGA) or local density approximation (LDA) in the study of systems containing correlated electrons by introducing the on-site Coulomb repulsion U. Results presented here are consistent for values of U in the interval from 4 to 8 eV, in a reasonable range compared to other similar cobaltates,\cite{cacoo_khomskii,bacoo_vpardo} to describe correctly the insulating behavior of the material and the localized nature of its electronic structure.
The calculations were fully converged
with respect to the k-mesh and R$_{mt}$K$_{max}$.
Values used for the k-mesh were 6$\times$6$\times$6 sampling of the full Brillouin zone for electronic structure calculations, and
21$\times$21$\times$21 for the transport properties.~R$_{mt}$K$_{max}$= 6.0
is chosen for all the calculations. Selected muffin tin radii were
the following: 1.82 a.u. for Co, 2.28 a.u. for Sr, and 1.61 a.u. for O.
Based on scalar relativistic basis functions, spin orbit coupling (SOC) effects were included
in a second-variational procedure.\cite{singh}
\section{Results}
\subsection{Structure}
The crystal structure (see Table \ref{positions}) of this phase was refined by Rietveld analysis of the NPD data and published by us \cite{cristina} in the R32 space group (no.~155), Z = 3, at room temperature and at 5 K, starting from the model defined by Harrison \textit{et al.}\cite{harrison} It contains two strontium, three cobalt, and three oxygen atoms in the asymmetric unit. The refinement of the occupancy factor for the Co atoms leads to a significant reduction of its contents for Co1, whereas Co2 and Co3 remained stoichiometric. The oxygen occupancy was slightly deficient for O2 and fully stoichiometic for O1 and O3. The refined crystallographic formula was SrCo$_{0.78(1)}$O$_{2.48(2)}$. According to this formula, the average oxidation state for Co is 3.79(1)+. The presence of Co$_3$O$_4$, segregated from the main phase during the synthesis process, was quantified as 2\%.
Both Sr atoms are 8-fold coordinated by O atoms ($\langle$Sr1-O$\rangle$ = 2.592(2) \AA, $\langle$Sr2-O$\rangle$ = 2.751(2) \AA) in irregular coordination. The atoms Sr1 and Sr2 are disposed in columns parallel to the c direction. Two cobalt atoms, Co2 and Co3 (site 6c) are octahedrally coordinated by oxygen atoms ($\langle$Co2-O$\rangle$ = 1.896(6) \AA, $\langle$Co3-O$\rangle$ = 1.905(6) \AA) and the third, Co1 (site 3b) occupies a distorted trigonal prism ($\langle$Co1-O$\rangle$ = 1.922(2) \AA~and O-Co1-O = 76.8(4)$^\circ$). The crystal structure consists of isolated, infinite chains of face-sharing CoO$_6$ polyhedra running along the c direction forming a trigonal lattice in the ab plane as can be seen in Fig.~\ref{struct}a). The repeat unit for Co-O species consists of four distorted octahedra sharing faces intermingled with prismatically-coordinated Co1 atoms (see Fig.~\ref{struct}b),c)). The inter-octahedral cobalt-cobalt distances are: Co2-Co3 = 2.39(3) \AA, Co2-Co2 = 2.54(3) \AA. The face-sharing prismatic/octahedra Co1-Co3 distance is 2.53(2) \AA. These results \cite{cristina} are comparable with those obtained by Harrison \textit{et al.}\cite{harrison} for the Sr$_6$Co$_5$O$_{15}$ phase, although in our case we are in the presence of a more severely Co deficient compound, with a slightly smaller cell volume of 968.3 \AA$^3$ (969.6 \AA$^3$ for Sr$_6$Co$_5$O$_{15}$ \cite{harrison}) as corresponding to a higher average oxidation state for Co cations. The in-plane distance between Co chains is around 5.62 \AA~in the ``ideal'' structure and is flanked, in the refined structure, by 5.54 and 5.75 \AA~(measured between different pairs of Co atoms belonging to two neighboring chains).
\begin{figure}
\includegraphics[width=\columnwidth,draft=false]{paper_fig1.eps}
\caption{(Color online) a) Top view of the structure of Sr$_6$Co$_5$O$_{15}$ showing the hexagonal symmetry of the ab plane, and the 6-fold coordination of the Co atoms by O atoms. The in-plane distance between Co chains is significantly larger than the Co-Co in-chain distance, leading to the structural quasi-one-dimensionality. Sr atoms are not shown for simplicity. b) Schematic picture of the structure of the CoO$_3$ chains in Sr$_6$Co$_5$O$_{15}$ showing the polyhedral environment of the different Co cations that occur along them. In the unit cell, formed by 5 Co atoms, 4 of them are situated in a distorted octahedron (blue color) and one is in a trigonal prismatic environment (yellow color). c) Detail of the face-sharing arrangement along the chains.}\label{struct}
\end{figure}
\begin{table}[h!]
\caption{Atomic positional parameters for Sr$_6$Co$_5$O$_{15}$ ``H'' phase after Rietveld refinement of NPD data at 5K. The space group of our compound is R32 (no. 155) and the lattice parameters are a=~b=~9.4740 \AA ~and c=~12.3606 \AA.}\label{positions}
\begin{center}
\begin{tabular}{c c c c c}
\hline
\hline
& Crystallographic& \\
Atom & position & Coordinates \\
\hline
Sr1 & 9e & (0.6437,0.0000,0.5000) \\
Sr2 & 9d & (0.3210,0.0000,0.0000) \\
Co1 & 3b & (0.0000,0.0000,0.5000) \\
Co2 & 6c & (0.0000,0.0000,0.1030) \\
Co3 & 6c & (0.0000,0.0000,0.2960) \\
O1 & 9d & (0.8436,0.0000,0.0000) \\
O2 & 18f & (0.4946,0.6728,0.4785) \\
O3 & 18f & (0.8436,-0.0240,0.6088) \\
\hline
\end{tabular}
\end{center}
\end{table}
\subsection{An ionic model}
Being the compound a correlated insulating oxide,\cite{iwasaki} an image based on an ionic point charge model (PCM) can give us a crude estimate of the possible electronic configuration of the material. We will use such a model to describe the charge distribution of the cations along the Co chain. We will consider possible ionic configurations for the different Co ions along the chain and calculate their total energy, just based on the electrostatic repulsion, simplifying to take into account only the first neighbor contribution, and neglecting other energetic terms.
Taking the usual valencies for Sr and O, the average valence for Co in the ideal stoichiometric compound Sr$_6$Co$_5$O$_{15}$ is +3.6. Following the PCM we have just described, the valencies of the Co ions can be distributed in two isoenergetic ways in order to minimize the Coulomb repulsion: i) 4 Co$^{4+}$ (d$^{5}$)+ 1 Co$^{2+}$ (d$^7$) ; ii) 3 Co$^{4+}$ (d$^5$)+ 2 Co$^{3+}$(d$^6$). Various ionic arrangements have been considered in the literature. According to the structure determined by Harrison \textit{et al.},\cite{harrison} Sun \textit{et al.}\cite{sunand} proposed 4 Co$^{4+}$+ 1 Co$^{2+}$ is the most suitable model because the polyhedral chain with 4 octahedral sites and one trigonal prismatic allows Co$^{4+}$ and Co$^{2+}$ to be located in different sites.
Whangbo \textit{et al.}\cite{srcoo_magn} also suggested the 4 Co$^{4+}$ (d$^{5}$)+ 1 Co$^{2+}$ (d$^7$) model using a H\"uckel tight binding calculation. This would be what we called solution i). Instead, Sugiyama \textit{et al.} \cite{sugiyama} proposed the existence of at least one non-magnetic atom in the chain for all the members of the series A$_{n+2}$Co$_{n+1}$O$_{3n+3}$: as n increases from 1 up to infinity, the Co valence increases from +3 and approaches +4 (e.g. for n=1, the charge distribution in the unit cell is 2 Co$^{3+}$; for n=2, 2 Co$^{3+}$+ 1 Co$^{4+}$ and in our case for n=4, 2 Co$^{3+}$+ 3 Co$^{4+}$).
This, we called solution ii). In addition, in Ref.~[\onlinecite{sugiyama}] is suggested that the spin distribution for the 2 Co$^{3+}$ ions in the chain is: a high spin state (HSS) with S=2 for the Co within a trigonal prismatic environment and a low spin state (LSS) with S=0 for the octahedral one. Meanwhile, the Co$^{4+}$ ions are all in a LSS with S=1/2 and located in the remaining octahedra.
Both solutions are equivalent energetically from an oversimplified ionic picture, but can be easily distinguished because solution ii) could lead to two non-magnetic Co atoms, whereas solution i) will have all the atoms being magnetic.
Our ab initio calculations confirm that the ground state electronic structure can be well described by an ionic model with 3 Co$^{4+}$ and 2 Co$^{3+}$ cations (solution ii) of our PCM). Below, we will give further details of the electronic structure beyond this simple ionic model.
\subsection{Electronic structure calculations}
Table~\ref{momentos_tabla} shows the magnetic moments of each Co cation in the structure obtained for U= 4.8 eV. They are consistent with the ionic distribution that would predict two non-magnetic Co$^{3+}$:~d$^6$ cations to occur along the chain. Co1 is a Co$^{4+}$:~d$^5$ cation with a magnetic moment of 1 $\mu_B$. Co2 is also a Co$^{4+}$:~d$^5$ cation and Co3 is close to a Co$^{3+}$:~d$^6$ configuration. The details can be understood looking at the magnetic interactions in the chain (see Fig.~\ref{acoplos}). The magnetic coupling between Co1 and Co2 is mediated by a Co3 (non magnetic). The overlap between Co3 and Co2 d-orbitals motivates a charge transfer that explains the magnetic moments obtained for them (lowered from 1 for Co2 (Co$^{4+}$:~d$^5$) and risen from 0 for Co3 (Co$^{3+}$:~d$^6$)). The Co$^{4+}$ cations are in a LSS (S=1/2) in agreement with Ref.~[\onlinecite{sugiyama}]. However, for our compound, both the Co$^{3+}$ atoms are non-magnetic (LSS) and located in octahedra.
\begin{table}[h!]
\caption{Projection of the spin magnetic moments of Co atoms in the Sr$_6$Co$_5$O$_{15}$ ground state.}\label{momentos_tabla}
\begin{center}
\begin{tabular}{c c c c c}
\hline
\hline
Atom & & Magnetic Moment & \\
\hline
Co1 & & 1.0$~\mu_B$ \\
Co2 & & -0.8$~\mu_B$ \\
Co3 & & -0.2$~\mu_B$ \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{figure}
\includegraphics[width=0.90\columnwidth,draft=false]{paper_fig2.eps}
\caption{(Color online) Magnetic couplings in the unit cell. We show the FM coupling between neighbor Co2 atoms and the AFM one between Co2 and Co1 mediated by a non-magnetic Co3.}\label{acoplos}
\end{figure}
A more realistic description of the electronic structure of the material is given by the partial density of states (DOS) plots of the various Co atoms in the structure (see Fig.~\ref{dos}). The material is an insulator, with a d-d gap of about 0.5 eV, for this particular value of U (4.8 eV). For Co1, a Co$^{4+}$:~d$^5$ cation in a trigonal prismatic environment, we can see a full d$_{z^2}$ level (being z the Co-chain axis) fully occupied, a hole in an xy-plane orbital (x$^2$-y$^2$, xy) at about 2 eV above the Fermi level and the higher-lying d$_{xz}$, d$_{yz}$ which remains unoccupied at higher energy, spin split by about 1 eV. The bands coming from Co2 (Co$^{4+}$:~d$^5$), in an octahedral environment, present a spin splitting of about 1 eV of the e$_g$ bands, located at 2 eV above the Fermi level. In this case, the magnetic moment points along the minority spin direction (the hole in the t$_{2g}$ multiplet is in the majority spin channel). The unoccupied t$_{2g}$ band of Co2 presents a double peak structure at about 1 eV above the Fermi level. We can observe an approximate d$^6$ DOS for Co3. Due to the hybridization between Co3 and Co2 d-orbitals, a double peak structure arises, showing a density of unoccupied t$_{2g}$ states for Co3 much smaller than for Co2 at about 1 eV above the Fermi level, consistent with the small magnetic moment of Co3.
\begin{figure}
\includegraphics[width=\columnwidth,draft=false]{paper_fig3.eps}
\caption{(Color online) Partial spin-polarized DOS of Co1, Co2, and Co3 atoms. Fermi energy is represented by the solid vertical line at zero. Co1 and Co2 are close to a d$^{5}$ electronic structure. Co3 is closer to a d$^{6}$ configuration with a little unoccupied t$_{2g}$ character.}\label{dos}
\end{figure}
Two features can be observed in these plots: i) the strongly localized nature of the electrons, with very narrow bands, less than 0.5 eV wide. The band widths are always smaller than the typical energies involved: both the Hund's rule (1 eV for the e$_g$ bands of the Co$^{4+}$:~d$^5$ cations) and crystal field splitting, ii) the different crystal field environments of the Co cations (octahedral for Co2 and Co3, trigonal prismatic for Co1), that lead to well different splittings with the e$_g$ states of Co3 being highest in energy, at $\approx$ 3~eV above the Fermi level for this U value chosen of 4.8 eV.
From these results, we can roughly sketch the electronic structure of Sr$_6$Co$_5$O$_{15}$: its unit cell is formed by three magnetic Co$^{4+}$:~d$^5$ cations (one Co1 in a trigonal prismatic environment and two Co2 in an octahedral environment) and two non-magnetic Co$^{3+}$:~d$^6$ atoms (Co3) in an octahedral environment.
\subsection{Magnetic properties}
We have performed LDA+U calculations for several values of the on-site Coulomb repulsion term. Since we are dealing with an insulating d$^5$/d$^6$ system, a value of U between 4 and 8 eV is reasonable to describe it correctly.\cite{ylvisaker} The magnetic ground state solution we will describe below (magnetic moments of Co atoms in the ground state are written in Table~\ref{momentos_tabla}) is the most stable for the said range of values of the on-site Coulomb repulsion.
Starting from the electronic structure described above, we can understand the magnetic couplings in the unit cell. Two couplings can be considered (see Fig.~\ref{acoplos}): one ferromagnetic (J$_{FM}$) direct exhange between nearest neighbor magnetic Co2:~d$^5$ cations and another one antiferromagnetic (J$_{AFM}$) between Co1:~d$^5$ and Co2:~d$^5$ mediated by a non-magnetic cation Co3:~d$^6$, that acts in a similar way to O anions in the oxygen-mediated superexchange in perovskites. Both these couplings can be understood in terms of the Goodenough-Kanamori-Anderson rules.\cite{goodenough_book} We can use our total energy calculations to describe and quantify the magnetic interactions in the unit cell (schematically depicted in Fig.~\ref{acoplos}). In order to give an estimate of the couplings along the chain, we can fit the total energies resulting from various possible collinear magnetic configurations to a Heisenberg model, in the form $H= \frac{1}{2}\sum^{}_{i,j}J_{ij}S_iS_j$. Calculations reveal the following values for the coupling constants J$_{FM}\approx220K$ and J$_{AFM}\approx6K$ (of opposite sign). As expected, the FM coupling is stronger than the AFM one, that occurs between second-neighbor cations mediated by a non-magnetic ion. The small AFM coupling is consistent with the fact that no long-range magnetic order along the Co-chains is observed experimentally above 4~K. Because of this peculiar magnetic arrangement, no 1D FM order (along the Co-chains) is observed, but only a short-ranged FM coupling between Co2 cations survives at high temperature. The total ordered moment is 1 $\mu_B$ per unit cell.
Figure 4 shows the magnetic susceptibility as a function of the temperature at 100 Oe. At 32 K, a kink can be noticed in the ZFC curve, which can be ascribed to the N\'eel temperature (T$_N$). Also, this is consistent with the significant increase of $\chi$ below this temperature observed in the FC curve. This magnetic transition could be related to an interchain 2D AFM interaction in the triangular lattice of the ab plane as in the other members of the series.\cite{sugiyama} Confirming this picture, a large negative Curie-Weiss temperature ($\theta= -109$ K) is obtained as in the n=2 and n=3 compounds.\cite{sugiyama} The T$_N$ value given by Sugiyama \textit{et al.}\cite{sugiyama2} for Sr$_6$Co$_5$O$_{15}$ is about 70 K, in agreement with the general magnetic behavior of the series. According to them, this temperature is due to the transition from a 1D FM order along the chains to a 2D AFM ordered state. In addition, no clear anomalies in the susceptibility vs. temperature curve for the members with n=1, 2, 3, and 5 are observed in Ref.~[\onlinecite{sugiyama}]. In our case, as we have seen above, there is no 1D FM order along the chains. Also, reasoning in terms of the hexagonal planes, non-magnetic planes formed by Co3 atoms intercalate between the magnetic ones. All this contributes to the lowering of T$_N$ with respect to the rest of the series.
\begin{figure}
\includegraphics[width=8cm,draft=false]{paper_fig4.eps}
\caption{(Color online) Magnetic susceptibility vs. temperature measured at H=100 Oe under ZFC and FC conditions. The N\'eel temperature that can be identified from the curves is about 32 K. The inset shows the variation of FC inverse susceptibility at low temperatures. $\theta$ represents the Curie-Weiss temperature.}\label{pablo}
\end{figure}
From our calculations, we can also obtain the value of the effective magnetic moment per formula unit and compare it with our experimental findings. Hence, taking the usual expressions for the square effective paramagnetic moment of each magnetic Co, $\mu_{eff}^2= [g_l^2l(l+1)+g_s^2s(s+1)]~\mu_B^2$ and considering l= 0 or 1, $\mu\in$[5.2, 6.7]~$\mu_B$ for the whole unit cell formed by 5 Co cations. Depending on the value of the orbital angular momentum (see below the details of our calculations), that would be the range of possible values for $\mu_{eff}$. This is consistent with the 5.6 $\mu_B$ value obtained by Sun el al,\cite{sunand} (if the orbital angular momenta were negligible), and it agrees with our experimental magnetic moment of 6.9 $\mu_B$ per formula (if the orbital angular momenta were aligned with the magnetization). This value was obtained from the linear fitting of $\chi^{-1}$ vs T curve in the paramagnetic region (see inset of Fig. 4).
\subsection{A peculiar quasi-one-dimensional oxide}
The magnetic properties of the material can be placed into the context of the other members of the same structural series.\cite{sugiyama} Keeping in mind the T$_N$ dependence with the in-plane distance for the quasi-one-dimensional cobaltates shown in Ref.~[\onlinecite{sugiyama2}] we can see the T$_N$ value for Sr$_6$Co$_5$O$_{15}$ of about 70 K, sensibly higher than the one obtained experimentally by us (see Fig.~\ref{pablo}) and others.\cite{sunand}
Both the end members of the series have in common an Ising-type behavior,\cite{cacoo_khomskii,bacoo_vpardo} with moments aligned along the chain direction with large values of the magnetocrystalline anisotropy. This quasi-one-dimensionality is somehow not observed in Sr$_6$Co$_5$O$_{15}$. We have studied the system by introducing SOC in the calculations with the magnetization lying along different crystallographic directions. However, none of these directions can align the orbital and magnetic moments of all the Co atoms at the same time (see Table~\ref{momentos_sot}). Large values of the orbital angular momenta are obtained for the Co atoms in the structure when the magnetization is set along different directions: the preferred direction for orienting their moments is different for each magnetic Co (Co1 and Co2) in the unit cell. For them, the orbital angular momentum is parallel to the spin moment, as a result of the Hund's third rule. For Co1, the degenerate x$^2$-y$^2$ and xy levels (where the hole resides) can form a linear combination of eigenstates with l$_z$=2,\cite{cacoo_khomskii} so the Co$^{4+}$:d$^5$ cation in a prismatic environment is susceptible of developing a ground state with a large value of the orbital angular momentum. On the other hand, Co2 (Co$^{4+}$:~d$^5$) can develop an l$_z$= 1 eigenstate since the t$_{2g}$ multiplet acts as an effective l= 1 multiplet.\cite{epr,stevens,enough,lacroix,eschrig} The ground-state quantization axis will be related to the local environment which is rotated for the two Co2 atoms in the unit cell. This explains why we have different l$_z$ values even for the two equivalent Co2 atoms in the structure. Consequently, this spin-system cannot be described as an Ising-type one due to the canting of the moments of the various magnetic ions with respect to each other. Such canting can be understood according to the different Co environments and as a local orientation of the moments along its particular symmetry axis. The values are summarized in Table \ref{momentos_sot}, and help understand the measured $\mu_{eff}$ value.
Another difference with the other compounds in the series is that, in this case, we do not have a strong in-chain FM coupling for all the Co atoms in the unit cell. Thus, the magnetic properties of Sr$_6$Co$_5$O$_{15}$ cannot be understood as FM spin chains because of the AFM coupling that (though weak) occur within them and the two non-magnetic Co atoms per unit cell.
Clearly, the magnetic properties of this compound differ from the other members of the series making it less quasi-one-dimensional.
\begin{table
\caption{Projection of the orbital angular momenta of Co atoms along the magnetization axis for different directions of the magnetization (in $\mu_{B}$ units).} \label{momentos_sot}
\begin{ruledtabular}
\begin{tabular}{lcccccccc}
\multicolumn{9}{c}{ Atom \hspace{1cm} l$_{z}$ for various magnetization directions} \\ \cline{1-2} \cline{3-9}
& & (111) &(101)&(110)& (011) &(100)& (010)& (001)\\
\hline
Co1 & & 0.99 & 0.75 & 0.66 & 0.72 & 0.41 & 0.36 & 0.44 \\
Co2(a) & & -0.05 & -0.15 & -0.17 & -0.19 & -0.22 & -0.28 & -0.31 \\
Co2(b) & & -0.05 & -0.20 & -0.18 & -0.17 & -0.28 & -0.29 & -0.27\\
\end{tabular}
\end{ruledtabular}
\end{table}
\subsection{Transport properties.}
We have calculated the thermoelectric power dependence with the temperature to further analyze the system properties and the magnetic ground state. This has been done taking our band structure calculations within a semiclassical approach based on the Boltzmann transport theory through the BoltzTraP code.\cite{boltztrap} A dense grid of 10000 k points in the full Brillouin zone has been used to obtain convergence. Taking the conductivity ($\sigma$) and Seebeck coefficient ($S$) calculated for both spin channels, the total thermopower has been obtained according to the two-current model expression:\cite{singh_nacoo}
\begin{equation}
S=\frac{\sigma'(\uparrow)S(\uparrow)+\sigma'(\downarrow)S(\downarrow)}{\sigma'(\uparrow)+\sigma'(\downarrow)}
\end{equation}
where $\sigma'$=~$\sigma/\tau$, within the constant scattering time ($\tau$) approximation. For the sake of comparison, we present the calculations obtained for the ground state AFM spin configuration shown in Fig.~\ref{acoplos} and also for a FM solution (which is higher in energy according to our calculations). We present the data calculated for both spin configurations at U= 4.8 eV, together with the experimental values of the Seebeck coefficient taken from Ref.~[\onlinecite{iwasaki}] (see Fig.~\ref{thermopower}). The results for the AFM solution fit the experimental values nicely, both in order of magnitude of the thermopower and also on the observed non-activated evolution with the temperature. However, the FM one differs clearly from the experiment, giving further evidence of the validity of our description of the electronic structure of the compound.
\begin{figure}
\includegraphics[width=\columnwidth,draft=false]{paper_fig5.eps}
\caption{(Color online) Experimental \cite{iwasaki} and calculated temperature dependence of the thermopower. The results for both the FM and AFM spin-configurations are plotted.}\label{thermopower}
\end{figure}
\section{Summary}
We have synthesized Sr$_6$Co$_5$O$_{15}$, measured its magnetic properties and performed ab initio calculations. The material is structurally quasi-one-dimensional and can be identified as a member of the structural series A$_{n+2}$B$'$B$_n$O$_{3n+3}$ of hexagonal quasi-one-dimensional Co oxides. It is an insulating antiferromagnet with T$_N$= 32 K, which does not correspond to the magnetic properties expected for a member of the series with that value of the in-plane Co-Co distance. This is due to its peculiar electronic structure properties: i) the in-chain couplings are not purely FM. AFM couplings (though weak) occur within the chains. ii) This, together with the existence of two non-magnetic Co atoms in the unit cell, reduces the inter-chain magnetic couplings, responsible of the higher T$_N$ in the other hexagonal quasi-one-dimensional cobaltates. iii) The preferred orientation of the orbital angular momenta is non-collinear, in contrast to the strong Ising-type behavior found in Ca$_3$Co$_2$O$_6$ and BaCoO$_3$. In addition, transport properties calculations support our understanding of its electronic structure and magnetic properties.
This anomalous behavior makes Sr$_6$Co$_5$O$_{15}$ less quasi-one-dimensional than expected due to its structure and helps understand the intricate structure-property relations in strongly correlated electron systems.
\section{Acknowledgments}
The authors thank the CESGA (Centro de Supercomputaci\'on de Galicia) for the computing facilities and the Ministerio de Educaci\'{o}n y Ciencia (MEC) for the financial support through the project MAT2009-08165. Authors also thank the Ministerio de Ciencia e Innovaci\'on (MICINN) for the project MAT2007-60536 and the Xunta de Galicia for the project INCITE08PXIB236053PR. A.~S. Botana thanks MEC for a FPU grant.
|
{
"timestamp": "2011-06-03T02:02:32",
"yymm": "1009",
"arxiv_id": "1009.3736",
"language": "en",
"url": "https://arxiv.org/abs/1009.3736"
}
|
\section{Introduction}\label{sec:introduction}
Graphical Gaussian models have attracted a lot of recent interest. In
these models an observed random vector $Y=(Y_1,\dots,Y_p)$ is assumed
to follow a multivariate normal distribution
$\mathcal{N}_p(\mu,\Sigma)$, where~$\mu$ is the mean vector and
$\Sigma$ the positive definite covariance matrix. Each model is
associated with an undirected graph $G=(V,E)$ with vertex set
$V=\{1,\dots,p\}$, and defined by requiring that for each nonedge
$(j,k)\notin E$, the variables $Y_j$ and $Y_k$ are conditionally
independent given all the remaining variables $Y_{\setminus\{j,k\}}$.
Here, $\setminus\{j,k\}$ denotes the complement $V\setminus\{j,k\}$.
Such pairwise conditional independence holds if and only if
$\Sigma_{jk}^{-1}=0$; see \cite{lauritzen} for this fact and general
background on graphical models. Therefore, inferring the graph
corresponds to inferring the nonzero elements of $\Sigma^{-1}$.
Classical solutions to the model selection problem include
constraint-ba\-sed approaches that test the model-defining conditional
independence constraints, and score-based searches that optimize a
model score over a~set of graphs. A review of this work can be found
in \cite{drtonperlman}. Recently, however, penalized likelihood
approaches based on the one-norm~of the concentration matrix
$\Sigma^{-1}$ have become increasingly popular. Meinshau\-sen and B\"uhlmann \citeyear{meinshausen}
proposed a method that uses lasso regressions of each variable $Y_j$ on
the remaining variables $Y_{\setminus
j}:=Y_{\setminus\{j\}}$. In subsequent work, \cite{yuan} and \cite{banerjee}
discuss the computation of the exact solution to the convex
optimization problem arising from the likelihood penalization. Finally,
\cite{friedmanhastie} developed the \textit{graphical lasso (glasso)},
which is a computationally efficient algorithm that maximizes the
penalized log-likelihood function through coordinate-descent. The
theory that accompanies these algorithmic developments supplies
high-dimensional consistency properties under assumptions of graph
sparsity; see, for example, Ravikumar et
al.~(\citeyear{ravikumar}).\looseness=-1
Inference of a graph can be significantly impacted, however, by
deviations from normality. In particular, contamination of a handful
of variables in a~few experiments can lead to a drastically wrong
graph. Applied work thus often proceeds by identifying and removing
such experiments before data analysis, but such outlier screening can
become difficult with large data sets. More importantly, removing
entire experiments as outliers may discard useful information from the
uncontaminated variables they may contain.
The existing literature on robust inference in graphical models is
fairly limited. One line of work concerns constraint-based approaches
and adopts robustified statistical tests [\cite{kalisch}]. An approach
for fitting the model associated with a given graph using a robustified
likelihood function is described in \cite{miyamura}. In some cases
simple transformations of the data may be effective at minimizing the
effect of outliers or contaminated data on a~small scale. A normal
quantile transformation, in particular, appears to be effective in many
cases.
In this paper we extend the scope of robust inference by providing a
tool for robust model selection that can be applied with highly
multivariate data. We build upon the \textit{glasso} of
\cite{friedmanhastie}, but model the data using multivariate
$t$-distributions. Using the EM algorithm, the \textit{tlasso} methods
we propose are only slightly less computationally efficient than the
\textit{glasso} but cope rather well with contaminated data.
The paper is organized as follows. In Section \ref{sec:glasso} we
review maximization of the penalized Gaussian log-likelihood function
using the \textit{glasso}. In Section~\ref{sec:t-dist} we introduce
the classical multivariate $t$-distribution and describe maximization
of the (unpenalized) log-likelihood using the EM algorithm. In
Sec\-tion~\ref{sec:tlasso} we combine the two techniques into the \textit{tlasso}
to maximize the penalized log-likelihood in the multivariate $t$ case.
In Section \ref{sec:alasso} we introduce an\vadjust{\eject} alternative multivariate
$t$-distribution and describe how inference can be done using
stochastic and variational EM. In Section \ref{sec:simulations} we
compare the \textit{glasso} to our $t$-based methods on simulated data. Finally, in
Section~\ref{sec:gene-expression-data} we analyze two different gene expression
data sets using the competing methods. Our findings are summarized in
Section \ref{sec:discussion}.\
\section{Graphical Gaussian models and the graphical lasso}\label{sec:glasso}
Suppose we observe a sample of $n$ independent random vectors
$Y_1,\dots, Y_n\in\mathbb{R}^p$ that are distributed according to the
multivariate normal distribution $\mathcal{N}_p(\mu,\Sigma)$.
Likelihood inference about the covariance matrix $\Sigma$ is based on
the log-likelihood function\vspace*{-2pt}
\[
\ell(\Sigma) = -\frac{np}{2}\log (2\pi)-\frac{n}{2}\log\det(\Sigma) -
\frac{n}{2} \operatorname{tr}(S\Sigma^{-1}),\vspace*{-2pt}
\]
where the empirical covariance matrix\vspace*{-2pt}
\[
S = (s_{jk})= \frac{1}{n} \sum_{i=1}^n (Y_i-\bar Y)(Y_i-\bar Y)^T\vspace*{-2pt}
\]
is defined based on deviations from the sample mean $\bar Y$. Let
$\Theta = (\theta_{jk})= \Sigma^{-1}$ denote the ($p\times
p$)-concentration matrix. In penalized likelihood methods a~one-norm
penalty is added to the log-likelihood function, which effectively
performs model selection because the resulting estimates of $\Theta$
may have entries that are exactly zero. Omitting irrelevant factors and
constants, we are led to the problem of maximizing the function
\begin{equation} \label{eq:pen-lik}
\log \det(\Theta) - \operatorname{tr}(S\Theta) - \rho \|\Theta\|_1
\end{equation}
over the cone of positive definite matrices, where $\|\Theta\|_1$ is
the sum of the absolute values of the entries of $\Theta$. The
multiplier $\rho$ is a positive tuning parameter. Larger values of
$\rho$ lead to more entries of $\Theta$ being estimated as zero.
Cross-validation or information criteria can be used to tune $\rho$.
The \textit{glasso} is an iterative method for solving the convex
optimization problem with the objective function in (\ref{eq:pen-lik}).
Its updates operate on the covariance matrix $\Sigma$. In each step one
row (and column) of the symmetric matrix $\Sigma$ is updated based on a
partial maximization of (\ref{eq:pen-lik}) in which all but the
considered row (and column) of $\Theta$ are held fixed. This partial
maximization is solved via coordinate-descent as briefly reviewed next.
Partition off the last row and column of $\Sigma=(\sigma_{jk})$ and $S$
as
\[
\Sigma =
\pmatrix{
\Sigma_{\setminus p,\setminus p} & \Sigma_{\setminus p,p} \vspace*{1pt} \cr
\Sigma_{\setminus p,p}^T & \sigma_{pp}},\qquad
S= \pmatrix{
S_{\setminus p,\setminus p} & S_{\setminus p,p}\vspace*{1pt} \cr
S_{\setminus p,p}^T & s_{pp}}.
\]
Then, as shown in \cite{banerjee}, partially maximizing
$\Sigma_{\setminus p,p}$ with $\Sigma_{\setminus p,\setminus p}$ held
fixed yields $\Sigma_{\setminus p,p}=\Sigma_{\setminus p,\setminus p}
\beta^*$, where $\beta^*$ minimizes
\[
\|(\Sigma_{\setminus p,\setminus p})^{1/2} \beta - (\Sigma_{\setminus
p,\setminus p})^{-1/2} S_{\setminus p,p} \|^2 + \rho \|\beta\|_1
\]
with respect to $\beta\in\mathbb{R}^{p-1}$. The \textit{glasso} finds
$\beta^*$ by coordinate descent in each\vadjust{\eject} of the coordinates
$j=1,\dots,p-1$, using the updates
\[
\beta^*_j = \frac{T (s_{jp}-\sum_{k<p,k \ne j} \sigma_{kj}
\beta^*_k, \rho )}{\sigma_{jj}},
\]
where $T(x,t)=\operatorname{sgn}(x)(|x|-t)_+$. The algorithm then cycles
through the rows and columns of $\Sigma$ and $S$ until convergence.
The diagonal elements are simply $\sigma_{pp}=s_{pp} + \rho$. See
\cite{friedmanhastie} for more details on the method.
\section{Graphical models based on the $t$-distribution}\label{sec:t-dist}
\subsection{Classical multivariate $t$-distribution}
The classical multivariate $t$-dis\-tribution $t_{p,\nu}(\mu,\Psi)$ on
$\mathbb{R}^p$ has Lebesgue density
\begin{equation} \label{eq:tdensity}
f_\nu(y;\mu,\Psi) = \frac{\Gamma ((\nu+p)/2) |\Psi|^{-1/2}}{(\pi
\nu )^{p/2} \Gamma
(\nu/2) [1 + \delta_y (\mu, \Psi)/\nu]^{(\nu+p)/2}}
\end{equation}
with $\delta_y(\mu,\Psi) = (y-\mu)^T \Psi^{-1} (y-\mu)$ and
$y\in\mathbb{R}^p$. The vector $\mu\in\mathbb{R}^p$ and the positive
definite matrix $\Psi=(\psi_{jk})$ determine the first two moments of
the distribution. If $Y\sim t_{p,\nu}(\mu,\Psi)$ with $\nu > 2$
degrees of freedom, then the expectation is $\mathbb{E}[Y]=\mu$ and the
covariance matrix is $\mathbb{V}[Y]=\nu/(\nu-2)\cdot\Psi$. From here
on we will always assume $\nu>2$ for the covariance matrix to exist.
For notational convenience and to illustrate the parallels with the
Gaussian model, we define $\Theta= (\theta_{jk})=\Psi^{-1}$.
\begin{figure}[b]
\includegraphics{410f01.eps}
\caption{Graph representing the process generating a multivariate
$t$-random vector $Y$ from a latent Gaussian random vector $X$ and a
single latent Gamma-divisor.}\label{fig:standard}
\end{figure}
If $X\sim\mathcal{N}_p(0,\Psi)$ is a multivariate normal random vector
independent of the Gamma-random variable $\tau\sim\Gamma(\nu/2,\nu/2)$,
then $Y=\mu+X/\sqrt{\tau}$ is distributed according to
$t_{p,\nu}(\mu,\Psi)$; see \cite{kotz}, Chapter~1. This scale-mixture
representation, illustrated in Figure \ref{fig:standard}, allows for
easy sampling. It also clarifies how the use of $t$-distributions
leads to more robust inference because extreme observations can arise
from small values of $\tau$. An additional useful fact is that the
conditional distribution of $\tau$ given $Y$ is again a
Gamma-distribution, namely,
\begin{equation} \label{eq:tau-given-Y}
(\tau\vert Y) \sim \Gamma\biggl( \frac{\nu + p}{2}, \frac{\nu +
\delta_{Y}(\mu,\Psi)}{2}\biggr).
\end{equation}
Let $G=(V,E)$ be a graph with vertex set $V=\{1,\dots,p\}$. We define
the associated graphical model for the $t$-distribution by requiring
that $\theta_{jk}=0$ for indices $j\not=k$ corresponding to a nonedge
$(j,k)\notin E$. This mimics the Gaussian model in that zero
constraints are imposed on the inverse of the covariance matrix.
However, in a $t$-distribution this no longer corresponds to
conditional independence, and the density $f_\nu(y;\mu,\Psi)$ does not
factor according to the graph. The conditional dependence manifests
itself, in particular, in conditional variances in that even if
$\theta_{jk}=0$,\vspace*{-1pt}
\[
\mathbb{V}[Y_j|Y_{\setminus j}]\not=
\mathbb{V}\bigl[Y_j|Y_{\setminus\{j,k\}}\bigr].\vspace*{-1pt}
\]
For a simple illustration of this inequality, let $\Psi$ be a diagonal
matrix. Then\vspace*{-2pt}
\[
\mathbb{V}[Y_j|Y_{\setminus j}] = \mathbb{E}[X_j^2/\tau|Y_{\setminus j}]
= \frac{1}{\theta_{jj}}\cdot\mathbb{E}[\tau^{-1}|Y_{\setminus j}] =
\frac{1}{\theta_{jj}}\cdot\frac{\nu +
\delta_{Y_{\setminus j}}(\mu_{\setminus j},\Psi_{\setminus j,\setminus
j})}{\nu + p-3},\vspace*{-2pt}
\]
which can be shown by taking iterated conditional expectations, and
using that\vspace*{-5pt}
\[
\mathbb{E}[X_j^2|Y_{\setminus j},\tau]= \mathbb{E}[X_j^2|X_{\setminus
j},\tau]= \mathbb{V}[X_j|X_{\setminus j}]=\frac{1}{\theta_{jj}}\vspace*{-2pt}
\]
and that $\tau$ given $Y_{\setminus j}$ has a Gamma-distribution;
recall (\ref{eq:tau-given-Y}). Clearly, $ \mathbb{V}[Y_j|Y_{\setminus
j}]$ depends on all $Y_k$, $k\not= j$.
Despite the lack of conditional independence, the following property
still holds (proved in the \hyperref[appendix]{Appendix}).\vspace*{-2pt}
\begin{proposition} \label{thm:1}
Let $X \sim \mathcal{N}_p(0,\Theta^{-1})$, where $\theta_{jk}=0$ for
pairs of indices $j\not= k$ that correspond to nonedges in the graph $G$.
Let $\tau$ be independent of~$X$ and follow any distribution on the positive real numbers with $\mathbb{E}[1/\tau] < \infty$ and define
$Y=\mu + X/\sqrt{\tau}$. If two nodes $j$ and $k$ are separated by a set
of nodes~$C$ in $G$, then $Y_j$ and $Y_k$ are conditionally uncorrelated
given $Y_{C}$.\vspace*{-2pt}
\end{proposition}
The edges in the graph indicate the allowed conditional independencies
in the latent Gaussian vector $X$. According to Proposition
\ref{thm:1}, however, we may also interpret the graph in terms of the
observed variables $Y_j$. The zero conditional correlations entail
that mean-square error optimal prediction of variable $Y_j$ can be
based on the variables $Y_k$ that correspond to neighbors of the node
$j$ in the graph, which is a very appealing property.\vspace*{-2pt}
\subsection{EM algorithm for estimation}\label{sec:em}
The lack of density factorization properties complicates likelihood
inference with $t$-distributions. However, the EM algorithm provides a
way to circumvent this issue. Equipped with the normal-Gamma
construction, we treat $\tau$ as a hidden variable and use that the
conditional distribution of $Y$ given $\tau$ is
$\mathcal{N}_p(\mu,\Psi/\tau)$. We now outline the EM algorithm for
the $t$-distribution assuming the degrees of freedom $\nu$ to be known.
If desired, $\nu$ could also be estimated in a line search that is best
based on the actual $t$-likelihood [Liu and Rubin~\citeyear{liurubin}].
Consider an $n$-sample $Y_1,\dots,Y_n$ drawn from
$t_{p,\nu}(\mu,\Psi)$. Let $\tau_1,\dots,\tau_n$ be an associated
sequence of hidden Gamma-random variables. Observation of the $\tau_i$
would lead to the following complete-data log-likelihood function for
$\mu$ and $\Theta=\Psi^{-1}$:\vspace*{-5pt}
\begin{eqnarray}
\ell_{\mathrm hid}(\mu,\Theta |Y,\tau) &\propto& \frac{n}{2} \log\det(\Theta)
- \frac{1}{2} \operatorname{tr}\Biggl(\Theta \sum_{i=1}^n \tau_i Y_i
Y_i^T\Biggr)\nonumber\\ [-11pt]\\ [-11pt]
&&{} + \mu^T \Theta \sum_{i=1}^n \tau_i Y_i - \frac{1}{2} \mu^T
\Theta \mu \sum_{i=1}^n \tau_i,\nonumber\vspace*{-2pt}
\end{eqnarray}
where, with some abuse, the symbol $\propto$ indicates that irrelevant
additive constants are omitted. The complete-data sufficient
statistics\vspace*{-2pt}
\[
S_{\tau} = \sum_{i=1}^n \tau_i,\qquad S_{\tau Y} = \sum_{i=1}^n \tau_i
Y_i,\qquad S_{\tau YY} = \sum_{i=1}^n \tau_i Y_i Y_i^T\vspace*{-2pt}
\]
are thus linear in $\tau$. We obtain the following EM algorithm for
computing the maximum likelihood estimates of $\mu$ and $\Psi$:
\begin{description}
\item[E-step:] The E-step is simple because\vspace*{-2pt}
\begin{equation} \label{eq:etau}
\mathbb{E}[\tau | Y] = \frac{\nu + p}{\nu + \delta_Y(\mu,\Psi)}.\vspace*{-2pt}
\end{equation}
Given current estimates $\mu^{(t)}$ and $\Psi^{(t)}$, we compute in the
$(t+1)$st iteration\vspace*{-2pt}
\[
\tau_i^{(t+1)} = \frac{\nu + p}{\nu+\delta_Y(\mu^{(t)},\Psi^{(t)})}.\vspace*{-2pt}
\]
\item[M-step:] Calculate the updated estimates\vspace*{-2pt}
\begin{eqnarray}\label{eq:mut+1}
\mu^{(t+1)} &=& \frac{\sum_{i=1}^n \tau_i^{(t+1)}
Y_i}{\sum_{i=1}^n \tau_i^{(t+1)}},\\[-1pt]
\Psi^{(t+1)} &=& \frac{1}{n} \sum_{i=1}^n \tau_i^{(t+1)} \bigl[
Y_i-\mu^{(t+1)}\bigr] \bigl[ Y_i-\mu^{(t+1)}\bigr]^T.\vspace*{-1pt}
\end{eqnarray}
\end{description}
\section{Penalized inference in $t$-distribution models}\label{sec:tlasso}
Model selection in graphical $t$-models can be performed, in principle,
by any of the classical constraint- and score-based methods. In
score-based searches through the set of all undirected graphs on $p$
nodes, however, each model would have to be refit using an iterative
method such as the algorithm from Section \ref{sec:em}. The penalized
likelihood approach avoids this problem.
Like in the Gaussian case, we put a one-norm penalty on the elements
of~$\Theta$ and wish to maximize the penalized log-likelihood function
\begin{equation}
\ell_{\rho,\mathrm{obs}}(\mu,\Theta|Y) = \sum_{i=1}^n \log
f_\nu(Y_i;\mu,\Theta^{-1}) -
\rho\|\Theta\|_1,
\end{equation}
where $f_\nu$ is the $t$-density from (\ref{eq:tdensity}). To achieve
this, we will use a modified version of the EM algorithm taking into
account the one-norm penalty.
We treat $\tau$ as missing data. In the E-step of our algorithm, we
calculate the conditional expectation of the penalized complete-data
log-likelihood
\begin{equation}\label{pcdl}
\ell_{\rho,\mathrm{hid}}(\mu,\Theta | Y,\tau)
\propto \frac{n}{2} \log |\Theta| - \frac{n}{2} \operatorname{tr} (\Theta
S_{\tau YY}(\mu)) -\rho \|\Theta\|_1
\end{equation}
with
\[
S_{\tau YY}(\mu) = \frac{1}{n} \sum_{i=1}^n \tau_i (Y_i -
\mu)(Y_i- \mu)^T.
\]
Since $\ell_{\rho,\mathrm{hid}}(\mu,\Theta | Y,\tau)$ is again linear in
$\tau$, the E-step takes the same form as in Section \ref{sec:em}. Let
$\mu^{(t)}$ and $\Theta^{(t)}$ be the estimates after the $t$th
iteration, and~$\tau_i^{(t+1)}$ the conditional expectation of $\tau_i$
calculated in the $(t+1)$st E-step. Then in the M-step of our
algorithm we wish to maximize
\[
\frac{n}{2} \log |\Theta| - \frac{n}{2} \operatorname{tr} \bigl(\Theta
S_{\tau^{(t+1)} YY}(\mu)\bigr) -\rho \|\Theta\|_1
\]
with respect to $\mu$ and $\Theta$. Differentiation with respect to
$\mu$ yields $\mu^{(t+1)}$ from~(\ref{eq:mut+1}) for any value of
$\Theta$. Therefore, $\Theta^{(t+1)}$ is found by maximizing
\begin{equation} \label{eq:hidglasso}
\frac{n}{2} \log |\Theta| - \frac{n}{2} \operatorname{tr} \bigl(\Theta S_{\tau^{(t+1)}
YY}\bigl(\mu^{(t+1)}\bigr)\bigr) -\rho \|\Theta\|_1.
\end{equation}
The quantity in (\ref{eq:hidglasso}), however, is exactly the objective
function maximized by the \textit{glasso}.
Iterating the E- and M-steps just described, we obtain what we call the
\textit{tlasso} algorithm. Since the one-norm penalty forces some elements of
$\Theta$ exactly to zero, the \textit{tlasso} performs model selection
and parameter estimation in a way that is similar to structural EM
algorithms [\cite{friedman}]. Convergence to a stationary point is
guaranteed in the penalized version of the EM algorithm
[\cite{mclachlan}, Chapter 1.6]; typically a~local maximum is found.
Note also that the maximized log-likelihood function is not concave,
and so one finds oneself in the usual situation of not being able to
give any guarantees about having obtained a global maximum.
\section{Alternative model}\label{sec:alasso
\subsection{Specification of the alternative $t$-model}
The \textit{tlasso} from Section \ref{sec:tlasso} performs particularly
well when a small fraction of the observations are contaminated (or
otherwise extreme). In this case, these observations are downweighted
in entirety, and the gain from reducing the effect of contaminated
nodes outweighs the loss from throwing away good data from other nodes.
In high-dimensional data sets, however, the contamination, or other
deviation form normality, may be in small parts of many observations.
Downweighting entire observations may then no longer achieve the
desired results. We will demonstrate this later in simulations (see
the bottom panel of Figure \ref{fig:sims}).
To handle the above situation better, we consider an alternative
extension of the univariate $t$-distribution, illustrated in
Figure \ref{fig:alternative}. Instead of one divisor~$\tau$ per
$p$-variate observation, we draw $p$ divisors $\tau_j$. For $j=1,
\ldots ,p$, let $\tau_j \sim \Gamma(\nu/2,\nu/2)$ be independent of
each other and of $X \sim \mathcal{N}_p(0,\Psi)$. We then say that the
random vector $Y$ with coordinates $Y_j = \mu_j + X_j/\sqrt{\tau_j}$
follows an alternative multivariate $t$-distribution; in symbols $Y
\sim t^*_{p,\nu}(\mu,\Psi)$.
\begin{figure}
\includegraphics{410f02.eps}
\caption{Graph representing the process generating a $t^*$-random
vector $Y$ from a latent Gaussian random vector $X$ and independent latent
Gamma-divisors.}\label{fig:alternative}
\end{figure}
Unlike for the classical multivariate $t$-distribution, the covariance
matrix~$\mathbb{V}[Y]$ is no longer a constant multiple of
$\Psi=(\psi_{jk})$ when $Y \sim t^*_{p,\nu}(\mu,\Psi)$. Clearly, the
coordinate variances are still the same, namely,
\[
\mathbb{V}[Y_j]=\frac{\nu}{\nu-2} \cdot \psi_{jj},
\]
but the covariance between $Y_j$ and $Y_k$ with $j\not=k$ is now
\[
\frac{\nu\Gamma((\nu -1)/2)^2}{2\Gamma(\nu/2)^2}
\cdot\psi_{jk} \le
\frac{\nu}{\nu-2} \cdot \psi_{jk}.
\]
The same matrix $\Psi$ thus implies smaller correlations (by the same
constant multiple) in the $t^*$-distribution. This reduced dependence
is not surprising in light of the fact that now different and
independent divisors appear in the different coordinates. Despite the
decrease in marginal correlations, the result of
Proposition \ref{thm:1} does not hold for conditional correlations in
the alternative model. That is, $\Psi^{-1}_{jk}=0$ does not imply
$Y_j$ and $Y_k$ are conditionally uncorrelated given
$Y_{\setminus\{j,k\}}$. Interpretation of the graph in the alternative
model is thus limited to considering edges to represent the allowed
conditional dependencies in the latent multivariate normal
distribution.
\begin{figure}
\includegraphics{410f03.eps}
\vspace*{-5pt}
\caption{Sample correlation of $Y_1$ and $Y_2$ for observations with
$Y_3$ in a window of size 0.01.}\label{fig:condcor}\vspace*{-5pt}
\end{figure}
The following simulation confirms the result and illustrates the
effect. We consider a $t^*_{3,3}(0,\Theta^{-1})$ distribution with
\[
\Theta =
\left[\matrix{
1&0&-0.5\cr
0&1&-0.5\cr
-0.5&-0.5&1\cr
}\right]
\]
and draw independent samples until we have 500,000 observations with
$x < Y_3 < x +0.01$ for $120$ values of $x$ in the range $(-6,6)$. The
sample correlations of $Y_1$ and $Y_2$ given the varying values of
$Y_3$ are shown in Figure \ref{fig:condcor}.
\subsection{Alternative tlasso}
Inference in the alternative model presents some difficulties because
the likelihood function is not available explicitly. The complete-data
log-likelihood function $\ell^*_{\rho,\mathrm{hid}}(\mu,\Theta|Y,\tau)$,
however, is simply the product of the evaluations of $p$
Gamma-densities ($\tau$ being a vector now) and a multivariate normal
density. We can thus implement an EM-type procedure if we are able to
compute the conditional expectation of $\ell^*_{\rho,\mathrm{hid}}(\mu,\Theta|Y,\tau)$ given $Y=(Y_1,\dots,Y_n)$. This time we treat
the $p$ random variables $(\tau_{i1}, \ldots ,\tau_{ip})$ as hidden for
each observation $i=1,\dots,n$. Unfortunately, the conditional
expectation is intractable. It can be estimated, however, using Markov
Chain Monte Carlo.
The complete-data log-likelihood function is equal to
\begin{equation}\label{altpcdl}
\ell^*_{\rho,\mathrm{hid}} (\mu , \Theta, | Y, \tau)
\propto \frac{n}{2} \log |\Theta|- \frac{n}{2} \operatorname{tr} (\Theta
S^*_{\tau YY}(\mu) )
-\rho \|\Theta\|_1,
\end{equation}
where
\[
S^*_{\tau YY}(\mu) = \frac{1}{n} \sum_{i=1}^n
D\bigl(\sqrt{\tau_i}\bigr) (Y_i - \mu)(Y_i- \mu)^TD\bigl(\sqrt{\tau_i}\bigr)
\]
and $D(\sqrt{\tau_i})$ is the diagonal matrix with
$\sqrt{\tau_i}=\sqrt{\tau_{i1}}, \ldots ,\sqrt{\tau_{ip}}$ along the
diagonal. The trace in (\ref{altpcdl}) is linear in the entries of the
matrix $\sqrt{\tau_i}\sqrt{\tau_i}^T$. A Gibbs sampler for estimating
the conditional expectation of this matrix given $Y$ cycles through the
coordinates indexed by $j=1, \ldots ,p$ and draws, in its $m$th
iteration, a number $\tau_{ij}^{(m)}$ from the conditional distribution
of $\tau_{ij}$ given $(\tau_{i \setminus j},Y)$. This full conditional
has density
\begin{equation} \label{eq:full-conditional}
f(\tau_{ij}|\tau_{i \setminus j},Y_i) = C(\alpha,\beta,\gamma)\cdot
\tau_{ij}^{\alpha
- 1} \exp \bigl\{-\tau_{ij} \beta - \sqrt{\tau_{ij}} \gamma \bigr\}
\end{equation}
with
\begin{equation}\label{eq:exactgibbs}
\hspace*{30pt}\alpha = \frac{\nu+1}{2}, \qquad \beta = \frac{\nu +
(Y_{ij}-\mu_j)^2\theta_{jj}}{2}, \qquad \gamma = (Y_{ij}-\mu_j)\Theta_{j
\setminus j}X_{ i\setminus j},
\end{equation}
and normalizing constant $C(\alpha,\beta,\gamma)$. This constant can
always be expressed using hypergeometric functions, but, as we detail
below, much simpler formulas can be obtained for the small integer
degrees of freedom $\nu$ that are of interest in practice. The simpler
formulas are obtained by partial integration.
From $\beta$ and $\gamma$ in (\ref{eq:exactgibbs}), form the ratio
$\gamma'=\gamma/(2\sqrt{\beta})$. In order to sample from the
distribution in (\ref{eq:full-conditional}), we may draw from
\begin{equation} \label{eq:full-cond-beta11}
f_{\alpha,\gamma'}(t) = C(\alpha,\gamma')\cdot
t^{\alpha
- 1} \exp \bigl\{-t - \sqrt{t} 2\gamma' \bigr\}
\end{equation}
and divide the result by $\beta$. For our default of $\nu=3$, that is,
$\alpha=2$, we thus need to sample from
\begin{equation} \label{eq:full-cond-beta1}
f_{\gamma}(t) = C(\gamma)\cdot
t \exp \bigl\{-t - \sqrt{t} 2\gamma \bigr\} .
\end{equation}
Writing $\Phi$ for the cumulative distribution function of the standard
normal distribution, the normalizing constant becomes
\begin{equation}\label{eq:erfc}
1/C(\gamma) = 1+
\gamma^2-\gamma(2\gamma^2+3)\sqrt{\pi}\exp\{\gamma^2\}\bigl(1- \Phi
\bigl(\gamma\sqrt{2}\bigr)\bigr).
\end{equation}
For $\gamma=0$, the density $f_\gamma(t)$ is a $\Gamma(2,1)$ density.
For moderate $\gamma$, we are thus led to the following rejection
sampling procedure to draw from $f_\gamma$.
Let $g_{\delta}$ be the density of a $\Gamma(2,\delta)$ distribution.
Rejection sampling using the family of densities $g_{\delta}$ as
instrumental densities proceeds by drawing a~proposal $T \sim
\Gamma(2,\delta)$ and a uniform random variable $U \sim
\mathcal{U}(0,1)$ and either accept if $U \le
f(T)/(M_{\delta}g_{\delta}(T))$ or repeat the process until acceptance.
Here,~$M_{\delta}$ is a suitable multiplier such that $f(t) \le
M_{\delta} g_{\delta}(t)$ for all $t\ge 0$.
An important ingredient to the rejection sampler is the parameter
$\delta$, which we choose as follows. In the case $\gamma < 0$, the
density $g_{\delta}$ has a heavier tail than~$f$ provided that
$\delta<1$. Focusing on the case $\alpha=2$, we have that for a given
$\delta<1$ the smallest $M$ such that $f(t) \le M g_{\delta}(t)$ for
all $t\ge 0$ is
\[
M_{\delta}= C \cdot\frac{1}{\delta^{2}} \exp \biggl\{
\frac{\gamma^2}{(1-\delta)} \biggr\}.
\]
Varying $\delta$, the multiplier $M_{\delta}$ is minimized at
\[
\delta= 1 + \frac{\gamma^2 - \sqrt{\gamma^4+8
\gamma^2}}{4}.
\]
If $\gamma>0$, then\vadjust{\goodbreak} setting $\delta=1$ yields a heavy enough tail and
$M_{\delta}=C$.
The rejection sampling performs draws from the exact conditional
distribution $f(\tau_{ij}|\tau_{i \setminus j},Y)$. We find it works
very well for data with not too extreme contamination such as, for
instance, in the original as well as bootstrap data from the
application discussed in Section \ref{sec:isoprenoid-pathway}. When
applied to data with very extreme observations $Y_{ij}$, however, one
is faced with larger positive values of $\gamma$. In this case the
instrumental densities $g_\delta$ provide a poor approximation to the
target density $f_\gamma$, and the acceptance probabilities in the
rejection sampling step become impractically low.
For $\gamma >1$, we thus use an alternative rejection procedure. Make
the transformation $s=\sqrt{t}$. We then wish to sample from
\[
h_{\gamma}(s) = 2 C(\gamma)\cdot s^3 \exp \{-s^2 - s 2\gamma
\} .
\]
Any $\Gamma(\alpha,\delta)$ distribution has a heavier tail than the
target distribution $h_{\gamma}(s)$. While it is not possible to find
an analytical solution for the optimal $\alpha$ and~$\delta$, letting
$\alpha=1$ and $\delta=(\gamma+1)/2$ yields acceptance probabilities
between $40\%$ and $50\%$ for most plausible values of $\gamma$. Since
this alternative procedure will only be needed occasionally, these
acceptance problems are adequate. Using this hybrid approach yields
overall acceptance probabilities greater than $98\%$ for the data with
extreme contamination described in Section \ref{sec:galactose}.
Returning to the iterations of the overall sampler, we calculate
$\sqrt{\tau_i}\sqrt{\tau_i}^T$ at the end of each cycle through the $p$
nodes, and then take the average over~$M$ iterations. This solves the
problem of carrying out one E-step, and we obtain the following
stochastic penalized EM algorithm, which we call the Monte Carlo
$t^*$\textit{-lasso} (or $t^*_{\mathrm MC}$\textit{-lasso} for short):
\begin{description}
\item[E-step:] Given current estimates $\mu^{(t)}$ and $\Psi^{(t)}$,
compute ${(\sqrt{\tau_i}\sqrt{\tau_i}^T)}^{(t+1)}$ by averaging the matrices
obtained in some large number $M$ of Gibbs sampler iterations, as
described above.
\item[M-step:] Calculate the updated estimates
\[
\mu_j^{(t+1)} = \frac{\sum_{i=1}^n \tau_{ij}^{(t+1)}
Y_{ij}}{\sum_{i=1}^n \tau_{ij}^{(t+1)}}.
\]
Use these and ${(\sqrt{\tau_i}\sqrt{\tau_i}^T)}^{(t+1)}$ to compute the
matrix $S^*_{\tau^{(t+1)} YY}(\mu^{(t+1)})$ to be plugged into the
trace term in (\ref{altpcdl}). Maximize the resulting penalized
log-likelihood function using the \textit{glasso}.
\end{description}
\subsection{Variational approximation}\label{sec:variational}
The above Monte Carlo procedure loses much of
the computational efficiency of the classical \textit{tlasso} from
Section~\ref{sec:tlasso}, however, and can be prohibitively expensive
for large $p$. For large problems, we turn instead to variational
approximations of the conditional densi\-ty~$f (\tau_i|Y_i)$ of the
vector $\tau_i$ given the observed vector $Y_i$.
The variational approach proceeds by approximating the conditional
density $f(\tau_{ij}|Y_i)$ by a factoring distribution. In our
context, however, it is easier to approximate the joint density
$f(\tau_i,Y_i)=f(\tau_i)f(Y_i|\tau_i)$ instead. The first term is
already in product form because we are assuming the individual divisor
$\tau_{ij}$ to be independent in the model formulation, and the second
term is the density of the multivariate normal distribution
\[
\mathcal{N}_p\bigl(\mu,D\bigl(1/\sqrt{\tau_i}\bigr)\Theta^{-1}
D\bigl(1/\sqrt{\tau_i}\bigr)\bigr).
\]
We approximate this normal distribution by a member of the set of
multivariate normal distributions with diagonal covariance matrix.
Application of this naive mean field procedure, that is, choosing a
distribution by minimizing Kullback--Leibler divergence, leads to the
approximating distribution
\begin{equation} \label{eq:meanfield}
\mathcal{N}_p\bigl(\mu,D\bigl(1/\sqrt{\tau}\bigr)\bar{\Theta}^{-1} D\bigl(1/\sqrt{\tau}\bigr)\bigr),
\end{equation}
where $\bar{\Theta}$ is the diagonal matrix with the same diagonal
elements as $\Theta$ [\cite{wainwrightjordan}, Chapter 5]. Writing
$q^*(Y|\tau)$ for the density of the distribution in
(\ref{eq:meanfield}), our approximation thus has the fully factoring
form $q^*_{\tau_i,Y_i}(\tau_i,Y_i)=f(\tau_i)q^*(Y_i|\tau_i)$. The
resulting conditional distribution also factors as
\[
q^*(\tau_i|Y_i)= \prod_{j=1}^p g(\tau_{ij}|Y_{ij}),
\]
where $g(\tau_{ij}|Y_{ij})$ is the density of the Gamma-distribution
$\Gamma(\alpha_{ij},\beta_{ij})$, with its parameters corresponding to
the quantities $\alpha$ and $\beta$ in (\ref{eq:exactgibbs}).
In conclusion, the variational E-step consists of calculating, for each
observation $Y_i$, the expectations
\[
\mathbb{E}_g[\tau_{ij} | Y_{ij}] =
\frac{\alpha_{ij}}{\beta_{ij}}, \qquad
\mathbb{E}_g\bigl[\sqrt{\tau_{ij}} | Y_{ij}\bigr] =
\frac{\Gamma(\alpha_{ij}+1/2)}{\Gamma(\alpha_{ij})\sqrt{\beta_{ij}}},
\]
and $\mathbb{E}[\sqrt{\tau_j}\sqrt{\tau_k} | Y_{i} ] =
\mathbb{E}_g[\sqrt{\tau_j}|Y_{ij}] \mathbb{E}_g[\sqrt{\tau_k} | Y_{ik}
]$. These values\vspace*{1pt} are then substituted into (\ref{altpcdl}). The M-step
is the same as in the $t^*_{\mathrm{MC}}$\textit{-lasso}.
The effect of the variational approximation is that the weight for
node~$j$ in observation $i$ is based solely on the squared deviation from
the mean, $(Y_{ij} - \mu_j)^2$ and the conditional variance
$1/\theta_{jj}$. For a given deviation from the mean, the larger the
conditional variance of the node, the smaller the weight given to that
node in that observation. But unlike in the $t^*_{\mathrm{MC}}$\textit{-lasso},
no consideration is given to deviation from the conditional
mean of the node in question given the rest. Some relevant information
is therefore not being used, but in our simulations the effect was not
noticeable.
The resulting variational $t^*$\textit{-lasso} ($t^*_{\mathrm{var}}$\textit{-lasso})
is only slightly more expensive than the \textit{tlasso} and,
despite the relatively crude approximation in the variational E-step,
performs well compared with the $t^*_{\mathrm{MC}}$\textit{-lasso}. Because
of this, we will use exclusively the $t^*_{\mathrm{var}}$\textit{-lasso}
when considering the alternative model in the simulations in the next
section.
\section{Simulation results}\label{sec:simulations}
\subsection{Procedure}
We used simulated data to compare the three procedures \textit{glasso},
\textit{tlasso} and $t^*_{\mathrm{var}}$\textit{-lasso} as follows. We generated a random $100\times100$
sparse inverse covariance (or dispersion) matrix $\Theta$ according to
the following procedure:
\begin{enumerate}[(a)]
\item[(a)] Choose each lower-triangular element of $\Theta$
independently to be
$-1$, $0$ or $1$ with probability $1\%$, $98\%$ and $1\%$, respectively.
\item[(b)] For $j>k$ set $\theta_{kj}=\theta_{jk}$. \item[(c)] Define
$\theta_{kk}=1+h$ where $h$ is the number of nonzero elements in
the $k$th row of $\Theta$.
\end{enumerate}
The final step ensures a strictly diagonally dominant, and thus
positive-definite matrix. To strengthen the partial correlations, we
reduced the diagonal elements by a common factor. We made this factor
as large as possible while maintaining positive-definiteness and
stability for inversion. For these particular matrices, fixing a
minimum eigenvalue of $0.6$ worked well.
We then generated $n=50$ observations from the
$\mathcal{N}_{100}(0,\Theta^{-1})$ distribution and ran each of the
three procedures with a range of values for the one-norm tuning
parameter $\rho$. To compare how well the competing methods recovered
the true edges, we drew ROC curves. We ran this whole process 250
times and then repeated the entire computation, drawing data from
$t_{100,3}(0,\Theta^{-1})$ and then $t^*_{100,3}(0,\Theta^{-1})$
distributions.
Simulating from $t$-distributions produces extreme observations, but a
more realistic setting might be one in which normal data is
contaminated in some fashion. For instance, consider broken probes or
misread observations in a~large gene expression microarray. Suppose
the contaminated data are not so extreme as to be manually screened or
otherwise identified as obvious outliers. To simulate this phenomenon,
we generate normal data as above, but randomly contaminated $2\%$ of
the values with data generated from independent univariate
$\mathcal{N}(\mu^*,0.2)$ random variables, where $\mu^*$ is equal to
$2.5$ times the largest diagonal element of $\Theta^{-1}$. These
contaminated values will be similar in magnitude to the $2\%$ tail of
the original $\mathcal{N}_{100}(0,\Theta^{-1})$ distribution and
therefore difficult to identify.
Finally, we would like to compare our developed $t$-procedures with
simpler approaches to robust inference. There are many ways to obtain
robust estimates of the covariance matrix, but these usually require
$n>p$. Instead we obtain a robust estimate for the marginal
covariances and variances using the procedure of \cite{kalisch}. Since
this is not guaranteed to result in a positive definite matrix, we add
a constant, $c$, to the diagonal elements of the matrix, where $c$ is
the minimum constant necessary to ensure the resulting matrix is
nonnegative definite. We then use this robust estimate of the
covariance matrix as input into the \textit{glasso} and refer to this
procedure as the \textit{robust glasso}.
\subsection{Results}
Our \textit{tlasso} and $t^*_{\mathrm{var}}$\textit{-lasso} are
computationally more expensive, since they call the \textit{glasso} at
each M-step. But in our simulations, the algorithms converge quickly.
If we run through multiple increasing values of the tuning parameter
$\rho$ for the one-norm penalty, it may take about $15$--$30$ EM
iterations for the initial small value of $\rho$, but only 2 or 3
iterations for later values, as we can ``warm start'' at the previous
output. But even in the initial run, two iterations typically lead to
a drastic improvement (in the $t$ likelihood) over the \textit{glasso}.
The only caveat is that the function being maximized by the \textit{tlasso}
methods is not guaranteed to be unimodal. We thus started in
several places, and let the algorithm run for longer than probably
necessary in practice. We did not observe drastically different
results from different starting places. Nonetheless, since we are not
guaranteed to find a global maximum, the statistical performances of
the \textit{tlasso} and $t^*_{\mathrm{var}}$\textit{-lasso} may, in principle, be understated here (and, of course,
the computational efficiency overstated).
\begin{figure}
\includegraphics{410f04.eps}
\vspace*{-5pt}
\caption{ROC curves depicting the performances of the four methods
under four different types of data. Each curve is an average over 250
simulations.} \label{fig:sims}\vspace*{-6pt}
\end{figure}
In the worst case scenario for our procedures relative to the \textit{glasso}---when
the data is normal and we assume $t$-distributions with
$3$ degrees of freedom---almost no statistical efficiency is lost. In
the numerous simulations we have run using normal data, the \textit{tlasso} and
\textit{glasso} do an essentially equally good job of
recovering the true graph (see Figure \ref{fig:sims}). The $t^*_{\mathrm{var}}$\textit{-lasso} performs
surprisingly well at small to moderate false discovery rates. The
\textit{robust glasso} is based on a less efficient estimator and does not
perform as well as the other procedures.
For data generated from a classical $t$-distribution with $3$ degrees
of freedom, the \textit{tlasso} provides drastic improvement over the
\textit{glasso} at the low false positive rates that are of practical interest.
The assumed normality and the occasional extreme observation lead to
numerous false positives when using the \textit{glasso}. Therefore,
there is very little computational---and little or no
statistical---downside to assuming $t$-distributions, but significant
statistical upside. Interestingly, the $t^*_{\mathrm{var}}$\textit{-lasso}
performs about as well as the \textit{tlasso}. The \textit{robust
glasso} outperforms the purely Gaussian procedure at low false positive
rates, since it is less susceptible to the most extreme
observations.
In the third case, with data generated from the alternative
$t$-distribution with $3$~degrees of freedom, only the $t^*_{\mathrm{var}}$\textit{-lasso}
is able to recover useful information without substantial noise.
The occasional large values are too extreme for the normal model to
explain and downweighting entire observations, as is done by the \textit{tlasso},
discards too much information when there are extreme values
scattered throughout the data. The \textit{robust glasso} offers only
a small improvement over the \textit{glasso}.
With the contaminated data, the $t^*_{\mathrm{var}}$\textit{-lasso} does not
perform as well in this case as it does with $t^*$ data. The extreme
values are not downweighted as much and, thus, the signals are noisier.
It still performs far better, however, than either of the other
methods, and is able to recover valuable information in a case where
manual\ screening of outliers would be very difficult. The
\textit{robust glasso} does not perform as well as the $t^*_{\mathrm{var}}$\textit{-lasso},
but offers a clear improvement over the \textit{glasso}
and might be a useful alternative.
\subsection{Notes on simulation}
The simulations show that the \textit{tlasso} performs very similarly
to the \textit{glasso} even with normal data. While one would expect a~model based on the $t$-distribution to fare better with normal data
than a~normal model would with $t$ data, the fact that there is almost
no statistical loss from the model misspecification is at first a bit
surprising. The similarity of the results can be explained, however,
by comparing the two procedures. In effect, the only difference is that
the \textit{tlasso} inputs a weighted sample covariance matrix into the
\textit{glasso} procedure; one can then think of the \textit{glasso} as
the \textit{tlasso}\vadjust{\goodbreak} with all weights set to~one.
As noted in Section \ref{sec:em}, these weights are the conditional
expectations of~$\tau$, which are, from equation (\ref{eq:etau}),
\begin{equation}
\tau_i^{(t+1)}= \mathbb{E}[\tau_i | Y_i] = \frac{\hat{\nu} + p}{\hat{\nu}
+ \delta_{Y_i}(\mu^{(t)},\Psi^{(t)})},
\end{equation}
where $\hat{\nu}$ is our estimate or assumption of the unknown degrees
of freedom. If $Y \sim t_{p,\nu}(\mu,\Psi)$ and $\nu > 4$, then
$\delta_Y(\mu,\Psi)/p$ is distributed according to the
$\mathcal{F}_{p,\nu}$ distribution [\cite{kotz}, Chapter 3]. Thus,
starting with the true values of $\mu$ and $\psi$, the variance of the
inverse weights is
\[
\mathbb{V} \biggl[ \frac{\hat{\nu} + \delta_Y(\mu,\Psi)}{\hat{\nu} + p}
\biggr]= \frac{2 p \nu^2 (p+\nu -2)}{(\nu-2)^2(\nu-4)(\hat{\nu}+p)^2}.
\]
For normal data (i.e., $\nu=\infty$), the variance is
$2p/(\hat{\nu}+p)^2$ and goes to $0$ very quickly as $p$ gets large, no
matter the assumed value of $\hat{\nu}$. If our current estimate of
$\Theta$ is reasonably close to the true $\Theta$, then the
observations will likely have very similar weights and the weighted
covariance matrix will be very close to the sample covariance matrix.
For $t$ data, the above variance tends to $2 \nu^2/(\nu-2)^2(\nu-4)$
for large $p$; so no matter how many variables we have, the
distribution of the inverse weights will have positive variance and the
\textit{tlasso} and \textit{glasso} estimates are less likely to agree.
\section{Gene expression data} \label{sec:gene-expression-data}
\subsection{Galactose utilization}\label{sec:galactose}
We consider data from microarray experiments with yeast strands
[\cite{gasch}]. As in \cite{drtonrichardson}, we limit this
illustration to $8$ genes involved in galactose utilization. An
assumption of normality is brought into question, in particular, by the
fact that in $11$ out of $136$ experiments with data for all 8 genes,
the measurements for 4 of the genes were abnormally large negative
values. In order to assess the impact of this handful of outliers, we
run each algorithm, adjusting the penalty term $\rho$ such that a graph
with a given number of edges is inferred. Somewhat arbitrarily we focus
on the top 9 edges. We do this once with all 136 experiments and then
again excluding the $11$ potential outliers.
As seen in Figure \ref{fig:gascht9}, the \textit{glasso} infers very
different graphs, with only 3 edges in common. When the ``outliers''
are included, the \textit{glasso} estimate in
Figure \ref{fig:gascht9}(a) has the 4 nodes in question fully
connected; when they are excluded, no edges among the 4 nodes are
inferred. The \textit{tlasso} does not exhibit this extreme behavior.
As seen in Figure \ref{fig:gascht9}(b), it recovers almost the same
graph in each case (7 out of 9 edges shared). When run with all the
data, the $\tau$ estimate is very small (${\sim}0.04$) for each of the
$11$ questionable observations compared with the average $\tau$
estimate of~$1.2$.\vadjust{\goodbreak} The graph in Figure \ref{fig:gascht9}(c) shows the
results from the $t^*_{\mathrm{MC}}$\textit{-lasso} which performs just as
well as the \textit{tlasso}. The $t^*_{\mathrm{var}}$\textit{-lasso} also
recovered 7 edges in both graphs (not shown) and infers relationships
similar to those found by the $t^*_{\mathrm{MC}}$\textit{-lasso}.
\begin{figure}
\includegraphics{410f05.eps}
\caption{Top 9 recovered edges: \textup{(a)} \textit{glasso},
\textup{(b)} \textit{tlasso}, \textup{(c)} $t^*_{\mathrm{MC}}$\textit{-lasso}. Dashed
edges were recovered only when including the outliers; dotted only when
excluding them; solid in both cases.}\label{fig:gascht9}
\end{figure}
Figure \ref{fig:heatmap} illustrates the flexibility of the weighting
schemes of the various procedures. Both $t^*$ procedures downweight
the 11 potential outliers observations for the 4 nodes in question, but
not for the other nodes. Thus, the alternative version is able to
extract information from the ``uncontaminated'' part of the $11$
observations while downweighting the rest. In this particular case,
with 125 other observations, downweighting the outliers is of primary
importance, and, thus, the increased flexibility of the $t^*_{\mathrm{MC}}$\textit{-lasso} over the \textit{tlasso} does not make much of a difference in the inferred graphs. This
might not be the case with a higher contamination level.
\subsection{Isoprenoid pathway}\label{sec:isoprenoid-pathway}
We next consider gene expression data for the isoprenoid pathways of
\textit{
Arabidopsis thaliana} discussed in \cite{wille}. Gene expressions were
measured in 118 Affymetrix microarrays for 39 genes. While the data
set described in the above section had clear deviations from normality,
the data described in this section has no obvious deviations that stand
out in exploratory plots.
\begin{figure}
\includegraphics{410f06.eps}
\vspace*{-5pt}
\caption{From left to right, inverse weights from \textit{tlasso},
followed by normalized gene expression data, and inverse weights from $t^*_{\mathrm{MC}}$\textit{-lasso} and $t^*_{\mathrm{var}}$\textit{-lasso}. Rows correspond to
genes and columns to observations. Lighter shades indicate larger values.
The \textit{tlasso} uses only one weight per observation and so must weight
each gene the same. All plots show the same subset of data including $11$
potential outliers.}\label{fig:heatmap}\vspace*{-5pt}
\end{figure}
Two approaches were considered in \cite{wille}. The first
(\textit{GGM1}) fit a~Gaussian graphical model using BIC and backward
selection to obtain a network with 178 edges. This number was deemed
too large for interpretation, and the authors considered instead only
the 31 edges found in at least $80\%$ of bootstrapped samples. The
second approach (\textit{GGM2}) tests the conditional independence of each pair of genes given a
third gene. An edge is drawn only if a test of conditional
independence is rejected for each other gene in the network. This
approach is advocated in the paper and appears to find a network with
better biological interpretation. The graph is shown in
Figure \ref{fig:isoprenoid1}, where shaded nodes indicate the so-called
MEP pathway.
\begin{figure}[b]
\includegraphics{410f07.eps}
\caption{A reproduction of the graph produced by Wille
et al. Solid
undirected edges are those found by the model selection procedure; dotted
arrows show the metabolic pathway.}\label{fig:isoprenoid1}\vspace*{3pt}
\end{figure}
Our approach is modeled after \textit{GGM1}. We used the $t^*_{\mathrm{var}}$\textit{-lasso} and increasing values or $\rho$ to find a path of models to test.
For each chosen model, we ran the $t^*_{\mathrm{var}}$\textit{-lasso} again,
but this time without penalty on the allowed edges. Since the $t^*$
likelihood is unavailable, we use leave-one-out cross-validation to
find the model with the lowest mean squared prediction error. Since
the exact conditionals from the alternative distribution are not
available in explicit form, we perform the cross-validation as follows:
\begin{enumerate}[(b)]
\item[(a)] Estimate $\Theta$ using all but one observation.
\item[(b)]
In the remaining\vadjust{\goodbreak} observation, estimate the values of the latent
normal variables for all but one of the coordinates in the same manner as
the variational E-step of Section \ref{sec:variational}.
\item[(c)] Predict the remaining normal value.
\item[(d)] Scale the
normal value by the expectation of $1/\sqrt{\tau}$.
\end{enumerate}
We remark that we also experimented with leaving out a larger fraction
of the observations as suggested in the work of \cite{shao}, but this
led to similar conclusions in the present example.
The cross-validation procedure gave a network with 122 edges. To
reduce to the graph size found by \textit{GGM2}, we took 500
bootstrapped samples of the data, fixing the parameter $\rho$ found in
cross-validation, and only included those edges found in more than
$98.5\%$ of the samples. For comparison, we also ran the above
procedure using the \textit{glasso}, but keeping $98\%$ of the samples
to obtain the same-sized graph.
We believe our procedure infers a graph that compares favorably (in
terms of biological interpretation) with that found by \textit{GGM2}.
Like \textit{GGM2}, we find a connection between AACT2 and the group
MK, MPDC1 and FPPS2; \textit{GGM1} found AACT2 to be disconnected from
the rest of the graph despite its high correlation with these three
genes. In the MEP pathway, our approach and \textit{GGM2} find similar
structure; compare Figures \ref{fig:isoprenoid1} and
\ref{fig:isoprenoid2}.
While our approach finds the key relationships identified in Wille
et al., it achieves this with fewer ``cross-talk'' edges between the two
pathways. The authors discuss plausible interpretations for such
interactions between the pathways, but a graph with less cross-talk
might be closer to the scientists' original expectation
(Figures \ref{fig:isoprenoid1} and \ref{fig:isoprenoid2}). It is worth
noting that the \textit{glasso} procedure performs better than \textit{GGM1}, with edge inclusion being far less sensitive to the particular
bootstrapped sample. The \textit{glasso} also finds the key
relationships of \textit{GGM2}. We also ran the \textit{tlasso}, which
gave results similar to the \textit{glasso} and with the $t^*_{\mathrm{MC}}$\textit{-lasso},
which behaved similar to the $t^*_{\mathrm{var}}$\textit{-lasso}. We do
not show these results here.
\begin{figure}
\includegraphics{410f08.eps}
\caption{Graphs recovered by bootstrapping
procedure with target
graph size of 43 using \textup{(a)} the \textit{glasso} and \textup{(b)} $t^*_{\mathrm{var}}$\textit{-lasso}.
The graph shows the key relationships identified previously, but
with fewer ``cross-talk'' edges.}\label{fig:isoprenoid2}
\end{figure}
\section{Discussion}\label{sec:discussion}
Our proposed \textit{tlasso} and $t^*_{\mathrm{var}}$\textit{-lasso}
algorithms are simple and effective methods for robust inference in
graphical models. Only slightly more computationally expensive than the
\textit{glasso}, they can offer great gains in statistical efficiency.
The \textit{alternative t} distribution is more flexible than the
classical $t$ and is generally preferred. We find that the simple
variational E-step is an efficient way to estimate the graph in the
alternative case, but also explored more sophisticated Monte Carlo
approximations.
We assumed $\nu=3$ degrees of freedom in our various \textit{tlasso}
and $t^*$\textit{-lasso} runs. As suggested in prior work on
$t$-distribution models, estimation of the degrees of freedom can be
done efficiently by a line search based on the observed log-likelihood
function in the classical model.
\begin{figure}
\includegraphics{410f09.eps}
\caption{The alternative $t$ model places an upper bound on the
correlation between two variables. This bound increases with $\nu$,
but is fairly restrictive for the small degrees of freedom we
consider.} \label{fig:maxcorr}
\end{figure}
In the alternative model, the choice of $\nu$ puts an explicit upper
bound on the maximum correlation between two variables, the upper bound
increasing quickly with $\nu$ (see Figure \ref{fig:maxcorr}). This
makes inference of the degrees of freedom potentially more relevant
than with the classical model, as an alternative model with small $\nu$
might not be a good fit for highly correlated variables. In order to
select $\nu$, a line search based on the hidden log-likelihood function
can be employed. For further flexibility, we may also allow the
degrees of freedom to vary with each node. That is, we could let the
divisors $\tau_j \sim \Gamma(\nu_j/2,\nu_j/2)$ be independent
$\Gamma$-divisors with possible different degrees of freedom $\nu_j$.
This leads to similar conditionals in the Gibbs sampler and the
resulting procedure is thus no more complicated. Nevertheless, for the
purposes of graph estimation, our experience and related literature
suggest that not much is lost by considering only a few small values
for the degrees of freedom. For instance, running the $t^*_{\mathrm{var}}$\textit{-lasso} procedure in Section \ref{sec:isoprenoid-pathway}
using $\nu=5$ produces a very similar result with one additional
cross-talk edge.
In the last section we used cross-validation to choose the one-norm
tuning parameter $\rho$. The likelihood is not available explicitly
for the $t^*$-distribution and so we cannot easily use information
criteria for the $t^*$\textit{-lasso}. Cross-validation often tends to
pick more edges than is desirable, however, when the goal is inference
of the graph and not optimal prediction. An interesting but
potentially difficult problem for future research would be to develop
rules for choosing $\rho$ that control an edge inclusion error rate;
compare \cite{banerjee}; \cite{meinshausen}.
Throughout the paper, we have penalized all the elements of $\Theta$.
One alternative is to remove the penalty from the diagonal elements of
$\Theta$, since we expect all these to be nonzero. This leads to
smaller estimated partial correlations, and we found it to result in
less stable behavior of the \textit{tlasso} in the sense of the number
of edges decreasing rather suddenly as $\rho$ increases.
Finally, we remark that other normal scale-mixture models could be
treated in a~similar fashion as the $t$-distribution models we
considered in this paper. However, the use of $t$-distributions is
particularly convenient in that it is rather robust to various types of
misspecification, involves only the choice of the degrees of freedom
parameters for the distribution of Gamma-divisors, and maintains good
efficiency when data are Gaussian.
\begin{appendix}
\section*{Appendix}\label{appendix}
\begin{pf*}{Proof of Proposition \ref{thm:1}}
According to standard graphical model theory [\cite{lauritzen}], it
suffices to show that $Y_j$ and $Y_k$ are conditionally uncorrelated
given $Y_{V\setminus\{j,k\}}$. Partition $V$ into $a=\{j,k\}$ and
$b=V\setminus \{j,k\}$. For a given value of~$\tau$,
\[
(Y_a |Y_b, \tau) \sim N_2\bigl(\mu_a - \Theta_{a,a}^{-1} \Theta_{a,b} (Y_b-
\mu_b), \Theta_{a,a}^{-1}/\tau \bigr)
\]
and
\[
(Y_j|Y_{k\,\cup\,b},\tau) \sim N\bigl(\mu_i -
\theta_{jj}^{-1}\Theta_{j,k\,\cup\,
b} (Y_{k\,\cup\,b}- \mu_{k\,\cup\,b}), \theta_{jj}^{-1}/\tau\bigr).
\]
Since $\theta_{jk}=0$,
\[
\mathbb{E}[Y_j|Y_{k \,\cup\, b},\tau] = \mu_j - \theta_{jj}^{-1}\Theta_{j,b} (Y_{b}-\mu_{b})=\mathbb{E}[Y_j|Y_b,\tau]
\]
for any value of $\tau$. Therefore,
\[
\mathbb{E}[Y_j|Y_{k \,\cup\,b}] =
\mathbb{E}[\mathbb{E}[Y_j|Y_{k\,\cup\,
b},\tau]|Y_{k\, \cup\, b}] =
\mathbb{E}[\mathbb{E}[Y_j|Y_b,\tau]|Y_b] = \mathbb{E}[Y_j|Y_b],
\]
which implies that $Y_j$ and $Y_k$ are conditionally uncorrelated given $Y_b$.
\end{pf*}
\end{appendix}
|
{
"timestamp": "2011-08-10T02:01:28",
"yymm": "1009",
"arxiv_id": "1009.3669",
"language": "en",
"url": "https://arxiv.org/abs/1009.3669"
}
|
\section{\label{sec:intro}Introduction}
Development of CMOS compatible processes of formation of germanium quantum dot (QD) dense arrays on the (001) silicon surface as well as multilayer Ge/Si epitaxial heterostructures on their basis is a challenging task of great practical significance \cite{Report_01-303,Pchel_Review,Wang-properties,Smagina,Wang-Cha,WG-near_IR,Photonic_crystal,QDFET,QDIP-Wang,QDIP-Wang1,Sabelnik-2,Dvur-IR-20mcm,QDIP-MIS,QDIP-Wang2}.
An important direction of applied researches in this area is the development of highly efficient monolithic far and mid infrared detector arrays which could be produced by a standard CMOS technology \cite{QDIP-Wang,QDIP-Wang1,Sabelnik-2,Dvur-IR-20mcm,QDIP-MIS,QDIP-Wang2}. Such detectors have to combine high perfection (uniformity, sensitivity, operating life, etc.) with high yield and low production price. A requirement of CMOS compatibility of technological processes imposes a hard constraint on conditions of all phases of the QD array manufacturing starting from the stage of preparation of a clean Si surface for Ge/Si heterostructure deposition: on the one hand, formation of a photosensitive layer must be one of the latest operations of the whole device production cycle because otherwise the structure with QDs would be destroyed by further high temperature annealings; from the other hand, high temperature processes during Ge/Si heterostructure formation on the late phase of the detector chip production would certainly wreck the readout circuit formed on the crystal. Therefore, lowering of the array formation temperature down to the values of $\lesssim 450^\circ$C\footnote{As well as decreasing of the wafer annealing temperatures and times during the clean Si(001) surface preparation.} is strongly required \cite{Report_01-303,Sabelnik-2}, and the Ge QD arrays meeting this requirement are referred to as CMOS compatible ones.
In addition to the requirement of the low temperature of a Ge QD array formation,
both high density of the germanium nanoclusters ($> 10^{11}$~cm$^{-2}$) and high uniformity of the cluster shapes and sizes (dispersion $<$ 10\,\%) in the arrays are necessary for employment of such structures in CMOS IR detectors \cite{Dvur-IR-20mcm}.
The molecular beam epitaxy (MBE) is known to be the main technique of formation of Ge/Si heterostructures with QDs \cite{Pchel_Review,Brunner}. A high density of the self-assembled hut clusters can be obtained in the MBE process of the Ge/Si(001) structure formation when depositing germanium on the Si(001) substrate heated to a temperature $T_{\rm gr}\lesssim 550^\circ$C. In this case the lower is the temperature of the silicon substrate during the Ge deposition the higher is the density of the clusters at the permanent quantity of the deposited Ge \cite{Yakimov,Jin}. For example, the density of the Ge clusters in the array was $6\times 10^{11}$~cm$^{-2}$ at $T_{\rm gr} = 360^\circ$C and the effective thickness of the deposited germanium layer\footnote{I.e. the Ge coverage or, in other words, the thickness of the Ge film measured by the graduated in advance film thickness monitor with the quartz sensor installed in the MBE chamber.} $h_{\rm Ge} = 8~{\rm\AA}$; the cluster density of only $\sim 2\times 10^{11}$\,cm$^{-2}$ was obtained at $T_{\rm gr} = 530^\circ$C and the same value of $h_{\rm Ge}$ \cite{classification}.
There is another approach to obtaining dense cluster arrays. The authors of Refs.~\cite{Smagina,Ion_irradiation_1,Dvur_irrad,Ion_irradiation} reached the cluster density of $\sim 9\times 10^{11}$\,cm$^{-2}$ using the pulsed irradiation of the substrate by a low-energy Ge$^+$ ion beam during the MBE growth of the Ge/Si(001) heterostructures at $T_{\rm gr}$ as high as $570^\circ$C.
Obtaining of the arrays of the densely packed Ge QDs on the Si(001) surface is an important task but the problem of formation of uniform arrays of the Ge clusters is much more challenging one. The process of Ge/Si(001) heterostructure formation with the Ge QD dense arrays and predetermined electrophysical and photoelectric parameters cannot be developed until both of these tasks are solved. The uniformity of the cluster sizes and shapes in the arrays determines not only the widths of the energy spectra of the charge carrier bound states in the QD arrays \cite{Smagina} but in a number of cases the optical and electrical properties of both the arrays themselves and the device structures produced on their basis \cite{Electrolumin}. To find an approach to the improvement of the Ge QD array uniformity on the Si(001) surface it is necessary to carry out a detailed morphological investigation of them.
This article presents the results of our recent investigations of several important issues of the Ge dense array formation and growth. We have studied the array nucleation phase (the transition from 2D growth of the wetting layer (WL) to 3D formation of the QD array when the nuclei of both species of huts---pyramids and wedges \cite{classification}---begin to arise on the $(M\times N)$ patches of WL). We have identified by STM the nuclei of both species, determined their atomic structure \cite{classification,Hut_nucleation} and observed the moment of appearance the first generation of the nuclei. We have investigated with high spatial resolution the peculiarities of each species of huts and their growth and derived their atomic structures \cite{Hut_nucleation,atomic_structure}. We have concluded that the wedge-like huts form due to a phase transition reconstructing the first atomic step of the growing cluster when dimer pairs of its second atomic layer stack up; the pyramids grow without such phase transitions. In addition, we have come to conclusion that wedges contain vacancy-type defects on the penultimate terraces of their triangular facets \cite{Hut_nucleation} which may decrease the energy of addition of new atoms to these facets and stimulate the quicker growth on them than on the trapezoidal ones and rapid elongation of wedges. We have shown also comparing the structures and growth of pyramids and wedges that shape transitions between them are very unlikely \cite{Hut_nucleation,atomic_structure}. Finally, we have explored the array evolution during MBE right up to the end of its life when most of clusters coalesce and start forming a nanocrystalline 2D layer.
Below, we present these results in detail.
\section{Methods, equipment and conditions of experiments}
\label{sec:setup}
The experiments were made using an integrated ultrahigh vacuum instrument \cite{classification} built on the basis of the Riber surface science center with the EVA~32 molecular beam epitaxy chamber connected to the STM GPI-300 ultrahigh vacuum scanning tunnelling microscope \cite{gpi300,STM_GPI-Proc,STM_calibration}.
This equipment allows us to carry out the STM study of samples at any phase of a substrate surface preparation and MBE growth. The samples can be transferred into the STM chamber for the examination and moved back into the MBE vessel for further processing as many times as required never leaving the UHV ambient and preserving the required cleanness for STM investigations with atomic resolution and MBE growth.
Initial substrates were 8$\times$8 mm$^{2}$ squares cut from the specially treated commercial B-doped CZ Si$(100)$ wafers ($p$-type, $\rho\,= 12~\rm\Omega\,$cm).
After washing and chemical treatment following the standard procedure described elsewhere \cite{phase_transition_ru,STM_RHEED} (which included washing in ethanol, etching in the mixture of HNO$_3$ and HF and rinsing in the deionized water), the silicon substrates were
mounted on the molybdenum STM holders and inflexibly clamped with the tantalum fasteners. The STM holders were placed in the holders for MBE made of molybdenum with tantalum inserts. Then the substrates were
loaded into the airlock and transferred into the preliminary annealing chamber where outgassed at the temperature of around $565^\circ$C and the pressure of about $5\times 10^{-9}$ Torr for about 24 hours. After that the substrates were moved for final treatment into the MBE chamber evacuated down to about $10^{-11}$\,Torr. There were two stages of annealing in the process of substrate heating in the MBE chamber\,--- at $\sim 600^\circ$C for $\sim 5$ minutes and at $\sim 800^\circ$C for $\sim 3$ minutes \cite{classification}. The final annealing at the temperature greater than $900^\circ$C was carried out for nearly $ 2.5$ minutes with the maximum temperature of about $ 925^\circ$C ($\sim 1.5$~minutes). Then the temperature was rapidly lowered to about $ 750\,^\circ$C. The rate of the further cooling was around $0.4^\circ$C/s that corresponded to the ``quenching'' mode applied in \cite{STM_RHEED}. The pressure in the MBE chamber grew to nearly $2\times 10^{-9}$ Torr during the deoxidization process.
The surfaces of the silicon substrates were completely purified of the oxide film as a result of this treatment; more data on the morphology of the prepared Si(001) clean surfaces can be found in Refs.~\cite{phase_transition_ru,STM_RHEED,our_Si(001)_en}.
\begin{figure}
\includegraphics[width=0.4\textwidth]{fig_1a_color}(a)
\includegraphics[width=0.4\textwidth]{fig_1b_color}(b)
\caption{\label{fig:nucleation}
STM image of Ge wetting layer on Si(001): (a) before cluster nucleation, $h_{\rm Ge}=4.4$~\r{A} ($U_{\rm s}=-1.86$~V, $I_{\rm t}=100$~pA); (b) arising nuclei of pyramidal (1) and wedgelike (2) huts, $h_{\rm Ge}=5.1$~\r{A} ($U_{\rm s}=+1.73$~V, $I_{\rm t}=150$~pA).}
\end{figure}
Ge was deposited directly on the deoxidized Si(001) surface from the source with the electron beam evaporation\footnote{The Si source was switched off during the experiments.}. The Ge deposition rate was about $0.15$~\r{A}/s; the effective Ge film thickness $h_{\rm Ge}$ was varied from 4~\r{A} to 15~\r{A} for different samples. The deposition rate and $h_{\rm Ge}$ were measured by the XTC film thickness monitor with the graduated in advance quartz sensor installed in the MBE chamber. The substrate temperature $T_{\rm gr}$ was $360^\circ$C during Ge deposition; the pressure in the MBE chamber did not exceed $10^{-9}$ Torr. The rate of the sample cooling down to the room temperature was approximately $0.4^\circ$C/s after the deposition.
The samples were heated by Ta radiators from the rear side in both preliminary annealing and MBE chambers. The temperature was monitored with chromel-alumel and tungsten-rhenium thermocouples in the preliminary annealing and MBE chambers, respectively. The thermocouples were mounted in vacuum near the rear side of the samples and {\it in situ} graduated beforehand against the IMPAC~IS\,12-Si pyrometer which measured the sample temperature through chamber windows.
The atmosphere composition in the MBE camber was monitored using the SRS~RGA-200 residual gas analyzer before and during the process.
After Ge deposition and cooling, the prepared samples were moved for analysis into the STM chamber in which the pressure did not exceed $10^{-10}$ Torr. The STM tip was {\it ex situ} made of the tungsten wire and cleaned by ion bombardment \cite{W-tip} in a special UHV chamber connected to the STM one. The images were obtained in the constant tunneling current ($I_{\rm t}$) mode at the room temperature. The STM tip was zero-biased while the sample was positively or negatively biased ($U_{\rm s}$) when scanned in empty or filled states imaging mode.
Original firmware \cite{gpi300,STM_GPI-Proc,STM_calibration} was used for data acquisition; the STM images were processed afterwords using the WSxM software \cite{WSxM}.
\section{\label{sec:results}Experimental data and structural models}
\subsection{\label{sec:formation}Array and hut cluster nucleation}
\begin{figure}
\includegraphics[width=0.4\textwidth]{fig_2a_color}(a)
\includegraphics[width=0.4\textwidth]{fig_2b_color}(b)
\caption{\label{fig:nuclei}
STM image of Ge wetting layer on Si(001): (a) $c(4\times 2)$~$(c)$ and $p(2\times 2)$ $(p)$ reconstructions within the $(M\times N)$ patches, $h_{\rm Ge}=6,0$~\r{A}, $U_{\rm s}= +1.80$~V, $I_{\rm t}=80$~pA; (b) new formations arise on the $(M\times N)$ patches due to nucleation of Ge pyramid (1) and wedge (2), $h_{\rm Ge}=6,0$~\r{A}, $U_{\rm s}= +2.60$~V, $I_{\rm t}=80$~pA.}
\end{figure}
\begin{figure}
\includegraphics[width=0.215\textwidth]{fig_3a}(a)
\includegraphics[width=0.215\textwidth]{fig_3b}(b)
\includegraphics[width=0.215\textwidth]{fig_3c_color}(c)
\caption{\label{fig:models}Models of nuclei of Ge hut clusters corresponding to the images given in Fig.~\ref{fig:nuclei}(b): (a) a pyramid,
(b) a wedge [1 is the wetting layer in the plots (a) and (b)]; (c) the models superimposed on the image given in Fig.~\ref{fig:nuclei}(b), the numbering is the same as in Fig.~\ref{fig:nuclei}(b).
}
\end{figure}
\begin{figure}
\includegraphics[width=0.4\textwidth]{fig_4a_color}(a)
\includegraphics[width=0.4\textwidth]{fig_4b_color}(b)
\caption{\label{fig:dots}
STM image of Ge wetting layer on Si(001): (a) $h_{\rm Ge} = 5.4$~\r{A} ($U_{\rm s}=+1.80$~V, $I_{\rm t}=100$~pA) and (b) $h_{\rm Ge} = 6.0$~\r{A}
($U_{\rm s}=+2.50$~V, $I_{\rm t}=80$~pA). Examples of characteristic features are numbered as follows: nuclei of pyramids (1) and wedges (2) [1 ML high over WL], small pyramids (3) and wedges (2) [2 ML high over WL, a $\rm \Gamma$-like wedge \cite{classification} is observed in the image (a)], 3 ML high pyramids (5) and wedges (6).}
\end{figure}
Investigating an evolution of the hut arrays we have arrived at a conclusion that a moment of an array nucleation during MBE precedes a moment of formation of the first hut on the WL.\footnote{Or, in other words, it foreruns a moment of formation of the first \{105\} faceted cluster with the height-to-width ratio of 1:10 on WL.} It is not a paradox. Hut cluster arrays nucleate when the first hut nuclei arise on the $(M\times N)$ patch of the wetting layer. This process is illustrated by Fig.~\ref{fig:nucleation}. An image (a) demonstrates a typical STM micrograph of the WL with the $(M\times N)$ patched structure ($h_{\rm Ge}=4.4$~\r{A}). This image does not demonstrate any feature which might be interpreted as a hut nucleus \cite{Hut_nucleation}. Such features first arise at the coverages $\sim 5$\,\r{A}: they are clearly seen in the image (b), which demonstrates a moment of the array birth ($h_{\rm Ge}=5.1$~\r{A}), and numbered by `1' for the pyramid nucleus and `2' for the wedge one (several analogous formations can be easily found by the readers on different patches). However, no hut clusters are seen in this picture.
Our interpretation is based on the results reported in Ref.~\cite{Hut_nucleation} which evidenced that there are two different types of nuclei on Ge wetting layer which evolve in the process of Ge deposition to pyramidal and wedge-like hut clusters.
Having assumed that nuclei emerge on WL as combinations of dimer pairs and/or longer chains of dimers in epitaxial configuration \cite{epinucleation} and correspond to the known structure of apexes specific for each hut species \cite{classification,atomic_structure}
we have investigated WL patches, 1 monolayer (ML) high formations on them and clusters of different heights (number of steps) over WL. As a result, we succeeded to select two types of formations different in symmetry and satisfying the above requirements, which first appear at a coverage of $\sim 5$~\r{A} and then arise on WL during the array growth. We have interpreted them as hut nuclei, despite their sizes are much less than those predicted by the first principle calculations \cite{hut_stability}, and traced their evolution to huts.
The nuclei formation is illustrated by Fig.~\ref{fig:nuclei}. The surface structure of the $(M\times N)$ patches is shown in the micrograph (a). The letter `$c$' indicates the
$c(4\times 2)$ reconstructed patch, `$p$' shows a patch with the $p(2\times 2)$ reconstruction \cite{Iwawaki_initial,Iwawaki_dimers}. Both reconstructions are always detected simultaneously that means they are very close (or degenerate) by energy. The image (b) shows two adjacent patches reconstructed by the born nuclei: `1' and `2' denote the pyramid (a formation resembling a blossom) and wedge nuclei respectively \cite{Hut_nucleation}. Their structural models derived from many STM images \cite{classification,atomic_structure,Hut_nucleation} are presented in Fig.~\ref{fig:models}(a, b) and superimposed on the images of the nuclei in Fig.~\ref{fig:models}(c). Note that both types of nuclei arise at the same moment of the MBE growth. It means that they are degenerate by the formation energy. An issue why two different structures, rather than one, arise to relief the WL strain remains open, however.
It is necessary to remark here that the nuclei are always observed to arise on sufficiently large WL patches. There must be enough room for a nucleus on a single patch. A nucleus cannot be housed on more than one patch. So, cluster nucleation is impossible on little (too narrow or short) patches (Fig.~\ref{fig:nuclei}(b)).
The hut nucleation goes on during the array further evolution. Fig.~\ref{fig:dots}
illustrates this process. An array shown in Fig.~\ref{fig:dots}(a) ($h_{\rm Ge} = 5.4$~\r{A}) consists of 1-ML nuclei (`1' and `2'), 2-ML and 3-ML pyramids and wedges (`3' and `5', `4' and `6' mark pyramids and wedges respectively).\footnote{Hereinafter, the cluster heighs are counted from the WL top.}
Fig.~\ref{fig:dots}(b) ($h_{\rm Ge} = 6.0$~\r{A}) demonstrates the simultaneous presence of nuclei (`1' and `2') and 2-ML huts (`3' and `4') with the growing much higher clusters.
Hut cluster nucleation on the WL surface continues until the final phase of the array life. This peculiarity distinguishes low-temperature growth mode from the high-temperature one \cite{classification}.
\subsection{\label{sec:structure}Structural models}
\begin{figure}
\includegraphics[width=0.33\textwidth]{fig_5a}(a)
\includegraphics[width=0.45\textwidth]{fig_5b}(b)
\caption{\label{fig:pyramids} Top views of the pyramidal QDs consisting of (a) 2 and (b) 6 monoatomic steps and (001) terraces on the wetting layer (1, 2 and 3 designate wetting layer, the first and the second layers of the clusters respectively). }
\end{figure}
\begin{figure}
\includegraphics[width=0.21\textwidth]{fig_6a}(a)
\includegraphics[width=0.30\textwidth]{fig_6b}(b)
\includegraphics[width=0.4\textwidth]{fig_6c}(c)
\caption{\label{fig:wedges} Growth of a wedge-like cluster:
(a) reconstruction of the first layer of a forming wedge during addition of epi-oriented dimer pairs of the second (001) terrace;
plots of atomic structures of a Ge wedge-shaped hut clusters composed by (b) 2 and (c) 6 monoatomic steps and (001) terraces on the wetting layer (the numbering is the same as in Fig.~\ref{fig:pyramids}; d marks a defect arisen because of one translation uncertainty of the left dimer pair position).
}
\end{figure}
It is commonly adopted that the hut clusters grow by successive filling the (001) terraces of the $\{105\}$ faces by the dimer rows \cite{Kastner}. However, formation of the sets of steps and terraces requires the hut base sides to be parallel to the $<$100$>$ directions. The pyramid nucleus satisfies this requirement, its sides aline with \textless 100\textgreater. Thus the pyramids grow without phase transition when the second and subsequent layers are added (Fig.~\ref{fig:pyramids}). Only nucleus-like structures of their apexes are rotated $90^{\circ}$ with respect to the rows on previous terraces to form the correct epitaxial configuration when the heights are increased by 1\,ML, but this rotation does not violate the symmetry of the previous layers of the cluster.
A different scenario of growth of the wedge-like clusters have been observed. Two base sides the wedge nucleus does not aline with $<$100$>$ (Fig.~\ref{fig:models}(b)). The ridge structure of a wedge is different from the nucleus structure presented in Fig.~\ref{fig:models}(b) \cite{classification,Hut_nucleation,atomic_structure}. It was shown in Ref.~\cite{Hut_nucleation} that the structure of the wedge-like cluster arise due to rearrangement of rows of the first layer in the process of the second layer formation (Fig.~\ref{fig:wedges}(a)).
The phase transition in the first layer generates the base with all sides directed along the $<$100$>$ axes which is necessary to give rise to the $\{105\}$ faceted cluster.
After the transition, the elongation of the elementary structure is possible only along a single axis which is determined by the symmetry (along the arrows in Fig.~\ref{fig:wedges}(a)). A formed 2-ML wedge is plotted in Fig.~\ref{fig:wedges}(b). A structure of the 6-ML wedge appeared as a result of further in-height growth is shown in
Fig.~\ref{fig:wedges}(c). The ridge structures of the 2-ML and 6-ML wedges is seen to coincide, which is not the case for different cluster heights. A complete set of the wedge ridges for different cluster heights can be obtained by filling the terraces by epi-oriented pairs of dimers.
It should be noted also that according to the proposed model the wedge-like clusters always contain point defects on the triangular (short) facets. The defects are located in the upper corners of the facets and caused by uncertainty of one translation in the position a dimer pair which forms the penultimate terrace of the triangular facet (Fig.~\ref{fig:wedges}). The predicted presence of these defects removes the degeneracy of the facets and hence
an issue of the pyramid symmetry violation which occurs if the pyramid-to-wedge transition is assumed (this issue was discussed in detail in Ref.~\cite{classification}). In addition,
the vacancy-type defects may decrease the energy of addition of new atoms to the triangular facets and stimulate the quicker growth on them than on the trapezoidal ones and rapid elongation of wedges.
These defects are absent on the facets of the pyramidal huts.
Their triangular facets are degenerate. Therefore, as it follows from our model, the trapezoidal and triangular facets of the wedge
are not degenerate with respect to one another even at very beginning of cluster growth. The wedges can easily elongate by growing on the
triangular facets faster than on trapezoidal ones. Pyramids, having degenerate facets, cannot elongate and grow
only in height outrunning wedges. This explains greater heights of pyramids \cite{classification}.
Analyzing the deduced structural models of pyramids and wedges, as well as their behaviour during the array nucleation and growth, we have come to conclusion that shape transitions between the clusters of different species are prohibited \cite{classification,Hut_nucleation,atomic_structure}.
\subsection{\label{sec:facets}Facets}
The presented models allowed us to
deduce a structure of the $\{105\}$ facets (Fig.~\ref{fig:face}(a)). This model resulting from the above simple crystallographic consideration corresponds to the paired dimers (PD) \cite{Mo} rather than more recent rebonded step (RS) model \cite{Fujikawa,Facet-105} which is now believed to improve the previous PD model by Mo {\it et al}.
A direct STM exploration of the $\{105\}$ facets confirms the derived model.
Being superposed with the empty state STM image of the cluster $\{105\}$ facet it demonstrates an excellent agreement with the experiment (Fig.~\ref{fig:face}(b)).
A typical STM image of the QD facet is presented in Fig.~\ref{fig:facet}. Characteristic distances on the facets are as follows: $\sim 10.5$~\r{A} in the \textless 100\textgreater~directions (along the corresponding side of the base) and $\sim 14$~\r{A} in the normal (\textless 051\textgreater) directions.
The facets are composed by structural units which are outlined by ellipses in Fig.~\ref{fig:facet}(a) and can be arranged along either [110] or [1${\overline 1}$0] direction on the (001) plane. We have interpreted them as pairs of dimers. Their positional relationship is obviously seen in the 3D micrograph presented in Fig.~\ref{fig:facet}(b).
Dangling bonds of the derived $\{105\}$-PD facets, due to high chemical activity,
may stimulate Ge atom addition and cluster growth. Thus
less stability and higher activity of the $\{105\}$-PD facets compared to the \linebreak
Ge(105)/Si(105)-RS plane, which is usually adopted in the literature for simulation of hut $\{105\}$ facets, may cause fast completion of hut terraces during epitaxy and be responsible (or even be necessary) for hut formation and growth.
\begin{figure}[h]
\includegraphics[width=0.49\textwidth]{fig_7a}(a)
\includegraphics[width=0.3\textwidth]{fig_7b}(b)
\caption{\label{fig:face}
(a) A structural model of the $\{105\}$ facet of hut clusters derived from the plots given in Figs.~\ref{fig:pyramids} and~\ref{fig:wedges} corresponds to the PD (pairs of dimers) model \cite{Mo}, S$_{\rm A}$ and S$_{\rm B}$ are commonly adopted designations of the monoatomic steps \cite{Chadi}, atoms situated on higher terraces are shown by larger circles.
(b) The schematic of the facet superimposed on its STM image ($4.3\times 4.4$~nm, $U_{\rm s}= +3.0$~V, $I_{\rm t}=100$~pA), the [100] direction is parallel to the corresponding base side, the steps rise from the lower right to the upper left corner.
}
\end{figure}
\begin{figure}
\includegraphics[width=0.215\textwidth]{fig_8a}(a)
\includegraphics[width=0.215\textwidth]{fig_8b-color}(b)
\caption{\label{fig:facet}
(a) 2D and (b) 3D STM images of the same area on Ge hut cluster facet ($h_{\rm Ge}=10$~\r{A}, $T_{\rm gr}=360^{\circ}$C,
$U_{\rm s}=+2.1$~V, $I_{\rm t}=80$~pA). The sides of the cluster base lie along the [100] direction; structural units revealed on the free surfaces of the (001) terraces and interpreted as paired dimers are marked out. }
\end{figure}
\subsection{\label{sec:density}Cluster density and fractions}
\begin{figure}
\includegraphics[width=0.215\textwidth]{fig_9a}(a)
\includegraphics[width=0.215\textwidth]{fig_9b}(b)
\caption{\label{fig:density} (a)
Density and (b) fraction of the Ge clusters in the arrays formed at $T_{\rm gr} = 360^\circ$C ($\square$~marks the pyramids, $\blacksquare$~designates the wedges, $\Circle$~is the total density).}
\end{figure}
Fig.~\ref{fig:density}(a) plots the dependence of the cluster density on $h_{\rm Ge}$ for different clusters in the arrays. It is seen that the density of wedges rises starting from $D_{\rm w}\approx 1,8\times 10^{11}$~cm$^{-2}$ at the beginning of the three-dimensional growth of Ge (the estimate is obtained by data extrapolation to $h_{\rm Ge}= 5$~\r{A}) and reaches the maximum of $\sim 5\times 10^{11}$~cm$^{-2}$ at $h_{\rm Ge} \sim 8$~\r{A}, the total density of clusters at this point $D_{\rm \Sigma} \sim 6\times 10^{11}$~cm$^{-2}$ is also maximum. Then both $D_{\rm w}$ and $D_{\rm \Sigma}$ slowly go down until the two-dimensional growth of Ge starts at $h_{\rm Ge} \sim 14$~\r{A} and $D_{\rm \Sigma}\approx D_{\rm w} \sim 2\times 10^{11}$~cm$^{-2}$ (the contribution of pyramids $D_{\rm p}$ to $D_{\rm \Sigma}$ becomes negligible---~$\sim 3\times 10^{10}$~cm$^{-2}$---at this value of $h_{\rm Ge}$).
The pyramid density exponentially drops as the value of $h_{\rm Ge}$ grows ($D_{\rm p} \approx 5\times 10^{11} \exp\{-2.0\times 10^7\,h_{\rm Ge}\}$, $h_{\rm Ge}$ is measured in centimeters). The maximum value of $D_{\rm p} \approx 1.8\times 10^{11}$~cm$^{-2}$ obtained from extrapolation to $h_{\rm Ge} = 5$~\r{A} coincides with the estimated initial value of $D_{\rm w}$.
The graphs of cluster fractions in the arrays versus $h_{\rm Ge}$ are presented in Fig.~\ref{fig:density}(b). Portions of pyramids and wedges initially very close ($\sim 50\%$ at $h_{\rm Ge}\sim 5$~\AA) rapidly become different as $h_{\rm Ge}$ rises. The content of pyramids monotonically falls. The fraction of the wedge-like clusters is approximately $57\%$ at the early stage of the array growth ($h_{\rm Ge} = 6$~\r{A}) and becomes $82\%$ at $h_{\rm Ge} = 8$~\r{A}. At further growth of the array, the content of the wedges reaches the saturation at the level of approximately $88\%$ at $h_{\rm Ge} = 10$~\r{A}.
The inference may be made from this observation that contrary to the intuitively expected from the consideration of symmetry, the wedge-like shape of the clusters is energetically more advantageous than the pyramidal one, and the more advantageous the more Ge atoms (and the more the number of terraces) constitute the cluster.
The probability of nucleation appears to be close to 1/2 for both wedges-like and pyramidal clusters at the initial stage of the array formation and low growth temperatures. Then, as the array grows, the formation of pyramids becomes hardly probable and most of them, which have already formed, vanish whereas the nucleation and further growth of wedges continues. The Ge pyramids on the Si(001) surface turned out to be less stable objects than the wedges.
Notice also that at $T_{\rm gr} = 360^\circ$C and the flux of Ge atoms ${\rm d} h_{\rm Ge}/{\rm d}t = 0.15$~\r{A}/s, the point $h_{\rm Ge} = 10$~\r{A} is particular. Not only the fraction of pyramids saturates at this point but the array in whole has the most uniform sizes of the clusters composing it (Figs.~\ref{fig:dots}, \ref{fig:arrays} and \ref{fig:finish}). This is concluded by us not only on the basis of analysis of the STM images of the Ge/Si(001) arrays but also from the data of the Raman scattering by the Ge/Si heterostructures with different low-temperature arrays of Ge quantum dots \cite{our_Raman_en,Raman_conf}. We refer to such arrays as optimal.
\subsection{\label{sec:life}Array life cycle}
\begin{figure}
\includegraphics[width=0.215\textwidth]{fig_10a-color}(a)
\includegraphics[width=0.215\textwidth]{fig_10b-3d-color}(b)
\includegraphics[width=0.215\textwidth]{fig_10c-color}(c)
\includegraphics[width=0.215\textwidth]{fig_10d-color}(d)
\caption{\label{fig:arrays}
STM 2D and 3D micrographs of Ge hut cluster dense arrays at different coverages ($T_{\rm gr} = 360^\circ$C): (a),\,(b) $h_{\rm Ge}=8$~\r{A} [(a)~$50.6\times 49.9$\,nm, $w$ is the wetting layer, $U_{\rm s}=+2.0$~V, $I_{\rm t}=80$~pA; (b) $U_{\rm s}=+2.0$~V, $I_{\rm t}=100$~pA]; (c),\,(d) $h_{\rm Ge}=10$~\r{A} [(c),\,(d) $U_{\rm s}=+2.1$~V, $I_{\rm t}=100$~pA].}
\end{figure}
\begin{figure}
\includegraphics[width=0.215\textwidth]{fig_11a-color}(a)
\includegraphics[width=0.215\textwidth]{fig_11b-color}(b)
\includegraphics[width=0.215\textwidth]{fig_11c-color}(c)
\includegraphics[width=0.215\textwidth]{fig_11d-color}(d)
\caption{\label{fig:finish}
STM topographs of Ge hut cluster dense arrays at different coverages ($T_{\rm gr} = 360^\circ$C): (a),\,(b) $h_{\rm Ge}=14$\,\r{A} [(a) $U_{\rm s}=+1.75$\,V, $I_{\rm t}=80$\,pA, (b) $U_{\rm s}=+3.0$\,V, $I_{\rm t}=100$\,pA]; (c),\,(d) $h_{\rm Ge}=15$\,\r{A} [(c) $U_{\rm s}=+2.0$\,V, $I_{\rm t}=120$\,pA, (d)~$20.3\times 20.4$\,nm, $U_{\rm s}=+3.6$\,V, $I_{\rm t}=120$\,pA], $w$ indicates the wetting layer patches, $i$ shows a distorted small Ge island 3\,ML high over WL.}
\end{figure}
A qualitative model accounting for the presence of the particular point at the low-temperature array growth is simple. The case is that at low enough temperatures of the array growth, the new Ge cluster nucleation competes with the process of growth of earlier formed clusters. The height of the dominating wedge-like clusters is observed to be limited by some value depending on $T_{\rm gr}$.\footnote{Note that we did not observe a height limitation of pyramids. We suppose that they may give rise to one of the types of array defects---huge clusters---which sometimes appear among ordinary huts \cite{defects_ICDS-25}.} At small $h_{\rm Ge}$, Ge clusters are small enough and the distances between them are large enough compared to the Ge atom (or dimer) diffusion (migration) length on the surface for nucleation of new clusters on the Ge wetting layer in the space between the clusters (Figs.~\ref{fig:dots}, ~\ref{fig:arrays}(a),(b)). At $h_{\rm Ge} = 10$~\r{A} and the above ${\rm d} h_{\rm Ge}/{\rm d}t$ values, the equilibrium of parameters (cluster sizes and distances between them, diffusion length at given temperature, Ge deposition rate, etc.) sets in, the rate of new cluster nucleation is decreased and the abundant Ge atoms are mainly spent to the growth of the available clusters (Fig.~\ref{fig:arrays}(c),(d)). After the clusters reach their height limit and in spite of it, Ge atoms continue to form up their facets. As soon as most of the clusters reach the height limit, nucleation of new clusters becomes energetically advantageous again and the nucleation rate rises. The second phase of clusters appears on the wetting layer and fills whole its free surface as $h_{\rm Ge}$ is increased (Fig.~\ref{fig:finish}). Further increase of $h_{\rm Ge}$ results in two-dimensional growth mode. It is clear now why the array is the most homogeneous (optimal) at $T_{\rm gr} = 360^\circ$C and $h_{\rm Ge} = 10$~\r{A} whereas the dispersion of the cluster sizes is increased at higher and lower values of $h_{\rm Ge}$ because of the small clusters containing in the array. It is clear also that the optimal array will appear at different value of $h_{\rm Ge}$ when $T_{\rm gr}$ or ${\rm d} h_{\rm Ge}/{\rm d}t$ are different.
As it follows from the data presented in this section and Section~\ref{sec:formation} the Ge hut array evolution and life cycle goes through three main phases: at $T_{\rm gr}=360^{\circ}$C, the array nucleates at $h_{\rm Ge} \sim 5$\,\r{A} (Fig.~\ref{fig:nucleation}), it reaches ripeness and optimum to $h_{\rm Ge} \sim 10$\,\r{A} (Fig.~\ref{fig:arrays}) and finishes its evolution at $h_{\rm Ge} \sim 14$\,\r{A} by filling whole the surface (Fig.~\ref{fig:finish}). Most of clusters start coalescing (Fig.~\ref{fig:finish}(b)) and 2D growth begins at greater $h_{\rm Ge}$ (Fig.~\ref{fig:finish}(c)).
Nevertheless, free areas of WL still remain even at $h_{\rm Ge} = 15$\,\r{A} (Fig.~\ref{fig:finish}(d)). The structure of the parches (`$w$') stays the same as in the beginning of the array formation although the WL regions are surrounded by large huts. Small 3D islands (`$i$'), although very distorted, are still recognizable on WL between the large huts. The hut nucleation on WL goes on even at as high coverages as 15\,\r{A} when virtually total coalescence of the mature huts have already happened.
\section{Conclusion}
\label{sec:conclusion}
In summary, we have studied the array nucleation phase and identified the nuclei of both hut species, determined their atomic structure and observed the moment of appearance of the first generation of the nuclei on WL. We have investigated with high spatial resolution the peculiarities of each species of huts and their growth and derived their atomic structures. We have concluded that the wedge-like huts form due to a phase transition reconstructing the first atomic step of the growing cluster when dimer pairs of its second atomic layer stack up; the pyramids grow without phase transitions. In addition, we have come to conclusion that wedges contain vacancy-type defects on the penultimate terraces of their triangular facets which may decrease the energy of addition of new atoms to these facets and stimulate the quicker growth on them than on the trapezoidal ones and rapid elongation of wedges. We have shown also comparing the structures and growth of pyramids and wedges that shape transitions between them are impossible. And finally, we have explored the array evolution during MBE right up to the concluding phase of its life when most clusters coalesce and start forming a nanocrystalline 2D layer.
|
{
"timestamp": "2011-01-17T02:01:38",
"yymm": "1009",
"arxiv_id": "1009.3831",
"language": "en",
"url": "https://arxiv.org/abs/1009.3831"
}
|
\section{Introduction}
The transformation of spiral galaxies to lenticulars in the cluster
environment over the last $\sim$5 Gyr is now well established
\citep{but78, but84, dre97}, but the processes responsible for
bringing about this change are still the subject of vigorous
discussion. One set of mechanisms centres on the removal of gas from
spiral galaxies in the cluster environment, whether through
collisional sweeping \citep{spi51, val90}, ram-pressure stripping of
disk gas \citep{gun72, cow77, nul82, qui00}, or the removal of
large-scale gas reservoirs in `strangulation' or `starvation'
scenarios \citep{lar80, bal00, bow04}, with the latter postulated to
result in anaemic spirals \citep{bek02}. Many studies have looked at
the effect of tidal interactions either in the general cluster
environment \citep{nog86, lav88, hen96}, or in cluster sub-units
\citep{her95, bar96, bek99, gne03}. Some have concluded that tidal
effects lead to galaxy-wide star formation \citep{byr90}; this is one
of the possible consequences of repeated high-velocity galaxy-galaxy
encounters, sometimes termed `harassment' \citep{moo99, mih04}.
Low-velocity tidal encounters, on the other hand, tend to drive gas
into the central regions of galaxies leading to nuclear star formation
(SF) and the build-up of bulges \citep{ken87, mih92, ion04}. It is
also possible that the bulk of the evolutionary activity took place
through pre-processing in galaxy groups \citep{zab98} prior to the
assembly of these groups into the present-day clusters.
It should be noted that, given the range of processes likely to be
operating, it is not clear {\it a priori} whether the predominant
effect of the cluster environment will be to suppress (through gas
removal or exhaustion) or enhance (through, e.g., tidal triggering) total SF
rates in disk galaxies. Observational studies have adduced evidence
supporting both possibilities, with suppressed SF being found by both
\citet{bal98} and \citet{has98}, whereas others \citep{don90, mos93, gav94,
biv97, mos98, gav98} conclude that cluster spirals have SF activity similar
to or enhanced in comparison with the field population.
A comprehensive survey of the observational work undertaken in this
area is beyond the scope of this paper, but it is illustrative to
consider some of the approaches that have been adopted. One route is
to focus on the properties of galaxies in intermediate-redshift
clusters, as exemplified by \citet{mor07} who studied the SF activity
in clusters at redshifts of $\sim$0.5. They identify a population of
disk galaxies with young stellar populations but no ongoing SF, which
they take as evidence for gradual curtailment of SF through
`strangulation'or similar processes. However, they also identify a
density threshold for intra-cluster gas, above which a single passage
of even a large galaxy can lead to gas stripping and an abrupt
transformation to a gas-depleted lenticular with no star formation.
A second approach is to use the statistical properties of very large
numbers of low-redshift galaxies now made available by the
Sloan Digital Sky Survey (SDSS). \citet{par07} present a study of the
colours and morphologies of $>$300,000 SDSS galaxies within
$z\sim$0.1, which they correlate with the local number density
around each of the galaxies. They find the
fraction of galaxies with early morphological type to be a
monotonically increasing function of this number density, and
propose tidal processes as the dominant mechanism of galaxy
transformation.
A third method, and the one adopted in the present study, is to look
in great detail at the SF properties (rates and spatial
distributions) of spiral galaxies in the nearest clusters. An
excellent example of this approach is the study of the Virgo cluster
undertaken by \citet{koo04b, koo04a} using spatially resolved SF
mapping based on \Ha\ narrow-band imaging. They find that many of the
Virgo cluster spiral galaxies show outer truncation of their SF, in
comparison with a sample of field galaxies, and some show
centrally-enhanced SF. They conclude that a combination of
ram-pressure stripping and tidally-induced SF are required to
explain these observations.
The present paper is part of a study that is applying techniques
similar to those of \citet{koo04b, koo04a} to eight other nearby
galaxy clusters. The sample definition is presented in the first
paper of this series \citep{tho08} which also describes the
observations (broad- and narrow-band CCD photometry) and the data
reduction process. The present paper contains an analysis of the
total SF properties (rates and \Ha\ equivalent widths) of the cluster
galaxies, which are compared with a field galaxy
sample derived from the \Ha\ Galaxy Survey \citep{jam04}, henceforth
\Ha GS. Future papers will look at more detailed properties of SF
within these galaxies, e.g. concentration indices and radial
distributions.
\section{Data}
\subsection{The comparison samples}
Global parameters for all observed cluster galaxies are given in Table
2 of \citet{tho08}. However, for a robust comparison with the \Ha GS
field data, the cluster data are restricted to two well defined
subsamples. The first of these is a complete sample of all Sa--Sc
galaxies in six of the eight survey clusters (Abell 400, 426, 569,
779, 1367, 1656) as surveyed by the Objective Prism Survey
\citep[henceforth OPS;][]{mos00,mos05}. The second contains all
emission line galaxies (ELGs) detected by the OPS in all eight
clusters (i.e. the above six clusters plus Abell 262 and 347) which
excludes some of the galaxies with emission lines of lower equivalent
width \citep{tho08}.
A full discussion of the completeness of the ELG sample as a function
of \Ha\ flux, equivalent width (EW) and surface brightness is given in
\citet{tho08}. To summarise, all three factors affect the detectability
of galaxies by the OPS, with surface brightness being the most important.
The ELG sample becomes significantly incomplete below an EW of 2~nm, and
below an \Ha\ flux of 3.2$\times 10^{-17}$~W~m$^{-2}$. However, the cleanest
detectability threshold is given by \Ha\ surface brightness with a limit
of 4$\times$10$^{-20}$~W~m$^{-2}$~arcsec$^{-2}$.
An essential requirement for this project is consistent morphological
classifications across the different samples used. For many of the
cluster galaxies, no literature classifications were available, and
these were provided by one of the authors (MW) working from plate
material as explained in \citet{mos00}. These classifications were
done on the revised de Vaucouleurs system \citep{dev59,dev74}, and
intercomparisons were performed where possible, with galaxies with
classifications given in the UGC \citep{nil73} and RC3 \citep{dev91}
catalogues. These comparisons revealed no systematic offsets and a
scatter of about 1 $T$-type in classification, similar to the scatter
found from blind repeats of the same galaxies. For the field sample,
classifications were again on the de Vaucouleurs system, taken
directly from the RC3 or UGC.
\citet{mos06} suggests that the late type (Sa and later) cluster galaxy
population has an infalling component with higher velocity dispersion
than the earlier types, as well as an asymmetric velocity dispersion
relative to the cluster mean. On the other hand, the early type
(E-S0/a) galaxies are consistent with a virially relaxed population
with a Gaussian velocity distribution. \citet{mos06} therefore
determines revised cluster mean velocities and velocity dispersions
from only the early type objects, using biweight estimators of scale
and central location. Assuming that the distribution of galaxies
follows the mass distribution of the cluster and that the system is
spherically symmetric, and following \citet{lew02}, Moss estimates
virial radii for all clusters using:
\begin{equation}
r_{vir} \simeq 3.5\sigma(1+z)^{-1.5},
\end{equation}
where $r_{vir}$ is in units of Mpc for $\sigma$ in units of 1000~km~s$^{-1}$.
This gives values 40\% larger than the $R_{200}$ cluster radius often used
as a proxy for the virial radius \citep[e.g. by][]{fin05} for standard
cosmological parameters ($H_0=$70~km~s$^{-1}$~Mpc$^{-1}$,
$\Omega_{\Lambda}$=0.7, $\Omega_{0}$=0.3); both definitions give a direct
proportionality between virial radius and cluster velocity dispersion.
Table 1 of \citet{mos06} shows revised values of cluster mean
velocity, $\overline{v}$ and velocity dispersion, $\sigma$, along with
virial radius, $r_{vir}$, for each sample cluster. These were used to
combine the sample galaxies, from six or eight individual clusters,
into one ensemble cluster, where radial distances from the centre are
normalised by the virial radius, and velocities relative to the
cluster mean are normalised by the cluster's velocity dispersion.
Galaxies within the two subsamples are restricted to those with
velocities within 3$\sigma$ of the revised mean cluster velocity,
$\overline{v}$. The data are also split into cluster galaxies, which
lie within 1 $r_{vir}$ of the cluster centre, and supercluster field
galaxies lying beyond 1 $r_{vir}$. The Sa--Sc sample comprises 105
galaxies of which 5 have no detectable \Ha\ emission. The ELG sample
comprises 115 objects, all of which have been detected in emission in
the current CCD data. Known AGN were excluded by searching the
NASA/IPAC Extragalactic Database (NED) and removing all galaxies
classified as Seyfert (Sy, with any numerical subtype) or LINER. Two
galaxies lying close to bright stars have also been excluded.
Galaxies in the cluster and supercluster subsamples are also
classified, based only on their $R$ band images, as to whether they show
signs of tidal disturbance. Objects exhibiting strong tidal features
and/or obvious distortion are classified as $T$, tidally disturbed;
those with less obvious warps, probable tidal tails and/or some
disturbance are given a classification of \textit{T:}, probably
disturbed; an asymmetric appearance or slight distortion of outer
spiral arms leads to a classification of \textit{T::}, possibly
disturbed; and galaxies with no sign of tidal disturbance are assigned
no value in this category.
The field data are taken from \Ha GS, a study of the SF properties of
327 field galaxies of all spiral and irregular types (S0a - Im, barred
and unbarred), with apparent magnitudes brighter than m$_B=$15.5,
recession velocities less than 3000~km~s$^{-1}$ and diameters between
1.7 and 6.0 arcminutes. The full \Ha GS sample includes a large number
of low-luminosity galaxies, but the subsample studied here was
restricted to galaxies brighter than M$_{B} \sim-$18.5. All galaxies
classified as AGN in NED (26 in total in \Ha GS) were excluded from
the sample, as was done for the cluster sample, but it is worth noting
that a study of the mean \Ha\ emission profiles of the \Ha GS galaxies
\citep{jam09} found that central unresolved components are not common,
and typically contribute less than 10\% of the total emission-line
flux when they are present. Thus even if some low-level AGN are
included in the cluster or field samples (as will almost certainly be
the case), the effect on amounts and spatial distributions of emission
line flux should be small. The \Ha GS survey does include some Virgo
cluster and group galaxies, which have also been excluded from the
comparison sample. The final field comparison sample includes 65
galaxies, 50 of which are of types Sa--Scd, with 4 of type S0a and the
remaining 11 being late-type spirals, Sd-Sm. Absolute $B$-band
magnitudes for the field data were taken uncorrected from \Ha GS and
have been corrected for internal and external extinction. Internal
extinction corrections $A_B$ were calculated following the methods of
\citet{dev91}, including both inclination- and type-dependence as
follows:
$$A_B = \alpha(T)log(a/b)$$
where $(a/b)$ is the major-to-minor axis ratio, here evaluated at the
$\mu_R=$24 isophote, and
$$\alpha(T)=1.5-0.03(T-5)^2.$$
For galaxies with T$<$0, no internal extinction correction was applied,
and for peculiar galaxies and those without spiral subtypes, a mean
spiral correction of 1.3 mag was used. Although the resulting
overall shapes of the cluster and field M$_{B}$ distributions differ, they are
well matched in mean and range.
Comparing the morphological type distributions of the cluster and
field samples, a larger fraction of early type and fewer late type
spirals are seen in the cluster environment, as expected from the
morphology-density relation. The cluster ELG sample also contains
three type categories not covered by the \Ha GS survey. Three ELGs
have types E--S0 (S0$-$), 15 are classified as peculiar (Pec), which
lie outside the Hubble sequence, and 23 are spirals of uncertain type
(S...). The ELG sample also includes two galaxies with no type
information, which are excluded from much of the following analysis.
\Ha GS observations are restricted to galaxies with major-to-minor
axis ratios of less than or equal to 4.0, in order to exclude highly
inclined objects ($i \gtrsim 81^{\circ}$), where extinction effects are
likely to be strongest. However, a Kendall rank test on the complete
cluster Sa--Sc sample shows no significant dependence ($\tau = -0.09$,
probability = 0.16) of EW on inclination within this survey, and
therefore no axis ratio cut is applied to the cluster samples.
\subsection{Observational data}
The data used are derived from CCD images taken in broad-band $R$ and
narrow-band \Ha\ filters, using the 1.0 m Jacobus Kapteyn Telescope
(JKT) and the 2.6 m Nordic Optical Telescope (NOT), both situated on
the island of La Palma. The instrumentation used on both the JKT and
NOT and the resulting cluster galaxy photometry are described fully in
\citet{tho08}, and the equivalent for the comparison field sample in
\citet{jam04}, so the details will not be repeated here. It should be
noted that the line fluxes and equivalent widths used are for the
\Ha\ line and the neighbouring \NII\ lines. The narrow-band imaging
was continuum-subtracted using appropriately-scaled and aligned
$R$-band images, which generally gives good results but inevitably
gives substantial errors for galaxies with low equivalent width
emission.
\section{Distribution of equivalent widths}
\label{sec:ewd}
\begin{figure}
\includegraphics[width=87mm]{ewt_F.eps}
\caption{Global EW vs. morphological type for the full field
comparison sample. Solid lines show the 2$\sigma$ limits of the
field population. Typical errors in EW are 10--15\% for high EW ($>
2$ nm) and 25--35\% for low EW ($< 2$ nm) galaxies
(\citealt{sha02}).}
\label{fig:biw_F}
\end{figure}
\begin{figure}
\includegraphics[width=87mm]{ewt_Cin2.eps}
\includegraphics[width=87mm]{ewt_Cout2.eps}
\caption{Distribution of EW with Hubble Type for the cluster Sa--Sc sample
split into disturbed (open triangles) and undisturbed (filled
triangles) objects. Solid lines show the 2$\sigma$ field limits from
Figure \ref{fig:biw_F}. Typical errors in EW are $\sim$5--15\% for
high EW ($> 2$nm), 15--25\% for moderate EW (1--2 nm) and 25--100\%
for low EW ($<$ 1 nm) objects.}
\label{fig:biw_Sac}
\end{figure}
Figure \ref{fig:biw_F} shows the distribution of total EW values with
type for the comparison field sample, where the total EW for each
galaxy is taken as the EW within the $R=$ 24 mag/sq. arcsec isophotal
radius ($r_{24}$). A biweight estimator method is used to calculate
the mean EW and standard deviation for each type and the 2$\sigma$
limits of the field data are plotted as solid lines in the figure.
The majority of field points lie within these limits. It is worth
noting here that the field galaxy with the highest EW in the field
sample, and which lies well beyond the 2$\sigma$ limit, is the Sbc
galaxy UGC5786 (NGC 3310). This is a well-studied example of a local
UV-bright starburst with a complex peculiar morphology
(\citealt{kin93}; \citealt{con00}), including a ``bow and arrow''
structure in the outer regions (\citealt{wal67};
\citealt{bal81,ber84,mul95}) most likely caused by a recent
merger with a smaller galaxy (\citealt{bal81,mul96};
\citealt{con00}).
Figure \ref{fig:biw_Sac} shows similar plots for the cluster and
supercluster Sa--Sc samples, where again the solid lines show the 2$\sigma$
limits generated from the field sample, and galaxies are split into
disturbed (open points) and undisturbed (filled points) objects on
each plot. It can be seen from Figure \ref{fig:biw_Sac}a that a
substantial fraction of cluster galaxies lie beyond the 2$\sigma$
field limits ($\sim$36\% compared to only 5\% of field Sa-Sc
galaxies). Ten cluster Sa--Sc galaxies (13.5\% of the total) appear to
have EW reduced compared to the field sample. Three of these galaxies
have no detected \Ha\ emission, however, 9 of the low emission objects are
Sb spirals for which the range in field EW values is surprisingly
small.
More significant are the 17 galaxies (23\%), particularly of earlier
Sa--Sb types, with EW values beyond the upper field limits. This suggests
that the cluster environment is causing an enhancement of star
formation in some spirals. There is also some evidence that galaxies
with enhanced emission may preferentially be disturbed, with nearly
59\% of enhanced galaxies showing clear signs of tidal disturbance
compared to less than 23\% of galaxies with no enhancement in EW
($\chi^{2}$ probability $< 0.05$). The supercluster field sample on
the other hand is similar to the true field population, with only two
galaxies lying significantly above the 2$\sigma$ limits, and both of these
have a disturbed appearance.
\begin{figure}
\includegraphics[width=87mm]{ewt_elgcl.eps}
\includegraphics[width=87mm]{ewt_elgsc.eps}
\caption{As Figure \ref{fig:biw_Sac} but showing the distribution of
EW with Hubble Type for the ELG sample. Dashed lines show assumed
limits for ELG types not included in the field sample (see text for
details). Typical EW errors are the same as Figure
\ref{fig:biw_Sac}.}
\label{fig:biw_elg}
\end{figure}
Figure \ref{fig:biw_elg} shows the distribution of EW values with type for
the cluster ELG sample, again split into cluster and supercluster
field objects. Here the solid lines again show the 2$\sigma$ limits
for the field sample. For the peculiar galaxies and
spirals of uncertain type, the upper limit is taken as the Sbc value
(shown as a dashed line) as this is the highest 2$\sigma$ value in the
field sample and should therefore provide a conservative limit for
these categories, such that any peculiar or unknown spiral galaxies
with EW values above this point can also be considered enhanced. For the
three E--S0 (S0$^-$) galaxies in the ELG sample it is assumed that the
upper field limit is the same as that for the S0--S0/a (S0$^+$)
objects. Again this is likely to be higher than the true value as
E--S0 types generally have very little or no current star formation.
Once again, Figure \ref{fig:biw_elg}a shows an increase of EW for a
number of cluster galaxies, particularly of types Sa--Sb, and
this enhanced emission also extends to earlier types with seven of 11
S0--S0/a objects (64\%) having EW values enhanced relative to the field. It
can also be seen that a few late type Sc and later galaxies as well as
some peculiar and unclassifiable spiral galaxies appear to have enhanced
emission. In contrast, the supercluster galaxies in Figure
\ref{fig:biw_elg}b are almost indistinguishable from the true field
sample.
The ELG sample is biased towards objects with more luminous, higher
surface brightness \Ha\ emission, and so includes fewer galaxies with
low EW. The low emission objects seen in the Sa--Sc sample are,
therefore, not included in the ELG dataset. However, of the galaxies
included in the ELG sample, some 34 objects (38\%) have enhanced EW
values, of which 25 ($\sim$74\%) are classified as disturbed. The
non-enhanced ELG objects also have a relatively high proportion of
disturbed galaxies, with around 61\% showing signs of tidal
disturbance. Even excluding peculiar galaxies, which are, as expected,
all disturbed, this figure still stands at 53\%, much higher than the
23\% disturbed galaxies seen amongst the Sa--Sc non-enhanced
objects. This suggests that disturbed galaxies preferentially have
brighter \Ha\ emission, even for non-enhanced objects.
Assigning values 0--3 to the tidal disturbance categories from
undisturbed (no rank) to definitely disturbed ($T$) allows a Kendall
rank test to be carried out on the complete Sa--Sc sample. This shows
a substantial correlation ($\tau = 0.39$) of EW with tidal disturbance
which is significant at the $>5\sigma$ level. Similarly, a K-S test is
performed to compare the distribution of EW values in the disturbed
and undisturbed samples. This gives a probability of only
$1.1\times10^{-5}$ that the data are drawn from the same distribution,
showing that the disturbed and undisturbed objects have significantly
different EW distributions at $>4\sigma$ level. The cumulative
distributions with EW for the disturbed (red) and undisturbed (black)
Sa--Sc galaxies are shown in Figure \ref{fig:ew_cp}.
\begin{figure}
\vspace{-20mm}
\includegraphics[width=87mm]{ew_cp.eps}
\caption{Cumulative distributions with EW for disturbed (red) and
undisturbed (black) galaxies. A K-S test suggests the distributions are
significantly different.}
\label{fig:ew_cp}
\end{figure}
\citet{koo04a} also detect a number of spirals with star formation rates
enhanced by up to a factor of 3. They find that these are generally
lower luminosity galaxies (M$_{B} > -18$), for which they lack a good
field comparison sample. These authors therefore restrict their sample
to galaxies brighter than M$_{R_{24}} = -19.5$ (M$_{B} \sim -18.5$) to
avoid a possible luminosity bias. \citet{ken84} and \citet{bos01} also
find that lower luminosity galaxies tend to have higher EW values.
For the full Sa--Sc cluster and supercluster data, a Kendall rank test
shows a moderate ($\tau = 0.24$) but significant (3.7$\sigma$)
dependence of EW on $R$ band magnitude. This result is in agreement
with, for example, \citet{gav96}, who find a significant
anti-correlation between galaxy mass and specific star formation rate.
The current sample, however, has a limiting magnitude of M$_{B}\sim
-18.5$, equivalent to the cut made by \citet{koo04a}, and therefore
includes only relatively bright galaxies. The mean magnitude of the
cluster sample is also slightly brighter than that of the field
galaxies. The correlations found in the present study, and by
\citet{gav96}, between EW and galaxy luminosity would thus predict
overall {\em lower} EW values for the cluster sample, and cannot be
used to explain the enhanced emission of cluster Sa--Sc galaxies when
compared to the field.
\section{Mean EW values}
Figure \ref{fig:sacmean} shows the mean EW values with type for the Sa--Sc
cluster (stars), supercluster (crosses) and field (squares)
samples. Means are calculated using biweight estimators to reduce the
influence of outliers and the error bars shown give the standard error
on the mean, $\sigma/\sqrt{n}$. The cluster data are also split into
disturbed (open triangles) and undisturbed (closed triangles)
galaxies. This split is not done for the supercluster sample due to the small
numbers of objects involved (only 31 galaxies in total), however, the
mean EW values for the full supercluster sample are generally consistent
with the field data.
\begin{figure}
\includegraphics[width=87mm]{Sacmean.eps}
\caption{Mean EW values with type for the Sa--Sc cluster sample (stars), the
field (squares), and the supercluster field (crosses). The cluster
sample is also split into disturbed (open triangles) and undisturbed
(closed triangles) galaxies.}
\label{fig:sacmean}
\end{figure}
It can be seen from Figure \ref{fig:sacmean} that the mean EW values for the
full cluster data seem to be reduced in general compared to the field
sample, with the exception of Sa galaxies, but this is only
significant for Sab and Sb types. The undisturbed objects follow the
same trend, and with the exception of Sbc types, where only a single
object is undisturbed, the undisturbed mean EW values are very similar to
those for the full sample. The disturbed objects, on the other hand,
show a very different trend, with mean EW values for Sa--Sab galaxies
greatly enhanced above both total cluster and field values. For later
types, however, the mean disturbed EW values are close to, and consistent
with, both the full and undisturbed cluster samples. Although the
highest EW enhancements are seen for the disturbed early type spirals,
the fraction of Sa--Sab galaxies that are tidally disturbed is only
21\%, compared to 48\% and 45\% for Sb--Sbc and Sc galaxies
respectively.
\begin{figure}
\includegraphics[width=87mm]{elgcmean_new.eps}
\caption{Mean EW values with type for the cluster ELG sample (stars)
split into
disturbed (open triangles) and undisturbed (filled triangles)
galaxies. The field data are shown as filled squares. Points for the
supercluster subset are not shown, because of the small number
of galaxies in this category; see Fig.~\ref{fig:biw_elg}. }
\label{fig:elgcmean}
\end{figure}
A similar plot is shown in Figure \ref{fig:elgcmean} for the cluster
ELG sample. Here all E--S0/a cluster galaxies are grouped into a
single bin and compared to the field mean S0/a EW. In order to
increase the numbers of objects in the cluster later type bins, all
Sc--Irr galaxies have also been grouped together. Spirals of
unclassifiable type are not included in this figure. The undisturbed
cluster ELGs have EW values comparable to the field data, however, the
ELG sample is biased towards galaxies with brighter \Ha\ emission, so
the true mean EW values may be somewhat lower. Once again, the
disturbed galaxy sample shows significantly enhanced mean EW values
for Sa--Sab spirals, but this also extends further to even earlier
types (E--S0/a). The sample of peculiar galaxies, all of which are
disturbed, has a mean EW greater than any field type. The increases of
mean EW in specific Hubble types in Figs. \ref{fig:sacmean} and
\ref{fig:elgcmean} illustrate that the difference in the overall EW
distributions of the disturbed and undisturbed galaxies, shown in
Fig. \ref{fig:ew_cp}, cannot be attributed simply to differences in
the morphological makeup of the disturbed and undisturbed samples.
Disturbance seems to affect EW in galaxies of a given type, at least
for early types.
It is of interest to consider possible reasons for this enhancement in EW
being apparent only for early types. Two explanations can be proposed.
The first is that enhanced emission tends to take place in the densest
cluster regions, where we expect galaxies to be stripped and generally have
lower disk emission, thus appearing as early-type galaxies. The second is
that the scatter in EW values for late-type unstripped galaxies is much
larger than for early types, and may therefore tend to mask any effect.
\section{Clustercentric radial distribution of \Ha\ EW}
\label{sec:ccrd}
\citet{koo04a} conclude that truncation of the star forming disk, via
ICM--ISM stripping, is the dominant process affecting galaxies in
clusters. If this is correct then the mean EW of cluster spiral
galaxies should decrease towards the cluster centre.
\begin{figure}
\includegraphics[width=87mm]{vradew_all.eps}
\caption{\Ha\ EW as a function of clustercentric distance for the
Sa--Sc sample. Large points show mean distances and EW values for bins of
equal numbers of galaxies.}
\label{fig:vrad}
\end{figure}
Studies using the 2dF Galaxy Redshift Survey (\citealt{lew02}) and
SDSS (\citealt{gom03}) have suggested that there is a transition in
star formation activity at a characteristic density corresponding to
the local density at $\sim$1 virial radius, although it is difficult
to disentangle this from the known morphology--density relation
(\citealt{bos06}). \citet{gav06} also find evidence that the average
\Ha\ EW of luminous spirals in the Virgo and Coma + Abell 1367 cluster
samples decreases in the inner $\sim$1 virial radius, although the
binning of the data results in very few points within 1~$r_{vir}$ (two
points for Virgo, one for Coma + A1367) such that it is not possible
to trace any gradual variation within the cluster
itself. \citet{yua05}, however, studied the star formation properties
of 184 bright cluster galaxies in the $z\sim0.08$ cluster A2255 and
found that, although there is a slight trend for the specific star
formation rates of early-type galaxies to decrease towards the cluster
centre, the inner late-type galaxies, in fact, tend to have higher
star formation rates.
Figure \ref{fig:vrad} shows the EW values of the Sa--Sc sample plotted
against distance in virial radii from the composite cluster
centre. Individual galaxies are plotted as black stars, and the red
triangles show the mean distance and equivalent width in 5
approximately equal bins, calculated using a biweight estimator. The
error bars show the standard error in EW for each bin. The mean EW values in
the cluster and supercluster Sa--Sc sample are lower than the overall
field mean for the Sa--Sc sample, however, this is likely to be due to
the different morphological mix in the cluster and field
samples. Figure \ref{fig:vrad}, however, shows no change in the star
formation rate of late type galaxies with clustercentric radius. This
suggests that stripping alone cannot dominate the transformation of
spiral galaxies in these clusters.
It is tempting to suggest that the lack of an observed trend between
decreasing star formation and proximity to the cluster centre may be
due to field interlopers projected towards the inner parts of the
cluster. \citet{mos00}, however, find that contamination by field
galaxies accounts for only $\sim$20\% of spirals within 0.5 Abell
radii (which corresponds to roughly 0.5 $r_{vir}$ for the composite
cluster), but will be more important outside this radius. Field
contamination would therefore likely increase any such trend and
cannot account for the flat distribution observed.
\begin{figure}
\includegraphics[width=87mm]{vradew_dst.eps}
\caption{As Figure \ref{fig:vrad}, but split into disturbed (open
points) and undisturbed (closed points) galaxies.}
\label{fig:vradd}
\end{figure}
\begin{figure}
\includegraphics[width=87mm]{vradew_elg.eps}
\caption{As Figure \ref{fig:vrad}, but for the ELG sample.}
\label{fig:vradelg}
\end{figure}
Given the observed differences in star formation activity for
disturbed and undisturbed galaxies, it is also of interest to study
the clustercentric radial distributions of EW values for each sample
separately. This is shown in Figure \ref{fig:vradd}. Here individual
undisturbed galaxies are shown as before with black stars, whilst the
disturbed population have open points. As with previous plots, the
mean values for the undisturbed galaxies are plotted as filled
triangles whilst open triangles represent the means for the binned
disturbed objects. Figure \ref{fig:vradd} shows that, as with the
complete sample, no change is seen in mean EW with distance from the
cluster centre for the undisturbed sample. As expected, the means for
the disturbed galaxies are higher than those for the undisturbed
objects, but, although the mean star formation appears to be slightly
higher at intermediate clustercentric distances, the points and error
bars are still consistent with a flat distribution. The same lack
of a significant trend is found for the ELG sample, shown in Fig.
\ref{fig:vradelg}.
This result is initially surprising, given the number of studies
finding negative correlations between emission-line strength and the
number density of the local environment. This dates back at least as
far as \citet{ost60}, who found emission lines to be less prevalent in
elliptical galaxies in dense clusters than in Virgo cluster
ellipticals. The same trend was found for larger samples of cluster
galaxies, including spiral galaxies, by \citet{gis78} and
\citet{dre85}. Most recently, \citet{vul10} conclude that the average
SF rate in 604 galaxies within 16 intermediate-redshift clusters vary
systematically with environment, even at fixed galaxy mass.
However, other studies have found different results, more in line
with those found here. These include \citet{biv97}, who found that
their overall conclusions regarding the correlation between
emission-line strength and environment depended critically on a
systematic bias resulting from the different effects of
magnitude-limited selection on field and cluster samples. Once this
had been corrected for, they found no difference between the emission
line properties of field and cluster galaxies of a given morphological
type. This result was confirmed by \citet{mos05}, who further discuss
the selection effect analysis of \citet{biv97} and find results
similar to those of the present paper for 379 galaxies in low-redshift
galaxies, with Objective Prism measures of \Ha\ emission. Finally,
\citet{car01} and \citet{rin05} both conclude that the distributions
of total \Ha\ EW values for galaxies with significant SF show no
difference between samples selected within or outside the virial
radius of their host clusters.
\section{Distribution of disturbed cluster galaxies}
A comparison of the cumulative distributions of the clustercentric distances of
disturbed vs. undisturbed Sa--Sc galaxies suggests that the radial
distributions of these samples are similar. A K-S test gives a
probability of 0.89 that they are drawn from the same parent
distribution.
Figure \ref{fig:vhd}, however, shows a rather different conclusion
based on the distribution of galaxy velocities within their
clusters. The normalised velocity dispersion is shown for all Sa--Sc
sample galaxies within 1~$r_{vir}$ of the composite cluster centre
(open histogram). The sample is also split into disturbed and
undisturbed galaxies, and their distributions are plotted separately
in the filled red histograms in the centre and top panels
respectively. The undisturbed galaxies appear to be fairly centrally
peaked, while the disturbed galaxies show a much flatter
distribution. The bottom panel in Figure \ref{fig:vhd} shows the
normalised velocity distribution for the early type elliptical and
lenticular galaxies in the six Sa--Sc sample clusters. These objects
were used to calculate the cluster means and dispersions that were
then employed to normalise the later type population, and they
therefore have a mean normalised velocity of 0.0, with a standard
deviation of 1.0.
\begin{figure}
\includegraphics[width=128mm]{vhist_early.eps}
\caption{Normalised velocity distribution for Sa--Sc sample galaxies
(top and centre), scaled to the cluster mean. Open histograms show the
full cluster Sa--Sc sample, with filled histograms showing the disturbed
(centre) and undisturbed (top) populations. The bottom plot shows the
normalised velocity distribution for the early type population in the six
clusters included in the Sa--Sc sample.}
\label{fig:vhd}
\end{figure}
The mean and dispersion of the velocity distribution have been
calculated for each Sa--Sc sample using biweight estimators. The
undisturbed sample has a mean of $0.11 \pm 0.14$ with a dispersion of
1.02, and is therefore consistent with the early type population used
to calculate the cluster velocity distributions (K-S probability =
0.77, data folded about $(v-\langle v \rangle )/\sigma_{v} = 0$). For
the disturbed galaxies, however, although the mean value of $0.09 \pm
0.30$ is still consistent with the composite cluster mean, the
dispersion has a much higher value of 1.45. This is, in fact, likely
to be a lower limit to the dispersion of disturbed galaxies as the
$3\sigma$ velocity cut applied in the selection of the sample means
that galaxies with higher velocity deviations would have been
omitted. Even so, a K-S test shows that the velocity distributions of
the disturbed Sa--Sc sample and the early type population are
different at $\gtrsim2\sigma$ significance. The distribution of
undisturbed galaxies drops off before the $3\sigma$ limit. Comparing
the velocities of the disturbed and undisturbed Sa--Sc galaxies, a K-S
test gives a probability of 0.06 that these are drawn from the same
distribution. The marginally higher velocity dispersion observed for
the disturbed galaxies ($\sim\sqrt{2}$ greater than the undisturbed
sample) is suggestive of an infalling population, and a similar result
was found for the larger sample of cluster galaxies studied by
\citet{mos06}. It should also be noted that the disturbed galaxy
Sa--Sc sample studied here has a somewhat later mean type ($T=$2.96,
cf. 2.12 for the undisturbed galaxies) which may have some bearing on
the interpretation of this result.
\section{Summary}
\label{sec:summary}
Comparison of the global \Ha\ EW values of cluster, supercluster and
field galaxies has identified a population of cluster galaxies
(particularly early type spirals and lenticulars) with enhanced star
formation compared to their field counterparts. These objects are also
more likely to have a disturbed appearance than non-enhanced galaxies.
Tidal disturbance is found to be correlated with higher \Ha\ EW at
$>5\sigma$ (Kendall rank test). A K-S test also shows that the
distributions of EW values in disturbed and undisturbed populations
are significantly different ($>4\sigma$). This disturbance is seen in
the stellar component and hence is indicative of tidal effects, rather
than, e.g., ram-pressure stripping. A number of galaxies with
unusually weak line emission are also seen in the cluster Sa--Sc
sample. The supercluster samples, on the other hand, appear very
similar to the field.
Comparing the mean EW values for each type suggests that star
formation may be reduced in general for undisturbed cluster galaxies
across most types. The disturbed galaxies, however, have mean EW
values well above those for field early type spirals, but for later
types they are consistent with the field and undisturbed cluster
samples. The sample of highly disturbed, peculiar galaxies included in
the ELG sample has a mean EW higher than any other field or cluster
type. These results suggest that galaxy--galaxy interactions and
mergers may play a significant role in the evolution of cluster
spirals.
A study of the clustercentric radial distribution of \Ha\ EW also
shows no correlation between EW and distance from the cluster
centre. This suggests that stripping alone, which would lead to a
gradual decrease of star formation towards the cluster centre, cannot
be solely responsible for the transformation of spiral galaxies in
these clusters, and adds further weight to the argument that tidal
interactions between galaxies may also be important.
An investigation of the distribution of disturbed and undisturbed
galaxies within the cluster shows that, although there appears to be
no difference in distribution as a function of clustercentric
distance, the disturbed galaxies have a marginally higher velocity
dispersion that may indicate an infalling population.
Future papers in this series will look in more detail at the distribution
of SF activity within the galaxies studied in the present paper. This
analysis will initially use concentration indices as a measure of the
compactness of SF, to probe the prevalence of outer truncation and
centrally-concentrated starbursts. Radial light profiles will then be
used in a more detailed study of these processes.
\begin{acknowledgements}
This research
has made use of the NASA/IPAC Extragalactic Database (NED) which is
operated by the Jet Propulsion Laboratory, California Institute of
Technology, under contract with the National Aeronautics and Space
Administration. The referee is thanked for many helpful suggestions.
CFB, PAJ and MW dedicate this paper to the memory of our respected
and sadly-missed colleague, Chris Moss.
\end{acknowledgements}
\bibliographystyle{bibtex/aa}
|
{
"timestamp": "2010-09-22T02:02:19",
"yymm": "1009",
"arxiv_id": "1009.4104",
"language": "en",
"url": "https://arxiv.org/abs/1009.4104"
}
|
\section{Introduction}
The main goal of this note is to introduce a new and more general
fractional sum operator that unify and extend the discrete
fractional operators used in fractional calculus.
Looking to the literature of discrete fractional difference
operators, two approaches are found (see, \textrm{e.g.}, \cite{4,comNuno:Rui:Z}):
one using the $\Delta$ point of view (sometimes called the forward fractional difference approach),
another using the $\nabla$ perspective (sometimes called the backward fractional
difference approach). Here we introduce a new operator, making use of the symbol
${_{\gamma}}\diamondsuit$ (\textrm{cf.} Definition~\ref{diamond}).
When $\gamma = 1$ the ${_{\gamma}}\diamondsuit$ operator is reduced to the $\Delta$ one;
when $\gamma = 0$ the ${_{\gamma}}\diamondsuit$ operator
coincides with the corresponding $\nabla$ fractional sum.
The work is organized as follows. In Section~\ref{sec:2}
we review the basic definitions of the discrete fractional calculus.
Our results are then given in Section~\ref{sec:3}:
we introduce the fractional diamond sum
(Definition~\ref{diamond}) and prove its main properties.
We end with Section~\ref{sec:4} of conclusions and future perspectives.
\section{Preliminaries}
\label{sec:2}
Here we only give a very short introduction to the basic definitions
in discrete fractional calculus. For more on the subject we refer
the reader to \cite{2,3,7}.
We begin by introducing some notation used throughout.
Let $a$ be an arbitrary real number and $b = a + k$
for a certain $k \in\mathbb{N}$ with $k \ge 2$.
Let $\mathbb{T}= \{a, a + 1, \ldots, b\}$.
According with \cite{8}, we define the
factorial function
$$
t^{(n)} = t(t-1)(t-2)\ldots(t-n+1),
\quad n\in\mathbb{N}.
$$
Also in agreement with the same authors \cite{9},
we define
$$
t^{\overline{n}}=t(t+1)(t+2)\ldots(t+n-1),
\quad n\in\mathbb{N},
$$
and $t^{\overline{0}}=1$.
Extending the two above definitions from an integer $n$ to an arbitrary
real number $\alpha$, we have
\begin{equation*}
t^{(\alpha)}=\frac{\Gamma(t+1)}{\Gamma(t+1-\alpha)}
\text{ \ and \ }
t^{\overline{\alpha}}=\frac{\Gamma(t+\alpha)}{\Gamma(t)},
\end{equation*}
where $\Gamma$ is the Euler gamma function.
Throughout the text we shall use the standard notations
$\sigma(s)=s+1$ and $\rho(s)=s-1$
of the time scale calculus in $\mathbb{Z}$ \cite{8,9}.
\begin{definition}[\cite{12}]
\label{delta}
The discrete delta fractional sum operator is defined by
\begin{equation*}
(\Delta_a^{-\alpha}f)(t)
=\frac{1}{\Gamma(\alpha)}\sum_{s=a}^{t-\alpha}(t-\sigma(s))^{(\alpha-1)}f(s),
\end{equation*}
where $\alpha>0$. Here $f$ is defined for $s=a \mod (1)$ and
$\Delta_a^{-\alpha}f$ is defined for $t=(a+\alpha) \mod (1)$.
\end{definition}
\begin{remark}
Given a real number $a$ and $b = a + k$, $k \in \mathbb{N}$,
$\sum_{s=a}^{b} g(s) = g(a) + g(a+1) + \cdots + g(b)$.
\end{remark}
\begin{remark}
Let $\mathbb{N}_t=\{t,t+1,t+2,\ldots\}$.
We note that $\Delta_a^{-\alpha}$ maps functions defined on
$\mathbb{N}_a$ to functions defined on $\mathbb{N}_{a+\alpha}$.
\end{remark}
Analogously to Definition~\ref{delta},
one considers the discrete nabla fractional sum operator:
\begin{definition}[\cite{4}]
\label{nabla}
The discrete nabla fractional sum operator is defined by
\begin{equation*}
(\nabla_a^{-\beta}f)(t)
=\frac{1}{\Gamma(\beta)}\sum_{s=a}^{t}(t-\rho(s))^{\overline{\beta-1}}f(s),
\end{equation*}
where $\beta>0$. Here $f$ is defined for $s=a \mod (1)$ and
$\nabla_a^{-\beta}f$ is defined for $t=a \mod (1)$.
\end{definition}
\begin{remark}
Let $\mathbb{N}_a=\{a,a+1,a+2,\ldots\}$.
The operator $\nabla_a^{-\beta}$ maps functions defined
on $\mathbb{N}_a$ to functions defined on $\mathbb{N}_{a}$.
The fact that $f$ and $\nabla_a^{-\beta} f$ have the same domain,
while $f$ and $\Delta_a^{-\alpha} f$ do not, explains why some
authors prefer the nabla approach.
\end{remark}
The next result gives a relation between the delta fractional sum
and the nabla fractional sum operators.
\begin{lemma}[\cite{4}]
\label{nabladelta}
Let $0\leq m-1<\nu\leq m$, where $m$ denotes an
integer. Let $a$ be a positive integer, and $y(t)$ be defined on
$t \in \mathbb{N}_a=\{a,a+1,a+2,\ldots\}$. The following statement holds:
$\left(\Delta_a^{-\nu} y\right)(t+\nu)=\left(\nabla_a^{-\nu} y\right)(t)$,
$t\in \mathbb{N}_{a}$.
\end{lemma}
\section{Main Results}
\label{sec:3}
We introduce a general discrete diamond-gamma fractional sum operator
by using a convex combination of the delta
and nabla fractional sum operators.
\begin{definition}
\label{diamond}
The diamond-$\gamma$ fractional operator
of order $(\alpha,\beta)$ is given,
when applied to a function $f$ at point $t$, by
\begin{equation*}
\left(_{\gamma}\diamondsuit_a^{-\alpha,-\beta}f\right)(t)
=\gamma \left(\Delta_a^{-\alpha}f\right)(t+\alpha)
+(1-\gamma) \left(\nabla_a^{-\beta}f\right)(t),
\end{equation*}
where $\alpha>0$, $\beta>0$, and $\gamma\in[0,1]$.
Here, both $f$ and $_{\gamma}\diamondsuit_a^{-\alpha,-\beta}f$
are defined for $t=a \mod (1)$.
\end{definition}
\begin{remark}
Similarly to the nabla fractional operator,
our operator $_{\gamma}\diamondsuit_a^{-\alpha,-\beta}$
maps functions defined on $\mathbb{N}_a$ to functions defined on $\mathbb{N}_a$,
$\mathbb{N}_a=\{a,a+1,a+2,\ldots\}$ for $a$ a given real number.
\end{remark}
\begin{remark}
The new diamond fractional operator of Definition~\ref{diamond} gives,
as particular cases, the operator of Definition~\ref{delta}
for $\gamma=1$,
$$
\left(_{1}\diamondsuit_a^{-\alpha,-\beta}f\right)(t)
=\left(\Delta_a^{-\alpha}f\right)(t+\alpha),
\quad t\equiv a \mod (1),
$$
and the operator of Definition~\ref{nabla} for $\gamma=0$,
$$
\left(_{0}\diamondsuit_a^{-\alpha,-\beta}f\right)(t)
=\left(\nabla_a^{-\beta}f\right)(t),
\quad t\equiv a \mod (1).
$$
\end{remark}
The next theorems give important properties of the
new, more general, discrete fractional operator
$_{\gamma}\diamondsuit_a^{-\alpha,-\beta}$.
\begin{theorem}
\label{thm:SUM}
Let $f$ and $g$ be real functions defined on $\mathbb{N}_a$,
$\mathbb{N}_a=\{a,a+1,a+2,\ldots\}$ for $a$ a given real number.
The following equality holds:
\begin{equation*}
\left(_{\gamma}\diamondsuit_a^{-\alpha,-\beta} (f+g)\right)(t)
=\left({_{\gamma}\diamondsuit}_a^{-\alpha,-\beta}f\right)(t)
+\left({_{\gamma}\diamondsuit}_a^{-\alpha,-\beta}g\right)(t).
\end{equation*}
\end{theorem}
\begin{proof}
The intended equality follows from the definition
of diamond-$\gamma$ fractional sum of order $(\alpha,\beta)$:
\begin{equation*}
\begin{split}
(_{\gamma}\diamondsuit_a^{-\alpha,-\beta}(f+g))(t)
&=\gamma(\Delta_a^{-\alpha}(f+g))(t+\alpha)
+(1-\gamma)(\nabla_a^{-\beta}(f+g))(t)\\
&=\frac{\gamma}{\Gamma(\alpha)}\sum_{s=a}^{t}(t+\alpha-\sigma(s))^{(\alpha-1)}(f(s)
+g(s))+\frac{1-\gamma}{\Gamma(\beta)}\sum_{s=a}^{t}(t-\rho(s))^{\overline{\beta-1}}(f(s)+g(s))\\
&=\frac{\gamma}{\Gamma(\alpha)}\sum_{s=a}^{t}(t+\alpha-\sigma(s))^{(\alpha-1)}f(s)
+\frac{\gamma}{\Gamma(\alpha)}\sum_{s=a}^{t}(t+\alpha-\sigma(s))^{(\alpha-1)}g(s)\\
&\qquad +\frac{1-\gamma}{\Gamma(\beta)}\sum_{s=a}^{t}(t-\rho(s))^{\overline{\beta-1}}f(s)
+\frac{1-\gamma}{\Gamma(\beta)}\sum_{s=a}^{t}(t-\rho(s))^{\overline{\beta-1}}g(s)\\
&= \left[\frac{\gamma}{\Gamma(\alpha)}\sum_{s=a}^{t}(t+\alpha-\sigma(s))^{(\alpha-1)}f(s)
+\frac{1-\gamma}{\Gamma(\beta)}\sum_{s=a}^{t}(t-\rho(s))^{\overline{\beta-1}}f(s)\right]\\
&\qquad + \left[\frac{\gamma}{\Gamma(\alpha)}\sum_{s=a}^{t}(t+\alpha-\sigma(s))^{(\alpha-1)}g(s)
+\frac{1-\gamma}{\Gamma(\beta)}\sum_{s=a}^{t}(t-\rho(s))^{\overline{\beta-1}}g(s)\right]\\
&=(_{\gamma}\diamondsuit_a^{-\alpha,-\beta}f)(t) + (_{\gamma}\diamondsuit_a^{-\alpha,-\beta}g)(t).
\end{split}
\end{equation*}
\end{proof}
\begin{theorem}
\label{thm:const}
Let $f(t)=k$ on $\mathbb{N}_a$, $k$ a constant.
The following equality holds:
$$
(_{\gamma}\diamondsuit_a^{-\alpha,-\beta}f)(t)
=\gamma\frac{\Gamma(t-a+1+\alpha) k}{\Gamma(\alpha+1)\Gamma(t-a+1)}
+(1-\gamma)\frac{\Gamma(t-a+1+\beta) k}{\Gamma(\beta+1)\Gamma(t-a+1)}\, .
$$
\end{theorem}
\begin{proof}
By definition of diamond-$\gamma$ fractional
sum of order $(\alpha,\beta)$, we have
\[\begin{split}
(_{\gamma}&\diamondsuit_a^{-\alpha,-\beta}k)(t)
=\gamma(\Delta_a^{-\alpha}k)(t+\alpha)+(1-\gamma)(\nabla_a^{-\beta}k)(t)
=\frac{\gamma}{\Gamma(\alpha)}\sum_{s=0}^t
k(t+\alpha-\sigma(s))^{(\alpha-1)}
+\frac{1-\gamma}{\Gamma(\beta)}\sum_{s=0}^t
k(t-\rho(s))^{\overline{\beta-1}}\\
&=\gamma\frac{\Gamma(t-a+1+\alpha)}{\alpha\Gamma(\alpha)\Gamma(t-a+1)}k+
(1-\gamma)\frac{\Gamma(t-a+1+\beta)}{\beta\Gamma(\beta)\Gamma(t-a+1)}k
=\gamma\frac{\Gamma(t-a+1+\alpha)}{\Gamma(\alpha+1)\Gamma(t-a+1)}k+
(1-\gamma)\frac{\Gamma(t-a+1+\beta)}{\Gamma(\beta+1)\Gamma(t-a+1)}k.
\end{split}
\]
\end{proof}
\begin{corollary}
\label{m:f:cor}
Let $f(t) \equiv k$
for a certain constant $k$. Then,
\begin{equation}
\label{Miller_Constant}
(\Delta_a^{-\alpha}f)(t+\alpha)
=\frac{\Gamma(t-a+1+\alpha)}{\Gamma(\alpha+1)\Gamma(t-a+1)}k.
\end{equation}
\end{corollary}
\begin{proof}
The result follows from Theorem~\ref{thm:const}
choosing $\gamma=1$ and recalling that
$(_{1}\diamondsuit_a^{-\alpha,-\beta}k)(t)
= (\Delta_a^{-\alpha}k)(t+\alpha)$.
\end{proof}
\begin{remark}
In the particular case when $a=0$, equality
\eqref{Miller_Constant} coincides with the result
of \cite[Sect.~5]{12}.
\end{remark}
The fractional nabla result analogous to Corollary~\ref{m:f:cor}
is easily obtained:
\begin{corollary}
If $k$ is a constant, then
\begin{equation*}
(\nabla_a^{-\beta}k)(t)
=\frac{\Gamma(t-a+1+\beta)}{\Gamma(\beta+1)\Gamma(t-a+1)}k.
\end{equation*}
\end{corollary}
\begin{proof}
The result follows from Theorem~\ref{thm:const}
choosing $\gamma=0$ and recalling that
$(_{0}\diamondsuit_a^{-\alpha,-\beta}k)(t)
=(\nabla_a^{-\beta}k)(t)$.
\end{proof}
\begin{theorem}
\label{thm:10}
Let $f$ be a real valued function and
$\alpha_1$, $\alpha_2$, $\beta_1$, $\beta_2>0$. Then,
\begin{equation*}
\left(_{\gamma}\diamondsuit_a^{-\alpha_1,
-\beta_1}\left(_{\gamma}\diamondsuit_a^{-\alpha_2,-\beta_2}f\right)\right)(t)
=\gamma\left({_{\gamma}}\diamondsuit_a^{-(\alpha_1+\alpha_2),
-(\beta_1+\alpha_2)} f\right)(t)+(1-\gamma)\left(
{_{\gamma}}\diamondsuit_a^{-(\alpha_1+\beta_2),-(\beta_1+\beta_2)} f\right)(t).
\end{equation*}
\end{theorem}
\begin{proof}
Direct calculations show the intended relation:
\begin{equation*}
\begin{split}
(_{\gamma}&\diamondsuit_a^{-\alpha_1,-\beta_1}(_{\gamma}\diamondsuit_a^{-\alpha_2,-\beta_2}f))(t)
=\gamma(\Delta_a^{-\alpha_1}(_{\gamma}\diamondsuit_a^{-\alpha_2,-\beta_2}f))(t+\alpha_1)
+ (1-\gamma)(\nabla_a^{-\beta_1}(_{\gamma}\diamondsuit_a^{-\alpha_2,-\beta_2}f))(t)\\
&=\gamma^2 (\Delta_a^{-\alpha_1}\left(\Delta_a^{-\alpha_2}f\right))(t+\alpha_1+\alpha_2)
+ \gamma(1-\gamma)\left(\Delta_a^{-\alpha_1}\left(\nabla_a^{-\beta_2}f\right)\right)(t+\alpha_1)
+ (1-\gamma)\gamma\left(\nabla_a^{-\beta_1}\left(\Delta_a^{-\alpha_2}f\right)\right)(t+\alpha_2)\\
&\quad + (1-\gamma)^2\left(\nabla_a^{-\beta_1}\left(\nabla_a^{-\beta_2}f\right)\right)(t)\\
&=\gamma^2\left(\Delta_a^{-(\alpha_1+\alpha_2)}f\right)(t+\alpha_1+\alpha_2)
+ \gamma(1-\gamma)\left(\Delta_a^{-\alpha_1}\left(\Delta_a^{-\beta_2}f\right)\right)(t+\alpha_1+\beta_2)\\
&\quad + (1-\gamma)\gamma\left(\nabla_a^{-\beta_1}\left(\nabla_a^{-\alpha_2}f\right)\right)(t)
+(1-\gamma)^2 \left(\nabla_a^{-(\beta_1+\beta_2)}f\right)(t)\\
&=\gamma^2\left(\Delta_a^{-(\alpha_1+\alpha_2)}f\right)(t+\alpha_1+\alpha_2)
+ \gamma(1-\gamma)\left(\Delta_a^{-(\alpha_1+\beta_2)}f\right)(t+\alpha_1+\beta_2)\\
&\quad + (1-\gamma)\gamma\left(\nabla_a^{-(\beta_1+\alpha_2)}f\right)(t)
+(1-\gamma)^2\left(\nabla_a^{-(\beta_1+\beta_2)}f\right)(t)\\
&=\gamma\left[\gamma\left(\Delta_a^{-(\alpha_1+\alpha_2)}f\right)(t+\alpha_1+\alpha_2)
+ (1-\gamma)\left(\nabla_a^{-(\beta_1+\alpha_2)}f\right)(t)\right]\\
&\quad + (1-\gamma)\left[\gamma\left(\Delta_a^{-(\alpha_1+\beta_2)}f\right)(t+\alpha_1+\beta_2)
+ (1-\gamma)\left(\nabla_a^{-(\beta_1+\beta_2)}f\right)(t)\right].
\end{split}
\end{equation*}
\end{proof}
\begin{remark}
If $\gamma=0$, then
$\left(_{\gamma}\diamondsuit_a^{-\alpha_1,-\beta_1}(_{\gamma}\diamondsuit_a^{-\alpha_2,-\beta_2}f)\right)(t)
=\left(\nabla_a^{-(\beta_1+\beta_2)}f\right)(t)$.
\end{remark}
\begin{remark}
If $\gamma=1$, then
$\left(_{\gamma}\diamondsuit_a^{-\alpha_1,-\beta_1}(_{\gamma}\diamondsuit_a^{-\alpha_2,-\beta_2}f)\right)(t)
=\left(\Delta_a^{-(\alpha_1+\alpha_2)}f\right)(t+\alpha_1+\alpha_2)$.
\end{remark}
\begin{remark}
If $\alpha_1=\alpha_2=\alpha$ and $\beta_1=\beta_2=\beta$, then
$\left(_{\gamma}\diamondsuit_a^{-\alpha,-\beta}(_{\gamma}\diamondsuit_a^{-\alpha,-\beta}f)\right)(t)
= \left({_{\gamma}}\diamondsuit_a^{-\alpha,-\beta}f\right)(t)$.
\end{remark}
We now prove a general Leibniz formula.
\begin{theorem}[Leibniz formula]
\label{thm:ProductRule}
Let $f$ and $g$ be real valued functions, $0<\alpha,~\beta<1$.
For all $t$ such that $t=a \mod (1)$, the following equality holds:
\begin{multline}
\label{eq:GLF}
\left(_{\gamma}\diamondsuit_a^{-\alpha,-\beta}(fg)\right)(t)
=\gamma\sum_{k=0}^\infty\binom{-\alpha}{k}\left[\left(\nabla^k
g\right)(t)\right] \cdot\left[\left(\Delta_a^{-(\alpha+k)}f\right)(t+\alpha+k)\right]\\
+(1-\gamma)\sum_{k=0}^\infty\binom{-\beta}{k}\left[\left(\nabla^k
g\right)(t)\right]\left[\left(\Delta_a^{-(\beta+k)}f\right)(t+\beta + k)\right],
\end{multline}
where
$$
\binom{u}{v}=\frac{\Gamma(u+1)}{\Gamma(v+1)\Gamma(u-v+1)}.
$$
\end{theorem}
\begin{proof}
By definition of the diamond fractional sum,
\begin{equation*}
\begin{split}
\left(_{\gamma}\diamondsuit_a^{-\alpha,-\beta}(fg)\right)(t)
&=\gamma \left(\Delta_a^{-\alpha}(fg)\right)(t+\alpha)
+(1-\gamma)\left(\nabla_a^{-\beta}(fg)\right)(t)\\
&=\frac{\gamma}{\Gamma(\alpha)}\sum_{s=a}^{t}(t+\alpha-\sigma(s))^{(\alpha-1)}f(s)g(s)
+\frac{1-\gamma}{\Gamma(\beta)}\sum_{s=a}^{t}(t-\rho(s))^{\overline{\beta-1}}f(s)g(s).
\end{split}
\end{equation*}
By Taylor's expansion of $g(s)$ \cite{1},
$$
g(s)=\sum_{k=0}^\infty \frac{(s-t)^{\overline{k}}}{k!}
(\nabla^k g)(t)=\sum_{k=0}^\infty (-1)^k\frac{(t-s)^{(k)}}{k!}(\nabla^k g)(t).
$$
Substituting the Taylor series of $g(s)$ at $t$,
\begin{multline*}
\left(_{\gamma}\diamondsuit_a^{-\alpha,-\beta}(fg)\right)(t)
=\frac{\gamma}{\Gamma(\alpha)}\sum_{s=a}^{t}(t+\alpha-\sigma(s))^{(\alpha-1)}f(s)
\left[\sum_{k=0}^\infty (-1)^k(t-s)^{(k)}\frac{(\nabla^k g)(t)}{k!}\right]\\
+\frac{1-\gamma}{\Gamma(\beta)}\sum_{s=a}^{t}(t-\rho(s))^{\overline{\beta-1}}f(s)
\left[\sum_{k=0}^\infty
(-1)^k(t-s)^{(k)}\frac{(\nabla^k g)(t)}{k!}\right].
\end{multline*}
Since
\begin{equation*}
\begin{split}
(t+\alpha-\sigma(s))^{(\alpha-1)}(t-s)^{(k)}&=(t+\alpha-\sigma(s))^{(\alpha+k+1)},\\
(t-\rho(s))^{\overline{\beta-1}}(t-s)^{(k)}&=(t+\beta-\sigma(s))^{(\beta+k+1)},
\end{split}
\end{equation*}
and $\displaystyle\sum_{s=t-k+1}^{t}(t-s)^{(k)}=0$, we have
\begin{multline*}
\left(_{\gamma}\diamondsuit_a^{-\alpha,-\beta}(fg)\right)(t)
=\frac{\gamma}{\Gamma(\alpha)}\sum_{k=0}^\infty
(-1)^k\frac{(\nabla^k g)(t)}{k!} \sum_{s=a}^{t-k}(t+\alpha-\sigma(s))^{(\alpha+k-1)}f(s)\\
+\frac{1-\gamma}{\Gamma(\beta)}\sum_{k=0}^\infty
(-1)^k\frac{(\nabla^k g)(t)}{k!}
\sum_{s=a}^{t-k}(t+\beta-\sigma(s))^{(\beta+k-1)}f(s).
\end{multline*}
Because
$$
(-1)^k=\frac{\Gamma(-\alpha+1)\Gamma(\alpha)}{\Gamma(-\alpha+k+1)\Gamma(k+\alpha)}
=\frac{\Gamma(-\beta+1)\Gamma(\beta)}{\Gamma(-\beta+k+1)\Gamma(k+\beta)}
$$
and $k!=\Gamma(k+1)$, the above expression becomes
\begin{equation*}
\begin{split}
\left(_{\gamma}\diamondsuit_a^{-\alpha,-\beta}(fg)\right)(t)
&=\frac{\gamma}{\Gamma(\alpha)} \sum_{k=0}^\infty (\nabla^k g)(t)
\binom{-\alpha}{k}\cdot\left[\frac{1}{\Gamma(k+\alpha)}
\sum_{s=a}^{t-k}(t+\alpha-\sigma(s))^{(\alpha+k-1)}f(s)\right]\\
&\qquad+\frac{1-\gamma}{\Gamma(\beta)} \sum_{k=0}^\infty (\nabla^k g)(t)
\binom{-\beta}{k}\left[\frac{1}{\Gamma(k+\beta)}
\sum_{s=a}^{t-k}(t+\beta-\sigma(s))^{(\beta+k-1)}f(s)\right]\\
&=\gamma\sum_{k=0}^\infty \binom{-\alpha}{k}(\nabla^k g)(t)
(\Delta_a^{-(\alpha+k)}f)(t+\alpha+k)\\
&\qquad+(1-\gamma)\sum_{k=0}^\infty
\binom{-\beta}{k}(\nabla^k g)(t)(\Delta_a^{-(\beta+k)}f)(t+\beta+k).
\end{split}
\end{equation*}
\end{proof}
\begin{remark}
Choosing $\gamma=0$ in our Leibniz formula \eqref{eq:GLF}, we obtain that
$$(\nabla_a^{-\beta}(fg))(t)
=\sum_{k=0}^\infty\binom{-\beta}{k}\left[(\nabla^k
g)(t)\right]\left[(\Delta_a^{-(\beta+k)}f)(t+\beta+k)\right].
$$
\end{remark}
\begin{remark}
Choosing $\gamma=1$ in our Leibniz formula \eqref{eq:GLF}, we obtain that
\begin{equation}
\label{LeibnizDelta}
(\Delta_a^{-\alpha}(fg))(t+\alpha)
=\sum_{k=0}^\infty\binom{-\alpha}{k}\left[(\nabla^k g)(t)\right]
\left[(\Delta_a^{-(\alpha+k)}f)(t+\alpha+k)\right].
\end{equation}
As a particular case of \eqref{LeibnizDelta}, let $a=0$.
Then, recalling Lemma~\ref{nabladelta},
we obtain the Leibniz formulas of \cite{5}.
\end{remark}
\section{Conclusion}
\label{sec:4}
The discrete fractional calculus is a subject under
strong current research
(see, \textrm{e.g.}, \cite{6,7,10,Goodrich1,Goodrich2} and references therein).
Two versions of the discrete fractional calculus, the delta and the nabla,
are now standard in the fractional theory.
Motivated by the diamond-alpha dynamic derivative on time scales \cite{11,13,14}
and the fractional derivative of \cite{withBasia:Spain2010},
we introduce here a combined diamond-gamma fractional sum
of order (alpha, beta), as a linear combination
of the delta and nabla fractional sum operators
of order alpha and beta, respectively.
The new operator interpolates between the delta and nabla cases,
reducing to the standard fractional delta operator when $\gamma =1$
and to the fractional nabla sum when $\gamma =0$.
Using the discrete fractional diamond sum here proposed,
one can now introduce the discrete fractional diamond difference
in the usual way. It is our intention to generalize the new
discrete diamond fractional operator
to an arbitrary time scale $\mathbb{T}$
(\textrm{i.e.}, to an arbitrary nonempty
closed set of the real numbers). Another line of research,
to be addressed elsewhere, consists to investigate
the usefulness of modeling with fractional diamond equations
and study corresponding fractional variational principles.
\newpage
\leftline{\bf\ Acknowledgments}
\vskip 10 pt
This work is part of the first author's Ph.D. project, carried out at
the University of Aveiro under the framework of the Doctoral
Programme \emph{Mathematics and Applications} of Universities of
Aveiro and Minho, and was partially presented during the
\emph{3rd Conference on Nonlinear Science and Complexity} (NSC10),
Cankaya University, Ankara, 26-29 July, 2010.
The financial support of the Polytechnic Institute
of Viseu and \emph{The Portuguese Foundation for Science and
Technology} (FCT), through the ``Programa de apoio \`{a}
forma\c{c}\~{a}o avan\c{c}ada de docentes do Ensino Superior
Polit\'{e}cnico'', Ph.D. fellowship SFRH/PROTEC/49730/2009, is here
gratefully acknowledged. The authors were also supported by FCT through the
\emph{Center for Research and Development in Mathematics and Applications} (CIDMA).
\vskip 20 pt
|
{
"timestamp": "2010-09-21T02:03:57",
"yymm": "1009",
"arxiv_id": "1009.3883",
"language": "en",
"url": "https://arxiv.org/abs/1009.3883"
}
|
\section{\label{Intro} Introduction}
The evaluation of tensorial Feynman integrals with $n$ external legs is an important
technical ingredient of perturbative quantum field theoretical calculations with
Feynman diagrams.
They are needed in particular for the fast and efficient numerical evaluation of next-to-leading
order contributions at high energy colliders. Special attention is concentrated these days on experiments performed at the LHC;
{for a snapshot on related activities see~\cite{Binoth:2010ra}.
Of course, there is an unlimited variety of other reasons to use tensor reductions of Feynman integrals with quite diverse requirements in detail,
and a unique all-purpose, final approach does not exist.
}
For a recent overview see~\cite{Denner:THHH2009} and references therein.
The first systematic approach to reduce {tensor components} to
{a basis of} scalar 1-point to 4-point integrals in generic dimension $d=4-2 \varepsilon$
for $n \le 4$ is the Passarino-Veltman reduction~\cite{Passarino:1978jh}, obtained by solving a system of linear equations.
In fact, this is a unique basis.
{We use, with some sophistication, Davydychev's approach~\cite{Davydychev:1991va}, where $n$-point tensor coefficients are represented in terms of scalar Feynman integrals.
For tensors of rank $R$ they are defined in space-time dimensions up to $4-2\epsilon+2R$, with an additional modification: propagators may appear with higher powers.
These integrals are complicated objects, and an important step towards their evaluation is the application of dimensional recurrence relations, derived for $L$-loop functions in~\cite{Tarasov:1996br}.
They have been systematically worked out for $L=1$ in~\cite{Fleischer:1999hq}, and
in a subsequent article~\cite{Fleischer:2003rm} the evaluation of scalar integrals in $d$ dimensions
with powers $1$ of the scalar propagators is advocated.
Alternatively, the straightforward derivation of representations in the generic dimension $4-2\epsilon$, or finally just in four dimensions, by means of recurrence relations
introduces coefficients containing inverse Gram determinants, which may become small in some
kinematical domains and thus raise numerical problems.
For $n \le 4$ this problem was not very severe
\cite{Devaraj:1997es}.
Serious problems arise, however, for $n$-point Feynman integrals with $n \geq 5$.
{In that case the choice of the tensor basis is not unique and the freedom may be used to
completely avoid the appearance of}
inverse Gram determinants for $n\geq 5$~\cite{Bern:1992em,Bern:1993kr,Bern:1994zx,Bern:1994cg,Campbell:1996zw,Binoth:1999sp,Denner:2002ii}.
On the contrary, for $n<5$
one has to find explicit methods to stabilize the numerics for vanishing
or small Gram determinants.
{In view of these facts, one may wonder if the approach of Davydychev can be worked out with an optimization of the handling of exceptional (small or vanishing) Gram determinants.}
This is what has been achieved in the present work.
For non-exceptional kinematics the
$g^{\mu \nu}$ tensor, considering $5$-point functions, is redundant and may be
expressed in terms of $4$-momenta.
A nicely compact algebraic result is obtained due to this ansatz after applying \emph{symmetrized} dimensional recurrences
, but inverse Gram determinants of $5$-point as well as of $4$-point functions are introduced.
In an earlier attempt~\cite{Fleischer:2007ff,Diakonidis:2008ij,Gluza:2009mj}, tensor ranks until $R=3$ were presented \emph{without} inverse Gram determinants of the $5$-point
function using the \emph{algebra of the signed minors}~\cite{Melrose:1965kb},
but a generalization to higher ranks was not evident.
In the present work we first apply a particular recursion, which was obtained in~\cite{Diakonidis:2009fx}.
It reduces the tensor rank from $R$ to $R-1$ and the further reduction
can be arranged in a systematic manner without introducing inverse 5-point Gram determinants.
The 4-point tensor coefficients, which are in fact, due to
\cite{Davydychev:1991va},
higher-dimensional $4$-point functions with higher powers of the scalar propagators,
were reduced by lengthy algebraic calculations
to higher-dimensional $4$-point functions with powers $1$ of the scalar propgators plus tensor coefficients of 3-point functions.
The latter higher-dimensional $4$-point functions are tensor coefficients of the $g^{\mu_1\mu_2}\cdots g^{\mu_{2l-1}\mu_{2l}}$ terms of 4-point tensors.
At this stage inverse 4-point Gram determinants are avoided completely.
The recursions of all remaining 3-point functions may be performed
{simply} \'{a} la~\cite{Fleischer:1999hq}.
{This is
the way the numerics for this article was performed. Nevertheless, the same approach
as above can also be applied to the $3$-point functions, leaving only
higher dimensional $3$-point functions with powers $1$ of the scalar propagators, thereby avoiding inverse Gram determinants of the $3$-point function.}
Finally, one has to calculate the higher dimensional $4$-point functions with
powers $1$ of the scalar propagators.
This may be done by direct evaluation, see e.g.~\cite{Binoth:2002xh}, or by further reduction to simpler integrals, which in general, however, introduces inverse powers of 4-point Gram determinants.
For small Gram determinants we therefore derive a relatively simple \emph{analytic expansion} in positive powers of the Gram determinant.
Such an infinite series was, to our knowledge, first proposed in equation~(36) of
\cite{Fleischer:2003rm}, but was not numerically applied so far.
In fact, this expansion applies for higher-dimensional integrals with powers $1$
of the scalar propagators only and would not be appropriate for a representation of tensor
coefficients in their original form.
{
Another approach to the problem of small Gram determinants was chosen in sect. 5.4 of~\cite{Denner:2005nn},
where
relations between different 3- and 4-point tensor coefficients are exploited
and the full set is calculated with increasing iterations.
The $n^{th}$ iteration
requires all $3$-point coefficients of rank $n$.
In contrary to that our expansion is
concerned only with the subset of the $g^{\mu_1\mu_2}\cdots g^{\mu_{2l-1}\mu_{2l}}$
coefficients of the 4-point tensors, which are approximated only by the corresponding
subset of the $3$-point tensor coefficients.
Indeed, our series expansion of the higher dimensional 4-point functions
is useful only since these are embedded in expressions which are already free of
inverse Gram determinants .}
{
In~\cite{Fleischer:2003rm} it was shown that the integrals under consideration
can be expressed in terms of multiple hypergeometric functions.
One can apply their series expansion or, alternatively,
one could use their representations in terms of $1$-dimensional integrals also given in~\cite{Fleischer:2003rm}.
These, in general, present the integrals in rather different domains of phase space, including the case of small Gram determinants of the $4$-point function.
Thus our approach offers a variety of options to adjust to the given kinematical
situation. For the time being we only use our series expansion applying
Pad\'{e} approximants.}
This turns out to be very efficient and allows to obtain high precision for the numerical values of the
tensor coefficients of the $4$-point function.
In an example, we demonstrate that the combination of representations for non-exceptional kinematics with this expansion covers the complete phase-space from medium to vanishing Gram determinants.
In view of the importance of stable numerics for tensor reductions, it would be welcome to have one or more complete opensource programs for this task, including the treatment of small Gram determinants.
To our knowledge, none is presently available.
Following the approach of this article, a C++ program is under development to close this gap~\cite{c++yundin:2010bb}.
The article is organized as follows.
In sect.~\ref{Tshift}, some definitions and basic formulae are recalled.
{Sect.~\ref{LargeGrams} describes a compact analytical tensor reduction of 5-point functions with non-exceptional kinematics.} In sect.~\ref{5to4}
the $5$-point functions up to rank $R=5$ are reduced to
$4$-point tensor coefficients in terms of
4-point integrals in higher dimensions and with higher powers of the scalar propagators.
In sect.~\ref{4togeneric}, we reduce these 4-point integrals in several steps.
In sect.~\ref{DiffQuo} the integrals are reduced to 4-point integrals in higher dimensions
with powers $1$ of the scalar propagators plus 3-point tensor coefficients. The results
are given in eqns.~\eqref{I4id+2},~\eqref{want1},~\eqref{fulld3} and~\eqref{fulld4}.
{Indeed, these eqns. are the central point of our approach since
they allow to proceed further in different directions.
One might e.g. apply the general method of calculating higher
dimensional integrals of~\cite{Fleischer:2003rm}.
With app.~\ref{App}, alternatively a reduction to the Passarino-Veltman basis is straightforward.
In subsect.~\ref{Gram}, we recall how to expand the
higher-dimensional integrals with powers $1$ of the scalar propagators for the case of vanishing
or small Gram determinants, the result being given in~(\ref{final})}.
The symmetrized recursion relations, useful
for the $5$-point functions with non-exceptional Gram determinants
are presented in sect.~\ref{LaGra}.
Additionally, in sect.~\ref{Simplify}
we give some relations which may be useful for an analytic simplification of
Feynman diagrams.
We end with conclusions in sect.~\ref{conclude}.
Appendix~\ref{App} contains a list of dimensional recurrences and app.~\ref{Bpp} collects divergent parts of higher-dimensional integrals. A numerical example is discussed in app.~\ref{Num}.
Appendix~\ref{app-nota} contains notations and some relevant algebraic relations.
\section{\label{Tshift}Tensor integrals in terms of integrals in shifted dimensions}
A tensorial Feynman integral with $n$ external legs is shown in fig.~\ref{fig-n-point} and is defined as
\begin{eqnarray}
\label{definition}
I_{n,\{\nu_j\}}^{\mu_1\cdots\mu_R} &=&~\frac{{(2 \pi \mu)}^{4-d} }{i {\pi}^2}~\int d^d k~~\frac{\prod_{r=1}^{R} k^{\mu_r}}{\prod_{j=1}^{n}c_j^{\nu_j}},
\end{eqnarray}
with denominators $c_j$, having \emph{indices} $\nu_j$ and \emph{chords}
$q_j$,
\begin{eqnarray}\label{propagators}
c_j &=& (k-q_j)^2-m_j^2 +i \epsilon.
\end{eqnarray}
Here, we use the generic dimension $d=4-2\epsilon$ and $\mu =1$.
Reducing the tensors to $1$- to $4$-point scalar functions $I_n^d$, in general their
expansions in terms of $\varepsilon$ is needed. The first expansion terms
can be expressed in terms of Euler dilogarithmic (or simpler) functions~\cite{'tHooft:1978xw,Denner:1991qq,vanOldenborgh:1990yc,Hahn:1998yk,Ellis:2007qk}.
\begin{figure}[bt]
\begin{center}
\includegraphics[width=.39\textwidth]{1-loop-n-point-kinematics
\end{center}
\caption[Momenta flow of the $n$-point function.]{%
\label{fig-n-point}
Momenta flow of the $n$-point function.}
\end{figure}
The six-point tensor integrals may be expressed in terms of five-point tensor functions~\cite{Fleischer:1999hq,Diakonidis:2008ij,Binoth:2005ff,Denner:2005nn}:
\begin{eqnarray}\label{tensor6general}
I_6^{\mu_1 \dots \mu_{R-1} \mu} =
- \sum_{s=1}^{6}
I_5^{\mu_1 \dots \mu_{R-1} ,s } \bar{Q}_s^{\mu},
\end{eqnarray}
where the auxiliary vectors $\bar{Q}_s$ are
\begin{eqnarray}
\bar{Q}_s^{\mu}&=&\sum_{i=1}^{6} q_i^{\mu} \frac{{0s\choose 0i}_6}{{0\choose 0}_6}~~~,~~~ s=1 \dots 6.
\label{Q6}
\end{eqnarray}
A similar formula exists also for five-point tensor integrals~\cite{Diakonidis:2009fx}:
\begin{eqnarray}
I_5^{\mu_1 \dots \mu_{R-1} \mu} =I_5^{\mu_1 \dots \mu_{R-1}} Q_0^{\mu} - \sum_{s=1}^{5}
I_4^{\mu_1 \dots \mu_{R-1},s } Q_s^{\mu}.
\label{tensor5general}
\end{eqnarray}
The auxiliary vectors here are:
\begin{eqnarray}
\label{Qs}
Q_s^{\mu}&=&\sum_{i=1}^{5} q_i^{\mu} \frac{{s\choose i}_5}{\left( \right)_5},~~~ s=0, \dots, 5.
\end{eqnarray}
{For later use, we introduce also
\begin{eqnarray}\label{3.11}
Q_s^{t,\mu}&=&\sum_{i=1}^{5} q_i^{\mu} \frac{ {st\choose it}_5}{{t\choose t}_5}~ \nonumber \\
&=&Q_s^{\mu}~~-\frac{{s\choose t}_5}{{t\choose t}_5}Q_t^{\mu} ,
\\\label{3.12}
Q_s^{tu,\mu}&=&\sum_{i=1}^{5} q_i^{\mu} \frac{ {stu\choose itu}_5}{{tu\choose tu}_5} \nonumber \\
&=&Q_s^{u,\mu}-\frac{{su\choose tu}_5}{{tu\choose tu}_5}Q_t^{u,\mu},
\end{eqnarray}
and
\begin{equation}
g^{\mu\, \nu}=2 \sum_{i,j=1}^{5} \frac{{i\choose j}_5}{\left( \right)_5} \, q_i^{\mu}\, q_j^{\nu} .
\label{gmunu}
\end{equation}
}
In fact,~\eqref{tensor6general} is essentially the same formula as~\eqref{tensor5general}, except
that ${\left( \right)}_5$ is replaced by ${0\choose 0}_6$ etc. and ${00\choose 0i}_6=0$.
With the definition
\begin{eqnarray}
\label{gram}
Y_{ij}=-(q_i-q_j)^2+m_i^2+m_j^2,
\end{eqnarray}
the \emph{modified Cayley determinant} of a topology
with internal lines $1 \cdots n $ becomes
\begin{eqnarray}\label{gram1}
()_n~\equiv~
\begin{vmatrix}
0 & 1 & 1 &\ldots & 1 \\
1 & Y_{11} & Y_{12} &\ldots & Y_{1n} \\
1 & Y_{12} & Y_{22} &\ldots & Y_{2n} \\
\vdots & \vdots & \vdots &\ddots & \vdots \\
1 & Y_{1n} & Y_{2n} &\ldots & Y_{nn}
\end{vmatrix}
.
\end{eqnarray}
One chord may be chosen arbitrarily to vanish, $q_n=0$, and then this object is the Gram determinant: \footnote{Usually we will use indices $s,t,\cdots = 1,\cdots, n$ for labelling internal lines, and indices $i,j,\cdots = 1,\cdots, n-1$ for labelling the (non-vanishing) chords.}
\begin{eqnarray}\label{gram11}
{()_n|_{q_n=0}} &=&{ -~\det ~ G_{n-1},}
\\\label{gram12}
{G_{n-1,ik}}&=& {2q_i q_k,~~ i,k=1,\dots n-1 .}
\end{eqnarray}
The Gram determinant is independent of the internal masses.
The \emph{signed minors}~\cite{Melrose:1965kb} are denoted as follows:
\begin{eqnarray}\label{gram2}
\left(
\begin{array}{ccc}
j_1 & j_2 & \cdots j_m\\
k_1 & k_2 & \cdots k_m\\
\end{array}
\right)_n .
\end{eqnarray}
They are determinants, labeled by those rows $j_1,j_2,\cdots j_m$ and columns $k_1,k_2,\cdots k_m$ which have been
excluded from the definition of the Gram determinant $()_n$, with sign
\begin{eqnarray}
\label{eq-modc}
\mathrm{sign}
\begin{pmatrix}
j_1 & j_2 & \cdots & j_m\\
k_1 & k_2 & \cdots & k_m\\
\end{pmatrix}_n
=
(-1)^{j_1+j_2+ \cdots +j_m+k_1+k_2+ \cdots +k_m} \cdot S(j_1, j_2 \cdots j_m) \cdot
S(k_1 , k_2 , \cdots k_m).
\end{eqnarray}
Here
$S(j_1 , j_2 \cdots j_m)$
gives the sign of permutations needed to place the indices in increasing order. \footnote{The definitions are related to similar ones used in the literature, see app.~\ref{app-nota}.}
We have e.g.
\begin{eqnarray}\label{gram3}
\Delta_n=
\begin{vmatrix
Y_{11} & Y_{12} &\ldots & Y_{1n} \\
Y_{12} & Y_{22} &\ldots & Y_{2n} \\
\vdots & \vdots &\ddots & \vdots \\
Y_{1n} & Y_{2n} &\ldots & Y_{nn}
\end{vmatrix
= {0\choose 0}_n.
\label{mcd}
\end{eqnarray}
Applying Davydychev's method~\cite{Davydychev:1991va}, one expresses the tensor integrals $I_n^{\mu_1 \dots \mu_{R}}$ by scalar Feynman integrals
$I_{n,i\cdots}^{(d)}$ in higher dimensions $d$ and with higher indices $\nu_i$.
We reproduce here integrals with rank $R\leq 5$:
\begin{eqnarray}
\label{tensor1}
I_n^{\mu} & =& \int \frac{d^d k}{{i\pi}^{d/2}} k^{\mu} \prod_{j=1}^{n} \, {c_j^{-1}}
\nonumber \\
&=& - ~ \sum_{i=1}^{n} \, q_i^{\mu} \, I_{n,i}^{[d+]} ,
\\
\label{tensor2}
I_{n}^{\mu\, \nu}& =& \int \frac{d^d k}{{i\pi}^{d/2}} k^{\mu} \, k^{\nu} \, \prod_{j=1}^{n} \, {c_j^{-1}}
\nonumber \\
&=&
\sum_{i,j=1}^{n} \, q_i^{\mu}\, q_j^{\nu} \, n_{ij} \, \, I_{n,ij}^{[d+]^2} -\frac{1}{2}
\, g^{\mu \nu} \, I_{n}^{[d+]} ,
\\
\label{tensor3}
I_{n}^{\mu\, \nu\, \lambda}& =& \int \frac{d^d k}{{i\pi}^{d/2}} k^{\mu} \, k^{\nu} \, k^{\lambda} \, \prod_{j=1}^{n} \, {c_j^{-1}}
\nonumber \\
&=&
-~ \sum_{i,j,k=1}^{n} \, q_i^{\mu}\, q_j^{\nu}\, q_k^{\lambda} \, n_{ijk} \, \, I_{n,ijk}^{[d+]^3}
+\frac{1}{2} \sum_{i=1}^{n} g^{[\mu \nu} q_i^{\lambda]} I_{n,i}^{[d+]^2} ,
\nonumber \\ \\
\label{tensor4}
I_{n}^{\mu\, \nu\, \lambda\, \rho} &=&
\int \frac{d^d k}{{i\pi}^{d/2}} k^{\mu} \, k^{\nu} \, k^{\lambda} \, k^{\rho} \, \prod_{j=1}^{n} \, {c_j^{-1}}
\nonumber \\ \nonumber \\
&=&
\sum_{i,j,k,l=1}^{n} \, q_i^{\mu}\, q_j^{\nu}\, q_k^{\lambda}
\, q_l^{\rho}\, n_{ijkl} \, \, I_{n,ijkl}^{[d+]^4}
-\frac{1}{2} \sum_{i,j=1}^{n} g^{[\mu \nu} q_i^{\lambda} q_j^{\rho]}
\, n_{ij} I_{n,ij}^{[d+]^3}
+\frac{1}{4} g^{[\mu \nu} g^{\lambda \rho]} I_{n}^{[d+]^2} ,
\nonumber \\ \\
\label{tensor5}
I_{n}^{\mu\, \nu\, \lambda\, \rho\, \sigma}&=&
\int \frac{d^d k}{i\pi^{d/2}} k^{\mu} \, k^{\nu} \, k^{\lambda} \, k^{\rho} \,
k^{\sigma} \,
\prod_{j=1}^{n} \, {c_j^{-1}}
\nonumber\\
&=&
-~ \sum_{i,j,k,l,m=1}^{n} \, q_i^{\mu}\, q_j^{\nu}\, q_k^{\lambda}
\, q_l^{\rho}\, q_m^{\sigma}\, n_{ijklm} \, \, I_{n,ijklm}^{[d+]^5}
+\frac{1}{2} \sum_{i,j,k=1}^{n} g^{[\mu \nu} q_i^{\lambda} q_j^{\rho} q_k^{\sigma]}
\, n_{ijk} I_{n,ijk}^{[d+]^4}
\nonumber\\&&
-~\frac{1}{4} \sum_{i=1}^{n} g^{[\mu \nu} g^{\lambda \rho} q_i^{\sigma]} I_{n,i}^{[d+]^3} .
\end{eqnarray}
\newpage
The following symmetrized tensors are used:
\begin{eqnarray}
\label{G2V1}
g^{[\mu \nu} q_i^{\lambda]}&=& g^{\mu \nu} \, q_i^{\lambda} \,+ g^{\mu \lambda}
\, q_i^{\nu} \,+ \, g^{\nu \lambda} \, q_i^{\mu} ,
\\
\label{G2V2}
g^{[\mu \nu} q_i^{\lambda} q_j^{\rho]}&=&\, g^{\mu \nu} \, q_i^{\lambda}
\, q_j^{\rho} \,+ g^{\mu \lambda} \, q_i^{\nu} \, q_j^{\rho} \,+
\, g^{\nu \lambda} \, q_i^{\mu} \, q_j^{\rho} \,+
\, g^{\mu \rho} \, q_i^{\nu}
\, q_j^{\lambda} \,+ g^{\nu \rho} \, q_i^{\mu} \, q_j^{\lambda} \,+
\, g^{\lambda \rho} \, q_i^{\mu} \, q_j^{\nu} ,
\nonumber \\ \\ ~~
\label{G2G2}
g^{[\mu \nu} g^{\lambda \rho]}~~&=&\, g^{\mu \nu} \,g^{\lambda \rho} \,+ g^{\mu \lambda} \, g^{\nu \rho} \,+
\, g^{\mu \rho} \, g^{\nu \lambda} ,
\\
\label{G2V3}
g^{[\mu \nu} q_i^{\lambda} q_j^{\rho} q_k^{\sigma]}&=&
g^{\mu \nu} q_i^{\lambda} q_j^{\rho} q_k^{\sigma}+g^{\mu \lambda} q_i^{\nu} q_j^{\rho} q_k^{\sigma}+
g^{\mu \rho} q_i^{\nu} q_j^{\lambda} q_k^{\sigma}+g^{\mu \sigma} q_i^{\nu} q_j^{\lambda} q_k^{\rho}+
g^{\nu \lambda} q_i^{\mu} q_j^{\rho} q_k^{\sigma}
\nonumber \\
&&+~
g^{\nu \rho} q_i^{\mu} q_j^{\lambda} q_k^{\sigma}+
g^{\nu \sigma} q_i^{\mu} q_j^{\lambda} q_k^{\rho}+g^{\lambda \rho} q_i^{\mu} q_j^{\nu} q_k^{\sigma}+
g^{\lambda \sigma} q_i^{\mu} q_j^{\nu} q_k^{\rho}+g^{\rho \sigma} q_i^{\mu} q_j^{\nu} q_k^{\lambda},~~~~
\\
g^{[\mu \nu} g^{\lambda \rho} q_i^{\sigma]}
&=&
g^{[\mu \nu} g^{\lambda \rho]} q_i^{\sigma}~+g^{[\mu \nu} g^{\lambda \sigma]} q_i^{\rho}+
g^{[\mu \nu} g^{\rho \sigma]} q_i^{\lambda}+g^{[\mu \sigma} g^{\lambda \rho]} q_i^{\nu}~+
g^{[\nu \sigma} g^{\lambda \rho]} q_i^{\mu}.
\label{G4V1}
\end{eqnarray}
The scalar integrals are:
\begin{eqnarray}
\label{eq:Inij}
I_{p, \, i\,j \,k\cdots} ^{[d+]^l,stu \cdots} = \int \frac{d^{[d+]^l}k}{i\pi^{[d+]^l/2}} \prod_{r=1}^{n} \, \frac{1}{c_r^{1+\delta_{ri} + \delta_{rj}+\delta_{rk}+\cdots
-\delta_{rs} - \delta_{rt}-\delta_{ru}-\cdots}} ,
\end{eqnarray}
where $[d+]^l=4-2 \varepsilon+2 l $.
The index $p$ is the number of propagators of the $p$-point function.
Note that equal lower and upper indices cancel.
The coefficients $n_{ij}, n_{ijk}$ and $n_{ijkl}$ etc. in~(\ref{tensor2}) to~(\ref{tensor5}) were introduced in~\cite{Diakonidis:2008ij}.
They stand for the product of factorials of the number
of equal indices: e.g. $n_{iiii}=4!, n_{ijii}=3!, n_{iijj}=2! 2!, n_{ijkk}=2!,n_{ijkl}=1!$;
the indices $i,j,k,l$ are assumed here to be different from each other.
The following relations are of particular relevance
for the successive application of recurrence relations to reduce higher-dimensional integrals:
\begin{eqnarray}\label{ndef1}
n_{ij}&=&{\nu}_{ij} ,
\nonumber\\
n_{ijk}&=&{\nu}_{ij} {\nu}_{ijk} ,
\nonumber\\
n_{ijkl}&=&{\nu}_{ij} {\nu}_{ijk} {\nu}_{ijkl},
\nonumber\\
n_{ijklm}&=&{\nu}_{ij} {\nu}_{ijk} {\nu}_{ijkl}{\nu}_{ijklm},
\end{eqnarray}
and:
\begin{eqnarray}
\label{nudef1}
{\nu}_{ij}&=&1+{\delta}_{ij} ,
\nonumber\\
{\nu}_{ijk}&=&1+{\delta}_{ik}+{\delta}_{jk} ,
\nonumber\\
{\nu}_{ijkl}&=&1+{\delta}_{il}+{\delta}_{jl}+{\delta}_{kl}.
\nonumber\\
{\nu}_{ijklm}&=&1+{\delta}_{im}+{\delta}_{jm}+{\delta}_{km}+{\delta}_{lm}.
\end{eqnarray}
In a second step, one may choose to express the higher-dimensional scalar integrals in terms of the generic scalar integrals
The algorithm
is based on recurrence relations with shifts of dimension $d \geq 4-2\varepsilon$ and indices $\nu_s\geq 1$,
\begin{eqnarray}
\label{eq:RR1
\left( \right)_n \nu_s \left( \mathbf{s^{+}} I_{n}^{(d+2)}\right)
&=&
- {s \choose 0}_n I_n^{(d)} + \sum_{t=1}^{n} {s \choose t}_n \left( \mathbf{t^{-}} I_{n}^{(d)} \right),
\end{eqnarray}
or with a shift of dimension $d$:
\begin{eqnarray}
\left( \right)_n (d-\sum_{s=1}^{n}\nu_s+1) I_n^{(d+2)}
&=&
{{0 \choose 0}_n} I_n^{(d)}
- \sum_{t=1}^n {0 \choose t}_n \left( \mathbf{t^{-}} I_{n}^{(d)} \right).
\label{eq:RR2}\end{eqnarray}
These relations hold for arbitrary index sets $\{\nu_s\}$.
The integrals $\mathbf{s^{+}} I_{n}^{(d)}$ and $\mathbf{t^{-}} I_{n}^{(d)}$ are obtained from $ I_{n}^{(d)}$ by replacing $\nu_s \rightarrow (\nu_s + 1)$ and $\nu_t \rightarrow (\nu_t - 1)$, respectively. For more explicit expressions see app.~\ref{App} .
\section{\label{LargeGrams}
An efficient reduction of $5$-point tensor integrals
}
The reduction of $5$-point tensor integrals to $4$-point tensor integrals at non-exceptional momenta
may be performed by iterative application of~\eqref{tensor5general}.
This was exemplified in~\cite{Diakonidis:2009fx}, and an opensource Fortran code \texttt{olotic}~\cite{olotic:2010aa} is available.
In this sect., we derive a very compact, explicit representation of the tensor coefficients for 5-point functions in a minimal basis, chosen to be free of the metric tensor.
This will rely on an exploitation of~\eqref{gmunu} and
\eqref{trick}, a specifically useful relation of the \emph{algebra of the signed minors},
and applying the \emph{symmetrized} dimensional recurrences of sect.~\ref{LaGra}
We investigate the $5$-point tensor integrals step by step.
For the tensor of rank $R=1$ we get from~\eqref{tensor5general}
\begin{eqnarray}\label{3.1}
I_5^{\mu}=I_5 \cdot Q_0^{\mu} -\sum_{s=1}^{5} I_4^s \cdot Q_s^{\mu},
\end{eqnarray}
and $I_5$ may be taken from~\eqref{scalar4p}.
Similarly for the tensor of rank $2$,
\begin{eqnarray}\label{3.2}
I_5^{\mu \nu}=I_5^{\mu} \cdot Q_0^{\nu} -\sum_{s=1}^{5} I_4^{\mu, s} \cdot Q_s^{\nu} ,
\end{eqnarray}
with four-point integrals from~\eqref{tensor1} and~\eqref{A511},
\begin{eqnarray}\label{3.3}
I_4^{\mu,s}&=&-\sum_{i=1}^{5} q_i^{\mu} I_{4,i}^{[d+],s}, \\\label{3.32}
I_{4,i}^{[d+],s}&=&-\frac{ {0s\choose is}_5}{{s\choose s}_5} I_4^s+ \sum_{t=1}^{5}
\frac{ {ts\choose is}_5}{{s\choose s}_5} I_3^{st} ,
\end{eqnarray}
such that we can write the tensor of rank $R=2$ as
\begin{eqnarray}\label{3.4}
I_5^{\mu \nu}=I_5^{\mu} \cdot Q_0^{\nu} -\sum_{s=1}^{5} \left\{Q_0^{s,\mu} I_4^s-
\sum_{t=1}^{5} Q_t^{s, \mu} I_3^{st} \right\}
Q_s^{\nu}.
\end{eqnarray}
Compared to~\eqref{tensor2}, this representation and the following ones are free of the metric tensor.
{Further, the compactness relies on the use of the auxiliary vectors $Q_s^{\mu},Q_t^{s, \nu}$ instead of the chords $q_i^{\mu}$.}
{The tensor of rank $R=3$ deserves a bit more effort,}
\begin{eqnarray}\label{3.138}
I_5^{\mu \nu \lambda}=I_5^{\mu\nu} \cdot Q_0^{\lambda} -\sum_{s=1}^{5} I_4^{\mu\nu, s} \cdot Q_s^{\lambda}.
\end{eqnarray}
The corresponding $4$-point function
reads now due to~\eqref{tensor3} and with~\eqref{A522}
\begin{eqnarray}
\label{TwoTy}
I_4^{\mu \nu ,s}&=&\sum_{i,j=1}^{5} q_i^{\mu} q_j^{\nu} {\nu}_{ij} I_{4,ij}^{[d+]^2,s}-\frac{1}{2} g^{\mu \nu}I_{4}^{[d+],s},
\\\label{TwoTy2}
{\nu}_{ij} I_{4,ij}^{[d+]^2,s}&=&-\frac{ {0s\choose js}_5}{{s\choose s}_5} I_{4,i}^{[d+],s}+
\frac{ {is\choose js}_5}{{s\choose s}_5}I_{4}^{[d+],s}+
\sum_{t=1}^{5}
\frac{ {ts\choose js}_5}{{s\choose s}_5} I_{3,i}^{[d+],st}.
\end{eqnarray}
Observe that for $i,j=s$
the integrals $I_{4,i}^{[d+],s}$ and $I_{4,ij}^{[d+]^2,s}$ vanish (due to vanishing signed minors)
such that indeed {a formal} summation over all five values of $i,j$ is possible.
\newpage
Now we use identity~\eqref{r2},
\begin{eqnarray}
\frac{ {is\choose js}_5}{{s\choose s}_5}&=&\frac{ {i\choose j}_5}{{\left( \right)}_5}
-\frac{ {s\choose i}_5 {s\choose j}_5 }{{\left( \right)}_5 {s\choose s}_5} \nonumber \\
&=&
\frac{ {i\choose j}_5}{{\left( \right)}_5}-\frac{{\left( \right)}_5}{{s\choose s}_5} Q_s^i Q_s^j,
\label{trick}
\end{eqnarray}
where the $Q_s^i$ are the vector components of $Q_s^{\mu}$, see~\eqref{Qs}.
Performing summation over $i,j$ in~(\ref{TwoTy}), the first term on the right hand side of
~\eqref{trick} yields
$\frac{1}{2} g^{\mu \nu } I_4^{[d+],s}$ (see~\eqref{gmunu}) and thus cancels against the last term in~(\ref{TwoTy}). Thus
we can write the 4-point tensor of rank $R=2$ as
\begin{eqnarray}\label{3.7}
I_4^{\mu \nu ,s}&=&\sum_{i,j=1}^{5} q_i^{\mu} q_j^{\nu} J_{4,ij}^{s},
\\\label{3.111}
J_{4,ij}^{s}&=&-\frac{ {0s\choose js}_5}{{s\choose s}_5} I_{4,i}^{[d+],s}-\frac{ {s\choose i}_5 {s\choose j}_5 }{{\left( \right)}_5 {s\choose s}_5}I_{4}^{[d+],s}+\sum_{t=1}^{5} \frac{ {ts\choose js}_5}{{s\choose s}_5} I_{3,i}^{[d+],st},
\end{eqnarray}
where the metric tensor has again disappeared compared to~\eqref{tensor2} and instead $q_i^{\mu} q_j^{\nu}$ contribute
for $i,j=s$. A compact notation can now be used with~\eqref{3.11} and~\eqref{3.12}:
\begin{eqnarray}\label{3.13}
I_4^{\mu \nu ,s}=Q_0^{s,\mu} Q_0^{s,\nu} I_{4}^{s}-\frac{{\left( \right)}_5}{{s\choose s}_5}
Q_s^{\mu} Q_s^{\nu}I_{4}^{[d+],s}
+\sum_{i=1}^{5} q_i^{\mu} q_j^{\nu}R_{3,ij}^{[d+]^2,s} ,
\end{eqnarray}
with $R_{3,ij}^{[d+]^2,s}$ the scratched version of~\eqref{wanty}.
Inserting~\eqref{3.13} in~\eqref{3.138} yields the compact expression for $I_5^{\mu\nu\lambda}$, free of the metric tensor.
Next, for the tensor of rank $R=4$ of the $5$-point function,
\begin{eqnarray}\label{3.42}
I_5^{\mu \nu \lambda\rho}=
I_5^{\mu\nu\lambda} \cdot Q_0^{\rho}
-\sum_{s=1}^{5} I_4^{\mu\nu\lambda, s} \cdot Q_s^{\rho},
\end{eqnarray}
we need the $4$-point function of rank $R=3$, according to~\eqref{tensor3}:
\begin{eqnarray}\label{3.14}
I_4^{\mu \nu \lambda ,s}&=&-\sum_{i,j,k=1}^{5} q_i^{\mu} q_j^{\nu} q_k^{\lambda} n_{ijk}
I_{4,ijk}^{[d+]^3,s}+\frac{1}{2}\sum_{i=1}^{5} g^{[\mu \nu} q_i^{\lambda ]}I_{4,i}^{[d+]^2,s},
\\
n_{ijk}I_{4,ijk}^{[d+]^3,s}&=&-\frac{{0s\choose is}_5 {0s\choose js}_5 {0s\choose ks}_5}
{{s\choose s}_5^3}I_{4}^s+
\left\{\frac{{is\choose js}_5}{{s\choose s}_5} I_{4,k}^{[d+]^2,s}+
(j \leftrightarrow k) + (i \leftrightarrow k) \right\}+R_{3,ijk}^{[d+]^3,s}.
\label{fullxs}
\end{eqnarray}
Here we have used for $I_{4,ijk}^{[d+]^3,s}$, instead of~\eqref{A533}, the symmetrized form~\eqref{fullx} with
$R_{3,ijk}^{[d+]^3,s}$ being the scratched version of~\eqref{fullxr}.
Applying again~\eqref{trick} we obtain the analogue of~\eqref{3.13},
\begin{eqnarray}\label{3.35}
I_4^{\mu \nu \lambda ,s}&=&\sum_{i,j,k=1}^{5} q_i^{\mu} q_j^{\nu} q_k^{\lambda} J_{4,ijk}^{s},
\\\label{3.351}
J_{4,ijk}^{s}&=&\frac{{0s\choose is}_5{0s\choose js}_5{0s\choose ks}_5}{{s\choose s}_5^3}I_{4}^s+\frac{1}{{\left( \right)}_5} \left\{\frac{ {s\choose i}_5 {s\choose j}_5 }{ {s\choose s}_5}I_{4,k}^{[d+]^2,s}+(j \leftrightarrow k) + (i \leftrightarrow k) \right\}-R_{3,ijk}^{[d+]^3,s}.
\end{eqnarray}
For the following it also pays to introduce
\begin{eqnarray}
J_4^{\mu, s}&=&\sum_{i=1}^{5} q_i^{\mu} I_{4,i}^{[d+]^2,s} \nonumber \\
&=&-Q_0^{s,\mu} I_4^{[d+]} +
\sum_{t=1}^{5} Q_t^{s,\mu} I_3^{[d+],st},
\label{J4V}
\end{eqnarray}
so that finally
\begin{eqnarray}\label{3.352}
I_4^{\mu \nu \lambda ,s }=Q_0^{s,\mu} Q_0^{s,\nu}Q_0^{s,\lambda} I_4^s+
\frac{{\left( \right)}_5}{{s\choose s}_5} Q_s^{[\mu}Q_s^{\nu}J_4^{\lambda, s]}
-\sum_{i,j,k=1}^{5} q_i^{\mu} q_j^{\nu} q_k^{\lambda} R_{3,ijk}^{[d+]^3,s}.
\end{eqnarray}
with $R_{3,ijk}^{[d+]^3,s}$ the scratched version of~\eqref{fullxr}.
This finishes the determination of $I_5^{\mu \nu \lambda\rho}$.
Finally, for the tensor of rank $R=5$ of the $5$-point function,
\begin{eqnarray}\label{3.52}
I_5^{\mu \nu \lambda\rho\sigma}=
I_5^{\mu\nu\lambda\rho} \cdot Q_0^{\sigma}
-\sum_{s=1}^{5} I_4^{\mu\nu\lambda\rho, s} \cdot Q_s^{\sigma},
\end{eqnarray}
we need the scratched tensor of
rank $R=4$ of the $4$-point function. The corresponding symmetrized tensor coefficients are
taken from~\eqref{fully} and~\eqref{I451} by scratching. We begin with
the term $I_4^{[d+]^2}$ from~\eqref{fully}.
Using again~\eqref{trick} we have
\begin{eqnarray}
&&
\frac{{is\choose ls}_5}{{s\choose s}_5}\frac{{js\choose ks}_5}{{s\choose s}_5}+
\frac{{js\choose ls}_5}{{s\choose s}_5}\frac{{is\choose ks}_5}{{s\choose s}_5}+
\frac{{ks\choose ls}_5}{{s\choose s}_5}\frac{{is\choose js}_5}{{s\choose s}_5}=
\frac{{i\choose l}_5}{{\left( \right)}_5}\frac{{j\choose k}_5}{{\left( \right)}_5}+
\frac{{j\choose l}_5}{{\left( \right)}_5}\frac{{i\choose k}_5}{{\left( \right)}_5}+
\frac{{k\choose l}_5}{{\left( \right)}_5}\frac{{i\choose j}_5}{{\left( \right)}_5} \nonumber \\
-&&\frac{{\left( \right)}_5}{{s\choose s}_5}\left\{
\frac{{i\choose l}_5}{{\left( \right)}_5} Q_s^j Q_s^k+
\frac{{j\choose k}_5}{{\left( \right)}_5} Q_s^i Q_s^l+
\frac{{j\choose l}_5}{{\left( \right)}_5} Q_s^i Q_s^k+
\frac{{i\choose k}_5}{{\left( \right)}_5} Q_s^j Q_s^l+
\frac{{k\choose l}_5}{{\left( \right)}_5} Q_s^i Q_s^j+
\frac{{i\choose j}_5}{{\left( \right)}_5} Q_s^k Q_s^l \right\}\nonumber \\
+&& 3 \frac{{\left( \right)}_5^2}{{s\choose s}_5^2} Q_s^i Q_s^j Q_s^k Q_s^l.
\label{ijkls}
\end{eqnarray}
The first term on the right hand side of~\eqref{ijkls} yields after summation over $i,j,k,l$
\begin{eqnarray}\label{3.357}
\sum_{ijkl=1}^5 q_i^{\mu} q_j^{\nu} q_k^{\lambda} q_l^{\rho}
\left\{
\frac{{i\choose l}_5}{{\left( \right)}_5}\frac{{j\choose k}_5}{{\left( \right)}_5}
+
\frac{{j\choose l}_5}{{\left( \right)}_5}\frac{{i\choose k}_5}{{\left( \right)}_5}
+
\frac{{k\choose l}_5}{{\left( \right)}_5}\frac{{i\choose j}_5}{{\left( \right)}_5} \right\}
=
\frac{1}{4} g^{[ \mu \nu} g^{\lambda \rho ]}.
\end{eqnarray}
The same contribution comes directly from~\eqref{tensor4}. Finally,
the second term of~\eqref{tensor4} contributes with the term $\sim I_4^{[d+]^2,s}$
of~\eqref{I451}
and again the first term on the right hand side of~\eqref{trick}
\begin{eqnarray}\label{3.358}
-\frac{1}{2} \sum_{i,j=1}^5 g^{[ \mu \nu} q_i^{\lambda} q_j^{\rho ]} \frac{{i\choose j}_5}{{\left( \right)}_5} \cdot
I_4^{[d+]^2,s}=-\frac{1}{2} g^{[ \mu \nu} g^{\lambda \rho ]} \cdot I_4^{[d+]^2,s},
\end{eqnarray}
i.e. the terms $g^{[ \mu \nu} g^{\lambda \rho ]} \cdot I_4^{[d+]^2,s}$ cancel.
The second
term on the right hand side of~\eqref{ijkls} yields after summation over $i,j$
\begin{eqnarray}\label{3.359}
&&
-~\frac{{\left( \right)}_5}{{s\choose s}_5}\sum_{i,j,k,l=1}^5 q_i^{\mu} q_j^{\nu} q_k^{\lambda} q_l^{\rho}
\nonumber \\
&&~~~
\left\{
\frac{{i\choose l}_5}{{\left( \right)}_5} Q_s^j Q_s^k+
\frac{{j\choose k}_5}{{\left( \right)}_5} Q_s^i Q_s^l+
\frac{{j\choose l}_5}{{\left( \right)}_5} Q_s^i Q_s^k+
\frac{{i\choose k}_5}{{\left( \right)}_5} Q_s^j Q_s^l+
\frac{{k\choose l}_5}{{\left( \right)}_5} Q_s^i Q_s^j+
\frac{{i\choose j}_5}{{\left( \right)}_5} Q_s^k Q_s^l \right\} \cdot I_4^{[d+]^2,s}
\nonumber \\
&=&
-~\frac{1}{2}\frac{{\left( \right)}_5}{{s\choose s}_5} g^{[\mu \nu} Q_s^{\lambda} Q_s^{\rho ]}
\cdot I_4^{[d+]^2,s}.
\end{eqnarray}
A contribution of this type also comes from the second term of~\eqref{tensor4}
with the term $\sim I_4^{[d+]^2,s}$ of~\eqref{I451}, but now the second part on
the right hand side of~\eqref{trick} is
\begin{eqnarray}\label{3.36}
\frac{1}{2}\frac{{\left( \right)}_5}{{s\choose s}_5} g^{[\mu \nu} Q_s^{\lambda} Q_s^{\rho ]}\cdot I_4^{[d+]^2,s},
\end{eqnarray}
which means that also these terms cancel. Finally there remains the last term in~\eqref{ijkls},
which does not cancel but contains no $g^{\mu \nu}$
\begin{eqnarray}\label{3.388}
3 \frac{{\left( \right)}_5^2}{{s\choose s}_5^2} Q_s^{\mu}Q_s^{\nu}Q_s^{\lambda}Q_s^{\rho}
\cdot I_4^{[d+]^2,s}.
\end{eqnarray}
The contributions from~\eqref{fully} of the type $I_{4,i}^{[d+]^2,s}$
- after cancelling the first term on the right hand side
of~\eqref{I451} -
can also be written in a compact manner,
\begin{eqnarray}\label{3.361}
\frac{{\left( \right)}_5}{{s\choose s}_5} Q_s^{[\mu}Q_s^{\nu} J_4^{\lambda,s} Q_0^{s,\rho ]},
\end{eqnarray}
where the symmetrization in the tensor indices
is understood.
Introducing like in~\eqref{J4V}
\begin{eqnarray}\label{3.362}
J_3^{\mu, st}&=&\sum_{i=1}^{5} q_i^{\mu} I_{3,i}^{[d+]^2,st} \nonumber \\
&=&-Q_0^{st,\mu} I_3^{[d+],st}+
\sum_{u=1}^5 Q_u^{st,\mu} I_2^{[d+],stu},
\end{eqnarray}
we can finally write
\begin{eqnarray}\label{3.363}
I_4^{\mu \nu \lambda \rho, s}&&=Q_0^{s,\mu} Q_0^{s,\nu}Q_0^{s,\lambda}Q_0^{s,\rho} I_4^s+
3 \frac{{\left( \right)}_5^2}{{s\choose s}_5^2} Q_s^{\mu}Q_s^{\nu}Q_s^{\lambda}Q_s^{\rho}
\cdot I_4^{[d+]^2,s}
+\frac{{\left( \right)}_5}{{s\choose s}_5} Q_{s}^{[\mu}Q_{s}^{\nu}J_4^{\lambda,s } Q_0^{s,\rho ]} \nonumber \\
&&-\sum_{\substack{t=1 \\ t \neq s}}^{5}\left\{ Q_0^{st,\mu} Q_0^{st,\nu} Q_0^{st,\lambda} I_3^{st}+
\frac{1}{{st\choose st}_5} \left[
{t\choose t}_5 Q_s^{[\mu}Q_s^{t,\nu}J_3^{\lambda,st ]}+
{s\choose s}_5 Q_t^{[\mu}Q_t^{s,\nu}J_3^{\lambda,st ]} \right] \right\} Q_t^{s,\rho} \nonumber \\
&&-Q_0^{s,\rho}\sum_{i,j,k=1}^{5} q_i^{\mu} q_j^{\nu} q_k^{\lambda} R_{3,ijk}^{[d+]^3,s}
+\sum_{\substack{t=1 \\ t \neq s}}^{5} Q_t^{s,\rho}\sum_{i,j,k=1}^{5} q_i^{\mu} q_j^{\nu} q_k^{\lambda} R_{2,ijk}^{[d+]^3,st}.
\end{eqnarray}
The $ J_4^{\lambda,s}$ defined in~\eqref{J4V} occurs again, the
$R_{3,ijk}^{[d+]^3,s}$ is given in~\eqref{fullxr}, and $R_{2,ijk}^{[d+]^3,st}$ in~\eqref{fullx23}.
With the expression for $I_4^{\mu \nu \lambda \rho, s}$, free of the metric tensor, we complete the rewriting of $I_5^{\mu \nu \lambda\rho\sigma}$ with~\eqref{3.52}.
{It is remarkable that all the coefficients of the $g^{\mu \nu}$ terms of the four-point functions in
\eqref{tensor2}-\eqref{tensor4} completely cancel in a way that the remaining tensor coefficients are much simpler than the original ones.
This is achieved due to the symmetrization of the recurrence relations given in sect.~\ref{LaGra}
and would have been seen less easily with the ``standard'' recursions of app.~\ref{App}. }
{
The new representations for the tensors may be useful in several respects.
First of all we have here an extremely compact notation, due to the use of auxiliary vectors $Q_s^{\mu}$, which is not evident at the outset.
Further, the representations may be used for a completely independent programming and thus for stringent numerical cross checks.
The latter one is an important aspect because there are not too many opportunities for that in case of the 5-point and 6-point functions.
Finally, the auxiliary vectors $Q_s^{\mu}$ have some specific properties so that they may be used for simplifying manipulations with physical amplitudes, see sect.~\ref{Simplify}. }
\vspace{0.5cm}
\section{\label{5to4}Reduction of $5$-point tensor coefficients }
The purpose of this sect. is to express the $5$-point tensor coefficients
in terms of $4$-point tensor coefficients, which will be evaluated in Sec.
\ref{4togeneric} in such a way that also the case of inverse sub-Gram determinants
can be dealt with in an elegant manner. The difference to the former sect. is
the fact that we avoid all inverse Gram determinants, $1/{\left( \right)}_5$ as well as
$1/{\left( \right)}_4$. {In this case we have to keep the $g^{\mu \nu}$-terms.}
\subsection{\label{degree01}Scalar and vector integrals}
For the \emph{scalar} 5-point function $I_5$, we use the recurrence relation~\eqref{eq:RR2}:
\begin{eqnarray}
(d-4) \left( \right)_5 I_{5}^{[d+]}={0\choose 0}_5 I_{5}
-\sum_{s=1}^{5} {0\choose s}_5 I_{4}^{s} .
\label{scalargn}
\end{eqnarray}
The integral $I_{5}^{[d+]}$ is finite for $d=4$, and we get in this limit:
\begin{eqnarray}
I_{5} \equiv E =
\frac{1}{{0\choose 0}_5}\sum_{s=1}^{5} {0\choose s}_5 I_{4}^{s},
\label{scalar4p}
\end{eqnarray}
i.e. the scalar $5$-point function is expressed in the limit $d \to 4$
in terms of scalar $4$-point functions, which are obtained by
scratching in the five terms of the sum the $s^{th}$ scalar propagator,
respectively.
This was already derived in~\cite{Melrose:1965kb}, see Eq.~(6.1) there. See also~\cite{Petersson:1965zz}.
The tensor $n$-point integral
of rank $R=1$ in~(\ref{tensor1}) can be expressed by integrals $I_{n,i}^{[d+]}$,
and we obtain quite similarly
\begin{eqnarray}\label{i5vc1}
I_{n}^{\mu}&=&\sum_{i=1}^{n} \, q_i^{\mu} E_i,
\\\label{i5vc2}
E_i&\equiv&
-I_{n,i}^{[d+]}
\nonumber \\
&=& (d+1-n) \frac{{0\choose i}_n}{{0\choose 0}_n} I_{n}^{[d+]}-
\frac{1}{{0\choose 0}_n} \sum_{s=1}^{n} {0i\choose 0s}_n I_{n-1}^{s},
\label{first}
\end{eqnarray}
where again for $n=5$ in the limit $d \to 4$ the scalar integral $I_{5}^{[d+]}$
disappears:
\begin{eqnarray}\label{i5vc1a}
E_i &=& \sum_{s=1}^{5} E_i^s,
\\\label{i5vc1b}
E_i^s &=& -\frac{ {0i\choose 0s}_5}{{0\choose 0}_5} I_{4}^{s}.
\end{eqnarray}
\subsection{\label{central}The recursion formulae}
For the general case, we use as a starting point~\eqref{tensor5general}.
In order to solve this eqn. recursively, we multiply it with ${0\choose 0}_5$
\footnote{Throughout the present work we assume ${0\choose 0}_5 \ne 0$. In case of vanishing
{and/or small ${0\choose 0}_5$ see the discussion in Sec.~2.2. of~\cite{Fleischer:1999hq}). }}
and use the identity
\begin{eqnarray}
{0\choose 0}_5 {s\choose i}_5 = {0s\choose 0i}_5 \left( \right)_5 + {0\choose i}_5 {s\choose 0}_5 .
\label{product}
\end{eqnarray}
The first term on the right-hand-side can cancel already a Gram determinant $\left( \right)_5$, and the second
one transforms a vector $Q_s^{\mu}$ into a vector $Q_0^{\mu}$.
As a result, we get from~\eqref{tensor5general} the general form
\begin{eqnarray}
{0\choose 0}_5 I_5^{\mu_1 \dots \mu_{R-1} \mu} =
T^{\mu_1 \cdots \mu_{R-1}}
Q_0^{\mu} - \sum_{s=1}^{5}I_4^{\mu_1 \cdots \mu_{R-1},s }
{\bar{Q}}_s^{0,\mu} ,
\label{starter}
\end{eqnarray}
with:
\begin{eqnarray}
T^{\mu_1 \dots \mu_{R-1}}={0\choose 0}_5 I_5^{\mu_1 \dots \mu_{R-1} }-\sum_{s=1}^{5} {s\choose 0}_5
I_4^{\mu_1 \dots \mu_{R-1},s },
\label{tensorT}
\end{eqnarray}
and
\begin{eqnarray}\label{barQ}
{\bar{Q}}_s^{0,\mu}&=&\sum_{i=1}^{5} q_i^{\mu} {0s\choose 0i}_n,~~~ s=1, \dots, 5.
\end{eqnarray}
The barred vectors are free of the inverse Gram determinant $()_5$.
Evidently, the reduction of $I_4^{\mu_1 \cdots \mu_{R-1},s }$ is also free of $()_5$, and
we have to care only about the product $T^{\mu_1 \cdots \mu_{R-1}} Q_0^{\mu}$.
The following observation will prove to be useful:
$T^{\mu_1 \dots \mu_{R-1}}$ contains general tensor structures as given in~\eqref{tensor1}--\eqref{tensor5} with chords $q_i$ and the metric tensor.
In fact, when calculating the $5$-point tensor recursively, we keep
at this stage the $4$-point tensor as given there.
With the $5$-point tensor of rank $R=1$, given in~(\ref{first}) above, the recursion is started.
In order to cancel $1/()_5$, in each recursive step a term ${s\choose i}_5$
will be generated and summed over with the corresponding chord $q_i$.
We will apply to such terms the identity
\begin{eqnarray}
{s\choose i}_5 \frac{{0\choose j}_5} {\left( \right)_5}=
-{0i\choose sj}_5 + {s\choose 0}_5 \frac{{i\choose j}_5}{\left( \right)_5} .
\label{cancel}
\end{eqnarray}
The ratio ${0\choose j}_5 / ()_5$ comes from $Q_0^{\mu}$, see~\eqref{Qs} .
In the first term of the right-hand-side of~(\ref{cancel})
the Gram determinant $\left( \right)_5$ has cancelled and the second term
yields a $g^{\mu \nu}$ contribution according to~\eqref{gmunu}.
The metric tensors in the original $T^{\mu_1 \dots \mu_{R-1}}$
remain unchanged.
From the following examples the scheme will become more evident.
\subsection{\label{degree2}The tensor integral of rank $R=2$}
Equation~(\ref{starter}) reads for the tensor of rank $R=2$:
\begin{eqnarray}
{0\choose 0}_5 I_5^{\mu \nu} =
\left[{0\choose 0}_5 I_5^{\mu}-\sum_{s=1}^{5} {s\choose 0}_5
I_4^{\mu,s } \right] Q_0^{\nu} - \sum_{s=1}^{5}I_4^{\mu,s }
{\bar{Q}}_s^{0,\nu} .
\label{starter2}
\end{eqnarray}
The square bracket, a special case of~(\ref{tensorT}) for $R=2$, will be rewritten now.
We use~(\ref{tensor1}) and~(\ref{i5vc1b}) for $d=4$ and insert the reduction~(\ref{A511})
with $l=1$:
\begin{eqnarray}\label{starter2a}
T^{\mu} &=& \sum_{s=1}^5 T^{\mu,s},
\\\label{starter2b}
T^{\mu,s}&=&
\sum_{i=1}^{5} q_i^{\mu} \left\{{0\choose 0}_5 E_i^s + {s\choose 0}_5 I_{4,i}^{[d+],s} \right\}
\nonumber \\
&=&\sum_{i=1}^{5} q_i^{\mu} \left\{-{0s\choose 0i}_5 I_4^s+{s\choose 0}_5
\left[-{0s\choose is}_5 I_{4}^{s} +\sum_{t=1,t \ne s}^{5} {ts\choose is}_5 I_{3}^{st} \right]
\frac{1}{{s\choose s}_5} \right\} .
\label{CancEi}
\end{eqnarray}
Using further
\begin{eqnarray}
{s\choose 0}_5{0s\choose is}_5=
{s\choose i}_5 {0s\choose 0s}_5 -{s\choose s}_5 {0s\choose 0i}_5 ,
\label{Zauberei1}
\end{eqnarray}
we see the cancellation of $E_i^s$. Additionally, it is
\begin{eqnarray}
{s\choose 0}_5 {ts\choose is}_5=
{s\choose i}_5 {ts\choose 0s}_5 -{s\choose s}_5 {ts\choose 0i}_5.
\label{Zauberei3}
\end{eqnarray}
Here the ${s\choose s}_5$ term cancels and the remaining factor ${ts\choose 0i}_5$
is antisymmetric in $s,t$, yielding a vanishing contribution after summation over
$s,t$.
With~(\ref{A401}), reintroducing $I_4^{[d+],s}$, we obtain
\begin{eqnarray}\label{bra1a}
T^{\mu,s}&=&\sum_{i=1}^{5} q_i^{\mu} T_i^{s} ,
\\
T_i^{s}&=& - {s\choose i}_5 I_{4}^{[d+],s} .
\label{bra1}
\end{eqnarray}
Here we observe the first occurrence of a term ${s\choose i}_5$, as mentioned in sect.~\ref{central}.
Using~(\ref{cancel}) and the notation
\begin{eqnarray}
I_{4}^{\mu\, \nu\,}= \sum_{i,j=1}^{5} \, q_i^{\mu}\, q_j^{\nu} E_{ij} +
g^{\mu \nu} E_{00} ,
\label{final2}
\end{eqnarray}
we finally get, taking into account~(\ref{tensor1}) for $n=4$, the expressions for the tensor coefficients:
\begin{eqnarray}\label{E00}
E_{00}&\equiv&
\sum_{s=1}^5 E_{00}^s
\nonumber\\&=&
- \sum_{s=1}^5 \frac{1}{2} \frac{1}{{0\choose 0}_5} {s\choose 0}_5 I_4^{[d+],s},
\\
E_{ij} &\equiv& \sum_{s=1}^5 E_{ij}^s
\nonumber\\\label{Exy}
&=&
\sum_{s=1}^5 \frac{1}{{0\choose 0}_5} \left[{0i\choose sj}_5 I_4^{[d+],s}+
{0s\choose 0j}_5 I_{4,i}^{[d+],s} \right].
\label{Eij}
\end{eqnarray}
The functions $I_4^{[d+],s}, I_{4,i}^{[d+],s}$ will be further treated in sect.~\ref{4togeneric}.
A comparison shows that the tensor coefficients $E_{ij}$ given in Eqs. (3.10)--(3.12) and (A.22) of~\cite{Diakonidis:2008ij} are much more involved.
\subsection{\label{appT3}Reduction of integrals with rank $R=3$}
For the tensor integral of rank $R=3$, Eq.~(\ref{starter}) reads:
\begin{eqnarray}
{0\choose 0}_5 I_5^{\mu \nu \lambda} =
\left[{0\choose 0}_5 I_5^{\mu \nu}-\sum_{s=1}^{5} {s\choose 0}_5
I_4^{\mu \nu,s } \right] Q_0^{\lambda} - \sum_{s=1}^{5}I_4^{\mu \nu,s }
{\bar{Q}}_s^{0,\lambda}.
\label{starter3}
\end{eqnarray}
Investigating the square bracket, i.e. the tensor~(\ref{tensorT}) for $R=3$,
we see that the corresponding $g^{\mu \nu}$ term vanishes.
Indeed, from~(\ref{E00}) and~(\ref{tensor2}) we have:
\begin{eqnarray}
{0\choose 0}_5 E_{00}^s+\frac{1}{2}{s\choose 0}_5 I_4^{[d+],s} =0.
\label{rank3cancel}
\end{eqnarray}
This is interesting in view of our general scheme, which was described in
Sec.~\ref{central}:
Since there is no vector $q_i$ in this contribution, no ${s\choose i}_5$
is produced, and if we assume that no inverse Gram ${\left( \right)}_5$ should occur
in this case, the contribution must vanish.
Further, from~(\ref{Exy}),~(\ref{tensor2}) and~(\ref{A522}) we obtain
\begin{eqnarray} \label{rank3cancel0}
T^{\mu \nu} &=& \sum_{s=1}^5 T^{\mu \nu,s},
\\\label{rank3cancela}
T^{\mu \nu,s} &=& \sum_{i,j=1}^{5} q_i^{\mu} q_j^{\nu} T^s_{ij},
\end{eqnarray}
and
\begin{eqnarray}\label{rank3cancelb}
T^s_{ij}&=&
{0\choose 0}_5 E_{ij}^s -
{s\choose 0}_5 {\nu}_{ij} I_{4,ij}^{[d+]^2,s} \nonumber \\
&=&
{s\choose s}_5 \left[{0i\choose sj}_5 I_4^{[d+],s}+
{0s\choose 0j}_5 I_{4,i}^{[d+],s} \right]
\nonumber \\
&&-~{s\choose 0}_5 \left[
-{0s\choose js}_5 I_{4,i}^{[d+],s} +{is\choose js}_5 I_{4}^{[d+],s}+
\sum_{t=1,t \ne s, i}^{5} {ts\choose js}_5 I_{3,i}^{[d+],st}
\right]\frac{1}{{s\choose s}_5}.
\label{toobtain}
\end{eqnarray}
With~(\ref{Zauberei1}) and
\begin{eqnarray}\label{canc3r}
{s\choose 0}_5 {is\choose js}_5=
{s\choose i}_5 {0s\choose js}_5 +{s\choose s}_5 {0i\choose sj}_5 ,
\label{Zauberei2}
\end{eqnarray}
we see that the complete term $E_{ij}^s$ cancels. As above we use again~(\ref{Zauberei3})
and with the same arguments as before
we see that only ${s\choose i}_5$-type terms remain such that~(\ref{cancel})
can be used again to cancel the Gram determinant.
Before collecting all contributions, we would like to point out
that, after the above manipulations, the expressions are in general not
explicitly symmetric in their indices, although
the original integral \emph{is} symmetric in $\mu, \nu, \lambda$.
Consequently, our result must also be symmetric in the indices $i,j,k$,
however, after summation over $s$ and $t$.
For an explicit example see also the discussion after~(\ref{wantx}).
If there is no explicit symmetry before summation over $s$ and $t$ it may be useful to
symmetrize the result.
With this in mind, collecting all contributions, we have
\begin{eqnarray}\label{}
T_{ij}^{s}=&&
\left\{-\left[ {s\choose i}_5 {0s\choose js}_5+ {s\choose j}_5 {0s\choose is}_5 \right] I_{4}^{[d+],s}
\right.
\nonumber \\
&&\left. ~~+\sum_{t=1,t \ne s, i}^{5}\left[ {s\choose i}_5 {ts\choose js}_5+ {s\choose j}_5 {ts\choose is}_5 \right]
\frac{d-2}{2}I_{3}^{[d+],st}
\right\}\frac{1}{{s\choose s}_5}.
\label{squarebr}
\end{eqnarray}
To obtain this result, the vector integral $I_4^{\mu}$, represented by tensor coefficients $I_{4,i}^{[d+],s}$, and the vector integral $I_3^{\mu}$, represented by tensor coefficients $I_{3,i}^{[d+],st}$ in~(\ref{toobtain}),
have been reduced to scalar 2-,3-, and 4-point integrals in generic dimension $d$ by means of
(\ref{A511}) and~(\ref{A312}).
{The 2-point functions cancel here.}
Further we need the identity
\begin{eqnarray} \label{tuzeroa}
{s\choose s}_5 {0st\choose 0st}_5 = {0s\choose 0s}_5 {st\choose st}_5 - {ts\choose 0s}_5^2,
\end{eqnarray}
and in order to get rid of the vector indices in the $2$-point functions, we need the relation
\begin{eqnarray}
\left[{ts\choose 0s}_5 {ust\choose jst}_5-{ts\choose js}_5 {ust\choose 0st}_5\right]{s\choose s}_5
=\left[{ts\choose 0s}_5{us\choose js}_5-{ts\choose js}_5{us\choose 0s}_5\right]{st\choose st}_5,
\label{tuzero}
\end{eqnarray}
which shows that after cancellation of ${st\choose st}_5$, Eq.~(\ref{tuzero}) is antisymmetric in
$t$ and $u$ such that it can be effectively considered to vanish after summation over $t$ and $u$.
This
allows finally to introduce $I_{3}^{[d+],st}$ according to~(\ref{A301}) into~\eqref{squarebr}.
There is a further subtlety concerning~(\ref{squarebr}).
The ultraviolet (UV) divergency
\begin{eqnarray}\label{eq-uv2}
I_{3,\mathrm{UV}}^{[d+],st} = -\frac{1}{2 \varepsilon},
\end{eqnarray}
when combined with $\frac{d-2}{2}=1-\varepsilon$, yields a constant finite contribution $\frac{1}{2}$.\footnote{See the discussion after~(\ref{A301}).}
Since, however,
\begin{eqnarray}
\sum_{t=1}^{5}{ts\choose is}_5=0,
\label{zerosum}
\end{eqnarray}
this term does \emph{not} contribute and we can put $d=4$.
In that case~(\ref{squarebr}) reads
\begin{eqnarray}
T^s_{ij}={s\choose i}_5 I_{4,j }^{[d+]^2,s}+{s\choose j}_5 I_{4,i }^{[d+]^2,s} ,
\label{bra2}
\end{eqnarray}
to be compared with~(\ref{bra1}).
According to our general scheme, each $q_i$
generates a factor ${s\choose i}_5$, the further factor being a higher-dimensional
integral with index (indices) being the same as in the remaining chords.
In fact,
(\ref{bra1}) is a vector coefficient so that no additional index is available
and thus the higher-dimensional integral cannot carry an index.
We just mention that, due to~(\ref{zerosum}) and~(\ref{A511}), also the integral $I_{4,i }^{[d+]^2,s}$
is UV and infrared (IR-) finite.
Applying~(\ref{cancel}) in~(\ref{starter3}), we obtain products
${s\choose 0}_5 I_{4,i }^{[d+]^2,s}$, for which we can write, using
~(\ref{Zauberei1}),~(\ref{Zauberei3}),~(\ref{Zauberei2}) and~(\ref{A401}) and setting $d \rightarrow d+2$:
\begin{eqnarray}
{s\choose 0}_5 I_{4,i }^{[d+]^2,s}={0s\choose 0i}_5 I_{4 }^{[d+],s}-
{s\choose i}_5 (d-1) I_{4 }^{[d+]^2,s}.
\label{I4idx}
\end{eqnarray}
Collecting all the contributions, our final result for the tensor of rank $R=3$ can be written as follows:
\begin{eqnarray}
I_{4}^{\mu\, \nu\, \lambda}&&= \sum_{i,j,k=1}^{5} \, q_i^{\mu}\, q_j^{\nu} \, q_k^{\lambda}
E_{ijk}+\sum_{k=1}^5 g^{[\mu \nu} q_k^{\lambda]} E_{00k},
\label{Exyz0}
\end{eqnarray}
with
\begin{eqnarray}
\label{Exyz1}
E_{00j} &\equiv& \sum_{s=1}^5 E_{00j}^s
\nonumber\\
&=& \sum_{s=1}^5 \frac{1}{{0\choose 0}_5} \left[\frac{1}{2} {0s\choose 0j}_5 I_4^{[d+],s}- \frac{d-1}{3} {s\choose j}_5 I_4^{[d+]^2,s} \right] ,
\\
E_{ijk} &\equiv& \sum_{s=1}^5 E_{ijk}^s
\nonumber\\
&=&- \sum_{s=1}^5\frac{1}{{0\choose 0}_5} \left\{ \left[{0j\choose sk}_5 I_{4,i}^{[d+]^2,s}+
(i \leftrightarrow j)\right]+{0s\choose 0k}_5 {\nu}_{ij} I_{4,ij}^{[d+]^2,s} \right\}.
\label{Exyz2}
\end{eqnarray}
In ~(\ref{Exyz1}), we can put $d=4$ because, similar to the discussion by means of~(\ref{zerosum}), it is
\begin{eqnarray}
\sum_{s=1}^5 {s\choose j}_5=0.
\label{SumZero}
\end{eqnarray}
Another possibility to argue uses that, due to~(\ref{A401}), $l=2$, the $d-1$ cancels.
The rank $R=3$ tensors were also treated in~\cite{Diakonidis:2008ij}.
We proved there successfully the cancellation of $1/()_5$, although the corresponding formulae were quite a bit longer than here: see Eqs. (3.41)--(3.42), (3.30)--(3.33), (3.40) in~\cite{Diakonidis:2008ij}. For the rank $R>3$, however, the tensor reduction would become really
awkward with the older approach.
\subsection{\label{appT4}Reduction of integrals with rank $R=4$}
For the tensor integral of rank $R=4$, Eq. ~(\ref{starter}) reads:
\begin{eqnarray}
{0\choose 0}_5 I_5^{\mu \nu \lambda \rho} =
\left[{0\choose 0}_5 I_5^{\mu \nu \lambda}-\sum_{s=1}^{5} {s\choose 0}_5
I_4^{\mu \nu \lambda,s } \right] Q_0^{\rho} - \sum_{s=1}^{5}I_4^{\mu \nu \lambda,s }
{\bar{Q}}_s^{0,\rho}.
\label{starter4}
\end{eqnarray}
Here $I_5^{\mu \nu \lambda}$ is given in~(\ref{Exyz0}) to~(\ref{Exyz2}), $I_4^{\mu \nu \lambda,s}$
in~(\ref{tensor3}), taken at $n=4$. In a similar manner we decompose the square
bracket in~(\ref{starter4}):
\begin{eqnarray}\label{eq-t5mnl}
T_{5}^{\mu\, \nu\, \lambda}&=& \sum_{s=1}^5 T_{5}^{\mu\, \nu\, \lambda,s},
\\
T_{5}^{\mu\, \nu\, \lambda,s}&=& \sum_{i,j,k=1}^{5} \, q_i^{\mu}\, q_j^{\nu} \, q_k^{\lambda}
T_{ijk}^s+\sum_{i=1}^5 g^{[\mu \nu} q_i^{\lambda]} T_{00i}^s,
\label{T3}
\end{eqnarray}
according to which:
\begin{eqnarray}
T_{00i}^s&=&\left[{0\choose 0}_5 E_{00i}^s-\frac{1}{2}{s\choose 0}_5 I_{4,i}^{[d+]^2,s}\right]\nonumber\\
&=& \frac{1}{2} {s\choose i}_5 I_4^{[d+]^2,s}.
\label{T00k}
\end{eqnarray}
Obviously, the tensor coefficient $E_{00k}^s$ has been completely eliminated - as observed before in~\eqref{CancEi}.
As in~(\ref{Exyz0}), we now write
\begin{eqnarray}
I_{4}^{\mu\, \nu\, \lambda \rho}&&= \sum_{i,j,k,l=1}^{5} q_i^{\mu} q_j^{\nu} q_k^{\lambda} q_l^{\rho}
E_{ijkl}+\sum_{i,j=1}^5 g^{[\mu \nu} q_i^{\lambda} q_j^{\rho]} E_{00ij}+ g^{[\mu \nu} g^{\lambda \rho]}
E_{0000}.
\label{Ewxyz0}
\end{eqnarray}
Proceeding as before, using~(\ref{cancel}), from~(\ref{T00k}) we obtain:
\begin{eqnarray}\label{Ew0000a}
E_{0000} &=& \sum_{s=1}^5 E_{0000}^s,
\\
E_{0000}^s&=& \frac{1}{4} \frac{{s\choose 0}_5}{{0\choose 0}_5} I_4^{[d+]^2,s}.
\label{Ew0000}
\end{eqnarray}
We remark that since~(\ref{T00k}) is summed over $s$, all (constant UV-) divergent contributions
from $I_4^{[d+]^2,s}$ can be dropped;
see also the discussion at the end of sect.~\ref{appT3}.
In the next step we calculate $T_{ijk}^s$:
\begin{align}\label{eq-tsijk}
T_{ijk}^s
= &{0\choose 0}_5 E_{ijk}^s+{s\choose 0}_5 {\nu}_{ij}{\nu}_{ijk}I_{4,ijk}^{[d+]^3,s}
\notag
\\
= &\frac{1}{{s\choose s}_5} \left\{ {s\choose s}_5\left[-{0i\choose sk}_5 I_{4,j }^{[d+]^2,s}-{0j\choose sk}_5 I_{4,i }^{[d+]^2,s}-
{0s\choose 0k}_5 {\nu}_{ij}I_{4,ij}^{[d+]^2,s}\right]
\right.
\notag \\
& +~\left. {s\choose 0}_5 \left[{is\choose ks}_5 I_{4,j }^{[d+]^2,s}+{js\choose ks}_5
I_{4,i }^{[d+]^2,s}-{0s\choose ks}_5 {\nu}_{ij}I_{4,ij}^{[d+]^2,s}+\sum_{t=1}^5
{ts\choose ks}_5 {\nu}_{ij}I_{3,ij}^{[d+]^2,st} \right]
\right\}
\notag \\
= &\left\{
{s\choose i}_5 {0s\choose ks}_5 I_{4,j }^{[d+]^2,s}+{s\choose j}_5 {0s\choose ks}_5 I_{4,i }^{[d+]^2,s}-
{s\choose k}_5 {0s\choose 0s}_5 {\nu}_{ij}I_{4,ij }^{[d+]^2,s}
\right.
\notag \\
&+~ \left. {s\choose k}_5 \sum_{t=1}^5{ts\choose 0s}_5 {\nu}_{ij}I_{3,ij }^{[d+]^2,st}
\right\}\frac{1}{{s\choose s}_5},
\end{align}
where again~(\ref{Zauberei1}),~(\ref{Zauberei3}) and~(\ref{Zauberei2}) have been applied
and ${\nu}_{ijk}I_{4,ijk}^{[d+]^3,s}$ has been replaced by means of~(\ref{A533}).
Again we observe that the complete tensor of lower rank (here $E_{ijk}^s$) cancels.
After
further lengthy manipulations and subsequent symmetrization,
the following analogue of~(\ref{bra1}) and~(\ref{bra2}) can be verified:
\begin{eqnarray}
T^s_{ijk}=-
\left\{
{s\choose i}_5 {\nu}_{jk}I_{4,jk }^{[d+]^3,s}+{s\choose j}_5 {\nu}_{ik}I_{4,ik }^{[d+]^3,s}
+{s\choose k}_5 {\nu}_{ij}I_{4,ij }^{[d+]^3,s}
\right\}.
\label{bra3}
\end{eqnarray}
Using again~(\ref{cancel}), we can immediately write down the pure spatial components:
\begin{eqnarray}
E_{ijkl} &\equiv& \sum_{s=1}^5 E_{ijkl}^s
\nonumber\\
&=& \sum_{s=1}^5\frac{1}{{0\choose 0}_5} \left\{ \left[
{0k\choose sl}_5 {\nu}_{ij}I_{4,ij }^{[d+]^3,s}+(i \leftrightarrow k)+(j \leftrightarrow k)\right]
+{0s\choose 0l}_5{n}_{ijk} I_{4,ijk}^{[d+]^3,s} \right\}.
\label{Ewxyz3}
\end{eqnarray}
For the mixed terms $E^s_{00ij}$, i.e. those containing the metric tensor, we have contributions
from different origins.
From $T_{00k}^s$ (see~(\ref{T00k})) we get
\begin{eqnarray}\label{eq-non}
-\frac{1}{2}\sum_{k,l=1}^5 g^{[\mu \nu} q_k^{\lambda]} q_l^{\rho}{0k\choose sl}_5 I_{4 }^{[d+]^2,s}.
\end{eqnarray}
From~(\ref{bra3}), we get
\begin{eqnarray}\label{eq-non2}
&&-\sum_{i,j,k,l=1}^{5} q_i^{\mu} q_j^{\nu} q_k^{\lambda} q_l^{\rho}{s\choose 0}_5\left\{
\frac{{i\choose l}_5}{{\left( \right)}_5} {\nu}_{jk}I_{4,jk }^{[d+]^3,s}+
\frac{{j\choose l}_5}{{\left( \right)}_5} {\nu}_{ik}I_{4,ik }^{[d+]^3,s}+
\frac{{k\choose l}_5}{{\left( \right)}_5} {\nu}_{ij}I_{4,ij }^{[d+]^3,s}
\right\}.\nonumber \\
=&&-\frac{1}{2}{s\choose 0}_5
\left\{g^{\mu \rho}\sum_{j,k=1}^5 q_j^{\nu} q_k^{\lambda}{\nu}_{jk}I_{4,jk }^{[d+]^3,s}+
g^{\nu \rho}\sum_{i,k=1}^5 q_i^{\mu} q_k^{\lambda}{\nu}_{ik}I_{4,ik }^{[d+]^3,s}+
g^{\lambda\rho}\sum_{i,j=1}^5 q_i^{\mu} q_j^{\nu} {\nu}_{ij}I_{4,ij }^{[d+]^3,s} \right\}.\nonumber \\
\end{eqnarray}
Finally, there is a contribution from the second term of~(\ref{tensor3}):
\begin{eqnarray}\label{eq-n1}
-\frac{1}{2} \sum_{i,l=1}^{5} (\, g^{\mu \nu} \, q_i^{\lambda} \,+ g^{\mu \lambda} \, q_i^{\nu} \,+
\, g^{\nu \lambda} \, q_i^{\mu} \, ) q_l^{\rho} {0s\choose 0l}_5 I_{4,i}^{[d+]^2,s} .
\end{eqnarray}
Collecting these contributions without symmetrization we have:
\begin{eqnarray}\label{eq-n2}
E_{00ij} &=& \sum_{s=1}^5 E_{00ij}^s,
\\
E^s_{00ij}&=&-\frac{1}{4 {0\choose 0}_5} \left\{{0i\choose sj}_5 I_{4 }^{[d+]^2,s}+
{0s\choose 0j}_5 I_{4,i }^{[d+]^2,s}+{s\choose 0}_5{\nu}_{ij}I_{4,ij }^{[d+]^3,s} \right\} .
\label{E00ij}
\end{eqnarray}
A general comment is in order at this place:
The only UV divergent term in~(\ref{E00ij})
is $I_{4 }^{[d+]^2,s}$, which comes from~(\ref{T00k}).
We see, however, that due to
(\ref{SumZero}) this (constant) term does not contribute when $T^s_{00k}$ is summed over $s$.
Thus, the UV divergent part can be dropped in $I_{4 }^{[d+]^2,s}$
and as a consequence it does also not appear in~(\ref{E00ij}).
This, after all, is only an
expression of the fact that the original tensor integral under consideration is finite.
\subsection{\label{appT5}Reduction of integrals with rank $R=5$}
For the tensor integral of rank $R=5$, Eq.~(\ref{starter}) reads:
\begin{eqnarray}
{0\choose 0}_5 I_5^{\mu \nu \lambda \rho \sigma} =
\left[{0\choose 0}_5 I_5^{\mu \nu \lambda \rho}-\sum_{s=1}^{5} {s\choose 0}_5
I_4^{\mu \nu \lambda \rho,s } \right] Q_0^{\sigma} - \sum_{s=1}^{5}I_4^{\mu \nu \lambda \rho,s }
{\bar{Q}}_s^{0,\sigma}.
\label{starter5}
\end{eqnarray}
Writing the square bracket in~(\ref{starter5}) as
\begin{eqnarray}
\label{s5a}
T_{5}^{\mu\, \nu\, \lambda \rho}
&=& \sum_{s=1}^5 T_{5}^{\mu\, \nu\, \lambda \rho,s},
\\
T_{5}^{\mu\, \nu\, \lambda \rho,s}
&=&
\sum_{i,j,k,l=1}^{5} q_i^{\mu} q_j^{\nu} q_k^{\lambda} q_l^{\rho}
T_{ijkl}^s+\sum_{i,j=1}^5 g^{[\mu \nu} q_i^{\lambda} q_j^{\rho]} T_{00ij}^s+ g^{[\mu \nu} g^{\lambda \rho]}
T_{0000}^s,
\label{T4}
\end{eqnarray}
we, first of all, observe that the $g^{[\mu \nu} g^{\lambda \rho]}$ term vanishes.
Indeed,
from~(\ref{Ew0000}) and ~(\ref{tensor5}) we have:
\begin{eqnarray}
T^s_{0000}={0\choose 0}_5 E_{0000}^s-\frac{1}{4}{s\choose 0}_5 I_4^{[d+]^2,s} =0.
\label{rank5cancel}
\end{eqnarray}
Again we have a situation like in~(\ref{rank3cancel}): There is no vector
$q_i$, and no ${s\choose i}_5$ is produced. Thus, no inverse Gram determinant appears
since this term vanishes.
The next term, $T^s_{00ij}$, is calculated similarly as is scetched in
sect.~\ref{appT3} with a result generalizing~(\ref{T00k}):
\begin{eqnarray}
T_{00ij}^s &=& \left[{0\choose 0}_5 E_{00ij}^s+\frac{1}{2}{s\choose 0}_5
{\nu}_{ij} I_{4,ij}^{[d+]^3,s}\right]
\nonumber \\
&=&- \frac{d}{8}\left\{ {s\choose i}_5 I_{4,j}^{[d+]^3,s}+
{s\choose j}_5 I_{4,i}^{[d+]^3,s} \right\} .
\label{T00ij}
\end{eqnarray}
In contrary to the discussion at the end of sect. {\ref{appT4}, summing ~(\ref{T00ij}) over $s$,
the UV divergence of the integrals $I_{4,i}^{[d+]^3,s}$ does not drop out since
$I_{4,i}^{[d+]^3,s}=0$ for $s=i$. The corresponding divergence cancels in this case
against a divergence coming from the last term of~(\ref{tensor4}).
In analogy to~(\ref{bra1}),~(\ref{bra2}) and~(\ref{bra3}) we also have
\begin{eqnarray}
T^s_{ijkl}=
{s\choose i}_5 {n}_{jkl}I_{4,jkl}^{[d+]^4,s}+{s\choose j}_5 {n}_{ikl}I_{4,ikl}^{[d+]^4,s}
+ {s\choose k}_5 {n}_{ijl}I_{4,ijl}^{[d+]^4,s}+{s\choose l}_5 {n}_{ijk}I_{4,ijk}^{[d+]^4,s} .
\label{bra4}
\end{eqnarray}
It is interesting to note that a second chain of tensor coefficients has developed for
the square bracket tensor $T^{\mu_1 \dots \mu_{R-1}}$~(\ref{tensorT})
which follows the same rule when proceeding to
higher ranks, namely~(\ref{T00k}) and~(\ref{T00ij}) to be compared with the chain
(\ref{bra1}),~(\ref{bra2}),~(\ref{bra3}) and~(\ref{bra4}).
The complete tensor of rank $R=5$~\eqref{tensor5} now reads
\begin{eqnarray}\label{compl5a}
I_{5}^{\mu\, \nu\, \lambda \rho \sigma} &=& \sum_{s=1}^5 I_{5}^{\mu\, \nu\, \lambda \rho \sigma,s} ,
\\
I_{5}^{\mu\, \nu\, \lambda \rho \sigma,s}
&=& \sum_{i,j,k,l,m=1}^{5} q_i^{\mu} q_j^{\nu} q_k^{\lambda} q_l^{\rho} q_m^{\sigma} E_{ijklm}^s+
\sum_{i,j,k=1}^5 g^{[\mu \nu} q_i^{\lambda} q_j^{\rho} q_k^{\sigma]} E_{00ijk}^s
+
\sum_{i=1}^5
~g^{[\mu \nu} g^{\lambda \rho} q_i^{\sigma]} E_{0000i}^s .
\nonumber \\
\label{compl5}
\end{eqnarray}
Using~(\ref{barQ}) and~(\ref{cancel}), we obtain for the pure spatial part
\begin{eqnarray}
E_{ijklm}^s &=&
-\frac{1}{{0\choose 0}_5} \left\{ \left[
{0l\choose sm}_5 {n}_{ijk}I_{4,ijk}^{[d+]^4,s}+(i \leftrightarrow l)+(j \leftrightarrow l)+
(k \leftrightarrow l)\right]
+{0s\choose 0m}_5{n}_{ijkl} I_{4,ijkl}^{[d+]^4,s} \right\}.
\nonumber \\
\label{Ewxyz5}
\end{eqnarray}
Next we consider again the mixed terms and begin with $E_{00ijk}$.
From~(\ref{T00ij}) we have
\begin{eqnarray}
\frac{1}{2}\sum_{i,j,k=1}^5 g^{[\mu \nu} q_i^{\lambda} q_j^{\rho]} q_k^{\sigma}
\left[ {0i\choose sk}_5 I_{4,j}^{[d+]^3,s}+{0j\choose sk}_5 I_{4,i}^{[d+]^3,s} \right].
\label{Mix51}
\end{eqnarray}
From~(\ref{bra4}) we get:
\begin{eqnarray}
&&\sum_{i,j,k,l.m=1}^{5} q_i^{\mu} q_j^{\nu} q_k^{\lambda} q_l^{\rho} q_m^{\sigma}
{s\choose 0}_5\left\{
\frac{{i\choose m}_5}{{\left( \right)}_5} {n}_{jkl}I_{4,jkl}^{[d+]^4,s}+
\frac{{j\choose m}_5}{{\left( \right)}_5} {n}_{ikl}I_{4,ikl}^{[d+]^4,s}
\right. \nonumber \\
&&\left. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+ \frac{{k\choose m}_5}{{\left( \right)}_5} {n}_{ijl}I_{4,ijl}^{[d+]^4,s}+
\frac{{l\choose m}_5}{{\left( \right)}_5} {n}_{ijk}I_{4,ijk}^{[d+]^4,s}
\right\}\nonumber \\
=&&\frac{1}{2}{s\choose 0}_5
\left\{g^{\mu \sigma}\sum_{j,k,l=1}^5 q_j^{\nu} q_k^{\lambda} q_l^{\rho}{n}_{jkl}I_{4,jkl}^{[d+]^4,s}+
g^{\nu \sigma}\sum_{i,k,l=1}^5 q_i^{\mu} q_k^{\lambda} q_l^{\rho}{n}_{ikl}I_{4,ikl}^{[d+]^4,s}
~\right. \nonumber \\
&&\left. ~~~~~~~~~~~~~~~~~~~~
+ g^{\lambda\sigma}\sum_{i,j,l=1}^5 q_i^{\mu} q_j^{\nu} q_l^{\rho} {n}_{ijl}I_{4,ijl}^{[d+]^4,s}+
g^{\rho\sigma}\sum_{i,j,k=1}^5 q_i^{\mu} q_j^{\nu} q_k^{\lambda }{n}_{ijk}I_{4,ijk}^{[d+]^4,s}
\right\}.
\label{Mix52}
\end{eqnarray}
There is a $4$-point contribution from the second term of~(\ref{tensor4}):
\begin{eqnarray}
\frac{1}{2} \sum_{i,j,k=1}^{5} g^{[\mu \nu} \, q_i^{\lambda} \, q_j^{\rho]} q_k^{\sigma}
{0s\choose 0k}_5 {\nu}_{ij} I_{4,ij}^{[d+]^3,s} .
\label{Mix53}
\end{eqnarray}
In~(\ref{Mix51}) and ~(\ref{Mix53}) we have the tensor structure
$g^{[\mu \nu} q_i^{\lambda} q_j^{\rho]} q_k^{\sigma}$,
and in~(\ref{compl5}) the structure $g^{[\mu \nu} q_i^{\lambda} q_j^{\rho} q_k^{\sigma]}$.
In order to identify them, we can make
use of the fact that the tensor $I_5^{\mu \nu \lambda \rho \sigma}$ is symmetric in all indices
due to which we have only to count the number of terms in the structures to be compared:
e.g.
(\ref{G2V3}) contains ten terms, while~(\ref{G2V2}) contains six terms.
Thus, replacing
(\ref{G2V2}), multiplied by $q_k^{\sigma}$,
by~(\ref{G2V3}) we have to introduce a
factor taking care of the ratio
of the numbers of terms in each of them.
Similarly this applies for~(\ref{Mix52}).
In this way we obtain
\begin{eqnarray}
E^s_{00ijk}=\frac{1}{2 {0\choose 0}_5} \left\{
\frac{3}{5}\left[\frac{d}{4}{0i\choose sk}_5 I_{4,j}^{[d+]^3,s}+\frac{d}{4}{0j\choose sk}_5 I_{4,i}^{[d+]^3,s}+
{0s\choose 0k}_5 {\nu}_{ij}I_{4,ij}^{[d+]^3,s}\right]+
\frac{2}{5} {s\choose 0}_5{n}_{ijk}I_{4,ijk}^{[d+]^4,s} \right\} .
\nonumber \\
\label{E00ijk}
\end{eqnarray}
This concludes the $E_{00ijk}$ and
finally we have to collect the contributions to $E_{0000i}$. They come from
the last term of~(\ref{tensor4}) and~(\ref{T00ij}) with the result
\begin{eqnarray}
E^s_{0000i}=-\frac{1}{4 {0\choose 0}_5}\frac{1}{5} \left\{
{0s\choose 0i}_5 I_{4}^{[d+]^2,s}+d
{s\choose 0}_5 I_{4,i}^{[d+]^3,s} \right\}.
\label{E0000i}
\end{eqnarray}
}
We will not proceed here further, but by now it may be evident to the reader how to treat tensors of higher rank.
One remark, however, is in order: identifying the $I_{4,i \cdots}^{[d+]^l}$ in
\eqref{tensor1} -~\eqref{tensor5} as $4$-point tensor coefficients, the above tensor coefficients
of the $5$-point functions should be equivalent to (6.17) -(6.21) of~\cite{Denner:2005nn}.
Nevertheless, working with higher dimensional $4$-point functions and in particular
using the \emph{algebra of the signed minors} appears advantageous to us.
\section{\label{4togeneric}Calculation of higher-dimensional $4$-point functions}
In the foregoing sects. the $5$-point tensor coefficients $I_{5,i\cdots}^{[d+]^{R-r}}$ have been
rewritten in terms of $4$-point tensor coefficients.
The factor $1/()_5$ has been completely avoided.
In detail we have:
\begin{itemize}
\item Tensors with $R=2$:
\\The tensor coefficients
$E_{00}, E_{ij}$ are expressed by $I_{4}^{[d+],s}, I_{4,i}^{[d+],s} $.
\item Tensors with $R=3$:
\\ The tensor coefficients
$E_{00k}, E_{ijk}$ are expressed by $I_{4}^{[d+],s}, I_{4}^{[d+]^2,s}, I_{4,i}^{[d+]^2,s}, I_{4,ij}^{[d+]^2,s} $.
\item Tensors with $R=4$:
\\ The tensor coefficients
$E_{0000}, E_{00ij}, E_{ijkl}$ are expressed by $I_{4}^{[d+]^2,s}, I_{4,i}^{[d+]^2,s}, I_{4,ij}^{[d+]^3,s}, I_{4,ijk}^{[d+]^3,s}$.
\item Tensors with $R=5$:
\\ The tensor coefficients
$E_{0000i}, E_{00ijk}, E_{ijklm}$ are expressed by $I_{4}^{[d+]^2,s}$, $I_{4,i}^{[d+]^3,s}$, $I_{4,ij}^{[d+]^3,s}$, $I_{4,ijk}^{[d+]^4,s}$, $I_{4,ijkl}^{[d+]^4,s}$.
\end{itemize}
It is our goal to find a representation of these integrals which is suited
for the most problematic cases occurring in practical calculations,
namely for
vanishing sub-Gram determinants $()_4={s\choose s}_5 $.
In the numerics we will make use of opensource programs for the calculation of few \emph{master integrals}, chosen here to be the scalar 1-point to 4-point functions in generic dimension $d=4-\varepsilon$,
in standard
notation the integrals $A_0$, $B_0$, $C_0$ and $D_0$.
They are available from e.g. the LoopTools/FF package~\cite{Hahn:1998yk,vanOldenborgh:1990yc} or from the QCDloop/FF package~\cite{Ellis:2007qk,vanOldenborgh:1990yc}.
For this purpose,
we have to reduce dimension and indices of the above integrals.
This may be done
by recurrence relations~\eqref{eq:RR1} and~\eqref{eq:RR2},
given in detail in app.~\ref{App} . In each
recursion step an inverse power of $()_4$ is generated, which causes numerical
problems for small $()_4$ although the original integrals $ I_{4,i\cdots}^{[d+]^{l},s}$
are finite and well-behaved there.
We proceed in two steps.
In subsect.~\ref{highdim4}, an intermediate step, we manage to write the integrals
in the form:
\begin{eqnarray}\label{eq-i4andz}
I_{4,ij\cdots k}^{[d+]^l,s} \sim \frac{{0s\choose ks}_5}{{s\choose s}_5} \left[I_{4,ij\cdots}^{[d+]^{l-1},s}-Z_{4,ij\cdots}^{[d+]^{l-1},s}\right]+R_{4,ij\cdots k}.
\end{eqnarray}
Here $Z_{4,ij\cdots}^{[d+]^{l-1},s}$ is constructed such that
in the limit $()_4 \rightarrow 0$
it has the same value as $I_{4,ij\cdots}^{[d+]^{l-1},s}$, i.e.
the first term, the difference quotient
$\left[I_{4,ij\cdots}^{[d+]^{l-1},s}-Z_{4,ij\cdots}^{[d+]^{l-1},s}\right] / {s\choose s}_5$,
stays finite in this limit.
Further, dimension and indices
are reduced.
The remainder $R_{4,ij\cdots k}$ does no contain an inverse $()_4$.
In a second step, in subsect.~\ref{DiffQuo}, we will eliminate the inverse $()_4$
in the difference quotient.
\subsection{\label{highdim4}%
{Difference quotients with $1/()_4$}}
In this subsect. we derive optimized, compact expressions, where the appearance of
possible singular $1/()_4$-terms is reduced as much as possible.
We will treat the singular behaviour using the fact that the integrals are exactly known
in the limit ${\left( \right)}_n \rightarrow 0$.
In fact, if ${\left( \right)}_n=0$, due to~\eqref{eq:RR2}
the $n$-point integrals degenerate to integrals with scratched propagators:
\begin{eqnarray}
\lim_{()_n \rightarrow 0} I_{n,i\cdots}^{(d)} = \sum_{t=1}^n \frac{{t\choose 0}_n}{{0\choose 0}_n}~~ {\bf t^-}I_{n,i\cdots}^{(d)}.
\label{ItoZ0}
\end{eqnarray}
Accordingly we define objects
which converge in the limit $()_4 \rightarrow 0$ to the corresponding
tensor coefficients, taken in that limit:
\begin{eqnarray}\label{z4d}
Z_4^{(d),s}&=&\sum_{t=1}^5 \frac{{ts\choose 0s}_5}{{0s\choose 0s}_5} I_3^{(d),st},
\\
Z_{4,i}^{(d),s}&=&
\sum_{t=1,t \ne i}^5 \frac{{ts\choose 0s}_5}{{0s\choose 0s}_5} I_{3,i}^{(d),st}+
\frac{{is\choose 0s}_5}{{0s\choose 0s}_5} I_4^{(d),s} ,
\label{Z4id}
\\
{\nu}_{ij}Z_{4,ij}^{(d),s}&=&
\sum_{t=1,t \ne i,j}^5 \frac{{ts\choose 0s}_5}{{0s\choose 0s}_5} {\nu}_{ij}I_{3,ij}^{(d),st}+
\frac{{is\choose 0s}_5}{{0s\choose 0s}_5} I_{4,j}^{(d),s}+
\frac{{js\choose 0s}_5}{{0s\choose 0s}_5} I_{4,i}^{(d),s} ,
\label{Z4ij}
\\\label{Z4ijk}
{\nu}_{ij}{\nu}_{ijk} Z_{4,ijk}^{(d),s}&=&
\sum_{t=1,t \ne i,j,k}^5 \frac{{ts\choose 0s}_5}{{0s\choose 0s}_5}
{\nu}_{ij}{\nu}_{ijk} I_{3,ijk}^{(d),st}
\nonumber \\
&&+~
\frac{{ks\choose 0s}_5}{{0s\choose 0s}_5} {\nu}_{ij}I_{4,ij}^{(d),s}
+
\frac{{js\choose 0s}_5}{{0s\choose 0s}_5} {\nu}_{ik}I_{4,ik}^{(d),s}
+
\frac{{is\choose 0s}_5}{{0s\choose 0s}_5} {\nu}_{jk}I_{4,jk}^{(d),s} .
\end{eqnarray}
Eq.~(\ref{A401}) reads in this notation
\begin{eqnarray}
I_4^{(d+2),s}
&=&{0s\choose 0s}_5 \frac{1}{{s\choose s}_5}\left[I_4^{(d),s}
-
\sum_{t=1}^5 \frac{{ts\choose 0s}_5}{{0s\choose 0s}_5} I_3^{(d),st} \right]\frac{1}{d-3}
\nonumber \\
&\equiv&
{0s\choose 0s}_5 \frac{1}{{s\choose s}_5}\left[I_4^{(d),s}-Z_4^{(d),s}
\right]\frac{1}{d-3} ,
\label{I4d+}
\end{eqnarray}
such that indeed for ${s\choose s}_5 \rightarrow 0$ the $I_4^{(d+2),s}$ remains finite.
We need this relation with $d=4-2 \varepsilon$ and $d=[d+]=6-2\varepsilon$ for tensors of rank $R=2,3$ and rank $R=3,4,5$, respectively.
The next integral is $I_{4,i}^{(d),s}$.
The recursion of integrals with one index, $I_{4,i}^{(d),s}$, is~\eqref{A511}.
To rewrite~\eqref{A511} in a similar manner as~\eqref{I4d+}, we evaluate the
right hand side of~\eqref{A511}, replacing $I_4$ by $Z_4$ :
\begin{eqnarray}\label{i4deqa}
\frac{{0s\choose is}_5}{{s\choose s}_5} Z_4^{(d),s}
- \sum_{t=1}^5 \frac{{ts\choose is}_5}{{s\choose s}_5} I_3^{(d),st}
&=&
\sum_{t=1}^5 \frac{1}{{s\choose s}_5} \left[ {0s\choose is}_5 \frac{{ts\choose 0s}_5}{{0s\choose 0s}_5}
-{ts\choose is}_5\right] I_3^{(d),st}
\nonumber \\
&=& -~\frac{1}{{0s\choose 0s}_5}\sum_{t=1}^5 {0st\choose 0si}_5 I_3^{(d),st},
\end{eqnarray}
where the latter eqn. is due to
\begin{eqnarray}\label{i4deq}
{0s\choose is}_5 {ts\choose 0s}_5 - {0s\choose 0s}_5 {ts\choose is}_5
= -{s\choose s}_5 {0st\choose 0si}_5.
\end{eqnarray}
The factor ${1} / {s\choose s}_5$ has cancelled and we obtain the analogue to~\eqref{I4d+}:
\begin{eqnarray}
I_{4,i}^{(d+2),s}
&=&
-\frac{{0s\choose is}_5}{{s\choose s}_5} \left[I_4^{(d),s}-Z_4^{(d),s}\right]+
\frac{1}{{0s\choose 0s}_5}\sum_{t=1}^5 {0st\choose 0si}_5 I_3^{(d),st}
\nonumber \\
&=&
\frac{1}{{0s\choose 0s}_5}\left[-{0s\choose is}_5 (d-3) I_4^{(d+2),s}
+\sum_{t=1}^5 {0st\choose 0si}_5 I_3^{(d),st}\right].
\label{I4id+2}
\end{eqnarray}
In the first line of~(\ref{I4id+2}) we have introduced a difference quotient
which can in the next step be replaced, due to~(\ref{I4d+}), by $I_4^{(d+2),s}$, i.e.
an integral of the same dimension as the original integral on the left hand side.
In fact the second line of~(\ref{I4id+2})
is already our final result for this type of tensor coefficient.
We need this result for the 5-point tensors of
rank $R=2$
{with $d=4-2 \varepsilon$ (generic dimension), for $R=3, 4$ with $d=\left[d+\right]$, and for $R=5$ with $d=\left[d+\right]^2$.
Eqn.~\eqref{I4id+2} demonstrates our principle as described above.}
In the tensor integrals of higher rank more complicated difference quotients appear and
will be dealt with in the next subsect.~\ref{DiffQuo}.
The procedure of calculation is the same as before.
In order to obtain, e.g., ${\nu}_{ij} I_{4,ij}^{\left[d+\right]^2,s}$,
we calculate the right hand side of~\eqref{A522} (for $l=2$), replacing $I_{4,i}^{\left[d+\right]}$ by
$Z_{4,i}^{\left[d+\right]}$. Using again relation~\eqref{i4deq} we find
\begin{eqnarray}
{\nu}_{ij} I_{4,ij}^{[d+]^2,s}=-\frac{{0s\choose js}_5}{{s\choose s}_5} \left[I_{4,i}^{[d+],s}-Z_{4,i}^{[d+],s}\right]+
\frac{1}{{0s\choose 0s}_5}\left[{0si\choose 0sj}_5 I_4^{[d+],s}+\sum_{t=1,t \ne i}^5 {0st\choose 0sj}_5 I_{3,i}^{[d+],st}\right]. \nonumber \\
\label{d3}
\end{eqnarray}
Next we obtain from~(\ref{A533})
\begin{eqnarray}
{\nu}_{ij}{\nu}_{ijk} I_{4,ijk}^{[d+]^3,s}=
&&-\frac{{0s\choose ks}_5}{{s\choose s}_5} {\nu}_{ij}
\left[I_{4,ij}^{[d+]^2,s}-Z_{4,ij}^{[d+]^2,s}\right] \nonumber \\
&&+\frac{1}{{0s\choose 0s}_5}\left[{0si\choose 0sk}_5 I_{4,j}^{[d+]^2,s}+
{0sj\choose 0sk}_5 I_{4,i}^{[d+]^2,s}
+\sum_{t=1,t \ne i,j}^5 {0st\choose 0sk}_5 {\nu}_{ij} I_{3,ij}^{[d+]^2,st}\right], \nonumber \\
\label{I4ijkd+3}
\end{eqnarray}
and with~(\ref{A555})
\begin{eqnarray}
{n}_{ijkl}I_{4,ijkl}^{[d+]^4,s}
&=&-\frac{{0s\choose ls}_5}{{s\choose s}_5} {\nu}_{ij} {\nu}_{ijk}
\left[I_{4,ijk}^{[d+]^3,s}-Z_{4,ijk}^{[d+]^3,s}\right]
+\frac{1}{{0s\choose 0s}_5}\left[{0sk\choose 0sl}_5 {\nu}_{ij}I_{4,ij}^{[d+]^3,s}+ \right.
\nonumber \\
&&\left. {0sj\choose 0sl}_5 {\nu}_{ik}I_{4,ik}^{[d+]^3,s}+
{0si\choose 0sl}_5 {\nu}_{jk}I_{4,jk}^{[d+]^3,s}
+\sum_{t=1,t \ne i,j,k}^5 {0st\choose 0sl}_5 {\nu}_{ij} {\nu}_{ijk}I_{3,ijk}^{[d+]^3,st}\right]. \nonumber \\
\label{I4ijkld+4}
\end{eqnarray}
We now have collected all contributions to higher-dimensional integrals
with an ${s\choose s}_5$ in the denominator in
such a way that also the numerator vanishes for ${s\choose s}_5=0$,
see~(\ref{ItoZ0}). These results are only a rewriting of the
recurrence relations, but they make the finiteness of the integrals at ${s\choose s}_5=0$
manifest.
They will be a starting point to find a final representation, which
is truly optimal for kinematical points around $()_4=0$.
In the second line of
(\ref{I4id+2})
we observe that
there are no explicit inverse Gram determinants anymore.
In the following we will show that this also holds for integrals
with any number of indices.
\subsection{\label{DiffQuo}%
{Reduction of the difference quotients}}
In app.~\ref{App} we reproduce a list of the
recurrence relations needed for the evaluation
of the $5$-point functions. In fact, since all tensor coefficients
of the $5$-point functions have been reduced to higher-dimensional $4$-point functions,
we need only the recursions for the latter.
When applying these formulae to $5$-point functions,
we have to identify ${\left( \right)_4}={s\choose s}_5$ and $I_{4}^{[d+]}=I_{5}^{[d+],s}$, etc.
In the present sect. we will drop the index $s$ in the Gram determinant and in the upper indices of
the integrals.
We now discuss the
higher-dimensional $4$-point functions needed
for the different tensor ranks of the $5$-point functions.
For the tensor of rank $R=2$~(\ref{Exy}) we need $I_4^{[d+]}$ and $I_{4,i}^{[d+]}$ given
in~(\ref{I4d+}) and~(\ref{I4id+2}).
In the spirit of our approach they are already
in the final form.
For the tensor of rank $R=3$,~(\ref{Exyz1}) and~(\ref{Exyz2}), we further need
$I_4^{[d+]^2}$, $I_{4,i}^{[d+]^2}$ and ${\nu}_{ij} I_{4,ij}^{[d+]^2}$.
These are given in
(\ref{I4d+}),~(\ref{I4id+2}) and~(\ref{d3}).
In fact, the first two are already in the
final form, while in the last one a new difference quotient appears.
Our general approach to cancel Gram determinants is, first of all, to use the recurrence
relations "backward", i.e. to express a $4$-point function of dimension $d$ by one of
dimension $d+2$, multiplied by a Gram determinant, plus a sum over $3$-point functions.
The factorized Gram determinant can be cancelled and for the collected sum over
$3$-point functions the algebra of Cayley determinants allows to combine them such that
again the Gram determinant factorizes and can be cancelled.
With the notation
$\left( \right) \equiv {\left( \right)}_4$ we obtain
\begin{eqnarray}
\frac{{0\choose 0}}{\left( \right)} \left[I_{4,i}^{[d+]}-Z_{4,i}^{[d+]}\right]=
-(d-2)\left[\frac{{0\choose i}}{{0\choose 0}}(d-1)I_4^{[d+]^2}-\frac{1}{{0\choose 0}}
\sum_{t=1}^4 {0t\choose 0i}I_3^{[d+],t}\right] ,
\label{DQ1}
\end{eqnarray}
and from~(\ref{d3})
\begin{eqnarray}
{\nu}_{ij} I_{4,ij}^{[d+]^2}= && ~~\frac{{0\choose i}}{{0\choose 0}}\frac{{0\choose j}}{{0\choose 0}}
(d-2)(d-1)I_4^{[d+]^2}+\frac{{0i\choose 0j}}{{0\choose 0}}I_{4}^{[d+]}\nonumber \\
&&-\frac{{0\choose j}}{{0\choose 0}}\frac{d-2}{{0\choose 0}}\sum_{t=1}^4 {0t\choose 0i}I_3^{[d+],t} ~~~+
\frac{1}{{0\choose 0}}\sum_{t=1}^4 {0t\choose 0j}I_{3,i}^{[d+],t} .
\label{want1}
\end{eqnarray}
This is the form we wanted to obtain.
The higher-dimensional $3$-point functions can
be calculated by means of the recurrence relations given in app.~\ref{sub3}, reducing them to
scalar functions in generic dimension.
If $\left( \right)$ is not small, the same applies for the higher-dimensional
$4$-point functions, in particular ~(\ref{A401});
otherwise
we will use~(\ref{z4d}) by setting up an expansion in the small
Gram determinant.
This will be done in subsect.~\ref{Gram}.
It is worth mentioning that we can deal with $3$-point functions like $I_{3,i}^{(d),t}$
in ~(\ref{want1}) in the same manner as we dealt with the $4$-point functions:
\begin{eqnarray}
I_{3,i}^{[d+],t}=-\frac{{0t\choose it}}{{0t\choose 0t}}(d-2) I_{3}^{[d+],t}+\frac{1}{{0t\choose 0t}}
\sum_{u=1}^4 {0tu\choose 0ti}I_2^{tu} ,
\label{indices}
\end{eqnarray}
to be compared with~\eqref{I4id+2}.
This allows to handle the $3$-point functions in case ${t\choose t}=0$, for which
case~(\ref{A301}) does not work - expanding in small ${t\choose t}$ if needed by the
use of~(\ref{ItoZ0}).
The ${0t\choose 0t}$, however, vanishes for an infrared $3$-point
function - thus we have to assume here that ${t\choose t}$ and ${0t\choose 0t}$ don't vanish
simultaneously in order to be able to apply at least one of them; this is also implicitly assumed for the case of the $4$-point functions.
Exploiting this approach systematically, it can be achieved in general
that the indices are carried, like in~\eqref{want1} and~\eqref{indices}, only by the Cayley determinants,
multiplied by scalar integrals in higher dimension. This property might become
useful for further analytical evaluation of the original Feynman diagrams, in performing partial sums over indices explicitly where needed.
We point out that due to the powers of $d$ in front of higher-dimensional integrals we have
to take into account finite rational contributions arising from the divergencies of the integrals;
see
app.~\ref{Bpp} for a list of examples.
For the tensors of rank $R=4$(see~(\ref{Ew0000}),~(\ref{Ewxyz3}) and~(\ref{E00ij}))
we further need $I_{4,ij}^{[d+]^3}$ and ${\nu}_{ij}{\nu}_{ijk} I_{4,ijk}^{[d+]^3}$.
The tensor with two indices was treated in~(\ref{want1}) and we have only to shift
the dimension: $d \rightarrow d+2$.
Much more involved is now the calculation of $I_{4,ijk}^{[d+]^3}$. The crucial point of
our approach is to obtain here an expression for the following difference quotient with
the envisaged properties:
\begin{eqnarray}
\frac{{0\choose 0}}{\left( \right)} {\nu}_{ij} \left[I_{4,ij}^{[d+]^2}-Z_{4,ij}^{[d+]^2}\right]=&&
d\frac{{0\choose i}}{{0\choose 0}}\frac{{0\choose j}}{{0\choose 0}}(d-3)(d+1)I_4^{[d+]^3}+
(d-3)\frac{1}{{0\choose 0}}{0i\choose 0j}I_4^{[d+]^2} + 2 {\nu}_{ij}I_{4,ij}^{[d+]^3} \nonumber \\
&&+\frac{1}{\left( \right)}
\left\{
d\frac{{0\choose i}}{{0\choose 0}}\frac{{0\choose j}}{{0\choose 0}}(d-3)
\sum_{t=1}^4 {t\choose 0}I_3^{[d+]^2,t}+\frac{1}{{0\choose 0}}{0i\choose 0j}\sum_{t=1}^4 {t\choose 0}I_3^{[d+],t} \right. \nonumber \\
&&- \left. (d-2)\frac{{0\choose j}}{{0\choose 0}} \sum_{t=1}^4 {0t\choose 0i}I_3^{[d+],t}+
\sum_{t=1}^4 {0t\choose 0j}I_{3,i}^{[d+],t} \right. \nonumber \\
&&-\sum_{t=1}^4 {t\choose 0}{\nu}_{ij} I_{3,ij}^{[d+]^2,t}
- \sum_{t=1}^4 {t\choose i}I_{3,j}^{[d+]^2,t}-
\left. \sum_{t=1}^4 {t\choose j}I_{3,i}^{[d+]^2,t} \right\} .
\label{DQ3}
\end{eqnarray}
By construction, the $4$-point functions have no explicit inverse Gram determinant anymore. It is interesting
that the formerly calculated $I_{4,ij}^{[d+]^3}$ (see~(\ref{want1})) enters here as a whole.
The remaining task is now to show that in the sum of $3$-point functions in~(\ref{DQ3}) a
Gram determinant $\left( \right)$ factorizes and thus cancels its overall factor $1/\left( \right)$. Indeed this is so.
After a tremendeous amount of cancellations one gets the result:
\begin{eqnarray}
\frac{{0\choose 0}}{\left( \right)} {\nu}_{ij} \left[I_{4,ij}^{[d+]^2}-Z_{4,ij}^{[d+]^2}\right]&&=
\frac{{0\choose i}}{{0\choose 0}}\frac{{0\choose j}}{{0\choose 0}}(d-1)d(d+1)I_4^{[d+]^3}+
(d-1)\frac{1}{{0\choose 0}}{0i\choose 0j}I_4^{[d+]^2}
\nonumber \\
&&-~\frac{(d-1)d}{{0\choose 0}}\frac{{0\choose j}}{{0\choose 0}}\sum_{t=1}^4 {0t\choose 0i}I_{3}^{[d+]^2,t}+
\frac{d-1}{{0\choose 0}}\sum_{t=1}^4 {0t\choose 0j}I_{3,i}^{[d+]^2,t}.
\label{DQ3simp}
\end{eqnarray}
The ${\nu}_{ij} I_{4,ij}^{[d+]^3}$ from~(\ref{want1}) ($d \rightarrow d+2$)
has now been explicitly inserted since it has the same structure as the final result.
Adding all contributions, using~(\ref{I4id+2}), we finally have
\begin{eqnarray}
{\nu}_{ij}{\nu}_{ijk} I_{4,ijk}^{[d+]^3}=&&
-\frac{{0\choose i}}{{0\choose 0}}
\frac{{0\choose j}}{{0\choose 0}}\frac{{0\choose k}}{{0\choose 0}}(d-1)d(d+1)I_4^{[d+]^3}
-\frac{{0i\choose 0j}{0\choose k}+{0i\choose 0k}{0\choose j}+{0j\choose 0k}{0\choose i}}
{{0\choose 0}^2}(d-1)I_4^{[d+]^2} \nonumber \\
&&+\frac{{0\choose j}}{{0\choose 0}}\frac{{0\choose k}}{{0\choose 0}}\frac{(d-1)d}{{0\choose 0}}
\sum_{t=1}^4 {0t\choose 0i}I_{3}^{[d+]^2,t}
-\frac{{0\choose k}}{{0\choose 0}}\frac{d-1}{{0\choose 0}}
\sum_{t=1}^4 {0t\choose 0j}I_{3,i}^{[d+]^2,t}\nonumber \\
&&+\sum_{t=1}^4 \frac{{0i\choose 0k}{0t\choose 0j}+
{0j\choose 0k}{0t\choose 0i}}{{0\choose 0}^2}I_{3}^{[d+],t}
+\frac{1}{{0\choose 0}}
\sum_{t=1,t \ne i,j}^4 {0t\choose 0k} {\nu}_{ij} I_{3,ij}^{[d+]^2,t}.
\label{fulld3}
\end{eqnarray}
It is obvious where the various contributions come from: Those being proportional to $-{0\choose k}/{0\choose 0}$
come from~(\ref{DQ3simp}), while all the others come from the second part of~(\ref{I4ijkd+3}).
We could indeed list these contributions separately without inserting the second part explicitly.
However, in this way the coefficient of $I_4^{[d+]^2}$ gets contributions from both terms which
combine to make the resulting coefficient explicitly symmetric in all indices.
The symmetry of the $3$-point functions is not so easily seen. Neverheless, numerically, it might
be faster not to combine these terms but just to retain what had been derived.
Also here, as discussed in~(\ref{indices}), we can replace the tensor-
$3$-point functions:
\begin{eqnarray}
{\nu}_{ij}I_{3,ij}^{[d+]^2,t}=&&~~~ \frac{{0t\choose it}}{{0t\choose 0t}}\frac{{0t\choose jt}}{{0t\choose 0t}}
(d-1)d~I_3^{[d+]^2,t}~~~~~~+\frac{1}{{0t\choose 0t}}{0ti\choose 0tj}I_{3}^{[d+],t} \nonumber \\
&&-\frac{{0t\choose jt}}{{0t\choose 0t}}\frac{d-1}{{0t\choose 0t}}\sum_{u=1}^4 {0tu\choose 0ti}I_2^{[d+],tu}
+\frac{1}{{0t\choose 0t}}\sum_{u=1}^4 {0tu\choose 0tj}I_{2,i}^{[d+],tu}.
\label{indiyes}
\end{eqnarray}
At this point, we observe a simple rule
of how to obtain~(\ref{fulld3}): replace in~(\ref{want1}) $d \rightarrow d+2$ and multiply with
$ -(d-1)\frac{{0\choose k}}{{0\choose 0}}$, where $(d-1)$ is to be chosen such that all factors $(d+i)$
increase by steps of 1 (see also~(\ref{want1}) ). In this manner we increase simulaneously
the dimension and the number of indices. Then one has to add the second part of~(\ref{I4ijkd+3}).
Finally, for the tensor of rank $R=5$ we need $I_{4,ijkl}^{[d+]^4}$. Due to the above rule
we need not again perform a complicated calculation, rather we apply the rule
as proceeding from~(\ref{want1}) to~(\ref{fulld3}), increasing simultaneously
dimension and number of indices:
we have to shift in~(\ref{fulld3}) $d \rightarrow d+2$, muliply with $-d\frac{{0\choose l}}{{0\choose 0}}$
and add the second part of~(\ref{I4ijkld+4}).
We obtain:
\begin{eqnarray}
&&{\nu}_{ij}{\nu}_{ijk} {\nu}_{ijkl} I_{4,ijkl}^{[d+]^4}=
\frac{{0\choose i}}{{0\choose 0}}
\frac{{0\choose j}}{{0\choose 0}}\frac{{0\choose k}}{{0\choose 0}}\frac{{0\choose l}}{{0\choose 0}}
d(d+1)(d+2)(d+3)I_4^{[d+]^4} \nonumber \\
&&+\frac{{0i\choose 0j}{0\choose k}{0\choose l}+{0i\choose 0k}{0\choose j}{0\choose l}+{0j\choose 0k}{0\choose i}{0\choose l}+
{0i\choose 0l}{0\choose j}{0\choose k}+{0j\choose 0l}{0\choose i}{0\choose k}+{0k\choose 0l}{0\choose i}{0\choose j}}
{{0\choose 0}^3} d(d+1)I_4^{[d+]^3}\nonumber \\
&&+\frac{{0i\choose 0l}{0j\choose 0k}+{0j\choose 0l}{0i\choose 0k}+{0k\choose 0l}{0i\choose 0j}}
{{0\choose 0}^2}I_4^{[d+]^2} \nonumber \\
&&-\frac{{0\choose j}}{{0\choose 0}}\frac{{0\choose k}}{{0\choose 0}}\frac{{0\choose l}}{{0\choose 0}}
\frac{d(d+1)(d+2)}{{0\choose 0}}\sum_{t=1}^4 {0t\choose 0i}I_{3}^{[d+]^3,t}
+\frac{{0\choose k}}{{0\choose 0}}\frac{{0\choose l}}{{0\choose 0}}\frac{d(d+1)}{{0\choose 0}}
\sum_{t=1}^4 {0t\choose 0j}I_{3,i}^{[d+]^3,t}\nonumber \\
&&-\frac{d}{{0\choose 0}^3}\sum_{t=1}^4
\left[{0i\choose 0k}{0t\choose 0j}+{0j\choose 0k}{0t\choose 0i}\right]{0\choose l}I_{3}^{[d+]^2,t}\nonumber \\
&&-\frac{d}{{0\choose 0}^3}\sum_{t=1}^4 \left[{0j\choose 0l}{0t\choose 0i}{0\choose k}+
{0i\choose 0l}{0t\choose 0j}{0\choose k}+{0k\choose 0l}{0t\choose 0i}{0\choose j}\right]I_{3}^{[d+]^2,t} \nonumber \\
&&+\frac{1}{{0\choose 0}^2}\sum_{t=1}^4 \left[{0j\choose 0l}{0t\choose 0k}I_{3,i}^{[d+]^2,t}+
{0i\choose 0l}{0t\choose 0k}I_{3,j}^{[d+]^2,t}+{0k\choose 0l}{0t\choose 0j}I_{3,i}^{[d+]^2,t}\right] \nonumber \\
&&-\frac{{0\choose l}}{{0\choose 0}}\frac{d}{{0\choose 0}}\sum_{t=1}^4{0t\choose 0k}{\nu}_{ij}I_{3,ij}^{[d+]^3,t}
+\frac{1}{{0\choose 0}}
\sum_{t=1,t \ne i,j}^4 {0t\choose 0l} {\nu}_{ij} {\nu}_{ijk} I_{3,ijk}^{[d+]^3,t}.
\label{fulld4}
\end{eqnarray}
Again we, first of all, mention that the $3$-point function $I_{3,ijk}^{[d+]^3,t}$
appearing here can as well be calculated like~(\ref{indices}) and~(\ref{indiyes}),
essentially by taking over the $4$-point result, shifting $d \rightarrow d+1$:
\begin{eqnarray}
{\nu}_{ij}{\nu}_{ijk} I_{3,ijk}^{[d+]^3,t}=&&
-\frac{{0t\choose it}}{{0t\choose 0t}}
\frac{{0t\choose jt}}{{0t\choose 0t}}\frac{{0t\choose kt}}{{0t\choose 0t}}d(d+1)(d+2)I_3^{[d+]^3,t}
-\frac{{0ti\choose 0tj}{0t\choose kt}+{0ti\choose 0tk}{0t\choose jt}+{0tj\choose 0tk}{0t\choose it}}
{{0t\choose 0t}^2}dI_3^{[d+]^2,t} \nonumber \\
&&+\frac{{0t\choose jt}}{{0t\choose 0t}}\frac{{0t\choose kt}}{{0t\choose 0t}}\frac{d(d+1)}{{0t\choose 0t}}
\sum_{u=1}^4 {0tu\choose 0ti}I_{2}^{[d+]^2,tu}
-\frac{{0t\choose kt}}{{0t\choose 0t}}\frac{d}{{0t\choose 0t}}
\sum_{u=1}^4 {0tu\choose 0tj}I_{2,i}^{[d+]^2,tu}\nonumber \\
&&+\sum_{u=1}^4 \frac{{0ti\choose 0tk}{0tu\choose 0tj}+
{0tj\choose 0tk}{0tu\choose 0ti}}{{0t\choose 0t}^2}I_{2}^{[d+],tu}
+\frac{1}{{0t\choose 0t}}
\sum_{u=1,t \ne i,j}^4 {0tu\choose 0tk} {\nu}_{ij} I_{2,ij}^{[d+]^2,tu}.
\label{fullt3}
\end{eqnarray}
In fact, having a closer look at our $4$-point tensor coefficients, we observe that
apart from the higher
dimensional $4$-point functions, all other terms are $3$-point tensor coefficients,
occasionally of higher tensor rank.
Eq.~\ref{fulld4} has been obtained by an educated guess.
Indeed, a step by step derivation would have been extremely tedious if one
would have had the courage at all to try the calculation.
Of course one needs a verification by numerical checks:
among others we found for non-exceptional Gram determinants an agreement
with LoopTools of typically more than ten decimals;
see sect.~\ref{Num} for some details.
Concerning these non-exceptional Gram determinants, we just mention that we
evaluate~(\ref{want1}),~(\ref{fulld3}) and~(\ref{fulld4}) by means of the recurrence relations
of app.~\ref{App}. For further details see Sec.~\ref{LaGra}.
\subsection{\label{Gram}%
Expansion of $I_4^{[d+]^L}$ for small Gram determinants}
The tensor coefficients in~(\ref{tensor1}) to~(\ref{tensor4}), in particular~\eqref{want1},
(\ref{fulld3}) and~(\ref{fulld4}),
have been expressed in terms of $4$-point functions in higher dimensions:
$I_4^{[d+]},I_4^{[d+]^2},I_4^{[d+]^3}$ and $I_4^{[d+]^4}$.
In our approach only these integrals can cause problems for small Gram determinants and
therefore, finding a special approach for their calculation, will finalize the problem
of calculating the $4$-poin tensor coefficients. We start from
the fact that for exactly vanishing Gram determinants~(\ref{z4d}) yields a finite value for $4$-point functions of any dimension and, taking into account higher orders, we set up an infinite
series in terms of powers of the Gram determinant:\footnote{The series is not a Taylor series {because the expansion coefficients are not the derivatives of} $I_4^{[d+]^L}$ at $()=0$.}
\begin{eqnarray}
I_4^{[d+]^L}=\sum_{j=0}^{\infty} r^j ~~ I_{4,j}^L,
~~~L=1,\cdots,4,
\label{wish}
\end{eqnarray}
where the coefficients $I_{4,j}^L$ have to be determined and
\begin{eqnarray}\label{rr}
r = \frac{{\left( \right)}}{{0\choose 0}} .
\end{eqnarray}
We use the observation that the recurrence relation with shift of dimension~(\ref{eq:RR2}) (see also~\ref{A401}) for the scalar higher-dimensional $4$-point
functions can be written as follows:
\begin{eqnarray}
\label{Z4d1l}
I_4^{[d+]^l}&=&Z_4^{[d+]^l}+\frac{{\left( \right)}}{{0\choose 0}}\left[(2 l +1) -2 \varepsilon \right]I_4^{[d+]^{(l+1)}},
~~~l=1,\cdots
\end{eqnarray}
where we re-wrote~(\ref{z4d}) as follows:
\begin{eqnarray}
Z_4^{[d+]^l}=\frac{1}{{0\choose 0}}\sum_{t=1}^4 {t\choose 0} I_3^{[d+]^l,t},
~~~l=1,\cdots
\label{Z4dla}
\end{eqnarray}
From now on we assume $l>0$, so that the integrals are infrared finite and the leading singularity in $\varepsilon$ is at most of the order $1/\varepsilon$.
Integrals in generic dimension ($l=0$) will be discussed at the end of the section.
We treat the finite and divergent parts separately:
\begin{eqnarray}
\label{Z4dl}
{I_4^{[d+]^l} }&=& {F_4^{[d+]^l} + \frac{D_4^{[d+]^l}}{\varepsilon} + {\cal O}(\varepsilon^2) ,}
\\
\label{Z4dlb}
{Z_4^{[d+]^l}} &=& {Z_{4F}^l + \frac{Z_{4D}^l}{\varepsilon} + {\cal O}(\varepsilon^2)}.
\end{eqnarray}
The first few iterations of~\eqref{Z4d1l} are
\begin{eqnarray}
\label{Z4d3l}
I_4^{[d+]^l}
&=&
Z_4^{[d+]^l}+\frac{{\left( \right)}}{{0\choose 0}}
c_{l+1}
I_4^{[d+]^{(l+1)}}
\\
&=&
Z_4^{[d+]^l}
+
\frac{{\left( \right)}}{{0\choose 0}}
c_{l+1}
\left\{
Z_4^{[d+]^{l+1}}+\frac{{\left( \right)}}{{0\choose 0}}
c_{l+2}
I_4^{[d+]^{(l+2)}}
\right\}
\nonumber \\
&=&
Z_4^{[d+]^l}
+
\frac{{\left( \right)}}{{0\choose 0}}
c_{l+1}
Z_4^{[d+]^{l+1}}
+
\frac{{\left( \right)^2}}{{0\choose 0}^2}
c_{l+1} c_{l+2
\left\{
Z_4^{[d+]^{l+2}}+\frac{{\left( \right)}}{{0\choose 0}}
c_{l+3}
I_4^{[d+]^{(l+3)}}
\right\}
\nonumber \\
&=& \cdots ~=~
Z_4^{[d+]^l}
+
\sum_{i=1}^{\infty} \frac{{\left( \right)^i}}{{0\choose 0}^i} \left[ \prod_{j=1}^{i} c_{l+j}\right] ~~Z_4^{[d+]^{l+i}} ,
\end{eqnarray}
with
\begin{eqnarray}
c_{l+j} &=& 2(l+j)-1-2\varepsilon .
\end{eqnarray}
The 4-point functions are expressed in terms of an infinite power series in $\left( \right)/{0\choose 0}$ with higher-dimensional 3-point functions in the expansion coeffients.
For the finite and divergent part {of~(\ref{Z4dl})} we get to lowest order
\begin{eqnarray}
\label{finite}
F_4^{[d+]^l}&=&Z_{4F}^l+\frac{{\left( \right)}}{{0\choose 0}} \left[(2 l +1) F_4^{[d+]^{(l+1)}}
-2 D_4^{[d+]^{(l+1)}} \right] ,
\\
\label{divergent}
D_4^{[d+]^l}&=&Z_{4D}^l+\frac{{\left( \right)}}{{0\choose 0}}\left[(2 l +1)D_4^{[d+]^{(l+1)}}\right].
\end{eqnarray}
Now the second part in ~(\ref{finite}) is proportional to ${\left( \right)}$ and
can be considered as a correction term for small Gram determinants, where also a
proper approximation for $F_4^{[d+]^{(l+1)}}$ has to be chosen.
{There are two simple choices.
One has just to set in~\eqref{finite} $F_4^{[d+]^{(l+1)}}=Z_{4F}^{l+1}$ and neglect the unknown higher order terms, or one selects a close kinematical point with $()=0$ and uses $F_4^{[d+]^{(l+1)}}=Z_{4F}^{l+1}|_{()=0}$.
We come back to these two alternatives later.
}
To calculate higher
order corrections, we perform now iterations.
Defining the correction term as
\begin{eqnarray}
{\delta Z_{4F,i}^l=\frac{{\left( \right)}}{{0\choose 0}}\left[(2 l +1) Z_{4F,i}^{(l+1)}-2~ D_4^{[d+]^{(l+1)}} \right], ~~~i=0,1,2 \cdots ,}
\label{Correction}
\end{eqnarray}
the iterative scheme then reads:
\begin{eqnarray}
{Z_{4F,i}^l=Z_{4F}^l+\delta Z_{4F,(i-1)}^l, ~~~ i=1,2, \cdots}
\label{iteration}
\end{eqnarray}
The index $i$ counts the highest power of ${\left( \right)}_4$ and the series $Z_{4F,i}^l$
is supposed to converge for growing $i$ towards $F_4^{[d+]^l}$.
As a condition of applicability of the iteration we can obviously use
\begin{eqnarray}
\frac{\delta Z_{4F,i}^l}{Z_{4F}^l} \sim
\frac{{\left( \right)}}{{0\choose 0}} \times \mathrm{scale} \ll 1,
\label{CondRec}
\end{eqnarray}
where the scale has dimension of a squared mass.
It is worth to perform the first few steps in the iteration explicitly:
\begin{eqnarray}\label{eq-fewsteps}
Z_{4F,i}^l&=&Z_{4F}^l+\frac{{\left( \right)}}{{0\choose 0}}\left[(2l+1)Z_{4F,(i-1)}^{(l+1)}-2 D_4^{[d+]^{(l+1)}}\right]
\\ \nonumber
&=&Z_{4F}^l+\frac{{\left( \right)}}{{0\choose 0}} (2l+1) \left\{ Z_{4F}^{(l+1)}+
\frac{{\left( \right)}}{{0\choose 0}} \left[(2l+3) Z_{4F,(i-1)}^{(l+2)}-2 D_4^{[d+]^{(l+2)}}\right]\right\}-2\frac{{\left( \right)}}{{0\choose 0}} D_4^{[d+]^{(l+1)}}
\\ \nonumber
&=& F_4^{[d+]^l} + \mathcal{O}(r^{i}) .
\end{eqnarray}
Performing $i$ steps in the iteration, we have
\begin{eqnarray}
Z_{4F,{i}}^{L}&=&\sum_{j=0}^{i-1} a_j^L r^j Z_{4F}^{(L+j)}+a_i^L r^i Z_{4F,0}^{(L+i)}
-2 \sum_{j=0}^{i-1} a_j^L r^{j+1} D_4^{[d+]^{(L+j+1)}},
\label{finitesum}
\end{eqnarray}
where { $r$ is given in~\eqref{rr} }
and
\begin{eqnarray}\label{eq-aj}
a_j^L=2^j \frac{\Gamma(L+j+\frac{1}{2})}{\Gamma(L+\frac{1}{2})}.
\end{eqnarray}
The $\Gamma(z)$ is the Euler Gamma function.
In~(\ref{finitesum}) we have to define yet $Z_{4F,0}^{(L+i)}$.
Strictly speaking for $Z_{4F,0}^{(L+i)}=F_4^{[d+]^{L+i}}$ we have $Z_{4F,i}^{(l)}=F_4^{[d+]^{l}}$
($i=1,2 \cdots$)
but in order to evaluate~(\ref{finitesum}) we have to choose an appropriate approximation.
Taking $i \rightarrow \infty$, however, and assuming convergence of the series, we have
\begin{eqnarray}
{F_4^{[d+]^L}=\sum_{j=0}^{\infty} a_j^L r^j Z_{4F}^{(L+j)}-2\sum_{j=0}^{\infty} a_j^L r^{j+1} D_4^{[d+]^{(L+j+1)}},}
\label{infinitesum}
\end{eqnarray}
i.e. in this limit the term with $Z_{4F,0}^{(L+i)}$ drops out. The choice of an approximation
for $Z_{4F,0}^{(L+i)}$ thus can influence only the first few partial sums of~\eqref{infinitesum}.
In fact, as will be seen in Sec.~\ref{Num} in an example, the convergence is quite good for
moderate $r$ and after a few steps the result is not very much dependent on the approximant
of $Z_{4F,0}^{(L+i)}$.
It remains to deal with the last term in~(\ref{finitesum}), i.e. $D_4^{[d+]^{(L+j+1)}}$.
As in~(\ref{finitesum}),
we define a partial sum
\begin{eqnarray}
Z_{4D,{i}}^{L}=&&\sum_{j=0}^{i} a_j^L r^j Z_{4D}^{(L+j)} ,
\label{Divfinitesum}
\end{eqnarray}
and get corresponding to~(\ref{infinitesum})
\begin{eqnarray}
D_4^{[d+]^L}=\sum_{j=0}^{\infty} a_j^L r^j Z_{4D}^{(L+j)}.
\label{Dinfinitesum}
\end{eqnarray}
In order to avoid in~(\ref{finitesum}) higher order terms coming from $D_4^{[d+]^{(L+j+1)}}$,
i.e. higher than those contained in the finite part, we re-write
\begin{eqnarray}
\sum_{j=0}^{i-1} a_j^L r^{j+1} D_4^{[d+]^{(L+j+1)}}= \sum_{j=0}^{i-1} a_j^L r^{j+1}Z_{4D,i-1-j}^{(L+j+1)}
+{\cal O}(r^{(i+1)}) .
\label{Div4}
\end{eqnarray}
E.g. for $j=i-1$ on the right hand side of~(\ref{Div4}) there contributes a term $r^{i}Z_{4D,0}^{(L+i)}$, where $i$ is the highest power of $r$ which occurs also in the finite part of~(\ref{finitesum}).
Some algebra yields
\begin{eqnarray}\label{somealg}
\sum_{j=0}^{i-1} a_j^L r^{j+1}Z_{4D,i-1-j}^{(L+j+1)}=\frac{1}{2} \sum_{j=0}^{i} a_j^L b_j^L r^{j}Z_{4D}^{(L+j)} ,
\end{eqnarray}
with coefficients
\begin{eqnarray}\label{digamma}
b_j^L
&=& \sum_{k=L+1}^{L+j} \frac{2}{2k-1}
\nonumber \\
&=&
\psi(L+j+\frac{1}{2})-\psi(L+\frac{1}{2}) ,
\end{eqnarray}
such that as final expression in terms of an infinite sum, we can write the solution of~\eqref{wish}:
\begin{eqnarray}
I_4^{[d+]^L}=\sum_{j=0}^{\infty} a_j^L r^j \left[Z_{4}^{(L+j)}-b_j^L Z_{4D}^{(L+j)}\right],~~~~
L=0, \cdots 4.
\label{final}
\end{eqnarray}
The $\psi(z)$ is the logarithmic derivative of the Gamma function (digamma function).
It is interesting to note that~(\ref{final}) is also valid for $L=0$. Possible additional infrared
divergent terms are then contained for $j=0$ in $Z_{4}^{(0)}$. Comparing~(\ref{final})
with~(\ref{wish}), we see that our goal is achieved.
{Having now obtained the compact expression~(\ref{final}),
it remains to discuss how to calculate the $Z_{4}^l$.
Quite naturally one first calculates the needed
$I_3^{[d+]^l,t}$ by means of recurrence~(\ref{A301}) and sums over $t$ according to~(\ref{Z4dla}).
This is possible since in general for $\left( \right)_4=0$, ${t\choose t}_4 \ne 0$
for $t=0,\cdots,4$.
Thus in order to evaluate~\eqref{final}, what is needed at the end, are
the $I_3^t, ~t=1, \cdots, 4$ , $I_2^{t,u}, ~t,u=1, \cdots, 4, t \ne u$
and $4$ $1$-point funcions for the kinematical point
under consideration, i.e. we need $14$ master integrals.
Applying~(\ref{A301}) in order to get the finite parts of $I_3^{[d+]^l,t}$, one also has to calculate the divergent parts of the higher-dimensional $2$- and $3$-point functions $I_2^{[d+]^{l-1},tu}$ and
$I_3^{[d+]^{l-1},t}$.
These have been discussed in app.~\ref{Bpp},
but only for low values of $l$.
They are needed for quite large $l$ ($l \sim 10$ and larger), and for larger $l$
the analytic expressions blow up considerably.
Further, for $l > 6$ the
analytic cancellation of the occurring Gram determinants is hard to perform.
So, we preferred to work numerically.
Amazingly, the situation
is different for the $2$-point functions. Without any problem we can produce with Mathematica
any higher divergences with recurrence ~(\ref{A211}), cancelling thereby the ${tu\choose tu}$ Gram
determinants.
For details see the discussion in sect.~\ref{sub4}.
To remain numerically as accurate as possible we use these analytic expressions when
calculating the divergences of the $3$-point functions numerically by recursion, starting
with $D_3^{[d+]}(t)=-\frac{1}{2}$, see~(\ref{D3t}).
As mentioned above, this has to be done for $l=1,\cdots ,l_{max}-1$.}
As a result, in a practical calculation one has the objects
\begin{eqnarray}
{Z_{4F,{i}}^{L},~~ i=0, \dots ,l_{max}-L,}
\label{I4L}
\end{eqnarray}
{which are a sequence of approximations for the finite part of the integrals $I_4^{[d+]^L}$,
i.e. $F_4^{[d+]^L}$.} In app.~\ref{Num} we give an example for their numerical evaluation.
At this point we stress that formulae {\eqref{want1},~\eqref{fulld3} and~\eqref{fulld4}}
are free of indexed inegrals $I_{4,i \cdots}^{[d+]^l}$ and thus enable a new access to the
calculation of tensor $4$-point functions. Relation~\eqref{final} was already obained
in~\cite{Fleischer:2003rm}, see (36) there: for $n$-point functions ($n=1,2,3,4$)
of arbitrary dimension \emph{with generic indices} $\nu_i=1$ this series was
derived in similar manner as above and a general scheme was developed of how
to find analyic continuations to kinematical domains, where this series does not converge.
E.g., if ${0\choose 0}_4$ is small, $r$ in~\eqref{rr} is large. For this case
\cite{Fleischer:2003rm} contains the description of how to
modify the procedure of solving the recursion such that the expansion parameter is small,
i.e. how to obtain from the recursion relation a series in $\frac{1}{r}$, see eqn. (23) ibid.
Beyond that, from this series
the $d$-dimensional $n$-point functions are obtained iteratively in terms of multiple
hypergeometric series with ratios of different signed minors as arguments.
For the 4-point function, e.g., the generalized hypergeometric functions
$_2F_1$, Appell function $F_1$ and the Lauricella-Saran funcion $F_S$ appear, see e.g.~(98) of~\cite{Fleischer:2003rm}.
Transformation formulae of the generalized
hypergeometric functions allow to extend their applicability to different domains
of the phase space.
In particular in our situation
we have to deal with integrals of dimension $d \ge 6 - 2 \varepsilon$. The hypergeometric functions in
\cite{Fleischer:2003rm} are also expressed in terms of 1-dimensional integrals and inspection
shows that for the large dimensions these are particularly well suited for numerical
evaluation (see eqns.~(78) and~(96)).
We leave this for further study.
{Another} attempt to perform the described series of approximations was undertaken in~\cite{Giele:2004ub}; see Eq.~(5) there.
A specific example was studied, namely forward light-by-light scattering through a massless fermion loop.
The approach was then not further followed.
\section{\label{LaGra}
Symmetrized recurrence relations
}
So far in the former sect. we were concerned with the evaluation of the $4$-point tensor coefficients for small Gram determinants.
If, however, the Gram determinant is not small there are other ways of doing the reduction.
In fact the ''standard'' Passarino-Veltman~\cite{Passarino:1978jh} reduction is one possibility.
This is, however, not a unique procedure.
While in~\cite{Diakonidis:2009fx} a systematic application of recursion relations of
type ~(\ref{tensor5general}) was performed for all tensor $n$-point functions, here we
take a different point of view, namely to arrange the tensor coefficients in~(\ref{tensor1}) to
(\ref{tensor4}) for the $4$-point functions in such a way that a possible analytic
simplification of the tensor as a whole is achieved.
We begin with $I_4^{\mu\nu}$ defined in~\eqref{tensor2}, containing as most complicated object $I_{4,ij}^{[d+]^2}$.
This integral, represented also by~\eqref{want1}, may be reduced by~\eqref{eq:RR1},
\begin{eqnarray}
{\nu}_{ij} I_{4,ij}^{[d+]^2}= && \frac{{0\choose i}{0\choose j}}{{\left( \right)}^2}I_{4}+
\frac{{i\choose j}}{{\left( \right)}}I_{4}^{[d+]}+R_{3,ij}^{[d+]^2},
\label{wantx}
\end{eqnarray}
with
\begin{eqnarray}
R_{3,ij}^{[d+]^2}&&=-\frac{{0\choose j}}{{\left( \right)}}\sum_{t=1}^4
\frac{{t\choose i}}{{\left( \right)}}I_{3}^{t}+
\sum_{t=1}^4\frac{{t\choose j}}{{\left( \right)}}I_{3,i}^{[d+],t}\nonumber \\
&&=-\sum_{t=1}^4\frac{{0\choose i} {t\choose j}
+{0\choose j}{t\choose i} -{t\choose i}{t\choose j}
\frac{{t\choose 0}}{{t\choose t}}}{{\left( \right)}^2}I_3^{t}
+\frac{1}{{\left( \right)}}\sum_{t,u=1}^4\frac{{t\choose j}}{{t\choose t}}
{ut\choose it}I_2^{tu}.
\label{wanty}
\end{eqnarray}
The first observation of interest here is the symmetry in the indices $i,j$. Only the last
term is not obviously symmetric. As was mentioned earlier, the symmetry is
in general seen only after summation over $s,t$, which we can exemplify here.
We use the relation
\begin{eqnarray}\label{eq-wa-4}
{t\choose j}{ut\choose it}={t\choose i}{ut\choose jt}+{t\choose t}{tu\choose ji}.
\end{eqnarray}
Inserting the left hand side into~(\ref{wanty}), the first term
on the right hand side has just exchanged indices
$i,j$. In the second contribution ${t\choose t}$ cancels due to which the sum over $s,t$
vanishes since ${tu\choose ji}$ is antisymmetric in $t,u$.
In~(\ref{wantx}) there remains as higher-dimensional integral $I_{4}^{[d+]}$, which is evaluated
according to~(\ref{A401}). Often it is as well used as ``master integral'' since it is UV and
IR finite. Having a look at~(\ref{tensor2}) we see that the second amplitude of the rank $R=2$
tensor is also just $I_{4}^{[d+]}$. This allows the following way of writing for this tensor.
Similarly as in~\cite{Diakonidis:2009fx} we introduce \footnote{$G^{\mu \nu}$ here differs
from the definition (24) in~\cite{Diakonidis:2009fx} by a factor of 2.
}
\begin{eqnarray}\label{eq-wa-5}
G^{\mu \nu}&=&g^{\mu \nu}-2 \sum_{i,j=1}^4 q_i^{\mu} q_j^{\nu}
\frac{{i\choose j}}{{\left( \right)}}
\nonumber \\
&=&\frac{8 v^{\mu} v^{\nu}}{{\left( \right)}} ,
\label{Gg}
\end{eqnarray}
with
\begin{eqnarray}\label{eq-wa-6}
v^{\mu}={\varepsilon}^{\mu \lambda \rho \sigma} (q_1-q_4)_{\lambda}(q_2-q_4)_{\rho}(q_3-q_4)_{\sigma}
,
\end{eqnarray}
and $v^2=\frac{1}{8}{\left( \right)}$. This allows to drop $I_{4}^{[d+]}$ in~(\ref{wantx})
and to replace $g^{\mu \nu}$ in~(\ref{tensor2}) by $G^{\mu \nu}$:
\begin{eqnarray}\label{eq-i4alt}
I_4^{\mu\nu}
&=&
\sum_{i,j=1}^{4} \, q_i^{\mu}\, q_j^{\nu} \,
\left[
\frac{{0\choose i}{0\choose j}}{{\left( \right)}^2}I_{4}
-\sum_{t=1}^4\frac{{0\choose i} {t\choose j}
+{0\choose j}{t\choose i} -{t\choose i}{t\choose j}
\frac{{t\choose 0}}{{t\choose t}}}{{\left( \right)}^2}I_3^{t}
+\frac{1}{{\left( \right)}}\sum_{t,u=1}^4\frac{{t\choose j}}{{t\choose t}}
{ut\choose it}I_2^{tu}
\right]
\nonumber \\
&&-~ \frac{1}{2}
\, G^{\mu \nu} \, I_{4}^{[d+]} .
\end{eqnarray}
This representation may become
advantageous in the analytic evaluation of diagrams since $G^{\mu \nu}$, contracted with
a (proper difference of) chord(s), vanishes. Remember that any external momentum may be
written as a sum of chords. For the tensor of rank $R=2$ this corresponds to (20) of
\cite{Diakonidis:2009fx}.
We will derive here the corresponding relations for the higher tensors as well.
Proceeding to $I_4^{\mu\nu\lambda}$,~(\ref{tensor3}), we ,first of all, need in the second term on the right hand side
$I_{4,i}^{[d+]^2}$, (\eqref{A511} with $l=2$):
\begin{eqnarray}
I_{4,i}^{[d+]^2}=-\frac{{0\choose i}}{{\left( \right)}}I_{4}^{[d+]}+
\sum_{t=1}^4\frac{{t\choose i}}{{\left( \right)}}I_{3}^{[d+],t},
\label{I447}
\end{eqnarray}
where again $I_{4}^{[d+]}$ appears and $I_{3}^{[d+],t}$ is given in~(\ref{A301}).
From the recursions of subsect.~\ref{sub2} we obtain
\begin{eqnarray}
&&{\nu}_{ij}{\nu}_{ijk} I_{4,ijk}^{[d+]^3}=-\frac{{0\choose i}{0\choose j}{0\choose k}}
{{\left( \right)}^3}I_{4}+\left\{
\frac{{i\choose j}}{{\left( \right)}} I_{4,k}^{[d+]^2}+
(j \leftrightarrow k) + (i \leftrightarrow k) \right\}+R_{3,ijk}^{[d+]^3},
\label{fullx}
\end{eqnarray}
where we have introduced an abbreviation for the remaining
$3$- and $2$-point functions:
\begin{eqnarray}
R_{3,ijk}^{[d+]^3} =&&
-\frac{{0\choose k}}{{\left( \right)}}R_{3,ij}^{[d+]^2}-
\sum_{t=1}^4\frac{{t\choose k}}{{\left( \right)}}\frac{{0t\choose jt}}{{t\choose t}}I_{3,i}^{[d+],t}-
\sum_{t=1}^4 \frac{{t\choose i}{t\choose j}{t\choose k}}{{\left( \right)}^2{t\choose t}} I_{3}^{[d+],t}
+\sum_{t,u=1}^4\frac{{t\choose k}{ut\choose jt}}{{\left( \right)}{t\choose t}}
I_{2,i}^{[d+],tu} \nonumber \\
=&&\sum_{t=1}^4\frac{{0\choose i}{0\choose j}{t\choose k}+{0\choose i}{t\choose j}{0\choose k}+
{t\choose i}{0\choose j}{0\choose k}} {{\left( \right)}^3}I_{3}^{t}
\nonumber \\
-&&\sum_{t=1}^4\frac{{0\choose i}{t\choose j}{t\choose k}+{t\choose i}{0\choose j}{t\choose k}+
{t\choose i}{t\choose j}{0\choose k}-{t\choose i}{t\choose j}{t\choose k}\frac{{t\choose 0}}{{t\choose t}}}{{\left( \right)}^3} \frac{{t\choose 0}}{{t\choose t}}I_{3}^{t}
-\sum_{t=1}^4 \frac{{t\choose i}{t\choose j}{t\choose k}}{{\left( \right)}^2{t\choose t}} I_{3}^{[d+],t}
\nonumber \\
-&&\sum_{t,u=1}^4\frac{\left[{0\choose j}{t\choose k}+{0\choose k}{t\choose j}-{t\choose j}{t\choose k}
\frac{{t\choose 0}}{{t\choose t}}\right]}{{\left( \right)}^2}\frac{{ut\choose it}}{{t\choose t}}I_{2}^{tu}
+\sum_{t,u=1}^4\frac{{t\choose k}{ut\choose jt}}{{\left( \right)}{t\choose t}}
I_{2,i}^{[d+],tu} .
\label{fullxr}
\end{eqnarray}
In~(\ref{fullx}), the $\frac{{i\choose j}}{{\left( \right)}}I_{4}^{[d+]}$ of~(\ref{wantx}) together
with part of the last contribution in~(\ref{A322}) has now been absorbed in $\frac{{i\choose j}}{{\left( \right)}} I_{4,k}^{[d+]^2}$: in~(\ref{A533}) only two terms of this type appear explicitely,
i.e. this form has no obvious symmetry. Further, in~(\ref{fullx}) and~(\ref{fullxr})
only the $4$- and $3$- point functions are explicitly symmetric in the indices $i,j,k$. In
order to demonstrate symmetry also for the $2$-point functions one would have to reduce
also $I_{2,i}^{[d+],tu}$, which is given in app.~\ref{App}.
We may now combine the results and simplify $I_4^{\mu\nu\lambda}$ correspondingly:
The $I_{4,k}^{[d+]^2}$ in~(\ref{fullx})
can be combined with the second part of~(\ref{tensor3}), i.e. it can be dropped in~(\ref{fullx}),
and in~(\ref{tensor3}) the $g^{\mu \nu}$ must then be replaced by $G^{\mu \nu}$.
(\ref{fullx}) has also been identified with a complete reduction of~(\ref{fulld3}).
For $I_4^{\mu\nu\lambda\rho}$~(\ref{tensor4}), we start from~(\ref{A555}):
\begin{eqnarray}
{n}_{ijkl} I_{4,ijkl}^{[d+]^4}
&=&
\frac{{0\choose i}{0\choose j}{0\choose k}{0\choose l}}
{{\left( \right)}^4}I_{4}-\frac{{0\choose l}}{{\left( \right)}}\left\{
\frac{{i\choose j}}{{\left( \right)}} I_{4,k}^{[d+]^2}+
(j \leftrightarrow k) + (i \leftrightarrow k) +R_{3,ijk}^{[d+]^3}\right\}
\nonumber \\
&&
+\left\{
\frac{{i\choose l}}{{\left( \right)}}\frac{{j\choose k}}{{\left( \right)}}+
\frac{{j\choose l}}{{\left( \right)}}\frac{{i\choose k}}{{\left( \right)}}+
\frac{{k\choose l}}{{\left( \right)}}\frac{{i\choose j}}{{\left( \right)}} \right\}I_4^{[d+]^2}
\\
&&
-\left\{
\frac{{i\choose l}}{{\left( \right)}}\frac{{0\choose k}}{{\left( \right)}}I_{4,j}^{[d+]^2}+
\frac{{j\choose l}}{{\left( \right)}}\frac{{0\choose k}}{{\left( \right)}}I_{4,i}^{[d+]^2}+
\frac{{k\choose l}}{{\left( \right)}}\frac{{0\choose j}}{{\left( \right)}}I_{4,i}^{[d+]^2}\right\}
\nonumber \\\nonumber
&&
+\frac{{i\choose l}}{{\left( \right)}}\sum_{t=1}^4 \frac{{t\choose k}}{{\left( \right)}}I_{3,j}^{[d+]^2,t}
+\frac{{j\choose l}}{{\left( \right)}}\sum_{t=1}^4 \frac{{t\choose k}}{{\left( \right)}}I_{3,i}^{[d+]^2,t}
+\frac{{k\choose l}}{{\left( \right)}}\sum_{t=1}^4 \frac{{t\choose j}}{{\left( \right)}}I_{3,i}^{[d+]^2,t}+
\sum_{t=1}^4 \frac{{t\choose l}}{{\left( \right)}}n_{ijk}I_{3,ijk}^{[d+]^3,t}.
\label{fully}
\end{eqnarray}
As can be seen, the ${n}_{ijkl} I_{4,ijkl}^{[d+]^4}$ can be expressed in a form which mainly contains
terms which also occur in
\begin{eqnarray}
{\nu}_{ij} I_{4,ij}^{[d+]^3}=-\frac{{0\choose j}}{{\left( \right)}}I_{4,i}^{[d+]^2}+
\frac{{i\choose j}}{{\left( \right)}}I_4^{[d+]^2}+ \sum_{t=1}^4 \frac{{t\choose j}}{{\left( \right)}}I_{3,i}^{[d+]^2,t},
\label{I451}
\end{eqnarray}
see~(\ref{A522}).
The $n_{ijk}I_{3,ijk}^{[d+]^3,t}$
can be written similarly like~(\ref{fullx}):
\begin{eqnarray}
&&{\nu}_{ij}{\nu}_{ijk} I_{3,ijk}^{[d+]^3,t}=-\frac{{0t\choose it}{0t\choose jt}{0t\choose kt}}
{{t\choose t}^3}I_{3}^t+\left\{ \left[
\frac{{i\choose j}}{{\left( \right)}}- \frac{{t\choose i}{t\choose j}}{{\left( \right)}{t\choose t}}\right]I_{3,k}^{[d+]^2,t}+
(j \leftrightarrow k) + (i \leftrightarrow k) \right\}+R_{2,ijk}^{[d+]^3,t}, \nonumber \\
\label{fullx3}
\end{eqnarray}
where $R_{2,ijk}^{[d+]^3,t}$ collects the remaining $2$- and $1$-point functions and is obtained
from~(\ref{fullxr}) as follows: all $3$-point functions are replaced by $2$-point functions
($3 \rightarrow 2$) and the $2$-point functions are replaced by $1$-point functions ($2 \rightarrow 1$).
All summation indices $u$ must be replaced by $v$ and summation indices $t$ must be replaced by $u$.
Finally in all determinants and integrals columns, lines and propagators $t$ must be scratched - like
in~(\ref{fullx3}), i.e.
\begin{eqnarray}\label{fullx23}
R_{2,ijk}^{[d+]^3,t} =
-\frac{{0t\choose kt}}{{t\choose t}}R_{2,ij}^{[d+]^2,t}-
\sum_{u=1}^4\frac{{ut\choose kt}}{{t\choose t}}\frac{{0tu\choose jtu}}{{tu\choose tu}}I_{2,i}^{[d+],tu}-
\sum_{u=1}^4 \frac{{ut\choose it}{ut\choose jt}{ut\choose kt}}{{t\choose t}^2{tu\choose tu}} I_{2}^{[d+],tu}
+\sum_{u,v=1}^4\frac{{ut\choose kt}{vtu\choose jtu}}{{t\choose t}{tu\choose tu}}
I_{1,i}^{[d+],tuv} , \nonumber \\
\end{eqnarray}
and
\begin{eqnarray}\label{fullx22}
R_{2,ij}^{[d+]^2,t}=-\frac{{0t\choose jt}}{{t\choose t}}\sum_{u=1}^4
\frac{{ut\choose it}}{{t\choose t}}I_{2}^{tu}+
\sum_{u=1}^4\frac{{ut\choose jt}}{{t\choose t}}I_{2,i}^{[d+],tu}.
\end{eqnarray}
To indicate how the various tensor components may be combined to simplify the result,
we only count here the number of contributions of a certain type. The second term
in~\eqref{tensor4} contains, according to~\eqref{G2V2}, $6$ terms of type~\eqref{I451}.
Each type of the $3$ terms in~\eqref{I451} is also contained in the other tensor
components in~\eqref{tensor4}:
\begin{itemize}
\item $\frac{{0\choose j}}{{\left( \right)}}I_{4,i}^{[d+]^2}$ occurs $6$ \rm{times} in~\eqref{fully},
\item $\frac{{i\choose j}}{{\left( \right)}}I_4^{[d+]^2}$ occurs $3$ \rm{times} in~\eqref{fully}
and occurs $3$ \rm{times} in the $3^{rd}$ term of~\eqref{tensor4},
\item $\sum_{t=1}^4 \frac{{t\choose j}}{{\left( \right)}}I_{3,i}^{[d+]^2,t}$ occurs $3$ \rm{times} in~\eqref{fully} and occurs $3$ \rm{times} in~\eqref{fullx3}.
\end{itemize}
Thus, rewriting~\eqref{Gg} as
\begin{eqnarray}\label{eq23}
\sum_{i,j=1}^4 q_i^{\mu} q_j^{\nu}
\frac{{i\choose j}}{{\left( \right)}}=\frac{1}{2} \left( g^{\mu \nu}-G^{\mu \nu} \right),
\end{eqnarray}
one can convince oneself that $g^{\mu \nu}$ cancels and it remains $G^{\mu \nu}$, which after
contraction with a chord drops out.
As we have seen, the above treatment of tensor coefficients requires at least
one step of iteration of the recursion relation, because of which it is applicable only for
non-vanishing Gram determinants. For these, however, it is very useful when we consider
$4$-point functions obtained by scratching one line of a $5$-point function. In this
case it is indeed possible to perform a cancellation of terms, which above still have
the factor $G^{\mu \nu}$. Also for this case, which we dealt with in sect.~\ref{LargeGrams},
the above presentation of the $4$-point tensor coefficients can be applied
- only scratching of one propagator has to be taken into account.
Beyond that the present approach results in reductions which make the
symmetry in the indices $i,j,k,l$ more transparent and as a consequence yield
certain blocks which can be calculated separately and combined to yield the complete
tensor coefficients.
\section{\label{Simplify}Analytic simplifications for contractions of tensors with chords}
Before a numerical program for calculations is set up, it turns out to be
advantageous to simplify Feynman diagrams analytically. A standard example
is the following.
If in the numerator of a Feynman integral a scalar
product $q_i \cdot k$ of a chord and an integration momentum occurs,
this {product} is usually expressed in terms of the difference of two
scalar propagators which can be cancelled against propagators in the
denominator. Already in~\cite{Diakonidis:2009fx} an alternative was
indicated, making use of the fact that the contraction of a vector
of the type~(\ref{Qs}) with a chord yields a simple expression:
\begin{eqnarray}
q_i \cdot Q_0 =\sum_{j=1}^{n-1} q_i q_j \frac{{0\choose j}_n}{{\left(\right)}_n}=-\frac{1}{2}\left(
Y_{in}-Y_{nn} \right), ~~~i=1, \dots , n-1,
\label{Scalar1}
\end{eqnarray}
and
\begin{eqnarray}
q_i \cdot Q_s =\sum_{j=1}^{n-1} q_i q_j \frac{{s\choose j}_n}{{\left( \right)}_n}=\frac{1}{2}\left(
{\delta}_{is}-{\delta}_{ns}\right), ~~~i=1, \dots , n-1, ~~~s=1, \dots n.
\label{Scalar2}
\end{eqnarray}
In~(\ref{Scalar1}) and~(\ref{Scalar2}) $q_n=0$ is assumed since only in this case
\begin{eqnarray}\label{Scalar2a}
q_i \cdot q_j =\frac{1}{2} \left[Y_{ij}-Y_{in}-Y_{nj}+Y_{nn} \right],
\end{eqnarray}
which is needed for their derivations.
Thus, if the reduction relation~(\ref{tensor5general}) for $I_5^{\mu_1 \dots \mu_{R-1} \mu}$ is contracted
with a $q_{i,\mu}$, we can advantageously apply~(\ref{Scalar1}) and~(\ref{Scalar2}) with $n=5$.
In case one considers a process with $5$ external legs, one can choose from the very beginning
$q_5=0$ in the tensor integrals. If, however, the $5$-point tensor is obtained by
reducing a $6$-point tensor,
cases with $q_5 \ne 0$ will occur.
In order to be able to
apply~(\ref{Scalar1}) and~(\ref{Scalar2}), it is recommended to perform a shift of the
integration momentum like $k \rightarrow k+ q_5$, i.e. $q_i \rightarrow q_i -q_5$. Such
a shift is not a problem at all, nevertheless it is interesting to see how this
shift can be implemented in the formalism.
The scalar integrals and the signed minors
are invariant under the shift.
We exemplify this for $I_n^{\mu\nu}$, writing
\begin{eqnarray}
-\sum_{i,j=1}^n (q_i-q_n)^{\mu}(q_j-q_n)^{\nu} {\nu}_{ij}I_{n,ij}^{[d+]^2}=
&&-\sum_{i,j=1}^n q_i^{\mu}q_j^{\nu} {\nu}_{ij}I_{n,ij}^{[d+]^2}+
q_n^{\mu}\sum_{i,j=1}^n q_j^{\nu} {\nu}_{ij}I_{n,ij}^{[d+]^2}
\nonumber \\
&&
+~
q_n^{\nu}\sum_{i,i=1}^n q_i^{\nu} {\nu}_{ij}I_{n,ij}^{[d+]^2}
-q_n^{\mu}q_n^{\nu} \sum_{i,j=1}^n {\nu}_{ij}I_{n,ij}^{[d+]^2} ,
\label{Inijd21}\end{eqnarray}
with
\begin{eqnarray}
{\nu}_{ij} I_{n,ij}^{[d+]^2}&=&-\frac{{0\choose j}_n}{\left( \right)_n} I_{n,i}^{[d+]} +
\sum_{t=1,t \ne i}^{n} \frac{{t\choose j}_n}{\left( \right)_n} I_{n-1,i}^{[d+],t} +
\frac{{i\choose j}_n}{\left( \right)_n} I_{n}^{[d+]}.
\label{Inijd2}
\end{eqnarray}
The sums
\begin{eqnarray}\label{eq-wa-83r}
\sum_{j}^n {\nu}_{ij}I_{n,ij}^{[d+]^2}&=&-I_{n,i}^{[d+]},
\\\label{eq-wa-83q}
\sum_{i=1}^n {\nu}_{ij}I_{n,ij}^{[d+]^2}&=&-I_{n,j}^{[d+]} ,
\\\label{eq-wa-83s}
\sum_{i,j=1}^n {\nu}_{ij}I_{n,ij}^{[d+]^2} &=&I_n
\end{eqnarray}
can be obtained by making use of
\begin{eqnarray}\label{eq-wa-8}
\sum_{i=1}^n {0\choose i}_n &=& {\left( \right)}_n ,
\\\label{eq-wa-9}
\sum_{i=1}^n {s\choose i}_n &=& 0.
\end{eqnarray}
Thus one has
\begin{eqnarray}\label{eq-wa-90}
\sum_{i,j=1}^n q_i^{\mu}q_j^{\nu} {\nu}_{ij}I_{n,ij}^{[d+]^2}=\sum_{i,j=1}^{n-1} {q^{'}}_i^{\mu}{q^{'}}_i^{\nu} {\nu}_{ij}I_{n,ij}^{[d+]^2}-q_n^{\mu}\sum_{i=1}^n q_i^{\nu}I_{n,i}^{[d+]}-
q_n^{\nu}\sum_{i=1}^n q_i^{\mu} I_{n,i}^{[d+]}-q_n^{\mu}q_n^{\nu}I_n ,
\end{eqnarray}
using the abbreviation ${q^{'}}_i^{\mu}=q_i^{\mu}-q_n^{\mu}$,
and with
\begin{eqnarray}\label{eq-wa-80}
-\sum_{i=1}^n q_i^{\mu}I_{n,i}^{[d+]}=-\sum_{i=1}^{n-1}{q^{'}}_i^{\mu}I_{n,i}^{[d+]}+q_n^{\mu} I_n,
\end{eqnarray}
derived in the same manner, with~(\ref{tensor1}) and~(\ref{tensor2})
the standard result of shifting the integration momentum is obtained.
The point is that the extra contributions obtained by the shift contain only integrals which
were needed already in the unshifted integral so that the shift does not require the calculation
of any new integral, see~(\ref{Inijd2}).
Assume now again that we are dealing with the $5$-point tensor and have $q_5=0$. The above trick
to avoid an increase of the tensor rank can be applied to~(\ref{starter}) as well:
contracting with $q_{i,\mu} $ the first term yields the contribution $q_i \cdot Q_0$
given in~(\ref{Scalar1}), for the second term we have
to find a formula for the scalar product $q_i \cdot {\bar{Q}}_s^{0}$. Indeed,
\begin{eqnarray}\label{eq-wa-81}
q_i {\bar{Q}}_s^{0}&=&\sum_{j=1}^{4} q_i q_j {0s\choose 0j}_5
\nonumber \\
&=& \frac{1}{2}
\left[{0\choose 0}_5 \left({\delta_{is}}-{\delta_{5s}}\right)+{s\choose 0}_5 \left(Y_{i5}-Y_{55}\right)\right].
\end{eqnarray}
For the first term in~(\ref{starter}) also another possibility exists, provided a
contraction with a further vector is available. In such a case the first term
on the right hand side of
(\ref{cancel}), which shows up explicitly in the tensor components
$E_{ij \dots}$, yields a double-sum like
\begin{eqnarray}\label{eq-wa-82}
\sum_{i,j=1}^{4} (q_a \cdot q_i) (q_b \cdot q_j) {0i\choose sj}_5=\frac{1}{2}q_a \cdot q_b{s\choose 0}_5
+\frac{1}{4}{\left(\right)}_5\left(Y_{b5}-Y_{55} \right)\left({\delta}_{as}-{\delta}_{5s}\right).
