|
|
|
|
@ -12,7 +12,6 @@
|
|
|
|
|
\usepackage{hyperref}
|
|
|
|
|
\usepackage{bm}
|
|
|
|
|
\usepackage{authblk}
|
|
|
|
|
%\usepackage{ulem}
|
|
|
|
|
|
|
|
|
|
\definecolor{qidong}{HTML}{000080}
|
|
|
|
|
\long\def\qidong#1{{\color{qidong}#1}}
|
|
|
|
|
@ -81,11 +80,11 @@
|
|
|
|
|
\maketitle
|
|
|
|
|
|
|
|
|
|
\abstract{
|
|
|
|
|
In this paper, we prove the existence of a crystallization transition for a family of hard-core particle models on periodic graphs in arbitrary dimensions.
|
|
|
|
|
In this paper, we prove the existence of a crystallization transition for a family of hard-core particle models on periodic graphs in dimension $d\ge2$.
|
|
|
|
|
We establish a criterion under which crystallization occurs at sufficiently high densities.
|
|
|
|
|
The criterion is more general than that in [Jauslin, Lebowitz, Comm. Math. Phys. {\bf 364}:2, 2018], as it allows models in which particles do not tile the space in the close-packing configurations, such as discrete hard-disk models.
|
|
|
|
|
To prove crystallization, we prove that the pressure is analytic in the inverse of the fugacity for large enough complex fugacities, using Pirogov-Sinai theory.
|
|
|
|
|
One of the main tools used for this result is the definition of a local density, based on a discrete generalization of Voronoi cells.
|
|
|
|
|
One of the main new tools used for this result is the definition of a local density, based on a discrete generalization of Voronoi cells.
|
|
|
|
|
We illustrate the criterion by proving that it applies to two examples: staircase models and the radius 2.5 hard-disk model on the square lattice.
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
@ -97,7 +96,7 @@ Crystallization is a very well-known phenomenon.
|
|
|
|
|
From a physical point of view, it has been studied extensively, and much of what has been observed can be understood using a combination of effective models and numerical simulations \cite{WJ57,AW57,PM86,St88,Mc10,BK11,Isobe2015}.
|
|
|
|
|
However, from a mathematical point of view, there is still much to do.
|
|
|
|
|
Even proving the existence of crystalline phases in (somewhat) realistic particle models can pose significant challenges \cite{Gaunt1965,Do68,HP74,Ba82,EB05,Hales2005,Theil2006,Mainini2014,Jauslin2018,Mazel2018,Mazel2019,Mazel2020,Mazel2021}.
|
|
|
|
|
One model which has received a considerable amount of attention for being both simple to state and somewhat realistic (as well as having applications to other fields such as coding theory \cite{Cohn2017,Viazovska2017}) is the hard-sphere model \cite{Adams1972,Berryman1983,Isobe2015}, in which particles are represented as identical spheres that interact via the constraint that no two spheres can overlap.
|
|
|
|
|
One model which has received a considerable amount of attention for being both simple to state and somewhat realistic (as well as having applications to other fields such as coding theory \cite{Cohn2017,Viazovska2017}) is the hard-sphere model \cite{Adams1972,Berryman1983,Isobe2015}, in which particles are represented as identical spheres that interact via the constraint that no two can overlap.
|
|
|
|
|
However, proving thermodynamic properties of this model has been a huge challenge: even proving that there is crystallization at zero temperature remained open for centuries, from the formulation of the problem by Kepler to the computer-assisted proof by Hales \cite{Hales2005}.
|
|
|
|
|
To this day, sphere packing problems are still the subject of active research, with recent breakthroughs in eight and twenty-four dimensions \cite{Cohn2017,Viazovska2017}.
|
|
|
|
|
At positive temperature, the problem of crystallization in the hard-sphere model is still wide open.
|
|
|
|
|
@ -114,70 +113,71 @@ We will call these Hard-Core Lattice Particle (HCLP) models.
|
|
|
|
|
This paper builds upon \cite{Jauslin2017,Jauslin2018}, in which a class of HCLP models was considered, which satisfy a \emph{non-sliding} condition as well as a \emph{tiling} condition.
|
|
|
|
|
The non-sliding condition roughly means that, at high densities, neighboring particles are locked into place, and cannot \emph{slide} with respect to one another.
|
|
|
|
|
The tiling condition states that it is possible to tile the lattice with the supports of the particles.
|
|
|
|
|
(Technically, the condition in \cite{Jauslin2017,Jauslin2018} is stronger than just non-sliding and tiling, but those are the important aspects of the condition.)
|
|
|
|
|
(Technically, the condition in \cite{Jauslin2017,Jauslin2018} is stronger than just non-sliding and tiling, but those are its important aspects.)
|
|
|
|
|
Whereas the former condition is necessary for crystallization, the latter is purely technical.
|
|
|
|
|
In fact, Mazel, Stuhl, and Suhov, in their extensive review of lattice regularizations of hard-disk models \cite{Mazel2018,Mazel2019,Mazel2020,Mazel2021}, have constructed an infinite class of lattice regularizations of the hard-disk model that do not slide, but that do not tile the plane because of the presence of interstitial space in between the disks.
|
|
|
|
|
In fact, Mazel, Stuhl, and Suhov, in their extensive review of lattice regularizations of hard-disk models \cite{Mazel2018,Mazel2019,Mazel2020,Mazel2021}, have constructed an infinite class of lattice regularizations of the hard-disk model for which they proved crystallization.
