prompt stringlengths 0 256 | answer stringlengths 1 256 |
|---|---|
\infty D^\alpha \tilde h_j(x)\cdot b_j.
\end{equation}
Next, since the range of $\widetilde H$ is the subset of $T(X)$,
\[
H=T^{-1}\cdot P\cdot \widetilde H.
\]
From here and \eqref{equ4.32} we obtain, for all $\alpha\in{\mbf Z}_+^n$, $|\alpha|\le k$, $ | x\in{\mbf R}^n$,
\begin{equation}\label{equ4.33}
H(\delta_x^\alpha)=\sum_{j=1}^\infty D^\alpha \tilde h_j(x)\cdot (T^{-1}\cdot P)(b_j)
\end{equation}
(convergence in $X$).
Finally, we set
\begin{equation}\label{equ4.34}
h_j:=\|(T^{-1}\cdot P)(b_j)\|\cdot\ |
tilde h\quad {\rm and}\quad v_j:=\frac{(T^{-1}\cdot P)(b_j)}{\|(T^{-1}\cdot P)(b_j)\|},\quad j\in{\mbf N}.
\end{equation}
Then all $v_j\in X$ are of norm one. In turn, all $h_j\in C_b^{k,\omega}({\mbf R}^n)$ and due to \eqref{equ4.32a},
\eqref{equ4.34} an | d the properties of $T$ and $P$ for all $j\in{\mbf N}$,
\[
\|h_j\|_{C_b^{k,\omega}({\mbf R}^n)}\le \|T^{-1}\|\cdot\|P\|\cdot\|\tilde h_j\|_{C_b^{k,\omega}({\mbf R}^n)}\le 2\cdot \|T\|\cdot\|T^{-1}\|\cdot\|P\|\cdot\|H\|\le 32\cdot\lambda^2\cdot\|H\|.
\]
Mo |
reover, by \eqref{equ4.33}, for all $\alpha\in{\mbf Z}_+^n$, $|\alpha|\le k$, $x\in{\mbf R}^n$,
\[
H(\delta_x^\alpha)=\sum_{j=1}^\infty D^\alpha h_j(x)\cdot v_j,
\]
as required.
The proof of Theorem \ref{teor1.10} is complete.
\end{proof}
\section{Pro | ofs of Theorem \ref{te1.4} and Corollary \ref{cor1.10}}
\subsection{Proof of Theorem \ref{te1.4}\,(1)}
\begin{proof}
Let $\Lambda_{n,k}:=\{\alpha\in {\mbf Z}_+^n\, :\, |\alpha|\le k\}$. We set
\begin{equation}\label{eq4.23}
M_{n,k}:=\bigl(\Lambda_{n,k}\ |
times{\mbf R}^n\bigr)\sqcup \bigl(\bigl(\Lambda_{n,k}\setminus\Lambda_{n,k-1}\bigr)\times\bigl(({\mbf R}^n\times{\mbf R}^n)\setminus\Delta_n\bigr)\bigr),
\end{equation}
where $\Delta_n:=\{(x,y)\in{\mbf R}^n\times{\mbf R}^n\, :\, x=y\}$.
Space $M_{n,k}$ ha | s the natural structure of a $C^\infty$ manifold, in particular, it is a locally compact Hausdorff space.
By $C_b(M_{n,k})$ we denote the Banach space of bounded continuous functions on $M_{n,k}$ equipped with supremum norm. Let us define a linear map $\ma |
thcal I: C_b^{k,\omega}({\mbf R}^n)\rightarrow C_b(M_{n,k})$ by the formula\medskip
\begin{equation}
\mathcal I(f)(m)=\left\{
\begin{array}{lll}
\displaystyle D^\alpha f(x)&{\rm if}&m=(\alpha,x)\in\Lambda_{n,k}\times {\mbf R}^n\\
\\
\displaystyle \frac{D^\ | alpha f(x)-D^\alpha f(y)}{\omega(\|x-y\|)}&{\rm if}&m=(\alpha, (x,y))\in \bigl(\Lambda_{n,k}\setminus\Lambda_{n,k-1}\bigr)\times\bigl({\mbf R}^n\times{\mbf R}^n\setminus\Delta_n\bigr),
\end{array}
\right.
\end{equation}
\begin{Proposition}\label{prop5.1}
$ |
\mathcal I$ is a linear isometric embedding.
\end{Proposition}
\begin{proof}
The statement follows straightforwardly from the definitions of the involved spaces.
\end{proof}
Since $\mathcal I\bigl(C_b^{k,\omega}({\mbf R}^n)\bigr)$ is a closed subspace of | $C_b(M_{n,k})$, the Hahn-Banach theorem implies that the adjoint map
\begin{equation}\label{eq4.25}
\mathcal I^*:\bigl(C_b(M_{n,k})\bigr)^*\rightarrow \bigl(C_b^{k,\omega}({\mbf R}^n)\bigr)^*
\end{equation}
of $\mathcal I$ is surjective of norm one.
Simi |
larly, $\mathcal I$ maps $C_0^{k,\omega}({\mbf R}^n)$ isometrically into the Banach subspace $C_0(M_{n,k})\subset C_b(M_{n,k})$ of continuous functions on $M_{n,k}$ vanishing at infinity. Thus the adjoint of $\mathcal I_0:=\mathcal I |_{C_0^{k,\omega}({\mb | f R}^n)}$ is the surjective map of norm one
\begin{equation}\label{eq4.26}
\mathcal I_0^*:\bigl(C_0(M_{n,k})\bigr)^*\rightarrow \bigl(C_0^{k,\omega}({\mbf R}^n)\bigr)^*.
\end{equation}
According to the Riesz representation theorem (see, e.g.,\cite{DS}), $\ |
bigl(C_0(M_{n,k})\bigr)^*$ is isometrically isomorphic to the space of countably additive regular Borel measures on $M_{n,k}$ with the norm being the total variation of measure. In what follows we identify these two spaces.
In the proof we use the followi | ng result.
\begin{Proposition}\label{prop4.2}
If $\omega$ satisfies condition \eqref{omega2}, then $C_0^{k,\omega}({\mbf R}^n)$ is weak$^*$ dense in $C_b^{k,\omega}({\mbf R}^n)$.
\end{Proposition}
\begin{proof}
Let $\{L_{NN}\}_{N\in{\mbf N}}$ be finite ra |
nk bounded linear operators on $C_b^{k,\omega}({\mbf R}^n)$ defined by \eqref{eq3.18}. According to Lemma \ref{lem3.5} for each $f\in C_b^{k,\omega}({\mbf R}^n)$ the sequence $\{L_{NN}f\}_{N\in{\mbf N}}$ weak$^*$ converges to $f$. Moreover, each $L_{NN}f\i | n C^\infty({\mbf R}^n)$, cf. Lemma \ref{lem2.2}\,(a). We set
\begin{equation}\label{e4.27}
\hat f_N:=\rho_N\cdot L_{NN}f,
\end{equation}
see section~4.3. Then $\hat f_N$ is a $C^\infty$ function with compact support on ${\mbf R}^n$ satisfying, due to Lemma |
s \ref{norm} and \ref{lem2.2}\,(b), the inequality
\begin{equation}\label{e4.28}
\|\hat f_N\|_{C_b^{k,\omega}({\mbf R}^n)}\le C_N^2\|f\|_{C_b^{k,\omega}({\mbf R}^n)},
\end{equation}
where $\lim_{N\rightarrow\infty}C_N=1+c_{k,n}\cdot 4\sqrt n \cdot (k+1)\cd | ot\lim_{t\rightarrow\infty}\frac{1}{\omega(t)}$.
