From ec3406fb7ae7141b0b163f74f3b24620fb41342b Mon Sep 17 00:00:00 2001
From: Ian Jauslin <jauslin@ias.edu>
Date: Tue, 19 Sep 2017 16:14:31 +0000
Subject: Update to v0.1

Fixed: deleted erroneous reference to Kramers and Wannier.

Fixed: changed boundary conditions.

Fixed: definition of \mathbb S(X).

Fixed: miscellaneous minor tweaks, formatting and typos.

Added: informal description of Gaunt-Fisher configurations.

Changed: title.

Changed: reformulate main theorem.
---
 Jauslin_Lebowitz_2017b.tex | 327 +++++++++++++++++++++++++--------------------
 1 file changed, 184 insertions(+), 143 deletions(-)

(limited to 'Jauslin_Lebowitz_2017b.tex')

diff --git a/Jauslin_Lebowitz_2017b.tex b/Jauslin_Lebowitz_2017b.tex
index f0ff71d..e3e846d 100644
--- a/Jauslin_Lebowitz_2017b.tex
+++ b/Jauslin_Lebowitz_2017b.tex
@@ -12,9 +12,9 @@
 
 \hbox{}
 \hfil{\bf\LARGE
-High-fugacity expansion and crystalline ordering\par
+High-fugacity expansion, Lee-Yang zeros and\par
 \vskip10pt
-\hfil for non-sliding hard-core lattice particle systems
+\hfil order-disorder transitions in hard-core lattice systems
 }
 \vskip80pt
 
@@ -46,22 +46,24 @@ We establish existence of order-disorder phase transitions for a class of ``non-
 \pagestyle{plain}
 
 \section{Introduction}
-\indent One of the most interesting open problems in the theory of equilibrium statistical mechanics, is to prove the existence of order-disorder phase transitions in continuum particle systems. While such fluid-crystal transitions are ubiquitous in real systems and are observed in computer simulations of systems with  effective pair potentials, there are no proofs, or even good heuristics, for showing this mathematically. A paradigmatic example of this phenomenon is the fluid-crystal transition for hard spheres in 3 dimensions, observed in simulations and experiments~\-\cite{WJ57,AW57,PM86,IK15}. Whereas, in 2 dimensions, crystalline states are ruled out by the Mermin-Wagner theorem~\-\cite{Ri07}, it is believed that there are other transitions for hard discs~\-\cite{BK11} (see~\-\cite{St88} or~\-\cite[section~8.2.3]{Mc10} for a review), though none have, as of yet, been proven. Such transitions are purely geometric. They are driven by entropy and depend only on the density, that is, on the volume fraction taken up by the hard particles.
+\indent One of the most important open problems in the theory of equilibrium statistical mechanics, is to prove the existence of order-disorder phase transitions in continuum particle systems. While such fluid-crystal transitions are ubiquitous in real systems and are observed in computer simulations of systems with  effective pair potentials, there are no proofs, or even good heuristics, for showing this mathematically. A paradigmatic example of this phenomenon is the fluid-crystal transition for hard spheres in 3 dimensions, observed in simulations and experiments~\-\cite{WJ57,AW57,PM86,IK15}. Whereas, in 2 dimensions, crystalline states are ruled out by the Mermin-Wagner theorem~\-\cite{Ri07}, it is believed that there are other transitions for hard discs~\-\cite{BK11} (see~\-\cite{St88} or~\-\cite[section~8.2.3]{Mc10} for a review), though none have, as of yet, been proven. Such transitions are purely geometric. They are driven by entropy and depend only on the density, that is, on the volume fraction taken up by the hard particles.
 \bigskip
 
-\indent The situation is different for lattice systems, where there are many examples for which such entropy-driven transitions have been proven. A simple example is that of hard ``diamonds'' on the square lattice (see figure~\-\ref{fig:shapes}{\it a}), which is a model on $\mathbb Z^2$ with nearest-neighbor exclusion. As was shown by Dobrushin~\-\cite{Do68}, this model transitions from a low-density disordered state to a high-density crystalline phase, where the even or odd sublattice is preferentially occupied. The heuristics of this transition had been understood earlier (the hard diamond model related to the 0-temperature limit of the antiferromagnetic Ising model for which the exponential of the magnetic field plays the role of the fugacity~\-\cite{BK73,LRS12}), for instance by Gaunt and Fisher~\-\cite{GF65}, who extrapolated a low- and high-fugacity expansion of the pressure $p(z)$ to find a singularity at a critical fugacity $z_c>0$. A similar analysis was carried out for the nearest neighbor exclusion on $\mathbb Z^3$ by Gaunt~\-\cite{Ga67}.
+\indent The situation is different for lattice systems, where there are many examples for which such entropy-driven transitions have been proven. A simple example is that of hard ``diamonds'' on the square lattice (see figure~\-\ref{fig:shapes}{\it a}), which is a model on $\mathbb Z^2$ with nearest-neighbor exclusion. As was shown by Dobrushin~\-\cite{Do68}, this model transitions from a low-density disordered state to a high-density crystalline phase, where the even or odd sublattice is preferentially occupied. The heuristics of this transition had been understood earlier (the hard diamond model is related to the 0-temperature limit of the antiferromagnetic Ising model for which the exponential of the magnetic field plays the role of the fugacity~\-\cite{BK73,LRS12}), for instance by Gaunt and Fisher~\-\cite{GF65}, who extrapolated a low- and high-fugacity expansion of the pressure $p(z)$ to find a singularity at a critical fugacity $z_t>0$. A similar analysis was carried out for the nearest neighbor exclusion on $\mathbb Z^3$ by Gaunt~\-\cite{Ga67}.
 
 \indent The low-fugacity expansion in powers of the fugacity $z$ dates back to Ursell~\-\cite{Ur27} and Mayer~\-\cite{Ma37}. Its radius of convergence was bounded below by Groeneveld~\-\cite{Gr62} for positive pair-potentials and by Ruelle~\-\cite{Ru63} and Penrose~\-\cite{Pe63} for general pair-potentials.
 
-\indent The high-fugacity expansion is an expansion in powers of the inverse fugacity $y\equiv z^{-1}$. As far as we know, it was first considered by Gaunt and Fisher~\-\cite{GF65} for the hard diamond model, without any indication of its having a positive radius of convergence.
+\indent The high-fugacity expansion is an expansion in powers of the inverse fugacity $y\equiv z^{-1}$. As far as we know, it was first considered by Gaunt and Fisher~\-\cite{GF65} for the hard diamond model, without any indication of its having a positive radius of convergence, or that its coefficients are finite in the thermodynamic limit beyond the first 9 terms.
 \bigskip
 
 \indent In this paper we prove, using an extension of Pirogov-Sinai theory~\-\cite{PS75,KP84}, that the high-fugacity expansion has a positive radius of convergence for a class of hard-core lattice particle systems in $d\geqslant 2$ dimensions. We call these {\it non-sliding} models. In addition, we show that these systems exhibit high-density crystalline phases, which, combined with the convergence of the low-fugacity expansion proved in~\-\cite{Gr62,Ru63,Pe63}, proves the existence of an order-disorder phase transition for these models. A preliminary account of this work, without proofs, is in~\-\cite{JL17b}.
 \bigskip
 
-\indent {\it Non-sliding} models are systems of identical hard particles which have a finite number of maximal density perfect coverings of the infinite lattice, and are such that any defect in a covering leaves an amount of empty space that is proportional to its size, and that a particle configuration is characterized by its defects (this will be made precise in the following). This class includes many of the models for which crystallization has been proved, namely the hard diamond (see figure~\-\ref{fig:shapes}{\it a}) model discussed above, as well as the hard cross model (see figure~\-\ref{fig:shapes}{\it b}), which corresponds to the third-nearest-neighbor exclusion on $\mathbb Z^2$, and the hard hexagon model on the triangular lattice (see figure~\-\ref{fig:shapes}{\it c}), which corresponds to the nearest-neighbor exclusion on the triangular lattice.
+\indent {\it Non-sliding} models are systems of identical hard particles which have a finite number $\tau$ of maximal density perfect coverings of the infinite lattice, and are such that any defect in a covering (a defect appears where a particle configuration differs from a perfect covering) leaves an amount of empty space that is proportional to its size, and that a particle configuration is characterized by its defects (this will be made precise in the following). This class includes all of the models for which crystallization has been proved, like the hard diamond (see figure~\-\ref{fig:shapes}{\it a}) model discussed above, as well as the hard cross model (see figure~\-\ref{fig:shapes}{\it b}), which corresponds to the third-nearest-neighbor exclusion on $\mathbb Z^2$, and the hard hexagon model on the triangular lattice (see figure~\-\ref{fig:shapes}{\it c}), which corresponds to the nearest-neighbor exclusion on the triangular lattice.
 
-\indent The hard cross model was studied by Heilmann and Pr\ae stgaard~\-\cite{HP74}, who gave a sketch of a proof that it has a crystalline high-density phase. Eisenberg and Baram~\-\cite{EB05} computed the first 6 terms of the high-fugacity expansion for this model, and conjectured that it should have a {\it first-order} order-disorder phase transition. We will prove the convergence of the high-fugacity expansion, and reproduce Heilmann and Pr\ae stgaard's result, but will stop short of proving the order of the phase transition, for which new techniques would need to be developed. We will also extend this result to the hard cross model on a fine lattice, although the present techniques do not allow us to go to the continuum.
+\indent In~\-\cite{GF65}, the first 13 terms of the low-fugacity expansion and the first 9 terms of the high-fugacity expansion were computed, from which Gaunt and Fisher predicted a phase transition at the point where both expansions, suitably extrapolated, meet.
+
+\indent The hard cross model was studied by Heilmann and Pr\ae stgaard~\-\cite{HP74}, who gave a sketch of a proof that it has a crystalline high-density phase. Eisenberg and Baram~\-\cite{EB05} computed the first 13 terms of the low-fugacity expansion and the first 6 terms of the high-fugacity expansion for this model, and conjectured that it should have a {\it first-order} order-disorder phase transition. We will prove the convergence of the high-fugacity expansion, and reproduce Heilmann and Pr\ae stgaard's result, but will stop short of proving the order of the phase transition, for which new techniques would need to be developed. We will also extend this result to the hard cross model on a fine lattice, although the present techniques do not allow us to go to the continuum.
 