|
|
|
|
|
These models are non-sliding but do not tile the plane because of the presence of interstitial space in between the disks.
|
|
|
|
|
|
|
|
|
|
In the present paper, we will extend \cite{Jauslin2017,Jauslin2018} by relaxing the tiling condition and prove, using Pirogov-Sinai theory \cite{Borgs1989,Pirogov1975,Zahradnk1984}, that a much larger class of non-sliding HCLP models crystallize.
|
|
|
|
|
In particular, all the models studied by \cite{Jauslin2017,Jauslin2018} fall within our general framework.
|
|
|
|
|
In addition, we can treat models that do not tile the space, such as a model of hard disks of radius 2.5 on the square lattice; see Figure \ref{fig:octagon}.
|
|
|
|
|
We will discuss some explicit examples in Section \ref{sec:examples} of this paper (see Figures \ref{fig:staircase} and \ref{fig:octagon}), but the framework is rather general, and also applies to models in more than two dimensions.
|
|
|
|
|
We will discuss some explicit examples in Section \ref{sec:examples} of this paper (see Figures \ref{fig:staircase} and \ref{fig:octagon}), but the framework is rather general, and applies to models in two and more dimensions.
|
|
|
|
|
(In one dimension, there are no phase transitions, and thus no crystallization.)
|
|
|
|
|
In addition, we have simplified the condition in \cite{Jauslin2017,Jauslin2018}, so the current work makes that criterion for proving crystallization in a HCLP model more easily usable.
|
|
|
|
|
\bigskip
|
|
|
|
|
|
|
|
|
|
Let us be more specific on the condition under which we will prove crystallization.
|
|
|
|
|
Let us be more specific on the class of models for which we will prove crystallization.
|
|
|
|
|
In this introduction, we will not give formal definitions, which can be found in Section \ref{sec:model} below (see in particular Assumption \ref{assumption}).
|
|
|
|
|
We treat HCLP systems on periodic graphs in any dimension.
|
|
|
|
|
The crux of the condition concerns the close-packing configurations, which are configurations of particles that maximize the density.
|
|
|
|
|
The most important part of the condition is that we require the number of close-packing configurations to be finite; see Figure \ref{fig:ground_state} for an example.
|
|
|
|
|
We consider HCLP systems on periodic graphs in any dimension $d\ge2$.
|
|
|
|
|
The crux of the criterion concerns the close-packing configurations, which are configurations of particles that maximize the density.
|
|
|
|
|
The most important part of the criterion is that we require the number of close-packing configurations to be finite; see Figure \ref{fig:ground_state} for an example.
|
|
|
|
|
This excludes \emph{sliding} models: if one can slide particles around without lowering the density, then the number of close-packings will be infinite, as is the case for instance in the $2\times2$-square model studied in \cite{Hadas2022}.
|
|
|
|
|
|
|
|
|
|
The argument we will use is based on controlling {\it defects} in close-packing configurations: if the density is sufficiently high (but not maximal), then the typical configurations will look similar to the close-packing ones.
|
|
|
|
|
In \cite{Jauslin2017,Jauslin2018}, the models considered have close-packing configurations that tile the space, so defects could be defined using sites in the lattice that are not covered by particles.
|
|
|
|
|
Since we allow non-tiling models, defining defects is more involved.
|
|
|
|
|
To do so, we decompose the lattice into {\it generalized Voronoi cells} (see Definition \ref{def:voronoi} below for a formal definition) which assign each and every point in the lattice to its nearest particles.
|
|
|
|
|
To do so, we decompose the space into {\it generalized Voronoi cells} (see Definition \ref{def:voronoi} below for a formal definition) which assign each and every point in the lattice to its nearest particles.
|
|
|
|
|
Thus, whereas the particles do not tile the space, the Voronoi cells cover it (with the caveat that we define Voronoi cells in such a way that they may overlap).
|
|
|
|
|
Defects are then defined from the size of Voronoi cells: in close-packing configurations, the cells are all the same size, so when the configuration deviates from a close-packing, cells will expand.
|
|
|
|
|
To quantify this, we introduce a {\it local density} for every particle in the configuration; see Definition \ref{def:local_density}.
|
|
|
|
|
We then identify defects by finding particles whose local density is lower than in the close-packing configurations.
|
|
|
|
|
|
|
|
|
|
To make this work without too many complications, we must impose some extra restrictions on the system.
|
|
|
|
|
In addition, we impose some extra restrictions on the system.
|
|
|
|
|
For one, we require that the close-packing configurations be distinct enough, in the sense that two different close-packings cannot merge seamlessly.
|
|
|
|
|
In addition, we assume that whenever a particle does not belong to a close-packing configuration, it must impose a dip in the local density, and this dip cannot occur {\it arbitrarily far} from the particle.
|
|
|
|
|
These two are important assumptions, without which the discussion would fail dramatically.
|
|
|
|
|
In addition, we assume that whenever a particle does not belong to a close-packing configuration, it constrains the local density to dip, and this dip cannot occur {\it arbitrarily far} from the particle.
|
|
|
|
|
These two are important assumptions, without which the proof would fail dramatically.
|
|
|
|
|
In addition to these, we impose additional constraints, which make our arguments easier, but could, in principle, be relaxed in future work, without changing the method too much.
