Update to v0.2:
Fixed: corrected the leading order in the asymptotics for \rho_1 in
theorem 1.2.
Fixed: The clusters in the cluster expansion are multi-sets, not sets.
Fixed: The Ursell function in lemma 3.3 needs to be divided by the
multiplicity of each element.
Fixed: In the proofs of lemma 3.4 and theorem 1.2, when individuating a
polymer from a cluster, one needs to sum over its multiplicity so
as to avoid overcounting.
Fixed: Wrong estimate in the proof of lemma 3.4 which changes some of the
constants.
Fixed: Missing a smallness condition on \alpha in the proof of lemma 3.4.
Fixed: Wrong argument for the analyticity of \zeta in the proof of
theorem 1.2.
Fixed: miscellaneous minor tweaks, reformulations, clarifications and
typos.
This commit is contained in:
Ian Jauslin
parent
ec3406fb7a
commit
f76337d739
25
Changelog
25
Changelog
@ -1,3 +1,28 @@
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0.2:
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* Fixed: corrected the leading order in the asymptotics for \rho_1 in
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theorem 1.2.
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* Fixed: The clusters in the cluster expansion are multi-sets, not sets.
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* Fixed: The Ursell function in lemma 3.3 needs to be divided by the
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multiplicity of each element.
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* Fixed: In the proofs of lemma 3.4 and theorem 1.2, when individuating a
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polymer from a cluster, one needs to sum over its multiplicity so
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as to avoid overcounting.
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* Fixed: Wrong estimate in the proof of lemma 3.4 which changes some of the
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constants.
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* Fixed: Missing a smallness condition on \alpha in the proof of lemma 3.4.
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* Fixed: Wrong argument for the analyticity of \zeta in the proof of
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theorem 1.2.
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* Fixed: miscellaneous minor tweaks, reformulations, clarifications and
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typos.
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0.1:
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* Fixed: deleted erroneous reference to Kramers and Wannier.
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@ -1,8 +1,6 @@
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\documentclass[point_reset_at_theo]{ian}
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\usepackage{amssymb}
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\usepackage[utf8]{inputenc}
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\usepackage[russian]{babel}
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\usepackage{graphicx}
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\usepackage{largearray}
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\usepackage{dsfont}
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@ -56,12 +54,12 @@ We establish existence of order-disorder phase transitions for a class of ``non-
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\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.
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\bigskip
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\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}.
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\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{JL17}.
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\bigskip
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\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.
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\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~\-\cite{Do68} (see figure~\-\ref{fig:shapes}{\it a}) model discussed above, as well as the hard cross model~\-\cite{HP74} (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~\-\cite{Ba82} (see figure~\-\ref{fig:shapes}{\it c}), which corresponds to the nearest-neighbor exclusion on the triangular lattice.
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\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.
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\indent The hard diamond model was studied by Gaunt and Fisher~\-\cite{GF65}, in which 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.
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\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.
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@ -81,16 +79,17 @@ We establish existence of order-disorder phase transitions for a class of ``non-
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\end{figure}
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\subsection{Hard-core lattice particle models}\label{sec:model}
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\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
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\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 $\omega\subset\mathbb R^d$ (we need not assume much about $\omega$, because we will mainly consider its intersections with the lattice), and forbid the supports of different particles from intersecting. In the examples mentioned above, the shapes would be a diamond, a cross or a hexagon (see figure~\-\ref{fig:shapes}). Note that $\omega$ may, in some cases be an open set, whereas in others, it might include a portion of its boundary (see section~\-\ref{section:non_sliding} for details). We define the grand-canonical partition function of the system at activity $z>0$ on any bounded $\Lambda\subset\Lambda_\infty$ as
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\begin{equation}
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\Xi_\Lambda(z)=\sum_{X\subset\Lambda}z^{|X|}\prod_{x\neq x'\in X}\varphi(x,x')
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\label{Xi}
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\end{equation}
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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 $|X|$:
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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 $\Omega(\Lambda)$. We define $N_{\mathrm{max}}$ as the maximal number of particles:
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\begin{equation}
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|X|\leqslant N_{\mathrm{max}}\leqslant |\Lambda|
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N_{\mathrm{max}}:=\mathrm{max}\{|X|,\ X\subset\Lambda\}
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.
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\end{equation}
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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 $\frac12<r<\frac1{\sqrt 2}$.
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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 $\frac12<r<\frac1{\sqrt 2}$.
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\bigskip
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\indent We will discuss the properties of the finite-volume {\it pressure} of hard-core particles systems, defined as
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@ -158,7 +157,7 @@ where $y\equiv z^{-1}$ and
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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).
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\label{blog}
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\end{equation}
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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
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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 the coefficients $c_k(\Lambda)$ diverge 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
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\begin{equation}
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Q_\Lambda(1)=\frac14|\Lambda|^2
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,\quad
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@ -174,7 +173,7 @@ Note that the pressure for this system, given by
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p(y)-\rho_m\log z=\log\left(\frac{1+\sqrt{1+4z}}2\right)-\frac12\log z
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=\log\left(\sqrt{1+\frac14y}+\frac12\sqrt y\right)
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\end{equation}
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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.
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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. There are examples in higher dimensions of sliding models for which the pressure is not analytic in $y$, and which are not crystalline at high fugacities (see, for example, \cite{GD07}).
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\bigskip
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\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.
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@ -214,7 +213,7 @@ which, in particular, implies that
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c_k(\Lambda)=-\frac1k\left(\frac1{|\Lambda|}\sum_{i=1}^{N_{\mathrm{max}}}\xi_i^k(\Lambda)\right)
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.
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\end{equation}
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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 $k$.
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When taking the thermodynamic limit, $kc_k$ 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 $\xi^k$ grows at most exponentially in $k$.
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\bigskip
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{\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 $\Lambda\to\Lambda_\infty$.
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@ -236,12 +235,12 @@ A {\it perfect covering} is defined as a particle configuration $X\in\Omega(\Lam
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\theoname{Definition}{sliding}\label{def:sliding}
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A hard-core lattice particle system is said to be {\it non-sliding} if the following hold.
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\begin{itemize}
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\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).
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\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
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\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). (Here, when we use the word `lattice', we do not intend a discrete subgroup of $\mathbb R^d$ but a discrete periodic subset of $\mathbb R^d$; the sets $\mathcal L_i$ will be called `sublattices' in the following, even though they may not have any group structure.)
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\item Given a bounded {\it connected} particle configuration $X\in\Omega(\Lambda_\infty)$ (that is, a configuration in which the union $\bigcup_{x\in X}\sigma_x$ is connected), we define $\mathbb S(X)$, roughly (see~\-(\ref{bbS}) for a formal definition), as the set of particle configurations $X'$ that
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\begin{itemize}
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\item contain $X$,
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\item are such that every $x'\in X'\setminus X$ is adjacent to $X$,
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\item leave no empty sites adjacent to $X$.
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\item leave no empty sites adjacent to $\bigcup_{x\in X}\sigma_x$.
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\end{itemize}
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(see figures~\-\ref{fig:cross_unique1} and~\-\ref{fig:cross_unique2}):
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\begin{equation}
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@ -256,22 +255,22 @@ A {\it perfect covering} is defined as a particle configuration $X\in\Omega(\Lam
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\endtheo
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\bigskip
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{\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.
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{\bf Remark}: In non-sliding models, every defect (recall that a defect appears where a configuration differs from a perfect covering) 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 subset of a perfect covering. The position of the particles surrounding it uniquely determines which one of the perfect coverings it is a subset of.
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\bigskip
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\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
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\indent In addition, we make the following assumption about the geometry of $\Lambda$: $\Lambda$ is {\it bounded}, 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
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\begin{equation}
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\Lambda=\bigcup_{x\in S}\sigma_x
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.
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\label{tiled}
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\end{equation}
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Note that the choice of $\mu$ will not play any role in the thermodynamic limit.
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The choice of $\mu$ will not play any role in the thermodynamic limit.
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\bigskip
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\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
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\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, roughly (see~\-(\ref{Omeganu}) for a formal definition),
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\begin{itemize}
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\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,
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\item the particles that neighbor the boundary must not neighbor an empty site.
