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\section{Introduction}\label{Sec_Setting}
Random fields under local maps are defined and analyzed in different fields of probability and statistics. In these studies a random field is typically a stochastic process whose variables take values in a subset of the real numbers, and which has a geometric index set such as the infinite lattice, a graph with infinite vertex set, or the Euclidean space. It is of interest to understand the behavior of the given random field with distribution $\mu$ under application of a map $T$ that acts on infinite-volume realizations of the process, and investigate resulting properties of the image process whose distribution we denote by $\mu'$.
Relevant deterministic maps $T$ typically have local-dependence windows. They may also be generalized to stochastic kernels that act locally. In this general setup let us call $\mu$ the {\em first-layer measure}, and the measure $\mu'$, which we shall be mostly interested in, the {\em second-layer measure}.
This setup is of theoretical interest, but also occurs in many applications of natural sciences, engineering, and statistics. We mention thinning transformations of Poisson point processes (in which points from a realization are omitted according to local rules)~\cite{bremaud1979optimal,brown1979position,isham1980dependent,rolski1991stochastic,last1993dependent,ball2005poisson,moller2010thinning,blaszczyszyn2019determinantal}, transformations in image analysis~\cite{MR1100283,MR1344683}, and renormalization group transformations in statistical mechanics (where a physical system like a ferromagnet is considered on increasingly large scales by means of maps which forget details on small scales)~\cite{griffiths1978position,griffiths1979mathematical,EnFeSo93}. Processes appearing as local maps in this way can also be viewed as generalizations of the much used {\em hidden Markov models}~\cite{MR1323178}.
Hidden Markov models are images under local kernels of an underlying {\em first-layer} Markov chain, and appear when noisy observations shall be modeled, and so the generalization is made to a situation with spatial index sets, and more complex dependence structures.
It was discovered first in the context of such renormalization group transformations that strictly local transformations acting on a spatially Markovian random field $\mu$ indexed by a lattice, may result in singularities in the image measure $\mu'$. Two concrete examples for this are provided by the low-temperature Ising model under a block average transformation, or the projection to a sublattice~\cite{MR1012855,EnFeSo93,bricmont1998renormalization,le2013almost,MR3830302}. The original motivation of such renormalization group transformations was, suggested by heuristic schemes of theoretical physics, to understand the iterated {\em coarse-graining dynamics} of the level of Hamiltonians to investigate critical behavior~\cite{wilson1974renormalization}.
The singularity in the second-layer measure $\mu'$ means, that $\mu'$ loses not only the spatial Markov property of $\mu$ under the map $T$ (which is less surprising), but it even loses the more general {\em Gibbs property}. This is more severe, it means that the infinite second-layer system acquires internal long-range dependence, and in particular does not posses a well-behaved Hamiltonian with good summability properties of its interaction potentials anymore. The singular long-range dependence appears on the level of finite-volume conditional probabilities of the image measure, which are not quasilocal functions of their conditioning. Put equivalently, finite-volume sub-systems depend on their boundary conditions arbitrarily far away, and their behavior cannot be described by kernels that are continuous in product topology. This may cause standard theory of infinite-volume states, including the variational principle, to fail, see the examples in~\cite{kulske2004relative}.
A variety of examples have been studied ever since, where non-Gibbsian behavior was proved to occur with different mechanisms, but always in regimes of sufficiently strong coupling, where the first-layer measure differs much from independence. (Having said this, there are known examples that show that the range of temperatures where non-Gibbsian behavior in the image system occurs, may be larger than the critical temperature for the first-layer system~\cite{haggstrom2003gibbs}) Moreover, examples have been found where, the set of discontinuity points is even of full measure w.r.t.~$\mu'$ itself, which is the strongest form of singularity~\cite{kulske2004relative,EnErIaKu12,JaKu16,bergmann2020dynamical}.
Main subclasses of relevant transformations which have been studied were projections in terms of a variety of deterministic maps~\cite{van2017decimation,le2013almost,kulske2017fuzzy,MR3648046,MR3961240}, and stochastic time-evolutions~\cite{le2002short,van2002possible,kulske2007spin,EnErIaKu12,roelly2013propagation,FeHoMa14,JaKu16,kraaij2021hamilton}, in various underlying geometries of lattice models, mean-field models, Kac-models, and models in the continuum~\cite{JaKu16}.
\subsubsection*{Informal result: Even independent fields may become non-Gibbs under projections}
In the present paper we provide a new and simple example that shows that a natural local transform of range $1$ can produce singularities, even when it is applied to an {\em independent field}. In our example we chose as the first-layer field the i.i.d.~Bernoulli lattice field $\mu_p$ on the integer lattice, with state space $\{0,1\}^{Z^d}$, and occupation probability $p\in [0,1)$. The Bernoulli lattice field in itself is studied in site-percolation, where one asks for existence of infinite clusters and refined connectedness properties~\cite{grimmett1999percolation}. It also drives more complex processes, in statistical mechanics of disordered systems~\cite{grimmett1997percolation,MR1766342,MR2252929}, and elsewhere in probability
~\cite{adler1991bootstrap,MR2283880,MR3161674,MR3156983,jahnel2020probabilistic}
and its application.
We then study the second-layer measure $\mu_p'$ that appears as an image under application of the concrete range-one map $T$ that is defined by removing from a realization of occupied sites the {\em occupied isolated sites}. $T$ is a projection map as it satisfies $T^2=T$, and we will call it {\em the projection to non-isolates}. Hence it keeps from a realization of occupied sites only the occupied clusters of size of at least two. This includes the infinite cluster, in case there is one, i.e., in the percolation regime of large enough $p$. We may also view $T$ as a simple smoothing transformation, as isolated 'dust' of occupied sites (or 'pixels') is forgotten under the map.
What to expect for the second-layer measure $\mu_p'$? As the $\mu_p$-probability that a given site is isolated equals
$p(1-p)^{2d}$, the map $T$ seems non-invasive, in particular for probabilities $p$ close to $1$. In particular the removed sites do no percolate in this regime. So one might naively conjecture that the second-layer measure shall not be much affected and well-behaved, and in particular is representable as a Gibbsian distribution with quasilocal conditional probabilities.
As a main message of this paper, we prove that this is not the case: We show that $\mu'_p$ is spatially {\em non-Markovian} and {\em non-quasilocal}, when $p<1$ is large, see Theorem~\ref{thm_large_p}. We hope that this result is of some interest for the percolation community. We complement this result by proving regularity of the projected measure when $p\geq 0$
is small enough see, Theorem~\ref{thm_small_p}. This implies the existence of a Gibbs-non Gibbs transition driven by $p$.
Let us finally discuss our map $T$ that projects to non-isolates (and removes the isolates) from a dual perspective. Namely, $T$ has a natural companion map $T^*$ that is again a projection map. $T^*$ does precisely the opposite, it projects to the isolated sites (and removes the non-isolates).
There is independent interest in the action of $T^*$ to the iid Bernoulli lattice field, for the reason that it produces the {\em thinned Bernoulli lattice field}, in which all occupied sites are separated. This thinned Bernoulli lattice field
is relevant, as it is the lattice analogue of the well-known and much studied Mat\'ern process in the continuum~\cite{matern1960spatial,moller2010perfect,baccelli2012extremal}. The latter by definition is derived from a first-layer Poisson process in Euclidean space by removing all points in the realization that have at least one point in the Euclidean ball of radius one.
Clearly, the second-layer measures of both maps $T,T^*$ acting jointly on the same Bernoulli lattice-field realization appear in a natural coupling. As a first guess one may conjecture from this that, either both second-layer measures are Gibbs, or both are non-Gibbs. We warn the reader that this is too naive, not only on the level of proofs, but also the statement may be false. We leave the analysis of the companion process, the thinned Bernoulli lattice field,
to another study.
The paper is organized as follows. In Section~\ref{Sec_Setting} we present the setting and our main results and non-Gibbsianness and Gibbsianness for the Bernoulli field under the removal-of-isolates transformation. In Section~\ref{Sec_Setting} we present the corresponding proofs.
\section{Setting and main results}\label{Sec_Setting}
To define our process we start from the $\Omega=\{0,1\}^{{\Bbb Z}^d}$-valued i.i.d.~Bernoulli field $\mu_p$ with parameter $p\in [0,1]$. We consider realizations of the Bernoulli field under the application of the transformation
$T:\Omega\rightarrow \Omega$
given by
$$
(T\omega)_x:=\omega'_x=\omega_x\Bigl(1-\prod_{y\in \partial x}(1-\omega_y)\Bigr),
$$
where $\partial x$ denotes the set of nearest neighbors of $x\in{\Bbb Z}^d$ in ${\Bbb Z}^d$, equipped with the usual neighborhood structure. In words, $T$ is the projection to the non-isolates, see Figure~\ref{Pix_Trans} for an illustration. The image measure under the transformation $$\mu'_p:=\mu_p\circ T^{-1}$$ is supported on the subset $\Omega':=T(\Omega)$ of sites that obey the non-isolation constraint.
\begin{figure}[!htpb]
\centering
\begin{subfigure}{0.45\textwidth}
\input{Pix_Bern.tex}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\input{Pix_Trans.tex}
\end{subfigure}
\caption{Realization of a Bernoulli field (left) and its image under the transformation $T$ (right).}
\label{Pix_Trans}
\end{figure}
Intuitively the application of $T$ should not change the measure very much at large $p$ close to $1$, where a typical configuration consists of a large percolating cluster and very few isolated sites. In this regime one may view $T$ as a cleansing transformation that wipes away the smallest dust of isolated sites, and keeps the apparent main parts. From the definition of $T$ as a local map it is also obvious that for variables at sites of graph distance greater equal than $3$ are independent, under $\mu'_p$, so that one could expect that $\mu'_p$ is a nicely behaved measure.
Recall that a {\em specification} $\gamma=(\gamma_\Lambda)_{\Lambda\Subset{\Bbb Z}^d}$ is a consistent and proper family of conditional probabilities, i.e., for all $\Lambda\subset\Delta\Subset{\Bbb Z}^d$, $\omega_\Lambda\in \Omega_\Lambda:=\{0,1\}^\Lambda$ and $\hat\omega\in \Omega$, we have that $\int_{\Omega}\gamma_\Delta(\d \tilde\omega|\hat\omega)\gamma_\Lambda(\omega_\Lambda|\tilde\omega)=\gamma_\Delta(\omega_\Lambda|\hat\omega)$, and for all $\omega_{\Lambda^{\rm c}}\in \Omega_{\Lambda^{\rm c}}$ we have $\gamma_\Lambda(\omega_{\Lambda^{\rm c}}|\hat\omega)={\mathds 1}\{\omega_{\Lambda^{\rm c}}=\hat\omega_{\Lambda^{\rm c}}\}$. A specification is called {\em quasilocal}, if for all $\Lambda\Subset{\Bbb Z}^d$ and $\hat\omega_\Lambda \in\Omega_\Lambda$, the mapping $\omega\mapsto\gamma_\Lambda(\hat\omega_\Lambda|\omega)$ is continuous with respect to the product topology on $\Omega$. We say that $\gamma$ is a specification for some random field $\mu$ on $\Omega$, if it satisfies the DLR equations, i.e., for all $\Lambda\Subset{\Bbb Z}^d$ and $\omega_\Lambda\in \Omega_\Lambda$, we have that $\int_{\Omega}\mu(\d \tilde\omega)\gamma_\Lambda(\omega_\Lambda|\tilde\omega)=\mu(\omega_\Lambda)$.
Our present result shows that this is not the case in the whole parameter regime, but the following is true: For large $p$, $\mu'_p$ is not spatially Markovian, it is not even a Gibbs measure in the sense of existence of a version of its finite-volume conditional probabilities which is continuous with respect to the product topology on $\Omega'$. More precisely we have the following theorem.
\begin{thm}(Non-Gibbsianness for large $p$)\label{thm_large_p} Consider the image measure $\mu'_p$ of the Bernoulli field on ${\Bbb Z}^d$ under the map to the non-isolates in lattice dimensions $d\geq 2$. Then, there is $p_c(d)<1$ such that for $p\in (p_c(d),1)$, there is no quasilocal specification
$\gamma'$ for $\mu'_p$.
\end{thm}
The result shows that Gibbsian descriptions of thinning processes of various types derived from Bernoulli or Poissonian fields are by no means obvious, and that more research on such processes in discrete and continuous setups is necessary.
In order to complement the above non-Gibbsianness result, let us also present the following theorem on the existence of a quasilocal specification for $\mu'_p$ for small values of $p$.
\begin{thm}(Gibbsianness for small $p$)\label{thm_small_p} Consider the image measure $\mu'_p$ of the Bernoulli field on ${\Bbb Z}^d$ under the map to the non-isolates in lattice dimensions $d\geq 1$. Then,
for $p< 1/(2d)$, there exists a quasilocal specification $\gamma'$ for $\mu'_p$.
\end{thm}
As we can see from the proofs, for $p< 1/(2d)$, in fact the continuity of $\gamma'$ is even exponentially fast.
In summary, the statements of Theorems~\ref{thm_large_p} and \ref{thm_large_p} indicate a phase-diagram of Gibbsianness of thinned Bernoulli fields under the local non-isolation constraint as exhibited in Figure~\ref{fig_1}.
\begin{figure}[!htpb]
\centering
\begin{tikzpicture}[scale=1]
\draw (-0.1,0) -- (10,0);
\draw (-0.1,1) -- (0,1);
\draw (-0.1,2) -- (0,2);
\draw (-0.1,3) -- (0,3);
\draw (0,0) -- (0,3);
\draw [fill=green,opacity=0.3](0,0) rectangle (10,1);
\draw [fill=red,opacity=0.3](0,2) rectangle (10,3);
\draw [fill=green,opacity=0.3](0,2.98) rectangle (10,3);
\node at (-0.3,3) {$1$};
\node at (-0.6,2) {$p_c(d)$};
\node at (-0.6,1) {$\frac{1}{2d}$};
\node at (-0.3,0) {$0$};
\node at (5,0.5) {Gibbsianness};
\node at (5,2.5) {non-Gibbsianness};
\end{tikzpicture}
\caption{Illustration of Gibbs-non-Gibbs transitions in $p$ for the thinned Bernoulli field under non-isolation constraint.}
\label{fig_1}
\end{figure}
In the following section, we present the proofs.
\section{Proofs}\label{Sec_Proofs}
The key idea of the proof is to first re-express conditional probabilities of $\mu'_p$ in finite volumes in terms of a first-layer constraint model in which occupied sites have to be isolated. Indeed, for large $p$, there are two distinct groundstates given by the (shifted) checkerboard configurations. We can leverage a Peierls' argument in order to show that the first-layer constraint model exhibits a phase transition of translational symmetry breaking. We note that the argument works even though there is {\em no} spin-flip symmetry in the system. The translational symmetry breaking gives rise to a point of discontinuity for which we subsequently show that it is present for any system of finite-volume conditional probabilities for $\mu'_p$, i.e., it is an essential discontinuity. The converse case of small $p$ can be handled by arguments using Dobrushin uniqueness techniques.
\subsection{Proof of Theorem~\ref{thm_large_p}: Non-Gibbianness}
\label{Sec_Proofs_High_p_s}
The main ingredient for the proof is to exhibit one non-removable bad configuration for conditional probabilities. For this, we will use the so-called two-layer view, in which one needs to understand
the Bernoulli field conditional on a fixed image configuration. We choose as the image configuration the all empty configuration, for which the first-layer measure becomes
the Bernoulli field $\mu_p$ {\em conditional on isolation}.
We proceed as follows. In Section~\ref{Sec_Pei} we exhibit a phase transition for the latter model at large $p$, in which translation symmetry is broken, which can be selected via suitable shapes of loophole-volumes. The technique is based on a (slightly non-standard) Peierls argument.
In Section~\ref{Sec_Non} we then show how this implies non-Gibbsianness of the image measure. This is based on the proof that jumps of conditional probabilities occur for certain suitably chosen local patterns, which allow to make a transparent connection to the first-layer model in suitable connected boxes, where the Peierls argument from Section~\ref{Sec_Pei} was made to work.
\subsubsection{Translational-symmetry breaking via a Peierls argument for the conditional first-layer model}\label{Sec_Pei}
For the purpose of showing that the empty configuration is bad for any specification, we will analyze the following particular finite-volume first-layer measures, and we will restrict to particular volumes $\Lambda$. Namely, let us consider finite volumes $\Lambda$ that have a {\em shape of type $0$}, and put fully occupied boundary conditions all $1$ outside of $\Lambda$. By this we mean that $\Lambda$ has a shape which allows to put the checkerboard groundstate of zeros and ones inside $\Lambda$ for which the origin obtains the value $0$ such that one obtains a configuration compatible with the boundary condition, see Figure~\ref{Pix_Types} for an illustration. We define
\begin{equation*}
\begin{split}
\nu_{\Lambda}( \omega_{\Lambda} ):=
\frac{\mu_{p,\Lambda}( \omega_{\Lambda} 1_{T(\omega_{\Lambda}1_{\Lambda^c})|_{\Lambda}=0_{\Lambda}} )}{
\mu_{p,\Lambda}(T(\sigma_{\Lambda}1_{\Lambda^c})|_{\Lambda}=0_{\Lambda} )
},
\end{split}
\end{equation*}
where $\mu_{p,\Lambda}$ is the Bernoulli product measure in $\Lambda$. Hence, by definition $\nu_{\Lambda}$ is the Bernoulli measure conditioned on isolation of ones inside $\Lambda$, where the isolation constraint remembers also the fully occupied boundary condition.
A similar definition is made for volumes of type $1$. For us, large boxes $B_L$ centered around the origin with sidelength $2L$ with a loophole boundary, will be useful, see the illustration in Figure~\ref{Pix_Loop}.
For such type-$0$ boxes $B_L$ we will show in this section that
\begin{equation}\label{eq_0}
\begin{split}
&\sup_{L}\nu_{B_L}(\omega_0=1)\leq \epsilon(p),\qquad \text{ with }\qquad\lim_{p\uparrow 1}\epsilon(p)=0.
\end{split}
\end{equation}
\begin{figure}[!htpb]
\centering
\input{Pix_Loop.tex}
\caption{Illustration of a type-0 volume with loophole boundary.
}
\label{Pix_Loop}
\end{figure}
This means that, with large probability, the origin copies the information from the boundary condition. Similarly, we will prove that the spin at the origin for the box shifted by a lattice unit vector $e$ satisfies
\begin{equation}\label{eq_1}
\begin{split}
&\sup_{L}\nu_{B_L+e}(\omega_0=0)\leq \epsilon(p),\qquad \text{ with }\qquad\lim_{p\uparrow 1}\epsilon(p)=0.
\end{split}
\end{equation}
This is an essential step as it proves that the shape of the volume $B_L$ induces a phase transition for the first-layer constrained model, and there is breaking of translational symmetry. To complete the proof of essential badness of the empty configuration on the second layer, we will however need to go one step further, and connect
to the measure on the second layer. This will be done in Lemma~\ref{lem_NonGibbs} below.
Note that configurations of the model are energetically equivalent under a lattice shift. They are not equivalent under the site-wise {\em spin-flip} that exchanges zeros and ones, much unlike the Ising antiferromagnet in zero external field. Therefore, the Peierls argument we are about to give has to be different from the one for the Ising ferromagnet or antiferromagnet.
Namely, the Peierls argument we will present involves suitable lattice shifts of parts of configurations, while the standard more straightforward Peierls argument for the Ising model involves spin-flips.
Consider the nearest-neighbor graph with vertex set ${\Bbb Z}^d$. Consider, for a spin configuration $\omega$, the set of sites
$$\Gamma(\omega):=\{x \in {\Bbb Z}^d\colon \text{ there exists } y\in \partial x \text{ such that }\omega_x=\omega_y=0 \}.$$
Note that there is a one-to-one correspondence between configurations $\omega$ that satisfy the neighborhood constraint, and sets $\Gamma(\omega)$. Note that outside of $\Gamma(\omega)$, the configuration $\omega$ looks like one of the two groundstates formed by the two possible checkerboard configurations of zeros and ones. Indeed, each site $x\not\in \Gamma(\omega)$ has the property that either $\omega_x=0$ and all the neighbors are $1$ (by definition of a contour),
or $\omega_x=1$ and all the neighbors are $0$ (as the model contains the isolation-constraint).
Further note that not all possible subsets of ${\Bbb Z}^d$ can occur as $\Gamma$, because of the isolation constraint of ones.
The connected (in the sense of graph-distance) components $\gamma$ of these sets $\Gamma$ are called {\em contours}.
To visualize this, consider a star-shaped contour that is built from flipping the one site from one to zero starting from
a checkerboard configuration, see Figure~\ref{Pix_MinCont}. This yields the minimal contour which has $2d+1$ sites. In two dimensions, e.g., it is possible that different contours can be reached from each other via the diagonal. Note that each $\gamma$ that is a contour of a configuration, must be surrounded by ones in nearest neighbor sense in the configuration. These ones must be surrounded by nearest neighbors which carry zeros, by the isolation constraint of ones. Hence the contour specifies the configuration up to sites with graph distance two.
The complement of a finite contour $\gamma$ has one infinite component, and finitely many finite components (the internal ones). Each of these components are labelled by one of the two labels $1$ (or $0$ respectively) determined whether, given $\gamma$, the component admits a configuration obtained by substituting the infinite-volume checkerboard configurations in which the origin in ${\Bbb Z}^d$ obtains a $1$ (or a $0$ respectively), see Figure~\ref{Pix_Types}.
\begin{figure}[!htpb]
\centering
\begin{subfigure}{0.45\textwidth}
\input{Pix_Type0.tex}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\input{Pix_Type1.tex}
\end{subfigure}
\caption{Illustration of the type-0 (left) and type-1 (right) groundstates, i.e., checkerboard configurations. Dots indicate occupied sites. The origin is indicated as $o$. }
\label{Pix_Types}
\end{figure}
A configuration is uniquely determined by its set of compatible contours. Contours $\gamma$ themselves are labelled by $1$ (or $0$) according to the two possible labels of their outer connected components. Contours are compatible when they arise from an allowed configuration, which is the case when the types of checkerboards on shared connected components of the complements match.
\begin{figure}[!htpb]
\centering
\input{Pix_MinCont.tex}
\caption{Illustration of a minimal contour in blue containing the origin. }
\label{Pix_MinCont}
\end{figure}
Now, suppose that $\gamma$ is a contour in a finite volume $\Lambda$. We decompose the volume in terms of the contour and the connected components of its complement
$$\Lambda=\gamma\cup V_0\cup_{i=1,\dots,k}V_i.$$
Here $V_0$ is the outer connected component of the complement of $\gamma$ intersected with $\Lambda$. (Note that the intersection with $\Lambda$ may produce several connected components, but this poses no difficulty.) The sets $(V_i)_{i\geq 1}$ are the interior connected components of the complement of $\gamma$. We also write
$$\overline\gamma=\gamma\cup_{i=1,\dots,k}V_i$$
for the sites that are contained in the support of $\gamma$ or surrounded by $\gamma$.
\medskip
We now prove the statements~\eqref{eq_0} and~\eqref{eq_1} via a Peierls estimate. It suffices
to treat the first case~\eqref{eq_0}, as the case~\eqref{eq_1} is similar. We start with a union bound
over contours surrounding the origin
\begin{equation*}
\begin{split}
\nu_{B_L}(\omega_0=1)&\leq\nu_{B_L}(\omega\colon \exists \gamma \text{ such that }
\Gamma(\omega)\ni \gamma \text{ and } \overline\gamma \ni 0)\leq \sum_{\gamma\colon \overline\gamma \ni 0 }
\nu_{B_L}(\omega\colon\Gamma(\omega)\ni \gamma ).
\end{split}
\end{equation*}
With a slight abuse of notation, we here write $\gamma \in \Gamma$ to indicate that $\gamma$ is a contour in $\Gamma$.
The main point for the Peierls estimate in our non-flip invariant situation, which is formulated in the following lemma,
will be the construction of compatible configurations after removal of contours. Let $|\gamma|$ denote the number of vertices in $\gamma$.
\begin{lem}\label{lem_Pei}
There exists a Peierls constant $\tau=\tau(p)$ with $\lim_{p\uparrow 1}\tau(p)=\infty$, such that for all $\gamma$ we have that
\begin{equation*}
\begin{split}
&\nu_{B_L}(\omega\colon \Gamma(\omega)\ni \gamma)\leq e^{-\tau |\gamma|}.