\end{eqnarray}
Further sums are obtained if the $4$-point tensors are contracted. These are all of the
type ${is\choose js}_5$, i.e. with line $s$ scratched. We just list a few of them:
\begin{eqnarray}\label{eq-wa-83}
\sum_{j=1}^{4}q_a \cdot q_j {0s\choose js}_5 &&=-\frac{1}{2} \left[
{s\choose 0}_5\left({\delta}_{as}-{\delta}_{5s}\right)+
{s\choose s}_5\left(Y_{a5}-Y_{55} \right)\right], \\\label{eq-wa-83a}
\sum_{i,j=1}^{4} q_i \cdot q_j {0s\choose is}_5{0s\choose js}_5 &&=\frac{1}{2} {s\choose s}_5 \left[
{0s\choose 0s}_5+Y_{55}{s\choose s}_5+2 {s\choose 0}_5{\delta}_{5s}\right],
\\\label{eq-wa-83v}
\sum_{i,j=1}^{4} q_i \cdot q_j {is\choose js}_5 &&=\frac{3}{2} {s\choose s}_5, \\\label{eq-wa-83b}
\sum_{i,j=1}^{4} (q_a \cdot q_i) (q_b \cdot q_j) {is\choose js}_5 &&=\frac{1}{2}q_a \cdot q_b{s\choose s}_5
-\frac{1}{4}{\left(\right)}_5\left({\delta}_{ab}{\delta}_{as}+{\delta}_{5s}\right),
\end{eqnarray}
and for $3$-point functions
\begin{eqnarray}\label{eq-wa-84}
\sum_{j=1}^{4}q_a \cdot q_j {ts\choose js}_5
&&=\frac{1}{2} \left\{{s\choose s}_5\left[
(1-{\delta}_{as}){\delta}_{at}-(1-{\delta}_{5s}){\delta}_{5t}\right]\right.
\nonumber \\
&&\left. ~~~~~~-{t\choose a}_5
(1-{\delta}_{at}){\delta}_{as}+{t\choose 5}_5(1-{\delta}_{5t}){\delta}_{5s} \right\} , \\\label{eq-wa-851}
\sum_{i,j=1}^{4} q_i \cdot q_j{ts\choose is}_5 {ts\choose js}_5 &&=\frac{1}{2} {s\choose s}_5
{st\choose st}_5 ,
\\\label{eq-wa-852}
\sum_{i,j=1}^{4} q_i \cdot q_j{ts\choose is}_5 {0s\choose js}_5 &&=\frac{1}{2} {s\choose s}_5
\left\{{0s\choose ts}_5-{s\choose s}_5(1-{\delta}_{5s}){\delta}_{5t}+{t\choose 5}_5 (1-{\delta}_{5t}){\delta}_{5s}\right\} ,
\\\label{eq-wa-854}
\sum_{i,j=1}^{4} q_i \cdot q_j{ist\choose jst}_5 &&={st\choose st}_5 .
\end{eqnarray}
Even a quadrupel sum appears:
\begin{eqnarray}\label{eq-wa-85}
\sum_{i,j,k,l=1}^{4}(q_i \cdot q_j)(q_k \cdot q_l){0i\choose sl}_5{ts\choose js}_5{ts\choose ks}_5
=\frac{1}{4} {s\choose 0}_5{s\choose s}_5{st\choose st}_5.
\end{eqnarray}
In fact, there are many more such sums. Our conclusion here is that in every scalar, which is obtained by
contraction with chords the appearing sums can be evaluated analytically in order to yield
compact expressions. This is due to the fact that the indices of the chords
$i,j, c\dots $ are carried by signed minors while the integrals don't necessarily
carry indices anymore.
\section{\label{conclude}Conclusions}
We have developed a new approach to reduce tensorial one-loop $n$-point Feynman integrals based
on an algebraic method elaborated in earlier papers.
The approach is worked out
up to 6-point tensors with rank $R \leq 6$
and a rule is found how to extend the method to higher ranks.
The first step was to reduce $5$-point tensors up to rank $5$ to $4$-point tensor
coefficients given in terms of higher-dimensional, indexed $4$-point functions.
The latter are expressed in terms of higher-dimensional integrals $I_4^{[d+]^L}$, $L=1, \cdots ,4$,
plus $3$-point tensor coefficients in~\eqref{I4id+2},~\eqref{want1},~\eqref{fulld3} and~\eqref{fulld4}.
So far no Gram determinants $\left( \right)_4$ occur.
Inverse powers of $\left( \right)_4$ do occur if the integrals $I_4^{[d+]^L}$ are reduced to
standard $A_0,B_0,C_0,D_0$ functions in generic dimension. For small $\left( \right)_4$
this is avoided by using the expansion~\eqref{final} in positive powers of $\left( \right)_4$.
Application of Pad\'{e}-approximants based on the $\varepsilon$-algorithm
to this expansion allows to calculate the
$I_4^{[d+]^L}$ in a simple manner to such a precision
that the complete phase space is covered with high numerical precision.
{A numerical opensource code of the formulae derived in this article is under development.}
As a matter of fact,~\eqref{final} is a special case of the general
method developed in~\cite{Fleischer:2003rm}. Apart from this special series expansion,
the integrals $I_4^{[d+]^L}$ are expressed in~\cite{Fleischer:2003rm}
in terms of multiple hypergeometric functions $_2F_1$, Appell function $F_1$ and Lauricella-Saran function
$F_S$.
In this context, it is a crucial property of eqns.~\eqref{I4id+2},~\eqref{want1},~\eqref{fulld3} and~\eqref{fulld4} to be free of integrals $I_{4,i\cdots}^{[d+]^L}$ with indices larger than one.
Using the notion of special functions allows a variety of options to adjust to various kinematical situations, and it might be interesting to explore their potential for a further improvement of numerical programs.
\section*{Acknowledgements}
We would like thank F. Campanario, Th. Diakonidis, B. Tausk and V. Yundin for useful discussions and V. Yundin for a careful reading of the manuscript.
{We are grateful to A. Denner for some clarifying discussion.}
J.F. thanks DESY for kind hospitality.
Work supported in part by Sonderforschungsbereich/Trans\-re\-gio SFB/TRR 9 of DFG
``Com\-pu\-ter\-ge\-st\"utz\-te Theoretische Teil\-chen\-phy\-sik"
and by the European Community's Marie-Curie Research Trai\-ning Network
MRTN-CT-2006-035505
``HEPTOOLS''.
|
{
"timestamp": "2011-02-17T02:02:20",
"yymm": "1009",
"arxiv_id": "1009.4436",
"language": "en",
"url": "https://arxiv.org/abs/1009.4436"
}
|
\section{Introduction}
During the last thirty years high performance computing (HPC) has become an increasingly-important tool in scientific research.
HPC studies enhance understanding of experimental findings, allow researchers to test theories on model systems, and even make it possible to investigate phenomena which cannot be investigated via classical experiments.
One class of computer experiments is of special interest: molecular dynamics (MD) simulations.
MD is used to simulate materials on an atomic (or coarser-grained) level using various interaction models.
Through advances in compute capabilities and algorithms, MD simulations have gradually expanded their range of applicability from modeling tiny systems of a few hundred atoms for up to a few thousand time steps, to performing short multi-billion atom simulations or multi-billion time-step simulations of smaller systems.
While this is already impressive in itself, a single cubic centimeter of matter contains on the order of $10^{23}$ atoms, and to model only one second of its time propagation, $10^{15}$ time steps (typically a femtosecond each) would be required.
Therefore, the interest in accelerating MD simulations is unstinting and of great interest for many computational scientists.
Easily programmable graphics cards (GPUs) represent a disruptive technology development that allows radical departure from recent years' gradual improvements in MD simulation speed.
By harnessing the compute capability of GPUs, MD practitioners will be able to simulate much larger systems for much longer simulated times.
GPUs represent a jump in the performance-to-cost ratio of at least a factor of five.
GPUs also achieve more flops-per-watt than corresponding CPU hardware, making next-generation GPU-based HPC supercomputers more feasible from an operating energy cost perspective.
The CUDA programming language is currently the most widely used programming model for GPUs.
Since its introduction, many scientific programmers have used CUDA to write extremely fast software, thereby enabling previously-impossible investigations.
Among those are also a number of MD codes which have shown speed-ups of 5-100x over existing CPU-based codes.
In this paper, we present our own implementation of a GPU-MD code called LAMMPS$_{\rm CUDA}$, which is introduced as an extension to the widely used MD code LAMMPS\cite{LAMMPS}.
With its 26 different force fields, LAMMPS$_{\rm CUDA}$ can model atomic, polymeric, biological, metallic, granular, and coarse-grained systems up to 20 times faster than a modern quad core workstation by harnessing a modern GPU.
At the same time it offers unprecedented multi-GPU support for an MD code.
By providing very effective scaling of simulations on up to hundreds of GPUs, LAMMPS$_{\rm CUDA}$ enables scientists to harness the full power of the world's most advanced supercomputers, such as the world's fastest supercomputer, the Tianhe-1A at the Chinese National Supercomputing Center in Tianji\cite{top500Nov2010}.
We start with a description of the design objectives of our implementation and an overview of the features of LAMMPS$_{\rm CUDA}$.
Then we discuss aspects of our GPU implementations of LAMMPS's pair force calculations.
Performance results are then presented for various MD simulations on single GPUs.
This is followed by a discussion of strategies that enable GPU-based MD codes to scale well on systems with many GPUs.
We also report and analyze LAMMPS$_{\rm CUDA}$ performance results on NCSA's Lincoln cluster, using up to 256 GPUs.
Parameters of the benchmark simulations are listed in Appendix \ref{sec:app_simulations} and hardware configuration are given in Appendix \ref{sec:app_hardware}.
\section{Design objectives, features and usage}
\label{sec:design}
Numerous GPU-MD codes have been under development during the past several years.
Some of those are new codes (HOOMD\cite{HOOMD}, AceMD\cite{ACEMD}), others are extensions or modifications of existing codes (e.g. NAMD\cite{NAMD}, Amber\cite{AMBER}, LAMMPS\cite{LAMMPS-GPU}).
Most of these projects are of limited scope and cannot compete with the rich feature sets of legacy CPU-based MD codes.
This is not surprising considering the amount of development time which has been spent on the existing codes; many of them have been under development for more than a decade.
Furthermore, some of these GPU-MD codes have been written to accelerate specific compute-intensive tasks, limiting the need to implement a broad feature set.
Our goal is to provide a GPU-MD code that can be used for simulation of a wide array of materials classes (e.g. glasses, semiconductors, metals, polymers, biomolecules, etc.) across a range of scales (atomistic, coarse-grained, mesoscopic, continuum).
LAMMPS can perform such simulations on CPU-based clusters.
It is a classical MD code that been under development since the mid 1990s, is freely-available, and includes a very rich feature set.
Since building such simulation software from scratch would be an enormous task, we instead leverage the tremendous effort that has gone into LAMMPS, and enable it to harness the compute power of GPUs.
We have written a LAMMPS "package" that can be built along with the existing LAMMPS software, thereby preserving LAMMPS' rich feature set for users while yielding tremendous computational speedups.
Other important LAMMPS features include an extensive scripting system for running simulations, and a simple-to-extend and modular code infrastructure that allows for easy integration of new features.
Most importantly it has an MPI-based parallelization infrastructure that exhibits good scaling behavior on up to thousands of nodes.
Finally, starting with an existing code like LAMMPS and building GPU versions of functions and classes one by one allows for easy code verification.
\noindent Our objectives can be summarized as follows (in order of decreasing priority):
\begin{enumerate}[(i)]
\item maintain the rich feature set and flexibility of LAMMPS,
\item achieve the highest possible speed-ups,
\item allow good parallel scalability on large GPU-based clusters,
\item minimize code changes,
\item write the code so that it is easy to maintain,
\item include GPU support for the full list of LAMMPS capabilities,
\item make the GPU capabilities easy for LAMMPS users to invoke.
\end{enumerate}
\noindent All of these design objectives have implications for design decisions, yet in many cases they are competing objectives.
For example objective (i) implies that the different operations of a simulation have to be done by different modules, and that the modules have to be able to be used in any combination requested by the user.
This in turn means that data, such as the particle positions, are loaded multiple times during a single simulation step from the device memory, which results in a considerably negative effect on the performance of the simulation.
Another slight performance hit is caused by the use of templates for the implementation of pair forces and communication routines.
While this greatly enhances maintainability, it adds some computational overhead.
By keeping full compatibility with LAMMPS we were able to minimize the GPU-related changes that users will need to make to existing input scripts.
In order to use LAMMPS$_{\rm CUDA}$ it is often enough to add the line ``accelerator cuda'' at the beginning of an existing input script.
This triggers use of GPUs for all GPU-enabled features in LAMMPS$_{\rm CUDA}$, while falling back to the original CPU version for all others.
Another big influence on design decisions comes from the limiting factors of the targeted architecture.
Since those have been discussed in detail elsewhere\cite{CUDA-PG}, here we only list the most important factors:
\begin{enumerate}[(a)]
\item in order to use the full GPU, thousands or even tens of thousands of threads are needed,
\item data transfer between the host and the GPU is slow,
\item the ratio of device memory bandwidth to computational peak performance is much smaller than on a CPU,
\item latencies of the device memory are large,
\item random memory accesses on the GPU are serialized.
\item 32 threads are executed in parallel
\end{enumerate}
Considering (b), we decided to minimize data transfers between device and host by running as many parts of the simulation as possible on the GPU.
This distinguishes our approach from other GPU extensions of existing MD codes, where only the most computationally-expensive pair forces are calculated on the GPU.
A work-flow chart of our implementation is shown in Figure \ref{fig:workflow}.
\begin{figure}[h!tb]
\begin{center}
\includegraphics[width=0.25\textwidth]{workflow}
\end{center}
\vspace{-0.5cm}
\caption{LAMMPS$_{\rm CUDA}$ work-flow, dashed boxes are done on the CPU, while solid boxes are done on the GPU.}\label{fig:workflow}
\end{figure}
Currently LAMMPS$_{\rm CUDA}$ supports 26 pair force styles; long range coulomb interactions via a particle-particle/particle-mesh (PPPM) algorithm; NVE, NVT, and NPT integrators; and a number of LAMMPS ``fixes''.
In addition, pair force calculations on the GPU can be overlapped with bonded interactions and long range coulomb interactions if those are evaluated on the CPU.
All of the bond, angle, dihedral, and improper forces available in the main LAMMPS program can be used.
Simulations can be performed in single (32 bit floats) and double (64 bit floats) precision, as well as in a mixed precision mode, where only the force calculation is done in single precision while the time integration is done in double precision.
In addition to the requirements of LAMMPS, only the CUDA toolkit (available for free from NVIDIA) is needed.
Currently only NVIDIA GPUs with a compute capability of 1.3 or higher are supported.
This includes GeForce 285, Tesla C1060 as well as GTX480 and Fermi C2050 GPUs.
The package is available under the GNU Public License and can be downloaded from {\tt http://code.google.com/p/gpulammps/}, where detailed installation instructions and feature lists can be found.
LAMMPS$_{\rm CUDA}$, which is encapsulated in the USER-CUDA package of LAMMPS, should not be confused with LAMMPS' ``GPU'' package, which has some overlapping capabilities (see Figures \ref{fig:bench_system_size} and \ref{fig:scaling}) and is also available from the same website.
\section{Pair Forces}
We analyzed two variants of short range force calculations: a cell list approach and a neighbor list approach.
While most CPU-based MD codes use a neighbor list approach for the force calculation, it has been suggested\cite{HarvestingCellLists} that the cell list approach is better suited for GPU implementations.
\subsection{Cell list approach}
The idea of the cell list approach is a spatial decomposition of the simulation box into a regular grid of small sub-cells, with a maximum number of atoms per cell.
Because LAMMPS uses neighbor lists, additional effort is required to re-order the existing data structures for the GPU calculation and to convert the data back into the original LAMMPS format for every usual computation not done on the GPU.
In order to implement this idea on the GPU, we associate every \textit{cell} with a CUDA thread \textit{block} and have each of the $n_{cell\_nmax}$ \textit{threads} of it calculate the forces for one \textit{particle} in the cell.
Furthermore it is necessary to choose the cell size $c$ (see Fig.
\ref{fig:cell_lists}(a)) and the maximum number of atoms per cell $n_{cell\_nmax}$.
For a given force cut-off radius $r_c$, we choose $c \approx 2 \: r_c$ in order to keep the average distance between particles in the cell $\approx r_c$ and to limit the frequency of re-assigning atoms to their cells.
Also, $c$ should be large enough to contain at least 32 particles in order to not to leave GPU threads idle.
Accordingly, $n_{cell\_nmax}$ is automatically chosen as a multiple of 32, depending on the particle density.
\begin{figure}[h!]
\caption{Cell list approach}
\subfigure[ 2D depiction of cell lists]
{
\label{fig:part_cell}
\includegraphics[width=0.2\textwidth]{particles_cell}
}
\subfigure[ cell update pattern]
{
\label{fig:cell_neighbor_pattern}
\includegraphics[width=0.2\textwidth]{cell_neighbor_pattern}
}
\label{fig:cell_lists}
\end{figure}
When performing the force calculations in the cell list approach, at least two more optimizations can be used.
The first is to use Newton's third law $\vec F_{ab} = - \vec F_{ba}$ to save half of the force calculation time.
In 2D, forces need to be explicitly computed for only 4 of 8 the neighboring cells, with the other 4 obtained via Newton's third law during other cells' updates.
Figure \ref{fig:cell_lists}(b) depicts an example of such an update pattern, with the explicitly-computed neighbors of cell E connected with cell E by a solid black line, and the other 4 neighbors of cell E connected with cell E by solid gray lines.
Every cell then follows this pattern, and the interactions between all neighboring cells are then considered exactly once, as verified for cell E.
In 3D, only 13 of the 26 neighboring cells are explicitly considered.
Note, however, that not every selection of 13 neighboring cells fulfills the required periodicity.
Execution of GPU thread blocks can be in any order, whether in sequence or in parallel.
Therefore, write conflicts may occur.
For example, in Figure \ref{fig:cell_lists}(b), cell A and cell D might try to update the forces in cell B at the same time.
In order to avoid such a write conflict and a resulting error in the calculation, the code has been written to execute only non-interfering groups of cells simultaneously.
If only one neighbor shell needs to be considered, there are six such groups in 2D and 18 such groups in 3D.
This does not significantly affect performance since $N$ groups are executed, each in approximately $\frac{1}{N}$ of the original time.
The second optimization is the use of shared memory for the positions of the particles in the neighboring cells.
If a cell contains more atoms than will fit in shared memory, the particles have to be loaded to shared memory in groups one after another.
For a more detailed discussion on this topic, see \cite{LarsBA}.
\subsection{Neighbor list approach}
In designing a neighbor list approach that uses blocks of threads, it becomes clear that there are two main ways that the force calculation work can be divvied up among the threads.
The first possibility is to use one thread per atom (TpA), where the thread loops over all of the neighbors of the given atom.
The second possibility is to use one block per atom (BpA), where each of the threads in the block loop over its designated portion of the neighbors of the given atom.
In the following, pseudo-code for both algorithms are given:\newline\newline
\noindent TpA algorithm:
\begin{lstlisting}
1 i = blockId*ThreadsPerBlock+threadId;
2 load(i) // coalesced access
3 for(jj = 0; jj<numneigh[i];
4 jj++) {
5 j <- neighbors[i][jj]
6 load(j) // random access
7 ftmp+=calcPairForce(i,j)
8 }
9
10 ftmp -> f[i] // coalesced access
\end{lstlisting}
\noindent BpA algorithm:
\begin{lstlisting}
1 i = blockId
2 load(i) // coalesced access
3 for(jj = 0; jj<numneigh[i];
4 jj+=ThreadsPerBlock) {
5 j <- neighbors[i][jj]
6 load(j) // random access
7 ftmp+=calcPairForce(i,j)
8 }
9 reduce(ftmp)
10 ftmp -> f[i] // coalesced access
\end{lstlisting}
Both algorithms ostensibly have the same number of instructions; however, when considering looping it becomes clear that the BpA algorithm requires the execution of a larger total number of lines of code.
The BpA algorithm also requires the relatively expensive reduction of $ftmp$ that is not required by the TpA algorithm.
BpA also requires use of a much larger total number of blocks.
For further clarification, Table \ref{tab:BpA-TpA-NExec} lists the number of times each line of code is executed, taking into account the number of blocks used, and considering that 32 threads of each block are executed in parallel.
\begin{table}
\caption{\label{tab:bf} Number of executions per line for the BpA and TpA algorithms}
\label{tab:BpA-TpA-NExec}
\begin{center}
\begin{tabular}{@{\hspace{1em}}c@{\hspace{1em}}|@{\hspace{1em}}c@{\hspace{1em}}c@{\hspace{1em}}}
\hline\hline
Lines & TpA & BpA \\
1,2,9,10: & natoms/32 & natoms\\
5,6,7: & (natoms/32)*nneigh & natoms*(nneigh/32)\\
\hline\hline
\end{tabular}
\end{center}
\end{table}
While this seems to indicate that TpA would always be faster, one has to take into account cache usage as well.
In order to reduce random accesses in the device memory while loading the neighbor atoms (limiting factor (e)), one can cache the positions using the texture cache.
(We also tested global cache on Fermi GPUs, but it turns out to be slower due to its cache line size of 128 bytes.)
This strategy improves the speed of both algorithms considerably, but it helps BpA more than TpA.
The underlying reason is that less atoms are needed simultaneously with BpA than with TpA.
As a result, BpA allows for better memory locality, and therefore the re-usage of data in the cache is increased (assuming atoms are spatially ordered).
Revisiting table \ref{tab:BpA-TpA-NExec} makes it evident that this better cache usage becomes increasingly important with an increasing number of neighbors, corresponding to an increased pair cutoff distance.
Consequently, one can expect a crossover cutoff for each type of pair force interaction, where TpA is faster for smaller cutoffs and BpA for larger.
Unfortunately it is hard to predict where this cutoff lies.
It not only depends on the complexity of the given pair force interaction, but also on the hardware architecture itself.
Therefore, a short test is the best way to determine the crossover cutoff.
The timing ratios shown in Figure \ref{fig:bench_BpAvsTpA} indicate that the force calculation time can depend significantly on the use of BpA or TpA.
Generally the differences are larger when running in single precision than when running in double precision.
For the LJ system, an increase of the cutoff from $2.5\sigma_0$ to $5.0\sigma_0$ can turn the 30\% TpA advantage into a 30\% BpA advantage.
Therefore we decided to implement both algorithms, and allow dynamic selection of the faster algorithm using a built-in mini benchmark during the setup of the simulation.
This ensures that the best possible performance is achieved over a wide range of cutoffs.
While our particular findings are true only for NVIDIA GPUs, one can expect that similar results would be found on other highly parallel architectures with comparable ratios of cache to computational power.
\begin{figure}[t]
\includegraphics[width=0.45\textwidth, ]{bench_BpAvsTpA}
\caption{Computation time comparison of BpA and TpA algorithms for two different benchmark systems in single and double precision.
The ratio of the force calculation time using the TpA algorithm and the force calculation time using the BpA algorithm is shown as a function of the cutoff.
System: LJ (32k atoms), Silicate (12k atoms) (see Appendix \ref{sec:app_simulations}); Hardware: CL (see Appendix \ref{sec:app_hardware})}
\label{fig:bench_BpAvsTpA}
\end{figure}
We tested the cell list approach and the neighbor list approach for a small LJ system as a function of cutoff radius (see Fig. \ref{fig:cell_neigh}).
For the neighbor list approach, three distinct regions can be seen, as labeled in the Figure.
For regions Ia and Ib the TpA method is faster than the BpA method.
Above $r_c=3.5\sigma$ the BpA algorithm is faster than the TpA method, so the code automatically switches to BpA, resulting in a different slope for region II.
The two different slopes in Ia and Ib are most likely a result of the limited texture cache size.
For small cutoffs, most neighbors fit into the texture cache, facilitating efficient re-usage of data.
But at some point the collective number of neighbors becomes large enough that the texture cache can no longer be used efficiently.
This changes the scaling behavior as a function of the cutoff radius.
\subsection{Comparing the cell and neighbor list approaches}
\label{sec:comp_neigh_cell}
Figure \ref{fig:cell_neigh} clearly demonstrates that the cell-list-based force evaluation is considerably slower than the neighbor-list-based approach for all cut-offs.
In this section, we will make a simple argument why this is not only true for the above example, but has to be expected in general.
Figure \ref{fig:cell_neigh} is based on an earlier program version that still featured both cell and neighbor lists.
Due to the weak performance of the cell list approach, we have completely dropped it and have focused our efforts on the optimization of the neighbor-list-based force calculation.
While further improvements might have been possible for the cell list approach as well, the superiority of the neighbor list approach appears to be inevitable, as explained below.
\begin{figure}[t]
\includegraphics[width=0.45\textwidth]{cutoff_binning_neighbor}
\caption{ Computation time comparison of cell-list-based and neighbor-list-based force calculations for a small LJ system, $n \approx 20,000$ particles.
In region I ($r_c<3.5\sigma$) the TpA algorithm is used, while in region II the BpA algorithm is used.
(The faster algorithm is automatically selected for each region.)
In sub-region Ia texture cache is used effectively by the TpA algorithm, but in region Ib the cache must be flushed frequently.
The jumps in the cell list curve are caused by the GPU requirement of $n_{cell\_nmax}$ being a multiple of 32 and the resulting unsteady proportion of started threads versus those that are actually needed. System: LJ (see Appendix \ref{sec:app_simulations}); Hardware: WS$_A$ (see Appendix \ref{sec:app_hardware})}
\label{fig:cell_neigh}
\end{figure}
Obviously, the time $t$ for processing a single interaction force consists of two parts: memory access (i.e. reading the other atom's position) and the evaluation of the force formula.
Na\"{\i}vely one might assume that the total time $T$ needed for all force calculations equals $t$ times the number of interactions.
However, both the cell and the neighbor list algorithms first load all potential interaction partners to check if they are within the cut-off radius $r_c$.
Whenever one thread finds an atom close enough ($r \leq r_c$) and evaluates the force formula, the other threads processing interactions with $r > r_c$ have to wait until every thread in the warp has completed its calculations.
Therefore, the time for both memory access and for the evaluation of the force formula scale with the number of possible interaction partners $N$, i.e. it is reasonable to say $T = t \cdot N$.
Still both factors depend on which algorithm is chosen (cell or neighbor list).
To determine which is faster, we examine the ratio of their computational times:
\begin{equation}
\frac { T_{\text{cell}} } { T_{\text{neigh}} }
=
\frac { t_{\text{cell}} } { t_{\text{neigh}} }
\cdot
\frac { N_{\text{cell}} } { N_{\text{neigh}} } \; .
\end{equation}
For geometric reasons, $\frac { N_{\text{cell}} } { N_{\text{neigh}} } = \frac{14 \cdot 3}{4 \pi} \approx 3.3 > 1$.
Figure \ref{fig:cell_benchmark_explanation}(a) illustrates the 3D situation in a 2D sketch.
The cell list approach requires loading all atom positions from $\frac{3^3 - 1}{2} + 1 = 14$ surrounding (cubic) cells, each of edge length $c = 2 r_c$, while the neighbor list includes only atoms within a sphere of radius $c$.
The cell list approach is wasteful in the sense that many non-interacting atoms are loaded into memory.
Since the cell list approach requires the loading of roughly 3.3 times more data into memory than the neighbor list approach, and since the time for the evaluation of the force formula is the same in both cases, the cell list approach can only be faster if its memory access time is smaller.
This could be possible due to coalesced memory accesses that can be done in the cell list approach.
In order to find out whether this is realistic, we model the situation with two parameters:
\begin{itemize}
\item $\alpha$: the factor by which the coalesced memory accesses are faster than random accesses ($\alpha > 1$).
\item $\gamma$: the fraction of $t$ which is assumed to be spent on memory accesses ($0 \leq \gamma \leq 1$).
\end{itemize}
Clearly, the cell lists need both high $\alpha$ and high $\gamma$ in order to gain the advantage with their faster memory accesses.
With a little algebra (see appendix \ref{sec:calc_comp}), we can quantify some limits for $\alpha$ and $\gamma$.
In order to make the cell list method viable, its memory accesses have to be at least 3.3 times faster than the memory accesses used in the neighbor list method, and at least 70\% of the total neighbor list force calculation time has to be spent on memory accesses.
In Figure \ref{fig:alphagamma} $\alpha(\gamma)$ is shown for $T_{\text{neigh}} = T_{\text{cell}}$.
While $\gamma > 70$\% is not unrealistic for computation of inexpensive pair forces, the use of texture reads in the pair force kernels limits the advantage of coalesced memory accesses considerably (i.e. decreasing $\alpha$), thus making the cell list approach always slower than the neighbor list approach.
In practice, the product of $\alpha$ and $\gamma$ is always below the solid line in Figure \ref{fig:alphagamma}, making the neighbor list approach the preferred alternative.
\begin{figure}[h!]
\subfigure[]
{
\label{fig:particles_cell_size_a}
\includegraphics[width=0.2\textwidth]{particles_cell_size_a}
}
\subfigure[]
{
\label{fig:alphagamma}
\includegraphics[width=0.23\textwidth, clip=]{celllist-alpha-gamma}
}
\caption{(a) Required memory elements, depicted in 2D \;\; (b) $\gamma$ is the fraction of the total force calculation time spend on memory accesses.
$\alpha$ is the factor by which coalesced memory accesses are faster than random memory accesses.
Along the solid line the cell-list-based force calculation is as fast as a neighbor-list-based pair force calculation.
The dotted lines are the asymptotic limits ($\gamma \approx 70\%$, $\alpha \approx 3.3$).}
\label{fig:cell_benchmark_explanation}
\end{figure}
\section{Single node performance}
\begin{figure*}[t]
\includegraphics[width=0.85\textwidth, ]{Bench-Precision-Overview}
\caption{Typical speed-up when using a single GTX 470 GPU versus a Quad-Core Intel i7 950 for various system classes.
Systems: see Figure (see Appendix \ref{sec:app_simulations}); Hardware WS$_B$ (see Appendix \ref{sec:app_hardware})}
\label{fig:bench_single_gpu}
\end{figure*}
\begin{figure}[tb]
\includegraphics[width=0.45\textwidth]{bench_size_lj}
\caption{Performance in number of atom-steps per second of a GTX 470 GPU and a single core of an i7 950 CPU.
LAMMPS$_{\rm CUDA}$ approaches its maximum performance only for system sizes larger than 200,000 particles.
The same system was also run using HOOMD version 0.9.1 and the ``GPU'' package of LAMMPS.
The CPU curve has also been plotted with a scaling factor of 40 to make it easier to see.
System: LJ (see Appendix \ref{sec:app_simulations}); Hardware: WS$_B$ (see Appendix \ref{sec:app_hardware})}\label{fig:bench_system_size}
\end{figure}
To assess the possible performance gains of harnessing GPUs, we have performed benchmark simulations of several important classes of materials.
Both the regular CPU version of LAMMPS and LAMMPS$_{\rm CUDA}$ were run on our workstation B (WS$_B$) with an Intel i7 950 quad core processor and a GTX 470 GPU from NVIDIA.
Simulations on the GPU were carried out in single, double, and mixed precision.
We compare the loop times for 10,000 simulation steps.
The results shown in Figure \ref{fig:bench_single_gpu} are proof of an impressive performance gain.
Even in the worst-case scenario, a granular simulation (which has extremely few interactions per particle), the GPU is 5.3 times as fast as the quad core CPU when using single precision and 2.0 times as fast in double precision.
In the best-case scenario the speed-up reaches a factor of 13.5 for the single precision simulation of a silicate glass involving long range coulomb interactions.
Single precision calculations are typically twice as fast as double precision calculations, while mixed precision is somewhere in between.
It is worthy to note that this factor of two between single and double precision is reached on consumer grade GeForce GPUs, despite the fact that their double precision peak performance is only 1/8th of their single precision peak performance.
This is a strong sign that LAMMPS$_{\rm CUDA}$ is memory bound.
Generally, the speed-up increases with the complexity of the interaction potential and the number of interactions per particle.
Additionally the speed-up also depends on the system size.
As stated in section \ref{sec:design} the GPU needs many threads in order to be fully utilized.
This means that the GPU cannot reach its maximum performance when there are relatively few particle-particle interactions.
This point is illustrated in Figure \ref{fig:bench_system_size}, where the number of atom-steps per second is plotted as a function of the system size.
As can be seen, at least 200,000 particles are needed to fully utilize the GPU for this Lennard-Jones system.
In contrast the CPU core is already nearly saturated with only 1,000 particles.
All systems used to produce Figure \ref{fig:bench_single_gpu} were large enough to saturate the GPU.
We have also plotted the performance curves for the GPU-MD program HOOMD (version 0.9.1) and the ``GPU'' package of LAMMPS in Figure \ref{fig:bench_system_size} for comparison purposes.
The characteristics of HOOMD are very similar to LAMMPS$_{\rm CUDA}$.
It reaches its top performance at about 200,000 particles.
Interestingly HOOMD is somewhat slower than LAMMPS$_{\rm CUDA}$ at very high particle counts, while it is significantly faster at system sizes of 16,000 particles and below.
This can probably be explained by the fact that HOOMD is a single GPU code, whereas LAMMPS$_{\rm CUDA}$ has some overhead due to its multi-GPU capabilities.
The ``GPU'' package of LAMMPS reaches its maximum performance at about 8,000 particles.
While it is faster than LAMMPS$_{\rm CUDA}$ for smaller systems (and even faster than HOOMD for fewer than 2,000 particles), it is significantly slower than LAMMPS$_{\rm CUDA}$ and HOOMD for this LJ system at large system sizes.
The reason is most likely that the ``GPU'' package of LAMMPS only off-loads the pair force calculations and the neighbor list creation to the GPU, while the rest of the calculation (e.g. communication, time integration, thermostats) is performed on the CPU.
This requires a lot of data transfers over the PCI bus, which reduces overall performance and sets an upper limit on the speed-up.
On the other hand, at very low particle counts the CPU is very efficient at doing these tasks that are less computationally demanding and memory bandwidth limited.
While a GPU has a much higher bandwidth to the device memory than does the CPU to the RAM, the whole data set can fit into the cache of the CPU for small system sizes.
So for the smallest system sizes, the CPU can handle these tasks more efficiently than the GPU, leading to the higher performance of the ``GPU'' package for small system sizes.
\section{Scaling}
\begin{table*}
\begin{tabular}{p{0.08\textwidth}p{0.45\textwidth}p{0.45\textwidth}}
&
\begin{center}{\large \bf LJ}\end{center}&
\begin{center}{\large \bf Silicate (cutoff)}\end{center}\\
\begin{center}\begin{rotate}{90} \hspace{1cm}{\large \bf \hspace{0.5cm} Weak Scaling} \end{rotate}\end{center}
&
\includegraphics[width=0.40\textwidth, ]{lj-melt-fixed-per-node3}
&
\includegraphics[width=0.40\textwidth, ]{silicate-cut-fixed-per-node}
\\
\begin{center}\begin{rotate}{90} \hspace{1cm}{\large \bf \hspace{0.5cm} Strong Scaling} \end{rotate}\end{center}
&
\includegraphics[width=0.40\textwidth, ]{lj-melt-fixed-size3}
&
\includegraphics[width=0.40\textwidth, ]{silicate-cut-fixed-size}
\\
\end{tabular}
\captionof{figure}{Multi-node scaling comparison for a fixed system size setup (strong scaling), and a constant number of atoms per node setup (weak scaling).
High parallel efficiency is harder to achieve for the strong scaling case. The insets show the parallel efficencies. Each node includes 2 GPUs and 2 Quad-Core CPUs.
Systems: LJ, silicate (cutoff) (see Appendix \ref{sec:app_simulations}); Hardware: Lincoln (see Appendix \ref{sec:app_hardware})}
\label{fig:scaling}
\end{table*}
In order to simulate large systems within a reasonable wall clock time, modern MD codes allow parallelization over multiple CPUs.
LAMMPS's spatial decomposition strategy was specifically chosen to enable this parallelization, allowing LAMMPS to run efficiently on modern HPC hardware.
Depending on the simulated system, it has been shown to have parallel efficiencies \footnote{We define atom-steps per second as number of simulated steps $s$ times the number of atoms $n$ divided by the wall clock time $t$: $k=s \cdot n \cdot t^{-1}$.
Let $k_s$ denote the atom-steps per second for a single CPU run, and let $k_m$ denote the atom-steps per second for an $m$ CPUs run.
Parallel efficiency, $p$, is then the ratio of $k_s$ to $k_m$ multiplied by $m$: $p=\frac{k_s}{k_m}m$.}
of 70\% to 95\% for up to several ten thousand CPU cores.
To split the work between the available CPUs, LAMMPS's spatial decomposition algorithm evenly divides the simulation box into as many sub-boxes as there are processors.
MPI is used for communication between processors.
During the run, each processor packs particle data into buffers for those particles that are within the interaction range of neighboring sub-boxes.
Each buffer is sent to the processor associated with each neighboring sub-box, while the corresponding data buffers from other processors are received and unpacked.
While the execution time of most parts of the simulation should in principle scale very well with the number of processors, communication time is a major exception.
With an increasing number of processors, the fraction of the total simulation time which is used for inter-processor communication increases.
This is already bad enough for CPU-based codes, where switching from 8 to 128 processors typically doubles or triples the relative portion of communication.
But for GPU-based codes, the situation is even worse since the compute-intensive parts of the simulation are executed much faster (typically by a factor of 20 to 50 times).
It is therefore understandable why it is essential to perform as much of the simulation as possible on the GPU.
Consider the following example: in a given CPU simulation, 90\% of the simulation time is spent on computing particle interaction forces.
Running only that part of the calculation on the GPU, and assuming a 20-fold speed-up in computing the forces, the overall speed-up is only a factor of 6.9.
If we then assume that with an increasing number of processors the fraction of the force calculation time drops to 85\% in the CPU version, then the overall speed-up would be only a factor of 5.2.
On top of the usual parallel efficiency loss of the CPU code, additional parallel efficiency is lost for the GPU-based code if only calculating the pair forces on the GPU.
If one processes the rest of the simulation on the GPU as well, the picture gets somewhat better.
Most of the other parts of the simulation are bandwidth bound, i.e. typical speed-ups are around 5.
Taking the same numbers as before yields an overall speed-up of 15.4 and 13.8, respectively.
So if parts of the code that are less optimal for the GPU are also ported, not only will single node performance be better, but the code should also scale much better.
While the above numbers are somewhat arbitrary, they illustrate the general trend.
In order to minimize the processing time on the host, as well as minimize the amount of data sent over the PCI bus, LAMMPS$_{\rm CUDA}$ builds the communication buffers on the GPU.
The buffers are then transferred back to the host and sent to the other processors via MPI.
Similarly, received data packages are transferred to the GPU and only opened there.
Actual measurements have been performed on NCSA's Lincoln cluster, where up to 256 GPUs on 128 nodes were used (see Figure \ref{fig:scaling}).
We compare weak and strong scaling behavior of LAMMPS$_{\rm CUDA}$ versus the CPU version of LAMMPS for two systems: LJ and silicate (cutoff).
In the weak scaling benchmark, the number of atoms per node is kept fixed, such that the system size grows with increasing number of nodes.
In this way, the approximate communication-to-calculation ratio should remain fairly constant, and the GPUs avoid underutilization issues.
In the strong scaling benchmark, the total number of atoms is kept fixed regardless of the number of nodes used.
This is done in order to see how much a given fixed-size problem can be accelerated.
Note that in Figure \ref{fig:scaling}, we plot the number of quad-core CPUs rather than the number of individual cores.
Please also note that in general the Lincoln-cluster would not be considered a GPU-``based'' cluster since the number of GPUs per node is relatively small and two GPUs share a single PCIe2.0 8x connection.
This latter issue represents a potential communication bottleneck since there are synchronization points in the code prior to data exchanges.
Consequently, both GPUs on a node attempt to transfer their buffers at the same time through the same PCIe connection.
On systems where each of the (up to four) GPUs of a node has its own dedicated PCIe2.0 16x slot, the required transfer time would be as little as one fourth of the time on Lincoln, thus allowing for even better scaling.
Since Lincoln is not intended for large-scale simulations, it features only a single data rate (SDR) InfiniBand connection with a network bandwidth that can become saturated when running very large simulations.
Nevertheless, Figure \ref{fig:scaling} shows that very good scaling is achieved on Lincoln.
There, the number of atom-steps per second (calculated by multiplying the number of atoms in the system by the number of executed time-steps, and dividing by the total execution time) is plotted against the number of GPUs and quad-core CPUs that were used.
We tested two different systems: a standard Lennard-Jones system (density 0.84~$\sigma^{-3}$, cutoff 3.0~$\sigma_0$), and a silicate system that uses the Buckingham potential and cutoff coulombic interactions (density 0.09~{\AA}$^{-3}$, cutoff 15~{\AA}).
While keeping the number of atoms per node constant, the scaling efficiency of LAMMPS$_{\rm CUDA}$ is comparable to that of regular CPU-based LAMMPS.
Even at 256 GPUs (128 nodes), a 65~\% scaling efficiency is achieved for the Lennard-Jones system that includes 500,000 atoms per node.
And a surprising 103~\% scaling efficiency is achieved for the silicate system run on 128 GPUs (64 nodes) and 34,992 atoms per node.
This means that for the silicate system, 128 GPUs achieved more than 65 times as many atom-steps per second than 2 GPUs.
In this case, a measured parallel efficiency slightly greater than unity is probably due to non-uniformities in the timing statistics caused by other jobs running on Lincoln at the same time.
We were also able to run this Lennard-Jones system with LAMMPS's ``GPU'' package.
As already seen in the single GPU performance, the GPU package is about a factor of three slower than LAMMPS$_{\rm CUDA}$ for this system.
The poorer single GPU performance leads to slightly better scaling for LAMMPS's GPU package.
Comparing the absolute performance of LAMMPS$_{\rm CUDA}$ with LAMMPS at 64 nodes gives a speed-up of 6 for the Lennard-Jones system and a speed-up of 14.75 for the silicate system.
Translating that to a comparison of GPUs versus single CPU cores means speed-ups of 24 and 59, respectively.
Such larger speed-ups are observed up to approximately 8 nodes (16 GPUs) in the strong scaling scenario, where we ran fixed-size problems of 2,048,000 Lennard-Jones atoms and 139,968 silicate atoms on an increasing number of nodes.
With 32 GPUs (16 nodes) the number of atoms per GPU gets so small (64,000 and 4,374 atoms, respectively) that the GPUs begin to be underutilized, leading to much lower parallel efficiencies (see Figure \ref{fig:bench_system_size}).
At the same time, the amount of MPI communication grows significantly.
In fact, for the silicate system with its large 15~{\AA} cutoff, each GPU starts to request not only the positions of atoms in neighboring sub-boxes, but also positions of atoms in next-nearest neighbor sub-boxes.
This explains the sharp drop in parallel efficency seen at 32 GPUs.
In consequence, 256 GPUs cannot simulate the fixed-size silicate system significantly faster than 16 GPUs.
On the other hand, those 16 GPUs on 8 nodes are faster than all 1024 cores of 128 nodes when using the regular CPU version of LAMMPS.
We also tested the ``GPU'' package of LAMMPS for strong scaling on the Lennard-Jones system. For this test, its parallel efficency is lower than that of LAMMPS$_{\rm CUDA}$ up to 32 GPUs.
For more than 32 GPUs, the ``GPU'' package shows stronger scaling than LAMMPS$_{\rm CUDA}$.
This can be ascribed to LAMMPS$_{\rm CUDA}$'s faster single node computations and subsequently higher communication-to-computation ratio.
(Note that each of the versions of LAMMPS discussed here have the same MPI communication costs.)
In LAMMPS$_{\rm CUDA}$, the time for the MPI data transfers actually reaches 50~\% of the total runtime when using 256 GPUs.
A simple consideration explains why the MPI transfers are a main obstacle for better scaling.
Since the actual transfer of data cannot be accelerated using GPUs, it constitutes the same absolute overhead as with the CPU version LAMMPS.
Considering that the rest of the code runs 15 to 60 times faster on a process-by-process basis, it is obvious that if 1~\% to 5~\% of the total time is spent on MPI transfers in the CPU LAMMPS code, communication can become the dominating time factor when using the same number of GPUs with LAMMPS$_{\rm CUDA}$.
\begin{figure}[tb]
\includegraphics[width=0.45\textwidth]{bench_mpi_lj}
\caption{Portion of runtime spent on MPI transfers, pair force calculations, and other computations in a weak scaling benchmark. The increase of the total runtime is caused almost completely by the increase in the MPI transfer times. System: LJ (see Appendix \ref{sec:app_simulations}); Hardware: Lincoln (see Appendix \ref{sec:app_hardware})}\label{fig:bench_mpi}
\end{figure}
That the MPI transfer time is indeed the main cause of the poor weak scaling performance can be shown by profiling the code.
Figure \ref{fig:bench_mpi} shows the total simulation time of the Lennard-Jones system versus the number of GPUs used.
It is broken down into the time needed for the pair force calculation, a lower estimate of the MPI transfer times and the rest.
The lower estimate of the MPI transfer time does not include any GPU$\leftrightarrow$host communication.
It only consists of the time needed to perform the MPI send and receive operations while updating the positions of atoms residing in neighboring sub-boxes.
All other MPI communication is included in the ``other'' time.
Clearly, almost all of the increase in the total time needed per simulation step can be attributed to the increase in the MPI communication time.
Furthermore at 64 GPUs a sharp increase in the MPI communication time is observed.
We presume that this can be attributed to the limited total network bandwidth of the single data rate InfiniBand installed in Lincoln.
Considering the relatively modest communication requirements of an MD simulation (at least for this simple Lennard-Jones system), this finding illustrates how important high throughput network connections are for GPU clusters.
In order to somewhat mitigate this problem, we have started to implement LAMMPS$_{\rm CUDA}$ modifications that will allow a partial overlap of force calculations and communication.
Preliminary results suggest that up to three quarters of the MPI communication time can be effectively hidden by that approach.
\begin{figure}[tb]
\includegraphics[width=0.45\textwidth]{bench_melt-billion-particles}
\caption{Loop time for 100 time-steps of a one billion particle Lennard-Jones system. On Lincoln, 288 GPUs were used. On the BlueGene/L system (with 32K processors and 64K processors) and RedStorm (with 10K processors), the regular CPU-based version of LAMMPS was used. System: LJ (see Appendix \ref{sec:app_simulations}); Hardware: Lincoln (see Appendix \ref{sec:app_hardware})}\label{fig:bench_billion}
\end{figure}
As a further example of what is possible with LAMMPS$_{\rm CUDA}$, we performed another large-scale simulation.
Using 288 GPUs on Lincoln, we ran a one billion particle Lennard-Jones system (Density: 0.844, Cutoff: 2.5~$\sigma$).
This simulation requires about 1~TB of aggregate device memory.
To the best of our knowledge, this is the largest MD simulation run on GPUs to date.
In Figure \ref{fig:bench_billion} loop times for 100 time-steps are shown for Lincoln, Red Storm (a Cray XT3 machine with 10368 processors sited at Sandia National Laboratories), and BlueGene/L (a machine with 65536 processor sited at Lawrence Livermore National Laboratory).
The data for the latter two machines was taken from the LAMMPS homepage (http://lammps.sandia.gov/bench.html).
Using 288 GPUs, Lincoln required 28.7~s to run this benchmark, landing between Red Storm using 10,000 processors (25.1~s) and the BlueGene/L machine using 32K processors (30.2~s).
\section {Conclusion}
In this paper we have presented our own implementation of a general purpose GPU-MD code that we call LAMMPS$_{\rm CUDA}$.
This code already supports 26 different force field types.
We discussed multiple approaches for performing pair force calculations and concluded that an adaptive neighbor-list-based approach yields the best results.
Specifically, we have shown that the cell list approach is generally slower.
If running on a quad-core workstation with a single GPU, users can expect a 5x to 14x reduction in time-to-solution by harnessing the GPU, depending on the simulated system class (i.e. biomolecular, polymeric, granular, metallic, semiconductor).
With a strong focus on scalability, LAMMPS$_{\rm CUDA}$ can efficiently use the upcoming generation of GPU-based hybrid clusters, such as Tianhe-1A, Nebulae and Tsubame 2.0 (the first, third, and fourth fastest supercomputers on the November 2010 Top500 list).
By performing scaling benchmarks on up to 256 GPUs, LAMMPS$_{\rm CUDA}$ was shown to achieve general speed-ups of 20x to 60x using the latest generation of C1060s versus modern CPU cores, again depending on the simulated system class.
These numbers imply that using LAMMPS$_{\rm CUDA}$ on a 32 node system with 4 GPUs per node can achieve the same overall speed as the original CPU version of LAMMPS on a conventional CPU-based cluster with 1024 nodes.
\section {Acknowledgments}
This work was partially supported by the National Center for Supercomputing Applications by providing access to the Lincoln GPU cluster.
Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy�s National Nuclear Security Administration under contract DE-AC04-94AL85000.
|
{
"timestamp": "2011-03-08T02:01:52",
"yymm": "1009",
"arxiv_id": "1009.4330",
"language": "en",
"url": "https://arxiv.org/abs/1009.4330"
}
|
\section{Introduction}
\label{Introduction}
In recent years, companies in various industries have been able to significantly improve
their inventory management processes through the integration of information technology into
their forecasting and replenishment systems, and by sharing demand-related information
with their supply chain partners, \cite{Aviv}. However, despite the benefits resulting
from the implementation of the above practices, inefficiencies still persist and are reflected in related costs.
The bullwhip effect, defined as the increase in variability along the supply chain, is
a frequent and expensive phenomenon identified as a key driver of inefficiencies associated with Supply Chain Management (SCM).
It distorts the demand signals, which causes instability in the supply chain, and increases the cost of supplying
end-customer demand.
\cite{Forrester58} was the first to popularize this phenomenon. Inspired by Forrester's work, several
researchers have studied the bullwhip effect. \cite{Sterman}
used the Beer Game, the most popular simulation of a simple production and distribution system,
to demonstrate that the bullwhip effect is a significant problem with important
managerial consequences. It results in unnecessary costs in supply chains such as inefficient use of production,
distribution and storage capacity, recruitment and training costs, increased inventory and
poor customer service levels (\cite{Metters} and \cite{Lee97b}).
\cite{Lee97a,Lee97b} identified four main causes of the bullwhip effect: demand forecasting, order batching, price fluctuation and supply shortages.
Of these, demand forecasting is recognised as one of the most important since the inventory system is directly affected by the forecasting technique chosen.
Three popular forecasting methods are commonly used: the Minimum Mean Squared Error (MMSE), Moving Average (MA) and Exponential Smoothing (ES).
\cite{Chen00a} quantify the bullwhip effect considering the MA forecast method for a simple two-stage supply chain and a first-order autoregressive demand process, AR(1). The authors show that the bullwhip effect is in part due to the effects of demand forecasting. Therefore, given complete access to customer demand information for each stage of the supply chain, the bullwhip effect can be significantly reduced. However, they also show that the bullwhip effect will exist even when demand information is shared by all stages of the supply chain and all stages use the same forecasting technique and the same inventory policy. In similar work \cite{Chen00b} quantify the bullwhip effect considering this time the ES forecast and two different demand processes: AR(1) demand process and a demand
process with a linear trend. In both works, the authors recognize an important limitation of their results: the models considers only non-optimal forecasting methods. The authors justify this limitation saying that ES and MA are commonly used in practice. Users are in general less familiar and less satisfied with more sophisticated methods like time series techniques.
\cite{Zhang04a} investigates the impact of MMSE, MA and ES forecasting methods
on the bullwhip effect for a simple inventory system in which AR(1)
demand process describes the customer demand and an Order-Up-To (OUT) inventory policy is used.
The study shows that different forecasting methods lead to bullwhip effect measures with
distinct properties in relation to lead-time and the underlying parameters of the demand
process. The author shows that MMSE forecasting method leads to the lowest
inventory cost. This result is not surprising since MMSE method is optimal when the
demand model is known to be an AR(1) process. On the other hand, if the
demand structure is not well known, the MA or ES method may
perform better than the MMSE method because they are more flexible.
Another aspect studied in relation of the bullwhip effect is the demand process. A variety of time-series demand models have appeared in the literature of inventory control and SCM. By far, the AR(1) process is the most frequently
adopted demand model to study the bullwhip effect (\cite{Chen00a,Chen00b}, \cite{Lee97a,Lee97b} and \cite{Zhang04a}).
Recent works use more sophisticated time series models like ARMA and ARIMA \citep{Box} to have more realistic demand models.
\cite{Luong} use an AR(2) and a general AR(p) model; \cite{Duc} use an ARMA(1,1) model. In all these models
an analytical derivation of the bullwhip effect measure is presented and the effects of the autoregressive coefficient on the bullwhip effect is investigated.
\cite{Zhang04b} uses an ARMA(p,q) model to study the demand evolution in supply chains. The author shows that the order history preserves the autoregressive structure of the demand. Zhang's work identifies an important application of this result relating to the quantification of the bullwhip effect. In this paper, inspired by Zhang's work, we study the theoretical and practical issues in order to measure the bullwhip effect for a generalized demand process. In addition, we programmed a function in \textsf{R} \citep{R}, called \texttt{SCperf}\footnote{See the supplementary material}, which implements the bullwhip effect and others supply chain performance variables.
It is well known that measuring the bullwhip effect is difficult in practice but the \texttt{SCperf} function overcomes this
problem thanks to the help of an \textsf{R} function (\texttt{ARMAtoMA}) which converts an ARMA process into an infinite moving average process. As far as practical applications are concerned, the economic implications of this phenomenon on the inventory cost have been considered.
Our contributions to this subject can be described as follows: first, this study hopes to improve the understanding of time series techniques. Second, we show that for certain types of demand processes the use of the optimal forecasting procedure that minimizes the mean squared forecasting error leads to significant reduction in the safety stock level. This highlights the potential economic benefits resulting from the use of this time series analysis. Finally, the \texttt{SCperf} function leads to a simple but powerful tool which can be helpful for the study of this phenomenom and other supply chain research problems.
The structure of our paper is as follows. The next section presents the inventory model. Section \ref{ARMApq case} presents a general ARMA(p,q) case with ARMA(1,1), MA(q), AR(p), AR(1) and AR(2) as particular cases. Next the economic implications are shown. The final section summarizes the main results of the research.
\section{Inventory model}
\label{Inventory model}
In this paper we consider a simple supply chain model for a single item and an OUT
inventory policy in which the retailer determines a target level or OUT level
and, for every review period, places an order sufficient to bring the inventory position back to this level.
As did \cite{Chen00b}, we consider that the ordered quantity made in period $t$ is received at the start of period $t+L$ where
$L$ is defined to be a fixed lead time plus the review period, i.e., $L$ is the lead time plus $1$.
For instance, in the case of zero lead time, $L=1$. Shortages are back-ordered and no fixed ordering cost exists.
In the remainder of the paper $L$ will call the lead time. This choise is made for sake of brevity, and should not create confusion.
The sequence of events during a replenishment cycle for each period $t$ can be described as follows: the retailer
receives orders made $L$ periods ago; the demand $d_t$ is observed and satisfied; the retailer
observes the new inventory level and finally places an order $O_t$ to the supplier. As a consequence of this sequence of events, the ordered
quantity can be written as:
\begin{equation} O_t=S_t-S_{t-1}+d_t,
\quad \label{O}
\end{equation}
where $S_t$ represents the OUT level in period $t$, i.e., the inventory position at the beginning of period
$t$. Note that in the above expression, we have implicitly assumed that the order quantity can be negative, i.e., returning items are allowed at no costs.
This unpleasant feature is needed for tractability. However, the free-return assumption
becomes negligible when the demand mean is sufficiently large. Further detail
about this assumption can be found in \cite{Lee00} and \cite{ChenLee}.
Under the OUT policy, the OUT level $S_t$ can be estimated from the observed demand as:
\begin{equation} S_t=\hat{D}^L_t+z\hat{\sigma}^L_t,
\quad \label{order_up}
\end{equation}
where $\hat{D}^L_t=\sum_{\tau=1}^{L}\hat{d}_{t+\tau}$ is an estimate of the mean demand over $L$ periods after period
$t$, $z$ is the safety factor which is a fixed constant chosen
to meet a required service level and $\hat{\sigma}^L_t=\sqrt{Var(D_t^L-\hat{D}_t^L)}$ is an estimate of
the standard deviation of $L$ periods forecast error. An OUT policy of this form is optimal when the
demand came from a normal distribution and there is no setup or fixed
order cost.
As \cite{Chen00b} mention, if the retailer follows an OUT policy of the form $S_t=D^L+z\sigma^L$,
where $D^L$ is the known mean and $\sigma^L$ is the standard deviation of the demand over $L$ periods, then
the OUT level in any period is constant and, consequently, the order is equal to the last observed
demand. Therefore, there is no bullwhip effect. However, these
values are, in general, unknown and the retailer must estimate them using some forecasting
technique. Note that the introduction of forecasting values in the calculation of $S_t$ is one of the main
causes for the variability increase along the supply chain or, in other words, the bullwhip effect.
The demand forecast is performed here by using the MMSE method. It was
shown that, for an ARMA process, the MMSE forecast for period $t+\tau$ is the conditional mean given the observed information\footnote{Box and Jenkins, 1970, pp.128.}.
Let $\digamma_t=\{d_t,d_{t-1}, ....\}$ be the information set which represents
all the information available until period $t$. Hence, the demand forecast for $\tau$ periods ahead is given by $E(d_{t+\tau}|\digamma_t)$.
In order to quantify the bullwhip effect we combine (\ref{O}) and (\ref{order_up}) to rewrite the order quantity as:
\begin{equation} O_t=(\hat{D}^L_t-\hat{D}^L_{t-1})+z(\hat{\sigma}^L_t-\hat{\sigma}^L_{t-1})+d_t.
\quad \label{order}
\end{equation}
We show later in the paper (see Lemma \ref{lem1}) that the standard deviation of lead-time forecast error remains constant over time for an ARMA(p,q) demand process. Hence, $\hat{\sigma}^L_t=\hat{\sigma}^L_{t-1}$ and the order quantity given in (\ref{order}) becomes
\begin{equation} O_t=(\hat{D}^L_t-\hat{D}^L_{t-1})+d_t.
\quad \label{order1}
\end{equation}
Let $M$ be the measure for the bullwhip effect. Since $M$ can be obtained from
the ratio between the unconditional variance of the order process to that of the demand process, we have
\begin{equation} M=\frac{Var(O_t)}{Var(d_t)}.
\quad \label{bullwhip}
\end{equation}
Note that $M$ is calculated by using the variances from both side of Equation (\ref{order1}). The fact that
$M=1$ means that there is no variance amplification, while $M>1$ means that the bullwhip effect is present. On the other hand,
$M<1$ means that the orders are smoothed if compared with the demand. The last case is less common since it is unlikely to have a situation where stages up the supply chain have a better representation of the customer demand than the first stage (i.e., the retailer).
In what follows, the corresponding bullwhip effect measure is derived for a general ARMA(p,q) demand
process and some particular cases are discussed. Since the calculation is complex, we cannot always express
this measure in a closed form. In this context, the \texttt{SCperf} function was developed to overcome this computational difficulty.
\section{ARMA(p,q) case}
\label{ARMApq case}
The demand process, $d_t$, seen by the retailer, is described by a stationary ARMA(p,q) process
as follows\footnote{Our representation differs from some works where the MA model is written
with negative coefficients, i.e., $d_t=\mu+\phi_1{d_{t-1}}+\cdots+\phi_p{d_{t-p}}+\epsilon_t-\theta_1\epsilon_{t-1}-
\cdots-\theta_q\epsilon_{t-q}$. We chose this representation to be in accordance with the \textsf{R}
software which was used to implement the bullwhip effect.}:
\begin{equation}d_t=\mu+\phi_1{d_{t-1}}+\cdots+\phi_p{d_{t-p}}+\epsilon_t+\theta_1\epsilon_{t-1}+\cdots+\theta_q\epsilon_{t-q},
\quad \label{armapq}
\end{equation}
where $\mu$ is a nonnegative constant, $\epsilon_t$ is i.i.d. normally distributed, with mean zero and variance
$\sigma_\epsilon^2$, $p$ is the autoregressive order of the process, $q$ is the moving average order of the process, $\phi_j$ is the autoregressive coefficient, and $\theta_j$ denotes the moving average coefficient. It is often useful to express (\ref{armapq}) in terms of
the lag operator, B, where $B^kd_t = d_{t-k}$. In order to do so, let $\phi(B)=1-\phi_1B-\cdots-\phi_pB^p$ and $\theta(B)=1+\theta_1B+\cdots+\theta_qB^q$. Hence, the demand process in (\ref{armapq}) can be expressed as:
$$\phi(B)d_t=\mu+\theta(B)\epsilon_t,$$
where $\phi(B)$ and $\theta(B)$ are known as the autoregressive and the moving average polynomials in the lag operator of degree $p$ and $q$.
If we substitute the lag operator by a constant $z$, we get the characteristic equations:
$$\phi(z)=1-\phi_1z-\phi_2z^2-\cdots-\phi_pz^p$$
and
$$\theta(z)=1+\theta_1z+\theta_2z^2+\cdots+\theta_qz^q.$$
The process is called the autoregressive process of order $p$,
AR(p), if $\theta(z)=1$ and a moving average process of order $q$, MA(q), if $\phi(z)=1$. We assume that the process described in (\ref{armapq}) is invertible and covariance stationary, i.e., the roots of the equations $\theta(z)=0$ and $\phi(z)=0$ must be outside the unit circle. To avoid the problem of parameter redundancy, it is assumed that the two characteristic equations share no common roots.
It is important to note that the constant $z$ in the above equations is different from the constant used to define the safety factor. We have chosen this notation to be in accordance with time series notation and we hope that this will not cause any future confusion. Using stationarity and taken expectations in (\ref{armapq}) directly it can be found that the mean of ARMA(p,q) demand process is defined by
\begin{equation}\mu_d=\frac{\mu}{1-\phi_1-\cdots-\phi_p}.
\quad \label{mean}
\end{equation}
It is known from time series theory that a stationary ARMA(p,q) demand process under the above conditions
can be written as an infinite moving average process of its errors, $MA(\infty)$, that is,
\begin{equation} d_t=\mu_d+\Sigma_{j=0}^\infty\psi_j\epsilon_{t-j},
\quad \label{MAinf}
\end{equation}
where $\mu_d$ is defined as in Equation (\ref{mean}) and the sequence $\{\psi_j\}$ in (\ref{MAinf}) is determined by the relation
$\psi(z)=\sum_{j=0}^{\infty}\psi_jz^j=\frac{\theta(z)}{\phi(z)}$, or equivalently by the identity
$$(\psi_0+\psi_1z+\psi_2z^2+\cdots)(1-\phi_1z-\phi_2z^2-\cdots-\phi_pz^p)=(1+\theta_1z+\theta_2z^2+\cdots+\theta_qz^q).$$
Equating coefficients of $z^j$, $j=0, 1, . . .,$ we find that
\begin{equation} \psi_j=\sum_{k=1}^{p}\phi_k\psi_{j-k}+\theta_j \mbox{ for } j \geq 1,
\quad \label{particular_psi}
\end{equation}
where $\theta_0=1$, $\theta_j=0$ for $j > q$, and $\psi_j=0$ for $j<0$. Note that equation (\ref{particular_psi}) is a recursive equation.
Therefore, the $\psi$-weights satisfy the homogeneous difference equation given by
\begin{equation} \psi_j-\sum_{k=1}^{p}\phi_k\psi_{j-k}=0, \mbox{ }j\geq max(p, q + 1),
\quad \label{general_psi}
\end{equation}
with initial conditions given by equation (\ref{particular_psi}). From homogeneous difference equation
theory the general solution for equation (\ref{general_psi}) can be read off directly as:
\begin{equation}\psi_j =c_1z_1^{-j} +\cdots+c_rz_p^{-j},
\quad \label{sol_psi}
\end{equation}
where $z_1,..,z_p$ are distinct roots of the polynomial $\phi(z)$ and $c_k$, for $k=1, 2, . . . , p$ are constants which depend on the initial
conditions.\footnote{In the case of the repeated root, the solution is different. See \cite{Shumway} for a brief and heuristic account of the topic. For
details about homogeneous difference equation theory the reader is referred to \cite{Mickens87}.} Now, from equation (\ref{MAinf}), the variance of the demand process can be expressed as:
\begin{equation} \sigma_d^2=\sigma^2_\epsilon\sum_{j=0}^\infty\psi^2_j.
\quad \label{vardemand}
\end{equation}
It is important to note that the $MA(\infty)$ representation depends on an infinite number of parameters and,
consequently, it is not directly useful in practical applications. On the other hand, \cite{Zhang04b}, using the $MA(\infty)$ representation, shows a property,
called by the author ARMA-in-ARMA-out (AIAO), which reveals that the order history preserves the autoregressive structure of the demand
and transforms its moving average structure according to a simple algorithm\footnote{\citealt[pp. 197]{Zhang04b}}. As the author remarks, the practical value of the AIAO property lies in its ability to make simpler the measuring of the bullwhip effect.
\begin{prop}\citep{Zhang04b} The retailer's demand process can be represented by an $MA(\infty)$ process with
respect to the retailer's full information shocks $\epsilon_t$, as in equation (\ref{MAinf}). Hence, the retailer's order $O_t$ to its supplier is
given by:
\begin{equation}
O_t=\mu_d+\sum_{j=0}^L\psi_j\epsilon_t+\sum_{j=1}^\infty\psi_{L+j}\epsilon_{t-j}
\quad \label{prop1}
\end{equation}
where the $\psi_j=0$ for $j<0$, $\psi_0=1$, and $\psi_j=\sum_{k=1}^{p}\phi_k\psi_{j-k}+\theta_j$ for $j \geq 1$.
\end{prop}
\pf See \cite{Zhang04b}. \hfill $\square$
\begin{prop} For a stationary ARMA(p,q) demand process, the measure for the bullwhip effect is defined by:
\begin{equation} M=1+\frac{2\sum_{i=0}^L\sum_{j=i+1}^L\psi_i\psi_j}{\sum_{j=0}^\infty\psi_j^2},
\quad \label{BE}
\end{equation}
where the $\psi_j=0$ for $j<0$, $\psi_0=1$, and $\psi_j=\sum_{k=1}^{p}\phi_k\psi_{j-k}+\theta_j$ for $j \geq 1$.
\end{prop}
\pf Taking the variance of the order quantity, Equation (\ref{prop1}), we have
$Var(O_t)=\sigma_\epsilon^2(\sum_{j=0}^L\psi_j)^2+\sigma_\epsilon^2\sum_{j=1}^\infty\psi_{L+j}^2
=\sigma_\epsilon^2(\sum_{j=0}^\infty\psi_j^2+2 \sum_{i=0}^L \sum_{j=i+1}^L\psi_i\psi_j)$. We complete the proof by substituting this result and (\ref{vardemand}) in (\ref{bullwhip}). \hfill $\square$
\begin{prop} The bullwhip effect increases when the lead-time $L$ increases if and only if $\psi_{L+1}\sum_{j=0}^{L}\psi_j>0$.
\end{prop}
\pf From equation (\ref{BE}), it is straightforward to see that the bullwhip effect exists, i.e., $M>1$, if and only if $\sum_{i=0}^L\sum_{j=i+1}^L\psi_i\psi_j>0$.
Let $g(L)=\sum_{i=0}^L\sum_{j=i+1}^L\psi_i\psi_j$ and $\triangle g(L)=g(L+1)-g(L)$. Then
\noindent $\triangle g(L)=\sum_{i=0}^{L+1}\sum_{j=i+1}^{L+1}\psi_i\psi_j-\sum_{i=0}^L\sum_{j=i+1}^L\psi_i\psi_j
=\psi_0(\sum_{j=1}^{L+1}\psi_j-\sum_{j=1}^L\psi_j)+\cdots+\psi_{L-1}(\sum_{j=L}^{L+1}\psi_j-\psi_L)+\psi_L\psi_{L+1}
=\psi_{L+1}\sum_{j=0}^L\psi_j$. Hence, $\triangle g(L)>0$ if and only if $\psi_{L+1}\sum_{j=0}^L\psi_j>0$. Hence, $g(L)$ is a non-decreasing function of the lead-time $L$ if and only if $\psi_{L+1}\sum_{j=0}^L\psi_j>0$. \hfill $\square$
\subsection{ARMA(1,1) case}
The stationary ARMA(1,1) demand process is described as follow:
\begin{equation}
d_t=\mu+\phi{d_{t-1}}+\epsilon_t+\theta\epsilon_{t-1}.
\quad \label{arma11}
\end{equation}
Stationarity and invertible conditions impose $|\phi|<1$
and $|\theta|<1$. It can be shown that the mean and variance of the demand process are
$\mu_d=\frac{\mu}{1-\phi_1}$ and $\sigma_d^2=\frac{(1+\theta^2+2\phi\theta)\sigma_\epsilon^2}{1-\phi^2}$, respectively.
\begin{prop} For a stationary ARMA(1,1) demand process the measure for the bullwhip effect is defined by:
\begin{equation}
M(L,\phi,\theta)=1+\frac{2(\phi+\theta)(1-\phi^L)}{(1-\phi)(1+\theta^2+2\phi\theta)}\left[1-\phi^{L+1}+\theta\phi(1-\phi^{L-1})\right].
\quad \label{bullwhip_arma11}
\end{equation}
\end{prop}
\pf Since the AR polynomial associated with (\ref{arma11}) is $\phi(z)=1-\phi z$, and its root, say $z_1$, is $z_1=\phi^{-1}$,
then the general solution for the $\psi$-weights can be written directly from equation (\ref{sol_psi}) as $\psi_j =c\phi^j$. From (\ref{particular_psi})
we find that the initial conditions are $\psi_0=1$ and $\psi_1= \phi+\theta$, which
combining with the general solution, results in $c=(\phi+\theta)/\phi$. Hence, $\psi_j =(\phi+\theta)\phi^{j-1}$ for $j\geq 1$.
Since we know $\psi_j$, we can rewrite the follow relations as:
\begin{eqnarray*}
\sum_{i=0}^L\sum_{j=i+1}^L\psi_i\psi_j&=&\psi_0\sum_{j=1}^L\psi_j+\sum_{i=1}^L\sum_{j=i+1}^L\psi_i\psi_j\\
&=&(\phi+\theta)\frac{1-\phi^L}{1-\phi}+\frac{\phi(\phi+\theta)^2(1-\phi^L)(1-\phi^{L-1})}{(1-\phi)(1-\phi^2)}\\
&=&\frac{(\phi+\theta)(1-\phi^L)}{(1-\phi)(1-\phi^2)}\left[1-\phi^{L+1}+\theta\phi(1-\phi^{L-1})
\right]
\end{eqnarray*}
and
$$\sum_{j=0}^\infty\psi^2_j=\frac{1+\theta^2+2\phi\theta}{1-\phi^2}.$$
Substituting the two above results in equation (\ref{BE}) we complete the proof. \hfill $\square$
Using a generalized formula for the variance ratio, we get a similar expression to that obtained by \cite{Duc}.
There are two other results found by the above authors which are easily verified.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.1]{fig1.eps}
\caption{Bullwhip generated with ARMA(1,1) demand process when L=1}\label{fig1}
\end{center}
\end{figure}
\bigskip
\begin{prop} The bullwhip effect exists, i.e, $M(L,\phi,\theta)>1$, if and only if, $\phi+\theta>0$.
\end{prop}
\pf \citealt[pp. 248-249]{Duc}. \hfill $\square$
\begin{prop} The bullwhip effect, measured by $M(L,\phi,\theta)$, has the following properties.
(a) If $\phi>0$, the bullwhip effect increases as $L$ increases.
(b) If $-\theta<\phi<0$ and $L$ is an odd number, the larger $L$ is, the smaller the bullwhip effect is.
(c) If $-\theta<\phi<0$ and $L$ is an even number, the larger $L$ is, the larger the bullwhip effect is.
\end{prop}
\pf \citealt[pp. 249]{Duc}. \hfill $\square$
\medskip
In conclusion the bullwhip effect occurs only when the sum of the AR parameter and the MA parameter is larger than zero ( See Figure~\ref{fig1}) and it does not always increase when the lead time $L$ increases. In fact, if $\phi+\theta>0$ and $\phi>0$ the bullwhip effect increases when the lead-time increase. However, if $-\theta<\phi<0$ and $L$ is an odd number, the bullwhip effect becomes smaller as $L$ becomes larger; if $-\theta<\phi<0$ and $L$ is an even number, the bullwhip effect becomes larger as $L$ becomes larger. Figure~\ref{fig2} represents situations where these facts are observed.
\begin{figure}[t]
\begin{center} \includegraphics[scale=0.4]{fig2.eps}
\caption{Effect of the AR coefficient on BE for different values of theta}\label{fig2}
\end{center}
\end{figure}
\subsection{MA(q) case}
The MA(q) demand process can be written as
$$d_t=\mu+\sum_{j=0}^{q}\theta_j\epsilon_{t-j}=\mu+(1+\theta_1B+\cdots+\theta_qB^q)\epsilon_t=\mu+\theta(B)\epsilon_t.$$
Since $\theta(B)$ is finite, no restrictions on
the $MA$ parameters are needed to ensure stationarity. Considering $q \rightarrow \infty$ the
infinite $MA$ representation is written as:
$$d_t=\mu_d+\sum_{j=0}^{\infty}\psi_j\epsilon_{t-j},$$
where $\psi_j=\theta_j$ for $j=0,1,..,q$ and $\psi_j=0$ for $j>q$. It
can be easily seen that $\mu_d=\mu$ and $\sigma_d^2=(1+\theta_1^2+\cdots+\theta_q^2)\sigma_\epsilon^2$.
Since the above demand process is i.i.d. the OUT level, $S_t$, is constant across all periods. Hence,
from Equation (\ref{O}), $O_t=d_t$, consequently, the bullwhip ratio equals one.
\subsection{AR(p) case}
The stationary AR(p) demand process is described as follow:
\begin{equation*}
d_t=\mu+\phi_1{d_{t-1}}+\cdots+\phi_p{d_{t-p}}+\epsilon_t
\end{equation*}
Assume that the AR parameters are such that $\{d_t\}$ is stationary. It is straightforward to verify that the $MA(\infty)$
representation is
$$d_t=\mu_d+\psi(B)\epsilon_t,$$
where $\mu_d$ is defined as in (\ref{mean}) and
$\psi(B)=\phi^{-1}(B)$. The $\psi$-weights in the $MA(\infty)$ representation
of $d_t$ are found directly from (\ref{sol_psi}) and it can be shown that the constants
are expressed by:
\begin{equation} c_i =\frac{z_i^{p-1}}{\prod_{k=1 k\neq i}^p (z_i-z_k)},
\quad \label{sol_c}
\end{equation}
where the constants terms $c_i$ sum to the unity,
$c_1+\cdots+c_p=1$, see \citealt[pp. 33-36,]{Hamilton} for details.
\subsection{AR(1) case}
The stationary AR(1) demand process is described as follows:
\begin{equation}
d_t=\mu+\phi{d_{t-1}}+\epsilon_t.
\quad \label{ar1}
\end{equation}
Stationarity condition imposes $|\phi|<1$. Using stationarity it can be shown that
the mean and the variance of the process are $\mu_d=\frac{\mu}{1-\phi_1}$ and
$\sigma^2_d=\frac{\sigma_\epsilon^2}{1-\phi_1^2}$, respectively.
\begin{prop} For a stationary AR(1) demand process the measure for the bullwhip effect is defined by:
\begin{equation} M(L,\phi)=1+\frac{2\phi(1-\phi^L)(1-\phi^{L+1})}{1-\phi}
\quad \label{bullwhip_ar1}
\end{equation}
\end{prop}
\pf As in the ARMA(1,1) case, the AR polynomial associated with
(\ref{ar1}) is $\phi(z)=1-\phi z$, and the root, say,
$z_1$, is $z_1=\phi^{-1}$. Using (\ref{sol_psi}) the general
solution is $\psi_j=c(z_1)^{-j}= c\phi_1^j$ with $\psi_0=1$ and $\psi_1 =\phi$ as
initial conditions. Combining the general solution with the initial conditions we find
$\psi_j=\phi^j$. Since $\psi_j=\phi^j$, Equation (\ref{BE}) can be expressed as:
\begin{equation}
M(L,\phi)=1+\frac{2\sum_{i=0}^L\sum_{j=i+1}^L\phi^i\phi^j}{\sum_{j=0}^\infty\phi^{2j}},
\quad \label{BE_ar1}
\end{equation}
where
\begin{eqnarray*}
\sum_{i=0}^L\sum_{j=i+1}^L\phi^i\phi^j&=&\sum_{i=0}^L\sum_{k=0}^{L-i-1}\phi^i\phi^{k+i+1}=\frac{\phi}{1-\phi}\sum_{i=0}^L\phi^{2i}(1-\phi^{L-i})\\
&=&\frac{\phi}{1-\phi} \left[\frac{1-\phi^{2(L+1)}}{1-\phi^2}-\frac{\phi^L(1-\phi^{L+1})}{1-\phi}\right]\\
&=&\frac{\phi}{1-\phi}\left[\frac{(1-\phi^L)(1-\phi^{L+1})}{1-\phi^2} \right]
\end{eqnarray*}
and $\sum_{j=0}^\infty\phi^{2j}=\frac{1}{1-\phi^2}$.
Substituting the two above results in (\ref{BE_ar1}) complete the proof. \hfill $\square$
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.4]{fig3.eps}
\caption{Relationship between the bullwhip effect and demand autocorrelation} \label{fig3}
\end{center}
\end{figure}
\begin{prop} For a stationary AR(1) demand process the bullwhip effect, measured by Equation (\ref{bullwhip_ar1}),
has the following properties:
(a) The bullwhip effect exists, i.e, $M(L,\phi)>1$, if and only if $\phi>0$.
(b) For $\phi > 0$, a longer lead-time leads to a more significant bullwhip effect.
\end{prop}
\pf Since $1-\phi>0$, $1-\phi^L>0$ and $1-\phi^{L+1}>0$ for $|\phi|<1$, it is
straightforward to see that $M(L,\phi)>1$, if and only if $\phi>0$. Let $f(L,\phi)=\phi(1-\phi^L)(1-\phi^{L+1})$
and $\triangle f(L)\equiv f(L+1,\phi)-f(L,\phi)$. Then, $\triangle f(L)=(1-\phi^2)(1-\phi^{L+1})\phi^{L+1}$.
It can be easily seen that $\triangle f(L)$ is an increasing function with respect to $L$ since $\phi>0$. Hence, the bullwhip effect, i.e,
$M(L,\phi)$, increases as $L$ increases since $\phi>0$. \hfill $\square$
\medskip
\noindent Figure \ref{fig3} depicts how the bullwhip effect generated by AR(1) demand process increases for different lead-time values, $L=1,...,6$.
We can observe that the increase of the lead-time has a strong impact on the bullwhip effect when $\phi>0.5$ and a less significant one when $\phi$ is positive and near zero and one. Therefore, as it was already noted by \cite{Zhang04a}, reduction on the lead-time can reduce the bullwhip effect if the demand autocorrelation is positive and away from zero and unity in the case of AR(1) demand process.
\subsection{AR(2) case}
The stationary AR(2) demand process satisfies:
\begin{equation}
d_t=\mu+\phi_1{d_{t-1}}+\phi_2{d_{t-2}}+\epsilon_t
\quad \label{BE_ar2}
\end{equation}
In the AR(2) case, stationarity implies that the roots
of $\phi(z)=0$ lie outside the unit circle or, equivalently,
the parameters $\phi_1$ and $\phi_2$ must lie in the triangular
region restricted by $\phi_1+\phi_2<1$, $\phi_2-\phi_1<1$ and
$|\phi_2|<1$. It can be shown that for a stationary $AR(2)$ demand process the mean and variance of the demand are
$\frac{\mu}{1-\phi_1-\phi_2}$ and $\frac{(1-\phi_2)\sigma_\epsilon^2}{(1+\phi_2)[(1-\phi_2)^2-\phi_1^2]}$, respectively.
\begin{prop} Let $z_1$ and $z_2$ be the solutions for the characteristic equation defined by the AR(2) process. For
a stationary AR(2) demand process the $\psi$-weights are defined by:
$$\psi_j =\frac{z_2^{1+j}-z_1^{1+j}}{z_1z_2(z_2-z_1)}$$
\end{prop}
\pf From Equation (\ref{general_psi}), the general solution for $\psi_j$-weights
for an AR(2) process is described by:
\begin{equation} \psi_j =c_1(z_1)^{-j}+c_2(z_2)^{-j}
\quad \label{sol_psiAR2}
\end{equation}
where
\begin{equation} z_1=\frac{-\phi_1+\sqrt{\phi_1^2+4\phi_2}}{2\phi_2},
\quad \label{z1}
\end{equation}
and
\begin{equation} z_2=\frac{-\phi_1-\sqrt{\phi_1^2+4\phi_2}}{2\phi_2}
\quad \label{z2}
\end{equation}
are the solutions for the characteristic equation
$1-\phi_1z-\phi_2z^2=0$. On the other hand, from Equation (\ref{sol_c}), the values of the constants are given by:
\begin{equation} c_1=\frac{z_1^{-1}}{z_1^{-1}-z_2^{-1}}
\quad \label{c1}
\end{equation}
and
\begin{equation} c_2=-\frac{z_2^{-1}}{z_1^{-1}-z_2^{-1}}
\quad \label{c2}
\end{equation}
Finally by replacing (\ref{z1}), (\ref{z2}), (\ref{c1}) and (\ref{c2}) in (\ref{sol_psiAR2}) we find the result. \hfill $\square$
\medskip
\noindent Note that the solution for the $\psi_j$-weights are a
function of the roots of the AR polynomial. In the AR(2) case,
the roots can be real if $\phi_1^2+4\phi_2>0$, or complex if
$\phi_1^2+4\phi_2<0$. In both cases, from a computational point of
view, the solution for the $\psi_j$-weights can be found and,
therefore, we can get a measure for the bullwhip effect. Since an explicit form
for the measure for the bullwhip effect is difficult to obtain, we investigated the relation of the
autoregressive coefficients and lead-time by numerical experimentation. For
an analytical derivation the reader is referred to \cite{Luong}.
When $\phi_1<0$, the bullwhip effect does not exist for $\phi_2\leq 0$ and for $\phi_2>0$, $\phi_2-\phi_1<1$.
On the other hand, when $\phi_1>0$ the bullwhip effect always exists for $\phi_2>0$, $\phi_1+\phi_2<1$ and for
$\phi_2<0$, $\phi_1+\phi_2<1$. The pattern shown when the lead-time is equal to one does not seem to be the same when the lead-time increases. Using the function \texttt{SCperf}, it can be verified that the there is no bullwhip effect when $\phi_1<0$ and $\phi_2\leq 0$ and always does when $\phi_1>0$,
$\phi_2>0$ and $\phi_1+\phi_2<1$. In the last case, we observe that the bullwhip effect increases when the lead-time $L$ increases, see Table \ref{tab1paper1}.
Table \ref{tab1paper1} also shows that there is no clear relation between the autoregressive parameters and the bullwhip effect when they have different signs. In these situations the bullwhip effect may or may not exist depending on the values of $\phi_1$, $\phi_2$ and $L$, and it does not always increase when lead-time increases. These remarks confirm the results pointed out by \cite{Luong}.
\ctable[ caption={Bullwhip effect generated for different AR(2)
demand process.*}, label=tab1paper1, pos=!tbp, ]{lrrrr}
{\tnote[*]{SL=0.95}}
{\FL\multicolumn{1}{l}{}&\multicolumn{1}{c}{L}&\multicolumn{1}{c}{AR(c(-0.2,0.7))}&\multicolumn{1}{c}{AR(c(0.6,-0.4))}&\multicolumn{1}{c}{AR(c(0.7,0.2))}\NN
\ML &$ 1$&$0.886667$&$1.822857$&$1.315000$\NN &$
2$&$1.222133$&$1.735086$&$1.842850$\NN &$
3$&$0.970805$&$1.170277$&$2.512887$\NN &$
4$&$1.379174$&$0.917179$&$3.291280$\NN &$
5$&$1.051166$&$0.949074$&$4.141105$\NN &$
6$&$1.450366$&$1.060235$&$5.035836$\NN &$
7$&$1.097494$&$1.117111$&$5.953552$\NN &$
8$&$1.464249$&$1.103809$&$6.877221$\NN &$
9$&$1.117408$&$1.072652$&$7.793541$\NN
&$10$&$1.447477$&$1.059437$&$8.692330$\NN \LL }
In conclusion, when both first-order and second-order AR parameters are positive, the bullwhip effect exists and it increases as lead-time goes up.
However, when the AR parameters have different signs the behaviour of the bullwhip effect is not clear. The bullwhip effect does not always exist and it is not always correct that the bullwhip effect necessarily increases when lead-time increases.
\section{Economic implications}
\label{Economic implications}
An important economic application of the use of time series methods can be seen in the safety stock level, which is the amount of
inventory that the retailer needs to keep in order to protect himself against deviations from average demand during lead time.
Let $SS=z\sigma_d\sqrt{L}$ and $SSLT=z\hat{\sigma}^L_t$ be two safety stock measures. The former
is traditionally used in some operational research manuals and it is based on the standard deviation of the demand over $L$ periods, the latter is the safety stock as defined in (\ref{order_up}) and it is based on the standard deviation of $L$ periods forecast error.
\citealt[pp. 271,]{Chen00b} pointed out that SSLT will be greater than SS, i.e., using time series analysis, the retailer will hold more safety stock to achieve the same service level. According to the authors this is because SS captures only the uncertainty due to the random error $\epsilon$ and SSLT captures this uncertainty plus the uncertainty due to the fact that the mean demand $D_t^L$ is estimated by $\hat{D}^L_t$, in our case using the MMSE forecasting method. We show by numerical experiments that for some special cases $SSLT$ is lower than $SS$ regarding lead-time and service level.
Using the \texttt{SCperf} function, it was verified that for ARMA and AR cases, high values on AR parameters and small values of lead-time result in lower $SSLT$. However, in general, there is a lead-time value for which this situation is reversed. Table \ref{tab2paper1} shows the safety stock levels SS and SSLT generated by $ARMA(0.95,0.4)$ demand
process and service level equal to $0.95$ for ten different values of lead-time, $L=1,..,10$.
For instance, for $L=2$ we have $SS=10.3$ and $SSLT=4.2$, a difference of $6$ units
which represents a saving of $59.2\%$ over SS. Note that this difference decreases when the lead-time increases until $L=6$ where we have SSLT larger than SS.
It is difficult to know for which value of lead-time SSLT becomes larger than SS. In general, it depends on the AR parameters of the demand.
For negative values of the AR parameters, it occurs for lower values of lead-time. Nevertheless, for the AR(2) case the AR parameters present a more complex relation with the performance of the SSLT.
When the first-order and second-order AR parameters are positive, the pattern is the same as the AR and ARMA case, that is, SSLT
becomes larger than SS for high values of lead-time. Moreover, when the first-order and second-order AR parameters have different signs, it is difficult to determine when the SSLT is better than SS as a measure for the safety stock level.
\ctable[ caption={Bullwhip, SS and SSLT generated by
ARMA(0.95,0.4) demand process.*}, label=tab2paper1, pos=!tbp, ]{lrrrr} {\tnote[*]{SCperf(0.95,0.4,L,0.95)}} {\FL\multicolumn{1}{l}{}&\multicolumn{1}{c}{L}&\multicolumn{1}{c}{Bullwhip}&\multicolumn{1}{c}{SS}&\multicolumn{1}{c}{SSLT}\NN
\ML
&$ 1$&$1.13711$&$ 7.299$&$ 1.645$\NN
&$ 2$&$1.44321$&$10.323$&$ 4.201$\NN
&$ 3$&$1.89270$&$12.643$&$ 7.304$\NN
&$ 4$&$2.46294$&$14.598$&$10.817$\NN
&$ 5$&$3.13393$&$16.322$&$14.652$\NN
&$ 6$&$3.88802$&$17.879$&$18.745$\NN
&$ 7$&$4.70970$&$19.312$&$23.048$\NN
&$ 8$&$5.58531$&$20.645$&$27.522$\NN
&$ 9$&$6.50289$&$21.898$&$32.137$\NN
&$10$&$7.45199$&$23.082$&$36.867$\NN
\LL
}
Table \ref{tab2paper1} shows that there is a benefit resulting from the use of SSLT instead of SS as
a measure for the safety stock level when regarding the lead-time. This benefit was verified for special demand processes where the AR parameters are high.
Moreover, if for those lead-time values where SSLT is smaller than SS, we consider the service level, it is verified that SSLT is always smaller than SS when the service level increases.
Table \ref{tab3paper1} presents SSLT and SS generated by the same demand process for $L=1,2,3$ and ten different values
of service level, $SL=0.9,0.91,...,0.99$. Note that when considering the service level, the difference between SS and SSLT increases for larger values of service level differently when lead-time is regarded. For instance, for $L=1$ and $SL=0.97$ we have $SS=8.35$ and $SSLT=1.88$. There is a difference of $6.47$ units which represents a saving of $77.46\%$ over SS.
All of these facts suggest that there is a potential benefit resulting from the use of time series analysis when regarding the lead-time
for some demand processes and, in this context, the benefit is even greater when the service level is considered.
On the other hand, the relationship between the bullwhip effect measure and the safety stock level is more complex. Although Table \ref{tab2paper1} shows a positive relation between the bullwhip effect and the safety stock level, this relationship is not completely clear as can be seen using the SCperf function for the $AR(2)$ case when $\phi_1=-0.2$ and $\phi_2=0.7$.
In conclusion, when inventory cost and service level are of primary concern the MMSE forecast should be used since it leads in some cases to lowest safety stock level. Although the MMSE forecasting requires more computational effort, the \texttt{SCperf} function implements this method in an easy way.
\begin{table}\caption{SS and SSLT generated by
different demand processes}\label{tab3paper1}
\begin{center}
\scalebox{0.7}{
\begin{tabular}{lcccccccccc}
\hline
\multicolumn{1}{l}{\bfseries Models}&
\multicolumn{1}{c}{\bfseries Service Level}&
\multicolumn{1}{c}{\bfseries }&
\multicolumn{2}{c}{\bfseries L=1}&
\multicolumn{1}{c}{\bfseries }&
\multicolumn{2}{c}{\bfseries L=2}&
\multicolumn{1}{c}{\bfseries }&
\multicolumn{2}{c}{\bfseries L=3}
\NN
\cline{2-2} \cline{4-5} \cline{7-8} \cline{10-11}
& SL & & SS & SSLT & & SS & SSLT& & SS & SSLT\NN
\hline
&$0.90$&&$ 5.687$&$1.282$&&$ 8.043$&$3.273$&&$ 9.850$&$ 5.691$\NN
&$0.91$&&$ 5.950$&$1.341$&&$ 8.414$&$3.424$&&$10.305$&$ 5.954$\NN
&$0.92$&&$ 6.235$&$1.405$&&$ 8.818$&$3.588$&&$10.800$&$ 6.239$\NN
&$0.93$&&$ 6.549$&$1.476$&&$ 9.262$&$3.769$&&$11.343$&$ 6.553$\NN
&$0.94$&&$ 6.899$&$1.555$&&$ 9.757$&$3.971$&&$11.950$&$ 6.904$\NN
$ARMA(0.95,0.4)$&$0.95$&&$ 7.299$&$1.645$&&$10.323$&$4.201$&&$12.643$&$ 7.304$\NN
&$0.96$&&$ 7.769$&$1.751$&&$10.987$&$4.471$&&$13.456$&$ 7.774$\NN
&$0.97$&&$ 8.346$&$1.881$&&$11.803$&$4.803$&&$14.456$&$ 8.352$\NN
&$0.98$&&$ 9.114$&$2.054$&&$12.889$&$5.245$&&$15.785$&$ 9.120$\NN
&$0.99$&&$10.323$&$2.326$&&$14.599$&$5.941$&&$17.881$&$10.330$\NN
\hline
\end{tabular}
}
\end{center}\end{table}
\section{Summary}
\label{Summary}
In this paper we quantify the bullwhip effect using Zhang's result for a stationary
ARMA(p,q) demand process which admits an $MA(\infty)$ representation.
It is well known that measuring the bullwhip effect is difficult in practice. We show that using
a generalized form of this measure, the computation of this ratio is simplified if compared with traditional recursive procedures.
In some particular cases we obtain explicit formulas for this ratio.
The \texttt{SCperf} function was programmed in \textsf{R} which implements the bullwhip effect.
We have evidenced that the use of this function makes possible accurate
estimations of the bullwhip effect and other supply chain performance variables. We point out that no approximation is required. Moreover, we show that for certain types of demand processes the use of MMSE considered in the model leads to a significant reduction in the safety
stock level regarding lead-time and service level. All of these observations highlight the potential economic benefits
resulting from the use of time series analysis but it depends on the underlying demand process. For instance, if we consider an ARMA(1,1) demand processes with a high AR parameter, the use of time series techniques leads to a significant reduction in
the safety stock level but this is not the case when a low AR parameter is considered.
The \texttt{SCperf} function leads to a simple but powerful tool which gives exact analytical solutions to a
set of supply chain equations, opening up a whole new range of research opportunities.
Moreover, since the function presented in this paper is easy to use, it might be used to
complement other managerial decision support tools. Finally, the code is given,
which makes, together with the fact that \textsf{R} is freeware, the whole research
reproducible by everyone. It may also be modified for specific tasks.
\section*{Acknowledgements}
The author thanks Alvaro Veiga and Pat Doody for their valuable comments on earlier versions of this paper and Brigid Crowley
for a language review. This research was supported by Brazilian State Science Foundation (CAPES) grant and, in part, by the Centre for Innovation in Distributed
Systems (CIDS - Ireland).
|
{
"timestamp": "2010-10-20T02:03:49",
"yymm": "1009",
"arxiv_id": "1009.3977",
"language": "en",
"url": "https://arxiv.org/abs/1009.3977"
}
|
\section{Case Studies}
\label{sec:case_studies}
This section briefly presents two case studies where we
use the new model checking commands. The analysis has been performed
on a 2.4GHz Intel\textsuperscript{\textregistered} Core 2 Duo processor with
2 GB of RAM.
\subsection{A Network of Medical Devices}
We apply the new Real-Time Maude commands
on a Real-Time Maude model of an interlock protocol for
a small network or medical devices, integrating an X-ray machine,
a ventilator machine, and a controller. The example was proposed by Lui
Sha, and the Real-Time Maude model is explained in \cite{phuket08}.
The ventilator machine helps a sedated patient to breathe during a surgery.
An X-ray can be taken during the surgery by pushing a button.
To allow an X-ray to be taken without blurring the picture,
the ventilator must be briefly turned off. Within a certain time bound, the X-ray
must be taken and then the ventilation machine must be restarted.
Furthermore, the ventilation machine should not be stopped too often.
The model also addresses
nondeterministic message delays and
clock \emph{drifts}.
In this model, all events take place when some ``timer''
expires or when a message arrives. Therefore, as proved in~\cite{wrla06}, the
system can be analyzed using the \emph{maximal}
time sampling strategy which advances time until the next timer expires,
so that the analyses remain sound and complete.
One time unit in the specification corresponds to one millisecond
in the case study.
\paragraph{Bounded Response Analysis.}
One requirement in this model is that
``the ventilation machine should not pause for more
than two seconds at a time.'' This can be expressed by the
bounded response formula
\[ \Box \;(\mbox{``machine is pausing''} \;\longrightarrow\;
\Diamond_{\leq 2 sec} \:\mbox{``machine is breathing''}). \]
In order to analyze this property, we first define two
state propositions, @isPausing@ and @isBreathing@, in the
expected way: @isPausing@ holds for states in which the ventilation machine
is not breathing, while @isBreathing@ holds when the ventilation machine is breathing.
The bounded response property is model checked using the
following Real-Time Maude command:
\small
\begin{alltt}
Maude> \emph{(br initState |= isPausing => <>le( 2000 ) isBreathing .)}
\end{alltt}
\normalsize
\noindent The result of this command is a path representing a counterexample
to the validity of the property:
\footnotesize
\begin{alltt}
Property not satisfied
Counterexample path:
\char123< ct : Controller | \texttt{clock} : 0, lastPauseTime : 0 >
< u : User | pushButtonTimer : 0, pushInterval : 60000 >
< vm : VentMachine | state : breathing >
< xr : X-ray | state : idle >\char125
=>[pushButton]
\char123< ct : Controller | \texttt{clock} : 0, lastPauseTime : 0 >
< u : User | pushButtonTimer : 60000, pushInterval : 60000 >
< vm : VentMachine | state : breathing >
< xr : X-ray | state : idle >
dly(pushButton,0,50,10)\char125
=>[dlyMsgArrives]
...
=>[idle]
\char123< ct : Controller | \texttt{clock} : 44000/21, lastPauseTime : 3000 >
< u : User | pushButtonTimer : 1220000/21, pushInterval : 60000 >
< vm : VentMachine | state : stopBreathing(9000/7)>
< xr : X-ray | state : idle >\char125
=>[tick]
\char123< ct : Controller | \texttt{clock} : 11000/3, lastPauseTime : 3000 >
< u : User | pushButtonTimer : 170000/3, pushInterval : 60000 >
< vm : VentMachine | state : stopBreathing(0)>
< xr : X-ray | state : idle >\char125
\end{alltt}
\normalsize
\noindent The result shows that the bounded response requirement does not hold.
This is due to the fact that the
ventilation machine may pause for 2.22 seconds, since its
internal clock is a little slow (see \cite{phuket08}).
A counterexample path is therefore produced, of which we display here only a part, showing the sequence of rules
that have been applied to reach a state where the clock
added internally to the system reaches a @clock@ value greater than 2000.
The analysis took less than a second to perform.
A similar analysis can be done to check whether the ventilation machine cannot pause
for more than 2.5 seconds. Since this property holds, the execution of the
bounded response command will simply not stop, since the
state space reachable from the initial state is not finite (i.e. due to the controller @clock@ attribute, which
just increases as time advances).
\paragraph{Minimum Separation Analysis.}
Another requirement says that the ventilator cannot pause more than once in ten minutes.
That is, the \emph{minimum separation} between two pauses is ten minutes. This property can be model checked
in Real-Time Maude as follows:
\small
\begin{alltt}
Maude> \emph{(ms initState |= isPausing separated by >= 600000 .)}
\footnotesize
Property not satisfied
Counterexample path:
\char123< ct : Controller | \texttt{clock} : 0, lastPauseTime : 0 >
< u : User | pushButtonTimer : 0, pushInterval : 60000 >
< vm : VentMachine | state : breathing >
< xr : X-ray | state : idle >\char125
=>[pushButton]
...
=>[stopBreathing]
\char123< ct : Controller | \texttt{clock} : 5951000/9, lastPauseTime : 663000 >
< u : User | pushButtonTimer : 530000/9, pushInterval : 60000 >
< vm : VentMachine | state : stopBreathing(2000)>
< xr : X-ray | state : wait(2500/3)>\char125
\end{alltt}
\normalsize
\noindent The requirement does not hold and a counterexample path is produced in
less than 10 secs, leading to a state where the internal
@Clock@ object reaches a @clock@ value smaller than $600000$, while its status is @off@.
\subsection{A Four-Way Traffic Intersection System}
In this section, we analyze a bounded response property of an object-oriented
Real-Time Maude model of a distributed fault-tolerant
four-way traffic light controller for cars and pedestrians
described in~\cite{traffic-light}.
The traffic light system for the 4-way intersection is designed as a collection of autonomous concurrent
objects that interact with each other by asynchronous message passing.
The system is highly parametric: ten different parameters can be specified for an initial state, such
as the presence of failures or emergency vehicles in the environment.
Each 4-way intersection has two roads crossing in two directions:
east-west (@EW@ in the specification) and north-south (@NS@ in the specification).
Each road has its own traffic lights. Each pedestrian light has a button that
can be pushed by a pedestrian in order to get the green light and cross the street.
The behavior of the four-way intersection is as expected.
We focus on the requirement
that ``no pedestrian should wait for more than five minutes'' to cross a road. This corresponds to
the bounded response formula
\[ \Box \;(\mbox{``pedestrian pushes the button''} \;\longrightarrow\;
\Diamond_{\leq 5 min} \:\mbox{``pedestrian light is green''}). \]
In order to analyze this property, we use the state propositions @buttonPushed@
and @pedLightGreen@ that take as parameter the direction of the crosswalk.
In less than 3 minutes, we successfully verified that the pedestrian
does not have to wait for more than 15 time units by executing the following Real-Time Maude command
(a time unit corresponds
to 15 seconds):
\small
\begin{alltt}
Maude > \emph{(br init("Imoan", minGreenTime + 2, minRedTime, 0, 0, 0, 1, 1, false, 0)
|= buttonPushed(NS) => <>le( 15 ) pedLightGreen(NS) .)}
Property satisfied
\end{alltt}
\normalsize
\noindent Furthermore, executing the same command, but for 14 time units, returned a counterexample.
\section{Related Work}
\label{sec:related}
There are several works determining decidable fragments of timed
temporal logics (e.g., \cite{bouyer:phd,quaknine:mtl}) in order to support
model checking algorithms for real-time systems. The tools
\textsc{Kronos}~\cite{kronos} and \textsc{REDLIB}~\cite{wang:redlib}
are two TCTL (timed CTL) model checkers for timed automata. The
popular timed-automaton-based tool
\textsc{Uppaal}~\cite{uppaalTutorial} provides model checking only for
a ``reachability subset'' of TCTL that does not include bounded
response or minimum separation.
The contrast to our work is already explained in the introduction.
Whereas the timed automaton formalism is quite restrictive
for the exact purpose of achieving decidability of analyses,
Real-Time Maude, and even its flat object-oriented subset considered in this paper,
is a much more expressive model. The cost of this expressiveness is of course
that most properties are in general undecidable for Real-Time Maude. So also for
the model checking commands in this paper, which are not guaranteed to terminate for
many Real-Time Maude models. Furthermore, since for dense time, Real-Time Maude
executes the tick rules according to a time sampling strategy, we must also
prove that, even when terminating, our model checking analyses are both sound and complete,
using, e.g., the techniques in~\cite{wrla06}.
Another obvious difference is that
we are covering only a fairly small, but important, subset of a MTL.
\newcommand{\ignore}[1]{}
\ignore{
In this paper we deal with model checking MTL properties for real-time
systems. Decidability of MTL over finite timed words was shown
in~\cite{quaknine:mtl}. First steps towards robust model-checking of
nested MTL properties for timed automata is described
in~\cite{Bouyer:robustmodel-checking}. An overview on linear-time
temporal logics for real-time systems, covering subsets and
extensions of MTL and including some decidability and undecidability
results (for the pointwise and for the continuous semantics), can be
found in \cite{bouyer:phd}. The complexity of model-checking formulas
of MTL, MITL, and TCTL over restricted sets of timed paths is the
topic of~\cite{raskin:restricted}.
Related work on model checking timed properties of real-time systems
include \emph{timed-automata based} tools for TCTL, a timed extension
of CTL.
\textsc{Kronos}~\cite{kronos} is a verification tool for real-time
systems developed at \textsc{Verimag}. In \textsc{Kronos}, components
of real-time systems are modeled by timed automata. Correctness
requirements are expressed in TCTL. The model checking algorithm
implemented in \textsc{Kronos} for checking TCTL for timed automata is
based upon a \emph{symbolic} representation of the infinite state
space by sets of linear constraints.
\textsc{Uppaal}~\cite{uppaalTutorial} is an integrated tool
environment for modeling, validation, and verification of real-time
systems. Systems are specified as networks of timed automata, extended
with data types (bounded integers, arrays, etc.). The tool offers
different alternatives for state space representation, e.g., using
difference bound matrices, under-, and over-approximations. It
provides an efficient \emph{symbolic} model checking procedure for a
subset of non-nested TCTL properties. Nested TCTL properties are not
supported by \textsc{Uppaal}.
\textsc{REDLIB}~\cite{wang:redlib} is a library constructed out of the
TCTL model checker \textsc{RED}. It uses BDD-like diagrams for the
efficient representation and manipulation of dense-time state-spaces.
Full TCTL model checking is provided for timed automata.
}
\section{Concluding Remarks}
\label{sec:conclusion}
This paper has explained how we have enriched
the important class of flat object-oriented Real-Time Maude models
with model checking features for bounded response and
minimum separation properties.
Object-oriented Real-Time Maude
specifications capture many systems
that cannot be specified as timed automata; indeed, all advanced
Real-Time Maude applications have been so specified. It is therefore
not surprising that the model checking problems we address are undecidable
in general. Therefore, our model checking analyses may fail to terminate, although they will
terminate if the properties do \emph{not} hold. Furthermore,
our model checking commands are executed with a selected time sampling strategy,
so that only a subset of all possible behaviors are analyzed. Hence,
our analyses may be incomplete or unsound. Nevertheless, for object-oriented
specifications we have identified easily checkable conditions that ensure
soundness and completeness of (untimed) model checking. Further on the positive
side, we have shown that (with reasonable assumptions on the treatment of dense time),
our model checking analyses terminate when the reachable state space is finite.
The implementation of our model checking procedures
follows a transformational approach that takes advantage of Maude's high
performance search command by transforming an MTL model
checking problem into checking the validity of an invariant property.
We proved the correctness of these transformations under mild conditions,
such as tick-invariance and time divergence.
The model checking commands have been integrated into Real-Time Maude and
have been successfully used to model check
a small network of medical devices~\cite{phuket08}, as well as on a larger model of a
traffic intersection system~\cite{traffic-light}.
The present work is just our first foray into
model checking metric temporal logic properties for Real-Time Maude specifications.
Much work remains ahead. First of all, we should extend the class
of MTL formulas we can model check, and extend the classes of
Real-Time Maude models for which such model checking can be performed.
For example, if the present techniques could be extended to
\emph{non-flat} (or \emph{hierarchical} ``Russian dolls'') object-oriented
Real-Time Maude specifications, then we would get for free model checkers
for these properties for both behavioral AADL models and hierarchical Ptolemy II DE models.
We should also extend the commands to analyze only paths up to a certain duration,
so that the reachable state space becomes finite.
The correctness proofs in this paper all deal with correctness w.r.t.\ the executed paths.
We must of course further investigate the soundness and completeness of
such analyses w.r.t.\ all possible behaviors of a system.
\paragraph{Acknowledgments.} We thank the anonymous reviewers for very
helpful comments on a previous version of this paper, and
gratefully acknowledge financial support by the Research Council of Norway
through the Rhytm project, and by the Research Council of Norway and the
German Academic Exchange Service (DAAD) through the DAADppp project
"Hybrid Systems Modeling and Analysis
with Rewriting Techniques (HySmart)."
\ignore{
The general transformational approach for model checking MTL properties,
presented in this paper, can be applied to other classes of
MTL properties. We plan for the future to extend this approach to
other MTL properties, such as classes of nested bounded until
(i.e., $(a\ \textit{U}_{\leq r_b}\ b)\ \textit{U}_{\leq r_c}$ ), or properties
that are required by other case studies. It would be also
interesting to extend the bounded response and
minimum separation commands to general interval
bounds on the temporal operator appearing in the formula.
Furthermore, since the model checking procedures are available
in Real-Time Maude as new commands, it will be easy and
interesting to test them on more case studies.
}
\section{Correctness of the Transformations}
\label{sec:proof}
In this section we give the correctness proofs for the
$\textit{BR}$-transformation and the $\textit{MS}$-transformation.
\else
\section{Correctness of Bounded Response Model Checking}
\label{sec:proof}
In this section we give the correctness proof for
our bounded response model checking. The correctness proof for minimum separation,
which we omit due to lack of space, is quite similar, and can be found
in an extended version of this paper~\cite{MSproof}.
\fi
To increase readability, in the following we use the notation
$\pi\models\phi$ instead of $\mathcal{R},L_{\Pi},\pi\models\phi$ if
$\cal R$ and $L_\Pi$ are clear from the context.
\iflong
\subsection{Bounded Response}
\fi
The following lemma states that the
\textit{BR}-transformation only adds some observators to the original
systems, without modifying its behavior.
\begin{lem}
\label{lem:protectedBR}
Let $\cal R$ be a real-time rewrite theory,
$L_{\Pi}$ with $p,q\in\Pi$ a labeling function for $\cal R$, and
let $\{t_0\}$ be an initial state for $\cal R$. Let $\widetilde{\cal
R}$, $\widetilde{L}_{\Pi}$, and $\{\widetilde{t}_0\}$ be the result of the
$\textit{BR}$-transformation applied to $\cal R$, $L_{\Pi}$, and $t_0$.
Then
for each path
$
\{t_0\} \stackrel{r_0}{\rightarrow} \{t_1\}
\stackrel{r_1}{\rightarrow} \ldots
$
in $\cal R$ there is a path
$
\{\widetilde{t}_0\} \stackrel{r_0}{\rightarrow} \{\widetilde{t}_1\}
\stackrel{r_1}{\rightarrow} \ldots
$
in $\widetilde{\cal{R}}$ such that,
for all $i$, there exists $t_i'$ with $\widetilde{t}_i =
t_i\
t_i'$ and vice versa,
for all paths
$
\{\widetilde{t}_0\} \stackrel{r_0}{\rightarrow} \{\widetilde{t}_1\}
\stackrel{r_1}{\rightarrow} \ldots
$
in $\widetilde{\cal R}$ there is a path
$
\{t_0\} \stackrel{r_0}{\rightarrow} \{t_1\}
\stackrel{r_1}{\rightarrow} \ldots
$
in $\cal R$ such that, for all $i$, $\widetilde{t}_i =
t_i\
t_i'$ for some $t_i'$.
\end{lem}
\begin{proof}
Adding the clock class and a clock object to the initial state
does not affect
the original part of the state, and defining @mte@ of the additional clocks to be
the infinity value @INF@ ensures that the new clocks don't modify the timed behavior of the
(original) system.
Furthermore, the transformation replaces each original rule by a
number of new rules, such that (1) each new rule acts on the
original state part as the original rule, and (2) for each original
rule and each extended state to which the original rule is
applicable there is exactly one new rule that is applicable. (1)
assures that the new rewrites yield the same result for the original
part of the state and (2) assures that no original paths are blocked by the new
rules. Thus the transformation does not modify the original
behavior.\\[2ex]
``$\rightarrow$'': Let $\{t_0\}
\stackrel{r_0}{\rightarrow} \{t_1\}
\stackrel{r_1}{\rightarrow} \ldots $ be a path of $\cal
R$. We define
$$\widetilde{t}_i =
t_i
\texttt{ < } c_{\textit{BR}} \texttt{ : Clock | clock : }x_i\texttt{, status : }y_i \texttt{ >}$$
\noindent for all $i$ with $\; x_i \in \mathbb{T}_{\cal{R},\texttt{Time}}\;$ and $\;y_i
\in \mathbb{T}_{\cal{R},\texttt{OnOff}}\;$ given inductively as
follows:
\begin{itemize}
\item $x_0 = 0$, and $y_0=\texttt{on}$ if $p \in L_{\Pi}(\{t_0\}) \land
q \not\in L_{\Pi}(\{t_0\})$ and $y_0=\texttt{off}$ otherwise.
\item For all $i$, if there is a tick rule yielding the rewrite $\{t_i\}
\stackrel{r_i}{\rightarrow} \{t_{i+1}\}$, then we distinguish
between the following cases:
\begin{itemize}
\item If $y_i = \texttt{on}$ and $x_i\leq r$, then we define
$y_{i+1} = \texttt{on}$ and $x_{i+1} = x_i + r_i$. \newline
Note that with the
definition of the \texttt{delta} equation we have
$\{\widetilde{t}_i\}\stackrel{r_i}{\rightarrow}\{\widetilde{t}_{i+1}\}$.
\item If $y_i = \texttt{on}$ and $x_i > r$, then we define
$y_{i+1} = \texttt{on}$ and $x_{i+1} = x_i$. \newline
Note that with the
definition of \texttt{delta} we have
$\{\widetilde{t}_i\}\stackrel{r_i}{\rightarrow}\{\widetilde{t}_{i+1}\}$.
\item Else, if $y_i = \texttt{off}$, we define
$y_{i+1} = \texttt{off}$ and $x_{i+1} = x_i$. \newline
With the
definition of the \texttt{delta} equation we have
$\{\widetilde{t}_i\}\stackrel{r_i}{\rightarrow}\{\widetilde{t}_{i+1}\}$.
\end{itemize}
\item For all $i$, otherwise there is an instantaneous rule
$t\; @=>@\; t' @ if @cond\;$ or $\;@{@ t@} => {@t'@} if @cond$, yielding the
rewrite $\{t_i\} \stackrel{r_i}{\rightarrow}\{t_{i+1}\}$ with $r_i
= 0$.
\begin{itemize}
\item If $y_i = \texttt{on}$ and $\{t_{i+1}\} \texttt{ |= } q \texttt{ =/= true}$
then we set $y_{i+1} = \texttt{on}$ and $x_{i+1}=x_i$. \newline
Note that
the first replacement of the original rule yields
$\{\widetilde{t}_i\}\stackrel{r_i}{\rightarrow}\{\widetilde{t}_{i+1}\}$.
\item If $y_i = \texttt{on}$ and $\{t_{i+1}\} \texttt{ |= } q$
then we set $y_{i+1} = \texttt{off}$ and $x_{i+1}=x_i$. \newline
Note that
the second replacement of the original rule yields
$\{\widetilde{t}_i\}\stackrel{r_i}{\rightarrow}\{\widetilde{t}_{i+1}\}$.
\item If $y_i = \texttt{off}$, $\{t_{i+1}\} \texttt{ |= } p$, and
$\{t_{i+1}\} \texttt{ |= } q \texttt{ =/= true}$ then we set
$y_{i+1} = \texttt{on}$ and $x_{i+1}=0$. \newline
Note that the third
replacement of the original rule yields
$\{\widetilde{t}_i\}\stackrel{r_i}{\rightarrow}\{\widetilde{t}_{i+1}\}$.
\item Else, if $y_i = \texttt{off}$ and either $\{t_{i+1}\} \texttt{
|= } q$ or $\{t_{i+1}\} \texttt{ |= } p \texttt{ =/= true}$ then
we set $y_{i+1} = \texttt{off}$ and $x_{i+1}=x_i$. \newline
Note that the fourth
replacement of the original rule yields
$\{\widetilde{t}_i\}\stackrel{r_i}{\rightarrow}\{\widetilde{t}_{i+1}\}$.
\end{itemize}
\end{itemize}
Above we made use of the fact that by definition for each $i$, the
corresponding labeling $\widetilde{L}_{\Pi}(\{\widetilde{t}_i\})$
in $\widetilde{\cal{R}}$ is equal to
$L_{\Pi}(\{t_i\})$. Clearly, all $\{\widetilde{t}_i\}$
are states of $\widetilde{\cal R}$. Especially,
$\{\widetilde{t}_0\}$ results from $\{t_0\}$ by the $\textit{BR}$-transformation. Thus
$\{\widetilde{t}_0\} \stackrel{r_0}{\rightarrow}
\{\widetilde{t}_1\} \stackrel{r_1}{\rightarrow} \ldots$ is a
path of
$\widetilde{\cal R}$.\\[2ex]
``$\leftarrow$'': Given a path $\{\widetilde{t}_0\}
\stackrel{r_0}{\rightarrow} \{\widetilde{t}_1\}
\stackrel{r_1}{\rightarrow} \ldots$ of $\widetilde{\cal R}$ such that
$$\widetilde{t}_i =
t_i
\texttt{ < } c_{\textit{BR}} \texttt{ : Clock | clock : }x_i\texttt{, status : }y_i \texttt{ >}$$
\noindent for each $i$, we show that $\{t_0\}
\stackrel{r_0}{\rightarrow} \{t_1\}
\stackrel{r_1}{\rightarrow} \ldots $ is a path of $\cal
R$.
\begin{itemize}
\item For all $i$, if
$\{\widetilde{t}_i\}\stackrel{r_i}{\rightarrow}\{\widetilde{t}_{i+1}\}$
can be gained by a tick rule in $\widetilde{\cal R}$, then clearly also
$\{t_i\}\stackrel{r_i}{\rightarrow}\{t_{i+1}\}$
can be gained by a tick rule in $\cal R$.
\item Otherwise if
$\{\widetilde{t}_i\}\stackrel{r_i}{\rightarrow}\{\widetilde{t}_{i+1}\}$
can be gained by an instantaneous rule in $\widetilde{\cal R}$, then
the original rule which got replaced by the above one yields
$\{t_i\}\stackrel{r_i}{\rightarrow}\{t_{i+1}\}$
in $\cal R$.
\end{itemize}
\end{proof}
The following lemma clarifies the semantics of the bounded response
property: On the one hand, if along a path after a $p$ event $r$ time
long no $q$ event occurs, then the path is a counterexample for the
property. On the other hand, if a path violates the bounded response
property, then either after a $p$ event $r$ time long no $q$ event
occurs, or the path is time-convergent and violates the unbounded
property $\Box\ (p\ \rightarrow (\Diamond\ q))$.
\begin{lem}
\label{lem:br_eq}
Let $\cal{R}$ be a real-time rewrite theory, $L_{\Pi}$ with $p,q\in\Pi$ a labeling
function for $\cal R$, and $\pi = \{t_0\} \stackrel{r_0}{\rightarrow} \{t_1\}
\stackrel{r_1}{\rightarrow} \ldots $ a path of $\cal R$.
Then
\[
\left[ \exists i,j.\ 0\leq i < j \ \land\ (\pi^i \models p) \
\land\ \left(\forall i\leq k \leq j.\ \pi^k\not\models q\right) \ \land \
\sum_{k=i}^{j-1}r_k > r\right] \quad \longrightarrow \quad
\left[\pi \not\models
\Box\ (p\ \rightarrow (\Diamond_{\leq r}\ q)) \right]
\]
and
\[
\begin{array}{l}
\left[ \pi \not\models
\Box\ (p\ \rightarrow (\Diamond_{\leq r}\ q))\right]
\ \ \longrightarrow \ \
\left[ \exists i,j.\ 0\leq i < j \ \land\ (\pi^i \models p) \
\land\ \left(\forall i\leq k \leq j.\ \pi^k\not\models q\right) \ \land \
\sum_{k=i}^{j-1}r_k > r\right]\\
\phantom{\left[ \pi \not\models
\Box\ (p\ \rightarrow (\Diamond_{\leq r}\ q))\right]
\ \ \longrightarrow \ \
}
\lor \ \left[\pi\not\models \Box\ (p\ \rightarrow (\Diamond\ q))\right]\ .
\end{array}
\]
\end{lem}
\begin{proof}
For the first implication, due to the semantics of MTL the following holds:
\[
\begin{array}{ll}
\exists i,j.\ 0\leq i < j \ \land\ (\pi^i \models p) \
\land\ \left(\forall i\leq k \leq j.\ \pi^k\not\models q\right) \ \land \
\sum_{k=i}^{j-1}r_k > r \ . & \rightarrow \\
\exists i.\ (\pi^i \models
p)\ \land\
\forall j\geq i.\ \left(\sum_{k=i}^{j-1}r_k \leq r \rightarrow
\pi^j\not\models q\right) & \rightarrow \\
\exists i.\ (\pi^i \models
p)\ \land\
(\pi^i \not\models \Diamond_{\leq r}\ q) & \rightarrow \\
\exists i.\ \pi^i \models
\neg (p\ \rightarrow
(\Diamond_{\leq r}\ q)) & \rightarrow \\
\pi \not\models
\Box\ (p\ \rightarrow (\Diamond_{\leq r}\ q)) \ .
\end{array}
\]
For the other direction,
\[
\begin{array}{ll}
\pi \not\models
\Box\ (p\ \rightarrow (\Diamond_{\leq r}\ q))
& \rightarrow \
\exists i.\ \pi^i \models
\neg (p\ \rightarrow
(\Diamond_{\leq r}\ q)) \\
& \rightarrow\
\exists i.\ (\pi^i \models
p)\ \land\
(\pi^i \not\models \Diamond_{\leq r}\ q) \\
& \rightarrow\
\exists i.\ (\pi^i \models
p)\ \land\
\forall j\geq i.\ \left(\sum_{k=i}^{j-1}r_k \leq r \rightarrow
\pi^j\not\models q\right) \ .
\end{array}
\]
Let $i$ be such an index with $\pi^i \models p$ and $\forall j\geq i.\
(\sum_{k=i}^{j-1}r_k \leq r \rightarrow \pi^j\not\models q)$. If
$\pi\not\models \Box\ (p\ \rightarrow (\Diamond\ q))$ then we are
ready. So assume $\pi\models \Box\ (p\ \rightarrow (\Diamond\
q))$, implying that there is a smallest index $l\geq i$ with
$\pi^l\models q$. From the above it follows that $\sum_{k=i}^l r_k > r$.
Note that by definition $r>0$ and thus $l>i$. Let $j=l-1$. From the
minimality of $l$ we first conclude that $\forall i\leq k \leq j.\
\pi^k\not\models q$. From the minimality of $l$ we furthermore
conclude that the rewrite
$\{\widetilde{t}_j\}\rightarrow\{\widetilde{t}_l\}$ is an
instantaneous step, and thus $\sum_{k=i}^j r_k = \sum_{k=i}^l r_k >
r$. That means,
\[
\exists i,j.\ 0\leq i < j
\ \land\ (\pi^i \models p) \ \land\ \left(\forall i\leq k \leq j.\
\pi^k\not\models q\right) \ \land \ \sum_{k=i}^{j-1}r_k > r \ .
\]
\end{proof}
The following main theorem formalizes the correctness of our transformation:
Firstly, if the bounded response property holds, then the model
checking algorithms will not provide any counterexample. Secondly, if
the bounded response model checking algorithm does not find any
counterexample, and if there are no time-convergent counterexamples,
then the property holds.
\begin{theorem}
\label{lem:extensionBR}
Let $\cal R$ be a real-time rewrite theory, $L_{\Pi}$ a labeling function
for $\cal R$ with $p,q\in\Pi$, and $\{t_0\}$ an initial state of $\cal
R$. Let $\widetilde{\cal{R}}$, $\widetilde{L}_{\Pi}$, and $\{\widetilde{t}_0\}$
be the result of the $\textit{BR}$-transformation applied to $\cal R$,
$L_{\Pi}$, and $\{t_0\}$. Then
$$
{\cal R},L_{\Pi},\{t_0\} \models \Box\ (p\ \rightarrow
(\Diamond_{\leq r}\ q)) \quad \longrightarrow \quad \widetilde{\cal
R},\widetilde{L}_{\Pi},\{\widetilde{t}_0\} \models \Box\
(\textit{clock}(c_{\textit{BR}}) \leq r),
$$
and
$$
\widetilde{\cal
R},\widetilde{L}_{\Pi},\{\widetilde{t}_0\} \models (\Box\ (p \rightarrow (\Diamond\ q))) \land (\Box\
(\textit{clock}(c_{\textit{BR}}) \leq r))
\quad \longrightarrow \quad
{\cal R},L_{\Pi},\{t_0\} \models \Box\ (p\ \rightarrow
(\Diamond_{\leq r}\ q)) ,
$$
where $\textit{clock}(c_{\textit{BR}})$ denotes the value of the @clock@
attribute of the clock object $c_{\textit{BR}}$.
\end{theorem}
\begin{proof}
For the first statement we show that
\[
\widetilde{\cal
R},\widetilde{L}_{\Pi},\{\widetilde{t}_0\} \not\models \Box\
(\textit{clock}(c_{\textit{BR}}) \leq r)
\]
implies
\[
{\cal R},L_{\Pi},\{t_0\}
\not\models \Box\ (p\ \rightarrow (\Diamond_{\leq r}\ q))\ .
\]
Thus assume $ \widetilde{\cal R},\widetilde{L}_{\Pi},\{\widetilde{t}_0\}
\not\models \Box\ (\textit{clock}(c_{\textit{BR}}) \leq r)$. That means,
there exists a path $\widetilde{\pi} = \{\widetilde{t}_0\}
\stackrel{r_0}{\rightarrow} \{\widetilde{t}_1\}
\stackrel{r_1}{\rightarrow} \ldots $ of $\widetilde{\cal R}$ with
$\widetilde{t}_i =
t_i
\texttt{ < } c_{\textit{BR}} \texttt{ : Clock | clock : }x_i\texttt{, status : }y_i \texttt{ >}$
and a smallest index $j$ such that
$x_j>r$. Since the clock value is initially $0$ and it increases
only due to tick rules if the clock is on, the clock must have been
switched on at some point before $j$. Furthermore, since $j$ is
minimal, the clock is continuously on from the last point where it
was switched on till $\widetilde{t}_j$.
Assume $i<j$ to be the smallest index such that the clock is
continuously on from $\widetilde{t}_i$ till $\widetilde{t}_j$.
Either $i$ is $0$ and the initial state satisfies $p\land \neg q$
and $x_i=0$, or $i>0$ and the rewrite from the $(i-1)$th state to
the $i$th state switched the clock from off to on and reset it to
$0$. In the latter case the corresponding rewrite has the condition
that $p\land\neg q$ holds in the $i$th state. Thus $p\land\neg q
\land x_i=0$ holds in state $\widetilde{t}_i$. The clock was kept
on from state $\widetilde{t}_i$ till state $\widetilde{t}_j$. The
only rules yielding this behavior are the tick rules increasing the
clock value with the duration of the rewrite, and instantaneous
rules assuring the invariance of $\neg q$ and letting the clock
value untouched. Due to tick-invariance, tick rules cannot
cause any change in the validity of the propositions, and $\neg q$ holds
all the way from the $i$th till the $j$th state. Furthermore, the
clock value at state $j$ is the sum of the durations of the rewrites
from the $i$th to the $j$th state. Thus
\[
\exists i,j.\ 0\leq i < j \ \land\ (\widetilde{\pi}^i \models p) \
\land\ \left(\forall i\leq k \leq j.\ \widetilde{\pi}^k\not\models
q\right) \ \land \ \sum_{k=i}^{j-1}r_k > r
\]
holds and with Lemma \ref{lem:br_eq} we get $ \widetilde{\pi}
\not\models \Box\ (p\ \rightarrow (\Diamond_{\leq r}\ q))$.
Using Lemma~\ref{lem:protectedBR} we conclude that there is also a
path $\pi$ of $\cal R$ such that
$
\pi \not\models
\Box\ (p\ \rightarrow (\Diamond_{\leq r}\ q))
$
and thus
$
{\cal R}, L_{\Pi},\{t_0\} \not\models
\Box\ (p\ \rightarrow (\Diamond_{\leq r}\ q))$.\\[2ex]
For the second statement assume that
\[
{\cal R},L_{\Pi},\{t_0\}
\not\models \Box\ (p\ \rightarrow (\Diamond_{\leq r}\ q))
\]
holds. We show that it implies
\[
\widetilde{\cal
R},\widetilde{L}_{\Pi},\{\widetilde{t}_0\} \not\models (\Box (p
\rightarrow (\Diamond\ q))) \land (\Box\ (\textit{clock}(c_{\textit{BR}})
\leq r))\ .
\]
Due to the assumption there exists a path $\pi = \{t_0\}
\stackrel{r_0}{\rightarrow} \{t_1\} \stackrel{r_1}{\rightarrow}
\ldots $ of $\cal R$ violating $\Box\ (p\ \rightarrow
(\Diamond_{\leq r}\ q))$. Now, either $ \widetilde{\cal
R},\widetilde{L}_{\Pi},\{\widetilde{t}_0\} \not\models \Box (p
\rightarrow (\Diamond\ q))$ and we are ready, or due to
Lemma~\ref{lem:protectedBR} there exists a path $\widetilde{\pi}
= \{\widetilde{t}_0\} \stackrel{r_0}{\rightarrow}
\{\widetilde{t}_1\} \stackrel{r_1}{\rightarrow} \ldots $ of
$\widetilde{\cal R}$ also violating $\Box\ (p\ \rightarrow
(\Diamond_{\leq r}\ q))$. With Lemma~\ref{lem:br_eq} we get
\[
\exists i,j.\ 0\leq i < j \ \land\ (\widetilde{\pi}^i \models p) \
\land\ \left(\forall i\leq k \leq j.\ \widetilde{\pi}^k\not\models
q\right) \ \land \ \sum_{k=i}^{j-1}r_k > r.
\]
Let $i$ and $j$ be the smallest indices satisfying the above
condition.
\begin{itemize}
\item If $i=0$ then by the fact that $\widetilde{\pi}^i\models p
\land \neg q$ we have by definition that the clock in $\widetilde{t}_0$ is on
and has the value $0$.
\item If $i>0$ and for all $n<i$, $\widetilde{t}_n$ does not satisfy
$p\land\neg q$, then by definition of the initial state the clock is
initially off and the clock does not get switched
on until the $(i-1)$th state, thus the clock is off in the $(i-1)$th state.
\item If $i>0$ and there is an $n<i$ with $\widetilde{t}_n$
satisfying $p\land\neg q$, then from the minimality of $i$ we conclude that
there is a minimal $n\leq m < i$ such that $\widetilde{t}_m$
satisfies $q$. From the minimality of $m$ we conclude that
$\{\widetilde{t}_{m-1}\}\stackrel{r_{m-1}}{\rightarrow}\{\widetilde{t}_m\}$
is due to an instantaneous rule, which, by definition, switches
the clock off.
\end{itemize}
Thus either $i=0$ and the clock is on in $\widetilde{t}_i$ with value $0$, or $i>0$
and the clock is off in state $\widetilde{t}_{i-1}$. Furthermore,
in the latter case the $(i-1)$th state satisfies $\neg p \lor q$
(otherwise $i$ would not be minimal), and the rewrite
$\{\widetilde{t}_{i-1}\}\stackrel{r_i}{\rightarrow}\{\widetilde{t}_i\}$ is
due to an instantaneous rule, which, again by definition, switches
the clock on and resets its value to $0$.
We get that the clock is on with value $0$ in $\widetilde{t}_i$. As $\neg q$
holds all the way from the $i$th till the $j$th state, the clock
remains on from the $i$th till the $j$th state. The rewrites of
$\widetilde{\cal R}$ assure that the clock value in state $\widetilde{t}_j$ is the
duration $\sum_{k=i}^{j-1}r_k$ that is by assumption larger than
$r$, what was to be shown.
\end{proof}
The following lemma states that finiteness of the state space is
preserved under the $\textit{BR}$-transformation, implying that our bounded
response model checking algorithm terminates for finite-space systems.
\begin{lem}
\label{lem:finiteBR}
Given a real-time rewrite theory $\cal R$, a labeling function
$L_{\Pi}$ of $\cal R$ with $p,q\in\Pi$, an initial state $\{t_0\}$ of
$\cal R$, and a fixed time sampling strategy, and furthermore, assuming that
\begin{itemize}
\item there are only finitely many states reachable in $\cal R$ from
initial state $\{t_0\}$ with the given time sampling, i.e., the
set
\[
\{\{t_i\}\ |\ \pi = \{t_0\}\stackrel{r_0}{\rightarrow}\{t_1\}\stackrel{r_1}{\rightarrow}\ldots \ \in Paths(\mathcal{R})_{t_0},\ i\in\mathbb{N}
\}
\]
is finite, and
\item the number of different rewrite durations in all possible paths
in $\cal R$ from $\{t_0\}$ under the given time sampling is finite, i.e., the
set
\[
\{r_i\ | \
\pi = \{t_0\}\stackrel{r_0}{\rightarrow}\{t_1\}\stackrel{r_1}{\rightarrow}\ldots \ \in Paths(\mathcal{R})_{t_0},\ i\in\mathbb{N} \}
\]
is finite,
\end{itemize}
then the bounded response model checking algorithm for
$\cal{R}$ using the same sampling strategy terminates.
\end{lem}
\begin{proof}
Assume that the above conditions hold. Notice that the bounded
response model checking algorithm always terminates if the set of
reachable states of the $\textit{BR}$-transformation
(from its initial state and under the given time sampling) is finite.
Since all instantaneous rules in the $\textit{BR}$-transformation
$\widetilde{\cal R}$ either leave the clock value untouched or reset
the clock value to $0$, the finiteness of the state space is
preserved under the instantaneous rules of $\widetilde{\cal R}$.
For the tick rules, on the one hand, if the clock value gets larger
than the bound $r$ in the bounded response formula, then the model
checking algorithm finds a counterexample and thus terminates. On
the other hand, since there are only finitely many possible rewrite
durations, there are only finitely many possible clock values
less than or equal to $r$. So if the clock value never exceeds $r$
than the reachable state space of the $\textit{BR}$-transformation remains
finite and the algorithm terminates in this case, too.
\end{proof}
\iflong
\subsection{Minimum Separation}
\input{correctness_ms}
\fi
\section{Model Checking MTL Properties of Object-Oriented Specifications}
\label{sec:implementation}
Real-Time Maude currently does not support MTL model checking. However, some
MTL formulas can already be model checked in Real-Time Maude
using the \emph{time-bounded} search and LTL model checking commands.
For example, we
can model check the
time-bounded
until property $\mathcal{R},L_{\Pi},t_0
\models p \;U_{\leq r}\; q$, for $p$ and $q$ \emph{state} properties from $\Pi$,
using the time-bounded model checking
command
\small
\begin{alltt}
(mc t0 |=t \(p\) U \(q\) in time <= \(r\) .)
\end{alltt}
\normalsize
\noindent We can also analyze the properties $\mathcal{R},L_{\Pi},t_0 \models \Box_{\leq r}\; p$
and $\mathcal{R},L_{\Pi},t_0 \models \Diamond_{\leq r}\; p$ in a similar way.
In this paper we present analysis algorithms for the following two
classes of MTL formulae:
\begin{enumerate}
\item \emph{Bounded response:}
$\Box\ (p \rightarrow (\Diamond_{\leq r}\ q))$
\item \emph{Minimum separation:} $\Box\ (p \rightarrow
(p\ \textit{W}\ (\Box_{\leq r}\ \neg p)))$
\end{enumerate}
We propose to transform an MTL model checking problem
$\mathcal{R},L_{\Pi}, t_0 \models \varphi$
into an untimed LTL model
checking problem
$\widetilde{\mathcal{R}},\widetilde{L}_{\Pi}, \widetilde{t_0} \models \widetilde{\varphi}$.
Both transformations add a \emph{clock} to the system: for model checking
bounded response properties, this clock measures the time since $p$
held without $q$ holding in the meantime; for minimum separation
properties, the clock measures the distance between two
non-consecutive $p$-states. We take care not to increase the clocks
``unnecessarily,'' so that if the state space reachable from
$t_0$ in $\mathcal{R}$ is finite, then the state space reachable from
$\widetilde{t_0}$ in $\widetilde{\mathcal{R}}$ remains finite, under reasonable
time-divergence assumptions on the executions.
We assume that our specifications are \emph{tick-invariant}~\cite{wrla06}
with regard to the state propositions occurring in the formula, i.e.,
a tick step does not change the valuation of the atomic propositions
occurring in the formula.
Most systems, including the two case studies in the paper,
satisfy tick-invariance, since the state propositions usually do
not involve the value of clock and timer attributes in the system.
\subsection{Bounded response: $\Box\ (p \rightarrow \Diamond_{\leq r}\ q)$}
\label{sec:bounded}
A bounded response property states that the system always reacts to a
request $p$ with an action $q$ within time $r$. For example, in our
medical devices case study,
the ventilation machine, helping a sedated patient to breathe,
should not be stopped for more than two seconds at a time; that is, each
state in which the machine is pausing must be followed by a state in
which the machine is breathing in two seconds or less.
The MTL
model checking problem
$$\mathcal{R},L_{\Pi},t_0 \models \Box\: (p
\longrightarrow \Diamond_{\leq r}\: q)$$
\noindent for $p,q\in\Pi$ state
propositions, can be transformed into the untimed model checking problem
$$\widetilde{\mathcal{R}}_r,\widetilde{L}_{\Pi},\widetilde{t_0} \models
\Box\: (p\longrightarrow \Diamond\: q) \;\wedge\;
\Box\:(\textit{clock}(c_{\textit{BR}}) \leq r)$$
\noindent where $\textit{clock}(c_{\textit{BR}})$ is the value of a ``clock''
that measures the time since $p$ held without $q$ holding in the
meantime. For real-time rewrite theories having only time-divergent
paths we could skip the first condition $\Box\: (p\longrightarrow
\Diamond\: q)$, that assures, that we also consider all
relevant time-convergent paths as possible counterexamples.
We add a ``clock'' $c_{\textit{BR}}$ to
the system, and update it as follows:
\begin{itemize}
\item[i)] If the clock $c_{\textit{BR}}$ is turned off, and a state satisfying
$p\wedge \neg q$ is reached, then the clock is set to 0 and is turned on.
\item[ii)] The clock is turned off when a state satisfying $q$ is reached.
\item[iii)] A clock that is on is increased according to the elapsed time
in the system.
\end{itemize}
For the very useful class of ``flat'' object-oriented specifications
formalized according to the guidelines in~\cite{journ-rtm}---all advanced
Real-Time Maude applications have been so specified---we can automate
the transformation from $\mathcal{R},L_{\Pi},t_0, p, q, r$ to
$\widetilde{\mathcal{R}},\widetilde{L}_{\Pi},\widetilde{t_0}$
as follows:
\begin{enumerate}
\item \label{lab:br_state} Add the following class for the clock:
\small
\begin{alltt}
class Clock | \texttt{clock} : Time, status : OnOff .
sort OnOff . ops on off : -> OnOff [ctor] .
\end{alltt}
\normalsize
\item \label{lab:br_init} Add a clock object to the initial state $\texttt{\char123}t_0\texttt{\char125}$, so that the initial state
becomes
\small
\begin{alltt}
\texttt{\char123}\(t_0\) <\! \(c_{\textit{BR}}\)\! : Clock\! |\! \texttt{clock}\! :\! 0, status\! :\! \(x\)\! >\texttt{\char125}
\end{alltt}
\normalsize
\noindent where $c_{\textit{BR}}$ is a constant of sort \texttt{Oid} and $x$ is @on@ if
$p\in L(\texttt{\char123}t_0\texttt{\char125})$ and
$q\not\in L(\texttt{\char123}t_0\texttt{\char125})$, and is @off@ otherwise.
Note that $p\in L(\texttt{\char123}t_0\texttt{\char125})$ can be
checked in Maude by checking whether
$\texttt{\char123}t_0\texttt{\char125}
@ |= @ p = @true@$.
\item \label{lab:br_tick} We keep Real-Time-Maude's object-oriented tick rule and extend
the functions \texttt{delta} and \texttt{mte} to clocks as follows, ensuring that
\texttt{mte} is not affected by the new clock object:
\small
\begin{alltt}
eq delta(< \(c_{\textit{BR}}\) : Clock | status : on, \texttt{clock} : T >, T') =
< \(c_{\textit{BR}}\) : Clock | \texttt{clock} : if T <= \(r\) then T + T' else T fi > .
eq delta(< \(c_{\textit{BR}}\) : Clock | status : off >, T') = < \(c_{\textit{BR}}\) : Clock | > .
eq mte(< \(c_{\textit{BR}}\) : Clock | >) = INF .
\end{alltt}
\normalsize
\noindent Notice that the @delta@ function ensures that
the clock value never increases more than necessary,
preserving \emph{finiteness} of the reachable state space from the initial state.
\item \label{lab:br_inst} Each \emph{instantaneous} rule
$t\; @=>@\; t' @ if @cond\;$ or $\;@{@ t@} => {@t'@} if @cond$
in $\mathcal{R}$ is replaced
by the rules:
\small
\begin{alltt}
\texttt{\char123}\(t\) REST < \(c_{\textit{BR}}\) : Clock | status : on >\texttt{\char125}
=> \texttt{\char123}\(t'\) REST < \(c_{\textit{BR}}\) : Clock | >\texttt{\char125} if \texttt{\char123}\(t'\) REST\texttt{\char125} |= \(q\) =/= true and \(cond\)
\end{alltt}
\normalsize
\noindent (if the clock is on, then it continues to stay on if a state
satisfying $\neg q$ is reached);
\small
\begin{alltt}
\texttt{\char123}\(t\) REST < \(c_{\textit{BR}}\) : Clock | status : on >\texttt{\char125}
=> \texttt{\char123}\(t'\) REST < \(c_{\textit{BR}}\) : Clock | status : off >\texttt{\char125} if \texttt{\char123}\(t'\) REST\texttt{\char125} |= \(q\) and \(cond\)
\end{alltt}
\normalsize
\noindent (if the clock is on, then it is turned off when a state satisfying $q$ is reached);
\small
\begin{alltt}
\texttt{\char123}\(t\) REST < \(c_{\textit{BR}}\) : Clock | status : off >\texttt{\char125}
=> \texttt{\char123}\(t'\) REST < \(c_{\textit{BR}}\) : Clock | \texttt{clock} : 0, status : on >\texttt{\char125}
if \texttt{\char123}\(t'\) REST \texttt{\char125} |= \(p\) and \texttt{\char123}\(t'\) REST\texttt{\char125} |= \(q\) =/= true and \(cond\)
\end{alltt}
\normalsize
\noindent (if the clock is off, then it is set to 0 and turned on when a state
satisfying $p\wedge \neg q$ is reached);
\small
\begin{alltt}
\texttt{\char123}\(t\) REST < \(c_{\textit{BR}}\) : Clock | status : off >\texttt{\char125}
=> \texttt{\char123}\(t'\) REST < \(c_{\textit{BR}}\) : Clock | >\texttt{\char125}
if \texttt{\char123}\(t'\) REST\texttt{\char125} |= \(q\) or \texttt{\char123}\(t'\) REST\texttt{\char125} |= \(p\) =/= true and \(cond\)
\end{alltt}
\normalsize
\noindent (if the clock is off, then it continues to stay off if a state
satisfying $q\vee \neg p$ is reached).
In the above rules @REST@ is a variable of sort @Configuration@ that
does not appear in the
original rule. @REST@ matches the ``other'' objects
and messages in the state.
\end{enumerate}
\noindent Summarizing, the \emph{\textit{BR}-transformation}
transforms a real-time rewrite theory $\cal R$, a labeling function
$L_{\Pi}$ of $\cal R$ with $p,q\in\Pi$, an initial state $t_0$ of $\cal R$,
and a bounded response formula $ \Box\: (p
\longrightarrow \Diamond_{\leq r}\: q)$ into the
triplet $\widetilde{\cal R}$, $\widetilde{L}_{\Pi}$, and $\widetilde{t}_0$ by
\begin{itemize}
\item transforming $\cal R$ into $\widetilde{\cal R}$ according to the
points \ref{lab:br_state}, \ref{lab:br_tick}, and \ref{lab:br_inst} above;
\item transforming $L_{\Pi}$ into $\widetilde{L}_{\Pi}$ by adapting its domain to
the transformed state space, but letting the labeling otherwise
unchanged, i.e., $L_{\Pi}(\{t\}) = \widetilde{L}_{\Pi}(\{t\ o\})$ for all states
$t$ of $\cal R$ and all \texttt{Clock} instances $o$;
\item extending the initial state $t_0$ according to point
\ref{lab:br_init} above, yielding $\widetilde{t}_0$.
\end{itemize}
\noindent The validity of the bounded response property $\Box\ (p \rightarrow
\Diamond_{\leq r}\ q)$ is equivalent to
$\Box\ (p \rightarrow
\Diamond\ q)$ \emph{and}
the clock value being less than
or equal to $r$ in each reachable state of the transformed module. The latter property
can be defined as an atomic proposition
\small
\begin{alltt}
op clock`<=_ : Time -> Prop [ctor] .
eq \char123REST <\! \(c_{\textit{BR}}\)\! :\! Clock\! |\! clock\! :\! T1\! >\char125 |= clock <= T2 = (T1 <= T2) .
\end{alltt}
\normalsize
\noindent and hence bounded response can be analyzed using Real-Time Maude's untimed
LTL model checking features. We have implemented the above model transformation
in Real-Time Maude. We have also implemented a bounded response model checking
command in the tool based on this transformation. However, for pragmatic reasons,
we do \emph{not} model check the property $\widetilde{\mathcal{R}},\widetilde{L}_{\Pi},\widetilde{t_0} \models
\Box\: (p\longrightarrow \Diamond\: q) \;\wedge\;
\Box\:(\textit{clock}(c_{\textit{BR}}) \leq r)$. Instead, we have observed the unsurprising
fact that, with time sampling strategy executions, all our large Real-Time Maude
applications are modeled as time-diverging theories. In these cases, bounded
response reduces to checking $\widetilde{\mathcal{R}},\widetilde{L}_{\Pi},\widetilde{t_0} \models
\Box\:(\textit{clock}(c_{\textit{BR}}) \leq r)$, which can be analyzed by the following search command
that searches for a state in which the clock value is greater than
$r$:
\small
\begin{alltt}
(utsearch [1] \texttt{\char123}\(t_0\) <\! \(c_{\textit{BR}}\)\! :\! Clock\! |\! \texttt{clock}\! :\! 0, status\! :\! \(x\)\! >\texttt{\char125} =>*
\texttt{\char123}C:Configuration <\! \(c_{\textit{BR}}\)\! :\! Clock | \texttt{clock}\! :\! T:Time\! >\texttt{\char125} such that T:Time > \(r\) .)
\end{alltt}
\normalsize
\noindent where $x$ is @on@ if $p\in L(\{t_0\})$ and $q\not\in
L(\{t_0\})$, and is @off@ otherwise. The practical difference is
that, whereas the LTL model checking does not terminate when the state
space reachable from $t_0$ in $\mathcal{R}$ is infinite, the above
search command provides a \emph{semi-decision} procedure for the
invalidity of the bounded response property. For an example of the benefit of this
time-divergence-assuming implementation, consider the bounded response
analysis of the medical systems example in
Section~\ref{sec:case_studies}. The reachable state space is infinite
because of the clock used in the original model; hence any direct LTL
model checking would not terminate, but we see that our bounded
response command indeed returns a counterexample falsifying the
bounded response property.
In our tool, the bounded response model checking command (for the automatic
\textit{BR}-transformation and the execution the Real-Time Maude search) is written with
syntax
\small
\begin{alltt}
(br \(t_0\) |= \(p\) => <>le( \(r\) ) \(q\) .)
\end{alltt}
\normalsize
\subsection{Minimum Separation: $\Box\ (p \rightarrow (p\ \textit{W}\
\ \Box_{\leq r}\ \neg p))$}
Given a real-time rewrite theory $\cal R$ with a labeling function
$L_{\Pi}$, $p\in\Pi$, all runs of $\cal R$
are made up of a sequence of blocks for which $p$ and $\neg p$ hold
alternatingly (see Figure~\ref{fig:chain}). The minimum separation
property requires that each $\neg p$-block occurring after a $p$-block
must have a minimum duration $r$. I.e., if the run for which we check
the property starts with a $p$-block, then all $\neg p$-blocks of the
run must have a duration at least $r$. Otherwise, if the run starts
with a $\neg p$-block, then the same holds for all $\neg p$-blocks
except the first one at the beginning of the run.
\begin{figure}[htb]
\centering
\begin{tikzpicture}[]
\node[] (p0) at (0,0) {};
\node[] (p1) at (2,0) {};
\node[] (p2) at (5,0) {};
\node[] (p3) at (6,0) {};
\node[] (p4) at (10,0) {};
\node[] (p4b) at (9.5,0) {};
\node[] (p5) at (13,0) {};
\path[thick,|-|] (p0) edge node[below] {$\underbrace{\hspace*{1.7cm}}_{p}$} (p1)
edge node[below] {\hspace*{1.7cm}$\underbrace{\hspace*{3cm}}_{\neg p}$} (p2)
edge node[below] {\hspace*{4.7cm}$\underbrace{\hspace*{1cm}}_{p}$} (p3)
edge node[below] {\hspace*{5.7cm}$\underbrace{\hspace*{4cm}}_{\neg p}$} (p4);
\path[thick,->] (p4b) edge node[below] {${\ldots}$} (p5);
\node[] (pp1) at (1.75,0.3) {};
\node[] (pp2) at (5,0.3) {};
\node[] (pp3) at (5.75,0.3) {};
\node[] (pp4) at (10,0.3) {};
\path[<->] (pp1) edge node[above] {$\geq r$} (pp2);
\path[<->] (pp3) edge node[above] {$\geq r$} (pp4);
\end{tikzpicture}
\begin{tikzpicture}[]
\node[] (p0) at (0,0) {};
\node[] (p1) at (1.5,0) {};
\node[] (p2) at (4,0) {};
\node[] (p3) at (6,0) {};
\node[] (p4) at (9,0) {};
\node[] (p5) at (11,0) {};
\node[] (p5b) at (10.75,0) {};
\node[] (p6) at (13,0) {};
\path[thick,|-|] (p0) edge node[below] {$\underbrace{\hspace*{1.2cm}}_{\neg p}$} (p1)
edge node[below] {\hspace*{1.2cm}$\underbrace{\hspace*{2.5cm}}_{p}$} (p2)
edge node[below] {\hspace*{3.7cm}$\underbrace{\hspace*{2cm}}_{\neg p}$} (p3)
edge node[below] {\hspace*{5.7cm}$\underbrace{\hspace*{3cm}}_{p}$} (p4)
edge node[below] {\hspace*{8.7cm}$\underbrace{\hspace*{2cm}}_{\neg p}$} (p5);
\path[thick,->] (p5b) edge node[below] {${\ldots}$} (p6);
\node[] (pp2) at (3.75,0.3) {};
\node[] (pp3) at (6,0.3) {};
\node[] (pp4) at (8.75,0.3) {};
\node[] (pp5) at (11,0.3) {};
\path[thick,<->] (pp2) edge node[above] {$\geq r$} (pp3);
\path[thick,<->] (pp4) edge node[above] {$\geq r$} (pp5);
\end{tikzpicture}
\caption{The form of runs satisfying the minimum separation property
$\Box\ (p \rightarrow (p\ \textit{W}\ \ \Box_{\leq r}\ \neg
p))$. The $p$- and $\neg p$-blocks may also be infinite.}
\label{fig:chain}
\end{figure}
We transform the MTL model checking problem
$$\mathcal{R},L_{\Pi},t_0 \models \Box\ (p \rightarrow (p\ \textit{W}\ \ \Box_{\leq r}\ \neg p))$$
\noindent into the untimed model checking
problem
$$\widetilde{\mathcal{R}},\widetilde{L}_{\Pi},\widetilde{t_0} \models
\Box\:(status(c_{\textit{MS}}) = @on@ \;\vee\; \textit{clock}(c_{\textit{MS}}) \geq r)$$
\noindent where $\textit{clock}(c_{\textit{MS}})$ is the value of a ``clock'' that
measures the time duration since we saw a $p$-state. That means, to model
check minimum separation properties, we add a ``clock'' $c_{\textit{MS}}$ to the
system, which is initially turned off and set to $r$: in this way we
ensure that an eventual initial $\neg p$-block does not cause a violation of the property.
We update the clock as follows:
\begin{enumerate}
\item[i)] If we move from a $p$-state to a $\neg p$-state, then the clock is turned on and reset to 0.
\item[ii)] The clock is turned off when a state satisfying $p$ is reached.
\item[iii)] A clock that is on is increased according to the elapsed time
in the system.
\end{enumerate}
We can automate the transformation to search for counterexamples of a minimum separation
property of the above form as follows:
\begin{enumerate}
\item \label{lab:ms_state} Add the same class for the clock as in Section~\ref{sec:bounded}:
\small
\begin{alltt}
class Clock | \texttt{clock} : Time, status : OnOff .
\end{alltt}
\normalsize
\item \label{lab:ms_init} Add a clock object to the initial state
$\texttt{\char123}t_0\texttt{\char125}$, yielding
\small
\begin{alltt}\small
\texttt{\char123}\(t_0\) <\! \(c_{\textit{MS}}\)\! :\! Clock\! |\! \texttt{clock}\! :\! \(r\), status\! :\! off\! >\texttt{\char125}
\end{alltt}
\normalsize
\noindent where $c_{\textit{MS}}$ is a constant of sort \texttt{Oid}.
\item \label{lab:ms_tick} We keep Real-Time-Maude's object-oriented tick rule and extend
the function \texttt{delta} and \texttt{mte} to clocks exactly
as in Section~\ref{sec:bounded}.
\item \label{lab:ms_inst} Each \emph{instantaneous} rule $t\; @=>@\;
t' @ if @cond$ or $\;@{@ t@} => {@t'@} if @cond$
in $\mathcal{R}$ is replaced
by the rules:
\small
\begin{alltt}\small
\texttt{\char123}\(t\) REST < \(c_{\textit{MS}}\) : Clock | status : on >\texttt{\char125}
=> \texttt{\char123}\(t'\) REST < \(c_{\textit{MS}}\) : Clock | >\texttt{\char125} if \texttt{\char123}\(t'\) REST\texttt{\char125} |= \(p\) =/= true and \(cond\)
\end{alltt}
\normalsize
\noindent (if the clock is on, then it continues to stay on, if a state
satisfying $\neg p$ is reached);
\small
\begin{alltt}
\texttt{\char123}\(t\) REST < \(c_{\textit{MS}}\) : Clock | status : on >\texttt{\char125}
=> \texttt{\char123}\(t'\) REST < \(c_{\textit{MS}}\) : Clock | status : off >\texttt{\char125} if \texttt{\char123}\(t'\) REST\texttt{\char125} |= \(p\) and \(cond\)
\end{alltt}
\normalsize
\noindent (if the clock is on, then it is turned off when a state satisfying $p$ is reached);
\small
\begin{alltt}
\texttt{\char123}\(t\) REST < \(c_{\textit{MS}}\) : Clock | status : off >\texttt{\char125}
=> \texttt{\char123}\(t'\) REST < \(c_{\textit{MS}}\) : Clock | >\texttt{\char125}
if (\texttt{\char123}\(t\) REST\texttt{\char125} |= \(p\) =/= true or \texttt{\char123}\(t'\) REST\texttt{\char125} |= \(p\)) and \(\)\(cond\)
\end{alltt}
\normalsize
\noindent (the clock remains off, if either we are in a state satisfying $\neg p$ or we move
to a state satisfying $p$; the first condition is needed to avoid switching the clock on in initial $\neg p$-blocks);
\small
\begin{alltt}
\texttt{\char123}\(t\) REST < \(c_{\textit{MS}}\) : Clock | status : off >\texttt{\char125}
=> \texttt{\char123}\(t'\) REST < \(c_{\textit{MS}}\) : Clock | status : on, \texttt{clock} : 0 >\texttt{\char125}
if \texttt{\char123}\(t\) REST\texttt{\char125} |= \(p\) and \texttt{\char123}\(t'\) REST\texttt{\char125} |= \(p\) =/= true and \(cond\)
\end{alltt}
\normalsize
\noindent (if the clock is off, and we move from a state satisfying $p$ to a
state satisfying $\neg p$, then the clock is turned on and reset to 0).
Again, \texttt{REST} is a variable of sort \texttt{Configuration} that
does not appear in the original rule.
\end{enumerate}
\noindent The \emph{\textit{MS}-transformation} therefore
transforms a real-time rewrite theory $\cal R$, a labeling function
$L_{\Pi}$ with $p\in\Pi$, an initial state $t_0$ of $\cal R$, a state proposition $p$, and a time value $r$
into the
triple $\widetilde{\cal R}$, $\widetilde{L}_{\Pi}$, and $\widetilde{t}_0$ by
\begin{itemize}
\item transforming $\cal R$ into $\widetilde{\cal R}$ according to the
points \ref{lab:ms_state}, \ref{lab:ms_tick}, and \ref{lab:ms_inst} above;
\item transforming $L_{\Pi}$ into $\widetilde{L}_{\Pi}$ by adapting its domain to
the transformed state space, but letting the labeling otherwise
unchanged, i.e., $\widetilde{L}_{\Pi}(\{t\ o\}) = L_{\Pi}(\{t\})$ for all states
$t$ of $\cal R$ and all \texttt{Clock}s $o$;
\item extending the initial state $t_0$ according to point
\ref{lab:ms_init} above, yielding $\widetilde{t}_0$.
\end{itemize}
\noindent Checking the minimum separation property $\Box\ (p
\rightarrow (p\ \textit{W}\ \ \Box_{\leq r}\ \neg p))$ is
equivalent to checking that the validity of $p$ implies that the clock
value is larger than or equal to $r$ in each state in the transformed
module. The violation of the latter can be checked by the following
search command that searches for a state in which the clock is off (which
implies that $p$ holds) and the clock value is smaller than $r$:
\small
\begin{alltt}
(utsearch [1] \texttt{\char123}\(t_0\) < \(c_{\textit{MS}}\) : Clock | \texttt{clock} : \(r\), status : off >\texttt{\char125} =>*
\texttt{\char123}C:Configuration < \(c_{\textit{MS}}\) : Clock | \texttt{clock} : T:Time, status : off >\texttt{\char125}
such that T:Time < \(r\) .)
\end{alltt}
\normalsize
The above \textit{MS}-transformation has been integrated in Real-Time
Maude, and model checking the
above minimum separation property can be done with the Real-Time Maude
command
\small
\begin{alltt}
(ms \(t_0\) |= \(p\) separated by >= \(r\) .)
\end{alltt}
\normalsize
\section{Introduction}
Real-Time Maude~\cite{journ-rtm} is a formal specification language and
a high-performance simulation and model checking
tool that extends the rewriting-logic-based
Maude system~\cite{maude-book} to support the formal specification
and analysis of \emph{real-time} systems.
Real-Time Maude differs from timed-automaton-based tools, such as
{\sc Uppaal}~\cite{uppaalTutorial} and {\sc Kronos}~\cite{kronos},
by emphasizing ease and expressiveness of specification
over algorithmic decidability of key properties. In particular,
Real-Time Maude supports the definition of any computable data type,
unbounded data structures, different communication models, and so on.
Because of its expressiveness, Real-Time Maude has been successfully
applied to a wide range of advanced state-of-the-art applications that are beyond the pale
of timed automata, including
the OGDC density control~\cite{ogdc-tcs} and LMST topology control~\cite{mike-wsn}
protocols for wireless sensor networks,
the CASH scheduling algorithm with capacity sharing
features that require unbounded queues~\cite{fase06}, the AER/NCA
active networks multicast protocol~\cite{aer-journ}, and the NORM multicast protocol
developed by the IETF~\cite{norm-paper}. Real-Time Maude's natural model of time, together with its
expressiveness, also makes it ideal as a semantic framework in which
real-time modeling languages can be given a formal semantics; such languages
then also get Real-Time Maude's
formal analysis capabilities essentially for free. Languages with a Real-Time Maude
semantics include: a timed extension of the Actor model~\cite{rtactors-in-rtmaude},
the Orc web services orchestration language~\cite{musab-orc07},
a language
developed at DoCoMo laboratories for handset applications~\cite{musab-fase09},
a behavioral subset of the avionics standard AADL~\cite{aadl-fmoods},
the visual model transformation language e-Motions~\cite{emotions-wrla10},
real-time model transformations in MOMENT2~\cite{fase10},
and a subset of Ptolemy II discrete-event models~\cite{icfem09}.
Real-Time Maude is particularly suitable to model real-time systems in
an \emph{object-oriented} style, and the paper~\cite{journ-rtm} identifies
some useful specification techniques for object-oriented
real-time systems. All the concrete applications mentioned above, and many of
the language semantics applications, are specified in an object-oriented
way using those techniques.
Real-Time Maude provides a spectrum of analysis methods, including
simulation through timed rewriting, untimed temporal logic model
checking, and (unbounded or time-bounded) search for reachability
analysis. However, up to know, Real-Time Maude has lacked the ability
to model check \emph{timed} (or \emph{metric}) temporal logic
properties. Such properties are obviously very important in many
real-time systems. For example, in case of an accident the airbag must
not just inflate \emph{eventually}, but within very tight time bounds.
For timed automata, such metric temporal logic model checking is
decidable\footnote{for finite behaviour, see, e.g., \cite{bouyer:phd}}, and
implemented in the {\sc Kronos} tool~\cite{kronos}.
For the much more expressive Real-Time Maude formalism,
supporting metric temporal logic checking, is obviously a much harder task.
This paper reports on our first attempts at providing metric temporal logic
model checking for Real-Time Maude.
We have taken the following pragmatic choices:
\begin{enumerate}
\item Supporting the model checking of
only a few classes of metric temporal logic properties, namely, the ones that
were needed in the above-mentioned applications. These
properties are:
\begin{itemize} \item \emph{Bounded response}: each $p$-state must be
followed by a $q$-state within time $r$ (where $p$ and $q$ are state
propositions). One example of a bounded response property is
``whenever the ventilator assisting the patient's breathing is
turned off, it must be turned on within 5 seconds''.
\item \emph{Minimum separation}: there must be at least time $r$
between two non-consecutive $p$-states. For example, ``the
ventilator should be turned on continuously for at least two minutes
between two pauses.''
\end{itemize}
\item
Supporting such model checking only for flat object-oriented models
specified according to the guidelines mentioned above. But as already said,
this class of systems includes all
the concrete Real-Time Maude
applications listed above.
\end{enumerate}
What is gained by restricting the classes of systems and properties is
\emph{efficiency}. Instead of implementing the model checking
algorithms from scratch, we pursue a \emph{transformational} approach,
where we take advantage of Maude's high performance analysis commands
and transform a metric model checking problem $\mathcal{R}, L, t_o
\models \phi$ into a problem $\widetilde{\mathcal{R}}, \widetilde{L},
\widetilde{t_o} \models \widetilde{\phi}$ that can be analyzed by
Real-Time Maude's efficient search and LTL model checking commands. Our transformations add a
clock which measures, respectively, the time since the earliest
$(p\wedge \neg q)$-state that has not been followed by a $q$-state
(for bounded response) and the last time since we saw a $p$-state (for
minimum separation). An important property is that -- under
reasonable time-divergence assumptions about the executions with the
selected time sampling strategy -- if the original reachable state
space is finite, then the model checking commands are guaranteed to
terminate. Furthermore, our model checking commands
are semi-decision procedures for the invalidity of the metric properties
for time-diverging systems.
The transformations have been implemented in Real-Time
Maude and the corresponding model checking commands have been made
available in the tool. We have applied the new commands on two case
studies, one on the safe interoperation of medical
devices~\cite{phuket08} and one on a fault-tolerant controller for
traffic lights in an intersection~\cite{traffic-light}.
We prove the correctness of the transformation under reasonable
assumptions, such as the real-time rewrite theory
being
\emph{tick-invariant}~\cite{wrla06}. Since real-time rewrite
theories do not have a ``region-automaton''-like discrete quotient,
for dense time Real-Time Maude uses \emph{time sampling strategies} to
execute the tick rules. That is, in model checking analyses for
dense-time models, only a subset of all possible behaviors are
analyzed. Therefore, Real-Time Maude analyses are in general not
(both) sound and complete; however, for object-oriented specifications
we have identified easily checkable conditions that guarantee
soundness and completeness of our analyses also for dense-time
systems~\cite{wrla06}.
This paper is organized as follows. Section~\ref{sec:preliminaries}
introduces Real-Time Maude and metric temporal~logic.
Section~\ref{sec:implementation} presents the properties that we
address and the corresponding transformations, whose correctness is
proved in Section~\ref{sec:proof}. Section~\ref{sec:case_studies}
shows two case studies of metric temporal logic model checking in
Real-Time Maude. Section~\ref{sec:related} discusses related work,
and Section~\ref{sec:conclusion} gives some
concluding remarks.
\section{Logic}
\subsection{Metric Temporal Logic}\label{sec:mtl}
\emph{Linear temporal logic (LTL)}~\cite{pnueli:ltl}
allows us to describe properties of paths of a given system.
The states are labeled with elements from a finite set $\Pi$ of atomic
propositions. Besides propositions and the usual Boolean
operators, LTL formulae can be built using the temporal \emph{until}
operator. Intuitively, the formula $p\ \textit{U}\ q$ (``$p$ until $q$'')
is satisfied by a path if the property $q$ becomes valid within an
arbitrary but finite number of steps and the property $p$ constantly
holds on the path before. As syntactic sugar we define
$\Diamond\ p$ (``eventually $p$'', defined as $\textit{true}\ \textit{U}\
p$) that is satisfied by a path if $p$ holds somewhere on the path, and
$\Box\ p$ (``globally $p$'', defined as $\neg (\textit{true}\
\textit{U}\ (\neg p))$) expressing that $p$ holds on the whole path. The
\emph{weak until} operator $p\ \textit{W}\ q$ is
defined as $(p\ \textit{U}\ q) \lor (\Box\ p)$.
For time-critical systems we need more expressive power to state that
some actions should happen \emph{within some time bounds}. There are
different extensions of LTL to capture also timed properties
(see~\cite{real-time-logics} for an overview). In this paper, we use
the extension \emph{metric temporal logic (MTL)}~\cite{koyMTL}, that
adds time interval bounds to the temporal operators.
For the until operator, the formula $p\ \textit{U}_{[t_1,t_2]}\
q$ states that $p\ \textit{U}\ q$ holds and, furthermore, $q$ occurs within
the time interval $[t_1,t_2]$.
Formulae of MTL are built using the following abstract syntax:
\[
\varphi \quad ::= \quad \textit{true} \quad | \quad p \quad |
\quad \neg \varphi \quad | \quad \varphi \land \varphi \quad |
\quad \varphi\ \textit{U}_{[t_1,t_2]}\ \varphi
\]
with $p\in\Pi$ and either $t_1,t_2\in\mathbb{R}$ with $t_1\leq t_2$ and
$t_2>0$, or $t_1\in\mathbb{R}$ and $t_2=\infty$. Note that
$\textit{U}_{[0,\infty]}$, for which we just write $\textit{U}$,
corresponds to the unbounded until of LTL. Besides the
usual Boolean operators $\lor,\rightarrow,\ldots$ we define as
syntactic sugar $\Diamond_{[t_1,t_2]}\ \varphi$ as $\textit{true}\
\textit{U}_{[t_1,t_2]}\ \varphi$ and $\Box_{[t_1,t_2]}\ \varphi$ as
$\neg(\textit{true}\ \textit{U}_{[t_1,t_2]} (\neg \varphi))$.
If the
lower bound $t_1$ is $0$, we use the notation $\varphi_1\ \textit{U}_{\leq
t_2}\ \varphi_2$, and analogously for the other operators.
Given a real-time rewrite theory $\cal{R}$,
the set of \emph{states} is defined as $\mathbb{T}_{\Sigma/E,\texttt{GlobalSystem}}$.
A set $\Pi$ of (possibly
parametric) {\em atomic propositions} on those states can
be defined equationally in a protecting extension $(\Sigma \cup \Pi,E \cup D)
\supseteq (\Sigma,E)$, and give rise to a {\em labeling function}
$L_{\Pi} : \mathbb{T}_{\Sigma/E,\texttt{GlobalSystem}} \rightarrow \mathcal{P}(\Pi)$
in the obvious way~\cite{maude-book}.
Adapting the pointwise semantics for MTL given in~\cite{real-time-logics},
we can define satisfaction of MTL formulas for real-time
rewrite theories over timed paths as follows:
\begin{definition}
Let $\cal{R}$ be a real-time rewrite theory, $L_{\Pi}$ a labeling
function on $\cal R$, and $\pi = t_0
\stackrel{r_0}{\rightarrow} t_1 \stackrel{r_1}{\longrightarrow}
\ldots $ a timed path in $\cal R$.
The satisfaction relation of an MTL formula $\phi$ for the
path $\pi$ in $\cal R$
is then defined recursively as follows:\\[1ex]
\begin{tabular}[h]{lp{11cm}}
${\cal R},L_{\Pi},\pi\models\textit{true}$ & always holds\\
${\cal R},L_{\Pi},\pi\models p$ & iff $p\in L_{\Pi}(t_0)$\\
${\cal R},L_{\Pi},\pi\models \neg \varphi$ & iff
${\cal R},L_{\Pi},\pi\not\models\varphi$\\
${\cal R},L_{\Pi},\pi\models \varphi_1\land\varphi_2$ & iff
${\cal R},L_{\Pi},\pi\models\varphi_1$ and ${\cal R},L_{\Pi},\pi\models\varphi_2$\\
${\cal R},L_{\Pi},\pi\models \varphi_1\ \textit{U}_{[r_a,r_b]}\ \varphi_2$ &
iff there exists a $j\in\mathbb{N}$ such that
${\cal R},L_{\Pi},\pi^j\models\varphi_2$, \newline
${\cal R},L_{\Pi},\pi^i \models \varphi_1$ for all
$0\leq i < j$, and $r_a \leq \sum_{k=0}^{j-1}r_k \leq r_b$.
\end{tabular}\\[1ex]
For a state $t_0$ of sort $\mathtt{GlobalSystem}$, the
satisfaction relation of an MTL formula $\phi$ for the state
$t_0$ in $\cal R$ is defined as:
\[
\mathcal{R},L_{\Pi},t_0 \models \phi \;\iff
\; \forall \pi\in Paths(\mathcal{R})_{t_0} \quad
\mathcal{R},L_{\Pi},\pi \models \phi
\]
\end{definition}
\section{Preliminaries}
\label{sec:preliminaries}
\input{real-time-maude}
\input{logic}
\input{implementation}
\input{correctness}
\input{case_studies}
\input{conclusion}
\ifworking
{\begin{center} \Large \bf Still \thetodo\ things to fix! \end{center}}
\fi
\bibliographystyle{eptcs}
\subsection{Real-Time Maude}
\label{sec:rtm}
In Real-Time Maude~\cite{journ-rtm},
real-time systems are modeled by a set of \emph{equations} and
\emph{rewrite rules}. The rewrite rules are divided into \emph{instantaneous}
rules, that model changes that are assumed to take zero time, and
\emph{tick} rules that model time advance.
Formally, a Real-Time Maude \emph{timed module} specifies a \emph{real-time
rewrite theory}~\cite{OlvMesTCS} of the form
$\mathcal{R}=(\Sigma, E, \mathit{IR}, \mathit{TR})$, where:
\begin{itemize}
\item $(\Sigma, E)$ is a \emph{membership equational
logic}~\cite{maude-book} theory with $\Sigma$ a
signature\footnote{That is, $\Sigma$ is a set of declarations of
\emph{sorts}, \emph{subsorts}, and \emph{function symbols}.} and
$E$ a set of {\em confluent and terminating conditional equations}.
$(\Sigma, E)$ specifies the system's state space as an algebraic
data type, and must contain a specification of a sort @Time@
modeling the (discrete or dense) time domain. We denote by
$\mathbb{T}_{\cal{R},\texttt{s}}$ all ground terms of sort
\texttt{s}.
\item $\mathit{IR}$
is a set of (possibly conditional) \emph{labeled instantaneous (rewrite)
rules} specifying
the system's \emph{instantaneous} (i.e., zero-time) local transitions, written
$@crl [@l@] : @t@ => @t'@ if @cond$,
where $l$ is a \emph{label}.
Such a rule specifies a \emph{one-step transition}
from an instance of $t$ to the corresponding
instance
of $t'$.
The rules are applied \emph{modulo} the
equations~$E$.\footnote{$E$ is a union $E'\cup
A$, where $A$ is a set of equational axioms such as associativity,
commutativity, and identity, so that deduction is performed \emph{modulo}
$A$. Operationally, a
term is reduced to its
$E'$-normal form modulo $A$ before any rewrite rule is applied.}
\item $\mathit{TR}$ is a set of \emph{tick (rewrite) rules}, written
with syntax
\vspace{-1mm}
\begin{alltt}
crl [\(l\)] : \texttt{\char123}\(t\)\texttt{\char125} => \texttt{\char123}\(t'\)\texttt{\char125} in time \(\tau\) if \(cond\) .
\end{alltt}
\vspace{-1mm}
that model time elapse. @{_}@ is a
built-in
constructor of sort \texttt{GlobalSystem}, and
$\tau$ is a term of sort @Time@ that denotes the \emph{duration}
of the rewrite.
\end{itemize}
The initial state must be a ground term of sort @GlobalSystem@ and
must be reducible to a term of
the form @{@$t$@}@ using the equations in the specification.
The form of the tick rules ensures that time advances uniformly in the whole system.
Following~\cite{OlvMesTCS}, we write $t \stackrel{r}{\rightarrow}
t'$ when $t$ can be rewritten into $t'$ in time $r$ by a
\emph{one-step rewrite}. Note that instantaneous steps have
duration $0$. A \emph{(timed) path} $\pi$ in $\cal R$ is an infinite
sequence
$$
\pi = t_0 \stackrel{r_0}{\rightarrow} t_1
\stackrel{r_1}{\rightarrow} t_2\ldots
$$
such that either
\begin{itemize}
\item for all $i\in\mathbb{N}$, $t_i\xrightarrow{r_i} t_{i+1}$ is a one-step
rewrite in $\mathcal{R}$;
or
\item there exists a $k\in\mathbb{N}$ such that $\;t_i\xrightarrow{r_i}
t_{i+1}$ is a one-step rewrite in $\mathcal{R}$ for all $0\leq i <
k$, there is no one-step rewrite from $t_k$ in $\mathcal{R}$,
and $t_j = t_k$ and $r_{j-1} = 0$ for each $j>k$.
\end{itemize}
We denote by $Paths(\mathcal{R})_{t_0}$
the set of all timed paths of $\cal R$ starting in $t_0$. We call a path $ \pi = t_0 \stackrel{r_0}{\rightarrow} t_1
\stackrel{r_1}{\rightarrow} t_2\ldots$ \emph{time-divergent} iff for
all $r\in\mathbb{R}$ there is an $i\in\mathbb{N}$ such that $\sum_{k=0}^i r_k
> r$. Paths that are not time-divergent are called
\emph{time-convergent}.
We define $\pi^k = t_k
\stackrel{r_{k}}{\rightarrow} t_{k+1}
\stackrel{r_{k+1}}{\rightarrow} \ldots$. A term $t'$ is
\emph{reachable} from $t_0$ in $\cal R$ in time $r$ iff there is a
path $\pi = t_0 \stackrel{r_0}{\rightarrow} \ldots
\stackrel{r_{k-1}}{\rightarrow} t_k \ldots$ with $t_k = t'$ and
$r=\sum_{i=0}^{k-1}r_i$.
\smallskip
The Real-Time Maude syntax is fairly intuitive; we refer
to~\cite{maude-book} for a detailed description. For example, a
function symbol $f$ is declared with the syntax \texttt{op }$f$ @:@
$s_1$ \ldots $s_n$ @->@ $s$, where $s_1\:\ldots\:s_n$ are the sorts
of its arguments, and $s$ is its (value) \emph{sort}. Equations are
written with syntax @eq@ $t$ @=@ $t'$, and @ceq@ $t$ @=@ $t'$ @if@
\emph{cond} are conditional equations. The mathematical variables in
such statements are declared with the keywords {\tt var} and {\tt
vars}.
In \emph{object-oriented} Real-Time Maude modules, a \emph{class} declaration
\begin{alltt}
class \(C\) | \(\mbox{$att_1$}\) : \(\mbox{$s_1$}\), \dots , \(\mbox{$att_n$}\) : \(\mbox{$s_n$}\) .
\end{alltt}
declares a class $C$ with attributes $att_1$ to $att_n$ of
sorts $s_1$
to $s_n$, respectively. An {\em object\/} of class $C$ in a state is
represented as a term
$@<@\: O : C \mid att_1: val_1, ... , att_n: val_n\:@>@$
of sort @Object@, where $O$, of sort @Oid@, is the
object's
\emph{identifier}, and where $val_1$ to
$val_n$ are the current values of the attributes $att_1$ to
$att_n$, respectively.
In a \emph{concurrent} object-oriented
system, the
state
is a term of
sort @Configuration@. It has
the structure of a \emph{multiset} made up of objects and messages.
Multiset union for configurations is denoted by a juxtaposition
operator (empty
syntax) that is declared associative and commutative, so that rewriting is
\emph{multiset
rewriting} supported directly in Real-Time Maude.
The dynamic behavior of concurrent
object systems is axiomatized by specifying its
transition patterns by rewrite rules. For example,
the rule
{\small
\begin{alltt}
rl [l] : m(O,w) < O : C | a1 : 0, a2 : y, a3 : w > =>
< O : C | a1 : T, a2 : y, a3 : y + w > dly(m'(O'),x) .
\end{alltt}
}
\noindent defines a parametrized family of transitions
(one for each substitution instance), which can be applied
whenever the attribute @a1@ of an
object {\tt O} of class @C@ has the value @0@, with the effect of altering
the attributes @a1@ and @a3@ of the object. Moreover,
a message @m@, with
parameters @O@ and @w@, is read and consumed, and a new message @m'(O')@ is sent
\emph{with delay} @x@ (see~\cite{journ-rtm}).
``Irrelevant'' attributes, such as @a2@, need not be mentioned
in a rule.
A \emph{flat} (or \emph{non-hierarchical}) object-oriented specification is one
where all rewrites happen in the ``outermost'' configuration; that is, no attribute value
$t$ rewrites to some $t'\not = t$.
The specification of time-dependent behavior of object-oriented real-time systems follows the
techniques given in~\cite{journ-rtm}.
Time elapse is modeled by the tick rule
{\small
\begin{alltt}
var C : Configuration . var T : Time .
crl [\(tick\)] : \texttt{\char123}C\texttt{\char125} => \texttt{\char123}delta(C, T)\texttt{\char125} in time T if T <= mte(C) [nonexec] .
\end{alltt}
}
\noindent The function @delta@ defines the effect of time elapse
on a configuration, and the function @mte@ defines the
maximum amount of time that can elapse before some action must take
place. These functions distribute over the objects and messages
in a configuration
and must be defined for all single objects and messages to
define the timed behavior of a system. The tick rule advances time
\emph{nondeterministically} by \emph{any} amount @T@ less than or equal to
@mte(C)@. To execute
such rules, Real-Time Maude offers a choice of
\emph{time sampling strategies}, so that only \emph{some} moments in time are
visited. The choice of such strategies includes:
\begin{itemize}
\item Advancing time by a fixed amount $\Delta$ in each application of
a tick rule.
\item The \emph{maximal} strategy, that advances
time to the next moment when some action must be taken, as defined by @mte@.
This corresponds to \emph{event-driven
simulation}.
\end{itemize}
\paragraph{Formal Analysis.}
A Real-Time Maude specification is \emph{executable}, under reasonable conditions,
and the tool offers a variety of formal analysis
methods. The \emph{rewrite} command simulates \emph{one} fair
behavior of the system \emph{up to a certain duration}.
The \emph{search}
command uses a breadth-first strategy to analyze all possible
behaviors of the system, by checking whether a state matching a
\emph{pattern} and satisfying
a \emph{condition} can be reached from the
initial state. Such a pattern typically describes the \emph{negation} of an
invariant, so that the search succeeds iff the invariant is violated.
The command which searches for $n$ states
satisfying the \emph{pattern} search criterion has syntax
\small
\begin{alltt}
(utsearch [\(n\)] \(t\) =>* \(pattern\) such that \(cond\) .)
\end{alltt}
\normalsize
Real-Time Maude also extends Maude's \emph{linear temporal logic model
checker}
to check whether
each behavior, possibly up to a certain time bound,
satisfies a temporal logic
formula.
\emph{State propositions} are terms of sort @Prop@, and their
semantics should be
given by (possibly conditional) equations of the form
\small
\begin{alltt}
\texttt{\char123}\(statePattern\)\texttt{\char125} |= \(prop\) = \(b\)
\end{alltt}
\normalsize
\noindent for $b$ a term of sort @Bool@, which defines the state
proposition $prop$ to hold in all states $@{@t@}@$ where $@{@t@}@$
\verb+|=+ $prop$ evaluates to @true@. We use the notation $\Pi$ for
the set of propositions and $L_\Pi$ for the (implicit) labeling
function assigning to each state the set of propositions that hold in the state. A temporal
logic \emph{formula} is constructed by state propositions and the
Boolean and temporal logic operators discussed in Section~\ref{sec:mtl}.
The time-bounded model checking command has syntax
\small
\begin{alltt}
(mc \(t\) |=t \(\mathit{formula}\) in time <= \(\tau\) .)
\end{alltt}
\normalsize
\noindent for initial state $t$ and temporal logic formula $\mathit{formula}$ .
Since the model checking commands execute
tick rules
according to the chosen time sampling strategy, only a subset of all possible
behaviors is analyzed. Therefore, Real-Time Maude analyses are in general
\emph{incomplete} for a given property.
However, in~\cite{wrla06} we have given easily checkable conditions for
ensuring that Real-Time Maude analyses are indeed sound and complete.
It is also worth remarking that in the rest of the paper, we implicitly consider
the different analyses w.r.t.\ Real-Time Maude executions. That is, for dense time,
by ``a
rewrite theory $\cal R$'' in the following sections we typically mean
the real-time rewrite theory $\mathcal{R}^{tss}$ that has been obtained
from an original time-nondeterministic real-time rewrite theory $\mathcal{R}$
by applying the theory transformation corresponding to using the time sampling strategy $tss$
when executing the tick rules~\cite{journ-rtm}.
|
{
"timestamp": "2010-09-23T02:00:59",
"yymm": "1009",
"arxiv_id": "1009.4264",
"language": "en",
"url": "https://arxiv.org/abs/1009.4264"
}
|
\section{Introduction and statements of the main results}
In this paper we address the following
problem: {\em classify all
homogeneous K\"{a}hler\ manifolds (\emph{h.K.m.} for short) which admit a K\"{a}hler\ immersion
into a given finite or infinite dimensional complex space form}.
\vskip 0.3cm
A K\"{a}hler\ immersion $f:(M, g)\rightarrow (S, g_S)$
from a K\"{a}hler\ manifold $(M , g)$ into a complex space form $(S, g_S)$
is a holomorphic map such that $f^*g_S=g$
(here $g$ and $g_S$ denote the K\"{a}hler\ metrics on $M$ and $S$ respectively).
\vskip 0.3cm
Recall that there are three types,
up to homotheties,
of complex space forms $(S, g_S)$
according to the sign of
their constant holomorphic sectional curvature:
\begin{itemize}
\item
the complex Euclidean space
$\mathbb{C}^N$, $N\leq\infty$,
with the flat metric denoted by $g_{0}$.
Here ${\mathbb{C}}^{\infty}$ is
the complex Hilbert space
$\ell^2(\mathbb{C})$
consisting of
sequences $z_j, j=1\dots, z_j\in {\mathbb{C}}$
such that
$\sum_{j=1}^{+\infty}|z_j|^2<+\infty$.
\item
the complex hyperbolic space
${\mathbb{C}}H^N$, $N\leq\infty$,
namely the unit ball in $\mathbb{C} ^N$ ($\sum_{j=1}^{N}|z_j|^2<1$)
endowed with the hyperbolic metric
$g_{hyp}$ of
holomorphic sectional curvature being $-4$, whose associated K\"{a}hler\
form $\omega_{hyp}$ is given by:
\begin{equation}\label{omegahyp}
\omega_{hyp}=-\frac{i}{2}\partial\bar\partial\log(1- \sum_{j=1}^{N}|z_j|^2).
\end{equation}
\item
the complex projective space
${\mathbb{C}}P^N, N\leq\infty$,
with the Fubini--Study metric $g_{FS}$
of holomorphic sectional curvature being $4$.
If $\omega_{FS}$ denotes the K\"{a}hler\ form associated to $g_{FS}$
then, in homogeneous
coordinates $[Z_0,\dots, Z_{N}]$,
$\omega_{FS}=\frac{i}{2}\partial\bar\partial\log \sum_{j=0}^{N}
|Z_j|^2$.
\end{itemize}
\vskip 0.3cm
\noindent
{\em Notation.} When we speak about the K\"{a}hler\ manifold $\mathbb{C}^N$ (resp. $\mathbb{C} H^N$ or $\mathbb{C} P^N$)
without mentioning the K\"{a}hler\ metric we will always mean $\mathbb{C}^N$ (resp. $\mathbb{C} H^N$ or $\mathbb{C} P^N$) equipped
with the metric $g_0$ (resp. $g_{hyp}$, $g_{FS}$).
\vskip 0.3cm
Note that, once that a K\"{a}hler\ immersion into a complex space form $(S, g_S)$ is given,
then all other K\"{a}hler\ immersions can be obtained by composing it with a unitary transformation
of $(S, g_S)$. This is due to the following celebrated rigidity theorem due to E. Calabi \cite{Ca53} which will be of constant use throughout this paper.
\vskip 0.3cm
\noindent
{\bf Theorem (Calabi's rigidity theorem)}
{\em Let $f:(M, g)\rightarrow (S, g_S)$ and
$\tilde f:(M, g)\rightarrow (S, g_S)$ be two K\"{a}hler\ immersions into the same complex space form $(S,
g_S)$.
Then there exists a unitary transformation $U$ of $(S, g_S)$ such that $f=U\circ\tilde f$.}
\vskip 0.3cm
\subsection{Immersions in $\mathbb{C}^N$ and $\mathbb{C} H^N$}
In the following two theorems we give a complete solution of our problem
when the ambient space is $\mathbb{C}^N$ or $\mathbb{C} H^N$, $N\leq\infty$.
In order to state our result
note that the map $f_n:{\mathbb{C}}H^n\rightarrow l^2({\mathbb{C}})$
given by:
\begin{equation}\label{calhyp}
z=(z_1,\dots z_n)\stackrel{f_n}{\mapsto}
(\dots, \sqrt{\frac{(|j|-1)!}{j!}}z_1^{j_1}\cdots z_n^{j_{n}}, \dots )
\end{equation}
is a K\"{a}hler\ immersion of ${\mathbb{C}}H^n$ into $l^2({\mathbb{C}})$, i.e. $f_n^*g_0=g_{hyp}$,
(see \cite{Ca53}), where
$|j|=j_1+\dots+j_n$
and $j!=j_1! \cdots j_{n}!$.
\begin{thm}\label{Flat-Case}
Let $(M, g)$ be a $n$-dimensional h.K.m..
\begin{itemize}
\item [(a)]
If $(M, g)$ can be K\"{a}hler\ immersed into $\mathbb{C} ^N$, $N<\infty$, then
$(M, g)=\mathbb{C}^n$;
\item [(b)]
if $(M, g)$ can be K\"{a}hler\ immersed into $\ell^2(\mathbb{C})$, then
$(M, g)$ equals
$$\mathbb{C} ^k
\times{\mathbb{C}}H^{n_1}_{\lambda_1}\times\cdots \times {\mathbb{C}}H^{n_r}_{\lambda_r},$$
where $k+n_1+\cdots +n_r=n$, $\lambda_j$, $j=1,\dots , r$
are positive real numbers and $\mathbb{C} H^{n_j}_{\lambda_j}=(\mathbb{C} H^{n_j}, \lambda_jg_{hyp}),\ j=1,\dots , r$ (hence $\mathbb{C} H^n_{1}=\mathbb{C} H^n$).
\end{itemize}
Moreover, in case (a) (resp. case (b)) the immersion is given, up to a unitary transformation of $\mathbb{C} ^N$ (resp. $\ell^2 (\mathbb{C}))$, by the linear inclusion $\mathbb{C}^n\hookrightarrow \mathbb{C} ^N$
(resp. by
$(f_0, f_1,\dots ,f_r)$,
where $f_0$ the linear inclusion
$\mathbb{C} ^k\hookrightarrow \ell^2(\mathbb{C})$
and each $f_j:{\mathbb{C}}H^{n_j}\rightarrow \ell^2({\mathbb{C}})$
is
$\sqrt{\lambda_j}$ times
the map (\ref{calhyp})).
\end{thm}
\begin{thm}\label{Negative-case}
Let $(M, g)$ be a $n$-dimensional h.K.m..
Then if $(M, g)$ can be K\"{a}hler\ immersed into $\mathbb{C} H^N$, $N\leq\infty$, then
$(M, g)=\mathbb{C} H^n$
and the immersion is given,
up to a unitary transformation of $\mathbb{C} H^N$,
by the linear inclusion $\mathbb{C} H^n\hookrightarrow \mathbb{C} H^N$
\end{thm}
\begin{rmk}\rm
Since a K\"{a}hler\ immersion is minimal, an alternative proof of (1) in Theorem \ref{Flat-Case} when $N<\infty$
follows by the work of
A. J. Di Scala \cite{DS02}.
\end{rmk}
\begin{rmk}\rm\label{assertion2}
Assertion (2) in Theorem \ref{Flat-Case}
is a generalization to arbitrary h.K.m. of Theorem 3.3 in \cite{DL07}
where the first and the third authors proved that a bounded
symmetric domain which can be K\"{a}hler\ immersed into $\ell ^2 (\mathbb{C})$ has necessarily rank one.
Actually, the method of the present paper, when applied to bounded symmetric domains, provides
us with an alternative and more elegant proof of
this result (cfr. Remark \ref{simpleproof} below).
\end{rmk}
\vskip 0.3cm
\subsection{Immersion in $\mathbb{C} P^N$}
There exists a large class (cfr. Conjecture 1 below) of h.K.m. which can be K\"{a}hler\ immersed into $\mathbb{C} P^{N}$.
In this paper a K\"ahler metric $g$ on a complex manifold $M$ will be called {\em projectively induced} if there exists an immersion $f:M \rightarrow \mathbb{C}P^{N}$, $N\leq\infty$, such that $f^*g_{FS}=g$.
An obvious necessary condition for $g$ to be projectively induced is that its associated K\"{a}hler\ form $\omega$ is integral
i.e. it represents the first Chern class $c_{1}(L)$ in $H^2(M, \mathbb{Z})$
of a holomorphic line bundle $L\rightarrow M$. Indeed $L$ can be taken as the pull-back of the hyperplane line bundle on $\mathbb{C} P^N$
whose first Chern class is given by $\omega_{FS}$.
Notice that if $\omega$ is an exact form (e.g. when $M$ is contractible) then $\omega$ is obviously
integral since its second cohomology class vanishes.
Other (less obvious) conditions are expressed by the following theorem and its corollary which represent our first result
about projectively induced K\"{a}hler\ metrics.
\vskip 0.3cm
\begin{thm}\label{necessary}
Assume that a h.K.m. $(M, g)$ admits a K\"ahler immersion $f : M \rightarrow \mathbb{C} P^{N}$, $N\leq\infty$. Then
$M$ is simply-connected and $f$ is injective.
\end{thm}
\begin{cor}\label{corolnecessary} Let $(M, g)$ be a complete and locally h.K.m..
Assume that $f: (M, g)\rightarrow \mathbb{C} P^N$, $N\leq\infty$, is a K\"{a}hler\ immersion.
Then $(M, g)$ is a h.K.m..
\end{cor}
When the dimension of the ambient space is finite, i.e. $(S, g_S)=\mathbb{C} P^N$, $N<\infty$,
$M$ is forced to be compact and a proof of Theorem \ref{necessary} is
well-known by the work of M. Takeuchi \cite{TA78}.
In this case he also provides a complete classification of all compact
h.K.m. which can be K\"{a}hler\ immersed into $\mathbb{C} P^N$
by making use of the representation theory of semisimple Lie groups.
Viceversa, it is not hard to see that
if a {\em compact} K\"{a}hler\ manifold can be K\"{a}hler\ immersed into $\mathbb{C} P^{\infty}$
then it can also be K\"{a}hler\ immersed into $\mathbb{C} P^N$ with $N<\infty$.
We believe that, up to homotheties, {\em any} simply-connected h.K.m. such that its associated K\"{a}hler\ form is integral can be K\"{a}hler\ immersed into $\mathbb{C} P^N$,
with $N\leq\infty$. This is expressed by the
following conjecture.
\vskip 0.3cm
\noindent
{\bf Conjecture 1:} {\em
Let $(M, g)$ be a simply-connected h.K.m. such that its associated K\"{a}hler\ form $\omega$ is integral.
Then there exists $\lambda_0 \in \mathbb{R}^+$ such that $\lambda_0 g$ is projectively induced.
}
\vskip 0.3cm
The integrality of $\omega$ in the conjecture is important since there exist simply-connected h.K.m. $(M, \omega)$ such that $\lambda \omega$ is not integral for any $\lambda \in \mathbb{R}^+$ (take, for example, $(M, g) = (\mathbb{C}P^1, g_{FS}) \times (\mathbb{C}P^1, \sqrt{2}g_{FS})$).
Observe also that there exist simply-connected (even contractible) h.K.m. $(M, g)$ such that $\omega$ is an integral form but
$g$ is not projectively induced.
In order to describe such an example we recall the following result (see Theorem 2 in \cite{LZ09}).
\vskip 0.3cm
\noindent
{\bf Theorem A.}
Let $g_B$ be the Bergman metric of an irreducible
Hermitian symmetric space of noncompact type $\Omega$.
Then $\lambda g_B$
is projectively induced if and only $\lambda\gamma$ belongs to $W(\Omega)\setminus \{0\}$, where $\gamma$ denotes the genus of $\Omega$ and $W(\Omega)$ its Wallach set.
\vskip 0.3cm
It turns out (see Corollary $4.4$ p. 27 in \cite{AR95} and references therein)
that $W(\Omega)$ consists only of real numbers and depends on two of the domain's invariants, denoted by $a$ (strictly positive natural number) and $r$ (the rank of $\Omega$).
More precisely we have
\begin{equation}\label{wallachset}
W(\Omega)=\left\{0,\,\frac{a}{2},\,2\frac{a}{2},\,\dots,\,(r-1)\frac{a}{2}\right\}\cup \left((r-1)\frac{a}{2},\,\infty\right).
\end{equation}
The set $W_d=\left\{0,\,\frac{a}{2},\,2\frac{a}{2},\,\dots,\,(r-1)\frac{a}{2}\right\}$ and the interval $W_c= \left((r-1)\frac{a}{2},\,\infty\right)$
are called respectively the {\em discrete} and {\em continuous} part of the Wallach set of the domain
$\Omega$.
Observe that when $r=1$, namely $\Omega$ is the complex hyperbolic space $\mathbb{C} H^n$,
then $g_B=(n+1)g_{hyp}$.
In this case (and only in this case) $W_d=\{0\}$ and $W_c=(0, \infty)$.
If $\operatorname{rank} (\Omega)=r\geq 2$
and $0<\lambda < \frac{a}{2\gamma}$
it follows by Theorem A that
$\lambda g_B$ is not projectively induced
and its associated K\"{a}hler\ form $\lambda \omega_B$ is integral (since $\Omega$ is contractible).
This provides us with the desired example.
\vskip 0.3cm
Notice also that from Theorem A it follows that the only irreducible bounded symmetric domain where $\lambda g_B$
is projectively induced for all $\lambda >0$ is the complex hyperbolic space.
In the following theorem, which represents our last result, we generalize this fact to
any homogeneous bounded domain (h.b.d. for short).
This will be a key ingredient in the proof of Theorem \ref{Flat-Case}.
\begin{thm}\label{thmsmall}
Let $(\Omega, g)$ be a $n$-dimensional h.b.d..
The metric $\lambda g$ is projectively induced for all
$\lambda >0$
if and only if
\begin{equation} \label{eqn:direct_prod}
(\Omega, g) ={\mathbb{C}}H^{n_1}_{\lambda_1}\times\cdots \times {\mathbb{C}}H^{n_r}_{\lambda_r},
\end{equation}
where $n_1+\cdots +n_r=n$, $\lambda_j$, $j=1,\dots , r$
are positive real numbers and
$\mathbb{C} H^{n_j}_{\lambda_j}=(\mathbb{C} H^{n_j}, \lambda_jg_{hyp}),\ j=1,\dots , r$.
\end{thm}
\vskip 0.3cm
\noindent
The paper contains another section dedicated to the proofs of our main results.
\vskip 0.3cm
\noindent
{\bf Aknowledgments:}
The second and third author would like to thank {\em Politecnico of Torino} for the wonderful hospitality
in their research stays in January 2010.
\section{Proof of the main results}
The basic ingredient for the proof of our results
is the following solution due to
J. Dorfmeister and K. Nakajima \cite{DN88} of the fundamental conjecture on h.K.m..
\vskip 0.3cm
\noindent
{\bf Theorem FC}
{\em A h.K.m. $(M, g)$ is the total space of a holomorphic
fiber bundle over a h.b.d. $\Omega$ in which the fiber ${\mathcal F} ={\mathcal E} \times {\mathcal C}$
is (with the
induced K\"ahler metric) the K\"{a}hler\ product of a flat homogeneous K\"ahler manifold ${\mathcal E}$ and a
compact simply-connected homogeneous K\"ahler manifold ${\mathcal C}$.}
\vskip 0.3cm
In order to prove Theorem \ref{Flat-Case}
recall that complete connected totally geodesic submanifolds of $\mathbb{R}^n$ are affine subspaces $p + \mathbb{W}$,
where $p \in \mathbb{R}^n$ and $\mathbb{W} \subset \mathbb{R}^n$ is a vector subspace.
We need the following result from \cite{AD03} which we include here for completeness.
\begin{lem} \label{parallel} Let $G$ be a connected Lie subgroup of
isometries of the Euclidean space $\mathbb{R}^n$. Let $G.p = p + \mathbb{V}$ and $G.q = q + \mathbb{W}$ be two totally geodesic $G$-orbits.
Then $\mathbb{V} = \mathbb{W}$, i.e. $G.p$ and $G.q$ are parallel affine subspaces of $\mathbb{R}^n$.
\end{lem}
\noindent
{\bf Proof. }
We can assume that $p = 0 \in \mathbb{R}^n$ and that $p,q$ are the points that realize the distance between both orbits
$G\cdot p$, $G\cdot q$, i.e.
$\operatorname{dist}(p,q) = \operatorname{dist}(G\cdot p,G\cdot q)$. Let $\gamma(t)=tq$ be the geodesic
that realizes the distance between $q$ and $\mathbb{V}$. So the vector $q$
is perpendicular to any $G$-orbit $G_t=G \cdot \gamma(t)$ $t \in
{\mathbb R}$. Let $X = x^*$ be any Killing vector field of $G$ and $\operatorname{Exp}(tX)$
its associated one-parameter group of isometries. Define
$h:I\times \mathbb{R }\rightarrow \mathbb{R}^n$ by $h_s(t):=\operatorname{Exp}(sX) \cdot
\gamma(t)$. Note that $X ( h_s(t)) = \frac{\partial h}{\partial
s}$ and that, for a fixed $s$, $h_s(t)$ is a geodesic.
Let $A_t$ be the shape operator at the point $\gamma(t)$ of the
orbit $G \cdot \gamma(t)$ in the direction of $\dot \gamma (t)$.
Define $f(t) := - \langle A_t(X ( \gamma(t))) ,X (\gamma(t))
\rangle =
\langle \frac{D}{\partial s}\frac{\partial h}{\partial t},
X(h_s(t)) \rangle \mid_{s=0}$. We have
$$\frac{d}{dt}f(t) = \langle \frac{D}{\partial t}
\frac{D}{\partial s}\frac{\partial h}{\partial t}, X(h_s(t))
\rangle \mid_{s=0} + \langle \frac{D}{\partial s} \frac{\partial
h}{\partial t}, \frac{D}{\partial t}X(h_s(t)) \rangle \mid_{s=0}
$$
$$ = \langle \frac{D}{\partial s}
\frac{D}{\partial t}\frac{\partial h}{\partial t}, X(h_s(t))
\rangle \mid_{s=0}
+ \langle \frac{D}{\partial t}\frac{\partial h}{\partial s},
\frac{D}{\partial t}X (h_s(t)) \rangle \mid_{s=0} $$
$$ = \| \nabla_{\dot{\gamma}(t)}
(X(\gamma(t))) \| ^2 .$$
Since $f(0)=0$ because $G \cdot p$ is totally geodesic, we get
$$f(1)= - \langle A_t(X( q)), X( q) \rangle \geq 0 \, .$$ Hence $A_1$ is
negative definite and since $G \cdot q$ is totally geodesic, any Killing
vector field $X$ is parallel along $\gamma(t)$. We can write
$\operatorname{Exp}(sX) \cdot p = e^{s\overline{X}}(p-c)+c+sd$, where $\overline{X}$ is the
projection of $X$ into ${\mathfrak{so}}_n$, $d \in \ker (\overline{X})$ and
$c \in \ker (\overline{X})^{\perp}$. Then a Killing vector field $X$ is
parallel along $\gamma(t)$ if and only if $q \in \ker (\overline{X})$.
Thus $\operatorname{Exp}(sX) \cdot q = q + \operatorname{Exp}(sX) \cdot p $ which implies that
$$\mathbb{V} = T_p (G\cdot p) \subset T_q (G\cdot q) = \mathbb{W} \, .$$
Reversing the role of $ \mathbb{V}$ and $\mathbb{W}$ the same argument yields $\mathbb{W} \subset \mathbb{V}$.
This completes the proof of the lemma. $\Box$
\vskip 0.3cm
\noindent
{\bf Proof of Theorem \ref{Flat-Case}.}
\noindent
Assume that there exists a K\"ahler immersion $f: M \rightarrow \mathbb{C}^N$. By Theorem FC
and by the fact that a h.b.d. is contractible
we get that $M = \mathbb{C}^k\times \Omega$ as a complex manifold since,
by the maximum principle, the fiber ${\mathcal F}$ cannot contain a compact manifold. Let $M = G/K$ be the homogeneous realization of $M$ (so the metric
$g$ is $G$-invariant).
It follows again by Theorem FC that there exists $L \subset G$ such that the $L$-orbits are the fibers of the fibration $\pi: M = G/K \rightarrow \Omega = G/L $. Let $F_p, F_q$ be the fibers over $p,q \in \Omega$. We claim that $f(F_p)$ and $f(F_q)$ are parallel affine subspaces of $\mathbb{C}^N$.
Indeed, by Calabi's rigidity $f(F_p)$ and $f(F_q)$ are affine subspaces of $\mathbb{C}^N$ since both $F_p$ and $F_q$ are flat K\"{a}hler\ manifolds of $\mathbb{C}^n$.
Moreover, Calabi rigidity theorem implies the existence of a morphism of groups $\rho: G \rightarrow Iso_{\mathbb{C}}(\mathbb{C}^N) = \mathrm{U}(\mathbb{C}^N) \ltimes \mathbb{C}^N$ such that
$f(g\cdot x) = \rho(g)f(x)$
for all $g \in G, x \in M$.
Let $W_{p,q}$ be the affine subspace generated by $f(F_p)$ and $f(F_q)$. Since both $f(F_p)$ and $f(F_q)$ are $\rho(L)$-invariant it follows that $W_{p,q}$ is also $\rho(L)$-invariant. Indeed, for any $g \in L$ the isometry $\rho(g)$ is an affine map and so must preserve the affine space generated by $f(F_p)$ and $f(F_q)$. Observe that $W_{p,q}$ is a finite dimensional complex Euclidean space, $\rho(L)$ acts on $W_{p,q}$
and $f(F_p)$ and $f(F_q)$ are two complex totally geodesic orbits in $W_{p,q}$. Then, by Lemma \ref{parallel}, we get that $f(F_p)$ and $f(F_q)$ are parallel affine subspaces of $W_{p,q}$ and hence of $\mathbb{C}^N$. Since $p,q \in \Omega$ are two arbitrary points it follows that $f(M)$ is a K\"ahler product. Thus $M = \mathbb{C}^{k}\times \Omega$ is a K\"ahler product of homogeneous K\"ahler manifolds. Using again the fact $M$ can be K\"{a}hler\ immersed into $\mathbb{C}^N$ it follows that the h.b.d. $\Omega$ can be K\"{a}hler\ immersed into $\mathbb{C}^N$.
If one denotes by $\varphi$ this immersion and by
$g_{\Omega}$ the homogeneous
K\"{a}hler\ metric of $\Omega$,
it follows that the map $\sqrt{\lambda}\varphi$ is a K\"{a}hler\ immersion of $(\Omega, \lambda g_{\Omega})$ into $\mathbb{C}^N$. Therefore, by Theorem 14 in \cite{Bo47}, $\lambda g_{\Omega}$ is projectively induced for all $\lambda >0$ and
Theorem \ref{thmsmall} yields
$$(M, g) = \mathbb{C} ^k
\times {\mathbb{C}}H^{n_1}_{\lambda_1}\times\cdots \times {\mathbb{C}}H^{n_r}_{\lambda_r},$$
where $k+n_1+\cdots +n_r=n$ and $\lambda_j$, $j=1,\dots , r$
are positive real numbers.
If the dimension $N$ of the ambient space $\mathbb{C}^N$ is finite then
$M =\mathbb{C}^n$ since there cannot exist a K\"{a}hler\ immersion of
$(\mathbb{C} H^{n_j}, \lambda_j g_{hyp})$
into $\mathbb{C}^N$, $N<\infty$ (see \cite{Ca53})
and this proves (a).
The last part of Theorem \ref{Flat-Case}
is a consequence of Calabi's rigidity theorem together with
Lemma 3.1 in \cite{DL07}
which asserts that
a K\"{a}hler\ map $f: M \times M' \rightarrow \mathbb{C} ^N$, $N\leq\infty$,
from a product $M \times M'$ of two K\"{a}hler\
manifolds is a product, i.e. $f(p,q) = (f_1(p),f_2(q))$ where
$f_1:M \rightarrow \mathbb{C} ^N$ and $f_2:M' \rightarrow \mathbb{C} ^N$ are
K\"{a}hler\ maps.
$\Box$
\begin{rmk}\label{simpleproof}\rm
As we have already pointed, Theorem \ref{thmsmall}, which is an important step in the
proof of the Theorem \ref{Flat-Case},
is a straightforward consequence of Theorem A above when the h.K.m. is a bounded symmetric domain.
Therefore the last part of Theorem \ref{Flat-Case} provides an alternative proof of Theorem 3.3 in \cite{DL07}
without the use of Calabi's diastasis function (cfr. Remark \ref{assertion2}).
\end{rmk}
\vskip 0.3cm
In order to prove Theorem \ref{Negative-case} we need the following lemma.
\begin{lem}\label{lemmazedda1}
If a K\"{a}hler\ manifold $(M, g)$ can be K\"ahler immersed into $\mathbb{C} H^N$, $N\leq\infty$, then it can also be
K\"ahler immersed into $\ell^2(\mathbb{C})$.
\end{lem}
\noindent
{\bf Proof. }
Let $f$ be the K\"{a}hler\ immersion of $(M, g)$ into $\mathbb{C} H^N$. If $N<\infty$ then the map $f_n\circ f : (M, g)\rightarrow \ell ^2(\mathbb{C})$,
where $f_n$ is given by (\ref{calhyp}), is a K\"{a}hler\ immersion. If $N=\infty$,
it follows by (\ref{omegahyp}) in the introduction that $\Phi =-\log (1-\sum_{j=1}^{\infty}|\phi_j|^2)
= \sum_{k=1}^{\infty}(\sum_{j=1}^{\infty}|\phi_j|^2)^k$ is a K\"{a}hler\ potential
for the metric $g$, i.e. $\frac{i}{2}\partial\bar\partial \Phi =\omega$, where $\omega$ is the K\"{a}hler\ form associated to the metric $g$
and the $\phi_j$'s are the components of $f$.
Hence $\Phi =\sum_{j=1}^{\infty}|h_j|^2$
for suitable holomorphic functions $h_j$, $j=1, 2, \dots$ on $M$ and
the map $h= (\dots, h_j, \dots):(M, g)\rightarrow \ell ^2(\mathbb{C})$ is the desired K\"{a}hler\ immersion.
\hspace*{\fill}$\Box$
\vskip 0.3cm
\noindent
{\bf Proof of Theorem \ref{Negative-case}.}
If a h.K.m. $(M, g)$ can be
K\"{a}hler\ immersed into $\mathbb{C} H^{N}$,
$N\leq\infty$,
then, by Lemma \ref{lemmazedda1} it can also be
K\"ahler immersed into $\ell^2(\mathbb{C})$.
By Theorem \ref{Flat-Case},
$(M, g)$ is then a K\"{a}hler\ product of complex space forms, namely
$$(M, g)=\mathbb{C} ^k
\times {\mathbb{C}}H^{n_1}_{\lambda_1}\times\cdots \times {\mathbb{C}}H^{n_r}_{\lambda_r}.$$
Then the conclusion follows by the fact that $\mathbb{C}^k$ cannot be K\"{a}hler\ immersed into
$\mathbb{C} H^N$ for all $N\leq\infty$ (see \cite{Ca53}), by Calabi's rigidity theorem
and by Theorem 2.11 in \cite{AD03}
which shows that there are not K\"{a}hler\ maps
from a product $M \times M'$ of K\"{a}hler\ manifolds into ${\mathbb{C}}
H^{N}$, $N\leq\infty$, (the proof in \cite{AD03} is given for $N<\infty$ but it
extends without any substantial change
to the infinite dimensional case).
$\Box$
\vskip 0.3cm
\noindent
{\bf Proof of Theorem \ref{necessary}.}
Theorem FC and the fact that a h.b.d. is contractible imply that $M$ is a {\em complex}
product $\Omega \times {\mathcal F}$, where ${\mathcal F}={\mathcal E}\times {\mathcal C}$ is a K\"{a}hler\ product of a flat K\"{a}hler\ manifold ${\mathcal E}$ K\"{a}hler\ embedded into $(M, g)$ and a simply-connected h.K.m. ${\mathcal C}$.
We claim that ${\mathcal E}$ is simply-connected and hence $M=\Omega\times {\mathcal E}\times {\mathcal C}$ is simply-connected.
In order to prove our claim notice that
${\mathcal E}$ is the K\"{a}hler\ product $\mathbb{C}^k\times T_1\times\cdots\times T_s$, where $T_j$ are flat complex tori.
So one needs to show that each $T_j$ reduces to a point.
If, by a contradiction, the dimension of one of this tori, say $T_{j_0}$ is not zero, then by composing
the K\"{a}hler\ immersion of $T_{j_0}$ in $(M, g)$
with the immersion $f:M\rightarrow \mathbb{C} P^N$
we would get a K\"{a}hler\ immersion of
$T_{j_0}$ into $\mathbb{C} P^N$ in contrast with a well-known result of Calabi \cite{Ca53} (see also Lemma 2.2 in \cite{TA78}).
In order to prove that $f$ is injective we first observe that, by Calabi's rigidity theorem, $f(M)$ is still a h.K.m..
Then, by the first part of the theorem, $f(M)\subset\mathbb{C} P^{N}$ is simply-connected. Moreover, since $M$ is complete and $f: M \rightarrow f(M)$ is a local isometry, it is a covering map (see, e.g., Lemma 3.3 p. 150 in \cite{DC92}) and hence injective.$\Box$
\vskip 0.3cm
\noindent
{\bf Proof of Corollary \ref{corolnecessary}.}
Let $\pi:\tilde M\rightarrow M$ be the universal covering map.
Then $(\tilde M, \tilde g)$ is a h.K.m. and, by Theorem \ref{necessary},
$f\circ\pi :\tilde M\rightarrow \mathbb{C} P^n$ is injective.
Therefore $\pi$ is injective,
and since it is a covering map,
it defines a holomorphic isometry between $(\tilde M, \tilde g)$
and $(M, g)$.
$\Box$
\vskip 0.3cm
\noindent
{\bf Proof of Theorem \ref{thmsmall}.}
First we find a global potential of the homogeneous K\"ahler metric $g$
on the domain $\Omega$ following Dorfmeister \cite{D85}.
By \cite[Theorem 2 (c)]{D85},
there exists a split solvable Lie subgroup
$S \subset \mathrm{Aut}(\Omega, g)$
acting simply transitively on the domain $\Omega$.
Taking a reference point $z_0 \in \Omega$,
we have a diffeomorphism
$S \owns s \overset{\sim}{\mapsto} s \cdot z_0 \in \Omega$,
and
by the differentiation, we get the linear isomorphism
$\gs := \mathrm{Lie}(S) \owns X
\overset{\sim}{\mapsto} X \cdot z_0 \in T_{z_0}\Omega \equiv \mathbb{C}^n$.
Then the evaluation of the K\"ahler form $\omega$ on $T_{z_o}\Omega$
is given by
$\omega(X\cdot z_o, Y \cdot z_0) = \beta([X,Y])\,\,
(X, Y \in \gs)$
with a certain linear form $\beta \in \gs^*$.
Let $j : \gs \to \gs$ be the linear map defined in such a way that
$(jX) \cdot z_0 = \sqrt{-1} (X \cdot z_0)$ for $X \in \gs$.
We have
$\Re g(X \cdot z_0,\,Y \cdot z_0) = \beta([jX, Y])$
for $X, Y \in \gs$,
and the right-hand side defines a positive inner product on $\gs$.
Let $\mathfrak{a}$ be the orthogonal complement of $[\gs, \gs]$ in $\gs$
with respect to the inner product.
Then $\mathfrak{a}$ is a commutative Cartan subalgebra of $\gs$.
Define $\gamma \in \mathfrak{a}^*$ by
$\gamma(C) := -4 \beta (jC)\,\,\,(C \in \mathfrak{a})$,
and we extended $\gamma$ to $\gs = \mathfrak{a} \oplus [\gs, \gs]$ by the zero-extension.
Keeping the diffeomorphism between $S$ and $\Omega$ in mind,
we define a positive smooth function $\Psi$ on $\Omega$ by
$$ \Psi((\exp X) \cdot z_0) = e^{-\gamma(X)} \,\,\, (X \in \gs). $$
From the argument in \cite[pp. 302--304]{D85},
we see that
\begin{equation} \label{eqn:globalpotential}
\omega = \frac{i}{2}\partial\bar\partial\log \Psi.
\end{equation}
It is known that
there exists a unique kernel function $\tilde{\Psi} : \Omega \times \Omega \to \mathbb{C}$
such that (1) $\tilde{\Psi}(z,z) = \Psi(z)$ for $z \in \Omega$
and (2) $\tilde{\Psi}(z,w)$ is holomorphic in $z$ and anti-holomorphic in $w$
(cf. \cite[Proposition 4.6]{I99}).
Let us observe that the metric $g$ is projectively induced
if and only if
$\tilde{\Psi}$ is a reproducing kernel of a Hilbert space
of holomorphic functions on $\Omega$.
Indeed,
if $f : \Omega \to \mathbb{C} P^N\,\,(N \le \infty)$ is a K\"ahler immersion
with
$f(z) = [\psi_0(z) : \psi_1(z) : \cdots ]\,\,(z \in \Omega)$
its homogeneous coordinate expression,
then we have
$\omega = \frac{i}{2} \partial\bar\partial\log \sum_{j=0}^{N}|\psi_j|^2$.
Comparing (\ref{eqn:globalpotential}) with it,
we see that
there exists a holomorphic function $\phi$ on $\Omega$ for which
$\Psi = |e^{\phi}|^2 \sum_{j=0}^N |\psi_j|^2$.
By analytic continuation,
we obtain
$\tilde{\Psi}(z,w) = e^{\phi(z)} \overline{e^{\phi(w)}} \sum_{j=0}^N \psi_j(z) \overline{\psi_j(w)}$
for $z, w \in \Omega$.
For any $z_1, \dots, z_m \in \Omega$ and $c_1, \dots, c_m \in \mathbb{C}$,
we have
\begin{align*}
\sum_{p,q=1}^m c_p \bar{c}_q\tilde{\Psi}(z_p,z_q)
&= \sum_{p,q=1}^m c_p \bar{c}_q e^{\phi(z_p)} \overline{e^{\phi(z_q)}} \sum_{j=0}^N \psi_j(z_p) \overline{\psi_j(z_q)} \\
&= \sum_{j=0}^N |\sum_{p=1}^m c_p e^{\phi(z_p)}\psi_j(z_p)|^2 \ge 0.
\end{align*}
Thus the matrix $(\tilde{\Psi}(z_p,z_q))_{p,q} \in \mathrm{Mat}(m,\mathbb{C})$ is always
a positive Hermitian matrix.
Therefore $\tilde{\Psi}$ is a reproducing kernel of a Hilbert space
(see \cite[p. 344]{Ar50}).
On the other hand,
if $\tilde{\Psi}$ is a reproducing kernel of a Hilbert space
$\mathcal{H} \subset \mathcal{O}(\Omega)$,
then by taking an orthonormal basis $\{\psi_j\}_{j=0}^N$ of $\mathcal{H}$,
we have a K\"ahler immersion $f : M \owns z \mapsto [\psi_0(z) : \psi_1(z) : \cdots] \in \mathbb{C} P^N$
because we have $\Psi(z) = \tilde{\Psi}(z,z) = \sum_{j=0}^N |\psi_j(z)|^2$.
Note that there exists no point $a \in \Omega$ such that $\psi_j(a) = 0$ for all $1 \le j \le N$
since $\Psi(z) = \sum_{j=0}^N |\psi_j(z)|^2$ is always positive.
The condition for $\tilde{\Psi}$ to be a reproducing kernel is described in \cite{I99}.
In order to apply the results,
we need a fine description of the Lie algebra $\gs$ with $j$
due to Piatetskii-Shapiro \cite{PS69}.
Indeed, it is shown in \cite[Chapter 2]{PS69} that
the correspondence between the h.b.d. $\Omega$ and the structure of $(\gs, j)$
is one-to-one up to natural equivalence.
For a linear form $\alpha$ on the Cartan algebra $\mathfrak{a}$,
we denote by $\gs_{\alpha}$ the root subspace
$\set{X \in \gs}{[C,X] = \alpha(C)X \,\, (\forall C \in \mathfrak{a})}$
of $\gs$.
The number $r := \dim \mathfrak{a}$
is nothing but the rank of $\Omega$.
Thanks to \cite[Chapter 2, Section 3]{PS69},
there exists a basis
$\{\alpha_1, \dots, \alpha_r\}$
of $\mathfrak{a}^*$
such that
$\gs = \gs(0) \oplus \gs(1/2) \oplus \gs(1)$ with
\begin{align*}
\gs(0) &= \mathfrak{a} \oplus
\sideset{\ }{^{\oplus}}\sum_{1 \le k < l \le r} \gs_{(\alpha_l - \alpha_k)/2}, \quad
\gs(1/2) = \sideset{\ }{^{\oplus}}\sum_{1 \le k \le r} \gs_{\alpha_k /2}, \\
\gs(1) &= \sideset{\ }{^{\oplus}}\sum_{1 \le k \le r} \gs_{\alpha_k}
\oplus
\sideset{\ }{^{\oplus}}\sum_{1 \le k < l \le r} \gs_{(\alpha_l + \alpha_k)/2}.
\end{align*}
If $\{A_1, \dots, A_r\}$ is the basis of $\mathfrak{a}$ dual to $\{\alpha_1, \dots, \alpha_r\}$,
then $\gs_{\alpha_k} = \mathbb{R} jA_k$.
Thus $\gs_{\alpha_k}\,\,(k=1, \dots, r)$ is always one dimensional,
whereas other root spaces $\gs_{\alpha_k/2}$ and $\gs_{(\alpha_l \pm \alpha_k)/2}$ may be $\{0\}$.
Since $\{\alpha_1, \dots, \alpha_r\}$ is a basis of $\mathfrak{a}^*$,
the linear form $\gamma \in \mathfrak{a}^*$ is written as
$\gamma = \sum_{k=1}^r \gamma_k \alpha_k$
with unique $\gamma_1, \dots, \gamma_r \in \mathbb{R}$.
Since $j A_k \in \gs_{\alpha_k}$,
we have
\begin{align*}
\gamma_k = \gamma(A_k) = -4 \beta (jA_k) = -4 \beta([A_k, jA_k]) = 4 \beta([jA_k, A_k])
\end{align*}
and the last term equals $4 g (A_k \cdot z_0, A_k \cdot z_0)$.
Thus we get $\gamma_k >0$.
For $\epsilon = (\epsilon_1, \dots, \epsilon_r) \in \{0,1\}^r$,
put
$q_k(\epsilon) := \sum_{l>k} \epsilon_l \dim \gs_{(\alpha_l- \alpha_k)/2}
\,\,\,(k=1, \dots, r)$.
Define
$$
\mathfrak{X}(\epsilon) := \set{(\sigma_1, \dots, \sigma_r) \in \mathbb{C}^r}
{\begin{aligned}
\sigma_k &> q_k(\epsilon) /2 \quad (\epsilon_k = 1)\\
\sigma_k &= q_k(\epsilon) /2 \quad (\epsilon_k = 0)
\end{aligned}},
$$
and $\mathfrak{X} := \bigsqcup_{\epsilon \in \{0,1\}^r} \mathfrak{X}(\epsilon)$.
By \cite[Theorem 4.8]{I99},
$\tilde{\Psi}$ is a reproducing kernel if and only if
$\underline{\gamma} := (\gamma_1, \dots, \gamma_r)$
belongs to $\mathfrak{X}$.
We denote by $W(g)$ the set of $\lambda >0$ for which $\lambda g$ is projectively induced.
Since
the metric $\lambda g$ corresponds to the parameter $\lambda \underline{\gamma}$,
we see that $\lambda g$ is projectively induced if and only if $\lambda \underline{\gamma} \in \mathfrak{X}$.
Namely we obtain
$$
W(g) = \set{\lambda >0}{\lambda \underline{\gamma} \in \mathfrak{X}},
$$
and the right-hand side is considered in \cite{I10}.
Put $q_k = \sum_{l>k} \dim \gs_{(\alpha_l - \alpha_k)/2}$ for $k=1, \dots, r$.
Then \cite[Theorem 15]{I10} tells us that
$$
W(g) \cup \{0\} \subset \set{\frac{q_k}{2\gamma_k}}{k=1, \dots, r} \cup (c_0, +\infty),
$$
where $c_0 := \max \set{\frac{q_k}{2\gamma_k}}{k=1, \dots, r}$.
Now assume that $\lambda g$ is projectively induced for all $\lambda >0$.
Then we have $c_0 = 0$, so that $\dim \gs_{(\alpha_l-\alpha_k)/2} = 0$ for all $1 \le k < l \le r$.
In this case, we see that $\gs$ is a direct sum of ideals
$\gs_k := j \gs_{\alpha_k} \oplus \gs_{\alpha_k/2} \oplus \gs_{\alpha_k}
\,\,\,(k=1, \dots, r),$
which correspond to the hyperbolic spaces $\mathbb{C} H^{n_k}$ with $n_k = 1 + (\dim_{\alpha_k/2})/2$
(\cite[pp. 52--53]{PS69}).
Therefore the Lie algebra $\gs$ corresponds to the direct product
$\mathbb{C} H^{n_1} \times \cdots \times \mathbb{C} H^{n_r}$,
which is biholomorphic to $\Omega$ because the homogeneous domain
$\Omega$ also corresponds to $\gs$.
Hence (\ref{eqn:direct_prod}) holds and Theorem 4 is verified.
\hspace*{\fill}$\Box$
\vskip 0.3cm
{\small
|
{
"timestamp": "2010-09-22T02:01:39",
"yymm": "1009",
"arxiv_id": "1009.4045",
"language": "en",
"url": "https://arxiv.org/abs/1009.4045"
}
|
\section{Introduction}
\label{sec:introduction}
The theory of inflation represents a cornerstone of the standard model
of modern cosmology~\cite{Guth:1980zm, Linde:1981mu,
Albrecht:1982wi,Linde:1983gd} (for a review, see \emph{e.g.~}
Refs.~\cite{Linde:2007fr, Mukhanov:1990me, Martin:2003bt,
Martin:2004um, Martin:2007bw}). By definition, it is a phase of
accelerated expansion which is supposed to take place in the very
early universe, somewhere between the electroweak to the Grand Unified
Theory energy scales, \emph{i.e.~} between $\sim 10^3\, \mbox{GeV}$ and $\sim
10^{15}\, \mbox{GeV}$~\cite{Guo:2010mm}. Inflation allows us to understand
several puzzles which plagued the pre-inflationary standard model and
that could not be understood otherwise. Without inflation, the
standard model of cosmology would remain incomplete and highly
unsatisfactory. The most spectacular achievement of inflation is that,
combined with quantum mechanics, it provides a convincing mechanism
for the origin of the cosmological fluctuations (the seeds of galaxies
and of Cosmic Microwave Background - CMB -
anisotropies)~\cite{Mukhanov:1981xt,Hawking:1982cz,Starobinsky:1982ee,
Guth:1982ec,Bardeen:1983qw} and it predicts that their spectrum
should be almost scale invariant (\emph{i.e.~} equal power on all spatial
scales)~\cite{Stewart:1993bc,Mukhanov:1990me,Liddle:1994dx} which is
fully consistent with the observations~\cite{Martin:2006rs}. This part
of the scenario is particularly remarkable since it combines general
relativity and quantum mechanics.
\par
However, the physical nature of the inflaton (the field driving
inflation) and its relation with the standard model of particle
physics and its extensions remain elusive. Moreover the shape of its
potential is not known except, of course, that it must be sufficiently
flat. This is not so surprising since, as mentioned above, the
inflationary mechanism is supposed to take place at energy scales larger
than typically $\sim 1 \mbox{TeV}$, in a regime where particle physics is not
known and has not been tested at accelerators. Another crucial aspect
of the inflationary scenario is how it ends and how it is connected to
the subsequent hot big bang phase. It is believed that, after the
slow-roll period, the field reaches the bottom of its potential,
oscillates and decays into
radiation~\cite{Turner:1983he,Kofman:1997yn,Bassett:2005xm,
Mazumdar:2010sa}. In this way, inflation is smoothly connected to
the radiation-dominated epoch. However, the energy density at which
the radiation-dominated era starts is not accurately known, although some new
constraints on the reheating have recently been obtained in
Refs.~\cite{Martin:2010kz,Nakayama:2008wy,Kuroyanagi:2009br}.
\par
Despite the fact that it has become a cornerstone of modern cosmology,
inflation is not as observationally constrained as the other
components of the standard model. To improve on this situation,
full numerical approaches can be put in place in order to use, in an
optimal way, the astrophysical data now at our disposal
~\cite{Ringeval:2007am, Salopek:1988qh, Grivell:1999wc, Leach:2000yw,
Adams:2001vc, Makarov:2005uh, Bird:2009pq}. This should allow
investigations on the ``fine structure'' of the inflationary
scenario. This program is particularly timely since new high-accuracy
astrophysical observations, such as the European Space Agency Planck
data~\cite{Lamarre:2003zh}, among others, will be released soon. They
will provide an unprecedented window of opportunity to learn about
inflation.
\par
In this article, we are concerned with the question of how to evaluate
the performance of a given inflationary model to explain the data as
compared with others. This problem can be dealt within Bayesian
inference~\cite{Trotta:2008qt} (see \emph{e.g.~} Ref.~\cite{Ballesteros:2007te}
for an application to inflationary model comparison). In fact,
Bayesian statistics can be used at two levels. The first level is to
determine which model parameter values are favoured by the data within
a given inflationary model, and this for all models. To this end, one
needs to compute the model's predictions for the relevant observables,
such as the CMB, the galaxy power spectra, etc., and then use the
experimental data to extract the posterior probability distributions
of the model parameters given the data and the theoretical priors. The
second level is to use Bayesian inference for model comparison. At
this level, one has to calculate, for each model, the global
likelihood (also known as the evidence, or model likelihood) which is
obtained by integrating the usual likelihood over all of the model
parameters' values, weighted by their prior probability
distribution. The resulting quantity can be used to compute the
posterior probability of the model, given the available data, thus
updating our prior belief in each of the inflationary models in light
of the observations. The Bayesian approach to model comparison has the
advantage of automatically incorporating a quantitative notion of
``Occam's razor'', \emph{i.e.~} more complex inflationary models are assigned a
larger posterior probability only if their complexity is effectively
required to explain the data.
\par
On the practical side, these two levels in Bayesian inference can be
implemented by adopting appropriate numerical algorithms to integrate
the power spectrum for a given inflationary model. This has been
routinely available for several years now and, in this paper, we use
the public code \texttt{FieldInf}~\cite{Ringeval:2005yn,
Martin:2006rs, Ringeval:2006}. This inflationary code is then
coupled with a CMB perturbation code, such as
\texttt{CAMB}~\cite{Lewis:1999bs}, and then linked with an appropriate
algorithm capable of delivering both the posterior distributions for
each model's parameters as well as the Bayesian evidence of each
model. The evidence is computed using the
publicly available \texttt{MultiNest}
code~\citep{Feroz:2007kg,Feroz:2008xx,Trotta:2008bp}, which implements
the nested sampling algorithm, employed as an add-on sampler to
\texttt{CosmoMC}~\citep{Lewis:2002ah}.
\par
On the theoretical side, one has to choose classes of scenarios that
are representative of the inflationary landscape and that one wishes
to analyze. In this article, we focus on large and small field models
for reasons specified in the following. The reheating stage is
described via the reheating parameter as introduced in
Refs.~\cite{Martin:2006rs, Martin:2010kz}. Moreover, since the
choice of priors is always relevant in problems of model
comparison, we have paid particular attention to their physical motivation and we
carefully investigate this question both for the parameters describing
the inflationary potential and for the reheating.
\par
This article is organized as follows. In the next section,
Sec.~\ref{sec:infpert}, we present the models studied, paying special
attention to the reheating part and the so-called reheating
parameter. In Sec.~\ref{sec:bayesianevidence}, we recall the
definition of the Bayesian evidence, describing in detail how the
priors on the free parameters characterizing each scenario are
chosen. We also explain how its calculation is implemented
numerically. Finally, in Sec.~\ref{sec:discussion}, we present our
results and discuss their physical implications. Readers already
familiar with the inflationary models, techniques and methods can directly jump to
Sec.~\ref{sec:priors}. Perhaps the most important outcome of our
article is that it sketches a general method which allows us to
quantify and determine the ``best'' model of inflation (within the
list of models considered here).
\section{Inflationary Cosmological Perturbations}
\label{sec:infpert}
In this section, after having briefly recalled how the theory of
cosmological perturbations of quantum-mechanical origin allows us to
derive the inflationary predictions, we present the scenarios studied
here, discuss our choice of parametrization and motivate it based on
physical considerations.
\subsection{Choosing the Inflationary Potential}
\label{subsec:infpot}
In order to compare inflation with various astrophysical observables,
one must first determine the power spectrum of the density
perturbations defined by the following expression:
\begin{equation}
{\cal P}_{\zeta}(k)\equiv \frac{k^3}{2\pi ^2}
\left\vert \zeta_{\boldsymbol k}\right\vert ^2
\label{Pzeta}
\end{equation}
where $\zeta_{\boldsymbol k}$ is the comoving curvature perturbation in
Fourier space and is a conserved quantity on super-Hubble length
scales~\cite{Mukhanov:1990me,Schwarz:2001vv,
Martin:2003bt,Martin:2004um,Martin:2007bw}.
This power spectrum depends on the shape of the inflaton's
potential, and thus, on its free parameters which have to be
specified. It is common to describe the landscape of possible single
field inflationary models with three different archetypal classes:
large field models, small field models and hybrid inflation. This
simple approach is based on the following considerations. Any inflaton
potential $V(\phi)$ can always be Taylor expanded as
\begin{equation}
V\left(\phi\right)=V_0\pm \alpha \left(\frac{\phi}{M_\mathrm{Pl}}\right)^2+\cdots .
\end{equation}
According to the value of the coefficients of the expansion, one
obtains different classes of models. If the constant term $V_0$
vanishes, then one obtains a large field
model~\cite{Linde:1983gd,Linde:1984st}. Instead of restricting
ourselves to a massive scenario, a simple generalization is to
consider an arbitrary power index $p$, not necessarily fixed to
$p=2$~\cite{Silverstein:2008sg}. If the constant term is not zero,
then one obtains a small field
model~\cite{Linde:1981mu,Albrecht:1982wi} (with a negative second
term) or an effective hybrid model~\cite{Linde:1993cn,Copeland:1994vg}
(with a positive second term). Again, instead of considering only a
quadratic term, it is more generic to let the power index
unspecified. This leads to the three classes mentioned before.
\par
An important question is whether the other terms of the Taylor
expansion are under control. This has led to a debate on the question
of whether vacuum expectation values of $\phi$ larger than the Planck
mass are meaningful or
not~\cite{Lyth:1998xn,Linde:2005ht,Linde:2007fr}. In the simple
approach used here, we do not take part in this discussion and
consider sub- as well as super-Planckian vacuum expectation
values. Moreover, hybrid inflation is an intrinsic multiple field
scenario (with the above potential, inflation could not actually stop)
which cannot always be described by a single field
approach~\cite{Clesse:2009ur, Clesse:2010iz}. Indeed, in a multiple
field model, the presence of entropy perturbations can cause the
evolution of $\zeta_{\boldsymbol k}$ on large scales and this effect can modify
the power spectrum during the pre-heating stage. Since this type of
effect is model-dependent, it must be studied for each scenario and,
for this reason, it is wiser, in a first step, to focus on simpler
models. For this reason, we will consider in the following only the
large and small field scenarios having, respectively, the following
potentials:
\begin{equation} \label{eq:large_field}
V(\phi)=M^4\left(\frac{\phi}{M_\mathrm{Pl}}\right)^p \text{ (large field),}
\end{equation}
and
\begin{equation}\label{eq:small_field}
V(\phi)=M^4\left[1-\left(\frac{\phi}{\mu}\right)^p\right]
\text{ (small field)}.
\end{equation}
Of course, this has to be considered as a first step towards a more
complete scan of the inflationary landscape. The large field model is
characterized by two parameters, the energy scale $M$ and the power
index $p$. The small field potential is characterized by three
parameters, $M$, $\mu $ and $p$. We come back to the issue of the
prior distributions to assign to each parameter in
section~\ref{sec:priors}.
\subsection{Describing the Reheating}
\label{subsec:reheating}
In order to compare an inflationary model with observations, we also
need to take into account the reheating stage which takes place after
the end of inflation and before the onset of the radiation-dominated
era. This is compulsory since one needs to know the actual value of a
physical wavenumbers during inflation from its observed value
today. For instance, the amplitude of the power spectrum $P_*$ is
measured at a given wavenumber, typically $k_*/a_0=0.05 \mbox{Mpc}^{-1}$,
where $a_0$ denotes the present-day scale factor. During inflation,
the corresponding physical wavenumber is stretched back to
\begin{equation}
\dfrac{k_*}{a} = \dfrac{k_*}{a_0} (1+z_\mathrm{end}){\rm e}^{N_{\rm end}-N},
\end{equation}
where $z_\mathrm{end}$ is the redshift at which inflation ended, $N_{\rm
end}$ the total number of e-folds during inflation and $N\equiv \ln
a$ the number of e-folds at the time considered during inflation. The
quantity $k_*/a$ is uncertain precisely due to the existence of the
reheating. Assuming instantaneous transitions between inflation,
reheating, radiation and matter era, one can simplify
\begin{equation}
\label{eq:redend}
1+z_\mathrm{end} = (1+z_\mathrm{eq}) \left(\dfrac{\rho_\ureh}{\rho_\ueq} \right)^{1/4}
\dfrac{a_\mathrm{reh}}{a_\mathrm{end}}\,,
\end{equation}
where ``reh'' and ``eq'' respectively stands for the end of reheating
and the equality between the energy density of radiation and
matter. The so-called reheating parameter
$R_\urad$~\cite{Martin:2006rs,Martin:2010kz} describes the evolution of
the Universe during the reheating stage and is defined by
\begin{equation}
\label{eq:Rraddef}
R_\urad \equiv \dfrac{a_\mathrm{end}}{a_\mathrm{reh}} \left( \dfrac{\rho_\uend}{\rho_\ureh}
\right)^{1/4},
\end{equation}
such that Eq.~(\ref{eq:redend}) becomes
\begin{equation}
1+z_\mathrm{end} = \dfrac{1}{R_\urad} \left( \dfrac{\rho_\uend}{\rho_{\radnow}} \right)^{1/4},
\end{equation}
where $\rho_{\radnow}$ is the energy density of radiation today\footnote{The
density parameter of radiation today is $\Omega_{\ur_0} \simeq 2.471
\times 10^{-5} h^2$.}. As a result, $R_\urad$ encodes all of our
ignorance on how the reheating influences the observable inflationary
power spectra. In fact, it is for inflation what the optical depth
$\tau$ is for CMB observations. The latter encodes how much
reionisation of the universe affects the measured CMB anisotropies
(independently of the details of the reionisation history, at least at
first order) while $R_\urad$ plays a similar role for the reheating. As
it should be clear from Eq.~(\ref{eq:Rraddef}), $R_\urad$ quantifies the
deviation from a reheating era which would be radiation-like.
\par
In fact, as discussed in Ref.~\cite{Martin:2010kz},
Eq.~(\ref{eq:Rraddef}) can be recast into various equivalent forms. In
terms of the number of e-folds during reheating $\Delta
N=N_\mathrm{reh} - N_\uend =\ln(a_\mathrm{reh}/a_\mathrm{end})$, one has
\begin{equation}
\label{eq:Rrad}
\ln R_\urad = \frac{\Delta N}{4}\left(-1+3\overline{w}_{\mathrm{reh}}\right),
\end{equation}
where $\overline{w}_{\mathrm{reh}}$ stands for the mean equation of state
parameter
\begin{equation}
\overline{w}_{\mathrm{reh}}\equiv \frac{1}{\Delta N}\int _{N_\uend}^{N_{\mathrm{reh}}}
\dfrac{P(n)}{\rho(n)}\, \dd n.
\end{equation}
Here $P(n)$ and $\rho(n)$ are the instantaneous total pressure and
energy density of the universe during reheating. This description is
completely general since no assumption about the physical properties
of the effective fluid dominating the matter content of the universe
during reheating has been made. One can also express $\Delta N$ in
terms of $\overline{w}_\ureh$ such that
\begin{equation}
\label{eq:Rradw}
\ln R_\urad = \frac{1-3\overline{w}_{\mathrm{reh}}}{12(1+\overline{w}_{\mathrm{reh}})}\ln \left(
\frac{\rho_\ureh}{\rho_\uend}\right).
\end{equation}
As expected, one can verify explicitly that $R_\urad=1$ if
$\overline{w}_{\mathrm{reh}}=1/3$.
\section{Bayesian Model Comparison}
\label{sec:bayesianevidence}
In this section, we briefly review Bayesian model comparison, which we
adopt to compare the performance of our inflationary models (for
further details, see \emph{e.g.~} \cite{Trotta:2008qt}). As a preliminary
remark, we notice that if one seeks to determine the most economical
description of the inflationary potential in light of the available
data, Bayesian model comparison is well suited, in that
classical statistics only allows to reject hypotheses, not to confirm
them (see also Ref.~\cite{Liddle:2007fy} for alternative model selection
criteria). Therefore, while some simpler models might become ruled
out in a classical sense (\emph{i.e.~} their parameter space can become
completely constrained by the data, until no viable region remains),
classical statistics does not allow one to rank the remaining models
in any way. Bayesian model comparison, with its natural inclusion of
the Occam's razor effect, is therefore the only available tool to
quantify in a self-consistent way our preference for a specific model.
\subsection{The Bayesian evidence}
\label{subsec:evid}
Bayesian model comparison aims at computing the posterior probability
of a model in view of the available data. The fundamental idea behind
the procedure is that ``economic'' models that fit well the data are
rewarded for their predictivity, while models with a large number of
free parameters that turn out not to be required by the data are
penalized for the wasted parameter space. Therefore, in a Bayesian
sense, the ``best'' model is the one that achieves the best compromise
between quality of fit and simplicity. One of the attractive features
of Bayesian model comparison is that it automatically embodies a
quantitative version of Occam's razor, i.e., the principle of
simplicity.
\par
Here and in the following, by ``model'' we denote a choice of
inflationary potential, together with a specification of its free
parameters, $\Theta_j$, {\em and} of their prior probability
distribution, $p(\Theta_j|\mathcal{M}_j)$. The specification of the prior is
fundamental for model comparison, as the prior shape and range
influence the Occam's razor effect. From Bayes' theorem, the
posterior probability of model $\mathcal{M}_j$ given the data $d$,
$p(\mathcal{M}_j|d)$, is related to the Bayesian evidence (or model
likelihood) $p(d|\mathcal{M}_j)$ by
\begin{eqnarray} \label{eq:postM}
p(\mathcal{M}_j|d)&=&\frac{p(d|\mathcal{M}_j)p(\mathcal{M}_j)}{p(d)}\, ,
\end{eqnarray}
where $p(\mathcal{M}_j)$ is the prior belief in model $\mathcal{M}_j$. In
Eq.~\eqref{eq:postM}, $p(d)=\sum_i p(d|\mathcal{M}_i)p(\mathcal{M}_i)$ is a
normalization constant (where the sum runs over all available known
models $\mathcal{M}_i$, $i=1,\dots, N$) and
\begin{equation} \label{eq:Bayesian_evidence}
p(d|\mathcal{M}_j)=\int \mathrm{d}
\Theta_j\, p(d|\Theta_j, \mathcal{M}_j) p(\Theta_j | \mathcal{M}_j)
\end{equation}
is the Bayesian evidence, where $p(d|\Theta_j, \mathcal{M}_j)$ is the
likelihood. The Bayesian evidence is thus the average likelihood under
the prior, and is the central quantity for Bayesian model comparison.
\par
Given two competing models, $\mathcal{M}_0$ and $\mathcal{M}_1$, the posterior odds
among them are given by \begin{equation} \frac{p(\mathcal{M}_0 | d)}{p(\mathcal{M}_1 | d)} =
B_{01} \frac{p(\mathcal{M}_0)}{p(\mathcal{M}_1)} , \end{equation} where we have introduced the
factor $B_{01}$ as defined as the ratio of the models' evidences
\begin{eqnarray}
B_{01}&\equiv&\frac{p(d|\mathcal{M}_0)}{p(d|\mathcal{M}_1)}\, .
\end{eqnarray}
The Bayes factor thus updates our relative state of belief in two
models from the prior odds to the posterior odds. Large values of
$B_{01}$ denote a preference for $\mathcal{M}_0$, and small values of
$B_{01}$ denote a preference for $\mathcal{M}_1$. The ``Jeffreys' scale''
(Table~\ref{Tab:Jeff}) gives an empirical prescription for translating
the values of $B_{01}$ into strengths of belief.
\begin{table}[t]
{\begin{tabular}{l l l} \hline
$|\ln B_{01}|$ & Odds & Strength of evidence \\\hline
$<1.0$ & $\lesssim 3:1$ & Inconclusive \\
$1.0$ & $\sim 3:1$ & Weak evidence \\
$2.5$ & $\sim 12:1$ & Moderate evidence \\
$5.0$ & $\sim 150:1$ & Strong evidence \\
\hline
\end{tabular}}
\caption{Empirical scale for evaluating the strength of evidence when
comparing two models, $\mathcal{M}_0$ versus $\mathcal{M}_1$ (so-called
``Jeffreys' scale'', here slightly modified following the
prescriptions given in \cite{Gordon:2007xm,Trotta:2008qt}). The
right-most column gives our convention for denoting the different
levels of evidence above these thresholds.\label{Tab:Jeff} }
\end{table}
Given two or more models, specified in terms of their parametrization
{\em and} priors on the parameters, it is straightforward (although
sometimes computationally challenging) to compute the Bayes
factor. Depending on the problem at hand,
semi-analytical~\cite{Trotta:2005ar,Heavens:2007ka} and
numerical~\cite{Mukherjee:2005wg,Feroz:2007kg,Feroz:2008xx,
Kilbinger:2009by,Serra:2007id} techniques are available. In the
usual case where the prior over models is taken to be non-committal
(\emph{i.e.~} $p(\mathcal{M}_j) = 1/N$), the model with the largest Bayes factor
ought to be preferred. Thus the computation of $B_{01}$ allows to
select one (or a few) promising model(s) from a set of known
models. This framework has recently been extended to evaluate the
probability that the set of known models is incomplete, see
Ref.~\cite{March:2010ex}.
\par
Finally, we can also summarize our findings in terms of posterior
probability for the entire class of models being considered here,
large field or small field. From Bayes' theorem, the posterior
probability for \emph{e.g.~} the small field class (SF) is given by
\begin{equation} \label{eq:Post}
p(\text{SF} | d) = \sum_{i=1}^{n_\text{SF}} \frac{p(d |
\text{SF}_i) p(\text{SF}_i)}{p(d)},
\end{equation}
where
\begin{equation}
p(d) =
\sum_{i=1}^{n_\text{SF}} p(d | \text{SF}_i) p(\text{SF}_i) +
\sum_{j=1}^{n_\text{LF}} p(d | \text{LF}_j) p(\text{LF}_j)
\end{equation}
and $n_\text{SF} = 3$ is the number of small field models considered
in the class, while $n_\text{LF} = 6$ is the number of large field
models, as explained in the next section. Regarding the choice of
priors for the models, in view of comparing the viability of large
field and small field inflation, it is natural to divide equally the
prior probability between the two classes, and then further subdivide
it equally among the models in each class, so that $p(\text{SF}_j) =
1/(2 n_\text{SF})$ and $p(\text{LF}_j) = 1/(2 n_\text{LF})$. For
reasons that shall become clear below, it will be convenient to
consider the Bayes factor between the various models and the large
field model with $p=2$ (LF$_2$), and it is therefore useful to divide
both the numerator and the denominator of Eq.~\eqref{eq:Post} by the
evidence of LF$_2$, obtaining:
\begin{align} \label{eq:Post_2} p(\text{SF} | d) & =
\frac{\sum_i^{n_\text{SF}} B_{i*} p(\text{SF}_i)}{\sum_i^{n_\text{SF}}
B_{i*} p(\text{SF}_i)
+ \sum_j^{n_\text{LF}} B_{j*} p(\text{LF}_j) } \\
& =\frac{\langle B_{i*} \rangle_\text{SF}}{\langle B_{i*} \rangle_\text{SF}
+ \langle B_{i*} \rangle_\text{LF}} \\
& =\left(1+ \frac{\langle B_{i*} \rangle_\text{LF}}{\langle B_{i*}
\rangle_\text{SF}}\right)^{-1},
\end{align}
where we have defined
\begin{align}
\langle B_{i*} \rangle_\text{SF} &
\equiv \frac{1}{n_\text{SF}}\sum_{i=1}^{n_\text{SF}} B_{i*}, \label{eq:BavSF}\\
\langle B_{i*} \rangle_\text{LF} &
\equiv \frac{1}{n_\text{LF}}\sum_{i=1}^{n_\text{LF}} B_{i*} \label{eq:BavLF}
\end{align}
and in the above $B_{i*}$ denotes the Bayes factor between model $i$
and the LF$_2$ model.
\par
\par
It is also instructive to consider the Bayesian complexity associated
with each model, defined as~\cite{Kullback:1951}
\begin{equation} \label{eq:Cb}
\mathcal{C}_\ub = -2 \left[\mathcal{D}_\text{KL} \left(P,\pi\right)- \widehat{\mathcal{D}_\text{KL}} \right],
\end{equation}
where, here, $\pi$ denotes the prior distribution and
$\mathcal{D}_\text{KL}\left(P,\pi\right)$ is the Kullback-Leiber divergence between the
posterior $P$ and the prior, $\pi$, namely
\begin{equation}
\mathcal{D}_\text{KL} \left(P,\pi\right) \equiv \int p\left(\theta\vert d\right)
\log \frac{p\left(\theta\vert d\right)}{\pi(\theta)}{\rm d}\theta.
\end{equation}
In Eq.~\eqref{eq:Cb}, $\widehat{\mathcal{D}_\text{KL}}$ denotes a point estimate for
the KL divergence. It has been shown
in~\cite{Kunz:2006mc,Trotta:2008qt} that the Bayesian complexity
measures the number of model parameters that the data can
constrain. Evaluated together with the evidence, the complexity helps
to assess whether the parametrization of a model is excessive for the
constraining power of the available data (for details,
see~\cite{Kunz:2006mc}). The complexity can be expressed
as
\begin{equation}
\label{eq:complexity_chisq}
\mathcal{D}_\text{KL} = \langle \chi^2 \rangle -
\widehat{\chi}^2,
\end{equation}
where $\chi^2 \equiv -2 \ln {\mathcal L}$ and the
expectation value is taken with respect to the posterior. The second
term, $\widehat{\chi}^2$ is a plug-in estimate that can be taken to be
for example the best-fit $\chi^2$ value or the value of the $\chi^2$
at the posterior mean. Here we adopt the best-fit value,
following~\cite{Kunz:2006mc}.
\par
As mentioned above, the evidence is computed using the publicly
available {\tt MultiNest}
code~\citep{Feroz:2007kg,Feroz:2008xx,Trotta:2008bp}, which implements
the nested sampling algorithm. The gist of nested sampling is that
the multi-dimensional evidence integral of
Eq.~\eqref{eq:Bayesian_evidence} is recast into a one-dimensional
integral. This is accomplished by defining the prior volume $x$ as
${\rm d} x \equiv p(\Theta_j | \mathcal{M}_j){\rm d}
\Theta_j $ so that
\begin{equation} \label{eq:def_prior_volume}
x(\lambda) = \int_{{\mathcal L}(\Theta_j)>\lambda} p(\Theta_j | \mathcal{M}_j) {\rm d}
\Theta_j,
\end{equation}
where the integral is over the parameter space enclosed by the
iso-likelihood contour ${\mathcal L}(\Theta_j) = \lambda$. So $x(\lambda)$
gives the volume of parameter space above a certain level $\lambda$
of the likelihood (for a specific model $\mathcal{M}_j$). Then the Bayesian evidence,
Eq.~\eqref{eq:Bayesian_evidence}, can be written as
\begin{equation} \label{eq:nested_integral}
p(d | \mathcal{M}_j) = \int_0^1 {\mathcal L}(x) {\rm d} x,
\end{equation}
where ${\mathcal L}(x)$ is the inverse of
Eq.~\eqref{eq:def_prior_volume}. Samples from ${\mathcal L}(x)$ can be
obtained by drawing uniformly samples from the likelihood volume
within the iso-contour surface defined by $\lambda$. The standard
deviation on the value of the log evidence can be estimated as
$(H/n_\text{live})^{1/2}$, where $H$ is the negative relative entropy
and $n_\text{live}$ is the number of live points adopted, which in
our case is $n_\text{live} = 1000$ (see~Ref.~\cite{Feroz:2007kg} for
details). We have checked that our evidence values are robust (within
error bars) if one increases $n_\text{live}$ to $5000$. The posterior
distributions have also been cross-checked with standard
Metropolis--Hastings Markov--Chain--Monte--Carlo (MCMC).
\subsection{Choice of Priors}
\label{sec:priors}
\begin{table*}
\begin{tabular*}{0.85\textwidth}{@{\extracolsep{\fill}}| l | c c c | c c c c c c | } \hline
Parameter & \multicolumn{3}{c|}{Small field models, Eq.~\eqref{eq:small_field}} & \multicolumn{6}{c|}{Large field models, Eq.~\eqref{eq:large_field} } \\
& SFI$_s$ & SFI$_l$ & SFI$_f$ & LFI$_p$ & LFI$_{2/3}$ & LFI$_1$ & LFI$_2$ & LFI$_3$ & LFI$_4$ \\ \hline
Normalization, $\ln P_*$ & \multicolumn{3}{c|}{$[2.7\times 10^{-10}, 4.0\times 10^{-10}]$} & \multicolumn{6}{c|}{$[2.7\times 10^{-10}, 4.0\times 10^{-10}]$} \\
Exponent, $p$ & \multicolumn{3}{c|}{$[2.4, 10]$} & $[0.2, 5]$ & $2/3$ & $1$& $2$ & $3$ & $4$\\
Vacuum expectation, $\log(\mu/M_\mathrm{Pl})$ & $[-1,0]$ & $[0,2]$ & $[-1,2]$ & \multicolumn{6}{c|}{Not applicable}\\
Reheating, $\ln R$ & \multicolumn{3}{c|}{$[-46, 15]$} & \multicolumn{6}{c|}{$[-46, 15]$} \\ \hline
$n$ & 4 & 4 & 4 & 3 & 2 & 2 & 2 & 2 & 2 \\
\hline
\end{tabular*}
\caption{Inflationary models considered in this analysis and priors on their parameters. All priors are taken to
be uniform (\emph{i.e.~} flat) in the variable and range specified, see the text
for a detailed justification. In the last row, $n$ is the number of free
parameters related to the inflationary sector. }
\label{tab:models}
\end{table*}
Since our aim is to evaluate the evidence of large and small field
models, it is absolutely crucial to choose well-motivated priors for
the parameters describing the potential. In order to see why it is so,
it is instructive to consider the evidence of a simple, one-parameter
toy case, where there is only one single parameter $\theta$, whose
prior density under model $\mathcal{M}$ is given by $p(\theta | \mathcal{M})$. We
shall further assume that the likelihood is much more sharply peaked
than the prior (\emph{i.e.~} the quantity $\theta$ has been well measured), so
that $p(\theta) \approx \text{const.}$ in the range $\delta\theta$
where the likelihood ${\mathcal L}(\theta)$ is appreciably different from
zero. Then the evidence of model $\mathcal{M}$,
Eq.~\eqref{eq:Bayesian_evidence}, is approximately equal to
\begin{equation}
\label{eq:penalty}
p(d | \mathcal{M} ) \approx
{\mathcal L}(\theta_\text{ML}) \, \delta\theta \, p(\theta_\text{ML} | \mathcal{M}),
\end{equation}
where $\theta_\text{ML}$ is the value that maximizes the likelihood
function. Since the prior must be normalized, $p(\theta_\text{ML} |
\mathcal{M}) \approx 1/\Sigma$, where $\Sigma$ is the characteristic width of
the prior. Therefore one finds that $p(d | \mathcal{M} ) \propto
\Sigma^{-1}$, \emph{i.e.~} the evidence scales inversely proportionally to the
width of the prior. The term $\delta\theta/\Sigma$ is the so-called
``Occam's factor'', which penalizes models with a large ``wasted''
parameter space under the prior, \emph{i.e.~} models for which the
characteristic width of the likelihood is much smaller than that of
the prior, $\delta\theta/\Sigma \ll 1$. Hence the \emph{a priori}
plausible range of parameter values determines the strength of the
Occam's penalty term, and for this reason it has to be carefully
chosen on the basis of physical considerations\footnote{Notice that
parameters which are unconstrained by the data are not penalized by
the Occam's factor, \emph{i.e.~} if the likelihood's width is similar to the
prior range, then $\delta\theta/\Sigma \sim 1$ and the Occam's
factor effect vanishes.}.
\par
Going back to the potentials \eqref{eq:large_field} and
\eqref{eq:small_field}, we notice that the parameter $M$, common to
both classes of models, is {\em a priori} unknown, and is
observationally determined by the overall normalization of the power
spectrum, $P_*$. Since the \emph{scale} of $M$ is unknown, it is
appropriate to adopt a prior flat on $\ln M$, to reflect the fact that
we are giving equal a priori probability to all orders of magnitude
within some suitably chosen lower and upper limits. A flat prior on
$\ln M$ is equivalent to a flat prior on $\ln P_*$, and therefore in
our numerical sampling we swap $\ln M$ for $\ln P_*$ as a fundamental
parameter. Since the overall power spectrum normalization is common to
all models, the precise range of values under the prior for $\ln P_*$
becomes irrelevant (as long as the range is sufficiently wide to
encompass the support of the likelihood), as all models share the same
Occam's razor penalty from this common parameter. In practice, we
chose $\ln P_* \in [2.7 \times 10^{-10}, 4.0 \times 10^{-10}]$, but
because of the above argument the Bayes factor between our models
would remain unchanged even if this range was arbitrarily enlarged.
\par
For large field models, we chose to adopt a flat prior in the range
$0.2<p<5$. The lower limit is arbitrarily chosen to encompass all
proposed large field potentials having a fractional
power~\cite{McAllister:2008hb,Kallosh:2010ug}. In principle, one could
imagine an arbitrarily small $p$ (which would suggest the use of a
Jeffreys' prior, instead) but, up to now, there is no theoretical
motivation to do so. On the other hand, there is no strong theoretical
reason not to consider a model with, say, $p=7$. However, we know that
the data already strongly disfavour models with $p>5$ (as a matter of
fact, even models with $p>3$ are disfavoured~\cite{Martin:2010kz}) and
therefore one expects that the evidence of models with $p>5$ (fixed)
would be strongly disfavoured. Furthermore, if one wanted to enlarge
the prior range to $p>5$ it would be easy to rescale the evidence to
account for the enlarged parameter space, since the likelihood is
close to 0 for $p>5$. This would lead to a larger Occam's penalty and
thus to a lower evidence, see Eq.~\eqref{eq:penalty}.
\par
For small field models, we have chosen a flat prior $p\in [2.4,10]$
as our representative class since $p=2$ is a very special
case. As discussed in Ref.~\cite{Martin:2006rs}, approaching the value
$p=2$ is numerically tricky and we have chosen the lower bound as the
closest, but different, possible value of $p>2$. Models with
$p<2$~\cite{Alabidi:2005qi, Alabidi:2008ej} might in principle be
included but would constitute another class of models since this would
require to cross the $p=2$ barrier. Moreover, models with negative
$p$ correspond to very different physical regimes. For instance, the
model with $p=-4$ is nothing but the Coulomb potential of brane
inflation and was analyzed in detail in~\cite{Lorenz:2007ze}. For the
reasons detailled in Sec.~\ref{sec:introduction}, and at this stage of
the analysis, we do not include those cases. The upper bound
for $p$ has been chosen typically an order of magnitude
higher. Theoretically, as already mentioned above, small field models
are archetypal of inflationary potentials which can be Taylor expanded
in the (small) field values, in units of a given vacuum expectation
value $\mu$. As a result, too large values of $p$ would appear quite
unnatural. Concerning $\mu$, its scale is a priori unknown and,
therefore, we have chosen a flat prior on $\log\left(\mu/M_\mathrm{Pl}\right)$
in the range $[-1,2]$. On one hand, if one has a theoretical prejudice
of viewing the small field models as representative of Taylor expanded
potential (as was done above), and in particular in the supersymmetric
framework, one would expect $\mu < M_\mathrm{Pl}$ to keep the supergravity
corrections under control. On the other hand, other theoretical
approaches do not forbid super-Planckian vacuum expectation
values~\cite{Brax:2005jv} since one can always consider that this
potential is obtained, not from a Taylor expansion but exactly from a
more fundamental theory. The corrections would therefore not be
controlled by the ratio of the vacuum expectation value to the Planck
mass but by the ratio of the energy density to the Planck
density. Hence our prior range is chosen in such a way as to extend
above the Planck mass. Concerning the boundary values, in the limit
$\mu/M_\mathrm{Pl}\gg 1$, one can show that the two first slow-roll parameters,
and hence all observable predictions, do not longer depend on both
$\mu$ and $p$. As a result, it is straightforward to show from
Eq.~(\ref{eq:penalty}) that the corresponding Bayes factors would be
unchanged for larger values of $\mu$. In the limit $\mu/M_\mathrm{Pl}\ll 1$,
the first slow-roll parameter becomes tiny and the second one becomes
$\mu$-independent such that, again, the observable predictions, and
thus the likelihood and the evidence, are no longer sensitive to
$\mu$.
\par
Finally, in addition to the two broad classes of large field and small
field models, we have introduced in our model space finer subdivisions
leading to more specific model classes. Motivated by the above prior
discussion, it is natural to further distinguish between small field
models allowing super-Planckian expectation values [\emph{i.e.~} with $\log
(\mu/M_\mathrm{Pl}) > 0$] from the ones that do not [$\log(\mu/M_\mathrm{Pl}) < 0$]. In the
large field class, we have also singled out some models having a
peculiar interest such as the genuine chaotic massive inflation model
($p=2$), monodromy inflation ($p=2/3$), linear inflation ($p=1$), and
the self-interacting potential ($p=3$ or $p=4$). Of course, one must
restrict oneself to the positive part of the potential when
necessary. Therefore we consider a total of 9 classes of models.
\par
Having parametrized the evolution of the universe during the
reheating in the previous section, one must now discuss the choice of
the prior on the reheating parameter. As shown in
Ref.~\cite{Martin:2006rs}, instead of working with $R_\urad$ introduced
in Eq.~\eqref{eq:Rraddef}, it is more convenient to work with the
rescaled reheating parameter $R$ defined by
\begin{equation}
\label{eq:Rdef}
R\equiv R_\urad\frac{\rho_\uend^{1/4}}{M_\mathrm{Pl}}\,.
\end{equation}
As we recap in the appendix~\ref{sec:R}, $R_\urad$ exhibits trivial
correlations with the normalisation of the power spectrum $P_*$ which
can be easily removed by considering $R$ instead. Notice that once the
inflationary model is specified, $\rho_\uend$ is known and
Eq.~(\ref{eq:Rdef}) is nothing but a rescaling. Clearly, the order of
magnitude of the different physical quantities appearing in
Eqs.~(\ref{eq:Rraddef}) and (\ref{eq:Rdef}) is unknown and this
suggests that we choose a flat prior on $\ln R$. The next step is to
determine the prior boundaries. In fact, using the expression of
$R_\urad$ given before, one also has
\begin{equation}
\ln R=\frac{1-3 \overline{w}_\ureh}{12(1+\overline{w}_\ureh)}\ln \left(
\frac{\rho_\ureh}{M_\mathrm{Pl}^4}\right)+\frac{1+3 \overline{w}_\ureh}
{6(1+ \overline{w}_\ureh)}\ln
\left(\frac{\rho_\uend}{M_\mathrm{Pl}^4}\right).
\end{equation}
Positivity energy conditions in General Relativity imposes that
$\overline{w}_\ureh$ cannot exceed unity and we want to separate inflation from
reheating such that $\overline{w}_\ureh$ cannot be less than $-1/3$. Moreover,
$\rho_\unuc<\rho_\ureh<\rho_\uend$, where $\rho_\unuc$ is the energy density at
Big-Bang Nucleosynthesis (BBN), which we take to be $\rho_\unuc^{1/4} =
10\, \mbox{MeV}$, this implies that $-46<\ln R<15 + (1/3)
\ln\left(\rho_\uend/M_\mathrm{Pl}^4\right)$. Since there is no preferred value for
$\ln R$, we initially take the maximal possible theoretically
allowed range $[-46,15]$. However, for each given model parameter
values, we then reject all $\ln R$ values not satisfying the
consistency bound $\ln R<15 + (1/3)
\ln\left(\rho_\uend/M_\mathrm{Pl}^4\right)$. Finally, notice that this description
of reheating via the $\ln R$ parameter and its prior range is common
to all models.
\par
To conclude the discussion on priors, we have chosen flat priors on
the standard cosmological parameters centered around their currently
measured values, \emph{i.e.~} for the density parameter of baryons $\Omega_\ub
h^2$, of dark matter $\Omega_\udm h^2$, the angular size of the sound
horizon at last scattering $\theta$ and the optical depth $\tau$. We
also marginalize over the amplitude of the unresolved SZ signal with a
flat prior in the range $A_\text{SZ} \in [0, 2]$. These prior choices
do not impact on our evidence result for the inflationary models as
all models share the same standard cosmological parameters and their
respective priors. We moreover assume throughout a flat universe as
predicted by cosmic inflation.
\par
The models we consider and the priors on the relevant inflationary
parameters are summarized in Table~\ref{tab:models}.
\section{Results and Discussion}
\label{sec:discussion}
\begin{figure*}
\begin{center}
\includegraphics[width=0.95\textwidth,clip=true]{gant}
\caption{Results for the Bayes factor between different inflationary
models considered in the analysis. The names of the models are
specified on the left of the figure. The Bayes factor are computed
taking massive large field model as the reference model, and the
results are given in the column on the right of the plot. The dotted
vertical lines indicate the thresholds of weak, moderate and strong
evidence, as per Table~\ref{Tab:Jeff}.}
\label{fig:evidence}
\end{center}
\end{figure*}
In this section, we present our model comparison results for the
classes of models described above. Concerning the data, we have used
the seven years Wilkinson Microwave Anisotropies Probe (WMAP7)
data~\cite{Komatsu:2010fb, Larson:2010gs, Jarosik:2010iu} complemented
with the Hubble Space Telescope (HST) constraints on the Hubble
constant today, $H_0 = 74.2 \pm 3.6$ km/s/Mpc~\cite{Riess:2009pu}. Our
findings are summarized in Fig.~\ref{fig:evidence}, where we show the
Bayes factors for each model, computed with respect to the large field
model with $p=2$.
\par
Within the class of large field models, we can see that models with $p
\geq 3$ are disfavoured, at the ``weak evidence level'' for $p=3$ and
at the ``strong evidence'' level for $p=4$. Clearly, one can conclude
that models with even larger (and fixed) values of $p$ would be even
more strongly disfavoured, so that they can be effectively ruled
out. We have chosen the large field $p=2$ model as our ``reference
model'' (the one with respect to which the Bayes factors are computed)
because it plays the role of a watershed point: large field models
with shallower potentials are preferred by the Bayesian evidence, with
$p=1$ and $p=2/3$ gathering slightly more than ``weak evidence'' in
their favour. However, the evidence is not strong enough to allow one
to conclude a definite preference for these models. The more generic
large field model with $p\in [0.2,5]$ is also weakly preferred over
LF$_2$, and this despite the extra parameter of the former, which
incurs an Occam's razor penalty. As expected, the performance of this
model, as measured by the evidence, falls in between the steep
potentials ($p>2$) and the shallower ones ($p<2$).
\par
Moving on to small field models, we remark that their overall
performance is superior to our reference large field model LF$_2$, but
quite comparable to the shallower large field models, despite the fact
that small field models have one or even two parameters more than
large field models. The very best models of inflation are small
field. Within the error bars, the evidence cannot distinguish between
a model with an upper cutoff at $M_\mathrm{Pl}$ and one that allows $\mu$ to go
above the Planck mass. Models with purely super-Planckian expectation
values are only very slightly disfavoured, by about 1 unit in the
log evidence. Therefore we can conclude that the data are presently
not sufficient to distinguish between the two scenarios.
\par
Further insight in the model comparison outcome can be garnered by
investigating simultaneously the Bayesian complexity and the evidence
(or the Bayes factor) of the models considered here (see
Ref.~\cite{Kunz:2006mc} for further details about the interpretation
of the complexity). The Bayesian complexity,
Eq.~\eqref{eq:complexity_chisq}, has been computed for each model from
a pure MCMC run whose convergence has been monitored by using the
R statistics implemented in \texttt{CosmoMC}~\citep{Lewis:2002ah}. The
chains have been stopped as soon as the estimated errors were below
$3\%$, which corresponds to a total number of samples ranging from
$5\times 10^4$ to $4\times 10^5$ depending on the underlying
inflationary model. The variance of our complexity estimate is
obtained from the variance of four sub-chains of equal length randomly
selected from the post burn-in samples. Both quantities are displayed
in Fig.~\ref{fig:complexity}, where the horizontal axis gives the
value of the number of input parameters for each model (both
inflationary and cosmological) minus the Bayesian complexity, which we
denote by the symbol $\Delta\mathcal{C}_\ub$. A value of $\Delta\mathcal{C}_\ub$ close to zero
means that the model parameters are well constrained by the data,
while $\Delta\mathcal{C}_\ub > 0$ gives an estimate of the effective number of
parameters remaining unconstrained by the data.
The value of $\Delta \mathcal{C}_\ub$ for the large field models with $p>2$ is
generally smaller, and reaches $\Delta\mathcal{C}_\ub \approx 0$ for $p=4$, the
model with the lowest evidence. This is a consequence of the tension
between these models and the data, which leads to the reheating
parameter becoming more and more constrained as $p$ increases: for
$p=4$, we find a 2$\sigma$ lower limit $\ln R > -2.1$, thus
leading to an increase in the value of the complexity by about 1
unit. Since the models with $p=3$ and $p=4$ have the smallest Bayes
factor while exhibiting values of $\Delta\mathcal{C}_\ub$ close to 0 (meaning that
all of their free parameters are well constrained), we can conclude
that those models are genuinely disfavoured by the data. On the other
hand, for the models having a similar Bayes factor,
Fig.~\ref{fig:complexity} shows that the larger number of free
parameters in the small field models corresponds to an increase in the
number of unconstrained parameters $\Delta\mathcal{C}_\ub$ with respect to its
value for the large field models with $p\leq 1$. This indicates that
the extra inflationary parameters in the small field class are not
being constrained by the data. Therefore we are led to conclude that
while a slight preference for small field models is beginning to
accumulate, it is too early to be able to conclusively favour small
field models over large field ones. It is expected that Planck data
will be able to conclusively pass judgement on this issue.
A consistent picture emerges when one considers the Bayesian
complexity of the two models with the largest number of parameters in
each class, namely SFI$_f$ and LFI$_p$, with 4 and 3 inflationary
parameters, respectively. For both cases, we find a similar
complexity, $\mathcal{C}_\ub \simeq 5.9$, which suggests that current data can
constrain up to approximately $2$ inflationary parameters. This is
because our models have all $N=5$ non-inflationary parameters in
common, including the SZ amplitude, and 4 of them are well constrained
and contribute approximately 4 units to the Bayesian complexity. This
leads to the conclusion that WMAP7 data are still insufficiently
powerful to fully constrain the whole inflationary sector as
parametrized in this work (see also Refs.~\cite{Kawasaki:2009yn,
Parkinson:2010zr}).
\begin{figure}
\begin{center}
\includegraphics[width=0.5\textwidth,clip=true]{complexity}
\caption{Bayes factor versus the effective number of unconstrained
parameters ($\Delta \mathcal{C}_\ub$) for all large and small field models. The
steeper LFI models are genuinely disfavoured by the data, as all of
their free parameters are well constrained. Small field models being
favoured by the evidence still have unconstrained parameters, and
therefore it is too early to conclusively rule out shallower ($p<2$)
large field models, despite the fact that they exhibit a slightly
smaller Bayes factor. }
\label{fig:complexity}
\end{center}
\end{figure}
\par
We can also evaluate the posterior probability for the entire class of
small field scenarios. From Eq.~\eqref{eq:Post_2} we find
\begin{equation}
p\left({\rm SF}\vert d\right)\simeq 0.77\pm 0.03,
\end{equation}
and, therefore, $p\left({\rm LF}\vert d\right)\simeq 0.23 \pm
0.03$. Therefore, the probability of the small field scenario has
risen from 50\% in the prior to 77\% in the posterior. This represents
posterior odds of $\sim 3:1$ in favour of small field inflation, as
compared with large field inflation. Although, as explained above,
this shift in the odds is by no means conclusive, it does represent an
indication that large field inflation is getting increasingly under
pressure from the data~\cite{Boyanovsky:2009xh}.
Finally, it is important to assess the robustness of our results with
respect to reasonable changes in our choice of models' priors. Our
choice to divide the prior probability equally between the LF class
and the SF class reflects the desire to compare both classes of models
on an equal footing \emph{a priori}. Another natural choice for the
models' prior would be to split the prior mass equally among models,
i.e. to assign $p({\rm SF}_i) = p({\rm LF}_i) = 1/(n_{\rm SF} + n_{\rm
LF})$. This choice would however result in prior odds of 2:1 in
favour of the LF class, which seems contrived, given that it arises
solely from the fact that we have double as many LF models as SF
models. Even with this (unfair to the SF class) prior choice, the
posterior probability for SF would be $p({\rm SF} | d) \simeq 0.6$ [up
from an initial prior probability $p({\rm SF}) = 1/3$], so our result
of a (slight) preference for SF models stands.
Finally, we notice that our result is robust with respect to the
inclusion of further models under either the SF or LF class, provided
such models are disfavoured by the data (as they would be e.g.~for
$p>5$ in the LF class). Inclusion of such highly disfavoured models
would result in their Bayes factors with respect to $\rm LF_2$ being
close to $0$, hence the average values defined in
Eqs.~\eqref{eq:BavSF} and ~\eqref{eq:BavLF} would simply be rescaled
by the new (larger) number of models in each class. However, the
posterior probability of SF models only depends on the ratio of the
average Bayes factors [see Eq.~\eqref{eq:Post_2}], hence such
rescaling factors would largely cancel out (for a detailed discussion
of this rearrangement of prior probability in a similar context, see
Ref.~\cite{March:2010ex}). This holds true provided the overall number
of models in each class is not widely different. We do not have any
reason to believe that this should be the case. However, if one of
the model classes truly had a much larger number of potential models
in it, one would have to carefully reconsider the choice of giving
both classes equal \emph{a priori} mass: after all, a class of models
with a smaller number of physically distinct possibilities in it is
\emph{a priori} more predictive than a class with a large number of
possible distinct models.
\section{Conclusion}
To summarize, this article presented the first calculation of the
Bayesian evidence for different classes of inflationary scenarios,
explaining from first principles how physically meaningful priors
could be derived for the fundamental parameters of the models. Among
the models studied here, small field models appear to be favoured,
albeit still in a fairly mild way. This result must be viewed as a
first step towards a more exhaustive exploration of the inflationary
landscape. With the techniques introduced here and the high accuracy
CMB data soon available, we have paved the way to the identification
of the best inflationary scenario.
\begin{acknowledgments}
We would like to thank Patrick Peter and Jean-Philippe Uzan for
useful discussions. RT would like to thank the Office of the Mayor
of the City of Paris for partial support and the Institut
d'Astrophysique de Paris (IAP) for hospitality. This work is
partially supported by the Belgian Federal Office for Science,
Technical and Cultural Affairs, under the Inter-university
Attraction Pole Grant No. P6/11
\end{acknowledgments}
|
{
"timestamp": "2011-03-21T01:01:35",
"yymm": "1009",
"arxiv_id": "1009.4157",
"language": "en",
"url": "https://arxiv.org/abs/1009.4157"
}
|
\section{Introduction}
In this decade exciting experimental results have been obtained in heavy
baryon spectroscopy. During these years, the ${1\over 2}^+$ and
${1\over 2}^-$ antitriplet states, $\Lambda_c^+,~\Xi_c^+,~\Xi_c^0$
and $\Lambda_c^+ (2593)$,\\
$\Xi_c^+(2790),~\Xi_c^0(2790)$ and the ${1\over 2}^+$
and ${3\over 2}^+$ and sextet states,
$\Omega_c^\ast,\Sigma_c^\ast,\Xi_c^\ast$ have been observed in experiments
\cite{Rstp01}. Among the s--wave bottom hadrons, only
$\Lambda_b,~\Sigma_b,~\Sigma_b^\ast,~\Xi_b$ and $\Omega_b$ have been discovered.
Moreover, in recent years many new states have been observed by BaBar and
BELLE collaborations, such as,
$X(3872),~Y(3930),~Z(3930),~X(3940),~Y(4008),~Z_1^+(4050),$\\
$Y(4140),~X(4160),~Z_2(4250),~Y(4260),~Y(4360),~Z^+(4430),$
and $Y(4660)$ which remain unidentified.
Of course, establishing these states is a remarkable progress in hadron
physics. It is expected that LHC, the world's largest--highest--energy particle
accelerator, will open new horizons in the discovery of the excited bottom
baryon sates \cite{Rstp02}. The experimental progress on heavy hadron
spectroscopy stimulated intensive theoretical studies in this respect (for a
review see \cite{Rstp03,Rstp04} and references therein). A detailed theoretical
study of experimental results on hadron spectroscopy and various weak and
strong decays can provide us with useful information about the quark
structure of new hadrons at the hadronic scale.
This scale belongs to the nonperturbative sector of QCD. Therefore, for
calculation of the form factors in weak decays and coupling constants in
strong decays, some nonperturbative methods are needed. Among many nonperturbative
methods, QCD sum rules \cite{Rstp05} is more reliable and predictive. In the
present work, we calculate the strong coupling constants of light
pseudoscalar mesons with sextet and antitriplet baryons, in light cone version
of the QCD sum rules (LCSR) method (for a review, see \cite{Rstp06}).
Note that some of the strong coupling constants have already been studied in
\cite{Rstp07,Rstp08,Rstp09} in the same framework.
The outline of this article is as follows. In section 2, we demonstrate how
coupling constants of pseudoscalar mesons with heavy baryons can be
calculated. In this section, the LCSR for the heavy baryon--pseudoscalar meson
coupling constants are also derived using the most general form of the baryon
currents. Section 3 is devoted to the numerical analysis and a comparison of
our results with the existing predictions in the literature.
\section{Light cone QCD sum rules for the coupling constants of
pseudoscalar mesons with heavy baryons}
Before presenting the detailed calculations for the strong coupling
constants of pseudoscalar mesons with heavy baryons, we would like to make few
remarks about the classification of heavy baryons. Heavy baryons with a
single heavy quark belong to either $SU(3)$ antisymmetric $\bar{3}_F$ or
symmetric $6_F$ flavor representations. Since we consider the ground states, the total spin of the two light quarks must one for $6_F$
and zero for $\bar{3}_F$, due to the symmetry property of their colors and
flavors, as a result of which we can write $J^P={1\over 2}^+ / {3\over 2}^+$
for $6_F$ and $J^P={1\over 2}^+$ for $\bar{3}_F$. Graphically, $6_F$ and
$\bar{3}_F$ representations are given in Fig. (1), where
$\alpha$, $\alpha +1$, $\alpha +2$ determine the charges of baryons
$(\alpha=-1$ or $ 0)$, and the asterix $(\ast)$ denote $J^P={3\over 2}^+$
states. In this work, we will consider only $J^P={1\over 2}^+$ states.
After this preliminary remarks, we proceed by calculating the strong
coupling constants of pseudoscalar mesons with heavy baryons within the
LCSR. For this purpose, we start by considering the following correlation
function:
\begin{eqnarray}
\label{estp01}
\Pi^{(ij)} = i \int d^4x e^{ipx} \left< {\cal P}(q) \left| {\cal T} \left\{
\eta^{(i)} (x) \bar{\eta}^{(j)} (0) \right\} \right| 0 \right>~,
\end{eqnarray}
where ${\cal P}(q)$ is the pseudoscalar--meson with
momentum $q$, $\eta$ is the interpolating current for the heavy baryons and
${\cal T}$ is the time ordering operator. Here, $i=1,~j=1$ describes the
sextet--sextet, $i=1,~j=2$ corresponds to sextet--triplet, and $i=2,~j=2$
describes triplet--triplet transitions. For convenience we shall denote
$\Pi^{(11)} = \Pi^{(1)}$, $\Pi^{(12)} = \Pi^{(2)}$ and $\Pi^{(22)} =
\Pi^{(3)}$.
The sum rules for the coupling constants of pseudoscalar mesons with heavy baryons
can be obtained by calculating the correlation function
(\ref{estp01}) in two different ways, namely, in terms of the hadrons and in
terms of quark gluon degrees of freedom, and then matching these two
representations.
Firstly, we calculate the correlation function (\ref{estp01}) in terms of
hadrons. Inserting complete sets of hadrons with the same quantum numbers
in the interpolating currents and isolating the ground states, we obtain
\begin{eqnarray}
\label{estp02}
\Pi^{(ij)} = {\left< 0 \left| \eta^{(i)}(0) \right|
B_2(p) \right> \left< B_2(p) {\cal P}(q) \left| \right.
B_1(p+q) \right> \left< B_1(p+q) \left| \bar{\eta}^{(j)}(0) \right|
0 \right> \over \left( p^2-m_2^2 \right) \left[(p+q)^2-m_1^2\right]}
+ \cdots~,
\end{eqnarray}
where $\left| B_2(p) \right>$ and $\left| B_1(p+q) \right>$ are the ${1\over 2}$ states,
and $m_2$ and $m_1$ are their masses, respectively. The dots in
Eq. (\ref{estp02}) describe contributions of the higher states and continuum. It follows from Eq. (\ref{estp02}) that in order to calculate the
correlation function in terms of hadronic parameters, the matrix elements entering to
Eq. (\ref{estp02}) are needed. These matrix elements are defined
in the following way:
\begin{eqnarray}
\label{estp03}
\left< 0 \left| \eta^{(i)} \right| B(p) \right> \!\!\! &=& \!\!\! \lambda_i u(p) ~, \nonumber \\
\left< B(p+q) \left| \eta^{(j)} \right| 0 \right> \!\!\! &=& \!\!\! \lambda_j \bar{u}(p+q)~, \nonumber \\
\left< B(p) {\cal P}(q) \left| \right. B(p+q) \right> \!\!\! &=& \!\!\! g \bar{u}(p) i\gamma_5
u(p+q)~,
\end{eqnarray}
where $\lambda_i$ and $\lambda_j$ are the residues of the heavy baryons,
$g$ is the coupling constant of pseudoscalar meson with heavy baryon and $u$
is the Dirac bispinor.
Using Eqs. (\ref{estp02}) and (\ref{estp03}) and performing summation
over spins of the baryons, we obtain the following representation of the
correlation function from the hadronic side:
\begin{eqnarray}
\label{estp04}
\Pi^{(ij)} \!\!\! &=& \!\!\! i {\lambda_i \lambda_j g \over \left( p^2-m_2^2 \right)
\left[(p+q)^2-m_1^2\right]} \Big\{ \rlap/q\rlap/p \gamma_5 + \mbox{other structures} \Big\},\nonumber\\
\end{eqnarray}
where we kept the structure which leads to a more reliable result.
In order to calculate the correlation function from QCD side, the forms of
the interpolating currents for the heavy baryons are needed. The general
form of the interpolating currents for the heavy spin ${1\over 2}$ sextet
and antitriplet baryons can be written as (see for example \cite{E.Bagan}),
\begin{eqnarray}
\label{estp05}
\eta_Q^{(s)} \!\!\! &=& \!\!\! - {1\over \sqrt{2}} \epsilon^{abc} \Big\{ \Big( q_1^{aT}
C Q^b \Big) \gamma_5 q_2^c + \beta \Big( q_1^{aT} C \gamma_5 Q^b \Big) q_2^c -
\Big[\Big( Q^{aT} C q_2^b \Big) \gamma_5 q_1^c + \beta \Big( Q^{aT} C
\gamma_5 q_2^b \Big) q_1^c \Big] \Big\}~, \nonumber\\
\label{estp06}
\eta_Q^{(anti-t)} \!\!\! &=& \!\!\! {1\over \sqrt{6}} \epsilon^{abc} \Big\{ 2 \Big( q_1^{aT}
C q_2^b \Big) \gamma_5 Q^c + 2 \beta \Big( q_1^{aT} C \gamma_5 q_2^b \Big) Q^c
+ \Big( q_1^{aT} C Q^b \Big) \gamma_5 q_2^c + \beta \Big(q_1^{aT} C
\gamma_5 Q^b \Big) q_2^c \nonumber \\
&+& \!\!\! \Big(Q^{aT} C q_2^b \Big) \gamma_5 q_1^c +
\beta \Big(Q^{aT} C \gamma_5 q_2^b \Big) q_1^c \Big\}~,
\end{eqnarray}
where $a,,b,c$ are the color indices and $\beta$ is an arbitrary parameter. It should also be noted that the general form of interpolating currents for light
spin 1/2 baryons was introduced in \cite{Y.Chang} and
$\beta=-1$ corresponds to the Ioffe current \cite{B.L.Ioffe}. The quark fields $q_1$ and
$q_2$ for the sextet and antitriplet are presented in Table 1.
\begin{table}[h]
\renewcommand{\arraystretch}{1.3}
\addtolength{\arraycolsep}{-0.5pt}
\small
$$
\begin{array}{|l|c|c|c|c|c|c|c|c|c|}
\hline \hline
& \Sigma_{b(c)}^{+(++)} &\Sigma_{b(c)}^{0(+)} &\Sigma_{b(c)}^{-(0)} & \Xi_{b(c)}^{-(0)'} &\Xi_{b(c)}^{0(+)'} &\Omega_{b(c)}^{-(0)} &\Lambda_{b(c)}^{0(+)} &
\Xi_{b(c)}^{-(0)}& \Xi_{b(c)}^{0(+)} \\ \hline
q_1&u &u & d&d &u &s &u&d &u \\
q_2&u & d& d&s & s&s &d &s &s\\
\hline \hline
\end{array}
$$
\caption{The quark flavors $q_1$ and $q_2$ for the baryons in the sextet and
the antitriplet representations}
\renewcommand{\arraystretch}{1}
\addtolength{\arraycolsep}{-1.0pt}
\end{table}
As has already been noted, in order to calculate the coupling constants of
pseudoscalar mesons with heavy baryons entering to sextet and
antitriplet representation, the calculation of the correlation function
from QCD part is needed. Before calculating it, we follow the approach given
in \cite{Rstp10,Rstp11,Rstp12,Rstp13,Rstp14} and try to find relations among
invariant functions involving coupling constants of pseudoscalar mesons with
sextet and antitriplet baryons.
We will show that the correlation functions responsible for coupling
constants of pseudoscalar mesons (P) with sextet--sextet (SS),
sextet--antitriplet (SA) and antitriplet-antitriplet (AA) baryons can each
be represented in terms of only one invariant function. Of course, the form
of the invariant functions for the couplings SSP, SAP and AAP are different
in the general case. It should be noted here that the relations presented
below are all structure independent.
We start our discussion by considering the sextet--sextet transition,
concretely. Consider the $\Sigma_b^0 \rightarrow \Sigma_b^0 \pi^0$ transition. The
invariant function for this transformation can be written in the following
form
\begin{eqnarray}
\label{estp07}
\Pi^{\Sigma_b^0 \rightarrow \Sigma_b^0 \pi^0} = g_{\pi\bar{u}u} \Pi_1^{(1)}(u,d,b) +
g_{\pi\bar{d}d} \Pi_1^{'(1)}(u,d,b) + g_{\pi\bar{b}b} \Pi_2^{(1)}(u,d,b)~,
\end{eqnarray}
where the interpolating current of $\pi^0$ meson is written as
\begin{eqnarray}
\label{nolabel}
J_{\pi^0} = \sum_{u,d} g_{\pi \bar{q}q}\bar{q}\gamma_5 q~. \nonumber
\end{eqnarray}
Obviously, the relations $g_{\pi \bar{u}u} = -g_{\pi \bar{d}d}={1\over \sqrt{2}}$, $g_{\pi
\bar{b}b} = 0$ hold for the $\pi^0$ meson. The invariant functions
$\Pi_1,~\Pi_1^{'}$ and $\Pi_2$ describe the radiation of $\pi^0$
meson from $u,~d$ and $b$ quarks of $\Sigma_b^0$ baryon, respectively, and
they can formally be defined as:
\begin{eqnarray}
\label{estp08}
\Pi_1^{(1)}(u,d,b) \!\!\! &=& \!\!\! \left< \bar{u}u \left| \Sigma_b^0 \bar{\Sigma}_b^0 \right| 0
\right>~, \nonumber \\
\Pi_1^{'(1)}(u,d,b) \!\!\! &=& \!\!\! \left< \bar{d}d \left| \Sigma_b^0 \bar{\Sigma}_b^0 \right| 0
\right>~, \nonumber \\
\Pi_2^{(1)}(u,d,b) \!\!\! &=& \!\!\! \left< \bar{b}b \left| \Sigma_b^0 \bar{\Sigma}_b^0 \right| 0
\right>~.
\end{eqnarray}
It follows from the definition of the interpolating current of $\Sigma_b$
baryon that it is symmetric under the exchange $u \lrar d$, hence
$\Pi_1^{'(1)}(u,d,b) = \Pi_1^{(1)}(d,u,b)$. Using this relation, we
immediately get from Eq. (\ref{estp07}) that,
\begin{eqnarray}
\label{estp09}
\Pi^{\Sigma_b^0 \rightarrow \Sigma_b^0 \pi^0} = {1\over \sqrt{2}} \Big[
\Pi_1^{(1)}(u,d,b) - \Pi_1^{(1)}(d,u,b) \Big]~,
\end{eqnarray}
and one can easily see that in the $SU_2(2)_f$ limit, $\Pi^{\Sigma_b^0 \rightarrow
\Sigma_b^0 \pi^0} = 0$.
The invariant function responsible for the $\Sigma_b^+ \rightarrow \Sigma_b^+
\pi^0$ transition can be obtained from the $\Sigma_b^0 \rightarrow \Sigma_b^0
\pi^0$ case by making the replacement $d \rightarrow u$, and using
$\Sigma_b^0=-\sqrt{2} \Sigma_b^+$, from which we get,
\begin{eqnarray}
\label{estp10}
4 \Pi_1^{(1)}(u,d,b) = -2 \left< \bar{u}u \left| \Sigma_b^+ \bar{\Sigma}_b^+ \right| 0
\right>~.
\end{eqnarray}
Appearance of the factor 4 on the left hand side is due to the fact that
each $\Sigma^+_b$ contains two $u$ quark, hence there are 4 possible ways for
radiating $\pi^0$ from the $u$ quark.
Making use of Eq.(\ref{estp09}), we get
\begin{eqnarray}
\label{estp11}
\Pi^{\Sigma_b^+ \rightarrow \Sigma_b^+ \pi^0} = \sqrt{2} \Pi_1^{(1)}(u,u,b).
\end{eqnarray}
The invariant function describing $\Sigma_b^- \rightarrow \Sigma_b^- \pi^0$ can
easily be obtained from the $\Sigma_b^0 \rightarrow \Sigma_b^0 \pi^0$ transition by
making the replacement $u \rightarrow d$ and taking into account $\Sigma_b^0 (u\rightarrow
d) = \sqrt{2} \Sigma_b^-$. Performing calculation similar to the previous
case, we get
\begin{eqnarray}
\label{estp12}
\Pi^{\Sigma_b^- \rightarrow \Sigma_b^- \pi^0} = \sqrt{2} \Pi_1^{(1)}(d,d,b).
\end{eqnarray}
Now, let us proceed to obtain the results for the invariant function
involving $\Xi_b^{'-(0)} \rightarrow \Xi_b^{'-(0)} \pi^0$ transition. The invariant
function for this transition can be obtained from the $\Sigma_b^0 \rightarrow
\Sigma_b^0 \pi^0$ case using the fact that $\Xi_b^{'0} = \Sigma_b^0
(d\rightarrow s)$ and $\Xi_b^{'-} = \Sigma_b^0
(u\rightarrow s)$. As a result, we obtain
\begin{eqnarray}
\label{estp13}
\Pi^{\Xi_b^{'0} \rightarrow \Xi_b^{'0} \pi^0} \!\!\! &=& \!\!\! {1\over \sqrt{2}}
\Pi_1^{(1)}(u,s,b)~, \nonumber \\
\Pi^{\Xi_b^{'-} \rightarrow \Xi_b^{'-} \pi^0} \!\!\! &=& \!\!\! - {1\over \sqrt{2}}
\Pi_1^{(1)}(d,s,b)~.
\end{eqnarray}
Obtaining relations among the invariant functions involving charged $\pi^\pm$
mesons requires more care. In this respect, we start by considering the
matrix element $\left< \bar{d}d \left|\Sigma_b^0 \bar{\Sigma}_b^0 \right| 0 \right>$,
where $d$ quarks from the $\Sigma_b^0$ and $\bar{\Sigma}_b^0$ from the
final $\bar{d}d$ state, and $u$ and $b$ quarks are the spectators. The
matrix element $\left< \bar{u}d \left|\Sigma_b^+ \bar{\Sigma}_b^0 \right| 0
\right>$describes the case where $d$ quark from $\bar{\Sigma}_b^0$ and $u$
quark from $\Sigma_b^+$ form the $\bar{u}d$ state and the remaining $u$ and
$b$ are being again the spectators. One can expect from this observation
that these matrix elements should be proportional to each other and
calculations confirm this expectation. So,
\begin{eqnarray}
\label{estp16}
\Pi^{\Sigma_b^0 \rightarrow \Sigma_b^+ \pi^-} \!\!\! &=& \!\!\! \left< \bar{u}d \left| \Sigma_b^+
\bar{\Sigma}_b^0 \right| 0 \right> = -\sqrt{2} \left< \bar{d}d \left| \Sigma_b^0
\bar{\Sigma}_b^0 \right| 0 \right> = -\sqrt{2} \Pi_1^{(1)}(d,u,b)~.
\end{eqnarray}
Making the replacement $u \lrar d$ in Eq. (\ref{estp16}), we obtain
\begin{eqnarray}
\label{estp17}
\Pi^{\Sigma_b^0 \rightarrow \Sigma_b^- \pi^+} \!\!\! &=& \!\!\! \left< \bar{d}u \left| \Sigma_b^-
\bar{\Sigma}_b^0 \right| 0 \right> = \sqrt{2} \left< \bar{u}u \left| \Sigma_b^0
\bar{\Sigma}_b^0 \right| 0 \right> = \sqrt{2} \Pi_1^{(1)}(u,d,b)~.
\end{eqnarray}
In estimating the coupling constants of SSP, SAP and AAP, it is enough to
consider the $\Sigma_b^0 \rightarrow \Sigma_b^0 P$, $\Xi_b^{'0} \rightarrow \Xi_b^0 P$ and
$\Xi_b^0 \rightarrow \Xi_b^0 P$ transitions, respectively. All remaining
transitions can be obtained from these transitions with the help of the
appropriate transformations among quark fields. Relations among the invariant functions of the charmed baryons can easily be obtained by making the
the
replacement $b \rightarrow c$ and adding to charge of each baryon a positive unit
charge.
Performing similar calculations, one can obtain rest of the required
expressions from the correlation functions in terms of the invariant
function $\Pi_1$, involving $\pi$, $K$ and $\eta$ mesons describing
sextet--sextet, sextet--antitriplet and antitriplet--antitriplet
transitions. In the present work, we neglect the mixing between $\eta$ and $\eta^{'}$
mesons.
It should also be noted here that all coupling constants for the SSP, SAP and AAP
are described only by one invariant function in each class of transitions,
but the forms of the invariant functions in each group of transitions are
different.
The invariant function $\Pi_1$ responsible for the $\Sigma_b^0 \rightarrow
\Sigma_b^0 P$, $\Xi_b^{'0} \rightarrow \Xi_b^0 P$ and $\Xi_b^0 \rightarrow \Xi_b^0 P$
transitions can be calculated in
deep Euclidean region, $-p^2 \rightarrow +\infty$ and $-(p+q)^2 \rightarrow +\infty$ using the
operator product expansion (OPE) in terms of the distribution amplitudes (DA's) of
the pseudoscalar mesons and light and heavy quark operators. Up to twist--4
accuracy, the matrix elements $\left< P(q) \left| \bar{q}(x) \Gamma q(0) \right| 0
\right>$ and $\left< P(q) \left| \bar{q}(x) G_{\mu\nu} q(0) \right| 0 \right>$, where
$\Gamma$ is any arbitrary Dirac matrix, are determined in terms of the DA's
of the pseudoscalar mesons, and their explicit expressions are given in
\cite{Rstp15,Rstp16,Rstp17}.
The light and heavy quark propagators are calculated in
\cite{Rstp18}, and
\cite{Rstp20}, respectively.
Using expressions of these propagators and
definitions of DA's for the pseudoscalar mesons, the correlation function
can be calculated from the QCD side, straightforwardly.
Equating the coefficients of the structure $\rlap/q\rlap/p \gamma_5$ of the
representation of the correlation function from hadronic and theoretical
sides, and applying the Borel transformation with respect to the variables
$p^2$ and $(p+q)^2$ in order to suppress the contributions of the higher
states and continuum, we obtain the following sum rules for the strong
coupling constants of the pseudoscalar mesons with sextet and antitriplet
baryons:
\begin{eqnarray}
\label{estp20}
g^{(i)} = {1 \over \lambda_1^{(i)} \lambda_2^{(i)}} e^{{m_1^{(i)2}
\over M_1^2} + {m_2^{(i)2}
\over M_2^2} }\, \Pi_1^{(i)}~,
\end{eqnarray}
where, $i=1,~2$ and $3$ for sextet--sextet, sextet--antitriplet and antitriplet--antitriplet, respectively and $M_1^2$ and $M_2^2$ are the Borel masses corresponding to
the initial and the final baryons. Since the masses of the initial and final
baryons are practically equal to each other, we take $M_1^2=M_2^2=2 M^2$;
and $\lambda_1^{(i)}$ and $\lambda_2^{(i)}$ are the residues of the initial and
final baryons, respectively, which are calculated in \cite{Rstp21}.
The explicit expressions for $\Pi_1^{(i)}$ are quite lengthy and we do not
present all of them here. As an example, we only present the explicit expression of the $\Pi_1^{(1)}$, which is given as:
\begin{eqnarray}
&&e^{m_Q^2/M^2 - m_{\cal P}^2/M^2} \Pi_1^{(1)} (u,d,b) = \nonumber \\
&&{(1-\beta)^2\over 32 \sqrt{2} \pi^2} M^4 m_Q^3 f_{\cal P}
\Big[ I_2 -
m_Q^2 I_3 \Big] \phi_\eta(u_0)
+ {(1-\beta^2)\over 64 \sqrt{2} \pi^2} M^4 m_Q^2 \mu_{\cal P}
\Big\{ \Big(i_3({\cal T},1) - 2 i_3({\cal T},v)\Big) I_2 \nonumber \\
&-& \!\!\! 2 m_Q^2 \Big[i_3({\cal T},1) - 2 i_3({\cal T},v) + (1-\widetilde{\mu}_{\cal P}^2)
\phi_\sigma(u_0) \Big] I_3 \Big\} \nonumber \\
&-& \!\!\!
{(1-\beta)^2\over 128 \sqrt{2} \pi^2} M^2 m_{\cal P}^2 m_Q f_{\cal P}
\Big\{ m_Q^2 \mathbb{A}(u_0) I_2 -
2 \Big( i_2({\cal V}_\parallel,1) - 2 i_2({\cal V}_\perp,1) \Big)
I_1 \nonumber \\
&-& \!\!\! 2 m_Q^2 \Big[ i_2({\cal A}_\parallel,1) -
2 \Big( i_2({\cal V}_\parallel,1) - i_2({\cal V}_\perp,1) + i_2({\cal
A}_\parallel,v)\Big) \Big]I_2\Big\} \nonumber \\
&-& \!\!\! {(1-\beta^2)\over 96 \sqrt{2} \pi^2} M^2
\Big[ 3 m_{\cal P}^2 m_Q^2 \mu_{\cal P} \Big( 2 m_Q^2
I_3 - I_2 \Big)
\Big( i_2({\cal T},1) - 2 i_2({\cal T},v) \Big) - 4 \langle \bar d d \rangle f_{\cal P} \pi^2
\phi_\eta(u_0) \Big] \nonumber \\
&+& \!\!\! {(1-\beta^2)\over 384 \sqrt{2} M^6} m_{\cal P}^2 m_Q^4 m_0^2 f_{\cal P}
\langle \bar d d \rangle \mathbb{A}(u_0) \nonumber \\
&-& \!\!\! {1\over 2304 \sqrt{2} M^4} m_Q^2 m_0^2 \langle \bar d d \rangle \Big\{
(1-\beta^2) m_{\cal P}^2 f_{\cal P}
\Big[ 5 \mathbb{A}(u_0) + 12 \Big(i_2({\cal A}_\parallel,1) +
i_2({\cal V}_\parallel,1) - 2 i_2({\cal A}_\parallel,v) \Big) \Big] \nonumber \\
&-& \!\!\! 8 m_Q \mu_{\cal P} (1-\widetilde{\mu}_{\cal P}^2)
[3 + \beta (2 + 3 \beta)] \phi_\sigma(u_0) \Big\} \nonumber \\
&-& \!\!\! {1\over 864 \sqrt{2} M^2} m_Q \langle \bar d d \rangle \Big\{9 (1-\beta^2) m_Q f_{\cal P}
\Big( m_{\cal P}^2 \mathbb{A}(u_0) + m_0^2 \phi_\eta(u_0) \Big) +
2 [5 + \beta (4 + 5 \beta)] m_0^2 \mu_{\cal P} (1-\widetilde{\mu}_{\cal
P}^2) \phi_\sigma(u_0) \Big\} \nonumber \\
&-& \!\!\! {1\over 576 \sqrt{2}} \Big\{(1-\beta^2) f_{\cal P} \langle \bar d d \rangle
\Big[ 6 m_{\cal P}^2 \mathbb{A}(u_0) - 12 m_{\cal P}^2
\Big(i_2({\cal A}_\parallel,1) +
i_2({\cal V}_\parallel,1) - 2 i_2({\cal A}_\parallel,v) \Big) +
m_0^2 \phi_\eta(u_0) \Big] \nonumber \\
&+& \!\!\! 8[3 + \beta (2 + 3 \beta)] m_Q \mu_{\cal P}
(1-\widetilde{\mu}_{\cal P}^2) \langle \bar d d \rangle \phi_\sigma(u_0) \Big\}
\end{eqnarray}
where $I_n$ is defined as:
\begin{eqnarray}
\label{nolabel}
I_n \!\!\! &=& \!\!\! \int_{m_Q^2}^\infty ds
{e^{m_Q^2/M^2 - s/M^2}\over s^n}~,\nonumber
\end{eqnarray}
and other parameters and functions as well as the way of continuum subtraction are given in \cite{Rstp11}.
To shorten the equation, we have ignored the light quark masses as well as terms containing gluon condensates in the above equation, but we
take into account
their contributions in numerical analysis.
\section{Numerical results}
In this section, we present the numerical results of the sum rules for strong
coupling constants of pseudoscalar mesons with sextet and antitriplet heavy
baryons, which are obtained in the previous section. The main input
parameters of LCSR are DA's for the pseudoscalar mesons which are given in
\cite{Rstp15,Rstp16,Rstp17,Rstp18}. The other input parameters entering to
the sum rules are $\langle \bar q q \rangle = -(0.24 \pm 0.001)^3~GeV^3$, $m_0^2 = (0.8 \pm
0.2)~GeV^2$ \cite{Rstp22}, $f_\pi = 0.131~GeV$, $f_K = 0.16~GeV$ and
$f_\eta = 0.13~GeV$ \cite{Rstp15}.
The sum rules for the SSP, SAP and AAP coupling constants have three
auxiliary parameters: Borel mass parameter $M^2$, continuum threshold $s_0$
and the arbitrary parameter $\beta$ which exists in the expression in the
expression for the interpolating currents. Obviously, the result for any
measurable physical quantity, being coupling constant in the present case,
should be independent on them. Therefore, our primary goal is to find such
regions of these parameters, where coupling constants exhibits no
dependence.
The upper limit of $M^2$ is determined by requiring that the continuum and
higher states contributions should be small compared to the total dispersion
integral. The lower limit can be obtained from the condition that the
condensate terms with highest dimensions contributes smaller compared to the
sum of all terms.
These two conditions leads to the working region, $15~GeV^2 \le M^2 \le 30~GeV^2$ for the bottom baryons and
$4~GeV^2 \le M^2 \le 12~GeV^2$ for the charmed ones.
The continuum threshold is not totally arbitrary but it depends on the energy of the first excited state with the same quantum numbers as the interpolating current.
We choose it in the domain between $s_0=(m_B + 0.5)^2~GeV^2$
and $s_0=(m_B + 1)^2~GeV^2$. As an example, let us consider the
$\Xi_b^{'0} \rightarrow \Xi_b^{'0} \pi^0$ transition. In Fig. (2), the dependence of
the strong coupling constant for the $\Xi_b^{'0} \rightarrow \Xi_b^{'0} \pi^0$
transition on $M^2$ is considered at different fixed values of $\beta$ and
a fixed value of $s_0$. We observe from this figure that the coupling
constant has a good stability in the ``working region" of $M^2$. In Fig. (3), we present the dependence of the
strong coupling constant for the $\Xi_b^{'0} \rightarrow \Xi_b^{'0} \pi^0$
transition on $\cos\theta$ at several fixed values of $s_0$
and at $M^2= 22.5~GeV^2$, where the angle $\theta$ is determined from
$\beta=\tan\theta$. From this figure, we see that the dependence of the coupling constant on $s_0$ diminishes when the higher values of the continuum threshold
are chosen from the considered working region. From this figure, we also observe that the strong coupling constant for the $\Xi_b^{'0} \rightarrow \Xi_b^{'0} \pi^0$ decay
becomes very large near the end points ($\cos\theta=\pm1$) and have zeros at some finite values of the $\cos\theta$. This behavior can be explained as follows.
From Eq. (\ref{estp20}) we see that the coupling constant is proportional to $ {1 \over \lambda_1^{(i)} \lambda_2^{(i)}} \Pi_1^{(i)}$. In general, zero's of the nominator and
denominator does not coincide since the OPE is truncated. In other words, calculations are not exact. For this reason, these points and any region between them are not
reliable regions for determination of physical quantities and suitable regions for $\cos\theta$ should be far from these regions.
It follows from Fig. (3) that in the region, $-0.5 \le
\cos\theta \le + 0.3$, the coupling constant seems to be insensitive to the
variation of $\cos\theta$. Here, we should also stress that our numerical results lead to the working region, $-0.6 \le
\cos\theta \le + 0.5$ common for masses of all heavy spin 1/2 baryons, which includes the working region of $\cos\theta$ for the coupling constant.
This region lie also inside the more wide interval of $\cos\theta$ obtained from analysis of the masses of the non strange heavy baryons
in \cite{E.Bagan,E.Bagan2,E.Bagan3}.
In general, the working region
of $\cos\theta$ for masses and coupling constants can be different, but in some cases as occur in our problem these regions coincide.
Similar analysis for the strong coupling constants of the light pseudoscalar
mesons with sextet and antitriplet heavy baryons are performed and the
results are presented in Tables (2), (3) and (4). In these Tables we also
present the predictions for the coupling constants coming from the Ioffe
currents when $\beta=-1$.
The errors in the values of the coupling constants presented in the
Tables (2), (3) and (4) include uncertainties coming from the variations of the
$s_0$, $\beta$ and $M^2$ as well as those coming from the other input parameters.
\begin{table}[t]
\renewcommand{\arraystretch}{1.3}
\addtolength{\arraycolsep}{-0.5pt}
\small
$$
\begin{array}{|l|r@{\pm}l|r@{\pm}l||l|r@{\pm}l|r@{\pm}l|}
\hline \hline
\multirow{2}{*}{$g^{\mbox{\small{\,channel}}}$} &\multicolumn{4}{c||}{\mbox{Bottom Baryons}} &
\multirow{2}{*}{$g^{\mbox{\small{\,channel}}}$} &\multicolumn{4}{c|}{\mbox{Charmed Baryons}} \\
& \multicolumn{2}{c}{\mbox{~General current~}} &
\multicolumn{2}{c||}{\mbox{~Ioffe current~}} & &
\multicolumn{2}{|c}{\mbox{~General current~}} &
\multicolumn{2}{c|}{\mbox{~Ioffe current~}} \\ \hline
g^{\Xi_b^{'0} \rightarrow \Xi_b^{'0} \pi^0} &~~~~~~~9.0&3.0 &~~~~~7.3&2.6 &
g^{\Xi_c^{'+} \rightarrow \Xi_c^{'+} \pi^0} &~~~~~~ 4.0&1.4 &~~~~ 3.0&1.1 \\
g^{\Sigma_b^0 \rightarrow \Sigma_b^- \pi^+} & 17.0&6.1 & 13.0&4.5 &
g^{\Sigma_c^+ \rightarrow \Sigma_c^0 \pi^+} & 8.0&2.8 & 4.1&1.5 \\
g^{\Xi_b^{'0} \rightarrow \Sigma_b^+ K^-} & 19.0&6.7 & 10.0&3.6 &
g^{\Xi_c^{'+} \rightarrow \Sigma_c^{++} K^-} & 9.0&3.4 & 3.0&1.0 \\
g^{\Omega_b^- \rightarrow \Xi_b^{'0} K^-} & 21.0&6.8 & 12.3&4.4 &
g^{\Omega_c^0 \rightarrow \Xi_c^{'+} K^-} & 9.0&3.4 & 5.6&1.9 \\
g^{\Sigma_b^+ \rightarrow \Sigma_b^+ \eta_1} & 12.5&4.4 & 8.7&3.1 &
g^{\Sigma_c^{++} \rightarrow \Sigma_c^{++} \eta_1} & 6.0&2.2 & 2.8&1.0 \\
g^{\Xi_b^{'0} \rightarrow \Xi_b^{'0} \eta_1} & 5.3&1.9 & 3.6&1.3 &
g^{\Xi_c^{'+} \rightarrow \Xi_c^{'+} \eta_1} & 2.6&0.9 & 0.7&0.2 \\
g^{\Omega_b^- \rightarrow \Omega_b^- \eta_1} & 26.0&7.4 & 20.0&5.5 &
g^{\Omega_c^0 \rightarrow \Omega_c^0 \eta_1} & 11.0&3.8 & 9.3&3.4 \\
\hline \hline
\end{array}
$$
\caption{The values of the strong coupling constants $g$ for the transitions
among the sextet and sextet heavy baryons with pseudoscalar mesons.}
\renewcommand{\arraystretch}{1}
\addtolength{\arraycolsep}{-1.0pt}
\end{table}
\begin{table}[h]
\renewcommand{\arraystretch}{1.3}
\addtolength{\arraycolsep}{-0.5pt}
\small
$$
\begin{array}{|l|r@{\pm}l|r@{\pm}l||l|r@{\pm}l|r@{\pm}l|}
\hline \hline
\multirow{2}{*}{$g^{\mbox{\small{\,channel}}}$} &\multicolumn{4}{c||}{\mbox{Bottom Baryons}} &
\multirow{2}{*}{$g^{\mbox{\small{\,channel}}}$} &\multicolumn{4}{c|}{\mbox{Charmed Baryons}} \\
& \multicolumn{2}{c}{\mbox{~General current~}} &
\multicolumn{2}{c||}{\mbox{~Ioffe current~}} & &
\multicolumn{2}{|c}{\mbox{~General current~}} &
\multicolumn{2}{c|}{\mbox{~Ioffe current~}} \\ \hline
g^{\Xi_b^{'0} \rightarrow \Xi_b^0 \pi^0} &~~~~~~~7.5&2.6 &~~~~~6.1&2.2 &
g^{\Xi_c^{'+} \rightarrow \Xi_c^+ \pi^0} &~~~~~~ 3.1&1.1 &~~~~ 2.0&0.7 \\
g^{\Sigma_b^- \rightarrow \Lambda_b^0 \pi^-} & 15.0&4.9 & 11.5&3.9 &
g^{\Sigma_c^0 \rightarrow \Lambda_c^+ \pi^-} & 6.5&2.4 & 5.6&1.8 \\
g^{\Sigma_b^0 \rightarrow \Xi_b^0 \bar{K}^0} & 11.5&3.9 & 8.9&3.1 &
g^{\Sigma_c^+ \rightarrow \Xi_c^+ \bar{K}^0} & 5.0&1.7 & 3.7&1.3 \\
g^{\Omega_b^- \rightarrow \Xi_b^- \bar{K}^0} & 17.0&4.5 & 13.5&4.8 &
g^{\Omega_c^0 \rightarrow \Xi_c^0 \bar{K}^0} & 6.5&2.3 & 3.0&1.1 \\
g^{\Xi_b^{'0} \rightarrow \Xi_b^- K^+} & 12.0&4.3 & 9.8&3.5 &
g^{\Xi_c^{'+} \rightarrow \Xi_c^0 K^+} & 4.5&1.6 & 2.1&0.8 \\
g^{\Xi_b^{'0} \rightarrow \Xi_b^0 \eta_1} & 16.0&5.6 & 12.0&4.3 &
g^{\Xi_c^{'+} \rightarrow \Xi_c^+ \eta_1} & 6.7&2.4 & 4.3&1.5 \\
\hline \hline
\end{array}
$$
\caption{The values of the strong coupling constants $g$ for the transitions
among the sextet and antitriplet heavy baryons with pseudoscalar mesons.}
\renewcommand{\arraystretch}{1}
\addtolength{\arraycolsep}{-1.0pt}
\end{table}
\begin{table}[h]
\renewcommand{\arraystretch}{1.3}
\addtolength{\arraycolsep}{-0.5pt}
\small
$$
\begin{array}{|l|r@{\pm}l|r@{\pm}l||l|r@{\pm}l|r@{\pm}l|}
\hline \hline
\multirow{2}{*}{$g^{\mbox{\small{\,channel}}}$} &\multicolumn{4}{c||}{\mbox{Bottom Baryons}} &
\multirow{2}{*}{$g^{\mbox{\small{\,channel}}}$} &\multicolumn{4}{c|}{\mbox{Charmed Baryons}} \\
& \multicolumn{2}{c}{\mbox{~General current~}} &
\multicolumn{2}{c||}{\mbox{~Ioffe current~}} & &
\multicolumn{2}{|c}{\mbox{~General current~}} &
\multicolumn{2}{c|}{\mbox{~Ioffe current~}} \\ \hline
g^{\Xi_b^0 \rightarrow \Xi_b^0 \pi^0} &~~~~~~~1.0&0.3 &~~~~~4.0&1.4 &
g^{\Xi_c^+ \rightarrow \Xi_c^+ \pi^0} &~~~~~~ 0.70&0.22 &~~~~ 2.7&0.9 \\
g^{\Xi_b^- \rightarrow \Lambda_b^0 K^-} & 1.5&0.5 & 5.2&1.8 &
g^{\Xi_c^0 \rightarrow \Lambda_c^+ K^-} & 0.9&0.3 & 2.2&0.7 \\
g^{\Xi_b^0 \rightarrow \Xi_b^0 \eta_1} & 0.6&0.2 & 2.9&1.0 &
g^{\Xi_c^+ \rightarrow \Xi_c^+ \eta_1} & 0.07&0.02 & 0.26&0.08 \\
g^{\Lambda_b^0 \rightarrow \Lambda_b^0 \eta_1} & 1.0&0.3 & 4.0&1.1 &
g^{\Lambda_c^+ \rightarrow \Lambda_c^+ \eta_1} & 0.75&0.24 & 1.9&0.66 \\
\hline \hline
\end{array}
$$
\caption{The values of the strong coupling constants $g$ for the transitions
among the antitriplet and antitriplet heavy baryons with pseudoscalar mesons.}
\renewcommand{\arraystretch}{1}
\addtolength{\arraycolsep}{-1.0pt}
\end{table}
\newpage
We see from these Tables that, there is substantial difference between the
predictions of the general current and the Ioffe current, especially for the
strong coupling constants of the antitriplet--antitriplet heavy baryons
with pseoduscalar mesons, which can be explained as follows. As a result of
the analysis of the dependence of the coupling constants on $\cos\theta$
we see that the value $\beta=-1$ belongs to the unstable region. Therefore, a
prediction at this point of $\beta$ is not reliable.
Finally, we compare our results with those existing in literature. In
various works, the coupling constant $\Sigma_c \rightarrow \Lambda_c \pi$ is
estimated to be
\begin{eqnarray}
\label{nolabel}
g^{\Sigma_c \rightarrow \Lambda_c \pi} = \left\{ \begin{array}{l}
8.88,~\mbox{\rm \cite{Rstp23}~(relativistic three--quark model)}~,\\
6.82,~\mbox{\rm \cite{Rstp24}~(light--front quark model)},\\
10.8 \pm 2.2,~\mbox{\rm \cite{Rstp09}~(LCSR)}~,\\
6.5 \pm 2.4,~\mbox{\rm (our result)~(LCSR)}~.
\end{array} \right. \nonumber
\end{eqnarray}
We see that, within errors our result is close to the results of
\cite{Rstp09,Rstp23,Rstp24}. The coupling constant for the $\Xi_Q \Xi_Q \pi$
transition LCSR is estimated to have the values
$g^{\Xi_c \rightarrow \Xi_c \pi} = 1.0 \pm 0.5$ and $g^{\Xi_b \rightarrow \Xi_b
\pi} = 1.6 \pm 0.4$, which are slightly larger compared to our
predictions. Finally, the coupling constant $g^{\Sigma_c \rightarrow \Sigma_c
\pi}$ is calculated in \cite{Rstp09} and it is obtained that
$g^{\Sigma_c \rightarrow \Sigma_c \pi} = -8.0 \pm 1.7$, which is in quite a
good agreement with our prediction.
In conclusion, the strong coupling constants of light pseudoscalar mesons with
sextet and antitriplet heavy baryons are studied within LCSR. It is shown
that, all coupling constants for the sextet--sextet, sextet--antitriplet and
antitriplet--antitriplet transitions are described by only one invariant
function in each class.
\section*{Acknowledgment}
The authors thank to A. Ozpineci for useful discussions.
\bAPP{A}{}
Here in this appendix, we present the expressions of the correlation functions
in terms of invariant function $\Pi_1^{(i)}$ involving
$\pi$, $K$ and $\eta$ mesons.
\begin{itemize}
\item Correlation functions describing pseudoscalar mesons with
sextet--sextet baryons.
\end{itemize}
\begin{appeeq}} \def\eaeeq{\end{appeeq}
\label{nolabel}
{1\over \sqrt{2}} \Pi^{\Sigma_b^+ \rightarrow \Sigma_b^0 \pi^+ } \!\!\! &=& \!\!\!
\Pi^{\Xi_b^{'0} \rightarrow \Sigma_b^0 \bar{K}^0 } =
\Pi^{\Sigma_b^0 \rightarrow \Xi_b^{'0} K^0 } =
\Pi_1^{(1)}(d,u,b)~, \nonumber \\
\Pi^{\Xi_b^{'0} \rightarrow \Xi_b^{'-} \pi^+ } \!\!\! &=& \!\!\!
\Pi_1^{(1)}(d,s,b)~, \nonumber \\
{1\over \sqrt{2}} \Pi^{\Sigma_b^- \rightarrow \Sigma_b^0 \pi^- } \!\!\! &=& \!\!\!
\Pi^{\Xi_b^{'-} \rightarrow \Sigma_b^0 K^- } =
\Pi^{\Sigma_b^0 \rightarrow \Xi_b^{'-} K^+ } =
\Pi_1^{(1)}(u,d,b)~, \nonumber \\
\Pi^{\Xi_b^{'-} \rightarrow \Xi_b^{'0} \pi^- } \!\!\! &=& \!\!\!
\Pi_1^{(1)}(u,s,b)~, \nonumber \\
{1\over \sqrt{2}} \Pi^{\Xi_b^{'0} \rightarrow \Sigma_b^+ K^- } \!\!\! &=& \!\!\!
{1\over \sqrt{2}} \Pi^{\Sigma_b^+ \rightarrow \Xi_b^{'0} K^+ } =
{\sqrt{6} \over 2} \Pi^{\Sigma_b^+ \rightarrow \Sigma_b^+ \eta_1} =
\Pi_1^{(1)}(u,u,b)~, \nonumber \\
{1\over \sqrt{2}} \Pi^{\Omega_b^- \rightarrow \Xi_b^{'0} K^- } \!\!\! &=& \!\!\!
{1\over \sqrt{2}} \Pi^{\Xi_b^{'0} \rightarrow \Omega_b^- K^+ } =
{1\over \sqrt{2}} \Pi^{\Omega_b^- \rightarrow \Xi_b^{'-} \bar{K}^0 } =
{1\over \sqrt{2}} \Pi^{\Xi_b^{'-} \rightarrow \Omega_b^- K^0 } =
- {\sqrt{6}\over 4} \Pi^{\Omega_b^- \rightarrow \Omega_b^- \eta_1}
\Pi_1^{(1)}(s,s,b)~, \nonumber \\
{1\over \sqrt{2}} \Pi^{\Xi_b^{'-} \rightarrow \Sigma_b^- \bar{K}^0 } \!\!\! &=& \!\!\!
{1\over \sqrt{2}} \Pi^{\Sigma_b^- \rightarrow \Xi_b^{'-} K^0 } =
{\sqrt{6} \over 2} \Pi^{\Sigma_b^- \rightarrow \Sigma_b^- \eta_1} =
\Pi_1^{(1)}(d,d,b)~, \nonumber \\
\Pi^{\Xi_b^{'0} \rightarrow \Xi_b^{'0} \eta_1} \!\!\! &=& \!\!\! {1\over \sqrt{6}} \Big[\Pi_1^{(1)}(u,s,b) - 2 \Pi_1^{(1)}(s,u,b)\Big]~, \nonumber \\
\Pi^{\Xi_b^{'-} \rightarrow \Xi_b^{'-} \eta_1} \!\!\! &=& \!\!\! {1\over \sqrt{6}} \Big[\Pi_1^{(1)}(d,s,b) - 2 \Pi_1^{(1)}(s,d,b)\Big]~, \nonumber \\
\eaeeq
\begin{itemize}
\item Correlation functions responsible for the transitions of the
sextet--antitriplet baryons.
\end{itemize}
\begin{appeeq}} \def\eaeeq{\end{appeeq}
\label{nolabel}
\sqrt{2} \Pi^{\Xi_b^{'0} \rightarrow \Xi_b^0 \pi^0 } \!\!\! &=& \!\!\!
\Pi^{\Xi_b^{'0} \rightarrow \Xi_b^- \pi^+ } =
\Pi^{\Xi_b^{'-} \rightarrow \Xi_b^0 K^- } =
\Pi_1^{(2)}(u,s,b)~, \nonumber \\
- \sqrt{2} \Pi^{\Xi_b^{'-} \rightarrow \Xi_b^- \pi^0 } \!\!\! &=& \!\!\!
\Pi^{\Xi_b^{'-} \rightarrow \Xi_b^0 \pi^- } =
\Pi_1^{(2)}(d,s,b)~, \nonumber \\
\Pi^{\Sigma_b^0 \rightarrow \Lambda_b^0 \pi^0 } \!\!\! &=& \!\!\! {1\over \sqrt{2}} \Big[\Pi_1^{(2)}(u,d,b) + \Pi_1^{(2)}(d,u,b)\Big]~, \nonumber \\
\Pi^{\Sigma_b^- \rightarrow \Lambda_b^0 \pi^- } \!\!\! &=& \!\!\!
- \Pi^{\Sigma_b^0 \rightarrow \Xi_b^- K^+ } =
\Pi_1^{(2)}(u,d,b)~, \nonumber \\
- {1\over \sqrt{2}} \Pi^{\Sigma_b^+ \rightarrow \Lambda_b^0 \pi^+ } \!\!\! &=& \!\!\!
- \Pi^{\Sigma_b^0 \rightarrow \Xi_b^0 \bar{K}^0 } =
- \Pi^{\Xi_b^{'0} \rightarrow \Lambda_b^0\bar{K}^0 } =
\Pi^{\Xi_b^{'0} \rightarrow \Xi_b^- K^+ } =
\Pi_1^{(2)}(d,u,b)~, \nonumber \\
- {1\over \sqrt{2}} \Pi^{\Sigma_b^- \rightarrow \Xi_b^- \bar{K}^0 } \!\!\! &=& \!\!\!
- {1\over \sqrt{2}} \Pi^{\Sigma_b^- \rightarrow \Lambda_b^0 K^- } =
\Pi_1^{(2)}(d,d,b)~, \nonumber \\
{1\over \sqrt{2}} \Pi^{\Omega_b^- \rightarrow \Xi_b^- \bar{K}^0 } \!\!\! &=& \!\!\!
{1\over \sqrt{2}} \Pi^{\Omega_b^- \rightarrow \Xi_b^0 K^- } =
\Pi_1^{(2)}(s,s,b)~, \nonumber \\
\Pi^{\Sigma_b^+ \rightarrow \Lambda_b^0 K^+ } \!\!\! &=& \!\!\! - \sqrt{2} \Pi_1^{(2)}(u,u,b)~, \nonumber \\
\Pi^{\Xi_b^{'0} \rightarrow \Xi_b^0 \eta_1} \!\!\! &=& \!\!\! {1\over \sqrt{6}} \Big[\Pi_1^{(2)}(u,s,b) + 2 \Pi_1^{(2)}(s,u,b)\Big]~, \nonumber \\
\Pi^{\Xi_b^{'-} \rightarrow \Xi_b^- \eta_1} \!\!\! &=& \!\!\! {1\over \sqrt{6}} \Big[\Pi_1^{(2)}(d,s,b) + 2 \Pi_1^{(2)}(s,d,b)\Big]~, \nonumber \\
\Pi^{\Sigma_b^0 \rightarrow \Lambda_b^0 \eta_1} \!\!\! &=& \!\!\! {1\over \sqrt{6}} \Big[\Pi_1^{(2)}(u,d,b) - \Pi_1^{(2)}(d,u,b)\Big]~. \nonumber
\eaeeq
\begin{itemize}
\item Correlation functions appearing in the
antitriplet--antitriplet pseudoscalar meson transitions.
\end{itemize}
\begin{appeeq}} \def\eaeeq{\end{appeeq}
\label{nolabel}
\sqrt{2} \Pi^{\Xi_b^0 \rightarrow \Xi_b^0 \pi^0 } \!\!\! &=& \!\!\!
\Pi^{\Xi_b^0 \rightarrow \Xi_b^- \pi^+ } =
\Pi_1^{(3)}(u,s,b)~, \nonumber \\
- \sqrt{2} \Pi^{\Xi_b^- \rightarrow \Xi_b^- \pi^0 } \!\!\! &=& \!\!\!
\Pi^{\Xi_b^- \rightarrow \Xi_b^0 \pi^- } =
\Pi_1^{(3)}(d,s,b)~, \nonumber \\
\Pi^{\Lambda_b^0\rightarrow \Lambda_b^0 \pi^0 } \!\!\! &=& \!\!\! {1\over \sqrt{2}} \Big[\Pi_1^{(3)}(u,d,b) - \Pi_1^{(3)}(d,u,b)\Big]~, \nonumber \\
\Pi^{\Xi_b^0 \rightarrow \Lambda_b^0\bar{K}^0 } \!\!\! &=& \!\!\! \Pi_1^{(3)}(u,u,b)~, \nonumber \\
\Pi^{\Xi_b^- \rightarrow \Lambda_b^0 K^- } \!\!\! &=& \!\!\! - \Pi_1^{(3)}(u,d,b)~, \nonumber \\
\Pi^{\Xi_b^0 \rightarrow \Xi_b^0 \eta_1} \!\!\! &=& \!\!\! {1\over \sqrt{6}} \Big[\Pi_1^{(3)}(u,s,b) - 2 \Pi_1^{(3)}(s,u,b)\Big]~, \nonumber \\
\Pi^{\Xi_b^- \rightarrow \Xi_b^- \eta_1} \!\!\! &=& \!\!\! {1\over \sqrt{6}} \Big[\Pi_1^{(3)}(d,s,b) - 2 \Pi_1^{(3)}(s,d,b)\Big]~, \nonumber \\
\Pi^{\Lambda_b^0\rightarrow \Lambda_b^0 \eta_1} \!\!\! &=& \!\!\! {1\over \sqrt{6}} \Big[\Pi_1^{(3)}(d,u,b) + \Pi_1^{(3)}(u,d,b)\Big]~. \nonumber
\eaeeq
The expressions for the charmed baryons can easily be obtained by making the
replacement $b \rightarrow c$ and adding to charge of each baryon a positive unit
charge.
\renewcommand{\thehran}{\thesection.\arabic{hran}}
|
{
"timestamp": "2010-12-07T02:01:35",
"yymm": "1009",
"arxiv_id": "1009.3658",
"language": "en",
"url": "https://arxiv.org/abs/1009.3658"
}
|
\section{Introduction}
Lepton-flavor violation (LFV),
if observed in a future experiment, is an evidence of new
physics beyond the standard model,
because the lepton-flavor number is conserved in
the standard model.
Since the processes
are theoretically free from the non perturbative hadronic effects
they provide accurate predictions for the decay rates and the
branching ratios (Br) of these processes. Furthermore, they are
theoretically rich as they carry considerable information about
the free parameters of the used model. On the other hand, the
experimental work which has been done regarding these decays
motivates their theoretical studies. For instance, experimental
prospect for $\mu \to e \gamma$ is promising with the recent
commencement of the MEG experiment which will probe Br$(\mu \to e
\gamma) \approx 10^{-13}$ two orders of magnitude beyond the
current limit. B factories search for the decay mode $\tau \to
\ell_i \ell_j \bar{\ell_j}$ at the $e^+ e^-$ experiment with
upper limits in the range Br$(\tau \to\ell_i \ell_j
\bar{\ell_j})\le (2-8)\times 10^{-8}$~\cite{Aubert:2003pc}.
Searches for $\tau \to \mu\mu\bar{\mu}$ can be performed at the
Large Hadron Collider (LHC) where $\tau$ leptons are copiously
produced from the decays of $W$, $Z$, $B$ and $D$, with anticipated
sensitivities to Br$(\tau \to \mu\mu\bar{\mu}) \approx
10^{-8}$~\cite{Giffels:2008ar}. The decay $\mu \to ee\bar{e}$ of
which there is a strict bound Br$(\mu \to ee\bar{e})\le 10^{-12}$
is a strong constraint on the parameter
space~\cite{Bellgardt:1987du}.
The present experimental upper limits for the branching ratios of
$\ell \to \ell_i \ell_j \bar{\ell}_j$ and $\ell \to \ell^{\prime}\gamma$ decays are given
by~\cite{Aubert:2003pc,Bellgardt:1987du} \begin{eqnarray} \label{llld} {\rm
Br}(\tau \to \ell_i \ell_j \bar{\ell}_j) \sim 10^{-8},\qquad {\rm
Br}({\mu \to \bar{e}ee}) \sim 10^{-12},
\end{eqnarray} and \cite{Aubert:2005ye, Aubert:2005wa,Brooks:1999pu}
\begin{eqnarray}
\label{lllm} {\rm Br}({\tau \to \mu\gamma}) & < & 6.8 \times 10^{-8},\\ \nonumber
{\rm Br}({\tau \to e\gamma}) & < & 1.1 \times 10^{-7},\\ \nonumber
{\rm Br}({\mu \to e\gamma}) & < & 1.2
\times 10^{-11}. \end{eqnarray}
Within the SM, the Brs of LFV decays are extremely
small. On the other hand, the difference between the experimental
value of the muon anomalous magnetic moment $a_\mu = (g-2)/2$ and
its SM prediction is given
by\cite{Bennett:2006fi,Yao:2006px,Miller:2007kk}
\begin{eqnarray}
\label{gm2} \Delta a_\mu = a^{\rm exp}_\mu - a^{\rm SM}_\mu =
(29.5 \pm 8.8)\times 10^{-10},
\end{eqnarray}
with a discrepancy of 3.4 $\sigma$. In spite of the substantial
progress in both experimental and theoretical sides, the
situation is not completely clear yet. However, the possibility
that the present discrepancy may arise from the errors in the
determination of the hadronic leading-order contribution to
$\Delta a_\mu$ seems to be unlikely as argued in
Ref.~\cite{Passera:2008jk}. There are many
attempts, in the literature, to explain this discrepancy through considering new
physics beyond SM~\cite{Bigi:1985jq,Czarnecki:2001pv,Ellis:2007fu}.
One of the possibilities for physics beyond the Standard Model is
the four-color symmetry between quarks and leptons introduced by Pati-Salam~\cite{Pati:1974yy}. The prediction of the
existence of gauge leptoquarks, which are rather heavy according
to the current available data, is a direct consequence of this
symmetry.
The current bounds on the leptoquarks production are set by
Tevatron, LEP and HERA~\cite{Wang:2004cj}. Tevatron experiments
have set limits on the scalar leptoquarks masses $M_{LQ} > $ 242
GeV. On the other hand, the limits that have been set by LEP and
HERA experiments are model dependent. The search for these novel
particles will be continued at the CERN LHC. Preliminary
studies at the LHC experiments, ATLAS~\cite{Mitsou:2004hm} and CMS~\cite{Abdullin:1999im}, indicate that clear signals can be
observed for masses up to 1.2 TeV.
Our aim in this paper is to analyze the branching ratios for all
processes given in Eqs.(\ref{llld})- (\ref{lllm}) in the context
of the LQ model. These LFV processes are generated at loop level
through exchanging scalar LQ particles which transmit the lepton
flavour mixing from the Yukawa couplings to the observed charged
lepton sector. Previous studies of such decays were performed
extensively by theorists~\cite{Okada:1999zk}. In the present study of
these decay channels, the light scalar leptoquark effects
to $\ell \to \ell_i \ell_j \bar{\ell}_j$ are discussed in detail,
namely the contributions of the
photon and Z boson penguins and box diagrams.
Also, we include the predictions for $\ell \to \ell_i \ell_i \bar{\ell}_i$
channels correlated with $\ell \to \ell^{\prime} \gamma$ rates which are
interesting within the framework we use. Furthermore, we take into
account $(g-2)_\mu$, $\mu-e$ conversion and $\pi \to e \nu_e, \mu \nu_\mu$
constraints imposed on the input parameter space.
This is carried here by considering the parametrization introduced in~\cite{Benbrik:2008si} for the case of the $\ell \to \ell_i \ell_j
\bar{\ell}_j$ decays.\\
The paper is organized as follows: In Section \ref{formalism}, we
list the relevant terms of the scalar leptoquark Lagrangian to the LFV
decays and the analytical expressions of the
scalar leptoquark contributions to $a_\mu$
and $\ell \to \ell^{\prime} \gamma$
decays. The analytical results of the LFV decays $\ell \to
\ell_i \ell_j \bar{\ell}_j$ will be presented in Sec.III.
In Sec.IV, we derive the constraints that can be imposed on some
leptoquark Yukawa couplings obtained using $\mu - e $ conversion.
The numerical results for $\tau$ and $\mu$ decays will be
presented in Sec.V. Finally, Sec.VI will be devoted to the
conclusion.
\section{Leptoquark Basics}
\label{formalism}
\subsection{Scalar Leptoquark Interactions}
In this section we list the relevant terms of the scalar
leptoquark Lagrangian to our LFV decay modes. We consider
isosinglet scalar leptoquarks. The effective Lagrangian that describes
the leptoquark interactions in the mass basis can be written as~\cite{CHH1999,Lagr1}:
\begin{eqnarray}
{\mathcal{L}}_{LQ}
&=&
\label{lag}
\overline{u^c_a} \bigg(h^{'}_{ai} \Gamma_{k,S_R} P_L+h_{ai}\Gamma_{k,S_L} P_R \bigg) e_i S^*_k
+\overline{e_{j}} \bigg( h^{'*}_{aj} \Gamma^\dagger_{S_R,k} P_R+h^*_{aj} \Gamma^\dagger_{S_L,k} P_L
\bigg) u^c_a S_k
\\\nonumber
&-& e Q_{(u^c)} A_\mu \overline{u^c_a} \gamma^\mu u^c_a - ieQ_{S} A_\mu S^*_{k} \!\stackrel{\leftrightarrow}{\partial^\mu}\! S_{k}
+ ieQ_{S} \tan\theta_W Z_\mu S^*_{k} \!\stackrel{\leftrightarrow}{\partial^\mu}\! S_{k}
\\\nonumber
&-& \frac{e }{s_W c_W} Z_\mu \overline{u^c_a}\gamma^\mu
\bigg( (T_{3(u^c)} - Q_{(u^c)} s^2_W) P_R - Q_{(u^c)} s^2_W P_L \bigg)
u^c_a,
\end{eqnarray}
where $k = 1,2$ are the leptoquark indices, $T_3 = -1/2$, $Q_{u^c}
= -2/3$ are quark's isospin and electric charge respectively, $Q_S
= -1/3 $ is the electric charge of the scalar leptoquarks $S_k$,
$a$ is up-type quark flavor indices, $i,j$ are lepton flavor
indices, $c_{W} = \cos\theta_W$ and $s_{W} = \sin\theta_W$.
The $\Gamma_{k, S_{L(R)}}$ are elements of leptoquark mixing matrix
that bring $S_{L(R)}$ to the mass eigenstate basis $S_k$:
\begin{eqnarray}
S_L = \Gamma^{\dagger}_{S_L, k} S_k, \qquad S^*_R = \Gamma_{k,S_R}
S^*_k,
\end{eqnarray}
Here $S_{L(R)}$ denotes the field associated with the
$\overline{e_{j}} P_{L(R)} u^c_a$ terms in
${\mathcal{L}}_{LQ}$~\cite{CHH1999}. Note that in the no-mixing
case ($\Gamma=1$), $S_{1(2)}$ reduce to $S_{L(R)}$ which are called
chiral leptoquarks as they only couple to quarks and leptons in
certain chirality structures.
Finally, the couplings $h$ and $h^{\prime}$ are 3 by 3 matrices that give
rise to various LFV processes and must be subjected to the
experimental constraints. In this work we do not intend to explore
the effects of all possible leptoquark interactions.
Instead, we try to demonstrate that a simple scalar leptoquark
model can provide rich and interesting LFV phenomena.
\subsection{Muon anomalous magnetic moment $(g-2)_{\mu}$}
\begin{figure}
\begin{center}
\vspace{2cm}
\input{fd_LQ.tex}
\vspace{-14.2cm} \caption{Feynman diagrams contributing to $\ell
\to \ell^\prime \gamma$,
${S}_k$ denotes the scalar leptoquark with $k=1,2$ and $u^c_a$ denotes up-type quark with $a=1,2,3$. }
\label{fig:gm2-diagrams}
\end{center}
\end{figure}
The LQ interaction can generate muon anomalous magnetic moment and
resolve the discrepancy between theoretical and experimental
results. The corresponding one-loop diagrams are shown in
Fig.~\ref{fig:gm2-diagrams}(a)- \ref{fig:gm2-diagrams}(b)
where $\ell=\ell^\prime = \mu$.
The extra contribution to $a_\mu $ arising from the LQ model due to
quark and scalar leptoquark one-loop contribution is given by
\begin{eqnarray}
a^{LQ}_{\mu} &=&\nonumber - \frac{N_c m^2_\mu}{8 \pi^2} \sum_{a=1}^{3}
\sum_{k=1}^{2}
\frac{1}{M^2_{S_k}} \bigg[ \big(|h_{a\mu} \Gamma_{k,S_L}|^2 + |h'_{a\mu} \Gamma_{k,S_R}|^2 \big)
\big( Q_{(u^c)} F_{2}(x_{ka})- Q_{S} F_{1}(x_{ka}) \big) \\&& -
\frac{m_{(u^c_a)}}{m_\mu} {\rm Re} \big(h'_{a\mu} h^*_{a\mu} \Gamma^{+}_{S_R,k}\Gamma_{k,S_L} \big)
\big(Q_{(u^c)} F_{3}(x_{ka}) - Q_{S} F_{4}(x_{ka}) \big) \bigg],
\label{eq:a_LQ}
\end{eqnarray}
In the above expression, $N_c = 3$, $Q_{S} = -1/3$, $Q_{u^c} =
-2/3$. The kinematic loop functions $F_{i}$ $(i=1,...,4)$ depend
on the variable $x_{ka} = m^2_{(u^c_a)} / M^2_{S_k}$, their
expressions are given in the appendix B.
Clearly, the use of leptoquark contribution to saturate the deviation
shown in Eq.(\ref{gm2}) leads to constraint leptoquark masses
$M_{S_{k}}$ (k=1,2), mixing angle
$\theta_{LQ}$ and the Yukawa couplings ($h_{a\mu}$, $h^{(\prime)}_{a\mu}$).
\subsection{$\ell \to \ell^{\prime} \gamma$}
In this subsection, we give the expression for the amplitude of
$\ell \to \ell'\gamma$ which is generated by exchange of scalar leptoquark. According to the gauge invariance, the amplitude can be written
as:
\begin{eqnarray}
\label{ampge} i{\mathcal{M}}^{\gamma}&=&ie\bar{u}(p_2) \bigg(
F^{\gamma}_{2RL} P_{L} + F^{\gamma}_{2LR} P_R\bigg)
(i\sigma_{\mu\nu}q^\nu)
u(p_1)\varepsilon^{\mu *}_{\gamma},
\end{eqnarray}
where $\varepsilon_\gamma$ is the polarization vector and $q= p_1
- p_2$ is the momentum transfer. For the amplitude of leptoquark
exchange at one-loop level, as depicted in
Fig.~\ref{fig:gm2-diagrams} with $\ell \neq \ell^{\prime}$, we have
\begin{eqnarray}
\label{lepq} F^{\gamma}_{2LR} &=&\nonumber \frac{N_c}{16
\pi^2}\sum_{a=1}^{3} \sum_{k=1}^{2}
\frac{1}{M^2_{S_k}}\Bigg[
\big(m_\ell h'_{a\ell} h^{'*}_{a\ell'} \Gamma^{\dagger}_{S_R,k}
\Gamma_{k,S_R} + m_{l'} h_{a\ell} h^{*}_{a\ell'}
\Gamma^{\dagger}_{S_L,k} \Gamma_{k,S_L} \big)
\\\nonumber &&\times
\big(Q_{(u^c)}F_{2}(x_{ka})
- Q_{S}F_{1}(x_{ka})\big)
\\ &&- m_{(u^c_a)}
\big(h_{a\ell} h^{'*}_{a\ell'} \Gamma^{\dagger}_{S_R,k}
\Gamma_{k,S_L}\big) \big(Q_{(u^c)}F_{3}(x_{ka}) - Q_{S}
F_4 (x_{ka})\big)\Bigg],
\\
F^{\gamma}_{2RL} &=& F^{\gamma}_{2LR} ( h
\leftrightarrow h', R \leftrightarrow L),
\end{eqnarray}
with $x_{ka} = m^2_{(u^c_a)} / M^2_{S_k}$.
The branching ratio of
$\ell \to \ell' \gamma$ is given by:
\begin{eqnarray}
{\rm Br}(\ell \to \ell' \gamma) &=&\frac{\alpha_{em}}{4
\Gamma(\ell)} \frac{(m^2_\ell - m^2_{\ell'})^3}{ m^3_\ell } \bigg(
|F^{\gamma}_{2LR}|^2 + |F^{\gamma}_{2RL}|^2 \bigg),
\end{eqnarray}
In our numerical calculations we analyze the Brs of the decays
under consideration by using the total decay widths of the
decaying leptons $\Gamma(\ell)$.
\section{$\ell^- \to \ell^-_i \ell^-_j \ell^+_j$}
In this section, we present the analytical results for the LFV
$\tau$ decay into three leptons with different flavor within
leptoquark model. Next, we give the analytical results relative
to the branching ratios of $\tau^- \to \ell^-_i \ell^-_j \ell^+_j$
(the analogous results in the muon sector can be obtained by means
of a simple generalization.)
We perform a complete one-loop calculation of the $\tau$ decay
width for all six possible channels, $\tau^- \to \mu^- \mu^-
\mu^+$, $\tau^- \to e^- e^- e^+$, $\tau^- \to \mu^- \mu^+ e^-$,
$\tau^- \to e^- e^+ \mu^-$, $\tau^- \to \mu^- \mu^- e^+$ and
$\tau^- \to \mu^+ e^- e^-$. The contribution generated by the
$\gamma$-, Z-penguins and box diagrams are presented here
separately. Throughout this section we follow closely the notation
and thr way of presentation of \cite{Hisano:1995cp}.
First, we define the amplitude for $\tau^-(p) \to \ell^-_i
(p_1)\ell^-_j(p_2) \ell^+_j (p_3)$ decays as the sum of the
various contributions,
\begin{eqnarray}
\label{tot} {\mathcal{A}}(\tau^- \to \ell^-_i \ell^-_j \ell^+_j) =
{\mathcal{A}}_{\gamma -penguin} + {\mathcal{A}}_{Z-penguin} +
{\mathcal{A}}_{box}.
\end{eqnarray}
In the following subsections, we present the results for these
contributions in terms of some convenient form factors.
\begin{figure}
\begin{center}
\vspace{3cm}
\input{fd1_LQ.tex}
\vspace{-8.2cm}
\caption{ Photon (a) and Z-penguin (b) and box (c) Feynman
diagrams contributing to $\ell^- \to \ell^-_i \ell^-_j \ell^+_j$,
${S}_k$ are the scalar leptoquark $k=1,2$, $u^c_a$ are type-up quark
with $a=1,2,3$. The (d) $(\mu-e)$ conversion Feynman diagram. }
\label{fig:tau3m-diagrams}
\end{center}
\end{figure}
\subsection{The $\gamma$-penguin contributions}
Diagrams in which a photon is exchanged are referred as
$\gamma$-penguin diagrams and are shown in
Figs.~\ref{fig:tau3m-diagrams}(a) and \ref{fig:tau3m-diagrams}(b) when
$V=\gamma$. The
amplitude of $\tau^-(p) \to \ell^-_i(p_1) \ell^-_j(p_2)
\ell^+_j(p_3)$ decays can be written as
\begin{eqnarray}
\label{ampgez} i{\mathcal{A}}_{\gamma- penguin}&=& \bar{u}(p_1)
\Big[ q^2 \gamma_\mu (T^L_1 P_L + T^R_1 P_R) + i m_{\tau}
\sigma_{\mu\nu} q^\nu (T^L_2 P_L + T^R_2 P_R)\Big]u(p) \\ \nonumber
&\times& \frac{e^2}{q^2} \bar{u}(p_2) \gamma^\mu v(p_3),
\end{eqnarray}
where $q$ is the photon momentum and $e$ is the electric charge.
The photon-penguin amplitude has two contributions, one from
Fig.~\ref{fig:tau3m-diagrams}(a) and the other from
Fig.~\ref{fig:tau3m-diagrams}(b) diagrams respectively as can be
seen from the structure of the form factors,
\begin{eqnarray}
T^{L,R}_{i} = T^{(a)L,R}_{i} + T^{(b)L,R}_{i}, \qquad {\rm i =
1,2}
\end{eqnarray}
\begin{eqnarray}
T^{(a)L}_{1} &=&-\frac{N_c Q_{(u^c)}}{16 \pi^2}\sum_{a=1}^{3} \sum_{k=1}^{2}\frac{1}{M^2_{S_k}} h^{\prime}_{a\tau} h^{\prime *}_{ai} \Gamma^{\dagger}_{S_R, k} \Gamma_{k,S_R} F_5 (x_{ka}),\\
T^{(a)L}_{2} &=& - \frac{N_c Q_{(u^c)}}{16 \pi^2}\sum_{a=1}^{3} \sum_{k=1}^{2}\frac{1}{M^2_{S_k}}\Bigg[ h_{a\tau} h^*_{ai} \Gamma^{\dagger}_{S_L,k} \Gamma_{k, S_L} F_1 (x_{ka}) + h^{\prime}_{a\tau} h^{\prime *}_{ai} \Gamma^{\dagger}_{S_R,k} \Gamma_{k, S_R} \frac{m_{i}}{m_{\tau}} F_1 (x_{ka}) \nonumber\\
&+& h^{\prime}_{a\tau} h^{*}_{ai} \Gamma^{\dagger}_{S_L, k} \Gamma_{k,S_R} \frac{m_{u_a}}{m_\tau}F_3 (x_{ka})\bigg]\\
T^{(a)R}_{i} &=& T^{(a)L}_{i} (h \leftrightarrow h', R
\leftrightarrow L).
\end{eqnarray}
and,
\begin{eqnarray}
T^{(b)L}_{1} &=&-\frac{N_c Q_S}{16 \pi^2}\sum_{a=1}^{3} \sum_{k=1}^{2}\frac{1}{M^2_{S_k}} h^{\prime}_{a\tau} h^{\prime *}_{ai} \Gamma^{\dagger}_{S_R, k} \Gamma_{k,S_R} F_6 (x_{ka}),\\
T^{(b)L}_{2} &=&\frac{N_c Q_S}{16 \pi^2}\sum_{a=1}^{3} \sum_{k=1}^{2}\frac{1}{M^2_{S_k}}\Bigg[ h_{a\tau} h^{*}_{ai} \Gamma^{\dagger}_{S_L,k} \Gamma_{k, S_L} F_2 (x_{ka}) + h^{\prime}_{a\tau} h^{\prime *}_{ai} \Gamma^{\dagger}_{S_R,k} \Gamma_{k, S_R} \frac{m_{i}}{m_{\tau}} F_2 (x_{ka}) \nonumber\\ &+& h^{\prime}_{a\tau} h^{*}_{ai} \Gamma^{\dagger}_{S_L,k}\Gamma_{k, S_R} \frac{m_{u_a}}{m_\tau}F_4 (x_{ka})\bigg]\\
T^{(b)R}_{i} &=& T^{(b)L}_{i} (h \leftrightarrow h', R
\leftrightarrow L).
\end{eqnarray}
where $x_{ka} = m^2_{u_a}/M^2_{S_k}$. Note that we have not
neglected any of the fermion masses. The analytical expressions for the
loop functions $F_{i}$ ($i = 1,...,6$) are given in appendix B.
\subsection{The $Z$-penguin contributions}
In addition to the photon penguin diagrams discussed in the
previous subsection, there are other types of penguin diagrams in
which the $Z$ boson is exchanged as shown in
Figs.~\ref{fig:tau3m-diagrams}(a)-\ref{fig:tau3m-diagrams}(b). The amplitude in this case
can be written as
\begin{eqnarray}
\label{ampge1} i{\mathcal{A}}_{Z- penguin}&=& \frac{i e^2}{m_Z^2
c^2_W s^2_W}\bar{u}(p_1) \gamma_\mu \big(Z^L P_{L} + Z^R P_R\big)
u(p)\\\nonumber &\times& \bar{u}(p_2) \gamma^\mu \big(g_{L} P_{L} + g_R
P_R\big) v(p_3) ,
\end{eqnarray}
As before, the coefficient $Z^{L(R)}$ can be written as a sum of
two terms from Feynman diagrams in
Fig.~\ref{fig:tau3m-diagrams}(a) and
Fig.~\ref{fig:tau3m-diagrams}(b):
\begin{eqnarray}
Z^{L,R} = Z^{(a)L,R} + Z^{(b)L,R}
\end{eqnarray}
where,
\begin{eqnarray}
\label{Zcou}
Z^{(a)L} &=&-\frac{N_c}{16 \pi^2}\sum_{a=1}^{3} \sum_{k=1}^{2}\frac{1}{M^2_{S_k}} h^{\prime}_{a\tau} h^{\prime *}_{ai} \Gamma^{\dagger}_{S_R,k} \Gamma_{k, S_R} \bigg[ 2 C_{R} F_8(x) - m^2_{u_a} C_{L} F_{7}(x_{ka})\bigg],\\
Z^{(a)R} &=& Z^{(a)L}(h^\prime \rightarrow h, R \leftrightarrow L).\\
Z^{(b)L} &=&-\frac{N_c}{16 \pi^2}\sum_{a=1}^{3} \sum_{k=1}^{2}\frac{1}{M^2_{S_k}} h^{\prime}_{a\tau} h^{\prime *}_{ai}\Gamma^{\dagger}_{S_R, k} \Gamma_{k, S_R} \bigg[ 2 Q_S \tan\theta_W \bigg]F_{8}(x_{ka}),\\
Z^{(b)R} &=& Z^{(b)L}(h^\prime \rightarrow h, R \leftrightarrow
L).