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\item every site $x\in\mathcal L_\nu$ such that $\Delta(\sigma_x,\Lambda_\infty\setminus\Lambda)\leqslant 1$, is occupied by a particle,
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\item the particles that neighbor the boundary must not neighbor an empty site in $\Lambda_\infty$.
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\end{itemize}
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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
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\begin{equation}
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@ -283,8 +282,9 @@ Thus, defining $\mathbb B_\nu(\Lambda):=\{x\in\mathcal L_\nu\cap\Lambda,\ \Delta
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\forall x\in\mathbb B_\nu(\Lambda),\ \Delta(\sigma_x,\mathcal E_{\Lambda_\infty}(X\cup\mathbb X_\nu(\Lambda)))>1
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\}
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.
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\label{Omeganu}
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\end{equation}
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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.
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We choose these particular boundary conditions in order to make the discussion below simpler. Certain types of more general boundary conditions would presumably lead to infinite volume measures which are convex combinations 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.
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\bigskip
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\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
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@ -318,14 +318,14 @@ Note that the 1-point correlation function is the local density. In addition, we
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\bigskip
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\theoname{Theorem}{crystallization and high-fugacity expansion}\label{theo:main}
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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 $|y|<y_0$.
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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 extremal Gibbs states. The $\nu$-th Gibbs state, obtained from the boundary condition labeled by $\nu$, is invariant under the translations of the sublattice $\mathcal L_\nu$. 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 $|y|<y_0$.
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\bigskip
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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 $x\Lambda_\infty$,
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These Gibbs states are {\it crystalline}: having picked the boundary condition $\nu$, the particles are much more likely to be on the $\mathcal L_\nu$ sublattice than the others: for every $x\in\Lambda_\infty$,
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\begin{equation}
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\rho_1^{(\nu)}(x)=
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\left\{\begin{array}{ll}
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\rho_m+O(y)&\mathrm{if\ }x\in\mathcal L_\nu\\[0.3cm]
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1+O(y)&\mathrm{if\ }x\in\mathcal L_\nu\\[0.3cm]
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O(y)&\mathrm{if\ not}
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.
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\end{array}\right.
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@ -343,7 +343,7 @@ Note that the 1-point correlation function is the local density. In addition, we
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\indent In this section, we present several examples of non-sliding hard-core lattice particle models.
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\bigskip
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|
||||
\point Let us start with the hard diamond model, or rather, a generalization to the ``hyperdiamond'' model $d\geqslant 2$-dimensions, which is equivalent to the nearest neighbor exclusion on $\mathbb Z^d$. It is formally defined by specifying the lattice $\Lambda_\infty=\mathbb Z^d$ and the hyperdiamond shape $\omega\subset\mathbb R^2$ (see figure~\-\ref{fig:shapes}{\it a}):
|
||||
\point Let us start with the hard diamond model, or rather, a generalization to the ``hyperdiamond'' model in $d\geqslant 2$-dimensions, which is equivalent to the nearest neighbor exclusion on $\mathbb Z^d$. It is formally defined by specifying the lattice $\Lambda_\infty=\mathbb Z^d$ and the hyperdiamond shape $\omega\subset\mathbb R^d$ (see figure~\-\ref{fig:shapes}{\it a}):
|
||||
\begin{equation}
|
||||
\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\}
|
||||
.
|
||||
@ -355,7 +355,7 @@ Note the adjunction of the point $(0,\cdots,0,1)$, whose absence would prevent t
|
||||
\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.
|
||||
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 (connected, here, refers to the set $\sigma_{x_1}\cup\sigma_{x_2}$) 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 from the position of a single one of its particles.
|
||||
\bigskip
|
||||
|
||||
\begin{figure}
|
||||
@ -384,8 +384,8 @@ and, for $p\in\{2,3,4,5\}$,
|
||||
\end{equation}
|
||||
with $v_2=(1,0)$, $v_3=(0,1)$, $v_4=(-1,0)$ and $v_5=(0,-1)$. The $\mathcal L_{2p-1}$ are related to $\mathcal L_1$ by translations, as are the $\mathcal L_{2p}$ related to $\mathcal L_2$, and $\mathcal L_2$ is mapped to $\mathcal L_1$ by the vertical reflection. Let us now check the non-sliding property. We first introduce the following definitions: two crosses at $x,x'$ whose supports are connected and disjoint are said to be (see figure~\-\ref{fig:cross_pair_classify})
|
||||
\begin{itemize}
|
||||
\item {\it left-packed} if $x-x'\in\{(1,2),(-2,1),(-1,-2),(2,-1)\}$
|
||||
\item {\it right-packed} if $x-x'\in\{(2,1),(-1,2),(-2,-1),(1,-2)\}$
|
||||
\item {\it left-packed} if $x-x'\in\{(1,2),(-2,1),(-1,-2),(2,-1)\}\subset\mathcal L_1$
|
||||
\item {\it right-packed} if $x-x'\in\{(2,1),(-1,2),(-2,-1),(1,-2)\}\subset\mathcal L_2$
|
||||
\item {\it stacked} if $x-x'\in\{(3,0),(0,3),(-3,0),(0,-3)\}$.
|
||||
\end{itemize}
|
||||
Now, consider a connected configuration of crosses $X$.
|
||||
@ -452,7 +452,25 @@ Now, consider a connected configuration of crosses $X$.
|
||||
\bigskip
|
||||
|
||||
\subsection{The GFc model}
|
||||
\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.
|
||||
\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. GFcs give us a formal way of defining the notion of a {\it defect}, which was left imprecise until now. 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
|
||||
|
||||
\theoname{Lemma}{GFc mapping}\label{lemma:GFc}
|
||||
@ -485,28 +503,6 @@ Now, consider a connected configuration of crosses $X$.
|
||||
\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
|
||||
@ -514,7 +510,7 @@ As was explained above, GFcs consist of empty sites and the particles that surro
|
||||
\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 $x$:
|
||||
\indent Given $x\in\Lambda$, let $\partial_X(x)$ denote the set of sites covered by particles neighboring $x$ which do not themselves cover $x$:
|
||||
\begin{equation}
|
||||
\partial_X(x):=\bigcup_{\displaystyle\mathop{\scriptstyle y\in X}_{\Delta(\sigma_y,x)=1}}\sigma_y.
|
||||
\end{equation}
|
||||
@ -539,7 +535,7 @@ As was explained above, GFcs consist of empty sites and the particles that surro
|
||||
(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 $\kappa_k$. 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 $\Gamma_i$:
|
||||
\indent By construction, for every $i\in\{1,\cdots,n\}$, each hole of $\Gamma_i$ (we recall that the holes of $\Gamma_i$ are denoted by $\hat\Gamma_{i,j}$) contains 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
|
||||
@ -552,9 +548,9 @@ As was explained above, GFcs consist of empty sites and the particles that surro
|
||||
\end{equation}
|
||||
where $X\cap\Gamma_i$ is the restriction of the particle configuration to $\Gamma_i$, and $\underline\mu_i$ is the map from $\mathcal H(\hat\Gamma_i)$ to $\{1,\cdots,\tau\}$ defined by
|
||||
\begin{equation}
|
||||
\underline\mu_i(\hat\Gamma_{i,j})=\mu_{k(\hat\Gamma_{i,h_{\Gamma_i}})}.
|
||||
\underline\mu_i(\hat\Gamma_{i,j})=\mu_{k(\hat\Gamma_{i,j})}.
|
||||
\end{equation}
|
||||
The set of quadruplets thus constructed is a set of GFcs, in the sense of definition~\-\ref{def:GFc}.
|
||||
The set of quadruplets thus constructed is a set of GFcs, in the sense of definition~\-\ref{def:GFc}, that is, $\underline\gamma(X)\subset\mathfrak C_\nu(\Lambda)$.