\end{split}
\end{equation*}
\end{lem}
\begin{proof}Define the activity of a contour $\gamma\subset B_L$
to be the natural weight of the zeros prescribed by it in the Bernoulli measure, i.e.,
$$
\rho(\gamma):=(1-p)^{|\gamma|}.
$$
For any configuration $\omega_U$ in a finite volume $U$
we write for its weight in the Bernoulli field
$$
R(\omega_U):=\prod_{x\in U }p^{\omega_x}(1-p)^{1-\omega_x}.
$$
In particular $\rho(\gamma)=R(0_\gamma)$, where we use the short-hand notation $0_B$ to indicate the all-zero configuration in the volume $B$. Then, we may write
\begin{equation*}
\begin{split}
&\nu_{B_L}(\omega: \Gamma(\omega)\ni \gamma )=\frac{\rho(\gamma)Z_{V_0}\prod_{i=1}^k Z_{V_i}}{Z_{B_L}},
\end{split}
\end{equation*}
where $Z_{V_i}$, for $i=0,1,\dots,k$ denotes the partition functions over all configurations in the volumes $V_i$ which are compatible with $\gamma$, with the isolation constraint on the ones, with weights provided by the Bernoulli measure for all sites in $V_i$.
\paragraph{Case 1.} Suppose that $\gamma$ only contains interior connected components of type $0$. This means that there is no typechange when going from the outside to the inside. Then we may remove $\gamma$, i.e., continue the checkerboard configuration outside of $\gamma$ to where $\gamma$ used to be.
This means that we will assign to each $\omega$ for which $\Gamma(\omega)\ni \gamma$ the reference
configuration $(\omega_{\Lambda \backslash\gamma}\omega^0_{\gamma})$ which appears in the partition function $Z_{\Lambda}$ to lower bound the latter. Here $\omega^0_{\gamma}$ denotes the type-0 checkerboard configuration on $\gamma$.
It is important to note that this removal keeps all other contributions from exterior and interior components compatible.
Hence, we immediately arrive at a lower bound
\begin{equation*}
\begin{split}
&Z_{B_L}\geq R(\omega^0_\gamma)Z_{V_0}\prod_{i=1}^k Z_{V_i}.
\end{split}
\end{equation*}
We may write $\rho(\gamma)=R(\omega^0_\gamma)((1-p)/p)^{N^\text{repl}}$, where $N^\text{repl}$ denotes the number of replacements of a zero by a one on the support of the contour. By definition of a contour each site in $\gamma$ has a neighbor which is zero. Therefore it will replaced itself by a one, or a neighbor of it will be replaced by a one. Hence $N^\text{repl}\geq|\gamma|/(2d +1)$. Thus, we have the desired estimate
\begin{equation*}
\begin{split}
&\nu_{B_L}(\omega: \Gamma(\omega)\ni \gamma )\leq \big((1-p)/p\big)^{|\gamma|/(2d +1)}.
\end{split}
\end{equation*}
\paragraph{Case 2.} Suppose now that $\gamma$ additionally also contains interior volumes of type $1$ (the bad type that does not agree with the boundary outside $\Lambda$), which we will denote by $W_j$, $j=1,\dots,l$. Writing $V_i$ for the interior components of type-0, we have
\begin{equation*}
\begin{split}
&\nu_{B_L}(\omega: \Gamma(\omega)\ni \gamma )=\frac{\rho(\gamma)Z_{V_0}\prod_{i=l+1}^k Z_{V_i}\prod_{j=1}^l Z_{W_j}}{Z_{B_L}},
\end{split}
\end{equation*}
where all partition functions are sums over compatible configurations in the respective connected components, such that the total configuration contains the contour $\gamma$.
The difficulty of this case is that the removal of $\gamma$ does not immediately create compatible configurations. However, it does so after the shift of each of the internal volumes of wrong checkerboard subtypes $W_j$ in one of the $2d$ possible (positive or negative) lattice directions $e$. Let us explain the details now.
Our comparison configuration will now be equal to the type-$0$ checkerboard on the following set $\gamma_e$ which describes the appropriate modification of $\gamma$, obtained by shifts of the internal components,
\begin{equation}\label{Con_Mov}
\begin{split}
\gamma_e:=\Bigl(\gamma \backslash\bigcup_{j=1}^l(W_j+e) \Bigr)\cup
\bigcup_{j=1}^lW_j \backslash (W_j+e).
\end{split}
\end{equation}
Note that there is the volume preservation $|\gamma_e|=|\gamma|$. For each $\omega$ for which $\Gamma(\omega)\ni \gamma$, the reference configuration will then be
\begin{equation}\label{Ref_Con}
\begin{split}
(\omega^0_{\gamma_e},\omega_{\cup_{i=l+1}^k V_i},(\theta_e \omega)_{\cup_{j=1}^l (W_j+e)}),
\end{split}
\end{equation}
where $\theta_e$ represents the shift of the configuration by $e$. We see from the definition that $\omega$ will not be modified on the external component and the internal components of good type. It will however be shifted by $e$ on the internal components of bad type, and it will be a checkerboard of good type on the modified contour $\gamma_e$. Note that this configuration really occurs in $Z_{\Lambda}$ as it satisfies the isolation constraint on the ones. Hence it can be used to lower bound the partition function which gives us
\begin{equation*}
\begin{split}
&Z_\Lambda\geq R(\omega^0_{\gamma_e})
Z_{V_0}\prod_{i=l+1}^k Z_{V_i}\prod_{j=1}^l Z_{W_j},
\end{split}
\end{equation*}
where we have used the shift invariance for the internal partition functions $Z_{W_j}$ (those with the bad types).
Now, the proof is finished once we show that there is a dimension-dependent constant $c_d>0$ such that
\begin{equation}\label{cd}
\begin{split}
&\rho(\gamma)\leq R(\omega^0_{\gamma_e})\big((1-p)/p\big)^{c_d |\gamma|}.
\end{split}
\end{equation}
We denote by $S^0$ the occupied sites of $\omega^0$ (the good checkerboard configuration).
The idea is to use the fact that any connected (in graph distance) subset of ${\Bbb Z}^d$ hits at least a positive fraction of $S^0$ to conclude the inequality
\begin{equation}\label{cd2}
\begin{split}
&|\gamma_e \cap S^0|\geq c_d |\gamma_e|=c_d |\gamma|.
\end{split}
\end{equation}
Then, the Inequality~\eqref{cd} would follow immediately from that.
However, there is the small problem with that argument since, while $\gamma$ by definition is connected (w.r.t.~graph distance), the modified set $\gamma_e$ may have obtained isolated sites, see Figure~\ref{Pix_Iso}.
\begin{figure}[!htpb]
\centering
\begin{subfigure}{0.325\textwidth}
\input{Pix_Iso_l.tex}
\end{subfigure}
\begin{subfigure}{0.325\textwidth}
\input{Pix_Iso_m.tex}
\end{subfigure}
\begin{subfigure}{0.325\textwidth}
\input{Pix_Iso_r.tex}
\end{subfigure}
\caption{Illustration of a configuration with one contour $\gamma$ (left in green), where the outside configuration is of type 0 and the inside configuration is of (bad) type 1. Moving the inside configuration by $-e_1$ (middle) creates a (good) configuration also inside the contour $\gamma_e$ as described in~\eqref{Con_Mov} (middle in green), however the shift also creates isolated zeros. On the right, in the large connected component of $\gamma_e$, sites are indicated in dashed blue, which can be flipped from unoccupied to occupied, as in the configuration presented in~\eqref{Ref_Con}, and therefore create an energetically preferable configuration.}
\label{Pix_Iso}
\end{figure}
This may occur if the only neighbor of such a site is eaten up by a translated interior component. We will now explain that this is not a real problem as the number of those unwanted sites is too small to spoil our desired bound~\eqref{cd2}.
Indeed, consider the case ${\Bbb Z}^2$ and $e=-e_1$ for visualization. Then, in each fixed row, the sets $\gamma$ and $\gamma_e$ contain the same number of sites. To each loss site (on the left) there corresponds a site in the set $\bigcup_{k=1}^lW_l \backslash (W_l+e)$ (which is added to the contour) and which is not isolated but has a nearest neighbor in $\gamma$. From this it follows that $\gamma_e$ still has a nearest-neighbor-connected component $\tilde\gamma_e$ that is at least of size $|\gamma|/2$. Now, already $\tilde\gamma_e$ being connected, it hits a fraction (which we call $\tilde c_d>0$) of $S^0$ and we arrive at
\begin{equation}\label{cd3}
\begin{split}
&|\gamma_e \cap S^0|\geq |\tilde \gamma_e \cap S^0|
\geq \tilde c_d |\tilde \gamma_e| \geq c_d|\gamma|,
\end{split}
\end{equation}
where $c_d=\tilde c_d/2$. Note that we may regard $\tilde\gamma_e$, as obtained in this process, as a contour, and its removal in the line of Case 1.
This proves the claim in the general form and hence proves the Peierls estimate.
\end{proof}
\subsubsection{Non-removable discontinuities for $\mu_p'$ via the relation to Bernoulli fields conditioned on isolation}\label{Sec_Non}
\begin{proof}[Proof of Theorem~\ref{thm_large_p}] Suppose that $\gamma'$ is a any specification for $\mu_p'$.
Consider a square $Q$ of sidelength $3$ around the origin. It will play the role of an observation window. Recall that we write $\omega'$ for the second-layer variables, and we write $\omega$ for the first-layer variables.
We will draw the assumption that
$$\omega'_{Q^c}\mapsto \gamma'_Q(\d \omega'_Q|\omega'_{Q^c})$$
is continuous at $\omega'_{Q^c}=0'_{Q^c}$ to a contradiction, by exhibiting a finite jump size.
We use the notation $1'_{B}$ for the all-one configuration in the volume $B$. A single-site observation window would not show the phenomenon, as a conditioning $0'_{0^c}$ for the model conditional on non-isolation forces the origin to be $0'$ due to the non-isolation constraint.
For the purpose of showing the persistence of jumps on $Q$, we consider the loophole volumes $B_L$ and $B_L+e$ as above, and choose for each $L$ cubes $C_L$ that contain both $B_L$ and $B_L+e$ with a layer of sufficiently large finite thickness. Thickness two will do.
In the first step, we
consider conditional probabilities of the second-layer measure $\mu_p'=T\mu_p$ of the form
\begin{equation*}
\begin{split}
&\mu_p'(\omega'_Q| \omega'_{C_L\backslash Q}).
\end{split}
\end{equation*}
We want to show that they are essentially discontinuous at the empty conditioning, so they cannot come from a quasilocal specification $\gamma'$. The latter argument will be discussed below in the second step, let us now discuss how to obtain the essential discontinuity. There is a slightly tricky part, as we need to go to higher volumes than single-sites, and need to take care of the constraints very carefully.
In order to do consider
\begin{equation*}
\begin{split}
&\mu_p'(\omega'_Q| 0'_{B_L\backslash Q}1'_{C_L\backslash B_L})
\end{split}
\end{equation*}
where $\omega'_Q\in \{0,1\}^Q$.
It is useful to compare with the empty configuration on the second layer
\begin{equation*}
\begin{split}
&\frac{\mu_p'(\omega'_Q| 0'_{B_L\backslash Q}1'_{C_L\backslash B_L})}
{\mu_p'(0'_Q| 0'_{B_L\backslash Q}1'_{C_L\backslash B_L})}=\frac{\mu_p'(\omega'_Q 0'_{B_L\backslash Q}1'_{C_L\backslash B_L})}
{\mu_p'(0'_Q 0'_{B_L\backslash Q}1'_{C_L\backslash B_L})}.
\end{split}
\end{equation*}
The denominators never vanish, as the configuration that appears in the denominator on the right-hand side obeys the non-isolation constraint.
Given the boundary condition $0'_{B_L\backslash Q}$, the non-isolation constraint on the second layer limits the configurations to non-isolated configurations on the cube $Q$ in order to have a non-zero measure.
We will take as a useful local pattern on $Q$, the following second-layer reference configuration, which is most easily visualized in two dimensions where it looks as follows
$$\omega'^*_Q=
\begin{array}{rrr}
0 & 1 & 0 \\
1 & 1 & 1 \\
0 & 1 & 0 \\
\end{array}.$$
It is clearly compatible with the second-layer constraint as $\omega'^*_Q 0'_{Q^c}$ contains
no isolated ones.
In general dimensions $d$ we choose it analogously, namely as the checkerboard groundstate of type-0 but with an additional $1$ at the origin, i.e.,
\begin{equation}\label{Eq_Om}
\begin{split}
(\omega'^*_Q)_i=
\begin{cases}
\omega^0_i & i \in Q\backslash 0 \\
1 & i=0.
\end{cases}
\end{split}
\end{equation}
We have the following useful observation. Given $\omega'^*_Q$, the underlying Bernoulli-field configuration $\omega_Q$ from which it appears as $T$-image, must take the same values on $Q$, independently of $\omega'_{Q^c}$.
To understand the following steps it will be helpful to make a notational distinction and write $\omega^*_Q=\omega'^*_Q$ when we refer to the same configuration as a {\em first-layer} configuration. With this we have
\begin{equation*}
\begin{split}
&\frac{\mu_p'(\omega'^*_Q 0'_{B_L\backslash Q}1'_{C_L\backslash B_L})}
{\mu_p'(0'_Q 0'_{B_L\backslash Q}1'_{C_L\backslash B_L})}=\frac{\mu_{p,C_L}(T\sigma_{C_L}=\omega'^*_Q 0'_{B_L\backslash Q}1'_{C_L\backslash B_L})}
{\mu_{p,C_L}(T\sigma_{C_L}=0'_Q 0'_{B_L\backslash Q}1'_{C_L\backslash B_L})}.
\end{split}
\end{equation*}
Next, we are aiming for a reformulation that involves only quantities of the first-layer model with isolation constraint in the {\em whole volume} $B_L$ including $Q$. For this we perform some manipulations.
We split the numerator as follows,
\begin{equation*}
\begin{split}
&\mu_{p,C_L}(T\sigma_{C_L}=\omega'^*_Q 0'_{B_L\backslash Q}1'_{C_L\backslash B_L})=\mu_{p,Q}(\omega^*_Q) \mu_{p,C_L \backslash Q}\Bigl(T(\omega^*_Q,\sigma_{C_L\backslash Q})\big|_{C_L\backslash Q} =0'_{B_L\backslash Q}1'_{C_L\backslash B_L}\Bigr).
\end{split}
\end{equation*}
Now, we change the middle site on $Q$ on the right-hand side from $1$ to $0$ to obtain a first-layer configuration that obeys the isolation constraint on $Q$. For this, we first write the simple identity
$$\mu_{p,C_L}(\omega^*_Q)=\tfrac{p}{1-p}
\mu_{p,C_L}(\omega^0_Q).$$
It is important to note that we may also replace
$$T_{C_L}(\omega^*_Q,\sigma_{C_L\backslash Q})\big|_{C_L\backslash Q}
=T_{C_L}(\omega^0_Q,\sigma_{C_L\backslash Q})\big|_{C_L\backslash Q},$$
which is possible as the middle site of $Q$ has no influence on the values of the restriction of
$T_{C_L}(\omega^0_Q,\sigma_{C_L\backslash Q})$ to $Q^c$.
So we arrive
at
\begin{equation*}
\begin{split}
&\mu_{p,C_L}(T\sigma_{C_L}=\omega'^*_Q 0'_{B_L\backslash Q}1'_{C_L\backslash B_L})=\frac{p}{1-p}
\mu_{p,C_L}(\sigma_Q=\omega^0_Q, T_{C_L}(\omega^0_Q,\sigma_{C_L\backslash Q})\big|_{C_L\backslash Q}
=0'_{B_L\backslash Q}1'_{C_L\backslash B_L}).
\end{split}
\end{equation*}
Now we have achieved our goal, as we may recognize that
\begin{equation*}
\begin{split}
&\frac{\mu_{p,C_L}(\sigma_Q=\omega^0_Q,
T_{C_L}(\omega^0_Q,\sigma_{C_L\backslash Q})\big|_{C_L\backslash Q}
=0'_{B_L\backslash Q}1'_{C_L\backslash B_L})}
{\mu_{p,C_L}(T\sigma_{C_L}=0'_Q 0'_{B_L\backslash Q}1'_{C_L\backslash B_L})}=\nu_{B_L}(\sigma_Q=\omega^0_Q),
\end{split}
\end{equation*}
with the conditional first-layer measure $\nu_{B_L}$ that is conditioned on isolation, in the whole volume $B_L$, as defined in the beginning of Section~\ref{Sec_Pei}. Indeed, the denominator on the l.h.s.~is the partition function of the conditional measure, up to the terms on $C_L\backslash B_L$ on which the first-layer configuration is frozen, and which cancel against those in the numerator. In particular there is no $C_L$-dependence.
We have thus proved the following {\em unfixing lemma.}
\begin{lem}\label{lem_NonGibbs}
The second-layer conditional probabilities and the first-layer model under the non-isolation constraint satisfy the following relation
\begin{equation*}
\begin{split}
&\frac{\mu_p'(\sigma_Q'=\omega'^*_Q| 0'_{B_L\backslash Q}1'_{C_L\backslash B_L})}
{\mu_p'(\sigma_Q'=0'_Q| 0'_{B_L\backslash Q}1'_{C_L\backslash B_L})}=\frac{p}{1-p}\nu_{B_L}(\sigma_Q=\omega^0_Q),
\end{split}
\end{equation*}
where $\omega^0_Q$ denotes the checkerboard groundstate and $\omega'^*_Q$ is defined in~\eqref{Eq_Om}.
The same relation holds for the shifted volume $B_L+e$.
\end{lem}
Note that this representation is particularly nice, as we are reduced to the discussion of the first-layer model in the full volume $B_L$ (and not a volume reduced by $Q$) where we have the Peierls estimate to our disposition.
In particular, by the Peierls estimate we have that
$$\nu_{B_L}(\sigma_Q=\omega^0_Q)\geq 1-\sum_{j\in Q}
\mu_{B_L}(\sigma_j\neq \omega^0_j)\geq 1-|Q|\epsilon(p),$$
while
$$\nu_{B_L+e}(\sigma_Q=\omega^0_Q)\leq \epsilon(p).$$
In the final step, we bring the arbitrarily chosen specification $\gamma'$ into play, with the aim to show that it must inherit a discontinuity at the empty configuration, too. For any pattern $\omega'_{\Bbb Q}$ in the observation window $Q$, we bound the infimum over perturbations of the empty configuration outside the volume $\Delta_L:=B_L\cap B_{L+e}$ via
\begin{equation*}
\begin{split}
\mu_p'(\omega'_Q| 0'_{B_L\backslash Q}1'_{C_L\backslash B_L})&=\int \gamma'_Q(\omega'_Q| 0'_{B_L\backslash Q}1'_{C_L\backslash B_L}
\tilde\omega'_{C_L^c})\,\,\mu_p'(\d\tilde\omega'_{C_L^c}| \omega'_{C_L\backslash Q}=0'_{B_L\backslash Q}1'_{C_L\backslash B_L})\cr
&\geq \inf_{\omega'_{\Delta_L^c}}
\gamma'_Q(\omega'_Q| 0'_{\Delta_L\backslash Q}
\omega'_{\Delta_L^c})=:a_L(\omega'_Q).
\end{split}
\end{equation*}
Similar arguments give that
\begin{equation*}
\begin{split}
\mu_p'(\omega'_Q| 0'_{B_L+e\backslash Q}1'_{C_L \backslash B_L+e}) &\geq a_L(\omega'_Q)\cr
\mu_p'(\omega'_Q| 0'_{B_L\backslash Q}1'_{C_L\backslash B_L}) &\leq \sup_{\omega'_{\Delta_L^c}}
\gamma_Q(\omega'_Q| 0'_{\Delta_L\backslash Q}
\omega'_{\Delta_L^c})=:b_L(\omega'_Q)\cr
\mu_p'(\omega'_Q| 0'_{B_L+e\backslash Q}1'_{C_L\backslash B_L+e})&\leq b_L(\omega'_Q).
\end{split}
\end{equation*}
Now consider specifically the patterns $0'_Q$ and $\omega'^*_Q$ and note that
\begin{equation}
\begin{split}\label{uppereast}
&\frac{p}{1-p}\nu_{B_L}(\sigma_Q=\omega^0_Q)=\frac{\mu_p'(\omega'^*_Q| 0'_{B_L\backslash Q}1'_{C_L\backslash B_L})}
{\mu_p'(0'_Q| 0'_{B_L\backslash Q}1'_{C_L\backslash B_L})}\leq \frac{b_L(\omega'^*_Q)}{a_L(0_Q)}
\end{split}
\end{equation}
and
\begin{equation}
\begin{split}\label{lowereast}
&\frac{p}{1-p}\nu_{B_L+e}(\sigma_Q=\omega^0_Q)=\frac{\mu_p'(\omega'^*_Q| 0'_{B_L+e\backslash Q}1'_{C_L\backslash B_L+e})}
{\mu_p'(0'_Q| 0'_{B_L+e\backslash Q}1'_{C_L\backslash B_L+e})}
\geq \frac{a_L(\omega'^*_Q)}{b_L(0_Q)},
\end{split}
\end{equation}
and remark that the denominators are uniformly bounded against zero.
Now, by the Peierls estimate presented in Lemma~\ref{lem_Pei} in previous section, we have lower bounds on the left-hand side of~\eqref{uppereast} and upper bounds on the left-hand side of~\eqref{lowereast}. These contradict the continuity assumption on the specification $\gamma'$, i.e., that the right-hand sides have the same limit as $L\uparrow\infty$.
This proves the discontinuity of the specification kernel $\gamma'_Q$ for any arbitrary specification $\gamma'$, at the fully empty configuration.
\end{proof}
\subsection{Proof of Theorem~\ref{thm_small_p}: Gibbsianness}
\label{Sec_Proofs_Low_p_s}
In this section, we construct a continuous specification $\gamma'$ for $\mu_p'$ for small $p$. The main ingredient is an application of the Dobrushin-uniqueness bound and the backward-martingale theorem. In the first step, we construct the conditional probabilities in finite volumes.
For this we use the following notation. For $\Lambda\subset{\Bbb Z}^d$ we denote by $\Lambda^c:={\Bbb Z}^d\setminus \Lambda$ its {\em complement} and by $\partial_-\Lambda:=\{x\in \Lambda\colon \text{there exists }y\in \Lambda^c\text{ with }y\sim_{{\Bbb Z}^d} x\}$ its {\em interior boundary}. The set $\Lambda^o:=\Lambda\setminus\partial_-\Lambda$ denotes the {\em interior} and $\bar\Lambda:=((\Lambda^c)^o)^c$ the {\em extension} of $\Lambda$. Moreover, $\partial_+\Lambda:=\bar\Lambda\setminus \Lambda$ then denotes the {\em outer boundary} of $\Lambda$.
\subsubsection{The specification}
Let us consider, for any (large) finite volume $\Delta\Subset{\Bbb Z}^d$, conditional probabilities of $\mu'_p$ inside $\Delta$ also given a (first-layer) boundary condition $\omega$ outside $\Delta$. More precisely, let $\Lambda\subset\Delta$, $\omega'=\omega'_\Lambda\omega'_{\Delta\setminus\Lambda}\omega'_{\Delta^c}\in \Omega'$ and let $\omega$ be such that $\omega_{\Delta^c}\in T^{-1}(\omega'_{(\Delta^o)^c})$, then
\begin{equation}\label{Eq0}
\begin{split}
\gamma'_{\omega_{\Delta^c}, \Lambda}&(\omega'_\Lambda|\omega'_{\Delta\setminus \Lambda})
:=\frac{
\sum_{\tilde\omega_{\Delta}}\mu_p(\tilde\omega_{\Delta}){\mathds 1}\{T_\Delta(\tilde\omega_{\Delta} \omega_{\Delta^c})=\omega'_\Delta\}}{
\sum_{\tilde\omega_{\Delta\setminus\Lambda^o}}\mu_p(\tilde\omega_{\Delta\setminus\Lambda^o}){\mathds 1}\{T_{\Delta \setminus \Lambda}(\tilde\omega_{\Delta\setminus\Lambda^o} \omega_{\Delta^c})=\omega'_{\Delta\setminus\Lambda}\}
}\\
=&\frac{
\sum_{\tilde\omega_{\Delta\setminus\Lambda^o}}\mu_p(\tilde\omega_{\Delta\setminus\Lambda^o}){\mathds 1}\{T_{\Delta \setminus \Lambda}(\tilde\omega_{\Delta\setminus\Lambda^o} \omega_{\Delta^c})=\omega'_{\Delta\setminus\Lambda}\}
\sum_{\tilde\omega_{\Lambda^o}}\mu_p(\tilde\omega_{\Lambda^o}){\mathds 1}\{T_\Lambda(\tilde\omega_{\Lambda^o}\tilde\omega_{\Delta \setminus \Lambda^o} )=\omega'_{ \Lambda} \}}{
\sum_{\tilde\omega_{\Delta\setminus\Lambda^o}}\mu_p(\tilde\omega_{\Delta\setminus\Lambda^o}){\mathds 1}\{T_{\Delta \setminus \Lambda}(\tilde\omega_{\Delta\setminus\Lambda^o} \omega_{\Delta^c})=\omega'_{\Delta\setminus\Lambda}\}
}\\
=&\frac{
\sum_{\tilde\omega_{\Delta\setminus\Lambda^o}}\mu_p(\tilde\omega_{\Delta\setminus\Lambda^o}){\mathds 1}\{T_{\Delta \setminus \Lambda}(\tilde\omega_{\Delta\setminus\Lambda^o} \omega_{\Delta^c})=\omega'_{\Delta\setminus\Lambda}\}
f_{\omega'_\Lambda}(\tilde\omega_{\partial_- \Lambda\cup\partial_+ \Lambda }) }{
\sum_{\tilde\omega_{\Delta\setminus\Lambda^o}}\mu_p(\tilde\omega_{\Delta\setminus\Lambda^o}){\mathds 1}\{T_{\Delta \setminus \Lambda}(\tilde\omega_{\Delta\setminus\Lambda^o} \omega_{\Delta^c})=\omega'_{\Delta\setminus\Lambda}\}
},
\end{split}
\end{equation}
where we wrote $T_\Lambda(\omega)$ instead of $(T(\omega))_\Lambda$ and
$$
f_{\omega'_\Lambda}(\omega_{\partial_- \Lambda\cup\partial_+ \Lambda }):= \sum_{\tilde\omega_{\Lambda^o}}\mu_p(\tilde\omega_{\Lambda^o}){\mathds 1}\{T_{\Lambda}(\tilde\omega_{\Lambda^o}\omega_{\partial_- \Lambda\cup\partial_+ \Lambda })=\omega'_{ \Lambda} \}
$$
is a local function. We have the following consistency result.
\begin{lem}\label{lem_specification}
Assume that, given $\Lambda \Subset {\Bbb Z}^d$ and $\omega' \in \Omega'$, $\lim_{\Delta \uparrow {{\mathbb Z}^d}} \gamma'_{\omega_{\Delta^c}, \Delta}(\omega'_\Lambda|\omega'_{\Delta\setminus \Lambda})=:\gamma'_\Lambda(\omega'_\Lambda | \omega'_{\Lambda^c})$ exists and is independent of $\omega_{\Delta^c} \in T^{-1}(\omega'_{(\Delta^o)^c})$. Then, $\gamma'$ is a specification for $\mu'_p$.