 \indent The hard hexagon model on the triangular lattice was shown to be exactly solvable by Baxter~\-\cite{Ba80,Ba82}, and to be crystalline at high densities. The exact solution provides an (implicit) expression for the pressure $p(z)$, from which the high-fugacity expansion can be obtained, as shown by Joyce~\-\cite{Jo88}.
 \bigskip
@@ -78,59 +80,29 @@ We establish existence of order-disorder phase transitions for a class of ``non-
   \label{fig:shapes}
 \end{figure}
 
-\subsection{Non-sliding hard-core lattice particle models}\label{sec:model}
-\indent Consider a $d$-dimensional lattice $\Lambda_\infty$, which we consider as a graph, that is, every vertex of $\Lambda_\infty$ has a set of {\it neighbors}. We denote the graph distance on $\Lambda_\infty$ by $\Delta$, in terms of which $x,x'\in\Lambda_\infty$ are neighbors if and only if $\Delta(x,x')=1$. We will consider systems of identical particles on $\Lambda_\infty$ with hard core interactions. We will represent the latter by assigning a {\it support} to each particle, which is a connected and bounded subset $\omega\subset\mathbb R^d$, and forbid the supports of different particles from overlapping. In the examples mentioned above, the shapes would be a diamond, a cross or a hexagon (see figure~\-\ref{fig:shapes}). We define the grand-canonical partition function of the system at activity $z>0$ on any bounded $\Lambda\subset\Lambda_\infty$ as
+\subsection{Hard-core lattice particle models}\label{sec:model}
+\indent Consider a $d$-dimensional lattice $\Lambda_\infty$. We consider $\Lambda_\infty$ as a graph, that is, every vertex of $\Lambda_\infty$ is assigned a set of {\it neighbors}. We denote the graph distance on $\Lambda_\infty$ by $\Delta$, in terms of which $x,x'\in\Lambda_\infty$ are neighbors if and only if $\Delta(x,x')=1$. We will consider systems of identical particles on $\Lambda_\infty$ with hard core interactions. We will represent the latter by assigning a {\it support} to each particle, which is a connected and bounded subset $\omega\subset\mathbb R^d$, and forbid the supports of different particles from overlapping. In the examples mentioned above, the shapes would be a diamond, a cross or a hexagon (see figure~\-\ref{fig:shapes}). We define the grand-canonical partition function of the system at activity $z>0$ on any bounded $\Lambda\subset\Lambda_\infty$ as
 \begin{equation}
   \Xi_\Lambda(z)=\sum_{X\subset\Lambda}z^{|X|}\prod_{x\neq x'\in X}\varphi(x,x')
   \label{Xi}
 \end{equation}
-in which $X$ is a particle configuration in $\Lambda$ (that is, a set of lattice points $x\in\Lambda$ on which particles are located), $|X|$ is the cardinality of $X$, and, denoting $\omega_x:= \{x+y,\ y\in\omega\}$, $\varphi(x,x')\in\{0,1\}$ enforces the hard core repulsion: it is equal to 1 if and only if $\omega_{x}\cap\omega_{x'}=\emptyset$. In the following, a subset $X\subset\Lambda_\infty$ is said to be a {\it particle configuration} if $\varphi(x,x')=1$ for every $x\neq x'\in X$, and we denote the set of particle configurations in $\Lambda$ by $\Omega(\Lambda)$. Note that the sum over $X$ is a finite sum, since the hard-core repulsion imposes a bound on $|X|$:
-\begin{equation}
-  |X|\leqslant N_{\mathrm{max}}.
-\end{equation}
-In addition, note that several different shapes can, in some cases, give rise to the same partition function. For example, the hard diamond model is equivalent to a system of hard disks of radius $r\in(\frac12,\frac1{\sqrt 2})$.
-\bigskip
-
-\indent Our main result concerns hard-core lattice particles that satisfy the {\it non-sliding} property, which, roughly, means that the system admits only a finite number of perfect coverings, that any defect in a covering induces an amount of empty space that is proportional to its volume, and that any particle configuration is entirely determined by its defects. More precisely, defining $\sigma_x$ as the set of lattice sites that are covered by a particle at $x$:
-\begin{equation}
-  \sigma_x:=\omega_x\cap\Lambda_\infty
-  \label{sigma}
-\end{equation}
-given a particle configuration $X\in\Omega(\Lambda)$, we define the set of {\it empty} sites as those that are not covered by any particle:
+in which $X$ is a particle configuration in $\Lambda$ (that is, a set of lattice points $x\in\Lambda$ on which particles are placed), $|X|$ is the cardinality of $X$, and, denoting $\omega_x:= \{x+y,\ y\in\omega\}$ ($\omega_x$ is the support of the particle {\it located} at $x$), $\varphi(x,x')\in\{0,1\}$ enforces the hard core repulsion: it is equal to 1 if and only if $\omega_{x}\cap\omega_{x'}=\emptyset$. In the following, a subset $X\subset\Lambda_\infty$ is said to be a {\it particle configuration} if $\varphi(x,x')=1$ for every $x\neq x'\in X$, and we denote the set of particle configurations in $\Lambda$ by $\Omega(\Lambda)$. Note that the sum over $X$ is a finite sum, since the hard-core repulsion imposes a bound on $|X|$:
 \begin{equation}
-  \mathcal E_\Lambda(X):=\{y\in\Lambda,\quad \forall x\in X,\ y\not\in\sigma_x\}
-  \label{mcE}
+  |X|\leqslant N_{\mathrm{max}}\leqslant |\Lambda|
 \end{equation}
-A {\it perfect covering} is defined as a particle configuration $X\in\Omega(\Lambda_\infty)$ that leaves no empty sites: $\mathcal E_{\Lambda_\infty}(X)=\emptyset$.
-\bigskip
-
-\theoname{Definition}{sliding}\label{def:sliding}
-  A hard-core lattice particle system, as defined above, is said to be {\it non-sliding} if the following hold.
-  \begin{itemize}
-    \item There exists $\tau>1$, a {\it periodic} perfect covering $\mathcal L_1$, and a finite family $(f_2,\cdots,f_\tau)$ of isometries of $\Lambda_\infty$ such that, for every $i$, $\mathcal L_i\equiv f_i(\mathcal L_1)$ is a perfect covering (see figure~\-\ref{fig:cross_packing} for an example).
-    \item Given a bounded {\it connected} particle configuration $X\in\Omega(\Lambda_\infty)$ (that is, a configuration in which the union of particle supports $\bigcup_{x\in X}\sigma_x$ is connected), we define $\mathbb S(X)$ as the set of particle configurations $X'$ that contain $X$ and {\it isolate} it from the rest of $\Lambda_\infty$ without leaving any empty space (see figures~\-\ref{fig:cross_unique1} and~\-\ref{fig:cross_unique2}):
-    \begin{equation}
-      \mathbb S(X):=\{X'\in\Omega(\Lambda_\infty),\ X'\supset X,\ \forall x\in X,\ \forall x'\in X',\ \Delta(\mathcal E_{\Lambda_\infty}(X'),\sigma_x)>1,\ \Delta(\sigma_x,\sigma_{x'})\leqslant 1\}
-    \end{equation}
-    in which, we recall, $\Delta$ denotes the graph distance on $\Lambda_\infty$. In order to be non-sliding, a model must be such that, for every bounded connected $X$, $\mathbb S(X)=\emptyset$, or, $\forall X'\in\mathbb S(X)$, there exists a unique $\mu\in\{1,\cdots,\tau\}$ such that $X'\subset\mathcal L_\mu$.
-  \end{itemize}
-\endtheo
-\bigskip
-
-{\bf Remark}: In non-sliding models, every defect induces an amount of empty space proportional to its size, because any connected particle configuration $X$ that is not a subset of any perfect covering must have $\mathbb S(X)=\emptyset$, which means that there must be some empty space next to it. In addition, a particle configuration is determined by the empty space and the particles surrounding it, since the remainder of the particle configuration consists of disconnected groups, each of which is the subsets of a perfect covering. The position of the particles surrounding it determines uniquely which one of the perfect coverings it is a subset of.
+where $|\Lambda|$ is the number of sites in $\Lambda$. In addition, note that several different shapes can, in some cases, give rise to the same partition function. For example, the hard diamond model is equivalent to a system of hard disks of radius $r$ with $\frac12<r<\frac1{\sqrt 2}$.
 \bigskip
 
-\indent Our main result is that the finite-volume {\it pressure} of non-sliding hard-core particles systems, defined as
+\indent We will discuss the properties of the finite-volume {\it pressure} of hard-core particles systems, defined as
 \begin{equation}
 p_\Lambda(z):=\frac1{|\Lambda|}\log \Xi_\Lambda(z)
 \label{p}
 \end{equation}
-satisfies
+and its infinite-volume limit
 \begin{equation}
-  p(z):=\lim_{\Lambda\to\Lambda_\infty}p_\Lambda(z)=\rho_m\log z+f(y)
+  p(z):=\lim_{\Lambda\to\Lambda_\infty}p_\Lambda(z)=:\rho_m\log z+f(y)
 \end{equation}
-in which $\rho_m$ is the maximal density $\rho_m=\lim_{\Lambda\to\Lambda_\infty}\frac{N_{\mathrm{max}}}{|\Lambda|}$ and $f$ is an analytic function of $y\equiv\frac1z$ for small values of $y$. The expansion of $f$ in powers of $y$ is called the {\it high-fugacity expansion} of the system. Note that the infinite-volume pressure $p(z)$ does not depend on the boundary conditions~\-\cite{Ru99}.
-\bigskip
+in which $y\equiv z^{-1}$ and $\rho_m$ is the maximal density $\rho_m=\lim_{\Lambda\to\Lambda_\infty}\frac{N_{\mathrm{max}}}{|\Lambda|}$. In particular, we will focus on the analyticity properties of $f(y)$. When $f(y)$ is analytic for small values of $y$, the system is said to admit a convergent {\it high-fugacity} expansion.
 
 \subsection{Low-fugacity expansion}\label{sec:low_fugacity}
 \indent The main ideas underlying the high-fugacity expansion come from the low-fugacity expansion, which we will now briefly review. It is an expansion of $p_\Lambda$ in powers of the fugacity $z$, and its formal derivation is fairly straightforward: defining the {\it canonical} partition function as
@@ -144,7 +116,7 @@ as the number of particle configurations with $k$ particles, (\ref{Xi}) can be r
 \end{equation}
 Injecting~\-(\ref{Xi_z}) into~\-(\ref{p}), we find that, formally,
 \begin{equation}
-  p_\Lambda(z)=\sum_{k=1}^\infty z^kb_k(\Lambda)
+  p_\Lambda(z)=\sum_{k=1}^\infty b_k(\Lambda)z^k
   \label{p_z}
 \end{equation}
 with
@@ -152,10 +124,10 @@ with
   b_k(\Lambda):=\frac1{|\Lambda|}\sum_{n=1}^k\frac{(-1)^{n+1}}n\sum_{\displaystyle\mathop{\scriptstyle k_1,\cdots,k_n\geqslant 1}_{k_1+\cdots+k_n=k}}Z_\Lambda(k_1)\cdots Z_\Lambda(k_n).
   \label{blog}
 \end{equation}
-As was shown in~\-\cite{Gr62,Ru63,Pe63}, there is a remarkable cancellation that eliminates the terms in $b_k(\Lambda)$ that diverge as $\Lambda\to\Lambda_\infty$, so that $b_k(\Lambda)\to b_k$ when $\Lambda\to\Lambda_\infty$. This becomes obvious when the $b_k(\Lambda)$ are expressed as integrals over Mayer graphs. In addition, the radius of convergence $R(\Lambda)$ of~\-(\ref{p_z}) converges to $R>0$, which is at least as large as the radius of convergence of $\sum_{k=1}^\infty b_kz^k$.
+As was shown in~\-\cite{Ur27,Ma37,Gr62,Ru63,Pe63}, there is a remarkable cancellation that eliminates the terms in $b_k(\Lambda)$ that diverge as $\Lambda\to\Lambda_\infty$, so that $b_k(\Lambda)\to b_k$ when $\Lambda\to\Lambda_\infty$. This becomes obvious when the $b_k(\Lambda)$ are expressed as integrals over Mayer graphs. In addition, the radius of convergence $R(\Lambda)$ of~\-(\ref{p_z}) converges to $R>0$, which is at least as large as the radius of convergence of $\sum_{k=1}^\infty b_kz^k$ (for positive pair potentials, $R$ is {\it equal} to the radius of convergence~\-\cite{Pe63}).
 