|
|
|
|
|
One of these is that we impose that different close-packing configurations are all related to each other by isometries, which ensures that the local density will be the same in different close-packings.
|
|
|
|
|
One of these is that we impose that different close-packing configurations be related to each other by isometries, which ensures that the local density will be the same in different close-packings.
|
|
|
|
|
In addition, we exclude the possibility that the local density could exceed the total density.
|
|
|
|
|
This can happen in certain models \cite{Hales2010}, and this would break a number of arguments made in our proof.
|
|
|
|
|
\bigskip
|
|
|
|
|
|
|
|
|
|
Under this condition, we prove that crystallization occurs at sufficiently high densities.
|
|
|
|
|
To do so, we follow the same philosophy as in \cite{Jauslin2018}, and prove that the model has a convergent {\it high-fugacity expansion}, that is, an analytic expansion in the inverse of the fugacity (an expansion in $e^{-\mu}$ where $\mu$ us the chemical potential).
|
|
|
|
|
To do so, we follow the same philosophy as in \cite{Jauslin2018}, and prove that the model has a convergent {\it high-fugacity expansion}, that is, an analytic expansion in the inverse of the fugacity (an expansion in $e^{\mu}$ where $\mu$ is the chemical potential).
|
|
|
|
|
The idea of a high fugacity expansion for HCLP models dates back, at least, to Gaunt and Fisher \cite{Gaunt1965} (see also \cite{Do68,HP74}), and was systematized in \cite{Jauslin2018}.
|
|
|
|
|
The present work is a continuation of \cite{Jauslin2018}, and we extend the treatment of such expansions to a much wider class of models.
|
|
|
|
|
In particular, we prove that the Lee-Yang zeros \cite{YL52,LY52} are all located inside a finite-radius disk in the complex fugacity plane.
|
|
|
|
|
Combining this with a classical Mayer expansion argument \cite{Ma37,Ur27,Ru63}, we thus prove that the Lee-Yang zeros lie in a finite annulus in the complex fugacity plane.
|
|
|
|
|
|
|
|
|
|
To prove the convergence of the high-fugacity expansion, we use Pirogov-Sinai theory \cite{Borgs1989,Pirogov1975,Zahradnk1984}, which allows us to balance the costs coming from the drops in the density caused by defects with the entropy gains the defects produce.
|
|
|
|
|
To prove the convergence of the high-fugacity expansion, we use Pirogov-Sinai theory \cite{Borgs1989,Pirogov1975,Zahradnk1984}, which allows us to balance the entropy gains the defects produce with the costs coming from the dips in the density caused by the defects.
|
|
|
|
|
\bigskip
|
|
|
|
|
|
|
|
|
|
The rest of this paper is structured as follows.
|
|
|
|
|
In Section \ref{sec:model}, we define the model more precisely, state the condition under which we will prove crystallization (see Assumption \ref{assumption}), and state our main results.
|
|
|
|
|
These are the staircase models, and the 12th nearest neighbor exclusion.
|
|
|
|
|
In Section \ref{sec:GFc}, we map the HCLP particle model to a {\it contour} model.
|
|
|
|
|
Following \cite{Jauslin2017,Jauslin2018}, we call these contours \emph{Gaunt-Fisher configurations}.
|
|
|
|
|
These formalize the notion of \emph{defect} mentioned above, which really should be understood as \emph{Gaunt-Fisher configurations}.
|
|
|
|
|
These formalize the notion of \emph{defect} mentioned above.
|
|
|
|
|
In Section \ref{sec:peierls}, we prove the crucial estimate that will allow Pirogov-Sinai theory to work for our system: the {\it Peierls condition}.
|
|
|
|
|
Roughly, that states that the cost of a Gaunt-Fisher configuration is exponentially large in its size, which will allow us to control the entropy of contours.
|
|
|
|
|
Roughly, it states that the cost of a defect is exponentially large in its size, which will allow us to control the entropy of contours.
|
|
|
|
|
In Section \ref{sec:Pirogov_Sinai}, we carry out the Pirogov-Sinai analysis.
|
|
|
|
|
Our approach is similar to that of Zahradn\'ik \cite{Zahradnk1984}, and readers unfamiliar with Pirogov-Sinai theory may want to study that reference to understand the philosophy behind the method (see also the textbook \cite{Friedli2017}).
|
|
|
|
|
Finally, in Section \ref{sec:examples}, we discuss some explicit examples of models for which we prove Assumption \ref{assumption}.
|
|
|
|
|
Our approach is similar to that of Zahradn\'ik \cite{Zahradnk1984} (see also the textbook \cite{Friedli2017}).
|
|
|
|
|
Finally, in Section \ref{sec:examples}, we discuss some explicit examples of models for which we prove Assumption \ref{assumption}: the staircase models, and the 12th nearest neighbor exclusion.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\section{Model and main result}\label{sec:model}
|
|
|
|
|
|
|
|
|
|
Let $\Lambda_{\infty}$ be a periodic graph embedded in $\R^{d}$.
|
|
|
|
|
Let $\Lambda_{\infty}$ be a periodic graph embedded in $\R^{d}$ with $d\ge 2$.
|
|
|
|
|
For example, $\Lambda_{\infty}$ could be $\mathbb Z^d$, the triangular lattice, or the honeycomb lattice (which is not, strictly speaking, a lattice, but rather a periodic graph).
|
|
|
|
|
Denote by $\mathrm{d}_{\Lambda_{\infty}}$ the (usual) graph distance on $\Lambda_{\infty}$.