\end{lem}
\begin{proof}
First note that for any $\omega'\in \Omega'$ we can estimate,
\begin{equation}\label{Eq_o}
\begin{split}
\bigl\lvert \mu'_p(\omega'_\Lambda | \omega'_{\Delta\setminus \Lambda}) - \gamma'_\Lambda(\omega'_\Lambda | \omega'_{\Lambda^c})\bigr \rvert
&\leq \sup_{\omega_{\partial_+\Delta}} \bigl \lvert \gamma'_{\omega_{\partial_+\Delta}, \Delta}(\omega'_\Lambda | \omega'_{\Delta\setminus \Lambda}) - \gamma'_\Lambda(\omega'_\Lambda | \omega'_{\Lambda^c})\bigr \rvert,
\end{split}
\end{equation}
where the supremum is taken over suitable boundary configurations compatible with $\omega'$, since $\mu'_p(\omega'_\Lambda | \omega'_{\Delta\setminus \Lambda})$ can be written as an integral with respect to $\gamma'_{\cdot, \Delta}(\omega'_\Lambda | \omega'_{\Delta\setminus \Lambda})$. In particular, under our assumptions, the right-hand side of Equation~\ref{Eq_o} tends to zero as $\Delta$ tends to ${\Bbb Z}^d$.
Now, consider a cofinal sequence $\Delta_n \uparrow \Lambda^c$ and let $({\cal F}'_{\Delta_n})_{n \in {\Bbb N}}$ denote the canonical filtration on $\Omega'$. Note that the sequence of random variables $\mu'_p(\omega'_\Lambda | {\cal F}'_{\Delta_n})$ that are $\mu'_p$-almost surely defined as $\mu'_p(\omega'_\Lambda | {\cal F}'_{\Delta_n})(\omega') = \mu'_p(\omega'_\Lambda | \omega'_{\Delta_n \setminus \Lambda})$, is a uniformly-integrable martingale adapted to $({\cal F}'_{\Delta_n})_{n \in {\Bbb N}}$. Since $\sigma \left( \bigcup_{n} {\cal F}'_{\Delta_n} \right) = {\cal F}'_{\Lambda^c}$, by L\'evy's zero-one law, $(\mu'_p(\omega'_\Lambda | {\cal F}'_{\Delta_n}))_{n \in {\Bbb N}}$ converges $\mu'_p$-almost surely and in $L^1$ towards $\mu'_p(\omega'_\Lambda | {\cal F}'_{\Lambda^c})$ as $n$ tends to infinity. But this implies that for any $\tilde\omega'_\Lambda$, we can pick $n$ sufficiently large such that
\begin{equation*}
\begin{split}
&\int\mu'_p(\omega')|\mu'_p(\tilde\omega'_\Lambda|\omega'_{\Lambda^c})-\gamma'_\Lambda(\tilde\omega'_\Lambda|\omega'_{\Lambda ^c})|\\
&\le \int\mu'_p(\omega')|\mu'_p(\tilde\omega'_\Lambda|\omega'_{\Lambda^c})-\mu'_p(\tilde\omega'_\Lambda|\omega'_{\Delta_n\setminus\Lambda})|+ \int\mu'_p(\omega')|\mu'_p(\tilde\omega'_\Lambda|\omega'_{\Delta_n\setminus\Lambda})-\gamma'_\Lambda(\tilde\omega'_\Lambda|\omega'_{\Lambda ^c})|<\varepsilon,
\end{split}
\end{equation*}
where we used L\'evy's zero-one law in the first summand and the bound~\eqref{Eq_o} in the second summand on the right-hand side. But this implies that $\int\mu'_p(\omega'_{\Lambda^c})\gamma'_\Lambda(\tilde\omega'_\Lambda|\omega'_{\Lambda ^c})=\mu'_p(\tilde\omega_\Lambda')$ and hence $\gamma'$ is a specification for $\mu'_p$.
\end{proof}
\subsubsection{Transformations into first-layer constraint models}
In order to establish the conditions of Lemma~\ref{lem_specification} for sufficiently small $p$, we employ the Dobrushin uniqueness theorem for the first-layer constraint model as defined in~\eqref{eq_2nd layer1}.
For this, first note that we can uniquely identify $\omega'$ with the subset of its occupied sites in ${\Bbb Z}^d$ and with some notational abuse $\bar\omega'\subset{\Bbb Z}^d$ of $\omega'$ is a {\em fixed area} in the sense that, under $T$, there is no choice for the Bernoulli field in how to realize $\omega'$. Recall that $\omega'$ consist of clusters of size at least two and $\bar\omega'$ then consists of clusters of size at least two surrounded by unoccupied sites, see Figure~\ref{Pix_Fix} for an illustration.
\begin{figure}[!htpb]
\centering
\input{Pix_Fix.tex}
\caption{Illustration of the fixed area (black and white dots) based on a thinned configuration (black dots). The thinned configuration is surrounded by unoccupied sites (white dots).}
\label{Pix_Fix}
\end{figure}
In view of this, we introduce the following specification associated to the {\em first-layer constraint model} on $\Omega$
\begin{equation}\label{eq_2nd layer1}
\gamma^{S}_{\Delta}(\omega_\Delta|\omega_{\Delta^c})
:=\frac{
\mu_p(\omega_{\Delta\cap S}){\mathds 1}\{\omega_{\Delta\cap S}\omega_{\Delta^c\cap S}\text{ is $T$-feasible on } \Delta\cap S\}}{
\sum_{\tilde{\omega}_{\Delta\cap S}}\mu_p(\tilde{\omega}_{\Delta\cap S}){\mathds 1}\{\tilde{\omega}_{\Delta\cap S}\omega_{\Delta^c\cap S}\text{ is $T$-feasible on } \Delta\cap S \}}.
\end{equation}
Here, $S\subset{\Bbb Z}^d$ is an {\em unfixed area} that is arbitrary at this stage, $\Delta\Subset {\Bbb Z}^d$ and any configuration $\omega\in \Omega$ is called {\em $T$-feasible} on a set $\Delta\cap S$ if all occupied sites of $\omega$ in $\Delta\cap S$ have no neighboring occupied sites in $\bar\Delta\cap S$. In particular, with this definition,
\begin{equation*}
\gamma'_{\omega_{\Delta^c}, \Lambda}(\omega'_\Lambda|\omega'_{\Delta\setminus \Lambda})=\gamma^{S}_{\Delta}(f_{\omega'_\Lambda}|\omega_{\Delta^c})
\end{equation*}
for the particular choice of the unfixed area given by
$S=S(\omega'_{\Delta\setminus \Lambda})=(\Delta\setminus \Lambda^o)\setminus \bar\omega_{\Delta\setminus \Lambda}'$.
Here we used that, in the fixed area $\bar\omega_{\Delta\setminus \Lambda}'$, the Bernoulli field is completely determined by $\omega_{\Delta\setminus \Lambda}'$ and hence the corresponding factor cancels in~\eqref{Eq0}. The following result verifies the conditions of Lemma~\ref{lem_specification} for sufficiently small $p$.
\begin{lem}\label{lem_Dob}
Let $p<1/(2d)$. Then, for any $\Lambda \Subset {\Bbb Z}^d$ and $\omega' \in \Omega'$, the limit $\lim_{\Delta \uparrow {{\mathbb Z}^d}} \gamma^{S(\omega'_{\Delta\setminus\Lambda})}_\Delta(f_{\omega'_\Lambda}|\omega_{\Delta^c})$ exists and is independent of $\omega_{\Delta^c} \in T^{-1}(\omega'_{(\Delta^o)^c})$.
\end{lem}
\begin{proof}
We use the Dobrushin-uniqueness approach for the specification $\gamma_\Delta^S$ as defined in~\eqref{eq_2nd layer1}, where $S\subset{\Bbb Z}^d$ is any unfixed area.
Consider the Dobrushin matrix
$$
C_{ij}(p) = \max_{\omega_{j^c} = \tilde{\omega}_{j^c}}\|\gamma^S_i(\cdot |\omega_{i^c})
-\gamma^S_i(\cdot |\tilde{\omega}_{i^c})\|_{\text{TV}}
$$ for $i,j\in S$, where complements are defined in $S$. Note that the exterior boundary $\partial_+S$ of $S$ consists of unoccupied sites, see Figure~\ref{Pix_Fix}. We have $C_{ij}(p)=0$ unless $i$ and $j$ are neighbors in $S$. Otherwise,
\begin{align*}
C_{ij}(p)=\tfrac{1}{2}\max_{\omega_{j^c} = \tilde{\omega}_{j^c}}(|\gamma^S_i(0|\omega_{i^c})
-\gamma^S_i(0 |\tilde{\omega}_{i^c})|+ |\gamma^S_i(1|\omega_{i^c})
-\gamma^S_i(1 |\tilde{\omega}_{i^c})|)=p,
\end{align*}
where the maximum is realized when $\omega_{j^c}$ is unoccupied while $\omega_{j} $ is unoccupied and $\tilde{\omega}_{j}$ is occupied.
In particular, for the Dobrushin criterion, we have
$$
c(p)=\sup_{i\in S}\sum_{j \sim i}C_{ij}(p)\le 2dp,
$$
independent of $S$. By~\cite[Theorem 8.7]{Ge11}, for all $p< 1/(2d)$ and $S$, $\gamma_\Delta^S$ admits a unique infinite-volume Gibbs measure $\mu^S$. Finally, using the remark made above~\cite[Equation 8.25]{Ge11}, $\gamma^{S}_\Delta(f_{\omega'_\Lambda}|\omega_{\Delta^c})$ converges uniformly in $\omega$ towards $\mu^{S}(f_{\omega'_\Lambda})$, which finishes the proof.
\end{proof}
\subsubsection{Quasilocality of the specification}
What remains to be done in order to finish the proof of Theorem~\ref{thm_small_p} is to establish quasilocality for the specification. Let $s$ denote the $\ell_\infty$ metric on ${\Bbb Z}^d$ and define $s(\Lambda,\Delta)=\inf\{s(i,j)\colon i\in \Lambda, j\in \Delta\}$.
\begin{lem}\label{lem_Quasi}
For $p<1/(2d)$ there exist constants $C,c>0$ such that for all $\Lambda\subset\Delta\Subset{\Bbb Z}^d$ and all configurations $\omega'$ and $\tilde\omega'$ with $\omega'_\Delta=\tilde\omega'_\Delta$ we have that
\begin{equation*}
|\gamma'_\Lambda(\omega'_\Lambda|\omega'_{\Lambda^c})-\gamma'_\Lambda(\omega'_\Lambda|\tilde\omega'_{\Lambda^c})|\leq C |\Lambda| e^{-c s(\Lambda, \Delta^c)}.
\end{equation*}
In particular, the specification $\gamma'$ is quasilocal.
\end{lem}
\begin{proof}
We use the representation of $\gamma'_\Lambda(\omega'_\Lambda|\omega'_{\Lambda^c})$ in terms of the unique infinite-volume Gibbs measure $\mu^{S}(f_{\omega'_\Lambda})$ as presented in the proof of Lemma~\ref{lem_Dob}. This representation exists since we work in the Dobrushin-uniqueness regime. Now, for the quasilocality, we use the criterion~\cite[Remark 8.26]{Ge11} applied to~\cite[Theorem 8.20]{Ge11}. More precisely, since $p<1/(2d)$, by~\cite[Theorem 8.20]{Ge11}, for $S\cap\Delta=S'\cap \Delta$, we have that
\begin{equation*}
|\mu^{S}(f_{\omega'_\Lambda})-\mu^{S'}(f_{\omega'_\Lambda})|\le D(\Lambda,\Delta),
\end{equation*}
where $D(\Lambda,\Delta)=\sum_{i\in \Lambda, j\in \Delta^c}\big(\sum_{n\ge 0}C^n\big)_{i,j}$ with $C^n=C^n(p)$ the $n$-th power of the Dobrushin matrix as presented in the proof of Lemma~\ref{lem_Dob}. Now choose $c>0$ sufficiently small such that $p \mathrm e^c<1/(2d)$, then, by~\cite[Remark 8.26]{Ge11},
\begin{equation*}
D(\Lambda,\Delta)\le |\Lambda|(1-2dp\mathrm e^c)^{-1}\mathrm e^{-c d(\Lambda,\Delta^c)}.
\end{equation*}
This finishes the proof.
\end{proof}
\section*{Acknowledgements}
We thank Nils Engler for inspiring discussions. This work was funded by the German Research Foundation under Germany's Excellence Strategy MATH+: The Berlin Mathematics Research Center, EXC-2046/1 project ID: 390685689 and the German Leibniz Association via the Leibniz Competition 2020.
\begin{scriptsize}
|
{
"timestamp": "2021-09-30T02:02:30",
"yymm": "2109",
"arxiv_id": "2109.13997",
"language": "en",
"url": "https://arxiv.org/abs/2109.13997"
}
|
\section{Introduction}\label{intro}
This paper is part of a series~\cite{MMS,MSMS,Bxu} aiming to obtain an asymptotic enumeration of finite Cayley graphs. However, the main players in this paper are not finite Cayley graphs, but finite transitive groups. Our results on finite transitive groups can then be used to make a considerable step towards the enumeration problem of Cayley graphs and thus getting closer to solving an outstanding question of Babai and Godsil, see~\cite{BaGo} or~\cite[Conjecture~3.13]{Go2}.
Let $G$ be a finite transitive group on $\Omega$, let $\alpha\in \Omega$ and let $G_\alpha$ be the stabilizer in $G$ of the point $\alpha$. The orbits of $G_\alpha$ on $\Omega$ are said to be the \textbf{\textit{suborbits}} of $G$ and their cardinalities are said to be the \textbf{\textit{subdegrees}} of $G$. In this paper, we are concerned in finite transitive groups having many subdegrees equal to $1$ or $2$. In particular, we are interested in the ratio
$${\bf I}_\Omega(G):=\frac{|\{\omega\in \Omega\mid \omega \textrm{ lies in a }G_\alpha\textrm{-orbit of cardinality at most two}\}|}{|\Omega|}.$$
As $G$ is transitive on $\Omega$, the value of ${\bf I}_\Omega(G)$ does not depend on $\alpha$. Clearly, $0<{\bf I}_\Omega(G)\le 1$.
\begin{theorem}\label{thrm:main1}
Let $G$ be a finite transitive group on $\Omega$, let $\alpha\in \Omega$ and let $G_\alpha$ be the stabilizer in $G$ of the point $\alpha$. If
${\bf I}_\Omega(G)>\frac{5}{6},$
then ${\bf I}_G(G_\alpha)=1$, that is, each suborbit of $G$ has cardinality at most $2$.
\end{theorem}
It turns out that finite transitive groups $G$ with ${\bf I}_\Omega(G)=1$ are classified by a classical result of Bergman and Lenstra~\cite{BL}. The result of Bergman and Lenstra is rather general and applies to arbitrary (i.~e.~not necessarily finite) groups. The proof of~\cite[Theorem~1]{BL} is very beautiful and it is based on certain equivalence relations; also the strengthening of Isaacs~\cite{isaacs} of the theorem of Bergman and Lenstra has a remarkably ingenious proof.
From~\cite[Theorem~1]{BL}, finite transitive groups with ${\bf I}_\Omega(G)=1$ can be partitioned in three families
\begin{enumerate}[(a)]
\item finite transitive groups $G$ where the stabilizer $G_\alpha$ has order $1$,
\item finite transitive groups $G$ where the stabilizer $G_\alpha$ has order $2$,
\item finite transitive groups $G$ admitting an elementary abelian normal $2$-subgroup $N$ with $|N:G_\alpha|=2$.
\end{enumerate}
In the first family, each suborbit of $G$ has cardinality $1$, that is, $G$ acts regularly on $\Omega$. In the second family, since $G_\alpha$ has cardinality $2$, each orbit of $G_\alpha$ has cardinality at most $2$. In the third family, since $N\unlhd G$, the orbits of $N$ on $\Omega$ form a system of imprimitivity for the action of $G$; as $|N:G_\alpha|=2$, the blocks of this system of imprimitivity have cardinality $2$ and hence all orbits of $G_\alpha$ have cardinality at most $2$.
Theorem~\ref{thrm:main1} shows that, with respect to the operator ${\bf I}_\Omega(G)$, there is a gap between $5/6$ and $1$. The value $5/6$ is special: there exist finite transitive groups attaining the value $5/6$.
\begin{theorem}\label{thrm:main2}
Let $G$ be a finite transitive group on $\Omega$, let $\alpha\in \Omega$ and let $G_\alpha$ be the stabilizer in $G$ of the point $\alpha$. If
${\bf I}_\Omega(G)=\frac{5}{6},$
then there exists an elementary abelian normal $2$-subgroup $N$ of $G$ with $|V:G_\alpha|=|G_\alpha|=4$.
Moreover, let $e_1,e_2,e_3,e_4$ be a basis of $V$, regarded as a $4$-dimensional vector space over the field with $2$ elements, with $G_\alpha=\langle e_1,e_2\rangle$, let $H:=G/{\bf C}_G(V)$ where ${\bf C}_G(V)$ is the centralizer of $V$ in $G$, and let $K$ be the stabilizer of the subspace $W$ in $\mathrm{GL}(V)$. Then, $H$ is $K$-conjugate to one of the following two groups:
$$
\left\langle
\begin{pmatrix}
0& 0& 0& 1\\
1& 1& 0& 0\\
0& 0& 1& 0\\
1& 0& 0& 1
\end{pmatrix},
\begin{pmatrix}
1& 1& 1& 1\\
0& 0& 1& 0\\
0& 1& 0& 0\\
0& 0& 0& 1
\end{pmatrix}
\right\rangle,\,\,\,
\left\langle
\begin{pmatrix}
0& 0& 0& 1\\
1& 1& 0& 0\\
0& 0& 1& 0\\
1& 0& 0& 1
\end{pmatrix},
\begin{pmatrix}
1& 1& 1& 1\\
0& 0& 1& 0\\
0& 1& 0& 0\\
0& 0& 0& 1
\end{pmatrix},
\begin{pmatrix}
1& 0& 0& 0\\
0& 1& 0& 0\\
1& 1& 0& 1\\
1& 1& 1& 0
\end{pmatrix}
\right\rangle.
$$
The first group has order $12$ and is isomorphic to the alternating group of degree $4$ and the second group has order $24$ and is isomorphic to the symmetric group of degree $4$.
Conversely, if $G$ is a finite group containing an elementary abelian normal $2$-subgroup $V:=\langle e_1,e_2,e_3,e_4\rangle$ of order $16$ and $H:=G/{\bf C}_G(V)$ is as above, then the action of $G$ on the set $\Omega$ of the right cosets of $\langle e_1,e_2\rangle$ gives rise to a finite permutation group of degree $4|G:V|$ with ${\bf I}_\Omega(G)=5/6$.
\end{theorem}
Theorem~\ref{thrm:main2} classifies the finite transitive groups attaining the bound $5/6$.
Before discussing our motivation for proving Theorems~\ref{thrm:main1} and~\ref{thrm:main2}, we make some speculations. A computer search among the transitive groups $G$ of degree at most $48$ with the computer algebra system \texttt{magma}~\cite{magma} reveals that, if ${\bf I}_\Omega(G)>1/2$, then ${\bf I}_\Omega(G)=(q+1)/2q$, for some $q\in\mathbb{Q}$ with $2q\in\mathbb{N}$. We pose this as a conjecture.
\begin{conjecture}\label{conj}
{\rm Let $G$ be a finite transitive group on $\Omega$. If ${\bf I}_\Omega(G)>1/2$, then ${\bf I}_\Omega(G)=(q+1)/2q$, for some $q\in\mathbb{Q}$ with $2q\in\mathbb{N}$.}
\end{conjecture}
If true, Conjecture~\ref{conj} establishes a permutation analogue with a classical problem in finite group theory. Let $G$ be a finite group and let $${\bf I}(G):=\{x\in G\mid x \textrm{ has order at most 2}\}.$$ Miller~\cite{Miller} in 1905 has shown that, if ${\bf I}(G)>3/4$, then each element of $G$ has order at most $2$ and hence $G$ is an elementary abelian $2$-group. In this regard, Theorem~\ref{thrm:main1} can be seen as a permutation analogue of the theorem of Miller, with the only difference that the ratio $3/4$ in the context of abstract groups has to bump up to $5/6$ in the context of permutation groups. Miller has also classified the finite groups $G$ with ${\bf I}(G)=3/4$. Therefore, Theorem~\ref{thrm:main2} can be seen as a permutation analogue of the classification of Miller. The theorem of Miller has stimulated a lot of research; for instance, Wall~\cite{Wall} has classified all finite groups $G$ with ${\bf I}(G)>1/2$. In his proof, Wall uses the Frobenius-Schur formula for counting involutions. An application of this classification shows that, if ${\bf I}(G)>1/2$, then ${\bf I}(G)=(q+1)/2q$, for some positive integer $q$. Therefore, in Conjecture~\ref{conj}, we believe that the same type of result holds for the permutation analogue ${\bf I}_\Omega(G)$, but allowing $q$ to be an element of $\{x/2\mid x\in\mathbb{N}\}$. As a wishful thinking, we also pose the following problem.
\begin{problem}\label{problema}
{\rm
Classify the finite transitive groups $G$ acting on $\Omega$ with ${\bf I}_\Omega(G)>1/2$.}
\end{problem}
Liebeck and MacHale~\cite{LieMac} have generalized the results of Miller and Wall in yet another direction. Indeed, Liebeck and MacHale have classified the finite groups $G$ admitting an automorphism inverting more than half of the elements of $G$. (The classical results of Miller and Wall can be recovered by considering the identity automorphism.) Then, this classification has been pushed even further by Fitzpatrick~\cite{fitzpatrick} and Hegarty and MacHale~\cite{hegarty}, by classifying the finite groups $G$ admitting an automorphism inverting exactly half of the elements of $G$. An application of this classification shows that, if $\alpha$ is an automorphism of $G$ inverting more than half of the elements of $G$, then the proportion of elements inverted by $\alpha$ is $(q+1)/2q$, for some positive integer $q$. Yet again, another analogue with Theorems~\ref{thrm:main1} and~\ref{thrm:main2}, with Conjecture~\ref{conj} and with Problem~\ref{problema}. We observe that a partial generalization of this type of results in the context of association schemes is in~\cite{MZ}.