 \subsection{High-fugacity expansion}
-\indent The low-fugacity expansion is obtained by perturbing around the vacuum state by adding particles to it. The high-fugacity expansion will be obtained by perturbing perfect coverings by introducing {\it defects}. Single-particle defects come with a cost $y\equiv z^{-1}$, which is, effectively, the fugacity of a hole. The main idea, due to Gaunt and Fisher~\-\cite{GF65}, is to carry out a low-activity expansion for the defects, which is similar to the low-fugacity expansion described above. Let us go into some more detail in the example of the hard diamond model.
+\indent The low-fugacity expansion is obtained by perturbing around the vacuum state by adding particles to it. The high-fugacity expansion will be obtained by perturbing perfect coverings by introducing {\it defects}. Single-particle defects, corresponding to removing one particle from a perfect covering, come with a cost $y\equiv z^{-1}$, which is, effectively, the fugacity of a hole. The main idea, due to Gaunt and Fisher~\-\cite{GF65}, is to carry out a cluster expansion for the defects, which is similar to the low-fugacity expansion described above. Let us go into some more detail in the example of the hard diamond model.
 \bigskip
 
 \indent We will take $\Lambda$ to be a $2n\times 2n$ torus, which can be completely packed with diamonds (see figure~\-\ref{fig:diamond_packing}). The number of perfect covering configurations is
@@ -178,7 +150,7 @@ in terms of which
 \end{equation}
 (we factor $\tau$ out because $Q_\Lambda(0)=\tau$ and we wish to expand the logarithm in~\-(\ref{p}) around 1). We thus have, formally
 \begin{equation}
-  p_\Lambda(y)=\frac1{|\Lambda|}\log\tau+\rho_m\log z+\sum_{k=1}^{N_{\mathrm{max}}} y^kc_k(\Lambda)
+  p_\Lambda(y)=\frac1{|\Lambda|}\log\tau+\rho_m\log z+\sum_{k=1}^{N_{\mathrm{max}}} c_k(\Lambda)y^k
   \label{p_y}
 \end{equation}
 where $y\equiv z^{-1}$ and
@@ -186,28 +158,29 @@ where $y\equiv z^{-1}$ and
   c_k(\Lambda):=\frac1{|\Lambda|}\sum_{n=1}^k\frac{(-1)^{n+1}}{n\tau^n}\sum_{\displaystyle\mathop{\scriptstyle k_1,\cdots,k_n\geqslant 1}_{k_1+\cdots+k_n=k}}Q_\Lambda(k_1)\cdots Q_\Lambda(k_n).
   \label{blog}
 \end{equation}
-The first 9 $c_k(\Lambda)$ are reported in~\-\cite[table~\-XIII]{GF65} and, as for the low-fugacity expansion, there is a remarkable cancellation that ensures that these coefficients converge to $c_k$ as $\Lambda\to\Lambda_\infty$. {\it But}, unlike the low-fugacity expansion, there is no {\it systematic} way of exhibiting this cancellation for general hard-core lattice particle systems. In fact there are many example of systems in which $c_2(\Lambda)$ diverges as $\Lambda\to\Lambda_\infty$, like the nearest-neighbor exclusion model in 1 dimension (which maps, exactly, to the 1-dimensional monomer-dimer model), for which
+The first 9 $c_k(\Lambda)$ are reported in~\-\cite[table~\-XIII]{GF65} and, as for the low-fugacity expansion, there is a remarkable cancellation that ensures that these coefficients converge to a finite value $c_k$ as $\Lambda\to\Lambda_\infty$. {\it But}, unlike the low-fugacity expansion, there is no {\it systematic} way of exhibiting this cancellation for general hard-core lattice particle systems. In fact there are many example of systems in which $c_2(\Lambda)$ diverges as $\Lambda\to\Lambda_\infty$, like the nearest-neighbor exclusion model in 1 dimension (which maps, exactly, to the 1-dimensional monomer-dimer model), for which
 \begin{equation}
-  Q_\Lambda(2)=\frac1{192}(|\Lambda|^2-4)|\Lambda|^2
-  ,\quad
   Q_\Lambda(1)=\frac14|\Lambda|^2
   ,\quad
+  Q_\Lambda(2)=\frac1{192}(|\Lambda|^2-4)|\Lambda|^2
+  ,\quad
   c_1(\Lambda)=\frac1{8}|\Lambda|
   ,\quad
   c_2(\Lambda)=-\frac1{192}|\Lambda|(5|\Lambda|^2+4).
+  \label{monomer_dimer_div}
 \end{equation}
-Note that the pressure for this model is
+Note that the pressure for this system, given by
 \begin{equation}
   p(y)-\rho_m\log z=\log\left(\frac{1+\sqrt{1+4z}}2\right)-\frac12\log z
   =\log\left(\sqrt{1+\frac14y}+\frac12\sqrt y\right)
 \end{equation}
-which is not an analytic function of $y\equiv z^{-1}$ at $y=0$ (though it is an analytic function of $\sqrt y$). Clearly, this model does not satisfy the non-sliding property.
+is not an analytic function of $y\equiv z^{-1}$ at $y=0$ (though it is an analytic function of $\sqrt y$). Clearly, this model does not satisfy the non-sliding property.
 \bigskip
 
-\indent Our approach, in this paper, is to prove that, for non-sliding models, the pressure is analytic in a disk around $y=0$, thus proving the validity of the Gaunt-Fisher expansion for non-sliding systems.
+\indent One of our goals, in this paper, is to prove that, for non-sliding models, the pressure is analytic in a disk around $y=0$, thus proving the validity of the Gaunt-Fisher expansion for non-sliding systems.
 \bigskip
 
-{\bf Remark}: Before moving to our main result, let us note that, at finite temperature, lattice gases of particles with a {\it bounded} pair potential $\varphi$ that admit a convergent low-fugacity expansion (for example for summable potentials) also admit a high-fugacity expansion. This follows immediately from the Kramers-Wannier~\-\cite{KW41} duality of the corresponding Ising model, which implies that
+{\bf Remark}: Let us note that, at finite temperature, lattice gases of particles with a {\it bounded} pair potential $\varphi$ that admit a convergent low-fugacity expansion (for example for summable potentials) also admit a high-fugacity expansion. This follows immediately from the spin-flip symmetry of the corresponding Ising model, which implies that
 \begin{equation}
   p_\Lambda(z)=
   \log(ze^{-\frac12\alpha})
@@ -216,12 +189,12 @@ which is not an analytic function of $y\equiv z^{-1}$ at $y=0$ (though it is an
   e^\alpha:=e^{\beta\sum_{x\in\Lambda}\varphi(|x|)}
   \label{particle_hole}
 \end{equation}
-The radius of convergence $R_y$ of the expansion in $y$ is therefore related to the radius $R_z$ of convergence of the expansion in $z$: $R_y=R_ze^{-\alpha}$. This coincides, at sufficiently high temperature, with the results of Gallavotti, Miracle-Sole and Robinson~\-\cite{GMR67}, who prove analyticity for small values of $\frac z{1+z}$.
+The radius of convergence $\tilde R(\Lambda)$ of the expansion in $y$ is therefore related to the radius $R(\Lambda)$ of convergence of the expansion in $z$: $\tilde R(\Lambda)=R(\Lambda)e^{-\alpha}$. This coincides, at sufficiently high temperature, with the results of Gallavotti, Miracle-Sole and Robinson~\-\cite{GMR67}, who prove analyticity for small values of $\frac z{1+z}$. (A similar result holds for bounded many-particle interactions.)
 
 \subsection{High-fugacity expansion and Lee-Yang zeros}
-\indent As was first remarked by Lee and Yang~\-\cite{YL52,LY52}, a powerful tool to study the thermodynamic properties of a system is to compute the positions of the roots of the partition function as a function of the fugacity $z$, called the {\it Lee-Yang zeros} of the model. In particular, the logarithm of the partition function and, consequently, the pressure, diverge at the Lee-Yang zeros, so when such roots lie on the positive real axis, they signal the presence of a phase transitions. Let us denote the set of Lee-Yang zeros of a hard-core lattice particle system by $\{\xi_1(\Lambda),\cdots,\xi_{N_{\mathrm{max}}}(\Lambda)\}$. The convergence of the low-fugacity expansion within its radius of convergence $R_z>0$ implies that every Lee-Yang zero satisfies $|\xi_i(\Lambda)|\geqslant R_z$, and that this inequality is sharp. Similarly, when the high-fugacity expansion has a positive radius of convergence $R_y>0$, every Lee-Yang zero must satisfy
+\indent As was pointed out by Lee and Yang~\-\cite{YL52,LY52}, a powerful tool to study the equilibrium properties of a system is via the positions of the roots of the partition function as a function of the fugacity $z$, called the {\it Lee-Yang zeros} of the system. In particular, the logarithm of the partition function and, consequently, the pressure, diverge at the Lee-Yang zeros, so whenever the limiting density of the roots approaches the positive real axis, this signals the presence of a phase transition. Let us denote the set of Lee-Yang zeros of a hard-core lattice particle system by $\{\xi_1(\Lambda),\cdots,\xi_{N_{\mathrm{max}}}(\Lambda)\}$. The convergence of the low-fugacity expansion within its radius of convergence $R(\Lambda)>0$ implies that every Lee-Yang zero satisfies $|\xi_i(\Lambda)|\geqslant R(\Lambda)$, and that this inequality is sharp. Similarly, when the high-fugacity expansion has a positive radius of convergence $\tilde R(\Lambda)>0$, every Lee-Yang zero must satisfy
 \begin{equation}
-  R_z\leqslant |\xi_i(\Lambda)|\leqslant R_y^{-1}
+  R(\Lambda)\leqslant |\xi_i(\Lambda)|\leqslant \tilde R(\Lambda)^{-1}
 \end{equation}
 and these inequalities are sharp. In addition, writing the partition function as
 \begin{equation}
@@ -244,37 +217,83 @@ which, in particular, implies that
 When taking the thermodynamic limit, $c_k$ is proportional to the average of the $k$-th power of $\xi$ weighted against the limiting distribution of Lee-Yang zeros. Thus, the high-fugacity expansion converges if and only if the average of $\xi^k$ divided by $k$ grows at most exponentially in $k$.
 \bigskip
 