|
||||
\bigskip
|
||||
|
||||
\begin{figure}
|
||||
@ -563,12 +559,12 @@ As was explained above, GFcs consist of empty sites and the particles that surro
|
||||
\label{fig:contour_nested}
|
||||
\end{figure}
|
||||
|
||||
\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
|
||||
\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 the first nested GFc 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 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
|
||||
The GFcs in $\underline\gamma(X)$ (see~\-(\ref{GFcs_X})) are compatible, but not necessarily external to each other. Roughly, the idea is to keep the GFcs that are external to each other, 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 each hole 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
|
||||
\begin{equation}
|
||||
\frac{\Xi^{(\nu)}_\Lambda(\underline z)}{\mathbf z_\nu(\Lambda)}=
|
||||
\sum_{\underline\gamma\subset\mathfrak C_\nu(\Lambda)}
|
||||
@ -581,18 +577,18 @@ As was explained above, GFcs consist of empty sites and the particles that surro
|
||||
\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}. Note that $\hat\Gamma_{\gamma,j}$ is obviously bounded and simply connected. It is also tiled, since, as is readily checked,
|
||||
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, connected and $\Lambda_\infty\setminus\hat\Gamma_{\gamma,j}$ is connected. It is also tiled, since, as is readily checked,
|
||||
\begin{equation}
|
||||
\hat\Gamma_{i,j}
|
||||
\hat\Gamma_{\gamma,j}
|
||||
=
|
||||
\bigcup_{x\in\mathcal L_{\underline\mu_i(\hat\Gamma_{i,j})}\cap\hat\Gamma_{i,j}}\sigma_x
|
||||
\bigcup_{x\in\mathcal L_{\underline\mu_i(\hat\Gamma_{\gamma,j})}\cap\hat\Gamma_{\gamma,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}):
|
||||
\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 $\nu$. (The denominator $\mathbf z_\nu$ also has a different index from the numerator, although this is not a problem since $\mathbf z_\nu$ and $\mathbf z_{\underline\mu_\gamma}$ are rather explicit.) In order to obtain a model of GFcs (which are not necessarily external to each other), 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}):
|
||||
\begin{equation}
|
||||
\frac{\Xi^{(\nu)}_\Lambda(\underline z)}{\mathbf z_\nu(\Lambda)}=
|
||||
\sum_{\underline\gamma\subset\mathfrak C_\nu(\Lambda)}
|
||||
@ -605,7 +601,7 @@ As was explained above, GFcs consist of empty sites and the particles that surro
|
||||
\right)
|
||||
\label{Xiexternal2}
|
||||
\end{equation}
|
||||
in which $\zeta_\nu^{(\underline z)}(\gamma)$ is defined in~\-(\ref{zeta}). We then rewrite $\Xi_{\hat\Gamma_{\gamma,j}}^{(\nu)}(\underline z)$ using~\-(\ref{Xiexternal2}), iterate, and, noting that, if $\Gamma_{\gamma,j}$ does not contain GFcs, then $\Xi_{\hat\Gamma_{\gamma,j}}^{(\nu)}(z)=\mathbf z_\nu(\hat\Gamma_{\gamma,j})$, we find~\-(\ref{XiGFc}).
|
||||
in which $\zeta_\nu^{(\underline z)}(\gamma)$ is defined in~\-(\ref{zeta}). We then rewrite $\Xi_{\hat\Gamma_{\gamma,j}}^{(\nu)}(\underline z)$ using~\-(\ref{Xiexternal2}), iterate, and, noting that, if $\hat\Gamma_{\gamma,j}$ does not contain GFcs, then $\Xi_{\hat\Gamma_{\gamma,j}}^{(\nu)}(\underline z)=\mathbf z_\nu(\hat\Gamma_{\gamma,j})$, we find~\-(\ref{XiGFc}).
|
||||
\qed
|
||||
|
||||
\subsection{Cluster expansion of the GFc model}
|
||||
@ -622,36 +618,37 @@ As was explained above, GFcs consist of empty sites and the particles that surro
|
||||
a(\gamma)
|
||||
\label{cvcd}
|
||||
\end{equation}
|
||||
in which $\gamma'\not\sim\gamma$ means that $\gamma'$ and $\gamma$ are {\it not} compatible (that is, their supports overlap), then
|
||||
in which $\gamma'\not\sim\gamma$ means that $\gamma'$ and $\gamma$ are {\it not} compatible (that is, the union of their supports is connected), then
|
||||
\begin{equation}
|
||||
\frac{\Xi_\Lambda^{(\nu)}(\Lambda)}{\mathbf z_\nu(\Lambda)}
|
||||
=\exp\left(
|
||||
\sum_{\underline\gamma\subset\mathfrak C_\nu(\Lambda)}
|
||||
\sum_{\underline\gamma\sqsubset\mathfrak C_\nu(\Lambda)}
|
||||
\Phi^T(\underline\gamma)
|
||||
\prod_{\gamma\in\underline\gamma}\zeta_\nu^{(\underline z)}(\gamma)
|
||||
\right)
|
||||
\label{ce}
|
||||
\end{equation}
|
||||
in which $\Phi^T$ is the {\it Ursell function}, defined as
|
||||
$\underline\gamma\sqsubset\mathfrak C_\nu(\Lambda)$ means that $\underline\gamma$ is a multiset (a multiset is similar to a set except for the fact that an element may appear several times in a multiset, in other words, a multiset is an unordered tuple) with elements in $\mathfrak C_\nu(\Lambda)$, and $\Phi^T$ is the {\it Ursell function}, defined as
|
||||
\begin{equation}
|
||||
\Phi^T(\gamma_1,\cdots,\gamma_n):=
|
||||
\frac1{N_{\underline\gamma}!}
|
||||
\sum_{\mathfrak g\in\mathcal G^T(n)}\prod_{\{j,j'\}\in\mathcal E(\mathfrak g)}(\Phi(\gamma_j,\gamma_{j'})-1)
|
||||
\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 $\gamma\in\mathfrak C_\nu(\Lambda)$,
|
||||
where $\Phi(\gamma_j,\gamma_{j'})\in\{0,1\}$ is equal to 1 if and only if $\Gamma_{\gamma_j}\cup\Gamma_{\gamma_{j'}}$ is {\it disconnected}, $\mathcal G^T(n)$ is the set of connected graphs on $n$ vertices and $\mathcal E(\mathfrak g)$ is the set of edges of $\mathfrak g$, and, if $n_{\gamma_i}$ is the multiplicity of $\gamma_i$ in $(\gamma_1,\cdots,\gamma_n)$, then $N_{\underline\gamma}!\equiv\prod_{j=1}^n(n_{\gamma_j}!)^{\frac1{n_{\gamma_j}}}$. In addition, for every $\gamma\in\mathfrak C_\nu(\Lambda)$,
|
||||
\nopagebreakaftereq
|
||||
\begin{equation}
|
||||
\sum_{\underline\gamma'\subset\mathfrak C_\nu(\Lambda)}
|
||||
\sum_{\underline\gamma'\sqsubset\mathfrak C_\nu(\Lambda)}
|
||||
\left|
|
||||
\Phi^T(\gamma\cup\underline\gamma')
|
||||
\Phi^T(\{\gamma\}\sqcup\underline\gamma')
|
||||
\prod_{\gamma'\in\underline\gamma'}
|
||||
\left(\zeta_\nu^{(\underline z)}(\gamma')e^{d(\gamma')}\right)
|
||||
\right|
|
||||
\leqslant
|
||||
e^{a(\gamma)}
|
||||
.
|
||||
\label{ce_remainder}
|
||||
\end{equation}
|
||||
where $\sqcup$ denotes the union operation in the sense of multisets.
|
||||
\endtheo
|
||||
\restorepagebreakaftereq
|
||||
\bigskip
|
||||
@ -676,11 +673,19 @@ As was explained above, GFcs consist of empty sites and the particles that surro
|
||||
d(\gamma):=-\xi|\Gamma_\gamma|\log\alpha>0
|
||||
\label{a}
|
||||
\end{equation}
|
||||
and
|
||||
\begin{equation}
|
||||
\delta=\varsigma\alpha^{1-(\theta+\xi)}
|
||||
,\quad
|
||||
\varsigma=\mathrm{max}\left(e^{2\cst c{cst:z}},\ 1+2\mathfrak n(e^{2\frac{\cst c{cst:z}}{\mathfrak n}}+1)\right).
|
||||
\label{delta_sigma}
|
||||
\end{equation}
|
||||
in which
|
||||
\begin{equation}
|
||||
\alpha:=e^1|z|^{-\rho_m(1+\mathcal N)^{-1}}\ll1
|
||||
.
|
||||
\alpha:=\varsigma e^{\chi}|z|^{-\rho_m(1+\mathcal N)^{-1}}\ll1
|
||||
\label{alpha}
|
||||
\end{equation}
|
||||
in which $\chi$ is the coordination number of $\Lambda_\infty$, that is, the maximal number of neighbors each vertex in $\Lambda_\infty$ has.