We now discuss our original motivation for proving Theorems~\ref{thrm:main1} and~\ref{thrm:main2}. A \textbf{\textit{digraph}} $\Gamma$ is an ordered pair $(V,E)$ with $V$ a finite non-empty set of vertices, and $E$ is a subset of $V\times V$, representing the arcs. A \textbf{\textit{graph}} $\Gamma$ is a digraph $(V,E)$, where the binary relation $E$ is symmetric. An automorphism of a (di)graph is a permutation on $V$ that preserves the set $E$.
\begin{definition}{\rm
Let $R$ be a group and let $S$ be a subset of $R$. The \textbf{\emph{Cayley digraph}} $\mathop{\Gamma}(R,S)$ is the digraph with $V=R$ and $(r,t) \in E$ if and only if $tr^{-1} \in S$.
The Cayley digraph is a graph if and only if $S=S^{-1}$, that is, $S$ is an inverse-closed subset of $R$.}
\end{definition}
The problem of finding graphical regular representations (GRRs) for groups has a long history. Mathematicians have studied graphs with specified automorphism groups at least as far back as the 1930s, and in the 1970s there were many papers devoted to the topic of finding GRRs (see for example \cite{babai11,Het,Im1, Im2,Im3,NW1,NW2,NW3,Wat}), although the ``GRR" terminology was coined somewhat later.
\begin{definition}{\rm
A \textbf{\emph{digraphical regular representation}} (DRR) for a group $R$ is a digraph whose full automorphism group is the group $R$ acting regularly on the vertices of the digraph.
Similarly, a \textbf{\emph{graphical regular representation}} (GRR) for a group $R$ is a graph whose full automorphism group is the group $R$ acting regularly on the vertices of the graph.}
\end{definition}
It is an easy observation that when $\mathop{\Gamma}(R,S)$ is a Cayley (di)graph, the group $R$ acts regularly on the vertices as a group of graph automorphisms. A DRR (or GRR) for $R$ is therefore a Cayley (di)graph on $R$ that admits no other automorphisms.
The main thrust of much of the work through the 1970s was to determine which groups admit GRRs. This question was ultimately answered by Godsil in~\cite{God}. The corresponding result for DRRs was proved by a much simpler argument by Babai~\cite{babai11}.
Babai and Godsil made the following conjecture. (Given a finite group $R$, $2^{{\bf c}(R)}$ denotes the number of inverse-closed subsets of $R$. See Definition~\ref{defeq:2} for the definition of generalized dicyclic group.)
\begin{conjecture}[\cite{BaGo}; Conjecture 3.13, \cite{Go2}]
{\rm
If $R$ is not generalised dicyclic or abelian of exponent greater than $2$, then for almost all inverse-closed subsets $S$ of $R$, $\mathop{\Gamma}(R,S)$ is a GRR. In other words,
$$\lim_{|R| \to \infty} \min\left\{ \frac{|\{S \subseteq R: S=S^{-1},\,\mathop{\mathrm{Aut}}(\mathop{\Gamma}(R,S))=R\}|}{2^{{\bf c}(R)}}: R\text{ admits a GRR}|\right\} =1.$$}
\end{conjecture}
From Godsil's theorem~\cite{God}, as $|R|\to \infty$, the condition ``$R$ admits a GRR" is equivalent to ``$R$ is neither a generalised dicyclic group, nor abelian of exponent greater than $2$."
The corresponding conjecture for Cayley digraphs (which does not require any families of groups to be excluded) was proved by Morris and the author in~\cite{MSMS}. Our current strategy for proving the conjecture of Babai and Godsil is to use the proof of the corresponding conjecture for Cayley digraphs as a template and extend the work in~\cite{MSMS} in the context of undirected Cayley graphs. This strategy so far has been rather successful and in~\cite{MMS,Bxu} the authors have already adapted some of the arguments in~\cite{MSMS} for undirected graphs.
One key tool in~\cite{MSMS} is an elementary observation of Babai.
\begin{lemma}\label{lemma1}
Let $G$ be a finite transitive group properly containing a regular subgroup $R$. Then there are at most $2^{\frac{3|\Omega|}{4}}$ Cayley digraphs $\Gamma$ on $R$ with $G\leq \mathop{\mathrm{Aut}}(\Gamma)$.
\end{lemma}
The proof of this fact is elementary, see for instance~\cite[Lemma~1.8]{MSMS}. Observe that the number of Cayley digraphs on $R$ is the number of subsets of $R$, that is, $2^{|R|}$. Therefore, Lemma~\ref{lemma1} says that, given $G$ properly containing $R$, only at most $2^{|R|-\frac{|R|}{4}}$ of these Cayley digraphs admit $G$ as a group of automorphisms. This gain of $|R|/4$ is one of the tools in~\cite{MSMS} for proving the Babai-Godsil conjecture on Cayley digraphs.
To continue our project of proving the Babai-Godsil conjecture for Cayley graphs, we need an analogue of Lemma~\ref{lemma1} for Cayley graphs. Observe that the number of Cayley graphs on $R$ is the number of inverse-closed subsets of $R$. We denote this number with $2^{{\bf c}(R)}$. It is not hard to prove (see for instance~\cite[Lemma~$1.12$]{MMS}) that $${\bf c}(R)=\frac{|R|+|{\bf I}(R)|}{2},$$
where ${\bf I}(R)=\{x\in R\mid x^2=1\}$.
To obtain this analogue one needs to investigate finite transitive groups having many suborbits of cardinality at most $2$. Therefore, our investigation leads to the following result.
\begin{theorem}\label{thrm:main3}
Let $G$ be a finite transitive group properly containing a regular subgroup $R$. Then one of the following holds
\begin{enumerate}[(a)]
\item the number of Cayley graphs $\Gamma$ on $R$ with $G\leq \mathop{\mathrm{Aut}}(\Gamma)$ is at most $2^{{\bf c}(R)-\frac{|R|}{96}}$,
\item $R$ is abelian of exponent greater than $2$,
\item $R$ is generalized dicyclic (see Definition~$\ref{defeq:2}$).
\end{enumerate}
\end{theorem}
\subsection{Notation}\label{notation}
In this section, we establish some notation that we use throughout the rest of the paper.
Given a subset $X$ of permutations from $\Omega$, we use an exponential notation for the action on $\Omega$ and hence, in particular, given $\omega\in \Omega$, we let
$$\omega^X:=\{\omega^x\mid x\in X\},$$
where $\omega^x$ is the image of $\omega$ under the permutation $x$.
Similarly, we let
$$\mathrm{Fix}_\Omega(X):=\{\omega\in \Omega\mid \omega^x=\omega,\,\forall x\in X\}.$$
Let $G$ be a transitive permutation group on $\Omega$.
For each positive integer $i$ and for each $\omega\in \Omega$, we let
\begin{equation}\label{notation:1}
\Omega_{\omega,i}:=\{\delta\in \Omega\mid |\delta^{G_\omega}|=i\}.
\end{equation}
Clearly,
\begin{equation}\label{eq:omega0}
\Omega=\Omega_{\omega,1}\cup\Omega_{\omega,2}\cup\Omega_{\omega,3}\cup\cdots
\end{equation} and the non-empty sets in this union form a partition of $\Omega$.
When $i:=1$, we have $$\Omega_{\omega,1}=\{\delta\in \Omega\mid G_\omega\textrm{ fixes }\delta\},$$ that is, $\Omega_{\omega,1}$ is the set of fixed points of $G_\omega$ on $\Omega$. It is well-known that $\Omega_{\omega,1}$ is a block of imprimitivity for the action of $G$ on $\Omega$, see for instance~\cite[1.6.5]{dixonmortimer}. Since this fact will play a role in what follows, we prove it here; this will also be helpful for setting up some additional notation. Let ${\bf N}_{G}(G_\omega)$ be the normalizer of $G_\omega$ in $G$. As ${\bf N}_G(G_\omega)$ contains $G_\omega$, the ${\bf N}_G(G_\omega)$-orbit containing $\omega$ is a block of imprimitivity for the action of $G$ on $\Omega$. Therefore it suffices to prove that $\Omega_{\omega,1}$ is the ${\bf N}_G(G_\omega)$-orbit containg $\omega$, that is, $\Omega_{\omega,1}=\omega^{{\bf N}_G(G_\omega)}=\{\omega^g\mid g\in {\bf N}_G(G_\omega)\}$. If $g\in {\bf N}_G(G_\omega)$, then $G_\omega=G_\omega^g=G_{\omega^g}$ and hence $G_\omega$ fixes $\omega^g$, that is, $\omega^g\in \Omega_{\omega,1}$. Conversely, let $\alpha\in \Omega_{\omega,1}$. As $G$ is transitive on $\Omega$, there exists $g\in G$ with $\alpha=\omega^g$. Thus $\omega^g\in \Omega_{\omega,1}$ and $G_\omega$ fixes $\omega^g$. This yields $G_\omega=G_{\omega^g}=G_\omega^g$ and $g\in{\bf N}_G(G_\omega)$. Therefore, $\alpha=\omega^g$ lies in the ${\bf N}_G(G_\omega)$-orbit containg $\omega$.
We let
\begin{equation*
d:=|\Omega_{\omega,1}|.
\end{equation*}
As $G$ is transitive on $\Omega$, $d$ does not depend on $\omega$. From the previous paragraph, we deduce that $d$ divides $|\Omega_{\omega,i}|$, for each positive integer $i$. We define
\begin{equation}\label{def:xi}
x_i:=\frac{|\Omega_{\omega,i}|}{|\Omega_{\omega,1}|}=\frac{|\Omega_{\omega,i}|}{d}\in\mathbb{N}.
\end{equation}
In particular, $x_1:=1$ and, from~\eqref{eq:omega0} and~\eqref{def:xi}, we have
\begin{equation*
|\Omega|=d\sum_{i}x_i.
\end{equation*}
\begin{definition}\label{defeq:2}{\rm
Let $A$ be an abelian group of even order and of exponent greater than $2$, and let $y$ be an involution of $A$. The generalised dicyclic group ${\rm Dic}(A, y, x)$ is the group $\langle A, x\mid x^2=y, a^x=a^{-1},\forall a\in A\rangle$. A group is called \textit{\textbf{generalised dicyclic}} if it is isomorphic to some ${\rm Dic}(A, y, x)$. When $A$ is cyclic, ${\rm Dic}(A, y, x)$ is called a dicyclic or generalised quaternion group.
}
\end{definition}
\section{Lemmata}\label{sec:lemmata}
In this section we use the notation established in Section~\ref{notation}.
\begin{lemma}\label{lemma:-4}
Let $G$ be a finite permutation group on a set $\Omega$ and let $\alpha\in \Omega$. If
$$\frac{|\Omega|}{2}<|\Omega_{\alpha,1}|+|\Omega_{\alpha,2}|<|\Omega|,$$
then
\begin{enumerate}[(a)]
\item\label{eq:lemma-41}$\Omega=\Omega_{\alpha,1}\cup\Omega_{1,2}\cup\Omega_{\alpha,4}$ (in particular, $\Omega_{\alpha,i}=\emptyset$, for every positive integer $i$ with $i\notin\{1,2,4\}$);
\item\label{eq:lemma-42} for every $\beta\in \Omega_{\alpha,4}$, $\Omega_{\alpha,2}\cap\Omega_{\beta,2}\ne\emptyset$;
\item\label{eq:lemma-43}for every $\beta\in\Omega_{\alpha,4}$ and for every $\omega\in \Omega_{\alpha,2}\cap\Omega_{\beta,2}$, we have
$G_\omega=(G_\alpha\cap G_\omega)(G_\beta\cap G_\omega)$.
\end{enumerate}
\end{lemma}
\begin{proof}
As $|\Omega_{\alpha,1}|+|\Omega_{\alpha,2}|<|\Omega|$, by~\eqref{eq:omega0}, we get that $\Omega_{\alpha,1}\cup\Omega_{\alpha,2}$ is strictly contained in $\Omega$. Therefore,
let $\beta\in \Omega\setminus(\Omega_{\alpha,1}\cup\Omega_{\alpha,2})$.
Since $\beta\notin \Omega_{\alpha,1}\cup\Omega_{\alpha,2}$, we have
\begin{equation}\label{eye-1}
|G_\alpha:G_\alpha\cap G_\beta|=|G_\beta:G_\alpha\cap G_\beta|>2.
\end{equation} See Figure~\ref{figureeye0}.
\begin{figure}[!ht]
\begin{tikzpicture}[node distance=1.3cm]
\node at (0,0) (A0) {$G_\alpha$};
\node[right of=A0] (A1) {};
\node[right of=A1] (A2) {$G_\beta$};
\node[below of=A1] (A3) {$G_\alpha\cap G_\beta$};
\draw[-] (A0) -- node[left]{$>2$}(A3);
\draw[-] (A2) -- node[right]{$>2$}(A3);
\end{tikzpicture}
\caption{}\label{figureeye0}
\end{figure}
From this we deduce
\begin{equation}\label{eye:2}\Omega_{\alpha,1}\cap \Omega_{\beta,1}=\Omega_{\alpha,2}\cap \Omega_{\beta,1}=\Omega_{\alpha,1}\cap \Omega_{\beta,2}=\emptyset.
\end{equation}
Indeed, if for instance $\omega\in \Omega_{\alpha,1}\cap\Omega_{\beta,2}$, then $|\omega^{G_\alpha}|=1$ and $|\omega^{G_\beta}|=2$. Therefore, $|G_\alpha:G_\alpha\cap G_\omega|=1$ and $|G_\beta:G_\beta\cap G_\omega|=2$. As $|G_\alpha:G_\alpha\cap G_\omega|=1$, we get $G_\alpha=G_\omega$. Now, as $|G_\beta:G_\beta\cap G_\omega|=2$ and $G_\alpha=G_\omega$, we get $2=|G_\beta:G_\beta\cap G_\omega|=|G_\beta:G_\beta\cap G_\alpha|$, which contradicts~\eqref{eye-1}. Therefore $\Omega_{\alpha,1}\cap \Omega_{\beta,2}=\emptyset$. The proof for all other equalities in~\eqref{eye:2} is similar.
From~\eqref{eye:2}, we obtain
\begin{equation}\label{eye:3}
(\Omega_{\alpha,1}\cup\Omega_{\alpha,2})\cap (\Omega_{\beta,1}\cup\Omega_{\beta,2})=\Omega_{\alpha,2}\cap\Omega_{\beta,2}.
\end{equation}
Recall that, by hypothesis, $|\Omega_{\alpha,1}\cup\Omega_{\alpha,2}|>|\Omega|/2$. Using this together with~\eqref{eye:3}, we get
\begin{align}\label{eye:4}
|\Omega_{\alpha,2}\cap\Omega_{\beta,2}|&=|(\Omega_{\alpha,1}\cup\Omega_{\alpha,2})\cap (\Omega_{\beta,1}\cup\Omega_{\beta,2})|\\\nonumber
&=|\Omega_{\alpha,1}\cup\Omega_{\alpha,2}|+ |\Omega_{\beta,1}\cup\Omega_{\beta,2}|-|(\Omega_{\alpha,1}\cup\Omega_{\alpha,2})\cup (\Omega_{\beta,1}\cup\Omega_{\beta,2})|\\\nonumber
&\ge |\Omega_{\alpha,1}\cup\Omega_{\alpha,2}|+ |\Omega_{\beta,1}\cup\Omega_{\beta,2}|-|\Omega|\\\nonumber
&>\frac{|\Omega|}{2}+\frac{|\Omega|}{2}-|\Omega|=0.
\end{align}
From~\eqref{eye:4}, we deduce $\Omega_{\alpha,2}\cap\Omega_{\beta,2}\ne\emptyset$. Let $\omega\in \Omega_{\alpha,2}\cap \Omega_{\beta,2}$. In particular, $|\omega^{G_\alpha}|=|\omega^{G_\beta}|=2$. This means that $|G_\alpha:G_\alpha\cap G_\omega|=|G_\beta:G_\beta\cap G_\omega|=2$. Since $|G_\alpha|=|G_\beta|=|G_\omega|$, we get that $G_\alpha\cap G_\omega$ and $G_\beta\cap G_\omega$ have both index $2$ in $G_\omega$. Suppose $G_\alpha\cap G_\omega=G_\beta\cap G_\omega$. Then
$$G_\alpha\cap G_\omega=G_\beta\cap G_\omega=G_\alpha\cap G_\beta\cap G_\omega\le G_\alpha\cap G_\beta$$
and hence
$$|G_\alpha:G_\alpha\cap G_\beta|\le |G_\alpha:G_\alpha\cap G_\omega|=2.$$
However, this contradicts~\eqref{eye-1}. Therefore, $G_\alpha\cap G_\omega$ and $G_\beta\cap G_\omega$ are two distinct subgroups of $G_\omega$ having index $2$. This yields
\begin{equation}\label{eye:5}G_\omega=(G_\alpha\cap G_\omega)(G_\beta\cap G_\omega),
\end{equation}
for each $\omega\in \Omega_{\alpha,2}\cap\Omega_{\beta,2}$.
From~\eqref{eye:5} and from the fact that $|G_\omega:G_\alpha\cap G_\omega|=|G_\omega:G_\beta\cap G_\omega|=2$, we see that $(G_\alpha\cap G_\omega)\cap (G_\beta\cap G_\omega)=G_\alpha\cap G_\beta\cap G_\omega$ has index $4$ in $G_\omega$. Since $|G_\omega|=|G_\alpha|=|G_\beta|$, we get that $G_\alpha\cap G_\beta\cap G_\omega$ has also index $4$ in $G_\alpha$ and in $G_\beta$.
Since $G_\alpha\cap G_\beta\cap G_\omega\le G_\alpha\cap G_\beta$, we get that $|G_\alpha:G_\alpha\cap G_\beta|=|G_\beta:G_\alpha\cap G_\beta|$ divides $|G_\alpha:G_\alpha\cap G_\beta\cap G_\omega|=4$. As $|G_\alpha:G_\alpha\cap G_\beta|=|G_\beta:G_\alpha\cap G_\beta|>2$, we get
$G_\alpha\cap G_\beta\cap G_\omega= G_\alpha\cap G_\beta$
and
$$|G_\alpha:G_\alpha\cap G_\beta|=|G_\beta:G_\alpha\cap G_\beta|=4.$$ We have summarized this paragraph in Figure~\ref{figureeye1}. In other words, $\beta\in \Omega_{\alpha,4}$.
\begin{figure}[!ht]
\begin{tikzpicture}[node distance=2.5cm]
\node at (0,0) (A0) {$G_\omega=(G_\alpha\cap G_\omega)(G_\beta\cap G_\omega)$};
\node[left of=A0] (A1) {};
\node[left of=A1](AA1){$G_\alpha$};
\node[right of=A0] (A2) {};
\node[right of=A2](AA2){$G_\beta$};
\node[below of=A1] (A3) {$G_\alpha\cap G_\omega$};
\node[below of=A2] (A4){$G_\beta\cap G_\omega$};
\node[below of=A0] (B){};
\node[below of=B] (A5){$G_\alpha\cap G_\beta$};
\draw[-] (A0) -- node[left]{2}(A3);
\draw[-] (AA1) -- node[left]{2}(A3);
\draw[-] (A0) -- node[right]{2}(A4);
\draw[-] (AA2) -- node[right]{2}(A4);
\draw[-] (A3) -- node[left]{2}(A5);
\draw[-] (A4) -- node[right]{2}(A5);
\end{tikzpicture}
\caption{}\label{figureeye1}
\end{figure}
Since $\beta$ is an arbitrary element in $\Omega\setminus(\Omega_{\alpha,1}\cup\Omega_{\alpha,2})$, we have proven part~\eqref{eq:lemma-41}. Now, as $\Omega_{\alpha,4}=\Omega\setminus(\Omega_{\alpha,1}\cup\Omega_{\alpha,2})$, part~\eqref{eq:lemma-42} follows from~\eqref{eye:4} and part~\eqref{eq:lemma-43} follows from~\eqref{eye:5}.
\end{proof}
\begin{lemma}\label{lemma:4esteso}
Let $G$ be a finite permutation group on a set $\Omega$ and let $\alpha\in \Omega$. If $\Omega=\Omega_{\alpha,1}\cup\Omega_{\alpha,2}\cup\Omega_{\alpha,4}$ and $|\Omega_{\alpha,1}|=|\Omega_{\alpha,4}|$, then $\Omega_{\alpha,1}\cup\Omega_{\alpha,4}$ is a block of imprimitivity for $G$. Moreover, ${\bf N}_G(G_\alpha)={\bf N}_G(G_\beta)$.
\end{lemma}
\begin{proof}
Let $\beta\in\Omega_{\alpha,4}$. As $\Omega_{\beta,1}\subseteq \Omega_{\alpha,4}$ and as $\Omega_{\beta,1}$ and $\Omega_{\alpha,4}$ have the same cardinality, we deduce $\Omega_{\alpha,4}=\Omega_{\beta,1}$. Analogously, $\Omega_{\beta,4}=\Omega_{\alpha,1}$.
Let $g\in G$ with $\beta=\alpha^g$. Now, we have
$$(\Omega_{\alpha,4})^g=\Omega_{\alpha^g,4}=\Omega_{\beta,4}=\Omega_{\alpha,1}.$$
Analogously, $\Omega_{\alpha,1}^g=\Omega_{\alpha,4}$.
So,
$$\Omega_{\alpha,1}^g=\Omega_{\alpha,4}\hbox{ and }\Omega_{\alpha,4}^g=\Omega_{\alpha,1}.$$
Therefore, $(\Omega_{\alpha,1}\cup\Omega_{\alpha,4})^g=\Omega_{\alpha,1}\cup\Omega_{\alpha,4}$ and $g^2$ fixes setwise $\Omega_{\alpha,1}$ and $\Omega_{\alpha,4}$.
Since $\Omega_{\alpha,1}$ is a block of imprimitivity for $G$ with setwise stabilizer ${\bf N}_G(G_\alpha)$, we deduce $g^2\in {\bf N}_G(G_\alpha)$. Set $T:=\langle {\bf N}_G(G_\alpha),g\rangle$.
Since $G_\alpha$ fixes setwise $\Omega_{\alpha,1}\cup\Omega_{\alpha,4}$, we deduce that $G_\alpha$ fixes setwise also $\Omega_{\alpha,4}=\Omega_{\beta,1}$. Now, for every $x\in {\bf N}_G(G_\alpha)$, we have
$$\Omega_{\alpha,1}^{g^{-1}\alpha g}=(\Omega_{\alpha,1}^{g^{-1}})^{xg}=\Omega_{\beta,1}^{xg}=(\Omega_{\beta,1}^x)^g=\Omega_{\beta,1}^g=\Omega_{\alpha,1}.$$
Thus $g^{-1}xg$ fixes setwise $\Omega_{\alpha,1}$ and hence $g^{-1}xg\in {\bf N}_G(G_\alpha)$. This yields $${\bf N}_G(G_\beta)={\bf N}_G(G_{\alpha^g})=({\bf N}_G(G_\alpha))^g={\bf N}_G(G_\alpha).$$
As $g$ normalizes ${\bf N}_G(G_\alpha)$, we have $T={\bf N}_G(G_\alpha)\langle g\rangle$ and
$$\alpha^T=(\alpha^{{\bf N}_G(G_\alpha)})^{\langle g\rangle}=\Omega_{\alpha,1}^{\langle g\rangle}=\Omega_{\alpha,1}\cup\Omega_{\alpha,4}.$$
Now, since $T$ is an overgroup of $G_\alpha$ and since $\Omega_{\alpha,1}\cup\Omega_{\alpha,4}$ is the $T$-orbit containing $\alpha$, we deduce that $\Omega_{\alpha,1}\cup\Omega_{\alpha,4}$ is a block of imprimitivity for $G$.
\end{proof}
We now need two rather technical lemmas, at first they seem out of context, but their relevance is pivotal in the proof of Lemma~\ref{lemma:-3}. We could phrase Lemma~\ref{lemma:44esteso} in a purely group theoretic terminology, but it is easier to state in our opinion using some terminology from graph theory.
\begin{lemma}\label{lemma:44esteso}
Let $G$ be a group, let $X$ be an elementary abelian $2$-subgroup of $G$, let $Y$ be a $G$-conjugate of $X$ with $Z:=X\cap Y$ having index $4$ in $X$ and in $Y$. Let $\Lambda_X:=\{X_1,X_2,X_3\}$ and $\Lambda_{Y}:=\{Y_1,Y_2,Y_3\}$ be the collection of the proper subgroups of $X$ and $Y$, respectively, properly containing $Z$.
Let $\Gamma$ be the bipartite graph having vertex set $\Lambda_X\cup\Lambda_Y$, where a pair $\{X_i,Y_j\}$
is declared to be adjacent if $X_iY_j$ is a subgroup of $G$ conjugate to $X$ via an element of $G$.
If $\Gamma$ has at least $6$ edges, then $X$ commutes with $Y$.
\end{lemma}
\begin{proof}
Suppose that
\begin{center}
$(\ast)\quad$ there exist two distinct vertices of $\Gamma$ having valency at least $2$.