\end{eqnarray}
the coefficients $C_{L(R)}$ and $g_{L(R)}$ denote Z boson coupling
to charged leptoquark S and charged leptons $l_{L(R)}$,
respectively and they are given by
\begin{eqnarray}
g_{L(R)} &=& T_{3L(R)} - Q_{em} \sin^2\theta_W,\\
C_{L(R)} &=& T_{3L(R) (u^c)} - Q_{(u^c)} \sin^2\theta_W,
\end{eqnarray}
where $T_{3L(R)}$ and $Q_{em}$ represent weak isospin and electric
charge of $l_{L(R)}$, respectively. The loop functions $F_i$ (i=7,8) are presnted
in the appendix B.
\subsection{The box contribution}
The amplitude corresponding to the box-type diagram shown in Fig~.\ref{fig:tau3m-diagrams}(c)
can be expressed as,
\begin{eqnarray}
\label{ampge21} i{\mathcal{A}}_{box}\nonumber &=& B^L_1 [\bar{u}(p_1)
\gamma^\mu P_L u(p)][\bar{u}(p_2) \gamma_\mu P_L v(p_3)] + B^R_1
[\bar{u}(p_1) \gamma^\mu P_R u(p)][\bar{u}(p_2) \gamma_\mu P_R
v(p_3)] \\ \nonumber &+& B^L_2 [\bar{u}(p_1) \gamma^\mu P_L
u(p)][\bar{u}(p_2) \gamma_\mu P_R v(p_3)] + B^R_2 [\bar{u}(p_1)
\gamma^\mu P_R u(p)][\bar{u}(p_2) \gamma_\mu P_L v(p_3)]\\\nonumber&+&
B^L_3 [\bar{u}(p_1) P_L u(p)][\bar{u}(p_2) P_L u(p)] + B^R_3
[\bar{u}(p_1) P_R u(p)][\bar{u}(p_2) P_R v(p_3)]\\\nonumber &+& B^L_4
[\bar{u}(p_1) \sigma^{\mu \nu}P_L u(p)][\bar{u}(p_2) \sigma_{\mu
\nu} P_L v(p_3)] \\ &+& B^R_4 [\bar{u}(p_1) \sigma^{\mu \nu} P_R
u(p)][\bar{u}(p_2) \sigma_{\mu \nu} P_R v(p_3)].
\end{eqnarray}
where
\begin{eqnarray}
B^{L,R}_i = B^{(c)L,R}_i \qquad i = 1,...,4
\end{eqnarray}
with,
\begin{eqnarray}
B^{(c)L}_1 &=& \frac{N_c}{32\pi^2} \sum^3_{a,a'=1}\sum^{2}_{k,k'=1} \widetilde{D}_0 (m^2_{u_a},m^2_{u_{a'}}, m^2_{S_k}, m^2_{S_{k^\prime}}) h^{\prime}_{a\tau} h^{\prime}_{a'j}h^{\prime *}_{a i} h^{\prime *}_{a' j}| \Gamma^\dagger_{S_R,k} \Gamma_{k^\prime,S_R}|^2 ,\\
B^{(c)L}_2 &=& \frac{N_c}{64\pi^2} \sum^3_{a,a'=1}\sum^{2}_{k,k'=1} h^{\prime}_{a \tau}h_{a'j} \Gamma_{k,S_R} \Gamma_{k^\prime,S_L} \bigg[h^{*}_{a'j} h^{\prime *}_{ai}\Gamma^\dagger_{S_R,k'} \Gamma^\dagger_{S_L,k} \widetilde{D}_0 (m^2_{u_a},m^2_{u_{a'}}, m^2_{S_k}, m^2_{S_{k^\prime}}) \nonumber\\ && - m_{u_a} m_{u_{a'}} h^{\prime *}_{a^\prime j} h^{*}_{ai}\Gamma^\dagger_{S_R,k} \Gamma^\dagger_{S_L,k^\prime} D_0 (m^2_{u_a},m^2_{u_{a'}}, m^2_{S_k}, m^2_{S_{k^\prime}})\bigg],\\
B^{(c)L}_3 &=& \frac{N_c}{16\pi^2}
\sum^3_{a,a'=1}\sum^{2}_{k,k'=1} m_{u_a} m_{u_a'}
h'_{a'j}h'_{a\tau} h^{*}_{ai} h^{*}_{a'j} \Gamma^\dagger_{S_L,k'}
\Gamma_{k,S_R}\Gamma^\dagger_{S_L,k} \Gamma_{k',S_R}\nonumber\\ &&
\,\,\,\,\,\,\,\,\,\,\,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\times
D_0(m^2_{u_a},m^2_{u_{a'}}, m^2_{S_k}, m^2_{S_{k'}})
\\
B^{(c)L}_4 &=& 0,\\
B^{(c)R} &=& B^{(c)L}(h^\prime \leftrightarrow h, R
\leftrightarrow L).
\end{eqnarray}
Again the loop functions $D_0$ and $\widetilde{D}_0$ are given in the appendix B.\\
By collecting all the formulas, the Branching ratios of $\tau^- \to \ell^-_i \ell^-_j \ell^+_j$ can
be written in terms of the different form factors as
\begin{eqnarray}
\rm{Br}(\tau^- \to \ell^-_i \ell^-_j \ell^+_j)& = &\nonumber
\frac{\alpha^2 m^5_{\tau}}{32 \pi \Gamma_\tau} \Bigg[ |T^L_1|^2 +
|T^R_1|^2 + \frac{2}{3}\bigg(|T^L_2|^2 + |T^R_2|^2\bigg)
\bigg(8\log\bigg(\frac{m_{\tau}}{2m_i}\bigg) {- 11}\bigg)\\\nonumber
&-&2 (T^L_1 T^{R*}_2 + T^L_2 T^{R*}_1 + {\rm h.c}) + {\frac{1}{3
m^4_Z s^4_W c^4_W}}\bigg({2}\big(|Z^L g_L|^2 + |Z^R g_R|^2\big)
\\\nonumber &+& |Z^L g_R|^2 + |Z^R g_L|^2 \bigg) + \frac{1}{6}
\big(|B^L_1|^2 + |B^R_1|^2) + \frac{1}{3} \big(|B^L_2|^2 +
|B^R_2|^2)
\\\nonumber &+& \frac{1}{24} \big(|B^L_3|^2 + |B^R_3|^2) + \frac{1}{3}\big( T^L_1 B^{L*}_1 + T^L_1 B^{L*}_2 + T^R_1 B^{R*}_1 + T^R_1 B^{R*}_2 +{\rm h.c}\big) \\\nonumber &-& \frac{2}{3}\big( T^R_2 B^{L*}_1 + T^L_2 B^{R*}_1 + T^L_2 B^{R*}_2 + T^R_2 B^{L*}_2 +{\rm h.c}\big)\\\nonumber &+& \frac{1}{3}\big( B^L_1 Z^{*}_L g_L + B^R_1 Z^{*}_R g_R + B^L_2 Z^{*}_L g_R + B^R_2 Z^{*}_R g_L + {\rm h.c}\big)\\\nonumber &+& \frac{1}{3}\big[ 2( T^L_1 Z^{*}_L g_L + T^R_1 Z^{*}_R g_R) + T^L_1 Z^{*}_L g_R + T^R_1 Z^{*}_R g_L + {\rm h.c}\big]\\ &+& \frac{1}{3}\big[ -4( T^R_2 Z^{*}_L g_L + T^L_2 Z^{*}_R g_R) {-2 (T^L_2 Z^{*}_R g_L + T^R_2 Z^{*}_L g_R + {\rm h.c})}\big] \Bigg]
\end{eqnarray}
where $\Gamma_\tau$ is the total decay width of $\tau$. All the form factors are real.
\section{$\mu -e $ conversion }
$\mu-e$ conversion in the
muonic atoms is one of the interesting charged LFV process that
can occur in many candidates of physics beyond the SM. Accurate
calculation of the $\mu- e$ conversion rate is essential to
compare the sensitivity to the LFV interactions in different
nuclei~\cite{Kitano:2002mt}.
In this section, we discuss the constraints that can be imposed on
the scalar leptoquark couplings using $\mu-e$ conversion
rate. The dominant contribution to the $\mu- e$ conversion rate
is obtained through considering the tree diagram shown in
Fig.~\ref{fig:tau3m-diagrams}(d) which leads to the effective Lagrangian
\begin{eqnarray}
\mathcal L_{eff}^{(u_a)} & = & \sum_{a=1}^{3} \sum_{k=1}^{2}
-\frac{1}{M^2_{S_k}}\Bigg[ \frac{1}{2} h_{a2} h^{*}_{a1}
\Gamma^{\dagger}_{S_L,k} \Gamma_{k,S_L} (\bar{e}\mathcal
\gamma^{\mu}P_L \mu )(\bar{u}_a\mathcal \gamma _{\mu}P_L u_a )
\nonumber\\&+&\frac{1}{8} h_{a2} h^{'*}_{a1}
\Gamma^{\dagger}_{S_R,k} \Gamma_{k,S_L} (\bar{e}\sigma^{\mu\nu}P_R
\mu )(\bar{u}_a \sigma_{\mu\nu}P_R u_a) \nonumber\\&-&\frac{1}{2}
h_{a2} h^{'*}_{a1} \Gamma^{\dagger}_{S_R,k} \Gamma_{k,S_L}
(\bar{e}P_R \mu )(\bar{u}_a P_R u_a )+ (h \leftrightarrow h', R
\leftrightarrow L)\Bigg], \end{eqnarray}
where we have used Fierz transformation for chiral fermions.
$P_{R,L}=(1\pm \gamma^5) / 2$, $u_{a}$ are light and heavy type-up
quarks and $\sigma$ matrix is defined by
$\sigma^{\mu\nu}=\frac{i}{2}[\gamma^{\mu},\gamma^{\nu}]$. The
operators involving $\bar{u}_a \gamma_\mu \gamma_5 u_a$,
$\bar{u}_a \gamma_5 u_a$, or $\bar{u}_a \sigma_{\mu \nu} u_a$
do not contribute to the coherent conversion processes and thus we
can drop them and write
\begin{eqnarray}
\mathcal L_{eff}^{(u_a)} & = & \sum_{a=1}^{3} \ \Bigg[ \left(
C^{(u_a)}_{VR} \; \bar{e}\mathcal \gamma^{\mu}P_R \mu +
C^{(u_a)}_{VL} \; \bar{e}\mathcal \gamma^{\mu} P_L \mu \right)
\bar{u}_a\gamma_{\mu}u_a
\nonumber\\
&+& \left( C^{(u_a)}_{SR} \; \bar{e} P_L \mu + C^{(u_a)}_{SL}
\; \bar{e} P_R \mu \right) \bar{u}_a u_a ~ \Bigg] ~.
\label{eq:weakscaleL}
\end{eqnarray}
where we have defined \begin{eqnarray} C^{(u_a)}_{VR}&=&- h_{a2}
h^*_{a1}\sum_{k}\frac{1}{2
M^2_{S_K}} \Gamma^{\dagger}_{S_L,k} \Gamma_{k,S_L}\nonumber\\
C^{(u_a)}_{SR}&=& \frac{1}{2}h_{a2} h^{'*}_{a1} \sum_{k}\frac{1}{
M^2_{S_K}} \Gamma^{\dagger}_{S_R,k}
\Gamma_{k,S_L}\label{coeff}\end{eqnarray}
$C^{(u_a)}_{VL}$ and $C^{(u_a)}_{SL}$ can be obtained by the
the exchange $h \leftrightarrow h', R \leftrightarrow L$ in
Eq.(\ref{coeff}).
The next step for the calculation of $\mu-e$ conversion is
to match the Lagrangian in Eq.(\ref{eq:weakscaleL}) to the
Lagrangian at the nucleon level.
Hence we integrate out the
heavy quarks~\cite{Cirigliano:2009bz} and so the effective
Lagrangian in Eq.(\ref{eq:weakscaleL}) becomes
\begin{eqnarray}\label{lagrangian}
\mathcal L_{\rm eff}^{(u)}&=&
\left( C^{(u)}_{VR} \; \bar{e}\mathcal
\gamma^{\mu}P_R \mu +
C^{(u)}_{VL} \; \bar{e}\mathcal \gamma^{\mu} P_L \mu \right)
\bar{u}\gamma_{\mu}u
\nonumber\\
&+&
\left( C^{(u)}_{SR} \; \bar{e} P_L \mu
+ C^{(u)}_{SL} \; \bar{e} P_R \mu \right)
\bar{u}u.
\end{eqnarray}
Then, the effective Lagrangian (\ref{lagrangian}) is matched
to the nucleon level Lagrangian~\cite{Kosmas:2001mv} through
the following replacements of the operators~\cite{Kitano:2002mt,Cirigliano:2009bz}:
\begin{eqnarray}
\bar{u}u &\rightarrow& G_{S}^{(u,N)}
{\bar{\psi}}_N {\psi}_N
\nonumber\\
\bar{u}\gamma_{\mu} u &\rightarrow& f_{VN}^{(u)} \;
{\bar{\psi}}_N \gamma_{\mu} {\psi}_N \;, \label{eq:replace}
\end{eqnarray}
where $N$ represents each nucleon ($N=p,n$), $\psi_N$ are the
nucleon fields, and $G,f$ are given by~\cite{Kitano:2002mt,Cirigliano:2009bz}
\begin{equation}
f_{Vp}^{(u)}=2,\,\,\,\,\,\,\, \,\,\,\, f_{Vn}^{(u)}=1,\,\,\,\,\,
\,\,G_{S}^{(u,p)}= 5.1,\,\,\,\, \,\,G_{S}^{(u,n)}= 4.3
\end{equation}
Finally, the Lagrangian at nucleon level can be written as
\begin{eqnarray}
\mathcal L_{eff}^{(N)}&=& \sum_{N=p,n} \Bigg[ \left(
\tilde{C}^{(N)}_{VR} \; \bar{e} \gamma^{\mu} P_R \mu +
\tilde{C}^{(N)}_{VL} \; \bar{e} \gamma^{\mu} P_L \mu \right) \
\bar{\psi}_N \gamma_{\mu} \psi_N
\nonumber\\
&+&
\, \left( \tilde{C}^{(N)}_{SR} \; \bar{e} P_L
\mu + \tilde{C}^{(N)}_{SL} \; \bar{e} P_R \mu \right) \;
\bar{\psi}_N \psi_N + h.c.\Bigg] \; .
\end{eqnarray}
where we have introduced the following redefinitions for the vector quantities:
\begin{eqnarray}
\tilde{C}^{(p)}_{VR} &=& C^{(u)}_{VR} \; f^{(u)}_{Vp} \\
\tilde{C}^{(n)}_{VR} &=& C^{(u)}_{VR} \; f^{(u)}_{Vn} \\
\tilde{C}^{(p)}_{VL} &=& C^{(u)}_{VL} \; f^{(u)}_{Vp} \\
\tilde{C}^{(n)}_{VL} &=& C^{(u)}_{VL} \; f^{(u)}_{Vn} ~,
\end{eqnarray}
while the scalar ones read:
\begin{eqnarray}
\tilde{C}^{(p)}_{SR} &=& C^{(u)}_{SR} \; G^{(u,p)}_{S}
\\
\tilde{C}^{(n)}_{SR} &=& C^{(u)}_{SR} \; G^{(u,n)}_{S}
\\
\tilde{C}^{(p)}_{SL} &=& C^{(u)}_{SL} \;
G^{(u,p)}_{S}\\
\tilde{C}^{(n)}_{SL} &=& C^{(u)}_{SL} \; G^{(u,n)}_{S} \ .
\end{eqnarray}
In order to calculate the $\mu - e$ conversion amplitude we need
to calculate the matrix elements of $\bar{\psi}_N \psi_N $ and
$\bar{\psi}_N \gamma_{\mu}\psi_N $ of the transition between the
initial and the final states of nucleus~\cite{Kitano:2002mt,Cirigliano:2009bz}:
\begin{eqnarray}
\langle A,Z|\bar{\psi}_p\psi_p|A,Z\rangle&=&Z\rho^{(p)} \nonumber\\
\langle A,Z|\bar{\psi}_n\psi_n|A,Z\rangle&=&(A-Z)\rho^{(n)} \nonumber\\
\langle A,Z|\bar{\psi}_p\gamma^0\psi_p|A,Z\rangle &=&Z\rho^{(p)} \nonumber\\
\langle A,Z|\bar{\psi}_n\gamma^0\psi_n|A,Z\rangle &=&(A-Z)\rho^{(n)} \nonumber\\
\langle A,Z|\bar{\psi}_N\gamma^i\psi_N|A, Z \rangle &=&0 \; .
\end{eqnarray}
where $|A,Z\rangle$ represents the nuclear ground state, with $A$
and $Z$ are the mass and atomic number of the isotope
respectively, while $\rho^{(p)}$ and $\rho^{(n)}$ are the proton
and neutron densities respectively.
Finally, the $\mu-e $
conversion rate is given by~\cite{Cirigliano:2009bz}:
\begin{eqnarray}
\Gamma_{conv} &=& \frac{m_{\mu}^{5}}{4} \left| 4 \left(
\tilde{C}^{(p)}_{SR} S^{(p)} + \tilde{C}^{(n)}_{SR} \; S^{(n)}
\right) + 4\tilde{C}^{(p)}_{VR} \; V^{(p)} + 4\tilde{C}^{(n)}_{VR}
\;V^{(n)} \right|^2
\nonumber \\
&+& \frac{m_{\mu}^{5}}{4} \left| 4 \left( \tilde{C}^{(p)}_{SL}
S^{(p)} + \tilde{C}^{(n)}_{SL} \; S^{(n)} \right) +
4\tilde{C}^{(p)}_{VL} \;V^{(p)} + 4\tilde{C}^{(n)}_{VL} \;
V^{(n)} \right|^2
\end{eqnarray}
where $ V^{(N)}, S^{(N)}$ are dimensionless integrals
representing the overlap of electron and muon wave functions
weighted by appropriate combinations of protons and neutron
densities~\cite{Kitano:2002mt}. For phenomenological applications,
it is useful to normalize the conversion rate to the muon capture
rate through the quantity:
\begin{equation}
B_{\mu- e} (Z) \equiv \frac{ \Gamma_{conv}(Z,A)
}{\Gamma_{capt} (Z,A) }~.
\end{equation}
The current bounds on $B_{\mu- e} $ for Titanium atom and Gold atom obtained by
SINDRUM collaboration are respectively $B_{\mu- e} (Ti)< 4.3\times 10^{-12}$~\cite{Dohmen:1993mp}, $B_{\mu- e} (Au)< 7\times 10^{-13}$
~\cite{Bertl:2006up} both at 90\%CL.
The numerical values of $ V^{(N)}, S^{(N)}$
and $\Gamma_{capt}$ for Titanium and Gold atoms are listed
In Table~\ref{TabTiAuData}.
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
Nucleus & $S^{(p)} [m_\mu^{5/2}]$ & $S^{(n)} [m_\mu^{5/2}]$
& $V^{(p)} [m_\mu^{5/2}]$& $V^{(n)} [m_\mu^{5/2}]$& $\Gamma_{capture}[10^6 s^{-1}]$ \\
\hline
$\text{Ti}^{48}_{22}$ & 0.0368 & 0.0435 & 0.0396 & 0.0468& 2.59 \\
$\text{Au}^{197}_{79}$& 0.0614 & 0.0918 & 0.0974 & 0.146 & 13.07 \\
\hline
\end{tabular}
\end{center}
\caption{Data taken from Tables I and VIII of \cite{Kitano:2002mt}.}
\label{TabTiAuData}
\end{table}
\section{Numerical results and discussion}
Let us now proceed to analyse and discuss our numerical results.
The quark masses are evaluated at the energy scale $\mu =
300$~GeV~\cite{Gray:1990}, which is the typical leptoquark mass
scale used in this work,
\begin{eqnarray}
m_t = 161.4 \,{\rm GeV}, \quad m_c = 0.55\, {\rm GeV},\quad
m_u = 11.4 \times 10^{-3}\,{\rm GeV},
\end{eqnarray}
while we use the following values for ~\cite{pdg}
\begin{eqnarray}
\alpha_{em} = 1/137.0359, \quad M_W = 80.45 \,{\rm GeV} , \quad
M_Z = 91.1875 \,{\rm GeV}.
\end{eqnarray}
We assume as in Ref.~\cite{Davidson:1993qk}, that all the couplings $h$ and $h'$ are real and equal to each other~\cite{CHH1999},
\begin{eqnarray}
h = h^\prime = h^*.
\end{eqnarray}
We use leptoquark mass splitting $\Delta= 500$ GeV in our
analysis, where $\Delta$ is defined as $\sqrt{M^2_{S_2} -
M^2_{S_1}}$. Consequently, the remaining parameters in the
leptoquark model are the mass of the light scalar leptoquark
$M_{S_1}$, the mixing angle $\theta_{LQ}$, and the couplings
$h_{a\ell}$ (a =u, c and t).
Also, we assume that the scalar leptoquark may
explain the discrepancy $\Delta a_\mu$ between the experimentally
measured muon $(g-2)_\mu$ and its SM prediction Eq.~(\ref{gm2})
and hence this condition restricts the possible range of the
parameters. We have performed a scan over all the input
parameters, $M_{S_1} \le 1500$ GeV, $-1 \le \sin\theta_{LQ} \le 1$
and $\alpha_{em} \le |h_{q\mu}|^2 \le 1 $. Then, after imposing
all the existing constraints arising from $\pi$ leptonic
decays and direct search, we select all sets of the input
parameters producing the same values for $(g-2)_\mu$ at 1$\sigma$
range of data.
\begin{figure}
\begin{picture}(320,260)
\put(-60,0){\mbox{\psfig{file=Figgm2_2.eps,height=3.4in,width=3.2in}}}
\put(33,-10){\makebox(0,0)[bl]{\large{$M_{S_1}$ (GeV)}}}
\put(-60,115){\makebox(0,0)[bl]{\rotatebox{90}{\large{$|h_{a\mu}|^2$
}}}}
\put(-27,170){\makebox(0,0)[bl]{\rotatebox{70}{{Charm-quark
}}}} \put(30,120){\makebox(0,0)[bl]{\rotatebox{45}{{Top-quark}}}}
\put(160,0){\mbox{\psfig{file=Figgm2_1.eps,height=3.4in,width=3.2in}}}
\put(165,115){\makebox(0,0)[bl]{\rotatebox{90}{\large{$\sin2\theta_{LQ}$
}}}} \put(253,-10){\makebox(0,0)[bl]{\large{$M_{S_1}$(GeV)}}}
\put(226,30){\makebox(0,0)[bl]{{Top-quark}}}
\put(210,150){\makebox(0,0)[bl]{\rotatebox{90}{{Charm-quark
}}}}
\end{picture}
\vspace{0.6cm} \caption{ The allowed regions on the
($M_{S_1}-|h_{a\mu}|^2$) plane (left) and on the
($M_{S_1}-\sin2\theta_{LQ}$) plane (right) for top-quark (red color) and
charm-quark (green color) contributions,
taking into account $a^{LQ}_\mu = \Delta a_{\mu}$
at 1$\sigma$.} \label{fig-gm2}
\end{figure}
In Fig.~\ref{fig-gm2}, we show the allowed regions for different
type-up quarks contributions which are compatible with $a^{LQ}_\mu =
\Delta a_{\mu}$ at 1$\sigma$ range of data where the red color (green color)
region corresponds to top-quark (charm-quark) contribution respectively.
As can be seen from the left panel of Fig.~\ref{fig-gm2},
the dominant contribution is
around $\sin2\theta_{LQ} \sim 0.7$ for both top and charm quarks.
In addition, we find that, sizeable scalar leptoquark effects to
the $(g-2)_\mu$ at 1$\sigma$ are obtained for the values of $M_{S_1}$
which satisfy $M_{S_1} \approx 1$ TeV for top quark contribution and
$M_{S_1} \approx 400$ GeV for charm quark contribution. We note also
that it is not possible to use the up quark loop contributions
alone for LQ, since the couplings $|h_{u\mu}|$ are strongly
constrained by the $\pi$ leptonic decays. It has been found that the
LHC has the potential to discover light scalar LQ with a mass up to
1.2 TeV and where the Yukawa coupling are equal to the electromagnetic
coupling \cite{Belyaev:2005ew}.
In order to find the constraints on the combination of LQ
couplings we require that each individual LQ coupling
contribution to the branching ratio does not exceed the
experimental current limits on the
Br($\ell\to\ell^{\prime}\gamma$) ~(\ref{lllm}) and $\mu-e$
conversion in nuclei. The latter process is used to set the
strongest constraints on the product $h_{u\mu}h_{ue}$ which
involve the first generation, since, in this case, the process is
induced at tree-level. On the other hand, the
$\ell\to\ell^{\prime}\gamma$ decays which are induced at one-loop
by the photon-penguins, Z-penguins and box diagrams [see
Figs.~\ref{fig:tau3m-diagrams}(a)-\ref{fig:tau3m-diagrams}(c)],
allow us to constrain the complementary combinations of the
couplings involving the second and third generation of quarks,
namely $h_{a\ell}h_{a\ell^\prime}$, where a = c, t. The $\mu-e$
conversion process can be also used to set constraints on the
second and third quark generations.
However, we stress that the bounds from the $\mu-e$ conversion
are suffering from being model-dependent due to the non perturbative
calculations of the nuclear form factors, while the bounds which are obtained
from the $\ell\to\ell^{\prime}\gamma$ decays are not. Taking
into account $(g-2)_\mu$ constraint, and experimental upper limits on
the $(\mu-e)_{Ti, Au}$ conversion
rates~\cite{Dohmen:1993mp,Bertl:2006up} at 90\%CL., we obtain the
following upper bounds
\begin{equation}
h_{u\mu}h_{ue} \leq 4.38\times 10^{-6} \label{mue11} \end{equation}
for Titanium atom while for Gold atom the bound reads
\begin{equation}
h_{u\mu}h_{ue} \leq 6.25\times 10^{-6} \label{mue22}
\end{equation}
Clearly, the bounds obtained for both Titanium and Gold atoms are
of the same order and severely constraint the product of
the leptoquark couplings $h_{u\mu}h_{u\mu}$. Our results are
consistent with the effective Hamiltonian and approximation used
in Ref.~\cite{Davidson:1993qk}. The products of the couplings
$h_{a\ell}h_{a\ell^\prime}$ where a=c,t and $\ell,\ell^\prime =
\tau, \mu, e$, appear only at the one-loop level contribution to
the $\tau\to (\mu,e) \gamma$ and $\mu\to e\gamma$ decays.
Therefore, they could be larger if they are compared with
$h_{u\mu}h_{ue}$ ones without violating the experimental upper
limits on the branching ratios. The bounds obtained on these
combinations of couplings in the muon sector are given by
\begin{equation}
h_{c\mu}h_{ce} \leq 1.22\times 10^{-3}, \quad
h_{t\mu}h_{te} \leq 5.73\times 10^{-3}
\end{equation}
while for the tau sector they become
\begin{eqnarray}
h_{c\tau}h_{c\mu} &\leq& 4.78\times 10^{-3}, \quad
h_{t\tau}h_{t\mu} \leq 8.13\times 10^{-3},\\
h_{c\tau}h_{ce} &\leq& 7.8\times 10^{-1}, \quad h_{t\tau}h_{te}
\leq 8.0\times 10^{-1},
\end{eqnarray}
In the following studies of the $\tau$ and $\mu$ LFV processes,
we will use the parameter
space which is discussed above.
We start by investigating the LFV $\tau$
and $\mu$ decay processes generated by the same LQ-scalar
interactions as those of $(g-2)_\mu$. At one-loop level, the
LQ-scalar gives contributions to the $\tau^- \to \mu^-
\mu^-\mu^+$, $\tau^- \to e^- e^- e^+$,
by means of the so-called $\gamma$- and Z- penguins and box diagrams.
\begin{figure}[H]
\begin{picture}(320,260)
\put(10,0){\mbox{\psfig{file=Figtmg1.eps,height=3.4in,width=3.2in}}}
\put(173,210){\makebox(0,0)[bl]{\large{ \bf(a)}}}
\put(103,-10){\makebox(0,0)[bl]{\large{$M_{S_1}$ (GeV)}}}
\put(65,203){\makebox(0,0)[bl]{\large{Current limit}}}
\put(10,88){\makebox(0,0)[bl]{\rotatebox{90}{\large{Br$(\tau\to
\mu\mu\bar{\mu})$ }}}}
\put(70,160){\makebox(0,0)[bl]{{Top-quark}}}
\put(55,20){\makebox(0,0)[bl]{\rotatebox{90}{{Charm-quark}}}}
\put(240,0){\mbox{\psfig{file=Figteg2.eps,height=3.4in,width=3.2in}}}
\put(404,210){\makebox(0,0)[bl]{\large{ \bf(b)}}}
\put(333,-10){\makebox(0,0)[bl]{\large{$M_{S_1}$(GeV)}}}
\put(240,100){\makebox(0,0)[bl]{\rotatebox{90}{\large{Br$(\tau\to
ee\bar{e})$ }}}}
\put(300,206){\makebox(0,0)[bl]{\large{Current limit}}}
\put(326,170){\makebox(0,0)[bl]{{Top-quark}}}
\put(285,20){\makebox(0,0)[bl]{\rotatebox{90}{{Charm-quark}}}}
\put(120,-290){\mbox{\psfig{file=Figmeg2.eps,height=3.4in,width=3.2in}}}
\put(210,-300){\makebox(0,0)[bl]{\large{$M_{S_1}$ (GeV)}}}
\put(110,-200){\makebox(0,0)[bl]{\rotatebox{90}{\large{Br$(\mu\to
ee\bar{e})$ }}}}
\put(295,-73){\makebox(0,0)[bl]{\large{ \bf(c)}}}
\put(184,-78){\makebox(0,0)[bl]{\large{Current limit}}}
\put(187,-210){\makebox(0,0)[bl]{{Top-quark}}}
\put(165,-270){\makebox(0,0)[bl]{\rotatebox{90}{{Charm-quark}}}}
\end{picture}
\vspace{11cm} \caption{ Scatter plots of (a) Br($\tau \to
\mu\mu\bar{\mu}$), (b) Br($\tau \to ee\bar{e}$) and (c) Br($\mu
\to ee\bar{e}$) as a function of light leptoquark mass $M_{S_1}$ for
top-quark (red color) and charm-quark (green color) contributions.
The horizontal lines of each plot
are the current limits of $\tau$ and $\mu$ LFV decay branching ratios.
}
\label{fig-tmg}
\end{figure}
\begin{figure}[H]
\begin{picture}(320,260)
\put(10,0){\mbox{\psfig{file=Figtmg2.eps,height=3.4in,width=3.2in}}}
\put(165,215){\makebox(0,0)[bl]{\large{\bf(a)}}}
\put(90,-10){\makebox(0,0)[bl]{\large{Br$(\tau \to \mu\gamma)$}}}
\put(57,203){\makebox(0,0)[bl]{\large{Current limit}}}
\put(10,100){\makebox(0,0)[bl]{\rotatebox{90}{\large{Br$(\tau\to
\mu\mu\bar{\mu})$ }}}}
\put(60,130){\makebox(0,0)[bl]{{Top-quark}}}
\put(155,20){\makebox(0,0)[bl]{\rotatebox{45}{{Charm-quark}}}}
\put(240,0){\mbox{\psfig{file=Figteg1.eps,height=3.4in,width=3.2in}}}
\put(343,-10){\makebox(0,0)[bl]{\large{Br$(\tau\to e \gamma)$}}}
\put(390,217){\makebox(0,0)[bl]{\large{\bf(b)}}}
\put(285,206){\makebox(0,0)[bl]{\large{Current limit}}}
\put(290,150){\makebox(0,0)[bl]{{Top-quark}}}
\put(370,20){\makebox(0,0)[bl]{\rotatebox{45}{{Charm-quark}}}}
\put(238,100){\makebox(0,0)[bl]{\rotatebox{90}{\large{Br$(\tau\to
ee\bar{e})$ }}}}
\put(120,-290){\mbox{\psfig{file=Figmeg1.eps,height=3.4in,width=3.2in}}}
\put(203,-300){\makebox(0,0)[bl]{\large{Br$(\mu\to e \gamma)$}}}
\put(106,-200){\makebox(0,0)[bl]{\rotatebox{90}{\large{Br$(\mu\to
ee\bar{e})$ }}}}
\put(165,-80){\makebox(0,0)[bl]{\large{Current limit}}}
\put(265,-73){\makebox(0,0)[bl]{\large{\bf(c)}}}
\put(215,-210){\makebox(0,0)[bl]{{Top-quark}}}
\put(230,-274){\makebox(0,0)[bl]{\rotatebox{23}{{Charm-quark}}}}
\end{picture}
\vspace{11cm} \caption{
Correlations between (a) Br$(\tau\to \mu\mu\bar{\mu})$ and Br$(\tau\to \mu\gamma)$, (b) Br$(\tau\to ee\bar{e})$ and Br$(\tau\to e\gamma)$, (c) Br$(\mu\to ee\bar{e})$ and Br$(\mu\to e\gamma)$. for
top-quark (red color) and charm-quark (green color) contributions
The vertical and horizontal lines correspond to the upper limits
of $\tau$ and $\mu$ LFV decay branching ratios.}
\label{fig-teg}
\end{figure}
In Figs~\ref{fig-tmg}, we present our predictions for the branching
ratios of ($\tau \to \mu\mu\bar{\mu}$) (a) and ($\tau \to
ee\bar{e}$) (b) as a function of light LQ mass $M_{S_1}$ for
top-quark (red color) and charm-quark (green color) contributions.
These plots have origin in $M_{S_1} = 300$ GeV which roughly corresponds to
the exclusion limit obtained at HERA~\cite{Wang:2004cj} for leptoquark masses
with couplings of electromagnetic strenght. As we can
see the main contribution comes from the top-quark contribution
and can reach 2.47 $\times 10^{-8}$ for {\rm Br}($\tau \to 3\mu$)
and 5.66 $\times 10^{-8}$ for {\rm Br}($\tau \to 3e$)
which are comparable with the
present bounds. We find that the main contribution to
$\tau \to ee\bar{e}$ and $\tau \to \mu\mu\bar{\mu}$ decays is produced from
the photon-penguins diagrams which were not taken into account in
Ref.~\cite{Davidson:1993qk}. In fact, for large LQ mass $(m_q \ll
M_{S_1} )$, the photon-penguins are proportional to
$h^2\log(m_q/M_{S_1})/M^2_{S_1}$ which were known as log
enhancement in the literature~\cite{Kuno:1999jp}. On the other
hand, the naive expectation of Z-penguin and box diagrams leads to
that they are of orders ${\cal{O}}(h^2 m^2_q/M^4_{S_1})$ and
${\cal{O}}(h^4 m^2_q/M^4_{S_1})$, respectively.
The same considerations regarding log enhancements hold for the
$\tau\to e \mu^- \mu^+$ and $\tau\to \mu e^-e^+$ processes.
However, the uppper limits on the Brs of $\tau\to e \mu^- \mu^+$ and $\tau\to \mu e^-e^+$ could be of ${\cal{O}}(10^{-8})$ and the order in size
is
{\rm Br}($\tau \to 3e$) $>$
{\rm Br}($\tau\to \mu^- e^-e^+$) $>$
{\rm Br}($\tau\to e^- \mu^- \mu^+$) $>$
{\rm Br}($\tau \to 3\mu$). Since, $\tau\to e^- e^- \mu^+$ and $\tau\to \mu^- \mu^- e^+$ are induced by box diagrams then they are expected to be small.
On the contrary, since the
current bound on the $\mu \to e \gamma$ decay imposes very strong
constraints on the related couplings, the predicted {\rm Br}($\mu
\to 3e$) is rather too small to be observed.
In Fig.~\ref{fig-teg}, we show the correlations between Br($\tau
\to 3\mu$) and Br($\tau \to \mu\gamma$) in the upper left panel,
(b) Br($\tau \to 3e$) and Br($\tau \to e\gamma$) in the upper
right panel, and (c) Br($\mu \to 3e$) and Br($\mu \to e\gamma$) in
the lower panel. We observe that it is possible to accommodate
both $\tau \to 3\ell$ and $\tau \to \ell\gamma$ branching ratios
for certain choices of LQ parameters. This leads to simple
correlation like
\begin{eqnarray}
\frac{Br(\tau\to3\ell)}{Br(\tau\to \ell\gamma)} \approx {\cal{O}}(10^{-1}), \qquad \frac{Br(\mu\to 3e)}{Br(\mu\to e\gamma)} \approx {\cal{O}}(10^{-3})
\end{eqnarray}
for top-quark contribution, which is in agreement with the ratio expected by the dominance of the Penguin-type Fig.\ref{fig:tau3m-diagrams}(a)-(b).
\section{Conclusion}
We have studied the muon anomalous magnetic moment, lepton flavor
violating muon and tau decays $\ell \to \ell_i \ell_j
\bar{\ell}_j$ and $\ell\to \ell^{\prime} \gamma$ that are generated by scalar LQ
interactions. We have found that scalar LQ can explain the
discrepancy between the experimental value of $(g-2)_\mu$ and its
standard model prediction without any contradictions with the
experimental bound of LFV tau decay processes. The present
experimental limits are used to constrain the leptoquark parameter
space. We set equal couplings
and obtain the upper limits
of the different product of leptoquark couplings by confronting
LFV observable with experimental results.
Our prediction
is that $\tau\to 3\mu, 3e, e2\mu $ and $\tau\to \mu 2e$
get the leading contributions from the so-called photon-penguin diagrams and could be of ${\cal{O}}
(10^{-8})$ which can be accessible by the presents
experiments and the future linear colliders, such as ILC. On the
contrary, the current bounds on LFV impose very strong constraints
on the Br$(\mu\to ee\bar{e})$ and the ratio is too small to be
observed in the near future. Hence any observation of LFV
processes in the charged lepton sector, which are being probed
with ever increasing sensitivity, would unambiguously point to
non-standard interactions. Indeed, such indirect observations taken in isolation may not imply much on the exact nature of new
physics. But a study of possible correlations of its effects on
different independently measured charged LFV observable might
provide a powerful cross-check and lead to identification of new
physics through LHC/LFV synergy.
\section*{ACKNOWLEDGEMENTS}
We would like to thank the Abdus Salam International Centre for Theoretical
Physics (ICTP) for good hospitality and acknowledge the considerable help of
the High Energy Section. This work was done at the high energy
section within the framework of the associate Scheme.
We would like also thank Chuan-Hung Chen for useful discussions and comments.
R.B. was supported by National Cheng Kung University
Grant No. HUA 97-03-02-063.
|
{
"timestamp": "2010-09-21T02:03:58",
"yymm": "1009",
"arxiv_id": "1009.3886",
"language": "en",
"url": "https://arxiv.org/abs/1009.3886"
}
|
\section*{Author Summary}
The understanding of the effective functionality that governs the enzymatic self-organized processes in cellular conditions is a crucial topic in the post-genomic era. A number of measures have been proposed for the functionality and correlations between biochemical time series. However, functional correlations do not imply effective connectivity and most synchronization measures do not distinguish between causal and non-causal interactions. In recent studies, Transfer Entropy (TE) has been proposed as a rigorous, robust and self-consistent method for the causal quantification of the functional information flow among nonlinear processes. Here, we have used TE to establish the effective functional connectivity of yeast glycolysis under dissipative conditions. Concretely, we have applied this method for a quantification of how much the temporal evolution of the activity of one enzyme helps to improve the future prediction of another. In the enzymatic activities, the oscillatory patterns of the metabolic products might have causal information which can be appropriately read-out by the TE. We have performed numerical studies of yeast glycolysis under dissipative conditions and found the emergence of a new kind of dynamical functional structure, characterized by changing connectivity flows and a metabolic invariant that constrains the activity of the irreversible enzymes.
\newpage
\section*{Introduction}
Yeast glycolysis is one of the most studied dissipative pathways of the cell; it was the first metabolic system in which spontaneous oscillations were observed \cite{duysens1957,chance1964}, and the study of these rhythms allowed the construction of the first dynamic model where the kinetics of an enzyme was explicitly considered \cite{goldbeter1972,goldbeter1973}. More concretely, the main instability-generating mechanism in the yeast glycolysis is based on the self-catalytic regulation of the enzyme phosphofructokinase \cite{goldbeter1972,boiteux1975,goldbeter2007}.
Glycolysis is the central pathway of glucose degradation which is implied in relevant metabolic processes, such as the maintenance of cellular redox states, the provision of ATP for membrane pumps and protein phosphorylation, biosynthesis, etc; and its activity is linked to a high variety of important cellular processes, e.g., glycolysis has a long history in cancer cell biology \cite{bagheri2006} and cell proliferation \cite{almeidaa2010}, there is a correlation between brain aerobic glycolysis and Amyloid-$\beta$ plaque deposition which might precede the clinical manifestations of the Alzheimer disease \cite{vlassenkoa2010}, the glycolytic inhibition abrogates epileptogenesis \cite{garriga2006}, and glycolysis is also related with oxidative stress \cite{colussi2006} and apoptosis \cite{nika2003}.
Over the last 30 years a large number of different studies focused on different molecular mechanisms allowing for the emergency of self-organized glycolytic patterns \cite{termonia1981,dano1999,wolf2000,reijenga2002,madsen2005,olsen2009}. Nevertheless, despite the intense advance of the knowledge of these metabolic structure, we still lack a quantitative description in cellular conditions of the effective functional structure and the causal effects among the enzymes.
In this paper, to go a next step further in the understanding of the relationship between the classical topological structure and functionality we have analyzed the effective connectivity of yeast glycolysis, which in inter-enzyme interactions accounts for the influence that the activity of one enzyme has on the future of another \cite{gerstein1969,friston1994,fujita2007,mukhopadhyay2007,pahle2008}.
For this purpose, we considered a yeast glycolytic model described by a system of three delay-differential equations in which there is an explicit consideration of the speed functions of the three irreversible enzymes hexokinase, phosphofructokinase and pyruvatekinase. These enzymatic activity functions were previously modeled and tested experimentally by other different groups \cite{viola1982,goldbeter1972,markus1980}.
We have obtained time series of enzymatic activity under different sources of the glucose input flux. The data corresponded to a typical quasi-periodic route to chaos which is in agreement with experimental conditions \cite{delafuente1996a}. The dynamics of the glycolytic system changes substantially trough this route, which allows for a better comparison of the enzymatic processes in periodic, quasi-periodic and chaotic conditions.
Using the non-linear analysis techniques such as Transfer Entropy \cite{schreiber2000} and Mutual Information \cite{cover1991}, we have analyzed the glycolytic series and quantified the effective connectivity of the enzymes.
The results show that in the numerical analysis of yeast glycolysis, under dissipative conditions, a effective functional structure emerges which is characterized by changing connectivity flows and a metabolic invariant that constraints the activity of the irreversible enzymes.
\section*{Results}
The monitoring of the fluorescence of NADH in glycolyzing baker$'$s yeast under sinusoidal glucose input flux, have shown that quasi-periodic time patterns are common at low amplitudes of the input and for high amplitudes chaotic behaviours emerge \cite{markus1985a,markus1985b}.
In order to simulate these metabolic processes, the system is considered under periodic input flux with a sinusoidal source of glucose $\mathrm{S}=\mathrm{S_0}+ \mathrm{A} \sin (\omega t)$. Assuming the experimental value of $\mathrm{S_0}=6$mM/h \cite{markus1984}, after dividing by $\mathrm{K}_{\mathrm{m}2}$ (the Michaelis constant of phosphofructokinase, see for more details Materials and Methods) we have obtained the normalized input flux $\mathrm{S_0}=0.033$ Hz.
Under these conditions, a wide range of different types of dynamic patterns can emerge as a function of the control parameter, hereafter the amplitude A of the sinusoidal glucose input flux \cite{delafuente1996a,delafuente1999,delafuente1996b}. In particular, it is observed a quasi-periodic route to chaos (cf. left panel in Fig. \ref{fig2}); thus for A$=0.001$ the biochemical oscillator exhibits a periodic pattern (Figure 2a). An increment of the amplitude to A$=0.005$ provokes a Hopf bifurcation generating another fundamental frequency, as a consequence, quasi-periodic behaviors emerge (Figure 2b). Above A$=0.021$, complex quasi-periodic oscillations appear (Figure 2c). After a new Hopf bifurcation the originated dynamical behavior is not particularly stable and small perturbations produce deterministic chaos (A$=0.023$, Figure 2d), as predicted by Ruelle and Takens \cite{ruelle1971}. This route is in agreement with experimental conditions \cite{delafuente1996a}.
To go a next step further in the understanding of the relationship between the classical topological structure and effective functionality we have analyzed by means of non-linear statistical tools the catalytic patterns belonging to this scenario to chaos, and for each transition represented in the Figure 2 we have obtained three time series corresponding to the variables $\alpha$, $\beta$ and $\gamma$ (12 in total), which denote respectively the normalized concentrations of glucose-6-phosphate, fructose 1-6-bisphosphate and pyruvate.
\subsection*{Effective functionality}
Transfer Entropy (TE) quantifies the reduction in uncertainty that one variable has on its own future when adding another. This measure allows for a calculation of the functional influence in terms of effective connectivity between two variables \cite{schreiber2000}.
The analysis of the glycolytic data by means of the TE method are shown in Table I. The 4D vectors in square brackets correspond to the results obtained for the 4 different amplitudes of the considered glucose input flux, A=[0.001;0.005;0.021;0.023].
The values of functional influence are ranging in $0.58\leq TE\leq 1.00$, with mean$=0.79$ and standard deviation=$0.12$, what indicates in general terms a high effective connectivity in the enzymatic system.
The minimum value 0.58 corresponded to the causality flow between E$_{3}$ and E$_{2}$ when a simple periodic behavior emerges. However, the functional connectivity from E2 to E3 shows the maximum value, achieved in all considered conditions of the glucose input flux.
The glycolytic effective connectivity is illustrated in
the right panel of Fig. 2. The arrows width is proportional to
the TE between pairs of enzymes. The values change trough
the quasi-periodic route to chaos, remarked from E3 to E2 by
black dashed circles, [0.58;0.84;0.61;0.66].
In all cases analyzed, the values of TE present a maximum statistical significance (pvalue=0).
\subsection*{Total Information flows and the functional invariant}
Next, we have measured the total information flow, defined as the total outward of Transfer Entropy arriving to one enzyme minus the total inward. Positive values mean that that enzyme is a source of causality flow and negative flows are interpreted as sinks or targets. The results of the total information flows are shown in Table II (pvalue=0). The maximum source of total transfer information (0.41) corresponds to the E2 enzyme (phosphofructokinase) for A=0.021, when complex quasi-periodic oscillations appear in the glycolytic system.
For all conditions the enzyme E$_{2}$ (phosphofructokinase) is the main source of effective influence and the enzyme E$_{3}$ (pyruvatekinase) a sink, which could be interpreted as a target from a point of view based on its effective functionality. The enzyme E$_{1}$ (hexokinase) is less constrained, and it has a flow close to zero for all conditions.
The attributed role to each enzyme, namely E$_{2}$ the source, E$_{3}$ the sink and E$_{1}$ no-constrained is an invariant and preserved trough the whole route to chaos.
\subsection*{Functional Synchronization}
Time correlations allows for
quantification about how much two time series are statistically
independent.
According to that, we have measured the time pairwise correlations
in the enzymatic system, and the corresponding results are shown in Table III. The main finding is that E$_{2}$ and E$_{3}$ are highly
synchronized (correlation=0.90,
pvalue=0) and E$_{1}$ is
anti-synchronized with both E$_{2}$ and E$_{3}$
(respectively, correlation equals -0.65 and -0.66, pvalue=0).
These values of time
correlations were almost constant trough the quasi-periodic route to chaos and established that the activities of E$_{2}$ and E$_{3}$ are grouped to the same function, being activated at similar time and
oppositely to E$_{1}$.
\subsection*{Redundancy and uncertainty reduction}
The Mutual Information (MI) quantifies how much the knowledge of one variable reduces the entropy or uncertainty of the another \cite{cover1991}.
The analysis of the glycolytic data by means of this method
are shown in Table IV.
The high values of MI (close to 0.50) proved a high informative redundancy between the pairs of enzymes. So, the number of bits of information transferred from one enzyme to another is much larger than the actually needed.
The values in the principal diagonal of Table IV represent the uncertainty for each variable. We have found these values gradually descending, H(E$_{1}$)=[1.00;1.00;1.00;1.00], H(E$_{2}$)=[0.85;0.84;0.85;0.86] and
H(E$_{2}$)= [0.76;0.74;0.76;0.78],
which is indicative of the uncertainty in the enzymatic activity patterns belonging to E$_{1}$, E$_{2}$ and E$_{3}$ is reduced monotonously for all analyzed conditions.
The values of MI have a maximum statistical significance (pvalue=0).
Finally, we have computed the Mutual Information between the glucose input
fluxes and the activity patterns of the different enzymes. In all cases, the MI was equal to zero, proving that the oscillations of
the glucose were statistical independent of the glucose-
6-phosphate, fructose 1-6-biphosphate and pyruvate, products of the main irreversible enzymes of glycolysis.
\section*{Discussion}
In this paper we have quantified essential aspects of the effective
functional connectivity among the main glycolytic enzymes
in dissipative conditions.
First, we have computed under different source of glucose
the causality flows in the metabolic system. This
level of the functional influence accounts for the contribution
of each enzyme to the generation of the different catalytic behavior and adds a directionality in the influence
interactions between enzymes.
The results show that the flows of functional connectivity change significantly during the different metabolic transitions analyzed, exhibiting high values of transfer entropy, and
in all considered cases, the enzyme
phosphofructokinase (E$_{2}$) is the main source of effective causality flow; the
pyruvatekinase (E$_{3}$) is the main sink of information flow; the
hexokinase (E$_{1}$) has a quasi-zero flow, meaning that, the total
information arriving to E$_{1}$ goes out to either E$_{2}$ or E$_{3}$.
The maximum source of total transfer information
(0.41) corresponds to the E$_{2}$ enzyme (phosphofructokinase) at the edge of chaos,
when complex quasi-periodic oscillations emerge (cf. Fig. 2).
This finding seems to be consistent with other studies which
show that when a dynamical system operates in the frontier between order (periodic behavior) and chaos its complexity
is maximal \cite{kaufmann1991,bertschinger2004}.
The
level of influence in terms of causal interactions between the enzymes is not always the same but varies
depending on the substrate fluxes and the dynamic characteristics
emerging in the system. In addition to the glycolitic topological
structure characterized by the
specific location of enzymes, substrates, products and regulatory
metabolites there is an functional structure of information
flows which is dynamic and exhibit notable variations of the causal interactions.
Another aspect of the glycolitic functionality was observed during the quantification of the Mutual Information, which measures how
much the uncertainty about the one enzyme is reduced by
knowing the other; we found that the uncertainty for E$_{1}$, E$_{2}$ and E$_{3}$ monotonously decreased for all the values of the periodic glucose input-flux.
Second, the numerical results show that for all analyzed cases the maximum effective connectivity corresponds to the
Transfer Entropy from E$_{2}$ to E$_{3}$, indicating the biggest information flow in the multi-enzyme instability-generating
system. This is also corroborated by the measure
of correlation between the different pairs of series which
shows that E$_{2}$ and E$_{3}$ are highly correlated, or synchronized
(correlation=0.90 pvalue=0) and E$_{1}$ is anti-correlated with
both E$_{2}$ and E$_{3}$ (respectively, correlation=-0.65 pvalue=0 and
correlation=-0.66 pvalue=0). The values of time correlations
establish that the activities of E$_{2}$ and E$_{3}$ are grouped
to the same function, being activated at similar time and oppositely
to E$_{1}$.
Third, our analysis allows for a hierarchical classification
in terms of what glycolytic enzyme is improving
the future prediction of what others, and the results reveals
in a quantitative manner that the enzyme E$_{2}$ (phosphofructokinase) is the major source of
causal information and represents the key-core of glycolysis.
The second in importance is the E$_{3}$ (pyruvatekinase).
From
the biochemical point of view the E$_{2}$ (phosphofructokinase)
has been commonly considered as a major checkpoint in the
control of glycolysis \cite{serrano1989,heinisch1996}. The main reason for this generalized
belief is that this enzyme exhibits a complex regulatory behavior that reflects its
capacity to integrate many different signals \cite{stryer1995}; from a dissipative
point of view, this enzyme catalyzes a reaction very far
from equilibrium and its self-catalytic regulation it has been
considered the main instability-generating mechanism for the
emergence of oscillatory patters in glycolysis \cite{goldbeter2007}. The functional studies presented here confirm in a quantitative manner that
the E$_{2}$ (phosphofructokinase) is the key-core of the pathway, and our results
make stronger and expand the classical biochemical studies of
glycolysis.
Forth, the dynamics of the glycolytic system
changes substantially trough the quasi-periodic route to chaos
when the amplitude of the input-flux varies. However, the hierarchy
obtained by transfer entropy, E$_{2}$ the flow, E$_{3}$ the sink
and E$_{1}$ a quasi-zero flow, is preserved during this route and
seems to be an invariant. This functional
invariant of a metabolic process may be important for the
understanding of functional enzymatic constraints in cellular
conditions; but this issue requires other additional studies.
Finally, we want to emphasize that Transfer Entropy as a
quantitative measure of effective causal connectivity can be a very useful
tool in studies of enzymatic processes that operate far from
equilibrium conditions. Moreover, many experimental observations have shown that the oscillations in the enzymatic activity seem to represent one of the most striking manifestations of the metabolic dynamic behaviors, of not only qualitative but also quantitative importance in cells (further details in Appendix III).
Transfer Entropy is
able to detect the directed exchange of causality flows among
the irreversible enzymes which might allow for a rigorous quantification
of the effective functional connectivity of many dissipative
metabolic processes in both normal and pathological cellular
conditions.
The TE method applied to our numerical studies of yeast glycolisis shows the emergence of a new kind of dynamical functional structure which is characterized by changing connectivity flows and a metabolic invariant that constrains the activity of the irreversible enzymes.
The understanding the effective connectivity of the
metabolic dissipative structures is crucial to address the
functional dynamics of cellular life.
\section*{Methods}
\subsection*{Model}
In Fig. \ref{fig1} are represented the main enzymatic processes of yeast glycolysis (the irreversible stages) with the enzymes arranged in series.
When the metabolite S (glucose) feeds the system, it is transformed by the first enzyme E$_{1}$ (hexokinase) into the product P$_{1}$ (glucose-6-phosphate). The enzymes E$_{2}$ (phosphofructokinase) and E$_{3}$ (pyruvatekinase) are allosteric, and transform the substrates P$'_{1}$ (fructose 6-phosphate) and P$'_{2}$ (phosphoenolpyruvate) in the products P$_{2}$ (fructose 1-6-bisphosphate) and P$_{3}$ (pyruvate), respectively. The step P$_2$ $\rightarrow$ P$'_2$ represents reversible activity processes, reflected in the dynamic system by the functional variable $\beta'$.
A part of P$_{1}$ does not continue in the metabolic system, and is removed with a rate constant of q$_{1}$ which is related with the activity of pentose phosphate pathway; likewise, q$_{2}$ is the rate constant for the sink of the product P$_{3}$ which is related with the activity of pyruvate dehydrogenase complex.
The main instability-generating mechanism in yeast glycolysis is the self-catalytic regulation of the enzyme E$_{2}$ (phosphofructokinase), specifically, the positive feed-back exerted by the reaction products, the ADP and fructose-1,6-bisphosphate \cite{goldbeter1972,boiteux1975,goldbeter2007}. From a strictly biochemical point of view, E$_{2}$ is also considered the main regulator enzyme of glycolysis \cite{stryer1995}. The second irreversible stage for its regulatory importance is catalyzed by the enzyme E$_{3}$ (pyruvatekinase) which is inhibited by the ATP reaction product \cite{stryer1995}. Finally, the third irreversible process corresponds to the first stage the enzyme E$_{1}$ (hexokinase) which is dependent on the ATP.
In the determination of the enzymatic kinetics of the enzyme E$_{1}$ (hexokinase) the equation of the reaction speed dependent on glucose and ATP has been used \cite{viola1982}. The speed function of the allosteric enzyme E$_{2}$ (phosphofructokinase) was developed in the framework of the concerted transition theory \cite{goldbeter1972}. The reaction speed of the enzyme E$_{3}$ (pyruvatekinase), dependent on ATP and phospoenolpyruvate, was also constructed on the allosteric model of the concerted transition \cite{markus1980}.
To study the kinetics of the dissipative glycolytic system we have considered normalized concentrations; $\alpha$, $\beta$ and $\gamma$ denoted respectively the normalized concentrations of P$_{1}$, P$_{2}$ and P$_{3}$. For a spatially homogeneous system the time-evolution is described by the following three delay differential equations:
\begin{eqnarray}
\frac{\mathrm{d} \alpha}{\mathrm{d} t} &=&\mathrm{z}_1 \sigma_1 \phi_1(\mu)-\sigma_2 \phi_2(\alpha,\beta)-\mathrm{q}_1 \alpha \nonumber \\
\frac{\mathrm{d} \beta}{\mathrm{d} t} &=&\mathrm{z}_2 \sigma_2 \phi_2(\alpha,\beta)-\sigma_3 \phi_3(\beta,\beta',\mu) \nonumber \\
\frac{\mathrm{d} \gamma}{\mathrm{d} t} &=&\mathrm{z}_3 \sigma_3 \phi_3(\beta,\beta',\mu)-\mathrm{q}_2 \gamma
\label{alpha-beta-gamma}
\end{eqnarray}
where the functional variables $\beta'$ and $\mu$ reflect the normalized concentrations of P$'_{2}$ (phosphoenolpyruvate) and ATP respectively. The three main enzymatic functions are the following:
\begin{eqnarray}
\phi_1(\mu)&=& \frac{\mu \mathrm{S}\mathrm{K}_{\mathrm{d}3}} {\left(\mathrm{K}_{3}\mathrm{K}_{2}+\mu\mathrm{K}_{\mathrm{m}1} \mathrm{K}_{\mathrm{d}3} + \mathrm{S}\mathrm{K}_{2}+\mu \mathrm{S}\mathrm{K}_{\mathrm{d}3}\right)} \nonumber \\
\phi_2(\alpha,\beta)&=&\frac{\alpha \left(1+\alpha\right)\left(1+\mathrm{d}_1\beta\right)^2}{\mathrm{L}_1\left(1+\mathrm{c}\alpha\right)^2+\left(1+\alpha\right)^2\left(1+\mathrm{d}_1\beta\right)^2} \nonumber \\
\phi_3(\beta,\beta',\mu)&=&\frac{\mathrm{d}_2\beta'\left(1+\mathrm{d}_2\beta'\right)^3}{\mathrm{L}_2\left(1+\mathrm{d}_3\mu\right)^4+\left(1+\mathrm{d}_2\beta\right)^4}
\label{phis}
\end{eqnarray}
and
\begin{eqnarray}
\beta'&=&\mathrm{f}(\beta(t-\lambda_1)) \nonumber \\
\mu&=&\mathrm{h}(\beta(t-\lambda_2)).
\label{fh}
\end{eqnarray}
The constants $\sigma_1$, $\sigma_2$ and $\sigma_3$ correspond to the maximum activity of E$_{1}$, E$_{2}$ and E$_{3}$ ($V_{\mathrm{m}1}$, $V_{\mathrm{m}2}$ and $V_{\mathrm{m}3}$) divided by the Michaelis constants of each enzyme, respectively $\mathrm{K}_{\mathrm{m}1}$, $\mathrm{K}_{\mathrm{m}2}$ and $\mathrm{K}_{\mathrm{m}3}$. The constants z's are defined as $\mathrm{z}_1=\mathrm{K}_{\mathrm{m}1} /\mathrm{K}_{\mathrm{m}2}$, $\mathrm{z}_2=\mathrm{K}_{\mathrm{m}2} /\mathrm{K}_{\mathrm{m}3}$ and $\mathrm{z}_3=\mathrm{K}_{\mathrm{m}3} /\mathrm{K}_{\mathrm{d}3}$, with $\mathrm{K}_{\mathrm{d}3}$ representing the dissociation constant of P$_{2}$ by E$_{3}$. The constants d's are $\mathrm{d}_1=\mathrm{K}_{\mathrm{m}3}/\mathrm{K}_{\mathrm{d}2}$, $\mathrm{d}_2=\mathrm{K}_{\mathrm{m}3}/\mathrm{K}_{\mathrm{d}3}$ and $\mathrm{d}_3=\mathrm{K}_{\mathrm{d}3}/\mathrm{K}_{\mathrm{d}4}$, with $\mathrm{K}_{\mathrm{d}4}$ representing the dissociation constant of ATP; $\mathrm{L}_1$ and $\mathrm{L}_2$ are respectively the allosteric constant of E$_{2}$ and E$_{3}$; c is the non-exclusive binding coefficient of the substrate P$_{1}$. More details about parameter values and experimental references are given in Appendix I.
From the dissipative point of view the essential enzymatic stages are those that correspond to the biochemical irreversible processes \cite{ebeling1986} and to simplify the model, we did not consider the intermediate part of glycolysis belonging to the enzymatic reversible stages. In this way, the functions f and h are supposed to be the identity function. Thus,
\begin{eqnarray}
\beta'&= &\beta(t-\lambda_1)\nonumber \\
\mu&=&\gamma(t-\lambda_2)
\label{betamu}
\end{eqnarray}
The initial functions present a simple harmonic oscillation in the following form:
\begin{eqnarray}
\alpha_0(t) &=& A+B \sin(2\pi/P)\nonumber \\
\beta_0(t) &=& C+D \sin(2\pi/P)\nonumber \\
\gamma_0(t) &=& E+F \sin(2\pi/P)
\label{functiont0}
\end{eqnarray}
with $A=26$, $B=12$, $C=12$, $D=10$, $E=7$, $F=6$ and $P=534$.
The dependent variables $\alpha$, $\beta$ and $\gamma$ were normalized dividing them by $\mathrm{K}_{\mathrm{m}2}$, $\mathrm{K}_{\mathrm{m}3}$ and $\mathrm{K}_{\mathrm{d}3}$, and the parameters $\lambda_1$ and $\lambda_2$ are time delays affecting the independent variable (see for more details the Appendix II).
The numerical integration of the system was performed with the package ODE Workbench, which created by Dr. Aguirregabiria is part of the Physics Academic Software. Internally this package uses a Dormand-Prince method of order 5 to integrate differential equations. Further information at http://www.webassign.net/pas/ode/odewb.html.
This model has been exhaustively analyzed before, revealing a notable richness of emergent temporal structures which included the three main routes to chaos, as well as a multiplicity of stable coexisting states, see for more details \cite{delafuente1996a,delafuente1999,delafuente1996b}.
\subsection*{Transfer Entropy}
TE allows for a quantification of how much the temporal evolution of the activity of one enzyme helps to improve the future prediction of another. The oscillatory patterns of the biochemical metabolites might have information which can be read-out by the TE.
For a convenient derivation, let generally assume that each of the pairs of enzymatic activity is represented by the two time series $X\equiv \{x_t\}_{t=1}^T$ and $Y\equiv \{y_t\}_{t=1}^T$ . Here, $x_t$ is the state value of the variable $X$ in time $t$, and similarly for $y_t$. Let $I(X^P,Y^P\rightarrow X^F)=-\sum_{x_{t+1},x_t,y_t}P(x_{t+1},x_t,y_t)\log_2 P(x_{t+1}|x_t,y_t)$ be the amount of information required to predict the future of $X$ ($X^F$) known both the pasts of $X$ and $Y$ ($X^P$ and $Y^P$). Analogously, let $I(X^P\rightarrow X^F)=-\sum_{x_{t+1},x_t}P(x_{t+1},x_t)\log_2 P(x_{t+1}|x_t)$ be the amount of information required to predict the future of $X$ known only its past. The difference $I(X^P\rightarrow X^F)-I(X^P,Y^P\rightarrow X^F)$ is by definition the transfer entropy from $Y$ to $X$, denoted by $\mathrm{TE}_{Y\rightarrow X}$. It quantifies the amount of information in digits that $Y$ adds to the predictability of $X$.
Rewriting the conditional probabilities as the joint probability divided by its marginal, one obtains an explicit form for the Transfer Entropy:
\begin{eqnarray}
TE_{Y\rightarrow X}=
\sum_{x_{t+1},x_t,y_t}P(x_{t+1},x_t,y_t)\log_2\left(\frac{P(x_{t+1},x_t,y_t)P(x_t)}{P(x_t,y_t)P(x_{t+1},x_t)}\right).
\label{TE}
\end{eqnarray}
The formula (\ref{TE}) is fully equivalent to the Mutual Information between $X^F$ and $Y^P$ conditioned to $X^P$. Thus,
$\mathrm{TE}_{Y\rightarrow X}\equiv I(X^F,Y^P|X^P)$, and consequently, Transfer Entropy says about how much information the inclusion of $Y^P$ adds to the prediction of $X^F$ only considering $X^P$, ie. $I(X^F,Y^P|X^P)=H(X^F|X^P)-H(X^F|X^P,Y^P)$. Therefore, TE is fully quantifying the information flows between pairs of variables. The values of TE were normalized between 0 and 1.
It is important to remark that the TE from $X$ to $Y$ is different to the one from $Y$ to $X$, ie. the effective connectivity is asymmetric, adding a directionality in time which accounts for a particular case of \textit{directed graphs}, the graph of information flows between pairs of enzymes.
Alternatively to the Transfer Entropy, effective connectivity can be obtained using Granger Causality \cite{granger1969}, which makes emphasis on how much from the past of one variable the predictability of its future is improved by adding the past of another variable. Recently, it has been proved that in the case of Gaussian variables both Transfer Entropy and Granger Causality are measuring exactly the same \cite{barnett2009}. Therefore, the information flows based on Transfer Entropy and the Granger causality interactions coincide for Gaussian variables.
\section*{Mutual Information and Redundancy}
MI quantifies how much the knowledge of one variable reduces the entropy or uncertainty of another. Therefore, MI says about how much information the two variables are sharing. The strongest point of the MI is that it extends functionality to high order statistics \cite{cover1991}.
Its definition is $MI(X,Y)=H(X)-H(X|Y)$, where $H(X|Y)=H(X,Y)-H(Y)$ is the conditional entropy of $X$ given $Y$. It accounts for the remaining uncertainty in $X$ knowing the variable $Y$. We referred $H(X)$ and $H(X,Y)$ as respectively the joint and marginal (Shanon) entropies.
For statistical independent $X$ and $Y$ variables one has $MI(X,Y)=0$. The other limit satisfies $MI(X,X)=H(X)$, because of $H(X|Y)=0$. Therefore, the MI of two variables is bounded and satisfies that $0\leq MI(X,Y)\leq H(X)$. High values of MI mean that the redundancy in information between the two variables is large. The values of MI were normalized between 0 and 1.
\section*{Number of bins vs Statistical significance}
For all the probabilities used in both Transfer Entropy (TE) and Mutual Information (MI) we used a number of bins of 10. As it is well-known, the calculation of these probabilities is sensitive to the number of bins. Instead of tuning it as a control parameter to compute the probabilities, we preferred explored the statistical significance of the computed values. This was achieved by comparing both the TE and MI between the two series of enzymatic activity, say X and Y, with the values obtained when considering a random permutation of Y, what we called, the shuffled Y.
The values of both TE and MI shown in Tables I and III
were larger than those calculated in the shuffled situation (for both TE and MI, pvalue=0, for 50 different samples).
\section*{Acknowledgments}
JMC is funded by the Spanish Ministerio de Ciencia e Innovacion, programa Ramon y Cajal, and from Junta de Andalucia, grants P09-FQM-4682 and P07-FQM-02725. I.M. De la Fuente acknowledges useful advises and suggestions from Prof. J. Veguillas.
\newpage
|
{
"timestamp": "2011-01-18T02:01:40",
"yymm": "1009",
"arxiv_id": "1009.3627",
"language": "en",
"url": "https://arxiv.org/abs/1009.3627"
}
|
\section{Introduction}\label{sec:intro}
Primordial non-Gaussianity provides cosmology one of the precious few
connections between primordial physics and the present-day universe. Standard
inflationary theory, with a single slowly rolling scalar field, predicts that
the spatial distribution of structures in the universe today is very nearly
Gaussian random (e.g.\
\cite{maldacena,Acquaviva:2002ud,Creminelli:2003iq,Lyth_Rodriguez,Seery_Lidsey};
for excellent recent reviews, see \cite{Chen_AA,Komatsu_CQG}). Departures
from Gaussianity, barring contamination from systematic errors or late-time
non-Gaussianity due to secondary processes, would be a violation of this
standard inflationary assumption. Constraining or detecting primordial
non-Gaussianity is therefore an important basic test of the standard
cosmological model.
Most of the study of non-Gaussianity in the literature to date has been
carried out assuming the magnitude of departure from Gaussianity is
scale-independent
(e.g.\ \cite{Komatsu_Spergel,Verde_CMBLSS,Scoccimarro:2003wn}). However, the
assumption that $f_{\rm NL}$ is constant for a wide range of scales could be an
over-simplification, since the primordial cosmic perturbations were presumably
produced from the time-dependent dynamics in the early universe. In
particular, single-field inflationary models with interactions, along with
most multi-field models, generically produce scale-dependent
non-Gaussianity. It is therefore not surprising that scale-dependence of
non-Gaussianity has been discussed in the community in recent years
\cite{Salopek,Falk_Ran_Sre,Luo_Schramm,Gangui_etal,Wang_Kam,Bartolo:2004if,see2, Chen2005,Liguori2006,Chen2006,LoVerde,Chen2008,Sefusatti2009,Kumar2010,Byrnes2010,bryb,wandsb, Riotto2010,Huang2010}. Notably,
the parameterization of the scale-dependent non-Gaussianity in our analysis is
applicable to the curvaton \cite{mol,lin5,enq5,ly5,moro} and the modulated
reheating scenarios \cite{lev,mati}, which are of great interest for their
potentially observable scale-dependent non-Gaussianity\footnote{For instance,
when the observed perturbations originate from the single
curvaton field, the ``running'' (with scale) of the non-Gaussianity
parameter is proportional to the third derivative of the curvaton potential,
$V'''$ \cite{chris5,chris6,huang}. Given that this third derivative is not
tightly constrained from the observed power spectrum, it can potentially
lead to observable {\it and} scale-dependent non-Gaussianity. Therefore,
constraints on the running of non-Gaussianity can be a powerful probe of the
origin of the primordial curvature perturbations.}.
Motivated by such inflationary models that predict detectable scale-dependent
non-Gaussianity, as well as a desire to have an easily usable basis for studying
those models, we present a novel scale-dependent ansatz for primordial
non-Gaussianity: we promote the parameter $f_{\rm NL}$ to a free
function of wavenumber $f_{\rm NL}(k)$. We define our model (Sec.~\ref{sec:NG}),
predict clustering bias of dark matter halos in our model
(Sec.~\ref{sec:NG_bias}), obtain an upper bound on the accuracy with which
these new parameters could be measured with a future large-scale structure
survey (Sec.~\ref{sec:forecasts}), and compare our model with other
parameterizations of non-Gaussianity in the literature (Sec.~\ref{sec:PC}).
\section{Scale dependent non-Gaussianity}
\label{sec:NG}
The most commonly discussed model of non-Gaussianity, often referred to as the
local model, is defined via \cite{Komatsu_Spergel}
\eqn{eq:localNG}{
\Phi(x)=\phi_G(x)+f_{\rm NL}(\phi_G(x)^2-\langle \phi_G (x)^2 \rangle ).
}
Here, $\Phi$ denotes the primordial curvature perturbations (Bardeen's
gauge-invariant potential), $\phi_G(x)$ is a Gaussian random field, and the
constant $f_{\rm NL}$ is the non-Gaussianity parameter. The local model has been
much studied, in part because it is the first two terms of the most general
local form of non-Gaussianity \cite{Babich_shape}.
In Fourier space, Eq.~(\ref{eq:localNG}) becomes
\begin{equation}
\Phi(k)=\phi_G(k)+f_{\rm NL}\int \frac{d^3 k'}{(2 \pi)^3}\phi_G(k')\phi_G(k-k').
\label{eq:localNG_kspace}
\end{equation}
(Hereafter, we omit the subscript $G$ on the Gaussian distribution when it is
clear from context.) In this paper, we study a model that generalizes Eq.~(\ref{eq:localNG_kspace}) -- we allow $f_{\rm NL}$ to vary with $k$ as well, while
assuming isotropy and homogeneity (so $f_{\rm NL}(${\boldmath $k$}$)=f_{\rm NL}(k)$). The
gravitational potential in the new model is defined via
\begin{equation}
\Phi(k)=\phi(k)+f_{\rm NL}(k)\int \frac{d^3 k'}{(2 \pi)^3}\phi(k')\phi(k-k').
\label{eq:fnlk_kspace}
\end{equation}
As mentioned above, this form of non-Gaussianity is expected in curvaton or
modulated reheating scenarios (see e.g.\ Ref.~\cite{chris5}, where this form
explicitly appears in the study of these models).
Note that this new ansatz is {\it not} local, which is clear when we transform
back into real space:
\begin{equation}
\Phi(x)=\phi+f_{\rm NL}(x)*(\phi(x)^2-\langle \phi (x)^2 \rangle ),
\label{eq:fnlk_realspace}
\end{equation}
where $*$ represents convolution and $x$ denotes a three-dimensional spatial
coordinate. These primordial
perturbations $\Phi(k)$ are related to the present-time (z=0) smoothed linear
overdensity $\delta_R$ by the Poisson equation:
\begin{equation}
\delta_R(k)=\frac{2}{3}
\frac{k^2 T(k)}{H_0^2 \Omega_{m}} \tilde{W}_R(k) \Phi(k)
\equiv \mathcal{M}_R(k) \Phi(k);
\label{eq:overdensity}
\end{equation}
where $T(k)$ is the matter transfer function, $H_0$ is the Hubble constant,
$\Omega_m$ is the matter density relative to critical today, and
$\tilde{W}_R(k)$ is the Fourier transform of the top-hat filter with radius
$R$. The smoothing spatial scale $R$ is related to the smoothing mass scale
$M$ via
\begin{equation}
M = {4 \over 3} \pi R^3 \rho_{m, 0},
\label{eq:mass_scale}
\end{equation}
where $ \rho_{m, 0}$ is the matter energy density today. The choice of mass
scale is discussed further in section \ref{sec:constraints}.
The bispectrum in our generalized model becomes
\begin{equation}
B_{\phi}(k_1, k_2, k_3) = 2[f_{\rm NL}(k_1) P_\phi(k_2)P_\phi(k_3) + {\rm perm.}],
\label{eq:fnlk_bispec}
\end{equation}
where $P_\phi$ is the power spectrum of potential fluctuations.
This reduces to the familiar expression $B(k_1, k_2, k_3) = 2 f_{\rm NL} (P_\phi(k_1)P_\phi(k_2) + {\rm perm.})$
when $f_{\rm NL}$ is a constant.
Notice the difference between our ansatz for the scale-dependent $f_{\rm NL}(k)$ (which has the corresponding bispectrum Eq.~(\ref{eq:fnlk_bispec})) and
the particular form of scale-dependent non-Gaussianity, discussed elsewhere in the
literature, which is defined as $f_{\rm NL}(k_1,k_2,k_3)\equiv B_\phi(k_1, k_2, k_3)/[2
P_\phi(k_1)P_\phi(k_2) + {\rm perm.}]$ (\cite{Byrnes2010,wandsb,bryb}). The two forms are inequivalent, and either
form can be borne out in realistic inflationary models; however, given that
our form lives in a lower-dimensional $k$-space, it is easier to simulate it
numerically \cite{Shandera2010} or treat it with the Fisher matrix analysis, as we
do in this paper.
\section{Non-Gaussianity and Bias}
\label{sec:NG_bias}
\subsection{The effect of a non-vanishing bispectrum on bias}
Dalal et al.\ \cite{Dalal} found, analytically and numerically, that the bias of dark
matter halos acquires strong scale dependence if $f_{\rm NL} \neq 0$:
\begin{equation}
b(k)=b_0 + f_{\rm NL}(b_0-1)\delta_c\, \frac{3\Omega_mH_0^2}{a\,g(a) T(k)c^2 k^2}.
\label{eq:bias}
\end{equation}
Here, $b_0$ is the usual Gaussian bias (on large scales, where it is
constant), $\delta_c\approx 1.686$ is the collapse threshold, $a$ is the scale
factor, $\Omega_m$ is the matter density relative to the critical density, $H_0$ is the
Hubble constant, $k$ is the wavenumber, $T(k)$ is the transfer function, and
$g(a)$ is the growth suppression factor\footnote{The usual linear growth
$D(a)$, normalized to be equal to $a$ in the matter-dominated epoch, is
related to the suppression factor $g(a)$ via $D(a)=ag(a)$, where $g(a)$ is
normalized to be equal to unity deep in the matter-dominated epoch.}. This
result has been confirmed by other researchers using a variety of methods,
including the peak-background split
\cite{Afshordi_Tolley,MV,Slosar_etal,Schmidt_Kam}, perturbation theory
\cite{McDonald,Taruya08,GP}, and numerical (N-body) simulations
\cite{Grossi,Desjacques_Seljak_Iliev,PPH}. Astrophysical measurements of the
scale dependence of the large-scale bias, using galaxy and quasar clustering
as well as the cross-correlation between the galaxy density and CMB
anisotropy, have recently been used to impose constraints on $f_{\rm NL}$ already
comparable to those from the cosmic microwave background (CMB) anisotropy
\cite{Slosar_etal, Afshordi_Tolley}, giving $f_{\rm NL}=28\pm 23$ ($1\sigma$), with
some dependence on the assumptions made in the analysis \cite{Slosar_etal}.
In the future, constraints on $f_{\rm NL}$ are expected to be on the order of a few
\cite{Dalal,Carbone,Sartoris,Cunha_NG}. The sensitivity of the large-scale
bias to other models of primordial non-Gaussianity has not yet been
investigated much (though see analyses in e.g.~\cite{Desjacques_gnl,MV09}).
Following the MLB formula \cite{Grinstein:1986en,MLB1986}, one can express
the two point correlation function of dark matter halos,
$\xi_h(\boldsymbol{x}_1,{\boldsymbol x}_2 )$, in terms of certain
configurations of the correlation functions of the underlying density field,
$\xi_R^{(N)}$. In the high-threshold limit ($\nu \gg 1$), this becomes:
\begin{eqnarray}
\xi_h(\boldsymbol{x}_1,{\boldsymbol x}_2 ) &=&\xi_h(x_{12}) \nonumber\\&=&
-1+\exp
\left(
\sum_{N=2}^{\infty} \sum_{j=1}^{N-1} \frac{\nu^N}{\sigma_R^N} \frac{1}{j! (N-j)!}\xi_R^{(N)}
\left[
\begin{array}{cc}
\boldsymbol{x}_1,...,\boldsymbol{x}_1,& \boldsymbol{x}_2,...,\boldsymbol{x}_2 \\
j~ {\rm times}&(N-j)~{\rm times}
\end{array}
\right]
\right);
\label{eq:Grinstein-Wise}
\end{eqnarray}
where $x_{ij}=|\boldsymbol{x}_i-\boldsymbol{x}_j|,\nu=\delta_c/\sigma_R$
represents the peak height, and ${\xi_R}^{(n)}(r)$ is the $n$-point
correlation function of the underlying matter density smoothed with a top-hat
filter of radius $R$. Keeping the terms up to the three-point correlation
function, which would be reasonable for the observationally allowed range of
$f_{\rm NL}$, the expansion series gives us the halo correlation function in terms of the field correlation functions:
\begin{equation}
\xi_h(x_{12})= \frac{\nu^2}{\sigma_R^2}
{\xi_R^{(2)}}(\boldsymbol{x}_1,\boldsymbol{x}_2)+ \frac{\nu^3}{\sigma_R^3}
{\xi_R^{(3)}}(\boldsymbol{x}_1,\boldsymbol{x}_1,\boldsymbol{x}_2).
\label{eq:ksi_MV}
\end{equation}
The Fourier transform of the real-space correlation function -- the power
spectrum -- is given, to the same expansion order as Eq.~(\ref{eq:ksi_MV}), by
\begin{equation}
\label{ph1}
P_h(k)= \frac{\nu^2}{\sigma_R^2}
P_R(k)+ \frac{\nu^3}{\sigma_R^3}
\int \frac{d^3 q}{(2 \pi)^3}
B_R(k,q,|\boldsymbol{k}-\boldsymbol{q}|)+\ldots
\end{equation}
The first term on the right-hand side includes the familiar (Gaussian) bias
$b={\nu}/{\sigma_R}$ (in the high-peak limit for which the MLB formula is valid)
for the Gaussian fluctuations. The effects of non-Gaussianity on the
galaxy bias are represented by the second term, including the bispectrum
$B_R$, which vanishes for the Gaussian fluctuations.
\subsection{From the bispectrum to bias}
If we denote the full bias of dark matter halos by $b+\Delta b$, where
$b$ represents the bias for the Gaussian fluctuations and $\Delta b$ is the
non-Gaussian correction, then
\begin{equation}
\frac{P_h}{P_R}=b^2\left(1+\frac{\Delta b}{b}\right)^2,
\label{eq:bL}
\end{equation}
where $P_h$ and $P_R$ are the power spectra of halos and dark matter,
respectively. The non-Gaussian correction to the linear peak bias to the
leading order becomes
\begin{equation}
\frac{\Delta b}{b} (k) =\frac{\nu}{\sigma_R} \,\frac{1}{2 P_R(k)} \int \frac{d^3
q}{(2 \pi)^3} B_R(k,q,|\boldsymbol{k}-\boldsymbol{q}|),
\end{equation}
where $B_R$ is the matter bispectrum on scale $R$.
Hence, the non-Gaussian correction
$\Delta b(k)$ can be expressed in terms of the primordial potential fluctuations as (\cite{MV}):
\begin{equation}
{\Delta b\over b} (k) =
{\delta_c\over D(z)}\, {1\over 8\pi^2 \sigma_R^2 \mathcal{M}_R(k)}
\int_0^{\infty} dk_1 k_1^2 \mathcal{M}_R(k_1)\, \int ^{1}_{-1}d\mu \mathcal{M}_R(k_2)
{B_{\phi}(k_1, k_2, k)\over P_{\phi}(k)}.
\label{eq:MV}
\end{equation}
We perform the integration over all triangles. The triangles' sides are
$k_1$, $k_2$, and $k$; the cosine of the angle opposite $k_2$ is $\mu$, so $k_2^2 = k_1^2 +
k^2 + 2k_1k\mu$. $\mathcal{M}_R (k)$ is the same function defined in
Eq.~(\ref{eq:overdensity}), and the time dependence of the critical
threshold for collapse is given as $\delta_c(z) = \delta_c/D(z)$, with
$\delta_c=1.686$.
\subsubsection {Constant $f_{\rm NL}$}
Eq.~(\ref{eq:MV}) leads to the famous scale-dependent
bias formula in the case of a constant $f_{\rm NL}$. For this model, the bispectrum is
\begin{equation}
B_\phi(k_1, k_2, k_3) = 2f_{\rm NL}\, [P_\phi(k_1)P_\phi(k_2) + {\rm perm.}].
\label{eq:local_bispec}
\end{equation}
Through Eq.~(\ref{eq:MV}), this leads to the result
\begin{eqnarray}
{\Delta b\over b} (k) &=& \nonumber
{\delta_c\over D(z)}\, {2f_{\rm NL}\over 8\pi^2 \sigma_R^2 \mathcal{M}_R(k)}
\int dk_1 k_1^2 \mathcal{M}_R(k_1)P_\phi(k_1)
\int d\mu \mathcal{M}_R(k_2) \left [ {P_\phi(k_2)\over P_\phi(k)} + 2\right ]\\[0.3cm]
&\equiv & {2f_{\rm NL} \delta_c\over D(z)}\,{\mathcal{F}(k)\over \mathcal{M}_R(k)},
\label{eq:dboverb_fnlconst}
\end{eqnarray}
where
\begin{equation}
\mathcal{F}(k)\equiv {1\over 8\pi^2 \sigma_R^2}
\int dk_1 k_1^2 \mathcal{M}_R(k_1)P_\phi(k_1)
\int d\mu \mathcal{M}_R(k_2) \left [ {P_\phi(k_2)\over P_\phi(k)} + 2\right ].
\end{equation}
Note that there is a factor of $2$ in Eq.~(\ref{eq:dboverb_fnlconst}) because we can exchange the order of integration of terms corresponding to $k_1$
and $k_2$.
Finally, we rewrite Eq.~(\ref{eq:dboverb_fnlconst}) by defining
\begin{eqnarray}
\mathcal{F}_1(k)&\equiv& {1\over 8\pi^2 \sigma_R^2 \mathcal{M}_R(k) P_\phi(k)}
\int dk_1 k_1^2 \mathcal{M}_R(k_1)P_\phi(k_1)
\int d\mu \mathcal{M}_R(k_2) P_\phi(k_2)
\label{eq:F1_dummy}
\\[0.2cm]
\mathcal{F}_2(k)&\equiv& {2\over 8\pi^2 \sigma_R^2 \mathcal{M}_R(k)}
\int dk_1 k_1^2 \mathcal{M}_R(k_1)P_\phi(k_1)
\int d\mu \mathcal{M}_R(k_2).
\label{eq:F2_dummy}
\end{eqnarray}
Then, for constant $f_{\rm NL}$,
\begin{equation}
{\Delta b\over b} (k) =
{2f_{\rm NL} \delta_c\over D(z)}\,\left [\mathcal{F}_1(k) + \mathcal{F}_2(k)\right ],
\end{equation}
and the derivative with respect to $f_{\rm NL}$ is
\begin{equation}
{\partial\over\partial f_{\rm NL}}\left [{\Delta b\over b} (k)\right ]
= {2 \delta_c\over D(z)}\,\left [\mathcal{F}_1(k) + \mathcal{F}_2(k)\right ].
\end{equation}
\subsubsection{Scale-dependent $f_{\rm NL}$}
Now we repeat the analysis of the previous section, but we allow $f_{\rm NL}(k)$ to be an
arbitrary function of scale, adopting the ansatz in
Eq.~(\ref{eq:fnlk_kspace}). We still assume homogeneity, so
$f_{\rm NL}(\vec{k})=f_{\rm NL}(k)$. The bispectrum is given by
\begin{equation}
B_{\phi}(k_1, k_2, k_3) = 2[f_{\rm NL}(k_1) P_\phi(k_2)P_\phi(k_3) + {\rm perm.}].
\end{equation}
Here, the triangle condition always holds, so that (for example) $k_1 =
|\vec{k_2}+\vec{k_3}|$. Following Eq.~(\ref{eq:MV}), we get
\begin{eqnarray}
{\Delta b\over b} (k)& =&
{\delta_c\over D(z)}\, {2\over 8\pi^2 \sigma_R^2 \mathcal{M}_R(k)}
\int dk_1 k_1^2 \mathcal{M}_R(k_1)P_\phi(k_1)
\nonumber \\[0.2cm]
&\times &\int d\mu \mathcal{M}_R(k_2) \left [ f_{\rm NL}(k){P_\phi(k_2)\over P_\phi(k)} + 2f_{\rm NL}(k_2)\right ].
\end{eqnarray}
This looks like Eq.~(\ref{eq:dboverb_fnlconst}) -- but this time, $f_{\rm NL}(k)$ is a function, not a constant. Thus, to find the derivative of $\Delta b/ b (k)$ with respect to the relevant parameters, we must parametrize $f_{\rm NL}(k)$ in a way that is valid for any general form of $f_{\rm NL}(k)$. We consider the
piecewise-constant (in wavenumber) parametrization where $f_{\rm NL}(k)$ is equal to $f_{\rm NL}^i$ in the $i$th wavenumber bin:
\begin{equation}
f_{\rm NL}^i\equiv f_{\rm NL}(k_i).
\label{eq:fnl_piecewise}
\end{equation}
The derivative
of $\Delta b/ b (k)$ with respect to these $f_{\rm NL}^i$ is:
\begin{eqnarray}
{\partial\over\partial f_{\rm NL}^j}\left [{\Delta b\over b} (k_i)\right ] &=&
{\delta_c\over D(z)}\, {2\over 8\pi^2 \sigma_R^2
\mathcal{M}_R(k)}\times\nonumber \\[0.3cm]
&& \left [
\delta_{ij}{1\over P_\phi(k)}
\int dk_1 k_1^2 \mathcal{M}_R(k_1)P_\phi(k_1)
\int d\mu \mathcal{M}_R(k_2)P_\phi(k_2) +\right .\\[0.2cm]
&&\left . +2\int_{k_2\in k_j} dk_1 k_1^2 \mathcal{M}_R(k_1)P_\phi(k_1)
\int d\mu \mathcal{M}_R(k_2)
\right ],\nonumber
\end{eqnarray}
where $\delta_{ij}$ is the Kronecker delta function. Note that the last
integral over $k_2$ only goes over the $j$th wavenumber bin.
This derivative can be rewritten more concisely as
\begin{equation}
{\partial\over\partial f_{\rm NL}^j}\left [{\Delta b\over b} (k_i)\right ] =
{2\delta_c\over D(z)}\left [\delta_{ij}\mathcal{F}_1(k) + \mathcal{F}^j_2(k)\right ].
\label{eq:fnlk_deriv}
\end{equation}
The functions $\mathcal{F}_1$ and $\mathcal{F}_2$ are defined as in
Eqs.~(\ref{eq:F1_dummy}) and (\ref{eq:F2_dummy}), except that the superscript
in $F^j_2$ indicates that the integral over $k_2$ is to be executed only over
the $j$th wavenumber bin.
\section{Forecasted measurements of the scale-dependent nongaussianity}
\label{sec:forecasts}
\subsection{Fisher Matrix Analysis}
\label{sec:constraints}
With an expression for
$\partial/\partial f_{\rm NL}^j[(\Delta b/ b) (k_i)]$ in hand
(Eq.~(\ref{eq:fnlk_deriv})), we can calculate the Fisher information matrix for
the parameters $f_{\rm NL}^j$ that describe the piecewise-constant $f_{\rm NL}(k)$. The Fisher matrix, in turn, allows us to forecast the extent to which the scale-dependent non-Gaussianity could
be measured in future galaxy surveys.
We consider measurements of the power spectrum $P_h(k)$ of dark matter halos (galaxies
or clusters, for example) averaged over thin spherical shells in $k$-space. The variance of $P_h(k)\equiv
P_h$ in each shell is \cite{FKP}
\begin{equation}
\label{power_error}
\sigma^2_{P_h} = \frac{2 P_h^2}{V_{\rm shell} \; V_{\rm survey}}
\left( \frac{1 + nP_h}{nP_h} \right)^2 = \frac{(2 \pi P_h)^2}{k^2 dk \;
V_{\rm survey}} \left( \frac{1 + nP_h}{nP_h} \right)^2,
\end{equation}
where $V_{\rm shell} = 4\pi k^2 dk/(2\pi)^3$ is the volume of the shell in
Fourier space (we are ignoring redshift distortion effects for simplicity here).
Therefore, the Fisher matrix for measurements of $P_h(k, z)$ is
\cite{Tegmark97}
\begin{equation}
F_{ij} = \sum_m V_m \int_{k_{\rm min}}^{k_{\rm max}}
\frac{\partial P_h(k, z_m)}{\partial p_i} \frac{\partial P_h(k, z_m)}{\partial
p_j}
\,\frac{1}{\left[ P_h(k, z_m) + \displaystyle\frac{1}{n} \right]^2}\, \frac{k^2 dk}{(2\pi)^2},
\label{PowerFisher}
\end{equation}
where $V_m$ is the comoving
volume of the $m$-th redshift bin, each redshift bin is centered on $z_m$, and we have summed over all redshift bins.
We adopt $k_{\rm min}=10^{-4}\,h^{-1}\,{\rm Mpc}$, and we choose $k_{\rm
max}$ as a function of $z$ so that $\sigma(\pi/(2k_{\rm max}), z) = 0.5$
\cite{SeoEisenstein2003}, which leads to $k_{\rm max}(z= 0)
\approx 0.1h\,{\rm Mpc}^{-1}$.
Finally, $p_{i}$ are the parameters of interest; in our case, these are the
$f_{\rm NL}^i$.
We assume a flat universe and a fiducial model of zero non-Gaussianity:
$f_{\rm NL}(k)=0=f_{\rm NL}^i$. We include six cosmological parameters in our Fisher
matrix aside from the $f_{\rm NL}^i$: Hubble's constant $H_0$; physical dark matter
and baryon densities $\Omega_{\rm cdm} h^2$ and $\Omega_{\rm b} h^2$; equation
of state of dark energy $w$; the log of the scalar amplitude of the matter
power spectrum, $\log A_s$; and the spectral index of the matter power
spectrum, $n_s$ . Fiducial values of these parameters correspond to their
best-fit WMAP7 values \cite{wmap7}. We also added the forecasted cosmological
parameter constraints from the CMB experiment Planck by adding its Fisher
matrix as a prior (W.\ Hu, private communication). Note that the CMB prior
does {\it not} include CMB constraints on non-Gaussianity; the CMB constraints
on $f_{\rm NL}(k)$ will be separately studied in a future
work. Finally, in addition to the cosmological parameters and
the $f_{\rm NL}^i$, we include five Gaussian bias parameters
in our Fisher matrix -- one $b_0(z)$ for each redshift bin. The fiducial
values of these parameters are set by the relations $b_0(z =0) = 2.2$, and
$b_0(z) = b_0(z = 0) / D(z)$.
We already have the derivatives of $b(k)$ with respect to each of the
$f_{\rm NL}^i$, so the derivative of $P_h(k)$ with respect to the $f_{\rm NL}^i$ is just
\begin{equation}
\label{BiasDerivative}
\frac{\partial P_{h}(k)}{\partial f_{\rm NL}^i}
= 2\, \frac{\partial b(k)}{\partial f_{\rm NL}^i}\, b(k) P_{\rm mat}(k);
\end{equation}
$P_{\rm mat}(k)$ is the $\Lambda$CDM matter power spectrum, easily
obtained from a numerical code such as CAMB. Since we only consider
information from large scales ($k\leq k_{\rm max} \approx 0.1\,h\,{\rm Mpc}^{-1}$), we do not
model the small amount of nonlinearity present at the high-$k$ end of these
scales.
We assume a future survey covering one-quarter of the sky (about 10,000
square degrees) out to $z = 1$, and find constraints for a set of 20 $f_{\rm NL}^i$
uniformly spaced in $\log k$ in the range $10^{-4} \leq k/(h\,{\rm Mpc}^{-1})\leq 1$, with a smoothing scale of
$M_{\rm smooth}=10^{14}M_\odot$.
Fig.~\ref{fig:results} shows the resulting unmarginalized (left panel) and marginalized
(right panel) constraints on the parameters $f_{\rm NL}^i$.
For both sets of constraints, we first marginalized over the other
cosmological parameters.\footnote{ Using six cosmological parameters along
with five $b_0(z)$ and 20 $f_{\rm NL}^i$ led us into some issues with
floating-point errors and numerical precision. The $31 \times 31$ Fisher
matrix we obtained was rather ill-conditioned and difficult to invert
reliably using 64-bit precision; we were eventually forced to move to
128-bit precision in order to accurately marginalize over the cosmological
parameters. } The $f_{\rm NL}^i$ have most of their degeneracy
among themselves; a plot showing the fully unmarginalized constraints on the
$f_{\rm NL}^i$ would not look much different than the left panel of Fig.~\ref{fig:results}. Note that, while some of the $f_{\rm NL}^i$ have support at
$k> k_{\rm max}(z = 1) \approx 0.2\,h\,{\rm Mpc}^{-1}$, we only use
information about those (and other) parameters coming from $k<k_{\rm max}$.
The constraints vary considerably as a function of the $k$ at which these
parameters are defined. The best-constrained $f_{\rm NL}^i$ corresponds to the
$10^{-0.8} < k <10^{-0.6}$ bin, and it has an estimated unmarginalized error
of $\sigma(f_{\rm NL}^{16}) = 7.3$; for comparison, the worst-constrained $f_{\rm NL}^i$,
which corresponds to the largest scale (smallest $k$) bin, has an
unmarginalized error well over one billion.
As expected, the marginalized constraints for the best-constrained parameters
are much weaker than the unmarginalized constraints -- even the best-measured
$f_{\rm NL}^i$ has an estimated marginalized error of $6 \times 10^2$. In general,
dependence of the constraints on the value of $k$ is determined by two
competing factors: as $k$ increases, there is a larger number of modes, each
with a smaller signal (given by the smaller nongaussian bias $\Delta
b$). The best-constrained $k$ is also affected by the fact that only
information out to $k=k_{\rm max}=0.1h\,{\rm Mpc}^{-1}$ is assumed from the galaxy
survey. In particular, we have checked that if we unrealistically assume
information to be available at all $k$ (instead of at $k< k_{\rm max}$)
without modeling the nonlinearities, the unmarginalized constraints on
$f_{\rm NL}^{i}$ improve monotonically with increasing $k$. Therefore, the raw
signal-to-noise ratio in $f_{\rm NL}^{i}$ increases with $k$. To
further demonstrate the effect of the choice of $k_{\rm max}(z)$, we also
plotted the errors obtained with the condition $\sigma(\pi/(2k_{\rm max}),
z) = 0.15$, which yields $k_{\rm max} (z = 0) \approx 0.03$.
\begin{figure}[t]
\begin{center}
\subfigure[Unmarginalized errors]{\includegraphics[width= 2.9in]{unmarginalized_fnlk_errors.pdf}}
\subfigure[Marginalized errors]{\includegraphics[width= 2.9in]{marginalized_fnlk_errors.pdf}}
\caption{Estimated unmarginalized (left panel) and marginalized (right
panel) constraints on piecewise-constant parameters $f_{\rm NL}^i$
assuming a future galaxy survey covering one-quarter of the sky out
to $z = 1$, with average number density of $2\times10^{-4}$
gal/Mpc$^3$. For comparison, the green line is the constraint found for a constant $f_{\rm NL}$
using the same survey assumptions, and the red histograms are the constraints found with a lower $k_{\rm max}$ (see text for details). While the
individual parameters $f_{\rm NL}^i$ are poorly constrained as expected,
their few best linear combinations -- the principal components --
are well measured; see the next section and text for details.
}
\label{fig:results}
\end{center}
\end{figure}
The smoothing mass scale chosen for this analysis (see
Eq.~(\ref{eq:overdensity})) has a small but noticeable effect on the constraints
yielded. Figure \ref{fig:massdep} shows that, in the case of the
unmarginalized errors, the $k$ at which non-Gaussianity is best constrained
decreases as the smoothing mass scale increases. (The behavior of the
marginalized errors is more complicated due to correlations in errors between
neighboring $f_{\rm NL}^i$.) Since the mass scale is proportional to
the physical scale (to the third power), this means that best-constrained $k$
decreases with increasing smoothing scale $R$, which is exactly what we should
expect. We remind the reader that while a survey filtered at some scale
$M_{\rm smooth}$ contains objects roughly more massive than this scale, in
practice the near-exponentially falling mass function implies that the number
density is dominated with $M\simeq M_{\rm smooth}$ halos.
\begin{figure}[t]
\begin{center}
\subfigure[Unmarginalized errors]{\includegraphics[width= 2.9in]{unmarginalized_mass_scales.pdf}}
\subfigure[Marginalized errors]{\includegraphics[width= 2.9in]{marginalized_mass_scales.pdf}}
\caption{ Estimated constraints obtained from future surveys with the
same parameters as the previous figure at different mass smoothing
scales $M_{\rm smooth}$ (labeled as $M$ in the legend). In other
words, these are errors for a survey with halos of $M\gtrsim M_{\rm smooth}$.
}
\label{fig:massdep}
\end{center}
\end{figure}
\section{Projection and Principal Components}
\label{sec:PC}
\subsection{Constraining other $f_{\rm NL}(k)$ models}
\label{sec:other_models}
Once the Fisher matrix $F$ has been obtained for the set of parameters
$f_{\rm NL}^i$, it is quite simple to find the best possible constraints on the
$f_{\rm NL}^i$ that could be obtained from a future galaxy redshift survey. By
projecting this Fisher matrix into another basis (see Appendix
\ref{app:projection}), it is also possible to find the constraints on any
arbitrary $f_{\rm NL}(k)$ without calculating a new Fisher matrix from scratch. A
trivial example can be found in Appendix \ref{app:projection}, where we find that
the estimated error on a constant $f_{\rm NL}$, assuming the same future survey as in the previous section, is
$\sigma(f_{\rm NL}) = 2.1$. (Note that this forecasted constraint is on a par
with the error expected from Planck, where $\sigma(f_{\rm NL}) \sim 5$.)
For another, scale-dependent example, consider the simple form of
non-Gaussianity analogous to the conventional parameterization of the power
spectrum
\begin{equation}
f_{\rm NL}(k) = f_{\rm NL}^* \left( k \over k_* \right)^{n_{\rm NG}},
\label{eq:runningfnl}
\end{equation}
where $k_*$ is an arbitrary fixed parameter, leaving $f_{\rm NL}^*$ and $n_{\rm NG}$
as the parameters of interest in this model. ($k_*$ is generally chosen to
minimize degeneracy between $f_{\rm NL}^*$ and $n_{\rm NG}$ for the observable of
interest. We have set $k_* = 0.165 h\,{\rm Mpc}^{-1}$, close to the optimal value in our
case; in CMB analysis, the optimal value is lower, around $0.06 h\,{\rm Mpc}^{-1}$.) The partial
derivatives of our basis of $f_{\rm NL}^i$ with respect to these parameters are:
\begin{eqnarray}
{\partial f_{\rm NL}^i \over \partial f_{\rm NL}^*} &=& \left( k \over k_* \right)^{n_{\rm NG}};
\\[0.2cm]
{\partial f_{\rm NL}^i \over \partial n_{\rm NG}} &=& f_{\rm NL}^* \left( k \over k_* \right)^{n_{\rm NG}} \log \left( k \over k_* \right).
\label{eq:sefusattipartials}
\end{eqnarray}
Starting in a basis of 20 $f_{\rm NL}^i$ evenly spaced in log $k$, we project down
to a basis of $f_{\rm NL}^*$ and $n_{\rm NG}$ in order to forecast constraints on the two new
parameters from a survey covering one-quarter of the sky out to $z=1$. We are
using the same limits of integration as in Section \ref{sec:constraints}, along with
the fiducial values $f_{\rm NL}^* = 50$ and $n_{\rm NG} = 0$. The forecasted
constraints on these parameters, marginalized over each other, are
$\sigma_{f_{\rm NL}^*} = 1.7$ and $\sigma_{n_{\rm NG}} = 0.58$. Despite a
superficial similarity between this model and the model used by Sefusatti et
al.\ in \cite{Sefusatti2009}, the two models are quite different, and our
results cannot be compared. The model used in \cite{Sefusatti2009} is a
function of three arguments, $k_1, k_2$, and $k_3$:
\begin{equation}
f_{\rm NL}(k_1, k_2, k_3) = f_{\rm NL}^* \left ({K \over k_*}\right )^{n_{NG}},
\label{eq:sefusatti_fnl}
\end{equation}
where $K = (k_1 k_2 k_3)^{1/3}$. This leads to a bispectrum of the form found
in Eq.~(\ref{eq:local_bispec}), but with $f_{\rm NL}(k_1, k_2, k_3)$ in place of $f_{\rm NL}$,
whereas our bispectrum is of the less-factorizable form
Eq.~(\ref{eq:fnlk_bispec}).
Another example we consider is the form of non-Gaussianity in which the
running on $f_{\rm NL}$ itself has running; that is, the case in which $n_{\rm NG}$
is a function of $k$. A simple case of this would be $f_{\rm NL}$ of the
form\footnote{Analogous parameterization for the power spectrum and its
motivations are discussed in \cite{kev2}.}
\begin{equation}
f_{\rm NL}(k) = e^{A k^B}.
\label{eq:runningrunning}
\end{equation}
Projecting the Fisher matrix down from the original basis $f_{\rm NL}^i$ to the
parameters $A$ and $B$, with fiducial values of $A = \log 50$ and $B = 0$, we
obtain forecasted constraints of $\sigma_A = 1.0$ and $\sigma_B = 0.15$. (In
this case, the survey characteristics and bounds of integration are the same
as in the previous example.)
\subsection{Principal components and relation to local and equilateral models}
\begin{figure}[t]
\begin{center}
\includegraphics[width=6in]{naivepc-0.pdf} \\[-0.5cm]
\includegraphics[width=6in]{naivepc-1.pdf} \\[-0.5cm]
\includegraphics[width=6in]{naivepc-2.pdf} \\[-0.5cm]
\includegraphics[width=6in]{special_naivepc-3.pdf}
\end{center}
\caption{The first four principal components of $f_{\rm NL}(k)$. The PCs,
$e^{(j)}(k)$, are eigenvectors of the Fisher matrix for the $f_{\rm NL}^i$, and
are ordered from the best-measured one ($j=0$) to the worst-measured one
($j=19$) for the assumed fiducial survey. }
\label{fig:unsplinedPC}
\end{figure}
We now represent a general function $f_{\rm NL}(k)$ in terms of
principal components (PCs). In this approach, the {\it data} determine which
particular modes of $f_{\rm NL}(k)$ are best or worst measured. The PCs also
constitute a useful form of data compression, so that one can keep only a few of
the best-measured modes to make inferences about the function
$f_{\rm NL}(k)$. Finally, the PCs will also enable us to measure the degree of
similarity between our scale-dependent ansatz and the local and equilateral
forms of non-Gaussianity.
\begin{figure}[t]
\begin{center}
\includegraphics[width= 6.2in]{pc_errors.pdf}
\caption{ RMS error on each principal component, along with the
cumulative error. }
\label{fig:cumerrs}
\end{center}
\end{figure}
It is rather straightforward to start from the covariance matrix for the
piecewise constant parameters $f_{\rm NL}^i$ and obtain the PCs of $f_{\rm NL}(k)$. The PCs are weights in wavenumber with amplitudes that
are uncorrelated by construction, and they are ordered
from the best-measured ($i=0$) to the worst-measured ($i=19$) for the
assumed fiducial survey. The construction of the PCs is described in Appendix
\ref{app:PC}. A few of these PCs of $f_{\rm NL}(k)$ are shown in
Fig.~\ref{fig:unsplinedPC}. For example, the best-measured PC has most of its
weight around $k=10^{-0.4}\,h\,{\rm Mpc}^{-1}$, which agrees with sensitivities of
piecewise-constant parameters shown in Fig.~\ref{fig:results}. The sensitivity is not greatest at the largest value of $k$ ($1\,h\,{\rm Mpc}^{-1}$) because we assumed cosmological information from $k\leq k_{\rm
max}=0.1\,h\,{\rm Mpc}^{-1}$. We checked that information available at a higher
$k_{\rm max}$ would shift the ``sweet spot" of sensitivity to higher wavenumbers.
The error in the best-measured PC is 4.8; however, the error in the next-best
measured PCs are 18.3 and 27.4, and the accuracy rapidly drops off from
there. Thus, the first three or four PCs should be enough for any conceivable
application. The error in each PC is plotted on a logarithmic scale in figure
\ref{fig:cumerrs}, along with the cumulative error $\sigma_{\rm cum}$, which
is defined as
\begin{equation}
{1 \over \sigma^2_{\rm cum}} = \sum_{i} {1 \over \sigma^2_i}.
\label{eq:cum_error}
\end{equation}
Each PC $e^{(j)}(k)$ has its own associated bispectrum (see Eq.~(\ref{eq:fnlk_bispec})):
\begin{equation}
\label{eq:pc_bispec}
B^{(j)}(k_1, k_2, k_3) =
2[ e^{(j)}(k_1) P(k_2)P(k_3) + e^{(j)}(k_3) P(k_1)P(k_2) + e^{(j)}(k_2) P(k_3)P(k_1) ].
\end{equation}
(As always, $k_1, k_2,$ and $k_3$ have a triangle relation: $ k_3 = | \vec{k_2} - \vec{k_1} | $.)
We would like to test the similarity of these bispectra to
those that have already been discussed in the literature. We can do this by
using a distance measure between bispectra, defined by `cosines'
developed in \cite{Babich_shape}. A cosine near unity implies that the two
bispectra have very similar shapes, and a cosine near zero implies the
opposite. The cosine is defined as
\begin{equation}
\label{cosine}
\cos(B_1, B_2) = \frac{B_1
\cdot B_2} { \sqrt{ \left( B_1 \cdot B_1 \right) \left( B_2 \cdot B_2
\right) }},
\end{equation}
where the inner product between two bispectra, $B_1 \cdot B_2$, is
\cite{Sefusatti2009}
\begin{equation}
\label{innerprod}
B_1 \cdot B_2 = \sum_{k_1, k_2, k_3} \frac{ B_1(k_1,
k_2, k_3) B_2(k_1, k_2, k_3) } { \Delta^2 B(k_1, k_2, k_3) }.
\end{equation}
The (Gaussian) variance of the bispectrum is
\begin{equation}
\label{gaussvar}
\Delta^2 B(k_1, k_2, k_3)
= \frac{1}{N_T} P(k_1)P(k_2)P(k_3)
\sim \frac{1}{N_T} (k_1 k_2 k_3)^{-3},
\end{equation}
where $N_T$ is the number of distinct triangular configurations of
$k_{1,2,3}$, and $P(k) \sim k^{-3}$ is the primordial curvature perturbation
power spectrum. (The overall constant is irrelevant, since it cancels out in
Eq.~(\ref{cosine}).)
\begin{table}[t]
\centering
\begin{tabular}
{ | c | c | c | }
\hline
& Local cosine & Equilateral cosine \\
\hline
$B^{(0)}$ & 0.669 & 0.074 \\
\hline
$B^{(1)}$ & 0.040 & 0.000 \\
\hline
$B^{(2)}$ & 0.099 & 0.030 \\
\hline
$B^{(3)}$ & 0.189 & 0.037 \\
\hline
\end{tabular}
\caption{Cosines of the first four
principal-component derived bispectra with the local bispectrum and the equilateral
bispectrum. A cosine near unity implies that the two bispectra have very
similar shapes, and a cosine near zero implies the opposite. Note that the zeroth PC, which is by far the best
measured (see Fig.\ \protect\ref{fig:cumerrs}), has a much larger overlap with the local model than with the
equilateral, as expected. }
\label{unsplinedcosines}
\end{table}
We first compare our bispectra Eq.~(\ref{eq:pc_bispec}) to the local model with
a constant $f_{\rm NL}$, whose bispectrum is (see Eqs.~(\ref{eq:localNG}) and
(\ref{eq:local_bispec}))
\begin{equation}
\label{squeezedB}
B_{\rm local}(k_1, k_2, k_3) \propto \frac{1}{k_1^3 k_2^3} + \frac{1}{k_1^3 k_3^3} + \frac{1}{k_2^3 k_3^3}.
\end{equation}
Most of the power of $B_{\rm local}$ is in so-called ``squeezed'' triangles,
in which one side is much smaller than the other two (comparable) sides, $k_1
<< k_2 \approx k_3$.
Another form for the bispectrum much discussed in the literature is the
``equilateral" bispectrum
\begin{equation}
\label{equiB}
B_{\rm equi}(k_1, k_2, k_3) = - \frac{2}{(k_2 k_1 k_3)^2} - B_{\rm local}(k_1, k_2, k_3) + \frac{1}{k_1 k_2^2 k_3^3} + \frac{1}{k_3 k_1^2 k_2^3} + \text{permutations}.
\end{equation}
In contrast with $B_{\rm local}$, most of the power of $B_{\rm equi}$ is in
triangles where $k_1 \approx k_2 \approx k_3$; hence the name ``equilateral''.
Table
\ref{unsplinedcosines} lists the cosines of the first few principal-component
derived bispectra with the local bispectrum and the equilateral bispectrum.
The form of Eq.~(\ref{eq:pc_bispec}) suggests that the PC-derived bispectra $B^{(j)}$ will have
more in common with the local bispectrum than the equilateral one. However,
it is initially conceivable that some $e^{(j)}(k)$ might exist which would yield
a bispectrum of the form in Eq.~(\ref{equiB}) when substituted into
Eq.~(\ref{eq:pc_bispec}) -- but in Appendix \ref{sec:ProofAppendix}, we
prove that \textit{no} such function exists. Thus, the only guarantees for the cosines of the $B^{(j)}$
are that the cosine of $B^{(0)}$ -- the bispectrum corresponding to the best-measured PC -- will
be large with the local model, and that none of the $B^{(j)}$ have a very large
cosine with the equilateral model. We expect the former because our model
looks like the local model; we expect the latter because of the proof in
Appendix \ref{sec:ProofAppendix}. Table \ref{unsplinedcosines} bears out this
expectation. The small cosines with the equilateral form of non-Gaussianity are also unsurprising because equilateral non-Gaussianity is expected to have a strongly suppressed signal in the non-Gaussian halo bias \cite{MV09}.
\section{Conclusions}
\label{sec:concl}
In this paper we have suggested a new phenomenological model of primordial
nongaussianity by generalizing the local model (parametrized with a constant
parameter $f_{\rm NL}$) to a scale-dependent, non-local class of models. There are
multiple ways to do this, and our choice was to write the Newtonian potential
as
\begin{equation}
\Phi(x)= \phi_G(x)+f_{\rm NL}(x)*(\phi_G(x)^2-\langle \phi_G (x)^2 \rangle ),
\end{equation}
where the convolution in real space corresponds to multiplication in
$k$-space, featuring an arbitrary function $f_{\rm NL}(k)$. Explicit calculations
show that such a form of the scale dependent $f_{\rm NL}$ is borne out in inflationary
models \cite{Salopek,Gangui_etal,chris5,chris6,huang}.
We calculated the bispectrum and bias of dark matter halos in this class of
models, following the formalism valid for high peaks
\cite{Grinstein:1986en,MLB1986}. We then specialized in the
piecewise-constant (in wavenumber) parametrization of $f_{\rm NL}(k)$ which, for the
case of narrow enough $k$-bins, recovers any arbitrary function. We used
forecasted constraints from an intermediate-future galaxy survey to calculate
errors on individual parameters $f_{\rm NL}^i$ (see Fig.~\ref{fig:results}) and
briefly studied dependence on the smoothing scale (Fig.~\ref{fig:massdep}).
We further calculated the principal components of $f_{\rm NL}(k)$, and thus
identified the best-measured configurations (in wavenumber) of this function
(see Fig.~\ref{fig:unsplinedPC}). While the sensitivity increases with
increasing $k$, restricting the survey information to scales where linear
perturbation theory is valid imposes a ``sweet spot'' in sensitivity of
$k\sim 0.1h\,{\rm Mpc}^{-1}$. We then calculated the overlap of the best-measured
principal components with two familiar classes of non-Gaussian models: local
($f_{\rm NL}={\rm const}$) and equilateral models, using a cosine measure between
the bispectra suggested in \cite{Babich_shape}. We found the expected result:
the best measured component overlaps much more with the local model (which our model generalizes) than with the equilateral one.
One immediate utility of our results is an easy adaptation to specific models of
non-Gaussianity predicted by classes of inflationary models.
If one wants to forecast the
accuracy with which parameters of a specific model of $f_{\rm NL}(k)$-style non-Gaussianity will be measured, neither
the halo bias nor the Fisher matrix needs to be calculated from scratch.
Instead, our formalism makes it possible to obtain these
forecasts by performing a simple linear projection to our piecewise-constant model; this
procedure is described in Appendix \ref{app:projection} and illustrated with a
few examples.
In future investigations, it will be interesting to consider specific
inflationary models, projecting down to specific forms for $f_{\rm NL}(k)$. It will
also be important to test how well the observable effects of scale-dependent
non-Gaussianity, studied here using the theoretical ansatz from
Eq.~(\ref{eq:Grinstein-Wise}), agree with numerical simulations; the first such
investigations, for select specific forms of $f_{\rm NL}(k)$, are now being done
\cite{Shandera2010}. Finally, it will be interesting to see how one can
optimally select objects in the universe (i.e.\ their mass) to probe
information about scale-dependence of non-Gaussianity. While in
Fig.~\ref{fig:massdep} we showed scaling of the best-determined scale of
$f_{\rm NL}(k)$ with the smoothing mass scale applied to the density field, a more
complete analysis might use the Halo Occupation Distribution (HOD) approach to
relate the content of dark matter halos to their mass.
\section{Acknowledgements}
We thank Chris Byrnes and Sarah Shandera for useful discussions, and the anonymous referee for constructive comments. AB and DH
are supported by DOE OJI grant under contract DE-FG02-95ER40899, NSF under
contract AST-0807564, and NASA under contract NNX09AC89G. KK is supported in
part by the Michigan Center for Theoretical Physics. DH and KK would like to thank the
Aspen Center for Physics where this project germinated, and DH also
acknowledges the generous hospitality of Centro de Ciencias de Benasque ``Pedro
Pascual''.
|
{
"timestamp": "2011-01-18T02:00:23",
"yymm": "1009",
"arxiv_id": "1009.4189",
"language": "en",
"url": "https://arxiv.org/abs/1009.4189"
}
|
\section{Introduction}
\subsection{The model}
\label{1p1}
The Hubbard model, with nearest-neighbor hopping $t$ and on-site repulsion $U
$ has been widely used to study the effects of correlations, as a simplified
model to describe compounds of transition metals and other systems. However,
one expects that in any system, in general the hopping between two sites
depend on the occupation of these two sites, which leads to the presence of
bond-charge interactions (also called correlated hopping terms) in the
Hamiltonian [such as $X$ in Eq. (\ref{hamil}) and $\sum_{\sigma ,\langle
ij\rangle }(c_{i\sigma }^{\dagger }c_{j\sigma }+{\rm H.c.})n_{i-\sigma
}n_{j-\sigma }$]. For example, in simple systems with one relevant orbital
per site, one would expect that when electrons are added to one site, the
screening of the core charge increases, and as a consequence, the wave
function of the orbital expands and the hopping to the nearest sites
increases. In general, any one-band effective model derived from more
complex Hamiltonians to describe the low-energy physics of some system, contains bond-charge interactions.
In fact, the generalized Hubbard model with correlated hopping terms has been derived and used to
describe the low-energy physics of intermediate valence systems \cite{foglio},
organic compounds \cite{kive,baeri,gamm,zhang,stra,br}, a Hubbard model
including lattice vibrations \cite{phon}, cuprate superconductors \cit
{schu,simon,opt}, and more recently optical lattices \cite{duan,good,kest}.
This is particularly interesting because the parameters can be tuned
experimentally in a wide range \cite{kest,duan2}. First-principles
calculations in transition-metal complexes suggest that the correlated
hopping terms can be large \cite{impu}.
As we shall see, the presence of bond-charge interaction leads to
qualitatively new physics. One example is that in two dimensions $d$-wave
pairing correlations, which are already present in the Hubbard model \cit
{scala} are strongly enhanced in the generalized Hubbard model for the
cuprates \cite{lili}, and one obtains $d$-wave superconductivity already at
the mean-field level \cite{dw}. In one dimension (1D), field-theoretical
\cite{jaka,bos} and numerical \cite{bos,topo} results show the presence of a
spontaneously dimerized bond-ordering wave (BOW) and a phase with dominant
triplet superconducting correlations at large distances that are absent in
the ordinary Hubbard model. For special values of the parameters, the model
with two- and three-body interactions has been solved exactly by the Bethe
ansatz \cite{igor}.
The simplest model with bond-charge interaction has been proposed by Hirsch
motivated by his theory of hole superconductivity \cite{hirsch,hir2}. The
Hamiltonian can be written as
\begin{equation}
H=-t\sum_{\sigma =\uparrow ,\downarrow ,\langle ij\rangle }(c_{i\sigma
}^{\dagger }c_{j\sigma }+{\rm H.c.})+U\sum_{i}n_{i\uparrow }n_{i\downarrow
}+X\sum_{\sigma ,\langle ij\rangle }(c_{i\sigma }^{\dagger }c_{j\sigma }
{\rm H.c.})(n_{i-\sigma }+n_{j-\sigma }). \label{hamil}
\end{equation
In 1D, the model can display a phase with dominant singlet superconducting
(SS) correlations, even for positive $U$ \cite{bos,japa,tll,bulka}. For $X=t
, the model has been solved exactly and there is a metal-insulator
transition for increasing $U$ \cite{exac,boer,flux}. More recently, the role of
entanglement in this quantum phase transition has been studied \cite{entro,agm}.
The response of the system to an applied magnetic flux
indicates that the metallic phase is not superconducting \cite{flux}.
However, the ground state is highly degenerate in this phase at it is
difficult to predict from the exact solution what happens when the degeneracy is
lifted by a small but finite $X-t$. In any case, standard bosonization \cit
{jaka,bos,japa} and numerical studies \cite{tll} have provided a general
physical picture of the behavior of the model, except at half filling. In
this case, taking as usual only the leading terms in the lattice constant $a
, $X$ disappears in the bosonization treatment [it enters as $X\cos (\nu \pi
)$, where $\nu $ is the filling fraction \cite{japa,bos}]. Therefore,
standard field theory predicts a SS phase for $U<0$, and a spin-density wave
(SDW) phase for $U>0$, as in the usual Hubbard model. However, a charge
insulator-metal transition driven by $X$ at finite $U_{c}>0$ has been found
numerically \cite{sup-ins} and later a quantum phase diagram has been
derived which includes a BOW phase \cite{cola}.
Recently, new numerical studies of the model at arbitrary filling identify
regions of phase separation for $X > 0.5 t$ \cite{ari2}.
\subsection{Phase diagram at half filling}
\label{1p2}
\begin{figure}[tbp]
\includegraphics[width=14cm]{hirf1.eps}
\caption{Phase diagram of the model at half filling obtained from the method
of level crossings. Squares: charge transition. Solid circles: spin
transition. The unit of energy is taken as $t=1$.}
\label{pd}
\end{figure}
The quantum phase diagram for $\nu =1/2$ has been obtained by a combination
of different numerical techniques \cite{cola,ari2}. In Fig. \ref{pd} we reproduce
the phase diagram obtained by the method of topological transitions
\cite{topo,abb}. These transitions correspond to jumps in the charge and spin
Berry phases which signal the corresponding transitions between the
thermodynamic phases, and coincide with a corresponding crossing of excited
levels, justified on the basis of conformal-field theory \cite{abb,naka}.
These quantities are also related to charge and spin localization indicators
\cite{loc1,loc2,pss,toro} used for example to characterize valence-bond-solid
states in quantum spin chains \cite{nato}. While changes in the Berry phase
are proportional to changes in polarization, the spin Berry phase tensor
provides a geometric characterization of the ferrotoroidic moment \cite{toro}.
The critical values of $U$ for the charge ($U_{c}$) and spin ($U_{s}$)
transition have been calculated in systems of up to $L=14$ lattice sites,
and extrapolated to the thermodynamic limit using a parabola in $1/L^{2}$.
Fig. \ref{pd} displays the extrapolated values for $0\leqslant X\leqslant 1$.
It is important to note that this interval can be extended to the whole
real axis using symmetry properties of the Hamiltonian \cite{tll}. A change
of phase of half of the sites [$c_{i\sigma }^{\dagger }\rightarrow
(-1)^{i}c_{i\sigma }^{\dagger }$], interchanges the signs of $t$ and $X$, so
that $H(-t,-X,U)\equiv H(t,X,U)$. Combining this with an electron-hole
transformation ($c_{i\sigma }^{\dagger }\rightarrow c_{i\sigma }$), one
obtains at half filling
\begin{equation}
H(t-2X,-X,U)\equiv H(t,X,U)\equiv H(2X-t,X,U). \label{sym}
\end{equation
Then, if the critical ratio $u(x)$ ($U_{c}/t$ or $U_{s}/t$) with $x=X/t$, $t>0$,
is known in the interval $0\leqslant x<1/2$, it can be
extended to negative values of $X$ using the first Eq. (\ref{sym}).
Similarly, the second Eq. (\ref{sym}) maps the interval $1/2<x\leqslant 1$
onto $x\eqslantgtr 1$. Explicitly
\begin{equation}
{\rm if }x<0{\rm , \; }u(x)=(1-2x)u\left( \frac{-x}{1-2x}\right) {\rm ; \; if
x\eqslantgtr 1{\rm , \; }u(x)=(2x-1)u\left( \frac{x}{2x-1}\right).
\label{rel}
\end{equation
It is interesting to note that the end points $x\rightarrow \pm \infty $ are
mapped onto $x=1/2.$
For $U>U_{c}$ ($U<U_{s}$), the system has a charge (spin) gap. For $U<U_{c}
, the system is in the SS phase, while for $U>U_{s}$, the system is in the
SDW phase, according to the dominant correlation functions at large
distances. In between, for $U_{c}<U<U_{s}$, one has the fully gapped BOW
phase. For small values of the interactions, the dominant correlation
functions in each phase can be understood from field theory \cite{jaka,bos}.
For $X>t/2$, the SS phase displays incommensurate correlations \cite{cola},
which can be qualitatively understood using a mean field approximation in
one of the terms obtained from bosonization \cite{cola}, leading to a
commensurate-incommensurate transition, with some similarities to the
physics of the Hubbard model when a large next-nearest-neighbor hopping is
added \cite{ttpu}, and some spin systems \cite{nerse}. The spontaneously
dimerized BOW phase has also been also found in the Hubbard model with
alternative on-site energies \cit
{abb,fab1,fab2,torio,manma,rap,bat1,dimer,tinca}, where it displays
ferroelectricity \cite{abb,bat1,dimer}.
The presence of the BOW phase in this model was
first predicted using field theory
and bosonization \cite{fab1,fab2}.
For $X<t/2$, the numerical results for the charge transition are consistent
with $U_{c}=0$, as in the ordinary Hubbard model \cite{cola}. The accuracy
of the results are not enough to establish if there is a kink or not at
X=t/2$, $U_{c}=0$. Arianna Montorsi has found that a good fit of the
numerical results for $t/2\leqslant X\leqslant t$ is \cite{ari}
\begin{equation}
{\rm if }x\eqslantgtr 1/2{\rm , \; }u_{c}=4\sqrt{2x-1}{\rm , \; } \label{am}
\end{equation
which is consistent with a kink at $x-1/2=u_{c}=0$, and has the nice
property that when it is extended analytically to $x\eqslantgtr 1$, it
satisfies the second symmetry relation (\ref{rel}). The value $U_{c}=4t$ for
$X=1$, is consistent with the exact solution \cite{exac,boer,flux} To our
knowledge, no justification of Eq. (\ref{am}) exists so far.
It has been verified that the spin transition is of Kosterlitz-Thouless
type \cite{cola}. In contrast to $U_{c}$, the critical value for the spin
transition $U_{s}(X)$ represented in Fig. \ref{pd} is smooth. For small
values of $X$, it increases as $X^{2}$, while for $X\sim t/2$
there is an inflection point. For $X<t$, $U_{s}>U_{c}$. While for $X=t$,
U_{s}=U_{c}=4t$, the second Eq. (\ref{rel}) implies that for $X>t$, also
U_{s}>U_{c}$. Therefore, there is no crossing between $U_{c}(X)$ and
U_{s}(X)$ at $X=t$. The fact that there is a finite value of $U_{s}(1/2)$
and the second Eq. (\ref{rel}) imply that $U_{s}(X)$ grows linearly with $X$
for $X\rightarrow +\infty $, in contrast to the $\sqrt{X}$ behavior for the
charge transition predicted by Eq. (\ref{am}).
\subsection{Previous field-theoretical results}
\label{1p3}
As stated in Section \ref{1p1}, standard continuum limit field theory and
bosonization fails at half filling because $X$ disappears from the $g_{i}$
coupling constants \cite{japa,bos}. In Ref. \cite{cola} we have calculated
vertex corrections to these $g_{i}$ using second order perturbation theory
in the bond-charge interaction $X$. The approach is similar to that done by
Tsuchiizu and Furusaki for the Hubbard model extended with nearest-neighbor
repulsion \cite{japan}, but for our Hamiltonian, Eq. (\ref{hamil}) it is not
necessary to introduce a low-energy cutoff. This approach led to the
following critical $U$ at the spin transition
\begin{equation}
U_{s}=\frac{8X^{2}}{\pi (t-X)} \label{usve}
\end{equation
This function lies below the numerical points in Fig. \ref{pd}, but seems to
represent correctly the limit $X\rightarrow 0$. However, unfortunately the
prediction of this approach for the charge transition is $U_{c}\sim U_{s}/2$
for small $X$, instead of $U_{c}=0$ found numerically. In addition, with
vertex corrections only, it is not possible to explain the nature of the
incommensurate SS phase for $X>t/2$.
In Ref. \cite{cola} we have also considered in the bosonized theory,
a term in next to
leading order in the lattice parameter $a$, which couples charge and spin in
a mean-field approximation. However, this approximation is questionable, and
the quantitative agreement between the analytical and numerical results for the charge
transition is poor.
In this work we include all terms of next to leading order in $a$, and
include them in a renormalization group (RG) treatment. This approach is
superior to perturbation theory in $X$ (as included in Ref. \cite{cola}
through vertex corrections). Retaining all these terms leads to a lengthy
algebra, but unfortunately selecting only a few of them, breaks the SU(2)
symmetry and leads to wrong results. Since our approach is a weak coupling
one, we restrict our study to $X<t/2$, which seems to be the more realistic
regime of parameters. Our effort is rewarded by an excellent agreement with
the numerical results for both critical values of $U$ at the corresponding
transitions.
\section{The field-theoretical approach}
\subsection{The continuum limit}
\label{2p1}
In order to construct the low-energy field theory for the Hamiltonian Eq.
\ref{hamil}), we suposse that both $U$ and $X$ are small. Therefore,
in the Fourier development of fermion operators we retain only the modes near $-k_F$ and
$k_F$,
where $k_F$ is the Fermi wave vector. Introducing a cutoff $\Lambda<< 1/a$,
where $a$ is the lattice parameter, and calling
$L$ the lenght of the system, the local annihilation operator $c_{n \sigma}$ can be written as:
\begin{eqnarray}
c_{n \sigma}&=&\sqrt{\frac{a}{L}} \sum^{\frac{\pi}{a}}_{k=-\frac{\pi}{a}} e^{ik n a} c_{k\sigma}\nonumber\\&\sim& \sqrt{\frac{a}{L}}\left[ e^{-ik_F na} \sum_{-\Lambda<k+k_F<\Lambda} e^{i(k+k_F)n a} c_{k\sigma}+
e^{ik_F na} \sum_{-\Lambda<k-k_F<\Lambda} e^{i(k-k_F)n a} c_{k\sigma}\right]\equiv
\nonumber\\
&&
\sqrt{a} \left[ e^{-ik_F n}
\psi_{\sigma-}(x=na)+e^{ik_F na} \psi_{\sigma+}(x=na)\right]
\label{fermion}
\end{eqnarray}
in the last step we have introduced
the left and right fermionic fields $\psi_{\sigma-}(x)$ and $\psi_{\sigma+}(x)$ respectively, by replacing the discrete lattice index $n$ by a continuous variable $x \sim na$.
This is possible because of the very small change undergone by sums in the second line of the previous equation,
when one goes from site $n$ to $n+1$.
Now we can undertake a gradient expansion for $H$ by making the replacement
\begin{eqnarray}
\psi_{\sigma\pm}[x=(n+1)a)]\rightarrow\psi_{\sigma\pm}(x=na)+ a
\partial_x \psi_{\sigma \pm}(x))
\label{discretder}
\end{eqnarray}
in all the terms of Eq. (\ref{hamil}). For the hopping operator we obtain
\begin{eqnarray}
(c_{n\sigma }^{\dagger }c_{n+1\sigma }+c_{n+1\sigma }^{\dagger }c_{n\sigma
}) &\sim& 2ia(-1)^{n}(\psi _{\sigma -}^{\dagger }\psi _{\sigma +}-\psi
_{\sigma +}^{\dagger }\psi _{\sigma -})+ \nonumber \\
&a^{2}&i[(\psi _{\sigma +}^{\dagger }\partial _{x}\psi _{\sigma +}-\partial
_{x}\psi _{\sigma +}^{\dagger }\psi _{\sigma +}-\psi _{\sigma -}^{\dagger
}\partial _{x}\psi _{\sigma -}+\partial _{x}\psi _{\sigma -}^{\dagger }\psi
_{\sigma -})+ \nonumber \\
&(-1)^{n}&\partial _{x}(\psi _{\sigma -}^{\dagger }\psi _{\sigma +}-\psi
_{\sigma +}^{\dagger }\psi _{\sigma -})]+O(a^{3}), \label{conthopp}
\end{eqnarray
where we have used $k_F=\pi /(2a)$. The number operator becomes
\begin{eqnarray}
n_{n{\sigma }} &\sim &
a \bigg[\rho_{\sigma +}(x)+\rho_{\sigma -}(x))
\nonumber\\
&+&(-1)^n\left(\psi _{\sigma +}^{\dagger
}(x)\psi _{\sigma -}(x)+\psi _{\sigma -}^{\dagger }(x)\psi
_{\sigma +}(x)\right)\bigg]
\end{eqnarray}
with $\rho _{\sigma +}=\psi _{\sigma +}^{\dagger }\psi _{\sigma +}$ and
\rho _{\sigma -}=\psi _{\sigma -}^{\dagger }\psi _{\sigma -}$.
By replacing $\sum_n$ by $\int \frac{dx}{a}$ and taking into account that
the integration of terms with an oscillating $(-1)^n$ prefactor vanish,
one obtains for the Hubbard Hamiltonian [corresponding to the first
two terms of Eq. (\ref{hamil})] the following form:
\begin{eqnarray}
H_U&=&
i v_F \int dx \left\{\psi _{\sigma +}^{\dagger }\partial _{x}\psi _{\sigma +}-\partial
_{x}\psi _{\sigma +}^{\dagger }\psi _{\sigma +}-\psi _{\sigma -}^{\dagger
}\partial _{x}\psi _{\sigma -}+\partial _{x}\psi _{\sigma -}^{\dagger }\psi
_{\sigma -}\right\}+\nonumber\\
&&\int dx \sum_\sigma\bigg\{ \frac{g_{4\perp}}{2} (\psi^{\dagger}_
\sigma+}\psi_{\sigma+}\psi^{\dagger}_{\overline{\sigma}+}\psi_{\overline
\sigma}+} +\psi^{\dagger}_{\sigma-}\psi_{\sigma-}\psi^{\dagger}_{\overline
\sigma}-}\psi_{\overline{\sigma}-}) +g_{2\perp}
\psi^{\dagger}_{\sigma+}\psi_{\sigma+}\psi^{\dagger}_{\overline{\sigma
-}\psi_{\overline{\sigma}-} + \nonumber \\
&&g_{1\perp} \psi^{\dagger}_{\sigma+}\psi_{\sigma-}\psi^{\dagger}_{\overline
\sigma}-}\psi_{\overline{\sigma}+}+ \frac{g_{3\perp}}{2} (\psi^{\dagger}_
\sigma-}\psi_{\sigma+}\psi^{\dagger}_{\overline{\sigma}-}\psi_{\overline
\sigma}+}+h.c.)\bigg\}. \label{hhubgeo}
\end{eqnarray}
The first line corresponds to the usual free Dirac Hamiltonian with
$v_F=at$ the bare Fermi velocity
(later we shall use $v_F=a(t-X)$, the Hartree-Fock value \cite{bos})
The terms with prefactors $g_{1\perp }$, $g_{2\perp
}$, $g_{3\perp }$, $g_{4\perp }$ correspond to
backward, forward two branch, Umklapp and forward one branch, respectively.
While all these constants are equal to $a U$, they might run independently under a
renormalization group (RG) flow. Moreover, in units
in which $\hbar=1$ and $v_F=1$,
the couplings $g_i$
are dimensionless. Now, if a coupling constant $g_i$ has units $E^{d-\Delta_i}$, $\Delta_i$ is known as the scaling dimension of the corresponding operator $O_i$, where $d$
is the spacetime dimension ($2$ in our case) \cite{Polchinski}. Therefore all interactions in Eq. (\ref{hhubgeo})
have scaling dimension $2$, they are marginal operators.
It is known that depending on the sign of the $g_i$
which correspond to the charge or spin sector of the theory, they can become
marginally relevant or irrelevant. The first case leads to a charge
or spin gap \cite{giama} (see also Section \ref{2p4}).
Let us see how this situation is modified by inclusion of the correlated hopping term
[the last one in Eq. (\ref{hamil})]
The sum of the number operators in (\ref{hamil}) has the following gradient expansion:
\begin{eqnarray}
(n_{n\overline{\sigma}}+n_{i+1\overline{\sigma}})&\sim& 2 a (\rho_{\overline
\sigma} +}+\rho_{\overline{\sigma} -})+a^2 [\partial_x(\rho_{\overline{\sigm
} +}+\rho_{\overline{\sigma} -})-\nonumber
\\ &(-1)^n& (\partial_x(\psi^{\dagger}_
\overline{\sigma}-}\psi_{\overline{\sigma}+}+ \psi^{\dagger}_{\overline
\sigma}+}\psi_{\overline{\sigma}-}))]+O(a^3)
\label{nini+1cont}
\end{eqnarray}
multiplying (\ref{conthopp}) by (\ref{nini+1cont}) we see that the terms quadratic in $a$ are oscillating
and vanish under integration. This is the result anticipated in Section \ref{1p1}.
This means that no scattering $\it at$ the Fermi level is generated by the correlated hopping interaction.
We should include term up to $O(a^2)$ in the Hamiltonian. We obtain:
\begin{eqnarray}
H_{X} &=&i\int dx\sum_{\sigma }\bigg\{g_{4\perp }^{\prime }[-\partial
_{x}\psi _{\sigma +}^{\dagger }\psi _{\sigma +}\psi _{\overline{\sigma
+}^{\dagger }\psi _{\overline{\sigma }+}+\partial _{x}\psi _{\sigma
-}^{\dagger }\psi _{\sigma -}\psi _{\overline{\sigma }-}^{\dagger }\psi _
\overline{\sigma }-}-{\rm H.c.}] \nonumber \\
&&g_{2\perp }^{\prime }[-\partial _{x}\psi _{\sigma +}^{\dagger }\psi
_{\sigma +}\psi _{\overline{\sigma }-}^{\dagger }\psi _{\overline{\sigma
-}+\partial _{x}\psi _{\sigma -}^{\dagger }\psi _{\sigma -}\psi _{\overline
\sigma }+}^{\dagger }\psi _{\overline{\sigma }+}-{\rm H.c.}] \nonumber \\
&&g_{1\perp }^{\prime }[-\partial _{x}\psi _{\sigma +}^{\dagger }\psi
_{\sigma -}\psi _{\overline{\sigma }-}^{\dagger }\psi _{\overline{\sigma
+}+\partial _{x}\psi _{\sigma -}^{\dagger }\psi _{\sigma +}\psi _{\overline
\sigma }+}^{\dagger }\psi _{\overline{\sigma }-}-{\rm H.c.}] \nonumber \\
&&g_{3\perp }^{\prime }[-\partial _{x}\psi _{\sigma -}^{\dagger }\psi
_{\sigma +}\psi _{\overline{\sigma }-}^{\dagger }\psi _{\overline{\sigma
+}+\partial _{x}\psi _{\sigma +}^{\dagger }\psi _{\sigma -}\psi _{\overline
\sigma }+}^{\dagger }\psi _{\overline{\sigma }-}-{\rm H.c.}]\bigg\}
\label{hXgeo}
\end{eqnarray
where all $g_{i}^{\prime }=a^{2}X$. The essential differences between the field
theory given by Eq. (\ref{hXgeo}) and the one given by Eq. (\ref{hhubgeo})
is the non local nature of the interaction arising
from the derivatives. In $k$-space this corresponds to scattering of electrons which
are near but not $\it on$ the Fermi surface.
Note that the coupling constants $g_{i}^{\prime }$ has dimension of the inverse of energy.
Therefore each term of Eq. (\ref{hXgeo}) has
dimension 3, they are irrelevant. As only these irrelevant operators appear
in the low energy fermionic field theory of the correlated hopping term $H_{X}$,
one might be tempted to conclude that there is no contribution of
these terms to the physical behavior of the system, in contrast to the
numerical results discussed in Section \ref{1p2}. \ How could we account of
this situation with our field theoretical analysis? In fact,
from a renormalization group (RG) point of view, all the operators allowed by
symmetry should be included in the effective theory. The fact that operators
present in Eq. (\ref{hhubgeo}) were not obtained in the derivation of Eq. (\ref{hXgeo})
means that in the initial conditions, the different $g$ do not
depend on $X$. However, they could acquire an $X$ dependence by the
couplings between $g$ and $g^{\prime }$ when the RG flow evolves.
We can take advatage of the well known studies of thermal critical phenomena with
RG \cite{Amit} to further understand this issue.
For this case we know that the
irrelevance of an operator means that the critical exponents are not affected
by its presence in the Hamiltonian.
However the critical temperature does depend on this operator.
In our case the presence of irrelevant operators will be crucial to determine
the boundaries in parameter space of the different phases,
where the spin or charge gap opens.
Finally, we note that the SU(2) invariance of the ordinary Hubbard model implies that under RG
flow, $g_{2\perp }=g_{1\perp }$ remains \cite{giama}. Similarly, it is shown
in \ref{a1} that $g_{2\perp }^{\prime }=g_{1\perp }^{\prime }$ is
required to keep the SU(2) invariance of the full Hamiltonian.
\subsection{Bosonization}
\label{2p2}
Bosonization is a powerful technique to analyze interacting one-dimensional
fermionic systems \cite{giama}. Some of the interacting terms in the fermionic
Hamiltonian become free non-interacting
terms in the bosonic Hamiltonian. The remaining terms contain in
general cosines of the bosonic fields.
Their effect can be studied by a perturbative implementation of the RG method.
If in the RG flow the coefficient of a cosine decreases, the fixed point corresponds
to a trivial theory of free bosons with known properties.
When the RG flow goes to strong coupling the coefficient of a cosine increases. The fields are trapped in a minimum of the free energy
and the different phases can be characterized by
calculating the classical value at this minimum of the bosonic operators corresponding to the physical observables.
In our case, the RG analysis is more involved, but as we shall show, it leads to a tractable theory
and correct results.
Let us therefore resort to a bosonic representation of the fermionic theory of Eq. (\ref{hXgeo})
We use the following bosonization formula for the left ($-$) and right ($+$)
fermions\cite{bosonreview} :
\begin{eqnarray}
\psi _{\sigma \pm }(x) &=&\frac{F_{\sigma \pm }}{\sqrt{L}}\colon e^{\mp
i\phi _{\sigma \pm }}\colon \label{bosno} \\
&=&\frac{F_{\sigma \pm }}{\sqrt{2\pi \alpha }}e^{\mp i\phi _{\sigma \pm }}
\label{bosnotno}
\end{eqnarray}
Equation (\ref{bosno}) is normal ordered and therefore does not contain a
somewhat uncomfortable short range cutoff $\alpha $, $F$ is the Klein factor
and $L$ the length of the chain. Eq. (\ref{bosnotno}) arises from Eq. (\re
{bosno}) by the explicit expansion of $\phi _{\sigma \pm }$ in term of the
boson creations ($b_{\sigma \pm }$) and annihilations ($b_{\sigma \pm
}^{\dagger }$). It is given by \cite{giama,bosonreview}
\begin{equation}
\phi_{\sigma \pm}(x)=\mp \underbrace{ i\sum_{n_p>0}\frac{e^{\mp ipx-\alpha
p/2}}{\sqrt{n_p}}b^{\dagger}_{\sigma \pm}}_{\varphi^{\dagger}_{\sigma \pm}}
\pm \underbrace{ i\sum_{n_p>0}\frac{e^{\pm ipx-\alpha p/2}}{\sqrt{n_p}
b_{\sigma \pm}}_{\varphi_{\sigma \pm}}, \label{expphi}
\end{equation}
where $\varphi _{\sigma \pm }^{\dagger }$ ($\varphi _{\sigma \pm }$) are the
creation (annihilation) part of the field $\phi _{\sigma \pm }$ and $p=\frac
Ln_{p}}{2\pi }$. We introduce the charge and spin bosonic fields $\phi _{pc}$
and $\phi _{ps}$ ($p=+,-$)
\begin{equation}
\phi _{pc}=\frac{(\phi _{p\uparrow }+\phi _{p\downarrow })}{\sqrt{2}}{\rm , \;
}\phi _{ps}=\frac{(\phi _{p\uparrow }-\phi _{p\downarrow })}{\sqrt{2}}.
\label{cs}
\end{equation
We also introduce phase fields $\phi _{m}$ and $\theta _{m}$ ($m=\uparrow
,\downarrow ,c,s$)
\begin{equation}
\phi _{m}=\frac{\phi _{m+}+\phi _{m-}}{2}{\rm , \; }\theta _{m}=\frac{\phi
_{m-}-\phi _{m+}}{2} \label{thatphi}
\end{equation}
The line before the last in Eq. (\ref{hXgeo}) bosonizes as:
\begin{eqnarray}
g_{1\perp }^{\prime }) &\sum_{\sigma }&(-\partial _{x}\psi _{\sigma
+}^{\dagger }\psi _{\sigma -}\psi _{\overline{\sigma }-}^{\dagger }\psi _
\overline{\sigma }+}+\partial _{x}\psi _{\sigma -}^{\dagger }\psi _{\sigma
+}\psi _{\overline{\sigma }+}^{\dagger }\psi _{\overline{\sigma }-}-{\rm
H.c.})= \nonumber \\
&-&\frac{i}{(2\pi \alpha )^{2}}\sum_{\sigma }[e^{2i\phi _{\sigma }}\partial
_{x}\phi _{\sigma +}e^{-2i\phi _{\overline{\sigma }}}+e^{-2i\phi _{\sigma
}}\partial _{x}\phi _{\sigma -}e^{2i\phi _{\overline{\sigma }}}]-{\rm H.c.}=
\nonumber \\
&-&\frac{4\sqrt{2}i}{(2\pi \alpha )^{2}}\cos (2\sqrt{2}\phi _{s})\partial
_{x}\phi _{c}=-\frac{4\sqrt{2}i}{L^{2}}\colon \cos (2\sqrt{2}\phi
_{s})\colon \partial _{x}\phi _{c} \label{g1pbos}
\end{eqnarray
In the last line we have normal ordered the cosine using Eqs. (\ref{bosno}) and (\ref{bosnotno}).
We also have:
\begin{eqnarray}
g_{3\perp }^{\prime }) &\sum_{\sigma }&[-\partial _{x}\psi _{\sigma
-}^{\dagger }\psi _{\sigma +}\psi _{\overline{\sigma }-}^{\dagger }\psi _
\overline{\sigma }+}+\partial _{x}\psi _{\sigma +}^{\dagger }\psi _{\sigma
-}\psi _{\overline{\sigma }+}^{\dagger }\psi _{\overline{\sigma }-}-h.c.]=
\nonumber \\
&\frac{4\sqrt{2}i}{(2\pi \alpha )^{2}}&\cos (2\sqrt{2}\phi _{c})\partial
_{x}\phi _{c}=\frac{4\sqrt{2}i}{L^{2}}\colon \cos (2\sqrt{2}\phi _{c})\colon
\partial _{x}\phi _{c} \label{g3pbos}
\end{eqnarray
The bosonization of $g_{2\perp }^{\prime }$ and $g_{4\perp }^{\prime }$
terms is a little more subtle. It is convenient to come back to the lattice
version of the derivate with respect to $x$ as is given in Eq. (\ref{discretder}). We have:
\begin{eqnarray}
g_{2\perp }^{\prime })\sum_{\sigma }\Bigg[ &-&\frac{\psi _{\sigma
+}^{\dagger }(x+a)-\psi _{\sigma +}^{\dagger }(x)}{a}\psi _{\sigma +}\psi _
\overline{\sigma }-}^{\dagger }\psi _{\overline{\sigma }-}+\frac{\psi
_{\sigma -}^{\dagger }(x+a)-\psi _{\sigma -}^{\dagger }(x)}{a}\psi _{\sigma
-}\psi _{\overline{\sigma }+}^{\dagger }\psi _{\overline{\sigma }+}
\nonumber \\
&+&\psi _{\overline{\sigma }-}^{\dagger }\psi _{\overline{\sigma }-}\psi
_{\sigma +}^{\dagger }\frac{\psi _{\sigma +}(x+a)-\psi _{\sigma +}(x)}{a
-\psi _{\overline{\sigma }+}^{\dagger }\psi _{\overline{\sigma }+}\psi
_{\sigma -}^{\dagger }\frac{\psi _{\sigma -}(x+a)-\psi _{\sigma -}(x)}{a
\Bigg]= \nonumber \\
\frac{1}{a}\sum_{\sigma }\Bigg[\Bigg( &-&\psi _{\sigma +}^{\dagger
}(x+a)\psi _{\sigma +}(x)+\psi _{\sigma +}^{\dagger }(x)\psi _{\sigma +}(x+a
\Bigg)\psi _{\overline{\sigma }-}^{\dagger }\psi _{\overline{\sigma }-}
\nonumber \\
&+&\Bigg(\psi _{\sigma -}^{\dagger }(x+a)\psi _{\sigma -}(x)-\psi _{\sigma
-}^{\dagger }(x)\psi _{\sigma -}(x+a)\Bigg)\psi _{\overline{\sigma
+}^{\dagger }\psi _{\overline{\sigma }+}\Bigg] \label{g2latt}
\end{eqnarray
We use Eq. (\ref{bosno}) to bosonize the first term into:
\begin{eqnarray}
&\frac{1}{a}&\Bigg(\psi _{\sigma +}^{\dagger }(x)\psi _{\sigma +}(x+a)-\psi
_{\sigma +}^{\dagger }(x+a)\psi _{\sigma +}(x)\Bigg)=\frac{1}{La}\Bigg
\colon e^{i\phi _{\sigma +}(x)}\colon \colon e^{-i\phi _{\sigma
+}(x+a)}\colon -{\rm H.c.}\Bigg]= \nonumber \\
&\frac{1}{La}&\Bigg[e^{-i\overbrace{(\varphi _{\sigma +}^{\dagger
}(x+a)-\varphi _{\sigma +}^{\dagger }(x))}^{\simeq a\partial _{x}\varphi
_{\sigma +}^{\dagger }+\frac{a^{2}}{2}\partial _{x}^{2}\varphi _{\sigma
+}^{\dagger }}}e^{(-i(\varphi _{\sigma +}(x+a)-\varphi _{\sigma +}(x))}\bigg
-\frac{Li}{2\pi a}\bigg)-{\rm H.c.}\Bigg]\simeq \nonumber \\
&-\frac{1}{2\pi a^{2}}&\Bigg[i\bigg(1-ia\partial _{x}\varphi _{\sigma
+}^{\dagger }-i\frac{a^{2}}{2}\partial _{x}^{2}\varphi _{\sigma +}^{\dagger
}-\frac{a^{2}}{2}(\partial _{x}\varphi _{\sigma +}^{\dagger })^{2}\bigg) \nonumber \\
&\times&
\bigg(1-ia\partial _{x}\varphi _{\sigma +}-i\frac{a^{2}}{2}\partial
_{x}^{2}\varphi _{\sigma +}-\frac{a^{2}}{2}(\partial _{x}\varphi _{\sigma
+})^{2}\bigg)-{\rm H.c.}\Bigg]= \nonumber \\
&-\frac{i}{2\pi a^{2}}&\Bigg[2\bigg(1-a^{2}\partial _{x}\varphi _{\sigma
+}^{\dagger }\partial _{x}\varphi _{\sigma +}-\frac{a^{2}}{2}(\partial
_{x}\varphi _{\sigma +})^{2}-\frac{a^{2}}{2}(\partial _{x}\varphi _{\sigma
+}^{\dagger })^{2}\bigg)\Bigg]= \nonumber \\
&\frac{i}{\pi a^{2}}&\Bigg[-1+\frac{a^{2}}{2}\colon (\partial _{x}\phi
_{\sigma +})^{2}\colon \Bigg] \label{term1}
\end{eqnarray}
In the second equality we have used the identity \cite{bosonreview}
$e^{A}e^{B}=e^{B}e^{A}e^{[A,B]}$ being the commutator a c number which could
be calculated by the explicit expansion given in Eq. (\ref{expphi})
with $\alpha =0$. The commutator becomes:
\begin{eqnarray}
&&\lbrack \varphi _{\sigma +}(x),\varphi _{\sigma +}^{\dagger }(x+a)]
=\sum_{p,p^{\prime }>0}\frac{e^{ipx}e^{-ip^{\prime }(x+a)}}{\sqrt
n_{p}n_{p}^{\prime }}}\underbrace{[b_{p},b_{p^{\prime }}^{\dagger }]
_{\delta _{p,p^{\prime }}} \nonumber \\
&&=\sum_{p}\frac{e^{-ipa}}{n_{p}}=-\log (1-e^{-i \frac{2\pi a}{L}})
\underbrace{\simeq }_{L>>a}-\log (i\frac{2\pi a}{L}).
\label{conm0}
\end{eqnarray}
Then,
\begin{eqnarray}
&&e^{[\varphi _{\sigma +}(x),\varphi _{\sigma +}^{\dagger }(x+a)]} =-\frac{i
}{2\pi a}, \label{conm}
\end{eqnarray}
the value previously used.
Proceeding in a similar way one finds (to be used later)
\begin{equation}
e^{[\varphi _{\sigma -}(x),\varphi _{\sigma -}^{\dagger }(x\pm a)]}=\pm
\frac{iL}{2\pi a}. \label{conm2}
\end{equation}
The term into the parenthesis in the last line of (\ref{g2latt}) can be
bosonized by similar steps. The result is:
\begin{equation}
\frac{1}{a}\Bigg(\psi _{\sigma -}^{\dagger }(x+a)\psi _{\sigma -}(x)-\psi
_{\sigma -}^{\dagger }(x)\psi _{\sigma -}(x+a)\Bigg)=\frac{i}{\pi a^{2}
\Bigg[-1+\frac{a^{2}}{2}\colon (\partial _{x}\phi _{\sigma -})^{2}\colon
\Bigg] \label{term2}
\end{equation
Taking into account that the normal ordered densities bozonize as:
\[
\colon \rho _{\sigma \pm }\colon =\colon \psi _{\sigma \pm }^{\dagger }\psi
_{\sigma \pm }\colon =-\frac{1}{2\pi }\colon \partial _{x}\phi _{\sigma \pm
}\colon
\
we obtain the bosonized expression of Eq. (\ref{g2latt}):
\begin{eqnarray}
&\frac{i}{2(\pi a)^{2}}&\sum_{\sigma }\Bigg[(\partial _{x}\phi _{\sigma
+}+\partial _{x}\phi _{\sigma -})-\frac{a^{2}}{2}\Bigg((\partial _{x}\phi
_{\sigma +})^{2}\partial _{x}\phi _{\overline{\sigma }-}+(\partial _{x}\phi
_{\sigma -})^{2}\partial _{x}\phi _{\overline{\sigma }+}\Bigg)\Bigg]
\nonumber \\
=\frac{i}{2(\pi a)^{2}} &&\Bigg[\sqrt{2}\bigg(\partial _{x}\phi
_{c+}+\partial _{x}\phi _{c-}\bigg)- \nonumber \\
&\frac{a^{2}}{2\sqrt{2}}&\bigg((\partial _{x}\phi _{c-})^{2}\partial
_{x}\phi _{c+}+\partial _{x}\phi _{c+}\bigg((\partial _{x}\phi
_{s-})^{2}-2\partial _{x}\phi _{s-}\partial _{x}\phi _{s+}\bigg)+ \nonumber
\\
&&\partial _{x}\phi _{c-}\bigg((\partial _{x}\phi _{c+})^{2}-2\partial
_{x}\phi _{s-}\partial _{x}\phi _{s+}+(\partial _{x}\phi _{s+})^{2}\bigg
\Bigg)\Bigg] \label{g2boso}
\end{eqnarray
Quite similar steps leads to the bosonization of the $g_{4}^{\prime }$ term.
We find:
\begin{eqnarray}
&\frac{i}{2(\pi a)^{2}}&\sum_{\sigma }\Bigg[(\partial _{x}\phi _{\sigma
+}+\partial _{x}\phi _{\sigma -})-\frac{a^{2}}{2}\Bigg((\partial _{x}\phi
_{\sigma +})^{2}\partial _{x}\phi _{\overline{\sigma }+}+(\partial _{x}\phi
_{\sigma -})^{2}\partial _{x}\phi _{\overline{\sigma }-}\Bigg)\Bigg]
\nonumber \\
=\frac{i}{2(\pi a)^{2}} &&\Bigg[\sqrt{2}\bigg(\partial _{x}\phi
_{c+}+\partial _{x}\phi _{c-}\bigg)- \nonumber \\
&\frac{a^{2}}{2\sqrt{2}}&\bigg((\partial _{x}\phi _{c-})^{3}+(\partial
_{x}\phi _{c+})^{3}-\partial _{x}\phi _{c-}(\partial _{x}\phi
_{s-})^{2}-\partial _{x}\phi _{c+}(\partial _{x}\phi _{s+})^{2}\Bigg)\Bigg]
\label{g4boso}
\end{eqnarray
Collecting the different pieces, going to an imaginary time $\tau =it$ and
defining complex space-time coordinates ($z=v_{F}\tau +ix$,$\overline{z
=v_{F}\tau -ix$), where $v_{F}=a(t-X)$ is the Fermi velocity (starting from
a Hartree-Fock decoupling \cite{bos}) we obtain the following expression for
the part of the action proportional to $X$.
\begin{equation}
S_{X}=a(G_{1}^{\prime }\int d^{2}rO_{1}^{\prime }(r)+G_{3}^{\prime }\int
d^{2}rO_{3}^{\prime }{d^{2}r}+G_{2}^{\prime }\int d^{2}rO_{2}^{\prime }
d^{2}r}+G_{4}^{\prime }\int d^{2}rO_{4}^{\prime }{d^{2}r)} \label{Szzvar}
\end{equation
where $G_{\alpha }^{\prime }=g_{\perp \alpha }^{\prime }/(a\pi v_{F})$ and
d^{2}r=v_{F}dxd\tau $. The different operators in Eq. (\ref{Szzvar}) are:
\begin{eqnarray}
O_{1}^{\prime } &=&i\frac{2\pi \sqrt{2}}{L^{2}}\colon \cos (2\sqrt{2}\phi
_{s})\colon (\partial _{z}\phi _{c-}-\partial _{\overline{z}}\phi _{c+})
\nonumber \\
O_{3}^{\prime } &=&-i\frac{2\pi \sqrt{2}}{L^{2}}\colon \cos (2\sqrt{2}\phi
_{c})(\partial _{z}\phi _{c-}-\partial _{\overline{z}}\phi _{c+})\colon
\nonumber \\
O_{2}^{\prime } &=&i\Bigg[\underbrace{\frac{\sqrt{2}}{2\pi a^{2}}\bigg
\partial _{\overline{z}}\phi _{c+}-\partial _{z}\phi _{c-}\bigg)
_{O_{2.1}^{\prime }}+\frac{1}{4\sqrt{2}\pi }\bigg(\underbrace{\colon
(\partial _{z}\phi _{c-})^{2}\partial _{\overline{z}}\phi _{c+}\colon
_{O_{2.2}^{\prime }} \nonumber \\
&+& \underbrace{\partial _{\overline{z}}\phi
_{c+}\colon (\partial _{z}\phi _{s-})^{2}\colon }_{O_{2.3}^{\prime }}
\underbrace{2\partial _{\overline{z}}\phi _{c+}\partial _{z}\phi
_{s-}\partial _{\overline{z}}\phi _{s+}}_{O_{2.4}^{\prime }}
\nonumber \\
&-&\bigg(\underbrace{\partial _{z}\phi _{c-}\colon (\partial _{\overline{z
}\phi _{c+})^{2}\colon }_{O_{2.5}^{\prime }}+\underbrace{2\partial _{z}\phi
_{c-}\partial _{z}\phi _{s-}\partial _{\overline{z}}\phi _{s+}
_{O_{2.6}^{\prime }}+\underbrace{\partial _{z}\phi _{c-}\colon (\partial _
\overline{z}}\phi _{s+})^{2}\colon }_{O_{2.7}^{\prime }}\bigg)\Bigg)\Bigg]
\nonumber \\
O_{4}^{\prime } &=&i\Bigg[\frac{\sqrt{2}}{2a^{2}\pi }\bigg(\partial _
\overline{z}}\phi _{c+}-\partial _{z}\phi _{c-}\bigg)- \nonumber \\
&\frac{1}{4\sqrt{2}\pi }&\bigg((\partial _{z}\phi _{c-})^{3}-(\partial _
\overline{z}}\phi _{c+})^{3}-\partial _{z}\phi _{c-}(\partial _{z}\phi
_{s-})^{2}+\partial _{\overline{z}}\phi _{c+}(\partial _{\overline{z}}\phi
_{s+})^{2}\Bigg)\Bigg] \label{defO'}
\end{eqnarray}
To obtain Eq. (\ref{Szzvar}) we have:
\begin{enumerate}
\item Included the normal order of each bosonic operator assuming that the
original fermionic operators were already normal ordered. This is a
prerequisite for the bosonization to work \cite{bosonreview}.
\item Taken into account that $\partial_x=i(\partial_z-\partial_{\overline {
}})$ and
\item that the right and left bosons depend on $\overline{z}$ and $z$
respectively. I.e. $\phi _{m+}(\overline{z})$ and $\phi _{m-}({z})$, ($m=c$
or $s$).
\end{enumerate}
This last fact arises when the explicit time dependence of the bosonic
creation and annihilation operator is deduced from the Heisenberg equations of motion
using the free bosonic Hamiltonian $H_{0}=v_{F}\sum_{k>0}kb_{mk+}^{\dagger
}b_{mk+}+v_{F}\sum_{k>0}kb_{mk-}^{\dagger }b_{mk-}$. One obtains $b_{mk\pm
}(\tau )=\exp ^{-v_{F}\tau k}b_{mk\pm }(0)$, and $b_{mk\pm }^{\dagger }(\tau
)=\exp ^{v_{F}\tau k}b_{mk\pm }^{\dagger }(0)$. Plugging these expressions
in equations like (\ref{expphi}) one obtains:
\begin{eqnarray}
\phi _{m+}(\overline{z}) &=&\underbrace{-i\sum_{n_{p}>0}\frac{e^{\overbrace
-ipx+pv_{F}\tau }^{p\overline{z}}}}{\sqrt{n_{p}}}b_{mp+}^{\dagger }
_{\varphi _{m+}^{\dagger }}+\underbrace{i\sum_{n_{p}>0}\frac{e^{\overbrace
ipx-pv_{F}\tau }^{-p\overline{z}}}}{\sqrt{n_{p}}}b_{mp+}}_{\varphi _{m+}}
\nonumber \\
\phi _{m-}(z) &=&\underbrace{-i\sum_{n_{p}>0}\frac{e^{\overbrace
-ipx-pv_{F}\tau }^{-pz}}}{\sqrt{n_{p}}}b_{mp-}}_{\varphi _{m-}}+\underbrace
i\sum_{n_{p}>0}\frac{e^{\overbrace{ipx+pv_{F}\tau }^{pz}}}{\sqrt{n_{p}}
b_{mp-}^{\dagger }}_{\varphi _{m-}^{\dagger }} \label{phizvarz}
\end{eqnarray
The total action is $S=S_{H}+S_{X}$, where $S_{H}$ is the usual bosonized
version of the Hubbard model of Eq. (\ref{hhubgeo})
\begin{eqnarray}
S_{H} &=&\frac{1}{2\pi }\int \partial _{z}\phi _{c-}\partial _{\overline{z
}\phi _{c+}d^{2}r+\frac{1}{2\pi }\int \partial _{z}\phi _{s-}\partial _
\overline{z}}\phi _{s+}d^{2}r+ \nonumber \\
&G_{1}&\int d^{2}rO_{1}(r)+G_{3}\int d^{2}rO_{3}{d^{2}r}+G_{2c}\int
d^{2}rO_{2c}{d^{2}r}+G_{2s}\int d^{2}rO_{2s}{d^{2}r}+ \nonumber \\
&&\delta v_{s}\int d^{2}rO_{4s}{d^{2}r}+\delta v_{c}\int d^{2}rO_{4c}{d^{2}r}
\label{Szzvarg}
\end{eqnarray
with
\begin{eqnarray}
O_{1} &=&\frac{2\pi }{L^{2}}\colon \cos (2\sqrt{2}\phi _{s})\colon
\label{o1} \\
O_{3} &=&\frac{2\pi }{L^{2}}\colon \cos (2\sqrt{2}\phi _{c})\colon
\label{o3} \\
O_{2c} &=&\frac{1}{4\pi }\partial _{z}\phi _{c-}\partial _{\overline{z}}\phi
_{c+} \label{O2c} \\
O_{2s} &=&-\frac{1}{4\pi }\partial _{z}\phi _{s-}\partial _{\overline{z
}\phi _{s+} \label{O2s} \\
O_{4c} &=&\frac{1}{8\pi }\bigg((\partial _{\overline{z}}\phi
_{c+})^{2}+(\partial _{{z}}\phi _{c-})^{2}\bigg) \label{O4c} \\
O_{4s} &=&-\frac{1}{8\pi }\bigg((\partial _{\overline{z}}\phi
_{s+})^{2}+(\partial _{{z}}\phi _{s-})^{2}\bigg) \label{O4s}
\end{eqnarray}
$\delta v_{c}$ ($\delta v_{s}$) renormalize the charge (spin) velocity.
Operators $O_{2c}$ and $O_{2s}$ (and $O_{4c}$ and $O_{4s}$) appear together
in the bosonized theory of the Hubbard model Eq. (\ref{hhubgeo}), where only interaction
between electron of different spin are taken into account. They are
independent operators in the general case where interaction between electron
of the same spin are included. For the Hubbard model, the values of all
couplings are $G_{\alpha }=$ $Ua/(\pi v_{F}).$
\subsection{The renormalization group equations}
\label{2p3}
Following Ref. \cite{cardy}, the RG equations for the coupling constants
\Gamma _{\alpha }$ ($G_{\alpha }$ or $G_{\alpha }^{\prime }$) present in the
action is
\begin{equation}
\frac{d\Gamma _{\gamma }}{dl}=(d-\Delta _{\gamma })\Gamma _{\gamma }
-\frac{S_{d}\lambda ^{\Delta _{\alpha }+\Delta _{\beta }-\Delta _{\gamma }-d}}{2}
\sum\limits_{\alpha \beta }C_{\alpha \beta }^{\gamma
}\Gamma _{\alpha }\Gamma _{\beta }, \label{rgcar}
\end{equation
where $d=2$ is the spacetime dimension of the system, $\Delta _{\gamma }$ is the
scaling dimension of the operator related with $\Gamma _{\gamma }$, $S_{d}$
is the area of a sphere of unit radius in $d$ dimensions ($2\pi $ in our
case), the $C_{\alpha \beta }^{\gamma }$ are the coefficients
of the following short-distance Operator Product Expansion (OPE):
\begin{equation}
O_{\alpha }^{\prime }(r)O_{\beta }^{\prime }(r^{\prime })=\sum_{\gamma
}C_{\alpha \beta }^{\gamma }\frac{O_{\gamma }(\frac{r+r^{\prime }}{2})}{\mid
r-r^{\prime }\mid ^{\Delta _{\alpha }+\Delta _{\beta }-\Delta _{\gamma }}}
\mbox{more
irrelevant operators,} \label{opec}
\end{equation}
and $\lambda$ is a number of order one, which comes from our definition of the short distance
cutoff as $a/\lambda$ (the exponent of $\lambda$ in Eq. (\ref{rgcar}) comes from the integral of
Eq. (\ref{opec}) with respect to $\mid r-r^{\prime }\mid$ in $d$ space-time dimensions).
The OPE's between two $O_{\alpha }$ operators are already known from the RG
equations of the ordinary Hubbard model \cite{giama}. The OPE's between one
$O_{\alpha }$ and one $O_{\beta }^{\prime }$ operator give another $O_{\gamma
}^{\prime }$ and have a prefactor $G_{\alpha }G_{\beta }^{\prime }\sim UX$.
We note that for small $X$, this product is of order $X^{3}$ on the spin
transition and of higher order or negligible on the charge transition. We
have neglected these OPE's. This is partially justified by that fact that
they generate $O_{\gamma }^{\prime }$ operators which are irrelevant, while
as we show below the OPE's between two $O_{\alpha }^{\prime }$ operators
generate marginal $O_{\alpha }$ operators. A deeper justification in given
on symmetry grounds: expressing the first Eq. (\ref{sym}) in terms of the
Hartree-Fock hopping $\tilde{t}=t-X$ [which is invariant under the
transformation $c_{i\sigma }^{\dagger }\rightarrow (-1)^{i}c_{i\sigma }$]
one has
\begin{equation}
H(\tilde{t},-X,U)\equiv H(\tilde{t},X,U). \label{sym2}
\end{equation
This means that for small $X$, there can be no terms of order $UX$ in the
action which correct the Hartree-Fock results. Therefore, the generated
operators in the OPE's between one $O$ and one
$O_{\alpha }^{\prime }$ should introduce corrections of higher order.
Now let us discuss the different operators that could arise from the OPE's
between two $O_{\alpha }^{\prime }$ operators. There are some cases where
these OPE's give operators of dimension $4$ or higher. This is for example
the case of the OPE between $O_{2.3}$ and $O_{2.7}$ or in general between
two operators included in $O_{2}$
which contain less than two fields in common.
There are other cases where the denominator does not depend only on the
distance between the two points under consideration but have factors of the
form $(z^{\prime }-z)^{-2}+(\overline{z}^{\prime }-\overline{z})^{-2}$. This
gives rise to a periodic function in the relative angle and the integral in
the angular part of $r-r^{\prime }$ [which was performed to arrive at
Eq. (\ref{rgcar})] vanishes. This is the cases of the OPE between $O_{1}^{\prime }$
and $(O_{2.4}^{\prime }+O_{2.6}^{\prime })$. Finally there are cases which
produce operators of the form (\ref{O2c}) and (\ref{O2s}). They simply
renormalize the charge or spin velocity and will not be taken into account
in our treatment
The remaining OPE's are displayed in the \ref{OPESO}. From this appendix we have:
\begin{eqnarray}
C_{2^{\prime }1^{\prime }}^{1} &=&C_{1^{\prime }2^{\prime }}^{1}=\frac{1}
2\pi }{\rm , \; }C_{2^{\prime }3^{\prime }}^{3}=C_{3^{\prime }2^{\prime
}}^{3}=-\frac{1}{2\pi } \nonumber \\
C_{2^{\prime }2^{\prime }}^{2s} &=&C_{2^{\prime }2^{\prime }}^{2c}=\frac{1}
\pi }. \label{Gs}
\end{eqnarray}
From Eqs. (\ref{rgcar}) and (\ref{Gs}), the usual RG equations for the
Hubbard model become modified as follows. For the charge sector
\begin{equation}
\frac{dG_{2c}}{dl}=G_{3}^{2}-\lambda ^{2}(G_{2}^{\prime })^{2}{\rm , \; }\frac
dG_{3}}{dl}=G_{2c}G_{3}+\lambda ^{2}G_{2}^{\prime }G_{3}^{\prime },
\label{ch}
\end{equation
for the spin sector
\begin{equation}
\frac{dG_{2s}}{dl}=-G_{1}^{2}-\lambda ^{2}(G_{2}^{\prime })^{2}{\rm , \;
\frac{dG_{1}}{dl}=-G_{2s}G_{1}-\lambda ^{2}G_{2}^{\prime }G_{1}^{\prime },
\label{sp}
\end{equation
and in addition
\begin{equation}
\frac{dG_{\alpha }^{\prime }}{dl}=-G_{\alpha }^{\prime }. \label{gp}
\end{equation}
\subsection{Analysis of the RG equations}
\label{2p4}
Taking into account the initial conditions, Eq. (\ref{gp}) can be integrated
immediately giving
\begin{equation}
G_{\alpha }^{\prime }=\frac{aX}{\pi v_{F}}\exp (-l). \label{gp2}
\end{equation
Replacing this equation in Eqs. (\ref{ch}), one obtains two coupled
differential equations for the charge sector
\begin{equation}
\frac{dG_{2c}}{dl}=G_{3}^{2}-Ae^{-2l}{\rm , \; }\frac{dG_{3}}{dl
=G_{2c}G_{3}+Ae^{-2l}{\rm , \; with \;}A=\left( \frac{\lambda aX}
{\pi v_{F}}\right)^{2}, \label{ch2}
\end{equation}
with the initial conditions $G_{2c}(l=0)=G_{3}(l=0)=Ua/(\pi v_{F})$.
It is known that for $A=0$, the flux continues along the separatrix
$G_{2c}=G_{3}$, and goes to infinite $G_{i}$ (charge gap) if $U>0$,
and to $G_{2c}=G_{3}=0$ (gapless case) if $U<0$. While an analytical solution for
$A\neq 0$ seems not possible, it is clear that the effect of $A$ is to push
the flux perpendicularly to the separatrix, favoring larger
$G_3$ and smaller $G_{2c}$. This does not modify the
final result that the critical value of $U$ which separates the regions of
diverging or vanishing $G_{3}(l\rightarrow +\infty )$ is $U_{c}=0$. We have
confirmed this by a numerical study of Eqs. (\ref{ch2}).
However, as a difference with the Hubbard model for which the flux is on the
separatrix, in our case, for $U<U_c$ (when $G_3$ flows to zero),
$G_{2c}$ converges to a negative value. This leads to a correlation exponent \cite{giama}
$K_c \sim 1- G_{2c}$ larger than 1. As a consequence, the singlet superconducting (SS)
correlation functions, which decay as $d^{-1/K_c}$ at large distance $d$ dominate over the
charge density wave (CDW) ones, which decay as $d^{-K_c}$ \cite{jaka,bos,giama}.
In the Hubbard model, for $U<0$, $K_c =1$ and both SS and CDW correlations decay as $1/d$.
For the spin sector, the RG equations become
\begin{equation}
\frac{dG_{2s}}{dl}=-G_{1}^{2}-Ae^{-2l}{\rm , \; }\frac{dG_{1}}{dl
=-G_{1}G_{2s}-Ae^{-2l},. \label{rgs1}
\end{equation
with the initial conditions $G_{2s}(0)=G_{1}(0)=Ua/(\pi v_{F})$.
It is clear that the flux of the RG equations remains on the
separatrix $G_{1}(l)=G_{2s}(l)$. Therefore, both equations (\ref{rgs1})
reduce to the same equation for $G_{1}=G_{2s}=G$. Changing variable
$z=\sqrt{A}e^{-l}$, this equation takes the form
\begin{equation}
z\frac{dG}{dz}=G^{2}+z^{2}. \label{rgs2}
\end{equation
Its solution is given in terms of Bessel functions
\begin{equation}
G(z)=\frac{z[Y_{1}(z)+CJ_{1}(z)]}{Y_{0}(z)+CJ_{0}(z)}, \label{rgsol}
\end{equation
where the constant $C$ is determined by the initial condition $G(\sqrt{A})=G_{\rm ini}$, giving
\begin{equation}
C=\frac{G_{\rm ini}Y_{0}(\sqrt{A})-\sqrt{A}Y_{1}(\sqrt{A})}{-G_{\rm ini}J_{0}(\sqrt{A})
\sqrt{A}J_{1}(\sqrt{A})}. \label{c}
\end{equation}
Mathematically, for $l\rightarrow \infty $ $(z\rightarrow 0)$,
Eq. (\ref{rgsol}) converges to zero. However, it may happen that $G(z_{d})$ diverges
for some intermediate value $z_{d}$ ($0<z<\sqrt{A}$),
jumping from $-\infty$ to $+\infty$ as $z$ decreases.
This means
physically that at an intermediate scale determined by $z_d$,
the solution flowed to the strong coupling fixed point at which a
spin gap opens.
The limiting value of $z_{d}$ for which such a behavior
takes place corresponds to $z_{d}\rightarrow 0$. Since for small values of
the argument $Y_{0}(z)\sim (2/\pi )\ln (z)$ and $J_{0}(z)\sim 1$,
a diverging $G(z_d)$ for $z_d \rightarrow 0$ implies a zero in the denominator of Eq. (\ref{rgsol}), and the
initial conditions should be such that $C$ also diverges
in this special case.
For small values of $A$
(as we have assumed in our whole treatment), there is no divergence in $G(z)$
if $C$ is negative. From this reasoning and Eq. (\ref{c}), we obtain the
following condition for the opening of a spin gap:
\begin{equation}
G_{\rm ini}<\frac{\sqrt{A}J_{1}(\sqrt{A})}{J_{0}(\sqrt{A})}\simeq \frac{A}{2}+\frac
A^{2}}{16}, \label{gl}
\end{equation
where the last member was obtained from a series expansion of the Bessel
functions.
From Eq. (\ref{gl}) and using $G_{\rm ini}=Ua/(\pi v_{F})$, we obtain the following critical
value of $U$ for the opening of the spin gap
\begin{equation}
U_{s}=\pi v_{F}\frac{\sqrt{A}J_{1}(\sqrt{A})}{J_{0}(\sqrt{A})}. \label{usrg}
\end{equation}
or approximately
\begin{equation}
U_{s}\simeq \frac{(\lambda aX)^{2}}{2\pi v_{F}}+\frac{(\lambda aX)^{4}}{8(\pi v_{F})^{3}}.
\label{usrgap}
\end{equation}
The final path taken by the RG flow
in each sector, determine the nature of the resulting phases.
As discussed above, for $U<U_c$, SS correlations dominate. For $U>U_s$ spin-spin correlations are
the largest at large distances as in the usual Hubbard model \cite{jaka,bos,giama}.
The phase in between, for $U_c<U<U_s$ is characterized by the presence of both gaps, and the
RG flow in each sector leads to
$G_{3} \rightarrow +\infty $ and $G_{1} \rightarrow -\infty $.
To minimize the respective cosine terms in the action [See Eqs. (\ref{Szzvarg}), (\ref{o1}) and (\ref{o3})],
the fields are frozen at the values $2\sqrt{2}\phi _{c}=\pi$ and $2\sqrt{2}\phi _{s}=0$.
As a consequence, the system has a spontaneously dimerized bond-ordering-wave (BOW) phase
with long range order. The order parameter
which takes a finite value on this phase is \cite{jaka,bos}
\begin{equation}
O_{BOW} = \sum_{i \sigma}(-1)^i (c_{i+1,\sigma }^{\dagger }c_{i\sigma }+{\rm H.c.})
\sim \sin(\sqrt{2}\phi _{c}) \cos(\sqrt{2}\phi _{s}).
\label{bow}
\end{equation}
\section{Comparison with the numerical results}
\label{s3}
The field theoretical result for the charge transition $U_{c}=0$ obtained in
the previous Section, agrees with the numerical results, presented in
Section \ref{1p2}. As explained in Section \ref{1p3}, this result is not
obtained if the initial values of the couplings of the Hubbard model (
G_{2c}(0)$ and $G_{3}(0)$) are corrected by vertex corrections in second
order in $X$ before bosonizing. We do not have a physical explanation for
this.
\begin{figure}[tbp]
\includegraphics[width=14cm]{hirf2.eps}
\caption{Solid circles (squares): critical value of the spin (charge) transition $U_s$ ($U_c$) as a function of $X$
obtained from the method of level crossings. Full line: solution of the RG equations given by
Eq. (\ref{usrg}). The dash dotted line at $U_c=0$ signal the boundary between the gapped and gapless charge phase
as given by Eq. (\ref{ch2}). The unit of energy is taken as $t=1$.}
\label{usf}
\end{figure}
To compare the critical value of $U$ for the spin transition, we take
$\lambda =4$, which using Eq. (\ref{usrgap}) and recalling that $v_{F}=a(t-X)
, leads to the same result as that obtained from vertex corrections Eq. (\re
{usve}) for small $X$. This leads to $A=\{4X/[\pi (t-X)]\}^{2}$. Replacing
this result in Eq. (\ref{usrg}), we obtain the function $U_{s}(X)$ that is
represented in Fig. \ref{usf}. The agreement with the numerical result up to
$X/t\sim 0.4$ is excellent.
\section{Summary and discussion}
\label{s4}
We have studied a field theory for the Hubbard model with small bond-charge
interaction $X$ at half filling.
While usually, it is enough to consider in the action only
terms linear in the lattice parameter $a$, in our case it is necessary
to include terms of order $a^2$
to obtain meaningful results at half
filling. These terms can be classified in a similar way as the linear
ones in terms of different processes in a "$g$-ology" treatment (forward one
branch, forward two branch, backward and Umklapp) but contain derivatives of
the fields in the space direction.
We have obtained the RG equations of the different couplings using Operator
Product Expansions. While the treatment of the new terms is awkward, most of
them should be retained to keep the spin SU(2) invariance of the model.
According to the dominant correlations at large distances, the phases of the
model can be classified as a singlet superconducting (SS) one for $U<U_{c}$,
a bond ordering wave (BOW) for $U_{c}<U<U_{s}$ and a spin density wave (SDW)
for $U>U_{s}$. The boundaries between the phases correspond to a charge
transition for on site repulsion $U=U_{c}$ and a spin transition at $U=U_{s}
, which correspond respectively to the opening of a charge gap and a closing
of the spin gap as $U$ increases. For the former transition we obtain
$U_{c}=0$ in agreement with previous numerical studies for $X<0.5$ \cite{cola}.
With only one adjustable parameter, we also obtain a very good agreement
with the numerical results for $U_{s}$ if $X\lesssim 0.4t.$
To explain accurately the dependence of $U_c$ for $X>0.5$,
it is necessary to go beyond our approach, possibly including more irrelevant operators.
As stated in Section \ref{1p1}, the model is an effective one-band model for
a variety of physical systems, in particular optical lattices \cit
{duan,good,kest,duan2}. In these systems, $U$ can be varied over the whole
range, including its sign, through tuning of the external magnetic field $B
. It is also possible to change $X/t$ by 20\%. Therefore, adjusting the
filling to one particle per site, it seems in principle possible to tune the
parameters in such a way the ground state of the system is in any of the
three phases: SS, BOW or SDW.
\section*{Acknowledgments}
We thank Pascal Simon and Luming Guan for useful discussions. We are
partially supported by CONICET, Argentina. This work was partially supported
by PIP 11220080101821 and 11220090100392 of CONICET, and PICT 2006/483, PICT 1647 and PICT R1776 of the ANPCyT.
|
{
"timestamp": "2010-11-30T02:04:26",
"yymm": "1009",
"arxiv_id": "1009.4113",
"language": "en",
"url": "https://arxiv.org/abs/1009.4113"
}
|
\section{Introduction}
\label{sec:intro}
Shaped and characterized femtosecond pulses are in widespread demand amongst the quantum control community \cite{Goswami2003,Ohmori2009,Dantus2004}. Through tailoring the phase, amplitude or polarization of the control pulse, the evolution of a quantum state may be manipulated in order to steer it towards a desired outcome. A typical scenario is the design of optical fields to control molecular motion, including the prospect of achieving site-specific chemistry and intramolecular rearrangements. During the last two decades, many impressive results \cite{Bonacic-Koutecky2006,Levis2001,Weinacht1999,Monmayrant2006} have arisen from technological breakthroughs in the generation of arbitrarily tailored pulses \cite{Monmayrant2010}.
Two principal active pulse-shaping techniques for ultrashort pulses are at the disposal of the experimentalist: a spatial light modulator (SLM) placed in the Fourier plane of a $4f$ zero-dispersion line \cite{Weiner2000,Monmayrant2004} or an acousto-optic programmable dispersive filter (AOPDF) \cite{Verluise2000}. Extensive studies of the $4f$ line have extended its available wavelength range and characterized its behaviour. In particular, it is now well known both experimentally and theoretically that such devices lead to spatio-temporal coupling effects, whereby the shaped electric field is dependent on the spatial position in the beam \cite{Danailov1989,Wefers1995,Wefers1996,Dorrer1998,Tanabe2002}. These studies have more recently been extended to the focal volume after a lens \cite{Sussman2008,Frei2009}.
By contrast, the AOPDF --- a newer technology within the control field --- has been less well characterized. Its first application entailed the corrective shaping of ultrashort pulses before an amplifier in order to improve compression of the amplified output \cite{Seres2003}. More recently, an angular dispersion effect which could affect such a laser chain has been presented \cite{Borzsonyi2010}. Further to this application, the AOPDF's shaping versatility, together with the large spectral range spanned (from the UV \cite{Coudreau2006} through the visible \cite{Monmayrant2005} to the near IR \cite{Verluise2000,Pittman2002}) renders it a valuable tool for control experiments \cite{Form2008}. In particular, the first--excited-state transitions of many organic and inorganic molecules lie in the UV wavelength range; hence the development of a practical UV pulse shaper is a great challenge and active field within the community. Very recently, interesting results have been obtained using shaped ultraviolet pulses \cite{Tseng2009, Roth2009,Greenfield2009}, and the variety of implementations of AOPDFs and SLMs is constantly increasing \cite{Monmayrant2010}. Amongst these contenders, the UV AOPDF based upon a KDP crystal is a good candidate, since it is versatile and tunable on a broad spectral range (\unit[250-410]{nm}) matching typical molecular electronic absorption bands \cite{Weber2010}. Nonetheless, to date no complete characterization has been performed of the spatio-temporal characteristics of AOPDF-shaped pulses --- in particular in the UV range. Indeed, some sources even assert AOPDFs to be entirely free of such effects \cite{Lee2009}, in contrast to the much maligned $4f$ line. At least one distortion, however, has already been identified: a lateral displacement which depends on the acoustic wave profile in the crystal \cite{Krebs2010}.
In this paper, we have undertaken the complete characterization of the space-time coupling effects produced by the AOPDF using spatially and spectrally resolved Fourier-transform interferometry (SSI) \cite{Monmayrant2010}. SSI is an interferometric technique that entails a relative measurement of the spectral phase between a reference and unknown pulse --- it thus lends itself to the measurement of the transfer function of a pulse shaper. As a metrology tool, SSI is suited to low pulse energies since it does not necessitate any nonlinear processes. (In the event that knowledge of the spectral phase of the input pulse \emph{per se} is required, absolute pulse characterization techniques may be applied \cite{Baum2004,Kane1994}.) This technique provides spatio-temporal resolution of the shaped pulses; it thus facilitates a comprehensive quantitative analysis of the ubiquitous spatio-temporal coupling induced by the AOPDF together with an explanation and numerical description of the physical mechanism. Our analysis encompasses a range of pulse shapes that are of the broadest utility to the control community.
\section{Methods}
\label{sec:methods}
\begin{figure}
\centering
\includegraphics[width = 0.9\columnwidth]{McCabe200910_Fig1.jpg}
\caption{The AOPDF (Fastlite Dazzler\texttrademark) SI characterization setup. The pulse shaper is placed in one arm of an interferometer. The unknown and reference arms are recombined at the entrance slit to a two-dimensional spectrometer with a slight angle and variable delay. The imaging spectrometer measures the resultant interference fringes, from which the relative spectral phase may be extracted. The spectrometer measures a spatially resolved spectrum along the slit axis $x$. A cylindrical lens focusses the beams onto the entrance slit of the spectrometer along the non-imaged spatial axis. A half-waveplate rotates the polarization in the reference beam arm.}
\label{fig:layout}
\end{figure}
The ultrafast source used for these experiments is an ultraviolet (UV) pulse train generated from a chirped-pulse amplified Ti:sapphire laser (CPA) \cite{Backus1998} via subsequent nonlinear interactions. The \unit[800]{nm} pulses are combined with their second harmonic at \unit[400]{nm} in order to generate the sum frequency at $\lambda_0 = \unit[267]{nm}$. Typical characteristics of the UV source are \unit[2]{$\mu$J} pulses with \unit[2]{nm} full-width at half-maximum (FWHM) bandwidth at the \unit[1]{kHz} repetition rate of the master laser. A typical beam width is around \unit[1-2]{mm}. Spatially resolved cross-correlation measurements of the UV pulses indicate a \unit[250]{fs} pulse duration without significant spatial wavefront distortion. (The pulse bandwidth would support a transform-limited duration of around \unit[50]{fs}; the difference is attributable to dispersive effects within the nonlinear crystals of the source.)
AOPDF pulse shapers are based on the dispersive propagation of light within an acousto-optic crystal. An incident ordinary optical wave interacts with a collinear acoustic wave, resulting in the diffraction of the optical wave onto the extraordinary axis. The spectral phase of a femtosecond optical pulse may be shaped via manipulation of the diffraction location for each spectral component along the length of the birefringent crystal; meanwhile the amplitude may be modulated via the size of the acoustic wave \cite{Kaplan2002}. A commercial AOPDF (the Fastlite Dazzler\texttrademark T-UV-260-410/T2), based on a \unit[75]{mm} KDP crystal designed for use at UV wavelengths, is employed for these experiments \cite{Coudreau2006,Weber2010}. The programmable temporal window is essentially fixed by the length of the crystal and the difference in refractive index of the crystal axes, and is about \unit[7]{ps} for this apparatus. A part of this window (for example, \unit[3]{ps} for a shaping window of three times the FWHM bandwidth according to the parameters given above) is required to self-compensate the natural dispersion induced by the KDP itself; if necessary this could be obviated by means of an external compressor.
The performance of the AOPDF is characterized using SSI \cite{Monmayrant2010,Tanabe2002}. The AOPDF is placed in one arm of an interferometer and its shaped output interferes with an unshaped reference arm (see Fig.\ \ref{fig:layout}). Since the AOPDF rotates polarization, a half-waveplate was placed in the reference beam arm. The two arms are combined with a small angle and a controllable relative delay at the entrance slit of a home-built imaging spectrometer \cite{Austin2009}. Since the spectrometer employs a two-dimensional detector, it is able to make a measurement of the spectrum as a function of position along the slit (aligned parallel with the plane of diffraction of the AOPDF). The detector is a charge-coupled device (CCD) camera (EHD Imaging UK-1158UV) with a pixel size of \unit[6.45]{$\mu$m}, and the spectrometer has an optical resolution of \unit[0.08]{nm} and \unit[40]{$\mu$m} along the spectral and spatial axes respectively. In order to increase signal, a cylindrical lens focusses the beams onto the entrance slit along the orthogonal spatial axis (i.e.\ along the non-imaged axis of the beam). The ensuing interferograms are detected with single-shot sensitivity.
The 2D interferogram measured by the spectrometer [see Fig.\ \ref{fig:data}(a)] is
\begin{align}
\label{eq:SI}
S(x,\omega) & = \modbr{A_\textrm{s}(x,\omega) e^{i\phi_\textrm{s}(x,\omega)} + A_\textrm{r}(x,\omega) e^{i\sqbr{\phi_\textrm{r}(x,\omega) + \omega \tau + k_x x}}}^2 \notag \\
& = \modbr{A_\textrm{s}(x,\omega)}^2 + \modbr{A_\textrm{r}(x,\omega)}^2 \notag \\ &+ \modbr{A_\textrm{s}(x,\omega)} \modbr{A_\textrm{r}(x,\omega)} \notag \\ & \times \cos \sqbr{\phi_\textrm{s}(x,\omega) - \phi_\textrm{r}(x,\omega) - \omega \tau - k_x x}.
\end{align}
Here $\tau$ is the time delay between the two pulses and $k_x$ is the difference between the transverse components of the propagation vectors (such that their subtended angle is $\theta = k_x/\modbr{\mathbf{k}}$). $A_\textrm{s}$, $A_\textrm{r}$, $\phi_\textrm{s}$ and $\phi_\textrm{r}$ denote the spatio-spectral amplitude and phase of the shaped (s) and reference (r) pulse respectively.
\begin{figure}
\centering
\includegraphics [width = \columnwidth]{McCabe200910_Fig2.pdf}
\caption{The Fourier filtering process. (a) A raw interferogram measured by the spectrometer camera. (b) A two-dimensional Fourier transform is performed. An a.c.\ term is filtered out within the Fourier domain. (c) An inverse two-dimensional Fourier transform of this term isolates the final term of equation \ref{eq:SI}. The mapping onto calibrated frequency and position axes is calculated. (d) Extracted phase difference $\phi_\textrm{s}(x,\omega) - \phi_\textrm{r}(x,\omega)$, modulo $2\pi$. A subsequent procedure calibrates the camera pixels into physical units of frequency and position.}
\label{fig:data}
\end{figure}
In order to extract a measurement of the spectral phase added by the AOPDF, this interferogram is Fourier transformed along both spatial and spectral dimensions. One of the a.c.\ terms is filtered out and inverse Fourier transformed with the carrier frequency removed. This isolates the final summand of equation \ref{eq:SI}, which contains the phase difference $\phi_\textrm{s}(x,\omega) - \phi_\textrm{r}(x,\omega)$ \cite{Takeda1982}. The spatial and spectral carriers, $k_x$ and $\tau$, are chosen in order to separate the a.c.\ and d.c.\ terms in the Fourier transform whilst ensuring that the fringe period is greater than the spectrometer resolution. In order to be able to handle complex temporal structure, a predominantly spatial carrier of $\theta \approx \unit[3]{mrad}$ and $\tau \approx 0$, giving rise to predominantly spatial fringes, is employed for these experiments. Typical data treated according to this process are shown in Fig.\ \ref{fig:data}.
In order to calibrate the intrinsic added second- and higher-order phase associated with the two arms of the interferometer (and specifically the dispersion of the beamsplitter and waveplate), an SSI measurement was taken with the AOPDF removed. The extracted relative higher-order phase varied by less than \unit[0.4]{rad} over the extent of the imaged spectrum.
\section{Results}
\label{sec:results}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{|l|c|}
\hline
Pulse shape & $H(\omega)$ \\
\hline
Pulse delay & $\exp\sqbr{-2(\omega-\omega_0)^2/\Delta\omega^2 + i\omega \tau}$ \\
$N$-pulse train & $\sum_{n=1}^N \exp(i \omega\tau_n)$\\
Chirps & $\exp \sqbr{-2(\omega-\omega_0)^2/\Delta\omega^2 + i\br{\omega-\omega_0}^2 \phi^{(2)}/2}$\\
$\pi$-step & $\exp\left\{ i \arctan \sqbr{(\lambda-\lambda_0)/\Delta\lambda_{\textrm{step}}} \right\}$ \\
\hline
\end{tabular}
\end{center}
\caption{Transfer function, $H(\omega)$, for the pulse shapes presented within Section \ref{sec:results}, where $\omega_0 = 2\pi c/\lambda_0$ is the central angular frequency. For the pulse delay and chirped-pulse cases, a narrowed spectral bandwidth of \mbox{$\Delta\lambda = (\lambda_0^2/2\pi c)\Delta\omega = \unit[1]{nm}$} was employed. All other parameters are defined in the text.}
\label{tab:transfer-fns}
\end{table}
A series of different phase and amplitude profiles that are of broad utility within control experiments were programmed into the AOPDF. In each case, it was verified that the device applied the correct complex
transfer function $H(\omega)$, such that the input and shaped output pulses were related by $E_\textrm{out}(\omega)= H(\omega)E_\textrm{in}(\omega)$. The spatial resolution of our system enabled this verification to be performed independently at all points in the beam.
It was also possible to test systematically for any spatial or spatio-temporal distortions caused by the AOPDF. In all cases, exactly one such distortion was detected: a frequency-dependent lateral
displacement of the output proportional to the applied group delay. Pulse shapes that entailed a range of group delays across the spectral bandwidth featured a corresponding spatio-spectral coupling in the output. In all cases, the effect was consistent with a coupling speed of \unit[0.25]{mm/ps} (i.e.\ a relative lateral displacement of the shaped pulse of \unit[0.25]{mm} per picosecond shift in the AOPDF diffraction window). No other spatial or spatio-temporal distortions were detected.
The precision of the measurements was as follows. In measuring the zeroth and first-order phase components of the AOPDF transfer function, the dominant source of error was instability of the interferometer, typically \unit[0.5]{fs} over the approximately fifteen-minute durations of the data acquisition runs. For measuring higher-order phase terms, as well as the amplitude of the AOPDF transfer function, the two most significant sources of error were camera shot noise and shot-to-shot fluctuations in the UV source itself. These limited the root-mean-square precision of the phase and intensity measurements to \unit[0.2]{rad} and \unit[10]{\%} respectively. These figures apply to regions where the intensity is greater than \unit[10]{\%} of the peak. It was verified that the presence of the AOPDF did not increase the size of any phase or intensity fluctuations.
The demonstration of such spatio-temporal coupling effects --- well known and studied for the case of $4f$-line pulse shaping --- gives important information to experimentalists wishing to use AOPDFs in a control experiment. These results are presented individually below. For each case, a mathematical expression for the transfer function employed is presented in Table \ref{tab:transfer-fns}.
\subsection{Pulse delay}
\label{sec:delays}
\begin{figure}
\centering
\includegraphics [width = 0.9\columnwidth]{McCabe200910_Fig3.pdf}
\caption{Spatio-temporal coupling for a single optical pulse as the diffraction position within the AOPDF crystal is varied. The central beam position along the spatial axis of the spectrometer is plotted as a function of delay $\tau$. A linear dependence is observed with a best-fit gradient of \mbox{\unit[0.249 $\pm$ 0.012]{mm/ps}}.}
\label{fig:delays}
\end{figure}
For the first experiment, an acoustic wave was launched inside the AOPDF that was designed to diffract a single optical pulse within the KDP crystal. The location of the acoustic wave was scanned along the length of the crystal in order to vary the pulse delay $\tau$. The acoustic wave was tailored in order to pre-compensate for the dispersion of the crystal, and the performance of this compensation was verified via the SI measurements. The pulse spectral FWHM intensity bandwidth was also narrowed using the AOPDF to $\Delta\lambda = (\lambda_0^2/2\pi c)\Delta\omega = \unit[1]{nm}$, where $c$ is the speed of light. This reduced the length of acoustic wave required to compensate for the crystal dispersion to \unit[2]{ps}, enabling a greater range of delays to be accessed without clipping the acoustic wave on the edges of the crystal.
The measured delays were found to be in agreement with the target delays to within an error of \unit[2]{\%}. The central beam position of the diffracted pulse was observed to vary linearly with delay with a coupling speed of \mbox{\unit[0.249 $\pm$ 0.012]{mm/ps}}. Not other variation, in either amplitude or phase, was identified. The results are presented in Fig.\ \ref{fig:delays}. This behaviour was also confirmed with a direct measurement of the beam position on a CCD camera.
\subsection{Pulse train}
\label{sec:train}
\begin{figure}
\centering
\includegraphics [width = \columnwidth]{McCabe200910_Fig4.pdf}
\caption{The reconstructed spatio-temporal amplitude distribution of a train of three pulses each separated by \unit[1.5]{ps}. The reconstructed pulse train exhibits a linear spatio-temporal coupling effect that is consistent with the \unit[0.25]{mm/ps} best-fit gradient observed for the pulse delay experiments (superimposed dotted line).}
\label{fig:train}
\end{figure}
Next, various trains of pulses with zero added second- and higher-order phase were prepared, with varying numbers of pulses ranging from two to thirteen. This entailed a sequence of acoustic waves localized at different points along the length of the AOPDF crystal.
A typical reconstructed spatio-temporal intensity distribution is shown in Fig.\ \ref{fig:train} for a train of three pulses separated by \unit[1.5]{ps} (such that $N=3$, $\tau_1 = \unit[-1.5]{ps}$, $\tau_2 = \unit[0]{ps}$ and $\tau_3 = \unit[1.5]{ps}$ according to the expression of Table \ref{tab:transfer-fns}). The pulse separation was verified to within \unit[1]{\%}. In order to make the most accurate measurement possible, the full temporal window of the pulse shaper was employed. The spatio-temporal coupling subsequently resulted in a worsened alignment for the third pulse in the train, concomitantly reducing the fringe visibility. This accounts for the apparent reduction in intensity for the final pulse in Fig.\ \ref{fig:train}.
A pronounced linear spatio-temporal coupling is observed in the reconstruction. The results are quantitatively consistent with the coupling speed observed during the pulse delay experiments (Section \ref{sec:delays}), as evinced by the superimposed best-fit line with the same coupling-speed gradient of \unit[0.249]{mm/ps}.
\subsection{Chirps}
\label{sec:chirps}
\begin{figure}
\centering
\includegraphics [width = \columnwidth]{McCabe200910_Fig5.pdf}
\caption{Spatio-spectral coupling effects for a series of chirped pulses ($\phi^{(2)}$ parameters as shown). (a) The reconstructed spatio-spectral intensities of pulses of different chirps. A spatio-spectral tilt is observed that is more significant for the more strongly chirped pulses and that changes sign with the sign of the chirp. (b) The extracted spatio-spectral coupling as a function of chirp (`+') together with a calculated best-fit coupling speed of \mbox{\unit[0.252 $\pm$ 0.004]{mm/ps}} (solid line). This value is in close agreement with the measurement of Section \ref{sec:delays}; the reconstructed pulse was otherwise found to be free of further spatio-temporal coupling effects. The vertical axis shows the change in central position of the beam across the spectral bandwidth of the pulse.}
\label{fig:chirp}
\end{figure}
A range of different pulses were prepared using the AOPDF bearing different chirps --- i.e.\ the parameter $\phi^{(2)}$ in Table \ref{tab:transfer-fns}. The AOPDF temporal shaping window allowed values within the range $\unit[-100000]{fs^2} \leq \phi^{(2)} \leq \unit[100000]{fs^2}$ to be assayed. A narrowed pulse bandwidth of $\Delta\lambda = \unit[1]{nm}$ was once again employed.
The extracted $\phi^{(2)}$ second-order polynomial phase coefficients matched the programmed values to within \unit[6]{\%}. The spatio-spectral intensities are shown in Fig.\ \ref{fig:chirp}(a) for a selection of $\phi^{(2)}$ values. A spatio-spectral tilt is observed that is stronger for more strongly chirped pulses and changes sign as the sign of the chirp is reversed [see the dashed lines of \mbox{Fig.\ \ref{fig:chirp}(a)}]. This observation has important consequences for control experiments with regard to spatial alignment with the sample. Besides the spatio-spectral tilt illustrated in Fig.\ \ref{fig:chirp}(a), however, the reconstructed pulse was found to reproduce the programmed pulse with good fidelity.
The spatio-spectral tilts for a range of chirps were extracted numerically and plotted in Fig.\ \ref{fig:chirp}(b). Since spectral chirp is intrinsically a frequency-dependent group delay, the best-fit gradient of these points can be related to a group-delay--dependent displacement via the corresponding chirped-pulse temporal duration. This fit took into account an intrinsic spatio-spectral tilt present on the reference beam corresponding to a \unit[0.35]{mm} shift in beam centre across the spectral bandwidth. The best-fit coupling speed for these experiments was \mbox{\unit[0.252 $\pm$ 0.004]{mm/ps}}, in very close agreement with Sections \ref{sec:delays} and \ref{sec:train}. This demonstrates that one single underlying physical mechanism is responsible for the different spatio-temporal coupling effects.
\subsection{$\pi$-step}
\label{sec:pi-step}
\begin{figure}
\centering
\includegraphics [width = \columnwidth]{McCabe200910_Fig6.pdf}
\caption{Spatio-spectral coupling effects for $\pi$ phase-steps of varying sharpnesses. (a) The reconstructed spatio-spectral intensities of a series of $\pi$ phase-steps with sharpnesses as indicated. A spatial shift is observed at the step frequency that is more pronounced for sharper steps. (b) Observed lateral displacement of the notch (data points) together with a calculation derived from the measured group delay at the phase step and a \unit[0.25]{mm/ps} spatio-temporal coupling speed (solid line). An example spectral phase across a slice through the middle of the pulse is shown inset (crosses) together with a fit of the function in Table \ref{tab:transfer-fns} (solid line).}
\label{fig:pi}
\end{figure}
The final experiment entailed the preparation of a transform-limited pulse with a $\pi$ phase-step at its central frequency as per the expression in Table \ref{tab:transfer-fns}. Phase steps of a range of sharpnesses, $\Delta\lambda_{\textrm{step}}$, were prepared. The results are shown in Fig.\ \ref{fig:pi}(a). A typical measured spectral phase across the centre of the pulse is shown inset in \mbox{Fig.\ \ref{fig:pi}(b)}. In general, the retrieved phase matched the programmed one with regard to the parameters of Table \ref{tab:transfer-fns}. The sharpest measured step sizes were of the order on \unit[0.08]{nm}; however, this was commensurate with the resolution of the spectrometer.
An important spatio-temporal coupling effect is observed in the reconstructed spectral intensities. A local spatial displacement occurs in the spectrum at the $\pi$-step frequency, resulting in a `notch' in the reconstructed spatio-spectral intensity [see arrow in \mbox{Fig.\ \ref{fig:pi}(a)]}. The size of the notch increases with the sharpness of the phase step. This spatio-temporal coupling effect has previously only been reported in a $4f$ zero-dispersion line \cite{Dorrer1998}; this study reveals similar behaviour for an AOPDF-based device.
This notch effect may once again be reconciled with a group-velocity--dependent displacement of the beam. The steep phase gradient at the location of the $\pi$ step is equivalent to a local group-delay term in the spectral phase, with a sharper step implying a steeper gradient in the spectral phase and hence a larger group delay. A group-velocity--dependent displacement therefore shifts spectral components spanned by $\Delta\lambda_{\textrm{step}}$ by an amount dependent on the step sharpness. A related effect in pixellated SLM pulse shapers is the complete spectral hole that appears for a sharpness equal to the spectral resolution of the device \cite{Wohlleben2004}. As the step sharpness is further increased in these AOPDF experiments, the $\pi$-step group delay will eventually exceed the temporal window of the crystal (which is inversely proportional to the AOPDF spectral resolution), and a spectral hole, rather than a notch, will be formed as a consequence.
This argument is supported by the calculations presented in Fig.\ \ref{fig:pi}(b) based on these experimental data. In this figure, the notch sizes for each image within Fig.\ \ref{fig:pi}(a) were extracted and plotted as a function of step sharpness (data points). The local group-delay terms at the phase step were calculated according to $\phi^{(1)} = \frac{\partial \phi \br{\omega}}{\partial \omega} \approx \frac{\pi}{\Delta \omega}$ where $\Delta \omega_{\textrm{step}} = (2\pi c/\lambda_0^2) \Delta\lambda_{\textrm{step}}$. They were then multiplied by the \unit[0.25]{mm/ps} coupling speed previously observed (solid line), and the resultant calculation shows good agreement with experiment (solid line). Once again, the results are found to be quantitatively consistent with the same spatio-temporal coupling effect as above, reinforcing the evidence for a single underlying physical mechanism for all of these manifestations.
\section{Discussion}
\label{sec:discussion}
Section \ref{sec:results} presented a spatially resolved SSI analysis of a range of different pulse shapes: a single transform-limited pulse with a variable delay, a train of pulses, chirped pulses and pulses with a $\pi$ phase-step at the centre of their spectrum. These pulse shapes lie at the heart of many ultrafast quantum control experiments and this study represents the first complete investigation of spatio-temporal coupling effects performed for an AOPDF pulse shaper. In each case, spatio-temporal and spatio-spectral couplings were observed in the reconstructed field. Each effect was shown to be consistent with a single effect that took the form of a group-delay--dependent position of the shaped pulses, as mentioned previously \cite{Krebs2010}. Incidentally, the time-to-space mapping produced by this coupling means that the spatio-spectral intensity profile of the Dazzler output pulse resembles a spectrogram, assuming that the Dazzler input pulse is near transform-limited so that the only contribution to the group-delay in the output arises from the Dazzler itself. Furthermore, each coupling was consistent with a coupling speed of \unit[0.25]{mm/ps}. No further spatio-temporal coupling effects were identified, and the AOPDF was otherwise found to reproduce the programmed pulse shapes faithfully. In particular, no significant angular dispersion effects (as reported by B\"{o}rzs\"{o}nyi \emph{et al.~} \cite{Borzsonyi2010}) were found. This is to be expected since B\"{o}rzs\"{o}nyi \emph{et al.~} only found this to be significant at high repetition rates where the acoustic-wave energy dissipation gave rise to thermal effects.
The results above highlight the need for experimentalists to pay close attention to these coupling issues during the design of control experiments based on an AOPDF pulse shaper. Such concerns have been studied extensively for the more widespread $4f$-line shapers, with coupling speed ranging from \unit[0.083]{mm/ps} \cite{Monmayrant2004} through \unit[0.145]{mm/ps} \cite{Wefers1996} to \unit[0.595]{mm/ps} \cite{Tanabe2002} already reported in the literature. For the $4f$-line geometry, the coupling speed $v$ is related to the available temporal shaping window $T$ and the input beam waist $\Delta x_\textrm{in}$ by $\modbr{v} = \Delta x_\textrm{in}/T$ \cite{Monmayrant2010}. The coupling speed reported here of \unit[0.25]{mm/ps} is therefore non-negligible by comparison.
It is thus apparent that a single spatio-temporal coupling mechanism within the AOPDF accounts for all the manifestations reported in Section \ref{sec:results}. In order to explain the physical nature of this group-delay--dependent displacement, it is necessary to consider a couple of effects present within the Dazzler: the birefringent and geometrical walk-off effects of the diffracted relative to the undiffracted beam, and the fact that each optical wavelength within the ultrafast pulse is diffracted at a given position in the AOPDF. These two effects combine to lead to a natural spatial chirp, with a coupling speed as quantified above.
To recapitulate, the birefringent walk-off concerns the phenomenon that the intensity distribution of a beam in an anisotropic crystal drifts away from the direction of the wave vector. The angle between the Poynting vector (which defines the direction of energy transport) and the $k$-vector is called the walk-off angle. Spatial walk-off occurs only for a beam with extraordinary polarization, which sees a refractive index $n_{\textrm{e}}$ during its propagation that depends on the angle between $\boldsymbol{k}$ and the optical axes. This angle depends on the crystal and parameters of the optical pulse; for the KDP crystal in this experiment, at \unit[268]{nm}, the walk-off angle is $\alpha \simeq \unit[32]{mrad}$. The geometrical walk-off, meanwhile, concerns the fact that during Bragg diffraction the beam is deviated by an angle corresponding to the phase-matching condition. For this experiment, this deviation is $\theta= \unit[-5.2]{mrad}$. It should be noted that both the geometric and birefringent walk-offs actually vary as a function of wavelength; however, this effect is negligible for the pulse bandwidth employed.
Thus the spatio-temporal effect can simply be seen as a shift $\delta x$ in the position of the diffracted beam that could be expressed as $\delta x =L\tan(\theta+\alpha)$, where $L$ is the distance of propagation along the extraordinary axis. The coupling speed is thus determined by $v = \delta x/T$, being a function of the walkoff-induced shift and the temporal window, rather than the input beam waist as for the case of a $4f$ line. This experiment employs a crystal of length \unit[75]{mm} such that the maximum shift is calculated as \unit[2]{mm}. Considering the fact that the temporal window available at this wavelength is $T = \unit[7.7]{ps}$, this implies an expected group-delay--dependent displacement of \mbox{\unit[$0.260 \pm 0.005$]{mm/ps}}, which is in very close agreement with our experimental measurements.
The birefringent and geometric walkoff effects are therefore confirmed as the single physical cause for the spatio-temporal coupling effect reported in the AOPDF pulse shaper. This coupling has important consequences for the application of AOPDF-shaped pulses to control experiments, since the displacement of the control pulses with a variation of pulse parameters may result in a worsened alignment with the target. One possible solution is to translate a lens before the AOPDF in order to bring the geometric plane of overlap of the spatially shifted output pulses into alignment with the gaussian focal plane \cite{Krebs2010}. Another might be to extend the walk-off compensation methods developed in non-linear optics be using a double-pass setup or a second crystal \cite{Smith1998}. It should be noted that the coupling speed depends on the parameters of the ultrafast pulses as well as the choice of crystal (indeed, the walk-off effects in TeO$_2$, which is used for AOPDFs in the IR wavelength range, are significantly less than in KDP); thus the calculation should be repeated along the lines above in order to make an informed choice of shaper in light of individual experimental tolerances for coupling effects.
\section{Conclusion}
\label{sec:conclusion}
In this paper, we have presented a systematic study of spatio-temporal coupling in an AOPDF pulse shaper that operates at UV wavelength ranges via spatially resolved phase and amplitude analysis of the shaped pulses. Such coupling effects have been widely studied for $4f$ zero dispersion lines due to the importance of the ramifications for control experiments. The AOPDF is an increasingly popular alternative shaping device thanks to its versatility, compactness, ease of alignment and wide wavelength range. Until now, however, its spatio-temporal coupling effects have not been comprehensively studied for a range of complex pulse shapes of interest to the control community.
We have discovered that there is one single significant effect at kilohertz repetition rates: a group-delay--dependent displacement of the shaped output. Further to this one effect, the AOPDF was found to produce faithfully the desired pulse shape. This coupling effect was manifested differently in the measured pulse depending on the class of pulse shape employed; however, in each case the coupling effect may be described by the same mechanism with consistent quantitative agreement. We have explained the physical origin of this mechanism and have shown excellent agreement between its calculated and measured values. Finally, we have identified some approaches that may allow the impact of this spatio-temporal coupling to be minimized during applications to control experiments.
\begin{acknowledgements}
The authors are grateful to N.\ Forget and A.\ Wyatt for useful discussions as well as to E.\ Baynard and S.\ Faure for technical assistance. This work was supported by the Marie Curie Initial Training Network (grant no.\ CA-ITN-214962-FASTQUAST), EPSRC grants EP/H000178/1 and EP/G067694/1, and Alliance (PHC/British Council).
\end{acknowledgements}
|
{
"timestamp": "2010-10-29T02:02:18",
"yymm": "1009",
"arxiv_id": "1009.4293",
"language": "en",
"url": "https://arxiv.org/abs/1009.4293"
}
|
\section{Introduction}
In\ this paper we introduce a new problem, which we call the \textit{Split
Variational Inequality Problem} (SVIP). Let $H_{1}$ and $H_{2}$ be two real
Hilbert spaces$.$ Given operators $f:H_{1}\rightarrow H_{1}$ and
$g:H_{2}\rightarrow H_{2},$ a bounded linear operator $A:H_{1}\rightarrow
H_{2}$, and nonempty, closed and convex subsets $C\subseteq H_{1}$ and
$Q\subseteq H_{2},$ the SVIP is formulated as follows
\begin{gather}
\text{find a point }x^{\ast}\in C\text{ such that }\left\langle f(x^{\ast
}),x-x^{\ast}\right\rangle \geq0\text{ for all }x\in C\label{eq:vip}\\
\text{and such that}\nonumber\\
\text{the point }y^{\ast}=Ax^{\ast}\in Q\text{ solves }\left\langle g(y^{\ast
}),y-y^{\ast}\right\rangle \geq0\text{ for all }y\in Q. \label{eq:svip
\end{gather}
When Looked at separately, (\ref{eq:vip}) is the classical \textit{Variational
Inequality Problem} (VIP) and we denote its solution set by $SOL(C,f)$. The
SVIP constitutes a pair of VIPs, which have to be solved so that the image
$y^{\ast}=Ax^{\ast},$ under a given bounded linear operator $A,$ of the
solution $x^{\ast}$ of the VIP in $H_{1}$, is a solution of another VIP in
another space $H_{2}$.
SVIP is quite general and should enable split minimization between two spaces
so that the image of a solution point of one minimization problem, under a
given bounded linear operator, is a solution point of another minimization
problem. Another special case of the SVIP is the \textit{Split Feasibility
Problem} (SFP) which had already been studied and used in practice as a model
in intensity-modulated radiation therapy (IMRT) treatment planning; see
\cite{CBMT, CEKB}.
We consider two approaches to the solution of the SVIP. The first approach is
to look at the product space $H_{1}\times H_{2}$ and transform the SVIP
(\ref{eq:vip})--(\ref{eq:svip}) into an equivalent \textit{Constrained VIP}
(CVIP) in the product space. We study this CVIP and devise an iterative
algorithm for its solution, which becomes applicable to the original SVIP via
the equivalence between the problems. Our new iterative algorithm for the
CVIP, thus for the SVIP, is inspired by an extension of the extragradient
method of Korpelevich \cite{Korpelevich}. In the second approach we present a
method that does not require the translation to a product space. This
algorithm is inspired by the work of Censor and Segal \cite{CS08a} and Moudafi
\cite{Moudafi}.
Our paper is organized as follows. In Section \ref{sec:Preliminaries} we
present some preliminaries. In Section \ref{sec:Algorithm} the algorithm for
the constrained VIP is presented. In Section \ref{sec:SVIP} we analyze the
SVIP and present its equivalence with the CVIP in the product space. In
Section \ref{sec:Direct SVIP} we first present our method for solving the
SVIP, which does not rely on any product space formulation, and then prove
convergence. In Section \ref{sec:applications} we present some applications of
the SVIP. It turns out that in addition to helping us solve the SVIP, the CVIP
unifies and improves several existing problems and methods where a VIP has to
be solved with some additional constraints. Relations of our results to some
previously published work are discussed in detail after Theorems \ref{th:cvip}
and \ref{Theorem1}.
\section{Preliminaries\label{sec:Preliminaries}}
Let $H$ be a real Hilbert space with inner product $\langle\cdot,\cdot\rangle$
and norm $\Vert\cdot\Vert,$ and let $D$ be a nonempty, closed and convex
subset of $H$. We write $x^{k}\rightharpoonup x$ to indicate that the sequence
$\left\{ x^{k}\right\} _{k=0}^{\infty}$ converges weakly to $x,$ and
$x^{k}\rightarrow x$ to indicate that the sequence $\left\{ x^{k}\right\}
_{k=0}^{\infty}$ converges strongly to $x.$ For every point $x\in H,$\ there
exists a unique nearest point in $D$, denoted by $P_{D}(x)$. This point
satisfie
\begin{equation}
\left\Vert x-P_{D}\left( x\right) \right\Vert \leq\left\Vert x-y\right\Vert
\text{\textit{ }for all}\mathit{\ }y\in D.
\end{equation}
The mapping $P_{D}$ is called the metric projection of $H$ onto $D$. We know
that $P_{D}$ is a nonexpansive operator of $H$ onto $D$, i.e.
\begin{equation}
\left\Vert P_{D}\left( x\right) -P_{D}\left( y\right) \right\Vert
\leq\left\Vert x-y\right\Vert \text{\textit{ }for all}\mathit{\ }x,y\in H.
\end{equation}
The metric projection $P_{D}$ is characterized by the fact that $P_{D}\left(
x\right) \in D$ and
\begin{equation}
\left\langle x-P_{D}\left( x\right) ,P_{D}\left( x\right) -y\right\rangle
\geq0\text{ for all }x\in H,\text{ }y\in D, \label{eq:ProjP1
\end{equation}
and has the propert
\begin{equation}
\left\Vert x-y\right\Vert ^{2}\geq\left\Vert x-P_{D}\left( x\right)
\right\Vert ^{2}+\left\Vert y-P_{D}\left( x\right) \right\Vert ^{2}\text{
for all }x\in H,\text{ }y\in D. \label{eq:ProjP2
\end{equation}
It is known that in a Hilbert space $H$
\begin{equation}
\Vert\lambda x+(1-\lambda)y\Vert^{2}=\lambda\Vert x\Vert^{2}+(1-\lambda)\Vert
y\Vert^{2}-\lambda(1-\lambda)\Vert x-y\Vert^{2} \label{eq:ConvexComb
\end{equation}
for all $x,y\in H$ and $\lambda\in\lbrack0,1].$
The following lemma was proved in \cite[Lemma 3.2]{Takahashi}.
\begin{lemma}
\label{Lemma:Takahashi} Let $H$ be a Hilbert space and let $D$ be a nonempty,
closed and convex subset of $H.$ If the sequence $\left\{ x^{k}\right\}
_{k=0}^{\infty}\subset H$ is \texttt{Fej\'{e}r-monotone} with respect to $D,$
i.e., for every $u\in D,
\begin{equation}
\Vert x^{k+1}-u\Vert\leq\Vert x^{k}-u\Vert\text{ for all }k\geq0,
\end{equation}
then $\left\{ P_{D}\left( x^{k}\right) \right\} _{k=0}^{\infty}$ converges
strongly to some $z\in D.$
\end{lemma}
The next lemma is also known (see, e.g., \cite[Lemma 3.1]{Nadezhkina}).
\begin{lemma}
\label{Lemma:Schu} Let $H$ be a Hilbert space, $\left\{ \alpha_{k}\right\}
_{k=0}^{\infty}$ be a real sequence satisfying $0<a\leq\alpha_{k}\leq b<1$ for
all $k\geq0,$ and let $\left\{ v^{k}\right\} _{k=0}^{\infty}$ and $\left\{
w^{k}\right\} _{k=0}^{\infty}$ be two sequences in $H$ such that for some
$\sigma\geq0$
\begin{equation}
\limsup_{k\rightarrow\infty}\Vert v^{k}\Vert\leq\sigma,\text{ and
\limsup_{k\rightarrow\infty}\Vert w^{k}\Vert\leq\sigma.
\end{equation}
I
\begin{equation}
\lim_{k\rightarrow\infty}\Vert\alpha_{k}v^{k}+(1-\alpha_{k})w^{k}\Vert=\sigma,
\end{equation}
the
\begin{equation}
\lim_{k\rightarrow\infty}\Vert v^{k}-w^{k}\Vert=0.
\end{equation}
\end{lemma}
\begin{definition}
Let $H$ be a Hilbert space, $D$ a closed and convex subset of $H,$ and let
$M:D\rightarrow H$ be an operator. Then $M$ is said to be \texttt{demiclosed}
at $y\in H$ if for any sequence $\left\{ x^{k}\right\} _{k=0}^{\infty}$ in
$D$ such that $x^{k}\rightharpoonup\overline{x}\in D$ and $M(x^{k})\rightarrow
y,$ we have $M(\overline{x})=y.$
\end{definition}
Our next lemma is the well-known Demiclosedness Principle \cite{Browder}.
\begin{lemma}
Let $H$ be a Hilbert space, $D$ a closed and convex subset of $H,$ and
$N:D\rightarrow H$ a nonexpansive operator. Then $I-N$ ($I$ is the identity
operator on $H$) is \texttt{demiclosed} at $y\in H.$
\end{lemma}
For instance, the orthogonal projection $P$ onto a closed and convex set is a
demiclosed operator everywhere because $I-P$ is nonexpansive \cite[page
17]{Goebel+Reich}.
The next property is known as the \textit{Opial condition} \cite[Lemma
1]{Opial}. It characterizes the weak limit of a weakly convergent sequence in
Hilbert space.
\begin{condition}
(\textbf{Opial) }For any sequence $\left\{ x^{k}\right\} _{k=0}^{\infty}$ in
$H$ that converges weakly to $x$
\begin{equation}
\liminf_{k\rightarrow\infty}\Vert x^{k}-x\Vert<\liminf_{k\rightarrow\infty
}\Vert x^{k}-y\Vert\text{ for all }y\neq x.
\end{equation}
\end{condition}
\begin{definition}
Let $h:H\rightarrow H$ be an operator and let $D\subseteq H.$
(i) $h$ is called \texttt{inverse strongly monotone (ISM)} with constant
$\alpha$ on $D\subseteq H$ i
\begin{equation}
\langle h(x)-h(y),x-y\rangle\geq\alpha\Vert h(x)-h(y)\Vert^{2}\text{ for all
}x,y\in D.
\end{equation}
(ii) $h$ is called \texttt{monotone} on $D\subseteq H$ i
\begin{equation}
\langle h(x)-h(y),x-y\rangle\geq0\text{ for all }x,y\in D.
\end{equation}
\end{definition}
\begin{definition}
An operator $h:H\rightarrow H$ is called \texttt{Lipschitz continuous} on
$D\subseteq H$ with constant $\kappa>0$ i
\begin{equation}
\Vert h(x)-h(y)\Vert\leq\kappa\Vert x-y\Vert\text{\ for all\ }x,y\in D.
\end{equation}
\end{definition}
\begin{definition}
Let $S:H\rightrightarrows2^{H}\mathcal{\ }$be a point-to-set operator defined
on a real Hilbert space $H$. $S$ is called a \texttt{maximal monotone
operator} if $S$ is \texttt{monotone}, i.e.
\begin{equation}
\left\langle u-v,x-y\right\rangle \geq0,\text{ for all }u\in S(x)\text{ and
for all }v\in S(y),
\end{equation}
and the graph $G(S)$ of $S,
\begin{equation}
G(S):=\left\{ \left( x,u\right) \in H\times H\mid u\in S(x)\right\} ,
\end{equation}
is not properly contained in the graph of any other monotone operator.
\end{definition}
It is clear that a monotone operator $S$ is maximal if and only if, for each
$\left( x,u\right) \in H\times H,$ $\left\langle u-v,x-y\right\rangle \geq0$
for all $\left( v,y\right) \in G(S)$ implies that $u\in S(x).$
\begin{definition}
Let $D$ be a nonempty, closed and convex subset of $H.$ The \texttt{normal
cone} of $D$ at the point $w\in D$ is defined b
\begin{equation}
N_{D}\left( w\right) :=\{d\in H\mid\left\langle d,y-w\right\rangle
\leq0\text{ for all }y\in D\}. \label{eq:normal-c
\end{equation}
\end{definition}
Let $h$ be an $\alpha$-ISM operator on $D\subseteq H,$ let $N_{D}\left(
w\right) $ be the normal cone of $D$ at a point $w\in D$, and define the
following point-to-set operator
\begin{equation}
S(w):=\left\{
\begin{array}
[c]{cc
h(w)+N_{D}\left( w\right) , & w\in C,\\
\emptyset, & w\notin C.
\end{array}
\right. \label{eq:maximal-S
\end{equation}
In these circumstances, it follows from \cite[Theorem 3]{Rockafellar76} that
$S$ is maximal monotone. In addition, $0\in S(w)\ $if and only if $w\in$
$SOL(D,h).$
For $T:H\rightarrow H$, denote by $\operatorname*{Fix}(T)$ the fixed point set
of $T,$ i.e.
\begin{equation}
\operatorname*{Fix}(T):=\{x\in H\mid T(x)=x\}.
\end{equation}
It is well-known tha
\begin{equation}
x^{\ast}\in SOL(C,f)\Leftrightarrow x^{\ast}=P_{C}(x^{\ast}-\lambda f(x^{\ast
})), \label{eq:fix-vip
\end{equation}
i.e., $x^{\ast}\in\operatorname*{Fix}(P_{C}(I-\lambda f)).$ It is also known
that every nonexpansive operator $T:H\rightarrow H$ satisfies, for all
$(x,y)\in H\times H,$ the inequality
\begin{equation}
\langle(x-T(x))-(y-T(y)),T(y)-T(x)\rangle\leq(1/2)\Vert(T(x)-x)-(T(y)-y)\Vert
^{2}\text{
\end{equation}
and therefore we get, for all $(x,y)\in H\times\operatorname*{Fix}(T),$
\begin{equation}
\langle x-T(x),y-T(x)\rangle\leq(1/2)\Vert T(x)-x\Vert^{2};\text{ }
\label{eq:Ne(Crombez)
\end{equation}
see, e.g., \cite[Theorem 3]{Crombez06} and \cite[Theorem 1]{Crombez}.
In the next lemma we collect several important properties that will be needed
in the sequel.
\begin{lemma}
\label{lemma:Mod-proj} Let $D\subseteq H$ be a nonempty, closed and convex
subset and let $h:H\rightarrow H$ be an $\alpha$-ISM operator on $H$. If
$\lambda\in\lbrack0,2\alpha],$ then\smallskip\
(i) the operator $P_{D}(I-\lambda h)$ is nonexpansive on $D.$
If, in addition, for all $x^{\ast}\in SOL(D,h),
\begin{equation}
\langle h(x),P_{D}(I-\lambda h)(x)-x^{\ast}\rangle\geq0\text{ for all\ }x\in
H, \label{eq:2.24
\end{equation}
then$\smallskip$ the following inequalities hold:
(ii) for all $x\in H$ and $q\in\operatorname*{Fix}(P_{D}(I-\lambda h)),
\begin{equation}
\langle P_{D}(I-\lambda h)(x)-x,P_{D}(I-\lambda h)(x)-q\rangle\leq0;
\label{QFNE
\end{equation}
(iii) for all $x\in H$ and $q\in\operatorname*{Fix}(P_{D}(I-\lambda h)),
\begin{equation}
\left\Vert P_{D}(I-\lambda h)(x)-q\right\Vert ^{2}\leq\left\Vert
x-q\right\Vert ^{2}-\left\Vert P_{D}(I-\lambda h)(x)-x\right\Vert ^{2}.
\label{eq:2.26
\end{equation}
\end{lemma}
\begin{proof}
(i) Let $x,y\in H.$ The
\begin{align}
\Vert P_{D}(I-\lambda h)(x)-P_{D}(I-\lambda h)(y)\Vert^{2} & =\Vert
P_{D}(x-\lambda h(x))-P_{D}(y-\lambda h(y))\Vert^{2}\nonumber\\
& \leq\Vert x-\lambda h(x)-(y-\lambda h(y))\Vert^{2}\nonumber\\
& =\Vert(x-y)-\lambda(h(x)-h(y))\Vert^{2}\nonumber\\
& =\Vert x-y\Vert^{2}-2\lambda\langle x-y,h(x)-h(y)\rangle\nonumber\\
& +\lambda^{2}\Vert h(x)-h(y)\Vert^{2}\nonumber\\
& \leq\Vert x-y\Vert^{2}-2\lambda\alpha\Vert h(x)-h(y)\Vert^{2}\nonumber\\
& +\lambda^{2}\Vert h(x)-h(y)\Vert^{2}\nonumber\\
& =\Vert x-y\Vert^{2}+\lambda(\lambda-2\alpha)\Vert h(x)-h(y)\Vert
^{2}\nonumber\\
& \leq\Vert x-y\Vert^{2}.
\end{align}
(ii) Let $x\in H$ and $q\in\operatorname*{Fix}(P_{D}(I-\lambda h)).$ The
\begin{align}
& \langle P_{D}(x-\lambda h(x))-x,P_{D}(x-\lambda h(x))-q\rangle\nonumber\\
& =\langle P_{D}(x-\lambda h(x))-x+\lambda h(x)-\lambda h(x),P_{D}(x-\lambda
h(x))-q\rangle\nonumber\\
& =\langle P_{D}(x-\lambda h(x))-(x-\lambda h(x)),P_{D}(x-\lambda
h(x))-q\rangle\nonumber\\
& -\lambda\langle h(x),P_{D}(x-\lambda h(x))-q\rangle.
\end{align}
By (\ref{eq:ProjP1}), (\ref{eq:fix-vip}) and (\ref{eq:2.24}), we ge
\begin{equation}
\langle P_{D}(x-\lambda h(x))-x,P_{D}(x-\lambda h(x))-q\rangle\leq0.
\end{equation}
(iii) Let $x\in H$ and $q\in\operatorname*{Fix}(P_{D}(I-\lambda h)).$ The
\begin{align}
\left\Vert q-x\right\Vert ^{2} & =\left\Vert (P_{D}(I-\lambda
h)(x)-x)-(P_{D}(I-\lambda h)(x)-q)\right\Vert ^{2}\nonumber\\
& =\left\Vert P_{D}(I-\lambda h)(x)-x\right\Vert ^{2}+\left\Vert
P_{D}(I-\lambda h)(x)-q\right\Vert ^{2}\nonumber\\
& -2\langle P_{D}(I-\lambda h)(x)-x,P_{D}(I-\lambda h)(x)-q\rangle.
\end{align}
By (ii), we ge
\begin{equation}
-2\langle P_{D}(I-\lambda h)(x)-x,P_{D}(I-\lambda h)(x)-q\rangle\geq0.
\end{equation}
Thus
\begin{equation}
\left\Vert q-x\right\Vert ^{2}\geq\left\Vert P_{D}(I-\lambda
h)(x)-x\right\Vert ^{2}+\left\Vert P_{D}(I-\lambda h)(x)-q\right\Vert ^{2
\end{equation}
o
\begin{equation}
\left\Vert P_{D}(I-\lambda h)x-q\right\Vert ^{2}\leq\left\Vert q-x\right\Vert
^{2}-\left\Vert P_{D}(I-\lambda h)x-x\right\Vert ^{2},
\end{equation}
as asserted.
\end{proof}
Equation (\ref{QFNE}) means that the operator $P_{D}(I-\lambda h)$ belongs to
the class of operators called the $\mathcal{T}$-class. This class
$\mathcal{T}$ of operators was introduced and investigated by Bauschke and
Combettes in \cite[Definition 2.2]{BC01} and by Combettes in \cite{Co}.
Operators in this class were named \textit{directed operators }by Zaknoon
\cite{Z} and further studied under this name by Segal \cite{Seg08} and by
Censor and Segal \cite{CS08, CS08a, CS09}. Cegielski \cite[Def. 2.1]{Ceg08}
studied these operators under the name \textit{separating operators}. Since
both \textit{directed }and\textit{\ separating }are key words of other,
widely-used, mathematical entities, Cegielski and Censor have recently
introduced the term \textit{cutter operators} \cite{cc11}. This class
coincides with the class $\mathcal{F}^{\nu}$ for $\nu=1$ \cite{Crombez} and
with the class DC$_{\boldsymbol{p}}$ for $\boldsymbol{p}=-1$ \cite{mp08}. The
term \textit{firmly quasi-nonexpansive} (FQNE) for $\mathcal{T}$-class
operators was used by Yamada and Ogura \cite{Yamada} because every
\textit{firmly nonexpansive} (FNE) mapping \cite[page 42]{Goebel+Reich} is
obviously FQNE.
\section{An algorithm for solving the constrained variational inequality
problem\label{sec:Algorithm}}
Let\textit{ }$f:H\rightarrow H$, and let $C$ and $\Omega$ be nonempty, closed
and convex subsets of $H$. The \textit{Constrained} \textit{Variational
Inequality Problem} (CVIP) is:
\begin{equation}
\text{find }x^{\ast}\in C\cap\Omega\text{ such that }\left\langle f(x^{\ast
}),x-x^{\ast}\right\rangle \geq0\text{ for all }x\in C. \label{eq:cvip
\end{equation}
The iterative algorithm for this CVIP, presented next, is inspired by our
earlier work \cite{CGR,CGR2} in which we modified the extragradient method of
Korpelevich \cite{Korpelevich}. The following conditions are needed for the
convergence theorem.
\begin{condition}
\label{Condition:a} $f$ is monotone on $C$.
\end{condition}
\begin{condition}
\label{Condition:b} $f$ is Lipschitz continuous on $H$ with constant
$\kappa>0.$
\end{condition}
\begin{condition}
\label{Condition:c} $\Omega\cap SOL(C,f)\neq\emptyset.$
\end{condition}
Let $\left\{ \lambda_{k}\right\} _{k=0}^{\infty}\subset\left[ a,b\right]
$\ for some $a,b\in(0,1/\kappa)$, and let \textit{ }$\left\{ \alpha
_{k}\right\} _{k=0}^{\infty}\subset\left[ c,d\right] $ for some\textit{
}$c,d\in(0,1)$. Then the following algorithm generates two sequences that
converge to a point $z\in\Omega$ $\cap$ SOL$(C,f),$ as the convergence theorem
that follows shows.
\begin{algorithm}
\label{alg:SubExt4SVIP}$\left. {}\right. $
\textbf{Initialization:} Select an arbitrary starting point $x^{0}\in H$.
\textbf{Iterative step:} Given the current iterate $x^{k},$ comput
\begin{equation}
y^{k}=P_{C}(x^{k}-\lambda_{k}f(x^{k})),
\end{equation}
construct the half-space $T_{k}$ the bounding hyperplane of which supports $C$
at $y^{k},
\begin{equation}
T_{k}:=\{w\in H\mid\left\langle \left( x^{k}-\lambda_{k}f(x^{k})\right)
-y^{k},w-y^{k}\right\rangle \leq0\},
\end{equation}
and then calculate the next iterate b
\begin{equation}
x^{k+1}=\alpha_{k}x^{k}+(1-\alpha_{k})P_{\Omega}\left( P_{T_{k}
(x^{k}-\lambda_{k}f(y^{k}))\right) . \label{eq:3.4
\end{equation}
\end{algorithm}
\begin{theorem}
\label{th:cvip}Let\textit{ }$f:H\rightarrow H$, and let $C$ and $\Omega$ be
nonempty, closed and convex subsets of $H$. Assume that Conditions
\ref{Condition:a}--\ref{Condition:c} hold, and let $\left\{ x^{k}\right\}
_{k=0}^{\infty}$ and $\left\{ y^{k}\right\} _{k=0}^{\infty}$ be any two
sequences generated by Algorithm \ref{alg:SubExt4SVIP} with $\left\{
\lambda_{k}\right\} _{k=0}^{\infty}\subset\left[ a,b\right] $\textit{\ for
some }$a,b\in(0,1/\kappa)$\textit{ and }$\left\{ \alpha_{k}\right\}
_{k=0}^{\infty}\subset\left[ c,d\right] $ for some\textit{ }$c,d\in(0,1)$.
Then $\left\{ x^{k}\right\} _{k=0}^{\infty}$ and $\left\{ y^{k}\right\}
_{k=0}^{\infty}$ converge to the same point $z\in\Omega\cap SOL(C,f)$ an
\begin{equation}
z=\lim_{k\rightarrow\infty}P_{\Omega\cap SOL(C,f)}(x^{k}).
\end{equation}
\end{theorem}
\begin{proof}
For the special case of fixed $\lambda_{k}=\tau$ for all $k\geq0$ this theorem
is a direct consequence of our \cite[Theorem 7.1]{CGR2} with the choice of the
nonexpansive operator $S$ there to be $P_{\Omega}$. However, a careful
inspection of the proof of \cite[Theorem 7.1]{CGR2} reveals that it also
applies to a variable sequence $\left\{ \lambda_{k}\right\} _{k=0}^{\infty}$
as used here.
\end{proof}
To relate our results to some previously published works we mention two lines
of research related to our notion of the CVIP. Takahashi and Nadezhkina
\cite{Nadezhkina} proposed an algorithm for finding a point $x^{\ast
\in\operatorname*{Fix}(N)\cap$SOL$(C,f),$ where $N:C\rightarrow C$ is a
nonexpansive operator. The iterative step of their algorithm is as
follows.\textbf{ }Given the current iterate $x^{k},$ comput
\begin{equation}
y^{k}=P_{C}(x^{k}-\lambda_{k}f(x^{k}))
\end{equation}
and the
\begin{equation}
x^{k+1}=\alpha_{k}x^{k}+(1-\alpha_{k})N\left( P_{C}(x^{k}-\lambda_{k
f(y^{k}))\right) .
\end{equation}
The restriction $P_{\Omega}|_{C}$ of our $P_{\Omega}$ in (\ref{eq:3.4}) is, of
course, nonexpansive, and so it is a special case of $N$ in \cite{Nadezhkina}.
But a significant advantage of our Algorithm \ref{alg:SubExt4SVIP} lies in the
fact that we compute $P_{T_{k}}$ onto a half-space in (\ref{eq:3.4}) whereas
the authors of \cite{Nadezhkina} need to project onto the convex set $C.$
Bertsekas and Tsitsiklis \cite[Page 288]{BT} consider the following problem in
Euclidean space: given $f:R^{n}\rightarrow R^{n}$, polyhedral sets
$C_{1}\subset R^{n}$ and $C_{2}\subset R^{m},$ and an $m\times n$ matrix $A$,
find a point $x^{\ast}\in C_{1}$ such that $Ax^{\ast}\in C_{2}$ an
\begin{equation}
\left\langle f(x^{\ast}),x-x^{\ast}\right\rangle \geq0\text{ for all }x\in
C_{1}\cap\{y\mid Ay\in C_{2}\}.
\end{equation}
Denoting $\Omega=A^{-1}(C_{2})$, we see that this problem becomes similar to,
but not identical with a CVIP. While the authors of \cite{BT} seek a solution
in SOL$(C_{1}\cap\Omega,f),$ we aim in our CVIP at $\Omega\cap$SOL$(C,f).$
They propose to solve their problem by the method of multipliers, which is a
different approach than ours, and they need to assume that either $C_{1}$ is
bounded or $A^{T}A$ is invertible, where $A^{T}$ is the transpose of $A.$
\section{The split variational inequality problem as a constrained variational
inequality problem in a product space\label{sec:SVIP}}
Our first approach to the solution of the SVIP (\ref{eq:vip})--(\ref{eq:svip})
is to look at the product space $\boldsymbol{H}=H_{1}\times H_{2}$ and
introduce in it the product set $\boldsymbol{D}:=C\times Q$ and the se
\begin{equation}
\boldsymbol{V:}=\{\mathbf{x}=(x,y)\in\boldsymbol{H}\mid Ax=y\}.
\end{equation}
We adopt the notational convention that objects in the product space are
represented in boldface type. We transform the SVIP (\ref{eq:vip
)--(\ref{eq:svip}) into the following equivalent CVIP in the product space:
\begin{align}
\text{Find a point }\boldsymbol{x}^{\ast} & \in\boldsymbol{D}\cap
\boldsymbol{V},\text{ such that }\left\langle \boldsymbol{h}(\boldsymbol{x
^{\ast}),\boldsymbol{x}-\boldsymbol{x}^{\ast}\right\rangle \geq0\text{
}\nonumber\\
\text{for all }\boldsymbol{x} & =(x,y)\in\boldsymbol{D},
\label{eq:c-as-svip
\end{align}
where $\boldsymbol{h}:\boldsymbol{H}\rightarrow\boldsymbol{H}$ is defined b
\begin{equation}
\boldsymbol{h}(x,y)=(f(x),g(y)).
\end{equation}
A simple adaptation of the decomposition lemma \cite[Proposition 5.7, page
275]{BT} shows that problems (\ref{eq:vip})--(\ref{eq:svip}) and
(\ref{eq:c-as-svip}) are equivalent, and, therefore, we can apply Algorithm
\ref{alg:SubExt4SVIP} to the solution of (\ref{eq:c-as-svip}).
\begin{lemma}
A point $\boldsymbol{x}^{\ast}=(x^{\ast},y^{\ast})$ solves (\ref{eq:c-as-svip
) if and only if $x^{\ast}$ and $y^{\ast}$ solve (\ref{eq:vip
)--(\ref{eq:svip}).
\end{lemma}
\begin{proof}
If $(x^{\ast},y^{\ast})$ solves (\ref{eq:vip})--(\ref{eq:svip}), then it is
clear that $(x^{\ast},y^{\ast})$ solves (\ref{eq:c-as-svip}). To prove the
other direction, suppose that $(x^{\ast},y^{\ast})$ solves (\ref{eq:c-as-svip
). Since (\ref{eq:c-as-svip}) holds for all $(x,y)\in\boldsymbol{D}$, we may
take $(x^{\ast},y)\in\boldsymbol{D}$ and deduce tha
\begin{equation}
\left\langle g(Ax^{\ast}),y-Ax^{\ast}\right\rangle \geq0\text{ for all }y\in
Q.
\end{equation}
Using a similar argument with $(x,y^{\ast})\in\boldsymbol{D,}$ we ge
\begin{equation}
\left\langle f(x^{\ast}),x-x^{\ast}\right\rangle \geq0\text{ for all }x\in C,
\end{equation}
which means that $(x^{\ast},y^{\ast})$ solves (\ref{eq:vip})--(\ref{eq:svip}).
\end{proof}
Using this equivalence, we can now employ Algorithm \ref{alg:SubExt4SVIP} in
order to solve the SVIP. The following conditions are needed for the
convergence theorem.
\begin{condition}
\label{Condition:a2} $f$ is monotone on $C$ and $g$ is monotone on $Q$.
\end{condition}
\begin{condition}
\label{Condition:b2} $f$ is Lipschitz continuous on $H_{1}$ with constant
$\kappa_{1}>0$ and $g$ is Lipschitz continuous on $H_{2}$ with constant
$\kappa_{2}>0.$
\end{condition}
\begin{condition}
\label{Condition:c2} $\boldsymbol{V}\cap SOL(\boldsymbol{D},\boldsymbol{h
)\neq\emptyset.$
\end{condition}
Let $\left\{ \lambda_{k}\right\} _{k=0}^{\infty}\subset\left[ a,b\right]
$\ for some $a,b\in(0,1/\kappa)$, where $\kappa=\min\{\kappa_{1},\kappa_{2
\}$, and let $\left\{ \alpha_{k}\right\} _{k=0}^{\infty}\subset\left[
c,d\right] $ for some\textit{ }$c,d\in(0,1)$. Then the following algorithm
generates two sequences that converge to a point $\boldsymbol{z
\in\boldsymbol{V}\cap SOL(\boldsymbol{D},\boldsymbol{h}),$ as the convergence
theorem given below shows.
\begin{algorithm}
\label{alg:SubExt4SVIP2}$\left. {}\right. $
\textbf{Initialization:} Select an arbitrary starting point $\boldsymbol{x
^{0}\in\boldsymbol{H}$.
\textbf{Iterative step:} Given the current iterate $\boldsymbol{x}^{k},$
comput
\begin{equation}
\boldsymbol{y}^{k}=\boldsymbol{P}_{\boldsymbol{D}}(\boldsymbol{x}^{k
-\lambda_{k}\boldsymbol{h}(\boldsymbol{x}^{k})),
\end{equation}
construct the half-space $\boldsymbol{T}_{k}$ the bounding hyperplane of which
supports $\boldsymbol{D}$ at $\boldsymbol{y}^{k},
\begin{equation}
\boldsymbol{T}_{k}:=\{\boldsymbol{w}\in\boldsymbol{H}\mid\left\langle \left(
\boldsymbol{x}^{k}-\lambda_{k}\boldsymbol{h}(\boldsymbol{x}^{k})\right)
-\boldsymbol{y}^{k},\boldsymbol{w}-\boldsymbol{y}^{k}\right\rangle \leq0\},
\end{equation}
and then calculat
\begin{equation}
\boldsymbol{x}^{k+1}=\alpha_{k}\boldsymbol{x}^{k}+(1-\alpha_{k})\boldsymbol{P
_{\boldsymbol{V}}\left( \boldsymbol{P}_{\boldsymbol{T}_{k}}(\boldsymbol{x
^{k}-\lambda_{k}\boldsymbol{h}(\boldsymbol{y}^{k}))\right) . \label{eq:5.8
\end{equation}
\end{algorithm}
Our convergence theorem for Algorithm \ref{alg:SubExt4SVIP2} follows from
Theorem \ref{th:cvip}.
\begin{theorem}
Consider $f:H_{1}\rightarrow H_{1}$ and $g:H_{2}\rightarrow H_{2},$ a bounded
linear operator $A:H_{1}\rightarrow H_{2}$, and nonempty, closed and convex
subsets $C\subseteq H_{1}$ and $Q\subseteq H_{2}$. Assume that Conditions
\ref{Condition:a2}--\ref{Condition:c2} hold, and let $\left\{ \boldsymbol{x
^{k}\right\} _{k=0}^{\infty}$ and $\left\{ \boldsymbol{y}^{k}\right\}
_{k=0}^{\infty}$ be any two sequences generated by Algorithm
\ref{alg:SubExt4SVIP2} with $\left\{ \lambda_{k}\right\} _{k=0}^{\infty
}\subset\left[ a,b\right] $\textit{\ for some }$a,b\in(0,1/\kappa)$\textit{,
where }$\kappa=\min\{\kappa_{1},\kappa_{2}\}$, \textit{and let} $\left\{
\alpha_{k}\right\} _{k=0}^{\infty}\subset\left[ c,d\right] $ for
some\textit{ }$c,d\in(0,1)$. Then $\left\{ \boldsymbol{x}^{k}\right\}
_{k=0}^{\infty}$ and $\left\{ \boldsymbol{y}^{k}\right\} _{k=0}^{\infty}$
converge to the same point $\boldsymbol{z}\in\boldsymbol{V}\cap
SOL(\boldsymbol{D},\boldsymbol{h})$ an
\begin{equation}
\boldsymbol{z}=\lim_{k\rightarrow\infty}\boldsymbol{P}_{\boldsymbol{V}\cap
SOL(\boldsymbol{D},\boldsymbol{h})}(\boldsymbol{x}^{k}).
\end{equation}
\end{theorem}
The value of the product space approach, described above, depends on the
ability to \textquotedblleft translate\textquotedblright\ Algorithm
\ref{alg:SubExt4SVIP2} back to the original spaces $H_{1}$ and $H_{2}.$
Observe that due to \cite[Lemma 1.1]{Pierra} for $\boldsymbol{x
\mathbf{=}(x,y)\in\boldsymbol{D,}$ we have $\boldsymbol{P}_{\boldsymbol{D
}(\boldsymbol{x})=(P_{C}(x),P_{Q}(y))$ and a similar formula holds for
$\boldsymbol{P}_{\boldsymbol{T}_{k}}.$ The potential difficulty lies in
$\boldsymbol{P}_{\boldsymbol{V}}$ of (\ref{eq:5.8}). In the finite-dimensional
case, since $\boldsymbol{V}$ is a subspace, the projection onto it is easily
computable by using an orthogonal basis. For example, if $U$ is a
$k$-dimensional subspace of $R^{n}$ with the basis $\{u_{1},u_{2},...,u_{k
\}$, then for $x\in R^{n},$ we hav
\begin{equation}
P_{U}(x)=\sum\limits_{i=1}^{k}\frac{\left\langle x,u_{i}\right\rangle }{\Vert
u_{i}\Vert^{2}}u_{i}.
\end{equation}
\section{Solving the split variational inequality problem without a product
space\label{sec:Direct SVIP}}
In this section we present a method\ for solving the SVIP, which does not need
a product space formulation as in the previous section. Recalling that
$SOL(C,f)$ and $SOL(Q,g)$ are the solution sets of (\ref{eq:vip}) and
(\ref{eq:svip}), respectively, we see that the solution set of the SVIP i
\begin{equation}
\Gamma:=\Gamma(C,Q,f,g,A):=\left\{ z\in SOL(C,f)\mid Az\in SOL(Q,g)\right\}
.
\end{equation}
Using the abbreviations $T:=P_{Q}(I-\lambda g)$ and $U:=P_{C}(I-\lambda f),$
we propose the following algorithm.
\begin{algorithm}
\label{Alg-SVIP}$\left. {}\right. $
\textbf{Initialization:} Let $\lambda>0$ and select an arbitrary starting
point $x^{0}\in H_{1}$.
\textbf{Iterative step:} Given the current iterate $x^{k},$ comput
\begin{equation}
x^{k+1}=U(x^{k}+\gamma A^{\ast}(T-I)(Ax^{k})),
\end{equation}
where $\gamma\in(0,1/L)$, $L$ is the spectral radius of the operator $A^{\ast
}A$, and $A^{\ast}$ is the adjoint of $A$.
\end{algorithm}
The following lemma, which asserts Fej\'{e}r-monotonicity, is crucial for the
convergence theorem.
\begin{lemma}
\label{lemma:Fejer} Let $H_{1}$ and $H_{2}$ be real Hilbert spaces and let
$A:H_{1}\rightarrow H_{2}$ be a bounded linear operator. Let $f:H_{1
\rightarrow H_{1}$ and $g:H_{2}\rightarrow H_{2}$ be\ $\alpha_{1}$-ISM and
$\alpha_{2}$-ISM operators on $H_{1}$ and $H_{2},$ respectively, and set
$\alpha:=\min\{\alpha_{1},\alpha_{2}\}$. Assume that $\Gamma\neq\emptyset$ and
that $\gamma\in(0,1/L)$. Consider the operators $U=P_{C}(I-\lambda f)$ and
$T=P_{Q}(I-\lambda g)$ with $\lambda\in\lbrack0,2\alpha]$. Then any sequence
$\left\{ x^{k}\right\} _{k=0}^{\infty},$ generated by Algorithm
\ref{Alg-SVIP}, is Fej\'{e}r-monotone with respect to the solution set
$\Gamma$.
\end{lemma}
\begin{proof}
Let $z\in\Gamma.$ Then $z\in SOL(C,f)$ and, therefore, by (\ref{eq:fix-vip})
and Lemma \ref{lemma:Mod-proj}(i), we ge
\begin{align}
\left\Vert x^{k+1}-z\right\Vert ^{2} & =\left\Vert U\left( x^{k}+\gamma
A^{\ast}(T-I)(Ax^{k})\right) -z\right\Vert ^{2}\nonumber\\
& =\left\Vert U\left( x^{k}+\gamma A^{\ast}(T-I)(Ax^{k})\right)
-U(z)\right\Vert ^{2}\nonumber\\
& \leq\left\Vert x^{k}+\gamma A^{\ast}(T-I)(Ax^{k})-z\right\Vert
^{2}\nonumber\\
& =\left\Vert x^{k}-z\right\Vert ^{2}+\gamma^{2}\left\Vert A^{\ast
}(T-I)(Ax^{k})\right\Vert ^{2}\nonumber\\
& +2\gamma\left\langle x^{k}-z,A^{\ast}(T-I)(Ax^{k})\right\rangle .
\end{align}
Thu
\begin{align}
\left\Vert x^{k+1}-z\right\Vert ^{2} & \leq\left\Vert x^{k}-z\right\Vert
^{2}+\gamma^{2}\left\langle (T-I)(Ax^{k}),AA^{\ast}(T-I)(Ax^{k})\right\rangle
\nonumber\\
& +2\gamma\left\langle x^{k}-z,A^{\ast}(T-I)(Ax^{k})\right\rangle .
\label{P0
\end{align}
From the definition of $L$ it follows, by standard manipulations, that
\begin{align}
\gamma^{2}\left\langle (T-I)(Ax^{k}),AA^{\ast}(T-I)(Ax^{k})\right\rangle &
\leq L\gamma^{2}\left\langle (T-I)(Ax^{k}),(T-I)(Ax^{k})\right\rangle
\nonumber\\
& =L\gamma^{2}\left\Vert (T-I)(Ax^{k})\right\Vert ^{2}. \label{P1
\end{align}
Denoting $\Theta:=2\gamma\left\langle x^{k}-z,A^{\ast}(T-I)(Ax^{k
)\right\rangle $ and using (\ref{eq:Ne(Crombez)}), we obtai
\begin{align}
\Theta & =2\gamma\left\langle A(x^{k}-z),(T-I)(Ax^{k})\right\rangle
\nonumber\\
& =2\gamma\left\langle A(x^{k}-z)+(T-I)(Ax^{k})-(T-I)(Ax^{k}),(T-I)(Ax^{k
)\right\rangle \nonumber\\
& =2\gamma\left( \left\langle T(Ax^{k})-Az,(T-I)(Ax^{k})\right\rangle
-\left\Vert (T-I)(Ax^{k})\right\Vert ^{2}\right) \nonumber\\
& \leq2\gamma\left( (1/2)\left\Vert (T-I)(Ax^{k})\right\Vert ^{2}-\left\Vert
(T-I)(Ax^{k})\right\Vert ^{2}\right) \nonumber\\
& \leq-\gamma\left\Vert (T-I)(Ax^{k})\right\Vert ^{2}. \label{P2
\end{align}
Applying (\ref{P1}) and (\ref{P2}) to (\ref{P0}), we see tha
\begin{equation}
\left\Vert x^{k+1}-z\right\Vert ^{2}\leq\left\Vert x^{k}-z\right\Vert
^{2}+\gamma(L\gamma-1)\left\Vert (T-I)(Ax^{k})\right\Vert ^{2}. \label{P3
\end{equation}
From the definition of $\gamma,$ we ge
\begin{equation}
\left\Vert x^{k+1}-z\right\Vert ^{2}\leq\left\Vert x^{k}-z\right\Vert ^{2},
\label{eq:x_k-z
\end{equation}
which completes the proof.
\end{proof}
Now we present our convergence result for Algorithm \ref{Alg-SVIP}.
\begin{theorem}
\label{Theorem1} Let $H_{1}$ and $H_{2}$ be real Hilbert spaces and let
$A:H_{1}\rightarrow H_{2}$ be a bounded linear operator. Let $f:H_{1
\rightarrow H_{1}$ and $g:H_{2}\rightarrow H_{2}$ be\ $\alpha_{1}$-ISM and
$\alpha_{2}$-ISM operators on $H_{1}$ and $H_{2},$ respectively, and set
$\alpha:=\min\{\alpha_{1},\alpha_{2}\}$. Assume that $\gamma\in(0,1/L)$.
Consider the operators $U=P_{C}(I-\lambda f)$ and $T=P_{Q}(I-\lambda g)$ with
$\lambda\in\lbrack0,2\alpha]$. Assume further that $\Gamma\neq\emptyset$ and
that, for all $x^{\ast}\in SOL(C,f),
\begin{equation}
\langle f(x),P_{C}(I-\lambda f)(x)-x^{\ast}\rangle\geq0\text{ for all\ }x\in
H.
\end{equation}
Then any sequence $\left\{ x^{k}\right\} _{k=0}^{\infty},$ generated by
Algorithm \ref{Alg-SVIP}, converges weakly to a solution point $x^{\ast
\in\Gamma$.
\end{theorem}
\begin{proof}
Let $z\in\Gamma.$ It follows from (\ref{eq:x_k-z}) that the sequence $\left\{
\left\Vert x^{k}-z\right\Vert \right\} _{k=0}^{\infty}$ is monotonically
decreasing and therefore convergent, which shows, by (\ref{P3}), that
\begin{equation}
\lim_{k\rightarrow\infty}\left\Vert (T-I)(Ax^{k})\right\Vert =0. \label{P4
\end{equation}
Fej\'{e}r-monotonicity implies that $\left\{ x^{k}\right\} _{k=0}^{\infty}$
is bounded, so it has a weakly convergent subsequence $\left\{ x^{k_{j
}\right\} _{j=0}^{\infty}$ such that $x^{k_{j}}\rightharpoonup x^{\ast}$. By
the assumptions on $\lambda$ and $g,$ we get from Lemma \ref{lemma:Mod-proj
(i) that $T$ is nonexpansive. Applying the demiclosedness of $T-I$ at $0$ to
(\ref{P4}), we obtai
\begin{equation}
T(Ax^{\ast})=Ax^{\ast}, \label{P5
\end{equation}
which means that $Ax^{\ast}\in SOL(Q,g)$. Denot
\begin{equation}
u^{k}:=x^{k}+\gamma A^{\ast}(T-I)(Ax^{k})\text{.
\end{equation}
The
\begin{equation}
u^{k_{j}}=x^{k_{j}}+\gamma A^{\ast}(T-I)(Ax^{k_{j}}).
\end{equation}
Since $x^{k_{j}}\rightharpoonup x^{\ast},$ (\ref{P4}) implies that $u^{k_{j
}\rightharpoonup x^{\ast}$ too. It remians to be shown that $x^{\ast}\in
SOL(C,f)$. Assume, by negation, that $x^{\ast}\notin SOL(C,f),$ i.e.,
$Ux^{\ast}\neq x^{\ast}.$ By the assumptions on $\lambda$ and $f,$ we get from
Lemma \ref{lemma:Mod-proj}(i) that $U$ is nonexpansive and, therefore, $U-I$
is demiclosed at $0$. So, the negation assumption must lead t
\begin{equation}
\lim_{j\rightarrow\infty}\left\Vert U(u^{k_{j}})-u^{k_{j}}\right\Vert \neq0.
\end{equation}
Therefore there exists an $\varepsilon>0$ and a subsequence $\left\{
u^{k_{j_{s}}}\right\} _{s=0}^{\infty}$ of $\left\{ u^{k_{j}}\right\}
_{j=0}^{\infty}$ such tha
\begin{equation}
\left\Vert U(u^{k_{j_{s}}})-u^{k_{j_{s}}}\right\Vert >\varepsilon\text{ for
all }s\geq0. \label{formula morethanDelta
\end{equation}
Inequality (\ref{eq:2.26}) now yields, for all $s\geq0,
\begin{align}
\left\Vert U(u^{k_{j_{s}}})-U(z)\right\Vert ^{2} & =\left\Vert
U(u^{k_{j_{s}}})-z\right\Vert ^{2}\leq\left\Vert u^{k_{j_{s}}}-z\right\Vert
^{2}-\left\Vert U(u^{k_{j_{s}}})-u^{k_{j_{s}}}\right\Vert ^{2}\nonumber\\
& <\left\Vert u^{k_{j_{s}}}-z\right\Vert ^{2}-\varepsilon^{2}.
\label{formula withsqdelta
\end{align}
By arguments similar to those in the proof of Lemma \ref{lemma:Fejer}, we hav
\begin{equation}
\left\Vert u^{k}-z\right\Vert =\left\Vert \left( x^{k}+\gamma A^{\ast
}(T-I)(Ax^{k})\right) -z\right\Vert \leq\left\Vert x^{k}-z\right\Vert .
\label{P6
\end{equation}
Since $U$ is nonexpansive
\begin{equation}
\left\Vert x^{k+1}-z\right\Vert =\left\Vert U(u^{k})-z\right\Vert
\leq\left\Vert u^{k}-z\right\Vert . \label{P7
\end{equation}
Combining (\ref{P6}) and (\ref{P7}), we ge
\begin{equation}
\left\Vert x^{k+1}-z\right\Vert \leq\left\Vert u^{k}-z\right\Vert
\leq\left\Vert x^{k}-z\right\Vert , \label{P8
\end{equation}
which means that the sequence $\{x^{1},u^{1},x^{2},u^{2},\ldots\}$ is
Fej\'{e}r-monotone with respect to $\Gamma.$ Since $x^{k_{j_{s}
+1}=U(u^{k_{j_{s}}})$, we obtai
\begin{equation}
\left\Vert u^{k_{j_{s+1}}}-z\right\Vert ^{2}\leq\left\Vert u^{k_{j_{s}
}-z\right\Vert ^{2}.
\end{equation}
Hence $\left\{ u^{k_{j_{s}}}\right\} _{s=0}^{\infty}$ is also
Fej\'{e}r-monotone with respect to $\Gamma.$ Now, (\ref{formula withsqdelta})
and (\ref{P8}) imply tha
\begin{equation}
\left\Vert u^{k_{j_{s+1}}}-z\right\Vert ^{2}<\left\Vert u^{k_{j_{s}
}-z\right\Vert ^{2}-\varepsilon^{2}\text{ for all }s\geq0,
\end{equation}
which leads to a contradiction. Therefore $x^{\ast}\in SOL(C,f)$ and finally,
$x^{\ast}\in\Gamma$. Since the subsequence$\ \left\{ x^{k_{j}}\right\}
_{j=0}^{\infty}$ was arbitrary, we get that $x^{k}\rightharpoonup x^{\ast}.$
\end{proof}
Relations of our results to some previously published works are as follows. In
\cite{CS08a} an algorithm for the Split Common Fixed Point Problem (SCFPP) in
Euclidean spaces was studied. Later Moudafi \cite{Moudafi} presented a similar
result for Hilbert spaces. In this connection, see also \cite{Masad+Reich}.
To formulate the SCFPP, let $H_{1}$ and $H_{2}$ be two real Hilbert spaces.
Given operators $U_{i}:H_{1}\rightarrow H_{1}$, $i=1,2,\ldots,p,$ and
$T_{j}:H_{2}\rightarrow H_{2},$ $j=1,2,\ldots,r,$ with nonempty fixed point
sets $C_{i},$ $i=1,2,\ldots,p$ and $Q_{j},$ $j=1,2,\ldots,r,$ respectively,
and a bounded linear operator $A:H_{1}\rightarrow H_{2}$, the SCFPP is
formulated as follows
\begin{equation}
\text{find a point }x^{\ast}\in C:=\cap_{i=1}^{p}C_{i}\text{ such that
}Ax^{\ast}\in Q:=\cap_{j=1}^{r}Q_{j}.
\end{equation}
Our result differs from\ those in \cite{CS08a} and \cite{Moudafi} in several
ways. Firstly, the spaces in which the problems are formulated. Secondly, the
operators $U$ and $T$ in \cite{CS08a} are assumed to be firmly
quasi-nonexpansive (FQNE; see the comments after Lemma \ref{lemma:Mod-proj}
above), where in our case here only $U$ is FQNE, while $T$ is just
nonexpansive. Lastly, Moudafi \cite{Moudafi} obtains weak convergence for a
wider class of operators, called demicontractive. The iterative step of his
algorithm is
\begin{equation}
x^{k+1}=(1-\alpha_{k})u^{k}+\alpha_{k}U(u^{k}),
\end{equation}
where $u^{k}:=x^{k}+\gamma A^{\ast}(T-I)(Ax^{k})$ for $\alpha_{k}\in(0,1).$ If
$\alpha_{k}=1,$ which is not allowed there, were possible, then the iterative
step of \cite{Moudafi} would coincide with that of \cite{CS08a}.
\subsection{A parallel algorithm for solving the multiple set split
variational inequality problem}
We extend the SVIP to the \textit{Multiple Set Split Variational Inequality
Problem} (MSSVIP), which is formulated as follows. Let $H_{1}$ and $H_{2}$ be
two real Hilbert spaces. Given a bounded linear operator $A:H_{1}\rightarrow
H_{2}$, functions $f_{i}:H_{1}\rightarrow H_{1},$ $i=1,2,\ldots,p,$ and
$g_{j}:H_{2}\rightarrow H_{2},$ $j=1,2,\ldots,r$, and nonempty, closed and
convex subsets $C_{i}\subseteq H_{1},$ $Q_{j}\subseteq H_{2}$ for
$i=1,2,\ldots,p$ and $j=1,2,\ldots,r$, respectively, the Multiple Set Split
Variational Inequality Problem (MSSVIP) is formulated as follows
\begin{equation}
\left\{
\begin{array}
[c]{l
\text{find a point }x^{\ast}\in C:=\cap_{i=1}^{p}C_{i}\text{ such that
}\left\langle f_{i}(x^{\ast}),x-x^{\ast}\right\rangle \geq0\text{ for all
}x\in C_{i}\\
\text{and for all }i=1,2,\ldots,p, \text{ and such that}\\
\text{the point }y^{\ast}=Ax^{\ast}\in Q:=\cap_{i=1}^{r}Q_{j}\text{ solves
}\left\langle g_{j}(y^{\ast}),y-y^{\ast}\right\rangle \geq0\text{ for all
}y\in Q_{j}\text{ }\\
\text{and for all }j=1,2,\ldots,r.
\end{array}
\right. \label{eq:mssvip
\end{equation}
For the MSSVIP we do not yet have a solution approach which does not use a
product space formalism. Therefore we present a simultaneous algorithm for the
MSSVIP the analysis of which is carried out via a certain product space. Let
$\Psi$ be the solution set of the MSSVIP
\begin{equation}
\Psi:=\left\{ z\in\cap_{i=1}^{p}SOL(C_{i},f_{i})\mid Az\in\cap_{i=1
^{r}SOL(Q_{j},g_{j})\right\} .
\end{equation}
We introduce the spaces $\boldsymbol{W}_{1}:\mathbf{=}H_{1}$ and
$\boldsymbol{W}_{2}:=H_{1}^{p}\times H_{2}^{r},$ where $r$ and $p$ are the
indices in (\ref{eq:mssvip}). Let $\left\{ \alpha_{i}\right\} _{i=1}^{p}$
and $\left\{ \beta_{j}\right\} _{j=1}^{r}$ be positive real numbers. Define
the following sets in their respective spaces
\begin{align}
\boldsymbol{C} & \mathbf{:}=H_{1}\text{ \ and \ \ \label{eq:prod}}\\
\boldsymbol{Q} & \mathbf{:}=\left( \prod_{i=1}^{p}\sqrt{\alpha_{i}
C_{i}\right) \times\left( \prod_{j=1}^{r}\sqrt{\beta_{j}}Q_{j}\right) ,
\end{align}
and the operator
\begin{equation}
\boldsymbol{A}\mathbf{:}=\left( \sqrt{\alpha_{1}}I,\ldots,\sqrt{\alpha_{p
}I,\sqrt{\beta_{1}}A^{\ast},\ldots,\sqrt{\beta_{r}}A^{\ast}\right) ^{\ast},
\end{equation}
where $A^{\ast}$ stands for adjoint of $A$. Denote $U_{i}:=P_{C_{i}}(I-\lambda
f_{i})$ and $T_{j}:=P_{Q_{j}}(I-\lambda g_{j})$ for $i=1,2,\ldots,p$ and
$j=1,2,\ldots,r$, respectively$.$ Define the operator $\boldsymbol{T
:\boldsymbol{W}_{2}\mathbf{\rightarrow}\boldsymbol{W}_{2}$ b
\begin{align}
\boldsymbol{T}\mathbf{(}\boldsymbol{y}\mathbf{)} & =\boldsymbol{T}\left(
\begin{array}
[c]{c
y_{1}\\
y_{2}\\
\vdots\\
y_{p+r
\end{array}
\right) \nonumber\\
& =\left( \left( U_{1}\left( y_{1}\right) \right) ^{\ast},\ldots,\left(
U_{p}\left( y_{p}\right) \right) ^{\ast},\left( T_{1}\left(
y_{p+1}\right) \right) ^{\ast},\ldots,\left( T_{r}(y_{p+r})\right) ^{\ast
}\right) ^{\ast}, \label{eq:fat
\end{align}
where $y_{1},y_{2},...,y_{p}\in H_{1}$ and $y_{p+1},y_{p+2},...,y_{p+r}\in
H_{2}$.
This leads to an SVIP with just two operators $\boldsymbol{F}$ and
$\boldsymbol{G}$ and two sets $\boldsymbol{C}$ and $\boldsymbol{Q},$
respectively, in the product space, when we take $\boldsymbol{C
\mathbf{=}H_{1}$, $\boldsymbol{F}\equiv\boldsymbol{0},$ $\boldsymbol{Q
\mathbf{\subseteq}\boldsymbol{W}_{2}$, $\boldsymbol{G}\mathbf{(
\boldsymbol{y}\mathbf{)}=\left( f_{1}(y_{1}),f_{2}(y_{2})\ldots,f_{p
(y_{p}),g_{1}(y_{p+1}),g_{2}(y_{p+2}),\ldots,g_{r}(y_{p+r})\right) ,$ and the
operator $\boldsymbol{A}:H_{1}\mathbf{\rightarrow}\boldsymbol{W}_{2}$. It is
easy to verify that the following equivalence holds
\begin{equation}
x\in\Psi\text{ if and only if }\boldsymbol{A}x\in\boldsymbol{Q}\mathbf{.
\end{equation}
Therefore we may apply Algorithm \ref{Alg-SVIP}
\begin{equation}
x^{k+1}=x^{k}+\gamma\boldsymbol{A}^{\ast}(\boldsymbol{T}-\boldsymbol{I
)(\boldsymbol{A}x^{k})\text{ for all }k\geq0,\label{itstepinPS
\end{equation}
to the problem (\ref{eq:prod})--(\ref{eq:fat}) in order to obtain a solution
of the original MSSVIP. We translate the iterative step (\ref{itstepinPS}) to
the original spaces $H_{1}$ and $H_{2}$ using the relatio
\begin{equation}
\boldsymbol{T}\mathbf{(}\boldsymbol{A}x)=\left( \sqrt{\alpha_{1}
U_{1}(x),\ldots,\sqrt{\alpha_{p}}U_{p}(x),\sqrt{\beta_{1}}AT_{1
(x),\ldots,\sqrt{\beta_{r}}AT_{r}(x)\right) ^{\ast
\end{equation}
and obtain the following algorithm.
\begin{algorithm}
\label{Nirits alg}$\left. {}\right. $
\textbf{Initialization:}$\ $Select an arbitrary starting point $x^{0}\in
H_{1}$.
\textbf{Iterative step: }Given the current iterate $x^{k},$ comput
\begin{equation}
x^{k+1}=x^{k}+\gamma\left( \sum_{i=1}^{p}\alpha_{i}(U_{i}-I)(x^{k
)+\sum_{j=1}^{r}\beta_{j}A^{\ast}(T_{j}-I)(Ax^{k})\right) ,
\end{equation}
where $\gamma\in(0,1/L),$ with $L=\sum_{i=1}^{p}\alpha_{i}+\sum_{j=1}^{r
\beta_{j}\Vert A\Vert^{2}$.
\end{algorithm}
The following convergence result follows from Theorem \ref{Theorem1}.
\begin{theorem}
\label{Theorem1*} Let $H_{1}$ and $H_{2}$ be two real Hilbert spaces and let
$A:H_{1}\rightarrow H_{2}$ be a bounded linear operator. Let $f_{i
:H_{1}\rightarrow H_{1},$ $i=1,2,\ldots,p,$ and $g_{j}:H_{2}\rightarrow
H_{2},$ $j=1,2,\ldots,r$, be $\alpha$-ISM operators on nonempty, closed and
convex subsets $C_{i}\subseteq H_{1},$ $Q_{j}\subseteq H_{2}$ for
$i=1,2,\ldots,p,$ and $j=1,2,\ldots,r$, respectively. Assume that\textbf{
}$\gamma\in(0,1/L)$ and $\Psi\neq\emptyset$\textbf{.} Set $U_{i}:=P_{C_{i
}(I-\lambda f_{i})$ and $T_{j}:=P_{Q_{j}}(I-\lambda g_{j})$ for $i=1,2,\ldots
,p$ and $j=1,2,\ldots,r$, respectively, with $\lambda\in\lbrack0,2\alpha]$.
If, in addition, for each $i=1,2,\ldots,p$ and $j=1,2,\ldots,r$ we have
\begin{equation}
\langle f_{i}(x),P_{C_{i}}(I-\lambda f_{i})(x)-x^{\ast}\rangle\geq0\text{ for
all\ }x\in H \label{eq:cond1
\end{equation}
for all $x^{\ast} \in SOL(C_{i},f_{i})$ and
\begin{equation}
\langle g_{j}(x),P_{Q_{j}}(I-\lambda g_{j})(x)-x^{\ast}\rangle\geq0\text{ for
all\ }x\in H, \label{eq:cond2
\end{equation}
for all $x^{\ast} \in SOL(C_{i},f_{i})$, then any sequence $\left\{
x^{k}\right\} _{k=0}^{\infty},$ generated by Algorithm \ref{Alg-SVIP},
converges weakly to a solution point $x^{\ast}\in\Psi$.
\end{theorem}
\begin{proof}
Apply Theorem \ref{Theorem1} to the two-operator SVIP in the product space
setting with $U=\boldsymbol{I}:H_{1}\rightarrow H_{1}$, $\operatorname*{Fix
U=\boldsymbol{C}\mathbf{,}$ $T=\boldsymbol{T}:\boldsymbol{W}\rightarrow
\boldsymbol{W}\mathbf{,}$ and $\operatorname*{Fix}T=\boldsymbol{Q}$.
\end{proof}
\begin{remark}
Observe that conditions (\ref{eq:cond1}) and (\ref{eq:cond2}) imposed on
$U_{i}$ and $T_{j}$ for $i=1,2,\ldots,p$ and $j=1,2,\ldots,r$, respectively,
in Theorem \ref{Theorem1*}, which are necessary for our treatment of the
problem in a product space, ensure that these operators are firmly
quasi-nonexpansive (FQNE). Therefore, the SVIP under these conditions may be
considered a \texttt{Split Common Fixed Point Problem} (SCFPP), first
introduced in \cite{CS08a}, with $\boldsymbol{C}\mathbf{,}$ $\boldsymbol{Q
\mathbf{,}$ $\boldsymbol{A}$ and $\boldsymbol{T}:\boldsymbol{W}_{2
\rightarrow\boldsymbol{W}_{2}$ as above, and the identity operator
$\boldsymbol{I}:\boldsymbol{C}\rightarrow\boldsymbol{C}$.\textbf{ }Therefore,
we could also apply \cite[Algorithm 4.1]{CS08a}. If, however, we drop these
conditions, then the operators are nonexpansive, by Lemma \ref{lemma:Mod-proj
(i), and the result of \cite{Moudafi} would apply.
\end{remark}
\section{Applications\label{sec:applications}}
The following problems are special cases of the SVIP. They are listed here
because their analysis can benefit from our algorithms for the SVIP and
because known algorithms for their solution may be generalized in the future
to cover the more general SVIP. The list includes known problems such as the
Split Feasibility Problem (SFP) and the Convex Feasibility Problem\textit{
}(CFP). In addition, we introduce two new \textquotedblleft
split\textquotedblright\ problems that have, to the best of our knowledge,
never been studied before. These are the Common Variational Inequality Point
Problem (CVIPP) and the Split Zeros Problem (SZP).
\subsection{The split feasibility and convex feasibility problems}
The Split Feasibility Problem (SFP) in Euclidean space is formulated as
follows:
\begin{equation}
\text{find a point }x^{\ast}\text{ such that }x^{\ast}\in C\subseteq
R^{n}\text{ and }Ax^{\ast}\in Q\subseteq R^{m},\label{eq:sfp
\end{equation}
where $C\subseteq R^{n},$ $Q\subseteq R^{m}$ are nonempty, closed and convex
sets, and $A:R^{n}\rightarrow R^{m}$ is given. Originally introduced in Censor
and Elfving \cite{CE}, it was later used in the area of intensity-modulated
radiation therapy (IMRT) treatment planning; see \cite{CEKB,CBMT}. Obviously,
it is formally a special case of the SVIP obtained from (\ref{eq:vip
)--(\ref{eq:svip}) by setting $f\equiv g\equiv0.$ The Convex Feasibility
Problem\textit{ }(CFP) in a Euclidean space\textit{ }is:
\begin{equation}
\text{find a point }x^{\ast}\text{ such that }x^{\ast}\in\cap_{i=1}^{m
C_{i}\neq\emptyset,
\end{equation}
where $C_{i},$ $i=1,2,\ldots,m,$ are nonempty, closed and convex sets in
$R^{n}.$ This, in its turn, becomes a special case of the SFP by taking in
(\ref{eq:sfp}) $n=m,$ $A=I$ and $C=\cap_{i=1}^{m}C_{i}.$ Many algorithms for
solving the CFP have been developed; see, e.g., \cite{BB96, CZ97}. Byrne
\cite{Byrne} established an algorithm for solving the SFP\textit{, }called the
CQ-Algorithm, with the following iterative step
\begin{equation}
x^{k+1}=P_{C}\left( x^{k}+\gamma A^{t}(P_{Q}-I)Ax^{k}\right)
,\label{eq:cq-alg
\end{equation}
which does not require calculation of the inverse of the operator $A,$ as in
\cite{CE}, but needs only its transpose $A^{t}$. A recent excellent paper on
the multiple-sets SFP which contains many references that reflect the
state-of-the-art in this area is \cite{lopezetal10}.
It is of interest to note that looking at the SFP from the point of view of
the SVIP enables us to find the minimum-norm solution of the SFP, i.e., a
solution of the for
\begin{equation}
x^{\ast}=\operatorname{argmin}\{\Vert x\Vert\mid x\text{ solves the SFP
(\ref{eq:sfp})}\}.
\end{equation}
This is done, and easily verified, by solving (\ref{eq:vip})--(\ref{eq:svip})
with $f=I$ and $g\equiv0.$
\subsection{The common variational inequality point problem}
The Common Variational Inequality Point Problem (CVIPP), newly introduced
here, is defined in Euclidean space as follows. Let $\left\{ f_{i}\right\}
_{i=1}^{m}\ $be a family of functions from $R^{n}$ into itself and let
$\left\{ C_{i}\right\} _{i=1}^{m}$ be nonempty, closed and convex subsets of
$R^{n}$ with $\cap_{i=1}^{m}C_{i}\neq\emptyset$. The CVIPP is formulatd as
follows:
\begin{align}
\text{find a point }x^{\ast} & \in\cap_{i=1}^{m}C_{i}\text{ such that
}\left\langle f_{i}(x^{\ast}),x-x^{\ast}\right\rangle \geq0\text{ }\nonumber\\
\text{for all }x & \in C_{i}\text{, }i=1,2,\ldots,m.
\end{align}
This problem can be transformed into a CVIP in an appropriate product space
(different from the one in Section \ref{sec:SVIP}). Let $R^{mn}$ be the
product space and define $\boldsymbol{F}:R^{mn}\rightarrow R^{mn}$ b
\begin{equation}
\boldsymbol{F}\left( (x_{1},x_{2},...,x_{m})^{t}\right) =\left(
(f_{1}(x_{1}),...,f_{m}(x_{m}))^{t}\right) ,
\end{equation}
where $x_{i}\in R^{n}$ for all $i=1,2,\ldots,m.$ Let the diagonal set in
$R^{mn}$ be
\begin{equation}
\boldsymbol{\Delta}:=\{\boldsymbol{x}\in R^{mn}\mid\boldsymbol{x
\mathbf{=}(a,a,...,a),\text{ }a\in R^{n}\}
\end{equation}
and define the product se
\begin{equation}
\boldsymbol{C}:=\Pi_{i=1}^{m}C_{i}.
\end{equation}
The CVIPP in $R^{n}$ is equivalent to the following CVIP in $R^{mn}$
\begin{align}
\text{find a point }\boldsymbol{x}^{\ast} & \in\boldsymbol{C}\cap
\mathbf{\Delta}\text{ such that }\left\langle \boldsymbol{F}(\boldsymbol{x
^{\ast}),\boldsymbol{x-x}^{\ast}\right\rangle \geq0\text{ }\nonumber\\
\text{for all }\boldsymbol{x} & =(x_{1},x_{2},...,x_{m})\in\boldsymbol{C}.
\end{align}
So, this problem can be solved by using Algorithm \ref{alg:SubExt4SVIP} with
$\Omega=\boldsymbol{\Delta}.$ A new algorithm specifically designed for the
CVIPP appears in \cite{cgrs10}.
\subsection{The split minimization and the split zeros problems}
From optimality conditions for convex optimization (see, e.g., Bertsekas and
Tsitsiklis \cite[Proposition 3.1, page 210]{BT}) it is well-known that if
$F:R^{n}\rightarrow R^{n}$ is a continuously differentiable convex function on
a closed and convex subset $X\subseteq R^{n},$ then $x^{\ast}\in X$ minimizes
$F$ over $X$ if and only i
\begin{equation}
\langle\nabla F(x^{\ast}),x-x^{\ast}\rangle\geq0\text{ for all }x\in X,
\label{eq:bert
\end{equation}
where $\nabla F$ is the gradient of $F$. Since (\ref{eq:bert}) is a VIP, we
make the following observation. If $F:R^{n}\rightarrow R^{n}$ and
$G:R^{m}\rightarrow R^{m}$ are continuously differentiable convex functions on
closed and convex subsets $C\subseteq R^{n}$ and $Q\subseteq R^{m},$
respectively, and if in the SVIP we take $f=\nabla F$ and $g=\nabla G,$ then
we obtain the following \textit{Split Minimization Problem }(SMP)
\begin{gather}
\text{find a point }x^{\ast}\in C\text{ such that }x^{\ast
=\operatorname{argmin}\{f(x)\mid x\in C\}\\
\text{and such that}\nonumber\\
\text{the point }y^{\ast}=Ax^{\ast}\in Q\text{ solves }y^{\ast
=\operatorname{argmin}\{g(y)\mid y\in Q\}.
\end{gather}
The \textit{Split Zeros Problem} (SZP), newly introduced here, is defined as
follows. Let $H_{1}$ and $H_{2}$ be two Hilbert spaces. Given operators
$B_{1}:H_{1}\rightarrow H_{1}$ and $B_{2}:H_{2}\rightarrow H_{2},$ and a
bounded linear operator $A:H_{1}\rightarrow H_{2}$, the SZP is formulated as
follows
\begin{equation}
\text{find a point }x^{\ast}\in H_{1}\text{ such that }B_{1}(x^{\ast})=0\text{
and }B_{2}(Ax^{\ast})=0. \label{eq:SZP
\end{equation}
This problem is a special\ case of the SVIP if $A$ is a surjective operator.
To see this, take in (\ref{eq:vip})--(\ref{eq:svip}) $C=H_{1}$, $Q=H_{2},$
$f=B_{1}$ and $g=B_{2},$ and choose $x:=x^{\ast}-B_{1}(x^{\ast})\in H_{1}$ in
(\ref{eq:vip}) and $x\in H_{1}$ such that $Ax:=Ax^{\ast}-B_{2}(Ax^{\ast})\in
H_{2}$ in (\ref{eq:svip}).
The next lemma shows when the only solution of an SVIP is a solution of an
SZP. It extends a similar result concerning the relationship between the
(un-split) zero finding problem and the VIP.
\begin{lemma}
Let $H_{1}$ and $H_{2}$ be real Hilbert spaces, and $C\subseteq H_{1}$ and
$Q\subseteq H_{2}$ nonempty, closed and convex subsets. Let $B_{1
:H_{1}\rightarrow H_{1}$ and $B_{2}:H_{2}\rightarrow H_{2}$ be $\alpha$-ISM
operators and let $A:H_{1}\rightarrow H_{2}$ be a bounded linear operator.
Assume that $C\cap\{x\in H_{1}\mid B_{1}(x)=0\}\neq\emptyset$ and that
$Q\cap\{y\in H_{2}\mid B_{2}(y)=0\}\neq\emptyset$, and denot
\begin{equation}
\Gamma:=\Gamma(C,Q,B_{1},B_{2},A):=\left\{ z\in SOL(C,B_{1})\mid Az\in
SOL(Q,B_{2})\right\} .
\end{equation}
Then, for any $x^{\ast}\in C$ with $Ax^{\ast}\in Q,$ $x^{\ast}$ solves
(\ref{eq:SZP}) if and only if $x^{\ast}\in\Gamma$.
\end{lemma}
\begin{proof}
First assume that $x^{\ast}\in C$ with $Ax^{\ast}\in Q$ and that $x^{\ast}$
solves (\ref{eq:SZP}). Then it is clear that $x^{\ast}\in\Gamma.$ In the other
direction, assume that $x^{\ast}\in C$ with $Ax^{\ast}\in Q$ and that
$x^{\ast}\in\Gamma.$ Applying (\ref{eq:ProjP2}) with $C$ as $D$ there,
$(I-\lambda B_{1})\left( x^{\ast}\right) \in H_{1},$ for any $\lambda
\in(0,2\alpha]$, as $x$ there, and $q_{1}\in C\cap\operatorname*{Fix
(I-\lambda B_{1}),$ with the same $\lambda,$ as $y$ there, we ge
\begin{align}
& \left\Vert q_{1}-P_{C}(I-\lambda B_{1})\left( x^{\ast}\right) \right\Vert
^{2}+\left\Vert (I-\lambda B_{1})\left( x^{\ast}\right) -P_{C}(I-\lambda
B_{1})\left( x^{\ast}\right) \right\Vert ^{2}\nonumber\\
& \leq\left\Vert (I-\lambda B_{1})\left( x^{\ast}\right) -q_{1}\right\Vert
^{2},
\end{align}
and, similarly, applying (\ref{eq:ProjP2}) again, we obtai
\begin{align}
& \left\Vert q_{2}-P_{Q}(I-\lambda B_{2})\left( Ax^{\ast}\right)
\right\Vert ^{2}+\left\Vert (I-\lambda B_{2})\left( Ax^{\ast}\right)
-P_{Q}(I-\lambda B_{2})\left( Ax^{\ast}\right) \right\Vert ^{2}\nonumber\\
& \leq\left\Vert (I-\lambda B_{2})\left( Ax^{\ast}\right) -q_{2}\right\Vert
^{2}.
\end{align}
Using the characterization of (\ref{eq:fix-vip}), we ge
\begin{equation}
\left\Vert q_{1}-x^{\ast}\right\Vert ^{2}+\left\Vert (I-\lambda B_{1})\left(
x^{\ast}\right) -x^{\ast}\right\Vert ^{2}\leq\left\Vert (I-\lambda
B_{1})\left( x^{\ast}\right) -q_{1}\right\Vert ^{2
\end{equation}
an
\begin{equation}
\left\Vert q_{2}-Ax^{\ast}\right\Vert ^{2}+\left\Vert (I-\lambda B_{2})\left(
Ax^{\ast}\right) -x^{\ast}\right\Vert ^{2}\leq\left\Vert (I-\lambda
B_{2})\left( Ax^{\ast}\right) -q_{2}\right\Vert ^{2}.
\end{equation}
It can be seen from the proof of Lemma \ref{lemma:Mod-proj}(i) that the
operators $I-\lambda B_{1}$ and $I-\lambda B_{2}$ are nonexpansive for every
$\lambda\in\lbrack0,2\alpha]$, so with $q_{1}\in C\cap\operatorname*{Fix
(I-\lambda B_{1})$ and $q_{2}\in Q\cap\operatorname*{Fix}(I-\lambda B_{2}),
\begin{equation}
\left\Vert (I-\lambda B_{1})\left( x^{\ast}\right) -q_{1}\right\Vert
^{2}\leq\left\Vert x^{\ast}-q_{1}\right\Vert ^{2
\end{equation}
an
\begin{equation}
\left\Vert (I-\lambda B_{2})\left( Ax^{\ast}\right) -q_{2}\right\Vert
^{2}\leq\left\Vert Ax^{\ast}-q_{2}\right\Vert ^{2}.
\end{equation}
Combining the above inequalities, we obtai
\begin{equation}
\left\Vert q_{1}-x^{\ast}\right\Vert ^{2}+\left\Vert (I-\lambda B_{1})\left(
x^{\ast}\right) -x^{\ast}\right\Vert ^{2}\leq\left\Vert x^{\ast
-q_{1}\right\Vert ^{2
\end{equation}
an
\begin{equation}
\left\Vert q_{2}-Ax^{\ast}\right\Vert ^{2}+\left\Vert (I-\lambda B_{2})\left(
Ax^{\ast}\right) -x^{\ast}\right\Vert ^{2}\leq\left\Vert Ax^{\ast
-q_{2}\right\Vert ^{2}.
\end{equation}
Hence, $\left\Vert (I-\lambda B_{1})\left( x^{\ast}\right) -x^{\ast
}\right\Vert ^{2}=0$ and $\left\Vert (I-\lambda B_{2})\left( Ax^{\ast
}\right) -Ax^{\ast}\right\Vert ^{2}=0.$ Since $\lambda>0,$ we get that
$B_{1}(x^{\ast})=0$ and $B_{2}(Ax^{\ast})=0$, as claimed$\medskip$
\end{proof}
\textbf{Acknowledgments}. This work was partially supported by Award Number
R01HL070472 from the National Heart, Lung and Blood Institute. The content is
solely the responsibility of the authors and does not necessarily represent
the official views of the National Heart, Lung and Blood Institute or the
National Institutes of Health. The third author was partially supported by the
Israel Science Foundation (Grant 647/07), by the Fund for the Promotion of
Research at the Technion and by the Technion President's Research
Fund.\bigskip
\section{Introduction}
In\ this paper we introduce a new problem, which we call the \textit{Split
Variational Inequality Problem} (SVIP). The connection of SVIP to inverse
problems and many relevant references to earlier work are presented in Section
\ref{sec:sip}. Let $H_{1}$ and $H_{2}$ be two real Hilbert spaces$.$ Given
operators $f:H_{1}\rightarrow H_{1}$ and $g:H_{2}\rightarrow H_{2},$ a bounded
linear operator $A:H_{1}\rightarrow H_{2}$, and nonempty, closed and convex
subsets $C\subseteq H_{1}$ and $Q\subseteq H_{2},$ the SVIP is formulated as
follows
\begin{gather}
\text{find a point }x^{\ast}\in C\text{ such that }\left\langle f(x^{\ast
}),x-x^{\ast}\right\rangle \geq0\text{ for all }x\in C\label{eq:vip}\\
\text{and such that}\nonumber\\
\text{the point }y^{\ast}=Ax^{\ast}\in Q\text{ and solves }\left\langle
g(y^{\ast}),y-y^{\ast}\right\rangle \geq0\text{ for all }y\in Q.
\label{eq:svip
\end{gather}
When looked at separately, (\ref{eq:vip}) is the classical \textit{Variational
Inequality Problem} (VIP) and we denote its solution set by $SOL(C,f)$. The
SVIP constitutes a pair of VIPs, which have to be solved so that the image
$y^{\ast}=Ax^{\ast},$ under a given bounded linear operator $A,$ of the
solution $x^{\ast}$ of the VIP in $H_{1}$, is a solution of another VIP in
another space $H_{2}$.
SVIP is quite general and should enable split minimization between two spaces
so that the image of a solution point of one minimization problem, under a
given bounded linear operator, is a solution point of another minimization
problem. Another special case of the SVIP is the \textit{Split Feasibility
Problem} (SFP) which had already been studied and used in practice as a model
in intensity-modulated radiation therapy (IMRT) treatment planning; see
\cite{CBMT, CEKB}.
We consider two approaches to the solution of the SVIP. The first approach is
to look at the product space $H_{1}\times H_{2}$ and transform the SVIP
(\ref{eq:vip})--(\ref{eq:svip}) into an equivalent \textit{Constrained VIP}
(CVIP) in the product space. We study this CVIP and devise an iterative
algorithm for its solution, which becomes applicable to the original SVIP via
the equivalence between the problems. Our new iterative algorithm for the
CVIP, thus for the SVIP, is inspired by an extension of the extragradient
method of Korpelevich \cite{Korpelevich}. In the second approach we present a
method that does not require the translation to a product space. This
algorithm is inspired by the work of Censor and Segal \cite{CS08a} and Moudafi
\cite{Moudafi}.
Our paper is organized as follows. The connection of SVIP to inverse problems
and many relevant references to earlier work are presented in Section
\ref{sec:sip}. In Section \ref{sec:Preliminaries} we present some
preliminaries. In Section \ref{sec:Algorithm} the algorithm for the
constrained VIP is presented. In Section \ref{sec:SVIP} we analyze the SVIP
and present its equivalence with the CVIP in the product space. In Section
\ref{sec:Direct SVIP} we first present our method for solving the SVIP, which
does not rely on any product space formulation, and then prove convergence. In
Section \ref{sec:applications} we present some applications of the SVIP. It
turns out that in addition to helping us solve the SVIP, the CVIP unifies and
improves several existing problems and methods where a VIP has to be solved
with some additional constraints. Further relations of our results to
previously published work are discussed in detail after Theorems \ref{th:cvip}
and \ref{Theorem1}.
\section{\textbf{The split variational inequality problem as a methodology for
inverse problems\label{sec:sip}}}
Following the case which has already been studied and used in practice as a
model in intensity-modulated radiation therapy (IMRT) treatment planning; see
\cite{CBMT, CEKB}, a prototypical \textit{Split Inverse Problem}\textbf{
}(SIP)\ concerns a model in which there are two spaces $X$ and $Y$ and there
is given a bounded linear operator $A:X\rightarrow Y.$ Additionally, there are
two inverse problems involved, one inverse problem denoted IP$_{1}$ formulated
in the space $X$ and another inverse problem IP$_{2}$ formulated in the space
$Y.$ The Split Inverse Problem (SIP) is the following
\begin{gather}
\text{find a point }x^{\ast}\in X\text{ that solves IP}_{1}\text{ }\\
\text{such that }\nonumber\\
\text{the point }y^{\ast}=Ax^{\ast}\in Y\text{ solves IP}_{2}\text{.
\end{gather}
Many models of inverse problems can be cast in this framework by choosing
different inverse problems for IP$_{1}$ and IP$_{2}$. The Split Convex
Feasibility Problem (SCFP) first published in \textit{Numerical Algorithms}
\cite{CE} is the first instance of a SIP\ in which the two problems IP$_{1}$
and IP$_{2}$ are CFPs each. This was used for solving an inverse problem in
radiation therapy treatment planning in \cite{CEKB}. More work on the SCFP can
be found in \cite{byrne02, CEKB, Dan-Gao, Moudafi, qx05, ssl08, xu06, Xu,
yang04, zhl09, zy05}. Two candidates for IP$_{1}$ and IP$_{2}$ that come to
mind are the mathematical models of the Convex Feasibility Problem (CFP) and
the problem of constrained optimization. In particular, the CFP formalism is
in itself at the core of the modeling of many inverse problems in various
areas of mathematics and the physical sciences; see, e.g., \cite{cap88} and
references therein for an early example. Over the past four decades, the CFP
has been used to model significant real-world inverse problems in sensor
networks, in radiation therapy treatment planning, in resolution enhancement,
in wavelet-based denoising, in antenna design, in computerized tomography, in
materials science, in watermarking, in data compression, in demosaicking, in
magnetic resonance imaging, in holography, in color imaging, in optics and
neural networks, in graph matching and in adaptive filtering, see
\cite{cccdh11} for exact references to all the above. More work on the CFP can
be found in \cite{Byrne, byrne04, cdh10}.
It is therefore natural to investigate if other inversion models for IP$_{1}$
and IP$_{2}$, besides CFP, can be embedded in the SIP methodology. For
example, CFP in the space $X$ and constrained optimization in the space $Y$?
In this paper we make a step in this direction by formulating a SIP with
Variational Inequality Problems (VIP) in each of the two spaces of the SIP.
Since, as is well-known, both CFP and constrained optimization are special
cases of VIP, our newly-proposed SVIP covers the earlier SCFP and allows for
new SIP situations. Such new situations are described in Section
\ref{sec:applications} below.
\section{Preliminaries\label{sec:Preliminaries}}
Let $H$ be a real Hilbert space with inner product $\langle\cdot,\cdot\rangle$
and norm $\Vert\cdot\Vert,$ and let $D$ be a nonempty, closed and convex
subset of $H$. We write $x^{k}\rightharpoonup x$ to indicate that the sequence
$\left\{ x^{k}\right\} _{k=0}^{\infty}$ converges weakly to $x,$ and
$x^{k}\rightarrow x$ to indicate that the sequence $\left\{ x^{k}\right\}
_{k=0}^{\infty}$ converges strongly to $x.$ For every point $x\in H,$\ there
exists a unique nearest point in $D$, denoted by $P_{D}(x)$. This point
satisfie
\begin{equation}
\left\Vert x-P_{D}\left( x\right) \right\Vert \leq\left\Vert x-y\right\Vert
\text{\textit{ }for all}\mathit{\ }y\in D.
\end{equation}
The mapping $P_{D}$ is called the metric projection of $H$ onto $D$. We know
that $P_{D}$ is a nonexpansive operator of $H$ onto $D$, i.e.
\begin{equation}
\left\Vert P_{D}\left( x\right) -P_{D}\left( y\right) \right\Vert
\leq\left\Vert x-y\right\Vert \text{\textit{ }for all}\mathit{\ }x,y\in H.
\end{equation}
The metric projection $P_{D}$ is characterized by the fact that $P_{D}\left(
x\right) \in D$ and
\begin{equation}
\left\langle x-P_{D}\left( x\right) ,P_{D}\left( x\right) -y\right\rangle
\geq0\text{ for all }x\in H,\text{ }y\in D, \label{eq:ProjP1
\end{equation}
and has the propert
\begin{equation}
\left\Vert x-y\right\Vert ^{2}\geq\left\Vert x-P_{D}\left( x\right)
\right\Vert ^{2}+\left\Vert y-P_{D}\left( x\right) \right\Vert ^{2}\text{
for all }x\in H,\text{ }y\in D. \label{eq:ProjP2
\end{equation}
It is known that in a Hilbert space $H$
\begin{equation}
\Vert\lambda x+(1-\lambda)y\Vert^{2}=\lambda\Vert x\Vert^{2}+(1-\lambda)\Vert
y\Vert^{2}-\lambda(1-\lambda)\Vert x-y\Vert^{2} \label{eq:ConvexComb
\end{equation}
for all $x,y\in H$ and $\lambda\in\lbrack0,1].$
The following lemma was proved in \cite[Lemma 3.2]{Takahashi}.
\begin{lemma}
\label{Lemma:Takahashi} Let $H$ be a Hilbert space and let $D$ be a nonempty,
closed and convex subset of $H.$ If the sequence $\left\{ x^{k}\right\}
_{k=0}^{\infty}\subset H$ is \texttt{Fej\'{e}r-monotone} with respect to $D,$
i.e., for every $u\in D,
\begin{equation}
\Vert x^{k+1}-u\Vert\leq\Vert x^{k}-u\Vert\text{ for all }k\geq0,
\end{equation}
then $\left\{ P_{D}\left( x^{k}\right) \right\} _{k=0}^{\infty}$ converges
strongly to some $z\in D.$
\end{lemma}
The next lemma is also known (see, e.g., \cite[Lemma 3.1]{Nadezhkina}).
\begin{lemma}
\label{Lemma:Schu} Let $H$ be a Hilbert space, $\left\{ \alpha_{k}\right\}
_{k=0}^{\infty}$ be a real sequence satisfying $0<a\leq\alpha_{k}\leq b<1$ for
all $k\geq0,$ and let $\left\{ v^{k}\right\} _{k=0}^{\infty}$ and $\left\{
w^{k}\right\} _{k=0}^{\infty}$ be two sequences in $H$ such that for some
$\sigma\geq0$
\begin{equation}
\limsup_{k\rightarrow\infty}\Vert v^{k}\Vert\leq\sigma,\text{ and
\limsup_{k\rightarrow\infty}\Vert w^{k}\Vert\leq\sigma.
\end{equation}
I
\begin{equation}
\lim_{k\rightarrow\infty}\Vert\alpha_{k}v^{k}+(1-\alpha_{k})w^{k}\Vert=\sigma,
\end{equation}
the
\begin{equation}
\lim_{k\rightarrow\infty}\Vert v^{k}-w^{k}\Vert=0.
\end{equation}
\end{lemma}
\begin{definition}
Let $H$ be a Hilbert space, $D$ a closed and convex subset of $H,$ and let
$M:D\rightarrow H$ be an operator. Then $M$ is said to be \texttt{demiclosed}
at $y\in H$ if for any sequence $\left\{ x^{k}\right\} _{k=0}^{\infty}$ in
$D$ such that $x^{k}\rightharpoonup\overline{x}\in D$ and $M(x^{k})\rightarrow
y,$ we have $M(\overline{x})=y.$
\end{definition}
Our next lemma is the well-known Demiclosedness Principle \cite{Browder}.
\begin{lemma}
Let $H$ be a Hilbert space, $D$ a closed and convex subset of $H,$ and
$N:D\rightarrow H$ a nonexpansive operator. Then $I-N$ ($I$ is the identity
operator on $H$) is \texttt{demiclosed} at $y\in H.$
\end{lemma}
For instance, the orthogonal projection $P$ onto a closed and convex set is a
demiclosed operator everywhere because $I-P$ is nonexpansive \cite[page
17]{Goebel+Reich}.
The next property is known as the \textit{Opial condition} \cite[Lemma
1]{Opial}. It characterizes the weak limit of a weakly convergent sequence in
Hilbert space.
\begin{condition}
(\textbf{Opial) }For any sequence $\left\{ x^{k}\right\} _{k=0}^{\infty}$ in
$H$ that converges weakly to $x$
\begin{equation}
\liminf_{k\rightarrow\infty}\Vert x^{k}-x\Vert<\liminf_{k\rightarrow\infty
}\Vert x^{k}-y\Vert\text{ for all }y\neq x.
\end{equation}
\end{condition}
\begin{definition}
Let $h:H\rightarrow H$ be an operator and let $D\subseteq H.$
(i) $h$ is called \texttt{inverse strongly monotone (ISM)} with constant
$\alpha$ on $D\subseteq H$ i
\begin{equation}
\langle h(x)-h(y),x-y\rangle\geq\alpha\Vert h(x)-h(y)\Vert^{2}\text{ for all
}x,y\in D.
\end{equation}
(ii) $h$ is called \texttt{monotone} on $D\subseteq H$ i
\begin{equation}
\langle h(x)-h(y),x-y\rangle\geq0\text{ for all }x,y\in D.
\end{equation}
\end{definition}
\begin{definition}
An operator $h:H\rightarrow H$ is called \texttt{Lipschitz continuous} on
$D\subseteq H$ with constant $\kappa>0$ i
\begin{equation}
\Vert h(x)-h(y)\Vert\leq\kappa\Vert x-y\Vert\text{\ for all\ }x,y\in D.
\end{equation}
\end{definition}
\begin{definition}
Let $S:H\rightrightarrows2^{H}\mathcal{\ }$be a point-to-set operator defined
on a real Hilbert space $H$. $S$ is called a \texttt{maximal monotone
operator} if $S$ is \texttt{monotone}, i.e.
\begin{equation}
\left\langle u-v,x-y\right\rangle \geq0,\text{ for all }u\in S(x)\text{ and
for all }v\in S(y),
\end{equation}
and the graph $G(S)$ of $S,
\begin{equation}
G(S):=\left\{ \left( x,u\right) \in H\times H\mid u\in S(x)\right\} ,
\end{equation}
is not properly contained in the graph of any other monotone operator.
\end{definition}
It is clear that a monotone operator $S$ is maximal if and only if, for each
$\left( x,u\right) \in H\times H,$ $\left\langle u-v,x-y\right\rangle \geq0$
for all $\left( v,y\right) \in G(S)$ implies that $u\in S(x).$
\begin{definition}
Let $D$ be a nonempty, closed and convex subset of $H.$ The \texttt{normal
cone} of $D$ at the point $w\in D$ is defined b
\begin{equation}
N_{D}\left( w\right) :=\{d\in H\mid\left\langle d,y-w\right\rangle
\leq0\text{ for all }y\in D\}. \label{eq:normal-c
\end{equation}
\end{definition}
Let $h$ be an $\alpha$-ISM operator on $D\subseteq H,$ define the following
point-to-set operator
\begin{equation}
S(w):=\left\{
\begin{array}
[c]{cc
h(w)+N_{D}\left( w\right) , & w\in C,\\
\emptyset, & w\notin C.
\end{array}
\right. \label{eq:maximal-S
\end{equation}
In these circumstances, it follows from \cite[Theorem 3]{Rockafellar76} that
$S$ is maximal monotone. In addition, $0\in S(w)\ $if and only if $w\in$
$SOL(D,h).$
For $T:H\rightarrow H$, denote by $\operatorname*{Fix}(T)$ the fixed point set
of $T,$ i.e.
\begin{equation}
\operatorname*{Fix}(T):=\{x\in H\mid T(x)=x\}.
\end{equation}
It is well-known tha
\begin{equation}
x^{\ast}\in SOL(D,h)\Leftrightarrow x^{\ast}=P_{D}(x^{\ast}-\lambda h(x^{\ast
})), \label{eq:fix-vip
\end{equation}
i.e., $x^{\ast}\in\operatorname*{Fix}(P_{D}(I-\lambda h)).$ It is also known
that every nonexpansive operator $T:H\rightarrow H$ satisfies, for all
$(x,y)\in H\times H,$ the inequality
\begin{equation}
\langle(x-T(x))-(y-T(y)),T(y)-T(x)\rangle\leq(1/2)\Vert(T(x)-x)-(T(y)-y)\Vert
^{2}\text{
\end{equation}
and therefore we get, for all $(x,y)\in H\times\operatorname*{Fix}(T),$
\begin{equation}
\langle x-T(x),y-T(x)\rangle\leq(1/2)\Vert T(x)-x\Vert^{2};\text{ }
\label{eq:Ne(Crombez)
\end{equation}
see, e.g., \cite[Theorem 3]{Crombez06} and \cite[Theorem 1]{Crombez}.
In the next lemma we collect several important properties that will be needed
in the sequel.
\begin{lemma}
\label{lemma:Mod-proj} Let $D\subseteq H$ be a nonempty, closed and convex
subset and let $h:H\rightarrow H$ be an $\alpha$-ISM operator on $H$. If
$\lambda\in\lbrack0,2\alpha],$ then\smallskip\
(i) the operator $P_{D}(I-\lambda h)$ is nonexpansive on $D.$
If, in addition, for all $x^{\ast}\in SOL(D,h),
\begin{equation}
\langle h(x),P_{D}(I-\lambda h)(x)-x^{\ast}\rangle\geq0\text{ for all\ }x\in
H, \label{eq:2.24
\end{equation}
then$\smallskip$ the following inequalities hold:
(ii) for all $x\in H$ and $q\in\operatorname*{Fix}(P_{D}(I-\lambda h)),
\begin{equation}
\langle P_{D}(I-\lambda h)(x)-x,P_{D}(I-\lambda h)(x)-q\rangle\leq0;
\label{QFNE
\end{equation}
(iii) for all $x\in H$ and $q\in\operatorname*{Fix}(P_{D}(I-\lambda h)),
\begin{equation}
\left\Vert P_{D}(I-\lambda h)(x)-q\right\Vert ^{2}\leq\left\Vert
x-q\right\Vert ^{2}-\left\Vert P_{D}(I-\lambda h)(x)-x\right\Vert ^{2}.
\label{eq:2.26
\end{equation}
\end{lemma}
\begin{proof}
(i) Let $x,y\in H.$ The
\begin{align}
\Vert P_{D}(I-\lambda h)(x)-P_{D}(I-\lambda h)(y)\Vert^{2} & =\Vert
P_{D}(x-\lambda h(x))-P_{D}(y-\lambda h(y))\Vert^{2}\nonumber\\
& \leq\Vert x-\lambda h(x)-(y-\lambda h(y))\Vert^{2}\nonumber\\
& =\Vert(x-y)-\lambda(h(x)-h(y))\Vert^{2}\nonumber\\
& =\Vert x-y\Vert^{2}-2\lambda\langle x-y,h(x)-h(y)\rangle\nonumber\\
& +\lambda^{2}\Vert h(x)-h(y)\Vert^{2}\nonumber\\
& \leq\Vert x-y\Vert^{2}-2\lambda\alpha\Vert h(x)-h(y)\Vert^{2}\nonumber\\
& +\lambda^{2}\Vert h(x)-h(y)\Vert^{2}\nonumber\\
& =\Vert x-y\Vert^{2}+\lambda(\lambda-2\alpha)\Vert h(x)-h(y)\Vert
^{2}\nonumber\\
& \leq\Vert x-y\Vert^{2}.
\end{align}
(ii) Let $x\in H$ and $q\in\operatorname*{Fix}(P_{D}(I-\lambda h)).$ The
\begin{align}
& \langle P_{D}(x-\lambda h(x))-x,P_{D}(x-\lambda h(x))-q\rangle\nonumber\\
& =\langle P_{D}(x-\lambda h(x))-x+\lambda h(x)-\lambda h(x),P_{D}(x-\lambda
h(x))-q\rangle\nonumber\\
& =\langle P_{D}(x-\lambda h(x))-(x-\lambda h(x)),P_{D}(x-\lambda
h(x))-q\rangle\nonumber\\
& -\lambda\langle h(x),P_{D}(x-\lambda h(x))-q\rangle.
\end{align}
By (\ref{eq:ProjP1}), (\ref{eq:fix-vip}) and (\ref{eq:2.24}), we ge
\begin{equation}
\langle P_{D}(x-\lambda h(x))-x,P_{D}(x-\lambda h(x))-q\rangle\leq0.
\end{equation}
(iii) Let $x\in H$ and $q\in\operatorname*{Fix}(P_{D}(I-\lambda h)).$ The
\begin{align}
\left\Vert q-x\right\Vert ^{2} & =\left\Vert (P_{D}(I-\lambda
h)(x)-x)-(P_{D}(I-\lambda h)(x)-q)\right\Vert ^{2}\nonumber\\
& =\left\Vert P_{D}(I-\lambda h)(x)-x\right\Vert ^{2}+\left\Vert
P_{D}(I-\lambda h)(x)-q\right\Vert ^{2}\nonumber\\
& -2\langle P_{D}(I-\lambda h)(x)-x,P_{D}(I-\lambda h)(x)-q\rangle.
\end{align}
By (ii), we ge
\begin{equation}
-2\langle P_{D}(I-\lambda h)(x)-x,P_{D}(I-\lambda h)(x)-q\rangle\geq0.
\end{equation}
Thus
\begin{equation}
\left\Vert q-x\right\Vert ^{2}\geq\left\Vert P_{D}(I-\lambda
h)(x)-x\right\Vert ^{2}+\left\Vert P_{D}(I-\lambda h)(x)-q\right\Vert ^{2
\end{equation}
o
\begin{equation}
\left\Vert P_{D}(I-\lambda h)(x)-q\right\Vert ^{2}\leq\left\Vert
q-x\right\Vert ^{2}-\left\Vert P_{D}(I-\lambda h)(x)-x\right\Vert ^{2},
\end{equation}
as asserted.
\end{proof}
Observe that, under the additional condition (\ref{eq:2.24}), Equation
(\ref{QFNE}) means that the operator $P_{D}(I-\lambda h)$ belongs to the class
of operators called the $\mathcal{T}$-class. This class $\mathcal{T}$ of
operators was introduced and investigated by Bauschke and Combettes in
\cite[Definition 2.2]{BC01} and by Combettes in \cite{Co}. Operators in this
class were named \textit{directed operators }by Zaknoon \cite{Z} and further
studied under this name by Segal \cite{Seg08} and by Censor and Segal
\cite{CS08, CS08a, CS09}. Cegielski \cite[Def. 2.1]{Ceg08} studied these
operators under the name \textit{separating operators}. Since both
\textit{directed }and\textit{\ separating }are key words of other,
widely-used, mathematical entities, Cegielski and Censor have recently
introduced the term \textit{cutter operators} \cite{cc11}. This class
coincides with the class $\mathcal{F}^{\nu}$ for $\nu=1$ \cite{Crombez} and
with the class DC$_{\boldsymbol{p}}$ for $\boldsymbol{p}=-1$ \cite{mp08}. The
term \textit{firmly quasi-nonexpansive} (FQNE) for $\mathcal{T}$-class
operators was used by Yamada and Ogura \cite{Yamada} because every
\textit{firmly nonexpansive} (FNE) mapping \cite[page 42]{Goebel+Reich} is
obviously FQNE.
\section{An algorithm for solving the constrained variational inequality
problem\label{sec:Algorithm}}
Let\textit{ }$f:H\rightarrow H$, and let $C$ and $\Omega$ be nonempty, closed
and convex subsets of $H$. The \textit{Constrained} \textit{Variational
Inequality Problem} (CVIP) is:
\begin{equation}
\text{find }x^{\ast}\in C\cap\Omega\text{ such that }\left\langle f(x^{\ast
}),x-x^{\ast}\right\rangle \geq0\text{ for all }x\in C. \label{eq:cvip
\end{equation}
The iterative algorithm for this CVIP, presented next, is inspired by our
earlier work \cite{CGR,CGR2} in which we modified the extragradient method of
Korpelevich \cite{Korpelevich}. The following conditions are needed for the
convergence theorem.
\begin{condition}
\label{Condition:a} $f$ is monotone on $C$.
\end{condition}
\begin{condition}
\label{Condition:b} $f$ is Lipschitz continuous on $H$ with constant
$\kappa>0.$
\end{condition}
\begin{condition}
\label{Condition:c} $\Omega\cap SOL(C,f)\neq\emptyset.$
\end{condition}
Let $\left\{ \lambda_{k}\right\} _{k=0}^{\infty}\subset\left[ a,b\right]
$\ for some $a,b\in(0,1/\kappa)$, and let \textit{ }$\left\{ \alpha
_{k}\right\} _{k=0}^{\infty}\subset\left[ c,d\right] $ for some\textit{
}$c,d\in(0,1)$. Then the following algorithm generates two sequences that
converge to a point $z\in\Omega$ $\cap$ SOL$(C,f),$ as the convergence theorem
that follows shows.
\begin{algorithm}
\label{alg:SubExt4SVIP}$\left. {}\right. $
\textbf{Initialization:} Select an arbitrary starting point $x^{0}\in H$.
\textbf{Iterative step:} Given the current iterate $x^{k},$ comput
\begin{equation}
y^{k}=P_{C}(x^{k}-\lambda_{k}f(x^{k})),
\end{equation}
construct the half-space $T_{k}$ the bounding hyperplane of which supports $C$
at $y^{k},
\begin{equation}
T_{k}:=\{w\in H\mid\left\langle \left( x^{k}-\lambda_{k}f(x^{k})\right)
-y^{k},w-y^{k}\right\rangle \leq0\},
\end{equation}
and then calculate the next iterate b
\begin{equation}
x^{k+1}=\alpha_{k}x^{k}+(1-\alpha_{k})P_{\Omega}\left( P_{T_{k}
(x^{k}-\lambda_{k}f(y^{k}))\right) . \label{eq:3.4
\end{equation}
\end{algorithm}
\begin{theorem}
\label{th:cvip}Let\textit{ }$f:H\rightarrow H$, and let $C$ and $\Omega$ be
nonempty, closed and convex subsets of $H$. Assume that Conditions
\ref{Condition:a}--\ref{Condition:c} hold, and let $\left\{ x^{k}\right\}
_{k=0}^{\infty}$ and $\left\{ y^{k}\right\} _{k=0}^{\infty}$ be any two
sequences generated by Algorithm \ref{alg:SubExt4SVIP} with $\left\{
\lambda_{k}\right\} _{k=0}^{\infty}\subset\left[ a,b\right] $\textit{\ for
some }$a,b\in(0,1/\kappa)$\textit{ and }$\left\{ \alpha_{k}\right\}
_{k=0}^{\infty}\subset\left[ c,d\right] $ for some\textit{ }$c,d\in(0,1)$.
Then $\left\{ x^{k}\right\} _{k=0}^{\infty}$ and $\left\{ y^{k}\right\}
_{k=0}^{\infty}$ converge weakly to the same point $z\in\Omega\cap SOL(C,f)$
an
\begin{equation}
z=\lim_{k\rightarrow\infty}P_{\Omega\cap SOL(C,f)}(x^{k}).
\end{equation}
\end{theorem}
\begin{proof}
For the special case of fixed $\lambda_{k}=\tau$ for all $k\geq0$ this theorem
is a direct consequence of our \cite[Theorem 7.1]{CGR2} with the choice of the
nonexpansive operator $S$ there to be $P_{\Omega}$. However, a careful
inspection of the proof of \cite[Theorem 7.1]{CGR2} reveals that it also
applies to a variable sequence $\left\{ \lambda_{k}\right\} _{k=0}^{\infty}$
as used here.
\end{proof}
To relate our results to some previously published works we mention two lines
of research related to our notion of the CVIP. Takahashi and Nadezhkina
\cite{Nadezhkina} proposed an algorithm for finding a point $x^{\ast
\in\operatorname*{Fix}(N)\cap$SOL$(C,f),$ where $N:C\rightarrow C$ is a
nonexpansive operator. The iterative step of their algorithm is as
follows.\textbf{ }Given the current iterate $x^{k},$ comput
\begin{equation}
y^{k}=P_{C}(x^{k}-\lambda_{k}f(x^{k}))
\end{equation}
and the
\begin{equation}
x^{k+1}=\alpha_{k}x^{k}+(1-\alpha_{k})N\left( P_{C}(x^{k}-\lambda_{k
f(y^{k}))\right) .
\end{equation}
The restriction $P_{\Omega}|_{C}$ of our $P_{\Omega}$ in (\ref{eq:3.4}) is, of
course, nonexpansive, and so it is a special case of $N$ in \cite{Nadezhkina}.
But a significant advantage of our Algorithm \ref{alg:SubExt4SVIP} lies in the
fact that we compute $P_{T_{k}}$ onto a half-space in (\ref{eq:3.4}) whereas
the authors of \cite{Nadezhkina} need to project onto the convex set $C.$
Various ways have been proposed in the literature to cope with the inherent
difficulty of calculating projections (onto closed convex sets) that do not
have a closed-form expression; see, e.g., He, Yang and Duan \cite{hyd10}, or
\cite{cgr-oms}.
Bertsekas and Tsitsiklis \cite[Page 288]{BT} consider the following problem in
Euclidean space: given $f:R^{n}\rightarrow R^{n}$, polyhedral sets
$C_{1}\subset R^{n}$ and $C_{2}\subset R^{m},$ and an $m\times n$ matrix $A$,
find a point $x^{\ast}\in C_{1}$ such that $Ax^{\ast}\in C_{2}$ an
\begin{equation}
\left\langle f(x^{\ast}),x-x^{\ast}\right\rangle \geq0\text{ for all }x\in
C_{1}\cap\{y\mid Ay\in C_{2}\}.
\end{equation}
Denoting $\Omega=A^{-1}(C_{2})$, we see that this problem becomes similar to,
but not identical with a CVIP. While the authors of \cite{BT} seek a solution
in SOL$(C_{1}\cap\Omega,f),$ we aim in our CVIP at $\Omega\cap$SOL$(C,f).$
They propose to solve their problem by the method of multipliers, which is a
different approach than ours, and they need to assume that either $C_{1}$ is
bounded or $A^{t}A$ is invertible, where $A^{t}$ is the transpose of $A.$
\section{The split variational inequality problem as a constrained variational
inequality problem in a product space\label{sec:SVIP}}
Our first approach to the solution of the SVIP (\ref{eq:vip})--(\ref{eq:svip})
is to look at the product space $\boldsymbol{H}=H_{1}\times H_{2}$ and
introduce in it the product set $\boldsymbol{D}:=C\times Q$ and the se
\begin{equation}
\boldsymbol{V}:=\{\mathbf{x}=(x,y)\in\boldsymbol{H}\mid Ax=y\}.
\end{equation}
We adopt the notational convention that objects in the product space are
represented in boldface type. We transform the SVIP (\ref{eq:vip
)--(\ref{eq:svip}) into the following equivalent CVIP in the product space:
\begin{align}
\text{Find a point }\boldsymbol{x}^{\ast} & \in\boldsymbol{D}\cap
\boldsymbol{V},\text{ such that }\left\langle \boldsymbol{h}(\boldsymbol{x
^{\ast}),\boldsymbol{x}-\boldsymbol{x}^{\ast}\right\rangle \geq0\text{
}\nonumber\\
\text{for all }\boldsymbol{x} & =(x,y)\in\boldsymbol{D},
\label{eq:c-as-svip
\end{align}
where $\boldsymbol{h}:\boldsymbol{H}\rightarrow\boldsymbol{H}$ is defined b
\begin{equation}
\boldsymbol{h}(x,y)=(f(x),g(y)).
\end{equation}
A simple adaptation of the decomposition lemma \cite[Proposition 5.7, page
275]{BT} shows that problems (\ref{eq:vip})--(\ref{eq:svip}) and
(\ref{eq:c-as-svip}) are equivalent, and, therefore, we can apply Algorithm
\ref{alg:SubExt4SVIP} to the solution of (\ref{eq:c-as-svip}).
\begin{lemma}
A point $\boldsymbol{x}^{\ast}=(x^{\ast},y^{\ast})$ solves (\ref{eq:c-as-svip
) if and only if $x^{\ast}$ and $y^{\ast}$ solve (\ref{eq:vip
)--(\ref{eq:svip}).
\end{lemma}
\begin{proof}
If $(x^{\ast},y^{\ast})$ solves (\ref{eq:vip})--(\ref{eq:svip}), then it is
clear that $(x^{\ast},y^{\ast})$ solves (\ref{eq:c-as-svip}). To prove the
other direction, suppose that $(x^{\ast},y^{\ast})$ solves (\ref{eq:c-as-svip
). Since (\ref{eq:c-as-svip}) holds for all $(x,y)\in\boldsymbol{D}$, we may
take $(x^{\ast},y)\in\boldsymbol{D}$ and deduce tha
\begin{equation}
\left\langle g(Ax^{\ast}),y-Ax^{\ast}\right\rangle \geq0\text{ for all }y\in
Q.
\end{equation}
Using a similar argument with $(x,y^{\ast})\in\boldsymbol{D,}$ we ge
\begin{equation}
\left\langle f(x^{\ast}),x-x^{\ast}\right\rangle \geq0\text{ for all }x\in C,
\end{equation}
which means that $(x^{\ast},y^{\ast})$ solves (\ref{eq:vip})--(\ref{eq:svip}).
\end{proof}
Using this equivalence, we can now employ Algorithm \ref{alg:SubExt4SVIP} in
order to solve the SVIP. The following conditions are needed for the
convergence theorem.
\begin{condition}
\label{Condition:a2} $f$ is monotone on $C$ and $g$ is monotone on $Q$.
\end{condition}
\begin{condition}
\label{Condition:b2} $f$ is Lipschitz continuous on $H_{1}$ with constant
$\kappa_{1}>0$ and $g$ is Lipschitz continuous on $H_{2}$ with constant
$\kappa_{2}>0.$
\end{condition}
\begin{condition}
\label{Condition:c2} $\boldsymbol{V}\cap SOL(\boldsymbol{D},\boldsymbol{h
)\neq\emptyset.$
\end{condition}
Let $\left\{ \lambda_{k}\right\} _{k=0}^{\infty}\subset\left[ a,b\right]
$\ for some $a,b\in(0,1/\kappa)$, where $\kappa=\min\{\kappa_{1},\kappa_{2
\}$, and let $\left\{ \alpha_{k}\right\} _{k=0}^{\infty}\subset\left[
c,d\right] $ for some\textit{ }$c,d\in(0,1)$. Then the following algorithm
generates two sequences that converge to a point $\boldsymbol{z
\in\boldsymbol{V}\cap SOL(\boldsymbol{D},\boldsymbol{h}),$ as the convergence
theorem given below shows.
\begin{algorithm}
\label{alg:SubExt4SVIP2}$\left. {}\right. $
\textbf{Initialization:} Select an arbitrary starting point $\boldsymbol{x
^{0}\in\boldsymbol{H}$.
\textbf{Iterative step:} Given the current iterate $\boldsymbol{x}^{k},$
comput
\begin{equation}
\boldsymbol{y}^{k}=\boldsymbol{P}_{\boldsymbol{D}}(\boldsymbol{x}^{k
-\lambda_{k}\boldsymbol{h}(\boldsymbol{x}^{k})),
\end{equation}
construct the half-space $\boldsymbol{T}_{k}$ the bounding hyperplane of which
supports $\boldsymbol{D}$ at $\boldsymbol{y}^{k},
\begin{equation}
\boldsymbol{T}_{k}:=\{\boldsymbol{w}\in\boldsymbol{H}\mid\left\langle \left(
\boldsymbol{x}^{k}-\lambda_{k}\boldsymbol{h}(\boldsymbol{x}^{k})\right)
-\boldsymbol{y}^{k},\boldsymbol{w}-\boldsymbol{y}^{k}\right\rangle \leq0\},
\end{equation}
and then calculat
\begin{equation}
\boldsymbol{x}^{k+1}=\alpha_{k}\boldsymbol{x}^{k}+(1-\alpha_{k})\boldsymbol{P
_{\boldsymbol{V}}\left( \boldsymbol{P}_{\boldsymbol{T}_{k}}(\boldsymbol{x
^{k}-\lambda_{k}\boldsymbol{h}(\boldsymbol{y}^{k}))\right) . \label{eq:5.8
\end{equation}
\end{algorithm}
Our convergence theorem for Algorithm \ref{alg:SubExt4SVIP2} follows from
Theorem \ref{th:cvip}.
\begin{theorem}
Consider $f:H_{1}\rightarrow H_{1}$ and $g:H_{2}\rightarrow H_{2},$ a bounded
linear operator $A:H_{1}\rightarrow H_{2}$, and nonempty, closed and convex
subsets $C\subseteq H_{1}$ and $Q\subseteq H_{2}$. Assume that Conditions
\ref{Condition:a2}--\ref{Condition:c2} hold, and let $\left\{ \boldsymbol{x
^{k}\right\} _{k=0}^{\infty}$ and $\left\{ \boldsymbol{y}^{k}\right\}
_{k=0}^{\infty}$ be any two sequences generated by Algorithm
\ref{alg:SubExt4SVIP2} with $\left\{ \lambda_{k}\right\} _{k=0}^{\infty
}\subset\left[ a,b\right] $\textit{\ for some }$a,b\in(0,1/\kappa)$\textit{,
where }$\kappa=\min\{\kappa_{1},\kappa_{2}\}$, \textit{and let} $\left\{
\alpha_{k}\right\} _{k=0}^{\infty}\subset\left[ c,d\right] $ for
some\textit{ }$c,d\in(0,1)$. Then $\left\{ \boldsymbol{x}^{k}\right\}
_{k=0}^{\infty}$ and $\left\{ \boldsymbol{y}^{k}\right\} _{k=0}^{\infty}$
converge weakly to the same point $\boldsymbol{z}\in\boldsymbol{V}\cap
SOL(\boldsymbol{D},\boldsymbol{h})$ an
\begin{equation}
\boldsymbol{z}=\lim_{k\rightarrow\infty}\boldsymbol{P}_{\boldsymbol{V}\cap
SOL(\boldsymbol{D},\boldsymbol{h})}(\boldsymbol{x}^{k}).
\end{equation}
\end{theorem}
The value of the product space approach, described above, depends on the
ability to \textquotedblleft translate\textquotedblright\ Algorithm
\ref{alg:SubExt4SVIP2} back to the original spaces $H_{1}$ and $H_{2}.$
Observe that due to \cite[Lemma 1.1]{Pierra} for $\boldsymbol{x
\mathbf{=}(x,y)\in\boldsymbol{D,}$ we have $\boldsymbol{P}_{\boldsymbol{D
}(\boldsymbol{x})=(P_{C}(x),P_{Q}(y))$ and a similar formula holds for
$\boldsymbol{P}_{\boldsymbol{T}_{k}}.$ The potential difficulty lies in
$\boldsymbol{P}_{\boldsymbol{V}}$ of (\ref{eq:5.8}). In the finite-dimensional
case, since $\boldsymbol{V}$ is a subspace, the projection onto it is easily
computable by using an orthogonal basis. For example, if $U$ is a
$k$-dimensional subspace of $R^{n}$ with the basis $\{u_{1},u_{2},...,u_{k
\}$, then for $x\in R^{n},$ we hav
\begin{equation}
P_{U}(x)=\sum\limits_{i=1}^{k}\frac{\left\langle x,u_{i}\right\rangle }{\Vert
u_{i}\Vert^{2}}u_{i}.
\end{equation}
\section{Solving the split variational inequality problem without a product
space\label{sec:Direct SVIP}}
In this section we present a method\ for solving the SVIP, which does not need
a product space formulation as in the previous section. Recalling that
$SOL(C,f)$ and $SOL(Q,g)$ are the solution sets of (\ref{eq:vip}) and
(\ref{eq:svip}), respectively, we see that the solution set of the SVIP i
\begin{equation}
\Gamma:=\Gamma(C,Q,f,g,A):=\left\{ z\in SOL(C,f)\mid Az\in SOL(Q,g)\right\}
.
\end{equation}
Using the abbreviations $T:=P_{Q}(I-\lambda g)$ and $U:=P_{C}(I-\lambda f),$
we propose the following algorithm.
\begin{algorithm}
\label{Alg-SVIP}$\left. {}\right. $
\textbf{Initialization:} Let $\lambda>0$ and select an arbitrary starting
point $x^{0}\in H_{1}$.
\textbf{Iterative step:} Given the current iterate $x^{k},$ comput
\begin{equation}
x^{k+1}=U(x^{k}+\gamma A^{\ast}(T-I)(Ax^{k})),
\end{equation}
where $\gamma\in(0,1/L)$, $L$ is the spectral radius of the operator $A^{\ast
}A$, and $A^{\ast}$ is the adjoint of $A$.
\end{algorithm}
The following lemma, which asserts Fej\'{e}r-monotonicity, is crucial for the
convergence theorem.
\begin{lemma}
\label{lemma:Fejer} Let $H_{1}$ and $H_{2}$ be real Hilbert spaces and let
$A:H_{1}\rightarrow H_{2}$ be a bounded linear operator. Let $f:H_{1
\rightarrow H_{1}$ and $g:H_{2}\rightarrow H_{2}$ be\ $\alpha_{1}$-ISM and
$\alpha_{2}$-ISM operators on $H_{1}$ and $H_{2},$ respectively, and set
$\alpha:=\min\{\alpha_{1},\alpha_{2}\}$. Assume that $\Gamma\neq\emptyset$ and
that $\gamma\in(0,1/L)$. Consider the operators $U=P_{C}(I-\lambda f)$ and
$T=P_{Q}(I-\lambda g)$ with $\lambda\in\lbrack0,2\alpha]$. Then any sequence
$\left\{ x^{k}\right\} _{k=0}^{\infty},$ generated by Algorithm
\ref{Alg-SVIP}, is Fej\'{e}r-monotone with respect to the solution set
$\Gamma$.
\end{lemma}
\begin{proof}
Let $z\in\Gamma.$ Then $z\in SOL(C,f)$ and, therefore, by (\ref{eq:fix-vip})
and Lemma \ref{lemma:Mod-proj}(i), we ge
\begin{align}
\left\Vert x^{k+1}-z\right\Vert ^{2} & =\left\Vert U\left( x^{k}+\gamma
A^{\ast}(T-I)(Ax^{k})\right) -z\right\Vert ^{2}\nonumber\\
& =\left\Vert U\left( x^{k}+\gamma A^{\ast}(T-I)(Ax^{k})\right)
-U(z)\right\Vert ^{2}\nonumber\\
& \leq\left\Vert x^{k}+\gamma A^{\ast}(T-I)(Ax^{k})-z\right\Vert
^{2}\nonumber\\
& =\left\Vert x^{k}-z\right\Vert ^{2}+\gamma^{2}\left\Vert A^{\ast
}(T-I)(Ax^{k})\right\Vert ^{2}\nonumber\\
& +2\gamma\left\langle x^{k}-z,A^{\ast}(T-I)(Ax^{k})\right\rangle .
\end{align}
Thu
\begin{align}
\left\Vert x^{k+1}-z\right\Vert ^{2} & \leq\left\Vert x^{k}-z\right\Vert
^{2}+\gamma^{2}\left\langle (T-I)(Ax^{k}),AA^{\ast}(T-I)(Ax^{k})\right\rangle
\nonumber\\
& +2\gamma\left\langle x^{k}-z,A^{\ast}(T-I)(Ax^{k})\right\rangle .
\label{P0
\end{align}
From the definition of $L$ it follows, by standard manipulations, that
\begin{align}
\gamma^{2}\left\langle (T-I)(Ax^{k}),AA^{\ast}(T-I)(Ax^{k})\right\rangle &
\leq L\gamma^{2}\left\langle (T-I)(Ax^{k}),(T-I)(Ax^{k})\right\rangle
\nonumber\\
& =L\gamma^{2}\left\Vert (T-I)(Ax^{k})\right\Vert ^{2}. \label{P1
\end{align}
Denoting $\Theta:=2\gamma\left\langle x^{k}-z,A^{\ast}(T-I)(Ax^{k
)\right\rangle $ and using (\ref{eq:Ne(Crombez)}), we obtai
\begin{align}
\Theta & =2\gamma\left\langle A(x^{k}-z),(T-I)(Ax^{k})\right\rangle
\nonumber\\
& =2\gamma\left\langle A(x^{k}-z)+(T-I)(Ax^{k})-(T-I)(Ax^{k}),(T-I)(Ax^{k
)\right\rangle \nonumber\\
& =2\gamma\left( \left\langle T(Ax^{k})-Az,(T-I)(Ax^{k})\right\rangle
-\left\Vert (T-I)(Ax^{k})\right\Vert ^{2}\right) \nonumber\\
& \leq2\gamma\left( (1/2)\left\Vert (T-I)(Ax^{k})\right\Vert ^{2}-\left\Vert
(T-I)(Ax^{k})\right\Vert ^{2}\right) \nonumber\\
& \leq-\gamma\left\Vert (T-I)(Ax^{k})\right\Vert ^{2}. \label{P2
\end{align}
Applying (\ref{P1}) and (\ref{P2}) to (\ref{P0}), we see tha
\begin{equation}
\left\Vert x^{k+1}-z\right\Vert ^{2}\leq\left\Vert x^{k}-z\right\Vert
^{2}+\gamma(L\gamma-1)\left\Vert (T-I)(Ax^{k})\right\Vert ^{2}. \label{P3
\end{equation}
From the definition of $\gamma,$ we ge
\begin{equation}
\left\Vert x^{k+1}-z\right\Vert ^{2}\leq\left\Vert x^{k}-z\right\Vert ^{2},
\label{eq:x_k-z
\end{equation}
which completes the proof.
\end{proof}
Now we present our convergence result for Algorithm \ref{Alg-SVIP}.
\begin{theorem}
\label{Theorem1} Let $H_{1}$ and $H_{2}$ be real Hilbert spaces and let
$A:H_{1}\rightarrow H_{2}$ be a bounded linear operator. Let $f:H_{1
\rightarrow H_{1}$ and $g:H_{2}\rightarrow H_{2}$ be\ $\alpha_{1}$-ISM and
$\alpha_{2}$-ISM operators on $H_{1}$ and $H_{2},$ respectively, and set
$\alpha:=\min\{\alpha_{1},\alpha_{2}\}$. Assume that $\gamma\in(0,1/L)$.
Consider the operators $U=P_{C}(I-\lambda f)$ and $T=P_{Q}(I-\lambda g)$ with
$\lambda\in\lbrack0,2\alpha]$. Assume further that $\Gamma\neq\emptyset$ and
that, for all $x^{\ast}\in SOL(C,f),
\begin{equation}
\langle f(x),P_{C}(I-\lambda f)(x)-x^{\ast}\rangle\geq0\text{ for all\ }x\in
H_{1}. \label{eq:5.9
\end{equation}
Then any sequence $\left\{ x^{k}\right\} _{k=0}^{\infty},$ generated by
Algorithm \ref{Alg-SVIP}, converges weakly to a solution point $x^{\ast
\in\Gamma$.
\end{theorem}
\begin{proof}
Let $z\in\Gamma.$ It follows from (\ref{eq:x_k-z}) that the sequence $\left\{
\left\Vert x^{k}-z\right\Vert \right\} _{k=0}^{\infty}$ is monotonically
decreasing and therefore convergent, which shows, by (\ref{P3}), that
\begin{equation}
\lim_{k\rightarrow\infty}\left\Vert (T-I)(Ax^{k})\right\Vert =0. \label{P4
\end{equation}
Fej\'{e}r-monotonicity implies that $\left\{ x^{k}\right\} _{k=0}^{\infty}$
is bounded, so it has a weakly convergent subsequence $\left\{ x^{k_{j
}\right\} _{j=0}^{\infty}$ such that $x^{k_{j}}\rightharpoonup x^{\ast}$. By
the assumptions on $\lambda$ and $g,$ we get from Lemma \ref{lemma:Mod-proj
(i) that $T$ is nonexpansive. Applying the demiclosedness of $T-I$ at $0$ to
(\ref{P4}), we obtai
\begin{equation}
T(Ax^{\ast})=Ax^{\ast}, \label{P5
\end{equation}
which means that $Ax^{\ast}\in SOL(Q,g)$. Denot
\begin{equation}
u^{k}:=x^{k}+\gamma A^{\ast}(T-I)(Ax^{k})\text{.
\end{equation}
The
\begin{equation}
u^{k_{j}}=x^{k_{j}}+\gamma A^{\ast}(T-I)(Ax^{k_{j}}).
\end{equation}
Since $x^{k_{j}}\rightharpoonup x^{\ast},$ (\ref{P4}) implies that $u^{k_{j
}\rightharpoonup x^{\ast}$ too. It remains to be shown that $x^{\ast}\in
SOL(C,f)$. Assume, by negation, that $x^{\ast}\notin SOL(C,f),$ i.e.,
$Ux^{\ast}\neq x^{\ast}.$ By the assumptions on $\lambda$ and $f,$ we get from
Lemma \ref{lemma:Mod-proj}(i) that $U$ is nonexpansive and, therefore, $U-I$
is demiclosed at $0$. So, the negation assumption must lead t
\begin{equation}
\lim_{j\rightarrow\infty}\left\Vert U(u^{k_{j}})-u^{k_{j}}\right\Vert \neq0.
\end{equation}
Therefore, there exists an $\varepsilon>0$ and a subsequence $\left\{
u^{k_{j_{s}}}\right\} _{s=0}^{\infty}$ of $\left\{ u^{k_{j}}\right\}
_{j=0}^{\infty}$ such tha
\begin{equation}
\left\Vert U(u^{k_{j_{s}}})-u^{k_{j_{s}}}\right\Vert >\varepsilon\text{ for
all }s\geq0. \label{formula morethanDelta
\end{equation}
Condition (\ref{eq:5.9}) justifies the use of Lemma \ref{lemma:Mod-proj} by
supplying (\ref{eq:2.24}). Therefore, inequality (\ref{eq:2.26}) now yields,
for all $s\geq0,
\begin{align}
\left\Vert U(u^{k_{j_{s}}})-U(z)\right\Vert ^{2} & =\left\Vert
U(u^{k_{j_{s}}})-z\right\Vert ^{2}\leq\left\Vert u^{k_{j_{s}}}-z\right\Vert
^{2}-\left\Vert U(u^{k_{j_{s}}})-u^{k_{j_{s}}}\right\Vert ^{2}\nonumber\\
& <\left\Vert u^{k_{j_{s}}}-z\right\Vert ^{2}-\varepsilon^{2}.
\label{formula withsqdelta
\end{align}
By arguments similar to those in the proof of Lemma \ref{lemma:Fejer}, we hav
\begin{equation}
\left\Vert u^{k}-z\right\Vert =\left\Vert \left( x^{k}+\gamma A^{\ast
}(T-I)(Ax^{k})\right) -z\right\Vert \leq\left\Vert x^{k}-z\right\Vert .
\label{P6
\end{equation}
Since $U$ is nonexpansive
\begin{equation}
\left\Vert x^{k+1}-z\right\Vert =\left\Vert U(u^{k})-z\right\Vert
\leq\left\Vert u^{k}-z\right\Vert . \label{P7
\end{equation}
Combining (\ref{P6}) and (\ref{P7}), we ge
\begin{equation}
\left\Vert x^{k+1}-z\right\Vert \leq\left\Vert u^{k}-z\right\Vert
\leq\left\Vert x^{k}-z\right\Vert , \label{P8
\end{equation}
which means that the sequence $\{x^{1},u^{1},x^{2},u^{2},\ldots\}$ is
Fej\'{e}r-monotone with respect to $\Gamma.$ Since $x^{k_{j_{s+1}
}=U(u^{k_{j_{s}}})$, we obtai
\begin{equation}
\left\Vert u^{k_{j_{s+1}}}-z\right\Vert ^{2}\leq\left\Vert u^{k_{j_{s}
}-z\right\Vert ^{2}.
\end{equation}
Hence $\left\{ u^{k_{j_{s}}}\right\} _{s=0}^{\infty}$ is also
Fej\'{e}r-monotone with respect to $\Gamma.$ Now, (\ref{formula withsqdelta})
and (\ref{P8}) imply tha
\begin{equation}
\left\Vert u^{k_{j_{s+1}}}-z\right\Vert ^{2}<\left\Vert u^{k_{j_{s}
}-z\right\Vert ^{2}-\varepsilon^{2}\text{ for all }s\geq0,
\end{equation}
which leads to a contradiction. Therefore $x^{\ast}\in SOL(C,f)$ and finally,
$x^{\ast}\in\Gamma$. Since the subsequence$\ \left\{ x^{k_{j}}\right\}
_{j=0}^{\infty}$ was arbitrary, we get that $x^{k}\rightharpoonup x^{\ast}.$
\end{proof}
Relations of our results to some previously published works are as follows. In
\cite{CS08a} an algorithm for the Split Common Fixed Point Problem (SCFPP) in
Euclidean spaces was studied. Later Moudafi \cite{Moudafi} presented a similar
result for Hilbert spaces. In this connection, see also \cite{Masad+Reich}.
To formulate the SCFPP, let $H_{1}$ and $H_{2}$ be two real Hilbert spaces.
Given operators $U_{i}:H_{1}\rightarrow H_{1}$, $i=1,2,\ldots,p,$ and
$T_{j}:H_{2}\rightarrow H_{2},$ $j=1,2,\ldots,r,$ with nonempty fixed point
sets $C_{i},$ $i=1,2,\ldots,p,$ and $Q_{j},$ $j=1,2,\ldots,r,$ respectively,
and a bounded linear operator $A:H_{1}\rightarrow H_{2}$, the SCFPP is
formulated as follows
\begin{equation}
\text{find a point }x^{\ast}\in C:=\cap_{i=1}^{p}C_{i}\text{ such that
}Ax^{\ast}\in Q:=\cap_{j=1}^{r}Q_{j}.
\end{equation}
Our result differs from\ those in \cite{CS08a} and \cite{Moudafi} in several
ways. Firstly, the spaces in which the problems are formulated. Secondly, the
operators $U$ and $T$ in \cite{CS08a} are assumed to be firmly
quasi-nonexpansive (FQNE; see the comments after Lemma \ref{lemma:Mod-proj}
above), where in our case here only $U$ is FQNE, while $T$ is just
nonexpansive. Lastly, Moudafi \cite{Moudafi} obtains weak convergence for a
wider class of operators, called demicontractive. The iterative step of his
algorithm is
\begin{equation}
x^{k+1}=(1-\alpha_{k})u^{k}+\alpha_{k}U(u^{k}),
\end{equation}
where $u^{k}:=x^{k}+\gamma A^{\ast}(T-I)(Ax^{k})$ for $\alpha_{k}\in(0,1).$ If
$\alpha_{k}=1,$ which is not allowed there, were possible, then the iterative
step of \cite{Moudafi} would coincide with that of \cite{CS08a}.
\subsection{A parallel algorithm for solving the multiple set split
variational inequality problem}
We extend the SVIP to the \textit{Multiple Set Split Variational Inequality
Problem} (MSSVIP), which is formulated as follows. Let $H_{1}$ and $H_{2}$ be
two real Hilbert spaces. Given a bounded linear operator $A:H_{1}\rightarrow
H_{2}$, functions $f_{i}:H_{1}\rightarrow H_{1},$ $i=1,2,\ldots,p,$ and
$g_{j}:H_{2}\rightarrow H_{2},$ $j=1,2,\ldots,r$, and nonempty, closed and
convex subsets $C_{i}\subseteq H_{1},$ $Q_{j}\subseteq H_{2}$ for
$i=1,2,\ldots,p$ and $j=1,2,\ldots,r$, respectively, the Multiple Set Split
Variational Inequality Problem (MSSVIP) is formulated as follows
\begin{equation}
\left\{
\begin{array}
[c]{l
\text{find a point }x^{\ast}\in C:=\cap_{i=1}^{p}C_{i}\text{ such that
}\left\langle f_{i}(x^{\ast}),x-x^{\ast}\right\rangle \geq0\text{ for all
}x\in C_{i}\\
\text{and for all }i=1,2,\ldots,p, \text{ and such that}\\
\text{the point }y^{\ast}=Ax^{\ast}\in Q:=\cap_{i=1}^{r}Q_{j}\text{ solves
}\left\langle g_{j}(y^{\ast}),y-y^{\ast}\right\rangle \geq0\text{ for all
}y\in Q_{j}\text{ }\\
\text{and for all }j=1,2,\ldots,r.
\end{array}
\right. \label{eq:mssvip
\end{equation}
For the MSSVIP we do not yet have a solution approach which does not use a
product space formalism. Therefore we present a simultaneous algorithm for the
MSSVIP the analysis of which is carried out via a certain product space. Let
$\Psi$ be the solution set of the MSSVIP
\begin{equation}
\Psi:=\left\{ z\in\cap_{i=1}^{p}SOL(C_{i},f_{i})\mid Az\in\cap_{i=1
^{r}SOL(Q_{j},g_{j})\right\} .
\end{equation}
We introduce the spaces $\boldsymbol{W}_{1}:\mathbf{=}H_{1}$ and
$\boldsymbol{W}_{2}:=H_{1}^{p}\times H_{2}^{r},$ where $r$ and $p$ are the
indices in (\ref{eq:mssvip}). Let $\left\{ \alpha_{i}\right\} _{i=1}^{p}$
and $\left\{ \beta_{j}\right\} _{j=1}^{r}$ be positive real numbers. Define
the following sets in their respective spaces
\begin{align}
\boldsymbol{C} & \mathbf{:}=H_{1}\text{ \ and \ \ \label{eq:prod}}\\
\boldsymbol{Q} & \mathbf{:}=\left( \prod_{i=1}^{p}\sqrt{\alpha_{i}
C_{i}\right) \times\left( \prod_{j=1}^{r}\sqrt{\beta_{j}}Q_{j}\right) ,
\end{align}
and the operator
\begin{equation}
\boldsymbol{A}\mathbf{:}=\left( \sqrt{\alpha_{1}}I,\ldots,\sqrt{\alpha_{p
}I,\sqrt{\beta_{1}}A^{\ast},\ldots,\sqrt{\beta_{r}}A^{\ast}\right) ^{\ast},
\end{equation}
where $A^{\ast}$ stands for adjoint of $A$. Denote $U_{i}:=P_{C_{i}}(I-\lambda
f_{i})$ and $T_{j}:=P_{Q_{j}}(I-\lambda g_{j})$ for $i=1,2,\ldots,p$ and
$j=1,2,\ldots,r$, respectively$.$ Define the operator $\boldsymbol{T
:\boldsymbol{W}_{2}\mathbf{\rightarrow}\boldsymbol{W}_{2}$ b
\begin{align}
\boldsymbol{T}\mathbf{(}\boldsymbol{y}\mathbf{)} & =\boldsymbol{T}\left(
\begin{array}
[c]{c
y_{1}\\
y_{2}\\
\vdots\\
y_{p+r
\end{array}
\right) \nonumber\\
& =\left( \left( U_{1}\left( y_{1}\right) \right) ^{\ast},\ldots,\left(
U_{p}\left( y_{p}\right) \right) ^{\ast},\left( T_{1}\left(
y_{p+1}\right) \right) ^{\ast},\ldots,\left( T_{r}(y_{p+r})\right) ^{\ast
}\right) ^{\ast}, \label{eq:fat
\end{align}
where $y_{1},y_{2},...,y_{p}\in H_{1}$ and $y_{p+1},y_{p+2},...,y_{p+r}\in
H_{2}$.
This leads to an SVIP with just two operators $\boldsymbol{F}$ and
$\boldsymbol{G}$ and two sets $\boldsymbol{C}$ and $\boldsymbol{Q},$
respectively, in the product space, when we take $\boldsymbol{C
\mathbf{=}H_{1}$, $\boldsymbol{F}\equiv\boldsymbol{0},$ $\boldsymbol{Q
\mathbf{\subseteq}\boldsymbol{W}_{2}$, $\boldsymbol{G}\mathbf{(
\boldsymbol{y}\mathbf{)}=\left( f_{1}(y_{1}),f_{2}(y_{2})\ldots,f_{p
(y_{p}),g_{1}(y_{p+1}),g_{2}(y_{p+2}),\ldots,g_{r}(y_{p+r})\right) ,$ and the
operator $\boldsymbol{A}:H_{1}\mathbf{\rightarrow}\boldsymbol{W}_{2}$. It is
easy to verify that the following equivalence holds
\begin{equation}
x\in\Psi\text{ if and only if }\boldsymbol{A}x\in\boldsymbol{Q}\mathbf{.