|
||||
\bigskip
|
||||
|
||||
In addition, there exists $\cst C{cst:deriv_Xi}\in(0,\xi)$ such that, for every $i\in\{1,\cdots,\mathfrak n\}$, and every $\mu\in\{1,\cdots,\tau\}$
|
||||
@ -696,11 +701,11 @@ As was explained above, GFcs consist of empty sites and the particles that surro
|
||||
\endtheo
|
||||
\bigskip
|
||||
|
||||
{\bf Remark}: The value of $z_0$ depends on the model. It is worked out rather explicitly in the proof, and appears as a smallness condition on $\alpha$, which is made explicit in~\-(\ref{assum_alpha1}), (\ref{assum_alpha2}), (\ref{assum_alpha3}), and~\-(\ref{assum_alpha4}). In these equations, we use the notation $\alpha\ll(\cdots)$ to mean ``there exists a constant $c>0$ such that if $\alpha<c(\cdots)$''.
|
||||
{\bf Remark}: The value of $z_0$ depends on the model. It is worked out rather explicitly in the proof, and appears as a smallness condition on $\alpha$, which is made explicit in~\-(\ref{assum_alpha1}), (\ref{assum_alpha2}), (\ref{assum_alpha3}), (\ref{assum_alpha4}), (\ref{assum_alpha5}) and~\-(\ref{assum_alpha6}). In these equations, we use the notation $\alpha\ll(\cdots)$ to mean ``there exists a small constant $c>0$ such that if $\alpha<c(\cdots)$''.
|
||||
\bigskip
|
||||
|
||||
\indent\underline{Proof}:
|
||||
We will prove this lemma along with the following inequality: $\exists \varsigma>0$ such that, for every $\mu\in\{1,\cdots,\tau\}$
|
||||
We will prove this lemma along with the following inequality: for every $\mu\in\{1,\cdots,\tau\}$
|
||||
\begin{equation}
|
||||
\left|\frac{\Xi_{\Lambda}^{(\mu)}(z)}{\Xi_{\Lambda}^{(\nu)}(z)}\right|
|
||||
\leqslant
|
||||
@ -742,14 +747,15 @@ As was explained above, GFcs consist of empty sites and the particles that surro
|
||||
\leqslant e^{2\cst c{cst:z}}\leqslant \varsigma e^{|\partial\Lambda|}
|
||||
\label{assumK_0}
|
||||
\end{equation}
|
||||
provided
|
||||
since, by~\-(\ref{delta_sigma}),
|
||||
\begin{equation}
|
||||
\varsigma\geqslant e^{2\cst c{cst:z}}
|
||||
.
|
||||
\label{bound_varsigma}
|
||||
\end{equation}
|
||||
So, to conclude this argument, it suffices to prove that $|\Lambda\cap\mathcal L_\mu|$ is independent of $\mu$. This follows from the fact that $\Lambda$ is {\it tiled} (see~\-(\ref{tiled})). In fact, we will show that for every $x\in\Lambda_\infty$, $|\mathcal L_\mu\cap\sigma_x|=1$ for any $\mu$, which, by~\-(\ref{tiled}) implies that $|\Lambda\cap\mathcal L_\mu|=\rho_m|\Lambda|$. We proceed in two steps, by first showing that $|\mathcal L_\mu\cap\sigma_x|$ is smaller than $2$, and then that it is larger than 0.
|
||||
\begin{itemize}
|
||||
\item To prove that $|\mathcal L_\mu\cap\sigma_x|<2$, we show that if $y,y'\in\mathcal L_\mu\cap\sigma_x$, then $\sigma_y\cap\sigma_{y'}\neq\emptyset$. Indeed, since $y\in\sigma_x$, writing $y'=x+\upsilon\in\sigma_x$, by translating by $\upsilon$, we find that $\sigma_{y'}\equiv\sigma_{x+\upsilon}\ni y+\upsilon\in\sigma_y$. Therefore, $|\mathcal L_\mu\cap\sigma_x|\leqslant 2$.
|
||||
\item To prove that $|\mathcal L_\mu\cap\sigma_x|<2$, we show that if $y,y'\in\mathcal L_\mu\cap\sigma_x$, then $\sigma_y\cap\sigma_{y'}\neq\emptyset$. Indeed, since $y\in\sigma_x$, writing $y'=x+\upsilon\in\sigma_x$, by translating by $\upsilon$, we find that $\sigma_{y'}\equiv\sigma_{x+\upsilon}\ni y+\upsilon\in\sigma_y$. Therefore, $|\mathcal L_\mu\cap\sigma_x|<2$.
|
||||
\item Finally, if $|\mathcal L_\mu\cap\sigma_x|=0$, then, since $\mathcal L_\mu$ is periodic, the density of $\mathcal L_\mu$ would be $<\rho_m$, which contradicts the fact that the $\mathcal L_i$ are related to each other by isometries.
|
||||
\end{itemize}
|
||||
All in all, $|\mathcal L_\mu\cap\sigma_x|=1$,which concludes the proof of~\-(\ref{assumK_0}).
|
||||
@ -760,46 +766,46 @@ As was explained above, GFcs consist of empty sites and the particles that surro
|
||||
|
||||
\subpoint By~\-(\ref{zeta}) and~\-(\ref{assumK}),
|
||||
\begin{equation}
|
||||
|\zeta_\nu^{(\underline z)}(\gamma)|\leqslant e^{2\cst c{cst:z}}\varsigma\frac{|z|^{|X_\gamma|}}{|z|^{\rho_m|\Gamma_\gamma|}}e^{|\Gamma_\gamma|}
|
||||
.
|
||||
|\zeta_\nu^{(\underline z)}(\gamma)|\leqslant e^{2\cst c{cst:z}}\varsigma^{h_{\Gamma_\gamma}}\frac{|z|^{|X_\gamma|}}{|z|^{\rho_m|\Gamma_\gamma|}}e^{\chi|\Gamma_\gamma|}
|
||||
\label{bound_zeta_pre}
|
||||
\end{equation}
|
||||
By definition~\-\ref{def:GFc}, in every configuration $X_\gamma$, every particle must be in contact with at least one empty site. Therefore, the fraction $\psi_\gamma(X_\gamma)$ of empty sites in $\Gamma_\gamma$ must satisfy
|
||||
in which $\chi$ is the coordination number of $\Lambda_\infty$ ($\chi$ appears because, for any set $A\subset\Lambda_\infty$, $|\partial A|\leqslant\chi|\partial(\Lambda_\infty\setminus A)|$). By definition~\-\ref{def:GFc}, in every configuration $X_\gamma$, every particle must be in contact with at least one empty site. Therefore, the fraction $\psi_\gamma(X_\gamma)$ of empty sites in $\Gamma_\gamma$ must satisfy
|
||||
\begin{equation}
|
||||
\psi_\gamma(X_\gamma):=\frac{|\mathcal E_{\Gamma_\gamma}(X_\gamma)|}{|\Gamma_\gamma|}\geqslant
|
||||
\frac1{\mathcal N+1}
|
||||
\end{equation}
|
||||
(recall that $\mathcal E_{\Gamma_\gamma}(X_\gamma)$ is the number of empty sites~\-(\ref{mcE}), and $\mathcal N$ is the maximal volume occupied by particles that neighbor a site~\-(\ref{mcN})). Therefore,
|
||||
(recall that $|\mathcal E_{\Gamma_\gamma}(X_\gamma)|$ is the number of empty sites~\-(\ref{mcE}), and $\mathcal N$ is the maximal volume occupied by particles that neighbor a site~\-(\ref{mcN})). Therefore,
|
||||
\begin{equation}
|
||||
|X_\gamma|
|
||||
=
|
||||
\rho_m|\Gamma_\gamma|(1-\psi_\gamma(X_\gamma))\leqslant\rho_m|\Gamma_\gamma|\frac{\mathcal N}{\mathcal N+1}.