\end{center} By symmetry, without loss of generality, we suppose that these two vertices are in $\Lambda_X$. Thus suppose that $X_i,X_j\in \Lambda_X$ have valency at least $2$ in $\Gamma$.
Let $Y_{i_1}$ and $Y_{i_2}$ be two neighbours of $X_i$ in $\Gamma$. Then, by definition, $X_iY_{i_1}$ and $X_iY_{i_2}$ are both subgroups of $G$ conjugate to $X$. Therefore, $X_iY_{i_1}$ and $X_iY_{i_2}$ are elementary abelian $2$-groups and hence $X_i$ commutes with both $Y_{i_1}$ and $Y_{i_2}$. Since $\langle Y_{i_1},Y_{i_2}\rangle=Y$, we deduce that $X_i$ commutes with $Y$.
Arguing as in the paragraph above with $X_i$ replaced by $X_j$, we deduce that $X_j$ commutes with $Y$. Therefore, $X=\langle X_i,X_j\rangle$ commutes with $Y$.
Now, it is elementary to see that every bipartite graph on six vertices, with parts having cardinality $3$ and having at least $6$ edges has the property $(\ast)$.
\end{proof}
Recally that a graph $\Gamma$ is said to be vertex-transitive if its automorphism group acts transitively on the vertices of $\Gamma$. Given a vertex $\omega$ of $\Gamma$, we denote by $\Gamma(\omega)$ the neighbourhood of $\omega$ in $\Gamma$.
\begin{lemma}\label{lemma:444esteso}
Let $\Gamma$ be a finite vertex-transitive graph having valency $2$, let $V$ be the set of vertices of $\Gamma$, let $\omega_1,\omega_2$ be two adjacent vertices of $\Gamma$ and let $W$ be a subset of $V$ containing $\omega_1$ and $\omega_2$ and with the property that, for any two distinct vertices $\delta_1,\delta_2$ in $W$, $V\setminus (\Gamma(\delta_1)\cup\Gamma(\delta_2))\subseteq W$. Then either $W=V$ or $|V|\le 6$.
\end{lemma}
\begin{proof}Since $\Gamma$ is vertex-transitive of valency $2$, $\Gamma$ is a disjoint union $s$ of cycles of the same length $\ell$. If $\ell\ge 7$ or if $\Gamma$ is disconnected, that is, $s\ge 2$, it can be easily checked that $W=V$.
\end{proof}
\begin{lemma}\label{lemma:-3}
Let $G$ be a finite permutation group on a set $\Omega$ and let $\alpha\in \Omega$. If
$$\frac{|\Omega|}{2}<|\Omega_{\alpha,1}|+|\Omega_{\alpha,2}|<|\Omega|,$$
then one of the following holds
\begin{enumerate}[(a)]
\item\label{lemma:-30}$|\Omega_{\alpha,1}|+|\Omega_{\alpha,2}|< 5|\Omega|/6$, or
\item\label{lemma:-31}
\begin{enumerate}[(i)]
\item\label{lemma:-322}$|\Omega_{\alpha,4}|\le 2|\Omega_{\alpha,1}|$, and
\item\label{lemma:-32}$G_\alpha$ is an elementary abelian $2$-group, and
\item\label{lemma:-33}$G_\alpha$ commutes with $G_\beta$, for every $\beta\in \Omega_{\alpha,4}$, and
\item\label{lemma:-34}$\langle G_\alpha,G_\beta\rangle=G_\alpha\times G_\beta$ is an elementary abelian normal $2$-subgroup of $G$ of order $16$, for every $\beta\in \Omega_{\alpha,4}$.
\end{enumerate}
\end{enumerate}
\end{lemma}
\begin{proof}
From Lemma~\ref{lemma:-4}, $\Omega=\Omega_{\alpha,1}\cup\Omega_{\alpha,2}\cup \Omega_{\alpha,4}$. Moreover, for each $\beta\in \Omega_{\alpha,4}$, we have shown that $G_\alpha$ contains a proper subgroup (namely, $G_\alpha\cap G_\omega$, for each $\omega\in \Omega_{\alpha,2}\cap\Omega_{\beta,2}$) strictly containg $G_\alpha\cap G_\beta$. This implies that the permutation group, $P$ say, induced by $G_\alpha$ in its action on the suborbit $\beta^{G_\alpha}$ is a $2$-group. (Indeed, if $G_\alpha$ induces the alternating group $\mathrm{Alt}(4)$ or the symmetric group $\mathrm{Sym}(4)$ on $\beta^{G_\alpha}$, then $G_\alpha$ acts primitively on $\beta^{G_\alpha}$ and hence $G_\alpha\cap G_\beta$ is maximal in $G_\alpha$.) Clearly, this $2$-group $P$ must be either cyclic of order $4$, or elementary abelian of order $4$, or dihedral of order $8$.
We have drawn in Figure~\ref{figureeye-2} the lattice of subgroups of the cyclic group of order $4$, the elementary abelian group of order $4$ and the dihedral group of order $8$: the dark colored nodes indicate the lattice of subgroups between the whole group and the stabilizer of a point.
Figure~\ref{figureeye-2} shows that, given $G_\alpha$ and $G_{\alpha}\cap G_\beta$, we only have one choice for $G_\alpha\cap G_\omega$ when $P$ is cyclic of order $4$ or dihedral of order $8$, whereas we have at most three choices for $G_\alpha\cap G_\omega$ when $P$ is elementary abelian of order $4$.
\begin{figure}[!ht]
\begin{tikzpicture}[node distance=1cm,inner sep=0pt]
\node[minimum size=2mm,circle, fill=black] at (-1,0) (A0) {};
\node[minimum size=2mm,circle,fill=black,below of=A0] (A1) {};
\node[minimum size=2mm,circle,fill=black,below of=A1] (A2) {};
\node[minimum size=2mm,circle, fill=black] at (3,0) (B0) {};
\node[minimum size=2mm,circle,fill=black,below of=B0] (B1) {};
\node[minimum size=2mm,circle,fill=black,below of=B1] (B2) {};
\node[minimum size=2mm,circle,fill=black,left of=B1] (B3) {};
\node[minimum size=2mm,circle,fill=black,right of=B1] (B4) {};
\node[minimum size=2mm,circle, fill=black] at (7,0) (C0) {};
\node[minimum size=2mm,circle,fill=lightgray,below of=C0] (C1) {};
\node[minimum size=2mm,circle,fill=black,left of=C1] (C2) {};
\node[minimum size=2mm,circle,fill=lightgray,right of=C1] (C3) {};
\node[minimum size=2mm,circle,fill=lightgray,below of=C1] (C4) {};
\node[minimum size=2mm,circle,fill=black,left of=C4] (C5) {};
\node[minimum size=2mm,circle,fill=lightgray,left of=C5] (C6) {};
\node[minimum size=2mm,circle,fill=lightgray,right of=C4] (C7) {};
\node[minimum size=2mm,circle,fill=lightgray,right of=C7] (C8) {};
\node[minimum size=2mm,circle,fill=lightgray,below of=C4] (C9) {};
\draw[-] (A0) -- (A1);
\draw[-] (A1) -- (A2);
\draw[-] (B0) -- (B1);
\draw[-] (B0) -- (B3);
\draw[-] (B0) -- (B4);
\draw[-] (B2) -- (B1);
\draw[-] (B2) -- (B3);
\draw[-] (B2) -- (B4);
\draw[-] (C0) -- (C1);
\draw[-] (C0) -- (C2);
\draw[-] (C0) -- (C3);
\draw[-] (C4) -- (C1);
\draw[-] (C4) -- (C2);
\draw[-] (C4) -- (C3);
\draw[-] (C9) -- (C5);
\draw[-] (C9) -- (C6);
\draw[-] (C9) -- (C7);
\draw[-] (C9) -- (C8);
\draw[-] (C9) -- (C4);
\draw[-] (C2) -- (C5);
\draw[-] (C2) -- (C6);
\draw[-] (C3) -- (C7);
\draw[-] (C3) -- (C8);
\end{tikzpicture}
\caption{}\label{figureeye-2}
\end{figure}
Given $\beta\in \Omega_{\alpha,4}$, let $$\mathcal{S}_{\alpha,\beta}:=\{G_\omega\mid \omega\in \Omega_{\alpha,2}\cap \Omega_{\beta,2}\}.$$
Observe that in the set $\mathcal{S}_{\alpha,\beta}$ we are collecting point stabilizers and not elements of $\Omega$ and hence different elements $\omega_1,\omega_2$ of $\Omega$ can give rise to the same element of $\mathcal{S}_{\alpha,\beta}$ when $G_{\omega_1}=G_{\omega_2}$.
We claim that
\begin{equation}\label{eye:6}
|\mathcal{S}_{\alpha,\beta}|\le
\begin{cases}
3&\textrm{when the permutation group induced by $G_\alpha$ on $\beta^{G_\alpha}$ or by $G_\beta$ on $\alpha^{G_\beta}$}\\
&\textrm{ is not an elementary abelian $2$-group of order $4$},\\
9&\textrm{otherwise}.
\end{cases}
\end{equation}
This claim follows from the paragraphs above and from Figure~\ref{figureeye-2}. Indeed, from Lemma~\ref{lemma:-4} part~\eqref{eq:lemma-43}, for each $X\in \mathcal{S}_{\alpha,\beta}$, there exists a proper subgroup $A$ of $G_\alpha$ and a proper subgroup $B$ of $G_\beta$ with $G_\alpha\cap G_\beta<A$, $G_\alpha\cap G_\beta<B$ and $X=AB$. Observe that we have at most $3$ choices for $A$ and at most $3$ choices for $B$ and hence at most $9$ choices for $X$. Moreover, as long as the permutation group induced on the corresponding orbit is not elementary abelian, we actually have only one choice for either $A$ or $B$ yielding at most $3$ choices for $X$.
\smallskip
For each $X\in\mathcal{S}_{\alpha,\beta}$, let $\mathcal{S}_X:=\{\omega\in \Omega_{\alpha,2}\cap\Omega_{\beta,2}\mid G_\omega=X\}$. From Section~\ref{notation} and from the notation therein, we have $|\mathcal{S}_X|=|\Omega_{\omega,1}|=d$. From this and from the definition of $\mathcal{S}_{\alpha,\beta}$, we obtain
\begin{equation}\label{eye:11}
|\Omega_{\alpha,2}\cap\Omega_{\beta,2}|=\left|\bigcup_{X\in\mathcal{S}_{\alpha,\beta}}\mathcal{S}_X\right|= \sum_{X\in\mathcal{S}_{\alpha,\beta}}|\mathcal{S}_X|= |\mathcal{S}_{\alpha,\beta}|d.
\end{equation}
From part~\eqref{eq:lemma-41} of Lemma~\ref{lemma:-4}, we have $\Omega=\Omega_{\alpha,1}\cup\Omega_{\alpha,2}\cup\Omega_{\alpha,4}$. From this, we immediately get $\Omega_{\beta,2}\subseteq \Omega\setminus\Omega_{\alpha,1}=\Omega_{\alpha,2}\cup\Omega_{\alpha,4}$ and hence
$\Omega_{\alpha,2}\cup\Omega_{\beta,2}\subseteq \Omega_{\alpha,2}\cup\Omega_{\alpha,4}$. Therefore,
\begin{align}\label{eye:10}
|\Omega_{\alpha,2}\cap\Omega_{\beta,2}|&=|\Omega_{\alpha,2}|+|\Omega_{\beta,2}|-|\Omega_{\alpha,2}\cup\Omega_{\beta,2}|\\\nonumber
&\ge |\Omega_{\alpha,2}|+|\Omega_{\beta,2}|-|\Omega_{\alpha,2}\cup\Omega_{\alpha,4}|\\\nonumber
&= |\Omega_{\alpha,2}|+|\Omega_{\beta,2}|-|\Omega_{\alpha,2}|-|\Omega_{\alpha,4}|\\\nonumber
&=|\Omega_{\alpha,2}|-|\Omega_{\alpha,4}|.
\end{align}
Now, dividing both sides of~\eqref{eye:11} and~\eqref{eye:10} by $|\Omega_{\alpha,1}|=d$, by recalling~\eqref{def:xi} and by rearranging the terms, we obtain
\begin{equation}\label{eye:30}
x_2\le |\mathcal{S}_{\alpha,\beta}|+x_4.
\end{equation}
\smallskip
We now suppose that part~\eqref{lemma:-30} does not hold and we show that part~\eqref{lemma:-322},~\eqref{lemma:-32},~\eqref{lemma:-33} and~\eqref{lemma:-34} are satisfied. In particular, we work under the assumption that
$$|\Omega_{\alpha,1}|+|\Omega_{\alpha,2}|\ge \frac{5|\Omega|}{6}.$$
As $|\Omega|=d(x_1+x_2+x_4)$, $|\Omega_{\alpha,1}|+|\Omega_{\alpha,2}|=d(x_1+x_2)$ and $x_1=1$, the inequality $|\Omega_{\alpha,1}|+|\Omega_{\alpha,2}|\ge 5|\Omega|/6$ gives
$$5x_4\le 1+x_2.$$
Now,~\eqref{eye:30} yields $5x_4\le 1+x_2\le 1+|\mathcal{S}_{\alpha,\beta}|+x_4$, that is, $4x_4\le 1+|\mathcal{S}_{\alpha,\beta}|$. From~\eqref{eye:6}, we deduce that $x_4\le 2$. This already shows part~\eqref{lemma:-322}.
When $x_4=2$, we deduce $|\mathcal{S}_{\alpha,\beta}|\ge 7$ and hence~\eqref{eye:6} yields that the permutation groups induced by $G_\alpha$ on $\beta^{G_\alpha}$ and by $G_\beta$ on $\alpha^{G_\beta}$ are both elementary abelian $2$-groups of order $4$. Since this argument does not depend upon $\beta\in\Omega_{\alpha,4}$, we have shown that $G_\alpha$ acts as an elementary abelian group on each of its orbits of cadinality $4$. Since all other orbits of $G_\alpha$ have cardinality $1$ or $2$, we deduce that $G_\alpha$ acts as an elementary abelian $2$-group on each of its orbits and hence $G_\alpha$ is an elementary abelian $2$-group. This shows part~\eqref{lemma:-32}, under the additional assumption that $x_4=2$. Moreover, as $|\mathcal{S}_{\alpha,\beta}|\ge 7$, Lemma~\ref{lemma:44esteso} applied with $X:=G_\alpha$ and $Y:=G_\beta$ gives that $G_\alpha$ and $G_\beta$ commute with each other. This shows that part~\eqref{lemma:-33} is satisfied. To prove part~\eqref{lemma:-34} we use Lemma~\ref{lemma:444esteso}. Let $\Gamma$ be the graph having vertex set $V$, the set of conjugates of $G_\alpha$ in $G$, that is, $$V:=\{G_\omega\mid\omega\in \Omega\}.$$ Then $|V|=1+x_2+x_4$. We declare two vertices $G_{\omega_1}$ and $G_{\omega_2}$ of $\Gamma$ adjacent if $G_{\omega_1}\cap G_{\omega_2}$ has index $4$ in $G_{\omega_1}$ (and hence also in $G_{\omega_2}$). Clearly, the action of $G$ by conjugation gives rise to a vertex-transitive action of $G$ on $\Gamma$. As $x_4=2$, $\Gamma$ has valency $2$. Let $W$ be the collection of all vertices $G_\omega$ of $\Gamma$ with $G_\alpha \cap G_\beta\le G_\omega$. Clearly, $G_\alpha,G_\beta\in W$ and, from Lemma~\ref{lemma:-4} part~\eqref{eq:lemma-43}, for any two distinct vertices $G_{\delta_1}$ and $G_{\delta_2}$ of $\Gamma$ contained in $W$, we have that $$\Omega_{\delta_1,2}\cap\Omega_{\delta_2,2}=V\setminus (\Gamma(G_{\delta_1})\cup\Gamma(G_{\delta_2}))\subseteq W.$$ From this, Lemma~\ref{lemma:444esteso} gives that either $W=V$ or $|V|\le 6$. The second alternative gives $x_2=|V|-1-x_4\le 3$, which contradicts the fact that $5x_4\le 1+x_2$. Therefore, $W=V$ and hence $G_\alpha\cap G_\beta\le G_\omega$, for every $\omega\in \Omega$. Thus $G_\alpha\cap G_\beta=1$ and hence $G_\alpha G_\beta=G_\alpha\times G_\beta$ is an elementary abelian $2$-group of order $16$. To prove that $G_\alpha\times G_\beta\unlhd G$ it suffices to apply again this argument to the collection $W$ of all vertices $G_\omega$ of $\Gamma$ with $G_\omega\le G_\alpha \times G_\beta$.
In particular, in the rest of the proof we work under the assumption $x_4=1$.
When $x_4=1$, we may refine some of the inequalities above. Indeed, when $x_4=1$, we have $\Omega_{\alpha,4}=\Omega_{\beta,1}$, because both sets have the same cardinality and $\Omega_{\beta,1}\subseteq \Omega_{\alpha,4}$. From this it follows $\Omega_{\alpha,2}=\Omega_{\beta,2}$. Therefore, from~\eqref{eye:11}, we get
$$dx_2=|\Omega_{\alpha,2}|=|\Omega_{\alpha,2}\cap\Omega_{\beta,2}|=d|\mathcal{S}_{\alpha,\beta}|.$$
Now, the inequality $5=5x_4\le 1+x_2$ implies $|\mathcal{S}_{\alpha,\beta}|=x_2\ge 4$. Again, we may use~\eqref{eye:6} to deduce that the permutation groups induced by $G_\alpha$ on $\beta^{G_\alpha}$ and by $G_\beta$ on $\alpha^{G_\beta}$ are both elementary abelian $2$-groups of order $4$.
This, as above, yields that $G_\alpha$ is an elementary abelian $2$-group, that is, part~\eqref{lemma:-32} holds.
From Lemma~\ref{lemma:-4} part~\eqref{eq:lemma-43}, $G_\alpha\cap G_\beta\le G_\omega$, for every $\omega\in \Omega_{\alpha,2}\cap \Omega_{\beta,2}$. In particular, $G_\alpha\cap G_\beta$ fixes pointwise $\Omega_{\alpha,2}\cap\Omega_{\beta,2}$. As $\Omega_{\alpha,2}\cap \Omega_{\beta,2}=\Omega_{\alpha,2}$, we deduce that $G_\alpha\cap G_\beta$ fixes pointwise $\Omega_{\alpha,2}$. Since $G_\alpha\cap G_\beta$ fixes pointwise also $\Omega_{\alpha,1}$ and $\Omega_{\beta,1}=\Omega_{\alpha,4}$, we obtain that $G_\alpha\cap G_\beta$ fixes pointwise $\Omega_{\alpha,1}\cup\Omega_{\alpha,2}\cup\Omega_{\alpha,4}=\Omega$. Thus $G_\alpha\cap G_\beta=1$ and $|G_\alpha|=4$. Observe also that when $x_4=1$, the hypothesis of Lemma~\ref{lemma:4esteso} are satisfied and hence ${\bf N}_G(G_\alpha)={\bf N}_G(G_\beta)$. Therefore $G_\beta$ normalizes $G_\alpha$. This gives that the commutator subgroup $[G_\alpha,G_\beta]$ lies in $G_\alpha\cap G_\beta=1$, that is, $G_\alpha$ commutes with $G_\beta$. This shows that part~\eqref{lemma:-33} is satisfied. Now, as $\Omega_{\alpha,2}=\Omega_{\beta,2}$, Lemma~\ref{lemma:-4} part~\eqref{eq:lemma-43} yields $G_\omega\le G_\alpha\times G_\beta$, for every $\omega\in \Omega_{\alpha,2}$. Therefore, $G_{\alpha}\times G_\beta$ contains $G_\omega$, for every $\omega\in \Omega$. Thus $$G_\alpha\times G_\beta=\langle G_\omega\mid\omega\in \Omega\rangle\unlhd G$$
and $G_\alpha\times G_\beta$ has order $16$. Thus part~\eqref{lemma:-34} is satisfied.
\end{proof}
We need one final preliminary lemma, with a somehow different flavour. We denote by $C_2$ and $C_4$ the cyclic groups of order $2$ and $4$, respectively, we denote by $Q_8$ the quaternion group of order $8$ and we denote by $D_8$ the dihedral group of order $4$.
\begin{lemma}\label{lemma:diff}
Let $R$ be a finite group, let $U$ be a proper subgroup of $R$ and let $r\in U$ be a central involution of $R$. Let $\tau:R\to R$ be the permutation defined by
$$
x\mapsto x^\tau:=
\begin{cases}
x&\textrm{when }x\in U,\\
xr&\textrm{when }x\in R\setminus U.
\end{cases}
$$
Then one of the following holds
\begin{enumerate}[(a)]
\item\label{eq:diff1}the number of inverse-closed subsets $S$ of $R$ with $S^\tau=S$ is at most $2^{{\bf c}(R)-\frac{|R|}{48}}$,
\item\label{eq:diff2}$R$ is generalized dicyclic,
\item\label{eq:diff22}$R\cong C_4\times C_2^\ell$, for some non-negative integer $\ell$.
\end{enumerate}
\end{lemma}
\begin{proof}
Let $\iota:R\to R$ be the permutation defined by $x^\iota=x^{-1}$, for every $x\in R$, and let $T:=\langle\iota,\tau\rangle$. Observe that a subset $S$ of $R$ is inverse-closed and $\tau$-invariant if and only if $S$ is $T$-invariant. In particular, the number of inverse-closed subsets $S$ of $R$ with $S^\tau=S$ is $2^\kappa$, where $\kappa$ is the number of orbits of $T$ on $R$. To compute $\kappa$ we use the orbit-counting lemma, which says that
\begin{equation}\label{ocl}
\kappa=\frac{1}{|T|}\sum_{t\in T}|\mathrm{Fix}_R(t)|.
\end{equation}
Observe that
\begin{align}\label{anuw}
\mathrm{Fix}_R(1)&:=R,\\\nonumber
\mathrm{Fix}_R(\iota)&:={\bf I}(R),\\\nonumber
\mathrm{Fix}_R(\tau)&:=U,\\\nonumber
\mathrm{Fix}_R(\iota\tau)&:={\bf I}(U)\cup\{x\in R\setminus U\mid x^2=r\}.\nonumber
\end{align}
Observe that $\iota\tau=\tau\iota$ and $\iota^2=\tau^2=1$. Therefore $T$ is an elementary abelian $2$-group of order at most $4$.
Observe that $\tau\ne 1$, because $U$ is a proper subgroup of $R$ and $r\ne 1$. If $\iota=1$, then $R$ is an elementary abelian $2$-group and $T=\langle \tau\rangle$. Thus~\eqref{ocl} and~\eqref{anuw} yield
\begin{align*}
\kappa&=\frac{1}{2}\left(|R|+|U|\right)\le\frac{|R|}{2}+\frac{|R|}{4}= \frac{3|R|}{4}\\
&=|R|-\frac{|R|}{4}={\bf c}(R)-\frac{|R|}{4}.
\end{align*}
Therefore, part~\eqref{eq:diff1} holds and the proof follows in this case. Suppose now $\iota=\tau$. This means that $U$ is an elementary abelian $2$-subgroup of $R$ and $x^{-1}=xr$, for every $x\in R\setminus U$. In other words, all elements in $U$ square to $1$ and all elements in $R\setminus U$ square to $r$. Let $\bar{R}:=R/\langle r\rangle$ and let us use the ``bar'' notation for the subgroups and for the elements of $\bar{R}$. Consider the function
$$( \cdot,\cdot):\bar{R}\times\bar{R}\to \langle r\rangle$$
defined by $( x\langle r\rangle, y\langle r\rangle)=x^{-1}y^{-1}xy$, for every $x,y\in R$. Similarly, consider the function
$$q:\bar{R}\to\langle r\rangle$$
defined by $q(x\langle r\rangle)=x^2$. It is not hard to see that, regarding $\bar{R}$ as a vector space over the field with $2$ elements, $(\cdot,\cdot)$ is a bilinear form and $q$ is a quadratic form polarizing to $(\cdot,\cdot)$, that is, $$q(\bar{x}\bar{y})q(\bar{x})q(\bar{y})=(\bar{x},\bar{y}),$$
for every $\bar{x},\bar{y}\in \bar{R}$.
Using this terminology, we have that each element of $\bar{U}$ is totally singular and each element of $\bar{R}\setminus\bar{U}$ is non-degenerate. From the classification of the quadratic forms over finite fields, we have $|\bar{R}:\bar{U}|\in \{2,4\}$. When $|\bar{R}:\bar{U}|=2$, we deduce that
$R$ is an abelian group isomorphic to the direct product $C_4\times C_2^\ell$, for some $\ell\ge0$. In particular, part~\eqref{eq:diff22} holds. When $|\bar{R}:\bar{U}|=4$, we deduce that
$R\cong Q_8\times C_2^\ell$, for some $\ell\ge0$. In particular, $R$ is generalized dicyclic and part~\eqref{eq:diff2} holds. For the rest our our argument, we may suppose that
$\tau\ne\iota\ne1$.