\end{equation}
Therefore we may apply Algorithm \ref{Alg-SVIP}
\begin{equation}
x^{k+1}=x^{k}+\gamma\boldsymbol{A}^{\ast}(\boldsymbol{T}-\boldsymbol{I
)(\boldsymbol{A}x^{k})\text{ for all }k\geq0, \label{itstepinPS
\end{equation}
to the problem (\ref{eq:prod})--(\ref{eq:fat}) in order to obtain a solution
of the original MSSVIP. We translate the iterative step (\ref{itstepinPS}) to
the original spaces $H_{1}$ and $H_{2}$ using the relatio
\begin{equation}
\boldsymbol{T}\mathbf{(}\boldsymbol{A}x)=\left( \sqrt{\alpha_{1}
U_{1}(x),\ldots,\sqrt{\alpha_{p}}U_{p}(x),\sqrt{\beta_{1}}AT_{1
(x),\ldots,\sqrt{\beta_{r}}AT_{r}(x)\right) ^{\ast
\end{equation}
and obtain the following algorithm.
\begin{algorithm}
\label{Nirits alg}$\left. {}\right. $
\textbf{Initialization:}$\ $Select an arbitrary starting point $x^{0}\in
H_{1}$.
\textbf{Iterative step: }Given the current iterate $x^{k},$ comput
\begin{equation}
x^{k+1}=x^{k}+\gamma\left( \sum_{i=1}^{p}\alpha_{i}(U_{i}-I)(x^{k
)+\sum_{j=1}^{r}\beta_{j}A^{\ast}(T_{j}-I)(Ax^{k})\right) ,
\end{equation}
where $\gamma\in(0,1/L),$ with $L=\sum_{i=1}^{p}\alpha_{i}+\sum_{j=1}^{r
\beta_{j}\Vert A\Vert^{2}$.
\end{algorithm}
The following convergence result follows from Theorem \ref{Theorem1}.
\begin{theorem}
\label{Theorem1*} Let $H_{1}$ and $H_{2}$ be two real Hilbert spaces and let
$A:H_{1}\rightarrow H_{2}$ be a bounded linear operator. Let $f_{i
:H_{1}\rightarrow H_{1},$ $i=1,2,\ldots,p,$ and $g_{j}:H_{2}\rightarrow
H_{2},$ $j=1,2,\ldots,r$, be $\alpha$-ISM operators on nonempty, closed and
convex subsets $C_{i}\subseteq H_{1},$ $Q_{j}\subseteq H_{2}$ for
$i=1,2,\ldots,p,$ and $j=1,2,\ldots,r$, respectively. Assume that\textbf{
}$\gamma\in(0,1/L)$ and $\Psi\neq\emptyset$\textbf{.} Set $U_{i}:=P_{C_{i
}(I-\lambda f_{i})$ and $T_{j}:=P_{Q_{j}}(I-\lambda g_{j})$ for $i=1,2,\ldots
,p$ and $j=1,2,\ldots,r$, respectively, with $\lambda\in\lbrack0,2\alpha]$.
If, in addition, for each $i=1,2,\ldots,p$ and $j=1,2,\ldots,r$ we have
\begin{equation}
\langle f_{i}(x),P_{C_{i}}(I-\lambda f_{i})(x)-x^{\ast}\rangle\geq0\text{ for
all\ }x\in H_{1} \label{eq:cond1
\end{equation}
for all $x^{\ast}\in SOL(C_{i},f_{i})$ and
\begin{equation}
\langle g_{j}(x),P_{Q_{j}}(I-\lambda g_{j})(x)-x^{\ast}\rangle\geq0\text{ for
all\ }x\in H_{2}, \label{eq:cond2
\end{equation}
for all $x^{\ast}\in SOL(C_{i},f_{i})$, then any sequence $\left\{
x^{k}\right\} _{k=0}^{\infty},$ generated by Algorithm \ref{Alg-SVIP},
converges weakly to a solution point $x^{\ast}\in\Psi$.
\end{theorem}
\begin{proof}
Apply Theorem \ref{Theorem1} to the two-operator SVIP in the product space
setting with $U=\boldsymbol{I}:H_{1}\rightarrow H_{1}$, $\operatorname*{Fix
U=\boldsymbol{C}\mathbf{,}$ $T=\boldsymbol{T}:\boldsymbol{W}\rightarrow
\boldsymbol{W}\mathbf{,}$ and $\operatorname*{Fix}T=\boldsymbol{Q}$.
\end{proof}
\begin{remark}
Observe that conditions (\ref{eq:cond1}) and (\ref{eq:cond2}) imposed on
$U_{i}$ and $T_{j}$ for $i=1,2,\ldots,p$ and $j=1,2,\ldots,r$, respectively,
in Theorem \ref{Theorem1*}, which are necessary for our treatment of the
problem in a product space, ensure that these operators are firmly
quasi-nonexpansive (FQNE). Therefore, the SVIP under these conditions may be
considered a \texttt{Split Common Fixed Point Problem} (SCFPP), first
introduced in \cite{CS08a}, with $\boldsymbol{C}\mathbf{,}$ $\boldsymbol{Q
\mathbf{,}$ $\boldsymbol{A}$ and $\boldsymbol{T}:\boldsymbol{W}_{2
\rightarrow\boldsymbol{W}_{2}$ as above, and the identity operator
$\boldsymbol{I}:\boldsymbol{C}\rightarrow\boldsymbol{C}$.\textbf{ }Therefore,
we could also apply \cite[Algorithm 4.1]{CS08a}. If, however, we drop these
conditions, then the operators are nonexpansive, by Lemma \ref{lemma:Mod-proj
(i), and the result of \cite{Moudafi} would apply.
\end{remark}
\section{Applications\label{sec:applications}}
The following problems are special cases of the SVIP. They are listed here
because their analysis can benefit from our algorithms for the SVIP and
because known algorithms for their solution may be generalized in the future
to cover the more general SVIP. The list includes known problems such as the
Split Feasibility Problem (SFP) and the Convex Feasibility Problem\textit{
}(CFP). In addition, we introduce two new \textquotedblleft
split\textquotedblright\ problems that have, to the best of our knowledge,
never been studied before. These are the Common Solutions to Variational
Inequalities Problem (CSVIP) and the Split Zeros Problem (SZP).
\subsection{The split feasibility and convex feasibility problems}
The Split Feasibility Problem (SFP) in Euclidean space is formulated as
follows:
\begin{equation}
\text{find a point }x^{\ast}\text{ such that }x^{\ast}\in C\subseteq
R^{n}\text{ and }Ax^{\ast}\in Q\subseteq R^{m}, \label{eq:sfp
\end{equation}
where $C\subseteq R^{n},$ $Q\subseteq R^{m}$ are nonempty, closed and convex
sets, and $A:R^{n}\rightarrow R^{m}$ is given. Originally introduced in Censor
and Elfving \cite{CE}, it was later used in the area of intensity-modulated
radiation therapy (IMRT) treatment planning; see \cite{CEKB,CBMT}. Obviously,
it is formally a special case of the SVIP obtained from (\ref{eq:vip
)--(\ref{eq:svip}) by setting $f\equiv g\equiv0.$ The Convex Feasibility
Problem\textit{ }(CFP) in a Euclidean space\textit{ }is:
\begin{equation}
\text{find a point }x^{\ast}\text{ such that }x^{\ast}\in\cap_{i=1}^{m
C_{i}\neq\emptyset,
\end{equation}
where $C_{i},$ $i=1,2,\ldots,m,$ are nonempty, closed and convex sets in
$R^{n}.$ This, in its turn, becomes a special case of the SFP by taking in
(\ref{eq:sfp}) $n=m,$ $A=I$ $Q=R^{n}$ and $C=\cap_{i=1}^{m}C_{i}.$ Many
algorithms for solving the CFP have been developed; see, e.g., \cite{BB96,
CZ97}. Byrne \cite{Byrne} established an algorithm for solving the
SFP\textit{, }called the CQ-Algorithm, with the following iterative step
\begin{equation}
x^{k+1}=P_{C}\left( x^{k}+\gamma A^{t}(P_{Q}-I)Ax^{k}\right) ,
\label{eq:cq-alg
\end{equation}
which does not require calculation of the inverse of the operator $A,$ as in
\cite{CE}, but needs only its transpose $A^{t}$. A recent excellent paper on
the multiple-sets SFP which contains many references that reflect the
state-of-the-art in this area is \cite{lopezetal10}.
It is of interest to note that looking at the SFP from the point of view of
the SVIP enables us to find the minimum-norm solution of the SFP, i.e., a
solution of the for
\begin{equation}
x^{\ast}=\operatorname{argmin}\{\Vert x\Vert\mid x\text{ solves the SFP
(\ref{eq:sfp})}\}.
\end{equation}
This is done, and easily verified, by solving (\ref{eq:vip})--(\ref{eq:svip})
with $f=I$ and $g\equiv0.$
\subsection{The common solutions to variational inequalities problem}
The Common Solutions to Variational Inequalities Problem (CSVIP), newly
introduced here, is defined in Euclidean space as follows. Let $\left\{
f_{i}\right\} _{i=1}^{m}\ $be a family of functions from $R^{n}$ into itself
and let $\left\{ C_{i}\right\} _{i=1}^{m}$ be nonempty, closed and convex
subsets of $R^{n}$ with $\cap_{i=1}^{m}C_{i}\neq\emptyset$. The CSVIP is
formulated as follows:
\begin{align}
\text{find a point }x^{\ast} & \in\cap_{i=1}^{m}C_{i}\text{ such that
}\left\langle f_{i}(x^{\ast}),x-x^{\ast}\right\rangle \geq0\text{ }\nonumber\\
\text{for all }x & \in C_{i}\text{, }i=1,2,\ldots,m.
\end{align}
This problem can be transformed into a CVIP in an appropriate product space
(different from the one in Section \ref{sec:SVIP}). Let $R^{mn}$ be the
product space and define $\boldsymbol{F}:R^{mn}\rightarrow R^{mn}$ b
\begin{equation}
\boldsymbol{F}\left( (x^{1},x^{2},\ldots,x^{m})\right) =(f_{1}(x^{1
),\ldots,f_{m}(x^{m})),
\end{equation}
where $x^{i}\in R^{n}$ for all $i=1,2,\ldots,m.$ Let the diagonal set in
$R^{mn}$ be
\begin{equation}
\boldsymbol{\Delta}:=\{\boldsymbol{x}\in R^{mn}\mid\boldsymbol{x
\mathbf{=}(a,a,\ldots,a),\text{ }a\in R^{n}\}
\end{equation}
and define the product se
\begin{equation}
\boldsymbol{C}:=\Pi_{i=1}^{m}C_{i}.
\end{equation}
The CSVIP in $R^{n}$ is equivalent to the following CVIP in $R^{mn}$
\begin{align}
\text{find a point }\boldsymbol{x}^{\ast} & \in\boldsymbol{C}\cap
\boldsymbol{\Delta}\text{ such that }\left\langle \boldsymbol{F
(\boldsymbol{x}^{\ast}),\boldsymbol{x-x}^{\ast}\right\rangle \geq0\text{
}\nonumber\\
\text{for all }\boldsymbol{x} & =(x^{1},x^{2},\ldots,x^{m})\in
\boldsymbol{C}.
\end{align}
So, this problem can be solved by using Algorithm \ref{alg:SubExt4SVIP} with
$\Omega=\boldsymbol{\Delta}.$ A new algorithm specifically designed for the
CSVIP appears in \cite{cgrs10}.
\subsection{The split minimization and the split zeros problems}
From optimality conditions for convex optimization (see, e.g., Bertsekas and
Tsitsiklis \cite[Proposition 3.1, page 210]{BT}) it is well-known that if
$F:R^{n}\rightarrow R^{n}$ is a continuously differentiable convex function on
a closed and convex subset $X\subseteq R^{n},$ then $x^{\ast}\in X$ minimizes
$F$ over $X$ if and only i
\begin{equation}
\langle\nabla F(x^{\ast}),x-x^{\ast}\rangle\geq0\text{ for all }x\in X,
\label{eq:bert
\end{equation}
where $\nabla F$ is the gradient of $F$. Since (\ref{eq:bert}) is a VIP, we
make the following observation. If $F:R^{n}\rightarrow R^{n}$ and
$G:R^{m}\rightarrow R^{m}$ are continuously differentiable convex functions on
closed and convex subsets $C\subseteq R^{n}$ and $Q\subseteq R^{m},$
respectively, and if in the SVIP we take $f=\nabla F$ and $g=\nabla G,$ then
we obtain the following \textit{Split Minimization Problem }(SMP)
\begin{gather}
\text{find a point }x^{\ast}\in C\text{ such that }x^{\ast
=\operatorname{argmin}\{f(x)\mid x\in C\}\\
\text{and such that}\nonumber\\
\text{the point }y^{\ast}=Ax^{\ast}\in Q\text{ and solves }y^{\ast
}=\operatorname{argmin}\{g(y)\mid y\in Q\}.
\end{gather}
The \textit{Split Zeros Problem} (SZP), newly introduced here, is defined as
follows. Let $H_{1}$ and $H_{2}$ be two Hilbert spaces. Given operators
$B_{1}:H_{1}\rightarrow H_{1}$ and $B_{2}:H_{2}\rightarrow H_{2},$ and a
bounded linear operator $A:H_{1}\rightarrow H_{2}$, the SZP is formulated as
follows
\begin{equation}
\text{find a point }x^{\ast}\in H_{1}\text{ such that }B_{1}(x^{\ast})=0\text{
and }B_{2}(Ax^{\ast})=0. \label{eq:SZP
\end{equation}
This problem is a special\ case of the SVIP if $A$ is a surjective operator.
To see this, take in (\ref{eq:vip})--(\ref{eq:svip}) $C=H_{1}$, $Q=H_{2},$
$f=B_{1}$ and $g=B_{2},$ and choose $x:=x^{\ast}-B_{1}(x^{\ast})\in H_{1}$ in
(\ref{eq:vip}) and $x\in H_{1}$ such that $Ax:=Ax^{\ast}-B_{2}(Ax^{\ast})\in
H_{2}$ in (\ref{eq:svip}).
The next lemma shows when the only solution of an SVIP is a solution of an
SZP. It extends a similar result concerning the relationship between the
(un-split) zero finding problem and the VIP.
\begin{lemma}
Let $H_{1}$ and $H_{2}$ be real Hilbert spaces, and $C\subseteq H_{1}$ and
$Q\subseteq H_{2}$ nonempty, closed and convex subsets. Let $B_{1
:H_{1}\rightarrow H_{1}$ and $B_{2}:H_{2}\rightarrow H_{2}$ be $\alpha$-ISM
operators and let $A:H_{1}\rightarrow H_{2}$ be a bounded linear operator.
Assume that $C\cap\{x\in H_{1}\mid B_{1}(x)=0\}\neq\emptyset$ and that
$Q\cap\{y\in H_{2}\mid B_{2}(y)=0\}\neq\emptyset$, and denot
\begin{equation}
\Gamma:=\Gamma(C,Q,B_{1},B_{2},A):=\left\{ z\in SOL(C,B_{1})\mid Az\in
SOL(Q,B_{2})\right\} .
\end{equation}
Then, for any $x^{\ast}\in C$ with $Ax^{\ast}\in Q,$ $x^{\ast}$ solves
(\ref{eq:SZP}) if and only if $x^{\ast}\in\Gamma$.
\end{lemma}
\begin{proof}
First assume that $x^{\ast}\in C$ with $Ax^{\ast}\in Q$ and that $x^{\ast}$
solves (\ref{eq:SZP}). Then it is clear that $x^{\ast}\in\Gamma.$ In the other
direction, assume that $x^{\ast}\in C$ with $Ax^{\ast}\in Q$ and that
$x^{\ast}\in\Gamma.$ Applying (\ref{eq:ProjP2}) with $C$ as $D$ there,
$(I-\lambda B_{1})\left( x^{\ast}\right) \in H_{1},$ for any $\lambda
\in(0,2\alpha]$, as $x$ there, and $q_{1}\in C\cap\operatorname*{Fix
(I-\lambda B_{1}),$ with the same $\lambda,$ as $y$ there, we ge
\begin{align}
& \left\Vert q_{1}-P_{C}(I-\lambda B_{1})\left( x^{\ast}\right) \right\Vert
^{2}+\left\Vert (I-\lambda B_{1})\left( x^{\ast}\right) -P_{C}(I-\lambda
B_{1})\left( x^{\ast}\right) \right\Vert ^{2}\nonumber\\
& \leq\left\Vert (I-\lambda B_{1})\left( x^{\ast}\right) -q_{1}\right\Vert
^{2},
\end{align}
and, similarly, applying (\ref{eq:ProjP2}) again, we obtai
\begin{align}
& \left\Vert q_{2}-P_{Q}(I-\lambda B_{2})\left( Ax^{\ast}\right)
\right\Vert ^{2}+\left\Vert (I-\lambda B_{2})\left( Ax^{\ast}\right)
-P_{Q}(I-\lambda B_{2})\left( Ax^{\ast}\right) \right\Vert ^{2}\nonumber\\
& \leq\left\Vert (I-\lambda B_{2})\left( Ax^{\ast}\right) -q_{2}\right\Vert
^{2}.
\end{align}
Using the characterization of (\ref{eq:fix-vip}), we ge
\begin{equation}
\left\Vert q_{1}-x^{\ast}\right\Vert ^{2}+\left\Vert (I-\lambda B_{1})\left(
x^{\ast}\right) -x^{\ast}\right\Vert ^{2}\leq\left\Vert (I-\lambda
B_{1})\left( x^{\ast}\right) -q_{1}\right\Vert ^{2
\end{equation}
an
\begin{equation}
\left\Vert q_{2}-Ax^{\ast}\right\Vert ^{2}+\left\Vert (I-\lambda B_{2})\left(
Ax^{\ast}\right) -x^{\ast}\right\Vert ^{2}\leq\left\Vert (I-\lambda
B_{2})\left( Ax^{\ast}\right) -q_{2}\right\Vert ^{2}.
\end{equation}
It can be seen from the proof of Lemma \ref{lemma:Mod-proj}(i) that the
operators $I-\lambda B_{1}$ and $I-\lambda B_{2}$ are nonexpansive for every
$\lambda\in\lbrack0,2\alpha]$, so with $q_{1}\in C\cap\operatorname*{Fix
(I-\lambda B_{1})$ and $q_{2}\in Q\cap\operatorname*{Fix}(I-\lambda B_{2}),
\begin{equation}
\left\Vert (I-\lambda B_{1})\left( x^{\ast}\right) -q_{1}\right\Vert
^{2}\leq\left\Vert x^{\ast}-q_{1}\right\Vert ^{2
\end{equation}
an
\begin{equation}
\left\Vert (I-\lambda B_{2})\left( Ax^{\ast}\right) -q_{2}\right\Vert
^{2}\leq\left\Vert Ax^{\ast}-q_{2}\right\Vert ^{2}.
\end{equation}
Combining the above inequalities, we obtai
\begin{equation}
\left\Vert q_{1}-x^{\ast}\right\Vert ^{2}+\left\Vert (I-\lambda B_{1})\left(
x^{\ast}\right) -x^{\ast}\right\Vert ^{2}\leq\left\Vert x^{\ast
-q_{1}\right\Vert ^{2
\end{equation}
an
\begin{equation}
\left\Vert q_{2}-Ax^{\ast}\right\Vert ^{2}+\left\Vert (I-\lambda B_{2})\left(
Ax^{\ast}\right) -x^{\ast}\right\Vert ^{2}\leq\left\Vert Ax^{\ast
-q_{2}\right\Vert ^{2}.
\end{equation}
Hence, $\left\Vert (I-\lambda B_{1})\left( x^{\ast}\right) -x^{\ast
}\right\Vert ^{2}=0$ and $\left\Vert (I-\lambda B_{2})\left( Ax^{\ast
}\right) -Ax^{\ast}\right\Vert ^{2}=0.$ Since $\lambda>0,$ we get that
$B_{1}(x^{\ast})=0$ and $B_{2}(Ax^{\ast})=0$, as claimed$\medskip$
\end{proof}
\textbf{Acknowledgments}. This work was partially supported by a United
States-Israel Binational Science Foundation (BSF) Grant number 200912, by US
Department of Army award number W81XWH-10-1-0170, by Israel Science Foundation
(ISF) Grant number 647/07, by the Fund for the Promotion of Research at the
Technion and by the Technion President's Research Fund.\bigskip
|
{
"timestamp": "2011-08-11T02:01:36",
"yymm": "1009",
"arxiv_id": "1009.3780",
"language": "en",
"url": "https://arxiv.org/abs/1009.3780"
}
|
\section{Introduction}
The present acceleration of the universe expansion has been well
established through numerous and complementary cosmological
observations \cite{Riess}. One explanation for the cosmic
acceleration is the dark energy (DE), an exotic energy with negative
pressure. Although the nature and cosmological origin of DE is still
enigmatic at present, a great variety of models have been proposed
to describe the DE (for review see \cite{Padmanabhan}). One of
interesting issues in modern cosmology is the thermodynamical
description of the accelerating universe driven by the DE.
In black hole physics, it was found that black holes emit Hawking
radiation with a temperature proportional to their surface gravity
at the event horizon and they have an entropy which is one quarter
of the area of the event horizon \cite{Hawking}. The temperature,
entropy and mass of black holes satisfy the first law of
thermodynamics \cite{Bardeen}. It was shown that the Einstein
equation can be derived from the first law of thermodynamics by
assuming the proportionality of entropy and the horizon area
\cite{Jacobson}.
The relation between the Einstein equation and the first law of
thermodynamics has been generalized to the cosmological context. It
was shown that by applying the Clausius relation $-{\rm d}E=T_A{\rm
d} S_A$ to the apparent horizon $\tilde{r}_{\rm A}$, the Friedmann
equation in the Einstein gravity can be derived if we take the
Hawking temperature $T_{\rm A}=1/(2\pi \tilde{r}_{\rm A})$ and the
entropy $S_{\rm A}=A/4$ on the apparent horizon, where $A$ is the
area of the horizon \cite{Cai05}. The equivalence between the first
law of thermodynamics and Friedmann equation was also found for
gravity with Gauss-Bonnet term, the Lovelock gravity theory and the
braneworld scenarios \cite{Cai05,Akbar,Sheykhi1}.
Note that in thermodynamics of apparent horizon in the standard
Friedmann-Robertson-Walker (FRW) cosmology, the geometric entropy is
assumed to be proportional to its horizon area, $S_A={A}/{4}$
\cite{Cai05}. However, this definition for the entropy can be
modified from the inclusion of quantum effects. For instance in
quantum tunneling formalism, taking into account the quantum back
reaction effects in the spacetime found by conformal field theory
methods and using the second law of thermodynamics, the corrections
to the both semiclassical Bekenstein-Hawking area law ($S_{\rm
BH}=A/4$) and Friedmann equation can be obtained \cite{Banerjee}.
The quantum corrections provided to the entropy-area relationship
lead to the curvature correction in the Einstein-Hilbert action and
vice versa \cite{Suj}. The power-law quantum correction to the
horizon entropy motivated by the entanglement of quantum fields
between inside and outside of the horizon is given by \cite{Saurya}
\begin{equation}
S_{\rm A}=\frac{A}{4}\left[1-K_{\alpha}
A^{1-\frac{\alpha}{2}}\right],\label{ec}
\end{equation}
where we take $c=k_B=\hbar=G=1$. Also $\alpha$ is a dimensionless
parameter and
\begin{equation}
K_\alpha=\frac{\alpha(4\pi)^{\frac{\alpha}{2}-1}}{(4-\alpha)r_c^{2-\alpha}},
\end{equation}
where $r_c$ is the crossover scale. Note that in the case of
$\alpha=0=K_{\alpha}$, Eq. (\ref{ec}) reduces to the well-known
Bekenstein-Hawking entropy-area relation $S_{\rm A}=S_{\rm BH}=A/4$.
Besides the first law of thermodynamics, a lot of attention has been
paid to the generalized second law (GSL) of thermodynamics in the
accelerating universe driven by the DE
\cite{Izquierdo1,Sadjadi07,Zhou07,Gong07,Sheykhi2,Karami1,Radicella}.
The GSL of thermodynamics like the first law is an accepted
principle in physics. According to the GSL, the entropy of matter
inside the horizon plus the entropy of the horizon do not decrease
with time \cite{Bekenstein,Davies,Izquierdo2,Wang1}.
Here, we would like to examine whether the power-law corrected
entropy (\ref{ec}) together with the matter field entropy inside the
apparent horizon will satisfy the GSL of thermodynamics. To be more
general we will consider an interacting viscous DE with dark matter
(DM) and radiation. The observations indicate that the universe
media is not a perfect fluid and the viscosity is concerned in the
evolution of the universe (see \cite{Ren1} and references therein).
This paper is organized as follows. In section 2, using the Clausius
relation we derive the modified Friedmann equation corresponding to
the power-law corrected entropy (\ref{ec}). In section 3, we study
the interacting viscous DE with DM and radiation in a non-flat
modified FRW universe. In section 4, we investigate the effect of
the power-law correction term to the entropy on the dynamics of DE.
Section 5 is devoted to conclusions. In Appendix A, we investigate
the validity of the GSL of gravitational thermodynamics with
power-law corrected entropy for the universe enclosed by the
apparent horizon.
\section{Clausius relation and modified Friedmann equation}
In the framework of FRW metric,
\begin{equation}
{\rm d}s^2 = h_{ij}{\rm d}x^i{\rm d}x^j + \tilde{r}^2{\rm
d}\Omega^2,
\end{equation}
where $\tilde{r}(t) = a(t)r$, $x^i = (t, r)$ and $h_{ij}$ =
diag($-1, a^2/(1 - kr^2)$), $i,j=0,1$, by setting
\begin{equation}
f:=h^{ij}\partial_{i}\tilde{r}\partial_{j}\tilde{r}=1-\left(H^2+\frac{k}{a^2}\right)\tilde{r}^2=0,\label{f}
\end{equation}
the location of the apparent horizon in the FRW universe is
obtained as \cite{Poisson}
\begin{equation}
\tilde{r}_{\rm A}=H^{-1}(1+\Omega_k)^{-1/2}.\label{ra}
\end{equation}
Here $\Omega_{k}=k/(a^2H^2)$ and $k=0,1,-1$ represent a flat, closed
and open universe, respectively. The Hawking temperature on the
apparent horizon is given by \cite{Cai05}
\begin{equation}
T_{\rm A}=\frac{1}{2\pi \tilde{r}_{\rm
A}}\left(1-\frac{\dot{\tilde{r}}_{\rm A}}{2H\tilde{r}_{\rm A}}
\right),\label{TA1}
\end{equation}
where $\frac{\dot{\tilde{r}}_{\rm A}}{2H\tilde{r}_{\rm A}}<1$
ensure that the temperature is positive.
To derive the modified Friedmann equation corresponding to the
power-law corrected entropy (\ref{ec}) we start with the Clausius
relation \cite{Jacobson}
\begin{equation}
-{\rm d}E=T_{\rm A}{\rm d}S_{\rm A},
\end{equation}
where $-{\rm d}E$ is the amount of energy crossing the apparent
horizon during the infinitesimal time interval ${\rm d}t$ in which
the radius of the apparent horizon is assumed to be fixed, i.e.
$\dot{\tilde{r}}_{\rm A} = 0$ \cite{Cai09}. This yields $T_{\rm
A}=1/(2\pi \tilde{r}_{\rm A})$.
Following \cite{Cai5} we have
\begin{equation}
-{\rm d}E=4\pi\tilde{r}_{\rm A}^3(\rho+p)H{\rm d}t,\label{dE}
\end{equation}
where $\rho$ and $p$ are the energy density and pressure of the
fluid, respectively, inside the universe and satisfy the energy
conservation law
\begin{equation}
\dot{\rho}+3H(\rho+p)=0.\label{econs}
\end{equation}
From Eqs. (\ref{ec}), (\ref{ra}), (\ref{TA1}) and using
$A=4\pi\tilde{r}_{\rm A}^2$ one can obtain
\begin{equation}
T_{\rm A}{\rm d}S_{\rm A}=T_{\rm A}\frac{\partial S_{\rm
A}}{\partial A}~{\rm
d}A=-\left(\dot{H}-\frac{k}{a^2}\right)\left\{1-\frac{\alpha}{2}\left[\left(H^2+\frac{k}{a^2}\right)^{1/2}r_c\right]^{\alpha-2}\right\}\tilde{r}_{\rm
A}^3H{\rm d}t.\label{TdS}
\end{equation}
Equating (\ref{dE}) with (\ref{TdS}) and using (\ref{econs}) gives
\begin{equation}
2H\left(\dot{H}-\frac{k}{a^2}\right)\left\{1-\frac{\alpha}{2}
\left[\left(H^2+\frac{k}{a^2}\right)^{1/2}r_c\right]^{\alpha-2}\right\}=\frac{8\pi}{3}\dot{\rho}.\label{feq1}
\end{equation}
Integrating with respect to cosmic time $t$ we get the modified
Friedmann equation
\begin{equation}
H^2+\frac{k}{a^2}-r_c^{-2}\left[r_c^{\alpha}\left(H^2+\frac{k}{a^2}\right)^{\alpha/2}-1\right]=\frac{8\pi}{3}\rho,
\label{FRW1}
\end{equation}
which in the absence of correction term, i.e. $\alpha=0$, it
recovers the well-known first Friedmann equation in the standard FRW
cosmology.
\section{Interacting viscous DE, DM and radiation}
Here we consider a non-flat FRW universe containing the DE, DM and
radiation. Hence the first modified Friedmann equation (\ref{FRW1})
corresponding to the power-law corrected entropy (\ref{ec}) takes
the form
\begin{equation}
H^2+\frac{k}{a^2}-r_c^{-2}\left[r_c^{\alpha}\left(H^2+\frac{k}{a^2}\right)^{\alpha/2}-1\right]
=\frac{8\pi}{3}(\rho_D+\rho_m+\rho_r),\label{eqf1}
\end{equation}
where $\rho_D$, $\rho_m$ and $\rho_r$ are the energy density of
DE, DM and radiation, respectively.
Using the following definitions
\begin{equation}
\Omega_{D}=\frac{8\pi \rho_{D}}{3H^2},~~~\Omega_{m}=\frac{8\pi
\rho_{m}}{3H^2},~~~\Omega_{r}=\frac{8\pi
\rho_{r}}{3H^2},\label{Omega1}
\end{equation}
\begin{eqnarray}
\Omega_\alpha&=&(Hr_c)^{-2}\left[(Hr_c)^{\alpha}(1+\Omega_{k})^{\alpha/2}-1\right],
\nonumber\\
&=&(1+\Omega_k)\left(\frac{\tilde{r}_A}{r_c}\right)^2\left[\left(\frac{\tilde{r}_A}{r_c}\right)^{-\alpha}-1\right]
,\label{Omega2}
\end{eqnarray}
one can rewrite Eq. (\ref{eqf1}) as
\begin{equation}
1+\Omega_k=\Omega_D+\Omega_m+\Omega_r+\Omega_\alpha.\label{eqf2}
\end{equation}
Here, we consider a viscous model of DE. In an isotropic and
homogeneous FRW universe, the dissipative effects arise due to the
presence of bulk viscosity in cosmic fluids. The DE with bulk
viscosity has a peculiar property to cause accelerated expansion of
phantom type in the late evolution of the universe
\cite{Brevik,Ren2}. Note that the total energy density still
satisfies the conservation law (\ref{econs}) where
\begin{equation}
\rho=\rho_D+\rho_m+\rho_r,\label{rho}
\end{equation}
\begin{equation}
p=\tilde{p}_D+p_r,\label{p1}
\end{equation}
and
\begin{equation}
\tilde{p}_D=p_D-3H\xi,
\end{equation}
is the effective pressure of the DE and $\xi$ is the bulk viscosity
coefficient \cite{Ren1,Ren2}. Note that $p_r=\rho_r/3$ and the DM is
pressureless, i.e. $p_m=0$. Here like \cite{Sheykhi3}, if we assume
$\xi=\varepsilon\rho_D H^{-1}$, where $\varepsilon$ is a constant
parameter, then the total pressure yields
\begin{equation}
p=(\omega_D-3\varepsilon)\rho_D+\frac{1}{3}\rho_r,\label{p2}
\end{equation}
where $\omega_D=p_D/\rho_D$ is the equation of state (EoS) parameter
of the viscous DE.
We further assume that the viscous DE, DM and radiation interact
with each other. Recently the scenario in which the DE interacts
with DM and radiation has been introduced to resolve the cosmic
triple coincidence problem \cite{triple}. In the presence of
interaction, the energy conservation laws for the viscous DE, DM and
radiation are not separately hold and we have
\begin{equation}
\dot{\rho_D}+3H\rho_D(1+\omega_D)=9H^2\xi-Q,\label{continD}
\end{equation}
\begin{equation}
\dot{\rho_m}+3H\rho_m=Q',\label{continm}
\end{equation}
\begin{equation}
\dot{\rho_r}+4H\rho_r=Q-Q',\label{continr}
\end{equation}
where $Q$ and $Q'$ stand for the interaction terms.
Taking a time derivative in both sides of Eq. (\ref{eqf1}), and
using Eqs. (\ref{Omega1}), (\ref{Omega2}), (\ref{eqf2}),
(\ref{continD}), (\ref{continm}), (\ref{continr}) and
$\xi=\varepsilon\rho_D H^{-1}$, the EoS parameter of interacting
viscous DE can be obtained as
\begin{equation}
\omega_D=-\frac{1}{3\Omega_D}\left\{2\left(\frac{\dot{H}}{H^2}-\Omega_k\right)\left[1-\left(\frac{\alpha}{2}\right)
\frac{\Omega_\alpha+(Hr_c)^{-2}}{1+\Omega_k}\right]
+3\Omega_m+4\Omega_r\right\}+3\varepsilon-1.\label{wlambda}
\end{equation}
The deceleration parameter is given by
\begin{equation}
q=-\left(1+\frac{\dot{H}}{H^2}\right).\label{q1}
\end{equation}
Substituting the term $\dot{H}/H^2$ from (\ref{wlambda}) into
(\ref{q1}) yields
\begin{equation}
q=\frac{(1+\Omega_k)}{2\left[1+\Omega_k-\frac{\alpha}{2}\Big(\Omega_\alpha+(Hr_c)^{-2}\Big)\right]}\Big[3\Omega_D(1+\omega_D-3\varepsilon)
+3\Omega_m+4\Omega_r\Big]-(1+\Omega_k).\label{q2}
\end{equation}
Using Eq. (\ref{eqf2}) one can rewrite (\ref{q2}) as
\begin{equation}
q=\frac{(1+\Omega_k)}{2\left[1+\Omega_k-\frac{\alpha}{2}\Big(\Omega_\alpha+(Hr_c)^{-2}\Big)\right]}\Big[1+\Omega_{k}+\Omega_{\alpha}(\alpha-3)
+\alpha(Hr_c)^{-2}+3\Omega_D(\omega_D-3\varepsilon)+\Omega_r\Big].\label{q3}
\end{equation}
\section{The effect of the power-law correction term to the entropy
on the dynamics of DE}
Here to see how the power-law correction term to the entropy
(\ref{ec}) influence the dynamics of DE in our selected model for
the universe, we need to incorporate a specific form of the DE model
as well as the interaction terms between DE, DM and radiation. To do
this we consider the power-law entropy-corrected version of the
holographic DE (HDE) model. The HDE model is motivated by the
holographic principle \cite{Hooft}. Following \cite{Li}, the HDE
density is given by
\begin{equation}
\rho_{D}=3c^2M^2_PL^{-2},\label{HDE}
\end{equation}
where $c$ is a dimensionless constant, $M_P$ is the reduced Planck
Mass $M_P^{-2}=8\pi$ with $G=1$ and $L$ is the IR cut-off.
Indeed, the definition and derivation of the HDE density depends on
the Bekenstein-Hawking entropy-area relation $S_{\rm BH} = A/4$,
where $A\sim L^2$ is the area of horizon. Taking into account the
power-law correction (\ref{ec}) to the Bekenstein-Hawking entropy,
which appears in dealing with the entanglement of quantum fields
between in and out the horizon, the HDE density is modified
accordingly. This modification yields the energy density of the
so-called ``power-law entropy-corrected HDE'' (PLECHDE) as
\cite{sheyjam}
\begin{equation}\label{rhoPLECHDE}
\rho _{D }=3c^2M_{P}^{2}L^{-2}-\beta M_{P}^{2}L^{-\alpha},
\end{equation}
where $\beta$ is a dimensional constant. In the special case
$\beta=0$, the above equation yields the well-known HDE density
(\ref{HDE}).
From definition $\rho_{D}=3M_P^2H^2\Omega_{D}$ and using Eq.
(\ref{rhoPLECHDE}), we get
\begin{equation}
L=\frac{c}{H}\left(\frac{\gamma_c}{\Omega_{D}}\right)^{1/2},\label{HLPLECHDE}
\end{equation}
where
\begin{eqnarray}
\gamma_c = 1 - \frac{\beta}{3c^2}L^{2-\alpha}.\label{gammacPLECHDE}
\end{eqnarray}
If we consider the apparent horizon as an IR cut-off,
$L=\tilde{r}_A$, in a non-flat FRW universe, then taking a
derivative of Eq. (\ref{rhoPLECHDE}) with respect to cosmic time $t$
yields
\begin{equation}
\frac{\dot{\rho}_D}{\rho_D}=\left(\frac{\alpha-2}{\gamma_c}-\alpha\right)\frac{\dot{\tilde{r}}_A}{\tilde{r}_A}.
\label{rhodotPLECHDE1}
\end{equation}
Taking a time derivative of Eq. (\ref{ra}) gives
\begin{equation}
\dot{\tilde{r}}_A=\frac{\Omega_k-\frac{\dot{H}}{H^2}}{(1+\Omega_k)^{3/2}}.\label{radotPLECHDE}
\end{equation}
Taking a time derivative of Eq. (\ref{eqf1}) and using Eqs.
(\ref{Omega2}), (\ref{continm}) and (\ref{continr}) gives
\begin{equation}
\frac{\dot{H}}{H^2}=\Omega_k+\frac{\frac{8\pi}{3H^3}(\dot{\rho}_D+Q)-3\Omega_m-4\Omega_r}{2-\alpha\Big(\frac{\tilde{r}_{\rm
A}}{\tilde{r}_{\rm A_0}}\Big)^{2-\alpha}},\label{HdotH2PLECHDE}
\end{equation}
where following \cite{Saurya,Dvali} we take $r_c=\tilde{r}_{\rm
A_0}$. In what follows, following Cruz et al. in \cite{triple} we
assume
\begin{equation}
Q=3b^2H(\rho_D+\rho_m+\rho_r),\label{Q1}
\end{equation}
\begin{equation}
Q'=3b'^2H(\rho_D+\rho_m+\rho_r),\label{Q2}
\end{equation}
with the coupling constants $b^2$ and $b'^2$.
Using Eqs. (\ref{radotPLECHDE}), (\ref{HdotH2PLECHDE}) and
(\ref{Q1}) one can rewrite Eq. (\ref{rhodotPLECHDE1}) as
\begin{equation}
\frac{\dot{\rho}_D}{3H\rho_D}=\frac{\Big(\frac{\alpha-2}{\gamma_c}-\alpha\Big)\Big[\Omega_m+\frac{4}{3}\Omega_r-b^2(1+\Omega_k-\Omega_{\alpha})\Big]}
{\Big(\frac{\alpha-2}{\gamma_c}-\alpha\Big)\Omega_D+\Big[2-\alpha\Big(\frac{\tilde{r}_A}{\tilde{r}_{A_0}}\Big)^{2-\alpha}
\Big](1+\Omega_k)}.\label{rhodotPLECHDE2}
\end{equation}
Substituting Eqs. (\ref{Q1}) and (\ref{rhodotPLECHDE2}) in
(\ref{HdotH2PLECHDE}) gives
\begin{eqnarray}
\frac{\dot{H}}{H^2}=\Omega_k+\frac{3b^2(1+\Omega_k-\Omega_{\alpha})-3\Omega_m-4\Omega_r
}{\Big(\frac{\alpha-2}{\gamma_c}-\alpha\Big)\Big(\frac{\Omega_D}{1+\Omega_k}\Big)+\Big[2-\alpha\Big(\frac{\tilde{r}_{\rm
A}}{\tilde{r}_{\rm A_0}}\Big)^{2-\alpha}\Big]}
.\label{HdotH2PLECHDE2}
\end{eqnarray}
Inserting Eq. (\ref{HdotH2PLECHDE2}) in (\ref{wlambda}) and using
Eq. (\ref{Omega2}) yields the EoS parameter of the interacting
viscous PLECHDE as
\begin{eqnarray}
\omega_D=-1+3\varepsilon-b^2\left(\frac{1+\Omega_k-\Omega_{\alpha}}{\Omega_D}\right)
~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
+\frac{\Big(\frac{\alpha-2}{\gamma_c}-\alpha\Big)\Big[b^2(1+\Omega_k-\Omega_{\alpha})-\Omega_m-\frac{4}{3}\Omega_r\Big]}
{\Big(\frac{\alpha-2}{\gamma_c}-\alpha\Big)\Omega_D+\Big[2-\alpha\Big(\frac{\tilde{r}_{\rm
A}}{\tilde{r}_{\rm
A_0}}\Big)^{2-\alpha}\Big](1+\Omega_k)}.\label{wDPLECHDE}
\end{eqnarray}
Replacing Eq. (\ref{HdotH2PLECHDE2}) into (\ref{q1}) gives the
deceleration parameter as
\begin{eqnarray}
q=-1-\Omega_k-\frac{3b^2(1+\Omega_k-\Omega_{\alpha})-3\Omega_m-4\Omega_r
}{\Big(\frac{\alpha-2}{\gamma_c}-\alpha\Big)\Big(\frac{\Omega_D}{1+\Omega_k}\Big)+\Big[2-\alpha\Big(\frac{\tilde{r}_{\rm
A}}{\tilde{r}_{\rm A_0}}\Big)^{2-\alpha}\Big]}.\label{qPLECHDE}
\end{eqnarray}
In the absence of correction term ($\alpha=0=\beta$), from Eqs.
(\ref{Omega2}) and (\ref{gammacPLECHDE}) we have $\Omega_{\alpha}=0$
and $\gamma_c=1$, respectively. If we also consider a spatially flat
FRW universe ($\Omega_k=0$), Eq. (\ref{ra}) shows that the apparent
horizon is same as the Hubble horizon, i.e. $\tilde{r}_A=H^{-1}$,
and $L=\tilde{r}_A=H^{-1}$. Now if we take
$\varepsilon=b^2=\Omega_r=0$ then Eq. (\ref{wDPLECHDE}) yields the
pressureless DE, i.e. $\omega_{D}=0$, where its EoS behaves like the
dust (or dark) matter. This result has been already obtained by Hsu
\cite{Hsu} for the HDE model with the IR cut-off $L=H^{-1}$. Also
from Eq. (\ref{qPLECHDE}) we obtain $q=1/2$. Therefore, choosing the
Hubble horizon as the IR cut-off $L=H^{-1}$ for the HDE model yields
a wrong EoS parameter and cannot drive the universe to accelerated
expansion.
Whereas in the presence of the power-law correction term, for the
flat FRW universe with $L=\tilde{r}_A=H^{-1}$ from Eq.
(\ref{HLPLECHDE}) we have $\gamma_c=\Omega_D/c^2$. Now if we take
$\varepsilon=b^2=\Omega_r=0$ then Eqs. (\ref{wDPLECHDE}) and
(\ref{qPLECHDE}) for the present time, $\tilde{r}_{\rm
A}=\tilde{r}_{\rm A_0}$, reduce to
\begin{equation}
\omega_{D_0}=-1-\left(\frac{1}{\Omega_{D_0}}-1\right)\left(1-\frac{1}{1-c^2-\Big(\frac{\alpha}{2-\alpha}\Big)\Omega_{D_0}}\right),
\label{omegaD0}
\end{equation}
\begin{equation}
q_0=-1+\frac{3(1-\Omega_{D_0})}{(2-\alpha)(1-c^2)-\alpha\Omega_{D_0}},\label{q0}
\end{equation}
where the subscript ``0'' denotes the present values of the
quantities. Taking $\Omega_{D_0}=0.73$ \cite{Riess} and $c=0.818$
\cite {Li6} then Eqs. (\ref{omegaD0}) and (\ref{q0}) yield
\begin{equation}
\omega_{D_0}=-1.021+\frac{0.480}{0.624-\alpha},\label{omegaD00}
\end{equation}
\begin{equation}
q_0=-\left(\frac{\alpha+0.139}{\alpha-0.624}\right).\label{q00}
\end{equation}
The above relations show that for $\alpha>0.624$ we have
$\omega_{D_0}<-1$ and $q_0<0$. Here $\omega_{D_0}<-1$ clears that
the PLECHDE with the IR cut-off $L=H^{-1}$ behaves like phantom DE.
Recent astronomical data indicates that the EoS parameter
$\omega_{D_0}$ at the present lies in a narrow strip around
$\omega_{D_0} = -1$ and is quite consistent with being below this
value \cite{Komatsu}. On the other hand, the satisfaction of the GSL
of gravitational thermodynamics for the universe with the power-law
corrected entropy (\ref{ec}) implies that $\alpha<2$ (see Appendix
A). Hence for the PLECHDE model with $0.624<\alpha<2$, the
identification of IR cut-off with the Hubble horizon $L=H^{-1}$, can
derive a phantom accelerating universe which is compatible with the
observations.
\section{Conclusions}
Here, we considered a power-law quantum correction to the entropy of
the dynamical apparent horizon motivated by the entanglement of
quantum fields between inside and outside of the horizon. Using the
Clausius relation we obtained the modified Friedmann equation. For a
non-flat modified FRW universe filled with an interacting viscous DE
with DM and radiation, we obtained the EoS parameter of interacting
viscous DE as well as the deceleration parameter. We studied the
effect of the power-law correction term to the entropy on the
dynamics of DE in our selected model for the universe. Interestingly
enough, we found that for the PLECHDE model which is the power-law
entropy-corrected version of the HDE model, the identification of IR
cut-off with Hubble horizon, $L=H^{-1}$, can lead to a phantom
accelerating universe. This is in contrast to the ordinary HDE where
$\omega_D=0$ if one chooses $L=H^{-1}$. Furthermore, we investigated
the validity of the GSL of thermodynamics on the apparent horizon.
We found out that the GSL of thermodynamics with power-law corrected
entropy-area relation $S_{\rm A}=\frac{A}{4}\left[1-K_{\alpha}
A^{1-\frac{\alpha}{2}}\right]$ is satisfied for $\alpha<2$.
\\
\\
\noindent{\textbf{Acknowledgements}\\ The authors thank the
anonymous referee for a number of valuable suggestions. The authors
also thank Professor Rabin Banerjee for useful discussions. The work
of K. Karami has been supported financially by Research Institute
for Astronomy and Astrophysics of Maragha, Iran.
\begin{appendix}
\section{GSL with power-law corrected entropy}
Here, we study the validity of the GSL of thermodynamics for the
power-law entropy-corrected Friedmann equation. According to the
GSL, entropy of the viscous DE, DM and radiation inside the horizon
plus the entropy associated with the horizon must not decrease in
time.
Taking a time derivative in both sides of Eq. (\ref{ra}), and using
Eqs. (\ref{econs}), (\ref{eqf1}), (\ref{Omega1}), (\ref{Omega2}),
(\ref{eqf2}), (\ref{rho}) and (\ref{p2}), one can get
\begin{equation}
{\dot{\tilde{r}}_{\rm
A}}=\frac{(1+\Omega_k)^{-1/2}}{2\left[1+\Omega_k-\frac{\alpha}{2}\Big(\Omega_\alpha+(Hr_c)^{-2}\Big)\right]}\Big[3\Omega_D(1+\omega_D-3\varepsilon)
+3\Omega_m+4\Omega_r\Big].\label{radot1}
\end{equation}
Using Eq. (\ref{q2}) one can rewrite (\ref{radot1}) as
\begin{equation}
{\dot{\tilde{r}}_{\rm
A}}=\frac{1+\Omega_k+q}{(1+\Omega_k)^{3/2}}.\label{radot2}
\end{equation}
From Eqs. (\ref{ec}) and (\ref{TA1}), the evolution of the apparent
horizon entropy is obtained as
\begin{equation}
T_{\rm A}\dot{S}_{\rm A}=4\pi H{\tilde{r}}_{\rm
A}^3(\rho+p)-2\pi{\tilde{r}}_{\rm A}^2{\dot{\tilde{r}}}_{\rm
A}(\rho+p).\label{em}
\end{equation}
The entropy of the universe including the viscous DE, DM and
radiation inside the dynamical apparent horizon can be related to
its energy and pressure in the horizon by Gibb's equation
\cite{Izquierdo2}
\begin{equation}
T{{\rm d}S}={\rm d}(\rho V)+p~{\rm d}V=V{\rm d}\rho+(\rho+p){\rm
d}V,\label{Gibbs}
\end{equation}
where ${\rm V}=4\pi \tilde{r}_{\rm A}^3/3$ is the volume of the
universe enclosed by the dynamical apparent horizon $\tilde{r}_{\rm
A}$. Following \cite{Sheykhi2,Karami1}, we assume that the
temperature $T$ of the universe enclosed by the dynamical apparent
horizon should be in equilibrium with the Hawking temperature
$T_{\rm A}$ associated with the dynamical apparent horizon, so we
have $T = T_{\rm A}$. Therefore from Eq. (\ref{Gibbs}) one can
obtain
\begin{equation}
T_{\rm A}\dot{S}=4\pi{\tilde{r}}_{\rm A}^2{\dot{\tilde{r}}_{\rm
A}}(\rho+p) -4\pi H{\tilde{r}}_{\rm A}^3(\rho+p),\label{en}
\end{equation}
where $S=S_D+S_m+S_r$ is the entropy in the universe containing the
viscous DE, DM and radiation. Finally, adding Eqs. (\ref{em}) and
(\ref{en}), the GSL due to the different contributions of the
viscous DE, DM, radiation and dynamical apparent horizon can be
obtained as
\begin{equation}
T_{\rm A}\dot{S}_{\rm {tot}}=2\pi{\tilde{r}}_{\rm
A}^2{\dot{\tilde{r}}_{\rm A}}(\rho+p),\label{eu}
\end{equation}
where $S_{\rm tot}=S+S_{\rm A}$ is the total entropy.
From Eqs. (\ref{rho}), (\ref{p2}) and using (\ref{Omega1}) one can
obtain
\begin{equation}
\rho+p=\frac{H^2}{8\pi}\Big[3\Omega_D(1+\omega_D-3\varepsilon)
+3\Omega_m+4\Omega_r\Big].\label{rhop1}
\end{equation}
Using Eq. (\ref{q2}) one can rewrite (\ref{rhop1}) as
\begin{equation}
\rho+p=\frac{H^2}{4\pi
}\left(\frac{1+\Omega_k+q}{1+\Omega_k}\right)\left[1+\Omega_k-\frac{\alpha}{2}\left(\Omega_\alpha+(Hr_c)^{-2}\right)\right].\label{rhop2}
\end{equation}
Substituting Eqs. (\ref{ra}), (\ref{radot2}) and (\ref{rhop2}) into
(\ref{eu}) yields the GSL as
\begin{equation}
T_{\rm A}\dot{S}_{\rm
{tot}}=\frac{(1+\Omega_k+q)^2}{2(1+\Omega_{k})^{7/2}}\left[1+\Omega_k-\frac{\alpha}{2}\left(\Omega_\alpha+(Hr_c)^{-2}\right)\right],\label{GSL3}
\end{equation}
which can be rewritten by the help of Eq. (\ref{q2}) as
\begin{equation}
T_{\rm A}\dot{S}_{\rm
{tot}}=\frac{\Big[3\Omega_D(1+\omega_D-3\varepsilon)
+3\Omega_m+4\Omega_r\Big]^2}{8(1+\Omega_{k})^{3/2}\left[1+\Omega_k-\frac{\alpha}{2}\Big(\Omega_\alpha+(Hr_c)^{-2}\Big)\right]}.\label{GSL4}
\end{equation}
By substituting $\Omega_\alpha$ from Eq. (\ref{Omega2}) in the above
equation we get
\begin{equation}
T_{\rm A}\dot{S}_{\rm
{tot}}=\frac{\Big[3\Omega_D(1+\omega_D-3\varepsilon)
+3\Omega_m+4\Omega_r\Big]^2}{8(1+\Omega_{k})^{5/2}\left[1-\frac{\alpha}{2}\left(\frac{\tilde{r}_{\rm
A}}{r_c}\right)^{2-\alpha}\right]}.\label{GSL5}
\end{equation}
According to Eq. (\ref{GSL5}), the validity of GSL, i.e. $T_{\rm
A}\dot{S}_{\rm {tot}}>0$, depends on the sign of the expression
$\left[1-\frac{\alpha}{2}\left(\frac{\tilde{r}_{\rm
A}}{r_c}\right)^{2-\alpha}\right]$ appearing in the denominator.
Hence the GSL is hold when
\begin{equation}
1-\frac{\alpha}{2}\left(\frac{\tilde{r}_{\rm
A}}{r_c}\right)^{2-\alpha}=1-\frac{\alpha}{2}\left(\frac{\tilde{r}_{\rm
A}}{\tilde{r}_{{\rm A}_0}}\right)^{2-\alpha}>0,\label{con}
\end{equation}
where following \cite{Saurya,Dvali} we identify the crossover scale
$r_c$ with the present value of the apparent horizon
$\tilde{r}_{{\rm A}_0}$. Since $\tilde{r}_{\rm A}/\tilde{r}_{{\rm
A}_0}$ tends to zero in the far past and its today value is
$\tilde{r}_{\rm A}/\tilde{r}_{{\rm A}_0}=1$, hence for $0\leq
\tilde{r}_{\rm A}/\tilde{r}_{{\rm A}_0}\leq 1$ the condition
(\ref{con}) is satisfied when $\alpha<2$. Therefore the GSL with the
power-law corrected entropy (\ref{ec}) is respected for $\alpha<2$.
\end{appendix}
|
{
"timestamp": "2011-08-23T02:00:57",
"yymm": "1009",
"arxiv_id": "1009.3833",
"language": "en",
"url": "https://arxiv.org/abs/1009.3833"
}
|
\section{Introduction}
Magnetic fields in the Sun and late-type stars produce active regions in the photosphere by affecting the transfer of energy and momentum in the outermost convective layers. These features consist of dark spots, bright faculae, and an enhanced network of magnetic flux tubes with radii down to about 100 km and kG field strength. Solar active regions can be studied in details thanks to a spatial resolution down to $\sim 100$~km and a time resolution better than $\sim 1$~s.
In distant stars, we lack spatial resolution (with the exception of the supergiant $\alpha$ Orionis, see \cite[Dupree 2010]{Dupree10}), so we must apply indirect techniques to map their photospheres. The most successful is Doppler Imaging that produces a two-dimensional map of the surface from a sequence of high-resolution line profiles, sampling the different rotation phases of the star (see, e.g., \cite[Strassmeier 2009, 2010]{Strassmeier09,Strassmeier10}). Its application requires that the rotational broadening of the spectral lines exceeds the macroturbulence by at least a factor of $4-5$, implying a minimum $ v \sin i \sim 10-15$~km~s$^{-1}$ in dwarf stars. Therefore, late-type stars that rotate as slowly as the Sun cannot be imaged by this technique. For those stars, we derive the distribution of brightness inhomogeneities from the rotational modulation of the flux produced by cool spots and bright faculae that come into and out of view as the star rotates. By comparing successive rotations, it is possible to study the time evolution of the active regions.
Another method exploits the tiny light modulations produced by the occultation of starspots by a transiting giant planet moving in front of the disc of its star (e.g. \cite[Silva-Valio et al. 2010; Wolter et al. 2009]{SilvaValioetal10,Wolteretal09}). It is a specialized version of the eclipse mapping technique, developed for the active components of close binary stars (e.g., \cite[Collier Cameron 1997; Lanza et al. 1998)]{CollierCameron97,Lanzaetal98}. Its application will be reviewed by \cite[Silva-Valio (2010)]{SilvaValio10}.
Stellar differential rotation and activity cycles can be studied also through techniques of time series analysis applied to sequences of seasonal optical photometry; see, e.g., \cite[Messina \& Guinan (2003); Koll\'ath \& Ol\'ah (2009); and Ol\'ah (2010)]{MessinaGuinan03,KollathOlah09,Olah10}.
In this review, I shall briefly report on the modelling of the light curves of some planet-hosting stars as observed by the space experiment CoRoT.
\section{Space-borne optical photometry}
Space-borne optical photometry has provided time series for several late-type stars spanning from several days up to several months thanks to the space missions MOST \cite[(Walker et al. 2003)]{Walkeretal03}, CoRoT \cite[(Auvergne et al. 2009)]{Auvergneetal09}, and Kepler \cite[(see, e.g., Haas et al. 2010; Ciardi et al. 2010)]{Haasetal10,Ciardietal10}.
MOST (the Microvariability and Oscillation of Stars satellite) has a telescope of 15~cm aperture that can observe a given target for up to $40-60$ days reaching a photometric precision of $50-100$ ppm (parts per million) on the brightest stars ($V \leq 4$). It has observed $\epsilon$~Eridani \cite[(Croll et al. 2006)]{Crolletal06} and k$^1$~Ceti \cite[(Walker et al. 2007)]{Walkeretal07} whose light curves have been modelled to extract information on stellar differential rotation.
CoRoT (Convection, Rotation and Transits) is a space experiment devoted to asteroseismology and the search for extrasolar planets through the method of transits. With its 27-cm aperture telescope, it can observe up to 12,000 targets per field for intervals of 150 days searching for transit signatures in the light curves. Its white-band light curves have a sampling of 32 or 512~s and a bandpass ranging from 300 to 1100~nm. A photometric accuracy of $\sim 100$~ppm is achieved for a G or K-type star of $V \sim 12$ by integrating the flux over individual orbits of the satellite. CoRoT provides some chromatic information on the light variability of its brightest targets ($V < 14.5$).
Kepler, launched in March 2009, has a telescope of 95~cm aperture and is continuously monitoring $\sim 100,000$ dwarfs in a fixed field of view for a time interval of at least 3.5 years, searching for planetary transits. Its accuracy reaches $\sim 30$~ppm in 1 hour integration on a G2V target of $V \sim 12$.
Stars with transiting giant planets allow us to derive the inclination of the stellar rotation axis, which is important to modelling their light curves for stellar activity studies (see, e.g., \cite[Mosser et al. 2009]{Mosseretal09}). Specifically, by fitting the transit light curve, we can measure the inclination of the planetary orbit along the line of sight, which is equal to the inclination of the stellar rotation axis if the spin and the orbital angular momentum are aligned. This hypothesis can be tested with the so-called Rossiter-McLaughlin effect, i.e., the apparent anomaly in the radial velocity of the star observed during the planetary transit that allows us to measure the angle between the projections of the stellar spin and the orbital angular momentum on the plane of the sky \cite[(Winn et al. 2005)]{Winnetal05}.
Another advantage of stars with transiting planets is that their fundamental parameters have been well determined because accurate stellar masses and radii are needed to derive accurate planetary parameters from the transit modelling.
\section{Light curve modelling}
Spot models of the light curves of late-type stars observed by MOST and CoRoT have already been published, while the modelling of Kepler light curves has just started \cite[(e.g., Brown et al. 2010)]{Brownetal10}. MOST time series were exploited to measure stellar differential rotation by fitting the wide-band light modulation with a few circular spots with fixed contrast. Thanks to such assumptions, it was possible to derive the latitudes of the spots and measure the variation of the angular velocity vs. latitude
\cite[(Croll et al. 2006; Walker et al. 2007)]{Crolletal06,Walkeretal07}.
For a generalization of that approach with the CoRoT light curves, see \cite[Mosser et al. (2009)]{Mosseretal09}.
In the Sun, active regions consist not only of cool spots but also of bright faculae whose contrast is maximum close to the limb and minimum at the disc centre. Moreover, the optical variability of the Sun is dominated by several active regions at the same time making a model based on a few spots poorly suitable to reproduce its active region pattern.
\cite[Lanza et al. (2007)]{Lanzaetal07} used the time series of the Total Solar Irradiance (TSI, e.g., \cite[Fr\"ohlich \& Lean 2004]{FrohlichLean04}) as a proxy for the solar optical light curve to test different modelling approaches assuming that the active regions consist of dark spots and bright faculae with fixed contrasts and in a fixed area proportion. A model with a continuous distribution of active regions and the maximum entropy regularization, to warrant the uniqueness and stability of the solution, is the most suitable and reproduces the distribution of the area of the sunspot groups vs. longitude with a resolution better that $\sim 50^{\circ}$, as well as the variation of the total spotted area vs. time. It allows a highly accurate reproduction of the TSI variations with a typical standard deviation of the residuals of $\sim 30-35$ ppm for time intervals of 14 days. However, the value of the faculae-to-spotted area ratio is a critical parameter because it affects the derived distribution of the active regions vs. longitude (see, \cite[Lanza et al. 2007]{Lanzaetal07}, for a detailed discussion).
The maximum entropy spot model tested in the case of the Sun has been applied to CoRoT light curves to derive the distribution of the stellar active regions vs. longitude and the variation of their total area vs. time. In general, information on the latitudes of stellar active regions cannot be extracted from a one-dimensional data set such as an optical light curve. Moreover, since the inclination of the rotation axis of the stars with transiting planets is generally close to $90^{\circ}$, it is impossible to constrain the spot latitudes because the duration of the transit of a spot across the stellar disc is independent of its latitude.
\section{Results from CoRoT light curves}
\cite[Lanza et al. (2009a)]{Lanzaetal09a} model the out-of-transit light curve of CoRoT-2, a G7V star with a giant planet with a mass of 3.3 Jupiter masses and an orbital period of 1.743~days \cite[(Alonso et al. 2008; Bouchy et al. 2008)]{Alonsoetal08,Bouchyetal08}. Since the spot pattern is evolving rapidly, they model individual intervals of 3.15 days along a sequence of 142~days. The star has a light curve amplitude of 0.06 mag, i.e., about 20 times that of the Sun at the maximum of the 11-yr cycle. Solar-like contrasts for the spots and the faculae are adopted.
The distribution of the spotted area vs. longitude and time is plotted in Fig.~4 of \cite[Lanza et al. (2009a)]{Lanzaetal09a}, here reproduced in Fig.~\ref{lanza_fig1}. The active regions are mainly found within two active longitudes, initially separated by $\sim 180^{\circ}$. The longitude initially around $0^{\circ}$ does not migrate, i.e., it rotates with the same period as the adopted reference frame, while the other longitude, initially around $180^{\circ}$, migrates backward, i.e., it rotates slower than the reference frame by $\sim 0.9$ percent. Individual active regions also migrate backward as they evolve, i.e., they rotate slower than the active longitude to which they belong, with a maximum difference of $\approx 2$ percent. The relative migration of the active longitudes can be interpreted as a consequence of their different latitudes on a differentially rotating star, yielding a lower limit of 0.9 percent for the amplitude of the differential rotation \cite[(Lanza et al. 2009a)]{Lanzaetal09a}. On the other hand, the backward migration of the individual active regions can be regarded as analogous to the braking of the rotation of solar active regions as they evolve because the relative amplitude of the angular velocity variation is remarkably similar (e.g., \cite[Zappal\`a \& Zuccarello 1991; Sch\"ussler \& Rempel 2005]{ZappalaZuccarello91,SchuesslerRempel05}).
Other authors have suggested a greater amplitude for the differential rotation of CoRoT-2, up to $\sim 8$ percent, from the migration of individual active regions \cite[(see Fr\"ohlich et al. 2009; Huber et al. 2010)]{Froehlichetal09,Huberetal10}. \cite[Savanov (2010)]{Savanov10} notices that the active regions appear in an alternate way in the two active longitudes, suggesting a short-term flip-flop phenomenon, reminiscent of the flip-flop cycles in some active stars that, however, have timescales of several years \cite[(Berdyugina \& Tuominen 1998; Berdyugina 2005)]{BerdyuginaTuominen98,Berdyugina05}.
\begin{figure}[]
\begin{center}
\includegraphics[width=1.7in,height=3.8in,angle=90]{lanza.ps}
\caption{The isocontours of the spot filling factor vs. longitude and time for CoRoT-2 after \cite[Lanza et al. (2009a)]{Lanzaetal09a}. The longitude reference frame rotates with the star with a period of 4.5221 days and the longitude increases in the same direction of the stellar rotation. The longitude scale has been extended beyond $0^{\circ}$ and $360^{\circ}$ (marked by the vertical dashed lines) to help follow the migration of the spots. In the electronic version, different colours indicate different relative filling factors of the starspots; from the minimum to the maximum: green, light green, light blue, blue, light pink, pink, and red. }
\label{lanza_fig1}
\end{center}
\end{figure}
The variation of the total spotted area shows remarkable oscillations with a cycle of $ \sim 29$ days (see Fig.~6 of \cite[Lanza et al. 2009a]{Lanzaetal09a}). In the Sun, short-term oscillations of the total spotted area have been observed close to the maximum of some of the 11-yr cycles and are called Rieger cycles. They have periods around 160 days, i.e., about five times longer than the cycles observed in CoRoT-2 \cite[(Oliver et al. 1998; Krivova \& Solanki 2002; Zaqarashvili et al. 2010)]{Oliveretal98,KrivovaSolanki02,Zaqarashvilietal10}. \cite[Lou (2000)]{Lou00} suggests that they may be due to hydromagnetic Rossby-type waves trapped in the solar convection zone. Since the wave frequency is proportional to the rotation frequency of the star, the expected period is close to that observed in CoRoT-2 because this star rotates five times faster than the Sun.
The approach introduced for CoRoT-2 has been applied to other late-type planet-hosting stars, viz. CoRoT-4 \cite[(Lanza et al. 2009b)]{Lanzaetal09b}, CoRoT-6 \cite[(Lanza et al. 2010a)]{Lanzaetal10a}, and CoRoT-7 \cite[(Lanza et al. 2010b)]{Lanzaetal10b}. From the migration of their active longitudes, a lower limit for the amplitude of their latitudinal differential rotation has been obtained. The results are listed in Table~\ref{lanza_table1}, together with those derived from MOST photometry and those by \cite[Mosser et al. (2009)]{Mosseretal09} for two of the CoRoT asteroseismic targets. In Table~\ref{lanza_table1}, the columns from left to right list the name of the star, its effective temperature $T_{\rm eff}$, its mean rotation period $P_{\rm rot}$, the relative amplitude of the differential rotation $\Delta \Omega / \Omega$, and the references.
A comparison with the differential rotation amplitudes as derived from Doppler Imaging or the Fourier transform of the spectral line profiles (\cite[Reiners 2006]{Reiners06}) shows that the values derived from the migration of the active longitudes are generally smaller than those expected in stars with the same effective temperature and rotation rates, by a typical factor of $2-3$. This suggests that their active regions are mostly localized at low latitudes, as in the case of the Sun.
\begin{table}
\begin{center}
\caption{Stellar differential rotation from spot modelling of space-borne photometry.}
\label{lanza_table1}
{\scriptsize
\begin{tabular}{|c|c|c|c|c|}
\hline
Star & $T_{\rm eff}$ & $P_{\rm rot}$ & $\Delta \Omega / \Omega $ & References \\
& (K) & (days) & & \\
\hline
$\epsilon $~Eridani & 4830 & 11.45 & $0.11 \pm 0.03^{*}$ & \cite[Croll et al. (2006)]{Crolletal06} \\
CoRoT-7 & 5275 & 23.64 & $0.058 \pm 0.017$ & \cite[Lanza et al. (2010b)]{Lanzaetal10b} \\
k$^{1}$~Ceti & 5560 & 8.77 & $0.09 \pm 0.006^{*}$ & \cite[Walker et al. (2007)]{Walkeretal07} \\
CoRoT-2 & 5625 & 4.52 & $\sim 0.009-0.08$ & \cite[Lanza et al. (2009a)]{Lanzaetal09a}; \\
& & & & \cite[Fr{\"o}hlich et al.(2009)]{Frohlichetal09} \\
HD~175726 & 6030 & 3.95 & $\approx 0.40^{*}$ & \cite[Mosser et al. (2009)]{Mosseretal09} \\
CoRoT-6 & 6090 & 6.35 & $0.12 \pm 0.02$ & \cite[Lanza et al. (2010a)]{Lanzaetal10a} \\
CoRoT-4 & 6190 & 9.20 & $0.057 \pm 0.015$ & \cite[Lanza et al. (2009b)]{Lanzaetal09b} \\
HD~181906 & 6360 & 2.71 & $\approx 0.25^{*}$ & \cite[Mosser et al. (2009)]{Mosseretal09} \\
\hline
\end{tabular}
}
\end{center}
\vspace{1mm}
\scriptsize{
{\it Note:}\\
$^{*}$Value of the $K$ coefficient estimated for a solar-like differential rotation law $P(\phi) = P_{\rm eq}/(1-K \sin^{2} \phi)$, where $\phi $ is the latitude, $P(\phi)$ the rotation period at latitude $\phi$, and $P_{\rm eq}$ the rotation period at the equator. \\
}
\end{table}
\section{The possible case for a magnetic star-planet interaction}
The planets of CoRoT-2, CoRoT-4, and CoRoT-6 are hot Jupiters, i.e., giant planets orbiting within 0.15 AU from their host stars. They interact tidally and possibly magnetically with their stars, which may lead to observable effects on stellar activity (see, e.g., \cite[Cuntz et al. 2000; Lanza 2008; 2009]{Cuntzetal00,Lanza08,Lanza09}). Current evidence of star-planet magnetic interaction (hereafter SPMI) is limited to the modulation of the chromospheric flux with the orbital phase of the planet in a few stars and in some seasons \cite[(Shkolnik et al. 2005, 2008)]{Shkolniketal05,Shkolniketal08}. Evidence of a coronal flux enhacement is much more controversial (cf. \cite[Kashyap et al. 2008]{Kashyapetal08} and \cite[Poppenhaeger et al. 2010]{Poppenhaegeretal10}), although some possible cases have been presented \cite[(Saar et al. 2008; Pillitteri et al. 2010)]{Saaretal08,Pillitterietal10}. SPMI features in the photosphere have been proposed for $\tau$ Bootis \cite[(Walker et al. 2008)]{Walkeretal08}, CoRoT-2 \cite[(Pagano et al. 2009)]{Paganoetal09}, and, possibly, HD~192263 \cite[(Santos et al. 2003)]{Santosetal03}. The mean rotation of $\tau$ Boo is synchronized with the orbital motion of its giant planet, so a modulation of its optical flux with the orbital period of the planet cannot be unambiguously attributed to SPMI.
Nevertheless, \cite[Walker et al. (2008)]{Walkeretal08} found an active region on the star which lasted for at least $\sim 500$ stellar rotations, i.e., 5 years, always leading the subplanetary meridian by $\sim 70^{\circ}$. The persistence of such a feature strongly suggests a connection with the planet.
\cite[Lanza et al. (2009b)]{Lanzaetal09b} suggest a similar phenomenon in the other synchronous system CoRoT-4, finding an active region located at the subplanetary longitude that has persisted for $\sim 70$ days. Even more intriguing is the case of CoRoT-6, a non-synchronous system with a planetary orbital period of 8.886 days and a mean stellar rotation period of 6.35 days. Assuming a longitude reference frame rotating with the mean stellar rotation period, the maximum of the spot filling factor in several active regions occurs when they cross a meridian at $-200^{\circ}$ from the subplanetary meridian. The probability of a chance occurrence is only $\sim 0.8$ percent (\cite[Lanza et al. 2010a]{Lanzaetal10a}).
It is difficult to find a mechanism for the allegedly supposed influence of the planet on the formation of stellar active regions. \cite[Lanza (2008)]{Lanza08} conjectured that the reconnection of the stellar coronal field with the magnetic field of the planet may induce a longitudal dependence of the hydromagnetic dynamo action in the star,
provided that some of the spot magnetic flux tubes come from the subphotospheric layers, as suggested by \cite[Brandenburg (2005)]{Brandenburg05}. Nevertheless, further observations of the photospheric SPMI are needed firstly to confirm the reality of the phenomenon and secondly to derive its dependence on stellar and planetary parameters.
|
{
"timestamp": "2010-09-22T02:02:15",
"yymm": "1009",
"arxiv_id": "1009.4099",
"language": "en",
"url": "https://arxiv.org/abs/1009.4099"
}
|
\section{Introduction}
The structure of the $^9$Be nucleus has been considered as a prototype
of cluster structure in nuclei. It has been described theoretically by
many different cluster models and several experiments have been
performed in order to understand its structure and decay
mechanisms. Much effort has been devoted to the $5/2^-$ state due to
its astrophysical importance in the formation of $^9$Be, $^{12}$C and
heavier elements in stellar nucleosynthesis\cite{pap07,alv08,bur10}.
$^9$Be and $^9$B are mirror nuclei that decay into $\alpha\alpha n$
and $\alpha \alpha p$ respectively. Therefore their structures are
expected to be very similar. We present here the results of
two-dimensional energy correlations after the three-body decay of
several $^9$Be resonances. Moreover the same kind of results for the
$5/2^-$-resonance of $^9$B are shown in order to show the
similarities.
\section{Theoretical Description Of Three-Body Resonances}
We employ here the complex-scaled hyperspherical adiabatic expansion
method to solve the Faddeev equations which describe our three-body
system\cite{nie01}. The angular part of the Hamiltonian is first
solved keeping fixed the value of the hyperradius $\rho$. Its
eigenvalues serve as effective potentials while the eigenfunctions,
$\Phi_{nJM}$ are used as a basis to expand the total wave-function
$\Psi^{JM} = \frac{1}{\rho^{5/2}}\sum_n f_n (\rho) \Phi_{nJM}
(\rho,\Omega)\;. $ The $\rho$-dependent expansion coefficients, $f_n
(\rho)$, are the hyperradial wave functions obtained from the coupled
set of hyperradial equations\cite{nie01}.
We consider a two-body interaction able to reproduce the low-energy
scattering properties of the two different pairs of particles in our
three-body system. Ali-Bodmer $\alpha-\alpha$ potential\cite{ali66}
and Coulomb potential between $\alpha$-particles are considered. The
$\alpha$-nucleon interaction is taken from Cobis et
al\cite{cob97}. The $^{9}$Be- and $^9$B-resonances are of three-body
character at large-distances, where they decay into two
$\alpha$-particles and one neutron or proton, but this is not
necessarily the case at short-distances.\cite{gar10} We use the
three-body model at all distances because the decay properties only
require the proper description of the emerging three particles. A
three-body short-range potential of the form $V_{3b} =
S\exp(-\rho^2/b^2)$ is included to adjust the corresponding
small-distance part of the effective potential. The correct resonance
energies, which are all-decisive for decay details as evident in the
probability for tunneling through a barrier, are then correctly
reproduced.
The eigenvalues of the angular Hamiltonian for fixed hyperradius,
serve as adiabatic potentials\cite{alv10}. Each of them correspond to
a specific combination of quantum numbers, i.e., partial-wave momenta
between two particles in each Jacobi system. Usually few angular
eigenvalues are needed for achieving convergence. At small distances
the potentials have wells that support the bound states and
resonances.
\section{Energy Distributions}
The energy distribution is the probability for finding a given
particle at a given energy. It can be measured experimentally and is
the only information that allows us to study the decay path. From the
theoretical point of view, this information is contained in the
large-distance part of the wave function, which must therefore be
computed accurately. The Zeldovic regularized Fourier transform of the
wave function gives the energy distributions\cite{fed04}. The
resonance wave functions change sometimes substantially from small to
large distances. This dynamic evolution allows a better understanding
of the decays details\cite{alv08}.
The decay mechanisms depend on the resonance properties and can be
either sequential or direct or a mixture. In our case, all of the
resonances can decay sequentially via $^8$Be($0^+$), i.e. it is
allowed by angular momentum conservation. One of the adiabatic
components is related to the $^8$Be+$n$ structure and approaches the
complex energy of the $^8$Be($0^+$) resonance. This component will give
the sequential contribution\cite{alv10}.
\subsection{$^9$Be}
\begin{figure}[th]
\vspace*{-3pt}
\centerline{\psfig{file=dalitz_a_n.ps,width=6.2cm,angle=270}}
\vspace*{-3pt}
\caption{Dalitz plots for a) the $5/2^-$-resonance of $^9$Be at
2.43~MeV of excitation energy, b) the $1/2^-$-resonance of $^9$Be at
2.82~MeV, c) the $5/2^+$-resonance of $^9$Be at 3.03~MeV, d) the
$3/2^+$-resonance of $^9$Be at 4.69~MeV, e) the $3/2^-$-resonance of
$^9$Be at 5.59~MeV. The upper panels show on the X-axis the neutron
energy divided by the maximum possible, and on the Y-axis the
$\alpha$ energy divided by the maximum possible. The lower panels
show on X and Y the energy of the two $\alpha$-particles divided by
the maximum. The sequential decay via $^8$Be($0^+$) has been removed.}
\label{fig9be}
\end{figure}
Fig. \ref{fig9be} corresponds to the two-dimensional energy
correlations (or Dalitz plots) of the three fragments of $^9$Be after
the decay. The two possibilities, $\alpha-n$ and $\alpha-\alpha$, are
shown and the $J^\pi$ of each state is labeled in the figure. The
energies are given in units of their maximum values for each case,
i.e. $5/9E_{res}$ for the $\alpha$-particles and $8/9E_{res}$ for the
neutrons. In all the cases we have removed the sequential decay via
$^8$Be($0^+$). It is important to remark that the results are directly
comparable to measured distributions.
The first thing that we observe in fig.~\ref{fig9be} is that all the
plots are symmetric between $\alpha$'s energies. This symmetry is
necessary since the $\alpha$'s are identical particles. The graphs
corresponding to $\frac{5}{2}^-$ and $\frac{3}{2}^-$, are very similar
to each other. First, we do not observe zeroes in the Dalitz
plots.\cite{fyn09} This means that angular momentum conservation does
not forbid any energy combination. Second, we see that the density
increases towards higher $\alpha$ energies. This is due to the Coulomb
repulsion, which enlarges the charged particles energies while reduces
that of the neutron. The other two cases show more structure and have
zero probability points or curves.
\subsection{$^9$Be vs $^9$B}
\begin{figure}[th]
\vspace*{-3pt}
\centerline{\psfig{file=9be9b.ps,width=3.6cm,angle=270}}
\vspace*{-3pt}
\caption{Dalitz plots for a) the $5/2^-$-resonance of $^9$Be at
2.43~MeV of excitation energy, b) the $5/2^-$-resonance of $^9$B at
2.34~MeV. The sequential decay via $^8$Be($0^+$) has been removed.}
\label{fig9b}
\end{figure}
In our description the only difference between $^9$Be and $^9$B is the
existing Coulomb interaction between the $\alpha$-particle and the
proton in $^9$B. The structure of the resonances in both nuclei are
expected to be very similar. Fig. \ref{fig9b} shows the comparison
between the Dalitz plots corresponding to the $5/2^-$-resonance of
$^9$Be and $^9$B. The patterns are, in fact, almost indistinguishable.
\section{Summary And Conclusions}
We have described $^9$Be and $^9$B resonances as three clusters by
means of the complex-scaled hyperspherical adiabatic expansion method,
including short-range and Coulomb interactions. The two-dimensional
energy correlations of the decaying fragments are shown for the
low-lying resonances of $^9$Be. We compare one of them ($5/2^-$) to
the corresponding one in $^9$B and find, as we expected, an almost
identical distribution. Our distributions are open to experimental
tests.
\section*{Acknowledgements}
This work was partly supported by funds provided by DGI of MEC (Spain)
under the contracts FIS2008-01301 and FPA2007-62216. R.A.R. acknowledges
support from Ministerio de Ciencia e Innovaci\'on (Spain) under the
``Juan de la Cierva'' program.
|
{
"timestamp": "2010-09-22T02:01:32",
"yymm": "1009",
"arxiv_id": "1009.4029",
"language": "en",
"url": "https://arxiv.org/abs/1009.4029"
}
|
\section{Introduction}
\label{Intro}
The stability of polymer thin films is an important research subject
in polymer physics and materials science. On the one hand, the aim may be to obtain
a stable film, as in coatings and lubrification. On the other hand,
soft films are used for microstructuring,
where they are destabilized to yield well-designed patterns
that are used e.g. as a mould for further microfabrication processes.
In both cases it is crucial to understand the stabilizing and destabilizing
mechanisms that prevail in polymer films, which can be either internal
(Van der Waals forces due to reduced dimensions, internal stresses,
decomposition in mixtures) or external (external stresses, external fields).
In recent studies on spin-coated polymer films it became apparent that
thin films are prone to store residual stresses \cite{Croll1979,Reiter:2005}.
Such stresses are created due to the fast evaporation
process of the spin-coating process: as evaporation is fast,
the polymer chains do not have the time to reach their equilibrium
configurations and in the final, glassy state the film
has frozen-in non-equilibrium configurations that give rise to stresses.
If these stresses are not relaxed, e.g. by ageing or tempering the films,
they influence the film stability as recently shown
in dewetting experiments and discussed theoretically
\cite{Reiter:2005,Fretigny,Raphael:2006.1,Reiter:2007.2,FZER1}.
There it has been shown that stresses increase the initial
dewetting velocity and also strongly influence the long time dynamics
of the dewetting films.
In case the film does not dewet, the stresses may still lead to destabilization
\cite{Raphael:2006.3}, as they should give rise
to an Asaro-Tiller-Grinfeld instability \cite{asaro:72,grinfeld:86,grinfeld:93}.
This mechanism has been proposed for stressed solids
in contact with their melt or for solids which evolve via
surface diffusion. Its origin is the fact that the solid can relax
stress and lower its energy
by creating surface undulations.
For polymer thin films
the interplay
between residual stresses and other, e.g.~externally applied,
destabilization forces constitutes an interesting question
of importance for all further manipulations of freshly spin-coated films.
In this work we reformulate the energy approach usually used to describe the
Grinfeld instability in a way that highlights the connection
with other known instabilities in thin films.
We use the bulk elastostatic equations together with
a time-dependent kinematic boundary condition at the free interface.
A direct coupling term between the height of the polymer film
and the displacement field arises, which has not been discussed before
as it is less relevant in atomic solids. In polymer films, however, this coupling
should be present and important. Moreover this term establishes the connection
to other elastic instabilities, namely to a buckling-like instability
under compressive stress and, in the case of an externally applied field, to the elasto-electric instability
investigated by Sharma {\it et al.}~\cite{SharmaPRL,monch2001,Sharmalong,Sharma08}.
Finally the growth rate of the height of the polymer film
is derived in case of simultaneous action of stress and external field.
This result is briefly compared to recent experiments
concerning the electrohydrodynamic instability
of very viscous (high molecular weight) spin-coated
thin polymer films heated above the glass transition \cite{Barbero09,SteinerEPL}.
The work is organized as follows:
First, in section \ref{Grin_elastic} we recall the classical, energy-based
formulation of the Grinfeld mechanism.
In section \ref{nonlin}, we start from a nonlinear elastic theory, derive
the bulk elastic equations and investigate in section \ref{elastsharma}
the stability under stretch/compression.
In section \ref{grindiffuse} we show that by allowing surface diffusion
via a kinematic boundary condition for the height of the film,
the Grinfeld result is regained in a well-defined limit.
The coupling between height and displacement via the kinematic boundary
condition can influence the classical Grinfeld instability for intermediate stresses.
In section \ref{grinE} we add the external electric field to our description.
We regain the instability discussed in Sharma {\it et al.}~\cite{SharmaPRL}
in a certain limit. Moreover the full growth rate of the film height is calculated and
its consequences for experiments are briefly discussed.
\section{Grinfeld instability - classical way of calculation;
effects of boundary condition at the substate}
\label{Grin_elastic}
To start with we briefly review the classical treatment of the Grinfeld instability
of an elastic medium under uniaxial stress \cite{Nozbook,cantat:98}.
Usually a semi-infinite solid is investigated, but in view of the thin
film geometry we allow for a finite thickness $h_0$ of the film.
The known results for the semi-infinite case can then be obtained
by performing the limit $h_0$ towards infinity.
As the dynamics of this instability is energy-driven
-- the system can lower its energy by creating surface undulations --
all the information needed to describe the system is contained in the (free) energy of the system,
which has an elastic part, $E_{{\rm el}}$, and a surface part, $E_{{\rm surf}}$.
We chose the coordinate system in such a way that the free surface is at $z=0$,
see the sketch of the geometry in Fig.~\ref{fig1}.
\begin{figure}[t]
\centering
\vspace{.2cm}
\includegraphics[width=0.48\textwidth]{fig1.eps}
\caption{\label{fig1} Sketch of the geometry. The thickness of the film
spans from $-h_0$ to $\zeta(x)$ in $z$-direction. There is uniaxial
stress in $x$-direction, which can be either compressive (as shown) or tensile.}
\end{figure}
For simplicity we assume a plane strain situation \cite{Maugis}
where the uniaxial prestress $\sigma_0$ is taken along the $x$-axis.
Consequently, we consider an undulation of the surface along $x$ given by
\begin{eqnarray}
\zeta(x)=\varepsilon A\cos(kx)\,.
\end{eqnarray}
We assume either an infinite system or periodic boundary conditions in
$x$-direction. $k$ is the wave number of the perturbation,
$A$ its amplitude and $\varepsilon$ a small book-keeping parameter used in the following
when dealing with expansions.
The elastic energy of an linearly elastic solid can be
written via the stress field $\sigma_{\alpha\beta}$ as \cite{landau_el}
\begin{eqnarray}
E_{\rm el}=\dfrac{1}{2 E}\int
\left[ \left(1 + \nu \right) \sigma_{\alpha \beta}^{2} - \nu \sigma_{\alpha \alpha} \sigma_{\beta \beta} \right] dx\,dz\,,
\end{eqnarray}
with $E$ the elastic or Young's modulus.
Summation convention is implied for indices occurring twice ($\alpha,\beta=1..3$).
Using a plane strain approximation, one gets
\begin{eqnarray}\label{Eelplanestr}
E_{\rm el}=\dfrac{1}{2 \bar{E}}\int
\left[ \left(1 + \bar{\nu} \right) \sigma_{ij}^{2} - \bar{\nu} \sigma_{ii} \sigma_{jj} \right] dx\,dz\,,
\end{eqnarray}
where now $i,j=1..2$
($1\leftrightarrow x,2\leftrightarrow z$) and
$\bar{E}=\frac{E}{1-\nu^{2}}$ and $\bar{\nu}=\frac{\nu}{1-\nu}$.
Assuming incompressibility, i.e. a Poisson's ratio of $\nu=1/2$,
one gets
$\bar{E}=\frac{4}{3}E$, $\bar{\nu}=1$.
We also will use the shear modulus $G$ later on and note the known relations,
$G=\frac{E}{2(1+\nu)}=\frac{1-\nu^2}{2(1+\nu)}\bar{E}=\bar{E}/4$.
As the system is invariant
in $y$-direction, $E_{{\rm el}}$ has units of energy per unit length.
The second energy in the problem is the surface energy
\begin{eqnarray}\label{Esurf}
E_{{\rm surf}}=\gamma\int\left(\sqrt{1+\zeta'(x)^2}-1\right)dx\,,
\end{eqnarray}
where $\gamma$ is the surface tension and $E_{{\rm surf}}$ is measured
with respect to the state of a flat surface.
To evaluate the elastic energy, one has to solve the elastostatic problem.
The prestress is
uniaxial along the $x$-axis and given
by $\sigma^0_{xx}=\sigma_0$, $\sigma^0_{zz}=0$ and $\sigma^0_{xz}=\sigma^0_{zx}=0$.
Note that $\sigma_0<0$ holds for the case of a compressive stress
and $\sigma_0>0$ in case of a tensile stress.
Undulations of the surface will give rise to an additional
relaxational stress $\tilde{\sigma}_{ij}$.
The total stress $\sigma_{ij}=\sigma^0_{ij}+\tilde{\sigma}_{ij}$
has to fulfill the Cauchy equilibrium equation
\begin{eqnarray}\label{eqeq}
\nabla_i\sigma_{ij}=0
\end{eqnarray}
and the compatibility equation
\begin{eqnarray}
\nabla^2(\sigma_{xx}+\sigma_{zz})=0\,
\end{eqnarray}
where $\nabla=(\partial_x,\partial_z)$. As the prestress $\sigma^0_{ij}$ trivially
fulfills these equations, we
introduce the Airy stress function $\chi(x,z)$ for the relaxational stress
via the known relations \cite{landau_el,Maugis}
\begin{eqnarray}\label{stress_via_chi}
\tilde{\sigma}_{xx}=\frac{\partial^2\chi}{\partial z^2}\,\,\,,\,
\tilde{\sigma}_{zz}=\frac{\partial^2\chi}{\partial x^2}\,\,\,,\,
\tilde{\sigma}_{xz}=-\frac{\partial^2\chi}{\partial x \partial z}\,\,\,.\,
\end{eqnarray}
The equilibrium equation is then automatically fulfilled and the compatibility
reduces to $\nabla^2\nabla^2\chi=0$.
This biharmonic equation has to be solved with the following boundary conditions
(BC).
At the free surface $z=\zeta(x)$, the normal-normal component of stress has to
balance the surface tension,
while the shear stress has to vanish.
With $\hat{\mathbf{n}}$ and $\hat{\mathbf{t}}$ denoting the unit vectors normal and tangential to the surface, respectively,
the BC at the free surface read
\begin{eqnarray}
\hat{n}_{i}\sigma_{ij}\hat{n}_{j}=
\gamma\,\dfrac{\zeta''(x)}{[1+\zeta'(x)^{2}]^{3/2}}\,\,,\,\,\,\,
\label{BCgrin}
\hat{t}_{i}\sigma_{ij}\hat{n}_{j}=0 \,,
\end{eqnarray}
or explicitly
\begin{eqnarray}
(\sigma_0+\tilde{\sigma}_{xx})\zeta'^{2}-2\tilde{\sigma}_{xz}\zeta'+\tilde{\sigma}_{zz}
&=&
\gamma\,\dfrac{\zeta''}{[1+\zeta'^{2}]^{1/2}},\quad\nonumber\\
\label{BCfree}
-\tilde{\sigma}_{xz}\zeta'^{2}+\zeta' \left( \tilde{\sigma}_{zz}-\sigma_0-\tilde{\sigma}_{xx}\right)+\tilde{\sigma}_{xz}
&=&0\,.
\end{eqnarray}
Note that all the stresses in Eqs.~(\ref{BCgrin}, \ref{BCfree})
have to be evaluated at the interface, i.e. at $z=\zeta(x)$.
At the bottom surface $z=-h_{0}$, where $h_{0}$ is the
film thickness,
we impose vanishing normal displacement
\begin{eqnarray}\label{BCbot1}
u_z=0 \quad{\rm at}\quad z=-h_{0}\,,
\end{eqnarray}
meaning that the film is not allowed to detach from the substrate.
As the second BC, we study two possibilities, depending on the preparation of the system:
First, to study the case of possible slippage at the
lower interface, one prescribes
\begin{eqnarray}\label{BCbot2}
{\rm slip\,\,BC:}\quad\quad\sigma_{xz}=0 \quad{\rm at}\quad z=-h_{0}\,,
\end{eqnarray}
implying vanishing shear stress at the bottom
(or equivalently a vanishing force on the lower surface of the film
in $x$-direction, i.e. no traction force).
This condition will be called 'slip BC' in the following.
A second relevant situation, applying to
the case where the polymer film is rigidly attached to the lower surface,
will be referred to as 'fixed BC',
\begin{eqnarray}\label{BCbotfixed}
{\rm fixed\,\,BC:}\quad\quad u_x=0 \quad{\rm at}\quad z=-h_{0}\,.
\end{eqnarray}
We will see that these two different BC, slip vs.~fixed, have a qualitative influence on
the instabilities discussed in the following.
{\bf Slip BC at the bottom:} The solution of the elastostatic problem
with the slip BC at the bottom, Eqs.~(\ref{BCbot1}, \ref{BCbot2}),
is the Airy stress function Eq.~(\ref{chisol}) given in appendix \ref{det}.
The coefficients occurring therein
have to be determined by the BC at the free surface:
one calculates the stresses via
Eqs.~(\ref{stress_via_chi}), evaluates them at the free surface
$z=\zeta(x)$ and expands in powers of $\epsilon$.
From the BC at the free surface, Eqs.~(\ref{BCfree}),
one then determines the coefficients in the Airy stress function
at order $\mathcal{O}(\epsilon)$, yielding Eqs.~(\ref{asbs}).
The problem is now solved at linear order in the undulation,
and we can study the corresponding energy of the system.
The elastic energy will change due to the undulation-induced
relaxational stress $\tilde{\sigma}_{ij}$. This change,
$\Delta E_{\rm{el}}=E_{\rm{el}}-E^0_{\rm{el}}$,
explicitly reads
\begin{eqnarray}\label{en_el}
\Delta E_{\rm{el}}&=&\frac{1}{2\bar{E}}\int \bigg[
\left(\sigma_0+\tilde{\sigma}_{xx}\right)^2+\tilde{\sigma}_{zz}^2+4\tilde{\sigma}_{xz}^2 \nonumber\\
&&\hspace{1.4cm}-2(\tilde{\sigma}_{xx} +\sigma_0)\tilde{\sigma}_{zz}-\sigma_0^2\,\bigg] dx\,dz\,.\,\,\,\,\,
\end{eqnarray}
With the Airy stress function determined, the stress field can be evaluated.
The integrations in Eq.~(\ref{en_el}) have first to be performed
over the film thickness, $\int_{-h_{0}}^{\zeta(x)} dz$.
Then one usually averages over $x$, assuming periodic boundary conditions:
by writing $\langle E\rangle$ it is understood that one has
averaged like $\frac{k}{2\pi}\int_{0}^{2\pi/k}\,dx$.
Note that due to this averaging the contribution in $\mathcal{O}(\epsilon)$
vanishes.
To leading order $\mathcal{O}(\epsilon^{2})$ one calculates
\begin{eqnarray}
\langle\Delta E_{\rm{el},s}\rangle\hspace{-1mm}&=&\hspace{-1mm}
-\frac{A^2}{2}\frac{k\left[\sigma_{0}^{2}+k^{2}\gamma^{2}
+\left(\sigma_{0}^{2}-k^{2}\gamma^{2} \right)\cosh\left(2 h_0 k \right)\right]}
{\bar{E}\left(2h_0k+\sinh\left(2 h_0 k\right)\right)}\,.\,\,\,\,\nonumber\\
\end{eqnarray}
The surface energy is directly calculated from
Eq.~(\ref{Esurf}) and yields in order $\mathcal{O}(\epsilon^{2})$
\begin{eqnarray}
\langle E_{{\rm surf}}\rangle=\gamma\langle\frac{1}{2}\zeta'(x)^2\rangle=\frac{\gamma}{4} A^2 k^2\,.\,\,\,
\end{eqnarray}
Note that the averaged quantities, $\langle\Delta E\rangle$, have
units of energy per unit area.
Let us now briefly discuss the obtained result.
To regain the classical limit of an semi-infinite elastic half space
one performs the limit $h_{0}\rightarrow\infty$.
The change in total energy,
$\langle \Delta E_{{\rm tot}}\rangle=\langle \Delta E_{\rm{el}}+E_{{\rm surf}}\rangle$,
then reduces to
\begin{eqnarray}\label{DEhinf}
\langle \Delta E_{{\rm tot},s}\rangle=-\frac{A^2}{4}\left(\frac{2\sigma_0^2}{\bar{E}}k-\gamma k^2\right)\,.\,\,\,
\end{eqnarray}
As becomes apparent, the
prestress
leads to an {\it decrease} of the energy.
The stress enters quadratically, thus both compressive and tensile
stress trigger the instability which makes it
different from buckling instabilities \cite{landau_el}, see also section \ref{elastsharma}.
The surface tension acts against the instability and stabilizes high
wave numbers, see the second term in Eq.~(\ref{DEhinf}).
As we are predominantly interested in polymer thin films, let us perform
the opposite limit, $h_0 k\ll1$.
This amounts to saying that the modulation wavelength is large compared
to the film thickness.
In this 'thin film' limit, the change in total energy reads
\begin{eqnarray}\label{DEhthin}
\langle \Delta E_{{\rm tot},s}\rangle
=-\frac{A^2}{4}\left(\frac{\sigma_0^2}{\bar{E}h_0}\left[1+\frac{2}{3}(h_0 k)^2\right]-\gamma k^2\right).\,\,\,\,\,\,
\end{eqnarray}
Again, the prestress is destabilizing, independent of its sign.
However, there is no wave number dependence of the destabilizing term to leading order.
The same result was recently obtained in the framework of
a lubrication approximation \cite{Raphael:2006.3}.
{\bf Fixed BC at the bottom:} In this case
the Airy stress function given by Eqs.~(\ref{chisolf}, \ref{afbf})
has to be used.
For the change in the elastic energy this results in
\begin{eqnarray}
&&\hspace{-5mm}\langle\Delta E_{\rm{el},f}\rangle\nonumber\\
&=&
-\frac{A^2}{2} \frac{k\left[ (\sigma_0^2-k^2 \gamma^{2}) \sinh\left(2 h_0 k\right)
+ 2 h_0 k(\sigma_0^2+k^2 \gamma^{2}) \right]}
{\bar{E} \left(2 h_0^2 k^2+\cosh (2 h_0 k)+1\right)}\,.\,\,\,\,\nonumber\\
\end{eqnarray}
To cross-check, in the classical limit of a semi-infinite elastic half space
one again recovers Eq.~(\ref{DEhinf}). This is expected
as for a half space the BC at the bottom
should not be important.
In contrast, in the thin film limit $h_0 k\ll1$ one gets
\begin{eqnarray}\label{DEhthinfix}
\langle \Delta E_{{\rm tot},f}\rangle
=-\frac{A^2}{4} \left(\frac{4\sigma_0^2 }{\bar{E}h_0} (h_0 k)^2-\gamma k^2\right).\,\,\,\,\,\,
\end{eqnarray}
Note that the prestress still lowers the energy, but now has the same wave number
dependence as the contribution from surface tension.
Thus only above a threshold,
\begin{eqnarray}\label{thresh_fixed}
\sigma_0>\sqrt{\frac{\gamma\bar E}{4h_0}}\,,
\end{eqnarray}
the prestress can destabilize the system.
We have seen that the total energy of the system can be lowered
by surface undulations in all the cases discussed above.
To establish these favorable undulations, it needs a mechanism that allows rearrangements to occur.
In the classical case of a solid in contact with its vapor, this is achieved by
melting-crystallization processes at the surface.
This results in a velocity of the boundary $v_{MC}=\Gamma\Delta E$
\cite{grinfeld:86}, where $\Gamma$ is a mobility.
A second possible mechanism -- on which we would like to focus here
in view of polymers -- is {\it surface diffusion}.
If atoms or vacancies (in case of a solid) or polymer chains
(in case of polymer films) feel an inhomogeneous chemical potential
at the surface, they will diffuse. As a result, the boundary will move with
a velocity $v_{D}=-Mk^2\Delta E$,
with a mobility coefficient $M$ \cite{asaro:72}.
Note the second order spatial derivative stemming from the diffusion
process and reflected in the $k^2$-dependence of $v_{D}$.
Much more is known about the Asaro-Tiller-Grinfeld-instability, for which
we refer to the literature.
For the nonlinear evolution beyond the instability, see
\cite{nozieres:93,Kohlert:03} for analytical work and \cite{Kassner:01}
for phase-field modeling.
Concerning experiments, very clean realizations of this instability have been observed
in Helium crystals \cite{torii:92} and single crystal polymer films \cite{Berrehar92}.
\section{Stretched elastic solid: nonlinear bulk formulation}
\label{nonlin}
To properly describe finite stresses in a thin polymer film, one
has to use a nonlinear elasticity formulation.
Let us assume that the film was originally in a stress-free state, described by coordinates
$\mathbf{X}=X_{i}\mathbf{e_{i}}$. Then we stretch (or compress) the film,
for simplicity uniaxially in the $x$-direction by a factor $\lambda>1$ ($\lambda<1$).
This state will be described by coordinates $\mathbf{x}=x_{i}\mathbf{e_{i}}$ and considered
as the {\it base state}. This state will be under uniaxial stress $\sigma_{xx}^0$, see below.
Finally the film is brought in close contact with the substrate (either still permitting for slip,
or perfectly fixed to it, see the two BCs discussed in the last section) and we let it evolve.
This current state will be described
by coordinates $\mathbf{\tilde{x}}=\tilde{x}_{i}\mathbf{e_{i}}$.
Note that we discuss here only the simple situation where the film is
attached {\it after} the stretch. The situation where the polymers
attach to the substrate while the film is stretched (which probably better corresponds to
the situation during spin-coating) is more involved as the uniaxiality
is broken due to the presence of the substrate, cf. Ref.~\cite{ken89}.
The total deformation gradient from $\mathbf{X}$ to $\mathbf{\tilde{x}}$
reads
\begin{eqnarray}
\mathbf{F}=\frac{\partial\mathbf{\tilde{x}}}{\partial\mathbf{X}}
=\frac{\partial\mathbf{\tilde{x}}}{\partial\mathbf{x}}\cdot\frac{\partial\mathbf{x}}{\partial\mathbf{X}}
=:\mathbf{F_{2}}\cdot\mathbf{F_{1}}\,.
\end{eqnarray}
Here
\begin{eqnarray}
\mathbf{F_{1}}= \mathrm{Diag}(\lambda, \lambda^{-1/2},\lambda^{-1/2})
\end{eqnarray}
describes the stretching (compression) of the film by a factor
$\lambda>1$ ($\lambda<1$). Note that this step must be described in the nonlinear regime,
as stresses are finite.
The second tensor (with $\mathbf{I}$ the identity),
\begin{eqnarray}
\mathbf{F_{2}}=\mathbf{I}+\nabla\mathbf{u}\,,
\end{eqnarray}
introduces the usual linear displacement gradient tensor $\nabla \mathbf{u}=(\partial_ju_i)_{ij}$ in the current state
with respect to the stretched state.
As we are only interested in the stability of the base state,
here a linearized theory is enough for our purposes.
As usual we denote with $\mathbf{B}=\mathbf{F} \cdotp \mathbf{F}^{\mathrm{T}}$
and $\mathbf{C}=\mathbf{F}^{\mathrm{T}} \cdotp \mathbf{F} $
the left and right Cauchy-Green tensors.
As $\mathbf{B}$ is in Eulerian frame we adopt it for the stresses.
$\mathbf{C}$ is in Lagrangian frame and is more convenient for the energy definition.
Using a Neo-Hookean elastic solid \cite{Macosko} , the Cauchy stress tensor is defined as
\begin{eqnarray}\label{Cauchy}
\sigma = G \mathbf{B}-P \mathbf{I}\,.
\end{eqnarray}
It describes the stress after a deformation in the current configuration.
$P$ is a Lagrangian multiplier (an effective pressure having
units of $[{\rm Pa}]$)
that ensures the incompressibility condition.
In the base state,
from Eq.~(\ref{Cauchy}) one directly gets $\sigma_{\alpha\beta}^0=0$ except for
\begin{eqnarray}\label{rel_sig_lambda}
\sigma_{xx}^0=G\left(\lambda^2-\lambda^{-1}\right)\,.
\end{eqnarray}
This establishes a connection between the stretch factor $\lambda$ and the prestress $\sigma_0$.
Now, let us consider again a plane deformation with respect to the prestretched base state.
Evaluating the Cauchy stress tensor
in linear order in the displacement gradient, imposing plane strain and using incompressibility,
one arrives at the bulk equations
\begin{eqnarray}
G \left( \lambda^{2} \partial_{x}^2u_{x} + \lambda^{-1} \partial_{z}^2u_{x} \right) - \partial_{x}P &=& 0, \nonumber\\
G \left( \lambda^{2} \partial_{x}^2u_{z} + \lambda^{-1} \partial_{z}^2u_{z} \right) - \partial_{z}P &=& 0.
\end{eqnarray}
Note the asymmetry introduced by $\lambda\neq1$, i.e. the prestretch.
All quantities can be expressed either in the base state $\mathbf{x}$ or in the current state
$\mathbf{\tilde{x}}$ - as deformations $\mathbf{u}$ are small, they amount to the same expressions.
For $\lambda=1$ one regains the classical
elastostatic equation for an incompressible solid,
$G \nabla^2 \mathbf{u}+\nabla P=0\,,$
where $\mathbf{u}$ is the displacement field.
The elastic energy density, $\rho_{\rm el}$, for the Neo-Hookean elastic solid reads
\begin{eqnarray}
\label{Neo-Hookean}
\rho_{\rm el}= \dfrac{G}{2} \left( \mathrm{Tr}(\mathbf{C}) - 3 \right)\,.
\end{eqnarray}
Here we did not include the pressure as a Langrangian multiplier (giving rise
to a term $+P(\det(\mathbf{B})-1)$), as incompressibility is imposed when solving the bulk equations,
see the next section.
Note that for plane strain and small deformations
one regains Eq.~(\ref{Eelplanestr}) to second order in displacement gradients, i.e.
$\rho_{\rm el}=\dfrac{1}{2 \bar{E}}\left[ \left(1 + \bar{\nu} \right) \sigma_{ij}^{2} - \bar{\nu} \sigma_{ii} \sigma_{jj} \right].$
Now we have established the equations for a nonlinear prestretch and a subsequent linear theory.
Note, however, that the Neo-Hookean model should not be used for $\lambda$-values too far from $\lambda=1$.
Otherwise effects of e.g.~the crosslink length must be taken into account
and one should use more realistic models like the Mooney-Rivlin solid \cite{Macosko}.
We will now investigate the stability of the prestressed base state
with respect to surface undulations:
(i) for the purely elastic case, (ii) in the presence of surface diffusion,
making a connection with the classical Grinfeld instability, (iii)
in the presence of an electric field normal to the free surface, regaining and generalizing results obtained
previously \cite{SharmaPRL,HeEPL} and finally (iv) with both surface diffusion and applied electric field.
\section{Stretched elastic solid: solutions for surface modulations}
\label{elastsharma}
In this and the following section we solve the elastic bulk equations
and show how surface diffusion can be incorporated within this approach
in a generic way to regain and generalize the Grinfeld result.
We use the same boundary conditions as introduced in section \ref{Grin_elastic}, i.e.
Eq.~(\ref{BCfree}) for the free surface and either
the slip BC or
the fixed BC at the bottom.
The stability of the base state can be studied by the ansatz
\begin{eqnarray}
u_x(x,z,t) &=& u_x(z) e^{ikx+st} \nonumber\\
u_z(x,z,t) &=& u_z(z) e^{ikx+st} \nonumber\\
P(x,z,t) &=& p(z) e^{ikx+st}
\end{eqnarray}
where the amplitudes
are small perturbations of order $\mathcal{O}(\varepsilon)$ in height perturbations, see Eq.~(\ref{heightperturb}) below.
Note that we allowed for a temporal dependence which will be used only in the following sections.
Using incompressibility, $iku_x+\partial_z u_z = 0 $, one obtains
a single decoupled equation for $u_z$ given by
\begin{eqnarray}\label{uzeq}
k^4 \lambda ^3 u_z-k^2 \left(\lambda ^3+1\right) u_z''+u_z^{(4)}=0\,.
\end{eqnarray}
With this equation solved, one easily obtains $u_x$ from incompressibility
and the pressure from
$p(z) = G \left(\frac{u_{z}'''(z)}{\lambda k^2}-\lambda ^2 u_{z}'(z)\right)$.
The general solution
of Eq.~(\ref{uzeq}) reads (for $\lambda\neq1$ \footnote{Note that the case $\lambda=1$ is singluar as it yields
only one wavenumber and additional solutions like $z\sinh(z)$.})
\begin{eqnarray}\label{gensol_uz}
u_z(z)&=&\epsilon\sum_{i=1,2}\big\{A_i\cosh \left[ k_i \left( z + h_0 \right) \right]\nonumber\\
&&\quad\quad\quad+B_i\sinh \left[ k_i \left( z + h_0 \right) \right]\big\}\,,
\end{eqnarray}
with
\begin{eqnarray}
k_1=k\,,\,\,\,{\rm and}\,\,k_2=l=k\lambda^{3/2}\,.
\end{eqnarray}
Imposing the BCs at the substrate yields
\begin{eqnarray}
u_z(-h_0) &=& 0 \Leftrightarrow A_2 = -A_1 \nonumber\\
\sigma_{xz}(-h_0) &=& 0 \Leftrightarrow A_2 = 0 ,\hspace{1cm}\quad({\rm slip\,\,BC})\nonumber\\
u_x(-h_0) &=& 0 \Leftrightarrow B_2 = - \dfrac{k}{l} B_1,\quad({\rm fixed\,\,BC})\,.\nonumber
\end{eqnarray}
As before, we parameterize the upper free interface of the thin polymer film by a harmonic function
with small amplitude of order $\mathcal{O}(\varepsilon)$
\begin{eqnarray}\label{heightperturb}
z= h(x,t) = \epsilon h e^{ikx+st}.
\end{eqnarray}
The normal vector of this surface reads $\hat{\mathbf{n}} = (-ik h(x,t), 0 , 1)$ at first order.
Thus at the free interface, cf. Eqs.~(\ref{BCgrin}), the BCs read
$\sigma_{ij} \hat{n}_{j} = - \gamma k^2 h(x,t) \hat{n}_{i}$.
They fix the remaining unknown coefficients and
one obtains $A_{1,s}=A_{2,s}=0$ and
\begin{eqnarray}
B_{1,s} &=& - h \dfrac{2 k \left(G \left(l^2-k^2\right) \cosh (h_0 l)+k^2 l \gamma \lambda \sinh (h_0 l)\right)}{G
\left(k^2+l^2\right) g^+(k,l)}\,, \nonumber\\
B_{2,s} &=& h \dfrac{k^3 \gamma \lambda \sinh (h_0 k)+G \left(l^2-k^2\right) \cosh (h_0 k)}{ l G \, g^+(k,l) }\,
\end{eqnarray}
in case of the slip BC at the bottom;
for the fixed BC
\begin{eqnarray}
A_{1,f} &=& h \dfrac{G \left(l^2-k^2 \right) v(k,l,k) + k^3 \gamma \lambda w(k,l) }
{G \left(k^4+6 k^2 l^2+l^4 -\left(k^2+l^2\right) \, f(k,l) \right)},\,\,\,\nonumber\\
B_{1,f} &=& - h \dfrac{G \left(l^2-k^2 \right) w(k,l) + \gamma \lambda k^3 v(k,l,l)}
{G \left(k^4+6 k^2 l^2+l^4 -\left(k^2+l^2\right) \, f(k,l) \right)}\quad\,\,\,\,
\end{eqnarray}
and $A_{2,f}=-A_{1,f}$, $B_{2,f} = - \dfrac{k}{l} B_{1,f}$.
We introduced the following abbreviations
\begin{eqnarray}
g^{\pm}(k,l) &=& \pm\sinh \left[ \left(l + k \right) h_0 \right] \left(l -k \right) \nonumber\\
&&+ \sinh \left[ \left(l - k \right) h_0 \right] \left(l + k \right)\,, \nonumber \\
f(k,l) &=& \cosh \left[ \left(l + k \right) h_0 \right] \left(l -k \right)^2 \nonumber\\
&&+ \cosh \left[ \left(l - k \right) h_0 \right] \left(l + k \right)^2\,, \nonumber \\
v(k,l,m) &=& \left( l^2 + k^2 \right) \cosh \left[ k h_0 \right] - 2 m^2 \cosh \left[ l h_0 \right]\,,\nonumber \\
w(k,l) &=& \left( l^2 + k^2 \right) \sinh \left[ k h_0 \right] - 2 k l \sinh \left[ l h_0 \right]\,.
\end{eqnarray}
With the general solution obtained, we can now investigate whether the base state is stable or unstable.
According to Eq.~(\ref{heightperturb}), solutions with non-zero wavenumber, if they exist, correspond to
surface undulations.
The condition for nontrivial solutions to exist can be written
as
\begin{eqnarray}\label{condntsol}
h(x,t)=u_z(x,z=0,t)
\end{eqnarray}
or $\epsilon h=u_z(0)$.
Namely, for consistency the displacement at the surface
must equal the height perturbation. An alternative formulation would have been
to write down the system of BCs as a $4\times4$-matrix equation and looking for nontrivial
solutions via the zeros of the determinant.
With $u_z$ known, Eq.~(\ref{condntsol}) can be written as
\begin{eqnarray}
hZ(k)=0
\end{eqnarray}
with a function of wave number $Z(k)$. If one finds wave numbers $k^*$ with $Z(k^*)=0$,
periodic solutions exist; otherwise
$hZ(k)=0$ implies $h=0$ and the film stays flat.
Explicitly, for the two considered BCs one gets
\begin{eqnarray}
\label{F_slip}
Z_{s}(k) &=& 4 k^3 l G \cosh\left(l h_0 \right) \sinh\left(k h_0 \right)\nonumber \\
& & -\sinh\left(l h_0 \right) \left(l^2 + k^2 \right)^2 G \cosh\left(k h_0 \right) \nonumber
\\ & & -\sinh\left(l h_0 \right) k^3 \gamma \lambda \left(l^2 - k^2 \right) \sinh\left(k h_0 \right), \\
\label{F_fix}
Z_{f}(k) &=& 4 G k^2 l \left( l^2+k^2 \right)+\frac{G}{2}(r\left(-l,k \right) - r\left(l,k \right)) \nonumber \\
& & + \frac{1}{2} k^3 \gamma \lambda \left( l^2-k^2 \right) g^{-}\left(k,l \right)\,,
\end{eqnarray}
with
$r\left(k,l \right) =(k+l)^2 (k^3 + 3 k l^2 - k^2 l + l^3)\cosh\left[h_0 \left(k - l\right) \right]$.
For both BCs, nontrivial solutions do not exist under tension, $\lambda>1$, as one would expect.
Buckling occurs under compression, but only for non-physical values,
namely for $\lambda<\lambda_c\simeq0.03$
for a typical surface tension of $\gamma = 0.5 h_{0} \bar{E}$.
For such high compressions, the Neo-Hookean law is no longer a good description.
Moreover, the assumption that the film stayed flat in the first step (from $\mathbf{X}$ to $\mathbf{x}$,
i.e. before attaching to the substrate) is not valid anymore - the film would have buckled long before.
Indeed the threshold for buckling for two free surfaces should be lower
than for the BC that the film does not detach from the substrate surface, Eq.~(\ref{BCbot1}).
Thus we can conclude that the film stays flat for all reasonable values of $\lambda$, $G$ and $\gamma$.
Note, however, that films can be unstable if they are swollen
{\it in the presence} of the substrate, cf.~Refs.~\cite{TanakaNat87,ken89}.
\section{Adding surface diffusion - the Grinfeld instability again and corrections}
\label{grindiffuse}
In the last section we investigated the stability of the base state
with respect to in-plane stresses and found that the purely elastic system is stable.
Here we add the effects of diffusion of polymer chains
close to the film surface due to stress relaxation-induced changes in the chemical potential.
As a consequence the system
can produce undulations by {\it diffusive transport} of material,
in addition to possible elastic displacements. We show that
one regains the Grinfeld instability in a well-defined limit.
The overall result is more general as it comprises corrections to the Grinfeld mechanism,
see below.
If we allow for surface diffusion, Eq.~(\ref{condntsol}) has to be modified in order
to allow for this dynamics.
For the height modulation $h=h(x,t)$ one can write
\begin{eqnarray}\label{surdiffus}
\partial_t h&=&\partial_t u_{z|z=0}-(\partial_t u_{x|z=0})(\partial_x h)\nonumber\\
&&+M\partial_x^2\left(\delta \mu_{|z=0}\right)\,.
\end{eqnarray}
The first two terms on the r.h.s.~stem from the standard kinematic BC
at a free surface, usually written as $\partial_t h=v_z-v_x\partial_x h$ with $h$ the height of the surface
and $(v_x,v_z)=\partial_t(u_x,u_z)$ the fluid velocity \cite{Bankoff:1997}.
The second term is purely nonlinear and can be
neglected in the following linear analysis.
The last term on the r.h.s.~represents the surface diffusion (note that in three dimensions
$\partial_x^2$ has to be replaced by the surface Laplacian \cite{Spencer91}).
It will smoothen gradients in the chemical potential, which is given by
\begin{eqnarray}\label{muwithgam}
\delta\mu=\delta E_{\rm el}-\gamma\kappa\,.
\end{eqnarray}
$\kappa$ is the mean curvature of the surface, given at $\mathcal{O}(\epsilon)$ by
$\kappa=\partial_x^2 h$. $\delta E_{\rm el}$ is the change in elastic energy density due to the surface undulation,
compared to the flat surface.
The coefficient $M$ is a mobility \cite{Mullins,Saul} and explicitly reads
$M=\frac{D n_s V^2}{k_B T}$,
where $k_B T$ is the thermal energy, $V$ is a microscopic volume
(of the polymer chain in our case),
$D$ is the surface diffusion coefficient
and $n_s$ is the surface density of diffusing objects.
Note that in the view of recent experiments on spin-cast polymer melts,
we here allow for a finite chain mobility (at least close to the free surface),
although we assumed a purely elastic behavior of the film. A generalization
of our approach to the more adequate viscoelastic case will be the subject of a forthcoming study.
Eq.~(\ref{surdiffus}) for the dynamics of the surface undulation
is further motivated in appendix \ref{motkinBC}.
The terms
arising naturally from the kinematic BC
are commonly not included in the treatment of the
Grinfeld instability, as in the usual context one concentrates
on the diffusive transport of atoms or vacancies.
Taking the coupling to the displacement into account -- if extended objects like polymers
are diffusing -- corrections to the 'classical' Grinfeld behavior arise:
the time derivative in $\partial_t u_{z|z=0}$ leads to a renormalization of the growth rate $s(k)$
of the height perturbations $h(x,t) = \epsilon h e^{ikx+st}$.
In view of this, in the following we will sometimes compare the 'classical' Grinfeld and the 'kinematic' case.
In the previous section we have already calculated the general solution for the
displacements. Thus the stress tensor is also known and
using Eq.~(\ref{Neo-Hookean}) one gets
the changes in the elastic energy $\delta E_{\rm el} = \rho_{\rm el} - \rho_{\rm el}^{0}$
with respect to the base state
\begin{eqnarray}
\delta E_{el,s}
&=& -\frac{\epsilon G \left(l^2-k^2\right) \cos (k x)}{k^2 \lambda } \nonumber \\
& &\hspace{-1cm}\cdot
\left(B_{1,s} k \cosh \left[k\left(h_0+z\right)\right]
+B_{2,s} l \cosh \left[l \left(h_0+z\right)\right]\right),\qquad\\
\delta E_{el,f} &=&- \frac{\epsilon G \left(l^2-k^2\right) \cos (k x)}{k^2 \lambda } \nonumber \\
& &\hspace{-1cm}\cdot\,\Big(
A_{1,f} \left( k \sinh \left[k (h_0+z)\right]-l \sinh\left[l (h_0+z)\right] \right)\nonumber \\
& &\hspace{-0.8cm}+ B_{1,f} k \left(\cosh \left[k (h_0+z)\right]-\cosh\left[l (h_0+z)\right]\right) \Big).\,
\end{eqnarray}
For the surface energy, as before Eq.~(\ref{Esurf}) yields
$E_{\rm surf}=\epsilon \cos\left(k x \right) e^{st}\gamma k^2 h$.
Now we can proceed in two ways:
{\bf Classical calculation, nonlinear case:}
First we can use the classical Grinfeld argument,
i.e.~we integrate from $-h_0$ to $h(x)$ over the film thickness
and average over the assumed periodic $x$-direction to obtain
$\Delta E_{\rm el}=\langle\int \delta E_{\rm el} \, \mathrm{d}z\rangle_{x}$.
Upon averaging the linear order in $\epsilon$ vanishes. At $\mathcal{O}(\epsilon^2)$, one gets
to leading order in $k$
\begin{eqnarray}
\delta E_{el,s} &=& -\frac{h^2 \epsilon ^2 \left(\lambda ^3-1\right)^2 \left(3 \lambda ^3+1\right) G }{16 h_0
\lambda \left(\lambda ^3+1\right)^2} + \mathcal{O}(k^{2})\,,\,\,\, \\
\delta E_{el,f} &=& -\frac{h^2 k^2 \epsilon ^2 h_0 \left(\lambda ^3-1\right)^2 G}{4
\lambda } + \mathcal{O}(k^{4})\,.
\end{eqnarray}
Let us compare to the result obtained in section \ref{Grin_elastic}.
In the limit $\lambda=1\pm\delta$ with $\delta\ll 1$ and using $\delta=\sigma_0/(3G)$ as
implied by Eq.~(\ref{rel_sig_lambda}) in this limit, one gets including the surface energy
\begin{eqnarray}\label{NLGslip}
\delta E_{{\rm tot},s} &=& -\dfrac{h^2}{4}\left(\dfrac{\sigma_0^2}{\bar{E} h_0} \left[1 + \left(h_0 k \right)^2\right]
-\gamma k^2\right), \\
\delta E_{{\rm tot},f} &=& -\dfrac{h^2}{4}\left(\dfrac{4 \sigma_0^2}{\bar{E} h_0}\left(h_0 k \right)^2-\gamma k^2\right).
\end{eqnarray}
Note that in leading order this is exactly Eqs.~(\ref{DEhthin}, \ref{DEhthinfix}).
The correction $\left(h_0 k \right)^2\ll 1$ in Eq.~(\ref{NLGslip})
has a slightly different prefactor as in Eq.~(\ref{DEhthin}),
which is due to the fact that the fully linear calculation from section \ref{Grin_elastic}
is only correct for {\it infinitesimal} stresses.
{\bf Consistent calculation at order $\mathcal{O}(\epsilon)$:}
The use of an averaging in the Grinfeld calculation seems not necessary to us.
We will thus determine the growth rate of surface undulations by using
\begin{eqnarray}
\partial_t h&=&\partial_t u_{z|z=0}+M\partial_x^2\left(\delta \mu_{|z=0}\right)\,.
\end{eqnarray}
The l.h.s. and the first term on the r.h.s. are of first order in $\epsilon$.
Thus it is sufficient to determine the change of the chemical potential
at this order, i.e. evaluating $\delta\mu$ at the surface.
The full growth rates obtained by this equation are given
by Eqs.~(\ref{fullss})-(\ref{fullsfkin})
in appendix \ref{det}
for the slip and the fixed BC, respectively.
In the thin film limit $h_0 k\ll 1$, one obtains
\begin{eqnarray}\label{ssk}
s_s (k) &\simeq& M k^2 \dfrac{G (\lambda^3-1)^2}{2 h_0 \lambda (1+\lambda^3)} \nonumber\\
&&\hspace{-1cm}+ M k^4 \bigg[ \dfrac{G h_0 (\lambda^3-1)^2}{6 \lambda}
- \gamma \left( 1 + \dfrac{\lambda^3 - 1}{2 \left( 1 +\lambda^3 \right)} \right)\bigg]\,\,\,
\end{eqnarray}
for the slip BC at the bottom and neglecting the $\partial_t u_{z|z=0}$ term at the free surface.
Including the kinematic term yields
\begin{eqnarray}\label{sskkin}
s_{s,kin} (k) &\simeq& M k^2 \dfrac{G (\lambda^3-1)^2}{h_0 \lambda (3+\lambda^3)} \nonumber \\
& &\hspace{-1.7cm} + M k^4 \bigg[ \dfrac{G h_0 (4+3\lambda^3+\lambda^6) (\lambda^3-1)^2}{3 \lambda (3+\lambda^3)^2}
- 4 \gamma\dfrac{ (1+\lambda^3)^2 }{(3+\lambda^3)^2}\bigg].\nonumber\\
\end{eqnarray}
For the fixed BC we get in both cases
\begin{eqnarray}\label{ssf}
s_{f} (k) &\simeq& M k^4 \left[\dfrac{G h_0}{\lambda} (\lambda^3-1)^2 -\gamma\right]\,.
\end{eqnarray}
Let us first discuss the limit of small stresses asgain, $\lambda=1\pm\delta$ with $\delta=\sigma_0/(3G)\ll 1$.
Both Eqs.~(\ref{ssk}, \ref{sskkin}) yield at leading order in the stress
$s(k)\simeq M k^2 \frac{\sigma_0^2}{\bar E h_0}$.
Except for a factor of $4$, at leading order this is exactly Eq.~(\ref{DEhthin}).
The same is true for the fixed BC and Eq.~(\ref{DEhthinfix}).
Hence in the low-stress and low-wave number limit, our results obtained for the dynamic BC at the free surface
are identical to those obtained by the energy-based calculation in section \ref{Grin_elastic}
in the following sense: $s(k)$ are growth rates as calculated from a dynamical equation
for the surface undulation.
When comparing to the Grinfeld calculation, there too one
has to impose a diffusion dynamics driven by the decrease in energy.
One can write
$\partial_t A=-M k^2 \frac{\partial}{\partial A}\langle \Delta E_{{\rm tot}}\rangle$,
with $E_{{\rm tot}}\propto A^2$.
The variational derivative with respect to $A$
yields a factor of 2.
Taking into account that in the energy approach one
has averaged over $\langle \cos^2(kx)\rangle$ yields another factor of 2,
which explains the differing prefactors.
However, one should note that using the spatial averaging process
implies a calculation order $\mathcal{O}(\varepsilon^2)$,
while our method is $\mathcal{O}(\varepsilon)$.
\begin{figure}[t]
\centering
\vspace{.2cm}
\includegraphics[width=0.48\textwidth]{fig2.eps}
\caption{\label{ssfig} Growth rates as a function of reduced wave number
for different stretch factors $\lambda$, corresponding to different prestress.
Panel a) displays the case of the slip BC at the bottom. Without stress, $\lambda=1$, the system is stable.
Any compression ($\lambda<1$) or stretch ($\lambda>1$) lead to an instability, but with differing rates.
$\lambda=1.1$ and $\lambda = 0.9$ correspond both to a prestress $|\sigma_0|/\bar{E}\simeq 0.075$.
Solid lines have been obtained with the kinematic BC at the free surface, pointed lines just with
the surface diffusion.
Panel b) shows the case of the fixed BC at the bottom.
An instability only occurs beyond critical $\lambda$-values.
Parameters: $\gamma = 0.5 h_{0} \bar{E}$.
}
\end{figure}
\begin{figure}[t]
\centering
\vspace{.2cm}
\includegraphics[width=0.48\textwidth]{fig3.eps}
\caption{\label{kmax} The fastest growing wavenumber $k_{max}$ as given by the maximum
of the growth rate $\bar{s}_{s,kin}$ shown in Fig.~\ref{ssfig}a) as a function of rescaled prestress
$|\sigma_0|/\bar{E}$. Solid lines are obtained for $\gamma=0.7h_0\bar E$, dashed lines for
$\gamma=0.5h_0\bar E$.
}
\end{figure}
Let us now discuss the effect of finite stretches and of the kinematic contribution.
At leading order in the stress, both Eqs.~(\ref{ssk}, \ref{sskkin}) reduce
to the Grinfeld result.
However, in next order in the stress Eq.~(\ref{ssk}) yields
$-\frac{2}{3} M k^2 \frac{\sigma_0^3}{{\bar E}^2 h_0}$,
while the kinematic version yields
$+\frac{1}{3} M k^2 \frac{\sigma_0^3}{{\bar E}^2 h_0}$.
First, this shows that the symmetry with respect to the sign of the stress,
i.e. whether it is due to stretch ($\sigma_0>0$) or compression ($\sigma_0<0$),
is broken by the elastic nonlinearity. Second, the sign of the correction
is sensitive to whether the kinematic BC at the free surface is important (e.g.~for diffusion of
extended objects like polymers in a network) or not.
To compare to a real system, we use the following parameter values as suggested by Ref.~\cite{Barbero09}:
$h_{0} = 140 {\rm nm}$ for the thickness of the film and $\bar{E}=5\cdot 10^{5} \mathrm{Pa}$ for the modulus.
For the surface tension we use the value for polystyrene, $\gamma_{PS}\simeq30\cdot10^{-3} {\rm N}{\rm m}^{-1}$.
Fig.~\ref{ssfig} displays the full growth rates, Eqs.~(\ref{fullss})-(\ref{fullsfkin})
in appendix \ref{det}, as a function of reduced wave number $kh_0$.
Note that we renormalized $\bar s(k)=(h_0^3/(M\bar E))s(k)$.
Fig.~\ref{ssfig}a) displays the case of the slip BC at the bottom,
with (solid curves) and without (dotted curves)
accounting for the kinematic BC at the free surface.
Finite stresses lead to a Grinfeld instability. Growth rates differ whether compression
($\lambda<0$) or extension ($\lambda>0$) is considered. In case of the fixed BC at the bottom,
see Fig.~\ref{ssfig}b), there exists a threshold stress beyond which the system becomes unstable.
For the chosen surface tension, $\gamma = 0.5 h_{0} \bar{E}$ in reduced units, the system
destabilizes for $\lambda>\lambda_{1,c}\simeq1.387$ and $\lambda<\lambda_{2,c}\simeq0.426$.
Note that a (symmetric) threshold stress also occurred in the linear model, cf.~Eq.~(\ref{thresh_fixed}).
Fig.~\ref{kmax} displays the dependence of the fastest growing wavenumber on
the prestress $|\sigma_0|/\bar{E}$ as obtained from $\lambda$ by Eq.~(\ref{rel_sig_lambda}).
One clearly sees the asymmetry with respect to compression/stretch for finite stresses.
To summarize, in the last two sections we proposed a general framework that includes the Grinfeld instability
as well as possible buckling.
The possibility of buckling is due to the coupling
of surface undulations and the displacement field via a kinemtic BC at the free surface.
One gets corrections to the Grinfeld instability, as contained in the full growth rates given in
appendix \ref{det}.
However, in the small wave-number limit and for thin films, the leading order
terms are identical with the classical result. For finite stresses the $\pm$-symmetry
with respect to stresses predicetd by the linear Grinfeld-theory is no longer valid.
In the next section we use the developed framework to study the simultaneous action of
in-plane residual stress and an electric field, both acting as destabilizing factors
for elastic films.
\section{Addition of external electric field}
\label{grinE}
Recently the instability of polymeric liquids \cite{SteinerEPL,Barbero09} and elastomers \cite{Sharma08}
in an external electric field acting normal to the film surface has been investigated
experimentally. In Ref.~\cite{Barbero09}, it has been found that the instability is faster
for freshly spin-casted films than for aged films. This suggests that stresses in
the fresh films due to the nonequilibrium production process may be involved
in the destabilization.
In view of this we generalize the developed approach to the case where
an external electric field is acting normal to the surface,
in addition to the stress
in $x$-direction.
The electrostatic part will be closely related to previous studies of
elastic instabilities \cite{SharmaPRL,monch2001,Sharmalong,Sharma08}
due to forces normal to the surface (Van der Waals or electric field).
Related studies have been undertaken in Refs.~\cite{Yang05,Yang06}.
However, there the thin film was regarded as conductive, the external stress
was imposed externally (implying that the base state with applied field
was fixed at $\sigma_{xx}^0=\sigma_0$ rather than $\sigma_{xx}^0=\sigma_0+F$ as
in our case with $F$ the additional contribution from the electric field, see below)
and the kinematic BC ({\it i.e.} the coupling of film height and displacement field)
at the free surface was not taken into account.
\begin{figure}[t]
\centering
\vspace{.2cm}
\includegraphics[width=0.48\textwidth]{fig4.eps}
\caption{\label{sketchE} Sketch of the capacitor geometry.
There is uniaxial stress in $x$-direction (which could also be tensile).
An external electric field in $z$-direction (for the unperturbed film)
is imposed by externally applying a voltage $V$ between the electrodes of distance $d$.}
\end{figure}
Let us assume that the polymer film
is brought into a parallel plate capacitor, see the sketch in Fig.~\ref{sketchE}.
A voltage difference $V$ is applied over the distance
of the two plates $d$ (the lower plate is at $z=-h_0$, the upper one at $z=d-h_0$).
The gap may be filled with any dielectric.
In view of the experiments in Ref.~\cite{Barbero09}, we take $\epsilon_1\simeq2.5$
(polystyrene) as the dielectric constant of the polymer film
and $\epsilon_2=1$, i.e. the gap is filled with air.
The electric field will introduce a stress at the polymer-air interface.
Let us introduce the Maxwell stress tensor
\begin{eqnarray}
T^{(k)}_{ij}=\epsilon_k \epsilon_0 \left( E^{(k)}_i E^{(k)}_j-\dfrac{1}{2} \left(E^{(k)}\right)^2
\delta_{ij}\right)\,,
\end{eqnarray}
where the index $k=1$ denotes the polymer film and $k=2$ the gap.
The BC at the free surface, cf. Eq.~(\ref{BCgrin}), now reads
\begin{eqnarray}
\label{Max_tensor}
\hat{\mathbf{n}}\cdot\left( \sigma^{(1)}- \sigma^{(2)}+T^{(1)} - T^{(2)} \right)\cdot\hat{\mathbf{n}}
=\gamma\partial_{xx}h\,,
\end{eqnarray}
where we wrote only the linear order expression for the surface tension.
We can put $\sigma^{(2)}=0$ (or to a constant pressure value that is not important),
$\sigma^{(1)}=\sigma$
and define an electrostatic 'pressure' (strictly speaking a normal stress) by
\begin{eqnarray}\label{pEdef}
p_E(h)=\hat{\mathbf{n}}\cdot\left(T^{(1)}- T^{(2)}\right)\cdot\hat{\mathbf{n}}\,.
\end{eqnarray}
Note that this electrostatic stress depends on the film thickness, see below.
The BC finally reads
$\hat{\mathbf{n}}\cdot \sigma\cdot \hat{\mathbf{n}}= \gamma \partial_x^2 h - p_E(h)$.
We now have to evaluate the additional contribution from the electric field.
We can again solve the problem by a perturbative method by writing
$\mathbf{E}^{(i)}=\mathbf{E}^{(i)}_{0}+\mathbf{E}^{(i)}_{1}$, where
$\mathbf{E}^{(i)}_{1}$ is the first order correction due to undulations.
To lowest order, one has to satisfy that the normal dielectric displacement is continuous,
$\epsilon_1 E^{(1)}_{0,z}=E^{(2)}_{0,z}$.
Second, we have $V=h_0 E^{(1)}_{0,z}+(d-h_0)E^{(2)}_{0,z}$, $E^{(i)}_{0,x}=0$ and thus one gets
$E^{(1)}_{0,z}=\epsilon_2 V/(h_0+\left(d-h_0 \right) \epsilon_1)$,
$E^{(2)}_{0,z}=\frac{\epsilon_1}{\epsilon_2} E^{(1)}_{0,z}$.
In the next order, we have to solve Maxwell's equations
\begin{eqnarray}\label{Maxwell}
\partial_z E_{1,x}^{(i)} - \partial_x E_{1,z}^{(i)}=0\,,\,\,\,
\partial_z E_{1,z}^{(i)} + \partial_x E_{1,x}^{(i)}=0\,,
\end{eqnarray}
with the BCs
\begin{eqnarray}
E_{1,x}^{(1)}(z=-h_0)=0\,&,&\,\,\,E_{1,x}^{(2)}(z=d-h_0)=0\,,\nonumber\\
\hat{\mathbf{n}}\cdot(\epsilon_2 \mathbf{E}^{(2)}-\epsilon_1 \mathbf{E}^{(1)})=0
\,&,&\,\,\,\hat{\mathbf{t}}\cdot(\mathbf{E}^{(2)}-\mathbf{E}^{(1)})=0\,.
\end{eqnarray}
These BC state that the field has to be perpendicular to the conductive electrodes and
that at the film surface one has continuity in the normal displacement and the tangential field.
Assuming $E^{(i)}_{1,z}\propto \cos(k x)$, the system is readily solved yielding
the field components given by Eqs.~(\ref{Ecomp}) in appendix \ref{det},
in agreement with Ref.~\cite{OnukiPA1995}.
Evaluating the normal-normal component of the Maxwell stress,
for the electrostatic pressure as defined in Eq.~(\ref{pEdef}) above
we get to leading order
\begin{eqnarray}
-p_E(\zeta)&=& F + Y(k)\,\zeta\,,
\end{eqnarray}
where
\begin{eqnarray}
\label{Fdef}
F&=&-p_E(0)=\frac{1}{2}\frac{\epsilon_0 \epsilon_1 \epsilon_2 (\epsilon_1-\epsilon_2)V^2}
{\left(\epsilon_2 h_0 + \left(d-h_0\right) \epsilon_1 \right)^2}\,,\,\\
\label{Ydef}
Y(k)&=&\frac{-2 k p_E(0)(\epsilon_1-\epsilon_2)}{\left[\epsilon_1 \tanh\left( \left(d-h_0\right) k\right)+\epsilon_2 \tanh\left(h_0 k\right)\right]}\,.
\end{eqnarray}
Note that both $F$ and $Y$ are strictly positive, $F, Y>0$.
As one has
$\hat{\mathbf{n}}\cdot (T^{(1)}-T^{(2)})\cdot\hat{\mathbf{t}}=0$, the tangential BC at the free surface is unchanged by the electric field.
In the base state, the contribution of the electric field will be
an isotropic pressure \cite{SharmaPRL,Sharmalong}, given by $F=-p_E(\zeta=0)$.
Concerning the displacements relative to the base state, the procedure is completely
analogous to the one in the previous sections.
Only the BC at the free surface,
and the chemical potential have to be changed
accordingly to include the electric stresses.
In the chemical potential, Eq.~(\ref{muwithgam}), we have to add the contribution due to
the electric stress by writing
\begin{eqnarray}\label{potchi}
\delta\mu&=&\delta E_{\rm el}-\gamma\kappa+p_E(\zeta)\nonumber\\
&=&\delta E_{\rm el}+\gamma k^2 \zeta -F-Y(k) \zeta\,.
\end{eqnarray}
As $F$ is a constant, its contribution to surface diffusion vanishes.
The general solution for the displacement field, Eq.~(\ref{gensol_uz}),
with the BCs at the substrate already imposed, is still valid.
One only has to determine the coefficients
fulfilling the new BC at the free interface.
These coefficients,
$B_{1,s}^{E}$, $B_{2,s}^{E}$ and $A_{1,f}^{E}$, $B_{1,f}^{E}$ can be obtained
from the respective solutions without field
by the simple substitution
\begin{eqnarray}
\gamma \rightarrow \left(\gamma - \frac{Y(k)}{k^2}\right)\,.
\end{eqnarray}
This rescaling of $\gamma$ permits
to obtain also the functions $Z_{s}^{E}(k)$, $Z_{f}^{E}(k)$
that determine the stability of the flat base state in the presence
of a field, as well as the growth
rates $s_{s}^{E}$ and $s_{f}^{E}$.
The obtained expressions are very general. Although unsightly they
contain the physics of buckling, the elasto-electric instability, the
Grinfeld-instability and surface diffusion in an applied electric field.
\begin{figure}[t]
\centering
\vspace{.2cm}
\includegraphics[width=0.48\textwidth]{fig5.eps
\caption{\label{Y_instab} Stability diagram for the electric field-induced instability.
The curves display the electric contribution to stress in reduced units,
$Y_0 h_0/\bar E$, as a function of reduced wave number $h_0 k$
for the slip and the fixed BC at the bottom, as indicated.
Solid lines correspond to the stress-free case, $\lambda =1.0$.
Dashed lines are for finite stretch, $\lambda =1.1 $,
and dotted lines for finite compression $\lambda=0.9$.
Parameters: $\gamma = 0.5 h_{0} \bar{E}$.
}
\end{figure}
Let us first discuss the case without surface diffusion.
One expects to get an instability for $Y>Y_c$ as described by Sharma {\it et al.} \cite{SharmaPRL}.
Note, however, that in case of an applied electric field $Y(k)$ is $k$-dependent, while
in Ref.~\cite{SharmaPRL} a Van der Waals-interaction with a contactor was studied,
where $Y$ is a constant.
To compare we write $Y\left(k\right)$ as a function of $Y_{0}$ where
$Y_0 = \lim\limits_{\substack{k \to 0}} Y\left(k\right)$,
\begin{eqnarray}
Y\left(k\right) = \frac{k Y_0 (d \epsilon_1+h_0 (\epsilon_2-\epsilon_1))}
{\epsilon_1 \tanh (k (d-h_0))+\epsilon_2 \tanh (h_0 k)}\,.
\end{eqnarray}
One gets the following conditions for instabilities
\begin{eqnarray}
\label{Y_slip}
Y_0 &=& \dfrac{- Z_s\left(k\right) \left(\epsilon_1 \tanh\left[k \left(d - h_0\right)\right]
+ \epsilon_2 \tanh\left[k h_0\right]\right)}{\lambda k^2 \left(l^2-k^2\right)
\sinh\left[k h_0\right]\sinh\left[l h_0\right]
\left(\epsilon_1 d + \left(\epsilon_1 - \epsilon_2 \right)h_0\right)}, \nonumber\\
&&\\
\label{Y_fix}
Y_0&=& \dfrac{2 Z_f\left(k\right) \left(\epsilon_1 \tanh\left[k \left(d - h_0\right)\right]
+ \epsilon_2 \tanh\left[k h_0\right]\right)}{\lambda k^2 \left(l^2-k^2\right) g^{-}\left(k,l\right)
\left(\epsilon_1 d + \left(\epsilon_1 - \epsilon_2 \right)h_0\right)}
\end{eqnarray}
for the slip and the fixed BC, respectively.
Fig.~\ref{Y_instab} represents stability diagrams for the elasto-electric instability,
for both BCs as given by Eqs.~(\ref{Y_slip}), (\ref{Y_fix}). For high enough $Y_0$, there exist
solutions with finite $k$. However, as $Y(k)$ depends on $k$ and one has a complicated
dependence on both $\lambda$ and $\gamma$, we could not obtain simple formulas for the threshold.
For $Y\left(k\right) = {\rm const}$ (as for Van der Waals-interactions)
we find the same result as given by Ref.~\cite{Pan09} (slip BC)
and as given in Ref.~\cite{Sharma08,HeEPL} (fixed BC).
We can observe the following general trends due to finite stretches:
considering the small wave number branch,
to get an undulation with the same small wave number,
$Y_{0} (\lambda = 0.9) >Y_{0} (\lambda = 1.0) >Y_{0} (\lambda = 1.1) $.
Thus compression acts stabilizing and tension destabilizing on small wave numbers.
On the other hand, for the large wave number branch one has
$Y_{0} (\lambda = 0.9) <Y_{0} (\lambda = 1.0) <Y_{0} (\lambda = 1.1) $,
thus tension is stabilizing and compression destabilizing.
Let us now look at the case with surface diffusion.
In the thin film limit $h_0k\ll1$ one gets for the slip BC
\begin{eqnarray}\label{ss_lambda}
s^{E}_{s} (k) &\simeq& s_s(k)+ M k^2 Y_0 \dfrac{\left( 3 \lambda^3 + 1 \right) }
{2 (1+\lambda^3)} \nonumber \\
&&+ M k^4 \dfrac{Y_0h_0\lambda^3/3}{2 \left(\lambda^3 + 1 \right)+(3 \lambda^3 + 1)Y_2},\\
\label{ss_kin_lambda}
s^{E}_{s,kin} (k) &\simeq& M k^2 \dfrac{G}{h_0 \lambda}
\dfrac{G \left(\lambda^3-1 \right)^2 + h_0 Y_0 \lambda \left( 1 + 3 \lambda^2 \right) }
{G \left(3 + \lambda^3 \right) - h_0 Y_0 \lambda},\qquad
\end{eqnarray}
excluding and including the effects of the kinematic BC, respectively.
In the latter case we show only the leading order contribution in $k^2$.
For the fixed BC one gets
\begin{eqnarray}
\label{sf_lambda}
s^{E}_{f} (k) &\simeq& s_{f}(k) +M k^2 Y_0 \nonumber\\
&&\hspace{-1cm}+ M k^4 \left(Y_0 \dfrac{h_0^2 \left(\lambda^3-1\right)}{2} + Y_2 \right), \,\,\,\\
\label{sf_kin_lambda}
s^{E}_{f,kin} (k) &\simeq& s_{f,kin} (k) + M k^2 Y_0 \nonumber\\
&& \hspace{-1cm}+ M k^4 \left( Y_0 h_0^2 \left(\lambda^3-1\right) + Y_0^2 \dfrac{h_0^3 \lambda}{3 G } + Y_2 \right)\,,
\end{eqnarray}
where we introduced $Y_2=\frac{1}{2}\frac{d^2}{dk^2}Y(k)_{|k=0}$.
In the limit of small stresses,
$\lambda=1\pm\delta$ with $\delta=\sigma_0/(3G)\ll 1$,
to lowest order $k^2$ and up to third order in stress one gets
\begin{eqnarray}
\label{ss_sigma}
s^{E}_{s} (k) &\simeq& M k^2
\bigg[ Y_0 + \sigma_0 \dfrac{Y_0}{\bar{E}} \nonumber\\
&&\quad\quad\quad + \sigma_0^2 \dfrac{3 \bar{E}-2\bar{Y_0}}{3 \bar{E}^2 h_0}
- 2 \sigma_0^3 \dfrac{9 \bar{E}+10\bar{Y_0}}{27\bar{E}h_0} \bigg],\\
\label{ss_kin_sigma}
s^{E}_{s,kin} (k) &\simeq& M k^2
\bigg[ Y_0 \dfrac{\bar{E}}{\bar{E}-\bar{Y_0}} +
\sigma_0 Y_0 \dfrac{6\bar{E}-5\bar{Y_0}}{3 \left(\bar{E}-\bar{Y_0}\right)^2}\nonumber\\
&&\hspace{-1cm} + \sigma_0^2 \frac{9\bar{E}^3-12 \bar{E}^2 \bar{Y_0}-12 \bar{E} \bar{Y_0}^2 +16 \bar{Y_0}^3 }
{9 \bar{E} h_0 \left( \bar{E}-\bar{Y_0}\right)^3}\nonumber\\
&&\hspace{-1cm} + \sigma_0^3 \frac{9\bar{E}^4-76 \bar{E}^3 \bar{Y_0}+116 \bar{E}^2 \bar{Y_0}^2-64
\bar{E} \bar{Y_0}^3 +16 \bar{Y_0}^4 }
{27 \bar{E}^2 h_0 \left( \bar{E}-\bar{Y_0}\right)^4}\bigg],\nonumber\\
\end{eqnarray}
where we introduced $\bar{Y_0}=h_0Y_0$, and
\begin{eqnarray}
\label{sf_kin_sigma}
s^{E}_{f} (k) = s^{E}_{f,kin} (k)&\simeq& M k^2 Y_0\,.
\end{eqnarray}
\begin{figure}[t]
\centering
\vspace{.2cm}
\includegraphics[width=0.45\textwidth]{fig6.eps}
\caption{\label{Growth_rate}
Growth rates as a function of reduced wave number
for different stretch factors $\lambda$, corresponding to different prestress, and with finite voltage applied
normal to the film.
Panel a) displays the case of the slip BC at the bottom. Without stress, $\lambda=1$,
the system is slightly unstable due to field-induced diffusion.
Compression ($\lambda<1$) or stretch ($\lambda>1$) lead to a more pronounced instability,
but with differing rates.
Solid lines have been obtained with the kinematic BC at the free surface,
pointed only with surface diffusion.
Panel b) shows the case of the fixed BC at the bottom, where finite
stresses only lead to small corrections.
Parameters: $\gamma = 0.5 h_{0} \bar{E}$, $\epsilon_1=2.5$, $\epsilon_2=1$, $d/h_0=5$, $h_0Y_0=0.001\bar E$.
}
\end{figure}
For the slip BC, Eqs.~(\ref{ss_lambda}, \ref{ss_kin_lambda})
display a coupling between prestress and the applied electric stress at the order $\mathcal{O}\left(k^2\right)$.
Hence the application of the field breaks the $\pm\sigma_0$-symmetry
already in lowest order in stress.
In the small stress limit
this coupling is linear like $Y_0\sigma_0$.
As one usually has $h_0Y_0\ll\bar{E}$ (otherwise the elasto-electric instability takes over),
for small wavelengths one might be driven to the conclusion
that the coupling between the electric field and the stress like $Y_{0} \sigma_0$
implies that compression acts stabilizing while stretch acts destabilizing.
However, the destabilizing contribution from the electric field, $M k^2 Y_0$,
is usually dominating and thus $\bar{s}^{E}>\bar{s}$,
compare e.g. Figs.~\ref{ssfig}a) and \ref{Growth_rate}a).
Thus the influence of the coupling is observed rather
beyond the maximum of the growth rate, cf.
Fig.~\ref{Growth_rate}a),
as a gap between the curves with compression and stretch.
However, in case of $\sigma_0\ll h_0Y\ll\bar E$, the two destabilizing forces
do not add and the growth rate under compression is indead slightly smaller
than the growth rate of an unstressed film.
As in the case without field one observes finite stress effects yielding
positive contributions like $+\sigma_0^3$ with the kinetic BC and negative ones
like $-\sigma_0^3$ with the non-kinetic version. Positive and negative prestress have thus
opposite effects and these effects depend on the kinematic BC.
Fig.~\ref{Growth_rate}a) displays the general growth rates for the slip BC.
For $\lambda<1$ ($\sigma_0<0$),
the kinematic version yields a smaller growth rate than the non-kinematic version.
However, for $\lambda>1$ ($\sigma_0 > 0$) the opposite is true.
Fig.~\ref{Growth_rate}b) shows that for the fixed BC finite stress
only leads to higher order corrections, since the leading order destabilization is
$+k^2 Y_0$ while the stress contributions are proportional to $k^4$.
Visible differences between $\bar{s}_{f,kin}\left(k\right)$
and $\bar{s}_{f}\left(k\right)$ appear only for rather high stretch factors, namely
$\lambda\gtrsim1.5$ or $\lambda \lesssim 0.7$ for the chosen surface tension.
For Fig.~\ref{Growth_rate} we used again parameters as suggested by Ref.~\cite{Barbero09},
namely a electrode distance of
$d = 5 h_{0}$, dielectric constants $\epsilon_2=2.5$ and $\epsilon_{1}=1$ for the PS film and the air gap, respectively, and
a voltage of $V=16\mathrm{V}$.
In reduced units this leads to $\gamma \simeq 0.5 h_{0} \bar{E}$, $Y_{0} h_{0} \simeq 0.0013 {\bar{E}}$.
Let us briefly discuss the relation of this work to the
experiments of Ref.~\cite{Barbero09}. There it has been found that
freshly produced films, that are supposedly stressed due to the nonequilibrium
preparation process of spin-coating, have faster growth rates
than aged films -- which had time to relax residual stresses.
This is in accordance with our findings that the two destabilization
mechanisms, the Grinfeld mechanism and the electric force acting on the free surface
of the film, in general join forces. However, in Ref.~\cite{Barbero09} it has
been found that the wave number of the instability
is smaller for fresh films than for aged films.
This is in contrast to our calculations, as in the general case
the unstable wave numbers increase
with stress, see also Fig.~\ref{Growth_rate}.
There are several possible reasons for this discrepancy,
like viscoelastic effects in the film, inhomogeneities, crust formation
due to spin-coating \cite{deGennes02}, etc.
As a next step we plan to generalize the approach proposed here to the
viscoelastic case, to come closer to these experiments.
Another interesting point is that the compressive-tensile symmetry
holding for the stress in case of the Grinfeld instability is broken
in several ways:
i) by finite stresses, ii) due to the kinematic BC,
i.e.~the coupling of film height and displacement field, and
iii) due to the presence of the external electric field.
While the effect is of order $\sigma_0^3$ in the absence of an electric field,
it is of order $Y_0\sigma_0$ in the presence of field.
Thus especially in an external electric field, surface undulations
have noticeably different growth rates and
this may be used experimentally to determine
whether stresses in thin films are compressive or tensile.
\begin{table}
{
\renewcommand{\arraystretch}{1.7}
\begin{tabular}{| l | c | c | c |c|}
\hline
system & BC & & destabilization $s\simeq$
& \hspace{1mm} Eq. \hspace{1mm} \\ \hline\hline
semi-$\infty$ \hspace{1mm} & & & $M k^2\frac{\sigma_0^2}{\bar E}k$
& \hspace{1mm} (\ref{DEhinf}) \hspace{1mm} \\ \hline
thin film \hspace{1mm} & \hspace{1mm}slip \hspace{1mm} & & $M k^2\frac{\sigma_0^2}{\bar E h_0}$
& \hspace{1mm} (\ref{DEhthin}) \hspace{1mm} \\ \hline
thin film & fixed & & $M k^2 \frac{4 \sigma_0^2}{\bar E h_0}(h_0k)^2$
& \hspace{.2mm} (\ref{DEhthinfix})\hspace{1mm} \\ \hline
thin film & slip & \hspace{1mm} el.field \hspace{1mm} & $M k^2\left(Y+\frac{h_0Y\sigma_0+\sigma_0^2}{\bar E h_0}\right)$
& \hspace{1mm} (\ref{ss_sigma}) \hspace{1mm} \\ \hline
thin film & kin,slip & \hspace{1mm} el.field \hspace{1mm} & $M k^2\left(Y+\frac{2h_0Y\sigma_0+\sigma_0^2}{\bar E h_0}\right)$
& \hspace{1mm} (\ref{ss_kin_sigma}) \hspace{1mm} \\ \hline
thin film & fixed & el.field & $M k^2Y$
& \hspace{1mm} (\ref{sf_kin_sigma}) \hspace{1mm} \\ \hline
\end{tabular}
}
\caption{\label{T2} Summary of the leading order destabilization terms
in the growth rate of surface undulations, $s(k)$, for
different BCs at the bottom and with or without electric field.
Both stress and electric field are assumed small, $\sigma_0,h_0 Y\ll\bar E$.}
\end{table}
\section{Conclusions and perspective}
\label{Concl}
We have studied the instability of a polymer film
under the simultaneous action of internal stress and an
externally applied electric field.
For this purpose we formulated a general framework that
has a very rich phenomenology:
in absence of surface diffusion the system is stable against buckling
but displays an electrically induced instability towards periodic undulations.
In case that the polymer chains are able to diffuse close to the surface
due to gradients in the chemical potential,
the Grinfeld mechanism becomes active, as well as a destabilizing contribution
induced by the external electric field. The growth rates of surface undulations
are sensitive to the boundary conditions at the bottom
and have a rich phenomenology, see Table \ref{T2}.
Our approach also highlights the importance of the coupling between the height
of the film's surface and the displacement field inside the film,
which naturally arises from the kinematic boundary condition at the film surface.
This coupling has been neglected in previous studies.
Its consequences can be seen as finite stress
corrections to the Grinfeld instability, and analogously for the
electric instability. Moreover, this coupling establishes the connection between
the above mentioned elasto-electric instabilities and the Grinfeld-like diffusive instabilities,
as becomes apparent from the general growth rates of height fluctuations.
These growth rates
have been calculated as a function of internal stress, electric field, mobility of the
chains and surface tension. It is shown that both destabilizing factors,
internal stress and electric field, generally add.
The relevance for recent experiments on spin-cast
thin polymer films has been only briefly discussed.
A generalization to the viscoelastic case, and possibly also including more structural details
of spin-cast film, is needed to account for these experiments.
In turn, as the experiments can measure separately
the most unstable wavelength and the growth rate,
they could give direct access to the internal stress and to
the mobility of polymers in thin films,
which both are of technological importance.
In particular, as the electric field makes the breakage of the
compressive-tensile symmetry of the Grinfeld instability induced by the
coupling to the displacement field noticeable,
careful measurements of the growth rates could be used to determine
the nature of the stresses, i.e. whether they are compressive or tensile.
The authors would like to thank Ken Sekimoto for stimulating discussions.
|
{
"timestamp": "2011-02-23T02:01:57",
"yymm": "1009",
"arxiv_id": "1009.4066",
"language": "en",
"url": "https://arxiv.org/abs/1009.4066"
}
|
\section{Introduction}
It is the instinct of every mathematician that whenever an infinite
object is defined in terms of a finite object one should be able to
describe various apparently infinite properties of the infinite object
in finitely-many terms. For example, when one considers infinite random
walks on a finite digraph it is satisfying to be able to describe the
asymptotic properties of these random walks in terms of the
finitely-many eigenvalues of the adjacency matrix.
In this paper we consider infinite partially ordered
sets (posets) associated to finite
directed graphs. The level poset of a graph that we introduce is an
infinite voltage graph closely related to the finite voltage graphs
studied by Gross and Tucker. It is a natural question to consider whether
a level poset has Eulerian intervals,
that is, every non-singleton interval satisfies the Euler-Poincar\'e relation.
One method to form Eulerian posets is via the doubling operations.
This corresponds to a standard trick widely used
in the study of network flows.
We extend these operations to level posets.
We look at questions that are often asked in the study of Eulerian
posets: verifying Eulerianness,
finding sufficient conditions which imply the order complex
is shellable and describing the flag numbers. Usually these questions are
aimed at a specific family of finite posets and
explicit answers are given. Here we instead look at infinitely-many intervals
defined by a single finite directed graph.
When the underlying graph of a level poset is strongly connected,
we show that it is enough to verify the Eulerian condition for intervals
up to a certain rank. This bound is linear in terms of
the two parameters period and index of the level poset.
Furthermore, for these Eulerian level posets we also
obtain that their order must be even.
The order two Eulerian poset is the classical butterfly poset.
See Example~\ref{example_order_4}
for an order $4$ example.
To show that a level poset has shellable intervals
we introduce the vertex shelling order condition.
This condition is an instance of
Kozlov's $CC$-labelings
and it implies shellability.
Furthermore,
we prove it is enough to verify this condition
for intervals whose length is bounded by the sum of
period and the index.
This is still a large task. However, we automate
it using the algebra of walks, reducing the problem of
computing powers of a certain matrix modulo an ideal.
See Example~\ref{example_order_4_shelling}
for such a calculation.
In this example we conclude that the
order complexes of the intervals are
not just homotopic equivalent
to spheres, but homeomorphic to them.
The ${\bf c}{\bf d}$-index is an invariant encoding
the flag $f$-vector of an Eulerian poset which
removes all linear relations among the flag $f$-vector entries.
It is a non-commutative homogeneous polynomial
in the two variables ${\bf c}$ and ${\bf d}$. For level Eulerian posets
there are infinitely-many intervals. We capture this information
by summing all the ${\bf c}{\bf d}$-indicies. This gives a non-commutative
formal power series which we call the ${\bf c}{\bf d}$-series. We show that the
${\bf c}{\bf d}$-series is a rational non-commutative generating function.
See Theorem~\ref{theorem_cd_rational}.
Recall that the infinite butterfly poset has the property
that the ${\bf c}{\bf d}$-index of any length $m+1$ interval
equals ${\bf c}^{m}$.
In our order $4$ example
of a level Eulerian poset,
there are intervals of length $m+1$ whose
${\bf c}{\bf d}$-index is the sum of every degree $m$ ${\bf c}{\bf d}$-monomial.
See Corollary~\ref{corollary_sum_of_monomials}.
In the concluding remarks we end with some open questions.
\section{Preliminaries}
\subsection{Graded, Eulerian and half-Eulerian posets}
A partially ordered set $P$ is {\em graded} if it has a unique minimum element
$\widehat{0}$, a unique maximum element $\widehat{1}$ and a rank function $\rho:P\rightarrow
{\mathbb N}$ such that $\rho(\widehat{0})=0$ and for every cover relation
$x \prec y$
we have $\rho(y)-\rho(x)=1$. The rank of $\widehat{1}$ is called the
{\em rank} of the
poset.
For two elements
$x \leq y$ in $P$ define
the rank difference
$\rho(x,y)$ by $\rho(y) - \rho(x)$.
Given a graded poset $P$ of rank $n+1$ and a subset
$S\subseteq \{1,\ldots,n\}$, define the {\em
$S$-rank selected subposet of $P$} to be the poset
$P_{S} = \{ x \in P\::\: \rho(x) \in S\} \cup \{\widehat{0},\widehat{1}\}.$
The {\em flag $f$-vector}
$(f_{S}(P)\: :\: S \subseteq \{1, \ldots, n\})$
of $P$ is
the $2^n$-dimensional vector whose entry
$f_S(P)$ is the number of maximal chains in $P_S$. For further details
about graded posets, see Stanley~\cite{Stanley_EC_1}.
A graded partially ordered set $P$ is {\em Eulerian} if every interval
$[x,y]$ in $P$ of rank at least $1$
satisfies $\sum_{x \leq z \leq y} (-1)^{\rho(z)}=0$.
Equivalently, the M\"obius function $\mu$ of the poset $P$
satisfies $\mu(x,y) = (-1)^{\rho(x,y)}$.
Classical examples of Eulerian posets include the face lattices
of polytopes and the Bruhat order of a Coxeter group.
The {\em horizontal double} $D_{\leftrightarrow}(P)$
of a graded poset $P$ is obtained by
replacing each element $x\in P - \{\widehat{0},\widehat{1}\}$ by two copies $x_{1}$ and
$x_{2}$ and preserving the partial order of the original poset $P$, that
is, we set $x_{i}<y_{j}$ in $D_{\leftrightarrow}(P)$
if and only if $x<y$ holds in $P$.
Following Bayer and Hetyei~\cite{Bayer_Hetyei_E,Bayer_Hetyei_G},
we call a graded poset $P$ {\em half-Eulerian} if its horizontal double
is Eulerian. The following lemma appears
in~\cite[Proposition~2.2]{Bayer_Hetyei_E}.
\begin{lemma}[Bayer--Hetyei]
\label{lemma_h_double}
A graded partially ordered set $P$ is half-Eulerian if and only if for every
non-singleton interval $[x,y]$ of $P$
$$
\sum_{x < z < y}
(-1)^{\rho(x,z) -1}
=
\left\{ \begin{array}{c l}
1 & \text{ if $\rho(x,y)$ is even,} \\
0 & \text{ if $\rho(x,y)$ is odd.}
\end{array} \right.
$$
\end{lemma}
As noted in~\cite[Section~4]{Bayer_Hetyei_G}, every graded poset $P$
gives rise to a half-Eulerian poset via the
``vertical doubling'' operation.
\begin{definition}
Given a graded poset $P$,
the {\em vertical double} of $P$ is the set
$D_{\updownarrow}(P)$ obtained by replacing each $x\in P- \{\widehat{0},\widehat{1}\}$ by two
copies $x_{1}$ and $x_{2}$, with $u<_{D_{\updownarrow}(P)} v$ in $Q$
exactly when one of the following conditions hold:
\begin{itemize}
\item[(i)] $u=\hat{0}$, $v\in P-\{\hat{0}\}$;
\item[(ii)] $u\in P-\{\hat{1}\}$, $v=\hat{1}$;
\item[(iii)] $u=x_{1}$ and $v=x_{2}$ for some $x\in
P-\{\hat{0},\hat{1}\}$; or
\item[(iv)] $u=x_{i}$ and $v=y_{j}$ for some $x, y\in
P-\{\hat{0},\hat{1}\}$, with $x<_P y$.
\end{itemize}
\end{definition}
\begin{lemma}[Bayer--Hetyei]
\label{lemma_v_double}
For a graded poset $P$,
the vertical double
$D_{\updownarrow}(P)$ is a half-Eulerian poset.
\end{lemma}
\subsection{Shelling the order complex of a graded poset}
Recall a simplicial complex $\Delta$ is a family of subsets
({\em faces}) of a finite vertex set $V$ satisfying $\{v\} \in \Delta$
for all $v \in V$ and if $\sigma \in \Delta$ and $\tau\subseteq \sigma$
then $\tau\in \Delta$. Maximal faces are called {\em facets}.
In this paper we will only consider order
complexes of graded posets. The {\em order complex $\Delta(P)$}
of a graded poset $P$ is the simplicial complex with vertex set
$P - \{\widehat{0},\widehat{1}\}$ whose faces are the chains
contained in $ P- \{\widehat{0},\widehat{1}\}$, that is,
$$ \Delta(P)
=
\{\{x_{1},x_{2}, \ldots, x_{k}\}
\:\: : \:\:
\widehat{0} < x_{1} < x_{2} < \cdots < x_{k} < \widehat{1}\} . $$
A simplicial complex is {\em pure} if every facet has the same dimension.
For a graded poset $P$ of rank $n+1$, the order complex
$\Delta(P)$
is pure of dimension $n-1$. A pure simplicial complex $\Delta$ is
{\em shellable} if there is an ordering $F_{1},F_{2},\ldots,F_{t}$ of its
facets such that for every $k\in \{2,\ldots,t\}$
the collection of faces of $F_{k}$ contained in some earlier $F_{i}$
is itself a
pure simplicial complex of dimension $\dim(\Delta)-1$. Equivalently,
there exists a face $R(F_{k})$ of $F_{k}$, called the facet restriction,
not contained in any earlier
facet such that every face $\sigma\subseteq F_{k}$ not contained in any
earlier $F_{i}$ contains $R(F_{k})$.
A complex being shellable implies
it is homotopy equivalent to a wedge of spheres of the same
dimension as the complex. For further details,
we refer the reader to the articles of Bj\"orner
and Wachs~\cite{Bjorner,Wachs}.
A shelling of the order complex of a graded poset is usually found by
labeling the cover relations in the maximal chains
of $P$. The first such labelings were the {\em
$CL$-labelings} introduced by Bj\"orner and
Wachs~\cite{Bjorner_Wachs_0,Bjorner_Wachs_1}. In this paper we will
consider a special example of Kozlov's {\em
$CC$-labelings}~\cite{Kozlov}, which were rediscovered
independently by Hersh and Kleinberg (see the Introduction
of~\cite{Babson_Hersh}).
\subsection{Periodicity of nonnegative matrices}
We will need a few facts regarding sufficiently high powers
of nonnegative square matrices. Unless noted otherwise, all statements cited
in this subsection may be found in the monograph of Sachkov and
Tarakanov~\cite[Chapter~6]{Sachkov_Tarakanov}.
The {\em underlying digraph $\Gamma(A)$} of a square matrix
$A=(a_{i,j})_{1\leq i,j\leq n}$ with nonnegative entries is the directed
graph on the vertex set $\{1,2,\ldots,n\}$ with $(i,j)$ being an edge if
and only if $a_{i,j}>0$.
Here and in the rest of the paper we use the notation
$A^{k}=(a^{(k)}_{i,j})_{1\leq i,j\leq n}$.
Given a vertex $i$ such that there is a
directed walk of positive length from $i$ to $i$, the
{\em period $d(i)$}
of the vertex $i$ is the greatest common divisor of all
positive integers $k$ satisfying $a^{(k)}_{i,i}>0$.
The period is constant on
strong components of~$\Gamma(A)$.
\begin{lemma}
\label{lemma_period_constant}
If the vertices $i\neq j$ of $\Gamma(A)$ belong to the same strong
component then $d(i)=d(j)$.
\end{lemma}
The matrix $A$ is {\em indecomposable} or {\em
irreducible} if for any $i,j\in \{1,2,\ldots,n\}$ there is a $t>0$ such
that the $(i,j)$ entry of $A^t$ is positive.
It is easy to
see that $A$ is indecomposable if and only if its
underlying digraph is strongly connected, that is, for any pair of vertices
$i$ and $j$ there is a directed walk from $i$ to~$j$. As
a consequence of Lemma~\ref{lemma_period_constant}
all vertices of the underlying
graph of an indecomposable matrix~$A$ have the same period. We call this
number the {\em period of the indecomposable matrix $A$}. Given an
$n \times n$
indecomposable matrix of period $d$, for each $i=1,2,\ldots,n$
there exists an integer $t_{0}(i)$ such that $a^{(k d)}_{i,i}>0$ holds for
all $k\geq t_{0}(i)$. Using
this observation is easy to show the following
theorem. See~\cite[Theorem~6.2.2 and Lemma~6.2.3]{Sachkov_Tarakanov}.
\begin{theorem}
\label{theorem_indecomposable-block-matrix}
Let $A$ be an $n\times n$ indecomposable nonnegative matrix of period
$d$. If $i$ is a fixed vertex of the digraph $\Gamma(A)$ then for any
other vertex $j$ there is a unique integer $r_{j}$
such that $0 \leq r_{j} \leq d-1$
and the following two statements hold:
\begin{itemize}
\item[(1)] $a^{(s)}_{i,j}>0$ implies $s \equiv r_{j} \bmod d$,
\item[(2)] there is a positive $t(j)$ such that
$a^{(k d + r_{j})}_{i,j}>0$ for
all $k \geq t(j)$.
\end{itemize}
Setting $j\in C_r$ if and only if $r_{j}=r$ provides a partitioning
$\{1,2,\ldots,n\}=\biguplus_{q=0}^{d-1} C_{q-1}$. Replacing $i$ with an
arbitrary fixed vertex results in the same ordered list of subclasses,
up to a cyclic rotation of the indices.
\end{theorem}
Ordering the elements of the set $\{1,2,\ldots,n\}$ in such a way that
the elements of each block $C_q$ form a consecutive sublist results in
a block matrix of the form
\begin{equation}
\label{equation_block-matrix}
A =
\begin{pmatrix}
0 &Q_{0,1} & 0 & \cdots & 0\\
0 & 0 & Q_{1,2} & \cdots & 0\\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & Q_{d-2,d-1}\\
Q_{d-1,0} & 0 & 0 & \cdots & 0\\
\end{pmatrix} ,
\end{equation}
where $Q_{q,q+1}$ occupies the rows
indexed with $C_q$ and the columns indexed by $C_{q+1}$ (here we set
$C_{d} = C_{0}$). The block matrix in~\eqref{equation_block-matrix} is
the {\em canonical form} of the indecomposable matrix $A$ with period
$d$. By Theorem~\ref{theorem_indecomposable-block-matrix} there
is a $t>0$ such that the canonical form of $A^{t d + 1}$ is similar to the
one given in~\eqref{equation_block-matrix} with the additional property that
all entries in the blocks $Q_{q,q+1}$ are strictly positive.
An $n\times n$ matrix $A$ with nonnegative entries is {\em primitive}
if there is a $\gamma>0$ such that all entries of~$A^{\gamma}$ are
positive. The smallest $\gamma$ with the above property is the {\em
exponent} of the primitive matrix $A$.
It is straightforward to see that a
nonnegative matrix is primitive if and only if it is indecomposable and
{\em aperiodic}, i.e., its period $d$ equals $1$. There is a quadratic
upper bound on the exponent of a primitive matrix due to Holladay and
Varga~\cite{Holladay_Varga}. See \cite[Theorem~6.2.10]{Sachkov_Tarakanov}.
\begin{theorem}[Holladay--Varga]
\label{theorem_exp-primitive}
The exponent $\gamma$ of an $n\times n$ primitive matrix satisfies
$$
\gamma \leq n^{2} - 2n + 2.
$$
\end{theorem}
The matrix operator we are about to introduce will be frequently used
in our paper and makes also stating the next few results easier.
\begin{definition}
Given a matrix $A$ with nonnegative entries, let
$\operatorname{Bin}(A)$ denote the binary matrix formed by replacing each
nonzero entry of $A$ with $1$. We call the resulting matrix the {\em binary
reduction} of $A$.
\end{definition}
Thus a matrix $A$ is primitive if there exists a power
$k$ such that $\operatorname{Bin}(A^{k}) = J$, where
$J$ is the matrix consisting of all $1$'s.
A generalization of Theorem~\ref{theorem_indecomposable-block-matrix}
may be found in the work of Heap and Lynn~\cite{Heap_Lynn}.
See also~\cite{Ptak,Ptak_Sedlacek,Rosenblatt}.
\begin{theorem}
\label{theorem_decomposable-block-matrix}
Given any non-negative square matrix $A$ there exists
integers $d$ and $\gamma$ such that
$\operatorname{Bin}(A^{t+d}) = \operatorname{Bin}(A^{t})$ for all $t \geq \gamma$.
\end{theorem}
The smallest integer $\gamma$ such that
for all $t \geq \gamma$ we have
$\operatorname{Bin}(A^{t+d}) = \operatorname{Bin}(A^{t})$
is known as the {\em index} of the matrix.
For a primitive matrix this is the exponent.
For estimates on the period $d$ and
the index~$\gamma$, we refer the reader to~\cite{Heap_Lynn}.
Here we only wish to emphasize the following
immediate generalization of Theorem~\ref{theorem_exp-primitive}.
See~\cite[Equation~(1.4)]{Heap_Lynn}.
\begin{theorem}
\label{theorem_exp-indecomposable}
Let $A$ be an $n\times n$ indecomposable matrix with nonnegative entries
having period $d$.
An upper bound for the index $\gamma$ of $A$
is $\gamma \leq (q^2-2q+2)d+2r$,
where $n = q d + r$ with $0 \leq r < d-1$.
\end{theorem}
Let $\Gamma$ be any digraph on a vertex set $V$ such that its edge set
$E$ is a subset of $V\times V$, i.e., $\Gamma$ may have loops but no
multiple edges. Recall the {\em adjacency matrix} $A$ of $\Gamma$ is a
$|V|\times |V|$ matrix whose rows and columns are indexed by the
vertices. The entry in the row indexed by $u\in V$ and in the column
$v\in V$ is $1$ if $(u,v)\in E$, and zero otherwise. Clearly $\Gamma$
is the underlying graph of its adjacency matrix. We may extend
the above notions of period and aperiodicity from matrices to digraphs
by defining the period of a digraph to be the period of its adjacency
matrix. See for instance~\cite{Perrin_Schutzenberger}. In particular, a
directed graph is aperiodic if and only if it is
strongly connected and there is no $k>1$ that divides the length of
every directed cycle.
\section{Level posets}
\begin{definition}
A partially ordered set $P$ is a {\em level poset} if the set of its
elements is of the form $V\times {\mathbb Z}$ for some finite nonempty
set $V$, the projection
onto the second coordinate is a rank function, and for any $u,v\in V$
and $i\in {\mathbb Z}$ we have $(u,i)<(v,i+1)$ if and only if
$(u,0)<(v,1)$ holds.
\end{definition}
\begin{figure}[t]
\setlength{\unitlength}{1.2mm}
\begin{center}
\begin{picture}(40,50)(0,0)
\put(0,25){
$\displaystyle
\begin{pmatrix}
1 & 1 & 1 & 0 \\
1 & 0 & 1 & 1 \\
0 & 1 & 0 & 1 \\
1 & 0 & 1 & 1
\end{pmatrix}
$}
\end{picture}
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\end{picture}
\end{center}
\caption{An adjacency matrix and its associated level poset.
Note that this is an example of a level Eulerian poset.}
\label{figure_main}
\end{figure}
Informally speaking, the Hasse diagram of a level poset
can be thought of
as a graph containing
a copy of the same vertex set $V$ at each ``level'' such that the portion
of the Hasse diagram containing the edges between elements of rank $i$
and rank $i+1$ may be obtained by vertically shifting the edges in the
Hasse diagram between the elements of rank $0$ and $1$. An example of a
level poset is shown in Figure~\ref{figure_main}. (The meaning of the term
Eulerian in this context will be explained in Section~\ref{sec:Eulerian}.)
Clearly it is sufficient to know the cover relations of the form
$(u,0)\prec (v,1)$ to obtain a complete description of a level
poset. Introducing the digraph $G$ with vertex set $V$ and edge set
$E:=\{(u,v)\in V\times V\::\: (u,0)<(v,1)\}$, we obtain a digraph
representing a relation $E\subseteq V\times V$ on the vertex set $V$, i.e., a
digraph with no multiple edges but possibly containing loops. We call
$G$ the {\em underlying graph} of the level poset $P$ and $P$ the {\em
level poset of $G$}. The poset $P$ and the digraph $G$
determine each other uniquely.
\begin{lemma}
\label{lemma_walks}
Let $P$ be a level poset on $V\times {\mathbb Z}$ and let $G$ be its underlying
digraph. Then for all $i,j\in{\mathbb Z}$ and for all $u,v\in V$ we have
$(u,i)<(v,j)$ in $P$ if and only if $i<j$ and there is a walk
$u=u_{0} \rightarrow u_{1} \rightarrow \cdots\rightarrow u_{j-i}=v$
of length $j-i$ from $u$ to $v$ in $G$.
\end{lemma}
The straightforward verification is left to the reader.
\begin{remark}
{\rm
By directing all the edges upwards in the Hasse diagram of $P$,
we obtain the {\em (right) derived graph} of the voltage
graph obtained from $G$ by assigning the voltage $1\in {\mathbb Z}$ to
each directed edge. For a
detailed discussion of the theory of voltage graphs, we refer the reader
to the work of Gross and Tucker~\cite{Gross_Tucker}. The
classical theory of voltage graphs focuses on the case where the
voltages belong to a finite group. Here we have to consider
${\mathbb Z}$, that is,
the simplest possible infinite group.
}
\end{remark}
Since the underlying digraph of a level poset $G$ is
uniquely determined by its adjacency matrix, every level poset is
uniquely determined by the adjacency matrix of its underlying
digraph. For brevity, we will use the term {\em underlying
matrix $M$} for ``adjacency matrix of the underlying digraph'' of a level
poset $P$, and the term {\em level poset of $M$} for
``level poset of the digraph whose adjacency matrix is~$M$''. The order
of the rows and columns of the
adjacency matrix corresponds to the
order of vertices at the same level read from the left to the
right in a Hasse diagram of the corresponding level
poset. For any square matrix $M$ whose rows and columns are
indexed with elements of a set $V$, we will use the notation $M_{u,v}$
for the entry in the row indexed by $u\in V$ and in the column indexed
by $v\in V$.
Using Lemma~\ref{lemma_walks} we may describe the partial order of $P$ in
terms of its underlying matrix $M$ as follows.
\begin{corollary}
\label{corollary_walks}
Given a level poset $P$ with underlying matrix $M$, we have
$(u,i)<(v,j)$ in $P$ if and only if $i<j$ and $M^{j-i}_{u,v}>0$ hold.
\end{corollary}
Using the operation $\operatorname{Bin}$ we may rephrase Corollary~\ref{corollary_walks} as
follows.
\begin{corollary}
Given a level poset $P$ with underlying matrix $M$, we have
$(u,i)<(v,j)$ in $P$ if and only if $i<j$ and $\operatorname{Bin}(M^{j-i})_{u,v}=1$ hold.
\end{corollary}
Clearly the underlying digraph of a level poset is strongly connected if
and only if the underlying matrix $M$ is indecomposable. Equivalently,
for any pair of vertices $u,v\in V$ such that $u\neq v$ there is a $p>0$ such
that the adjacency matrix $M$ satisfies $\operatorname{Bin}(M^{p})_{u,v}=1$.
If $|V|>1$ then the adjacency matrix $M$ of a
strongly connected digraph must also satisfy $\operatorname{Bin}(M^{p})_{u,u}=1$ for some
$p$, given any $u\in V$.
Having a strongly connected underlying digraph is
neither a necessary nor sufficient condition for the Hasse diagram of a
level poset (considered as an undirected graph) to be a connected
graph. An example of a connected level poset whose underlying digraph is
not strongly connected is the level poset with the underlying adjacency
matrix
$M=
\begin{pmatrix}
0 & 1 \\ 0 & 1
\end{pmatrix}$.
For level posets with strongly connected underlying digraphs, a
necessary and sufficient condition for the connectivity of their Hasse
diagram may be stated using the notion of aperiodic
graphs.
A digraph on $n$ vertices is {\em aperiodic} if and only if
the underlying adjacency matrix $M$ is primitive.
\begin{theorem}
Assume that $P$ is the level poset of a strongly connected digraph
$G$. Then the Hasse diagram of $P$ is connected if and only
$G$ is aperiodic.
\end{theorem}
\begin{proof}
Assume first $G$ is aperiodic and that its adjacency matrix $M$
satisfies $\operatorname{Bin}(M^{p})=J$. Clearly $\operatorname{Bin}(M^{n})=J$ for all
$n\geq p$. Given any $(u,i)$ and $(v,j)$ in $P$ there is a directed walk of
length $p+\max(i,j)-i$ from $u$ to $u$ and a directed walk of
length $p+\max(i,j)-j$ from $v$ to $u$ in~$G$. The first walk lifts to a
walk from $(u,i)$ to $(u,\max(i,j)+p)$ in the Hasse diagram of $P$,
whereas the second walk lifts to a walk from $(v,j)$ to
$(u,\max(i,j)+p)$ in the Hasse diagram. Thus we may walk from $(u,i)$ to
$(v,j)$ by first walking along the edges of the walk from $(u,i)$ to
$(u,\max(i,j)+p)$ and then following the edges of the walk from $(v,j)$ to
$(u,\max(i,j)+p)$ backwards.
Assume next that $G$ is not aperiodic. Let $k>1$ be an integer dividing
the length of every cycle. It is easy to see that
we may color the vertex set $V$ of $G$ using $k$ colors in such a way
that $(u,v)$ is an edge only if $u$ and $v$ have different colors. This
coloring may be lifted to the Hasse diagram of $P$ by setting the color
of $(u,i)$ to be the color of $u$ for each $(u,i)\in P$. There is no
walk between elements of the same color in the Hasse diagram of $P$.
\end{proof}
Let $\alpha = (\alpha_{1}, \ldots, \alpha_{r})$ be a composition
of~$m$, that is,
$\alpha_{1}, \ldots, \alpha_{r}$ are positive integers
whose sum is~$m$.
Let $S$ be the associated subset
of $\{1, \ldots, m-1\}$, that is,
$$ S = \{\alpha_{1}, \alpha_{1}+\alpha_{2},
\ldots,
\alpha_{1} + \cdots + \alpha_{r-1}\} . $$
The flag $f$-vector entry $S$ of any interval
$[(u,i),(v,i+m)] \cong [(u,0),(v,m)]$ in
a level poset may computed using its underlying adjacency matrix as
follows.
\begin{lemma}
\label{lemma_flag_f}
Let $P$ be a level poset whose underlying digraph has vertex set $V$
of cardinality $n$.
Let $F_{S}$ be the $n \times n$ matrix whose $(u,v)$ entry is
$f_{S}([(u,0),(v,m)])$ if $(u,0) \leq (v,m)$ in $P$
and $0$ otherwise. Then the matrix~$F_{S}$
is given by
$$ F_{S} = \operatorname{Bin}(M^{\alpha_{1}}) \cdot
\operatorname{Bin}(M^{\alpha_{2}}) \cdots
\operatorname{Bin}(M^{\alpha_{r}}) , $$
where $(\alpha_{1}, \ldots, \alpha_{r})$
is the composition associated with the subset
$S \subseteq \{1, \ldots, m-1\}$.
\end{lemma}
Note that every interval $[(u,0),(v,m)]$ in $P$ is isomorphic to all
intervals of the form $[(u,i),(v,i+m)]$ where $i\in {\mathbb Z}$
is an arbitrary integer.
\section{Level Eulerian posets}
\label{sec:Eulerian}
\begin{definition}
\label{definition_level_Eulerian}
We call a level poset $P$ a {\em level Eulerian poset} if every interval
is Eulerian.
\end{definition}
As a consequence of Lemma~\ref{lemma_flag_f}
we have the following condition for Eulerianness.
\begin{lemma}
A level poset is Eulerian
if and only if its adjacency matrix $M$
satisfies
\begin{equation}
\sum_{i=0}^{p} (-1)^{i} \cdot \operatorname{Bin}(M^{i}) \cdot \operatorname{Bin}(M^{p-i}) = 0
\label{equation_level_Eulerian}
\end{equation}
holds for all $p \geq 1$.
\label{lemma_level_Eulerian}
\end{lemma}
As it was noted in~\cite[Lemma~4.4]{Ehrenborg_k-Eulerian}
and~\cite[Lemma~2.6]{Ehrenborg_Readdy_b},
a graded poset of
odd rank is Eulerian if all of its proper intervals
are Eulerian. Thus it suffices to verify the condition
in Lemma~\ref{lemma_level_Eulerian} for even integers $p$. As a consequence
of Theorem~\ref{theorem_decomposable-block-matrix}, equation
\eqref{equation_level_Eulerian} only needs to be verified for
finitely-many values of $p$.
\begin{theorem}
Let $P$ be the level poset of an $n\times n$
indecomposable matrix $M$ with period $d$ and
index~$\gamma$.
Then $P$ is level Eulerian if and only if $M$ satisfies
the Eulerian condition~\eqref{equation_level_Eulerian}
for $p<2\gamma+4d$. For odd $d$, the bound for $p$ may be improved to
$p<2\gamma+2d$.
\label{theorem_Euler_bound}
\end{theorem}
\begin{proof}
We introduce $\Sigma(p)$ as a shorthand for $\sum_{i=0}^{p} (-1)^{i}
\cdot \operatorname{Bin}(M^{i}) \cdot \operatorname{Bin}(M^{p-i})$.
We wish to calculate
$\Sigma(p+2d)-\Sigma(p)$ for an arbitrary $p\geq 2\gamma$.
For $i=0,1,\ldots,\gamma-1$, the term $(-1)^{i} \cdot \operatorname{Bin}(M^{i})
\cdot \operatorname{Bin}(M^{p-i})$ in $\Sigma (p)$ cancels with the term $(-1)^{i} \cdot
\operatorname{Bin}(M^{i}) \cdot \operatorname{Bin}(M^{p+2d-i})$ in $\Sigma(p+2d)$
since $i<\gamma (d)$ and $p\geq
2\gamma$ imply $p-i\geq \gamma$.
For $i=\gamma, \gamma+1,\ldots, p$, the term $(-1)^{i} \cdot
\operatorname{Bin}(M^{i}) \cdot \operatorname{Bin}(M^{p-i})$ in $\Sigma (p)$
cancels with the term
$(-1)^{i} \cdot \operatorname{Bin}(M^{i+2d}) \cdot \operatorname{Bin}(M^{p-i})$ in $\Sigma (p+2d)$
since $\operatorname{Bin}(M^{i})=\operatorname{Bin}(M^{i+2d})$ for $i\geq
\gamma$.
After these cancellations, we obtain
\begin{equation}
\label{equation_p2d}
\Sigma(p+2d)-\Sigma(p)=\sum_{i=\gamma}^{\gamma+2d-1} (-1)^{i}
\cdot \operatorname{Bin}(M^{i}) \cdot \operatorname{Bin}(M^{p+2d-i})\quad\mbox{for $p\geq 2\gamma$}.
\end{equation}
If $d$ is odd, then the right-hand side of~\eqref{equation_p2d} is
zero. Indeed, for each $i$ satisfying $\gamma\leq i\leq \gamma+d-1$,
the term $(-1)^{i} \cdot \operatorname{Bin}(M^{i}) \cdot \operatorname{Bin}(M^{p+2d-i})$ cancels with
$(-1)^{d+i} \cdot \operatorname{Bin}(M^{d+i}) \cdot \operatorname{Bin}(M^{p+d-i})$ since $(-1)^d=-1$,
$\operatorname{Bin}(M^{i})=\operatorname{Bin}(M^{d+i})$ and $\operatorname{Bin}(M^{p+2d-i})=\operatorname{Bin}(M^{p+d-i})$.
This concludes the proof of the theorem in the case when $d$ is odd.
Assume from now on that $d$ is even. Substituting any $p\geq 2\gamma+2d$
in~\eqref{equation_p2d} yields
\begin{align*}
\Sigma(p+2d)-\Sigma(p)
&=\sum_{i=\gamma}^{\gamma+2d-1} (-1)^{i} \cdot \operatorname{Bin}(M^{i}) \cdot
\operatorname{Bin}(M^{p+2d-i})\\
&=\sum_{i=\gamma}^{\gamma+2d-1} (-1)^{i} \cdot \operatorname{Bin}(M^{i}) \cdot \operatorname{Bin}(M^{p-i})
=\Sigma(p)-\Sigma(p-2d).\\
\end{align*}
We obtain that
\begin{equation}
\label{equation_sigma_prec}
\Sigma(p+2d)-\Sigma(p)=\Sigma(p)-\Sigma(p-2d) \quad\mbox{holds for
$p\geq 2\gamma+2d$.}
\end{equation}
Therefore, if we verify that $\Sigma(p)=0$ holds for $p\leq
2\gamma+4d$, the equality $\Sigma(p)=0$ for $p>2(\gamma+d)$ may
be shown by induction on $p$ using~\eqref{equation_sigma_prec}.
\end{proof}
As a consequence of Theorems~\ref{theorem_exp-primitive} and~\ref{theorem_Euler_bound}
we obtain the following upper bound.
\begin{corollary}
To determine whether the level poset $P$ of an $n\times n$
primitive
binary matrix $M$ is a level Eulerian poset one must only verify
the Eulerian condition~\eqref{equation_level_Eulerian}
for $p\leq 2n^2-4n+6$.
\end{corollary}
\begin{remark}
{\rm
For a general $n \times n$
adjacency matrix it seems hard to give a better than
exponential estimate as a function of $n$ for the bounds
given in Theorem~\ref{theorem_Euler_bound}. However, for indecomposable
matrices we may still obtain a polynomial estimate using
Theorem~\ref{theorem_exp-indecomposable}.
}
\end{remark}
\begin{example}
{\rm
The simplest level Eulerian poset is the butterfly poset
whose underlying adjacency matrix is
$$ M = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} . $$
This matrix has exponent $\gamma = 1$ and hence by
Theorem~\ref{theorem_Euler_bound} it is enough
to verify the Eulerian
condition~\eqref{equation_level_Eulerian} for $p=2$.
}
\label{example_order_2}
\end{example}
\begin{example}
{\rm
Consider the level poset shown in Figure~\ref{figure_main}. Its underlying
adjacency matrix $M$ satisfies
$$
\operatorname{Bin}(M^{2})=
\begin{pmatrix}
1&1&1&1\\
1&1&1&1\\
1&0&1&1\\
1&1&1&1\\
\end{pmatrix} $$
and $\operatorname{Bin}(M^{3})=J$.
Thus $M$ is primitive and the exponent is given by
$\gamma=3$.
Theorem~\ref{theorem_Euler_bound} gives the bound $p < 8$.
Hence to show that this matrix
produces a level Eulerian poset, we need verify
the Eulerian condition~\eqref{equation_level_Eulerian}
for the three values $p = 2, 4, 6$, which is a straightforward task.
}
\label{example_order_4}
\end{example}
Starting from the butterfly poset, for each $n\geq 2$ we may construct
a level Eulerian poset whose underlying digraph has $n$ vertices by
repeatedly using the following lemma.
The drawback to this construction is that it does
not add any more strongly connected components to the underlying graph.
\begin{lemma}
Let $M$ be a $n \times n$ matrix whose poset is level Eulerian
and let $\vec{v}$ be a column vector of~$M$.
Then\\[2mm]
(i)
the level poset of the transpose matrix $M^{T}$ is
also level Eulerian.
\\
(ii)
the $(n+1) \times (n+1)$ matrix
$$ \begin{pmatrix} M & \vec{v} \\ 0 & 0 \end{pmatrix} $$
is also level Eulerian.
\end{lemma}
If we restrict our attention to level posets with strongly
connected underlying digraphs, we obtain the following restriction
on the order.
\begin{theorem}
Let $P$ be a level Eulerian poset whose underlying matrix $M$
is indecomposable.
Then the order of the matrix $M$ is even.
\label{theorem_even_order}
\end{theorem}
\begin{proof}
Let $d$ and $\gamma$ be respectively the period and index
of the matrix $M$.
Let $\delta$ be the least multiple of $d$ which is greater than or equal
to $\gamma$. (An upper bound for $\delta$ is $\gamma + d-1$.)
By reordering the vertices of the graph $G$,
we may assume that the matrix $M$ has the block form
given in equation~\eqref{equation_block-matrix}.
Hence $M^{\delta}$ is also a block matrix.
Since $\delta \geq \gamma$, each block in
$\operatorname{Bin}(M^{\delta})$ is either the zero matrix or the matrix $J$
of all ones.
Since $\delta$ is a multiple of $d$,
the matrix $\operatorname{Bin}(M^{\delta})$ has the form
\begin{equation}
\label{equation_M_to_delta}
\operatorname{Bin}(M^{\delta})
=
\begin{pmatrix}
J&0&\cdots&0\\
0&J&\cdots&0\\
\vdots&\vdots&\ddots&\vdots\\
0&0&\cdots&J\\
\end{pmatrix} ,
\end{equation}
where the $q$th block is a $c_{q}\times c_{q}$ square matrix
whose entries are all $1$'s.
Apply the trace to
the Eulerian condition~\eqref{equation_level_Eulerian}
for $p=2\delta$
and consider this equation modulo~$2$.
Recall $\operatorname{trace}(A B) = \operatorname{trace}(B A)$
holds for any pair of square matrices,
and in particular, it holds for
$A = \operatorname{Bin}(M^{i})$ and
$B = \operatorname{Bin}(M^{2\delta-i})$.
Hence the Eulerian condition~\eqref{equation_level_Eulerian}
collapses to
$$ \operatorname{trace}\left({\operatorname{Bin}(M^{\delta})}^{2}\right) \equiv 0 \bmod 2 . $$
Note that the square of the
matrix $\operatorname{Bin}(M^{\delta})$ is given by
$$
{\operatorname{Bin}(M^{\delta})}^{2}
=
\begin{pmatrix}
c_{0} \cdot J&0&\cdots&0\\
0&c_{1} \cdot J&\cdots&0\\
\vdots&\vdots&\ddots&\vdots\\
0&0&\cdots&c_{q-1} \cdot J\\
\end{pmatrix} . $$
Hence the trace of the above matrix is
$\sum_{q=0}^{d-1} c_{q}^{2} \equiv \sum_{q=0}^{d-1} c_{q} \bmod 2$.
Hence we conclude that the order of $M$ is an even number.
\end{proof}
\section{Level half-Eulerian posets}
In analogy to level Eulerian posets
(Definition~\ref{definition_level_Eulerian}) we define level
half-Eulerian posets as follows.
\begin{definition}
\label{definition_level_half-Eulerian}
A level poset $P$ is said to be a
{\em level half-Eulerian poset} if every interval
is half-Eulerian.
\end{definition}
In analogy to the horizontal doubling operation introduced
in~\cite{Bayer_Hetyei_E,Bayer_Hetyei_G}, we define the {\em horizontal
double $D_{\leftrightarrow}(P)$ of a level poset $P$} as the poset
obtained by replacing each $(u,i)\in P$ by two copies $(u_{1},i)$, $(u_{2},i)$
and preserving the partial order of $P$, i.e., setting $(u_{k},i)<(v_{l},j)$
in $D_{\leftrightarrow}(P)$ if and only if $(u,i)<(v,j)$ holds in $P$.
\begin{proposition}
The horizontal double $D_{\leftrightarrow}(P)$ of a level poset $P$ is a
level poset. In particular, if the underlying adjacency matrix of $P$ is
$M$ then the underlying adjacency matrix $D_{\leftrightarrow}(M)$ of
$D_{\leftrightarrow}(P)$ is
\begin{equation}
\label{equation_horizontal_doubling_matrix}
D_{\leftrightarrow}(M)=
\begin{pmatrix}
M & M \\ M & M
\end{pmatrix} .
\end{equation}
\end{proposition}
The straightforward verification is left to the reader. As an immediate
consequence of Lemma~\ref{lemma_h_double}, we obtain the following corollary.
\begin{corollary}
\label{corollary_h_double}
A level poset $P$ is half-Eulerian if and only if its horizontal
double $D_{\leftrightarrow}(P)$ is level Eulerian.
\end{corollary}
Lemma~\ref{lemma_level_Eulerian} has the following half-Eulerian
analogue.
\begin{lemma}
A level poset is half-Eulerian
if and only if its adjacency matrix $M$
satisfies
\begin{equation}
\sum_{i=1}^{p-1} (-1)^{i-1} \cdot \operatorname{Bin}(M^{i}) \cdot \operatorname{Bin}(M^{p-i}) =
\left\{
\begin{array}{ll}
J&\mbox{if $p$ is even,}\\
0&\mbox{if $p$ is odd,}\\
\end{array}
\right.
\label{equation_level_half-Eulerian}
\end{equation}
for all $p \geq 1$.
\label{lemma_level_half-Eulerian}
\end{lemma}
Directly from
Lemmas~\ref{lemma_h_double} and~\ref{lemma_flag_f},
a level poset $P$ is half-Eulerian if and only if its underlying
adjacency matrix $M$ satisfies~\eqref{equation_level_half-Eulerian} for
all $p>0$. It is straightforward to show
directly that the adjacency matrix
$M$ of a level poset $P$ satisfies~\eqref{equation_level_half-Eulerian}
for a given $p>0$ if and only if the matrix $D_{\leftrightarrow}(M)$
given in~\eqref{equation_horizontal_doubling_matrix}
satisfies~\eqref{equation_level_Eulerian} for
the same $p$. As a consequence, we only need to verify
\eqref{equation_level_half-Eulerian} for values of $p$ up to the bound stated
in Theorem~\ref{theorem_Euler_bound}.
Unfortunately
the natural generalization of the vertical doubling operation to a level
poset by replacing each element $(u,i)$ of a level poset $P$
with two copies $(u_{1},2i)$ and $(u_{2},2i+1)$,
setting $(u_{1},2i)<(u_{2},2i+1)$ for each $u$,
and setting $(u_{2},2i+1)<(v_{1},2j)$ whenever
$(u,i)<(v,j)$ does not work.
This operation does not result in
a level poset because
the cover relations between levels
$2i$ and $2i+1$ would be different from the cover relations between
levels $2i-1$ and $2i$.
However, we may perform this operation, take two
copies, shift the Hasse diagram of one of the copies
one step up, and finally intertwine the two copies.
In short, consider the level poset with the adjacency matrix
\begin{equation}
\label{equation_vertical_doubling_matrix}
D_{\updownarrow}(M) = \begin{pmatrix} 0&I\\ M&0 \end{pmatrix} ,
\end{equation}
where $M$ is the $n\times n$ underlying adjacency matrix
of the level poset $P$ and $I$ is the $n\times n$ identity matrix.
\begin{definition}
Let $P$ be a level poset with adjacency matrix $M$.
Define the {\em vertical double $D_{\updownarrow}(P)$}
of $P$ to be the level poset whose
adjacency matrix $D_{\updownarrow}(M)$ is given
by~\eqref{equation_vertical_doubling_matrix}.
\end{definition}
As a direct consequence of Lemma~\ref{lemma_v_double}, we have the following
corollary.
\begin{corollary}
\label{corollary_v_double}
The vertical double $D_{\updownarrow}(P)$ of an arbitrary level poset
$P$ is level half-Eulerian.
\end{corollary}
Combining Corollaries~\ref{corollary_h_double}
and~\ref{corollary_v_double}, we obtain the
following statement.
\begin{corollary}
\label{corollary_h_v_double}
For any square binary matrix $M$, the matrix
$$
D_{\leftrightarrow}(D_{\updownarrow} (M))=
\begin{pmatrix}
0&I&0&I\\
M&0&M&0\\
0&I&0&I\\
M&0&M&0
\end{pmatrix}
$$
is the adjacency matrix of a level Eulerian poset.
\end{corollary}
\begin{remark}
{\rm
The vertical doubling operation induces a widely used
operation on the underlying digraph. If $G$ is a digraph with adjacency
matrix $M$
then $D_{\updownarrow}(M)$ is the adjacency matrix of the digraph
$D_{\updownarrow}(G)$ obtained from $G$ as follows.
\begin{enumerate}
\item Replace each vertex $u$ of $G$ with two copies $u_{1}$ and $u_{2}$.
\item The edge set of $D_{\updownarrow}(G)$ consists of all edges of the
form $u_{1}\rightarrow u_{2}$ and of all edges of the form $u_{2}\rightarrow
v_{1}$ where $u\rightarrow v$ is an edge in $G$.
\end{enumerate}
Introducing a graph identical to or very similar to $D_{\updownarrow}(G)$
is often used in the study of network flows.
These type of constructions also appear in proofs of the
vertex-disjoint path variant of Menger's theorem
as a way to reduce the
study of vertex capacities to that of edge capacities.
}
\end{remark}
Every half-Eulerian poset arising as a vertical double of a level poset
has an even number of elements at each level, and each canonical block of its
underlying adjacency matrix has even period. This observation may be
complemented by the following analogue of
Theorem~\ref{theorem_even_order}
for half-Eulerian posets.
\begin{theorem}
\label{theorem_odd_order}
Let $P$ be a level half-Eulerian poset
whose underlying matrix $M$ is primitive.
Then the order of the matrix $M$ is odd.
\end{theorem}
\begin{proof}
Since $M$ is primitive,
let $\gamma$ be the exponent of the matrix $M$.
Recall that $\operatorname{Bin}(M^{\gamma})=\operatorname{Bin}(M^{\gamma+1})=\cdots=\operatorname{Bin}(M^{2\gamma})=J$.
Hence we may rewrite the half-Eulerian
condition~\eqref{equation_level_half-Eulerian}
for $p=2\gamma$
as
\begin{equation}
\label{equation_p-even}
X J + (-1)^{\gamma-1} J^{2} + J X = J,
\end{equation}
where
$X=\sum_{i=1}^{\gamma-1} (-1)^{i-1} \cdot \operatorname{Bin}(M^{i})$.
Similarly, the half-Eulerian condition~\eqref{equation_level_half-Eulerian}
for $p=2\gamma+1$ yields
\begin{equation}
\label{equation_p-odd}
X J - J X = 0.
\end{equation}
Combining equations~\eqref{equation_p-even} and~\eqref{equation_p-odd}
modulo $2$
yields that
$J^{2} \equiv J \bmod 2$.
Since $J^{2} = n J$,
the order $n$ must be odd.
\end{proof}
It is interesting to note that
unlike the proof of Theorem~\ref{theorem_even_order},
the trace operation does not appear
in the argument for
Theorem~\ref{theorem_odd_order}.
\section{Shellable level posets}
Labelings that induce a shelling of the order complex
of a graded poset
have a vast literature. In the case of level posets it is natural to
seek a labeling that may be defined in a uniform fashion for the order
complex of every interval. The next definition is an example of such a
uniform labeling.
\begin{definition}
\label{definition_vertex_shelling}
Let $G$ be a directed graph on the vertex set $V$. A linear
order on $V$ is a {\em vertex shelling order} if for any $u,v\in V$ and
every pair of walks
$u=v_{0}\rightarrow v_{1} \rightarrow \cdots\rightarrow v_{k}=v$
and $u=v_{0}'\rightarrow v_{1}'\rightarrow \cdots\rightarrow v_{k}'=v$
of the same length such that $v_{1}'<v_{1}$ holds, there is a $j\in [1,k-1]$
and a vertex $w\in V$ such that $w<v_{j}$ holds and $v_{j-1}\rightarrow w$ and
$w\rightarrow v_{j+1}$ are edges of $G$.
\end{definition}
The term ``vertex shelling order'' is justified by the following
result.
\begin{theorem}
\label{theorem_vertex_shelling}
Let $P$ be a level poset and $<$ be a vertex shelling order on the
vertex set of the underlying digraph of $P$.
Associate to each
maximal chain $(u,i)=(v_{i},i)\prec (v_{i+1},i+1)\prec \cdots \prec
(v_{j-1},j-1)\prec (v_{j},j)=(v,j)$ in the interval $[(u,i),(v,j)]$
the word $v_{i} \cdots v_{j}$.
Then ordering the maximal chains of the interval $[(u,i),(v,j)]$
in $P$ by increasing lexicographic order of
the associated words is a shelling
of the order complex $\Delta([(u,i),(v,j)])$.
\end{theorem}
\begin{proof}
The maximal chains of an interval $[(u,i),(v,j)]$ in a level poset $P$
are in a one-to-one correspondence with walks $u=v_{i}\rightarrow
v_{i+1}\rightarrow \cdots\rightarrow v_{j-1}\rightarrow v_{j}=v$
of length $j-i$ in the underlying digraph. Assume that the maximal chain encoded
by the word $v_{i}\cdots v_{j}$ is preceded by the chain encoded by
the word $v_{i}^{\prime} \cdots v_{j}^{\prime}$.
Let $k \in [i+1,j-1]$ be the least
index such that $v_{k}^{\prime} \neq v_{k}$ and let $l \in [k+1,j]$
be the least index such
that $v_{l} = v_{l}^{\prime}$.
Since $v_{i} \cdots v_{j}$ is preceded by $v_{i}^{\prime} \cdots
v_{j}^{\prime}$ in the lexicographic order,
we must have $v_{k}^{\prime} < v_{k}$.
As a consequence of Definition~\ref{definition_vertex_shelling} applied to the walks
$v_{k-1}\rightarrow \cdots\rightarrow v_{l}$ and
$v_{k-1}^{\prime} \rightarrow \cdots \rightarrow v_{l}^{\prime}$,
there is an $m\in [k,l-1]$ and a vertex $w$
such that $w<v_m$ holds and $v_{m-1}\rightarrow w$ and
$w \rightarrow v_{m+1}$ are edges of the underlying digraph.
The maximal chain
associated to the word $v_{i}v_{i+1}\cdots v_{m-1} w v_{m+1}\cdots v_{j}$
precedes the maximal chain associated to $v_{i}\cdots v_{j}$, the
intersection of the two chains has codimension one, and contains the
intersection of the chain associated to $v_{i}\cdots v_{j}$ with the
chain associated to $v_{i}^{\prime} \cdots v_{j}^{\prime}$.
\end{proof}
\begin{remark}
{\rm
The shelling order used in Theorem~\ref{theorem_vertex_shelling} is induced by
labeling each cover relation $(u,i)\prec(v,i+1)$ by the vertex $v$. This
is an example of Kozlov's {\em $CC$-labelings}~\cite{Kozlov},
discovered independently by Hersh and Kleinberg.
See the Introduction of~\cite{Babson_Hersh}.
Even if the linear order
$<$ is not a vertex shelling order, labeling each cover relation
$(u,i) \prec (v,i+1)$ by the vertex $v$ and
using the linear order $<$ to
lexicographically order the maximal chains in each interval induces an
{\em $FA$-labeling}, as defined by Billera and
Hetyei~\cite{Billera_Hetyei_planar}.
See also \cite{Billera_Hetyei_flag}.
Essentially the same labelings were
used by Babson and Hersh~\cite{Babson_Hersh} to construct a discrete
Morse matching, a technique
which helps determine the homotopy type of the order
complex of an arbitrary graded poset. We refer the reader to the above
cited sources for further information.
}
\end{remark}
In analogy to Theorem~\ref{theorem_Euler_bound} the
vertex shelling order condition
needs to be verified only for finitely-many
values of $k$. We prove this for strongly connected digraphs.
\begin{theorem}
\label{theorem_vertex_shelling_bound}
Let $P$ be the level poset of an indecomposable $n\times n$ matrix $M$
with period~$d$ and index~$\gamma$.
Then the underlying digraph~$G$ of $P$ is vertex
shellable if and only if it satisfies the vertex shelling order
condition stated in
Definition~\ref{definition_vertex_shelling} for $k\leq \gamma+d$.
\end{theorem}
\begin{proof}
Assume $G$ satisfies the condition stated in Definition~\ref{definition_vertex_shelling}
for $k\leq \gamma+d$ and let $k$ be the least integer for which the
condition is violated. We must
have $k\geq \gamma+d+1$. Consider any pair of walks
$u=v_{0}\rightarrow v_{1}\rightarrow \cdots\rightarrow v_{k}=v$ and
$u=v_{0}'\rightarrow v_{1}'\rightarrow \cdots\rightarrow v_{k}'=v$ such that
$v_{1}'<v_{1}$. As in the proof of Theorem~\ref{theorem_even_order}, let
$\delta\leq \gamma + d-1$ be the least multiple of $d$ which is greater
than or equal to $\gamma$. After rearranging the rows and columns if
necessary, $\operatorname{Bin}(M)$ takes the form given in~\eqref{equation_block-matrix} and
$\operatorname{Bin}(M^{\delta})$ takes the form given in~\eqref{equation_M_to_delta}.
As in~\eqref{equation_block-matrix}, we may assume that the block
$Q_{q,q+1}$ occupies the rows indexed by $C_{q}$ and the columns indexed
by $C_{q+1}$. The vertex set of $G$ is the
disjoint union of the sets $C_{0},\ldots, C_{d-1}$ and every edge starting
in $C_q$ ends in~$C_{q+1}$. Since there is a walk of length $k-1$ from
$v_{1}'$ to $v_{k}=v$ and there is a walk of length $k-\delta-1$ from
$v_{\delta+1}$
to $v_{k}$, it follows that $v_{1}'$ and $v_{\delta+1}$ belong to the same
set $C_q$. As a consequence of equation~\eqref{equation_M_to_delta}, there is a
walk $v_{1}'\rightarrow v_{2}''\rightarrow \cdots\rightarrow
v_{\delta}''\rightarrow v_{\delta+1}$ of length $\delta$ from $v_{1}'$ to
$v_{\delta+1}$. Note that $\delta+1\leq \gamma+d<k$. By the minimality
of $k$, the walks $v_{0}\rightarrow v_{1}\rightarrow \cdots\rightarrow
v_{\delta}\rightarrow v_{\delta+1}$ and
$v_{0}\rightarrow v_{1}'\rightarrow v_{2}''\rightarrow
\cdots \rightarrow v_{\delta}''\rightarrow v_{\delta+1}$ still
satisfying $v_{1}'<v_{1}$ cannot violate the
vertex shelling order
condition stated in
Definition~\ref{definition_vertex_shelling}. Hence there is a $j\in [1,\delta]$
and a vertex $w\in V$ such that $w<v_{j}$ holds and $v_{j-1}\rightarrow w$ and
$w\rightarrow v_{j+1}$ are edges of $G$, and the pair of walks
$u=v_{0}\rightarrow v_{1}\rightarrow \cdots\rightarrow v_{k}=v$ and
$u=v_{0}'\rightarrow v_{1}'\rightarrow \cdots\rightarrow v_{k}'=v$ do not
violate the vertex shelling order condition,
in contradiction with our assumption.
\end{proof}
The verification whether a linear order on the vertices of a digraph is
a vertex shelling order may be automated by introducing the algebra
of walks.
\begin{definition}
Let $G$ be a digraph with edge set $E$ on the vertex set $V$.
Assume $G$ has no multiple edges. The {\em algebra of walks
$\walks{{\mathbb Q}}{G}$} is the quotient of the free non-commutative algebra
over ${\mathbb Q}$ generated by the set of variables $\{x_{u,v}\::\: (u,v)\in E\}$
by the ideal generated by the set of monomials
$\{x_{u_{1},v_{1}}x_{u_{2},v_{2}}\::\:v_{1}\neq u_{2} \}$.
\end{definition}
A vector space basis for $\walks{{\mathbb Q}}{G}$ may be given by $1$,
which labels the trivial walk, and all
monomials $x_{v_{0},v_{1}}x_{v_{1},v_{2}}\cdots x_{v_{k-1},v_{k}}$
such that $v_{0}\rightarrow v_{1}\rightarrow \cdots\rightarrow v_{k}$ is a walk in $G$.
\begin{notation}
We introduce $x_{v_{0},v_{1},\ldots, v_{k}}$ as a
shorthand for $x_{v_{0},v_{1}}x_{v_{1},v_{2}}\cdots x_{v_{k-1},v_{k}}$.
\end{notation}
\begin{theorem}
\label{theorem_walks}
Let $G$ be a digraph on the vertex set $V$
of cardinality $n$ having no multiple edges and
let $<$ be a linear order on $V$. Let $I_{<}$
be the ideal in $\walks{{\mathbb Q}}{G}$
generated by all monomials $x_{v_{0},v_{1}}x_{v_{1},v_{2}}$ such that
there is a vertex $v_{1}^{\prime} < v_{1}$
such that $v_{0} \rightarrow v_{1}^{\prime} \rightarrow v_{2}$
is a walk in $G$.
Let $Z=(z_{u,v})_{u,v\in V}$ be the
$n \times n$ matrix whose rows and columns are indexed by the vertices of
$G$ such that $z_{u,v}=x_{u,v}$ if $(u,v)$ is an edge and it is zero
otherwise. If over the
ring $\walks{{\mathbb Q}}{G}/I_{<}$ every entry in
every power of the matrix $Z$ is a single monomial or zero,
then $<$ is a vertex shelling order.
\end{theorem}
\begin{proof}
We first calculate the powers of the matrix $Z$ over the ring
$\walks{{\mathbb Q}}{G}$. It is straightforward to see by induction on $k$
that the entry $z^{(k)}_{u,v}$ in $Z^k$ is the sum of all monomials
of the form $x_{v_{0},v_{1},\ldots ,v_{k}}$ where
$u=v_{0}\rightarrow v_{1}\rightarrow \cdots \rightarrow v_{k}=v$
is a walk of length $k$ from $u$ to $v$.
The effect of factoring by the ideal $I_{<}$
may be easily described by introducing
the following {\em flip operators} $\sigma_{i}$ for $i\geq 1$.
Given a monomial $x_{v_{0},v_{1},\ldots ,v_{k}}$,
set
$$ \sigma_{i}(x_{v_{0},v_{1},\ldots ,v_{k}})
=
x_{v_{0},v_{1},\ldots,v_{i-1},v_{i}^{\prime},v_{i+1},\ldots,v_{k}} , $$
if $v_{i}^{\prime}$
is the least vertex in the linear order
such that $v_{i-1} \rightarrow v_{i}^{\prime} \rightarrow v_{i+1}$
is a walk in the digraph.
Clearly a monomial belongs to $I_{<}$ if and only if
is not fixed by some $\sigma_{i}$. The order $<$ induces a lexicographic
order on all walks of length $k$ from $u$ to $v$. Applying a flip
$\sigma_{i}$ to a monomial $x_{v_{0},v_{1},\ldots,v_{k}}$
either leaves the monomial unchanged or replaces it with a monomial
that represents a lexicographically smaller walk of length $k$ from $u$
to $v$. In particular, the monomial representing the lexicographically
least walk of length $k$ from $u$ to $v$ does not belong to $I_{<}$ and so
$z^{(k)}_{u,v}\neq 0$ in $\walks{{\mathbb Q}}{G}/I_{<}$ if there is a walk
of length $k$ from $u$ to $v$.
Assume first that each $z^{(k)}_{u,v}$ contains at most one monomial that
does not belong to $I_{<}$. As noted above, in this case $z^{(k)}_{u,v}$
contains exactly one monomial not belonging to $I_{<}$ and this monomial
represents the lexicographically least walk of length $k$ from $u$ to
$v$. Consider a pair of walks
$u=v_{0}\rightarrow v_{1}\rightarrow \cdots\rightarrow v_{k}=v$ and
$u=v_{0}'\rightarrow v_{1}'\rightarrow \cdots\rightarrow v_{k}'=v$
satisfying $v_{1}'<v_{1}$. Since
$u=v_{0}\rightarrow v_{1}\rightarrow \cdots\rightarrow v_{k}=v$ is not the lexicographically
least walk of length $k$ from $u$ to $v$, the monomial
$x_{v_{0},v_{1},\ldots,v_{k}}$ must belong to
$I_{<}$. Thus there is a $j\in [1,k-1]$ and a vertex $w<v_{j}$ such that
$$ \sigma_{j}(x_{v_{0},v_{1},\ldots,v_{k}})
=
x_{v_{0},v_{1},\ldots,v_{j-1},w,v_{j+1},\ldots,v_{k}} . $$
This $j$ and $w$ show that the
vertex shelling order
condition stated in
Definition~\ref{definition_vertex_shelling} is satisfied.
Assume that $<$ is a vertex shelling order and, by way of
contradiction, assume that $z^{(k)}_{u,v}$ contains at least two monomials
$x_{v_{0},v_{1},\ldots,v_{k}}$ and
$x_{v_{0},v_{1}',\ldots, v_{k-1}',v_{k}}$ not belonging to $I_{<}$.
Without loss of generality we may assume $v_{0}=v_{0}', v_{1}=v_{1}',
\ldots, v_{i-1}=v_{i-1}'$ and $v_{i}'<v_{i}$. Applying the
vertex shelling order condition given
to the pair of walks $v_{i-1}\rightarrow
v_{i}\rightarrow \cdots\rightarrow v_{k}$ and $v_{i-1}\rightarrow
v_{i}'\rightarrow \cdots\rightarrow v_{k}$, we obtain a $j\in
[i,k-1]$ and a vertex $w$ such that $w<v_{j}$ and $v_{j-1}\rightarrow w$
and $w\rightarrow v_{j+1}$ are edges. But then
$\sigma_{j}(x_{v_{0},v_{1},\ldots,v_{k}})\neq
x_{v_{0},v_{1},\ldots,v_{k}}
$
and $x_{v_{0},v_{1},\ldots,v_{k}}$ does not belong to $I_{<}$.
\end{proof}
Using Theorem~\ref{theorem_vertex_shelling_bound}, the proof of Theorem~\ref{theorem_walks}
may be modified to show the following.
\begin{proposition}
\label{proposition_walks_bound}
Let $M$ be an indecomposable matrix having
period $d$ and index $\gamma$. When applying
Theorem~\ref{theorem_walks} to decide whether an order of the vertices is a
vertex shelling order, one needs to verify the condition on the matrices
$Z^{k}$ only for $k \leq \gamma+d$.
\end{proposition}
For a shellable Eulerian posets we can conclude more.
\begin{theorem}
If $P$ is an Eulerian poset of rank $n+1$ whose order complex
$\Delta(P - \{\widehat{0},\widehat{1}\})$ is shellable, then
the order complex is homeomorphic to an $n$-dimensional sphere.
\label{theorem_shellable_Eulerian}
\end{theorem}
\begin{proof}
Since every interval of rank $2$ in an Eulerian poset
is a diamond, every subfacet of the order complex
is contained in exactly two facets. Hence the
order complex $\Delta(P - \{\widehat{0},\widehat{1}\})$ is a pseudo-manifold
without boundary.
Let $F_{1}, \ldots, F_{t}$ be a shelling of the order complex.
Since the reduced Euler characteristic equals
the M\"obius function $\mu(P)$, which in turn equals $(-1)^{n+1}$
as $P$ is Eulerian,
the order complex is homotopy equivalent to one sphere.
Hence there is one facet that changes the topology
during shelling.
We can move this facet to be last facet of the shelling,
that is, $F_{t}$.
Hence the previous facets form
a contractible complex.
The shelling implies that
the complex $F_{1} \cup \cdots \cup F_{t-1}$ is collapsible to a point.
Lastly, the complex $F_{1} \cup \cdots \cup F_{t-1}$ is
a pseudo-manifold with boundary.
By a result of J.H.C.\ Whitehead~\cite[Theorem~1.6]{Forman},
such a pseudo-manifold is homeomorphic to an $n$-dimensional ball.
The result follows by gluing back the last facet $F_{t}$ along
the common boundary.
\end{proof}
\begin{example}
{\rm
Consider again the level poset shown in Figure~\ref{figure_main}.
See also Example~\ref{example_order_4}.
Consider the linear
order $1<2<3<4$ where $i$ is the vertex associated to row (and column)
$i$. We claim this is a vertex shelling order. By
Proposition~\ref{proposition_walks_bound} we need to check that all entries of
$Z^k$ are zero or monomials for $k\leq 4$.
Direct calculation shows
$$
Z=
\begin{pmatrix}
x_{1,1} & x_{1,2} & x_{1,3} & 0\\
x_{2,1} & 0 & x_{2,3} & x_{2,4}\\
0 & x_{3,2} & 0 & x_{3,4}\\
x_{4,1} & 0 & x_{4,3} & x_{4,4}
\end{pmatrix}
,\quad
Z^{2}=
\begin{pmatrix}
x_{1,1,1} & x_{1,1,2} & x_{1,1,3} & x_{1,2,4}\\
x_{2,1,1} & x_{2,1,2} & x_{2,1,3} & x_{2,3,4}\\
x_{3,2,1} & 0 & x_{3,2,3} & x_{3,2,4}\\
x_{4,1,1} & x_{4,1,2} & x_{4,1,3} & x_{4,3,4}
\end{pmatrix}
$$
$$
Z^{3}=
\begin{pmatrix}
x_{1,1,1,1} & x_{1,1,1,2} & x_{1,1,1,3} & x_{1,1,2,4}\\
x_{2,1,1,1} & x_{2,1,1,2} & x_{2,1,1,3} & x_{2,1,2,4}\\
x_{3,2,1,1} & x_{3,2,1,2} & x_{3,2,1,3} & x_{3,2,3,4}\\
x_{4,1,1,1} & x_{4,1,1,2} & x_{4,1,1,3} & x_{4,1,2,4}
\end{pmatrix}
\quad\mbox{and}
$$
$$
Z^{4}=
\begin{pmatrix}
x_{1,1,1,1,1} & x_{1,1,1,1,2} & x_{1,1,1,1,3} & x_{1,1,1,2,4}\\
x_{2,1,1,1,1} & x_{2,1,1,1,2} & x_{2,1,1,1,3} & x_{2,1,1,2,4}\\
x_{3,2,1,1,1} & x_{3,2,1,1,2} & x_{3,2,1,1,3} & x_{3,2,1,2,4}\\
x_{4,1,1,1,1} & x_{4,1,1,1,2} & x_{4,1,1,1,3} & x_{4,1,1,2,4}
\end{pmatrix} . $$
Thus we conclude that each interval is shellable
and hence by Theorem~\ref{theorem_shellable_Eulerian}
that the order complex of each interval is
homeomorphic to a sphere.
}
\label{example_order_4_shelling}
\end{example}
We conclude this section with an example of a level Eulerian poset that
has a strongly connected underlying digraph and non-shellable intervals.
\begin{example}
{\rm
Consider the level poset $P$ whose underlying graph has the adjacency matrix
$$
M=
\begin{pmatrix}
1 & 1 & 0\\
0 & 0 & 1\\
1 & 0 & 0\\
\end{pmatrix}
. $$
We then have
$$
\operatorname{Bin}(M^{2})=
\begin{pmatrix}
1 & 1 & 1\\
1 & 0 & 0\\
1 & 1 & 0\\
\end{pmatrix} ,
\quad
\operatorname{Bin}(M^{3})=
\begin{pmatrix}
1 & 1 & 1\\
1 & 1 & 0\\
1 & 1 & 1\\
\end{pmatrix}
$$
and
$\operatorname{Bin}(M^{4})=J$.
Thus $M$ is primitive and the exponent is $\gamma=4$.
By the half-Eulerian
analogue of Theorem~\ref{theorem_Euler_bound}, to check whether $P$ is
half-Eulerian we only need to verify the half-Eulerian
condition~\eqref{equation_level_half-Eulerian}
for $p<10$. Furthermore, just as
in the Eulerian case, we only need to
check~\eqref{equation_level_half-Eulerian}
holds for even values of $p$.
We leave this to the reader as an exercise. The level poset $P$ is
half-Eulerian, so its horizontal double $D_{\leftrightarrow}(P)$ is
Eulerian. Let $1$ denote the vertex corresponding
to the first row in $M$
and consider the interval $[(1,0),(1,3)]$ in
$D_{\leftrightarrow}(P)$. The order complex of this interval has two
connected components and is thus not shellable.
}
\end{example}
\section{The ${\bf a}{\bf b}$- and ${\bf c}{\bf d}$-series of level and level Eulerian posets}
Level Eulerian posets are infinite in nature, so
one must encode their face incidence data using a non-commutative
series.
For a reference on non-commutative formal power series,
see~\cite[Section~6.5]{Stanley_EC_2}.
In this section we review the notions of the flag $h$-vector,
the ${\bf a}{\bf b}$-index for finite posets
and,
in the case the poset is Eulerian,
the ${\bf c}{\bf d}$-index.
We extend these notions to the ${\bf a}{\bf b}$-series and
${\bf c}{\bf d}$-series of a level Eulerian poset.
The main result of this section is that
the ${\bf c}{\bf d}$-series of any level
Eulerian poset is a rational generating function.
For a finite graded poset $P$ of rank $m+1$,
the flag $f$-vector has $2^{m}$
entries. When the poset $P$ is Eulerian, there are linear relations
among these entries known as the generalized
Dehn--Sommerville relations~\cite{Bayer_Billera}.
They describe a subspace whose dimension
is given by the $m$th Fibonacci number. The ${\bf c}{\bf d}$-index offers
an explicit bases for this subspace. In order to describe it, we
begin by defining the flag $h$-vector and the ${\bf a}{\bf b}$-index.
The {\em flag $h$-vector} of the poset $P$
is defined by the invertible relation
$$ h_{S} = \sum_{T \subseteq S} (-1)^{|S-T|} \cdot f_{T} . $$
Hence the flag $h$-vector encodes the same information as the
flag $f$-vector.
Let ${\bf a}$ and ${\bf b}$ be two non-commutative variables
each of degree one.
For $S$ a subset of $\{1, \ldots, m\}$ define the
${\bf a}{\bf b}$-monomial $u_{S} = u_{1} u_{2} \cdots u_{m}$
by letting $u_{i} = {\bf b}$ if $i \in S$ and
$u_{i} = {\bf a}$ otherwise. The {\em ${\bf a}{\bf b}$-index} of the poset $P$
is defined by
$$ \Psi(P)
=
\sum_{S} h_{S} \cdot u_{S} , $$
where the sum is over all subsets
$S \subseteq \{1, \ldots, m\}$.
Bayer and Klapper~\cite{Bayer_Klapper} proved that
for an Eulerian poset $P$ the ${\bf a}{\bf b}$-index can be written
in terms of the non-commutative variables ${\bf c} = {\bf a} + {\bf b}$ and
${\bf d} = {\bf a}{\bf b} + {\bf b}{\bf a}$ of degree
one and two, respectively.
There are several proofs of this fact in the
literature~\cite{Ehrenborg_k-Eulerian,Ehrenborg_Readdy_homology,Stanley_d}.
When $\Psi(P)$ is written in terms of ${\bf c}$ and ${\bf d}$,
it is called the {\em ${\bf c}{\bf d}$-index} of the poset $P$.
Another way to approach the ${\bf a}{\bf b}$-index is by
chain enumeration. For a chain in the poset $P$
$c = \{\widehat{0} = x_{0} < x_{1} < \cdots < x_{k+1} = \widehat{1}\}$
define its weight to be
$$ \operatorname{wt}(c)
=
({\bf a}-{\bf b})^{\rho(x_{0},x_{1}) - 1}
\cdot
{\bf b}
\cdot
({\bf a}-{\bf b})^{\rho(x_{1},x_{2}) - 1}
\cdot
{\bf b}
\cdots
{\bf b}
\cdot
({\bf a}-{\bf b})^{\rho(x_{k},x_{k+1}) - 1} . $$
The ${\bf a}{\bf b}$-index is then given by
\begin{equation}
\Psi(P) = \sum_{c} \operatorname{wt}(c) ,
\label{equation_chain_definition}
\end{equation}
where the sum ranges over all chains $c$ in the poset $P$.
For a level poset $P$ and two vertices $x$ and $y$ in the underlying
digraph, set $\Psi([(x,i),(y,j)])$ to be zero if
$(x,i) \not\leq (y,j)$. Define the {\em ${\bf a}{\bf b}$-series
$\Psi_{x,y}$} of the level poset $P$
to be the non-commutative formal power series
$$ \Psi_{x,y}
=
\sum_{m \geq 0} \Psi([(x,0),(y,m+1)]) . $$
Since the $m$th term in this sum is homogeneous of degree $m$,
the sum is well-defined.
Finally, for a level poset $P$ let $\Psi$ be the matrix
whose $(x,y)$ entry is
the ${\bf a}{\bf b}$-series $\Psi_{x,y}$.
Our goal is to show
that the ${\bf a}{\bf b}$-series $\Psi_{x,y}$
is a rational non-commutative formal power series.
Let $K(t)$ denote the matrix
$$ K(t) = M + \operatorname{Bin}(M^{2}) \cdot t + \operatorname{Bin}(M^{3}) \cdot t^{2}
+ \cdots , $$
and let $K_{x,y}(t)$ denote the $(x,y)$ entry of the matrix $K(t)$.
Observe $K_{x,y}(t)$ is the generating function having the
coefficient of $t^{m-1}$ to be $1$ if there is a walk of
length $m$ from
the vertex $x$ to the vertex $y$ and zero otherwise.
To prove the main result of this section we need
the following classical result
due to
Skolem~\cite{Skolem},
Mahler~\cite{Mahler} and
Lech~\cite{Lech}.
The formulation here is
the same as that given
in~\cite[Chapter~4, Exercise~3]{Stanley_EC_1}).
In fact, since we are only dealing with integer coefficients,
it is sufficient to use
Skolem's original result~\cite{Skolem}.
\begin{theorem}[Skolem--Mahler--Lech]
Let $\sum_{n \geq 0} a_{n} \cdot t^{n}$ be a rational generating
function and let
$$ b_{n}
=
\left\{ \begin{array}{c l}
1 & \text{ if } a_{n} \neq 0 , \\
0 & \text{ if } a_{n} = 0 .
\end{array} \right. $$
Then the generating function
$\sum_{n \geq 0} b_{n} \cdot t^{n}$ is rational.
\label{theorem_Skolem_Mahler_Lech}
\end{theorem}
\begin{lemma}
The generating function $K_{x,y}(t)$ is rational.
\end{lemma}
\begin{proof}
Let $G(t)$ denote the matrix
$$ G(t) = M + M^{2} \cdot t + M^{3} \cdot t^{2} + \cdots
= M \cdot (I - M \cdot t)^{-1} $$
and let $G_{x,y}(t)$ denote the $(x,y)$ entry of this matrix.
Clearly $G_{x,y}(t)$ is the generating function for
the number of walks from the vertex $x$ to the vertex $y$
where the coefficient of $t^{m-1}$ is the number of walks
of length $m$.
Furthermore, it is clear that $G_{x,y}(t)$ is
a rational function.
Hence by Theorem~\ref{theorem_Skolem_Mahler_Lech}
the result follows.
\end{proof}
When the underlying digraph is strongly connected
and has period $d$, it easy to observe that
$K_{x,y}(t)$ is the rational function $t^{r}/(1-t^{d})$
minus a finite number of terms, where the lengths of the walks
from $x$ to $y$ are congruent to $r$ modulo $d$.
\begin{theorem}
The ${\bf a}{\bf b}$-series $\Psi_{x,y}$ is
a rational generating function in the non-commutative variables
${\bf a}$ and ${\bf b}$.
\label{theorem_ab_rational}
\end{theorem}
\begin{proof}
We first restrict ourselves to
summing weights of chains
which has length $k+1$
in the level poset,
that is, after excluding
the minimal and maximal element,
those chains consisting of $k$ elements.
The matrix enumerating such
chains is given by the product
$$ K({\bf a}-{\bf b}) \cdot {\bf b} \cdot K({\bf a}-{\bf b}) \cdot {\bf b} \cdots
\cdot {\bf b} \cdot K({\bf a}-{\bf b})
=
K({\bf a}-{\bf b}) \cdot ({\bf b} \cdot K({\bf a}-{\bf b}))^{k} . $$
Summing over all $k \geq 0$, we obtain
\begin{equation}
\Psi = K({\bf a}-{\bf b}) \cdot (I - {\bf b} \cdot K({\bf a}-{\bf b}))^{-1} .
\label{equation_Psi}
\end{equation}
Hence each entry of the matrix $\Psi$ is
a rational generating function in ${\bf a}$ and ${\bf b}$.
\end{proof}
We turn our attention to the ${\bf c}{\bf d}$-index of level Eulerian posets.
\begin{theorem}
For a level Eulerian poset
the ${\bf a}{\bf b}$-series $\Psi_{x,y}$ is
a rational generating function in the non-commutative variables
${\bf c}$ and ${\bf d}$.
\label{theorem_cd_rational}
\end{theorem}
We call the resulting generating function
guaranteed in Theorem~\ref{theorem_cd_rational}
the {\em ${\bf c}{\bf d}$-series}.
\begin{proof}[Proof of Theorem~\ref{theorem_cd_rational}]
Observe equation~\eqref{equation_Psi}
is equivalent to
\begin{equation}
\Psi
=
K({\bf a}-{\bf b})
+
K({\bf a}-{\bf b}) \cdot {\bf b} \cdot \Psi .
\label{equation_Psi_1}
\end{equation}
Consider the involution that
exchanges the variables ${\bf a}$ and ${\bf b}$. Note that this involution
leaves series expressed in ${\bf c}$ and ${\bf d}$ invariant. Apply
this involution to equation~\eqref{equation_Psi_1} gives
\begin{equation}
\Psi
=
K({\bf b}-{\bf a}) + K({\bf b}-{\bf a}) \cdot {\bf a} \cdot \Psi .
\label{equation_Psi_2}
\end{equation}
Add the two equations~\eqref{equation_Psi_1}
and~\eqref{equation_Psi_2} and divide by $2$.
\begin{equation}
\Psi
=
(K({\bf a}-{\bf b}) + K({\bf b}-{\bf a}))/2
+
(K({\bf a}-{\bf b}) \cdot {\bf b} + K({\bf b}-{\bf a}) \cdot {\bf a})/2 \cdot \Psi .
\label{equation_Psi_1_2}
\end{equation}
Divide the generating function $K(t)$ into its even, respectively odd,
generating function, that is,
let
$$ K_{0}(t) = \frac{ K(\sqrt{t}) + K(-\sqrt{t}) }{2}
\:\:\:\: \text{ and } \:\:\:\:
K_{1}(t) = \frac{ K(\sqrt{t}) - K(-\sqrt{t}) }{2 \cdot \sqrt{t}} . $$
We have
$K(t) = K_{0}(t^{2}) + K_{1}(t^{2}) \cdot t$
and
$K(-t) = K_{0}(t^{2}) - K_{1}(t^{2}) \cdot t$.
Note that
\begin{eqnarray*}
(K({\bf a}-{\bf b}) + K({\bf b}-{\bf a}))/2
& = &
K_{0}({\bf c}^{2} - 2 \cdot {\bf d}) , \\
(K({\bf a}-{\bf b}) \cdot {\bf b} + K({\bf b}-{\bf a}) \cdot {\bf a})/2
& = &
K_{0}({\bf c}^{2} - 2 \cdot {\bf d}) \cdot {\bf c}
+
K_{1}({\bf c}^{2} - 2 \cdot {\bf d}) \cdot (2 \cdot {\bf d} - {\bf c}^{2}) .
\end{eqnarray*}
The result now follows since $K_{0}$ and $K_{1}$ are rational
generating functions and
we can solve for $\Psi$ in
equation~\eqref{equation_Psi_1_2}.
\end{proof}
Bayer and Hetyei~\cite{Bayer_Hetyei_G}
proved that
the ${\bf a}{\bf b}$-index of a half-Eulerian poset
is a polynomial in the two variables ${\bf a}$ and $({\bf a}-{\bf b})^{2}$.
We now show this also holds for the rational
series of a level half-Eulerian poset.
Define the algebra morphism $f_{\leftrightarrow}$ on
${\mathbb R}\langle\langle {\bf a},{\bf b} \rangle\rangle$
by $f_{\leftrightarrow}({\bf a}-{\bf b}) ={\bf a}-{\bf b}$
and $f_{\leftrightarrow}({\bf b}) = 2{\bf b}$.
It is then easy to observe from
the chain definition~\eqref{equation_chain_definition}
of the ${\bf a}{\bf b}$-index that for any poset~$P$
the ${\bf a}{\bf b}$-index of the poset~$P$ and its horizontal double
are related by
$\Psi(D_{\leftrightarrow}(P)) = f_{\leftrightarrow}(\Psi(P))$.
\begin{corollary}
The ${\bf a}{\bf b}$-series $\Psi_{x,y}$ of a level half-Eulerian poset
is a rational generating function in the non-commutative variables
${\bf a}$ and $({\bf a}-{\bf b})^{2}$.
\end{corollary}
\begin{proof}
Consider the horizontal double of the level half-Eulerian poset.
Its ${\bf a}{\bf b}$-series is a rational generating function
in terms of ${\bf a}+{\bf b} = {\bf c}$ and $({\bf a}-{\bf b})^{2} = {\bf c}^{2} - 2 {\bf d}$.
The result follows by applying the inverse morphism
$f_{\leftrightarrow}^{-1}$ to this
rational series.
\end{proof}
We similarly define the algebra morphism $f_{\updownarrow}$
by
$f_{\updownarrow}({\bf a}-{\bf b}) = ({\bf a}-{\bf b})^{2}$
and
$f_{\updownarrow}({\bf b}) = {\bf b} ({\bf a}-{\bf b}) + ({\bf a}-{\bf b}) {\bf b} + {\bf b}^{2}
= {\bf a}{\bf b} + {\bf b}{\bf a} + {\bf b}^{2}$.
By the chain definition~\eqref{equation_chain_definition}
of the ${\bf a}{\bf b}$-index we can conclude that
$\Psi(D_{\updownarrow}(P)) = f_{\updownarrow}(\Psi(P))$.
We end this section by
presenting the corresponding results for horizontal-
and vertical-doubling of a level poset.
The proof is straightforward and hence omitted.
\begin{proposition}
Let $P$ a level poset with underlying matrix $M$.
Then we have
$$
\Psi(D_{\leftrightarrow}(P))
=
\begin{pmatrix}
f_{\leftrightarrow}(\Psi(P)) & f_{\leftrightarrow}(\Psi(P)) \\
f_{\leftrightarrow}(\Psi(P)) & f_{\leftrightarrow}(\Psi(P))
\end{pmatrix}
$$
and
$$
\Psi(D_{\updownarrow}(P))
=
\begin{pmatrix}
{\bf a} \cdot f_{\updownarrow}(\Psi(P)) &
I + {\bf a} \cdot f_{\updownarrow}(\Psi(P)) \cdot {\bf a} \\
f_{\updownarrow}(\Psi(P)) &
f_{\updownarrow}(\Psi(P)) \cdot {\bf a} \\
\end{pmatrix} ,
$$
where the two morphisms
$f_{\leftrightarrow}$ and $f_{\updownarrow}$
are applied entrywise to the matrices.
\end{proposition}
\section{Computing the ${\bf c}{\bf d}$-series}
The recursions~\eqref{equation_Psi_1},
\eqref{equation_Psi_2} and~\eqref{equation_Psi_1_2}
are not very practical for explicitly computing the
${\bf c}{\bf d}$-series of a level Eulerian poset.
In this section we offer a different method
to show the ${\bf c}{\bf d}$-series has a given expression
based upon the coalgebraic techniques
developed in~\cite{Ehrenborg_Readdy_coproducts}.
Define a derivation
$\Delta : {\mathbb R}\langle\langle {\bf a},{\bf b} \rangle\rangle
\longrightarrow
{\mathbb R}\langle\langle {\bf a},{\bf b},{\bf t} \rangle\rangle$
by
$\Delta({\bf a}) = \Delta({\bf b}) = {\bf t}$, $\Delta(1) = 0$
and require that it satisfy the product rule
$\Delta(u \cdot v) = \Delta(u) \cdot v + u \cdot \Delta(v)$.
It is straightforward to verify that this derivation is
well-defined. Observe that the coefficient of a monomial $u$
in $\Delta(v)$ is zero unless $u$ contains exactly one ${\bf t}$.
Note that for a formal power series $u$ without constant term
we have that
$$ \Delta\left( \frac{1}{1-u} \right)
=
\frac{1}{1-u} \cdot \Delta(u) \cdot \frac{1}{1-u} , $$
since
$\Delta(u^{m}) = \sum_{i=0}^{m-1} u^{i} \cdot \Delta(u) \cdot u^{m-1-i}$
and then by summing over all $m$.
When restricting the derivation $\Delta$ to non-commutative polynomials
${\mathbb R}\langle {\bf a},{\bf b} \rangle$, it becomes equivalent to the coproduct
on ${\bf a}{\bf b}$-polynomials introduced by Ehrenborg and Readdy
in~\cite{Ehrenborg_Readdy_coproducts}.
To see this fact, observe that the subspace of
${\mathbb R}\langle {\bf a},{\bf b},{\bf t} \rangle$ spanned by monomials
containing exactly one ${\bf t}$ is isomorphic to
${\mathbb R}\langle {\bf a},{\bf b} \rangle \otimes {\mathbb R}\langle {\bf a},{\bf b} \rangle$
by mapping the variable ${\bf t}$ to the tensor sign, that is,
$u \cdot {\bf t} \cdot v \longmapsto u \otimes v$.
We need two properties of the derivation $\Delta$.
The first is that the ${\bf a}{\bf b}$-index is a coalgebra homomorphism,
that is, for a poset $P$ we have
\begin{equation}
\Delta(\Psi(P))
=
\sum_{\widehat{0} < x < \widehat{1}}
\Psi([\widehat{0},x]) \cdot {\bf t} \cdot \Psi([x,\widehat{1}]) .
\label{equation_coalgebra_morphism}
\end{equation}
See~\cite[Proposition~3.1]{Ehrenborg_Readdy_coproducts}.
Applying~\eqref{equation_coalgebra_morphism}
to all the rank $m+1$ intervals of a level poset, we have that
$$ \Delta(\Psi_{m})
=
\sum_{i=0}^{m-1} \Psi_{i} \cdot {\bf t} \cdot \Psi_{m-1-i} , $$
where $\Psi_{m}$ denotes the degree $m$ terms of
the ${\bf a}{\bf b}$-series $\Psi$.
The second property of the derivation
is that when restricting the derivation
to ${\bf a}{\bf b}$-polynomials of degree $n$, the kernel of the map
is spanned by $({\bf a}-{\bf b})^{n}$.
See~\cite[Lemma~2.2]{Ehrenborg_Readdy_coproducts}.
We now prove the main result of this section.
It is a method to recognize the ${\bf a}{\bf b}$-series matrix of a level poset.
\begin{theorem}
The $n \times n$ matrix $\Psi$ of the ${\bf a}{\bf b}$-series
of a level poset
is the unique solution to the equation system
\begin{eqnarray}
\left. \Psi\right|_{{\bf a} = t, {\bf b} = 0}
& = &
K(t) ,
\label{equation_Delta_1} \\
\Delta(\Psi)
& = &
\Psi \cdot {\bf t} \cdot \Psi .
\label{equation_Delta_2}
\end{eqnarray}
\label{theorem_Delta}
\end{theorem}
\begin{proof}
Let $\Gamma$ be a solution to the two
equations~\eqref{equation_Delta_1} and~\eqref{equation_Delta_2}.
Write $\Gamma$ as the sum
$\sum_{m \geq 0} \Gamma_{m}$ where the entries
of the matrix $\Gamma_{m}$ are homogeneous of degree $m$.
By induction on $m$ we will prove that
$\Gamma_{m}$ is equal to $\Psi_{m}$, the $m$th homogeneous
component of the matrix $\Psi$. The base case $m=0$ is as follows.
$$ \Gamma_{0} = \left.\Gamma\right|_{{\bf a}={\bf b}=0}
= \left.K(t)\right|_{t=0} = M = \Psi_{0} . $$
Now assume the statement is true for all values less than $m$.
Observe that the $m$th component of
equation~\eqref{equation_Delta_2} is
$$ \Delta(\Gamma_{m})
=
\sum_{i=0}^{m-1}
\Gamma_{i} \cdot {\bf t} \cdot \Gamma_{m-1-i}
=
\sum_{i=0}^{m-1}
\Psi_{i} \cdot {\bf t} \cdot \Psi_{m-1-i}
=
\Delta(\Psi_{m}) . $$
Hence the difference $\Gamma_{m} - \Psi_{m}$ is
a constant matrix $N$ times the ${\bf a}{\bf b}$-polynomial $({\bf a}-{\bf b})^{m}$.
However, the matrix $N$ is zero since
$\left. \Gamma_{m}\right|_{{\bf b}=0} = \left. \Psi_{m}\right|_{{\bf b}=0}$,
proving that $\Gamma_{m}$ is equal to $\Psi_{m}$,
completing the induction.
\end{proof}
\begin{example}
{\rm
Consider the level Eulerian poset in Figure~\ref{figure_main}.
We claim that its ${\bf c}{\bf d}$-series matrix $\Psi$ is given by
$$
\Psi
=
\begin{pmatrix}
\frac{1}{1 - {\bf c} - {\bf d}} &
\frac{1}{1 - {\bf c} - {\bf d}} \cdot {\bf c} + 1 &
\frac{1}{1 - {\bf c} - {\bf d}} &
\frac{1}{1 - {\bf c} - {\bf d}} - 1
\\[2 mm]
\frac{1}{1 - {\bf c} - {\bf d}} &
\frac{1}{1 - {\bf c} - {\bf d}} \cdot {\bf c} &
\frac{1}{1 - {\bf c} - {\bf d}} &
\frac{1}{1 - {\bf c} - {\bf d}}
\\[2 mm]
{\bf c} \cdot \frac{1}{1 - {\bf c} - {\bf d}} &
{\bf c} \cdot \frac{1}{1 - {\bf c} - {\bf d}} \cdot {\bf c} + 1 &
{\bf c} \cdot \frac{1}{1 - {\bf c} - {\bf d}} &
{\bf c} \cdot \frac{1}{1 - {\bf c} - {\bf d}} + 1
\\[2 mm]
\frac{1}{1 - {\bf c} - {\bf d}} &
\frac{1}{1 - {\bf c} - {\bf d}} \cdot {\bf c} &
\frac{1}{1 - {\bf c} - {\bf d}} &
\frac{1}{1 - {\bf c} - {\bf d}}
\end{pmatrix} .
$$
It is straightforward to check
the first condition in Theorem~\ref{theorem_Delta}:
$$
\left. \Psi\right|_{{\bf a} = t, {\bf b} = 0}
=
\left. \Psi\right|_{{\bf c} = t, {\bf d} = 0}
=
\begin{pmatrix}
\frac{1}{1 - t} &
\frac{1}{1 - t} &
\frac{1}{1 - t} &
\frac{1}{1 - t} - 1
\\[2 mm]
\frac{1}{1 - t} &
\frac{1}{1 - t} - 1 &
\frac{1}{1 - t} &
\frac{1}{1 - t}
\\[2 mm]
\frac{1}{1 - t} - 1 &
\frac{1}{1 - t} - t &
\frac{1}{1 - t} - 1 &
\frac{1}{1 - t}
\\[2 mm]
\frac{1}{1 - t} &
\frac{1}{1 - t} - 1 &
\frac{1}{1 - t} &
\frac{1}{1 - t}
\end{pmatrix}
=
K(t) .
$$
To verify the second condition, define the four vectors
$$
x = \begin{pmatrix} 1 \\ 1 \\ {\bf c} \\ 1 \end{pmatrix} ,
\:\:\:\:
y = \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix} ,
\:\:\:\:
z = \begin{pmatrix} 1 & {\bf c} & 1 & 1 \end{pmatrix}
\:\:\:\: \text{ and } \:\:\:\:
w = \begin{pmatrix} 0 & 1 & 0 & 0 \end{pmatrix} ,
$$
and the matrix
$$
A = \begin{pmatrix}
0 & 1 & 0 & -1 \\
0 & 0 & 0 & 0 \\
0 & 1 & 0 & 1 \\
0 & 0 & 0 & 0
\end{pmatrix} .
$$
We have the following relations between
this matrix and these vectors:
$$
z \cdot {\bf t} \cdot x = 2 \cdot {\bf t} + {\bf c} {\bf t} + {\bf t} {\bf c}
= \Delta({\bf c} + {\bf d}),
\:\:
A \cdot x = 2 \cdot y,
\:\:
z \cdot A = 2 \cdot w
\:\: \text{and} \:\:
A \cdot {\bf t} \cdot A = 0 . $$
Furthermore, the derivative $\Delta$ acts as follows
$$ \Delta(x) = 2 \cdot {\bf t} \cdot y,
\:\:
\Delta(z) = 2 \cdot {\bf t} \cdot w
\:\: \text{and} \:\:
\Delta(A) = 0 . $$
Observe now that
$$ \Psi = x \cdot \frac{1}{1 - {\bf c} - {\bf d}} \cdot z + A . $$
Hence we have the following calculation
\begin{eqnarray*}
\Psi \cdot {\bf t} \cdot \Psi
& = &
\left(x \cdot \frac{1}{1 - {\bf c} - {\bf d}} \cdot z + A\right)
\cdot {\bf t} \cdot
\left(x \cdot \frac{1}{1 - {\bf c} - {\bf d}} \cdot z + A\right) \\
& = &
x \cdot \frac{1}{1 - {\bf c} - {\bf d}} \cdot z \cdot {\bf t} \cdot
x \cdot \frac{1}{1 - {\bf c} - {\bf d}} \cdot z \\
& &
+
x \cdot \frac{1}{1 - {\bf c} - {\bf d}} \cdot z \cdot A \cdot {\bf t}
+
{\bf t} \cdot A \cdot x \cdot \frac{1}{1 - {\bf c} - {\bf d}} \cdot z
+
A \cdot {\bf t} \cdot A \\
& = &
x \cdot \frac{1}{1 - {\bf c} - {\bf d}} \cdot \Delta({\bf c} + {\bf d})
\cdot \frac{1}{1 - {\bf c} - {\bf d}} \cdot z \\
& &
+
x \cdot \frac{1}{1 - {\bf c} - {\bf d}} \cdot 2 \cdot w \cdot {\bf t}
+
{\bf t} \cdot 2 \cdot y \cdot \frac{1}{1 - {\bf c} - {\bf d}} \cdot z \\
& = &
x \cdot \Delta\left(\frac{1}{1 - {\bf c} - {\bf d}}\right) \cdot z
+
x \cdot \frac{1}{1 - {\bf c} - {\bf d}} \cdot \Delta(z)
+
\Delta(x) \cdot \frac{1}{1 - {\bf c} - {\bf d}} \cdot z \\
& = &
\Delta\left(x \cdot \frac{1}{1 - {\bf c} - {\bf d}} \cdot z\right) \\
& = &
\Delta\left(\Psi\right) ,
\end{eqnarray*}
proving our claim.
}
\end{example}
As a corollary to this example we obtain
an interesting Eulerian poset
whose ${\bf c}{\bf d}$-index has all of its coefficients to be $1$.
\begin{corollary}
The ${\bf c}{\bf d}$-index of the interval
$[(1,0),(1,m+1)]$ in the level poset
in Figure~\ref{figure_main}
is the sum of all ${\bf c}{\bf d}$-monomials of degree $m$.
\label{corollary_sum_of_monomials}
\end{corollary}
\section{Concluding remarks}
Given a non-commutative rational formal power series in the variables
${\bf a}$ and ${\bf b}$ which can be expressed in terms of ${\bf c}$ and
${\bf d}$, is it necessarily a non-commutative rational formal power series
in the variables ${\bf c}$ and ${\bf d}$? In other words, is the
following equality true
$$ F_{\text{rat}}\langle\langle {\bf a},{\bf b} \rangle\rangle
\cap
F\langle\langle {\bf c},{\bf d} \rangle\rangle
=
F_{\text{rat}}\langle\langle {\bf c},{\bf d} \rangle\rangle , $$
where $F$ is a field?
It is clear that right-hand side of the above
is contained in the left-hand side.
Corollary~\ref{corollary_sum_of_monomials}
suggests a question about the existence of Eulerian posets.
For which subsets $M$ of ${\bf c}{\bf d}$-monomials of degree $m$
is there an Eulerian poset whose ${\bf c}{\bf d}$-index is the sum
of the monomials in~$M$?
The two extreme cases
${\bf c}^{m}$ and $\sum_{\deg(w) = m} w$
both arrive from level Eulerian posets.
An open question is if the eigenvalues or
other classical matrix invariants carry information
about the corresponding level poset,
such as if the level poset is Eulerian or shellable.
\section*{Acknowledgments}
The first author was partially funded by National Science Foundation grant
DMS-0902063.
The authors thank the Department of Mathematics
at the University of Kentucky for funding a research visit
for the second author to the University of Kentucky,
where part of this research was carried out.
\newcommand{\journal}[6]{{\sc #1,} #2, {\it #3} {\bf #4} (#5), #6.}
\newcommand{\book}[4]{{\sc #1,} ``#2,'' #3, #4.}
|
{
"timestamp": "2010-09-21T02:01:09",
"yymm": "1009",
"arxiv_id": "1009.3585",
"language": "en",
"url": "https://arxiv.org/abs/1009.3585"
}
|
\section{Introduction}
Scalar fields on a curved background have received considerable attention because of their relationship with Bosonic string theory \cite{21}. One normally focuses on the quantum properties of string theory
(such as the absence of the conformal anomaly only if the dimension of the target space exceeds four), but it is both interesting and important to have an understanding of the classical canonical structure
of this model if one is to truly comprehend the implications of the quantization procedure. In this paper we undertake the task of applying Dirac's analysis of constrained systems \cite{1,2,3,4,5,6}
to the problem of $N$ scalar fields on a curved two dimensional manifold. We focus in particular on the first class constraints that appear and what they tell us about the gauge invariance present in the theory.
A number of novel features arise.
Generally, in any discussion of metric fields on a two dimensional space, the action for the metric is ignored as the Einstein-Hilbert (EH) action $\sqrt{-g}g^{\mu\nu}R_{\mu\nu}(g_{\alpha\beta})$ in two
dimensions (2D), when treated as a function of the metric $g_{\mu\nu}$ alone (the second order form), is a pure surface term and has no dynamical degrees of freedom. We note though that this lack of dynamics
does not mean that it cannot be quantized; this has been studied in refs. \cite{30,31} using a BRST analysis. There has also been a discussion of the canonical structure of the first order EH action
in 2D \cite{7}. The first class constraints that occur have been shown to imply that there is an an invariance under the gauge transformation
\begin{equation}
g_{\mu \nu} \rightarrow g_{\mu \nu} +\omega_{\mu\nu}
\end{equation}
which is consistent with there being no degrees of freedom present in the action.
Normally when a matter field is coupled with a gauge field (eg. the electron is coupled to a photon), any gauge invariance present in the uncoupled gauge field action is respected by the action in which
the coupling is present. In this case however, the coupling of $N$ scalars $f^a, (a=1,2...N)$ to the metric $g_{\mu\nu}$ through the Lagrangian
\begin{equation}
\mathcal{L}_f = \frac{1}{2} \sqrt{-g}\, g^{\mu\nu} \partial_\mu f^a \partial_\nu f^a,
\end{equation}
while being diffeomorphism invariant, does not respect the symmetry of eq. (1). In this paper, we first address the problem of disentangling how supplementing the second order order
EH action in 2D by the action of eq. (2) alters the constraint structure of the theory and thereby leads to a new gauge
invariance that is distinct from that of eq. (1).
The problem of reconciling the gauge invariance present in the action for the free gauge field with that occurring when it is coupled to a matter field becomes even more interesting when the free gauge action
is the first order EH action in 2D. We first note that this action, $\sqrt{-g}g^{\mu\nu}R_{\mu\nu}(\Gamma^{\lambda}_{\alpha\beta}),$ is not equivalent to the second order form, unlike what occurs in $D>2$
dimensions \cite{14,15}. This is because the affine connection $\Gamma^{\lambda}_{\mu\nu}$ is no longer given by the Christoffel symbol
\begin{equation}
\left\lbrace \begin{array}{c}
\lambda \\
\mu\nu
\end{array} \right\rbrace = \frac{1}{2} g^{\lambda\sigma} \left(g_{\sigma\mu ,\nu} + g_{\sigma\nu ,\mu} - g_{\mu\nu , \sigma}\right)
\end{equation}
but rather
\begin{equation}
\Gamma_{\mu\nu}^\lambda =
\left\lbrace \begin{array}{c}
\lambda \\
\mu\nu
\end{array} \right\rbrace + \delta_\mu^\lambda\xi_\nu + \delta_\nu^\lambda\xi_\mu -
g_{\mu\nu} \xi^\lambda
\end{equation}
(where $\xi^\lambda$ is an arbitrary vector) when solving the equation of motion for $\Gamma_{\mu\nu}^\lambda$. We first consider the implication of having this extra field arising in the model. We then review
analysis \cite{8,9,10,11,12,13} which shows that the canonical structure of the first order EH action in 2D shows that there are no physical degrees of freedom in the model despite it not being topological,
and that the first class constraint that arise result in a novel gauge transformation
\begin{equation}
\delta h^{\mu\nu} = - \left(\epsilon^{\mu\rho} h^{\nu\sigma} + \epsilon^{\nu\rho}h^{\mu\sigma}\right)\omega_{\rho\sigma}
\end{equation}
\begin{equation}
\delta G_{\mu\nu}^\lambda = - \epsilon^{\lambda\rho} \omega_{\mu\nu ,\rho} - \epsilon^{\rho\sigma}
\left(G_{\mu\rho}^\lambda \omega_{\nu\sigma} + G_{\nu\rho}^\lambda \omega_{\mu\sigma}\right)
\end{equation}
where $\epsilon^{01} = -\epsilon^{10} = 1$, $\epsilon^{00} = \epsilon^{11} = 0$, $h^{\mu\nu} = \sqrt{-g}\, g^{\mu\nu}$
and $G_{\mu\nu}^\lambda = \Gamma_{\mu\nu}^\lambda - \frac{1}{2} \left(\delta_\mu^\lambda \Gamma_{\rho\nu}^\rho + \delta_\nu^\lambda \Gamma_{\rho\mu}^\rho\right)$.
This is distinct from the manifest diffeomorphism invariance present. We then address the problem of seeing how the first class constraints that lead to eqs. (5,6) are modified when the free action for
$h^{\mu\nu},G_{\mu\nu}^\lambda$ is supplemented by
\begin{equation}
\mathcal{L}_f = \frac{1}{2} h^{\mu\nu} \partial_\mu f^a\,\partial_\nu f^a.
\end{equation}
A number of interesting features arise in the course of applying the Dirac constraint formalism to these two models in which a scalar field propagates on a curved surface. First of all, when there are $N$
scalar fields, the constraints and their associated gauge conditions combine to leave just $2N-4$ dynamical degrees of freedom in the theory. Secondly, when one considers either the first or second order EH
action to be the action for the gauge field coupled to the scalar matter field, the number of first class constraints in each generation is not the same. For $N=1$, there are in the case of the second order EH
action, three primary and two secondary first class constraints, while with the first order EH action there are three primary and secondary first class constraints and two tertiary first class constraints.
Consequently, when using these constraints to find the gauge invariance that they imply to be present in the initial action, one finds that the techniques of both C (refs. \cite{16,17}) and of HTZ
(refs. \cite{18,19}) do not lead to a unique gauge transformation. Neither diffeomorphism invariance not conformal invariance are implied by these first class constraints; indeed for the first order action
the gauge generator derived from the first class constraints implies that the scalar field and affine connections mix under a gauge transformations.
In the next two sections we present a canonical analysis of a scalar field on a curved background, using the second, then the first, order EH action for the metric, including a discussion of the gauge
transformations implied by the first class constraints. In appendix, the way in which the first class constraints can be used to find the generator of the gauge transformation is outlined, using both the approach
of C \cite{16,17} and of HTZ \cite{18,19}.
\section{Second order EH Action and Scalar Fields}
We begin by first reviewing how the second order EH action in 2D can be treated using the Dirac constraint formalism \cite{7}, despite it being a topological theory.
We then couple the metric to a scalar field and consider how this affects the gauge invariance of eq. (3).
The second order EH action is
\begin{equation}
S_{EH} = \int dx \sqrt{-g}\, g^{\mu\nu} R_{\mu\nu}
\end{equation}
where
\begin{equation}
R_{\mu\nu} = \Gamma^\lambda_{\mu\nu ,\lambda} - \Gamma^\lambda_{\lambda\mu ,\nu} + \Gamma_{\lambda\sigma}^\lambda \Gamma_{\mu\nu}^\sigma -
\Gamma_{\sigma\mu}^\lambda \Gamma_{\lambda\nu}^\sigma
\end{equation}
and $\Gamma_{\mu\nu}^\lambda = \left\lbrace \begin{array}{c}
\lambda \\
\mu\nu \end{array}\right\rbrace$.
In any dimension \cite{20}
\begin{align}
\sqrt{-g}& \, g^{\mu\nu} \left(\Gamma_{\mu\nu ,\lambda}^\lambda - \Gamma_{\lambda\mu , \nu}^\lambda \right)\nonumber \\
&= \left(\sqrt{-g}\, g^{\mu\nu} \Gamma_{\mu\nu}^\lambda\right)_{,\lambda} -
\left(\sqrt{-g}\, g^{\mu\nu} \Gamma_{\lambda\mu}^\lambda\right)_{,\nu} \nonumber \\
&-2\sqrt{-g}\, g^{\mu\nu} \left(\Gamma_{\lambda\sigma}^\lambda \Gamma_{\mu\nu}^\sigma - \Gamma_{\sigma\mu}^\lambda \Gamma_{\lambda\nu}^\sigma\right)
\end{align}
and hence if surface terms are discarded, then $S_{EH}$ can be replaced by the non-covariant action
\begin{equation}
S_{\Gamma\Gamma}^{(2)} = - \int dx \sqrt{-g}\, g^{\mu\nu} \left(\Gamma_{\lambda\sigma}^\lambda \Gamma_{\mu\nu}^\sigma - \Gamma_{\sigma\mu}^\lambda \Gamma_{\lambda\nu}^\sigma\right).
\end{equation}
It is this form of the action that was used by Dirac in the analysis of the canonical structure of the EH action in 4D \cite{22}. (See also refs \cite{32,33}.)
We too will use it as the initial action for analyzing the EH action in 2D.
In 2D, eq. (11) becomes
\begin{align}
= \frac{1}{2} \int dx &(-g)^{-3/2} \left[ g_{11,0}\left(g_{01}g_{00,1} - g_{00} g_{01,1}\right)\right. \\
& + g_{00,0} \left(g_{11} g_{01,1} - g_{01}g_{11,1}\right) \nonumber \\
& \left. + g_{01,0} \left(g_{00} g_{11,1} - g_{11}g_{00,1}\right)\right]. \nonumber
\end{align}
If one were to choose conformal coordinates so that $g_{00} = -g_{11} = \rho(x)$, $g_{01} = 0$ as in \cite{21}, then $S_{\Gamma\Gamma}$ vanishes. However, if $g_{01} \neq 0$ then
$S_{\Gamma\Gamma}$ is amenable to canonical analysis \cite{7}. However, it becomes apparent that $S_{\Gamma\Gamma}$ itself is a surface term if we adopt the coordinates \cite{22}
\begin{equation}
\delta = \frac{-\sqrt{-g}}{g_{11}}\;\;\;, \rho = \frac{g_{01}}{g_{11}}\;\;\;\;, g_{11}
\end{equation}
so that
\begin{align}
S_{\Gamma\Gamma}^{(2)} &= \int dx \frac{1}{\delta^2} \left(\delta_{,0}\rho_{,1} - \rho_{,0}\delta_{,1}\right)\nonumber \\
&= \int dx \left[\left(\frac{\rho_{,0}}{\delta}\right)_{,1} - \left(\frac{\rho_{,1}}{\delta}\right)_{,0} \right].
\end{align}
We will not employ the variables $\delta$ and $\rho$ in our canonical analysis; they simply serve to simplify the demonstration that $S^{(2)}_{\Gamma\Gamma}$ is a surface term. They do appear in ref. \cite{25} though.
From eq. (12), we find the primary constraints
\begin{subequations}
\begin{eqnarray}
\chi^{11}&=& \pi^{11} - \frac{1}{2(-g)^{3/2}} \left(g_{01} g_{00,1} -
g_{00} g_{01,1}\right)\\
\chi^{00}&=& \pi^{00} - \frac{1}{2(-g)^{3/2}} \left(g_{11} g_{01,1} -
g_{01} g_{11,1}\right)\\
\chi^{01}&=& \pi^{01} - \frac{1}{2(-g)^{3/2}} \left(g_{00} g_{11,1} -
g_{11} g_{01,1}\right)
\end{eqnarray}
\end{subequations}
where $(\pi^{11}, \pi^{00}, \pi^{01})$ are the canonical momenta conjugate to $(g_{11}, g_{00}, g_{01})$ respectively. (If one were to simply discard the action of eq.(12)
because of its topological nature,
then we would merely have $\chi^{11}=\pi^{11},\,\chi^{00}=\pi^{00}$ and $\chi^{01}=\pi^{01}$.) The Poisson Bracket (PB) of any two of these
constraints vanishes. Furthermore, the canonical Hamiltonian vanishes. Consequently there are three primary first class constraints and no secondary constraints associated
with $S_{\Gamma\Gamma}^{(2)}$ using any of the techniques of refs. \cite{16,17,18,19} one finds the generator of gauge transformations to be
\begin{equation}
G = \int dx \left[ \omega_{11} \chi^{11} + \omega_{00}\chi^{00} + \omega_{01}\chi^{01}\right]
\end{equation}
which results in
\begin{equation}
\delta g_{\mu\nu} = \omega_{\mu\nu}
\end{equation}
as in eq. (3). Eq. (17) also would follow from just taking $\chi^{11}=\pi^{11},\,\chi^{00}=\pi^{00}$ and $\chi^{01}=\pi^{01}$, as is appropriate if were to discard
the action all together because of it being topological.
We note that with these first class constraints of eq. (15) and the three associated gauge conditions,
these are six restrictions on the six canonical variables ($g_{\mu\nu}$ and $\pi^{\mu\nu}$) in phase space, leaving no physical degrees of freedom.
Supplementing $S_{\Gamma\Gamma}^{(2)}$ with the action for a massless scalar field $f$ \cite{23}
\begin{equation}
S_f = \frac{1}{2} \int dx \sqrt{-g} \,g^{\mu\nu} f_{,\mu}f_{,\nu}
\end{equation}
we find that the momentum conjugate to $f$ is
\begin{equation}
p = \sqrt{-g} \left(g^{00} f_{,0} + g^{01}f_{,1}\right) = \frac{1}{\sqrt{-g}} \left(-g_{11} f_{,0} + g_{01}f_{,1}\right)
\end{equation}
so that the part of the canonical Hamiltonian arising from $S_f$ in eq. (18) is
\begin{equation}
\mathcal{H}_c = \delta S+\rho I\!\!P
\end{equation}
where $S$ and $I\!\!P$ are two new secondary constraints
\begin{align}
S &= \frac{1}{2}\left(p^2 + f_{,1}^2\right)\\
I\!\!P &= pf_{,1} .
\end{align}
We note that although only the combinations $\delta$ and $\rho$ enter both eqs. (14) and (20), all three components of $h^{\mu\nu}$ appear in
the initial action of eqs. (11) and (18). These three must be all included be as fields in the canonical analysis. In ref. \cite{21}, a special "conformal gauge''
was used to dispense with the "conformal factor'' contribution to the action of eq. (18), reducing the number of independent components of the metric from three to two.
However, choosing a "gauge'' at the outset of any canonical analysis is inconsistent with Dirac's procedure \cite{1,2,3,4,5}.
Using test functions as in ref. \cite{24} we find
\begin{subequations}
\begin{eqnarray}
\left\lbrace S(x), S(y) \right\rbrace &=& \left( - I\!\!P (x) \partial^y_1 + I\!\!P (y)\partial^x_1\right)
\delta (x-y) \nonumber \\
&=& \left\lbrace I\!\!P(x), I\!\!P(y)\right\rbrace \\
\left\lbrace I\!\!P(x), S(y) \right\rbrace &=& \left( - S(x) \partial^y_1 + S(y)\partial^x_1\right)
\delta (x-y) \nonumber \\
&=& \left\lbrace S(x), I\!\!P(y)\right\rbrace
\end{eqnarray}
\end{subequations}
and thus no tertiary constraints arise.
With eqs. (15,21,22) we see that there are now five first class constraints, which when combined with five associated gauge conditions,
leaves us with ten restrictions on the eight variables in phase space $g_{\mu\nu},\,f$ and their associated momenta).
If the single scalar field $f$ in eq.(6) were replaced by $N$ scalars $f^a~~(a=1,2...N)$ in an $O(N)$ symmetric fashion,
there still would be ten constraints in phase space, but there would now be $2N+6$ variables, leaving $2N-4$ net physical degrees of freedom.
Only if $N>2$ are there true physical degrees of freedom.
The general form of the gauge generator for $S_{\Gamma\Gamma}^{(2)} + S_f$, when using the HTZ approach \cite{18,19}, is
\begin{align}
G_{HTZ} = \int dx(A_{11} \chi^{11} &+ A_{00} \chi^{00} + A_{01} \chi^{01}\nonumber \\
&+ B_S S + B_{I\!\!P} I\!\!P)
\end{align}
with $(A_{11}, A_{00}, A_{01})$ being found in terms of $B_S$ and $B_{I\!\!P}$ by using eq. (A5). (In ref. \cite{25} no
consistent way of deriving the generator of gauge transformations was used; its form is merely postulated.)
Together, eqs. (15, 20, 23) lead to eq. (A5) being satisfied to order $S$ and $I\!\!P$ provided
\begin{subequations}
\begin{eqnarray}
(B_{I\!\!P})_{,0} &+& B_S \left( -\frac{\sqrt{-g}}{g_{11}}\right)_{,1} - (B_S)_{,1} \left(- \frac{\sqrt{-g}}{g_{11}}\right) \nonumber \\
&+& B_{I\!\!P}\left( \frac{g_{01}}{g_{11}}\right)_{,1}
-(B_p)_{,1} \left( \frac{g_{01}}{g_{11}}\right) \nonumber \\
&+& \frac{1}{g_{11}}\left( \frac{g_{01}}{g_{11}} A_{11} - A_{01}\right) = 0
\end{eqnarray}
and \nonumber \\
\begin{eqnarray}
(B_S)_{,0} &+& B_S \left( \frac{g_{01}}{g_{11}}\right)_{,1} - (B_S)_{,1} \frac{g_{01}}{g_{11}} \\
&+& B_{I\!\!P} \left( -\frac{\sqrt{-g}}{g_{11}}\right)_{,1}-(B_{I\!\!P})_{,1} \left( -\frac{\sqrt{-g}}{g_{11}}\right) \nonumber \\
&+& \left[- \frac{\sqrt{-g}}{g_{11}^2} - \frac{g_{00}}{2g_{11}\sqrt{-g}} \right]A_{11} \nonumber \\
\qquad &-& \frac{1}{2\sqrt{-g}} A_{00} + \frac{g_{01}}{g_{11}\sqrt{-g}} A_{01} =0. \nonumber
\end{eqnarray}
\end{subequations}
As there are only two secondary constraints following from three primary constraints, eq. (25) does not uniquely fix $A_{00}$, $A_{11}$ and $A_{01}$ in terms of
$B_S$ and $B_p$.
In any case, eq. (25) is difficult to deal with, so we will employ the approach of C which involves equations of the form of eq. (A12). In this approach, the
form of the primary constraints that are used affects the form of the gauge generator \cite{26}. We find it most convenient to use as primary constraints
expressions suggested by the momenta conjugate to $\rho$, $\delta$ and $g_{11}$ under a canonical transformation:
\begin{subequations}
\begin{eqnarray}
\overline{\chi}^\rho &=& 2\chi^{00} g_{01} + \chi^{01} g_{11}\\
\overline{\chi}^\delta &=& 2\chi^{00} \sqrt{-g} \\
\overline{\chi}^{11} &=& \chi^{11} + \chi^{00} \left(\frac{ g_{00}}{g_{11}}\right) + \chi^{01} \left(\frac{ g_{01}}{g_{11}}\right)
\end{eqnarray}
\end{subequations}
so that
\begin{equation}
\left\lbrace \overline{\chi}^\rho, \mathcal{H}_c \right\rbrace = -I\!\!P,\;\;\;
\left\lbrace \overline{\chi}^\delta , \mathcal{H}_c \right\rbrace = -S,\;\;\;
\left\lbrace \overline{\chi}^{11}, \mathcal{H}_c \right\rbrace = 0.
\end{equation}
In eq. (A12), derived by using the approach of C \cite{16,17}, we take
\begin{equation}
G_1^\rho = \overline{\chi}^\rho
\end{equation}
so that
\begin{equation}
G_0^\rho + \left\lbrace G_1^\rho , H_T\right\rbrace = \rm{p.c.}\nonumber
\end{equation}
which leads to
\begin{align}
G_0^\rho(x) &= I\!\!P (x) + \int dy \left[\alpha_{\rho\rho} (x-y) \overline{\chi}^\rho (y)\right. \nonumber \\
&+ \left. \alpha_{\rho\delta} (x-y) \overline{\chi}^\delta (y) + \alpha_{\rho 11} (x-y) \overline{\chi}^{11}(y)\right].
\end{align}
In turn, we must now have by eq. (A12)
\begin{equation}
\left\lbrace G_0^\rho , H_T \right\rbrace = \rm{p.c.}
\end{equation}
which fixes
\begin{align}
\int dx & \epsilon^\rho (x) G_0^\rho (x) \nonumber \\
&= \int dx \left[ \epsilon^\rho I\!\!P + \overline{\chi}^\rho \left(
\epsilon_{,1}^\rho \left(\frac{g_{01}}{g_{11}}\right) - \epsilon^\rho \left(\frac{g_{01}}{g_{11}}\right)_{,1}\right)\right. \nonumber \\
&+ \left. \overline{\chi}^\delta \left(\epsilon_{,1}^\rho
\left(\frac{-\sqrt{-g}}{g_{11}}\right) - \epsilon^\rho \left(\frac{-\sqrt{-g}}{g_{11}}\right)_{,1}\right)\right].
\end{align}
So also, if
\begin{equation}
G_1^\delta = \overline{\chi}^\delta
\end{equation}
then eq. (A12) leads to
\begin{align}
&\int dx \epsilon^\delta (x) G_0^\delta (x) = \int dx \left[ \epsilon^\delta S + \overline{\chi}^\delta \left( \epsilon_{,1}^\delta \left(
\frac{g_{01}}{g_{11}}\right)\right.\right.\\
&- \left.\left. \epsilon^\delta \left(\frac{g_{01}}{g_{11}}\right)_{,1}\right) + \overline{\chi}^\rho
\left(\epsilon_{,1}^\delta \left(\frac{-\sqrt{-g}}{g_{11}}\right) - \epsilon^\delta \left(\frac{-\sqrt{-g}}{g_{11}}\right)_{,1}\right)\right];\nonumber
\end{align}
we finally obtain the full generator
\begin{align}
&G_C = \int dx \left\{\epsilon^\rho I\!\!P + \epsilon^\delta S + \epsilon^{11} \overline{\chi}^{11}\right.\\
&+ \overline{\chi}^\rho \left(
\epsilon_{,1}^\rho \left(\frac{g_{01}}{g_{11}}\right) - \epsilon^\rho
\left(\frac{g_{01}}{g_{11}}\right)_{,1} \right. \nonumber \\
&\quad + \left. \epsilon_{,1}^\delta
\left(\frac{-\sqrt{-g}}{g_{11}}\right) - \epsilon^\delta \left(\frac{-\sqrt{-g}}{g_{11}}\right)_{,1}\right)\nonumber \\
&+ \overline{\chi}^\delta \left(
\epsilon_{,1}^\delta \left(\frac{g_{01}}{g_{11}}\right) - \epsilon^\delta
\left(\frac{g_{01}}{g_{11}}\right)_{,1}\right. \nonumber \\
&\quad +\left. \epsilon_{,1}^\rho
\left(\frac{-\sqrt{-g}}{g_{11}}\right) - \epsilon^\rho \left(\frac{-\sqrt{-g}}{g_{11}}\right)_{,1}\right)\nonumber \\
&+ \left. \dot{\epsilon}^\rho \overline{\chi}^\rho + \dot{\epsilon}^\delta \overline{\chi}^\delta\right\} \nonumber
\end{align}
by eq. (A10).
A third approach is to find the gauge generator, again using the HTZ approach of eq. (A5), but this time
employing the primary constraints of eq. (26) so that
\begin{equation}
\overline{G}_{HTZ} = \int dx \left( \overline{A}_\rho\overline{\chi}^p + \overline{A}_\delta\overline{\chi}^\delta +
\overline{A}_{11}\overline{\chi}^{11} + \overline{B}_S S+ \overline{B}^{I\!\!P} I\!\!P\right)
\end{equation}
in place of eq. (24). Eq. (A5) results in
\begin{subequations}
\begin{align}
\frac{\partial \overline{B}_S}{\partial t} - \overline{A}_\delta &+ \overline{B}_S \left(\frac{g_{01}}{g_{11}}\right)_{,1}
- \overline{B}_{S,1} \left(\frac{g_{01}}{g_{11}}\right) \nonumber \\
&+ \overline{B}_{I\!\!P} \left(\frac{-\sqrt{-g}}{g_{11}}\right)_{,1} - \overline{B}_{I\!\!P,1} \left(\frac{-\sqrt{-g}}{g_{11}}\right) = 0\\
\intertext{\rm{and}}
\nonumber \\
\frac{\partial \overline{B}_{I\!\!P}}{\partial t} - \overline{A}_\rho &+ \overline{B}_{I\!\!P} \left(\frac{g_{01}}{g_{11}}\right)_{,1}
- \overline{B}_{I\!\!P,1} \left(\frac{g_{01}}{g_{11}}\right)\nonumber\\
&+ \overline{B}_S \left(\frac{-\sqrt{-g}}{g_{11}}\right)_{,1} - \overline{B}_{S,1} \left(\frac{-\sqrt{-g}}{g_{11}}\right) = 0.
\end{align}
\end{subequations}
From eqs. (34) and (36) we see that $G_C = \overline{G}_{HTZ}$.
With the generator $G_{HTZ}$ of eq. (24), we find that
\begin{align}
\delta f &= \left\lbrace f, G_{HTZ}\right\rbrace\nonumber \\
&= B_S p + B_{I\!\!P} f_{,1}
\end{align}
which by eq. (19) becomes
\begin{equation}
= B_S \sqrt{-g}\, g^{00} f_{,0} + \left(B_S \sqrt{-g}\, g^{01} + B_{I\!\!P}\right)f_{,1}.
\end{equation}
This is identical to the diffeomorphism transformation
\begin{equation}
\delta f = \eta^0 f_{,0} + \eta^1f_{,1}
\end{equation}
provided
\begin{align}
B_S &= - \frac{\sqrt{-g}}{g_{11}} \eta^0\\
B_{I\!\!P} &= \eta^1 + \frac{g_{01}}{g_{11}} \eta^0.
\end{align}
Eq. (25) cannot be uniquely solved for $A_{11}$, $A_{00}$ and $A_{01}$ in terms of $B_S$ and $B_{I\!\!P}$, but a particular solution with $B_S$ and $B_{I\!\!P}$ given by eqs. (40, 41) is
\begin{subequations}
\begin{align}
A_{11} &= 2g_{01} \eta_{,1}^0 + 2g_{11} \eta_{,1}^1 + \eta^0g_{11,0} + \eta^1 g_{11,1}\\
A_{00} &= 2g_{01} \eta_{,0}^1 + 2g_{00} \eta_{,0}^0 + \eta^1 g_{00,1} + \eta^0 g_{00,0}\\
A_{01} &= g_{00} \eta_{,1}^0 + g_{01} \left( \eta_{,0}^0 + \eta_{,1}^1\right) + g_{11} \eta^1_{,0} \nonumber \\
&\quad+ \eta^0 g_{01,0} + \eta^1 g_{01,1}.
\end{align}
\end{subequations}
These expressions are consistent with $\delta g_{\mu\nu} = \left\lbrace g_{\mu\nu}, G_{HTZ}\right\rbrace$ giving the diffeomorphism transformation
\begin{equation}
\delta g_{\mu\nu} = g_{\mu\rho} \eta^\rho_{,\nu} + g_{\nu\rho} \eta_{,\mu}^\rho + \eta^\rho g_{\mu\nu ,\rho}.
\end{equation}
An additional solution to eq. (25) is
\begin{equation}
B_S = B_{I\!\!P} = 0
\end{equation}
\begin{equation}
A_{00} = \Lambda g_{00},\quad A_{11} = \Lambda g_{11}, \quad A_{01} = \Lambda g_{01}
\end{equation}
so that
\begin{equation}
\delta g_{\mu\nu} = \left\lbrace g_{\mu\nu} G_{HTZ}\right\rbrace = \Lambda g_{\mu\nu}.
\end{equation}
This is the Weyl conformal (scale) invariance. The transformations generated by $G_{HTZ}$ has also been found in ref. [23], and can also be
found using $G_C$ and $\overline{G}_{HTZ}$.
We now consider gauge invariance in two dimensions when a massless scalar field is coupled to the metric and the EH action is first
order. Some aspects of this action were considered in ref. \cite{12}.
\section{First Order EH Action and Scalar Fields}
In $d$ dimensions, the action of eq. (8) can be written
\begin{equation}
S_{hG} = \int d^dx\, h^{\mu\nu} \left(G_{\mu\nu ,\lambda}^\lambda + \frac{1}{d-1} G_{\lambda\mu}^\lambda G_{\sigma\nu}^\sigma - G_{\sigma\mu}^\lambda G_{\lambda\nu}^\sigma\right).
\end{equation}
We begin by examining the equations of motion that follow from this form of the first order EH action before considering its canonical structure.
From eq. (47), the equations of motion for $G_{\mu\nu}^\lambda$ is
\begin{align}
h^{\mu\nu}_{,\lambda} - \frac{1}{d-1} &\left(\delta_\lambda^\mu h^{\nu\alpha} + \delta_\lambda^\nu h^{\mu\alpha}\right) G_{\alpha\beta}^\beta\nonumber \\
&+ G_{\lambda\alpha}^\mu h^{\nu\alpha} + G_{\lambda\alpha}^\nu h^{\mu\alpha} = 0
\end{align}
from which it follows immediately that
\begin{equation}
G_{\alpha\beta}^\beta = -\frac{1}{2} \left(\frac{d-1}{d-2}\right)h_{\rho\sigma}h^{\rho\sigma}_{,\alpha}.
\end{equation}
Substitution of eq. (49) into eq. (48) gives
\begin{align}
h_{,\lambda}^{\mu\nu} + \frac{1}{2(d-2)} &\left(\delta_\lambda^\mu h^{\nu\alpha} + \delta_\lambda^\nu h^{\mu\alpha}\right) h_{\rho\sigma}h^{\rho\sigma}_{\;,\alpha}\nonumber \\
&+ G_{\lambda\alpha}^\mu h^{\nu\alpha} + G_{\lambda\alpha}^\nu h^{\mu\alpha} = 0
\end{align}
which when combined with equations for $h^{\nu\lambda}_{\;,\mu}$ and $h^{\lambda\mu}_{\;,\nu}$ leads to
\begin{align}
G_{\mu\nu}^\lambda &= \frac{1}{2} h^{\lambda\rho}\left( h_{\mu\rho ,\nu} + h_{\nu\rho , \mu} - h_{\mu\nu , \rho}\right) \nonumber \\
&- \frac{1}{2(d-2)} h_{\mu\nu}h^{\lambda\rho}h_{\alpha\beta}h^{\alpha\beta}_{\;,\rho} .
\end{align}
For $d\neq2$, this is equivalent to having $\Gamma_{\mu\nu}^\lambda = \left\lbrace \begin{array}{c}
\lambda \\
\mu\nu \end{array}\right\rbrace$.~
From eqs. (49, 51) it is apparent that $d = 2$ dimensions is special. If $d = 2$, then eq. (48) leads to a consistency condition on the equations of
motion for $G_{\mu\nu}^\lambda$
\begin{equation}
h_{\mu\nu} h^{\mu\nu}_{\;,\lambda} = \frac{1}{\Delta} \Delta_{,\lambda} = 0 \quad (\Delta \equiv \det h^{\mu\nu})
\end{equation}
in place of eq. (49). Eq. (52) is consistent with
\begin{equation}
\Delta = (\det h^{\mu\nu}) = - (-\det g_{\mu\nu})^{\frac{d}{2}-1}
\end{equation}
when $d = 2$.
If now we set
\begin{equation}
G_{\mu\nu}^\lambda = \frac{1}{2} h^{\lambda\rho}\left( h_{\mu\rho ,\nu} + h_{\nu\rho , \mu} - h_{\mu\nu , \rho}\right) + h_{\mu\nu} X^\lambda
\end{equation}
where $X^\lambda$ is an arbitrary vector, then
\begin{align}
&- \left(\delta_\lambda^\mu h^{\nu\alpha} + \delta_\lambda^\nu h^{\mu\alpha}\right) G_{\alpha\beta}^\beta + G_{\lambda\alpha}^\mu h^{\nu\alpha} +
G_{\lambda\alpha}^\nu h^{\mu\alpha}\nonumber \\
&\qquad= - \frac{1}{2}\left(\delta_\lambda^\mu h^{\nu\alpha} + \delta_\lambda^\nu h^{\mu\alpha}\right) h^{\sigma\rho}h_{\sigma\rho ,\alpha} - h^{\mu\nu}_{\;,\lambda}
\end{align}
and hence eq. (54) satisfies eq. (48) provided eq. (52) is also satisfied. Arbitrariness is also present in $\Gamma_{\mu\nu}^\lambda$ \cite{14,15} when $d = 2$ if the equation
of motion for $\Gamma_{\mu\nu}^\lambda$ that follows from the first order form of the EH action in terms of $\Gamma_{\mu\nu}^\lambda$ and $g_{\mu\nu}$ is
solved to give eq. (3). Substitution of eq. (3) into the first order form of the EH action in terms of $\Gamma_{\mu\nu}^\lambda$ and $g_{\mu\nu}$ \
yields the second order form of the two dimensional EH action with all dependence on the arbitrary vector $\xi^\lambda$ dropping out. In contrast,
substitution of eq. (54) into eq. (47) with $d = 2$ leads to
\begin{align}
\int dx^2 & [ h^{\mu\nu} \left(G_{\mu\nu ,\lambda}^\lambda + G_{\lambda\mu}^\lambda G_{\sigma\nu}^\sigma - G_{\sigma\mu}^\lambda G_{\lambda\nu}^\sigma\right)] \\
&= \int dx^2\left[ \left(2X^\lambda + \frac{1}{2\Delta} h^{\lambda\rho} \Delta_{,\rho} + h^{\lambda\rho}h^{\sigma\tau}h_{\rho\sigma ,\tau}\right)_{,\lambda} \right. \nonumber \\
&\qquad - \frac{1}{\Delta} X^\lambda \Delta_{,\lambda} + \frac{1}{4\Delta^2} h^{\mu\nu} \Delta_{,\mu}\Delta_{,\nu}\nonumber \\
&\qquad + \frac{1}{4} h^{\mu\nu} h_{,\mu}^{\alpha\beta} h_{\alpha\beta ,\nu} + \frac{1}{2} h_{\mu\nu} h^{\alpha\mu}_{\;\,,\beta} h^{\beta\nu}_{\;\;,\alpha}\Bigg].\nonumber
\end{align}
Upon dropping the total derivatives in eq. (56), we see that $X^\lambda$ remains as a Lagrange multiplier that ensures that eq. (52) is satisfied.
Thus the role of $X^\lambda$ in eq. (54) is different from that of $\xi^\lambda$ in eq. (3).
We now perform a canonical analysis of $S_{hG}$ when $d = 2$. In order to do this we rewrite eq. (47) as
\begin{align}
S_{hr} &= \int d^2x \bigg[ -G_{00}^0 h_{,0} - 2G_{01}^0h_{,0}^1 - G_{11}^0h^{11}_{,0}\\
&\qquad\qquad - G_{00}^1 (h_{,1} + 2hG_{01}^0 + 2h^1G_{11}^0)\nonumber \\
&\qquad\qquad - 2G_{01}^1 (h_{,1}^1 - hG_{00}^0 + h^{11}G_{11}^0)\nonumber \\
&\qquad\qquad - G_{11}^1 (h_{,1}^{11} - 2h^1G_{00}^0 - 2h^{11}G_{01}^0)\bigg] .\nonumber \\
&\qquad\qquad\qquad\qquad (h = h^{00}, \quad h^1 = h^{01})\nonumber
\end{align}
From eq. (57) it is apparent that the momenta conjugate to $(h, h^1, h^{11})$ are
\begin{equation}
\pi = -G_{00}^0,\quad \pi_1 = -2G_{11}^0,\quad \pi_{11} = -G_{11}^0
\end{equation}
respectively. The momenta conjugate to the ``Lagrange multiplier'' fields $(\xi^1 = G_{00}^1,\quad \xi = 2G_{01}^1,\quad
\xi_1 = G_{11}^1)$ are zero; these primary constraints lead to the secondary constraints
\begin{subequations}
\begin{eqnarray}
\phi_1 &=& h_{,1} - h\pi_1 - 2h^1\pi_{11}\\
\phi &=& h_{,1}^1 + h\pi - h^{11}\pi_{11}\\
\phi^1 &=& h_{,1}^{11} + 2h^1\pi + h^{11}\pi_{1}.
\end{eqnarray}
\end{subequations}
(These fields $\xi^1,\xi,\xi_1$ are in fact treated as degrees of freedom, and are not merely Lagrange multipliers as is done in refs. \cite{34,35}.)
This constraint structure leads to the gauge transformation of eqs. (4, 5) \cite{7,8,9,10,11,12}. We see that despite the fact that $G^1_{\mu\nu}$ is a ``Lagrange multiplier``
field, its transformation under eq. (5) is not merely an arbitrary shift, demonstrating why it needs to be treated as a dynamical variable whose associated canonical momentum vanishes.
Under this transformation
\begin{equation}
\delta\Delta = 0
\end{equation}
and, according to eq. (54),
\begin{align}
\delta X^\mu &= \delta\left( h^{\mu\nu} G_{\lambda\nu}^\lambda - \frac{1}{2\Delta} h^{\mu\nu}\Delta_{,\nu}\right)\\
&= - h^{\mu\nu}\epsilon^{\lambda\sigma}\omega_{\nu\lambda ,\sigma} + \epsilon^{\mu\nu}\omega_{\nu\lambda} h^{\lambda\sigma}G_{p\sigma}^\rho\nonumber\\
& \qquad -h^{\mu\nu}G^\lambda_{\rho\nu}\epsilon^{\rho\sigma}\omega_{\lambda\sigma} - \frac{1}{2\Delta} \left(\epsilon^{\mu\lambda} h^{\sigma\nu} +
\epsilon^{\nu\lambda}h^{\sigma\mu}\right)\Delta_{,\nu}.\nonumber
\end{align}
Let us now supplement the action of eq. (47) with $d = 2$ by
\begin{equation}
S_f = \frac{1}{2} \int dx^2\, h^{\mu\nu} f_{,\mu}f_{,\nu}.
\end{equation}
The canonical momenta if $h^{\mu\nu}$, $G_{\mu\nu}^\lambda$ and $f$ are all independent fields given by
\begin{equation}
p = \frac{\partial\mathcal{L}}{\partial f_{,0}} = hf_{,0} + h^1f_{,1}
\end{equation}
\begin{equation}
\Pi_\lambda^{\mu\nu} = \frac{\partial\mathcal{L}}{\partial G_{\mu\nu ,0}^{\lambda}} =0
\end{equation}
as well as ($\pi$, $\pi_1$ and $\pi_{11}$).
The canonical Hamiltonian is
\begin{equation}
\mathcal{H}_C = \frac{1}{h} \Sigma + \left(\frac{-h^1}{h}\right) I\!\!P + \xi^1\phi_1 + \xi\phi + \xi_1\phi^1,
\end{equation}
where
\begin{equation}
\Sigma = \frac{1}{2}(p^2 - \Delta f_{,1}^2)
\end{equation}
and $I\!\!P$ is given in eq. (22). We now will show that $\phi^1,\,\phi,\,\phi_1,\,\\ I\!\!P$ and $\Sigma$ are all first class constraints.
The primary constraints
\begin{equation}
\Pi_1^{\mu\nu} = 0
\end{equation}
are first class; they lead to the secondary first class constraints
\begin{equation}
\phi_1 = \phi = \phi^1 = 0.
\end{equation}
One can show that
\begin{equation}
\left\lbrace \phi_1, \phi^1\right\rbrace = 2\phi\qquad , \left\lbrace \phi, \phi^1\right\rbrace = \phi^1, \qquad \left\lbrace \phi_1, \phi\right\rbrace = \phi_1
\end{equation}
\begin{equation}
\left\lbrace \phi_1, \Delta\right\rbrace = \left\lbrace \phi, \Delta \right\rbrace = \left\lbrace \phi^1, \Delta\right\rbrace = 0
\end{equation}
\begin{equation}
\Delta_{,1} = h \phi^1 + h^{11}\phi_1 - 2h^1 \phi,
\end{equation}
and, by using test functions as in ref. \cite{24},
\begin{equation}
\left\lbrace \Sigma(x), \Sigma(y)\right\rbrace = (\Delta(x) I\!\!P(x) \partial^y_1 - \Delta(y) I\!\!P(y)\partial_1^x)\delta(x-y)
\end{equation}
\begin{subequations}
This is not identical to the algebra of eq. (23a) unless $\Delta=1$. In addition we have
\begin{align}
&\left\lbrace \Sigma(x), I\!\!P(y)\right\rbrace \nonumber \\
&\quad=\left[(-\Sigma(x)\partial^y_1 + \Sigma(y)\partial_1^x) + \frac{1}{2} f_{,1}^2\Delta_{,1}\right]\delta(x-y)\\
&\left\lbrace I\!\!P(x), \Sigma(y)\right\rbrace \nonumber \\
&\quad= \left[-\Sigma(x)\partial^y_1 + \Sigma(y)\partial_1^x - \frac{1}{2} f_{,1}^2\Delta_{,1}\right]\delta(x-y)
\end{align}
\end{subequations}
Only if $\Delta_{,1}=0$ does eq. (73) reduce to the algebra of eq. (23b)
for the tertiary first class constraints $\Sigma$ and $I\!\!P$.
As was the case when we considered coupling $N$ scalars to the metric field in section 2, the EH action by itself has no net physical degrees of freedom,
while with the $N$ scalar fields there are $2N-4$ net physical degrees of freedom.
If the equation of motion were invoked so that by eq. (52) $\Delta$ would be constant, then $h$, $h^1$ and $h^{11}$ would not be independent, nor by eq. (71) would
$\phi_1$, $\phi$ and $\phi^1$. However, we will not impose this condition so that all components of $h^{\mu\nu}$ are independent.
(One could also ensure that $\Delta$ is constant by using a Lagrange multiplier.)
Using the HTZ approach, \cite{18,19} the generator of a gauge transformation is, by eq. (A2), of the form
\begin{equation}
G = \int dx (a^1\Pi_1 + a\Pi + a_1\Pi^1 + b^1\phi_1 + b\phi + b_1\phi^1 + c_\Sigma \Sigma + c_{I\!\!P}I\!\!P)
\end{equation}
where $\Pi_1$, and $\Pi$ and $\Pi^1$ are the momenta conjugate to $\xi^1$, $\xi$ and $\xi_1$ respectively. By eqs. (65, 69-73) it follows that
\begin{align}
&\left\lbrace G, \int dy \mathcal{H}_c \right\rbrace = \int dx \left\lbrace -a^1\phi_1 - a\phi - a_1\phi^1\right.\nonumber \\
&+ (b^1\xi - b\xi^1)\phi_1+2 (b^1\xi_1 - b_1\xi^1)\phi + (b\xi_1 - b_1\xi)\phi^1\nonumber\\
&+ \frac{1}{h^2} (bh + 2b_1h^1)\Sigma\nonumber\\
&+ \frac{1}{h^2}\left[-hh^1b-h^2b^1 + (hh^{11} - 2h^{1{^2}})b_1\right]I\!\!P\nonumber \\
&+ \left[\Delta\left(c_{\Sigma,1}(\frac{1}{h}) - c_\Sigma(\frac{1}{h})_{,1}\right) + c_{I\!\!P,1}(\frac{h^1}{h}) - c_{I\!\!P,1}(\frac{h^1}{h})_{,1}\right]I\!\!P\nonumber \\
&+ \left[ c_{\Sigma,1}(\frac{h^1}{h}) - c_\Sigma(\frac{h^1}{h})_{,1} - c_{I\!\!P,1}(\frac{1}{h}) + c_{I\!\!P}(\frac{1}{h})_{,1}\right] \Sigma\nonumber \\
& \left. - \frac{1}{2} \Delta_{,1}f_{,1} \left(\frac{h^1}{h} c_\Sigma + \frac{1}{h} c_{I\!\!P}\right)\right\rbrace
\end{align}
provided we ignore possible dependence of $(a^1, a, a_1)$ and $(b^1, b, b_1)$ on dynamical variables. (In the HTZ approach, ($c_\Sigma , c_{I\!\!P}$) are chosen to be
independent of dynamical variables.)
Eq. (A5) to orders $\Sigma$ and $I\!\!P$ respectively gives
\begin{equation}
\frac{\partial c_\Sigma}{\partial t} + \left[ + c_{\Sigma ,1} (\frac{h^1}{h}) - c_{\Sigma} (\frac{h^1}{h})_{,1} - c_{I\!\!P ,1} (\frac{1}{h}) + c_{I\!\!P} (\frac{1}{h})_{,1} \right]
\end{equation}
\begin{equation}
+ \frac{1}{h^2} (bh + 2b_1h^1) = 0\nonumber
\end{equation}
\begin{equation}
\frac{\partial c_{I\!\!P}}{\partial t} + \left[ \Delta\left( c_{\Sigma ,1} (\frac{1}{h}) - c_{\Sigma} (\frac{1}{h})_{,1}\right) + c_{I\!\!P ,1} (\frac{h^1}{h}) - c_{I\!\!P}
(\frac{h^1}{h})_{,1}\right]
\end{equation}
\begin{equation}
+ \frac{1}{h^2} [-hh^1b -h^2b^1 + (hh^{11} - 2h^{1^{2}})b_1] = 0\nonumber
\end{equation}
which relate $(b^1, b, b_1)$ to ($c_\Sigma , c_{I\!\!P}$). These equations are altered when ($c_\Sigma , c_{I\!\!P}$) depend on $(h, h^1, h^{11})$ by terms linear in
$(\xi^1, \xi , \xi_1)$.
We find that much like eq. (38)
\begin{equation}
\delta f = \left\lbrace f,G\right\rbrace = (c_\Sigma h) f_{,0} + (c_\Sigma h^1 + c_{I\!\!P})f_{,1}
\end{equation}
which reduce to eq. (39) provided $c_\Sigma$ and $c_{I\!\!P}$ acquire dependence on $h^1$ and $h$,
\begin{equation}
c_\Sigma = \eta^0/h
\end{equation}
\begin{equation}
c_{I\!\!P} = \eta^1 - h^1 \eta^0/h.
\end{equation}
If $c_\Sigma$ and $c_{I\!\!P}$ have this form, then eqs. (76) and (77) acquire extra contributions on the left side of
\begin{equation}
- \frac{\eta^0\xi}{h} - \frac{2h^1\eta^0\xi_1}{h^2}
\end{equation}
and
\begin{equation}
\frac{h^1\eta^0}{h} \xi + \eta^0\xi^1 + \frac{1}{h^2} (2h^{1^{2}} - hh^1)\eta^0\xi_1
\end{equation}
respectively. Upon substituting eqs. (79, 80) into eqs. (76, 77) when supplemented by eqs. (81, 82) we find two equations for $b$, $b_1$ and $b^1$ that are consistent
with taking
\begin{align}
b &= \eta_{,0}^0 + \eta_{,1}^1 + \eta^0\xi\\
b_1 &= \frac{1}{2h^1}\left(\eta^0 h_{,0} + \eta^1 h_{,1} - 2h^1 \eta_{,1}^0 - 2h\eta_{,0}^0 \right) + \eta^0\xi_1\\
b^1 &= \frac{1}{h^1}\left(\eta^1_{,0} h^1 - \eta_{,0}^0 h^{11}\right)+ \frac{h^{11}}{2hh^1} \left(\eta^1h_{,1} + \eta^0h_{,0}\right)\\
& \qquad \qquad - \frac{1}{h} (\eta^1 h_{,1}^1 + \eta^0 h_{,0}^1) + \eta^0\xi^1.\nonumber
\end{align}
With ($b, b_1, b^1$) given by eqs. (83-85) we find that
\begin{align}
\delta h &= \left\{h,G\right\} = -h \eta_{,0}^0 + h\eta_{,1}^1 + \eta^0 h_{,0} + \eta^1h_{,1} \\
& \qquad\qquad\qquad - 2h^1 \eta_{,1}^0+\eta^0 (h\xi + 2h^1\xi_1)\nonumber\\
\delta h^1 &= \left\{h^1,G\right\} = -h \eta_{,0}^1 + \eta^1h_{,1}^1 + \eta^0 h_{,0}^1 - h^{11}\eta_{,1}^0\\
& \qquad\qquad\qquad\qquad + \eta^0 (-h\xi^1 + h^{11}\xi_1)\nonumber\\
\delta h^{11} &= \left\{h^{11},G\right\} = -2h^1 \eta_{,0}^1 + h^{11} \eta_{,0}^0 - h^{11}\eta_{,1}^1 \\
& + h^{11}_{,0}\eta^0 + h^{11}_{,1}\eta^1-\frac{1}{h}(\Delta_{,0} \eta^0 + \Delta_{,1}\eta^1) \nonumber\\
&\qquad\qquad\qquad + \eta^0 (-2h^1\xi^1 - h^{11}\xi). \nonumber
\end{align}
From eq. (43), under a diffeomorphism transformation
\begin{equation}
\delta h^{\mu\nu} = h^{\mu\lambda} \theta_{,\lambda}^\nu + h^{\nu\lambda} \theta_{,\lambda}^\mu - (h^{\mu\nu}\theta^\lambda)_{,\lambda}
\end{equation}
which is the transformation of eqs. (86-88) provided
\begin{equation}
\theta^\lambda = -\eta^\lambda \;\;,\qquad \Delta_{,0} = \Delta_{,1} = 0\;\; {\rm{and}}\;\;\xi^1 = \xi = \xi_1 = 0.\nonumber
\end{equation}
An additional solution to eqs. (76, 77) is
\begin{equation}
c_\Sigma = c_{I\!\!P} = 0 \;,\qquad b = \frac{-2b_1h^1}{h}\;, b^1 = \frac{h^{11}b_1}{h}
\end{equation}
so that
\begin{equation}
b^1\phi_1 + b\phi + b_1\phi^1 = \frac{b_1}{h} \Delta_{,1}\;,
\end{equation}
and hence
\begin{equation}
\delta h^{\mu\nu} = \left\{ h^{\mu\nu},G\right\} = 0.
\end{equation}
Finding the variation of $G_{\mu\nu}^\lambda$ requires knowing the coefficients ($a^1, a, a_1$) in eq. (74). These are found by considering
these terms in eq. (A5) proportional to ($\phi^1, \phi, \phi_1$). By eq. (75), these are respectively given by
\begin{subequations}
\begin{align}
\frac{\partial b_1}{\partial t} &- a_1 + (b\xi_1 - b_1\xi) - \frac{1}{2} f_{,1}^2 (h^1c_\Sigma + c_{I\!\!P}) = 0\\
\frac{\partial b}{\partial t} &- a + 2(b^1\xi_1 - b_1\xi^1) + f_{,1}^2 \frac{h^1}{h}(h^1c_\Sigma + c_{I\!\!P}) = 0\\
\frac{\partial b^1}{\partial t}&- a^1 +(b^1\xi - b\xi^1) \\
&-\frac{1}{2} f_{,1}^2 \frac{h^{11}}{h}(h^1c_\Sigma + c_{I\!\!P}) = 0 \nonumber
\end{align}
\end{subequations}
provided we ignore terms in $\left\{G, \mathcal{H}_c\right\}$ that are linear in ($\phi^1, \phi , \phi_1$) on account of the dependency of
($b^1, b, b_1$) on ($h, h^1, h^{11}$) following from eqs. (76, 77). If one were to supplement eqs. (92, 93) with terms
\begin{equation}
\phi^1 \left\{ b_{1,}\phi^1\xi_1 + \phi\xi + \phi_1\xi^1\right\} + \phi\left\{ b, \phi^1\xi_1 + \phi\xi + \phi_1\xi^1\right\}\nonumber
\end{equation}
\begin{equation}
+ \phi_1 \left\{ b^1, \phi^1\xi_1 + \phi\xi + \phi_1\xi^1\right\}
\end{equation}
in order to take into account the dependency of ($b_1, b, b^1$) on ($h, h^1, h^{11}$), and use eqs. (83-85) for ($b_1, b, b^1$), one
encounters ill defined PBs of the form $\left\{h_{,0}, \pi\right\}$ indicating a breakdown of the HTZ procedure for finding the generator of a gauge transformation
that leads to eq. (A5).
However, it is possible to overcome this shortcoming of the HTZ approach for finding the generator of a gauge transformation. If instead of eqs. (A3), one were
to take the change in a dynamical variable $A$ to be given by
\begin{equation}
\delta A = \nu^{a_{i}} \left\{ A, \gamma_{a_{i}}\right\}
\end{equation}
so that $\nu^{a_{i}}$ is not affected when one computes the PB,
then the change in the extended action of eq. (A1) would be
\begin{align}
\delta S_E = & \int dt \bigg[- v^{a{_{i}}} \big(\left\{\gamma_{a_{i}},p^j\right\}\dot{q}_j - \left\{\gamma_{a_{i}},q_j\right\}\dot{p}^i\\
&- \left\{ \gamma_{a_{i}},q_j\right\} \frac{\partial H_c}{\partial q_i} - \left\{ \gamma_{a_{i}},p^j\right\} \frac{\partial H_c}{\partial p^j} \nonumber\\
& - U^{a_{j}} \left\{ \gamma_{a_{i}},\gamma_{a_{j}}\right\} \big) - \delta U^{a_{i}} \gamma_{a_{i}}\bigg]\nonumber
\end{align}
provided we do an integration by parts, dropping the surface term. Eq. (96) further reduces to
\begin{align}
\delta S_E = \int dt & \bigg[- v^{a{_{i}}} \big(\frac{\partial\gamma_{a_{i}}}{\partial q_j}\dot{q}_j + \frac{\partial\gamma_{a_{i}}}{\partial p^j}\dot{p}^j\\
&- \left\{ \gamma_{a_{i}},H_c + U^{a_{j}} \gamma_{a_{j}}\right\} \big) - \delta U^{a_{i}}\gamma_{a_{j}}\bigg]\nonumber
\end{align}
as $u^{a_{j}}$ is not dynamical; a further integration by parts without keeping the surface terms leads to
\begin{equation}
\delta S_E = \int dt \Bigg[ + \gamma_{a_{i}} \frac{D\nu^{a_{i}}}{Dt} + \nu^{a_{i}} \left\{\gamma_{a_{i}}, H_c + U^{a_{j}}\gamma_{a_{j}}\right\} - \delta U^{a_{i}}\gamma_{a_{i}}\Bigg]
\end{equation}
which is almost identical to eq. (A4). However, the coefficients $\nu^{a_{i}}$ are not involved in the evaluation of any PBs.
For the system we have been considering, we can employ eq. (98) to find the gauge transformation of a dynamical variable $A$
\begin{align}
\delta A &= \overline{a}^1 \left\{ A,\Pi_1\right\} + \overline{a} \left\{ A,\Pi\right\} + \overline{a}_1 \left\{ A,\Pi^1\right\}\\
&\qquad\qquad + \overline{b}^1 \left\{ A,\phi_1\right\} + \overline{b} \left\{ A,\phi\right\} + \overline{b}_1 \left\{ A,\phi^1\right\}\nonumber \\
& \qquad\qquad\qquad + \overline{c}_\Sigma \left\{ A,\Sigma \right\}+ \overline{c}_{I\!\!P} \left\{ A,I\!\!P\right\}.\nonumber
\end{align}
Eq. (98), when used in the same way eq. (A4) has been used by HTZ \cite{18,19} fixes ($\overline{b}^1, \overline{b}, \overline{b}_1$) in terms of
($\overline{c}_{\Sigma}, \overline{c}_{I\!\!P}$) by eqs. (76, 77) and in turn determines ($\overline{a}^1, \overline{a}, \overline{a}_1$) by
eqs. (92, 93).
We find that, for example, that eq. (95) leads to
\begin{equation}
\hspace{-6cm}\delta G_{01}^1 = \overline{a}\left\{\frac{1}{2} \xi, \Pi\right\}
\end{equation}
which, by eq. (93b) becomes
\begin{equation}
\qquad = \frac{1}{2} \left[ \frac{\partial\overline{b}}{\partial t} + 2 (\overline{b}^1 \xi_1 - \overline{b}_1 \xi^1) + f_{,1}^2 \frac{h^1}{h}(h^1 \overline{c}_\Sigma + \overline{c}_{I\!\!P})\right].
\end{equation}
Eqs. (79, 80, 83-85) in turn show that eq. (101) reduces to
\begin{subequations}
\begin{eqnarray}
\delta G_{01}^1 &=& \frac{1}{2} \left[ \left(\eta_{,0}^0 + \eta_{,0}^1 + 2\eta^0 G_{01}^1\right)_{,0} + 2( \frac{1}{h^1} (\eta_{,0}^1 h^1 - \eta_{,0}^0h^{11})\right.\nonumber\\
&+& \frac{h^{11}}{2hh^1}(\eta^1 h_{,1}+ \eta^0 h_{,0}) - \frac{1}{h} (\eta^1 h^1_{,1} + \eta^0 h^1_{,0}))G_{11}^1\nonumber\\
&-& \left(\frac{1}{h^1}\right) \left(\eta^0 h_{,0}+ \eta^1 h_{,1} - 2h\eta^0_{,1} - 2h\eta_{,0}^0\right)G_{00}^1\nonumber\\
&+& \left. f_{,1}^2 \frac{h^1}{h}\eta^1\right]
\end{eqnarray}
Similarly, we find that
\begin{eqnarray}
&\delta G_{00}^1& = \overline{a}_1\left\{\xi^1,\Pi_1\right\}\nonumber\\
&=& \frac{\partial\overline{b}_1}{\partial t} + (\overline{b}\xi_1 - \overline{b}_1 \xi) - \frac{1}{2} f_{,1}^2 (h^1c_\Sigma + c_{I\!\!P})
\end{eqnarray}
and
\begin{eqnarray}
\delta G_{11}^1&=& \overline{a}^1\left\{\xi_1,\Pi^1\right\}= \frac{\partial\overline{b}^1}{\partial t} + (\overline{b}^1\xi - \overline{b} \xi^1)\nonumber \\
&-& \frac{1}{2} f_{,1}^2 \frac{h^{11}}{h}(h^1c_\Sigma + c_{I\!\!P})
\end{eqnarray}
\end{subequations}
Eqs. (102) have a term proportional to $f_{,1}^2$; similarly by eqs. (95, 66), $\delta G_{00}^0$ has a term proportional to $-\frac{1}{2} h^{11} c_\Sigma f_{,1}^2$. It is
apparent that $\delta G_{\mu\nu}^\lambda$ always has a contribution proportional to $f_{,1}^2$. This mixing of the affine connection and scalar field under a gauge transformation
is somewhat unusual. The change in $G_{\mu\nu}^\lambda$ under a diffeomorphism is
\begin{align}
\delta G_{\mu\nu}^\lambda &= - G_{,\mu\nu}^\lambda + \frac{1}{2}\left(\delta_\mu^\lambda \theta_{,\nu\rho}^\rho + \delta_\nu^\lambda \theta_{,\mu\rho}^\rho\right)
- \theta^\rho G_{\mu\nu ,\rho}^\lambda\nonumber \\
&+ G_{\mu\nu}^\rho \theta_{,\rho}^\lambda - \left( G_{\mu\rho}^\lambda \theta_{,\nu}^\rho +
G_{\nu\rho}^\lambda \theta_{,\mu}^\rho\right)
\end{align}
which does not mix $G_{\mu\nu}^\lambda$ and $f_{,1}$.
It is also possible to use the approach of [16,17] to find the gauge generator associated with $S_{hG} + S_f$ when $d = 2$. In eq. (A12), $N = 2$ since there are
tertiary constraints. With $G_2 = \Pi^1$ and $\mathcal{H}_c$ given by eqs. (65), it follows from
\begin{equation}
G_1 + \left\{ G_2, H_c\right\} \approx p.c.
\end{equation}
that
\begin{align}
G_1(x) &= \phi^1(x) + \int dy [\alpha^1 (x-y)\Pi_1(y)\nonumber \\
&+ \alpha(x-y)\Pi(y)+\alpha_1(x-y)\Pi^1(y)];
\end{align}
next
\begin{equation}
G_0 + \left\{ G_1, H_c\right\} \approx p.c.
\end{equation}
leads to
\begin{align}
G_0 &= \int dy \left[\beta^1(x-y)\Pi_1(y) \right.\nonumber\\
& + \beta(x-y)\Pi(y) +\beta_1(x-y)\Pi^1(y)\nonumber \\
& \left. + \alpha^1(x-y)\phi_1(y) + \alpha(x-y)\phi(y) + \alpha_1(x-y)\phi^1(y)\right]\nonumber \\
&+ 2\xi^1(x)\phi(x) + \xi(x)\phi^1(x)\\
& \quad - \frac{2h^1(x)\Sigma(x)}{h^2(x)} + \left( \frac{2h^{1^{2}}(x) - h(x)h^{11}(x)}{h^2(x)}\right)I\!\!P(x).\nonumber
\end{align}
The final condition
\begin{equation}
\left\{ G_0, H_c\right\} \approx p.c.
\end{equation}
is satisfied to orders $\Sigma$, $I\!\!P$, $\phi^1$, $\phi$ and $\phi_1$ respectively provided
\begin{subequations}
\begin{align}
&\frac{\alpha}{h} + \frac{2h^1\alpha_1}{h^2} + \frac{4\xi^1}{h} + \frac{6 h^1 \xi}{h^2} + \left(\frac{8h^{1^{2}}- 2hh^{11}}{h^3}\right)\xi_1\nonumber \\
&- 2\left(\frac{h^{1^{2}}}{h^2}\right)_{,1} \frac{1}{h} + \left(\frac{hh^{11}}{h}\right)_{,1} \frac{1}{h^2} = 0
\end{align}
\begin{align}
&-\alpha^1 - \frac{h^1\alpha}{h} + \left(\frac{-2h^{1^{2}}+ hh^{11}}{h^2}\right)\alpha_1 - \frac{4h^1\xi^1}{h}\nonumber \\
&+ \left(\frac{-6h^{1^{2}}+ 3hh^{11}}{h^2}\right)\xi+ \left(\frac{-8h^{1^{2}}+ 6hh^{11}}{h^3}\right)(h^1\xi_1) \nonumber \\
&- \frac{h^{11}}{h} \left(\frac{h^1}{h}\right)_{,1} + \left(\frac{2h^{1^{2}}- hh^{11}}{h^2}\right)_{,1} \left(\frac{h^1}{h}\right) = 0
\end{align}
\end{subequations}
\begin{subequations}
\begin{align}
&-\beta_1 + \alpha\xi_1 - \alpha_1\xi + 2\xi^1\xi_1 - \xi^2 + \frac{h^{11}}{2h} f_{,1}^2 \\
&\qquad\qquad\qquad+ \left\{\alpha_{,1}H_c\right\} = 0\nonumber\\
&-\beta + 2(\alpha^1\xi_1 - \alpha_1\xi^1 -\xi\xi^1) - \frac{h^1h^{11}}{h^2} f_{,1}^2 \\
&\qquad\qquad\qquad+ \left\{\alpha,H_c\right\} = 0\nonumber\\
&-\beta^1 + \alpha^1\xi - \alpha\xi^1 -2\xi^{1^{2}} + \frac{h^{11^{2}}f_{,1}^3}{2h^2}\\
&\qquad\qquad\qquad+ \left\{\alpha^1,H_c\right\} = 0.\nonumber
\end{align}
\end{subequations}
In exactly, the same way we find that if $G_2 = \Pi$, then
\begin{align}
G_1 &= \phi + \int dy (\alpha^1 \Pi_1 + \alpha\Pi + \alpha_1\Pi^1)\\
G_0 &= \int dy \left[\beta^1 \Pi_1 + \beta\Pi + \beta_1\Pi^1 + \alpha^1\phi_1 + \alpha\phi + \alpha_1\phi^1\right]\nonumber\\
&\qquad\qquad + \xi^1\phi_1 - \xi_1\phi^1 - \frac{1}{h} (\Sigma - h^1I\!\!P)
\end{align}
with
\begin{subequations}
\begin{align}
&\frac{\alpha}{h} + \frac{2h^1\alpha_1}{h^2} + \frac{\xi}{h} = 0\\
&-\frac{\alpha^1}{h} - \frac{h^1\alpha}{h} + \frac{-2h^{1^{2}} + hh^{11}}{h^2} \alpha_1\\
&\qquad\qquad\qquad - 2\xi^1 - \frac{h^1\xi}{h} = 0\nonumber
\end{align}
\end{subequations}
\begin{subequations}
\begin{align}
&- \beta_1 + \xi_1\alpha - \xi\alpha_1 + \xi\xi_1 + \left\{\alpha_1,H_c\right\} = 0\\
&- \beta + 2\left( \xi_1\alpha^1 - \xi^1\alpha_1 + 2\xi^1\xi_1\right) + \left\{\alpha,H_c\right\} = 0\\
&- \beta^1 + \left( \xi\alpha^1 - \xi^1\alpha + \xi\xi^1\right) + \left\{\alpha^1,H_c\right\} = 0.
\end{align}
\end{subequations}
Finally, if $G_2 = \Pi_1$, then we find that
\begin{equation}
G_1 = \phi_1 + \int dy\left[ \alpha^1 \Pi_1 + \alpha\Pi + \alpha_1\Pi^1\right]
\end{equation}
\begin{align}
G_0 &= \int dy \left[ \beta^1\Pi_1 + \beta\Pi + \beta_1\Pi^1 + \alpha^1\phi_1 + \alpha\phi + \alpha_1\phi^1\right]\nonumber\\
&\qquad\qquad - \xi\phi_1 - 2\xi_1\phi + I\!\!P
\end{align}
and so
\begin{subequations}
\begin{align}
&\frac{\alpha}{h}+ \frac{2h^1}{h^2}\alpha_1 - \frac{2\xi_1}{h} + \left(\frac{1}{h}\right)_{,1} = 0\\
&- \alpha^1 + \alpha_1 \left( \frac{-2h^{1^{2}} + hh^{11}}{h^2}\right) - \frac{h^1}{h} \alpha+ \xi \nonumber \\
&+ \frac{2h^1}{h}\xi_1 - \left(\frac{h^1}{h}\right)_{,1} = 0
\end{align}
\end{subequations}
\begin{subequations}
\begin{align}
&- \beta_1 + \xi_1\alpha - \xi\alpha_1 - 2\xi_1^2 - \frac{1}{2} f_{,1}^2
+ \left\{\alpha_1,H_c\right\} = 0\\
&- \beta + 2\xi_1\alpha^1 - 2\xi^1\alpha_1 - 2\xi\xi_1 + \frac{h^1}{h} f_{,1}^2\nonumber \\
&\qquad\qquad\qquad\qquad+\left\{\alpha,H_c\right\} = 0\\
&- \beta^1 + \xi\alpha^1 - \xi^1\alpha + 2\xi^1\xi_1 - \xi^2 - \frac{h^{11}}{2h} f_{,1}^2\nonumber \\
&\qquad\qquad\qquad\qquad+\left\{\alpha^1,H_c\right\} = 0.
\end{align}
\end{subequations}
In the instance where $G_2 = \Pi^1$, the two conditions of eqs. (109a,b) do not fix $\alpha^1$, $\alpha$ and $\alpha_1$ uniquely; however eqs.
(110a,b,c) do determine $\beta^1$, $\beta$ and $\beta_1$ in terms of $\alpha^1$, $\alpha$ and $\alpha$. This lack of uniqueness in the gauge
generator is a consequence of there being but two tertiary first class constraints following from the three primary first class constraints. The
same pattern is repeated when $G_2 = \Pi$ (eqs. (113, 114)) and $G_2 = \Pi$, (eqs. (117, 118)). In each case though, $\beta^1$, $\beta$ and $\beta_1$
depend on $f_{,1}^2$ in such a way that the transformation $\delta G_{\mu\nu}^\lambda$ depends on $f_{,1}^2$ as was the case when the HTZ approach to
finding a gauge generator was used.
\section{Discussion}
In this paper we have closely followed the Dirac constraint formalism \cite{1,2,3,4,5,6} to analyze the gauge structure of a two dimensional massless scalar
field in curved space. Though it has long been recognized that this is related to the Bosonic string \cite{21} and that this is a system involving constraints, it does not appear that
a full constraint analysis has been performed on this system. It always appears that some fields have been eliminated by choosing to work in a ``convenient'' gauge before
the constraints are identified, or that the generator of gauge transformations is postulated rather than derived from the first class constraints (see for example
ref. \cite{25}).
In this analysis we have included the EH action in second order form \cite{7}, even though it normally is dropped since it does not contain any dynamical degrees of freedom.
This suggests that we also consider the first order EH action whose canonical structure in the absence of matter leads to a gauge invariance generated
by the first class constraints that appears distinct from diffeomorphism invariance, and which accounts for the absence of dynamical degrees of freedom \cite{8,9,10,11,12,13}. (We
might also look at other actions for the two dimensional metric field be considered, such as the Weyl scalar invariant action which involves a vector field \cite{27}.)
One peculiarity in our canonical analysis is that by adding the scalar field $f$, two degrees of freedom are added in phase space, but this also results in two more first class
constraints (either $S$ and $I\!\!P$ or $\Sigma$ and $I\!\!P$ for the second order and first order EH actions respectively) which when combined with the
associated gauge conditions, leads to a negative number of degrees of freedom ($-2$) in phase space. This issue was raised but not satisfactorily resolved in
ref. \cite{12}. If there are $N$ scalars $f^a$ and the kinetic term for these scalars were $O(N)$ symmetric, then there are ten restrictions on $2N+6$ fields in phase space,
leaving $2N-4$ independent degrees of freedom. There are also $2N-4$ net degrees of freedom when using the first order form of the EH action.
The problem with having an unexpected number of degrees of freedom (especially when $N=1$) is implicit in all discussions of the canonical structure of the Bosonic string
that we have encountered in the literature (see for example ref. \cite{25}) but no satisfactory resolution of the problem has been provided. In particular, if $N=1$,
it would seem that the first class constraints of eqs. (21,22), or eqs. (22,66) would require imposing a gauge fixing that would over determine $f$ and its conjugate momentum $p$.
For $N=26$ there is a positive number of degrees of freedom (48) even after a gauge is chosen and this problem of over determination of $f^{(a)}$ and $p^{(a)},(a=1...26)$ does not arise.
Consequently, the Bosonic string does not suffer from this particular inconsistency. In fact though, one should not be surprised that if $N=1$ there are no degrees of freedom associated with the scalar $f$,
as the equation of motion for $h^{\mu\nu}$ that follows from eq. (7) is $(\partial_{\mu}f)(\partial_{\nu}f)=0$ which implies that $f$ does not propagate. The equation of motion that follows from $g_{\mu\nu}$
in eq. (2) is $\partial_{\mu}\partial_{\nu}f-\frac{1}{2}g_{\mu\nu}g^{\alpha\beta}\partial_{\alpha}f\partial_{\beta}f=0$ which has the same implications. For $N>1$fields, $f^{(a)}$ is not necessarily a constant
in order to satisfy the equations of motion for the metric.
Our analysis displays some interesting features of the approaches of C and HTZ to finding the gauge generator from the first class constraints. First of all, it
is apparent from our discussion of the gauge generator when the EH action is second order that the actual form of the generator is dependent on how the constraints
are chosen. When using the method of C, which form of the primary constraints is chosen is important (as was pointed out in ref. \cite{26}) while the form of the gauge
generator found using the approach of HTZ is different when different linear combinations of constraints of the highest order are employed.
The diffeomorphism invariance manifestly present in the initial Lagrangian is only recovered when using the second order form of the EH action if the gauge parameters
associated with the secondary constraints are field dependent (which is contrary to the HTZ approach). There is also a residual symmetry occurring in this case. This
additional symmetry resulting from the gauge generator is the Weyl scale symmetry. Thus both diffeomorphism invariance and Weyl scale invariance are gauge symmetries.
The HTZ formalism, when applied to first order form of the EH action plus the action for a scalar field, yields the diffeomorphism transformation for the scalar field only if the gauge
parameters associated with the tertiary constraints are again field dependent. The resulting equations for the gauge parameters associated with primary constraints
involves ill defined PBs that can be avoided by slightly modifying the HTZ procedure. When this is done, the resulting gauge transformation is unusual as it mixes the affine
connection and the scalar field in an non-polynomial fashion. We have attempted unsuccessfully to find such a gauge invariance directly from the action
given in eqs. (47, 62).
Of course, once the canonical structure of these models is disentangled, their quantization is to be considered. This may have implications for Bosonic string theory.
\begin{acknowledgement}
We would like to thank S.V. Kuzmin and N. Kiriushcheva for helpful discussions and R. Macleod for encouragement.
\end{acknowledgement}
|
{
"timestamp": "2011-11-08T02:03:18",
"yymm": "1009",
"arxiv_id": "1009.3578",
"language": "en",
"url": "https://arxiv.org/abs/1009.3578"
}
|
\section{Introduction}
\noindent
Let $f\:\CDach\rightarrow \CDach$ be a rational map on the Riemann sphere $\CDach$ of degree $\ge 2$. As usual we call a point $p\in \CDach$ a {\em critical point} of $f$ if near $p$ the map $f$ is not a local homeomorphism. A {\em postcritical point} is any point obtained as an image of a critical point under forward iteration of $f$. So if we denote by $\operatorname{crit}(f)$ the set of critical points of $f$ and by $f^n$ the $n$-th iterate of $f$, then the set of postcritical points of $f$ is given by
\begin{equation*}
\operatorname{post}(f):=\bigcup_{n\geq 1} \{f^n(c):c\in \operatorname{crit}(f)\}.
\end{equation*}
It is a fundamental fact in complex dynamics that much information on the dynamics of $f$ can be deduced from the structure of the orbits of critical points. A very strong assumption in this respect is that each such orbit is finite, i.e., that the set $\operatorname{post}(f)$ is a finite set. In this case the map $f$ is called {\em postcritically-finite}.
A characterization of such maps is due to Thurston. The framework for his investigations
was the setting of branched covering maps of $2$-spheres. These are continuous maps $f\: S^2 \rightarrow S^2$ on an oriented
$2$-sphere $S^2$ that near each point can be written as $z\mapsto z^d$ after suitable
orientation-preserving coordinate changes in source and target.
For such maps the sets of critical and postcritical points can be defined in the same way as for rational maps on $\CDach$.
A {\em Thurston map} is a branched covering map
$f\: S^2 \rightarrow S^2$ with a finite set of postcritical points that is not a homeomorphism. In this paper we study Thurston maps that are expanding in an appropriate sense.
By definition this means that there exists a Jordan curve $\mathcal{C}$ in $S^2$ containing the set of postcritical points of $f$ such that the complementary components of $f^{-n}(\mathcal{C})$
become uniformly small as $n\to \infty$.
For rational Thurston maps this is satisfied if
and only if $f$ has no periodic critical points. It is also equivalent to requiring that the Julia set $\mathcal{J}(f)$
of $f$ is the whole Riemann sphere (see Proposition~\ref{prop:rationalexpch}).
Note that in contrast to the rational case, in general an expanding Thurston map may have periodic
critical points; see Example \ref{ex:barycentric}.
\subsection{Results} One of our goals is to obtain a description of the dynamics of an expanding Thurston map in terms of finite combinatorial data. A very general setting which allows one to address this question is the recently developed theory of self-similar group actions (see \cite{Ne}, in particular Sect.\ 6). Our approach is more concrete and based on the existence of $f$-invariant Jordan curves $\mathcal{C}\subset S^2$, i.e., Jordan curves with $f(\mathcal{C})\subset \mathcal{C}$. We also require that $\operatorname{post}(f)\subset \mathcal{C}$, because under this condition the preimages $f^{-n}(\mathcal{C})$ of $\mathcal{C}$ under the iterates $f^n$ of $f$ form the $1$-skeleton of a cell decomposition of $S^2$ that allows one to recover $f^n$ as a ``cellular map" in essentially combinatorial terms (see the discussion below).
Some of our main results are about existence and uniqueness of such Jordan curves $\mathcal{C}$.
For rational Thurston maps we have the following statement.
\begin{theorem}\label{thm:main0}
Let $f\colon\CDach \rightarrow \CDach$
be a rational Thurston map with Julia set
$\mathcal{J}(f)=\CDach$. Then for each sufficiently large
$n\in \N$ there exists a quasicircle $\mathcal{C}\subset
\CDach$ with $\operatorname{post}(f) \subset\mathcal{C}$ and $f^n(\mathcal{C})\subset \mathcal{C}$.
\end{theorem}
If a curve $\mathcal{C}$ is invariant for some iterate $f^n$, then one cannot expect it to be invariant for some other iterate $f^k$ unless $k$ is a multiple of $n$ (see Remark~\ref{rem:ndep}). So in general, the curve $\mathcal{C}$ in the previous theorem (also in Theorem~\ref{thm:main} below) will depend on $n$.
For the definition of a quasicircle see
Section~\ref{sec:cc-quasicircle}.
Theorem~\ref{thm:main0} is a special case of a more general fact.
\begin{theorem}
\label{thm:main}
Let $f\colon S^2\to S^2$ be an expanding Thurston map, and $\mathcal{C}\subset S^2$ be a Jordan curve with $\operatorname{post}(f)\subset \mathcal{C}$. Then for
each sufficiently large $n\in \N$ there exists a Jordan curve $\widetilde{\mathcal{C}}$ that is invariant for $f^n$ and isotopic to $\mathcal{C}$ rel.\ $\operatorname{post}(f)$.
\end{theorem}
See Section~\ref{sec:thmaps} for a discussion of isotopies and related terminology. Since $\widetilde{\mathcal{C}}$ is isotopic to $\mathcal{C}$ rel. $\operatorname{post}(f)$,
it will also contain the set
$\operatorname{post}(f)$.
The curve $\widetilde \mathcal{C}$ is actually a quasicircle if $S^2$ is equipped with a suitable metric (see Theorem~\ref{thm:Cquasicircle} below).
An obvious question is whether
one can always find a Jordan curve $\mathcal{C}\subset
S^2$ with $\operatorname{post}(f) \subset\mathcal{C}$ that is invariant under the map $f$ itself, and hence invariant under all iterates $f^n$.
It turns out that this is not true in general (Example~\ref{ex:noinvCC}), but one can give a necessary and sufficient condition for the existence of such an invariant curve.
\begin{theorem}[Existence of invariant curves]
\label{thm:exinvcurvef}
Let $f\colon S^2\to S^2$ be an expanding Thurston map.
Then the following conditions are equivalent:
\begin{enumerate}[{\upshape(i)}]
\smallskip
\item
\label{item:ex_invC1}
There exists an $f$-invariant Jordan curve $ \widetilde \mathcal{C}\subset
S^2$ with $\operatorname{post}(f)\subset\widetilde \mathcal{C}$.
\smallskip
\item
\label{item:ex_invC2}
There exist Jordan curves $\mathcal{C}, \mathcal{C}'\subset S^2$ with
$\operatorname{post}(f)\subset \mathcal{C}, \mathcal{C}'$ and $ \mathcal{C}'\subset f^{-1}(\mathcal{C})$, and an isotopy $H\colon S^2\times I\to S^2$ rel.\ $\operatorname{post}(f)$
with $H_0 = \id_{S^2}$ and $H_1(\mathcal{C})= \mathcal{C}'$ such that
the map
\begin{equation*}
\widehat{f}:= H_1 \circ f \text{ is combinatorially expanding
for } \mathcal{C}'.
\end{equation*}
\end{enumerate}
Moreover, if \eqref{item:ex_invC2} is true, then there exists an $f$-invariant Jordan curve $\widetilde \mathcal{C}\subset S^2$ with $\operatorname{post}(f)\subset \widetilde \mathcal{C}$ that is isotopic to $\mathcal{C}$ rel.\ $\operatorname{post}(f)$ and isotopic to $ \mathcal{C}'$ rel.\
$f^{-1}(\operatorname{post}(f))$.
\end{theorem}
See Definition~\ref{def:combexp} for the notion of an combinatorially expanding
Thurston map (Definition~\ref{def:connectop} and \eqref{def:dk} are also relevant here).
The condition of combinatorial expansion in (ii) is easy to
check in general (see Remark~\ref{rem:expcheck}~(a),
Proposition~\ref{prop:nscombexp}, and the examples discussed in
Section~\ref{sec:constructc}). One can actually give a criterion for
the existence of
an invariant curve in a given isotopy class rel.\ $\operatorname{post}(f)$ or rel.\
$f^{-1}(\operatorname{post}(f))$ (see Remark~\ref{rem:expcheck}~(c) and Proposition~\ref{prop:invC_invC1}). Moreover, if
an $f$-invariant Jordan curve $\mathcal{C}$ exists, then it is the Hausdorff
limit of a sequence of Jordan curves $\mathcal{C}^n$ that can be obtained
from a simple iterative procedure
(Proposition~\ref{prop:invCit}).
Our existence results are complemented by the following uniqueness statement for invariant Jordan curves.
\begin{theorem} \label{thm:uniqc}
Let $f\: S^2\rightarrow S^2$ be an expanding Thurston map and
$\mathcal{C}$ and $\mathcal{C}'$ be $f$-invariant Jordan curves in $S^2$ that both contain the set $\operatorname{post}(f)$.
Then $\mathcal{C}=\mathcal{C}'$ if and only if $\mathcal{C}$ and $\mathcal{C}'$ are isotopic rel.\ $f^{-1}(\operatorname{post}(f))$.
\end{theorem}
As a consequence one can prove that if $\#\operatorname{post}(f)=3$, then there are at most finitely many $f$-invariant Jordan curves $\mathcal{C}\subset S^2$ with $\operatorname{post}(f)\subset \mathcal{C}$ (Corollary~\ref{cor:finitepost3}). In the general case, there are at most finitely many such curves $\mathcal{C}$ in a given
isotopy class rel.\ $\operatorname{post}(f)$ (Corollary~\ref{cor:finiterelP}). In
general, a Thurston map may have infinitely many such invariant
curves $\mathcal{C}$ (Example~\ref{ex:infty_C}).
Expanding Thurston maps are abundant and include specific maps on $\CDach$ such as
$f(z)=1-2/z^2$ or $f(z)=1+({\mathbf{\imath}}-1)/z^4$. More examples can be found in Section~\ref{sec:examples-two-tile}. A large class of well-understood Thurston maps are
{\em Latt\`es maps}. These are rational maps obtained as quotients of conformal torus endomorphisms (note that the terminology is not uniform and some authors use the term
Latt\`es map with a slightly different meaning). We will discuss an explicit Latt\`es map in detail below.
A general method for producing Thurston maps is given by
Pro\-po\-si\-tion~\ref{prop:rulemapex} in combination
with Corollary~\ref{cor:combexp1}; conversely, as follows from Theorem~\ref{thm:main}, at least some iterate of every expanding Thurston map can be obtained from this construction.
If $f\:S^2\rightarrow S^2$ is an expanding Thurston map and $\mathcal{C}\subset S^2$ a Jordan curve with $\operatorname{post}(f) \subset\mathcal{C}$, then for each $n\in \N_0$ one can define an associated cell decomposition $\mathcal{D}^n=\mathcal{D}^n(f,\mathcal{C})$ of $S^2$ (Section~\ref{sec:tiles}). Its $0$-skeleton is the set $f^{-n}(\operatorname{post}(f))$, and its $1$-skeleton
the set
of $f^{-n}(\mathcal{C})$ (see Section~\ref{s:celldecomp} for the terminology). In general, these cell decompositions $\mathcal{D}^n$ are not compatible for different levels $n$, but if the curve $\mathcal{C}$ is $f$-invariant, then $\mathcal{D}^{n+1}$ is a refinement of $\mathcal{D}^n$
for each $n\in \N$, and the pair $(\mathcal{D}^{1}, \mathcal{D}^0)$ gives
a {\em cellular Markov partition} (Definition~\ref{def:cellular}) for $f$ that determines the combinatorics of all cell decompositions $\mathcal{D}^n$.
This cellular Markov partition for $f$ is of a particular type, namely coming from a {\em two-tile subdivision rule} (see Section~\ref{sec:subdivisions}).
The main consequence of Theorem~\ref{thm:main} is that we get such a
two-tile subdivision rule for some iterate $F=f^n$
of every expanding Thurston map. This essentially allows one to describe the map $F$ in terms of finite combinatorial data.
\begin{cor}\label{cor:subdivnlarge} Let $f\: S^2 \rightarrow S^2$ be an expanding Thurston map.
Then for each sufficiently large
$n$ there exists a two-tile subdivision rule that is realized by $F=f^n$.
\end{cor}
If we allow more general cellular Markov partitions, it seems very likely that not only an iterate of $f$, but $f$ itself allows a cellular Markov partition.
\begin{conj*}\label {}
Every expanding Thurston map admits a cellular Markov partition.
\end{conj*}
We believe that even though we were not able to settle the conjecture, the methods established in this paper will be useful for answering this question.
We will see that an expanding Thurston map $f\:S^2\rightarrow S^2$ induces a natural class of metrics on $S^2$ that we call {\em visual
metrics} (see Section~\ref{sec:expansion}). Each visual metric $d$ has
an associated {\em expansion factor} $\Lambda>1$ and is characterized
by the geometric property that for cells $\sigma, \tau$ in the cell
decompositions $\mathcal{D}^n=\mathcal{D}^n(f,\mathcal{C})$ we have $\diam_d(\sigma)\asymp
\Lambda^{-n}$ if $\sigma$ has dimension $\ge 1$ and $\operatorname{dist}_d(\sigma,
\tau)\gtrsim \Lambda^{-n}$ if $\sigma\cap \tau=\emptyset$
(Lemma~\ref{lem:expoexp}). Any two visual metrics are {\em snowflake
equivalent}, and the class of visual metrics for $f$ and any iterate
of $f$ are the same.
The properties of visual metrics are essential in proving the following statement (see Section~\ref{sec:symdym} for the relevant definitions).
\begin{theorem} \label{thm:expThfactor}
Let $f\: S^2\rightarrow S^2$ be an expanding Thurston map. Then $f$ is a factor
of the left-shift $\Sigma\: J^\omega\rightarrow J^\omega $ on the space $J^\omega$ of all sequences in a finite set $J$ of cardinality $\#J=\deg(f)$.
\end{theorem}
An immediate consequence of this theorem and its proof is the fact
that the periodic points of an expanding Thurston map $f\: S^2\rightarrow S^2$
are dense in $S^2$ (Corollary~\ref{cor:perdense}).
The proof of Theorem~\ref{thm:expThfactor} is a simple adaption of the proof of a similar statement in \cite[Thm.~3.4]{Ka03a}. The basic idea seems to go back to \cite{Jo} (see also \cite {Prz}).
Our choice of the term ``visual metric'' is motivated by the close relation of this concept to the notion of a visual metric on the boundary of a Gromov
hyperbolic space. For each expanding Thurston map $f\:S^2\rightarrow S^2$ one can construct a Gromov hyperbolic graph $\mathcal{G}$ whose boundary at
infinity $\partial_\infty \mathcal{G}$ can be identified with $S^2$ so that
the class of visual metrics on $\partial_\infty \mathcal{G}$
in the sense of Gromov hyperbolic spaces is identical to the class of visual metrics for $f$ in our sense (see Remark~\ref{rem:gromovgraph} for more explanation). In this paper we will not pursue this point of view further though (see \cite{HP} for an exposition of similar ideas).
It is possible to describe the range of possible expansion factors
of visual metrics for an expanding Thurston map $f$. If $d$ a visual metric with expansion factor $\Lambda>1$, then a ``snow-flaking'' $d^\alpha$
with $\alpha\in (0,1)$ results in a visual metric with the smaller expansion factor $\Lambda^\alpha$. So the relevant problem is
to find the supremum of all such expansion factors. This supremum is given by a {\em combinatorial expansion factor} $\Lambda_0(f)$ that can in principle be computed from combinatorial data (see Proposition~\ref{prop:exp} for the definition of $\Lambda_0(f)$).
\begin{theorem}\label{thm:visexpfactors} Let $f\:S^2\rightarrow S^2$ be an expanding Thurston map with combinatorial expansion factor $\Lambda_0(f)$.
\begin{itemize}
\smallskip
\item[(i)] If $\Lambda$ is the expansion factor of a visual metric
for $f$, then $1<\Lambda\le \Lambda_0(f)$.
\smallskip
\item[(ii)] Conversely, if $1<\Lambda<\Lambda_0(f)$, then there exists a visual metric $d$ for $f$ with expansion factor $\Lambda$. Moreover, the visual metric $d$ can be chosen to have the following additional property:
For every $x\in S^2$ there exists a neighborhood $U_x$ of $x$ such that
\begin{equation} \label{simmetric}
d(f(x), f(y))=\Lambda d(x,y)\text{ for all } y\in U_x.
\end{equation} \end{itemize}
\end{theorem}
In particular, if $f$ is an expanding Thurston map, then we can always find a visual metric $d$ so that $f$ scales
the metric $d$ by a constant factor at each point.
The example
of the Latt\`es map $g\:\CDach\rightarrow \CDach$ discussed below illustrates this statement. If we equip $\CDach$ with a suitable visual metric for $g$ (a flat orbifold metric with four conical singularities), then $g$ behaves like a piecewise
similarity map, where distances are scaled by the factor $\Lambda=2$.
The combinatorial expansion factor $\Lambda_0(f)$ is invariant under topological conjugacy (Proposition~\ref{prop:expfacinv}) and well-behaved under iteration (see \eqref{eq:wellbeh}).
The invariant curve
$\mathcal{C}$ in Theorem~\ref{thm:main} equipped with (the restriction of) a visual metric is a
quasicircle. This follows from the following general fact applied to the map $f^n$.
\begin{theorem}
\label{thm:Cquasicircle} Let $f\: S^2\rightarrow S^2$ be an expanding Thurston map, and $\mathcal{C}\subset S^2$ be a Jordan curve with $\operatorname{post}(f)\subset \mathcal{C}$. If $\mathcal{C}$ is $f$-invariant, then $\mathcal{C}$ equipped with (the restriction of) a visual metric for $f$ is a quasicircle.
\end{theorem}
Properties of $f$ are encoded in the geometry of $(S^2,d)$, where $d$ is a visual metric. For example, $(S^2,d)$ is a {\em doubling metric space} if and only if $f$ has no periodic critical points (see Section~\ref{sec:periodic}). One can also recognize when $f$ is topologically conjugate to a rational map.
\begin{theorem}\label{thm:qsrational}
Let $f\:S^2\rightarrow S^2$ be an expanding Thurston map, and $d$ a visual metric for $f$.
Then $(S^2,d)$ is quasisymmetrically equivalent to $\CDach$
if and only if $f$ is topologically conjugate to a rational map.
\end{theorem}
Here $\CDach$ is equipped with the chordal metric $\sigma$. For the
definition of quasisymmetric maps see Section~\ref{sec:cc-quasicircle}.
If $f\: \CDach\rightarrow \CDach$ is an expanding Thurston map and rational, then the last theorem implies that the chordal metric $\sigma$ on $\CDach$ is quasisymmetrically equivalent to a visual metric; in general, $\sigma$ is not a visual metric itself.
Thurston studied the question when a given Thurston map is represented by a conformal dynamical system
from a slightly different viewpoint. He asked when
a Thurston map $f\:S^2\rightarrow S^2$ is in a suitable sense {\em equivalent} (see Definition~\ref{def:Thequiv}) to a rational map and obtained a necessary and sufficient condition \cite{DH}. For expanding Thurston maps his notion of equivalence actually means the same as topological conjugacy of the maps (Theorem~\ref{thm:exppromequiv}).
The proof of Theorem \ref{thm:qsrational} does not use Thurston's theorem
mentioned above. Indeed, none of our results relies on this fact. Thus, our methods
possibly provide a different approach for its proof.
It is not clear how useful Theorem~\ref{thm:qsrational}
is for deciding whether an explicitly given expanding Thurston map is
topologically conjugate to a rational map.
It likely that our techniques can be used to formulate a more efficient criterion, but
we will not pursue this further here. We content ourselves with a
simple statement that easily follows from our results.
\begin{theorem}\label{thm:3postrat}
Let $f\:S^2\rightarrow S^2$ be a Thurston map with $\#\operatorname{post}(f)=3$. Then $f$ is Thurston equivalent to a rational Thurston map. If the map $f$ is expanding, then $f$ is topologically conjugate to a rational Thurston map if and only if $f$ has no periodic critical points.
\end{theorem}
Theorem~\ref{thm:main} can be used to study the topological and measure theoretic dynamics of an expanding Thurston map $f\:S^2\rightarrow S^2$ under iteration. For example, we have $h_{top}(f)=\log(\deg(f))$, where $h_{top}(f)$ is the topological entropy of $f$ (Corollary~\ref{cor:topent}).
The following statement gives information of the statistical behavior of $f$ under iteration.
\begin{theorem} \label{thm:maxentr0}
Let $f\: S^2 \rightarrow S^2$ be an expanding Thurston map. Then there exists a unique measure $\mu$
of maximal entropy for $f$. The map $f$ is mixing for $\mu$.
\end{theorem}
This theorem follows from results due to Ha\"\i ssinsky-Pilgrim \cite[Thm.~3.4.1]{HP}. We will present a different proof
and give an explicit description of $\mu$ in terms of the
cell decompositions $\mathcal{D}^n(F,\mathcal{C})$, where $\mathcal{C}$ is an invariant curve as in Theorem~\ref{thm:main} and $F=f^n$
(see Proposition~\ref{prop:exmeasure} and Theorem~\ref{thm:nuF}).
If the map $f$ has no periodic critical points, then the measure $\mu$ is {\em Ahlfors regular}. More precisely, if $d$ is a visual metric
for $f$ with expansion factor $\Lambda>1$, then for all balls
with small radius $r$ we have
$$\mu(B_d(x,r))\asymp r^Q,$$
where $Q=\log(\deg(f))/\log(\Lambda)$ (Proposition~\ref{prop:Ahlforsreg}). In particular, this number $Q$ is the Hausdorff dimension of $(S^2, d)$.
\subsection{Latt\`es maps} \label{sec:Lattes}\index{Latt\`{e}s map}
The simplest expanding Thurston maps are \defn{Latt\`{e}s
maps}.
They were the first examples of rational maps whose Julia set is the
whole sphere.
We will remind the reader of the construction by discussing a particular Latt\`es map in detail (see \cite{La} and
\cite{Mi06}).
The unit square $[0,1]^2\subset \R^2\cong \C$ can be conformally mapped to the upper half-plane in $\CDach$ such that
the vertices $0,1, 1 + {\mathbf{\imath}}, {\mathbf{\imath}}$ of the square
correspond to the points $0,1,\infty,-1$, respectively. By Schwarz reflection we can extend this to a map
$\wp\colon
\C\to \CDach$. Up to post-composition by a M\"{o}bius transformation this map is the classical
\defn{Weierstra\ss\ $\wp$-function}; it is doubly periodic with respect to
the lattice $L :=2\Z^2$ and gives a double branched covering
map of the torus $\T^2:=\C/L$ to
the sphere $\CDach$.
Consider the map
\begin{equation*}
\psi\colon \C\to \C,
\quad u \mapsto \psi(u):= 2 u.
\end{equation*}
One can check that there is a well-defined and unique map $g\colon \CDach\to
\CDach$ such that the diagram
\begin{equation}
\label{eq:Lattes}
\xymatrix{
\C \ar[r]^\psi \ar[d]_{\wp} &
\C \ar[d]^{\wp}
\\
\CDach \ar[r]^g & \CDach
}
\end{equation}
commutes. The map $g$ is rational, in fact
$$g(z)=4\frac{z(1-z^2)}{(1+z^2)^2} \quad \text{ for } z\in \CDach. $$
The Julia set of $g$ is the whole
sphere.
\ifthenelse{\boolean{nofigures}}{}{
\begin{figure}
\centering
\includegraphics[scale=0.5]{mapg.eps}
\begin{picture}(10,10)
\put(-122,19){$\scriptstyle 0$}
\put(0,-5){$\scriptstyle 1$}
\put(-132,140){$\scriptstyle -1$}
\put(-5,119){$\scriptstyle \infty$}
\put(-172,-3){$\scriptstyle 1\mapsto 0$}
\put(-240,5){$\scriptstyle \mapsto 1$}
\put(-300,18){$\scriptstyle 0\mapsto 0$}
\put(-313,80){$\scriptstyle \mapsto -1$}
\put(-300,148){$\scriptstyle -1\mapsto 0$}
\put(-233,135){$\scriptstyle \mapsto 1$}
\put(-177,120){$\scriptstyle \infty \mapsto 0$}
\put(-170,55){$\scriptstyle \mapsto -1$}
\put(-148,80){$\scriptstyle g$}
\put(-230, 65){$\scriptstyle \mapsto \infty$}
\end{picture}
\caption{The Latt\`{e}s map $g$.}
\label{fig:mapg}
\end{figure}
}
One can describe $g$ geometrically as follows. There is an
essentially unique orbifold metric on $\CDach$ (see, for example,
\cite[App.~E]{Mi}, and \cite[App.~A] {McM}
for the terminology) with four conical singularities whose pull-back by $\wp$ is the Euclidean metric on $\C$.
Geometrically, the sphere equipped with this metric looks like a pillow.
In general, a \emph{pillow} is
a metric space obtained from glueing two identical Euclidean
polygons together along their boundary, equipped with the induced path-metric.
In our case,
the upper and lower half-planes in $\CDach$ equipped with the orbifold metric are
isometric to copies of the square $[0,1]^2$.
If we glue two copies of this square along their boundaries, then we obtain
the pillow.
We color one of these squares, say the upper half-plane,
white, and the other square, the lower half-plane, black.