|
||||
\end{equation}
|
||||
Therefore,
|
||||
Therefore, by~\-(\ref{alpha}), (\ref{bound_varsigma}) and~\-(\ref{bound_zeta_pre}), and using the fact that $h_{\Gamma_\gamma}\leqslant|\Gamma_\gamma|$,
|
||||
\begin{equation}
|
||||
|\zeta_\nu^{(\underline z)}(\gamma)|\leqslant
|
||||
\varsigma^2\left(e^1|z|^{-\rho_m\frac1{\mathcal N+1}}\right)^{|\Gamma_\gamma|}
|
||||
\equiv\varsigma^2\alpha^{|\Gamma_\gamma|}
|
||||
\varsigma\left(\varsigma e^\chi|z|^{-\rho_m\frac1{\mathcal N+1}}\right)^{|\Gamma_\gamma|}
|
||||
\equiv\varsigma\alpha^{|\Gamma_\gamma|}
|
||||
.
|
||||
\label{bound_zeta}
|
||||
\end{equation}
|
||||
Thus,
|
||||
Thus, by~\-(\ref{a}),
|
||||
\begin{equation}
|
||||
|\zeta_\nu^{(\underline z)}(\gamma)|e^{a(\gamma)+d(\gamma)}\leqslant\varsigma^2\alpha^{(1-(\theta+\xi))|\Gamma_\gamma|}
|
||||
|\zeta_\nu^{(\underline z)}(\gamma)|e^{a(\gamma)+d(\gamma)}\leqslant\varsigma\alpha^{(1-(\theta+\xi))|\Gamma_\gamma|}
|
||||
\label{bound_zetaead}
|
||||
\end{equation}
|
||||
which proves the first of~\-(\ref{cvcd}) with $\delta\equiv\varsigma^2\alpha^{1-(\theta+\xi)}$, which, provided
|
||||
which proves the first inequality in~\-(\ref{cvcd}) with $\delta\equiv\varsigma\alpha^{1-(\theta+\xi)}$, which, provided
|
||||
\begin{equation}
|
||||
\alpha\ll\varsigma^{-2(1-(\theta+\xi))^{-1}}
|
||||
\alpha\ll\varsigma^{-(1-(\theta+\xi))^{-1}}
|
||||
\label{assum_alpha1}
|
||||
\end{equation}
|
||||
is small.
|
||||
satisfies $\delta\ll 1$.
|
||||
\bigskip
|
||||
|
||||
\subpoint By~\-(\ref{bound_zetaead}),
|
||||
\subpoint We now turn to the second inequality in~\-(\ref{cvcd}). By~\-(\ref{bound_zetaead}),
|
||||
\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
|
||||
\varsigma
|
||||
\sum_{\displaystyle\mathop{\scriptstyle\gamma'\in\mathfrak C_\nu(\Lambda)}_{\gamma'\not\sim\gamma}}
|
||||
\alpha^{(1-(\theta+\xi))|\Gamma_{\gamma'}|}.
|
||||
\end{equation}
|
||||
@ -807,10 +813,10 @@ As was explained above, GFcs consist of empty sites and the particles that surro
|
||||
\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
|
||||
\varsigma
|
||||
(\chi+1)|\Gamma_\gamma|\sum_{\ell=1}^\infty \chi^{2\ell}\alpha^{(1-(\theta+\xi))\ell}
|
||||
\end{equation}
|
||||
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
|
||||
($(\chi+1)|\Gamma_\gamma|$ 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}
|
||||
@ -819,14 +825,14 @@ As was explained above, GFcs consist of empty sites and the particles that surro
|
||||
\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
|
||||
\varsigma
|
||||
\cst c{cst:sumup}
|
||||
|\Gamma_{\gamma'}|
|
||||
|\Gamma_\gamma|
|
||||
\label{bound_entropy1}
|
||||
\end{equation}
|
||||
for some constant $\cst c{cst:sumup}>0$. If, in addition,
|
||||
\begin{equation}
|
||||
\alpha\ll e^{-\varsigma^2\cst c{cst:sumup}\theta^{-1}}
|
||||
\alpha\ll e^{-\varsigma\cst c{cst:sumup}\theta^{-1}}
|
||||
\label{assum_alpha3}
|
||||
\end{equation}
|
||||
then this implies~\-(\ref{cvcd}).
|
||||
@ -841,7 +847,7 @@ As was explained above, GFcs consist of empty sites and the particles that surro
|
||||
=
|
||||
\sum_{\gamma'\in\mathfrak C_\mu(\Lambda)}
|
||||
\frac{\partial \zeta_\mu^{(\underline z)}(\gamma')}{\partial\log z(\mathfrak x_i)}
|
||||
\sum_{\underline\gamma\subset\mathfrak C_\mu(\Lambda)}\Phi^T(\gamma'\cup\underline\gamma)\prod_{\gamma\in\underline\gamma}\zeta_\mu^{(\underline z)}(\gamma)
|
||||
\sum_{\underline\gamma\sqsubset\mathfrak C_\mu(\Lambda)}\Phi^T(\{\gamma'\}\sqcup\underline\gamma)\prod_{\gamma\in\underline\gamma}\zeta_\mu^{(\underline z)}(\gamma)
|
||||
\end{equation}
|
||||
so, by~\-(\ref{ce_remainder}),
|
||||
\begin{equation}
|
||||
@ -954,13 +960,14 @@ As was explained above, GFcs consist of empty sites and the particles that surro
|
||||
\alpha^\xi 3e^1
|
||||
\cst c{cst:iso}^{(d)}
|
||||
d!
|
||||
\varsigma^2
|
||||
\varsigma
|
||||
\cst c{cst:sumup}
|
||||
.
|
||||
\end{equation}
|
||||
which, provided
|
||||
\begin{equation}
|
||||
\alpha\leqslant \left(3e^1\cst c{cst:iso}^{(d)}d!\varsigma^2\cst c{cst:sumup}\right)^{-(\xi-\cst C{cst:deriv_Xi})^{-1}}
|
||||
\alpha\leqslant \left(3e^1\cst c{cst:iso}^{(d)}d!\varsigma\cst c{cst:sumup}\right)^{-(\xi-\cst C{cst:deriv_Xi})^{-1}}
|
||||
\label{assum_alpha5}
|
||||
\end{equation}
|
||||
implies~\-(\ref{bound_deriv}).