The paragraph above can be summarized by saying that $T=\langle\iota,\tau\rangle$ has order $4$ and hence, from~\eqref{anuw},~\eqref{ocl} becomes
\begin{align}\label{ocl1}
\kappa&=
\frac{1}{4}
\left(
|\mathrm{Fix}_R(1)||+
|\mathrm{Fix}_R(\iota)|+
|\mathrm{Fix}_R(\tau)|+
|\mathrm{Fix}_R(\tau\iota)|
\right)\\\nonumber
&=\frac{1}{4}\left(
|R|+|{\bf I}(R)|+|U|+|{\bf I}(U)|+|\{x\in R\setminus U\mid x^2=r\}|\right)\\\nonumber
&\le
\frac{1}{4}\left(
|R|+|{\bf I}(R)|+|U|+|{\bf I}(R)|+|\{x\in R\setminus U\mid x^2=r\}|\right)\\\nonumber
&=\frac{|R|+|{\bf I}(R)|}{2}-\left(
\frac{|R|}{4}-\frac{|U|}{4}-\frac{|\{x\in R\setminus U\mid x^2=r\}|}{4}
\right)\\\nonumber
&={\bf c}(R)-\left(
\frac{|R|}{4}-\frac{|U|}{4}-\frac{|\{x\in R\setminus U\mid x^2=r\}|}{4}
\right).\nonumber
\end{align}
Set $\mathcal{S}:=\{x\in R\setminus U\mid x^2=r\}$.
If $\mathcal{S}=\emptyset$, then the proof follows immediately from~\eqref{ocl1}, indeed, part~\eqref{eq:diff1} holds true. Therefore, for the rest of the proof we suppose $$\mathcal{S}\ne\emptyset.$$
To conclude we divide the proof in various cases.
Suppose first that $|R:U|=2$. Let $x\in \mathcal{S}$ and observe that $R=U\cup Ux$. Now, a computation yields
$$\mathcal{S}=\{ux\mid u\in U, u^x=u^{-1}\}.$$
When $\mathcal{S}=Ux$, the action of $x$ on $U$ by conjugation is an automorphism of $U$ inverting each element of $U$. Therefore $U$ is abelian and $R$ is generalized dicyclic. Hence part~\eqref{eq:diff2} holds. When $\mathcal{S}\subsetneq Ux$, the result of Liebeck and MacHale~\cite{LieMac} shows that the automorphism $x$ can invert at most $3/4$ of the elements of $U$ and hence $|\mathcal{S}|\le 3|U|/4=3|R|/8$. Now,~\eqref{ocl1} gives $\kappa\le {\bf c}(R)-|R|/32$; hence part~\eqref{eq:diff1} holds and the proof follows. Therefore, for the rest of the proof we may suppose
\begin{equation}\label{bound}|R:U|\ge3.
\end{equation}
When $|\mathcal{S}|\le 3|R|/4-|U|/2$, from~\eqref{ocl1} and~\eqref{bound}, we deduce
\begin{align*}
\kappa&\le {\bf c}(R)-\left(
\frac{|R|}{4}-\frac{|U|}{4}-\frac{3|R|}{16}+\frac{|U|}{8}\right)\\
&= {\bf c}(R)-\left(
\frac{|R|}{16}-\frac{|U|}{8}\right)\\
&\le{\bf c}(R)-\left(
\frac{|R|}{16}-\frac{|R|}{24}\right)={\bf c}(R)-
\frac{|R|}{48}
\end{align*} and the proof follows. Therefore, for the rest of the proof, we suppose $$|\mathcal{S}|> 3|R|/4-|U|/2.$$
Let $u$ be an arbitrary element of $U$. Then $u\mathcal{S}\subseteq R\setminus U$ and hence $\mathcal{S}\cup u\mathcal{S}\subseteq R\setminus U$. Therefore
\begin{align}\label{ocl2}
|\mathcal{S}\cap u\mathcal{S}|&=
|\mathcal{S}|+|u\mathcal{S}|-|\mathcal{S}\cup u\mathcal{S}|=2|\mathcal{S}|-|\mathcal{S}\cup u\mathcal{S}|\\\nonumber
&\ge 2|\mathcal{S}|-(|R|-|U|)>\frac{3|R|}{2}-|U|-(|R|-|U|)=\frac{|R|}{2}.\nonumber
\end{align}
Now, let $ux\in\mathcal{S}\cap u\mathcal{S}$. Then $x\in\mathcal{S}$ and hence
$$r=(ux)^2=uxux=uu^xx^2=uu^xr.$$
Therefore $u^x=u^{-1}$. Now, repeating the argument above with $y\in \mathcal{S}\cap u\mathcal{S}$, we deduce $u^y=u^{-1}$ and hence $xy^{-1}\in {\bf C}_R(u)$. Since we have $|\mathcal{S}\cap u\mathcal{S}|$ choices for $y$,~\eqref{ocl2} implies $|{\bf C}_R(u)|>|R|/2$ and hence $R={\bf C}_R(u)$. Since $u$ is an arbitrary element of $U$, we deduce that $U$ is a central subgroup of $R$.
Since $u^x=u^{-1}$, for every $u\in U$ and for every $ux\in \mathcal{S}\cap u\mathcal{S}$, and since $U$ is contained in the center of $R$, we deduce that $U$ has exponent $2$. Since $U$ is a central subgroup of $R$ of exponent $2$, we now have an easier description of for $\mathcal{S}$, that is,
$$\mathcal{S}=\{x\in R\mid x^2=r\}.$$
Now that we know that $U$ has exponent $2$, we consider the quotient group $\bar{R}:=R/\langle r\rangle$. Observe that each element of $\bar U$ is an involution. Assume that $\bar{R}$ is not an elementary abelian $2$-group. Then, the theorem of Miller~\cite{Miller} yields $|{\bf I}(\bar{R})|\le 3|\bar{R}|/4$. In particular, the number of involutions in $\bar{R}\setminus \bar{U}$ is at most $3|\bar{R}|/4-|\bar{U}|$. Since each element in $\bar{\mathcal{S}}$ is an involution and since $\bar{\mathcal{S}}\subseteq \bar{R}\setminus \bar{U}$, we deduce $|\mathcal{S}|\le 3|R|/4-|U|$. Using this inequality in~\eqref{ocl1}, we get
$$\kappa\le {\bf c}(R)-\frac{|R|}{16},$$
part~\eqref{eq:diff1} holds
and the proof follows in this case. It remains to consider the case that $\bar{R}$ is an elementary abelian $2$-group.
For this remaining case, we consider the bilinear form
$$( \cdot,\cdot):\bar{R}\times\bar{R}\to \langle r\rangle$$
defined by $( x\langle r\rangle, y\langle r\rangle)=x^{-1}y^{-1}xy$, for every $x,y\in R$, and its quadratic form
$$q:\bar{R}\to\langle r\rangle$$
defined by $q(x\langle r\rangle)=x^2$. Again, we use the classification of the quadratic forms over finite fields. Using this terminology, $\bar{U}$ is totally isotropic and contained in the kernel of the the bilinear form $(\cdot,\cdot)$ and the elements of $\bar{\mathcal{S}}$ are non-degenerate. Using this information we obtain that $R$ is isomorphic to one of the following groups
\begin{itemize}
\item $C_4\times C_2^\ell$, for some $\ell\ge 0$,
\item $\underbrace{D_8\circ D_8\circ \cdots \circ D_8}_{t\textrm{ times}}\times C_2^\ell$, for some $\ell\ge 0$ and $t\ge 1$,
\item $Q_8\circ\underbrace{D_8\circ D_8\circ \cdots \circ D_8}_{(t-1)\textrm{ times}}\times C_2^\ell$, for some $\ell\ge 0$ and some $t\ge 1$,
\item $C_4\circ \underbrace{D_8\circ D_8\circ \cdots \circ D_8}_{t\textrm{ times}}\times C_2^\ell$, for some $\ell\ge 0$ and some $t\ge 1$.
\end{itemize}
In the first case, an explicit computation gives $|\mathcal{S}|=|R|/2$. Hence~\eqref{ocl1} gives
\begin{align*}
\kappa&\le{\bf c}(R)-\left(\frac{|R|}{4}-\frac{|U|}{4}-\frac{|R|}{8}\right)=
{\bf c}(R)-\left(\frac{|R|}{8}-\frac{|U|}{4}\right)\\
&\le{\bf c}(R)-\left(\frac{|R|}{8}-\frac{|R|}{16}\right)
={\bf c}(R)-\frac{|R|}{16}
\end{align*}
and part~\eqref{eq:diff1} holds. In the second case, an explicit computation gives $|\mathcal{S}|=(2^t-1)|R|/2^{t+1}\le |R|/2$. Therefore we may argue as in the previous case and we obtain that part~\eqref{eq:diff1} holds. In the third case, an explicit computation gives $|\mathcal{S}|=(2^t+1)|R|/2^{t+1}$. When $t=1$, $R\cong Q_8\times C_2^\ell$ is generalized dicyclic and hence part~\eqref{eq:diff2} hods. When $t\ge 2$, we have $|\mathcal{S}|\le 5|R|/8$ and hence~\eqref{ocl1} gives
\begin{align*}
\kappa&\le{\bf c}(R)-\left(\frac{|R|}{4}-\frac{|U|}{4}-\frac{5|R|}{32}\right)=
{\bf c}(R)-\left(\frac{3|R|}{32}-\frac{|U|}{4}\right)\\
&\le{\bf c}(R)-\left(\frac{3|R|}{32}-\frac{|R|}{16}\right)
={\bf c}(R)-\frac{|R|}{32}.
\end{align*}
Thus, we obtain that part~\eqref{eq:diff1} holds. In the forth (and last) case, an explicit computation gives $|\mathcal{S}|=|R|/2$. Therefore we may argue as in the first case and we obtain that part~\eqref{eq:diff1} holds.
\end{proof}
\section{Proof of Theorems~\ref{thrm:main1} and~\ref{thrm:main2}}
In this section, using Section~\ref{sec:lemmata} we prove both Theorems~\ref{thrm:main1} and~\ref{thrm:main2}. Thus, let $G$ be a finite transitive permutation group on $\Omega$ with $${\bf I}_\Omega(G)\ge \frac{5}{6}.$$
If ${\bf I}_\Omega(G)=1$, then there is nothing to prove and hence we may suppose that ${\bf I}_\Omega(G)<1$. Let $\alpha\in \Omega$. From Lemma~\ref{lemma:-4}, we have
$$\Omega=\Omega_{\alpha,1}\cup\Omega_{\alpha,2}\cup\Omega_{\alpha,4}.$$
Since ${\bf I}_\Omega(G)<1$, $\Omega_{\alpha,4}\ne\emptyset$. Let $\beta\in \Omega_{\alpha,4}$. From Lemma~\ref{lemma:-3},
$$V:=G_\alpha\times G_\beta$$
is an elementary abelian normal $2$-subgroup of $G$ of order $16$. Let $e_1,e_2,e_3,e_4$ be a basis of $V$, regarded as a vector space over the field with $2$ elements, and with $G_\alpha=\langle e_1,e_2\rangle$. Let $H:=G/{\bf C}_G(V)$ and $W:=G_\alpha$. Clearly, $H\le \mathrm{GL}(V)\cong\mathrm{GL}_4(2)$. Now, consider the action of $H$ on the $2$-dimensional subspaces of $V$ and consider $O:=\{W^h\mid h\in H\}$, the $H$-orbit containing $W$. Clearly,
$$\frac{|\Omega_{\alpha,1}\cup\Omega_{\alpha,2}|}{|\Omega|}=\frac{|\{U\in O\mid |W:W\cap U|\le 2\}|}{|O|}.$$
Observe that the right hand side of this equality can be easily computed with the help of a computer. With the computer algebra system~\texttt{magma}~\cite{magma}, we have computed all the subgroups of $\mathrm{GL}_4(2)$. Then, we have selected only the subgroups $H$ with the property that
$$V=\langle W^h\mid h\in H\rangle\hbox{ and }\bigcap_{h\in H}W^h=0.$$
(This selection is due to the fact that $V=\langle G_\alpha^g\mid g\in G\rangle$ and that $G_\alpha$ is core-free in $G$.)
Then, for each such subgroup $H$, we have computed the orbit $O=W^H$ and we have computed the ratio $\frac{|\{U\in O\mid |W:W\cap U|\le 2\}|}{|O|}$. We have checked that in all cases this ratio is at most $5/6$. In particular, Theorem~\ref{thrm:main1} is proved. Moreover, we have checked that this ratio is $5/6$ if and only if $H$ is given in the statement of Theorem~\ref{thrm:main2}. Since this construction can be reversed, we also obtain the converse implication for Theorem~\ref{thrm:main2}.
\section{Proof of Theorem~\ref{thrm:main3}}
Let $G$ be a finite transitive group properly containing a regular subgroup $R$. Since $R$ acts regularly, we may identify the domain of $G$ with $R$. Now, the number of Cayley graphs $\mathop{\Gamma}(R,S)$ on $R$ with $G\le \mathrm{Aut}(\mathop{\Gamma}(R,S))$ is the number of inverse-closed subsets $S$ of $R$ left invariant by $G_1$, where $G_1$ is the stabilizer of the point $1\in R$ in $G$. In particular, to prove Theorem~\ref{thrm:main3}, we need to estimate the number of inverse-closed subsets of $R$ that are union of $G_1$-orbits.
Suppose first that $${\bf I}_R(G)=1.$$ Since $R$ is properly contained in $G$, from the theorem of Bergman and Lenstra mentioned in Section~\ref{intro}, we have two cases to consider
\begin{itemize}
\item $|G_1|=2$,
\item $G$ contains an elementary abelian normal $2$-subgroup $N$ with $|N:G_1|=2$.
\end{itemize}
Assume first that $|G_1|=2$. Let $\varphi\in G_1\setminus\{1\}$. From the Frattini argument, $G=RG_1$ and hence $|G:R|=2$. This gives $R\unlhd G$ and hence $\varphi$ acts by conjugation on $R$ as a group automorphism. Now, from~\cite[Lemma~$2.7$]{Bxu} or~\cite[Theorem~1.13]{MMS}, we have that
\begin{enumerate}[(a)]
\item the number of $\varphi$-invariant inverse-closed subsets of $R$ is at most $2^{{\bf c}(R)-\frac{|R|}{96}}$, or
\item $R$ is abelian of exponent greater than $2$ and $\varphi$ is the automorphism of $R$ mapping each element to its inverse, or
\item $R$ is generalized dicyclic and $\varphi$ is an automorphism of $R$ with $x^\varphi\in \{x,x^{-1}\}$, for every $x\in R$.
\end{enumerate}
In particular, the proof of Theorem~\ref{thrm:main3} follows in this case.
Assume next that $G$ contains an elementary abelian normal $2$-subgroup $N$ with $|N:G_1|=2$. Since $R$ acts transitively, $G=RN$. Moreover, since $R$ acts regularly, $G=RG_1$ and $R\cap G_1=1$. Thus $|R\cap N|=|N|/|G_1|=2$. Let $r$ be a generator of $R\cap N$. Since $\langle r\rangle=R\cap N\unlhd R$, $r$ is a central involution of $R$. Let $U:={\bf N}_R(G_1)$. Since ${\bf N}_R(G_1)$ is a block of imprimitivity for $G$, $U={\bf N}_R(G_1)$ is also a block of imprimitivity for the regular action of $R$ and hence $U$ is a subgroup of $R$. As $G_1\ne1 $ because $R$ is properly contained in $G$, we deduce that $U$ is a proper subgroup of $R$. Now, $G_1$ fixes pointwise $U$ and, for every $x\in R\setminus U$, we have
$$x^{G_1}=\{x,xr\}.$$
Let $\tau:R\to R$ be the permutation defined by
$$
x\mapsto x^\tau:=
\begin{cases}
x&\textrm{when }x\in U,\\
xr&\textrm{when }x\in R\setminus U.
\end{cases}
$$
We have shown that $S\subseteq R$ is $G_1$-invariant if and only if $S$ is $\langle\tau\rangle$-invariant. Therefore, the proof of this case follows from Lemma~\ref{lemma:diff}.
To conclude the proof of Theorem~\ref{thrm:main3}, it remains to consider the case that $${\bf I}_R(G)\ne 1.$$ From Theorem~\ref{thrm:main1}, we have ${\bf I}_R(G)\le 5/6$. Recall that ${\bf I}(R)=\{x\in R\mid x^2=1\}.$ We define
\begin{align*}
&a:=|\Omega_{R,1}\cap {\bf I}(R)|,&&b:=|\Omega_{R,1}\cap (R\setminus {\bf I}(R))|,\\
&c:=|\Omega_{R,2}\cap {\bf I}(R)|,&&d:=|\Omega_{R,2}\cap (R\setminus {\bf I}(R))|,\\
e&:=|(R\setminus (\Omega_{R,1}\cup \Omega_{R,2}))\cap {\bf I}(R)|,&&f:=|(R\setminus (\Omega_{R,1}\cup \Omega_{R,2}))\cap (R\setminus {\bf I}(R))|.
\end{align*}
As ${\bf I}_R(G)\le 5/6$, we deduce
\begin{equation}\label{final?}
\frac{|R|}{6}\le|R\setminus(\Omega_{R,1}\cup\Omega_{R,2})|=e+f.
\end{equation}
Let $\iota:R\to R$ be the permutation defined by $x^\iota:=x^{-1}$, for every $x\in R$, and let $T:=\langle \iota,G_1\rangle$. Now, the number of $G_1$-invariant inverse-closed subsets of $R$ is exactly the number of $T$-invariant subsets of $R$. Moreover, the number of $T$-invariant subsets of $R$ is $2^\kappa$, where $\kappa$ is the number of orbits of $T$ on $R$.
The group $T$ has
\begin{itemize}
\item orbits of cardinality $1$ on $\Omega_{R,1}\cap {\bf I}(R)$,
\item orbits of cardinality $2$ on $\Omega_{R,1}\cap (R\setminus {\bf I}(R))$,
\item orbits of cardinality $2$ on $\Omega_{R,2}\cap {\bf I}(R)$,
\item orbits of cardinality at least $2$ on $\Omega_{R,2}\cap (R\setminus {\bf I}(R))$,
\item orbits of cardinality at least $3$ on $(R\setminus(\Omega_{R,1}\cup\Omega_{R,2}))\cap {\bf I}(R)$,
\item orbits of cardinality at least $4$ on $(R\setminus(\Omega_{R,1}\cup\Omega_{R,2}))\cap (R\setminus {\bf I}(R))$.
\end{itemize}
All of these assertions are trivial except, possibly, the last one. Indeed, if $x\in (R\setminus(\Omega_{R,1}\cup\Omega_{R,2}))\cap (R\setminus {\bf I}(R))$, then $x$ is not an involution and the $G_1$-orbit $x^{G_1}$ has cardinality at least $3$. As
$$(x^{G_1})^{-1}=(x^{-1})^{G_1},$$
we deduce that $|x^T|$ has even cardinality and hence $|x^T|$ is at least $4$.
Summing up, we have
\begin{align*}
\kappa&\le a+\frac{b}{2}+\frac{c}{2}+\frac{d}{2}+\frac{e}{3}+\frac{f}{4}=a+c+e+\frac{b}{2}+\frac{d}{2}+\frac{f}{2}-\left(\frac{c}{2}+\frac{2e}{3}+\frac{f}{4}\right)\\
&=\frac{|R|+|{\bf I}(R)|}{2}-\left(\frac{c}{2}+\frac{2e}{3}+\frac{f}{4}\right)
={\bf c}(R)-\left(\frac{c}{2}+\frac{2e}{3}+\frac{f}{4}\right)\\
&\le{\bf c}(R)-\left(\frac{2e}{3}+\frac{f}{4}\right)\le {\bf c}(R)-\left(\frac{e}{4}+\frac{f}{4}\right)\\
&\le{\bf c}(R)-\frac{|R|}{24},
\end{align*}
where in the last inequality we have used~\eqref{final?}. This concludes the proof of Theorem~\ref{thrm:main3}.
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\end{document}
|
{
"timestamp": "2021-09-29T02:25:36",
"yymm": "2109",
"arxiv_id": "2109.13882",
"language": "en",
"url": "https://arxiv.org/abs/2109.13882"
}
|
\section{Introduction}
The scattering between kinks has become a very popular research topic in recent decades because of its astonishing properties \cite{Manton2004, Shnir2018, Kevrekidis2019}. The study of the collisions between kinks and antikinks in the $\phi^4$ model was initially addressed in the seminal references \cite{Sugiyama1979, Campbell1983, Anninos1991}. As it is well known, only two different scattering channels arise: \textit{bion formation} (where kink and antikink collide and bounce back over and over emitting radiation in every impact) and \textit{kink reflection} (where kink and antikink collide and bounce back a finite number of times before moving away). These two channels are predominant, respectively, for low and large values of the initial collision velocity. In these studies emerges the fascinating property that the two previously mentioned channels are infinitely interlaced in the transition of these regimes, giving rise to a fractal structure embedded in the final versus initial velocity diagram. The kink reflection windows included in this region involve scattering processes where kink and antikink collide and bounce back a finite number of times before definitely escaping away. This kink dynamics could have important consequences on physical applications where the presence of these topological defects allows the understanding of certain non-linear phenomena. Kinks (and topological defects in general) have been employed in a wide variety of physical disciplines, such as Condensed Matter \cite{Eschenfelder1981,Jona1993,Strukov}, Cosmology \cite{Vilenkin1994,Vachaspati2006}, Optics \cite{Mollenauer2006,Schneider2004,Agrawall1995}, molecular systems \cite{Davydov1985,Bazeia1999}, Biochemistry \cite{Yakushevich2004}, etc.
The appearance of a fractal structure in the velocity diagram describing the kink scattering for the $\phi^4$ model is based on the existence of an internal vibrational mode (the shape mode) associated to the kink solutions. The presence of this massive mode together with the zero mode triggers the \textit{resonant energy transfer mechanism}, which allows the redistribution of the energy between the kinetic and vibrational energy pools when the kinks collide. In a usual scattering event the kink and the antikink approach each other and collide. A certain amount of kinetic energy is transferred to the shape mode, such that kink and antikink become wobblers (kinks whose shape modes are excited), which try to escape from each other. If the kinetic energy of each wobbler is not large enough both of them end up approaching and colliding again. This process can continue indefinitely or finish after a finite number of collisions. In this last case, enough vibrational energy is returned back to the zero mode as kinetic energy, which allows the wobblers to escape. This mechanism and other related phenomena have been thoroughly analyzed in a large variety of models \cite{Shiefman1979, Peyrard1983, Goodman2005, Gani1999, Malomed1989, Gani2018, Gani2019,Simas2016,Gomes2018,Bazeia2017b,Bazeia2017a, Bazeia2019, Adam2019, Romanczukiewicz2018, Adam2020, Mohammadi2020, Yan2020,Romanczukiewicz2017, Weigel2014, Gani2014, Bazeia2018b, Lima2019, Marjaheh2017, Belendryasova2019, Zhong2020, Bazeia2020c, Christov2019, Christov2019b, Christov2020,Halavanau2012, Romanczukiewicz2008, Alonso2018, Alonso2018b,Alonso2017, Alonso2019, Alonso2020, Alonso2021, Ferreira2019, Goodman2002, Goodman2004,Malomed1985,Malomed1992, Saadatmand2015,Saadatmand2018, Manton1997,Adam2018,Adam2019b,Adam2020b,Dorey2011,Dorey2018,Mohammadi2021b,Campos2020,Blanco-Pillado2021}, revealing the enormous complexity of these events and the difficulty in explaining this phenomenon analytically. The \textit{collective coordinate approach} has been used to accomplish this task for decades, reducing the field theory to a finite dimensional mechanical system, where the separation between the kinks and the wobbling amplitudes associated to the shape modes are promoted to dynamical variables. This method has been progressively improved, see for example \cite{Sugiyama1979,Takyi2016, Kevrekidis2019,Pereira2020} and references therein, and recently, a reliable description of the kink scattering in the $\phi^4$ model has been achieved in the reflection-symmetric case \cite{Manton2021}, by introducing in this scheme the removal of a coordinate singularity in the moduli space and choosing the appropriate initial conditions.