-{\bf Remark}: For bounded potentials, using the Kramers-Wannier duality in~\-(\ref{particle_hole}), we find that the Lee-Yang zeros all lie in an annulus of radii $R_z$ and $e^\alpha/R_z$. Note that if one were to consider a hard-core model as the limit of a bounded repulsive potential, the hard-core limit would correspond to taking $\alpha\to\infty$. This implies that some zeros move out to infinity. This does not, however, imply that in the hard-core limit $\Xi_\Lambda(y)$ vanishes for $y=0$: indeed the distribution of Lee-Yang zeros does not approach the hard-core limit continuously, as is made obvious by the fact that the number of Lee-Yang zeros for finite potentials is $|\Lambda|$, whereas it is $N_{\mathrm{max}}$ in the hard-core limit.
+{\bf Remark}: As noted earlier, for bounded potentials, we find that the Lee-Yang zeros all lie in an annulus of radii $R(\Lambda)$ and $e^\alpha/R(\Lambda)$. Note that if one were to consider a hard-core model as the limit of a bounded repulsive potential, the hard-core limit would correspond to taking $\alpha\to\infty$. This implies that some zeros move out to infinity and that the radius of convergence of the high-fugacity expansion tends to 0 as $\alpha\to\infty$. This does not, however, imply that in the hard-core limit $\Xi_\Lambda(y)$ vanishes for $y=0$: indeed the distribution of Lee-Yang zeros does not approach the hard-core limit continuously, as is made obvious by the fact that the number of Lee-Yang zeros for finite potentials is $|\Lambda|$, whereas it is $N_{\mathrm{max}}$ in the hard-core limit. Instead, when a hard-core particle system has a convergent high-fugacity expansion, there is a bound on the remaining zeros which remains finite as $\Lambda\to\Lambda_\infty$.
+
+\subsection{Definitions and results}
+\indent We focus on the class of hard-core lattice particle models that satisfy the {\it non-sliding} property, which, roughly, means that the system admits only a finite number of perfect coverings, that any defect in a covering induces an amount of empty space that is proportional to its volume, and that any particle configuration is entirely determined by its defects. More precisely, defining $\sigma_x$ as the set of lattice sites that are covered by a particle located at $x$:
+\begin{equation}
+  \sigma_x:=\omega_x\cap\Lambda_\infty
+  \label{sigma}
+\end{equation}
+given a particle configuration $X\in\Omega(\Lambda)$, we define the set of {\it empty} sites as those that are not covered by any particle:
+\begin{equation}
+  \mathcal E_\Lambda(X):=\{y\in\Lambda,\quad \forall x\in X,\ y\not\in\sigma_x\}
+  \label{mcE}
+\end{equation}
+A {\it perfect covering} is defined as a particle configuration $X\in\Omega(\Lambda_\infty)$ that leaves no empty sites: $\mathcal E_{\Lambda_\infty}(X)=\emptyset$.
+\bigskip
+
+\theoname{Definition}{sliding}\label{def:sliding}
+  A hard-core lattice particle system is said to be {\it non-sliding} if the following hold.
+  \begin{itemize}
+    \item There exists $\tau>1$, a {\it periodic} perfect covering $\mathcal L_1$, and a finite family $(f_2,\cdots,f_\tau)$ of isometries of $\Lambda_\infty$ such that, for every $i$, $\mathcal L_i\equiv f_i(\mathcal L_1)$ is a perfect covering (see figure~\-\ref{fig:cross_packing} for an example).
+    \item Given a bounded {\it connected} particle configuration $X\in\Omega(\Lambda_\infty)$ (that is, a configuration in which the union of particle supports $\bigcup_{x\in X}\sigma_x$ is connected), we define $\mathbb S(X)$ as the set of particle configurations $X'$ that
+    \begin{itemize}
+      \item contain $X$,
+      \item are such that every $x'\in X'\setminus X$ is adjacent to $X$,
+      \item leave no empty sites adjacent to $X$.
+    \end{itemize}
+    (see figures~\-\ref{fig:cross_unique1} and~\-\ref{fig:cross_unique2}):
+    \begin{equation}
+      \mathbb S(X):=
+      \{X'\in\Omega(\Lambda_\infty),\ X'\supset X,\ \Delta(\mathcal E_{\Lambda_\infty}(X'),{\textstyle\bigcup_{x\in X}\sigma_x})> 1,\ \forall x'\in X', \Delta(\sigma_{x'},{\textstyle\bigcup_{x\in X}\sigma_x})\leqslant 1\}
+      \label{bbS}
+    \end{equation}
+    in which, we recall, $\Delta$ denotes the graph distance on $\Lambda_\infty$. In order to be non-sliding, a model must be such that, for every bounded connected $X$, $\mathbb S(X)=\emptyset$, or, $\forall X'\in\mathbb S(X)$, there exists a unique $\mu\in\{1,\cdots,\tau\}$ such that $X'\subset\mathcal L_\mu$.
+  \nopagebreakafteritemize
+  \end{itemize}
+\restorepagebreakafteritemize
+\endtheo
+\bigskip
+
+{\bf Remark}: In non-sliding models, every defect induces an amount of empty space proportional to its size because any connected particle configuration $X$ that is not a subset of any perfect covering must have $\mathbb S(X)=\emptyset$, which means that there must be some empty space next to it. In addition, a particle configuration is determined by the empty space and the particles surrounding it, since the remainder of the particle configuration consists of disconnected groups, each of which is the subsets of a perfect covering. The position of the particles surrounding it uniquely determines which one of the perfect coverings it is a subset of.
+\bigskip
 
-\subsection{Main result}
-\indent Our main result concerns the partition function and correlation functions of non-sliding hard-core lattice particle systems, with the following boundary conditions. First of all, we require the set $\Lambda$ to be {\it tiled}, by which we mean that there must exist $\mu\in\{1,\cdots,\tau\}$ and a set $S\subset\Lambda_\infty$ such that
+\indent In addition, we make the following assumption about the geometry of $\Lambda$: $\Lambda$ is {\it bounded}, {\it simply connected} (that is, it is connected and $\Lambda_\infty\setminus\Lambda$ is connected), and {\it tiled}, by which we mean that there must exist $\mu\in\{1,\cdots,\tau\}$ and a set $S\subset\mathcal L_\mu$ such that
 \begin{equation}
   \Lambda=\bigcup_{x\in S}\sigma_x
-  ,\quad
-  \sigma_x\cap\sigma_{x'}=\emptyset,\ \forall x\neq x'\in S
   .
   \label{tiled}
 \end{equation}
-For $\nu\in\{1,\cdots,\tau\}$, we define $\Omega_\nu$ as the set of particle configurations such that
+Note that the choice of $\mu$ will not play any role in the thermodynamic limit.
+\bigskip
+
+\indent Given such a $\Lambda$, we will consider the following boundary conditions. Given $\nu\in\{1,\cdots,\tau\}$ (which is not necessarily equal to the $\mu$ with which we tiled $\Lambda$), we define $\Omega_\nu(\Lambda)$ as the set of particle configurations such that
 \begin{itemize}
-  \item if the complement of $\Lambda$ were covered by a $\nu$-covering, then the particles in $\Lambda$ would not overlap with those outside $\Lambda$,
-  \item the space left empty by the configuration must not neighbor the boundary of $\Lambda$:
+  \item every site $x\in\mathcal L_\nu$ that neighbors the boundary, that is, $\Delta(\sigma_x,\Lambda_\infty\setminus\Lambda)\leqslant 1$, is occupied by a particle,
+  \item the particles that neighbor the boundary must not neighbor an empty site.
 \end{itemize}
+Thus, defining $\mathbb B_\nu(\Lambda):=\{x\in\mathcal L_\nu\cap\Lambda,\ \Delta(\sigma_x,\Lambda_\infty\setminus\Lambda)\leqslant 1\}$ as the set of sites in $\mathcal L_\nu$ that neighbor the boundary, and $\mathbb X_\nu(\Lambda):=\mathcal L_\nu\setminus\Lambda$, we define
 \begin{equation}
-  \Omega_\nu(\Lambda):=\{X\subset\Lambda,\quad
-  \forall x\in\mathcal L_\nu\setminus\Lambda,\ \forall x'\in X,\quad
-  \varphi(x,x')=1,\quad
-  \Delta(\mathcal E_\Lambda(X),\sigma_x)>1\}
+  \Omega_\nu(\Lambda):=
+  \{
+    X\subset\Lambda,\quad
+    X\supset\mathbb B_\nu(\Lambda)
+    ,\quad
+    \forall x\in\mathbb B_\nu(\Lambda),\ \Delta(\sigma_x,\mathcal E_{\Lambda_\infty}(X\cup\mathbb X_\nu(\Lambda)))>1
+  \}
+  .
 \end{equation}
-in which, we recall, $\Delta$ denotes the graph distance on $\Lambda_\infty$. We chose this particular boundary condition in order to make the discussion below simpler. More natural boundary conditions would presumably be treatable as well, though they might complicate the proof somewhat. In addition, we generalize the notion of fugacity by allowing it to depend on the position of the particle: given a function $\underline z:\Lambda_\infty\to[0,\infty)$, we define the partition function with fugacity $\underline z$ and boundary condition $\nu$ as
+We choose these particular boundary conditions in order to make the discussion below simpler. More general boundary conditions would presumably lead to infinite volume measures which are superpositions of those induced by the boundary conditions considered here. For example, for the hard diamond model with periodic or open boundary conditions, we would get a limiting state which is a $\frac12$-$\frac12$ superposition of the even and odd states.
+\bigskip
+
+\indent Allowing the fugacity to depend on the position of the particle, we define the partition function with fugacity $\underline z:\Lambda_\infty\to[0,\infty)$ and boundary condition $\nu$ as
 \begin{equation}
   \Xi_\Lambda^{(\nu)}(\underline z)=\sum_{X\in\Omega_\nu(\Lambda)}\left(\prod_{x\in X}\underline z(x)\right)\prod_{x\neq x'\in X}\varphi(x,x')
   .
   \label{Xi_nu}
 \end{equation}
-\bigskip
-
-\indent Since the infinite-volume pressure is independent of the boundary condition, it can be recovered from $\Xi_\Lambda^{(\nu)}(\underline z)$ by setting $\underline z(x)\equiv z$. By allowing the fugacity to depend on the position of the particle, we can compute the {\it $\mathfrak n$-point truncated correlation functions} of the system with $\nu$-boundary conditions at fugacity $z$, defined as
+Since the infinite-volume pressure is independent of the boundary condition, it can be recovered from $\Xi_\Lambda^{(\nu)}(\underline z)$ by setting $\underline z(x)\equiv z$. By allowing the fugacity to depend on the position of the particle, we can compute the {\it $\mathfrak n$-point truncated correlation functions} of the system with $\nu$-boundary conditions at fugacity $z$, defined as
 \begin{equation}
   \rho_{n,\Lambda}^{(\nu)}(\mathfrak x_1,\cdots,\mathfrak x_n):=
   \left.\frac{\partial^{\mathfrak n}}{\partial\log\underline z(\mathfrak x_1)\cdots\partial\log\underline z(\mathfrak x_{\mathfrak n})}
@@ -288,30 +307,33 @@ as well as its infinite-volume limit
   \rho_{n,\Lambda}^{(\nu)}(\mathfrak x_1,\cdots,\mathfrak x_n)
   .
 \end{equation}
-Note that the 1-point correlation function is the local density.
+Note that the 1-point correlation function is the local density. In addition, we define the {\it average density} as
+  \begin{equation}
+    \rho:=\lim_{\Lambda\to\Lambda_\infty}\frac1{|\Lambda|}\sum_{x\in\Lambda}\rho_{1,\Lambda}^{(\nu)}(x)
+    .
+  \end{equation}
 \bigskip
 