Clearly, sequence $\{D^\alpha \hat f_N\}_{N\in{\mbf N}}$ converges pointwise to $D^\alpha f$ for all $\alpha\in {\mbf Z}_+^n$, $|\alpha|\le k$. Also, due to condition \eqref{omega2} all $\hat f_N\in C_0^{ |
k,\omega}({\mbf R}^n)$. Hence, according to Proposition \ref{prop3.1}, sequence $\{\hat f_N\}_{N\in{\mbf N}}$ weak$^*$ converges to $f$. This shows that $C_0^{k,\omega}({\mbf R}^n)$ is weak$^*$ dense in $C_b^{k,\omega}({\mbf R}^n)$.
\end{proof}
Next, let | $i^*: \bigl(C_b^{k,\omega}({\mbf R}^n)\bigr)^*\rightarrow \bigl(C_0^{k,\omega}({\mbf R}^n)\bigr)^*$ be the linear surjective map of norm one adjoint to the isometrical embedding
$i: C_0^{k,\omega}({\mbf R}^n)\hookrightarrow C_b^{k,\omega}({\mbf R}^n)$.
\b |
egin{C}\label{cor4.3}
Restriction of $i^*$ to $G_b^{k,\omega}({\mbf R}^n)$ is injective.
\end{C}
\begin{proof}
Proposition \ref{prop4.2} implies that functions in $C_0^{k,\omega}({\mbf R}^n)$ regarded as linear functionals on $G_b^{k,\omega}({\mbf R}^n)$ s | eparate the points of $G_b^{k,\omega}({\mbf R}^n)\, (\subset \bigl(C_b^{k,\omega}({\mbf R}^n)\bigr)^*)$. If $i^*(v)=0$ for some $v\in G_b^{k,\omega}({\mbf R}^n)$, then
\[
0=(i^*(v))(f)=f(v)\quad {\rm for\ all}\quad f\in C_0^{k,\omega}({\mbf R}^n).
\]
Hen |
ce, $v=0$.
\end{proof}
We set
\[
\tilde\delta_x^\alpha:=i^*(\delta_x^\alpha),\quad |\alpha|\le k,\ x\in{\mbf R}^n.
\]
By definition, maps $\phi_\alpha: {\mbf R}^n\rightarrow \bigl(C_b^{k,\omega}({\mbf R}^n)\bigr)^*$, $x\mapsto\delta_x^\alpha$, $|\alpha|\le | k$, are continuous and bounded and so are the maps $i^*\circ\phi_\alpha:{\mbf R}^n\rightarrow \bigl(C_0^{k,\omega}({\mbf R}^n)\bigr)^*$.
\begin{Proposition}\label{prop4.4}
The range of $\mathcal I_0^*$ coincides with $i^*(G_b^{k,\omega}({\mbf R}^n))$.
\e |
nd{Proposition}
\begin{proof}
Let $\mu\in \bigl(C_0(M_{n,k})\bigr)^*$ be a countably additive regular Borel measures on $M_{n,k}$. We set, for all admissible $\alpha$ and all Borel measurable sets $U\subset M_{n,k}$,
\[
\mu_\alpha^1(U)=\mu\bigl(U\cap (\{ | \alpha\}\times{\mbf R}^n)\bigr)\quad {\rm and}\quad
\mu_\alpha^2(U)=\mu\bigl(U\cap \bigl(\{\alpha\}\times \bigl(({\mbf R}^n\times{\mbf R}^n)\setminus\Delta_n\bigr)\bigr)\bigr).
\]
Then $\mu=\sum_{\alpha,j}\mu_\alpha^j$.
Let us show that each $\mathcal I_ |
0^*(\mu_\alpha^j)$ belongs to $i^*(G_b^{k,\omega}({\mbf R}^n))$. Indeed, for
$j=1$ consider the Bochner integral
\begin{equation}\label{eq4.27}
J(\mu_\alpha^1):=\int_{x\in{\mbf R}^n}i^*(\phi_\alpha(x))\, d\mu_\alpha^1(x)=i^*\left(\int_{x\in{\mbf R}^n}\phi_ | \alpha(x)\, d\mu_\alpha^1(x)\right).
\end{equation}
Since $\phi_\alpha$ is continuous and bounded, the above integral is well-defined and its value is an element of $i^*(G_b^{k,\omega}({\mbf R}^n))$. By the definition of the Bochner integral, for each $f\i |
n C_0^{k,\omega}({\mbf R}^n)$,
\[
(J(\mu_\alpha^1))(f)=\int_{x\in{\mbf R}^n}\bigl(i^*(\phi_\alpha(x))\bigr)(f)\,d\mu_\alpha^1(x)=\int_{x\in{\mbf R}^n} D^\alpha f(x)\,d\mu_\alpha^1(x)=:\bigl(\mathcal I_0^*(\mu_\alpha^1)\bigr)(f).
\]
Hence,
\[
\mathcal I_0^* | (\mu_\alpha^1)=J(\mu_\alpha^1)\in i^*(G_b^{k,\omega}({\mbf R}^n)).
\]
Similarly, for $\alpha\in\Lambda_{n,k}\setminus\Lambda_{n,k-1}$ we define
\begin{equation}\label{eq4.28}
J(\mu_\alpha^2):=\int_{z=(x,y)\in ({\mbf R}^n\times{\mbf R}^n)\setminus\Delta_n}\ |
frac{i^*(\phi_\alpha(x))-i^*(\phi_\alpha(y))}{\omega(\|x-y\|)} d\mu_\alpha^2(z).
\end{equation}
Then, as before, we obtain that
\[
\mathcal I_0^*(\mu_\alpha^2)=J(\mu_\alpha^2)\in i^*(G_b^{k,\omega}({\mbf R}^n)).
\]
Thus we have established that the range | of $\mathcal I_0^*$ is a subset of $i^*(G_b^{k,\omega}({\mbf R}^n))$. Since the map $\mathcal I_0^*$ is surjective and its range contains all $\tilde\delta_x^\alpha$, it must contain $i^*(G_b^{k,\omega}({\mbf R}^n))$ as well.
This completes the proof of |
the proposition.
\end{proof}
In particular, we obtain that $\bigl(C_0^{k,\omega}({\mbf R}^n)\bigr)^*=i^*(G_b^{k,\omega}({\mbf R}^n))$, i.e., by the inverse mapping theorem $i^*$ restricted to $G_b^{k,\omega}$ maps it isomorphically onto $\bigl(C_0^{k,\ome | ga}({\mbf R}^n)\bigr)^*$.
Let us show that if $\lim_{t\rightarrow\infty}\omega(t)=\infty$, then $i^*$ is an isometry.
Assume, on the contrary, that for some $v\in G_b^{k,\omega}({\mbf R}^n)$,
\begin{equation}\label{eq4.31}
\|i^*(v)\|_{(C_0^{k,\omega}({ |
\mbf R}^n))^*}<\|v\|_{G_b^{k,\omega}({\mbf R}^n)}.
\end{equation}
Let $f\in C_0^{k,\omega}({\mbf R}^n)$, $\|f\|_{C_b^{k,\omega}({\mbf R}^n)}=1$, be such that
\[
v(f)=\|v\|_{G_b^{k,\omega}({\mbf R}^n)}.
\]
Let $\{f_N\}_{N\in{\mbf N}}\subset C_0^{k,\omega}({ | \mbf R}^n)$,
$\|f_N\|_{C_b^{k,\omega}({\mbf R}^n)}\le C_N^2$, $N\in{\mbf N}$, be the sequence of Proposition \ref{prop4.2} weak$^*$ converging to $f$. Observe that $\lim_{N\rightarrow\infty} C_N=1$ due to the above condition for $\omega$. Then from \eqref |
{eq4.31} and \eqref{e4.28} we obtain
\[
\begin{array}{l}
\displaystyle
\|v\|_{G_b^{k,\omega}({\mbf R}^n)}=v(f)=\lim_{N\rightarrow\infty} v(f_N)=\lim_{N\rightarrow\infty} \bigl(i^*(v)\bigr)(f_N)\medskip\\
\displaystyle
\le \|i^*(v)\|_{(C_0^{k,\omega}({\mbf | R}^n))^*}\cdot\varlimsup_{N\rightarrow\infty} \|f_N\|_{C_b^{k,\omega}({\mbf R}^n)}\le \|i^*(v)\|_{(C_0^{k,\omega}({\mbf R}^n))^*}<\|v\|_{G_b^{k,\omega}({\mbf R}^n)},
\end{array}
\]
a contradiction proving that $i^*$ is an isometry.