As previously mentioned, after the first collision the initially unexcited kink and antikink become wobblers, so in a $n$-bounce scattering process the subsequent $n-1$ collisions can be understood as scattering processes between two wobblers. This observation justifies an intrinsic interest on the collision between these objects. The evolution of a single wobbler has been studied by employing perturbation expansion schemes by different authors, see \cite{Barashenkov2009,Barashenkov2018,Segur1983} and references therein. The scattering between wobblers in the $\phi^4$ model has been discussed in \cite{Alonso2021b} for a space reflection symmetric scenario. This situation is relevant in the original kink scattering problem where the mirror symmetry is preserved. The goal of these investigations is to bring insight into the resonant energy transfer mechanism by means of numerical analysis of the scattering solutions derived from the corresponding the Klein-Gordon partial differential equations. In this context it is worthwhile mentioning that the scattering of wobblers in the double sine-Gordon model has been studied by Campos and Mohammadi \cite{Campos2021}.
In this paper we shall continue with this line of research by investigating the asymmetric scattering between wobblers in two different scenarios, which are considered representative of this context. The scattering processes addressed in previous works involve wobblers which evolve with the same phase. This implies that a constructive interference between the shape modes associated to each wobbler takes place at the collision. In this work we propose the analysis of the scattering between wobblers with opposite phases, such that now a destructive interference between the vibrational modes occurs at the impact. The second scenario is described by the collision between a wobbler and an unexcited kink. This allows us to monitor the transfer of the vibrational energy from the wobbler to the kink. We will show that the fractal structures ruled by the resonance phenomenon in these two cases display very different patterns.
The organization of this paper is as follows: in Section \ref{sec:2} the theoretical background of the $\phi^4$ model together with the analytical description of kinks and wobblers is introduced. The kink-antikink scattering is also discussed, which allows us to describe the numerical setting employed to study the problem. Section~\ref{sec:3} is dedicated to study the scattering between wobblers with opposite phase, whereas the collision between a wobbler and an unexcited kink is addressed in Section \ref{sec:4}. Finally, some conclusions are drawn in Section~\ref{sec:5}.
\section{The $\phi^4$ model: kinks and wobblers}
\label{sec:2}
The dynamics of the $\phi^4$ model in (1+1) dimensions is governed by the action
\begin{equation}\label{action}
S=\int d^2 x \,\, {\cal{L}}(\partial_{\mu}\phi, \phi) \hspace{0.5cm},
\end{equation}
where the Lagrangian density ${\cal{L}}(\partial_{\mu}\phi, \phi)$ is of the form
\begin{equation}\label{lagrangiandensity}
{\cal{L}}(\partial_{\mu}\phi, \phi) = \frac{1}{2} \,\partial_\mu \phi \, \partial^\mu \phi - V(\phi) \hspace{0.5cm} \mbox{with} \hspace{0.5cm} V(\phi) = \frac{1}{2} (\phi^2 -1)^2 \hspace{0.5cm}.
\end{equation}
The use of dimensionless field and coordinates, as well as Einstein summation convention, are assumed in expressions (\ref{action}) and (\ref{lagrangiandensity}). Here, the Minkowski metric $g_{\mu\nu}$ has been set as $g_{00}=-g_{11}= 1$ and $g_{12}=g_{21}=0$. Therefore, the non-linear Klein-Gordon partial differential equation
\begin{equation}
\frac{\partial^2 \phi}{\partial t^2} - \frac{\partial^2 \phi}{\partial x^2} = -2\phi(\phi^2-1)
\label{pde}
\end{equation}
characterizes the time-dependent solutions of this model. The energy-momentum conservation laws imply that the total energy and momentum
\begin{equation}
E[\phi] = \int dx \Big[ \frac{1}{2} \Big( \frac{\partial \phi}{\partial t} \Big)^2 + \frac{1}{2} \Big( \frac{\partial \phi}{\partial x} \Big)^2 + V(\phi) \Big] \hspace{0.5cm}, \hspace{0.5cm}
P[\phi] = - \int dx\, \frac{\partial \phi}{\partial t} \, \frac{\partial \phi}{\partial x} \hspace{0.5cm}, \label{invariants}
\end{equation}
are system invariants. The kinks/antikinks ($+/-$)
\begin{equation}
\phi_{\rm K}^{(\pm)}(t,x;x_0,v_0) = \pm \tanh \left[\frac{x-x_0-v_0 t}{\sqrt{1-v_0^2}}\right] \label{travelingkink}
\end{equation}
are travelling solutions of (\ref{pde}), whose energy density is localized around the kink center $x_C=x_0+v_0 t$ (the value where the field profile vanishes). The parameter $v_0$ can be interpreted as the kink velocity. As it is well known, the solutions (\ref{travelingkink}) are topological defects because they asymptotically connect the two elements of the set of vacua ${\cal M}=\{-1,1\}$. These solutions have a normal mode of vibration. When this mode is excited the size of these solutions (called \textit{wobbling kinks} or \textit{wobblers}) periodically oscillates with frequency $\omega=\sqrt{3}$. This fact has been numerically checked and has been analytically proved in the linear regime. The spectral problem
\[
{\cal H} \psi_{\omega^2}(x) = \omega^2 \psi_{\omega^2}(x)
\]
of the second order small fluctuation operator associated with the static kink/antikink,
\begin{equation}
{\cal H} = - \frac{d}{dx^2} + 4-6\,{\rm sech}^2 (x-x_0), \label{hessian}
\end{equation}
involves the shape mode
\[
\psi_{\omega^2=3}(x;x_0)= \, {\rm sinh}\, (x-x_0) \, {\rm sech}^2 (x-x_0)
\]
with eigenvalue $\omega^2=3$. The discrete spectrum of the operator (\ref{hessian}) is completed with the presence of a zero mode
\[
\psi_{\omega^2=0}(x;x_0)= \, {\rm sech}^2 (x-x_0) = \left. \frac{\partial \phi_K^{(+)}}{\partial x}\right|_{t=0,v_0=0} \hspace{0.5cm},
\]
whereas the continuous spectrum emerges on the threshold value $\omega^2=4$.
As a result of this linear analysis, the expression
\begin{equation}
\phi_{\rm W}^{(\pm)}(t,x;x_0,v_0,\omega,a,\delta) = \pm \tanh \left[ \frac{x-x_0 - v_0 t}{\sqrt{1-v_0^2}} \right] + a \sin(\omega t+\delta){\rm{sech}} \left[ \frac{x-x_0 - v_0 t}{\sqrt{1-v_0^2}} \right] \tanh\left[ \frac{x-x_0 - v_0 t}{\sqrt{1-v_0^2}} \right] \label{wobbler}
\end{equation}
can be considered a good approximation of a traveling wobbler in the linear regime $a\ll 1$. Note that $\phi_{\rm W}^{(-)}(t,x)$ describes a \textit{wobbling antikink} (or \textit{antiwobbler}).
The maximum deviation of the wobbler (\ref{wobbler}) from the kink (\ref{travelingkink}) takes place at the points
\begin{equation}
x_M^{(\pm)} = x_C \pm \sqrt{1-v_0^2} \,\,{\rm arccosh}\, \sqrt{2} \hspace{0.5cm} , \label{points}
\end{equation}
where the relation
\[
\left| \phi_{\rm W} (x_M^{(\pm)}) - \phi_{\rm K} (x_M^{(\pm)})\right| = \frac{1}{2} \, |a|
\]
holds. An optimized strategy to measure the wobbling amplitude of a traveling wobbler in a numerical scheme is to monitor the profile of these solutions at the points (\ref{points}). By using fourth order perturbation theory in the expansion parameter $a$, it has been proved that $a$ depends on time, $a=a(t)$, and decays following the expression
\begin{equation}
|a(t)|^2 = \frac{|a(0)|^2}{1+\omega \,\xi_I\, |a(0)|^2 t}, \label{amplitude}
\end{equation}
where $\xi_I$ is a constant. However, when the initial wobbling amplitude $a(0)$ is small, the decay is very slow and becomes appreciable only after a long time $t\sim |a(0)|^{-2}$ \cite{Barashenkov2009,Barashenkov2018}.
The scattering between a kink and an antikink has been thoroughly analyzed in the physical and mathematical literature during the last decades. In this case, a kink and antikink which are well separated are pushed together with initial collision velocity $v_0$. Taking into account the spatial reflection symmetry of the system the kink can be located at the left of the antikink or vice versa. For very small values of the time $t$ (with respect to the impact time), the previous scenario is characterized by the concatenation
\begin{equation}
\Phi_{KK}(t,x;x_0,v_0) = \phi_K^{(\pm)}(t,x;x_0,v_0) \cup \phi_K^{(\mp)}(t,x;-x_0,-v_0) \label{configuration01}
\end{equation}
for $x_0\gg 0$, where we have introduced the notation
\begin{equation}
\phi_K^{(\pm)}(t,x;x_0,v_0) \cup \phi_K^{(\mp)}(t,x;-x_0,-v_0)\equiv \left\{ \begin{array}{ll} \phi_K^{(\pm)}(t,x;x_0,v_0) & \mbox{if } x\leq 0, \\[0.2cm] \phi_K^{(\mp)}(t,x;-x_0,-v_0) & \mbox{if } x> 0 . \end{array} \right. \label{concadef}
\end{equation}
The initial separation distance between the kink and the antikink is equal to $2x_0$. The configuration (\ref{configuration01}) defines the initial conditions of the scattering problem. As it is well known, there exist two different scattering channels in this case: (1) \textit{bion formation}, where kink and antikink end up colliding and bouncing back over and over, and (2) \textit{kink reflection}, where kink and antikink collide, bounce, and finally recede with respective final velocities $v_{f,L}$ and $v_{f,R}$ in the opposite direction in which they were initially traveling. These scattering regimes are predominant, respectively, for low and high values of the initial velocity $v_0$. In Figure \ref{fig:VelDiaAmp000} the two previously mentioned final velocities $v_{f,L}$ and $v_{f,R}$ are plotted as a function of the initial collision velocity $v_0$.
\begin{figure}[htb]
\centerline{\includegraphics[height=3.5cm]{veldiaamp000}}
\caption{\small Final versus initial velocity diagram for the kink-antikink scattering. The final velocity of the bion is assumed to be zero in this context. The color code is used to specify the number of bounces suffered by the kinks before escaping.} \label{fig:VelDiaAmp000}
\end{figure}
From the spatial reflection symmetry exhibited by the initial configuration (\ref{configuration01}) it is clear that $v_{f,L}=-v_{f,R}$ and that the velocity of a bion must be zero. Therefore, the velocity diagram in Figure \ref{fig:VelDiaAmp000} is symmetric with respect to the $v_0$-axis. In the next sections we shall address asymmetric scattering events where this symmetry is lost and $|v_{f,L}|\neq |v_{f,R}|$ in general. The fascinating property found in this scattering problem is that the transition between the two previously mentioned regimes is ruled by a fractal structure where the bion formation and the kink reflection regimes are infinitely interlaced. The kink reflection windows included in this initial velocity interval involve scattering processes where kink and antikink collide and bounce back a finite number of times exchanging energy between the zero and shape modes before definitely moving away. These processes involve the so called \textit{resonant energy transfer mechanism}.
For the previously mentioned $n$-bounce processes (with $n\geq 2$) it is clear that after the first impact the subsequent collisions correspond to scattering processes between wobblers because, in general, the collision between kinks causes the excitation of their shape modes. Taking into account the spatial reflection symmetry of the problem, the wobbling amplitudes and phases of the colliding wobblers are equal. Therefore, these events are characterized by an initial configuration of the form
\begin{equation}
\Phi_{WW}(t,x;x_0,v_0,\omega,a,\delta) =
\phi_W^{(\pm)}(t,x; x_0,v_0,\omega,a,\delta) \cup
\phi_W^{(\mp)}(t,x; -x_0,-v_0,\omega,a,\delta) \,. \label{configuration02}
\end{equation}
This scattering problem has been numerically studied in \cite{Alonso2021b}. By mirror symmetry, it can be assumed that the phases of the shape modes of the traveling wobblers are also the same at the impact time, so a constructive interference takes places in the collision. As a consequence, it was found that the fractal pattern enlarges and becomes more complex as the value of the initial wobbling amplitude $a$ increases. Another interesting property in this context is the emergence of isolated 1-bounce windows, which are not present in the original kink-antikink scattering. It is clear that the scattering between wobblers characterized by the initial configuration (\ref{configuration02}) is extremely relevant to study the resonant energy transfer mechanism in this problem. However, because of the spatial reflection symmetry of this type of processes, wobblers transfer the same amount of energy to each other at the collision, that is, the scattered wobblers travel away with the same final speeds and wobbling amplitudes. In this work we are interested in analyzing more general scattering events where the energy transfer mechanism becomes asymmetric with respect to the traveling wobblers.
The first type of processes which could involve novel properties in this framework is the collision between two wobblers with opposite phase. This scenario can be characterized by the initial configuration
\begin{equation}
\Phi_{W \widetilde{W}}(t,x;x_0,v_0,\omega,a,\delta) =
\phi_W^{(\pm)}(t,x; x_0,v_0,\omega,a,\delta) \cup
\phi_W^{(\mp)}(t,x; -x_0,-v_0,\omega,a,\pi+\delta) \, . \label{configuration04}
\end{equation}
We have employed the notation $W\widetilde{W}$ as subscript of $\Phi$ in (\ref{configuration04}) to simply emphasize that the wobblers have different initial phases and to distinguish this configuration from (\ref{configuration02}). In this case it is assumed that the wobblers evolve preserving a phase difference of $\pi$, giving place to a destructive interference in the excitation of the shape modes of each wobbler when they collide. It is expected that the final versus initial velocity diagrams associated to these scattering events will be affected by this fact and that they will be very different from those found in the constructive interference scenario (\ref{configuration02}), analyzed in \cite{Alonso2021b}.
Another important situation which deserves attention is the scattering between a wobbler and a kink. These asymmetric events can be characterized by the initial configuration
\begin{equation}
\Phi_{WK}(t,x;x_0,v_0,\omega,a,\delta) =
\phi_W^{(\pm)}(t,x; x_0,v_0,\omega,a,\delta) \cup
\phi_K^{(\mp)}(t,x; -x_0,-v_0) \, , \label{configuration03}
\end{equation}
where without loss of generality the non-excited antikink/kink $\phi_K^{(\mp)}(t,x; -x_0,-v_0)$ has been placed to the right of the wobbler/antiwobbler. This situation allows to analyze how the vibrational energy is transferred to the non-excited kink in a better way than in the previous contexts.
In order to study the scattering between kinks and wobblers in the two previously described scenarios, in the present work we shall employ numerical approaches based on the discretization of the partial differential equation (\ref{pde}) with different initial conditions determined by the configurations (\ref{configuration04}) and (\ref{configuration03}). The particular numerical scheme used here is a fourth-order explicit finite difference algorithm implemented with fourth-order Mur boundary conditions, which has been designed to address non-linear Klein-Gordon equations, see the Appendix in \cite{Alonso2021}. The linear plane waves are absorbed at the
boundaries in this numerical scheme avoiding that radiation is reflected in the simulation contours. To rule out the presence of spurious phenomena attributable to the use of a particular numerical algorithm, a second numerical procedure is used to validate the results. This double checking has been carried out by means of an energy conservative second-order finite difference algorithm with Mur boundary conditions.
As previously mentioned, the initial settings for our scattering simulations are described by single solutions (kinks or wobblers) which are initially well separated, and are pushed together with initial collision velocity $v_0$. This situation is characterized by the concatenation (\ref{configuration04}) for the scattering between wobblers with opposite phase and by (\ref{configuration03}) for the scattering between a wobbler and a kink, both of them with $x_0\gg 0$. These configurations verify the partial differential equation (\ref{pde}) in a very approximate way for very small values of the time when $x_0\gg 0$ and $a\ll 1$. Therefore, $\Phi(t=0)$ and $\frac{\partial \Phi}{\partial t}(t=0)$ provide the initial conditions of our scattering problem.
In particular, our numerical simulations have been carried out in a spatial interval $x\in [-100,100]$ where the centers of the single solutions are initially separated by a distance $d=2x_0=30$. Simulations have been performed for $v_0\in [0.04,0.9]$ with initial velocity step $\Delta v_0=0.001$, which is decreased to $\Delta v_0=0.00001$ in the resonance interval.
At this point it is worthwhile mentioning that the expression (\ref{wobbler}) is only an approximation of the exact wobbler solution. When this expression is employed as initial condition in the Klein-Gordon equation (\ref{pde}) a small amount of radiation is emitted for a very small period of time. In this time interval the approximate solution (\ref{wobbler}) decays to the exact wobbler. When considering a traveling wobbler, this radiation emission can cause a very small change in its velocity. This effect takes place when $\delta\neq 0,\pi$ in the expression (\ref{wobbler}) and it is maximized for $\delta= \pm \pi/2$. In order to avoid this effect we shall implement initial conditions by setting $\delta=0$ in the configurations (\ref{configuration04}) and (\ref{configuration03}). By taking this restriction we guarantee that the traveling wobbler involved in (\ref{configuration03}) continues to move with velocity $v_0$ after the initial radiation emission. As mentioned above, this effect is very small and unnoticeable in the final versus initial velocity diagrams. However, we shall analyze the velocity difference of the resulting wobblers and in this context it is better to avoid this influence. On the other hand, for the values $\delta=0$ or $\delta=\pi$ the decay of the approximation (\ref{wobbler}) to the real wobbler induces a very small variation in its wobbling amplitude. This effect also is very small and does not affect the global properties of the scattering processes discussed in this paper. An alternative scheme to implement initial configurations (\ref{configuration03}) with non-vanishing initial phases is to find an approximately equivalent configuration with vanishing phase. This can be obtained, for example, taking into account that
\[
\phi_W^{(\pm)}(t,x; x_0,v_0,\omega,a,\delta) = \phi_W^{(\pm)}(t-\textstyle\frac{\delta}{\omega},x; x_0-\frac{\delta v_0}{\omega},v_0,\omega,a,0)\hspace{0.3cm} .
\]
\section{Scattering between wobblers with opposite phases}
\label{sec:3}
In this section we shall analyze the asymmetric scattering between two wobblers whose shape modes have the same amplitude but they have opposite phases with respect to our inertial system, which is located at the center of mass. In this context, a wobbler and an anti-wobbler approach each other with initial velocity $v_0$ and $-v_0$, respectively. They evolve preserving the phase difference of $\pi$ and collide giving place to a destructive interference between the shape modes of the involved wobblers. Figure~\ref{fig:VelDiaContrafaseAmp020} shows the final versus initial velocity diagrams for three representative values of the initial wobbling amplitude, $a=0.04$, $a=0.1$, and $a=0.2$.
\begin{figure}[h]
\centerline{\includegraphics[height=3.5cm]{veldiacontrafaseamp004}}
\medskip
\centerline{\includegraphics[height=3.5cm]{veldiacontrafaseamp010}}
\medskip
\centerline{\includegraphics[height=3.5cm]{veldiacontrafaseamp020}}
\caption{\small Final versus initial velocity diagram for the scattering between two wobblers with opposite phase for the values of the wobbling amplitude $a=0.04$, $a=0.1$ and $a=0.2$. The color code is used to specify the number of bounces suffered by the kinks before escaping. The black curves determine the final velocity of the bion formed for low initial velocities.} \label{fig:VelDiaContrafaseAmp020}
\end{figure}
Unsurprisingly, the only scattering channels to emerge in this new scenario are still bion formation and kink reflection. As before, the former is predominant for small values of the initial velocity $v_0$, while the latter is found for large values. However, these velocity diagrams display some important differences regarding the scattering of wobblers addressed in \cite{Alonso2021b}, where the corresponding shape modes have the same phase and a constructive interference occurs in the collision. In this new context, the destructive interference avoids the emergence of isolated 1-bounce windows (at least for non-extreme values of $a$), as can be observed in Figures~\ref{fig:VelDiaContrafaseAmp020} and \ref{fig:VelDiaContrafaseFractal}. The suppression of this mechanism implies that the fractal structure width does not grow.
In Figure \ref{fig:VelDiaContrafaseFractal} the evolution of the fractal pattern can be visualized as the value of the initial amplitude $a$ increases. First, we can observe that the value of the critical velocity $v_c$ varies very slowly as the initial amplitude $a$ grows. For instance, $v_c \approx 0.2601$ for $a=0.02$, whereas $v_c\approx 0.2681$ for $a=0.2$, following a linear dependence in $a$ for intermediate values. Second, it can be seen that the 2-bounce windows are deformed as the value of $a$ increases and get broken up in smaller 2-bounce windows. The first 2-bounce window shown in Figure \ref{fig:VelDiaContrafaseFractal} for $a=0.04$ can be used to illustrate this mechanism. This window gets distorted when $a=0.12$ and split into two pieces for $a=0.14$. In turn, one of these pieces is divided again into two new 2-bounce windows for $a=0.20$. Third, spontaneous generation of $n$-bounce windows with $n\geq 2$ can also be identified in the sequence of graphics included in Figure~\ref{fig:VelDiaContrafaseFractal}. For instance, for $a=0.12$ a small 3-bounce window spontaneously emerges in the interval $[0.21585,0.2174]$, which was occupied by the bion formation regime for previous values of $a$. Subsequently, this window is split into two parts, resulting in a 2-bounce window in the middle for $a=0.14$, which is surrounded by new $n$-bounce windows. This new 2-bounce window gets bigger as $a$ increases and finally splits into two new 2-bounce windows once more, as you can see from the graphics for $a=0.20$. This window generation mechanism could explain the clustering of 2-bounce windows that arise around the $v_0=0.2566$ value for $a=0.20$.
\begin{figure}[htb]
\centerline{\includegraphics[height=2.5cm]{veldiacontrafasefractalamp004}}
\medskip
\centerline{\includegraphics[height=2.5cm]{veldiacontrafasefractalamp010}}
\medskip
\centerline{\includegraphics[height=2.5cm]{veldiacontrafasefractalamp012}}
\medskip
\centerline{\includegraphics[height=2.5cm]{veldiacontrafasefractalamp014}}
\medskip
\centerline{\includegraphics[height=2.5cm]{veldiacontrafasefractalamp020}}
\caption{\small Evolution of the fractal pattern found in the velocity diagrams associated to the scattering between wobblers with opposite phases as a function of the initial wobbling amplitude $a$.} \label{fig:VelDiaContrafaseFractal}
\end{figure}
Another important characteristic of this type of scattering processes is that the final velocities of the scattered wobblers are different. This behavior is not surprising because the initial configuration (\ref{configuration04}) is not symmetric. Recall that the initial wobbling phases of the colliding wobblers are different. This velocity difference is very small, and therefore not noticeable in the velocity diagrams shown in Figure~\ref{fig:VelDiaContrafaseAmp020}. In order to emphasize this feature we define the magnitude
\begin{equation}
\Delta v_f = |v_{f,R}|-|v_{f,L}| \, , \label{deltavf}
\end{equation}
as the difference between the final speed $|v_{f,R}|$ of the rightward traveling wobbler and the final speed $|v_{f,L}|$ of the leftward traveling wobbler. Positive values of $\Delta v_f$ imply that the wobbler scattered to the right travels faster than the wobbler scattered to the left, whereas negative values describe the reverse situation. In Figure~\ref{fig:VelocityDifferenceContrafase}, the magnitude $\Delta v_f$ is plotted as a function of the initial velocity $v_0$ and the wobbling amplitude $a$. There, we can see that $\Delta v_f$ has oscillating behavior, which means that there are alternating initial velocity windows in which the wobbler traveling from the left travels faster than the wobbler traveling from the right and vice versa. The amplitudes of the oscillations exhibited by $\Delta v_f$ grow as the value of the parameter $a$ increases. This is reasonable because the vibrational energy stored in the shape mode is greater for bigger values of $a$ and the resonant energy transfer mechanism may deflect a greater amount of this energy to the kinetic energy pool. However, the most remarkable property exhibited by Figure~\ref{fig:VelocityDifferenceContrafase} is that the zeroes of $\Delta v_f$, the initial velocity values for which the two wobblers disperse with the same velocity, are approximately independent of the initial amplitude $a$. This behavior is precisely followed for sufficiently large values of $v_0$, where the effect of the resonance regime is not noticed (approximately for $v_0\geq 0.3$ in Figure~\ref{fig:VelocityDifferenceContrafase}).
\begin{figure}[htb]
\centerline{\includegraphics[height=3.3cm]{velocitydifferencecontrafase}}
\caption{\small Final velocity difference $\Delta v_f$ of the scattered wobblers as a function of the collision velocity $v_0$ and the initial wobbling amplitude $a$ for the scattering of two wobblers with opposite phase. Recall that $\Delta v_f=0$ for $a=0$ due to spatial reflection symmetry. For the sake of clarity, $n$-bounce processes with $n\geq 2$ have not been included in the plot. The vertical dashed lines mark the zeroes $\widetilde{v}_k$ of the final velocity difference $\Delta v_f$.} \label{fig:VelocityDifferenceContrafase}
\end{figure}
In Table \ref{ZerosContrafase}, the zeros $\widetilde{v}_k$ of the final velocity difference $\Delta v_f$ (explicitly computed for the case $a=0.04$) are shown in the non-resonance regime. The values $\widetilde{v}_k$ correspond to the nodes of the oscillations found in Figure~\ref{fig:VelocityDifferenceContrafase}, which have been remarked by means of vertical dashed lines.