|
|
|
|
|
Our interest is in Hard-Core Lattice Particle (HCLP) systems on $\Lambda_{\infty}$, which we formalize as follows.
|
|
|
|
|
@ -191,7 +191,7 @@ Formally, given any $\Lambda\subseteq\Lambda_{\infty}$, we define the set of par
|
|
|
|
|
\Omega(\Lambda):=\set{X\subseteq\Lambda\mid\omega_{x}\cap\omega_{x'}=\emptyset\text{ for all }x\ne x'\in X}.
|
|
|
|
|
\end{equation}
|
|
|
|
|
We will study this system in the grand canonical ensemble:
|
|
|
|
|
if $\Lambda$ is finite, we define the partition function at fugacity $z$ ($:= e^{\beta\mu}$, where $\mu$ is the chemical potential and $\beta$ the inverse temperature) as
|
|
|
|
|
if $\Lambda$ is finite, we define the partition function at fugacity $z$ ($:= e^{\mu}$, where $\mu$ is the chemical potential) as
|
|
|
|
|
\begin{equation}
|
|
|
|
|
\Xi_{z}(\Lambda):=\sum_{X\in\Omega(\Lambda)}z^{\abs{X}},
|
|
|
|
|
\end{equation}
|
|
|
|
|
@ -203,7 +203,7 @@ Let
|
|
|
|
|
\rho_{\max}:=\lim_{\Lambda\Uparrow\Lambda_{\infty}}\rho_{\max}(\Lambda)
|
|
|
|
|
\label{rhomax}
|
|
|
|
|
\end{equation}
|
|
|
|
|
be the maximal density and its infinite-volume limit.
|
|
|
|
|
be the maximal density and its infinite-volume limit (the limits here are taken in the sense of van Hove).
|
|
|
|
|
Finally, define the finite-volume pressure of the system as
|
|
|
|
|
\begin{equation}
|
|
|
|
|
p_{z}(\Lambda):=\frac{1}{\abs{\Lambda}}\log\Xi_{z}(\Lambda)
|
|
|
|
|
@ -218,14 +218,14 @@ Our main result is that, provided the model satisfies a \emph{non-sliding} condi
|
|
|
|
|
\subsection{Assumption on the model}
|
|
|
|
|
To specify the assumption on the model, we will need a few definitions.
|
|
|
|
|
|
|
|
|
|
First, we will assume that $\Lambda_\infty$ is such that the boundary of any connected set is connected in a coarse-grained sense, which we will now define.
|
|
|
|
|
First, we will assume that $\Lambda_\infty$ is such that the boundary of any simply connected set is connected in a coarse-grained sense, which we will now define.
|
|
|
|
|
|
|
|
|
|
\begin{definition}\label{def:rconnected}
|
|
|
|
|
Two points $x,y\in\Lambda_\infty$ are \emph{neighbors} if and only if $d_{\Lambda_\infty}(x,y)\le1$, which gives us a natural notion of connectedness in $\Lambda_\infty$.
|
|
|
|
|
In addition, a set $S\subset\Lambda_\infty$ is said to be $r$-connected if $\forall x,y\in S$, there exists a path $x\equiv x_0,x_1,\cdots,x_N\equiv y\in S$ such that $d_{\Lambda_\infty}(x_i,x_j)\le r$.
|
|
|
|
|
In addition, a set $S\subset\Lambda_\infty$ is said to be $r$-connected if $\forall x,y\in S$, there exists a path $x\equiv x_0,x_1,\cdots,x_N\equiv y$ in $S$ such that $d_{\Lambda_\infty}(x_i,x_j)\le r$.
|
|
|
|
|
\end{definition}
|
|
|
|
|
|
|
|
|
|
We will assume that $\Lambda_\infty$ is such that there exists $\mathcal R_0\in\mathbb N$ such that the interior and exterior boundaries (see Definition \ref{def:boundaries}) of any simply connected set are $\mathcal R_0$-connected (see Assumption \ref{assumption} below).
|
|
|
|
|
We will assume that $\Lambda_\infty$ is such that there exists $\mathcal R_0\in\mathbb N$ such that the interior and exterior boundaries (see Definition \ref{def:boundaries}) of any simply connected set are $\mathcal R_0$-connected (see Item \ref{asm:lattice} of Assumption \ref{assumption} below).
|
|
|
|
|
This is a very weak assumption that was shown to hold for a very large class of graphs \cite{Timar2013} including $\mathbb Z^d$ (for which $\mathcal R_0=d$), the triangular lattice (for which $\mathcal R_0=1$), and the honeycomb lattice (for which $\mathcal R_0=3$).
|
|
|
|
|
See Figure \ref{fig:boundary_rconnected} for an example.
|
|
|
|
|
|
|
|
|
|
@ -245,7 +245,7 @@ See Figure \ref{fig:boundary_rconnected} for an example.
|
|
|
|
|
\begin{figure}
|
|
|
|
|
\hfil\includegraphics[width=4cm]{boundary_rconnected.pdf}
|
|
|
|
|
|
|
|
|
|
\caption{An example of a subset of $\mathbb Z^2$ and its boundary: the boundary is 2-connected.}
|
|
|
|
|
\caption{An example of a subset of $\mathbb Z^2$ and its exterior boundary: the boundary is 2-connected.}
|
|
|
|
|
\label{fig:boundary_rconnected}
|
|
|
|
|
\end{figure}
|
|
|
|
|
|
|
|
|
|
@ -254,7 +254,7 @@ Now, let us define the notion of ground states, which could also be called \emph
|
|
|
|
|
\begin{definition}[ground state]
|
|
|
|
|
A \emph{ground state} in $\Lambda$ is a configuration $X\in\Omega(\Lambda)$ that maximizes the density: $\abs{X}=\abs{\Lambda}\rho_{\max}(\Lambda)$.