The proof of Theorem \r |
ef{te1.4}\,(1) is complete.
\end{proof}
\subsection{Proof of Theorem \ref{te1.4}\,(2)}
\begin{proof}
By the hypotheses of the theorem there exists a weak$^*$ continuous operator $T\in Ext(C_b^{k,\omega}(S);C_b^{k,\omega}({\mbf R}^n))$ such that $T(C_0^{k,\ | omega}(S))\subset C_0^{k,\omega}({\mbf R}^n)$. This implies that there is a bounded linear projection of the geometric preduals of the corresponding spaces
$P: G_b^{k,\omega}({\mbf R}^n)\rightarrow G_b^{k,\omega}(S)$ such that $P^*=T$.
Let $q_S: C_b^{k,\o |
mega}({\mbf R}^n)\rightarrow C_b^{k,\omega}(S)$ and $q_{S0}: C_0^{k,\omega}({\mbf R}^n)\rightarrow C_0^{k,\omega}(S)$ denote the quotient maps induced by restrictions of functions on ${\mbf R}^n$ to $S$. Finally, let $i: C_0^{k,\omega}({\mbf R}^n)\rightar | row C_b^{k,\omega}({\mbf R}^n)$ and
$i_S: C_0^{k,\omega}(S)\rightarrow C_b^{k,\omega}(S)$ be the bounded linear maps corresponding to inclusions of the spaces. Note that $i$ is an isometric embedding and $i_S$ is injective of norm $\le 1$.
\begin{Lm}\label |
{lem5.5}
$T_0:=T|_{C_0^{k,\omega}(S)}:C_0^{k,\omega}(S)\rightarrow C_0^{k,\omega}({\mbf R}^n)$ is a bounded linear map between Banach spaces.
\end{Lm}
\begin{proof}
For $f\in C_0^{k,\omega}(S)$ we have
\[
\|T_0f\|_{C_0^{k,\omega}({\mbf R}^n)}=\|(T\circ i_ | S)(f)\|_{C_b^{k,\omega}({\mbf R}^n)}\le\|T\|\cdot\|i_S(f)\|_{C_b^{k,\omega}(S)}\le \|T\|\cdot\|f\|_{C_0^{k,\omega}(S)},
\]
as required.
\end{proof}
Now, we have the following two commutative diagrams of adjoints of the above bounded linear maps (one corre |
sponding to upward arrows and another one to downward arrows):
\begin{equation}\label{equ5.38}
\begin{array}{cccc}
\bigl(C_b^{k,\omega}({\mbf R}^n)\bigr)^*&\stackrel{i^*}{\longrightarrow}&\bigl(C_0^{k,\omega}({\mbf R}^n)\bigr)^*\smallskip\\
_{\mbox{{\tin | y $q_S^*$}}}\!\uparrow\ \ \ \downarrow\mbox{{\tiny $T^*$}}&&_{\mbox{{\tiny $q_{S0}^*$}}}\!\uparrow\ \ \ \downarrow\mbox{{\tiny $T_0^*$}}
\\
\bigl(C_b^{k,\omega}(S)\bigr)^*&\stackrel{i_S^*}{\longrightarrow}&\bigl(C_0^{k,\omega}(S)\bigr)^*.
\end{array}
\end |
{equation}
Here $T^*\circ q_S^*=(q_S\circ T)^*={\rm id}$ and $T_0^*\circ q_{S0}^*=(q_{S0}\circ T_0)^*={\rm id}$, maps $q_S^*$ and $q_{S0}^*$ are isometric embeddings and
map $i^*$ is surjective.
Note that $i^*|_{G_b^{k,\omega}({\mbf R}^n)}:G_b^{k,\omega} | ({\mbf R}^n)\rightarrow \bigl(C_0^{k,\omega}({\mbf R}^n)\bigr)^*$ is an isomorphism by the first part of the theorem. Also, by the definition of $P$ (see \eqref{proj} in section~3.3 above),
\begin{equation}\label{equ5.39}
q_S^*\circ (T^*|_{G_b^{k,\omega}({ |
\mbf R}^n)})=P.
\end{equation}
Let us show that the map
\[
I:=i_S^*\circ(T^*|_{G_b^{k,\omega}(S)}):G_b^{k,\omega}(S)\rightarrow \bigl(C_0^{k,\omega}(S)\bigr)^*
\]
is an isomorphism.\smallskip
(a) Injectivity of $I$: If $I(v)=0$ for some $v\in G_b^{k,\omeg | a}(S)$, then by the commutativity of \eqref{equ5.38} and by \eqref{equ5.39},
\[
0=(q_{S0}^*\circ i_S^*)(T^*v)=(i^*\circ q_S^*)(T^*v)=i^*(Pv)=i^*(v).
\]
Since $i^*$ is injective, the latter implies that $v=0$, i.e., $I$ is an injection.\medskip
(b) Surject |
ivity of $I$: Let $v\in \bigl(C_0^{k,\omega}(S)\bigr)^*$. Since $T_0^*$ is surjective, there exists $v_1\in \bigl(C_0^{k,\omega}({\mbf R}^n)\bigr)^*$ such that $T_0^*(v_1)=v$. Further, since
$i^*|_{G_b^{k,\omega}({\mbf R}^n)}:G_b^{k,\omega}({\mbf R}^n)\rig | htarrow \bigl(C_0^{k,\omega}({\mbf R}^n)\bigr)^*$ is an isomorphism, there exists $v_2\in G_b^{k,\omega}({\mbf R}^n)$ such that $i^*(v_2)=v_1$.
Now, by the commutativity of \eqref{equ5.38},
\[
v=(T_0^*\circ i^*)(v_2)=(i_S^*\circ T^*)(v_2)=(i_S^*\circ (T^*\ |
circ q_S^*)\circ T^*)(v_2)=(i_S^*\circ T^*)(Pv_2)=I(Pv_2),
\]
i.e., $I$ is a surjection.
So $I$ is a bijection and therefore by the inverse mapping theorem it is an isomorphism.
This completes the proof of the second part of Theorem \ref{te1.4}.
\end{pro | of}
\subsection{Proof of Corollary \ref{cor1.10}}
\begin{proof}
Let $X\subset C_0^{k,\omega}({\mbf R}^n)$ be the closure of the space of $C^\infty$ functions with compact supports on ${\mbf R}^n$. Assume, on the contrary, that there exists $f\in C_0^{k,\om |
ega}({\mbf R}^n)\setminus X$. Then there exists a functional $\lambda\in \bigl(C_0^{k,\omega}({\mbf R}^n)\bigr)^*$ such that
$\lambda|_X=0$ and $\lambda(f)=1$.
Let $i^*: \bigl(C_b^{k,\omega}({\mbf R}^n)\bigr)^*\rightarrow \bigl(C_0^{k,\omega}({\mbf R}^n) | \bigr)^*$ be the adjoint of the isometrical embedding $i:C_0^{k,\omega}({\mbf R}^n)\hookrightarrow C_b^{k,\omega}({\mbf R}^n)$. According to the arguments of the proof of Theorem \ref{te1.4},
$i^*|_{G_b^{k,\omega}({\mbf R}^n)}:G_b^{k,\omega}({\mbf R}^n)\r |
ightarrow \bigl(C_0^{k,\omega}({\mbf R}^n)\bigr)^*$ is an isomorphism. Hence, for $\tilde\lambda:=(i^*|_{G_b^{k,\omega}({\mbf R}^n)})^{-1}(\lambda)$ we have $g(\tilde\lambda)=0$ for all $g\in X$ and $f(\tilde\lambda)=1$. Observe that $X$ is weak$^*$ dense | in $C_0^{k,\omega}({\mbf R}^n)$ (see the proof of Proposition \ref{prop4.2}). Thus $X$ separates the points of $G_b^{k,\omega}({\mbf R}^n)$.