 \indent Our main result is summarized in the following theorem.
 \bigskip
 
 \theoname{Theorem}{crystallization and high-fugacity expansion}\label{theo:main}
-  Consider a non-sliding hard-core lattice particle system and a boundary condition $\nu\in\{1,\cdots,\tau\}$. For any $\mathfrak n\geqslant 1$ and $\mathfrak x_1,\cdots,\mathfrak x_{\mathfrak n}\in\Lambda_\infty$, both $p(z)-\rho_m\log z$ and the $n$-point truncated correlation function $\rho_n^{(\nu)}(\mathfrak x_1,\cdots,\mathfrak x_n)$ are analytic functions of $y\equiv z^{-1}$, with a positive radius of convergence $y_0$.
+  Consider a non-sliding hard-core lattice particle system. There exists $y_0>0$ such that, if $|y|<y_0$, then there are $\tau$ distinct Gibbs states. The $\nu$-th Gibbs state, obtained from the boundary condition labeled by $\nu$, is invariant under the translations of the sub-lattice $\mathcal L_\nu$. In addition, for any boundary condition $\nu\in\{1,\cdots,\tau\}$, any $\mathfrak n\geqslant 1$ and $\mathfrak x_1,\cdots,\mathfrak x_{\mathfrak n}\in\Lambda_\infty$, both $p(z)-\rho_m\log z$ and the $n$-point truncated correlation function $\rho_n^{(\nu)}(\mathfrak x_1,\cdots,\mathfrak x_n)$ are analytic functions of $y$ for $|y|<y_0$.
   \bigskip
 
-  Furthermore, defining the average density as
-  \begin{equation}
-    \rho:=\lim_{\Lambda\to\Lambda_\infty}\frac1{|\Lambda|}\sum_{x\in\Lambda}\rho_{1,\Lambda}^{(\nu)}(x)
-  \end{equation}
-  both $p+\rho_m\log(\rho_m-\rho)$ and $\rho_n^{(\nu)}(\mathfrak x_1,\cdots,\mathfrak x_n)$ are analytic functions of $\rho_m-\rho$, with a positive radius of convergence.
-  \bigskip
-  
-  In addition, if $|y|$ is sufficiently small, then the particles are much more likely to be on the $\mathcal L_\nu$ sublattice than the others: for every $x\in\mathcal L_\nu$ and $x'\in\Lambda_\infty\setminus\mathcal L_\nu$,
+  These Gibbs states are {\it crystalline}: having picked the boundary condition $\nu$, the particles are much more likely to be on the $\mathcal L_\nu$ sub-lattice than the others: for every $x\Lambda_\infty$,
   \begin{equation}
-    \rho_1^{(\nu)}(x)=\rho_m+O(y)
-    ,\quad
-    \rho_1^{(\nu)}(x')=O(y).
+    \rho_1^{(\nu)}(x)=
+    \left\{\begin{array}{ll}
+      \rho_m+O(y)&\mathrm{if\ }x\in\mathcal L_\nu\\[0.3cm]
+      O(y)&\mathrm{if\ not}
+      .
+    \end{array}\right.
     \label{crystallization}
   \end{equation}
+  \bigskip
+
+  Finally, both $p+\rho_m\log(\rho_m-\rho)$ and $\rho_n^{(\nu)}(\mathfrak x_1,\cdots,\mathfrak x_n)$ are analytic functions of $\rho_m-\rho$, with a positive radius of convergence.
 \endtheo
 \bigskip
 
@@ -326,11 +348,12 @@ Note that the 1-point correlation function is the local density.
   \omega=\left\{(x_1,\cdots,x_d)\in(-1,1)^d,\ {\textstyle\sum_{i=1}^n}|x_i|<1\}\cup\{(0,\cdots,0,1)\right\}
   .
 \end{equation}
-Note the adjunction of the point $(0,\cdots,0,1)$, whose absence would prevent the existence of any perfect covering (see figure~\-\ref{fig:diamond_packing}). The notion of {\it connectedness} in $\Lambda_\infty$ is defined as follows: two points are connected if and only if they are at distance 1 from each other. There are 2 perfect coverings (see figure~\-\ref{fig:diamond_packing}):
+Note the adjunction of the point $(0,\cdots,0,1)$, whose absence would prevent the existence of any perfect covering (see figure~\-\ref{fig:diamond_packing}), and implies that each hyperdiamond covers two sites. The notion of {\it connectedness} in $\Lambda_\infty$ is defined as follows: two points are connected if and only if they are at distance 1 from each other. There are 2 perfect coverings (see figure~\-\ref{fig:diamond_packing}):
 \begin{equation}
   \mathcal L_1=\{(x_1,\cdots,x_d)\in\mathbb Z^d,\ x_1+\cdots+x_d\mathrm{\ even}\}
   ,\quad
   \mathcal L_2=\{(x_1,\cdots,x_d)\in\mathbb Z^d,\ x_1+\cdots+x_d\mathrm{\ odd}\}
+  \label{coverings_diamonds}
 \end{equation}
 which are related to each other by the translation by $(0,\cdots,0,1)$. Finally, this model satisfies the non-sliding condition because any pair $x_1,x_2\in\mathbb Z^d$ of hyperdiamonds whose supports are disjoint and connected are both in the same sublattice: $(x_1,x_2)\in\mathcal L_1^2\cup\mathcal L_2^2$, and the distinct sublattices do not overlap $\mathcal L_1\cap\mathcal L_2=\emptyset$. Connected hyperdiamond configurations are, therefore, always subsets of $\mathcal L_1$ or of $\mathcal L_2$, and one can find which one it is by from the position of a single one of its particles.
 \bigskip
@@ -344,12 +367,14 @@ which are related to each other by the translation by $(0,\cdots,0,1)$. Finally,
 \point Let us now consider the hard-cross model (see figure~\-\ref{fig:shapes}{\it b}), for which $\Lambda_\infty=\mathbb Z^2$, and
 \begin{equation}
   \omega=\textstyle\left\{(n_x+x,n_y+y),\ (x,y)\in(-\frac12,\frac12)^2,\ (n_x,n_y)\in\{-1,0,1\}^2,\ |n_x|+|n_y|\leqslant 1\right\}.
+  \label{shape_cross}
 \end{equation}
 There are $10$ perfect coverings (see figure~\-\ref{fig:cross_packing}):
 \begin{equation}
   \mathcal L_1=\{(n_x+2n_y,2n_x-n_y),\ (n_x,n_y)\in\mathbb Z^2\}
   ,\quad
   \mathcal L_2=\{(-n_x+2n_y,2n_x+n_y),\ (n_x,n_y)\in\mathbb Z^2\}
+  \label{coverings_cross}
 \end{equation}
 and, for $p\in\{2,3,4,5\}$,
 \begin{equation}
@@ -367,7 +392,7 @@ Now, consider a connected configuration of crosses $X$.
 \begin{itemize}
   \item If $|X|=1$, then $\mathbb S(X)$ (see definition~\-\ref{def:sliding}) consists of the two configurations in figure~\-\ref{fig:cross_unique1}, each of which is the subset of a unique sublattice $\mathcal L_\mu$.
   \item If $X$ contains at least one pair $x,x'\in X$ of stacked crosses, which, without loss of generality, we assume satisfies $x-x'=(-3,0)$, then one of the two sites $x+(1,1)$ or $x+(2,1)$ cannot be covered by any other cross (see figure~\-\ref{fig:cross_sliding}{\it a}), which implies that $\mathbb S(X)=\emptyset$.
-  \item If every pair of crosses in $X$ is either left- or right-packed,and there exists at least one triplet $x,x',x''\in X$ whose supports are connected and disjoint, and is such that $x,x'$ is right-packed and $x,x''$ is left-packed. Without loss of generality, we assume that $x-x'=(2,1)$ and $x-x''=(-1,-2)$ (see figure~\-\ref{fig:cross_sliding}{\it b}) or $x-x''=(-2,1)$ (see figure~\-\ref{fig:cross_sliding}{\it c}). In the former case, the site $x+(-1,1)$ cannot be covered by any other crosses. In the latter case, one of the three sites $x+(-1,-2)$, $x+(0,-2)$ or $x+(1,-2)$ cannot be covered by any other cross. Thus, $\mathbb S(X)=\emptyset$.
+  \item We now assume that every pair of crosses in $X$ is either left- or right-packed, and there exists at least one triplet $x,x',x''\in X$ whose supports are connected and disjoint, and is such that $x,x'$ is right-packed and $x,x''$ is left-packed. Without loss of generality, we assume that $x-x'=(2,1)$ and $x-x''=(-1,-2)$ (see figure~\-\ref{fig:cross_sliding}{\it b}) or $x-x''=(-2,1)$ (see figure~\-\ref{fig:cross_sliding}{\it c}). In the former case, the site $x+(-1,1)$ cannot be covered by any other crosses. In the latter case, one of the three sites $x+(-1,-2)$, $x+(0,-2)$ or $x+(1,-2)$ cannot be covered by any other cross. Thus, $\mathbb S(X)=\emptyset$.
   \item Finally, suppose that every pair of crosses is left-packed (the case in which they are all right-packed is treated identically). Let $Y$ be a pair of left-packed crosses, $\mathbb S(Y)$ consists of a single configuration, depicted in figure~\-\ref{fig:cross_unique2}, which is a subset of a unique sublattice $\mathcal L_\mu$. Since there is a unique way of isolating each left-packed pair in $X$, there is a single way of isolating $X$, that is, $\mathbb S(X)$ consists of a single configuration, which is the union over left-packed pairs $Y$ in $X$ of the unique configuration in $\mathbb S(Y)$, and is, therefore, a subset of a unique sublattice $\mathcal L_\mu$.
 \end{itemize}
 \bigskip
@@ -427,31 +452,9 @@ Now, consider a connected configuration of crosses $X$.
 \bigskip
 