|
|
|
|
|
Taking the limit $\Lambda\Uparrow\Lambda_{\infty}$ in the sense of van Hove, the ground states tend to limiting configurations in $\Omega(\Lambda_{\infty})$.
|
|
|
|
|
An (infinite-volume) ground state is denoted by $\mathcal L^\#$ where $\#$ takes values in $\mathcal G$.
|
|
|
|
|
An (infinite-volume) ground state is denoted by $\mathcal L^\#$ where $\#$ takes values in a set which we denote by $\mathcal G$.
|
|
|
|
|
In other words, $\mathcal G$ is a set of indices, each of which specifies a ground state.
|
|
|
|
|
\end{definition}
|
|
|
|
|
|
|
|
|
|
@ -262,14 +262,14 @@ See Figure \ref{fig:ground_state} for an example.
|
|
|
|
|
|
|
|
|
|
\begin{figure}
|
|
|
|
|
\hfil\includegraphics[width=4.8cm]{3staircase_ground_state.pdf}
|
|
|
|
|
\caption{A section of one of the ground states for the 3-staircase model; see Section \ref{subsec:n_staircases}.}
|
|
|
|
|
\caption{A portion of one of the ground states for the 3-staircase model; see Section \ref{subsec:n_staircases}.}
|
|
|
|
|
\label{fig:ground_state}
|
|
|
|
|
\end{figure}
|
|
|
|
|
|
|
|
|
|
We will assume that $\mathcal{G}$ is finite and that the ground states are periodic.
|
|
|
|
|
Moreover, we will assume that the different ground states are related to each other by \emph{species-preserving isometries}, which are invertible transformations of $\R^{d}$ that preserve the shapes of particles:
|
|
|
|
|
We will assume that $\mathcal{G}$ is finite (see Item \ref{asm:finitely_many_ground_states} of Assumption \ref{assumption}).
|
|
|
|
|
Moreover, we will assume that the different ground states are related to each other by \emph{species-preserving isometries} (see Item \ref{asm:isometry} of Assumption \ref{assumption}), which are invertible transformations of $\R^{d}$ that preserve the shapes of particles:
|
|
|
|
|
|
|
|
|
|
\begin{definition}[species-preserving isometry]
|
|
|
|
|
\begin{definition}[species-preserving isometry]\label{def:isometry}
|
|
|
|
|
A species-preserving isometry is a Euclidean transformation $\psi$ satisfying the following properties:
|
|
|
|
|
\begin{enumerate}
|
|
|
|
|
\item the restriction $\restr{\psi}{\Lambda_{\infty}}$ induces a graph automorphism of $\Lambda_{\infty}$ (in particular, it is an isometry with respect to the graph distance $\mathrm{d}_{\Lambda_{\infty}}$);
|
|
|
|
|
@ -306,7 +306,7 @@ Given $X\in\Omega(\Lambda_{\infty})$, define the local density at $x\in X$ as
|
|
|
|
|
\end{equation}
|
|
|
|
|
Accordingly, the maximum local density of the model is
|
|
|
|
|
\begin{equation}
|
|
|
|
|
\rho_{\max}^{\loc}:=\sup_{\substack{X\in\Omega(\Lambda_{\infty})\\X\ni\vec{0}}}\rho_{X}(\vec{0}).
|
|
|
|
|
\rho_{\max}^{\loc}:=\sup_{\substack{X\in\Omega(\Lambda_{\infty})\\x\in X}}\rho_{X}(x).
|
|
|
|
|
\end{equation}
|
|
|
|
|
\end{definition}
|
|
|
|
|
|
|
|
|
|
@ -316,7 +316,7 @@ Examples of the computation of $\rho_{\mathrm{max}}^{\mathrm{loc}}$ is provided
|
|
|
|
|
Notice that $\rho_{\max}^{\loc}=\rho_{\max}$ in the case of a tiling model, in which all sites are covered in the ground states.
|
|
|
|
|
For general models, however, this is not always the case.
|
|
|
|
|
For instance, in the hard-sphere model, it is possible to have localized configurations in which the local density (defined as the inverse of the volume of the standard Voronoi cell in the continuum) exceeds the close-packing density \cite{Hales2010}.
|
|
|
|
|
In this paper, we will only consider models for which the equality $\rho_{\max}^{\loc}=\rho_{\max}$ holds, but without requiring the tiling property.
|
|
|
|
|
In this paper, we will only consider models for which the equality $\rho_{\max}^{\loc}=\rho_{\max}$ holds (see Item \ref{asm:density_max_local} of Assumption \ref{assumption}), but without requiring the tiling property.
|
|
|
|
|
\end{remark}
|
|
|
|
|
|
|
|
|
|
Next, we introduce the notion of $\#$-correctness, which is inspired by the construction of the (graph) Voronoi dual \cite{Honiden2010} and will later be used to identify defects in a configuration.