Since $g(\tilde\lambda)=0$ for all $g\in X$, the latter implies that $\tilde\lambda=0$, a contradiction with $f(\ti |
lde\lambda)=1$.
This shows that $X=C_0^{k,\omega}({\mbf R}^n)$.
Clearly, $X$ is separable (it contains, e.g., the dense countable set of functions of the form $\rho_N\cdot p$, $N\in{\mbf N}$, where $p$ are polynomials with rational coefficients and $\{ | \rho_N\}_{N\in{\mbf N}}$ is a fixed sequence of $C^\infty$ cut-off functions weak$^*$ converging in $C_b^{k,\omega}({\mbf R}^n)$ to the constant function $f= 1$).
This completes the proof of the corollary.
\end{proof}
\section{Proof of Theorem \ref{te1.1 |
1}}
\subsection{Proof of Theorem \ref{te1.11} for Weak $k$-Markov Sets}
First, we recall some results proved in \cite{BB1,BB2, B}.
(1) If $S\in {\rm Mar}_k^*({\mbf R}^n)$, then a function $f\in C_b^{k,\omega}(S)$ has derivatives of order $\le k$ at each w | eak $k$-Markov point $x\in S$, i.e., there exists a (unique) polynomial $T_x^k(f)\in\cP_{k,n}$ such that
\[
\lim_{y\to x}\frac{|f(y)-T_x^k(f)(y)|}{\|y-x\|^k}=0.
\]
If $T_x^k(f)(z):=\sum_{|\alpha|\le k} \frac{c_\alpha}{\alpha !} (z-x)^\alpha$, $\alpha\in{\ |
mbf Z}_+^n$, then $c_\alpha$ is called the partial derivative of order $|\alpha|$ at $x$ and is denoted as $D_S^\alpha f (x)$.\medskip
(2) If $\tilde f\in C_b^{k,\omega}({\mbf R}^n)$ is such that $\tilde f|_S=f$, then the Taylor polynomial $T_x^k(\tilde | f)$ of order $k$ of $\tilde f$ at $x$ coincides with $T_x^k(f)$.\medskip
(3) The analog of the the classical Whitney-Glaeser theorem holds:
A function $f\in C(S)$ belongs to $C_b^{k,\omega}(S)$ if and only if it has derivatives of order $\le k$ at each |
weak $k$-Markov point $x\in S$ and there exists a constant $\lambda>0$ such that for all weak $k$-Markov points $x,y\in S$, $z\in\{x,y\}$
\begin{equation}\label{equ6.40}
\begin{array}{l}
\displaystyle
\max_{|\alpha|\le k}|D^\alpha_S f(x)|\le \lambda\quad | \text{and}\\
\\ \displaystyle \max_{|\alpha|\le k}\frac{|D_S^\alpha \bigl(T_x^k(f)-T_y^k(f)\bigr)(z)|}{\|x-y\|^{k-|\alpha|}}\le\lambda\cdot\omega(\|x-y\|).
\end{array}
\end{equation}
Moreover,
\[
\|f\|_{C_b^{k,\omega}(S)}\approx\inf\lambda
\]
with consta |
nts of equivalence depending only on $k$ and $n$.\medskip
(4)
There exists a bounded linear extension operator $T: C_b^{k,\omega}(S)\to C_b^{k,\omega}({\mbf R}^n)$ of finite depths
\[
(Tf)(x):=
\left\{
\begin{array}{ccc}
\displaystyle \sum_{i=1}^\infty \ | lambda_i(x)f(x_i)&{\rm if}&x\in{\mbf R}^n\setminus S
\\
f(x)&{\rm if}& x\in S,
\end{array}
\right.
\]
where all $\lambda_i\in C^\infty({\mbf R}^n)$ and have compact supports in ${\mbf R}^n\setminus S$, all $x_i\in S$ and for each $x\in{\mbf R}^n$ the numbe |
r of nonzero terms in the above sum is at most ${n+k\choose n}\cdot w$, where $w$ is the order of the Whitney cover of ${\mbf R}^n\setminus S$.
The construction of $T$ repeats that of the Whitney-Glaeser extension operator \cite{Gl}, where instead of jets | $T_x^k(f)$ of $f\in C_b^{k,\omega}(S)$ at weak $k$-Markov points $x\in S$ (forming a dense subset of $S$) one uses polynomials of degree $k$ interpolating $f$ on certain subsets of cardinality ${n+k\choose n}$ close to $x$. In particular, as in the case |
of the Whitney-Glaeser extension operator, we obtain that $T\in Ext(C_b^{k,\omega}(S);C_b^{k,\omega}({\mbf R}^n))$ for all moduli of continuity $\omega$. Also, by the construction, if $f\in C_b^{k,\omega}(S)$ is the restriction to $S$ of a $C^\infty$ func | tion with compact support on ${\mbf R}^n$, then $Tf\in C_b^{k,\omega}({\mbf R}^n)$ has compact supports in all closed $\delta$-neihbourhoods of $S$ (i.e., sets $[S]_\delta:=\{x\in{\mbf R}^n\, :\, \inf_{y\in S}\|x-y\|\le\delta\}$, $\delta>0$).
\begin{proof} |
[Proof of Theorem \ref{te1.11} for $S\in {\rm Mar}_k^*({\mbf R}^n)$]
Let $\rho\in C^\infty({\mbf R}^n)$, $0\le \rho\le 1$, be such that $\rho|_{[S]_{1}}=1$, $\rho|_{{\mbf R}^n\setminus [S]_{3}}=0$
and for some $C_{k,n}\in{\mbf R}_+$ (depending on $k$ and $ | n$ only)
\begin{equation}\label{rho1}
\sup_{x\in\mathbb R^n}|D^\alpha\rho(x)|\le C_{k,n}\quad {\rm for\ all}\quad \alpha\in\mathbb Z_+^n.
\end{equation}
(E.g., such $\rho$ can be obtained by the convolution of the indicator function of $[S]_2$ with a fixed |
radial $C^\infty$ function with support in the unit Euclidean ball of ${\mbf R}^n$ and with $L_1({\mbf R}^n)$ norm one.) We define a new extension operator by the formula
\begin{equation}\label{eq6.42}
\widetilde Tf=\rho\cdot Tf,\qquad f\in C_b^{k,\omega} | (S).
\end{equation}
\begin{Lm}\label{lem6.1}
Operator $\widetilde T\in Ext(C_b^{k,\omega}(S);C_b^{k,\omega}({\mbf R}^n))$ for all moduli of continuity $\omega$.
\end{Lm}
\begin{proof}
We equip $C_b^{k,\omega}({\mbf R}^n)$ with equivalent norm \[
\|f\|_{C_b |
^{k,\omega}({\mbf R}^n)}':=\max\left\{\|f\|_{C_b^{k,\omega}({\mbf R}^n)},|f|_{C_b^{k,\omega}({\mbf R}^n)}'\right\},
\]
where $|f|_{C_b^{k,\omega}({\mbf R}^n)}'$ is defined similarly to $|f|_{C_b^{k,\omega}({\mbf R}^n)}$ but with the supremum taken over all | $x\ne y$ such that $\|x-y\|\le 1$, see \eqref{eq3}--\eqref{eq5}. (Note that the constants of equivalence between these two norms depend on $\omega$.)
Now, using word-by-word the arguments of Lemma \ref{norm} with $\rho_\ell$ replaced by $\rho$, $\ell$ re |
placed by $1$, and $c_{k,n}$ replaced by $C_{k,n}$ we obtain for some constant $C=C(k,n,\omega)$ and all $h\in C_b^{k,\omega}({\mbf R}^n)$,
\begin{equation}\label{eq6.43}
\|\rho\cdot h\|_{C_b^{k,\omega}({\mbf R}^n)}'\le C\cdot\|h\|_{C_b^{k,\omega}({\mbf R} | ^n)}'.