 \subsection{The GFc model}
-\indent In this subsection, we map the particle system to a GFc model. The precise definition of the set of GFcs is given in definition~\-\ref{def:GFc}. It is a bit technical, and the readers that are interested in the proof of lemma~\-\ref{lemma:GFc} are invited to skip definition~\-\ref{def:GFc} and come back to it as it appears in the lemma. An example is provided in figure~\-\ref{fig:contour_example}.
-\bigskip
-
-\theoname{Definition}{Gaunt-Fisher configurations}\label{def:GFc}
-  Given a connected subset $\Gamma\subset\Lambda$, we denote the {\it exterior} of $\Gamma$ by $\Gamma_0$, and its {\it holes} by $\mathcal H(\Gamma)\equiv\{\hat\Gamma_1,\cdots,\hat\Gamma_{h_{\Gamma}}\}$ with $h_{\Gamma}\geqslant 0$. Formally, $\hat\Gamma_0,\cdots,\hat\Gamma_{h_\Gamma}$ are the connected components of $\Lambda_\infty\setminus\Gamma$, and $\hat\Gamma_0$ is the only unbounded one.
-  \bigskip
-  
-  Given $\nu\in\{1,\cdots,\tau\}$, a GFc is a quadruplet $\gamma\equiv(\Gamma_\gamma,X_\gamma,\nu,\underline\mu_\gamma)$ in which $\Gamma_\gamma$ is a {\it connected} and {\it bounded} subset of $\Lambda$, $X_\gamma\in\Omega(\Gamma_\gamma)$, and $\underline\mu_\gamma$ is a map $\mathcal H(\Gamma_\gamma)\to\{1,\cdots,\tau\}$, and satisfies the following condition. Let $\mathfrak X_\gamma$ denote the particle configuration obtained by covering the exterior and holes of $\Gamma_\gamma$ by particles:
-  \begin{equation}
-    \mathfrak X_\gamma:=
-    \left(\mathcal L_\nu\cap\hat\Gamma_{\gamma,0}\right)\cup
-    \left(\bigcup_{j=1}^{h_{\Gamma_\gamma}}\left(\mathcal L_{\underline\mu_\gamma(\hat\Gamma_{\gamma,j})}\cap\hat\Gamma_{\gamma,j}\right)\right)
-    .
-  \end{equation}
-  A quadruplet $\gamma$ is a GFc if
-  \begin{itemize}
-    \item The particles in $X_\gamma$ are entirely contained inside $\Gamma_\gamma$ and those in $\mathfrak X_\gamma$ do not intersect $\Gamma_\gamma$: $\forall x\in X_\gamma$, $\sigma_x\subset\Gamma_\gamma$ and $\forall x'\in\mathfrak X_\gamma$, $\sigma_x\cap\Gamma_\gamma=\emptyset$.
-    \item for every $x\in X_\gamma$, there exists $y\in\sigma_x$ (recall~\-(\ref{sigma})) and $y'\in\mathcal E_\Lambda(X_\gamma\cup\mathfrak X_\gamma)$ (recall~\-(\ref{mcE})) such that $y$ and $y'$ are neighbors,
-    \item for every $x\in \mathfrak X_\gamma$ and $y\in\sigma_x$ and every $y'\in\mathcal E_\Lambda(X_\gamma\cup\mathfrak X_\gamma)$, $y$ and $y'$ are {\it not} neighbors.
-  \end{itemize}
-  We denote the set of GFcs by $\mathfrak C_\nu(\Lambda)$.
-\endtheo
+\indent We start by mapping the particle system to a model of Gaunt-Fisher configurations. This step is analogous to the contour mapping in the Peierls argument~\-\cite{Pe36}, which we will now briefly recall. Consider the two-dimensional ferromagnetic Ising model. Having fixed a boundary condition in which every spin on the boundary is up, one can represent any spin configuration as a collection of {\it contours}, which correspond to the interfaces of the regions of up and down spins. Since these boundaries are unlikely at low temperatures, the effective activity of a contour is low. We wish to adapt this construction to non-sliding hard-core lattice systems. Defining boundaries in this context is more delicate than in the Ising model, due to the necessity of constructing a model of contours that does not have any long range interactions. We will identify boundaries by focusing on empty space, and define GFcs as the connected components of the union of the empty space and the supports of the particles surrounding it. The mapping is given in the following lemma.
 \bigskip
 
-
 \theoname{Lemma}{GFc mapping}\label{lemma:GFc}
   The partition function~\-(\ref{Xi_nu}) can be rewritten as
   \begin{equation}
@@ -461,7 +464,7 @@ Now, consider a connected configuration of crosses $X$.
     \prod_{\gamma\in\underline\gamma}\zeta_\nu^{(\underline z)}(\gamma)
     \label{XiGFc}
   \end{equation}
-  where
+  where $\mathfrak C_\nu(\Lambda)$ is the set of GFcs, defined in definition~\-\ref{def:GFc} below, $\Phi(\gamma,\gamma')\in\{0,1\}$ is equal to 1 if and only if $\Gamma_\gamma$ and $\Gamma_{\gamma'}$ are disconnected,
   \begin{equation}
     \mathbf z_\nu(\Lambda):=\prod_{x\in\Lambda\cap\mathcal L_\nu}z(x)
     \label{bfz}
@@ -478,27 +481,49 @@ Now, consider a connected configuration of crosses $X$.
       {\Xi_{\hat\Gamma_{\gamma,j}}^{(\nu)}(\underline z)}
     \label{zeta}
   \end{equation}
-   and $\Phi(\gamma,\gamma')\in\{0,1\}$ is equal to 1 if and only if $\Gamma_\gamma$ and $\Gamma_{\gamma'}$ are disjoint.
+  in which we used the following definition. Given a connected subset $\Gamma\subset\Lambda$, we denote the {\it exterior} of $\Gamma$ by $\hat\Gamma_0$, and its {\it holes} by $\mathcal H(\Gamma)\equiv\{\hat\Gamma_1,\cdots,\hat\Gamma_{h_{\Gamma}}\}$ with $h_{\Gamma}\geqslant 0$. Formally, $\hat\Gamma_0,\cdots,\hat\Gamma_{h_\Gamma}$ are the connected components of $\Lambda_\infty\setminus\Gamma$, and $\hat\Gamma_0$ is the only unbounded one.
+\endtheo
+\bigskip
+
+As was explained above, GFcs consist of empty sites and the particles that surround them. The following definition follows somewhat naturally from the proof of lemma~\-\ref{lemma:GFc} below.
+\bigskip
+
+\theoname{Definition}{Gaunt-Fisher configurations}\label{def:GFc}
+  Given $\nu\in\{1,\cdots,\tau\}$, a GFc is a quadruplet $\gamma\equiv(\Gamma_\gamma,X_\gamma,\nu,\underline\mu_\gamma)$ in which $\Gamma_\gamma$ is a {\it connected} and {\it bounded} subset of $\Lambda$, $X_\gamma\in\Omega(\Gamma_\gamma)$, and $\underline\mu_\gamma$ is a map $\mathcal H(\Gamma_\gamma)\to\{1,\cdots,\tau\}$, and satisfies the following condition. Let $\mathfrak X_\gamma$ denote the particle configuration obtained by covering the exterior and holes of $\Gamma_\gamma$ by particles:
+  \begin{equation}
+    \mathfrak X_\gamma:=
+    \left(\mathcal L_\nu\cap\hat\Gamma_{\gamma,0}\right)\cup
+    \left(\bigcup_{j=1}^{h_{\Gamma_\gamma}}\left(\mathcal L_{\underline\mu_\gamma(\hat\Gamma_{\gamma,j})}\cap\hat\Gamma_{\gamma,j}\right)\right)
+    .
+  \end{equation}
+  A quadruplet $\gamma$ is a GFc if
+  \begin{itemize}
+    \item The particles in $X_\gamma$ are entirely contained inside $\Gamma_\gamma$ and those in $\mathfrak X_\gamma$ do not intersect $\Gamma_\gamma$: $\forall x\in X_\gamma$, $\sigma_x\subset\Gamma_\gamma$ and $\forall x'\in\mathfrak X_\gamma$, $\sigma_x\cap\Gamma_\gamma=\emptyset$.
+    \item for every $x\in X_\gamma$, $\Delta(\sigma_x,\mathcal E_\Lambda(X_\gamma\cup\mathfrak X_\gamma))=1$ (recall that $\Delta$ is the graph distance on $\Lambda_\infty$, $\sigma_x$ is the support of the particle at $x$~\-(\ref{sigma}), and $\mathcal E_\Lambda(X_\gamma\cup\mathfrak X_\gamma)$ is the set of sites left uncovered by the configuration $X_\gamma\cup\mathfrak X_\gamma$~\-(\ref{mcE})),
+    \item for every $x\in \mathfrak X_\gamma$, $\Delta(\sigma_x,\mathcal E_\Lambda(X_\gamma\cup\mathfrak X_\gamma))>1$.
+  \end{itemize}
+  We denote the set of GFcs by $\mathfrak C_\nu(\Lambda)$.
 \endtheo
 \bigskip
 
+
 \indent\underline{Proof}:
   We will first map particle configurations to a set of GFc, then extract the most external ones, and conclude the proof by induction.
   \bigskip
 
-  \point{\bf GFcs.} To a configuration $X\in\Omega_\nu(\Lambda)$, we associate a set of {\it external GFcs}, in the following way, which is reminiscent of how Peierls' contours are associated to Ising spin configurations. See figure~\-\ref{fig:contour_example} for an example.
+  \point{\bf GFcs.} To a configuration $X\in\Omega_\nu(\Lambda)$, we associate a set of {\it external GFcs}. See figure~\-\ref{fig:contour_example} for an example.
   \bigskip
 