We divide each of these two squares in
$4$ smaller squares of half the side length, and color the $8$ small squares in a checkerboard
fashion
black and white. If we map one such small white square to the large white square
by a Euclidean similarity, then this map
extends by reflection to the whole pillow. There are obviously many different ways to color and map the small
squares.
If we do this in an appropriate way as indicated in Figure \ref{fig:mapg}, then we obtain the map $g$.
The vertices where four small squares intersect are the critical points of
$g$. They are mapped by $g$ to the set $\{1,\infty,-1\}$, which in turn is mapped to
$\{0\}$. The point $0$ is a fixed point of $g$. Hence $g$ is a postcritically-finite
map with $\operatorname{post}(g)=\{0,1,\infty,-1\}$. So the postcritical points of $g$ are the vertices of the pillow, which are, in more technical terms, the conical singularities of our orbifold metric. For the map $g$ we can take $n=1$ in
Theorem~\ref{thm:main}, and as the quasicircle $\mathcal{C}$
the extended real line $\widehat{\R}=\R\cup\{\infty\}$.
Note that $\widehat{\R}$ contains the set $\{0,1,\infty,-1\}$ of postcritical points of $g$.
The set $g^{-1}(\widehat{\R})$ is the set of all edges of the small squares on the left hand
side of Figure \ref{fig:mapg}, and so the tiling in the picture is determined by $g^{-1}(\widehat{\R})$.
Theorem \ref{thm:main} enables us to give a combinatorial description
as in this example
for some iterate of {\em every} expanding Thurston map
$f\:S^2\rightarrow S^2$.
Indeed, suppose $F=f^n$ is an iterate of $f$ for which there exists an invariant Jordan curve $\mathcal{C}\subset S^2$ with
$\operatorname{post}(f)=\operatorname{post}(F)\subset \mathcal{C}$. Then we get a natural cell decomposition $\mathcal{D}^0=\mathcal{D}^0(F,\mathcal{C})$ of $S^2$, consisting of the points in
$\operatorname{post}(F)$ as vertices, the closures of the components of $\mathcal{C}\setminus \operatorname{post}(F)$ as edges, and the closures of the two components of $S^2\setminus \mathcal{C}$ as $2$-dimensional cells or tiles. The cell-decomposition $\mathcal{D}^0$ pulls back under $F$ (see Lemma~\ref{lem:pullback}) to a cell decomposition
$\mathcal{D}^1=\mathcal{D}^1(F,\mathcal{C})$ of $S^2$ such that $F$ is cellular for $(\mathcal{D}^1, \mathcal{D}^0)$, i.e., for each cell $c$ in $\mathcal{D}^1$ the map $F|c$ is a homeomorphism of $c$ onto a cell in $\mathcal{D}^0$. Moreover, since $\mathcal{C}$ is $F$-invariant, $\mathcal{D}^1$ is a refinement of $\mathcal{D}^0$.
The pair $(\mathcal{D}^1, \mathcal{D}^0)$ together with a normalization
essentially determines $F$ uniquely (see Proposition~\ref{prop:rulemapex} and Lemma~\ref{lem:labeluniq}). In this sense, $F$ is described by finite combinatorial data.
\subsection{Related work}
Part of this paper has overlap with work by other researchers, notably Ha\"\i ssinsky-Pilgrim \cite{HP}, and Cannon-Floyd-Parry \cite{CFP07}.
Theorem~\ref{thm:main0} was announced by the first author during an Invited Address at the AMS Meeting at Athens, Ohio, in March 2004, where he gave a short outline of the proof. After the talk he was informed by W.~Floyd that related results had independently obtained by Cannon-Floyd-Parry (which later appeared as \cite{CFP07}).
Theorem~\ref{thm:qsrational} has already been published by Ha\"\i ssinsky-Pilgrim
as part of a more general statement \cite[Thm.~4.2.11]{HP}. Special cases go back to work by the second author \cite{Me02} and unpublished joint work by B.~Kleiner and the first author. The current more general version seems to have emerged after a
visit of the first author at the University of Indiana at Bloomington in
February 2003.
During this visit the first author explained concepts of quasiconformal geometry to K.~Pilgrim and his joint work with B.~Kleiner on Cannon's conjecture in geometric group theory. K.~Pilgrim in turn pointed out Theorem~\ref{thm:exppromequiv} and the ideas for its proof to the first author. After this visit versions of Theorem~\ref{thm:qsrational}
with an outline for the proof were found independently by K.~Pilgrim and the first author.
A proof of Theorem~\ref{thm:qsrational}
was discovered soon afterwards by the authors using ideas from \cite{Me02}
(see \cite{Me08} for an argument along similar lines) in combination with Theorem~\ref{thm:main}.
\subsection{Outline of the paper and main ideas}
The paper is organized as follows.
After fixing some notation in Section~\ref{sec:not}, we review Thurston maps and some basic related concepts in Section~\ref{sec:thmaps}. We also give a precise definition of an {\em expanding Thurston map} (Definition~\ref{def:exp}).
We then collect general facts about cell decompositions in Section~\ref{s:celldecomp}. In particular, we introduce the concept of a {\em cellular Markov partition} (Definition~\ref{def:cellular}). We will later show in Section~\ref{sec:symdym} that under some additional assumptions a continuous map on a compact metric space is a factor of a subshift of finite type if it has a cellular Markov partition (Proposition~\ref{prop:subshift}).
In Section~\ref{sec:2spherecd} we specialize
to cell decompositions on $2$-spheres. Lem\-ma~\ref{lem:constrmaps} shows how to construct branched covering maps and Thurston maps from cell decompositions.
One can pull-back a cell decomposition $\mathcal{D}$ of a $2$-sphere by a branched covering map $f$ if the vertex set ${\bf V}$ of $\mathcal{D}$ contains the critical values of $f$ (Lemma~\ref{lem:pullback}). Since the set $\operatorname{post}(f)$ contains the critical values of all
iterates of a Thurston map $f$, this applies to all iterates $f^n$ if $\operatorname{post}(f)\subset {\bf V}$.
We use this in Section~\ref{sec:tiles} to show that if $f\: S^2\rightarrow S^2$ is a Thurston map and $\mathcal{C}\subset S^2$ a Jordan curve with
$\operatorname{post}(f)\subset \mathcal{C}$, then we obtain a natural sequence $\mathcal{D}^n=\mathcal{D}^n(f,\mathcal{C})$ of cell decompositions of $S^2$. These cell decompositions are our most important technical tool for studying Thurston maps. Their properties are summarized in Proposition~\ref{prop:celldecomp}.
Simple applications are Proposition~\ref{prop:post2}, which gives a classification of all Thurston maps $f$ with $\#\operatorname{post}(f)=2$ up to Thurston equivalence, and Corollary~\ref{cor:no<3} showing that for an expanding Thurston map $f$ we have $\#\operatorname{post}(f)\ge 3$.
In general, the cell decompositions $\mathcal{D}^n=\mathcal{D}^n(f,\mathcal{C})$ are not compatible for different levels $n$ unless the Jordan curve $\mathcal{C}$ is $f$-invariant. To overcome the ensuing problems, we introduce the concept of an {\em $n$-flower} $W^n(p)$ of a vertex $p$ in the cell decomposition $\mathcal{D}^n$ (Section~\ref{sec:flowers}).
The set $W^n(p)$ is formed by the interiors of all cells in $\mathcal{D}^n$ that meet $p$ (see Definition~\ref{def:flower} and
Lemma~\ref{lem:flowerprop}). An important fact is that while in general
a component of the preimage $f^{-n}(K)$ of a small connected
set $K$ will not be contained in an $n$-tile (i.e., a $2$-dimensional cell in $\mathcal{D}^n$), it is always contained in an $n$-flower
(Lemma~\ref{lem:preimsmall}).
In Section~\ref{sec:flowers} we also define a quantity $D_n=D_n(f,\mathcal{C})$ that measures the combinatorial expansion rate of a Thurston map. It is given by the minimal number of $n$-tiles needed to form a connected set joining ``opposite sides" of $\mathcal{C}$ (see \eqref{def:dk} and Definition~\ref{def:connectop}).
Visual metrics for expanding Thurston maps are introduced in Section~\ref{sec:expansion}. Their most important properties are stated in Proposition~\ref{prop:visualsummary} and
Lemma~\ref{lem:expoexp}. In particular, if $f\:S^2\rightarrow S^2$ is an expanding Thurston map, and $d$ a visual metric, then the $d$-diameters of cells in $\mathcal{D}^n$ will approach $0$ at an exponential rate as $n\to \infty$. This implies that lifts of paths under $f^n$ shrink to $0$ exponentially fast if $n\to \infty$ (Lemma~\ref{lem:liftpathshrinks}). This fact is of crucial importance. It implies that every expanding Thurston map is a factor
of a shift operator (see Section~\ref{sec:symdym} where
Theorem~\ref{thm:expThfactor} is proved).
The exponential shrinking of lifts will also be used in Section~\ref{sec:iso} to show that if two expanding Thurston maps are Thurston equivalent, then they are topologically conjugate
(Theorem~\ref{thm:exppromequiv}).
The idea for the proof is to lift an initial isotopy repeatedly to obtain a sequence of isotopies that form a ``tower". If the Thurston maps are expanding, then the diameters of the tracks of these isotopies shrink fast enough so that
the isotopies converge to a time independent homeomorphism that provides the desired conjugacy.
In this section we also prove some results on isotopies of Jordan curves.
In the next Section~\ref{sec:Thurtoncurves} we study the cell decomposition $\mathcal{D}^n=\mathcal{D}^n(f,\mathcal{C})$ under the additional assumption that $\mathcal{C}$ is $f$-invariant. In this case $\mathcal{D}^{n+k}$ is a refinement of $\mathcal{D}^n$ for all $n,k\in \N_0$; on a more intuitive level, each cell in any
of the cell decompositions $\mathcal{D}^n$ is ``subdivided" by cells on a higher level. Moreover, the pair
$(\mathcal{D}^{n+k}, \mathcal{D}^n)$ is a cellular Markov partition for $f$
(Proposition~\ref{prop:invmarkov}). If
a Thurston map has an invariant Jordan curve $\mathcal{C}$ with $\operatorname{post}(f)\subset \mathcal{C}$, then it can
be described by a {\em two-tile subdivision rule} (Definition~\ref{def:subdivcomb}) as discussed in Section~\ref{sec:subdivisions}. The main result here is Proposition~\ref{prop:rulemapex} that gives a general method for constructing Thurston maps with invariant curves from a two-tile subdivision rule (see Definition~\ref{def:labeldecomp} and the following discussion).
We call $f$ {\em combinatorially expanding} for an invariant curve $\mathcal{C}$ if there exists a number
$n_0\in \N$ such that $D_{n_0}(f,\mathcal{C})\ge 2$ (Definition~\ref{def:combexp}). This means that there exists $n_0\in \N$ so that no $n_0$-tile joins opposite sides of $\mathcal{C}$. In this case the numbers $D_n=D_n(f,\mathcal{C})$ grow at an exponential rate as $n\to \infty$ (Lemma~\ref{lem:submult}).
It is easy to see that if $f$ is an expanding Thurston map, then
$D_n(f,\mathcal{C})\to \infty$ as $n\to \infty$. In particular, if $\mathcal{C}$ is $f$-invariant, then $f$ is combinatorially expanding for $\mathcal{C}$. The converse is not true in general,
but in Section~\ref{sec:combexp} we show that every combinatorially expanding Thurston map is (Thurston) equivalent to an expanding Thurs\-ton map (Proposition~\ref{prop:combexp} and Corollary~\ref{cor:combexp1}).
The intuitive reason for his is that if the diameters of the tiles
in $\mathcal{D}^n(f,\mathcal{C})$ fail to shrink to zero as $n\to \infty$, then one can ``correct" the map $f$ so that this becomes true without affecting the combinatorics of the cell decompositions $\mathcal{D}^n(f,\mathcal{C})$.
It is somewhat cumbersome to implement this idea. We do this by introducing an equivalence relation that forces descending sequences of $n$-tiles to shrink to points as $n\to \infty$. We then invoke Moore's theorem (Theorem~\ref{thm:moore}) to show that the quotient space of the original $2$-sphere $S^2$ by this relation is also a $2$-sphere $\widetilde S^2$. The original Thurston map $f\: S^2\rightarrow S^2$ descends to a map
$\widetilde f\: \widetilde S^2\rightarrow \widetilde S^2$ and one can show that
$\widetilde f$ is an expanding Thurston map that is equivalent to the original map $f$.
Section~\ref{sec:graphs} contains some auxiliary statements on graphs. The main result is Lemma~\ref{prop:isotopicpath} that gives a sufficient criterion when a Jordan curve can be isotoped into the $1$-skeleton of a cell decomposition of a $2$-sphere.
Existence and uniqueness results for invariant Jordan curves
are proved in Section~\ref{sec:constructc}. This section constitutes the center of the present work. Here we establish Theorems~\ref{thm:main},
\ref{thm:exinvcurvef}, and \ref{thm:uniqc}, and Corollary~\ref{cor:subdivnlarge}. Among these statements
Theorem~\ref{thm:main} is the most difficult to prove. It is based on Lemma~\ref{prop:isotopicpath} and Theorem~\ref{thm:exinvcurvef}.
Roughly speaking,
the idea for the proof of Theorem~\ref{thm:main} can be summarized as follows. Let $f\:S^2\rightarrow S^2$ be an expanding Thurston map, and $\mathcal{C}\subset S^2$ be a Jordan curve with $\operatorname{post}(f)\subset \mathcal{C}$. Consider the cell decompositions $\mathcal{D}^n=\mathcal{D}^n(f,\mathcal{C})$ of $S^2$ induced by $f$ and $\mathcal{C}$. If $n$ is large enough, one can find a homeomorphism $\psi$ of $S^2$ that maps $\mathcal{C}$ into the
$1$-skeleton $f^{-n}(\mathcal{C})$ of $\mathcal{D}^n$ and is isotopic to the identity on $S^2$ by an isotopy that fixes the points in $\operatorname{post}(f)$. Essentially, this follows from the fact that $f$ is expanding
and so the $1$-skeleton $f^{-n}(\mathcal{C})$ of $\mathcal{D}^n$ forms a fine ``grid'' in $S^2$ for $n$ large. This grid contains $\operatorname{post}(f)$ and allows us to trace the curve $\mathcal{C}$ closely.
The map $\widehat F=\psi\circ f^n$ will be a Thurston map
with an invariant curve
$\widehat \mathcal{C}=\psi(\mathcal{C})\supseteq \operatorname{post}(\widehat F)=\operatorname{post}(f)$. Moreover, if $n$ is large enough, then $\widehat F$ is combinatorially expanding. Actually, we can even assume that
$\widehat F$ is expanding, because if necessary, the map can be ``corrected" to have this property (Corollary~\ref{cor:combexp1}).
The expanding Thurston maps $F=f^n$ and $\widehat F$ are Thurston equivalent and hence topologically conjugate by Theorem~\ref{thm:exppromequiv}. Since the map $\widehat
F$ has the invariant Jordan curve $\widehat \mathcal{C}$, one obtains an $F$-invariant curve with the desired properties as an image of $\widehat \mathcal{C}$ under a homeomorphism that conjugates $\widehat F$ and $F$.
In Section~\ref{sec:cc-quasicircle} we review the notion of a
quasicircle and prove Theorem~\ref{thm:Cquasicircle}. We also show
that if $f\:S^2\rightarrow S^2$ is an expanding Thurston map, $S^2$ is
equipped with a visual metric $d$ for $S^2$, and the cell
decompositions $\mathcal{D}^n(f,\mathcal{C})$, $n\in \N_0$, are obtained from an
$f$-invariant Jordan curve $\mathcal{C}$, then the edges in these cell
decompositions are
uniform quasiarcs and the boundaries of tiles are uniform quasicircles
(Proposition~\ref{prop:arc}).
If $f\colon S^2\rightarrow S^2$ is an expanding Thurston map and $S^2$ is
equipped with a visual metric $d$ for $f$, then $(S^2, d)$ has some
properties that are important in the analysis of metric spaces. For
example, $(S^2, d)$ is linearly locally connected
(Proposition~\ref{prop:annLLC}). Moreover,
$(S^2, d)$ is a doubling metric space if and only if $f\: S^2\rightarrow S^2$ has no periodic critical points (Theorem~\ref{thm:perdoub}). Actually, the absence of periodic critical points even implies that $(S^2, d)$ is Ahlfors regular
(Proposition~\ref{prop:Ahlforsreg}).
In Section~\ref{sec:combexpfac} we revisit visual metrics. We introduce the combinatorial expansion factor $\Lambda_0(f)$ associated with an expanding Thurston map $f$ and prove Theorem~\ref{thm:visexpfactors}.
The topic of Section~\ref{sec:rational-maps} are rational Thurston maps on the Riemann sphere $\CDach$. For such maps the notion of expansion can be characterized in more familiar terms (Proposition~\ref{prop:rationalexpch}). Here we prove
Theorems~\ref{thm:main0}, \ref{thm:qsrational}, and \ref{thm:3postrat}.
In
Section~\ref{sec:entropy} we study the dynamics of an expanding Thurston map from a measure theoretic point of view. The main result is Theorem~\ref{thm:nuF},
which gives an explicit description of the unique measure of maximal entropy.
We list some open problems in the final Section \ref{sec:probrem}.
\medskip \noindent
{\bf Acknowledgements.} The authors would like to thank J.~Cannon, W.~ Floyd, P.~Ha\"{\i}ssinsky, B.~Kleiner, W.~Parry, K.~Pilgrim, J.~Souto, D.~Sullivan, Q.~Yin, and M.~Zieve for interesting discussions and useful comments.
\subsection{Examples of Thurston maps}
\label{sec:examples}
Throughout the paper we consider several examples of Thurston maps to
illustrate various phenomena. We list them here with a short description
for easy reference. The relevant terms used in these descriptions are later defined in the body of the paper.
\smallskip
A Latt\`{e}s map was considered in Section~\ref{sec:Lattes}.
The map
in Example~\ref{ex:z2-1} is $f_1(z)=z^2-1$; it realizes a two-tile
subdivision rule that is not combinatorially expanding.
In Example~\ref{ex:barycentric} there are two maps $f_2$ and $\widetilde{f}_2$ that both
realize the barycentric subdivision rule. The map $f_2$ is a
rational map, but it is not expanding (i.e., its Julia set is not the whole
Riemann sphere $\CDach$). The map $\widetilde{f}_2$ however is
expanding. It is an example of an expanding Thurston map with
periodic critical points.
The map $f_3$ in Example~\ref{ex:obstructed_map} (realizing a certain
two-tile subdivision rule) is an obstructed map. This means
$f_3$ is not Thurston equivalent to a rational map.
The map $f_4$ in Example~\ref{ex:2x3} is again not Thurston
equivalent to a rational map. While somewhat easier than the map $f_3$ in Example
\ref{ex:obstructed_map}, it is less generic, since $f_4$ has
a parabolic orbifold, whereas $f_3$ has a hyperbolic
orbifold. The map $f_4$ realizes the $2$-by-$3$
subdivision rule. With respect to a suitable visual metric for
$f_4$, the sphere $S^2$ consists of two copies of a Rickman's
rug.
In Example~\ref{ex:R_mario3} a whole class of maps is
considered. The first one is the map $f_5(z)=1-2/z^2$ which realizes a simple
two-tile subdivision rule. By ``adding flaps'' we obtain the other
maps. All these maps are rational; in fact they are given by an
explicit formula, which makes them easy to understand and visualize.
In Example~\ref{ex:exp_notcexp} we consider a Thurston map $f$ that is
not combinatorially expanding, yet Thurston equivalent
to an expanding Thurston map $g$. This shows that the
sufficient condition in Proposition~\ref{prop:combexp} is not
necessary.
In Example~\ref{ex:invC} we illustrate the main ideas of
Section~\ref{sec:constructc}. In particular, we show how for a specific
map $f$
an $f$-invariant curve $\widetilde{\mathcal{C}}$ with $\operatorname{post}(f)\subset
\widetilde{\mathcal{C}}$ is constructed; see Figure~\ref{fig:invC_constr}.
In Example~\ref{ex:infty_C} we show that the Latt\`{e}s map $g$ from
Section~\ref{sec:Lattes} has
infinitely many distinct $g$-invariant curves $\mathcal{C}$ with
$\operatorname{post}(g)\subset \mathcal{C}$.
In Example~\ref{ex:noinvCC} we consider an expanding Thurston map $f$
for which no $f$-invariant Jordan curve $\mathcal{C}$
with $\operatorname{post}(f)\subset \mathcal{C}$ exists.
In Remark~\ref{rem:ndep} we show that the $f^n$-invariant curve
$\widetilde{\mathcal{C}}$ given by Theorem~\ref{thm:main} will in general
depend on $n$.
In Example~\ref{ex:Cit} we use another Latt\`{e}s map to illustrate
an iterative construction of invariant curves; see
Figure~\ref{fig:Cit}.
Example \ref{ex:Cinv_notcexp} shows what can happen if one of the
necessary conditions in the iterative procedure for producing invariant curve
is violated. Namely, the ``limiting object'' $\widetilde{\mathcal{C}}$ is not
a Jordan curve anymore. The map used to illustrate this phenomenon is
again a Latt\`{e}s map.
In Example~\ref{ex:rect} (again a Latt\`{e}s map) we obtain a
non-trivial (in particular non-smooth) invariant
curve that is rectifiable.
In Example~\ref{ex:notattained} we exhibit an expanding Thurston map
$f$
for which no visual metric with expansion factor $\Lambda$ equal to the
combinatorial expansion factor $\Lambda_0(f) $ exists. This shows that
part (ii) in Theorem~\ref{thm:visexpfactors} cannot be improved.
\section{Notation}\label{sec:not}
\noindent
We denote by $\N=\{1,2,\dots\}$ the set of natural numbers, and by $\N_0=\{0,1, 2. \dots \}$ the set of natural numbers including $0$. The symbol
${\mathbf{\imath}}$ stands for the imaginary unit in the complex plane $\C$. We define
$\D:=\{z\in \C:|z|<1\}$ as the open unit disk in $\C$.
Let $(X,d)$ be a metric space, $a\in X$ and $r>0$. We denote by
$B_d(a,r)=\{x\in X:d(a,x)< r\}$ and by
$\overline B_d(a,r)=\{x\in X: d(a,x)\le r\}$ the open and the
closed ball of radius $r$ centered at $a$, respectively. If $A,B\subset X$, we
let $\diam_d(A)$ be the diameter, $\overline A$ be the closure of $A$
in $X$, and
$$ \operatorname{dist}_d(A,B)=\inf\{d(x,y): a\in A, y\in B\}$$
be the distance of $A$ and $B$.
If $p\in X$, we let $\operatorname{dist}_d(p, A)=\operatorname{dist}_d(\{p\}, A)$.
If $\epsilon>0$ then
$$\mathcal{N}^\epsilon_d(A):=\{ x\in X: \operatorname{dist}_d(x,A)<\epsilon\}$$
is the open $\epsilon$-neighborhood of $A$.
We drop the subscript $d$ in $B_d(a,r)$, etc., if the metric
$d$ is is clear from the context.
The cardinality of a set $M$ is denoted by $\#M$ and the identity map on $M$ by $\id_M$. If $f\: M \rightarrow M$ is a map, then
$f^n$ for $n\in \N$ is the $n$-th iterate of $f$. We set $f^0:=\id_M$.
Two non-negative quantities $a$ and $b$ are said to be \defn{comparable}
if there is a constant $C\ge 1$ depending on some obvious ambient parameters such that
\begin{equation*}
\frac{1}{C}a\leq b\leq C a.
\end{equation*}
We then write $a\asymp b$.
The constant $C$ is refered to by $C(\asymp)$. We write
$a\lesssim b$ or $b\gtrsim a$,
if there is a constant $C>0$ such that $a\leq C b$. We refer to the constant as $C(\lesssim)$ or $C(\gtrsim)$.
\section{Thurston maps}\label{sec:thmaps}
\noindent
Let $S^2$ be a topological $2$-sphere with a fixed orientation, and $f\:S^2\rightarrow S^2$ be a continuous map. Then $f$ is called a \defn{branched covering map}\index{branched covering map} of
$S^2$ if we can write it locally as the map $z\mapsto z^d$ for some $d\in \N$ after
orientation-preserving homeomorphic changes of
coordinates in domain and range. More precisely, we require that for each point $p\in S^2$
there exists $d\in \N$, open neighborhoods $U$ of $p$ and $V$ of $q=f(p)$, open neighborhoods $U'$ and $V'$ of $0\in \CDach$ and orientation-preserving homeomorphisms $\varphi\:U\rightarrow U'$ and $\psi\:V\rightarrow V'$ with $\varphi(p)=0$ and $\psi(q)=0$ such that
$$(\psi \circ f\circ \varphi^{-1})(z)=z^d$$
for all $z\in U'$.
The integer $d=\colon\!\!\!\deg_f(p)\geq 1$ is uniquely determined by $f$ and $p$, and
called the \defn{local degree}\index{local degree} of the map $f$ at
$p$. A point $c\in S^2$ with
$\deg_f(c)\geq 2$ is called a
\defn{critical point}\index{critical point} of $f$, and a point that has a critical
point as a preimage a {\em critical value}.\index{critical value} The set of all critical points is denoted
by $\operatorname{crit}(f)$.\index{crit@$\operatorname{crit}(f)$} Obviously, if $f$ is a branched covering map on $S^2$, then $\operatorname{crit}(f)$ only consists of isolated points and is hence a finite subset of $S^2$. Moreover, $f$ is an open and surjective mapping, and {\em finite-to-one}, i.e., every point has finitely many preimages under $f$. More precisely, if $\deg(f)$ is the topological degree of $f$, then
$$ \sum_{p\in f^{-1}(q)}\deg_f(p)=\deg(f)$$
for every $q\in S^2$ (see \cite[Sect.~2.2]{Ha}). In particular, if $q$ is not a critical value of $f$, then $q$ has precisely $\deg(f)$ preimages.
For $n\in \N_0$ we denote by $f^n$ the $n$-th iterate of $f$ (where $f^0:=\id_{S^2}$). If $f$ is a branched covering map on $S^2$, then the same is true for $f^n$ and we have $\deg(f^n)=\deg(f)^n$ and
\begin{equation}\label{eq:critpfn}
\operatorname{crit}(f^n)=\operatorname{crit}(f)\cup f^{-1}(\operatorname{crit}(f))\cup \dots \cup f^{-(n-1)}(\operatorname{crit} (f)).
\end{equation}
The {\em set of postcritical points}\index{postcritical point} of $f$ is defined as
\begin{equation*}
\operatorname{post}(f):=\bigcup_{n\in \N} \{f^n(c):c\in \operatorname{crit}(f)\}.
\end{equation*}
\index{post@$\operatorname{post}(f)$}
If the cardinality $\#\operatorname{post}(f)$ is finite, then $f$ is called {\em postcritically-finite}.\index{postcritically-finite}
For $n\in \N$ we have $\operatorname{post} (f^n)=\operatorname{post} (f)$ and
$f^n(\operatorname{crit}(f^n))\subset \operatorname{post} (f)$.
The last inclusion implies that away from $\operatorname{post}(f)$ each iterate $f^n$ is a covering map and all ``branches of the inverse of $f^n$'' are defined; more precisely, if $U\subset S^2\setminus \operatorname{post}(f)$ is a path connected and simply connected set, $q\in U$, and $p\in S^2$ a point with $f^n(p)=q$, then there exists
a unique continuous map $g\: U\rightarrow S^2$ with
$g(q)=p$ and $f^n\circ g=\id_U$. Informally, we refer to such a right inverse of $f^n$ as a ``branch of $f^{-n}$''.
We can now record the definition of the main object of investigation in this paper.
\begin{definition}[Thurston maps]\label{def:f}
A \defn{Thurston map}\index{Thurston map} is a branched covering map $f\: S^2\rightarrow S^2$ of a $2$-sphere $S^2$ with $\deg(f)\ge 2$ and finite set of postcritical points. \end{definition}
There are no Thurston maps with $\#\operatorname{post}(f)\in \{0,1\}$ (see Remark~\ref{rem:no01}), and all Thurston maps
with $\#\operatorname{post}(f)=2$ are Thurston equivalent (see below) to a map $z\mapsto z^k$, $k\in \Z\setminus\{-1,0,1\}$, on the Riemann sphere (see Proposition~\ref{prop:post2}).
Let $f\:S^2\rightarrow S^2$ be a Thurston map and $\mathcal{C}$ be a Jordan curve in $S^2$ with $\operatorname{post}(f)\subset \mathcal{C}$.
We fix a metric $d$ on $S^2$ that induces the standard topology on $S^2$.
For $n \in \N$ we denote by $\operatorname{mesh}(f,n,\mathcal{C})$ the supremum of the diameters of all connected components of the set $f^{-n}(S^2\setminus \mathcal{C})$.
\begin{definition}[Expansion]
\label{def:exp}
A Thurston map $f\: S^2\rightarrow S^2$ is called
\defn{expanding}\index{expanding|textbf}\index{Thurston map!expanding} if there exists a Jordan curve $\mathcal{C}$ in $S^2$ with $\operatorname{post}(f)\subset \mathcal{C}$ and
\begin{equation} \label{def:expanding}
\lim_{n\to \infty} \operatorname{mesh}(f,n,\mathcal{C}) =0.
\end{equation}
\end{definition}
As we will see later, the set $f^{-n}(S^2\setminus \mathcal{C})$ has only finitely many components, so the supremum in the definition of $\operatorname{mesh}(f,n,\mathcal{C})$ is actually a maximum.
We will also prove (see Lemma~\ref{lem:exp_ind_C}) that if the relation \eqref{def:expanding} is satisfied for one Jordan curve $\mathcal{C}\supset \operatorname{post}(f)$, then it actually holds for every such curve.
Note further that this really is a topological property, as it is independent of the choice of the metric on $S^2$ if it induces the given topology on $S^2$.
Our notion of expansion for a Thurston map is equivalent to
a similar concept of expansion introduced by Ha\"\i ssinky-Pilgrim (see
\cite[Sect.~2.2] {HP} and Proposition~\ref{prop:expequivexp}).
If the Thurston map is a rational map $f\: \CDach\rightarrow \CDach$ on the
Riemann sphere $\CDach$, then one can show that $f$ is expanding if
and only if $f$ does not have periodic critical points if and only
if its Julia set is equal to $\CDach$ (see
Section~\ref{sec:rational-maps} for more details).
To define a suitable notion of equivalence for Thurston maps
we recall the definition of an isotopy between spaces.
Let $I=[0,1]$, and $X$ and $Y$ be topological spaces. An {\em isotopy between $X$ and $Y$}\index{isotopy}
is a continuous map $\phi\: X\times I \rightarrow Y$ such that each
map $\phi_t:=\phi(\cdot, t)$, $t\in I$, is a homeomorphism of $X$ onto $Y$.
For a subset $A$ of $X$, we say $\phi$ is an isotopy {\em relative}
to $A$ (abbreviated ``$\phi$ is an isotopy rel.\ $A$'') if
$\phi_t(a)=\phi_0(a)$ for all $a\in A$ and $t\in I$. So this means that the image of each point in $A$ remains fixed during the isotopy. Two homeomorphisms $\varphi, \psi\: X\rightarrow Y$ are called {\em isotopic rel.~$A$} if there exists an isotopy $\phi\: X\times I \rightarrow Y$ rel.~$A$
with $\phi_0=\varphi$ and $\phi_1=\psi$.
\begin{definition}[Thurston equivalence]\label{def:Thequiv}
Two Thurston maps $f\: S^2\rightarrow S^2$ and $g\:\widehat S^2\rightarrow \widehat S^2$ are called \defn{(Thurs\-ton)
equivalent}\index{Thurston equivalent|textbf} if there exist homeomorphisms
$h_0,h_1\:S^2\rightarrow \widehat S^2 $ that are isotopic rel.\ $\operatorname{post}(f)$ and satisfy
$ h_0\circ f = g\circ h_1$.
\end{definition}
Here $\widehat S^2$ is another $2$-sphere. Often $S^2=\widehat S^2$,
but sometimes it is important to distinguish the spheres on which
the Thurston maps are defined.
For equivalent Thurston maps as in Definition~\ref{def:Thequiv} we have the following commutative diagram:
\begin{equation}\label{Thequiv1}
\xymatrix{
S^2 \ar[r]^{h_1} \ar[d]_f & \widehat S^2 \ar[d]^g
\\
S^2 \ar[r]^{h_0} & \widehat S^2.
}
\end{equation}
Note that in this situation
\begin{equation}\label{eq:postfg}
\operatorname{post}(g)=h_0(\operatorname{post}(f))=h_1(\operatorname{post}(f)).
\end{equation}
Indeed, it is clear that
\begin{equation}\label{critptsfg}
\operatorname{crit}(g)=h_1(\operatorname{crit}(f)).\end{equation}
Moreover, since $h_0|\operatorname{post}(f)=h_1|\operatorname{post}(f)$ and $f^n(\operatorname{crit}(f))\subset\operatorname{post} (f)$
for all $n\in \N$, the relation \eqref{critptsfg} inductively implies
$$g^n(\operatorname{crit}(g))=h_0(f^n(\operatorname{crit}(f)))=h_1(f^n(\operatorname{crit}(f)))$$ for all $n\in \N$. Hence
\begin{align*}
\operatorname{post}(g)=\bigcup_{n\in \N}g^n(\operatorname{crit}(g))=\bigcup_{n\in \N}
h_0(f^n(\operatorname{crit}(f)))
\\
=h_0(\operatorname{post}(f))=h_1(\operatorname{post}(f))
\end{align*}
as desired.
We call the maps $f$ and $g$ \defn{topologically
conjugate}\index{topologically conjugate|textbf} if there exists a
homeomorphism $h\: S^2\rightarrow \widehat S^2 $ such that $h\circ f = g\circ
h$. We will see later (see Theorem~\ref{thm:exppromequiv}) that if two
expanding Thurston maps are equivalent, then they are topologically
conjugate.
If $S^2$ and $\widehat S^2$ are $2$-spheres, $f\:S^2\rightarrow S^2$ is an expanding Thurston map and
$g\:\widehat S^2\rightarrow \widehat S^2$ is a topologically conjugate to $f$, then $g$ is also an expanding Thurston map. So the notion of an expanding Thurston map is invariant under topological conjugacy.
\section{Cell decompositions}\label{s:celldecomp}
\noindent
In this section we review some facts about cell decompositions. Most of this material is fairly standard.
For the purpose of the present paper we could have restricted ourselves to cell decompositions of subsets of a $2$-sphere, but it is more transparent to discuss the topic in greater generality. At first reading the reader may want to skim through this section to pick up relevant definitions and statements.
A crucial concept introduced in this section is the notion of a ``cellular Markov partition'' (Definition~\ref{def:cellular}) of a map. One can show that if a map admits a cellular Markov partition, then its dynamics is linked to symbolic dynamics; in particular, it is a factor of a subshift of finite type (see Proposition~\ref{prop:subshift}).
In the following $X$ will always be a locally compact Hausdorff space. A {\em
(closed topological) cell\index{cell} of dimension $n=\dim(c)\in \N$} in $X$ is a set $c\subset X$ that is homeomorphic to the closed unit ball $\overline \B^n$ in $\R^n$. We denote by $\partial c$ the set of
points corresponding to $\partial \overline \B^n$ under such a homeomorphism between $c$ and $\overline \B^n$. This is independent of the homeomorphism chosen, and the set $\partial c$ is well-defined. We call $\partial c$ the {\em boundary} and $\inte(c)=c\setminus \partial c$ the {\em interior} of $c$. Note that boundary and interior
of $c$ in this sense will in general not agree with
the boundary and interior of $c$ regarded as a subset of the topological space $X$.
A {\em cell of dimension $0$} is $X$ is a set $c\subset X$ consisting of a single point. We set $\partial c=\emptyset$ and $\inte(c)=c$ in this case.
\begin{definition}[Cell decompositions]\label{def:celldecomp}
Suppose that $\mathcal{D}$ is a collection of cells in a locally compact Hausdorff space $X$. We say that $\mathcal{D}$ is a
{\em cell decomposition}\index{cell!decomposition} of $X$
provided the following conditions are satisfied:
\begin{itemize}
\smallskip
\item[(i)]
the union of all cells in $\mathcal{D}$ is equal to $X$,
\smallskip
\item[(ii)] we have $\inte(\sigma)\cap \inte(\tau)= \emptyset$, whenever $\sigma,\tau\in \mathcal{D}$, $\sigma\ne \tau$,
\smallskip
\item[(iii)] if $\tau\in \mathcal{D}$, then $\partial \tau$ is a union of cells in $\mathcal{D}$,
\smallskip
\item[(iv)] every point in $X$ has a neighborhood that meets only finitely many cells in $\mathcal{D}$.
\end{itemize}
\end{definition}
If $\mathcal{D}$ is a collection of cells in some ambient space $X$, then we call $\mathcal{D}$ a {\em cell complex}\index{cell complex} if $\mathcal{D}$ is a cell decomposition of the underlying set
$$|\mathcal{D}|=\bigcup\{c:c\in \mathcal{D}\}. $$
Suppose $\mathcal{D}$ is a cell decomposition of $X$.
By (iv), every compact subset of $X$ can only meet finitely
many cells in $\mathcal{D}$. In particular, if $X$ is compact, then $\mathcal{D}$ consists of only finitely many cells. Moreover, for each $\tau \in \mathcal{D}$, the set
$\partial \tau$ is compact and hence equal to a finite union of cells in $\mathcal{D}$.
It follows from basic dimension theory that if
\ $\dim(\tau)=n$,
then $\partial \tau$ is equal to a union of cells in $\mathcal{D}$ that have dimension $n-1$.
The
union $X^n$ of all cells in $\mathcal{D}$ of dimension $\le n$ is called the
$n$-{\em skeleton}\index{n@$n$-!skeleton}\index{skeleton} of the cell decomposition.
It is useful to set $X^{-1}=\emptyset$. It follows from the local
compactness of $X$ and property (iv) of a cell decomposition that
$X^n$ is a closed subset of $X$ for each $n\in \N_0$.
By the last remark in the previous paragraph, we have $\partial \tau\subset X^{n-1}$ for each $\tau \in \mathcal{D}$ with $\dim(\tau)=n$.
\begin{lemma} \label{lem:uniondisjint}
Let $\mathcal{D}$ be a cell decomposition of $X$. Then for each $n\in \N_0$ the $n$-skeleton $X^n$ is equal to the disjoint union of the sets $\inte(c)$, $c\in \mathcal{D}$, $\dim(c)\le n$.
Moreover, $X$ is equal to the disjoint union of the sets $\inte(c)$, $c\in \mathcal{D}$.
\end{lemma}
\begin{proof} We show the first statement by induction on $n\in \N_0$.
Since $\inte(c)=c$ for each cell $c$ in $\mathcal{D}$ of dimension $0$, it is clear that $X^0$ is the disjoint union of the interiors of all cells $c\in \mathcal{D}$ with $\dim(c)=0$.
Suppose that the statement is true for $X^n$, and let $p\in X^{n+1}$ be arbitrary.
If $p\in X^n$, then $p$ is contained in the interior of a cell $c\in \mathcal{D}$ with $\dim(c)\le n$ by induction hypothesis. In the other case, $p\in X^{n+1}\setminus X^{n}$, and so there exists $c\in \mathcal{D}$ with $\dim(c)=n+1$ and $p\in c$. Since $\partial c\subset X^n$, it follows that $p\in c\setminus \partial c=\inte(c)$. So $X^{n+1}$ is the union of the interiors of all cells $c$ in $\mathcal{D}$ with $\dim(c)\le n+1$. This union is disjoint, because distinct cells in a cell decomposition have disjoint interiors.
The second statement follows from the first, and the obvious fact that $X=\bigcup_{n\in \N_0}X^n$.
\end{proof}
The lemma immediately implies that if $\tau\in \mathcal{D}$ and $\dim(\tau)=n$,
then each point $p\in \inte(\tau)$ is an interior point of $\tau$ regarded as a subset of the topological space $X^n$.
\begin{lemma}\label{lem:celldecompint} Let $\mathcal{D}$ be a cell decomposition of $X$.
\begin{itemize}
\smallskip
\item[(i)]
If $\sigma$ and $\tau$ are two distinct cells in $\mathcal{D}$ with $\sigma\cap \tau\ne \emptyset$, then one of the following statements holds: $\sigma\subset \partial \tau$, $\tau \subset \partial \sigma$, or
$\sigma\cap \tau =\partial \sigma \cap \partial\tau$ and this intersection
consists of cells in $\mathcal{D}$ of dimension strictly less than $\min\{\dim(\sigma), \dim(\tau)\}$.
\smallskip
\item[(ii)] If $\sigma,\tau_1, \dots, \tau_n$ are cells in $\mathcal{D}$ and $\inte(\sigma)\cap (\tau_1\cup \dots \cup \tau_n)\ne \emptyset$, then $\sigma\subset \tau_i$ for some $i\in\{1,\dots, n\}$.
\end{itemize}
\end{lemma}
\begin{proof} (i) We may assume that $l=\dim(\sigma)\le
m=\dim (\tau)$, and prove the statement by induction on $m$.
The case $m=0$ is vacuous and hence trivial. Assume that the statement is true whenever both cells have dimension $<m$.
If $l=m$ then by definition of a cell decomposition $\inte(\sigma)$ is disjoint from $\tau \subset \inte(\tau) \cup X^{m-1}$, and similarly
$\inte (\tau)\cap \sigma=\emptyset$. Hence
$\sigma\cap \tau =\partial \tau \cap \partial \sigma$. Moreover, both sets $\partial \sigma$ and $\partial \tau$ consist of finitely many cells in $\mathcal{D}$ of dimension $\le m-1$. Applying the
induction hypothesis to pairs of these cells, we see that
$\partial \tau \cap \partial \sigma$ consists of cells of dimension $<m$ as desired.
If $l<m$, then $\sigma \subset X^{m-1}$ and so $\sigma \cap \inte (\tau)=\emptyset$. This shows that $\sigma\cap \tau=\sigma\cap \partial \tau$. Moreover, we have $\partial \tau=c_1\cup \dots \cup c_s$, where $c_1, \dots , c_s$ are cells of dimension $m-1$.
So we can apply the induction hypothesis to the pairs
$(\sigma, c_i)$. If $\sigma=c_i$ or $\sigma \subset \partial c_i$ for some $i$, then $\sigma \subset \partial \tau$; we cannot have $c_i\subset \partial \sigma$, because $c_i$ has dimension $m-1$, and $\partial \sigma$ is a set of topological dimension $<m-1$. So if none of the first possibilities occurs, then $\sigma \cap c_i=\emptyset $ or $\sigma \cap c_i=\partial \sigma\cap\partial c_i$ and this set consists of
cells of dimension $<l$ (by induction hypothesis) contained in
$\partial c_i\subset c_i\subset \partial \tau$ for all $i$. In this case $\sigma\cap \tau=\partial \sigma \cap \partial \tau$, and this sets consists of cells of dimension $<l$ as desired. The claim follows.
\smallskip
(ii) There exists $i\in \{1,\dots, n\}$ with $\inte(\sigma)\cap \tau_i\ne \emptyset$. By the alternatives in (i) we then must have $\sigma=\tau_i$ or $\sigma\subset \partial\tau_i$. Hence
$\sigma\subset \tau_i$.
\end{proof}
\begin{lemma}\label{lem:conncomp} Let $A\subset X$ be a closed set, and $U\subset X\setminus A$ a nonempty open and connected set. If $\partial U\subset A$, then $U$ is a connected component of $X\setminus A$.
\end{lemma}
\begin{proof} Since $U$ is a nonempty connected set in the complement of $A$, this set is contained in a unique connected component $V$ of $X\setminus A$.
Since $\partial U\subset A\subset X\setminus V$, we have $V \cap \overline U=V\cap U=U$ showing that $U$ is relatively open and closed in $V$.
Since $U\ne \emptyset$ and $V$ is connected, it follows that $U=V$ as desired. \end{proof}
\begin{lemma} \label{lem:opencells}
Let $\mathcal{D}$ be a cell decomposition of $X$ with $n$-skeleton $X^n$, $n\in \{-1\}\cup\N_0$.
Then for each $n\in \N_0$ the nonempty connected components of $X^{n}\setminus X^{n-1}$ are precisely the sets $\inte(\tau)$, $\tau\in \mathcal{D}$, $\dim(\tau)=n$.\end{lemma}
\begin{proof} Let $\tau$ be a cell in $\mathcal{D}$ with $\dim(\tau)=n$.
Then
$\inte(\tau)$ is a connected set contained in $X^n\setminus X^{n-1}$ that is relatively open with respect to $X^n$. Its relative boundary is a subset of $\partial \tau$ and hence contained in the closed set $X^{n-1}$. It follows by Lemma~\ref{lem:conncomp} that $\inte(\tau)$ is equal to a component $V$ of $X^n\setminus X^{n-1}$.
Conversely, suppose that $V$ is a nonempty connected component of
$X^n\setminus X^{n-1}$. Pick a point $p\in V$. Then $p$ lies
in the interior of a unique cell $\tau\in\mathcal{D}$ with $\dim(\tau)=n$. It follows from the first part of the proof that $V=\inte(\tau)$.
\end{proof}
\begin{definition}[Refinements]\label{def:ref}
Let $\mathcal{D}'$ and $\mathcal{D}$ be two cell decomposition of the space $X$. We
say that $\mathcal{D}'$ is a {\em
refinement}\index{cell!decomposition!refinement}\index{refinement of
cell decomposition} of $\mathcal{D}$ if
the following two conditions are satisfied:
\begin{itemize}
\smallskip
\item[(i)] For every cell $\sigma\in \mathcal{D}'$ there exits a cell
$\tau\in \mathcal{D}$ with $\sigma\subset \tau$.
\smallskip
\item[(ii)] Every cell $\tau\in \mathcal{D}$ is the union of all cells
$\sigma\in \mathcal{D}'$ with $\sigma\subset \tau$.
\end{itemize}
\end{definition}
It is easy to see that if $\mathcal{D}'$ is a refinement of $\mathcal{D}$ and $\tau \in \mathcal{D}$, then the cells $\sigma\in \mathcal{D}'$ with $\sigma\subset \tau$ form a cell decomposition of $\tau$. Moreover, every cell in $\mathcal{D}'$ arises in this way. So roughly speaking, the refinement $\mathcal{D}'$ of the cell decomposition $\mathcal{D}$ is obtained by decomposing each cell in $\mathcal{D}$ into smaller cells. We informally refer to this process as {\em subdividing} the cells in $\mathcal{D}$ by the smaller cells in $\mathcal{D}'$.
\begin{lemma}\label{lem:mincell} Let $\mathcal{D}'$ and $\mathcal{D}$ be two cell decompositions of $X$, and $\mathcal{D}'$ be a refinement of $\mathcal{D}$.
Then for every cell $\sigma\in \mathcal{D}'$ there exists a minimal
cell $\tau \in \mathcal{D}$ with $\sigma\subset \tau$, i.e., if $\widetilde\tau\in \mathcal{D}$ is another cell with $\sigma\subset \widetilde\tau$, then $\tau \subset \widetilde\tau$. Moreover, $\tau$ is the unique cell with $\inte(\sigma)\subset \inte (\tau)$.
\end{lemma}
\begin{proof} First note that if $\sigma\in \mathcal{D}'$, $\tau_1, \dots, \tau_n\in \mathcal{D}$ and
$$\inte(\sigma)\cap( \tau_1\cup \dots \cup \tau_n)\ne \emptyset, $$ then $\sigma\subset \tau_i$ for some $i\in \{1, \dots, n\}$. Indeed, by definition of a refinement the union of all cells in $\mathcal{D}'$ contained in some $\tau_i$ covers $\tau_1\cup \dots \cup \tau_n$. Hence this union meets $\inte(\sigma)$. It follows from Lemma~\ref{lem:celldecompint} (ii) that $\sigma$ is contained in one of these cells from $\mathcal{D}'$ and hence in one of the cells $\tau_i$.
Now if $\sigma\in \mathcal{D}'$ is arbitrary, then $\sigma$ is contained in some cell of $\mathcal{D}$ by definition of a refinement, and hence in a cell $\tau\in \mathcal{D}$ of minimal dimension.
Then $\tau$ is minimal among all cells in $\mathcal{D}$ containing $\sigma$. Indeed, let $\widetilde\tau\ne \tau$ be another cell in $\mathcal{D}$ containing $\sigma$. We want to show that $\tau\subset\widetilde \tau$.
One of the alternatives in Lemma~\ref{lem:celldecompint} (i) occurs.
If
$\tau\subset \partial \widetilde\tau \subset \widetilde\tau$ we are done. The second
alternative, $\widetilde\tau\subset \partial \tau$, is impossible, since $\tau $ has minimal dimension
among all cells containing $\sigma$.
The third alternative leads to leads to $\sigma\subset \tau\cap\widetilde\tau=\partial \tau\cap \partial \widetilde\tau$, where the latter intersection
consists of cells in $\mathcal{D}$ of dimension
$<\dim(\tau)$. By the first part of the proof $\sigma$ is contained in one of these cells, again contradicting the definition of $\tau$. Hence $\tau$ is minimal.
We have $\inte(\sigma)\subset \inte(\tau)$; for otherwise $\inte(\sigma)$ meets $\partial \tau$ which is a union of cells in $\mathcal{D}$. Then $\sigma$ would be contained in one of these cells by the first part of the proof. This contradicts the minimality of $\tau$.
Finally, it is clear that $\tau\in \mathcal{D}$ is the unique cell with
$\inte(\sigma)\subset\inte( \tau)$, because distinct cells in a cell decomposition have disjoint interior.
\end{proof}
\begin{definition}[Cellular maps and cellular Markov partitions]\label{def:cellular}
Let $\mathcal{D}'$ and $\mathcal{D}$ be two cell decompositions of $X$, and $f\: X\rightarrow
X$ be a continuous map. We say that $f$ is {\em
cellular}\index{cellular map} for $(\mathcal{D}', \mathcal{D})$ if the following condition is satisfied:
\begin{itemize}
\smallskip
\item[] If $\sigma\in \mathcal{D}'$ is arbitrary, then $f(\sigma)$ is a cell in $\mathcal{D}$ and $f|\sigma$ is a homeomorphism of $\sigma$ onto $f(\sigma)$.
\end{itemize}
If $f$ is cellular with respect to $(\mathcal{D}',\mathcal{D})$ and $\mathcal{D}'$ is a refinement of $\mathcal{D}$, then the pair $(\mathcal{D}', \mathcal{D})$ is called a {\em cellular Markov partition}\index{cellular Markov partition} for $f$.
\end{definition}
\begin{lemma}\label{lem:markoviso}
Let $\mathcal{D}'$ and $\mathcal{D}$ be cell decompositions of $X$, and $f\: X\rightarrow X$ be a continuous map that is cellular for
$(\mathcal{D}', \mathcal{D})$.
Suppose that $\sigma'\in \mathcal{D}'$, $\tau\in \mathcal{D}$, and $\tau\subset f(\sigma')$.
Then there exists $\tau'\in \mathcal{D}'$ with
$\tau'\subset \sigma'$ and $f(\tau')=\tau$.
\end{lemma}
\begin{proof} Note that $f|\sigma'$ is a homeomorphism of $\sigma'$ onto $\sigma=f(\sigma')$. Pick a point $q\in \inte(\tau)$.
Then there exists a point $p\in \sigma'$ with $f(p)=q$, and a cell $\tau'\in \mathcal{D}'$ with $p\in \inte(\tau')$. Then $\inte(\tau')$ meets $\sigma'$ and so $\tau'\subset \sigma'$ (Lemma~\ref{lem:celldecompint}~(ii)).
Moreover, $f(\tau')$ is a cell in $\mathcal{D}$ with $q=f(p)\in
\inte(f(\tau'))$. It follows that $\tau=f(\tau')$.
\end{proof}
\begin{prop} \label{prop:inftychain}
Let $\mathcal{D}'$ and $\mathcal{D}$ be two cell decompositions of $X$, and $f\: X\rightarrow X$ be a continuous map.
If $(\mathcal{D}', \mathcal{D})$ is a cellular Markov partition for $f$, then
there exist unique cell decompositions $\mathcal{D}^n$ of $X$ for $n\in \N_0$ such that
\begin{itemize}
\smallskip
\item[(i)] $\mathcal{D}^0=\mathcal{D}$, $\mathcal{D}^1=\mathcal{D}'$, and $\mathcal{D}^{n+1}$ is a refinement of $\mathcal{D}^n$ for $n\in \N_0$,
\smallskip
\item[(ii)] each pair $(\mathcal{D}^{n+1}, \mathcal{D}^n)$, $n\in \N_0$, is a cellular Markov partition for $f$. \end{itemize}
\end{prop}
Note that this implies that $\mathcal{D}^{n+k}$ is a refinement of $\mathcal{D}^n$, and $f^k$ is cellular with respect to $(\mathcal{D}^{n+k}, \mathcal{D}^n)$ for all $n,k\in \N_0$. So $(\mathcal{D}^{n+k}, \mathcal{D}^n)$ is a cellular Markov partition for $f^k$.
\begin{proof} The cell decompositions $\mathcal{D}^n$ are constructed inductively.
Let $\mathcal{D}^0=\mathcal{D}$ and $\mathcal{D}^1=\mathcal{D}'$.
The idea for constructing the refinement $\mathcal{D}^2$ of $\mathcal{D}^1$ is very simple: we want to decompose a cell $\sigma\in \mathcal{D}^1$ into cells in a similar way, as the cell $f(\sigma)\in \mathcal{D}^0$ is decomposed by the cells $\sigma'\subset f(\sigma)$ in $\mathcal{D}^1$. Accordingly, we define the set $\mathcal{D}^2$ as
$$ \mathcal{D}^2=\{ (f|\sigma')^{-1}(\sigma): \sigma,\sigma'\in \mathcal{D}^1
\text{ and } \sigma\subset f(\sigma')\}. $$
Then $\mathcal{D}^2$ consists of cells. Indeed, if
$\sigma,\sigma'\in \mathcal{D}^1$
and $ \sigma\subset f(\sigma')$, then $f|\sigma'$ is a homeomorphism of $\sigma'$ onto $f(\sigma')\in \mathcal{D}^0$.
Hence the preimage $\lambda:=(f|\sigma')^{-1}(\sigma)$ of the cell $\sigma$ under this homeomorphism is a cell. Note that the cell $\sigma=f(\lambda)$ is uniquely determined by $\lambda$, but $\sigma'$ in general is not.
We now show that $\mathcal{D}^2$ is a cell decomposition of $X$
by verifying the conditions (i)--(iv) of Definition~\ref{def:celldecomp}.
{\em Condition}~(i): Let $x\in X$ be arbitrary. Then there exists $\sigma'\in \mathcal{D}^1$ with $x\in \sigma'$. The set $f(\sigma')$ is a cell in $\mathcal{D}^0$. Since $\mathcal{D}^1$ is a refinement of $\mathcal{D}^0$, there exists a cell $\sigma\in \mathcal{D}^1$ with $f(x)\in \sigma\subset f(\sigma').$ Then $(f|\sigma')^{-1}(\sigma)$ is a cell in $\mathcal{D}^2$
that contains $x$. It follows that the cells in $\mathcal{D}^2$ cover $X$.
{\em Condition}~(ii):
Let $\lambda_1, \lambda_2\in \mathcal{D}^2$ be arbitrary, and assume that $\inte(\lambda_1)\cap \inte(\lambda_2)\ne \emptyset$. We have to show that $\lambda_1=\lambda_2$.
We have $\inte(f(\lambda_i))=f(\inte(\lambda_i))$ for $i=1,2$.
So $f(\lambda_1)$ and $f(\lambda_2)$ are cells in $\mathcal{D}^1$ with a common interior point. Hence $\sigma:=f(\lambda_1)=f(\lambda_2)\in
\mathcal{D}^1$.
By definition of $\mathcal{D}^2$ there exist cells $\sigma_i\in \mathcal{D}^1$ with $\sigma \subset f(\sigma_i)$ and $\lambda_i=(f|\sigma_i)^{-1}(\sigma)$ for $i=1,2$. Let $\tau$ be the
minimal cell in $\mathcal{D}^0$ that contains $\sigma$. Then $\sigma\subset \tau\subset f(\sigma_1)\cap f(\sigma_2)$. By
Lemma~\ref{lem:markoviso} there exist cells $\widetilde \sigma_i\in \mathcal{D}^1$ with $\widetilde \sigma_i\subset \sigma_i$ and $f(\widetilde \sigma_i)=\tau$ for $i=1,2$.
By Lemma~\ref{lem:mincell} we have
$\inte(\sigma)\subset \inte(\tau)$. Applying the homeomorphism
$(f|\sigma_i)^{-1}$ to both sets in this inclusion, we obtain $\inte(\lambda_i)\subset \inte(\widetilde \sigma_i)$ for $i=1,2$.
It follows that $\widetilde \sigma_1$ and $\widetilde \sigma_2$
are cells in $\mathcal{D}^1$ with a common interior point. Hence $\widetilde \sigma_1=\widetilde \sigma_2$, and so
$$\lambda_1=(f|\sigma_1)^{-1}(\sigma)=
(f|\widetilde \sigma_1)^{-1}(\sigma)=(f|\widetilde \sigma_2)^{-1}(\sigma)
=(f|\sigma_2)^{-1}(\sigma)=\lambda_2$$ as desired.
{\em Condition}~(iii):
Let $\lambda\in \mathcal{D}^2$ and $x\in \partial \lambda$ be arbitrary. Then there exist cells $\sigma, \sigma'\in \mathcal{D}^1$
with $\sigma\subset f(\sigma')$ and $\lambda=(f|\sigma')^{-1}(\sigma)$. Moreover, $\sigma=f(\lambda)$ and so
$f(x)\in \partial \sigma$. By definition of a cell decomposition there exists a cell $\widetilde \sigma\in \mathcal{D}^1$ with $f(x)\subset \widetilde \sigma \subset \partial \sigma$. Then $\widetilde \lambda=
(f|\sigma')^{-1}(\widetilde \sigma)$ is a cell in $\mathcal{D}^2$ with
$x\in \widetilde \lambda\subset \partial \lambda$. It follows that
$\partial \lambda$ is a union of cells in $\mathcal{D}^2$.
\smallskip
We have verified conditions (i)--(iii) in the definition of a cell
decomposition. Before we prove the last condition (iv), we will
first show that $\mathcal{D}^2$ has the required properties of a refinement.
Indeed, if $\lambda\in \mathcal{D}^2$ and $\sigma,\sigma'\in\mathcal{D}^1$ are such that $\sigma \subset f(\sigma')$ and
$\lambda=(f|\sigma')^{-1}(\sigma)$, then
$$\lambda\subset (f|\sigma')^{-1}(f(\sigma'))=\sigma'.$$
So every cell in $\mathcal{D}^2$ is contained in a cell in $\mathcal{D}^1$.
Moreover, let $\sigma'\in \mathcal{D}^1$ and $x \in \sigma'$ be arbitrary. Then $f(\sigma')$ is a cell in $\mathcal{D}^0$ containing
$f(x)$. Since $\mathcal{D}^1$ is a refinement of $\mathcal{D}^0$ there exists $\sigma\in \mathcal{D}^1$ with $f(x)\in \sigma \subset f(\sigma')$.
Then $\lambda=(f|\sigma')^{-1}(\sigma)$ is a cell in $\mathcal{D}^2$ with $x\in \lambda\subset \sigma'$. It follows that every cell in $\mathcal{D}^1$ is a union of cells in $\mathcal{D}^2$.
Moreover, this is a finite union. Indeed, if $\sigma'\in \mathcal{D}^1$, then $f|\sigma'$ induces a bijection of the cells in $\mathcal{D}^2$ contained in $\sigma'$ and the cells in $\mathcal{D}^1$ contained in $f(\sigma')\in \mathcal{D}^0$. Since the latter set is finite, the former is finite as well.
\smallskip
{\em Condition~}(iv): This can now easily be established.
If $p\in X$ is arbitrary, then there exists a neighborhood $U$ of $p$ that meets only finitely many cells in $\mathcal{D}^1$. Every cell in $\mathcal{D}^2$ that meets $U$ must be contained in one of these finitely many cells from $\mathcal{D}^1$. Since every cell in $\mathcal{D}^1$ contains only finitely many cells
in $\mathcal{D}^2$, it follows that $U$ meets only finitely many cells
in $\mathcal{D}^2$.
We have proved that $\mathcal{D}^2$ is cell decomposition of $X$ that is a refinement of $\mathcal{D}^1$. It immediately follows from the definition of $\mathcal{D}^2$ that $f$ is cellular with respect to
$(\mathcal{D}^2, \mathcal{D}^1)$. Therefore, $(\mathcal{D}^2, \mathcal{D}^1)$ is a cellular Markov
partition for $f$.
To show uniqueness of $\mathcal{D}^2$ suppose that $\widetilde {\mathcal{D}}^2$ is another cell decomposition of $X$ such that
$(\widetilde {\mathcal{D}}^2, \mathcal{D}^1)$ is a cellular Markov partition for $f$.
Then $\widetilde {\mathcal{D}}^2\subset \mathcal{D}^2$. Indeed, let $\lambda\in \widetilde {\mathcal{D}}^2$ be arbitrary. Since $\widetilde {\mathcal{D}}^2$ is a refinement of $\mathcal{D}^1$, there exists a cell
$\sigma'\in \mathcal{D}^1$ with $\lambda\subset \sigma'$.
Moreover, $\sigma=f(\lambda)$ is a cell in $\mathcal{D}^1$ and
$\sigma\subset f(\sigma')$. Since $f|\sigma'$ is a homeomorphism of $\sigma'$ onto $f(\sigma')$ it follows that
$$\lambda =(f|\sigma')^{-1}(\sigma)\in \mathcal{D}^2. $$
If the inclusion $\widetilde {\mathcal{D}}^2\subset \mathcal{D}^2$ were strict,
then there would be a cell in $\mathcal{D}^2$ whose interior would be disjoint from the interior of all the cells in $\widetilde {\mathcal{D}}^2$. This is impossible, because $\widetilde {\mathcal{D}}$ is a cell decomposition of $X$ and so the interiors of the cells in
$\widetilde {\mathcal{D}}$ form a cover of $X$. So $\widetilde {\mathcal{D}}^2= \mathcal{D}^2$.
\smallskip
We have shown the existence and uniqueness of a cell decomposition $\mathcal{D}^2$ of $X$ with the desired properties.
Now $\mathcal{D}^3$ is constructed from $(\mathcal{D}^2, \mathcal{D}^1)$ in the same way as $\mathcal{D}^2$ was constructed from $(\mathcal{D}^1, \mathcal{D}^0)$.
Continuing in this manner we get the desired existence and uniqueness of the cell decompositions $\mathcal{D}^n$.
\end{proof}
\begin{rem} \label{rem:combinatorics} The main idea of the previous proof can be summarized by saying that if the cell decompositions $\mathcal{D}^n$ and $\mathcal{D}^{n-1}$ have already been defined for some $n\in \N$, then one obtains the elements in $\mathcal{D}^{n+1}$ by subdividing the cells $\sigma\in \mathcal{D}^n$ in the same way as the images $f(\sigma)$ in $\mathcal{D}^{n-1}$ are subdivided by the cells in $\mathcal{D}^{n}$.
From this description it is clear that the ``combinatorics"
of the cells in the sequence $\mathcal{D}^n$, $n\in \N_0$, that is, their inclusion and intersection pattern, is determined by
the pair $(\mathcal{D}^1, \mathcal{D}^0)$ and by the assignment $c\in \mathcal{D}^1\mapsto f(c)\in \mathcal{D}^0$. Such an assignment of a cell in $\mathcal{D}^0$ to each cell in $\mathcal{D}^1$ is related to the concept of a ``labeling" (see Definition~\ref{def:labeldecomp}). So for
the combinatorics of the decompositions $\mathcal{D}^n$ the only relevant information on the map $f$ is its induced
``labeling" $c\in \mathcal{D}^1\mapsto f(c)\in \mathcal{D}^0$.
It is not hard, but somewhat tedious, to formulate a precise statement based on a suitable notion of
``combinatorial equivalence" for such sequences of cell decompositions (see the related Definition~\ref{def:compiso} where we define the notion of an isomorphism between cell complexes). We will not do this, because it would not add anything of substance, but content ourselves with the
intuitive statement that the ``combinatorics" of the sequence $\mathcal{D}^n$, $n\in \N_0$,
is determined by the pair $(\mathcal{D}^1, \mathcal{D}^0)$, and the assignment $c\in \mathcal{D}^1 \mapsto f(c)\in \mathcal{D}^0$.
\end{rem}
\section{Cell decompositions of $2$-spheres}
\label{sec:2spherecd}
\noindent
In this section we study cell decompositions
of $2$-spheres\index{cell!decomposition!
of $2$-sphere} and their relation to postcritically-finite branched covering maps. We first review some standard concepts and results from plane topology (see \cite{Mo} for more details).
Let $S^2$ be a $2$-sphere. An {\em arc} $\alpha$ in $S^2$ a homeomorphic image of the unit interval $[0,1]$. The points corresponding corresponding to $0$ and $1$ under such a homeomorphism are called the {\em endpoints} of $\alpha$. They are the unique points $p\in \alpha$ such that $\alpha\setminus \{p\}$ is connected. If $p$ is an {\em interior point} of $\alpha$, i.e., a point in $\alpha$ distinct from the endpoints, then there exist arbitrarily small open neighborhoods $W$ of $p$ such that
$W\setminus \alpha$ has precisely two open connected components $U$ and $V$.
A {\em closed Jordan region} $X$ in $S^2$ is
a homeomorphic image of the closed unit disk $\overline \D$. The boundary $\partial X$ of a closed Jordan region $X\subset S^2$ is a {\em Jordan curve}, i.e., the homeomorphic image of the unit circle $\partial \D$.
If $J\subset S^2$ is a Jordan curve, then by the Sch\"onflies Theorem there exists a homeomorphism $\varphi\: S^2 \rightarrow \CDach$ such that $\varphi(J)=\partial \D$. In particular,
the set $S^2\setminus J$ has two connected component, both homeomorphic to $\D$. Note that arcs and closed Jordan region are cells of dimension $1$ and $2$, respectively.
Let $\mathcal{D}$ be a cell decomposition of $S^2$. Since the topological dimension of $S^2$ is equal to $2$, no cell in $\mathcal{D}$ can have dimension $>2$.
We call the $2$-dimensional cells in $\mathcal{D}$ the {\em tiles}, and the $1$-dimensional cells in $\mathcal{D}$ the {\em edges} of $\mathcal{D}$. The {\em vertices} of $\mathcal{D}$ are the points $v\in S^2$ such that $\{v\}$ is a cell in $\mathcal{D}$ of dimension $0$. So there is a somewhat subtle distinction between vertices and cells of dimension $0$: a vertex is an element of $S^2$, while a cell of dimension $0$ is a subset of $S^2$ with one element.
If $c$ is a cell in $\mathcal{D}$, we denote by $\partial c$ the boundary and by
$\inte(c)$ the interior of $c$ as introduced in the beginning of Section~\ref{s:celldecomp}. Note that
for edges and $0$-cells $c$ this is different from the boundary and the interior of $c$ as a subset of the topological space $S^2$.
We always assume that the sphere $S^2$ is {\em oriented}, i.e., one of the two generators
of the singular homology group $H_2(S^2)\cong \Z$ (with coefficients in $\Z$) has been chosen as the {\em fundamental class} of $S^2$.
The orientation on $S^2$ induces an orientation on every Jordan region $X\subset S^2$ which in turn
induces an orientation on $\partial X$ and on every arc $\alpha \subset \partial X$.
This can be made precise by considering
the fundamental homology classes representing orientations.
For example, if $X$ is a closed Jordan region in $S^2$, then the fundamental class of $S^2$ maps to a generator of $H_2(X,\partial X)$ under
the natural isomorphism
$$ H_2(S^2)\cong H_2(S^2, S^2\setminus \inte(X))
\cong H_2(X,\partial X)\cong \Z$$
induced by the inclusion map and excision. Hence we get an induced orientation on $X$.
On a more intuitive level, an orientation of an arc is just a selection of one of the endpoints as the {\em initial point} and the other endpoint as the {\em terminal point}.
Let $X\subset S^2$ be a Jordan region in the oriented $2$-sphere $S^2$ equipped with the induced orientation. If $\alpha\subset \partial X$ is an arc with a given orientation, then we say that $X$ lies {\em to the left} or {\em to the right} of $\alpha$ depending
on whether the orientation on $\alpha$ induced by the orientation of $X$ agrees with the given orientation on $\alpha$ or not.
Similarly, we say that with a given orientation of $\partial X$ the Jordan region $X$ lies to the left or right of $\partial X$.
To describe orientations, it is useful to introduce the notion of a
flag. By definition a {\em flag}\index{flag}
in $S^2$ is a triple $(c_0,c_1, c_2)$, where $c_i$ is an $i$-dimensional cell for $i=0,1,2$, $c_0\subset \partial c_1$, and $c_1\subset \partial c_2$. So a flag in $S^2$ is a closed Jordan region $c_2$ with an arc $c_1$ contained in its boundary, where the point in $c_0$ is an distinguished endpoint of $c_1$. We orient the arc $c_1$ so that the
point in $c_0$ is the initial point in $c_1$. The flag is called {\em positively-} or {\em negatively-oriented} (for the given orientation on $S^2$) depending on whether $c_2$ lies to the left or to the right of the oriented arc $c_1$.
A positively-oriented flag determines the orientation on $S^2$ uniquely. The standard orientation on $\CDach$ is the one for which the {\em standard flag}
$(c_0', c'_1, c'_2)$ is positively-oriented, where
$c_0'=\{0\}$, $c_1'=[0,1]\subset \R$, and
$$c'_2=\{z\in \C: 0\le \text{Re}(z)\le 1,\ 0\le
\text{Im}(z) \le \text{Re}(z)\}. $$
Since edges and tiles
in a cell decomposition $\mathcal{D}$ of $S^2$ are arcs and closed Jordan regions, respectively, it makes sense to speak of oriented edges and tiles in
$\mathcal{D}$. A {\em flag in $\mathcal{D}$} is a flag $(c_0,c_1, c_2)$, where $c_0,c_1,c_2$ are cells in $\mathcal{D}$. If $c_i$ are $i$-dimensional cells in $\mathcal{D}$ for $i=0,1,2$, then $(c_0,c_1,c_2)$ is a flag in $\mathcal{D}$ if and only if $c_0\subset c_1\subset c_2$.
Cell decompositions of $S^2$ have additional properties that we summarize in the next lemma.
\begin{lemma}\label{lem:specprop}
Let $\mathcal{D}$ be a cell decomposition of $S^2$. Then it has the following properties:
\smallskip
\begin{itemize}
\item[(i)] There are only finitely many cells in $\mathcal{D}$.
\smallskip
\item[(ii)] The tiles in $\mathcal{D}$ cover $S^2$.
\smallskip
\item[(iii)] Let $X$ be a tile in $\mathcal{D}$. Then there exists a number $k\in \N$, $k\ge 2$, such that $X$ contains precisely $k$ edges $e_1, \dots, e_k$ and $k$ vertices
$v_1, \dots, v_k$ in $\mathcal{D}$. Moreover, these edges and vertices lie on the boundary $\partial X$ of $X$, and we have
$$\partial X=e_1\cup \dots \cup e_k. $$ The indexing of these vertices and edges can be chosen
such that $v_j\in \partial e_j\cap \partial e_{j+1}$
for $j=1, \dots, k$ (where $e_{k+1}:=e_1$).
\smallskip
\item[(iv)] Every edge $e\in \mathcal{D}$ is contained in the boundary of precisely two tiles $\mathcal{D}$. If $X$ and $Y$ are these tiles, then $\inte(X)\cup \inte(e)\cup \inte(Y)$ is an open
set.
\smallskip
\item[(v)] Let $v$ be a vertex of $\mathcal{D}$. Then there exists a number $d\in \N$, $d\ge 2$, such that $v$ is contained in precisely $d$ tiles
$X_1, \dots , X_{d}$, and $d$ edges $e_1, \dots, e_{d}$ in $\mathcal{D}$.
We have $v\in \partial X_j$ and $v\in \partial e_j$ for each $j=1, \dots, d$. Moreover, the indexing of these tiles and edges can be chosen
such that $e_j\subset \partial X_j\cap \partial X_{j+1}$
for $j=1, \dots, d$ (where $X_{d+1}:=X_1$).
\smallskip
\item[(vi)] The $1$-skeleton of $\mathcal{D}$ is connected and equal to the union of all edges in $\mathcal{D}$.
\end{itemize}
\end{lemma}
Note that property (iii) actually holds for all tiles (i.e., $2$-dimensional cells) in every cell decomposition.
If the boundary of a tile $X$ is subdivided into vertices and edges as
in (iii), we say that $X$ is a {\em (topological) $k$-gon}.
If the edge $e$ and the tiles $X$ and $Y$ are as in
(iv), then there exists a unique orientation of $e$ such that $X$ lies to the left and $Y$ to the right of $e$.
We say that the cells $\{v\}, e_1, \dots, e_d, X_1, \dots, X_d$ as in
(v) form the {\em cycle}\index{cycle of vertex}
of the vertex $v$ and call $d$ the {\em length} of the
cycle.\index{length of cycle}
We refer to $X_1, \dots, X_d$ as the tiles and to $e_1, \dots, e_d$ as the
edges of the cycle.
\ifthenelse{\boolean{nofigures}}{}{
\begin{figure}
\centering
\begin{overpic}
[width=9cm,
tics =20]{cycle.eps}
\put(92,49){$X_1$}
\put(63,84){$X_2$}
\put(19,76){$X_3$}
\put(5,33){$X_4$}
\put(35,2){$X_5$}
\put(76,9){$X_6$}
\put(80,60){$e_1$}
\put(49,78){$e_2$}
\put(18,61){$e_3$}
\put(18,25){$e_4$}
\put(49,7){$e_5$}
\put(80,25){$e_6$}
\put(54,42){$v$}
\end{overpic}
\caption{Cycle of a vertex $v$.}
\label{fig:cycle}
\end{figure}
}
\begin{proof} (i) By Definition~\ref{def:celldecomp}~(iv) this follows from the compactness of $S^2$ and the fact that every point in $S^2$ has a neighborhood that meets only finitely many cells in $\mathcal{D}$.
(ii) The set consisting of all vertices and the union of all edges
has empty interior (in the topological sense) by (i) and Baire's theorem. Hence the union of all tiles is a dense set in $S^2$. Since this union is also closed by (i), it is all of $S^2$.
(iii) Let $X$ be a tile in $\mathcal{D}$. Then $\inte(X)$ does not meet any edge or vertex, and $\partial X$ is a union of
edges and vertices. Since there are only finitely many vertices, $\partial X$ must contain an edge, and hence at least two vertices. Suppose $v_1, \dots, v_k$, $k\ge 2$, are all the
vertices on $\partial X$. Since $\partial X$ is a Jordan curve, we can choose the indexing of these vertices so that $\partial X$ is a union of arcs $\alpha_j$ with pairwise disjoint interior
such that $\alpha_j$ has the endpoints $v_j$ and $v_{j+1}$ for $j=1, \dots, k$, where $v_{k+1}=v_1$. Then for each $j=1, \dots, k$ the set $\inte(\alpha_j)$ is connected and lies in the $1$-skeleton of the cell decomposition $\mathcal{D}$, it is disjoint from the $0$-skeleton and has boundary contained in the $0$-skeleton. It follows from Lemma~\ref{lem:conncomp} and Lemma~\ref{lem:opencells} that there exists an edge
$e_j$ in $\mathcal{D}$ with $\inte(e_j)=\inte(\alpha_j)$. Hence
$\alpha_j=e_j$, and so $\alpha_j$ is an edge in $\mathcal{D}$.
It is clear that $\partial X$ does not contain other edges in $\mathcal{D}$. The statement follows.
(iv) Let $e$ be an edge in $\mathcal{D}$. Pick $p\in\inte(e)$. By (ii) the point $p$ is contained in some tile $X$ in $\mathcal{D}$.
By Lemma~\ref{lem:celldecompint} we have $e\subset X$. On the other hand, $\inte(X)$ is disjoint from each edge and so $e\subset \partial X$. It follows from the Sch\"onflies theorem
that the set $X$ does not contain a neighborhood of $p$.
Hence every neighborhood of $p$ must meet tiles distinct from $X$. Since there are only finitely many tiles, it follows that
there exists a tile $Y$ distinct from $X$ with $p\in Y$. By the same reasoning as before, we have $e\subset \partial Y$.
Let $q\in \inte(e)$ be arbitrary. Then there exist arbitrarily small open neighborhoods $W$ of $q$ such that $W\setminus \inte(e)$ consists of two connected components $U$ and $V$. If $W$ is small enough, then $U$ and $V$
do not meet $\partial X$. Since $q\in\overline {\inte(X)}$, one of the sets, say $U$, meets $\inte(X)$, and so $U\subset \inte(X)$. We can also assume that the set $W$ is small enough so that it does not meet $\partial Y$ either. By the same reasoning, $U$ or $V$ must be contained in $\inte(Y)$, and, since $\inte(X)\cap \inte(Y)=\emptyset$, we have $V\subset \inte(Y)$.
Hence
$\inte(X)\cup \inte(e)\cup \inte(Y)$ is a neighborhood of each point in $ \inte(e)$ which implies that this set
is open.
Suppose that $Z$ is another tile in $\mathcal{D}$ with $e\subset \partial Z$.
Since
$X\cup Y$ contains an open neighborhood for $p$, there exists a point $x\in \inte(Z)$ near $p$ with $x\in X\cup Y$, say $x\in X$. Since the interior of a
tile is disjoint from all other cells, we conclude $X=Z$.
This shows the uniqueness of $X$ and $Y$.
(v) Let $v$ be a vertex of $\mathcal{D}$. If an edge $e$ in $\mathcal{D}$ contains $v$, then $v$ is an endpoint of $e$ and we orient $e$ so that $v$ is the initial point of $e$. By (ii) there exists a tile $X_1$ in $\mathcal{D}$ with $v\in X_1$. Then $v\in \partial X_1$,
and so by (iii) there exist two edges in $\partial X_1$ that contain $v$. For one of these oriented edges, denote it by $e_1$, the tile $X_1$ will lie on the right of $e_1$. Then $v\in e_1\subset \partial X_1$ and $X_1$ will lie on the left of the other oriented edge. By (iv) there exists a unique tile $X_2\ne X_1$
with $e_1\subset \partial X_2$. Then $X_2$ will lie on the left of $e_1$. By (iii) there exists a unique edge $e_2\subset \partial X_2$ distinct from $e_1$ with $v\in e_2$. The tile $X_2$ will lie on the right of $e_2$.
We can continue in this manner to obtain tiles
$X_1, X_2, \dots$ and edges $e_1, e_2, \dots$ that
contain $v$ and satisfy $X_j\ne X_{j+1}$, $e_j\ne e_{j+1}$, and $e_j\subset \partial X_j\cap \partial X_{j+1}$ for all $j\in \N$. Moreover, $X_j$ will lie on the right and $X_{j+1}$ on the left of the oriented edge $e_j$. Since
there are only finitely many tiles, there exists a smallest number $d\in \N$ such that the tiles $X_1, \dots, X_d$ are all distinct and $X_{d+1}$ is equal to one of the tiles $X_1, \dots, X_d$. Since $X_1\ne X_2$, we have $d\ge 2$.
Moreover, $X_{d+1}=X_1$. To see this we argue by contradiction and assume that $X_{d+1}$ is equal to one
of the tiles $X_2, \dots, X_{d}$
say $X_{d+1}=X_j$. Note that $X_d\ne X_{d+1}$, so
$2\le j\le d-1$. Then $e=e_{d}$ is an edge with $v\in e$ that is contained in $\partial X_d$ and in
$\partial X_{d+1}=\partial X_{j}$. Hence $e=e_{j-1}$ or $e=e_{j}$.
Since $X_{d+1}=X_j$ lies on the left of $e=e_d$, we must have
$e=e_{j-1}$. Then $e$ is contained in the boundary of the three distinct tiles $X_{j-1}, X_j, X_d$ which is impossible by (iv). So indeed $X_{d+1}=X_1$.
By a similar reasoning we can show that the edges $e_1, \dots, e_d$ are all distinct. Indeed, suppose $e=e_j=e_k$, where $1\le j<k\le d$. Then $k>j+1$ and $e$ is contained in the boundary
of the three distinct tiles $X_{j}, X_{j+1}, X_k$ which is again absurd.
To show that there are no other edges and tiles containing
$v$ note that by (iii) the set
$$U=\inte(X_1)\cup \inte(e_1)\cup \inte(X_2)\cup \dots \cup \inte(e_d)\cup \inte(X_{d+1})$$
is open. Moreover, its boundary $\partial U$ consists
of the point $v$ and a closed set
$$A\subset \bigcup_{j=1}^d \partial X_j $$
disjoint from $\{v\}$. Hence $v$ is an isolated boundary point of $U$ which implies that $W=U\cup\{v\}$ is an open neighborhood of $v$.
If $c$ is an arbitrary cell in $\mathcal{D}$ with $v\in c$ and $c\ne \{v\}$, then $v\in \overline{\inte(c)}$. This implies that $\inte(c)$ meets $U$. Since interiors of distinct cells in $\mathcal{D}$ are disjoint, this is only possible if $c$ is equal to one of the edges $e_1, \dots, e_d$ or one of the tiles $X_1, \dots, X_d$. The statement follows.
(vi) By (v) every vertex is contained in an edge. Hence the $1$-skeleton $E$ of $\mathcal{D}$ is equal to the union of all edges in $\mathcal{D}$. To show that $E$ is connected, let
$x,y\in E$ be arbitrary. Since the tiles in $\mathcal{D}$ cover $S^2$, there exist
tiles $X$ and $Y$ with $x\in X$ and $y\in Y$. The interior of each tile is disjoint from the $1$-skeleton $E$, and so $x\in \partial X$ and $y\in \partial Y$.
Since $S^2$ is connected, there exist tiles $X_1, \dots, X_N$ in $\mathcal{D}$ such that
$X_1=X$, $X_N=Y$, and
$X_i\cap X_{i+1}\ne \emptyset$ for $i=1, \dots, N-1$.
The interior of a tile meets no other tile. Hence $\partial X_i\cap \partial X_{i+1}\ne \emptyset$ for $i=1, \dots, N-1$. Since each set $\partial X_i$ is connected, it follows that
$$K=\partial X_1\cup \dots \cup \partial X_N$$ is a connected subset of $E$
containing $x$ and $y$. Hence $E$ is connected.
\end{proof}
Let $d\in \N$, $d\ge 2$, and the tiles $X_j$ and edges $e_j$
for $j\in \N$
be as defined in the proof of statement (v) of the previous lemma. Then we showed that $X_{d+1}=X_1$, but it is useful to point out that actually $X_j=X_{j+d}$ and $e_{j} =e_{j+d}$ for all $j\in \N$.
Indeed we have seen that $X_{d+1}=X_1$. Moreover,
$e_1, e_{d}, e_{d+1}$ are edges in $\mathcal{D}$ that contain $v$ and are contained in the boundary of the tile $X_1=X_{d+1}$. Since there are only two such edges, $e_1\ne e_d$, and $ e_{d}\ne e_{d+1}$, we conclude that $e_{d+1}=e_1$. Then $e_1=e_{d+1}$ is an edge contained in the boundary of the tiles $X_1, X_2, X_{d+2}$. Since there are precisely two tiles containing an edge in its boundary, $X_1\ne X_2$ and $X_1=X_{d+1}\ne X_{d+2}$ it follows that $X_{d+2}=X_2$.
If we continue in this manner, shifting all indices by $1$ in each step, we see that $e_{d+2}=e_2$, $X_{d+3}=X_3$,
etc., as claimed.
Note that if we choose the indexing of the edges $e_j$ and $X_j$ as in the proof of
statement (v) of the previous lemma, then for each $j\in \N$
the flag $(\{v\}, e_j, X_{j+1})$ in $\mathcal{D}$ is positively-oriented, and flag
$(\{v\}, e_j, X_{j})$ is negatively-oriented. In other words, if $e_j$ is oriented so that $v$ is the initial point of $e_j$, then $X_{j+1}$ lies to the left and $X_j$ lies to the right of $e_j$.
\begin{lemma} \label{lem:constrmaps}
Let $\mathcal{D}'$ and $\mathcal{D}$ be cell decompositions of $S^2$, and
$f\: S^2\rightarrow S^2$ be a cellular map for $(\mathcal{D}', \mathcal{D})$ such that $f|X$ is orientation-preserving for each tile $X$ in $\mathcal{D}'$.
\begin{itemize}
\smallskip
\item[(i)] Then $f$ is a branched covering map on $S^2$. Each critical point of $f$ is a vertex of $\mathcal{D}'$.
\smallskip
\item[(ii)] If in addition each vertex in $\mathcal{D}$ is also a vertex in $\mathcal{D}'$, then every point in $\operatorname{post}(f)$ is a vertex of $\mathcal{D}$. In particular, $f$ is postcritically-finite, and hence a Thurston map
if $f$ is not a homeomorphism.
\end{itemize}
\end{lemma}
\begin{proof} (i) We will show that for each point $p\in S^2$, there exists $k\in \N$, an orientation-preserving homeomorphism $\varphi$ of the open unit disk $\D=\{z\in \C:|z|<1\}$ onto a neighborhood $W'$ of $p$,
and an orientation-preserving homeomorphism $\psi$ of a neighborhood $W\supset f(W')$ of $q=f(p)$ onto $\D$ such that $\varphi(0)=p$, $\psi(q)=0$, and
$$ (\psi\circ f\circ \varphi)(z)=z^k$$
for all $z\in \D$. The desired relation between the points and maps can be represented by the
commutative diagram
\begin{equation}\label{eq:digr}
\xymatrix{
p\in W' \ar[r]^{f} & q\in W \ar[d]^{\psi}
\\
0\in \D \ar[u]^{\varphi} \ar[r]^{z\mapsto z^k} & 0\in \D.
}
\end{equation}
We will use the fact that
if $f$ is an orientation-preserving local
homeomorphism near $p$, then we can take $k=1$ and can always find suitable homeomorphisms $\varphi$ and $\psi$.
Let $p\in S^2$ be arbitrary.
Since $S^2$ is the disjoint union of the interior of the cells in $\mathcal{D}'$, the point $p$ is contained in the interior of a tile or an edge in $\mathcal{D}'$, or is a vertex of $\mathcal{D}'$. Accordingly, we consider three cases.
\smallskip
{\em Case 1.} There exists a tile $X'\in \mathcal{D}'$ with $p\in \inte(X')$. Then $W':=\inte(X')$ is an open neighborhood of $p$, and $f|W'$ is an orientation-preserving homeomorphism of $W'=\inte(X')$ onto $W:=\inte(X)$, where $X=f(X')\in \mathcal{D}$. Hence $f$ is a orientation-preserving local homeomorphism near $p$.
\smallskip
{\em Case 2.} There exists an edge $e'\in \mathcal{D}'$ with $p\in \inte(e')$. By Lem\-ma~\ref{lem:specprop}~(iv)
there exist distinct tiles $X',Y'\in \mathcal{D}'$ such that $e\subset \partial X'\cap \partial Y'$. Then $W'=\inte(X')\cup \inte(e')\cup \inte(Y')$ is an open neighborhood of $p$. Since $f$ is cellular, $X=f(X')$ and $Y=f(Y')$ are tiles in $\mathcal{D}$ and $e=f(e')$ is an edge in $\mathcal{D}$. Moreover, $e\subset \partial X\cap \partial Y$.
We orient $e'$ so that $X'$ lies to the left and $Y'$
to the right of $e'$. Since $f$ is orientation-preserving if restricted to cells in $\mathcal{D}'$, the tile $X$ lies to the left, and $Y$ to the right of the image $e$ of $e'$. In particular, $X\ne Y$, and so the sets
$\inte(X),\inte(e),\inte(Y)$ are pairwise disjoint, and their union is open. Since $f$ is cellular and hence a homeomorphism if restricted to cells (and interior of cells), it follows that
$f|W'$ is a homeomorphism of $W'$ onto the open set
$W=\inte(X)\cup \inte(e)\cup \inte(Y)$.
Moreover, it is clear that $f|W'$ is orientation-preserving.
Since $W'$ is open and contains $p$, the map $f$ is a orientation-preserving local homeomorphism near $p$.
\smallskip
{\em Case 3.} The point $p$ is a vertex of $\mathcal{D}'$.
As in the proof of Lem\-ma~\ref{lem:specprop}~(v) we can choose
tiles $X'_j\in \mathcal{D}'$ and edges $e'_j\in \mathcal{D}$ for $j\in \N$ that contain $p$ and satisfy
$X'_j\ne X'_{j+1}$, $e'_j\ne e'_{j+1}$, and $e'_j\subset \partial X'_j\cap \partial X'_{j+1}$ for all $j\in \N$. There exists $d'\in \N$
such that $X'_{d'+1}=X'_1$ and such that the tiles $X'_1, \dots, X'_{d'}$ and the edges $e'_1, \dots, e'_{d'}$ are all distinct and such that
$$W'=\{p\}\cup \inte(X'_1)\cup \inte(e'_1)\cup \inte(X'_2)\cup \dots \cup \inte(e'_{d'})$$
is an open neighborhood of $p$.
Moreover, by the remark following
Lemma~\ref{lem:specprop}, we know that $X'_j=X'_{j+d'}$ and $e'_j=e'_{j+d'}$ for all $j\in \N$.
Define $X_j=f(X'_j)$ and $e_j=f(e'_j)$ for $j\in \N$.
Since $f$ is cellular for $(\mathcal{D}',\mathcal{D})$, the set $X_j$ is a tile and $e_j$ an edge in $\mathcal{D}$. Moreover, $e_j\subset \partial X_j\cap \partial X_{j+1}$ for
$j\in \N$. Since $X'_j$ and $X'_{j+1}$ are distinct tiles containing the edge $e'_{j+1}$ in their boundaries, it follows by an argument as in Case~2 above that $X_j\ne X_{j+1}$ for $j\in \N$. Moreover, since $e'_{j}$ and $e'_{j+1}$ are distinct edges in $\mathcal{D}'$ contained in $X'_{j+1}$, and $f|X_{j+1}' $ is a homeomorphism, we also have $e_{j}\ne e_{j+1}$ for $j\in \N$.
As in the proof of Lem\-ma~\ref{lem:specprop}~(v) we see that there exists a number $d\in \N$, $d\ge 2$,
such that $X_{d+1}=X_1$, and such the tiles $X_1, \dots, X_d$ and the edges $e_1, \dots, e_d$ are all distinct. Moreover,
$$W=\{q\}\cup \inte(X_1)\cup \inte(e_1)\cup \inte(X_2)\cup \dots \cup \inte(e_d)$$ is an open neighborhood of $q=f(p)$, and $X_i=X_{j+d}$ and $e_j=e_{j+d}$ for all $j\in \N$.
The periodicity properties of the indexing of the tiles $X'_j$ and $X_j$ imply that $d\le d'$ and that $d$ is a divisor of $d'$. Hence there exists $k\in \N$ such that $d'=kd$.
We now claim that after suitable coordinate changes
near $p$ and $q$, the map $f$ can be given the form $z\mapsto z^k$.
For $N\in \N$, $N\ge 2$, and $j\in \N$ define half-open line segments
$$ L_j^N=\{re^{2\pi {\mathbf{\imath}} j/N}: 0\le r<1\}\subset \D $$ and sectors
$$\Sigma_j^N=\{re^{{\mathbf{\imath}} t}: 2\pi (j-1)/N\le t\le 2\pi j/N
\text{ and } 0\le r<1\}\subset \D. $$
We then construct a homeomorphism $\psi\: W \rightarrow \D$ with $\psi(q)=0$ as follows. For each $j=1, \dots, d$
we first map the half-open arc $\{q \}\cup \inte(e_j)$ homeomorphically to
the half-open line segment $L_j^d$.
Then
$q$ is mapped to $0$, so these maps are consistently defined for $q$. Since $X_j$ is a Jordan region, we can extend the homeomorphisms on $\{q\}\cup \inte(e_{j-1}) \subset \partial X_j$ and on $\{q\}\cup
\inte(e_j)\subset \partial X_j$ to a homeomorphism of
$$\{q\}\cup \inte(e_{j-1})\cup\inte(e_j)\cup \inte(X_j)$$ onto the sector $\Sigma_j^d$ for each $j=2, \dots, d+1$.
Since the sets
$$\{q\}, \inte(e_1), \dots, \inte(e_{d}), \inte(X_2), \dots, \inte(X_{d+1})=\inte(X_1)$$ are pairwise disjoint and have $W$ as a union,
these homeomorphisms paste together to a well-defined homeomorphism $\psi$ of $W$ onto $\D$.
Note that $\psi(q)=0$ and $\psi(X_j\cap W)=\Sigma_j^d$ for each $j=1, \dots, d$.
A homeomorphism $\varphi\: \D\rightarrow W'$ is defined as follows. If $z\in \D$ is arbitrary, then $z\in \Sigma^{d'}_j$ for some $j=1, \dots, d'$.
Hence $z^k\in \Sigma^{d}_j$, and so $\psi^{-1}(z^k)\in X_j\cap W$.
Since $f$ is a homeomorphism of $X'_j\cap W'$ onto $X_j\cap W$, it follows that $(f|X'_j)^{-1}(\psi^{-1}(z^k))$ is defined and lies in $X'_j\cap W'$.
We set
$$\varphi(z)=(f|X'_j)^{-1}(\psi^{-1}(z^k)).$$
It is straightforward to verify that $\varphi$ is well-defined and a homeomorphism of $\D$ onto $W'$ with
$\varphi(0)=p$. It follows from the definition of $\varphi$ that
$(\psi\circ f\circ \varphi)(z)=z^k$ for $z\in \D$, and so we have the diagram \eqref{eq:digr}.
We assume that the tiles $X'_j$ and the edges $e_j'$ are indexed by the procedure in the proof of Lem\-ma~\ref{lem:specprop}~(v). Then each flag $(\{p\},e'_j, X'_{j+1})$ is positively-oriented
(see the remark after the proof of Lem\-ma~\ref{lem:specprop}). Since $f|X'_j$ is orientation-preserving, this implies that the flag $(\{q\},e_j, X_{j+1})$ is also positively-oriented. We conclude that $\psi$ is orientation-preserving, since $\psi$
maps the positively-oriented flag $(\{q\},e_j, X_{j+1})$ in $S^2$ to the posi\-tively-oriented flag
$(\{0\}, L^d_j, \Sigma^d_{j+1})$ in $\CDach$.
As follows from its definition, the map $ \varphi$ is then also orientation-preserving. Hence
$\psi$ and $\varphi$ are local homeomorphisms as desired.
We have shown that in all cases the map $f$ has a local behavior as claimed. It follows that $f$ is a branched covering map. Moreover, we have seen that $f$ near each point is a local homeomorphism unless $p$ is a vertex of $\mathcal{D}'$. It follows that each critical point of $f$ is a vertex of $\mathcal{D}'$.
(ii) Suppose in addition that every vertex of $\mathcal{D}$ is also
a vertex of $\mathcal{D}'$. Let $p$ be a critical point of $f$. Then by (i) the point $p$ is a vertex of $\mathcal{D}'$. Since $f$ is cellular for $(\mathcal{D}',\mathcal{D})$, the point $f(p)$ is a vertex of
$\mathcal{D}$. Hence $f(p)$ is also a vertex of $\mathcal{D}'$, and we can apply the argument again, to conclude that $f^2(p)$ is a vertex of $\mathcal{D}$, etc. It follows that $\operatorname{post}(f)$ is a subset of the set of vertices of $\mathcal{D}$. In particular, $\operatorname{post}(f)$ is finite, and so $f$ is postcritically-finite.
\end{proof}
\begin{rem}\label{rem:dd'} Let the map $f\:S^2 \rightarrow S^2$ and the cell decompositions $\mathcal{D}'$ and $\mathcal{D}$
be as in the previous lemma, and let $p$ be a vertex in $\mathcal{D}'$. Then $q=f(p)$ is a vertex in $\mathcal{D}$. If $d'$ and $d$ are the lengths of the cycles of $p$ in $\mathcal{D}'$ and $q$ in $\mathcal{D}$, respectively, then $d'=d\deg_f(p)$. Moreover, the tiles and edges of the cycle of $q$ in $\mathcal{D}$ are the images under $f$ of the tiles and edges of the cycle of $p$ in $\mathcal{D}$. This was established in Case~3 of the proof of Lemma~\ref{lem:constrmaps}. \end{rem}
\begin{lemma} \label{lem:pullback} Let $f\: S^2\rightarrow S^2$ be a branched covering map and $\mathcal{D}$ a cell decomposition of $S^2$ such that every point in $f(\operatorname{crit}(f))$ is a vertex in $\mathcal{D}$. Then there
exists a unique cell decomposition $\mathcal{D}'$ of $S^2$ such that
$f$ is cellular with respect to $(\mathcal{D}', \mathcal{D})$.
\end{lemma}
\begin{proof} To show existence we define
$\mathcal{D}'$ to be the set of all cells $c\subset S^2$ such that $f(c)$ is a cell in $\mathcal{D}$ and $f|c$ is a homeomorphism of $c$ onto $f(c)$. It is clear that $\mathcal{D}'$ does not contain cells of dimension $>2$. As usual we call the cells $c$ in $\mathcal{D}'$ edges or tiles depending on whether $c$ has dimension $1$ or $2$, respectively. The vertices $p$ of $\mathcal{D}'$ are the
points in $S^2$ such that $\{p\}$ is a cell in $\mathcal{D}'$ of dimension $0$.
It is clear that the set of vertices of $\mathcal{D}'$ is equal to $f^{-1}({\bf V})$, where ${\bf V}$ is the set of vertices of $\mathcal{D}$.
To show that $\mathcal{D}'$ is a cell decomposition of $S^2$, we first establish two claims.
\smallskip
{\em Claim 1.} If $p\in S^2$ and $q=f(p)\in \inte(X)$ for some tile $X\in \mathcal{D}$, then there exists a unique tile $X'\in \mathcal{D}'$ with $p\in X'$.
In this case let $U=\inte(X)$. Then $U$ is an open and
simply connected set in the complement of ${\bf V}\supset f(\operatorname{crit}(f))$. Hence there exists a unique continuous map $g\:U\rightarrow U':=g(U)$ with
$f\circ g=\text{id}_U$ and $g(q)=p$. The map $g$ is a homeomorphism onto its image $U'$. Hence
$U'\subset S^2$ is open and simply connected.
We equip $S^2$ with some base metric inducing the standard topology. In the following metric terms will refer to this metric. Recall that
$\mathcal {N}^\epsilon(A)$ denotes the open $\epsilon$-neighborhood of a set $A\subset S^2$. Then $f$ has the following property:
for all $w\in S^2$ and all $\epsilon>0$, there exists $\delta>0$ such that
\begin{equation} \label{prope1}
f^{-1}(B(w,\delta)) \subset {\mathcal N}^\epsilon(f^{-1}(w)).
\end{equation}
Indeed, if for some $w\in S^2$ and $\epsilon>0$ there is no such $\delta$,
then there exists a sequence $\{z_i\}$ in
$ S^2 \setminus \mathcal{N}^\epsilon(f^{-1}(w))$ such that
$f(z_i)\in B(w, 1/i)$ for all $i\in \N$.
By passing to a subsequence, we may assume that
$z_i \to z\in S^2$. Then
$f(z)=\lim_{i\to \infty} f(z_i) = w$, while
$$\operatorname{dist}(z, f^{-1}(w))=\lim_{i\to \infty} \operatorname{dist}
(z_i,f^{-1}(w) )\ge \epsilon.$$ This is a contradiction showing
\eqref{prope1}.
We want to prove that $g$ has a continuous extension
to $\overline U=X$. For this it suffices to show that $\{g(w_i)\}$
converges whenever $\{w_i\}$
is a sequence in $U$ converging to a point $w\in \partial U$. Since
$g$ is a right inverse of $f$, it follows that the limit points of
$\{g(w_n)\}$ are contained in $f^{-1} (w)$. Since $f$ is
finite-to-one,
the point $w$ has finitely many preimages $z_1, \dots, z_m$ under
$f$.
We can choose $\epsilon>0$ so small that the sets
$B(z_i, \epsilon)$, $i=1, \dots, m$, are pairwise disjoint.
By \eqref{prope1} we can find $\delta>0$ such that
\begin{equation}\label{prope2}
f^{-1}(B(w, \delta))\subset \bigcup_{i=1}^m B(z_i, \epsilon).
\end{equation}
The set $\overline U=X$ is a closed Jordan region, and hence locally
connected. So there exists an open connected set $V\subset U$ such
that $\overline V$ is a neighborhood of $w$ in $\overline U$ and
$\overline V\subset B(w,\delta)$.
Then $g(V)$ is connected subset of $f^{-1}(B(w,\delta))$. Since the union on the right hand side of
\eqref{prope2} is disjoint, the set
$g(V)$ must be contained in one of the sets of this union, say
$g(V)\subset B(z_k, \epsilon)$.
Now $w_i \in V$ for sufficiently large $i$, and so
all limit points of $\{g(w_i)\}$ are contained in
$\overline {g(V)}\subset \overline B(z_k, \epsilon)$.
On the other hand, the only possible limit points of $\{g(w_i)\}$ are
$z_1, \dots, z_m$, and $z_k$ is the only one contained in
$\overline B(z_k, \epsilon)$. This implies
$\{g(w_i)\}\to z_k$. So $g$ has indeed a continuous extension to
$\bar U$. We also denote it by $g$.
It is clear that
\begin{equation}\label{prope3}
f \circ g=\text{id}_{\overline U}.
\end{equation}
This implies that $g$ is a homeomorphism of $\overline U=X$
onto its image $X':=g(\overline U)=\overline {g(U)}$.
Then $X'$ is a closed Jordan region, and
by \eqref{prope3} the map $f|X'$ is a homeomorphism of $X'$ onto
$\overline U=X$. Hence $X'$ is a tile in $\mathcal{D}'$ with $p\in g(U)\subset X'$.
So a tile $X'\in \mathcal{D}'$ containing $p$ exists. We want to show that it is the only tile in $\mathcal{D}'$ containing $p$. Indeed, suppose $Y'\in \mathcal{D}'$ is another tile with $p\in Y'$. Then $f(Y')$ is a tile in $\mathcal{D}$ containing the point $q=f(p)\in \inte(X)$.
Hence $f(Y')=X$, and so $f|Y'$ is a homeomorphism of $Y'$ onto $X$. Let $h=(f|X)^{-1}$.
Then $g$ and $h$ are both inverse branches of $f$
defined on the simply connected region $U$ with $g(q)=p=h(q)$. Hence $h$ and $g$ agree on $U$, and so by continuity also on $\overline U$. It follows that $X'=g(X)=
h(X)=Y'$ as desired.
\smallskip
{\em Claim 2.} If $p\in S^2$ and $q=f(p)\in \inte(e)$ for some edge $e\in \mathcal{D}$, then there exists a unique edge $e'\in \mathcal{D}'$, and precisely two distinct tiles $X'$ and $Y'$ in $\mathcal{D}'$ that contain $p$. Moreover, $e'\subset \partial X'\cup \partial Y'$.
By Lemma~\ref{lem:specprop} (iv) we know that that are precisely two distinct tiles $X,Y\in \mathcal{D}$ that contain $e$ in their boundary, and
that $U=\inte(X)\cup \inte(e)\cup \inte(Y)$
is an open and simply connected region in the complement of the set ${\bf V}\supset f(\operatorname{crit}(f))$.
Hence there exists a unique continuous map $g\:U\rightarrow S^2$ with $g(q)=p$ and $f\circ g=\id_{U}$.
As before one can show that the maps $g_1:=g|\inte(X)$ and $g_2:=g|\inte(Y)$ have continuous extensions to $X$ and $Y$, respectively.
We use the same notation $g_1$ and $g_2$ for these extensions. It is clear that $g_1|e=g_2|e$.
Moreover, $g_1$ is a homeomorphism of $X$ onto a closed Jordan region $X'=g_1(X)$ with inverse map
$f|X'$. In particular, $X'$ is a tile in $\mathcal{D}'$. Similarly, $Y'=
g_2(Y)$ is a tile in $\mathcal{D}'$.
The tiles $X'$ and $Y'$ are distinct, because $f$ maps them to different tiles in $\mathcal{D}$. Moreover, $e':=g_1(e)=g_2(e)$ is an edge in $\mathcal{D}'$ with $p\in e'\subset \partial X'\cap \partial Y'$.
It remains to prove the uniqueness part. If $\widetilde e$ is another edge in $\mathcal{D}'$ with $p\in \widetilde e$, then $f$ is a homeomorphism
of $\widetilde e$ onto $e$. Then $(f|\inte(e'))^{-1}$ and
$f(\inte(\widetilde e))^{-1}$ are right inverses of $f$ defined on the open arc $\inte(e)$ that both map $q$ to $p$. Hence these right inverses must agree on $\inte(e)$. By continuity this implies $(f|e')^{-1}=(f|\widetilde e)^{-1}$ on $e$, and so $e'=(f|e')^{-1}(e)=(f|\widetilde e)^{-1}(e)=\widetilde e$.
If $Z'$ is another tile in $\mathcal{D}'$ with $p\in Z'$, then $f$ maps $\partial Z'$ homeomorphically to the boundary $\partial f(Z')$ of the tile $f(Z')\in \mathcal{D}$. Moreover, $p\in \partial Z'$; for otherwise $f(p)$ would lie in the set $\inte(f(Z'))$ which is disjoint of
$e$.
It follows that there is an edge in $\mathcal{D}'$ that contains $p$ and is contained in the boundary of $\partial Z'$. Since this edge
in $\mathcal{D}'$ is unique, as we have just seen, we know that $e'\subset \partial Z'$.
Note that $X'\cup Y'\supset g(U)\supset \inte(e')$, and so
$X'\cup Y'$ contains an open neighborhood for each point
in $\inte(e')$. Since $e'\subset \partial Z'$ there exists a point $x\in \inte(Z')$ near $p$ with $x\in X'\cup Y'$, say $x\in X'$.
Then $f(x)$ is contained in the interior of the tile $f(Z')\in \mathcal{D}$. Since $x\in X'\cap Z'$ and $X'$ and $Z'$ are both tiles in $\mathcal{D}'$ we conclude $X'=Z'$ by the first claim. This concludes the proof of Claim 2.
\smallskip
Now that we have established the claims, we can show that
$\mathcal{D}$ is a cell decomposition of $S^2$ by verifying conditions (i)--(iv) of Definition~\ref{def:celldecomp}.
\smallskip
{\em Condition} (i): If $p\in S^2$ is arbitrary, then $f(p)$ is a vertex of $\mathcal{D}$ or $f(p)$ lies in the interior of an edge or in the interior of a tile in $\mathcal{D}$. In the first case $p$ is a vertex of $\mathcal{D}'$, and in the other two cases $p$ lies in cells in $\mathcal{D}'$ by Claim 1 and Claim 2.
It follows that the cells in $\mathcal{D}'$ cover $S^2$.
\smallskip
{\em Condition} (ii): Let $\sigma, \tau $ be cells in $\mathcal{D}'$ with $\inte(\sigma)\cap \inte(\tau)\ne \emptyset$. Then $f(\sigma)$ and $f(\tau)$ are cells in $\mathcal{D}$ with $\inte(f(\sigma))\cap \inte(f(\tau))\ne \emptyset$. Hence $\lambda=f(\sigma)=f(\tau)$.
In particular, $\sigma$ and $\tau$ have the same dimension.
If $\sigma$ and $\tau$ are both tiles, then $\sigma=\tau$ by Claim 1, because every point in $\inte(\sigma)\cap \inte(\tau)\ne \emptyset $ has an image under $f$ in $\inte(\lambda)$. Similarly, if $\sigma$ and $\tau$ are edges, then $\sigma=\tau$ by Claim 2.
If $\sigma$ and $\tau$ consist of vertices in $\mathcal{D}'$, then the relation
$\inte(\sigma)\cap \inte(\tau)\ne \emptyset$ trivially implies
$\sigma=\tau$.
\smallskip
{\em Condition} (iii): Let $c'\in \mathcal{D}'$ be arbitrary. Then $f|c'$ is a homeomorphism of $c'$ onto the cell $c=f(c')\in \mathcal{D}$. Note that $(f|c')^{-1}(\sigma)\in \mathcal{D}'$ whenever $\sigma\in \mathcal{D}$ and $\sigma\subset c$. Since
$\partial c'= (f|c')^{-1}(\partial c)$ and $\partial c$ is a union of cells in $\mathcal{D}$, it follows that $\partial c'$ is a union of cells in $\mathcal{D}'$.
\smallskip
{\em Condition} (iv): To establish the final property of a cell decomposition for $\mathcal{D}'$, we will show that $\mathcal{D}'$ consists of only finitely many
cells. Indeed, let $N_i\in \N$ be the number of cells of dimension $i$ in $\mathcal{D}$ for $i=0,1,2$.
Since the vertices in $\mathcal{D}'$ are the preimages of the vertices of $\mathcal{D}$, we have at most $\deg(f) N_0$ vertices in $\mathcal{D}'$.
Pick one point in the interior of each edge in $\mathcal{D}$. The set $M$ of these points consists of $N_1$ elements. If $q\in M$, then $q\notin{\bf V}\supset f(\operatorname{crit}(f))$, and so $q$ is not a critical value of $f$. Hence $\#f^{-1}(M)=N_1\deg(f)$. It follows from Claim 2 that each element of $f^{-1}(M)$ is contained in a unique
edge in $\mathcal{D}'$, and it follows from the definition of $\mathcal{D}$ that each edge in $\mathcal{D}'$
contains a unique point in $f^{-1}(M)$. Hence the number of edges in $\mathcal{D}'$ is equal to $\#f^{-1}(M)=N_1\deg(f)$.
Similarly, pick a point in the interior of each tile in $\mathcal{D}$ and let $M$ be the set of these points. Then $\#f^{-1}(M)=N_2\deg(f)$ and by the same reasoning as above based on Claim 1, we see
that the the number of tiles in $\mathcal{D}'$ is equal to $\#f^{-1}(M)=N_2\deg(f)$.
\smallskip
We have shown that $\mathcal{D}'$ is a cell decomposition of $S^2$.
It follows immediately from the definition of $\mathcal{D}'$ that $f$ is cellular with respect to $(\mathcal{D}', \mathcal{D})$.
\smallskip
To show uniqueness of $\mathcal{D}'$ suppose that $\widetilde {\mathcal{D}}$ is another cell decomposition such that $f$ is cellular with respect to $(\widetilde {\mathcal{D}}, \mathcal{D})$. Then by definition of $\mathcal{D}'$ every cell in $\widetilde {\mathcal{D}}$ also lies in $\mathcal{D}'$.
So we have $\widetilde {\mathcal{D}}\subset \mathcal{D}'$. If this inclusion were strict, then there would be a cell $c\in \mathcal{D}'$ whose interior $\inte(c)\ne \emptyset$ is disjoint from the interior of all cells in
$\widetilde {\mathcal{D}}$. This is impossible, since these interiors form a cover of $S^2$. Hence $\widetilde {\mathcal{D}}=\mathcal{D}'$.
\end{proof}
\begin{rem}\label{rem:no01} No Thurston maps $f\:S^2\rightarrow S^2$ with $\#\operatorname{post}(f)\in \{0,1\}$ exist. Indeed, suppose that $f$ is such a map. Then $U:=S^2\setminus \operatorname{post}(f)$ is simply connected, and so there
exists a continuous map $g\: U\rightarrow S^2$ with
$f\circ g=\id_U$.
If $\operatorname{post}(f)=0$, we have $U=S^2$ and so we conclude that $g$ is a homeomorphism onto its image.
This image must be all of $S^2$. Hence $g$ and $f$ are homeomorphisms, contradicting our assumption $\deg(f)\ge 2$ (see Definition~\ref{def:f}).
If $\#\operatorname{post}(f)=1$, we have $U=S^2\setminus\{p\}$ for some $p\in S^2$. Then by an argument as in the proof of Claim 1 in Lemma~\ref{lem:pullback}, one can show that $g$ has a continuous extension to the point $p$, and hence to $S^2$. If we denote this extension to $S^2$ also by $g$, then $f\circ g=\id_{S^2}$, and again we conclude that $f$ is a homeomorphism
and obtain a contradiction. \end{rem}
\section{Cell decompositions induced by Thurston maps}\index{cell!decomposition!induced by Thurston map}
\label{sec:tiles}
\noindent
Let $f\: S^2\rightarrow S^2$ be a Thurston map, and $\mathcal{C}\subset S^2$ be a Jordan curve such that $\operatorname{post}(f)\subset \mathcal{C}$.
In this section will show that the pair $(f,\mathcal{C})$ induces natural cell decompositions of $S^2$.
By the
Sch\"onflies theorem there are two closed Jordan regions
$\XOb, \XOw\subset S^2$\index{X0@$\XOw,\XOb$} whose boundary is $\mathcal{C}$. Our notation for these regions is suggested by the fact that we often think of
$\XOb$ as being assigned or carrying the color ``black", represented by the symbol ${\tt b}$,
and $\XOw$ as being colored ``white" represented by ${\tt w}$.
We will discuss this more precisely later in this section (see Lemma~\ref{lem:colortiles}).
The sets $ \XOb$ and $\XOw$ are topological cells of dimension $2$. We
call them \defn{tiles of order $0$} or $0$-{\em tiles}. The
postcritical
points of $f$ are on the boundary of
$\XOw$ and $\XOb$. We consider them as \defn{vertices} of $\XOw$ and
$\XOb$, and the closed arcs of $\mathcal{C}$ between vertices as the
\defn{edges} of the $0$-tiles. In this way, we think of $\XOw$
and $\XOb$ as topological $m$-gons where $m=\#\operatorname{post}(f)\ge 2$ (see Remark~\ref{rem:no01}). To emphasize
that these edges and vertices belong to $0$-tiles, we call them $0$-{\em edges} and
$0$-{\em vertices}.
A $0$-{\em cell} is a $0$-tile, a $0$-edge, or a set consisting of
a $0$-vertex. Obviously, the $0$-cells form a cell decomposition of $S^2$ that we denote by $\mathcal{D}^0=\mathcal{D}^0(f,\mathcal{C})$.
Since every point in $\operatorname{post}(f)$ is a vertex of $\mathcal{D}^0$, we can apply Lemma~\ref{lem:pullback} to obtain a unique cell decomposition $\mathcal{D}^1=\mathcal{D}^1(f,\mathcal{C})$ such that $f$ is cellular with respect to $(\mathcal{D}^1, \mathcal{D}^0)$. The vertices of $\mathcal{D}^1$ are precisely the points whose image is a vertex of
$\mathcal{D}^0$. In particular, since $f(\operatorname{post}(f))\subset \operatorname{post}(f)$, or equivalently
$\operatorname{post}(f)\subset f^{-1}(\operatorname{post}(f))$,
it follows that every point in $\operatorname{post}(f)$ is a vertex for $\mathcal{D}^1$.
Hence we can apply Lemma~\ref{lem:pullback} again and obtain a cell decomposition $\mathcal{D}^2=\mathcal{D}^2(f,\mathcal{C})$ such that
$f$ is cellular with respect to $(\mathcal{D}^2, \mathcal{D}^1)$.
Continuing in this manner, we obtain cell decompositions $\mathcal{D}^n=\mathcal{D}^n(f,\mathcal{C})$ of $S^2$ for $n\in \N_0$ such that $f$ is cellular for $(\mathcal{D}^{n+1}, \mathcal{D}^n)$ for all $n\in \N_0$.
We call the elements in $\mathcal{D}^n$ the {\em $n$-cells}\index{n@$n$-!cell}
for $(f,\mathcal{C})$, or simply $n$-cells if $f$ and $\mathcal{C}$ are understood.
We call $n$ the {\em order}\index{order of cell} of an $n$-cell. When we speak of
$n$-cells, then $n$ always refers to this order and not to the
dimension of the cell.
An $n$-cell of dimension $2$ is called an {\em $n$-tile},\index{n@$n$-!tile}
and an $n$-cell of dimension $1$ an {\em $n$-edge}.\index{n@$n$-!edge}
An $n$-{\em vertex}\index{n@$n$-!vertex}
is a point $p\in S^2$ such that
$\{p\}$ is an $n$-cell of dimension $0$.
With $f$ and $\mathcal{C}$ understood we denote the set of all $n$-tiles,
$n$-tiles, and $n$-vertices by $\X^n$,\index{Xn@$\X^n$}
$\E^n$,\index{En@$\E^n$} and ${\bf V}^n$,\index{Vn@${\bf V}^n$} respectively.
In the following proposition we summarize properties of the
cell decompositions $\mathcal{D}^n$.
\begin{prop} \label{prop:celldecomp}
Let $k,n\in \N_0$, let $f\: S^2\rightarrow S^2$ be a Thurston map, $\mathcal{C}\subset S^2$ be a Jordan curve with $\operatorname{post}(f)\subset \mathcal{C}$, $\mathcal{D}^n=\mathcal{D}^n(f,\mathcal{C})$, and $m=\#\operatorname{post}(f)$.
\smallskip
\begin{itemize}
\smallskip
\item[(i)] The map $f^k$ is cellular with respect to $(\mathcal{D}^{n+k}, \mathcal{D}^n)$. In particular, if $\tau$ is any
$(n+k)$-cell, then $f^k(\tau)$ is an $n$-cell, and $f^k|\tau$ is a homeomorphism of $\tau$ onto $f(\tau)$.
\smallskip
\item[(ii)] Let $\sigma$ be an $n$-cell. Then $f^{-k}(\sigma)$ is equal to the union of all
$(n+k)$-cells $\tau$ with $f^k(\tau)=\sigma$.
\smallskip
\item[(iii)] The $0$-skeleton of $\mathcal{D}^n$ is the set ${\bf V}^n=f^{-n}(\operatorname{post}(f))$, and we have ${\bf V}^n \subset {\bf V}^{n+k}$. The $1$-skeleton of $\mathcal{D}^n$ is equal to $f^{-n}(\mathcal{C})$.
\smallskip
\item[(iv)] We have $\#{\bf V}^n\le m \deg(f)^n$, $\#\E^n=m\deg(f)^n$, and
$\#\X^n=2\deg(f)^n$ for $n\in \N_0$.
\smallskip
\item[(v)] The $n$-edges are precisely the closures of the connected components of $f^{-n}(\mathcal{C})\setminus f^{-n}(\operatorname{post}(f))$. The $n$-tiles are precisely the closures of the connected components of $S^2\setminus f^{-n}(\mathcal{C})$.
\smallskip
\item[(vi)] Every $n$-tile is an $m$-gon, i.e., the number of $n$-edges and $n$-vertices contained in its boundary is equal to $m$.
\end{itemize}
\end{prop}
\begin{proof} (i) This immediately follows from the facts that $f$ is cellular with respect to $(\mathcal{D}^{n+1}, \mathcal{D}^n)$ for each $n$, and that compositions of cellular maps are cellular (if, as in our case, the obvious compatibility requirement for the cell decompositions involved is satisfied).
(ii) It follows from (i) and Lemma~\ref{lem:pullback} that $\mathcal{D}^{n+k}$ is the unique cell decomposition of $S^2$ such that
$f^k$ is cellular with respect to $(\mathcal{D}^{n+k}, \mathcal{D}^n)$.
Moreover, recall from the proof of
Lemma~\ref{lem:pullback} that a topological cell $c\subset S^2$ is an $(n+k)$-cell if and only if $f^k(c)$ is an $n$-cell and $f^k|c$ is a homeomorphism of $c$ onto $f^k(c)$.
This immediately implies the statement if $\sigma=\{q\}$, where $q$ is an $n$-vertex.
Suppose $\sigma$ is equal to an $n$-edge $e$.
Let $M$ be the union of all $(n+k)$-edges $e'$ with $f^k(e')= e$. It is clear that $M\subset f^{-k}(e)$. To see the converse inclusion, let $p\in f^{-k}(e)$ be arbitrary.
If $p\in f^{-k}(\inte(e))$, then from Claim 2 in the proof of Lemma~\ref{lem:pullback} it follows that there exists an $(n+k)$-edge $e'$ with $p\in e'$. Then $f^k(e')$ is an $n$-edge that contains $q=f^k(p)\in \inte(e)$. Hence $e=f^k(e')$, and so
$f^{-k}(\inte(e))\subset M$.
If $p\in f^{-k}(\partial e) $, then $q=f^k(p)\in \partial e$ is an $n$-vertex, and so $p$ is an $(n+k)$-vertex as we have seen. It follows from Case~3 in the proof of
Lemma~\ref{lem:constrmaps} that there exists an $(n+k)$-edge $e'$ that contains $p$ with $f^k(e')= e$. Hence $p\in M$. We conclude that $f^{-k}(e)=f^{-k}(\inte(e))\cup f^{-k}(\partial e)\subset M$, and so
$M=f^{-k}(e)$ as desired.
\smallskip
If $\sigma$ is equal to an $n$-tile $X$, let $M$ be
the union of all $(n+k)$-tiles $X'$ with $f^k(X')=X$.
Then $M\subset f^{-k}(X)$. For the converse inclusion let again $p\in f^{-k}(X)$ be arbitrary.
If $p\in f^{-k}(\inte(X))$, then by Claim 1 in the proof of Lemma~\ref{lem:pullback} there exists an $(n+k)$-tile with
$p\in X'$. Similarly as above we conclude $f^k(X')=X$, and so $p\in M$.
If $p\in f^{-k}(\partial X)$, then
$q=f^k(p)\in \partial X$, and so
there exists an $n$-edge $e\subset \partial X$ with $q\in e$. By what we have seen in the first part of the proof, there exists an $(n+k)$-edge $e'$ with $f^k(e')=e$ and
$p\in e'$. By Lemma~\ref{lem:specprop}~(iv) there exist two $(n+k)$-tiles $X'$ and $Y'$ with $e'\subset \partial X'\cap \partial Y'$. Then $f^k(X')$ and $f^k(Y')$ are $n$-tiles containing $e$ in their boundary. We know that near each point of $\inte(e')$ the map $f^k$ is an orientation-preserving homeomorphism. As in the proof of Lemma~\ref{lem:constrmaps} one can use this fact to show that $f^k(X')$ and $f^k(Y')$ are distinct $n$-tiles. Since there are only two $n$-tiles containing $e$ in their boundary, we conclude that one of the $n$-tiles $f^k(X')$ or $f^k(Y')$ is equal to $X$, say $f^k(X')=X$. Since $p\in e'\subset X'$, it follows that $p\in M$.
Hence $f^{-k}(X)= f^{-k}(\inte(X))\cup f^{-k}(\partial X)\subset M$, and we conclude that $M=f^{-k}(X)$ as desired.
(iii) The $0$-skeleton of $\mathcal{D}^n$ is the set ${\bf V}^n$ of all vertices of $\mathcal{D}^n$. By (ii) we know that ${\bf V}^n=
f^{-n}( {\bf V}^0)=f^{-n}(\operatorname{post}(f))$. Moreover,
$$f^{n+k}({\bf V}^n)\subset f^{n+k}(f^{-n}(\operatorname{post}(f)))\subset f^k(\operatorname{post}(f))\subset \operatorname{post}(f)={\bf V}^0, $$
and so ${\bf V}^n\subset {\bf V}^{n+k}$.
The $1$-skeleton of $\mathcal{D}^n$ is equal to the set consisting of all $n$-vertices and the union of all $n$-edges. As follows from (ii) this set is equal to the preimage of the $1$-skeleton
of $\mathcal{D}^0$ under the map $f^n$. Since the $1$-skeleton
of $\mathcal{D}^0$ is equal to $\mathcal{C}$, it follows that the
$1$-skeleton of $\mathcal{D}^n$ is equal to $f^{-n}(\mathcal{C})$.
(iv) Note that $\deg(f^n)=\deg(f)^n$, and that $\#{\bf V}^0=m$, $\#\E^0=m$, and $\#\X^0=2$. The statements about ${\bf V}^n,
\E^n$, and $\X^n$, then follow from the corresponding statement established in the last part of the proof of Lemma~\ref{lem:pullback}.
(v) This immediately follows from (iii) and Lemma~\ref{lem:opencells}.
(vi) If $X$ is an $n$-tile, then $f^n|X$ is a homeomorphism of
$X$ onto the $0$-tile $f^n(X)$. The $n$-vertices contained
in $X$ are precisely the preimages of the $0$-vertices
contained in $f^n(X)$; hence $X$ contains exactly $m=\#\operatorname{post}(f)$ $n$-vertices, and hence also the same number of $n$-edges (Lemma~\ref{lem:specprop}~(iii)). So every $n$-tile is an $m$-gon.
\end{proof}
Instead of an inequality for $\#{\bf V}^n$ as in (iv) one can easily give a precise
formula for this number; namely, if we set $d=\deg(f)$, and $m=\#\operatorname{post}(f)$,
then $\#\X^n=2d^n$ and $\E^n=md^n$. Moreover, by Euler's polyhedral formula we have
$$ \#\X^n-\#\E^n+\#{\bf V}^n=2, $$
and so
$$ \#{\bf V}^n= (m-2) d^n+2. $$
By property~(v) in the previous proposition we have
$$\operatorname{mesh}(f,n,\mathcal{C})=\max_{X\in \X^n}\diam(X),$$
and so if $\mathcal{C}$ is as in Definition~\ref{def:expanding}, then
$f$ is expanding\index{expanding}
if $$\lim_{n\to \infty} \max_{X\in \X^n}\diam(X)=0.$$
In other words, the Thurston map $f$ is expanding if there exists a Jordan curve $\mathcal{C}\supset \operatorname{post}(f)$ such that the diameter of the $n$-tiles for $f$ and $\mathcal{C}$ go to $0$ uniformly with $n$. This fact was the motivation behind our definition of an expanding Thurston map.
It is often useful, in particular in graphical representations, to assign to each tile
one of the two colors ``black" and ``white" represented by the symbols
${\tt b}$ and ${\tt w}$, respectively. To formulate this, we denote
by $\X^\infty$ the disjoint union of the sets $\X^n$, $n\in \N_0$ (for given $f$ and $\mathcal{C}$). More informally, $\X^\infty$ is the set of all tiles. Note that
in general, a set can be a tile for different levels $n$, so the same tile may be represented by multiple copies in $\X^\infty$ distinguished
by their levels $n$.
\begin{lemma}[Colors of tiles]\label{lem:colortiles}
There exits a map $L\: \X^\infty\rightarrow \{{\tt b}, {\tt w}\} $ with the following properties:
\smallskip
\begin{itemize}
\smallskip
\item[(i)] $L(X^0_{\tt b})={\tt b}$ and $L(X^0_{\tt w})={\tt w}$.
\smallskip
\item[(ii)] If $n,k\in \N_0$, $X^{n+k}\in \X^{n+k}$, and $X^n=f^k(X^{n+k})
\in \X^n$, then $L(X^{n})=L(X^{n+k})$.
\smallskip
\item[(iii)] If $n\in \N_0$, and $X^n$ and $Y^n$ are two distinct $n$-tiles
that have an $n$-edge in common, then $L(X^{n})\ne L(Y^{n})$.
\end{itemize}
Moreover, $L$ is uniquely determined by properties \textnormal{(i)} and
\textnormal{(ii)}.
\end{lemma}
So with the normalization (i) one can uniquely assign colors ``black" or ``white" to the tiles so that all iterates of $f$ are color-preserving as in (ii).
By (iii) colors of distinct $n$-tiles are different if they share an $n$-edge.
Our notion of colorings of tiles is related to the more general concept
of a {\em labeling} of cells in a cell decomposition (see Section~\ref{sec:subdivisions}, in particular Lemma~\ref{lem:labelexis}).
\begin{proof} To define $L$ we assign colors to the two $0$-tiles $X^0_{\tt b}$ and $X^0_{\tt w}$ as in (i). If $Z^n$ is an $n$-tile for some arbitrary level
$n\ge 0$, then $f^n(Z^n)$ is a $0$-tile (Proposition~\ref{prop:celldecomp}~(i)), and so it already has a color assigned. We set $L(Z^n):=L(f^n(Z^n))$.
This defines a map $L\: \X^\infty\rightarrow \{{\tt b}, {\tt w}\}$. By definition, $L$ has
property (i). To show (ii), assume that $n,k\in \N_0$ and
$X^{n+k}\in \X^{n+k}$. Then by Proposition~\ref{prop:celldecomp}~(i),
we have $X^n:=f^k(X^{n+k})\in \X^n$, and $f^{n+k}(X^{n+k}),f^n(X^n)\in \X^0$. So by definition of $L$ we have
$$ L(X^n)=L(f^n(X^n))=L(f^n(f^k(X^{n+k})))=L(f^{n+k}(X^{n+k}))=L(X^{n+k})$$
as desired.
Let $X^n$ and $Y^n$ be as in (iii). Then again by Proposition~\ref{prop:celldecomp}~(i), we have $f^n(X^n), f^n(Y^n)\in \X^0$.
Moreover, by the same argument as in Case~2 of the proof of
Lemma~\ref{lem:constrmaps} the $0$-tiles $f^n(X^n)$ and $f^n(Y^n)$
are distinct. So one of them is equal to $X^0_{\tt b}$, while the other one is equal to $X^0_{\tt w}$. In particular, $L(f^n(X^n))\ne L(f^n(Y^n))$, and so by definition of $L$ we have
$$ L(X^n)=L(f^n(X^n))\ne L(f^n(Y^n))=L(Y^n)$$
as desired. It follows that $L$ has the properties (i)--(iii).
It is clear that $L$ is uniquely determined by (i) and (ii).
\end{proof}
By using the cell decompositions $\mathcal{D}^n$ one can easily classify all Thurs\-ton maps with two postcritical points up to Thurston equivalence.
\begin{prop}\label{prop:post2} Let $f\:S^2\rightarrow S^2$ be a Thurston map with $\#\operatorname{post}(f)=2$. Then $f$ is Thurston equivalent to a map of the form $z\mapsto z^k$ on $\CDach$,
where $k\in \Z\setminus\{-1,0,1\}$.
\end{prop}
\begin{proof} Using some auxiliary conjugations if necessary, we may assume that $S^2=\CDach$ and $\operatorname{post}(f)=\{0,\infty\}$.
We pick $\mathcal{C}=\RDach\supset \operatorname{post}(f)$, and consider the cell decompositions $\mathcal{D}^n(f,\mathcal{C})$ of $\CDach$.
Since $ \operatorname{post}(f)\subset f^{-n}(\operatorname{post}(f))$, the points $0$ and $\infty$ are $n$-vertices for each $n\in \N_0$.
Consider the $1$-tiles $X_1, \dots, X_d$, and the $1$-edges
$e_1, \dots, e_d$ of the cycle of $0$, considered as a $1$-vertex, where $d$ is the length of the cycle.
If the indexing of these tiles and edges is as in Lemma~\ref{lem:specprop}~(v), then
\begin{equation}\label{Xee}
\partial X_j=e_{j-1}\cup e_j
\end{equation} for
$j=1, \dots, d$ (where $e_{0}=e_d$). This is true, because
$\#\operatorname{post}(f)=2$ and so
every $n$-tile is a $2$-gon. It shows that apart from
$0$ the edges $e_1, \dots, e_d$ have one other $1$-vertex
$p\ne 0$ in common.
It follows from Lemma~\ref{lem:specprop} (iv) that the set
$$U= \bigcup_{j=1}^d\inte(X_j) \cup \bigcup_{i=j}^d\inte(e_j)$$ is open, and \eqref{Xee} implies that
$\partial U=\{0,p\}$.
Hence $0$ and $p$ are isolated boundary points of $U$, and so the set $\overline U=U\cup\{0,p\}$ is open and closed.
We conclude that $\overline U=S^2$. Since $\infty$ is a $1$-vertex, and the set $U$ does not contain
any $1$-vertex, we must have $p=\infty$.
Since $f(\operatorname{post}(f))\subset \operatorname{post}(f)$, we have $f(0)=0$ or $f(0)=\infty$.
Assume that $f(0)=0$. Since $f$ is injective on $1$-tiles and $f(\infty)\in \{0,\infty\}$, we have $f(\infty)=\infty$. By an argument similar to (and simpler than) the one in the proof of Lemma~\ref{lem:constrmaps} we will show that one can find a homeomorphism
$\varphi\:\CDach\rightarrow \CDach$ that fixes $0$ and $\infty$
and satisfies $f(\varphi(z))=z^k$ for all $z\in \CDach$, where $k=d/2$. Note that $d$ must be even,
because $f$ maps the $1$-tiles $X_1, \dots, X_d$ alternately to the two $0$-tiles, i.e., the upper and the lower half-planes
in $\CDach$. By reindexing if necessary, we may assume that $X_1, X_3, \dots$ are mapped
to the upper and $X_2, X_4, \dots$ to the lower half-plane.
Define sectors in $\CDach$ by
$$\Sigma_j=\{re^{2\pi {\mathbf{\imath}} t}:(j-1)/d\le t\le j/d,\, r\ge 0\}\cup \{\infty\}$$
for $j=1, \dots, d$.
If $z\in \CDach$ is arbitrary, then $z\in \Sigma_j$ for some $j\in \{1, \dots, d\}$. Then $z^k$ lies in the upper or lower half-plane depending on whether $j$ is even or odd; hence
$(f|X_j)^{-1}(z^k)$ is defined. We put $\varphi(z):=(f|X_j)^{-1}(z^k)$. Then $\varphi$ is a well-defined homeomorphism
on $\CDach$ that fixes $0$ and $\infty$ and satisfies
$f(\varphi(z))=z^k$ for all $z\in \CDach$. This last identity implies that $\varphi$ is orientation-preserving, because the maps $f$ and $z\mapsto z^k$ are a local orientation-preserving homeomorphisms away from their critical points.
If $f(0)=\infty$, then we apply the above argument
to the map $z\mapsto 1/f(z)$. We conclude that in any case
there exists an orientation-preserving homeomorphism
$\varphi$ on $\CDach$ that fixes $0$ and $\infty$ and satisfies
$f(\varphi(z))=z^k$ for all $z\in \CDach$, where $k$ in a
non-zero integer. Actually, $k\not\in \{-1,1\}$, because otherwise $f$ would be a homeomorphism. Since every
orientation-preserving homeomorphism $\varphi$ on $\CDach$ fixing $0$ and $\infty$ is isotopic to $\id_{\CDach}$ rel.\ $\{0,\infty\}$ (see Lemma~\ref{lem:homeo}), it follows that $f$ is Thurston equivalent to the
map $z\mapsto z^k$, where $k\in \Z\setminus\{-1,0,1\}$.
\end{proof}
\begin{cor}\label{cor:no<3} If $f\:S^2\rightarrow S^2$ is an expanding Thurston map, then
$\#\operatorname{post}(f)\ge 3$.
\end{cor}
\begin{proof} We know that $\#\operatorname{post}(f) \ge 2$ (see Remark~\ref{rem:no01}).
The same reasoning as in the proof of Proposition~\ref{prop:post2} shows that if $f\: S^2\rightarrow S^2$ is a Thurston map with $\#\operatorname{post}(f)=2$, then each $n$-tile contains the set $\operatorname{post}(f)$. In particular,
$n$-tiles have a diameter uniformly bounded away from $0$, and so the map cannot be expanding. Therefore, if $f$ is expanding, then $\#\operatorname{post}(f)\ge 3$.
\end{proof}
Due to the last corollary, in the following we can restrict ourselves to the case of Thurston maps $f$
with $\#\operatorname{post}(f)\ge 3$.
\section{Flowers}\label{sec:flowers}
\noindent
In this section $f\:S^2\rightarrow S^2$ is a Thurston map with $\#\operatorname{post}(f)\ge 3$. We fix a Jordan curve $\mathcal{C}\subset S^2$ with
$\operatorname{post}(f)\subset \mathcal{C}$, and consider the cell decompositions $\mathcal{D}^n=\mathcal{D}^n(f,\mathcal{C})$ and use the terminology and notation of the previous section.
\begin{definition}[$n$-Flowers]
\label{def:flower}
Let $n\in \N_0$, and $p\in S^2$ be an $n$-vertex. Then the $n$-\defn{flower}\index{n@$n$-!flower}\index{flower} of $p$ is defined as
\begin{equation*}
W^n(p):= \bigcup\{\inte(c): c\in \mathcal{D}^n, \ p\in c\}.
\end{equation*}
\end{definition}
So the $n$-flower $W^n(p)$ of the $n$-vertex $p$ is the union of the
interiors of all cells in cycle of $p$ in $\mathcal{D}^n$ (see Figure
\ref{fig:cycle} as well as
Lemma~\ref{lem:specprop}~(v) and the discussion after this lemma).
The main reason why we introduced flowers is the following.
Consider a simply
connected domain $U\subset S^2$ not containing a postcritical
point of $f$ and branches $g_n$ of $f^{-n}$ defined on $U$. Then it may happen that the number of $n$-tiles
intersecting $g_n(U)$ is unbounded as $n\to \infty$, even if the diameter of $U$ is
small. For example, this happens when $f$ has a periodic critical point $p$ (see Section~\ref{sec:periodic}), and $U$ spirals around one of the points in the cycle generated by $p$. However
if $\diam (U)$ is sufficiently small, then $g_n(U)$ is always contained in one
$n$-flower as we shall see. We first prove some basic properties of flowers.
\begin{lemma} Let $n\in \N_0$, and $p\in S^2$ be an $n$-vertex. As in Lemma~\ref{lem:specprop} let $e_1, \dots, e_d$ be the $n$-edges and $X_1, \dots, X_d$ be the $n$-tiles of the cycle of $p$, where $d\in \N$, $d\ge 2$, is the length of the cycle.
\smallskip
\label{lem:flowerprop}
\begin{itemize}
\item[(i)] Then $d=2\deg_{f^n}(p)$, the set $W^n(p)$ is an open and simply connected
neighborhood of $p$ that contains no other $n$-vertex, and we have
\begin{multline}\label{eq:flowerrep}
W^n(p)
=\{p\}\cup \bigcup_{i=1}^d\inte(X_i) \cup \bigcup_{i=1}^d\inte(e_i)=\\
S^2\setminus \bigcup\{c\in \mathcal{D}^n : c\in \mathcal{D}^n, \ p\notin c\}.
\end{multline}
\smallskip
\item[(ii)] We have $\overline {W^n(p)}=X_1\cup \dots \cup X_d,$
and the set
$\partial W^n(p)$ is the union of all $n$-edges $e$ with
$p\notin e$ and
$e\subset \partial X_i$ for some $i\in \{1, \dots, d\}$.
\smallskip
\item[(iii)] If $c$ is an arbitrary $n$-cell, then either $p\in c$ and $c\subset \overline{W^n(p)}$, or $c\subset S^2\setminus W^n(p)$.
\end{itemize}
\end{lemma}
\begin{proof} (i) By Remark~\ref{rem:dd'} the length $d$ of the cycle of the vertex $p$ (in the cell decomposition $\mathcal{D}^n$) is a multiple $d=kd'$ of the length $d'$ of the cycle of the image point $q=f^n(p)$ (in the cell decomposition $\mathcal{D}^0$), where $k$ is the degree of $f^n$ at $p$. Since $d'=2$, we have $d=2\deg_{f^n}(p)$ as claimed.
The first equality in \eqref{eq:flowerrep} follows from Lemma~\ref{lem:specprop}~(v) . Based on this, the argument in Case~3 of the proof of Lemma~\ref{lem:constrmaps} shows that the set $W^n(p)$ is homeomorphic to
$\D$. Hence $W^n(p)$ is open and simply connected, and it follows
from the first equality in
\eqref{eq:flowerrep} that $W^n(p)$ contains no other $n$-vertex than $p$.
Let $M=S^2\setminus \bigcup\{c\in \mathcal{D}^n : c\in \mathcal{D}^n, \ p\notin c\}$. If $x\in W^n(p)$, then $x$ is an interior point in one of the cells $\tau$ forming the cycle of $p$. So if $c$ is any $n$-cell with $x\in c$, then $\tau \subset c$
by Lemma~\ref{lem:celldecompint}~(ii). This implies $p\in c$, and so $x\in M$ by definition of $M$. Hence $ W^n(p)\subset M$.
Conversely, if $x\in M$, let $\tau$ be an $n$-cell of smallest dimension that contains $x$. Obviously, $x\in \inte(\tau)$. On the other hand, the definition of $M$ implies that $p\in \tau$. Hence $\tau$ is a cell in the cycle of $p$,
and so $x\in W^n(p)$. We conclude $M\subset W^n(p)$, and so
$M=W^n(p)$ as desired.
(ii) Equation \eqref{eq:flowerrep} implies $\overline {W^n(p)}= X_1\cup \dots \cup X_n$.
Every point $x\in \partial W^n(p)$ is contained in one of the sets $\partial X_i$. Since $W^n(p)$ is open, the point $x$ is not contained in $\{p\}\cup \inte(e_{i-1})\cup\inte(e_i)\subset W^n(p)$ and hence in one of the $n$-edges $e$ in the boundary of $X_i$ distinct from $e_{i-1}$ and $e_i$; note that there exists such an edge, because each $n$-tile is an $m$-gon, where $m=\#\operatorname{post}(f)\ge 3$, and so contains more than two $n$-edges in its boundary. Then $p\notin e$, and $x$ is contained in an $n$-edge with the desired properties.
Conversely, if $e$ is an $n$-edge with $p\notin e$ and $e\subset \partial X_i$, then $e\subset S^2\setminus W^n(p)$ by \eqref{eq:flowerrep}, and $e\subset X_i\subset \overline{W^n(p)}$.
Hence $e\subset \partial W^n(p)$.
(iii) This follows from (i) and \eqref{eq:flowerrep}.
\end{proof}
Note that if we color tiles as in Lemma~\ref{lem:colortiles}, then the colors
of the tiles $X_1, \dots, X_d$ associated to an $n$-flower as in the previous lemma will alternate.
\begin{lemma}\label{lem:mapflowers}
Let $k,n\in \N_0$.
\begin{itemize}
\smallskip
\item[(i)] If $p\in S^2$ is an $(n+k)$-vertex, then $f^k(W^{n+k}(p))=W^n(q)$.
\smallskip
\item[(ii)] If $q\in S^2$ is an $n$-vertex, then the connected components of $f^{-k}(W^n(q))$ are the $(n+k)$-flowers $W^{n+k}(p)$, $p\in f^{-k}(q)$.
\smallskip
\item[(iii)] A connected set $K\subset S^2$ is contained in an $(n+k)$-flower if and only if $f^k(K)$ is contained in an $n$-flower.
\end{itemize}
\end{lemma}
\begin{proof} (i) It is clear that $q=f^k(p)$ is an $n$-vertex.
Let $e'_1, \dots, e'_{d'}$ be the $(n+k)$-edges and
$X'_1, \dots, X'_{d'}$ be the $(n+k)$-tiles in the cycle of $p$, and define $e_i=f^k(e'_i)$ and $X_i=f^k(X'_i)$ for $i=1, \dots, d'$.
Then from Remark~\ref{rem:dd'} it follows that
$e_1, \dots, e_{d'}$ are the $n$-edges and $X_1, \dots, X_{d'}$
are the $n$-tiles in the cycle of $q$ . Here we may have possible repetitions of edges and tiles.
Since the map $f^k$ is cellular for $(\mathcal{D}^{n+k}, \mathcal{D}^n)$,
we have $f^k(\inte(e'_i))=\inte(e_i)$ and $f^k(\inte(X'_i))=\inte(X_i)$ for all $i=1, \dots, d'$.
Using this and \eqref{eq:flowerrep} the statement
follows.
\smallskip
(ii) If $p\in f^{-1}(q)$, then $p$ is an $(n+k)$-vertex. By (i) the $(n+k)$-flower $W^{n+k}(p)$ is an open and connected subset of $f^{-k}(W^n(q))$.
Suppose that $x\in \partial W^{n+k}(p)$. Then by
Lemma~\ref{lem:flowerprop}~(ii) there exists
an $(n+k)$-tile $X'$, and an $(n+k)$-edge $e'$ with
$p\in X'$, $p\notin e'$, and $x\in e'\subset \partial X'$. Then $X=f^k(X')$ is an $n$-tile, $e=f^k(e')$ is an $n$-edge, and
$q\in X$, $f(x)\in e\subset \partial X$. Since
$f^k|X'$ is a homeomorphism of $X'$ onto $X$, we also have
$q\notin X$. Lemma~\ref{lem:flowerprop}~(ii) implies that
$f^k(x)\in \partial W^n(q)$, and so $f^k(x)\notin W^n(q)$, because
flowers are open sets.
We conclude that $x\in S^2\setminus f^{-k}(W^n(q))$, and so
$\partial W^{n+k}(p)\subset S^2\setminus f^{-k}(W^n(q))$. It now follows from
Lemma~\ref{lem:conncomp} that $W^{n+k}(p)$
is a connected component of $f^{-k}(W^n(q))$.
Conversely, suppose that $U$ is a connected component of $f^{-k}(W^n(q))$.
Then $U$ is an open set and so it meets the interior $\inte(X')$
of some
$(n+k)$-tile $X'$. Then $X=f^k(X')$ is an $n$-tile that meets
$W^n(q)$. Hence $q\in X$, and so there exists an $(n+k)$-vertex $p\in X'$ with $f^k(p)=q$.
Then by the first part of the proof, the set $W^{n+k}(p)$ is a connected component of $f^{-k}(W^n(q))$. Since $W^{n+k}(p)$ contains the set $\inte(X')$ and so meets $U$, we must have
$W^{n+k}(p)=U.$
\smallskip
(iii) Suppose $K$ is contained in the $(n+k)$-flower $W^{n+k}(p)$. Then by (i) the set $f^k(W^{n+k}(p))=W^n(f^k(p))$ is an $n$-flower and it contains $f^k(K)$.
Conversely, if $f^k(K)$ is contained in the $n$-flower $W^n(q)$, then $K$ is a connected set in $f^{-k}(W^n(q))$. Hence $K$ lies in a connected component of $f^{-k}(W^n(q))$, and hence in an $(n+k)$-flower by (ii).
\end{proof}
Similarly, as we defined an $n$-flower for an $n$-vertex, one can also define an {\em edge flower}
for an $n$-edge. These sets provide ``canonical" neighborhoods for $n$-vertices and $n$-edges
defined in terms of $n$-cells.
\begin{definition}[Edge flowers]
\label{def:edgeflower}
Let $n\in \N_0$, and $e$ be an $n$-edge. Then the
$n$-\defn{edge flower}\index{n@$n$-!edge flower}\index{edge flower}
of $e$ is defined as
\begin{equation*}
W^n(e):= \bigcup\{\inte(c) : c\in \mathcal{D}^n,\ c\cap e\ne \emptyset\}.
\end{equation*}
\end{definition}
We list some properties of edge flowers. They correspond to similar properties of $n$-flowers as
in Lemma~\ref{lem:flowerprop}. Note that in contrast to an $n$-flower, an edge flower will not be simply connected in general.
\begin{lemma}\label{lem:edgeflower}
Let $e$ be an $n$-edge whose boundary $\partial e$ consists of the $n$-vertices $u$ and $v$.
\begin{itemize}
\smallskip
\item[(i)] Then $W^n(e)$ is an open set containing $e$, and
\begin{equation} \label{eq:eflowerrep}
W^n(e)=W^n(u)\cup W^n(v)=S^2\setminus \bigcup\{c : c\in \mathcal{D}^n,\ c\cap e=\emptyset\}.\end{equation}
\item[(ii)] We have $ \overline{W^n(e)} = \bigcup \{X\in \X^n : X\cap e \neq \emptyset\}$
and
\begin{multline*}
\partial {W^n(e)}= \bigcup \{c \in \mathcal{D}^n : c\cap e=\emptyset \text{ and } \\ \text{
there exists $X\in \X^n$ with $X\cap e \ne \emptyset $ and $c\subset \partial X$}\},
\end{multline*}
where each $n$-cell $c$ in the last union either consists of one $n$-vertex or is an $n$-edge.
\smallskip
\item[(iii)] If $c$ is an arbitrary $n$-cell, then either $c\cap e\ne \emptyset$ and $c\subset \overline{W^n(e)}$, or $c\subset S^2\setminus W^n(e)$.
\end{itemize}
\end{lemma}
\begin{proof} (i) It follows from Lemma~\ref{lem:celldecompint}~(i) that an $n$-cell $c$ meets $e$ if and only if it contains one of the endpoints $u$ and $v$ of $e$. Hence $W^n(e)=W^n(u)\cup W^n(v)$ by the definition of flowers. By Lemma~\ref{lem:flowerprop}~(i) this implies that $W^n(e)$ is open, and, since $e$ is an edge in the cycles of $u$ and $v$, we also have
$$e=\{u\}\cup \inte(e)\cup\{v\}\subset W^n(u)\cup W^n(v)=W^n(e).$$
Let $M= S^2\setminus \bigcup\{c : c\in \mathcal{D}^n,\ c\cap e=\emptyset\}$. If an $n$-cell $c$ does not meet $e$, then it contains neither $u$ nor $v$. Hence
by \eqref{eq:flowerrep} we have
$S^2\setminus M\subset (S^2\setminus W^n(u))\cap
(S^2\setminus W^n(v))=S^2\setminus W^n(e) $,
and so $ W^n(e)\subset M$.
Conversely, let $x\in M$ be arbitrary, and $c$ be the unique $n$-cell $c$ such that $x\in \inte(c)$. Then $c\cap e\ne \emptyset $ and so $u\in c$ or $v\in c$. It follows that $x\in W^n(u)\cup W^n(v)=W^n(e)$.
We conclude that $M\subset W^n(e)$, and so $M=W^n(e)$ as claimed.
\smallskip
(ii) By Lemma~\ref{lem:celldecompint}~(i) an $n$-tile $X$ meets $e$ if and only if $X$ contains $u$ or $v$.
Hence by (i) and Lemma~\ref{lem:flowerbds}~(ii) we have
$$ \overline{W^n(e)}=\overline{W^n(u)}\cup \overline{W^n(v)}=\bigcup \{X\in \X^n : X\cap e \neq \emptyset\}$$ as desired.
For the second claim suppose that
$c$ is an $n$-cell and $X$ an $n$-tile with
$c\cap e=\emptyset$, $X\cap e\ne \emptyset$, and
$c\subset \partial X$. Then $c\subset S^2\setminus W^n(e)$ and $c$ must be an $n$-edge or consist of an $n$-vertex.
Moreover, $c\subset X\subset \overline {W^n(e)}$. It follows that $c\subset \partial W^n(e)$.
Conversely, let $x$ be a point in
$\partial W^n(e)$. Then by (i) the point
$x$ is also a boundary point of $W^n(u)$ or $W^n(v)$, say
$x\in \partial W^n(u)$.
By Lemma~\ref{lem:flowerprop}~(ii) there exists an $n$-edge $e'$ and an $n$-tile $X$ with $x\in e'$,
$u\in X$, $u\notin e'$ and $e'\subset\partial X$.
If $x$ is an $n$-vertex, we let $c=\{x\}$. Then $c$ is an $n$-cell and we have $c\cap e=\emptyset$, because $W^n(e)$ is an open neighborhood of $e$ and $c$ lies in $\partial W^n(e)\subset S^2\setminus W^n(e)$.
Moreover, $X\cap e\ne \emptyset $ and $c\subset e'\subset \partial X$. So $c$ is an $n$-cell with the desired properties containing $x$.
If $x$ is not a vertex we put $c=e'$.
Again if $c\cap e=e'\cap e=\emptyset$, then $c$ is an $n$-cell with the desired properties containing $x$.
The other case, where $e'\cap e\ne \emptyset$, leads to a contradiction. Indeed, then we have $v\in e'$. Moreover, since
$x$ is not a vertex, it follows that $x\in \inte(e')$; but then
$x\in \inte(e')\subset W^n(v)\subset W^n(e)$ which is impossible,
because $x\in \partial W^n(e)\subset S^2\setminus W^n(e)$.
\smallskip
(iii) If $c$ is an $n$-cell and $c\cap e=\emptyset$, then $c\subset S^2\setminus W^n(e)$. If $c\cap e\ne \emptyset$, then
$c$ contains $u$ or $v$, and so $c\subset \overline{W^n(u)}\cup \overline{W^n(v)}=\overline {W^n(e)}$.
\end{proof}
We fix a base metric on $S^2$ that induces the given topology.
We will define a constant $\delta_0>0$
such that any connected set of
diameter $< \delta_0$ (with respect to the base metric) is contained in a single $0$-flower.
There is a slight difference for the cases $\#\operatorname{post}(f)=3$ and $\#\operatorname{post}(f)\ge 4$.
In order to treat these two cases simultaneously, the following definition is useful.
\begin{definition}[Joining opposite sides]
\label{def:connectop}
A set $K\subset S^2$ \defn{joins opposite sides}\index{joining opposite sides|textbf}
of $\mathcal{C}$
if $\#\operatorname{post}(f)\geq 4$ and $K$ meets two disjoint $0$-edges, or
if $\#\operatorname{post}(f)=3$ and $K$ meets all three $0$-edges.
\end{definition}
We then define\index{d0@$\delta_0$}
\begin{multline}\label{defdelta}
\delta_0=\delta_0(f,\mathcal{C})=\inf\{\diam(K): K\subset S^2\text { is a set}\\ \text{joining opposite sides of } \mathcal{C}\}.
\end{multline}
Then $\delta_0>0$. For if $\#\operatorname{post}(f)=4$, then
$\delta_0$ is bounded below by the positive number
$$\min\{\operatorname{dist}(e,e'): e \text{ and } e' \text{ are disjoint }
0\text{-edges}\}. $$
If $\#\operatorname{post}(f)=3$ and we had $\delta_0=0$, then it would follow from a simple limiting argument that the three $0$-edges had a common point. This is absurd.
\begin{lemma}\label{lem:floweropp}
A connected set $K\subset S^2$ joins opposite sides of $\mathcal{C}$ if and only if $K$ is not contained in a single $0$-flower (of a $0$-vertex).
\end{lemma}
\begin{proof} If $K$ is contained in a $0$-flower $W^0(p)$,
where $p\in \mathcal{C}$ is a $0$-vertex, then
$K$ meets at most two $0$-edges, namely the ones that have the common endpoint $p$. So $K$ does not join opposite sides of $\mathcal{C}$.
Conversely, suppose $K$ does not join opposite sides of $\mathcal{C}$. We have to show that $K$ is contained in some $0$-flower. Note that $K$ cannot meet three distinct $0$-edges.
If $K$ does not meet any $0$-edge, then $K$ is contained in every $0$-flower. If $K$ meets only one $0$-edge $e$, then
$K$ is contained in the $0$-flowers $W^0(u)$ and $W^0(v)$,
where $u$ and $v$ are the endpoints of $e$.
If $K$ meets two edges, then these edges share a common endpoint $v\in {\bf V}^0=\operatorname{post}(f)$. This is always true if $\#\operatorname{post}(f)=3$ and follows from the fact that $K$ does not join opposite sides of $\mathcal{C}$ if $\#\operatorname{post}(f)\ge 4$. Moreover, $K$ cannot meet a third $0$-edge which implies that
$K\subset W^n(v)$.
\end{proof}
By the previous lemma every connected set $K\subset S^2$ with $\diam(K)<\delta_0$ is contained in a $0$-flower.
\begin{lemma}
\label{lem:preimsmall}
Let $n\in \N_0$, and $\delta_0>0$ be as in \eqref{defdelta}.
\begin{itemize}
\smallskip
\item[(i)] If $K\subset S^2$ is a connected set with $\diam (K)< \delta_0$, then each component of $f^{-n}(K)$ is contained in some $n$-flower.
\smallskip
\item[(ii)] If $\gamma\colon [0,1]\to S^2$ is a path such
that $\diam (\gamma)< \delta_0$, then each lift $\widetilde{\gamma}$ of
$\gamma$ by $f^n$ has an image that is contained in some $n$-flower.
\end{itemize}
\end{lemma}
Here by definition a {\em lift} of $\gamma$ by $f^n$ is any path $\widetilde{\gamma}\:[0,1]\rightarrow S^2$ with ${\gamma}=f^n\circ \widetilde\gamma$.
\begin{proof} (i) The set $K$ is contained in some $0$-flower
$W^0(p)$, $p\in {\bf V}^0$, by Lemma~\ref{lem:floweropp} and the definition of $\delta_0$. So if $K'$ is a component of $f^{-n}(K)$, then $K'$ is contained in a component of $f^{-n}(W^0(p))$, and hence in an $n$-flower by
Lemma~\ref{lem:mapflowers}~(ii).
(ii) The reasoning is exactly the same as in (i). The image of $\gamma$ is contained in some $0$-flower; by Lemma~\ref{lem:mapflowers}~(ii) this implies that the image of $\widetilde \gamma$ is contained in an $n$-flower.
\end{proof}
We will often have to estimate how many tiles are needed to connect certain
points. If we have a condition that is formulated ``at the top level'', i.e.,
for connecting points in $\mathcal{C}$, then the map $f^n$ can be used to translate this to $n$-tiles.
\begin{lemma}\label{lem:maptotop} Let $n\in \N_0$, and $K\subset S^2$ be a connected set.
If there exist two disjoint $n$-cells $\sigma$ and $\tau$ with $K\cap \sigma\ne \emptyset$ and
$K\cap \tau \ne \emptyset$, then $f^n(K)$ joins
opposite sides of $\mathcal{C}$.
\end{lemma}
\begin{proof} It suffices to show that $K$ is not contained in any $n$-flower, because then $f^n(K)$ is not contained in any $0$-flower (Lemma~\ref{lem:mapflowers}~(iii)) and so $f^n(K)$ joins opposite sides of $\mathcal{C}$ (Lemma~\ref{lem:floweropp}). We consider several cases.
\smallskip
{\em Case~1.} One of the cells is an $n$-vertex,
say $\sigma=\{v\}$, where $v\in {\bf V}^n$. Then $v\in K$, so the only $n$-flower that $K$ could possibly be contained in is $W^n(v)$, because no other $n$-flower contains the $n$-vertex $v$. But since $\sigma$ and $\tau$ are disjoint, we have $v\notin \tau$, and so $\tau \subset S^2\setminus W^n(v)$.
Hence $K\cap (S^2\setminus W^n(v))\ne \emptyset$, and so
$W^n(v)$ does not contain $K$.
\smallskip
{\em Case~2.} Suppose one of the cells is an $n$-edge,
say $\sigma=e\in \E^n$.
Then $e$ has two endpoints $u,v\in {\bf V}^n$. The only $n$-flowers that meet $e$ are $W^n(u)$ and $W^n(v)$; so these $n$-flowers are the only ones that could possibly contain $K$. But the set $W^n(e)=W^n(u)\cup W^n(v)$ does not contain
$K$, because $K$ meets the set $\tau$ which lies in the complement of $W^n(e)$.
\smallskip
{\em Case~3.} One of the cells in an $n$-tile, say $\sigma\in \X^n$. Then $K$ meets $\partial X$. Since $\partial X$ consists of $n$-edges, the set $K$ meets an $n$-edge disjoint from $\tau$. So we are reduced to Case~2.
\end{proof}
For $n\in \N_0$ we denote by $D_n$\index{Dn@$D_n$} the minimal number of $n$-tiles required to form a connected set joining opposite
sides of $\mathcal{C}$; more precisely,
\begin{multline} \label{def:dk}
D_n=\min\big\{N\in \N: \text{there exist } X_1,\dots, X_N\in \X^n \text{ such that}\\ K=\bigcup_{j=1}^N X_j \text{ is connected and joins
opposite sides of }
\mathcal{C} \big\}.
\end{multline}
Of course, $D_n$ depends on $f$ and the choice of $\mathcal{C}$. If we want to emphasize this dependence, we write $D_n=D_n(f,\mathcal{C})$.
From Lemma~\ref{lem:maptotop} we can immediately derive the following consequence.
\begin{lemma} \label{lem:flowerbds}
Let $n,k\in \N_0$. Every set of $(n+k)$-tiles whose union is connected
and meets two disjoint $n$-cells contains at least $D_k$ elements.
\end{lemma}
\begin{proof} Suppose $K$ is a union of $(n+k)$-tiles with the stated properties. Then the images of these tiles under $f^n$ are $k$-tiles and $f^n(K)$ joins opposite sides of $\mathcal{C}$ by Lemma~\ref{lem:maptotop}.
Hence there exist at least $D_k$ distinct $k$-tiles in the union forming $f^n(K)$ and hence at least $D_k$ distinct $(n+k)$-tiles in $K$.
\end{proof}
\begin{lemma} \label{lem:difflevel} There exists $M\in \N$ with the following property:
\begin{itemize}
\smallskip
\item[(i)] Each $n$-tile, $n\in \N$, can be covered by $M$ $(n-1)$-flowers.
\smallskip
\item[(ii)] Each $n$-tile, $n\in \N_0$, can be covered by $M$ $(n+1)$-flowers.
\end{itemize}
\end{lemma}
For easier formulation of this lemma and the subsequent proof, we assume for simplicity that a cover by {\em at most} $M$ element contains precisely $M$ elements. This can always be achieved by repetition of elements in the cover.
\begin{proof} (i) Let $\delta_0>0$ be as in \eqref{defdelta}. Then there exists $M\in \N$ such that each of the finitely many $1$-tiles $X$ is a union of $M$ connected sets $U\subset X$ with $\diam(U)<\delta_0$. If $Y$ is an arbitrary $n$-tile, $n\ge 1$, then $Z=f^{n-1}(Y)$ is a $1$-tile and $f^{n-1}|Y$ a homeomorphism of $Y$ onto $Z$.
Hence $Y$ is a union of $M$ sets of the form $(f^{n-1}|Y)^{-1}(U)$, where $U\subset Z$ is connected and $\diam(U)<\delta_0$.
Each set $(f^{n-1}|Y)^{-1}(U)$ is connected and so by Lemma~\ref{lem:preimsmall}~(i) it lies in an $(n-1)$-flower. Hence $Y$ can be covered by $M$ $(n-1)$-flowers.
(ii)
There exists $M\in \N$ such that each of the two $0$-tiles $X$ can be covered
by $M$ connected sets $U\subset X$
with $\diam (f(U))<\delta_0$. If $Y$ in an arbitrary $n$-tile, then $Z=f^n(Y)$ is a $0$-tile. By the same reasoning as above,
the set $Y$ is a union of $M$ sets of the form $(f^{n}|Y)^{-1}(U)$, where $U\subset Z$ is connected and $\diam(f(U))<\delta_0$.
Then $U'=(f^n|Y)^{-1}(U)$ is connected, and $f^{n+1}(U')=f(U)$ which implies $\diam(f^{n+1}(U'))<\delta_0$.
Hence by Lemma~\ref{lem:preimsmall}~(i) the set $U'$ is contained in some $(n+1)$-flower.
Since $M$ of the sets $U'$ cover $Y$, it follows that each $n$-tile can be covered by $M$ $(n+1)$-flowers.
\end{proof}
\begin{lemma}\label{lem:tileflower} Let $\mathcal{C}$ and $\widetilde
\mathcal{C}$ be two Jordan curves in $S^2$ that both contain
$\operatorname{post}(f)$. Then there exists a number $M$ such that each $n$-tile for $(f,\widetilde \mathcal{C})$ is covered by $M$ $n$-flowers for $(f,\mathcal{C})$.
\end{lemma}
\begin{proof} The proof is very similar to the proof of Lemma~\ref{lem:difflevel}.
Let $\delta_0=\delta_0(f,\mathcal{C})>0$ be the number as defined in \eqref{defdelta}. There exists a number $M$ such that each of the two $0$-tiles $X$ for $(f,\widetilde \mathcal{C})$ is a union of $M$ connected sets $U\subset X$ with $\diam(U)<\delta_0$. If $Y$ is an arbitrary $n$-tile for $(f,\widetilde \mathcal{C})$, then $Z=f^{n}(Y)$ is a $0$-tile for $(f,\widetilde \mathcal{C})$ and $f^{n}|Y$ is a homeomorphism of $Y$ onto $Z$.
Hence $Y$ is a union of $M$ sets of the form $(f^{n}|Y)^{-1}(U)$, where $U\subset Z$ is connected and $\diam(U)<\delta_0$.
Each set $(f^{n}|Y)^{-1}(U)$ is connected and so by Lemma~\ref{lem:preimsmall}~(i) it lies in an $n$-flower for $(f,\mathcal{C})$. Hence $Y$ can be covered by $M$ such $n$-flowers.
\end{proof}
%
%
\section{Expansion and visual metrics}
\label{sec:expansion}
\noindent Let $f\:S^2\rightarrow S^2$ be a Thurston map. Throughout this section we will assume that $f$ is expanding.
We will show that $S^2$ then carries a natural class of metrics that allows us to estimate the distance of points in terms of combinatorial data derived from the tiles in the cell decompositions defined in Section~\ref{sec:tiles}.
We fix a base metric on $S^2$ that induces the standard topology. The purpose of this is to be able to formulate some essentially topological properties (such as expansion of the
map $f$) in more convenient metric terms. Notation for metric terms will refer to this base metric unless otherwise indicated.
As we have seen in Section~\ref{sec:tiles}, the property of expansion
can equivalently be stated as
\begin{equation}
\label{eq:defexpXn}
\max_{X\in \X^n}\diam (X)\to 0 \text{ as } n\to \infty,
\end{equation}
where the tiles are defined with respect to a Jordan curve $\mathcal{C}\supset \operatorname{post}(f) $ as in Definition
\ref{def:exp}.
First we want to convince ourselves that
this is independent of the choice of the curve $\mathcal{C}$.
\begin{lemma}
\label{lem:exp_ind_C} Let $f\:S^2 \rightarrow S^2$ be an expanding\index{expanding}
Thurston map. Then
\begin{equation}\label{meshtozero}\lim_{n\to \infty} \operatorname{mesh}(f,n,\widetilde \mathcal{C})=0\end{equation}
for every Jordan curve $\widetilde \mathcal{C}\subset S^2$ with $\operatorname{post}(f)\subset\widetilde \mathcal{C}$.
\end{lemma}
\begin{proof} Since $f$ is expanding, there exists a Jordan
curve $\mathcal{C}\subset S^2$ with $\operatorname{post}(f)\subset \mathcal{C}$ such that \eqref{meshtozero} holds. Then
$$ \max_{X\in \X^n} \diam(X)=\operatorname{mesh}(f,n,\mathcal{C}) \to 0$$
as $n\to \infty$, where $\X^n$ is the set of $n$-tiles for $(f,\mathcal{C})$. Lemma~\ref{lem:flowerprop}~(ii) implies that
\begin{equation}\label{diamflower}
\diam(W^n(v))\le 2 \max_{X\in \X^n} \diam(X)
\end{equation} for each $n$-flower for $(f,\mathcal{C})$.
Let
$\widetilde \mathcal{C}\subset S^2$ be another Jordan curve with
$\operatorname{post}(f)\subset \widetilde \mathcal{C}$, and the number $M$ be as in Lemma~\ref{lem:tileflower}. Then each $n$-tile for $(f, \widetilde \mathcal{C})$ can be covered by $M$ $n$-flowers for $(f,\mathcal{C})$.
If a connected set is covered by a finite union of connected sets, then its diameter is bounded by the sum of the diameters of the sets in the union.
Combining this with \eqref{diamflower} and denoting the set of $n$-vertices for $(f,\mathcal{C})$ by ${\bf V}^n$, we conclude
that
\begin{eqnarray*} \operatorname{mesh}(f,n,\widetilde \mathcal{C})&=&
\max \{\diam(\widetilde X): \widetilde X \text{ is an $n$-tile for $(f, \widetilde \mathcal{C})$}\} \\ &\le& M \max_{p\in {\bf V}^n} \diam(W^n(p)) \\
&\le &
2M\max_{X\in \X^n} \diam(X)\\
&=& 2M\operatorname{mesh}(f,n,\mathcal{C}).
\end{eqnarray*}
Hence
$\operatorname{mesh}(f,n,\widetilde \mathcal{C})\to 0$ as $n\to \infty$ as desired.\end{proof}
Our definition of expansion is somewhat {\em ad hoc}, but it has the advantage that it relates to the geometry of tiles. An equivalent, and maybe more conceptual description
can be given in terms of the behavior of open covers of $S^2$ under pull-backs by the iterates of the map.
We start with some definitions. Let $\mathcal{U}$ be an open cover
of $S^2$. We define $\operatorname{mesh}(\mathcal{U})$ to be the supremum of all diameters of connected components of sets in $\mathcal{U}$. If $g\: S^2\rightarrow S^2$ is a continuous map, then the
{\em pull-back of $\mathcal{U}$ by $g$} is defined as
$$ g^{-1}(\mathcal{U})=\{V : V \text{ connected component of }
g^{-1}(U), \text{ where } U \in \mathcal{U}\}.$$
Obviously, $g^{-1}(\mathcal{U})$ is also an open cover of
$S^2$. Similarly we denote by $g^{-n}(\mathcal{U})$ the pull-back of
$\mathcal{U}$ by $g^n$.
\begin{prop}\label{prop:expequivexp}\index{expanding} Let $f\: S^2\rightarrow S^2$ be a Thurston map.
Then the following conditions are equivalent:
\begin{itemize}
\item[(i)] The map $f$ is expanding.
\smallskip
\item[(ii)] There exists $\delta_0>0$ with the following property: if $\mathcal{U}$ is a cover of $S^2$ by open and connected sets that satisfies $\operatorname{mesh}(\mathcal {U})<\delta_0$, then
$$\lim_{n\to\infty}
\operatorname{mesh} (f^{-n}(\mathcal {U}))=0.$$
\smallskip
\item[(iii)] There exists an open cover $\mathcal{U}$ of $S^2$ with
$$\lim_{n\to\infty}
\operatorname{mesh} (f^{-n}(\mathcal {U}))=0. $$
\smallskip
\item[(iv)] There exists an open cover $\mathcal{U}$ of $S^2$ with
the following property: for every open cover $\mathcal{V}$ of $S^2$
there exists $N\in N$ such that $f^{-n}(\mathcal{U})$ is finer than
$\mathcal{V}$ for every $n\in \N$ with $n>N$; i.e.,
for every set $U'\in f^{-n}(\mathcal {U})$ there exists a
set $V\in \mathcal{V}$ such that $U'\subset V$.
\end{itemize}
\end{prop}
Conditions (iii) is the notion of expansion as defined by
Ha\"\i ssinsky-Pilgrim (see \cite[Sect.~2.2]{HP}). Thus our notion of
expansion agrees with the one in \cite{HP}.
Condition (iv) is essentially a reformulation of (iii)
in purely topological terms without reference to the base metric on $S^2$ (which enters in the definition of the mesh of an open cover). One can reformulate (ii) in a similar spirit. If there exists a Jordan curve $\mathcal{C}\subset S^2$ with $\operatorname{post}(f)\subset \mathcal{C}$ and $f(\mathcal{C})\subset \mathcal{C}$, then expansion of the map $f$ can be characterized in yet another way (see Lemma~\ref{lem:charexpint}).
\begin{proof} We will show $\text{(i)}\Rightarrow \text{(ii)}\Rightarrow \text{(iii)}\Rightarrow \text{(i)}$ and $\text{(iii)}\Rightarrow \text{(iv)}
\Rightarrow \text{(iii)}$.
\smallskip \noindent
$\text{(i)}\Rightarrow \text{(ii)}$: Suppose $f$ is expanding. Pick a Jordan curve $\mathcal{C}\subset S^2$ with $\operatorname{post}(f)\subset \mathcal{C}$ and let $\delta_0>0$ be as in \eqref{defdelta} (note that $\#\operatorname{post}(f)\ge 3$ by Corollary~\ref{cor:no<3}). Suppose $\mathcal{U}$ is a cover of $S^2$ by open and connected sets that satisfies $\operatorname{mesh}(\mathcal {U})<\delta_0$. If $U\in \mathcal {U}$,
then $U$ is connected and $\diam(U)<\delta_0$. So if $V$ is an arbitrary connected component of $f^{-n}(U)$, then by Lemma~\ref{lem:preimsmall} the set $V$ is contained in an $n$-flower for $(f,\mathcal{C})$.
Hence
$$ \diam(V)\le 2 \operatorname{mesh}(f,n,\mathcal{C}), $$ which implies
$$\operatorname{mesh}(f^{-n}(\mathcal {U}) )\le2 \operatorname{mesh}(f,n,\mathcal{C}). $$ Since $f$ is an expanding Thurston map, we have $\operatorname{mesh}(f,n,\mathcal{C} )\to 0$, and so
$\operatorname{mesh}(f^{-n}(\mathcal {U})) \to 0$ as $n\to \infty$.
\smallskip \noindent
$\text{(ii)}\Rightarrow \text{(iii)}$: Obvious.
\smallskip \noindent
$\text{(iii)}\Rightarrow \text{(i)}$: Suppose $\mathcal {U}$ is an open cover of $S^2$ as in (iii).
Pick a Jordan curve $\mathcal{C}\subset S^2$ with $\operatorname{post}(f)\subset \mathcal{C}$, and let $\delta>0$ be a {\em Lebesgue number} for the cover
$\mathcal {U}$, i.e., every set $K\subset S^2$ with $\diam(K)<\delta$ is contained in
a set $U\in \mathcal {U}$. We can find a number $M\in \N$ such that each of the two $0$-tiles for $(f,\mathcal{C})$ can be written as a union of $M$ connected sets $V$ with $\diam(V)<\delta$. Then each such set $V$ is contained in a set $U\in \mathcal {U}$.
Now if $X$ is an arbitrary $n$-tile for $(f, \mathcal{C})$, then $Y=f^{n}(X)$ is a $0$-tile for $(f,\mathcal{C})$ and $f^{n}|X$ is a homeomorphism of $X$ onto $Y$.
Hence $X$ is a union of $M$ connected sets of the form $(f^{n}|Y)^{-1}(V)$, where $V\subset Y$ is connected and lies in a set $U\in \mathcal{U}$. Then
$(f^{n}|X)^{-1}(V)$ lies in a component of $f^{-n}(U)$, and so
$$\diam ((f^{n}|X)^{-1}(V))\le \operatorname{mesh}(f^{-n}(\mathcal {U}) ). $$
This implies
$$\diam (X)\le M \operatorname{mesh}(f^{-n}(\mathcal {U}) ).$$
Hence $$\operatorname{mesh}(f,n,\mathcal{C})\le M \operatorname{mesh}(f^{-n}(\mathcal {U})). $$ Since $
\operatorname{mesh}(f^{-n}(\mathcal {U}))\to 0$, we also have $\operatorname{mesh}(f,n,\mathcal{C})\to 0$
as $n\to \infty$.
Hence $f$ is expanding.
\smallskip \noindent
$\text{(iii)}\Rightarrow \text{(iv)}$: Suppose $\mathcal {U}$ is an open cover of $S^2$ as in (iii), and $\mathcal{V}$ is an arbitrary open cover of $S^2$. Let $\delta>0$ be a Lebesgue number for the cover
$\mathcal {V}$, i.e., every set $K\subset S^2$ with $\diam(K)<\delta$ is contained in
a set $V\in \mathcal {V}$. By (iii) we can find $N\in \N$ such that
$\operatorname{mesh}(f^{-n}(\mathcal {U}))<\delta$ for $n>N$. If $n>N$ and $U'$ is a set in
$f^{-n}(\mathcal {U})$, then $\diam(U')<\delta$ by definition of
$\operatorname{mesh}(f^{-n}(\mathcal {U}))$. Hence there exists $V\in \mathcal{V}$
such that $U'\subset V$.
\smallskip \noindent
$\text{(iv)}\Rightarrow \text{(iii)}$: Suppose $\mathcal{U}$
is an open cover of $S^2$ as in (iv). Then $\mathcal{U}$ also satisfies condition (iii); indeed, let $\epsilon>0$ be arbitrary, and
let $\mathcal{V}$ be the open cover of $S^2$ consisting of all open balls of radius $\epsilon/2$. Then $\diam(V)\le \epsilon$ for all $V\in \mathcal{V}$. Moreover, by (iv) there exists $N\in \N$ such that for $n>N$ every set in
$f^{-n}(\mathcal {U})$ is contained in a set in $\mathcal{V}$. In particular,
$\operatorname{mesh}(f^{-n}(\mathcal {U})) \le \epsilon $ for $n>N$.
This shows that $\mathcal{U}$ satisfies condition (iii).
\end{proof}
Let $\mathcal{C}\subset S^2$ be a Jordan curve with $\operatorname{post}(f)\subset \mathcal{C}$. We define $\widetilde D_n=\widetilde D_n(f,\mathcal{C})$ as the minimal number of tiles of order $l\geq n$ for $(f,\mathcal{C})$ required to
join opposite
sides of $\mathcal{C}$, i.e., the smallest number $N\in \N$ for which there are
tiles $X_i\in\bigcup_{l\geq n} \X^l$, $i=1,\dots, N$, such that
$K=\bigcup_{i=1}^N X_i$ is connected and
joins opposite sides of $\mathcal{C}$.
Note that the quantity $\widetilde D_n$
is a variant of the quantity $D_n$ defined in
\eqref{def:dk}. While the sets $K$ used to define $D_n$
are unions of tiles of order $n$, the sets $K$ in the definition of $\widetilde D_n$ are unions of tiles of order $k\ge n$; in particular, $D_k\ge \widetilde D_n$ for $k\ge n$.
\begin{lemma} \label{lem:Dtoinfty} Let $f\: S^2\rightarrow S^2$ be an expanding Thurston map, and $\mathcal{C}\subset S^2$ be a Jordan curve with $\operatorname{post}(f)\subset \mathcal{C}$. Let $D_n=D_n(f,\mathcal{C})$ and $\widetilde D_n=\widetilde D_n(f,\mathcal{C})$ for $n\in \N_0$.
Then $D_n\to \infty$ and $\widetilde D_n\to \infty$ as $n\to \infty$.
\end{lemma}
\begin{proof}
We know that $D_k\ge \widetilde D_n$ whenever $k\ge n$. So it suffices to show $\widetilde D_n\to \infty$ as $n\to \infty$.
Let $\delta_0>0$ be defined as in \eqref{defdelta} and
suppose $K=X_1\cup \dots \cup X_N$ is a connected union of tiles of order $\ge n$ that joins opposite sides of $\mathcal{C}$. Then
\begin{eqnarray*}
\delta_0 &\le& \diam (K)\,\le \, \sum_{i=1}^N\diam(X_i)\\
&\le& N \max_{i=1, \dots, N} \diam (X_i) \\
&\le & N \sup_{k\ge n} \operatorname{mesh}(f,k,\mathcal{C}).
\end{eqnarray*}
Putting $c_n:=\sup_{k\ge n} \operatorname{mesh}(f,k,\mathcal{C})$, we conclude that $N\ge \delta_0/c_n$, and so $\widetilde D_n\ge \delta_0/c_n$.
Since $f$ is expanding we have $\operatorname{mesh}(f,n,\mathcal{C})\to 0$ and so also $c_n\to 0$ as $n\to \infty$. This implies that $\widetilde D_n
\to \infty$ as desired.
\end{proof}
If $f$ is expanding and $\mathcal{C}$ is given, then in view of the last lemma, we can find a number $k_0=k_0(f,\mathcal{C})\in \N$ such that
\begin{equation}\label{def:k0}
\widetilde D_{k_0}=\widetilde D_{k_0}(f,\mathcal{C})\ge 10.
\end{equation}
This inequality will be useful in the following.
\begin{lemma}
\label{lem:Thiterates} Let $f\: S^2\rightarrow S^2$ be a Thurston map, $n\in \N$, and $F=f^n$. Then $F$ is a Thurston map with $\operatorname{post}(F)=\operatorname{post}(f)$.
The map $f$ is expanding if and only if $F$ is expanding.
\end{lemma}
\begin{proof} Since $f$ is a Thurston map, the map $F$ is a branched covering map on $S^2$
with $\operatorname{post}(F)=\operatorname{post}(f)$ (see Section~\ref{sec:thmaps}) and $\deg(F)=\deg(f)^n\ge 2$. Hence $F$ is also a Thurston map.
Fix a Jordan curve $\mathcal{C}\subset S^2$ with $\operatorname{post}(f)=\operatorname{post}(F)\subset \mathcal{C}$.
It follows from the definitions that
$$\operatorname{mesh}(F,k,\mathcal{C})=\operatorname{mesh}(f,nk,\mathcal{C}) $$ for all $k\in \N_0$.
If $f$ is expanding, then by Lemma~\ref{lem:exp_ind_C} we have
$\operatorname{mesh}(f,k,\mathcal{C}) \to 0$ as $k\to \infty$ which implies that
$$\operatorname{mesh}(F,k,\mathcal{C})=\operatorname{mesh}(f,nk,\mathcal{C})\to 0$$ as $k\to \infty$. Hence $F$ is expanding.
Conversely, suppose that $F$ is expanding. Then we know that
\begin{equation} \label{Fexp}
\lim_{k\to \infty} \operatorname{mesh}(F,k,\mathcal{C})= \lim_{k\to \infty} \operatorname{mesh}(f,nk,\mathcal{C})=0.
\end{equation}
Let the constant $M\ge 1$ be as in
Lemma~\ref{lem:difflevel} for the map $f$ and the Jordan curve $\mathcal{C}$. By an argument similar as in the proof of
Lemma~\ref{lem:exp_ind_C} one can show that
$$\operatorname{mesh}(f, l+1, \mathcal{C})\le 2M\operatorname{mesh}(f, l, \mathcal{C}) $$
for all $l\in \N_0$.
This implies
$$ \operatorname{mesh}(f, l, \mathcal{C}) \le (2M)^n \operatorname{mesh}(f, n \lfloor l/ n\rfloor, \mathcal{C})$$ for all $l\in \N_0$ and so by \eqref{Fexp} we have
$ \operatorname{mesh}(f,l, \mathcal{C})\to 0$ as $l\to \infty$. This shows that $f$ is expanding.
\end{proof}
\begin{definition}\label{def:mxy} Let $f\: S^2\rightarrow S^2$ be an expanding Thurston map, and $\mathcal{C}\subset S^2$ be a Jordan curve with $\operatorname{post}(f)\subset \mathcal{C}$, and $x,y\in S^2$.
For $x\ne y$ we define\index{m@$m_{f,\mathcal{C}}$}
\begin{multline*}
m_{f,\mathcal{C}}(x,y):=\max\{n\in \N_0: \text{there exist non-disjoint $n$-tiles } \\ \text{ $X$ and $Y$ for $(f,\mathcal{C})$ with $x\in X$, $y \in Y$}\}.
\end{multline*}
If $x=y$ we define $m_{f,\mathcal{C}}(x,x):=\infty$.
\end{definition}
Note that a maximal number $n$ as in the definition of $m_{f,\mathcal{C}}(x,y)$ for $x\ne y$ exists, because we know that for an expanding Thurston map the diameters of $n$-tiles tend to $0$ if $m\to \infty$.
We usually drop one or both subscripts in
$m_{f,\mathcal{C}}(x,y)$ if $f$ or $\mathcal{C}$ are clear from the context. A similar combinatorial quantity that is essentially equivalent to $m_{f,\mathcal{C}}(x,y)$ for $x\ne y$ is
\begin{multline*}
m'_{f,\mathcal{C}}(x,y):=\min\{n\in \N_0: \text{there exist disjoint $n$-tiles } \\ \text{ $X$ and $Y$ for $(f,\mathcal{C})$ with $x\in X$, $y \in Y$}\}
\end{multline*}
(see Lemma~\ref{lem:mprops} (v)).
In the next lemma we collect some of the properties of the function $m_{f,\mathcal{C}}$. For the proof the following terminology is useful. A {\em (finite) chain} in $S^2$
is a finite sequence $A_1,\dots, A_N$ of sets in $S^2$ such that $A_i\cap A_{i+1}\ne \emptyset$ for $i=1, \dots, N-1$. It {\em joins} two points $x,y\in S^2$ if $x\in A_1$ and $y\in A_N$. We say that this chain is {\em simple}
if there is no proper subsequence of $A_1, \dots, A_N$ that is also
chain joining $x$ and $y$.
If $K$ is a compact connected set in $S^2$, $\mathcal{U}$ an open cover of $K$, and $x,y\in K$, then one can always find a simple chain of sets in $\mathcal{U}$ joining $x$ and $y$.
\begin{lemma}\label{lem:mprops} Let $f\: S^2\rightarrow S^2$ be an expanding Thurston map, $\mathcal{C}\subset S^2$ be a Jordan curve with
$\operatorname{post} (f)\subset \mathcal{C}$, and $m=m_{f,\mathcal{C}}$. \smallskip
\begin{itemize}
\item[(i)] There exists a number $k_1>0$ such that
\begin{equation}\label{Gromovineq}
\min\{m(x,z), m(y,z)\}\le m(x,y)+k_1
\end{equation}
for all
$x,y,z\in S^2.$
\smallskip
\item[(ii)] We have
$$m(f(x),f(y))\geq m(x,y)-1$$ for all $x,y\in S^2$.
\smallskip
\item[(iii)] Let $\widetilde \mathcal{C}\subset S^2$ be another Jordan curve with $\operatorname{post}(f)\subset \widetilde \mathcal{C}$.
Then there exists a constant $k_2>0$ such that
\begin{equation}\label{Cind}
m(x,y)-k_2\leq m_{f,\widetilde\mathcal{C}}(x,y)\leq m(x,y)+k_2
\end{equation}
for all $x,y\in S^2$.
\smallskip
\item[(iv)]
Let $F=f^n$ be an iterate of $f$. Then there exists a constant $k_3>0$ such that \begin{equation*}
m(x,y)-k_3 \leq n\cdot m_{F,\mathcal{C}}(x,y) \leq
m(x,y),
\end{equation*}
for all $x,y\in S^2$.
\smallskip
\item[(v)] There exists a constant $k_4>0$ with the following property: if $x,y\in S^2$, $x\ne y$,
and $m'_{f,\mathcal{C}}(x,y)$ is the smallest number $ m'\in \N_0$ for which there
exist $m'$-tiles $X',Y'\in \mathcal{D}^{m'}(f, \mathcal{C})$ with $x\in X'$, $y\in Y'$ and
$X'\cap Y'=\emptyset$, then \begin{equation*}
m(x,y)-k_4 \leq m'_{f,\mathcal{C}}(x,y) \leq m(x,y)+1.
\end{equation*}
\end{itemize}
\end{lemma}
\begin{proof} We fix $k_0=k_0(f,\mathcal{C})$ as in \eqref{def:k0}.
Let $x,y\in S^2$ be arbitrary. In order to establish the desired inequalities we may always assume $x\ne y$. Unless otherwise stated, tiles will be for $(f,\mathcal{C})$.
(i) Let $m=m(x,y)\in
\N_0$ be as in Definition~\ref{def:mxy}. We can pick
$(m+1)$-tiles $X_0$
and $Y_0$ containing $x$ and $y$, respectively. Then
$ X_0\cap Y_0=\emptyset$ by definition of $m$.
Define $n:= m+k_0$, and let $z\in S^2$ be arbitrary. We claim that
$m(x,z)\le n \text{ or } m(y,z)\le n$.
Otherwise $m(x,z)\ge n+1$ and $m(y,z)\ge n+1$, and so
by Definition~\ref{def:mxy} there exist numbers $m_1,m_2\ge n+1$ and
$m_1$-tiles $X$ and $Z$ with $x\in X$, $z\in Z$ and $X\cap Z\ne \emptyset$, and $m_2$-tiles $Y$ and $Z'$ with $y\in Y$, $z\in Z'$ and $X\cap Z'\ne \emptyset$.
Then the set $K=X\cup Z\cup Z'\cup Y$ is connected and meets the disjoint
$(m+1)$-tiles $X_0$ and $Y_0$. Thus $f^{m+1}(K)$
joins opposite sides of $\mathcal{C}$ by Lemma~\ref{lem:maptotop}, and consists of four tiles of order
$ \ge n-m= k_0$. This contradicts \eqref{def:k0}, proving the claim.
So we have $m(x,z)\leq m+k_0$ or $m(y,z)\leq m+k_0$. This implies \eqref{Gromovineq} with the constant $k_1=k_0$ which is independent of $x$ and $y$.
\smallskip (ii) We may assume that
$m=m(x,y)\ge 1$. Then there are non-disjoint
$m$-tiles $X$ and $Y$ with $x\in X$ and $y\in Y$.
It follows that
$f(X)$ and $f(Y)$ are non-disjoint $(m-1)$-tiles with $f(x)\in f(X)$ and $f(y)\in f(Y)$. Hence $m(f(x),f(y))\geq m-1$ as desired.
\smallskip
(iii) Let $\widetilde m=m_{f,\widetilde C}(x,y)\in \N_0$. Then there exist $\widetilde m$-tiles $\widetilde X$ and $\widetilde Y$ for
$(f,\widetilde \mathcal{C})$ with $x\in \widetilde X$, $y\in \widetilde Y$, and
$\widetilde X\cap \widetilde Y\ne \emptyset$.
By Lemma~\ref{lem:tileflower} the sets $\widetilde X$ and $\widetilde Y$ are each contained in $M$ $\widetilde m$-flowers for $(f,\mathcal{C})$, where $M$ is independent
of $ \widetilde X$ and $\widetilde Y$. In particular, this implies that we can find a chain of at most $M$ such $\widetilde m$-flowers joining $x$ and $y$. Since any two tiles in the closure $\overline {W^n(v)}$ of an $n$-flower have the point $v$ in common, it follows that there exists a chain $X_1, \dots, X_N$ of $\widetilde m$-tiles for $(f,\mathcal{C})$ joining $x$ and $y$ with $N\le 2M$. Let $x_1:=x$, $x_N:=y$, and for $i=2, \dots, N-1$, pick a point $x_i\in X_i$.
Then $m(x_i, x_{i+1}) \ge \widetilde m$ for $i=1, \dots, N-1$. Hence by repeated application of (i) we obtain
\begin{eqnarray*}
\widetilde m &\le &\min\{m(x_i, x_{i+1}): i=1, \dots, N-1\}\\
&\le&
m(x_1,x_N)+Nk_1\,\le \, m(x,y)+2Mk_1.
\end{eqnarray*}
Since $2Mk_1$ is independent of $x$ and $y$, we get an upper bound as in \eqref{Cind}. A lower bound
is obtained by the same argument if we reverse the roles of $\mathcal{C}$ and $\widetilde \mathcal{C}$.
\smallskip
(iv) The map $F$ is also an expanding Thurston map, and we have
$\operatorname{post}(f)=\operatorname{post}(F)$ (see Lemma~\ref{lem:Thiterates}); so the Jordan curve $\mathcal{C}$ contains the set of postcritical points of $F$ and
$m_{F,\mathcal{C}}$ is defined. It follows from Proposition~\ref{prop:celldecomp}~(v) that the $m$-tiles for $(F,\mathcal{C})$ are precisely the $(nm)$-tiles for $(f,\mathcal{C})$,
In the ensuing proof we will only
consider tiles for $(f, \mathcal{C})$.
Let $m_F=m_{F,\mathcal{C}}(x,y)$ and $m=m(x,y)$; then there are non-disjoint $(nm_F)$-tiles $X$ and $Y$ with $x\in X$ and $y\in Y.$ So $m\ge nm_F$ which gives the desired upper bound.
We claim that on the other hand, we have
$m\le nm_F+k_3$, where $k_3=n+k_0-1$.
To see this assume that $$m\ge nm_F+k_3+1=n(m_F+1)+k_0.$$
Then we can find non-disjoint $m$-tiles $X$ and $Y$ with $x\in X$, $y\in Y$. Moreover, we can pick $n(m_F+1)$-tiles $X'$ and $Y'$ with $x\in X'$ and $y\in Y'$. By definition of $m_F$ we know that $X'\cap Y'=\emptyset$, so
$X'$ and $Y'$ are disjoint $n(m_F+1)$-tiles joined by the connected set $K=X\cup Y$. Hence by Lemma~\ref{lem:flowerbds} $K$ must consists of at least $$D_{m-n(m_F+1)}\ge \widetilde D_{k_0}\ge 10$$
$m$-tiles; but $K$ consists of only two such $m$-tiles. This is a contradiction showing the desired claim.
\smallskip (v) Let $m'=m'_{f,\mathcal{C}}(x,y)$ be defined as in (v). Then $m'\ge 1$, because the two $0$-tiles have nonempty intersection. So $m'-1\ge 0$, and there exist $(m'-1)$-tiles $X$ and $Y$ with $x\in X$ and $y\in Y$. Then $X\cap Y\ne \emptyset$ by definition of
$m'$, and so $m(x,y)\ge m'-1$.
Conversely, let $m=m(x,y)$. Suppose $m'< m-k_0$.
Then there exist $m'$-tiles $X'$ and $Y'$ with $X'\cap Y'=\emptyset$, $m$-tiles
$X$ and $Y$ with $X\cap Y\ne \emptyset$, and $x\in X\cap X'$, $y\in Y\cap Y'$.
Hence $K=X\cup Y$ is a union of two $m$-tiles joining the disjoint $m'$-tiles
$X'$ and $Y'$; but such a union must consist of at least at
$$D_{m-m'}\ge \widetilde D_{k_0}\ge 10$$
$m$-tiles by Lemma~\ref{lem:flowerbds}. This is a contradiction showing that $m-k_0\le m'$. So the claim is true with $k_4=k_0$.
\end{proof}
\begin{definition}[Visual metrics]\label{def:visual} Let $f\: S^2\rightarrow
S^2$ be an expanding Thurston map and $d$ be a metric on $S^2$. Then
$d$ is called a {\em visual metric}\index{visual metric|textbf}
(for $f$) if there exists a
Jordan curve $\mathcal{C}\subset S^2$ with $\operatorname{post}(f)\subset \mathcal{C}$, and constants
$\Lambda>1$,
and $A\ge 1$ such that
\begin{equation}\label{visual} (1/A) \Lambda^{-m(x,y)}\le d(x,y)\le A
\Lambda^{-m(x,y)}
\end{equation}
for all $x,y\in S^2$, where $m(x,y)=m_{f,\mathcal{C}}(x,y)$.
\end{definition}
Here we use the convention $\Lambda^{-\infty}=0$. The number $\Lambda$
is called the {\em expansion factor}\index{expansion factor}
of the metric $d$. It is easy to
see that the expansion factor of each visual metric is uniquely
determined; different visual metrics may have different expansion
factors.
Let $d$ and $d'$ be two metrics on a space $X$. They are called {\em bi-Lipschitz equivalent}\index{bi-Lipschitz} if there exists a constant
$C\ge 1$ such that
$$ (1/C) d(x,y)\le d'(x,y)\le Cd(x,y) $$
for all $x,y\in X$. They are called {\em snowflake equivalent}\index{snowflake equivalent}
if there exist constants $\alpha>0$ and $C\ge 1$ such that
$$(1/C) d(x,y)\le d'(x,y)^\alpha \le Cd(x,y) $$
for all $x,y\in X$. Obviously, bi-Lipschitz equivalence of two metrics implies their snowflake equivalence.
\begin{rem} \label{rem:gromovgraph} As briefly mentioned in the introduction, one can use an expanding Thurston map $f\:S^2\rightarrow S^2$ to define an infinite
graph $\mathcal{G}$ that is Gromov hyperbolic (see \cite{gh, BuS} for the definition of Gromov hyperbolic spaces and an explanation of the related terminology employed in the present remark). Namely, we consider
a Jordan curve $\mathcal{C}\subset S^2$ with $\operatorname{post}(f)\subset \mathcal{C}$ and the tiles in the cell decompositions $\mathcal{D}^n(f,\mathcal{C})$, $n\in \N_0$. The vertex set of $\mathcal{G}$
is the set of all tiles, and one connects two vertices by an edge in $\mathcal{G}$ if the corresponding tiles have nonempty intersection and levels that differ by at most $1$ (there are other reasonable ways to define the edges in $\mathcal{G}$). One can show that $\mathcal{G}$ is Gromov hyperbolic, and that the boundary at infinity $\partial_\infty \mathcal{G}$ of $\mathcal{G}$ can be identified
with $S^2$. Moreover, $m_{f,\mathcal{C}}(x,y)$ is essentially the Gromov product
of two points $x,y\in S^2\cong \partial_\infty \mathcal{G}$ (with one of the $0$-tiles chosen as a basepoint in $\mathcal{G}$). Then the notion of a visual
metric on $ \partial_\infty \mathcal{G}$ as in the theory of Gromov
hyperbolic spaces coincides with our notion of a visual metric on $S^2\cong
\partial_\infty \mathcal{G}$.
\end{rem}
In the following proposition we summarize properties of visual metrics.
\begin{prop}\label{prop:visualsummary}\index{visual metric}
Let $f\:S^2\rightarrow S^2$ be an expanding Thurston map.
\smallskip
\begin{itemize}
\item[(i)] There exist visual metrics for $f$.
\smallskip
\item[(ii)] Every visual metric induces the standard topology on $S^2$.
\smallskip
\item[(iii)] Let $d$ be a visual metric with expansion factor $\Lambda$. Then an inequality as in \eqref{visual} with the same expansion factor $\Lambda$ holds for every Jordan curve $\widetilde \mathcal{C}\subset S^2$ with $\operatorname{post}(f)\subset \widetilde \mathcal{C}$ if $m=m_{f,\widetilde \mathcal{C}}$ and $A$ is suitably chosen depending on $\widetilde \mathcal{C}$.
\smallskip
\item[(iv)] Any two visual metrics are snowflake equivalent; if they have the same expansion factor $\Lambda$, then they are bi-Lipschitz equivalent.
\smallskip
\item[(v)] A metric is a visual metric for any iterate $F=f^n$ if and only if it is a visual metric for $f$.
\end{itemize}
\end{prop}
\begin{proof} (i) Fix a Jordan curve $\mathcal{C}\subset S^2$ with
$\operatorname{post}(f)\subset \mathcal{C}$.
Recall that a function $q\colon
S^2\times S^2\to [0,\infty)$ is called a \defn{quasimetric} if it
has the symmetry property $q(x,y)=q(y,x)$,
satisfies the conditions $q(x,y)=0\Leftrightarrow x=y$, and the inequality
\begin{equation}
\label{eq:qmetric}
q(x,y)\leq K(q(x,z)+q(z,y)),
\end{equation}
holds for a constant $K\geq 1$ and all $x,y,z\in S^2$.
We now define a quasimetric $q$ on $S^2$. To this purpose, we fix $\Lambda>1$
and set
\begin{equation}
\label{eq:defq}
q(x,y):=\Lambda^{-m(x,y)},
\end{equation}
for $x,y\in S^2$, where $m(x,y)=m_{f,\mathcal{C}}(x,y)\in \N_0\cup \{\infty\}$ is as in
Definition~\ref{def:mxy}.
Symmetry and the property $q(x,y)=0 \Leftrightarrow x=y$ are
clear. The quasi-triangle inequality
(\ref{eq:qmetric}) follows from Lemma~\ref{lem:mprops} (i).
It is well known (see \cite[Prop.~14.5]{He}) that a
sufficient ``snowflaking'' of a quasimetric leads to a distance
function that is comparable to a metric. This means there is a
metric $d$ and $0<\epsilon<1$ such that $q^\epsilon\asymp d$. Then $d$ is a visual metric for $f$ (with expansion factor $\Lambda^\epsilon$).
(ii) Let $d$ be a visual metric for $f$ satisfying \eqref{visual}, and $d'$ our fixed ``base metric'' on $S^2$ that induces the standard topology of $S^2$. We have to show that if $x\in S^2$ and $\{x_i\}$ is a sequence in $S^2$, then $d(x_i,x)\to 0$
if and only if $d'(x_i, x)\to 0$ as $i\to \infty$.
Now by \eqref{visual} the relation $d(x_i,x)\to 0$ is obviously equivalent to $m_i:=m_{f,\mathcal{C}}(x_i,x)\to \infty$.
So if $d(x_i,x)\to 0$, then $m_i\to \infty$, and so for each $n\in \N_0$ we have $m_i\ge n$ for sufficiently large $i$.
For these $i$ we have $$ d'(x_i, x)\le 2 \sup_{k\ge n} \operatorname{mesh}(f,k,\mathcal{C})$$ as follows from the definition of $m_{f,\mathcal{C}}$. Since $f$ is expanding, we know that $\sup_{k\ge n} \operatorname{mesh}(f,k,\mathcal{C})\to 0$ as $n\to \infty$. Hence $d'(x_i, x)\to 0$ as $i\to \infty$.
Conversely, suppose that $d'(x_i, x)\to 0$ as $ i\to \infty$.
Let $n\in \N_0$ be arbitrary. Then $x$ lies in some $n$-flower
$W^n(v)$. Since flowers are open sets, we have $x_i\in W^n(v)$ for sufficiently large $i$.
For each of these $i$ we can find $n$-tiles $X$ and $Y$ with $x\in X$, $x_i\in Y$, and $v\in X\cap Y$. This implies $m_i\ge n$. Hence $m_i\to \infty$ as desired.
(iii) This follows from Lemma~\ref{lem:mprops} (iii).
(iv) This follows from (iii) and the definition of a visual metric.
(v) This follows from (iii) and Lemma~\ref{lem:mprops} (iv).
\end{proof}
The next lemma gives a geometric characterization of visual metrics.
\begin{lemma}
\label{lem:expoexp} Let $f\: S^2\rightarrow S^2$ be an expanding Thurston map, $\mathcal{C}\subset S^2$ be a Jordan curve with $\operatorname{post}(f)\subset \mathcal{C}$, and $d$ be a visual metric for $f$ with expansion factor
$\Lambda>1$. Then there exists a constant $C\ge 1$ such that
\begin{itemize}
\item[(i)] $\operatorname{dist}_d(\sigma, \tau)\ge (1/C)\Lambda^{-n}$ whenever $\sigma$ and $\tau$ are disjoint $n$-cells,
\smallskip
\item[(ii)] $ (1/C)\Lambda^{-n} \le \diam_d(\tau)\le C\Lambda^{-n}$ for all $n$-edges and all $n$-tiles $\tau$.
\end{itemize}
Conversely, if $d$ is a metric on $S^2$ satisfying conditions \textnormal{(i)} and
\textnormal{(ii)} for some constant $C\ge 1$, then $d$ is a visual metric with expansion factor $\Lambda>1$.
\end{lemma}
Here it is understood that $n$-cells are for $(f,\mathcal{C})$.
\begin{proof} By Proposition~\ref{prop:visualsummary} (iii) we may assume that $d$ satisfies \eqref{visual}, where $m=m_{f,\mathcal{C}}$.
(i) Let $k_0$ be defined as in \eqref{def:k0}, and let $\sigma$ and $\tau$ be disjoint $n$-cells. If $x\in \sigma$ and $y\in \tau$ are arbitrary, then $m=m(x,y)<n+k_0$. Indeed, if this were not the case, then we could find $(n+k)$-tiles
$X$ and $Y$ with $x\in X$, $y\in Y$, $X\cap Y\ne \emptyset$, and $k\ge k_0$. Then $K=X\cup Y$ is a connected set
meeting disjoint $n$-cells. Hence by Lemma~\ref{lem:maptotop} the set
$f^n(K)$ joins opposite sides of $\mathcal{C}$. On the other hand, $f^n(K)$ consists of two $k$-tiles $X'=f^n(X)$ and $Y'=f^n(Y)$, where $k\ge k_0$. This is impossible by definition of $k_0$.
Therefore, $d(x,y) \ge (1/A)\Lambda^{-n-k_0}$, and so
we get the desired bound $\operatorname{dist}(\sigma, \tau)\ge (1/C')\Lambda^{-n}$ with the constant $C'=C\Lambda^{k_0}$ that is independent of $n$, $\sigma$, and $\tau$.
(ii) If $x,y$ are points in some $n$-tile $X$, then $m(x,y)\ge n$. Since every $n$-edge is contained in an $n$-tile, this inequality is still true if $x$ and $y$ are contained in an $n$-edge.
Hence $d(x,y)\le A\Lambda^{-m(x,y)}\le A \Lambda^{-n}$, and so
$\diam(\tau)\le A \Lambda^{-n}$ whenever $\tau$ is an $n$-tile or $n$-edge, where the constant $A$ is as in \eqref{visual}.
A similar lower bound for the diameter of an $n$-edge or $n$-tile $\tau$ follows from (i) and the fact that every $n$-edge or $n$-tile contains two distinct $n$-vertices.
For the converse suppose that we have (i) and (ii). Let $x,y\in S^2$, $x\ne y$, be arbitrary, and $m=m_{f,\mathcal{C}}(x,y)$.
Then we can find $m$-tiles $X$ and $Y$ with $x\in X$, $y\in Y$ and $X\cap Y\ne \emptyset$.
By (ii) we have
$$ d(x,y)\le \diam(X)+\diam(Y) \lesssim \Lambda^{-m}. $$
We can also find $(m+1)$-tiles $X'$ and $Y'$ with $x\in X'$, $y\in Y'$. By definition of $m$ we then have and $X'\cap Y'=\emptyset$. Hence by (i)
$$d(x,y)\ge \operatorname{dist}(X',Y')\gtrsim \Lambda^{-m}. $$
Since the implicit multiplicative constants in the previous inequalities are independent of $x$ and $y$, it follows that $d$ is a visual metric.
\end{proof}
It is possible to establish this phenomenon of ``exponential shrinking'' for other types of sets. For example, we have
$$ \diam(W^n(v)) \le C \Lambda^{-n}$$ for every
$n$-flower for $(f,\mathcal{C})$ where the constant $C$ is independent of $n$ and $v$. Of particular importance will be exponential shrinking for lifts of paths.
\begin{lemma}
\label{lem:liftpathshrinks} Let $f\:S^2\rightarrow S^2$ be an expanding Thurston map, and
$d$ be a visual metric for $f$ with expansion factor $\Lambda>1$. Then for every path
$\gamma\colon [0,1]\to S^2$ there exists a
constant $A\ge 1$
with the following property: if $\widetilde \gamma$ is any lift of
$\gamma$ under $f^n$,
then
\begin{equation*}
\diam_d (\widetilde \gamma) \le A \Lambda^{-n}.
\end{equation*}
\end{lemma}
\begin{proof} Pick a Jordan curve $\mathcal{C}\subset S^2$ with $\operatorname{post}(f)\subset \mathcal{C}$, and let $\delta_0>0$ be as in \eqref{defdelta}. Then
we can break up
$\gamma$ into a finite number of paths $\gamma_i$, $i=1, \dots, N$,
traversed in successive order such that $\diam(\gamma_i)<\delta_0$
for all $i=1,\dots, N$. By Lemma~\ref{lem:preimsmall}~(ii) each lift of the pieces $\gamma_i$ is contained in one $n$-flower, and so the whole lift $\widetilde \gamma$ in $N$ $n$-flowers. Hence by
Lemma~\ref{lem:expoexp} we have $\diam (\widetilde\gamma) \leq C N
\Lambda^{-n}$ with a constant $C$ independent of $n$ and $\gamma$.
\end{proof}
In general the constant $A$ in the last lemma will dependent on $\gamma$, but the proof shows that we can take the same constant $A$ for a family of paths if there exists $N\in \N$ such that each
path can be broken up into at most $N$ subpaths of diameter
$<\delta_0$.
Let $f$ be an expanding Thurston map, and $\mathcal{C}\subset S^2$ be a Jordan curve
with $\operatorname{post}(f)\subset \mathcal{C}$.
It is useful to define neighborhoods of points by using the cells in our decompositions
$\mathcal{D}^n=\mathcal{D}^n(f, \mathcal{C})$. To define them let $x\in S^2$ and $n\in \N_0$, and set
\begin{multline}
\label{eq:defUk}
U^n(x)
= \bigcup\{Y\in \X^n: \text{there exists an $n$-tile $X$ with}
\\ \null \text{ $x\in X$ and }X\cap Y\neq\emptyset\}. \end{multline}
It is convenient to define $U^n(x)$ also for negative integers
$n$. We set $U^n(x)=U^0(x)=S^2$ for $n<0$.
It follows from Lemma~\ref{lem:expoexp} that the sets $U^n(x)$ resemble
metric balls very closely.
\begin{lemma}
\label{lem:UmB} Let $d$ be a visual metric for $f$ with expansion factor $\Lambda>1$. Then
there are constants $K\ge 1$ and $n_0\in \N_0$ with the following properties.
\smallskip
\begin{itemize}
\item[(i)]
For all $x\in S^2$ and all $n\in \Z$
\begin{equation*
B_d(x, r/K)\subset U^n(x)\subset B_d(x, Kr),
\end{equation*}
where $r=\Lambda^{-n}$.
\item[(ii)]
For all $x\in S^2$ and all $r>0$
\begin{equation*}
U^{n+n_0}(x)\subset B_d(x,r)\subset U^{n-n_0}(x),
\end{equation*}
where $n=\left\lceil-\log r/\log\Lambda\right\rceil$.
\end{itemize}
\end{lemma}
\begin{proof} (i) Let $m=m_{f,\mathcal{C}}$. If $y\in U^n(x)$, then $m(x,y)\ge n$, and
so $d(x,y)\lesssim \Lambda^{-n}=r$. This gives the inclusion
$U^n(x)\subset B(x,Kr)$ for a suitable constant $K$ independent of $x$ and $n$.
Conversely, suppose that $y\notin U^n(x)$. Then $n\ge 1$. If we pick $n$-tiles $X$ and $Y$ with $x\in X$ and $y\in Y$, then $X\cap Y=\emptyset $ by definition of $U^n(x)$. So by Lemma~\ref{lem:expoexp}~(ii) we have
$$d(x,y)\ge \operatorname{dist}(X,Y) \gtrsim \Lambda^{-n}= r. $$
Hence $B_d(x,r/K)\subset U^n(x)$ if $K$ is suitably large independent of $x$ and $r$.
(ii) Choose $n_0=\left\lceil\log K/\log \Lambda\right\rceil+1$, where $K$ is as in (i).
Then $\Lambda^{-n_0}\le1/( \Lambda K)$. Moreover,
$\Lambda^{-n}\le r\le \Lambda \Lambda^{-n}$, and so
$$K\Lambda^{-n-n_0}\le r\le
(1/K)\Lambda^{-n+n_0}. $$
The desired inclusion then follows from (i).
\end{proof}
\begin{lemma} \label{lem:quasiball}
Let $f\: S^2\rightarrow S^2$ be an expanding Thurston map, $\mathcal{C}\subset S^2$ be a Jordan curve with $\operatorname{post}(f)\subset \mathcal{C}$, and $d$ be a visual metric for $f$ with expansion factor
$\Lambda>1$. Then there exists a constant $C\ge 1$ with the following property:
for every $n$-tile $X$ for $(f,\mathcal{C})$ there exists a point $p\in X$ such that
$$B_d(p, (1/C)\Lambda^{-n})\subset X \subset B_d(p, C\Lambda^{-n}).$$
\end{lemma}
So if $S^2$ is equipped with a visual metric, then tiles are ``quasi-round'', and every tile contains points that are ``deep inside'' the tile.
\begin{proof} With a suitable constant $C$ independent of $n$, an inclusion of the form
$$X\subset B_d(p, C\Lambda^{-n})$$ holds for every $n$-tile $X$ and every point $p\in X$ as follows from Lemma~\ref{lem:expoexp} (i).
The main difficulty for an inclusion in the opposite direction is to find an appropriate point $p$. For this purpose let $k_0\in \N$ be the number defined in \eqref{def:k0}, and $X$ be an arbitrary $n$-tile.
Since $f$ is an expanding Thurston map, we have $\operatorname{post}(f)\ge 3$ (see Corollary~\ref{cor:no<3}), and so
$\partial X$ contains at least three distinct $n$-vertices
$v_1,v_2,v_3$. Using these vertices, we can find three arcs
$\alpha_1, \alpha_2,\alpha_3\subset \partial X$ with pairwise disjoint interior such that
$\partial X= \alpha_1\cup \alpha_2\cup \alpha_3$ and such
that $\alpha_i$ has the endpoints $v_i$ and $v_{i+1}$ for $i=1,2,3$, where $v_4=v_1$. In general $\alpha_i$ will not be an $n$-edge, but since it lies on $\partial X$, and its endpoints are $n$-vertices, it is the union of all the $n$-edges that it contains.
We now define
$$A_i=\bigcup_{x\in \alpha_i} U^{n+k_0}(x) $$
for $i=1,2,3$, where $U^{n+k_0}(x)$ is defined as in \eqref{eq:defUk}.
Then the set $A_i$ is the union of all $(n+k_0)$-tiles that meet an $(n+k_0)$-tile that has nonempty intersection with $\alpha_i$. In particular, $A_i$ is a closed set that contains $\alpha_i$.
We claim that the sets $A_1$, $A_2$, $A_3$ do not form a cover of $X$. To reach a contradiction suppose that
$X\subset A_1\cup A_2\cup A_3$. We can regard $X$ as a topological simplex with the sides $\alpha_i$, $i=1,2,3$.
Then the closed sets $A_1, A_2, A_3$ form a cover of $X$ such that each set $A_i$ contains the side $\alpha_i$ of the simplex for $i=1,2,3$. A well-known result due to Sperner \cite[p.~378]{AH} then implies that $A_1\cap A_2\cap A_3\ne \emptyset$.
Pick a point $x\in A_1\cap A_2\cap A_3$. Then by definition of $A_i$, there exist $(n+k_0)$-tiles $X_i$ and $Y_i$ with
$X_i\cap \alpha_i\ne \emptyset$, $x\in Y_i$, and $X_i\cap Y_i\ne \emptyset$, where $i=1,2,3$. Then the set
$$K=\bigcup_{i=1}^3(X_i\cup Y_i)$$
consists of at most six $(n+k_0)$-tiles, is connected, and meets each of the arcs $\alpha_1,\alpha_2, \alpha_3$.
Hence $K'=f^n(K)$ is a connected set that consists of at most
six $k_0$-tiles, and meets each of the the arcs $\beta_i=f^n(\alpha_i)$,
$i=1,2,3$. Note that each arc $\beta_i$ is the union of all $0$-edges edges that it contains. Hence for $i=1,2,3$ there exists a $0$-edge $e_i\subset \beta_i$ with $e_i\cap K' \ne \emptyset$. Since the arcs $\beta_1,\beta_2,\beta_3$ have pairwise disjoint interior, it follows that the $0$-edges $e_1,e_2,e_3$ are all distinct. So $K'$ is a connected set that meets three distinct $0$-edges. Hence it joins opposite sides of $\mathcal{C}$. So $K'$ should contain at least $D_{k_0}\ge \widetilde D_{k_0}\ge 10$ tiles of order $k_0$.
This is a contradiction, because $K'$ is a union of at most six $k_0$-tiles.
This proves the claim that the sets $A_1,A_2, A_3$ do not cover $X$, and we conclude that we can find a point
$$p\in X\setminus (A_1\cup A_2\cup A_3).$$
We claim that $U^{n+k_0}(p)\subset X$. If not, we could find a point
$y\in U^{n+k_0}(p)\setminus X$, and $(n+k_0)$-tiles
$U$ and $V$ with $p\in U$, $y\in V$, and $U\cap V\ne \emptyset$.
Then the connected set $U\cup V$ must meet $\partial X$, and hence one of the arcs $\alpha_i$; but then $p\in A_i$
by definition of $A_i$. This is a contradiction showing
the desired inclusion $U^{n+k_0}(p)\subset X$.
Using Lemma~\ref{lem:UmB}~(i) it follows that
$B_d(p,(1/C)\Lambda^{-n})\subset X$, where $C\ge 1$ is a constant independent of $n$ and $X$.
\end{proof}
\section{Symbolic dynamics}
\label{sec:symdym}
\noindent
Shift operators serve as important paradigms in symbolic dynamics.
Often a goal in understanding a dynamical system $(X,f)$ given by the iteration of a map $f$ on a space $X$ is to
link it to shift operators, or more generally, to shifts of finite type. For expanding Thurston maps this is accomplished by
Theorem~\ref{thm:expThfactor} stated in the introduction. The statement is essentially due to Kameyama (see \cite[Thm.~3.4]{Ka03a}). His notion of an expanding Thurston map is different from ours, but his proof carries over to
our setting with only minor modifications (see below). In this section we will also establish a related fact (see
Proposition~\ref{prop:subshift}) for maps with cellular Markov partitions as introduced in Section~\ref{s:celldecomp}. We start with some basic definitions.
Let $J$ be a finite set. We consider $J$ as an {\em alphabet} and
its elements as {\em letters} in this alphabet. A {\em word} is a finite sequence $w=i_1\dots i_n$, where $n\in \N_0$ and $i_1, \dots, i_n\in J$. For $n=0$ we interpret this as the {\em empty word} $\emptyset$. The number $n$ is called the {\em length} of the word
$w=i_1i_2\dots i_n$. The words of length $n$ can be identified
with $n$-tuples in $J$, i.e., elements of the Cartesian power $J^n$. The letters, i.e., the elements in $J$, are precisely the words of length $1$. If $w=i_1i_2\dots i_n$ and $w'=j_1\dots j_m$, then
we denote by $ww'=i_1i_2\dots i_nj_1\dots j_m$ the word obtained by concatenating $w$ and $w'$.
Let $J^*$ be the set of all words in the alphabet $J$. The {\em (left-)shift}
$\Sigma\: J^*\rightarrow J^*$ is defined by setting $\Sigma(i_1i_2\dots i_n)=i_2\dots i_n$
for a word $w=i_1i_2\dots i_n\in J^*$. We denote by $J^\omega$
the set of all sequences $(i_n)$ in $J$, where
the sequence elements $i_n\in J$ are indexed by $n\in \N$. More informally, we consider a sequence $s=(i_n)\in J^\omega$ as a ``word of infinite length" and write $s=i_1i_2\dots $. If
$s=(i_n)\in J^\omega$ and $n\in \N_0$, then we denote by $[s]_n\in J^*$ the
word $s_n=i_1\dots i_n$ consisting of the first $n$ elements of the sequence $s$.
The {\em \mbox{(left-)}shift} $\Sigma\: J^\omega \rightarrow J^ \omega$ is the map that assigns to each sequence $(i_n)\in J^\omega $ the sequence $(j_n)\in J^\omega$ with $j_n=i_{n+1}$ for all $n\in\N$. In our notation we do not distinguish the shifts on $J^*$ and $J^\omega$ and denote both maps by $\Sigma$.
Note that $[\Sigma(s)]_n=\Sigma([s]_{n+1})$ for all $s\in J^\omega$; indeed,
if $s=i_1i_2\dots$, then we have
$$[\Sigma(s)]_n=[i_2i_3\dots]_n=i_2\dots i_{n+1}=\Sigma(i_1\dots i_{n+1})=
\Sigma([s]_{n+1}). $$
If we equip
$J$ with the discrete topology, then $J^\omega$ carries a natural metrizable product topology. This topology is induced by the ultrametric $d$ given by
$d(s,s')=2^{-N}$ for $s=(i_n)\in J^\omega $ and $s'=(j_n) \in J^\omega$, $s\ne s'$, where $N=\min\{n:i_n\ne j_n\}$. In particular, two elements $s,s'\in J^\omega$ are close if and only if $s_n=s'_n$ for some large $n$. Equipped with this topology, the space $J^\omega$ is compact.
Let $T\: J\times J\rightarrow \{0,1\} $ be a map encoding ``allowed'' transitions between the letters, and let $J^\omega_T$ be the set of all sequences
$(i_n)$ in $J^\omega$ such that $T(i_n, i_{n+1})=1$ for all $n\in \N$.
Then $\Sigma(J^\omega_T)\subset J^\omega_T$, and so we can consider $\Sigma$
as a map on $J^\omega_T$. A {\em subshift of finite type}\index{subshift of finite type}
is a map of
the form $\Sigma_T:=\Sigma|J^\omega_T$ for some finite set $J$ and some
map $T\:J\times J\rightarrow \{0,1\} $.
Suppose that $X$ and $\widetilde X$ are topological spaces, and $f\:
X\rightarrow X$ and $\widetilde f\: \widetilde X\rightarrow \widetilde X$ are
continuous maps. We say that the dynamical system $(X,f)$ is a {\em
factor} of the dynamical system\index{factor of a dynamical system}
$(\widetilde X, \widetilde f)$ if there exists a surjective continuous map
$\varphi\: \widetilde X\rightarrow X$ such that $\varphi\circ \widetilde f=f\circ \varphi$.
The following proposition shows that under some mild additional
assumptions a map with a cellular Markov partition can be obtained as
a factor of a subshift of finite type. Since we know that at
least some iterate $f^n$ of an expanding Thurston map $f$ has a
cellular Markov partition as in this proposition, it follows
immediately that $f^n$ is a factor of a subshift of finite type.
Of course, by Theorem~\ref{thm:expThfactor}, which will be proved
below, an even stronger statement is true in this case.
\begin{prop}\label{prop:subshift} Let $(X,d)$ be a compact metric space,
$f\: X\rightarrow X$ be a continuous map with a cellular Markov partition $(\mathcal{D}', \mathcal{D})$, and $\mathcal{D}^n$ for $n\in \N_0$ be the cell decompositions of $X$ as given by
Proposition~\ref{prop:inftychain}.
Suppose that $$\lim_{n\to \infty} \,\max_{\tau\in \mathcal{D}^n}\,\diam(\tau)=0.$$
Then $f$ is a factor of a subshift of finite type. \end{prop}
\begin{proof}
Let $J=\mathcal{D}^1=\mathcal{D}'$. Note that since $X$ is compact, the set $J$ is finite. As above we let $J^\omega $ be the set of all sequences in $J$, and $\Sigma\:J^\omega\rightarrow J^\omega$ be the shift operator. Define the map $T\: J\times J\rightarrow \{0,1\}$ as follows: if $\sigma, \tau\in J=\mathcal{D}^1$, we put
$$
T(\sigma, \tau)=1 \text{ if $\tau\subset f(\sigma)$ and $T(\sigma, \tau)=0$ otherwise.} $$ So we have $T(\sigma, \tau)=1$ precisely
if $\tau\in \mathcal{D}^1$ is one of the cells into which $f(\sigma)\in \mathcal{D}^0$ is subdivided.
We want to prove that $(X,f)$ is a factor of $(J^\omega_T, \Sigma_T)$, where
$\Sigma_T=\Sigma|J^\omega_T$.
Let $\mathcal{S}$ be the set of all sequences $(\sigma_n)$, where $\sigma_n\in \mathcal{D}^n$ and $\sigma_{n+1}\subset \sigma_n$ for $n\in \N$.
Since $f^{n-1}$ is cellular for $(\mathcal{D}^n, \mathcal{D}^1)$, it follows that if $(\sigma_n)\in \mathcal{S}$,
then $(x_n)\in J^\omega $, where $x_n=f^{n-1}(\sigma_n)\in \mathcal{D}^1=J$ for $n\in \N$.
Moreover, for each $n\in \N$ we have $x_{n+1}=f^n(\sigma_{n+1})\subset f^n(\sigma_n)=f(x_n)$, and so
$T(x_n, x_{n+1})=1$.
It follows that $(x_n)\in J^\omega_T$.
In this way we get a map
$$\Phi\: \mathcal{S}\rightarrow J^\omega_T, \quad (\sigma_n)\in\mathcal{S} \mapsto\Phi[(\sigma_n)]:=(f^{n-1}(\sigma_n)). $$
The map $\Phi$ is a bijection. To show injectivity, suppose that $(\sigma_n)$ and
$ (\tau_n)$ are sequences in $\mathcal{S}$, and $\Phi[(\sigma_n)]=\Phi[(\tau_n)]$. Then $f^{n-1}(\sigma_n)=f^{n-1}(\tau_n)$ for all $n\in \N$. We show inductively that this implies $\sigma_n=\tau_n$ for all
$n\in \N$. Indeed, for $n=1$ we have $\sigma_1=\tau_1$ by definition of $\Phi$.
Suppose that $\sigma_n=\tau_n=:\lambda$. Since $f^n|\lambda$ is a homeomorphism
of $\lambda$ onto $f^n(\lambda)\in \mathcal{D}^0$, and $\sigma_{n+1}, \tau_{n+1}\subset \lambda$, we have
$$ \sigma_{n+1} = (f^n|\lambda)^{-1} (f^n(\sigma_{n+1}))=(f^n|\lambda)^{-1}
(f^n(\tau_{n+1}))=\tau_{n+1}. $$
Hence $(\sigma_n)=(\tau_n)$.
To show surjectivity of $\Phi$, let $(x_n)\in J^\omega_T$ be arbitrary. We define a sequence
of cells $\sigma_n\in \mathcal{D}^n$ with $f^{n-1}(\sigma_n)=x_n$ inductively as follows. Let $\sigma_1:=x_1\in \mathcal{D}^1$.
Suppose $\sigma_n$ is already defined such that $x_n=f^{n-1}(\sigma_n)$. Since $T(x_n, x_{n+1})=1$, we have $x_{n+1}\subset f(x_n)$. Pick a point $q\in \inte(x_{n+1})$. Since $f^{n}|\sigma_n$ is a homeomorphism of $\sigma_n$ onto $f(x_n)\in \mathcal{D}^0$, there exists a unique
point $p\in \sigma_n$ with $f^{n}(p)=q$. Since $\mathcal{D}^{n+1}$ is a refinement of $\mathcal{D}^n$,
the cells in $\mathcal{D}^{n+1}$ contained in $\sigma_n$ form a cell decomposition of $\sigma_n$. Hence $\sigma_n$ is the disjoint union of the interiors of the cells in $\mathcal{D}^{n+1}$ contained in
$\sigma_n$. So there exists a unique cell $\tau\in \mathcal{D}^{n+1}$ with $\tau\subset \sigma_{n}$ and $p\in \inte(\tau)$. Then $f^{n}(\tau) $ is a cell in $\mathcal{D}^1$ with
$q=f^{n}(p)\in \inte(f^{n}(\tau))$. Since $x_{n+1}$ is the unique cell in $\mathcal{D}^1$ containing $q$ in its interior, we must have $x_{n+1}=f^{n}(\tau)$. Now define $\sigma_{n+1}:=\tau$. Then $f^{n}(\sigma_{n+1})=x_{n+1}$
as desired.
Note that by construction $\sigma_{n+1}\subset \sigma_n$ for $n\in \N$.
Hence $(\sigma_n)\in \mathcal{S}$, and we have $\Phi[(\sigma_n)]=(x_n)$.
\smallskip
We define a map $\Psi\: \mathcal{S}\rightarrow X$ as follows: if $(\sigma_n)\in \mathcal{S}$, then
$\sigma_{n+1}\subset \sigma_n$ for $n\in \N$ and $\diam(\sigma_n)\to 0$ as $n\to \infty$
by our hypotheses. Hence the intersection $\bigcap_{n\in \N}\sigma_n$ contains a unique
point $p\in X$. Set $\Psi[ (\sigma_n)]=p$.
Now let $\varphi:=\Psi\circ \Phi^{-1}\: J^\omega_T\rightarrow X$. Then this map is continuous on
$J^\omega_T$. We sketch the proof for this, leaving the details, which can easily be filled in, to the reader. If $(x_n)$ and $(y_n)$ are points in $J^\omega_T$ that are ``close'', then there exists large $N\in \N$ such that $x_1=y_1, \dots, x_N=y_N$.
If $(\sigma_n)=\Phi^{-1}[(x_n)]$ and $(\tau_n)=\Phi^{-1}[(y_n)]$, then the argument used for establishing the injectivity of $\Phi$ shows that $\sigma_1=\tau_1, \dots, \sigma_N=\tau_N$. Hence if $p=\varphi[(x_n)]$ and $q=\varphi[(y_n)]$, then
$p,q\in \sigma_N=\tau_N\in \mathcal{D}^N$. By our hypotheses the diameter of a cell in $\mathcal{D}^N$ is small if $N$ is large. Hence $p$ and $q$ are close if $(x_n)$ and $(y_n)$ are close.
The continuity of $\varphi$ follows.
The map $\varphi$ is surjective. To see this let $p\in X$ be arbitrary. By the bijectivity of $\Phi$ it is enough to find a sequence $(\sigma_n)\in \mathcal{S}$ with $p\in \bigcap_{n\in \N}\sigma_n$. Appropriate cells $\sigma_n$ can be found as follows: Since the cells in $\mathcal{D}^1$ cover $X$, there exists $\sigma_1\in \mathcal{D}^1$ with $p\in \sigma_1$. Since $\mathcal{D}^2$ is a refinement of $\mathcal{D}^1$, the cells in $\mathcal{D}^2$ contained in $\sigma_1$ cover $\sigma_1$. Hence there exists a cell $\sigma_2\in \mathcal{D}^2$ with $p\in \sigma_2$ and $\sigma_2\subset\sigma_1$. Repeating this argument for $\sigma_2$, we can find a cell $\sigma_3\in \mathcal{D}^3$ with $\sigma_3\subset \sigma_2$, and $p\in \sigma_3$, etc.
In this way we get a sequence $(\sigma_n)$ as desired.
Finally, in order to show that $\varphi\circ \Sigma_T= f\circ \varphi=f\circ \Psi\circ \Phi^{-1}$, let $(x_n)\in \Sigma_T$ be arbitrary, $(\sigma_n)=\Phi^{-1}[(x_n)]$, and $p=\varphi[(x_n)]$ be the unique point
in the intersection $\bigcap_{n\in \N}\sigma_n$.
Define $\tau_n=f(\sigma_{n+1})$ for $n\in \N$. Then $\tau_n \in \mathcal{D}^n$ and $\tau_{n+1}\subset \tau_n$ for $n\in \N$. Hence $(\tau_n)\in \mathcal{S}$. If $(y_n)=\Phi[(\tau_n)]\in J^\omega_T$, then $y_n=f^{n-1}(\tau_n)=f^n(\sigma_{n+1})=x_{n+1}$.
Hence $(y_n)=S_T[(x_n)]$. Moreover, $f(p)\in \bigcap_{n\in \N} \tau_n$, and so
$f(p)=\varphi[(y_n)]=(\varphi\circ \Sigma_T)[(x_n)]$. On the other hand,
$f(p)=(f\circ \varphi)[(x_n)]$. The desired identity $\varphi\circ \Sigma_T= f\circ \varphi$ follows.
We conclude that $(X,f)$ is a factor of $(J^\omega_T,\Sigma_T)$, and hence a factor of a subshift of finite type. \end{proof}
\begin{rem} Let $\varphi\colon J^\omega_T \to X$ be as in the proof of the last proposition. If we define an equivalence relation $\sim$ on $J^\omega_T$ by
$$(x_n)\sim (y_n) \Longleftrightarrow \varphi[(x_n)] = \varphi[(y_n)]$$
for $(x_n),(y_n)\in J^\omega_T$, then the quotient
space
$\widetilde X:=J^\omega_T/\!\!\sim$ is homeomorphic to $X$ and the map $\widetilde f\:\widetilde X\rightarrow \widetilde X $ induced by $\Sigma_T$ on $\widetilde X$ is topologically conjugate to $f\: X\to X$.
\end{rem}
We are now ready to prove Theorem~\ref{thm:expThfactor}.
\begin{proof}[Proof of Theorem~\ref{thm:expThfactor}] Let $f\:S^2\rightarrow S^2$ be an expanding Thurston map, and $k:=\deg(f)\ge 2$. Fix a visual metric $d$ for $f$, and let $\Lambda>1$ be its expansion factor. In the following metric concepts refer to $d$. We consider tiles for $(f,\mathcal{C})$ and color them black and white as in Lemma~\ref{lem:colortiles}. Choose a basepoint $p\in S^2\setminus \operatorname{post}(f)$
in the interior of the white $0$-tile $\XOw$.
\smallskip \noindent
{\em Claim 1.} $\displaystyle
\sup_{x\in S^2} \operatorname{dist}(x, f^{-n}(p))\lesssim \Lambda^{-n}$, \\
where $C(\lesssim)$ is independent of $n$. In other words, the set $f^{-n}(p)$ forms a very dense net in $S^2$ if
$n$ is large.
To see this let $x\in S^2$ be arbitrary. Then $x$ lies in some $n$-tile $X^n$. If $X^n$ is white, then $X^n$ contains
a point in $f^{-n}(p)$ and so $\operatorname{dist}(x, f^{-n}(p))\le \diam(X^n)$.
If $X^n$ is black, then $X^n$ shares an edge with a white $n$-tile $Y^n$.
Then $Y^n$ contains a point in $f^{-n}(p)$, and so
$\operatorname{dist}(x, f^{-n}(p))\le \diam(X^n)+\diam(Y^n)$.
From the inequalities in both cases and Lemma~\ref{lem:expoexp} we
conclude
$$ \operatorname{dist}(x, f^{-n}(p))\lesssim \Lambda^{-n}, $$
where $C(\lesssim)$ is independent of $x$ and $n$.
Claim~1 follows.
\smallskip
None of the points in $S^2\setminus \operatorname{post}(f)$ is a critical value for any of the iterates $f^n$ of $f$. Moreover, each iterate $f^n$ is a covering
map $f^n\: S^2\setminus f^{-n}(\operatorname{post}(f))\rightarrow S^2\setminus \operatorname{post}(f)$. Since $p\in S^2\setminus \operatorname{post}(f)$, we have $f^{-n}(p)\subset S^2\setminus f^{-n}(\operatorname{post}(f))$ and
\begin{equation}\label{eq:preimcard}
\#f^{-n}(p)=\deg(f^n)=\deg(f)^n=k^n
\end{equation}
for $n\in \N$.
In particular, $$f^{-1}(p)\subset S^2\setminus f^{-1}(\operatorname{post}(f))\subset S^2\setminus \operatorname{post}(f),$$ and $\#f^{-1}(p)=k$.
Let $q_1, \dots, q_k\in S^2\setminus \operatorname{post}(f)$ be the points in $f^{-1}(p)$.
For $i=1, \dots, k$ we pick a path $\alpha_i\:[0,1]\rightarrow S^2\setminus \operatorname{post}(f)$
with $\alpha_i(0)=p$ and $\alpha_i(1)=q_i$.
Let $J:=\{1,\dots, k\}$, and consider the shift $\Sigma\: J^\omega \rightarrow J^\omega
$. We want to show that $f$ is a factor of $\Sigma$, i.e., that there exists a continuous and surjective map $\varphi\: J^\omega \rightarrow S^2$ with
$f\circ \varphi=\varphi\circ \Sigma$. In order to define $\varphi$, we first construct a suitable map $\psi$ that assigns to each word in $J^*$ a point in $S^2$.
\smallskip\noindent
{\em Definition of $\psi$.} The map $\psi\: J^*\rightarrow S^2$ will be defined inductively such that
$$ \psi(w)\in f^{-n}(p), $$ whenever $n\in \N_0$ and $w\in J^n\subset J^*$ is a
word of length $n$.
For the empty word $\emptyset$ we set $\psi(\emptyset)=p$, and for the
word consisting of the single letter $i\in J$ we set $\psi(i):=q_i\in f^{-1}(p)$.
Now suppose that $\psi$ has been defined for all words of length $\le n$, where
$n\in \N$.
Let $w$ be an arbitrary word of length $n+1$. Then $w=w'i$, where
$w'\in J^*$ is a word of length $n$ and $i\in J$. So $\psi(w')\in f^{-n}(p)$ is already defined. Since $f^n(\psi(w'))=p$ and $f^n\: S^2\setminus
f^{-n}(\operatorname{post}(f))\rightarrow S^2\setminus \operatorname{post}(f)$ is a covering map, the path
$\alpha_i$ has a unique lift with initial point $\psi(w')$, i.e., there exists
a unique path $\widetilde \alpha_i\: [0,1]\rightarrow S^2$
with $\widetilde \alpha_i(0)=\psi(w)$ and $f^n\circ \widetilde \alpha_i=
\alpha_i$. We now define
$ \psi(w):=\widetilde \alpha_i(1)$.
Note that then
$$f^{n+1}( \psi(w))=f^{n+1}(\widetilde \alpha_i(1))=f( \alpha_i(1))=f(q_i)=p.$$
Hence $\psi(w)\in f^{-(n+1)}(p)$.
This shows that a map $\psi\: J^*\rightarrow S^2$ with the desired properties
exists.
\smallskip \noindent
{\em Claim 2.} $f(\psi(w))=\psi(\Sigma(w))$ for all non-empty words $w\in
J^*$.
We prove this by induction on the length of the word $w$. If $w=i\in J$, then
$$f(\psi(w))=f(\psi(i))=f(q_i)=p=\psi(\emptyset)=\psi(\Sigma(i))=
\psi(\Sigma(w)).$$
So the claim is true for words of length $1$.
Suppose the claim is true for words of length $\le n$, where $n\in\N$.
Let $w$ be a word of length $n+1$. Then $w=w'i$, where $w'$ is a word of
length $n$ and $i\in J$.
Let $\widetilde \alpha_i$ be the path as above used in the definition of
$\psi(w)$. Define $\widetilde \beta_i:=f\circ \widetilde \alpha_i$. Then
$\widetilde \beta_i$ is a lift of $\alpha_i$ by $f^{n-1}$. By induction hypothesis its initial point is
$$\widetilde \beta_i(0)=f(\widetilde \alpha_i(0))=f(\psi(w'))=\psi(\Sigma(w')). $$
In other words, $\widetilde \beta_i$ is the unique
path as in the definition of $\psi$ used to determine $\psi(\Sigma(w')i)$ from $\psi(\Sigma(w'))$,
i.e., $\psi(\Sigma(w')i)=\widetilde \beta_i(1)$.
Hence
$$\psi(\Sigma(w))=\psi(\Sigma(w')i)=\widetilde\beta_i(1)=
f(\widetilde\alpha_i(1))=f(\psi(w'i))=f(\psi(w))$$
as desired, and Claim 2 follows.
\smallskip \noindent
{\em Claim 3.} For each $n\in \N$ the map $\psi|J^n\: J^n\rightarrow f^{-n}(p)$
is a bijection. In other words, the map $\psi$ provides a ``coding" of the points in $f^{-n}(p)$ by words of length $n$.
Again we prove this by induction on $n$. By definition of $\psi$ it is true for
$n=1$.
Suppose it is true for some $n\in \N$. It suffices to show
that the map $\psi|J^{n+1}\: J^{n+1} \rightarrow f^{-(n+1)}(p)$ is surjective, since both sets $J^{n+1}$ and $f^{-(n+1)}(p)$ have the same cardinality
$k^{n+1}$. So let $x\in f^{-(n+1)}(p)$ be arbitrary. Then
$f^n(x)\in f^{-1}(p)$, and so there exists $i\in J$ with $f^n(p)=q_i$.
Since $$x\in f^{-(n+1)}(p)\subset S^2\setminus f^{-(n+1)}(\operatorname{post}(f))\subset
S^2\setminus f^{-n}(\operatorname{post}(f)), $$ and $f^n
\: S^2\setminus
f^{-n}(\operatorname{post}(f))\rightarrow S^2\setminus \operatorname{post}(f)$ is a covering map, we can
lift the path $\alpha_i$ by $f^n$ to a path $\widetilde
\alpha_i\:[0,1]\rightarrow S^2$ whose terminal point is $x$ (to see this,
lift $\alpha_i$ traversed in opposite direction so that the initial point of the lift is $x$).
Then $f^n(\widetilde \alpha_i(0))=\alpha_i(0)=p$, and so
$\widetilde \alpha_i(0)\in f^{-n}(p)$. By induction hypothesis there exists a
word $w'\in J^n$ with $\psi(w')=\widetilde \alpha_i(0)$.
Then $\widetilde \alpha_i$ is a path as used to determine $\psi(w'i)$ from
$\psi(w')$. So if we set $w:=w'i\in J^{n+1}$, then
$$ \psi(w)=\psi(w'i)=\widetilde \alpha_i(1)=x.$$
This shows that $\psi|J^{n+1}\: J^{n+1} \rightarrow f^{-(n+1)}(p)$ is surjective. Claim~3 follows.
\smallskip \noindent
{\em Claim 4.} If $s\in J^\omega $, then the points $\psi([s]_n)$, $n\in \N$, form a Cauchy sequence in $S^2$ (recall that $[s]_n$ is the word consisting of the first $n$ elements of the sequence $s$).
By definition of $\psi$ the points $\psi([s]_n)$ and $\psi([s]_{n+1})$
are joined by a lift of one of the paths $\alpha_1, \dots, \alpha_k$ by
$f^n$. Hence by Lemma~\ref{lem:liftpathshrinks} we have
\begin{equation} \label{eq:Cauest}
d(\psi([s]_n), \psi([s]_{n+1})\lesssim \Lambda^{-n},
\end{equation}
where $C(\lesssim)$ is independent of $n$ and $s$.
Hence $(\psi([s]_n))$ is a Cauchy sequence.
\smallskip\noindent
{\em Definition of $\varphi$.} If $s\in J^\omega$, then by Claim~4 the limit
$$\varphi(s):= \lim_{n\to \infty} \psi([s]_n) $$
exists. This defines a map $\varphi\: J^\omega \rightarrow S^2$.
\smallskip\noindent
{\em Claim 5.} $f\circ \varphi = \varphi\circ \Sigma$.
To see this, let $s\in J^\omega $ be arbitrary. Note that
$\Sigma([s]_n)=[\Sigma(s)]_{n-1}$
for $n\in \N$. Hence by Claim 2 and the continuity of $f$ we have
$$f(\varphi(s))=\lim_{n\to \infty} f(\psi([s]_n))=\lim_{n\to \infty}
\psi(\Sigma([s]_n))=\lim_{n\to \infty}\psi([\Sigma(s)]_{n-1})=\varphi(\Sigma(s)).
$$ The claim follows.
\smallskip\noindent
{\em Claim 6.} The map $\varphi\: J^\omega \rightarrow S^2$ is continuous
and surjective.
Let $s\in J^\omega $ and $n\in \N$. Then \eqref{eq:Cauest}
shows that
\begin{equation}\label{eq:ssnclose}
d(\varphi(s), \psi([s]_n))\lesssim \sum_{l=n}^\infty \Lambda^{-l}\lesssim
\Lambda^{-n},
\end{equation}
where $C(\lesssim)$ is independent of $n$ and $s$.
Hence if $s,s'\in \Sigma$ and $[s]_n=[s']_n$, then
$$d(\varphi(s), \varphi(s'))\lesssim \Lambda^{-n}, $$
where $C(\lesssim)$ is independent of $n$, $s$, and $s'$.
The continuity of $\varphi$ follows from this; indeed, if $s$ and $s'$ are ``close" in $J^\omega$, then $[s]_n=[s']_n$ for some large $n$, and so the image points $\varphi(s)$ and $\varphi(s')$ are close in $S^2$.
Since $J^\omega $ is compact, the continuity of $\varphi$ implies that
the image $\varphi(J^\omega )$ is also compact and hence closed in $S^2$. The surjectivity of $\varphi$ will follow, if we can show that $\varphi$ has dense image in $S^2$.
To see this let $x\in S^2$ and $n\in \N$ be arbitrary.
Then by Claim~1 we can find a point $y\in f^{-n}(p)$ with
$d(x,y)\lesssim \Lambda^{-n}, $ where $C(\lesssim)$ is independent of $x$ and $n$.
Moreover, by Claim 3 there exists a word $w\in J^n$ with
$\psi(w)=y$. Pick $s\in J^\omega$ such that $[s]_n=w$. Then by
\eqref{eq:ssnclose} we have
$$d(x,\varphi( s))\le d(x,y)+d(y, \varphi(s))=d(x,y)+d(\psi([s]_n),\varphi(s))\lesssim \Lambda^{-n}, $$
where $C(\lesssim)$ is independent of the choices.
Hence
$$\sup_{x\in S^2}\operatorname{dist}(x, \varphi(J^\omega))\lesssim \Lambda^{-n}$$
for all $n$, where $C(\lesssim)$ is independent of $n$. This shows that
$\varphi(J^n)$ is dense in $S^2$ and the claim follows.
\smallskip
The theorem now follows from Claim 5 and Claim 6.
\end{proof}
The procedure that we employed to code
the elements in $f^{-n}(p)$ by words of length $n$ is well-known \cite[Sect.~5.2]{Ne}.
It is a standard fact in Complex Dynamics that the repelling periodic points
of a rational map on $\CDach$ are dense in its Julia set.
The following statement is an analog of this for expanding Thurston maps.
As we will see, it easily follows from the proof of Theorem~\ref{thm:expThfactor}.
\begin{cor}\label{cor:perdense}
Let $f\:S^2\rightarrow S^2$ be an expanding Thurston map.
Then the periodic points of $f$ are dense in $S^2$. \end{cor}
\begin{proof} We use the notation and setup of the proof of Theorem~\ref{thm:expThfactor}.
It suffices to show that if $x\in S^2$ and $n\in \N$ are arbitrary, then there exists a point $z\in S^2$ with $f^n(z)=z$ and
$d(x,z)\lesssim \Lambda^{-n}$. Here and in the following, $C(\lesssim)$
is independent of $x$ and $n$.
To find such a point $z$, we apply Claim~1 in the proof of
Theorem~\ref{thm:expThfactor} and conclude that there exists $y\in f^{-n}(p)$ with
$d(x,y)\lesssim \Lambda^{-n}$. By Claim~3 in this proof there exists a word
$w\in J^*$ of length $n$ such that $\psi(w)=y$. Let $s$ be the unique sequence obtained by periodic repetition of the letters in $w$, i.e., $s\in J^\omega$ is the unique sequence with $[s]_n=w$ and $\Sigma^n(s)=s$.
Put $z:=\varphi(s)$. Then Claim~5 in the proof of Theorem~\ref{thm:expThfactor} implies
$$ f^n(z)=f^n(\varphi(s))=\varphi(\Sigma^n(s))=\varphi(s)=z.$$
Moreover, by \eqref{eq:ssnclose} we have
$$d(y,z)=d(\psi(w), \varphi(s))=d(\psi([s]_n), \varphi(s))\lesssim \Lambda^{-n}, $$
and so
$$d(x,z)\le d(x,y)+d(y,z)\lesssim \Lambda^{-n}. $$
The statement follows. \end{proof}
\section{Isotopies}
\label{sec:iso}
\noindent
In this section we present some topological facts about isotopies that will be important throughout the paper.
Let $I=[0,1]$, and $X$ and $Y$ be topological spaces. Recall (see Section~\ref{sec:thmaps}) that an isotopy between $X$ and $Y$
is a continuous map $H\: X\times I \rightarrow Y$ such that each
map $H_t:=H(\cdot, t)$ is a homeomorphism of $X$ onto $Y$.
If $A\subset X$, then $H$ is an isotopy\index{isotopy} {\em relative to $A$} or {\em rel.\ $A$} if
$H_t(a)=H_0(a)$ for all $a\in A$ and $t\in I$.
If $\varphi\: X\rightarrow Y$ and $\psi\:X\rightarrow Y$ are homeomorphisms, we say that
$\varphi$ and $\psi$ are {\em isotopic (rel.\ $A\subset X$)} if there exists an
isotopy $H\: X\times I \rightarrow Y$ (rel.\ $A$) such that $H_0=\varphi$ and
$H_1=\psi$.
When we say that a family $H_t$ (where it is understood that $t\in I$) of homeomorphisms from $X$ onto $Y$ is an isotopy between $X$ and $Y$, we
consider $t$ as a variable
in $I$ and mean that the map $(x,t)\in X\times I\mapsto H_t(x)$
is an isotopy. This is a slightly imprecise, but convenient way of expression.
If $X=Y$ then $H_t$ is called an isotopy on $X$.
If $A,B,C\subset X$, then we say that $B$ is {\em isotopic to $C$ rel.\ $A$} or {\em $B$ can be isotoped into $C$ rel.\ $A$} if
there exists an isotopy $H\: X\times I\rightarrow X$ rel.\ $A$ with
$H_0=\id_X$ and $H_1(B)=C$. Note that this notion depends on the ambient space $X$ containing the sets $A$, $B$, $C$.
\subsection{Equivalent expanding Thurston maps are topologically conjugate}
\label{sec:topol-interl-1}
Recall that two Thurston maps $f\: S^2\rightarrow S^2$ and $g\: \widehat S^2\rightarrow \widehat S^2$ on $2$-spheres $S^2$ and $\widehat S^2$ are (Thurston)
equivalent (see Definition~\ref{def:Thequiv}) if there exist homeomorphisms
$h_0,h_1\:S^2\rightarrow \widehat S^2 $ that are isotopic rel.\ $\operatorname{post}(f)$ and satisfy that
$ h_0\circ f = g\circ h_1$.
We then have the commutative diagram:
\begin{equation}\label{Thequiv2}
\xymatrix{
S^2 \ar[r]^{h_1} \ar[d]_f & \widehat S^2 \ar[d]^g
\\
S^2 \ar[r]^{h_0} & \widehat S^2.
}
\end{equation}
The maps $f$ and $g$ are topologically conjugate if there exists a homeomorphism $h\: S^2\rightarrow \widehat S^2 $ such that $h\circ f = g\circ h$.
Obviously, the notion of Thurston equivalence is weaker than
topological conjugacy of the maps.
We will show that under the additional assumption that the maps are expanding, we can promote an equivalence between two Thurston maps to a
topological conjugacy. The idea for the proof of this statement uses well-known ideas in dynamics.
A statement very similar to Theorem~\ref{thm:exppromequiv}
below was proved by Kameyama \cite{Ka03a}. Since his notion of ``expanding'' is different from ours, we will present the details of the proof.
First, we state a lifting theorem that will be needed (see \cite[Lem.~4.3]{Ka03a}).
\begin{prop}[Isotopy lifting for branched covers]\label{prop:isotoplift}\index{isotopy!lift}
Let $f\: S^2\to S^2$ and $g\:\widehat S^2\rightarrow \widehat S^2$ be Thurston maps, and $h_0,\widetilde h_0\colon S^2\to\widehat S^2$ be homeomorphisms
such that $h_0|\operatorname{post}(f)=\widetilde h_0|\operatorname{post}(f)$ and $h_0\circ f = g\circ \widetilde h_0$. Suppose $H\:S^2\times I\rightarrow \widehat S^2$ is an isotopy
rel.\ $\operatorname{post}(f)$ with $H_0=h_0$.
Then the isotopy $H$ uniquely lifts to an isotopy $\widetilde{H}\:S^2\times I\rightarrow \widehat S^2$ rel.\ $f^{-1}(\operatorname{post}(f))$ such that $\widetilde{H}_0=\widetilde h_0$ and $g\circ \widetilde{H}_t=H_t\circ f $ for all
$t\in I$. \end{prop}
So if we set ${h}_1:={H}_{1}$ and $\widetilde{h}_1:=\widetilde{H}_{1}$, then we have the following commutative diagram:
\begin{equation*}
\xymatrix{
S^2 \ar[rr]^{\widetilde{H}\colon\widetilde{h}_0\simeq \widetilde{h}_1} \ar[dd]_f
& & \widehat S^2 \ar[dd]^g
\\ & & &
\\
S^2 \ar[rr]^{ H\colon h_0\simeq h_1}
& &\widehat S^2.
}
\end{equation*}
\begin{proof} By an argument similar to the one used to establish \eqref{eq:postfg} one can show that
\begin{equation}\label{postfg2}
h_0(\operatorname{post}(f))=\widetilde h_0(\operatorname{post}(f))=\operatorname{post}(g).
\end{equation} This implies
that $$H_t(\operatorname{post}(f))=\operatorname{post}(g)$$ for all $t\in I$.
Hence $H_t|S^2\setminus \operatorname{post}(f)$ is an isotopy
between $S^2\setminus \operatorname{post}(f)$ and $\widehat S^2\setminus \operatorname{post}(g)$.
Moreover, it follows from \eqref{postfg2} that
$$ \widetilde h_0(f^{-1}(\operatorname{post}(f)))= g^{-1}(\operatorname{post}(g)).$$ So the map
$\widetilde h_0|S^2\setminus f^{-1}(\operatorname{post}(f))$ can be considered as a lift
of $$H_0|S^2\setminus \operatorname{post}(f)=h_0|S^2\setminus \operatorname{post}(f)$$ by the
(non-branched) covering maps
$$f\: S^2\setminus f^{-1}(\operatorname{post}(f))\rightarrow S^2\setminus \operatorname{post}(f)$$
and $$g\: \widehat S^2\setminus g^{-1}(\operatorname{post}(g))\rightarrow \widehat S^2\setminus \operatorname{post}(g).$$
By the usual homotopy lifting theorem for covering maps (see \cite[p.~60, Prop.~1.30]{Ha})
the isotopy $H_t|S^2\setminus \operatorname{post}(f)$ lifts to a unique isotopy
$\widetilde{H}_t$ between $S^2\setminus f^{-1}(\operatorname{post}(f))$ and $\widehat S^2\setminus g^{-1}(\operatorname{post}(g))$ such that
$$\widetilde{H}_0=\widetilde h_0| S^2\setminus f^{-1}(\operatorname{post}(f))$$ and
$g\circ \widetilde{H}_t=H_t\circ f$ on $S^2\setminus f^{-1}(\operatorname{post}(f))$
for all $t\in I$.
Since $H_t$ is constant in $t$ on $\operatorname{post}(f)$, each map $\widetilde{H}_t$ has a continuous extension to $S^2$, also denoted by $\widetilde{H}_t$. Then $\widetilde{H}_t|f^{-1}(\operatorname{post}(f))$
does not depend on $t$. Moreover, each map $\widetilde{H}_t$ is a homeomorphism from $S^2$ onto $\widehat S^2$, because an inverse of $\widetilde{H}_t$ can be obtained by lifting the isotopy $H^{-1}_t$. So $\widetilde{H}_t$
is an isotopy between $S^2$ and $\widehat S^2$ rel.\ $f^{-1}(\operatorname{post}(f))$. It is clear that it has the desired properties.
\end{proof}
Note that if in the previous proposition $H$ is an isotopy relative to a set $M\subset S^2$
with $\operatorname{post}(f)\subset M$, then the lift
$\widetilde H$ is an isotopy rel.\ $f^{-1}(M)$.
Indeed, if $p\in f^{-1}(M)$, then $f(p)\in M$ and so
$$g(\widetilde H_t(p))=H_t(f(p))=h_0(f(p))=:\!q$$
for all $t\in I$. Thus $t\mapsto \widetilde H_t(p)$ is a path
contained in the finite set $g^{-1}(q)$ and hence a constant path.
For later use we record a simple lemma about preimages of sets.
\begin{lemma}
\label{lem:lifts_inverses}
Let $f\:X\rightarrow X$ and $g\:Y\rightarrow Y$ be maps
defined on some sets $X$ and $Y$, and $h_0,h_1\:X\rightarrow Y$ be bijections
with $g\circ h_1=h_0\circ f $. Then for every set $A\subset X$ we have
\begin{equation*}
g^{-1}(h_0(A))= h_1(f^{-1}(A)).
\end{equation*}
\end{lemma}
\begin{proof} Under the given assumptions, consider a set $A\subset X$, and let $y\in g^{-1}(h_0(A))\subset Y$ be arbitrary. Since $h_1$ is a bijective, there exists $x\in X$ with
$h_1(x)=y$. Then
$$h_0(f(x))=g(h_1(x))=g(y)\in h_0(A), $$
and so, since
$h_0\:X\rightarrow Y$ is also a bijection, we have $f(x)\in A$. This implies that $x\in f^{-1}(A)$ and $y=h_1(x)\in h_1(f^{-1}(A))$. Thus $g^{-1}(h_0(A))\subset h_1(f^{-1}(A))$.
For the other inclusion, let $y\in h_1(f^{-1}(A))$ be arbitrary. Then
$$g(y)\in (g\circ h_1)(f^{-1}(A))=(h_0\circ f)(f^{-1}(A))\subset
h_0(A), $$ and so
$y\in g^{-1}(h_0(A))$. Thus, $h_1(f^{-1}(A))\subset
g^{-1}(h_0(A))$, and the claim follows.
\end{proof}
The following lemma will be of crucial importance.
\begin{lemma}[Exponential shrinking of tracks of isotopies]
\label{lem:exp_shrink}$\quad$\\
Let $f\: S^2\to S^2$ and $g\:\widehat S^2\rightarrow \widehat S^2$ be
Thurston maps, and $H^n\:S^2\times I\rightarrow \widehat S^2$ be isotopies
rel.\ $\operatorname{post}(f)$ satisfying $g\circ H^{n+1}_t=H^n_t\circ f $ for
$n\in \N_0$ and $t\in I$.
If $g$ is expanding and $\widehat S^2$ is equipped with a visual metric for $g$,
then the tracks of the isotopies
$H^n$ shrink exponentially as $n\to \infty$; more precisely, if $d$ is a visual metric for $g$ with expansion factor $\Lambda>1$, then there exists a constant $C\ge 1$ such that \begin{equation}\label{eq:expshrink}
\sup_{x\in S^2}\diam_d( \{H^n_t(x):t\in I\})\leq C \Lambda^{-n}
\end{equation}
for all $n\in \N_0$.
\end{lemma}
\begin{proof}
For all
$n\in \N_0$ and $t\in I$ we have $g^n\circ H^n_t=H^0_t\circ f^n$; so
for fixed $x\in S^2$ and $n\in \N_0$ the path $t\mapsto H^n_t(x)$ in
$\widehat S^2$ is a lift of the path $t\mapsto H^0_t(f^n(x))$ by the
map $g^n$. Recall that in the proof of Lemma \ref{lem:liftpathshrinks}
we had to break up the path $\gamma$ into $N$ pieces $\gamma_j$ so
that $\diam (\gamma_j)<\delta_0$ (see also (\ref{defdelta})).
Since $H^0$ is uniformly continuous we can choose the number $N$
uniformly for all the paths $t\mapsto H^0_t(y)$, $y\in S^2$. Since
$g$ is expanding, Lemma~ \ref{lem:liftpathshrinks} then implies that
\begin{equation*}
\sup_{x\in S^2}\diam_d( \{H^n_t(x):t\in I\})\lesssim \Lambda^{-n}
\end{equation*}
for all $n\in \N$, where $C(\lesssim)$ is independent of $n$.
\end{proof}
\begin{theorem}[Thurston equivalence implies topological conjugacy]
\label{thm:exppromequiv}\index{Thurston equivalent}
Let $f\: S^2\to S^2$ and $g\:\widehat S^2\to \widehat S^2$ be
equivalent Thurs\-ton maps that are expanding. Then they are
topologically conjugate.\index{topologically conjugate}
More precisely,
if we have a Thurston equivalence between $f$ and $g$ as in
\eqref{Thequiv2}, then there exists a homeomorphism $h\:S^2\rightarrow
\widehat S^2$ such that $h$ is isotopic to $h_1$
rel.~$f^{-1}(\operatorname{post}(f))$ and satisfies $h\circ f=g\circ h$.
\end{theorem}
Since $\operatorname{post}(f)\subset f^{-1}(\operatorname{post}(f))$ and $h_0$ and $h_1$ are isotopic rel.\ $\operatorname{post}(f)$ this implies that $h$ is also isotopic to $h_0$ rel.\ $\operatorname{post}(f)$.
\begin{proof} The main idea of the proof is to lift a suitable initial isotopy repeatedly and use the fact that by Lemma \ref{lem:exp_shrink} the tracks of the isotopies shrink exponentially fast. The desired conjugacy is then obtained as a limit.
By assumption there exists an isotopy $H^0_t$ between $S^2$ and $\widehat S^2$
rel.\ $\operatorname{post}(f)$ such that $h_0\circ f=g\circ h_1$, where
$h_0=H^0_0$ and $h_1=H^0_1$.
By Proposition~\ref{prop:isotoplift} we can lift
the isotopy $H_t^0$ between $h_0$ and $h_1$ to an isotopy $H^1_t$
rel.\ $f^{-1}(\operatorname{post}(f))\supset \operatorname{post}(f)$ between $h_1$ and $h_2:=H^1_1$.
Note that the map $h_1$ plays two roles here: it is the endpoint $H^0_1$ of the initial isotopy $H_t^0$, and also a lift of $h_0$.
Repeating this argument we get homeomorphisms $h_n$
and isotopies $H^n_t$ between $S^2$ and $\widehat S^2$ rel.\ $f^{-n}(\operatorname{post}(f))\supset \operatorname{post}(f)$ for all
$n\in \N_0$ such $H_t^{n}\circ f =g \circ H_t^{n+1}$,
$H^n_0=h_{n} $ and $H^n_1=h_{n+1} $ for all $n\in \N_0$ and $t\in I$.
So we have the following ``infinite tower" of isotopies:
\begin{equation*}
\xymatrix{
{}\ar[d] & \overset{\vdots}{\phantom{X}} & {}\ar[d]
\\ S^2 \ar[rr]^{H^2\colon h_2\simeq h_3} \ar[dd]_f
& & \widehat S^2 \ar[dd]^g
\\ & &
\\ S^2 \ar[rr]^{H^1\colon h_1\simeq h_2} \ar[dd]_f
& & \widehat S^2 \ar[dd]^g
\\ & &
\\ S^2 \ar[rr]^{H^0\colon h_0\simeq h_1}
& & \widehat S^2
}
\end{equation*}
We want to show that for $n\to \infty$ the maps $h_n$ converge to a homeomorphism
$h_\infty$ that gives the desired topological conjugacy between $f$ and $g$.
To see this fix a visual metric $d$ on $\widehat S^2$, and assume that it has the
expansion factor $\Lambda>1$. Metric concepts on $\widehat S^2$ will refer to this metric in the following. Since $g$ is expanding, Lemma~ \ref{lem:exp_shrink} implies that
\begin{equation}\label{diamtracks}
\sup_{x\in S^2}\diam( \{H^n_t(x):t\in I\})\lesssim \Lambda^{-n}
\end{equation}
for all $n\in \N$, where $C(\lesssim)$ is independent of $n$.
In particular,
$$ \operatorname{dist}(h_{n+1},h_n):=\sup_{x\in S^2} \operatorname{dist}(h_n(x), h_{n+1}(x))
\lesssim \Lambda^{-n}$$ for all $n\in \N_0$, and so there is a continuous map
$h_\infty\: S^2\rightarrow \widehat S^2$ such that $h_n\to h_\infty$ uniformly on $S^2$ as $n\to \infty$. Since
$h_{n-1}\circ f= g\circ h_n$, we have
$h_{\infty}\circ f=g\circ h_\infty$.
The map $h_\infty$ is a homeomorphism. To see this we repeat the argument where we interchange the roles of $f$ and $g$. More precisely, we consider the isotopy $(H_t^0)^{-1}$ between $h_0^{-1}$ and $h_1^{-1}$. The corresponding tower of repeated lifts of this initial isotopy is given by the isotopies $(H^n_t)^{-1}$ between
$h_{n}^{-1}$
and $h_{n+1}^{-1}$. By the argument in the first part of the proof we see that the maps $h_n^{-1}$ converge to a continuous map
$k_\infty\: \widehat S^2\rightarrow S^2$ uniformly on $\widehat S^2$ as $n\to \infty$.
By uniform convergence we have
$k_\infty\circ h_\infty(x)=\lim_{n\to \infty} h_n^{-1}\circ h_n(x)=x$
for all $x\in S^2$. Hence $k_\infty\circ h_\infty=\text{id}_{S^2}$. Similarly,
$h_\infty\circ k_\infty=\text{id}_{\widehat S^2}$, and so $k_\infty$ is a continuous inverse
of $h_\infty$. Hence $h_\infty$ is a homeomorphism.
The conjugating map $h=h_\infty$ is isotopic to
$h_1$ rel.~$f^{-1}(\operatorname{post}(f))$. To see this we will define an isotopy rel.\ $f^{-1}(\operatorname{post}(f))$
that is obtained by concatenating (with suitable time change) the isotopies $H^1, H^2, \dots$
and take $h=h_\infty$ as the endpoint at time $t=1$. The precise
definition is as follows.
We break up the unit interval into intervals
\begin{equation*}
I=[0,1]= \left[0,\frac{1}{2}\right]\cup
\left[\frac{1}{2},\frac{3}{4}\right] \cup \dots \cup
\left[1-2^{-n},1 -2^{-n-1}\right]
\cup \dots \cup \{1\}.
\end{equation*}
The $n$-th interval in this union is denoted by $I^n=[1-2^{-n},1
-2^{-n-1}]$. Let $s_n\colon I^n \to I$, $s_n(t)= 2^{n+1}(t- (1
-2^{-n}))$, for $n\in \N_0$. We define
$H\: S^2\times I\rightarrow \widehat S^2$ by
\begin{align*}
H(x,t) := H^{n+1}(p,s_n(t))
\end{align*}
if $p\in S^2$ and $t\in I^n$ for some $n\in \N_0,$
and $H(p,t)=h(p)$ for $p\in S^2$ and $t=1$.
We claim that $H$ is indeed an isotopy between $h_1$ and $h$ rel.~$f^{-1}(\operatorname{post}(f))$.
Note that $H$ is well defined, $H_1=h$, and $H_{1-1/2^n}=h_{n+1}$ for $n\in \N_0$. Moreover, $H_t$ is a homeomorphism for each $t\in I$, and $H_t|f^{-1}(\operatorname{post}(f))$ does not depend on $t$.
To establish our claim, it remains to verify that $H$ is continuous. It is clear that
$H$ is continuous at each point $(p,t)\in S^2\times [0,1)$.
Moreover, as follows from the uniform convergence $h_n\to h$ as $n\to \infty$ and inequality \eqref{diamtracks}, we have $H_t\to H_1$ uniformly on $S^2$ as $t\to 1$. This together with the continuity of $h=H_1$ implies the continuity of $H$ at points $(p,t)\in S^2\times I$ with $t=1$.
\end{proof}
\begin{rem}\label{rem:betterandbetter}
The previous proof gives a procedure for approximating the conjugating map $h=h_\infty$. Indeed, as follows from the remark after the proof of Proposition~\ref{prop:isotoplift}, the map $H^n_t$ is constant in $t$ on $f^{-n}(\operatorname{post}(f))$
for all $n\in \N_0$. This implies that
$h_n=h_{n+1}=\dots =h_\infty$ on the set $f^{-n}(\operatorname{post}(f))$, and so the map $h_n$ sends the points in $f^{-n}(\operatorname{post}(f))$ to the ``right" points in $g^{-n}(\operatorname{post}(g))$. The isotopy $H_t^n$ then
deforms $h_n$ to a map $h_{n+1}$ such that the points in $f^{-(n+1)}(\operatorname{post}(f))$
have the correct images in $g^{-(n+1)}(\operatorname{post}(g))$ as well, etc.
Since by expansion the union of the sets
$$ \operatorname{post}(f)\subset f^{-1}(\operatorname{post}(f))\subset f^{-2}(\operatorname{post}(f))\subset \dots$$
is dense in $S^2$, this gives better and better approximations of limit map $h_\infty$.
\end{rem}
To record an immediate consequence of Theorem~\ref{thm:exppromequiv}, we introduce some terminology related to the notion of snowflake equivalent metrics defined in Section~\ref{sec:expansion}.
Let $(X,d_X)$ and $(Y, d_Y)$ be metric spaces.
A homeomorphism $h\: X\rightarrow Y$ is called a {\em snowflake equivalence} if there exist constants $\alpha>0$ and $C\ge 1$ such that
$$\frac 1C d_X(x,x')^\alpha \le d_Y(h(x),h(x'))\le Cd_X(x,x')^\alpha$$
for all $x,x'\in X$. The spaces $X$ and $Y$ are called {\em snowflake equivalent} if there exists a snowflake equivalence between $X$ and $Y$. Note that two metrics
$d$ and $d'$ on a space $X$ are snowflake equivalent as defined in Section~\ref{sec:expansion} if and only if the identity map $\id_X\:(X,d)\rightarrow (X,d')$ is a snowflake equivalence.
\begin{cor} \label{cor:conjisom} Let $f\:S^2\rightarrow S^2$ and
$g\:\widehat S^2\rightarrow \widehat S^2$ be expanding Thurs\-ton maps that are Thurston equivalent. Then $S^2$ equipped with any visual metric with respect to $f$ is snowflake equivalent to $\widehat S^2$ equipped with any visual metric with respect to $g$. Every homeomorphism $h\: S^2\rightarrow \widehat S^2$ satisfying $h\circ f=g\circ h$ is a snowflake equivalence.
\end{cor}
\begin{proof} By Theorem~\ref{thm:exppromequiv} we know that there exists a topological conjugacy between $f$ and $g$, i.e., a homeomorphism $h\: S^2\rightarrow \widehat S^2$ such that $h\circ f=g\circ h$.
Let $d$ be a visual metric on $S^2$ with respect to $f$, and $\widehat d$ be a visual metric on $\widehat S^2$ with respect to $g$. Let $\Lambda>1$ and $\widehat \Lambda>1$ be the expansion factors of $d$ and $\widehat d$, respectively.
It suffices to show that $h\: (S^2, d)\rightarrow (\widehat S^2, \widehat d)$ is a snowflake equivalence.
To see this pick a Jordan curve $\mathcal{C}\subset S^2$
with $\operatorname{post} (f)\subset \mathcal{C}$. Then $\widehat \mathcal{C}=h(\mathcal{C})$ is a Jordan curve
in $\widehat S^2$ with $\operatorname{post}(g)=h(\operatorname{post}(f))\subset \widehat \mathcal{C}$.
Since $h$ conjugates $f$ and $g$, it follows from Proposition~\ref{prop:celldecomp} (iii) and (v) or, alternatively, from the uniqueness statement in Lemma~\ref{lem:pullback} that for each $n\in \N_0$ the images of the
cells in the cell decomposition $\mathcal{D}^n:=\mathcal{D}^n(f,\mathcal{C})$ of $S^2$ under the map $h$ are precisely the cells in the cell decomposition $\widehat
\mathcal{D}^n=\mathcal{D}^n(g, \widehat \mathcal{C})$ of $\widehat S^2$; so we have
\begin{equation} \label{eq:CChCC}
\widehat \mathcal{D}^n=\{h(c): c\in \mathcal{D}^n\}
\end{equation} for all $n\in \N_0$.
This implies that
$$\widehat m(h(x),h(x'))=m(x,x')$$ for all $x,x'\in S^2$, where
$\widehat m=m_{g, \widehat \mathcal{C}}$ and $m=m_{f, \mathcal{C}}$ (recall Definition~\ref{def:mxy}).
Combining this with Proposition~\ref{prop:visualsummary} (iii) we see that
$$ \widehat d(h(x), h(x'))\asymp
\widehat \Lambda^{-\widehat m(h(x),h(x'))}= \widehat \Lambda^{-m(x, x')}= \Lambda^{-\alpha m(x, x')}\asymp d(x, x')^\alpha$$ for all $x,x'\in S^2$, where
$\alpha=\log(\widehat \Lambda)/\log(\Lambda)$ and the implicit multiplicative constants do not depend on $x$ and $x'$. It follows that $h$ is a snowflake equivalence.
\end{proof}
\subsection{Isotopies of Jordan curves}
\label{sec:isotopy-rel.-postf}\index{isotopy!of Jordan curve}
In the following $S^2$ is a $2$-sphere
equipped with a fixed base metric. It will be the ambient space for
all isotopies.
In this subsection we study the problem
when two Jordan curves $J$ and
$K$ on $S^2$ passing through a given finite set $P$ of points in the same order can be deformed into each other by an isotopy of $S^2$ rel.\ $P$.
If $\#P\le 3$ this is always the case (see
Lemma~\ref{lem:deform<4} below).
For $\#P\ge 4$ this is not always true as the example
in Figure \ref{fig:isotopcounter} shows. Here $K=S^1$ is the unit
circle and $P=\{1,{\mathbf{\imath}}, -1,-{\mathbf{\imath}}\}\subset S^1$. The Jordan curve $J$ (which
contains $P$) is drawn with a thick line. The curves $K=S^1$and $J$ are not
isotopic rel.\ $P$. In fact, $J$ may be obtained from $S^1$ by a
``Dehn twist" around the points $-{\mathbf{\imath}}$ and $1$. Note that in this example we
can make the Hausdorff distance (see \eqref{eq:def_Hausdorffd})
between $J$ and $S^1$ arbitrarily small.
\ifthenelse{\boolean{nofigures}}{}{
\begin{figure}
\centering
\begin{overpic}
[width=6cm,
tics=20]{isotoprelncounter.eps}
\put(70,86){$S^1$}
\put(91,45){$J$}
\put(40,90){$\scriptstyle{{\mathbf{\imath}}}$}
\put(-9,58){$\scriptstyle{-1}$}
\put(29,23){$\scriptstyle{-{\mathbf{\imath}}}$}
\put(84,58){$\scriptstyle{1}$}
\end{overpic}
\caption{$J$ is not isotopic to $S^1$ rel.\ $\{1,{\mathbf{\imath}}, -1, -{\mathbf{\imath}}\}$.}
\label{fig:isotopcounter}
\end{figure}
}
We will need the following statement.
\begin{prop} \label{prop:isotreln}
Suppose $J$ is a Jordan curve in $S^2$
and $P\subset J$ a set consisting of $n\ge 3$ distinct points $p_1, \dots, p_n,
p_{n+1}=p_1$ in cyclic order on $J$. For $n=1,\dots, n$ let
$\alpha_i$ be the unique arc on $J$ with endpoints $p_i$ and $p_{i+1}$
such that $\inte(\alpha_i)\subset J\setminus P$. Then there exists $\delta>0$ with the following property:
Let $K$ be another Jordan curve in $S^2$
passing through the points $p_1, \dots, p_n$ in cyclic order, and let $\beta_i$ for $i=1, \dots, n$ be the arc with endpoints $p_i$ and $p_{i+1}$ such that $\inte(\beta_i)\subset J\setminus P$.
If
\begin{equation*}
\beta_i \subset \mathcal{N}^\delta(\alpha_i)
\end{equation*}
for all $i=1, \dots, n$, then there exists an isotopy $H_t$ on
$S^2$ rel.\ $P$ such that $H_0=\id_{S^2}$ and
$H_1(J)=K$.
\end{prop}
In other words, if the arcs $\beta_i$ of the Jordan curve $K$ are contained in sufficiently small neighborhoods of the corresponding arcs
$\alpha_i$ of $J$, then one can deform $J$ into $K$ by an isotopy of
$S^2$ that keeps the points in $P$ fixed. Even though this statement
seems ``obvious'', a complete proof is surprisingly difficult and involved.
As we will see, the proof of this statement easily follows from two lemmas
in \cite{Bu}.
\begin{lemma} \label{twoarcs}
Let $\Om\subset S^2$ be a simply connected region, $p,q\in \Om$ distinct points,
and $\alpha$ and $\beta$ arcs in $\Om$ with endpoints $p$ and $q$. Then $\alpha$ is isotopic to $\beta$ rel.\ $\{p,q\} \cup
S^2\setminus\Om$.
\end{lemma}
So arcs in a simply connected region with the same endpoints can be deformed
into each other so that the endpoints and the complement of the region stay fixed. The lemma follows from \cite[p.~413, A.6 Thm.~(ii)]{Bu}).
\begin{lemma} \label{isotopylem}
Suppose we have two Jordan curves $J$ and $K$
as in Proposition~\ref{prop:isotreln} such that for each $i=1, \dots, n$
the arc $\alpha_i$ is isotopic to $\beta_i$ rel.\ $P$.
Then $J$ is isotopic to $K$ rel.\ $P$.
\end{lemma}
This is essentially \cite[p.~411, A.5 Thm.]{Bu}.
\begin{proof}[Proof of Proposition~\ref{prop:isotreln}]
For each arc $\alpha_i$ there exists a simply connected region $\Om_i$ that contains
$\alpha_i$ but does not contain any element of $P$ different from the endpoints of $\alpha_i$. There exists $\delta>0$ such that $\mathcal{N}^\delta(\alpha_i)\subset \Om_i$ for all
$i=1, \dots, n$.
Then by Lemma~\ref{twoarcs} every arc $\beta_i $ in $\mathcal{N}^\delta(\alpha_i)$ with the same endpoints as $\alpha_i$ can be isotoped to
$\alpha_i$ rel.\ $P$.
The proposition now follows from Lemma~\ref{isotopylem}.
\end{proof}
If $\#P\le 3$ in Proposition~\ref{prop:isotreln}, then $J$ can always be isotoped to $K$ rel.\ $P$.
\begin{lemma} \label{lem:deform<4} Suppose $J$ and $K$ are Jordan curves in $S^2$ and
$P\subset J\cap K$ is a set with $\#P\le 3$. Then $J$ is isotopic to $K$ rel.\ $P$. \end{lemma}
\begin{proof} Suppose first that $P$ consists of exactly three distinct points
$p_1$, $p_2$, $p_3$. Define the arcs $\alpha_i$ and $\beta_i$ as in Proposition~\ref{prop:isotreln}. Then
for each $i=1, 2,3$ the arcs $\alpha_i$ and $\beta_i$ have the same endpoints $p_i$ and $p_{i+1}$ and are contained the simply connected region $\Om_i=S^2\setminus\{p_{i+2}\}$, where indices are understood modulo $3$. Hence by Lemma~\ref{twoarcs} each arc
$\alpha_i$ is isotopic to $\beta_i$ rel.\ $P$.
Again Lemma~\ref{isotopylem} implies that $J$ is isotopic to $K$ rel.\ $P$.
If $\#P\le 2$, we may assume that $S^2=\CDach$. Then by applying the first part of the proof (by adding auxiliary points to $P$) one sees that both $J$ and $K$ are isotopic to circles on $\CDach$ rel.\ $P$. Hence $J$ is isotopic to $K$ rel.\ $P$. \end{proof}
\begin{lemma} \label{lem:homeo} Let $S^2$ and $\widehat S^2$ be oriented $2$-spheres, and $P\subset S^2$ be a set with $\#P\le 3$.
If $\alpha\: S^2\rightarrow\widehat S^2$ and $\beta\: S^2\rightarrow\widehat S^2$
are orientation-preserving homeomorphisms with $\alpha|P=\beta|P$, then $\alpha$ and $\beta$ are isotopic rel.\ $P$.
\end{lemma}
\begin{proof} The statement is essentially well-known. For the sake of completeness we will give a proof, but will leave some of the details to the reader. These details can easily be filled in by using the facts about isotopies that will be discussed
later before Proposition~\ref{prop:thurstonex}.
In the proof of the uniqueness part of this proposition, we will use very similar arguments.
By considering $\alpha\circ \beta^{-1}$ one
can reduce the lemma to the case where $S^2=\widehat S^2$ and
$\beta=\id_{S^2}$. Then $\alpha$ fixes the points in $P$, and we have to
show that $\alpha$ is isotopic to $\id_{S^2}$ rel.\ $P$. We first assume that $\#P=3$.
Pick a Jordan curve $K\subset S^2$ with $P\subset K$, and let $J=\alpha(K)$. Then $P\subset J\cap K$, and so by Lemma~\ref{lem:deform<4} the Jordan curve
$J$ can be isotoped into $K$ rel.\ $P$. This implies that $\alpha$ is isotopic rel.\ $P$ to a homeomorphism $\alpha_1$ on $S^2$
with $\alpha_1(K)=K$. Let $e$ be one of the three subarcs of $K$ determined $P$. Since $\alpha_1$ fixes the three points in
$P$, this map restricts to a homeomorphism of $e$ that does not move the endpoints
of $e$. Hence on $e$ the map $\alpha_1$ is isotopic to the identity on $e$ rel.\ $\partial e$. By pasting the isotopies on these arcs together,
one can find an isotopy $h\: K\times I\rightarrow K$ rel.\ $P$ such that
$h_0=\alpha_1|K$ and $h_1=\id_{K}$. One can extend $h$ to each of the complementary components of $K$ to obtain an isotopy $H\: S^2\times I\rightarrow S^2$ rel.\ $P$ such that $H_1=\id_{S^2} $ and $H(p,t)=h(p,t)$ for all $p\in K$ and $t\in I$. Then $\alpha_2:=H_0$ is a homeomorphism on $S^2$ that is isotopic to $\id_{S^2}$ rel.\ $P$ such that $\alpha_1|K=\alpha_2|K$.
This implies that $\alpha_1$ and $\alpha_2$ are isotopic rel.\ $K\supset P$. If $\sim$ indicates that two homeomorphisms on $S^2$ are isotopic rel.\ $P$, then we have
$\alpha\sim \alpha_1 \sim \alpha_2\sim \id_{S^2}$, and so
$\alpha\sim \id_{S^2}$ as desired.
If $\#P \le 2$, then we pick a set $P'\subset S^2$ with $\#P'=3$ and $P'\supset P$. By the first part of the proof it suffices to find an isotopy rel.\ $P$
of the given map $\alpha$ to a homeomorphism $\alpha'$ that fixes the points in $P'$. It is clear that such an isotopy can always be found; for an explicit construction one can assume that $S^2=\CDach$
and can then obtain the desired isotopy by post-composing $\alpha$ by a suitable continuous family of M\"obius transformations, for example.
\end{proof}
The following lemma will be crucial for the proof of the uniqueness statement on invariant Jordan curves.
In its proof we will use the following topological fact: if $D$ is a two-dimensional cell and $\varphi\: D
\rightarrow S^2$ is a continuous map such that $\varphi| \partial D$ is injective, then the set
$\varphi(\inte(D))$
contains one of the two complementary components of the Jordan curve $\varphi(\partial D)$.
Indeed, by applying the Sch\"onflies Theorem and using auxiliary homomorphisms
we can reduce to the case where $D= \overline \D$, $S^2=\CDach$, $\varphi|\partial \D=\id_{\partial \D}$, and $\infty\notin \varphi(D)$.
Then $\D\subset \varphi(\D)$. This follows from a simple degree argument and the statement can be generalized to higher dimensions; for an elementary exposition of this and related facts
in dimension $2$ see \cite{Bur}, in particular \cite[Cor.~3.5]{Bur}.
\begin{lemma} \label{lem:isoJcin1ske} Let $\mathcal{D}$ be a cell decomposition of $S^2$ with $1$-skeleton $E$ and vertex set ${\bf V}$, and suppose that every tile in $\mathcal{D}$ contains at least three vertices in its boundary. If $J$ and $K$ are Jordan curves that are both contained in $E$ and are isotopic rel.\ ${\bf V}$, then $J=K$. \end{lemma}
\begin{proof} Let $H\: S^2\times I\rightarrow S^2$ be an isotopy rel.\ ${\bf V}$
such that $H_0=\id_{S^2}$ and $H_1(J)=K$.
Note that if $M\subset S^2$ is a set disjoint from ${\bf V}$, then it remains disjoint from ${\bf V}$ during the isotopy, i.e., if $M\cap {\bf V}=\emptyset$, then $H_t(M)\cap {\bf V}=\emptyset$ for all
$t\in I$. This follows from the fact that each map $H_t$, $t\in [0,1]$, is a homeomorphism on $S^2$ with $H_t|{\bf V}=\id_{{\bf V}}$.
\smallskip
Let $e$ be an edge in $\mathcal{D}$. We claim that if $H_1(e)\subset E$, then $H_1(e)=e$.
First note that $H_1(e)$ is an edge in $\mathcal{D}$.
Indeed, since $\partial e\subset {\bf V}$ and the isotopy $H$ does not move vertices,
the arc $H_1(e)$ has the same endpoints as $e$. Moreover, $\inte(e)\cap {\bf V}=\emptyset$, and so $H_1(\inte(e))\cap {\bf V}=\emptyset$ by what we have just seen. So $H_1(\inte(e))$ is a connected set in the $1$-skeleton $E$ of $\mathcal{D}$ disjoint from the $0$-skeleton ${\bf V}$. By Lemma~\ref{lem:opencells} there exists an edge
$e'$ in $\mathcal{D}$ with $H_1(\inte(e))\subset \inte(e')$. Since the endpoints of $H_1(e)$ lie in
${\bf V}$, this implies that $e'=H_1(e)$.
To show that $e'=e$ we argue by contradiction and assume that $e\ne e'$. Then $e$ and $e'$ have the same endpoints, but no other points in common. Hence $\alpha=e\cup e'$ is a Jordan curve that contains two vertices, namely the endpoints of $e$ and $e'$, but no other vertices. Let $\Om_1$ and $\Om_2$ be the two open Jordan regions that form the complementary components of $\alpha$. Then both regions $\Om_1$ and $\Om_2$ contain vertices.
To see this note that the interior of every tile $X$ is a connected set disjoint from the $1$-skeleton $E$, and hence also disjoint from $\alpha$. Hence $\inte(X)$ is contained in $\Om_1$
or $\Om_2$. Moreover, since the union of the interiors of tiles is dense in $S^2$, both regions $\Om_1$ and $\Om_2$ must contain the interior of at least one tile.
Now consider $\Om_1$, for example, and pick a tile $X$ with $\inte(X)\subset \Om_1$.
Then by our hypotheses the set $X\subset \overline{\Om}_1=\Om_1\cup\alpha$ contains at least three vertices. Since only two of them can lie on $\alpha$, the set $\Om_1$ must contain a vertex. Similarly, $\Om_2$ must contain at last one vertex.
A contradiction can now be obtained from the fact that during the isotopy $H$ the set $\inte(e)$
remains disjoint from the set of vertices, but on the other hand it has to sweep out one of the domains $\Om_1$ or $\Om_2$ and hence it meets a vertex.
To make this rigorous, we apply the topological fact mentioned before the statement of the lemma.
Let $D$ be the quotient of the product space $e\times I $ obtained by identifying
all points $(u,t)$, $t\in I$, and by identifying
all points $(v,t)$, $t\in I$, where $u$ and $v$ are the two endpoints of $e$. Then $D$ is a two-dimensional cell. Since the isotopy $H$ does not move the points $u$ and $v$, the map $(p, t)\mapsto H_t(p)$ on $e\times I$ induces a continuous map $\varphi\: D \rightarrow S^2$.
Moreover, $\varphi | \partial D$ is a homeomorphism of $\partial \D$ onto $\alpha$.
Hence $\Om_1$ or $\Om_2$ is contained in the set
$$\varphi(\inte(D))=\bigcup_{t\in (0,1)} H_t(\inte(e)). $$
In particular, the set $\varphi(\inte(D))$ contains a vertex. This is a contradiction, because we know that
no set $H_t(\inte(e))$, $t\in I$, meets ${\bf V}$. Thus $H_1(e)=e$ as desired.
\smallskip
Having verified the statement about edges, it is now easy to see that $J=K$.
Indeed, $J$ is a union of edges in $\mathcal{D}$; to see this consider the components of the set
$J\setminus {\bf V}$. If $\gamma$ is such component, then $\overline \gamma \setminus \gamma\subset
{\bf V}$. Moreover, $\gamma$ is contained in the $1$-skeleton $E$, and does not meet the $0$-skeleton ${\bf V}$. Again by Lemma~\ref{lem:opencells} the set $\gamma$ must be contained in the interior $\inte(e)$
of some edge $e$. This is only possible if $\gamma=\inte(e)$. Hence $\overline \gamma=e$. Since $J$ is the union of the closures of these components $\gamma$, it follows that
$J$ is the union of edges $e$. For each such edge $e$ we have $H_1(e)\subset K\subset E$ and so $H_1(e)=e$ by the first part of the proof. This implies $J\subset K$. Since $J$ and $K$ are Jordan curves, the desired identity $J=K$ follows.
\end{proof}
\section{Thurston maps with invariant curves}
\label{sec:Thurtoncurves}
\noindent Let $f\: S^2\rightarrow S^2$ be a Thurston map, $\mathcal{C}\subset S^2$ be a Jordan curve with $\operatorname{post}(f)\subset \mathcal{C}$, and
$\mathcal{D}^n$ be the cell decomposition of $S^2$ given by the $n$-cells for $(f,\mathcal{C})$.
A set $M\subset S^2$ is called
{\em $f$-invariant}\index{invariant set} (or simply {\em invariant} if $f$ is understood) if
\begin{equation}
\label{eq:deffinv}
f(M)\subset M\quad \text{or equivalently} \quad M \subset f^{-1}(M).
\end{equation}
Since the set $\operatorname{post}(f)$ is $f$-invariant, we have
\begin{equation} \label {1postinv}
\operatorname{post}(f)\subset f^{-1}(\operatorname{post}(f))\subset f^{-2}(\operatorname{post}(f))\subset \dots
\end{equation}
We know (see Proposition~\ref{prop:celldecomp}~(iii)) that \eqref{1postinv} is equivalent to the inclusions
$${\bf V}^0\subset {\bf V}^1\subset {\bf V}^2\subset \dots $$
for the vertex sets of the cell decompositions $\mathcal{D}^n$.
In general, a similar inclusion chain will not hold for the $1$-skeleta $E^n:=f^{-n}(\mathcal{C})$ of $\mathcal{D}^n$, but if $\mathcal{C}$ is $f$-invariant, then we have
$$\mathcal{C}=E^0\subset E^1\subset E^2\subset \dots$$
Actually, more is true as the following statement shows.
\begin{prop}\label{prop:invmarkov} Let $k,n\in \N_0$, $f\: S^2\rightarrow S^2$ be a Thurston map, and $\mathcal{C}\subset S^2$ be an $f$-invariant Jordan curve with $\operatorname{post}(f)\subset \mathcal{C}$. Then we have:
\begin{itemize}
\smallskip
\item[(i)]$(\mathcal{D}^{n+k}, \mathcal{D}^k)$ is a cellular Markov partition for $f^n$.
\smallskip
\item[(ii)] Every $(n+k)$-tile $X^{n+k}$ is contained in a unique $k$-tile $X^k$.
\smallskip
\item[(iii)] Every $k$-tile $X^k$ is equal to the union of all $(n+k)$-tiles $X^{n+k}$
with $X^{n+k}\subset X^k$.
\smallskip
\item[(iv)] Every $k$-edge $e^k$ is equal to the union of all $(n+k)$-edges
$e^{n+k}$
with $e^{n+k}\subset e^k$.
\end{itemize}
\end{prop}
\begin{proof} (i) We know that the map $f^n$ is cellular for $(\mathcal{D}^{n+k}, \mathcal{D}^n)$ (Proposition~\ref{prop:celldecomp}); so we have to show that
$\mathcal{D}^{n+k}$ is a refinement of $\mathcal{D}^n$ (see Definition~\ref{def:ref}).
By the invariance of $\mathcal{C}$ we have
$E^{n+k}=f^{-(n+k)}(\mathcal{C})\supset E^k=f^{-k}(\mathcal{C})$, and so $S^2\setminus E^{n+k}\subset S^2\setminus E^k$.
To establish the first property of a refinement, we will show
that every $(n+k)$-cell is contained in some $k$-tile.
Let $\sigma$ be an arbitrary $(n+k)$-cell. If $\sigma$ is a
$(n+k)$-tile, then $\inte(\sigma)$ is a connected set in
$S^2\setminus E^{n+k}\subset S^2\setminus E^k$
and hence contained in the interior of a $k$-tile $\tau$ (see Proposition~\ref{prop:celldecomp}).
It follows that $\sigma=\overline {\inte(\sigma)}\subset \tau$.
If $\sigma $ is an $(n+k)$-edge or an $(n+k)$-vertex, then it is contained in an
$(n+k)$-tile (Lemma~\ref{lem:specprop}~(iv) and~(v)), and hence in some
$k$-tile by what we have just seen.
To establish the second property of a refinement, let $\tau$ be an arbitrary $k$-cell. We have to show that
the $(n+k)$-cells $\sigma$ contained in $\tau$ cover $\tau$.
If $\tau$ consists of a $k$-vertex $p$, then $p$ is also an $(n+k)$-vertex, and the statement is trivial.
If $\tau$ is a $k$-edge, consider the points in ${\bf V}^{n+k}$ that lie on $\tau$.
Note that this includes the elements of $\partial \tau \subset {\bf V}^k\subset {\bf V}^{n+k}$.
By using these points to partition $\tau$, we can find finitely many arcs $\alpha_1, \dots, \alpha_N$ such that
$\tau=\alpha_1\cup\dots\cup \alpha_N$, each arc $\alpha_i$ has endpoints in ${\bf V}^{n+k}\supset {\bf V}^k$ and has interior $\inte(\alpha_i)$ disjoint from
${\bf V}^{n+k}$. Then for each $i=1, \dots, N$ the set $\inte(\alpha_i)$ is a connected set in
$E^k\setminus {\bf V}^{n+k}\subset E^{n+k}\setminus {\bf V}^{n+k}$. It follows that $\inte(\alpha_i)$ and hence also $\alpha_i$ is contained in some $(n+k)$-edge $\sigma_i$ (Proposition~\ref{prop:celldecomp}~(v)).
Since the endpoints of $\alpha_i$ lie in ${\bf V}^k$, they cannot lie in $\inte(\sigma_i)$, and so they are also endpoints of $\sigma_i$. This implies that $\alpha_i=\sigma_i$. In particular, the $(n+k)$-edges $\sigma_1, \dots ,\sigma_N$ are contained in $\tau$ and form a cover of $\tau$. The statement follows in this case.
Finally, let $\tau$ be a $k$-tile. If $p\in\inte(\tau)$ is arbitrary, then $p$ is contained in an $(n+k)$-tile $\sigma$.
By the first part of the proof, $\sigma$ is contained in a $k$-cell. Since $\tau$ is the only $k$-cell that contains $p$, we must have $\sigma\subset \tau$.
This implies that the union of the $(n+k)$-cells contained in $\tau$ cover $\inte(\tau)$. On the other hand, this union consists of finitely many tiles and is hence a closed set.
It follows that the union also contains $\overline{\inte(\tau)}=\tau$.\smallskip
(ii) We have just seen that every $(n+k)$-tile $X^{n+k}$ is contained in a $k$-tile $X^k$. This tile is unique. For suppose $\widetilde X^k$ is another $k$-tile with $X^{n+k}\subset \widetilde X^k$.
Then
$$ \emptyset\ne \inte(X^{n+k})\subset \inte(X^k)\cap \inte(\widetilde X^k),$$
and so $X^k$ and $\widetilde X^k$ have common interior points. This implies
$X^k=\widetilde X^k$. \smallskip
(iii)--(iv): Both statements were established in the proof of (i).
\end{proof} \medskip
Let $f$ and $\mathcal{C}$ be as as in the previous proposition. Then by (i) the pair $(\mathcal{D}^1, \mathcal{D}^0)$ is a cellular Markov partition for $f$,
and this partition generates the cell decompositions $\mathcal{D}^n$
as in Proposition~\ref{prop:inftychain}.
If $X^n $ is any $n$-tile, then by (ii) there exist unique
$i$-tiles $X^i$ for $i=0, \dots, n-1$ such that
$$X^n\subset X^{n-1}\subset \dots\subset X^0. $$
We refer to the statements (iii) and (iv) informally by saying that
tiles and edges are ``subdivided" by tiles and edges of higher order. Let
$\mathcal{S}=\mathcal{S}(f,\mathcal{C})$ denote the set of all sequences $\{X^n\}$, where $X^n$ is an $n$-tile for $n\in \N_0$ and
\begin{equation*}
X^0\supset X^1\supset X^2\supset \dots \end{equation*}
Since tiles are subdivided by tiles of higher order, for each point
$p\in S^2$ we can find a sequence $\{X^n\}\in \mathcal{S}$ such that
$p\in \bigcap_n X^n$. Here it is understood that the intersection is taken over all $n\in \N_0$. In the following we use
a similar convention for intersections of sets labeled by some index $n$, $k$, etc., if the range of the indices is clear from the context.
In general, a sequence $\{X^n\}\in \mathcal{S}$ that contains a given point $p\in S^2$ is not unique. Moreover, the intersection $\bigcap_n X^n$ may contain more than
one point. It turns out that this gives a criterion when $f$ is expanding.
\begin{lemma} \label{lem:charexpint}
Let $f\: S^2\rightarrow S^2$ be a Thurston map, and $\mathcal{C}\subset S^2$ be an $f$-invariant Jordan curve with $\operatorname{post}(f)\subset \mathcal{C}$. Then
the map $f$ is expanding\index{expanding} if and
only if for each sequence $\{X^n\}\in \mathcal{S}(f,\mathcal{C})$ the intersection
$\bigcap_n X^n$ consists of precisely one point.
\end{lemma}
\begin{proof} Fix a metric on $ S^2$ that induces the standard topology on $S^2$. If $f$ is expanding and
$\{X^n\}\in \mathcal{S}:=\mathcal{S}(f,\mathcal{C})$, then
$\diam(X^n)\to 0$ as $n\to \infty$. Hence $\bigcap_n X^n$ cannot contain more than one point. On the other hand, this set is an intersection of
a nested sequence of nonempty compact sets and hence nonempty.
So the set $\bigcap_n X^n$ contains precisely one point.
For the converse direction suppose that $\bigcap_n X^n$ is a singleton set for each sequence $\{X^n\}\in \mathcal{S}$.
To establish that
$ f$ is expanding we have to show that
$$\lim_{n\to\infty} \max\,\{ \diam ( X): X \text{ is an $n$-tile} \}=0.$$
We argue by contradiction and assume that this is not the case. Then there exists $\delta>0$
such that $\diam( X )\ge \delta$ for some tiles $X$ of arbitrarily high order.
We define a descending sequence of tiles
$X^0\supset X^1\supset X^2\supset\dots$ as follows. Let $X^0$ be a $0$-tile such that
$X^0$ contains tiles $X$ of arbitrarily high order
with $\diam( X )\ge \delta$. Since
the tiles are subdivided by tiles of higher order, and so every tile is contained in one of the finitely many $0$-tiles (in our case there actually two $0$-tiles), there exists such a $0$-tile. Note that then
$\diam( X^0 )\ge \delta$. Moreover, among the finitely many $1$-tiles into which $X^0$ is subdivided there must be a $1$-tile $X^1\subset X^0$ such that $X^1$
contains tiles $X$ of arbitrarily high order
with $\diam(X )\ge \delta$. Again this implies that $\diam( X^1 )\ge \delta$.
Repeating this procedure we obtain a sequence $\{X^n\}\in \mathcal{S}$ such that
$\diam( X^n )\ge \delta$ for all $n\in \N_0$.
It is easy to see that this implies that the set $\bigcap_n X^n$ also has
diameter $\ge \delta>0$, and so it contains at least two points. This is a contradiction showing that $f$ is expanding.
\end{proof}
Let $f\:S^2\rightarrow S^2$ be a Thurston map, $\mathcal{C}\subset S^2$ be a Jordan curve with $\operatorname{post}(f)\subset S^2$, and assume that $\#\operatorname{post}(f)\ge 3$. Recall the definition of the numbers $D_n=D_n(f,\mathcal{C})$ in \eqref{def:dk}.
We know that $D_n\to \infty$ if $f$ is expanding (see Lemma~\ref{lem:Dtoinfty}). If $f$ is not necessarily expanding, but the Jordan curve $\mathcal{C}$ used in the definition of $D_n$ is invariant, then it follows from the previous discussion that the numbers $D_n$ are increasing, i.e., $D_{n+1}\ge D_n$ for all $n\in \N_0$. Moreover, one can show
exponential increase of the numbers $D_n$ under the additional assumption that there exists $n_0\in \N$ with $D_{n_0}\ge 2$. This is the content of the following lemma.
\begin{lemma} \label{lem:submult}
Let $f\: S^2\rightarrow S^2$ be a Thurston map with $\#\operatorname{post}(f)\ge 3$, let $\mathcal{C}\subset S^2$ be a Jordan curve with $\operatorname{post}(f)\subset \mathcal{C}$ and suppose that $\mathcal{C}$ is $f$-invariant, and let $D_n=D_n(f,\mathcal{C})$ for $n\in \N_0$. Then for all $n,k\in \N_0$ we have
$$ D_{n+k}\ge D_nD_k$$
if $\#\operatorname{post}(f)\ge 4$, and
\begin{equation} \label{weaksuper} D_{n+k}\ge D_n(D_k-1)+1\end{equation}
if $\#\operatorname{post}(f)=3$.
Moreover,
$$\alpha:=\lim_{n\to \infty}\frac1n \log(D_n)=\sup_{n\in \N}\frac1n \log(D_n)\le
\log(\deg(f)).$$
If in addition there exists $n_0\in \N$ with $D_{n_0}\ge 2$, then $\alpha>0$ and $D_n\to \infty$ as $n\to \infty$.
\end{lemma}
Before we prove this lemma
let us fix some terminology.
An $n$-\defn{chain} is a finite sequence of $n$-tiles $X_1,\dots,
X_N$, where $X_i\cap X_{i+1}\neq \emptyset$ for $i=1, \dots, N-1$. It \defn{joins} two disjoint sets $A$ and $B$ if $A\cap X_1\ne \emptyset$ and $B\cap X_N\ne \emptyset$. The chain joins two points $x$ and $y$ if it joins the sets $\{x\}$ and $\{y\}$. The $n$-chain is
called a \defn{simple chain} joining $A$ and $B$ if there is no proper subsequence of $X_1, \dots, X_N$ that is also a chain joining $A$ and $B$. If we put
$X_{-1}:=A$ and $X_{N+1}:=B$, then this is equivalent with the requirement that $X_i\cap X_j=\emptyset$ whenever $-1\le i <j\le N+1$ and $j-i\ge 2$.
\ifthenelse{\boolean{nofigures}}{}{
\begin{figure}
\centering
\begin{overpic}
[width=10cm,
tics=20]{proofDk.eps}
\put(93,15){$E$}
\put(49,17){$e=c_1$}
\put(44,22){${X_1}$}
\put(93,78){$\widetilde{E}$}
\put(44,78){$\widetilde{e}=c_{M+1}$}
\put(70,45){$W^k(c_1)$}
\put(24.6,53){$c_2$}
\put(33,51.5){$X_{s_2-1}$}
\put(31,60){$X_{s_2}$}
\put(24,28){$Y_1$}
\put(22,79){$X_N$}
\end{overpic}
\caption{The proof of Lemma \ref{lem:submult}.}
\label{fig:proofDk}
\end{figure}
}
\begin{proof}[Proof of Lemma \ref{lem:submult}]
{\em Case 1.} $\#\operatorname{post}(f)\geq 4$.
Let $X_1, \dots, X_N$ be a set of
$(n+k)$-tiles whose union is connected and joins opposite sides of
$\mathcal{C}$. We may assume that these tiles form a chain joining disjoint
$0$-edges $E$ and $\widetilde{E}$. To prove the desired inequality
we will break this chain into $M$ subchains $X_{s_i}, \dots,
X_{s_{i+1}-1}$, where $M\in \N$, $i=1, \dots, M$, and $s_1=1<s_2<\dots<s_{M+1}=N+1$.
The length of each subchain (i.e., the number
$s_{i+1}-s_i$) will be at least $D_n$. The number $M$ of
subchains will be at least $D_k$. Thus $N\geq D_n D_k$, and since the minimum over all $N$ is equal to $D_{n+k}$, the desired inequality follows.
To achieve the desired bound on the length we will ensure that
each subchain $X_{s_i}, \dots, X_{s_{i+1}-1}$
joins disjoint $k$-cells.
Then the length of such a subchain is at least $D_n$ by
Lemma~\ref{lem:flowerbds}.
To control the number of subchains, we will associate to each one
a $k$-tile $Y_i$. These $k$-tiles $Y_1,\dots, Y_M$ will
form a $k$-chain joining $E$ and $\widetilde E$, and hence opposite sides of $\mathcal{C}$. Thus $M$, which is
the number of $k$-tiles in this chain, as well as the number of subchains, is at
least $D_k$ (by definition of this quantity; see \eqref{def:dk}).
\smallskip
We now provide the details of the construction, which is illustrated
in Figure \ref{fig:proofDk}.
We will use auxiliary $k$-cells
$c_1,c_2, \dots$ of dimension $\le 1$.
If $c_i$ is $0$-dimensional, then $c_i$ consists of a
$k$-vertex $p_i$, and we let $W^k(c_i):=W^k(p_i)$ (see Definition~\ref{def:flower}).
If $c_i$ is $1$-dimensional, then $c_i$ is a $k$-edge and
$W^k(c_i)$ is the edge flower of
$c_i$ as in Definition~\ref{def:edgeflower}.
Since $\mathcal{C}$ is $f$-invariant, the cell decomposition $\mathcal{D}^{k}$ is a refinement of $\mathcal{D}^0$.
Hence there exist disjoint
$k$-edges $e\subset E$ and $\widetilde e\subset \widetilde E$ with
$X_1\cap e\ne \emptyset$ and $X_N\cap \widetilde e\ne \emptyset$.
For some number $M\in \N$ we will now inductively define $k$-cells $c_1, \dots, c_{M+1}$ of dimension $\le 1$,
$k$-tiles $Y_1, \dots, Y_M$, and indices $s_1=1< s_2<\dots <s_{M+1}=N+1$ with the following properties:
\begin{itemize}
\smallskip
\item[(i)] $c_1=e$, $c_{M+1}=\widetilde e$, and $c_{i}\cap c_{i+1}=\emptyset $ for $i=1, \dots, M$.
\smallskip
\item[(ii)] $c_{i}\cap Y_i\ne \emptyset$ for $i=1, \dots , M$, $c_{i+1}\subset \partial Y_i$
for $i=1, \dots , M-1$, and $\widetilde e\cap Y_M \ne \emptyset$.
\smallskip
\item[(iii)] $X_{s_{i}}, \dots, X_{s_{i+1}-1}$ is an $(n+k)$-chain
joining $c_{i}$ and $c_{i+1}$ for $i=1, \dots, M$.
\end{itemize}
Note that (i) and (ii) imply that $E\cap Y_1\supset e\cap Y_1\ne \emptyset$,
$\widetilde E\cap Y_M\supset \widetilde e\cap Y_M\ne \emptyset$, and $Y_i\cap Y_{i+1}\supset c_{i+1}\cap Y_{i+1}\ne \emptyset$
for $i=1, \dots, M-1$. Hence $Y_1, \dots, Y_M$ will be a $k$-chain joining the
$0$-edges $E$ and $\widetilde E$ as desired.
\smallskip
Let $s_1=1$ and $c_1=e$.
Suppose first that $\widetilde e$ meets $\overline{W^k(c_1)}$.
Since $\widetilde e$ is disjoint from $e=c_1$ and hence from
$W^k(c_1)$, the points in $\widetilde e\cap
\overline{W^k(c_1)}$ lie in $\partial W^k(c_1)$. By Lemma~\ref{lem:edgeflower}~(ii) there
exists a $k$-tile $Y_1$ that meets both $c_1$ and
$ \widetilde e\supset \widetilde e\cap
\overline{W^k(c_1)}$.
We let $M=1$, set $c_2=\widetilde e$, and stop the construction. We have all the desired properties (i)--(iii).
In the other case where $\widetilde e\cap \overline{W^k(c_1)}=\emptyset$ not all the $(n+k)$-tiles $X_1, \dots, X_N$ are contained in $\overline{W^k(c_1)}$. So there exists
a smallest index $s_2\ge 1$ such that $X_{s_2}$ meets
$S^2\setminus \overline{W^k(c_1)}$. Then $s_2>s_1=1$, because $X_1$ meets $e=c_1$ and is hence contained in $\overline{W^k(c_1)}$. To see this we use Lemma~\ref{lem:edgeflower}~(iii) and the fact that $X_1$ is contained in some $k$-tile. Moreover, for a similar reason we have $X_{s_2}\subset S^2
\setminus W^k(c_1)$. By definition of $s_2$ the set
$X_{s_{2}-1}$ is contained in $\overline{W^k(c_1)}$. Hence
every point in the nonempty intersection $X_{s_{2}-1}\cap X_{s_2}$ lies in $\partial W^k(c_1)$. In particular, by Lemma~\ref{lem:edgeflower}~(iii) there exists a $k$-cell $c_2\subset \partial W^k(e_1)$ of dimension $\le 1$ that has common points with both $X_{s_{2}-1}$ and $X_{s_2}$, and a $k$-tile $Y_1\subset \overline {W^k(c_1)}$ with $ c_1\cap Y_1\ne \emptyset$ and
$c_2\subset \partial Y_1$. Then $c_1\cap c_2=\emptyset=c_2\cap \widetilde e$, and the chain $X_{s_1}=X_1, \dots, X_{s_2-1}$ joins $c_1$ and $c_2$.
We can now repeat the construction as in the first step by using the chain $X_{s_2}, \dots, X_N$ that joins the disjoint $k$-cells $c_2$ and $\widetilde e$, etc. If in the process
one of the cells $c_i$ has dimension $0$, we invoke
Lemma~\ref{lem:flowerprop} (ii) and (iii) instead of Lemma~\ref{lem:edgeflower}~(ii) and (iii)
in the above construction.
The construction eventually stops, and it is clear that we obtain cells and indices with the desired properties.
\medskip
{\em Case~2.} $\#\operatorname{post}(F)=3$.
Let $E_1,E_2,E_3$ be the three $0$-edges.
Consider a connected union $K$ of $(n+k)$-tiles joining opposite
sides of $\mathcal{C}$ with $N=D_{n+k}$ elements. Then $K$ meets
$k$-edges contained in the $0$-edges, say $k$-edges $e_i\subset E_i$
for $i=1,2,3$. From $K$ we can extract a simple $(n+k)$-chain
joining $e_1$ and $e_2$ as well as another simple chain that
joins $e_3$ to one tile $X$ in the chain joining $e_1$ and
$e_2$. Starting from this ``center tile'' $X$, we can find three
simple $(n+k)$-chains that join $X$ to the edges $e_1,e_2,e_3$,
respectively, and have only the tile $X$ in common.
More precisely, for $i=1,2,3$ we can find $N_i\in \N_0$ and
$(n+k)$-chains $ X, X^i_1, \dots, X^i_{N_i}$ that join $X$ and
$e_i$. Here the first tile $X$ is the same in all chains and it
is understood that the chain consists only of $X$ if $N_i=0$.
Moreover, all the $(n+k)$-tiles
$$X, X^1_1, \dots, X^1_{N_1}, X^2_1, \dots, X^2_{N_2}, X^3_1, \dots, X^3_{N_3}$$
are pairwise distinct tiles from $K$. Thus their number is bounded
by the number of $(n+k)$-tiles in $K$. Since they still form a
connected set joining opposite sides of $\mathcal{C}$, we have
$N_1+N_2+N_3+1= N=D_{n+k}$.
Pick a $k$-tile $Y$ with $X\subset Y$, and consider the chain $X,
X^1_1, \dots , X^1_{N_1}$.
Suppose that $Y\cap e_1=\emptyset$. Since $X\subset Y$ we have
$N_1\ge 1$ and the chain $ X_1, \dots, X_{N_1}$ joins $Y$ and
$e_1$. Hence this chain or a subchain must also join a $k$-edge
$e \subset \partial Y$ and $e_1$.
Then $e\cap e_1=\emptyset$. As in the first part of
the proof we can find $k$-tiles $Y_1, \dots , Y_{M_1}$ joining
$e$ and $e_1$, where $M_1\in \N$ and $N_1\ge M_1
D_n$.
If $Y\cap e_1\neq\emptyset$, we set $M_1=0$ and do not define new
$k$-tiles. In any case we have that $Y, Y^1_1, \dots, Y^1_{M_1}$
is a chain joining $Y$ and $e_1$ (again we use the convention that
this chain consists only of $Y$ if $M_1=0$). We also have $N_1\ge
M_1 D_n$ (which is trivial if $M_1=0$).
Using a similar construction for the other indices $i=2,3$, we
obtain numbers $M_i\in \N_0$ for each $i=1, 2,3$ that satisfy $N_i\ge
M_iD_n$, and chains
$Y, Y^i_1, \dots, Y^i_{M_i}$ of $k$-tiles that join $Y$ and $e_i$.
The union of these $k$-tiles is a connected set joining opposite sides
of $\mathcal{C}$. Hence it contains at least $D_k$ distinct elements. On the other hand, the number of distinct $k$-tiles in the union is
at most $M_1+M_2+M_3+1$ (note that the
three chains may have other $k$-tiles in common apart from $Y$). Hence $D_k\le M_1+M_2+M_3+1$,
and it follows that
$$ D_n(D_k-1)+1\le D_n(M_1+M_2+M_3)+1\le N_1+N_2+N_3+1\le D_{n+k}, $$
which is the desired inequality (\ref{weaksuper}).
\medskip
In order to prove the remaining statements first note that inequality \eqref{weaksuper} is also true if $\operatorname{post}(f)=4$. A simple induction argument using \eqref{weaksuper} shows that if
$D_{N}\ge 2$ for some $N\in \N$, then
\begin{equation} \label{eq:supmult5}
D_{kN}\ge D_N^{k-1}+1
\end{equation} for all $k\in \N$.
For such $N$ let $k(n)=\lfloor n/N\rfloor$. Noting that the sequence $\{D_n\}$ is non-decreasing and using \eqref{eq:supmult5}, we obtain
\begin{eqnarray*} \liminf_{n\to \infty} \frac1n \log (D_n)&\ge&
\liminf_{n\to \infty} \frac1n \log (D_{k(n)N})\\ &\ge&
\liminf_{n\to \infty} \frac{k(n)-1}n \log (D_N)=
\frac1N \log (D_N). \end{eqnarray*}
This inequality is trivially true if $D_N=1$, and so
$$\liminf_{n\to \infty} \frac1n \log D_n\ge \sup_{n\in \N}\frac1n \log D_n. $$
On the other hand,
$$\limsup_{n\to \infty} \frac1n \log (D_n)\le \sup_{n\in \N}
\frac1n \log (D_n), $$
and so
$$\alpha=\lim_{n\to\infty}\frac1n \log (D_n)=\sup_{n\in \N} \frac1n \log (D_n). $$
Note that $D_n\le \#\X^n\le 2\deg(f)^n$ which implies $\alpha\le \log(\deg(f))$.
Finally, if there exists $n_0\in \N$ with $D_{n_0}\ge 2$, then
$$\alpha=\sup_{n\in \N} \frac1n\log(D_n) \ge \frac{1}{n_0}\log (D_{n_0})>0, $$ and it is clear from the definition of $\alpha$ that $D_n\to \infty$ as $n\to\infty$.
\end{proof}
The last lemma shows that if $f$ is a Thurston map with an invariant Jordan curve $\mathcal{C}\subset S^2$ with $\operatorname{post}(f)\subset \mathcal{C}$, and there exists $n_0\in \N$ such that $D_{n_0}=D_{n_0}(f,\mathcal{C})\ge 2$, then the numbers $D_n=D_n(f,\mathcal{C})$ are actually
exponentially increasing; indeed, if $\alpha>0$ is as in the lemma and $\epsilon>0$ is arbitrary, then
$D_n\gtrsim e^{(\alpha-\epsilon)n}$ for large $n$.
This situation will be important enough to warrant a separate definition.
\begin{definition}[Combinatorial expansion]\label{def:combexp}
Let $f\:S^2\rightarrow S^2$ be a Thurston map. We call $f$
\defn{combinatorially expanding}\index{combinatorially expanding|textbf}\index{expanding!combinatorially|textbf}
if $\operatorname{post}(f)\ge 3$, and there exists a
Jordan curve $\mathcal{C}\subset S^2$ that is $f$-invariant, satisfies $\operatorname{post}(f)\subset \mathcal{C}$, and for which
there is a number $n_0\in \N$ such that
$D_{n_0}(f,\mathcal{C})\ge 2$.
\end{definition}
If $f$ and $\mathcal{C}$ are as in the previous definition, then we say that
$f$ is {\em combinatorially expanding for $\mathcal{C}$}.
By definition of $D_n(f,\mathcal{C})$ the condition $D_{n_0}(f,\mathcal{C})\ge 2$ means that no single $n_0$-tile joins opposite sides of
$\mathcal{C}$. If a map $f$ as in Definition~\ref{def:combexp} is expanding and $\mathcal{C}\subset S^2$ is an $f$-invariant Jordan curve with $\operatorname{post}(f)\subset \mathcal{C}$, then $f$ is also combinatorially expanding for $\mathcal{C}$ (in this case $D_n(f,\mathcal{C})\to \infty $; see Lemma~\ref{lem:Dtoinfty}).
The converse is not true in general, since a combinatorially expanding Thurston map need not be expanding. However, we will see in Section~\ref{sec:combexp} that each combinatorially expanding Thurston is equivalent to
an expanding Thurston map with an invariant curve.
The condition of combinatorial expansion is indeed combinatorial in nature, because it can be verified just by knowing the combinatorics of the cell decompositions
$\mathcal{D}^n=\mathcal{D}^n(f,\mathcal{C})$, $n\in \N_0$. This in turn is determined by the combinatorics of
the pair $(\mathcal{D}^1, \mathcal{D}^0)$ and the map $c\in \mathcal{D}^1\mapsto f(c)\in \mathcal{D}^0$ (see Remark~\ref{rem:combinatorics}).
\smallskip
We finish this section with a lemma on combinatorially expanding Thurston maps
that will be useful later.
\begin{lemma} \label{lem:vinint}
Let $f\:S^2\rightarrow S^2$ be a Thurston map that is combinatorially expanding for the Jordan curve $\mathcal{C}\subset S^2$, and suppose that
$D_{n_0}(f,\mathcal{C})\ge 2$ where $n_0\in \N$.
\begin{itemize}
\smallskip
\item[(i)] If $n\in \N_0$ and $e$ is an $n$-edge, then there exists an $(n+n_0)$-vertex $p$ with $p\in \inte(e)$.
\smallskip
\item[(ii)] If $n\in \N_0$ and $X$ is an $n$-tile, then there exists an $(n+n_0)$-edge with
$\inte(e)\subset \inte(X)$, and an $(n+2n_0)$-vertex $p$ with $p\in \inte(X)$.
\end{itemize}
\end{lemma}
\begin{proof} In the previous statements and the ensuing proof it is understood that the term $k$-cell for $k\in \N_0$ refers to a cell in
$\mathcal{D}^k=\mathcal{D}^k(f,\mathcal{C})$.
(i) Suppose $e$ is an $n$-edge that does not contain $(n+n_0)$-vertices in its interior.
By Proposition~\ref{prop:invmarkov}~(iv) the $n$-edge $e$ is equal to the union of all $(n+n_0)$-edges contained in $e$. Thus $e$ must be an $(n+n_0)$-edge
itself. Let $u$ and $v$ be the endpoints of $e$, and $X$ be an $(n+n_0)$-tile
containing $e$ in its boundary. Then $K=X$ meets the two disjoint $n$-cells $\{u\}$ and $\{v\}$. Hence by Lemma~\ref{lem:flowerbds} the set $K$ should consist of at least $D_{n_0}=D_{n_0}(f,\mathcal{C})\ge 2$ $(n+n_0)$-tiles. This is a contradiction proving the statement.
(ii) Let $X$ be an $n$-tile. By
Proposition~\ref{prop:invmarkov}~(iii) we know that $X$ is the union of all $(n+n_0)$-tiles contained in $X$. In particular, there exists an $(n+n_0)$-tile $Y$
with $Y\subset X$. We claim that there exists an $(n+n_0)$-edge in the boundary of $Y$ that meets $\inte(X)$. Otherwise, $\partial Y\cap \inte(X)=\emptyset$, and as $Y\subset X$, we must have $\partial Y\subset \partial X$. Since both sets $\partial Y$ and $\partial X$ are Jordan curves, this is only possible if $\partial Y=\partial X$. Then $Y$ meets all $n$-vertices contained in $\partial X$, and two distinct $n$-vertices in particular. As in the proof of (i), this leads to a contradiction.
Hence there exists an $(n+n_0)$-edge $e$ with $e\cap \inte(X)\ne \emptyset$.
Since $\inte(X)$ is an open subset of $S^2$, we then also have $\inte(e)\cap \inte(X)\ne \emptyset$.
Since $\mathcal{D}^{n+n_0}$ is a refinement of $\mathcal{D}$, by Lemma~\ref{lem:mincell}
we know that there is a unique cell $\tau$ in $\mathcal{D}^n$ with $\inte(e)\subset \inte(\tau)$. Then
$\inte(\tau)\cap \inte(X)\ne \emptyset$, and so $X=\tau$ by Lemma~\ref{lem:uniondisjint}.
Hence $\inte(e)\subset \inte(X)$ as desired.
By (i) there exists an $(n+2n_0)$-vertex $p$ with $p\in \inte(e)$. Then we also have $p\in \inte(X)$ as desired.
\end{proof}
\section{Two-tile subdivision rules}
\label{sec:subdivisions}
\noindent We have seen how a Thurston map $f$ and a Jordan curve $\mathcal{C}\subset S^2$ with $\operatorname{post}(f)\subset \mathcal{C}$ can be used to define cell decompositions $\mathcal{D}^n=\mathcal{D}^n(f,\mathcal{C})$ of $S^2$. If $\mathcal{C}$ is $f$-invariant, then $(\mathcal{D}^1,\mathcal{D}^0)$ is a cellular Markov partition for $f$.
In this section we will see that this process can be reversed.
Starting with a pair $(\mathcal{D}^1, \mathcal{D}^0)$ of cell decompositions
of $S^2$, one can show that under suitable additional assumptions there exists a postcritically-finite branched covering map that is cellular for
$(\mathcal{D}^1, \mathcal{D}^0)$. As before we will refer to the elements in $\mathcal{D}^0$ as the $0$-cells and the elements in $\mathcal{D}^1$ as the $1$-cells, and speak of $1$-edges, $0$-tiles, etc.
The map $f$ is unique
up to Thurston equivalence if additional data is provided, namely a {\em labeling}
$\mathcal{D}^1\rightarrow \mathcal{D}^0$. The concept of a labeling extracts the relevant combinatorial properties of the map $\tau\in \mathcal{D}^1\mapsto f(\tau)\in \mathcal{D}^0$ if $f$ is cellular for $(\mathcal{D}^1,\mathcal{D}^0)$.
Here is the precise general definition.
\begin{definition}[Labelings]\label{def:labeldecomp}
Let $\mathcal{D}^1$ and $\mathcal{D}^0$ be cell complexes. Then a {\em
labeling}\index{labeling}
of $(\mathcal{D}^1, \mathcal{D}^0)$ is map
$L\: \mathcal{D}^1 \rightarrow \mathcal{D}^0$ satisfying the following conditions:
\begin{itemize}
\smallskip
\item[(i)] $\dim (L(\tau))=\dim(\tau)$ for all $\tau\in \mathcal{D}^1$,
\smallskip
\item[(ii)]
if $\sigma, \tau\in \mathcal{D}^1$ and $\sigma\subset \tau$, then $L(\sigma)\subset L(\tau)$.
\smallskip
\item[(iii)] if $\sigma, \tau,c\in \mathcal{D}^1$, $\sigma,\tau\subset c$ and $L(\sigma)=L(\tau)$, then $\sigma=\tau$.
\end{itemize}
\end{definition}
So a labeling is a map $L\: \mathcal{D}^1\rightarrow \mathcal{D}^0$ that preserves inclusions and dimensions of cells, and is ``injective on cells'' $c\in \mathcal{D}^1$ in the sense of (iii).
In particular, every cell of dimension $0$ in $\mathcal{D}^1$ is mapped to a cell of dimension $0$ in $\mathcal{D}^0$. If $v$ is a {\em vertex} in $\mathcal{D}^1$, i.e., if $\{v\}$ is a cell of dimension $0$ in $\mathcal{D}^1$, then we can write $L(\{v\})=\{w\}$, where
$w$ is a vertex in $\mathcal{D}^0$. We define $L(v)=w$. In the following we always assume that a labeling $L\: \mathcal{D}^1\rightarrow \mathcal{D}^0$ has been extended to the set of
vertices of $\mathcal{D}^1$ in this way; this will allow us to ignore the subtle distinction between vertices and cells of dimension $0$, i.e., sets consisting of one vertex.
If we have a labeling $L\: \mathcal{D}^1\rightarrow \mathcal{D}^0$ we should think of each element $\tau\in \mathcal{D}^1$ as ``carrying'' the label $L(\tau)\in \mathcal{D}^0$. In applications it is often
more intuitive and convenient to allow more general index sets $\mathcal{L}$ of the same cardinality $\mathcal{D}^0$ as labeling sets for the elements in $\mathcal{D}^1$.
In such situations we fix a bijection $\psi\: \mathcal{D}^0\rightarrow \mathcal{L}$ and call a map
$L'\: \mathcal{D}^1 \rightarrow \mathcal{L}$ a labeling if $\psi^{-1}\circ L'\: \mathcal{D}^1\rightarrow \mathcal{D}^0$ is a labeling in the sense of Definition~\ref{def:labeldecomp}.
We will discuss this further below.
Related to labelings is the concept of an isomorphism between cell complexes.
\begin{definition}[Isomorphisms of cell complexes]\label{def:compiso}
Let $\mathcal{D}$ and $\mathcal{D}'$ be cell complexes. A bijection
$\phi\: \mathcal{D}\rightarrow \mathcal{D}'$ is called an {\em isomorphism (of cell complexes)} if the following conditions are satisfied:
\begin{itemize}
\smallskip
\item[(i)] $\dim (\phi(\tau))=\dim (\tau)$ for all $\tau\in \mathcal{D}$,
\smallskip
\item[(ii)] if $\sigma,\tau \in \mathcal{D}$, then $\sigma\subset \tau$ if and only if
$\phi(\sigma)\subset \phi(\tau)$.
\end{itemize}
If we are given another cell complex $\mathcal{D}^0$ and labelings $L\: \mathcal{D}\rightarrow \mathcal{D}^0$ and
$L'\: \mathcal{D}'\rightarrow \mathcal{D}^0$, then an isomorphism $\phi\: \mathcal{D}\rightarrow \mathcal{D}'$ is called {\em label-preserving} if $L= L'\circ \phi$.
\end{definition}
Let $S^2$ be an oriented $2$-sphere, and
$\mathcal{D}$ be a cell decomposition of $S^2$. Recall (see Section~\ref{sec:2spherecd}) that a flag in $\mathcal{D}$ is a triple $(c_0,c_1,c_2)$, where $c_i$ is a cell in $\mathcal{D}$ of dimension $i$ for $i=0,1,2$ and $c_0\subset c_1 \subset c_2$. In this case $c_0=\{v\}$ consists of a vertex $v$ of $\mathcal{D}$ which must be one of the endpoints of $c_1$, and $c_1$ is oriented by considering $v$ as the initial and the other vertex in $\partial c_1$ as the terminal point of $c_1$.
The cell $c_2$ is one of the two tiles in $\mathcal{D}$ that contain $c_1$ in their boundary. The flag $(c_0,c_1,c_2)$ is positively-oriented if $c_2$ lies on the left of the oriented edge $c_1$
according to the given orientation of $S^2$.
If $L\:\mathcal{D}^1\rightarrow \mathcal{D}^0$ is a labeling of a pair $(\mathcal{D}^1, \mathcal{D}^0)$ of cell decompositions of $S^2$ and $(c_0,c_1,c_2)$ is a flag in
$\mathcal{D}^1$, then $(L(c_0),L(c_1),L(c_2))$ is a flag in $\mathcal{D}^0$. This follows from the definition of a labeling. So a labeling maps
``flags to flags''. We say that the labeling is {\em
orientation-preserving}\index{labeling!orientation-preserving}
if it maps positively-oriented flags in $\mathcal{D}^1$ to
positively-oriented flags in $\mathcal{D}^0$.
If $f\:S^2\rightarrow S^2$ is cellular for $(\mathcal{D}^1, \mathcal{D}^0)$, then $f$ induces a natural labeling $L\:\mathcal{D}^1\rightarrow \mathcal{D}^0$ given by $L(\tau)=f(\tau)$ for $\tau \in \mathcal{D}^1$. Moreover, if in addition $f|X$ is orientation-preserving for each tile $X$ in $\mathcal{D}^1$ (which is always true if $f$ is a branched covering map), then this labeling $L$ is
orientation-preserving.
If a labeling $L\:\mathcal{D}^1\rightarrow \mathcal{D}^0$ is given, then we say that a map
$f\:S^2\rightarrow S^2$ that is cellular for $(\mathcal{D}^1, \mathcal{D}^0)$ is {\em
compatible}\index{labeling!compatible}
with the labeling $L$ if $L(\tau)=f(\tau)$ for each $\tau \in \mathcal{D}^1$, i.e., if the labeling induced by $f$ is equal to the given labeling.
In the proof of the next proposition we need some simple facts about
homeomorphisms and isotopies. If $\alpha$ is an arc, then every homeomorphism $\varphi\: \alpha \rightarrow \alpha$ that fixes the endpoints of $ \alpha$ is isotopic to the identity
rel.~$\partial \alpha$. Indeed, we may assume that $\alpha$ is equal to the unit interval $I=[0,1]$.
Then $\varphi(0)=0$, $\varphi(1)=1$, and $\varphi$ is strictly increasing on $[0,1]$.
Define $H\: I\times I\rightarrow I$ by
$$H(s,t)=(1-t)\varphi(s)+ts$$ for $s,t\in I$. Then for each
$t\in I$, the map $H_t=H(\cdot,t)$ is strictly increasing on $I$. It follows that $H$ is an isotopy. We have
$H_t(0)=0$ and $H_t(1)=1$ for all $t\in I$, and
$H_0=\varphi$ and $H_1=\id_I$. Hence
$\varphi$ and $\id_I$ are isotopic rel.~$\partial I=\{0,1\}$ by the isotopy $H$.
Let $X\subset S^2$ be a closed Jordan region. If $h\: \partial X\times I\rightarrow \partial X$ is an isotopy with $h(\cdot, 0)=\id_{\partial X}$,
then there exists an isotopy $H\: X\times I\rightarrow X$ such that $H(\cdot, 0)=\id_X$ and $H(p,t)=h(p,t)$ for all $p\in \partial X$ and $t\in I$.
So an isotopy $h$ on the boundary of $X$ with $h_0=\id_{\partial X}$ can be extended to an isotopy $H$ on $X$
with $H_0=\id_X$.
To see this, we may assume that $X=\overline \D$. Then
$H$ is obtained from $h$ by radial extension; more precisely, we define $$H(re^{{\mathbf{\imath}} s},t)=rh(e^{{\mathbf{\imath}} s},t)$$
for all $r\in [0,1]$ and $s\in [0,2\pi]$. Then $H$ is well-defined and it is easy to see that $H$ is an isotopy with the desired properties. By using the Sch\"onfles theorem and a similar radial extension one can also show that if $X$ and $X'$ are closed Jordan regions in $S^2$, then every homeomorphism $\varphi\: \partial X\rightarrow \partial
X'$ extends to a homeomorphism $\Phi\: X\rightarrow X'$.
If $\varphi \:X\rightarrow X$ is a homeomorphism with
$\varphi|\partial X=\id_{\partial X}$, then $\varphi$ is isotopic to $\id_{X}$ rel.~$\partial X$. Indeed, again we may assume that $X=\overline \D$. Then we obtain the desired isotopy by the ``Alexander trick": for $z\in \overline \D$ and $t\in I$ we define
$ H(z,t)=t\varphi(z/t) $ if $|z|< t$, and $H(z,t)=z$ if $|z|\ge t$. It is easy to see that $H$ is an isotopy rel.\ $\partial \D$ with $H_0=\id_X$ and $H_1=\varphi$.
If $\varphi, \widetilde \varphi \:X\rightarrow X$ are two homeomorphism with
$\varphi|\partial X=\widetilde \varphi |\partial X$, then we can apply the previous remark to $\psi= \widetilde \varphi \circ \varphi^{-1}$ and conclude that
$\varphi$ and $\widetilde \varphi$ are isotopic rel.\ $\partial X$.
\begin{prop}\label{prop:thurstonex} Let $(\mathcal{D}^1, \mathcal{D}^0)$ be a pair of cell decompositions
of an oriented $2$-sphere $S^2$ with an
orientation-preserving
labeling. Assume that every vertex of $\mathcal{D}^0$ is also a vertex of $\mathcal{D}^1$.
Then there exists a postcritically-finite branched covering map $f\:S^2\rightarrow S^2$ that is cellular for $(\mathcal{D}^1, \mathcal{D}^0)$ and is compatible with the given labeling.
The map $f$ is unique up to Thurston equivalence.
\end{prop}
Note that $f$ is a Thurston map if $\deg(f)\ge 2$, i.e., if $f$ is not a homeomorphism.
\begin{proof} The existence of a map $f$ as desired follows from the well-known procedure of successive extensions on the skeleta of the
cell decomposition $\mathcal{D}^1$. Indeed, let $L\:\mathcal{D}^1\rightarrow \mathcal{D}^0$ be an orientation-preserving labeling. If $v\in S^2$ is a $1$-vertex (i.e., a vertex in $\mathcal{D}^1$), then $L(v)$ is a $0$-vertex (i.e., a vertex in $\mathcal{D}^0$).
Set $f(v)=L(v)$. This defines $f$ on the $0$-skeleton of $\mathcal{D}^1$.
To extend this to the $1$-skeleton of $\mathcal{D}^1$, let $e$ be an arbitrary $1$-edge. Then $e'=L(e)$ is a $0$-edge. Moreover, if $u$ and $v$ are the $1$-vertices that are the endpoints of $e$, then $u'=f(u)=L(u)$ and
$v'=f(v)=L(v)$ are distinct $0$-vertices contained in $e'$. Hence they are the endpoints of $e'$. So we can extend $f$ to $e$ by choosing a homeomorphism of $e$ onto $e'$ that agrees with $f$ on the endpoints of $e$. In this way we can continuously extend $f$ to the $1$-skeleton of $\mathcal{D}^1$ so that $f|\tau$ is a homeomorphism of $\tau$ onto $L(\tau)$ whenever
$\tau\in \mathcal{D}^1$ is a cell of dimension $\le 1$.
If $X$ is an arbitrary $1$-tile, then $\partial X$ is a subset of the $1$-skeleton of $\mathcal{D}^1$ and hence $f$ is already defined on $\partial X$. Then
$f|\partial X$ is a continuous mapping of $\partial X$ into the boundary $\partial X'$ of the $0$-tile $X'=L(X)$. The map $f|\partial X$ is injective. Indeed, suppose that $u,v\in \partial X$ and $f(u)=f(v)$. Then there exist unique $1$-cells $\sigma, \tau\subset \partial X$ of dimension $\le 1$ such that $u\in \inte(\sigma)$ and $v\in \inte(\tau)$. Then
$$f(u)=f(v)\in \inte(f(\sigma))\cap \inte(f(\tau))=\inte(L(\sigma))\cap \inte (L(\tau))$$
and so the $1$-cells $L(\sigma) $ and $L(\tau)$ must be the same. Since $L$ is a labeling and $\sigma, \tau \subset X\in \mathcal{D}^1$, it follows that $\sigma=\tau$.
As the map $f$ restricted to the $1$-cell $\sigma=\tau$ is injective, we conclude $u=v$ as desired.
Since every injective and continuous map of a Jordan curve into another Jordan curve is surjective, it follows that
$f|\partial X$ is a homeomorphism of $\partial X$ onto $\partial X'$. Hence $f$ can be extended to a homeomorphism of $X$ onto $X'$. These extensions on different $1$-tiles paste together to a continuous map $f\:S^2\rightarrow S^2$ that is cellular and is compatible with the given
labeling. Moreover, $f|X$ is orientation-preserving for each $1$-tile $X$ as follows from the fact that the labeling is orientation-preserving.
By Lemma~\ref{lem:constrmaps}~(i) and (ii) the map $f$ is a postcritically-finite branched covering map.
This shows that a map with the stated properties exists.
To show uniqueness suppose that $g\:S^2\rightarrow S^2$ is another such map. Then for each cell $\tau\in \mathcal{D}^1$, the maps $f|\tau$ and $g|\tau$ are homeomorphisms of $\tau$ onto $L(\tau)\in \mathcal{D}^0$.
Hence $\varphi_\tau=(g|\tau)^{-1}\circ (f|\tau)$ is a homeomorphism
of $\tau$ onto itself. The family $\varphi_\tau$, $\tau \in \mathcal{D}^1$, of these homeomorphism is obviously compatible under inclusions: if $\sigma,\tau \in \mathcal{D}^1$ and
$\sigma\subset \tau$, then $\varphi_\tau(p)=\varphi_\sigma(p)$ for all $p\in \sigma$.
Using this we can define a map $\varphi\:S^2\rightarrow S^2$ as follows. For $p\in S^2$ pick $\tau\in \mathcal{D}^1$ with $p\in \tau$. Then set $\varphi(p)=\varphi_\tau(p)$. The compatibility properties of the homeomorphisms $\varphi_\tau$ imply that $\varphi$ is well-defined. Indeed, suppose that $\tau, \tau'$ are cells in $\mathcal{D}^1$ with $p\in \tau\cap \tau'$. There exists a unique cell $\sigma\in \mathcal{D}^1$ with $p\in \inte(\sigma)$. It follows from Lemma~\ref{lem:celldecompint}~(ii) that
$\sigma\subset \tau\cap \tau'$.
Hence
$$\varphi_\tau(p)=\varphi_\sigma(p)=\varphi_{\tau'}(p). $$
It is clear that $g\circ \varphi =f$. Moreover, $\varphi|\tau=\varphi_\tau$ is a homeomorphism of $\tau$ onto itself whenever $\tau\in \mathcal{D}^1$. This implies that $\varphi$ is continuous as there only finitely many cells in $\mathcal{D}^1$, and that $\varphi$ is surjective. The map $\varphi$ is also injective as follows from the facts that $\varphi (\inte(\tau))=\inte(\tau)$ and that $\varphi |\tau$ is injective for each $\tau\in \mathcal{D}^1$, and that $S^2$ is the disjoint union of the sets $\inte(\tau)$, $\tau\in \mathcal{D}^1$.
Hence $\varphi\: S^2\rightarrow S^2$ is continuous and bijective, and so a homeomorphism of $S^2$ onto itself. Note that $f$ and $g$ agree
on the set ${\bf V}^1$ of $1$-vertices. Hence $\varphi$ is the identity on ${\bf V}^1$. Moreover, the sets of postcritical points of $f$ and $g$ are contained in the set of $0$-vertices and hence in ${\bf V}^1$.
So Thurston equivalence of $f$ and $g$ will follow, if we can show that $\varphi$ is isotopic to $\text{id}_{S^2}$ rel.~${\bf V}^1$.
This again follows from a procedure based on successive extension on skeleta of $\mathcal{D}^1$. Indeed, let $\E^1$ be the set of $1$-edges,
$E^1=\bigcup\{e: e\in \E^1\}$ be the $1$-skeleton of $\mathcal{D}^1$, and $e\in \E^1$ be an arbitrary $1$-edge. Since $\varphi$ is the identity on vertices in $\mathcal{D}^1$, the map $\varphi|e=\varphi_e$ is isotopic to $\id_e$ rel.~$\partial e$. Then these isotopies on edges paste together to an isotopy of $\varphi|E^1$ to $\id_{E^1}$ rel.~${\bf V}^1$. If $X$ is a tile in $\mathcal{D}^1$, then this isotopy is defined on $\partial X\subset E^1$, and we can extend it to an isotopy
of a homeomorphism on $X$ that agrees with $\varphi|\partial X$ on $\partial X$ to $\id_X$. These extensions on tiles $X$ paste together
to an isotopy $\Phi\: S^2 \times [0,1]\rightarrow S^2$ rel.~${\bf V}^1$ such that
$\Phi(\cdot, 1)=\id_{S^2}$ and
$\widetilde \varphi|E^1=\varphi|E^1$, where $\widetilde \varphi:=\Phi(\cdot, 0).$
For each tile $X\in \mathcal{D}^1$ the maps
$\varphi|X$ and $\widetilde \varphi|X$ are homeomorphisms of $X$ onto itself that agree on $\partial X\subset E^1$. As we have seen in the discussion before the statement of the proposition, this implies that $\varphi|X$ and $\widetilde \varphi|X$ are isotopic rel.\ $\partial X$. By pasting these isotopies on tiles together, we can find an isotopy $\Psi \: S^2 \times [0,1]\rightarrow S^2$ rel.~$E^1$ with
$\Psi(\cdot, 0)=\varphi$ and $\Psi(\cdot, 1)=\widetilde \varphi$.
The concatenation of the isotopies $\Psi$ and $\Phi$ gives the desired isotopy
rel.\ $ {\bf V}^1$ between $\varphi$ and $\id_{S^2}$.\end{proof}
As we know from Section~\ref{sec:tiles}, every Thurston map $f\:S^2\rightarrow S^2$ arises from cell decompositions $\mathcal{D}^1$ and $\mathcal{D}^0$ of $S^2$ as in the last proposition. This gives a useful description of a Thurston map in combinatorial terms. If one wants to study the dynamics of $f$, one is interested in the cell decompositions $\mathcal{D}^n$ obtained from pulling back $\mathcal{D}^0$ by $f^n$ as in Lemma~\ref{lem:pullback}. In general, in order to determine the combinatorics of the whole sequence $\mathcal{D}^n$, $n\in \N_0$ (i.e., the inclusion and intersection patterns of cells on {\em all} levels), is not enough to just know the pair $(\mathcal{D}^0, \mathcal{D}^1)$ and the labeling $\tau\in \mathcal{D}^1\mapsto f(\tau)\in \mathcal{D}^0$, but one also needs specific information on the {\em pointwise} mapping behavior of $f$ on the cells in $\mathcal{D}^1$. Indeed, suppose $g$ is another map that is cellular for $(\mathcal{D}^0, \mathcal{D}^1)$ and
induces the same labeling as $f$, i.e., $f(\tau)=g(\tau)$ for all $\tau\in \mathcal{D}^1$.
Let $\widetilde \mathcal{D}^n$ be the cell decomposition of $S^2$ obtained from $\mathcal{D}^0$ by pulling back by $g^n$. Then one can show (by an argument very similar to the considerations in the proof of Lemma~\ref{lem:invrealize} below) that $\mathcal{D}^n$ and $\widetilde \mathcal{D}^n$ are isomorphic cell complexes for {\em fixed} $n\in \N_0$ (see Definition~\ref{def:compiso}). In contrast, the intersection
patterns of corresponding cells in $\mathcal{D}^n$ and $\widetilde \mathcal{D}^n$
on {\em distinct} levels $n$ may be quite different.
The situation changes if $(\mathcal{D}^1, \mathcal{D}^0)$ is a cellular Markov partition for $f$, because then
the combinatorics of $\mathcal{D}^n$ is completely determined by
$(\mathcal{D}^1, \mathcal{D}^0)$ and the combinatorial data given by the labeling $\tau\in \mathcal{D}^1\mapsto f(\tau)\in \mathcal{D}^0$ (see Remark~\ref{rem:combinatorics}).
This suggests that if one wants to study Thurston maps as given by
Proposition~\ref{prop:thurstonex} from a purely combinatorial point of view, then one should add the additional assumption that $\mathcal{D}^1$ is a refinement of $\mathcal{D}^0$. We will restrict
ourselves to the case
where $\mathcal{D}^0$ contains only two tiles. Then all the relevant assumptions can be condensed into the following definition.
\begin{definition}[Two-tile subdivision rules]\label{def:subdivcomb} Let $S^2$ be a $2$-sphere.
A {\em two-tile subdivision rule}\index{two-tile subdivision rule}\index{subdivision}
for $S^2$ is a triple
$(\mathcal{D}^1, \mathcal{D}^0, L)$ of cell decompositions $\mathcal{D}^0$ and $\mathcal{D}^1$ of $S^2$
and an orientation-preserving labeling $L\: \mathcal{D}^0\rightarrow \mathcal{D}^1$. We assume that the cell decompositions
satisfy the following conditions:
\begin{itemize}
\smallskip
\item[(i)] $\mathcal{D}^0$ contains precisely two $0$-tiles.
\smallskip
\item[(ii)] $\mathcal{D}^1$ is a refinement of $\mathcal{D}^0$, and $\mathcal{D}^1$ contains more than two tiles.
\smallskip
\item[(iii)] If $k$ is the number of $0$-vertices, then $k\ge 3$ and every tile in $\mathcal{D}^1$ is a $k$-gon.
\smallskip
\item[(iv)] The length of the cycle of every $1$-vertex is even.
\end{itemize}
\end{definition}
Our concept of a two-tile subdivision rule is a special case of the more general concept of a \defn{subdivision rule}. See \cite{BS, CFP01, CFP06, CFKP, Me02} for work related to
subdivision rules. The reason for the name {\em two-tile} subdivision rule is that
the data given by $(\mathcal{D}^1,\mathcal{D}^0)$ determines how the {\em two} $0$-tiles are subdivided by the cells in $\mathcal{D}^1$, and this together with the labeling $L$ can be used to create a sequence of cell decomposition $\mathcal{D}^n$ where each cell $\tau\in \mathcal{D}^1$ is subdivided by the cells in $\mathcal{D}^2$ in the same as
as the cell $L(\tau)\in \mathcal{D}^0$ is subdivided by the $1$-cells, etc.
Our definition is tailored to generate Thurston maps, so a more accurate term would have been a ``two-tile subdivision rule generating a Thurston map'', but we chose the shorter term for brevity.
Conversely, suppose $f\:S^2 \rightarrow S^2$ is a Thurston map with $k:=\operatorname{post}(f)\ge 3$,
and $\mathcal{C}\subset S^2$ is an $f$-invariant curve with $\operatorname{post}(f)\subset \mathcal{C}$.
If $\mathcal{D}^0=\mathcal{D}^0(f,\mathcal{C})$, $\mathcal{D}^1=\mathcal{D}^1(f, \mathcal{C})$, and $L\: \mathcal{D}^1\rightarrow \mathcal{D}^0$
is the (orientation-preserving) labeling induced by $f$, then $(\mathcal{D}^1,\mathcal{D}^0,L)$ is a two-tile subdivision rule. This immediately follows from Proposition~\ref{prop:celldecomp} and Proposition~\ref{prop:invmarkov}.
Let $\mathcal{D}^0$ be a cell decomposition of $S^2$ with precisely two tiles $X$ and $Y$. Then necessarily $\partial X= \partial Y$. The set $\mathcal{C}:=\partial X= \partial Y$ is a Jordan curve which we call {\em the Jordan curve of $\mathcal{D}^0$}. Then $\mathcal{C}$ is the $1$-skeleton of $\mathcal{D}^0$ and all vertices and edges of $\mathcal{D}^0$ lie on $\mathcal{C}$. If $k$ is the number of these vertices on $\mathcal{C}$ and $\mathcal{D}^1$ is another cell decomposition of $S^2$,
then
a Thurston map that is cellular
for $(\mathcal{D}^1, \mathcal{D}^0)$ can only exist if each tile in $\mathcal{D}^1$ is a $k$-gon,
i.e., it contains exactly $k$ vertices and edges in its boundary. Moreover, the length of each vertex cycle in $\mathcal{D}^1$ has to be
even, because it must be an integer multiple of the length
of a vertex cycle in $\mathcal{D}^0$ which is always equal to $2$. This motivated conditions (iii) and (iv) in Definition~\ref{def:subdivcomb}.
The next proposition immediately follows from Proposition~\ref{prop:thurstonex} and
gives a large supply of Thurston maps.
\begin{prop}\label{prop:rulemapex} Let $(\mathcal{D}^1,\mathcal{D}^0, L)$ be a two-tile subdivision rule on $S^2$. Then there exists a Thurston map $f\:S^2\rightarrow S^2$ that is cellular for $(\mathcal{D}^0, \mathcal{D}^1)$ and is compatible with the labeling $L$. The map $f$ is unique up to Thurston equivalence.
Moreover, the Jordan curve $\mathcal{C}$ of $\mathcal{D}^0$ is
$f$-invariant and contains the set $\operatorname{post}(f)$.
\end{prop}
\begin{proof} The first part is just a special case
of Proposition~\ref{prop:thurstonex}. Note that $f$ is a Thurston map; indeed,
the number of $1$-tiles is equal to $2\deg(f)$, and also $>2$ by condition (ii) in Definition~\ref{def:subdivcomb}~(ii). So $\deg(f)\ge 2$.
Since $\mathcal{D}^1$ is a refinement
of $\mathcal{D}^0$, the $1$-skeleton $\mathcal{C}$ of $\mathcal{D}^0$ is contained in the $1$-skeleton of $\mathcal{D}^1$. Moreover, since
$f$ is cellular for $(\mathcal{D}^1,\mathcal{D}^0)$, this map sends the $1$-skeleton of $\mathcal{D}^1$ into the $1$-skeleton of $\mathcal{D}^0$. Hence $f(\mathcal{C})\subset \mathcal{C}$, and so $\mathcal{C}$ is $f$-invariant.
Each postcritical point of $f$ is a vertex of $\mathcal{D}^0$ and hence contained in $\mathcal{C}$. \end{proof}
If the map $f$ is as in Proposition~\ref{prop:rulemapex}, then we say that it {\em realizes}
the two-tile subdivision rule.
If, as usual, ${\bf V}^0$ denotes the set of vertices of $\mathcal{D}^0$ and ${\bf V}^1$ the set of vertices $\mathcal{D}^1$, we then
have $\operatorname{crit}(f)\subset {\bf V}^1$ and $\operatorname{post}(f)\subset {\bf V}^0$ (see
Lemma~\ref{lem:constrmaps}). Since the length of each cycle in $\mathcal{D}^0$ is $2$,
a vertex $v$ in ${\bf V}^1$ is a critical point of $f$ if and only if the length of the cycle of $v$ in $\mathcal{D}^1$ is $\ge 4$ (see Remark~\ref{rem:dd'}).
Hence if $f$ and $g$ both realize the subdivision rule, then $\operatorname{crit}(f)=\operatorname{crit}(g)\subset {\bf V}^1$. Moreover, since the orbit of
any point in ${\bf V}^1$ is completely determined by the labeling, we then also
have $\operatorname{post}(f)=\operatorname{post}(g)\subset {\bf V}^0$.
Theorem~\ref{thm:main} implies that every expanding Thurston map $f$ with $\#\operatorname{post}(f)\ge 3$ has an iterate $F=f^n$ that is obtained from a two-tile subdivision rule as in the previous proposition.
If one wants to discuss specific examples of Thurston maps that realize a given two-tile subdivision rule $(\mathcal{D}^1, \mathcal{D}^0, L)$, then it is convenient to represent the relevant data
in a compressed form. As we will see, the information on $L$ is completely determined by a pair of corresponding positively-oriented flags in
$\mathcal{D}^1$ and $\mathcal{D}^0$. See Lemma~\ref{lem:labeluniq} below for a precise statement.