\end{equation}
Since $T\in Ext(C_b^{k,\omega}(S);C_b^{k,\omega}({\mbf R}^n))$ for all moduli of continuity $\omega$, inequality \eqref{eq6.43} implies the required statement.
\end{proof}
Clearly, $\widetilde T$ is of finite depth. Moreover, if $f |
\in C_b^{k,\omega}(S)$ is the restriction to $S$ of a $C^\infty$ function with compact support on ${\mbf R}^n$, then $\widetilde Tf\in C_b^{k,\omega}({\mbf R}^n)$ and has compact support on ${\mbf R}^n$ due to the properties of operator $T$, (see part (4) | above). Finally, since the set of all $C_b^{k,1}(S)$ functions (i.e., for this space $\omega(t):=t$, $t\in{\mbf R}_+$) with compact supports on $S$ is dense in $C_0^{k,\omega}(S)$ (because $\omega$ satisfies condition \eqref{omega2}, see Corollary \ref{cor |
1.10}), the preceding property of $\widetilde T$ and Lemma \ref{lem6.1} imply that $\widetilde T(C_0^{k,\omega}(S))\subset C_0^{k,\omega}({\mbf R}^n)$. Therefore $\widetilde T$ satisfies the hypotheses of Theorem \ref{te1.4}\,(2) (weak$^*$ continuity of $\ | widetilde T$ follows from Theorem \ref{teo1.6}). This implies the required statement:
$\bigl(C^{k,\omega}_0(S)\bigr)^*$ is isomorphic to $G_b^{k,\omega}(S)$ for all $\omega$ satisfying \eqref{omega2} and all weak $k$-Markov sets $S$.
Now, $G_b^{k,\omega}( |
S)$ has the metric approximation property due to the Grothendieck result \cite[Ch.\,I]{G} (formulated before Remark
\ref{k} of section~1.4 above) because this space has the approximation property by Theorem \ref{te1.3}. Also, $C_0^{k,\omega}(S)$ has the me | tric approximation property because its dual has it, see, e.g., \cite[Th.\,3.10]{C}.
The proof of the theorem for $S\in {\rm Mar}_k^*({\mbf R}^n)$ is complete.
\end{proof}
\subsection{Proof of Theorem \ref{te1.11} in the General Case}
\begin{proof}
We r |
equire some auxiliary results.
Let $\widetilde\omega$ be the modulus of continuity satisfying
\begin{equation}\label{eq6.46}
\varlimsup_{t\rightarrow 0^+}\frac{\omega_o(t)}{\widetilde\omega(t)}<\infty.
\end{equation}
\begin{Lm}\label{lem6.2}
The restricti | on of the pullback map $H^*:C_b^k({\mbf R}^n)\rightarrow C_b^k({\mbf R}^n)$, $H^*f:=f\circ H$, to $C_b^{k,\tilde\omega}({\mbf R}^n)$ belongs to $\mathcal L(C_b^{k,\tilde\omega}({\mbf R}^n);C_b^{k,\tilde\omega}({\mbf R}^n))$.
\end{Lm}
\begin{proof}
We set |
$H=(h_1,\dots, h_n)$. Then by the hypothesis (a) of the theorem all $D^i h_j\in C_b^{k-1,\omega_o}({\mbf R}^n)$, where $\omega_o$ satisfies \eqref{equ1.8}.
Let $f\in C_b^{k,\tilde\omega}({\mbf R}^n)$. Then for each $\alpha\in{\mbf Z}_+^n$, $|\alpha|\le k$ | , by the Fa\`{a} di Bruno formula, see, e.g., \cite{CS}, we obtain
\begin{equation}\label{eq6.47}
(D^\alpha (f\circ H))(x)=\sum_{0<|\lambda|\le |\alpha|} D^\lambda f(H(x))\cdot P_\lambda\left(\bigl[D^\beta H(x)\bigr]_{0<|\beta|\le |\alpha|}\right);
\end{eq |
uation}
here $P_\lambda\left(\bigl[D^\beta H(x)\bigr]_{0<|\beta|\le |\alpha|}\right)$ are polynomials of degrees $\le |\alpha|$ without constant terms with coefficients in $\mathbb Z_+$ bounded by a constant depending on $k$ and $n$ only in variables $D^\ | beta h_j$, $0<|\beta|\le |\alpha|$, $1\le j\le n$. Since clearly $C_b^{0,\tilde\omega}({\mbf R}^n)$ is a Banach algebra with respect to the pointwise multiplication of functions, to prove the lemma it suffices to check that all $D^\lambda f(H(\cdot))$ and |
$D^\beta h_j$ belong to $C_b^{0,\tilde\omega}({\mbf R}^n)$. For $|\beta|=k$ this is true because $D^\beta h_j\in C_b^{0,\omega_o}({\mbf R}^n)\subset C_b^{0,\widetilde\omega}({\mbf R}^n)$ by the definition of $H$ and by condition \eqref{eq6.46}, while for | $1\le |\beta|\le k-1$ because $D^\beta h_j\in C_b^{0,1}({\mbf R}^n)$ which is continuously embedded into $C_b^{0,\widetilde\omega}({\mbf R}^n)$. Similarly, for $D^\lambda f(H(\cdot))$ with $1\le |\alpha|\le k-1$ this is true because of the continuous embe |
dding $C_b^{0,1}({\mbf R}^n)\hookrightarrow C_b^{0,\widetilde\omega}({\mbf R}^n)$ and because $H$ is Lipschitz, while for $|\lambda|=k$ by the definition of $f$ and the fact that $H$ is Lipschitz.
\end{proof}
Using this lemma we prove the following result | .
\begin{Lm}\label{lem6.3}
The operator $(H|_{S'})^*: C_b^{k,\widetilde\omega}(S)\rightarrow C_b^{k,\widetilde\omega}(S')$, $(H|_{S'})^*f:=f\circ H|_{S'}$, is well-defined and belongs to $\mathcal L(C_b^{k,\widetilde\omega}(S); C_b^{k,\widetilde\omega}(S') |
)$. Moreover, it is weak$^*$ continuous.\footnote{Here the weak$^*$ topologies are defined by means of functionals in $G_b^{k,\widetilde\omega}(\tilde S)$, where $\tilde S$ stands for $S'$ or $S$.}
\end{Lm}
\begin{proof}
Let $\tilde f\in C_b^{k,\widetilde\ | omega}({\mbf R}^n)$ be such that $\tilde f|_{S}=f$ and $\|\tilde f\|_{C_b^{k,\widetilde\omega}({\mbf R}^n)}=\|f\|_{C_b^{k,\widetilde\omega}(S)}$. Then by Lemma \ref{lem6.2} we have
\[
\begin{array}{l}
\displaystyle
(H|_{S'})^*f=f\circ H_{S'}=(\tilde f\circ |
H)|_{S'}=(H^*\tilde f)|_{S'}\in C_b^{k,\widetilde\omega}(S')\quad {\rm and}\\
\\
\displaystyle \|(H|_{S'})^*f\|_{C_b^{k,\widetilde\omega}(S')}\le \|H^*\|\cdot\|\tilde f\|_{C_b^{k,\widetilde\omega}({\mbf R}^n)}=\|H^*\|\cdot\|f\|_{C_b^{k,\widetilde\omega}(S | )}.
\end{array}
\]
This shows that the operator $(H|_{S'})^*: C_b^{k,\widetilde\omega}(S)\rightarrow C_b^{k,\widetilde\omega}(S')$ is well-defined and belongs to $\mathcal L(C_b^{k,\widetilde\omega}(S); C_b^{k,\widetilde\omega}(S'))$.