   \indent Given $x\in\Lambda$, let $\partial_X(x)$ denote the set of sites covered by particles neighboring $x$:
   \begin{equation}
-    \partial_X(x):=\{y\in\Lambda_\infty,\quad \exists y'\in X,\ y\in\sigma_{y'},\ \sigma_{y'}\mathrm{\ neighbors\ }x\}.
+    \partial_X(x):=\bigcup_{\displaystyle\mathop{\scriptstyle y\in X}_{\Delta(\sigma_y,x)=1}}\sigma_y.
   \end{equation}
-  Given $x\neq x'\in\mathcal E_\Lambda(X)$ (recall~\-(\ref{mcE})), $x$ and $x'$ are said to be {\it $X$-neighbors} if they are either neighbors, or if $\partial_X(x)\cap\partial_X(x')\neq\emptyset$. This defines a natural notion of an $X$-connected subset of $\mathcal E_\Lambda(X)$: $E\subset\mathcal E_\Lambda(X)$ is $X$-connected if there exists a path $x_0,\cdots,x_n\in E$ such that $x_0\equiv x$, $x_n\equiv x'$ and for $i\in\{1,\cdots,n\}$, $x_{i-1}$ and $x_i$ are $X$-neighbors. Now, let $\{E_1,\cdots,E_n\}$ denote the set of $X$-connected components of $\mathcal E_\Lambda(X)$. We associate a set $\Gamma_i$ to each $E_i$:
+  Consider the union of the set of empty sites and the particles neighboring it:
   \begin{equation}
-    \Gamma_i:=E_i\cup\left(\bigcup_{x\in E_i}\partial_X(x)\right).
-    \label{Gamma}
+    \mathbb U_\Lambda(X):=\mathcal E_\Lambda(X)\cup\left(\bigcup_{x\in\mathcal E_\Lambda(X)}\partial_X(x)\right)
+  .
   \end{equation}
-  By construction, distinct $\Gamma_i$'s are disjoint.
+  We denote the connected components of $\mathbb U_\Lambda(X)$ by $\Gamma_1,\cdots,\Gamma_n$. These will be the supports of the GFcs associated to the configuration.
   \bigskip
 
   \begin{figure}
@@ -507,14 +532,20 @@ Now, consider a connected configuration of crosses $X$.
     \label{fig:contour_example}
   \end{figure}
 
-  \indent We then denote the connected components of $\Lambda_\infty\setminus(\Gamma_1\cup\cdots\cup\Gamma_n)$ by $\{\kappa_1,\cdots,\kappa_m\}$. By construction, each $\kappa_i$ is covered by particles, which we denote by $X_i$, and
+  \indent We then denote the connected components of $\Lambda_\infty\setminus(\Gamma_1\cup\cdots\cup\Gamma_n)$ by $\{\kappa_1,\cdots,\kappa_m\}$. By construction, each $\kappa_i$ is covered by particles. We denote the particle configuration restricted to $\kappa_i$ by $X_i:=X\cap\kappa_i$. In addition, we define $\bar X_i$ as the union of $X_i$ and the particles that surround $\kappa_i$:
   \begin{equation}
     \bar X_i:=X_i\cup\{x\in X,\ \exists x'\in X_i,\ \Delta(\sigma_x,\sigma_{x'})=1\}\in\mathbb S(X_i)
   \end{equation}
-  (we recall that $\Delta$ is the graph distance on $\Lambda_\infty$, and that $\mathbb S$ was defined in definition~\-\ref{def:sliding}) so that, by the non-sliding condition, there exists a {\it unique} $\mu_i\in\{1,\cdots,\tau\}$ such that $\bar X_i\subset\mathcal L_{\mu_i}$. See figure~\-\ref{fig:contour_nested} for an example.
+  (we recall that $\mathbb S$ was defined in definition~\-\ref{def:sliding}). By the non-sliding condition, there exists a {\it unique} $\mu_i\in\{1,\cdots,\tau\}$ such that $\bar X_i\subset\mathcal L_{\mu_i}$. See figure~\-\ref{fig:contour_nested} for an example.
   \bigskip
 
-  \indent By construction, for every $i\in\{1,\cdots,n\}$, the holes of $\Gamma_i$, which, we recall, are denoted by $\hat\Gamma_{i,j}$, contain at least one of the $\kappa_k$. When a hole contains several $\kappa_k$'s, there is one that is {\it more external} than the others (see figure~\-\ref{fig:contour_nested}). Let us now make this idea more precise, and use it to define the GFc configuration associated to $X$. For every $i\in\{1,\cdots,n\}$ and $j\in\{0,\cdots,h_{\Gamma_i}\}$, we define $k(\hat\Gamma_{i,j})\in\{1,\cdots,m\}$ such that, for every $i'\in\{1,\cdots,n\}$ and $j'\in\{1,\cdots,h_{\Gamma_{i'}}\}$, $\kappa_{k(\hat\Gamma_{i,j})}\cap\hat\Gamma_{i',j'}\neq\emptyset$ if and only if $(i,j)=(i',j')$. We then define the set of GFcs associated to $X$ as the set of quadruplets
+  \indent By construction, for every $i\in\{1,\cdots,n\}$, the holes of $\Gamma_i$, which, we recall, are denoted by $\hat\Gamma_{i,j}$, contain at least one of the $\kappa_k$. In fact, for every $i\in\{1,\cdots,n\}$ and $j\in\{0,\cdots,h_{\Gamma_i}\}$ there exists a unique index $k(\hat\Gamma_{i,j})\in\{1,\cdots,m\}$ such that $\kappa_{k(\hat\Gamma_{i,j})}$ is contained inside $\hat\Gamma_{i,j}$ and is in contact with $\Gamma_i$:
+  \begin{equation}
+    \kappa_{k(\hat\Gamma_{i,j})}\subset\hat\Gamma_{i,j}
+    ,\quad
+    \Delta(\kappa_{k(\hat\Gamma_{i,j})},\Gamma_i)=1
+  \end{equation}
+  (see figure~\-\ref{fig:contour_nested}). We then define the set of GFcs associated to $X$ as the set of quadruplets
   \begin{equation}
     \underline\gamma(X)=\left\{\left(\Gamma_i,X\cap\Gamma_i,\ \mu_{k(\hat\Gamma_{i,0})},\ \underline\mu_i\right),\quad i\in\{1,\cdots,n\}\right\}
     \label{GFcs_X}
@@ -534,7 +565,7 @@ Now, consider a connected configuration of crosses $X$.
 
   \point{\bf External GFc model.} We have thus mapped $X$ to a model of GFcs. Note that the indices $\mu_\cdot$ must match up, that is, if a GFc is in the hole of another, its external $\mu$ must be equal to the $\mu$ of the hole it is in. This is a long range interaction between GFcs, which makes the GFc model difficult to study. Instead, we will map the system to a model of {\it external} GFcs, that do not have long range interactions. We introduce the following definitions: two GFcs $\gamma,\gamma'\in\mathfrak C_\nu(\Lambda)$ are said to be
   \begin{itemize}
-    \item {\it compatible} if their supports do not intersect, that is, $\Gamma_{\gamma}\cap\Gamma_{\gamma'}=\emptyset$,
+    \item {\it compatible} if their supports are disconnected, that is, $\Delta(\Gamma_{\gamma},\Gamma_{\gamma'})>1$,
     \item {\it external} if their supports are in each other's exteriors, that is, $\Gamma_{\gamma}\subset\hat\Gamma_{\gamma',0}$ and $\Gamma_{\gamma'}\subset\hat\Gamma_{\gamma,0}$.
   \end{itemize}
   The GFcs in $\underline\gamma(X)$ (see~\-(\ref{GFcs_X})) are compatible, but not necessarily external. Roughly, the idea is to keep the GFcs that are external, since those do not have long-range interactions (they all share the same external $\mu$, which is fixed to $\nu$ once and for all). At that point, the particle configuration in the exterior of all GFcs is fixed, and we are left with summing over configurations in the holes. The sum over configurations in holes is of the same form as~\-(\ref{Xi_nu}), with $\Lambda$ replaced by the hole, and the boundary condition by the appropriate $\underline\mu$. Following this, we rewrite~\-(\ref{Xi_nu}) as
@@ -550,7 +581,15 @@ Now, consider a connected configuration of crosses $X$.
     \right)
     \label{Xiexternal}
   \end{equation}
-  in which $\Phi_{\mathrm{ext}}(\gamma,\gamma')\in\{0,1\}$ is equal to 1 if and only if $\gamma$ and $\gamma'$ are {\it compatible} and {\it external}. We have, thus, rewritten the model as a system of external GFcs.
+  in which $\Phi_{\mathrm{ext}}(\gamma,\gamma')\in\{0,1\}$ is equal to 1 if and only if $\gamma$ and $\gamma'$ are {\it compatible} and {\it external}. Note that $\hat\Gamma_{\gamma,j}$ is obviously bounded and simply connected. It is also tiled, since, as is readily checked,
+  \begin{equation}
+    \hat\Gamma_{i,j}
+    =
+    \bigcup_{x\in\mathcal L_{\underline\mu_i(\hat\Gamma_{i,j})}\cap\hat\Gamma_{i,j}}\sigma_x
+    .
+  \end{equation}
+  
+  We have, thus, rewritten the model as a system of external GFcs.
   \bigskip
 
   \point{\bf GFc model.} The last factor in~\-(\ref{Xiexternal}) is similar to the left side of~\-(\ref{Xiexternal}), except for the fact that the boundary condition is $\underline\mu_\gamma(\hat\Gamma_{\gamma,j})$ instead of $\nu$. In order to obtain a model of GFcs (which are not necessarily external), we could iterate~\-(\ref{Xiexternal}), but, as was discussed earlier, this would induce long-range correlations. Instead, we introduce a trivial identity into~\-(\ref{Xiexternal}):
@@ -600,6 +639,7 @@ Now, consider a connected configuration of crosses $X$.
     \label{PhiT}
   \end{equation}
   where $\Phi(\gamma_j,\gamma_{j'})\in\{0,1\}$ is equal to 1 if and only if $\Gamma_{\gamma_j}$ and $\Gamma_{\gamma_{j'}}$ are {\it disjoint}, $\mathcal G^T(n)$ is the set of connected graphs on $n$ vertices and $\mathcal E(\mathfrak g)$ is the set of edges of $\mathfrak g$. In addition, for every $\gamma\in\mathfrak C_\nu(\Lambda)$,
+  \nopagebreakaftereq
   \begin{equation}
     \sum_{\underline\gamma'\subset\mathfrak C_\nu(\Lambda)}
     \left|
@@ -613,6 +653,7 @@ Now, consider a connected configuration of crosses $X$.
     \label{ce_remainder}
   \end{equation}
 \endtheo
+\restorepagebreakaftereq
 \bigskip
 