To describe labelings for pairs $(\mathcal{D}^1, \mathcal{D}^0)$ as in Definition~\ref{def:subdivcomb} we proceed
in the manner discussed after Definition~\ref{def:labeldecomp} and choose a more general index set $\mathcal{L}$ for the labeling of the elements
in $\mathcal{D}^0$ and $\mathcal{D}^1$.
To set this up, it is useful to introduce some terminology first.
Let $X$ be a closed Jordan region on $S^2$ with distinct points $v_0, \dots, v_{k-1}, v_{k}=v_0$, $k\ge 3$, on its boundary. Here we use
the cyclic group $\Z_k=\{0, 1,\dots,k-1\}=\Z/k\Z$ as an index set.
Suppose that the points $v_0, \dots, v_{k-1}$ are indexed such that if we start at $v_0$ and run through $\partial X$ with suitable orientation, then the points $v_0, \dots, v_{k-1}$ are traversed in successive order. If this is true and if with this orientation of $\partial X$ the region $X$ lies on the left, then we call the points $v_1, \dots, v_k$ in {\em cyclic}
order on $\partial X$, and otherwise, if $X$ lies on the right, in {\em anti-cyclic} order on $\partial X$.
If the points $v_0, \dots, v_{k-1}$ are in cyclic or anti-cyclic order on $\partial X$, then $\partial X$ is
decomposed into
unique arcs $e_0, \dots, e_{k-1}$; here $e_l$ for $l\in \Z_k$ is the unique subarc of $\partial X$ that has the endpoints $v_l$ and $v_{l+1}$, but does not contain any other of the points $v_i$, $i\in \Z_k\setminus\{l,l+1\}$. We say that the arcs $e_0, \dots, e_{k-1}$ are in {\em cyclic} or {\em anti-cyclic} order on
$\partial X$, if this is true for the points $v_0, \dots, v_{k-1}$, respectively.
Let $\mathcal{D}$ be a cell decomposition of $S^2$.
A chain of tiles $X_1, \dots, X_N$ in $\mathcal{D}$ is called an {\em $e$-chain} if for $i=1, \dots, N-1$ we have $X_i\ne X_{i+1}$ and there exists an edge $e_i$ in $\mathcal{D}$ with $e_i\subset \partial X_{i}\cap \partial X_{i+1}$. The $e$-chain {\em joins} the tiles $X$ and $Y$ if $X_1=X$ and $X_N=Y$.
If $X$ is an arbitrary tile if $\mathcal{D}$, then every tile $Y$ in $\mathcal{D}$ can be joined to $X$ by an $e$-chain. This follows from the fact that the union of the tiles $Y$that can be joined to $X$ is equal to $S^2$; indeed, this union is a nonempty closed set, and it is also open, as follows from Lemma~\ref{lem:specprop} (iv) and (v). Hence the union is all of $S^2$.
Similarly as in Lemma~\ref{lem:colortiles}, we will label the tiles in $\mathcal{D}$ by the two symbols ${\tt b}$ and ${\tt w}$, representing the two colors ``black" and ``white", respectively. So then each tile in $\mathcal{D}$ will carry one of these colors.
The following lemma will
be the basis for the construction of labelings.
\begin{lemma}\label{lem:labelexis}
Let $\mathcal{D}$ be a cell decomposition of $S^2$, and denote by ${\bf V}$
the set of vertices, by $\E$ the set of edges, and by $\X$ the set of tiles in $\mathcal{D}$.
Suppose that the length of the cycle of every vertex in $\mathcal{D}$ is even and that there exists $k\ge 3$ such that every tile in $\X$ is a $k$-gon.
Then for each positively-oriented flag $(c_0,c_1,c_2)$ in $\mathcal{D}$ there exist maps
$L_{{\bf V}}\: {\bf V}\rightarrow \Z_k$, $L_\E\: \E\rightarrow \Z_k$, and $L_\X\: \X\rightarrow \{{\tt b}, {\tt w}\}$ with the following properties:
\begin{itemize}
\smallskip
\item[(i)] $L_{{\bf V}}(p_0)=0$, where $c_0=\{p_0\}$, $L_\E(c_1)=0$, and $L_\X(c_2)={\tt w}$,
\smallskip
\item[(ii)] if $X, Y\in \X$ are two distinct tiles with a common edge on their boundaries, then
$L_\X(X)\ne L_\X(Y)$,
\smallskip
\item[(iii)] if $X$ is an arbitrary tile in $\X$, and $v_0, \dots, v_{k-1}$ are the vertices on its boundary indexed by $\Z_k$, then we can choose the indices of the vertices so that $L_{{\bf V}}(v_i)=i$ for each $i\in \Z_k$, and such that the order of the vertices on $\partial X$ is cyclic if $L_\X(X)={\tt w}$ and anti-cyclic if $L_\X(X)={\tt b}$,
\smallskip
\item[(iv)] if $e\in \E$ and $l=L_\E(e)$, then $L_{{\bf V}}(\partial e)=
\{l,l+1\}$,
\smallskip
\item[(v)] if $X$ is an arbitrary tile in $\X$, and $e_0, \dots, e_{k-1}$ are the edges on its boundary indexed by $\Z_k$, then we can choose the indices of the edges so that $L_{{\bf E}}(e_i)=i$ for each $i\in \Z_k$, and such that the order of the edges on $\partial X$ is cyclic if $L_\X(X)={\tt w}$ and anti-cyclic if $L_\X(X)={\tt b}$,
\smallskip
\item[(vi)] if $(\tau_0, \tau_1, \tau_2)$ is a flag in $\mathcal{D}$, then the flag is positively-oriented if and only if
there exists $l\in \Z_k$ such that $L_{\bf V}(\tau_0)=\{l\}$, $L_\E(\tau_1)=l$,
$L_\X(\tau_2)=\tt w$, or
$L_{\bf V}(\tau_0)=\{l\}$, $L_\E(\tau_1)=l-1$,
$L_\X(\tau_2)=\tt b$.
\end{itemize}
The maps $L_{{\bf V}}$, $L_\E$, and $L_\X$ are uniquely determined by the properties \textnormal{(i)--(iv)}.
\end{lemma}
Condition (ii) says that one of the tiles containing an edge is ``black'' and the other is ``white''.
Moreover, by (iii) and (v) we can index the vertices $v$ and edges $e$ on the boundary of a tile by the label $L_\X(v)\in \Z_k$ and $L_{\E}(e)\in \Z_k$, respectively, so that the vertices and edges of a white tile are in cyclic order and the
ones on a black tile are in anti-cyclic order. By (iv) the label $L_\E(e)$ of an edge $e\in \E$ is determined by the labels $L_{\bf V}(u)$ and
$L_{\bf V}(v)$ of the two endpoints of $e$ (here it is important that $k\ge 3$).
\begin{proof} We first establish the following statement.
{\em Claim.} Suppose that $J\subset S^2$ is a Jordan curve that does not contain
any vertex (in $\mathcal{D}$) and has the property that for every edge $e$
the intersection $e\cap J$ is either empty, or $e$ meets both components of $S^2\setminus J$ and $e\cap J$ consists of a single point. Then $J$ meets an even number of edges.
To see this pick one of the complementary components $U$ of $S^2\setminus J$, and let $d_1, \dots, d_n$, $n\in \N_0$, be the length of the cycles of the vertices contained in $U$ ( for $n=0$ we consider this as an empty list). Let $\E_J$ be the set of all edges that meet $J$ and $\E_U$ be the set of all edges contained in $U$. From our assumption on the intersection property of $J$ with edges it follows that an edge is contained in $U$ if and only if its two endpoints are in $U$, and it meets $J$ if and only if one endpoint is in $U$ and the other in $S^2\setminus \overline{U}$. Hence
$$d_1+\dots+d_n=\#\E_J+2\#\E_U,$$
because the sum on the left hand side counts every edge in $\E_J$ once, and every edge in $\E_U$ twice. Since all the numbers $d_1, \dots, d_n$ are even, it follows that the number $\#\E_J$ of edges that $J$ meets is also even, proving the claim. \smallskip
To show existence and uniqueness of the map $L_\X$ we proceed as follows. For every tile $Y$ there exits an $e$-chain
$Y_0=c_2, \dots, Y_N=Y$ of tiles joining the ``base tile'' $c_2$ to $Y$.
We put $L_\X(Y)={\tt w}$ or $L_\X(Y)={\tt b}$ depending on whether
$N$ is even or odd. It is clear that if this is well-defined, then it is the unique choice for $L_\X(Y)$. This follows from the normalization (i) and that fact that by (ii) the labels of tiles have to alternate along an $e$-chain.
To see that $L_\X$ is well-defined it is enough to show that if an
$e$-chain $X_0, X_1, \dots, X_{N}$ forms a cycle, i.e., if $X_0=
X_{N}$, then $N$ is even. To prove this we may make the additional assumption that $N\ge 3$ and that the chain is {\em simple}, i.e., that the tiles $X_1, \dots X_N$ are all distinct.
We can choose edges $e_i$ for $i=1, \dots, N$ such that
$e_i\subset \partial X_{i-1}\cap \partial X_{i}$. Then the edges $e_1, \dots, e_N$ are all distinct. For otherwise, $e_i=e_j$ for some
$1\le i<j\le N$. Then $e_i=e_j$ is contained in the boundary of the tiles $X_{i-1}, X_{i}, X_{j-1}, X_{j}$ which is impossible, because three of these tiles must be distinct (note that $N\ge 3$).
We now construct a Jordan curve $J$ as follows. For each edge $e_i$ pick a point $x_i\in \inte(e_i)$. Moreover, for $i=1,\dots, N$, we can choose an arc $\alpha_i\subset X_i$ with endpoints $x_{i}$ and $x_{i+1}$ such
that $\inte(\alpha_i)\subset \inte(X_i)$. Here $x_{N+1}:=x_1$.
Then $J=\alpha_1\cup \dots \cup \alpha_N$ is a Jordan curve that has properties as in the claim above. The curve $J$ meets the edges $e_1, \dots, e_N$ and no others. Hence $N$ is even. Thus
$L_\X$ is well-defined, and it satisfies property (ii) and is normalized as in (i). \smallskip
To show the existence of $L_{{\bf V}}$ it is useful to quickly recall some basic definitions from the homology and cohomology of chain complexes.
Denote by $\E_o$ the set of oriented edges in $\mathcal{D}$. Let
$C(\X)$ and $C(\E_o)$ be the free modules over $\Z_k$
generated by the sets $\X$ and $\E_o$, respectively.
So $C(\E_o)$, for example, is just the set of formal finite sums
$\sum a_ie_i$, where $a_i\in \Z_k$ and $e_i\in \E_o$. Note that in contrast to other commonly used definitions of chain complexes we have $e+\widetilde e\ne 0$ if $e$ and $\widetilde e$ are the same edges with opposite orientation.
There is a unique boundary operator $b\: C(\X)\rightarrow C(\E_o)$
that is a module homomorphism and satisfies
$$bX:=b(X)=\sum_{e\subset \partial X} e $$
for each tile $X$,
where the sum is extended over all oriented edges $e\subset \partial X$ so that $X$ lies on the left of $e$.
Let $e$ be an oriented edge and $X$ be the unique tile with $e\subset\partial X$ that is on the left of $e$. We put $\alpha(e)=1\in \Z_k$ or $\alpha(e)=-1\in \Z_k$ depending on whether $L_\X(X)={\tt w}$ ($X$ is a white tile) or $L_\X(X)={\tt b}$ ($X$ is a black tile). If $e$ and $\widetilde e$ are the same edges with opposite orientation, then $\alpha(e)+\alpha(\widetilde e)=0$ as follows from property (ii) of $L_\X$.
The map $\alpha$ can be uniquely extended to a homomorphism
$\alpha\: C(\E_o)\rightarrow \Z_k$. In the language of cohomology it is a ``cochain''. This cochain $\alpha$ is a cocycle, i.e.,
\begin{equation}\label{abX=0}
\alpha(b X)=\sum_{e\subset\partial X}\alpha(e)=\pm k=0\in \Z_k
\end{equation}
for every tile $X$,
considered as one of the generators of $C(\X)$.
Indeed, by our convention on the orientation of edges $e\subset \partial X$ in the above sum, for each such edge we get the same contribution $\alpha(e)$ in the sum, and so, since $X$ has $k$ edges, the sum is equal to $\pm k=0\in \Z_k$.
Consider an arbitrary closed edge path consisting of the oriented edges
$e_1, \dots, e_n$; so the terminal point of $e_i$ is the initial point of $e_{i+1}$ for $i=1, \dots, n$, where $e_{n+1}:=e_1$.
We claim that
\begin{equation}\label{alphacycle}
\sum_{i=1}^n\alpha(e_i)=0.
\end{equation}
Essentially, this is a consequence of the fact that we have
$H^1(S^2, \Z_k)=0$ for the first cohomology group of $S^2$ with coefficients in $\Z_k$. This implies that the cocycle $\alpha$ is a coboundary.
We will present a simple direct argument. To show \eqref{alphacycle} it is clearly enough to establish this
for simple closed edge paths, i.e., for closed edge paths where
the union of the edges forms a Jordan curve $J\subset S^2$. Let $U$ be
the complementary component of $S^2\setminus J$ so that $U$ lies on the left if we traverse $J$ according to the orientation given by the edges $e_i$. If $X_1, \dots, X_M$ are all the tiles contained in $\overline U$, then
$$b(X_1+\dots+X_M)=\sum_{e\subset \overline U}e, $$
where the sum is extended over oriented edges contained in $\overline U$. Each edge on $J$ is equal to one of the edges $e_i$ and it appears in the above sum exactly once and with the same orientation as $e_i$. All other edges in $\overline U$ appear twice and with opposite orientations.
Hence by \eqref{abX=0},
$$\sum_{i=1}^n \alpha(e_i)= \sum_{e\subset \overline U}\alpha(e)=
\sum_{i=1}^M\alpha(b X_i)=0. $$
We now define $L_{{\bf V}}\: {\bf V}\rightarrow \Z_k$ as follows.
For $v\in {\bf V}$ we can pick an edge path consisting of the oriented edges $e_1, \dots, e_n$ that joins the base point $p_0$ to $v$ (this list of edges may be empty if $v=p_0$).
The existence of such an edge path follows from the connectedness of the $1$-skeleton of $\mathcal{D}$ (see Lemma~\ref{lem:specprop}~(vi)).
Put
\begin{equation}\label{defphi}
L_{{\bf V}}(v)=\sum_{i=1}^n\alpha(e_i).
\end{equation}
This is well-defined, because
we have \eqref{alphacycle} for every closed edge path; we also have the normalization $L_{{\bf V}}(p_0)=0$.
The definition of $L_{{\bf V}}$ implies that if $e$ is an oriented edge, and $u$ is the initial and $v$ the terminal point of $e$, then
\begin{equation}\label{incrdecr}
L_{\bf V}(v)=L_{{\bf V}}(u)+\alpha(e).
\end{equation}
This means that if we go from the initial point $u$ of $e$ to the terminal point $v$, then the value of $L_{{\bf V}}$ is increased by $1$ or decreased by $-1$ depending on whether the tile on the left of $e$ is white or black. The desired property (iii) of $L_{{\bf V}}$ immediately follows from this.
This shows existence of $L_{{\bf V}}$. Conversely, every function
$L_{{\bf V}}$ with property (iii) must satisfy \eqref{incrdecr}. Together with the normalization $L_{{\bf V}}(p_0)=0$ this implies that $L_{{\bf V}}$ is given by the formula
\eqref{defphi}, and so we have uniqueness.
To define $L_\E$ note that if
$e\in \E$, then by (ii) we can choose a unique orientation for $e$ such that the tile on the left is ``white'', and the one one the right is ``black''. If $u$ is the initial and $v$ the terminal point of $e$ according to this orientation, and $L_{{\bf V}}(u)=l\in \Z_k$, then $L_{{\bf V}}(v)=l+1$.
Now set $L_\E(e):= l$. Then $L_\E$ has property (iv).
Moreover, we also have the normalization (i) for $L_\E$; indeed, is $c_1$ is oriented so that $p_0$ is the initial point of $c_1$, then $c_2$ lies on the left of $c_1$, because the flag $(c_0,c_1,c_2)$ is positively-oriented. Since $L_\X(c_2)={\tt w}$, the tile $c_2$ is white and so $L_\E(e)=L_{{\bf V}}(p_0)=0$. Uniqueness of $L_\E$ follows from (iii) and the uniqueness of $L_{{\bf V}}$.
We have proved (i)--(iv) and the uniqueness statement. It remains to establish (v) and (vi).
To show (v) let $X\in \X$ be arbitrary. Then by (iii) we can assume that the indexing of the $k$ vertices
$v_0, \dots v_{k-1}$ on $\partial X$ is such that $L_{\bf V}(v_i)=i$ for all $i\in \Z_k$, and that
$v_0, \dots v_{k-1}$ are met in successive order if we traverse $\partial X$. This implies
that for each $i\in \Z_k$ there exists a unique edge $e_i\subset \partial X$ in $\mathcal{D}$ with endpoints $v_i$ and $v_{i+1}$. Hence by (iv) we have $L_{\E}(e_i)=i$. Moreover, by (iii) the edges $e_0, \dots , e_{k-1}$ are in cyclic or anti-cyclic order on $\partial X$ depending on whether $L_{\X}(X)=\tt w$ or
$L_{\X}(X)=\tt b$. So (v) holds.
Finally, to see that (vi) is true, let $(\tau_0, \tau_1, \tau_2)$ be a flag in
$\mathcal{D}$. Then $\tau_0=\{u\}$ for some $u\in {\bf V}$. The vertex $u$ is the initial point of the oriented edge $\tau_1$. Let $v\in {\bf V}$
be the terminal point of $\tau_1$, and define $l=L_{\bf V}(u)$.
Depending on whether the flag is positively- or negatively-oriented, the vertex $v$ follows $u$ is cyclic or anti-cyclic
order on $\partial \tau_2$. So if the flag is positively-oriented,
then
by property (iii) we have $L_{\bf V}(v)=l+1$ if $L_\X(\tau_2)=\tt w$
and $L_{\bf V}(v)=l-1$ if $L_\X(\tau_2)=\tt b$. Property (iv) implies that
$L_{\E}(\tau_1)=l$ if $L_\X(\tau_2)=\tt w$ and
$L_{\E}(\tau_1)=l-1$ if $L_\X(\tau_2)=\tt b$.
So if $(\tau_0, \tau_1, \tau_2)$ is positively-oriented, then the cells in
this flag carry the labels $l$, $l$, $\tt w$, or $l$, $l-1$, $\tt b$, respectively.
Similarly, if $(\tau_0, \tau_1, \tau_2)$ is negatively-oriented, then we get the labels $l$, $l-1$, $\tt w$, or $l$, $l$, $\tt b$ for the cells in the flag. Statement (vi) follows from this. \end{proof}
\begin{lemma} \label{lem:labeluniq}
Let $(\mathcal{D}^1, \mathcal{D}^0)$ be a pair of cell decompositions of $S^2$
satisfying conditions \textnormal{(i)--(iv)} in
Definition~\ref{def:subdivcomb}, and let
$(c'_0,c'_1,c'_2)$ and $(c_0,c_1,c_2)$ be
positively-oriented flags
in $\mathcal{D}^1$ and $\mathcal{D}^0$, respectively.
Then there exists a unique orientation-preserving labeling $L\:\mathcal{D}^1\rightarrow \mathcal{D}^0$ with $(L(c'_0), L(c'_1), L(c'_2))=(c_0,c_1,c_2)$. \end{lemma}
In particular, $(\mathcal{D}^1, \mathcal{D}^0, L)$ is a two-tile subdivision rule.
\begin{proof} For $i=0,1$ denote by ${\bf V}^i, \E^i, \X^i $ the set of vertices, edges, and tiles of $\mathcal{D}^i$, respectively.
To describe the labeling for $(\mathcal{D}^1, \mathcal{D}^0)$ we proceed
in the manner discussed after Definition~\ref{def:labeldecomp} and choose a particular index set $\mathcal{L}$ for the labeling of the elements
in $\mathcal{D}^0$ and $\mathcal{D}^1$.
We let $\mathcal{L}$ be the set that consists of two disjoint copies of $\Z_k$ (one will be for the vertices, and one for the edges), and the set $\{\tt b, \tt w\}$, where again we think of $\tt w$ representing ``white'' and $\tt b$ representing ``black''.
We assign to $c_2\in \X^0$ the color ``white'', and ``black'' to the other tile in $\X^0$. We assign $0\in \Z_k$ to the $0$-vertex
$v_0\in c_0$. Then there is a unique way to assign labels in $\Z_k$ to the other vertices on $\mathcal{C}:=\partial c_2$ (and the corresponding cells of dimension $0$) such that if $v_0, v_1, \dots, v_{k-1}$ are the vertices indexed by their label, then they are in cyclic order on $\mathcal{C}$ as considered as the boundary of the white $0$-tile and in anti-cyclic order for the black $0$-tile.
Each $0$-edge $e$
is an arc on $\mathcal{C}$ with endpoints $v_l$ and $v_{l+1}$ for a unique $l\in\Z_k$. We label $e$ by $l$ (where $l$ is thought of to belong to
the second copy of $\Z_k$). Since $(c_0, c_1, c_2)$ is a positively-oriented flag, and $v_0$ is the initial point of $c_1$, the edge $c_1$ has the label $0$. All this is just a special case of Lemma~\ref{lem:labelexis}.
If in this way we assign to each element in $\mathcal{D}^0$ a label in $\mathcal{L}$, we get a bijection $\psi\: \mathcal{D}^0\rightarrow \mathcal{L}$. Note that if $(\tau_0, \tau_1,\tau_2)$ is any positively-oriented flag in $\mathcal{D}^0$, then its image under $\psi$
has the form $(l,l,\tt w)$ or $(l,l-1,\tt b)$ for some $l\in \Z_k$
(cf.~Lemma~\ref{lem:labelexis}~(v)).
For $\mathcal{D}^1$ we invoke Lemma~\ref{lem:labelexis} directly to set up a suitable map
$\varphi\:\mathcal{D}^1\rightarrow \mathcal{L}$. Since $\mathcal{D}^1$ satisfies the conditions of Lemma~\ref{lem:labelexis}, we can find maps $L_{\bf V}\:{\bf V}^1\rightarrow \Z_k$, $L_\E\: \E^1\rightarrow \Z_k$, and
$L_\X\: \X^1\rightarrow\{ \tt b, \tt w\}$ with the properties (ii)--(iv) stated in the lemma and the normalizations $L_{\bf V}(v'_0)=0$, where $c'_0=\{v'_0\}$, $L_\E(c'_1)=0$, and $L_\X(c'_2)=\tt w$.
The maps $L_{\bf V}$, $L_\E$, $L_\X$ induce a unique map $\varphi\: \mathcal{D}^1\to \mathcal{L}$ such that $\varphi(c)=L_\X(c)$ if $c$ is a $1$-tile, $\varphi(c)=L_\E(c)$ if $c$ is a $1$-edge, and
$\varphi(c)=L_{\bf V}(v)$ if $c=\{v\}$ consists of a $1$-vertex $v$.
Now define $L:=\psi^{-1}\circ \varphi:\ \mathcal{D}^1\rightarrow \mathcal{D}^0$. The map $L$ assigns to each $1$-cell $c$
the unique $0$-cell that has the same dimension as $c$ and carries the same label in $\mathcal{L}$ as $c$.
It follows immediately from the properties of the maps
$\psi$ and $\varphi$ that $L$ preserves dimensions, respects inclusions, and is injective on cells. Hence $L$ is a labeling
according to Definition~\ref{def:labeldecomp}.
By our normalizations the map $L$ sends the flag $(c'_0,c'_1,c'_2)$ to
$(c_0,c_1,c_2)$.
Moreover, $L$ is
orientation-preserving. Indeed, $\varphi$ maps the cells $\tau_0, \tau_1, \tau_2$ in a positively-oriented flag in $\mathcal{D}^1$ to
$l$, $l$, $\tt w$, or to $l$, $l-1$, $\tt b$, respectively, where $l\in \Z_k$. These triples correspond to
positively-oriented flags in
$\mathcal{D}^0$.
It follows that $L$ has the desired properties. \smallskip
To show uniqueness, we reverse the process. Given
$L$ with the stated properties, we use the same
map $\psi\: \mathcal{D}^0\rightarrow \mathcal{L}$ as above and define maps
$L_{\bf V}\: {\bf V}^1\rightarrow \Z_k$, $L_\E\:\E^1\rightarrow \Z_k$, $L_\X\: \X^1\rightarrow
\{\tt b, \tt w\}$ such that $L_\X(c)=(\psi\circ L)(c)$ if $c$ is a $1$-tile, $L_\E(c)=(\psi\circ L)(c)$ if $c$ is a $1$-edge, and such that
$L_{\bf V}(v)=(\psi\circ L)(c)$ if $c=\{v\}$ consists of a $1$-vertex $v$.
Then we have normalizations $L_{\bf V}(v'_0)=0$, $L_\E(c'_1)=0$, and
$L_\X(c'_2)=\tt w$ as in Lemma~\ref{lem:labelexis}~(i). If we can show that $L_{\bf V}$, $L_\E$,
$L_\X$ have the properties (ii)--(iv) in Lemma~\ref{lem:labelexis}, then the uniqueness of $L$ will follow from the corresponding uniqueness statement in this lemma.
To see this let $e\in \mathcal{D}^1$ be arbitrary, and $X,Y\in \mathcal{D}^1$ be the two tiles that contain $e$ in its boundary. Let $u,v\in {\bf V}^1$ be the two endpoints of $e$. We may assume that notation is chosen so that the flag $(\{u\},e,X)$ is
positively-oriented. Then
$(\{v\},e,Y)$ is also positively-oriented. It follows that
the images of these flags under $L$ are
positively-oriented. Since $L$ is injective on cells, and so $L(u)\ne L(v)$, this implies that
$L(X)\ne L(Y)$. So $L(X)$ and $L(Y)$ carry different colors (given by $\psi$) which implies that $X$ and $Y$ also carry
different colors by definition of $L_\X$. Hence $L_\X$ has property (ii) in Lemma~\ref{lem:labelexis}. By switching the notation for $u$ and $v$ and $X$ and $Y$ if necessary, we may assume that $X$ is a white tile. Since the flag $(\{L(u)\}, L(e), L(X))$
is positively-oriented, and $\phi(X)$ is white, it follows that for some $l\in \Z_k$ we have $\psi(L(u))=l$ and $\psi(L(e))=l$. Hence $L_{\bf V}(u)=l$ and $L_\E(e)=l$. Similarly, using that $L(Y)$ is black and that $(\{L(v)\}, L(e), L(Y))$ is positively-oriented, it follows that $L_{\bf V}(v)=l+1$.
In other words, if we run along an oriented edge $e$ in $\mathcal{D}^1$ so that a white tile lies on the left of $e$, then the label of the endpoints of $e$ (given by $L_{\bf V}$) is increased by one, and decreased by one if a black tile lies on the left. Hence $L_{\bf V}$ has the property (iii) in Lemma~\ref{lem:labelexis}.
Moreover, we also see that the label $L_\E(e)$ is related to the labels of its endpoints as in statement (iv) of Lemma~\ref{lem:labelexis}. The uniqueness of $L$ follows.
\end{proof}
Let $f$ be a map realizing a two-tile subdivision rule $(\mathcal{D}^1,\mathcal{D}^0,L)$. We want to show that the property of $f$ being combinatorially expanding for the Jordan curve $\mathcal{C}$ of $\mathcal{D}^0$
is independent of the realization. In contrast, this is not true for
expansion of the map (see Example~\ref{ex:barycentric}).
We require a lemma.
\begin{lemma} \label{lem:cexp_Cinv}
Let $f\: S^2\to S^2$ and $g\:\widehat S^2\rightarrow \widehat S^2$ be
Thurston maps. Suppose that $\# \operatorname{post}(f)\ge 3$, that
$\mathcal{C}\subset S^2$ is an $f$-invariant Jordan curve with
$\operatorname{post}(f)\subset \mathcal{C}$, and that $h_0,h_1\: S^2 \rightarrow \widehat S^2$
are orientation-preserving homeomorphisms satisfying
$h_0|\operatorname{post}(f)=h_1|\operatorname{post}(f)$, $h_0\circ f=g\circ h_1$, and
$h_0(\mathcal{C})=h_1(\mathcal{C})$.
Then $f$ is combinatorially expanding for $\mathcal{C}$ if and only if $g$ is combinatorially expanding for $\widehat \mathcal{C}:=h_0(\mathcal{C})=h_1(\mathcal{C})$.
\end{lemma}
\begin{proof} We have $\operatorname{post}(g)=h_0(\operatorname{post}(f))=h_1(\operatorname{post}(f))$
(see the proof of \eqref{eq:postfg}). Hence $\#\operatorname{post}(g)=\#\operatorname{post}(f)\ge 3$. Moreover, $\widehat \mathcal{C}\subset \widehat S^2$ is a Jordan curve
with $\operatorname{post}(g)\subset \widehat \mathcal{C}$. This curve is $g$-invariant, since
$$g(\widehat \mathcal{C})=g(h_0(\mathcal{C}))=h_1(f(\mathcal{C}))\subset h_1(\mathcal{C})=\widehat \mathcal{C}. $$
So the statement that $g$ is combinatorially expanding for
$\widehat \mathcal{C}$ is meaningful (see Definition~\ref{def:combexp}).
Pick an orientation of $\mathcal{C}$. By our assumptions the map $\varphi:=h_1^{-1}\circ h_0$ fixes the elements of $\operatorname{post}(f)$ pointwise and the Jordan curve $\mathcal{C}$ setwise. Since $\#\operatorname{post}(f)\ge 3$ and
$\operatorname{post}(f)\subset \mathcal{C}$, this implies that $\varphi$ preserves the orientation of $\mathcal{C}$. Since $\varphi$ is an orientation-preserving homeomorphism on $S^2$, the map $\varphi$ sends each of the
complementary components of $\mathcal{C}$ to itself. Thus, $\varphi$
is cellular for $\mathcal{D}^0:=\mathcal{D}(f,\mathcal{C})$ and we have $\varphi(c)=c$ for each cell $c\in \mathcal{D}^0$. As in the proof of Proposition~\ref{prop:thurstonex}, this implies that $\varphi$ is isotopic to
$\id_{S^2}$ rel.\ $\operatorname{post}(f)$. Hence $h_0=h_1\circ \varphi$ is isotopic
to $h_1=h_1\circ \id_{S^2}$ rel.\ $\operatorname{post}(f)$, and so there exists an isotopy $H^0\: S^2\times I\rightarrow \widehat S^2$ rel.\ $\operatorname{post}(f)$ with
$H^0_0=h_0$ and $H^0_1=h_1$.
As in the proof of Proposition~\ref{thm:exppromequiv}, based Proposition
\ref{prop:isotoplift} we can repeatedly lift the initial isotopy $H^0$. In this way we can find isotopies $H^n\: S^2\times I\rightarrow \widehat S^2$ rel.\ $\operatorname{post}(f)$ such that
$H^n_t\circ f=g\circ H^{n+1}_t$ and $H^{n+1}_0=H^n_1$ for all $n\in \N_0$ and $t\in I$. Note that $H^n$ for $n\ge 1$ is actually an isotopy rel.\ $f^{-1}(\operatorname{post}(f))\supset \operatorname{post}(f)$.
Define homeomorphisms $h_n:= H^n_0$ for
$n\in\N_0$ (note that for $n=0$ and $n=1$ these maps agree with our given maps $h_0$ and $h_1$). Then
$h_n\circ f=g\circ h_{n+1}$ , and so
\begin{equation} \label{eq:fngn}
h_0\circ f^n=g^n\circ h_n
\end{equation} for all $n\in \N_0$.
We have $h_n|\operatorname{post}(f)=h_0|\operatorname{post} (f)$ which implies
\begin{equation}\label{eq:hnsame}
h_n(\operatorname{post}(f))=\operatorname{post}(g)
\end{equation}
for all $n\in \N_0$. Moreover,
$h_n|f^{-1}(\operatorname{post}(f))=h_1|f^{-1}(\operatorname{post}(f))$ and so
\begin{equation} \label{eq:hnfgpre}
h_n(f^{-1}(\operatorname{post}(f)))=g^{-1}(\operatorname{post}(g))
\end{equation} for
$n\in \N_0$ as follows from \eqref{eq:hnsame} and
Lemma~\ref{lem:lifts_inverses}.
Our hypotheses imply that if $c$ is a cell in $\mathcal{D}^0(f,\mathcal{C})$, then $h_0(c)$ is a cell in
$\mathcal{D}^0(g,\widehat \mathcal{C})$. Since the set
$$\widehat \mathcal{D}^n:= \{ h_n(c): c\in \mathcal{D}^n(f,\mathcal{C})\}$$ is a cell decomposition of $\widehat S^2$, it follows from this and \eqref{eq:fngn}
that $g^n$ is cellular for $(\widehat \mathcal{D}^n, \mathcal{D}^0(g, \widehat \mathcal{C}))$. Since $g^n$ is also cellular for the pair $(\mathcal{D}^n(g,\mathcal{C}), \mathcal{D}^0(g, \widehat \mathcal{C}))$, the uniqueness statement in Lemma~
\ref{lem:pullback} implies that $\widehat \mathcal{D}^n=\mathcal{D}^n(g,\mathcal{C})$ for all
$n\in \N_0$. In other words, the $n$-cells for $(g,\widehat \mathcal{C})$ are precisely the images of the $n$-cells for $(f,\mathcal{C})$ under the homeomorphism $h_n$.
We also have
\begin{equation}\label{eq:varccc}
h_n(\mathcal{C})=\widehat \mathcal{C}
\end{equation}
for each $n\in \N_0$. This can be seen by induction on $n$ as follows.
The statement is true for $n=0$ and $n=1$ by our hypothesis and by the definition of $\widehat \mathcal{C}$. Assume that $h_n(\mathcal{C})=\widehat \mathcal{C}$ for some $n\in \N$. Then by
Lemma \ref{lem:lifts_inverses} and the induction hypotheses we have
$$J:=h_{n+1}(\mathcal{C})\subset h_{n+1}(f^{-1}(\mathcal{C}))= g^{-1}( h_n(\mathcal{C}))=g^{-1}(\widehat \mathcal{C}). $$ It follows from \eqref{eq:hnfgpre} that
$(H^{n}_t)\circ h_{n}^{-1}$ is an isotopy on $\widehat S^2$ rel.\ $g^{-1}(\operatorname{post}(g))$. It isotopes $\widehat \mathcal{C}=h_n(\mathcal{C})\subset g^{-1}(\widehat \mathcal{C})$ into $J=h_{n+1}(\mathcal{C})$ rel.\ $g^{-1}(\operatorname{post}(g))$.
So $\mathcal{C}$ and $J$ are Jordan curves contained in the $1$-skeleton $g^{-1}(\widehat \mathcal{C})$ of $\mathcal{D}^1(g,\widehat \mathcal{C})$ that
are isotopic relative to the set $g^{-1}(\operatorname{post}(g))$ of vertices of
$\mathcal{D}^1(g,\widehat \mathcal{C})$. Lemma~\ref{lem:isoJcin1ske} implies that $J=\widehat \mathcal{C}$, and
\eqref{eq:varccc} follows.
Now \eqref{eq:varccc} and \eqref{eq:hnsame} imply that a chain of $n$-tiles for $(f,\mathcal{C})$ joins opposite sides of
$\mathcal{C}$ if and only if their images under $h_n$ form a chain joining opposite sides of $\widehat \mathcal{C}$.
Since the images of the $n$-tiles for $(f,\mathcal{C})$ under $h_n$ are the precisely the $n$-tiles
for $(g, \widehat \mathcal{C})$, we have $D_n(f, \mathcal{C})=D_n(g, \widehat \mathcal{C})$ for each $n\in \N_0$. The statement follows. \end{proof}
Now we can show the desired realization independence of
combinatorial expansion.
\begin{lemma} \label{lem:invrealize}
Let $(\mathcal{D}^1,\mathcal{D}^0,L)$ be a two-tile subdivision rule on $S^2$ and $\mathcal{C}$ be the Jordan curve of $\mathcal{D}^0$. Suppose that the maps $f\:S^2\rightarrow S^2$ and $g\:S^2\rightarrow S^2$ both realize the subdivision rule and that $\#\operatorname{post}(f)=\#\operatorname{post}(g)\ge 3$.
Then $f$ is combinatorially expanding for $\mathcal{C}$ if and only if $g$ is combinatorially expanding for $\mathcal{C}$.
\end{lemma}
\begin{proof} Let ${\bf V}^0$ and
${\bf V}^1$ be the set of vertices of $\mathcal{D}^0$ and $\mathcal{D}^1$,
respectively. Then $P:=\operatorname{post}(f)=\operatorname{post}(g)\subset {\bf V}^0 \subset {\bf V}^1$.
It follows from the proof of the uniqueness part of Proposition~\ref{prop:thurstonex}
that there exists a homeomorphism $h_1\:S^2\rightarrow S^2$ isotopic to $\id_{S^2}$ rel.\ ${\bf V}^1\supset \operatorname{post}(f)=\operatorname{post}(g)$ that satisfies $f=g\circ h_1$. Moreover, $h_1(e)=e$ for each
edge $e$ in $\mathcal{D}^1$. Since $\mathcal{D}^1$ is a refinement of $\mathcal{D}^0$ and so the $1$-skeleton $\mathcal{C}$ of $\mathcal{D}^0$ is contained in the $1$-skeleton of $\mathcal{D}^1$, this implies $h_1(\mathcal{C})=\mathcal{C}$. Define $h_0=\id_{S^2}$. Since $h_1$ is isotopic to $\id_{S^2}$ rel.\ $P$ we have $h_1|P=\id_{S^2}|P=h_0|P$.
Moreover, $h_0\circ f=g\circ h_1$, $h_1(\mathcal{C})=\mathcal{C}=h_0(\mathcal{C})$, and both $h_0$ and $h_1$ are orientation-preserving homeomorphisms on $S^2$.
This shows that the hypotheses of Lemma~\ref{lem:cexp_Cinv} are satisfied (with $\widehat S^2=S^2$), and so $f$ is combinatorially expanding for $\mathcal{C}$ if and only if $g$ is combinatorially for $\widehat \mathcal{C}=h_0(\mathcal{C})=h_1(\mathcal{C})=\mathcal{C}$.
\end{proof}
Based on the previous lemma we say that a two-tile subdivision rule $(\mathcal{D}^1, \mathcal{D}^0,L)$ is {\em combinatorially expanding} if one (and hence each) map $f$ that realizes the subdivision rule is
combinatorially expanding for the Jordan curve $\mathcal{C}$ of $\mathcal{D}^0$; here we tacitly assume that the hypothesis $\#\operatorname{post}(f)\ge 3$ of the previous lemma is true.
\subsection{Examples of two-tile subdivision rules}
\label{sec:examples-two-tile}
We present some examples of two-tile subdivision rules. A first
example can be obtained from the map $g$ in Section
\ref{sec:Lattes} and the subdivision rule as indicated in Figure \ref{fig:mapg}.
\ifthenelse{\boolean{nofigures}}{}{
\begin{figure}
\centering
\begin{picture}(10,10)
\put(258,62){$\scriptstyle{0}$}
\put(210,62){$\scriptstyle{-1}$}
\put(264,107){$\scriptstyle{\infty}$}
\put(86,58){$\scriptstyle{0\mapsto -1}$}
\put(126,58){$\scriptstyle{1\mapsto 0}$}
\put(88,107){$\scriptstyle{\infty\mapsto \infty}$}
\put(36,44){$\scriptstyle{-1\mapsto 0}$}
\put(167,78){$\scriptstyle{p}$}
\end{picture}
\includegraphics[width=11cm]{examplez2-1.eps}
\caption{The two-tile subdivision rule for $z^2-1$.}
\label{fig:subdivz2-1}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=12cm]{poly1_7.eps}
\caption{Tiles of order $7$ for Example \ref{ex:z2-1}.}
\label{fig:tiles8_z2-1}
\end{figure}
}
\begin{ex}
\label{ex:z2-1}
Our next example is as follows. The white $0$-tile is the (closure
of the) upper half-plane, the black $0$-tile is the (closure of the)
lower half-plane in $\CDach$. The $0$-vertices are the points
$-1,0,\infty$. Thus the $0$-edges are $[-\infty, -1], [-1,0],
[0,\infty]$. The cell decomposition $\mathcal{D}^0$ is indicated to the
right of Figure \ref{fig:subdivz2-1}.
The white $1$-tiles are the first and third quadrant, the black
$1$-tiles are the second and forth quadrant. The $1$-vertices and their labelings are as
follows. The point $\infty$ is the only $1$-vertex labeled $\infty$,
the $1$-vertices $-1,1$ are labeled $0$, the $1$-vertex $0$ is
labeled $-1$. The cell decomposition is indicated to the left in
Figure \ref{fig:subdivz2-1}.
Here and in the following a point that
is marked ``$a\mapsto b$'' is a $1$-vertex $a$ that is also a $0$-vertex and that is labeled by the $0$-vertex $b$.
Thus the map realizing the two-tile subdivision rule
will map $a$ to $b$. Similarly ``$\mapsto b$'' marks a $1$-vertex
that is labeled by the $0$-vertex $b$; thus the
realizing map will map this $1$-vertex to $b$.
The pair $(\mathcal{D}^1,\mathcal{D}^0)$ together with this
orientation-preserving labeling $L$ is a two-tile subdivision rule. It is straightforward to check that we
can choose the map $p\colon S^2\to S^2$ that is generated by
$(\mathcal{D}^1,\mathcal{D}^0, L)$ according to Proposition \ref{prop:rulemapex} as
the map $f_1(z)=z^2-1$.
This two-tile subdivision rule is not combinatorially expanding. Namely, the point $\infty$
is the only preimage of itself by $p$. Thus every $n$-tile contains
$\infty$. Since the $n$-tiles cover
the whole sphere, there has to be an $n$-tile containing both
$0,\infty$. This shows that the subdivision rule in not combinatorially
expanding.
In fact, for all $n$ there exist two $n$-tiles that contain all postcritical points
$-1,0,\infty$.
Figure \ref{fig:tiles8_z2-1} shows the tiles of order $7$.
\end{ex}
\ifthenelse{\boolean{nofigures}}{}{
\begin{figure}
\centering
\includegraphics[width=11cm]{barycentric.eps}
\begin{picture}(10,10)
\put(-330,-10){$\scriptstyle{-1\mapsto 1}$}
\put(-164,-10){$\scriptstyle{\mapsto -1}$}
\put(-15,-10){$\scriptstyle{1\mapsto 1}$}
\put(-168,275){$\scriptstyle{\infty\mapsto 1}$}
\put(-257,142){$\scriptstyle{\mapsto -1}$}
\put(-83,142){$\scriptstyle{\mapsto -1}$}
\put(-177,120){$\scriptstyle{\mapsto \infty}$}
\put(-145,55){$\scriptstyle{\mapsto \infty}$}
\end{picture}
\caption{The barycentric subdivision rule.}
\label{fig:barycentric}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=11cm]{R_bary4.eps}
\caption{Tiles of order $4$ for the barycentric subdivision rule.}
\label{fig:bary4}
\end{figure}
}
\begin{ex}[The barycentric subdivision rule] \label{ex:barycentric}
We glue two equilateral triangles together along their boundaries to
form a polyhedral surface $S^2$ that is conformally equivalent to $\CDach$. The two triangles are the $0$-tiles. We can find a conformal equivalence of $S^2$ with $\CDach$ such that the triangles correspond to the upper and lower half-planes, and the vertices to the points $-1,1,\infty$. For convenience we identify the vertices
with $-1,1,\infty$; they are the
$0$-vertices. The $0$-edges are the three edges of the
triangles. The bisectors divide each triangle (each $0$-tile) into
$6$ smaller triangles. These $12$
small triangles are the $1$-tiles. The labeling of the $1$-vertices
is indicated in Figure \ref{fig:barycentric}.
Again we obtain a two-tile subdivision rule.
We can realize this subdivision rule by a map $f_2$ that {\em conformally} maps
$1$-tiles to the $0$-tiles. Under the indicated identification of $S^2$ with $\CDach$, the map is then given by
\begin{equation*}
f_2(z)=1- \frac{54 (z^2-1)^2}{(z^2+3)^3},
\end{equation*}
see \cite[Example 4.6]{CFKP}. The subdivision rule is combinatorially expanding. The map
$f_2$ however is not expanding. This follows from Proposition~\ref{prop:rationalexpch} as the point $1$ is both a critical and a fixed point of $r$. One can show that the Julia set of $f_2$ is a
Sierpi\'{n}ski carpet, i.e., a set homeomorphic to the standard
Sierpi\'{n}ski carpet.
It is possible however to choose a different realization of the two-tile subdivision rule with the given
labeling as in Figure \ref{fig:barycentric}
by a map $\widetilde{f}_2$ that is \emph{expanding}.
Namely, we use {\em affine} maps to map the $1$-tiles (the small triangles in the barycentric subdivision rule of the equilateral triangles) to the $0$-tiles.
In this case the $n$-tiles are Euclidean triangles for each $n\in \N$. The collection of all $n$-tiles is obtained from the $(n-1)$-tiles similarly as the $1$-tiles where constructed from the $0$-tiles: one subdivides each
Euclidean triangle representing an $(n-1)$-tile by its bisectors.
It is clear that the diameters of $n$-tiles tend to $0$ as $n\to \infty$. Hence $\widetilde{f}_2$ is expanding, and so this
map is an example of an expanding Thurston map with periodic critical points.
\end{ex}
\ifthenelse{\boolean{nofigures}}{}{
\begin{figure}
\centering
\begin{picture}(10,10)
\put(170,0){$\scriptstyle{a}$}
\put(10,-7){$\scriptstyle{a\mapsto a}$}
\put(100,-7){$\scriptstyle{\mapsto a}$}
\put(40,80){$\scriptstyle{\mapsto a}$}
\put(72,65){$\scriptstyle{\mapsto a}$}
\put(160,65){$\scriptstyle{\mapsto a}$}
\put(300,25){$\scriptstyle{\gamma}$}
\put(152,43){$\scriptstyle{\gamma_1}$}
\put(122,12){$\scriptstyle{\gamma_2}$}
\put(168,37){$\scriptstyle{f_3}$}
\end{picture}
\includegraphics[width=11cm]{thurstonexample.eps}
\caption{The subdivision rule for Example \ref{ex:obstructed_map}.}
\label{fig:obstructed_map}
\end{figure}
}
\begin{ex}
\label{ex:obstructed_map}
The next example is given by a subdivision rule similar to the one that was realized by the
map $g$ from Section \ref{sec:Lattes}; see Figure
\ref{fig:mapg}. Again we start with a sphere that is obtained by
glueing together two squares (these are the two $0$-tiles) along
their boundary. The four vertices are the $0$-vertices that divide
the common boundary into four $0$-edges. Each of the two squares, or
more precisely $0$-tiles, is divided into four squares
of half the side-length. These $8$ smaller squares are $1$-tiles. The edges of these squares
are $1$-edges. We slit the
sphere along one such $1$-edge (say in the white $0$-tile) and
glue in two small squares at the slit, as indicated
to the left
in Figure \ref{fig:obstructed_map}. In this way we obtain two additional $1$-tiles. Topologically we have subdivided
the white $0$-tile into six $1$-tiles; the black $0$-tile is
subdivided into four $1$-tiles. The labeling (meaning the
coloring) of the $1$-tiles can be seen from Figure
\ref{fig:obstructed_map}. Here only the $1$-vertices that are labeled by one specific
$0$-vertex have been indicated to keep the picture simple and avoid unnecessary detail.
Again we obtain a two-tile subdivision rule.
It is realized by a Thurston map $f_3$ that is combinatorially expanding. It is
not equivalent to a rational map, since $f_3$ has a
\emph{Thurston obstruction}.\index{Thurston map!obstructed}
In the present case, where $\#\operatorname{post}(f_3) = 4$ and $f_3$ has a
\defn{hyperbolic orbifold} (see for example \cite[Appendix
E]{Mi} or \cite[Appendix A]{McM}) a Thurston obstruction is given
by a Jordan curve $\gamma\subset S^2\setminus \operatorname{post}(f_3)$ with the following properties:
\begin{itemize}
\item The Jordan curve $\gamma$
is \defn{non-peripheral}, i.e., each component of
$S^2 \setminus \gamma$ contains two postcritical
points.
\item Each non-peripheral component $\gamma_j$ of $f_3^{-1}(\gamma)$
is homotopic to $\gamma$ in $S^2\setminus \operatorname{post}(f_3)$.
\item If $d_j$ is the degree of the map $f_3\colon \gamma_j \to
\gamma$, then we have
\begin{equation*}
\sum_j \frac{1}{d_j} \geq 1.
\end{equation*}
\end{itemize}
Thurston's theorem implies that a Thurston map $f$ with hyperbolic
orbifold and $\#\operatorname{post}(f)=4$ is equivalent to a rational map if and only if it has no
Thurston obstruction (see \cite{DH}). Figure
\ref{fig:obstructed_map} shows an
obstruction for the map $f_3$.
\end{ex}
\ifthenelse{\boolean{nofigures}}{}{
\begin{figure}
\centering
\begin{overpic}
[width=11cm,
tics=20]{lattes2x3.eps}
\put(59,11){$\scriptstyle{0}$}
\put(101,-1){$\scriptstyle{1}$}
\put(101,40){$\scriptstyle{\infty}$}
\put(59,54){$\scriptstyle{-1}$}
\put(39,-3){$\scriptstyle{1\mapsto 0}$}
\put(20,3){$\scriptstyle{\mapsto 1}$}
\put(-1,10){$\scriptstyle{0\mapsto 0}$}
\put(-7,26){$\scriptstyle{\mapsto -1}$}
\put(-5,40){$\scriptstyle{\mapsto 0}$}
\put(-5,54){$\scriptstyle{-1\mapsto -1}$}
\put(41,13){$\scriptstyle{\mapsto -1}$}
\put(41,26){$\scriptstyle{\mapsto 0}$}
\put(39,41){$\scriptstyle{\infty \mapsto -1}$}
\put(19,48){$\scriptstyle{\mapsto \infty}$}
\put(18,18){$\scriptstyle{\mapsto \infty}$}
\put(17.5,27.5){$\scriptstyle{\mapsto 1}$}
\put(49,24){$\scriptstyle{f_4}$}
\end{overpic}
\caption{The 2-by-3 subdivision rule.}
\label{fig:2x3}
\end{figure}
}
\begin{ex}
[The 2-by-3 subdivision rule]
\label{ex:2x3}
We present another example of an expanding Thurston map $f_4$ that is not (Thurston)
equivalent to a rational map. In a sense, this is the easiest
example. However, it has a \emph{parabolic orbifold}. Thus Thurston's
criterion, as explained in the last example, does not
apply.
The map $f_4$ will be a {\em Latt\`{e}s-type map} (see \cite{Qian} for precise definitions) and can be constructed in the same fashion as the
Latt\`{e}s map
$g$ from
Section \ref{sec:Lattes}; namely, define $\psi\colon \C \to \C$ by setting
$\psi(x+ y {\mathbf{\imath}}) = 2x+ 3 y{\mathbf{\imath}}$ for $x,y\in \R$. Then $f_4\:\CDach \rightarrow \CDach$
is the unique map that makes the diagram \eqref{eq:Lattes}
commutative (with $g$ replaced by $f_4$).
This map is
a realization of the two-tile subdivision rule shown in
Figure~\ref{fig:2x3}. We form a pillow $P$ by glueing two unit squares
together along their boundaries. The two squares, as well as the four common sides and vertices of the squares, form the
$0$-tiles, $0$-edges,
$0$-vertices, respectively.
Each of the two faces (i.e., squares) of the pillow is divided into
$6$ rectangles as shown in the figure. These $12$ rectangles are the
$1$-tiles. Their sides and vertices are the $1$-edges and $1$-vertices. The
coloring of $1$-tiles, as well as the labeling of the $1$-vertices, is
indicated on the left of Figure~\ref{fig:2x3}. The map $f_4$ sends each
of the 12 rectangles affinely to one of the two squares forming the faces of the pillow. This implies that each $n$-tile is a rectangle
with side lengths $1/2^n$ and $1/3^n$. In particular, $f_4$ is an expanding
Thurston map.
The fact that $f_4$ is not equivalent to a rational map is well-known (see \cite[Prop.~9.7]{DH}). An argument for this fitting into the framework of our present work can be sketched as follows. In our outline we will rely on some results and concepts that we will be discussed later on.
To reach a contradiction, suppose that $f_4$ is equivalent to a rational map
$R\: \CDach\rightarrow \CDach$. Then $R$ is a Thurston map with no periodic critical points and is hence expanding (Proposition~\ref{prop:rationalexpch}).
So by Theorem~\ref{thm:exppromequiv} the maps $f_4$ and $R$ are topologically conjugate. This in turns implies by Theorem~\ref{thm:qsrational} that if our pillow $P$ is equipped with a visual metric $d$
for $f_4$, then $(P, d)$ is quasisymmetrically equivalent to the standard $2$-sphere (i.e., $\CDach$ equipped with the chordal metric).
In particular, if $X^0$ is a $0$-tile (i.e.,
one of the faces of the pillow $P$) equipped with a visual metric $d$,
then it can be mapped into the standard $2$-sphere by a quasisymmetric map.
Now there are visual metrics for $f_4$ with expansion factor
$\Lambda=2$ (it is not hard to see this directly; it also follows from the general argument in the proof of Theorem~\ref{thm:visexpfactors} presented in Section~\ref{sec:combexpfac}; indeed, if $\mathcal{C}$ is the boundary of the pillow (which is $f_4$-invariant), then we have
$D_1=D_1(f_4,\mathcal{C})=2$ in \eqref{extraonL}).
If $d$ is such a metric, then $(X^0, d)$ is bi-Lipschitz equivalent
to a Rickman's rug
$R_\alpha$. Here by definition the {\em Rickman's rug}
\index{Rickman's rug} $R_\alpha$ for $0<\alpha<1$ is is the unit square
$[0,1]^2\subset \R^2$ equipped with the metric $d_\alpha$
given by
$$d_\alpha((x_1,y_1), (x_2,y_2)) =
\abs{x_1-x_2} + \abs{y_1-y_2}^\alpha$$ for $(x_1,y_1), (x_2,y_2)\in
[0,1]^2$. In our
case, $(X^0,d)$ is bi-Lipschitz equivalent to $R_\alpha$ with $\alpha= \log 2/\log 3$. It is well-known that
no quasisymmetric map can lower the
Hausdorff dimension
$$\dim_H(R_\alpha)=1+ \log 3/\log 2>2$$ of $R_\alpha$
(see \cite[Theorem
15.10]{He}); in particular, $R_\alpha$ and hence also $(X^0, d)$, cannot be mapped into the standard $2$-sphere by a quasisymmetric map.
This is a contradiction showing that $f_4$ is not Thurston equivalent to a rational map.
\end{ex}
\begin{ex}
\label{ex:R_mario3}
We conclude this section by giving a whole class of examples that are similar to
Example \ref{ex:obstructed_map}.
The map $f_6$ defined below will be used to illustrate the
construction of the invariant curve in Example \ref{ex:invC}, see
Figure \ref{fig:invC_constr}.
\smallskip
The simplest two-tile subdivision of this class is given as
follows. Consider a right-angled, isosceles
Euclidean triangle $T$ (thus its angles are $\pi/2, \pi/4,
\pi/4$). The perpendicular bisector of the hypotenuse
divides $T$ into two
triangles similar to $T$ (scaled by the factor $\sqrt{2}$).
Glue two copies of the triangle $T$ together along their boundaries to form a
pillow (i.e., a topological sphere) as before. The two faces of the
pillow (i.e., the two copies of $T$) are the $0$-tiles, and the common sides
and vertices of theses faces are the $0$-edges and $0$-vertices.
\ifthenelse{\boolean{nofigures}}{}{
\begin{figure}
\centering
\begin{overpic}
[width=11cm,
tics=20]{lattes244.eps}
\put(53,14){$\scriptstyle{-1}$}
\put(101,-1){$\scriptstyle{1}$}
\put(85,30){$\scriptstyle{\infty}$}
\put(39,-3){$\scriptstyle{1\mapsto -1}$}
\put(25,30){$\scriptstyle{\infty\mapsto 1}$}
\put(18,4){$\scriptstyle{0\mapsto \infty}$}
\put(-5,11){$\scriptstyle{-1\mapsto -1}$}
\put(45,18){$\scriptstyle{f_5}$}
\end{overpic}
\caption{The subdivision rule realized by $1-2/z^2$.}
\label{fig:lattes244}
\end{figure}
}
We divide each of the $0$-tiles (i.e., each face of the pillow) along the
perpendicular bisector of the hypotenuse. The four triangles thus
obtained are the $1$-tiles. Their vertices and sides are the
$1$-vertices and $1$-edges. If the labeling is as indicated on the right
of Figure \ref{fig:lattes244}, then we obtain a two-tile subdivision rule that
can be realized
by the map $f_5(z)= 1-2/z^2$ (this is actually a Latt\`{e}s map).
\smallskip
As in Example~\ref{ex:obstructed_map}, we can ``add a flap'' to modify
the subdivision rule. More precisely, we cut the pillow along the $1$-edge
that is the perpendicular bisector of
the hypotenuse of the white $0$-tile. We take a copy of the pillow, scale
it by the factor
$1/\sqrt{2}$, and cut it along one leg. We then glue the two sides of the slit to corresponding sides of the slit on the original pillow. This is
indicated on the left in Figure \ref{fig:triangle_flap}, where we
also show the coloring of $1$-tiles and the labeling of the
$1$-vertices.
\ifthenelse{\boolean{nofigures}}{}{
\begin{figure}
\centering
\begin{overpic}
[width=12cm,
tics=20]{lattesflap.eps}
\put(56,13){$\scriptstyle{\omega}$}
\put(100,-2){$\scriptstyle{1}$}
\put(85,25){$\scriptstyle{\infty}$}
\put(41,-2){$\scriptstyle{1\mapsto\omega}$}
\put(30,25){$\scriptstyle{\infty\mapsto 1}$}
\put(23,38){$\scriptstyle{\overline{\omega}\mapsto \omega}$}
\put(20,4.5){$\scriptstyle{0\mapsto \infty}$}
\put(-3,13){$\scriptstyle{\omega\mapsto \omega}$}
\put(46,18){$\scriptstyle{f_6}$}
\end{overpic}
\caption{Adding a flap.}
\label{fig:triangle_flap}
\end{figure}
\begin{figure}
\centering
\begin{picture}(10,10)
\put(167,-8){$\scriptstyle{1\mapsto \omega}$}
\put(320,97){$\scriptstyle{\infty\mapsto 1}$}
\put(167,184){$\scriptstyle{\omega\mapsto \omega}$}
\put(4,99){$\scriptstyle{\infty\mapsto 1}$}
\put(147,95){$\scriptstyle{0\mapsto \infty}$}
\put(217,95){$\scriptstyle{\overline{\omega}\mapsto \omega}$}
\end{picture}
\includegraphics[width=11cm]{R_mario3.eps}
\caption{The subdivision rule realized by $f_6$.}
\label{fig:R_mario31}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=10cm]{R_mario3_4.eps}
\caption{Tiles of order $4$ of the map $f_6$.}
\label{fig:R_mario3}
\end{figure}
}
\smallskip
We also show the same subdivision rule in Figure \ref{fig:R_mario31}. Here
the two triangles drawn with a thick line are the $0$-tiles, the white
is drawn to the right, the black to the left. Their edges are the
$0$-edges, their vertices the $0$-vertices.
To obtain a topological
sphere we have to match the two pairs of $0$-edges with the same
markings.
The white $0$-tile is
subdivided into four $1$-tiles, the black $0$-tile into two
$1$-tiles. The labeling of the $1$-vertices is shown in Figure
\ref{fig:R_mario3}. As before this yields a two-tile subdivision
rule. It can be realized by the rational map $f_6$ given by
\begin{equation*}
f_6(z)=1+\frac{\omega -1}{z^3},
\end{equation*}
where $\omega= e^{4\pi {\mathbf{\imath}}/3}$.
\smallskip
The previous example can be modified as follows. Instead of adding one
flap to the $1$-edge bisecting the white $0$-tile in Figure
\ref{fig:lattes244}, we can add $n$ flaps. Similarly we can glue in $m$
flaps at the $1$-edge bisecting the black $0$-tile. The resulting
subdivision rule can be realized by the rational map $f_7$ given by
\begin{equation*}
f_7(z) = 1+ \frac{\omega -1}{z^d},
\end{equation*}
where $d=n+m+2$ and $\omega= e^{2\pi{\mathbf{\imath}} \frac{n+1}{d}}$.
\smallskip
The map $f_6$ will be used to illustrate the construction of an
invariant Jordan curve $\mathcal{C}$ with $\operatorname{post}(f_6)\subset \mathcal{C}$. The tiles
of order $4$ are shown in Figure \ref{fig:R_mario3}.
\end{ex}
More examples can be found in \cite{CFKP} and \cite{Me02}. In
\cite{Me02} and \cite{Me09a} two-tile subdivision rules realizable by rational maps were used to show that certain self-similar surfaces are
\emph{quasispheres}, i.e., quasisymmetric images of unit sphere in
$\R^3$. More general examples of subdivisions can be found in \cite{CFP06b}.
The Figures \ref{fig:tiles8_z2-1}, \ref{fig:bary4}, and
\ref{fig:R_mario3} show \emph{symmetric conformal tilings}. This means that if
two tiles share an edge, they are conformal reflections of each other
along this edge. Then the tiling can be obtained by successive reflections, and so each tile encodes the information for the whole
tiling.
\section{Combinatorially expanding Thurston maps}
\label{sec:combexp}
\noindent
The purpose of this section is to establish the following fact.
\begin{prop}\label{prop:combexp} Let $F\:S^2\rightarrow S^2$ be a Thurston map that has an invariant Jordan curve $\mathcal{C}\subset S^2$ with $\operatorname{post}(F)\subset \mathcal{C}$. If $F$ is combinatorially expanding\index{combinatorially expanding}\index{expanding!combinatorially} for $\mathcal{C}$, then there exists an expanding Thurston map $\widetilde F\: \widetilde S^2\rightarrow \widetilde S^2$ that is Thurston equivalent to $F$ and has an
$\widetilde F$-invariant Jordan curve $\widetilde \mathcal{C}\subset \widetilde S^2$ with $ \operatorname{post}(\widetilde F)\subset \widetilde \mathcal{C}$.
Moreover, there exist orientation-preserving homeomorphisms $h_0,h_1\: S^2\rightarrow \widetilde S^2$ that are isotopic rel.~$\operatorname{post}(F)$ and satisfy $h_0\circ F=\widetilde F\circ h_1$ and $h_0(\mathcal{C})=\widetilde \mathcal{C}= h_1(\mathcal{C})$.
\end{prop}
So up to Thurston equivalence every {\em combinatorially expanding} Thurs\-ton map with an invariant Jordan curve can be promoted to an {\em expanding} Thurston map with an invariant curve.
The proof will occupy the rest of the section.
The idea is to introduce a suitable equivalence relation $\sim$ on the sphere $S^2$
on which $F$ acts and use Moore's Theorem (see Theorem~\ref{Moore}) to show that
the quotient space $S^2/\!\!\sim$ is also a $2$-sphere. The map $\widetilde F$ will then be the induced map on $S^2/\!\!\sim$.
So let $F\: S^2\rightarrow S^2$ be a Thurston map
as in the proposition. We fix an invariant $F$-Jordan curve $\mathcal{C}\subset S^2$ with $ \operatorname{post}(F)\subset \mathcal{C}$ for which $F$ is
combinatorially expanding. As before we denote
by $\X^n$, $\E^n$ and ${\bf V}^n$ the set of $n$-tiles, $n$-edges, and $n$-vertices, respectively, of the cell decomposition $\mathcal{D}^n=\mathcal{D}^n(F,\mathcal{C})$ defined in Section~\ref{sec:tiles}. A subset $\tau \subset S^2$ is called a {\em tile} if it is an $n$-tile for some $n\in \N_0$. We use the terms {\em edge}, {\em vertex}, {\em cell} in a similar way.
In particular, in this section the term ``cell'' will always be used with this specific meaning. We will use the term {\em topological cell} to refer to the more general notion of cells as defined in
Section~\ref{s:celldecomp}.
Since $\mathcal{C}$ is $F$-invariant, $\mathcal{D}^{n+k}$ is a refinement of $\mathcal{D}^n$ for all $n,k\in \N_0$. For each $X\in \X^{n+k}$ there exists a unique $Y\in \X^n$ with $X\subset Y$. Conversely, each $n$-tile $Y$ is equal to the union of all $(n+k)$-tiles contained in $Y$, and similarly each $n$-edge $e$ is equal to the union of all $(n+k)$-edges contained in $e$ (all this was proved in
Proposition~\ref{prop:invmarkov}). We will use this fact that cells are subdivided by cells of the same dimension and higher order repeatedly in the following.
As in the beginning of Section~\ref{sec:Thurtoncurves} we denote by $\mathcal{S}=\mathcal{S}(F,\mathcal{C})$ the set of all sequences $\{X^n\}$ with $X^n\in \X^n$ for $n\in \N_0$ and
\begin{equation*}
X^0\supset X^1\supset X^2\supset \dots \end{equation*}
We know (see Lemma~\ref{lem:charexpint}) that expansion of a Thurston map with an invariant curve is characterized by the condition that
$\bigcap_n X^n$ is always a singleton set if $\{X^n\}\in \mathcal{S}$. This may not be that case for our given map $F$, and so we want to identify all points in such an intersection $\bigcap_n X^n$. This will not lead to an equivalence relation, since transitivity may fail. As we will see, this issue is resolved if we define the relation as follows.
\begin{definition} \label{def:erel} Let $x,y\in S^2$ be arbitrary. We write
$ x\sim y$ if and only if for all $\{X^n\}, \{Y^n\}\in \mathcal{S}$
with $x\in \bigcap_n X^n$ and $ y\in
\bigcap_n Y^n$ we have $X^n\cap Y^n\neq \emptyset$ for
all $n\in \N_0$.
\end{definition}
Recall from \eqref{def:dk} that $D_n=D_n(F,\mathcal{C})$ denotes the minimal number
of $n$-tiles forming a connected set $K^n$ joining opposite sides of $\mathcal{C}$.
Since $F$ is combinatorially expanding for $\mathcal{C}$
(see Definition~\ref{def:combexp}), we have $\#\operatorname{post}(F)\ge 3$ and so the term
``joining opposite sides" is meaningful (see Definition~\ref{def:connectop}).
Moreover, there exists $n_0\in \N$ such that $D_{n_0}(F,\mathcal{C})\ge 2$, and so
by Lemma~\ref{lem:submult} we have $D_n=D_n(F,\mathcal{C})\to \infty$ as $n\to \infty$. In combination
with Lemma~\ref{lem:flowerbds} this implies that if
$\tau, \sigma$ are disjoint $k$-cells and $K^n$ is a connected set of $n$-tiles with $\sigma\cap K^n\ne \emptyset$ and $\tau\cap K^n\ne \emptyset$, then the number of tiles in $K^n$ tends to infinity and so cannot stay bounded as $n\to \infty$. We will use this fact in the proof of the following lemma.
\begin{lemma} \label{lem:aeq}
The relation $\sim$ is an equivalence relation on $S^2$.
\end{lemma}
\begin{proof} Reflexivity and symmetry of the relation $\sim$ are clear.
To show transitivity, let $x,y,z\in S^2$ be arbitrary and assume that
$x\sim y$ and $y\sim z$. Let $\{X^n\}, \{Z^n\}\in \mathcal{S}$
with $x\in \bigcap_n X^n$ and $ z\in
\bigcap_n Z^n$ be arbitrary. We have to show that $X^n\cap Z^n\ne \emptyset$ for all $n\in \N_0$.
If this is not the case, then there exists $n_0\in \N_0$ such that $X^{n_0}\cap Z^{n_0}= \emptyset$. To reach a contradiction, pick a sequence $\{Y^n\}\in \mathcal{S}$
with $y\in \bigcap_n Y^n$. Since $x\sim y$ and $y\sim z$, we have
$X^n\cap Y^n\ne \emptyset$ and $Y^n \cap Z^n\ne \emptyset$ for all $n\in \N_0$. Then $X^{n_0}\cap Y^n\supset X^{n}\cap Y^n\ne \emptyset$
and $Z^{n_0}\cap Y^n\supset Z^{n}\cap Y^n\ne \emptyset$
for all $n\ge n_0$.
So the $n$-tile $Y^n$ connects the disjoint $n_0$-tiles
$X^{n_0}$ and $Y^{n_0}$ for all $n\ge n_0$. As we discussed, this is impossible by Lemma~\ref{lem:flowerbds}.
\end{proof}
It is clear that $\sim$ is the ``smallest" equivalence relation such that
all points in an intersection $\bigcap_n X_n$ with $\{X^n\}\in \mathcal{S}$
are equivalent.
If $x\in S^2$ we denote by $[x]\subset S^2$ the equivalence class of $x$
with respect to the equivalence relation $\sim$, and by $$\widetilde S^2=S^2/\!\!\sim=\{[x]:x\in S^2\}$$ the quotient space of $S^2$ under $\sim$. So $\widetilde S^2$ consists of all equivalence classes of $\sim$. Such an equivalence class is both a point in
$\widetilde S^2$ and a subset of $S^2$. We equip $\widetilde S^2$ with the quotient topology. Then the quotient map $\pi\: S^2\rightarrow \widetilde S^2$, $x\in S^2\mapsto [x]$, is continuous.
\bigskip \noindent
{\bf The quotient space $\widetilde {S}^2$ is a topological $2$-sphere.} \noindent
Our first goal is to show that $\widetilde {S}$ is a topological $2$-sphere.
For the moment consider an arbitrary equivalence relation $\sim$ on the sphere $S^2$. We call it
{\em closed}\index{closed equivalence relation}
if $\{(x,y)\in S^2\times S^2: x\sim y\}$ is a closed subset of $S^2\times S^2$ (in the older literature the term
{\em ``upper semicontinuous"} is often used instead).
This is equivalent to the following condition:
If $\{x_n\}$ and $\{y_n\}$ are arbitrary convergent sequences in $S^2$ with $x_n\to x$ and $y_n\to y$ as $n\to \infty$, and $x_n\sim y_n$ for all $n\in \N$, then $x\sim y$. The equivalence classes of a closed equivalence relation on $S^2$ are closed and hence compact subsets of $S^2$.
We need the following key theorem (see \cite{Moo} for the original proof, \cite[p.~187, Thm.~1]{Da} for a stronger statement, and \cite[Supplement~1]{Ca} for a general discussion on the $2$-sphere recognition problem).
\begin{theorem}[Moore 1925] \label{Moore}\index{Moore's Theorem}
\label{thm:moore} Let $\sim$ be an equivalence relation on a $2$-sphere $S^2$.
Suppose that
\begin{itemize}
\smallskip
\item[\textnormal{(i)}] the equivalence relation $\sim$ is closed,
\smallskip
\item [\textnormal{(ii)}] each equivalence class of $\sim$ is a connected subset of $S^2$,
\smallskip
\item [\textnormal{(iii)}] the complement of each equivalence class of $\sim$ is a connected subset of $S^2$,
\smallskip
\item [\textnormal{(iv)}] there are at least two distinct equivalence classes.
\end{itemize}
Then the quotient space $S^2/\!\!\sim$ is homeomorphic to $S^2$.
\end{theorem}
Here it is understood that $S^2/\!\!\sim$ is equipped with the quotient topology.
To apply this theorem in our situation, we need some preparation.
\begin{lemma} \label{lem:erel}
Let $x,y\in S^2$ be arbitrary. Then the following conditions
are equivalent:
\begin{itemize}
\smallskip
\item[\textnormal{(i)}] $x\sim y$,
\smallskip
\item[\textnormal{(ii)}] there exist $\{X^n\}, \{Y^n\}\in \mathcal{S}$
with $x\in \bigcap_n X^n$, $ y\in
\bigcap_n Y^n$, and $X^n\cap Y^n\neq \emptyset$ for
all $n\in \N_0$,
\smallskip
\item[\textnormal{(iii)}]
for all cells $\sigma,\tau \subset S^2 $ with $x\in \sigma$, $y\in \tau$, we have $\sigma\cap \tau \ne \emptyset$.
\end{itemize}
\end{lemma}
\begin{proof} The implication (i)$\Rightarrow$(ii) is clear.
To show the reverse implication (ii)$\Rightarrow$(i),
we assume that there exist $\{X^n\}, \{Y^n\}\in \mathcal{S}$
with $x\in \bigcap_n X^n$, $ y\in
\bigcap_n Y^n$, and $X^n\cap Y^n\neq \emptyset$ for
all $n\in \N_0$. We claim that
if $\{U^n\}, \{V^n\}\in \mathcal{S}$ are two other sequences with
$x\in \bigcap_n U^n$ and $y\in \bigcap_n V^n$, then $U^n\cap V^n\ne \emptyset $
for all $n\in \N_0$. To reach a contradiction assume that $U^{n_0}\cap V^{n_0} = \emptyset$ for some $n_0\in \N_0$. We then have
$$U^{n_0}\cap X^{n}\supset \{x\} \ne \emptyset \quad \text{ and } \quad V^{n_0}\cap Y^{n}\supset \{y\} \ne \emptyset $$
for all $n\in \N$. Moreover, $X^n\cap Y^n\ne \emptyset$, and so for each $n\in \N_0$, the $K^n:=X^n\cup Y^n$ is connected, consists of two $n$-tiles, and meets the disjoint
$n_0$-tiles $U^{n_0}$ and $V^{n_0}$.
As before this contradicts Lemma~\ref{lem:flowerbds}. Hence $x\sim y$ as desired.
The implication (iii)$\Rightarrow$(i) is again clear. To prove
(i)$\Rightarrow$(iii), suppose that $x\sim y$. We argue by contradiction and assume that there exist cells $\sigma,\tau$ with
$x\in \sigma$, $y\in \tau$ and $\sigma\cap \tau=\emptyset$.
By subdividing the cells if necessary we may assume that
$\sigma$ and $\tau$ are cells on the same level $n_0$.
There are sequences $\{X^n\}, \{Y^n\}\in \mathcal{S}$
with $x\in \bigcap_n X^n$, $ y\in
\bigcap_n Y^{n}$, $\sigma \subset X^{n_0}$ and $\tau \subset Y^{n_0}$.
Since $x\sim y$, we have $X^n\cap Y^n\ne \emptyset$ for all $n$.
This implies that for $n\ge n_0$ the set $K^n=X^n\cup Y^n$ is connected and consists of at most two $n$-tiles. Moreover,
$$K^n\cap \sigma \supset X^n\cap \sigma\supset\{x\}\ne \emptyset,$$
and similarly, $K^n\cap \tau\ne \emptyset$. Hence $K^n$ connects the disjoint $n_0$-cells $\sigma$ and $\tau$. Since $F$ is combinatorially expanding, this is impossible by Lemma~\ref{lem:flowerbds} for large $n$. This gives the desired contradiction.
\end{proof}
We now describe the geometry of the equivalence classes of $\sim$.
This will be used in verifying the conditions in Theorem~\ref{Moore}.
\begin{lemma}\label{lem:eclass}Let $M\subset S^2$ be an arbitrary equivalence class with respect to $\sim$. Then for each $n\in \N_0$ there exists a simply connected region $\Om^n\subset S^2$ with the following properties: the
set $\overline {\Om}^{{\null}_{\scriptstyle n}}$ consists of $n$-tiles,
we have $\Om^{n+1}\subset \Om^{n}$ for all $n\in \N_0$, and
\begin{equation} \label{eq:Mbigcap}
M=\bigcap_{n}\Om^n =\bigcap_{n}\overline {\Om}^{{\null}_{\scriptstyle n}}.
\end{equation}
\end{lemma}
\begin{proof} We consider three cases. In each case it is enough to define the
sets $\Om^n$ for $n\ge m$, where $m\in \N_0$ is suitably chosen. We put
$\Omega^n=S^2$ for $n<m$.
We will establish \eqref{eq:Mbigcap} by showing that
$\bigcap_{n}\overline {\Om}^{{\null}_{\scriptstyle n}}\subset M$ and $M\subset \bigcap_{n}\Om^n$ in all cases. \smallskip
{\em Case 1.} $M$ contains a vertex.
In this case $M=[v]$ where $v$ is an $m$-vertex for some $m\in \N_0$.
Then $v$ is also an $n$-vertex for $n\ge m$.
Let $\Om^n=W^n(v)$ for $n\ge m$ be the $n$-flower of $v$
(see Definition~\ref{def:flower}). Then $\Om^n$ is a simply
connected region and $\overline {\Om}^{{\null}_{\scriptstyle n}}$ consists of $n$-tiles as follows from Lemma~\ref{lem:flowerprop} (i) and (ii). Moreover, the definition of an $n$-flower and Lemma~\ref{lem:mincell} imply that $\Om^m\supset \Om^{m+1}\supset \Om^{m+2} \dots$.
Let $x\in \bigcap_{n}\overline {\Om}^{{\null}_{\scriptstyle n}}$ be arbitrary. We want to show that $x\in M=[v]$.
If this is not the case, then there exists $n_0\ge m$ and
an $n_0$-tile $X^{n_0}$ such that $x\in X^{n_0}$, but $v\not\in X^{n_0}$.
On the other hand, since $x\in \overline {\Om}^{{\null}_{\scriptstyle n}}$, for each $n\ge n_0$
there exists an $n$-tile $Y^n$ such that $v,x\in Y^n$.
Then $Y^n$ meets the disjoint $n_0$-cells $\{v\}$ and $X^{n_0}$ for all $n\ge n_0$.
This is impossible by Lemma~\ref{lem:flowerbds}. It follows that $\bigcap_{n} \overline {\Om}^{{\null}_{\scriptstyle n}}\subset M$.
If $y\not \in \Om^n=W^n(v)$ for some $n\ge m$, then there exists an $n$-cell $\tau$ such that $v\not \in \tau$ and $y\in \tau$. Since $v$ is a vertex, this implies $v\not\sim y$ by Lemma~\ref{lem:erel}. So
$M=[v]\subset \bigcap_n\Om^n$, and we have
$$ \bigcap_n \overline {\Om}^{{\null}_{\scriptstyle n}} \subset M\subset \bigcap_n \Om^n, $$ showing that
the three sets in this inclusion chain are the same.
We conclude that
the sets $\Om^n$ have all the desired properties.
\smallskip
{\em Case 2.} $M$ does not contain a vertex, but meets an edge.
In this case we can find $m\in \N_0$, an $m$-edge $e^m$ and a point $x\in e^m$ such that $M=[x]$. By our assumption, $x$ is not a vertex and hence an interior point of each edge that contains $x$.
So $x\in \inte(e^m)$, and, since the $(m+1)$-edges subdivide $e^m$, there exists a unique $(m+1)$-edge $e^{m+1}$
such that $x\in \inte(e^{m+1})\subset e^{m+1}\subset e^m$. Repeating this procedure, we obtain
a nested sequence of $n$-edges $e^n$ for $n\ge m$ such that $x$ is an interior point of each $e^n$.
There exist precisely two distinct $n$-tiles $X^n$ and
$Y^n$ that contain $e^n$ on their boundaries. Define
$$\Om^n=\inte(X^n)\cup \inte(e^n) \cup \inte (Y^n). $$ Then $\Om^n$ is
the union of two disjoint open Jordan regions and an open arc contained in the boundary of both regions. Hence $\Om^n$ is simply connected. Moreover, $\overline {\Om}^{{\null}_{\scriptstyle n}}=X^n\cup Y^n$ consists of two tiles.
An interior point of $X^{m+1}$ close to $\inte(e^{m+1})$ belongs to $X^m$
or to $Y^m$, since $X^m\cup Y^m$ is a neighborhood of each point in $\inte(e^m)\supset \inte(e^{m+1})$. This is only possible if $X^{m+1}\subset X^m$ or $X^{m+1}\subset Y^m$ (see the first part of the proof of Lemma~\ref{lem:mincell}). We may assume that notation is chosen so that
$X^{m+1}\subset X^m$. We then must have $Y^{m+1}\subset Y^m$. Repeating this argument and switching notation for $X^n$ and $Y^n$ as necessary, we see that we may assume
$X^m\supset X^{m+1}\supset X^{m+2}\dots $ and
$Y^m\supset Y^{m+1}\supset Y^{m+2}\dots $. In particular, we have
$\Om^{n+1}\subset \Om^n$ for all $n\ge m$.
Let $z\in \bigcap_{n} \overline {\Om}^{{\null}_{\scriptstyle n}}$ be arbitrary. We want to show that $z\in M=[x]$. Suppose this is not true, and pick a sequence $\{Z^n\}\in \mathcal{S}$ with $z\in \bigcap_nZ^n$.
Since $x\in \bigcap_n X^n$, $x\in \bigcap_n Y^n$ and $x\not\sim z$ by assumption, by condition (ii) in Lemma~\ref{lem:erel} there exists $n_0\ge m$ such that $X^{n_0}\cap Z^{n_0}=\emptyset$ and $Y^{n_0}\cap Z^{n_0}=\emptyset$.
This is absurd, since $z\in Z^{n_0}\cap \overline {\Om}^{{\null}_{\scriptstyle n_0}}=Z^{n_0}\cap
(X^{n_0}\cup Y^{n_0})$. We conclude that $ \bigcap_{n} \overline {\Om}^{{\null}_{\scriptstyle n}}\subset M$.
Suppose $z\not\in \Om^{n_0}$ for some $n_0\ge m$. Then there
exists an $n_0$-tile or an $n_0$-edge $\tau$ with $z\in \tau$ that does not meet
the interior of $e^{n_0}$. Let $u$ and $v$ be the endpoints of $e^{n_0}$.
Since these points are vertices, they do not belong to $M$ by assumption. Hence the set
$$\bigcap_n e^n \subset \bigcap_n \overline {\Om}^{{\null}_{\scriptstyle n}}\subset M$$
does not contain $u$ or $v$ either. It follows that there exists $n_1\ge n_0$ such that $u,v\not\in \sigma:=e^{n_1}$. Then $x\in \sigma$, $z\in \tau$, and $\sigma\cap \tau=\emptyset$, because
$$ \sigma\cap \tau = e^{n_1}\cap \tau\subset (e^{n_0}\setminus \{u,v\})\cap \tau =\emptyset. $$ This implies that $x\not\sim z$ by condition~(iii) in Lemma~\ref{lem:erel}~(iii).
Therefore, $M\subset \bigcap \Om^n$, and so the sets $\Om^n$ have all the desired properties.
\smallskip
{\em Case 3.} $M$ does not meet any edge.
Pick a point $x\in M$ and a sequence $\{X^n\}\in \mathcal{S}$
with $x\in \bigcap_n X^n$. Define $\Om^n:=\inte(X^n)$ for $n\in \N_0$. Then
the sets $\Om^n$ are nested simply connected regions, and $\overline \Om^n=X^n$ is an $n$-tile. It follows from Lemma~\ref{lem:erel}~(ii) that $$\bigcap_n \overline {\Om}^{{\null}_{\scriptstyle n}}\subset [x]=M.$$
It remains to show that $[x]\subset \bigcap_n \Om^n$.
If this is not the case, then there exists a point $y\in S^2$ with $x\sim y$, and
$n_0\in \N_0$ such that $y\not \in \Om^{n_0}$. Since $\overline {\Om}^{{\null}_{\scriptstyle n_0}}$ is an $n_0$-tile, the boundary of $\Om^{n_0}$ is a union of $n_0$-edges. Since $M$ does not meet any edge, we have $y\not \in \overline {\Om}^{{\null}_{\scriptstyle n_0}}=X^{n_0}$.
Now pick a sequence $\{Y^n\}\in \mathcal{S}$
with $y\in \bigcap_n Y^n$. Then $Z^n=X^n\cap Y^n\ne \emptyset$
for all $n\in \N_0$.
The sets $Z^n$, $n\in \N_0$, are nonempty nested compact sets.
Hence there exists a point $z\in \bigcap_n Z^n$. Then $x\sim z$ and so $z\in M$.
On the other hand, we have $y\not \in X^{n_0}$ and so $X^{n_0}\ne Y^{n_0}$. Hence the intersection $Z^{n_0}$
consists of $n_0$-cells on the boundary of $X^{n_0}$ and of $Y^{n_0}$, and is hence contained in a union of $n_0$-edges. Since $z\in M\cap Z^{n_0}$ this means that $M$ meets an edge
contradicting our assumption.
\end{proof}
\begin{cor}
\label{lem:erelcor}
Each equivalence class $M$ of $\sim$ is a compact connected set with connected complement $S^2\setminus M$.
\end{cor}
\begin{proof} Let $M$ be an arbitrary equivalence class of $\sim$. By Lemma~\ref{lem:eclass} the set $M$ is the intersection of a nested
sequence of compact and connected sets. Hence $M$ is also compact
and connected.
The complement of an open simply connected set in $S^2$ is connected. So Lemma~\ref{lem:eclass} also shows that
the complement $S^2\setminus M$ of $M$ is a union of an increasing sequence of connected sets. Hence $S^2\setminus M$ is connected.
\end{proof}
\begin{lemma}
\label{lem:simusc} Let $\sim$ be the equivalence relation on $S^2$ as in Definition~\ref{def:erel}.
Then the quotient space $\widetilde S^2=S^2/\!\!\sim$ is homeomorphic to $S^2$.
\end{lemma}
\begin{proof} By Lemma~\ref{lem:aeq} our relation $\sim$ is indeed an equivalence relation. It remains to verify the conditions (i)--(iv) in
Theorem~\ref{thm:moore}.
{\em Condition} (i): Let $\{x_n\}$ and $\{y_n\}$ be convergent sequences with $x_n\rightarrow x$ and $y_n\rightarrow y$ as $n\to \infty$, and suppose that $x_n\sim y_n$ for all $n\in \N$. We have to show that $x\sim y$. Suppose this is not the case.
Then the equivalence classes $[x]$ and $[y]$ are disjoint.
By Lemma~\ref{lem:eclass} for sufficiently large $n$ there exist simply connected nested regions $U^n$ and $V^n$ such that
$$[x]=\bigcap_n U^n=\bigcap_n \overline U^{{}_{\scriptstyle n}} \text{ and }
[y]=\bigcap_n V^n=\bigcap_n \overline V^{{}_{\scriptstyle n}}. $$
Since $[x]$ and $[y]$ are disjoint, the sets $\overline U^{{}_{\scriptstyle n}}$ and $\overline V^{{}_{\scriptstyle n}}$ will also be disjoint for sufficiently large $n$, say
$\overline U^{{}_{\scriptstyle n_0}}\cap\overline V^{{}_{\scriptstyle n_0}} =\emptyset$. On the other hand, since $U^{n_0}\supset[x]$ and $V^{n_0}\supset[y]$ are open, there exists $n_1\in \N$ such that
$x_{n_1}\in U^{n_0}$ and $y_{n_1}\in V^{n_0}$.
Since $\overline U^{{}_{\scriptstyle n_0}}$ and $\overline V^{{}_{\scriptstyle n_0}}$ consist of $n_0$-tiles and are disjoint, this means that there exist $n_0$-tiles $\sigma$ and $\tau$
with $x_{n_1}\in \sigma$, $y_{n_1}\in \tau$ and $\sigma\cap \tau =\emptyset$. Hence $x_{n_1}\not \sim y_{n_1}$ by Lemma~\ref{lem:erel}. This is a contradiction. It follows that $\sim$ is closed.
{\em Conditions} (ii)+(iii): The statements were proved in Corollary~\ref{lem:erelcor}.
{\em Condition} (iv): There are at least two equivalence classes, because no two distinct vertices are equivalent by Lemma~\ref{lem:erel}, and each postcritical point of $F$ (there are at least
$3$ such points) is a vertex.
This shows that the conditions in Moore's theorem are satisfied, and so $\widetilde S^2$ is indeed a sphere.
\end{proof}
\bigskip \noindent
{\bf Quotients of cells and the induced cell decompositions on $\widetilde S^2$.}
\noindent We now study what happens to our cells under the quotient map $\pi\: S^2 \rightarrow
\widetilde S^2$.
If $A\subset S^2$ is an arbitrary set, we denote by
$\widetilde A$ its image under the projection map $\pi$.
So $\widetilde A=\pi(A)=\{[x]:x\in A\}$. We will see that if $\sigma$ is an arbitrary cell (i.e., an $n$-cell for $(F,\mathcal{C})$ and some $n\in \N_0$), then $\widetilde \sigma$ is a topological cell
of the same dimension (Lemma~\ref{lem:qcells}). Moreover, the images $\widetilde \sigma$ of the $n$-cells $\sigma$ form a cell decomposition of $\widetilde S^2$ (Lemma~\ref{lem:cdcd}).
Consider an equivalence class $M\subset S^2$ of $\sim$. Depending on the cases in the proof of
Lemma~\ref{lem:eclass} we associate to $M$ a nested
sequence $c^m\supset c^{m+1} \supset \dots$ of $n$-cells $c^n$, where $n\ge m\in \N$, as follows:
If $M=[v]$, where $v$ is an
$m$-vertex we set $c^n := \{v\}$ for all $n\geq m$. When $M$ does not
contain a vertex, but meets an $m$-edge we set $c^n:=e^n$ for all
$n\geq m$, where $e^m\supset e^{m+1} \supset \dots$ is the nested
sequence of edges from Case 2 of the proof of Lemma 11.5. Finally,
when $M$ does not meet an edge, we set $c^n=X^n$ for all $n\geq m$,
where $X^m\supset X^{m+1} \supset \dots$ is the nested sequence from
Case 3 of the proof of Lemma~\ref{lem:eclass}. Note that in all cases $ \bigcap c^n\subset M$.
The following lemma shows that the sequence $c^m\supset c^{m+1} \supset \dots$ can be used to give a criterion when a cell $\tau$ intersects the
equivalence class $M$.
\begin{lemma}
\label{lem:eq_intersections}
Let $M$ be an arbitrary equivalence class with the associated nested sequence of cells $c^m
\supset c^{m+1} \supset \dots$ defined as above.
If $\tau$ is an arbitrary cell, then $\tau \cap M\ne \emptyset$
if and only if
$ c^n \subset \tau$
for all sufficiently large $n$.
\end{lemma}
\begin{proof}
{\em Case~1.} $M=[v]$, where $v$ is an $m$-vertex.
Then $\tau \cap M\ne \emptyset$ if and only if $\tau \supset \{v\}=c^m=c^{m+1}= \dots $. This immediately follows from
Lemma~\ref{lem:erel}.
\smallskip
{\em Case~2.} $M$ does not contain a vertex, but meets an edge. Then $c^m\supset c^{m+1} \supset\dots$
is the nested sequence of edges $e^m\supset e^{m+1}\supset \dots$ as defined in the proof of Lemma~\ref{lem:eclass} in this case.
Then $\tau \cap M \ne \emptyset$ if and only if
$e^n\subset \tau$ for all large $n$.
Indeed, since $\emptyset \ne \bigcap_n e^n \subset M$, it is clear that the second condition implies the first one.
Conversely, suppose $\tau \cap M\ne \emptyset$.
By subdividing $\tau$ if necessary, we may assume that $\tau$ is a $k$-cell with $k\ge m$.
Then $\tau$ has nonempty intersection
with $\Om^k=\inte(X^k)\cup \inte(e^k) \cup \inte (Y^k)\supset M$, where the notation is as in Case~2 of the proof of Lemma~\ref{lem:eclass}.
Since the interiors of $k$-cells are pairwise disjoint, this
is only possible if $\tau$ is an $k$-edge and
$\tau=e^k$, or if $\tau$ is a $k$-tile and $\tau=X^k$ or $\tau=Y^k$. In any case $e^k\subset \tau$, and so $e^n\subset \tau $ for all large $n$.
\smallskip
{\em Case~3.} $M$ does not meet an edge. Then $c^m\supset c^{m+1} \supset\dots$
is the nested sequence of tiles $X^m\supset X^{m+1}\supset\dots$ as defined in the proof of Lemma~\ref{lem:eclass} in this case.
Then $\tau \cap M\ne \emptyset$ if and only if $X^n\subset \tau$ for large $n$. Again one direction is clear.
For the other direction we can again assume that $\tau$ is a $k$-cell with $k\ge m$. As we have seen in the proof of Case~3 of Lemma~\ref{lem:eclass}, the set $M$ lies in the interior of each tile $X^n$ .
So if $\tau\cap M\ne \emptyset$, then $\tau$ meets the interior of $X^k$. This is only possible if $\tau$ is a $k$-tile and $\tau=X^k$. Obviously, we then have
$X^n\subset \tau$ for large $n$.
\end{proof}
The following lemma states that if we pass to the quotient space $\widetilde S^2=S^2/\!\!\sim$, then we do not create ``new'' intersections or inclusions between cells.
\begin{lemma}\label{lem:nonewint} If $\sigma$ and $\tau$ are cells, then $\widetilde \sigma \cap \widetilde \tau =\widetilde {\sigma \cap \tau}$.
Moreover, we have $\widetilde \sigma\subset \widetilde \tau$ if and only if
$\sigma\subset \tau$. \end{lemma}
\begin{proof} The inclusion $\widetilde {\sigma \cap \tau}\subset \widetilde \sigma \cap \widetilde \tau$ is trivial.
For the other inclusion let $M$ be an arbitrary equivalence class and suppose that $M$ (considered as a point in $\widetilde S^2$) is an element of the set
$\widetilde \sigma \cap \widetilde \tau\subset \widetilde S^2$. Then $M$ (considered as a subset of $S^2$) meets both cells $\sigma$ and $\tau$.
By Lemma~\ref{lem:eq_intersections} there exist $n$-cells $c^n\subset M$
such that both $c^n\subset \sigma$ and $c^n\subset \tau$ for all sufficiently large $n$.
In particular, for such $n$ we have $M\cap \sigma\cap \tau \supset c^n \ne \emptyset$. Hence $M$ (now again considered as a point in $\widetilde S^2$) lies in $\widetilde {\sigma \cap \tau}$, and we have $ \widetilde\sigma \cap \widetilde \tau\subset \widetilde {\sigma \cap \tau}$ as desired.
In the second statement the implication $\sigma\subset \tau \Rightarrow
\widetilde \sigma\subset \widetilde \tau$ is trivial. For the other implication assume that $\widetilde \sigma\subset \widetilde \tau$. Let $k$ and $n$ be the orders of $\sigma$ and $\tau$, respectively. For the moment we make the additional assumption that $k\ge n$.
By Lemma~\ref{lem:vinint} there exists a vertex $v$ such that $v\in \inte(\sigma)$ (note that this is trivial if $\sigma$ is a $0$-dimensional cell).
Then $[v]\in \widetilde \sigma \subset \widetilde \tau$, and so $[v]\cap \tau \ne \emptyset$. Lemma~\ref{lem:erel}~(iii) implies that $v\in \tau$ showing that
\begin{equation}\label{eq:stint}
\inte(\sigma)\cap \tau \ne \emptyset.
\end{equation}
Since $k\ge n$, the cell decomposition $\mathcal{D}^k$ containing $\sigma$ is a refinement of the
cell decomposition $\mathcal{D}^n$ containing $\tau$.
Therefore, as we have seen in the first part of the proof of Lemma~\ref{lem:mincell}, the relation \eqref{eq:stint} forces the inclusion
$\sigma\subset \tau$.
If $k<n$, we subdivide $\sigma$ into cells of order $n$. By the previous argument, $\tau$ will contain each of these cells, and so we always have have $\sigma\subset \tau$ as desired.
\end{proof}
\begin{lemma}\label{lem:edgeinter} Let $M\subset S^2$ be an equivalence class and $E\subset S^2$ be a finite union of edges. Then $E\cap M$ is connected. \end{lemma}
\begin{proof} By subdividing the edges in the union representing $E$, we may assume that $E$ consists
of $k$-edges $\tau^k_1, \dots, \tau^k_N$, where $k$ is large enough.
Again we consider three cases for $M$ using the notation of the proof of Lemma~\ref{lem:eclass}.
\smallskip
{\em Case 1.} $M=[v]$, where $v$ is a vertex. Then, as we have seen in the proof of Case 1 of Lemma~\ref{lem:eq_intersections}, $\tau^k_i\cap M\ne \emptyset$ if and only if $v\in \tau^k_i$.
By passing to larger $k$ and subdividing the edges $\tau^k_i$ if necessary, we may assume that $v\in \tau^k_i$ if and only if $v$ is one of the two endpoints of
$\tau^k_i$. If $v$ is an endpoint of $\tau^k_i$, then
there is a unique nested sequence of
edges $\tau^k_i\supset \tau_i^{k+1}\supset \dots$
such that $\tau^n_i$ for $n\ge k$ is an $n$-edge with one of its endpoints equal to $v$.
It follows from the representation of $M$
as in Case 1 of the proof of Lemma~\ref{lem:eclass} that
$M\cap \tau_i^k=\bigcap_n \tau_i^n\supset\{v\}$. In particular,
this set is a point or an arc and hence connected.
It follows that $E\cap M$ is empty or consists of a union of connected sets that share the point $v$. Hence
$E\cap M$ is connected.
\smallskip
{\em Case 2.} $M$ meets an edge, but does not contain a vertex. We use notation as in
Case 2 of the proof of Lemma~\ref{lem:eclass}. We may assume that $k\ge m$, where $m$ is as in the proof of this lemma.
As we in this proof, the only $k$-edge that meets $M$
is $e^k$. So $E\cap M=
e^k\cap M$ or $E\cap M=\emptyset$, depending whether $e^k$ is among the edges $\tau^k_1, \dots, \tau^k_N$ or not; moreover, $e^k\cap M=\bigcap_n e^n$ as follows from the representation of $M$ as in Case~2 of the proof of Lemma~\ref{lem:eclass}. In particular, $e^k\cap M$ is a point or an arc, and hence connected.
It follows that $E\cap M$ is connected.
\smallskip
{\em Case 3.} $M$ does not meet an edge. Then $E\cap M=\emptyset$ is connected.
\end{proof}
For the first main result of this subsection we will need a $1$-dimensional version of Moore's Theorem. It can easily be derived from the topological characterization of arcs and topological circles (in equivalent form this as stated as two exercises in \cite[p.~21, Ex.~2 and 3]{Da}). We will give a simple direct proof for the convenience of the reader.
\begin{lemma}\label{lem:1dimMoore} Let $J$ be an arc or a topological circle, and $\sim$ be an equivalence relation on $J$.
Suppose that
\begin{itemize}
\smallskip
\item [\textnormal{(i)}] each equivalence class of $\sim$ is a compact and connected subset of $J$,
\smallskip
\item [\textnormal{(ii)}] there are at least two distinct equivalence classes.
\end{itemize}
Then the quotient space $\widetilde J=J/\!\! \sim$ is an arc or a topological circle, respectively.
\end{lemma}
\begin{proof} We may assume that $J$ is equal to the unit interval $[0,1]$ or
the unit circle $\partial \D\subset \C$. We denote by $[u]\subset J$ the equivalence class
of a point $u\in J$, and by $\pi\: J\rightarrow J/\!\!\sim $ the quotient map that sends each point $u\in J$ to $[u]$ (considered as a point
in $ J/\!\!\sim $). By our assumptions, in both cases $J=[0,1]$ and $J=\partial
\D$ each set $[u]$ is a subarc of $J$ (possibly degenerate).
This implies that the equivalence relation $\sim$ is closed. Indeed, suppose
$(u_n)$ and $(v_n)$ are two sequences in $J$ with $u_n\sim v_n$ for $n\in \N$, $u_n\to u\in J$ and $v_n\to v\in J$ as $n\to \infty$.
Let $\alpha_n:=[u_n]=[v_n]$ for $n\in \N_0$. If in the sequence $\alpha_1, \alpha_2, \dots $ one of the sets $\alpha_n$ appears infinitely often, then, by passing to a subsequence if necessary, we may assume that $\alpha:=\alpha_1=\alpha_2=\dots$. Then $(u_n)$ and $(v_n)$ are sequences in $\alpha$. Since $\alpha$ is a closed set, we conclude $u,v\in \alpha$, and so, since $\alpha$ is also an equivalence class, we have $u\sim v$ as desired.
If none of the sets appears $\alpha_n$ appears infinitely often in the sequence, then necessarily $\diam(\alpha_n)\to 0$ as $n\to \infty$, as $J$ does not carry infinitely many pairwise disjoint arcs with diameter bounded away from $0$. This implies $|u_n-v_n|\to 0$ as $n\to \infty$, and so $u=v$; again we have $u\sim v$ showing that $\sim $ is closed.
Since $\sim$ is closed, the quotient space $J/\!\!\sim$ is Hausdorff; this follows from the fact that the complement of the ``diagonal" in $J\times J$ is an open set, and hence a neighborhood for each of the points it contains. Moreover, the space
$J/\!\!\sim$ is also compact, since it is the continuous image of $J$ under the continuous map $\pi$. So $J/\!\!\sim$ is a compact Hausdorff space in both cases $J=[0,1]$ and $J=\partial \D$.
Now we restrict ourselves to the case $J=[0,1]$. For $u,v\in [0,1]$ we define
$[u]<[v]$ iff $u<v$. This is well-defined, and a total order for the equivalence
classes of $\sim$, i.e., for $u,v\in J$ we have precisely one of the cases $[u]<[v]$, $[v]<[u]$, or $[u]=[v]$.
We want to show that $J/\!\!\sim$ is homeomorphic to $[0,1]$. For this it suffices to find a surjective and continuous map $h\:[0,1]\rightarrow [0,1]$ such that $h(u)=h(v)$ for
$u,v\in [0,1]$ if and only if $u\sim v$. For then $h$ will descend to a
continuous bijection $\widetilde h$ of the compact Hausdorff space $J/\!\!\sim$ onto
$[0,1]$. Then $\widetilde h$ is a homeomorphism as desired.
We now construct $h$. By enumerating the sets $[u]$ for $u\in [0,1]\cap\Q$ we can find a sequence of closed subintervals $I_0, I_1, \dots$ of $[0,1]$
with $I_0=[0]$, $I_1=[1]$ such that the intervals $I_0, I_1, \dots$ are pairwise distinct equivalence classes for $\sim$ and such that
$$D:=\bigcup_{n} I_n$$
is dense in $[0,1]$. Note that by our hypotheses we have $I_0\ne I_1$. Moreover, whenever $x,y\in [0,1]$ and $[u]<[v]$, then there exists
$n\in \N_0$ with $[u]<I_n<[v]$; otherwise, each $q\in [x,y]\cap \Q$, and hence each point in $[u,v]$, would be contained in one of the sets $[u]$ and $[v]$;
this is impossible since $[u]$ and $[v]$ are closed disjoint sets with non-empty intersection with $[u,v]$, and $[u,v]$ is connected. So between any two distinct equivalence classes there is always one of the sets $I_n$; this and $I_0\ne I_1$ imply that the list $I_0, I_1, \dots $ is infinite.
We now inductively pick a number $y_n\in [0,1]$ for each set $I_n$ so
that $I_n<I_k$ if and only if $y_n<y_k$. This is done as follows.
Let $y_0=0$ and $y_1=1$, and suppose numbers $y_0, \dots, y_n$ with the desired properties have been chosen for the sets $I_0, \dots, I_n$, where $n\ge1$.
For some bijection $\phi\:\{0,\dots, n\}\rightarrow \{0,\dots, n\}$ with $\phi(0)=0$ and $\phi(n)=1$, we have
$$I_{\phi(0)}=I_0<I_{\phi(1)}<\dots <I_{\phi(n)}=I_1.$$
Then there exists a unique number $l\in \{0,\dots, n-1\}$ such
that $$I_{\phi(l)}<I_{n+1}<I_{\phi(l+1)}. $$
Define $y_{n+1}:=\frac12 (y_{\phi(l)}+y_{\phi(l+1)})$. It is clear that this
gives a correspondence $I_n\leftrightarrow y_n$ for $n\in \N_0$ as desired.
If $I_n<I_k$, then there exists $m>\max\{n,k\}$ such that $I_n<I_m<I_k$.
This and the definition of $y_n$ imply that the set $D':=\{y_n:n\in \N_0\}$
is dense in $[0,1]$; indeed, if we use the relation $<$ to successively order the values $y_0=0, y_1=1, y_2, \dots$, and $y_n<y_k$ are immediate neighbors in this ordering up to some point, then eventually another value $y_m$ will fall in the ``gap" $(y_n, y_k)$ cutting it in half, and so all gaps between the values $y_n$ will become small.
There exists a unique function $h\: D\rightarrow D'$ such that $h|I_n\equiv y_n$
for all $n\in \N_0$. Then $h$ is non-increasing, and we can extend it to a non-increasing function on $[0,1]$, also denoted $h$, by setting
$$h(u):=\sup\{h(t):t\in D\cap[0,u]\} \text{ for } u\in [0,1]. $$
Then $h$ is a non-increasing function on $[0,1]$ with a dense image in $[0,1]$.
Hence $h$ is continuous (at each point left-hand and right-hand limit exist and agree with the function value), and surjective.
It remains to verify that $h(u)=h(v)$ if and only if $u\sim v$.
Suppose first that $u,v\in [0,1]$ and $u\sim v$. If $u=v$, then $h(u)=h(v)$. If
$u\ne v$, then $[u]=[v]$ is a non-degenerate interval, and so it must be
among the intervals $I_0, I_1, \dots$ by density of $D$, say $[u]=[v]=I_n$. Then $h(u)=y_n=h(v)$ by definition of $h$.
Conversely, suppose that $u,v\in [0,1]$ and $u\not \sim v$. Without loss of generality $u<v$. Then $[u]<[v]$ and we can choose intervals $I_n$ and $I_k$ with $k,n\in \N_0$ such that $[u]<I_n<I_k<[v]$. Then the properties of $h$ imply
$$h(u)\le h|I_n= y_n<h|I_k=y_k\le h(v), $$
and so $h(u)\ne h(v)$ as desired. This completes the proof that $J/\!\!\sim=[0,1]/\!\!\sim$ is homeomorphic to $[0,1]$, and hence an arc.
Note that the images of the endpoints of $J$, i.e., $0$ and $1$, under the quotient map $\pi$ are the endpoints of the quotient arc $J/\!\!\sim$. Indeed, the endpoints of an arc
$\alpha$ are precisely the points $p\in\alpha$ for which
$\alpha\setminus\{p\}$ is connected. Note that $[0,1]\setminus [0]$ is a half-open interval and hence connected. Hence $$(J/\!\!\sim)\setminus \{\pi(0)\}=\pi([0,1]\setminus [0]) $$ is connected as the continuous image of the
connected set $[0,1]\setminus [0]$. So $\pi(0)$ is one of the endpoints of $J/\!\!\sim$, and a similar argument shows that $\pi(1)$ is the other endpoint.
We now turn to the case $J=\partial \D$. Then by our hypotheses we can pick points
$u,v\in \partial \D$ with $u\not\sim v$. In particular, $u\ne v$. Let $\alpha$ and $\beta$ be the
two subarcs of $\partial \D$ with endpoints $u$ and $v$.
Define $p=\pi(u)$, $q=\pi(v)$, $\widetilde \alpha=\pi(\alpha)$,
$\widetilde \beta=\pi(\beta)$. Then $p\ne q$, and our hypotheses imply that
$ \widetilde \alpha\cap \widetilde \beta=\{p,q\}$ (a connected set in $\partial \D$ that meets both $\alpha$ and $\beta$ must meet $u$ or $v$).
By the first part of the proof (consider $\sim$ restricted to $\alpha$ and $\beta$), the sets $ \widetilde \alpha$ and $\widetilde \beta$ are arcs, and they have the endpoints $p$ and $q$. So they have precisely their endpoints in common. Every compact Hausdorff space
that can be represented as a union of two such arcs is homeomorphic to
$\partial \D$. Hence $J=\partial \D/\!\!\sim$ is homeomorphic to
$\partial \D$ and thus a topological circle. \end{proof}
\begin{lemma}\label{lem:qcells} Let $\tau$ be a an edge or a tile. Then
$\widetilde \tau$ is an arc or a closed Jordan region, respectively. Moreover, $\partial \widetilde \tau=
\widetilde{\partial \tau}$.
\end{lemma}
Here $\partial \tau $ (and similarly $\partial \widetilde \tau$) refers as usual to the boundary of a cell $\tau$ as defined
in Section~\ref{s:celldecomp}. So $\partial \tau $ is the topological boundary of $\tau$ in $S^2$ if $\tau$ is a tile, and equal to the set consisting of the two endpoints of $\tau$ if $\tau$ is an edge. If $\tau$ is $0$-dimensional cell, i.e., a singleton set consisting of a vertex, then $\partial \tau=\emptyset$, and the
statement in the lemma is trivially true.
So the lemma can be formulated in an equivalent form by saying that
if $\tau\subset S^2$ is a cell (in one of the cell decompositions $\mathcal{D}^n(F,\mathcal{C}))$, then $\widetilde \tau\subset \widetilde S^2$ is a
cell (in the general topological sense) of the same dimension, and the boundary of $\widetilde \tau$ is the image of the boundary of $\tau$ under the quotient map.
\begin{proof} Suppose first that $\tau$ is an edge.
Then our equivalence relation $\sim$ on $S^2$ restricts to an equivalence relation on $\tau$ whose
quotient space can be identified with the subset
$\widetilde \tau$ of $\widetilde S^2$. The equivalence classes on $\tau$ have the form $\tau\cap M$, where $M\subset S^2$ is an equivalence class with respect to $\sim$.
Each set $\tau \cap M$ is compact, as $\sim$ is closed, and con\-nected by Lem\-ma~\ref{lem:edgeinter}. Moreover, $\tau$ meets a least two distinct equivalence classes, as its endpoints are vertices and hence not equivalent. Lemma~\ref{lem:1dimMoore} implies that $\widetilde \tau\subset \widetilde S^2$ is indeed an arc.
Let $u$ and $v$ be the two endpoints of $\tau$. Then $[u]\cap \tau$ is a compact connected subset of $\tau$ containing $u$. Hence this set is a subarc of $\tau$ with one endpoint equal to $u$. This implies that
the set $\tau\setminus [u]$ is connected, and so the set $\pi(\tau\setminus [u])=\widetilde \tau\setminus
\{\pi(u)\}$ is also connected. Therefore, $\pi(u)$ is an endpoint of $\widetilde \tau$ (a similar argument was presented in the proof of Lemma~\ref{lem:1dimMoore}). By the same reasoning we see that $\pi(v)$ is also an endpoint
of $\widetilde \tau$. Since $u$ and $v$ are distinct vertices, we have $u\not\sim v$ and so $\pi(u)\ne \pi(v)$. Hence $\partial \widetilde \tau
=\{\pi(u), \pi(v)\}= \pi(\{u,v\})= \widetilde {\partial \tau}$. \medskip
If $\tau$ is a tile, say an $n$-tile, then $\tau$ is a closed Jordan region whose boundary $J=\partial \tau$ is a topological circle consisting of finitely many edges. By the Sch\"{o}nflies Theorem we can write $
S^2$ as a disjoint union
$ S^2=\Omega_1\cup J \cup \Omega_2$,
where $\Omega_1$ and $\Omega_2$ are open Jordan regions bounded by $J$. Then $\tau$ coincides with one of the sets $\overline \Om_1$ or $\overline \Om_2$, say $\tau=\overline \Om_1$.
The set $\widetilde J\subset \widetilde S^2$ is also a topological circle as follows from the fact that $\sim$ is closed, Lemma~\ref{lem:edgeinter}, and Lemma~\ref{lem:1dimMoore}. So we can also write
$\widetilde S^2$ as a disjoint union
$\widetilde
S^2=D_1\cup \widetilde J \cup D_2$, where $D_1$ and $D_2$ are open Jordan regions in $\widetilde S^2$ bounded by $\widetilde J$.
If we take preimages under the quotient map $\pi\:S^2\rightarrow \widetilde S^2$, we get the disjoint union
$S^2=\pi^{-1}(D_1)\cup \pi^{-1}(\widetilde J)\cup
\pi^{-1}(D_2)$. Since point preimages under $\pi$, i.e., equivalence classes, are connected, the map
$\pi$ is {\em monotone}, and hence preimages of connected sets are connected \cite[p.~18, Prop.\ 1]{Da}. So the sets
$\pi^{-1}(D_1)$ and $\pi^{-1} (D_2)$ are open connected sets disjoint from $ \pi^{-1}(\widetilde J)\supset J$. It follows that each of the sets $\pi^{-1}(D_1)$ and $\pi^{-1} (D_2)$ is contained in one of the
regions $\Om_1$ and $\Om_2$.
These sets cannot be contained in the same region $\Om_i$. Indeed, if $\pi^{-1}(D_1)\cup \pi^{-1} (D_2)\subset \Om_1$, for example, then $\Om_2 \subset \pi^{-1}(\widetilde J)$, and so $\pi(\Om_2)\subset \widetilde J$. This means that every point in $\Om_2$ is equivalent to a point
in $J$. This is impossible, because $\Om_2$ contains the interior of an $n$-tile, and hence a $k$-vertex for some $k>n$ (see Lemma~\ref{lem:vinint}~(ii)). Such a vertex is not equivalent to any point in $J$ by Lemma~\ref{lem:erel}.
By what we have just seen, we may assume that indices are chosen such that $\pi^{-1}(D_1)\subset \Om_1$ and
$\pi^{-1}(D_2)\subset \Om_2$. Then
$ \overline D_1=D_1\cup \widetilde J\subset \pi(\overline \Om_1)$ and $\pi(\overline \Om_1)\cap D_2=\emptyset$. Hence $ \widetilde \tau=\pi(\overline \Om_1)=\overline
D_1$ is a closed Jordan region.
Moreover, $\partial \widetilde \tau=\partial D_1=\widetilde J= \widetilde{\partial \tau}$. \end{proof}
The next lemma summarizes some of the results in this subsection.
\begin{lemma} \label{lem:cdcd} Let $n,k\in \N_0$.
Then we have:
\begin{itemize}
\smallskip
\item[(i)] For each $\tau\in \mathcal{D}^n$ the set $\widetilde \tau$ is a topological cell in $\widetilde S^2$ of the same dimension as $\tau$.
\smallskip
\item[(ii)] For $\sigma, \tau\in \mathcal{D}^n$, we have $\widetilde \sigma=\widetilde \tau$ if and only if $\sigma=\tau$.
\smallskip
\item[(iii)] $\widetilde \mathcal{D}^n:=\{\widetilde \tau: \tau \in \mathcal{D}^n\}$ is a cell decomposition of $\widetilde S^2$.
\smallskip
\item[(iv)] The map $\tau \in \mathcal{D}^n\mapsto \widetilde \tau\in \widetilde \mathcal{D}^n$ is an isomorphism between the cell complexes $\mathcal{D}^n$ and $\widetilde \mathcal{D}^n$.
\smallskip
\item[(v)] $\widetilde \mathcal{D}^{n+k}$ is a refinement of $\widetilde \mathcal{D}^n$.
\end{itemize}
\end{lemma}
\begin{proof} (i) This follows from Lemma~\ref{lem:qcells}. \smallskip
(ii)--(iii)
Let $\sigma$ and $\tau$ be arbitrary
$n$-cells, and suppose that $\inte(\widetilde \sigma)\cap \inte(\widetilde \tau)\ne \emptyset$. Pick a point $p\in \inte(\widetilde \sigma)\cap \inte(\widetilde \tau)$. Then $p\in \widetilde \sigma \cap \widetilde \tau=\widetilde {\sigma \cap \tau}$ (see Lemma~\ref{lem:nonewint}), and so
there exists $x\in \sigma \cap \tau$ with $\pi(x)=p$.
Then $x\in \inte(\sigma)$, for otherwise
$x\in \partial \sigma$ and so $p=\pi(x)\in \partial
\widetilde \sigma$ (see Lemma~\ref{lem:qcells}),
contradicting the choice of $p$. Similarly, $x\in \inte(\tau)$. So $x\in \inte(\sigma)\cap \inte(\tau)$ which implies that $\sigma=\tau$. Statement (ii) follows.
This shows that the topological cells $\widetilde \tau$ for $\tau \in \mathcal{D}^n$ are all distinct, and no two have a common interior point. Moreover, there are finitely
many of these cells, and they cover $\widetilde S^2$, because the cells in $\mathcal{D}^n$ cover $S^2$.
Finally, for a cell $\widetilde \tau$ we have $\partial
\widetilde \tau=\widetilde {\partial \tau}$ (Lemma~\ref{lem:qcells}). Since $\partial \tau$ is a union of cells in $\mathcal{D}^n$, it follows that $\partial
\widetilde \tau$ is a union of cells in $\widetilde \mathcal{D}^n$. This shows that $\widetilde \mathcal{D}^n$ is a cell decomposition of $\widetilde S^2$. \smallskip
(iii) By (i) and (ii) the map $\tau \in \mathcal{D}^n\mapsto \widetilde \tau\in \widetilde \mathcal{D}^n$ is a bijection between $\mathcal{D}^n$ and $\widetilde \mathcal{D}^n$ that preserves dimensions of cells. By Lemma~\ref{lem:nonewint} the map also satisfies condition (ii)
in Definition~\ref{def:compiso}. Hence it is an isomorphism between cell complexes. \smallskip
(iv) This immediately follows from the definitions and the fact that $\mathcal{D}^{n+k}$ is a refinement of $\mathcal{D}^n$.
\end{proof}
\bigskip \noindent
{\bf The induced map $\widetilde F$ on $\widetilde S^2$.}
We will now show that $F$ induces a map on the sphere
$\widetilde S^2$ and study the properties of this map.
\begin{lemma} \label{lem:Find}
Suppose that $x,y\in S^2$ and $x\sim y$. Then $F(x)\sim F(y)$.
\end{lemma}
\begin{proof} Let $x,y\in S^2$ with $x\sim y$ be arbitrary. Pick $\{X^n\}, \{Y^n\}\in \mathcal{S}$ with $x\in \bigcap X^n$ and $y\in \bigcap_n Y^n$. Define $U^n=F(X^{n+1})$ and $V^n=F(Y^{n+1})$ for $n\in \N_0$. Then $U^n$ and $V^n$ are $n$-tiles, and so
$\{U^n\}, \{V^n\}\in \mathcal{S}$. Moreover, $F(x)\in \bigcap_n U^n$
and $F(y)\in \bigcap_n V^n$. Since $x\sim y$ we have $X^n\cap Y^n\ne\emptyset$
for all $n\in \N$. Hence
$$U^n\cap V^n=F(X^{n+1})\cap F(Y^{n+1})\supset F(X^{n+1}\cap Y^{n+1})\ne \emptyset$$
for all $n\in \N_0$. Lemma~\ref{lem:erel}~(ii) now implies that $F(x)\sim F(y)$ as desired.
\end{proof}
By the previous lemma the map $\widetilde F\: \widetilde S^2\rightarrow \widetilde S^2$ given by
$$ \widetilde F([x])=[F(x)] \quad \text{for} \quad x\in S^2$$
is well-defined. Then $ \widetilde F\circ \pi=\pi\circ F$, and it follows from the properties of the quotient topology that $\widetilde F$ is continuous.
In the following $\widetilde \mathcal{D}^n=\{\widetilde \tau: \tau \in \mathcal{D}^n(F,\mathcal{C})\}$ for $n\in \N_0$ will denote the cell decomposition of $\widetilde S^2$ as provided by Lemma~\ref{lem:cdcd}.
As the next lemma shows, the map ${\widetilde F}^n$ has injectivity properties similar to $ F^n$.
\begin{lemma}\label{lem:cellinj}
Let $\tau$ be a $n$-cell, $n\in \N$. Then
${\widetilde F}^n$ is a homeomorphism of $\widetilde \tau$ onto $\widetilde \sigma$, where $\sigma=F^n(\tau)$. In particular, $\widetilde F^n$ is cellular for $(\widetilde \mathcal{D}^n, \widetilde \mathcal{D}^0)$.
\end{lemma}
\begin{proof} Note that $F^n$ is a homeomorphism of $\tau$ onto $\sigma$. Hence
$$\widetilde F^n(\widetilde \tau)= (\widetilde F^n\circ \pi)(\tau)=(\pi\circ F^n)(\tau)=\widetilde \sigma$$
showing that ${\widetilde F}^n$ maps $\widetilde \tau$ onto $\widetilde \sigma$.
So it remains to show the injectivity of ${\widetilde F}^n$ on $\widetilde \tau$, or equivalently, that if $x,y\in \tau$ and $F^n(x)\sim F^n(y)$, then $x\sim y$. Since every vertex and every edge is contained in a tile, we may also assume that $\tau$ is an $n$-tile.
If $x,y\in \tau$ and $F^n(x)\sim F^n(y)$, then
we can pick sequences $\{X^k\}$ and $\{Y^k\}$ in $ \mathcal{S}$ such that $X^n=Y^n=\tau$ and $x\in \bigcap_kX^k$, $y\in \bigcap_k Y^k$.
Then $F^n(X^{k+n})$ and $F^n(Y^{k+n})$ are $k$-tiles for $k\in \N_0$. Moreover, the sequences $\{F^n(X^{k+n})\}$ and
$\{F^n(Y^{k+n})\}$ are in $\mathcal{S}$, and
$F^n(x)\in \bigcap_k F^n(X^{k+n})$, $F^n(y)\in \bigcap_k F^n(Y^{k+n})$. Since $F^n(x)\sim F^n(y)$,
this implies that $F^n(X^{k+n})\cap F^n(Y^{k+n})\ne \emptyset $ for all $k\in \N_0$. Since $X^{k+n}, Y^{k+n}\subset \tau$ for $k\ge 0$ and $F^n|\tau$ is injective, we conclude that $X^{k+n}\cap Y^{k+n}\ne \emptyset$ for $k\ge 0$. Since $X^n=Y^n=\tau$, we also have
$X^k=Y^k$ for $k=0, \dots , n-1$. Hence $X^k\cap Y^k\ne \emptyset$ for all $k\ge 0$. Lemma~\ref{lem:erel}~(ii) then shows that $x\sim y$ as desired.
The fact that $\widetilde F^n$ is cellular for $(\widetilde \mathcal{D}^n, \widetilde \mathcal{D}^0)$ follows from the first part of the proof and the fact that $F^n$ is cellular for $(\mathcal{D}^n, \mathcal{D}^0)$.
\end{proof}
\bigskip\noindent
{\bf The auxiliary homeomorphisms $h_0$ and $h_1$.}
We now want to define certain homeomorphisms
$h_0,h_1\: S^2\rightarrow \widetilde S^2$ that make the following diagram commutative:
\begin{equation*}
\xymatrix{
S^2 \ar[r]^{h_1} \ar[d]_F & \widetilde S^2 \ar[d]^{\widetilde F}
\\
S^2 \ar[r]^{h_0} & \widetilde S^2.
}
\end{equation*}
For the definition of $h_0$ recall that
$S^2$ is the union of two $0$-tiles $X^0_1$ and $X^0_2$ with common boundary $\mathcal{C}$.
The Jordan curve $\mathcal{C}$ is further decomposed into $k=\#\operatorname{post}(F)\ge 3$ $0$-edges and $0$-vertices.
If we consider the images of these cells of order $0$ under the quotient map $\pi$, then by Lemma~\ref{lem:cdcd} we get a cell decomposition $\widetilde
\mathcal{D}^0$ of $\widetilde S^2$. It contains two tiles $\widetilde X^0_1$ and $\widetilde X^0_2$ whose common boundary is the Jordan curve
$\widetilde \mathcal{C}=\pi(\mathcal{C})$ (this follows from the second statement in Lemma~\ref{lem:qcells}), and $k$ distinct vertices and edges on
$\widetilde \mathcal{C}$. There are no other cells in $\widetilde \mathcal{D}^0$.
By constructing successive extensions of maps
sending the $i$-skeleton of the cell decomposition $\mathcal{D}^0$ to the $i$-skeleton
of $\widetilde \mathcal{D}^0$ for $i=0,1,2$ (as in the proof of the first part of
Proposition~\ref{prop:thurstonex}), one sees that there
exists a (non-unique) homeomorphism $h_0\:S^2\rightarrow \widetilde S^2$ such that
$$ h_0(\tau)=\widetilde \tau$$ for all $0$-cells $\tau$. In particular, if $v$ is a $0$-vertex (i.e., a point in $\operatorname{post}(F)$), then $h_0(v)=\pi(v)$. We orient $\widetilde S^2$ so that $h_0$ is orientation-preserving.
Now let $\tau$ be an arbitrary $1$-cell in $S^2$. Then $F(\tau)$ is a $0$-cell, and by Lemma~\ref{lem:cellinj}
the map $\widetilde F|\widetilde \tau$ is a homeomorphism of $\widetilde \tau$ onto $\widetilde {F(\tau)}={h_0(F(\tau))}$. Hence the map
$$\varphi_{\tau}:=(\widetilde F|\widetilde \tau)^{-1}\circ h_0 \circ (F|\tau)$$ is well-defined and a homeomorphism from $\tau$ onto $\widetilde \tau$.
If $x\in \tau$, then $y=\varphi_\tau(x)$ is the unique point $y\in \widetilde \tau $ with $\widetilde F(y)=h_0(F(x))$.
If $\sigma$ and $\tau$ are two $1$-cells in $S^2$
with $\sigma\subset \tau$, then
$$ \varphi_\tau|\sigma=\varphi_\sigma. $$
Indeed, if $x\in \sigma$, then $y=\varphi_\sigma(x)
\in \widetilde \sigma\subset \widetilde\tau$ and
$\widetilde F(y)=h_0(F(x))$. Hence
$\varphi_\sigma(x)=y=\varphi_\tau(x)$ by the uniqueness property of $\varphi_\tau(x)$.
If a point $x\in S^2$ lies in two $1$-cells $\tau$ and $\tau'$, then $\varphi_\tau(x)=\varphi_{\tau'}(x)$. Indeed, there exists a unique $1$-cell $\sigma$ with $x\in \inte(\sigma)$. Then
$\sigma\subset \tau \cap \tau'$ by Lemma~\ref{lem:celldecompint}~(ii), and so by what we have just seen we conclude
$$ \varphi_\tau(x)=\varphi_\sigma(x)= \varphi_{\tau'}(x). $$
This allows us to define a map $h_1\: S^2\rightarrow \widetilde S^2$ as follows. If $x\in S^2$, pick a $1$-cell $\tau$ in $S^2$ with $x\in \tau$, and set
$$h_1(x)=\varphi_\tau(x).$$ Then $h_1$ is well-defined.
\begin{lemma}\label{lem:phi}
The map $h_1\: S^2\rightarrow\widetilde S^2$ is an
orientation-preserving homeomorphism of $S^2$ onto $\widetilde S^2$ satisfying $h_0\circ F=\widetilde F\circ h_1$ and
$h_1(\tau)=\widetilde \tau$ for each cell $\tau$ in $\mathcal{D}^1$.
\end{lemma}
\begin{proof} We have $h_1|\tau=\varphi_\tau$ for each $1$-cell $\tau$. Hence $h_1$ is continuous if restricted to an arbitrary $1$-cell, and hence continuous on $S^2$. The definition of $h_1$ and $\varphi_\tau$ imply that $h_0\circ F=\widetilde F\circ h_1$ and
$h_1(\tau)=\widetilde \tau$ for each $1$-cell $\tau \subset S^2$. We want to show next that $h_1$ is a
homeomorphism onto $\widetilde S^2$.
Surjectivity is clear, because $\widetilde S^2$ is equal to the union of all sets $\widetilde \tau=h_1(\tau)$, where $\tau$ runs through all $1$-cells.
To show injectivity, let $y\in \widetilde S^2$ be arbitrary. There exists a $1$-cell $\tau$ of minimal
dimension such that $y\in \widetilde \tau$. If $\sigma$ is any other $1$-cell with $y\in \widetilde \sigma$, then
$\tau\subset \sigma$. To see this, first note that by Lemma~\ref{lem:nonewint} we have $y\in \widetilde \tau \cap \widetilde \sigma =\widetilde {\tau\cap \sigma}$. Now if $\tau$ is not contained in $\sigma$, then it follows from Lemma~\ref{lem:celldecompint}~(i) that the set
$\tau \cap \sigma$ is a union of $1$-cells of dimension strictly less than the dimension of $\tau$. The image of one of these $1$-cells under the quotient map $\pi$ has to contain $y$, contradicting the definition of $\tau$.
Since $y\in \widetilde \tau =h_1(\tau)$, there exists a point $x\in \tau$ with $h_1(x)=y$.
We claim that $y$ has no other preimage under $h_1$. Indeed, suppose that $x'\in S^2$ and
$h_1(x')=y$. Pick a $1$-cell $\sigma$ with
$x'\in \sigma$. Then $y\in h_1(\sigma)=\widetilde \sigma$, and so $\tau\subset \sigma$.
Hence $x\in \sigma$. The map $h_1|\sigma=\varphi_\sigma$ is injective on $\sigma$. Since
$x,x'\in \sigma$ and $h_1(x)=y=h_1(x')$,
we conclude $x=x'$ as desired. The injectivity of $h_1$ follows.
It remains to show that $h_1$ is orientation-preserving. Pick some positively-oriented flag $(c_0,c_1,c_2)$ in $\mathcal{D}^0$. Since $\mathcal{D}^1$ is a refinement of $\mathcal{D}^0$, there exists a subflag in $\mathcal{D}^1$, i.e., a flag $(d_0,d_1, d_2)$ in $\mathcal{D}^1$ with
$d_i\subset c_i$ for $i=0,1,2$. Then $(d_0,d_1, d_2)$
is also positively-oriented. The image of
$(c_0,c_1,c_2) $ under $h_0$ is the flag
$(\widetilde c_0, \widetilde c_1,\widetilde c_2) $.
This flag is positively-oriented, because $h_0$ is orientation-preserving. The image of
$(d_0,d_1, d_2)$ under $h_1$ is the flag
$(\widetilde d_0, \widetilde d_1,\widetilde d_2) $. This is a subflag of $(\widetilde c_0, \widetilde c_1,\widetilde c_2)$ and hence positively-oriented.
It follows that $h_1$ maps one positively-oriented flag to a positively-oriented flag. Thus $h_1$ is
orientation-preserving.
\end{proof}
We are now ready to prove the main result of this section.
\begin{proof}[Proof of Proposition~\ref{prop:combexp}] Consider the map $\widetilde F\: \widetilde S^2\rightarrow \widetilde S^2$ defined earlier.
Since $\widetilde F=h_0\circ F\circ h_1^{-1}$,
the map $F$ is a branched covering map on $S^2$, and
$h_1$ and $h_0$ are orientation-preserving homeomorphisms, it follows that the map
$\widetilde F$ is a branched covering map on
$\widetilde S^2$. Moreover, we have
$\operatorname{crit}(\widetilde F)=h_1(\operatorname{crit}(F))$.
If $v$ is a $1$-vertex, then $\{v\}$ is a $1$-cell of dimension $0$, and so $h_1(\{v\})=\widetilde { \{v\}}=\{\pi(v)\}$. Hence $h_1(v)=\pi(v)$.
Since $F(\operatorname{crit}(F))\subset\operatorname{post}(F)$, every point in $\operatorname{crit}(F)$ is a $1$-vertex.
It follows that
\begin{eqnarray*}\operatorname{post} (\widetilde F)&=&\bigcup_{n\in \N}\widetilde F^n(\operatorname{crit}(\widetilde F))
=\bigcup_{n\in \N}\widetilde F^n(h_1(\operatorname{crit}(F)))\\
&=&\bigcup_{n\in \N}\widetilde F^n(\pi(\operatorname{crit}(F)))
=\bigcup_{n\in \N}(\pi\circ F^n)(\operatorname{crit}(F))=\pi(\operatorname{post}(F)).
\end{eqnarray*}
In particular, $\operatorname{post} (\widetilde F)$ is a finite set which implies that $\widetilde F$ is a Thurston map.
The Jordan curve $\widetilde {\mathcal{C}}=\pi(\mathcal{C})$ satisfies
$$\operatorname{post} (\widetilde F)=\pi(\operatorname{post}(F))\subset \pi(\mathcal{C})=\widetilde {\mathcal{C}}. $$
Since $F(\mathcal C)\subset \mathcal C$, we also have
$$ \widetilde F(\widetilde {\mathcal{C}})=
(\widetilde F\circ \pi)(\mathcal{C})=(\pi\circ F)
(\mathcal{C})\subset \pi(\mathcal{C})=
\widetilde {\mathcal{C}}. $$
This shows that $\widetilde {\mathcal{C}}$ is invariant
with respect to $ \widetilde F$ and contains the set of postcritical points of
$ \widetilde F$.
The Jordan curve $\mathcal{C}$ is the union of the $0$-edges $e$. By definition of $h_0$ we have
$h_0(e)=\widetilde e=\pi(e)$ for each such $0$-edge $e$ and hence $h_0(\mathcal{C})=\pi(\mathcal{C})=
\widetilde {\mathcal{C}}$.
Since $\mathcal{D}^1$ is a refinement of $\mathcal{D}^0$, the curve $\mathcal{C}$ can also be represented as a union of
$1$-edges. By Lemma~\ref{lem:phi} we have
$h_1(e)=\widetilde e=\pi(e)$ for each $1$-edge $e$, and hence
$h_1(\mathcal{C})=\pi(\mathcal{C})=
\widetilde {\mathcal{C}}$. Finally, every point in $\operatorname{post}(F)$ is both a $0$-vertex and a $1$-vertex.
This implies that $h_0(v)=\pi(v)=h_1(v)$ for all $v\in \operatorname{post}(F)$. So $h_0$ and $h_1$ are orientation-preserving homeomorphism $S^2\rightarrow \widetilde S^2$ that
map the Jordan curve $\mathcal{C}$ to the same image $\widetilde {\mathcal{C}}$ and agree on the finite set $\operatorname{post}(F)\subset \mathcal{C}$. It follows that $h_1^{-1}\circ h_0$ is cellular for $(\mathcal{D}^0,\mathcal{D}^0)$. As in the proof of the second part of Lemma~\ref{prop:thurstonex} this implies that
$h_1^{-1}\circ h_0$ is isotopic to $\id_{S^2}$ rel.~$\operatorname{post}(F)$, and hence $h_0$ and $h_1$ are isotopic rel.\ $\operatorname{post}(F)$. This implies that $F$ and $\widetilde F$ are Thurston equivalent.
It remains to show that $\widetilde F$ is expanding. Since $\widetilde \mathcal{C}
\subset \widetilde S^2$ is an $\widetilde F$-invariant Jordan curve with
$\operatorname{post} (\widetilde F)\subset \widetilde \mathcal{C}$, we can do this by verifying the condition in Lemma~\ref{lem:charexpint}.
Note that $\mathcal{D}^0(\widetilde F, \widetilde \mathcal{C})=\widetilde \mathcal{D}^0$. Moreover, since $\widetilde F^n$ is cellular for $(\widetilde \mathcal{D}^n, \widetilde \mathcal{D}^0)$, it follows from Lemma~\ref{lem:pullback} that $\mathcal{D}^n( \widetilde F, \widetilde \mathcal{C})=
\widetilde \mathcal{D}^n$ for all $n\in \N_0$.
So the $n$-tiles for $(\widetilde F,\widetilde \mathcal{C})$ are precisely
the sets $\widetilde X$, where $X$ is an $n$-tile on $S^2$ for $(F,\mathcal{C})$.
Moreover, for two tiles $X$ and $Y$ for $(F, \mathcal{C})$ we have $X\subset Y$ if and only if $\widetilde X\subset \widetilde Y$ (see Lemma~\ref{lem:nonewint}).
So if
$Z^0\supset Z^1\supset Z^2\dots$ is a nested sequences of $n$-tiles $Z^n$ for $(\widetilde F,\widetilde \mathcal{C})$, then we can find a corresponding nested sequences $\{X^n\}\in \mathcal{S}$ such that $Z^n=\widetilde X^n$ for all $n\in \N_0$. Thus, in order to establish that
$\widetilde F$ is expanding, it suffices to show that if $\{X^n\}\in \mathcal{S}$ is arbitrary, then the intersection
$\bigcap_n \widetilde X^n$ contains precisely one point.
It is clear that this intersection contains at least one point.
We argue by contradiction and assume that $\bigcap_n \widetilde X^n$ contains more that one point, or equivalently, there exist two distinct (and hence disjoint) equivalence classes $M$ and $N$ with respect to $\sim$ such that $M^n:=M\cap X^n\ne \emptyset$ and $N^n:=N\cap X^n \ne \emptyset$ for all $n\in \N_0$.
Since equivalence classes and tiles are compact, in this way we get descending sequences $M^0\supset M^1\supset \dots$ and $N^0\supset N^1\supset\dots$ of nonempty and compact sets. Hence
the sets $\bigcap_n M^n=M\cap \bigcap_n X^n$ and
$\bigcap_n N^n=N\cap \bigcap_n X^n$ are nonempty. So there exists points
$x\in M\cap \bigcap_n X^n$ and
$y\in N\cap \bigcap_n X^n$. Since $x$ and $y$ lie in different equivalence classes, they are not equivalent. On there other hand, we have $x,y\in
\bigcap_n X^n$ and $\{X^n\}\in \mathcal{S}$. Hence $x\sim y$ by Lemma~\ref{lem:erel}~(ii). This is a contradiction showing that $\widetilde F$ is expanding.
\end{proof}
\begin{cor}\label{cor:combexp1}
Let $F\:S^2\rightarrow S^2$ be a Thurston map that has an invariant Jordan curve $\mathcal{C}\subset S^2$ with $\operatorname{post}(F)\subset \mathcal{C}$. If $F$ is combinatorially expanding for $\mathcal{C}$, then there exists an orientation-preserving
homeomorphism $\phi\:S^2\rightarrow S^2$ that is isotopic to the identity on $S^2$ rel.~$\operatorname{post}(F)$ such that $\phi(\mathcal{C})=\mathcal{C}$ and $G=\phi\circ F$ is an expanding Thurston map.
\end{cor}
Note that $\operatorname{post}(G)=\operatorname{post}(F)$ and $G(\mathcal{C})=\mathcal{C}$. So the corollary says
that if a Thurston map $F$ is combinatorially expanding for an invariant Jordan curve $\mathcal{C}\subset \operatorname{post}(F)$, then by ``correcting" the map by post-composing with a suitable homeomorphism we can obtain an expanding Thurston map $G$ with the same invariant curve and the same set of postcritical points.
\begin{proof} Using the notation of Proposition~\ref{prop:combexp} define $\phi=h_1^{-1}\circ h_0$.
Then $\phi$ is isotopic to the identity on $S^2$ rel.~$\operatorname{post}(F)$ and $\phi(\mathcal{C})=\mathcal{C}$. Moreover,
$G=\phi\circ F=h_1^{-1} \circ \widetilde F \circ h_1$, and so $G$ is topologically conjugate to the expanding Thurston map $\widetilde F$, and hence itself an expanding Thurston map.
\end{proof}
\begin{cor}\label{cor:combexp2}
Let $(\mathcal{D}^1, \mathcal{D}^0,L )$ be a two-tile subdivision rule on $S^2$ suppose that it can be realized by a Thurston map $F\:S^2\rightarrow S^2$ that is combinatorially
expanding for the Jordan curve $\mathcal{C}$ of $\mathcal{D}^0$ and for which its set $\operatorname{post}(F)$ is equal to the set ${\bf V}^0$ of $0$-vertices. Then the subdivision rule with the labeling can be realized by an expanding Thurston map $G$. \end{cor}
In general we only have $\operatorname{post}(F)\subset{\bf V}^0$. The stronger assumption $\operatorname{post}(F)={\bf V}^0$ is not very essential in the corollary. We added it for convenience in order to avoid some cumbersome technicalities in the proof.
\begin{proof} Let $G$ and $\phi$ be as in Corollary~\ref{cor:combexp1}. Then $\operatorname{post}(F)=\operatorname{post}(G)={\bf V}^0$, and so $\mathcal{D}^0=\mathcal{D}^0(F,\mathcal{C})=\mathcal{D}^0(G,\mathcal{C})$. Since $\phi$ is isotopic to the identity, this map is orientation-preserving. Since $\phi(\mathcal{C})=\mathcal{C}$ and
$\phi$ is the identity on $\operatorname{post}(F)=\operatorname{post}(G)$, the map $\phi$ is cellular for $(\mathcal{D}^0, \mathcal{D}^0)$.
Since $F$ is cellular for $(\mathcal{D}^1,\mathcal{D}^0)$, the
map $G=\phi \circ F$ is cellular for $(\mathcal{D}^1, \mathcal{D}^0)$ and we have $G(c)=F(c)$ for each
cell $c\in \mathcal{D}^1$. Since $F$ realizes the subdivision rule with the given labeling, this shows that $G$ is also a realization. Since $G$ is expanding, the claim follows.
\end{proof}
Proposition \ref{prop:combexp} shows that for a Thurston map with
an invariant Jordan curve containing all its postcritical points,
combinatorial expansion is \emph{sufficient} for the existence of a
Thurston equivalent map that is expanding. The question arises whether
combinatorial expansion is \emph{necessary} as well. The answer
is negative, as the following example of a Thurston map $f$ will show. The map $f$ has an invariant Jordan curve $\mathcal{C}$ containing all its postcritical
points. It is not combinatorially expanding for $\mathcal{C}$ (and hence not
expanding), yet equivalent to an expanding Thurston map $g$.
\ifthenelse{\boolean{nofigures}}{}{
\begin{figure}
\centering
\begin{overpic}
[width=12cm,
tics=20]{exp_notcexp.eps}
\put(63,57){$f$}
\put(54,18){$g$}
\put(5,41){$\mathcal{C}$}
\put(18,3){$\widetilde{\mathcal{C}}$}
\put(80,41){$\mathcal{C}$}
\put(22.5,37){$h_1$}
\put(86,37){$h_0$}
\end{overpic}
\caption{The map $f$ is not combinatorially expanding, but equivalent to the expanding map $g$.}
\label{fig:exp_notcexp}
\end{figure}
}
\begin{ex}
\label{ex:exp_notcexp}
The map $f\colon S^2\to S^2$ is illustrated in the top of Figure
\ref{fig:exp_notcexp}.
It has four postcritical points, which are the vertices of the
pillow shown in the top right. The front shows the white $0$-tile,
the back is the black $0$-tile. The subdivision of the $0$-tiles is
indicated in the top left of the figure. Here we have cut the pillow
along three $0$-edges. The left shows the subdivision of the white
$0$-tile, the right the subdivision of the black $0$-tile. On
the left the preimages of one postcritical point (a $0$-vertex)
are given. So we only show the labels of $1$-vertices that the labeling sends
to the $0$-vertex on the bottom left of the
pillow.
The boundary of the pillow is the Jordan curve $\mathcal{C}$ containing all
postcritical points of $f$. It is invariant under $f$, but the map
$f$ is not combinatorially expanding for $\mathcal{C}$: for arbitrarily large $n$ there will
always be an $n$-tile (contained in the white $0$-tile) joining the
$0$-edges on the left and on the right.
We will see that $f$ is Thurston equivalent to the map $g\colon
\widetilde{S}^2\to \widetilde{S}^2$,
indicated in the bottom of Figure \ref{fig:exp_notcexp}. The map $g$
is a Latt\`{e}s map, obtained similarly as the example in
Section~\ref{sec:Lattes} as a quotient of $\psi \colon \C \to \C$, $\psi(z)=
3z$. It is clear that $g$ is expanding.
We fix a homeomorphism $h_0\colon S^2 \to \widetilde{S}^2$ that maps
the boundary of the pillow of $f$ (i.e., the curve $\mathcal{C}$) to the
boundary of the pillow of $g$, and the postcritical points of $f$
to the postcritical points of $g$. Let $\mathcal{D}^1$ and $\widetilde{\mathcal{D}}^1$
denote the cell decompositions of $S^2$ and $\widetilde{S}^2$, respectively,
into $1$-cells as indicated on the left of Figure
\ref{fig:exp_notcexp}.
As in the proof of
Proposition~\ref{prop:thurstonex} there is a homeomorphism $h_1\colon S^2\to
\widetilde{S}^2$ that maps the curve $\mathcal{C}\subset S^2$ to the curve
$\widetilde{\mathcal{C}}\subset \widetilde{S}^2$ indicated in the figure, is
cellular with respect to $(\mathcal{D}^1,\widetilde{\mathcal{D}}^1$), and satisfies
\begin{equation*}
g\circ h_1 = h_0 \circ f.
\end{equation*}
Obviously, the map $h_1$ is isotopic to $h_0$ rel.\ $\operatorname{post}(f)$. Thus
$f$ and $g$ are Thurston equivalent.
Note that $g$ has an invariant curve $\widetilde{\mathcal{C}}'$ (different
from $\widetilde{\mathcal{C}}$ of course) that contains all its postcritical
points.
\end{ex}
\section{Auxiliary results on graphs}
\label{sec:graphs}
\noindent
The main result in this section is Lemma~\ref{prop:isotopicpath} which gives a criterion when a
Jordan curve $\mathcal{C}$ in a $2$-sphere $S^2$ can be isotoped relative to a finite set $P\subset \mathcal{C}$ into the $1$-skeleton of a given cell decomposition $\mathcal{D}$ of $S^2$. We first discuss some facts about graphs that are
needed in the proof.
Since all the graphs we consider will be embedded in a $2$-sphere, we base the concept of a graph on a topological definition rather than a combinatorial one as usual.
A {\em (finite) graph}\index{graph} is a compact Hausdorff space $G$
equipped with a fixed cell decomposition $\mathcal{D}$ such that $\dim(c)\le
1$ for all $c\in \mathcal{D}$. The cells $c$ in $\mathcal{D}$ of dimension $1$ are
called the {\em edges} of the graph, and the points $v\in G$ such that
$\{v\}$ is a $0$-dimensional cell in $\mathcal{D}$ the {\em vertices} of the
graph.
An {\em oriented edge} $e$ in a graph is an edge, where one of the vertices in $\partial e$ has been chosen as the {\em initial point} and the other
vertex as the {\em terminal point} of $e$. An {\em edge
path}\index{edge path} in $G$
is a finite sequence $\alpha$ of oriented edges $e_1, \dots, e_N$
such that
the terminal point of $e_i$ is the initial point of $e_{i+1}$ for $i=1, \dots, N-1$.
We denote by $|\alpha|=e_1\cup \dots \cup e_N$ the underlying set of the edge path.
The edge path $\alpha$ {\em joins} the vertices $a,b\in G$ if the initial point of $ e_1$ is $a$ and the terminal point of $e_N$ is $b$. The number $N$ is called the {\em length} of the edge path. The edge path is called {\em simple} if
$e_i$ and $e_j$ are disjoint for
$1\le i<j\le N$ and $j-i\ge2$, and $e_i\cap e_j$ consists of precisely one point (the terminal point of $e_i$ and initial point of $e_j$) when $j=i+1$.
If the edge path $\alpha$ is simple, then $|\alpha|$
is an arc. The edge path is called a {\em loop} if the terminal point of $e_N$ is the initial point of $e_1$.
A graph is connected (as a topological space) if and only if any two vertices $a,b\in G$, $a\ne b$, can be joined by an edge path.
The {\em combinatorial distance} of two vertices $a$ and $b$ in a connected graph $G$ is defined as the minimal length of all edge paths joining the points (interpreted as $0$ if $a=b$). The vertices $a,b\in G$ are called {\em neighbors} if their combinatorial distance is equal to $1$, i.e., if there exists an edge $e$ in $G$ whose endpoints are $a$ and $b$. A vertex $q\in G$ is called a {\em cut point}
of $G$ if $G\setminus\{q\}$ is not connected. A vertex $q\in G$ is not a cut point if and only if all vertices $a,b\in G\setminus \{q\}$, $a\ne b$, can be joined by an edge path $\alpha$ with $q\notin |\alpha|$.
\begin{lemma}
\label{lem:simple}
Let $G$ be a connected graph without cut points. Then for all vertices
$a,b,p\in G$ with $a\ne b$ there exists a simple edge path $\gamma$ in $G$ with $p\in |\gamma|$ that joins $a$ and $b$. \end{lemma}
\begin{proof}
Since $G$ is connected, there exist edge paths in $G$ joining $a$ and $b$. By removing loops from such a path if necessary, we can also obtain such an edge path in $G$ that is simple.
Among all such simple
paths, there is one that contains a vertex with minimal combinatorial distance to $p$.
More precisely, there exists a simple edge path $\alpha$ in $G$ with endpoints $a$ and $b$, and a vertex $q\in |\alpha|$ such that the combinatorial distance $k\in \N_0$ of $q$ and $p$ is minimal among all combinatorial distances between $p$ and vertices on simple paths joining $a$ and $b$. If $k=0$ then $q=p$ and we can take $\gamma=\alpha$.
We will show that the alternative case $k\ge 1$ leads to a contradiction.
By definition of combinatorial distance, there exists an edge path joining
$q$ to $p$ consisting of $k\ge 1$ edges. The second vertex
$q'$ on this path as travelling from $q$ to $p$ is a neighbor of $q$
whose combinatorial distance to $p$ is $k-1$ and hence strictly smaller that the combinatorial distance of $q$ to $p$. In particular, $q'\not \in
|\alpha|$
by choice of $q$ and $\alpha$.
We will obtain the desired contradiction if we can show that there exists a simple edge path $\sigma$ in $G$ that joins $a$ and $b$ and passes through $q'$.
\ifthenelse{\boolean{nofigures}}{}{
\begin{figure}
\centering
\begin{overpic}
[width=10cm,
tics=20]{abpLemma.eps}
\put(0,53){$a$}
\put(15,45.5){$r$}
\put(45,1){$p$}
\put(52,47){$q$}
\put(44.5,33){$q'$}
\put(97,52){$b$}
\put(30,58){$\alpha$}
\put(71,50){$\sigma$}
\put(20,35){$\beta$}
\end{overpic}
\caption{Constructing a path through $a,b,p$.}
\label{fig:abp_path}
\end{figure}
}
For the construction of $\sigma$
we apply our assumption that $G$ has no cut points; in particular, $q$ is no cut point and hence there exists
an edge path $\beta$ with $q\notin |\beta|$ that joins
$q'$ to a vertex in the (nonempty) set $A=|\alpha|\setminus \{q\}$. We may assume
that $\beta$ is simple and that the endpoint $r\ne q'$ of $\beta$ is the only point in $|\beta|\cap A$.
Moreover, we may assume that $r$ lies between $a$ and $q$ on the path $\alpha$
(the argument in the other case where $r$ lies between $q$ and $b$ is similar). Now let $\sigma$ be the edge path obtained by traveling from $a$ to $r$ along $\alpha$, then from $r$ to $q'$ along $\beta$, then from $q'$ to $q$ along an edge
(this is possible since $q$ and $q'$ are neighbors), and finally from $q$
to $b$ along $\alpha$. See the illustration
in Figure~\ref{fig:abp_path}. Then $\sigma$ is a simple edge path in $G$ that
passes through $q'$ and has the endpoints $a$ and $b$. This gives the desired contradiction.
\end{proof}
Now let $S^2$ be a $2$-sphere, and $\mathcal{D}$ be a cell decomposition of $S^2$. We denote the set of tiles, edges, and vertices in $\mathcal{D}$ by $\X$, $\E$, and ${\bf V}$, respectively. In the following the term cell, tile, etc., refers to elements in these
sets.
Let $M\subset \X$ be a set of tiles. We denote by $|M|$ its underlying set; so
$$ |M|=\bigcup_{X\in M} X. $$
The set
\begin{equation}
\label{eq:def_GM}
G_M:=\bigcup_{X\in M} \partial X
\end{equation}
admits a natural cell
decomposition consisting of all cells contained in $G_M$.
Obviously, no such cell can be a tile, so with this cell
decomposition $G_M$ is a graph.
Similarly as in Section~\ref{sec:subdivisions}, we call a sequence
of tiles $X=X_1, \dots, X_N=Y$ an {\em $e$-chain}\index{e-chain@$e$-chain}
joining $X$ and $Y$ if for $i=1, \dots, N$ there exists an edge
$e_i$ with $e_i\subset \partial X_{i-1}\cap \partial X_i$. A set $M$ of
tiles is {\em $e$-connected}\index{e-connected@$e$-connected}
if every two tiles in $M$ can be joined by an $e$-chain.
\begin{lemma}
\label{lem:nocut}
Let $M\subset \X$ be a set of tiles that
is $e$-connected. Then the graph
$G_M$ is connected and has no cut points.
\end{lemma}
\begin{proof}
Let
$a,b\in G_M$ be arbitrary vertices with $a\ne b$. We can pick tiles $X$ and $Y$ in $M$ such that $a$ is a vertex in $X$ and $b$ is a vertex in $Y$. By assumption there exists an $e$-chain $X_1, X_1, \dots, X_N$ in $M$ with
$X_1=X$ and $X_N=Y$. The vertices of a tile $X_i$ lie in
$G_M$; they subdivide the Jordan curve $\partial X_i$ such that successive vertices on $\partial X_i$ are connected by an edge
and are hence neighbors in $G_M$. An edge path
$\alpha$ in $G_M$ joining $a$ and $b$ can now be obtained as follows: starting from $a\in \partial X_0$, use edges on the boundary
of $X_0$ to find an edge path in $G_M$ that joins $p_0=a$ to a vertex $p_1$ of $X_1$. This is possible, since $X_0$ and $X_1$ have a common edge and hence at least two common vertices. Then run from $p_1$ along
edges on $\partial X_1$ to a vertex $p_2$ of $X_2$, and so on. Once we arrived at a vertex $p_k$ of $X_k$, we can reach $b$ by running from $p_k$ to $p_{k+1}=b$ along edges on $X_k$.
In this way we obtain an edge path $\alpha$ in $G_M$ that joins $a$ and $b$.
A slight refinement of this argument also shows that we can construct the
path $\alpha$ so that it avoids any given vertex $q$ in $G_M$ distinct
from $a$ and $b$. Indeed, choose $p_0=a$ as before. Since
$X_0$ and $X_1$ have at least two vertices in common, we can pick a common vertex $p_1$ of $X_0$ and $X_1$ that is distinct from $q$.
There exists an arc on $\partial X_0$ (possibly degenerate) that does not contain $q$ and joins $p_0$ and $p_1$. This arc (if non-degenerate) consists of edges and if we follow these edges, we obtain an edge path in $G_M$ that does not contain $q$ and joins $p_0$ and $p_1$. In the same way we can find an edge path
in $G_M$ that avoids $q$ and joins $p_1$ to a vertex $p_2\in \partial X_1\cap \partial X_2$, and so on. Concatenating all these edge paths we get a path $\alpha$ as desired.
This shows that $G_M$ is connected and has no cut points.
\end{proof}
\begin{lemma}
\label{lem:paththrough}
Let $M\subset \X$ be a set of tiles that is $e$-connected,
and let $a,b,p\in |M|$ be distinct vertices. Then there
exists a simple edge path $\alpha$ in $G_M$ with $p\in |\alpha|$
that joins $a$ and $b$. \end{lemma}
In particular, this applies if $M$ consists of a single $e$-chain.
\begin{proof} This follows from Lemma~\ref{lem:nocut} and
Lemma~\ref{lem:simple}. \end{proof}
\begin{lemma}
\label{lem:echain} Let $\gamma \:J\rightarrow S^2$ be a path in $S^2$ defined on a closed interval $J\subset \R$ and
$M=M(\gamma)$ be the set of tiles having nonempty intersection with
$\gamma$. Then $M$ is $e$-connected.
\end{lemma}
\begin{proof} We first prove the following claim.
If $[a,b]\subset \R$,
$\alpha\:[a,b]\rightarrow S^2$ is a path, and $X$ and $Y$ are tiles with $\alpha(a)\in X$ and $\alpha(b)\in Y$, then there exists an $e$-chain $X_1=X, X_2,\dots, X_N=Y$ such that
$X_i\cap \alpha \ne \emptyset $ for all $i=1, \dots, N$.
In the proof of this claim, we call an $e$-chain $X_1, \dots, X_N$
{\em admissible} if $X_1=X$ and $X_i\cap \alpha \ne \emptyset $ for all $i=1, \dots, N$. So we want to find an admissible $e$-chain whose last tile is $Y$.
Let $T\subset [a,b]$ be the set of all points $t\in [a,b]$ for which there exists an
admissible $e$-chain $X_1, \dots, X_N$ with $\alpha(t)\in X_N$.
We first want to show that $b\in T$.
First note that the set $T$ is closed. Indeed, suppose that $(t_k)$ is a sequence in $T$
with $t_k\to t_\infty \in [a,b]$ as $k\to \infty$. Then for each $k\in \N$ there
exists an admissible $e$-chain $X^1_1, \dots , X^k_{N_k}$ with
$\alpha(t_k)\in X^k_{N_k}$. Define $Z_k=X^k_{N_k}$ to be the last
tile in this chain. Since there are only finitely many tiles, there exists
one tile, say $Z$, among the tiles $Z_1$, $Z_2$, $Z_3$, $\dots$ that appears infinitely often in this sequence. Since $\alpha(t_k)\to \alpha(t_\infty)$, we have $\alpha(t_k)\in Z$ for infinitely many $k$, and tiles are closed, we conclude $\alpha(t_\infty)\in Z$. By definition of $Z$ there exists an admissible $e$-chain $X_1, \dots, X_N$ with $X_N=Z$. Then $\alpha(t_\infty)\in Z=X_N$, and so $t_\infty\in T$.
Obviously, $a\in T$ and so $T$ is nonempty. Since $T$ is also closed, the set $T$ has a maximum, say
$m\in [a,b]$. We have to show that $m=b$;
we will see that the assumption $m<b$ leads to a contradiction.
Consider $p=\alpha(m)$. Then there exists an admissible $e$-chain $X_1, \dots, X_N$ with $p\in Z:=X_N$.
If $p\in \inte(Z)$, then $\alpha(t)\in Z$ and so $t\in T$ for $t\in(m,b]$ close to $m$. This is impossible by definition of $m$.
If $p$ does not belong to $\inte(Z)$, then $p$ must be a boundary point
of $Z$. Suppose first that $p$ is in the interior of an edge
$e\subset \partial Z$. By Lemma~\ref{lem:specprop}~(iv) there exists precisely one tile $Z'$ distinct from $Z$ such that $e\subset \partial Z'$.
Moreover, $Z\cup Z'$ is a neighborhood of $p$, and so
points $\alpha(t)$ with $t\in (m,b]$ close to $m$ belong to $Z$ or $Z'$.
Since $Z'$ contains $p$ and hence meets $\alpha$, and $Z$ and $Z'$ share an edge, $X_1, \dots, X_N, Z'$ is an admissible $e$-chain. It follows that
$t\in T$ for $t\in (m,b]$ close to $m$. Again this is impossible by definition of
$m$.
If $p$ is a boundary point
of $Z$, but not in the interior of an edge, then $p$ is a vertex.
The tiles in the cycle of $p$ form a neighborhood of $p$,
and so a point $\alpha(t)$ for some $ t\in (m,b]$ close to $m$ will belong to a tile $Z'$
in the cycle of $p$. It follows from Lemma~\ref{lem:specprop} that any two tiles in the cycle
of a vertex can be connected by an $e$-chain consisting of tiles in the cycle.
Hence there exits an $e$-chain $Z=Z_1, \dots, Z_K=Z'$ such that
$p\in Z_j$ for $j=1, \dots, K$.
In particular, $\alpha\cap Z_j\ne \emptyset $ for $j=1, \dots, K$, and so $X_1, \dots, X_N=Z=Z_1, \dots, Z_K=Y'$ is an admissible $e$-chain.
Since $\alpha(t)\in Z'=Z_K$, we have $t\in T$, again a contradiction.
We have exhausted all possibilities proving that $b\in T$ as desired. So there exists an admissible $e$-chain $X_1=X, \dots, X_N$ with $\alpha(b)\in X_N$. If $X_N=Y$, then we are done. If $X_N\ne Y$, then $\alpha(b)\in \partial X_N\cap \partial Y$, and so $\alpha(b)$ is an interior point of an edge $e$ with $e\subset \partial X_N\cap \partial Y$, or $\alpha(b)$ is a vertex. As in the first part of the proof, one then can extend the admissible $e$-chain $X_1=X, \dots, X_N$ to obtain an admissible $e$-chain whose last tile is $Y$. The claim follows.
The claim now easily implies the statement of the lemma. Indeed, pick
a point $a\in J$ and
fix a tile
$X\in \X$ with $\gamma(a)\in X$. Then $X\in M=M(\gamma)$. If $Y\in M$ is arbitrary, then there exists $b\in J$ with $\gamma(b)\in Y$. If $b\ge a$,
then we apply the claim to the path $\alpha=\gamma|[a,b]$, and if
$b\le a$ to the path $\alpha=\gamma|[b,a]$. This shows that we can
find an $e$-chain in $M$ that joins $Y\in M$ to the fixed tile
$X$. Hence any two tiles in $M$ can be joined by an $e$-chain in $M$.
\end{proof}
\medskip For the formulation of the next statement, we need a slight generalization of Definition~\ref{def:connectop}.
Let $\mathcal{C}\subset S^2$ be a Jordan curve, and $P\subset \mathcal{C}$ be a finite set with
$\#P\geq 3$. The points in $P$ divide $\mathcal{C}$ into subarcs
that have endpoints in $P$, but whose interiors
are disjoint from $P$.
We say that a set $K\subset S^2$ {\em joins opposite sides of $(\mathcal{C},P)$}\index{joining opposite sides}
if $k=\#\operatorname{post}(f)\ge 4$ and $K$ meets two of these arcs that are non-adjacent (i.e., disjoint), or if $k=\#\operatorname{post}(f)=3$ and $K$ meets all of these
arcs (in this case there are three arcs).
In the following proposition and its proof, metric notions refer to some fixed base metric on $S^2$.
\begin{lemma
\label{prop:isotopicpath}
Let $\mathcal{C}\subset S^2$ be a Jordan curve, and $P\subset \mathcal{C}$ a finite set with $k=\#P\geq 3$.
Then there exists
$\epsilon_0 >0$ with the following property:
Suppose that $\mathcal{D}$ is a cell decomposition of $S^2$ with vertex set $\mathbf{V}$ and $1$-skeleton $E$.
If $P\subset \mathbf{V}$ and
$$\max_{c\in \mathcal{D}} \diam (c)< \epsilon_0,$$ then there exists a Jordan curve $\mathcal{C}'\subset E$ that is isotopic to $\mathcal{C}$ rel.\ $P$ and so that
no tile in $\mathcal{D}$ joins opposite sides of $(\mathcal{C}', P)$. \end{lemma}
\begin{proof} Fix an orientation of $\mathcal{C}$ and let $p_1, \dots, p_k$ be the points in $P$ in cyclic order on $\mathcal{C}$. The
points in $P$ divide $\mathcal{C}$ into subarcs $\mathcal{C}_1, \dots, \mathcal{C}_k$ such that for $i=1, \dots, k$ the arc $\mathcal{C}_i$ has the endpoints $p_i$ and $p_{i+1}$ and has interior disjoint from $P$. Here and in the following
the index $i$ is understood modulo $k$, i.e., $p_{k+1}=p_1$, etc.
Note that $\mathcal{C}_i\cap \mathcal{C}_{i+1}=\{p_{i+1}\}$ for $i=1, \dots, k$.
There exists a number $\delta_0>0$ such that no set $K\subset S^2$ with $\diam(K) <\delta_0$ joins opposite sides of $(\mathcal{C},P)$ (this can be seen as in the discussion after \eqref{defdelta}).
Now choose $\delta>0$ as in Proposition~\ref{prop:isotreln} for $J=\mathcal{C}$ and $n=k$. We may assume that $3\delta<\delta_0$.
We break up $\mathcal{C}$ into
subarcs
\begin{equation}
\label{eq:alph_gamma}
\alpha_1, \gamma_1, \alpha_2, \gamma_2, \dots, \gamma_k,
\alpha_1,
\end{equation}
arranged in cyclic order on $\mathcal{C}$,
such that $p_i$ is an interior point of $\alpha_i$ and we have
$\alpha_i\subset
B(p_i, \delta/2)$ for each $i=1,\dots, k$. The arcs in (\ref{eq:alph_gamma}) have disjoint
interiors, and two arcs have an endpoint in common if and only if
there are adjacent in this cyclic order in which case they share one
endpoint. So each ``middle piece''
$\gamma_i$ does not contain any point from $P$ and is contained in the interior of
$\mathcal{C}_i$.
We choose $0<\epsilon_0<\delta/4$ so small that the distance
between non-adjacent arcs in (\ref{eq:alph_gamma})
is $\ge 10 \epsilon_0$ and so that
$$ \operatorname{dist}(p_i, \gamma_{i-1}\cup \gamma_i)\ge 10 \epsilon_0$$
for $i=1, \dots, k$.
\smallskip
Now suppose we have a cell decomposition $\mathcal{D}$ of $S^2$ such that $P$ is contained in the vertex set $\mathbf{V}$ of $\mathcal{D}$ and
$$\max_{c\in \mathcal{D}} \diam (c) < \epsilon_0.
$$
Our
goal is to find a Jordan curve $\mathcal{C}'\subset S^2$ consisting of arcs
$\mathcal{C}_i'$ that are unions of edges, have endpoints $p_i$ and
$p_{i+1}$, and satisfy
$$\mathcal{C}'_i\subset \mathcal{N}^\delta(\mathcal{C}_i) $$ for
$i=1, \dots, k$.
\smallskip
Let $\mathbf{A}_i$ be the set of all tiles intersecting $\alpha_i$ and
$\mathbf{C}_i$ be the set of all tiles intersecting $\gamma_i$, for
all $i=1,\dots, k$.
Recall that for a given set of tiles $M$, we denote by $\abs{M}$ the
union of tiles in $M$. Let $A_i:= \abs{\mathbf{A}_i}$ and $C_i=
\abs{\mathbf{C}_i}$.
Note that
\begin{equation*}
A_i \subset \mathcal{N}^{\epsilon_0}(\alpha_i) \quad
\text{and}
\quad
C_i \subset \mathcal{N}^{\epsilon_0}(\gamma_i).
\end{equation*}
Furthermore
\begin{eqnarray}
\label{eq:ABCDinNd}
A_i\cup C_i\cup A_{i+1}&\subset&
\mathcal{N}^{\epsilon_0}(\alpha_i)
\cup \mathcal{N}^{\epsilon_0}(\gamma_i)
\cup \mathcal{N}^{\epsilon_0}(\alpha_{i+1}) \notag
\\
&\subset &
B(p_i, \delta) \cup \mathcal{N}^{\epsilon_0}(\gamma_i) \cup
B(p_{i+1},\delta) \\ \notag
&\subset& \mathcal{N}^\delta(\mathcal{C}_i),
\end{eqnarray}
and the natural cyclic order of these sets is
\begin{equation}\label{setlist}
A_1, C_1, A_2, C_2,\dots,A_k, C_k, A_1.
\end{equation}
By choice of $\epsilon_0$ we have that if two of the sets in (\ref{setlist})
are not adjacent in the cyclic order, then their distance is $\ge
8\epsilon_0$ and so their intersection is empty. Moreover, for $i=1,
\dots, k$ the only one of these sets that contains $p_i$ is $A_i$. The
construction that follows is illustrated in
Figure~\ref{fig:pf_des_Grauens}. Here the two large dots represent two
points $p_i,p_{i+1}$ and the thick line the curve $\mathcal{C}$.
\ifthenelse{\boolean{nofigures}}{}{
\begin{figure}
\centering
\vspace{0.3cm}
\begin{overpic}
[width=12cm,
tics=20]{pf_des_Grauens.eps}
\put(14,33.5){${\scriptstyle B(p_i, \delta/2)}$}
\put(84,33.5){${\scriptstyle B(p_{i+1}, \delta/2)}$}
\put(8,15){${\scriptstyle p_i}$}
\put(89,15){${\scriptstyle p_{i+1}}$}
\put(50,33){${\scriptstyle \mathcal{C}_i}$}
\put(29,22){${\scriptstyle v_i}$}
\put(68,23){${\scriptstyle v_i'}$}
\put(19.5,2){${\scriptstyle v_{i-1}'}$}
\put(80.5,0){${\scriptstyle v_{i+1}}$}
\put(9,25){${\scriptstyle A_i}$}
\put(89,24){${\scriptstyle A_{i+1}}$}
\put(22,21.6){${\scriptstyle a_i}$}
\put(74,21.4){${\scriptstyle a_{i+1}}$}
\put(46,15){${\scriptstyle c_i}$}
\put(42,23){${\scriptstyle C_i}$}
\end{overpic}
\caption{Construction of the curve $\mathcal{C}'$.}
\label{fig:pf_des_Grauens}
\end{figure}
}
Consider the graphs $G_{A_i}$, $G_{C_i}$ of all edges in $A_i$, $C_i$,
see \eqref{eq:def_GM}. Note that there is at least one tile contained
in both $\mathbf{A}_i$ and $\mathbf{C}_i$, namely any tile containing
the point where $\alpha_i, \gamma_i$ intersect. Thus $G_{A_i}$ and
$G_{C_i}$, similarly $G_{C_i}$ and $G_{A_{i+1}}$, are not disjoint.
Pick a simple edge path $c_i'$ in $G_{C_i}$ connecting $A_i,
A_{i+1}$. Let $c_i$ be a subarc of $\abs{c_i'}$
whose initial point $v_i$ lies in $A_i$, whose
terminal point $v'_i$ lies $A_{i+1}$, and that has no other points
in common with $A_i$ and $A_{i+1}$.
The points $v_i$ and $v'_i$ are vertices. To see this, suppose that
$v_i$, for example, is not a vertex. Then, since $v_i\in c_i'$ and
$c_i'$ is an edge path, there exists an edge $e\subset c_i$ such that
$v_i\in \inte(e)$. But then necessarily $e\subset A_i$ and $v_i$ cannot
be the only point of $c_i$ that belongs to $A_i$.
\smallskip
Note that $v'_{i-1}\in C_{i-1}$ and $v_i\in C_i$ are distinct vertices in $A_i$.
Recall that $p_i\in A_i$.
Thus it follows from Lemma~\ref{lem:echain} and
Lemma~\ref{lem:paththrough} that there exists an arc $a_i\subset A_i$ with
$p_i\in a_i$ that
consists of edges and has the endpoints $v'_{i-1}$ and $v_i$.
Since $p_i\notin C_{i-1}\cup C_i$, we have $v'_{i-1}, v_i\ne p_i$, and so $p_i\in \inte(a_i)$.
If we arrange the arcs $a_i$ and $c_i$ in cyclic order
$$ a_1, c_1, a_2, c_2\dots, a_k, c_k, a_1,$$
then two of these arcs have nonempty intersection
if and only if they are adjacent in the order. If two arcs are
adjacent, then their intersection consists of a common endpoint.
Therefore, the set
$$\mathcal{C}':=a_1\cup c_1 \cup a_2\cup c_2 \cup\dots \cup a_k \cup c_k$$
is a Jordan curve that passes through the
points $p_1, \dots, p_k$. Moreover, $\mathcal{C}'$ consists of edges and is hence contained in the $1$-skeleton $E$ of $\mathcal{D}$.
By construction each vertex $p_i$ is an interior point of the arc
$a_i$. Thus it divides $a_i$ into two subarcs $a_i^-$ and $ a_i^+$
consisting of edges such that $p_i$ is a common endpoint of $a_i^-$ and $ a_i^+$, and such that $a^-_i$ shares an endpoint with $c_{i-1}$ and $ a_i^+$ one with $c_i$. Then $$\mathcal{C}'_i:=a^+_i\cup c_i\cup a^-_{i+1}$$ for $i=1,
\dots, k$ is an arc that consists of edges and has endpoints $p_i$ and
$p_{i+1}$. The arcs $\mathcal{C}'_1, \dots, \mathcal{C}'_k$ have pairwise disjoint interior. Moreover,
$${\mathcal{C}}'=\mathcal{C}'_1 \cup \dots \cup \mathcal{C}'_k, $$
and by \eqref{eq:ABCDinNd} we have
$$ \mathcal{C}'_i\subset A_{i}\cup C_i \cup A_{i+1}
\subset
\mathcal{N}^{\delta}(\mathcal{C}_i). $$
Hence by Proposition~\ref{prop:isotreln} and choice of $\delta$, the curve $\mathcal{C}'$ is isotopic to $\mathcal{C}$ rel.\ $P$.
It remains to show that no tile in $\mathcal{D}$ joins opposite sides of $(\mathcal{C}', P)$.
To see this, we argue by contradiction. Suppose that there exists a tile
$X$ in $\mathcal{D}$ that joins opposite
sides of $(\mathcal{C}',P)$. Then $K:= \mathcal{N}^\delta(X)$ joins opposite sides of $(\mathcal{C},P)$, since $\mathcal{C}'_i\subset \mathcal{N}^\delta(\mathcal{C}_i)$ for all $i=1, \dots, k$.
By choice of $\delta_0$ we then have
$$ \delta_0\le \diam(K)\le 2\delta+\diam(X)\le 2\delta+\epsilon_0<3\delta<\delta_0,$$
which is impossible.
\end{proof}
\section{Invariant curves}
\label{sec:constructc}
\noindent This section is central for this work. We will prove
existence and uniqueness results for invariant Jordan curves of expanding
Thurston maps.
We will also show that if an invariant Jordan curve exits, then it can be obtained from an iterative procedure. We start
by looking at a specific example that will illustrate some of the main ideas.
\ifthenelse{\boolean{nofigures}}{}{
\begin{figure}
\centering
\begin{tabular}{ccc}
\begin{overpic}
[width=2.4in,
tics=20]{invC0.eps}
\put(35,32){$\omega$}
\put(73,53){$1$}
\put(14,18){$\mathcal{C}^0$}
\end{overpic}
& &
\begin{overpic}
[width=2.4in,
tics=20]{invC1.eps}
\put(41.5,27){$\omega$}
\put(73,53){$1$}
\put(30,8){$\mathcal{C}^1$}
\end{overpic}
\\
\\
\begin{overpic}
[width=2.4in,
tics=20]{invC2.eps}
\put(31.3,27){$\omega$}
\put(73,43){$1$}
\put(37,5){$\mathcal{C}^2$}
\end{overpic}
& &
\begin{overpic}
[width=2.4in, tics=20]{invC3.eps}
\put(31.3,30){$\omega$}
\put(73,42){$1$}
\put(21.5,5){$\mathcal{C}^3$}
\end{overpic}
\\
\\
\begin{overpic}
[width=2.4in, tics=20]{invC4.eps}
\put(31.3,27){$\omega$}
\put(75,43){$1$}
\put(17,5){$\mathcal{C}^4$}
\end{overpic}
& &
\begin{overpic}
[width=2.4in, tics=20]{invC5.eps}
\put(31.3,25){$\omega$}
\put(75,42){$1$}
\put(35,2){$\widetilde{\mathcal{C}}$}
\end{overpic}
\end{tabular}
\caption{The invariant curve for Example \ref{ex:invC}.}
\label{fig:invC_constr}
\end{figure}
}
\begin{ex}
\label{ex:invC}
Let $S^2=\CDach $ and $f\: S^2 \rightarrow S^2$ be the map defined by
$f(z)= 1+ (\omega-1)/z^3$ for $z\in S^2$, where $\omega= e^{4\pi
{\mathbf{\imath}}/3}$. Note that $f=f_6$ is one of the maps considered in
Example~\ref{ex:R_mario3}. It realizes the subdivision rule shown in
Figure~\ref{fig:triangle_flap} and Figure~\ref{fig:R_mario31}.
Note that $f(z)= \tau(z^3)$, where
$\tau(w)= 1+ (\omega-1)/w$ is a M\"{o}bius transformation that maps
the upper half-plane to the half-plane above the line through the points
$\omega$ and $1$ (indeed, $\tau$ maps $0,1,\infty$ to $\infty,
\omega,1$, respectively).
We have $\operatorname{crit}(f)=\{0,\infty\}$ and $\operatorname{post}(f)=\{\omega, 1, \infty\}$.
One can obtain an $f$-invariant Jordan curve $\widetilde \mathcal{C}\subset S^2$ with $\operatorname{post}(f)\subset \widetilde \mathcal{C}$ as follows. We start by choosing a Jordan
curve $\mathcal{C}^0\subset S^2$ containing all postcritical points of $f$. Here we
let
$\mathcal{C}^0$ be the (extended) line through $\omega$ and $1$ (i.e., the circle on $\CDach$
through $\omega,1, \infty$).
Now consider $f^{-1}(\mathcal{C}^0)= \bigcup_{k=0,\dots, 5} R_k$, where
$$R_k=\{re^{{\mathbf{\imath}} k\pi/3} : 0\leq r\leq \infty\}$$ is the ray from $0$ through
the the sixth root of unity $e^{{\mathbf{\imath}} k\pi /3}$; see the top right in Figure
\ref{fig:invC_constr}. We choose a Jordan curve $\mathcal{C}^1\subset S^2$ such that
$$ \mathcal{C}^1\subset f^{-1}(\mathcal{C}^0),\
\operatorname{post}(f)\subset \mathcal{C}^1, \text{ and }
\mathcal{C}^1 \text{ is isotopic to $\mathcal{C}^0$ rel.\ $\operatorname{post}(f)$.}
$$
This is not always possible, but in our specific case
there is a unique Jordan curve $\mathcal{C}^1\subset f^{-1}(\mathcal{C}^0)$ with
$\operatorname{post}(f)\subset \mathcal{C}^1$, namely $\mathcal{C}^1=R_0\cup R_4$, the
union of the two rays through $\omega$ and through $1$.
Since $\#\operatorname{post}(f)= 3$, the requirement that $\mathcal{C}^1$ is isotopic to $\mathcal{C}^0$ rel.\ $\operatorname{post}(f)$ is automatic for our specific map $f$ by
Lemma \ref{lem:deform<4}.
Let $H\colon S^2\times I\to S^2$ be an isotopy rel.\ $\operatorname{post}(f)$
that deforms $\mathcal{C}^0$ to $\mathcal{C}^1$, i.e., $H_0=\id_{S^2}$ and $H_1(\mathcal{C}^0)=
\mathcal{C}^1$.
Given the data $\mathcal{C}^0$, $\mathcal{C}^1$, and $H$, there are two ways to obtain an
$f$-invariant Jordan curve isotopic to $\mathcal{C}^0$ rel.\ $\operatorname{post}(f)$ and
isotopic to $\mathcal{C}^1$ rel.\
$f^{-1}(\operatorname{post}(f))$.
For the first approach we consider the Thurston map
$\widehat{f}:= H_1\circ f$. Since $\mathcal{C}^1\subset f^{-1}(\mathcal{C}^0)$ we have $f(\mathcal{C}^1)\subset \mathcal{C}^0$, and so
$$\widehat{f}(\mathcal{C}^1)= (H_1\circ
f)(\mathcal{C}^1) \subset H_1(\mathcal{C}^0)=\mathcal{C}^1. $$ Thus $\mathcal{C}^1$ is
$\widehat f$-invariant. The two-tile subdivision rule given by
$\mathcal{D}^1=\mathcal{D}^1(\widehat{f}, \mathcal{C}^1)$, $\mathcal{D}^0=\mathcal{D}^0(\widehat{f},\mathcal{C}^1)$, and the labeling induced by $\widehat f$ is as in
Figure \ref{fig:R_mario31}. The map $\widehat{f}$ is combinatorially
expanding for $\mathcal{C}^1$; indeed, no $2$-tile for $(\widehat f, \mathcal{C}^1)$ joins opposite sides of $\mathcal{C}^1$. Thus by
Corollary \ref{cor:combexp1} there is a homeomorphism $\phi\colon
S^2\to S^2$ isotopic to the identity on $S^2$ rel.\ $\operatorname{post}(\widehat f)=
\operatorname{post}( f)$ such that
$\phi(\mathcal{C}^1)= \mathcal{C}^1$ and $g= \phi\circ \widehat{f}$ is
expanding. Since $f$ is also expanding (this follows from Proposition~\ref{prop:rationalexpch} below) and $g$ is Thurston equivalent to $f$, there is
a homeomorphism $h\colon S^2\to S^2$ such that $h\circ f= g\circ
h$ (Theorem~\ref{thm:exppromequiv}). Then $\widetilde{\mathcal{C}}:= h^{-1}(\mathcal{C}^1)$ is an $f$-invariant
Jordan curve containing $\operatorname{post}(f)$. In Theorem \ref{thm:exinvcurvef}
we give a general existence result, which is proved in the same
fashion.
For the second approach we use Proposition \ref{prop:isotoplift} to lift
$H=H^0$ by the map $f$ to an isotopy $H^1$ with $H^1_0=\id_{S^2}$. Then we lift $H^1$ to an isotopy $H^2$ with $H^2_0=\id_{S^2}$, etc. In this way, we find a sequence of isotopies $H^n$ and
inductively define $\mathcal{C}^{n+1}:= H_1^n(\mathcal{C}^n)$. We will see in Proposition
\ref{prop:invCit} that the sequence $\{\mathcal{C}^n\}$ of
Jordan curves converges in the Hausdorff sense to an $f$-invariant Jordan
curve $\widetilde{\mathcal{C}}$ containing all postcritical points of $f$ as
desired. This is illustrated in Figure \ref{fig:invC_constr}.
\smallskip
In fact in our example there is a \emph{unique} $f$-invariant Jordan
curve $\widetilde \mathcal{C}\subset \CDach$ with $\operatorname{post}(f)\subset\widetilde{\mathcal{C}}$.
To see this, note that since $\#\operatorname{post}(f)=3$, every such curve $\widetilde \mathcal{C}$ is isotopic rel.\ $\operatorname{post}(f)$ to the curve $\mathcal{C}^0$ chosen above. So we can find an isotopy $K\: S^2\times I\rightarrow S^2$ rel.\ $\operatorname{post}(f)$ with $K_0=\id_{S^2}$ and $K_1(\widetilde \mathcal{C})=\mathcal{C}^0$. By Proposition \ref{prop:isotoplift} we can lift $K$ to an isotopy $\widetilde K\: S^2\times I\rightarrow S^2$ rel.\ $f^{-1}(\operatorname{post}(f))$ with $\widetilde K_0=\id_{S^2}$ and
$K_t\circ f=f\circ \widetilde K_t$ for $t\in I$. Then by Lemma~\ref{lem:lifts_inverses} we have
$$\mathcal{C}':= \widetilde K_1(\widetilde \mathcal{C})\subset \widetilde K_1(f^{-1}(\widetilde \mathcal{C}))=f^{-1}(K_1(\widetilde \mathcal{C}))=f^{-1}(\mathcal{C}^0).$$
So $\mathcal{C}'$ is a Jordan curve in $S^2$ with $\mathcal{C}'\subset f^{-1}(\mathcal{C}^0)$ and $\operatorname{post}(f)\subset \mathcal{C}'$. Since $\mathcal{C}^1$ is the unique such curve, we conclude $\mathcal{C}'=\widetilde K_1(\widetilde\mathcal{C})=\mathcal{C}^1$. In particular, $\widetilde \mathcal{C}$ is isotopic to $\mathcal{C}^1$ rel.\ $f^{-1}(\operatorname{post}(f))$ by the isotopy $\widetilde K$.
So every $f$-invariant Jordan curve $\widetilde \mathcal{C}$ with $\operatorname{post}(f)\subset \widetilde \mathcal{C}$ lies in the same isotopy class rel.\ $f^{-1}(\operatorname{post}(f))$ as
$\mathcal{C}^1$. Hence by
Theorem \ref{thm:uniqc} (which we will prove momentarily) there is at most one such Jordan curve $\widetilde \mathcal{C}$. The uniqueness of $\widetilde \mathcal{C}$ follows.
\end{ex}
\subsection{Existence and uniqueness of invariant curves}
\label{subsec:exuniq}
We start with establishing uniqueness results.
\begin{proof} [Proof of Theorem~\ref{thm:uniqc}.] Let $f\:S^2\rightarrow S^2$ be an expanding Thurston map, and suppose that $\mathcal{C}$ and $\mathcal{C}'$ are $f$-invariant Jordan curves in $S^2$ that both contain the set $\operatorname{post}(f)$ and are isotopic rel.\ $f^{-1}(\operatorname{post}(f))$. We have to show that $\mathcal{C}=\mathcal{C}'$.
Under the given assumptions there exists an isotopy $H^0\: S^2\times
I\rightarrow S^2$ rel.\ $f^{-1}(\operatorname{post}(f))$ with $H^0_0=\id_{S^2}$ and $H^0_1(\mathcal{C})=\mathcal{C}'$. Since $\operatorname{post}(f)\subset f^{-1}(\operatorname{post}(f))$, the map $H^0$ is also an isotopy rel.\ $\operatorname{post}(f)$.
Hence by Proposition~\ref{prop:isotoplift} we can find an isotopy
$H^1\: S^2\times
I\rightarrow S^2$ rel.\ $f^{-1}(\operatorname{post}(f))$ with $H^1_0=\id_{S^2}$ and $f\circ H^1_t=H^0_t\circ f$ for all $t\in I$. Repeating this argument, we obtain isotopies $H^n\: S^2\times
I\rightarrow S^2$ rel.\ $f^{-1}(\operatorname{post}(f))$ with $H^n_0=\id_{S^2}$ and $f\circ H^{n+1}_t=H^n_t\circ f$ for all $t\in I$ and $n\in \N_0$.
{\em Claim:} $H^n_1(\mathcal{C})=\mathcal{C}'$ for $n\in \N_0$. To see this we use induction on $n$.
For $n=0$ the claim is true by choice of $H^0$.
Suppose that $H^n_1(\mathcal{C})=\mathcal{C}'$ for some $n\in \N_0$.
Then Lemma~\ref{lem:lifts_inverses}
and the identity $f\circ H^{n+1}_1=H^n_1\circ f$ imply that
$$H^{n+1}_1(f^{-1}(\mathcal{C}))=f^{-1}(H^n_1(\mathcal{C}))=f^{-1}(\mathcal{C}').$$ Since $\mathcal{C}$ and $\mathcal{C}'$ are $f$-invariant, we have the inclusions
$\mathcal{C}\subset f^{-1}(\mathcal{C})$ and $\mathcal{C}'\subset f^{-1}(\mathcal{C}')$. In particular,
$$\widetilde \mathcal{C}:= H^{n+1}_1(\mathcal{C})\subset H^{n+1}_1(f^{-1}(\mathcal{C}))=f^{-1}(\mathcal{C}')$$ is a Jordan curve contained in $f^{-1}(\mathcal{C}')$. Moreover, $\mathcal{C}$ and
$\widetilde \mathcal{C} $ are isotopic rel.\ $f^{-1}(\operatorname{post}(f))$ (by the isotopy $H^{n+1}$). Since $\mathcal{C}$ and $\mathcal{C}'$ are isotopic rel.\ $f^{-1}(\operatorname{post}(f))$ by our hypotheses, it follows that $\mathcal{C}'$ and $\widetilde \mathcal{C}$ are also isotopic rel.\ $f^{-1}(\operatorname{post}(f))$. Both sets are contained in $f^{-1}(\mathcal{C}')$.
Now $f^{-1}(\mathcal{C}')$ is the $1$-skeleton of the cell decomposition $\mathcal{D}^1(f, \mathcal{C}')$. This cell decomposition has the vertex set $f^{-1}(\operatorname{post}(f))$. Moreover, since $f$ is expanding, $\operatorname{post}(f)\ge 3$, and so every tile in $\mathcal{D}^1(f,\mathcal{C}')$ has at least three vertices. So the hypotheses of
Lemma~\ref{lem:isoJcin1ske} are satisfied and we conclude that
$\mathcal{C}'=\widetilde \mathcal{C}=H^{n+1}_1(\mathcal{C})$. The claim follows.
Fix a visual metric on $S^2$. Then the tracks of the isotopies $H^n$ shrink at an exponential rate as $n\to \infty$ (Lemma~\ref{lem:exp_shrink}). Since $H^n_0=\id_{S^2}$, it follows that $H^n_1\to \id_{S^2}$ uniformly as $n\to \infty$.
Since $H^n_1(\mathcal{C})=\mathcal{C}'$ for all $n\in \N_0$ by the claim, we conclude
$\mathcal{C}=\mathcal{C}'$ as desired.
\end{proof}
\begin{cor}[Inv.\ curves in a given isotopy class rel.\ $\operatorname{post}(f)$] \label{cor:finiterelP}
$\quad$
\noindent
Let $f\: S^2\rightarrow S^2$ be an expanding Thurston map and
$\mathcal{C}\subset S^2$ be a Jordan curve with $\operatorname{post}(f)\subset \mathcal{C}$. Then there are at most finitely many $f$-invariant Jordan curves $\widetilde \mathcal{C}\subset S^2 $ with $\operatorname{post}(f)\subset \widetilde \mathcal{C}$ that are isotopic to $\mathcal{C}$ rel.\ $\operatorname{post}(f)$.
\end{cor}
\begin{proof} Let $\widetilde \mathcal{C}$ be such an $f$-invariant Jordan curve.
Then there exists an isotopy $H\:S^2\times I\rightarrow S^2$ rel.\ $\operatorname{post}(f)$ with $H_0=\id_{S^2}$ and $H_1(\widetilde \mathcal{C})=\mathcal{C}$. Lifting $H$ we get an isotopy
$\widetilde H\: S^2\times I\rightarrow S^2$ rel.\ $f^{-1}(\operatorname{post}(f))$ such that
$\widetilde H_0=\id_{S^2}$ and $f\circ \widetilde H_t=H_t\circ f$ for all $t\in I$. Since $\widetilde \mathcal{C}$ is $f$-invariant, we have $\widetilde \mathcal{C}\subset f^{-1}(\widetilde \mathcal{C})$. So Lemma~\ref{lem:lifts_inverses} implies that
$$\widetilde H_1(\widetilde \mathcal{C})\subset \widetilde H_1(f^{-1}(\widetilde \mathcal{C}))= f^{-1}(H_1(\widetilde \mathcal{C}))=f^{-1}(\mathcal{C}).$$ Hence $\widetilde \mathcal{C}$ is isotopic rel.\ $f^{-1}(\operatorname{post}(f))$ to the Jordan curve
$\widetilde H_1(\widetilde \mathcal{C})$ that is contained in $f^{-1}(\mathcal{C})$.
Any such Jordan curve is a union of edges in the cell decomposition $\mathcal{D}^1(f, \mathcal{C})$ (see the last part of the proof of Lemma~\ref{lem:isoJcin1ske}). In particular, there are only finitely many distinct Jordan curves contained in $f^{-1}(\mathcal{C})$. This implies that there are only finitely many isotopy classes rel.\ $f^{-1}(\operatorname{post}(f))$ represented by curves $\widetilde \mathcal{C}$ satisfying the assumptions of the corollary. Since an $f$-invariant Jordan curve $\widetilde \mathcal{C}\subset S^2$ with $\operatorname{post}(f)\subset\widetilde \mathcal{C}$ is unique in its isotopy class rel.\ $f^{-1}(\operatorname{post}(f))$ by Theorem~\ref{thm:uniqc}, the statement follows.
\end{proof}
\begin{cor}\label{cor:finitepost3}
Let $f\: S^2\rightarrow S^2$ be an expanding Thurston map with $\operatorname{post}(f)=3$. Then there are at most finitely many $f$-invariant Jordan curves
$\widetilde \mathcal{C}\subset S^2$ with $\operatorname{post}(f)\subset \widetilde \mathcal{C}$.
\end{cor}
\begin{proof} Pick a Jordan curve $\mathcal{C}\subset S^2$ with $\operatorname{post}(f)\subset \mathcal{C}$.
Since we have $\#\operatorname{post}(f)=3$, by Lemma~\ref{lem:deform<4} every Jordan curve $\widetilde \mathcal{C}\subset S^2$ with $\operatorname{post}(f)\subset \widetilde\mathcal{C}$ is isotopic to $\mathcal{C}$ rel.\ $\operatorname{post}(f)$.
The statement now follows from Corollary~\ref{cor:finiterelP}.
\end{proof}
In contrast to the case $\#\operatorname{post}(f)=3$, expanding Thurston maps $f$ with $\#\operatorname{post}(f)\ge 4$ can have infinitely many distinct
invariant Jordan curves containing the set of postcritical points.
\ifthenelse{\boolean{nofigures}}{}{
\begin{figure}
\centering
\begin{overpic}
[width=12cm,
tics=20]{inf_invC.eps}
\put(42,26){$\wp$}
\put(32,42){$\R^2$}
\put(13,5){$G_\tau$}
\put(57,15){$\mathcal{C}_\tau$}
\end{overpic}
\caption{Invariant curves for the Latt\`{e}s map $g$.}
\label{fig:inf_invC}
\end{figure}
}
\begin{ex}[Infinitely many invariant curves]
\label{ex:infty_C}
Let $g$ be the Latt\`{e}s map from Section
\ref{sec:Lattes}. In the following it is advantageous to use real notation and consider the maps $\psi$ and $\wp$ used in the definition of $g$ as in \eqref{eq:Lattes} as maps on $\R^2$. Then
$\psi(u)= 2u$ for $u\in \R^2$. Moreover, for $u_1,u_2\in \R^2$ we have
$\wp(u_1)=\wp(u_2)$ if and only if $u_2=\pm u_1+\gamma$ for $\gamma\in L=2\Z^2$.
Recall that the extended real line $\widehat{\R}$ (which is the boundary
of the pillow) is $g$-invariant and contains $\{-1,0,1,\infty\}=
\operatorname{post}(g)$. Let $S=\partial Q$ be the boundary of the
unit square $Q=[0,1]^2\subset \R^2$, and $G$ be the standard square grid, i.e., the union of the
horizontal and vertical lines through the points in $\Z^2$. Then
$\wp|S$ is injective and $\wp(S)=\wp(G)=\widehat \R$. So the $g$-invariant curve $\widehat \R$ is obtained by mapping the boundary $S$ of the fundamental domain $Q$ of $\Z^2$ or the standard grid $G$ by $\wp$.
One can obtain other $g$-invariant Jordan curves by mapping the boundaries of other fundamental domains of $\Z^2$ or other grids by $\wp$.
To explain this, consider a $(2\times 2)$-matrix $\tau\in \text{SL}_2(\Z)$. We identify $\tau$ with the linear map $u\mapsto \tau u$ on $\R^2$ induced by left-multiplication of $u\in \R^2$ (considered as a column vector) by the matrix $\tau$. Define $Q_\tau:=\tau(Q)$,
$S_\tau:=\partial Q_\tau=\tau(S)$, and the corresponding grid $G_\tau=\tau(G)$. Since $\tau$ is a linear map and $\tau(L)=L$, it follows that
for $u_1,u_2\in \R^2$ we have $\wp(\tau(u_1))=\wp(\tau(u_2))$ if and only if
$u_2=\pm u_2+\gamma$, where $\gamma \in L$. This implies that $\wp|S_\tau$ is injective, and so $\mathcal{C}_\tau:=\wp(S_\tau)\subset \CDach$ is a Jordan curve.
Moreover, $\wp(G_\tau)=\wp(S_\tau)$. Since $\psi\circ \tau=\tau\circ \psi$,
we have
$$\psi(G_\tau)=\psi(\tau(G))=\tau(\psi(G))\subset \tau(G)=G_\tau,$$
and so
$$ g(\mathcal{C}_\tau)=g(\wp(G_\tau))=\wp(\psi(G_\tau))\subset \wp(G_\tau)=\mathcal{C}_\tau. $$
Hence $\mathcal{C}_\tau$ is $g$-invariant. Since $\Z^2\subset G_\tau$, we also have $\operatorname{post}(g)=\wp(\Z^2)\subset \wp(G_\tau)=\mathcal{C}_\tau$.
So $\mathcal{C}_\tau$ is an $g$-invariant Jordan curve that contains the set
$\operatorname{post}(g)$. An example of this construction is indicated in Figure \ref{fig:inf_invC}. The curve $\mathcal{C}_\tau$ is drawn
in thick on the right.
The curve $\mathcal{C}_\tau$ determines the grid $G_\tau$ uniquely; indeed,
one obtains generating vectors of the two lines in $G_\tau$ through
$0$ by locally lifting $\mathcal{C}_\tau$ near $\wp(0)=0\in \operatorname{post}(g)\subset \mathcal{C}_\tau$
to $0$ by the map $\wp$. The whole grid $G_\tau$ is obtained by translating these two lines by vectors in $\Z^2$.
This implies that the map $\tau \in \text{SL}_2(\Z)\mapsto \mathcal{C}_\tau$ is four-to-one; indeed, if $\tau, \sigma\in \text{SL}_2(\Z)$, then, as we have seen, $\mathcal{C}_\tau=\mathcal{C}_\sigma$ if and only if $G_\tau=G_\sigma$. On the other hand,
$G_\tau=G_\sigma$ if and only if $\sigma^{-1}\circ \tau\in \text{SL}_2(\Z)$ is one of the four rotations (by integer multiples of $\pi/2$) that preserve the grid $G$. In particular, there exist infinitely many $g$-invariant Jordan curves
$\widetilde \mathcal{C}\subset \CDach$ with $\operatorname{post}(g)\subset \widetilde \mathcal{C}$. \end{ex}
We now turn to existence results. As the following example shows, for
a Thurston map $f\:S^2\rightarrow S^2$ an $f$-invariant Jordan curve
$\mathcal{C}\subset S^2$ with $\operatorname{post}(f)\subset \mathcal{C}$ need not exist.
\begin{ex}\label{ex:noinvCC}
Consider the map $f\: \CDach \rightarrow \CDach$ defined by
$$f(z)={\mathbf{\imath}}\frac{z^4-{\mathbf{\imath}}}{z^4+{\mathbf{\imath}}}$$ for $z\in \CDach$.
The
critical points of $f$ are $0$ and $\infty$. The forward orbits of the critical points of $f$ under iteration can be represented by the so-called
\defn{ramification portrait}:
\begin{equation}
\label{eq:ramification_h}
\xymatrix @R=1pt{
0 \ar[r]^{4 : 1} & -{\mathbf{\imath}} \ar[dr] &
\\
& & 1 \ar@(r,u)[]
\\
\infty \ar[r]^{4 : 1} & {\mathbf{\imath}} \ar[ur] &
}
\end{equation}
The labels over the arrows indicate the local degree if it is different from $1$. It follows that
the set of postcritical
points of $f$ is given by $\operatorname{post}(f)=\{-{\mathbf{\imath}},1,{\mathbf{\imath}} \}$. So $f$ is a Thurston
map. This map is also expanding as follows from
Proposition~\ref{prop:rationalexpch} below.
\end{ex}
\begin{lemma}
\label{lem:g1noinvC} Let $f$ be the map from Example \ref{ex:noinvCC}.
Then there is no $f$-invariant Jordan curve $\widetilde{\mathcal{C}}\subset
\CDach $ with $\operatorname{post}(f)\subset \widetilde{\mathcal{C}}$.
\end{lemma}
\ifthenelse{\boolean{nofigures}}{}{
\begin{figure}
\centering
\begin{overpic}
[width=7cm,
tics=20]{ex_no_invC.eps}
\put(52,28){$1$}
\put(-1,29){$0$}
\put(99,29){$\infty$}
\put(46,5){$-{\mathbf{\imath}}$}
\put(50,43){${\mathbf{\imath}}$}
%
\put(22,19){$R_0$}
\put(12,43){$R_2$}
\put(12,7){$R_6$}
\end{overpic}
\caption{No invariant Jordan curve $\widetilde{\mathcal{C}}\supset \operatorname{post}$.}
\label{fig:no_invC}
\end{figure}
}
\begin{proof}
We have $f(z)= \varphi(z^4)$ for $z\in \CDach$, where
\begin{equation} \label{eq:mobius1}
\varphi(w)= {\mathbf{\imath}} \frac{w-{\mathbf{\imath}}}{w+{\mathbf{\imath}}}, \quad w\in \CDach,
\end{equation}
is
a M\"{o}bius transformation that maps the upper half-plane to the
unit disk (note that $\varphi$ maps $0,1,\infty$ to $-{\mathbf{\imath}}, 1,
{\mathbf{\imath}}$, respectively). Let $\mathcal{C}:= \partial \D$ be the unit circle. Then
\begin{equation*}
f^{-1}(\mathcal{C})=\bigcup_{k=0, \dots, 7} R_k,
\text{ where }
R_k= \{re^{{\mathbf{\imath}} k\pi/4} : 0\leq r\leq \infty\}.
\end{equation*}
The postcritical points
$-{\mathbf{\imath}},1,{\mathbf{\imath}}$ lie on distinct rays $R_k$. Two such rays have the points $0$ and $\infty$ in common and no other points. Thus
there is no Jordan curve in $f^{-1}(\mathcal{C})$ containing all postcritical
points, see Figure \ref{fig:no_invC}. As we will see in Theorem~\ref{thm:exinvcurvef}, the existence of such a Jordan curve is a necessary condition for the existence of an $f$-invariant Jordan curve $\widetilde \mathcal{C}\subset \CDach$ with $\operatorname{post}(f)\subset \widetilde \mathcal{C}$ (in our specific case
where
$\#\operatorname{post}(f)=3$ the choice of $\mathcal{C}$ does not matter since all Jordan curves that contain $\operatorname{post}(f)$ are isotopic rel.\ $\operatorname{post}(f)$). Hence there is no $f$-invariant Jordan curve $\widetilde \mathcal{C}\subset \CDach$ with $\operatorname{post}(f)\subset \widetilde \mathcal{C}$.
One can also see this by a simple argument directly.
Indeed,
suppose that $\widetilde{\mathcal{C}}\subset\CDach $ is a Jordan
curve with $\operatorname{post}(f)\subset \widetilde{\mathcal{C}}$ and
$f(\widetilde{\mathcal{C}})\subset \widetilde{\mathcal{C}}$. The unit circle
$\mathcal{C}=\partial \D$ is also a Jordan curve
containing the set $\operatorname{post}(f)$. Hence by Lemma~\ref{lem:deform<4} there exists an isotopy $H\: \CDach \times I\rightarrow
\CDach$ rel.\ $\operatorname{post}(f)$ such that $H_0=\id_{\CDach} $ and
$H_1(\widetilde{\mathcal{C}})=\mathcal{C}$.
By Proposition~\ref{prop:isotoplift} the isotopy $H$ can be lifted
to an isotopy $\widetilde H\: \CDach \times I\rightarrow \CDach$ rel.\
$\operatorname{post}(f)$
such that $\widetilde H_0=\id_{\CDach}$ and $H_t\circ f=f\circ
\widetilde H_t$ for all $t\in I$.
Since $\widetilde{\mathcal{C}}\subset
f^{-1}(\widetilde{\mathcal{C}})$, it follows from Lemma
\ref{lem:lifts_inverses} that $$\widetilde{H}_1(\widetilde{\mathcal{C}})
\subset \widetilde{H}_1(f^{-1}(\widetilde{\mathcal{C}})) =
f^{-1}(H_1(\widetilde{\mathcal{C}})) = f^{-1}(\mathcal{C}).$$
This means that
the Jordan curve $\mathcal{C}':= \widetilde{H}_1(\widetilde{\mathcal{C}})$ is contained
in $f^{-1}(\mathcal{C})$. Moreover, it contains all postcritical points,
since $\widetilde{\mathcal{C}}$ does, and the points in $\operatorname{post}(f)$ stay fixed under the isotopy $\widetilde{H}$. As we have seen above, no such Jordan curve exists and we get a contradiction as desired.
\end{proof}
By a similar (though somewhat more lengthy) argument one can show that
the Latt\`{e}s map $f(z)= \frac{{\mathbf{\imath}}}{2}(z + 1/z)$ does not have an
$f$-invariant curve $\mathcal{C}$ with $\operatorname{post}(f)\subset \mathcal{C}$. Another such
example can be found in \cite[Section 4]{CFP10}.
\smallskip
We now turn to the proof for the necessary and sufficient criterion for the existence of an invariant Jordan curve as formulated in
Theorem~\ref{thm:exinvcurvef}. Note that in (\ref{item:ex_invC2}) of this theorem the condition on $\widehat{f}$ is
meaningful.
Indeed, $\widehat f=H_1 \circ f$ is a branched cover of $S^2$ with
$\operatorname{crit}(\widehat f)=\operatorname{crit}(f)$. Since $H_1$ is isotopic to
$\id_{S^2}$ rel.~$\operatorname{post}(f)$ we have $\operatorname{post}(\widehat
f)=\operatorname{post}(f)$, and so $\widehat f$ is a Thurston map.
Furthermore, $ \mathcal{C}'$ is a Jordan curve with
$\operatorname{post}(\widehat f)=\operatorname{post}(f)\subset \mathcal{C}' $. Since $ \mathcal{C}'=H_1(\mathcal{C})\subset f^{-1}(\mathcal{C})$, we have that
$\widehat{f}( \mathcal{C}')= H_1\circ f ( \mathcal{C}')\subset H_1(\mathcal{C})= \mathcal{C}'$. Hence
$ \mathcal{C}'$ is invariant with respect to $\widehat{f}$, and it makes sense
to require that $\widehat{f}$ is combinatorially expanding for
$\mathcal{C}'$.
\begin{proof} [Proof of Theorem~\ref{thm:exinvcurvef}]
(i) $\Rightarrow$(ii): Suppose $\widetilde \mathcal{C}$
is as in (i). Then in (ii) we let $\mathcal{C}=\mathcal{C}'= \widetilde \mathcal{C}$, and the isotopy $H$ be such that $H_t=\id_{S^2}$ for all $t\in I$. Then
$ \mathcal{C}'= \widetilde \mathcal{C}\subset f^{-1}(\widetilde \mathcal{C})=f^{-1}(\mathcal{C})$, and $\widehat f=f$ is combinatorially expanding for the invariant curve $ \mathcal{C}'= \widetilde \mathcal{C}$, since $f$ is expanding.
(ii)$\Rightarrow$(i): Let $\mathcal{C}$, $\mathcal{C}'$, $H$, and $\widehat f$ be as in (ii), and define $\chi=H_1$.
As we have seen in the remark after the theorem, $\widehat f$ is a Thurston map
with $\operatorname{post} (\widehat f)=\operatorname{post}(f)$, and $ \mathcal{C}'$ is an $\widehat f$-invariant Jordan curve containing the set $\operatorname{post} (\widehat f)=\operatorname{post}(f)$.
Since $\widehat f$ is combinatorially expanding for
$ \mathcal{C}'$, Corollary~\ref{cor:combexp1} implies that there exists a homeomorphism $\phi\:S^2\rightarrow S^2$ that is isotopic to the identity rel.~$\operatorname{post}(\widehat f)=\operatorname{post}(f)$ such that
$\phi( \mathcal{C}')= \mathcal{C}'$ and $g=\phi\circ \widehat f$ is an expanding Thurston map.
Since $g=(\phi\circ \chi)\circ f$, and $\phi\circ \chi$ is isotopic to the identity on $S^2$ rel.~$\operatorname{post}(f)$, the maps $f$ and $g$ are Thurston equivalent.
If notation is as in \eqref{Thequiv1} (with $\widehat S^2=S^2$), then we can take $h_0=\phi\circ \chi$ and $h_1=\id_{S^2}$.
By Theorem~\ref{thm:exppromequiv} we can find a homeomorphism
$h\:S^2\rightarrow S^2$ that is isotopic to $h_1=\id_{S^2}$ rel.~$f^{-1}(\operatorname{post}(f))$ with $f\circ h=h\circ g$.
Let $ \widetilde \mathcal{C}=h (\mathcal{C}')$. Then
$ \widetilde \mathcal{C}$ is a Jordan curve in $S^2$ that is isotopic to $ \mathcal{C}'$ rel.~$f^{-1}(\operatorname{post}(f))$, and hence isotopic to $\mathcal{C}$ rel.\ $\operatorname{post}(f)$; in particular, $ \widetilde \mathcal{C}$ contains the set $\operatorname{post}(f)$. Moreover, $\widetilde \mathcal{C}$ is $f$-invariant, because we have
\begin{eqnarray*}
f(\widetilde \mathcal{C})&=&f(h(\mathcal{C}'))\,=\, h(g( \mathcal{C}'))\,=\,h(\phi(\widehat f( \mathcal{C}')))\\
&\subset &
h(\phi( \mathcal{C}'))\,=\,h( \mathcal{C}')= \widetilde \mathcal{C}.
\end{eqnarray*}
The proof is complete. \end{proof}
\begin{rems}\label{rem:expcheck} (a) The condition of combinatorial expansion in (ii) of
Theorem~\ref{thm:exinvcurvef} is combinatorial in nature and can easily be checked in principle. A simple sufficient criterion for this can be formulated as follows:
if no $1$-tile for $(f,\mathcal{C})$ joins opposite sides of $ \mathcal{C}'$, then $\widehat f$ is combinatorially expanding for $ \mathcal{C}'$.
To see this note that
$$\widehat f^{-1}( \mathcal{C}')=f^{-1}(H_1^{-1}( \mathcal{C}'))= f^{-1}(\mathcal{C}).$$
By Proposition~\ref{prop:celldecomp}~(v) this implies that the $1$-tiles for
$(\widehat f, \mathcal{C}')$ are precisely the $1$-tiles for $(f,\mathcal{C})$.
Hence if no $1$-tile for $(f,\mathcal{C})$ joins opposite sides of $ \mathcal{C}'$, then $D_1(\widehat f, \mathcal{C}')\ge 2$ and so $\widehat f$ is combinatorially expanding for $ \mathcal{C}'$. We will later formulate a necessary and sufficient condition for
combinatorial expansion of $\widehat f$ (see Proposition~\ref{prop:nscombexp}).
(b) The condition of combinatorial expansion in (ii) is independent of the chosen isotopy
$H$. More precisely, suppose $H^1,H^2\: S^2\times I\rightarrow S^2$ are two isotopies with $H^1_0=H^2_0=\id_{S^2}$ and $H^1_1(\mathcal{C})=H^2_1(\mathcal{C})=\mathcal{C}'$. Then $\widehat f_1=H^1_1\circ f$ is combinatorially expanding for $\mathcal{C}'$ if and only if $\widehat f_2=H^2_1\circ f$ is combinatorially
expanding for $\mathcal{C}'$. This follows immediately from
Lemma~\ref{lem:cexp_Cinv} (with $f=\widehat f_1$, $g=\widehat f_2$, $h_0=H^2_1\circ (H^1_1)^{-1}$, $h_1=\id_{S^2}$, and $\mathcal{C}=\mathcal{C}'$).
(c) Theorem~\ref{thm:exinvcurvef} can be slightly modified to give necessary and sufficient conditions for the existence of an invariant curve in a given isotopy class rel.\ $\operatorname{post}(f)$ or rel.\ $f^{-1}(\operatorname{post}(f))$. An existence statement for a given isotopy class rel.\ $f^{-1}(\operatorname{post}(f))$ is especially relevant in view of the complementary uniqueness statement given by Theorem~\ref{thm:uniqc}.
To formulate this precisely, let $\widehat \mathcal{C}\subset S^2$ be given a Jordan curve with $\operatorname{post}(f)\subset \widehat \mathcal{C}$. Then an $f$-invariant Jordan curve $\widetilde \mathcal{C}\subset S^2$ {\em isotopic to $\widehat \mathcal{C}$ rel.\ $\operatorname{post}(f)$} exists if and only if condition (ii) in Theorem~\ref{thm:exinvcurvef} is true
for a Jordan curve $\mathcal{C}$ {\em isotopic to $\widehat \mathcal{C}$ rel.\
$\operatorname{post}(f)$}. This immediately follows from the proof of this theorem.
Similarly, an $f$-invariant Jordan curve $\widetilde \mathcal{C}\subset S^2$ {\em isotopic to $\widehat \mathcal{C}$ rel.\ $f^{-1}(\operatorname{post}(f))$} exists if and only if condition (ii) in Theorem~\ref{thm:exinvcurvef} is true with the additional requirement that {\em $\mathcal{C}'$ is isotopic to $\widehat \mathcal{C}$ rel.\ $f^{-1}(\operatorname{post}(f))$}.
\end{rems}
Existence of an $f$-invariant Jordan curve $\widetilde \mathcal{C}$ in a given isotopy class rel.\
$f^{-1}(\operatorname{post}(f))$ is particularly interesting, because if there is such a curve $\widetilde \mathcal{C}$, then it is unique by Theorem~\ref{thm:uniqc}.
It is worthwhile to formulate some explicit conditions which guarantee existence in this. They are stated in the next proposition and are a slight variation of the condition given in Remark~\ref{rem:expcheck}~(c).
\begin{prop}[Existence of invariant curves rel.\ $f^{-1}(\operatorname{post}(f))$]
\label{prop:invC_invC1}
\null $\quad$
\noindent
Let $f\colon S^2 \to S^2$ be an expanding Thurston map and
$ \mathcal{C}\subset S^2$ be a Jordan curve with $\operatorname{post}(f)\subset
\mathcal{C} $. Then the following conditions are equivalent:
\begin{enumerate}[{\upshape(i)}]
\smallskip
\item
\label{item:invC_invC1_1}
There exists an $f$-invariant Jordan curve $\widetilde{\mathcal{C}}$ that is
isotopic to $ \mathcal{C}$ rel.\ $f^{-1}(\operatorname{post}(f))$.
\smallskip
\item
\label{item:invC_invC1_2}
There exists an isotopy $H\colon S^2\times I\to S^2$ rel.\
$f^{-1}(\operatorname{post}(f))$ with $H_0=\id_{S^2}$ and $H_1(\mathcal{C})\subset
f^{-1}(\mathcal{C})$ such that \begin{equation*}
\widehat{g}:= f \circ H_1
\end{equation*}
is combinatorially expanding for $\mathcal{C}$.
\smallskip
\item
\label{item:invC_invC1_3}
There exists an isotopy $H\colon S^2\times I\to S^2$ rel.\
$f^{-1}(\operatorname{post}(f))$ with $H_0=\id_{S^2}$ and $\mathcal{C}':=H_1( \mathcal{C})\subset
f^{-1}( \mathcal{C})$ such that
\begin{equation*}
\widehat{f}:=H_1 \circ f
\end{equation*}
is combinatorially expanding for $ \mathcal{C}'$.
\end{enumerate}
\end{prop}
\begin{proof}
\eqref{item:invC_invC1_1} $\Rightarrow$ \eqref{item:invC_invC1_2}:
Suppose that there exists an $f$-invariant Jordan $\widetilde{\mathcal{C}}$ with
$\operatorname{post}(f)\subset \widetilde \mathcal{C}$ that is isotopic to $\mathcal{C}$ rel.\ $f^{-1}(\operatorname{post}(f))$.
Then we can find an isotopy
$K\colon S^2\times I\to S^2$ rel.\
$f^{-1}(\operatorname{post}(f))$ with $K_0=\id_{S^2}$ and
$K_1(\widetilde{\mathcal{C}}) = \mathcal{C}$.
Let $\widetilde{K}$ be the lift of $K$ by $f$ according to
Proposition \ref{prop:isotoplift}, i.e., the isotopy rel.\
$f^{-2}(\operatorname{post}(f))\supset f^{-1}(\operatorname{post}(f))$ with $\widetilde{K}_0=\id_{S^2}$ and $K_t\circ f= f
\circ \widetilde{K}_t$ for $t\in I$. Define an isotopy $H\:S^2\times I\rightarrow S^2$ by setting $H_t = \widetilde{K}_t \circ
(K_t)^{-1}$ for $t\in I$. Then $H$ is an isotopy rel.\ $f^{-1}(\operatorname{post}(f))$ with
$H_0=\id_{S^2}$. Moreover,
\begin{align*}
H_1(\mathcal{C})&= (\widetilde{K}_1 \circ (K_1)^{-1})(\mathcal{C}) =
\widetilde{K}_1 (\widetilde{\mathcal{C}}) \subset
\widetilde{K}_1(f^{-1}(\widetilde{\mathcal{C}}))
\\
&= f^{-1}(K_1(\widetilde{\mathcal{C}}))= f^{-1}(\mathcal{C}).
\end{align*}
Here we used the $f$-invariance of $\widetilde{\mathcal{C}}$ (i.e.,
$\widetilde{\mathcal{C}}\subset f^{-1}(\widetilde{\mathcal{C}})$), as well as
Lemma~\ref{lem:lifts_inverses}. Note that
\begin{equation*}
\widehat{g} := f\circ H_1 = f\circ \widetilde{K}_1 \circ
(K_1)^{-1} = K_1 \circ f \circ (K_1)^{-1}.
\end{equation*}
Hence $\widehat{g}$ is a Thurston map with
$\operatorname{post}(\widehat{g})=\operatorname{post}(f)$, $\mathcal{C}$ is an $g$-invariant Jordan
curve with $\mathcal{C}\supset \operatorname{post}(g)$, and it follows from
Lemma~\ref{lem:cexp_Cinv} that $\widehat{g}$ is combinatorially
expanding for
$\mathcal{C}$.
\smallskip \noindent
\eqref{item:invC_invC1_2} $\Rightarrow$ \eqref{item:invC_invC1_3}:
Let $H$ and $\widehat g$ be as in \eqref{item:invC_invC1_2}, and define
$\widehat f:=H_1\circ f$ and $\mathcal{C}':=H_1(\mathcal{C})$. Then $\widehat f$ is
a Thurston map with $\operatorname{post}(\widehat{f})=\operatorname{post}(f)$, and we have
$$ \widehat f(\mathcal{C}')=\widehat f(H_1(\mathcal{C}))\subset \widehat f(f^{-1}(\mathcal{C}))
=H_1(f(f^{-1}(\mathcal{C})))= H_1(\mathcal{C})=\mathcal{C}'. $$ Hence $\mathcal{C}'$ is an
$\widehat f$-invariant Jordan curve with $\operatorname{post}(\widehat f)\subset \mathcal{C}'$.
Since
$$\widehat f\circ H_1 = H_1 \circ f \circ H_1= H_1 \circ \widehat g,$$
Lemma~\ref{lem:cexp_Cinv} implies that $\widehat f$ is combinatorially
expanding for
$\mathcal{C}'$.
\smallskip \noindent
\eqref{item:invC_invC1_3} $\Rightarrow$ \eqref{item:invC_invC1_1}:
If (iii) is true, then by Theorem~\ref{thm:exinvcurvef} there exists an
$f$-invariant Jordan curve $\widetilde \mathcal{C}$ with $\operatorname{post}(f)\subset
\widetilde \mathcal{C}$ that is isotopic to $\mathcal{C}'$ rel.\
$f^{-1}(\operatorname{post}(f))$. Since $\mathcal{C}'=H_1(\mathcal{C})$ and $H$ is an isotopy rel.\
$f^{-1}(\operatorname{post}(f))$, the curve $\mathcal{C}'$, and hence also $\widetilde \mathcal{C}$,
is isotopic to $\mathcal{C}$ rel.\ $f^{-1}(\operatorname{post}(f))$.
\end{proof}
We now turn to the proof of Theorem~\ref{thm:main}. We need the following auxiliary result.
\begin{lemma}\label{lem:isotopicpath} Let $f\: S^2 \rightarrow S^2$ be an expanding Thurston map,
and $\mathcal{C}\subset S^2$ be a Jordan curve with $\operatorname{post}(f)\subset \mathcal{C}$. Then for all sufficiently large $n$ there exists a Jordan curve $\mathcal{C}'\subset f^{-n}(\mathcal{C})$ that is isotopic to $\mathcal{C}$ rel.\ $\operatorname{post}(f)$. Moreover, $\mathcal{C}'$ can be chosen so that no $n$-tile for $(f,\mathcal{C})$ joins opposite sides of
$\mathcal{C}'$.
\end{lemma}
\begin{proof} We fix some base metric on $S^2$.
Let $P:=\operatorname{post}(f)$. Since $f$ is expanding, we have $k:=\#P=\#\operatorname{post}(f)\ge 3$
by Corollary~\ref{cor:no<3}. Pick $\epsilon_0>0$ as in Lemma~\ref{prop:isotopicpath}. Since $f$ is expanding, for large enough $n$ we have
$$ \operatorname{mesh}(f,n,\mathcal{C})=\max_{c\in \mathcal{D}^n(f,\mathcal{C})}\diam(c)<\epsilon_0.$$
For such $n$ consider the cell decomposition $\mathcal{D}=\mathcal{D}^n(f,\mathcal{C})$ of $S^2$. Its vertex set is the set $f^{-n}(\operatorname{post}(f))\supset \operatorname{post}(f)=P$ of $n$-vertices
and its $1$-skeleton is the set $f^{-n}(\mathcal{C})$.
Hence by Lemma~\ref{prop:isotopicpath} there exists a Jordan curve
$\mathcal{C}'\subset f^{-n}(\mathcal{C})$
that is isotopic to $\mathcal{C}$ rel.\ $P=\operatorname{post}(f)$ and so that no tile in $\mathcal{D}$, i.e.,
no $n$-tile for $(f,\mathcal{C})$, joins opposite side of $ \mathcal{C}'$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:main}.]
Let $f$ and $\mathcal{C}$ be as in the statement of the theorem. By Lemma~\ref{lem:isotopicpath} for sufficiently large $n\in \N$ there exists an
isotopy $H\:S^2\times I\rightarrow S^2$ rel.\ $\operatorname{post}(f)$ such that $H_0=\id_{S^2}$ and
$\mathcal{C}':=H_1(\mathcal{C})\subset f^{-n}(\mathcal{C})$ and such that no $n$-tile for $(f,\mathcal{C})$ joins opposite sides of $ \mathcal{C}'$.
For such $n$ let $F=f^n$. Then $F$ is an expanding Thurston map with $\operatorname{post}(F)=\operatorname{post}(f)$. Then $\mathcal{C}$ and $\mathcal{C}'$ are Jordan curves with $\operatorname{post}(F)\subset \mathcal{C},\mathcal{C}'$, and $H$ is an isotopy rel.\ $\operatorname{post}(F)$ that deforms $\mathcal{C}$ into $\mathcal{C}'\subset f^{-n}(\mathcal{C})=F^{-1}(\mathcal{C})$. Moreover,
$F=f^n$ is cellular for $(\mathcal{D}^n(f,\mathcal{C}), \mathcal{D}^0(f,\mathcal{C}))$. Hence by Lemma~\ref{lem:pullback} we have
$\mathcal{D}^1( F, \mathcal{C})=\mathcal{D}^n(f,\mathcal{C})$, and so the $1$-cells
for $(F, \mathcal{C})$ are precisely the $n$-cells for $(f,\mathcal{C})$.
So no $1$-tile for $(F,\mathcal{C})$ joins opposite side of
$ \mathcal{C}'$ and by Remark~\ref{rem:expcheck}~(a) the map $H_1\circ F$ is combinatorially expanding for $ \mathcal{C}'$.
This shows that condition (ii) in
Theorem~\ref{thm:exinvcurvef} is satisfied. Hence there exists a Jordan curve $\widetilde \mathcal{C}\subset S^2$ that is $F$-invariant and is isotopic to $\mathcal{C}$ rel.\ $\operatorname{post}(F)=\operatorname{post}(f)$ as desired.
\end{proof}
\begin{rem}\label{rem:ndep} In general,
the $f^n$-invariant Jordan curve $\widetilde{\mathcal{C}}$ as in Theorem~\ref{thm:main} will
depend on $n$, and one cannot expect that $\widetilde{\mathcal{C}}$ is invariant
for {\em all} sufficiently high iterates of $f$. To illustrate this, consider the map $f$ from
Example~\ref{ex:noinvCC} (see also Lemma~\ref{lem:g1noinvC}). Recall that $f(z)= \varphi(z^4)$ for $z\in \CDach$, where
$\varphi$ is as in \eqref{eq:mobius1}.
The M\"{o}bius transformation
$\varphi$ maps the extended real line $\widehat \R$ to the unit circle $\partial \D$, and $\partial \D$ to $\widehat \R$.
This implies the unit circle
$\widetilde \mathcal{C}:= \partial \D$ satisfies
$f^{2n}(\widetilde \mathcal{C})\subset \widetilde \mathcal{C}$ for every $n\in \N$. Note that $\operatorname{post}(f)=\{-{\mathbf{\imath}},1,{\mathbf{\imath}}\}\subset
\widetilde \mathcal{C}$. Thus $\widetilde \mathcal{C}$ is a Jordan curve with
$\operatorname{post}(f)\subset
\widetilde \mathcal{C}$ that is
invariant for every \emph{even} iterate $f^{2n}$.
On the other hand, for $n\in \N_0$ we have $f^{2n+1}(\partial \D)\subset \widehat \R$. Since $f^{2n+1}$ is a finite-to-one map, the set $f^{2n+1}(\partial \D)$ is infinite, and so we cannot
have $f^{2n+1}(\partial \D)\subset \partial \D$ (for otherwise, $f^{2n+1}(\partial \D)\subset \partial \D\cap \widehat \R=\{-1,1\}$).
Thus the unit circle $\partial \D=\widetilde \mathcal{C}$ is not invariant for
any \emph{odd} iterate of $f$.
\end{rem}
\begin{proof}[Proof of Corollary~\ref{cor:subdivnlarge}] Let
$f\:S^2\rightarrow S^2$ be an expanding Thurston map. It follows from
Theorem~\ref{thm:main} that for each sufficiently large $n\in \N$
there exists an
$f^n$-invariant Jordan curve $\widetilde \mathcal{C}\subset S^2$ with
$\operatorname{post}(f)=\operatorname{post}(f^n)\subset \widetilde \mathcal{C}$. For such $n$ let $F=f^n$,
$\mathcal{D}^0=\mathcal{D}^0(F,\widetilde \mathcal{C})$, and $\mathcal{D}^1(F,\mathcal{C})$. Define the
orientation-preserving labeling $L\:\mathcal{D}^1\rightarrow \mathcal{D}^0$ by $L(c)=F(c)$
for $c\in \mathcal{D}^1$. Then it is clear that $(\mathcal{D}^1, \mathcal{D}^0, L)$ is a
two-tile subdivision rule that is realized by $F$ (see
Definition~\ref{def:subdivcomb} and the following discussion).
\end{proof}
\subsection{Iterative construction of invariant curves}
\label{subsec:ittproc} $\quad$
\noindent
Given data as in Theorem
\ref{thm:exinvcurvef} (\ref{item:ex_invC2}), the
$f$-invariant curve $\widetilde{\mathcal{C}}$ can be obtained by an iterative
procedure. To explain this,
we first recall the definition of {\em Hausdorff convergence}
of sets.
Let $(X,d)$ be a compact metric space. If $A,B\subset X$ are subsets of
$X$, then their {\em Hausdorff distance}\index{Hausdorff distance}
is defined as
\begin{equation}
\label{eq:def_Hausdorffd}
\operatorname{dist}^H_d(A,B)=\inf\{\delta>0: A\subset \mathcal{N}_d^\delta (B)\text{
and } B \subset \mathcal{N}_d^\delta (A)\}.
\end{equation}
Assume $A$ and $A_n$ for $n\in \N$ are closed subsets of $X$. We say
that $A_n\to A$ as $n\to \infty$ {\em in the sense of Hausdorff
convergence}\index{Hausdorff convergence} if
$$ \lim_{n\to \infty}\operatorname{dist}^H_d(A_n,A)=0.$$
Note that in this case a point $x\in X$ lies in $A$ if and only if
there exists a sequence $(x_n)$ of points in $X$ such that $x_n\in
A_n$ for $n\in \N$ and $x_n\to x$ as $n\to \infty$.
Now let $f\: S^2\rightarrow S^2$ be an expanding Thurston map, $\mathcal{C},\mathcal{C}'\subset S^2$ be Jordan curves with $\operatorname{post}(f)\subset \mathcal{C},\mathcal{C}'$ and
$\mathcal{C}'\subset f^{-1}(\mathcal{C})$, and let $H\colon S^2\times
I\to S^2$ be an isotopy rel.\ $\operatorname{post}(f)$ that deforms $\mathcal{C}$ to
$\mathcal{C}'$, i.e., $H_0=\id_{S^2}$ and $H_1(\mathcal{C})=\mathcal{C}'$. For the moment we
do \emph{not} assume that the map $\widehat{f}= H_1\circ f$ is
combinatorially expanding for $\mathcal{C}'$.
Let $H^0:= H$. Using Proposition \ref{prop:isotoplift} repeatedly, we can find isotopies $H^n\colon S^2\times I\to S^2$ rel.\
$f^{-1}(\operatorname{post}(f))$ such that $H^n_0=\id_{S^2}$ and $f \circ H_t^n =
H_t^{n-1} \circ f$ for all $n\in \N$, $t\in I$.
Now define Jordan curves inductively by setting
$\mathcal{C}^0:= \mathcal{C}$, and $\mathcal{C}^{n+1}:= H^n_1(\mathcal{C}^n)$ for
$n\in\N_0$. Note that then $\mathcal{C}^1=\mathcal{C}'$.
To summarize, we start with the following data (for a given $f$):
\begin{itemize}
\smallskip
\item[] a Jordan curve $\mathcal{C}^0=\mathcal{C}\subset S^2$ with $\operatorname{post}(f)\subset \mathcal{C}^0$,
\smallskip \item[]
a Jordan curve $\mathcal{C}^1=\mathcal{C}'\subset S^2$ isotopic to $\mathcal{C}^0\subset S^2$ rel.\
$\operatorname{post}(f)$ with $\mathcal{C}^1\subset f^{-1}(\mathcal{C}^0)$,
\smallskip
\item[]
an isotopy $H^0\:S^2\times I\rightarrow S^2$ rel.\ $\operatorname{post}(f)$ such that
$H^0_0=\id_{S^2}$ and $H^0_1(\mathcal{C}^0)=\mathcal{C}^1$.
\end{itemize}
We then define inductively:
\begin{itemize}
\smallskip
\item[] isotopies $H^n\:S^2\times I\rightarrow S^2$ such that $H^n_0=\id_{S^2}$ and $f \circ H_t^n =
H_t^{n-1} \circ f$ for all $n\in \N$, $t\in I$,
\smallskip
\item[]
Jordan curves $ \mathcal{C}^{n+1}:= H^n_1(\mathcal{C}^n)$ for $n\in \N_0$.
\end{itemize}
Figure~\ref{fig:invC_constr} illustrates this procedure for
Example \ref{ex:invC}. Since this example is rather complicated and it is hard to grasp the involved isotopies, we present a simpler example
for the construction.
\ifthenelse{\boolean{nofigures}}{}{
\begin{figure}
\centering
\begin{overpic}
[width=12cm,
tics=20]{Citerate.eps}
\put(31,87.8){$H^0_1$}
\put(29.7,51.5){$H^1_1$}
\put(31,14){$\dots$}
\put(4,70){$\mathcal{C}^0$}
\put(39,70){$\mathcal{C}^1$}
\put(4,34){$\mathcal{C}^1$}
\put(39,34){$\mathcal{C}^2$}
\put(4,-2){$\mathcal{C}^2$}
\put(39,-2){$\widetilde{\mathcal{C}}$}
\end{overpic}
\caption{Iterative construction of an invariant curve.}
\label{fig:Cit}
\end{figure}
}
\begin{ex}\label{ex:Cit} Let $f$ be the Latt\`{e}s map obtained as in
(\ref{eq:Lattes}), where we choose
\begin{equation*}
\psi\colon \C\to \C, \quad u \mapsto \psi(u):=5u.
\end{equation*}
It is straightforward to check that the extended real line
$\mathcal{C}:=\widehat{\R}=\R\cup\{\infty\}$ is $f$-invariant and contains all
postcritical points $0,1,\infty,-1$ of $f$.
As in Figure \ref{fig:mapg} we represent the sphere as a
pillow, i.e., two squares glued together along their boundary. The
boundary of the pillow (i.e., the boundary along which the two squares
were glued together) represents the curve $\mathcal{C}$, the two
squares represent the $0$-tiles, one of which is colored white, the
other black.
The map $f$ can then be described as follows. Each
of the two sides of the pillow is divided into $5\times 5$ squares,
which are colored in a checkerboard fashion. The map $g$ acts by
mapping each small white square to the white side of the pillow, and each
small black square to the black side.
The two sides of the
pillow are the $0$-tiles with respect to $\mathcal{C}$; the $4$ vertices of
the pillow are the postcritical points in this model.
The small
squares are the $1$-tiles (for $(f,\mathcal{C})$). The coloring of the $0$-
and $1$-tiles corresponds to a labeling map $L_\X$ as in
Lemma~\ref{lem:labelexis}.
There exist $f$-invariant Jordan curves that are isotopic to $\mathcal{C}$
rel.\ $\operatorname{post}(f)$, but distinct from $\mathcal{C}$. The construction of one
such curve is illustrated in Figure~\ref{fig:Cit}. Namely, we set
$\mathcal{C}^0:= \mathcal{C}$. The Jordan curve $\mathcal{C}^1$ is shown in the top right, as
well as
in the middle left picture. In the latter picture, we see that $\mathcal{C}^1$
consists of $1$-edges, i.e., $\mathcal{C}^1\subset f^{-1}(\mathcal{C}^0)$. Moreover, there
exists an isotopy $H^0\colon S^2\times I\to S^2$ rel.\ $\operatorname{post}(f)$ that
deforms $\mathcal{C}^0$ to $\mathcal{C}^1$ (i.e., $H^0_0= \id_{S^2}$ and
$H^0_1(\mathcal{C}^0)=\mathcal{C}^1$). We also see here
how the black and the white $0$-tile are deformed by $H^0_1$;
namely, the four
small black squares in the top left of
Figure \ref{fig:Cit} are part of the image of the black $0$-tile
(which is at the back of the pillow) under $H^0_1$.
The Jordan curve $\mathcal{C}^2:= H^1_1(\mathcal{C}^1)$ consists of $2$-edges, i.e.,
$\mathcal{C}^2\subset g^{-2}(\mathcal{C}^0)$, see the bottom left.
The two pictures in the middle of Figure \ref{fig:Cit} indicate how
$H^1$ deforms $1$-tiles. Roughly speaking $H^1$ deforms
each black/white $1$-tile ``in the same fashion'' as $H^0$ deforms the
black/white $0$-tiles.
The curves $\mathcal{C}^n$
Hausdorff converge to $\widetilde{\mathcal{C}}$, which is a $g$-invariant
Jordan curve with $\operatorname{post}(g)\subset \widetilde{\mathcal{C}}$ (see
Lemma \ref{lem:CCnprop} (viii) and Proposition \ref{prop:invCit}).
\end{ex}
\smallskip
There is a conceptually different way to obtain $\mathcal{C}^{n+1}$ from
$\mathcal{C}^n$, which will be explained in detail in Remark
\ref{rem:C_arc_replace}. Loosely speaking, we replace each $n$-edge
$\alpha^n\subset \mathcal{C}^n$ by $(n+1)$-edges ``in the same fashion'' as
the $0$-edge $\alpha^0:= f^n(\alpha^0)\subset \mathcal{C}^0$ is replaced by
the arc $\beta^1\subset \mathcal{C}^1$ with the same
endpoints (which are postcritical points) as $\alpha^0$. Note that
$\beta^1=H^0_1(\alpha^0)$, and that $\beta^1$ consists of $1$-edges.
\medskip
To prepare the proof that under suitable conditions our iteration
process has an invariant Jordan curve as a Hausdorff limit,
we summarize some properties of the Jordan curves $\mathcal{C}^n$.
\begin{lemma} \label{lem:CCnprop}
Let $f\colon S^2\rightarrow S^2$ be an expanding Thurston map, and the
Jordan curves $\mathcal{C}^n$ for $n\in \N_0$ be defined as above.
Then the following statements are true:
\begin{enumerate}[{\upshape(i)}]
\smallskip
\item \label{CCprop1}
$ \mathcal{C}^{n+k}\subset f^{-k}(\mathcal{C}^{n})$ for
$n,k\in \N_0$.
\smallskip
\item \label{CCprop2}
$\mathcal{C}^{n+k}$ is isotopic to $
\mathcal{C}^n$ rel.\ $f^{-n}(\operatorname{post}(f))$ for $n,k\in \N_0$.
\smallskip
\item \label{CCprop3}
$\mathcal{C}^{n+k}\cap f^{-n}(\operatorname{post}(f))=
\mathcal{C}^n\cap f^{-n}(\operatorname{post}(f))$ for $n,k\in \N_0$.
\smallskip
\item \label{CCprop4}
$\operatorname{post}(f)\subset \mathcal{C}^n$ for $n\in \N_0$.
\smallskip
\item \label{CCprop4a}
For $n,k\in \N_0$ the curve $ \mathcal{C}^{n+k}$ consists of $n$-edges
for $(f, \mathcal{C}^k)$.
\smallskip
\item \label{CCprop5}
For $n\in \N$ the curve $\mathcal{C}^n$ is the unique Jordan curve in $S^2$
with $\mathcal{C}^n\subset f^{-1}(\mathcal{C}^{n-1})$ that is isotopic to $\mathcal{C}^1$ rel.\
$f^{-1}(\operatorname{post}(f))$.
\smallskip
\item \label{CCprop6}
The sequence $\mathcal{C}^n$, $n\in \N_0$, only depends on $\mathcal{C}^0$ and
$\mathcal{C}^1$ and not of the choice of the initial isotopy $H=H^0$ used
in the definition of the sequence.
\smallskip
\item \label{CCprop7}
As $n\to \infty$ the sets $\mathcal{C}^n$ Hausdorff converge to a closed
$f$-invariant set $\widetilde \mathcal{C}\subset S^2$ (i.e., $f(\widetilde
\mathcal{C})\subset \widetilde \mathcal{C})$ with $\operatorname{post}(f)\subset \widetilde \mathcal{C}$.
\end{enumerate}\end{lemma}
\begin{proof} In the following we use the isotopies $H^n$ as in the definition of the sequence $\mathcal{C}^n$, and set $h_n=H^n_1$ for $n\in \N_0$.
\smallskip
(i) It suffices to show that $\mathcal{C}^n \subset f^{-1}(\mathcal{C}^{n-1})$ for $n\in \N$. We prove this by induction on $n$; this is clear for $n=1$. Assume that
the statement holds for some $n\in \N$; so $\mathcal{C}^n\subset
f^{-1}(\mathcal{C}^{n-1})$. Since $h_{n}=H^n_1$ and $h_{n-1}=H^{n-1}_1$ are homeomorphisms with $f\circ h_{n}=h_{n-1}\circ f$,
we have $h_n(f^{-1}(\mathcal{C}^{n}))= f^{-1}(h_{n-1}(\mathcal{C}^{n}))$ by Lemma
\ref{lem:lifts_inverses}. Thus
\begin{equation*}
\mathcal{C}^{n+1}= h_n(\mathcal{C}^n) \subset h_n(f^{-1}(\mathcal{C}^{n-1}))=
f^{-1}(h_{n-1}(\mathcal{C}^{n-1}))= f^{-1}(\mathcal{C}^n),
\end{equation*}
and (i) follows.
\smallskip
(ii)--(iv) From the definition of $H^n$, the remark after the proof of Proposition~\ref{prop:isotoplift}, and induction on $n$ we conclude that $H^n$
is an isotopy rel.\ $f^{-n}(\operatorname{post}(f))$. Since $H^n_0=\id_{S^2}$ and $$f^{-n}(\operatorname{post}(f))\subset f^{-(n+k)}(\operatorname{post}(f)) $$ for $n,k\in \N_0$,
statements (ii) and (iii) immediately
follow from this by induction on $k$ for fixed $n$. Statement (iv) follows from (iii) (with $n=0$ and $k\in \N_0$ arbitrary) and the fact that $\operatorname{post}(f)\subset \mathcal{C}^0$.
\smallskip
(v) By (iv) we have $\#(f^{-n}(\operatorname{post}(f))\cap \mathcal{C}^{n+k})\ge
\#\operatorname{post}(f)\ge 3.$ In particular, the points in $f^{-n}(\operatorname{post}(f))$ that
lie on $\mathcal{C}^{n+k}$
subdivide this curve into arcs whose endpoints lie in $f^{-n}(\operatorname{post}(f))$ and whose interiors are disjoint from $f^{-n}(\operatorname{post}(f))$.
Let $\alpha\subset \mathcal{C}^{n+k}$ be one of these arcs. Then
we have $\inte(\alpha)\subset f^{-n}(\mathcal{C}^k)\setminus f^{-n}(\operatorname{post}(f))$ by (i), and
$\partial \alpha \subset f^{-n}(\operatorname{post}(f))$. Since by Proposition~\ref{prop:celldecomp}~(iii) the set $f^{-n}(\mathcal{C}^k)$ is the $1$-skeleton and the
set $f^{-n}(\operatorname{post}(f))$ the $0$-skeleton of the cell decomposition $\mathcal{D}^n(f, \mathcal{C}^k)$, we conclude from Lemmas~\ref{lem:conncomp} and \ref{lem:opencells} that $\alpha$ is an edge in $\mathcal{D}^n(f, \mathcal{C}^k)$, i.e., an $n$-edge for $(f,\mathcal{C}^k)$. Hence $\mathcal{C}^{n+k}$ consists of $n$-edges for
$(f,\mathcal{C}^k)$.
\smallskip
(vi) By (i) and (ii) we know that $\mathcal{C}^n$ for $n\in \N$ is a Jordan curve
with $\mathcal{C}^n\subset f^{-1}(\mathcal{C}^{n-1})$ that is isotopic to $\mathcal{C}^1$ rel.\ $f^{-1}(\operatorname{post}(f))$. Let $\widehat \mathcal{C}\subset f^{-1}(\mathcal{C}^{n-1})$ be another Jordan curve isotopic to $\mathcal{C}^1$ rel.\ $f^{-1}(\operatorname{post}(f))$. Then $\mathcal{C}^n$ and $\widehat \mathcal{C}$ are isotopic to each other rel.\ $f^{-1}(\operatorname{post}(f))$. Note that $f^{-1}(\mathcal{C}^{n-1})$ is the $1$-skeleton of the cell decomposition $\mathcal{D}^1(f, \mathcal{C}^{n-1})$ and $f^{-1}(\operatorname{post}(f))$ its set of vertices. Hence by
Lemma~\ref{lem:isoJcin1ske} we have $\widehat \mathcal{C}=\mathcal{C}^n$, and the uniqueness statement for $\mathcal{C}^n$ follows.
\smallskip
(vii) It follows from (vi) and induction on $n$ that $\mathcal{C}^n$ is uniquely determined by $\mathcal{C}^0$ and $\mathcal{C}^1$.
\smallskip
(viii) Pick some visual metric $d$ for $f$, and let $\Lambda >1$ be
the expansion factor of $d$. By Lemma~\ref{lem:exp_shrink} the
diameters of the tracks
of the isotopy $H^n$ are bounded by $C\Lambda^{-n}$, where $C$ is a
fixed constant. Since $H^n_0=\id_{S^2}$ and $\mathcal{C}^{n+1}=H^n_1(\mathcal{C}^n)$
for
$n\in \N_0$ this implies that $\operatorname{dist}_d^N(\mathcal{C}^n,\mathcal{C}^{n+1})\le
C\Lambda^{-n}$ for $n\in \N_0$. It follows that the sequence $\mathcal{C}^n$
is a Cauchy sequence with respect to Hausdorff distance. Now the space
of all non-empty closed subsets of a compact metric space is compact
if it is equipped with the Hausdorff distance. Hence there exists a
subsequence
of the sequence $\mathcal{C}^n$ that converges in the Hausdorff sense
to a non-empty closed set $\widetilde \mathcal{C}\subset S^2$. Since $\mathcal{C}^n$ is a Cauchy sequence,
it follows that $\mathcal{C}^n\to \widetilde \mathcal{C}$ as $n\to \infty$ in the Hausdorff
sense. Since $\operatorname{post}(f)\subset \mathcal{C}^n$ for all $n\in \N_0$ by (iv), we have
$\operatorname{post}(f)\subset \widetilde \mathcal{C}$.
It remains to show that $\widetilde \mathcal{C}$ is $f$-invariant. To see this, let $p\in \widetilde \mathcal{C}$ be arbitrary. Then there exists a sequence $(p_n)$
of points in $S^2$ such that $p_n\in \mathcal{C}^n$ for $n\in \N_0$ and $p_n\to p$ as $n\to \infty$. By continuity of $f$ we have $f(p_n)\to f(p) $ as $n\to \infty$. Moreover, (i) implies that $f(p_n)\in \mathcal{C}^{n-1}$ for $n\in \N$. Hence $f(p)\in \widetilde\mathcal{C}$, and so the set $\widetilde \mathcal{C}$ is indeed $f$-invariant.
\end{proof}
As an application of the preceding setup we prove a statement
that gives a necessary and sufficient condition for the map $\widehat{f}$
in Theorem~\ref{thm:exinvcurvef} to be combinatorially expanding.
\begin{prop}\label{prop:nscombexp}
Let $f\:S^2\rightarrow S^2$ be an expanding Thurston map and the isotopy $H^0\:S^2\times I\rightarrow S^2$ and Jordan curves $\mathcal{C}^n$ for $n\in \N_0$ be defined as above.
Then $\widehat{f}=H^0_1 \circ f$ is
combinatorially expanding for $\mathcal{C}^1=\mathcal{C}'$ if and only if there exists $n\in \N$ such
that no $n$-tile for $(f,\mathcal{C}^0)$ joins opposite sides
of $\mathcal{C}^n$.
\end{prop}
\begin{proof} Let $H^n$ for $n\in \N_0$ be the isotopies used in the definition of the curves $\mathcal{C}^n$. Set $h_n:= H_1^n$. Then $\widehat f=
h_0\circ f$, $\mathcal{C}^{n+1}=h_n(\mathcal{C}^n)$, and $h_n\circ f = f
\circ h_{n+1}$ for $n\in \N_0$. It follows by induction that for $n\in \N$ we have
$$
\widehat{f}^n =h_0\circ f \circ \dots \circ h_0\circ f= h_0 \circ f^n \circ h_{n-1} \circ \dots \circ h_1, $$
and so
$$ h_0 \circ f^n= \widehat{f}^n\circ h_1^{-1}\circ \dots h_{n-1}^{-1} . $$
Hence
$$ f^{-n}(\mathcal{C}^0)=f^{-n}(h_0^{-1}(\mathcal{C}^1))=
( h_{n-1}\circ \dots \circ h_1) ( \widehat{f}^{-n}(\mathcal{C}^1)). $$
Recall that the $n$-tiles for $(f,\mathcal{C}^0)$ are the closures of the
complementary components of $f^{-n}(\mathcal{C}^0)$, and the $n$-tiles for $(\widehat f, \mathcal{C}^1)$ the closures of the
complementary components of $\widehat {f}^{-n}(\mathcal{C}^1)$ (Proposition \ref{prop:celldecomp}~(v)). So from the previous identity we conclude that the
$n$-tiles for $(f,\mathcal{C}^0)$ are precisely the images of
the $n$-tiles for $(\widehat{f},\mathcal{C}^1)$ under the
homeomorphism $h_{n-1}\circ \dots \circ h_1$. Note that this
homeomorphism is isotopic to $\id_{S^2}$ rel. $\operatorname{post}(f)=\operatorname{post}(\widehat f)$ and maps $\mathcal{C}^1$ to $\mathcal{C}^n$. Thus no
$n$-tile for $(\widehat{f},\mathcal{C}^1)$ joins opposite
sides of $\mathcal{C}^1$ if and only if no $n$-tile for $(f,\mathcal{C}^0)$ joins opposite sides of $\mathcal{C}^n$.
Now $\widehat{f}$ is combinatorially expanding for $\mathcal{C}^1$ if and only if
there exists $n\in \N$ such that no $n$-tile for $(\widehat{f}, \mathcal{C}^1)$ joins opposite sides of $\mathcal{C}^1$. By what we have seen, this is the case if and only if there exists $n\in \N$ such that no $n$-tile for $(f,\mathcal{C}^0)$
joins opposite sides of $\mathcal{C}^n$.
\end{proof}
The Hausdorff limit $\widetilde \mathcal{C}$ of the curves $\mathcal{C}^n$
as provided by Lem\-ma~\ref{lem:CCnprop}~(viii) will not be a Jordan curve in general. The following proposition shows that this is the case
if the map $\widehat{f}=H^0_1 \circ f$ is combinatorially expanding for $\mathcal{C}^1$. Actually, one can also show that this condition is also necessary for $\widetilde \mathcal{C}$ to be a Jordan curve, but we will not present the proof for this statement as it is somewhat involved.
\begin{prop}[Iterative procedure to obtain an invariant curve]
\label{prop:invCit}
${}$
\noindent
Let $f\:S^2\rightarrow S^2$ be an expanding Thurston map and the isotopy $H^0\:S^2\times I\rightarrow S^2$ and Jordan curves $\mathcal{C}^n$ for $n\in \N_0$ be defined as above.
If $\widehat{f}=H^0_1 \circ f$ is combinatorially expanding for
$\mathcal{C}^1=\mathcal{C}'$, then
$\mathcal{C}^n$ Hausdorff converges to a
Jordan curve $\widetilde \mathcal{C}\subset S^2$ as $n\to \infty$.
In this case, the curve $\widetilde \mathcal{C}$ is $f$-invariant
and $\operatorname{post}(f)\subset \widetilde \mathcal{C}$. Furthermore $\widetilde{\mathcal{C}}$ is
isotopic to $\mathcal{C}^1$ rel.\ $f^{-1}(\operatorname{post}(f))$.
\end{prop}
\begin{proof} Suppose that $\widehat f=H^0_1\circ f$ is combinatorially expanding for $\mathcal{C}^1$. From Theorem
\ref{thm:exinvcurvef} it follows that there exists an $f$-invariant
Jordan curve $\widetilde{\mathcal{C}}\subset S^2$ with $\operatorname{post}(f)\subset \widetilde \mathcal{C}$ that is isotopic to $\mathcal{C}^0$ rel.\ $\operatorname{post}(f)$ and isotopic to $\mathcal{C}^1$ rel.\
$f^{-1}(\operatorname{post}(f))$. Let $K^0\colon S^2\times I\to S^2$ be an isotopy
rel.\ $\operatorname{post}(f)$ that deforms $\widetilde{\mathcal{C}}$ to $\mathcal{C}^0$;
so $K^0_0=\id_{S^2}$ and $K^0_1(\widetilde{\mathcal{C}})= \mathcal{C}^0$. Using
Proposition \ref{prop:isotoplift} repeatedly, we can find isotopies $K^n\: S^2\times I\rightarrow S^2$ rel.\ $f^{-1}(\operatorname{post}(f))$ with $K^n_0=\id_{S^2}$ such that
$f\circ K^n_1=K^{n-1}_1\circ f$ for $n\in \N$.
{\em Claim.} $ \widetilde \mathcal{C}^n:= K^n_1(\widetilde \mathcal{C})=\mathcal{C}^n$ for all $n\in \N_0$.
We prove this claim by induction on $n$; it follows from the choice of $K^0$ for
$n=0$. Assume that the statement is true for some $n\in \N_0$. Then
$K_1^{n}(\widetilde \mathcal{C})= \mathcal{C}^n$, and so by Lemma
\ref{lem:lifts_inverses} we have
$$\widetilde \mathcal{C}^{n+1}=K_1^{n+1}(\widetilde \mathcal{C})\subset
K_1^{n+1}(f^{-1}(\widetilde \mathcal{C}))=f^{-1}( K_1^{n}(\widetilde \mathcal{C}))=
f^{-1}(\mathcal{C}^n).$$
Since $K^{n+1}$ is an isotopy rel.\
$f^{-1}(\operatorname{post}(f))$, the curve $\widetilde \mathcal{C}^{n+1}$ is isotopic to
$\widetilde \mathcal{C}$ and hence to $\mathcal{C}^1$ rel.\ $f^{-1}(\operatorname{post}(f))$. So
Lemma \ref{lem:CCnprop}~\eqref{CCprop5} implies that
$\widetilde \mathcal{C}^{n+1}=\mathcal{C}^{n+1}$ proving the claim.
By Lemma \ref{lem:exp_shrink} the maps $K^n_1$ converge uniformly to the
identity on $S^2$. Hence $\mathcal{C}^n=K^n(\widetilde\mathcal{C})$ Hausdorff
converges to the Jordan curve $\widetilde{\mathcal{C}}$ as $n\to \infty$.
The statement follows.
\end{proof}
\begin{rem}
\label{rem:iterate_given_Ct} If $f\: S^2\rightarrow S^2$ is an expanding Thurston map, then
every $f$-invariant Jordan curve $\widetilde{\mathcal{C}}$ with
$\operatorname{post}(f)\subset \widetilde{\mathcal{C}}$ can be obtained by our iterative
procedure. Indeed, suppose that $\widetilde{\mathcal{C}}$ is such a curve.
Trivially, we can then take $\mathcal{C}=\mathcal{C}^0=\widetilde \mathcal{C}$, $\mathcal{C}'=\mathcal{C}^1=\widetilde \mathcal{C}$, and $H^0_t=\id_{S^2}$ for $t\in I$. Then $\mathcal{C}^n=\widetilde
\mathcal{C}$ for all $n\in \N_0$ and so $\mathcal{C}^n \to \widetilde \mathcal{C}$ as $n\to \infty$.
Actually, a much stronger statement is true. Namely, we can start
with {\em any} Jordan curve $\mathcal{C}$ in the same isotopy class rel.\
$\operatorname{post}(f)$ as $\widetilde \mathcal{C}$. Suppose that $\mathcal{C}$ is such a curve. First, we claim that then there exists a unique Jordan curve $\mathcal{C}'\subset f^{-1}(\mathcal{C})$ that is isotopic to $\widetilde \mathcal{C}$ rel.\ $f^{-1}(\operatorname{post}(f))$. To see this, let $K^0\colon S^2\times I \rightarrow
S^2$ be an isotopy rel.\ $\operatorname{post}(f)$ with $K^0_0=\id_{S^2}$ and
$K^0_1(\widetilde{\mathcal{C}})=\mathcal{C}$.
By Proposition
\ref{prop:isotoplift} we can lift $K^0$ by $f$ to an isotopy $K^1$ rel.\ $f^{-1}(\operatorname{post}(f))$ with $K^1_0=\id_{S^2}$ and $K^0_t
\circ f = f\circ K^1_t$ for all $t\in I$.
Then the Jordan curve $\mathcal{C}': = K^1_1(\widetilde{\mathcal{C}})$ satisfies
\begin{equation*}
\mathcal{C}'= K^1_1(\widetilde{\mathcal{C}}) \subset
K^1_1(f^{-1}(\widetilde{\mathcal{C}})) = f^{-1}(K^0_1(\widetilde{\mathcal{C}})) =
f^{-1}(\mathcal{C}).
\end{equation*}
Here we used $\widetilde{\mathcal{C}}\subset f^{-1}(\widetilde{\mathcal{C}})$ and Lemma \ref{lem:lifts_inverses}. This shows existence of a curve $\mathcal{C}'$ with the desired properties. Uniqueness of $\mathcal{C}'$ follows from
Lemma~\ref{lem:isoJcin1ske}.
Define $H\: S^2\times I\rightarrow S^2$ by setting
$H_t=K^1_t\circ (K^0_t)^{-1}$ for $t\in I$. Then $H$ is an isotopy rel.\ $\operatorname{post}(f)$ that deforms $\mathcal{C}^0:=\mathcal{C}$ into
$\mathcal{C}^1:=\mathcal{C}'$. Indeed, we have $ H_0=\id_{S^2}$ and
$$H_1(\mathcal{C}^0)=K^1_1( (K^0_1)^{-1}(\mathcal{C}))=K^1_1(\widetilde \mathcal{C})=\mathcal{C}'=\mathcal{C}^1. $$
Moreover,
$$\widehat{f}:= H_1\circ f=
K^1_1\circ (K^0_1)^{-1} \circ f = K^1_1 \circ f \circ
(K^1_1)^{-1}. $$ Thus it follows from Lemma \ref{lem:cexp_Cinv} that
$\widehat{f}$ is combinatorially expanding for $\mathcal{C}^1=\mathcal{C}'=K^1_1(\widetilde \mathcal{C})$.
Define the sequence $\{\mathcal{C}^n\}$ starting from $\mathcal{C}^0$ and $ \mathcal{C}^1$ as
before. From Proposition \ref{prop:invCit} it follows that as $n\to \infty$ the curves $\mathcal{C}^n$ Hausdorff
converge to an $f$-invariant Jordan curve that is isotopic to $\mathcal{C}^1$, and
hence isotopic to $\widetilde{\mathcal{C}}$, rel.\
$f^{-1}(\operatorname{post}(f))$. From Theorem \ref{thm:uniqc} it follows that the
unique such curve is $\widetilde{\mathcal{C}}$. Thus $\mathcal{C}^n\to\widetilde{\mathcal{C}}$ in the Hausdorff sense as $n\to \infty$.
\end{rem}
\medskip
\begin{rem}\label{rem:C_arc_replace} In the inductive definition of
$\mathcal{C}^{n+1}=H^n_1(\mathcal{C}^n)$ one
can construct $\mathcal{C}^{n+1}$ from $\mathcal{C}^n$ by an {\em edge replacement
procedure}
without explicitly knowing the isotopy
$H^n$. To explain this, suppose that $n\in \N$, and that $\mathcal{C}^n$ has
already been constructed (starting from given curves $\mathcal{C}^0$ and
$\mathcal{C}^1$). We know by Lemma~\ref{lem:CCnprop}~\eqref{CCprop4a}
that $\mathcal{C}^n$ consists of $n$-edges $\alpha^n$ for $(f,\mathcal{C}^0)$.
Then
$\mathcal{C}^{n+1}$ is obtained from $\mathcal{C}^n$ by replacing
each $n$-edge $\alpha^n\subset \mathcal{C}^n$ by a certain arc $\beta^{n+1}$
with the same endpoints as $\alpha^n$.
Indeed, we can set $\beta^{n+1}:=H^n_1(\alpha^n)\subset \mathcal{C}^{n+1}$. Then the union of these arcs $\beta^{n+1}$ is equal to $\mathcal{C}^{n+1}$. Moreover, since $H^n$ is an isotopy relative to the set $f^{-n}(\operatorname{post}(f))$ of $n$-vertices, and $\alpha^n$ is an $n$-edge for $(f,\mathcal{C}^0)$ and so has $n$-vertices as endpoints, the arcs $\alpha^n$ and $\beta^{n+1}$ have the same endpoints.
\smallskip
Now the arc $\beta^{n+1}$ is the unique arc in $f^{-n}(\mathcal{C}^1)$ that is isotopic
to $\alpha^n$ rel.\ $f^{-n}(\operatorname{post}(f))$. This property often allows
one to determine $\beta^{n+1}$ from $\alpha^n$ without knowing
$H^n$ explicitly.
To see that this characterization of $\beta^{n+1}$ holds, note that by Lem\-ma~\ref{lem:CCnprop} (i) we have $\beta^{n+1}\subset \mathcal{C}^{n+1}\subset f^{-1}(\mathcal{C}^n)$.
Moreover, $\beta^{n+1}=H^n_1(\alpha^n)$ is isotopic to $\alpha^n$ rel.\ $f^{-n}(\operatorname{post}(f))$.
Suppose $\widetilde \beta^{n+1}\subset f^{-n}(\mathcal{C}^1)$ is another arc
that is isotopic to $\alpha^n$ rel.\ $f^{-n}(\operatorname{post}(f))$.
Then the arcs $\beta^{n+1}$ and $\widetilde \beta^{n+1}$ have endpoints in
$f^{-n}(\operatorname{post}(f))$, but contain no other points in this set, since
this is true for $\alpha^n$. This and the inclusions
$\beta^{n+1},\widetilde \beta^{n+1}\subset f^{-n}(\mathcal{C}^1)$ imply that
$\beta^{n+1}$ and $\widetilde \beta^{n+1}$ are $n$-edges for $(f, \mathcal{C}^1)$
(see the argument in the proof of Lemma~\ref{lem:CCnprop}~\eqref{CCprop4a}).
Since
$\beta^{n+1}$ and $\widetilde \beta^{n+1}$ are isotopic relative to the set $f^{-n}(\operatorname{post}(f))$, which is the $0$-skeleton of $\mathcal{D}^n(f, \mathcal{C}^1)$, it follows from the first part of the proof of Lemma~\ref{lem:isoJcin1ske} that $\beta^{n+1}=\widetilde
\beta^{n+1}$ as desired.
\smallskip
As we have just seen, $\beta^{n+1}$ is an $n$-edge for $(f,\mathcal{C}^1)$.
Since $\beta^{n+1}$ has endpoints in the set
$f^{-n}(\operatorname{post}(f))\subset f^{-(n+1)}(\operatorname{post}(f))$ and $\beta^{n+1}\subset\mathcal{C}^{n+1}\subset
f^{-(n+1)}(\mathcal{C}^0)$, a similar argument also shows that $\beta^{n+1}$ consists of $(n+1)$-edges for $(f,\mathcal{C}^0)$.
\ifthenelse{\boolean{nofigures}}{}{
\begin{figure}
\centering
\begin{overpic}
[width=12cm,
tics=20]{edge_replace.eps}
\put(14,31){$\alpha^0\subset \mathcal{C}^0$}
\put(47,41){$H^0_1$}
\put(47,6){$H^n_1$}
\put(20,18.5){$f^n|\alpha^n$}
\put(17,-3){$\alpha^n\subset\mathcal{C}^n$}
\put(79,-3.4){$\beta^{n+1}\subset \mathcal{C}^{n+1}$}
\put(83,18.5){$f^n|\beta^{n+1}$}
\put(76,31){$\beta^1\subset\mathcal{C}^1$}
\end{overpic}
\caption{Iterative construction by replacing edges.}
\label{fig:replace_edge}
\end{figure}
}
\smallskip
One can look at the arc replacement procedure $\alpha^n \to \beta^{n+1}$ from yet another point of view. Since $\alpha^n$ is an $n$-edge for $(f,\mathcal{C}^0)$, the map $f^n|\alpha^n$ is a homeomorphism of $\alpha^n$
onto the $0$-edge
$\alpha^0:=f^n(\alpha^n)\subset \mathcal{C}^0$ for $(f, \mathcal{C}^0)$
(Proposition~\ref{prop:celldecomp}~(i)). The endpoints of $\alpha^0$ lie in $\operatorname{post}(f)$. Then
$\beta^1:=H^0_1(\alpha^0)$ is the unique subarc of $ \mathcal{C}^1$ that has
the same endpoints as $\alpha^0$, but contains no other points in
$\operatorname{post}(f)$ (here it is important that $\#(\mathcal{C}^1\cap\operatorname{post}(f))=\#\operatorname{post}(f)\ge 3)$. Since $f^n\circ H^n_1=H^0_1\circ f^n$ and $\beta^{n+1}=H^n_1(\alpha^n)$, the map $f^n|\beta^{n+1}$ is thus a homeomorphism of $\beta^{n+1}$ onto
$\beta^1$.
Often, this information (together with the fact that $\alpha^n$ and $\beta^{n+1}$ share endpoints) is enough to determine
$\beta^{n+1}$ uniquely.
We illustrate this procedure in Figure
\ref{fig:replace_edge}. Here the map $f$ (as well as the curves
$\mathcal{C}^0, \mathcal{C}^1,\dots$ and the isotopies $H^0,H^1,\dots$) are as in
Example \ref{ex:Cit}, see also Figure \ref{fig:Cit}.
\smallskip
For example, suppose that $\beta^1$ lies in {\em single} $0$-tile
$X^0$ for $(f, \mathcal{C}^0)$, i.e., in one of the Jordan regions bounded by
$\mathcal{C}^0$. This is not always true, but in Example \ref{ex:Cit} as well
as the Examples~\ref{ex:Cinv_notcexp} and \ref{ex:rect}
discussed below this is the case. Then there exists a unique
$n$-tile $X^n$ for $(f, \mathcal{C}^0)$ with $\alpha^n\subset \partial X^n$ and $f^n(X^n)=X^0$; if we assign colors to tiles for $(f, \mathcal{C}^0)$ as in Lemma~\ref{lem:colortiles},
then $X^n$ is the unique $n$-tile for $(f,\mathcal{C}^0)$ that contains
$\alpha^n$ in its boundary and has the same color as $X^0$.
Consider the arc $\widetilde \beta^{n+1}:=(f^n|X^n)^{-1}(\beta^1)\subset X^n$. Then
$\widetilde \beta^{n+1}$ has the same endpoints as
$(f^n|X^n)^{-1}(\alpha^0)=\alpha^n$ and is contained in $f^{-n}(\mathcal{C}^1)$.
Moreover, $\widetilde \beta^{n+1}$ is isotopic to $\alpha^n$ rel.\ $f^{-n}(\operatorname{post}(f))$; this easily follows from Lemma~\ref{twoarcs}, since our assumptions imply that one can find a suitable simply connected domain $\Om\subset S^2$ that contains $\widetilde \beta^{n+1}$ and $\alpha^n$ and no point in $f^{-n}(\operatorname{post}(f))$ except the endpoints of $\widetilde \beta^{n+1}$ and $\alpha^n$.
By what we have seen above, we conclude $\beta^{n+1}=\widetilde \beta^{n+1}$, and so
\begin{equation}\label{eq:arcreplacement}
\beta^{n+1}=(f^n|X^n)^{-1}(\beta^1).
\end{equation}
In the special case under consideration, this leads to a very convenient edge replacement procedure that can be summarized as follows:
Suppose the arc $\beta^1\subset \mathcal{C}^1$ corresponding to $\alpha^0=f^n(\alpha^n)\subset \mathcal{C}^0$ lies in a single $0$-tile $X^0$, and let $X^n$ be the $n$-tile that contains $\alpha^n$ its boundary and has the same color as $X^0$ (so that $f^n(X^n)=X^0)$. Then $\alpha^n$ is replaced by the arc $\beta^{n+1}$ in $X^n$ that corresponds to
$\beta^1\subset X^0$ under the homeomorphism $f^n|X^n$ of $X^n$ onto $X^0$. \end{rem}
\smallskip
The next example illustrates what happens if the map
$\widehat{f}$ in Proposition~\ref{prop:invCit} is not combinatorially expanding.
\begin{ex}
\label{ex:Cinv_notcexp}
Let $g$ be the Latt\`{e}s map obtained as in \eqref{eq:Lattes}, where
$$\psi\: \C\rightarrow \C, \quad u\mapsto \psi(u):=3u. $$ This map was
already considered in Example \ref{ex:exp_notcexp}. See also the
bottom of Figure \ref{fig:exp_notcexp}. We let $\mathcal{C}^0$ be the
boundary of the pillow. The curve $\mathcal{C}^1\subset g^{-1}(\mathcal{C}^0)$ is
drawn with a thick line
in the top left of Figure \ref{fig:Cinv_notexp}. Clearly there is an
isotopy $H^0$ rel.\ $\operatorname{post}(g)$ (the four vertices of the pillow) that
deforms $\mathcal{C}^0$ to $\mathcal{C}^1$. Note that $\widehat{g}=H^0_1\circ g$ is
not combinatorially expanding for $\mathcal{C}^1$, see Figure
\ref{fig:exp_notcexp}. Starting with the data $\mathcal{C}^0$, $\mathcal{C}^1$,
$H^0$, we inductively define Jordan curves
$\mathcal{C}^n$ as described before.
It is slightly more difficult than in Example \ref{ex:Cit} to see here how $\mathcal{C}^{n+1}$ evolves
from $\mathcal{C}^n$, since different $0$-edges
are replaces by different arcs (consisting of $1$-edges). Namely,
each $0$-edges that is drawn horizontally in Figure
\ref{fig:Cinv_notexp} is replaced by itself. Note that every
horizontal $1$-edge is mapped by $g$ to a horizontal $0$-edge, thus
is replaced by itself in the construction of $\mathcal{C}^2$ from $\mathcal{C}^1$
(see Remark \ref{rem:C_arc_replace}).
Then $\mathcal{C}^n\to
\widetilde{\mathcal{C}}$ as $n\to \infty$ in the Hausdorff sense, where
the set $\widetilde{\mathcal{C}}$
is indicated on the right of Figure
\ref{fig:Cinv_notexp}. In this case, $\widetilde{\mathcal{C}}$ is not a Jordan curve
and $S^2\setminus \widetilde{\mathcal{C}}$
has three components.
Of course, the ``self-intersections'' of the limit set
$\widetilde{\mathcal{C}}$ can be more complicated in general.
\end{ex}
\ifthenelse{\boolean{nofigures}}{}{
\begin{figure}
\centering
\begin{overpic}
[width=12cm,
tics=20]{Cnot_exp.eps}
\put(7,30){$\mathcal{C}^1$}
\put(-3.5,20){$\mathcal{C}^0$}
\put(54,2){$\widetilde{\mathcal{C}}$}
\end{overpic}
\caption[Example where $\widetilde{\mathcal{C}}$ is not a Jordan curve.]
{Since $\widehat{g}$ is not combinatorially expanding,
$\widetilde{\mathcal{C}}$ is not a Jordan curve.}
\label{fig:Cinv_notexp}
\end{figure}
}
\ifthenelse{\boolean{nofigures}}{}{
\begin{figure}
\centering
\begin{overpic}
[width=12cm,
tics=20]{5x5invC1.eps}
\put(-3,3){$\mathcal{C}^1$}
\put(54,3){$\widetilde{\mathcal{C}}$}
\end{overpic}
\caption{A non-trivial rectifiable invariant Jordan curve.}
\label{fig:not_rect_invC}
\end{figure}
}
We close this section with one more example.
It gives an example of a
non-trivial invariant curve that is rectifiable.
\begin{ex}
\label{ex:rect} Let $f$ be the map from Example
\ref{ex:Cit}, i.e., the Latt\`{e}s map obtained as in
(\ref{eq:Lattes}), where we choose $\psi\colon \C\to \C$, $u
\mapsto \psi(u):=5u$.
The curve $\mathcal{C}=\mathcal{C}^0$ is the boundary of the pillow as before. The
curve $\mathcal{C}^1$ (which is isotopic rel.\ $\operatorname{post}(f)$ by an isotopy
$H^0$) is the thick curve indicated in the left of Figure
\ref{fig:not_rect_invC}. Note that no $1$-tile for $(f,\mathcal{C}^0)$
connects opposite sides of $\mathcal{C}^1$. Thus the sequence of curves
$\{\mathcal{C}^n\}$, defined
as before, Hausdorff converges to an $f$-invariant Jordan curve
$\widetilde{\mathcal{C}}$ by Proposition \ref{prop:nscombexp} and
Proposition \ref{prop:invCit}.
We briefly explain the iterative construction of the curves
$\mathcal{C}^n$.
Note that the three $0$-edges on the top,
bottom, and right side of the pillow are deformed by $H^0$ to
themselves.
Recall that $f$
maps the lower left $1$-tile to the white $0$-tile by the map
$u\mapsto 5u$ and ``extends to other $1$-tiles by reflection''. Note
that $f$ maps all $1$-edges in $\mathcal{C}^1$ that are not on the left
$0$-edge, to one of the $0$-edges on
the top, right side, or bottom of the pillow. Thus they are replaced
by themselves when constructing $\mathcal{C}^2$ from $\mathcal{C}^1$, see Remark
\ref{rem:C_arc_replace}.
The resulting $f$-invariant Jordan curve $\widetilde{\mathcal{C}}$ is shown
on the right.
It is not hard to see that if the pillow is equipped with the flat
metric, then the $f$-invariant curve $\widetilde{\mathcal{C}}$ is rectifiable.
\end{ex}
\section{Invariant curves are quasicircles}
\label{sec:cc-quasicircle}
\noindent
Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. A map $h\: X \rightarrow Y$ is
called a {\em quasisymmetry}\index{quasisymmetry}
or {\em quasisymmetric} if it is a
homeomorphism and if there exists a homeomorphism
$\eta\:[0,\infty)\rightarrow [0,\infty)$ such that
$$ \frac {d_Y(h(u),h(v))}{d_Y(h(u),h(w))}\le \eta \left(
\frac {d_Y(u,v)}{d_Y(u,w)}\right)$$
for all $u,v,w\in X$, $u\ne w$. If we want to emphasize the distortion function $\eta$ here, then we call $f$ an {\em $\eta$-quasisymmetry} or
{\em $\eta$-quasisymmetric}.
The metric spaces $X$ and $Y$ are called {\em quasisymmetrically equivalent} if there exists a quasisymmetry $h\: X\rightarrow Y$. Two metrics $d$ and $d'$ on a space $X$ are called {\em quasisymmetrically equivalent}
if the identity map $\id_X\: (X,d)\rightarrow (X,d')$ is a quasisymmetry. Note that snowflake equivalence
in its various incarnations (see Section~\ref{sec:iso}) is stronger than quasisymmetric equivalence, since every snowflake equivalence between metric spaces is a quasisymmetry.
Suppose that $S$ is a {\em metric circle}, i.e., a metric space
homeomorphic to a circle. Then $S$ is called a {\em quasicircle}\index{quasicircle}
if it
is quasisymmetrically equivalent with the unit circle $\partial \D$ in
$\C$ (equipped with the Euclidean metric).
A metric space $X$ is called {\em doubling}\index{doubling}
if there exists $N\in \N$ such that every
open ball of radius $0<r\le 2 \diam(X)$ in $X$ can be covered by at most $N$ open balls of
radius $r/2$.
According to a theorem by Tukia and V\"ais\"al\"a \cite[p.~113, Thm.~4.9]{TV} a metric
circle $(S,d)$ is a quasicircle if and only if
\smallskip
\begin{itemize}
\item[(i)] $(S,d)$ is \defn{doubling},
\smallskip
\item[(ii)] $S$ satisfies the \defn{Ahlfors condition}:\index{Ahlfors condition} there is a constant
$K\ge 1$ such that for all points $x,y\in S$, $x\ne y$, we have
\begin{equation*}
\diam_d(\gamma)\le Kd(x,y)
\end{equation*}
for one of the subarcs $\gamma$ of $S$ with endpoints $x$ and $y$.
\end{itemize}
The {\em chordal metric}\index{chordal metric}
$\sigma$ on the Riemann sphere $\CDach$
is given by
$$ \sigma(z,w)=\frac {2|z-w|}{\sqrt {1+|z|^2} \sqrt {1+|w|^2}} $$
for $z,w\in \C$ and by an appropriate limit of this expression if $z=\infty$ or $w=\infty$.
If $J\subset \CDach$ is a Jordan curve, then, unless another metric is specified, we call $J$ a quasicircle if $J$ is a quasicircle if equipped with the chordal metric.
Since $J$ with the chordal metric is always a doubling metric space, a Jordan curve $J\subset \CDach$ is a
quasicircle if and only if $J$ satisfies the Ahlfors condition.
We are now ready to prove Theorem~\ref{thm:Cquasicircle} stated in the introduction.
\begin{proof}[Proof of Theorem \ref{thm:Cquasicircle}] Suppose $\mathcal{C}$ is an $f$-invariant Jordan curve as in the statement, and
let $d$ be a visual metric on $S^2$ with expansion factor $\Lambda>1$.
In the ensuing proof, we will consider edges for $(f,\mathcal{C})$.
Since $\mathcal{C}$ is $f$-invariant, edges are subdivided by edges of higher order (see Proposition~\ref{prop:invmarkov}~(iv)). The Jordan curve $\mathcal{C}$ is the union of all $0$-edges, so this implies that $\mathcal{C}$ is a union of $n$-edges for all
$n\in \N_0$.
If $n,k\in \N_0$ and $\widetilde e$ is an arbitrary $(n+k)$-edge with
$\widetilde e\subset \mathcal{C}$, then there exists a unique $n$-edge $e'$ with $\widetilde e\subset e'\subset \mathcal{C}$. To see this pick, $p\in \inte(\widetilde e)$. Then there exists an
$n$-edge $e'$ with $p\in e'\subset \mathcal{C}$. Since $e'$ is a union of $(n+k)$-edges, it follows from Lemma~\ref{lem:celldecompint}~(ii)
that $\widetilde e\subset e'$. Uniqueness of $e'$ is clear, because
$p$ is an interior point of each $n$-edge that contains $\widetilde e$, and distinct $n$-edges have disjoint interior.
If $e'$ is an $n$-edge, then the number of $(n+k)$-edges $\widetilde e$ contained in $e'$ is $\le \#\operatorname{post}(f)\deg(f)^k$; indeed, the images of these
$(n+k)$-edges $\widetilde e$ under the map $f^n$ are distinct $k$-edges,
and the number of $k$-edges is equal to $\#\operatorname{post}(f)\deg(f)^k$ (see
Lemma~\ref{lem:celldecompint}).
After these preliminaries, we are ready to show that $\mathcal{C}$ equipped with (the restriction of) $d$ is a quasicircle. We first establish that
$\mathcal{C}$ is doubling.
Let $x\in \mathcal{C}$, and $0<r\le 2\diam(\mathcal{C})$. In order to show that $\mathcal{C}$ is doubling, it suffices to cover $B_d(x,r)\cap \mathcal{C}$ by a controlled number
of sets of diameter $<r/4$.
It follows from Lemma~\ref{lem:expoexp} that we can find $n\in \N_0$ depending on $r$, and $k_0\in \N_0$ independent of $r$ and $x$ with the following properties:
\begin{itemize}
\smallskip
\item[(i)] $r\asymp \Lambda^{-n}$,
\smallskip
\item[(ii)] $\diam(e)\le r/4$ whenever $e$ is an $(k_0+n)$-edge,
\smallskip
\item[(iii)] $\operatorname{dist}(e,e')\ge r$ if $n-k_0\ge 0$ and $e$ and $e'$ are disjoint $(n-k_0)$-edges.
\end{itemize}
Let $E$ be the set of all $(n+k_0)$-edges contained in $\mathcal{C}$ that meet $B_d(x,r)$. Then the collection $E$ forms a cover of $\mathcal{C}\cap B_d(x,r)$ and consists of sets of diameter $<r/4$ by (ii).
Hence it suffices to find a uniform upper bound on $\#E$.
If $n< k_0$, then $\#E\le \#\operatorname{post}(f)\deg(f)^{2k_0}. $
Otherwise, $n-k_0\ge 0$. Then we can find an $(n-k_0)$-edge $e\subset \mathcal{C}$ with $x\in e$. Let $\widetilde e$ be an arbitrary edge in $E$. Again we can find an $(n-k_0)$-edge $e'\subset \mathcal{C}$ that contains $\widetilde e$.
There exists a point $y\in \widetilde e\cap B_d(x,r)$.
Hence $\operatorname{dist}(e,e')\le d(x,y)<r$. This implies $e\cap e'\ne \emptyset$ by (iii).
So whatever $\widetilde e\in E$ is, the $(n-k_0)$-edge $e'\subset \mathcal{C}$ meets the fixed $(n-k_0)$-edge $e$. This leaves at most three possibilities for $e'$, namely $e$, and the two ``neighbors`` of $e$ on $\mathcal{C}$. So there are three or less
$(n-k_0)$-edges that contain all the edges
in $E$. Since each $(n-k_0)$-edge contains at most
$\#\operatorname{post}(f)\deg(f)^{2k_0}$ edges of order $(n+k_0)$, it follows that
$\#E\le 3\#\operatorname{post}(f)\deg(f)^{2k_0}$. In any case we get the desired bound for $\#E$.
It remains to show the Ahlfors condition. Let $x,y\in
\mathcal{C}$ with $x\ne y$ be arbitrary, and let $n_0\ge 0$ be the smallest integer for which there exist $n_0$-edges $e_x\subset \mathcal{C}$ and $e_y\subset \mathcal{C}$ with $x\in e_x$, $y\in e_y$
and $e_x\cap e_y=\emptyset$. Note that $n_0$ is well-defined, because $f$ is expanding and so the diameter of $n$-edges approaches $0$ uniformly as $n\to \infty$.
Then by Lemma~\ref{lem:expoexp}~(i),
$$d(x,y)\gtrsim \Lambda^{-n_0}.$$ If $n_0=0$, then
$$\diam(\mathcal{C})\lesssim d(x,y)$$ and there is nothing to prove.
If $n_0\ge 1$, we can find $(n_0-1)$-edges $e'_x\subset \mathcal{C}$
and $e'_y\subset \mathcal{C}$ with $x\in e'_x$, $y\in e'_y$, and $e'_x\cap e'_y\ne \emptyset$. Then $e'_x\cup e'_y$ must contain
one of the subarcs $\gamma$ of $\mathcal{C}$ with endpoints $x$ and $y$.
Hence
$$ \diam (\gamma)\le \diam(e'_x)+\diam(e'_y)\lesssim \Lambda^{-n_0}\lesssim d(x,y).$$ Since the implicit multiplicative constants in the previous inequalities do not depend on $x$ and $y$, we get a bound as desired. \end{proof}
Since the class of visual metrics for $f$ and any of its iterates coincide (see Proposition~\ref{prop:visualsummary}~(v)), we may apply this theorem also to any iterate of $f$ with an invariant Jordan curve $\mathcal{C}\supset\operatorname{post}(f)$. In particular, the Jordan curve in Theorem~\ref{thm:main} is a quasicircle
if equipped with a visual metric for $f$.
A family of quasisymmetries (possibly defined on different spaces) is
called {\em uniformly quasisymmetric}\index{quasisymmetry!uniform}
if there exists a
homeomorphism $\eta\:[0,\infty]\rightarrow [0,\infty]$ such that each map in
the family is an $\eta$-quasi\-symmetry. Obviously, each finite
family of quasisymmetries is uniformly quasisymmetric. If $h$ is an
$\eta$-quasisymmetry, then $h^{-1}$ is an
$\widetilde\eta$-quasisymmetry, where $\widetilde \eta$ only depends
on $\eta$; actually, one can take $\widetilde \eta\: [0,\infty)\rightarrow
[0,\infty)$ defined by $\widetilde \eta(0)=0$ and $\widetilde
\eta(t)=1/\eta^{-1}(1/t)$ for $t>0$.
This implies that if a family of maps is uniformly quasisymmetric, then the family of inverse maps is also uniformly quasisymmetric.
If $X,Y,Z$ are metric spaces, and $h_1\: X\rightarrow Y$ is $\eta_1$-quasisymmetric and $h_2\: Y\rightarrow Z$ is $\eta_2$-quasisymmetric, then
$h_2\circ h_1$ is $\eta$-quasisymmetric, where $\eta=\eta_2\circ \eta_1$. Hence the family of all compatible compositions of maps in two
uniformly quasisymmetric families is again uniformly quasisymmetric.
An arc $\alpha$ equipped with some metric $d$ is called a {\em quasiarc}\index{quasiarc}
if there exits a quasisymmetry of the unit interval $[0,1]$ onto $(\alpha, d)$. It is known that $(\alpha, d)$ is a quasiarc if and only if $(\alpha, d)$ is doubling and there exists a constant $K\ge 1$ such that
$\diam_d(\gamma)\le K d(x,y)$, whenever $x,y\in \alpha$, $x\ne y$, and $\gamma$ is the subarc of $\alpha$ with endpoints $x$ and $y$ \cite{TV}.
So quasiarcs admit a similar geometric characterization as quasicircles.
A family of arcs is said to consist of {\em uniform quasiarcs}\index{quasiarc!uniform}
if there
exists a homeomorphism $\eta\:[0,\infty]\rightarrow [0,\infty]$ such that for
each arc $\alpha$ in the family there exists an $\eta$-quasisymmetry
$h\:[0,1]\rightarrow \alpha$. Similarly, a family of quasicircles is said to
consist of {\em uniform quasicircles}\index{quasicircle!uniform}
if there
exists a homeomorphism $\eta\:[0,\infty]\rightarrow [0,\infty]$ that for each
quasicircle $S$ in the family there exists an $\eta$-quasisymmetry
$h\:\partial \D \rightarrow S$. A family of quasicircles consists of uniform
quasicircles if and only if the geometric conditions characterizing
quasicircles, i.e.,
the doubling condition and the Ahlfors condition, holds with uniform
parameters. A similar statement is true for families of quasiarcs \cite{TV}.
We want to show that if the assumptions are as in Theorem~\ref{thm:Cquasicircle}, then all boundaries of tiles for $(f,\mathcal{C})$ are quasicircles and all edges for $(f,\mathcal{C})$ are quasiarcs. Actually, the family of all boundaries of tiles consists of uniform quasicircles and the family of all edges consists of uniform quasiarcs. One way to do this is to repeat the proof Theorem~\ref{thm:Cquasicircle} and show that the geometric conditions characterizing quasiarcs and quasicircles are true for the edges and boundaries of tiles with uniform constants. We choose a different approach that is based on the following lemma which is of independent
interest.
\begin{lemma}\label{lem:unifqs} Let $f\:S^2\rightarrow S^2$ be an expanding Thurston map, and $\mathcal{C}\subset S^2$ be an $f$-invariant Jordan curve with
$\operatorname{post}(f)\subset \mathcal{C}$. Suppose that $S^2$ is equipped with a visual metric $d$ for $f$ with expansion factor $\Lambda>1$,
and
denote by $\X^n$ for $n\in \N_0$ the set of $n$-tiles for $(f,\mathcal{C})$. Then there exists a constant $C\ge 1$ with the following property:
If $k,n\in \N_0$, $X^{n+k}\in \X^{n+k}$, and $x,y\in X^{n+k}$, then
\begin{equation}\label{eq:unifsim}
\frac1C d(x,y)\le \frac{d(f^n(x), f^n(y))}{\Lambda^{n}}\le Cd(x,y).
\end{equation}
In particular, the family $\mathcal{F}:=\{ f^n|X^{n+k}: k,n\in \N_0,\ X^{n+k}\in \X^{n+k}\}$
is uniformly quasisymmetric.
\end{lemma}
\begin{proof} In the following all cells will be for $(f,\mathcal{C})$. Let $m=m_{f,\mathcal{C}}$ be as in Definition~\ref{def:mxy}.
We know by Definition~\ref{def:visual} and by Lemma~\ref{lem:mprops}~(iii)
that $d(x,y)\asymp \Lambda^{-m(x,y)}$, whenever $x,y\in S^2$.
If $n\in \N_0$, then Lemma~\ref{lem:mprops}~(ii) implies that
$$m(f^n(x), f^n(y))\ge m(x,y)-n, $$ and so
$$ d(f^n(x), f^n(y))\lesssim \Lambda^n d(x,y). $$
Here the implicit multiplicative constants are independent of $x$ and $y$.
To obtain an inequality in the other direction, let $x,y\in X^{n+k}\in
\X^{n+k}$, where $n,k\in \N_0$, We may assume that $x\ne y$. Then by definition of
$m(x,y)$ we have $n+k\le m(x,y)<\infty$. Let $l:=m(x,y)+1\in \N$.
Since $l>n+k$, the $(n+k)$-tile $X^{n+k}$ is subdivided by tiles of order
$l$ (Proposition~\ref{prop:invmarkov}~(iii)). Hence there exist $l$-tiles
$X,Y\subset X^{n+k}$ with $x\in X$ and $y\in Y$.
Then $X\cap Y=\emptyset$ by definition of $m(x,y)$. Let $X':=f^n(X)$ and
$Y':=f^n(Y)$. Then by Proposition~\ref{prop:celldecomp}~(i) the sets $X'$ and $Y'$ are $(l-n)$-tiles. Since $f^n|X^{n+k}$ is injective, these tiles are disjoint, and we have $f^n(x)\in X'$ and $f^n(y)\in Y'$.
So from Lemma~\ref{lem:expoexp}~(i) we conclude that
$$ d(f^n(x), d^n(y))\ge \operatorname{dist}_d(X',Y')\gtrsim \Lambda^{-(l-n)}\asymp \Lambda^{n}\Lambda^{-m(x,y)}\asymp \Lambda^{-n} d(x,y). $$
Here the implicit multiplicative constants are again independent of $x$ and $y$, and we get the other desired inequality.
Inequality~\eqref{eq:unifsim} immediately implies that the family $\mathcal{F}$ is uniformly quasisymmetric. To see this, let $k,n\in \N_0$ and $X^{n+k}\in \X^{n+k}$. Then $f^n|X^{n+k}$ is a homeomorphism onto its image (see Proposition~\ref{prop:celldecomp}~(i)). Moreover, if $u,v,w\in X^{n+k}$, $u\ne w$, then by \eqref{eq:unifsim} we have
$$ \frac{d(f^n(u), f^n(v))}{d(f^n(u), f^n(w))}\le C^2 \frac{d(u,v)}{d(u,w)}.$$
Hence $f^n|X^{n+k}$ is $\eta$-quasisymmetric, where $\eta(t)=C^2t$ for $t\ge 0$. Since $\eta$ is independent of the chosen map,
the family $\mathcal{F}$ is uniformly quasisymmetric.
\end{proof}
\begin{prop} \label{prop:arc} Let $f\:S^2\rightarrow S^2$ be an expanding Thurston map and $\mathcal{C}\subset S^2$ be an $f$-invariant Jordan curve with
$\operatorname{post}(f)\subset \mathcal{C}$. Suppose that $S^2$ is equipped with a visual metric for $f$, and for $n\in \N_0$ denote by $\X^n$ the set of $n$-tiles and by $\E^n$ the set of $n$-edges
for $(f,\mathcal{C})$.
Then the family $ \{\partial X: n\in \N_0 \text{ and } X\in \X^n\}$ consists of
uniform quasicircles and the family
$\{e: n\in \N_0 \text{ and } e\in \E^n\}$ of uniform quasiarcs.
\end{prop}
In particular, edges for $(f,\mathcal{C})$ are quasiarcs and the boundaries of all tiles are quasicircles.
\begin{proof} By Theorem~\ref{thm:Cquasicircle} there exists a quasisymmetric map $h\: \partial \D \rightarrow \mathcal{C}$. Let $X$ be an arbitrary tile for
$(f,\mathcal{C})$, say an $n$-tile, where $n\in \N_0$. Then
$f^n|X$ is a homeomorphism of $X$ onto the $0$-tile $f^n(X)$ (Proposition~\ref{prop:celldecomp}~(i)), and so
$$ f^n(\partial X)=\partial f^n(X)=\mathcal{C}. $$
By Lemma~\ref{lem:unifqs} the map $f^n|X$, and hence also the map
$(f^n|X)^{-1}$, is a quasisymmetry. It follows that
$(f^n|X)^{-1}\circ h$ is a quasisymmetric map from $\partial \D$ onto
$\partial X$. Hence $\partial X$ is a quasicircle.
Actually, the family of these quasicircles $\partial X$ is uniform, since the family of all relevant maps $(f^n|X)^{-1}\circ h$ is uniformly quasisymmetric as follows from Lemma~\ref{lem:unifqs}.
\smallskip
The proof that the family $\{e: n\in \N_0 \text{ and } e\in \E^n\}$ consists of uniform quasiarcs runs along the same lines. First note that each $0$-edge is a subarc of $\mathcal{C}$, and hence corresponds to a subarc of $\partial \D$ under the quasisymmetry $h$. Since this subarc can be
mapped to the unit interval $[0,1]$ by a bi-Lipschitz homeomorphism,
each $0$-edge is quasisymmetrically equivalent to $[0,1]$ and hence a quasiarc.
Now let $e$ be an arbitrary edge for $(f,\mathcal{C})$, say an $n$-edge, where $n\in \N_0$. Then $f^n|e$ is a homeomorphism of $e$ onto the $0$-edge $f^n(e)$ (Proposition~\ref{prop:celldecomp}~(i)). Moreover, there exists an $n$-tile $X$ with $e\subset X$ (Lemma~\ref{lem:specprop}). Then $f^n|e$ is the restriction of the map $f^n|X$ to $e$, and it follows from Lemma~\ref{lem:unifqs} that $f^n|e$ is a quasisymmetry. Hence $e$ is quasisymmetrically equivalent to a $0$-edge and hence a quasiarc.
Lemma~\ref{lem:unifqs} actually implies that the family $\{f^n|e: n\in \N_0\text{ and } e\in \E^n\}$ is uniformly quasisymmetric. So each edge is quasisymmetrically equivalent to a $0$-edge by a quasisymmetry in a uniformly quasisymmetric family. Since there are only finitely many $0$-edges, this implies that the family of all edges for $(f,\mathcal{C})$ consists of uniform quasiarcs.
\end{proof}
A metric space $(X,d)$ is called {\em linearly locally
connected}\index{linearly locally connected ($LLC$)}
(often abbreviated as $LLC)$ if there exists a constant $C\ge 1$ such
that
the following two conditions are satisfied:
\begin{itemize}
\smallskip
\item[(i)] If $p\in X$, $r>0$, and $x,y\in B_d(x,r)$, $x\ne y$, then there exists a continuum $E\subset X$ with $x,y\in E$ and $E\subset B_d(p,Cr).$
\smallskip
\item[(ii)] If $p\in X$, $r>0$, and $x,y\in X\setminus B_d(x,r)$, $x\ne y$, then there exists a continuum $E\subset X$ with $x,y\in E$ and $E\subset
X\setminus B_d(p,r/C).$
\end{itemize}
The following proposition shows that a $2$-sphere equipped with a visual metric for an expanding Thurston map is linearly locally connected.
\begin{prop}\label{prop:annLLC}
Let $f\:S^2\rightarrow S^2$ be an expanding Thurston map, and suppose that $S^2$ is equipped with a visual
metric $d$ for $f$. Then the following statements are true:
\begin{itemize}
\smallskip
\item[(i)] There exists a constant $C'\ge 1$ such that any two points $x,y\in S^2$ can be joined by a path $\alpha$ in $S^2$ with
\begin{equation}\label{eq:bt}
\diam(\alpha)\le C'd(x,y).
\end{equation}
\smallskip
\item[(ii)]
There exists a constant $C\ge 1$ with the following property:
If $p\in S^2$, $r>0$, and $x,y\in \overline B(p,2r)\setminus B(p,r)$, then
there exists a path $\gamma$ in $S^2$ joining $x$ and $y$ with
\begin{equation}\label{eq:annLLC}\gamma\subset
\overline B(p,Cr)\setminus B(p,r/C).
\end{equation}
\smallskip
\item[(iii)] $(S^2,d)$ is linearly locally connected.
\end{itemize}
\end{prop}
The statement (i)--(iii) are not logically independent, but one can show
the implications $\text{(ii)}\Rightarrow \text{(iii)}\Rightarrow \text{(i)}.$
\begin{proof} Let $\Lambda>1$ be the expansion factor of $d$.
Then for some Jordan curve $\mathcal{C}\subset S^2$ with $\operatorname{post}(f)\subset \mathcal{C}$ we have $d(u,v)\asymp \Lambda^{-m_{f,\mathcal{C}}(u,v)}$ for $u,v\in S^2$ (see Definitions~\ref{def:mxy} and \ref{def:visual}). In the following all cells will be for $(f,\mathcal{C})$.
\smallskip
(i) Let $x,y\in S^2$, $x\ne y$, be arbitrary, and $n=m_{f,\mathcal{C}}(x,y)\in \N_0$.
Then there exist $n$-tiles $X$ and $Y$ with $x\in X$, $y\in Y$, and $X\cap Y\ne \emptyset$. Since $X$ and $Y$ are Jordan regions, we can find a path $\alpha$ in $X\cup Y$ that joins $x$ and $y$.
Then by Lemma~\ref{lem:expoexp} we have
$$\diam(\alpha)\le \diam (X)+\diam(Y)\lesssim \Lambda^{-n}\asymp d(x,y),
$$ where the implicit multiplicative constants are independent of $x$ and $y$. Statement (i) follows.
\smallskip
(ii) Let $p\in S^2$, $r>0$, and $x,y\in \overline B(p,2r)\setminus B(p,r)$. In the following all implicit multiplicative constant will be independent of these initial choices of $p$, $r$, $x$, and $y$.
Define
$$n:=\max\{m_{f,\mathcal{C}}(p,x), m_{f,\mathcal{C}}(p,y)\}+1.$$
Then
$$\Lambda^{-n}\asymp \min\{d(p,x), d(p,y)\}\asymp r. $$
Let $X,Y,Z$ be $n$-tiles with $x\in X$, $y\in Y$, and $p\in Z$.
Then by definition of $n$ we have
$X\cap Z=\emptyset$ and $Y\cap Z=\emptyset$.
Since $f$ is expanding, we can choose $k_0\in \N_0$ as in \eqref{def:k0}.
In particular, every connected set of $k_0$-tiles joining opposite sides of $\mathcal{C}$ must contain at least $10$ $k_0$-tiles.
Consider set the $U^{n+k_0}(p)$ as defined in \eqref{eq:defUk}.
Then $f^n(U^{n+k_0}(p))$ is connected, and consists of $k_0$-tiles. This set cannot join opposite sides of $\mathcal{C}$; for otherwise, we could find a connected set of six
$k_0$-tiles with this property (see the proof Lemma~\ref{lem:quasiball}
for a similar reasoning). This is impossible by definition of $k_0$.
Hence $f^n(U^{n+k_0}(p))$ is contained in a $0$-flower (Lemma~\ref{lem:floweropp}) which implies that $U^{n+k_0}(p)$ is contained in an $n$-flower (Lemma~\ref{lem:mapflowers}~(iii)). So there exists an $n$-vertex $v$ with $p\in U^{n+k_0}(p)\subset W^n(v)$. Since $Z$ contains $p$, this tile must be one of the $n$-tiles forming the cycle of $v$. So $v\in Z$, and
$v\notin X,Y$. This in turn implies that $X$ and $Y$ do not meet $W^n(v)$ (see Lemma~\ref{lem:flowerprop}~(iii)).
Pick a path $\alpha$ in $S^2$ that joins $x$ and $y$ and satisfies
\eqref{eq:bt}. By Lemma~\ref{lem:echain} we can find a set $M$ of $n$-tiles that forms an $e$-chain joining $X$ and $Y$ so that each tile in $M$ has non-empty intersection with $\alpha$.
Pick $n$-vertices $x'\in \partial X$, $y'\in \partial Y$. Since $X$ and $Y$ do not contain $v$, we have $x',y'\ne v$.
Consider the graph $G_M=\{\partial U: U\in M\}$. It consists of $n$-edges, is connected, has no cut points (Lemma~\ref{lem:nocut}), and contains $x'$ and $y'$ as vertices. Hence
there exists an edge path in $G_M$ joining $x'$ and $y'$ whose underlying
set $\beta$ does not contain $v$. Then this edge path does not contain any
edge in the cycle of $v$ and so $\beta\cap W^n(v)=\emptyset$.
Let $\gamma$ be the path in $S^2$ that is obtained by running from $x$ to $x'$ along some path in $X$, then from $x'$ to $y'$ along $\beta$, and then from $y'$ to $y$ along some path in $Y$. Then $\gamma$ joins $x$ and $y$.
Since the sets $X,Y,\beta$ have empty intersection with $W^n(v)$ and hence with $U^{n+k_0}(p)$, it follows that $\gamma\cap U^{n+k_0}(p)=\emptyset$.
So by Lemma~\ref{lem:UmB} we have
$$ \operatorname{dist}(p,\gamma)\gtrsim \Lambda^{-(n+k_0)}\asymp\Lambda^{-n}\asymp
r. $$
Hence there exits a constant $C_1\ge 1$ independent of the initial choices such that
$$\gamma\cap B(p, r/C_1)=\emptyset. $$
The set $\gamma$ can be covered by $n$-tiles that meet $\alpha$.
Since
$$ \diam (\alpha)\le C'd(x,y)\le 4C'r\lesssim r, $$ and
$$\max\{\diam(U): U\text{ is an $n$-tile}\}\lesssim \Lambda^{-n}\asymp r, $$ we conclude that
$$\diam (\gamma)\le \diam(\alpha)+2\max\{\diam(U): U\text{ is an $n$-tile}\}\lesssim r.$$
Since the initial point $x$ of $\gamma$ has distance $\le 2r$ from $p$, it follows that there exists a constant $C_2\ge 1$
independent of the initial choices such that
$\gamma\subset B(p, C_2r). $
Setting $C=\max\{C_1, C_2\}$, get the inclusion \eqref{eq:annLLC}.
\smallskip
(iii) To show that $(S^2, d)$ is linearly locally connected, we verify
the two relevant conditions; here we can use possibly different constants $C$ in each of the conditions.
Let $p\in X$, $r>0$, and $x,y\in B(p,r)$, $x\ne y$, be arbitrary.
Choose a path $\alpha$ as in \eqref{eq:bt}, and define $E:=\alpha$ and $C=2C'+1$. Then $x,y\in E$, and, since $\diam(\alpha)\le C'd(x,y)\le 2C'r$,
we have
$$E\subset B(p, r+\diam(\alpha))\subset B(p, Cr). $$
The first of the $LLC$-condition follows.
For the second condition, let $p\in X$, $r>0$, and $x,y\in X\setminus B(p,r)$ with $x\ne y$ be arbitrary. Let $\alpha$ be a path in $S^2$ joining
$x$ and $y$. If $\alpha\cap B(p,r)=\emptyset$, define $E:=\alpha$.
Then $E$ is a continuum with $x,y\in E$ and $E\subset X\setminus B(p,r)$.
If $\alpha$ meets $B(p,r)$, then, as we travel from $x$ to $y$ along $\alpha$, there exists a first point with $x'\in \overline B(p,2r)$. Note that if $x\in \overline B(p,2r)$, then
$x'=x$, and $d(p,x')=2r$ otherwise. In any case, $x'\in \overline B(p,2r)\setminus
B(p,r)$. Let $\alpha_x$ be the subpath of $\alpha$ obtained by traveling
along $\alpha$ starting from $x$ until we reach $x'$.
Then $\alpha_x\subset X\setminus B(p, r)$.
Traveling along $\alpha$ in the opposite direction starting from $y$, we define a point $y'\in \overline B(p,2r)\setminus
B(p,r)$ and a subpath $\alpha_y\subset X\setminus B(p, r)$ of $\alpha$ joining $y$ and $y'$ similarly.
Then $x',y'\in \overline B(p, 2r)\setminus B(p,r)$. Hence by (ii)
there exists a path $\gamma$ in $S^2$ that joins $x'$ and $y'$ and satisfies $\gamma\subset X\setminus B(p,r/C)$. Here $C\ge 1$ is a constant
independent of the initial choices.
Now define $E=\alpha_x\cup \gamma\cup \alpha_y$. Then $E$ is a continuum with $x,y\in E$ and $E\subset X\setminus B(p,r/C)$.
It follows that the second $LLC$ condition is satisfied as well.
\end{proof}
\section{Periodic critical points}
\label{sec:periodic}
\noindent Let $f\: S^2\rightarrow S^2$ be a branched covering map on a $2$-sphere $S^2$. A point $p\in S^2$ is called {\em periodic} if there exists $n\in \N$ such that $f^n(p)=p$. The smallest $n$ for which this is true
is called the {\em period} of the periodic point.
The point $p$ is called {\em pre-periodic} if there exists $n\in \N_0$ such that $f^n(p)$ is periodic.
The following lemma is well-known.
\begin{lemma} \label{lem:cycle}
Let $f\:S^2\rightarrow S^2$ be a branched covering map.
Then $f$ has no periodic critical points if and only if there exists $N\in \N$ such that
$$ \deg_{f^n}(p)\le N$$
for all $p\in S^2$ and all $n\in \N$.
\end{lemma}
\begin{proof}
Note that for $p\in S^2$ and $n\in \N$
we have
\begin{equation}\label{eq:degfn}
\deg_{f^n}(p)=\prod_{k=0}^{n-1}\deg_f(f^{k}(p))
\end{equation}
So if $p$ is a periodic critical point of period $l$, say, and $d=\deg_f(p)\ge 2$, then
$$\deg_{f^n}(p) \ge d^{ \lceil n/l \rceil} \ge 2^{n/l} \to \infty$$ as
$n\to \infty$. Hence $\deg_{f^n}(p)$ is not uniformly bounded.
If $f$ has no periodic critical point, then the orbit $p$, $f(p)$, $f^2(p)$, $\dots$ of a point $p\in S^2$ can contain each critical point at most once.
Hence by \eqref {eq:degfn}
we have
$$ \deg_{f^n}(p)\le N:=\prod_{c\in \operatorname{crit}(f)} \deg_f(c). $$
Note that the last product is finite, because $f$ has only finitely many critical points. \end{proof}
\begin{theorem} \label{thm:perdoub} Let $f\:S^2\rightarrow S^2$ be an expanding Thurston map.
Then $S^2$ equipped with a visual metric for $f$ is doubling if and only if $f$ has no periodic critical points.
\end{theorem}
Actually, if $f$ has no periodic critical points and $d$ is a visual metric, then $(S^2,d)$ is not only doubling, but
even Ahlfors regular (see Proposition~\ref{prop:Ahlforsreg}).
\begin{proof}
Assume first that $f$ has no periodic critical points. By Theorem~\ref{thm:main}
there exists an iterate $F=f^n$ and an $F$-invariant Jordan curve $\mathcal{C}\subset S^2$ with $\operatorname{post}(f)=\operatorname{post}(F)\subset \mathcal{C}$. Then $F$ is also an expanding Thurston map (Lemma~\ref{lem:Thiterates}) and it has no critical points as easily follows from \eqref{eq:critpfn}. It suffices to show that $S^2$ is doubling
with a visual metric for $F$, because the class of visual metrics for $f$ and $F$ agree (Proposition~\ref{prop:visualsummary} (v)).
Fix such a visual metric for $F$, and denote by $\Lambda>1$ its expansion factor. In the following cells
will refer to cells for $(F,\mathcal{C})$.
Then by
Lemma~\ref{lem:cycle}, there exists $N\in \N$ such that $\deg_{F^k}(p)\le N$ for all $p\in S^2$ and $k\in \N_0$. This implies that the closure of every $k$-flower consists of at most $2N$ tiles of order $k$
(see Lemma~\ref{lem:flowerprop}).
To establish that $S^2$ is doubling we now proceed similarly as in the proof of Theorem~\ref{thm:Cquasicircle}.
Let $x\in S^2$ and $0<r\le 2 \diam (S^2)$ be arbitrary.
We have to cover $B(x,r)$ by a controlled number of sets of diameter $<r/4$.
Again, using Lemma~\ref{lem:expoexp}, we can find $n\in \N_0$ depending on $r$, and $k_0\in \N_0$ independent of $r$ and $x$ with the following properties:
\begin{itemize}
\smallskip
\item[(i)] $r\asymp \Lambda^{-n}$,
\smallskip
\item[(ii)] $\diam(X)\le r/4$ whenever $X$ is an $(k_0+n)$-tile,
\smallskip
\item[(iii)] $\operatorname{dist}(X,Y)\ge r$ whenever $n-k_0\ge 0$ and $X$ and $Y$ are disjoint $(n-k_0)$-tiles.
\end{itemize}
Let $T$ be the set of all $(n+k_0)$-tiles that meet $B(x,r)$. Then the collection $T$ forms a cover of $B(x,r)$ and consists of sets of diameter $<r/4$ by (ii).
Hence it suffices to find a uniform upper bound on $\#T$, independent of $x$ and $r$.
If $n< k_0$, then $\#T\le 2\deg(F)^{2k_0} $ (see Proposition~\ref{prop:celldecomp}~(iv)) and we have such a bound.
Otherwise, $n-k_0\ge 0$. Pick an $(n-k_0)$-tile $X$ with $x\in X$.
If $Z$ is an arbitrary $(n+k_0)$-tile in $T$, then we can find a unique $(n-k_0)$-tile $Y$ that contains $Z$ (here we use that $\mathcal{C}$ is $F$-invariant and so each
tile is subdivided by tiles of higher order).
There exists a point $y\in Z\cap B(x,r)$.
Hence $\operatorname{dist}(X,Y)\le d(x,y)<r$. This implies $X\cap Y\ne \emptyset$ by (iii).
So whatever $Z\in T$ is, the corresponding $(n-k_0)$-tile $Y\supset Z$ meets the fixed $(n-k_0)$-tile $X$. Hence $Y$ must share an
$(n-k_0)$-vertex $v$ with $X$ which implies $Y\subset \overline{W^{n-k_0}(v)}$. Since $\overline{W^{n-k_0}(v)}$ consists of at most $2N$ tiles of order $(n-k_0)$, and the number of $(n-k_0)$-vertices in $X$ is equal to $\#\operatorname{post}(F)$, this leaves at most $2N\#\operatorname{post}(f)$ possibilities for $Y$.
Since every $(n-k_0)$-tile contains at most
$2\deg(f)^{2k_0}$ tiles of order $(n+k_0)$, it follows that
$\#T\le 4N\#\operatorname{post}(f)\deg(f)^{2k_0}$.
So we get a uniform bound as desired.
To show the other implication we use
the following fact about doubling spaces, which is easy to show: in every ball there cannot be too many pairwise disjoint smaller balls of the same radius. More precisely, for every $\eta\in (0,1)$ there is a number $K$ such that every ball open ball of radius $r$ contains at most $K$ pairwise disjoint open balls of radius $\eta r$.
Now suppose $f\: S^2\rightarrow S^2$ is an expanding Thurston map such that $S^2$ equipped with some visual metric is doubling. Pick a Jordan curve $\mathcal{C}\subset S^2$ with $\operatorname{post}(f)\subset \mathcal{C}$. In the following cells will be for $(f,\mathcal{C})$. Let $p\in S^2$ and $n\in\N$. In order to show that $f$ has no periodic critical points it suffices to give a uniform bound on $d=\deg_{f^n}(p)$ (see Lemma~\ref{lem:cycle}).
For this we may assume that $\deg_{f^n}(p)\ge 2$. Then $p$ is an $n$-vertex and the closure of the $n$-flower consists of precisely $2\deg_{f^n}(p)$ $n$-tiles. These $n$-tiles have pairwise disjoint interior and each interior contains a ball of radius $r\asymp \Lambda^{-n}$ (see Lemma~\ref{lem:quasiball}). On the other hand, $\diam(\overline{W^n(p)})\lesssim \Lambda^{-n}$. Since $S^2$ is doubling, it follows that the number of these tiles and hence $\deg_{f^n}(p)$ is uniformly bounded from above by a constant independent of $n$ and $p$.
Hence $f$ has no periodic critical points.
\end{proof}
\section{The combinatorial expansion factor}
\label{sec:combexpfac}
\noindent
In this section we will study the asymptotic rate at which the quantity
$D_k$ introduced in \eqref{def:dk} grows as $k\to \infty$. Note that $D_k$ depends on $f$ and $\mathcal{C}$. If we want to emphasize this dependence, we write
$D_k=D_k(f,\mathcal{C})$. We require the following lemma.
\begin{lemma}\label{lem:flowpow} Let $n\in \N_0$, $f\:S^2\rightarrow S^2$ be a Thurston map with $\#\operatorname{post}(f)\ge 3$, and $\mathcal{C}\subset S^2$ a Jordan curve
with $\operatorname{post}(f)\subset \mathcal{C}$. If there exists a connected set $K\subset S^2$ that joins
opposite sides of $\mathcal{C}$ and can be covered by $N$ $n$-flowers
for $(f,\mathcal{C})$, then $D_n(f,\mathcal{C})\le 4N$. \end{lemma}
\begin{proof} We assume first that $\#\operatorname{post}(f)=3$.
Let $K$ be as in the statement.
By picking a point from the intersection of $K$ with each of the three $0$-edges, we can find a set $\{x,y,z\}\subset K$ such that
$\{x,y,z\}$ meets opposite sides of $\mathcal{C}$.
Since $K$ is connected and can be covered by $N$ $n$-flowers, we can find $n$-vertices $v_1, \dots, v_N\in S^2$ such that
$x\in W^n(v_1)$, $y\in W^n(v_N)$ and
$W^n(v_i)\cap W^n(v_{i+1})\ne \emptyset $ for $i=1, \dots, N-1$.
It follows from Lemma~\ref{lem:flowerprop}~(i) that there exists
a chain consisting of $n$-tiles $X_1, \dots, X_{2N}$ joining $x$ and $y$ (recall the terminology discussed before Lemma~\ref{lem:mprops}).
Similarly, there exists a chain $X_1', \dots, X_{2N}'$ of $n$-tiles joining $x$ and $z$. The union $K'$ of the $n$-tiles in these two chains is a connected set consisting of at most $4N$ $n$-tiles.
It contains the set $\{x,y,z\}$ and hence joins opposite sides of $\mathcal{C}$. Thus $D_n(f,\mathcal{C})\le 4N$.
If $\#\operatorname{post}(f)\ge 4$, the proof is similar and easier. In this case we can find a set $\{x,y\}\subset K$ that joins opposite sides of $\mathcal{C}$.
By the same argument as before, we get the bound $D_n(f,\mathcal{C})\le 2N$. \end{proof}
\begin{prop} \label{prop:exp} Let $f\:S^2\rightarrow S^2$ be an expanding Thurston map, and $\mathcal{C}\subset S^2$ be a Jordan curve
with $\operatorname{post}(f)\subset \mathcal{C}$.
Then the limit
$$\Lambda_0(f)=\lim_{k\to \infty} D_k(f,\mathcal{C})^{1/k}$$
exists. Moreover, this limit is independent of $\mathcal{C}$ and we have $1<\Lambda_0(f)<\infty$.
\end{prop}
We call $\Lambda_0(f)$\index{L0@$\Lambda_0$}
the {\em combinatorial expansion factor}\index{combinatorial expansion factor}
of $f$.
\begin{proof} Set $D_k=D_k(f,\mathcal{C})$.
We will first show that there exists a constant
$C_1\ge 1$ such that
\begin{equation}\label{DDk} (1/C_1) D_k\le D_{k+1}\le C_1D_k
\end{equation}
for all $k\in \N_0$.
Indeed, if $k\in \N_0$ then there exists a connected set $K$ joining
opposite sides of $\mathcal{C}$ that consists of $D_k$ $k$-tiles (for $(f,\mathcal{C})$). By Lemma~\ref{lem:difflevel} we can cover $K$ by $MD_k$ $(k+1)$-flowers, where $M\in \N$ is independent of $k$. Hence by Lemma~\ref{lem:flowpow} we have
$D_{k+1}\le C D_k$, where $C=4M$. An inequality in the opposite direction follows from a similar argument again based on
Lemma~\ref{lem:difflevel} and Lemma~\ref{lem:flowpow}.
Now let $\widetilde{\mathcal{C}} \subset S^2$ be another Jordan curve that contains $\operatorname{post}(f)$. Set $\widetilde D_k=D_k(f,\widetilde{\mathcal{C}})$.
We will show that there exists
a constant $C_2\ge 1$ such that
\begin{equation}\label{DDtilde}(1/C_2) D_k\le \widetilde D_k\le C_2 D_k\end{equation} for all
$k\in \N_0$.
Let $\widetilde \delta_0=\delta_0(f,\widetilde \mathcal{C})>0$ be defined as in \eqref{defdelta} for $f$, $\widetilde \mathcal{C}$, and a base metric $d$ on $S^2$. Since $f$ is expanding, we there exists $k_0\in \N_0$ such that
$\diam(X)<\widetilde \delta_0/2$ whenever $X$ is
$k_0$-tile for $(f,\mathcal{C})$.
We can find a compact connected set $\widetilde K$ joining opposite sides of $\widetilde{\mathcal{C}}$ that consists of $\widetilde D_k$ $k$-tiles for $(f,\widetilde{\mathcal{C}})$. Then $\diam( \widetilde K)\ge \widetilde \delta_0$ and so $\widetilde K$ contains two points with $d(x,y)\ge \widetilde\delta_0$. There exist $k_0$-tiles $X$ and $Y$ for $(f,\mathcal{C})$ such that $x\in X$ and $y\in Y$. By choice of $k_0$ we have $X\cap Y=\emptyset$, and so
$\widetilde K$ joins $k_0$-tiles for $(f,\mathcal{C})$ that are disjoint.
Hence $f^{k_0}(\widetilde K)$ joins opposite sides of
$\mathcal{C}$ by Lemma~\ref{lem:maptotop}. Every $k$-tile
for $(f,\widetilde {\mathcal{C}})$ can be covered by $M$ $k$-flowers for $(f,\mathcal{C})$, where $M$ only depends on $\mathcal{C}$ and $\widetilde{\mathcal{C}}$ (Lemma~\ref{lem:tileflower}).
This and Lemma~\ref{lem:mapflowers} imply that if $k\ge k_0$, then we can cover $f^{k_0} (\widetilde K)$ by $M\widetilde D_k$ $(k-k_0)$-flowers for $(f,\mathcal{C})$.
So by Lemma~\ref{lem:flowpow} we have
$$D_{k-k_0}\le 4M \widetilde D_k,$$
and the first part of the proof implies
$$ D_k\le C_1^{k_0} D_{k-k_0}\le 2M C_1^{k_0}\widetilde D_k.$$
If $k\le k_0$ we get a similar bound from the inequalities $D_k\le 2\deg(f)^{k_0}$ and $\widetilde D_k\ge 1$. It follows that there exists a constant $C$ independent of $k$ such that
$$ D_k\le C \widetilde {D}_k$$
for all $k\in \N_0$. An inequality in the opposite direction is obtained by reversing the roles of $\mathcal{C}$ and $\widetilde {\mathcal{C}}$ and using an inequality similar to \eqref{DDk} for $\widetilde D_k$.
A consequence of \eqref{DDtilde} is that if the sequence
$ \{D_k(f,\mathcal{C})^{1/k}\}$ converges as $k\to \infty$, then $ \{D_k(f,\widetilde \mathcal{C})^{1/k}\}$ also converges and has the same limit.
So if the limit exists, it does not depend on $\mathcal{C}$. To show existence we may impose additional assumptions on $\mathcal{C}$, namely by Theorem~\ref{thm:main}, we may assume that $\mathcal{C}$ is invariant for some iterate $F=f^n$ of $f$. Since $F$ is a also an expanding Thurston map (Lemma~\ref{lem:Thiterates}), it follows from
Lemma~\ref{lem:Dtoinfty} and Lemma~\ref{lem:submult} that the limit
$$\Lambda_0(F,\mathcal{C})=\lim_{k\to\infty} D_k(F,\mathcal{C})^{1/k}$$ exists and that
$\Lambda_0(F,\mathcal{C})\in (1,\infty)$.
Since the $k$-tiles for $(F,\mathcal{C})$ are precisely the $(nk)$-tiles for $(f,\mathcal{C})$ we have $D_{nk}=D_k(F,\mathcal{C})$ for all $k\in \N_0$, and so
$$D_{nk}^{1/(nk)}=D_k(F,\mathcal{C})^{1/(nk)}\to \Lambda_0(f):=\Lambda_0(F,\mathcal{C})^{1/n}\in (1, \infty)$$
as $k\to \infty$. Invoking \eqref{DDk} it follows that
$D_k^{1/k}\to \Lambda_0(f)$ as $k\to \infty$. The proof is complete.
\end{proof}
If $F=f^n$ is an iterate of $f$, then, as was pointed out in the previous proof, we have
$$D_k(F,\mathcal{C})=D_{nk}(f,\mathcal{C})$$ whenever $k\in \N_0$ and $\mathcal{C}$ is a Jordan curve with $\operatorname{post}(f)\subset \mathcal{C}$.
This implies
\begin{equation}\label{eq:wellbeh}
\Lambda_0(F)=\Lambda_0(f^n)=\Lambda_0(f)^n.
\end{equation}
The combinatorial expansion factor is invariant under topological conjugacy as the next statement shows.
\begin{prop} \label{prop:expfacinv}
Suppose $f\:S^2\rightarrow S^2$ and
$g\:\widehat S^2\rightarrow \widehat S^2$ are expanding Thurs\-ton map that are topologically conjugate. Then
$\Lambda_0(f)=\Lambda_0(g)$.
\end{prop}
\begin{proof} By assumption there exists a homeomorphism $h\: S^2\rightarrow \widehat S^2$ such that $h\circ f=g\circ h$. Pick a Jordan curve $\mathcal{C}\subset S^2$ with $\operatorname{post}(f)\subset \mathcal{C}$ and let $\widehat \mathcal{C}= h(\mathcal{C})$. Then $\widehat \mathcal{C}$ is a Jordan curve that contains $\operatorname{post}(g)$, and, as in the proof of Corollary~\ref{cor:conjisom}, we have
$$\mathcal{D}^k(g, \widehat \mathcal{C})=\{h(c): c\in \mathcal{D}^k(f,\mathcal{C})\}$$ for $k\in \N_0$. This implies that $D_k(f,\mathcal{C})=D_k(g, \widehat \mathcal{C})$ for all $k\in \N_0$ and so
$$\Lambda_0(g)=\lim_{k\to \infty} D_k(g,\widehat \mathcal{C})^{1/k}=
\lim_{k\to \infty} D_k(f,\mathcal{C})^{1/k}=\Lambda_0(f) $$
as desired.
\end{proof}
For the proof of Theorem~\ref{thm:visexpfactors} we need the following lemma.
\begin{lemma}\label{adhoc} Let $k,n\in \N_0$, $f\: S^2 \rightarrow S^2$ be a Thurston map, and $\mathcal{C}\subset S^2$ be a Jordan curve with $\operatorname{post}(f)\subset \mathcal{C}$.
\begin{itemize}
\smallskip
\item[(i)] If $Z\subset S^2$ is a Jordan region such that $f^n|Z$ is a homeomorphism and $f(Z)$ is a $k$-tile, then $Z$ is an
$(n+k)$-tile.
\smallskip
\item[(ii)] If $X'$ is a $k$-tile and $p\in S^2$ is a point with $f^n(p)\in \inte(X')$, then there exists a unique $(n+k)$-tile with $p\in X$ and $f^n(X)=X'$.
\end{itemize}
\end{lemma}
\begin{proof} It is understood that tiles are for $(f,\mathcal{C})$. Note that $f$ is cellular for $(\mathcal{D}^{n+k}(f,\mathcal{C}), \mathcal{D}^{k}(f,\mathcal{C}))$, and the set $f^{-k}(\operatorname{post}(f))\supset \operatorname{post}(f)$ of vertices of $\mathcal{D}^{k}(f,\mathcal{C})$ contains the set $f^n(\operatorname{crit}(f^n))\subset \operatorname{post}(f)$
(the last inclusion follows from \eqref{eq:critpfn}). Hence we are in the situation of Lemma~\ref{lem:pullback} with
$\mathcal{D}=\mathcal{D}^{k}(f,\mathcal{C})$ and $\mathcal{D}'=\mathcal{D}^{n+k}(f,\mathcal{C})$.
Then (i) follows from the uniqueness statement of Lemma~\ref{lem:pullback} and the definition of $\mathcal{D}'$ in the first paragraph of the
proof of this lemma.
Moreover, under the assumptions of (ii) it follows from Claim 1 in the proof of Lemma~\ref{lem:pullback} that there exits a unique $(n+k)$-tile $X$ with $p\in X$. Then $f^n(X)$ is a $k$-tile
containing $f^n(p)\in \inte(X')$, and so $f^n(X)=X'$.
\end{proof}
\medskip
\begin{proof}[Proof of Theorem~\ref{thm:visexpfactors}] Fix a Jordan curve $\mathcal{C}\subset S^2$ with $\operatorname{post}(f)\subset \mathcal{C}$, and let $D_k=D_k(f,\mathcal{C})$ for $k\in \N_0$. In the following cells will be for $(f,\mathcal{C})$.
(i) The proof of the first part is easy. Suppose $d$ is a visual metric with expansion factor $\Lambda$. Then
there exists a constant $C\ge 1$ such that
$$ \diam(X)\le C \Lambda^{-k}$$
for all $k$-tiles (Lemma~\ref{lem:expoexp}~(ii)). Let
$\delta_0=\delta_0(f,\mathcal{C})>0$ be defined as in \eqref{defdelta} for
$f$, $\mathcal{C}$, and the metric $d$.
For each $k\in \N_0$ there exists a connected set joining opposite sides of $\mathcal{C}$ that consists of $D_k$ $k$-tiles. Hence
$$ \delta_0\le \diam(K)\le C D_k \Lambda^{-k}. $$
Taking the $k$th roots here and letting $k\to \infty$, it follows that
$\Lambda \le \Lambda_0(f)$ as desired.
\medskip
We break up the proof of (ii) in two parts (iia) and (iib).
\smallskip
(iia) We make the additional assumption that
the Jordan curve $\mathcal{C}$ is $f$-invariant and
$\Lambda>1$ satisfies
\begin{equation}\label{extraonL}
\Lambda\le D_1.
\end{equation}
In this case, we now proceed to construct a visual metric $d$ with expansion factor
$\Lambda$ that
satisfies \eqref{simmetric}. We first introduce some
terminology.
\smallskip
A \defn{tile chain} $P$ is a finite sequence of tiles $X_1,\dots,
X_N$, where $X_j\cap X_{j+1}\neq \emptyset$ for $j=1, \dots, N-1$.
Here we do
not
require the tiles to be of the same order.
A {\em subchain} of $P$ is a tile chain of the form $X_{j_1}, \dots, X_{j_s}$, where $1\le j_1<\dots<j_s\le N$.
The tile chain {\em joins} the
sets $A,B\subset S^2$ if $A\cap X_1\ne \emptyset$ and $B\cap X_N\ne \emptyset$. It joins the points $x,y\in S^2$ if it joins $\{x\}$ and $\{y\}$. Every chain joining two sets $A$ and $B$ contains a {\em simple} subchain joining $A$ and $B$, i.e., a chain that does not contain a proper subchain joining the sets. A chain $X_1, \dots, X_N$ joining two disjoint sets
$A$ and $B$ is simple if and only if $X_i\cap X_j=\emptyset$ whenever $0\le i,j\le N+1$ and $|i-j|\ge 2$, where $X_0=A$ and $X_{N+1}=B$.
Define the {\em weight} of a $k$-tile $X^k$ to be
\begin{equation}
w(X^k):=\Lambda^{-k},
\end{equation}
and the \defn{$w$-length} of a tile
chain $P$ consisting of the tiles $X_1,\dots, X_N$ as
\begin{equation*}
\operatorname{length}_w(P):=\sum_{j=1}^N w(X_j).
\end{equation*}
Now for $x,y\in S^2$ we define
\begin{equation}
\label{eq:defdF}
d(x,y):=\inf_P\operatorname{length}_w(P),
\end{equation}
where the infimum is taken over all tile chains $P$ joining $x$ and
$y$. Obviously, such tile chains exist and the infimum can be taken over simple tile
chains $P$.
{\em Claim 1.} The distance function $d$ is a visual metric with expansion factor $\Lambda$.
Symmetry and the triangle inequality immediately follow from the
definition of $d$. Since $f$ is expanding, we also have $d(x,x)=0$ for $x\in S^2$ .
Let $x,y\in S^2$ with $x\ne y$ be arbitrary, and define
$m=m(x,y)=m_{f,\mathcal{C}}(x,y)$ (see Definition~\ref{def:mxy}). Then there exist $m$-tiles $X$ and $Y$ with $x\in X$, $y\in Y$ and $X\cap Y\ne \emptyset$.
So $X,Y$ is a tile chain joining $x$
and $y$, and thus
\begin{equation*}
d(x,y)\leq w(X) + w(Y)=2\Lambda^{-m}.
\end{equation*}
To prove the claim, it remains to establish a lower bound $d(x,y)\ge (1/C)\Lambda^{-m}$
for a suitable constant $C$ independent of $x$ and $y$.
Pick $(m+1)$-tiles $X'$ and $Y'$ with $x\in X'$ and $y\in Y'$. Then
$X'\cap Y'= \emptyset$ by definition of $m$. Every tile chain joining
$x$ and $y$
contains a simple tile chain joining $X'$ and $Y'$.
So let $P$ be a simple tile chain joining $X'$ and $Y'$, and suppose it consists of the tiles $ X_1,\dots, X_N$. Let
$k\in \N_0 $ be the largest order of any tile in $P$. If $k\leq m+1$, then we get the favorable estimate
\begin{equation}\label {favorable}
\operatorname{length}_w(P)\ge \Lambda^{-k}\ge \Lambda^{-m-1}.
\end{equation}
Otherwise, $k>m+1$. We want to show that then we can replace the $k$-tiles
in $P$ by
$(k-1)$-tiles without increasing the $w$-length of the tile
chain. The construction is illustrated in Figure \ref{fig:pf_1_6}.
To see this, set $X_0=X'$, $X_{N+1}=Y'$, and
let $X_i$, where $1\le i\le N$, be the first $k$-tile in $P$. Since $P$ is a simple
chain joining $X'$ and $Y'$, the tile $X_i$ is not contained
in $X_{i-1}$ and so it has to meet $\partial X_{i-1}$. Since the order of $X_{i-1}$ is $<k$, we can find a $(k-1)$-edge $e\subset \partial X_{i-1}$
with $e\cap X_i\ne \emptyset$. Here and below we use the fact that $\mathcal{C}$ is $f$-invariant, and so cells of any order are subdivided by cells of higher order. Every
$(k-1)$-tile meets $e$ or is contained in the complement of the edge flower
$W^{k-1}(e)$ (see Lemma~\ref{lem:edgeflower}~(iii)). Since tiles of order $\le k-1$ are subdivided into tiles of order $k-1$, this implies that also every tile of order $\le k-1$ meets $e$ or is contained in the complement of $W^{k-1}(e)$.
\ifthenelse{\boolean{nofigures}}{}{
\begin{figure}
\centering
\begin{overpic}
[width=12cm,
tics=20]{pf_thm1_6.eps}
\put(9.4,14.7){$\scriptstyle{x}$}
\put(90.4,10.5){$\scriptstyle{y}$}
\put(8,0){$\scriptstyle{X'=X_0}$}
\put(27,0){$\scriptstyle{X_1}$}
\put(47,12){$\scriptstyle{X_{i-1}}$}
\put(55,12){$\scriptstyle{X_i}$}
\put(55,26){$\scriptstyle{e}$}
\put(60,26){$\scriptstyle{Z}$}
\put(65,33){$\scriptstyle{W^{k-1}(e)}$}
\put(67,15.5){$\scriptstyle{X_j}$}
\put(73.5,3.5){$\scriptstyle{X_N}$}
\put(85,0){$\scriptstyle{Y'=X_{N+1}}$}
\end{overpic}
\caption{Replacing $k$-tiles by $(k-1)$-tiles.}
\label{fig:pf_1_6}
\end{figure}
}
Now $P$ is simple and so no
tile in the ``tail'' $X_{i+1},\dots, X_N, X_{N+1}$ meets
$e$. Let $j'\in \N$ be the largest number such that $i\le j'\le N$ and all
tiles $X_i, \dots, X_{j'}$ are $k$-tiles. Then $X_{j'+1}$ has order
$\le k-1$. Since this tile does not meet $e$, it is contained in $S^2\setminus W^{k-1}(e)$, and so the tiles $X_i, \dots, X_{j'}$ form a chain of $k$-tiles joining $e$ and $S^2\setminus W^{k-1}(e)$. Let $j\in \N$ be the smallest number with $i\le j\le j'$ such that $X_j$ meets the complement of $W^{k-1}(e)$. Then $X_i, \dots, X_j$
is a chain $P^k$ of $k$-tiles joining $e$ and $S^2\setminus W^{k-1}(e)$.
In particular, $P^k$ joins two disjoint $(k-1)$-cells as follows from the definition of an edge flower (see Definition~\ref{def:edgeflower}).
Moreover, $X_j$ is the only tile in the chain $P^k$ that meets the complement
of $W^{k-1}(e)$.
Since $P^k$ joins disjoint $(k-1)$-cells, it follows from Lemma~\ref{lem:flowerbds} that $P^k$ has at least $D_1$ elements, and so by \eqref{extraonL},
\begin{equation*}
\operatorname{length}_w(P^k)\geq D_1 \Lambda^{-k}\geq \Lambda^{-k+1}.
\end{equation*}
Let $Z$ be the unique $(k-1)$-tile with $Z\supset X_j$. Then
$Z\cap X_{j+1}\ne \emptyset$. We also have $Z\cap e \ne \emptyset$; for otherwise $X_j\subset Z\subset S^2\setminus W^{k-1}(e)$. Then $j>i$ and $X_{j-1}$ meets $X_j$ and so the complement of $W^{k-1}(e)$ contradicting the definition of $j$.
So $Z\cap X_{i-1}\supset Z\cap e\ne \emptyset$.
Thus we can replace the subchain $P^k$ of $P$ by the single $(k-1)$-tile $Z$ to obtain a chain
$P'$ joining $X'$ and $Y'$. It satisfies
\begin{equation*}
\operatorname{length}_w(P')=\operatorname{length}_w(P)-\operatorname{length}_w(P^k) + w(Z) \leq \operatorname{length}_w(P).
\end{equation*}
By passing to a subchain of $P'$ we can find a simple chain $P''$
joining $X'$ and $Y'$ that contains fewer $k$-tiles than $P$ and satisfies $\operatorname{length}_w(P'')\le \operatorname{length}_w(P)$.
Continuing this process we can remove all $k$-tiles from the chain tile
joining $X'$ and $Y'$ without increasing its $w$-length. If $k-1>m+1$, we can repeat the process and remove the $(k-1)$-tiles without increasing the $w$-length, etc. In the end the we obtain a tile chain $\widetilde P$
joining $X'$ and $Y'$ that contains no tiles of order $>m+1$ and satisfies $\operatorname{length}_w(\widetilde P)\le \operatorname{length}_w(P)$.
Thus
\begin{equation*}
\operatorname{length}_w(P)\geq \operatorname{length}_w (\widetilde P)\ge \Lambda^{-m-1}.
\end{equation*}
This together with the previous estimate \eqref{favorable} implies
$$d(x,y)\ge (1/\Lambda)\Lambda^{-m}. $$
This is an inequality as desired, and so $d$ is indeed a visual metric with expansion factor $\Lambda$. In particular, this implies that $d$ induces the given topology on $S^2$ (see Proposition~\ref{prop:visualsummary}~(ii)). \smallskip
{\em Claim 2.} The visual metric $d$ as defined in \eqref{eq:defdF} has the expansion property \eqref{simmetric}.
We first show that
\begin{equation}\label{eq:uppexp}
d(f(x),f(y)) \leq\Lambda d(x,y),
\end{equation}
for all $x,y$ with $d(x,y)<1$. \smallskip
Indeed, suppose $x,y\in S^2$ with $d(x,y)<1$ are arbitrary.
Let $P$ be an arbitrary tile chain that joins $x$ and $y$ and consists of the tiles $X_1, \dots, X_N$.
Assume in addition that $P$ satisfies $\operatorname{length}_w(P)<1$. Then $P$ does not contain $0$-tiles and hence $f(X_1), \dots, f(X_N)$ is a tile chain joining $f(x)$ and $f(y)$. Calling the latter chain $f(P)$, we have $$\operatorname{length}_w (f(P))=\Lambda \operatorname{length}_w(P). $$
Taking the infimum over all such chains $P$ leads to the desired inequality \eqref{eq:uppexp}.
For an inequality in the other direction
we now consider two cases for $x\in S^2$.
{\em Case 1.}
$x\notin \operatorname{crit}(f).$
\noindent
Then we can find an open neighborhood $U$ of $x$ on which $f$ is a homeomorphism. Then $U'=f(U)$ is an open set containing $f(x)$.
We can choose $\epsilon>0$ and $\delta>0$ such that
$B_d(x,\delta)\subset U$,
$B_d(f(x),\epsilon)\subset U'$, and $f(B_d(x,\delta))\subset B_d(f(x),\epsilon)$.
Define $U_x=B(x,\delta)$, and let $y\in U_x$ be arbitrary. Then
$d(f(x), f(y))<\epsilon$. Consider a tile chain $P'$ joining
$f(x)$ and $f(y)$ whose $w$-length is close enough to $d(f(x), f(y))$ so that
$\operatorname{length}_w(P')<\epsilon$. By definition of the metric $d$, for every point $z$ that lies on a tile in $P'$, we have $d(f(x),z)\le \operatorname{length}_w(P')<\epsilon$. Hence $P'$ lies in $B_d(f(x), \epsilon)\subset U'$.
It follows that $(f|U)^{-1}$ is defined on every tile $X'$ in $P'$, and so by Lemma~\ref{adhoc}~(i) the Jordan region
$X=(f|U)^{-1}(X')$ is a tile contained in $U$. If $k$ is the order of $X'$, then $k+1$ is the order of $X$.
By considering these images of tiles in $P'$ under $(f|U)^{-1}$, we get a tile chain $P$ joining $x$ and $y$ with
$\operatorname{length}_w (P) = (1/\Lambda) \operatorname{length}_w(P')$. Taking the infimum over $P'$ we obtain
\begin{equation}\label{uppdbd}
d(x,y)\leq (1/\Lambda) d(f(x),f(y)),
\end{equation}
as desired.
{\em Case 2.} $x\in\operatorname{crit}(f)$.
Then $x\in f^{-1}(\operatorname{post}(f))$, and so $x$ is a $1$-vertex. Consider the flower $U=W^1(x)$, and its image $U'=f(W^1(x))=W^0(f(x))$. These are open neighborhoods of $x$ and $f(x)$, respectively, and the map
$f|U\setminus\{x\}$ is a (non-branched) covering map of $U\setminus\{x\}$ onto $U'\setminus \{f(x)\}$ (all this follows from the considerations in the proof of
Lemma~\ref{lem:constrmaps}).
Again we can choose $\epsilon>0$ and $\delta>0$ such that $B_d(x,\delta)\subset U$,
$B_d(f(x),\epsilon)\subset U'$, and $f(B_d(x,\delta))\subset B_d(f(x),\epsilon)$.
Define $U_x=B_d(x,\delta)$, and let $y\in U_x$ be arbitrary. In order to show \eqref{uppdbd} we may assume $x\ne y$. Then
$d(f(x), f(y))<\epsilon$ and $f(x)\ne f(y)$. Consider a tile chain $P'$
joining $f(x)$ and $f(y)$ consisting of tiles $X'_1,\dots, X'_N$. We can make the further assumptions that $X'_1$ is the only tile in this chain that contains $f(x)$ and that $\operatorname{length}_w(P')$ is close enough to $d(f(x), f(y))$ such that $\operatorname{length}_w (P')<\epsilon$. As before this implies that $P'$ lies in $U'$.
We now choose a path $\gamma\:[0,N]\rightarrow U'$ with the following properties:
\begin{itemize}
\smallskip
\item[(1)] $\gamma(0)=f(x)$, $\gamma(N)=f(y)$ and $\gamma(t)\ne f(x)$ for $t\ne 0$,
\smallskip
\item[(2)] $\gamma([i-1,i])\subset X'_i$ for $i=1, \dots, N$,
\smallskip
\item[(3)] $\gamma(i-1/2)\in \inte(X'_i)$ for $i=1, \dots, N$.
\end{itemize}
Since the tiles $X'_i$ are Jordan regions, such a path $\gamma$ can easily be obtained by first running from $f(x)$ in $X'_1$ to an interior point of $X'_1$, then to a point in $X'_1\cap X'_2$, then to an interior point of $X'_2$, etc., and finally to $f(y)\ne f(x)$ in $X_N$. Since $X'_1$ is the only tile in $P'$ containing $f(x)$, this can be done so that the path never meets $f(x)$ except in its initial point.
There exists a lift $\alpha$ of this path (under $f$) with endpoints $x$ and $y$, i.e., a path $\alpha\:[0,N]\rightarrow U$ with $\alpha(0)=x$,
$\alpha(N)=y$, and $f\circ \alpha=\gamma$. To obtain $\alpha$, lift
$\gamma|(0,N]$ under the covering map $f|U\setminus\{x\}$ such that the lift ends at $y$, and note that the lift has a unique continuous extension to $[0,N]$ by choosing $x$ to be its initial point.
Using this lift $\alpha$, we can construct a lift of our tile chain $P'$ as follows. Consider a tile $X'_i$ in $P'$ and
let $k_i$ be its order. Put $p_i:=\alpha(i-1/2) $ and $p_i':=\gamma(i-1/2)$.
Then $f(p_i)=p'_i\in \inte(X_i')$. By Lemma~\ref{adhoc}~(ii) there exists a unique $(k_i+1)$-tile $X_i$ with $p_i\in X_i$ and $f(X_i)=X_i'$.
Note that
$$\gamma((0,N])\subset U'\setminus\{f(x)\}=W^0(f(x))\setminus \{f(x)\}\subset S^2\setminus\operatorname{post}(f)$$ and that the map
$$f\: S^2\setminus f^{-1}(\operatorname{post}(f)) \rightarrow S^2\setminus \operatorname{post}(f)$$ is a
covering map. This implies that $\alpha|[i-1,i]$ is the unique lift of $\gamma|[i-1,i]$ with $\alpha(i-1/2)=p_i$. On the other hand, the path $\beta_i=(f|X_i)^{-1}\circ (\gamma|[i-1,i])$ is also a lift of $\gamma|[i-1,i] $ under $f$ with $\beta_i(i-1/2)=p_i$ by definition of $X_i$.
Hence $\beta_i=\alpha|[i-1,i]$ and so
$\alpha([i-1,i])\subset X_i$. It follows that $x=\alpha(0)\in X_1$, $y=\alpha(N)\in X_N$, and $X_i\cap X_{i+1}\supset \{\alpha(i)\}\ne \emptyset$
for $i=1, \dots, N-1$.
Therefore, the tiles $X_1, \dots, X_N$ form a tile chain $P$ joining $x$ and $y$. Since its tiles have an order by $1$ larger the order of the the corresponding tiles in $P'$, we have
$\operatorname{length}_w (P) = (1/\Lambda) \operatorname{length}_w (P')$. Taking the infimum over $P'$ we again obtain
inequality \eqref{uppdbd}
Combining \eqref{eq:uppexp} and \eqref{uppdbd} we see that every point $x\in S^2$ has a neighborhood $U_x$ such that \eqref{simmetric} holds.
This concludes the proof for the existence of the visual metric $d$
under the additional assumption in case (iia).
\medskip
(iib) We now consider the general case.
We can choose an iterate of $F=f^n$ of $f$ such that
$F$ has an $F$-invariant Jordan curve $\mathcal{C}\subset S^2$ with $\operatorname{post}(f)=\operatorname{post}(F)\subset \mathcal{C}$ (Theorem~\ref{thm:main}).
Note that $D_k(f,\mathcal{C})^{1/k}\to \Lambda_0(f)>\Lambda$. Hence
if $n$ is sufficiently large, what we may assume by passing to an iterate of $F$, we also have
$$ D_1(F,\mathcal{C}) = D_n(f,\mathcal{C})\ge \Lambda^n. $$
This means that $F$ is an expanding Thurston map satisfying the
assumptions in (iia).
Thus there exists a visual metric for $F$
with expansion factor $\Lambda^n$ that satisfies the corresponding
version of \eqref{simmetric}. We call this metric $\rho$ in order to
distinguish it from that metric $d$ that we are trying to
construct. Then $\rho$ is a visual metric for $F$ with expansion
factor $\Lambda^n$, and for each $x\in S^2$ there exists an open
neighborhood $U_x$ of $x$ such that
\begin{equation}\label{rhoexp}
\rho(F(x), F(y))=\rho(f^n(x), f^n(y))=\Lambda^n \rho(x,y)
\end{equation}
for all $y\in U_x$.
We now define $d$ as
\begin{equation}
\label{eq:defdf}
d(x,y)=\frac{1}{n}\sum_{i=0}^{n-1}\Lambda^{-i} \rho(f^{i}(x), f^{i}(y))
\end{equation}
for $x,y\in S^2$.
Then \eqref{simmetric} follows from the corresponding property
\eqref{rhoexp} with the same sets $U_x$, $x\in S^2$; indeed,
if $x\in S^2$ and $y\in U_x$ then by \eqref{rhoexp} we have
\begin{align*}
d(f(x),f(y)) &= \frac{1}{n} \sum_{i=0}^{n-1}\Lambda^{-i} \rho(f^{i+1}(x), f^{i+1}(y))
\\
& = \frac{1}{n}\biggl(\Lambda \sum_{i=0}^{n-2}\Lambda^{-(i+1)} \rho(f^{i+1}(x), f^{i+1}(y))+ \Lambda \rho(x,y)\biggr)
\\
& = \Lambda \frac{1}{n} \sum_{i=0}^{n-1}\Lambda^{-i} \rho(f^{i}(x), f^{i}(y))
\, =\, \Lambda d(x,y).
\end{align*}
It remains to show that $d$ is a visual metric for $f$ with expansion factor $\Lambda$. It is clear that $d$ is a metric on $S^2$.
Let $m=m_{f,\mathcal{C}}$ and $m_F=m_{F,\mathcal{C}}$ be as in Definition~\ref{def:mxy}. Since
$\rho$ is a visual metric for $F$ with expansion factor $\Lambda^n$ we have
$$ \rho(x,y)\asymp \Lambda^{-nm_F(x,y)}\asymp \Lambda^{-m(x,y)}$$ for
all $x,y\in S^2$ by Lemma~\ref{lem:mprops} (iv).
Hence
$$d(x,y)\ge
\frac{1}{n} \rho(x,y)\gtrsim \Lambda^{-m(x,y)} . $$
Moreover, by Lemma~\ref{lem:mprops} (ii) we have
$$ m(f^i(x), f^i(y))\ge m(x,y)-i$$
and so
$$ \rho(f^i(x), f^i(y))\asymp \Lambda^{-m(f^i(x),f^i(y))}\le
\Lambda^i\Lambda^{-m(x,y)}$$
for all $i\in \N_0$. Hence
$$d(x,y)\lesssim \frac 1n \sum_{i=0}^{n-1}\Lambda^{-m(x,y)}
= \Lambda^{-m(x,y)}. $$
It follows that $d(x,y)\asymp \Lambda^{-m(x,y)}$ for all $x,y\in S^2$
where $C(\asymp)$ is independent of $x$ and $y$. This shows that $d$ is a visual metric for $f$ with expansion factor
$\Lambda$.
\end{proof}
If $f\:S^2\rightarrow S^2$ is an expanding Thurston map and $\Lambda$ is the expansion factor of a visual metric for $f$, then by Theorem~\ref{thm:visexpfactors}~(i) we have $1<\Lambda\le \Lambda_0(f)$. On the other hand,
statement (ii) in this theorem only guarantees the existence of a visual metric
with expansion factor $\Lambda$ for $1<\Lambda< \Lambda_0(f)$. As the following example shows, this statement is optimal, since a visual metric with expansion factor $\Lambda=\Lambda_0(f)$ does not exist in general.
\begin{ex}\label{ex:notattained} In the following we will leave the verification of some details to the reader. The setup is very similar to Example~\ref{ex:infty_C}. We use real notation and consider the map $\wp\:\R^2\cong \C \rightarrow \CDach$ as defined in Section~\ref{sec:Lattes}. For $u_1,u_2\in \R^2$ we have
$\wp(u_1)=\wp(u_2)$ if and only if $u_2=\pm u_1+\gamma$ for $\gamma\in L=2\Z^2$. The critical points of $\wp$ are precisely the points in $\Z^2$. Note that
$\wp(\Z^2)=\{0,1, \infty, -1\}$ and $\wp(\frac12 \Z^2\setminus \Z^2)=\{\sqrt 2-1, 1-\sqrt 2, {\mathbf{\imath}},-{\mathbf{\imath}}\}$. Now let
$$A= \left(\begin{array}{cc} 2 & 2\\
0 & 2\end{array}\right).
$$
Then for $n\in \N$ we have
\begin{equation} \label{eq:Apower}
A^n= \left(\begin{array}{cc} 2^n & n2^{n}\\
0 & 2^n\end{array}\right) \text{ and } A^{-n}= \left(\begin{array}{cc} 2^{-n} & -n2^{-n}\\
0 & 2^{-n}\end{array}\right).
\end{equation}
Denote by $\psi\: \R^2\rightarrow \R^2$ the map induced by left-multiplication by $A$, i.e., $\psi(u)=Au$ for $u\in \R^2$, where $u\in \R^2$ is considered as a column vector.
Then there exists a well-defined and unique map $f\: \CDach \rightarrow \CDach$ such that the diagram
\begin{equation*}
\xymatrix{
\R^2 \ar[r]^\psi \ar[d]_{\wp} &
\R^2 \ar[d]^{\wp}
\\
\CDach \ar[r]^f & \CDach
}
\end{equation*}
commutes. The map $f$ is a branched covering map of $\CDach$ with
$$\operatorname{crit}(f)=\wp(\psi^{-1}(\Z^2)\setminus \Z^2) =\wp(\tfrac 12 \Z^2\setminus \Z^2) =\{\sqrt 2-1, 1-\sqrt 2, {\mathbf{\imath}},-{\mathbf{\imath}}\}, $$
and $$\operatorname{post}(f)=\wp(\Z^2)=\{0,1,\infty, -1\}. $$ Hence $f$ is a Thurston map.
Let $Q=[0,1]^2$ be the unit square and $G\subset \R^2$ by the standard grid (see Example~\ref{ex:infty_C}). Then $\mathcal{C}:=\wp(\partial Q)=\wp (G)$ is a Jordan curve in $\CDach$ with $\operatorname{post}(f)=\wp(\Z^2)\subset \mathcal{C}$.
Let $n\in \N$ and $G_n=\psi^{-n}(G)$. Then the $n$-tiles for $(f,\mathcal{C})$ are precisely the closures of the images of the complementary regions of $G_n$ in $\R^2$ under the map $\wp$, i.e., precisely the sets
given by $$X^n=\wp(\psi^{-n}(\alpha +Q)),$$ where $\alpha\in \Z^2$.
Indeed, suppose a set $X^n\subset \CDach$ has this form. Then $X^n$ is a Jordan region, since $\wp$ is injective on $\psi^{-n}(\alpha+Q)$.
Moreover, since $f^n\circ \wp =\wp\circ \psi^n$ and since $\wp|\alpha+Q$ is a homeomorphism of $\alpha+Q$ onto the $0$-tile $\wp(\alpha+Q)$, the map $f^n|X^n$ is a homeomorphism of the Jordan region $X^n$ onto a $0$-tile.
So by Lemma~\ref{adhoc}~(i), the set $X^n$ is indeed an $n$-tile. Since
these sets $X^n$ cover $\CDach$, there are no other $n$-tiles.
If a point $(x,y)\in \R^2$ (now considered as a row vector) lies in $\psi^{-n}(Q)$, then $-n2^{-n}\le x\le 2^{-n}$ and $0\le y\le 2^{-n}$ as follows from \eqref{eq:Apower}.