Further, the fact t |
hat the operator $H^*:C_b^{k,\widetilde\omega}({\mbf R}^n)\rightarrow C_b^{k,\widetilde\omega}({\mbf R}^n)$ is weak$^*$ continuous follows straightforwardly from Proposition \ref{prop3.1}, Lemma \ref{lem6.2} and the the Fa\`{a} di Bruno formula \eqref{eq6. | 47}.
Let $T\in Ext(C_b^{k,\widetilde\omega}(S);C_b^{k,\widetilde\omega}({\mbf R}^n))$ be the extension operator of finite depth (see section~1.3) and $q_{S'}: C_b^{k,\widetilde\omega}({\mbf R}^n)\rightarrow C_b^{k,\widetilde\omega}(S')$ be the quotient map |
induced by restrictions of functions on ${\mbf R}^n$ to $S'$. Then clearly, for all $f\in C_b^{k,\widetilde\omega}(S)$,
\[
(H|_{S'})^*f=f\circ H|_{S'}=((Tf)\circ H)|_{S'}=(q_{S'}\circ H^*\circ T)f.
\]
Therefore $(H|_{S'})^*=q_{S'}\circ H^*\circ T$. Here t | he operator $T$ is weak$^*$ continuous by Theorem \ref{teo1.6} and the operator $q_{S'}$ is weak$^*$ continuous because it is adjoint of the isometric embedding $G_b^{k,\widetilde\omega}(S')\hookrightarrow G_b^{k,\widetilde\omega}({\mbf R}^n)$. This impli |
es that the operator $(H|_{S'})^*$ is weak$^*$ continuous as well.
\end{proof}
We are ready to prove Theorem \ref{te1.11}.
Let $\widetilde T\in Ext(C_b^{k,\omega}(S');C_b^{k,\omega}({\mbf R}^n))$ be the extension operator of the first part of Theorem \ | ref{te1.11}, see \eqref{eq6.42}. We set (for $\widetilde \omega:=\omega$)
\begin{equation}\label{eq6.48}
E:=\widetilde T\circ (H|_{S'})^*.
\end{equation}
\begin{Lm}\label{lem6.4}
Operator $E\in Ext(C_b^{k,\omega}(S); C_b^{k,\omega}({\mbf R}^n))$, is weak$^ |
*$ continuous and maps $C_0^{k,\omega}(S)$ in $C_0^{k,\omega}(S')$.
\end{Lm}
\begin{proof}
The first two statements follow from the hypotheses of the theorem, Lemma \ref{lem6.3} and the fact that $\widetilde T$ is weak$^*$ continuous. So let us check the l | ast statement.
Let $f\in C_0^{k,\omega}(S)$ be the restriction of a $C^\infty$ function with compact support on ${\mbf R}^n$. Since $H|_{S'}:S'\rightarrow S$ is a proper map (by hypothesis (b) of the theorem), $(H|_{S'})^*f\in C_b^{k,\omega}(S')$ has com |
pact support. Moreover, since $f\in C_b^{k,\omega_o}(S)$, Lemma \ref{lem6.3} applied to $\widetilde\omega=\omega_o$ implies that $(H|_{S'})^*f\in C_b^{k,\omega_o}(S')$. Finally, since $\widetilde T\in Ext(C_b^{k,\omega_o}(S');C_b^{k,\omega_o}({\mbf R}^n) | )$ as well, $Ef\in C_b^{k,\omega_o}({\mbf R}^n)$ and has compact support (because $(H|_{S'})^*f$ has it). Due to condition \eqref{equ1.8} for $\omega_o$ we obtain from here that $Ef\in C_0^{k,\omega}({\mbf R}^n)$. Since the set of such functions $f$ is den |
se in $C_0^{k,\omega}(S)$ (see Corollary \ref{cor1.10}), $E$ maps $C_0^{k,\omega}(S)$ in $C_0^{k,\omega}({\mbf R}^n)$, as required.
\end{proof}
Now the result of the theorem follows from Lemma \ref{lem6.4} and Theorem \ref{te1.4}\,(2); that is,
$G_b^{k,\o | mega}(S)$ is isomorphic to $\bigl(C_0^{k,\omega}(S)\bigr)^*$ and so $G_b^{k,\omega}(S)$ and $C_0^{k,\omega}(S)$
have the metric approximation property (see the argument at the end of section~6.1 above).
The proof of the theorem is complete.
\end{proof}
|
\section{Introduction}
\label{sec:introduction}
In recent years there has been a resurgence of interest in the
properties of metastable states, due mostly to the studies of the
jammed states of hard sphere systems; see for revi | ews
Refs. \onlinecite{charbonneau16, baule16}. There are many topics to
study, including for example the spectrum of small perturbations
around the metastable state, i.e. the phonon excitations and the
existence of a boson peak, and whet |
her the Edwards hypothesis works
for these states. In this paper we shall study some of these topics in
the context of classical Heisenberg spin glasses both in the presence
and absence of a random magnetic
field. Here the metastable states which we stu | dy are just the minima of the Hamiltonian, and so are well-defined outside the mean-field limit. It has been known for some time that there are strong
connections between spin glasses and structural glasses
~\cite{tarzia2007glass,f |
ullerton2013growing, moore06}. It has been
argued in very recent work~\cite{baity2015soft} that the study of the
excitations in classical Heisenberg spin glasses provides the opportunity to
contrast with similar phenomenology in | amorphous
solids~\cite{wyart2005geometric, charbonneau15}. The minima and excitations about the minima in
Heisenberg spin glasses have been studied for many years \cite{bm1981,
yeo04, bm1982} but only in the absence of external fields.
|
In Sec. \ref{sec:models} we define the models to be studied as
special cases of the long-range one - dimensional $m$-component vector spin
glass where the exchange interactions $J_{ij}$ decrease with the
distance between the spins | at sites $i$ and $j$ as
$1/r_{ij}^{\sigma}$. The spin $\mathbf{S}_i$ is an $m$-component unit vector. $m=1$ corresponds to the Ising model, $m=2$ corresponds to the XY model and $m=3$ corresponds to the Heisenberg model. By tuning the parame |
ter $\sigma$, one can have
access to the Sherrington-Kirkpatrick (SK) model and on dilution to
the Viana-Bray (VB) model, and indeed to a range of universality
classes from mean-field-type to short-range type
\cite{leuzzi2 | 008dilute}, although in this paper only two special
cases are studied; the SK model and the Viana-Bray model. We intend
to study the cases which correspond to short-range models in a future
publication.
In Sec. \ref{sec:metastability} we have us |
ed numerical methods to learn about
the metastable minima of the SK model and the Viana Bray model. Our main procedure for finding the minima is
to start from a random configuration of
spins and then align each spin with the local field
produced by | its neighbors and the external random field, if present. The process is
continued until all spins are aligned with their local fields. This
procedure finds local minima of the Hamiltonian. In the thermodynamic
limit, the energy per spin $\varepsilon$ |
of these states reaches a characteristic
value, which is the same for almost all realization of the bonds and random external fields, but slightly dependent on the dynamical algorithm used for selecting the spin to be
flipped e.g. the ``polite'' or | ``greedy'' or Glauber dynamics or the sequential algorithm used in the numerical work in this paper
\cite{newman:99,parisi:95}.
In the
context of Ising spin glasses in zero random fields such states were first
studied by Parisi \cite{parisi:95} |
. For Ising spins these
dynamically generated states are an unrepresentative subset of the
totality of the one-spin flip stable metastable states, which in
general have a distribution of local fields $p(h)$ with $p(0)$ is
finite \cite{rober | ts:81}, whereas those generated dynamically are
marginally stable and have $p(h) \sim h$, just like that in the true
ground state \cite{yan:15}. Furthermore these states have a trivial
overlap with each other: $P(q)= \delta(q)$ \cite{parisi:95}; |
there is
no sign of replica symmetry breaking amongst them. Presumably to
generate states which show this feature one needs to start from
initial spin configurations drawn from a realization of the system at
a temperature where broken replic | a symmetry is already present before
the quench.