|
||||
\bigskip
|
||||
@ -1018,12 +1025,12 @@ As was explained above, GFcs consist of empty sites and the particles that surro
|
||||
\begin{equation}
|
||||
\log\left(\frac{\Xi_{\Lambda}^{(\mu)}(z)}{\Xi_{\Lambda}^{(\nu)}(z)}\right)
|
||||
=
|
||||
\sum_{\underline\gamma\subset\mathfrak C_\mu(\Lambda)}\Phi^T(\underline\gamma)\prod_{\gamma\in\underline\gamma}\zeta_\mu^{(z)}(\gamma)
|
||||
\sum_{\underline\gamma\sqsubset\mathfrak C_\mu(\Lambda)}\Phi^T(\underline\gamma)\prod_{\gamma\in\underline\gamma}\zeta_\mu^{(z)}(\gamma)
|
||||
-
|
||||
\sum_{\underline\gamma\subset\mathfrak C_\nu(\Lambda)}\Phi^T(\underline\gamma)\prod_{\gamma\in\underline\gamma}\zeta_\nu^{(z)}(\gamma)
|
||||
\sum_{\underline\gamma\sqsubset\mathfrak C_\nu(\Lambda)}\Phi^T(\underline\gamma)\prod_{\gamma\in\underline\gamma}\zeta_\nu^{(z)}(\gamma)
|
||||
\label{flip_ce}
|
||||
\end{equation}
|
||||
(we recall that $z^{|\Lambda\cap\mathcal L_\mu|}$ is independent of $\mu$ so the $\mathbf z_\mu(\Lambda)$ and $\mathbf z_\nu(\Lambda)$ factors cancel out). We then split these cluster expansions into {\it bulk} and {\it boundary} contributions, which are defined as follows. Let $\mathfrak C_\mu^{(|\Lambda|)}(\Lambda_\infty)$ denote the set of GFcs in $\Lambda_\infty$ whose upper-leftmost corner (if $d>2$, then this notion should be extended in the obvious way) is in $\Lambda$. Note that $\mathfrak C_\mu^{(|\Lambda|)}(\Lambda_\infty)$ only depends on $\Lambda$ through its cardinality $|\Lambda|$. We then write
|
||||
(we recall that $z^{|\Lambda\cap\mathcal L_\mu|}$ is independent of $\mu$ so the $\mathbf z_\mu(\Lambda)$ and $\mathbf z_\nu(\Lambda)$ factors cancel out). We then split these cluster expansions into {\it bulk} and {\it boundary} contributions, which are defined as follows. Let $\mathfrak C_\mu^{(|\Lambda|)}(\Lambda_\infty)$ denote the set of GFcs in $\Lambda_\infty$ whose upper-leftmost corner (if $d>2$, then this notion should be extended in the obvious way) is in $\Lambda$. Note that $\mathfrak C_\mu^{(|\Lambda|)}(\Lambda_\infty)$ only depends on $\Lambda$ through its cardinality $|\Lambda|$ (up to a translation). We then write
|
||||
\begin{equation}
|
||||
\sum_{\underline\gamma\subset\mathfrak C_\mu(\Lambda)}\Phi^T(\underline\gamma)\prod_{\gamma\in\underline\gamma}\zeta_\mu^{(z)}(\gamma)
|
||||
=
|
||||
@ -1036,20 +1043,20 @@ As was explained above, GFcs consist of empty sites and the particles that surro
|
||||
\begin{equation}
|
||||
\begin{array}{>\displaystyle l}
|
||||
\mathfrak B^{(|\Lambda|)}_\mu(\Lambda_\infty):=
|
||||
\sum_{\gamma'\in\mathfrak C_\mu^{(|\Lambda|)}(\Lambda_\infty)}
|
||||
\zeta_\mu^{(z)}(\gamma')
|
||||
\sum_{\underline\gamma\subset\mathfrak C_\mu(\Lambda_\infty)}
|
||||
\Phi^T(\gamma'\cup\underline\gamma)\prod_{\gamma\in\underline\gamma}\zeta_\mu^{(z)}(\gamma)
|
||||
\sum_{m=1}^\infty\sum_{\gamma'\in\mathfrak C_\mu^{(|\Lambda|)}(\Lambda_\infty)}
|
||||
(\zeta_\mu^{(z)}(\gamma'))^m
|
||||
\sum_{\underline\gamma\sqsubset\mathfrak C_\mu(\Lambda_\infty)\setminus\{\gamma'\}}
|
||||
\Phi^T(\{\gamma'\}^m\sqcup\underline\gamma)\prod_{\gamma\in\underline\gamma}\zeta_\mu^{(z)}(\gamma)
|
||||
\\[1cm]
|
||||
\mathfrak b^{(\Lambda)}_\mu(\Lambda_\infty):=
|
||||
\sum_{\gamma'\in\mathfrak C_\mu^{(|\Lambda|)}(\Lambda_\infty)}
|
||||
\zeta_\mu^{(z)}(\gamma')
|
||||
\sum_{\displaystyle\mathop{\scriptstyle\underline\gamma\subset\mathfrak C_\mu(\Lambda_\infty)}_{\gamma'\cup\underline\gamma\not\subset\mathfrak C_\mu(\Lambda)}}
|
||||
\Phi^T(\gamma'\cup\underline\gamma)\prod_{\gamma\in\underline\gamma}\zeta_\mu^{(z)}(\gamma)
|
||||
.
|
||||
\sum_{m=1}^\infty\sum_{\gamma'\in\mathfrak C_\mu^{(|\Lambda|)}(\Lambda_\infty)}
|
||||
(\zeta_\mu^{(z)}(\gamma'))^m
|
||||
\sum_{\displaystyle\mathop{\scriptstyle\underline\gamma\sqsubset\mathfrak C_\mu(\Lambda_\infty)\setminus\{\gamma'\}}_{(\{\gamma'\}^m\sqcup\underline\gamma)\not\sqsubset\mathfrak C_\mu(\Lambda)}}
|
||||
\Phi^T(\{\gamma'\}^m\sqcup\underline\gamma)\prod_{\gamma\in\underline\gamma}\zeta_\mu^{(z)}(\gamma)
|
||||
\end{array}
|
||||
\label{frakB}
|
||||
\end{equation}
|
||||
in which $\{\gamma'\}^m$ is the multiset with $m$ elements that are all equal to $\gamma'$.
|
||||
\bigskip
|
||||
|
||||
\subsubpoint The bulk terms cancel each other out. Indeed, we recall (see section~\-\ref{sec:model}) that there exists an isometry $F_{\mu,\nu}$ of $\Lambda_\infty$ such that $F_{\mu,\nu}(\mathcal L_\mu)=\mathcal L_\nu$. In addition, since $F_{\mu,\nu}$ is an isometry, it maps perfect coverings to perfect coverings, and this map is denoted by $f_{\mu,\nu}:\{1,\cdots,\tau\}\to\{1,\cdots,\tau\}$:
|
||||
@ -1065,10 +1072,11 @@ As was explained above, GFcs consist of empty sites and the particles that surro
|
||||
\begin{equation}
|
||||
\mathfrak B_\mu^{(|\Lambda|)}(\Lambda_\infty)
|
||||
=
|
||||
\sum_{\gamma'\in\mathfrak C_\nu^{(|F_{\mu,\nu}(\Lambda)|)}(F_{\mu,\nu}(\Lambda_\infty))}
|
||||
\zeta_\nu^{(z)}(\gamma')
|
||||
\sum_{\underline\gamma\subset\mathfrak C_\nu(F_{\mu,\nu}(\Lambda_\infty))}
|
||||
\Phi^T(\gamma'\cup\underline\gamma)\prod_{\gamma\in\underline\gamma}\zeta_\nu^{(z)}(\gamma)
|
||||
\sum_{m=1}^\infty\sum_{\gamma'\in\mathfrak C_\nu^{(|F_{\mu,\nu}(\Lambda)|)}(F_{\mu,\nu}(\Lambda_\infty))}
|
||||
(\zeta_\nu^{(z)}(\gamma'))^m
|
||||
\sum_{\underline\gamma\sqsubset\mathfrak C_\nu(F_{\mu,\nu}(\Lambda_\infty))\setminus\{\gamma'\}}
|
||||
\Phi^T(\{\gamma'\}^m\sqcup\underline\gamma)\prod_{\gamma\in\underline\gamma}\zeta_\nu^{(z)}(\gamma)
|
||||
\label{frakBrot}
|
||||
\end{equation}
|
||||
so, since $F_{\mu,\nu}(\Lambda_\infty)=\Lambda_\infty$ and $|F_{\mu,\nu}(\Lambda)|=|\Lambda|$,
|
||||
\begin{equation}
|
||||
@ -1081,16 +1089,16 @@ As was explained above, GFcs consist of empty sites and the particles that surro
|
||||
\end{equation}
|
||||
\bigskip
|
||||
|
||||
\subsubpoint Finally, we estimate the boundary term. First of all, since every cluster $\gamma'\cup\underline\gamma$ that is not a subset of $\mathfrak C_\mu(\Lambda)$ must contain at least one GFc that goes over the boundary of $\Lambda$,
|
||||
\subsubpoint Finally, we estimate the boundary term. First of all, since every cluster $\{\gamma'\}\sqcup\underline\gamma$ that is not a subset of $\mathfrak C_\mu(\Lambda)$ must contain at least one GFc that goes over the boundary of $\Lambda$,
|
||||
\begin{equation}
|
||||
\mathfrak b_\mu^{(\Lambda)}(\Lambda_\infty)
|
||||
\leqslant
|
||||
\sum_{\displaystyle\mathop{\scriptstyle\gamma'\in\mathfrak C_\nu(\Lambda_\infty)}_{\displaystyle\mathop{\scriptstyle\Gamma_{\gamma'}\cap\Lambda\neq\emptyset}_{\Gamma_{\gamma'}\cap(\Lambda_\infty\setminus\Lambda)\neq\emptyset}}}
|
||||
|\zeta_\mu^{(z)}(\gamma')|
|
||||
\sum_{\underline\gamma\subset\mathfrak C_\mu(\Lambda_\infty)}
|
||||
\left|\Phi^T(\gamma'\cup\underline\gamma)\prod_{\gamma\in\underline\gamma}\zeta_\mu^{(z)}(\gamma)\right|
|
||||
\sum_{\underline\gamma\sqsubset\mathfrak C_\mu(\Lambda_\infty)}
|
||||
\left|\Phi^T(\{\gamma'\}\sqcup\underline\gamma)\prod_{\gamma\in\underline\gamma}\zeta_\mu^{(z)}(\gamma)\right|
|
||||
\end{equation}
|
||||
so, by~\-(\ref{ce_remainder}),
|
||||
(for the purpose of an upper bound, we can reabsorb the sum over $m$ in~\-(\ref{frakB}) in the sum over $\underline\gamma$) so, by~\-(\ref{ce_remainder}),
|
||||
\begin{equation}
|
||||
|\mathfrak b_\mu^{(\Lambda)}(\Lambda_\infty)|
|
||||
\leqslant
|
||||
@ -1098,16 +1106,21 @@ As was explained above, GFcs consist of empty sites and the particles that surro
|
||||
|\zeta_\mu^{(z)}(\gamma')|
|
||||
e^{a(\gamma')}
|
||||
\end{equation}
|
||||
which, rewriting, as we did earlier $e^{a(\gamma')}=e^{-d(\gamma')}e^{a(\gamma')+d(\gamma')}$ and using $d(\gamma')\geqslant-\xi\log\alpha$, implies
|
||||
which, rewriting, as we did earlier $e^{a(\gamma')}=e^{-d(\gamma')}e^{a(\gamma')+d(\gamma')}$ and using $d(\gamma')\geqslant-\xi\log\alpha$, implies, similarly to the derivation of~\-(\ref{bound_entropy1}),
|
||||
\begin{equation}
|
||||
|\mathfrak b_\mu^{(\Lambda)}(\Lambda_\infty)|
|
||||
\leqslant
|
||||
\alpha^\xi\varsigma^2\cst c{cst:sumup}|\partial\Lambda|.