|
|
|
|
|
@ -346,35 +346,35 @@ Given $\mathcal R_2\in\mathbb N$ (we use the subscript ${}_{2}$ for reasons that
|
|
|
|
|
A particle $x\in X$ is
|
|
|
|
|
is said to be $(\#,\mathcal{R}_2)$-correct if
|
|
|
|
|
\begin{equation}
|
|
|
|
|
\textrm{for all } y\in\mathcal N_{X}^{(\mathcal R)}(x),\quad y\mathrm{\ is\ }\#\mathrm{-correct}.
|
|
|
|
|
\textrm{for all } y\in\mathcal N_{X}^{(\mathcal R_2)}(x),\quad y\mathrm{\ is\ }\#\mathrm{-correct}.
|
|
|
|
|
\end{equation}
|
|
|
|
|
In addition, $x\in X$ is said to be $\mathcal{R}_2$-incorrect if $x$ is not $(\#,\mathcal{R}_2)$-correct for any $\#\in\mathcal G$.
|
|
|
|
|
Finally, given a ground state $\#$, let $\mathcal{C}^{(\mathcal{R}_2)}_{\#}(X)$ denote the set of $(\#,\mathcal{R}_2)$-correct particles in $X$, and $\mathcal{I}^{(\mathcal{R}_2)}(X):=X\setminus\bigcup_{\#\in\mathcal{G}}\mathcal{C}^{(\mathcal{R}_2)}_{\#}(X)$ the set of $\mathcal{R}_2$-incorrect particles.
|
|
|
|
|
\end{definition}
|
|
|
|
|
|
|
|
|
|
We can now state the conditions we need to require of the models we consider here.
|
|
|
|
|
We can now state the criterion under which we will prove crystallization.
|
|
|
|
|
|
|
|
|
|
\begin{assumption}\label{assumption}
|
|
|
|
|
We require that the model satisfy the following properties:
|
|
|
|
|
\begin{enumerate}
|
|
|
|
|
\item $\Lambda_\infty$ is a periodic graph embedded in $\mathbb R^d$ with finite maximal coordination number (the number of neighbors is bounded), and is such that there exists $\mathcal R_0\in\mathbb N$ such that the interior and exterior boundaries (see Definition \ref{def:boundaries}) of any simply connected set are $\mathcal R_0$-connected; see Definition \ref{def:rconnected}.\label{asm:lattice}
|
|
|
|
|
\item $d\ge 2$, and $\Lambda_\infty$ is a periodic graph embedded in $\mathbb R^d$ with finite maximal coordination number (the number of neighbors is bounded), and is such that there exists $\mathcal R_0\in\mathbb N$ such that the interior and exterior boundaries (see Definition \ref{def:boundaries}) of any simply connected set are $\mathcal R_0$-connected; see Definition \ref{def:rconnected}.\label{asm:lattice}
|
|
|
|
|
|
|
|
|
|
\item There exist only finitely many ground states: $\mathcal{G}$ is finite and nonempty. \label{asm:finitely_many_ground_states}
|
|
|
|
|
|
|
|
|
|
\item
|
|
|
|
|
The ground states are related by species-preserving isometries: given $\#,\#'\in\mathcal{G}$, there exists a species-preserving isometry $\psi$ such that $\psi(\mathcal{L}^{\#})=\mathcal{L}^{\#'}$.\label{asm:isometry}
|
|
|
|
|
The ground states are related by species-preserving isometries (see Definition \ref{def:isometry}): given $\#,\#'\in\mathcal{G}$, there exists a species-preserving isometry $\psi$ such that $\psi(\mathcal{L}^{\#})=\mathcal{L}^{\#'}$.\label{asm:isometry}
|
|
|
|
|
|
|
|
|
|
\item The maximum density is equal to the maximum local density: $\rho_{\mathrm{max}}^{\mathrm{loc}}=\rho_{\mathrm{max}}$. \label{asm:density_max_local}
|
|
|
|
|
\item The maximum density is equal to the maximum local density: $\rho_{\mathrm{max}}^{\mathrm{loc}}=\rho_{\mathrm{max}}$ (see Definition \ref{def:local_density} and \eqref{rhomax}). \label{asm:density_max_local}
|
|
|
|
|
|
|
|
|
|
\item Different ground states cannot merge seamlessly: for any $X\in\Omega(\Lambda_{\infty})$, if $x\in X$ is $\#$-correct and $x'\in\mathcal{N}_{X}(x)$ is $\#'$-correct, then $\#=\#'$. \label{asm:imperfect_transition}
|
|
|
|
|
\item Different ground states cannot merge seamlessly: for any $X\in\Omega(\Lambda_{\infty})$, if $x\in X$ is $\#$-correct and $x'\in\mathcal{N}_{X}(x)$ is $\#'$-correct, then $\#=\#'$ (see Definition \ref{def:correct}). \label{asm:imperfect_transition}
|
|
|
|
|
|
|
|
|
|
\item A particle that is incorrect leads to a localized dip in the local density: there exist $\mathcal R_1,\mathcal S_1\in\mathbb N$ and $\epsilon>0$ such that, for all $X\in\Omega(\Lambda)$ and $x\in X$, if $x$ is $\mathcal R_1$-incorrect, then there exists $y\in X$ such that $d_{\Lambda_{\infty}}(x,y)\le\mathcal S_1$ and $\rho_{X}^{-1}(y)\ge\rho_{\mathrm{max}}^{-1}+\epsilon$.\label{asm:density_local_density}
|
|
|
|
|
\item A particle that is incorrect leads to a localized dip in the local density: there exist $\mathcal R_1,\mathcal S_1\in\mathbb N$ and $\epsilon>0$ such that, for all $X\in\Omega(\Lambda)$ and $x\in X$, if $x$ is $\mathcal R_1$-incorrect (see Definition \ref{def:R_correct}), then there exists $y\in X$ such that $d_{\Lambda_{\infty}}(x,y)\le\mathcal S_1$ and $\rho_{X}^{-1}(y)\ge\rho_{\mathrm{max}}^{-1}+\epsilon$.\label{asm:density_local_density}
|
|
|
|
|
\end{enumerate}
|
|
|
|
|
\end{assumption}
|
|
|
|
|
|
|
|
|
|
\begin{remark}
|
|
|
|
|
In the simplest cases, Item \ref{asm:density_local_density} holds for $\mathcal R_1=\mathcal S_1=0$.