The location of these points seems to depend mainly on the value of the wobbling phase when the collision between the wobblers occurs. This conjecture is heuristically supported by the following simple argument. Remember that $x_0$ denotes the initial position of the kink center, while $\omega$ represents the wobbling frequency. As previously discussed, the values $x_0=15$ and $\omega=\sqrt{3}$ have been implemented for our numerical simulations. Let $v_0$ be the initial velocity at which the wobblers are initially approaching. In the point particle approximation the collision would happen at the time $t_I=\frac{x_0}{v_0}$. We must bear in mind that there are several factors in the real dynamics which break the precision of this assumption. For example, the interaction between the kinks and/or wobblers can make the collision velocity vary (it is not a constant velocity $v_0$). We shall assume that the phase of the wobbler at the instant $t_I$ can be expressed as
\[
\varphi(v_0)= c(x_0) \, \frac{x_0}{v_0} \omega \sqrt{1-v_0^2} + \delta \, .
\]
where $c(x_0)$ is a correction factor which is included to incorporate the previously mentioned behavior. The main assumption in this case is that $c(x_0)$ does not depend on $v_0$. If we think about the initial impact velocity as a variable $v$, then it makes sense to consider $\varphi(v)=c(x_0) \, \frac{x_0}{v} \omega \sqrt{1-v^2} + \delta$.
Those phenomena depending only on the wobbling phase must exhibit a periodicity based on the relation
\begin{equation}
\varphi(v_0) - \varphi(v)= T\, k\, , \hspace{0.6cm} k\in \mathbb{Z}\,, \label{fase}
\end{equation}
where $T$ is the periodicity associated to our problem. In general, $T=2\pi$ but in the present scenario where we are interested in the zeroes $\widetilde{v}_k$ of $\Delta v_f$ the symmetry of the initial configuration leads to the choice $T=\pi$. From (\ref{fase}) we conclude that the discrete set of velocities
\begin{equation}
f_k(v_0,T)=\frac{v_0 \, x_0 \, \omega}{\sqrt{\frac{k^2 \, T^2\, v_0^2}{4 \,c(x_0)^2} - \frac{T}{c(x_0)} \, k \, v_0 \, x_0 \, \omega \sqrt{1-v_0^2} + x_0^2 \, \omega^2}} \,,\hspace{0.6cm} k\in \mathbb{Z},
\label{velocities}
\end{equation}
must share similar features. The nodes $\widetilde{v}_k$ of $\Delta v_f$ can be approximately figured out by using equation (\ref{velocities}). In Table \ref{ZerosContrafase} (third column) the values $V_{k}=f_k(v_0,\pi)$, obtained by using the formula (\ref{velocities}) taking as initial input $v_0=\widetilde{v}_0=0.301538$, are included. The value $c(x_0)$ has been adjusted to $c(x_0)= 0.465$. The comparison between the data allows us to conclude that the previous conjecture is satisfied at least for intermediate values of the initial velocity. Of course, the nonlinear nature of the problem makes the argument only an approximation to the actual behavior. This is clear for very large values of the collision velocity. In this regime the amplitudes of the oscillations of $\Delta v_f$ are very attenuated compared to intermediate values of $v_0$. For these cases radiation emission can play a predominant role in the scattering processes.
\begin{table}[htb]
\centerline{\begin{tabular}{|c|c|c|} \hline
$k$ & $\widetilde{v}_k$ & $V_k$ \\ \hline
0 & 0.301538 & 0.301538 \\
1 & 0.313991 & 0.313224\\
2 & 0.326057 & 0.325797 \\
3 & 0.340542 & 0.339354 \\
4 & 0.354757 & 0.354006 \\
5 & 0.371738 & 0.369877 \\
6 & 0.388575 & 0.387110 \\
\hline
\end{tabular} \hspace{0.2cm} \begin{tabular}{|c|c|c|} \hline
$k$ & $\widetilde{v}_k$ & $V_k$ \\ \hline
7 & 0.408291 & 0.405864 \\
8 & 0.428219 & 0.426323 \\
9 & 0.451464 & 0.448689 \\
10 & 0.475173 & 0.473188 \\
11 & 0.502713 & 0.500069 \\
12 & 0.531044 & 0.529591 \\
13 & 0.563882 & 0.562022 \\
\hline
\end{tabular} \hspace{0.2cm}
\begin{tabular}{|c|c|c|} \hline
$k$ & $\widetilde{v}_k$ & $V_k$ \\ \hline
14 & 0.597459 & 0.597606 \\
15 & 0.636259 & 0.636531 \\
16 & 0.675558 & 0.678857 \\
17 & 0.727740 & 0.724418 \\
18 & 0.770909 & 0.772667 \\ && \\ && \\ \hline
\end{tabular}}
\caption{Comparison between the zeros $\widetilde{v}_k$ of the final velocity difference $\Delta v_f$ and the values $V_k=f_k(v_0,\pi)$ obtained by using equation (\ref{velocities}) for the scattering between wobblers with opposite phase and initial wobbling amplitude $a=0.04$. } \label{ZerosContrafase}
\end{table}
At this point it is worthwhile mentioning that the zeroes $\widetilde{v}_k$ introduced in Table \ref{ZerosContrafase} have been computed when $\delta=0$ in the initial configuration (\ref{configuration04}). The particular location of these points depends on the initial phase $\delta$ introduced in (\ref{configuration04}), although it is clear that the same pattern is periodically reproduced for the values $\delta + k T$ with $k\in \mathbb{Z}$.
\begin{figure}[h]
\centerline{\includegraphics[height=4.2cm]{amplitudfinalcontrafase020kink1y2}}
\caption{\small Graphics of the final wobbling amplitudes of the wobblers scattered to the left and to the right as a function of the initial velocity $v_0$ and the initial amplitude $a$. For the sake of clarity, $n$-bounce processes with $n\geq 2$ have not been included. The vertical dashed lines mark the zeroes $\widetilde{v}_k$ of the final velocity difference $\Delta v_f$. The letters $L$ and $R$ label the smooth amplitude functions associated with wobblers traveling left and right, respectively. } \label{fig:AmplitudFinalContrafaseKink1y2}
\end{figure}
Once the final velocities of the scattered wobblers have been examined, we shall now analyze the behavior of the wobbling amplitude of these evolving topological defects.
In Figure~\ref{fig:AmplitudFinalContrafaseKink1y2}, the oscillation amplitudes of the wobblers moving to the left and to the right are represented as a function of the initial velocity $v_0$ and the initial amplitude $a$. There it can be seen that this magnitude follows an oscillating behavior with respect to the kink-antikink scattering. The variation of these oscillations grows as the parameter $a$ increases. Furthermore, the amplitudes of the resulting wobblers follow an antagonistic behavior. When the oscillation amplitude of the wobbler moving to the left reaches a maximum as a function of the initial velocity $v_0$, the oscillation amplitude of the wobbler moving to the right is minimized and vice versa. The asymmetry of the initial configuration (\ref{configuration04}) causes the wobblers to vibrate at different amplitudes in general. On the other hand, there are some points in the graphs shown in Figure~\ref{fig:AmplitudFinalContrafaseKink1y2} where the amplitudes of the two wobblers coincide. Surprisingly, these points coincide with the zeroes $\widetilde{v}_k$ of the final velocity difference $\Delta v_f$ (as we can observed by means of the vertical dashed lines plotted in Figure~\ref{fig:AmplitudFinalContrafaseKink1y2}). In conclusion, for the initial velocities $\widetilde{v}_k$ the scattered wobblers travel with the same velocity and vibrate with the same wobbling amplitude.
In order to explore the relation between the final velocity and the final wobbling amplitude of the scattered wobblers, we define the amplitude difference
\begin{equation}
\Delta a = \frac{1}{2} \left[ a_{f,R} - a_{f,L} \right]\, ,
\end{equation}
where $a_{f,R}$ and $a_{f,L} $ are, respectively, the final oscillation amplitudes of the wobblers moving to the right and to the left. $\Delta a > 0$ means that the wobbler scattered to the right vibrates strongly than that moving to the left, whereas $\Delta a < 0$ describes the opposite situation. Figure~\ref{fig:AmplitudYVelocidadContrafase} shows simultaneously the final velocity and the amplitude differences $\Delta v_f$ and $\Delta a$, as functions of the initial velocity $v_0$ for the particular value $a=0.10$. It can be seen that when a scattered wobbler gains more kinetic energy than the other, it obtains less vibrational energy, and vice versa. The values $\widetilde{v}_k$ are interpreted as the collision velocities for which the final velocities and the wobbling amplitudes of the scattered wobblers are the same.
\begin{figure}[htb]
\centerline{\includegraphics[height=3.4cm]{amplitudyvelocidadcontrafase}}
\caption{\small Graphics of $\Delta v_f$ (final velocity difference) and $\Delta a$ (final wobbling amplitude difference) as functions of the initial collision velocity $v_0$ for the scattering between wobblers with opposite phase with $a=0.10$. $n$-bounce processes with $n\geq 2$ have not been included. The vertical dashed lines mark the zeroes $\widetilde{v}_k$ of $\Delta v_f$. } \label{fig:AmplitudYVelocidadContrafase}
\end{figure}
Finally, another consequence of the asymmetry of these scattering events is that the bion (formed as a bound state between the two colliding wobblers) can now move with certain final non-vanishing velocity after the impact. This velocity will be very small and for this reason it is sometimes difficult to compute its magnitude numerically. In~Figure~\ref{fig:VelDiaContrafaseBionAmp010} the region of the velocity diagram introduced in Figure~\ref{fig:VelDiaContrafaseAmp020} for $a=0.10$ with $v_0\in [0.10,0.18]$ has been enlarged to illustrate the behavior of the bion velocity.
Again, we find an oscillating pattern, clearly seen in Figure~\ref{fig:VelDiaContrafaseBionAmp010} for the interval $v_0\in [0.13,0.16]$. Also, it turns out that the formula (\ref{velocities}) still governs this oscillating behavior. In the previously mentioned range of $v_0$, vertical dashed lines have been plotted to approximately mark the location of the nodes of the bion velocity. The values used correspond to the initial velocities $v_1 \approx 0.134$, $v_2 \approx 0.1364$, $v_3 \approx 0.1388$, $v_4 \approx 0.1415$, $v_5 \approx 0.1443$, $v_6 \approx 0.1475$, $v_7 \approx 0.1506$, $v_8 \approx 0.1538$, $v_9 \approx 0.1572$ and $v_{10} \approx 0.161$, which can be approximately reproduced by (\ref{velocities}).
\begin{figure}[htb]
\centerline{\includegraphics[height=3.4cm]{veldiacontrafasebionamp010}}
\caption{\small Final bion velocity as a function of the initial velocity $v_0$ in the interval $v_0\in [0.10,0.18]$ for the scattering between two wobblers with opposite phase and the initial wobbling amplitude $a=0.10$. The vertical dashed lines mark some of the nodes of the curve.} \label{fig:VelDiaContrafaseBionAmp010}
\end{figure}
\section{Scattering between a kink and a wobbler}
\label{sec:4}
In this section we shall study the scattering between a wobbler and a kink. This scenario is characterized by the concatenation (\ref{configuration03}). With the first choice of signs, this configuration describes a wobbler and an antikink which travel respectively with velocities $v_0$ and $-v_0$. The rightward traveling wobbler and the leftward traveling antikink approach each other, collide, and bounce back. As usual, the \textit{formation of a bion} and the \textit{reflection of the solutions} complete the list of possible scattering channels. In the reflection regime, the initially unexcited antikink becomes an anti-wobbler after the collision because, in general, the shape mode of this solution is excited. Therefore, after the impact two wobblers emerge moving away with different final velocities in our inertial system. The goal of this study is to analyze the transfer of the vibrational and kinetic energies between the resulting wobblers. The dependence of the final velocities of the scattered extended particles on the initial velocity $v_0$ has been graphically represented in Figure~\ref{fig:VelDiaAmp020} for the cases $a=0.04$, $a=0.1$, and $a=0.2$.
\begin{figure}[htb]
\centerline{\includegraphics[height=3.5cm]{veldiaamp004}}
\medskip
\centerline{\includegraphics[height=3.5cm]{veldiaamp010}}
\medskip
\centerline{\includegraphics[height=3.5cm]{veldiaamp020}}
\caption{\small Final versus initial velocity diagram for the wobbler-antikink scattering for the values of the wobbling amplitude $a=0.04$, $a=0.10$ and $a=0.20$. The color code is used to specify the number of bounces suffered by the kinks before escaping.} \label{fig:VelDiaAmp020}
\end{figure}
\begin{figure}[bht]
\centerline{\includegraphics[height=3cm]{velodiaisolated}}
\caption{\small Velocity diagrams for the wobbler-antikink scattering showing the emergence and location of the isolated 1-bounce windows as the value of the wobbling amplitude increases. The vertical dashed lines mark the values of $v_0$ at the center of the 1-bounce windows for the extreme case $a=0.2$. For the sake of clarity, $n$-bounce processes with $n\geq 2$ have not been included. } \label{fig:VelDiaIsolated}
\end{figure}
Some of the most relevant characteristics described in \cite{Alonso2021b} for the scattering between wobbling kinks are also found in this framework, such as the emergence of isolated 1-bounce windows and the growing complexity of the fractal pattern as the initial wobbling amplitude $a$ of the originally rightward-traveling wobbler increases. It is also worthwhile mentioning the presence of oscillations in the 1-bounce tail arising for large values of the initial velocity. However, these features are less accentuated in this scenario. The reason of this behavior lies in the fact that the constructive interference is maximized when the wobblers collide with the same wobbling phase. In particular, we can observe the existence of two isolated 1-bounce windows for the case $a=0.04$. They occupy approximately the region $[0.2458,0.2522] \cup [0.2607,0.2717]$. For $a=0.1$ six of these windows can be identified in $[0.2068,0.2085] \cup [0.2183,0.2214] \cup [0.2310,0.2358] \cup [0.2452,0.2521] \cup [0.2610,0.2709] \cup [0.2787,0.2925]$. Finally, the number of these windows explodes as the initial amplitude $a$ grows. This can be observed in the velocity diagram for $a=0.2$ in Figure \ref{fig:VelDiaAmp020}. Some of the widest 1-bounce windows in this case arise in the set of intervals $[0.2067,0.2108]\cup [0.2185,0.2234] \cup [0.2315,0.2375] \cup [0.2460,0.2535] \cup [0.2621,0.2717] \cup [0.2804,0.2928] \cup [0.3012,0.3179]$. From the previous list of 1-bounce windows, it can be verified that once an isolated 1-bounce window emerges its location is approximately fixed (although its width slightly grows) as the initial wobbling amplitude $a$ increases. This behavior can be checked in Figure \ref{fig:VelDiaIsolated}. Note that the deviation from the rule described above is a small translation of the center of these windows. In Figure \ref{fig:VelDiaIsolated} the vertical dashed lines mark the values of the initial velocity which determine the centers of the 1-bounce windows for the extreme case $a=0.2$. Once again, these velocities approximately follow relation (\ref{velocities}), which reveals that the role of the phase of the evolving shape mode is predominant in this phenomenon.
The velocity diagrams shown in Figure~\ref{fig:VelDiaAmp020} also have some distinctive properties of their own.
Because the scattering processes introduced in this section are asymmetric, the final velocities of the resulting wobblers are different, as well as their wobbling amplitudes. In order to illustrate this feature more clearly, the difference $\Delta v_f$ between the final speeds of the scattered wobblers is plotted for different values of the wobbling amplitude $a$ in Figure~\ref{fig:VelocityDifference}. For the sake of simplicity, only 1-bounce events have been included in Figure~\ref{fig:VelocityDifference}. As in the case of the scattering between wobblers with opposite phase discussed in Section \ref{sec:3}, the zeros of this function $\Delta v_f$ are approximately independent of the initial amplitude $a$ and, indeed, coincide with the zeroes $\widetilde{v}_k$ introduced in Table~\ref{ZerosContrafase} in Section~\ref{sec:3}. This behavior underlies the fact that the initially rightward wobbler defined in the configuration (\ref{configuration03}) has the same initial conditions as those given by the configuration (\ref{configuration04}).
\begin{figure}[h]
\centerline{\includegraphics[height=3.3cm]{velocitydifference}}
\caption{\small Final velocity difference $\Delta v_f$ of the scattered wobblers as a function of the collision velocity $v_0$ and the initial wobbling amplitude $a$ for the scattering of a wobbler and a kink. $n$-bounce processes with $n\geq 2$ have not been included. The vertical dashed lines mark the zeroes $\widetilde{v}_k$ of $\Delta v_f$ displayed in Table~\ref{ZerosContrafase}. } \label{fig:VelocityDifference}
\end{figure}
In Figure~\ref{fig:AmplitudFinalTotalKink} the final wobbling amplitudes of the scattered wobblers are plotted as a function of the initial velocity $v_0$ and the initial wobbling amplitude $a$. Recall that $a_L(v_0,a)$ and $a_R(v_0,a)$ represent, respectively, the final wobbling amplitudes of the resulting leftward and rightward traveling wobblers after the collision. We can observed that the shape modes of the scattered wobblers become excited and its amplitudes are similar as a function of the initial velocity, oscillating around the values found for the kink-antikink scattering events (with $a=0$). However, the amplitude of these oscillations is much bigger for the final rightward traveling wobbler.
\begin{figure}[htb]
\centerline{\includegraphics[height=3.5cm]{amplitudfinaltotalkink1}}\medskip
\centerline{\includegraphics[height=3.5cm]{amplitudfinaltotalkink2}}
\caption{\small Final wobbling amplitudes $a_L$ and $a_R$ of the wobblers scattered to the left (top) and to the right (bottom) as a function of the initial velocity $v_0$ and the initial wobbling amplitude $a$ of the colliding wobbler. The vertical dashed lines mark the zeroes $\widetilde{v}_k$ of $\Delta v_f$ displayed in Table \ref{ZerosContrafase}.} \label{fig:AmplitudFinalTotalKink}
\end{figure}
To illustrate the role of the the zeroes $\widetilde{v}_k$ of the final velocity difference $\Delta v_f$ shown in Table~\ref{ZerosContrafase} in this scenario, the functions $\Delta v_f$ and $\Delta a$ have been represented simultaneously for the case $a=0.10$ in Figure~\ref{fig:AmplitudFinal020Kink1y2}. As in the scattering between wobblers with opposite phase, the values $\widetilde{v}_k$ determine the initial velocities for which the final velocities and the final wobbling amplitudes are the same for the both scattered wobblers.
\begin{figure}[h]
\centerline{\includegraphics[height=3.5cm]{amplitudyvelocidad}}
\caption{\small Graphics of $\Delta v_f$ (final velocity difference) and $\Delta a$ (final wobbling amplitude difference) as a function of the initial collision velocity $v_0$ for the scattering between a wobbler and an antikink with $a=0.10$. $n$-bounce processes with $n\geq 2$ have not been included. The vertical dashed lines mark the zeroes $\widetilde{v}_k$ of $\Delta v_f$.} \label{fig:AmplitudFinal020Kink1y2}
\end{figure}
\section{Conclusions}
\label{sec:5}
This paper delves into the study on the scattering between wobbling kinks initially addressed in \cite{Alonso2021b}. Here, we have investigated the asymmetric scattering between kinks and wobblers (kinks whose shape mode is excited) in the standard $\phi^4$ model. In particular, two different scenarios in this context have been considered: (a) the scattering between wobblers with opposite phases, and (b) the scattering between a wobbler and an unexcited antikink. Both cases exhibit the usual bion formation and reflection regimes, which are infinitely interlaced forming a fractal structure embedded in the final versus initial velocity diagram. However, the first case involves a destructive interference of the shape modes in the collision. As a consequence, the growth in the complexity of the fractal pattern is smaller than that found in \cite{Alonso2021b}, where the colliding wobbling kinks travel with the same phase leading to a constructive interference at the impact. For example, the emergence of isolated 1-bounce windows is not found in this new case (at least for moderate values of the initial wobbling amplitude $a$), although the splitting of $n$-bounce widows is present. On the other hand, the kink scattering in the second scenario displays similar features (although more attenuated) than to those found in \cite{Alonso2021b}.
Due to the asymmetry of the initial configurations (\ref{configuration04}) and (\ref{configuration03}), the final velocities and wobbling amplitudes of the scattered wobblers are different in general. However, there is a sequence of initial velocities for which both the final velocities and wobbling amplitudes coincide. These values are almost independent of the initial wobbling amplitude $a$ when the initial wobbling phase considered in (\ref{configuration04}) and (\ref{configuration03}) is fixed. Besides, the values of these velocities very approximately follow the expression (\ref{velocities}). This means that the phase associated to the shape modes of the evolving wobblers at the collision instant plays a predominant role in the scattering properties of these objects.
Indeed, (\ref{velocities}) allows to obtain values of the initial velocities which share similar features. For example, this expression has been used in the second scenario to predict the location of the maxima of the isolated 1-bounce windows. Finally, it is also worthwhile mentioning the results displayed in Figures \ref{fig:AmplitudYVelocidadContrafase} and \ref{fig:AmplitudFinal020Kink1y2}. It can be verified that systematically when a scattered wobbler gains more kinetic energy than the other, it obtains less vibrational energy and vice versa.
The research introduced in the present work opens up some possibilities for future work. For example, the $\phi^6$ model implies a resonance regime similar to the $\phi^4$ model, although it does not present vibrational eigenstates in the second-order small fluctuation operator. The characteristics of scattered wobbling kinks can be analyzed to study their influence on the resonant energy transfer mechanism. Alternatively, you can build a model twin to the $\phi^6$ model that involves internal modes. By doing this, we could compare the scattering processes of the twin model with those of the standard $\phi^6$ model. In this way, it will be possible to examine the role that shape modes play in the collision process.
Furthermore, many other different topological defects (kinks in the double sine-Gordon model, deformed $\phi^4$ models, hybrid and hyperbolic models, etc.) could be studied in the new perspective presented here. Work in these directions is in progress.
\section*{Acknowledgments}
A. Alonso-Izquierdo acknowledges Spanish MCIN financial support under grant PID2020-113406GB-I0. He also acknowledges the Junta de Castilla y Le\'on for financial support under grants SA067G19. L.M. Nieto acknowledges Spanish MCIN financial support under grant PID2020-113406GB-I0. This research has made use of the high performance computing resources of the Castilla y Le\'on Supercomputing Center (SCAYLE, www.scayle.es), financed by the European Regional Development Fund (ERDF).
|
{
"timestamp": "2021-10-05T02:09:37",
"yymm": "2109",
"arxiv_id": "2109.13904",
"language": "en",
"url": "https://arxiv.org/abs/2109.13904"
}
|
"\\section{Introduction and Context}\n\\subsection{Historical Context}\n\nCircuit complexity is a br(...TRUNCATED)
| {"timestamp":"2021-09-30T02:00:12","yymm":"2109","arxiv_id":"2109.13917","language":"en","url":"http(...TRUNCATED)
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"\\section{Introduction}\nWe study the bit complexity of finding approximate solutions to the follow(...TRUNCATED)
| {"timestamp":"2021-09-30T02:00:42","yymm":"2109","arxiv_id":"2109.13956","language":"en","url":"http(...TRUNCATED)
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"\\section{Introduction}\nThe most generic family of regular, stationary, asymptotically flat, elect(...TRUNCATED)
| {"timestamp":"2022-04-12T02:42:24","yymm":"2109","arxiv_id":"2109.13961","language":"en","url":"http(...TRUNCATED)
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"\\section{Introduction}\nBiological imaging which precisely labels microscopic structures offers a (...TRUNCATED)
| {"timestamp":"2021-09-30T02:03:48","yymm":"2109","arxiv_id":"2109.14025","language":"en","url":"http(...TRUNCATED)
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"\n\\section{Introduction} \\label{sec:intro}\n\nCharacterizing young planetary systems is key to im(...TRUNCATED)
| {"timestamp":"2021-09-30T02:02:26","yymm":"2109","arxiv_id":"2109.13996","language":"en","url":"http(...TRUNCATED)
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"\\section{Introduction}\\label{intro}\n\n\nDuring the last years, a large number of young and inter(...TRUNCATED)
| {"timestamp":"2021-09-30T02:00:47","yymm":"2109","arxiv_id":"2109.13959","language":"en","url":"http(...TRUNCATED)
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"\\section{Introduction}\n\\label{sec:introduction}\n\nThe so-called \\textit{cooling function} $\\L(...TRUNCATED)
| {"timestamp":"2022-03-30T02:28:37","yymm":"2109","arxiv_id":"2109.13926","language":"en","url":"http(...TRUNCATED)
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