|
||||
\alpha^\xi\varsigma\cst c{cst:sumup}|\partial\Lambda|.
|
||||
\label{boundary}
|
||||
\end{equation}
|
||||
\bigskip
|
||||
|
||||
\subsubpoint Thus, inserting~\-(\ref{bulk}) and~\-(\ref{boundary}) into~\-(\ref{bulk_boundary}) and~\-(\ref{flip_ce}), we find that, provided~\-(\ref{assum_alpha3}) holds,
|
||||
\subsubpoint Thus, inserting~\-(\ref{bulk}) and~\-(\ref{boundary}) into~\-(\ref{bulk_boundary}) and~\-(\ref{flip_ce}), provided
|
||||
\begin{equation}
|
||||
2\alpha^\xi\varsigma\cst c{cst:sumup}\leqslant 1
|
||||
\label{assum_alpha6}
|
||||
\end{equation}
|
||||
we find that
|
||||
\begin{equation}
|
||||
\log\left(\frac{\Xi_{\Lambda}^{(\mu)}(z)}{\Xi_{\Lambda}^{(\nu)}(z)}\right)
|
||||
\leqslant
|
||||
@ -1116,11 +1129,12 @@ As was explained above, GFcs consist of empty sites and the particles that surro
|
||||
By combining this bound with~\-(\ref{taylor_remainder}) and~\-(\ref{taylor}), we find that~\-(\ref{assumK}) holds with
|
||||
\begin{equation}
|
||||
\varsigma=1+2\mathfrak n(e^{2\frac{\cst c{cst:z}}{\mathfrak n}}+1).
|
||||
\label{varsigma}
|
||||
\end{equation}
|
||||
\qed
|
||||
|
||||
\subsection{High-fugacity expansion}
|
||||
\indent We now conclude this section by summarizing the validity of the high-fugacity expansion as a standalone theorem, which is a simple consequence of lemmas~\-\ref{lemma:GFc} and~\-\ref{lemma:bound_zeta}, and showing how it implies theorem~\-\ref{theo:main}.
|
||||
\indent We now conclude this section by summarizing the validity of the high-fugacity expansion as a stand-alone theorem, which is a simple consequence of lemmas~\-\ref{lemma:GFc}, \ref{lemma:cluster_expansion} and~\-\ref{lemma:bound_zeta}, and showing how it implies theorem~\-\ref{theo:main}.
|
||||
\bigskip
|
||||
|
||||
\theoname{Theorem}{high-fugacity expansion}\label{theo:expansion}
|
||||
@ -1139,7 +1153,7 @@ As was explained above, GFcs consist of empty sites and the particles that surro
|
||||
|
||||
The partition function~\-(\ref{Xi_nu}) can be rewritten as
|
||||
\begin{equation}
|
||||
\frac{\Xi_\Lambda^{(\nu)}(\underline z)}{\mathbf z_\nu(\Lambda)}=\exp\left(\sum_{\underline\gamma\subset\mathfrak C_\nu(\Lambda)}\Phi^T(\underline\gamma)\prod_{\gamma\in\underline\gamma}\zeta_\nu^{(\underline z)}(\gamma)\right)
|
||||
\frac{\Xi_\Lambda^{(\nu)}(\underline z)}{\mathbf z_\nu(\Lambda)}=\exp\left(\sum_{\underline\gamma\sqsubset\mathfrak C_\nu(\Lambda)}\Phi^T(\underline\gamma)\prod_{\gamma\in\underline\gamma}\zeta_\nu^{(\underline z)}(\gamma)\right)
|
||||
\label{ce_theo}
|
||||
\end{equation}
|
||||
where $\mathbf z_\nu(\Lambda)$ and $\zeta_\nu^{(\underline z)}(\gamma)$ were defined in~\-(\ref{bfz}) and~\-(\ref{zeta}), and $\Phi^T$ was defined in~\-(\ref{PhiT}).
|
||||
@ -1147,10 +1161,10 @@ As was explained above, GFcs consist of empty sites and the particles that surro
|
||||
|
||||
In addition, (\ref{ce_theo}) is absolutely convergent: there exist $\epsilon,\cst C{cst:cvce}>0$, such that, for every $\gamma'\in\mathfrak C_\nu(\Lambda)$,
|
||||
\begin{equation}
|
||||
\sum_{\underline\gamma\subset\mathfrak C_\nu(\Lambda)}
|
||||
\sum_{\underline\gamma\sqsubset\mathfrak C_\nu(\Lambda)}
|
||||
\left|
|
||||
\Phi^T(\gamma'\cup\underline\gamma)
|
||||
\zeta_\nu^{(\underline z)}(\gamma')\prod_{\gamma'\in\underline\gamma}\zeta_\nu^{(\underline z)}(\gamma')
|
||||
\Phi^T(\{\gamma'\}\sqcup\underline\gamma)
|
||||
\zeta_\nu^{(\underline z)}(\gamma')\prod_{\gamma''\in\underline\gamma}\zeta_\nu^{(\underline z)}(\gamma'')
|
||||
\right|
|
||||
\leqslant
|
||||
\cst C{cst:cvce}\epsilon^{|\Gamma_\gamma|}
|
||||
@ -1171,14 +1185,14 @@ As was explained above, GFcs consist of empty sites and the particles that surro
|
||||
\begin{equation}
|
||||
p_\Lambda^{(\nu)}(z)=\frac1{|\Lambda|}\log\Xi_\Lambda^{(\nu)}=\frac1{|\Lambda|}\log\mathbf z_\nu(\Lambda)+
|
||||
\frac1{|\Lambda|}
|
||||
\sum_{\underline\gamma\subset\mathfrak C_\nu(\Lambda)}\Phi^T(\underline\gamma)\prod_{\gamma\in\underline\gamma}\zeta_\nu^{(z)}(\gamma).
|
||||
\sum_{\underline\gamma\sqsubset\mathfrak C_\nu(\Lambda)}\Phi^T(\underline\gamma)\prod_{\gamma\in\underline\gamma}\zeta_\nu^{(z)}(\gamma).
|
||||
\end{equation}
|
||||
Furthermore,
|
||||
\begin{equation}
|
||||
\log\mathbf z_\nu(\Lambda)=\rho_m|\Lambda|\log z
|
||||
.