Because the initial state is random, one would also expect for vector
spin glasses that the states reached after the quench from infinite
temperature would have only a trivial overlap with each |
other
\cite{newman:99} and this is indeed found to be the case in
Sec. \ref{sec:overlap}. We have studied the energy which is reached
in the quench for both the $m=2$ and $m=3$ SK models but for the case
of zero applied random field and | in both cases it is very close to
the energy $E_c$ which marks the boundary above which the minima
where spins are parallel to their local fields have trivial overlaps
with each other, while below it the minima have overlaps with full
broken |
replica symmetry features \cite{bm:81a, bm1981}. In
Ref. \onlinecite{bm1981} the number $N_S(\varepsilon)$ of minima of
energy $\varepsilon$ was calculated for the case of zero random field
in the SK model and in fact it is only for this | model and zero field
that the value of $E_c$ is available. That is why in
Sec. \ref{sec:marginal} only this case was studied numerically. The
work in Sec. \ref{sec:SKanalytic} was the start of an attempt to have
the same information |
in the presence of random vector fields.
The number of minima $N_S(\varepsilon)$ is exponentially large so it
is useful to study the complexity defined as $g(\varepsilon)= \ln
N_S(\varepsilon)/N$, where $N$ is the number of spins in the
| system. Despite the fact that minima exist over a large range of
values of $\varepsilon$ a quench by a particular algorithm seems to
reach just the minima which have a characteristic value of
$\varepsilon$. What is striking is that th |
is characteristic value is close
to the energy $E_c$ at which the minima would no longer have a
trivial overlap with each other but would start to acquire replica
symmetry breaking features, at least for the $m=2$ and $m=3$ SK
models in | zero field. The states reached in the quenches are usually
described as being marginally stable \cite{muller:15}. The
coincidence of the energy obtained in the numerical quenches with the
analytically calculated $E_c$ suggests that long-r |
ange correlations
normally associated with a continuous transition will also be found
for the quenched minima since such features are present in the
analytical work at $E_c$ \cite{bm:81a}. In the Ising case the field
distribution $p(h)$ pro | duced in the quench is very different from
that assumed when determining $E_c$, and the quenched state energy at
$\approx -0.73$ was so far below from the Ising value of $E_c=-0.672$
that the connection of its marginality to the onset of broken repli |
ca
symmetry has been overlooked. We believe that the identification of
the energy $E_c$ reached in the quench with the onset of replica
symmetry breaking in the overlaps of the minima is the most important
of our results.
In Sec. \ref{sec:SK | analytic} we present our analytical work on the
$m$-component SK model in the presence of an $m$-component random
field. It has been shown that in the mean-field limit
~\cite{sharma2010almeida} that under the application of a ra |
ndom
magnetic field, of variance $h_r^2$, there is a phase transition
line in the $h_r - T$ plane, the so-called de Almeida-Thouless (AT)
line, across which the critical exponents lie in the Ising AT
universality class. Below this li | ne, the ordered phase has full
replica symmetry breaking. This ordered phase is similar to the
Gardner phase expected in high-dimensional hard sphere systems
\cite{charbonneau16}. In Sec. \ref{sec:SKanalytic} we study the
minima o |
f the Heisenberg Hamiltonian in the presence of a random
vector field. In the presence of such a field the Hamiltonian no
longer has any rotational invariances so one might expect there to
be big changes in the excitations about the minimum | as there will be
no Goldstone modes in the system.
We start Sec. \ref{sec:SKanalytic} by studying the number of local
minima $N_S(\varepsilon)$ of the Hamiltonian which have energy per
spin of $\varepsilon$. The calculation within th |
e annealed
approximation, where one calculates the field and bond averages of
$N_S(\varepsilon)$ is just an extension of the earlier calculation
of Bray and Moore for zero random field \cite{bm1981}. When the
random field $h_r> h_{AT}$, w | here $h_{AT}$ is the field at which the
AT transition occurs, the complexity is zero, but $g(\varepsilon)$
becomes non-zero for $h_r < h_{AT}$. When it is non-zero, it is
thought better to average the complexity itself over the random
f |
ields and bonds so that one recovers results likely to apply to a
typical sample. We have attempted to calculate the quenched
complexity $g$ for the SK model in the presence of a random field.
The presence of this random field greatly | complicates the algebra
and the calculations in Sec. \ref{sec:quenched} and the Appendix
really just illustrate the problems that random fields pose when
determining the quenched average but do not overcome the algebraic
difficulties.
|
The annealed approximation is much simpler
and using it we have calculated the density of states $\rho(\lambda)$
of the Hessian matrix associated with the minimum for the SK model.
When $h_r > h_{AT}$ there is a gap
$\lambda_0$ in the s | pectrum below which there are no excitations.
$\lambda_0$ tends to zero as $h_r \to h_{AT}$. For $m \ge 4$,
$\rho(\lambda) \sim \sqrt{\lambda-\lambda_0}$ as $\lambda \to
\lambda_0$. For $m =3$ the square root singularity did not o |
ccur,
much to our surprise. For $h_r < h_{AT}$, the square-root singularity
applies for all $m > 2$ with $\lambda_0=0$. Thus in the low-field
phase, despite the fact that in the presence of the random fields
there are no continuous symmetr | ies in the system and hence no
Goldstone modes, there are massless modes present. In
Sec. \ref{sec:density} we present numerical work which shows that
even for $h_r < h_{AT}$ when the annealed calculation of the density of
stat |
es of the SK model cannot be exact, it nevertheless is in
good agreement with our numerical data.
We have also calculated in Sec. \ref{sec:spinglasssusceptibility} the zero temperature spin glass
susceptibility $\chi_{SG}$ for $h_ | r > h_{AT}$ for the SK model and
find that for all $m > 2$ it diverges to infinity as $h_r \to h_{AT}$
just as is found at finite temperatures \cite{sharma2010almeida}.
For the SK model, because the complexity is zero for $h_r > h_{AT}$,
the quen |
ch produces states sensitive to the existence of an AT
field. The quench then goes to a state which is the ground state or at
least one very like it. The AT field is a feature of the true
equilibrium state of the system, which in our cas | e is the state of
lowest energy. In Sec. \ref{sec:spinglasssusceptibility} we have
studied a ``spin glass susceptibility'' obtained from the minima
obtained in our numerical quenches and only for the SK model is there
evidence for a diverging |
spin glass susceptibility. For the VB model,
there is no sign of any singularity in the spin glass susceptibility
defined as an average over the states reached in our quench from
infinite temperature, but we cannot make any statement concerning | the
existence of an AT singularity in the true ground state. This is the
problem studied in Ref. \onlinecite{lupo:16}.
Finally in
Sec. \ref{sec:conclusions} we summarise our main results and make some suggestions for further research.
|
\section{Models}
\label{sec:models}
The Hamiltonians studied in this paper are generically of the form
\begin{equation}
\mathcal{H} = -m\sum_{\langle i, j \rangle} J_{ij} \mathbf{S}_i \cdot \mathbf{S}_j - \sqrt{m}\sum_i \mathbf{h}_i \cdot \mathbf{S}_i \, | ,
\label{Ham}
\end{equation}
where the $\mathbf{S}_i$, $i = 1, 2, \cdots, N$,
are classical $m$-component vector spins of unit length. This form of writing the Hamiltonian allows for easy comparison against a Hamiltonian where the spins are normalized to |
have length $\sqrt{m}$. We are particularly interested in Heisenberg spins, for which $m=3$.
The magnetic fields $h_i^\mu$, where $\mu$ denotes a Cartesian spin component,
are chosen to be
independent Gaussian random fields, uncorrelated between
sites, wi | th zero mean, which satisfy
\begin{equation}
[ h_i^\mu h_j^\nu]_{av} = h_r^2\, \delta_{ij}\, \delta_{\mu\nu} \, .
\label{hs}
\end{equation}
The notation $[\cdots]_{av}$ indicates an average over the quenched disorder and the magnetic fields.