This implies that if we equip $\CDach$ with the flat metric (see Section~\ref{sec:Lattes}), then for each $n$-tile $X^n$ we have $\diam(X^n)\lesssim
n2^{-n}$.
In particular,
$$\lim_{n\to \infty} \max\{ \diam(X^n): X^n \text{ is as $n$-tile for $(f,\mathcal{C})$}\} =0, $$
and so $f$ is an expanding Thurston map. Moreover, we also conclude that
$D_n:=D_n(f,\mathcal{C})\gtrsim 2^n/n$. Note that we actually have $\diam(X^n)\asymp
2^{-n}/n$ for each $n$-tile $X^n$. This implies that the flat metric on $\CDach$ is {\em not} a visual metric for $f$ (see Lemma~\ref{lem:expoexp}~(ii)).
If $X_i:=\wp(\psi^{-n}((0, -i)+Q))$ for $i=1, \dots , N:=
\lceil 2^n/n\rceil$, then $X_1, \dots, X_N$ is a chain of $n$-tiles joining opposite sides of $\mathcal{C}$. Hence $D_n\le N\lesssim 2^{-n}/n$.
It follows that
$D_n\asymp 2^n/n$, and so
$$\Lambda_0(f)=\lim_{n\to \infty} D_n^{1/n}=2. $$
If $\Lambda$ is the expansion factor of a visual metric, then, as we have seen in the proof Theorem~\ref{thm:visexpfactors}~(i), we must have
$1\lesssim D_n\Lambda^{-n}\asymp 2^n\Lambda^{-n}/n $. Hence
a visual metric for $f$ with expansion factor $\Lambda =\Lambda_0(f)=2$ does not exist.
\end{ex}
The map $f$ as defined above is not conformal and an example of a {\em Latt\`es-type map}.
Latt\`es and Latt\`es-type maps in connection with the combinatorial expansion factor are more systematically investigated in \cite{Qian}.
\section{Rational Thurston maps}
\label{sec:rational-maps}
\noindent
In this section we consider rational Thurston maps, i.e.,
rational maps $\R\: \CDach \rightarrow \CDach$ on the Riemann sphere $\CDach$ that are Thurston maps.
Metric notions on $\CDach$ will usually refer to the chordal metric $\sigma$ on $\CDach$.
We first establish a proposition that shows when a rational Thurston map is expanding.
In the statement of this proposition and its proof we will use some basic concepts of complex dynamics.
For the relevant definitions and general background on this subject, see \cite{CG} and \cite{Mi}.
\begin{prop} \label{prop:rationalexpch}
Let $R\:\CDach\rightarrow \CDach$ be a rational
Thurston map. Then the following conditions are equivalent:
\smallskip
\begin{itemize}
\item[(i)] $R$ is expanding.\index{expanding}
\smallskip
\item[(ii)] The Julia set of $R$ is equal to $\CDach$.
\smallskip
\item[(iii)] $R$ has no periodic critical points.
\end{itemize}
\end{prop}
\begin{proof} It suffices to establish the chain of implications $\text{(i)}\Rightarrow \text{(ii)}
\Rightarrow \text{(iii)} \Rightarrow \text{(i)}$.
$\text{(i)}\Rightarrow \text{(ii)}$: Suppose $R$ is expanding. Pick a Jordan curve
$\mathcal{C}\subset \CDach$ with $\operatorname{post}(R)\subset \mathcal{C}$, and let $U$ be an arbitrary
nonempty
open set. Since $R$ is expanding, the diameters of $n$-tiles for $(f,\mathcal{C})$ approach $0$ uniformly as $n\to \infty$. This implies that there exist $n\in\N_0$, and $n$-tiles $X,Y\subset U$ such that $X$ and $Y$ are distinct, but share an $n$-edge. Then $R^n(X)$ and $R^n(Y)$ are the two $0$-tiles. In particular,
$R^n(U)=\CDach$. So every nonempty open set $U$ is mapped to the whole Riemann sphere by a sufficiently high iterate of $R$. This implies that the Julia set $\mathcal{J}$ of $R$ is equal to $\CDach$.
For otherwise, the {\em Fatou set} $\mathcal {F}=\CDach\setminus \mathcal{J}$ of $R$ is a nonempty open subset of $\CDach$.
Since $\mathcal{F}$ is $R$-invariant and $\mathcal{J}\ne \emptyset$ and so $\mathcal{F}\ne \CDach$, we get $R^n(\mathcal{F})=\mathcal{F}\ne \CDach$ for all $n\in \N_0$. This
contradicts what we have just seen.
$\text{(ii)}\Rightarrow \text{(iii)}$: If the Julia set of $R$ is equal to $\CDach$, then its Fatou set is empty. This implies that $R$ cannot have periodic critical points, because a periodic critical point of a rational map
is part of a {\em super-attracting cycle} and belongs to the Fatou set.
$\text{(iii)}\Rightarrow \text{(i)}$: Suppose $R$ has no periodic critical points.
It is a known fact from complex dynamics that then there exists a conformal metric on
$\CDach$ with conical singularities in the points of $\operatorname{post}(R)$ such that for the norm $\Vert R'(z) \Vert$ of the derivative
with respect to this metric we have
$\Vert R'(z)\Vert\geq \rho>1$ for all $z\in \CDach$ (see \cite[Thm.~19.6]{Mi} or \cite[V.4.3.1]{CG}). More precisely, there exists a smooth function
$\lambda\: \CDach\setminus\operatorname{post}(R)\rightarrow (0,\infty)$ such that for each
$p\in \operatorname{post} (R)$ we have
\begin{equation}\label{conicsing}
\lambda(z)\asymp \frac{1}{\sigma(z,p)^{\alpha_p}}
\end{equation}
for all $z$ near $p$, where $\alpha_p\in (0,1)$.
Moreover, there exists $\rho>1$ such that
\begin{equation} \label{infiniexp}
\Vert R'(z)\Vert= \frac{\lambda(R(z))R^{\sharp}(z)}{\lambda(z)}\ge \rho
\end{equation} for all $z\in \CDach$,
where
$$ R^{\sharp}(z)=\frac{(1+|z|^2)|R'(z)|}{1+|R(z)|^2}$$ is the {\em spherical derivative}
of $R$ and the expressions have to be understood in a suitable limiting sense at singularities (there are at most finitely many).
Using $\lambda$ we can introduce a path metric on $\CDach$ defined by
$$d(x,y)=\inf_{\gamma} \int_\gamma \lambda\, ds$$
for $x,y\in \CDach$,
where integration is with respect to the spherical length element $ds$ and the infimum is
taken over all rectifiable paths $\gamma\:[0,1]\rightarrow \CDach$ with $\gamma(0)=x$ and $\gamma (1)=y$.
Since we have $\alpha_p\in (0,1)$ for the exponents in \eqref{conicsing}, it follows that that there exist constants $C\ge 1$ and $\epsilon\in (0,1)$ such that
$$\frac 1C \sigma(z,w)\le d(z,w)\le C\sigma(z,w)^{1-\epsilon} $$
for all $z,w\in \CDach$ (actually, we can take $\displaystyle\epsilon=\max_{p\in \operatorname{post}(R)}
\alpha_p$).
In particular, $d$
is a metric on $\CDach$ that induces the standard topology on $\CDach$.
Moreover,
$$\operatorname{length}_d(\gamma)=\int_\gamma\lambda\,ds$$ for all rectifiable paths $\gamma\:[0,1]\rightarrow \CDach$.
This together with \eqref{infiniexp} implies that if $\gamma\:[0,1]\rightarrow \CDach$ is a path, then
\begin{equation}\label{pathexp}
\operatorname{length}_d (R\circ \gamma)\ge \rho \operatorname{length}_d(\gamma).
\end{equation}
Now one can see that $R$ is expanding as follows. We pick a Jordan curve $\mathcal{C}\subset \CDach$
with $\operatorname{post}(R)\subset \mathcal{C}$ and consider the two
$0$-tiles $\XOw,\XOb\in \mathcal{D}^0(R, \mathcal{C})$. We assume that $\mathcal{C}$ has been chosen
so that any two points in one of the $0$-tiles can be joined by a path in the same tile of controlled $d$-length. More precisely, we require that
there exists a constant $L_0$ such that if $x,y\in X^0_i$ for $i\in
\{{\tt w},{\tt b}\}$, then there exists a rectifiable path $\beta $ in $X^0_i$ joining $x$ and $y$ such that
\begin{equation}\label{eq:uniflen}
\operatorname{length}_d(\beta )\le L_0.
\end{equation}
A Jordan curve $\mathcal{C}$ with this property can easily be obtained if we choose it
to consist of finitely many geodesic segments in the spherical metric (here we think of $\CDach$ as identified with the unit sphere in $\R^3$ by stereographic
projection). It is clear that then we easily get a uniform estimate as in \eqref{eq:uniflen} for points $x$ and $y$ that stay away from the finitely many singularities of the conformal density $\lambda$; if one of the points is close (or even equal) to such a singularity, then one has to choose the initial piece of the connecting curve $\beta$ so that
its initial piece runs away from the singularity ``radially". Since $\alpha_p<1$
for the exponent in \eqref{conicsing}, this leads to a favorable uniform estimate as in
\eqref{eq:uniflen}.
Now consider an $n$-tile $X\in \mathcal{D}^n(R, \mathcal{C})$, and let $u,v\in X$
be arbitrary. Then $R^n(X)$ is a $0$-tile, say $R^n(X)=\XOw$, and
$R^n|X$ is a homeomorphism of $X$ onto $\XOw$. Let $x=R^n(u)$ and
$y=R^n(v)$.
By choice of $\mathcal{C}$, we can find a path $\beta$ in $\XOw$ that joins
$x$ and $y$ and satisfies $\operatorname{length}_d(\beta)\le L_0. $
Then $\gamma:=(R^n|X)^{-1}\circ \beta$ is a path in $X$ joining $u$ and $v$.
We have $\beta =R^n\circ \gamma$, and so by applying \eqref{pathexp}
repeatedly we conclude
$$\operatorname{length}_d(\gamma)\le \frac 1{\rho^n}\operatorname{length}_d(\beta )\le L_0\rho^{-n}. $$
This implies $d(u,v)\le L_0 \rho^{-n}$. Since $u,v\in X$ were arbitrary, we have
$$\diam_d(X)\le L_0\rho^{-n}. $$
So, if as usual $\X^n$ denotes the set of all $n$-tiles for $(R,\mathcal{C})$, then
$$\max_{X\in \X^n}\diam_d(X)\le L_0\rho^{-n}. $$
Since $\rho>1$ it follows that
$$\lim_{n\to \infty} \max_{X\in \X^n}\diam_d(X)=0. $$
Hence $R$ is expanding. \end{proof}
For the proof of Theorem~\ref{thm:qsrational} we need some preparation.
Let $\mathcal{D}$ be a cell complex. A subset $\mathcal{D}'\subset \mathcal{D}$ is called a {\em subcomplex} of $\mathcal{D}$ if the following condition is true:
if $\tau\in \mathcal{D}'$, $\sigma \in \mathcal{D}$,
and $\sigma\subset \tau$, then $\sigma\in \mathcal{D}'$.
If $\mathcal{D}'$ is a subcomplex of $\mathcal{D}$, then the cells in $\mathcal{D}'$ form a cell decomposition of the underlying set
$$ |\mathcal{D}'|:=\bigcup\{c:c\in \mathcal{D}'\}. $$
Now suppose that $f\: \CDach \rightarrow \CDach$ is a rational Thurston map, and $\mathcal{C}\subset \CDach$ is a Jordan curve with $\operatorname{post}(f)\subset \mathcal{C}$.
Consider the cell decompositions $\mathcal{D}^n=\mathcal{D}^n(f,\mathcal{C})$ of
$\CDach$. If $\tau\in \mathcal{D}^n$, then $f^n|\tau$ is a homeomorphism of
the $n$-cell $\tau$ onto the $0$-cell $f^n(\tau)$. So the map $\tau\mapsto f^n(\tau)$ induces a labeling $\mathcal{D}^n\rightarrow \mathcal{D}^0$ (see Definition~\ref{def:labeldecomp}). We call this the
{\em natural labeling} on $\mathcal{D}^n$. Similarly, the map $\tau\mapsto f^n(\tau)$ induces a natural labeling on every subcomplex of $\mathcal{D}^n$.
\begin{lemma}\label{lem:confequiv}Let $n,m\in \N_0$, $\mathcal{D}$ be a subcomplex of $\mathcal{D}^n$, and $\mathcal{D}'$ be a subcomplex of $\mathcal{D}^m$ equipped with the natural labelings. If $\psi\: \mathcal{D}\rightarrow \mathcal{D}'$ is a label-preserving isomorphism, then there exists
a homeomorphism $h\: |\mathcal{D}|\rightarrow |\mathcal{D}'|$ such that
\begin{itemize}
\smallskip
\item[(i)] $h(\tau)=\psi(\tau)$ for each $\tau \in \mathcal{D}$,
\smallskip
\item[(ii)] $h$ maps $\inte(|\mathcal{D}|)$ conformally onto $\inte(|\mathcal{D}'|)$.
\end{itemize}
\end{lemma}
See Definition~\ref{def:compiso} for the notion of a label-preserving isomorphism of cell complexes.
Roughly speaking, the lemma says that combinatorial equivalence of two subcomplexes $\mathcal{D}$ and $\mathcal{D}'$
gives conformal equivalence of their underlying sets.
\begin{proof} If $\tau\in \mathcal{D}$, then $f^n(\tau)=f^m(\psi(\tau))$, because
$\psi$ is label-preserving. Hence we can define a homeomorphism
$h_\tau:=(f^m|\psi(\tau))^{-1}\circ (f^n|\tau)$ of $\tau $ onto $\psi(\tau)$.
It is clear that the maps $h_\tau$ are compatible under inclusions of cells: if $\sigma,\tau\in \mathcal{D}$ and $\sigma\subset \tau$, then $h_\tau|\sigma=h_\sigma$.
We now define a map $h\: |\mathcal{D}|\rightarrow |\mathcal{D}'|$ as follows. For $p\in |\mathcal{D}|$ pick $\tau\in \mathcal{D}$ with $p\in \tau$. Set $h(p):=h_\tau(p)$. By an argument as in the proof of of Proposition~\ref{prop:thurstonex} one can show that $h$ is well-defined and a homeomorphism of $|\mathcal{D}|$ onto $|\mathcal{D}'|$. Obviously, $h$ has property (i).
To establish property (ii) it suffices to show that $h$ is holomorphic near each interior point $p$ of $|\mathcal{D}|$. We consider several cases. If $p$ is an interior
point of a tile $X\in \mathcal{D}$, then this is clear by definition of $h$.
Suppose $p$ is an interior point of an $n$-edge $e$. Then the two $n$-tiles $X,Y\in \mathcal{D}^n$ that contain $e$ in their boundaries are in $\mathcal{D}$.
Similarly, if $X'=\psi(X)$, $Y'=\psi(Y)$, $e'=\psi(e)$, then $X'$ and $Y'$ are the two $m$-tiles that contain the $m$-edge $e'$ in their boundaries.
Let $X_0=f^n(X)=f^m(X')$, $Y_0=f^n(Y)=f^m(Y')$ and $e_0=f^n(e)=f^m(e')$, and set
$U=\inte(X)\cup \inte(e)\cup \inte (Y)$,
$U'=\inte(X')\cup \inte(e')\cup \inte (Y')$, and
$U_0=\inte(X_0)\cup \inte(e_0)\cup \inte (Y)_0$. It follows from the
considerations in the proof of Lemma~\ref{lem:pullback} that $f^n|U$ is a conformal map of $U$ onto $U_0$ and that $f^m$ is a conformal map of
$U'$ onto $U_0$. Hence $g=(f^m|U')^{-1}\circ (f^n|U)$ is a conformal map of $U$ onto $U'$. Obviously, $g=h|U$, and so $h$ is holomorphic near $p\in \inte(e)\subset U$.
Finally, if $p$ is an $n$-vertex, then there is an open neighborhood $U\subset |\mathcal{D}|$ of $p$ that contains no other $n$-vertex. By what we have seen, $h$ is holomorphic on $U\setminus \{p\}$. Since $h$ is continuous in $p$, the point $p$ is a removable singularity, and hence $h$ is holomorphic near $p$.
\end{proof}
The following lemma provides a version of Koebe's distortion theorem
for conformal maps on multiply connected regions. As we will see in the proof,
the statement can easily be reduced to some classical estimates.
\begin{lemma} \label{lem:distest} Let $\Om\subset \CDach$ be a region, and
$A,B\subset \Om$ continua. Then there exist constants $C=C(A,B,\Om)>0$ and $C'=C'(A,\Om)>0$ such that for all conformal maps $h\: \Om\rightarrow \Om':=h(\Om) $ whose image $\Om'$ is contained in a hemisphere of $\CDach$ we have
\begin{equation}\label{di0}
\frac 1{C} \diam(h(A))\le \diam(h(B))\le C \diam(h(A)),
\end{equation}
and
\begin{equation}\label{di1}
\operatorname{dist}(h(A), \partial \Om')\le C' \diam(h(A)).
\end{equation}
\end{lemma}
The condition that the image of $h$ is contained in a hemisphere prevents it from being too large. It is easy to see that without such an assumption the lemma is not true in general.
\begin{proof} Using auxiliary rotations, we may assume that $\Om\subset \C$ and $\Om'\subset \D$. Then the chordal and Euclidean metrics on $\Om'$ are bi-Lipschitz equivalent. So it suffices to prove the desired inequalities for the Euclidean instead of the chordal metric.
If $h\:\Om \rightarrow \Om'$ is conformal, and $D=D(z_0,r)\subset \Om$ is a Euclidean disk with $2D:=D(z_0, 2r)\subset \Om$, then Koebe's distortion theorem
\cite[Thm.~1.3]{Po} implies that $$\max_{z\in D}|h'(z)|\le C_0\min_{z\in D}|h'(z)|, $$
where $C_0\ge 1$ is a universal constant (one can actually take $C_0=81$).
If $z_1\in A$ and $z_2\in B$ are arbitrary, then there exists a chain $D_1, \dots , D_N$ of Euclidean disks with $z_1\in D_1$, $z_2\in D_N$ and $2D_i\subset \Om$ for $i=1, \dots, N$, where $N$ is bounded above by a constant only depending on $A$, $B$, $\Om$, but not on $z_1$ and $z_2$.
It follows that
$$\max_{z\in B}|h'(z)|\le C_1\min_{z\in A}|h'(z)|, $$
where $C_1$ only depends on $A,B,\Om$ (and not on $h$). Since
$$\diam(h(B))\le C_2 \max_{z\in B}|h'(z)|, $$
and
\begin{equation}\label{di3}
\min_{z\in A}|h'(z)|\le C_3 \diam(h(A)),
\end{equation}
where $C_2=C_2(B,\Om)$ and $C_3=C_3(A,\Om)$, it follows that
$$ \diam(h(B))\le C_4\diam(h(A)),$$
where $C_4=C_4(A,B,\Om)$. An inequality in the opposite direction follows by reversing the roles of $A$ and $B$.
By a standard estimate for conformal maps (see \cite[Cor.~1.4]{Po})
we have
$$\operatorname{dist}(h(z), \partial \Om')\le |h'(z)|\operatorname{dist}(z, \partial \Om)$$
for all $z\in \Om$. This
implies that
$$\operatorname{dist}(h(A), \partial \Om')\le C_5
\min_{z\in A}|h'(z)|, $$ where $C_5=C_5(A)$. This together with \eqref{di3} gives an inequality as in \eqref{di1}.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:qsrational}]
Suppose first that $f$ is an expanding Thurston map and that $f$ is topologically conjugate to a rational map. Then this rational map is also
a Thurston map, and it is expanding as one can see by an argument as in the proof of Corollary~\ref{cor:conjisom}. Moreover, the conjugating map will be a snowflake equivalence with respect to visual metrics, and in particular a quasisymmetry.
All this implies that we may actually assume that $f$ itself is an expanding rational Thurston map.
By Theorem~\ref{thm:main} we choose an iterate $F=f^n$ of $f$ that has an invariant Jordan curve $\mathcal{C}\subset \CDach$ with $\operatorname{post}(f)=\operatorname{post}(F)\subset \mathcal{C}$.
In the following all cells will be with respect to
$(F,\mathcal{C})$.
The map $F$ is also an expanding rational Thurston map (Lemma~\ref{lem:Thiterates}), and the class of visual metrics for $f$ and $F$ are the same (Proposition~\ref{prop:visualsummary}~(v)).
So for the desired implication of the theorem, it suffices to show that if $d$ is any visual metric for $F$, then $d$ is quasisymmetrically equivalent to the chordal metric $\sigma$.
The argument is now very similar
to the considerations in \cite{Me08}, which go back to \cite{Me02}. For the convenience of the reader we will provide full details. We will proceed in several steps. Metric notions will refer to the chordal metric.
\smallskip
Let $k\in \N_0$ and $X,Y\in \X^k$. Then
\begin{equation} \label{samesize}
X\cap Y\neq \emptyset \Rightarrow \diam (X)\asymp \diam (Y).
\end{equation}
Here $C(\asymp)$ is independent of $X, Y$, and the order $k$.
This is seen as follows. For $k\in \N_0$ and non-disjoint $k$-tiles $X$ and $Y$,
consider the complex $\mathcal{D}(X,Y)$, equipped with the natural labeling, consisting of all
$k$-cells $c$ for which there exists a $k$-tile $Z$ with $c\subset Z$ and $Z\cap (X\cup Y)\ne \emptyset$.
Obviously,
\begin{equation}\label{eq:defVkXY}
|\mathcal{D}(X,Y)|=\bigcup \{Z\in\X^k : Z\cap(X\cup Y)\neq \emptyset\}. \end{equation}
Let $\Om(X,Y)$ be the interior of $|\mathcal{D}(X,Y)|$. Then $\Om(X,Y)$ is a region containing $X$ and $Y$.
Suppose that $X',Y'$ is a pair of non-disjoint $m$-tiles, $m\in \N_0$. We call the complexes $\mathcal{D}(X',Y')$ and $\mathcal{D}(X,Y)$ equivalent if there exists a label-preserving
isomorphism $\psi\: \mathcal{D}(X',Y')\rightarrow \mathcal{D}(X,Y)$
with $\psi(X')=X$ and $\psi(Y')=Y$.
If $\mathcal{D}(X',Y')$ and $\mathcal{D}(X,Y)$ are equivalent, then
by Lemma~\ref{lem:confequiv} there exists a conformal map $h\: \Om(X',Y')\rightarrow \Om(X,Y)$ with
$h(X')=X$ and $h(Y')=Y$.
Since $F$ is an expanding rational Thurston map, it has no periodic critical points (Proposition~\ref{prop:rationalexpch}) and so the length of the cycle of each vertex is uniformly bounded (see
Lemma~\ref{lem:flowerprop}~(i) and Lemma~\ref{lem:cycle}).
This implies that the number of $k$-tiles, and hence the number of $k$-cells, in
$\mathcal{D}(X,Y)$ is uniformly bounded by a number independent of $X$, $Y$, and $k$.
Therefore, among the complexes
$\mathcal{D}(X,Y)$ there are only finitely many equivalence classes. Since $F$ is expanding, there are also only finitely many complexes
$\mathcal{D}(X,Y)$ such that $\Om(X,Y)$ is not contained in a hemisphere.
Hence we can find finitely many complexes
$\mathcal{D}(X_1, Y_1), \dots, \mathcal{D}(X_N, Y_N)$ such that
each complex $\mathcal{D}(X,Y)$ not in this list is
equivalent to one complex $\mathcal{D}(X_i,Y_i)$ and such that $\Om(X,Y)$ is contained in a hemisphere.
It follows from Lemma~\ref{lem:distest} (applied to
$A=X_i$, $B=Y_i$, $\Om=\Om(X_i,Y_i)$, and the conformal map $f\:\Om(X_i,Y_i)\rightarrow \Om(X,Y)$ produced by Lemma~\ref{lem:confequiv})
that
$\diam(X)\asymp \diam(Y)$ with $C(\asymp)$ independent of $X$, $Y$, and $k$.
\smallskip
As a consequence one immediately obtains the following fact. If $m,k\in \N_0$, $X\in\X^k$,
$Y\in \X^{k+m}$,
and $Y\subset X$, then
\begin{equation}\label{eq:diamXmkXm}
\diam (Y)\asymp \diam (X),
\end{equation}
where $C(\asymp)=C(m)$.
Indeed, it is clear that $\diam(Y)\le \diam(X)$. On the other hand, the number of $(m+k)$-tiles contained in
$X$ is bounded by a number only depending on $m$. This and \eqref{samesize} imply
that
$ \diam (Y)\asymp \diam (Z)$ whenever $Z\in \X^{k+m}$ and
$Z\subset X$, where $C(\asymp)=C(m)$.
Hence
$$ \diam(X)\le \sum_{Z\in \X^{k+m}, Z\subset X}
\diam(Z)\lesssim \diam(Y),$$
where $C(\lesssim)=C(m)$.
\smallskip
Let $k\in \N_0$ and $X,Y\in \X^k$. Then
\begin{equation}\label{XYdistest}
X\cap Y=\emptyset \Rightarrow \operatorname{dist}(X,Y)\gtrsim \diam (X),
\end{equation}
where $C(\gtrsim)$ is independent of $X,Y$, and $k$. The argument to see this is very similar to the one for \eqref{samesize}
Indeed, for $k\in \N_0$ and $X\in \X^k$ consider the cell complex $\mathcal{D}(X)$, equipped with the natural labeling, consisting of all
$k$-cells $c$ for which there exists a $k$-tile $Z$ with $c\subset Z$ and $Z\cap X\ne \emptyset$.
Then
\begin{equation}
|\mathcal{D}(X)|=\bigcup \{Z\in\X^k : X\cap Z\neq \emptyset\}. \end{equation}
If we define $\Om(X)$ to be the interior
of $|\mathcal{D}(X)|$, then $|\mathcal{D}(X)|$ is a region containing $X$.
If $m\in \N_0$ and $X'$ is an $m$-tile, then we call the complexes $\mathcal{D}(X')$ and $\mathcal{D}(X)$ equivalent if there exists a label-preserving
isomorphism $\psi\: \mathcal{D}(X')\rightarrow \mathcal{D}(X)$
with $\psi(X')=X$.
Again there only finitely many equivalence classes of the complexes $\mathcal{D}(X)$. Based on Lemma~\ref{lem:confequiv} and Lemma~\ref{lem:distest}, we conclude that
$$\operatorname{dist}(X, \partial \Om(X))\lesssim \diam(X), $$
where $C(\lesssim)$ does not depend on $X$ and $k$.
Now if $Y$ is a $k$-tile with $X\cap Y=\emptyset$, then $Y\cap\Om(X)=\emptyset$ and so
$$ \operatorname{dist}(X,Y)\le \operatorname{dist}(X,\partial \Om(X))\lesssim \diam(X)$$
as desired.
\smallskip
Since $F$ has no periodic critical points, the space $(\CDach, d)$ is
doubling (Theorem~\ref{thm:perdoub}). The Riemann sphere $\CDach$ is
connected, and $(\CDach, \sigma)$ is also doubling. Hence, in order
to establish that $\id_{\CDach}\colon (\CDach, d)\to (\CDach,
\sigma)$ is quasisymmetric, it is enough to show that the map
$\id_{\CDach}$ is \defn{weakly quasisymmetric},\index{weak quasisymmetry}\index{quasisymmetry!weak}
i.e., that there exists
a constant $H\ge 1$ such that
\begin{equation}
\label{eq:defweakqs}
d(x,y)\leq d(x,z) \Rightarrow\sigma(x,y)\leq H \sigma(x,z)
\end{equation}
for all $x,y,z\in \CDach$
(see \cite[Thm.~10.19]{He}).
Let $m(\cdot, \cdot)=m_{F,\mathcal{C}}$ be as in Definition~\ref{def:mxy}, and let $x,y,z\in \CDach$ with $d(x,y)\leq d(x,z)$
be arbitrary. We may assume that $x\ne y$. Since $d$ is a visual metric, there exists a constant $k_0\in \N$ independent of $x,y,z$ such that
\begin{equation*}
m(x,z) - k_0 \le m(x,y)=:m\in \N_0. \end{equation*}
By definition of $m$ we can find $m$-tiles $X$ and $Y$ with $x\in X$, $y\in Y$, and $X\cap Y\ne \emptyset$. We can also find $(m+k_0 +1)$-tiles $X'$ and $Z'$ with $x\in X'\subset X$ and $z\in Z'$. Then $X'\cap Z'=\emptyset$. Thus
\begin{alignat*}{2}
\sigma(x,y) & \leq \diam (X) + \diam (Y) \asymp \diam (X)
& & \quad\text{ by \eqref{samesize}}
\\
& \asymp \diam (X')
& & \quad \text{ by (\ref{eq:diamXmkXm})}
\\
& \lesssim \operatorname{dist}(X', Z')
& & \quad\text{ by \eqref{XYdistest}}
\\
& \leq\sigma(x,z).
\end{alignat*}
Here all implicit multiplicative constants can be chosen independently of $x,y,z$.
Hence $\id_{\CDach}\colon (\CDach, d)\rightarrow (\CDach, \sigma)$ is quasisymmetric.
This proves the first implication of the theorem.
\smallskip
For the converse direction suppose that $f\: S^2\rightarrow S^2$ is an expanding Thurston map, $d$ is a visual metric for $f$ on $S^2$, and that
there exists a quasisymmetry
$h\colon (S^2, d)\rightarrow (\CDach, \sigma)$. Since all visual metrics are snowflake and hence also quasisymmetrically equivalent, we may also assume that $d$ is a visual metric for $f$ satisfying \eqref{simmetric} in Theorem~\ref{thm:visexpfactors}.
The map $h^{-1}$ is also a quasisymmetry; so $h$ and $h^{-1}$ are $\eta$-quasi\-symmetric for some distortion function $\eta$.
We consider the conjugate $g=h\circ f\circ h^{-1}\colon \CDach \to \CDach$ of $f$ by $h$.
We claim that the family of iterates $\{g^n\}$, $n\in \N$, is uniformly quasiregular, i.e., each map $g^n$ is $K$-quasiregular with $K$ independent of $n$ (see \cite[Ch.1, Sec.~2]{Ri} for the definition of a $K$-quasiregular map). The reason is that with the metric $d$ satisfying \eqref{simmetric}, the map
$f$ is locally ``conformal'', and so the dilatation of $g^n=h\circ f^n\circ h^{-1}$ can be bounded by the dilatations of $h$ and $h^{-1}$, and hence by a constant independent of $n$.
To be more precise, let $n\in \N$, $u\in \CDach$, and for small $\epsilon>0$ consider points $v,w\in\CDach$ with $\sigma(u,v)=\sigma(u,w)=\epsilon$. Define $x=h^{-1}(u)$, $y=h^{-1}(v)$, $z=h^{-1}(w)$. By Theorem~\ref{thm:visexpfactors} we have that if $\epsilon>0$ is sufficiently small (depending on $u$ and $n$), then
\begin{eqnarray*}
\frac{\sigma(g^n (u), g^n( v))}{\sigma(g^n(u), g^n(w))}&=&\frac{\sigma(h(f^n (x)) , h(f^n (y)))}{\sigma(h(f^n (x)), h(f^n (z)))}\\
& \leq& \eta\left( \frac{d(f^n (x), f^n (y))}{d(f^n(x) , f^n (z))}\right)
\,=\, \eta\left(\frac{d(x,y)}{d(x,z)}\right)\\
& = &\eta\left( \frac{d(h^{-1}(u), h^{-1} (v))}{d(h^{-1}(u),h^{-1}( w))}\right)\le H:=\eta(\eta(1)).
\end{eqnarray*}
Hence
\begin{multline*}
H(g^n,u):= \\ \limsup_{\epsilon\to 0}
\max\biggl\{ \frac{\sigma(g^n(u), g^n(v))}
{\sigma( g^n (u), g^n (w))}: v,w\in \CDach,\, \sigma(u,v)=\sigma(u,w)=\epsilon\biggr\} \leq H
\end{multline*}
for all $u\in \CDach$ and $n\in \N$.
This inequality implies that $g^n$ is locally
$H$-quasiconformal on the set $\CDach\setminus \operatorname{crit}(g^n)$ (according to the so-called ``metric'' definition of quasiconformality; see \cite[Sect.~34]{Va}).
In particular, this implies that $g^n|\CDach\setminus \operatorname{crit}(g^n)$ is $K$-quasiregular with $K=K(H)$ independent of $n$. Since the finite set $\operatorname{crit}(g^n)$ is removable for quasiregularity (see \cite[Ch.~7, Sect.~1]{Ri}), we conclude that $g^n$ is $K$-quasiregular with $K$ independent of $n$.
So the family of iterates $\{g^n\}$ of $g$ is
uniformly quasiregular.
This implies that
$g$ is topologically conjugate to a rational map (see \cite[p.~508, Thm.~21.5.2]{IM}). Hence $f$ is also topologically conjugate to a rational map.
\end{proof}
In the previous proof we actually showed the following fact.
\begin{cor} \label{cor:visualqsmetric} Let $R\: \CDach \rightarrow \CDach$ be an expanding rational Thurston map. Then every visual metric for $R$ on $\CDach$ is quasisymmetrically equivalent to the chordal metric on $\CDach$.
\end{cor}
Our previous results now immediately give a proof of Theorem~\ref{thm:main0}.
\begin{proof}
[Proof of Theorem~\ref{thm:main0}]
Let $f\: \CDach \rightarrow \CDach$ be a Thurston map whose Julia set is the whole Riemann sphere. Then $f$ is expanding
(Proposition~\ref{prop:rationalexpch}) and so by Theorem~\ref{thm:main} there exist a Jordan curve $\mathcal{C}\subset \CDach$ with $\operatorname{post}(f)\subset \mathcal{C}$ that is invariant for some iterate $f^n$ of $f$. By
Theorem~\ref{thm:Cquasicircle} the curve $\mathcal{C}$ is a quasicircle if it is equipped with a visual metric for $f$. Corollary~\ref{cor:visualqsmetric}
implies that $\mathcal{C}$ is also a quasicircle in the usual sense, i.e., a quasicircle if equipped with the chordal metric on $\CDach$.
\end{proof}
\begin{proof}
[Proof of Theorem~\ref{thm:3postrat}] Suppose that $f\:S^2\rightarrow S^2$ is a Thurston map with $\#\operatorname{post}(f)=3$. We pick a Jordan curve
$\mathcal{C}\subset S^2$ with $\operatorname{post}(f)\subset \mathcal{C}$ and consider the cell decompositions
$\mathcal{D}^0:=\mathcal{D}^0(f, \mathcal{C})$ and $\mathcal{D}^1:=\mathcal{D}^1(f,\mathcal{C})$. Since
$\#\operatorname{post}(f)=3$, the tiles in $\mathcal{D}^0$ and $\mathcal{D}^1$ are {\em (topological) triangles}, i.e., they contain three vertices and edges in their boundaries.
Let $\Delta$ be a fixed equilateral Euclidean triangle of side-length $1$.
There exists a (non-unique) path metric $d_1$ on $S^2$ such that
each tile $X$ in $\mathcal{D}^1$ equipped with $d_1$ is isometric to $\Delta$ (here and below it is understood that the vertices of the triangles correspond to each other under the isometry).
Then, roughly speaking, the metric space $(S^2, d_1)$ is an (abstract) polyhedral surface that can be obtained
by gluing together different copies of $\Delta$, one for each tile in $\mathcal{D}^1$, according to the combinatorics of $\mathcal{D}^1$.
Similarly, there is a path metric $d_0$ on $S^2$ such that
each of the two tiles in $\mathcal{D}^0$ equipped with $d_0$ is isometric to
$\Delta$. The metric space $(S^2, d_0)$ is isometric to a ``pillow" consisting
of two copies of $\Delta$ glued together along their boundaries.
The map $f$ induces an orientation-preserving labeling $c\in \mathcal{D}^1\mapsto f(c)\in \mathcal{D}^0$. Note that if $X$ is a tile in $\mathcal{D}^1$, then
$X$ and $f(X)$ if equipped with $d_1$ and $d_0$, respectively, are both isometric to $\Delta$, and hence to each other. We can arrange this isometry so that it agrees with the map $f$ on the vertices of $X$.
By using these piecewise isometries on tiles, one can construct a unique map $g\:S^2\rightarrow S^2$ that is cellular for $(\mathcal{D}^1, \mathcal{D}^0)$ and compatible with the given labeling, and has the property
that $g$ restricted to any cell $c\in \mathcal{D}^1$ equipped with the metric $d_1$ is an isometry onto the corresponding cell $g(c)=f(c)\in \mathcal{D}^0$ equipped with $d_0$. Then $\operatorname{post}(f)=\operatorname{post}(g)$, and the maps $f$ and $g$ are
Thurston equivalent by Proposition~\ref{prop:rulemapex}.
The surface $(S^2, d_1)$ carries a unique conformal structure compatible with the polyhedral structure (away from the vertices, one uses
orientation-preserving local isometries to the Euclidean plane as chart maps on the surface; this determines the conformal structure uniquely, because each of the finitely many vertices is a ``removable singularity").
By the uniformization theorem there exists a conformal map $\alpha$ from
$(S^2, d_1)$ (considered as a Riemann surface) onto the Riemann
sphere $\CDach$.
Similarly, $(S^2, d_0)$ has a natural Riemann surface structure and there
is a conformal map $\beta\:(S^2, d_0)\rightarrow \CDach$.
By post-composing $\beta$ by suitable M\"obius transformation if necessary, we may assume that the maps $\alpha$ and $\beta$ are identical
on the $3$-element set $P=\operatorname{post}(f)=\operatorname{post}(g)$. Since $\alpha$ and $\beta$ are orientation-preserving,
these homeomorphisms are then isotopic rel.\ $P$ by Lemma~\ref{lem:homeo}.
The map $g\: (S^2, d_1)\rightarrow (S^2, d_0)$ is a local isometry away from the vertices of $\mathcal{D}^1$. So this map is a holomorphic map from the
Riemann surface $(S^2, d_1)$ to the Riemann surface $(S^2, d_0)$.
Hence $R:=\beta\circ g \circ \alpha^{-1}$ is a holomorphic map from
$\CDach$ to $\CDach$, and so a rational map.
Moreover, $R$ is a Thurston map (this follows from the argument used to establish
\eqref{eq:postfg}) and $R$ is Thurston equivalent to $g$ and hence to $f$.
The first part of the theorem follows.
Suppose in addition that $f$ is expanding and has no periodic critical points. Since the latter condition is invariant under Thurston equivalence, the rational Thurston
map $R$ constructed as above will then not have periodic critical points either, and is hence expanding by
Proposition~\ref{prop:rationalexpch}. Therefore, the maps $f$ and $R$ are topologically conjugate by
Theorem~\ref{thm:exppromequiv}.
Conversely, if $f$ is expanding and topologically conjugate to a rational
map $R$, then $R$ is an expanding Thurston map. Hence $R$ has no periodic critical points by Proposition~\ref{prop:rationalexpch}, which implies that $f$ cannot have periodic critical points either.
\end{proof}
\section{The measure of maximal entropy}
\label{sec:entropy}
\noindent
In this section we investigate measure-theoretic properties of the dynamical system given by iteration of a Thurston map. We first review same facts about topological and metric entropy. For more background on these topics see \cite{KH,Wa}.
In the following, $(X,d)$ is a compact metric space, and $g\:X\rightarrow X$ a continuous map.
For $n\in \N_0$, and $x,y\in X$ we define
\begin{equation}\label{eq:defdng}
d_g^n(x,y)=\max\{d(g^k(x), g^k(y)): k=0, \dots, n-1\}.\end{equation}
Then $d_g^n$ is a metric on $X$. Let $D(g,\epsilon,n)$ be the minimal number of
sets whose $d_g^n$-diameter is at most $\epsilon>0$ and whose union covers $X$.
One can show that the limit
$$h(g,\epsilon):=\lim_{n\to\infty}\frac 1n \log(D(g,\epsilon,n))$$
exists \cite[p.~109, Lem.~3.1.5]{KH}. Obviously, the quantity
$h(g,\epsilon)$ is non-increasing in $\epsilon$. One defines the {\em
topological entropy}\index{topological entropy}\index{entropy!topological}
of $g$ (see \cite[Sect.~3.1.b]{KH}) as
$$h_{top}(g):=\lim_{\epsilon\to 0} h(g,\epsilon)\in[0,\infty]. $$
If one uses another metric $d'$ on $X$, then one obtains the same quantity for $h_{top}(g)$ if $d'$ induces the same topology on $X$ as $d$ \cite[p.~109, Prop.~3.1.2]{KH}.
The topological entropy is also well-behaved under iteration. Indeed,
if $n\in \N$, then $h_{top}(g^n)=n h_{top}(g) $ \cite[p.~111, Prop.~3.1.7~(3)]{KH}.
We denote by ${\mathcal B}$ the $\sigma$-algebra of all Borel sets on $X$. A measure on $X$ is understood to be a Borel measure, i.e., one defined on ${\mathcal B}$. If $X$ is compact and A
measure $\mu$ is called $g$-{\em invariant}\index{invariant measure}\index{measure!invariant}
if
\begin{equation}\label{invariance}
\mu(g^{-1}(A))=\mu(A)
\end{equation} for all $A\in {\mathcal B}$. Note that by continuity of $g$, we have $g^{-1}(A)\in \mathcal{B}$ whenever $A\in \mathcal{B}$. We denote by $\mathcal{M}(X,g)$ the set of all $g$-invariant Borel probability measures on $X$.
If $\mu$ is a probability measure on a compact metric space
$X$, then it is {\em regular}.\index{measure!regular}
This means that for every $\epsilon>0$ and every Borel set $A\subset X$ there exists a compact set $K\subset A$ with $\mu(A\setminus K)<\epsilon$
({\em inner regularity}) and an open set $U\subset X$ with $A\subset U$ and $\mu(U\setminus A)<\epsilon$ ({\em outer regularity}). See \cite[Thm.~2.18]{Ru} for a more general result that contains this statement as a special case.
A {\em semi-algebra} ${\mathcal S}$ is a system of sets in $X$ satisfying the following properties: (i) $\emptyset \in {\mathcal S}$, (ii) $A\cap B\in {\mathcal S}$, whenever $A,B\in {\mathcal S}$,
and (iii) $X\setminus A$ is a finite union of disjoint sets in ${\mathcal S}$, whenever $A\in {\mathcal S}$. A semi-algebra
{\em generates} ${\mathcal B}$ if ${\mathcal B}$ is the smallest
$\sigma$-algebra containing ${\mathcal S}$.
Let $\mathcal{S}$ be a semi-algebra generating ${\mathcal B}$. If $\mu$ and $\nu$ are two measures on $X$ and $\mu(A)=\nu(A)$ for all $A\in \mathcal{S}$, then $\mu=\nu$. Similarly, in order to show that a measure $\mu$ is $g$-invariant it is enough to verify \eqref{invariance} for all sets $A$ in ${\mathcal S}$ (see \cite[p.~20, proof of Thm.~1.1]{Wa} for the simple argument on how to prove these statements).
Let $\mu\in \mathcal{M}(X,g)$. Then $g$ is called {\em ergodic}\index{ergodic}
for $\mu$ if for each set
$A\in {\mathcal B}$ with $g^{-1}(A)=A$ we have
$\mu(A)=0$ or $\mu(A)=1$. The map $g$ is called {\em mixing}\index{mixing}
for
$\mu$ if
\begin{equation}\label{mixing}
\lim_{n\to \infty} \mu(g^{-n}(A)\cap B)=\mu(A)\mu(B)
\end{equation}
for all $A,B\in {\mathcal B}$. Obviously, if $g$ is mixing for $\mu$, then $g$ is also ergodic. To establish mixing, one only has to verify \eqref{mixing} for sets $A$ and $B$ in a semi-algebra generating $\mathcal{B}$ (\cite[p.~41, Thm.~1.17~(iii)]{Wa}; note that the terminology in \cite{Wa} slightly differs from ours).
If $\mu,\nu\in \mathcal{M}(X,g)$, $g$ is ergodic for $\mu$, and and $\nu$ is absolutely continuous with respect to $\mu$, then $\nu=\mu$ \cite[p.~153, Rems.~(1)]{Wa}.
Our next goal is to define the metric entropy of $g$ for a measure
$\mu$. We will follow \cite[Sect.~4.3]{KH} with slight differences
in notation and terminology (see also \cite[Ch.~4]{Wa}).
Let $\mu\in \mathcal{M}(X,g)$. A {\em measurable partition} $\xi$ for $(X,\mu)$ is a countable collection $\xi=\{A_i:i\in I\}$ of sets in $\mathcal{B}$ such that
$\mu(A_i\cap A_j)=0$ for $i,j\in I$, $i\ne j$ and
$$ \mu\biggl(X\setminus \bigcup_{i\in I} A_i\biggr)=0. $$ Here $I$ is a countable (i.e., finite or countably infinite) index set.
The {\em symmetric difference} of two sets $A,B\subset X$ is defined as
$$A\Delta B=(A\setminus B)\cup (B\setminus A).$$
Two measurable partitions $\xi$ and $\eta$ for $(X,\mu)$ are called {\em equivalent}
if there exists a bijection between the sets of positive measure in $\xi$ and the sets of positive measure in $\eta$ such that corresponding sets have
a symmetric difference of vanishing $\mu$ measure. Roughly speaking, this means that the partitions are the same up to sets of measure zero.
Let $\xi=\{A_i:i\in I\}$ and $\eta=\{B_j:j\in J\}$ be measurable partitions of $(X,\mu)$.
Then
$$\xi \vee \eta:=\{ A_i\cap B_j: i\in I,\, j\in J\} $$
is also a measurable partition, called the {\em join} of $\xi$ and $\eta$. The join of finitely many measurable partitions is defined similarly.
Let
$$g^{-1}(\xi):=\{g^{-1}(A_i): i\in I\} $$
and for $n\in \N$ define
$$\xi^{n}_g=\xi\vee g^{-1}(\xi)\vee \dots \vee g^{-(n-1)}( \xi). $$
The {\em entropy} of $\xi$ (for given $g$) is
$$ H_\mu(g,\xi)=\sum_{i\in I} \mu(A_i)\log(1/\mu(A_i))\in [0,\infty]. $$
Here it is understood that the function $\phi(x)=x\log(1/x)$ is continuously extended to $0$ by setting $\phi(0)=0$.
One can show that if $H_\mu(g,\xi)<\infty$, then the quantities $H_\mu(g, \xi^{n}_g)$, $n\in \N_0$, are {\em subadditive} in the sense that $$H_\mu(g, \xi^{n+k}_g)\le H_\mu(g, \xi^{n}_g)+H_\mu(g, \xi^{k}_g)$$ for all $k,n\in \N_0$ \cite[p.~168, Prop.~4.3.6]{KH}. This implies (\cite[p.~87, Thm.~4.9]{Wa}; see also the argument in the last part of the proof of Lemma~\ref{lem:submult}) that
$$h_\mu(g,\xi):=\lim_{n\to\infty}\frac 1n H_\mu(g,\xi^n_g)\in [0,\infty)$$ exists and
we have
$$h_\mu(g,\xi)=\inf_{n\in \N}\frac 1n H_\mu(g,\xi^n_g). $$
The quantity $h_\mu(g, \xi)$ is called the {\em (metric) entropy of $g$ relative
to} $\xi$. The {\em (metric) entropy}\index{metric entropy}\index{entropy!metric}
of $g$ for $\mu$ is defined as
\begin{multline}h_\mu(g)=\sup\{ h_\mu(g,\xi): \xi \text { is a measurable}\\
\text {partition of }
(X,\mu) \text{ with } H_\mu(g,\xi)<\infty\}.
\end{multline}
In this definition it is actually enough to take the supremum over all finite measurable partitions $\xi$ (this easily follows from
``Rokhlin's inequality" \cite[p.~169, Prop.~4.3.10 (4)] {KH}).
We call a finite measurable partition $\xi$ a {\em generator}\index{generator}
for $(g,\mu)$ if
the following condition is true: Let $\mathcal{A}$ be the smallest $\sigma$-algebra containing all sets in the partitions $\xi_g^n$, $n\in \N$. Then for each Borel set $B\in \mathcal{B}$ there exists a set $A\in \mathcal{A}$ such that $\mu(A\Delta B)=0$. If for every set $B\in \mathcal{B}$ and for every $\epsilon>0$, there exists $n\in \N$
and a union $A$ of sets in $\xi_g^n$ with $\mu(A\Delta B)<\epsilon$, then $\xi$ is a generator for $(g,\mu)$.
If $\xi$ is a generator, then $h_\mu(g)=h_\mu(g,\xi)$ by the Kolmogorov-Sinai Theorem \cite[p.~95, Thm.~4.17]{Wa}.
If $\mu\in {\mathcal M}(g,X)$ and $n\in \N$, then \cite[pp.~171--172, Prop.~4.3.16~(4)]{KH}
$$h_\mu(g^n)=n h_\mu(g).$$
If $\alpha\in [0,1]$ and $\nu\in {\mathcal M}(g,X)$ is another measure, then \cite[p.~183, Thm.~8.1]{Wa}
$$ h_{\alpha \mu+(1-\alpha) \nu}(g)=\alpha h_{\mu}(g)+(1-\alpha)h_{\nu}(g). $$
The topological entropy is related to the metric entropy by the so-called variational principle. It states that
\cite[p.~188, Thm.~8.6]{Wa}
$$ h_{top}(g)=\sup \{h_\mu(g): \mu\in {\mathcal M}(g,X)\}. $$
A measure $\mu\in \mathcal{M}(g,X)$ for which
$ h_{top}(g)=h_\mu(g)$ is called a {\em measure of maximal entropy}.\index{measure!of maximal entropy}
Let $\widetilde X$ be another compact metric space. If $\mu$ is a
measure on $X$ and $\varphi\: X\rightarrow \widetilde X$ is continuous, then
the {\em push-forward } $\varphi_*\mu$ of $\mu$ by $\varphi$ is the
measure given by $\varphi_*\mu(A)=\mu(\varphi^{-1}(A))$ for all Borel
sets $A\subset \widetilde X$.
Suppose $\widetilde g\: \widetilde X\rightarrow \widetilde X$ is a continuous map, and $\mu \in \mathcal{M}(X,g)$ and $\widetilde \mu \in \mathcal{M}(\widetilde X, \widetilde g)$.
Then the dynamical system $(\widetilde X, \widetilde g,
\widetilde \mu)$ is called a {\em (topological) factor} of
$(X, g, \mu)$ if there exists a continuous map $\varphi\: X \rightarrow \widetilde X$ such that $\varphi_*\mu = \widetilde \mu$ and
$ \widetilde g\circ \varphi=\varphi\circ g$. In this case
$h_{\widetilde \mu}(\widetilde g)\le h_\mu(g)$ \cite[p.~171, Prop.~4.3.16]{KH}.
\medskip
Now let $S^2$ be a $2$-sphere and $f\: S^2\rightarrow S^2$ be an expanding Thurston map. Our goal is to describe a measure of maximal entropy for $f$ and show its uniqueness. Existence and uniqueness of such a measure can also be derived from work by
Pilgrim and Ha\"{\i}ssinsky \cite{HP} who used the so-called thermodynamical formalism for this purpose. We will present a direct elementary argument that has the advantage of giving additional insight into the dynamical behavior of $f$.
By Theorem~\ref{thm:main} we can fix a sufficiently high iterate $F=f^n$ of $f$ that has an $F$-invariant
Jordan curve $\mathcal{C}\subset S^2$ with $\operatorname{post}(f)=\operatorname{post}(F)\subset \mathcal{C}$. Then $F$ is also an expanding Thurston map (Lemma~\ref{lem:Thiterates}).
In the following we consider the cell decompositions $\mathcal{D}^k=\mathcal{D}^k(F,\mathcal{C})$ for $k\in \N_0$.
As usual a {\em cell} is a cell in any of the cell decompositions
$\mathcal{D}^k$, $k\in \N_0$, and the terms {\em tiles}, {\em edges}, and {\em vertices} are used in a similar way.
By Proposition~\ref{prop:invmarkov} the cell decomposition $\mathcal{D}^{m+k}$ is a refinement of $\mathcal{D}^k$ for $m,k\in \N_0$,
and cells are subdivided by cells of higher order.
We denote by
$\XOw$ and $\XOb$ the two $0$-tiles, and color the tiles for $(F,\mathcal{C})$ as in Lemma~\ref{lem:colortiles}. In particular, $\XOw$ is colored white and
$\XOb$ is colored black.
For $k\in \N_0$ let $w_k$ be the number of white and $b_k$ be the
number of black $k$-tiles contained in $\XOw$, and similarly let
$w'_k$ and $b'_k $ be the number of white and black $k$-tiles
contained in $\XOb$. Then considerations as in the last part of the
proof of Lemma~\ref{lem:pullback} imply that
\begin{equation}\label{eq:wbdeg}
w_k+w'_k= b_k+b'_k=\deg(F)^k.
\end{equation}
Note that $ b_1, w'_1\ne 0$. Indeed, suppose that $b_1=0$, for
example. Then $\XOw$ contains only white $1$-tiles. Let $X\subset \XOw$
be such a $1$-tile, $e\subset X$ be a $1$-edge with $e\subset \partial
X$, and $Y$ be the other $1$-tile containing $e$. Then $Y$ is black
and so $Y\subset \XOb$. Hence
$$e\subset X\cap Y\subset \XOw\cap \XOb= \partial \XOw. $$
Since $\partial X$ is a union of $1$-edges, it follows that $\partial
X\subset \partial \XOw$. As $\XOw$ and $X$ are Jordan regions and $X\subset
\XOw$, this is only possible if $X=\XOw$.
Hence $\XOw$ is a $1$-tile and $F|\XOw$ is a homeomorphism of $\XOw$
onto itself. Applying Lemma~\ref{adhoc}~(i) repeatedly, it follows
that $\XOw$ is a
$k$-tile for each $k\in \N_0$. This is impossible, because $F$ is
expanding and so the diameters of $k$-tiles approach $0$ as $k\to
\infty$.
Define
\begin{equation}\label{def:wb}
w:=\frac{b_1}{b_1+w'_1}, \; b:=\frac{w'_1}{b_1+w'_1}.
\end{equation}
Then $w,b>0$ and $w+b=1$.
It follows from \eqref{eq:wbdeg} for $k=1$ that the matrix
\begin{equation}\label{eq:matrixA}
A=\left(\begin{array}{cc} w_{1} & b_{1}\\
w'_{1} & b'_{1}\end{array}\right)
\end{equation}
has the eigenvalues $\lambda_1=\deg(F)$ and $\lambda_2=w_1-b_1$ with respective eigenvectors
$$v_1=
\left(\!\!\begin{array}{c} w\\
b\end{array}\!\!\right) \text{ and } v_2=
\left(\!\!\!\begin{array}{cc} &1\\
-\!\!\!\!\!&1\end{array}\!\!\right). $$
Here $|\lambda_2|=|w_1-b_1|<\lambda_1=\deg(F)$. Indeed,
since $1\le b_1\le \deg(F)$ and $0\le w_1\le \deg(F)$, we otherwise had
$w_1=0$ and $b_1=\deg(F)\ge 2$. Then the white $0$-tile contains only black $1$-tiles. Arguing similarly as in the discussion before the lemma, there can then be only one such tile, and so $b_1=1$. This is a contradiction.
The existence of a largest positive eigenvalue $\lambda_1$ for $A$
with a corresponding eigenvector with all positive
coordinates is an instance of the Perron-Frobenius Theorem (\cite[p.~52, Thm.~1.9.11]{KH}).
Let $k,l,m\in \N_0$ with $m\ge l\ge k$ be arbitrary. The map $F^k$ preserves colors of tiles, i.e., if $X^{m}$ is an $m$-tile, then $F^k(X^{m})$ is an $(m-k)$-tile with the same color as $X^{m}$. Moreover, if $Y^l$ is an $l$-tile, then it follows from Lemma~\ref{adhoc}~(i) that the map $X^{m}\mapsto F^{k}(X^{m})$ induces a bijection between the $m$-tiles contained in $Y^l$ and the $(m-k)$-tiles contained in the
$(l-k)$-tile
$Y^{l-k}:=F^k(Y^l)$.
If we use this for $m=k+1$ and $l=k$, then we see that a white $k$-tile contains
$w_1$ white and $b_1$ black $(k+1)$-tiles, and similarly each black
$k$-tile contains $w'_1$ white and $b'_1$ black $(k+1)$-tiles.
This leads to the identity
\begin{equation}\label{matrixrec} \left(\begin{array}{cc} w_{k+1} & b_{k+1}\\
w'_{k+1} & b'_{k+1}
\end{array} \right)=
\left(\begin{array}{cc} w_{k} & b_{k}\\
w'_{k} & b'_{k}
\end{array} \right) \left(\begin{array}{cc} w_{1} & b_{1}\\
w'_{1} & b'_{1}
\end{array} \right).
\end{equation} for $k\in \N_0$.
The following lemma is a consequence.
\begin{lemma} \label{lem:counttiles} For all $k\in \N$ we have
\begin {eqnarray*} w_k&=&w\deg(F)^k+b(w_1-b_1)^k, \quad
b_k\,=\, w\deg(F)^k-w(w_1-b_1)^k,\\
w'_k&=&b\deg(F)^k-b(w_1-b_1)^k,\quad \,\,
b'_k\,= \, b\deg(F)^k+w(w_1-b_1)^k.
\end{eqnarray*} \end{lemma}
Since $|w_1-b_1|<\deg(F)$, the terms with $\deg(F)^k$ in
these identities are the main terms for large $k$.
\begin{proof} This follows from \eqref{matrixrec} by induction.
\end{proof}
The following lemma will be instrumental in proving that each edge is a set of measure zero for the measure of maximal entropy of $F$.
\begin{lemma}\label{lem:coveredges}
There exists $1\le L<\deg(F)$ such that for all $k,m\in \N_0$ and each $m$-edge $e$ there exists a collection $M$ of $(m+k)$-tiles with $\#M\le CL^{k}$ such that $e$ is contained in the interior of the set $\bigcup_{X\in M} X$.
Here $C$ is independent of $k$.
\end{lemma}
The total number of $(m+k)$-tiles is $2\deg(F)^{m+k}$. So the lemma says that for large $k$, the $m$-edge $e$ can be covered by a substantially smaller number of $(m+k)$-tiles.
\begin{proof} It follows from Lemma~\ref{lem:quasiball} and Lemma~\ref{lem:expoexp}~(i) that one can find $k_0\in \N$ such that for every $s$-tile $X$, $s\in \N_0$, there exist two $(s+k_0)$-tiles $Y$ and $Z$, one white and one black, with
$Y\subset \inte(X)$ and $Z\subset \inte(X)$.
Every white $s$-tile contains $w_{k_0}$ white and $b_{k_0}$ black $(s+k_0)$-tiles, and every black $s$-tile contains $w_{k_0}'$ white and $b_{k_0}'$ black $(s+k_0)$-tiles. By \eqref{eq:wbdeg} we also know that
$w_{k_0}+w_{k_0}'=b_{k_0}+b_{k_0}'=\deg(F)^{k_0}$.
Now let $e$ be an arbitrary $m$-edge. For each
$l\in \N_0$ we will define certain collections of
$(m+lk_0)$-tiles $T_l$ whose union contains $e$. We will denote the number of white tiles in $T_l$ by $N^{\tt w}_l$, the number of black tiles in $T_l$ by $N^{\tt b}_l$, and define $N_l=\max\{N_l^{\tt w}, N^{\tt b}_l\}$. Then the number of tiles in $T_l$ is bounded by $2N_l$.
Let $T_0$ be the set of all $m$-tiles that meet $e$. Then the union of the tiles in
$T_0$ is the closure of the edge flower of $e$ and so it contains $e$ in its interior.
Suppose the collection $T_l$ has been constructed.
Then we subdivide each of the tiles
$U$ in $T_l$ into $(m+(l+1)k_0)$-tiles and remove one white and one black tile
$(m+(l+1)k_0)$-tile from the interior of $U$. We define $T_{l+1}$ as the collection of all tiles obtained in this way from tiles in $T_l$. Since $\inte(U)\cap e=\emptyset$ for each $U\in T_l$, the union of the tiles in $T_{l+1}$ still contains $e$
in its interior.
Then for the number of white tiles in $T_{l+1}$ we have the estimate
\begin{eqnarray*}N^{\tt w}_{l+1}&=& N^{\tt w}_l(w_{k_0}-1)+N^{\tt b}_l(w_{k_0}'-1) \\
&\le & N_l (w_{k_0}+w_{k_0}'-2) =N_l(\deg(F)^{k_0}-2).
\end{eqnarray*}
Similarly, $$N^{\tt b}_{l+1}\le N_l(\deg(F)^{k_0}-2), $$ and so
$$N_{l+1}\le N_l(\deg(F)^{k_0}-2).$$
Let $$L:=(\deg(F)^{k_0}-2)^{1/k_0}<\deg(F).$$
Then
$$\#T_l\le 2N_l\le 2N_0 L^{k_0l}$$
is a bound for the total number of tiles in $T_l$.
Now let $k\in \N$ be arbitrary. Let $l\in \N_0$ be the smallest number with $lk_0\ge k$. For each tile
$(m+lk_0)$-tile $U$ in $T_l$ we can pick a $(m+k)$-tile that contains $U$. Let $M$ be that collection of all $(m+k)$-tiles obtained in this way. Then the union of all tiles in $M$ contains $e$ in its interior and we have
$$ \#M\le \#T_l\le 2N_0 L^{k_0l}\le
2N_0 L^{k_0}L^{k}=CL^k,$$
where $C=2N_0L^{k_0}$. The claim follows.
\end{proof}
The constant $C$ in the previous lemma depends on $e$.
If we require the weaker property that the collection $M$ of $(m+k)$-tiles only covers $e$, then we can choose the collection so
that $\#M\le CL^k$ with a constant $C$ independent of $e$. Indeed, in this case, we can choose $T_0$ to consist of the two $m$-tiles $X$ and $Y$, one white and one black, that contain $e$ in their boundary. Then $N_0=1$ and this leads to an inequality of the desired type with a constant $C$ independent of $e$.
In the next lemma we obtain an upper bound for the topological entropy of $f$.
\begin{lemma} \label{lem:topent} $h_{top}(f)\le\log (\deg(f))$.
\end{lemma}
As we will see later, we actually have $h_{top}(f)=\log (\deg(f))$ (see Corollary~\ref{cor:topent}.
\begin{proof} Since $h_{top}(F)=nh_{top}(f)$ and $\deg(F)=\deg(f)^n$, it suffices to show that $h_{top}(F)\le \log(\deg(F))$.
To show that
$h_{top}(F)\le \log(\deg(F))$, we fix a base metric $d$ on $S^2$ and let $\epsilon>0$ be arbitrary.
Since $F$ is expanding, we can find $k_0\in \N_0$ such that
$\diam(X)\le \epsilon$ whenever $X\in \X^k$ for $k\ge k_0$.
Now if $k\in \N_0$ and $X\in \X^{k+k_0}$ is arbitrary, then
$F^i(X)$ is a $(k-i+k_0)$-tile for $i=0,1\dots, k$, and so
$\diam(F^i(X))<\epsilon$. This implies that the diameter of $X$
with respect to the metric $d^k_F$ (see \eqref{eq:defdng}) is $\le \epsilon$.
Since the number of $(k+k_0)$-tiles is equal to $2\deg(F)^{k+k_0}$ and these tiles form a cover of $S^2$, it follows that
$D(F,\epsilon,k)\le 2\deg(F)^{k+k_0}$, and so
$ h(\epsilon, F)\le \log(\deg(F))$. Letting $\epsilon\to 0$ we conclude
$h_{top}(F)\le \log(\deg(F))$ as desired.
\end{proof}
Since the curve $\mathcal{C}$ is $F$-invariant, we can restrict $F$ to $\mathcal{C}$ to obtain a map $F|\mathcal{C}\: \mathcal{C}\rightarrow \mathcal{C}$. The following lemma shows that the topological entropy of this restriction is strictly smaller than $\log (\deg(F))$.
\begin{lemma} \label{lem:topents} $h_{top}(F|\mathcal{C})<\log (\deg(F))$.
\end{lemma}
\begin{proof} The proof is very similar to the first part of the proof of Lem\-ma~\ref{lem:topent}. Again let $d$ be a base metric on $S^2$.
Since $\mathcal{C}$ consists of $\#\operatorname{post}(f)$ $0$-edges, by Lemma~\ref{lem:coveredges} we can cover $\mathcal{C}$ by a collection $M_k$ of
$k$-tiles, where $\#M_k\le CL^k$. Here $1\le L<\deg(F)$ and $C$ is independent of $k$. The $k$-edges in the boundaries
of the $k$-tiles in $M_k$ then form a cover of $\mathcal{C}$.
It is clear that each $k$-edge contained in $\mathcal{C}$ must belong to this collection.
Hence if $E_k$ is the set of all $k$-edges contained in $\mathcal{C}$, we have $\#E_k\le C'L^k$ with a constant $C'$ independent of $k$.
Now let $\epsilon>0$ be arbitrary.
Since $F$ is expanding, we can find $k_0\in \N_0$ such that
$\diam(X)\le \epsilon$ whenever $X\in \X^k$ for $k\ge k_0$.
Since every $k$-edge is contained in a $k$-tile, we also have $\diam(e)\le \epsilon$ whenever $e\in \E^k$ for $k\ge k_0$.
If $k\in \N_0$ and $e\in E_{k+k_0}$ is arbitrary, then
$F^i(e)$ is a $(k-i+k_0)$-edge for $i=0,1\dots, k$, and so
$\diam(F^i(e))<\epsilon$. This implies that the diameter of $e$
with respect to the metric $d^k_F$ is $\le \epsilon$.
It follows that
$D(F|\mathcal{C},\epsilon,k)\le \#E_{k+k_0}\le C'L^{k_0+k}$, and so
$ h(\epsilon, F)\le \log(L)$. Letting $\epsilon\to 0$ we conclude
$h_{top}(F)\le \log(L)<\log(\deg(F))$ as desired.
\end{proof}
In the following we let
$$E^\infty=\bigcup_{k\in \N_0}F^{-k}(\mathcal{C}). $$
Then $E^\infty$ is a Borel set.
Proposition~\ref{prop:celldecomp}~(iii) (applied to the map $F$) implies that $E^\infty$ is equal to union of all edges. Since every vertex is contained in an edge, the set $E^\infty$ also contains all vertices.
Moreover, we have
\begin{equation}\label{eq:CinftyFinv}
F^{-1}(E^\infty)=E^\infty.
\end{equation}
Indeed, since the preimage of every edge is a union of edges, it is clear
that $F^{-1}(E^\infty)\subset E^\infty$. For the other inclusion note that
$F(\mathcal{C})\subset \mathcal{C}$ and so
$$F(E^\infty)\subset \bigcup_{k\in \N_0}F(F^{-k}(\mathcal{C}))\subset E^\infty\cup F(\mathcal{C})=E^\infty.$$
\begin{lemma}\label{lem:generator}
Let $\mu$ be an $F$-invariant probability measure on $S^2$ with $\mu(E^\infty)=0$. Then for each $k\in \N$ the set $\X^k$ of $k$-tiles forms a measurable partition of $S^2$. It is equivalent to the partition $\xi^k_F$ where $\xi=\X^1$.
Moreover, $\xi=\X^1$ is a generator for $(F,\mu)$.
\end{lemma}
\begin{proof} Note that $\mu(E^\infty)=0$ implies that all edges are sets of $\mu$-measure zero. Since every vertex is contained in an edge, we also have $\mu(\{v\})=0$ for all vertices $v$. The $k$-tiles cover $S^2$ and two distinct $k$-tiles have only edges or vertices, i.e., a set of $\mu$-measure zero in common. Hence $\X^k$ is a measurable partition of $S^2$.
Let $X\in \X^k$ be arbitrary. Then for $i=1, \dots, k$ there exist unique $i$-tiles
$X_i$ with $X=X_k\subset X_{k-1}\subset \dots \subset X_1$.
Put $Y_i=F^{i-1}(X_i)$ for $i=1, \dots, k$. Then $Y_1, \dots, Y_{k}$ are $1$-tiles. We claim that
\begin{equation}
\label{eq:Xxikg}
X=Y_1\cap F^{-1}(Y_2)\cap \dots \cap F^{-(k-1)}(Y_{k}).
\end{equation}
To verify this, denote the right hand side in this equation by $\widetilde X$. Then it is clear that $X\subset \widetilde X$. We verify $X=\widetilde X$ by inductively showing that for any point $x\in \widetilde X$ we have
$x\in X_i$ for $i=1, \dots, k$, and so $x\in X_k=X$.
Indeed, since $\widetilde X \subset Y_1=X_1$ this is clear for $i=1$. Suppose $x\in X_i$ for some $i<k$. Then
$F^i|X_i$ is a homeomorphism of $X_i$ onto the $0$-tile $F^i(X_i)$. Moreover, $x\in X_i$, $X_{i+1}\subset X_i$ and $F^i(x)\in Y_{i+1}=F^i(X_{i+1})$. Hence by injectivity of $F^i$ on $X_i$ we have $x\in X_{i+1}$.
Equation~\eqref{eq:Xxikg} shows that every element in $\X^k$ belongs to $\xi^k_F$. This implies
that the measurable partitions $\X^k$ and $\xi^k_F$ are equivalent ($\xi^k_F$
may contain additional sets, but they have to be of measure zero).
To establish that $\xi$ is a generator, let $B\subset S^2$ be an arbitrary
Borel set and $\epsilon>0$. By what we have seen, it is enough to show that there exists $k\in \N$ and a union $A$ of $k$-tiles such that $\mu(A \Delta B)<\epsilon$.
By regularity of $\mu$
there exists a compact set $K\subset A$ and an open set $U\subset S^2$ with $K\subset A\subset U$ and $\mu(U\setminus K)<\epsilon$.
Since the diameter of tiles goes to $0$ uniformly with the order of the tiles, we can choose $k\in \N$ so large that every $k$-tile that meets $K$ is contained in the open neighborhood $U$ of $K$. Define
$$A=\bigcup\{X\in \X^k: X\cap K\ne \emptyset\}.$$ Then
$K\subset A\subset U$. This implies $A\Delta B\subset U\setminus K$, and so
$$\mu(A\Delta B)\le \mu(U\setminus K)<\epsilon$$ as desired.
The proof is complete.
\end{proof}
\begin{lemma}\label{lem:SgeneratesB}
Let $\mathcal{S}$ be the collection of all sets consisting of the empty set and the interiors of all cells. Then $\mathcal{S}$ is a semi-algebra generating the Borel $\sigma$-algebra $\mathcal{B}$ on $S^2$.
\end{lemma}
\begin{proof} We first show that $\mathcal{S}$ is a semi-algebra by verifying conditions (i)--(iii) of a semi-algebra.
\smallskip \noindent
{\em Condition} (i): By definition of $\mathcal{S}$ we have $\emptyset\in \mathcal{S}$.
\smallskip \noindent
{\em Condition} (ii): Let $A,B\in \mathcal{S}$. In order to show that $A\cap B\in \mathcal{S}$, we may assume that $A=\inte(\sigma)$ and $B=\inte(\tau)$, where $\sigma$ is a $k$-cell and $\tau$ is an $l$-cell, and $k\ge l$. Let $p\in \inte(\tau)$ be arbitrary. Then by Lemma~\ref{lem:uniondisjint} there exists a unique $k$-cell $c$ with $p\in \inte(c)$. Since $\mathcal{D}^k$ is a refinement of $\mathcal{D}^l$, there exists a unique $l$-cell $\tau'$ with $\inte(c)\subset \inte (\tau')$
(see Lemma~\ref{lem:mincell}). Then $\tau$ and $\tau'$ are both $l$-cells containing the point $p$ in their interiors. This implies that $\tau'=\tau$, and so $\inte(c)\subset \inte(\tau)$.
It follows that $\inte(\tau)$ can be written as a disjoint union of interiors of $k$-cells. This implies that either $A\cap B=\inte(\sigma)$ or $A\cap B=\emptyset$. In both cases, $A\cap B\in \mathcal{S}$.
\smallskip \noindent
{\em Condition} (iii): Let $A\in \mathcal{S}$ be arbitrary. If $A=\emptyset$, then $S^2\setminus A=S^2$, and so $S^2\setminus A$ is equal to the disjoint union of the interiors of the $0$-cells, and so a disjoint union of elements in $\mathcal{S}$.
If $A=\inte(\tau)$ where $\tau$ is a $k$-cell, then $S^2\setminus A$ is the disjoint union of the interiors of all $k$-cells distinct from $\tau$. Again $S^2\setminus A$ is a disjoint union of sets in $\mathcal{S}$.
So $\mathcal{S}$ is indeed a semi-algebra.
\smallskip \noindent
{\em $\mathcal{S}$ generates $\mathcal{B}$}: Let $\mathcal{A}$ be the smallest $\sigma$-algebra containing $\mathcal{S}$. Since $\mathcal{S}$
consists of Borel sets, we have $\mathcal{A}\subset \mathcal{B}$. So in order to show that $\mathcal{A}=\mathcal{B}$ is suffices to establish that $U\in \mathcal{A}$ for each open subset $U$ of $S^2$.
Let $p\in U$ be arbitrary. Then for each $k\in \N_0$ the point $p$ is contained in the interior of some $k$-cell. Since $F$ is expanding, the diameter of $k$-cells approaches $0$ as $k\to \infty$. Hence there exists a cell $c$ with $p\in \inte(c)\subset U$. This implies that $U$ as a union of elements in $\mathcal{S}$. Since for each $k\in \N_0$ there are only finitely many
$k$-cells, the collections $\mathcal{S}$ is countable, and so $U$ is a countable union of elements in $\mathcal{S}$. Hence $U\in \mathcal{A}$ as desired.
\end{proof}
\begin{prop}\label{prop:exmeasure} There exists a unique probability measure $\mu_F$ on $S^2$ such that for each $X\in \X^k$, $k\in \N_0$, we have \begin{equation}\label{tilemass}
\mu_F(X)=
\left\{
\begin{array} {c}
w \deg (F)^{-k}, \\ b \deg (F)^{-k}
\end{array}\right. \text {if}
\begin{array} {c} \text{ $X$ is white,}\\
\text{ $X$ is black.}
\end{array}
\end{equation}
We have $\mu_F(E^\infty)=0$. Moreover, the measure $\mu_F$ is $F$-invariant, $F$ is mixing for $\mu_F$, and
$h_{\mu_F}(F)=\log(\deg(F))$.
\end{prop}
So in particular, edges and vertices are sets of $\mu_F$-measure zero, and $F$ is ergodic for $\mu_F$.
\begin{proof} The proof proceeds in several steps.
\smallskip \noindent {\em Construction of $\mu_F$ and $\mu_F(E^\infty)=0$}:
For each $k$-tile $X$, $k\in \N_0$, put $w(X)=w (\deg F)^{-k}$ if $X$ is white and $w(X)=b (\deg F)^{-k}$ if $X$ is black.
If $X\in \X^k$ is white, then $w_1$ is the number of white $(k+1)$-tiles contained in $X$, and $b_1$ the number of black $(k+1)$-tiles contained in $X$. Since $w_1+w'_1=\deg( F)$, we have
\begin{align*}
\sum_{Y\in \X^{k+1}, Y\subset X} w(Y)&=
\frac{w_1w + b_1 b}{\deg (F)^{k+1}}=\frac{w_1b_1 + b_1 w'_1}{(b_1 + w'_1)\deg (F)^{k+1}}
\\
&=\frac{b_1}{(b_1 +w'_1)\deg( F)^k}
=w(X).
\end{align*}
A similar equation is also true for black $k$-tiles. If we iterate these identities, we get
\begin{equation}\label {martingale}
\sum_{Y\in \X^{k+m}, Y\subset X} w(Y)=w(X)
\end{equation}
for all $k,m\in \N_0$ and all $X\in \X^k$.
For $A\subset S^2$ we now define
\begin{equation}\label{defmu*}
\mu^*(A)=\inf_{{\mathcal U}} \sum_{X\in {\mathcal U}} w(X),
\end{equation}
where the infimum is taken over all covers ${\mathcal U}$ of $A$ by tiles (not necessarily of the same order). By subdividing tiles into tiles of high order and using \eqref{martingale}, one sees
that in the infimum in the definition of $\mu^*(A)$ it is enough to only consider covers by tiles whose orders exceed a given number $k$. Based on this and the fact that $\max_{x\in \X^k}\diam(X)\to 0$ as $k\to \infty$, it is clear that $\mu^*$ is a {\em metric outer measure}, i.e., if $A,B\subset S^2$ are sets with $\operatorname{dist}(A,B)>0$ (with respect to some base metric on $S^2$), then
$$\mu^*(A\cup B)=\mu^*(A)+\mu^*(B). $$
It is a known fact that the restriction of a metric outer measure to the $\sigma$-algebra of Borel sets is a measure. We denote this restriction of $\mu^*$ by $\mu_F$.
If $A\subset S^2$ is compact, then it is enough to consider only finite covers by tiles in \eqref{defmu*}. Indeed, suppose
${\mathcal U}=\{X_i: i\in \N\}$ is an infinite cover of the compact set $A\subset S^2$ by tiles, and let $\epsilon>0$ be arbitrary. It follows from Lemma~\ref{lem:coveredges} that for each $i\in \N$ we can find
a finite collection ${\mathcal U}_i$ of tiles such that
$ X_i\subset \inte(X'_i), $ where
$$ X'_i=\bigcup_{X\in {\mathcal U}_i}X$$ and
$$\sum_{X\in {\mathcal U}_i}w(X)\le w(X_i)+\epsilon/2^i. $$
Finitely many sets
$\inte(X'_{i_1}), \dots, \inte(X'_{i_m})$ will cover
$A$. Then
$$\mathcal{U}'= \mathcal{U}_{i_1}\cup \dots \cup
\mathcal{U}_{i_m}$$
is a finite collection of tiles that covers $A$, and we have
$$\sum_{X\in {\mathcal U}'}w(X)\le
\sum_{X\in {\mathcal U}}w(X)+\epsilon. $$
Since $\epsilon>0$ was arbitrary, we conclude that for compact sets $A$ we get the same infimum in \eqref{defmu*} if we restrict ourselves to finite covers by tiles.
A consequence of this is that $\mu_F(X)=w(X)$ for each $X\in \X^k$, $k\in \N_0$. Indeed, by definition of $\mu_F$ we obviously have $\mu_F(X)\le w(X)$.
For an inequality in the opposite direction, it is enough to consider an arbitrary finite cover ${\mathcal U}$ of $X$ by tiles. By subdividing the tiles in ${\mathcal U}$ if necessary, we may also assume that they all have the same order $l$ and that $l\ge k$. Since ${\mathcal U}$ is a cover of $X$ and $l$-tiles have pairwise disjoint interiors,
this implies that $Y\in {\mathcal U}$ whenever $Y\in \X^l$ and $Y\subset X$. Hence
$$w(X)=\sum_ {Y\in \X^{l}, Y\subset X} w(Y)\le \sum_ {Y\in {\mathcal U}} w(Y). $$
Taking the infimum over all ${\mathcal U}$ we get
$ w(X)\le \mu_F(X)$ as desired.
Since $\mu_F(X)=w(X)$ for all tiles $X$, we have \eqref{tilemass}. \smallskip
It follows from Lemma~\ref{lem:coveredges} and the definition of $\mu_F$, that if $e$ is an edge, then
$\mu_F(e)=0$. Since $E^\infty$ is the (countable) union of all edges, we have $\mu_F(E^\infty)=0$. This also shows that
$$\mu_F(S^2)=\sum_{X\in \X^0} \mu_F(X)=\sum_{X\in \X^0} w(X)=w+b=1, $$
and so $\mu_F$ is a probability measure.
\smallskip \noindent
{\em Uniqueness of $\mu_F$}:
Suppose that $\mu$ is another probability measure on $S^2$ satisfying the analog of \eqref{tilemass}. Then from Lemma~\ref{lem:coveredges} it follows that each edge is a set of $\mu$-measure zero. Hence $\mu_F(\inte(c))=\mu(\inte(c))$ for all cells $c$. Since the empty set together with the interiors of all cells form a semi-algebra $\mathcal{S}$ generating the Borel $\sigma$-algebra on $S^2$ (see Lemma~\ref{lem:SgeneratesB}), we conclude that $\mu=\mu_F$.
\smallskip \noindent
{\em $\mu_F$ is $F$-invariant}:
To show that $\mu_F$ is $F$-invariant, it is enough to verify that
\begin{equation}\label{eq:muFinvF}
\mu_F(F^{-1}(A))=\mu_F(A)\end{equation} for all sets $A$ in the semi-algebra $\mathcal{S}$. This is true if $A=\emptyset$.
Edges are sets of measure zero, and the preimage of an edge is a finite union of edges (see Proposition~\ref{prop:celldecomp}~(ii)). This implies that \eqref{eq:muFinvF} holds if $A=\inte(e)$ for an edge $e$, or if $A=\{v\}$ for a vertex $v$ (since every vertex is contained in an edge).
Moreover, if $X$ is a tile, then $X\setminus \inte(X)$ is a union of edges, and so we have $\mu(F^{-1}(\inte(X)))=\mu_F(F^{-1}(X))$ and $\mu_F(\inte(X))=\mu_F(X)$.
So in order to establish \eqref{eq:muFinvF}
it remains to show that
$$\mu_F(F^{-1}(X))= \mu_F(X)$$ for all tiles $X$.
To see this note that
if $X$ is a $k$-tile, then $F^{-1}(X)$ is a union of $\deg(F)$ $(k+1)$-tiles that have the same color as $X$. Since the intersection of any two distinct $(k+1)$-tiles is contained in a union of edges, and hence a set of measure zero, it follows that from \eqref{tilemass} that
\begin{equation*}
\mu_F(F^{-1}(X))=\deg(F) \frac{\mu_F(X)}{\deg( F)}=\mu_F(X).
\end{equation*}
\smallskip \noindent
{\em $F$ is mixing for $\mu_F$}: It suffices to show that for all sets $A$ and $B$ in the semi-algebra $\mathcal{S}$ we have
$$\mu_F(F^{-m}(A)\cap B)\to \mu_F(A)\mu_F(B)$$
as $m\to \infty$. Based on the fact that edges are sets of measure zero and that the preimage of each edge under $F$ is a finite union of edges, for this it suffices to show that for all tiles $X$ and $Y$ we have
$$\mu_F(F^{-m}(X)\cap Y)\to \mu_F(X)\mu_F(Y)$$
as $m\to \infty$.
So let $k,l,m\in \N_0$, $X=X^k\in \X^k$ and $Y=Y^l\in \X^l$ be arbitrary. We may assume that $m\ge l$.
Then $F^{-m}(X^k)$ is a union of $(m+k)$-tiles that have the same color as $X^k$. Since edges are sets of measure zero and the $(m+k)$-tiles subdivide the $l$-tile $Y^l$, it follows that
\begin{multline}
\label{book1} \mu_F(F^{-m}(X^k)\cap Y^l)=\\ \frac{\mu_F(X^k)}{\deg(F)^m}\cdot \#\{Z^{m+k}\in \X^{m+k}: Z^{m+k}\subset Y^l, \, F^m(Z^{m+k})=X^k\}.
\end{multline}
Let $X^0\in \X^0$ be the unique tile with $X^k\subset X^0$, and $Y^0:=F^l(Y^l)\in \X^0$. We assume that $X^0$ and $Y^0$ are both white; the other cases are similar.
Every $(m+k)$-tile $Z^{m+k}$ lies in a unique ``parent" $m$-tile $Z^m$. Since $Y^l$ is an $l$-tile and $m\ge l$, we have $Z^{m+k}\subset Y^l$ if and only if $Z^m\subset Y^l$. If $F^m(Z^{m+k})=X^k$, then $F^m(Z^m)$ is a $0$-tile containing $X^k$, and so
$F^m(Z^m)=X^0$.
Conversely, if $Z^m$ is an $m$-tile and $F^m(Z^m)=X^0$, then it follows from
Lemma~\ref{adhoc}~(i) that $Z^{m+k}:=(F^m|Z^m)^{-1}(X^k)$ is the unique $(m+k)$-tile with $Z^{m+k}\subset Z^m$ and $F^m(Z^{m+k})=X^k$. These statements imply that the map
$Z^{m+k}\mapsto Z^m$ that assigns to each $(m+k)$-tile $Z^{m+k}$ its unique parent $m$-tile $Z^m$ induces a bijection between the sets
$\{Z\in \X^{m+k}: Z^{m+k}\subset Y^l, \, F^m(Z^{m+k})=X^k\}$ and
$\{Z^m\in \X^m: Z^m\subset Y^l,\, F^m(Z^m)=X^0\}$.
Hence
\begin{multline} \#\{Z^{m+k}\in \X^{m+k}: Z^{m+k}\subset Y^l, \, F^m(Z^{m+k})=X^k\}=\\
\# \{Z^m\in \X^m: Z^m\subset Y^l,\, F^m(Z^m)=X^0\}.
\label{book2}
\end{multline}
Since $X^0$ is white, the last
quantity is equal to the number of white $m$-tiles contained in
$Y^l$. Applying the homeomorphism $F^l|Y^l$, we see that
this number is equal to $w_{m-l}$, the number of white $(m-l)$-tiles contained in the white $0$-tile $Y^0=F^l(Y^l)$.
Combing this with \eqref{book1}, \eqref{book2}, and
Lemma~\ref{lem:counttiles}, we get
$$\mu_F(F^{-m}(X)\cap Y)=\\ \frac{\mu_F(X)}{\deg(F)^m}\cdot w_{m-l}\to \frac{\mu_F(X)}{\deg(F)^l}\cdot w=\mu_F(X)\mu_F(Y)$$ as $m\to \infty$.
\smallskip\noindent
{\em The identity $h_{\mu_F}(F)=\log(\deg(F))$}:
According to Lemma~\ref{lem:generator} the measurable partition $\xi=\X^1$ is a generator for $(F,\mu_F)$, and so $h_{\mu_F}(F)=h_{\mu_F}(F,\xi)$. Moreover, for each $k\in \N$, the measurable partition $\xi_F^k$ is equivalent to the measurable partition $\X^k$ given by the $k$-tiles. Hence
\begin{eqnarray*}
H_{\mu_F}(F, \xi^k_F)&=&H_{\mu_F}(F, \X^k)=\sum_{X\in \X^k}
\mu_F(X)\log(1/ \mu_F(X))\\
&= &w\log(\deg(F)^k/w)+b\log(\deg(F)^k/b)\\
&=& k\log(\deg(F))+ w\log(1/w)+b\log(1/b).
\end{eqnarray*}
This implies
$$h_{\mu_F}(F)=h_{\mu_F}(F,\xi)=\lim_{k\to \infty} \frac 1k H_{\mu_F}(F, \xi^k_F)=\log(\deg(F)).
$$
The proof is complete.
\end{proof}
\begin{cor}\label{cor:topent} $h_{top}(f)=\log(\deg(f))$.
\end{cor}
\begin{proof} We know that $h_{top}(F)\le \log(\deg(F))$, as we have seen in the proof of Lemma~\ref{lem:topents}.
Moreover, we have
$h_{\mu_F}(F)=\log(\deg(F))$
by Proposition~\ref{prop:exmeasure}, and so $h_{top}(F)\ge \log(\deg(F))$ by the variational principle. It follows that $h_{top}(F)= \log(\deg(F))$.
Since $F=f^n$ and so $\deg(F)=\deg(f)^n$ and $h_{top}(F)=n h_{top}(f)$, the claim follows.
\end{proof}
\begin{theorem}
\label{thm:nuF}
The measure $\mu_F$ is the unique measure of maximal entropy\index{measure!of maximal entropy}
for $f$, i.e., the unique $f$-invariant probability measure $\mu_F$ with $h_{\mu_F}(f)=h_{top}(f)$. Moreover, $f$ is mixing for $\mu_F$.
\end{theorem}
In particular, $f$ is ergodic for $\mu_F$.
\begin{proof} We first show uniqueness.
So let $\nu$ be a probability measure that is $f$-invariant and satisfies
$h_\nu(f)=h_{top}(f)$. Then $\nu$ is $F$-invariant and satisfies
\begin{equation}\label{comptop}
h_\nu(F)=nh_\nu(f)=nh_{top}(f)=h_{top}(F)=\log(\deg(F)).
\end{equation}
We will show that this implies that $\nu =\mu_F$.
The proof proceeds in several steps.
We can write $\nu$ as a convex combination $\nu=\beta \nu_s+(1-\beta)\nu_a$, where $\nu_s$ is a probability measure that is singular with respect to $\mu_F$ and $\nu_a$ is a probability measure that is absolutely continuous with respect
to $\mu_F$. Since $\mu_F$ and $\nu$ are $F$-invariant, it follows from the
uniqueness of the decomposition of a measure into singular and absolutely continuous part that $\nu_s$ and $\nu_a$ are also $F$-invariant. Since $F$ is ergodic for $\mu_F$ and $\nu_a$ is $F$-invariant and absolutely continuous with respect to $\mu_F$, it follows that $\nu_a=\mu_F$.
If $\beta=0$, then $\nu=\mu_F$ and we are done.
If $\beta>0$, then we can use the equation \begin{eqnarray*}
\log(\deg(F))&=& h_\nu(F)\,=\, \beta h_{\nu_s}(F)+(1-\beta)h_{\mu_F}(F)\\
&=&
\beta h_{\nu_s}(F)+(1-\beta)\log(\deg(F)),
\end{eqnarray*}
to conclude that
$$h_{\nu_s}(F)=\log(\deg(F)). $$
We will show that this is impossible by proving that for every $F$-invariant
probability measure $\nu$ that is singular with respect to
$\mu_F$ we must have $$h_{\nu}(F)<\log(\deg(F)). $$
The uniqueness of $\mu_F$ will then follow.
So let $\nu$ be such a measure and consider the union $E^\infty$ of all edges. Assume first that $\nu(E^\infty)>0$. By \eqref{eq:CinftyFinv}
we can then write $\nu$ as a convex combination
$\nu=\alpha \nu_1+(1-\alpha)\nu_2$ of two $F$-invariant probability measures
$\nu_1$ and $\nu_2$, where $\alpha=\nu(E^\infty)$, $\nu_1$ is concentrated on $E^\infty$, and
$\nu_2$ on $S^2\setminus E^\infty$.
Since $\nu_1$ is $F$-invariant, we have $\nu_1(F^{-k}(\mathcal{C}))=\nu_1(\mathcal{C})$ for all $k\in \N_0$. On the other hand, $\mathcal{C}\subset F^{-k}(\mathcal{C})$, and so
$\nu_1(F^{-k}(\mathcal{C})\setminus\mathcal{C})=0$. This implies that $\nu_1(E^\infty\setminus \mathcal{C})=0$. So $\nu_1$ is actually concentrated on $\mathcal{C}$.
Therefore, by the variational principle and by Lemma~\ref{lem:topents} we have
$$h_{\nu_1}(F)=h_{\nu_1}(F|\mathcal{C})\le h_{top}(F|\mathcal{C})<\log(\deg(F)).$$
We also have $h_{\nu_2}(F) \le h_{top}(F)= \log(\deg(F))$, and so
$$h_\mu(F)=\alpha h_{\nu_1}(F)+(1-\alpha)h_{\nu_2}(F)<\log(\deg(F)). $$
In this case we are done.
In the remaining case we have $\nu (E^\infty)=0.$ Then by
Lemma~\ref{lem:generator} $\xi=\X^1$ is a generator for $(F,\nu)$, and so $h_\nu(F)=h_\nu(F,\nu)$. In particular,
$$h_\nu(F)=\lim_{k\to \infty}\frac 1k \sum_{X\in \X^k} \nu(X)\log(1/\nu(X)), $$
and the limit is given by the infimum of the sequence elements.
Since $\nu$ and $\mu_F$ are mutually singular, we can find a Borel set $A\subset S^2$ with $\nu(A)=1$ and $\mu_F(A)=0$. Using inner regularity of $\nu$ and outer regularity of $\mu_F$, for each $\epsilon>0$ we can find a compact set $K\subset S^2$ and an open set $U\subset S^2$ with $K\subset A\subset U$, $\nu(K)>1-\epsilon$ and $\mu_F(U)<\epsilon$.
If $k$ is sufficiently large, then we can cover the set $K$ by $k$-tiles contained in $U$.
Using this for smaller and smaller $\epsilon>0$, we conclude that for each $k\in \N$ we can find a set $M_k\subset \X^k$ such that for $A_k:=\bigcup \{X\in M_k\}$
we have $\nu( A_k)\to 1$ and $\mu_F(A_k)\to 0$ as
$k\to \infty$. Note that $\mu_F(X)\ge c\deg(F)^{-k}$ for each $X\in \X^k$, where $c>0$ is independent of $k$ and $X$. Hence
$$\#M_k\le \frac 1c \mu_F(A_k)\deg(F)^k. $$
We also have $\#\X^k=2\deg(F)^k$.
Note that if $M\subset \X^k$ is arbitrary and $A=\bigcup_{X\in M}X$, then by concavity of the function $x\mapsto \phi(x)= x \log(1/x)$, we have
\begin{multline*}
\sum_{X\in M} \nu(X)\log(1/\nu(X)) \le \#M\cdot \phi\biggl(
\frac1 {\#M} \sum_{X\in M}\nu(X)\biggr)\\=\nu(A) \log(\#M/\nu(A))\le \nu(A) \log(\#M)+1/e.
\end{multline*}
Hence
\begin{eqnarray*}
k\log(\deg(F))&=&kh_\nu(F)\\
&\le &\sum_{X\in \X^k} \nu(X)\log(1/\nu(X)) \\ &= &
\sum_{X\in M_k} \dots \,+\, \sum_{X\in \X^k\setminus M_k} \dots \\ &\le&
\nu(A_k) \log( \#M_k)+\nu(S^2\setminus A_k) \log(\#\X^k)+C_1\\ \le \ \nu (A_k)\log(\mu_F(A_k))\!\!\! \!\!\!\!\!\!\!\!\!\!
&&+\,(\nu(A_k) +
\nu(S^2\setminus A_k))\log( \deg(F)^k)+C_2\\ &= &
\nu (A_k)\log(\mu_F(A_k))+ k\log(\deg(F)) +C_2.\end{eqnarray*}
Here the constants $C_1$ and $C_2$ do not depend on $k$.
An inequality of this type is impossible as $$\nu (A_k)\log(\mu_F(A_k))\to -\infty$$ for $k\to \infty$.
This shows that if there is a measure of maximal entropy for $f$, then it has to agree with $\mu_F$. Actually, we proved that $\mu_F$ is the unique measure of maximal entropy for $F$.
We now show that $\mu_F$ is $f$-invariant and a measure of maximal entropy for $f$.
Indeed, the measure $f_*\mu_F$ is $F$-invariant and the triple $(S^2, F, f_*\mu)$
is a factor of $(S^2, F,\mu_F)$ by the map $f$. It follows that
$h_{f_*\mu}(F)\le h_\mu(F)$. Iterating this and noting that $f^n_*{\mu_F}=F_*\mu_F=\mu_F$ by $F$-invariance of $\mu_F$, leads to
$$h_{\mu_F}(F)=h_{f^n_*\mu_F}(F)\le h_{f^{n-1}_*\mu_F}(F)\le \dots\le
h_{f_*{\mu_F}}(F)\le h_{\mu_F}(F). $$
Hence $h_{f_*\mu_F}(F)= h_{\mu_F}(F)$, and so $f_*\mu_F$ is a measure of maximal entropy for $F$. By uniqueness of $\mu_F$ we have $f_*{\mu_F}=\mu_F$ showing that $\mu_F$ is $f$-invariant. From the computation \eqref{comptop} we see that
that $h_{\mu_F}(f)=\log(\deg(f))=h_{top}(f)$. So $\mu_F$ is a measure of maximal entropy for $f$, and by the first part of the proof we know that it is the unique such measure.
It remains to show that $f$ is mixing for $\mu_F$. Indeed,
since $F$ is mixing for $\mu_F$ (Proposition~\ref{prop:exmeasure}) and $\mu_F$ is $f$-invariant, we have that for all $m \in \{0, \dots, n-1\}$, and all Borel sets
$A,B\subset S^2$,
\begin{multline*} \mu_F(f^{-(nl+m)}(A)\cap B)=\mu_F(F^{-l}(f^{-m}(A))\cap B)\to \\
\mu_F(f^{-m}(A))\mu_F(B)=\mu_F(A)\mu_F(B)
\end{multline*} as
$l \to \infty$. Hence we get the desired relation
$$ \mu_F(f^{-k}(A)\cap B) \to \mu_F(A)\mu_F(B)$$
as $k\to \infty$. The proof is complete.
\end{proof}
A metric space $X$ is called \defn{(Ahlfors)}
$Q$\defn{-regular},\index{Ahlfors regular}
where $Q>0$, if there exists a (Borel) measure $\mu$ such that
$$ \frac 1C R^Q\le \mu(\overline B(x,R))\le C R^Q$$
for all closed balls $\overline B(x,R)$ with $x\in X$ and $0<R\leq \diam (X)$, where $C\ge 1$ is independent of the ball.
\begin{prop}
\label{prop:Ahlforsreg} Let $f\: S^2\rightarrow S^2$ be a an expanding Thurston map without periodic critical points, and $d$ be a visual metric with expansion factor $\Lambda$. Then $(S^2, d)$ is Ahlfors $Q$-regular with
$$Q=\frac{\log(\deg(f))}{\log(\Lambda)}.$$
\end{prop}
\begin{proof} Consider the iterate $F=f^n$ chosen as in the beginning of this section, and let $d$ be a visual metric for $f$ with expansion factor $\Lambda$. Then
$d$ is a visual metric for $F$ with expansion factor $\Lambda'=\Lambda^n$.
Consider an arbitrary closed ball $\overline B(x,R)$, where $x\in S^2$ and $0<R\le \diam(S^2)$. We use the sets $U^m(x)$ as defined in \eqref{eq:defUk} for the map $F$. Since $F$ does not have periodic critical points, the length of the cycle of each
vertex is uniformly bounded (see Lemma~\ref{lem:cycle} and
Lemma~\ref{lem:flowerprop}~(i)). This implies that each set $U^m(x)$ consists of
a uniformly bounded number of tiles. In particular, these sets are closed. Moreover, by Lemma~\ref{lem:UmB} (applied to $F$) we have
\begin{equation*}
U^{m+n_0}(x)\subset \overline B(x,R) \subset U^{m-n_0}(x),
\end{equation*}
where $m=\left\lceil -\log (R)/\log(\Lambda')\right\rceil$ and $n_0\in \N_0$ is a constant independent of the ball.
Noting that
$$ Q=\frac{\log(\deg(F))}{\log(\Lambda')}$$
and using \eqref{tilemass},
we get
\begin{eqnarray*} \mu_F(U^{m+n_0}(x))&\asymp &
\mu_F(U^{m-n_0}(x)) \asymp \deg(F)^{-m}\\
&\asymp& \exp(\log(R)\log(\deg(F))/\log(\Lambda'))=R^Q,
\end{eqnarray*}
and so $\mu_F(\overline B(x,R)) \asymp R^Q$. Here the constants $C(\asymp)$ are independent of the ball. The statement follows. \end{proof}
\section{Concluding remarks and open problems}
\label{sec:probrem}
\noindent The authors hope that the results in the present paper will serve as a foundation for future work in the area. Our line of investigation has been continued in \cite{Me09b, Me09c, Qian}. In the following we will discuss a selection of questions and open problems that are worth pursuing.
The basis of our combinatorial approach is the existence of cellular Markov
partitions for Thurston maps. We know by Corollary~\ref{cor:subdivnlarge} that such
a cellular Markov
partition (related to a two-tile subdivision rule) exists for every sufficiently high iterate of an expanding Thurston map. This leads to the following question:
\begin{prob}\label{prob1} Does every expanding Thurston map $f\:S^2\rightarrow S^2$ have a
cellular Markov partition? \end{prob}
In the introduction we conjectured that the answer should be
affirmative. To prove the conjecture, one essentially has to construct a
connected
graph $G\subset S^2$ with $\operatorname{post}(f)\subset G$ that is
$f$-invariant, i.e., $f(G)\subset G$. A different way to phrase the
problem is to ask whether $f$ (and not some iterate $f^n$) can be
described
by a (suitably defined) \emph{$k$-tile subdivision rule}. Here $k$ would be the number of
components of $S^2\setminus G$.
\smallskip
As we have seen, the dynamics of an expanding Thurston map $f$ generates a
class of visual metrics, and so a fractal geometry on the $2$-sphere on which it acts. This does not rule out that
the map $f$ can actually be described by a smooth model.
\begin{prob}\label{prob3} Is every expanding Thurston map $f\: S^2\rightarrow S^2$ topologically conjugate to a smooth
expanding Thurston map $g\: \CDach \rightarrow \CDach$? \end{prob}
This question was raised by K.~Pilgrim.
We expect that this is true for at least every sufficiently high iterate of $f$.
In our examples of rational Thurston maps, the coefficients were always algebraic numbers. In general one can ask:
\begin{prob}\label{prob3a} Is every expanding rational Thurston map
topologically conjugate to a rational Thurston map with
algebraic coefficients?
\end{prob}
J.~Souto posed this question after a talk by the first author. We expect a positive answer.
We know that every expanding Thurston map $f$ has iterates $f^n$ with invariant
curves as in Theorem~\ref{thm:main}.
This naturally leads to the question whether one can bound the order of this iterate in terms of some natural invariants of the map.
\begin{prob}
\label{prob:estimate_n} Let $f\: S^2\rightarrow S^2$ be an expanding
Thurston map. Is there a number $N_0\in \N$, depending on some natural data
such as $\deg(f)$, $\#\operatorname{post}(f)$, and $\Lambda_0(f)$ such that for all
$n\geq N_0$ there exists an $f^n$-invariant Jordan
curve $\mathcal{C}\subset S^2$ with $\mathcal{C}\subset\operatorname{post}(f)$?
\end{prob}
Recall that $\Lambda_0(f)$ denotes the combinatorial expansion factor
of $f$ (see Section~\ref{sec:combexpfac}).
According to one of our main results every expanding Thurston
map has an iterate that can be described by a two-tile subdivision rule
(Corollary~\ref{cor:subdivnlarge}).
If a map realizes a subdivision rule, it should be possible to extract all relevant information about the map from the combinatorial data. This underlying principle is the common thread for the next three problems.
Every two-tile subdivision rule is realized by a Thurston map
that is unique up to Thurston equivalence. In contrast, a Thurston map may realize combinatorially different two-tile subdivision rules. This leads
to the following question:
\begin{prob}
\label{prob:diff_sub_same_f} Suppose two
Thurston
maps $f$ and $g$ realize different two-tile subdivision rules. How can one decide from combinatorial data whether the maps are
Thurston equivalent?
\end{prob}
Of course, there are several simple necessary conditions, such as
$\deg(f)=\deg (g)$ and $\#\operatorname{post}(f)= \#\operatorname{post}(g)$, whose validity can easily
be checked from the subdivision rules. In addition, the maps have to have the same \emph{critical portrait} (see Example
\ref{ex:noinvCC}). A related question, namely when polynomials with
the same critical portrait are Thurston equivalent, is answered in
\cite{BN}.
\begin{prob}
\label{prob:comb_exp}
Let $f$ be an expanding Thurston map that realizes a two-tile
subdivision rule. Is there an effective way to compute the
combinatorial expansion factor $\Lambda_0(f)$ from the combinatorial description?
\end{prob}
Since $\Lambda_0(f)$ is defined
as a limit, a priori one cannot expect to find $\Lambda_0(f)$ by a finite
procedure. However, if $f$ is a Latt\`es or Latt\`es-type map
(see Example~\ref{ex:notattained}), then $\Lambda_0(f)$ is the smallest
absolute value of any eigenvalue of the matrix describing the underlying
torus endomorphism \cite{Qian}. In general, one may speculate that if $f$ realizes a two-tile subdivision rule with underlying cell decompositions $\mathcal{D}$ and $\mathcal{D}'$, then
$\Lambda_0(f)$ is related to the spectral radius of a matrix that is obtained from the incidence relations of the cells in $\mathcal{D}'$ and their images under $f$ in $\mathcal{D}$.
\begin{prob}
\label{prob:f_rational}
Let $f$ be an expanding Thurston map that realizes a two-tile
subdivision rule. Is it possible to decide from the subdivision rule whether $f$ is Thurston
equivalent to a rational map?
\end{prob}
As discussed in the introduction, Thurston proved a necessary and sufficient condition for a Thurston map to be equivalent to a rational map \cite{DH}. In principle, it is
possible to check this criterion if the map is given by a subdivision rule.
In practice, the sufficiency part is hard to verify, since it involves the behavior of the map on infinitely many homotopy classes of curves.
So in Problem~\ref{prob:f_rational} we really ask for a substantially simpler criterion, preferably based on the verification of a finite condition.
As we have seen (Theorem~\ref{thm:3postrat}), the case $\#\operatorname{post}(f)=3$ is completely understood. The case $\#\operatorname{post}(f)=4$ should also be rather accessible. For understanding the underlying issues it might be worthwhile to reprove Thurston's result by using a combinatorial approach
as in the present paper.
From Theorem \ref{thm:qsrational} it follows that for the solution of
Problem~\ref{prob:f_rational} one may equivalently
ask if one can decide from the subdivision rule whether
$S^2$ equipped with a visual metric $d$ is quasisymmetrically equivalent to the standard $2$-sphere (i.e., the unit sphere in $\R^3$).
This is closely related to Cannon's conjecture in Geometric
Group Theory which can be reformulated as a similar quasisymmetric equivalence problem (see \cite{Bo} for more background, and
\cite{Ca94} for a different approach).
The general problem when a metric
$2$-sphere $(S^2,d)$ is quasisymmetrically equivalent to the standard
$2$-sphere is a hard problem that is not completely understood
(for results in this direction, see \cite{BK}).
It is well-known that there are many analogies between Complex Dynamics
and the theory of Kleinian groups (informally refered to as {\em Sullivan's dictionary}). It seems very fruitful to explore
this systematically for Thurston maps and translate statements for these maps into corresponding group theoretic analogs. Expanding Thurston maps correspond to Gromov hyperbolic groups whose boundary at infinity is a $2$-sphere.
This is suggested, for example, by Theorem~\ref{thm:qsrational}, which can be seen as an analog of Cannon's Conjecture,
or by Remark~\ref{rem:gromovgraph}.
For Gromov hyperbolic groups whose boundary at infinity is a Si\'er\-pinski carpet
there is an analog of Cannon's conjecture---the Kapovich-Kleiner conjecture.
It predicts that these groups arise from some standard situation in hyperbolic geometry. It would be interesting to formulate an analog of this
for Thurston maps in the spirit of Theorem~\ref{thm:qsrational}.
\smallskip If $f\:S^2\rightarrow S^2$ is an expanding Thurston map without periodic critical points, then $S^2$ equipped with a visual metric $d$ is Ahlfors regular
(Proposition~\ref{prop:Ahlforsreg}). The infimal exponent $Q$ such that
$(S^2, d)$ is quasisymmetrically equivalent to an Ahlfors $Q$-regular
space is called the {\em (Ahlfors regular) conformal dimension} of $(S^2, d)$.
This is an important numerical invariant of the fractal geometry of $(S^2,d)$.
Actually, this conformal dimension only depends on $f$ and not on the choice of the visual metric $d$, since all visual metrics are quasisymmetrically equivalent.
\begin{prob} \label{prob4} Is it possible to determine the conformal
dimension of a $2$-sphere equipped with a
visual metric of an expanding Thurston map
$f$ in terms of dynamical data of $f$?\end{prob}
Bonk-Geyer-Pilgrim \cite{Bo} stated a conjecture expressing this conformal dimension in terms of eigenvalues of certain matrices related to the dynamics of $f$. One half of this conjecture has recently been proved
Ha\"\i ssinsky-Pilgrim \cite{HP08}.
To prove his characterization of rational maps among Thurston maps
\cite{DH}, Thurston used an
iteration procedure on a suitable Teichm\"uller space.
This idea may also be fruitful for studying some problems related
to our existence criteria for invariant Jordan curves. Namely, in view
of Theorem~\ref{thm:exinvcurvef} one is led to the general question how to find non-trivial sets
$K\subset S^2$ with $\operatorname{post}(f)\subset K$ (and maybe additional geometric
features) for which there exists an isotopy rel.\ $\operatorname{post}(f)$ that
deforms the set into a subset of its preimage $f^{-1}(K)$.
It may be possible to reformulate this as a fixed
point problem under iteration of a pull-back operation on a
suitable space.
As we discussed above, Problem \ref{prob1} is equivalent
to finding an $f$-invariant graph $G$ with suitable properties. If one does not insist on describing the map by a cellular Markov partition, then other
invariant sets $K$ may lead to other natural combinatorial descriptions
of the map or its iterates.
Actually, in many cases one can show the existence of an $f^n$-invariant arc
$A\subset S^2$ with $\operatorname{post}(f)\subset A$.
Slightly more general, one can ask if an invariant tree
$T\subset S^2$ with $\operatorname{post}(f)\subset T$ exists, i.e., a graph
$T\subset S^2$ (in the sense of Section \ref{sec:graphs}) such that
$S^2\setminus T$ is connected.
\begin{prob} \label{prob7}
Let $f\:S^2\rightarrow S^2$ be expanding Thurston map. Does there
exists a tree $T\subset S^2$ with $\operatorname{post}(f)\subset K$ that is
invariant under $f$ (or at least under a sufficiently high iterate
of $f$)?
\end{prob}
If such an invariant tree $T$ exists, then $\Om^0=S^2\setminus T$ is a simply connected
region that is subdivided by the complementary components $\Om^1$
of $f^{-1}(T)$. Moreover, on each component $\Om^1$ the map $f|\Om^1$ is a homeomorphism onto $\Om^0$.
A similar statement then also holds for the iterates of $f$.
Since there is only one open ``$0$-tile" $\Om^0$, the dynamics of $f$ is
described by a {\em one-tile subdivision rule}. This would simplify the combinatorial description of $f$; on the other hand, open tiles then are only simply connected regions and not necessarily open Jordan regions.
So the simplified combinatorial description comes at the price of a
more complicated geometry of tiles.
For postcritically finite polynomials such an invariant tree always
exists,
namely the \emph{Hubbard tree} introduced by Douady and Hubbard, see
\cite{DH84}. In fact the Hubbard tree determines the polynomial
uniquely up to conjugation by a map $z\mapsto az+b$, $a\in
\C\setminus\{0\}, b\in \C$, see \cite{Poi}.
Cannon-Floyd-Parry show in
\cite[Theorem 3.1]{CFP10} that a
sufficiently high iterate of each Latt\`{e}s map has an invariant
tree. Note however that ``Latt\`{e}s map'' is used there in a more
restrictive sense, i.e., it denotes a quotient of a \emph{double}
cover of a conformal torus endomorphism.
\smallskip
If one further relaxes the geometric requirements on ``tiles", then other
combinatorial descriptions of (expanding) Thurston maps
exist. The result presented in Section \ref{sec:symdym} can be reinterpreted in this vein. This is related
to the description of $f$ via ``limit spaces'' of its
\emph{iterated monodromy group} (see \cite{Ne} and the discussion below).
\smallskip
Yet another point of view to obtain combinatorial descriptions of Thurston maps is to consider sets that are ``dual" to invariant sets containing the postcritical set. To illustrate this,
we consider
the Latt\`{e}s map $g$ from Section \ref{sec:Lattes}, and again represent the sphere
$S^2$ as a pillow as in Figure \ref{fig:mapg}. Let $K$ be the subset of the pillow that is of the union of four Jordan curves, namely the two
curves consisting of the points at distance $1/3$ and at $2/3$, respectively, from the bottom edge of the pillow, and the
two closed curves of the points at distance $1/3$ and $2/3$ from the left
edge. Then $K$ is
$g$-invariant and for each $n\in \N_0$ the set $g^{-n}(K)$ is a graph that is the $1$-skeleton of a cell decomposition of the pillow. Moreover,
if $\mathcal{C}$ is the ($g$-invariant) boundary of the pillow, then
$g^{-n}(K)$ is a dual graph to $g^{-n}(\mathcal{C})$.
\smallskip
In this paper we mostly considered expanding Thurston maps. One may ask
whether combinatorial descriptions exist for more general types of maps.
\begin{prob}
\label{prob:non-expanding}
Let $f\:S^2 \rightarrow S^2 $ be a Thurston map (not necessarily
expanding). Is there a Jordan curve $\mathcal{C}\subset S^2$ with
$\operatorname{post}(f)\subset \mathcal{C}$ that is invariant for some iterate
$f^n$? Are there other natural decompositions of the sphere $S^2$
that are invariant (in a suitable sense) under $f$ or some iterate
$f^n$?
\end{prob}
When $f$ is a polynomial natural decompositions are obtained via
\emph{external rays} (see \cite{DH84}). Closely related are
\emph{Yoccoz puzzles} (see \cite{Hu} and \cite{Mi00}).
A lack
of such a description for rational maps is one of the main reasons
why their study is harder than the study of polynomials.
\smallskip
Similarly, one can ask if and how our results extend to
maps that are not postcritically-finite.
The theory of {\em coarse expanding
dynamical systems} developed in \cite{HP} should be relevant here.
\begin{prob}
\label{prob:invC_JS2}
Let $R\colon \CDach\to \CDach$ be a rational map (not necessarily
post\-crit\-i\-cally-finite) whose Julia set is the
whole Riemann sphere $\CDach$. Does there exist a natural
combinatorial description of $R$ or some iterate $R^n$?
\end{prob}
The rational maps $R\: \CDach \rightarrow \CDach$ of a given degree $d\ge 2$ form a complex manifold $\mathcal{R}_d$
of dimension $2d+1$.
M.\ Rees has shown \cite{Re} that the set of points in $\mathcal{R}_d$ where the
corresponding rational map $R$ has as Julia set equal to the whole sphere has
positive measure with respect to the natural measure class on
$\mathcal{R}_d$. These points/rational maps are obtained by a slight
perturbation of a certain expanding Thurston map $R_0\in \mathcal{R}_d$. It would be interesting to find combinatorial descriptions of expanding Thurston maps that change under such perturbations in a controlled
manner.
\smallskip
Our combinatorial approach seems quite relevant
in the study of the {\em iterated monodromy group}\index{iterated
monodromy group}
of a Thurston map $f\:S^2\rightarrow S^2$.
To state the basic definitions, let $G=\pi_1(S^2\setminus \operatorname{post}(f), p)$ be the fundamental
group of $S^2\setminus \operatorname{post}(f)$ represented by homotopy classes of loops
in $S^2\setminus \operatorname{post}(f)$ with initial point $p\in S^2\setminus \operatorname{post}(f)$.
Then there exists a natural action of $G$ on the disjoint union $T$ of the sets
$f^{-n}(p)$, $n\in \N_0$. Namely, if $x\in f^{-n}(p)\subset T$, and $g\in G$ is represented by a loop $\alpha$ starting at $p$, then we can lift $\alpha$ by the map $f^n$ to a path $\widetilde \alpha$ with initial point $x$. If $y$ is the endpoint of
$\widetilde \alpha$, then $y\in f^{-n}(p)$, and we define $g(x):=y$.
The action $G\curvearrowright T$ is not effective in general, i.e., there may be elements $g\in G$ that act as the identity on $T$. Let $H$ be the ineffective
kernel of this action, i.e., the normal subgroup of $G$ consisting of all elements
in $G$ acting as the identity on $T$. Then the {\em iterated monodromy
group} of $f$ is defined as the quotient group
$\Gamma=G/H$. The action of $G$ on $T$ induces a natural action $\Gamma\curvearrowright T$. By the same method
as in the proof of Theorem~\ref{thm:expThfactor} one can find a coding
of the elements in $T$ by words in a finite alphabet.
Then it is not hard to write down explicit recursive relations for the action
of suitably chosen generators of $\Gamma$ on these words.
In principle, this describes the group $\Gamma$ completely, but it seems hard
in specific cases to get a complete understanding of the structure of the group.
\begin{prob} \label{prob8} What is the
iterated monodromy group of an expanding
Thurston map? In particular, what is the
growth behavior of such a group?\end{prob}
The last question is interesting, because there is an example of a (non-expanding) Thurston map (namely, the polynomial $f(z)=z^2+{\mathbf{\imath}}$)
whose iterated monodromy group is a group of {\em intermediate growth}
\cite{BP}. Formerly, these groups had been regarded as rather exotic objects.
\smallskip
We conclude our discussion with some remarks on higher dimensions.
The concept of postcritically-finite maps has been generalized to
several complex variables and the dynamics of these maps has been
studied (see, for example, \cite{FS92, FS94, Jo}). A possible different direction is the theory of
quasiregular mappings on $\widehat \R^n=\R^n\cup \{\infty\}$.
\begin{prob} \label{prob11}Develop a theory of quasiregular maps in
higher dimensions that generalizes the theory of Thurston maps in
dimension $2$.
\end{prob}
It is not even clear what the basic definitions should be. It seems reasonable
to require that a {\em quasiregular Thurston map} $f\:\widehat \R^n \rightarrow
\widehat \R^n$ is {\em uniformly quasiregular} (i.e., there exists
$K\ge 1$ such that $f^n$ is $K$-quasiregular for each $n\in \N$) and
that the postcritical set $\operatorname{post}(f)$ of $f$ (i.e., the forward orbit
of the branch set of $f$) is ``small" and ``non-exotic". For example,
one could require that $\operatorname{post}(f)$ is an embedded cell complex in
$\widehat \R^n$
of codimension $2$.
|
{
"timestamp": "2010-09-21T02:01:44",
"yymm": "1009",
"arxiv_id": "1009.3647",
"language": "en",
"url": "https://arxiv.org/abs/1009.3647"
}
|
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\title{DB pairs and vanishing theorems}
\author{S\'ANDOR J KOV\'ACS\xspace}
\date{\today}
\thanks{\mythanks}
\address{\myaddress}
\email{skovacs@uw.edu\xspace}
\urladdr{http://www.math.washington.edu/$\sim$kovacs\xspace}
\subjclass[2010]{14J17}
\maketitle
\newcommand{Szab\'o-resolution\xspace}{Szab\'o-resolution\xspace}
\newcommand{\Delta}{\Delta}
\centerline{\sf In memoriam Professor Masayoshi Nagata}
\begin{abstract}
The main purpose of this article is to define the notion of \uj{Du~Bois} singularities
for pairs and proving a vanishing theorem using this new notion. The main vanishing
theorem specializes to a new vanishing theorem for resolutions of log canonial
singularities.
\end{abstract}
\section{Introduction}
The class of rational singularities is one of the most important classes of
singularities. Their essence lies in the fact that their cohomological behavior is
very similar to that of smooth points. For instance, vanishing theorems can be easily
extended to varieties with rational singularities. Establishing that a certain class
of singularities is rational opens the door to using very powerful tools on varieties
with those singularities.
Du~Bois (or DB) singularities are probably somewhat harder to appreciate at first,
but they are equally important. Their main importance comes from two facts: They are
not too far from rational singularities, that is, they share many of their
properties, but the class of \uj{Du~Bois} singularities is more inclusive than that of
rational singularities. For instance, log canonical singularities are \uj{Du~Bois}, but
not necessarily rational. The class of \uj{Du~Bois} singularities is also more stable
under degeneration.
Recently there has been an effort to extend the notion of rational singularities to
pairs. There are at least two approaches; Schwede and Takagi \cite{MR2492473} are
dealing with pairs $(X,\Delta)$ where $\rdown{\Delta}=0$ while Koll\'ar and Kov\'acs
\cite{KollarKovacsRP} are studying pairs $(X,\Delta)$ where $\Delta$ is reduced.
The main goal of this article is to extend the definition of \uj{Du~Bois} singularities to
pairs in the spirit of the latter approach.
Here is a brief overview of the paper.
In section~\ref{sec:rational-du-bois} some basic properties of rational and DB
singularities are reviewed, a few new ones are introduced, and the DB defect is
defined. In section~\ref{sec:pairs-gener-pairs} I recall the definition and some
basic properties of pairs, generalized pairs, and rational pairs. I define the notion
of a DB pair and the DB defect of a generalized pair and prove a few basic
properties. In section~\ref{sec:cohom-with-comp} I recall a relevant theorem from
Deligne's Hodge theory and derive a corollary that will be needed later. In
section~\ref{sec:db-pairs} one of the main results is proven. A somewhat weaker
version is the following. See Theorem~\ref{thm:wdb-is-db} for the stronger statement.
\begin{thm}
Rational pairs are DB pairs.
\end{thm}
This generalizes \cite[Theorem~S]{Kovacs99} and \cite[5.4]{MR1741272} to pairs. In
section \ref{sec:vanishing-theorems} I prove a rather general vanishing theorem for
DB pairs and use it to derive the following vanishing theorem for log canonical
pairs.
\begin{thm}\label{thm:main}
Let $(X, \Delta)$ be a \uj{$\mathbb{Q}$-factorial} log canonical pair, $\pi : \widetilde X
\to X$ a log resolution of $(X,
\Delta)$.
Let $\widetilde \Delta=\bigl(\pi^{-1}_*\rdown \Delta + \excnklt(\pi)\bigr)_{\red}$. Then
\begin{equation*}
{\mcR\!}^{i}\pi_* \, \scr{O}_{\widetilde X}(- \widetilde \Delta) = 0\quad\text{for $i>0$.}\\
\end{equation*}
\end{thm}
A philosophical consequence one might draw from this theorem is that log canonical
pairs are not too far from being rational. One may even view this a vanishing theorem
similar to the one in the definition of rational singularities cf.\
\eqref{def:rtl-sing}, \eqref{def:rat-sing-pairs} with a correction term as in
vanishing theorems with multiplier ideals. Notice however, that this is in a dual
form compared to Nadel's vanishing, and hence does not follow from that, especially
since the target is not necessarily Cohen-Macaulay.
Theorem~\ref{thm:main} is also closely related to Steenbrink's characterization of
normal isolated \uj{Du~Bois} singularities \cite[3.6]{SteenbrinkMixed} (cf.\
\cite[4.13]{DuBois81}, \cite[6.1]{Kovacs-Schwede09}).
A weaker version of this theorem was the corner stone of a recent result on extending
differential forms to a log resolution \cite{GKKP10}. For details on how this theorem
may be applied, see the original article. It is possible that the current theorem
will lead to a strengthening of that result.
\begin{demo}{\bf Definitions and Notation}\label{demo:defs-and-not}
Unless otherwise stated, all objects are assumed to be defined over $\mathbb{C}$, all
schemes are assumed to be of finite type over $\mathbb{C}$ and a morphism means a morphism
between schemes of finite type over $\mathbb{C}$.
If $\phi:Y\to Z$ is a birational morphism, then $\exc(\phi)$ will denote the
\emph{exceptional set} of $\phi$. For a closed subscheme $W\subseteq X$, the ideal
sheaf of $W$ is denoted by $\scr{I}_{W\subseteq X}$ or if no confusion is likely, then
simply by $\scr{I}_W$. For a point $x\in X$, $\kappa(x)$ denotes the residue field of
$\scr{O}_{X,x}$.
For morphisms $\phi:X\to B$ and $\vartheta: T\to B$, the symbol $X_T$ will denote
$X\times_B T$ and $\phi_T:X_T\to T$ the induced morphism. In particular, for $b\in
B$ we write $X_b = \phi^{-1}(b)$.
Of course, by symmetry, we also have the notation $\vartheta_X:T_X\simeq X_T\to X$
and if $\scr{F}$ is an $\scr{O}_X$-module, then $\scr{F}_T$ will denote the $\scr{O}_{X_T}$-module
$\vartheta_X^*\scr{F}$.
Let $X$ be a complex scheme (i.e., a scheme of finite type over $\mathbb{C}$) of dimension
n. Let $D_{\rm filt}(X)$ denote the derived category of filtered complexes of
$\scr{O}_{X}$-modules with differentials of order $\leq 1$ and $D_{\rm filt, coh}(X)$
the subcategory of $D_{\rm filt}(X)$ of complexes $\cx K$, such that for all $i$,
the cohomology sheaves of $Gr^{i}_{\rm filt}K^{{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}}$ are coherent cf.\
\cite{DuBois81}, \cite{GNPP88}. Let $D(X)$ and $D_{\rm coh}(X)$ denote the derived
categories with the same definition except that the complexes are assumed to have
the trivial filtration. The superscripts $+, -, b$ carry the usual meaning
(bounded below, bounded above, bounded). Isomorphism in these categories is
denoted by $\,{\simeq}_{\text{qis}}\,$. A sheaf $\scr{F}$ is also considered as a complex $\scr{F}^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}$ with
$\scr{F}^0=\scr{F}$ and $\scr{F}^i=0$ for $i\neq 0$. If $K^{{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}}$ is a complex in any of the
above categories, then $h^i(K^{{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}})$ denotes the $i$-th cohomology sheaf of
$K^{{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}}$.
The right derived functor of an additive functor $F$, if it exists, is denoted by
${\mcR\!} F$ and ${\mcR\!}^iF$ is short for $h^i\circ {\mcR\!} F$. Furthermore, $\mathbb{H}^i$,
$\mathbb{H}^i_{\rm c}$, $\mathbb{H}^i_Z$ , and $\scr{H}^i_Z$ will denote ${\mcR\!}^i\Gamma$,
${\mcR\!}^i\Gamma_{\rm c}$, ${\mcR\!}^i\Gamma_Z$, and ${\mcR\!}^i\scr{H}_Z$ respectively, where
$\Gamma$ is the functor of global sections, $\Gamma_{\rm c}$ is the functor of
global sections with proper support, $\Gamma_Z$ is the functor of global sections
with support in the closed subset $Z$, and $\scr{H}_Z$ is the functor of the sheaf of
local sections with support in the closed subset $Z$. Note that according to this
terminology, if $\phi\colon Y\to X$ is a morphism and $\scr{F}$ is a coherent sheaf on
$Y$, then ${\mcR\!}\phi_*\scr{F}$ is the complex whose cohomology sheaves give rise to the
usual higher direct images of $\scr{F}$.
We will often use the notion that a morphism ${f}: {\sf A}\to {\sf B}$ in a derived
category \emph{has a left inverse}. This means that there exists a morphism
$f^\ell: {\sf B}\to {\sf A}$ in the same derived category such that
$f^\ell\circ{f}:{\sf A}\to{\sf A}$ is the identity morphism of ${\sf A}$. I.e., $f^\ell$ is
a \emph{left inverse} of ${f}$.
Finally, we will also make the following simplification in notation. First observe
that if $\iota:\Sigma \hookrightarrow X$ is a closed embedding of schemes then $\iota_*$ is
exact and hence ${\mcR\!}\iota_*=\iota_*$. This allows one to make the following
harmless abuse of notation: If ${\sf A}\in\ob D(\Sigma)$, then, as usual for sheaves,
we will drop $\iota_*$ from the notation of the object $\iota_*{\sf A}$. In other
words, we will, without further warning, consider ${\sf A}$ an object in $D(X)$.
\end{demo}
\begin{ack}
I would like to thank Donu Arapura for explaining some of the intricacies of the
relevant Hodge theory to me\uj{, Osamu Fujino for helpful remarks,} and the referee
for useful comments.
\end{ack}
\section{Rational and {Du~Bois} singularities}\label{sec:rational-du-bois}
\begin{defn}\label{def:rtl-sing}
Let $X$ be a normal variety and $\phi :Y \rightarrow X$ a resolution of
singularities. $X$ is said to have \emph{rational} singularities if
${\mcR\!}^i\phi_*\scr{O}_Y=0$ for all $i>0$, or equivalently if the natural map $\scr{O}_X\to
{\mcR\!}\phi_*\scr{O}_Y$ is a quasi-isomorphism.
\end{defn}
\begin{comment}
A very useful property of rational singularities is that they are
\emph{Cohen-Macaulay}. We will define this notion next.
\begin{defn}\label{def:CM}
A finitely generated non-zero module $M$ over a noetherian local ring $R$ is called
\emph{Cohen-Macaulay} if its depth over $R$ is equal to its dimension. For the
defntion of depth and dimension we refer the reader to \cite{MR1251956}. The ring
$R$ is called \emph{Cohen-Macaulay} if it is a Cohen-Macaulay module over itself.
Let $X$ be a scheme and $x\in X$ a point. We say that $X$ has \emph{Cohen-Macaulay}
singularities at $x$ (or simply $X$ is \emph{CM} at $x$), if the local ring
$\scr{O}_{X,x}$ is Cohen-Macaulay.
If in addition, $X$ admits a dualizing sheaf $\omega_X$ which is a line bundle in a
neighbourhood of $x$, then $X$ is \emph{Gorenstein} at $x$.
The scheme $X$ is \emph{Cohen-Macaulay} (resp.\ \emph{Gorenstein}) if it is
\emph{Cohen-Macaulay} (resp.\ \emph{Gorenstein}) at $x$ for all $x\in X$.
\end{defn}
\end{comment}
\uj{Du~Bois} singularities are defined via Deligne's Hodge theory
We will need a little preparation before we can define them.
The starting point is \uj{Du~Bois}'s construction, following Deligne's ideas, of the
generalized de~Rham complex, which we call the \emph{\uj{Deligne-Du~Bois} complex}.
Recall, that if $X$ is a smooth complex algebraic variety of dimension $n$, then the
sheaves of differential $p$-forms with the usual exterior differentiation give a
resolution of the constant sheaf $\mathbb{C}_X$. I.e., one has a filtered complex of
sheaves,
$$
\xymatrix{%
\scr{O}_X \ar[r]^{d} & \Omega_X^1 \ar[r]^{d} & \Omega_X^2 \ar[r]^{d} & \Omega_X^3
\ar[r]^{d} & \dots \ar[r]^{d} & \Omega_X^n\simeq \omega_X,
}
$$
which is quasi-isomorphic to the constant sheaf $\mathbb{C}_X$ via the natural map $\mathbb{C}_X\to
\scr{O}_X$ given by considering constants as holomorphic functions on $X$. Recall that
this complex \emph{is not} a complex of quasi-coherent sheaves. The sheaves in the
complex are quasi-coherent, but the maps between them are not $\scr{O}_X$-module
morphisms. Notice however that this is actually not a shortcoming; as $\mathbb{C}_X$ is not
a quasi-coherent sheaf, one cannot expect a resolution of it in the category of
quasi-coherent sheaves.
The \uj{Deligne-Du~Bois} complex is a generalization of the de~Rham complex to singular
varieties. It is a complex of sheaves on $X$ that is quasi-isomorphic to the
constant sheaf $\mathbb{C}_X$. The terms of this complex are harder to describe but its
properties, especially cohomological properties are very similar to the de~Rham
complex of smooth varieties. In fact, for a smooth variety the \uj{Deligne-Du~Bois}
complex is quasi-isomorphic to the de~Rham complex, so it is indeed a direct
generalization.
The construction of this complex, $\FullDuBois{X}$, is based on simplicial
resolutions. The reader interested in the details is referred to the original article
\cite{DuBois81}. Note also that a simplified construction was later obtained in
\cite{Carlson85} and \cite{GNPP88} via the general theory of polyhedral and cubic
resolutions. An easily accessible introduction can be found in \cite{Steenbrink85}.
Other useful references are the recent book \cite{PetersSteenbrinkBook} and the
survey \cite{Kovacs-Schwede09}. We will actually not use these resolutions here. They
are needed for the construction, but if one is willing to believe the listed
properties (which follow in a rather straightforward way from the construction) then
one should be able follow the material presented here. The interested reader should
note that recently Schwede found a simpler alternative construction of (part of) the
\uj{Deligne-Du~Bois} complex that does not need a simplicial resolution \cite{MR2339829}.
For applications of the \uj{Deligne-Du~Bois} complex and \uj{Du~Bois} singularities other than
the ones listed here see \cite{SteenbrinkMixed}, \cite[Chapter 12]{Kollar95s},
\cite{Kovacs99,Kovacs00c,KSS10,KK10}.
The word ``hyperresolution'' will refer to either a simplicial, polyhedral, or cubic
resolution. Formally, the construction of $\FullDuBois{X}$ is the same regardless the
type of resolution used and no specific aspects of either types will be used.
The next theorem lists the basic properties of the \uj{Deligne-Du~Bois} complex:
\begin{thm}[{\cite{DuBois81}}]\label{defDB}
Let $X$ be a complex scheme of finite typ
. Then there exists a functorially defined object $\underline{\Omega}_X^{{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}} \in \Ob D_{\rm
filt}(X)$ such that using the notation
$$
\underline{\Omega}_X^ p \colon\!\!\!= Gr^{p}_{\rm filt}\, \underline{\Omega}_X^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }} [p],
$$
it satisfies the following properties
\begin{enumerate-p}
\item
$$
\underline{\Omega}_X^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }} \,{\simeq}_{\text{qis}}\, \mathbb{C}_{X}.
$$
\item $\underline{\Omega}_{(\_)}^{{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}}$ is functorial, i.e., if $\phi \colon Y\to
X$ is a morphism of complex schemes of finite type, then there exists a natural
map $\phi^{*}$ of filtered complexes
$$
\phi^{*}\colon \underline{\Omega}_X^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }} \to {\mcR\!}\phi_{*}\underline{\Omega}_Y^{{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}}
$$
Furthermore, $\underline{\Omega}_X^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }} \in \Ob \left(D^{b}_{\rm filt, coh}(X)\right)$ and if
$\phi$ is proper, then $\phi^{*}$ is a morphism in $D^{b}_{\rm filt, coh}(X)$.
\label{functorial}
\item Let $U \subseteq X$ be an open subscheme of $X$. Then
$$
\underline{\Omega}_X^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }} \big\vert_ U \,{\simeq}_{\text{qis}}\,\underline{\Omega}^{\,{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}}_U.
$$
\item If $X$ is proper, then there exists a spectral sequence degenerating at $E\uj{_1}$
and abutting to the singular cohomology of $X$:
$$
E\uj{_1}^{pq}={\mathbb{H}}^q \left(X, \underline{\Omega}_X^ p \right) \Rightarrow H^{p+q}(X, \mathbb{C}).
$$\label{item:Hodge}
\item If\/ $\varepsilon_{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}\colon X_{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}\to X$ is a hyperresolution, then
$$
\underline{\Omega}_X^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }} \,{\simeq}_{\text{qis}}\, {\mcR\!}{\varepsilon_{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}}_* \Omega^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}_{X_{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}}.
$$
In particular, $h^i\left(\underline{\Omega}_X^ p \right)=0$ for $i<0$.\label{item:1}
\item There exists a natural map, $\scr{O}_{X}\to \underline{\Omega}_X^ 0$, compatible with
(\ref{defDB}.\ref{functorial}).
\label{item:dR-to-DB}
\item If\/ $X$ is a normal crossing divisor in a smooth variety, then
$$
\underline{\Omega}_X^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }} \,{\simeq}_{\text{qis}}\,\Omega^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}_X.
$$
In particular,
$$
\underline{\Omega}_X^ p \,{\simeq}_{\text{qis}}\,\Omega^p_X.
$$
\label{item:8}
\item If\/ $\phi\colon Y\to X$ is a resolution of singularities, then
$$
\underline{\Omega}_X^{\dim X} \,{\simeq}_{\text{qis}}\, {\mcR\!}\phi_*\omega_Y.
$$
\item\label{item:exact-triangle} Let $\pi : \widetilde X \rightarrow X$ be a projective
morphism and $\Sigma \subseteq X$ a reduced closed subscheme such that $\pi$ is an
isomorphism outside of $\Sigma$. Let $E$ denote the reduced subscheme of $\widetilde
X$ with support equal to $\pi^{-1}(X)$.
Then for each $p$ one has an exact triangle of objects in the derived category,
$$
\xymatrix{ \underline{\Omega}^p_X \ar[r] & \underline{\Omega}^p_\Sigma \oplus {\mcR\!} \pi_* \underline{\Omega}^p_{\widetilde X}
\ar[r]^-{-}
& {\mcR\!} \pi_* \underline{\Omega}^p_E \ar[r]^-{+1} & .\\
}
$$
\item \label{item:2} Suppose $X=Y\cup Z$ is the union of two closed subschemes and
denote their intersection by $W\colon\!\!\!= Y\cap Z$. Then for each $p$ one has an
exact triangle of objects in the derived category,
$$
\xymatrix{ \underline{\Omega}^p_X \ar[r] & \underline{\Omega}^p_{Y} \oplus \underline{\Omega}^p_{Z}
\ar[r]^-{-} & \underline{\Omega}^p_W \ar[r]^-{+1} & .\\
}
$$
\end{enumerate-p}
\end{thm}
\begin{comment}
Naturally, one may choose $D=\emptyset$ and then it is simply omitted from the
notation. The same applies to $\underline{\Omega}_X^p\colon\!\!\!= Gr^{p}_{\rm filt}\,
\underline{\Omega}_X^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}[p]$.
\end{comment}
It turns out that the \uj{Deligne-Du~Bois} complex behaves very much like the de~Rham
complex for smooth varieties. Observe that (\ref{defDB}.\ref{item:Hodge}) says that
the Hodge-to-de~Rham (a.k.a.\ Fr\"olicher) spectral sequence works for singular
varieties if one uses the \uj{Deligne-Du~Bois} complex in place of the de~Rham complex.
This has far reaching consequences and if the associated graded pieces $\underline{\Omega}_X^ p$
turn out to be computable, then this single property leads to many applications.
Notice that (\ref{defDB}.\ref{item:dR-to-DB}) gives a natural map $\scr{O}_{X}\to
\underline{\Omega}^0_X$, and we will be interested in situations when this map is a
quasi-isomorphism. When $X$ is proper over $\mathbb{C}$, such a quasi-isomorphism implies
that the natural map
\begin{equation*}
H^i(X^{\rm an}, \mathbb{C}) \rightarrow H^i(X, \scr{O}_{X}) = \mathbb{H}^i(X, \DuBois{X})
\end{equation*}
is surjective because of the degeneration at $E_1$ of the spectral sequence in
(\ref{defDB}.\ref{item:Hodge}). Notice that this is the condition that is crucial for
Kodaira-type vanishing theorems cf.\ \cite[\S 9]{Kollar95s}.
Following \uj{Du~Bois}, Steenbrink was the first to study this condition and he christened
this property after \uj{Du~Bois}. It should be noted that many of the ideas that play
important roles in this theory originated from Deligne. Unfortunately the now
standard terminology does not reflect this.
\begin{defn}\label{def:db-sing}
A scheme $X$ is said to have \emph{\uj{Du~Bois}} singularities (or \emph{DB}
singularities for short) if the natural map $\scr{O}_{X}\to \underline{\Omega}^0_X$ from
(\ref{defDB}.\ref{item:dR-to-DB}) is a quasi-isomorphism.
\end{defn}
\begin{rem}
If $\varepsilon : X_{{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}} \rightarrow X$ is a hyperresolution of $X$ then $X$ has
DB singularities if and only if the natural map $\scr{O}_X \rightarrow {\mcR\!}
{\varepsilon_{{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}}}_* \scr{O}_{X_{{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}}}$ is a quasi-isomorphism.
\end{rem}
\begin{example}
It is easy to see that smooth points are DB and Deligne proved that normal crossing
singularities are DB as well cf.\ (\ref{defDB}.\ref{item:8}), \cite[Lemme
2(b)]{MR0376678}.
\end{example}
In applications it is very useful to be able to take general hyperplane sections. The
next statement helps with that.
\begin{prop}\label{prop:db-cx-of-hyper}
Let $X$ be a quasi-projective variety and $H\subset X$ a general member of a very
ample linear system. Then $\underline{\Omega}_H^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}\,{\simeq}_{\text{qis}}\, \underline{\Omega}_X^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}\otimes_L\scr{O}_H$.
\end{prop}
\begin{proof}
Let $\varepsilon_{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}: X_{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}\to X$ be a hyperresolution. Since $H$ is general,
the fiber product $X_{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}\times_XH\to H$ provides a hyperresolution of $H$. Then
the statement follows from (\ref{defDB}.\ref{item:1}) applied to both $X$ and $H$.
\end{proof}
We saw in (\ref{defDB}.\ref{item:1}) that $h^i\left(\underline{\Omega}_X^0 \right)=0$ for $i<0$. In
fact, there is a corresponding upper bound by \cite[III.1.17]{GNPP88}, namely that
$h^i\left(\underline{\Omega}_X^0 \right)=0$ for $i>\dim X$. It turns out that one can make a
slightly better estimate.
\begin{prop}[\protect{cf.\ \cite[13.7]{GKKP10}, \cite[4.9]{KSS10}}]
\label{prop:top-coh-vanishes}
Let $X$ be a positive dimensional variety (i.e., reduced). Then the $i^{\text{th}}$
cohomology sheaf of $\underline{\Omega}_X^p$ vanishes for all $i\geq \dim X$, i.e.,
$h^i(\underline{\Omega}_X^p)=0$ for all $p$ and for all $i\geq \dim X$.
\end{prop}
\begin{proof}
For $i>\dim X$ or $p>0$, the statement follows from \cite[III.1.17]{GNPP88}. The
case $p=0$ and $i=n:=\dim X$ follows from either \cite[13.7]{GKKP10} or
\cite[4.9]{KSS10}.
\end{proof}
\begin{comment}
Proof of above statement:
, so we
only need to prove the case when $p=0$ and $i=n:=\dim X$.
%
Let $S:=\Sing X$ and $\pi : \widetilde X \to X$ a log resolution with
exceptional divisor $E$ such that $\widetilde X\setminus E \simeq X\setminus
S$. Consider the exact triangle (\ref{defDB}.\ref{item:exact-triangle}),
\begin{equation}
\label{eq:14}
\xymatrix{%
\underline{\Omega}_X^0 \ar[r] & \underline{\Omega}_S^0 \oplus {\mcR\!}\pi_*\underline{\Omega}_{\widetilde
X}^0 \ar[r] & {\mcR\!}\pi_*\underline{\Omega}_{E}^0 \ar^-{+1}[r] & }.
\end{equation}
Notice that $\widetilde X$ is smooth and $E$ is an snc divisor, so they are both
DB and hence $\underline{\Omega}_{\widetilde X}^0\simeq \scr{O}_{\widetilde X}$ and
$\underline{\Omega}_{E}^0\simeq \scr{O}_{E}$.
%
Next, consider the long exact sequence of cohomology sheaves induced by the
above exact triangle,
\begin{multline*}
\xymatrix{%
\dots\ar[r] & h^{n-1}(\underline{\Omega}_{S}^0)\oplus {\mcR\!}^{n-1}\pi_*\scr{O}_{\widetilde
X}\ar[r]^-{\varsigma} & {\mcR\!}^{n-1}\pi_*\scr{O}_E\ar[r] & }\\
\xymatrix{%
\ar[r] &
h^{n}(\underline{\Omega}_{X}^0) \ar[r] &
h^{n}(\underline{\Omega}_{S}^0)\oplus {\mcR\!}^{n}\pi_*\scr{O}_{\widetilde X}. }
\end{multline*}
Since $\dim S<n$, \cite[III.1.17]{GNPP88} implies that
$h^{n}(\underline{\Omega}_{S}^0)=0$. Furthermore, as $\pi$ is birational, the
dimension of any fibre of $\pi$ is at most $n-1$ and hence
${\mcR\!}^{n}\pi_*\scr{O}_{\widetilde X}=0$. This implies that
$h^{n}(\underline{\Omega}_{X}^0)\simeq \coker\varsigma$.
On the other hand, it follows from the construction of the exact triangle
(\ref{eq:14}) that $\varsigma(0,{\_})$ is the map ${\mcR\!}^{n-1}\pi_*\scr{O}_{\widetilde X} \to
{\mcR\!}^{n-1}\pi_*\scr{O}_{E}$ induced by the short exact sequence
$$
0\to \scr{O}_{\widetilde X}(-E)\to \scr{O}_{\widetilde X}\to \scr{O}_E\to 0.
$$
Again, the same dimension bound as above implies that ${\mcR\!}^{n}\pi_*\scr{O}_{\widetilde
X}(-E)=0$, so we obtain that $\varsigma(0,\_ )$ is surjective, and then so is
$\varsigma$. Therefore, as desired, $h^{n}(\underline{\Omega}_{X}^0)\simeq
\coker\varsigma=0$.
\end{comment}
\noindent
Another, much simpler fact that will be used later is the following:
\begin{cor}
If $\dim X=1$, then $h^i(\underline{\Omega}_X^p)=0$ for $i\neq 0$. In particular $X$ is DB if and
only if it is semi-normal.
\end{cor}
\begin{comment}
\begin{demo}
{Observation}\label{obs:omega-mod-O} If $X$ is reduced, then the natural map
$\scr{O}_X\to\underline{\Omega}^0_X$ induces an embedding $\scr{O}_X\hookrightarrow h^0(\underline{\Omega}^0_X)$.
\end{demo}
\begin{proof}
Since $X$ is reduced, it is enough to prove this statement at the generic points
of $X$. At those points the statement is trivial as by (\ref{defDB}.\ref{item:1})
the morphism $\scr{O}_X\to h^0(\underline{\Omega}^0_X)$ is an isomorphism on $X\setminus\Sing X$.
\end{proof}
\end{comment}
\begin{proof}
The first statement is a direct consequence of \eqref{prop:top-coh-vanishes}. For
the last statement recall that the seminormalization of $\scr{O}_X$ is exactly
$h^0(\underline{\Omega}^0_X)$, and so $X$ is seminormal if and only if $\scr{O}_X\simeq h^0(\underline{\Omega}^0_X)$
\cite[5.2]{MR1741272} (cf.\ \cite[5.4.17]{Schwede06}, \cite[4.8]{MR2339829}, and
\cite[5.6]{MR2503989}).
\end{proof}
\begin{defn}\label{def:db-defect}
The \emph{DB defect} of $X$ is the mapping cone of the morphism $\scr{O}_X\to \underline{\Omega}^0_X$.
It is denoted by $\underline{\Omega}_X^\times$. As a simple consequence of the definition, one has
an exact triangle,
$$
\xymatrix{%
\scr{O}_X \ar[r] & \underline{\Omega}^0_X \ar[r] & \underline{\Omega}^\times_X \ar[r]^-{+1} & .
}
$$
Notice that $h^0(\underline{\Omega}_X^\times)\simeq h^0(\underline{\Omega}_X^0)/\scr{O}_X$ and
$h^i(\underline{\Omega}_X^\times)\simeq h^i(\underline{\Omega}_X^0)$ for $i>0$.
\end{defn}
\begin{prop}\label{prop:hyper-plane-cuts}
Let $X$ be a quasi-projective variety and $H\subset X$ a general member of a very
ample linear system. Then $\underline{\Omega}_H^\times\,{\simeq}_{\text{qis}}\, \underline{\Omega}_X^\times\otimes_L\scr{O}_H$.
\end{prop}
\begin{proof}
This follows easily from the definition and \ref{prop:db-cx-of-hyper}.
\end{proof}
The next simple observation explains the name of the DB defect.
\begin{lem}\label{lem:db-defect}
A variety $X$ is DB if and only if the DB defect of $X$ is acyclic, that is,
$\underline{\Omega}_X^\times\,{\simeq}_{\text{qis}}\, 0$.
\end{lem}
\begin{proof}
This follows directly from the definition.
\end{proof}
\begin{prop}\label{prop:DB-defect-for-unions}
Let $X=Y\cup Z$ be a union of closed subschemes with intersection $W=Y\cap Z$. Then
one has an exact triangle of the DB defects of $X,Y,Z$, and $W$:
\begin{equation*}
\xymatrix{ \underline{\Omega}^\times_X \ar[r] & \underline{\Omega}^\times_{Y} \oplus \underline{\Omega}^\times_{Z}
\ar[r]^-{-} & \underline{\Omega}^\times_W \ar[r]^-{+1} & .\\
}
\end{equation*}
\end{prop}
\begin{proof}
Recall that there is an analogous exact triangle (a.k.a.\ a short exact sequence)
for the structure sheaves of $X,Y,Z$, and $W$, which forms a commutative diagram
with the exact triangle of (\ref{defDB}.\ref{item:2}),
$$
\xymatrix{
\scr{O}_X \ar[d] \ar[r] & \scr{O}_{Y} \oplus \scr{O}_{Z}
\ar[d] \ar[r]^-{-} & \scr{O}_W \ar[d] \ar[r]^-{+1} & \\
\underline{\Omega}^0_X \ar[r] & \underline{\Omega}^0_{Y} \oplus \underline{\Omega}^0_{Z}
\ar[r]^-{-} & \underline{\Omega}^0_W \ar[r]^-{+1} & .\\
}
$$
Then the statement follows by the (derived category version of the) 9-lemma.
\end{proof}
\section{Pairs and generalized pairs}\label{sec:pairs-gener-pairs}
\subsection{Basic definitions}\label{ssec:basic-definitions}
For an arbitrary proper birational morphism, $\phi:Y\to X$, $\exc(\phi)$ stands for
the exceptional locus of $\phi$. A \emph{$\mathbb{Q}$-divisor} is a $\mathbb{Q}$-linear
combination of integral Weil divisors; $\Delta=\sum a_i\Delta_i$, $a_i\in\mathbb{Q}$,
$\Delta_i$ (integral) Weil divisor. For a $\mathbb{Q}$-divisor $\Delta$, its
\emph{round-down} is defined by the formula: $\rdown \Delta=\sum
\rdown{a_i}\Delta_i$, where $\rdown{a_i}$ is the largest integer not larger than
$a_i$.
A \emph{log variety} or \emph{pair} $(X,\Delta)$ consists of an equidimensional
variety (i.e., a reduced scheme of finite type over a field $k$) $X$ and an
effective $\mathbb{Q}$-divisor $\Delta\subseteq X$.
A morphism of pairs $\phi:(Y,B)\to (X,\Delta)$ is a morphism $\phi:Y\to X$ such
that $\phi(\supp B)\subseteq \supp\Delta$.
Let $(X,\Delta)$ be a pair with $\Delta$ a reduced integral divisor. Then
$(X,\Delta)$ is said to have \emph{simple normal crossings} or to be an \emph{snc
pair at $p\in X$} if $X$ is smooth at $p$, and there are local coordinates
$x_1,\dots,x_n$ on $X$ in a neighbourhood of $p$ such that $\supp\Delta\subseteq
(x_1\cdots x_n=0)$ near $p$. $(X,\Delta)$ is \emph{snc} if it is snc at every $p\in
X$.
A morphism of pairs $\phi:(Y,\Delta_Y)\to (X,\Delta)$ is a \emph{log resolution of
$(X,\Delta)$} if $\phi:Y\to X$ is proper and birational,
$\Delta_Y=\phi^{-1}_*\Delta$, and $(\Delta_Y)_{\red}+\exc(\phi)$ is an snc divisor
on $Y$.
Note that we allow $(X,\Delta)$ to be snc and still call a morphism with these
properties a log resolution. Also note that the notion of a log resolution is not
used consistently in the literature.
If $(X,\Delta)$ is a pair, then $\Delta$ is called a \emph{boundary} if
$\rdown{(1-\varepsilon)\Delta}=0$ for all $0<\varepsilon<1$, i.e., the coefficients
of all irreducible components of $\Delta$ are in the interval $[0,1]$. For the
definition of \emph{klt, dlt}, and \emph{lc} pairs see \cite{KM98}.
Let $(X, \Delta)$ be a pair and $\mu:X^{\rm m}\to X$ a proper birational
\uj{morphism.
Let $E=\sum a_iE_i$ be the discrepancy divisor, i.e., a linear combination of
exceptional divisors such that
$$
K_{X^{\rm m}}+\mu^{-1}_*\Delta \sim_{\mathbb{Q}} \mu^*(K_X + \Delta) + E
$$ and let
$\Delta^{\rm m}\colon\!\!\!= \mu^{-1}_*\Delta + \sum_{a_i\leq -1}E_i$. For an irreducible
divisor $F$ on a birational model of $X$ we define its discrepancy as its
coefficient in $E$. Notice that as divisors correspond to valuations, this
discrepancy is independent of the model chosen, it only depends on the divisor. A
\emph{non-klt place} of a pair $(X,\Delta)$ is an irreducible divisor $F$ over $X$
with discrepancy at most $-1$ and a \emph{non-klt center} is the image of any
non-klt place. $\excnklt(\mu)$ denotes the union of the loci of all non-klt places
of $\phi$.
Note that in the
literature, non-klt places and centers are often called log canonical places and
centers. For a more detailed and precise definition see
\cite[p.37]{Hacon-Kovacs10}.
Now if $(X^{\rm m}, \Delta^{\rm m})$ is as above, then it is a \emph{minimal dlt
model} of $(X,\Delta)$ if it is a dlt pair and the discrepancy of every
$\mu$-exceptional divisor is at most $-1$ cf.\ \cite{KK10}. Note that if
$(X,\Delta)$ is lc with a minimal dlt model $(X^{\rm m}, \Delta^{\rm m})$, then
$K_{X^{\rm m}}+\Delta^{\rm m} \sim_{\mathbb{Q}} \mu^*(K_X+\Delta)$.
\subsection{Rational pairs}
Recall the definition of a \emph{log resolution} from \eqref{ssec:basic-definitions}:
A morphism of pairs $\phi:(Y,\Delta_Y)\to (X,\Delta)$ is a \emph{log resolution of
$(X,\Delta)$} if $\phi:Y\to X$ is proper and birational,
$\Delta_Y=\phi^{-1}_*\Delta$, and $(\Delta_Y)_{\red}+\exc(\phi)$ is an snc divisor on
$Y$.
\begin{defn}\label{def:normal-pair}
Let $(X,\Delta)$ be a pair and $\Delta$ an integral divisor. Then $(X,\Delta)$ is
called a \emph{normal pair} if there exists a log resolution $\phi:(Y,\Delta_Y)\to
(X,\Delta)$ such that the natural morphism $\phi^\#:\scr{O}_X(- \Delta)\to
\phi_*\scr{O}_Y(-{\Delta_Y})$ is an isomorphism.
\end{defn}
\begin{defn}
\label{def:weakly-rtl-sing-pairs}
A pair $(X,\Delta)$ with $\Delta$ an integral divisor is called a \emph{weakly
rational pair} if there is a log resolution $\phi:(Y,\Delta_Y)\to (X,\Delta)$
such that the natural morphism $\scr{O}_X(- \Delta)\to {\mcR\!}\phi_*\scr{O}_Y(-{\Delta_Y})$
has a left inverse.
\end{defn}
\begin{lem}
Let $(X,\Delta)$ be a weakly rational pair. Then it is a normal pair.
\end{lem}
\begin{proof}
The $0^\text{th}$ cohomology of the left inverse of $\scr{O}_X(- \Delta)\to
{\mcR\!}\phi_*\scr{O}_Y(-{\Delta_Y})$ gives a left inverse of $\phi^\#:\scr{O}_X(- \Delta)\to
\phi_*\scr{O}_Y(-{\Delta_Y})$. As the morphism $\phi$ is birational, the kernel of the
left inverse of $\phi^\#$ is a torsion sheaf. However, since
$\phi_*\scr{O}_Y(-{\Delta_Y})$ is torsion-free, this implies that $\phi^\#$ is an
isomorphism.
\end{proof}
\begin{defn}\cite{KollarKovacsRP}
\label{def:rat-sing-pairs}
Let $(X,\Delta)$ be a pair where $\Delta$ is an integral divisor. Then $(X,\Delta)$
is called a \emph{rational pair} if there exists a log resolution
$\phi:(Y,\Delta_Y)\to (X,\Delta)$ such that
\begin{enumerate-cont}
\item $\scr{O}_X(-\Delta)\simeq \phi_*\scr{O}_Y(-{\Delta_Y})$, i.e., $(X,\Delta)$ is
normal,
\item ${\mcR\!}^i\phi_*\scr{O}_Y(-{\Delta_Y})=0$ for $i>0$, and
\item ${\mcR\!}^i\phi_*\omega_Y({\Delta_Y})=0$ for $i>0$.
\label{item:GR-vanishing-for-pairs}
\end{enumerate-cont}
\end{defn}
\begin{lem}
Let $(X,\Delta)$ be a pair where $\Delta$ is an integral divisor. Then it is a
rational pair if and only if it is a weakly rational pair and
${\mcR\!}^i\phi_*\omega_Y({\Delta_Y})=0$ for $i>0$.
\end{lem}
\begin{proof}
This follows directly from \cite[105]{KollarKovacsRP}.
\end{proof}
\begin{rem}\label{rem:rtl-is-relative}
Note that the notion of a \emph{rational pair} describes the ``singularity'' of the
relationship between $X$ and $\Delta$. From the definition it is not clear for
instance whether $(X,\Delta)$ being rational implies that $X$ has rational
singularities.
\end{rem}
\begin{rem}
If $\Delta=\emptyset$, then
(\ref{def:rat-sing-pairs}.\ref{item:GR-vanishing-for-pairs}) follows from
Grauert-Riemenschneider vanishing and $X$ is weakly rational if and only if it is
rational by \cite{Kovacs00b}.
\end{rem}
\begin{comment}
Example~\ref{ex:rtl-pairs} shows that the notion of rational pairs is quite general
in the sense that on any Cohen-Macaulay $\mathbb{Q}$-factorial variety $X$ there exists a
reduced hypersurface $\Sigma\subset X$ such that $(X,\Sigma)$ is a rational pair.
First we need two lemmata that are useful on their own.
\begin{lem}\label{lem:CM-implies-dual-vanishing}
Let $\phi:Y\to X$ be a projective birational morphism and assume that both $X$ and
$Y$ are Cohen-Macaulay schemes. Let $\scr{E}$ be a locally free sheaf on $Y$ and assume
that $\phi_*\left(\omega_Y\otimes\scr{E}\right)$ is locally free on $X$ and
${\mcR\!}^i\phi_*\left(\omega_Y\otimes\scr{E}\right)=0$ for $i>0$. Then $\phi_*(\scr{E}^*)$ is
locally free and ${\mcR\!}^i\phi_*(\scr{E}^*)=0$ for $i>0$. ($\scr{E}^*$ denotes the dual of
$\scr{E}$).
\end{lem}
\begin{proof}
Let $n=\dim X=\dim Y$. Then using Grothendieck duality, the fact that both $X$ and
$Y$ are CM, that ${\mcR\!}\phi_*\left(\omega_Y\otimes\scr{E}\right)\,{\simeq}_{\text{qis}}\,
\phi_*\left(\omega_Y\otimes\scr{E}\right)$, and that the latter is a locally free sheaf
one obtains that
\begin{multline*}
{\mcR\!}\phi_*\scr{E}^*\,{\simeq}_{\text{qis}}\, {\mcR\!}\phi_*{\mcR\!}\sHom(\omega_Y\otimes\scr{E}, \omega_Y)\,{\simeq}_{\text{qis}}\, \\
{\mcR\!}\phi_*{\mcR\!}\sHom(\omega_Y\otimes\scr{E}, \omega_Y^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }})[-n]\,{\simeq}_{\text{qis}}\,
{\mcR\!}\sHom({\mcR\!}\phi_*\left(\omega_Y\otimes\scr{E}\right), \omega_X^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }})[-n]\,{\simeq}_{\text{qis}}\, \\
{\mcR\!}\sHom(\phi_*\left(\omega_Y\otimes\scr{E}\right), \omega_X) \,{\simeq}_{\text{qis}}\,
\omega_X\otimes\left(\phi_*\left(\omega_Y\otimes\scr{E}\right)\right)^*.
\end{multline*}
So the statement is proven.
\end{proof}
\begin{lem}
Let $\phi:Y\to X$ be a projective birational morphism and assume that $X$ is
$\mathbb{Q}$-factorial. Let $A\subset Y$ be an ample divisor. Then there exists a natural
number $n\in\mathbb{N}$ such that $\phi_*\omega_Y(nA)$ is a line bundle on $X$ and
${\mcR\!}^i\phi_*\omega_Y(nA)=0$ for $i>0$.
\end{lem}
\begin{proof}
As $A$ is ample, it follows that ${\mcR\!}^i\phi_*\omega_Y(nA)=0$ for $i>0$ if $n\gg
0$. We only need to prove that there exists an $n\in\mathbb{N}$ such that
$\phi_*\omega_Y(nA)$ is a line bundle on $X$.
For $n\gg 0$ we have that $B_n\colon\!\!\!= K_Y+nA$ is an effective divisor. Consider
$\phi_*B_n$ the cycle theoretic push-forward of $B_n$ on $X$. As $X$ is
$\mathbb{Q}$-factorial there exists an $n\in\mathbb{N}$ such that $n(\phi_*A)=\phi_*(nA)$ is a
Cartier divisor.
\end{proof}
\begin{example}\label{ex:rtl-pairs}
Let $X$ be a CM $\mathbb{Q}$-factorial variety and $\phi:Y\to X$ a resolution of
singularities of $X$ such that $E\colon\!\!\!=\Ex(\phi)$ is a divisor with snc
singularities. Let $H$ be a sufficiently ample divisor on $Y$ which is a general
member of its linear system, so that $\Gamma\colon\!\!\!= E+H$ is also an snc divisor.
Since $H$ is sufficiently ample we may assume that ${\mcR\!}^i\phi_*\omega_Y(\Gamma)=0$
for $i>0$.
\end{example}
\end{comment}
\subsection{Generalized pairs}
\begin{defn}
A \emph{generalized pair} $(X,\Sigma)$ consists of an equidimensional variety
(i.e., a reduced scheme of finite type over a field $k$) $X$ and a subscheme
$\Sigma\subseteq X$. A morphism of generalized pairs $\phi:(Y,\Gamma)\to
(X,\Sigma)$ is a morphism $\phi:Y\to X$ such that $\phi(\Gamma)\subseteq \Sigma$.
A \emph{reduced generalized pair} is a generalized pair $(X,\Sigma)$ such that
$\Sigma$ is reduced.
The \emph{log resolution} of a generalized pair $(X, W)$ is a proper birational
morphism $\pi: \widetilde X\to X$ such that $\exc(\pi)$ is a divisor and
$\pi^{-1}W+\exc(\pi)$ is an snc divisor.
Let $X$ be a complex scheme and $\Sigma$ a closed subscheme whose complement in $X$
is dense. Then $(X_{{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}}, \Sigma_{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }})\to (X, \Sigma)$ is a \emph{good
hyperresolution} if $X_{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}\to X$ is a hyperresolution, and if
$U_{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}=X_{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}\times_X (X\setminus \Sigma)$ and $\Sigma_{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}=X_{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}\setminus
U_{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}$, then, for all $\alpha$, either $\Sigma_\alpha$ is a divisor with normal
crossings on $X_\alpha$ or $\Sigma_\alpha=X_\alpha$. Notice that it is possible
that $X_{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}$ has components that map into $\Sigma$. These component are contained
in $\Sigma_{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}$. For more details and the existence of such hyperresolutions
see \cite[6.2]{DuBois81} and \cite[IV.1.21, IV.1.25, IV.2.1]{GNPP88}. For a primer
on hyperresolutions see the appendix of \cite{Kovacs-Schwede09}.
\end{defn}
Let $(X,\Sigma)$ be a reduced generalized pair. Consider the \uj{Deligne-Du~Bois} complex
of $(X,\Sigma)$ defined by Steenbrink \cite[\S 3]{Steenbrink85}:
\begin{defn}
The \emph{\uj{Deligne-Du~Bois} complex} of the reduced generalized pair $(X,\Sigma)$ is
the mapping cone of the natural morphism $\varrho:\underline{\Omega}^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}_X\to \underline{\Omega}^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}_\Sigma$
twisted by $(-1)$. In other words, it is an object $\underline{\Omega}^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}_{X,\Sigma}$ in
$D_{\rm filt}(X)$ such that it completes $\varrho$ to an exact triangle:
\begin{equation}
\label{eq:11}
\xymatrix{%
\underline{\Omega}^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}_{X,\Sigma} \ar[r] & \underline{\Omega}^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}_{X} \ar[r] & \underline{\Omega}^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}_{\Sigma}
\ar[r]^-{+1} & . }
\end{equation}
The associated graded quotients of $\underline{\Omega}^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}_{X,\Sigma}$ will be denoted as usual:
$$
\underline{\Omega}^p_{X,\Sigma}\colon\!\!\!= Gr^p_{\rm filt}\underline{\Omega}^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}_{X,\Sigma}[p].
$$
Notice that the above triangle is in $D_{\rm filt}(X)$ and hence for all $p\in\mathbb{N}$
we obtain another exact triangle:
\begin{equation}
\label{eq:4}
\xymatrix{%
\underline{\Omega}^p_{X,\Sigma} \ar[r] & \underline{\Omega}^p_{X} \ar[r] & \underline{\Omega}^p_{\Sigma}
\ar[r]^-{+1} & .
}
\end{equation}
\end{defn}
\begin{example}
Let $(X,\Sigma)$ be an snc pair. Then $\underline{\Omega}^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}_{X,\Sigma}\,{\simeq}_{\text{qis}}\,
\Omega_X^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}(\log\Sigma)(-\Sigma)$.
\end{example}
\noindent
The \uj{Deligne-Du~Bois} complex of a pair is funtorial in the following sense:
\begin{prop}
Let $\phi:(Y,\Gamma)\to (X,\Delta)$ be a morphism of generalized pairs. Then there
exists a filtered natural morphism $\underline{\Omega}^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}_{X,\Sigma}\to
{\mcR\!}\phi_*\underline{\Omega}_{Y,\Gamma}^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}$.
\end{prop}
\begin{proof}
There exist compatible filtered natural morphisms $\underline{\Omega}_X^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}\to
{\mcR\!}\phi_*\underline{\Omega}_Y^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}$ and $\underline{\Omega}_\Sigma^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}\to {\mcR\!}\phi_*\underline{\Omega}_\Gamma^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}$ by
(\ref{defDB}.\ref{functorial}). They induce the following morphism between exact
triangles,
$$
\xymatrix{%
\underline{\Omega}^0_{X,\Sigma} \ar[r]\ar@{-->}[d] & \underline{\Omega}^0_{X} \ar[r]\ar[d] & \underline{\Omega}^0_{\Sigma}
\ar[r]^-{+1} \ar[d] & \\
{\mcR\!}\phi_*\underline{\Omega}^0_{Y,\Gamma} \ar[r] & {\mcR\!}\phi_*\underline{\Omega}^0_{Y} \ar[r] &
{\mcR\!}\phi_*\underline{\Omega}^0_{\Gamma} \ar[r]^-{+1} & , }
$$
and thus one obtains the desired natural morphism.
\end{proof}
It follows easily from the definition and \eqref{prop:top-coh-vanishes} that we have
the following bounds on the non-zero cohomology sheaves of $\underline{\Omega}_{X,\Sigma}^p$.
\begin{prop}\label{prop:top-coh-vanishes-rel}
Let $X$ be a positive dimensional variety. Then the $i^{\text{th}}$ cohomology
sheaf of $\underline{\Omega}_{X,\Sigma}^p$ vanishes for all $i\geq \dim X$, i.e.,
$h^i(\underline{\Omega}_{X,\Sigma}^p)=0$ for all $p$ and for all $i\geq \dim X$.
\end{prop}
\begin{proof}
This follows directly from \eqref{prop:top-coh-vanishes} using the long exact
cohomology sequence associated to (\ref{eq:4}).
\end{proof}
\subsection{DB pairs and the DB defect}
\begin{defn}\label{def:DB-defect}
Recall the short exact sequence for the restriction of regular functions from $X$
to $\Sigma$:
$$
\xymatrix{%
0 \ar[r] & \scr{I}_{\Sigma\subseteq X} \ar[r] & \scr{O}_X \ar[r] & \scr{O}_\Sigma \ar[r] & 0.
}
$$
By (\ref{defDB}.\ref{item:dR-to-DB}) there exist compatible natural maps $\scr{O}_X\to
\underline{\Omega}_X^0$ and $\scr{O}_\Sigma\to \underline{\Omega}_\Sigma^0$, and they induce a morphism between exact
triangles,
\begin{equation}
\label{eq:12}
\xymatrix{%
\scr{I}_{\Sigma\subseteq X} \ar[r]\ar@{-->}[d] & \scr{O}_X \ar[r]\ar[d] &
\scr{O}_\Sigma \ar[d] \ar[r]^-{+1} & \\
\underline{\Omega}^0_{X,\Sigma} \ar[r] & \underline{\Omega}^0_{X} \ar[r] & \underline{\Omega}^0_{\Sigma} \ar[r]^-{+1} & , }
\end{equation}
A reduced generalized pair $(X,\Sigma)$ will be called a \emph{DB pair} if the
natural morphism $\scr{I}_{\Sigma\subseteq X}\to \underline{\Omega}^0_{X,\Sigma}$ from (\ref{eq:12})
is a quasi-isomorphism.
\end{defn}
\begin{rem}
Note that just like the notion of a {rational pair}, the notion of a \emph{DB pair}
describes the ``singularity'' of the relationship between $X$ and $\Sigma$. From
the definition it is not clear for instance whether $(X,\Sigma)$ being DB implies
that $X$ has DB singularities. \ujj{I expect that this is
not true, but at the time of writing this article I do not know an example of an
irreducible $X$ and an appropriate $\Sigma\subset X$ such that $(X,\Sigma)$ is a
DB pair, but $X$ does not have DB singularities.}
\end{rem}
\begin{prop}\label{prop:rel-funtorial}
Let $\phi:(Y,\Gamma)\to (X,\Sigma)$ be a morphism of generalized pairs. Then there
exists a natural morphism $\underline{\Omega}^0_{X,\Sigma}\to {\mcR\!}\phi_*\underline{\Omega}_{Y,\Gamma}^0$ and a
commutative diagram,
\begin{equation*}
\xymatrix{%
\scr{I}_{\Sigma\subseteq X} \ar[d]\ar[r] & \underline{\Omega}^0_{X,\Sigma}\ar[d] \\
{\mcR\!}\phi_*\scr{I}_{\Gamma\subseteq Y} \ar[r] & {\mcR\!}\phi_*\underline{\Omega}_{Y,\Gamma}^0 \\
}
\end{equation*}
\end{prop}
\begin{proof}
Similarly to (\ref{eq:12}) and one obtains a commutative diagram for $(Y,\Gamma)$:
$$
\xymatrix{%
\scr{I}_{\Gamma\subseteq Y} \ar[r]\ar[d] & \scr{O}_Y \ar[r]\ar[d] & \scr{O}_\Gamma \ar[d]
\ar[r]^-{+1} & \\
\underline{\Omega}^0_{Y,\Gamma} \ar[r] & \underline{\Omega}^0_{Y} \ar[r] & \underline{\Omega}^0_{\Gamma}
\ar[r]^-{+1} & .
}
$$
Then $\phi$ induces a morphism between these diagrams:
$$
\xymatrix{%
\scr{I}_{\Sigma\subseteq X} \ar[rr]\ar[dd] \ar[rd] && \scr{O}_X \ar[rr]\ar'[d][dd]
\ar[rd] && \scr{O}_\Sigma \ar'[d][dd] \ar[rd]
\ar[r]^-{+1} & \\
& \underline{\Omega}^0_{X,\Sigma} \ar[rr]\ar[dd] && \underline{\Omega}^0_{X}
\ar[rr]\ar[dd] && \underline{\Omega}^0_{\Sigma} \ar[dd] \ar[r]^-{+1} & \\
{\mcR\!}\phi_* \scr{I}_{\Gamma\subseteq Y} \ar'[r][rr]\ar[rd] &&
{\mcR\!}\phi_* \scr{O}_\Gamma \ar'[r][rr]\ar[rd] &&
{\mcR\!}\phi_* \scr{O}_Y \ar[rd] \ar[r]^-{+1} & \\
& {\mcR\!}\phi_* \underline{\Omega}^0_{Y,\Gamma} \ar[rr] && {\mcR\!}\phi_* \underline{\Omega}^0_{Y} \ar[rr] &&
{\mcR\!}\phi_* \underline{\Omega}^0_{\Gamma} \ar[r]^-{+1} &
. }
$$
The front face of this diagram provides the one claimed in the statement.
\end{proof}
Similarly to \eqref{def:db-defect} we introduce the {DB defect} of the pair
$(X,\Sigma)$:
\begin{defn}\label{def:rel-db-defect} The
\emph{DB defect} of the pair $(X,\Sigma)$ is the mapping cone of the morphism
$\scr{I}_{\Sigma\subseteq X}\to \underline{\Omega}^0_{X,\Sigma}$. It is denoted by
$\underline{\Omega}_{X,\Sigma}^\times$. Again, one has the exact triangles,
\begin{gather}
\xymatrix{%
\scr{I}_{\Sigma\subseteq X} \ar[r] & \underline{\Omega}^0_{X,\Sigma} \ar[r] &
\underline{\Omega}^\times_{X,\Sigma} \ar[r]^-{+1} & .
}\label{eq:5}
\\
\intertext{and} %
\xymatrix{%
\underline{\Omega}^\times_{X,\Sigma} \ar[r] & \underline{\Omega}^\times_{X} \ar[r] & \underline{\Omega}^\times_{\Sigma}
\ar[r]^-{+1} & .
}\label{eq:6}
\end{gather}
And, again, one has that
\begin{equation}
\label{eq:8}
h^0(\underline{\Omega}_{X,\Sigma}^\times)\simeq
h^0(\underline{\Omega}_{X,\Sigma}^0)/\scr{I}_{\Sigma\subseteq X} \quad\text{and}\quad
h^i(\underline{\Omega}_{X,\Sigma}^\times)\simeq h^i(\underline{\Omega}_{X,\Sigma}^0) \quad\text{for $i>0$.}
\end{equation}
\end{defn}
\begin{lem}
\label{lem:db-defect-rel}
Let $(X,\Sigma)$ be a reduced generalized pair. Then the following are equivalent:
\begin{enumerate}
\item The pair $(X,\Sigma)$ is DB.
\label{item:3}
\item The DB defect of $(X,\Sigma)$ is acyclic, that is, $\underline{\Omega}_{X,\Sigma}^\times\,{\simeq}_{\text{qis}}\,
0$.
\label{item:6}
\item The induced natural morphism $\underline{\Omega}_X^\times\to
\underline{\Omega}_{\Sigma}^\times$ is a quasi-isomorphism.
\label{item:7}
\item The induced natural morphism $h^i(\underline{\Omega}_X^\times)\to
h^i(\underline{\Omega}_{\Sigma}^\times)$ is an isomorphism for all $i\in\mathbb{Z}$.
\label{item:4}
\item The induced natural morphism $h^i(\underline{\Omega}_X^0)\to
h^i(\underline{\Omega}_{\Sigma}^0)$ is an isomorphism for all $i\neq 0$ and a surjection
with kernel isomorphic to $\scr{I}_{\Sigma\subseteq X}$ for $i=0$.
\label{item:5}
\end{enumerate}
\end{lem}
\begin{subrem}
This statement also applies in the case when $\Sigma=\emptyset$, so it implies
\eqref{lem:db-defect}.
\end{subrem}
\begin{proof}
The equivalence of (\ref{lem:db-defect-rel}.\ref{item:3}) and
(\ref{lem:db-defect-rel}.\ref{item:6}) follows from (\ref{eq:5}), the equivalence
of (\ref{lem:db-defect-rel}.\ref{item:6}) and
(\ref{lem:db-defect-rel}.\ref{item:7}) follows from (\ref{eq:6}), the equivalence
of (\ref{lem:db-defect-rel}.\ref{item:7}) and
(\ref{lem:db-defect-rel}.\ref{item:4}) follows from the definition of
quasi-isomorphism, and the equivalence of (\ref{lem:db-defect-rel}.\ref{item:4})
and (\ref{lem:db-defect-rel}.\ref{item:5}) follows from the definition of the DB
defect $\underline{\Omega}_{X,\Sigma}^\times$ \eqref{def:rel-db-defect} and (\ref{eq:8}).
\end{proof}
Cutting by hyperplanes works the same way as in the absolute case:
\begin{prop}\label{prop:db-cx-of-hyper-rel}
Let $(X,\Sigma)$ be a reduced general pair where $X$ is a quasi-projective variety
and $H\subset X$ a general member of a very ample linear system. Then
$\underline{\Omega}_{H,H\cap\Sigma}^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}\,{\simeq}_{\text{qis}}\, \underline{\Omega}_{X,\Sigma}^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}\otimes_L\scr{O}_H$ and
$\underline{\Omega}_{H,H\cap\Sigma}^\times\,{\simeq}_{\text{qis}}\, \underline{\Omega}_{X,\Sigma}^\times\otimes_L\scr{O}_H$.
\end{prop}
\begin{proof}
This follows directly from \eqref{prop:db-cx-of-hyper}, (\ref{eq:11}), and
\eqref{prop:hyper-plane-cuts}.
\end{proof}
We also have the following adjunction type statement.
\begin{prop
\label{prop:DB-defect-of-a-union}
Let $X=(Y\cup Z)_{\red}$ be a union of closed reduced subschemes with intersection
$W=(Y\cap Z)_{\red}$. Then the DB defects of the pairs $(X,Y)$ and $(Z,W)$ are
quasi-isomorphic. I.e.,
$$\underline{\Omega}^\times_{X,Y}\,{\simeq}_{\text{qis}}\, \underline{\Omega}^\times_{Z,W}.$$
\end{prop}
\begin{proof}
Consider the following diagram of exact triangles,
\begin{equation*}
\xymatrix{%
%
\text{\phantom{mn}}
\underline{\Omega}^\times_{X,Y} \ar[d]^\alpha \ar[r] & \underline{\Omega}^\times_X \ar[d]^\beta \ar[r] &
\underline{\Omega}^\times_Y \ar[d]^\gamma \ar[r]^-{+1} & \\
\text{\phantom{mn}}\underline{\Omega}^\times_{Z,W} \ar[r] & \underline{\Omega}^\times_Z \ar[r] &
\underline{\Omega}^\times_W \ar[r]^-{+1} &, \\
}
\end{equation*}
where $\beta$ and $\gamma$ are the natural restriction morphisms and $\alpha$ is
the morphism induced by $\beta$ and $\gamma$ on the mapping cones. Then by
\cite[2.1]{KK10} there exists an exact triangle
\begin{equation*}
\xymatrix{%
{\sf Q} \ar[r] & \underline{\Omega}^\times_{Y} \oplus \underline{\Omega}^\times_{Z}
\ar[r] & \underline{\Omega}^\times_W \ar[r]^-{+1} & .
}
\end{equation*}
and a map $\sigma:\underline{\Omega}^\times_X\to {\sf Q}$ compatible with the above diagram such that
$\alpha$ is an isomorphism if and only if $\sigma$ is one. On the other hand,
$\sigma$ is indeed an isomorphism by \eqref{prop:DB-defect-for-unions} and so the
statement follows.
\end{proof}
\begin{comment}
Let $X=Y\cup Z$ be a union of closed subschemes with intersection $W=Y\cap Z$.
Suppose that there exists an integer $s\in\mathbb{N}$ such that the natural map
$h^i(\underline{\Omega}^0_Z)\to h^i(\underline{\Omega}^0_W)$ is an isomorphism for $i>s$ and surjective for
$i=s$.
\end{comment}
\section{Cohomology with compact support}\label{sec:cohom-with-comp}
Let $X$ be a complex scheme of finite type and $\iota:\Sigma\hookrightarrow X$ a closed
subscheme. Deligne's Hodge theory applied in this situation gives the following
theorem:
\begin{thm}\cite{MR0498552}%
\label{thm:hodge}
Let $X$ be a complex scheme of finite type, $\iota:\Sigma\hookrightarrow X$ a closed
subscheme and $j:U\colon\!\!\!= X\setminus \Sigma\hookrightarrow X$. Then
\begin{enumerate}
\item The natural composition map $j_{!}\mathbb{C}_{U}\to \scr{I}_{\Sigma\subseteq X} \to
\underline{\Omega}^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}_{X,\Sigma}$ is a quasi-isomorphism, i.e., $\underline{\Omega}^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}_{X,\Sigma}$ is a
resolution of the sheaf $j_{!}\mathbb{C}_{U}$.
\item The natural map $H_{\rm c}^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}(U,\mathbb{C})\to \mathbb{H}^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}(X, \underline{\Omega}^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}_{X,\Sigma})$ is
an isomorphism.
\item If in addition $X$ is proper, then the spectral sequence,
$$
E\uj{_1}^{p,q}= \mathbb{H}^q(X, \underline{\Omega}^p_{X,\Sigma}) \Rightarrow H_{\rm c}^{p+q}(U,\mathbb{C})
$$
degenerates at $E\uj{_1}$ and abuts to the Hodge filtration of Deligne's mixed Hodge
structure. \label{item:9}
\end{enumerate}
\end{thm}
\begin{proof}
Consider an embedded hyperresolution of $\Sigma\subseteq X$:
$$
\xymatrix{%
\text{\phantom{${{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}$}}\Sigma_{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }} \hskip-.5ex\ar[r]^-{\varrho_{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}}
\ar[d]_{\varepsilon_{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}\hskip-.5ex} & X_{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }} \ar[d]^{\varepsilon_{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}}
\hskip-1ex \\
\Sigma \ar[r]_\varrho & X
}
$$
Then by (\ref{defDB}.\ref{item:1}) and by definition $\underline{\Omega}_{X,\Sigma}^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }} \,{\simeq}_{\text{qis}}\,
{\mcR\!}{\varepsilon_{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}}_* \Omega^{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}_{X_{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}, \Sigma_{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}}$. The statements
then follow from \cite[8.1, 8.2, 9.3]{MR0498552}. See also \cite[IV.4]{GNPP88}.
\end{proof}
\begin{cor}\label{cor:surjectivity}
Let $X$ be a proper complex scheme of finite type, $\iota:\Sigma\hookrightarrow X$ a closed
subscheme and $j:U\colon\!\!\!= X\setminus \Sigma\hookrightarrow X$. Then the natural map
$$
H^i(X, \scr{I}_{\Sigma\subseteq X}) \to \mathbb{H}^i(X, \underline{\Omega}_{X,\Sigma}^0)
$$
is surjective for all $i\in \mathbb{N}$.
\end{cor}
\begin{proof}
By (\ref{thm:hodge}.\ref{item:9}) the natural composition map
$$
H^i_{\rm c}(U, \mathbb{C}) \to H^i(X, \scr{I}_{\Sigma\subseteq X}) \to \mathbb{H}^i(X,
\underline{\Omega}_{X,\Sigma}^0)
$$
is surjective. This clearly implies the statement.
\end{proof}
\section{DB pairs in nature}\label{sec:db-pairs}
\begin{prop}
\label{prop:pair-of-DBs-is-DB}
Let $(X,\Sigma)$ be a reduced generalized pair.
If either $X$ or $\Sigma$ is DB, then the other
one is DB if and only if $(X,\Sigma)$ is a DB pair.
\end{prop}
\begin{proof}
Consider the exact triangle (\ref{eq:6})
$$
\xymatrix{%
\underline{\Omega}^\times_{X,\Sigma} \ar[r] & \underline{\Omega}^\times_{X} \ar[r] & \underline{\Omega}^\times_{\Sigma}
\ar[r]^-{+1} & .
}
$$
Clearly, if one of the objects in this triangle is acyclic, then it is equivalent
that the other two are acyclic. Then the statement follows by \eqref{lem:db-defect}
and \eqref{lem:db-defect-rel}.
\end{proof}
As one expects it from a good notion of singularity, smooth points are DB. For pairs,
being smooth is replaced by being snc.
\begin{cor}
\label{cor:snc-is-db}
Let $(X,\Delta)$ be an snc pair. Then it is also a DB pair.
\end{cor}
\begin{proof}
This follows directly from \eqref{prop:pair-of-DBs-is-DB} cf.\
(\ref{defDB}.\ref{item:8}) \cite[3.2]{Steenbrink85}. It also follows from
\eqref{cor:lc-is-DB}.
\end{proof}
\begin{cor}\label{cor:lc-is-DB}
Let $(X,\Delta)$ be a log canonical pair and $\Lambda\subset X$ an effective
integral Weil divisor such that $\supp\Lambda\subseteq \supp\rdown\Delta$. Then
$(X,\Lambda)$ is a DB pair.
\end{cor}
\begin{proof}
By choice $\Lambda$ is a union of non-klt centers of the pair $(X,\Delta)$ and
hence by \cite[Theorem~1.4]{KK10} both $X$ and $\Lambda$ are DB. Then
$(X,\Lambda)$ is a DB pair by \eqref{prop:pair-of-DBs-is-DB}.
\end{proof}
\begin{thm}
\label{thm:wdb-is-db}
Let $(X,\Sigma)$ be a reduced generalized pair. Assume that the natural morphism
$\scr{I}_{\Sigma\subseteq X}\to\underline{\Omega}_{X,\Sigma}^0$ has a left inverse. Then $(X,\Sigma)$
is a DB pair.
\end{thm}
\begin{proof}
We will mimic the proof of \cite[1.5]{Kovacs00c}. The statement is local so we may
assume that $X$ is affine and hence quasi-projective
\begin{lem}\label{lem:local-surjectivity}
Assume that there exists a finite subset $P\subseteq X$ such that $(X\setminus
P,\Sigma\setminus P)$ is a DB pair. Then the induced morphism
$$
H^i_P(X, \scr{I}_{\Sigma\subseteq X}) \to \mathbb{H}^i_P(X, \underline{\Omega}_{X,\Sigma}^0)
$$
is surjective for all $i\in\mathbb{N}$.
\end{lem}
\begin{proof}
Let $\overline X$ be a projective closure of $X$ and let $\overline\Sigma$ be the closure of
$\Sigma$ in $\overline X$. Let $Q=\overline X\setminus X$, $Z=P\overset{_{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\ }}}\cup Q$, and $U=\overline
X\setminus Z=X\setminus P$. Consider the exact triangle of functors,
\begin{equation}
\label{eq:13}
\xymatrix{%
\mathbb{H}^0_Z({\overline X},\underline{\hskip 10pt} ) \ar[r] & \mathbb{H}^0({\overline X},\underline{\hskip 10pt} ) \ar[r] & \mathbb{H}^0(U,\underline{\hskip 10pt} )
\ar[r]^-{+1} &
}
\end{equation}
and apply it to the morphism $\scr{I}_{{\overline \Sigma}\subseteq {\overline X}} \to \underline{\Omega}_{{\overline
X},{\overline \Sigma}}^0$. One obtains a morphism of two long exact sequences:
$$
\hskip-1em\xymatrix{%
\mathbb{H}^{i-1}(U,\scr{I}_{{\overline \Sigma}\subseteq {\overline X}} ) \ar[d]^{\alpha_{i-1}}\ar[r] &
\mathbb{H}^{i}_Z({\overline X},\scr{I}_{{\overline \Sigma}\subseteq {\overline X}} ) \ar[d]^{\beta_i}\ar[r]
& \mathbb{H}^{i}({\overline X},\scr{I}_{{\overline \Sigma}\subseteq {\overline X}} ) \ar[d]^{\gamma_i}\ar[r]
&
\mathbb{H}^{i}(U,\scr{I}_{{\overline \Sigma}\subseteq {\overline X}} ) \ar[d]^{\alpha_i}\\
\mathbb{H}^{i-1}(U,\underline{\Omega}_{{\overline X},{\overline \Sigma}}^0 ) \ar[r] & \mathbb{H}^{i}_Z({\overline X},\underline{\Omega}_{{\overline
X},{\overline \Sigma}}^0 ) \ar[r] & \mathbb{H}^{i}({\overline X},\underline{\Omega}_{{\overline X},{\overline \Sigma}}^0
) \ar[r] & \mathbb{H}^{i}(U,\underline{\Omega}_{{\overline X},{\overline \Sigma}}^0 ) .}
$$
By assumption, $\alpha_i$ is an isomorphism for all $i$. By
\eqref{cor:surjectivity}, $\gamma_i$ is surjective for all $i$. Then by the
5-lemma, $\beta_i$ is also surjective for all $i$.
By construction $P\cap Q=\emptyset$ and hence
\begin{align*}
H^i_Z({\overline X}, \scr{I}_{\overline\Sigma\subseteq {\overline X}}) &\simeq H^i_P({\overline X},
\scr{I}_{\overline\Sigma\subseteq {\overline X}}) \oplus
H^i_Q({\overline X}, \scr{I}_{\overline\Sigma\subseteq {\overline X}}) \\
\mathbb{H}^i_Z({\overline X}, \underline{\Omega}_{{\overline X},\overline\Sigma}^0) &\simeq \mathbb{H}^i_P({\overline X}, \underline{\Omega}_{{\overline
X},\overline\Sigma}^0) \oplus \mathbb{H}^i_Q({\overline X}, \underline{\Omega}_{{\overline X},\overline\Sigma}^0)
\end{align*}
It follows that the natural map (which is also the restriction of $\beta_i$),
$$
H^i_P(\overline X, \scr{I}_{\overline\Sigma\subseteq \overline X}) \to \mathbb{H}^i_P(\overline X, \underline{\Omega}_{\overline
X,\overline\Sigma}^0)
$$
is surjective for all $i$
Now, by excision on local cohomology one has that
$$
H^i_P(\overline X, \scr{I}_{\overline\Sigma\subseteq \overline X}) \simeq H^i_P(X, \scr{I}_{\Sigma\subseteq X})
\quad\text{and}\quad
\mathbb{H}^i_P(\overline X, \underline{\Omega}_{\overline X,\overline\Sigma}^0) \simeq \mathbb{H}^i_P(X, \underline{\Omega}_{X,\Sigma}^0).
$$
and so \eqref{lem:local-surjectivity} follows.
\end{proof}
It is now relatively straightforward to finish the proof of \ref{thm:wdb-is-db}:
By taking repeated hyperplane sections and using \eqref{prop:db-cx-of-hyper-rel} we
may assume that there exists a finite subset $P\subseteq X$ such that $(X\setminus
P,\Sigma\setminus P)$ is a DB pair. Therefore we may apply \eqref{lem:local-surjectivity}.
By assumption, the natural morphism $\scr{I}_{\Sigma\subseteq X}\to\underline{\Omega}_{X,\Sigma}^0$
has a left inverse. This implies that applying any cohomology operator on this map
induces an injective map on cohomology. In particular, this implies that the
natural morphism
$$
H^i_P(X, \scr{I}_{\Sigma\subseteq X}) \to \mathbb{H}^i_P(X, \underline{\Omega}_{X,\Sigma}^0)
$$
is injective for all $i\in\mathbb{N}$. By \eqref{lem:local-surjectivity} they are also
surjective and hence an isomorphism. Thus the DB defect $\underline{\Omega}_{X,\Sigma}^\times$ is
such that all of its local cohomology groups are zero:
$$
\mathbb{H}^i_P(X,\underline{\Omega}_{X,\Sigma}^\times)=0\quad\text{for all $i$.}
$$
On the other hand, by assumption $\underline{\Omega}_{X,\Sigma}^\times$ is supported entirely on
$P$, so $\mathbb{H}^i(X\setminus P, \underline{\Omega}_{X,\Sigma}^\times)=0$ as well. However, then
$\mathbb{H}^i(X, \underline{\Omega}_{X,\Sigma}^\times)=0$ by the long exact sequence induced by
(\ref{eq:13}). Now $\dim P\leq 0$ so the spectral sequence that computes
hypercohomology from the sheaf cohomology of the cohomology of the complex
$\underline{\Omega}_{X,\Sigma}^\times$ degenerates and gives that for any $i\in \mathbb{N}$, $\mathbb{H}^i(X,
\underline{\Omega}_{X,\Sigma}^\times)=H^0(X, h^i(\underline{\Omega}_{X,\Sigma}^\times))$, so, since we assumed
that $X$ is affine, it follows that $h^i(\underline{\Omega}_{X,\Sigma}^\times)=0$ for all $i$.
Therefore $\underline{\Omega}_{X,\Sigma}^\times\,{\simeq}_{\text{qis}}\, 0$ and thus the statement is proven.
\end{proof}
\begin{cor}
\label{corr:rtl-is-DB}
Let $(X,\Delta)$ be a weakly rational pair. Then it is a DB pair.
\end{cor}
\begin{proof}
Let $\phi:(Y,\Delta_Y)\to (X,\Delta)$ be a log resolution \ujjj{such that $\gamma$
admits a left inverse $\gamma^\ell$}. Then by \eqref{prop:rel-funtorial} one has
the commutative diagram:
\begin{equation*}
\xymatrix{%
\scr{O}_X(-\Delta) \ar[d]^\gamma\ar[r] & \underline{\Omega}^0_{X,\Delta}\ar[d]^\alpha \\
{\mcR\!}\phi_*\scr{O}_Y(-\Delta_Y) \ar[r]^-\delta_-\,{\simeq}_{\text{qis}}\, \ar@/^/[u]^{\gamma^\ell}
& {\mcR\!}\phi_*\underline{\Omega}_{Y,\Delta_Y}^0 &
}
\end{equation*}
Recall that as $(Y,\Delta_Y)$ is an snc pair, it is also DB by
\eqref{cor:snc-is-db} and hence $\delta$ is a quasi-isomorphism.
\ujj{By assumption $(Y,\Delta_Y)$ is a weakly rational pair
so $\gamma$ admits a left inverse $\gamma^\ell$.} Then $\gamma^\ell\circ
\delta^{-1}\circ\alpha$ is a left inverse to $\scr{O}_X(-\Delta)\to \underline{\Omega}^0_{X,\Delta}$,
so the statement follows from \eqref{thm:wdb-is-db}.
\end{proof}
\begin{cor}
\label{cor:rtl-is-DB}
A rational pair is a DB pair.
\end{cor}
\begin{proof}
As a rational pair is also a weakly rational pair, this is straighforward from
\eqref{corr:rtl-is-DB}.
\end{proof}
\begin{cor}
\label{cor:dlt-is-DB}
Let $(X,\Delta)$ be a dlt pair and $\Lambda\subset X$ an effective integral Weil
divisor such that $\supp\Lambda\subseteq \supp\rdown\Delta$. Then $(X,\Lambda)$ is
a DB pair.
\end{cor}
\begin{proof}
A dlt pair is also an lc pair, so this follows from \eqref{cor:lc-is-DB}.
\end{proof}
\ujj{
\begin{proof}[Proof \#2]
If $(X,\Delta)$ is a dlt pair, then $(X,\Lambda)$ is a rational pair by
\cite[111]{KollarKovacsRP},
so this also follows from \eqref{cor:rtl-is-DB}.
\end{proof}
}
\begin{comment}
An example of a DB pair where neither $X$ nor $\Sigma$ is DB. STILL COMING
\end{comment}
\section{Vanishing Theorems}\label{sec:vanishing-theorems}
\noindent
The folowing is the main vanishing result of this paper. Note that a weaker version
of it appeared in \cite[13.4]{GKKP10}.
\begin{thm}\label{thm:vanishing}
Let $(X, \Sigma)$ be a DB pair and $\pi : \widetilde X \to X$ a proper birational
morphism with $E := \exc(\pi)$. Let $\widetilde\Sigma = E \cup \pi^{-1}\Sigma$ and
$\Upsilon := \overline{\pi(E) \setminus \Sigma}$, both considered with their induced
reduced subscheme structure. Further let $s\in\mathbb{N}$, $s>0$ such that
$h^i(\underline{\Omega}_{\Upsilon,\Upsilon\cap\Sigma}^\circ)=0$ for $i\geq s$. Then
$$
{\mcR\!}^{i}\pi_* \underline{\Omega}_{\widetilde X,\widetilde\Sigma}^0 = 0 \text{\qquad for all \, $i \geq s$.}
$$
\end{thm}
\begin{proof}
Let $\Gamma = \Sigma \cup \Upsilon$ and consider the exact triangle
(\ref{defDB}.\ref{item:exact-triangle}),
\begin{equation}
\label{eq:2}
\xymatrix{%
\underline{\Omega}_X^0 \ar[r] & \underline{\Omega}_\Gamma^0 \oplus
{\mcR\!}\pi_*\underline{\Omega}_{\widetilde X}^0 \ar[r] & {\mcR\!}\pi_*\underline{\Omega}_{\widetilde \Sigma}^0
\ar^-{+1}[r] & },
\end{equation}
which induces the long exact sequence of sheaves:
\begin{equation*}
\xymatrix{%
\ar[r] & h^i(\underline{\Omega}_X^0)\ar[r]^-{(\alpha^i,\sigma^i)} &
h^i(\underline{\Omega}_\Gamma^0)\oplus
{\mcR\!}^i\pi_*\underline{\Omega}_{\widetilde X}^0 \ar[r
&
{\mcR\!}^i\pi_*\underline{\Omega}_{\widetilde \Sigma}^0 \ar[r] &
h^{i+1}(\underline{\Omega}_X^0)\ar[r] &
}
\end{equation*}
By \eqref{lem:db-defect-rel} the natural morphism $\gamma^i:
h^i(\underline{\Omega}_X^0)\to h^i(\underline{\Omega}_{\Sigma}^0)$ is an isomorphism for $i>0$.
By \eqref{prop:DB-defect-of-a-union} and the assumption
we obtain that $h^i(\underline{\Omega}_{\Gamma,\Sigma}^\circ)=0$ for $i\geq s>0$ and hence the
natural morphism $\beta^i: h^i(\underline{\Omega}_\Gamma^0)\to
h^i(\underline{\Omega}_{\Sigma}^0)$ is an isomorphism
for $i\geq s$.
Using
the fact that $\gamma^i=\beta^i\circ\alpha^i$ we obtain that the morphism
$\alpha^i: h^i(\underline{\Omega}_X^0) \to h^i(\underline{\Omega}_\Gamma^0)$ is an isomorphism
for $i\geq s>0$ and hence the natural restriction map
$$
\varrho^i: {\mcR\!}^i\pi_*\underline{\Omega}^0_{\widetilde X} \to {\mcR\!}^i\pi_*\underline{\Omega}^0_{\widetilde \Sigma}
$$
is an isomorphism for $i\geq s$. This in turn implies that ${\mcR\!}^i\pi_* \underline{\Omega}_{\widetilde
X,\widetilde\Sigma}^0=0$ for $i\geq s$ as desired.
\end{proof}
\noindent
As a corollary, a slight generalization of \cite[13.4]{GKKP10} follows.
\begin{cor}\label{cor:vanishing-1}
Let $(X, \Sigma)$ be a DB pair and $\pi : \widetilde X \to X$ a log resolution of
$(X,\Sigma)$ with $E := \exc(\pi)$. Let $\widetilde\Sigma = E \cup \pi^{-1}\Sigma$ and
$\Upsilon := \overline{\pi(E) \setminus \Sigma}$, both considered with their induced
reduced subscheme structure. Then
$$
{\mcR\!}^{i}\pi_* \scr{I}_{\widetilde\Sigma\subseteq \widetilde X} = 0
\text{\qquad for all \, $i \geq \max \bigl(\dim \Upsilon,1 \bigr)$.}
$$
In particular, if $X$ is normal of dimension $n\geq 2$, then ${\mcR\!}^{n-1}\pi_*
\scr{I}_{\widetilde\Sigma\subseteq \widetilde X} = 0$.
\end{cor}
\begin{proof}
Let $s= \max \bigl(\dim \Upsilon, 1 \bigr)$. Then
$h^i(\underline{\Omega}_{\Upsilon,\Upsilon\cap\Sigma}^\circ)=0$ for $i\geq s$ by
\eqref{prop:top-coh-vanishes-rel}. As the pair $(\widetilde X,\widetilde \Sigma)$ is snc, it is
also DB and hence $\underline{\Omega}^0_{\widetilde X,\widetilde \Sigma}\,{\simeq}_{\text{qis}}\, \scr{I}_{\widetilde\Sigma\subseteq \widetilde X}$.
Therefore the statement follows from \eqref{thm:vanishing}.
\end{proof}
\noindent
We have a stronger result for log canonical pairs \ujjj{and for that we need the
following definition:
\begin{defn}\label{def:szabo-res}
A log resolution of a dlt pair $(Z,\Theta)$, $g:(Y, \Gamma)\to (Z,\Theta)$ is
called a \emph{Szab\'o-resolution}, if there exist $A,B$ effective $\mathbb{Q}$-divisors
on $Y$ without common irreducible components, such that $\supp (A+B)\subset \exc
(g)$, $\rdown{A}=0$, and
$$
K_{Y}+\Gamma \sim_{\mathbb{Q}} g^*(K_Z+\Theta) - A + B.
$$
\end{defn}
\begin{rem}
Every dlt pair admits a Szab\'o-resolution by \cite{MR1322695} (cf.\ \cite[2.44]{KM98}).
\end{rem}
}
\begin{cor}\label{cor:top-vanishing}
Let $(X, \Delta)$ be a \uj{$\mathbb{Q}$-factorial} log canonical pair, $\pi : \widetilde X
\to X$ a log resolution of $(X, \Delta)$,
\ujjj{and}
let $\widetilde \Delta=\bigl(\pi^{-1}_*\rdown \Delta + \excnklt(\pi)\bigr)_{\red}$. Then
\begin{equation*}
{\mcR\!}^{i}\pi_* \, \scr{O}_{\widetilde X}(- \widetilde \Delta) = 0\quad\text{for $i>0$.}\\
\end{equation*}
\end{cor}
\begin{proof}
\ujjj{%
First note that the statement is true if $(X,\Delta)$ is an snc pair and $\pi$ is
the blow up of $X$ along a smooth center. Indeed, if the center is a non-klt
center, then the statement is a direct consequence of the Kawamata-Viehweg
vanishing theorem and if the center is not a non-klt center, then this is a
Szab\'o-resolution and the statement follows as in the proof of
\cite[111]{KollarKovacsRP}. This implies the following:
\begin{sublem}\label{eq:15}
Let $\pi_i:(X_i,\Delta_i)\to (X,\Delta)$ for $i=1,2$ be two log resolutions of
$(X,\Delta)$ and let $\widetilde \Delta_i=\bigl((\pi_i^{-1})_*\rdown \Delta +
\excnklt(\pi_i)\bigr)_{\red}\subset X_i$. Then
\begin{equation*}
{\mcR\!}(\pi_1)_* \, \scr{O}_{X_1}(-\widetilde \Delta_1) \simeq
{\mcR\!}(\pi_2)_* \, \scr{O}_{X_2}(-\widetilde \Delta_2)
\end{equation*}
\end{sublem}
\begin{proof}
By \cite[Theorem 0.3.1(6)]{MR1896232} (cf.\ \cite[Theorem 3.8]{MR2180406}) the
induced birational map between $X_1$ and $X_2$ can be written as a sequence of
blowing ups and blowing downs along smooth centers. Then the statement follows
from the above observation and the definition of the $\widetilde \Delta$'s.
\end{proof}
}
\ujjj{Now we turn to proving the general case.}
Consider a minimal dlt model $\mu: (X^{\rm m},\Delta^{\rm m})\to (X, \Delta)$
\cite[3.1]{KK10}. Let $\Sigma\colon\!\!\!= \rdown\Delta\cup \mu(\exc(\mu))$ considered
with the induced reduced subscheme structure. From the definition of a minimal dlt
model it follows that $\Sigma$ is a union of non-klt centers of $(X,\Delta)$. Then
by \cite[Theorem~1.4]{KK10} both $X$ and $\Sigma$ are DB, and hence $(X,\Sigma)$ is
a DB pair by \eqref{prop:pair-of-DBs-is-DB}.
Since $(X^{\rm m},\Delta^{\rm m})$ is dlt, $(X^{\rm m},\rdown{\Delta^{\rm m}})$ is
a DB pair by \eqref{cor:lc-is-DB}.
Therefore,
$$
\underline{\Omega}_{X^{\rm m},\rdown{\Delta^{\rm m}}}^0\,{\simeq}_{\text{qis}}\, \scr{O}_{X^{\rm m}}(-\rdown{\Delta^{\rm m}}).
$$
By the definition of a minimal dlt model $\rdown{\Delta^{\rm
m}}=\bigl(\pi^{-1}\Sigma\bigr)_{\red}\supseteq \exc(\mu)$ and then it follows
from \eqref{thm:vanishing} that ${\mcR\!}^i\mu_*\scr{O}_{X^{\rm m}}(-\rdown{\Delta^{\rm
m}})=0$ for $i>0$ and hence
\begin{equation}
\label{eq:10}
{\mcR\!}\mu_*\scr{O}_{X^{\rm m}}(-\rdown{\Delta^{\rm m}}) \,{\simeq}_{\text{qis}}\, \scr{O}_{X}(-\rdown\Delta).
\end{equation}
\ujj{%
Next let $\sigma : \widehat X\to X$ be a log resolution of $(X,\Delta)$ that factors
through both $\pi$ and $\mu$. Then one has the following commutative diagram:
$$
\xymatrix{%
\widehat X \ar[r]^\tau \ar@{-->}[d]_\lambda \ar[rd]^\sigma &
\text{\vphantom{$\widehat X$}}{X^{\rm m}\hskip-1.2ex} \ar[d]^\mu \\
\widetilde X \ar[r]_\pi & \text{\vphantom{$\widetilde X$}}X
}
$$
Let $\widehat \Delta\subseteq \widehat X$ denote the strict transform of $\widetilde \Delta$
on $\widehat X$. It follows from the definition of a minimal dlt model that $\widehat
\Delta$ is also the strict transform of $\rdown{\Delta^{\rm m}}$ from $X^{\rm
m}$.
As both $(X^{\rm m}, \Delta^{\rm m})$ and $(\widetilde X,\widetilde \Delta)$ are dlt, $(X^{\rm
m},\rdown{\Delta^{\rm m}})$ and $(\widetilde X,\widetilde \Delta)$ are also rational by
\cite[111]{KollarKovacsRP},
so we have that
\begin{align*}
{\mcR\!}\tau_*\scr{O}_{\widehat X}(-\widehat\Delta) &\,{\simeq}_{\text{qis}}\, \scr{O}_{X^{\rm m}}(-\rdown{\Delta^{\rm
m}}), \text{ and}\\
{\mcR\!}\lambda_*\scr{O}_{\widehat X}(-\widehat\Delta) &\,{\simeq}_{\text{qis}}\, \scr{O}_{\widetilde X}(-\widetilde\Delta)
\end{align*}
Therefore, by (\ref{eq:10}),
\begin{multline*}
{\mcR\!}\pi_*\scr{O}_{\widetilde X}(-\widetilde\Delta) \,{\simeq}_{\text{qis}}\, {\mcR\!}\pi_*{\mcR\!}\lambda_*\scr{O}_{\widehat
X}(-\widehat\Delta) \,{\simeq}_{\text{qis}}\, {\mcR\!}\sigma_*\scr{O}_{\widehat X}(-\widehat\Delta) \,{\simeq}_{\text{qis}}\, \\
{\mcR\!}\mu_*{\mcR\!}\tau_*\scr{O}_{\widehat X}(-\widehat\Delta) \,{\simeq}_{\text{qis}}\, {\mcR\!}\mu_*\scr{O}_{X^{\rm
m}}(-\rdown{\Delta^{\rm m}}) \,{\simeq}_{\text{qis}}\, \scr{O}_{X}(-\rdown\Delta).
\end{multline*}
}%
\ujjj{%
Next let $\tau : \widehat X\to X^{\rm m}$ be the Szab\'o-resolution of $(X^{\rm
m},\Delta^{\rm m})$, $\sigma=\mu\circ\tau$, $\widehat \Delta=\tau^{-1}_*\rdown
{\Delta^{\rm m}}=\bigl(\sigma^{-1}_*\rdown {\Delta} +
\excnklt(\sigma)\bigr)_{\red} $, and $\lambda=\pi^{-1}\circ\sigma$.
Then by \cite[111]{KollarKovacsRP},
we have that
$$
{\mcR\!}\tau_*\scr{O}_{\widehat X}(-\widehat\Delta) \,{\simeq}_{\text{qis}}\, \scr{O}_{X^{\rm m}}(-\rdown{\Delta^{\rm
m}}),
$$
and hence, by (\ref{eq:10}),
\begin{multline*}
{\mcR\!}\sigma_*\scr{O}_{\widehat X}(-\widehat\Delta) \,{\simeq}_{\text{qis}}\, {\mcR\!}\mu_*{\mcR\!}\tau_*\scr{O}_{\widehat
X}(-\widehat\Delta) \,{\simeq}_{\text{qis}}\,\\ {\mcR\!}\mu_*\scr{O}_{X^{\rm m}}(-\rdown{\Delta^{\rm m}})
\,{\simeq}_{\text{qis}}\, \scr{O}_{X}(-\rdown\Delta).
\end{multline*}
The proof is finished by applying (\ref{eq:15}) to $\widehat X$ and $\widetilde X$. }
\end{proof}
\noindent
Finally, observe that \ujjj{\eqref{cor:top-vanishing}} implies that log canonical
singularities are not too far from being rational:
\begin{cor}
Let $X$ be a variety with log canonical singularities and $\pi:\widetilde X\to X$ a
resolution of $X$ with $E_{\lc}\colon\!\!\!= \excnklt(\pi)$. Then
$$
{\mcR\!}^{i}\pi_* \, \scr{O}_{\widetilde X}(- E_{\lc}) = 0\quad\text{for $i>0$.}
$$
\end{cor}
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\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR}
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|
{
"timestamp": "2011-01-14T02:02:19",
"yymm": "1009",
"arxiv_id": "1009.3536",
"language": "en",
"url": "https://arxiv.org/abs/1009.3536"
}
|
\section{Introduction}
The enigmatic Of?p stars are identified by a number of peculiar and outstanding observational properties. The classification was first introduced by Walborn (1972) according to the presence of C~{\sc iii} $\lambda 4650$ emission with a strength comparable to the neighbouring N~{\sc iii} lines. Well-studied Of?p stars are now known to exhibit recurrent, and apparently periodic, spectral variations (in Balmer, He~{\sc i}, C~{\sc iii} and Si~{\sc iii} lines) with periods ranging from days to decades, strong C~{\sc iii} $\lambda 4650$ in emission, narrow P Cygni or emission components in the Balmer lines and He~{\sc i} lines, and UV wind lines weaker than those of typical Of supergiants (see Naz\'e et al. 2010 and references therein).
Only 5 Galactic Of?p stars are known (Walborn et al. 2010): HD 108, HD 148937, HD 191612, NGC 1624-2 and CPD$-28^{\rm o} 2561$. Three of these stars - HD 108, HD 148937 and HD 191612 - have been studied in detail. In recent years, HD 191612 was carefully examined for the presence of magnetic fields (Donati et al. 2006), and was clearly detected. Recent observations, obtained chiefly within the context of the Magnetism in Massive Stars (MiMeS) Project (Martins et al. 2010; Wade et al., in prep) have furthermore detected magnetic fields in HD 108 and HD 148937, thereby confirming the view of Of?p stars as a class of slowly rotating, magnetic massive stars.
\section{HD 191612}
HD 191612 was the first Of?p star in which a magnetic field was detected (Donati et al. 2006). Subsequent MiMeS observations with ESPaDOnS@CFHT (Wade et al., in prep) confirm the existence of the field, and demonstrate the sinusoidal variability of the longitudinal field with the H$\alpha$ and photometric period of 537.6 d. As shown in Fig. 1, the longitudinal field, H$\alpha$ and photometric extrema occur simultaneously when folded according to the 537.6 d period. This implies a clear relationship between the magnetic field and the circumstellar envelope. We interpret these observations in the context of the oblique rotator model, in which the stellar wind couples to the kilogauss dipolar magnetic field, generating a dense, structured magnetosphere, resulting in all observables varying according to the stellar rotation period.
\begin{figure}
\centering
\includegraphics[width=3in,angle=-90]{gregMag2.ps}
\caption{Longitudinal field (top), H$\alpha$ EW (middle) and Hipparcos mag (bottom) of the Of?p star HD 191612, all phased according to the 537.6 d period. From Wade et al., in preparation.}
\label{mag_geo}
\end{figure}
\section{HD 108}
HD 108 was the second Of?p star in which a magnetic field was detected (Martins et al. 2010). Based on long-term photometric and spectroscopic monitoring, HD 108 is suspected to vary on a timescale of ~50-60 y (Naz\'e et al. 2001). The magnetic observations acquired by Martins et al. from 2007-2009 show at most a marginal increase of the longitudinal field during more than 2 years of observation. This supports the proposal that the variation timescale is in fact the stellar rotational period, and that HD 108 is a magnetic oblique rotator that has undergone extreme magnetic braking.
\section{HD 148937}
HD 148937 was recently observed intensely by the MiMeS Collaboration, resulting in the detection of circular polarisation within line profiles indicative of the presence of an organised magnetic field of kilogauss strength (Wade et al., in prep). Although the field is consistently detected in the observations, no variability is observed, in particular according to the 7.03 d spectral period. This result supports the proposal by Naz\'e et al. (2010) that HD 148937 is observed with our line-of-sight near the stellar rotational pole.
|
{
"timestamp": "2010-09-21T02:00:59",
"yymm": "1009",
"arxiv_id": "1009.3564",
"language": "en",
"url": "https://arxiv.org/abs/1009.3564"
}
|
\section{Introduction}
The early evolution of low mass, isolated protostellar cores in the pre-Class 0 stage is now understood as the interplay between gravity, rotation, magnetic fields, and radiative cooling. Strong magnetic fluxes have been observed in molecular clouds \citep[e.g.][]{1999ApJ...520..706C}. Recent simulations have focused on the changes endued by a magnetic field in the collapse, through either a 3D SPHR or AMR approach \citep{1994ApJ...432..720B,2004MNRAS.348L...1M, HW2004b, 2005MNRAS.362..369M,2005MNRAS.362..382M, 2006ApJ...641..949B, 2007MNRAS.377...77P, 2008A&A...477....9H, 2009ApJ...706L..46D}. It has been shown that magnetic fields slow the collapse timescale and brake the rotation of the initial core and of massive disc-like structures that subsequently form. Magnetic fields have also been shown to suppress fragmentation and the formation of bars and spiral waves. Furthermore, they facilitate the launching of molecular outflows.
Several theoretical problems have arisen from these simulations. First, from axisymmetric simulations outside of 6.7 AU, it is argued that Keplerian discs cannot form in an ideal MHD collapse \citep{2008ApJ...681.1356M}, except in the limit of very weak magnetic flux or high magnetic diffusivity whether numerical \citep{2010ApJ...716.1541K} or through strong ambipolar diffusion \citep{2009ApJ...698..922M}. Secondly, magnetic tension and pressure seem to efficiently suppress fragmentation in the early stages of protostellar collapse so that it is difficult to form multi-star systems.
With the ability to extend the magnetized collapse further, we can begin to examine questions such as how the Core Mass Function (CMF) is related to the Initial Mass Function of stars \citep[e.g.][]{2000ApJ...545..364M}, the nature of the molecular outflow on larger scales and whether fragmentation is suppressed even in later stages.
We present our results of the early collapse using 3D ideal and non-ideal magnetohydrodynamic simulations, and the first results of the later stages of the magnetized collapse and outflow. We are able to evolve the simulation an additional $10^4-10^5$ yr due to the implementation of the sink particle in the FLASH AMR code \citep{2010ApJ...713..269F}.
\section{Numerical Methods and Initial Conditions}
We model our a stellar core as in previous papers \citep[e.g.][]{2009ApJ...706L..46D, 2004MNRAS.355..248B}, by embedding a slightly over-critical 1 $M_\odot$ Bonnor-Ebert sphere \citep{1956MNRAS.116..351B,1955ZA.....37..217E} in a low density environment. We add to this density distribution a 10\% over-density to ensure collapse and a 10\% $m=2$ perturbation to break symmetry \citep[see e.g.][]{2009ApJ...706L..46D,2004MNRAS.355..248B}. The background is an isothermal, low density environment in pressure equilibrium with the sphere (the density is set by choosing a background that is 10 times warmer than the sphere). The box is roughly 10 times the size of the BE radius (0.81 pc in these models). We add to the sphere uniform rotation such that the ratio of rotational and gravitational energy is moderate ($\beta_\mathrm{rot}$=0.046, similar to the value used in \citet{2008A&A...477....9H}) and a constant $\beta_\mathrm{plasma} = 2c_s^2/v_\mathrm{A}^2 = 46.01$. In this model, one can relate the mass to flux ratio $\mu/\mu_0$, where $\mu_0$ is the critical mass to flux of $\mu_0 = (2\pi G^{1/2})^{-1}$, by $\mu/\mu_0 \simeq 0.74 c_s / v_\mathrm{A}$ = 3.5.
We use sink radii of 12.7 AU for longer runs, and radii of 3.2 AU to test the effect of sink particle size on the result. The accretion radius of a sink corresponds to 2.5 cells at the highest refinement level, and the critical gas density (beyond which gas can be accreted into particles) corresponds to the Jeans' density at the core temperature (20 K) of these cells. This gives $\rho_\mathrm{acc} = 3.69\times 10^{-12}~\mathrm{g~cm^{-3}}$ and $\rho_\mathrm{acc} = 5.91\times 10^{-11}~\mathrm{g~cm^{-3}}$ for 12.7 and 3.2 AU sinks respectively. Each Jeans' length is refined by at least 8 cells and de-refined for more than 32 cells. Our customized version of the FLASH AMR code is described in previous work \citep{2000ApJS..131..273F, 2006MNRAS.373.1091B, 2008MNRAS.391.1659D,2010ApJ...713..269F}.
\section{The Early Collapse Phase}
The advantage of \emph{not} using a sink particle is that we properly resolve the gas in the collapse. The disadvantage is that we are limited to pre-Class 0 times ($\approx 10^5$ yr). We compare the following early collapse models: i) ambipolar diffusion, ii) ideal MHD and iii) hydrodynamics (e.g.~$\bm{B}=0$).
The rotational properties of these early structures are shown in Figure 1a. We compare surface density averaged values of $v_\phi/v_r$ and $v_\phi/v_\mathrm{Kepler}$ to measure the degree of rotation in these early collapsed structures. The comparison is done at the maximum common central surface density ($\Sigma_z$). This is limited primarily by the ambipolar diffusion model which has the most constrained timestep. More evolved versions of ideal MHD and hydro models are included as thin lines in the graph, hinting at the evolution of the possible structures. The inward velocity has not halted (e.g. the flow is sub-Keplerian) in these early structures, however we are indeed seeing flattened rotationally dominated accretion discs (with associated outflows) being generated with sizes of 6-7 AU, facilitated by the suppression of gravitational instabilities. Bars produced in the hydrodynamic model have prevented a similar structure from forming by efficiently redistributing angular momentum.
To demonstrate the suppression of fragmentation further, we ran the simulation with 10 times more rotational energy in order to study early fragmentation ($\beta_\mathrm{rot}$ = 0.74, corresponding to high values seen in simulations of core formation in a turbulent medium \citep{2007MNRAS.382...73T}). Indeed our hydrodynamic collapse produces a wide binary with a separation of about 1000 AU. The ideal MHD model suppresses the fragmentation, as documented in previous studies \citep{2007MNRAS.377...77P, 2008A&A...477...25H}, resulting in a large bar and a central condensed structure. Meanwhile, the ambipolar diffusion model produces an intermediate result, a bar that has fragmented to form a binary. It perhaps through ambipolar diffusion that any ``fragmentation crisis'' can be solved. These results are discussed further in \citep{2009ApJ...706L..46D}.
\begin{figure}
\begin{tabular}{cc}
\includegraphics[width=6cm]{./figs/duffin_1a.eps} &
\includegraphics[width=6cm]{./figs/duffin_1b.ps} \\
Figure 1a & Figure 1b \\
\includegraphics[width=6cm]{./figs/duffin_1c.eps} &
\includegraphics[width=6cm]{./figs/duffin_1d.eps} \\
Figure 1c & Figure 1d
\end{tabular}
\caption{\label{fig:1}The early to late evolution of the protostellar core with ideal MHD. In a). \citep[adapted from][]{2009ApJ...706L..46D}, the rotation properties $v_\phi/v_r$ and $v_\phi/v_\mathrm{Kepler}$ of the early collapse (no sink particle). In b), contours of density (purple or medium grey, $3.3\times10^{-17}~\mathrm{g~cm^{-3}}$ and black, $1.33\times10^{-15}~\mathrm{g~cm^{-3}}$ at 2000 and 100 AU radii respectively) and $v_z$ (yellow or light grey, $1 km/s$) at the end of the simulation (with 3.2 AU sinks). In c), rotational properties of disk at the end of the simulation (with 3.2 AU sink). In d), mass evolution of different sink particle models, from left to right: 12 AU sink particle and a polytropic equation of state, 12 AU sink particle and 3.2 AU sink particle both with molecular cooling.
}
\end{figure}
\section{The Later Evolution: Sink Particles}
Using sink particles and ideal MHD, we were able to take the next step and evolve the collapse to much later times. The result is shown in Figure 1b, wherein evolution over an additional 30 kyr is achieved. The sink particle mass evolution is shown in Figure 1d for 12 AU sink particles (with molecular cooling or polytropic equation of state similar to that of \citet{2010arXiv1001.1404M}) and a smaller 3.2 AU sink particle with molecular cooling \citep{2006MNRAS.373.1091B}. By the end of the smaller sink collapse, the sink particle is $0.2 M_\odot$ and the outflow has grown to $10^4$ AU above the mid-plane of the disk. We are able to run the simulations of larger sinks further, and indeed these particles end up with nearly all of the core mass (80-90\%). This is much higher than estimates of theoretical models seeking to relate the Core Mass Function (CMF) to the Initial Mass Function (IMF) \citep[e.g.][]{2000ApJ...545..364M}. These results suggest that not much mass has in fact been cleared out by the outflow. This may be due to the fact that most mass must settle into the accretion disk before the outflow is launched, leaving less mass available to be cleared.
The accretion disk is represented by the ratio of $v_\phi/v_r$ in Figure 1c, showing a 2000 AU accretion disk, which appears to be dissipating by the end of the simulation (as seen in Figure 1d). There are two types of outflows, one lower speed ($\lesssim 1$ km/s) and a central, higher speed centrifugally driven wind.
The inner disk and central component of the outflow are warped and precessing, as shown in Figure 1b. The appearance of precessing warped discs and their outflow is extremely interesting and is related to the back reaction of the MHD outflow on the disk \citep[e.g.][]{2003ApJ...591L.119L}. Many of the qualitative properties of molecular outflows, including jet precession, clumpiness and an onion layered velocity structure \citep{2007prpl.conf..277P} are seen in our simulation, and occur naturally as a consequence of solving the gravito-magnetohydrodynamic equations.
\section{Summary}
In the early stages, accretion discs are small ($< 10$ AU), massive, flattened, rotationally dominated, and are held together by the magnetic field. As the collapse continues over an additional $10^5$ yr, the accretion disc and its associated outflow grow in size. The outflow torques and warps the disk leading to disc and jet outflow and precession respectively. Most material will be accreted by the star raising issues concerning the extent of protostellar feedback on stellar masses.
\bibliographystyle{apj}
|
{
"timestamp": "2010-09-23T02:02:49",
"yymm": "1009",
"arxiv_id": "1009.4445",
"language": "en",
"url": "https://arxiv.org/abs/1009.4445"
}
|
\section{Introduction}
Using an xuv attosecond pulse to photoemit electrons from gaseous or
solid targets into the electric field of a synchronized delayed
femtosecond (fs) infrared (ir) laser pulse provides a powerful tool
for investigating ultrafast electron dynamics by recording
ir-laser-streaked xuv PE spectra~\cite{Krausz09}. For strong laser
fields and sufficiently fast PEs, streaked PE spectra are
conveniently described in SFA~\cite{Lewenstein94}, i.e., by ignoring
the interaction of the residual ion with the released PE. In this
case, subject only to the ir laser electric field, the propagation
of PEs can be described in terms of analytically known ``Volkov"
states~\cite{Zhang09}. This leads to a delay-dependent energy shift
$\delta E_{COE}^{SFA}(\tau)=-k A_L(-\tau)$ of the center-of-energy
(COE) $E_{COE}=\omega_X-|\varepsilon_B|$ in the PE spectrum, where
$\omega_X$ is the xuv-pulse central frequency, $\varepsilon_B$ the
binding energy in the initial bound state, $k$ the PE asymptotic
momentum, and $A_{L}$ the vector potential of the ir pulse. We use
atomic units except where stated otherwise and define the delay
$\tau$ between the centers of the xuv and ir pulse as positive if
the xuv pulse precedes the ir pulse.
The interpretation of sub-fs temporal shifts in streaked PE spectra
is a matter of current debate. For example, the recent
measurement~\cite{Cavalieri07} of a relative delay of $\approx 110
\pm 70$~as between the ir-streaked xuv photoemission from localized
4f core levels and delocalized conduction-band (CB) states of a
W(110) surface was understood in the original
ref.~\cite{Cavalieri07} and a subsequent theoretical
work~\cite{Kazansky09} as the difference $\delta t_{CB-4f}
=t_{CB}-t_{4f}$ between the arrival times of CB and 4f core PEs at
the surface. This interpretation is based on the assumption that the
ir pulse does not penetrate the surface, such that CB and 4f
electrons that are released at the same time $-\tau$ by the
absorption of an xuv photon get streaked only upon arrival at the
surface, producing the COE shifts $E_{COE}=-kA_L(-\tau+t_{CB})$ and
$-kA_L(-\tau+t_{4f})$, respectively, in the PE spectra. According to
this two-step explanation (photorelease followed by streaking), the
total emission probability $P_{tot} =\int dE P(E,\tau)$ from a given
initial state would not depend on $\tau$.
In contrast, our analysis of experimental streaked photoemission
data for a tungsten~\cite{Cavalieri07} and rhenium
surface~\cite{private} indicates that $P_{CB}(\tau)$ oscillates with
$A_L(\tau)$, with an amplitude of $\approx 10$\% of the average
value. Furthermore, the continuity of the wavefunction and its
derivative at the surface, implies that an intense fs ir pulse
affects the PEs inside the solid, even if the ir electric field were
prohibited from penetrating the surface~\cite{Varro98}. We have
shown that this observed temporal shift can be reproduced within the
SFA and interpreted it as an interference effect in the emission
from different lattice sites~\cite{Zhang09}, observing that the SFA
cannot account for relative temporal shifts in the emission from
different levels of isolated atoms. A classical transport simulation
including the effect of (in)elastic collisions of released PEs with
tungsten cores on the propagation of PE inside the solid leads to
$\delta t_{CB-4f}=33$~as~\cite{Lemell09}. Thus, different
models~\cite{Cavalieri07,Kazansky09,Zhang09,Lemell09,Baggesen09}
strongly deviate with regard to the assumed attenuation of the
streaking ir electric field $E_{L}(t)$ inside the solid, ranging
from no penetration into the surface~\cite{Kazansky09} to
penetration depths of 30~\cite{Zhang09} and 85 layer
spacings~\cite{Lemell09} or larger than the electron mean free
path~\cite{Baggesen09}. The detailed modeling of the (relative)
delay in the photoemission from metal surfaces is further
complicated by the complex band structure and the ensuing difficulty
in assigning a group velocity to the motion of PE wave packets
inside the dispersive conduction band~\cite{Cavalieri07}, surface
charge accumulation, and the general concern that static
band-structure calculations and the assumption of an instantaneous
plasmon response (i.e., static image charge interactions) are
invalid at the as time scale. These shifts may be of particular
importance in the interpretation of streaking spectra for complex
targets, such as metals, and emphasize the need for more detailed
studies of streaked photoemission spectra.
It is of fundamental importance to first understand all
contributions to this temporal shift for simple systems. In this
work, we focus on the effect of simultaneous ir laser pulse and
Coulomb interactions on streaked photoemission spectra from the
prototypical ground state of a one-dimensional hydrogen atom. This
Coulomb-laser coupling effect was first investigated by Kroll and
Watson~\cite{Kroll73} in their study of laser-assisted atomic
scattering. It also affects the spectra of high harmonic generation,
multiphoton ionization, and laser-assisted xuv photoionization. For
example, the Coulomb interaction causes the xuv streaked PE spectra
to be right-left asymmetric~\cite{Smirnova08}. In RABITT
measurements (Reconstruction of Attosecond Beating by Interference
of Two photon Transition), these simultaneous ir laser pulse and
Coulomb interactions induce a so-called atomic phase which shifts
sideband intensities as a function of the delay between the xuv
pulse train and the ir
pulse~\cite{Veniard96,Toma02,Mauritsson05,Varju05}.
In this work, we demonstrate, numerically and analytically, how the
{\em coupling} of the ir laser pulse and the final-state Coulomb
interaction of the PE with the residual ion gives rise to a
significant temporal shift $\delta \tau$ in the COE of streaked PE
spectra with respect to those approximated in SFA. As we will show,
inclusion of this Coulomb-laser (CL) coupling alters both, the
amplitude and phase of the COEs in streaked PE spectra, leading to a
COE shift $\delta E_{COE}^{CL}(\tau)=-K A_L(\tau-\delta\tau)$ with
an oscillation amplitude $K>k$. Thus, $\delta\tau $ and the
streaking amplitude ratio $K/k$ i) help to reveal details of the PE
dynamics including the combined interaction of Coulomb and laser
forces and ii) converge to their SFA limits, $0$ and $1$,
respectively, at sufficiently large PE energies.
\begin{figure}[t]
\begin{center}
\includegraphics[width=1.0\columnwidth,keepaspectratio=true,
draft=false]{Fig1.eps} \vspace{-6mm} \caption{(Color online)
Streaked photoemission from 1D model hydrogen atoms. TDSE
calculations for xuv pulses with (a) $\hbar \omega_X=90$ and (b)
$25$~eV. (c) Corresponding centers-of-energy $\delta E_{COE}(\tau)$
for $\hbar \omega_X=90$ (solid line) and $25$~eV (dashed line).
To facilitate the identification of the relative temporal shifts
$\delta\tau$, $\delta E_{COE}(\tau, \hbar \omega_X=90$~eV) is
normalized to the $\hbar \omega_X=25$~eV result. (d) $\delta\tau$
and (e) oscillation amplitude relative to the SFA for TDSE (full
line) and eikonal approximation (dashed line) calculations.
\label{fig:coe_exuv} } \vspace{-6mm}
\end{center}
\end{figure}
We numerically solve the TDSE, including the electron-proton
interaction, and then compare our results for $P(E,\tau)$ with SFA
and eikonal approximation (EA) calculations for a large range of PE
kinetic energies. Strongly dependent on $\omega_X$, we find temporal
shifts $\delta\tau$ of more than 50~as. Fig.s~\ref{fig:coe_exuv} (a)
and (b) show the TDSE ir-streaked PE spectra $P(E,\tau)$ for
gaussian xuv pulses of length $\tau_{X}=300$~as with
$\hbar\omega_X=90$ and $25$~eV, respectively. The ir pulse is also
assumed to be a gaussian and has a peak intensity of
$I_{L}=2\times10^{12}$~W/cm$^2$, a carrier frequency
$\omega_{L}=1.6$~eV/$\hbar$, and a pulse length $\tau_L=5$~fs. These
spectra are shifted by $\delta\tau=60$~as, which becomes apparent in
the corresponding COE shifts $\delta E_{COE}(\tau) $ in
Fig.~\ref{fig:coe_exuv} (c). The solid curves in
Fig.s~\ref{fig:coe_exuv} (d) and (e) show $\delta \tau$ and the
ratio $K/k$ of streaking-oscillation amplitudes for a large range of
$\omega_X$. Within an EA
approach~\cite{Joachain83,Gersten75,Smirnova08}, we can trace
(details will be given further below) this $\omega_X$-dependent
temporal shift and the oscillation amplitude enhancement to the CL
coupling in the PE final state (dashed curves in
Fig.s~\ref{fig:coe_exuv} (d) and (e)). $\delta\tau$ and $K/k$, for
the TDSE and EA calculations, converge at large $\omega_X$ to their
respective SFA limits $\delta\tau=0$ and $K/k=1$ due to the
diminishing influence of the residual ion's Coulomb force at
increasing PE energies. Thus, $\delta\tau$ and $K/k-1$, are measures
for the combined action of the Coulomb and laser force on the PE
relative to the action of the ir laser force alone. Since $\delta
\tau<0$, the attractive Coulomb force does not delay the PE
emission, as one might intuitively expect. We also note that $K>k$
reveals a Coulomb-enhancement effects that is reminiscent of the
Coulomb potential's infinite range leading to well-understood
``Coulomb-Cusps" in energy-differential collision-induced PE
spectra~\cite{Thumm92}.
This article is organized as follows. In Sec. \ref{sec:tdse}, we
present numerical results based on the TDSE. In Sec.
\ref{sec:eikonal}, we adopt an EA to take into account the
simultaneous ir and Coulomb interactions of the PEs, and compare our
EA and TDSE results. In Sec. \ref{sec:polarization}, we examine the
effect of the polarization of the initial state on the streaked
spectrum. We conclude in Sec. V. In the appendix, we show that,
within the eikonal approximation, the obtained atomic phase in
RABITT is identical to the relative temporal shift in the streaked
PE spectrum.
\section{Time-Dependent Schr\"{o}dinger Equation for Streaking}
\label{sec:tdse}
The exact wavefunction of the one-dimensional model atom interacting
with the ir and xuv pulse is determined by the TDSE (in the length
gauge)
\begin{align}
\label{eq:tdse} i\frac{\partial}{\partial
t}\Psi(x,t)&=\left[-\frac{1}{2}\frac{d^2}{dx^2}+U(x)+V(x,t)\right]
\Psi(x,t),
\end{align}
where $U(x)$ is the Coulomb potential,
$V(x,t)=x\left[E_{L}(t)+E_X(t)\right]$ the interaction with the ir
and xuv pulse, and $E_{L(X)}$ the electric field of the ir (xuv)
pulse. Assuming single-photon ionization in a sufficiently weak xuv
pulse, and after splitting the exact wave function for the atom in
the combined xuv and ir electric fields according to
$\Psi(x,t)=\psi_g(x,t)+\delta\psi(x,t)$, (\ref{eq:tdse}) can be
replaced by two coupled equations~\cite{Kazansky07}
\begin{align}
\label{eq:init} i\frac{\partial}{\partial
t}\psi_g(x,t)&=\left[-\frac{1}{2}\frac{d^2}{dx^2}+U(x)+xE_{L}(t)\right]
\psi_g(x,t),
\displaybreak[0]\\
\label{eq:ex} i\frac{\partial}{\partial
t}\delta\psi(x,t)&=\left[-\frac12\frac{d^2}{dx^2}+U(x)+xE_{L}(t)\right]
\delta\psi(x,t)
\displaybreak[0]\nonumber\\
&\ \ \ +xE_X(t+\tau)\psi_g(x,t).
\end{align}
Equation (\ref{eq:init}) determines the evolution (polarization) of
the initial state in the ir field and (\ref{eq:ex}) the generation
of PE wave packets by the xuv pulse and their evolution in the ir
field. The electric fields $E_{L(X)}=-\partial A_{L(X)}(t)/\partial
t$ of the ir (xuv) pulses are derived from the vector potentials
$A_{L(X)}(t)=A_{L(X),0}\cos(\omega_{L(X)}t)e^{-2\log2
t^2/\tau^2_{L(X)}}$. Since $E_{L(X)}(t\rightarrow\pm\infty)=0$,
equations (\ref{eq:init}) and (\ref{eq:ex}) are subject to the
initial conditions
$\psi_g(x,t\rightarrow-\infty)=\psi(x)e^{-i\varepsilon_Bt}$ and
$\delta\psi(x,t\rightarrow-\infty)=0$. The ground-state initial wave
function $\psi(x)$ and energy $\varepsilon_B$ are obtained from
\begin{align}
\label{eq:gs}
\varepsilon_B\psi(x)&=\left[-\frac{1}{2}\frac{d^2}{dx^2}+U(x)\right]\psi(x).
\end{align}
We solve (\ref{eq:init})-(\ref{eq:gs}) numerically by wave-packet
propagation for times $|t|\le 2.5\tau_L$ with a step size $\Delta
t=0.2$ on a spatial grid with $|x| \le 2000$ and spacing $\Delta x
=0.25$. Assuming free-electron dispersion, $E=\frac12k^2$, we
calculate the ir-assisted xuv photoemission probability
\begin{align}
\label{eq:probtau}
P(E,\tau)=\left|\delta\tilde{\psi}(k,\tau,t\rightarrow\infty)\right|^2
\end{align}
and the corresponding COE
\begin{align}
E_{COE}(\tau)=\frac12\int dk\left|k \,
\delta\tilde{\psi}(k,\tau,t\rightarrow\infty)\right|^2/P_{tot}(\tau),
\end{align}
where $\delta\tilde{\psi}(k,\tau,t)$ is the Fourier transform of
$\delta\psi(x,t)$, and the total emission probability is
\begin{align}
P_{tot}(\tau)=\int dk
\left|\delta\tilde{\psi}(k,\tau,t\rightarrow\infty)\right|^2.
\end{align}
We model the target atom based on the soft-core Coulomb potential
\begin{align}
U(x)=-\frac{1}{\sqrt{x^2+a^2}},
\end{align}
and adjust the parameter $a=\sqrt{2}$ to the ground state binding
energy $\varepsilon_B=-13.6$~eV of the hydrogen atom. We refer to
the exact solution of (\ref{eq:init})-(\ref{eq:gs}) as ``TDSE
result" and retrieve the SFA results by ignoring $U(x)$ in
(\ref{eq:ex}). The comparison of TDSE and SFA results is shown in
Fig.s~\ref{fig:coe_exuv}(a)-(c). By dropping the laser interaction
$xE_L(t)$ in (\ref{eq:init}), we verified numerically that for the
given parameters the polarization of this initial state by the ir
pulse can be neglected~\cite{Kazansky07,Baggesen09}. The initial
state polarization effect on the streaked xuv PE spectrum is further
discussed in Sec.~\ref{sec:polarization}.
\section{Eikonal Approximation}
\label{sec:eikonal}
In order to trace the influence of the combined action of Coulomb
potential and ir pulse on the PE, we write the PE wave function as
\begin{align}
\label{eq:wave} \psi_k(x,t)=a_k(x,t)
e^{i[k+A_L(t)]x-ik^2t/2+iS_k(x,t)}
\end{align}
with a local phase $S_k(x,t)$. The real amplitude $a_k$ is not
important for the present investigation. In SFA, the phase
$S_k(x,t)$ is given by the Volkov phase
\begin{align}S_k^{SFA}(t)=k\int_t^{\infty} dt'
A_L(t')
\end{align}
and independent of $x$. In EA, and {\em without} the ir field, the
phase accumulated by the PE during its propagation in $U(x)$ from
the location $x$ at time $t$ to the electron detector is calculated
along the free-electron classical trajectory
$x^\prime(t',t,x)=x+k(t'-t)$~\cite{Joachain83}
\begin{align}
\label{eq:EA} S^{C}_k(x)=\int_{t}^{\infty}dt'
U[x'(t',t,x)]=\frac{1}{k}\int_x^{\infty}dx'U(x').
\end{align}
In the presence of the ir field, the free-electron classical
trajectory is modified by a laser-induced drift
\begin{align}
\Delta x(t,t')&=\int_t^{t'}dt'' A_L(t'')
\end{align}
to become
\begin{align}
x_L(t',t,x)=x^\prime(t',t,x)+\Delta x(t,t').
\end{align}
Replacing $x'$ with $x_L$ in (\ref{eq:EA}), we obtain the CL
phase~\cite{Gersten75,Smirnova08}
\begin{align}
S^{CL}_k(x,t)=\int_{t}^{\infty}dt' \, U[x_{L}(t',t,x)],
\end{align}
and an eikonal approximation to the local phase in (\ref{eq:wave})
\begin{align}
\label{eq:EAA} S^{EA}_k(x,t)=S_k^{SFA}(t)+ S^{CL}_k(x,t).
\end{align}
In typical streaking experiments and for this study, the ir
intensity ($\sim 10^{12}$ W/cm$^2$) is low enough for $\Delta
x(t,t')$ being a small deviation from $x^\prime(t',t,x)$. We thus
expand $S^{CL}_k$ about $x^\prime(t',t,x)$ and obtain to first
order in $\Delta x(t,t')$
\begin{align}
\label{eq:Gk} S^{CL}_k(x,t)&=S^{C}_k(x)-\int_{t}^{\infty}dt'
F[x^\prime(t',t,x)]\Delta x(t,t'),
\end{align}
with the Coulomb force
\begin{align}
F[x^\prime(t',t,x)]=-\frac{\partial
U[x^\prime(t',t,x)]}{\partial x^\prime}.
\end{align}
The first term in (\ref{eq:Gk}) is the laser-free eikonal Coulomb
phase. This term is independent of time as we explicitly indicate in
(\ref{eq:EA}). As we should show below this phase does not induce
any temporal shift in the streaked xuv spectrum, but changes the
transition probability. The second term in (\ref{eq:Gk}) includes
Coulomb scattering of the PE while it absorbs or releases ir
photons~\cite{Gersten75}. It is proportional to the ir vector
potential and causes a temporal shift in the streaked spectrum.
The numerical results for (\ref{eq:Gk}) in Fig.~\ref{fig:eikonal}
(a) resolve the spatial contributions to the CL coupling phase
$S^{CL}_k(x,t)-S^{C}_k(x)$. Keeping in mind that the EA is designed
for short PE de-Broglie wavelengths~\cite{Joachain83}, we confirmed
by comparison with full TDSE results (not shown) that
$S^{CL}_k(x,t)$ remains appropriate down to $k=1$, i.e., a PE energy
of $\approx 14$~eV~\cite{Smirnova08}. This supports the validity of
the EA for the range of PE kinetic energies in
Fig.s~\ref{fig:coe_exuv} and \ref{fig:eikonal}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=1.0\columnwidth,keepaspectratio=true,
draft=false]{Fig2.eps} \vspace{-6mm} \caption{(Color online) (a)
Time evolution of the CL coupling phase (see text).
Contribution to the PE streaking $\delta E^{EA}_{COE}(\tau)$ at
$x=0$ due to final state CL coupling at (b) $\hbar\omega_X=90$ and
(c) 25~eV. \label{fig:eikonal} }\vspace{-6mm}
\end{center}
\end{figure}
The transition amplitude for xuv photoemission from the initial
state $\psi_i$ to the final state $\psi_k$,
\begin{align}
\label{eq:tmatrix} T_k(\tau)=-i\int\!dt \,
\langle\psi^*_k(t)|xE_{X}(t+\tau)|\psi_i(t)\rangle,
\end{align}
provides the PE probability $P(E=k^2/2,\tau)=|T_k(\tau)|^2$ as an
alternative to (\ref{eq:probtau}). Neglecting the laser distortion
of the initial state (using $\psi_i(x,t) \approx
\psi_i(x)e^{-i\varepsilon_Bt}$) and employing the EA-approximated PE
wave function for $\psi_k(x,t)$, we obtain
\begin{align}
T^{EA}_k(\tau)=&-i\int\!dt\,dx \, a(x,t)e^{-i[k+A_L(t)]x}\, x\psi(x)
\nonumber\\
&\times E_X(t+\tau)e^{-iS^{EA}_k(x,t)}e^{-i(\varepsilon_B-k^2/2)t}.
\label{eq:T}
\end{align}
The COE of the spectrum for a free PE would be
$E_{COE}=k^2/2=\omega_X-|\varepsilon_B|$. The (local) energy shift
caused by the ir field and CL coupling in EA is given by
\begin{align}
\delta E^{EA}_{COE}(x,t)=\partial S^{EA}_k(x,t)/\partial t,
\end{align}
and does not depend on the time-independent laser-free eikonal phase
$S^C_k(x)$ in (\ref{eq:Gk}). For sub-fs xuv pulses, contributions to
the time integral (\ref{eq:T}) mainly arise near the center of the
xuv pulse at $t=-\tau$. Approximating $\delta
E^{EA}_{COE}(x,\tau)\approx\partial S(x,t=-\tau)/\partial t$, we
obtain
\begin{align}
\label{eq:EA_Ecoe} \delta E^{EA}_{COE}(x,\tau)=- k A_L(\tau) +
E_{COE}^{CL,1}(x,\tau) +E_{COE}^{CL,2}(x,\tau),
\end{align}
where
\begin{align} \label{eq:Ec1}
E_{COE}^{CL,1}(x,\tau)&=\frac{U(x)}{k}A_{L}(\tau),\\
\label{eq:Ec2}
E_{COE}^{CL,2}(x,\tau)&=-\frac{1}{k}\int_{x}^{\infty}dx'F(x')A_{L}
\left(\frac{x'-x}{k}-\tau\right)
\end{align}
are the two contributions to the CL shift,
$E_{COE}^{CL}=E_{COE}^{CL,1}+E_{COE}^{CL,1}$, shown in
Fig.s~\ref{fig:eikonal} (b) and (c) at $x=0$ for $\hbar \omega_X=90$
and $25$~eV.
According to (\ref{eq:Ec1}) and (\ref{eq:Ec2}),
$E_{COE}^{CL}(x,\tau)$ is proportional to $1/k$, while $\delta
E_{COE}^{SFA}(\tau)$ is proportional to $k$. Therefore, the CL
coupling effect decreases for increasing PE kinitic energies. As
shown in Fig.s~\ref{fig:eikonal} (b) and (c), the cancelation
between $E_{COE}^{CL,1}$ and $E_{COE}^{CL,2}$ becomes stronger and
further reduces the CL coupling with increasing $k$. Note that
$E_{COE}^{CL,1}(x,\tau)$ mainly increases the oscillation amplitude
of the COE in SFA, while $E_{COE}^{CL,2}(x,\tau)$ changes the
oscillation amplitude {\em and} induces a phase shift. We can thus
introduce a local temporal shift $\delta\tau(x)$ (relative to the
SFA phase) and a local oscillation amplitude $K(x)$ by rewriting
(\ref{eq:EA_Ecoe}) as
\begin{align}
\delta E_{COE}^{EA}(x,\tau)= K(x)A_{L}[\tau-\delta\tau(x)].
\end{align}
Fig.s \ref{fig:eikonal_exuv25} (a) and (b) show $\delta\tau(x)$ and
$K(x)$ as functions of $x$ for different $\omega_X$. $\delta\tau(x)$
can be positive or negative. The actual shift $\delta\tau$ in the
streaking spectrum is obtained by spatial integration according to
(\ref{eq:T}). However, as shown in Fig.~\ref{fig:eikonal_exuv25} (c)
and (d), $\delta E_{COE}^{EA}(x=0,\tau)$ agrees well with the TDSE
result, since the initial wave function $\psi(x)$ is localized at
$x=0$. Similarly, we find that the full TDSE results for
$\delta\tau$ (solid line in Fig.~\ref{fig:coe_exuv} (d)) and $K/k$
(solid line in Fig.~\ref{fig:coe_exuv} (e)) agree well with the EA
results $\delta\tau(x)$ (dashed line in Fig.~\ref{fig:coe_exuv} (d))
and $K(x)/k$ (dashed line in Fig.~\ref{fig:coe_exuv} (e)) evaluated
at $x=0$. This justifies approximating $\Delta
S_k^{EA}(x,t)=S^{EA}_k(x,t)-S^{C}_k(x)$ in (\ref{eq:EAA})
\begin{align}
\label{eq:Gk2} \Delta S^{EA}_k(x,t)&\approx
S_k^{SFA}(t)-\int_{t}^{\infty}dt'
F[x^\prime(t',t,x=0)]\Delta x(t,t') \nonumber\\
&\approx-\frac{KE_L(t+\delta\tau)}{\omega_L^2},
\end{align}
where, in the second line, the slow varying envelop approximation is
used.
\begin{figure}[t]
\begin{center}
\includegraphics[width=1.0\columnwidth,keepaspectratio=true,
draft=false]{Fig3.eps} \vspace{-6mm} \caption{(Color online) (a)
Local temporal shift $\delta\tau(x)$, and (b) local oscillation
amplitude $K(x)$ induced by the CL interaction in EA. Comparison of
the streaked COEs in SFA for $\hbar \omega_X=25$~eV with streaking
energies (c) in EA at $x=0$ and (d) from full TDSE calculations. The
PE is assumed to move to the right ($k>0$).
\label{fig:eikonal_exuv25}} \vspace{-6mm}
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=1.0\columnwidth,keepaspectratio=true,
draft=false]{Fig4.eps} \vspace{-6mm} \caption{(Color online)
Comparison of eikonal temporal shift $\delta\tau$ and oscillation
amplitude ratio $K/k$ at two $\omega_L$. \label{fig:coe_omega0.8}}
\vspace{-6mm}
\end{center}
\end{figure}
We notice from (\ref{eq:EA_Ecoe})-(\ref{eq:Ec2}) that the three
contributions to $\delta E^{EA}_{COE}(x,\tau)$ are equally
proportional to the ir electric field amplitude. Therefore,
$\delta\tau$ and $K/k$ do not depend on the intensity of the ir
pulse. However, reducing $\omega_L$, $|\delta\tau|$ increases and
$K/k$ decreases (Fig.~\ref{fig:coe_omega0.8}). This is consistent
with (\ref{eq:Ec2}): at smaller $\omega_L$, $A_L$ oscillates slower,
leading to less cancelation in the time integral and thus to larger
$\delta\tau$. Simultaneously, stronger cancelation between $\delta
E_{COE}^{CL,1}$ and $\delta E_{COE}^{CL,2}$, results in smaller
$K(x)$.
\section{Polarization of the Initial State by the IR Pulse}
\label{sec:polarization}
The effect of initial state polarization by the ir pulse on the
streaked xuv photoemission spectrum has been addressed
previously~\cite{Kazansky07,Smirnova06,Baggesen10}. In this section,
we analyze how it affects the temporal shift $\delta\tau$ and the
oscillation amplitude ratio $K/k$. We find that the significance of
the initial-state polarization depends on whether or not the laser
unperturbed initial state is energetically isolated from other
levels.
\begin{figure}[t]
\begin{center}
\includegraphics[width=1.0\columnwidth,keepaspectratio=true,
draft=false]{Fig5.eps} \vspace{-6mm} \caption{(Color online)
Polarization effect of the initial state on ir-streaked
photoemission from the ground state of 1d model hydrogen atoms for
$\hbar\omega_X=25$~eV. (a) Spectrogram with initial state
polarization. (b) Spectrogram without initial state polarization.
(c) Corresponding centers of energies $\delta E_{COE}(\tau)$.
\label{fig:fig5} } \vspace{-6mm}
\end{center}
\end{figure}
We first consider the non-degenerate case. For our one-dimensional
hydrogen atom, all levels are non-degenerate. In our TDSE
calculation, the initial-state polarization by the ir pulse can be
included (excluded) by keeping (dropping) the term $xE_L(t)$ in
(\ref{eq:init}). In Figs.~\ref{fig:fig5} and \ref{fig:fig6} we
compare the polarized (a) and unpolarized (b) spectrograms and their
corresponding centers of energy (c) for the ground state level and
the first excited state. Due to its large separation in energy from
all excited states, the effect of the polarization of the ground
state by the laser pulse on the spectrum is small. It slightly
increases the oscillation amplitude but barely changes the temporal
shift $\delta\tau$. In contrast, the first excited state, whose
binding energy is 6.34~eV, can be easily polarized due to its
laser-induced coupling to the second excited level at 3.64~eV. As
can be seen in Fig.~\ref{fig:fig6}, the polarization distorts the
spectrogram for negative delays where the ir pulse precedes the xuv
pulse. This distortion does not uniformly shift the spectrograms.
Therefore, $\delta\tau$ cannot be uniquely defined as a
delay-independent temporal shift.
\begin{figure}[t]
\begin{center}
\includegraphics[width=1.0\columnwidth,keepaspectratio=true,
draft=false]{Fig6.eps} \vspace{-6mm} \caption{(Color online) Same
as Fig.~\ref{fig:fig5}, but for the first excited state of the 1d
hydrogen atom. \label{fig:fig6} } \vspace{-6mm}
\end{center}
\end{figure}
For the degenerate case, we consider a space spanned by the
stationary wavefunctions $\psi_{200}(\br)$ and $\psi_{210}(\br)$ of
the real (3-dimensional) hydrogen atom. Under the influence of the
laser pulse, the wavefunction is
\begin{align}
\psi(\br,t)=\left[a_{200}(t)\psi_{200}(\br)
+a_{210}(t)\psi_{210}(\br)\right].
\end{align}
By shifting the energy scale such that the binding energies of the
two degenerate stationary states are
$\varepsilon_{200}=\varepsilon_{210}=0$, and by substituting
$\psi(\br,t)$ into the TDSE
\begin{align}
i\frac{\partial}{\partial
t}\psi(\br,t)=\left[H_{at}+zE_{L}(t)\right]\psi(\br,t),
\end{align}
we obtain the equations of motion for the coefficients $a_{200}(t)$
and $a_{210}(t)$,
\begin{align}
i\frac{d}{dt}a_{200}(t)&=\mu E_L(t)a_{210}(t),\\
i\frac{d}{dt}a_{210}(t)&=\mu E_L(t)a_{200}(t),
\end{align}
where $H_{at}$ is the atomic Hamiltonian and
$\mu=\langle\psi_{200}(\br)|z|\psi_{210}(\br)\rangle$ the
dipole-coupling matrix element. The above equations can be solved
analytically~\cite{Grossmann08},
\begin{align}
a_{200}(t)&=a_{200}^0\cos\left[\mu
A_L(t)\right]+ia_{210}^0\sin\left[\mu A_L(t)\right],\\
a_{210}(t)&=a_{210}^0\cos\left[\mu
A_L(t)\right]+ia_{200}^0\sin\left[\mu A_L(t)\right],
\end{align}
where $a_{200}^0$ and $a_{210}^0$ are the initial amplitudes at
$t_0\rightarrow-\infty$.
For example, the initial values $a_{200}^0=\pm1/\sqrt{2}$ and
$a_{210}^0=1/\sqrt{2}$ give the wavefunctions
\begin{align}
\label{eq:pm} \psi_{\pm}(\br,t)=\psi_{\pm}(\br)e^{\pm i\mu A_L(t)},
\end{align}
which evolve from the Stark states
\begin{align}
\psi_{\pm}(\br)=\frac{1}{\sqrt{2}}\left[\psi_{210}(\br)\pm\psi_{200}(\br)\right].
\end{align}
Similarly, for $a_{200}^0=1$ and $a_{210}^0=0$, we obtain the
wavefunction
\begin{align}
\label{eq:2s}
\psi_{2s}(\br,t)=\frac{1}{\sqrt{2}}\left[\psi_+(\br)e^{i\mu
A_L(t)}-\psi_-(\br)e^{-i\mu A_L(t)}\right],
\end{align}
which evolves from an initial 2s state, while $a_{200}^0=0$ and
$a_{210}^0=1$ results in a wavefunction that evolves from an
stationary 2p state,
\begin{align}
\label{eq:2p}
\psi_{2p}(\br,t)=\frac{1}{\sqrt{2}}\left[\psi_+(\br)e^{i\mu
A_L(t)}+\psi_-(\br)e^{-i\mu A_L(t)}\right].
\end{align}
Next, we calculate the ir-streaked spectrum using any of the
wavefunctions (\ref{eq:pm})-(\ref{eq:2p}) as the initial state in
(\ref{eq:tmatrix}). In order to disentangle temporal shifts induced
by i) the initial-state polarization (relative to an unpolarized
target) and ii) the Coulomb potential acting on the final PE state
(relative to the SFA, see section III), we neglect the final-state
distortion by the Coulomb potential and study initial-state
polarization effects within the SFA. In this polarization-effect
study, we hence use the Volkov wavefunction $\psi_{\bk}(\br,t)$ as
an approximation to the final state.
If the initial state evolves from a stationary 2s or 2p state
according to (\ref{eq:2s}) or (\ref{eq:2p}), the polarization causes
delay-dependent interferences between the two Stark states
$\psi_{\pm}$. This interference significantly changes the
energy-differential PE yield in the streaking trace in
Fig.~\ref{fig:fig7} b relative to the trace for an unpolarized
initial state in Fig.~\ref{fig:fig7} a. However, the interference
does not induce a relative temporal shift of the polarized relative
to the unpolarized spectrum, which is best seen in the centers of
energy of the two spectra in Fig.~\ref{fig:fig7} c.
This lack of an interference-induced temporal shift is explained by
the fact that the dipole expectation values
$\langle\psi_{2s}(\br,t)|z|\psi_{2s}(\br,t)\rangle$ and
$\langle\psi_{2p}(\br,t)|z|\psi_{2p}(\br,t)\rangle$ are zero at all
times, even though the ir-laser pulse mixes the stationary 2s and 2p
states.
The situation is different for the states (\ref{eq:pm}) that evolve
out of initial Stark states $\psi_{\pm}$. The comparison of the
energy-differential PE yields in the streaking trace for initial
states (\ref{eq:pm}) with and without including ir-laser-induced
initial-state polarization shows only very small, hardly noticeable,
differences (Fig.s~\ref{fig:fig8} (a-c)). However, temporal
shifts~\cite{Baggesen10} become noticeable in the corresponding
centers of energies. At a PE energy of 60~eV, they amount to 41~as
between an ir-laser polarized and unpolarized initial $\psi_{+}$
state (Fig.~\ref{fig:fig8} (d)) and to 82~as between polarized
initial $\psi_{+}$ and $\psi_{-}$ states (Fig.~\ref{fig:fig8} (e)).
These shifts originate in the permanent dipole moments of the Stark
states whose interaction with the ir-laser electric field shifts the
streaked spectra.
\begin{figure}[t]
\begin{center}
\includegraphics[width=1.0\columnwidth,keepaspectratio=true,
draft=false]{Fig7.eps} \vspace{-6mm} \caption{(Color online)
Photoelectron spectrum for an initial 2s state of hydrogen:
(a) Neglecting ir-laser-induced initial-state polarization by
setting $\mu=0$ in (\ref{eq:2s}).
(b) Including laser-induced initial-state polarization, using
$\mu=3$~a.u. in (\ref{eq:2s}).
(c) Corresponding centers of energies $E_{COE}(\tau)$. No
polarization-induced temporal shift is observed between the results
for polarized and unpolarized initial states. Similar results (not
shown) were obtained for initial 2p states, using (\ref{eq:2p}).
\label{fig:fig7} } \vspace{-6mm}
\end{center}
\end{figure}
\begin{figure}[b]
\begin{center}
\includegraphics[width=1.0\columnwidth,keepaspectratio=true,
draft=false]{Fig8.eps} \vspace{-6mm} \caption{(Color online)
Photoelectron spectra for the initial n=2 Stark states (\ref{eq:pm})
of hydrogen:
(a) For $\psi_+(\br,t)$, neglecting ir-laser-induced initial-state
polarization by setting $\mu=0$ in (\ref{eq:pm}).
(b) For $\psi_+(\br,t)$, including laser-induced initial-state
polarization, using $\mu=3$~a.u. in (\ref{eq:pm}).
(c) For $\psi_-(\br,t)$, including laser-induced initial-state
polarization, using $\mu=3$~a.u. in (\ref{eq:pm}).
(d,e) Corresponding centers of energies $ E_{COE}(\tau)$, showing a
relative temporal shift between the streaking traces of (d)
polarized and unpolarized initial $\psi_+(\br,t)$ states and (e)
polarized initial $\psi_+(\br,t)$ and $\psi_-(\br,t)$ states.
\label{fig:fig8} } \vspace{-6mm}
\end{center}
\end{figure}
\section{Conclusions}
\label{sec:conclusion}
We have shown how the simultaneous interaction of an xuv PE with the
electric field of a streaking ir laser pulse and the Coulomb
potential of the residual ion induces a specific
Coulomb-Laser-coupling phase and leads to an attosecond temporal
shift and amplitude enhancement in the oscillation of the streaked
PE spectrum. This shift and amplitude enhancement become significant
and observable as the xuv photon energy approaches the ionization
threshold. It can be explained semiclassically in terms of an added
Coulomb-phase factor in the PE wave function. This factor reveals
the origin of the observable temporal shift as a Coulomb-laser
coupling effect in the PE dynamics: the PE absorbs and releases ir
photons while moving subject to the ionic Coulomb force. The
analytical results obtained in EA show that the CL coupling induces
a temporal shift relative to $A_L$, thus relative to the SFA result.
For the experimental observation of $\delta\tau$ and $K/k$ as a
function of the PE kinetic energy, we suggest using xuv pulses with
tunable xuv photon energy to photoemit electrons from two levels
with a large energy separation~\cite{Schultze10}.
We have also examined the effect of ir-laser-induced polarization of
the initial state on the ir-streaked xuv PE spectrum. If the initial
state is not degenerate and has a large energetic separation from
all other states, its very small polarization does not noticeably
affect the PE spectrum. On the other hand, if the initial state can
easily be coupled to other states by the ir-laser pulse, its
polarization is important and, interestingly, does not uniformly
shift the spectrum. If the initial state has a permanent dipole
moment, such as the n=2 Stark states of hydrogen, there is a
relative temporal shift in the streaking traces i) for different
initial Stark states and ii) with and without inclusion of the
initial-state polarization.
\begin{acknowledgments}
We thank F. He for helpful discussions. This work was supported by
the NSF and the Division of Chemical Sciences, Office of Basic
Energy Sciences, Office of Energy Research, US~DOE. Some of the
numerical computations for this project were performed on the Beocat
cluster at Kansas State University.
\end{acknowledgments}
\begin{widetext}
|
{
"timestamp": "2010-09-21T02:03:59",
"yymm": "1009",
"arxiv_id": "1009.3887",
"language": "en",
"url": "https://arxiv.org/abs/1009.3887"
}
|
\section{Introduction}
Anomalous X-ray pulsars (AXPs) are pulsar-like objects, whose X-ray
luminosities are in excess of their rotational energy losses while
they show no binary signature, thus acquiring the name ``anomalous''
X-ray pulsars (Mereghetti 2008). AXPs, along with soft gamma-ray
repeaters (SGRs), are candidate magnetars, neutron stars powered
by strong magnetic field decay (Duncan \& Thompson 1992; Paczynski
1992). Alternative explanations for AXPs and SGRs involve a normal
neutron star accreting from a supernova fallback disk (Alpar 2001; Chatterjee et al. 2000).
It is then a very fundamental question to determine whether AXPs and
SGRs are magnetars or accretion-powered systems. To finally solve
this problem is not only helpful to understand the equation of state
at supra-nuclear densities, but also very meaningful to explain high
energy astrophysical phenomena (Xu 2007).
The magnetar model is prevailing in explaining bursts of AXPs and SGRs (Paczynski 1992; Thomspon \& Duncan 1995).
However, bursting behavior in accretion model is not absolutely impossible (Rothschild et al. 2002; Xu et al. 2006).
It is also possible that the magnetar field ($\sim 10^{14}-10^{15}\,\mathrm{G}$)
responsible for bursts is in higher multipole form
while a normal dipole component ($\sim 10^{12}-10^{13}\,\mathrm{G}$) interacts with the fallback disk (Eksi \& Alpar 2003; Ertan et al. 2007).
Observations in the optical/IR band are informative, e.g. a debris
disk is found around AXP 4U 0142+61 (Wang et al. 2006). The optical/IR observation of 4U 0142+61 can be explained uniformly
in an accretion fallback disk model (Ertan \& Cheng 2004; Ertan et al. 2007).
However, if the disk is passive, a fallback disk is also compitable with the magnetar scenario (Wang et al. 2006).
Therefore observations at other wavelengths are very necessary to understand the real nature of AXPs and SGRs, especially in gamma-rays.
The outer gap model (e.g. Cheng et al. 1986) is very successful, and high energy gamma-ray emissions of AXPs have been calculated and predicted
by Cheng \& Zhang (2001) in the magnetar domain, using the thick outer gap model (Zhang \& Cheng 1997).
The detailed calculations of Cheng \& Zhang (2001) predicted that Fermi/LAT should be able to detect gamma-ray emission of AXPs,
including 4U 0142+61, if they are magnetars. However, a recent Fermi/LAT observation of 4U 0142+61 has been reported,
which shows no detection (Sasmaz Mus \& Gogus 2010). Then there seems a conflict between theory and observation.
While adopting the thick outer gap model
(Zhang \& Cheng 1997), simple calculations show that AXPs are not high energy gamma-ray emitters if they are normal neutron stars accreting from
fallback disks. We suggest that Fermi/LAT observation of AXPs and SGRs can be applied to distinguish
between the magnetar model and the accretion model.
The non-detection of 4U 0142+61 may prefer the accretion model.
In \S 2 we compare theoretical predictions from the magnetar model with Fermi/LAT observation of AXP 4U 0142+61.
Discussions are presented in \S 3.
\section{Theoretical calculations in the magnetar model}
Zhang \& Cheng (1997) developed the thick outer gap model for long period pulsars.
The typical Lorentz factor is determined by equaling energy loss and gain.
The $\gamma-\gamma$ pair production threshold determines the size of the outer gap self-consistently.
If the X-ray photons are provided by surface thermal emission, the size of the outer gap is (eq.(24) in Zhang \& Cheng 1997)
\begin{equation}
f=4.5 P^{7/6} B_{12}^{-1/2} T_{6}^{-2/3} R_{6}^{-3/2},
\end{equation}
where $P$ is the pulsar rotation period, $B_{12}$ is the stellar magnetic field in units of $10^{12}\,\mathrm{G}$,
$T_{6}$ is the surface temperature in units of $10^6\,\mathrm{K}$, $R_{6}$ is the stellar radius in units of
$10^{6}\,\mathrm{cm}$. Here $f$ should be less than one for outer gap to exist.
In magnetar model for AXPs and SGRs, typical parameters are $P=7\,\mathrm{s}$,
$B=5\times 10^{14}\,\mathrm{G}$, $T=0.5\,\mathrm{keV}$. The stellar radius is chosen as $R=12\,\mathrm{km}$, which is moderate for
realistic equations of state (Lattimer \& Prakash 2007, fig 6 there)
(In Cheng \& Zhang 2001, the stellar radius is chosen as $15\,\mathrm{km}$).
The corresponding outer gap size is then $f=0.46$,
which means that if AXPs and SGRs are magnetars they should be high energy gamma-ray emitters.
On the other hand, if AXPs and SGRs are normal neutron stars
whose (dipolar) magnetic fields are $10^{12}-10^{13}\,\mathrm{G}$ (Alpar 2001; Chatterjee et al. 2000),
the corresponding outer gap size is $f=3-10$. Therefore if AXPs and SGRs are normal neutron stars accreting from fallback disks,
they will not radiate high energy gamma-rays\footnote{Ertan \& Cheng (2004) argued that accretion-powered system can also emit high energy gamma-rays
if the inner disk rotates faster than the neutron star. However, this criterion cannot be matched
for the debris disk around 4U 0142+61 either as a passive disk (Wang et al. 2006) or as a gaseous accretion disk
(Ertan et al. 2007).}. Thus Fermi/LAT observations of AXPs and SGRs can be helpful to distinguish between the magnetar model and the accretion model.
Sasmaz Mus \& Gogus (2010) reported Fermi/LAT observation of AXP 4U 0142+61.
With an exposure time of $31.7\,\mathrm{Ms}$, they find no detection of high energy gamma-ray
emission from 4U 0142+61 in both $0.2-1\,\mathrm{GeV}$ and $1-10\,\mathrm{GeV}$ band.
Observational upper limits and theoretical calculations in the magnetar model are shown in figure 1.
For 4U 0142+61, its parameters are $P=8.688\,\mathrm{s}$, $B=2.6\times 10^{14}\,\mathrm{G}$, $T=0.395\,\mathrm{keV}$
(from the McGill AXP/SGR catalog\footnote{http://www.physics.mcgill.ca/$\sim$pulsar/magnetar/main.html}).
The magnetic field is calculated from $B=6.4\times10^{19}\sqrt{P\dot{P}}$, which is 2 times larger than usually reported since
the polar magnetic field is more important in the case of pulsar radiation (Shapiro \& Teukolsky 1983).
The size of the outer gap for 4U 0142+61 is $f=0.96$.
The distance $d=2.5\,\mathrm{kpc}$ and solid angle $\Delta \Omega=1$ are used during the calculation. From figure 1,
the observational upper limits are below the theoretical calculations for large inclination angles ($60^{\circ}$, $75^{\circ}$)
or when the inner boundary of outer gap can extend to 10 stellar radii (Hirotani et al. 2003; Hirotani \& Shibata 2001).
\begin{figure}[!t]
\centering
\includegraphics[width=0.75\textwidth]{fig1.eps}
\caption{Fermi/LAT upper limits of AXP 4U 0142+61 compared with outer gap calculations in the magnetar domain.
The solid, dashed, and dotted lines are for inclination angle $45^{\circ}$, $60^{\circ}$, $75^{\circ}$ respectively
(Zhang \& Cheng 1997; Cheng \& Zhang 2001).
The dotdashed line takes into consideration that the inner boundary of outer gap may extend to 10 stellar radii
(Hirotani et al. 2003; Hirotani \& Shibata 2001).
The empty down triangle and filled down triangle are Fermi/LAT upper limits ($0.2-1\,\mathrm{GeV}$ and $1-10\,\mathrm{GeV}$)
from $2^{\circ}$ and $15^{\circ}$ extraction region respectively
(Sasmaz Mus \& Gogus 2010). The upper limits in $1-10\,\mathrm{GeV}$ are nearly coincide.}
\end{figure}
Possible reasons why Fermi/LAT has not seen the expected high energy gamma-rays from AXP 4U 0142+61 are:
\begin{enumerate}
\item Its radius is smaller than $12\,\mathrm{km}$;
\item Its distance is much larger than $2.5\,\mathrm{kpc}$;
\item The inclination angle is small, e.g. $45^{\circ}$;
\item Beaming of gamma-ray radiation;
\item The radiated high energy gamma-ray photons are absorbed due to internal or external matter.
\end{enumerate}
For order of magnitude estimations, the neutron star radius is often taken as $10\,\mathrm{km}$.
However, for realistic equations of state, this choice corresponds to a soft equation of state (Lattimer \& Prakash 2007).
For a stiff equation of state
the radius can be as large as $15\,\mathrm{km}$. A radius of $12\,\mathrm{km}$ is a moderate choice (Lattimer \& Prakash 2007, fig 6 there).
Neutron star equation of state studies (e.g. Tsuruta 2006) also prefer medium to stiff equations of state.
Figure 2 shows the model calculations for AXP 4U 0142+61 when the distance is 2 times larger, i.e. $5\,\mathrm{kpc}$, along with
Fermi/LAT sensitivity curve for $5\sigma$ detection (Atwood et al. 2009). Even when the distance is 2 times larger than we presently
employed, Fermi/LAT should also be able to detect the expected gamma-ray emission of 4U 0142+61. Also in figure 2, when the inclination angle is small,
e.g. $45^{\circ}$, its high energy radiation is decreased along with an increase in the low energy part
(cf. fig 4 in Cheng \& Zhang 2001).
Therefore if the inclination angle is small, although Fermi/LAT could not detect 4U 0142+61 in $(1-10)\,\mathrm{GeV}$ band,
it could detect 4U 0142+61 in $(0.1-1)\,\mathrm{GeV}$ and lower energy band. In Cheng \& Zhang (2001), the inclination angle determines the
inner boundary of the outer gap. Recent modeling indicates that the inner boundary may extend to 10 stellar radii
(Hirotani et al. 2003; Hirotani \& Shibata 2001). Employing this assumption, the corresponding model calculations are shown in
figure 1 and 2.
According to Cheng \& Zhang (2001) and references therein,
the solid angle for known gamma-ray pulsars ranges from $0.5-2.5$. Recent Fermi observations of gamma-ray pulsars also show a relatively
broad pulse profile (Ray \& Parkinson 2010). Therefore the beaming of gamma-ray radiation is not the key factor obscuring our observation
of gamma-ray emissions and this problem can be cleared with future Fermi/LAT observations of more AXPs and SGRs.
The magnetic field at the inner boundary of outer gap is $2.6\times 10^5\,\mathrm{G}$ for inclination angle $75^{\circ}$
(or $2.6\times 10^{11}\,\mathrm{G}$ when the inner boundary is chosen as 10 stellar radii). The absorption of high energy photons
is not significant at the inner boundary due the weakness of the magnetic field (Ruderman \& Sutherland 1975). For AXP 4U 0142+61,
it has a debris disk whose photon energy is typically $0.1-1\,\mathrm{eV}$ (Wang et al. 2006). The $\gamma-\gamma$ absorption
is negligible for GeV photons (Zhang \& Cheng 1997).
In conclusion, based on the thick outer gap model (Zhang \& Cheng 1997), for a variety of the parameter space in magnetar model,
Fermi/LAT should be able to detect the expected high energy gamma-ray emission from AXP 4U 0142+61.
This is in conflict with Sasmaz Mus \& Gogus (2010).
\begin{figure}[!t]
\centering
\includegraphics[width=0.75\textwidth]{fig2.eps}
\caption{Fermi/LAT integral sensitivity curve and model calculations for AXP 4U 0142+61.
The solid, dotted, and dotdashed lines are the same as those in figure 1, except that
the integral flux is shown instead of differential flux. The corresponding thick lines are model calculations
when the distance is 2 times larger, i.e. $5\,\mathrm{kpc}$. The thick dashed line is the Fermi/LAT sensitivity curve (Atwood et al. 2009).}
\end{figure}
\section{Discussions}
At the beginning of \S 2, we show that AXPs are not high energy gamma-ray emitters ($f$ larger than 1)
if they are normal neutron stars accreting from fallback disks. Therefore the non-detection
in a Fermi/LAT observation of AXP 4U 0142+61 can be naturally explained in the accretion model
for AXPs. The spectra energy distribution of 4U 0142+61 indicates an energy break at about $1\,\mathrm{MeV}$ (Sasmaz Mus \& Gogus 2010).
If hard X-ray emission of 4U 0142+61 originates from near the stellar surface,
the energy break is also at $1\,\mathrm{MeV}$ for a normal neutron star (Zhang \& Cheng 1997).
Of course the detailed origin of AXP hard X-ray emission needs further studies.
In the accretion model for AXPs (Alpar 2001; Chatterjee et al. 2000) (also for SGRs, if they are indeed one population),
the long period of AXPs is due to disk braking in the propeller phase. They are now
X-ray luminous since they have entered the accretion phase. The bursts of AXPs and SGRs may due to
accretion induced quakes (AIQs) (Xu et al. 2006; Xu 2007), or quakes and plate tectonics of neutron stars
(Rothschild et al. 2002). The accretion induced quake model of Xu et al. (2006) provides a link between
persistent emission and bursts. A hybrid model is also possible which the magnetar field is in
higher multipole form and the spin down is governed by a normal dipole component interacting with a fallback disk (Eksi \& Alpar 2003).
The recently reported low magnetic field SGR (SGR 0418+5927 with $B_{\mathrm{dipole}}<7.5\times 10^{12}\,\mathrm{G}$, Rea et al. 2010)
is consistent with the accretion model.
For AXP 4U 0142+61, as noted in section 2, it will not
emit high energy gamma-rays even if it is a magnetar, when its radius is $10\,\mathrm{km}$ instead of $12\,\mathrm{km}$.
Therefore future Fermi/LAT observations of more AXPs and SGRs are very necessary.
Outer gap predictions in the magnetar domain for other AXPs and SGRs are shown in figure 3.
Model calculations for three AXPs and one SGR are shown, using observational parameters from the McGill AXP/SGR online catalog.
For gamma-ray luminous and nearby sources, model calculations of AXP 1E 1547.0-5408 and AXP 1E 1048.1-5937 are well above the
Fermi/LAT sensitivity curve. Therefore future Fermi/LAT observations of these two sources are highly recommended.
Among other AXPs, some are not supposed to be high-energy gamma-ray emitters ($f$ larger than 1),
some have relatively low gamma-ray luminosities as shown for AXP XTE J1810-197 in figure 3,
some lies too far away from us. For the two candidate high energy gamma-ray emitting SGRs, SGR 1806-20 and SGR 1900+14,
they are too far away to be detected by Fermi/LAT, as shown for SGR 1806-20 in figure 3.
In conclusion, based on the thick outer gap model (Zhang \& Cheng 1997), the non-detection in a Fermi/LAT observation of
AXP 4U 0142+61 may prefer the accretion model. Future Fermi/LAT observations of AXP 1E 1547.0-5408 and AXP 1E 1048.1-5937
will help us make clear whether they are magnetars or not\footnote{During the submission of this paper, the Fermi-LAT collaboration
have published their observations for all known AXPs and SGRs (Abdo et al. 2010), where still no significant detection is reported.
This result is in favor of our conclusions.}.
\begin{figure}[!t]
\centering
\includegraphics[width=0.75\textwidth]{fig3.eps}
\caption{Model calculations for other AXPs and SGRs (Zhang \& Cheng 1997; Cheng \& Zhang 2001).
The inclination angle is chosen as $60^{\circ}$ and star raidus $10\,\mathrm{km}$.
The solid, dashed, dotdashed and dotted lines are for AXP 1E 1547.0-5408, AXP 1E 1048.1-5937, AXP XTE J1810-197 and SGR 1806-20, respectively.
The thick line is the Fermi/LAT sensitivity curve (Atwood et al. 2009).}
\end{figure}
\section*{Acknowledgments}
The authors would like to thank Dr. Stephen Justham for help in the English of this paper.
This work is supported by the National Natural Science Foundation
of China (Grant Nos. 10935001, 10973002), and the National Basic Research Program of China
(Grant No. 2009CB824800)
|
{
"timestamp": "2010-11-17T02:01:31",
"yymm": "1009",
"arxiv_id": "1009.3620",
"language": "en",
"url": "https://arxiv.org/abs/1009.3620"
}
|
\section{INTRODUCTION}
The inequalit
\begin{equation}
f\left( \frac{a+b}{2}\right) \leq \frac{1}{b-a}\dint\limits_{a}^{b}f(x)d
\leq \frac{f(a)+f(b)}{2} \label{1.1}
\end{equation
where $f:I\subset
\mathbb{R}
\rightarrow
\mathbb{R}
$ is a convex function defined on the interval $I$ of
\mathbb{R}
,$ the set of real numbers, and $a,b\in I$ with $a<b,$ is well known in the
literature as Hadamard's inequality.
For some recent results related to this classic inequality, see \cite{PP},
\cite{MEO}, \cite{MU}, \cite{SS}, and \cite{DA}, where further references
are given.
In \cite{HM}, Hudzik and Maligranda considered, among others, the class of
functions which are $s-$convex in the second sense. This class is defined as
following:
\begin{definition}
A function $f:[0,\infty )\rightarrow $
\mathbb{R}
$ is said to be $s-$convex in the second sense i
\begin{equation*}
f(\lambda x+(1-\lambda )y)\leq \lambda ^{s}f(x)+(1-\lambda )^{s}f(y)
\end{equation*
holds for all $x,y\in \lbrack 0,\infty ),$ $\lambda \in \lbrack 0,1]$ and
for some fixed $s$ $\in (0,1].$
\end{definition}
The class of $s-$convex functions in the second sense is usually denoted
with $K_{s}^{2}.$ It is clear that if we choose $s=1$ we have ordinary
convexity of functions defined on $[0,\infty ).$
In \cite{UK}, K\i rmac\i\ \textit{et al.}, proved the following inequalities
related to product of convex functions. These are given in the next theorems.
\begin{theorem}
Let $f,g:[a,b]\rightarrow
\mathbb{R}
,a,b\in \lbrack 0,\infty ),$ $a<b,$ be functions such that $g$ and $fg$ are
in $L^{1}([a,b]).$ If $f$ is convex and nonnegative on $[a,b],$ and if $g$
is $s-$convex on $[a,b]$ for some fixed $s$ $\in (0,1),$ the
\begin{equation}
\frac{1}{b-a}\dint\limits_{a}^{b}f(x)g(x)dx\leq \frac{1}{s+2}M(a,b)+\frac{1}
(s+1)(s+2)}N(a,b) \label{1.2}
\end{equation
wher
\begin{equation*}
M(a,b)=f(a)g(a)+f(b)g(b)\text{ and }N(a,b)=f(a)g(b)+f(b)g(a).
\end{equation*}
\end{theorem}
\begin{theorem}
Let $f,g:[a,b]\rightarrow
\mathbb{R}
,$ $a,b\in \lbrack 0,\infty ),$ $a<b,$ be functions such that $g$ and $fg$
are in $L^{1}([a,b]).$ If $f$ is $s_{1}-$convex and $g$ is $s_{2}-$convex on
$[a,b]$ for some fixed $s_{1},s_{2}\in (0,1),$ the
\begin{eqnarray}
\frac{1}{b-a}\dint\limits_{a}^{b}f(x)g(x)dx &\leq &\frac{1}{s_{1}+s_{2}+1
M(a,b)+B(s_{1}+1,s_{2}+1)N(a,b) \notag \\
&=&\frac{1}{s_{1}+s_{2}+1}\left[ M(a,b)+s_{1}s_{2}\frac{\Gamma (s_{1})\Gamma
(s_{2})}{\Gamma (s_{1}+s_{2}+1)}N(a,b)\right] \label{1.3}
\end{eqnarray}
\end{theorem}
\begin{theorem}
Let $f,g:[a,b]\rightarrow
\mathbb{R}
,$ $a,b\in \lbrack 0,\infty ),$ $a<b,$ be functions such that $g$ and $fg$
are in $L^{1}([a,b]).$ If $f$ is convex and nonnegative on $[a,b],$ and if
g $ is $s-$convex on $[a,b]$ for some fixed $s$ $\in (0,1),$ the
\begin{eqnarray}
&&2^{s}f(\frac{a+b}{2})g(\frac{a+b}{2})-\frac{1}{b-a}\din
\limits_{a}^{b}f(x)g(x)dx \label{1.4} \\
&\leq &\frac{1}{(s+1)(s+2)}M(a,b)+\frac{1}{s+2}N(a,b) \notag
\end{eqnarray}
\end{theorem}
For similar results, see the papers \cite{HM}, \cite{DF}.
In \cite{SS}, Dragomir defined convex functions on the co-ordinates as
follows and proved Lemma 1 related to this definiton:
\begin{definition}
Let us consider the bidimensional interval $\Delta =[a,b]\times \lbrack c,d]$
in
\mathbb{R}
^{2}$ with $a<b,$ $c<d.$ A function $f:\Delta \rightarrow
\mathbb{R}
$ will be called convex on the co-ordinates if the partial mappings
f_{y}:[a,b]\rightarrow
\mathbb{R}
,$ $f_{y}(u)=f(u,y)$ and $f_{x}:[c,d]\rightarrow
\mathbb{R}
,$ $f_{x}(v)=f(x,v)$ are convex where defined for all $y\in \lbrack c,d]$
and $x\in \lbrack a,b].$ Recall that the mapping $f:\Delta \rightarrow
\mathbb{R}
$ is convex on $\Delta $ if the following inequality holds,
\begin{equation*}
f(\lambda x+(1-\lambda )z,\lambda y+(1-\lambda )w)\leq \lambda
f(x,y)+(1-\lambda )f(z,w)
\end{equation*
for all $(x,y),(z,w)\in \Delta $ and $\lambda \in \lbrack 0,1].$
\end{definition}
\begin{lemma}
Every \ convex mapping $f:\Delta \rightarrow
\mathbb{R}
$ is convex on the co-ordinates, but converse is not general true.
\end{lemma}
A formal definition for co-ordinated convex functions may be stated as
follow [see \cite{MAL}]:
\begin{definition}
A function $f:\Delta \rightarrow \mathbb{R}$ is said to be convex on the
co-ordinates on $\Delta $ if the following inequality
\begin{align*}
& f(tx+(1-t)y,su+(1-s)w) \\
& \leq tsf(x,u)+t(1-s)f(x,w)+s(1-t)f(y,u)+(1-t)(1-s)f(y,w)
\end{align*
holds for all $t,s\in \lbrack 0,1]$ and $(x,u),(x,w),(y,u),(y,w)\in \Delta $.
\end{definition}
In \cite{SS}, Dragomir established the following inequalities:
\begin{theorem}
Suppose that $f:\Delta =$ $[a,b]\times \lbrack c,d]\rightarrow
\mathbb{R}
$ is convex on the co-ordinates on $\Delta .$ Then one has the inequalities
\begin{eqnarray}
&&f(\frac{a+b}{2},\frac{c+d}{2}) \notag \\
&\leq &\frac{1}{\left( b-a\right) \left( d-c\right) }\dint\limits_{a}^{b
\dint\limits_{c}^{d}f(x,y)dxdy \label{1.5} \\
&\leq &\frac{f(a,c)+f(a,d)+f(b,c)+f(b,d)}{4} \notag
\end{eqnarray}
\end{theorem}
Similar results, refinements and generalizations can be found in \cite{LA},
\cite{DAR1}, \cite{DAR2}, \cite{DAR3}, \cite{MSET} and \cite{OZ2}.
In \cite{DAR3}, Alomari and Darus defined $s-$convexity on $\Delta $ as
follows:
\begin{definition}
Consider the bidimensional interval $\Delta :=$ $[a,b]\times \lbrack c,d]$
in $[0,\infty )^{2}$ with $a<b$ and $c<d.$ The mapping $f:\Delta \rightarrow
\mathbb{R}
$ is $s-$convex on $\Delta $ i
\begin{equation*}
f(\lambda x+(1-\lambda )z,\lambda y+(1-\lambda )w)\leq \lambda
^{s}f(x,y)+(1-\lambda )^{s}f(z,w)
\end{equation*
holds for all $(x,y),$ $(z,w)\in \Delta $ with $\lambda \in \lbrack 0,1]$
and for some fixed $s$ $\in (0,1].$
\end{definition}
In \cite{DAR3}, Alomari and Darus proved the following lemma:
\begin{lemma}
Every $s-$convex mappings $f:\Delta :=$ $[a,b]\times \lbrack c,d]\subset
\lbrack 0,\infty )^{2}\rightarrow \lbrack 0,\infty )$ is $s-$convex on the
co-ordinates, but converse is not true in general .
\end{lemma}
In \cite{LAT}, Latif and Alomari established Hadamard-type inequalities for
product of two convex functions on the co-ordinates as follow:
\begin{theorem}
Let $\ f,g:\Delta :=$ $[a,b]\times \lbrack c,d]\subset
\mathbb{R}
^{2}\rightarrow \lbrack 0,\infty )$ be convex functions on the co-ordinates
on $\Delta $ with $a<b$ and $c<d.$ Then
\begin{eqnarray}
&&\frac{1}{\left( b-a\right) \left( d-c\right) }\dint\limits_{a}^{b}\din
\limits_{c}^{d}f(x,y)g(x,y)dxdy \label{1.6} \\
&\leq &\frac{1}{9}L(a,b,c,d)+\frac{1}{18}M(a,b,c,d)+\frac{1}{36}N(a,b,c,d)
\notag
\end{eqnarray
wher
\begin{eqnarray*}
L(a,b,c,d) &=&f(a,c)g(a,c)+f(b,c)g(b,c)+f(a,d)g(a,d)+f(b,d)g(b,d) \\
M(a,b,c,d) &=&f(a,c)g(a,d)+f(a,d)g(a,c)+f(b,c)g(b,d)+f(b,d)g(b,c) \\
&&+f(b,c)g(a,c)+f(b,d)g(a,d)+f(a,c)g(b,c)+f(a,d)g(b,d) \\
N(a,b,c,d) &=&f(b,c)g(a,d)+f(b,d)g(a,c)+f(a,c)g(b,d)+f(a,d)g(b,c)
\end{eqnarray*}
\end{theorem}
\begin{theorem}
Let $\ f,g:\Delta :=$ $[a,b]\times \lbrack c,d]\subset
\mathbb{R}
^{2}\rightarrow \lbrack 0,\infty )$ be convex functions on the co-ordinates
on $\Delta $ with $a<b$ and $c<d.$ The
\begin{eqnarray}
&&4f(\frac{a+b}{2},\frac{c+d}{2})g(\frac{a+b}{2},\frac{c+d}{2}) \label{1.7}
\\
&\leq &\frac{1}{\left( b-a\right) \left( d-c\right) }\dint\limits_{a}^{b
\dint\limits_{c}^{d}f(x,y)g(x,y)dxdy \notag \\
&&+\frac{5}{36}L(a,b,c,d)+\frac{7}{36}M(a,b,c,d)+\frac{2}{9}N(a,b,c,d)
\notag
\end{eqnarray
where $L(a,b,c,d),$ $M(a,b,c,d)$, $N(a,b,c,d)$ as in (\ref{1.6}).
\end{theorem}
The main purpose of this paper is to establish new inequalities like (\re
{1.6}) and (\ref{1.7}), but now for convex functions and $s-$convex
functions of $2-$variables on the co-ordinates.
\section{MAIN RESULTS}
\begin{theorem}
Let $\ f:\Delta :=$ $[a,b]\times \lbrack c,d]\subset \lbrack 0,\infty
)^{2}\rightarrow \lbrack 0,\infty )$ be convex function on the co-ordinates
and $g:\Delta :=$ $[a,b]\times \lbrack c,d]\subset \lbrack 0,\infty
)^{2}\rightarrow \lbrack 0,\infty )$ be $s-$convex function on the
co-ordinates with $a<b,$ $c<d$ and $f_{x}(y)g_{x}(y),$ $\
f_{y}(x)g_{y}(x)\in L^{1}[\Delta ]$ for some fixed $s\in (0,1].$ Then one
has the inequality
\begin{eqnarray}
&&\frac{1}{\left( d-c\right) \left( b-a\right) }\dint\limits_{a}^{b}\din
\limits_{c}^{d}f(x,y)g(x,y)dxdy \label{2.1} \\
&\leq &\frac{1}{\left( s+2\right) ^{2}}L(a,b,c,d)+\frac{1}{(s+1)\left(
s+2\right) ^{2}}M(a,b,c,d) \notag \\
&&+\frac{1}{(s+1)^{2}(s+2)^{2}}N(a,b,c,d) \notag
\end{eqnarray
wher
\begin{eqnarray*}
L(a,b,c,d) &=&\frac{1}{\left( s+2\right) ^{2}}\left( \left[
f(a,c)g(a,c)+f(b,c)g(b,c)\right] +\left[ f(a,d)g(a,d)+f(b,d)g(b,d)\right]
\right) \\
M(a,b,c,d) &=&\frac{1}{(s+1)\left( s+2\right) ^{2}}\left( \left[
f(a,c)g(b,c)+f(b,c)g(a,c)\right] +\left[ f(a,d)g(b,d)+f(b,d)g(a,d)\right]
\right) \\
&&+\frac{1}{(s+1)\left( s+2\right) ^{2}}\left( \left[
f(a,c)g(a,d)+f(b,c)g(b,d)\right] +\left[ f(a,d)g(a,c)+f(b,d)g(b,c)\right]
\right) \\
N(a,b,c,d) &=&\frac{1}{(s+1)^{2}(s+2)^{2}}\left( \left[
f(a,c)g(b,d)+f(b,c)g(a,d)\right] +\left[ f(a,d)g(b,c)+f(b,d)g(a,c)\right]
\right)
\end{eqnarray*}
\end{theorem}
\begin{proof}
Since $f$ is co-ordinated convex and $g$ is co-ordinated $s-$convex, from
Lemma 1 and Lemma 2, the partial mapping
\begin{eqnarray*}
f_{y} &:&[a,b]\rightarrow \lbrack 0,\infty ),\text{ }f_{y}(x)=f(x,y),\text{
y\in \lbrack c,d] \\
f_{x} &:&[c,d]\rightarrow \lbrack 0,\infty ),\text{ }f_{x}(y)=f(x,y),\text{
x\in \lbrack a,b]
\end{eqnarray*
are convex on $[a,b]$ and $[c,d],$ respectively, where $x\in \lbrack a,b],$
y\in \lbrack c,d]$. Similarly
\begin{eqnarray*}
g_{y} &:&[a,b]\rightarrow \lbrack 0,\infty ),\text{ }g_{y}(x)=g(x,y),\text{
y\in \lbrack c,d] \\
g_{x} &:&[c,d]\rightarrow \lbrack 0,\infty ),\text{ }g_{x}(y)=g(x,y),\text{
x\in \lbrack a,b]
\end{eqnarray*
are $s-$convex on $[a,b]$ and $[c,d],$ respectively, where $x\in \lbrack
a,b],$ $y\in \lbrack c,d].$
Using (\ref{1.2}), we can writ
\begin{eqnarray*}
\frac{1}{d-c}\dint\limits_{c}^{d}f_{x}(y)g_{x}(y)dy &\leq &\frac{1}{s+2
\left[ f_{x}(c)g_{x}(c)+f_{x}(d)g_{x}(d)\right] \\
&&+\frac{1}{(s+1)(s+2)}\left[ f_{x}(c)g_{x}(d)+f_{x}(d)g_{x}(c)\right]
\end{eqnarray*
That i
\begin{eqnarray*}
\frac{1}{d-c}\dint\limits_{c}^{d}f(x,y)g(x,y)dy &\leq &\frac{1}{s+2}\left[
f(x,c)g(x,c)+f(x,d)g(x,d)\right] \\
&&+\frac{1}{(s+1)(s+2)}\left[ f(x,c)g(x,d)+f(x,d)g(x,c)\right]
\end{eqnarray*
Dividing both sides by$\ (b-a)$ and integrating over $[a,b],$ we ge
\begin{eqnarray}
&&\frac{1}{\left( d-c\right) \left( b-a\right) }\dint\limits_{a}^{b}\din
\limits_{c}^{d}f(x,y)g(x,y)dxdy \label{2.2} \\
&\leq &\frac{1}{s+2}\left[ \frac{1}{b-a}\dint\limits_{a}^{b}f(x,c)g(x,c)dx
\frac{1}{b-a}\dint\limits_{a}^{b}f(x,d)g(x,d)dx\right] \notag \\
&&+\frac{1}{(s+1)(s+2)}\left[ \frac{1}{b-a}\din
\limits_{a}^{b}f(x,c)g(x,d)dx+\frac{1}{b-a}\dint\limits_{a}^{b}f(x,d)g(x,c)d
\right] \notag
\end{eqnarray
By applying (\ref{1.2}) to each term of right hand side of above inequality,
we hav
\begin{eqnarray*}
\frac{1}{b-a}\dint\limits_{a}^{b}f(x,c)g(x,c)dx &\leq &\frac{1}{s+2}\left[
f(a,c)g(a,c)+f(b,c)g(b,c)\right] \\
&&+\frac{1}{(s+1)(s+2)}\left[ f(a,c)g(b,c)+f(b,c)g(a,c)\right]
\end{eqnarray*
\begin{eqnarray*}
\frac{1}{b-a}\dint\limits_{a}^{b}f(x,d)g(x,d)dx &\leq &\frac{1}{s+2}\left[
f(a,d)g(a,d)+f(b,d)g(b,d)\right] \\
&&+\frac{1}{(s+1)(s+2)}\left[ f(a,d)g(b,d)+f(b,d)g(a,d)\right]
\end{eqnarray*
\begin{eqnarray*}
\frac{1}{b-a}\dint\limits_{a}^{b}f(x,c)g(x,d)dx &\leq &\frac{1}{s+2}\left[
f(a,c)g(a,d)+f(b,c)g(b,d)\right] \\
&&+\frac{1}{(s+1)(s+2)}\left[ f(a,c)g(b,d)+f(b,c)g(a,d)\right]
\end{eqnarray*
\begin{eqnarray*}
\frac{1}{b-a}\dint\limits_{a}^{b}f(x,d)g(x,c)dx &\leq &\frac{1}{s+2}\left[
f(a,d)g(a,c)+f(b,d)g(b,c)\right] \\
&&+\frac{1}{(s+1)(s+2)}\left[ f(a,d)g(b,c)+f(b,d)g(a,c)\right]
\end{eqnarray*
Using these inequalities in (\ref{2.2}), (\ref{2.1}) is proved, that is
\begin{eqnarray*}
&&\frac{1}{\left( d-c\right) \left( b-a\right) }\dint\limits_{a}^{b}\din
\limits_{c}^{d}f(x,y)g(x,y)dxdy \\
&\leq &\frac{1}{\left( s+2\right) ^{2}}\left( \left[
f(a,c)g(a,c)+f(b,c)g(b,c)\right] +\left[ f(a,d)g(a,d)+f(b,d)g(b,d)\right]
\right) \\
&&+\frac{1}{(s+1)\left( s+2\right) ^{2}}\left( \left[
f(a,c)g(b,c)+f(b,c)g(a,c)\right] +\left[ f(a,d)g(b,d)+f(b,d)g(a,d)\right]
\right) \\
&&+\frac{1}{(s+1)\left( s+2\right) ^{2}}\left( \left[
f(a,c)g(a,d)+f(b,c)g(b,d)\right] +\left[ f(a,d)g(a,c)+f(b,d)g(b,c)\right]
\right) \\
&&+\frac{1}{(s+1)^{2}(s+2)^{2}}\left( \left[ f(a,c)g(b,d)+f(b,c)g(a,d)\right]
+\left[ f(a,d)g(b,c)+f(b,d)g(a,c)\right] \right)
\end{eqnarray*
We can find the same result using by $f_{y}(x)$ and $g_{y}(x).$
\end{proof}
\begin{remark}
In (\ref{2.1}), if we choose $s=1$, (\ref{1.6}) is obtained.
\end{remark}
\begin{remark}
In (\ref{2.1}), if we choose $s=1$ and $f(x)=1$ which is convex, we get the
second inequality in (\ref{1.5})
\begin{equation*}
\frac{1}{\left( d-c\right) \left( b-a\right) }\dint\limits_{a}^{b}\din
\limits_{c}^{d}g(x,y)dxdy\leq \frac{g(a,c)+g(b,c)+g(a,d)+g(b,d)}{4}
\end{equation*}
\end{remark}
In the next theorem we will also make use of the Beta function of Euler
type, which is for $x,y>0$ defined a
\begin{equation*}
B(x,y)=\dint\limits_{0}^{1}t^{x-1}(1-t)^{y-1}dt=\frac{\Gamma (x)\Gamma (y)}
\Gamma (x+y)}
\end{equation*
and the Gamma function is defined a
\begin{equation*}
\Gamma (x)=\dint\limits_{0}^{\infty }t^{x-1}e^{-t}dt,\text{ for }x>0.
\end{equation*}
\begin{theorem}
Let $\ f:\Delta :=$ $[a,b]\times \lbrack c,d]\subset \lbrack 0,\infty
)^{2}\rightarrow \lbrack 0,\infty )$ be $s_{1}-$convex function on the
co-ordinates and $g:\Delta :=$ $[a,b]\times \lbrack c,d]\subset \lbrack
0,\infty )^{2}\rightarrow \lbrack 0,\infty )$ be $s_{2}-$convex functions on
the co-ordinates with $a<b,$ $c<d$ and $f_{x}(y)g_{x}(y),$
f_{y}(x)g_{y}(x)\in L^{1}[\Delta ]$ for some fixed $s_{1},s_{2}\in (0,1].$
Then one has the inequality
\begin{eqnarray}
&&\frac{1}{\left( d-c\right) \left( b-a\right) }\dint\limits_{a}^{b}\din
\limits_{c}^{d}f(x,y)g(x,y)dxdy \label{2.3} \\
&\leq &\frac{1}{\left( s_{1}+s_{2}+1\right) ^{2}}L(a,b,c,d)+\frac
B(s_{1}+1,s_{2}+1)}{s_{1}+s_{2}+1}M(a,b,c,d) \notag \\
&&+\left[ B(s_{1}+1,s_{2}+1)\right] ^{2}N(a,b,c,d) \notag \\
&=&\frac{1}{\left( s_{1}+s_{2}+1\right) ^{2}}\left[ L(a,b,c,d)+\frac
s_{1}s_{2}\Gamma (s_{1})\Gamma (s_{2})}{\Gamma (s_{1}+s_{2}+1)
M(a,b,c,d)\right. \notag \\
&&\left. +\left[ \frac{s_{1}s_{2}\Gamma (s_{1})\Gamma (s_{2})}{\Gamma
(s_{1}+s_{2}+1)}\right] ^{2}N(a,b,c,d)\right] \notag
\end{eqnarray
wher
\begin{eqnarray*}
L(a,b,c,d) &=&\left[ f(a,c)g(a,c)+f(b,c)g(b,c)+f(a,d)g(a,d)+f(b,d)g(b,d
\right] \\
M(a,b,c,d) &=&\left[ f(a,c)g(b,c)+f(b,c)g(a,c)+f(a,d)g(b,d)+f(b,d)g(a,d
\right] \\
&&+\left[ f(a,c)g(a,d)+f(b,c)g(b,d)+f(a,d)g(a,c)+f(b,d)g(b,c)\right] \\
N(a,b,c,d) &=&\left[ f(a,c)g(b,d)+f(b,c)g(a,d)+f(a,d)g(b,c)+f(b,d)g(a,c
\right]
\end{eqnarray*}
\end{theorem}
\begin{proof}
Since $f$ is co-ordinated $s_{1}-$convex and $g$ is co-ordinated $s_{2}-
convex, from Lemma 2, the partial mapping
\begin{eqnarray*}
f_{y} &:&[a,b]\rightarrow \lbrack 0,\infty ),\text{ }f_{y}(x)=f(x,y) \\
f_{x} &:&[c,d]\rightarrow \lbrack 0,\infty ),\text{ }f_{x}(y)=f(x,y)
\end{eqnarray*
are $s_{1}-$convex on $[a,b]$ and $[c,d],$ respectively, where $x\in \lbrack
a,b],$ $y\in \lbrack c,d].$ Similarly
\begin{eqnarray*}
g_{y} &:&[a,b]\rightarrow \lbrack 0,\infty ),\text{ }g_{y}(x)=g(x,y) \\
g_{x} &:&[c,d]\rightarrow \lbrack 0,\infty ),g_{x}(y)=g(x,y)
\end{eqnarray*
are $s_{2}-$convex on $[a,b]$ and $[c,d],$ respectively, where $x\in \lbrack
a,b],$ $y\in \lbrack c,d].$
Using (\ref{1.3}), we ge
\begin{eqnarray*}
\frac{1}{d-c}\dint\limits_{c}^{d}f_{x}(y)g_{x}(y)dy &\leq &\frac{1}
s_{1}+s_{2}+1}\left[ f_{x}(c)g_{x}(c)+f_{x}(d)g_{x}(d)\right] \\
&&+B(s_{1}+1,s_{2}+1)\left[ f_{x}(c)g_{x}(d)+f_{x}(d)g_{x}(c)\right]
\end{eqnarray*
Therefor
\begin{eqnarray*}
\frac{1}{d-c}\dint\limits_{c}^{d}f(x,y)g(x,y)dy &\leq &\frac{1}{s_{1}+s_{2}+
}\left[ f(x,c)g(x,c)+f(x,d)g(x,d)\right] \\
&&+B(s_{1}+1,s_{2}+1)\left[ f(x,c)g(x,d)+f(x,d)g(x,c)\right]
\end{eqnarray*
Dividing both sides of the above inequality by $(b-a)$ and integrating over
[a,b],$ we hav
\begin{eqnarray}
&&\frac{1}{\left( d-c\right) \left( b-a\right) }\dint\limits_{a}^{b}\din
\limits_{c}^{d}f(x,y)g(x,y)dxdy \label{2.4} \\
&\leq &\frac{1}{s_{1}+s_{2}+1}\left[ \frac{1}{b-a}\din
\limits_{a}^{b}f(x,c)g(x,c)dx+\frac{1}{b-a}\dint\limits_{a}^{b}f(x,d)g(x,d)d
\right] \notag \\
&&+B(s_{1}+1,s_{2}+1)\left[ \frac{1}{b-a}\dint\limits_{a}^{b}f(x,c)g(x,d)dx
\frac{1}{b-a}\dint\limits_{a}^{b}f(x,d)g(x,c)dx\right] \notag
\end{eqnarray
By applying (\ref{1.3}) to right side of (\ref{2.4}), and we proceed
similarly as in the proof of Theorem 7, we can writ
\begin{eqnarray*}
&&\frac{1}{\left( d-c\right) \left( b-a\right) }\dint\limits_{a}^{b}\din
\limits_{c}^{d}f(x,y)g(x,y)dxdy \\
&\leq &\frac{1}{\left( s_{1}+s_{2}+1\right) ^{2}}\left[
f(a,c)g(a,c)+f(b,c)g(b,c)+f(a,d)g(a,d)+f(b,d)g(b,d)\right] \\
&&+\frac{B(s_{1}+1,s_{2}+1)}{s_{1}+s_{2}+1}\left[
f(a,c)g(b,c)+f(b,c)g(a,c)+f(a,d)g(b,d)+f(b,d)g(a,d)\right] \\
&&+\frac{B(s_{1}+1,s_{2}+1)}{s_{1}+s_{2}+1}\left[
f(a,c)g(a,d)+f(b,c)g(b,d)+f(a,d)g(a,c)+f(b,d)g(b,c)\right] \\
&&+\left[ B(s_{1}+1,s_{2}+1)\right] ^{2}\left[
f(a,c)g(b,d)+f(b,c)g(a,d)+f(a,d)g(b,c)+f(b,d)g(a,c)\right]
\end{eqnarray*
That is
\begin{eqnarray*}
&&\frac{1}{\left( d-c\right) \left( b-a\right) }\dint\limits_{a}^{b}\din
\limits_{c}^{d}f(x,y)g(x,y)dxdy \\
&\leq &\frac{1}{\left( s_{1}+s_{2}+1\right) ^{2}}L(a,b,c,d)+\frac
B(s_{1}+1,s_{2}+1)}{s_{1}+s_{2}+1}M(a,b,c,d) \\
&&+\left[ B(s_{1}+1,s_{2}+1)\right] ^{2}N(a,b,c,d) \\
&=&\frac{1}{\left( s_{1}+s_{2}+1\right) ^{2}}\left[ L(a,b,c,d)+\frac
s_{1}s_{2}\Gamma (s_{1})\Gamma (s_{2})}{\Gamma (s_{1}+s_{2}+1)
M(a,b,c,d)\right. \\
&&\left. +\left[ \frac{s_{1}s_{2}\Gamma (s_{1})\Gamma (s_{2})}{\Gamma
(s_{1}+s_{2}+1)}\right] ^{2}N(a,b,c,d)\right]
\end{eqnarray*
which completes the proof.
\end{proof}
\begin{remark}
In (\ref{2.3}) if we choose $s_{1}=s_{2}=1,$ (\ref{2.3}) reduces to (\re
{1.6}).
\end{remark}
\begin{theorem}
Let $\ f:\Delta :=$ $[a,b]\times \lbrack c,d]\subset \lbrack 0,\infty
)^{2}\rightarrow \lbrack 0,\infty )$ be convex function on the co-ordinates
and $g:\Delta :=$ $[a,b]\times \lbrack c,d]\subset \lbrack 0,\infty
)^{2}\rightarrow \lbrack 0,\infty )$ be $s-$convex function on the
co-ordinates with $a<b,$ $c<d$ and $f_{x}(y)g_{x}(y),$ $f_{y}(x)g_{y}(x)\in
L^{1}[\Delta ]$ for some fixed $s\in (0,1].$ Then one has the inequality
\begin{eqnarray}
&&2^{2s+1}f(\frac{a+b}{2},\frac{c+d}{2})g(\frac{a+b}{2},\frac{c+d}{2})
\label{2.5} \\
&\leq &\frac{2}{\left( b-a\right) (d-c)}\dint\limits_{a}^{b}\din
\limits_{c}^{d}f(x,y)g(x,y)dxdy \notag \\
&&+\frac{5}{(s+1)(s+2)^{2}}L(a,b,c,d)+\frac{2s^{2}+6s+6}{(s+1)^{2}(s+2)^{2}
M(a,b,c,d) \notag \\
&&+\frac{2s+6}{(s+1)(s+2)^{2}}N(a,b,c,d) \notag
\end{eqnarray}
\end{theorem}
\begin{proof}
Since $f$ is co-ordinated convex and $g$ is co-ordinated $s-$convex, from
Lemma 1 and Lemma 2, the partial mapping
\begin{eqnarray}
f_{y} &:&[a,b]\rightarrow \lbrack 0,\infty ),\text{ }f_{y}(x)=f(x,y)
\label{2.6} \\
f_{x} &:&[c,d]\rightarrow \lbrack 0,\infty ),\text{ }f_{x}(y)=f(x,y) \notag
\end{eqnarray
are convex on $[a,b]$ and $[c,d],$ respectively, where $x\in \lbrack a,b],$
y\in \lbrack c,d]$. Similarly
\begin{eqnarray*}
g_{y} &:&[a,b]\rightarrow \lbrack 0,\infty ),\text{ }g_{y}(x)=g(x,y),\text{
y\in \lbrack c,d] \\
g_{x} &:&[c,d]\rightarrow \lbrack 0,\infty ),\text{ }g_{x}(y)=g(x,y),\text{
x\in \lbrack a,b]
\end{eqnarray*
are $s-$convex on $[a,b]$ and $[c,d],$ respectively, where $x\in \lbrack
a,b],$ $y\in \lbrack c,d].$
Using (\ref{1.4}) and multiplying both sides of the inequalities by $2^{s},$
we get
\begin{eqnarray}
&&2^{2s}f(\frac{a+b}{2},\frac{c+d}{2})g(\frac{a+b}{2},\frac{c+d}{2})
\label{2.7} \\
&&-\frac{2^{s}}{b-a}\dint\limits_{a}^{b}f(x,\frac{c+d}{2})g(x,\frac{c+d}{2
)dx \notag \\
&\leq &\frac{2^{s}}{(s+1)(s+2)}\left[ f(a,\frac{c+d}{2})g(a,\frac{c+d}{2
)+f(b,\frac{c+d}{2})g(b,\frac{c+d}{2})\right] \notag \\
&&+\frac{2^{s}}{s+2}\left[ f(a,\frac{c+d}{2})g(b,\frac{c+d}{2})+f(b,\frac{c+
}{2})g(a,\frac{c+d}{2})\right] \notag
\end{eqnarray
an
\begin{eqnarray}
&&2^{2s}f(\frac{a+b}{2},\frac{c+d}{2})g(\frac{a+b}{2},\frac{c+d}{2})
\label{2.8} \\
&&-\frac{2^{s}}{d-c}\dint\limits_{c}^{d}f(\frac{a+b}{2},y)g(\frac{a+b}{2
,y)dy \notag \\
&\leq &\frac{2^{s}}{(s+1)(s+2)}\left[ f(\frac{a+b}{2},c)g(\frac{a+b}{2},c)+f
\frac{a+b}{2},d)g(\frac{a+b}{2},d)\right] \notag \\
&&+\frac{2^{s}}{s+2}\left[ f(\frac{a+b}{2},c)g(\frac{a+b}{2},d)+f(\frac{a+b}
2},d)g(\frac{a+b}{2},c)\right] \notag
\end{eqnarray
Now, on adding (\ref{2.7}) and (\ref{2.8}), we get
\begin{eqnarray}
&&2^{2s+1}f(\frac{a+b}{2},\frac{c+d}{2})g(\frac{a+b}{2},\frac{c+d}{2})
\label{2.9} \\
&&-\frac{2^{s}}{b-a}\dint\limits_{a}^{b}f(x,\frac{c+d}{2})g(x,\frac{c+d}{2
)dx-\frac{2^{s}}{d-c}\dint\limits_{c}^{d}f(\frac{a+b}{2},y)g(\frac{a+b}{2
,y)dy \notag \\
&\leq &\frac{1}{(s+1)(s+2)}\left[ 2^{s}f(a,\frac{c+d}{2})g(a,\frac{c+d}{2
)+2^{s}f(b,\frac{c+d}{2})g(b,\frac{c+d}{2})\right] \notag \\
&&+\frac{1}{s+2}\left[ 2^{s}f(a,\frac{c+d}{2})g(b,\frac{c+d}{2})+2^{s}f(b
\frac{c+d}{2})g(a,\frac{c+d}{2})\right] \notag \\
&&+\frac{1}{(s+1)(s+2)}\left[ 2^{s}f(\frac{a+b}{2},c)g(\frac{a+b}{2
,c)+2^{s}f(\frac{a+b}{2},d)g(\frac{a+b}{2},d)\right] \notag \\
&&+\frac{1}{s+2}\left[ 2^{s}f(\frac{a+b}{2},c)g(\frac{a+b}{2},d)+2^{s}f
\frac{a+b}{2},d)g(\frac{a+b}{2},c)\right] \notag
\end{eqnarray
Applying (\ref{1.4}) to each term of right hand side of the above
inequality, we hav
\begin{eqnarray*}
&&2^{s}f(a,\frac{c+d}{2})g(a,\frac{c+d}{2}) \\
&\leq &\frac{1}{d-c}\dint\limits_{c}^{d}f(a,y)g(a,y)dy+\frac{1}{(s+1)(s+2)
\left[ f(a,c)g(a,c)+f(a,d)g(a,d)\right] \\
&&+\frac{1}{s+2}\left[ f(a,c)g(a,d)+f(a,d)g(a,c)\right]
\end{eqnarray*
\begin{eqnarray*}
&&2^{s}f(b,\frac{c+d}{2})g(b,\frac{c+d}{2}) \\
&\leq &\frac{1}{d-c}\dint\limits_{c}^{d}f(b,y)g(b,y)dy+\frac{1}{(s+1)(s+2)
\left[ f(b,c)g(b,c)+f(b,d)g(b,d)\right] \\
&&+\frac{1}{s+2}\left[ f(b,c)g(b,d)+f(b,d)g(b,c)\right]
\end{eqnarray*
\begin{eqnarray*}
&&2^{s}f(a,\frac{c+d}{2})g(b,\frac{c+d}{2}) \\
&\leq &\frac{1}{d-c}\dint\limits_{c}^{d}f(a,y)g(b,y)dy+\frac{1}{(s+1)(s+2)
\left[ f(a,c)g(b,c)+f(a,d)g(b,d)\right] \\
&&+\frac{1}{s+2}\left[ f(a,c)g(b,d)+f(a,d)g(b,c)\right]
\end{eqnarray*
\begin{eqnarray*}
&&2^{s}f(b,\frac{c+d}{2})g(a,\frac{c+d}{2}) \\
&\leq &\frac{1}{d-c}\dint\limits_{c}^{d}f(b,y)g(a,y)dy+\frac{1}{(s+1)(s+2)
\left[ f(b,c)g(a,c)+f(b,d)g(a,d)\right] \\
&&+\frac{1}{s+2}\left[ f(b,c)g(a,d)+f(b,d)g(a,c)\right]
\end{eqnarray*
\begin{eqnarray*}
&&2^{s}f(\frac{a+b}{2},c)g(\frac{a+b}{2},c) \\
&\leq &\frac{1}{b-a}\dint\limits_{a}^{b}f(x,c)g(x,c)dx+\frac{1}{(s+1)(s+2)
\left[ f(a,c)g(a,c)+f(b,c)g(b,c)\right] \\
&&+\frac{1}{s+2}\left[ f(a,c)g(b,c)+f(b,c)g(a,c)\right]
\end{eqnarray*
\begin{eqnarray*}
&&2^{s}f(\frac{a+b}{2},d)g(\frac{a+b}{2},d) \\
&\leq &\frac{1}{b-a}\dint\limits_{a}^{b}f(x,d)g(x,d)dx+\frac{1}{(s+1)(s+2)
\left[ f(a,d)g(a,d)+f(b,d)g(b,d)\right] \\
&&+\frac{1}{s+2}\left[ f(a,d)g(b,d)+f(b,d)g(a,d)\right]
\end{eqnarray*
\begin{eqnarray*}
&&2^{s}f(\frac{a+b}{2},c)g(\frac{a+b}{2},d) \\
&\leq &\frac{1}{b-a}\dint\limits_{a}^{b}f(x,c)g(x,d)dx+\frac{1}{(s+1)(s+2)
\left[ f(a,c)g(a,d)+f(b,c)g(b,d)\right] \\
&&+\frac{1}{s+2}\left[ f(a,c)g(b,d)+f(b,c)g(a,d)\right]
\end{eqnarray*
\begin{eqnarray*}
&&2^{s}f(\frac{a+b}{2},d)g(\frac{a+b}{2},c) \\
&\leq &\frac{1}{b-a}\dint\limits_{a}^{b}f(x,d)g(x,c)dx+\frac{1}{(s+1)(s+2)
\left[ f(a,d)g(a,c)+f(b,d)g(b,c)\right] \\
&&+\frac{1}{s+2}\left[ f(a,d)g(b,c)+f(b,d)g(a,c)\right]
\end{eqnarray*
Using these inequalities in (\ref{2.9}), we hav
\begin{eqnarray}
&&2^{2s+1}f(\frac{a+b}{2},\frac{c+d}{2})g(\frac{a+b}{2},\frac{c+d}{2})
\label{2.10} \\
&&-\frac{2^{s}}{b-a}\dint\limits_{a}^{b}f(x,\frac{c+d}{2})g(x,\frac{c+d}{2
)dx-\frac{2^{s}}{d-c}\dint\limits_{c}^{d}f(\frac{a+b}{2},y)g(\frac{a+b}{2
,y)dy \notag \\
&\leq &\frac{1}{(s+1)(s+2)}\frac{1}{\left( d-c\right) }\left[
\dint\limits_{c}^{d}f(a,y)g(a,y)dy+\dint\limits_{c}^{d}f(b,y)g(b,y)dy\right]
\notag \\
&&+\frac{1}{\left( s+2\right) }\frac{1}{\left( d-c\right) }\left[
\dint\limits_{c}^{d}f(a,y)g(b,y)dy+\dint\limits_{c}^{d}f(b,y)g(a,y)dy\right]
\notag \\
&&+\frac{1}{(s+1)(s+2)}\frac{1}{\left( b-a\right) }\left[ \din
\limits_{a}^{b}f(x,c)g(x,c)dx+\dint\limits_{a}^{b}f(x,d)g(x,d)dx\right]
\notag \\
&&+\frac{1}{\left( s+2\right) }\frac{1}{\left( b-a\right) }\left[
\dint\limits_{a}^{b}f(x,c)g(x,d)dx+\dint\limits_{a}^{b}f(x,d)g(x,c)dx\right]
\notag \\
&&+\frac{2}{(s+1)^{2}(s+2)^{2}}L(a,b,c,d)+\frac{2}{(s+1)\left( s+2\right)
^{2}}M(a,b,c,d) \notag \\
&&+\frac{2}{\left( s+2\right) ^{2}}N(a,b,c,d) \notag
\end{eqnarray
Now by applying (\ref{1.4}) to $2^{s}f(\frac{a+b}{2},y)g(\frac{a+b}{2},y),$
integrating over $[c,d]$, and dividing both sides by $(d-c),$ we get
\begin{eqnarray}
&&\frac{2^{s}}{(d-c)}\dint\limits_{c}^{d}f(\frac{a+b}{2},y)g(\frac{a+b}{2
,y)dy \label{2.11} \\
&&-\frac{1}{\left( b-a\right) (d-c)}\dint\limits_{a}^{b}\din
\limits_{c}^{d}f(x,y)g(x,y)dxdy \notag \\
&\leq &\frac{1}{(s+1)(s+2)}\left[ \frac{1}{(d-c)}\din
\limits_{c}^{d}f(a,y)g(a,y)dy+\frac{1}{(d-c)}\din
\limits_{c}^{d}f(b,y)g(b,y)dy\right] \notag \\
&&+\frac{1}{s+2}\left[ \frac{1}{(d-c)}\dint\limits_{c}^{d}f(a,y)g(b,y)dy
\frac{1}{(d-c)}\dint\limits_{c}^{d}f(b,y)g(a,y)dy\right] \notag
\end{eqnarray
Similarly by applying (\ref{1.4}) to $2^{s}f(x,\frac{c+d}{2})g(x,\frac{c+d}{
}),$ integrating over $[a,b]$, dividing both sides by $(b-a),$ we ge
\begin{eqnarray}
&&\frac{2^{s}}{(b-a)}\dint\limits_{a}^{b}f(x,\frac{c+d}{2})g(x,\frac{c+d}{2
)dx \label{2.12} \\
&&-\frac{1}{\left( b-a\right) (d-c)}\dint\limits_{a}^{b}\din
\limits_{c}^{d}f(x,y)g(x,y)dxdy \notag \\
&\leq &\frac{1}{(s+1)(s+2)}\left[ \frac{1}{\left( b-a\right)
\dint\limits_{a}^{b}f(x,c)g(x,c)dx+\frac{1}{\left( b-a\right)
\dint\limits_{a}^{b}f(x,d)g(x,d)dx\right] \notag \\
&&+\frac{1}{s+2}\left[ \frac{1}{\left( b-a\right) }\din
\limits_{a}^{b}f(x,c)g(x,d)dx+\frac{1}{\left( b-a\right)
\dint\limits_{a}^{b}f(x,d)g(x,c)dx\right] \notag
\end{eqnarray
By addition (\ref{2.11}) and (\ref{2.12}), we hav
\begin{eqnarray}
&&\frac{2^{s}}{(d-c)}\dint\limits_{c}^{d}f(\frac{a+b}{2},y)g(\frac{a+b}{2
,y)dy+\frac{2^{s}}{(b-a)}\dint\limits_{a}^{b}f(x,\frac{c+d}{2})g(x,\frac{c+
}{2})dx \notag \\
&&-\frac{2}{\left( b-a\right) (d-c)}\dint\limits_{a}^{b}\din
\limits_{c}^{d}f(x,y)g(x,y)dxdy \label{2.13} \\
&\leq &\frac{1}{(s+1)(s+2)}\left[ \frac{1}{(d-c)}\din
\limits_{c}^{d}f(a,y)g(a,y)dy+\frac{1}{(d-c)}\din
\limits_{c}^{d}f(b,y)g(b,y)dy\right. \notag \\
&&\left. +\frac{1}{\left( b-a\right) }\dint\limits_{a}^{b}f(x,c)g(x,c)dx
\frac{1}{\left( b-a\right) }\dint\limits_{a}^{b}f(x,d)g(x,d)dx\right] \notag
\\
&&+\frac{1}{s+2}\left[ \frac{1}{(d-c)}\dint\limits_{c}^{d}f(a,y)g(b,y)dy
\frac{1}{(d-c)}\dint\limits_{c}^{d}f(b,y)g(a,y)dy\right. \notag \\
&&\left. +\frac{1}{\left( b-a\right) }\dint\limits_{a}^{b}f(x,c)g(x,d)dx
\frac{1}{\left( b-a\right) }\dint\limits_{a}^{b}f(x,d)g(x,c)dx\right] \notag
\end{eqnarray
From (\ref{2.10}) and (\ref{2.13}) and simplifying we ge
\begin{eqnarray*}
&&2^{2s+1}f(\frac{a+b}{2},\frac{c+d}{2})g(\frac{a+b}{2},\frac{c+d}{2})\leq
\frac{2}{\left( b-a\right) (d-c)}\dint\limits_{a}^{b}\din
\limits_{c}^{d}f(x,y)g(x,y)dx \\
&&+\frac{4s+6}{(s+1)^{2}(s+2)^{2}}L(a,b,c,d)+\frac{2s^{2}+6s+6}
(s+1)^{2}(s+2)^{2}}M(a,b,c,d) \\
&&+\frac{2s^{2}+8s+6}{(s+1)^{2}(s+2)^{2}}N(a,b,c,d)
\end{eqnarray*}
\end{proof}
\begin{remark}
In (\ref{2.5}), if we choose $s=1,$ we obtained (\ref{1.7}).
\end{remark}
\begin{remark}
In (\ref{2.5}), if we choose $s=1$ and $f(x)=1$ which is convex, we have the
following Hadamard-type inequality like (\ref{1.5})
\begin{eqnarray*}
&&4g(\frac{a+b}{2},\frac{c+d}{2})-\frac{1}{\left( b-a\right) (d-c)
\dint\limits_{a}^{b}\dint\limits_{c}^{d}g(x,y)dx \\
&\leq &\frac{3\left[ g(a,c)+g(b,c)+g(a,d)+g(b,d)\right] }{4}
\end{eqnarray*}
\end{remark}
\begin{theorem}
Let $\ f,g:\Delta :=$ $[a,b]\times \lbrack c,d]\subset
\mathbb{R}
^{2}\rightarrow
\mathbb{R}
$ be convex function on the co-ordinates with $a<b,$ $c<d$ and
f_{x}(y)g_{x}(y),$ $\ f_{y}(x)g_{y}(x)\in L^{1}[\Delta ].$ Then one has the
inequality
\begin{eqnarray*}
&&\frac{1}{\left( b-a\right) ^{2}\left( d-c\right) ^{2}}\left[ f\left(
a,c\right) \dint\limits_{a}^{b}\dint\limits_{c}^{d}\left( x-b\right) \left(
y-d\right) g(x,y)dydx\right. \\
&&+f\left( b,c\right) \dint\limits_{a}^{b}\dint\limits_{c}^{d}\left(
a-x\right) \left( y-d\right) g(x,y)dydx+f\left( a,d\right)
\dint\limits_{a}^{b}\dint\limits_{c}^{d}\left( x-b\right) \left( c-y\right)
g(x,y)dydx \\
&&+f\left( b,d\right) \dint\limits_{a}^{b}\dint\limits_{c}^{d}\left(
a-x\right) \left( c-y\right) g(x,y)dydx+g\left( a,c\right)
\dint\limits_{a}^{b}\dint\limits_{c}^{d}\left( x-b\right) \left( y-d\right)
f(x,y)dydx \\
&&+g\left( b,c\right) \dint\limits_{a}^{b}\dint\limits_{c}^{d}\left(
a-x\right) \left( y-d\right) f(x,y)dydx+g\left( a,d\right)
\dint\limits_{a}^{b}\dint\limits_{c}^{d}\left( x-b\right) \left( c-y\right)
f(x,y)dydx \\
&&\left. +g\left( b,d\right) \dint\limits_{a}^{b}\dint\limits_{c}^{d}\left(
a-x\right) \left( c-y\right) f(x,y)dydx\right] \\
&\leq &\frac{1}{\left( b-a\right) (d-c)}\dint\limits_{a}^{b}\din
\limits_{c}^{d}f(x,y)g(x,y)dx+\frac{1}{9}L(a,b,c,d)+\frac{1}{18}M(a,b,c,d)
\frac{1}{36}N(a,b,c,d)
\end{eqnarray*
where $L(a,b,c,d),$ $M(a,b,c,d),$ $N(a,b,c,d)$ defined as in Theorem 6.
\end{theorem}
\begin{proof}
Since $f$ and $g$ are co-ordinated convex functions on the co-ordinates on
\Delta $, from the definition of co-ordinated convexity, we can writ
\begin{equation*}
f(ta+(1-t)b,sc+(1-s)d)\leq
tsf(a,c)+t(1-s)f(a,d)+s(1-t)f(b,c)+(1-t)(1-s)f(b,d)
\end{equation*
an
\begin{equation*}
g(ta+(1-t)b,sc+(1-s)d)\leq
tsg(a,c)+t(1-s)g(a,d)+s(1-t)g(b,c)+(1-t)(1-s)g(b,d)
\end{equation*}
holds for all $t,s\in \lbrack 0,1]$. By using the elementary inequality, if
e\leq f$ and $p\leq r,$ then $er+fp\leq ep+fr$ for all $e,f,p,r\in
\mathbb{R}
,$ we ge
\begin{eqnarray*}
&&f(ta+(1-t)b,sc+(1-s)d) \\
&&\times \left[ tsg(a,c)+t(1-s)g(a,d)+s(1-t)g(b,c)+(1-t)(1-s)g(b,d)\right] \\
&&+g(ta+(1-t)b,sc+(1-s)d) \\
&&\times \left[ tsf(a,c)+t(1-s)f(a,d)+s(1-t)f(b,c)+(1-t)(1-s)f(b,d)\right] \\
&\leq &\left[ f(ta+(1-t)b,sc+(1-s)d)g(ta+(1-t)b,sc+(1-s)d)\right] \\
&&+\left[ tsf(a,c)+t(1-s)f(a,d)+s(1-t)f(b,c)+(1-t)(1-s)f(b,d)\right] \\
&&\times \left[ tsg(a,c)+t(1-s)g(a,d)+s(1-t)g(b,c)+(1-t)(1-s)g(b,d)\right] .
\end{eqnarray*
By integrating the above integral on $\left[ 0,1\right] \times \left[ 0,
\right] $, with respect to $t$, $s$ and by taking into account the change of
variables $ta+(1-t)b=x,$ $(a-b)dt=dx$ and $sc+(1-s)d=y,$ $(c-d)ds=dy$, we
obtain the desired result.
\end{proof}
|
{
"timestamp": "2011-04-29T02:01:05",
"yymm": "1009",
"arxiv_id": "1009.4085",
"language": "en",
"url": "https://arxiv.org/abs/1009.4085"
}
|
\section{Introduction}
It has been known for a long time that rotation curves of neutral hydrogen
clouds in the outer regions of galaxies cannot be explained in terms of the
luminous matter content of the galaxies, at least within the context of
Newtonian gravity or of general relativity (see [1] for a review). In
addition, the velocity dispersion in galaxy clusters indicates a much greater
mass than would be inferred from luminous matter contained in the individual
galaxies. The most widely accepted explanation, based on the standard
gravitational theory, postulates that almost every galaxy hosts a large amount
of nonluminous matter, the so called gravitational dark matter, consisting of
unknown particles not included in the particle standard model, forming a halo
around the galaxy. The dark matter provides the needed gravitational field and
the mass required to match the observed galactic flat rotation curves in
galaxy clusters.
The dark matter problem arises because of a na\"{\i}ve Newtonian analysis of
the near constant tangential velocity of rotation up to distances far beyond
the luminous radius of the galaxies. This leads to the conclusion that the
energy density decreases with the distance as $r^{-2}$ and therefore that the
mass of galaxies increases as $m(r)\propto r$.
There are several proposals for the dark matter component, ranging from new
exotic particles such as those predicted by supersymmetry [2] to other less
exotic candidates such as massive neutrinos or even ordinary celestial bodies
such as Jupiter like objects. Several other analytic halo models exist in the
literature including those sourced by scalar fields. Fay [3] considered
modelling by Brans-Dicke massless scalar field while Matos, Guzm\'{a}n and
Nu\~{n}ez [4] considered massless minimally coupled scalar field with a
potential. A boson star formed by a self interacting massive scalar field with
quartic interaction potential as a model of galactic halo was investigated by
Colpi, Shapiro and Wasserman [5]. A similar boson star as a model of galactic
halo was first investigated by Lee and Koh [6]. A recent halo model is
discussed also in the braneworld theory [7]. Some authors [8] have considered
global monopoles as a candidate for galactic halo in the framework of the
scalar tensor theory of gravity.
Alternatively, explanation of the observed flat rotation curve is provided
through modification of Newtonian dynamics (in the region of very small
accelerations) [9,10]. Another attempt to resolve the dark matter problem lies
within the framework of Weyl conformal gravity suggested by Mannheim and
Kazanas [11], with the distinction that it does not postulate flat rotation
curve as an input but \textit{predict} it. Such a prediction is possible also
within the Chern Simmons gravity inspired by string theory [12,13].
In recent years, the amount of dark matter in the Universe has become known
more precisely: CMB anisotropy data indicate that about $85\%$ of total matter
in a galaxy is dark in nature. Big bang nucleosynthesis and some other
cosmological observations require that the bulk of dark matter be
non-baryonic, cold or warm, stable or long-lived and not interacting with
visible matter. However, despite long and intensive investigations, little is
as yet known about the nature of dark matter.
The purpose of the present work is to show that, with the input of flat
rotation curve and the assumption that dark matter be described as a perfect
fluid, the general theory of relativity not only accounts for the dark matter
component of galactic clusters but also suggests the spatial curvature of the
universe. The gravitational influence of an arbitrary dark matter component is
controlled by its stress tensor. In this context we note that, while the
existing approaches of explaining flat rotation curves are useful in their own
right, many of the models end up with predicting anisotropic dark matter fluid
stress tensor. On the other hand, no physical mechanism is stated explaining
why a spherical distribution should have such anisotropy. There is also no
observational evidence in support of it. Therefore, it seems more reasonable
to consider an isotropic perfect fluid distribution for dark matter because
predictions from such model at stellar and cosmic scales have been
observationally corroborated beyond any doubt.
We shall particularly study the general features of dark matter such as its
equation of state. The mass distribution in galactic haloes should provide a
direct probe into the nature of particles constituting the dark matter because
the inner structure of the halo is particularly sensitive to the dark matter
properties [14]. In this investigation, we shall work only from a fluid
perspective leaving the question about the particle identity of dark matter open.
\section{Gravitational field in the dark matter region}
Galactic rotational velocity profiles [15] of almost all the spiral galaxies
are characterized by a rapid increase from the galactic center, reaching a
nearly constant velocity from the nearby region of the galaxy far out to the
halo region. Our target is to exploit this observed feature to obtain the
space-time metric in the halo region, and analyze it. As mentioned already,
observational data suggest that the dark matter component in the galaxy
accounts for almost $85\%$ of its total mass. Naturally, luminous matter does
not contribute significantly to the total energy density of the galaxy,
particularly in the halo region. Therefore, we shall treat the matter in the
galactic halo region as a perfect fluid defined by stresses $T_{r
^{r}=T_{\theta}^{\theta}=T_{\phi}^{\phi}=p$, where $T_{\nu}^{\mu}$ is the
matter energy momentum tensor.
The general static spherically symmetric spacetime is represented by the
following metri
\begin{equation}
ds^{2}=-e^{\nu(r)}dt^{2}+e^{\lambda(r)}dr^{2}+r^{2}(d\theta^{2}+sin^{2}\theta
d\phi^{2}),
\end{equation}
where the functions$\ \nu(r)$ and $\lambda(r)$ are the metric potentials. Then
the Einstein field equations become ($c=1$)
\begin{equation}
e^{-\lambda}\left[ \frac{\lambda^{\prime}}{r}-\frac{1}{r^{2}}\right]
+\frac{1}{r^{2}}=8\pi G\rho
\end{equation}
\begin{equation}
e^{-\lambda}\left[ \frac{1}{r^{2}}+\frac{\nu^{\prime}}{r}\right] -\frac
{1}{r^{2}}=8\pi Gp
\end{equation}
\begin{equation}
\frac{1}{2}e^{-\lambda}\left[ \frac{1}{2}(\nu^{\prime})^{2}+\nu^{\prime
\prime}-\frac{1}{2}\lambda^{\prime}\nu^{\prime}+\frac{1}{r}({\nu^{\prime
}-\lambda^{\prime}})\right] =8\pi Gp.
\end{equation}
For a circular stable geodesic motion in the equatorial plane, the
consideration of flat rotation curve gives the condition (see Appendix
\begin{equation}
e^{\nu}=B_{0}r^{l},
\end{equation}
where $l$ is given by $l=2(v^{\phi})^{2}$ and $B_{0}$ is an integration
constant. The observed rotational curve profile in the region dominated by
dark matter is such that the rotational velocity $v^{\phi}$ becomes
approximately a constant with $v^{\phi}\sim10^{-3}$ ( 300km/s) for a typical
galaxy. The Eqs. (3) and (4) then lead to the following equatio
\begin{equation}
(e^{-\lambda})^{\prime}+\frac{ae^{-\lambda}}{r}=\frac{c}{r},
\end{equation}
wher
\begin{equation}
a=-\frac{4(1+l)-l^{2}}{2+l
\end{equation}
an
\begin{equation}
c=-\frac{4}{2+l}.
\end{equation}
An exact solution of Eq.(6) is given by
\begin{equation}
e^{-\lambda}=\frac{c}{a}+\frac{D}{r^{a}},
\end{equation}
where $D$ is an integration constant. Several observations follow from the
metric (1):
\textbf{(a)} Note that the space time metric given by Eq.(1) through Eqs. (5)
and (9) is an interior solution. This type of spacetime definitely cannot be
asymptotically flat neither can it have the form of a spacetime due to a
centrally symmetric black hole. What can be said is that this line element
describes the region where the tangential velocity of the test particles is
constant and that it has to be joined with the exterior region with other
types of space time. The extent of $v^{\phi}$= constant ends at some larger
distance where the region becomes asymptotically flat. In principle, the
constant $D$ should be obtained from the junction conditions but the galactic
boundary is not observationally defined yet. It is more likely that dark
matter distribution will not remain the same near the periphery of the halo.
\textbf{(b)} Inserting the above solution in Eqs. (2) - (4), one can readily
get the expressions for $\rho$, $p$ a
\begin{equation}
\rho= \frac{1}{8 \pi G } \left[ \frac{l(4-l)}{4+4l-l^{2}} r^{-2} -
\frac{D(6-l)(1+l)}{2+l} r^{l(2-l)/(2+l)}\right]
\end{equation}
\begin{equation}
p=\frac{1}{8\pi G}\left[ \frac{l^{2}}{4+4l-l^{2}}r^{-2
+D(1+l)r^{l(2-l)/(2+l)}\right]
\end{equation}
The most notable aspect is the presence of the last term in the expressions of
both energy density $\rho$ and pressure $p$ that increases with radial
distance (from galactic center). Note that according to the Newtonian theory,
which is supposed to be indistinguishable from general relativity in very weak
field, one should expect only the term $\rho_{\text{Newton}}=\frac{1}{8\pi
G}\frac{l}{r^{2}}$ in agreement with the Poisson equation, which integrates to
give the Newtonian mass $M(r)$ increasing linearly with $r$. But the input of
the constant tangential velocity leads to general relativistic corrections to
Newtonian expressions as evident in Eqs. (10), (11). The first term of the
right hand side of Eq.(10) gives the expected Newtonian term in the leading
order and to the same order a general relativistic correction term $\frac
{1}{8\pi G}\frac{5l^{2}}{4r^{2}}$. Since $l$ is small, the correction terms
are small as expected.
However, the second term in Eq.(10) has completely different nature than the
conventional feature of dark matter energy density; the corresponding mass
increases \textit{nonlinearly }with radial distance $r$, though a bit slowly
as $l$ is very small. This contribution will vanish if $D$ vanishes. We would
see that even $D=0$ is consistent with the flat rotation curve.
\textbf{(c)} The parameter $D$ is recognized [17] as the spatial curvature
(with a negative sign) of the universe. A comparison of the obtained space
time metric in the limit $l\rightarrow0$ with the static
Friedmann--Lem\^{a}itre-Robertson--Walker (FLRW) metric [ $ds^{2
=-dt^{2}+\frac{1}{1-kr^{2}}dr^{2}+r^{2}\left( d\theta^{2}+sin^{2}\theta
d\phi^{2}\right) $ ] leads to the identification that $D=-k$. Since our input
was only the flat rotation curve and we have not considered anything regarding
cosmological spatial curvature while deriving the metric, the appearance of
$D$ in the metric element is quite interesting. The solution thus may be
thought of dark matter induced space-time embedded in a static FLRW metric. In
general $l\neq0$ and hence $a$ is not equal to $-2$, and the interpretation of
spatial curvature does not seem at all evident.
An immediate question is how a spatially curved FLRW universe could be static.
If it is due to balance of the curvature term by some other fluid with
equation of state $\rho=-3p$, the question of stability will arise. It seems
that we have ended up with a static universe because right from the beginning
we looked for a static solution (all the metric element are considered time
independent). While working on a local problem (flat rotation curve), the
universe is usually considered at any particular epoch fixing the scale factor
$R(t)$ to be constant (often normalized to unity, $R(t_{0})=1$ at the present
epoch). The dynamicity of the FLRW metric is thus absent in local
gravitational phenomena. However, we see in the present work that the
curvature effect has appeared even in local gravitational phenomena. It would
certainly be an interesting but challenging task to study the stability
criterion in this configuration [17]. We reserve it for a separate communication.
\textbf{(d)} The equation of state parameter $\omega$ for the effective fluid
(dark matter plus \lq curvature fluid \rq), can be obtained directly from
Eqs.(10) and (11), which is given b
\begin{equation}
\frac{p}{\rho}=\omega=\frac{l^{2}r^{a}+D(1+l)(4+4l-l^{2})}{l(4-l)r^{a
-D(6-l)(1+l)(4+4l-l^{2})/(2+l)}.
\end{equation}
Since $l$ is small, effectively $\omega\simeq\frac{l^{2}r^{a}+4D}{4lr^{a
-12D}$. If the total matter in the flat rotation curve region of galaxies has
to be non-exotic, i.e. if the dark matter satisfies the known energy
conditions, $\omega$ for the total matter content must be positive. This is
achieved only if we set $l^{2}r^{a}/4<D<lr^{a}/3$. Note that our discussion is
restricted only in the region of flat rotation curve of galaxies for which $r$
is typically between few ten kpc to few hundred kpc and as mentioned already
$v^{\phi}$ is few hundred km/s. Hence positive $\omega$ implies that $D$ is
nearly zero, if not exactly. Figure 1 shows that variation of $\omega$ with
$D$ for a typical distance $r=200\;kpc$ and $v^{\phi}=300\;km/s$.
\begin{figure}[ptb]
\begin{center}
\vspace{0.5cm} \includegraphics[width=0.5\textwidth]{fig1.eps}
\end{center}
\caption{The variation of $\omega$ with $D$ for a typical distance $r=200$ in
$Kpc$.
\label{fig3
\end{figure}
As may be seen from the figure 1, the limit on $D$ is much stringent compared
to the (cosmological) observational restriction. The equation of state of the
dark matter component may be obtained by taking $D=0$ in the above equation
which give
\begin{equation}
\omega=\frac{l}{(4-l)}\simeq2.5\times10^{-7}\text{.
\end{equation}
Thus $p<<\rho$ for dark matter implying its non-relativistic nature, which is
a well known fact. Further $\omega$ is positive, which means that the fluid is
non-exotic. On the other hand, the equation of state of `curvature fluid 'is
found to be $\rho+3p=0$ from Eq.(12) in the limit $l\rightarrow0$, which is a
familiar result.
\textbf{(e)} Note that we can rewrite $e^{\lambda}$ \ in the standard
Schwarzschild for
\begin{equation}
e^{\lambda}=\left[ 1-\frac{2m(r)}{r}\right] ^{-1
\end{equation}
which is often convenient. Such a form has the advantage that it immediately
reveals not only the mass parameter $m(r)$ but also shows that the proper
radial length is larger than the Euclidean length because $r>2m(r)$. But most
importantly, this inequality is essential for signature protection, which
dictates that $e^{\lambda}>1$. This is a crucial condition to be satisfied by
any valid metric. Now, from the metric function (9), for $D=0$, we ge
\begin{equation}
e^{\lambda}=1+\frac{4l-l^{2}}{4}>1.
\end{equation}
This implies that the essential requirement is fulfilled for this value of $D
$.
\textbf{(f)} Let us rewrite the metric (1) for $D=0$ under the radial
rescalin
\begin{equation}
r=\sqrt{\frac{c}{a}}r^{\prime},
\end{equation}
which yield
\[
ds^{2}=-B_{0}^{\prime}r^{\prime l}dt^{2}+dr^{\prime2}+\left( \frac{c
{a}\right) r^{\prime2}(d\theta^{2}+sin^{2}\theta d\phi^{2}),\newlin
\]
\begin{equation}
\text{\ }B_{0}^{\prime}=B_{0}\left( \sqrt{\frac{c}{a}}\right) ^{l
\end{equation}
showing a surplus angle in the surface area given by
\begin{equation}
S_{1}=4\pi r^{2}.\frac{c}{a}=\frac{4\pi r^{2}}{1+2(v^{\phi})^{2}-(v^{\phi
})^{4}}.
\end{equation}
If the probe particles were photons, so that $v^{\phi}=1$, the surface area
would remain \textit{finite} but reduced to half of the spherical surface area
$S_{2}=4\pi r^{2}$. This is an interesting result, which distinguishes itself
from that in the massless scalar field model where $S\rightarrow\infty$ as
$v^{\phi}=1$ [4]. For a typical rotational velocity of $v^{\phi}=10^{-3}$ in
the galactic halo region, the difference of the two surface areas
\begin{equation}
S_{2}-S_{1}=4\pi r^{2}\left[ \frac{2(v^{\phi})^{2}-(v^{\phi})^{4
}{1+2(v^{\phi})^{2}-(v^{\phi})^{4}}\right]
\end{equation}
grows as $\sim10^{-6}$ in units of flat surface area, which indicates another
deviation from the massless scalar model in which it grows as $\sim10^{-12}$ [4].
\textbf{(g)} The Ricci scalar for the derived spacetime is given by
\begin{equation}
R=\frac{Da^{2}(4+l)-(aD+cr^{a})(l^{2}+2l+4)+4ar^{a}}{2ar^{2+a}
\end{equation}
As $l\rightarrow0$, $R=-6D$, once again suggesting that $D$ is the spatial
curvature. We plot $R$ vs $r$. One may note that the value of $R$ is small.
For small different values of $D$, one can not distinguish the variations of
$R$ with respect to $r$ ( see figure 2 ). But for higher values of $D$, figure
3 shows a sharp variation of $R$. Note that large values of $D$ imply
$\omega(r)<0$, i.e., dark matter has to be exotic in nature.
\begin{figure}[ptb]
\begin{center}
\vspace{0.5cm} \includegraphics[width=0.4\textwidth]{fig3.eps}
\end{center}
\caption{The variation of $R$ with $r$ in $Kpc$ for small different values of
D. We choose $v^{\phi}\sim10^{-3}$ (300km/s) for a typical galaxy.
\label{fig3
\end{figure}\begin{figure}[ptb]
\begin{center}
\vspace{0.5cm} \includegraphics[width=0.4\textwidth]{fig10.eps}
\end{center}
\caption{The variation of $R$ with $r$ in $Kpc$ for large different values of
D. We choose $v^{\phi}\sim10^{-3}$ (300km/s) for a typical galaxy.
\label{fig3
\end{figure}
\section{Other aspects}
\subsection{ \underline{\textbf{\ The total gravitational energy}}:}
One notes from equation (12) that the halo matter is not exotic in nature and
consequently, we expect attractive gravity in the halo. Following the
suggestion given by Lyndell - Bell et al [18] , we calculate the total
gravitational energy $E_{G}$ between two fixed radii , say, $r_{1}$ and
$r_{2}$:
\begin{align}
E_{G} & =M-E_{M}=4\pi\int_{r_{1}}^{r_{2}}[1-\sqrt{e^{\lambda(r)}}]\rho
r^{2}dr\newline\\
& =4\pi\int_{r_{1}}^{r_{2}}\left[ 1-\sqrt{\frac{1}{\frac{c}{a}+\frac{D
{r^{a}}}}\right] \left[ \frac{1}{8\pi G}\left( \frac{D(a-1)}{r^{a+2}
+\frac{(1-\frac{c}{a})}{r^{2}}\right) \right] r^{2}dr,
\end{align}
where
\begin{equation}
M=4\pi\int_{r_{1}}^{r_{2}}\rho r^{2}dr
\end{equation}
is the Newtonian mass given b
\begin{equation}
M=4\pi\int_{r_{1}}^{r_{2}}\rho r^{2}dr=\frac{1}{G}\left[ \frac{(1-\frac{c
{a})r}{2}-\frac{D}{2r^{a-1}}\right] _{r_{1}}^{r_{2}}.
\end{equation}
Thus we get the total gravitational energy as
\begin{align}
E_{G} & =\frac{1}{G}\left[ \frac{(1-\frac{c}{a})r}{2}-\frac{D}{2r^{a-1
}-(1-\frac{c}{a})\frac{rF[(0.5,\frac{1}{a});(1+\frac{1}{a});-\frac{ar^{a}D
{c}]}{\sqrt{\frac{c}{a}}}\right] _{r_{1}}^{r_{2}}\newline\nonumber\\
& +\frac{1}{G}\left[ D(1-a)r^{(-0.5a+1)}\frac{F[(-0.5+\frac{1}{a
,0.5);(0.5+\frac{1}{a});-\frac{cr^{a}}{aD}]}{\sqrt{D}a(-0.5+\frac{1}{a
)}\right] _{r_{1}}^{r_{2}}.
\end{align}
The figures 4 and 5 show that the total gravitational energy is small but
negative whether we choose $D$ non-zero or zero for arbitrary $r_{2}>r_{1}>0$.
Thus for the existence of non-exotic matter in the halo, we recommend the
value of $D\leq10^{-11}$. In the distant halo region we have taken, typically
, $r$ $\sim200$ kpc in the figures below.
\begin{figure}[ptb]
\begin{center}
\vspace{0.5cm}\includegraphics[width=0.5\textwidth]{fig5.eps}
\end{center}
\caption{The variation of $E_{G}$ with ~r~ in Kpc. The lower limit of
integration in equation (24) fixed at, say, $r_{1} = 100$ kpc while $r_{2}$ is
varied from 100 to 500 Kpc. We choose $G=1$, $D=10^{-11}$ and $v^{\phi
\sim10^{-3}$ ( 300km/s) for a typical galaxy.
\label{fig3
\end{figure}
\begin{figure}[ptb]
\begin{center}
\vspace{0.5cm}\includegraphics[width=0.4\textwidth]{fig6.eps}
\end{center}
\caption{The variation of $E_{G}$ with~ r ~in Kpc. The lower limit of
integration in equation (24) fixed at, say, $r_{1} = 100$ kpc while $r_{2}$ is
varied from 100 to 500 Kpc. We choose $G=1$ , $D=0$ and $v^{\phi}\sim10^{-3}$
( 300km/s) for a typical galaxy.
\label{fig3
\end{figure}
\subsection{ \underline{\textbf{\ Attraction}}:}
Now we study the geodesic equation given by
\begin{equation}
\frac{d^{2}x^{\alpha}}{d\tau^{2}}+\Gamma_{\alpha}^{\mu\gamma}\frac{dx^{\mu
}{d\tau}\frac{dx^{\gamma}}{d\tau}=0
\end{equation}
for a test particle that has been \textquotedblleft placed'\ at some radius
$r_{0}$. This yields the radial equatio
\begin{eqnarray}
\frac{d^{2}r}{d\tau^{2}}&=&-\frac{1}{2}\left[ \frac{c}{a}+\frac{D}{r^{a
}\right] \nonumber \\
&& \left[ \frac{Da}{r^{a+1}}\left( \frac{c}{a}+\frac{D}{r^{a
}\right) ^{-2}\left( \frac{dr}{d\tau}\right) ^{2}+B_{0}lr^{l-1}\left(
\frac{dt}{d\tau}\right) ^{2}\right] _{r=r_{0}},
\end{eqnarray}
\newline which is negative as the the quantity in the square bracket is
positive. Thus particles are attracted towards the center. This result is in
agreement with the observations i.e., gravity on the galactic scale is
attractive ( clustering , structure formation etc ).