|
||||
\end{equation}
|
||||
Now, by~\-(\ref{zeta}), $\zeta_\nu^{(z)}(\gamma)$ is a rational function of $y$, whose denominator tends to 1 as $y\to0$. It is, therefore, an analytic function of $y$ for small $y$. In addition, $p_\Lambda^{(\nu)}(z)$ converges in the $\Lambda\to\Lambda_\infty$ limit uniformly in $y$, indeed, splitting into bulk and boundary terms as in~\-(\ref{bulk_boundary}), we find that the bulk term $\frac1{|\Lambda|}\mathfrak B_\nu^{(|\Lambda|)}(\Lambda_\infty)$ is independent of $\Lambda$, and that the boundary term $\frac1{|\Lambda|}\mathfrak b_\nu^{(\Lambda)}(\Lambda_\infty)$ vanishes in the infinite-volume limit~\-(\ref{boundary}). Therefore,
|
||||
Now, by~\-(\ref{zeta}), $\zeta_\nu^{(z)}(\gamma)$ is a rational function of $y$, and, by~\-(\ref{cvcd}), it is bounded by 1 for small $y$, uniformly in $\gamma$. It is, therefore, an analytic function of $y$ for small $y$. In addition, $p_\Lambda^{(\nu)}(z)$ converges in the $\Lambda\to\Lambda_\infty$ limit uniformly in $y$, indeed, splitting into bulk and boundary terms as in~\-(\ref{bulk_boundary}), we find that the bulk term $\frac1{|\Lambda|}\mathfrak B_\nu^{(|\Lambda|)}(\Lambda_\infty)$ is independent of $\Lambda$, and that the boundary term $\frac1{|\Lambda|}\mathfrak b_\nu^{(\Lambda)}(\Lambda_\infty)$ vanishes in the infinite-volume limit~\-(\ref{boundary}). Therefore,
|
||||
\begin{equation}
|
||||
p(z)=
|
||||
\rho_m\log z
|
||||
@ -1192,40 +1206,40 @@ As was explained above, GFcs consist of empty sites and the particles that surro
|
||||
|
||||
\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)}
|
||||
\sum_{\underline\gamma\sqsubset\mathfrak C_\nu(\Lambda)}
|
||||
\frac{\partial^{\mathfrak n}}{\partial\log\underline z(\mathfrak x_1)\cdots\partial\log\underline z(\mathfrak x_{\mathfrak n})}
|
||||
\Phi^T(\underline\gamma)\prod_{\gamma\in\underline\gamma}
|
||||
\zeta_\nu^{(\underline z)}(\gamma)
|
||||
\end{equation}
|
||||
converges to
|
||||
\begin{equation}
|
||||
\sum_{\underline\gamma\subset\mathfrak C_\nu(\Lambda_\infty)}
|
||||
\sum_{\underline\gamma\sqsubset\mathfrak C_\nu(\Lambda_\infty)}
|
||||
\frac{\partial^{\mathfrak n}}{\partial\log\underline z(\mathfrak x_1)\cdots\partial\log\underline z(\mathfrak x_{\mathfrak n})}
|
||||
\Phi^T(\underline\gamma)\prod_{\gamma\in\underline\gamma}
|
||||
\zeta_\nu^{(\underline z)}(\gamma)
|
||||
\end{equation}
|
||||
uniformly in $y$, or, in other words, that their difference
|
||||
\begin{equation}
|
||||
\sum_{\gamma'\in\mathfrak C_\nu(\Lambda_\infty)\setminus\mathfrak C_\nu(\Lambda)}
|
||||
\sum_{\underline\gamma\subset\mathfrak C_\nu(\Lambda_\infty)}
|
||||
\sum_{m=1}^\infty\sum_{\gamma'\in\mathfrak C_\nu(\Lambda_\infty)\setminus\mathfrak C_\nu(\Lambda)}
|
||||
\sum_{\underline\gamma\sqsubset\mathfrak C_\nu(\Lambda_\infty)\setminus\{\gamma'\}}
|
||||
\frac{\partial^{\mathfrak n}}{\partial\log\underline z(\mathfrak x_1)\cdots\partial\log\underline z(\mathfrak x_{\mathfrak n})}
|
||||
\Phi^T(\gamma'\cup\underline\gamma)
|
||||
\zeta_\nu^{(\underline z)}(\gamma')
|
||||
\Phi^T(\{\gamma'\}^m\sqcup\underline\gamma)
|
||||
(\zeta_\nu^{(\underline z)}(\gamma'))^m
|
||||
\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 are not contained inside $\Lambda$. 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 $\{\gamma'\}\sqcup\underline\gamma$ 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)}
|
||||
\sum_{\underline\gamma\sqsubset\mathfrak C_\nu(\Lambda_\infty)}
|
||||
\left|
|
||||
\Phi^T(\gamma'\cup\underline\gamma)
|
||||
\Phi^T(\{\gamma'\}\sqcup\underline\gamma)
|
||||
\zeta_\nu^{(\underline z)}(\gamma')
|
||||
\prod_{\displaystyle\mathop{\scriptstyle\gamma\in\underline\gamma}_{\mathrm{vol}(\gamma'\cup\underline\gamma)\geqslant\mathrm{dist}(\mathfrak x_1,\Lambda_\infty\setminus\Lambda)}}
|
||||
\prod_{\displaystyle\mathop{\scriptstyle\gamma\in\underline\gamma}_{\mathrm{vol}(\{\gamma'\}\sqcup\underline\gamma)\geqslant\mathrm{dist}(\mathfrak x_1,\Lambda_\infty\setminus\Lambda)}}
|
||||
\zeta_\nu^{(\underline z)}(\gamma)
|
||||
\right|
|
||||
\end{equation}
|
||||
in which $\mathrm{vol}(\gamma'\cup\underline\gamma):=|\Gamma_{\gamma'}|+\sum_{\gamma\in\underline\gamma}|\Gamma_\gamma|$. By proceeding as in~\-(\ref{boundary}), we bound this contribution by
|
||||
in which $\mathrm{vol}(\{\gamma'\}\sqcup\underline\gamma):=|\Gamma_{\gamma'}|+\sum_{\gamma\in\underline\gamma}|\Gamma_\gamma|$. By proceeding as in~\-(\ref{boundary}), we bound this contribution by
|
||||
\begin{equation}
|
||||
\cst c{cst:corr}
|
||||
\alpha^{\xi\mathrm{dist}(\mathfrak x_1,\Lambda_\infty\setminus\Lambda)}
|
||||
@ -1233,12 +1247,12 @@ As was explained above, GFcs consist of empty sites and the particles that surro
|
||||
for some constant $\cst c{cst:corr}>0$, so it vanishes as $\Lambda\to\Lambda_\infty$. Furthermore, by the same argument, we show that the sum over $\underline\gamma$ in
|
||||
\begin{equation}
|
||||
\frac{\partial^{\mathfrak n}}{\partial\log\underline z(\mathfrak x_1)\cdots\partial\log\underline z(\mathfrak x_{\mathfrak n})}
|
||||
\sum_{\underline\gamma\subset\mathfrak C_\nu(\Lambda_\infty)}\Phi^T(\underline\gamma)\prod_{\gamma\in\underline\gamma}\zeta_\nu^{(z)}(\gamma)
|
||||
\sum_{\underline\gamma\sqsubset\mathfrak C_\nu(\Lambda_\infty)}\Phi^T(\underline\gamma)\prod_{\gamma\in\underline\gamma}\zeta_\nu^{(z)}(\gamma)
|
||||
\end{equation}
|
||||
is absolutely convergent, so
|
||||
\begin{equation}
|
||||
\frac{\partial^{\mathfrak n}}{\partial\log\underline z(\mathfrak x_1)\cdots\partial\log\underline z(\mathfrak x_{\mathfrak n})}
|
||||
\sum_{\underline\gamma\subset\mathfrak C_\nu(\Lambda)}\Phi^T(\underline\gamma)\prod_{\gamma\in\underline\gamma}\zeta_\nu^{(z)}(\gamma)
|
||||
\sum_{\underline\gamma\sqsubset\mathfrak C_\nu(\Lambda)}\Phi^T(\underline\gamma)\prod_{\gamma\in\underline\gamma}\zeta_\nu^{(z)}(\gamma)
|
||||
\end{equation}
|
||||
is analytic in $y$ for small $|y|$. Finally,
|
||||
\begin{equation}
|
||||
@ -1268,6 +1282,7 @@ We are grateful to Giovanni Gallavotti and Roman Koteck\'y for enlightening disc
|
||||
|
||||
\begin{thebibliography}{WWW99}
|
||||
\small
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||||
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