We shall stud |
y two models, the Sherrington-Kirkpatrick (SK) model and the Viana-Bray (VB) model. Both are essentially mean-field models. In the Sherrington-Kirkpatrick model, the bonds $J_{ij}$ couple all pairs of sites and are drawn from a Gaussian distribution with | zero mean and the variance $1/(N-1)$.
The Viana-Bray model can be regarded as a special case of a diluted one-dimensional model where the sites are arranged around a ring.
The procedure to determine the bonds $J_{ij}$ to get the diluted model is as speci |
fied in Refs. \onlinecite{leuzzi2008dilute,sharma2011phase,sharma2011almeida}.
The probability of there being a non-zero interaction between sites $(i,j)$ on the ring falls off with distance as a power-law, and when an
interaction does occur, its variance | is independent of $r_{ij}$. The mean number of non-zero bonds from a site is fixed to be $z$.
To generate the set of pairs $(i,j)$ that have an interaction with the desired probability
the spin $i$ is chosen randomly, and then $j \ (\ne i)$ is chosen at |
distance $r_{ij}$ with probability
\begin{equation}
p_{ij} = \frac{r_{ij}^{-2\sigma}}{\sum_{j\, (j\neq i)}r_{ij}^{-2\sigma}} \, ,
\end{equation}
where $r_{ij}=\frac{N}{\pi}\sin\left[\frac{\pi}{N}(i-j)\right]$ is the
length of the chord between the sites $ | i,j$ when all the sites are
put on a circle. If $i$ and $j$ are already connected, the process is
repeated until a pair which has not been connected before is
found. The sites $i$ and $j$ are then connected with an interaction
picked from a Gaussian inter |
action whose mean is zero and whose
standard deviation is set to $J \equiv 1$. This process is repeated
precisely $N_b = z N / 2 $ times. This procedure automatically gives
$J_{ii} = 0$. Our work concentrates on the case where the coordination number is f | ixed at $z=6$ to mimic the $3$-d cubic scenario.
The SK limit ($z=N-1, \sigma = 0$) is a special case of this model, as is the VB model which also has $\sigma = 0$, but
the coordination number $z$ has (in this paper) the value $6$. The advantage of th |
e one-dimensional long-range model for numerical studies is that by simply tuning the value of $\sigma$ one can mimic the properties of finite dimensional systems~\cite{leuzzi2008dilute,sharma2011phase,sharma2011almeida} and we have already done some work | using this device. However, in this paper we only report on our work on the SK and VB models.
\section{Numerical studies of the minima obtained by quenching}
\label{sec:metastability}
In this section we present our numerical studies of the minima of |
the
VB and SK models. We begin by describing how we found the minima
numerically. They are basically just quenches from infinite
temperature. In Sec. \ref{sec:overlap} we have studied the overlap
between the minima and we find that | the minima produced have only
trivial overlaps with one another. In Sec \ref{sec:marginal} we
describe our evidence that the minima of the SK model in zero field
have marginal stability as they have an energy per spin close to the
energy $E_c |
$ which marks the energy at which the minima starting to
have overlaps showing replica symmetry breaking features.
At zero temperature, the metastable states (minima) which we study are those obtained by aligning every spin along its local field direc | tion, starting off from a random initial state. In the notation used for our numerical work based on Eq. (\ref{Ham}) we iterate the equations
\begin{equation}
\mathbf{S}^{n+1}_i= \frac{\mathbf{H}^{n}_i}{|\mathbf{H}_i^{n}|},
\label{eq:parn}
\end{equation}
|
where the local fields after the $n$th iteration, $\mathbf{H}_i^{n}$, are given by
\begin{equation}
\mathbf{H}_i^n= \sqrt{m} \mathbf{h}_i+m \sum_j J_{ij} \mathbf{S}_j^{n}.
\label{eq:hdefn}
\end{equation}
For a given disorder
sample, a random configuration | of spins is first created which would be a possible spin configuration at infinite temperature. Starting from the
first spin and scanning sequentially all the way up to the $N^{th}$
spin, every spin is aligned to its local field according to
Eq.~(\ref{eq: |
parn}), this whole process constituting one
sweep. The vector $(\Delta \mathbf{S}_{1},\Delta \mathbf{S}_{2},\cdots,\Delta \mathbf{S}_{N})$ is computed by subtracting the
spin configuration before the sweep from the spin configuration
generated after the sw | eep. The quantity $\eta = \frac{1}{Nm}\sum_{\mu=1}^{m}\sqrt{\sum_{j=1}^{N}(\Delta S_{j\mu})^{2}}$ is a measure of how close the configurations before and
after the sweep are. The spin configurations are iterated over many sweeps until the value of $\eta$ f |
alls below $0.00001$, when the system is deemed to have converged to the metastable state described by Eq.~(\ref{eq:par}), which will be a minimum of the energy at zero temperature. Differing starting configurations usually generate different minima, at le | ast for large systems.
\subsection{Overlap distribution}
\label{sec:overlap}
\begin{figure}
\includegraphics[width=\columnwidth]{fig_VB.eps}
\caption{(Color online) The overlap distribution $P(q)$ for the VB model ($\sigma =0, z =6$, $h_r=0.6$) for t |
he minima generated by the prescription described in the text. $P(q)$ seems to be approaching a delta function as $N$ tends to infinity.}
\label{fig0}
\end{figure}
It is informative to study the overlaps between the various minima.
Consider the overlap b | etween two minima $A$ and $B$ defined as
\begin{equation}
q \equiv \frac{1}{N}\sum_{i}\mathbf{S}_i^{A} \cdot \mathbf{S}_i^{B}.
\end{equation}
Numerically, the following procedure is adopted. A particular realization of the bonds and fields is chosen. Choos |
ing a random initial spin configuration, the above algorithm is implemented and descends to a locally stable state. This generates a metastable spin state that is stored. One then chooses a second initial condition, and the algorithm is applied, which gene | rates a second metastable spin state which is also stored. One repeats this $N_{min}$ times generating in total $N_{min}$ metastable states (some or all of which might be identical). One then overlaps all pairs of these states, so there are $N_{pairs} = N_ |
{min}(N_{min}-1)/2$ overlaps which are all used to make a histogram. The whole process is averaged over $N_{samp}$ samples of disorder. Fig.~\ref{fig0} shows the overlap distribution of the metastable states obtained by the above prescription for the VB mo | del. The figure suggests that in the thermodynamic limit, the distribution of overlaps, $P(q) = \delta(q-q_0(h_r))$. In zero field we have
found that $q_0(h_r=0)=0$. Since we study only a finite system of $N$ spins, the delta function peak is broadened |
to a Gaussian centered around $q_0$ and of width $O(\frac{1}{\sqrt{N}})$. We studied also the SK model, for a range of values for the $h_r$ fields, and the data are consistent with $P(q)$ just having a single peak in the thermodynamic limit. This suggests | that the metastable states generated by the procedure of repeatedly putting spins parallel to their local fields starting from a random state always produces minima which have a $P(q)$ of the same type as would be expected for the paramagnetic phase.
N |
ewman and Stein \cite{newman:99} showed that for Ising spins in zero field that when one starts off from an initial state, equivalent to being at infinite temperature, and quenches to zero temperature one always ends up in a state with a trivial $P(q)=\del | ta(q)$, in agreement, for example with the study of Parisi \cite{parisi:95}. Our results for vector spin glasses seem exactly analogous to the Ising results.
\subsection{Marginal stability}
\label{sec:marginal}
In this subsection we shall focus on t |
he Ising, XY ($m = 2$) and
Heisenberg ($m=3$) SK models with zero random field. Parisi found for
the Ising case that when starting a quench from infinite temperature,
when the spins are just randomly up or down, and putting spins
parallel to | their local fields according to various algorithms, the
final state had an energy per spin $\varepsilon=-0.73$
\cite{parisi:95}. In their studies of one-spin flip stable spin
glasses in zero field, Bray and Moore \cite{bm1981,bm |
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