|
|
|
|
|
These are cases in which the local density is maximal if and only if a particle is $(\#,0)$-correct, which includes all of the tiling examples discussed in \cite{Jauslin2018}, as well as the staircase models discussed in \S\ref{subsec:n_staircases}.
|
|
|
|
|
These are cases in which the local density is maximal if and only if a particle is $(\#,0)$-correct (note that $(\#,0)$-correct is not the same as $\#$-correct, but this is not an important distinction), which includes all of the tiling examples discussed in \cite{Jauslin2018}, as well as the staircase models discussed in Section \ref{subsec:n_staircases}.
|
|
|
|
|
For these models, it is relatively straightforward to verify Item \ref{asm:density_local_density}.
|
|
|
|
|
However, there also may be situations in which the local density may only dip at a finite, but large distance $\mathcal S_1$, in which case proving the assumption may be more difficult.
|
|
|
|
|
To deal with such cases, we prove the following lemma, which provides an equivalent assumption that may be easier to verify.
|
|
|
|
|
@ -391,19 +391,20 @@ We require that the model satisfy the following properties:
|
|
|
|
|
(that is, the set of configurations that have a constant local density which is maximal is equal to the set of ground states).
|
|
|
|
|
\end{lemma}
|
|
|
|
|
|
|
|
|
|
This lemma is proved in \S\ref{sec:equiv_assumption}.
|
|
|
|
|
This lemma is proved in Section \ref{sec:equiv_assumption}.
|
|
|
|
|
It allows one to use our main result in situations where it is easier to prove that $\set{\mathcal L^\#\mid\#\in\mathcal G}=g_m$ than to prove Item \ref{asm:density_local_density} of Assumption \ref{assumption}.
|
|
|
|
|
\bigskip
|
|
|
|
|
|
|
|
|
|
We briefly describe how Assumption \ref{assumption} will enter our analysis.
|
|
|
|
|
\begin{itemize}
|
|
|
|
|
\item
|
|
|
|
|
Item \ref{asm:finitely_many_ground_states} allows us to control the number of ways a ground state can be perturbed, thus controlling the entropy of defects: if the number of ground states were infinite, there is the risk that there are too many ways to create defects, which could lead to defect-formation being likely; see Lemma \ref{lem:clusterexpansion}.
|
|
|
|
|
Item \ref{asm:finitely_many_ground_states} allows us to control the number of ways a ground state can be perturbed, thus controlling the entropy of defects: if the number of ground states were infinite, there is a risk that there are too many ways to create defects, which could lead to defect-formation being likely; see Lemma \ref{lem:clusterexpansion}.
|
|
|
|
|
|
|
|
|
|
\item
|
|
|
|
|
Item \ref{asm:isometry} allows us to control the ratio of partition functions of defects of different ground states, which is a standard step in Pirogov-Sinai theory; see Proposition \ref{prop:central_estimates}, specifically \eqref{bulkterm}.
|
|
|
|
|
|
|
|
|
|
\item
|
|
|
|
|
Item \ref{asm:density_max_local} excludes situations in which the local density can be smaller than $\rho_{\mathrm{max}}$, which would break the proof of Proposition \ref{prop:central_estimates}; see Lemma \ref{lemma:rholoc_cst}.
|
|
|
|
|
Item \ref{asm:density_max_local} excludes situations in which the local density can be larger than $\rho_{\mathrm{max}}$, which would break the proof of Proposition \ref{prop:central_estimates}; see also Lemma \ref{lemma:rholoc_cst}.
|
|
|
|
|
|
|
|
|
|
\item
|
|
|
|
|
Items \ref{asm:lattice} and \ref{asm:imperfect_transition} allow us to map the particle model to a \emph{contour} model in a one-to-one way: using this assumption, we can fully specify a defect-free region by only looking at the particles neighboring this region; see Lemma \ref{lemma:unique_Rcorrect} and Proposition \ref{prop:GFc}.
|
|
|
|
|
@ -422,16 +423,19 @@ Assumption \ref{assumption}, however, requires more of the model than simply tha
|
|
|
|
|
In principle, this suggests that the assumption can be relaxed.
|
|
|
|
|
For instance, the requirement in Item \ref{asm:lattice} that the graph $\Lambda_{\infty}$ is periodic is presumably not necessary, though this would require some changes in the argument, and it is not clear that the rest of the assumptions could be satisfied for aperiodic graphs.