\subsection{ \underline{\textbf{\ Stability:}}}
Let us define the four velocity $U^{\alpha}=\frac{dx^{\sigma}}{d\tau}$ for a
test particle moving solely in the space of the halo (restricting ourselves to
$\theta=\pi/2$), the equation $g_{\nu\sigma}U^{\nu}U^{\sigma}=-m_{0}^{2}$ can
be cast in a Newtonian form
\begin{equation}
\left( \frac{dr}{d\tau}\right) ^{2}=E^{2}+V(r)
\end{equation}
which give
\begin{equation}
V(r)=-\left[ E^{2}\left\{ 1-\frac{r^{-l}\left[ \frac{c}{a}+\frac{D}{r^{a
}\right] }{B_{0}}\right\} +\left[ \frac{c}{a}+\frac{D}{r^{a}}\right]
\left( 1+\frac{L^{2}}{r^{2}}\right) \right]
\end{equation
\begin{equation}
E=\frac{U_{0}}{m_{0}},L=\frac{U_{3}}{m_{0}},
\end{equation}
where the constants $E$ and $L$, respectively, are the conserved relativistic
energy and angular momentum per unit rest mass of the test particle. Circular
orbits are defined by $r=R=$constant. so that $\frac{dR}{d\tau}=0$ and,
additionally, $\frac{dV}{dr}\mid_{r=R}=0$. From these two conditions follow
the conserved parameters:
\begin{equation}
L=\pm\sqrt{\frac{l}{2-l}}R
\end{equation}
and using it in $V(R)=-E^{2}$, we get
\begin{equation}
E=\pm\sqrt{\frac{2B_{0}}{2-l}}R^{\frac{l}{2}}.
\end{equation}
The orbits will be stable if $\frac{d^{2}V}{dr^{2}}\mid_{r=R}<0$ and unstable
if $\frac{d^{2}V}{dr^{2}}\mid_{r=R}>0$. Putting the expressions for $L$ and
$E$ in $\frac{d^{2}V}{dr^{2}}\mid_{r=R}$, we obtain, after straightforward
calculations, the final result, viz.,
\begin{equation}
\frac{d^{2}V}{dr^{2}}\mid_{r=R}=-\left[\frac{2c l (1-2l)}{a(2-l) R^2}+\frac
D( 16 l +8l^2-4l^3+2l^4) }{(2-l)(2+l)^2}
R^\frac{l(2-l)}{2+l}\right]
\end{equation}
One may note that, for $D=0$ as well as for $D\neq0$, $\frac{d^{2}V}{dr^{2
}\mid_{r=R}<0$. So the circular orbits are always stable.
\section{ Conclusions:}
It is well known that observation of flat rotation curve suggests that a
substantial amount of non-luminous dark matter is hidden in the galactic halo.
Here we have found that the flat rotation curve suggests also the background
geometry of the universe. The space-time geometry we have obtained can be
interpreted as the one due to dark matter embedded in the static FLRW
universe. This is probably the first indication that the spatial curvature of
the universe can be obtained from a local gravitational phenomenon.
If we demand that matter in the flat rotation curve region be non-exotic
(i.e., obey the usual energy conditions), we obtain the result that the
universe should be nearly flat, if not exactly so, which is consistent with
modern cosmological observations.
The equation of state of the dark matter component has been obtained by
treating it as perfect fluid and the expressions for general relativistic
correction terms for the pressure and energy density over those obtained from
the Newtonian theory are also derived. The corrections are, however, small as expected.
The geodesic equation [Eq.(26)] of a test particle in the derived spacetime
suggests that the particle will be attracted towards the center. We have
quantified the attractive effect in the relativistic case by calculating the
total gravitational energy $E_{G}$ ( which is negative ) in the halo region.
We have also demonstrated the stability of the circular orbits in our
spacetime solution. Thus our solution satisfies two crucial physical
requirements - stability of circular orbits and attractive gravity in the halo
region. Investigation on other observational constraints on the model is underway.
\subsection*{Acknowledgments}
The authors are thankful to an anonymous referee for his/her insightful
comments that have led to significant improvements, particularly on the
interpretational aspects. FR is grateful to IMSc, Chennai and IUCAA, Pune for
providing research facilities. \newline
\subsection*{Appendix}
To derive tangential velocity of circular orbits, we start with the line
element
\[
ds^{2}=-e^{\nu(r)}dt^{2}+e^{\lambda(r)}dr^{2}+r^{2}(d\theta^{2}+sin^{2}\theta
d\phi^{2}).
\]
The Lagrangian for a test particle reads
\[
2\mathbf{\mathit{L}}=-e^{\nu(r)}\dot{t}^{2}+e^{\lambda(r)}\dot{r}^{2
+r^{2}(\dot{\theta}^{2}+sin^{2}\theta\dot{\phi}^{2}).
\]
We guess the conserved quantities, the energy $E=e^{\nu(r)}\dot{t}$, the
$\phi$-momentum $L_{\phi}=r^{2}sin^{2}\theta\dot{\phi}$, and the total angular
momentum, $L^{2}={L_{\theta}}^{2}+\left( \frac{L_{\phi}}{\sin\theta}\right)
^{2}$, with $L_{\theta}=r^{2}\dot{\theta}$. The radial motion equation can be
written as:
\[
\dot{r}^{2}+V(r)=0
\]
with the potential $V(r)$ given y
\[
V(r)=-e^{-\nu(r)}\left( e^{-\lambda(r)}E^{2}-\frac{L^{2}}{r^{2}}-1\right) .
\]
For circular orbits, we have the conditions, $\dot{r}=0$, $V_{r}=0$ and
$V_{rr}~>~0$. These imply the following expressions for the energy and total
momentum of the particles in circular orbits:
\[
E^{2}=\frac{2e^{2\nu(r)}}{2e^{\nu(r)}-r(e^{\nu(r)})_{r}
\
\[
L^{2}=\frac{r^{3}(e^{\nu(r)})_{r}}{2e^{\nu(r)}-r(e^{\nu(r)})_{r}}.
\]
The second derivative of the potential evaluated at the extrema
\[
V(r)_{rr}|_{\text{extrema}}=2\frac{\frac{r(e^{\nu(r)})_{rr}}{e^{\nu(r)}
+\frac{(e^{\nu(r)})_{r}}{e^{\nu(r)}}\left( 3-\frac{r(e^{\nu(r)})_{r}
{e^{\nu(r)}}\right) }{re^{\lambda(r)}\left( 2-\frac{r(e^{\nu(r)})_{r
}{e^{\nu(r)}}\right)
\]
Now, the tangential velocity
\[
(v^{\phi})^{2}=r^{2}e^{-\nu(r)}(\dot{\theta}^{2}+sin^{2}\theta\dot{\phi}^{2})
\]
can be obtained for particles in stable circular orbits as
\[
(v^{\phi})^{2}=\frac{r(e^{\nu(r)})_{r}}{2e^{\nu(r)}}.
\]
Using the tangential velocity to be constant for several radii, the above
expression yields
\[
e^{\nu}=B_{0}r^{l
\]
where $l$ is given by $l=2(v^{\phi})^{2}$ and $B_{0}$ is an integration constant.
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\end{document}
|
{
"timestamp": "2010-09-23T02:01:38",
"yymm": "1009",
"arxiv_id": "1009.3572",
"language": "en",
"url": "https://arxiv.org/abs/1009.3572"
}
|
\section{Introduction}
Secular aberration drift is an apparent alteration in the velocity of distant objects caused by the acceleration of the Solar System barycenter in the Milky Way. The proper motion field of extragalactic sources affected by the secular aberration drift should present a dipolar structure with amplitude 4--6~microseconds of arc per year ($\mu$as/yr) directed towards the Galactic center (see, e.g., Fanselow 1983, Bastian~1995, Eubanks et al.~1995, Sovers et al.~1998, Mignard 2002, Kovalevsky 2003, Kopeikin \& Makarov 2006).
Observations of extragalactic radio sources by very long baseline interferometry (VLBI) have been done since the late 1970s by various agencies, mainly the United States Navy and the National Aeronautic and Space Administration (NASA), and have been coordinated since 1998 by the International VLBI Service for Geodesy and Astrometry (IVS, Schl\"uter \& Behrend 2007). The main purpose of these observations is to monitor the Earth orientation and realize terrestrial and celestial reference frames.
One of the cornerstones of the VLBI technique is the accurate realization of the celestial reference system using distant objects that supposedly have no detectable proper motion (Feissel \& Mignard~1998). The most recent realization of the International Celestial Reference Frame (ICRF2, Fey et al.~2009) provides absolute radio source coordinates for 3,414 sources. For 1,448 ICRF2 sources observed in at least two sessions, the median formal error is 175~$\mu$as. For the most observed radio sources, the inflated position errors are 40~$\mu$as in each coordinate.
This catalog did not include proper motions of sources because the dominant proper motion for any source is the somewhat temporary one associated with internal structure changes that produce apparent motions that are an order of magnitude greater than the aberrational drift (e.g., Fey et al. 1997). Such proper motions are not constant with time and are about an order of magnitude higher than the 4--6~$\mu$as/yr proper motions expected from the secular aberration drift. However, we believe that the secular aberration drift can be detected even with source position errors of $\sim$100~$\mu$as and apparent motion formal errors of 10--100~$\mu$as/yr (e.g., Feissel-Vernier 2003).
VLBI measurement of the secular aberration drift is important for both astrometry and astrophysics. Indeed, it would give an independent estimate of the acceleration vector applied to the Solar System barycenter without refering to objects within the Galaxy, so it can be used to constrain the mass of the central regions of the Milky Way. Moreover, the secular aberration drift produces a slow deformation of the celestial reference frame axes, and, if not corrected in geodetic VLBI sofware packages, could lead to contamination in estimates of other geodetic parameters, such as source coordinates and the Earth's nutation and precession (Titov~2010).
Gwinn et al. (1997) have investigated the radio source velocity field obtained from the analysis of 16 years of VLBI data. Though the prime interest of their paper was to find an upper limit for the mass-energy of the cosmological gravitational wave background from the quadrupolar structure of the velocity field, their estimate of the Galactocentric acceleration is higher than expected by a factor of two with comparable standard errors. MacMillan~(2005) processed another seven~years of data but did not detect this effect. Using the OCCAM geodetic VLBI analysis software package, Titov~(2009) reports a statistically significant dipole harmonic, but its magnitude was far from the theoretical value.
The approach used by MacMillan (2005) and Titov (2009) consisted of a direct estimation of the Galactocentric acceleration without intermediate estimation of the proper motions. Our approach differs from these two. We used a three-step procedure that includes (i) a production of source coordinate time series from analysis of VLBI delays, (ii) a least-square fit of proper motions to source coordinate time series, including editing large anomalous proper motions, and (iii) the fit of the Galactocentric acceleration to proper motions. This three-step analysis helped to remove outliers that significantly biased the results of the previous analyses.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=8.5cm]{15718fg1.eps}
\includegraphics[width=8.5cm]{15718fg2.eps}
\end{center}
\caption{The distribution of sources vs. the declination ({\it Top}), the proper motion formal errors $\sigma=(\sigma_{\alpha\cos\delta}^2+\sigma_{\delta}^2)^{1/2}$ vs. the declination ({\it Bottom}).}
\label{fig00}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=8.5cm]{15718fg3.eps}
\end{center}
\caption{The proper motion in right ascension vs. the right ascension of 40 sources observed in more than 1,000 sessions.}
\label{fig20}
\end{figure}
\section{Aberration in proper motions}
Consider the Solar System barycenter moving on a quasi circular orbit at distance $\vec R$ of the Galactic center with an acceleration
\begin{equation}
\vec a=\frac{{\rm d}^2\vec R}{{\rm d}t^2}=-\frac{kM(R)}{R^3}\vec R,
\end{equation}
where $k$ is the constant of gravitation, and $M(R)$ the equivalent mass entering the problem. Using $kM=\omega^2R^3=V^2R$, one gets the acceleration $\vec a={\rm d}\vec V/{\rm dt}=-\omega V\vec u$, resulting in the aberration effect on the proper motion $\vec\mu$ of distant bodies as seen from the Solar System barycenter (e.g., Kovalevsky 2003)
\begin{equation} \label{abb}
\Delta\vec\mu=\frac{\omega V}{c}\vec u,
\end{equation}
wherein $c$ is the speed of light and $\vec u$ the unit vector pointing towards the Galactic center.
Recent estimates of the Galactic parameters give a distance to the Galactic center of $8.4\pm0.6$~kpc and a circular rotation speed of $254\pm16$~km/s (Reid et al.~2009). The expected acceleration $a=V^2/R$ equals $(2.5\pm0.5)\times10^{-13}$~km/s$^2$. This corresponds to an aberration of a distant body proper motion up to $5.0\pm1.0$~$\mu$as/yr.
Further accelerated motion of the whole Milky Way would produce an additional aberration. According to Doppler shift measurements of the Cosmic Background Explorer (COBE) satellite, which realizes a celestial reference frame based on the cosmic microwave background (CMB) radiation, the Local Group of galaxies moves at $\sim$630~km/s (Kogut et al.~1993). Although considered as linear, this motion should correspond to a slowly accelerated motion on longer time scales, of orbit radius of several Mpc. It would result in an aberration that is far too small to be investigated here.
For a distant body of equatorial coordinates $(\alpha,\delta)$, (\ref{abb}) also reads as
\begin{eqnarray} \label{dip}
\Delta\mu_{\alpha}\cos\delta&=&-d_1\sin\alpha+d_2\cos\alpha, \\
\Delta\mu_{\delta}&=&-d_1\cos\alpha\sin\delta-d_2\sin\alpha\sin\delta+d_3\cos\delta,
\end{eqnarray}
where the $d_i$ are the components of the acceleration vector in unit of the proper motion, and which corresponds to degree 1 spheroidal (or electric) development of (see, e.g., Mathews~1981, Mignard \& Morando 1990)
\begin{equation} \label{gen}
\vec\mu=\sum_{l,m}\left(a_{l,m}^E\vec Y_{l,m}^E+a_{l,m}^M\vec Y_{l,m}^M\right),
\end{equation}
where $d_1=a_{1,1}^E$, $d_2=a_{1,-1}^E$, $d_3=a_{1,0}^E$, and $Y_{l,m}^E$ and $Y_{l,m}^M$ are the vector spherical harmonics of electric and magnetic types of degree $l$ and order~$m$.
In addition to the aberration distortion, there may also be a small global rotation that can be described by the toroidal (or magnetic) harmonics of degree~1:
\begin{eqnarray} \label{rot}
\Delta\mu_{\alpha}\cos\delta&=&r_1\cos\alpha\sin\delta+r_2\sin\alpha\sin\delta-r_3\cos\delta, \\
\Delta\mu_{\delta}&=&-r_1\sin\alpha+r_2\cos\alpha,
\end{eqnarray}
where $r_i$ can be expressed in terms of vector spherical harmonics coefficients as $r_1=a_{1,1}^M$, $r_2=a_{1,-1}^M$, and $r_3=a_{1,0}^M$.
To investigate a possible quadrupolar anisotropy of the velocity field, we give the development of the degree 2 vector spherical harmonics (i.e., $l=2$ in Eq.~(\ref{gen})):
\begin{eqnarray} \label{quad}
\Delta\mu_{\alpha}\cos\delta&=&-(a_{2,2}^{E,\rm Re}\sin 2\alpha-a_{2,2}^{E,\rm Im}\cos 2\alpha)\cos\delta \nonumber \\
& &+(a_{2,1}^{E,\rm Re}\sin\alpha-a_{2,1}^{E,\rm Im}\cos\alpha)\sin\delta \nonumber \\
& &+(a_{2,2}^{M,\rm Re}\sin 2\alpha-a_{2,2}^{M,\rm Im}\cos 2\alpha)\sin\delta\cos\delta \nonumber \\
& &+(a_{2,1}^{M,\rm Re}\sin\alpha-a_{2,1}^{M,\rm Im}\cos\alpha)\cos2\delta \nonumber \\
& &-a_{2,0}^M\sin\delta\cos\delta, \\
\Delta\mu_{\delta}&=&-(a_{2,2}^{E,\rm Re}\cos 2\alpha+a_{2,2}^{E,\rm Im}\sin 2\alpha)\sin\delta\cos\delta \nonumber \\
& &-(a_{2,1}^{E,\rm Re}\cos\alpha+a_{2,1}^{E,\rm Im}\sin\alpha)\cos 2\delta \nonumber \\
& &+(a_{2,2}^{M,\rm Re}\cos 2\alpha+a_{2,2}^{M,\rm Im}\sin 2\alpha)\cos\delta \nonumber \\
& &-(a_{2,1}^{M,\rm Re}\cos\alpha+a_{2,1}^{M,\rm Im}\sin\alpha)\sin\delta \nonumber \\
& &+a_{2,0}^E\sin\delta\cos\delta.
\end{eqnarray}
\section{Results and discussion}
\subsection{Data processing}
We processed 5,030 sessions of the permanent geodetic and astrometric VLBI program since 1979, totalling 7,285,312 group delay measurements at 8.4~GHz. Radio source coordinates were estimated once per session, together with Earth orientation parameters and station coordinates. The cut-off elevation angle was set to 5$^{\circ}$. A priori zenith delays were determined from local pressure values (Saastamoinen 1972), which were then mapped to the elevation of the observation using the Vienna mapping functions (B\"ohm et al.~2006). Zenith wet delays were estimated as a continuous piecewise linear function at 30-min intervals. Troposphere gradients were estimated as 8-hr east and north piecewise functions at all stations except a set of 110 stations with short observational histories. Station heights were corrected for atmospheric pressure and oceanic tidal loading. The relevant loading quantities were deduced from surface pressure grids from the U.~S. NCEP/NCAR reanalysis project atmospheric global circulation model (Kalnay et al.~1996, Petrov \& Boy~2004) and from the FES~2004 ocean tide model (Lyard et al.~2004). No-net rotation (NNR) and translation constraints per session were applied to the positions of all stations, excluding Fort Davis (Texas), Pie Town (New Mexico), Fairbanks (Alaska), and the TIGO antenna at Concepci\'on, Chile because of strong non linear displacements (These two sites experienced post-seismic relaxation effects after large earthquakes on the Denali fault in 2003, and between Talca and Concepci\'on in early 2010). A priori precession and nutation comply with the IAU~2000/2006 resolutions, which include the nutation model of Mathews et al.~(2002), the improved precession model of Capitaine et al.~(2003b), and the non rotating origin-based coordinate transformation between terrestrial and celestial coordinate systems (Capitaine et al.~2003a). Usually, an NNR constraint is used to fix the ICRS axes. However, Titov (2010) argues that application of a tight NNR constraint may wipe out all systemtic effects in the proper motion of reference radio sources. Therefore, we tied the celestial frame to the ICRF2 using a loose NNR constraint uniformly applied for each session. More details are discussed later in Section~3.2. The calculations used the Calc~10.0/Solve~2010.05.21 geodetic VLBI analysis software package, which was developed and maintained at NASA Goddard Space Flight Center, and were carried out at the Paris Observatory IVS Analysis Center (Gontier et al.~2008).
\begin{table*}
\begin{center}
\begin{tabular}{lrrrrr}
\hline
\hline
\noalign{\smallskip}
& DR & DR1 & DR2 & DR3 & DR4 \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
No. sources & 555 & 268 & 257 & 40 & 515 \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
Dipole & & & \\
\noalign{\smallskip}
$d_1$ & $-0.7 \pm 0.8$ & $-1.9 \pm 1.0$ & $-1.6 \pm 0.9$ & $-3.4 \pm 1.0$ & $ 4.2 \pm 1.3$ \\
\noalign{\smallskip}
$d_2$ & $-5.9 \pm 0.9$ & $-7.1 \pm 1.1$ & $-6.0 \pm 1.0$ & $-6.2 \pm 1.2$ & $-4.6 \pm 1.3$ \\
\noalign{\smallskip}
$d_3$ & $-2.2 \pm 1.0$ & $-3.5 \pm 1.2$ & $-2.9 \pm 1.1$ & $-3.8 \pm 1.2$ & $ 1.4 \pm 1.7$ \\
\noalign{\smallskip}
Amplitude & $ 6.4 \pm 1.5$ & $ 8.1 \pm 1.9$ & $ 6.9 \pm 1.7$ & $ 8.0 \pm 2.0$ & $ 6.4 \pm 2.5$ \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
Rotation & & & \\
\noalign{\smallskip}
$r_1$ & $-2.6 \pm 0.9$ & $-3.2 \pm 1.2$ & $-2.7 \pm 1.1$ & $-1.9 \pm 1.3$ & $-4.3 \pm 1.4$ \\
\noalign{\smallskip}
$r_2$ & $ 0.4 \pm 1.0$ & $-0.6 \pm 1.2$ & $-0.3 \pm 1.1$ & $ 2.5 \pm 1.4$ & $-2.1 \pm 1.5$ \\
\noalign{\smallskip}
$r_3$ & $ 0.8 \pm 0.7$ & $-0.3 \pm 0.9$ & $ 0.0 \pm 0.8$ & $-0.5 \pm 0.9$ & $ 4.1 \pm 1.2$ \\
\noalign{\smallskip}
Amplitude & $ 2.8 \pm 1.5$ & $ 3.3 \pm 1.9$ & $ 2.7 \pm 1.7$ & $ 3.2 \pm 2.0$ & $ 6.3 \pm 2.4$ \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\multicolumn{2}{l}{Direction of the acceleration vector} & & & & \\
\noalign{\smallskip}
Right Ascension ($^\circ$) & $263\pm11$ & $255\pm11$ & $255\pm11$ & $241\pm12$ & $312\pm16$ \\
\noalign{\smallskip}
Declination ($^\circ$) & $-20\pm12$ & $-25\pm11$ & $-25\pm11$ & $-28\pm12$ & $13\pm21$ \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
Pre-fit wrms & 22.1 & 19.7 & 16.3 & 9.2 & 33.3 \\
\noalign{\smallskip}
Post-fit wrms & 21.7 & 18.9 & 15.7 & 7.5 & 32.7 \\
\noalign{\smallskip}
Reduced $\chi^2$ & 1.9 & 1.9 & 1.8 & 2.1 & 1.9 \\
\noalign{\smallskip}
\hline
\end{tabular}
\end{center}
\caption{The multipole coefficients ($\mu$as/yr) of the velocity field. Uncertainties are 1$\sigma$.}
\label{tab01}
\end{table*}
Before 1990, the general deficiency of the VLBI networks, including the number of observed sources and observing antennas per session, makes the VLBI products less reliable (see, e.g., Gontier et al. 2001, Malkin 2004, Feissel-Vernier et al. 2004, Lambert \& Gontier 2009 who reports interesting statistical results and remarks about the VLBI evolution over the past two decades). For this reason we removed data before 1990. A treatment of the full data base over 1979--2010 is nevertheless presented later for comparison.
In the coordinate time series, data points resulting from fewer than three reliable observations within a session were removed, and outliers were eliminated so that the $\chi^2$ is reasonably close to unity. Then, proper motions were computed by weighted least-squares for time series containing at least ten points and longer than ten~years. Weights were taken as the inverse of the squared formal error. A set of 39 sources showing significant non linear positional variations due to large-scale variations in their structure (including 3C84, 3C273B, 3C279, 3C345, 3C454.3, and 4C39.25) were isolated in the ICRF2 work and treated in such a manner that they did not perturb the geodetic solutions (Fey et al.~2009). We removed these 39 sources from our data set.
The final sample contains proper motions of 555 sources and is made available electronically. Figure~\ref{fig00} displays the distribution of sources and proper motion formal errors in declination. Near the polar areas, the number of sources decreases proportionally to the cosine of the declination. It also shows the nonuniformity of the sample and a lack of sources at declinations under $-40^{\circ}$. Figure~\ref{fig20} displays $\mu_{\alpha}\cos\delta$ versus $\alpha$ for 40 sources observed in more than 1,000 sessions (see Section 3.2 for details). The apparent motions of these sources are estimated very accurately thanks to a large number of observations. A systematic in $\sin\alpha$ of magnitude less than 10~$\mu$as clearly shows up. Tiny underlying aberrational drift is indicative even for a limited number of well-observed radio sources.
\subsection{Dipole}
This section comprises our results of the dipole component estimation. We start with a main solution including all 555 radio sources. Then we consider different subsets of radio sources to verify the robustness of the main solution.
First, dipole and rotation coefficients were fitted by weighted least-squares following Eqs.~(3)--(4) and (6)--(7) (Table~\ref{tab01}, column DR). Reported errors are standard formal errors. The fit produces correlations of $\sim$0.4 between $d_1$ and $r_2$ and between $d_2$ and $r_1$. Figure~\ref{fig01} displays both the proper motions of the 555 sources and the estimated dipole component of the velocity field. Within error bars, the dipole amplitude and direction agree with predictions from measurements of the Galactic parameters. The corresponding centrifugal acceleration of the Solar System barycenter is $(3.2\pm0.7)\times10^{-13}$~km/s$^2$. By fixing the distance of the Solar System to the Galactic center to 8.4~kpc, the DR solution yields a circular rotation speed of~276~km/s.
To test the robustness of our estimates, we fitted the parameters to various subsamples of sources, including sources with structure indices (Charlot~1990, Fey \& Charlot 2000) lower than 2.5 (DR1), thus keeping only the most compact sources only. Since structure indices were not available for all sources, the number of sources in the sample is considerably reduced. Nevertheless, the amplitude and the orientation of the dipole do not change significantly with respect to DR. Another test consisted of only keeping ICRF2 defining sources (DR2). Two hundred fifty seven of them have a sufficient observational history to pass our selection scheme successfully. Again, it does not change the amplitude and the orientation of the dipole significantly.
In all the above solutions, the rotation is about 5~$\mu$as/yr and is consistent with the ICRF2 error. If the rotation is fixed to zero, the estimated dipole is of $7.1\pm1.5$~$\mu$as/yr toward ($\alpha=263\pm9^{\circ}$, $\delta=-17\pm11^{\circ}$).
Owing to the uneven number of observations for the 555 radio sources, their proper motion standard errors vary by three orders of magnitude: from 5 to 5000~$\mu$as/yr. It is necessary to check whether the estimates of the dipole component are dominated by a small number of radio sources with small errors on proper motion. Another fit was therefore done to the 40 sources observed in more than 1,000 sessions and used to produce Fig.~\ref{fig20}~(DR3). These sources have formal errors of proper motions less than 15~$\mu$as. DR3 has the best weighted rms among all the solutions in Table~\ref{tab01}.
The last fit (DR4) was done to the remaining 515 sources observed in less than 1,000 sessions. DR3 and DR4 deviate from the main solution DR in opposite directions. Though the orientation of the dipole is mainly constrained by the 40 sources with the longest observational history, the magnitude of the dipole is closer to the theoretical values when the 515 sources are used.
We also made the adjustment without removing data before 1990. It yielded a dipole amplitude of $6.4\pm1.5$~$\mu$as/yr, directed towards ($\alpha=262\pm10^{\circ}$, $\delta=-15\pm12^{\circ}$). Though the amplitude is comparable to the one obtained over 1990--2010, the direction appeared changed by almost 5$^{\circ}$ in declination. It is likely due to a lack of sources observed in the southern hemisphere during the first decade of VLBI.
Overall, the estimate of the dipole effect obtained from the DR solution ($6.4\pm1.5$~$\mu$as/yr; $\alpha=263\pm11^{\circ}$, $\delta=20\pm12^{\circ}$) is considered confident and could be recommended for futher applications.
\subsection{NNR constraint}
Before going further, we address the problem of the constraint that was applied to the observation during the first step of the VLBI data reduction (i.e., production of source coordinate time series). Table~\ref{tab10} displays the amplitude and the orientation of the dipole fitted to time series obtained with different constraints. The first line is the DR solution for which the $\sigma$ of the NNR constraint is 2~as. Decreasing the $\sigma$ to a value close to the milli arcsecond or lower makes the dipole move off the Galactic center.
The last two lines of Table~\ref{tab10} mention solutions without NNR. Applying no constraint at all would cause degeneracy of the normal equation matrix for daily parameters. Therefore, in this solution, each source is tied to the ICRF2 position using a loose constraint of 2~as. We can see that such a constraint is equivalent to the NNR applied with the same $\sigma$.
\begin{table}
\begin{center}
\begin{tabular}{lrrrrr}
\hline
\hline
\noalign{\smallskip}
NNR & $\sigma$ & Amplitude & $\alpha$ & $\delta$ \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
Yes & 2~as & $6.4\pm1.5$ & $263\pm11$ & $-20\pm12$ \\
\noalign{\smallskip}
Yes & 2~mas & $5.2\pm0.7$ & $237\pm7$ & $22\pm8$ \\
\noalign{\smallskip}
No & 2~as & $6.5\pm1.5$ & $263\pm11$ & $-23\pm12$ \\
\noalign{\smallskip}
No & 2~mas & $3.3\pm0.5$ & $242\pm7$ & $22\pm7$ \\
\noalign{\smallskip}
\hline
\end{tabular}
\end{center}
\caption{The dipole amplitude ($\mu$as/yr) and orientation ($^{\circ}$) obtained with different constraints with uncertainties of 1$\sigma$.}
\label{tab10}
\end{table}
\begin{figure}
\begin{center}
\includegraphics[width=8.5cm]{15718fg4.eps}
\includegraphics[width=8.5cm]{15718fg5.eps}
\end{center}
\caption{({\it Top}) The proper motions of the 555 sources, and ({\it Bottom}) the estimated dipole component of the velocity field. The green line represents the equator of the Milky Way, whose center is indicated by the blue marker.}
\label{fig01}
\end{figure}
\subsection{Quadrupole}
In a second fit, we adjusted all parameters of Eqs.~(3)--(4) and (6)--(9) (Table~\ref{tab02}). Maximum correlations of 0.5 showed up between $d_1$ and $a_{2,1}^{M,\rm Re}$. Including of the quadrupole harmonics did not change the dipole and rotational effects. The amplitude of the aberrational effect increased by 0.7~$\mu$as/yr, and the right ascension of the vector moved off the theoretical value, but the declination of the vector direction almost coincided now with the theoretical prediction. Fixing the dipole and rotation parameters to the DR values and adjusting the quadrupole parameters only leads to $6.4\pm3.4$~$\mu$as.
The quadrupole component may come either from the Hubble constant anisotropy or the primordial gravitational waves (Kristian \& Sachs~1966, Pyne et al.~1996, Gwinn et al.~1997, Jaffe 2004). Though the Hubble constant anisotropy only affects $a_{2,0}^E$ and $a_{2,2}^{E,\rm Re}$ (Titov~2009), the primordial gravitational waves affect all the quadrupole terms in Table~\ref{tab02}. Whereas the resultant amplitude is less than 3$\sigma$ standard error, the only marginally statistically significant component $a_{2,1}^{E,\rm Re}=4.1\pm1.3$~$\mu$as/yr can be converted into the energy density of gravitational waves $\Omega_{\rm GW}$. Using Eq.~(11) of Gwinn et al.~(1997) for this component, one gets $\Omega_{\rm GW}=0.0042\pm0.0004\;h^{-2}$, where $h=H_0/(100~{\rm km/s})$ is the normalized Hubble constant. The squared proper motion of quasars at cosmologic distances are sensitive to gravitational waves of long wavelength, even comparable to the scale of the Universe, and proportional to $\Omega_{\rm GW}$ on a wide range of frequencies, from the inverse of the period of observations to Hubble time. Therefore, the value estimated above may indicate the upper limit of the gravitational waves density integrated over a range of frequencies less than 10$^{-9}$~Hz.
\subsection{Dependence on the redshift}
We also looked at the dipole and quadrupole amplitudes as functions of the redshift, available for 488 sources (Titov \& Malkin~2009). Each estimate incorporated a subsample containing 122 sources whose redshifts are between the values indicated by the $x$-axis ticks of Fig.~\ref{fig02}. The mean declination of each subsample is between 7.2$^{\circ}$ and 11.7$^{\circ}$. The dipole and quadrupole amplitudes do not present significant dependence on~$z$.
\begin{table}
\begin{center}
\begin{tabular}{lrr}
\hline
\hline
\noalign{\smallskip}
No. sources & 555 & \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
Dipole & & \\
\noalign{\smallskip}
$d_1$ & $ 0.7 \pm 0.9$ & \\
\noalign{\smallskip}
$d_2$ & $-6.2 \pm 1.0$ & \\
\noalign{\smallskip}
$d_3$ & $-3.3 \pm 1.0$ & \\
\noalign{\smallskip}
Amplitude & $ 7.1 \pm 1.7$ & \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
Rotation & & \\
\noalign{\smallskip}
$r_1$ & $-2.4 \pm 1.0$ & \\
\noalign{\smallskip}
$r_2$ & $ 0.4 \pm 1.1$ & \\
\noalign{\smallskip}
$r_3$ & $ 0.8 \pm 0.7$ & \\
\noalign{\smallskip}
Amplitude & $ 2.6 \pm 1.7$ & \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
Quadrupole & Re & Im \\
\noalign{\smallskip}
$a_{2,2}^E$ & $ 1.6 \pm 1.0$ & $-0.6 \pm 1.0$ \\
\noalign{\smallskip}
$a_{2,1}^E$ & $ 4.1 \pm 1.3$ & $-1.7 \pm 1.2$ \\
\noalign{\smallskip}
$a_{2,0}^E$ & $ 2.8 \pm 1.2$ & \\
\noalign{\smallskip}
$a_{2,2}^M$ & $-0.6 \pm 1.3$ & $ 2.4 \pm 1.3$ \\
\noalign{\smallskip}
$a_{2,1}^M$ & $ 2.0 \pm 1.1$ & $-0.6 \pm 1.1$ \\
\noalign{\smallskip}
$a_{2,0}^M$ & $ 0.7 \pm 0.9$ & \\
\noalign{\smallskip}
Amplitude & $ 6.4 \pm 3.6$ & \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\multicolumn{2}{l}{Direction of the acceleration vector} & \\
\noalign{\smallskip}
Right Ascension ($^\circ$) & $277\pm12$ & \\
\noalign{\smallskip}
Declination ($^\circ$) & $-28\pm10$ & \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
Pre-fit wrms & 22.1 & \\
\noalign{\smallskip}
Post-fit wrms & 21.6 & \\
\noalign{\smallskip}
Reduced $\chi^2$ & 1.9 & \\
\noalign{\smallskip}
\hline
\end{tabular}
\end{center}
\caption{The quadrupole coefficients ($\mu$as/yr) of the velocity field with uncertainties od 1$\sigma$.}
\label{tab02}
\end{table}
\begin{figure}
\begin{center}
\includegraphics[width=8.5cm]{15718fg6.eps}
\end{center}
\caption{The dipole (red disks) and quadrupole (blue triangles) amplitudes as functions of the redshift. Uncertainties are 1$\sigma$.}
\label{fig02}
\end{figure}
\section{Conclusion}
This study showed that VLBI has now accumulated accurate enough data to detect the Galactocentric acceleration through its effect on distant radio source positions. It turns out that the current definition of the celestial reference frame as epochless and based on the assumption that quasars have no detectable proper motions should be mitigated. In the future, VLBI realizations of the celestial reference system should correct source coordinates for this effect, possibly by providing source positions, together with a corrective formula.
The European optical astrometry mission Gaia (Perryman et al. 2001), scheduled for 2012, should be able to determine the components of the acceleration vector with a relative precision of 10\% (Mignard 2002). To improve the VLBI determination of the Galactocentric acceleration and to confirm the significance of the quadrupole systematics, more proper motions of extragalactic radio sources need to be measured over the next decade. Concentrating on sources showing a high positional stability and having a low structure index would reduce unwanted effects of intrinsic motion caused by the relativistic jets and other modification of the source structure. In addition, it is necessary to run a dedicated program to measure the redshift of the reference radio sources using large optical facilities, especially in the southern hemisphere (Maslennikov et al.~2010).
\begin{acknowledgements}
The authors are grateful to M. Eubanks, D. Jauncey, S. Klioner, S. Kopeikin, S. Kurdubov, C. Le Poncin-Lafitte, and C. Ma for useful discussions regarding theoretical issues and practical advices on the methods. A special thanks goes to O. Sovers for finding the early report by J. Fanselow mentioning the secular aberration drift. The authors also thank an anonymous referee who helped in improving the manuscript. This paper has been published with permission of the Chief Executive Officer of Geoscience Australia.
\end{acknowledgements}
|
{
"timestamp": "2011-04-13T02:00:58",
"yymm": "1009",
"arxiv_id": "1009.3698",
"language": "en",
"url": "https://arxiv.org/abs/1009.3698"
}
|
\section{Introduction}
In a variety of recently discovered materials, superconductivity apparently arises directly from the electron correlations themselves.
However, these materials are complex, and many material specific details
can play a role in the mechanism of superconductivity. The problem is greatly simplified in the weak coupling limit, where we recently showed \cite{Raghu2010} that an asymptotically exact treatment of the problem is possible, valid in cases in which the superconducting state emerges at low temperatures from a well-formed Fermi liquid. Nonetheless, even under these circumstances, the character of the superconducting state and the transition temperature depend in a complicated way on details of the band-structure, both near and far from the Fermi surface.
To the extent that there are basic principles at work underlying the mechanism of unconventional superconductivity, it would be a great advance to find simple model systems which exhibit such behavior. Here, we consider the possibility of unconventional superconductivity in some model systems with particularly simple electronic structures, where controlled theory is possible, and where, conceivably,
experimental tests of the theory are feasible. Specifically, we consider circumstances in which superconductivity may occur in a two dimensional electron gas (2DEG) in a high mobility
heterostructure.
Here,
due to the the stiffness of the lattice and the limited phase space for electron-phonon scattering, electron-phonon coupling is probably negligible, and
the single-particle dynamics can be treated accurately within a rotationally invariant effective mass approximation. Moreover, the strength of the correlations can, to a large extent, be tuned by varying the electron density.
The possibility of an electronic pairing mechanism in systems with rotational invariance was first put forth in a seminal paper by Kohn and Luttinger\cite{Kohn1965}.
Although $U$, the bare interactions among electrons are repulsive, there are {\it effective} attractive interactions that arise at $\mathcal{O}(U^2)$. Kohn and Luttinger focussed on the portion of the effective attractions
associated with the non-analyticities in $\chi(q)$, the particle-hole susceptibility, at momentum $q= 2k_F$ which reflect the sharpness of the Fermi surface at zero temperature.
More generally, what is required for this mechanism to work is strong $q$ dependence of $\chi(q)$ for $q \leq 2k_F$. Indeed, the Kohn Luttinger instability of a 3 dimensional rotationally invariant system results in the formation of a p-wave superconducting ground state due to the peak in
$\chi(q)$ near $q=0$. While this result is valid only in the weak-coupling regime where $U \ll E_F$, it is widely believed that the p-wave ground state obtained this way is adiabatically connected to the more realistic (and more strongly correlated) example of Helium-3\cite{Leggett1975}.
However, the Kohn-Luttinger effect is exponentially weaker in a rotationally invariant 2DEG\cite{Galitski2003}, due to
the fact
$\chi(q)$ is
independent of momentum for
momenta $q \le 2k_F$.
It was later shown
that at $\mathcal{O}(U^3)$, the 2DEG does exhibit
a pairing instability\cite{Chubukov1992}.
Still, at least in weak-coupling, electronically mediated superconductivity in the 2DEG is
negligible.
In this paper, we
show that by perturbing the 2DEG, it is possible to
significantly enhance the superconducting
transition temperature
by engineering circumstances in which instabilities
arise at $\mathcal{O}(U^2)$ in perturbation theory.
We present asymptotically exact\cite{Raghu2010} weak coupling solutions of the superconducting instability in several systems that are variants of the simplest, rotationally invariant 2DEG.
As a first example, we show that
partially spin-polarizing the 2DEG
produces
a non-unitary $p+ip$ superconductor.
Y. Kagan and A. Chubukov have previously addressed this problem using
an expansion in powers of the electron concentration
\cite{Kagan1989}, and their result reduces to ours in the weak coupling limit.
As a second example, we consider
the 2DEG in a semiconductor heterostructure quantum well with two populated subbands. We show that this system can possess both p-wave and d-wave ground states and present the phase diagram of this system.
This paper is organized as follows. In the next section, we review the method developed in Ref. \cite{Raghu2010} and discuss its straightforward generalization needed for the present context. In Section III, the effect of partially polarizing the 2DEG is studied. In Section
IV, we consider
two subbands in a 2DEG quantum well.
Technical details of the various calculations are presented in the Appendix. In a forthcoming paper\cite{RaghuII} we will consider a variety of slightly more complex situations pertinent to particular semiconductor heterostructures.
\section{Perturbative renormalization group treatment of superconductvity}
In this section, we review the prescription of Ref. \cite{Raghu2010} and discuss
its generalization
to the present context.
We integrate out high energy modes in two steps.
In the first step, we integrate out all
modes outside a narrow range of energies $\Omega$ about the Fermi energy. $\Omega$ is not a physical energy in the problem, but rather a calculational device. It is chosen large enough so that the interactions can be treated perturbatively but small enough that it can be set to zero in all non-singular expressions without causing significant error, {\it i.e.} it is chosen to satisfy the inequalities $\rho U^2 \gg \Omega \gg \mu\exp\{-[1/\rho U] \}$,
where $\rho$ is the density of states at the Fermi energy and $\mu$ is the Fermi energy. The effective interactions generated in the process then serve as the ``bare'' interactions in a second step, in which the remaining problem is solved using the perturbative renormalization group procedure of Shankar and Polchinski\cite{Shankar1994,Polchinski1992}. $T_c$ is, up to an unknown multiplicative constant, given by the energy scale, $T^*$, at which a relevant interaction grows to be of order 1. It was shown by careful analysis of perturbative expressions up to 4th order in the interaction strength that the resulting expression for $T^*$ is independent of
$\Omega$.
The analysis of Ref. \cite{Raghu2010} leads to the following prescription for computing the leading order asymptotic behavior of $T_c$ for weak interactions: First, compute the effective interaction in the Cooper channel at energy scale $\Omega$, $\Gamma^{(a)}(\hat k,\hat k^\prime)$, to second order in the interactions. Here, $\hat k$ and $\hat k^\prime$ denote points on the Fermi surface, and $\Gamma$ is the vertex for scattering a pair of electrons with momenta $\hat k$ and $-\hat k$ to states with momenta $\hat k^\prime$ and $-\hat k^\prime$, where if there are multiple band indices,
the subband index is implicitly determined depending on whether the momenta are on one Fermi surface or the other, and where there is a different matrix depending on whether the electron pair forms a spin singlet ($\Gamma^{(s)}$) or a spin triplet ($\Gamma^{(t)}$). We then construct the related dimensionless matrix
\begin{equation}
g^{(a)}_{\hat k,\hat k^\prime} \equiv
\rho \sqrt{\bar v/v(\hat k)}\Gamma^{(a)}(\hat k,\hat k^\prime) \sqrt{\bar v/v(\hat k^\prime)},
\end{equation}
where $v(\hat k)$ is the magnitude of the Fermi velocity on the Fermi surface of the corresponding subband, and $\rho$ is the total density of states
at the Fermi energy.
Manifestly, $g$ is a real, symmetric, hence Hermitian matrix, so it has a complete set of eigenstates and eigenvalues,
\begin{equation}
\sum_{\hat k^\prime}g^{(a)}_{\hat k,\hat k^\prime} \phi_{\hat k^\prime}^{(a,m)} =\lambda^{(a,m)}\phi_{\hat k}^{(a,m)}.
\end{equation}
Among all the possible solutions, we identify the most negative eigenvalue,
\begin{equation}
\lambda \equiv {\rm Min}\left[ \lambda^{(a,m)}\right], \lambda < 0
\end{equation}
Then,
\begin{equation}
T_c \sim \mu \exp[-1/\vert \lambda \vert ].
\end{equation}
\section{Partially polarized Fermi surface}
As a first example, we consider
a
partially spin polarized 2DEG
with short-ranged repulsive interactions:
\begin{eqnarray}
H&=&H_0 + H_1 \nonumber \\
H_0 &=& \sum_{\sigma} \int \frac{d^2 k}{(2 \pi)^2} E_{\sigma, \sigma'}(\bm k) \psi^{\dagger}_{\sigma}(\bm k) \psi_{\sigma'}(\bm k) \nonumber \\
H_1 &=&U \int \frac{d^2 k_1 d^2 k_2 d^2 k_3}{(2 \pi)^6}
\psi^{\dagger}_{\uparrow}(\bm k_1) \psi^{\dagger}_{ \downarrow}(\bm k_2) \psi_{ \downarrow} (\bm k_3) \psi_{\uparrow}(\bm k_4) \nonumber \\
\end{eqnarray}
where $ \bm k_4 = \bm k_1 + \bm k_2 - \bm k_3$,
\begin{equation}
E_{\sigma, \sigma'} = \epsilon_{\bm k} \delta_{\sigma, \sigma'} + \bm h \cdot \bm \tau_{\sigma \sigma'},
\end{equation}
and $\bm h$ is a mean-field
that
renders the ground state
spin-polarized. Such a partially-polarized system can occur in a narrow-well semiconductor heterostructure in the presence of a parallel magnetic field (in which case ${\bf h} = g \mu_B {\bf H}_{\parallel}$), or in a ferromagnetic phase with spontaneously broken symmetry, such as
probably occurs in the Hubbard model
away from half-filling in the strong-coupling limit $U \gg t$\cite{Pilati2010,Liinprep}. (However, in the latter case, it requires something of an intuitive leap to treat the residual interactions beyond those that produce the mean-field ${\bf h}$ as ``weak.'')
Since the Fermi surfaces are spin-polarized, singlet pairing
is suppressed,
so the leading superconducting instability will therefore be in the spin triplet
channel. We first consider the limit in which there is no spin-orbit coupling, in which case, the two particle scattering amplitude is a separate function for each spin-polarization. As derived in the appendix,
\begin{eqnarray}
\Gamma_{\uparrow} (\hat k, \hat q) = - U^2 \chi_{\downarrow}(\vec k - \vec q) \nonumber \\
\Gamma_{\downarrow} (\hat k, \hat q) = - U^2 \chi_{\uparrow}(\vec k - \vec q) \nonumber
\end{eqnarray}
where $\chi_{\sigma}$ is the contribution of spin $\sigma$ electrons to the susceptibility.
In the case of a rotationally invariant system with $\epsilon_{\vec k,\sigma} = k^2/2m +\sigma h$,
$v_{f,\sigma}(\hat k) =
k_{f,\sigma}/m$
and $\rho_{\sigma} = \rho = m/2 \pi$ is independent of the spin-polarization. Therefore the matrix $g_{\hat k, \hat q}$ defined in the previous section is
\begin{equation}
g^{\sigma}_{\hat k, \hat q} = \rho \Gamma_{\sigma} (\hat k, \hat k')
\end{equation}
The particle-hole susceptibility for this system has the following well-known
form (see the appendix):
\begin{equation}
\label{chi2d}
\chi_{\sigma}(\vec q) = \frac{\rho}{2 } \left[ 1 - \frac{{\rm Re} \sqrt{q^2 - (2 k_{F \sigma})^2}}{q} \right]
\end{equation}
Thus, $\chi_{\sigma}(q)$ is a constant for $q < 2k_{F \sigma}$, has a derivative discontinuity at $q = 2 k_{F \sigma}$, and vanishes as $1/q^2$ when $ q >> 2k_{F \sigma}$.
The rotational invariance of the problem
implies that the triplet eigenfunctions are of the form
\begin{equation}
\psi^{t,m}_{\sigma} (\hat k) = \psi(k_{F \sigma}) \cos{ \left( m \theta_{\hat k} \right)}
\end{equation}
where $m$ is an odd integer.
The eigenvalue problem for this system therefore
reduces to the integral expressions:
\begin{eqnarray}
\lambda^{(m, \uparrow)}&=& - \rho U^2 \int \frac{d \theta}{2 \pi} \chi_{\downarrow}\left( 2k_{F \uparrow} \left| \sin{\left(\theta/2 \right) } \right| \right) \cos{ \left( m \theta \right)} \nonumber \\
\lambda^{(m, \downarrow)} &=& - \rho U^2 \int \frac{d \theta}{2 \pi} \chi_{\uparrow}\left( 2k_{F \downarrow} \left| \sin{\left(\theta/2 \right) } \right| \right) \cos{ \left( m \theta \right)} \nonumber \\
\end{eqnarray}
where $\theta$ is the angle relative
between $\hat k$ and $\hat q$.
Without loss of generality, we suppose that $k_{F \downarrow} < k_{F \uparrow}$.
For any $\hat k, \hat q$ on the smaller (spin-down) Fermi surface, $\hat k - \hat q < 2k_{F \downarrow} < 2k_{F \uparrow}$
so the effective interaction,
$\sim \chi_{\uparrow}(\hat k - \hat q)$,
is a constant. Therefore, it follows that $\lambda_{m \downarrow} = 0$ for all $m$ or in other words, the smaller Fermi surface
has no superconducting instability to $\mathcal O(U^2)$. (
Presumably, $\lambda_{m,\downarrow} \sim \mathcal O(U^3)$. )
Conversely, the effective interaction between electrons on the larger (spin-up) Fermi surface is $\sim \chi_{\downarrow}(\hat k - \hat q)$, which {\it does} depend on the relative
position of the incoming and outgoing electrons on the Fermi surface.
Using Eq. \ref{chi2d}, the
above expression for the eigenvalue on the larger Fermi surface
becomes
\begin{equation}
\lambda_{m \uparrow}(\eta) = \frac{\rho^2 U^2}{\pi} \int_{\theta_c}^{\pi} d \theta \frac{ \sqrt{ \sin^2{\frac{\theta}{2} } - \eta^2} }{\sin{\frac{\theta}{2}} } \cos{\left( m \theta \right) }
\end{equation}
where $\eta = \left( k_{F \downarrow}/k_{F \uparrow} \right)$, $0 \leq \eta \leq 1$, and $\theta_c = 2 \sin^{-1}{\eta}$.
As can be seen from the equation above, $\lambda_{m \uparrow}(0) = \lambda_{m \uparrow}(1) = 0$. That is, in the limit where the Fermi surface is either completely polarized, or completely unpolarized, there is no superconducting instability to $\mathcal{O}(U^2)$. However, for intermediate values of the polarization, the integral above yields
\begin{equation}
\lambda_{1 \uparrow}(\eta) = - \rho^2 U^2 \eta \left( 1 - \eta \right)
\end{equation}
which is clearly negative for all intermediate values of $\eta$. This is
the main result of this section: by polarizing the Fermi surfaces in two dimensions, there is a significant enhancement of p-wave superconductivity. The optimal pairing strength occurs when $\eta = 1/2$, so that
\begin{equation}
{\rm Max}\left[\lambda_{m \uparrow}(\eta) \right] = \lambda_{1 \uparrow} (\eta = 0.5) = - \frac{ (\rho U)^2}{4}
\end{equation}
( Note that Eq. 7 of Ref. \cite{Kagan1989} reduces to this result in the limit of weak interaction. )
For completeness, we quote the next leading eigenvalue, which corresponds to the f-wave (i.e. $m=3$) solution:
\begin{equation}
\lambda_{3 \uparrow}(\eta) = - \rho^2 U^2 \eta \left[ 1 - \eta\left( 3 - 4 \eta^2 + 2 \eta^4 \right) \right]
\end{equation}
which is not symmetric about the point $\eta = 0.5$. \\
Weak but non-vanishing spin orbit coupling will generically change this situation, since superconductivity will be induced in the minority fluid by the proximity effect as soon as the majority fluid becomes superconducting. In 2D, this induced superconductivity will generally track the fundamental order parameter.
\section{Two subbands in a 2DEG}
\label{2subband}
In this section, we consider the case of a 2DEG in a semiconductor heterostructure having two subbands,
with Hamiltonian:
\begin{eqnarray}
H&=&H_0 + H_1 \nonumber \\
H_0 &=& \sum_{a = 1,2} \sum_{\sigma} \int \frac{d^2 k}{(2 \pi)^2} \epsilon_{\bm k, a} \psi^{\dagger}_{a, \sigma}(\bm k) \psi_{a, \sigma}(\bm k) \nonumber \\
H_1 &=& \sum_{a.. d} \sum_{\sigma, \sigma'} V^{\sigma, \sigma'}_{ab,cd} \int \frac{d^2 k_1 d^2 k_2 d^2 k_3}{(2 \pi)^6} [ \nonumber \\
&&\psi^{\dagger}_{a, \sigma}(\bm k_1) \psi^{\dagger}_{b, \sigma'} \psi_{c, \sigma'} (\bm k_3) \psi_{d, \sigma}(\bm k_4)]
\end{eqnarray}
where $a$ is the subband index and is used to distinguish the smaller $(a = 1)$ and larger ($a=2$) Fermi surface,
and $\epsilon_{\bm k,a} = k^2/2m + \delta_a$ with $\delta_1 = 0$ and $\delta_2 > 0$.
The interactions are assumed to be short-ranged, consisting of an intra-band repulsion $U$, an inter-band repulsion $V$, and an inter-band pair-hopping amplitude $J$. The interaction matrix
in the basis
$(1 \sigma 1 \sigma', 1 \sigma 2 \sigma', 2 \sigma 1 \sigma', 2 \sigma 2 \sigma')$
is thus
\begin{equation}
V^{\sigma , \sigma'}_{ab, cd} =
\left( \begin{array}{cccc}
U_{\sigma \sigma'} & 0 & 0 & J_{\sigma \sigma'} \\
0 & 0 & V_{\sigma \sigma'} & 0 \\
0 & V_{\sigma \sigma'} & 0 & 0 \\
J_{\sigma \sigma'} & 0 & 0 & U_{ \sigma \sigma'}
\end{array} \right)
\end{equation}
where
\begin{eqnarray}
U_{\sigma \sigma'} &=& U\left( 1 - \delta_{\sigma \sigma'} \right) \nonumber \\
J_{\sigma \sigma'} &=& J\left( 1 - \delta_{\sigma \sigma'} \right) \nonumber \\
V_{\sigma \sigma'} &=& V
\end{eqnarray}
As before,
rotational invariance enables us to label the eigenstates by the eigenvalue of the rotation operator,
\begin{equation}
\phi^{(m)}_{\hat k} = \phi_a^{(m)}
\cos{\left( m \theta \right)}
\end{equation}
where the complex amplitude $\phi_a^{(m)}$ depends only on the subband index associated with $\hat k$,
$\theta$ is the angle between $\hat k$ and an arbitrarily defined $x$ axis, and $m$ must be an even integer in the singlet channel
and an odd integer in the triplet channel.
Consequently, for each integer $m$, rotational symmetry reduces the eigenvalue problem to a $2\times 2$ problem,
\begin{equation}
\sum_{a,a^\prime}
\tilde g^{(m)}_{a,a^\prime}
\phi^{(m)}_{a^\prime}
= \lambda^{(m)}\phi^{(m)}_{a}
\end{equation}
where
\begin{equation}
\tilde g^{(m)}_{a,a^\prime}
\equiv \int_a
\frac {d\hat k}{2\pi}\int_{a^\prime}
\frac {d\hat k}{2\pi}
g^{(y)}_{\hat k,\hat k^\prime} e^{-im\theta}e^{im\theta^\prime},
\label{tildeg}
\end{equation}
where $y=s$ (singlet) for $m$ even and $y=t$ (triplet) for $m$ odd.
The most negative eigenvalue for fixed $m$ is
\begin{eqnarray}
\lambda^{(m)} =&&- \left(\frac{\tilde g^{(m)}_{1,1}+ \tilde g^{(m)}_{2,2}} 2\right) \\
&&-\sqrt{ \left(\frac { \tilde g^{(m)}_{1,1}- \tilde g^{(m)}_{2,2}} 2\right)^2+| \tilde g^{(m)}_{1,2}|^2}
\nonumber
\end{eqnarray}
We first consider the {\bf spin triplet channel} ($m$ odd) which is only a slight extension of the result obtained for a partially polarized Fermi surface. As shown in the Appendix, for odd $m$, the effective interaction is diagonal in the subband index, and depends on $U$, $V$, but not $J$:
\begin{eqnarray}
g^{(t)}_{1,1} (\hat k, \hat q) &=& - \rho U^2 \chi_{1,1} (\hat k - \hat q) - 2 \rho V^2 \chi_{2,2} (\hat k - \hat q) \nonumber\\
g^{(t)}_{2,2} (\hat k, \hat q) &=& -\rho U^2 \chi_{2,2} (\hat k - \hat q) - 2 \rho V^2 \chi_{1,1} (\hat k - \hat q) \nonumber \\
g^{(t)}_{1,2} (\hat k, \hat q) & = & 0
\end{eqnarray}
where
\begin{equation}
\chi_{a,b}( \bm k) = \int \frac{d^2 p}{(2 \pi)^2} \frac{f(\epsilon_{\bm p + \bm k, a}) - f(\epsilon_{\bm p, b})}{\epsilon_{\bm p + \bm k, a} - \epsilon_{\bm p, b}}
\end{equation}
is the particle-hole susceptibility generalized to the two band system.
The intraband susceptibilities
are precisely the same functions used before:
\begin{equation}
\label{chi2d2}
\chi_{a,a}(\vec q) = \frac{\rho}{2 } \left[ 1 - \frac{{\rm Re} \sqrt{q^2 - (2 k_{F a})^2}}{q} \right]
\end{equation}
with the subband index playing the role that the spin played in the previous section. Therefore, we may simply
transcribe the results found in the previous section to the present context. The band which forms the smaller Fermi surface ($a=1$) does not exhibit a superconducting instability to $\mathcal{O}(U^2)$.
The larger Fermi surface exhibits a triplet
p-wave instability with a pairing strength determined solely by $V$:
\begin{equation}
\lambda^{(1)}(\eta) = -4 \rho^2 V^2 \eta( 1- \eta)
\end{equation}
where $\eta = \left( k_{F 1}/ k_{F 2} \right)$.
In the {\bf spin-singlet channel}, the matrix $g$
has off-diagonal components:
\begin{eqnarray}
\label{singlet}
g^{(s)}_{1,1}(\hat k, \hat q) &=& \rho U_1
- 2\rho V^2 \chi_{2,2}(\hat k - \hat q)
\\
g^{(s)}_{2,2}(\hat k, \hat q) &=& \rho U_2
- 2 \rho V^2 \chi_{1,1}(\hat k - \hat q) \nonumber \\
g^{(s)}_{1,2}(\hat k, \hat q) &=& \rho U_{12}
+ \rho VJ \left[ \chi_{1,2}(\hat k + \hat q) + \chi_{1,2}( \hat k - \hat q) \right] \nonumber
\end{eqnarray}
where $U_{ab}$ are momentum independent interactions,
\begin{eqnarray}
U_1 &\equiv & U + U^2P_1(\Omega) + J^2 P_2(\Omega) + U^2 \chi_{1,1}(\hat k + \hat q)
\nonumber \\
U_2 &\equiv & U + U^2P_2(\Omega) + J^2 P_1(\Omega) + U^2 \chi_{2,2}(\hat k + \hat q)
\nonumber \\
U_{12} &\equiv& UJ \left[ P_1 (\Omega) + P_2(\Omega) \right]
\end{eqnarray}
where the particle-particle susceptibility,
\begin{eqnarray}
P_a (\Omega) &\equiv &
\int \frac{d^2 q}{(2 \pi)^2} \frac{ 2 f
(\epsilon_{\bm q,a}) -1 }{i \Omega - 2 \epsilon_{\bm q,a}} \nonumber \\
& \sim & \rh
\log \left[ \frac{E_F-\delta_a}{\Omega} \right] + \mathcal{O}(\Omega),
\end{eqnarray}
is a momentum-independent constant which diverges logarithmically at low energies.
Despite this divergence, the second order contributions to $U_\alpha$ are unimportant, since
they do not enter the gap equation for any $m\neq 0$,
and any s-wave solution is already
killed by the first order terms proportional to $U$.
For $m>0$ and even, the effective intra-band interaction depends only on the interaction $V$ and is non-zero only for the larger Fermi surface, whereas the inter-band interaction depends both on $V$ and $J$:
\begin{eqnarray}
\tilde g_{1,1}^{(m)} &= &0 \nonumber \\
\tilde g_{2,2}^{(m)} &=& -2V^2 \rho \int \frac{d \theta}{2 \pi} \chi_{1,1}\left( 2 k_{F2} \left | \sin \left( \theta/2 \right)\right | \right) \cos\left( m \theta \right) \nonumber \\
\tilde g_{1,2}^{(m)} &=& 2VJ \rho \int \frac{d \theta}{2 \pi} \chi_{1,2}
\left( k_{\theta} \right) \cos\left( m \theta \right) \nonumber \\
k_{\theta} &=& k_{F2} \sqrt{\left(1-\eta \right)^2 + 4 \eta \sin^2 \left( \theta/2 \right) } .
\end{eqnarray}
The explicit expression for the inter-band susceptibility $\chi_{1,2}(\bm q) $ is derived in the Appendix.
Since the $m=0$
eigenvalues are always positive,
the dominant singlet instability
is in the d-wave ($m=2$)
channel. The quantity $\tilde g_{2,2}^{(2)}$ is obtained by computing
\begin{eqnarray}
\tilde g_{2,2}^{(2)} &=& -2V^2 \rho^2 \int_{-\pi}^{\pi} \frac{d \theta}{2 \pi}
d \theta \frac{ {\rm Re} \sqrt{ \sin^2{\frac{\theta}{2} } - \eta^2} }{\sin{\frac{\theta}{2}} } \cos{\left( 2 \theta \right) } \nonumber \\
&=& -V^2 \rho^2 \eta \left(\eta-1 \right)\left(\eta^2+\eta-1 \right)
\end{eqnarray}
The interband interaction $\tilde g_{1,2}^{(2)}$ is also obtained using Eq. \ref{singlet}:
\begin{equation}
\tilde g_{1,2}^{(2)} = -\frac{V J \rho^2}{2 \pi} \Phi(\eta)
\end{equation}
where, for $0 \le x < 1$,
\begin{equation}
\Phi(x) = \frac{\pi x^4+2 \sin^{-1} x}{x^2}
- 2\frac {\sqrt{1-x^2}} x
\end{equation}
This function is discussed in detail in the Appendix. An important property of $\Phi(x)$ is that it is a
monotonically increasing function of $x$ for $0 \le x <1$ (see Fig. \ref{interbandchi}). Therefore, the effective interband scattering grows with $\eta$.
The pairing strength in the d-wave channel is obtained from these quantities via
\begin{equation}
\lambda^{(2)}(\eta) = \frac{\tilde g^{(2)}_{2,2}}{2} - \frac{1}{2} \sqrt{\left(\tilde g^{(2)}_{2,2} \right)^2 + 4 \left( \tilde g^{(2)}_{1,2}\right)^2 }
\end{equation}
\begin{figure*}
\includegraphics[width=5.5in]{phasediag.eps}
\caption{Phase diagram of a 2DEG having two subbands. (a) Phase diagram for fixed $ \eta\equiv k_{F1}/k_{F2} = 0.5$ as a function of the dimensionless couplings
$\rho V$ and $\rho J$. $U$ does not enter the problem except in that it is responsible for the suppression of s-wave pairing.
(b) Phase diagram for fixed
$\rho V=0.1$ as a function of $\eta$ and
$\rho J$.
c) The dimensionless strength of the pairing interaction in the p-wave (solid line) and d-wave (dashed line) channels for fixed $J=0$. d) Same as c), but for $J=0.2V$.
}
\label{phasediag}
\end{figure*}
Having derived closed form expressions for the p-wave and d-wave pairing strengths, we can
construct the phase diagram, shown in Fig. \ref{phasediag}. The phases are labeled according to the symmetry of the most negative eigenvalue, so
the phase boundaries
are the lines at which
$\lambda^{(2)}=\lambda^{(1)} < 0$. Since the d-wave and p-wave eigenvalues are both negative for all $0 \le \eta \le 1$, where one phase is stable, the other is metastable.
It would require different methods of analysis to completely characterize the phase competition. However, in weak coupling, the phase with the larger $|\lambda_m|$ has an exponentially larger $T_c$, and so gaps the entire Fermi surface at temperatures far above the putative transition temperature of the subdominant phase. Thus, a BCS mean-field treatment of this problem would suggest that at low temperatures, there is a direct, first order transition, or at most an exponentially narrow region of phase coexistence between the extremal pure d-wave and p-wave phases.
Since the pair-hopping term only affects spin-singlet superconductivity, and since $\vert \lambda^{(1)} \vert > \vert \tilde g_{2,2}^{(2)} \vert$, it follows that for $J=0$, the p-wave solution always remains the favored ground state, as can be seen from Fig. \ref{phasediag}(a). However, as the interband scattering is enhanced, the d-wave pairing strength grows. Since the interband scattering increases monotonically as a function of both $\eta$ and $J$, it is seen that for sufficiently large values of either parameter the p-wave superconductivity gives way to a d-wave ground state. In Figs. \ref{phasediag}(c-d), we show how the magnitude of $\lambda^{(1)}$ and $\lambda^{(2)}$ depend on $\eta$, from which one can see that $T_c$ is maximal in the p-wave channel when
$\eta=0.5$. However, when $J \ne 0$, the
d-wave channel grows monotonically with $\eta$ and ultimately overtakes the p-wave pairing strength
as $\eta \rightarrow 1$. Note, however, that $\eta$ can never
equal unity in this context, since it
is determined by the thickness of the quantum well.
\section{Discussion}
We have obtained analytical expressions for various unconventional
superconducting ground states of a clean 2DEG in the presence of weak, short-ranged repulsive interactions. Ultimately, to make contact with experiments involving real 2DEGs, we must take into account the Coulomb interactions.
In the small $r_s$ limit,
the Coulomb interactions are sufficiently well screened
that it may be reasonable to treat them as weak and short-ranged\footnote{A diagrammatic approach to the full Coulomb problem in 3D at small $r_s$ was explored in Ref. \cite{Chubukov1989}, but there are many subtleties which make this hard to extend}. We thus imagine we can relate the physical problem to a problem with short-ranged interactions and speculate on two ways in which unconventional superconductivity could be found in the 2DEG in physically realizable semiconductor heterostructures. (We shall present more complicated examples in a forthcoming publication. \cite{RaghuII})
In the first scenario, an in-plane magnetic field
is applied to partially polarize the
2DEG in a narrow quantum well. This system
is predicted to exhibit p-wave pairing with a transition temperature which is non-monotonic in the magnetic field. The optimal transition temperature is
obtained for a magnetic
field
at which the ratio of the distinct spin Fermi momenta is $\eta=1/2$. In the second scenario,
the 2DEG is confined to a relatively broad quantum well, and the density is tuned to the range in which
two transverse subbands
are occupied.
For fixed total electron density, the ratio, $\eta^2$, of densities in the two subbands increases with increasing thickness $w$ of the quantum well. When this ratio is small, a p-wave
groundstate arises, with a $T_c$ that rises sharply with increasing $\eta$ so long as $\eta < 1/2$. However, this gives rise
to a d-wave ground states above
a certain critical thickness.
Insight into the dependence of $V/J$ on the thickness, $w$, is obtained by considering the
Coulomb interactions. A simple estimate shows that
for $k_Fw \ll 1$, $V \sim e^2/ k_F$ and $ J \sim
V\left( w k_F \right)$.
Therefore, for thicker quantum wells, $J$ becomes
increasingly important and favors d-wave pairing whereas thinner quantum wells should exhibit p-wave pairing.
Depending on the ratio of $V/J$, the optimal $T_c$ occurs either for $\eta \approx 1/2$ (p-wave) or for the largest possible $\eta$ (d-wave). In both cases, $T_c \sim E_F \exp{\left[ -\alpha/ (\rho V)^2
\right]}$ where $\alpha$ is an
$\mathcal{O}(1)$ constant. We have found that for d-wave superconductivity in the 2 subband system,
values as low as $\alpha \sim 1$ are within reach.
Three practical considerations warrant mention. Due to the unconventional nature of the
superconductvity,
it is very fragile to even weak quenched disorder. Therefore, the results presented here are likely to be realized only in the purest samples with mean free paths exceeding the Fermi wavelength by several orders of magnitude. Furthermore, for small $r_s$, the plasma frequency is small compared to the Fermi energy, {\it i.e.} $\omega_p \sim \sqrt{r_s}E_F$, so even if it is reasonable to treat the interactions as short-ranged at low energies, this approximation is certainly not valid all the way to the Fermi energy. Finally, since the transition temperatures are exponentially low in the effective interactions,
ultimately the superconductivity studied here is likely to be observable only in the regime $r_s \sim 1$, where the long-range character of the Coulomb interaction may not be negligible, and where, even for short-range interactions, a well-controlled solution to the problem is unfeasible.
We therefore are forced rely on the hope that the asymptotic results smoothly extrapolate to the intermediate coupling regime, where it is conceivable that these states can be observed in experiment.
With these caveats, we turn to the most uncertain part of the discussion, and make the following {\it crude} quantitative estimate of $T_c$ based on our calculations: We identify $V$ with the Fourier transform of the Coulomb interaction evaluated at $k_F$, {\it i.e.} $V \approx e^2 \pi/k_F$, from which it follows that $\rho V \approx (r_s/4)$. Since we are going to extrapolate to $r_s \sim 1$ in any case, we simply ignore subtleties associated with the small value of $\omega_p$. Then,
$T_c \sim E_F\exp[-\alpha (4/r_s)^2]$, where, for optimal circumstances $\alpha \approx 1$.
{\bf Acknowledgements} { We would like to acknowledge important discussions with D. Scalapino in early stages of this project. We thank S. Das Sarma, E. Fradkin, D. Scalapino, and especially A. Chubukov for their insightful comments. This work was supported, in part, by NSF grant number DMR-0758356 at Stanford.}
|
{
"timestamp": "2010-09-22T02:00:48",
"yymm": "1009",
"arxiv_id": "1009.3600",
"language": "en",
"url": "https://arxiv.org/abs/1009.3600"
}
|
\section{Introduction}
\label{Introduction}
We consider compactification of $({\cal N}=1,D=11)$ supergravity on a 7-manifold $X^7$ with betti numbers ($b_0, b_1, b_2, b_3, b_3, b_2, b_1, b_0$) and define a generalized mirror symmetry
\begin{equation}
(b_0, b_1, b_2, b_3) \rightarrow (b_0, b_1, b_2 -\rho/2, b_3+\rho/2)
\label{mirror}
\end{equation}
under which
\begin{equation}
\rho(X^7) \equiv 7b_0-5b_1+3b_2 -b_3
\end{equation}
changes sign
\begin{equation}
\rho \rightarrow -\rho
\end{equation}
Generalized self-mirror theories are defined to be those for which $\rho$ vanishes. In the case of $G_2$ manifolds with $b_1=0$, Joyce \cite{Joyce:1996a,Joyce:1996b} refers to $\rho=0$ as an ``axis of symmetry''. For related work on mirror symmetry and Joyce-manifiolds, see \cite{Shatashvili:1994zw,Acharya:1997rh,Gaberdiel:2004vx}.
The massless sectors of these compactifications have
\begin{equation}
f=4(b_0+b_1+b_2+b_3)
\end{equation}
degrees of freedom. Interestingly enough, we shall see in section \ref{D=4} that the quantity $\rho$ also shows up in their on-shell trace anomaly \cite{Duff:1977ay,Duff:1993wm}
\begin{equation}
g_{\mu\nu}<T^{\mu\nu}>=A \frac{1}{32\pi^2}R^{*}{}^{\mu\nu\rho\sigma}R^{*}{}_{\mu\nu\rho\sigma}
\end{equation}
which is given by
\begin{equation}
A=-\frac{1}{24}\rho.
\end{equation}
Hence generalized self-mirror theories have vanishing anomaly with betti numbers\footnote{We assume that there is a $U(1)^{b_1}$ isometry, which will be the case for $X^{(8-{\cal N})}\times T^{({\cal N}-1)}$ with $X^{(8-{\cal N})}$ simply connected.}
\begin{equation}
(b_0, b_1, b_2, b_3)= (1, {\cal N}-1, n, 3n-5{\cal N}+12)
\end{equation}
and
\begin{equation}
f=16(n-{\cal N}+3)
\end{equation}
degrees of freedom, where $1 \leq {\cal N} \leq 8$ is the number of supersymmetries. If we denote the $D=11$ fields by $(g_{MN}; \psi_{M}; A_{MNP})$ and the corresponding $D=4$ fields by $(g_{\mu\nu}, {\cal A}_{\mu}, {\cal A}; \psi_{\mu}, \chi; A_{\mu\nu\rho}, A_{\mu\nu}, A_{\mu}, A)$, then the possible generalized self-mirror theories and their betti numbers are:
\begin{itemize}
\item{${\cal N}=1, n \geq 0$, $f=16(n+2)$}
$X^7: (1, 0, n, 3n+7, 3n+7, n, 0, 1)$
yielding 1 graviton $(g_{\mu\nu};\psi_{\mu}; A_{\mu\nu\rho})$ plus $n$ vector $(\chi; A_{\mu})$ plus $(3n+7)$ chiral $({\cal A}; \chi; A)$.
\item{${\cal N}=2, n \geq 0$, $f=16(n+1)$}
$X^6:(1, 0, n, 2n+2, n, 0, 1)$; $X^6 \times S^1: (1, 1, n, 3n+2, 3n+2, n, 1, 1)$
yielding 1 graviton $(g_{\mu\nu}, {\cal A}_{\mu}; 2\psi_{\mu}; A_{\mu\nu\rho})$ plus $n$ vector $({\cal A}; 2\chi; A_{\mu}, A)$ plus $n$ hyper $(2{\cal A}; 2\chi; 2A)$ plus 1 linear $({\cal A}; 2\chi; A_{\mu\nu}, 2A)$.
\item{${\cal N}=3, n \geq 3$, $f=16n$}
$X^5: (1, 0, n-1, n-1, 0, 1)$; $X^5 \times T^2: (1, 2, n, 3n-3, 3n-3, n, 2, 1)$
yielding 1 graviton $(g_{\mu\nu}, 2{\cal A}_{\mu}; 3\psi_{\mu}, \chi; A_{\mu\nu\rho}, A_{\mu})$ plus $(n-3)$ vector $(3{\cal A}; 4\chi; A_{\mu}, 3A)$ plus $2$ 2-form $(2{\cal A}; \chi; A_{\mu\nu}, A_{\mu}, 3A)$.
\item{${\cal N}=4, n\geq 6$, $f=16(n-1)$}
$X^4: (1, 0, n-3, 0, 1)$; $X^4 \times T^3: (1, 3, n, 3n-8, 3n-8, n, 3, 1)$
yielding 1 graviton $(g_{\mu\nu}, 3 {\cal A}_{\mu}, {\cal A}, 4 \psi_{\mu}, 4 \chi, A_{\mu\nu\rho}, 3 A_{\mu}, A)$ plus $(n - 6)$ vector $(3{\cal A}, 4\chi; A_{\mu}, 3A)$ plus $3$ 2-form $(2{\cal A}; 4 \chi; A_{\mu\nu},A_{\mu}, 3 A)$.
The case $n=25$ corresponds to $X^4=K3$ \cite{Duff:1983vj}.
\item{${\cal N}=5, n=6$, $f=64$}
$X^3: (1, 0, 0, 1)$; $X^3 \times T^4: (1, 4, 6, 5, 5, 6, 4, 1)$
yielding 1 graviton $(g_{\mu\nu}, 4{\cal A}_{\mu}, {\cal A}; 5\psi_{\mu}, 11\chi; A_{\mu\nu\rho}, 4A_{\mu\nu}, 6A_{\mu}, 5A)$.
\item{${\cal N}=6, n=11$, $f=128$}
$X^2: (1, 0, 1)$; $X^2 \times T^5: (1, 5, 11,15,15, 11, 5, 1)$
yielding 1 graviton $(g_{\mu\nu}, 5{\cal A}_{\mu}, 10{\cal A}; 6\psi_{\mu}, 16\chi; A_{\mu\nu\rho}, 5A_{\mu\nu},11A_{\mu},15A)$.
\item{${\cal N}=8, n=21$, $f=256$}
$T^7: (1, 7, 21, 35, 35, 21, 7, 1)$
yielding 1 graviton $(g_{\mu\nu}, 7{\cal A}_{\mu}, 28 {\cal A}; 8\psi_{\mu}, 56\chi; A_{\mu\nu\rho}, 7A_{\mu\nu},21 A_{\mu},35 A)$.
\end{itemize}
In listing these results, we simply record what the betti numbers of the compactifying 7-manifold would have to be, without attempting to prove in all cases that such manifolds actually exist. Of particular interest are the four cases
\begin{equation}
(b_0,b_1,b_2,b_3)=(1,{\cal N}-1,3{\cal N}-3, 4{\cal N}+3)
\end{equation}
with ${\cal N}=1,2,4,8$, namely the four ``curious'' supergravities, discussed in
\cite{Duff:2010b}:
$({\cal N}=1, n=0, f=32)$, $({\cal N}=2, n=3, f=64)$, $({\cal N}=4, n=9, f=128)$, $({\cal N}=8, n=21 ,f=256)$, which enjoy some remarkable properties\footnote{ The ${\cal N}=8, 4, 2, 1$ cases are related \cite{Cremmer:1979up,Ferrara:1989nm,Sen:1995ff,Gaberdiel:2004vx} to the orbifolds $T^7$, ${T^7}/{Z_2}$, ${T^7}/({Z_2 \times Z_2})$ , ${T^7}/(Z_2 \times Z_2 \times Z_2)$.}.
In section \ref{IIA} we note that the case of M-theory on $X^6 \times S^1$ with betti numbers $(b_0, b_1, b_2, b_3, b_3, b_2, b_1, b_0)$ is equivalent to Type IIA on $X^6$ with betti numbers $(c_0, c_1, c_2, c_3, c_2, c_1, c_0)$ related by
\begin{equation}
(b_0 ,b_1, b_2, b_3)= (c_0, c_0+c_1, c_1+c_2,c_2+c_3)
\end{equation}
and hence
\be
\rho(X^6 \times S^1)= \chi(X^6)
\ee
where $ \chi(X^6)$ is the Euler number of $X^6$
\be
\chi(X^6)=2c_0-2c_1+2c_2 -c_3.
\ee
The generalized mirror symmetry transformation (\ref{mirror}) then becomes
\be
(c_0, c_1, c_2, c_3) \rightarrow (c_0, c_1, c_2-\chi/2, c_3+\chi)
\ee
under which $\chi$ also changes sign
\begin{equation}
\chi \rightarrow -\chi.
\end{equation}
Further specializing to $X^6$=Calabi-Yau with betti numbers: $(1, 0, h^{11}, 2+2h^{21}, h^{11}, 0, 1)$ our generalized mirror symmetry reduces to the familiar interchange of hodge numbers $h^{11} \leftrightarrow h^{21}$ \cite{Shing:492595}. As for the trace anomaly,
\be
A=-\frac{\chi}{24}
\ee
and so in Euclidean signature
\begin{equation}
\int d^4x \sqrt{g}g_{\mu\nu}<T^{\mu\nu}>=-\frac{1}{24}\chi(M^4)\chi(X^6)=-\frac{1}{24}\chi(M^4 \times X^6)
\end{equation}
where $\chi(M^4)$ is the Euler number of spacetime.
The compactifications of $({\cal N}=1, D=10)$ supergravity on $X^6$ are just given by the NS sector of Type IIA. Their massless sectors have
\begin{equation}
f=4(2c_0+2c_1+2c_2+c_3)
\end{equation}
degrees of freedom. Their anomaly is given by
\begin{equation}
A=-\frac{1}{24}(65c_0-15c_1+c_2+c_3/2)
\end{equation}
which vanishes when
\begin{equation}
(c_0, c_1, c_2, c_3)= (1, 2{\cal N}-2, n, 30{\cal N}-95-n)
\end{equation}
and
\begin{equation}
f=4(26{\cal N}-97+3n).
\end{equation}
So the only possibility is:
\begin{itemize}
\item{${\cal N}=4$, $n=15$, $f=128$}
1 graviton $(g_{\mu\nu}, 3 {\cal A}_{\mu}, \Phi, 4 \psi_{\mu}, 4 \chi, A_{\mu\nu}, 3 A_{\mu})$ plus $3$ vector $(3{\cal A}; 4 \chi; A_{\mu}, 3 A)$ + 3 vector $({\cal A}_{\mu}, 4{\cal A}; 4 \chi; 2 A)$
\end{itemize}
Note that the field content of the ${\cal N}=4$ graviton and vector multiplets arising from compactification of $({\cal N}=1,D=10)$ on $T^6$ is different from those arising from $({\cal N}=2,D=10)$ on $X^4 \times T^2$ with $X^4$ betti numbers $(1,0,6,0,1)$. In particular the anomalies of each multiplet vanish separately. These two versions of ${\cal N}=4$ are the dual pair discussed in \cite{Sen:1995ff}. Note also that the $({\cal N}=1,D=10)$ vector multiplet $(A_M; \chi)$ appearing in the heterotic string yields the vector $({ A_{\mu}}; 4 \chi; 6A)$ on $T^6$ which separately has $A=0$ also.
In obtaining these results, we adopt the interpretation of \cite{Duff:1980qv} that assigns different anomalies to $A_{\mu\nu}$ and ${\cal A}$ even though they are naively dual\footnote{As may be seen even in the apparently simple example of abelian 1-forms in four dimensions, these dualities are quite subtle \cite{Witten:1995gf}.} to one another (each with $f=1$ ) and nonzero anomaly to $A_{\mu\nu\rho}$ (with $f=0$). This is controversial, with some authors agreeing \cite{Nicolai:1980td} and others taking the view that $A_{\mu\nu}$ has the same anomaly as ${\cal A}$ and that $A_{\mu\nu\rho}$ has vanishing anomaly \cite{Siegel:1980ax,Fradkin:1983tg,Grisaru:1984vk,Buchbinder:2008jf}. For the purposes of comparison, we give the results that the alternative view would yield in appendix \ref{dual}. In particular, for M on $X^7$ and Type IIA on $X^6$
one finds
\begin{equation}
A(M)=-\frac{1}{24}{(41b_0-19b_1-3b_2+b_3)}
\end{equation}
\begin{equation}
A(IIA)=-\frac{1}{24}{(-22c_0+22c_1+2c_2-c_3)}
\end{equation}
\begin{equation}
A(IIB)=\frac{1}{24}(26c_0-26c_1+2c_2-c_3)
\end{equation}
none of which seems to have any separate topological significance (although $A(IIA)-A(IIB)=-{\chi}/{12}$). All yield a nonzero result for ${\cal N} > 3$. It should be noted, moreover, that our interpretation is supported by string calculations \cite{Antoniadis:1992sa}.
Given the relation between trace anomalies and logarithmic corrections to black hole entropy
\cite{Fursaev:1994te,Solodukhin:2008dh,Banerjee:2010qc,Aros:2010jb,Dowker:2010bu}, one is tempted to conclude that these are absent in generalized self-mirror theories. The authors of \cite{Banerjee:2010qc}, however, do not reach this conclusion and it seems that there are still some unresolved issues.
Finally, in section \ref{Fermion} we introduce a fermionic mirror map
\begin{equation}
(b_0, b_1, b_2, b_3) \rightarrow (b_0, b_1+{\cal N'}-{\cal N}, b_2 -{\cal N'}+{\cal N}, b_3)
\label{eq:mirrorfermi}
\end{equation}
which preserves the number of spin 2, spin 1 and spin 0 but changes the number of spin 3/2 (from ${\cal N} $ to ${\cal N}'$) and spin 1/2, keeping $f$ fixed. Previously known examples \cite{Ferrara:2006yb,Ferrara:2008ap,Roest:2009sn} of fermionic mirror pairs are provided by $({\cal N}=6,{\cal N}'=2)$, $({\cal N}=4,{\cal N}'=2)$ and $({\cal N}=3,{\cal N}'=2)$ supergravity plus matter theories. Both members of a pair have exactly the same bosonic field content including interactions. Curiously, the partner with the higher supersymmetry is generalized self-mirror in the bosonic sense. In addition, we find a new two-parameter family with $({\cal N}=1,{\cal N}'=2)$.
\FloatBarrier
\section{M on $X^7$}
\label{D=4}
\subsection{Betti numbers}
Consider $X^{(8-{\cal N})}\times T^{({\cal N}-1)}$ compactification of $D=11$ supergravity with 128+128 degrees of freedom
\[
(g_{MN}, \Psi_{M}, A_{MNP})
\]
as shown in Table \ref{D=11}. We denote the betti numbers of $X^7$, $X^6$, $X^5$, $X^4$, $X^3$, $X^2$ by the letters $b$, $c$, $e$, $d$, $j$, $k$, respectively. The betti numbers of $S^1$ are $(1,1)$, of $T^2$ are $(1,2,1)$, of $T^3$ are $(1,3,3,1)$, of $T^4$ are $(1,4,6,4,1)$, of $T^5$ are $(1,5,10,10,5,1)$ of $T^7$ are $(1,7,21,35,21,7,1)$, so
\begin{equation}\label{bet2}
\begin{split}
X^7&:(b_0 ,b_1, b_2, b_3)\\
X^6 \times S^1&: (c_0, c_0+c_1, c_1+c_2,c_2+c_3)\\
X^5 \times T^2&: (e_0, 2e_0+e_1, e_0+2 e_1+e_2, e_1+3e_2)\\
X^4 \times T^3&: (d_0, 3d_0+d_1, 3d_0+3d_1+d_2, d_0+4d_1+3d_2)\\
X^3 \times T^4&: (j_0, 4j_0+j_1, 7j_0+4j_1, 5j_0+10j_1)\\
X^2 \times T^5&: (k_0, 5k_0+k_1, 11k_0+5k_1, 15k_0+10k_1).
\end{split}
\end{equation}
\FloatBarrier
\subsection{Trace anomalies}
\label{trace}
The fields in the massless sector of each $D=4$ theory will exhibit an on-shell Weyl anomaly \cite{Duff:1977ay,Duff:1993wm}
\begin{equation}
g_{\mu\nu}<T^{\mu\nu}>=A \frac{1}{32\pi^2}R^{*}{}^{\mu\nu\rho\sigma}R^{*}{}_{\mu\nu\rho\sigma}
\end{equation}
so that in Euclidean signature
\begin{equation}
\int d^4x \sqrt{g}g_{\mu\nu}<T^{\mu\nu}>=A \chi(M^4)
\end{equation}
where $\chi(M^4)$ is the Euler number of spacetime. We adopt the interpretation of \cite{Duff:1980qv} that assigns different anomalies to $A_{\mu\nu}$ and ${\cal A}$ even though they are naively dual to one another (each with one degree of freedom) and nonzero anomaly to $A_{\mu\nu\rho}$ (with zero degrees of freedom). Starting with a Lagrangian
\be
-\frac{1}{2}\phi \Delta \phi
\ee
the one-loop effective action is
\be
ln ({\rm det}~ \Delta)^{-1/2}.
\ee
The total trace of the stress tensor, which we refer to as the ``anomaly'' even when $\Delta$ is not conformal, is then given by the Schwinger-De Witt coefficients $B$, which in four spacetime dimensions are quadratic in the curvature. When the operator is the laplacian on $p$-forms $\Delta_p$, the corresponding coefficients $B_p$ obey
\be
\int d^4x (B_0-B_1+B_2-B_3+B_4) =\frac{1}{32\pi^2}\int d^4x *R*R=\chi(M^4)= b_0-b_1+b_2-b_3+b_4
\ee
The ghost-for-ghost procedure \cite{Siegel:1980jj} means that we have
\begin{equation}\label{bet1}
\begin{split}
p&=0: B_0\\
p&=1: B_1-2B_0\\
p&=2: B_2-2B_1+3B_0\\
p&=3: B_3-2B_2+3B_1-4B_0\\
p&=4: B_4-2B_3+3B_2-4B_1+5B_0\\
\end{split}
\end{equation}
Bearing in mind $B_p=B_{(4-p)}$, we find
\be
A_2-A_0=1
\ee
even though both describe one degree of freedom
and
\be
A_3=-2
\ee
\be
A_4=3
\ee
even though both describe zero degrees of freedom. In fact for $p \geq 3$
\be
A_p=(-1)^p(p-1).
\ee
The value of the $A$ coefficient for each supergravity field \cite{Duff:1977ay,Duff:1993wm} is given in Table \ref{D=11}. Remarkably, we find that the total anomaly depends on $\rho$
\begin{equation}
A=-\frac{1}{24}\rho(X^7).
\end{equation}
So the anomaly flips sign under generalized mirror symmetry and vanishes for generalized self-mirror theories. In the case of $({\cal N}=1,D=11)$ on $X^6 \times S^1$, or equivalently (Type IIA, $D=10$) on $X^6$,
\begin{equation}
A=-\frac{1}{24}\chi(X^6)
\end{equation}
and so in Euclidean signature
\begin{equation}
\int d^4x \sqrt{g}g_{\mu\nu}<T^{\mu\nu}>=-\frac{1}{24}\chi(M^4)\chi(X^6)=-\frac{1}{24}\chi(M^4 \times X^6)
\end{equation}
where $\chi(M^4)$ is the Euler number of spacetime. It would be interesting to see if this formula generalizes to other spacetime dimensions.
For $X^{(8-{\cal N})}\times T^{({\cal N}-1)}$ with ${\cal N} \geq 3$ the anomaly vanishes identically as shown in Table \ref{D=11}. Of particular interest are the four cases
\begin{equation}
(b_0,b_1,b_2,b_3)=(1,{\cal N}-1,3{\cal N}-3, 4{\cal N}+3)
\end{equation}
with ${\cal N}=1,2,4,8$, namely the four ``curious'' supergravities, discussed in
\cite{Duff:2010b}:
$({\cal N}=1, n=0, f=32)$, $({\cal N}=2, n=3, f=64)$, $({\cal N}=4, n=9, f=128)$, $({\cal N}=8, n=21 ,f=256)$, which enjoy some remarkable properties.
\begin{table}[h!]
$\begin{array}{llrrrrrrrrrrrrrrr}
&Field &f&360A&X^7& X^6 \times S^1&X^5 \times T^2&X^4 \times T^3 &X^3 \times T^4&X^2 \times T^5&T^7 \\
&&&&&&&\\
\bigskip
&&&&\\
g_{MN}&g_{\mu\nu}&2&848&b_0&c_0&e_0&d_0&j_0&k_0&1\\
~&{\cal A}_{\mu}&2&-52&b_1&c_0+c_1&2e_0+e_1&3d_0+d_1&4j_0+j_1&5k_0+k_1&7\\
~&{\cal A}&1&4&-b_1+b_3 &-c_0-c_1+c_2+c_3&-2e_0+3e_2&-2d_0+3d_1+3d_2&j_0+9j_1&10k_0+9k_1&28\\
\psi_{M}&\psi_{\mu}&2&-233&b_0+b_1&2c_0+c_1&3e_0+e_1&4d_0+d_1&5j_0+j_1&6k_0+k_1&8\\
~&\chi&2&7&b_2+b_3&c_1+2c_2+c_3&e_0+3e_1+4e_2&4d_0+7d_1+4d_2&11j_0+15j_1&26k_0+15k_1&56\\
A_{MNP}&A_{\mu\nu\rho}&0&-720&b_0&c_0&e_0&d_0&j_0&k_0&1\\
~&A_{\mu\nu}&2&364&b_1&c_0+c_1&2e_0+e_1&3d_0+d_1&4j_0+j_1&5k_0+k_1&7\\
~&A_{\mu}&2&-52&b_2&c_1+c_2&e_0+2e_1+e_2&3d_0+3d_1+d_2&6j_0+5j_1&11k_0+5k_1&21\\
~&A&1&4&b_3&c_2+c_3&e_1+3e_2&d_0+4d_1+3d_2&5j_0+10j_1&15k_0+5k_1&35\\
&&&&&\\
&&&&&&&\\
total ~A&&&&-\rho/24&-\chi/24&0&0&0&0&0&\\
\end{array}$
\caption{ $X^{(8-{\cal N})}\times T^{({\cal N}-1)}$ compactification of $D=11$ supergravity}
\label{D=11}
\end{table}
\FloatBarrier
\subsection{Multiplets}
Here we group the individual fields into supermultiplets as shown in Tables \ref{Y7} to \ref{Y0}, making use of table \ref{4}.
\begin{table}[h!]
$\begin{array}{lrrrrrrrrrrrrrrrr}
Field &360A &{\cal N}=8~ graviton &&{\cal N}=4 ~graviton&{\cal N}=4~gravitino &{\cal N}=4 ~vector_A & {\cal N}=4~ vector_{\cal A}\\
\bigskip
&&&&&\\
g_{\mu\nu}&848&{\bf 1}&&{\bf 1}&\\
{\cal A}_{\mu}&-52&{\bf 7}&&{\bf 3}&4.{\bf 1}&&\\
{\cal A}&4&{\bf 27}&&&4.{\bf 3}&9.{\bf 1}&{\bf 1}+{\bf 5} \\
\Phi&4&{\bf 1}&&{\bf 1}&\\
\psi_{\mu}&-233&{\bf 8}&&2.{\bf 2}&2.{\bf 2}\\
\chi&7&{\bf 8+48}&&2.{\bf 2}&10.{\bf 2}+2.{\bf 4}&6.{\bf 2}&2.{\bf 2}+2.{\bf 4}&\\
A_{\mu\nu\rho}&-720&{\bf 1}&&{\bf 1}\\
A_{\mu\nu}&364&{\bf 7}&&&4.{\bf 1}&&{\bf 3}\\
A_{\mu}&-52&{\bf 21}&&3.{\bf 1}&4.{\bf 3}&3.{\bf 1}&{\bf 3}\\
A&4&{\bf 35}&&{\bf 1}&4.{\bf 1}+4.{\bf 3}&3.{\bf 3}&3.{\bf 3}\\
&&&&&&&\\
{}~&&A=0&&A=3&A=0&A=0&A=-3\\
\end{array}$
\caption{ $({\cal N}=8,SO(7)) \rightarrow ({\cal N}=4,SO(3)) $ decomposition appropriate for M and Type IIA compactifications. }
\label{4}
\end{table}
\begin{table}[h!]
$\begin{array}{llrrrrrrrrrr}
{\cal N}=1~&multiplet &f&360A&{\cal N}=b_0+b_1 & {\cal N}=8 & {\cal N}=1 & \\
&&&&&&&\\
\bigskip
graviton&(g_{\mu\nu};\psi_{\mu}; A_{\mu\nu\rho})&2+2&-105&b_0&1&1\\
gravitino&({\cal A}_{\mu}; \psi_{\mu}) &2+2&-285&b_1&7&0 \\
&&&&&&&\\
vector&(\chi; A_{\mu})&2+2&-45&b_2&21&n\\
&&&&&&&\\
chiral &({\cal A}; \chi; A)&2+2&15&-b_1+b_3&28&3n+7\\
&&&&&&\\
linear&(\chi; A_{\mu\nu}, A)&2+2&375&b_1&7&0\\
&&&&&&\\
total ~f &&&&4(b_0+b_1+b_2+b_3)&256&16(n+2)\\
&&&&&&\\
total ~A &&&&-(7b_0-5b_1+3b_2-b_3)/24&0&0\\
\end{array}$
\caption{ The $D=4$ multiplets in an ${\cal N}$=1 basis. }
\label{Y7}
\end{table}
\begin{table}[h!]
$\begin{array}{llrrrrrrrrrrrrr}
{\cal N}=2~& multiplet &f&360A&{\cal N}=2c_0+c_1 & {\cal N}=8 & {\cal N}=2 & \\
&&&&&&&\\
\bigskip
graviton&(g_{\mu\nu}, {\cal A}_{\mu}; 2\psi_{\mu}; A_{\mu\nu\rho})&4+4&-390&c_0&1&1\\
gravitino&({\cal A}_{\mu}; \psi_{\mu}, \chi; A_{\mu}) &4+4&-330&c_1&6&0 \\
&&&&&&&\\
vector&({\cal A}, 2\chi; A_{\mu}, A)&4+4&-30&c_2&15&n\\
&&&&&&&\\
hyper&(2{\cal A}; 2\chi; 2A)&4+4&30&-c_0-c_1+c_3/2&3&n\\
&&&&&&\\
linear&({\cal A}; 2\chi; A_{\mu\nu}, 2A)&4+4&390&c_0+c_1&7&1\\
&&&&&&\\
total ~f &&&&4(2c_0+2c_1+2c_2+c_3)&256&16(n+1)\\
&&&&&&\\
total ~A &&&&-(2c_0-2c_1+2c_2-c_3)/24&0&0\\
\end{array}$
\caption{ The $D=4$ multiplets in an ${\cal N}$=2 basis.}
\label{Y6}
\end{table}
\begin{table}[h!]
$\begin{array}{llrrrrrrrrrr}
{\cal N}=3~& multiplet &f&360A&{\cal N}=3e_0+e_1 & {\cal N}=8 & {\cal N}=3 & \\
&&&&&&&\\
\bigskip
graviton&(g_{\mu\nu}, 2{\cal A}_{\mu}; 3\psi_{\mu}, \chi; A_{\mu\nu\rho}, A_{\mu})&8+8&-720&e_0&1&1\\
gravitino&({\cal A}_{\mu}, {\cal A}; \psi_{\mu}, 3\chi; 2A_{\mu}, A) &8+8&-360&e_1&5&0 \\
&&&&&&&\\
vector&(3{\cal A}; 4\chi; A_{\mu}, 3A)&8+8&0&-2e_0-e_1+e_2&3&n-3\\
&&&&&&\\
2-form&(2{\cal A}; 4\chi; A_{\mu\nu}, A_{\mu}, 3A)&8+8&360&2e_0+e_1&7&2\\
&&&&&&&\\
total ~f &&&&16(e_0+e_1+e_2)&256&16n\\
&&&&&&\\
total ~A &&&&0&0&0\\
\end{array}$
\caption{ The $D=4$ multiplets in an ${\cal N}$=3 basis.}
\label{Y5}
\end{table}
\begin{table}[h!]
$\begin{array}{llrrrrrrrrrr}
{\cal N}=4~ & multiplet&f&360A& {\cal N}=4d_0+d_1 & {\cal N}=8 & {\cal N}=4 & \\
&&&&&&&\\
\bigskip
graviton&(g_{\mu\nu}, 3 {\cal A}_{\mu}, {\cal A}, 4 \psi_{\mu}, 4 \chi, A_{\mu\nu\rho}, 3 A_{\mu}, A)&16+16&-1080&d_0&1&1\\
gravitino&({\cal A}_{\mu}, 3{\cal A}, \psi_{\mu}, 7 \chi, A_{\mu\nu}, 3A_{\mu},4A) &16+16&0&d_1&4&0 \\
&&&&&&&\\
vector&(3{\cal A}; 4 \chi; A_{\mu}, 3 A)&8+8&0&-3d_0+d_2&3&n-6\\
&&&&&\\
2-form &(2{\cal A}; 4 \chi; A_{\mu\nu},A_{\mu}, 3 A)&8+8&360&3d_0&3&3\\
&&&&&&&\\
total ~f &&&&16(2d_0+2d_1+d_2)&256&16(n-1)\\
&&&&&&\\
total ~A &&&&0&0&0\\
\end{array}$
\caption{ The $D=4$ multiplets in an ${\cal N}$=4 basis.}
\label{Y4}
\end{table}
\begin{table}[h!]
$\begin{array}{llrrrrrrrrrr}
{\cal N}=5~ & multiplet&f&360A&{\cal N}=5j_0+j_1&{\cal N}=8&{\cal N}=5 \\
&&&&&&&\\
\bigskip
graviton&(g_{\mu\nu}, 4{\cal A}_{\mu}, {\cal A}; 5\psi_{\mu}, 11\chi; A_{\mu\nu\rho}, 4A_{\mu\nu}, 6A_{\mu}, 5A)
&32+32&0&j_0&1&1\\
gravitino&({\cal A}_{\mu}, 9{\cal A}, \psi_{\mu}, 15 \chi, A_{\mu\nu}, 5A_{\mu}, 10A) &32+32&0&j_1&3&0 \\
&&&&&&&\\
total ~f &&&&64(j_0+j_1)&256&64\\
&&&&&&&\\
total ~A &&&&0&0&0\\
&&&&&&\\
\end{array}$
\caption{ The $D=4$ multiplets in an ${\cal N}$=5 basis.}
\label{Y3}
\end{table}
\begin{table}[h!]
$\begin{array}{llrrrrrrrrrr}
{\cal N}=6~ & multiplet&f&360A&{\cal N}=6k_0+k_1&{\cal N}=8&{\cal N}=6 \\
&&&&&&&\\
\bigskip
graviton&(g_{\mu\nu}, 5{\cal A}_{\mu}, 10{\cal A}; 6\psi_{\mu}, 26\chi; A_{\mu\nu\rho}, 5A_{\mu\nu},11A_{\mu},15A)
&64+64&0&k_0&1&1\\
gravitino&({\cal A}_{\mu}, 9{\cal A}, \psi_{\mu}, 15\chi, A_{\mu\nu}, 5A_{\mu},10A) &32+32&0&k_1&2&0 \\
&&&&&&&\\
total ~f &&&&64(2k_0+k_1)&256&128\\
&&&&&&&\\
total ~A &&&&0&0&0\\
\end{array}$
\caption{ The $D=4$ multiplets in an ${\cal N}$=6 basis.}
\label{Y2}
\end{table}
\begin{table}[h!]
$\begin{array}{llrrrrrrrrrr}
{\cal N}=8~ & multiplet&f&360A&{\cal N}=8 \\
&&&&&&&\\
\bigskip
graviton&(g_{\mu\nu}, 7{\cal A}_{\mu}, 28 {\cal A}; 8\psi_{\mu}, 56\chi; A_{\mu\nu\rho}, 7A_{\mu\nu},21 A_{\mu},35 A)
&256&0&1&\\
&&&&&&&\\
total ~f &&&&256\\
&&&&&&&\\
total ~A &&&&0&\\
\end{array}$
\caption{ The $D=4$ multiplets in an ${\cal N}$=8 basis.}
\label{Y0}
\end{table}
\FloatBarrier
\section{From $D=10$ on $X^6$}
\label{IIA}
\subsection{IIA}
Consider Type IIA in $D=10$. In the NS sector we have the fields
$(g_{MN}, \Phi; \psi_M, \chi; {A}_{MN})$ with $f=64+64$; in the RR sector we have the fields $({\cal A}_M;\psi_M, \chi; A_{MNP})$ also with $f=64+64$. We compactify on generic $X^6$ with independent betti numbers $(c_0,c_1,c_2,c_3)$ and on $T^6$ with $(1,6.15.20)$. The results for $NS$ and $RR$ separately and combined are shown in Table \ref{D=10A}.
\begin{table}[h!]
$\begin{array}{llrrrrrrrrrrrrrrr}
Field &f&360A&NS&T^6&RR&T^6&IIA&T^6\\
&&&&&&&\\
\bigskip
&&&&\\
g_{\mu\nu}&2&848&c_0&1&0&0&c_0&1\\
{\cal A}_{\mu}&2&-52&c_1&6&c_0&1&c_0+c_1&7\\
{\cal A}&1&4&-2c_0-2c_1+c_2+c_3&21&c_1&6&-2c_0-c_1+c_2+c_3&27\\
\Phi&1&4&c_0&1&0&0&c_0&1\\
\psi_{\mu}&2&-233&c_0+c_1/2&4&c_0+c_1/2&4&2c_0+c_1&8\\
\chi&2&7&c_1/2+c_2+c_3/2&28&c_1/2+c_2+c_3/2&28&c_1+2c_2+c_3&56\\
A_{\mu\nu\rho}&0&-720&0&0&c_0&1&c_0&1\\
A_{\mu\nu}&1&364&c_0&1&c_1&6&c_0+c_1&7\\
A_{\mu}&2&-52&c_1&6&c_2&15&c_1+c_2&21\\
A&1&4&c_2&15&c_3&20&c_2+c_3&35\\
&&&&&&&\\
total~f&&&2(2c_0+2c_1+2c_2+c_3)&128&2(2c_0+2c_1+2c_2+c_3)&128&4(2c_0+2c_1+2c_2+c_3)&256\\
total~A&&&(65c_0-15c_1+c_2+c_3/2)/24&0&(-67c_0+17c_1-3c_2+c_3/2)/24&0&-(2c_0-2c_1+2c_2-c_3)/24&0\\
\end{array}$
\caption{ $X^6$ compactification of $D=10$ Type IIA sugravity}
\label{D=10A}
\end{table}
\FloatBarrier
From Table \ref{D=10A} we have
\begin{equation}
\begin{split}
A(NS)&=\frac{1}{24}(65c_0-15c_1+c_2+c_3/2)\\
A(RR)&=\frac{1}{24}(-67c_0+17c_1-3c_2+c_3/2)\\
A(IIA)&=-\frac{1}{24}(2c_0-2c_1+2c_2-c_3)=-\frac{1}{24}\chi\\
\end{split}
\end{equation}
Now consider Type IIA on ${\tilde X}^6$, the mirror of $X^6$, with betti numbers
\be
(c_0,c_1,-c_0+c_1+c_3/2, 2c_0+2c_1+2c_2)
\ee
The NS sector remains unchanged and from Table \ref{D=10A'} we have
\begin{equation}
\begin{split}
\tilde A({ {NS}})&=\frac{1}{24}(65c_0-15c_1+c_2+c_3/2)\\
\tilde A({{RR}})&=\frac{1}{24}(-63c_0+13c_1+c_2-3c_3/2)/24\\
\tilde A({ {IIA}})&=\frac{1}{24}(2c_0-2c_1+2c_2-c_3)=\frac{1}{24}\chi\\
\end{split}
\end{equation}
\begin{table}[h!]
$\begin{array}{llrrrrrrrrrrrrrrr}
Field &f&360A&NS&T^6&RR&T^6&IIA&T^6\\
&&&&&&&\\
\bigskip
&&&&\\
g_{\mu\nu}&2&848&c_0&1&0&0&c_0&1\\
{\cal A}_{\mu}&2&-52&c_1&6&c_0&1&c_0+c_1&7\\
{\cal A}&1&4&-c_0-3c_1+2c_2+c_3/2&21&c_1&6&-c_0-2c_1+2c_2+c_3/2&27\\
\Phi&1&4&c_0&1&0&0&c_0&1\\
\psi_{\mu}&2&-233&c_0+c_1/2&4&c_0+c_1/2&4&2c_0+c_1&8\\
\chi&2&7&c_1/2+c_2+c_3/2&28&c_1/2+c_2+c_3/2&28&c_1+2c_2+c_3&56\\
A_{\mu\nu\rho}&0&-720&0&0&c_0&1&c_0&1\\
A_{\mu\nu}&1&364&c_0&1&c_1&6&c_0+c_1&7\\
A_{\mu}&2&-52&c_1&6&-c_0+c_1+c_3/2&15&-c_0+2c_1+c_3/2&21\\
A&1&4&-c_0+c_1+c_3/2&15&2c_0-2c_1+2c_2&20&c_0-c_1+2c_2+c_3/2&35\\
&&&&&&&\\
total~f&&&2(2c_0+2c_1+2c_2+c_3)&128&2(2c_0+2c_1+2c_2+c_3)&128&4(2c_0+2c_1+2c_2+c_3)&256\\
total~\tilde A&&&(65c_0-15c_1+c_2+c_3/2)/24&0&(-63c_0+13c_1+c_2-3c_3/2)/24&0&(2c_0-2c_1+2c_2-c_3)/24&0\\
\end{array}$
\caption{ ${\tilde X}^6$ compactification of $D=10$ Type IIA supergravity}
\label{D=10A'}
\end{table}
\subsection{{\cal N}=1}
The massless sectors of the $(\mathcal{N}=1,D=10)$ supergravity compactifications are given just by the NS sector
\begin{equation}
f=2(2c_0+2c_1+2c_2+c_3)
\end{equation}
and
\begin{equation}
A=-\frac{1}{24}(65c_0-15c_1+c_2+c_3/2).
\end{equation}
They have vanishing anomaly when
\begin{equation}
(c_0, c_1, c_2, c_3)= (1, 2{\cal N}-2, n, 30{\cal N}-95-n)
\end{equation}
and
\begin{equation}
f=4(26{\cal N}-97+3n)
\end{equation}
so the only possibility is:
\begin{itemize}
\item{${\cal N}=4$, $n=15$, $f=128$}
\end{itemize}
Next consider an $({\cal N}=1, D=10)$ vector multiplet $(A_M, \chi)$ with $f=8+8$ as shown in Table \ref{D=10'}.
\begin{table}[h!]
$\begin{array}{llrrrrrrrrrr}
&Field &f&360A&X^6&X^4 \times T^2& T^6\\
&&&&&&&\\
\bigskip
&&&&\\
A_{M}&A_{\mu}&2&-52&c_0&d_0&1\\
~&{A}_{}&1&4&c_1&2 d_0+d_1&6\\
\chi~&\chi&2&7&c_0+c_1/2&2d_0+d_1/2&4\\
&&&&&\\
total~ f&&&&2(2c_0+c_1)&2(4d_0+d_1)&16&\\
&&&&&&&\\
total~A&&&&(-3c_0+c_1/2)/24&(-2d_0+d_1/2)/24&0\\
\end{array}$
\caption{ Compactifications of ${\cal N}$=1 $D=10$ vector multiplet}
\label{D=10'}
\end{table}
The massless sectors of the vector compactifications have
\begin{equation}
f=2(2c_0+c_1)
\end{equation}
and
\begin{equation}
A=\frac{1}{24}(-3c_0+c_1/2).
\end{equation}
They have vanishing anomaly when
\begin{equation}
(c_0, c_1)= (1, 2{\cal N}-2)
\end{equation}
and
\begin{equation}
f=4{\cal N}.
\end{equation}
so the only possibility is:
\begin{itemize}
\item{${\cal N}=4$, $f=16$}
\end{itemize}
\FloatBarrier
\subsection{ IIB }
\label{IIB}
Consider Type IIB in $D=10$. In the NS sector we have the $D=10$ fields $(g_{MN}, \psi_M, B_{MN},\chi, \Phi)$; with $f=64+64$; in the R-R we have $(A_{MNPQ}{}^+, \psi, C_{MN},\chi, D)$ also with $f=64+64$. If IIB is T-dual to IIA, we might expect $A(IIB)=\chi/24$ on $X^6$ since $A(IIA)=-\chi/24$ on $X^6$ and $\tilde A({ {IIA}})=\chi/24$ on its mirror. But IIB is tricky: how do we assign four dimensional tensors coming from the self-dual 5-form in $D=10$? The authors of \cite{Cremmer:1997ct,Cremmer:1998px} address this problem in the case of $T^6$ by first writing the Lagrangian in $D=9$, where it coincides with that of IIA except $A_{MNP}$ is swapped for its dual $A_{MNPQ}$, and then compactifying on $T^5$. The results are shown in Table \ref{D=10B} and, assigning $360A=1080$ to $A_{\mu\nu\rho\sigma}$ as in section \ref {trace}, we find that the anomaly vanishes for IIB just as for IIA. Unfortunately, this trick does not generalize in a useful way for us, because $X^5 \times S^1$ has vanishing Euler number and is therefore not a good laboratory for testing mirror symmetry.
\begin{table}[h!]
$\begin{array}{llrrrrrrrrrrrrrrr}
Field &f&360A&NS&T^6&RR&T^6&IIB&T^6\\
&&&&&&&\\
\bigskip
&&&&\\
g_{\mu\nu}&2&848&&1&&0&&1\\
{\cal A}_{\mu}&2&-52&&6&&1&&7\\
{\cal A}&1&4&&21&&6&&27\\
\Phi&1&4&&1&&0&&1\\
\psi_{\mu}&2&-233&&4&&4&&8\\
\chi&2&7&&28&&28&&56\\
A_{\mu\nu\rho\sigma}&0&1080&&0&&1&&1\\
A_{\mu\nu\rho}&0&-720&&0&&5&&5\\
A_{\mu\nu}&1&364&&1&&11&&12\\
A_{\mu}&2&-52&&6&&15&&21\\
A&1&4&&15&&15&&30\\
&&&&&\\
total~f&&&&128&&128&&256\\
total~A&&&&0&&0&&0\\
\end{array}$
\caption{ $T^6$ compactification of $D=10$ Type IIB supergravity}
\label{D=10B}
\end{table}
\FloatBarrier
\section{Fermionic mirrors}
\label{Fermion}
The bosonic mirror map
\begin{equation}
(b_0, b_1, b_2, b_3) \rightarrow (b_0, b_1, b_2 -\rho/2, b_3+\rho/2)
\label{eq:mirrorbose}
\end{equation}
preserves the number of spin 3/2 and spin 1/2 but changes the number of spin 1 and spin 0
as in \autoref{boson}. Note incidentally that the number of fields of spin $(2, 3/2, 1, 1/2, 0)$ equals $(b_0, b_0+b_1, b_1+b_2, b_2+b_3, 2b_3)$=$(a_0, a_1, a_2, a_3, a_4)$ where the $a_i$ are the betti numbers of $X^7 \times S^1$. Since there are equal numbers of bosons and fermions, we must have $2a_0-2a_1+2a_2-2a_3+a_4=0$ which is just the vanishing of the euler number $\chi(X^7 \times S^1)$.
\begin{table}[h!]
$\begin{array}{lllllllllrrrrrrr}
Spin &&&X&X'\\
&&&&\\
2&&&b_0&b_0\\
3/2&&&b_0+b_1&b_0+b_1\\
1&&&b_1+b_2&b_1+b_2 -\rho/2\\
1/2&&&b_2+b_3&b_2+b_3\\
0&&&2b_3&2b_3+\rho \\
&&\\
&&&f=4(b_0+b_1+b_2+b_3)&f'=4(b_0+b_1+b_2+b_3)&\\
&&&\rho=(7-5b_1+3b_2-b_3)&\rho'=-(7-5b_1+3b_2-b_3)&\\
\end{array}$
\caption{ Bosonic mirror symmetry}
\label{boson}
\end{table}
We define a fermionic mirror map
\begin{equation}
(b_0, b_1, b_2, b_3) \rightarrow (b_0, b_1+{\cal N}'-{\cal N}, b_2 -{\cal N}'+{\cal N}, b_3)
\label{eq:mirrorfermi2}
\end{equation}
which preserves the number of spin 2, spin 1 and spin 0 but changes the number of spin 3/2 (from ${\cal N} $ to ${\cal N}'$) and spin 1/2, keeping $f$ fixed as in \autoref{fermion}.
\begin{table}[h!]
$\begin{array}{lllllllrrrrrrr}
Spin &&&X&X'\\
&&&&\\
2&&&b_0&b_0\\
3/2&&&b_0+b_1&b_0+b_1+{\cal N}'-{\cal N}\\
1&&&b_1+b_2&b_1+b_2\\
1/2&&&b_2+b_3&b_2+b_3-{\cal N}'+{\cal N}\\
0&&&2b_3&2b_3 \\
&&\\
&&&f=4(1+b_1+b_2+b_3)&f'=4(1+b_1+b_2+b_3)&\\
&&&\rho=(7-5b_1+3b_2-b_3)&\rho'=(7-5b_1+3b_2-b_3)-8(N'-{\cal N})&\\
\end{array}$
\caption{ Fermionic mirror symmtry}
\label{fermion}
\end{table}
However, we further require that each member of the pair have identical bosonic lagrangians. Previously known examples \cite{Andrianopoli:1996wf,Ferrara:2008ap,Roest:2009sn} are provided by $({\cal N}=6,{\cal N}'=2)$, $({\cal N}=4,{\cal N}'=2)$ and $({\cal N}=3,{\cal N}'=2)$ supergravity plus matter theories, as shown below. We also provide the relevant coset structure (after dualization). Curiously, the partner with the higher supersymmetry is generalized self-mirror in the bosonic sense. In addition, we find a new two-parameter family with $({\cal N}=1,{\cal N}'=2)$.
\begin{itemize}
\item{${\cal N}=6$, ${\cal N}'$=2}
${\cal N}=6$ with $(b_0,b_1,b_2,b_3)=(1,5,11,15)$ and Magic ${\cal N}'$=2 with $(b_0,b_1,b_2,b_3)=(1,1,15,15)$ as in \autoref{magic6}.
\begin{table}[h!]
$\begin{array}{lllllllrrrrrrr}
Spin &&&X&X'\\
&&&&\\
2&&&1&1\\
3/2&&&6&2\\
1&&&16&16\\
1/2&&&26&30\\
0&&&30&30 \\
&&&&\\
&&&f=128&f'=128\\
&&&\rho=0&\rho'=32\\
\end{array}$
\caption{$ {\cal N}=6$ and $ {\cal N}'=2$}
\label{magic6}
\end{table}
The relevant coset is
\be
\frac{SO^*(12)}{U(6)}
\ee
\item{${\cal N}=4$, ${\cal N'}$=2}
${\cal N}=4$ with $(b_0,b_1,b_2,b_3)=(1,3,5,7)$ and ${\cal N'}$=2 with $(b_0,b_1,b_2,b_3)=(1,1,7,7)$ as in \autoref{4}.
\begin{table}[h!]
$\begin{array}{lllllllrrrrrrr}
Spin &&&X&X'\\
&&&&\\
2&&&1&1\\
3/2&&&4&2\\
1&&&8&8\\
1/2&&&12&14\\
0&&&14&14 \\
&&&&\\
&&&f=64&f'=64\\
&&&\rho=0&\rho'=16\\
\end{array}$
\caption{$ {\cal N}=4$ and ${\cal N}'=2$}
\label{4}
\end{table}
The relevant coset is
\be
\frac{SL(2)}{U(1)} \times \frac{SO(6,2)}{SO(6) \times SO(2)}
\ee
\item{${\cal N}=3$, ${\cal N}'$=2}
${\cal N}=3$ with $(b_0,b_1,b_2,b_3)=(1,2,2,3)$ and ${\cal N}'$=2 with $(b_0,b_1,b_2,b_3)=(1,1,3,3)$ as in \autoref{3}.
\begin{table}[h!]
$\begin{array}{lllllllrrrrrrr}
Spin &&&X&X'\\
&&&&\\
2&&&1&1\\
3/2&&&3&2\\
1&&&4&4\\
1/2&&&5&6\\
0&&&6&6\\
&&&&\\
&&&f=32&f'=32\\
&&&\rho=0&\rho'=8\\
\end{array}$
\caption{ ${\cal N}=4$ and ${\cal N}'=2$}
\label{3}
\end{table}
The relevant coset is
\be
\frac{SU(3,1)}{SU(3) \times U(1)}
\ee
\item{${\cal N}=1$, ${\cal N}'$=2}
${\cal N}=1$ with $(b_0,b_1,b_2,b_3)=(1,0,n_1+1,2n_2+n_1)$ and ${\cal N}'$=2 with $(b_0,b_1,b_2,b_3)=(1,1,n_1, 2n_2+n_1)$ as in \autoref{1}.
\begin{table}[h!]
$\begin{array}{lllllllrrrrrrr}
Spin &&&X&X'\\
&&&&\\
2&&&1&1\\
3/2&&&1&2\\
1&&&n_1+1&n_1+1\\
1/2&&&2n_2+2n_1+1&2n_2+2n_1\\
0&&&4n_2+2n_1&4n_2+2n_1 \\
&&&&\\
&&&f=8(n_1+n_2+1)&f'=8(n_1+n_2+1)\\
&&&\rho=2(1+n_1-n_2)+8&\rho'=2(1+n_1-n_2)\
\end{array}$
\caption{ ${\cal N}=1$ and ${\cal N}'=2$}
\label{1}
\end{table}
The relevant coset is
\be
\frac{SU(1,n_1)}{U(n_1)} \times \frac{SU(2,n_2)}{SU(2) \times SU(n_2) \times U(1)}
\ee
and describes $\mathcal{N}=1$ supergravity plus $n_1+1$ vector mutiplets and $n_1+2n_2$ chiral paired with $\mathcal{N}=2$ supergravity plus $n_1$ vector multiplets and $n_2$ hypermutiplets. The $\mathcal{N}=1$ partner is bosonic self mirror when $n_2=n_1+5$ and the $\mathcal{N}=2$ partner is bosonic is self-mirror when $n_2=n_1+1$.
\end {itemize}
\section{Acknowledgments}
Conversations and correspondence with Leron Borsten, Duminda Dahanayke, Alessio Marrani, William Rubens, Ashoke Sen, Samson Shatashvili and Edward Witten are much appreciated. MJD is supported in part by the STFC under rolling Grant No. ST/G000743/1. S. F. is supported by the ERC Advanced Grant no 226455,``Supersymmetry, Quantum Gravity and Gauge Fields" and in part by DOE Grant DE-FG03-91ER40662. MJD is grateful for hospitality at the CERN theory division, where he was supported by the above ERC Advanced Grant.
\FloatBarrier
|
{
"timestamp": "2010-09-23T02:02:47",
"yymm": "1009",
"arxiv_id": "1009.4439",
"language": "en",
"url": "https://arxiv.org/abs/1009.4439"
}
|
\section{Introduction} \label{intro}
If the taste of a cake depends primarily on its ingredients, then the study of dwarf galaxies deserves a very careful attention. These objects are indeed believed to be the smallest baryonic counterparts to the dark matter building blocks in our Universe. It is established that dwarf galaxies are by far the most numerous galactic population, and they come in diverse flavours. Early-type dwarfs are predominantly old objects, with a regular, elliptical shape and an extremely low, if not absent, neutral (HI) gas content. The subclasses of this category are dwarf spheroidals, with a lower surface brightness, and dwarf ellipticals, which have higher central concentrations. On the other hand, late-type dwarfs are actively star-forming objects with substantial amounts of neutral gas and irregular shapes (see, e.g., \cite{grebel01}).
The most detailed information we have about dwarf galaxies comes directly from our closest neighbourhood. The Local Group (LG) contains two giant spirals and more than 50 dwarf members, for which decades of excellent data have revealed plenty of properties (e.g., \cite{mateo98,tolstoy09}). Photometry and spectroscopy have been performed on the closest companions of our Milky Way, and perhaps the most intriguing finding from over the last decade has been the extreme diversity in their star formation histories (SFHs). State-of-the-art observations and advanced analysis tools have allowed us to discover a wealth of information, for example that all of the LG dwarfs share a common, ancient population ($\gtrsim10$ Gyr, \cite{grebel04}). The extremes in this sample are, on the one hand, objects that only contain such old populations and, on the other hand, dwarfs that formed most of their stars at more recent times or those that have been experiencing a ``gasping'' star formation over their lives (\cite{marconi95}). LG dwarf galaxies all have a rather low metallicity content ([Fe/H] $\lesssim-1.0$) that is related to their luminosity (e.g., \cite{grebel03}). It is also interesting to analyze the shapes of their metallicity distribution functions (MDFs, e.g., \cite{koch06}), which are fairly well reproduced by theoretical models (e.g., \cite{lanfranchi04,marcolini08}). It has now been shown that many dwarfs, both of elliptical and irregular shape, contain population gradients, with the more metal-rich/younger stars being more centrally concentrated with respect to the more metal-poor/older ones. In some cases the subpopulations are also kinematically distinct (e.g., \cite{harbeck01,tolstoy04,battaglia06}).
In the last few years, many studies have started to look beyond the borders of the LG and have analyzed the properties of dwarf galaxies in nearby groups (e.g., \cite{kara02,trent02,kara04,sharina08,weisz08,bouchard09,dalcanton09}). We are particularly interested in the resolved stellar populations of these objects, since stars provide the record of a galaxy's past evolution and thus constitute a powerful tool to constrain its physical properties. Objects with distances within $\sim10$ Mpc from us are resolvable into individual stars, although the depth of the photometry decreases very rapidly and thus precludes us from reaching the exquisite quality of data we have for LG members. Moreover, as mentioned before, dwarf galaxies are numerous and heterogeneous systems, and in environments different to that of the LG, we can learn more about which factors shape their evolution. In this contribution, we present the first results stemming from the study of the resolved stellar populations of the Centaurus A (CenA) group dwarf members.
The CenA/M83 group is located at a mean distance of $\sim3.7$ Mpc and is composed of two subgroups, each one of which contains a giant galaxy (the peculiar elliptical CenA and the spiral M83) plus a total of about 60 dwarf companions. This group is believed to be a denser and maybe more dynamically evolved system than the LG (see, e.g., \cite{jerjen00a,rejkuba06}), thus making it a very appealing target for dwarf galaxy evolutionary studies and possible environmental effects.
\begin{table*}
\centering
\caption{Fundamental properties for our sample of early-type galaxies.}
\label{etd}
\begin{tabular}{lccccccccc}
\hline
\hline
Galaxy&RA&DEC&$T$&$D$&$A_{I}$&$M_{B}$&$\Theta$&$<$[Fe/H]$>_{med}$&Last SF\\
&(J2000)&(J2000)&&(Mpc)&&&&(dex)&(Gyr)\\
\hline
{KK189, CenA-dE1}&$13\,12\,45.0$&$-41\,49\,55$&$-3$&$4.42\pm0.33$&$0.22$&$-10.52$&$2.0$&$-1.52\pm0.20$&$\sim9$\\
{ESO269-66, KK190}&$13\,13\,09.2$&$-44\,53\,24$&$-5$&$3.82\pm0.26$&$0.18$&$-13.85$&$1.7$&$-1.21\pm0.33$&$2-3$\\
{KK197, SGC1319.1-4216}&$13\,22\,01.8$&$-42\,32\,08$&$-3$&$3.87\pm0.27$&$0.30$&$-12.76$&$3.0$&$-1.08\pm0.41$&$2-5$\\
{KKs55}&$13\,22\,12.4$&$-42\,43\,51$&$-3$&$3.94\pm0.27$&$0.28$&$-9.91$&$3.1$&$-1.56\pm0.10$&$-$\\
{KKs57}&$13\,41\,38.1$&$-42\,34\,55$&$-3$&$3.93\pm0.28$&$0.18$&$-10.07$&$1.8$&$-1.45\pm0.28$&$-$\\
{CenN}&$13\,48\,09.2$&$-47\,33\,54$&$-3$&$3.77\pm0.26$&$0.27$&$-10.89$&$0.9$&$-1.49\pm0.15$&$-$\\
\hline
\end{tabular}
\begin{list}{}{}
\item[Columns:] (1) name of the galaxy; (2-3) equatorial coordinates (units of right ascension are hours, minutes, and seconds, and units of declination are degrees, arcminutes, and arcseconds); (4) morphological type; (5) distance (derived with the TRGB method); (6) foreground extinction in $I$-band; (7) absolute $B$ magnitude; (8) tidal index (i.e., degree of isolation); (9) median metallicity and metallicity spread; (10) most recent epoch of significant star formation. The references for the reported values are \cite{kara07,crnojevic10,crnojevic10b}.
\end{list}
\end{table*}
\section{Data} \label{sec:0}
We use archival Hubble Space Telescope (HST) data taken with the Advanced Camera for Surveys (ACS) instrument, which were originally obtained to determine the distance of the targets with the tip of the red giant branch (TRGB) method (see \cite{kara07}). We use data for six early-type dwarfs (companions of CenA, of which four are dwarf spheroidals and two are dwarf ellipticals) and ten late-type dwarfs (five companions of CenA and five of M83). The general properties of our targets are listed in Tab. \ref{etd} and \ref{ltd}. For each of our target dwarfs, observations in the F606W (corresponding to the $V$-band) and F814W (corresponding to the $I$-band) filters are available. These reach approximately 2.5 magnitudes below the TRGB. The photometry on our optical data is run with the DOLPHOT package (\cite{dolphin02}), which we also use to perform extensive artificial star tests in order to quantify the photometric errors and incompleteness for the observations.
For three of our predominantly old early-type dwarf targets, near-infrared (NIR) data have been taken at the Very Large Telescope (VLT) with the Infrared Spectrometer And Array Camera (ISAAC). In combination with the excellent optical data, the images at these wavelengths (specifically, $J$- and $K$-band) help us to distinguish the populations belonging to our targets from the Galactic foreground (the CenA group lies at a galactic latitude of $b\sim20^{\circ}$), and thus to analyze in more detail the stars more luminous than the TRGB, indicative of an intermediate-age population (IAP, with ages in the range $\sim1-9$ Gyr).
\section{Early-type dwarfs} \label{sec:1}
\subsection{Metallicity distribution functions}
\begin{figure}
\centering
\resizebox{1\columnwidth}{!}
{\includegraphics{E269_cmdopt_proc.eps}}
\caption{\footnotesize{Optical (dereddened) CMD for one of the target early-type dwarfs (ESO269-66). The dominant feature of the CMD is a prominent RGB, with a small presence of luminous AGB stars above the TRGB. Representative photometric errorbars are plotted on the left side of the CMD. The green lines are stellar isochrones with a fixed age of 10 Gyr, and metallicity values of [Fe/H] $=-2.5$, $-1.2$ and $-0.4$.}}
\label{cmd_opt}
\end{figure}
\begin{figure}
\centering
\resizebox{1\columnwidth}{!}
{\includegraphics{E269_mdf_proc.eps}}
\caption{\footnotesize{MDF for one of the target early-type dwarfs (ESO269-66), derived via interpolation of isochrones at a fixed age (10 Gyr) and with varying metallicity. Overlaid (black line) is the MDF convolved with the observational errors. Also plotted in the upper left corner is the median error on the individual values of [Fe/H].}}
\label{mdf_opt}
\end{figure}
As can be seen from the example in Fig. \ref{cmd_opt}, the color-magnitude diagrams (CMDs) of our early-type dwarf targets show a prominent red giant branch (RGB), composed of old populations. Although this evolutionary stage suffers from the well known age-metallicity degeneracy, if we only consider a simple stellar population, a spread in metallicity has the effect of producing a broader RGB than a spread in age would do. Given the very little number of luminous AGB stars (indicative of an IAP), we can make the simplified assumption that the stars in our target galaxies are old (we choose a value of 10 Gyr), and assume that the width of their RGBs is entirely due to a range of metallicities (see, e.g., \cite{rejkuba05,harris07}). This is an approximation, but with our data we cannot resolve a possible age spread of a few Gyr at these old ages, which is likely to be present due to prolonged star formation. We use the Dartmouth stellar evolutionary models (\cite{dotter08}), which fit the CMDs of old globular clusters very well (e.g., \cite{glatt08}). We adopt isochrones with varying metallicities, from [Fe/H] $=-2.5$ to $-0.3$ in steps of 0.2 dex, and then interpolate among them to assign a metallicity value to each RGB star. We only do this for stars with magnitudes up to $\sim1.5$ mag fainter than the TRGB, because the photometric errors are smaller and the isochrones are more separated in this region. Moreover, we choose isochrones that have no $\alpha$-element enhancement, since we cannot constrain this. In this way we estimate the mean metallicity for our targets, all of which have a metal-poor stellar content, and draw their MDFs (an example is shown in Fig. \ref{mdf_opt}). We also compute the internal spreads in metallicity for the target dwarfs, after subtraction of the observational errors. Our results are listed in Tab. \ref{etd}.
While a small change ($\sim2$ Gyr) in the chosen age would lead to very small differences in our resulting metallicities, the presence of intermediate-age asymptotic giant branch (AGB) stars could influence the derived MDFs. No population younger than $\sim1$ Gyr is visible in the early-type dwarf CMDs, as can be seen from the absence of blue stars. Although a few luminous AGB stars are definitely present above the TRGB, the relative foreground contamination in this region of the CMD is significant. We give an estimate for the fraction of IAPs in our target objects, and we conclude that the resulting low fractions ($\lesssim20\%$) do not significantly change our results. This and other possible sources of uncertainties are discussed in detail in \cite{crnojevic10}.
We also find weak metallicity gradient as a function of galactic radius in some of our objects. Moreover, we arbitrarily divide the stellar samples into metal-poor and metal-rich (with respect to the median metallicity values), and check whether these subpopulations are similarly distributed within the galaxies. For the two most massive of the objects considered, the metal-rich stars are significantly more centrally concentrated, and a Kolmogorov-Smirnov test confirms that the subsamples are statistically distinct. More than two subpopulations might be present in these galaxies, but this is not resolvable in our observations.
\subsection{Intermediate-age populations} \label{sec:1:iaps}
\begin{figure}
\centering
\resizebox{1\columnwidth}{!}
{\includegraphics{E269_cmdnir_proc.eps}}
\caption{\footnotesize{NIR (dereddened) CMD for one of the target early-type dwarfs (ESO269-66). As opposed to the optical data, here the Galactic foreground contamination is confined to a vertical sequence (see Section \ref{sec:1:iaps} for details). Representative photometric errorbars are plotted along the CMDs. The green lines are isochrones with a fixed age of 10 Gyr and metallicity values of [Fe/H] $=-2.5$, $-1.2$ and $-0.4$.}}
\label{cmd_nir}
\end{figure}
We would now like to extract more information about the presence of IAPs in our targets and to estimate the latest epoch when star formation took place there. As mentioned before, it is difficult to constrain the amount of luminous AGB stars in our target galaxies given the amount of Galactic foreground. However, these stars are more luminous at NIR wavelenghts and, most importantly, in a NIR CMD the foreground is clearly confined to a vertical sequence with colors $0.3<J_0-K_0<1.0$ (compare, e.g., Fig. \ref{cmd_nir} to Fig. \ref{cmd_opt}). We thus have a powerful tool, when combining the deep and extremely well resolved optical HST images with the NIR data, to firmly identify candidate luminous AGB stars. We do this by cross-correlating the two stellar catalogs, and by selecting objects found above the TRGB in both datasets (for details, see \cite{crnojevic10b}). We can apply this technique to three of our target early-type dwarfs since we have NIR imaging data for them.
From our analysis, we are able to find the most recent episode of star formation by using the empirical relation between the absolute bolometric magnitude of the AGB candidates and their age (see \cite{rejkuba06}). Our results are reported in Tab. \ref{etd}. We compare the derived ratio of luminous AGB stars to RGB stars to the Maraston stellar evolutionary models (\cite{maraston05}), and estimate the IAP fractions for our targets. The resulting fractions are very low, between $5\%$ and $15\%$, thus confirming our previous estimates from the optical data. We note that these fractions are likely to be lower limits to the true ones, given our observational uncertainties and also the uncertainties in the stellar evolutionary models (see, e.g., \cite{melbourne10}).
Finally, we consider the resulting properties found for the whole sample of our target galaxies as a function of deprojected distance from CenA, and as a function of the tidal index (i.e., degree of isolation, positive for galaxies sitting in a dense environment and negative for isolated objects, see e.g. \cite{kara07}). We do not find any obvious correlation between metallicity or IAP fraction and the mentioned parameters, but this result could be biased by the small size of our sample.
\section{Late-type dwarfs} \label{sec:2}
\subsection{Recent star formation histories}
\begin{table*}
\centering
\caption{Fundamental properties for our sample of late-type galaxies.}
\label{ltd}
\begin{tabular}{lccccccccc}
\hline
\hline
Galaxy&RA&DEC&$T$&$D$&$A_{I}$&$M_{B}$&$\Theta$&$<$[Fe/H]$>$&$<$SFR$>$\\
&(J2000)&(J2000)&&(Mpc)&&&&(dex)&($10^{-2}$M$_{\odot}$yr$^{-1})$\\
\hline
{ESO381-18}&$12\,44\,42.7$&$-35\,58\,00$&$10$&$5.32\pm0.51$&$0.12$&$-12.91$&$-0.6$&$-1.40\pm0.13$&$0.35\pm0.26$\\
{ESO381-20}&$12\,46\,00.4$&$-33\,50\,17$&$10$&$5.44\pm0.37$&$0.13$&$-14.44$&$-0.3$&$-1.45\pm0.17$&$0.70\pm0.48$\\
{ESO443-09, KK170}&$12\,54\,53.6$&$-28\,20\,27$&$10$&$5.97\pm0.46$&$0.13$&$-11.82$&$-0.9$&$-1.46\pm0.14$&$0.08\pm0.07$\\
{IC4247, ESO444-34}&$13\,26\,44.4$&$-30\,21\,45$&$10$&$4.97\pm0.49$&$0.12$&$-14.07$&$1.5$&$-1.43\pm0.11$&$1.01\pm0.66$\\
{ESO444-78, UGCA365}&$13\,36\,30.8$&$-29\,14\,11$&$10$&$5.25\pm0.43$&$0.10$&$-13.11$&$2.1$&$-1.37\pm0.21$&$0.63\pm0.36$\\
\hline
{KK182, Cen6}&$13\,05\,02.9$&$-40\,04\,58$&$10$&$5.78\pm0.42$&$0.20$&$-12.48$&$-0.5$&$-1.46\pm0.21$&$0.10\pm0.14$\\
{ESO269-58}&$13\,10\,32.9$&$-46\,59\,27$&$10$&$3.80\pm0.29$&$0.21$&$-14.60$&$1.9$&$-0.98\pm0.20$&$6.79\pm3.93$\\
{KK196, AM1318-444}&$13\,21\,47.1$&$-45\,03\,48$&$10$&$3.98\pm0.29$&$0.16$&$-11.90$&$2.2$&$-1.43\pm0.25$&$0.20\pm0.11$\\
{HIPASS J1348-37}&$13\,48\,33.9$&$-37\,58\,03$&$10$&$5.75\pm0.66$&$0.15$&$-11.90$&$-1.2$&$-1.50\pm0.07$&$0.14\pm0.05$\\
{ESO384-16}&$13\,57\,01.6$&$-35\,20\,02$&$10$&$4.53\pm0.31$&$0.14$&$-13.17$&$-0.3$&$-0.97\pm0.15$&$0.60\pm0.23$\\
\hline
\end{tabular}
\begin{list}{}{}
\item[Columns:] (1) name of the galaxy (upper sample: M83 companions, lower sample: CenA companions); (2-3) equatorial coordinates (units of right ascension are hours, minutes, and seconds, and units of declination are degrees, arcminutes, and arcseconds); (4) morphological type; (5) distance (derived with the TRGB method); (6) foreground extinction in $I$-band; (7) absolute $B$ magnitude; (8) tidal index (i.e., degree of isolation); (9) lifetime average SFR; (10) mean metallicity. The references for the reported values are \cite{kara07}, \cite{bouchard09}, \cite{crnojevic10c}, \cite{crnojevic10d}.
\end{list}
\end{table*}
\begin{figure}
\centering
\resizebox{1\columnwidth}{!}
{\includegraphics{cmd_proc_381-20.eps}}
\caption{\footnotesize{Optical CMD for one of the target late-type dwarfs (ESO381-20). We overlay Padova isochrones with a fixed metallicity of [Fe/H] $=-1.4$ and varying ages. The ages are respectively: 4, 12, 35 and 80 Myr (magenta lines, from the blue to the red part of the CMD, indicating massive main sequence and helium-burning stars); 200, 550 Myr, and 1 Gyr (blue lines, helium-burning and AGB stars); 7 and 14 Gyr (red lines, RGB stars). We also report typical $1\sigma$ photometric uncertainties on the left side of the CMD.}}
\label{cmd_ltd}
\end{figure}
\begin{figure}
\centering
\resizebox{1\columnwidth}{!}
{\includegraphics{sfh_proc_381-20.eps}}
\caption{\footnotesize{\emph{Upper panel}. SFH derived for one of the late-type galaxies studied (ESO381-20). The SFR is plotted as a function of time, the oldest age being on the left side and the most recent time bin on the right edge of the (logarithmic) horizontal axis. The size of the time bins is variable, due to the different amount of information obtainable from each CMD for different stellar evolutionary stages. The black dashed line indicates the mean SFR over the whole galaxy's lifetime. \emph{Lower panel}. Metallicity as a function of time (with the same axis convention as above). The black dashed line represents the mean metallicity over the galaxy's lifetime (note that the metallicity evolution is poorly constrained).}}
\label{sfh}
\end{figure}
Unlike the early-type dwarfs considered in the previous Section, the late-type dwarfs of our sample show additional structure in their CMDs (see Fig. \ref{cmd_ltd} as an example), thus giving us the possibility to extract more information about their past histories. We divide our targets in two subsamples, namely the M83 and the CenA companions. As can be seen from Tab. \ref{ltd}, some of the dwarfs considered have a positive tidal index, while others have a negative tidal index. We are thus also able to look for environmental effects on their evolution.
We analyze our late-type dwarfs with the synthetic CMD modeling technique, which has proved to be an extremenly powerful tool in deriving the SFHs of dwarf galaxies in the LG and nearby groups (see, e.g., \cite{skillman03,dalcanton09}). Starting from Padova stellar isochrones (\cite{marigo08}, which also reproduce helium-burning stages, as opposed to the Dartmouth ones), we construct synthetic CMDs with a range of physical parameters and then compare them to the observed ones using a maximum likelihood algorithm in order to find the best-fitting solution (for more details, see \cite{cole07,crnojevic10c}). In this way we quantify the amount of stars produced by the galaxy in each different evolutionary stage, which corresponds to a specific age. Given that the only information for ages older than $\sim4-8$ Gyr is coming from the age-metallicity degenerate RGB stage, we are not able to resolve bursts in the star formation for these ages. However, the young, massive main sequence and helium-burning stars, and the intermediate-age luminous AGB stars allow us to reconstruct the star formation rate (SFR) in the past $1-2$ Gyr more accurately. With this technique, it is also possible to derive a mean metallicity estimate for the studied objects, but due to the shallow photometry and the degeneracies present in the CMD, its evolution with time is extremely uncertain. The results of our SFH recovery process are listed in Tab. \ref{ltd} for both subsamples of the galaxies, and show the typical behavior expected for late-type dwarfs.
We show an example of a resulting SFH (i.e., star formation rate, SFR, as a function of time) in Fig. \ref{sfh}. The derived SFHs for our sample show considerable diversity in their main features, with some of the target dwarfs having already formed most of their stellar content more than 8 Gyr ago, some others having been significantly active only at recent times, some being possibly in a transition phase from late-type to early-type morphology, and some having just experienced a strong starburst (for more details, see \cite{crnojevic10c,crnojevic10d}). Overall, the general trend seen in all the SFHs is the presence of long quiescent periods, interrupted by more or less extended (few tens to several hundreds Myr) episodes of enhanced star formation.
\subsection{Stellar spatial distributions}
As a final step, we divide the stellar content of each galaxy into subsamples with different ages and produce density maps for each of them. It is interesting to see a confirmation of previous studies (e.g., \cite{dohm97,glatt08,weisz08}), that the youngest stars tend to be concentrated in clumps close to the central regions of the galaxies, while the oldest stars are uniformely distributed all over their bodies. This stems from the fact that the young stars still have the imprint position of their birthplaces, while the old ones have had the time to migrate and redistribute within the galaxy.
We take advantage of the intrinsic properties of the blue helium-burning (BHeB, forming a vertical sequence with $V-I\sim0$, see Fig. \ref{cmd_ltd}) stars to estimate the star formation timescales. In this evolutionary stage, stars with different ages are very well separated in the CMD, as opposed to what is seen, for example, in the RGB phase. We can thus trace the spatially resolved SFH of a galaxy by considering stellar maps of BHeB stars with different ages. We find that star forming complexes have diameters of $\sim100$ pc and timescales of $\gtrsim100$ Myr, and we confirm that global bursts in a galaxy are long-lived (several hundreds of Myr) events, within which smaller, localized star forming regions form and dissolve (see also \cite{mcquinn09}).
Our results point towards a stochastic star formation mode for late-type dwarfs (see also \cite{weisz08}). We also look for possible relations between the physical properties of our dwarfs and their environment. There is no clear correlation between the lifetime average SFR with tidal index, or with deprojected distance from the closest giant galaxy, and the same conclusion is also valid for the metallicity. This is a reasonable result, if we consider that average properties depend on the whole galaxy's history, and we only have information about the current position of the objects within the group because their orbits are unknown. However, if we compute the ratio of the SFR in the last $\sim500$ Myr to the lifetime average value, we do find that it correlates with environment. Namely, dwarfs that are closest to the dominant group galaxy, and found in a denser environment, have a lower such ratio with respect to the more isolated objects (see also \cite{bouchard09}). Finally, we find that dwarfs located in denser regions will consume their HI gas content much faster than isolated ones, provided they continue to form stars at their lifetime average SFR.
\section{Discussion and conclusions} \label{concl}
We have studied a sample of dwarf galaxies in the nearby CenA/M83 group, an environment similar to the LG but denser (in terms of number of galaxies) and possibly more evolved. The results reported in this contribution can be now compared to what we know about our own LG.
In terms of metallicity content of their old populations, the early-type dwarfs studied are metal-poor and always show metallicity spreads, just as the LG companions. They moreover follow the luminosity-metallicity relation originally seen for the LG (see, e.g., \cite{grebel03}), and then also found to extend to other nearby environments (e.g., \cite{sharina08}). The shapes of the derived MDFs are overall very similar to the ones spectroscopically derived for LG dwarf members (e.g., \cite{koch06}), which implies the combination of supernovae-driven enrichment and galactic outflows, although we note that a more detailed comparison is not viable because of the intrinsic uncertainties of our method. Some of our target dwarfs also show the presence of distinct stellar populations, similarly to some LG dwarfs (e.g., Fornax, Sculptor, Sextans). The intriguing difference between the two groups emerges only when we look at the IAPs more carefully. Namely, an analysis of the luminous AGB stars tells us that the IAP fractions in CenA companions are smaller than what is found for Milky Way dwarf spheroidal companions and for M31 dwarf elliptical companions (the relative differences are not affected by model uncertainties). Our results are more similar to what is seen for dwarf spheroidal companions of M31 (see \cite{crnojevic10b}). This is one of the first studies where IAPs for dwarfs in nearby groups are investigated from their resolved stellar populations. Although our sample is still too small to draw firm conclusions, our results suggest that the CenA environment could somehow have played a role in the suppression of its companions' star formation. This is a very interesting result which certainly deserves more observational evidence.
The late-type dwarfs considered in our study all show complex and varied SFHs, with lifetime average SFRs of between $\sim10^{-3}$ and $\sim6 \times 10^{-2}$M$_\odot$yr$^{-1}$. If we compare these values to LG dwarf galaxies in the same luminosity range, we find that the CenA/M83 dwarfs have slightly higher values, comparable to what is found for the M81 group (a highly interacting nearby group) late-type dwarfs (see \cite{weisz08}). The global enhancements in SFR that we are able to resolve from the recent SFHs of our targets last for several hundreds of Myr, producing stars at a rate of $\sim2-3$ times the average SFR. We show that within these bursts, localized star forming regions have sizes of $\sim100$ pc and timescales of $\sim100$ Myr. We conclude that star formation in these objects is a stochastic process. Finally, also among late-type dwarfs we find a hint for environmental effects, in the sense that dwarfs in denser environments and closer to the dominant galaxy have had lower SFRs in the last $\sim500$ Myr, and are consuming their gas reservoirs faster than the isolated ones.
The CenA/M83 group has proved to be a very promising target, in which the resolved stellar populations of the dwarf companions can be analyzed to better understand the details of galactic evolution. The study presented in this contribution should thus be an appetizer, and stimulate the curiosity for the work that has started to be done for nearby groups of galaxies.
\vspace{-0.3cm}
\begin{acknowledgement}
DC wishes to thank the organizers of the conference for making this meeting stimulating and enjoyable. DC acknowledges travel support from the IMPRS and ARI/ZAH. DC is grateful to S. Pasetto and S. Jin for a careful reading of the manuscript and for their support.
\end{acknowledgement}
\vspace{-0.6cm}
|
{
"timestamp": "2010-09-23T02:00:16",
"yymm": "1009",
"arxiv_id": "1009.4198",
"language": "en",
"url": "https://arxiv.org/abs/1009.4198"
}
|
\section{Introduction}
In one-dimensional quantum systems, a completely different behavior
for the integer spin
chains from the half-integer spin chains was predicted by Haldane
\cite{Haldane1,Haldane2}.
The antiferromagnetic isotropic
spin-1 model introduced by Affleck, Kennedy, Lieb and Tasaki
(AKLT model) \cite{AKLT},
whose ground state can be exactly calculated, has been a useful toy model
to validate Haldane's prediction of the massive behavior
for integer spin chains.
Moreover, it lead to a deeper understanding for integer spin chains such as the
discovery of the special type of long-range order \cite{DR,Tasaki}.
The AKLT model has been generalized to
higher-spin models,
anisotropic models, etc
\cite{AAH,KSZ1,KSZ2,KSZ3,TS1,BY,GR,SR,TZX,TZXLN,KM,AHQZ}.
The Hamiltonians are essentially linear combinations
of projection operators with nonnegative coefficients,
and their ground states are called valence-bond-solid (VBS) state.
Recently, the VBS state was investigated
in perspective of
its relation to quantum information and experimental implementation
by means of optical lattices,
see Refs.~\cite{VMC,GMC} for example.
There are largely three types of representations for the ground state which
are equivalent to each other:
the Schwinger boson representation, the spin coherent
representation and the matrix product representation.
For isotropic higher-spin models,
the spin-spin correlation functions \cite{FH}
and the entanglement entropy \cite{XKHK,KX}
have been calculated by utilizing the spin coherent representation
and the properties of Legendre polynomials.
For the $q$-deformed spin-1 model,
spin-spin correlation functions were evaluated
\cite{KSZ1,KSZ2,KSZ3}
from the matrix product representation.
In this paper, we consider the ground state of
a $q$-deformed higher-integer-spin model
which was constructed recently in Ref.~\cite{M} ($q$-VBS state).
From its matrix product representation,
we analyze one and two point functions of the $q$-VBS
ground state.
We notice that a matrix, which is constructed from the matrix product
representation, plays a fundamental role in computing correlation functions,
especially spin-spin two point correlation functions.
Investigating the structure of the matrix in detail,
we obtain its eigenvalues and eigenvectors.
Utilizing the results, we determine the correlation
amplitudes and correlation lengths of the longitudinal and transverse
spin-spin correlation functions.
This paper is organized as follows.
In the next section, we briefly review
the quantum group $U_q(su(2))$, and investigate the
finite dimensional highest weight representation in terms of
Schwinger bosons.
In Section \ref{Sec:qVBS},
we precisely define the higher-spin generalization of the
$q$-deformed AKLT model on an $L$-site chain, and
rigorously derive its $q$-VBS ground state
in a matrix product form.
The squared norm of the state will be written
in terms of the trace of the $L$-th power of a matrix $G$,
which plays an important role in this paper.
In section \ref{Sec:spec-of-G},
we obtain the eigenvalues and eigenvectors of $G$.
Utilizing them,
we compute one and two point functions
in Section \ref{Sec:correlation}.
Especially, we determine the correlation amplitudes
and correlation lengths of the longitudinal and transverse
spin-spin correlation functions.
Section \ref{Sec:Conclusion}
is devoted to the conclusion of this paper.
\section{\label{Sec:} The quantum group $U_q(su(2))$ }
We introduce several notations,
fixing a real number $q$ throughout this paper.
Let us define the $q$-integer, $q$-factorial
and $q$-binomial coefficient for $N\in{\mathbb Z}_{\ge 0}$ as
\begin{align}
\begin{split}
[N]=\frac{q^N-q^{-N}}{q-q^{-1}}, \quad
[N]!=
\begin{cases}
\displaystyle \prod_{I=1}^{N} [I] & N\in {\mathbb N},\\
1 & N=0,
\end{cases}
\\
\left[ \begin{array}{c} N \\ K \end{array} \right]
=\begin{cases}
\displaystyle \frac{ [N]! }{ [K]! [N-K]! } & K=0,\dots,N,\\
0 & \rm otherwise,
\end{cases}
\end{split}
\end{align}
respectively.
The quantum group $U_q (su(2))$ \cite{Drinfeld,Jimbo}
is defined by generators $X^+,X^-$ and $H$
with relations
\begin{align}
\left[X^+,X^-\right] = \frac{q^H-q^{-H}}{q-q^{-1}},\quad
\left[H,X^{\pm}\right] =\pm 2 X^{\pm} .
\end{align}
The comultiplication is given by
\begin{align}
\Delta \left( X^\pm \right)
= X^\pm \otimes q^{H/2} + q^{-H/2} \otimes X^\pm, \quad
\Delta (H) = H\otimes \mathrm{Id} + \mathrm{Id} \otimes H .
\end{align}
$U_q (su(2))$ has the Schwinger boson representation,
where the generators are realized as
\begin{align}
X^+=a^\dagger b,\quad X^-=b^\dagger a ,\quad H=N_a-N_b,
\end{align}
with $q$-bosons $a$ and $b$ satisfying
\begin{align}
&a a^\dagger -q a^\dagger a = q^{-N_a}, \quad
b b^\dagger -q b^\dagger b = q^{-N_b}, \\
&[N_a,a] = -a ,\quad [N_a,a^\dagger] = a^\dagger, \quad
[N_b,b] = -b ,\quad [N_b,b^\dagger] = b^\dagger.
\end{align}
We denote the space where $(2j+1)$-dimensional
highest weight representation of $U_q(su(2))$ is realized by $V_j$.
The basis of $V_j$ is given by
\begin{align}
|j;m {\rangle}=\frac{(a^{\dagger})^{j+m}(b^{\dagger})^{j-m}}
{ \sqrt{ [j+m]! [j-m]! }}
|\mathrm{vac} {\rangle}, \ \ (m=-j, \dots, j). \label{basis}
\end{align}
The Weyl representation which we describe below,
is an equivalent representation to
the Schwinger boson representation,
and is efficient for practical calculation.
Let us denote the $q$-bosons $a$ and $b$ acting on the
$\alpha$-th site as $a_\alpha$ and $b_\alpha$.
The Weyl representation is to represent
$a_\alpha^\dagger$, $b_\alpha^\dagger$,
$a_\alpha$ and $b_\alpha$ on the space of polynomials
$\mathbb{C}[x_\alpha,y_\alpha]$ as
\begin{align}
a_\alpha^\dagger = x_\alpha,\quad
b_\alpha^\dagger = y_\alpha,\quad
a_\alpha = \frac{1}{x_\alpha}\frac{D_q^{x_\alpha}-D_{q^{-1}}^{x_\alpha}}{q-q^{-1}},
\ \
b_\alpha=\frac{1}{y_\alpha}\frac{D_q^{y_\alpha}-D_{q^{-1}}^{y_\alpha}}{q-q^{-1}},
\end{align}
where
\begin{align}
D_{p}^{x_\alpha}f(x_\alpha,y_\alpha)=f(px_\alpha,y_\alpha),
\ \ D_{p}^{y_\alpha}f(x_\alpha,y_\alpha)=f(x_\alpha,py_\alpha).
\end{align}
The generators of $U_q(su(2))$ are now represented as
\begin{align}
X^+_\alpha = \frac{x_\alpha}{y_\alpha}
\frac{ D_q^{y_\alpha} - D^{y_\alpha}_{q^{-1}} }
{ q-q^{-1} },\
X^-_\alpha = \frac{y_\alpha}{x_\alpha}
\frac{ D_q^{x_\alpha} - D^{x_\alpha}_{q^{-1}} }
{ q-q^{-1} },\
q^{H_\alpha} = D^{x_\alpha}_q D^{y_\alpha}_{q^{-1}}.
\end{align}
The tensor product of two irreducible representations has the
Clebsch-Gordan decomposition
\begin{align}
V_{S} \otimes V_{S} &= \bigoplus_{J=0}^{2S}V_J , \\
\label{CGdeco}
|S;m_1\rangle \otimes|S;m_2\rangle
&= \sum_{J=0}^{2S}
\q{\begin{array}{ccc}
S & S & J \\
m_1 & m_2 & m_1+m_2
\end{array}}
|J; m_1+m_2\rangle ,
\end{align}
where
\begin{align}
&\q{\begin{array}{ccc} S_1 & S_2 & J \\ m_1 & m_2 & m \end{array}}
= \delta_{m_1+m_2,m}(-1)^{S_1-m_1}
q^{m_1(m_1+m_2+1)+ \{ S_2(S_2+1)-S_1(S_1+1)-J(J+1) \} /2} \nn \\
&\times \sqrt{ \frac{[J+m]! [J-m]! [S_1-m_1]! [S_2-m_2]! [S_1+S_2-J]! [2J+1]
}{[S_1+m_1]! [S_2+m_2]! [S_1-S_2+J]! [S_2-S_1+J]! [S_1+S_2+J+1]! } }
\\
&\times \sum_{z=\mathrm{Max}(0,-S_1-m_1,J-S_2-m_1)}^
{\mathrm{Min}(J-m,S_1-m_1,S_2+J-m_1)}
\frac{ (- q^{m+J+1} )^z [S_1+m_1+z]! [S_2+J-m_1-z]!}{[z]! [J-m-z]! [S_1-m_1-z]! [S_2-J+m_1+z]!}, \nn
\end{align}
is the $q$-analog
of the Clebsch-Gordan coefficient \cite{KR}.
(The factor $q^{m_1 m_2/2}$ is missing in Ref.~\cite{KR}.)
This coefficient
is compatible with the inverse
of the decomposition \eqref{CGdeco}
\begin{align}
|J;m\rangle = \sum_{m_1+m_2=m}
\q{\begin{array}{ccc}
S & S & J \\
m_1 & m_2 & m_1+m_2
\end{array}}
|S;m_1\rangle \otimes|S;m_2\rangle .
\end{align}
For later purpose, we will also
investigate the Clebsch-Gordan decomposition
of $U_q(su(2))$ in terms of the Schwinger boson
or the Weyl representation.
Utilizing
\begin{align}
\Delta X^{\pm}_{\alpha\beta}
=& X^\pm_\alpha \otimes q^{H_\beta /2}
+ q^{-H_\alpha /2} \otimes X^\pm_\beta,
\end{align}
one can show that the highest weight vector
$v_J \in V_J \ ( \Delta X^+ v_J = 0)$
acting on the $\alpha$-th and $\beta$-th sites
is given by
\begin{align}
v_J = (x_\alpha x_\beta)^J
\prod_{\nu=1}^{2S-J}
(x_\alpha y_\beta -q^{2(\nu-S-1)}x_\beta y_\alpha ).
\end{align}
Moreover, we can show the following:
\begin{proposition}\label{X^-^nv_J}
\begin{align}
\begin{split}
\left( \Delta X^-_{\alpha\beta} \right)^n v_J
=& (x_\alpha x_\beta)^{J-n} q^{nS} [n]!
\sum_{\mu=0}^n q^{-2\mu S}
\q{\begin{array}{c} J \\ \mu \end{array} }
\q{\begin{array}{c} J \\ n-\mu \end{array} }
\left(x_\alpha y_\beta \right)^\mu
\left(x_\beta y_\alpha \right)^{n-\mu} \\
& \times
\prod_{\nu=1}^{2S-J}
\left( x_\alpha y_\beta -q^{2(\nu-S-1)}x_\beta y_\alpha \right) .
\end{split}
\label{vectorexpression}
\end{align}
\end{proposition}
A proof of this proposition
is given in Appendix \ref{Sec:proof}.
\begin{remark}
Let $n\ge 2J+1$.
Noting
\begin{align}
\q{\begin{array}{c} J \\ \mu \end{array}}
\q{\begin{array}{c} J \\ n-\mu \end{array}}
=0,
\end{align}
for $0 \le \mu\le n$,
one can see that
\begin{align}
\left( \Delta X^-_{\alpha\beta} \right)^n v_J = 0.
\end{align}
\end{remark}
\section{\label{Sec:qVBS}$q$-VBS state}
The model we treat in this paper
is an anisotropic integer spin-$S$ Hamiltonian
on an $L$-site chain with the periodic boundary condition
\begin{align}
&{\mathcal H}=\sum_{k\in {\mathbb Z}_L}
\sum_{J=S+1}^{2S}C_J(k,k+1) \left(\pi_J\right)_{k,k+1},
\label{hamiltonian}
\end{align}
where $C_J(k,k+1) > 0$, and $\left(\pi_J\right)_{k,k+1}$,
which acts on the $k$-th and $(k+1)$-th sites,
is the $U_q (su(2))$ projection operator
from $V_S \otimes V_S$ to $V_J$ as
\begin{align}
\begin{split}
\pi_J=&
\sum_{m_1,m_2, m_1', m_2'=-S}^S
\q{\begin{array}{ccc} S & S & J \\ m_1 & m_2 & m_1+ m_2 \end{array}}
\q{\begin{array}{ccc} S & S & J \\ m_1' & m_2' &
m_1'+ m_2' \end{array}}\\
&\quad\quad\times \delta_{m_1+m_2, m_1'+ m_2'}
|S; m_1' \rangle \langle S; m_1|
\otimes
|S; m_2' \rangle \langle S;m_2| .
\end{split}
\end{align}
The nonnegativity
\begin{align}\label{nonnegativity}
\langle \psi | \left(\pi_J\right)_{k,k+1} | \psi \rangle \ge 0
\quad \left( \text{for any vector } | \psi \rangle \right)
\end{align}
implies that all the eigenvalues of $\mathcal H$
are nonnegative.
(Of course, $\langle \psi |$ is
the Hermitian conjugate of $| \psi \rangle$.)
Moreover, we will see that the energy of
the ground state $|\Psi\rangle$ is zero:
\begin{align}\label{HPsi=0}
\mathcal H |\Psi\rangle =0.
\end{align}
Since we set $C_J(k,k+1)>0$,
we find that \eqref{HPsi=0} is equivalent to
\begin{align}\label{local-condition}
\left(\pi_J\right)_{k,k+1} | \Psi \rangle =0
\quad \left(\forall k\in {\mathbb Z}_L,\
\forall J\in \{ S+1,\dots,2S \} \right) ,
\end{align}
noting the nonnegativity \eqref{nonnegativity}.
From Proposition \ref{X^-^nv_J},
one observes that any vector in
$\bigoplus_{0\le J\le S}
V_J \subset V_S \otimes V_S $
of the $k$-th and $(k+1)$-th sites has the form
\begin{align}\label{local-form}
\sum_{0 \le A,B \le S} C_{A B}
x_k^{A} y_k^{S-A} x_{k+1}^{B} y_{k+1}^{S-B}
\prod_{m=1}^S
(q^m x_k y_{k+1} - q^{-m} y_k x_{k+1} ) ,
\end{align}
where $ C_{A B}$ does not depend on
$x_k,y_k,x_{k+1}$ or $y_{k+1} $.
Thus, the condition \eqref{local-condition}
imposes the restriction that $|\Psi\rangle$
has the form
\begin{align}
| \Psi {\rangle} = P\left(\{x_k\}_{k\in{\mathbb Z}_L},
\{y_k\}_{k\in{\mathbb Z}_L}\right)
\prod_{k\in{\mathbb Z}_L} \prod_{m=1}^S
(q^m x_k y_{k+1} - q^{-m} y_k x_{k+1} )
\end{align}
with some polynomials $P$
such that this form is consistent with \eqref{local-form}
for $\forall k\in{\mathbb Z}_L$.
The unique choice of $P$ with such consistency is a constant
(which can be set to be 1),
and we achieve the unique ground state
\begin{align}
| \Psi {\rangle} =
\prod_{k\in{\mathbb Z}_L} \prod_{m=1}^S
(q^m x_k y_{k+1} - q^{-m} y_k x_{k+1} ) .
\end{align}
In the Schwinger boson representation, we have
\begin{align}
| \Psi {\rangle}=\prod_{k\in{\mathbb Z}_L} \prod_{m=1}^S
(q^m a_k^{\dagger} b_{k+1}^{\dagger}-q^{-m} b_k^{\dagger} a_{k+1}^{\dagger})
|\mathrm{vac} {\rangle},
\label{schwingergroundstate}
\end{align}
which is a generalization of the $q=1$ case \cite{AAH}.
Note that each site have the correct spin value:
$N_k| \Psi \rangle =S| \Psi \rangle$ ($k \in {\mathbb Z}_L$)
where $N_k:=(N_{a_k}+N_{b_k})/2$.
Our ground state is
a $q$-deformation of the valence-bond-solid (VBS) state,
which we call $q$-VBS state, see figure \ref{q-VBS}.
\begin{figure}
\begin{center}
\includegraphics[height=25mm,clip]{AM_Pic.eps}
\caption{Conceptual figure of the $q$-VBS state.
Each line is a $q$-deformed valence bond,
and the circle $\bigcirc$ represents the $q$-symmetrization of
spin-1/2 particles $\bullet$ at each site.}
\label{q-VBS}
\end{center}
\end{figure}
The Schwinger boson representation
of the ground state \eqref{schwingergroundstate}
can be transformed into the following equivalent form
called the matrix product representation \cite{M}, which
generalizes the $q=1$ \cite{TS2} or $S=1$ \cite{KSZ1} case.
Noting \eqref{basis}, we have
\begin{align}\label{mpf}
| \Psi \rangle&=\mathrm{Tr}
[g_1 \star g_2 \star \cdots \star g_{L-1} \star g_L],
\end{align}
where
$g_k$ is an $(S+1)\times(S+1)$
vector-valued matrix
acting on the $k$-th site
whose element is given by
\begin{align}
\begin{split}
g_k(i,i')
&= (-1)^{S-i} q^{(i+ i'-S)(S+1)/2}
\sqrt{\left[\begin{array}{c} S \\ i \end{array} \right]
\left[ \begin{array}{c} S \\ i' \end{array} \right]
[S-i+ i']! [S+i- i']! } \ |S; i'-i\rangle_k \\
&=: h_{i i'} |S; i'-i \rangle_k , \quad\quad (0\le i, i'\le S) .
\end{split}
\end{align}
The symbol $\star$ for two
$(S+1)\times(S+1)$ vector-valued matrices
\begin{align}
x=
\left(\begin{array}{ccc}
|x_{00}\rangle & \cdots & |x_{0S}\rangle \\
\vdots & \ddots & \vdots \\
|x_{S0}\rangle & \cdots & |x_{SS}\rangle \\
\end{array}\right)
,\quad
y=
\left(\begin{array}{ccc}
|y_{00}\rangle & \cdots & |y_{0S}\rangle \\
\vdots & \ddots & \vdots \\
|y_{S0}\rangle & \cdots & |y_{SS}\rangle \\
\end{array}\right),
\end{align}
is defined by
\begin{align}
x\star y=
\left(\begin{array}{ccc}
\sum_{u=0}^{S} |x_{0u}\rangle\otimes |y_{u0}\rangle & \cdots &
\sum_{u=0}^{S} |x_{0u}\rangle\otimes |y_{uS}\rangle \\
\vdots & \ddots & \vdots \\
\sum_{u=0}^{S} |x_{Su}\rangle\otimes |y_{u0}\rangle & \cdots &
\sum_{u=0}^{S} |x_{Su}\rangle\otimes |y_{uS}\rangle \\
\end{array}\right),
\end{align}
which is associative.
For example, for $S=2$,
\begin{align}
g_k =
\left(\begin{array}{ccc}
h_{00} |2;0\rangle_k & h_{01} |2;1\rangle_k & h_{02} |2;2\rangle_k \\
h_{10} |2;-1\rangle_k & h_{11} |2;0\rangle_k & h_{12} |2;1\rangle_k \\
h_{20} |2;-2\rangle_k & h_{21} |2;-1\rangle_k & h_{22} |2;0\rangle_k
\end{array}\right),
\end{align}
and the product in the form \eqref{mpf}
is calculated as
\begin{align}
\begin{split}
& g_1 \star\cdots\star g_L \\
=& \left(\begin{array}{ccc}
\left( g_1 \star\cdots\star g_L \right) (0,0) &
\left( g_1 \star\cdots\star g_L\right) (0,1) &
\left( g_1 \star\cdots\star g_L\right) (0,2) \\
\left( g_1 \star\cdots\star g_L \right) (1,0) &
\left( g_1 \star\cdots\star g_L\right) (1,1) &
\left( g_1 \star\cdots\star g_L\right) (1,2) \\
\left( g_1 \star\cdots\star g_L \right) (2,0) &
\left( g_1 \star\cdots\star g_L\right) (2,1) &
\left( g_1 \star\cdots\star g_L\right) (2,2)
\end{array}\right),
\end{split}
\end{align}
with
\begin{align}
\begin{split}
& \left( g_1 \star\cdots\star g_L \right) (i,i') \\
=& \sum_{i_k=0,1,2}h_{ii_2} h_{i_2i_3} \cdots h_{i_{L-1}i_L} h_{i_L i'} \\
&\quad \ \times
|2;i_2-i\rangle_1 \otimes |2;i_3-i_2\rangle_2 \otimes\cdots\otimes
|2;i_L-i_{L-1}\rangle_{L-1} \otimes |2;i'-i_L\rangle_L .
\end{split}
\end{align}
Then the matrix product ground state \eqref{mpf} is
\begin{align}
\begin{split}
&\left( g_1 \star\cdots\star g_L \right) (0,0)+
\left( g_1 \star\cdots\star g_L \right) (1,1)+
\left( g_1 \star\cdots\star g_L \right) (2,2) \\
=
& \sum_{i_k=0,1,2}h_{i_1i_2} h_{i_2i_3} \cdots h_{i_{L-1}i_L} h_{i_Li_1} \\
&\ \times
|2;i_2-i_1\rangle_1 \otimes |2;i_3-i_2\rangle_2 \otimes\cdots\otimes
|2;i_L-i_{L-1}\rangle_{L-1} \otimes |2;i_1-i_L\rangle_L .
\end{split}
\end{align}
We define $g_k^\dagger$
by replacing each ket vector in the matrix $g_k$
by its corresponding bra vector:
\begin{align}
g^\dagger_k(i, i') = h_{i i'} \ _k \langle S; i'-i |.
\end{align}
For example, for $S=2$,
\begin{align}
g^\dagger_k =
\left(\begin{array}{ccc}
h_{00}\ _k \langle 2;0| & h_{01}\ _k \langle 2;1| & h_{02}\ _k \langle 2;2| \\
h_{10}\ _k \langle 2;-1| & h_{11}\ _k \langle 2;0| & h_{12}\ _k \langle 2;1| \\
h_{20}\ _k \langle 2;-2| & h_{21}\ _k \langle 2;-1| &h_{22}\ _k \langle 2;0|
\end{array}\right).
\end{align}
Now we introduce ``$G$ matrix'',
which will play an important role in our study.
Let us set an $(S+1)^2$ dimensional vector space $W$
and its dual orthogonal space $W^*$ as
\begin{align}
W=\bigoplus_{0 \le a,b \le S} {\mathbb C}|a,b {\rangle\!\rangle} , \quad
W^*=\bigoplus_{0 \le a,b \le S} {\mathbb C}{\langle\!\langle} a,b | . \quad
\end{align}
Here,
$\{|a,b {\rangle\!\rangle} \ | \ a,b=0, \dots, S \}$
($\{ {\langle\!\langle} a,b | \ | \ a,b=0, \dots, S \}$)
is an orthonormal (dual orthonormal) basis.
We define an $(S+1)^2 \times (S+1)^2$
matrix $G$ acting on the space $W$ as
\begin{align}
G_{(a,b;c,d)} &={\langle\!\langle} a,b | G | c,d {\rangle\!\rangle} =
g^\dagger (a,c) g(b,d) ,
\label{gmatrix}
\end{align}
or equivalently as
\begin{align}
G=g^\dagger \otimes g .
\end{align}
We also introduce $G_A$ for an operator $A$
acting on the one-site vector space $V_S$ as
\begin{align}
\left(G_A\right)_{(a,b;c,d)} &=
{\langle\!\langle} a,b | G_A | c,d {\rangle\!\rangle} =
g^\dagger (a,c) Ag(b,d).
\end{align}
Each element of the matrix $G$ can be expressed explicitly as
\begin{align}
G_{(a,b;c,d)} =
\delta_{c-a,d-b}T_{abcd} ,
\end{align}
where
\begin{align}
\begin{split}
&T_{abcd} = h_{ac} h_{bd} =
(-1)^{a+b} q^{(a+b+c+d-2S)(S+1)/2} \\
&\ \times
\sqrt{
\left[ \begin{array}{c} S \\ a \end{array} \right]
\left[ \begin{array}{c} S \\ b \end{array} \right]
\left[ \begin{array}{c} S \\ c \end{array} \right]
\left[ \begin{array}{c} S \\ d \end{array} \right]
[S-a+c]! [S+a-c]! [S-b+d]! [S+b-d]! }\ .
\end{split}
\end{align}
Each element of $G_A$ for $A=S^z, S^+$ and $S^-$,
which act on $|S;m\rangle$ as
\begin{align}
S^z |S;m\rangle =& m |S;m\rangle, \\
S^+ |S;m\rangle =& \sqrt{(S-m)(S+m+1)} |S;m+1\rangle, \\
S^- |S;m\rangle =& \sqrt{(S+m)(S-m+1)} |S;m-1\rangle,
\end{align}
can be also expressed as
\begin{align}
\left(G_{S^z}\right)_{(a,b;c,d)}=&\delta_{c-a,d-b}(d-b) T_{abcd},
\label{GSZ}\\
\left(G_{S^+}\right)_{(a,b;c,d)}=&\delta_{c-a,d-b+1}
\sqrt{(S-d+b)(S+d-b+1)} T_{abcd},
\label{GSPLUS} \\
\left(G_{S^-}\right)_{(a,b;c,d)}=&\delta_{c-a,d-b-1}
\sqrt{ (S+d-b)(S-d+b+1) } T_{abcd}
\label{GSMINUS}.
\end{align}
The squared norm of the ground state is calculated as
\begin{align}
\begin{split}
\langle\Psi|\Psi\rangle
=& \Tr \left[ g_1^\dagger \star \cdots \star g_L^\dagger \right]
\Tr \left[ g_1 \star\cdots\star g_L \right] \\
=& \Tr \left[\left( g_1^\dagger \star \cdots \star g_L^\dagger \right)
\otimes \left( g_1 \star \cdots \star g_L \right) \right] \\
=& \Tr \left[\left( g_1^\dagger \otimes g_1 \right) \star
\cdots \star \left( g_L^\dagger \otimes g_L \right) \right] \\
=& \Tr G^L.
\label{PsiPsi=TrGL}
\end{split}
\end{align}
Note that the elements of $g^\dagger_k \otimes g_k = G$
are no longer vectors,
and thus we can replace the symbol $\star$ by the usual product
in the third line of \eqref{PsiPsi=TrGL}.
The one point function $ \langle A \rangle$
of an operator $A$ can be written
in terms of $G$ and $G_A$ as
\begin{align}
\begin{split}
\langle A \rangle
=\frac{\langle\Psi|A_1|\Psi\rangle}{\langle\Psi|\Psi\rangle}
=\frac{\Tr \left[g_1^\dagger \star \cdots \star g_L^\dagger\right]
\Tr \left[A_1g_1 \star g_2 \star \cdots \star g_L \right] }
{\langle\Psi|\Psi\rangle}
= \frac{\Tr G_A G^{L-1} }{ \Tr G^L }, \label{onepoint}
\end{split}
\end{align}
where $A_kg_k$ is defined by
$\left(A_kg_k\right)(i, i')=A_k\left(g_k(i, i')\right)$.
In the same way, the two point function of $A$ and $B$
can also be written in terms of $G, G_A$ and $G_B$ as
\begin{align}
\langle A_1 B_r \rangle& = \left(\Tr G^L\right)^{-1}
\Tr G_A G^{r-2} G_B G^{L-r}. \label{twopoint}
\end{align}
Investigating the eigenvalues and eigenvectors of the matrix $G$
will be crucial for the analysis of correlation functions.
In the next section, we study the $G$ matrix in detail.
\section{\label{Sec:spec-of-G} Spectral structure of the $G$ matrix}
In Ref.~\cite{M}, we conjectured that the spectrum of $G$ is given by
\begin{align}
\lambda_\ell=(-1)^\ell\left(\q{S}!\right)^2
\q{\begin{array}{c} 2S+1 \\ S-\ell \end{array}},
\quad(\ell=0,1,\dots,S), \label{eigenvalues}
\end{align}
where the degree of the degeneracy of each $\lambda_\ell$
is $2\ell+1$.
One can easily find that
\begin{align}\label{0>1>cdots>S}
|\lambda_0| > |\lambda_1|> \cdots >|\lambda_S| .
\end{align}
In this section, we prove the conjecture
by giving an exact form for the eigenvector
corresponding to each eigenvalue. \\
First one observes that the
$G$ matrix has the following block diagonal structure:
\begin{align}
G=&\bigoplus_{-S\le j\le S}G^{ (j) },\quad G^{ (j) }\in {\rm End} W_j,\\
W=& \bigoplus_{-S\le j\le S}W_j, \quad
W_j=\begin{cases}
\displaystyle
\bigoplus_{0\le i \le S-j}{\mathbb C}|i,i+j{\rangle\!\rangle} & j\ge 0,\\
\displaystyle
\bigoplus_{0\le i \le S+j}{\mathbb C}|i-j,i {\rangle\!\rangle} & j<0.
\end{cases}
\end{align}
The size of each block $G^{ (j) }$ is $(S-|j|+1)\times(S-|j|+1)$.
Each element of $G^{ (j) }$ is
\begin{align}
\begin{split}
{\langle\!\langle} a,a+j|G^{ (j) }|c,c+j{\rangle\!\rangle}
=&(-1)^jq^{(a+c+j-S)(S+1)} \q{S-a+c}!\q{S+a-c}! \\
& \times \sqrt{
\q{\begin{array}{c} S \\ a \end{array}}
\q{\begin{array}{c} S \\ a+j \end{array}}
\q{\begin{array}{c} S \\ c \end{array}}
\q{\begin{array}{c} S \\ c+j \end{array}}
} \ .
\end{split}
\end{align}
We construct intertwiners among the $2S+1$ blocks
$G^{ (j) }\ (j=-S, \dots, S) $.
This helps us to construct
eigenvectors of each block from
another block with a smaller size.
(The same idea was used in Ref.~\cite{AKSS} to study
the spectrum of a multi-species exclusion process).
Let us define a family of linear operators
$\{I_j\}_{-S\le j\le-1,1\le j\le S}$ as
\begin{align}
I_j&\in
{\rm Hom}(W_j,W_{j-1}), \\
{\langle\!\langle} a,a+j-1 |I_j| c,c+j {\rangle\!\rangle}
&=
\begin{cases}
\displaystyle
q^{-a}\sqrt{ \frac{\q{a+j}\q{S-a-j+1}}{\q{j}\q{S-j+1}} } & c=a, \\
\displaystyle
-q^{1-a-j}\sqrt{ \frac{\q{a} \q{S-a+1}}{\q{j}\q{S-j+1}} } & c=a-1, \\
0 & \rm otherwise
\end{cases} \\
\nonumber
\end{align}
for $1\le j\le S$, and
\begin{align}
I_j&\in {\rm Hom}(W_j,W_{j+1}), \\
{\langle\!\langle} a-j-1,a |I_j| c-j,c {\rangle\!\rangle}
&=
\begin{cases}
\displaystyle
q^{-a}\sqrt{ \frac{\q{a-j}\q{S-a+j+1}}{\q{-j}\q{S+j+1}} } & c=a, \\
\displaystyle
-q^{1-a+j}\sqrt{ \frac{\q{a} \q{S-a+1}}{\q{-j}\q{S+j+1}} } & c=a-1, \\
0 & \rm otherwise .
\end{cases}\\
\nonumber
\end{align}
for $-S\le j\le -1$.
By direct calculation, one finds
\begin{proposition}
The matrix $I_j$ enjoys the intertwining relation
\begin{align}
\begin{split}
\label{intrel}
I_j G^{ (j) } =& G^{ (j-1) } I_j \quad {\rm for }\ 1\le j\le S,\\
I_j G^{ (j) } =& G^{ (j+1) } I_j \quad {\rm for }\ -S\le j\le -1.
\end{split}
\end{align}
\label{intertwining}
\end{proposition}
With the use of Proposition \ref{intertwining}, one can show the following:
\begin{theorem}\label{eigen-structure}
Each block $G^{ (j) }$ has
a simple (nondegenerated) spectrum
\begin{align}
{\rm Spec}\ G^{ (j) }= \{ \lambda_\ell \}_{|j| \le \ell\le S},
\end{align}
and the corresponding eigenvectors
are given by
\begin{align}
\label{edgeeigen}
|\lambda_{|j|} {\rangle\!\rangle}_j&=
\begin{cases}
\displaystyle
\sum_{0\le i\le S-\ell}q^{(\ell+1)i}
\sqrt{\frac{\q{S-\ell}! \q{i+\ell}! \q{S-i}!}{
[S]! [\ell]! [S -i-\ell]! [i]! } }
|i,i+\ell{\rangle\!\rangle} & j\ge 0, \\
\displaystyle
\sum_{0\le i\le S-\ell}q^{(\ell+1)i}
\sqrt{\frac{\q{S-\ell}! \q{i+\ell}! \q{S-i}!}{
[S]! [\ell]! [S -i-\ell]! [i]! } }
|i+\ell,i{\rangle\!\rangle} & j<0,
\end{cases}
\end{align}
for $\ell=|j|$, and
\begin{align}
\label{nakaeigen}
|\lambda_\ell{\rangle\!\rangle}_j&=
\begin{cases}
I_{j+1}|\lambda_\ell {\rangle\!\rangle}_{j+1}
=I_{j+1}I_{j+2}\cdots I_{\ell} |\lambda_\ell{\rangle\!\rangle}_\ell & j\ge0, \\
I_{j-1}|\lambda_\ell {\rangle\!\rangle}_{j-1}
=I_{j-1}I_{j-2}\cdots I_{-\ell} |\lambda_\ell{\rangle\!\rangle}_{-\ell} & j<0. \\
\end{cases}
\end{align}
for $|j|+1\le \ell \le S$.
\label{gmatrixstructure}
\end{theorem}
\begin{figure}[h]
\begin{align*}
\begin{array}{llllllllll}
W_S & |\lambda_S{\rangle\!\rangle}_S \\
\ \downarrow \scriptstyle{I}_S & \quad\text{\rotatebox[origin=c]{270}{$\mapsto$} } \\
W_{S-1}& |\lambda_S{\rangle\!\rangle}_{S-1} &
|\lambda_{S-1}{\rangle\!\rangle}_{S-1} \\
\ \downarrow\scriptstyle{I}_{S-1} & \quad\text{\rotatebox[origin=c]{270}{$\mapsto$} } & \quad\text{\rotatebox[origin=c]{270}{$\mapsto$} } \\
W_{S-2}& |\lambda_S{\rangle\!\rangle}_{S-2} &
|\lambda_{S-1}{\rangle\!\rangle}_{S-2} &
|\lambda_{S-2}{\rangle\!\rangle}_{S-2}
\vspace{2mm} \\
\ \vdots & \quad\vdots & \quad \vdots
& \quad\vdots \quad\quad\quad\quad\quad\ \ddots
\vspace{2mm} \\
W_2 & |\lambda_S{\rangle\!\rangle}_2 &|\lambda_{S-1}{\rangle\!\rangle}_2 &
|\lambda_{S-2}{\rangle\!\rangle}_2 \quad \ \dots
&\,|\lambda_2{\rangle\!\rangle}_2 \\
\ \downarrow\scriptstyle{I}_2 & \quad\text{\rotatebox[origin=c]{270}{$\mapsto$} } & \quad\text{\rotatebox[origin=c]{270}{$\mapsto$} }
& \quad\text{\rotatebox[origin=c]{270}{$\mapsto$} } & \quad\text{\rotatebox[origin=c]{270}{$\mapsto$} } \\
W_1 & |\lambda_S{\rangle\!\rangle}_1 & |\lambda_{S-1}{\rangle\!\rangle}_1 &
|\lambda_{S-2}{\rangle\!\rangle}_1 \quad \ \dots
&\,|\lambda_2{\rangle\!\rangle}_1&|\lambda_1{\rangle\!\rangle}_1 \\
\ \downarrow\scriptstyle{I}_1 & \quad\text{\rotatebox[origin=c]{270}{$\mapsto$} } & \quad\text{\rotatebox[origin=c]{270}{$\mapsto$} }
& \quad\text{\rotatebox[origin=c]{270}{$\mapsto$} } & \quad\text{\rotatebox[origin=c]{270}{$\mapsto$} } & \quad\text{\rotatebox[origin=c]{270}{$\mapsto$} } & \\
W_0 & |\lambda_S{\rangle\!\rangle}_0 &|\lambda_{S-1}{\rangle\!\rangle}_0 &
|\lambda_{S-2}{\rangle\!\rangle}_0 \quad \ \dots
&\, |\lambda_2{\rangle\!\rangle}_0 & |\lambda_1{\rangle\!\rangle}_0 &|\lambda_0{\rangle\!\rangle}_0 \\
\ \uparrow_{I_{-1}} & \quad \text{\rotatebox[origin=c]{90}{$\mapsto$} } & \quad \text{\rotatebox[origin=c]{90}{$\mapsto$} }
& \quad \text{\rotatebox[origin=c]{90}{$\mapsto$} } & \quad \text{\rotatebox[origin=c]{90}{$\mapsto$} } & \quad \text{\rotatebox[origin=c]{90}{$\mapsto$} } & \\
W_{-1} & |\lambda_S{\rangle\!\rangle}_{-1} & |\lambda_{S-1}{\rangle\!\rangle}_{-1} &
|\lambda_{S-2}{\rangle\!\rangle}_{-1} \quad \dots
&\, |\lambda_2{\rangle\!\rangle}_{-1}&|\lambda_1{\rangle\!\rangle}_{-1} \\
\ \uparrow_{I_{-2}} & \quad \text{\rotatebox[origin=c]{90}{$\mapsto$} } & \quad \text{\rotatebox[origin=c]{90}{$\mapsto$} }
& \quad \text{\rotatebox[origin=c]{90}{$\mapsto$} } & \quad \text{\rotatebox[origin=c]{90}{$\mapsto$} } \\
W_{-2} & |\lambda_S{\rangle\!\rangle}_{-2} &|\lambda_{S-1}{\rangle\!\rangle}_{-2} &
|\lambda_{S-2}{\rangle\!\rangle}_{-2} \quad \dots
&\, |\lambda_2{\rangle\!\rangle}_{-2}
\vspace{2mm} \\
\ \vdots & \quad\vdots & \quad \vdots
& \quad\vdots \quad\quad\quad\quad\quad\ \dotsd
\vspace{2mm} \\
W_{-S+2}& |\lambda_S{\rangle\!\rangle}_{-S+2} &
|\lambda_{S-1}{\rangle\!\rangle}_{-S+2} &
|\lambda_{S-2}{\rangle\!\rangle}_{-S+2} \\
\ \uparrow_{I_{-S+1}} & \quad \text{\rotatebox[origin=c]{90}{$\mapsto$} } & \quad \text{\rotatebox[origin=c]{90}{$\mapsto$} } \\
W_{-S+1}& |\lambda_S{\rangle\!\rangle}_{-S+1} &
|\lambda_{S-1}{\rangle\!\rangle}_{-S+1} \\
\ \uparrow_{I_{-S}} & \quad \text{\rotatebox[origin=c]{90}{$\mapsto$} } \\
W_{-S} & |\lambda_S{\rangle\!\rangle}_{-S} \\
\end{array}
\end{align*}
\caption{Structure of the eigenvectors of $G$ \eqref{gmatrix}.
}\label{structure}
\end{figure}
Figure \ref{structure} is helpful to understand
how the eigenvectors are constructed.
We prove this theorem below
for only $j\ge 0$
since one can show it for $j<0$ in the same way.
{\it Proof of Theorem \ref{eigen-structure}.}
First, by direct calculation given below,
we find that
$G^{(j)}$ has an eigenvalue $\lambda_j$
and its eigenvector is
$|\lambda_j{\rangle\!\rangle}_j$ defined by \eqref{edgeeigen}.
Each element of $ G^{ (j) }|\lambda_j {\rangle\!\rangle}_j $
is calculated as
\begin{align}
\begin{split}
& {\langle\!\langle} a, a+j|G^{ (j) }|\lambda_j {\rangle\!\rangle}_j \\
&= \sum_{0\le c\le S-j}(-1)^j
q^{(a+c+j-S)(S+1)} \q{S-a+c}!\q{S+a-c}! \\
& \quad\times \sqrt{
\q{\begin{array}{c} S \\ a \end{array}}
\q{\begin{array}{c} S \\ a+j \end{array}}
\q{\begin{array}{c} S \\ c \end{array}}
\q{\begin{array}{c} S \\ c+j \end{array}} }
q^{(j+1)c}\displaystyle
\sqrt{\frac{\q{S-j}! \q{c+j}! \q{S-c}!}{
\q{S}! \q{j}! \q{S-c-j}! \q{c}! } }
\\
&= (-1)^j q^{(a+j-S-1)(S+1)-(j+1)}
\sqrt{
\q{\begin{array}{c} S \\ a \end{array}}
\q{\begin{array}{c} S \\ a+j \end{array}}
\displaystyle \frac{\q{S-j}!}{\q{S}! \q{j}!}} \\
&\quad\times \q{S}!
\sum_{0\le c\le S-j } q^{(c+1)(S+j+2)}
\frac{\q{S-a+c}!\q{S+a-c}!}{\q{S-c-j}! [c]!} .
\end{split}
\end{align}
Using the formula
\begin{align}
\sum_{0\le k\le n}
\q{
\begin{array}{c} \alpha+n-k \\ n-k \end{array} }
\q{ \begin{array}{c} \beta+k \\ k \end{array} }
q^{k(\alpha+\beta+2)}
=\q{ \begin{array}{c} \alpha+\beta+n+1 \\ n \end{array} }
q^{n(1+\beta)},
\end{align}
we obtain
\begin{align}
\begin{split}
{\langle\!\langle} a, a+j|G^{ (j) }|\lambda_j {\rangle\!\rangle}_j
&= (-1)^j q^{(a+j-S-1)(S+1)-(j+1)}
\sqrt{
\q{\begin{array}{c} S \\ a \end{array}}
\q{\begin{array}{c} S \\ a+j \end{array}}
\displaystyle \frac{\q{S-j}!}{\q{S}! \q{j}!}} \\
&\quad\times \q{S}!\,
q^{-Sj+a(j-S)+S^2+2S+2}
\q{\begin{array}{c} 2S+1 \\ S-j \end{array}}
\q{S-a}!\q{a+j}!
\\
&= (-1)^j\left(\q{S}!\right)^2
\q{\begin{array}{c} 2S+1 \\ S-j \end{array}}
q^{(j+1)a}
\sqrt{\displaystyle
\frac{\q{S-j}!\q{a+j}!\q{S-a}!}{\q{S}!\q{j}!\q{S-a-j}!\q{a}!} }
\\
&= \lambda_j {\langle\!\langle} a, a+j |\lambda_j{\rangle\!\rangle}_j .
\end{split}
\end{align}
Note that the first element of $|\lambda_j{\rangle\!\rangle}_j$ is 1
by the definition \eqref{edgeeigen}:
$ {\langle\!\langle} 0,j | \lambda_j{\rangle\!\rangle}_j=1$.
Next, we show by induction that
$G^{(j)}$ has eigenvalues $\lambda_\ell$ ($j\le\ell\le S$)
and their corresponding eigenvectors are
given by $|\lambda_\ell{\rangle\!\rangle}_j$ defined by \eqref{nakaeigen}.
Suppose the theorem is true for $|\lambda_\ell {\rangle\!\rangle}_{j+1},
\ell=j+1, \dots,S \ (j \ge 0)$,
that is to say that the block
$G^{ (j+1) }$ has the eigenvalues $\lambda_\ell$ and their
corresponding eigenvectors $| \lambda_\ell {\rangle\!\rangle}_{j+1}$
$\big( G^{ (j+1) }|\lambda_\ell{\rangle\!\rangle}_{j+1}=
\lambda_\ell|\lambda_\ell{\rangle\!\rangle}_{j+1} $
with $|\lambda_\ell{\rangle\!\rangle}_{j+1} \neq 0\big)$
for $\ell=j+1,\dots,S$.
Additionally, suppose
that the first element of each $|\lambda_\ell{\rangle\!\rangle}_{j+1} $ is 1.
Using the intertwining relation \eqref{intrel},
one finds $G^{ (j) }I_{j+1}|\lambda_\ell{\rangle\!\rangle}_{j+1}
=\lambda_\ell I_{j+1}|\lambda_\ell{\rangle\!\rangle}_{j+1}$.
We also find that the first element of
$ I_{j+1}|\lambda_\ell{\rangle\!\rangle}_{j+1} $ is 1,
and thus $I_{j+1}|\lambda_\ell{\rangle\!\rangle}_{j+1}$ is nonzero.
Furthermore, thanks to
$\ell_1\neq\ell_2\Rightarrow\lambda_{\ell_1} \neq\lambda_{\ell_2}$,
the vectors $I_{j+1}|\lambda_\ell{\rangle\!\rangle}_{j+1}$ (${j+1\le\ell\le S}$)
are distinct (in other words, $I_{j+1}$ is injective).
We have already constructed the remaining
eigenvector of $G^{(j)}$ explicitly, which is
$| \lambda_j {\rangle\!\rangle}_j$ with
its eigenvalue $\lambda_j$ distinct from
$\lambda_\ell\ (j+1\le\ell\le S)$.
\qed
The conjecture for the eigenvalues
of the $G$ matrix that we exhibited in the beginning of this section
follows as a simple corollary of Theorem \ref{gmatrixstructure}.
Moreover, we constructed their eigenvectors
which are important for computing spin-spin correlation functions.
\begin{proposition}\label{prop-squared-norm}
The squared norm of $|\lambda_\ell{\rangle\!\rangle}_j$ is
\begin{align}
{}_j{\langle\!\langle} \lambda_\ell|\lambda_\ell{\rangle\!\rangle}_j
=q^{S(|j|+1) - \ell(\ell+1) }
\frac{ [S+\ell+1]! [\ell-|j|]! [S-\ell]! [|j|]! }
{ [S]! [\ell+|j|]! [S-|j|]! [2\ell+1] },
\label{squared-norm}
\end{align}
where we denote the transpose of $| \lambda_\ell {\rangle\!\rangle}_j$ by
${}_j {\langle\!\langle} \lambda_\ell|$.
\end{proposition}
We prove this proposition only for $j\ge 0$.
{\it Proof of Proposition \ref{prop-squared-norm}. }
One can easily show
that the product of intertwiners
(which is also an intertwiner)
has the following form by induction:
\begin{align}
\begin{split}
&{\langle\!\langle} a,a+j |
I_{j+1} I_{j+2}\cdots I_{\ell+1}I_{\ell}
| c,c+\ell {\rangle\!\rangle} \\
=&
(-1)^{a - c} q^{c j - a \ell}
\q{\begin{array}{c} \ell-j \\ a - c\end{array} }
\sqrt{
\frac{ [j]! [S-\ell]! [a]! [S-c]! [c+\ell]! [S-(a+j) ]! }
{ [\ell]! [S-j]! [c]! [S-a]! [a+j]! [ S -(c + \ell) ]! } } .
\end{split}
\end{align}
Then,
${}_j {\langle\!\langle} \lambda_\ell |\lambda_\ell{\rangle\!\rangle}_j
={}_\ell {\langle\!\langle} \lambda_\ell |
\left(I_{j+1} \cdots I_\ell\right)^{\rm T}
I_{j+1} \cdots I_\ell |\lambda_\ell{\rangle\!\rangle}_\ell $
is calculated as
\begin{align}
\begin{split}
&{}_j {\langle\!\langle} \lambda_\ell |\lambda_\ell{\rangle\!\rangle}_j
= \frac{\left([S-\ell]!\right)^2 [j]! }{ [S]! \left([\ell]!\right)^2 [S-j]! }
\sum_{0 \le a\le S-j \atop 0\le i,i'\le S }
(-1)^{i+i'} q^{( i+i')(\ell+j+1)-2a\ell} \\
& \quad\quad \times
\q{ \begin{array}{c} \ell-j \\ a-i \end{array} }
\q{ \begin{array}{c} \ell-j \\ a-i' \end{array} }
\frac{[S-i]! [i+\ell]! [S-i']! [i'+\ell]! [a]! [S-(a+j)]! }
{[i]! [S-(i+\ell)]! [i']! [S-(i'+\ell)]! [S-a]![a+j]!} .
\end{split}
\end{align}
The triple sum has the closed form
\begin{align}
q^{(j+1)S-\ell(\ell+1)}
\frac{ \left([\ell]!\right)^2 [\ell-j]![S+\ell+1]!}
{[S-\ell]! [j+\ell]! [2\ell+1]},
\end{align}
which finishes the proof.\qed
\section{\label{Sec:correlation}Spin-spin correlation functions}
In the last section, we investigated the
eigenvalues and eigenvectors of the $G$ matrix.
By utilizing Theorem \ref{gmatrixstructure}
and noting \eqref{0>1>cdots>S},
the one point function $\langle A \rangle$
can be represented as
\begin{align}
\langle A \rangle=\lambda_0^{-1}
\frac{ {}_0 {\langle\!\langle} \lambda_0 | G_A | \lambda_0 {\rangle\!\rangle}_0}
{{}_0 {\langle\!\langle} \lambda_0 |\lambda_0 {\rangle\!\rangle}_0}
\end{align}
in the thermodynamic limit $L\to\infty$.
As an application, we can calculate
the probability of finding $S^z=m$ value as
\begin{align}
\begin{split}
{\rm Prob}(S^z=m)
=& \big\langle\, |S;m \rangle \langle S;m | \, \big\rangle \\
=& \frac{[S+m]![S-m]!}{[2S+1]!}
\sum_{i=0}^{S} q^{(S+2)(2i-m-S)}
\left[
\begin{array}{c}
S \\
i-m
\end{array}
\right]
\left[
\begin{array}{c}
S \\
i
\end{array}
\right].
\end{split}
\end{align}
The two point function \eqref{twopoint} can be also represented as
\begin{align}
\langle A_1 B_r \rangle=\sum_{\ell=0}^S \lambda_\ell^{-2}
\left( \frac{\lambda_\ell}{\lambda_0} \right)^r
\sum_{j=-\ell}^\ell
\frac{{}_0 {\langle\!\langle} \lambda_0 |G_A | \lambda_\ell {\rangle\!\rangle}_j
{}_j {\langle\!\langle} \lambda_\ell |G_B | \lambda_0 {\rangle\!\rangle}_0}
{{}_0 {\langle\!\langle} \lambda_0 | \lambda_0 {\rangle\!\rangle}_0
{}_j {\langle\!\langle} \lambda_\ell | \lambda_\ell {\rangle\!\rangle}_j},
\label{twopointrep}
\end{align}
in the thermodynamic limit.
Inserting
\eqref{GSZ}, \eqref{GSPLUS}, \eqref{GSMINUS}, \eqref{eigenvalues},
\eqref{edgeeigen}, \eqref{nakaeigen} and \eqref{squared-norm}
into \eqref{twopointrep}, one finds
the large-distance ($r\to \infty$) behaviors
of the spin-spin correlation functions
$ \langle S_1^z S_r^z \rangle$
and $ \langle S_1^+ S_r^- \rangle $ are
\begin{align}
\langle S_1^z S_r^z \rangle&=-
\frac{[3] [S+2] }{q^{2S-2}
[S] ([2S+1] !)^2}
({}_0 {\langle\!\langle} \lambda_1|G_{S^z}| \lambda_0 {\rangle\!\rangle}_0)^2
\left(-\frac{[S] }{[S+2] } \right)^r, \\
\langle S_1^+ S_r^- \rangle&=-
\frac{[2] [3] [S+2] }{q^{3S-2} ([2S+1] ! [S] )^2}
({}_{-1} {\langle\!\langle} \lambda_1|G_{S^-}| \lambda_0 {\rangle\!\rangle}_0)^2
\left(-\frac{[S] }{[S+2] } \right)^r,
\end{align}
where
\begin{align}
\begin{split}
& \!\!\!\!\!\!
{}_0 {\langle\!\langle} \lambda_1|G_{S^z}| \lambda_0 {\rangle\!\rangle}_0
=\frac{q^{-S^2-S-1}}{q^S-q^{-S} }
\sum_{i, i'=0}^{S}(i- i')q^{(S+2)(i+ i')} \\
&\times \{q^{S+1}+q^{-S-1}-(q+q^{-1})q^{2 i'-S} \}
[S+i- i'] ! [S+ i'-i] !
\left[ \begin{array}{c} S \\ i \end{array} \right]
\left[ \begin{array}{c} S \\ i' \end{array} \right] ,
\end{split}
\\
\begin{split}
& \!\!\!\!\!\!
{}_{-1} {\langle\!\langle} \lambda_1|G_{S^-}| \lambda_0 {\rangle\!\rangle}_0 =
{}_0 {\langle\!\langle} \lambda_0|G_{S^+}| \lambda_1 {\rangle\!\rangle}_{-1} \\
&=-q^{-S^2-S/2+1/2} \sum_{i=0}^{S} \sum_{ i'=0}^{S-1}
q^{(S+2)i+(S+3) i'}
\sqrt{\left[\begin{array}{c} S \\ i'+1 \end{array} \right]
\left[ \begin{array}{c} S \\ i' \end{array} \right]}
\\
&\quad \times
\sqrt{(S+i- i') [S+i- i'] (S-i+ i'+1) [S-i+ i'+1]}
\\
&\quad \times
\sqrt{ [i'+1][S-i'] [S]^{-1} } [S+ i'-i] ! [S+i- i'-1] !
\left[ \begin{array}{c} S \\i \end{array}\right] .
\end{split}
\end{align}
Note that the terms with $(j,\ell)=(0,1)$ and $(-1,1)$
in \eqref{twopointrep} dominate
the large-distance behaviors of $\langle S^z_1 S^z_r \rangle$
and $\langle S^+_1 S^-_r \rangle$, respectively,
since
\begin{align}
{}_0{\langle\!\langle}\lambda_0|G_{S^z}|\lambda_0{\rangle\!\rangle}_0
={}_1{\langle\!\langle} \lambda_1|G_{S^z}|\lambda_0{\rangle\!\rangle}_0
={}_{-1}{\langle\!\langle} \lambda_1|G_{S^z}|\lambda_0{\rangle\!\rangle}_0 =0, \\
{}_0{\langle\!\langle}\lambda_0|G_{S^-}|\lambda_0{\rangle\!\rangle}_0
={}_1{\langle\!\langle} \lambda_1|G_{S^-}|\lambda_0{\rangle\!\rangle}_0
={}_0{\langle\!\langle} \lambda_1|G_{S^-}|\lambda_0{\rangle\!\rangle}_0 =0.
\end{align}
Both $ \langle S_1^z S_r^z \rangle$ and
$ \langle S_1^+ S_r^- \rangle$ exhibit exponential decay with
correlation length
\begin{align}
\zeta=\left( \ln \frac{[S+2] }{[S] } \right)^{-1},
\end{align}
generalizing the results for $q=1$ \cite{FH}
or $S=1$ \cite{KSZ1} case.
\section{Conclusion \label{Sec:Conclusion}}
In this paper, we investigated one and two point functions
of the $q$-VBS ground state of
an integer spin model (the $q$-deformed higher-spin AKLT model).
The formulation of correlation functions by use of the matrix
product representation of the ground state shows that the structure
of a matrix, which we call $G$ matrix, plays an important role.
We obtained the eigenvalues and eigenvectors of the
$G$ matrix with the help of constructing intertwiners connecting
different blocks of $G$.
Then we calculated the spin-spin correlation functions
by use of the eigenvalues and eigenvectors of the $G$ matrix,
and determined the correlation amplitudes and correlation lengths of the
longitudinal and transverse spin-spin correlation functions.
It is interesting to investigate other types of correlation functions.
For example, the entanglement entropy,
which is defined in terms of the reduced density matrix,
is a typical quantification of the entanglement
of quantum systems.
It is intriguing to calculate the entanglement entropy
for the $q$-deformed model and observe the change from the isotropic point
\cite{XKHK,FKR,KHH} (see also Refs.~\cite{KHK,KKKKT}
for other VBS states).
\section*{Acknowledgements}
The authors thank Atsuo Kuniba and Kazumitsu Sakai
for useful discussion.
CA also thanks Kirone Mallick and Andreas Schadschneider
for the very kind hospitality during his stay in Europe.
This work is supported by Grant-in-Aid for Young Scientists (B) 22740106
and Global COE program
``Education and Research Hub for Math-for-Industry.''
|
{
"timestamp": "2012-01-04T02:08:47",
"yymm": "1009",
"arxiv_id": "1009.4018",
"language": "en",
"url": "https://arxiv.org/abs/1009.4018"
}
|
\section{Introduction}
The theory of small deviations of random
functions is currently in intensive development. In this paper we
address small deviations of Gaussian random fields in $L_2$-norm.
Suppose we have a real-valued Gaussian random field $X(x)$, $x\in {\cal O}\subset
{\mathbb R}^d$, with zero mean and covariance function
$G_X(x,y)=EX(x)X(y)$ for $x,y\in{\cal O}$. Let $\mu$ be a finite measure
on ${\cal O}$. Set
\[
\|X\|_{\mu}=\|X\|_{L_2({\cal O};\mu)}=(\int\limits_{\cal O} X^2(x)\ \mu(dx))^{1/2}
\]
(the subscript $\mu$ will be omitted when $\mu$ is the Lebesgue measure)
and consider
\[
Q(X,\mu \ ;\varepsilon)={\bf P}\{\|X\|_{\mu}\leq \varepsilon \}.
\]
The problem is to evaluate the behavior of $Q(X,\mu\ ;\varepsilon) $ as
$\varepsilon \rightarrow 0$. Note that the case
$\mu(dx)=\psi(x)dx$, $\psi\in L_1({\cal O})$, can be easily reduced to the
Lebesgue case $\rho\equiv1$ replacing $X$ by the Gaussian field $X \sqrt{\rho}$.
In general case, by scaling, one can assume that $\mu({\cal O})=1$.
According to the well-known Karhunen-Lo\`eve expansion, we
have
$$\|X\|_{\mu}^2 \overset{d}{=} \sum_{n=1}^\infty \lambda_n\xi_n^2,$$
where $\xi_n$, $n\in\mathbb N$, are independent standard
normal r.v.'s, and $\lambda_n>0$, $n\in\mathbb N$,
$\sum\limits_n\lambda_n <\infty $, are the eigenvalues of the
integral equation
$$ \lambda f(x)=\int\limits_{\cal O}G_X(x,y)f(y)\mu(dy), \qquad x\in {\cal O}.
$$
Thus we arrive to the equivalent problem of studying the asymptotic behavior
of ${\bf P}\left\{\sum_{n=1}^\infty \lambda _n \xi_n^2\leq\varepsilon ^2 \right\}$
as $\varepsilon \to 0+ $. The answer heavily depends on the available information about the
eigenvalues $\lambda_n$.
The study of small deviation problem was initiated by Sytaya \cite{S}
and continued by a number of authors. See the history of the problem
and the summary of main results in two excellent reviews
\cite{Lf} and \cite{LiS}. The references to the latest results on
$L_2$-small deviation asymptotics can be found on the site
\cite{site}.\medskip
In this paper we continue to study the small deviation asymptotics of
a vast and important class of Gaussian random fields having the
covariance of ``tensor product'' type. It means that this covariance
can be decomposed in a product of ``marginal'' covariances depending
on different arguments. The classical examples of such fields are
the Brownian sheet and the Brownian pillow. The notion of tensor
products of Gaussian processes or Gaussian measures is known long
ago. Such Gaussian fields are also studied in related domains, see,
e.g., \cite{PW} and \cite{LP}. We recall briefly the construction
of these fields.
Suppose we have two Gaussian random functions $X(x)$, $x \in {\mathbb R}^m$,
and $Y(y)$, $y \in {\mathbb R}^n$, with zero means
and covariances $G_X(x,u)$, $x,u \in {\mathbb R}^m$, and
$G_Y(y,v)$, $y,v \in {\mathbb R}^n$, respectively. Consider
the new Gaussian function $Z(x,y)$, $x \in {\mathbb R}^m$, $y\in {\mathbb R}^n$,
which has zero mean and the covariance
$$G_Z((x,y),(u,v)) =G_X(x,u)G_Y(y,v).$$
Such Gaussian function obviously exists, and the integral operator
with the kernel $G_Z$ is the tensor product of two ``marginal''
integral operators with the kernels $G_X$ and $G_Y$.
Therefore we use in the sequel the notation $Z = X\otimes Y$ and
we call the Gaussian field $Z$ {\it the tensor product} of the
fields $X$ and $Y$. The generalization to the multivariate case
when obtaining the fields $\underset{j=1}{\overset{d}\otimes} X_j$
is straightforward.\medskip
The investigations of small deviations of Gaussian random functions of
this class were started in a classical paper \cite{Cs} where the logarithmic
$L_2$-small ball asymptotics was obtained for the Brownian sheet
${\mathbb W}_d={\mathbb W}_d(x_1,...,x_d)$ on the unit cube.
This result was later extended by Li \cite{Li} to some other random fields.
In a very interesting paper \cite{FT} the {\it exact}
asymptotics of small deviations in $L_2$-norm for the Brownian
sheet was obtained using the Mellin transform. However, it is not clear
if the method of \cite{FT} yields small deviation results for a more general
class of Gaussian fields.
A new approach developed in the paper \cite{KNN} is based on
abstract theorems describing the spectral asymptotics of tensor
products and of sums of tensor products for self-adjoint operators
in Hilbert space. This approach gave the opportunity to consider
quite general class of tensor products with the eigenvalues
$\lambda_n^{(j)}$ of marginal covariances having the so-called
{\it regular behaviour}:
$$\lambda_n^{(j)}\sim\frac {\varphi_j (n)} {n^{p_j}},\, n \to \infty,$$
where $p_j>1$ and $\varphi_j$ are some {\it slowly varying functions} (SVFs).\medskip
In this paper we consider the case where $\lambda_n^{(j)}$ have faster rate
of decreasing. To be more precise, we assume that the so-called {\it counting functions}
$${\cal N}_j(t)=\#\{n:\ \lambda_n^{(j)}>t\}$$
are SVFs. Such behavior of eigenvalues is typical for processes with smooth covariances,
see, e.g., \cite{Na1}.\medskip
The structure of the paper is as follows. In \S 2 we
present some auxiliary information about slowly varying functions.
Next, in \S 3 we prove new results on the spectral asymptotics
for tensor products of compact self-adjoint operators with slowly varying
counting functions. Then, in \S 4, using the result of \cite{Na1}, we
evaluate the small ball constants for the special rate of eigenvalues decay,
namely,
$${\cal N}(t) \sim \ln^p(1/t)\Phi(\ln(1/t)) \,,\qquad t\to 0+,$$
with $p\ge0$ and $\Phi$ being a SVF. Finally, we apply our theory to various
specific examples of Gaussian random fields.
\section{ Auxiliary information on SVFs}
We recall that a positive
function $\varphi(\tau)$, $\tau>0$, is called a {\it slowly
varying} function (SVF) at infinity if for any $c>0$
\begin{equation}\label{slow}
\varphi(c\tau)/\varphi(\tau)\to 1 \qquad \mbox{as}\quad
\tau\to+\infty.
\end{equation}
It is easily seen that any smooth positive function $\varphi$
satisfying $\tau\varphi'(\tau)/\varphi(\tau)\to 0$ as
$\tau\to+\infty$ is slowly varying. This test shows that the
functions equal to $\ln^p(\tau)\ln^{\varkappa}(\ln(\tau))$ for $\tau>>1$
are slowly varying.\medskip
We need some simple properties of SVFs. Their proofs can be found,
for example, in \cite{Se}.
\begin{prop}\label{SVF}
Let $\varphi$ be a SVF. Then following statements are true:
{\bf 1}. The relation (\ref{slow}) is uniform with respect to $c\in[a,b]$
for $0<a<b<+\infty$.
{\bf 2}. There exists an equivalent SVF\ $\widehat\varphi\in {\mathcal
C}^2({\mathbb R}_+)$ (i.e. $\frac {\widehat\varphi(\tau)}{\varphi(\tau)}\to1$ as
$\tau\to+\infty$) such that
\begin{equation}\label{equiv_slow}
\tau\cdot(\ln(\widehat\varphi))'(\tau)\to 0,\qquad
\tau^2\cdot(\ln(\widehat\varphi))''(\tau)\to 0,\qquad\tau\to+\infty.
\end{equation}
{\bf 3}. The function $\tau\mapsto\tau^p\varphi(\tau)$, $p>0$, up to equivalence
at infinity, is monotone for large $\tau$,
and its inverse function is $\tau\mapsto\tau^{1/p}\phi(\tau)$, where $\phi$ is a
SVF.
\end{prop}
For two nondecreasing and unbounded SVFs $\varphi$ and $\psi$, we define their {\it asymptotic convolution}
$$(\varphi\star\psi)(\tau)=\int\limits_1^{\tau} \varphi( \tau / \sigma)\,d\psi(\sigma).
$$
\begin{rem}
It is easy to see that the asymptotic convolution is connected with the Mellin convolution (see
\cite[\S 2]{KNN}) by the relation
$$(\varphi\star\psi)(\tau)=(\varphi*\psi_1)(\tau),\qquad \mbox{where}\qquad \psi_1(\tau)=\tau\psi'(\tau).$$
Therefore, basic properties of the asymptotic convolution could be extracted from \cite[Theorem 2.2]{KNN}.
However, for the reader's convenience, we give them with full proofs.
\end{rem}
\begin{tm}\label{convolution}
The following statements are true:
{\bf 1}. $(\varphi\star\psi)(\tau)\le \varphi(\tau)\psi(\tau)$.
{\bf 2}. $\varphi(\tau) = o((\varphi\star\psi)(\tau))$ as $\tau\to+\infty$.
{\bf 3}. Asymptotic symmetry:
$$(\varphi\star\psi)(\tau) = (\psi\star\varphi)(\tau) + O(\varphi(\tau)+\psi(\tau)),\qquad \tau\to+\infty.
$$
{\bf 4}. If $\varphi(\tau)=\widehat\varphi(\tau)\cdot(1+o(1))$ as $\tau\to+\infty$, then
$$(\varphi\star\psi)(\tau)=(\widehat\varphi\star\psi)(\tau)\cdot(1+o(1)),\qquad \tau\to+\infty.$$
{\bf 5}. $\varphi\star\psi$ is a nondecreasing SVF.
\end{tm}
\begin{rem}
Note that, by the statement {\bf 3}, the statements {\bf 2} and {\bf 4} hold true with replacing $\varphi$ by $\psi$.
\end{rem}
\begin{proof}
{\bf 1}. This fact is trivial, since $\varphi( \tau / \sigma)\le\varphi(\tau)$ for all
$\sigma\in\,[1,\tau]$.\medskip
{\bf 2}. By Proposition \ref{SVF}, part~{\bf 1}, for any $a>1$ we have
$$(\varphi\star\psi)(\tau)>\int\limits_1^a \varphi(\tau /\sigma)\,d\psi(\sigma)=
\varphi(\tau)(\psi(a)-\psi(1))\cdot(1+o(1)),\qquad \tau\to+\infty.
$$
Since $\psi$ is unbounded, the statement follows.\medskip
{\bf 3}. Integrating by parts and changing the variable, we obtain
$$(\varphi\star\psi)(\tau)=\varphi(1) \psi(\tau) - \varphi(\tau)\psi(1) + (\psi\star\varphi)(\tau).
$$
By {\bf 2}, the statement follows.\medskip
{\bf 4}. By {\bf 2} and {\bf 3}, for any $a>1$
$$(\varphi\star\psi)(\tau)\sim(\psi\star\varphi)(\tau)\sim
\int\limits_a^{\tau}\psi(\tau/\sigma)\,d\varphi(\sigma)\sim
\int\limits_1^{\frac {\tau}a}\varphi(\tau/\sigma)\,d\psi(\sigma),
\qquad\tau\to+\infty.$$
Given $\varepsilon>0$, one can take $a$ so large that
$1-\varepsilon<\frac {\widehat\varphi(\lambda)}{\varphi(\lambda)}<1+\varepsilon$ for $\lambda>a$, and the statement follows.\medskip
{\bf 5}. Due to {\bf 4} and to Proposition \ref{SVF}, part~{\bf 2}, we can assume $\varphi$ and $\psi$ smooth.
We have
$$\tau\cdot(\varphi\star\psi)'(\tau) = \tau\psi '(\tau) \varphi(1) +
\int\limits_1^\tau\frac{\tau}{\sigma}\cdot\varphi '(\tau/\sigma)\,d\psi(\sigma).
$$
By {\bf 2},
$$\tau\psi'(\tau)=o(\psi(\tau))=o((\varphi\star\psi)(\tau)),\qquad \tau\to+\infty.$$
Next, due to (\ref{equiv_slow}) we have for any $a>1$ and $\tau>a$
\begin{multline*}
\left|\int\limits_1^\tau\frac{\tau}{\sigma}\cdot\varphi '(\tau/\sigma)\,d\psi(\sigma)\right|\le
\left|\int\limits_1^{\frac {\tau}a}\frac{\tau}{\sigma}\cdot\varphi '(\tau/\sigma)\,d\psi(\sigma)\right|+
\left|\int\limits_{\frac {\tau}a}^\tau\frac{\tau}{\sigma}\cdot\varphi '(\tau/\sigma)\,d\psi(\sigma)\right|\le\\
\le \sup\limits_{\lambda\ge a}\left|\frac {\lambda\varphi'(\lambda)}{\varphi(\lambda)}\right|\cdot (\varphi\star\psi)(\tau)+\sup\limits_{\lambda\le a}|\lambda\varphi'(\lambda)|\cdot\psi(\tau).
\end{multline*}
By Proposition \ref{SVF}, part~{\bf 2}, given $\varepsilon>0$, one can take $a$ so large that
$\left|\frac {\lambda\varphi'(\lambda)}{\varphi(\lambda)}\right|<\varepsilon$ for $\lambda>a$.
This gives, subject to {\bf 2},
$\tau\cdot(\varphi\star\psi)'(\tau)=o((\varphi\star\psi)(\tau))$ as $\tau\to+\infty$, and the statement follows.
\end{proof}
{\bf Example 1}. Let
$$\varphi(\tau)=\ln^\alpha(\tau)\cdot\Phi(\ln(\tau)),\qquad
\psi(\tau)=\ln^\beta(\tau)\cdot\Psi(\ln(\tau)),
$$
where $\alpha,\ \beta\ge0$ while $\Phi$ and $\Psi$ are SVFs\footnote{If $\alpha=0$ (respectively, $\beta=0$), then we require
in addition that $\Phi$ (respectively, $\Psi$) is nondecreasing and unbounded.}.
Then, as $\tau\to+\infty$,
\begin{equation}\label{conv_pq}
(\varphi\star\psi)(\tau)\sim\frac {\Gamma(\alpha+1)\Gamma(\beta+1)}{\Gamma(\alpha+\beta+1)}\cdot\varphi(\tau)\psi(\tau).
\end{equation}
\begin{proof}
Changing the variable we obtain
$$(\varphi\star\psi)(\tau)=
\int\limits_0^{\ln(\tau)}(\ln(\tau)-s)^\alpha\,\Phi(\ln(\tau)-s)\,d\big[s^\beta\,\Psi(s)\big].
$$
First, let at least one of exponents $\alpha$ and $\beta$ be positive. By Theorem \ref{convolution}, part~{\bf 3}, one can assume that $\beta>0$. Then, substituting $s=\ln(\tau)\vartheta$ we get
\begin{multline*}
(\varphi\star\psi)(\tau)=\varphi(\tau)\psi(\tau)\times\\
\times\int\limits_0^1(1-\vartheta)^\alpha\,\frac {\Phi(\ln(\tau)(1-\vartheta))}{\Phi(\ln(\tau))}\cdot
\vartheta^{\beta-1}\,\frac {\beta\Psi(\ln(\tau)\vartheta)+\ln(\tau)\vartheta\Psi'(\ln(\tau)\vartheta)}{\Psi(\ln(\tau))}\,d\vartheta.
\end{multline*}
By Proposition \ref{SVF}, part {\bf 3}, for any $\varepsilon>0$ the function $T^{\varepsilon}\Phi(T)$
increases for large $s$. Hence for $0<z<1$, $T>0$ we have
$$\frac {\Phi(zT)}{\Phi(T)}=\frac 1{z^\varepsilon}\cdot\frac {(zT)^{\varepsilon}\Phi(zT)}{T^{\varepsilon}\Phi(T)}\le
\frac 1{z^\varepsilon}.
$$
This estimate and a similar estimate for $\Psi$ give us the majorant required in Lebesgue
Dominated Convergence Theorem. Passing to the limit in the integral we obtain as
$\tau\to+\infty$
$$(\varphi\star\psi)(\tau)\sim\varphi(\tau)\psi(\tau)\cdot\int\limits_0^1(1-\vartheta)^\alpha\cdot
\beta\vartheta^{\beta-1}\,d\vartheta,
$$
and we arrive at (\ref{conv_pq}).\medskip
Now let $\alpha=\beta=0$. Then for any $\delta\in\,]0,1[$
$$(\varphi\star\psi)(\tau)\ge\int\limits_0^{\delta\ln(\tau)}\Phi(\ln(\tau)-s)\,d\Psi(s)\ge
\Phi(\ln(\tau)(1-\delta))\cdot[\Psi(\ln(\tau)\delta)-\Psi(0)].
$$
By definition of SVF, we obtain
$$\liminf\limits_{\tau \to +\infty}\frac{(\varphi\star\psi)(\tau)}{\varphi(\tau)\psi(\tau)}\ge 1.
$$
Taking into account Theorem \ref{convolution}, part {\bf 1}, we arrive at (\ref{conv_pq}).
\end{proof}
{\bf Example 2}. Let
$$\varphi(\tau)= \exp(a^{\frac 1{p'}}\ln^{\frac 1p}(\tau))\,\ln^{\alpha}(\tau)\cdot\Phi(\ln (\tau)),\qquad
\psi(\tau)= \exp(b^{\frac 1{p'}}\ln^{\frac 1p}(\tau))\,\ln^{\beta}(\tau)\cdot\Psi(\ln (\tau)),
$$
where $a,b>0$, $\alpha,\beta\ge0$, $p>1$, $p'$ stands for the H\"older conjugate exponent,
while $\Phi$ and $\Psi$ are SVFs.
Then, as $\tau\to+\infty$,
\begin{equation}\label{conv_exp}
(\varphi\star\psi)(\tau)\sim\sqrt\frac{2\pi}{p-1}\ \frac
{a^{\alpha+\frac 12}b^{\beta+\frac 12}}{(a+b)^{\gamma +\frac 12}}\
\exp((a+b)^{\frac 1{p'}}\ln^{\frac 1p}(\tau))\,\ln^{\gamma}(\tau)\cdot \Phi(\ln (\tau))\Psi(\ln (\tau)),
\end{equation}
where $\gamma=\alpha+\beta+\frac 1{2p}$.\medskip
\begin{proof}
Similarly to Example 1, we have
\begin{multline*}
(\varphi\star\psi)(\tau) \sim {\frac {b^{\frac 1{p'}}}p}\,T^{\alpha+\beta+\frac 1p}\cdot\Phi(T)\Psi(T)\times\\
\times\int\limits_0^1 \exp( T^{\frac 1p}S(\vartheta))\cdot
(1-\vartheta)^\alpha\vartheta^{\beta-\frac 1{p'}}\cdot\frac {\Phi(T(1-\vartheta))}{\Phi(T)}\cdot
\frac {\Psi(T\vartheta)}{\Psi(T)} \,d\vartheta,
\end{multline*}
where $T=\ln(\tau)$,
$S(\vartheta) = a^{\frac 1{p'}}(1-\vartheta)^{\frac 1p} + b^{\frac 1{p'}}\vartheta^{\frac 1p}$.
Denote by $\vartheta_\star$ the maximum point of $S(\vartheta)$.
Then, using the Laplace method and Proposition \ref{SVF}, part~{\bf 1}, we have
$$(\varphi\star\psi)(\tau)\sim
{\frac {b^{\frac 1{p'}}}p}\,T^{\alpha+\beta+\frac 1{2p}}\cdot\Phi(T)\Psi(T)
\sqrt\frac{2\pi}{-S''(\vartheta_\star)}
\exp(T^{\frac 1p}S(\vartheta_\star))
(1-\vartheta_\star)^\alpha\vartheta_\star^{\beta-\frac 1{p'}},
$$
Direct calculation shows that
$$\vartheta^\star =\frac b{a+b}, \qquad S(\vartheta_\star) = (a+b)^{\frac 1{p'}},\qquad
S^{''}(\vartheta^\star) = -\frac 1{pp'} \frac {(a+ b)^{3-{\frac 1p}}}{ab},$$
and we arrive at (\ref{conv_exp}).
\end{proof}
\section{Spectral asymptotics for tensor products of compact
self-adjoint operators}
We recall that for a compact self-adjoint nonnegative operator
${\cal T}={\cal T}^*\ge0$ in a Hilbert space $H$ we denote by
$\lambda_n =\lambda_n({\cal T})$ its positive eigenvalues
arranged in a nondecreasing sequence and repeated according to
their multiplicity. Also we introduce the counting function
$${\cal N}(t)={\cal N}(t,{\cal T})=\#\{n: \lambda_n ({\cal T})>t\}.
$$
Note that in view of Proposition \ref{SVF}, part~{\bf 2}, any SVF
arising in an asymptotic formula can be assumed smooth.
\begin{tm}\label{ocenka}
Let ${\cal T}$ and $\widetilde {\cal T}$ be compact self-adjoint nonnegative
operators in Hilbert spaces $H$ and $\widetilde H$, respectively. Let
\begin{equation}\label{asymp}
{\cal N}(t)\sim \varphi(1/t),\qquad \widetilde {\cal N}(t)\equiv {\cal N}(t,\widetilde {\cal T})
\sim\widetilde\varphi(1/t),\qquad t\to+0,
\end{equation}
where $\varphi$ and $\widetilde\varphi$ are unbounded SVFs at infinity.
Then for any $\varepsilon>0$ the operator ${\cal T}\otimes\widetilde {\cal T}$
in the space $H\otimes\widetilde H$ satisfies the following estimates:
\begin{equation}\label{ocenka_oplus}
{\cal N}_{\otimes}(t)\equiv{\cal N}(t,{\cal T}\otimes\widetilde{\cal T})
\lessgtr \alpha_{\pm}(\varepsilon)
\cdot\left[ S(t,\varepsilon)+\widetilde S(t,\varepsilon)+
\int\limits_{\alpha_{\mp}(\varepsilon)/\varepsilon}^{\varepsilon\tau}
\varphi(\tau/\sigma)\,d\widetilde\varphi(\sigma)\right]
\end{equation}
uniformly with respect to $t>0$ (here $\tau$ stands for
$\alpha_{\pm}(\varepsilon)/t$). For $\alpha_{\mp}(\varepsilon)/\varepsilon >\varepsilon\tau$
the integral in (\ref{ocenka_oplus}) should be omitted.
In (\ref{ocenka_oplus}) $\alpha_{\pm}(\varepsilon)\to1$ as $\varepsilon\to+0$, and the functions
$S$, $\widetilde S$ satisfy the following relations as $t\to+0$:
\begin{equation}\label{S}
S(t,\varepsilon)\sim \varphi(1/t)\widetilde\varphi(1/\varepsilon);\qquad
\widetilde S(t,\varepsilon)=o(\widetilde\varphi(1/t)).
\end{equation}
\end{tm}
\begin{proof}
We establish the upper bound for ${\cal N}_{\otimes}(t)$.
The lower estimate can be proved in the same way. We have
$${\cal N}_{\otimes}(t) =\sum\limits_n {\cal N}(t/\widetilde\lambda_n)=
\sum\limits_{\widetilde\lambda_n<\varepsilon}{\cal N}(t/\widetilde\lambda_n)+S(t,\varepsilon),$$
\noindent where
$$S(t,\varepsilon)=\sum\limits_{\widetilde\lambda_n\ge\varepsilon}
{\cal N}(t/\widetilde\lambda_n).$$
The asymptotics (\ref{S}) for the last sum follows from Proposition \ref{SVF}, part~{\bf 1}.
Denote by $\widetilde\theta$ the inverse function for $\widetilde\varphi$.
Then the second relation in (\ref{asymp}) implies $\widetilde\lambda_n/\widetilde\theta(n)\to1$ as $n\to\infty$,
and we have
$$\alpha_-(\varepsilon)\widetilde\theta(n)\le\widetilde\lambda_n\le
\alpha_+(\varepsilon)\widetilde\theta(n)$$ for $\widetilde\lambda_n<\varepsilon$,
with $\alpha_{\pm}(\varepsilon)\to1$ as $\varepsilon\to+0$.
Using monotonicity of $\cal N$ we get
$$\sum\limits_{\widetilde\lambda_n<\varepsilon} {\cal N}(t/\widetilde\lambda_n)\le
\sum\limits_{\widetilde\theta(n)<\varepsilon\alpha_-^{-1}(\varepsilon)} {\cal N}(t/\alpha_+(\varepsilon)\widetilde\theta(n)),
$$
and by monotonicity of the function $n\mapsto {\cal N}(t/\alpha_+(\varepsilon)\widetilde\theta(n))$ we estimate
the sum by an integral:
\begin{equation}\label{q}
\sum\limits_{\widetilde\lambda_n<\varepsilon} {\cal N}(t/\widetilde\lambda_n)\le
{\cal N}\left(\frac {\alpha_-(\varepsilon)t} {\alpha_+(\varepsilon)\varepsilon}\right)+
\int\limits_0^{\varepsilon \alpha^{-1}_-(\varepsilon)}{\cal N}(t/\alpha_+(\varepsilon)\theta) (-d\widetilde\varphi(1/\theta)).
\end{equation}
The first term in (\ref{q}) is $O(\varphi(1/t))$. Therefore,
adding it to the term $S(t,\varepsilon)$ we obtain the term
$\alpha_+(\varepsilon)S(t,\varepsilon)$ with $\alpha_+(\varepsilon)\to1$ as $\varepsilon\to 0+$.
Further,
splitting the integral in (\ref{q}) and changing variables we obtain
\begin{equation}\label{qq}
\int\limits_{\varepsilon}^{+\infty}{\cal N}(s)\,d\widetilde\varphi(\alpha_+(\varepsilon)s/t)+
\int\limits_{\alpha_-(\varepsilon)/\varepsilon} ^{\varepsilon\tau}{\cal N}(\sigma/\tau)\,d\widetilde\varphi(\sigma).
\end{equation}
The first integral in (\ref{qq}) gives us the term $\widetilde S(t,\varepsilon)$. Integrating by parts
(note that for $s>\|T\|$ the integrand equals 0) we obtain
$$\widetilde S(t,\varepsilon)={\cal N}(\varepsilon)\widetilde\varphi(\alpha_+(\varepsilon)\varepsilon/t)
-\int\limits_{\varepsilon}^{\|T\|}\widetilde\varphi(\alpha_+(\varepsilon)s/t)\,d{\cal N}(s).
$$
By Proposition \ref{SVF}, part~{\bf 1},
$\frac {\widetilde\varphi(\alpha_+(\varepsilon)s/t)}{\widetilde\varphi(1/t)}\to1$ as $t\to 0+$
uniformly with respect to $s\in[\varepsilon,\|T\|]$ , and we arrive at (\ref{S}).
By the first relation in (\ref{asymp}) we can estimate $\cal N(\sigma/\tau)$ in the second integral by
$\alpha_+(\varepsilon)\varphi(\tau/\sigma)$ that gives the integral term in (\ref{ocenka_oplus}).
\end{proof}
\begin{tm}\label{spectr_asymp}
Let operators $\cal T$ and $\widetilde{\cal T}$ be as in Theorem $\ref{ocenka}$. Then
\begin{equation}\label{asymp_oplus}
{\cal N}_{\otimes}(t) \sim \phi(1/t)\equiv(\varphi\star\widetilde {\varphi})(1/t).
\end{equation}
\end{tm}
\begin{proof}
Fix arbitrary $\varepsilon >0$ and consider the estimates (\ref{ocenka_oplus}). By Theorem \ref{convolution}, part~{\bf 2},
we have
$$S(t,\varepsilon) = o((\varphi\star\widetilde{\varphi})(1/t)),\quad
\widetilde{S}(t,\varepsilon)=o((\varphi\star\widetilde{\varphi})(1/t)), \qquad t\to+0.$$
Further, the integral in the right-hand side of (\ref{ocenka_oplus}) differs from the
convolution $(\varphi\star\widetilde \varphi)(\tau)$ by the integrals
$$\int\limits_{\varepsilon \tau}^{\tau}\varphi(\tau/\sigma)\,d\widetilde{\varphi}(\sigma)=
O(\widetilde{\varphi}(\tau))=o((\varphi\star\widetilde{\varphi})(\tau)),\qquad \tau \to +\infty,$$
$$\int\limits_1^{\alpha_{\mp}/\varepsilon}\varphi(\tau/\sigma)\,d\widetilde{\varphi}(\sigma)=
O(\varphi(\tau))= o((\varphi\star\widetilde{\varphi})(\tau)), \qquad\tau \to +\infty,$$
(we recall that $\tau=\alpha_{\pm}(\varepsilon)/t$).
Due to Theorem \ref{convolution}, part~{\bf 5},
$(\varphi\star\widetilde{\varphi})(\tau) \sim(\varphi\star\widetilde{\varphi})(1/t)$, and hence
$$\limsup\limits_{t\to 0+}\frac{{\cal N}_{\otimes}(t)}{\phi(1/t)}\le \alpha_+(\varepsilon),\qquad
\liminf\limits_{t\to 0+}\frac{{\cal N}_{\otimes}(t)}{\phi(1/t)}\ge \alpha_-(\varepsilon),
$$
where $\phi$ is defined in (\ref{asymp_oplus}). Taking into account
that $\alpha_{\pm}(\varepsilon)\to 1$ as $\varepsilon \to 0+$, we arrive at
(\ref{asymp_oplus}).
\end{proof}
\section{Small ball asymptotics. Examples}
To connect the given asymptotic behavior of eigenvalues $\lambda_n$ with the
logarithmic decay rate for small deviations, we use the following statement, that is
slightly reformulated \cite[Theorem 2]{Na1}.
\begin{prop}
Let $(\lambda_n)$, $n\in\mathbb N$, be a
positive sequence with counting function ${\cal N}(t)$.
Suppose that
$${\cal N}(t)\sim \varphi(1/t), \qquad t\to 0+,
$$
where $\varphi$ is a function slowly varying at infinity.
Then, as $r\to 0+$,
\begin{equation}\label{vero}
\ln{\bf P}\left\{\sum_{n=1}^\infty\lambda_n\xi_n^2\le r
\right\} \sim -\frac 12 \int\limits_1^u \varphi(z)\,\frac {dz}z,
\end{equation}
where $u=u(r)$ satisfies the relation
\begin{equation}\label{ur}
\frac {\varphi(u)}{2u}\sim r, \qquad r\to 0+.
\end{equation}
\end{prop}
{\bf Example 3}. Let
$\varphi(\tau)=\ln^\alpha(\tau)\cdot\Phi(\ln(\tau))$, where $\alpha\ge0$ while $\Phi$ is a SVF\footnote{If $\alpha=0$,
then we require in addition that $\Phi$ is nondecreasing and unbounded.}. Then, as $\varepsilon\to 0+$,
\begin{equation}\label{smalldev}
\ln{\bf P}\left\{\sum_{n=1}^\infty\lambda_n\xi_n^2\le \varepsilon^2\right\} \sim
-\frac{2^\alpha}{\alpha+1}\,\varphi(1/\varepsilon)\ln(1/\varepsilon).
\end{equation}
\begin{proof}
Changing the variable $z=\frac u\sigma$ and using (\ref{conv_pq}), we observe that
$$\int\limits_1^u \varphi(z)\,\frac {dz}z=(\varphi\star\ln)(u)\sim
\frac 1{\alpha+1}\,\varphi(u)\ln(u),\qquad u\to+\infty.
$$
Next, direct calculation shows that $u=\frac {\varphi(1/r)}{2r}$ satisfies (\ref{ur}). Therefore, formula (\ref{vero}) gives, as $r\to 0+$,
$$\ln{\bf P}\left\{\sum_{n=1}^\infty\lambda_n\xi_n^2\le r\right\} \sim
-\frac 1{2(\alpha+1)} \,\varphi(1/r)\ln(1/r).
$$
Replacing $r$ by $\varepsilon^2$, we arrive at (\ref{smalldev}).
\end{proof}
{\bf Example 4}. Let
$\varphi(\tau)=\exp(a\ln^{\frac 1p}(\tau))\,\ln^\alpha(\tau)\cdot\Phi(\ln(\tau))$,
where $\alpha\ge0$, $p>1$ while $\Phi$ is a SVF. Then, as $\varepsilon\to 0+$,
\begin{multline}\label{smalldev1}
\ln{\bf P}\left\{\sum_{n=1}^\infty\lambda_n\xi_n^2\le \varepsilon^2\right\} \sim \\
\sim-\frac {2^{\alpha-\frac 1p}p}a\,\exp\Big[2\ln(1/\varepsilon)\cdot
\sum\limits_{k=1}^{[p']}c_k\big(\frac a{2^{\frac 1{p'}}}
\ln^{-\frac 1{p'}}(1/\varepsilon)\big)^k\Big]\,\ln^{\alpha+\frac 1{p'}}(1/\varepsilon)
\cdot\Phi(\ln (1/\varepsilon)),
\end{multline}
where
\begin{equation}\label{coeff}
c_1=1;\quad\qquad c_k=\frac 1{k!}\,\prod\limits_{m=0}^{k-2}\Big(\frac kp-m\Big)\quad\mbox{for} \quad k\ge2.
\end{equation}
In particular, if $p>2$ then
$$\ln{\bf P}\left\{\sum_{n=1}^\infty\lambda_n\xi_n^2\le \varepsilon^2\right\} \sim
-\frac {2^{-\frac 1p}p}a\,\varphi(\varepsilon^2)\ln^{\frac 1{p'}}(1/\varepsilon).
$$
\begin{proof}
Changing the variable we obtain
$$\int\limits_1^u\varphi(z)\,\frac {dz}z=T^{\alpha +1}\cdot\Phi(T)
\int\limits_0^1 \exp(a T^{\frac 1p}\vartheta^{\frac 1p})\cdot\vartheta^\alpha\cdot\frac {\Phi(T\vartheta))}{\Phi(T)}\,d\vartheta,
$$
where $T=\ln(u)$.
The Laplace method and the Lebesgue dominated convergence theorem give us the asymptotics
$$\int\limits_1^u\varphi(z)\,\frac {dz}z\sim \frac pa
\exp(a\ln^{\frac 1p} (u))\,\ln^{\alpha+\frac 1{p'}}(u)\cdot \Phi(\ln (u)),\qquad u\to+\infty$$
(we recall that $p'$ is the H\"older conjugate exponent for $p$).
Next, direct though cumbersome calculation shows that
$$u=\frac 1{2r}\,\exp\Big[\ln(1/r)\cdot
\sum\limits_{k=1}^{[p']}c_k\big(a\ln^{-\frac 1{p'}}(1/r)\big)^k\Big]\,\ln^{\alpha}(1/r)\cdot
\Phi(\ln (1/r))$$
with $c_k$ given by (\ref{coeff}) satisfies (\ref{ur}). Therefore, formula (\ref{vero}) gives,
as $r\to 0+$,
\begin{multline*}
\ln{\bf P}\left\{\sum_{n=1}^\infty\lambda_n\xi_n^2\le r\right\}
\sim-\frac pa\,ur\ln^{\frac 1{p'}}(u) \sim \\
\sim-\frac p{2a}\,\exp\Big[\ln(1/r)\cdot
\sum\limits_{k=1}^{[p']}c_k\big(a\ln^{-\frac 1{p'}}(1/r)\big)^k\Big]\,\ln^{\alpha+\frac 1{p'}}(1/r)
\cdot\Phi(\ln (1/r)).
\end{multline*}
Replacing $r$ by $\varepsilon^2$, we arrive at (\ref{smalldev1}).
\end{proof}
Turning to specific fields, we start with a stationary sheet
$${\cal R}^{\cal G}_d(x)=\underset{j=1}{\overset{d}\otimes} R^{{\cal G}_j}(x_j),\qquad
x=(x_1,\dots,x_d) \in [0,1]^d,
$$
where $R^{{\cal G}_j}$ are stationary Gaussian processes with zero mean-values and
the spectral densities
$$h_{R^{{\cal G}_j}}(\xi)=\exp(-{\cal G}_j(\xi)), \qquad \xi\in\mathbb R$$
(here ${\cal G}_j$ is even and ${\cal G}_j(\xi)\to+\infty$ as $\xi\to+\infty$).
Assume for simplicity that ${\cal G}$ is smooth and $\xi{\cal G}'(\xi)\to+\infty$ as $\xi\to+\infty$.
Then it is easy to check that the corresponding covariances $G_{R^{\cal G}}$ are smooth functions.
For instance, it is well known that
$${\cal G}(\xi)=|\xi|\quad\Longrightarrow\quad
G_{R^{\cal G}}(s,t)=\frac 1{\pi(1+(s-t)^2)};
$$
$${\cal G}(\xi)=\xi^2\quad\Longrightarrow\quad
G_{{\cal G}}(s,t)=\frac 1{2\sqrt{\pi}}\exp\left(-\frac {(s-t)^2}{4}\right).
$$
Small deviations of such processes in various $L_p$-norms in the case ${\cal G}(\xi)=|\xi|^\alpha$ were
considered in \cite{Na1}, \cite{AILZ}, \cite{Ku}.
\begin{prop}\label{exp_sheet}
{\bf 1}. Let ${\cal G}_j(\xi)\sim C_j\ln^p(\xi)$ as $\xi\to+\infty$, with $p>1$. Then, as
$\varepsilon\to 0+$,
\begin{multline}\label{log}
\ln {\bf P}\{\|{\cal R}^{\cal G}_d\|\le\varepsilon\}\sim -\frac p2
\Big(\pi^{d+1}{\mathfrak B}{\mathfrak C}\Big(\frac {p-1}2\Big)^{d-1}
\Big)^{-\frac 12}\times\\
\times\exp\Big[2\ln(1/\varepsilon)\cdot
\sum\limits_{k=1}^{[p']}c_k\Big(\frac {2\ln(1/\varepsilon)}{\mathfrak B}\Big)^{-\frac k{p'}}\Big]
\Big(\frac {2\ln(1/\varepsilon)}{\mathfrak B}\Big)^{1+\frac {d-3}{2p}},
\end{multline}
where
$${\mathfrak B}=\sum\limits_{j=1}^dC_j^{-\frac 1{p-1}};\qquad
{\mathfrak C}=\prod\limits_{j=1}^dC_j^{\frac 1{p-1}},
$$
while $c_k$ are given by (\ref{coeff}). In particular, if $p>2$ then
$$\ln {\bf P}\{\|{\cal R}^{\cal G}_d\|\le\varepsilon\}\sim -\frac p2
\Big(\pi^{d+1}{\mathfrak B}{\mathfrak C}\Big(\frac {p-1}2\Big)^{d-1}
\Big)^{-\frac 12}\exp\Big[{\mathfrak B}^{\frac 1{p'}}\Big(2\ln(1/\varepsilon)\Big)^{\frac 1p}\Big]
\Big(\frac {2\ln(1/\varepsilon)}{\mathfrak B}\Big)^{1+\frac {d-3}{2p}}.
$$
{\bf 2}. Let ${\cal G}_j(\xi)\sim\xi^q\Phi_j(\xi)$ as $\xi\to+\infty$, with $0<q\le1$ and
$\Phi_j$ being an SVF (if $q=1$ we require in addition that $\Phi_j(\xi)\to0$ as $\xi\to+\infty$).
Then, as $\varepsilon\to 0+$,
\begin{equation}\label{q<1}
\ln {\bf P}\{\|{\cal R}^{\cal G}_d\|\le\varepsilon\}\sim
-\frac {2^{\frac dq}\Gamma^d(\frac {q+1}q)}{\pi^d\Gamma(\frac dq+2)}\cdot
\ln^{\frac dq+1}(1/\varepsilon)\cdot\prod\limits_{j=1}^d\phi_j(\ln(1/\varepsilon)),
\end{equation}
where $\phi_j$, $j=1,\dots,d$, are SVFs depending only on $\Phi_j$ and on $q$; for $q=1$ we have $\phi_j(t)\to+\infty$ as $t\to+\infty$. In particular,
$$\Phi_j(\xi)=C_j\ln^p(\xi)\quad\Longrightarrow\quad
\ln {\bf P}\{\|{\cal R}^{\cal G}_d\|\le\varepsilon\}\sim
-\frac {\Gamma^d(\frac {q+1}q)}{\pi^d\Gamma(\frac dq+2)}\cdot
\Big(\frac {2q^p}{{\mathfrak C}}\Big)^{\frac dq}\cdot
\frac {\ln^{\frac dq+1}(1/\varepsilon)}{\ln^{\frac {p d}q}(\ln(1/\varepsilon))},
$$
where ${\mathfrak C}=\Big(\prod\limits_{j=1}^d C_j\Big)^{\frac 1d}$ (we recall that $p<0$ for
$q=1$).\medskip
{\bf 3}. Let ${\cal G}_j(\xi)\sim C_j\xi$ as $\xi\to+\infty$. Then, as $\varepsilon\to 0+$,
\begin{equation}\label{q=1}
\ln {\bf P}\{\|{\cal R}^{\cal G}_d\|\le\varepsilon\}\sim
-\frac {2^d}{\pi^d(d+1)!\,{\mathfrak C}}\cdot\ln^{d+1}(1/\varepsilon),
\end{equation}
where
$${\mathfrak C}=\prod\limits_{j=1}^d
\frac {{\bf K}(\mbox{\rm sech}(\pi/2C_j))}{{\bf K}(\tanh(\pi/2C_j))}$$
while $\bf K$ is the complete elliptic integral of the first kind, see, e.g., \cite[8.11]{GR}.\medskip
{\bf 4}. Let ${\cal G}_j(\xi)\sim\xi\Phi_j(\xi)$ as $\xi\to+\infty$, where $\Phi_j$ is an SVF.
Suppose in addition that ${\cal G}_j(\xi)$ is convex for large $\xi$ and $\Phi_j(\xi)\to+\infty$
as $\xi\to+\infty$. Then, as $\varepsilon\to 0+$,
\begin{equation}\label{q=1>}
\ln {\bf P}\{\|{\cal R}^{\cal G}_d\|\le\varepsilon\}\sim
-\frac 1{(d+1)!}\cdot\frac {\ln^{d+1}(1/\varepsilon)}{\prod\limits_{j=1}^d\ln(\phi_j(\ln(1/\varepsilon)))},
\end{equation}
where $\phi_j$, $j=1,\dots,d$ are SVFs depending only on $\Phi_j$, $\phi_j(t)\le\Phi_j(t)$ and
$\phi_j(t)\to+\infty$ as $t\to+\infty$. In particular,
$$\Phi_j(\xi)=C_j\ln^p(\xi)\quad\Longrightarrow\quad
\ln {\bf P}\{\|{\cal R}^{\cal G}_d\|\le\varepsilon\}\sim
-\frac 1{p^d(d+1)!}\cdot
\frac {\ln^{d+1}(1/\varepsilon)}{\ln^d(\ln(\ln(1/\varepsilon)))}
$$
(we recall that $p>0$).\medskip
{\bf 5}. Let $\ln({\cal G}_j(\xi))\sim q\ln(\xi)$ as $\xi\to+\infty$, with $q>1$. Then,
as $\varepsilon\to 0+$,
\begin{equation}\label{q>1}
\ln {\bf P}\{\|{\cal R}^{\cal G}_d\|\le\varepsilon\}\sim
-\frac {q^d}{(q-1)^d(d+1)!}\cdot\frac {\ln^{d+1}(1/\varepsilon)}{\ln^d(\ln(1/\varepsilon))}.
\end{equation}
{\bf 6}. Let $\ln({\cal G}_j(\xi))/\ln(\xi)\to+\infty$ as $\xi\to+\infty$. Then,
as $\varepsilon\to 0+$,
\begin{equation}\label{q=infty}
\ln {\bf P}\{\|{\cal R}^{\cal G}_d\|\le\varepsilon\}\sim
-\frac 1{(d+1)!}\cdot\frac {\ln^{d+1}(1/\varepsilon)}{\ln^d(\ln(1/\varepsilon))}.
\end{equation}
\end{prop}
\begin{rem}
General formulas (\ref{log})--(\ref{q=infty}) are new even for $d=1$. A particular case of purely power
dependence of ${\cal G}_j$ with $d=1$ was considered in \cite{Na1}. The recent preprint \cite{Ku} deals
with ${\cal G}_j(\xi)=C_j\xi^2$ for arbitrary $d$ but does not contain exact constant in (\ref{q>1}).
Note that in the superexponential case (parts {\bf 4}--{\bf 6}) the logarithmic small ball asymptotics
does not change if one multiplies ${\cal G}_j$ by a constant.
\end{rem}
\begin{proof}
{\bf 1}. It is shown in the remarkable paper \cite{W} that, if ${\cal G}_j(\xi)/\xi\to0$ as
$\xi\to+\infty$, then
\begin{equation}\label{Widom1}
\lambda_n^{(j)}\sim\exp(-{\cal G}_j(\pi n\cdot(1+o(1))))\qquad \mbox{as}\quad n\to\infty.
\end{equation}
In our case (\ref{Widom1}) implies
$${\cal N}_j(t)\sim \pi^{-1} \cdot\exp(C_j^{-\frac 1p}\ln^{\frac 1p}(1/t)),\qquad t\to 0+.
$$
Using Theorem \ref{spectr_asymp} and Example 2 successively $d-1$ times, we obtain that
$${\cal N}(t)\sim
\Big(\pi^{d+1}{\mathfrak C}{\mathfrak B}^{1+\frac {d-1}p}\Big(\frac {p-1}2\Big)^{d-1}
\Big)^{-\frac 12}
\exp({\mathfrak B}^{\frac 1{p'}}\ln^{\frac 1p}(1/t))\ln^{\frac {d-1}{2p}}(1/t).
$$
By (\ref{smalldev1}) we arrive at (\ref{log}).\medskip
{\bf 2}. It follows from (\ref{Widom1}) that in this case
$${\cal N}_j(t)\sim \pi^{-1} \cdot\ln^{\frac 1q}(1/t)\phi_j(\ln(1/t)),\qquad t\to 0+,
$$
and in a mentioned particular case
$${\cal N}_j(t)\sim \pi^{-1}\ \bigg(\frac {q^p}{C_j}
\cdot\frac{\ln(1/t)}{\ln^p(\ln(1/t))}\bigg)^{\frac 1q},\qquad t\to 0+.
$$
Using Theorem \ref{spectr_asymp} and Example 1 successively $d-1$ times, we obtain that
$${\cal N}(t)\sim \frac {\Gamma^d(\frac {q+1}q)}
{\pi^d\Gamma(\frac dq+1)\,}\cdot
\ln^{\frac dq}(1/t)\cdot\prod\limits_{j=1}^d\phi_j(\ln(1/t)).
$$
By (\ref{smalldev}) we arrive at (\ref{q<1}).\medskip
{\bf 3}. It follows from \cite{W}, that in this case
$${\cal N}_j(t)\sim\frac {{\bf K}(\tanh(\pi/2C_j))}{\pi\, {\bf K}(\mbox{sech}(\pi/2C_j))}\cdot\ln(1/t),
\qquad t\to 0+.
$$
Similarly to the case {\bf 2}, we arrive at (\ref{q=1}).\medskip
{\bf 4}. It is proved in \cite{W}, that, if ${\cal G}_j(\xi)$ is convex for large $\xi$ and
${\cal G}_j(\xi)/\xi\to+\infty$ as $\xi\to+\infty$, then
\begin{equation}\label{Widom2}
\ln(\lambda_n^{(j)})\sim-{\cal G}_j(\xi(n))\qquad \mbox{as}\quad n\to\infty,
\end{equation}
where $\xi(n)<n$ is a unique solution of the equation
$${\cal G}_j(\xi)=2n\ln\Big(\frac n{\xi}\Big).
$$
In this case we obtain from (\ref{Widom2})
$${\cal N}_j(t)\sim\frac {\ln(1/t)}{2\ln(\phi_j(\ln(1/t)))},\qquad t\to 0+.
$$
and in a mentioned particular case
$${\cal N}_j(t)\sim \frac 1{2p}\cdot\frac {\ln(1/t)}{\ln(\ln(\ln(1/t)))},\qquad t\to 0+.
$$
Similarly to the case {\bf 2}, we arrive at (\ref{q=1>}).
{\bf 5}. It follows from (\ref{Widom2}), that in this case
$${\cal N}_j(t)\sim\frac 1{2-\frac 2q} \cdot\frac {\ln(1/t)}{\ln(\ln(1/t))},\qquad t\to 0+.
$$
Similarly to the case {\bf 2}, we arrive at (\ref{q>1}).
{\bf 6}. It follows from (\ref{Widom2}), that in this case
$${\cal N}_j(t)\sim\frac 12 \cdot\frac {\ln(1/t)}{\ln(\ln(1/t))},\qquad t\to 0+.
$$
Similarly to the case {\bf 2}, we arrive at (\ref{q=infty}).
\end{proof}
Now we consider a smooth homogeneous sheet
$${\cal Z}^{({\mathfrak a},{\mathfrak b})}_d(x)=
\underset{j=1}{\overset{d}\otimes} Z^{({\mathfrak a}_j,{\mathfrak b}_j)}(x_j),\qquad
x\in [0,1]^d,
$$
where $Z^{({\mathfrak a}_j,{\mathfrak b}_j)}$ are Gaussian processes with zero mean-values and
the covariances
$$G_{Z^{({\mathfrak a}_j,{\mathfrak b}_j)}}(s,t)=
\frac{s^{{\mathfrak b}_j}t^{{\mathfrak b}_j}}{(s+t)^{{\mathfrak a}_j}}.
$$
Some properties of these processes were studied in \cite{LiS2} (for
${\mathfrak b}=\frac 12 ({\mathfrak a}+1)$) and \cite{AGKLS}.
\begin{prop}\label{hom_sheet}
Let ${\mathfrak a}_j>0$, and
${\mathfrak c}_j\equiv 2{\mathfrak b}_j-{\mathfrak a}_j+1>0$, $j=1,\dots,d$.
Then, as $\varepsilon\to 0+$,
\begin{equation}\label{laptev}
\ln {\bf P}\{\|{\cal Z}^{({\mathfrak a},{\mathfrak b})}_d\|\le\varepsilon\}\sim
-\frac {2^{2d}}{\pi^{2d}(2d+1)!\,{\mathfrak C}}\cdot\ln^{2d+1}(1/\varepsilon),
\end{equation}
where ${\mathfrak C}=\prod\limits_{j=1}^d {\mathfrak c}_j$.
\end{prop}
\begin{proof}
It is shown in \cite{La}, that
$${\cal N}_j(t)\sim \big(2\pi^2 {\mathfrak c}_j\big)^{-1} \cdot\ln^2(1/t),
\qquad t\to 0+.
$$
Similarly to the Proposition \ref{exp_sheet}, case {\bf 2}, we arrive at (\ref{laptev}).
\end{proof}
The next example deals with conventional Brownian sheet
\begin{equation}\label{sheet}
{\mathbb W}_d(x)=\underset{j=1}{\overset{d}\otimes} W_j(x_j),\qquad x\in [0,1]^d,
\end{equation}
where $W_j$ are Wiener processes.
As mentioned in the Introduction, the logarithmic asymptotics of small deviations
for the Brownian sheet in $L_2$-norm with respect to the Lebesgue measure was obtained
in \cite{Cs}. In \cite{KNN}, as a particular case of Proposition 5.1
(see also Theorem 8.2), this result was generalized to the case of absolutely continuous
measure with arbitrary continuous density. Now we are going to obtain the logarithmic
$L_2$-small ball asymptotics in the case of the discrete, degenerately self-similar measure
(see \cite{VSh}, \cite{NSh}). We recall briefly the construction of this measure (for $d=1$).\medskip
Let $0=\alpha_1<\alpha_2<\ldots<\alpha_M<\alpha_{M+1}=1$, $M\ge2$, be a partition
of the segment $[0,1]$. For some $m\in\{1,\dots,M\}$, denote by $a=\alpha_{m+1}-\alpha_m$
the length of $[\alpha_m,\alpha_{m+1}]$. Also we introduce a real number $\delta$ and
a real vector $(\beta_k)$, $k=1,\ldots,M$, such that
\begin{enumerate}
\item $0<\delta<1$;
\item $\beta_1=0$;\qquad $\chi_{\{m=M\}}\cdot\delta+\beta_M=1$;
\item $\beta_k<\beta_{k+1}$,\quad $k=1,\dots,M-1$;\qquad $\delta\beta_M+\beta_m<\beta_{m+1}$
\end{enumerate}
(for $m=M$ the last inequality is irrelevant).
It is shown in \cite[\S 2]{NSh} that under these conditions there exists a unique function $f$
such that ${\cal S}(f)=f$, where ${\cal S}$ is the simplest {\it similarity operator}
\begin{equation*}\label{eq:auxto}
{\cal S}[f](t)=\delta\cdot f(a^{-1}(t-\alpha_m))+
\sum\limits_{k=1}^M\beta_k\cdot\chi_{\,]\alpha_k,\alpha_{k+1}[}(t).
\end{equation*}
Moreover, $f$ increases on $[0,1]$, and $f(0)=0$, $f(1)=1$. The derivative $f'$ in the sence of
distributions is a discrete probability measure $\mu$ on $[0,1]$ with
a unique singular point $\widehat x=\frac{\alpha_m}{1-a}$. It is called {\it degenerately self-similar
measure}, generated by the set of parameters $M$, $m$, $\delta$, $(\alpha_k)$ and $(\beta_k)$.
\begin{prop}\label{brownian_sheet}
Let the measure $\mu$ on $[0,1]^d$ be the tensor product:
$$\mu(dx)=\underset{j=1}{\overset{d}\otimes}\mu_j(dx_j),
$$
where $\mu_j$, $j=1,\dots,d$, are degenerately self-similar measures, generated, respectively, by the
sets of parameters
$$ M_j,\quad m_j,\quad \delta_j,\quad (\alpha_k^{(j)}),\quad (\beta_k^{(j)}),\qquad j=1,\dots,d.
$$
Then, as $\varepsilon\to 0+$,
\begin{equation}\label{samopod}
\ln {\bf P}\{\|{\mathbb W}_d\|_{\mu}\le\varepsilon\}\sim
-\frac {2^d\,{\mathfrak C}}{(d+1)!}\cdot\ln^{d+1}(1/\varepsilon),
\end{equation}
where
$${\mathfrak C}=\prod\limits_{j=1}^d\frac {M_j-1}{\ln(a_j\delta_j)}
$$
(we recall that $a_j=\alpha^{(j)}_{m_j+1}-\alpha^{(j)}_{m_j}$).
\end{prop}
\begin{proof}
It is shown in \cite{NSh}, that
$${\cal N}_j(t)\sim \frac {M_j-1}{\ln(a_j\delta_j)}\cdot\ln(1/t),
\qquad t\to 0+.
$$
Similarly to the Proposition \ref{exp_sheet}, case {\bf 2}, we arrive at (\ref{samopod}).
\end{proof}
\begin{rem}
Note, analogously to Remark 6 in \cite{KNN}, that the replacement of any factor in (\ref{sheet}) by the
Brownian bridge, by the Ornstein--Uhlenbeck process or by similar process does not influence on the relation
(\ref{samopod}). For instance, the Brownian pillow ${\mathbb B}_d(x)=\underset{j=1}{\overset{d}\otimes}\ B(x_j)$ satisfies, as $\varepsilon\to0$, the relation
$$\ln {\bf P}\{\|{\mathbb B}_d\|_{\mu}\le\varepsilon\} \sim \ln {\bf P}\{\|{\mathbb W}_d\|_{\mu}\le\varepsilon\}.$$
\end{rem}
Now we consider the {\it isotropically integrated} Brownian sheet
$$({\mathbb W}_d)_{\mathfrak s}(x)=\underset{j=1}{\overset{d}\otimes} W_{\mathfrak s}(x_j),$$
where
$$W_{\mathfrak s}(t)\equiv W_{\mathfrak s}^{[b_1,\,\dots,\,b_{\mathfrak s}]}(t) =
(-1)^{b_1+\,\dots\,+b_{\mathfrak s}}\underbrace
{\int\limits_{b_{\mathfrak s}}^t\dots\int\limits_{b_1}^ {t_1}}_ {{\mathfrak s}} \
W (s)\ ds\ dt_1\dots$$
(here any $b_k$ equals either zero or one, $t\in[0,1]$; for various $j=1,\dots,d$ the multi-indices
$[b_1,\,\dots,\,b_{\mathfrak s}]$ can differ).
\begin{prop}
Let a measure $\mu$ on $[0,1]^d$ be as in Proposition $\ref{brownian_sheet}$.
Then, as $\varepsilon\to 0+$,
\begin{equation}\label{samopod1}
\ln {\bf P}\{\|({\mathbb W}_d)_{\mathfrak s}\|_{\mu}\le\varepsilon\}\sim
-\frac {2^d\,{\mathfrak C}}{(d+1)!}\cdot\ln^{d+1}(1/\varepsilon),
\end{equation}
where
$${\mathfrak C}=\prod\limits_{j=1}^d\frac {M_j-1}{\ln(a_j\delta_j^{2{\mathfrak s}+1})}.
$$
\end{prop}
\begin{proof}
This statement can be proved in the same way as Proposition \ref{brownian_sheet}.
\end{proof}
Finally, we can consider the fields-products corresponding to
essentially different marginal processes. We restrict ourselves
to a single example. On $[0,1]^2$ consider the Gaussian field
${\mathfrak R}(x_1)\otimes Z^{({\mathfrak a}, {\mathfrak b})}(x_2)$,
where ${\mathfrak R}$ is a stationary Gaussian process with zero mean-value and
the spectral density $h_{\mathfrak R}(\xi)=\frac 1{\Gamma(|\xi|)}$.
\begin{prop}
Let ${\mathfrak a}>0$, and ${\mathfrak c}\equiv 2{\mathfrak b}-{\mathfrak a}+1>0$. Then
$$\ln {\bf P}\{\|{\mathfrak R}\otimes Z^{({\mathfrak a}, {\mathfrak b})}\|\le\varepsilon\}\sim
-\frac 1{6\pi^2{\mathfrak c}}\cdot\frac {\ln^4(1/\varepsilon)}{\ln(\ln(\ln(1/\varepsilon)))}.
$$
\end{prop}
\begin{proof}
The asymptotics of marginal counting functions are calculated in Proposition \ref{exp_sheet},
part {\bf 4}, and in Proposition \ref{hom_sheet}. The result follows from Example 1 and (\ref{smalldev}).
\end{proof}
\bigskip
We are grateful to M.A.~Lifshits for useful comments and to Ya.Yu.~Nikitin for the constant encouragement.
We also thank T.~K\"uhn who kindly provided us with the text of preprint \cite{Ku}.\medskip
|
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"timestamp": "2010-11-18T02:01:53",
"yymm": "1009",
"arxiv_id": "1009.4412",
"language": "en",
"url": "https://arxiv.org/abs/1009.4412"
}
|
\section{Introduction}
Classical and quantum information processing network architectures utilize light (optical photons) for the transmission of information over extended distances, ranging from hundreds of meters to hundreds of kilometers~\cite{Agrawal2002,Duan2001}. The utility of optical photons stems from their weak interaction with the environment, large bandwidth of transmission, and resiliency to thermal noise due to their high frequency ($\sim 200~\text{THz}$). Acoustic excitations (phonons), though limited in terms of bandwidth and their ability to transmit information farther than a few millimeters, can be delayed and stored for significantly longer times and can interact resonantly with RF-microwave electronic systems~\cite{Pozar2004}. This complimentary nature of photons and phonons suggests hybrid phononic-photonic systems as a fruitful avenue of research, where a new class of \emph{optomechanical} circuitry could be made to perform a range of tasks out of reach of purely photonic and phononic systems. A building block of such a hybrid architecture would be elements coherently interfacing optical and acoustic circuits. The optomechanical translator we propose in this paper acts as a chip-scale \emph{transparent, coherent interface} between phonons and photons and fulfills a key requirement in such a program.
In the quantum realm, systems involving optical, superconducting, spin or charge qubits coupled to mechanical degrees of freedom~\cite{Cleland2004,Geller2005,rabl-2009,wallquist-2009-137,Chang2010,Stannigel2010} have been explored. The recent demonstration of coherent coupling between a superconducting qubit and a mechanical resonance by O'Connell, et al.~\cite{OConnell2010}, has provided an experimental backing for this vision and is the latest testament to the versatility of mechanics as a connecting element in hybrid quantum systems. In the specific case of phonon-photon state transfer, systems involving trapped atoms, ions, nanospheres~\cite{Parkins1999,Massoni2000,Orszag2002,Rodrigues2006,chang_cavity_2010}, and mechanically compliant optical cavity structures~\cite{Zhang2003} have all been considered. In these past studies, the state of an incoming light field is usually mapped onto the motional state of an atom, ion, or macroscopic mirror, through an exact timing of control pulses, turning on and off the interaction between the light and mechanical motion in a precise way. The ability to simultaneously implement phononic and photonic waveguides in optomechanical crystal (OMC) structures~\cite{Eichenfield2009} opens up the opportunity to implement a~\textit{traveling-wave} phonon-photon translator (PPT). Such a device, operating continuously, connects acoustic and optical waves to each other in a symmetric manner, and allows for on-the-fly conversion between phonons and photons without having to precisely time the information and control pulses. In effect, the problem of engineering control pulses is converted into a problem of engineering coupling rates.
Our proposal for a PPT is motivated strongly by recent work~\cite{Kippenberg2008,Favero2009} on radiation pressure effects in micro- and nano-scale mechanical systems~\cite{Chan2009,Eichenfield2009-zipper,Li2008,Lin2009,Wiederhecker2009,Roels2009,Safavi-Naeini2010a}. Furthermore, the concrete realization of a PPT is aided by the considerable advances made in the last decade in the theory, design and engineering of thin-film artificial quasi-2D (patterned membrane) crystal structures containing photonic~\cite{Painter1999,Painter1999a,Vuckovic2001,Takano2006,Song2005a,Notomi2005} and phononic~\cite{Olsson2009,Sanchez-Perez1998,Vasseur2007,Gu2006,Mohammadi2008a} ``band gaps''. Such systems promise unprecedented control over photons and phonons, and have been separately subject to extensive investigation. Their unification, in the form of OMCs which possesses a simultaneous phononic and photonic bandgap~\cite{Eichenfield2009,Eichenfield2009a,Safavi-Naeini2010,Pennec2010,Mohammadi2010}, and in which the interaction between the photons and phonons can be controlled, promises to further expand the capabilities of both photonic and phononic architectures and forms the basis of the proposed PPT implementation.
The outline of this paper is as follows. In Sections~\ref{sec:outline} and~\ref{sec:analysis} we introduce and study the PPT system as an abstraction, at first classically and then quantum mechanically. After introducing the basic system, its properties and its scattering matrix, we study the effects of quantum and classical noise on device operation. In Section~\ref{sec:implementation} we design and simulate a possible physical implementation of the system, utilizing recent results in simultaneous phononic-photonic bandgap materials~\cite{Safavi-Naeini2010}. Finally, in Section~\ref{sec:applications}, we demonstrate a few possible applications of the PPT. Focusing first on ``classical'' applications, we evaluate the performance of the PPT when used for the implementation of an optical delay line and wavelength converter. Finally, we show how such a system could be used in theory to do high fidelity quantum state transfer between optical and superconducting qubits.
\section{Outline of Proposed System} \label{sec:outline}
\begin{figure}[htbp]
\begin{center}
\scalebox{0.8}{\includegraphics{./Figure1.pdf}}
\caption{Full system diagram. Circles represent resonant modes, while rectangles represent waveguides. Blue is for photonics, beige is for phononics, a color-scheme followed in other parts of the paper. The coupling $h$ between the two optical modes is modulated by the intervening phonon resonance.}
\label{fig:full_system_diagram}
\end{center}
\end{figure}
The proposed PPT system, shown in Fig.~\ref{fig:full_system_diagram}, consists of a localized mechanical resonance ($b$) which couples the two optical resonances ($a_1$,$a_2$) of an optomechanical cavity via radiation pressure. External coupling to and from the mechanical resonance is provided by an acoustic waveguide, while each of the optical resonances are coupled to via separate optical waveguides. Multi-optical-mode optomechanical systems have been proposed and experimentally studied previously in the context of enhancing quantum back-action, reduced lasing threshold, and parametric instabilities~\cite{Braginsky2001,Zou2002,Ju2006,Tomes2009,Grudinin2009a,Dobrindt2010,Grudinin2010}. Here we use a two-moded optical cavity as it allows for the spatial filtering and separation of signal and pump optical beams while reducing the quantum noise in the system, as is explained below.
A general description of the radiation-pressure-coupling of the mechanical and optical degrees of freedom in such a structure is as follows. For each of the two high-$Q$ optical resonances of the cavity we associate an annihilation operator $\hat{a}_k$ and a frequency $\omega_k$ ($k=1,2$). Geometric deformation of the optomechanical cavity parameterized by $x$, changes the frequencies of the optical modes by $g_k(x)$. The deformation, due to the localized mechanical resonance with annihilation operator $\hat{b}$ and frequency $\Omega$, can be quantized and given by $\hat{x} = x_\text{ZPF} (\hat{b} + \hat{b}^\dagger)$. There is also a coupling between the two optical cavity modes given by $h(x)$, where for resonant intermodal mechanical coupling the cavity structure must be engineered such that $\Omega = \omega_2 - \omega_1$. In a traveling wave PPT device consisting of the two optical cavity resonances and a single mechanical resonance, the lower frequency cavity mode ($a_1$ in this case) is used as a 'pump' cavity which enables the inter-conversion of phonons in the mechanical resonance ($b$) to photons in the second, higher frequency, optical cavity mode ($a_2$) through a two-photon process in which pump photons are either absorbed or emitted as needed. The 'signals' representing the phonon and photon quanta to be exchanged will thus be contained in $b$ and $a_1$, respectively.
As described, the Hamiltonian of this system is,
\begin{eqnarray}
\hat{H} &=& \hbar \omega_1 \hat{a}^\dagger_1 \hat{a}_1 + \hbar \omega_2 \hat{a}^\dagger_2 \hat{a}_2 + \hbar \Omega \hat{b}^\dagger \hat{b} + \hbar(g_1 (\hat{x})\hat{a}^\dagger_1 \hat{a}_1 +g_2 (\hat{x}) \hat{a}^\dagger_2 \hat{a}_2)\nonumber\\
&& +\hbar h (\hat{x})(\hat{a}^\dagger_2 \hat{a}_1 + \hat{a}^\dagger_1 \hat{a}_2) + i \sqrt{2\kappa_{1,e}} E_{\text{pump}} (e^{-i\omega_L t}\hat{a}^\dagger_1 + e^{i\omega_Lt} \hat{a}_1),\label{eqn:full_total_Hamiltonian}
\end{eqnarray}
\noindent where we have added a classical optical pumping term of electric field amplitude $E_{\text{pump}}$ and frequency $\omega_L$. Optical pumping is performed through one of the optical waveguides with (field) coupling rate to the $a_1$ cavity resonance given by $\kappa_{1,e}$. In addition to the waveguide loading of each optical resonance ($\kappa_{k,e}$), the total optical loss rate of each cavity mode includes an intrinsic component ($\kappa_{k,i}$) of field decay due to radiation, scattering, and absorption. Similarly for the mechanical resonance, we have a field decay rate given by $\gamma = \gamma_e + \gamma_i$ which is a combination of waveguide loading and intrinsic losses. The constant parts of both $h(\hat{x})$ and $g_k(\hat{x})$ ($h(0)$, $g_k(0)$) can be eliminated by a change of basis and are thus taken to be zero. As discussed below, it is advantageous to choose a cavity structure symmetry in which $g_k(\hat{x}) = 0$ up to linear order in $\hat{x}$. In fact, we can generally assume that the mechanical displacements are small enough to make the linear order the only important term in the interaction. Assuming then a properly chosen cavity symmetry,
\begin{eqnarray}
g_k(\hat{x}) = g\cdot(\hat{b} + \hat{b}^\dagger) = 0\qquad\mbox{and}\qquad
h(\hat{x}) = h\cdot(\hat{b} + \hat{b}^\dagger),\nonumber
\end{eqnarray}
which yields a simplified Hamiltonian
\begin{eqnarray}
\hat{H} &=& \hbar \omega_1 \hat{a}^\dagger_1 \hat{a}_1 + \hbar \omega_2 \hat{a}^\dagger_2 \hat{a}_2 + \hbar \Omega \hat{b}^\dagger \hat{b}+ \hbar h (\hat{b} + \hat{b}^\dagger)(\hat{a}^\dagger_2 \hat{a}_1 + \hat{a}^\dagger_1 \hat{a}_2)\nonumber\\
&&+ i \sqrt{2\kappa_{1,e}} E_{\text{pump}} (e^{-i\omega_L t}\hat{a}^\dagger_1 + e^{i\omega_Lt} \hat{a}_1).\label{eqn:total_Hamiltonian}
\end{eqnarray}
\begin{figure}[htbp]
\begin{center}
\scalebox{0.9}{\includegraphics{./Figure2.pdf}}
\caption{Optical sidebands and processes for phonon-photon translation. The optical pump is located on sideband $\alpha_{1,0}$, on resonance with the first cavity mode at frequency $\omega_1$. There are three relevant optical sidebands to consider in the translation process, $\alpha_{2,+}$, $\alpha_{1,0}$ and $\alpha_{2,-}$. The inter-sideband photon scattering gives rise to phonon emission and absorption. The state transfer occurs through scattering between sidebands $\alpha_{2,+}$ and $\alpha_{1,0}$, whereas inter-sideband scattering between $\alpha_{1,0}$ and $\alpha_{2,-}$ can be thought of as phonon noise. Note that for the $g=0$ case considered here there are no sidebands at $\omega_1\pm\Omega$ for cavity mode $a_1$ and no sidebands at $\omega_1$ for mode $a_2$.}
\label{fig:doublecav_sidebands}
\end{center}
\end{figure}
Treating the system classically and approximately, we can write each intracavity photon and phonon amplitude, and their inputs (see Fig.~\ref{fig:doublecav_sidebands}) as a Fourier decomposition of a few relevant sidebands:
\begin{eqnarray}
a_1(t) &=& \alpha_{1,0}e^{-i\omega_1 t} + \alpha_{1,+} e^{-i(\omega_1+\Omega) t}+ \alpha_{1,-} e^{-i(\omega_1-\Omega) t} \\
a_2(t) &=& \alpha_{2,0}e^{-i\omega_1 t} + \alpha_{2,+} e^{-i(\omega_1+\Omega) t} + \alpha_{2,-} e^{-i(\omega_1-\Omega) t} \\
b(t) &=& \beta_{0} + \beta_{+} e^{-i\Omega t}\\
b_\text{in}(t) &=& \beta_{\text{in},+} e^{-i\Omega t}\\
a_\text{2,in}(t) &=& \alpha_{\text{in},+} e^{-i(\omega_1+\Omega) t}\\
b_\text{out}(t) &=&\beta_{\text{out},+} e^{-i\Omega t}\\
a_\text{2,out}(t) &=&\alpha_{\text{out},0}e^{-i\omega_1 t} + \alpha_{\text{out},+} e^{-i(\omega_1+\Omega) t}
+ \alpha_{\text{out},-}e^{-i(\omega_1-\Omega) t}.
\end{eqnarray}
The equations of motion arrived at from the system Hamiltonian (presented generally in the following section) then become algebraic relations between the $\alpha$ and $\beta$ sideband amplitudes. By ignoring the self-coupling term ($g=0$), pumping on-resonance with cavity mode $a_1$ ($\omega_L = \omega_1$), and engineering the optical cavity mode splitting for mechanical resonance ($\omega_2 = \omega_1 + \Omega$), we arrive at classical sideband amplitudes,
\begin{eqnarray}
\alpha_{1,+}&=& \alpha_{1,-} = \alpha_{2,0} = \beta_{0} = 0,\\
\alpha_{1,0} &=& \frac{\sqrt{2 \kappa_e}}{\kappa} E_{\text{pump}} + O(h\alpha_{k,\pm} \beta_+), \\
\alpha_{2,+} &=& -\frac{i h \alpha_{1,0}}{\kappa}\beta_{+} - \frac{\sqrt{2 \kappa_e}}{\kappa} \alpha_{\text{in},+},\\
\alpha_{2,-} &=& -\frac{i h \alpha_{1,0}}{\kappa+2 i \Omega}\beta_{+}^\ast,\\
\beta_{+} &=& -\frac{i h \alpha_{1,0}^\ast}{\gamma}\alpha_{2,+} - \frac{\sqrt{2\gamma_e}}{\gamma} \beta_{\text{in},+} -\frac{i h \alpha_{1,0}}{\gamma}\alpha_{2,-}^\ast.
\end{eqnarray}
From here we see that the central sideband amplitude of cavity mode $a_1$, $\alpha_{1,0}$, is proportional to the sum of a term containing the pump field $E_\text{pump}$ and terms containing products of the optical and mechanical sideband amplitudes. By increasing $E_\text{pump}$ the effect of the other sidebands on the pump resonance amplitude can be made negligible, and we assume here and elsewhere in this work that the pump sideband is generally left unaffected by the dynamics of the rest of the system. As desired the optical sideband which contains mechanical information is $\alpha_{2,+}$ since it is the only sideband directly proportional to $\beta_+$. The constant of proportionality between these two terms is seen to contain both $h$ and $\alpha_{1,0}$, demonstrating the role of the pump beam in the conversion process. Since coherent information transfer between the optics and mechanics is occurring between $\beta_+$ and $\alpha_{2,+}$, it is desirable to remove the effects of the lower energy photonic sideband, $\alpha_{2,-}$. This sideband can be made significantly smaller in magnitude than $\alpha_{2,+}$ in the sideband-resolved regime where $\Omega \gg \kappa$. A convenient way to visualize all of the processes in the system is shown in Figure~\ref{fig:doublecav_sidebands} where the photonic sideband amplitudes $\alpha_{2,\pm}$ and $\alpha_{1,0}$ are represented as ``energy levels'', with transitions between them being due to the emission and absorption of phonons.
From this approximate analysis it is clearly suggested that in a sideband resolved optomechanical system, a state-transfer process is possible between the phononic and photonic resonances, and the process is controlled by a pump beam~\cite{Zhang2003}. A more in-depth study of the system dynamics required to understand how such processes may be used for traveling wave phonon-photon conversion, and a full investigation of all relevant noise sources required to understand the applicability of such a system to quantum information, is carried out in the following section.
\section{Analysis} \label{sec:analysis}
A detailed treatment of the operation of a traveling phonon-photon translator is carried out in this section. At first, the dynamics of the system are simplified while still taking into account the noise processes related to the sideband $\alpha_{2,-}$. In this way one is left with an effective `beam-splitter' Hamiltonian, which describes the coherent interaction between the optics and mechanics, while the aforementioned noise processes are accounted for through an effective increase in the thermal bath temperature. This is followed by a treatment of the traveling-wave problem through a scattering matrix formulation, which provides insight into the role of the intracavity pump photon number ($|\alpha_{1,0}|^2$) and optimizing the state transfer efficiency.
\subsection{Simplified Dynamics of the System}
Starting from the Hamiltonian in equation~(\ref{eqn:total_Hamiltonian}), a set of Heisenberg-Langevin Equations can be written down,~\cite{GardinerZoller}:
\begin{eqnarray}
\dot{\hat{a}}_1 &=& - (\kappa_1 + i \omega_1) \hat{a}_1
- i h (\hat{b}+\hat{b}^\dagger) \hat{a}_2
+ \sqrt{2\kappa_{1,e}} E_{\text{pump}} e^{-i\omega_L t} - \sqrt{2\kappa_1} \hat{a}^\prime_\text{1,in}, \\
\dot{\hat{a}}_2 &=& - (\kappa_2 + i \omega_2 ) \hat{a}_2 -
i h (\hat{b}+\hat{b}^\dagger) \hat{a}_1
- \sqrt{2\kappa_2} \hat{a}^\prime_\text{2,in}, \\
\dot{\hat{b}} &=& -i \Omega \hat{b} -i h (\hat{a}^\dagger_2 \hat{a}_1 + \hat{a}^\dagger_1 \hat{a}_2)
- \gamma \hat{b} + \gamma \hat{b}^\dagger - \sqrt{2\gamma} \hat{b}^\prime_\text{in}.
\end{eqnarray}
The input coupling terms as written above include both external waveguide coupling and intrinsic coupling due to lossy channels (see Fig.~\ref{fig:full_system_diagram}). Separated, the intrinsic (with subscript $i$) and extrinsic (with no subscript) components look as follows,
\begin{eqnarray}
\sqrt{2\kappa_k} \hat{a}^\prime_\text{k,in} &=& \sqrt{2\kappa_{k,e}} \hat{a}_\text{k,in} + \sqrt{2\kappa_{k,i}} \hat{a}_\text{k,in,i},\\
\sqrt{2\gamma} \hat{b}^\prime_\text{in} &=& \sqrt{2\gamma} \hat{b}_\text{in} + \sqrt{2\gamma_i} \hat{b}_\text{in,i},\\
\kappa_k &=& \kappa_{k,i} + \kappa_{k,e},\\
\gamma &=& \gamma_i + \gamma_e.
\end{eqnarray}
As the fluctuations in the fields are of primary interest, each Heisenberg operator can be rewritten as a fluctuation term around a steady-state value,
\begin{eqnarray}
\hat{a}_1(t) &\rightarrow& (\alpha_1 +\hat{a}_1)e^{-i\omega_1 t},\\
\hat{a}_2(t) &\rightarrow& (\alpha_2 + \hat{a}_2)e^{-i\omega_2 t},\\
\hat{b}(t) &\rightarrow& (\beta + \hat{b})e^{-i\Omega t}.
\end{eqnarray}
Assuming that the pump beam is driven resonantly with $a_{1}$ ($\omega_L = \omega_1$), the $c$-number steady-state values are equal to $(\alpha_1, \alpha_2, \beta) = (\frac{\sqrt{2 \kappa_{1,e}}}{\kappa} E_{\text{pump}},0,0)$.
For the fluctuation dynamics, with $\Delta \equiv (\omega_{2}-\omega_{1}) - \Omega$, the resulting equations are:
\begin{eqnarray}
\dot{\hat{a} }_1 &=& - \kappa_1 \hat{a}_1 - i h \hat{a}_2 (\hat{b} e^{-i 2 \Omega t} + \hat{b}^\dagger) e^{-i\Delta t} -\sqrt{2\kappa} \hat{a}^\prime_\text{1,in}e^{i\omega_1 t} \label{eqn:HL_full_a1dot} \\
\dot{\hat{a} }_2 &=& - \kappa_2 \hat{a}_2 - i h ( \alpha_1 + \hat{a}_1) (\hat{b} + \hat{b}^\dagger e^{+i 2 \Omega t} )e^{+i\Delta t} -\sqrt{2\kappa} \hat{a}^\prime_\text{2,in}e^{i \omega_2 t} \label{eqn:HL_full_a2dot} \\
\dot{\hat{b}} &=&-\gamma (\hat{b} - \hat{b}^\dagger e^{+i 2 \Omega t} )-i h ( \alpha_1 + \hat a_1 ) \hat a^\dagger_2 e^{+i(\Delta + 2 \Omega)t}
\nonumber \\
&&- i h ( \alpha_1 + \hat a_1 )^\dagger \hat{a}_2 e^{-i\Delta t} - \sqrt{2\gamma} \hat{b}^\prime_\text{in}e^{i\Omega t} \label{eqn:HL_full_bdot}
\end{eqnarray}
Ignoring all the mechanically anti-resonant terms for now, and invoking the rotating wave approximation (RWA) valid when $\Delta \ll \Omega$ and $\Omega \gg |\alpha_1| h$, we arrive at the simplified set of fluctuation equations,
\begin{eqnarray}
\dot{\hat{a} }_1 &=& - \kappa_1 \hat{a}_1 - i h \hat{a}_2 \hat{b}^\dagger e^{-i\Delta t} -\sqrt{2\kappa} \hat{a}^\prime_\text{1,in} e^{i\omega_1 t}\\
\dot{\hat{a} }_2 &=& - \kappa_2 \hat{a}_2 - i h(\tilde \alpha_1 + \hat{a}_1) \hat{b}e^{+i\Delta t} -\sqrt{2\kappa} \hat{a}^\prime_\text{2,in} e^{i \omega_2 t} \\
\dot{\hat{b}} &=& -\gamma \hat{b} - i h(\tilde \alpha_1 + \hat a_1 )^\dagger \hat{a}_2 e^{-i\Delta t} -\sqrt{2\gamma} \hat{b}^\prime_\text{in}e^{i\Omega t}
\end{eqnarray}
\label{ss:spont_emission}
By ignoring all of the counter-rotating terms proportional to $e^{+i2\Omega t}$, we have also neglected the noise processes alluded to previously due to the $\alpha_{2,-}$ sideband. Of the mechanically anti-resonant terms which have been dropped, the terms proportional to $\alpha_1$ ($h \alpha_1\hat{b}^\dagger e^{+i 2 \Omega t}$ in equation~(\ref{eqn:HL_full_a2dot}) and $h \alpha_1 \hat a^\dagger_2 e^{+i 2 \Omega t}$ in equation~(\ref{eqn:HL_full_bdot})) are the largest and most significant in terms of error in the rotating wave approximation. These terms correspond to $\hat{a}_1 \hat{a}^\dagger_2 \hat{b}^\dagger$ + h.c. in the Hamiltonian and cause inter-sideband photon scattering between the pump, $\alpha_{1,0}$, and its lower frequency sideband, $\alpha_{2,-}$ as shown in Figure~\ref{fig:doublecav_sidebands}. This inter-sideband scattering process causes emission and absorption of phonons in the mechanical part of the PPT, thus in principle, even when the extrinsic phonon inputs are in the vacuum state, spontaneous scattering of photons from $\alpha_{1,0}$ to $\alpha_{2,-}$ may populate the mechanical cavity with a phonon.
This effect was studied in Refs.~\cite{Wilson-Rae2007,marquardt07} in the context of quantum limits to optomechanical cooling. Similar to that work, a master equation for the phononic mode with the $\omega_1-\Omega$ optical sideband adiabatically eliminated results in an additional \emph{phononic} spontaneous emission term given by,
\begin{equation}
\dot{\rho}_\text{b,spon} = \frac{G^2}{\kappa}\frac{1}{\left(\frac{2 \Omega}{\kappa}\right)^2 + 1}( 2 \hat{b}^\dagger \rho \hat{b} - \hat{b} \hat{b}^\dagger \rho - \rho \hat{b} \hat{b}^\dagger )
\end{equation}
where $G = h |\tilde \alpha^{\mathrm{ss}}_1|$. The master equation for the mechanics, found by tracing over all bath and optical variables, is of the form $\dot{\rho} = \dot{\rho}_\text{b,i} +\dot{\rho}_\text{b,spon} +\dot{\rho}_\text{b,e}-i/\hbar[H_b,\rho]$, where the terms on the right hand side of the equation are respectively the intrinsic phononic loss, the phonon spontaneous emission, the phonon-waveguide coupling, and the coherent evolution of the system. The first two terms can be lumped together into an effective intrinsic loss, $\dot{\rho}^\prime_\text{b,i}=\dot{\rho}_\text{b,i} +\dot{\rho}_\text{b,spon}$,
\begin{eqnarray}
\dot{\rho}^\prime_\text{b,i} &=& \gamma_i^\prime (\bar{n}^\prime + 1) ( 2 \hat{b} \rho \hat{b}^\dagger - \hat{b}^\dagger \hat{b} \rho - \rho \hat{b}^\dagger \hat{b})
+ \gamma_i^\prime \bar{n}^\prime ( 2 \hat{b}^\dagger \rho \hat{b} - \hat{b} \hat{b}^\dagger \rho - \rho \hat{b} \hat{b}^\dagger ),\label{eqn:effective_louivillian} \\
\gamma_i^\prime &=& \gamma_i (1-n_\text{spon}), \label{eqn:gamma_prime} \\
\bar{n}^\prime &=& \frac{\bar{n} + n_\text{spon}}{1-n_\text{spon}} \label{eqn:n_prime},
\end{eqnarray}
and
\begin{equation}
n_\text{spon} \equiv \frac{G^2}{\kappa \gamma_i} \frac{1}{\left(\frac{2 \Omega}{\kappa}\right)^2 + 1}.
\end{equation}
Hence, by assuming that the intrinsic loss phonon bath is at a modified temperature with occupation number $\bar{n}^\prime$, and changing the intrinsic phonon loss rate to $\gamma^\prime_i$, the spontaneous emission and intrinsic loss noise are lumped into one effective thermal noise Liouvillian for the mechanics. Note that it is possible in this model to have $\gamma^\prime_i$ negative when $n_\text{spon}>1$; however, the motional decoherence rate is always positive and given by $\gamma_i^\prime \bar{n}^\prime = \gamma_i (\bar{n} + n_\text{spon})$.
In Section~\ref{ss:scattering_matrix} we will see that the optimal state transfer efficiency is given by $G^2 = \gamma \kappa$ \footnote{The $\gamma$ in this case is in principle $\gamma_e + \gamma^\prime_i$ as opposed to $\gamma_e + \gamma_i$, and the equations must therefore be solved self-consistently. This is discussed in section~\ref{ss:spont_mod}.} in which case\begin{equation}
n_\text{spon} \approx \frac{\gamma}{\gamma_i} \left(\frac{\kappa}{2 \Omega}\right)^2,
\end{equation}
\noindent in the sideband resolved limit. In the case where $g \ne 0$ (recall this is the self-coupling radiation pressure term), photons scattering from the pump into the lower frequency sideband ($\omega_1 - \Omega$) can scatter into the $a_{1}$ cavity mode which is only detuned by $\Omega$. This is to be compared with the $g=0$ case considered thus far in which the detuning is $2\Omega$ for scattering into the $a_{2}$ mode. As such, for the $g \ne 0$ case there will be roughly four times the spontaneous emission noise, with $n^{g\ne0}_\text{spon} \approx {\gamma}/{\gamma_i} ({\kappa}/{\Omega})^2$.
A final simplification can be made by neglecting the fluctuations in the strong optical pump of cavity mode $a_{1}$. Considering that the fluctuations in the variables are all of the same order, and that $ \hat{a}_1$ always appears as $\alpha_1 + \hat{a}_1$ in the equations of motion for $\hat{b}$ and $\hat{a}_2$, we can ignore the dynamics of the pump fluctuations in the case where $|\alpha_1| \gg \avg{\op{a}{1}},\avg{\op{a}{2}}$ and $\avg{\op{b}{}}$. This is the undepleted pump approximation. Adiabatically removing the pump from the dynamics of the system yields a pump-enhanced optomechanical coupling $G = h |\alpha_1|$ between optical cavity mode $a_{2}$ and the mechanical resonance $b$. Dropping the subscript from the cavity mode $a_{2}$ and moving to a rotating reference frame results in the new effective Hamiltonian~\cite{Zhang2003},
\begin{equation}
\hat{H}_{\mathrm{eff}} = -\Delta \hat{b}^\dagger \hat{b}+
G \hat{a}^\dagger \hat{b} + G^\ast \hat{a} \hat{b}^\dagger.
\label{eqn:H_eff_start}
\end{equation}
\noindent The system diagram and symbol corresponding to this simplified model of the PPT are shown in Figure~\ref{fig:simplified_system_diagram}.
\begin{figure}[htbp]
\begin{center}
\scalebox{1}{\includegraphics{./Figure3.pdf}}
\caption{Simplified PPT system diagram and symbol. The coupling rate $G$ is proportional to $h$ and $\sqrt{n_1}$, where $n_1$ is the number of photons in the pump mode. }
\label{fig:simplified_system_diagram}
\end{center}
\end{figure}
\subsection{Scattering Matrix Formulation of the Phonon-Photon Translator}\label{ss:scattering_matrix}
To understand the properties of the PPT as a waveguide adapter, we begin with a study of its scattering matrix. Starting from the effective Hamiltonian given in equation~(\ref{eqn:H_eff_start}), the Heisenberg-Langevin equations of motion for the Hamiltonian~(\ref{eqn:H_eff_start}) are written under a Markov approximation in the frequency domain,
\begin{eqnarray}
- i \omega \tilde{a}(\omega) &=& -\kappa \tilde{a}(\omega) - i G \tilde{b}(\omega) - \sqrt{2 \kappa_e} \atlss{in} - \sqrt{2 \kappa_i} \atlss{in,i}, \label{eqn:freqdom1}\\
- i \omega \tilde{b}(\omega) &=& -(\gamma - i \Delta) \tilde{b}(\omega) - i G^\ast \tilde{a}(\omega) - \sqrt{2 \gamma_e} \btlss{in} - \sqrt{2 \gamma_i} \btlss{in,i}\label{eqn:freqdom2}.
\end{eqnarray}
The intrinsic noise terms $\atlss{in,i}$ and $\btlss{in,i}$ are the initial-state boson annihilation operators for the baths, while the extrinsic terms $\atlss{in}$ and $\btlss{in}$ are annihilation operators for the optical and mechanical guided modes for each respective waveguide. Since the effective Hamiltonian (\ref{eqn:H_eff_start}) has been used to derive equations (\ref{eqn:freqdom1}-\ref{eqn:freqdom2}), thus neglecting the counter-rotating terms present in the full system dynamics, the effects of phonon spontaneous emission noise is included separately. Following the discussion in Section~\ref{ss:spont_emission}, the effective Liouvillian~(\ref{eqn:effective_louivillian}) corresponds to replacing $\gamma_i$ with $\gamma_i^\prime$ in (\ref{eqn:freqdom2}) and using a Langevin force $\btlss{in,i}$ satisfying the relations,
\begin{eqnarray}
\avg{\btlssd{in,i}\btlssp{in,i}} &=&\bar{n}^\prime \delta(\omega-\omega^\prime), \\
\avg{\btlssp{in,i}\btlssd{in,i}} &=& (\bar{n}^\prime+1) \delta(\omega-\omega^\prime),
\end{eqnarray}
where $\gamma_i^\prime$ and $\bar{n}^\prime$ are given in equations (\ref{eqn:gamma_prime}-\ref{eqn:n_prime}). The intrinsic optical noise correlations are only due to vacuum fluctuations and given by $\avg{\atlssp{in,i}\atlssd{in,i}} = \delta(\omega-\omega^\prime)$.
To ensure efficient translation, competing requirements of matching and strong coupling between waveguide and resonator must be satisfied. This is similar to the problem of designing integrated optical filters using resonators and waveguides~\cite{Manolatou1999,Barclay2003}. From the above equations and the input-output boundary condition ~\cite{Gardiner1993,GardinerZoller}, we arrive at the matrix equation,
\begin{eqnarray}
\twovec{\atlss{out}}{\btlss{out}} &=& \mathcal{S} \twovec{\atlss{in}}{\btlss{in}} + \mathcal{N}\twovec{\atlss{in,i}}{\btlss{in,i}},
\end{eqnarray}
with scattering and noise matrices
\begin{equation}
\mathcal{S} = \twotwomat{s_{11}(\omega)}{s_{12}(\omega)}{s_{21}(\omega)}{s_{22}(\omega)}\qquad\mbox{and}\qquad
\mathcal{N} = \twotwomat{n_{11}(\omega)}{n_{12}(\omega)}{n_{21}(\omega)}{n_{22}(\omega)}.
\end{equation}
The elements of the scattering matrix $\mathcal{S}$ are
\begin{eqnarray}
s_{11}(\omega) &=& 1 - \frac{ 2 \kappa_e (\gamma -i(\Delta + \omega)) }{|G|^2 + (\gamma - i(\omega+\Delta))(\kappa - i \omega)},\\
s_{12}(\omega) &=& \frac{ 2 i G^\ast \sqrt{\gamma_e \kappa_e}}{ |G|^2 + (\gamma - i(\omega+\Delta))(\kappa - i \omega)},\\
s_{21}(\omega) &=& s_{12}^\ast (\omega),\\
s_{22}(\omega) &=& 1 - \frac{ 2 \gamma_e (\kappa -i\omega) }{|G|^2 + (\gamma - i(\omega+\Delta))(\kappa - i \omega)}.
\end{eqnarray}
Similar expressions are also found for the noise scattering matrix elements $n_{ij}(\omega)$, with their extrema reported below.
\begin{figure}[htbp]
\begin{center}
\scalebox{1}{\includegraphics{./Figure4.pdf}}
\caption{Phonon-photon scattering matrix amplitudes for $(\gamma_e,\gamma_i,\kappa_e,\kappa_i,G)=2\pi\times(10,1,2000,200,155.6) ~\text{MHz}$. (a) Plot of the frequency dependence of the optical reflection $s_{11}$. The broad over-coupled optical line is visible, along with the PPT feature near the center in the unshaded region. This unshaded region is shown in more detail in plots (b), (c), (d), and (e), showing the frequency dependence of the scattering matrix elements $s_{11}$, $s_{12}$, $s_{21}$ and $s_{22}$, respectively. In each plot, the curves (\textcolor{blue}{$-$}), (\textcolor{green}{$--$}) and (\textcolor{red}{$\cdot -$}) represent detunings of $\Delta = 0, 2\pi\times 200$ and $2\pi\times 400$ MHz respectively. }
\label{fig:scattering_matrix}
\end{center}
\end{figure}
In order to obtain efficient conversion the cavities must be over-coupled to their respective waveguides, ensuring that the phonon (photon) has a higher chance of leaking into the waveguide continuum modes than escaping into other loss channels. In this regime, $\kappa \approx \kappa_e$ and $\gamma \approx \gamma_e$. In the weak coupling regime, $G < \kappa$, the response of the system exhibits a maximum for $s_{12}$ and $s_{21}$ at $\omega=0$ and a minimum at the same point for $s_{11}$ and $s_{22}$. In fact, with realistic system parameters, only the weak-coupling regime leads to efficient translation. In strong-coupling ($G > \kappa$ and $\kappa \gg \gamma$) the photon is converted to a phonon at rate $G$, and then back to a photon before it has a chance to leave through the much slower phononic loss channel at rate $\gamma$, causing there to be significant reflections and reduced conversion efficiency. To find the optimal value of $G$ we consider the extrema given by
\begin{eqnarray}
|s_{11}|_\text{min} &=& \left| \frac{G^2 + \gamma \kappa_i - \gamma \kappa_e}{G^2 + \gamma \kappa_i + \gamma \kappa_e}\right|,\\
|s_{12}|_\text{max} &=& \left| \frac{2 G \sqrt{\gamma_e \kappa_e}}{G^2 + \gamma \kappa}\right|,\\
|s_{22}|_\text{min} &=& \left| \frac{G^2 + \kappa \gamma_i - \kappa \gamma_e}{G^2 + \kappa \gamma_i + \kappa \gamma_e}\right|.
\end{eqnarray}
In the over-coupled approximation and in the case where $\kappa_i=\gamma_i=0$, it is easy to see that the full translation condition $|s_{12}|_\text{max} = 1$ is achievable by setting $G$ equal to
\begin{equation}
G^\text{o} = \sqrt{\gamma \kappa}\label{eqn:matching_condition}.
\end{equation}
This result has a simple interpretation as a matching requirement. The photonic loss channel viewed from the phononic mode has a loss rate of $\frac{G^2}{\kappa}$. Matching this to the purely mechanical loss rate of the same phononic mode, $\gamma$, one arrives at $G^\text{o} = \sqrt{\gamma \kappa}$. The same argument can be used for the photonic mode, giving the same result. Under this matched condition, the linewidth of the translation peak in $|s_{12}|^2$ is simply
\begin{equation}
\gamma_\text{transfer} = \frac{4 |G^\text{o}|^2}{\kappa} = 2\gamma. \label{eqn:gamma_transfer}
\end{equation}
With intrinsic losses taken into account, either $|s_{11}|$ or $|s_{22}|$ (but not both) can be made exactly 0 by setting $G^2 = \gamma(\kappa_e - \kappa_i)$ or $G^2 = (\gamma_e - \gamma_i)\kappa$, respectively. The optimal state transfer condition, however, still occurs for $G^\text{o} = \sqrt{\gamma \kappa}$. The extremal values ($\omega = 0$) of the scattering matrix are in this case are,
\begin{eqnarray}
|s_{11}|^{\text{optimal}}_{\omega=0} &=& \frac{\kappa_i}{\kappa_e + \kappa_i}, \label{eqn:s11}\\
|s_{12}|^{\text{optimal}}_{\omega=0}&=& \sqrt{\frac{\gamma_e \kappa_e}{\gamma \kappa}},\\
|s_{22}|^{\text{optimal}}_{\omega=0} &=& \frac{\gamma_i}{\gamma_e + \gamma_i},
\end{eqnarray}
\noindent with corresponding noise matrix elements of,
\begin{eqnarray}
|n_{11}|^{\text{optimal}}_{\omega=0} &=& \frac{\sqrt{\kappa_i \kappa_e}}{\kappa_e + \kappa_i},\label{n_mat_1}\\
|n_{12}|^{\text{optimal}}_{\omega=0}&=& \sqrt{\frac{\gamma_i \kappa_e}{\gamma \kappa}},\label{n_mat_2}\\
|n_{21}|^{\text{optimal}}_{\omega=0}&=& \sqrt{\frac{\gamma_e \kappa_i}{\gamma \kappa}},\label{n_mat_3}\\
|n_{22}|^{\text{optimal}}_{\omega=0}&=& \frac{\sqrt{\gamma_i \gamma_e}}{\gamma_e + \gamma_i}.\label{n_mat_4}
\end{eqnarray}
For a set of parameters typical of an optomechanical crystal system, the magnitudes of the scattering matrix elements versus frequency are plotted in Figure~\ref{fig:scattering_matrix}. In these plots we have assumed resonant optical pumping of the $a_{1}$ cavity mode and considered several different detuning values $\Delta$. The normalized optical reflection spectrum ($|s_{11}|^2$) is shown in Figure~\ref{fig:scattering_matrix}(a), in which the broad optical cavity resonance can be seen along with a deeper, narrowband resonance that tunes with $\Delta$. This narrowband resonance is highlighted in Figure~\ref{fig:scattering_matrix}(b), showing that the optical reflection is nearly completely eliminated on resonance. Photons on resonance, instead of being reflected, are being converted into outgoing phonons as can be seen in resonance peak of $|s_{21}|^2$ shown in Figure~\ref{fig:scattering_matrix}(d). A similar reflection dip and transmission peak is visible for the phononic reflection ($s_{22}$) and phonon to photon translation ($s_{12}$) curves. It is also notable from Figure~\ref{fig:scattering_matrix} that for small detunings $\Delta$ of the system ($\Delta < \kappa, \Omega$) the resonant scattering matrix elements are only weakly affected and the translation process maintains its efficiency.
\subsubsection{Modifications to Matching Condition due to Counter-Rotating Terms}\label{ss:spont_mod}
As a consequence of the counter-rotating terms treated in Section~\ref{ss:spont_emission}, $\gamma$ is weakly dependent on $G$. In particular, by making the substitution $\gamma_i \rightarrow \gamma_i^\prime$, where $\gamma_i^\prime$ is given by equation~(\ref{eqn:gamma_prime}), the equation for the optimal $G$ becomes $\frac{|G^\text{o}|^2 }{\kappa}=\gamma_e + \gamma_i^\prime$, where $\gamma_i^\prime$ is itself dependent on $G$. Algebraic manipulations give us the desired value of $G$,
\begin{equation}
|G^\text{o}|^2 = \left(\gamma \kappa \right)\frac{\left(1+\left(\frac{2\Omega}{\kappa}\right)^2\right)}{\left(2+\left(\frac{2\Omega}{\kappa}\right)^2\right)}, \label{eqn:Go_mod}
\end{equation}
simplifying in the sideband-resolved limit to $G^\text{o} = \sqrt{\gamma \kappa}\left(1 - \left(\frac{\kappa}{2\Omega}\right)^2\right).$ The values for the scattering matrix elements given in equations (\ref{eqn:s11}-\ref{n_mat_4}) are also suitably modified by the substitutions $\gamma_i \rightarrow \gamma_i^\prime$ and $\gamma \rightarrow \gamma_e + \gamma_i^\prime$.
\subsubsection{Phononic Waveguide Losses}\label{app:waveguide_loss}
The issue of waveguide loss is one that is normally ignored in quantum optical systems where low-loss fiber or free-space links are readily available. On the phononic side of the systems studied here, the length of the waveguide and its intrinsic losses may be large enough for the waveguide attenuation factor to become important. Because of the negative effect of attenuation in applications involving optomechanical delay lines, and quantum state transfer, it is useful to model this loss and see how the scattering matrix elements are altered.
A model of the system shown in Figure~\ref{fig:waveguide-loss}(b), where the lossy waveguide is replaced by a single beam splitter. This is accomplished by first modeling the lossy waveguide as a large number $N$ of cascaded beam splitters, each reflecting $2\alpha \Delta z$ away from the main beam, with $L = \Delta z N$. This serial array of beam splitters can be combined into one, with reflectivity $\eta = e^{-2\alpha L}$ as $N\rightarrow \infty$.
The relation for $\hat{d}_\text{out}$ can be found by starting from the approximate scattering matrix relation for $\hat{b}_\text{out}$,
\begin{eqnarray}
\hat{b}_\text{out}(t) &=& s_{21}\hat{a}_\text{in}(t) + s_{22} \hat{b}_\text{in}(t) + n_{12} \hat{a}_\text{in,i}(t)+ n_{22}\hat{b}_\text{in,i}(t),
\end{eqnarray}
and using the beam splitter relations,
\begin{eqnarray}
\hat{d}_\text{out}(t) &=& \sqrt{\eta} ~b_\text{out} + \sqrt{1-\eta}~b_\text{wg,i,1}\\
\hat{b}_\text{in}(t) &=& \sqrt{\eta} ~d_\text{in} + \sqrt{1-\eta}~b_\text{wg,i,2}.
\end{eqnarray}
Assuming that the phonon cavity intrinsic loss bath is at the same temperature and uncorrelated with the phonon waveguide intrinsic loss baths, then one finds for translation through the lossy phonon waveguide,
\begin{eqnarray}
\hat{d}_\text{out}(t) &=& s^\prime_{21}\hat{a}_\text{in}(t) + s^\prime_{22} \hat{d}_\text{in}(t) + n^\prime_{12} \hat{a}_\text{in,i}(t)+ n^\prime_{22}\hat{b}_\text{in,i}(t),
\end{eqnarray}
where
\begin{eqnarray}
s^\prime_{21} &=&\sqrt{\eta}~s_{21},\\
s^\prime_{22} &=&\eta~s_{22},\\
n^\prime_{12} &=&\sqrt{\eta}~n_{12},\\
n^\prime_{22} &=&\sqrt{(1-\eta)\eta |s_{22}|^2 + \eta|n_{22}|^2 +1-\eta}.
\end{eqnarray}
The value of $s_{21}$ is simply reduced by a factor $\sqrt{\eta}$ due to the lossy waveguide. For propagation lengths short relative to the attenuation length of the lossy waveguide, $\alpha L \ll 1$, and the reduction in translation $\sqrt{\eta} \approx 1-\alpha L$ is small. The added noise due to the waveguide attenuation is contained in $n^\prime_{22}$, and is also seen to be small for $\alpha L \ll 1$.
\begin{figure}[htbp]
\begin{center}
\scalebox{1}{\includegraphics{./Figure5.pdf}}
\caption{(a) System diagram of a PPT connected to a lossy waveguide of length $L$ with attenuation factor $\alpha$. (b) An equivalent system diagram in which the waveguide is replaced by a beam splitter with a reflectivity $\eta$. }
\label{fig:waveguide-loss}
\end{center}
\end{figure}
\subsection{Effects of Thermal and Quantum Noise}\label{ss:noise}
In general, the type of noise which is relevant to the PPT depends on the conditions in which it is used. For example, when used as a bridge between RF-microwave photonics and optics for classical applications at room temperature, the thermal noise affecting the RF signal will be at a level which makes the quantum noise induced by spontaneous pump scattering irrelevant. On the other hand, when the system is used at cryogenic temperatures for connecting a superconducting circuit QED system to an optical system as described below, the quantum noise of the translation process itself becomes dominant. In what follows we analyze both sources of noise.
\subsubsection{Thermal Noise on the Optical Side}
The noise power on each of the output waveguide channels can be found by using the scattering matrix formulation described in Section~\ref{ss:scattering_matrix}. Here we evaluate the effects of thermal noise classically. When the system is used an optical drop filter, in which an optical beam is sent in and the optical reflection is measured, we find:
\begin{eqnarray}
a_\text{out}(t) &=& s_{11}(t) \convolution a_\text{in}(t) + s_{12}(t) \convolution b_\text{in}(t) \nonumber\\
&& + n_{11}(t) \convolution a_\text{in,i} (t)+n_{12}(t) \convolution b_\text{in,i}(t)
\end{eqnarray}
where $\convolution$ represents convolution. Assuming that there are no cross-correlations between the various input noise terms, we find the classical spectral density of the noise to be:
\begin{eqnarray}
S_\text{out}(\omega) = |s_{12}(\omega)|^2 \bar{n} + |n_{12}(\omega)|^2 (\bar{n} + n_\text{spon}).\label{eqn:Sout1}
\end{eqnarray}
For a system with mechanical frequency less than $10$~GHz, at room temperature ($T=300~\text{K}$) the corresponding spontaneous emission noise is much less than the thermal noise, $n_\text{spon} \ll \bar{n} \approx 10^3$, and $n_\text{spon}$ can be ignored.
Ignoring the quantum noise term for the moment, and evaluating equation (\ref{eqn:Sout1}) at $\omega = 0$ after substituting in equations (\ref{n_mat_1}-\ref{n_mat_4}), the total thermal noise power on the reflected optical signal is found to be,
\begin{equation}
P^N_{\text{o,out}} = \hbar \omega_2 \bar{n} \frac{\kappa_e}{\kappa} \pi B, \label{eqn:P_nbar}
\end{equation}
where $B=2\gamma$ is the bandwidth of the PPT. In the classical regime, $\bar{n} = kT_\text{bath} / (\hbar \Omega)$, and this equation reduces to $ P^N_{\text{o,out}} = 2\pi \gamma kT_\text{bath}({\omega_2}/{\Omega})({\kappa_e}/{\kappa}).$
This result has a simple interpretation. The noise power $\pi\gamma kT_\text{bath}$ is the standard thermal noise input on the phononic side of the PPT. The ratio $\omega_2/\Omega$ is a translation factor, arising from the fact that quantum-limited conversion of phonons to photons causes an increase in energy by the factor of the ratio of their frequencies. Finally, the factor $\kappa_e/\kappa$ is the extraction efficiency of photons from the optical mode to the waveguide.
Alternatively, one may define an equivalent optical temperature by setting $\pi \kappa_e k T_\text{o,eff} = P^N_{\text{o,out}}$, yielding $T_\text{o,eff} = ({2 Q_o}/{Q_m})T_\text{bath}$, where the $Q$'s represent the loaded quality factors of the optical and mechanical resonators. This last expression must be interpreted carefully, only in terms of a power equivalence, as spectrally the noise on the optical side of the PPT has a bandwidth of $2\gamma$, while thermal noise radiating from the cavity would have a bandwidth of $\kappa$.
\subsubsection{Phonon Spontaneous Emission Noise}
To calculate the effective increase in noise brought about by the spontaneous pump scattering and phonon emission process, we start from equation~(\ref{eqn:Sout1}) and use the spontaneous emission contribution $n_\text{spon}$ to find,
\begin{equation}
P^N_{\text{o,out}} = \left(\hbar \omega_2\right) \left(\bar{n}+ \frac{1}{\left(\frac{2 \Omega}{\kappa}\right)^2 + 1}\right)\left(\frac{\kappa_e}{\kappa}\right) \left(2\pi \gamma\right).
\end{equation}
The second term in the brackets is due to the spontaneous emission of phonons by the optical pump beam, and is added to the thermal noise exiting the optical side of the PPT. Put in terms of an effective contribution to the thermal \emph{photon} occupation number, the spontaneous pump scattering effectively adds,
\begin{eqnarray}
n_\text{o,spon} = \frac{2 \gamma}{\kappa} \left(\frac{\kappa}{2 \Omega}\right)^2,
\end{eqnarray}
\noindent thermal photons to the cavity. This equivalence is only in terms of total noise power emitted, as spectrally the noise is emitted over the $2\gamma$ bandwidth of the PPT resonance, not the entire optical cavity resonance as discussed above.
\section{Proposed On-Chip Implementation} \label{sec:implementation}
Up to this point in the analysis of the PPT, the discussion has been kept as general as possible. Such a system is, however, interesting only insofar as it is realizable, and we attempt here to establish the practicality of a PPT. Building upon recent experimental~\cite{Eichenfield2009} and theoretical work~\cite{Eichenfield2009a,Safavi-Naeini2010}, we provide the design of an optomechanical crystal formed in a silicon microchip that can realize a PPT system with high phonon-photon translation efficiency.
As previously mentioned, optomechanical crystals~\cite{Eichenfield2009} are engineered structures in which phonons and photons may be independently routed and their interactions controlled. In order to create a suitable OMC structure for the implementation of a PPT device one looks to a crystal lattice providing simultaneous phononic and photonic bandgaps for the guiding and co-localization of phonons and photons~\cite{Maldovan2006}. We have recently proposed~\cite{Safavi-Naeini2010} such an OMC system, formed from a silicon-on-insulator wafer and consisting of a patterned thin membrane of silicon. The proposed ``snowflake'' crystal lattice supports a phononic bandgap in the $5$-$10$~GHz mechanical frequency band and a photonic bandgap in the $1500$~nm optical wavelength band. This quasi-2D crystal structure was also shown to support low-loss photonic and phononic waveguides, optical resonances with radiation-limited $Q > 10^6$ co-localized with mechanical resonances of frequency $\Omega \approx 2\pi\times 10~\text{GHz}$, and a single quanta optomechanical interaction rate of $g \approx 2\pi \times 300~\text{kHz}$.
In this section we design an example OMC implementation of a PPT in the silicon snowflake crystal. We limit ourselves here to a two-dimensional (2D) crystal involving only 2D Maxwell's equations for transverse-electric (TE) polarized optical waves and in-plane elastic deformations of an infinitely thick slab. This simplifies the analysis and avoids some of the technical challenges related to achieving high optical $Q$s in a quasi-2D thin film structure, challenges which have already been studied and met elsewhere~\cite{Song2005a,Kuramochi2006,Safavi-Naeini2010}.
\subsection{Single and Double Cavity Systems}
\begin{figure}[htbp]
\begin{center}
\includegraphics{./Figure6.pdf}
\end{center}
\caption{(a) The electric field magnitude $|\mathbf E|$ of the L2 cavity photonic resonance at $\nu_o = 199~\text{THz}$. (b-e) The line defect optical waveguide structure, described in the text, which couples to the L2 cavity photonic resonance. The numerically simulated (c) acoustic bandstructure and (d) optical bandstructure of the line defect optical waveguide with $W=0.135a$. (e) Plot of the out-of-plane component of the magnetic field, $H_z$, of the guided optical mode at $X$-point of the bandstructure. (f) The displacement field $|\mathbf Q|$ of the L2 cavity phononic resonance at $\nu_m = 11.2~\text{GHz}$. (g-j) The line defect acoustic waveguide structure, described in the text, which couples to the L2 cavity phononic resonance. The numerically simulated (h) acoustic bandstructure and (i) photonic bandstructure of the line defect acoustic waveguide with $r_c = 0.82 r$. (j) The magnitude of the mechanical displacement field, $|\mathbf Q|$, of the guided acoustic mode at the $X$-point of the bandstructure. Calculations of the acoustic waveguide bandstructure are done using an FEM model~\cite{COMSOL2009}, while for the optical waveguide simulations, a plane-wave-expansion method was utilized~\cite{Johnson2001:mpb}.
\label{fig:cavities_and_waveguides}}
\end{figure}
The snowflake crystal consists of a hexagonal lattice of snowflake-shaped holes patterned into silicon as shown in Figure~\ref{fig:cavities_and_waveguides}. The snowflake lattice used here is characterized by a lattice constant $a=400~\text{nm}$, snowflake radius $r = 168~\text{nm}$, and width $w = 60~\text{nm}$. It possesses a full phononic bandgap from $8.6$ to $12.6$ GHz and a photonic pseudo-bandgap for TE optical waves from $180$ to $230$ THz. Defects in the crystal (features breaking the discrete translational symmetry of the underlying lattice) can support resonances with frequencies within the optical and mechanical bandgaps, leading to highly localized, strongly interacting resonances~\cite{Safavi-Naeini2010}. As an example, by filling an adjacent pair of snowflake-shaped holes in the crystal, a so-called ``L2'' cavity is formed which supports a localized photonic resonance at a frequency $\nu_o = 199~\text{THz}$ and a phononic resonance at a frequency $\nu_m = 11.2~\text{GHz}$. The defect cavity structure and finite-element-method (FEM)~\cite{COMSOL2009} simulated field profiles of these co-localized modes are shown in Figures~\ref{fig:cavities_and_waveguides}(a) and ~\ref{fig:cavities_and_waveguides}(f).
The L2 cavity forms the basis of a more complex double-optical-mode cavity structure with the desired symmetry properties for efficient phonon-to-photon translation. By placing two separate L2 cavities close to one another, at sufficiently small separations, even and odd optical and mechanical super-modes form with split mechanical and optical resonant frequencies. We choose to displace the cavities from each other in the $y$ direction, as shown in full system diagram of Figure~\ref{fig:full_system_only}. These super-modes of the coupled cavities are characterized with respect to their vector symmetry about the mirror symmetry transformation $\sigma_y (x,y) = (x,-y)$. We denote these super-modes $\mathbf E_\pm$ and $\mathbf Q_\pm$, where `$+$' denotes symmetric and `$-$' denotes anti-symmetric symmetry with respect to $\sigma_y$. The supermodes can be written approximately as,
\begin{equation}
\mathbf E_\pm = \frac{\mathbf E_a \pm \mathbf E_b}{\sqrt{2}}\qquad\mbox{and}\qquad\mathbf Q_\pm = \frac{\mathbf Q_a \pm \mathbf Q_b}{\sqrt{2}},
\end{equation}
\noindent where the subscripts $a,b$ label the individual cavity fields. The cavity separation (14 rows) shown in Figure~\ref{fig:full_system_only} is chosen such that the optical supermode frequency splitting is very nearly identical to the mechanical mode frequency of $\nu_m = 11.2~\text{GHz}$, as ascertained by FEM simulations. We focus here only on the mechanical mode of odd vector symmetry, $\mathbf Q_-$, since as we will show below, this is the mode which cross-couples the two optical super-modes to each other.
\subsection{Optomechanical Coupling Rates}
Optomechanical coupling (or acousto-optic scattering) arises from the coupling of optical cavity modes under deformations in the geometric structure. In the canonical form of radiation pressure, a mechanical deformation in the cavity induces a shift in the resonance frequency of a given cavity mode. The coefficient describing the cavity mode dispersion with mechanical displacement also quantifies the strength of the radiation pressure force that photons in the cavity mode exert back on the mechanical structure. More generally, mechanical deformations may couple one optical cavity mode to another.
The self-coupling and inter-modal couplings caused by a mechanical deformation are modeled by the position dependent interaction rates ${g}_k(\hat{x})$ and ${h}(\hat{x})$, respectively, in the Hamiltonian of eqn.~(\ref{eqn:full_total_Hamiltonian}). Both types of deformation-dependent optomechanical couplings may be calculated to first order using a variant of the Feynman-Hellmann perturbation theory, the Johnson perturbation theory~\cite{Johnson2002}, which takes into account moving boundaries in electromagnetic cavities and has been used successfully in the past to model optomechanical crystal cavities~\cite{Eichenfield2009,Eichenfield2009a}. The Hamiltonian, given to first-order by,
\begin{eqnarray}
\op{H}{} &=& \hbar \sum_{i} \omega_{i} \opdagger{a}{i}\op{a}{i} + \hbar \Omega \opdagger{b}{}\op{b}{}+ \frac{\hbar}{2} \sum_{i,j} g_{i,j} (\opdagger{b}{} + \op{b}{}) \opdagger{a}{i} \op{a}{j},
\end{eqnarray}
is then a generalization of that shown previously in equation (\ref{eqn:total_Hamiltonian}), with
\begin{eqnarray}
g_{i,j} &= &\frac{\omega_{i,j}}{2} \sqrt{\frac{\hbar}{2\omega_m t}}\frac{ \int dl \left(\mathbf{Q} \cdot \mathbf{n} \right) \left(\Delta \epsilon \mathbf{E}^{\parallel\ast}_{i}\cdot\mathbf{E}^\parallel_{j} - \Delta(\epsilon^{-1}) \mathbf{D}^{\perp\ast}_{i}\cdot \mathbf{D}^\perp_{j}\right)}{\sqrt{\int dA \rho |\mathbf{Q}|^2 \int dA \epsilon |\mathbf{E}_i|^2 \int dA \epsilon |\mathbf{E}_j|^2}},\label{eqn:gij}
\end{eqnarray}
where $\mathbf{E}$, $\mathbf{D}$, $\mathbf{Q}$ and $t$ are the optical mode electric field, optical mode displacement field, mechanical mode displacement field, and the thickness of the slab or thin-film. For convenience, in what follows we denote the overlap integral in equation (\ref{eqn:gij}) as $\bra{\mathbf E_i} \mathbf Q \ket{\mathbf E_j}$. Since the optical supermodes are nearly degenerate, we replace $\omega_{i,j}$ with either $\omega_+$ or $\omega_-$ with little error, and following the notation used previously denote cross-modal coupling as $h = g_{+,-}= g_{-,+}$.
For the optical and mechanical modes of a single L2 cavity (see Fig.~\ref{fig:cavities_and_waveguides}(a) and Fig.~\ref{fig:cavities_and_waveguides}(f)) the optomechanical self-coupling is numerically calculated using (\ref{eqn:gij}) to be $g=\bra{\mathbf E} \mathbf Q \ket{\mathbf E}/2\pi = 489~\text{kHz}$. Then using the symmetry selection rules in the overlap integrals, the only coupling terms involving the $\mathbf Q_-$ mode are the cross-coupling terms $\bra{\mathbf E_\pm} \mathbf Q_- \ket{\mathbf E_\mp}$. These terms are calculated to be $\bra{\mathbf E_+} \mathbf Q_- \ket{\mathbf E_-}=\bra{\mathbf E_-} \mathbf Q_- \ket{\mathbf E_+} = \bra{\mathbf E} \mathbf Q \ket{\mathbf E}/\sqrt{2}$ to good approximation. For the supermodes of our double L2 cavity system this yields a cross-coupling rate of $h/2\pi = 346~\text{kHz}$.
\subsection{Implementation of Waveguides}
A line-defect on an optomechanical crystal acts as a waveguide for light and sound~\cite{ref:Chutinan2000,ref:Johnson4,safavi-naeini10}. In principle then, the same waveguide may be used to shuttle both the photons and phonons around on an OMC microchip. However, due to the different properties of optical and acoustic excitations, in particular, their typically disparate quality factors (roughly $10^6$ for $200$~THz photons and $10^4$ for GHz phonons in silicon), the cavity loading requirements may be different for photons and phonons. As such, it is more convenient to implement two physically separate sets of waveguides, one for optics and the other for mechanics.
Our chosen line defect for optical waveguiding, shown in Fig. \ref{fig:cavities_and_waveguides}(b), consists of a row of
removed holes, with the rows above and below shifted toward one another by $W$ such that the distance between the
centers of the snowflakes across the line defect is $\sqrt{3}a - 2W$. Simulations of this line defect waveguide show
that there are no acoustic waveguide bands resonant with the localized mechanical L2 cavity modes of interest (see
Fig. \ref{fig:cavities_and_waveguides}(c)), and therefore that this waveguide will not load the mechanical part of the
L2 cavity system. Optically, this line defect has a single optical band crossing the frequencies of the localized
optical L2 cavity modes of interest (see Fig. \ref{fig:cavities_and_waveguides}(d)), providing the single-mode optical
waveguiding required for the PPT.
The line defect used for the acoustic waveguide consists of a row of holes which have been reduced in size, as shown in Fig. \ref{fig:cavities_and_waveguides}(g)). By shrinking the size of the central row of snowflake holes by $18\%$, a single-mode acoustic waveguide is formed which spectrally overlaps the localized mechanical resonances of the L2 cavity (see Fig. \ref{fig:cavities_and_waveguides}(h)), and allows for mechanical coupling to the PPT. This same line defect waveguide is reflective for optical photons in the band of localized optical resonances of the L2 cavity (see Fig. \ref{fig:cavities_and_waveguides}(i)), and thus is isolated from the optical part of the PPT. Below we discuss how both the optical and mechanical waveguides may be used to load the PPT resonant cavity.
\subsection{Cavity-Waveguide Coupling}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=15cm]{./Figure7.pdf}
\end{center}
\caption{(a) The full optomechanical crystal PPT system, consisting of a pair of coupled L2 defect cavities with acoustic and optical waveguide couplers. The waveguide coupling to the cavities, for both optics and acoustics, consists of a pair of horizontal line defect waveguides, one to each of the L2 cavities. The optical waveguides (highlighted in blue) are the two outer waveguides and the acoustic waveguides (highlighted in beige) are the two inner waveguides. The relevant symmetric and anti-symmetric optical (blue) and acoustic (beige) supermodes of the cavity-waveguide system are shown as envelope functions with the appropriate symmetry (dashed for symmetric and solid for anti-symmetric). The FEM-simulated (b) photonic ($|\avg{\mathbf S_o}_t|^2$) and (c) phononic ($|\avg{\mathbf S_m}_t|^2$) Poynting vectors of the lower cavity-waveguide structure, illustrating the selective optical loading of the lower waveguide and the selective mechanical loading of the upper waveguide on the cavity.\label{fig:full_system_only}}
\end{figure}
By bringing the optical waveguide near the L2 cavity, the optical cavity resonance is evanescently coupled to the guided modes of the line-defect, as shown in Fig.~\ref{fig:full_system_only}(b). Control over this coupling rate is achieved at a coarse level by changing the distance (number of unit cells) between the cavity and waveguide. For the structure considered here, a gap of 8 rows is sufficient to achieve a coupling rate $\kappa_e$ in the desired range. A fine tuning of the coupling rate is accomplished by adjusting the waveguide width parameter, with a value of $W = 0.135 a$ resulting in a loaded optical cavity $Q$-factor of $Q_\text{WG,o} \approx 3\times10^5$ (the corresponding external waveguide coupling rate is $\kappa_e/2\pi = 300~\text{MHz}$). Considering that intrinsic optical $Q$-factors as high as $3\times10^6$ have been achieved in microfabricated thin-film silicon photonic crystal cavities similar to the sort studied here~\cite{Takahashi2009}, the calculated optical waveguide loading should put such a cavity structure well into the over-coupled regime ($\kappa_e/\kappa_i \approx 10$). A short section in which $W$ is tapered is used to close off the waveguide on one side.
The same design procedure for the acoustic waveguide results in an evanescently coupled waveguide at a distance of only one row from the L2 cavity. Since the acoustic line-defect waveguide does not support Bloch modes at the optical cavity frequency, no additional loss is calculated for the optical cavity resonance. In this geometry the mechanical cavity loading is calculated to be $Q_\text{WG,m} = 1.3\times10^3$, corresponding to an extrinsic coupling rate $\gamma_e/2\pi = 4.4~\text{MHz}$. Taking $Q_i \approx 10^4$ as an achievable intrinsic mechanical $Q$-factor (we need a citation to the Berkeley group here for Si GHz resonators), such a loading also puts the mechanical system in the over-coupling regime ($\gamma_e/\gamma_i \approx 10$).
Simulations of the above cavity-waveguide couplings are performed using FEM~\cite{COMSOL2009} with absorbing boundaries at the ends of the waveguide. The resulting time-average electromagnetic Poynting vector $|\avg{\mathbf S_o}_t|^2 = |\mathbf E \times \mathbf H^\ast|/2$ of the optical field leaking from the L2 optical cavity resonance is plotted in Fig.~\ref{fig:full_system_only}(b), while the mechanical Poynting vector $|\avg{\mathbf S_m}_t|^2= |- \mathbf v \cdot \mathbf T|^2$ ($\mathbf v$ is the velocity field, and $\mathbf T$ the stress tensor) of the acoustic waves radiating from the mechanical mode of the L2 cavity is shown in Fig.~\ref{fig:full_system_only}(c). It is readily apparent from these two plots that the coupling of the two different waveguides to the L2 cavity act as desired; the acoustic radiation is coupled only to the phononic waveguide, and the optical radiation is coupled only to the photonic waveguide.
In order to individually address and out-couple from the even and odd symmetry cavity resonances of double-L2-cavity structure used to form the PPT, a pair of waveguides is used for each of the optical and mechanical couplings. As shown in the overall PPT design of Figure~~\ref{fig:full_system_only}(a), each of the L2 cavities has an acoustic and an optical waveguide coupled to them. Excitation of a pair of waveguides either in or out of phase would thus allow for coupling to the symmetric or anti-symmetric supermodes, respectively, of the double-L2-cavity. Similarly, spatial filtering (via an integrated directional coupler or waveguide filter for instance) of the output of a pair of waveguides would allow for the selective read-out of either the symmetric or anti-symmetric cavity modes. One could in principle utilize spectral filtering to perform the selective mode coupling; however, with the narrowband nature of the optical and mechanical supermode splittings, spatially independent channels of excitation and read-out may be a preferred option.
In summary, the OMC PPT as designed couples the symmetric and anti-symmetric optical modes of a double-L2-cavity system via a co-localized anti-symmetric mechanical resonance at frequency $\nu_m = 11.2~\text{GHz}$. In the notation of Section~\ref{sec:analysis}, the lower frequency symmetric optical mode is the pump mode (cavity mode $a_{1}$), the anti-symmetric mode is the signal mode or cavity mode $a_{2}$, and the anti-symmetric mechanical resonance is phonon mode $b$. Both optical modes are designed to have a resonant frequency in the near-IR around a frequency of $200$~THz, with a frequency splitting engineered to be equal to the mechanical frequency, $\omega_- - \omega_+ \approx\Omega=2\pi \times 11.2$~GHz. The numerically calculated waveguide and optomechanical coupling rates for this system are $(\kappa_e, \gamma_e, h) = 2\pi\times( 300, 4.4, 0.35)~\text{MHz}$, with the required number of intracavity pump photons for optimum operation ($G\approx\sqrt{\kappa_{e}\gamma_{e}}$) of such a PPT estimated to be only $|\alpha_{1,0}|^2 = 1.1\times10^4$ (assuming minimal intrinsic losses and $\kappa_e\approx\kappa$, $\gamma_e\approx\gamma$).
\section{Applications} \label{sec:applications}
At the simplest level, the extremely narrow optical response of the PPT, as shown in Figure~\ref{fig:scattering_matrix}, provides the opportunity for design and fabrication of filters with MHz-scale linewidths in the optical domain. By comparison, a purely passive optical design would require optical cavities with quality factors of $Q\approx10^8$. More generally, such a scheme demonstrates a promising aspect of optomechanics in the realm of optical information processing. In this section three examples applications of the PPT are studied in detail. The first two, optical delay lines and wavelength converters are further examples of optical information processing that is of considerable interest in both classical and quantum information processing. The last application, using the PPT to provide optical ``flying qubit'' capability to superconducting microwave quantum systems, is an example of how optomechanics can have a fundamental role in hybrid quantum system engineering.
\subsection{Delay Lines}
\begin{figure}[htbp]
\begin{center}
\includegraphics{./Figure8.pdf}
\caption{Schematic layout (left) and signal path (right) of a PPT used in an optical delay line/filter configuration in which a phonon mirror at the end of the acoustic output reflects outgoing phonons back into the PPT. The large optical delay is afforded by the acoustic path length of the signal in which acoustic waves propagate some $10^5$ times slower than photons. Optical filtering is provided by the narrow resonance bandwidth of the mechanical component of the PPT.}
\label{fig:delay_line}
\end{center}
\end{figure}
Efficient reversible conversion between traveling photons and phonons can be used to realize an optical delay line, as shown schematically in Figure~\ref{fig:delay_line}. In this geometry, the acoustic waveguide used to extract phonons from the PPT is terminated abruptly, forming an effective acoustic wave mirror which reflects phonons back toward the PPT after some propagation distance and delay. Resonant photons sent into the optical port of the PPT, will then re-emerge, reflected and delayed by twice the length of the acoustic waveguide. The delay line functionality comes from the inherent slowness of acoustic waves in comparison to electromagnetic waves (roughly a factor of $10^5$ for waves in silicon). For a similar reason, electro-acoustic piezoelectric structures are used to create the chip-scale RF-microwave filters found in many compact wireless communication devices~\cite{Lakin1995}.
The usefulness and bounds on the main characteristics of a PPT-based delay line, i.e., the total delay possible and the delay-bandwidth product, may be simply estimated without referring to a particular implementation of the system. The maximum possible delay is given by the lifetime of an excitation on the mechanical side of the system (cavity and waveguide), and is given by $1/\gamma_i$, which is limited by material properties of the mechanical system. The bandwidth of PPT conversion is given by eqn.~(\ref{eqn:gamma_transfer}), and is the total loss rate seen by the mechanical resonance, $2\gamma$. As such the delay-bandwidth product can at most be
\begin{equation}
\Delta \omega \tau \sim \frac{2\gamma}{\gamma_i},
\end{equation}
which is approximately twice the acoustic waveguide to mechanical resonance over-coupling ratio $\gamma_e/\gamma_i$ in the PPT. Manipulation of the acoustic waves within the delay waveguide itself, before conversion back to optics, may also be envisioned, enabling existing phononic information processing capabilities~\cite{Olsson2009} to be applied to optical signals.
\subsection{Wavelength Conversion}
\begin{figure}[htbp]
\begin{center}
\includegraphics{./Figure9.pdf}
\caption{(a) Schematic of a back-to-back PPT structure for optical wavelength convertsion/delay. For pure wavelength conversion, the phononic waveguide is vestigial, and can be removed by simply coupling the optical elements in both PPTs to the same mechanical resonant element. The simplified effective system diagram for such a device is shown in (b), where optical resonances $\hat{a}$ and $\hat{c}$ are coupled to the same mechanical resonance $\hat{b}$. The pump cavities for each system are omitted.}
\label{fig:ppt_wavelength_converter}
\end{center}
\end{figure}
Figure~\ref{fig:ppt_wavelength_converter}(a) shows the schematic for a PPT based device that can act simultaneously as a narrowband filter, a delay line and a wavelength converter. It consists of connecting serially by a common acoustic waveguide, two PPT devices operating at different optical but matched mechanical frequencies. Interestingly, if the only goal is to perform photon-to-photon wavelength conversion, one can omit the connecting acoustic waveguide. By placing the two PPTs ``on top of each other'', such that the optical cavities on both PPTs are coupled to the same mechanical resonance, as shown in Figure~\ref{fig:ppt_wavelength_converter}(b), optical wavelength conversion can be accomplished. Such a PPT geometry could be realized by either having photonic cavities with multiple modes, or using two photonic cavities coupled to the same mechanical mode. Such photon-to-photon conversion could even be taken to an extreme, allowing, for instance, optical-to-microwave wavelength conversion if one of the photonic cavities is a microwave cavity.
For the simplified wavelength conversion system of Fig.~\ref{fig:ppt_wavelength_converter}(b), the PPT matching condition of equation (\ref{eqn:matching_condition}) and noise analysis of Section~\ref{ss:noise} carry over with only minor adjustments. For the simplified wavelength conversion system the thermal noise is now split between the two optical channels ($\hat{a}$ and $\hat{c}$ of Fig.~\ref{fig:ppt_wavelength_converter}(b)), while the spontaneous emission noise in the system is approximately doubled (for similar optical cavities) due to the two uncorrelated spontaneous emission processes occurring from the optical pumping of each individual cavity. In a single element PPT, the optimal $G$ matches the pure mechanical damping of the mechanical resonance ($\gamma$) to the induced optomechanical loading of the mechanical resonance ($G^2/\kappa$) by the optical cavity. The matching condition for the simplified wavelength converter now must balance a mechanical resonance coupled on one side to an optical cavity with induced mechanical loading rate $G_a^2/\kappa_a$ and on the other side to a second optical cavity with induced loading rate $G_c^2/\kappa_c$. As such, assuming that $\gamma_i \ll G_k^2/\kappa_k$ we arrive at the \emph{photon-photon converter} matching condition,
\begin{equation}
\frac{G_a^2}{\kappa_a} = \frac{G_c^2}{\kappa_c}.
\end{equation}
The optomechanical system as described would act as a quantum-limited optomechanical wavelength converter. Finally, we note that this particular implementation of the wavelength converter could also function in a wider array of optomechanical platforms since there is no longer a need for phononic waveguides.
\subsection{Quantum State Transfer and Networking between Circuit QED and Optics}
\begin{figure*}[htbp]
\begin{center}
\scalebox{0.70}{\includegraphics{./Figure10.pdf}}
\caption{System diagram for quantum state transfer from optical to superconducting qubits. The two 3-port components are circulators with each input connected to the output directly clockwise from it. }
\label{fig:ciraczoller}
\end{center}
\end{figure*}
Two of the key requirements for a viable platform for quantum computation are the ability to do storage and communication of quantum information. For the case of superconducting phase qubits, promising theoretical proposals to provide
such functionality have involved electrical~\cite{Plastina2003} and mechanical resonators~\cite{Cleland2004,Geller2005}. Experimentally, an electromagnetic resonator quantum bus was demonstrated by Sillanp\"a\"a et al.~\cite{Sillanpaa2007} in 2007, while more recently O'Connell et al.~\cite{OConnell2010} demonstrated the strong coupling of a mechanical resonator to a superconducting qubit. Circuit QED (cQED) to date remains limited by the lack of a true long range state transfer mechanism, one which is readily available for the case of quantum-optical qubits, in the form of optical fibers and free-space links. Using the PPT system, one potentially could implement a version of the quantum state transfer protocol of Cirac et al.~\cite{Cirac1997} allowing for the high fidelity transfer of states between optical and superconducting qubits. Such a system would satisfy one of the original goals of hybrid quantum system~\cite{Tian2004} by interfacing a quantum optical and solid-state qubit.
By connecting the phononic waveguide of a PPT to a piezoelectric resonator strongly coupled to a superconducting qubit~\cite{Cleland2004,Geller2005,OConnell2010}, and connecting the PPT at its optical end to an optical cavity QED system, as shown in Figure~\ref{fig:ciraczoller}, the phonon-photon translator could be used as an intermediary in a state transfer protocol among two energy-disparate quantum systems. The optical system $(A)$ is composed of a Fabry-Perot cavity containing a three-level atom system in a $\Lambda$ configuration. As shown in~\cite{Cirac1997}, in the correct limit, this system may be modeled with effective Jaynes-Cummings (JC) Hamiltonian with a Rabi frequency $g_A(t)$ controlled by another beam. For the superconducting system $(B)$, a mechanical resonance is coupled to a phase qubit, with a bias current used to change the frequency of the resultant two-level system, which effectively changes the coupling rate between the qubit and mechanical resonator. This leads again to a system with an externally controllable Rabi frequency of $g_B(t)$. The Hamiltonian of each subsystem is then given by
\begin{equation}
\hat{H}_{j} = \hbar g_j(t) e^{-i \phi_j(t)} \hat{\sigma}_j \hat{c}^\dagger_j + \hbar g_j(t) e^{+i \phi_j(t)} \hat{\sigma}^\dagger_j \hat{c}_j\qquad(j=A,B),
\end{equation}
where $\hat{c}_j$ are the annihilation operators of the photonic or phononic resonances external to the PPT, and $\hat{\sigma}_j$ are the level lowering operators for the respective qubits to which they are coupled.
Each cavity mode, with annihilation operators $\hat{c}_j$, is coupled to its respective waveguide with a loss rate $\Gamma_j$. To characterize the PPT, the intrinsic losses in these cavities are ignored. Additionally, the phononic waveguide is assumed to be loss-less, though losses may be taken into account through the minor readjustment of the scattering parameters studied in Section~\ref{app:waveguide_loss}. Using the input-output boundary conditions~\cite{Gardiner1993,GardinerZoller} the frequency domain expression for the noise input into systems $A$ and $B$ are found to be,
\begin{eqnarray}
\tilde{c}_{A,\text{in}}(\omega) &=& \tilde{a}_\text{in,c}(\omega),\\
\tilde{c}_{B,\text{in}}(\omega) &=& s_{21}(\omega) \sqrt{\Gamma_A} \tilde{c}_A(\omega) + s_{21}(\omega) \tilde{a}_\text{in,c}(\omega)
+ s_{22}(\omega) \tilde{b}_\text{in,c}(\omega) \nonumber\\&&+ n_{12}(\omega) \tilde{a}_\text{in,i}(\omega)
+ n_{22}(\omega) \tilde{b}_\text{in,i}(\omega),
\end{eqnarray}
where $\hat{a}_{\text{in,c}}$ and $\hat{b}_{\text{in,c}}$ represent the noise being coupled into the system from the third input of each circulator in Figure~\ref{fig:ciraczoller}. To convert this equation to the time-domain, the convolution between various operators and scattering matrix elements must be taken. If the photon pulse is of sufficiently large temporal width, i.e., with bandwidth less than the bandwidth of the PPT, the frequency dependence of each scattering matrix element can be removed, replacing it with its extremal value assuming that the system is operating at resonance $(\Delta = 0)$. This requires that the coupling rates $g_j(t)$ should change slowly relative to the response of the PPT. Under this condition, the input-output relations in the time-domain are then
\begin{eqnarray}
\hat{c}_{A,\text{in}}(t) &=& \hat{a}_\text{in,c}(t), \\
\hat{c}_{B,\text{in}}(t) &=& s_{21} \sqrt{\Gamma_A} \hat{c}_A(t) + s_{21}\hat{a}_\text{in,c}(t) \nonumber\\&&+ s_{22} \hat{b}_\text{in,c}(t) + n_{12} \hat{a}_\text{in,i}(t)+ n_{22}\hat{b}_\text{in,i}(t).
\end{eqnarray}
In modeling the noise of the system, it is assumed that the optical noise inputs are in the vacuum state, and that the phononic noise is thermal with thermal phonon occupation numbers $\bar{n}$ and $\bar{n}^\prime$ for $\hat{b}_\text{in,c}(t)$ and $\hat{b}_\text{in,i}(t)$ , respectively, where the PPT phonon spontaneous emission noise is combined with the intrinsic thermal bath coupling of the mechanical mode in $\bar{n}^\prime$ as described in section~\ref{ss:spont_emission}.
Using standard operational methods of Quantum Stochastic Differential Equations~\cite{GardinerZoller,LesHouches1995}, the master equation describing the evolution of systems $A$ and $B$ is found to be,
\begin{eqnarray}
\dot{\rho} &=& \frac{1}{i\hbar} [ H_A + H_B, \rho] + \frac{\Gamma_A}{2}\mathcal{L}_{I, A}\rho +\frac{\Gamma_B}{2}\mathcal{L}_{I, B}\rho \nonumber\\&&
+ \frac{\Gamma_B}{2}\left(|s_{22}|^2 \bar{n} + |n_{22}|^2 \bar{n}^\prime \right)\mathcal{L}_{T,B}\rho\nonumber\\&&
+\sqrt{\Gamma_A\Gamma_B}|s_{21}| \left([ c_B^\dagger,c_A \rho] + [\rho c_A^\dagger,c_B]\right)\label{eqn:AB_mastereqn}
\end{eqnarray}
with the Liouvillians $\mathcal{L}_{I, j}$ and $\mathcal{L}_{T, j}$ are given by,
\begin{eqnarray}
\mathcal{L}_{I, j}\rho &=& 2 \hat{c}_j \rho \hat{c}_j^\dagger - \hat{c}_j^\dagger \hat{c}_j \rho - \rho \hat{c}_j^\dagger \hat{c}_j,\\
\mathcal{L}_{T, j}\rho &=& \mathcal{L}_{I, A}\rho
+2 \hat{c}_j^\dagger \rho \hat{c}_j - \hat{c}_j \hat{c}_j^\dagger \rho - \rho \hat{c}_j \hat{c}_j^\dagger.
\end{eqnarray}
\noindent The final term in the master equation~(\ref{eqn:AB_mastereqn}) is the cascading term \cite{Gardiner1993,LesHouches1995,Carmichael1993} which gives rise to the unidirectional coupling between the systems.
For the PPT, parameters typical to an OMC structure such as the one with scattering matrices plotted in Fig.~\ref{fig:scattering_matrix} are used, $(\gamma_e,\gamma_i,\kappa_e,\kappa_i,\Omega)=2\pi\times(10,1,2000,200,8000) ~\text{MHz}$. Equation~(\ref{eqn:Go_mod}) can be used to find the optimal matching optomechanical coupling rate, which for the assumed PPT parameters is $G^\text{o} = 155.4~\text{MHz}$. At this optomechanical coupling rate, the resonant noise and scattering matrix parameters of the PPT are $(s_{21},s_{22},n_{21},n_{22}) = (0.917, 0.074, 0.290, 0.262)$, with a spontaneous emission noise equivalent occupation number of $n_\text{spon} = 0.200$. Assuming that the PPT is cooled to the same cryogenic temperature of the superconducting qubit system ($T<100$~mK), the thermal bath component of the effective thermal occupancy of the PPT mechanical resonance can be neglected, and $\bar{n}^\prime \approx 0.251$.
The exact functional form of the $g_j(t)$ are found through numerical optimization. This was done by taking the pulse-shape to be a step smoothed by a sinusoidal function with a rise (fall) time of $t_{r}$ ($t_{f}$). Optimization on the state transfer fidelity for an ideal PPT ($s_{21} = 1$) and for circulators running in the direction shown in Fig.~\ref{fig:ciraczoller} (qubit transfer from optical to superconducting system) leads to rise and fall times of $t_A = 23.5~\mu\text{s}$ and $t_B = 16~\mu\text{s}$, which are within the modeled PPT's $(2\gamma)/2\pi=22~\text{MHz}$ bandwidth.
\begin{figure}[htb]
\begin{center}
\scalebox{0.8}{\includegraphics{./Figure11.pdf}}
\caption{The main plot shows the time variation of the fidelity $F_1$ found by numerically solving the quantum master equation for a version of the quantum state transfer protocol introduced by Cirac, et al.~\cite{Cirac1997} for the state transfer from the optical to superconducting qubit. PPT parameters of $(\gamma_e,\gamma_i,\kappa_e,\kappa_i,\Omega)=2\pi\times(10,1,2000,200,8000) ~\text{MHz}$ are used here to find the the optical ($F^{B,A}_1$; \textcolor{black}{$-\cdot$}) and superconducting ($F^{A,B}_1$; \textcolor{red}{$-\cdot$}) state-transfer fidelity curves for an initial state of $|1\rangle_{A}|0\rangle_{B}$. We see that a maximum fidelity $F^{A,B}_1$ of approximately 0.8 for the final state transfer is possible. The continuous lines (\textcolor{black}{$-$}) and (\textcolor{red}{$-$}) were found by quantum simulation of an ideal PPT, i.e. $\gamma_i=\kappa_i = 0$. In all cases analyzed the external system parameters were $(\Gamma_A,\Gamma_B,g^{\text{max}}_A,g^{\text{max}}_B) = 2\pi\times(50,5.0,5.0,1.0) ~\text{MHz}$. The inset plot shows the shape of the control pulse used for the optical (\textcolor{black}{$--$}) and superconducting (\textcolor{red}{$-\cdot$}) qubits.}
\label{fig:fidelity_plot}
\end{center}
\end{figure}
Putting this all together, in Fig.~\ref{fig:fidelity_plot} we plot estimates of the fidelity of the quantum state transfer between system $A$ and $B$ via the connecting PPT. The definition of fidelity used to calculate the state transfer efficiency is $F^{A,B}_j=\text{Tr}_A(|j_B\rangle\langle j_B|\rho)$, where $j=0,1,+$ represent respectively the ground $|0\rangle$, excited $|1\rangle$, and $|+\rangle =2^{-1/2}(|0\rangle+|1\rangle)$ states of the atomic and superconducting two-level systems. Under these conditions, and considering an ideal PPT ($s_{21} = 1$), states are transfered with fidelities $F^{A,B}_1=0.9983$, $F^{A,B}_+=0.9995$ and $F^{A,B}_0 = 1.00$. Taking into account the actual scattering and noise matrix values given above for the PPT, and accounting for the spontaneous emission noise of the PPT we find that the fidelities are reduced to $F^{A,B}_1=0.803$, $F^{A,B}_+=0.936$ and $F^{A,B}_0 = 0.983$. The inverse system, with circulators turning the opposite direction to transfer qubits from the superconducting to optical system was also studied, for which the same input pulses only time reversed, and yield fidelities $F^{B,A}_1 = 0.772$, $F^{B,A}_+ = 0.904$ and $F^{B,A}_0 = 0.983$.
\section{Summary}
We have introduced the concept and design of a traveling phonon-photon translator. We have shown that with a realistic set of parameters and the use of existing silicon optomechanical crystal technology, efficient and reversible conversion between phonons and photons should be possible. By characterizing the noise processes experienced by such a device, both classically and quantum mechanically, we have shown the utility of traveling phonon-photon translation to important problems in both classical optical communication and quantum information processing.
\section{Acknowledgments}
The authors wish to thank Darrick Chang, Thiago Alegre, Matt Eichenfield, Klemens Hammerer, and Peter Zoller for useful discussions. This work was supported by the DARPA/MTO ORCHID program through a grant from AFOSR, and the NSF through EMT grant no. 0622246 and CIAN grant no. EEC-0812072. ASN acknowledges support through NSERC of Canada.
\section*{References}
|
{
"timestamp": "2010-09-21T02:00:34",
"yymm": "1009",
"arxiv_id": "1009.3529",
"language": "en",
"url": "https://arxiv.org/abs/1009.3529"
}
|
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