 \indent We will now show that~\-(\ref{cvcd}) holds for an appropriate choice of $a$, $d$ and $\delta$.
@@ -670,19 +711,19 @@ Now, consider a connected configuration of crosses $X$.
   \bigskip
 
   \begin{figure}
-    \hfil\includegraphics[width=6cm]{assymmetry1.pdf}{\footnotesize\it a.}
-    \hfil\includegraphics[width=6cm]{assymmetry2.pdf}{\footnotesize\it b.}
+    \hfil\includegraphics[width=7.5cm]{assymmetry1.pdf}{\footnotesize\it a.}
+    \hfil\includegraphics[width=7.5cm]{assymmetry2.pdf}{\footnotesize\it b.}
     \caption{%
-      Two different boundary conditions for the hard-cross model. The set $\Lambda$ is outlined by the thick black line. The crosses that are drawn are those mandated by the boundary condition (the boundary condition stipulates that every cross that is in contact with the boundary must be of a specified phase), and the remaining available space in $\Lambda$ is colored gray. The partition function in the case of figure~\-{\it a} is
-      $$z^{16}(1+4y+10y^2+8y^3+y^4)$$
+      Two different boundary conditions for the hard-cross model. The set $\Lambda$ is outlined by the thick black line. The crosses that are drawn are those mandated by the boundary condition (the boundary condition stipulates that every cross that is in contact with the boundary must be of a specified phase and cannot be in contact with empty sites), and the remaining available space in $\Lambda$ is colored gray. In figure~\-{\it a}, $\Lambda$ can be tiled by the covering corresponding to the boundary condition, whereas it cannot in figure~\-{\it b}. The partition function in the case of figure~\-{\it a} is
+      $$z^{25}(1+y)$$
       whereas that in figure~\-{\it b} is
-      $$z^{16}(1+6y+18y^2+48y^3+43y^4+13y^5+y^6).$$
+      $$z^{25}(1+5y+14y^2+18y^3+9y^4+y^5).$$
     }
     \label{fig:assymmetry}
   \end{figure}
 
   
-  \point First of all, if $\Lambda$ is so small that it cannot contain a GFc, that is, $\mathcal C_\mu(\Lambda)=\emptyset$ for every $\mu\in\{1,\cdots,\tau\}$, then~\-(\ref{cvcd}) is trivially true, and
+  \point First of all, if $\Lambda$ is so small that it cannot contain a GFc, that is, $\mathfrak C_\mu(\Lambda)=\emptyset$ for every $\mu\in\{1,\cdots,\tau\}$, then~\-(\ref{cvcd}) is trivially true, and
   \begin{equation}
     \Xi_{\Lambda}^{(\mu)}(\underline z)=\mathbf z_\mu(\Lambda)=\prod_{x\in\Lambda\cap\mathcal L_\mu}z(x)
     .
@@ -762,14 +803,14 @@ Now, consider a connected configuration of crosses $X$.
     \sum_{\displaystyle\mathop{\scriptstyle\gamma'\in\mathfrak C_\nu(\Lambda)}_{\gamma'\not\sim\gamma}}
     \alpha^{(1-(\theta+\xi))|\Gamma_{\gamma'}|}.
   \end{equation}
-  We bound the number of GFcs $\gamma'$ that are {\it incompatible} with a fixed GFc $\gamma$ by the number of walks on $\Lambda_\infty$ of length $2|\Gamma_{\gamma'}|\equiv2\ell$ that intersect $\Gamma_\gamma$:
+  We bound the number of GFcs $\gamma'$ that are {\it incompatible} with a fixed GFc $\gamma$ by the number of walks on $\Lambda_\infty$ of length $2|\Gamma_{\gamma'}|\equiv2\ell$ that intersect or neighbor $\Gamma_\gamma$:
   \begin{equation}
     \sum_{\displaystyle\mathop{\scriptstyle\gamma'\in\mathfrak C_\nu(\Lambda)}_{\gamma'\not\sim\gamma}}e^{a(\gamma')+d(\gamma')}|\zeta_\nu^{(\underline z)}(\gamma')|
     \leqslant
     \varsigma^2
-    |\Gamma_\gamma|\sum_{\ell=1}^\infty \chi^{2\ell}\alpha^{(1-(\theta+\xi))\ell}
+    (\chi+1)|\Gamma_\gamma|\sum_{\ell=1}^\infty \chi^{2\ell}\alpha^{(1-(\theta+\xi))\ell}
   \end{equation}
-  in which $\chi$ is the degree of $\Lambda_\infty$, and, provided
+  in which $\chi$ is the coordination number (that is, the number of neighbors of each vertex) of $\Lambda_\infty$ ($(\chi+1)|\Gamma_\gamma|$ is a bound on the number of sites that intersect or neighbor $\Gamma_\gamma$). Now, provided
   \begin{equation}
     \alpha\ll\chi^{-2(1-(\theta+\xi))^{-1}}
     \label{assum_alpha2}
@@ -828,7 +869,7 @@ Now, consider a connected configuration of crosses $X$.
 	-
 	\mathds 1\left(\mathfrak x_i\in\mathcal L_\mu\cap\hat\Gamma_{\gamma',j}\right)
       \right)
-      \\[0.5cm]\hfill
+      \\[0.5cm]
       &+
       \sum_{j=1}^{h_{\Gamma_{\gamma'}}}\left(
 	\frac\partial{\partial\log z(\mathfrak x_i)}\log\left(
@@ -955,8 +996,8 @@ Now, consider a connected configuration of crosses $X$.
       \right|
       +
       \alpha^{\cst C{cst:deriv_Xi}}
-      .
     \right)
+    .
   \end{equation}
   Thus,
   \begin{equation}
@@ -1149,7 +1190,7 @@ Now, consider a connected configuration of crosses $X$.
   Furthermore, by lemma~\-\ref{lemma:cluster_expansion}, the sums over $\gamma'$ and $\underline\gamma$ in $\frac1{|\Lambda|}\mathfrak B_\nu^{(|\Lambda|)}(\Lambda_\infty)$ (see~\-(\ref{frakB})) are absolutely convergent, which implies that $p(z)-\rho_m\log z$ is an analytic function of $y$ for small value of $|y|$.
   \bigskip
 
-  \point By a similar argument, we show that the correlation functions are analytic by proving that
+  \point By a similar argument, we show that the correlation functions are analytic in $y$ for smallvalues of $|y|$ by proving that
   \begin{equation}
     \sum_{\underline\gamma\subset\mathfrak C_\nu(\Lambda)}
     \frac{\partial^{\mathfrak n}}{\partial\log\underline z(\mathfrak x_1)\cdots\partial\log\underline z(\mathfrak x_{\mathfrak n})}
@@ -1173,7 +1214,7 @@ Now, consider a connected configuration of crosses $X$.
     \prod_{\gamma\in\underline\gamma}\zeta_\nu^{(\underline z)}(\gamma)
     \label{remainder_correlations}
   \end{equation}
-  vanishes in the infinite-volume limit. It is straightforward to check (this is done in detail for the first derivative in the proof of lemma~\-\ref{lemma:bound_zeta}, see~\-(\ref{first_deriv})) that the derivatives of $\log\zeta_\nu^{(\underline z)}(\gamma)$ are bounded analytic functions of $y$, uniformly in $\gamma$, and are proportional to indicator functions that force $\Gamma_\gamma$ to contain each of the $\mathfrak x_i$ with respect to which $\zeta$ is derived. Therefore, the clusters $\gamma'\cup\underline\gamma$ that contribute are those which contain all the $\mathfrak x_i$ and that exit $\Lambda$. We can therefore bound~\-(\ref{remainder_correlations}) by
+  vanishes in the infinite-volume limit. It is straightforward to check (this is done in detail for the first derivative in the proof of lemma~\-\ref{lemma:bound_zeta}, see~\-(\ref{first_deriv})) that the derivatives of $\log\zeta_\nu^{(\underline z)}(\gamma)$ are bounded analytic functions of $y$, uniformly in $\gamma$, and are proportional to indicator functions that force $\Gamma_\gamma$ to contain each of the $\mathfrak x_i$ with respect to which $\zeta$ is derived. Therefore, the clusters $\gamma'\cup\underline\gamma$ that contribute are those which contain all the $\mathfrak x_i$ and that are not contained inside $\Lambda$. We can therefore bound~\-(\ref{remainder_correlations}) by
   \begin{equation}
     \sum_{\displaystyle\mathop{\scriptstyle\gamma'\in\mathfrak C_\nu(\Lambda_\infty)}_{\Gamma_{\gamma'}\ni\mathfrak x_1}}
     \sum_{\underline\gamma\subset\mathfrak C_\nu(\Lambda_\infty)}
@@ -1281,9 +1322,6 @@ doi:{\tt\color{blue}\href{http://dx.doi.org/10.1098/rsta.1988.0077}{10.1098/rsta
 \bibitem[KP86]{KP86}R. Koteck\'y, D. Preiss - {\it Cluster expansion for abstract polymer models}, Communications in Mathematical Physics, volume~\-103, issue~\-3, pages~\-491-498, 1986,\par\penalty10000
 doi:{\tt\color{blue}\href{http://dx.doi.org/10.1007/BF01211762}{10.1007/BF01211762}}.\par\medskip
  
-\bibitem[KW41]{KW41}H.A. Kramers, G.H. Wannier - {\it Statistics of the Two-Dimensional Ferromagnet. Part I}, Physical Review, volume~60, issue~3, pages~252-262, 1941,\par\penalty10000
-doi:{\tt\color{blue}\href{http://dx.doi.org/10.1103/PhysRev.60.252}{10.1103/PhysRev.60.252}}.\par\medskip
- 
 \bibitem[LRS12]{LRS12}J.L. Lebowitz, D. Ruelle, E.R. Speer - {\it Location of the Lee-Yang zeros and absence of phase transitions in some Ising spin systems}, Journal of Mathematical Physics, volume~\-53, issue~\-9, number~\-095211, 2012,\par\penalty10000
 doi:{\tt\color{blue}\href{http://dx.doi.org/10.1063/1.4738622}{10.1063/1.4738622}}, arxiv:{\tt\color{blue}\href{http://arxiv.org/abs/1204.0558}{1204.0558}}.\par\medskip
  
@@ -1295,6 +1333,9 @@ doi:{\tt\color{blue}\href{http://dx.doi.org/10.1063/1.1749933}{10.1063/1.1749933
  
 \bibitem[Mc10]{Mc10}B.M. McCoy - {\it Advanced Statistical Mechanics}, International Series of Monographs on Physics~\-146, Oxford University Press, 2010.\par\medskip
  
+\bibitem[Pe36]{Pe36}R. Peierls - {\it On Ising's model of ferromagnetism}, Mathematical Proceedings of the Cambridge Philosophical Society, volume~\-32, issue~\-3, pages~\-477-481, 1936,\par\penalty10000
+doi:{\tt\color{blue}\href{http://dx.doi.org/10.1017/S0305004100019174}{10.1017/S0305004100019174}}.\par\medskip
+ 
 \bibitem[Pe63]{Pe63}O. Penrose - {\it Convergence of Fugacity Expansions for Fluids and Lattice Gases}, Journal of Mathematical Physics, volume~\-4, issue~\-10, pages~\-1312-1320, 1963,\par\penalty10000
 doi:{\tt\color{blue}\href{http://dx.doi.org/10.1063/1.1703906}{10.1063/1.1703906}}.\par\medskip
  
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