|
|
|
|
|
More interestingly, there exist non-sliding models that violate Item \ref{asm:density_max_local}, but for which one nevertheless expects crystallization to occur at high densities.
|
|
|
|
|
In particular, the hard-sphere model in $\R^{3}$ is known to allow for local configurations to exceed the close-packing density but at the expense of the overall density \cite{Hales2010} and so $\rho_{\mathrm{max}}<\rho_{\mathrm{max}}^{\mathrm{loc}}$.
|
|
|
|
|
Thus, it may also be possible to weaken Item \ref{asm:density_local_density}, although our treatment of the Peierls condition in Lemma \ref{lem:Peierls} would need to be adapted accordingly.
|
|
|
|
|
Thus, it may also be possible to dispense of Item \ref{asm:density_max_local}, although our treatment of the Peierls condition in Lemma \ref{lem:Peierls} would need to be adapted accordingly.
|
|
|
|
|
|
|
|
|
|
Finally, note that the requirement $d\ge2$ is technically redundant: if $d=1$, then simply connected sets will certainly not have $\mathcal R_0$-connected boundaries.
|
|
|
|
|
However, it is worth emphasizing that our method will not work for $d=1$, as the Pirogov-Sinai analysis would fall through.
|
|
|
|
|
(Rightfully so, there are no crystalline phases in one dimension.)
|
|
|
|
|
\end{remark}
|
|
|
|
|
|
|
|
|
|
\begin{remark}[comparison of Assumption \ref{assumption} with \cite{Jauslin2018}]
|
|
|
|
|
The condition in Assumption \ref{assumption}, under which we prove crystallization, is more general than the condition in \cite{Jauslin2018}.
|
|
|
|
|
Obviously, we do not require the particle to tile $\Lambda_\infty$, but the generalization goes a little further.
|
|
|
|
|
For one thing, we do not require that the ground states be periodic, as this can be proved from Items \ref{asm:lattice} and \ref{asm:finitely_many_ground_states}; see Lemma \ref{lemma:periodic_ground_state} below.
|
|
|
|
|
For one, we do not require that the ground states be periodic, as this can be proved from Items \ref{asm:lattice} and \ref{asm:finitely_many_ground_states}; see Lemma \ref{lemma:periodic_ground_state} below.
|
|
|
|
|
In addition, the analog of Item \ref{asm:density_local_density} in \cite{Jauslin2018} requires the drop in the density to occur {\it right next} to the incorrect particle.
|
|
|
|
|
Here, we are more general and allow for the drop in density to occur farther away, which allows us to treat models such as the hard-disk model in Section \ref{sec:examples}.
|
|
|
|
|
Here, we are more general and allow for the drop in density to occur farther away, which allows us to treat models such as the one of hard disks of radius 2.5 in Section \ref{sec:disk}.
|
|
|
|
|
\end{remark}
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
@ -492,12 +496,11 @@ where $\indicator{x}$ denotes the characteristic function that $x\in X$.
|
|
|
|
|
|
|
|
|
|
\begin{remark}
|
|
|
|
|
We have computed a value for $z_0$, which is deferred to the appendix: see \eqref{estimate_z}.
|
|
|
|
|
This bound is quite far from optimal, but it is instructive to see how the parameters appearing in Assumption \ref{assumption} affect the radius of convergence.
|
|
|
|
|
This bound is quite far from optimal, but it is instructive to see how the parameters appearing in Assumption \ref{assumption} affect our estimate on the radius of convergence.
|
|
|
|
|
\end{remark}
|
|
|
|
|
|
|
|
|
|
As a consequence of Theorem \ref{thm:crystallization}, there are at least as many extremal Gibbs distributions as there are close-packing configurations ($\abs{\mathcal{G}}$), in all of which the translational symmetry of $\Lambda_{\infty}$ is broken.
|
|
|
|
|
Also, as we have noted, an intermediate result in the proof of Theorem \ref{thm:crystallization} is the construction of an expansion of $p(z)-\rho_{\max}\log z$ in powers of $z^{-1}$, which is shown to be absolutely convergent when $\abs{z}$ is sufficiently large.
|
|
|
|
|
We summarize the latter as a standalone theorem.
|
|
|
|
|
Also, as we have noted, an intermediate result in the proof of Theorem \ref{thm:crystallization} is the construction of an expansion of $p(z)-\rho_{\max}\log z$ in powers of $z^{-1}$, which is shown to be absolutely convergent when $\abs{z}$ is sufficiently large:
|
|
|
|
|
|
|
|
|
|
\begin{theorem}[analyticity] \label{thm:analyticity}
|
|
|
|
|
Under Assumption \ref{assumption}, $p(z)-\rho_{\max}\log z$ is analytic in $z^{-1}$ on $\set{z\in\C\mid\abs{z}>z_{0}}$.
|
|
|
|
|
@ -505,7 +508,7 @@ Under Assumption \ref{assumption}, $p(z)-\rho_{\max}\log z$ is analytic in $z^{-
|
|
|
|
|
|
|
|
|
|
Let us give a brief outline of the proof of Theorem \ref{thm:analyticity}, from which Theorem \ref{thm:crystallization} is proved.
|
|
|
|
|
The first step is to map the model to a contour model.
|
|
|
|
|
Here, a \emph{contour} will be called a \emph{Gaunt-Fisher configuration}, in honor of \cite{Gaunt1965}, and abbreviated GFc.
|
|
|
|
|
Here, a \emph{contour} will be called a \emph{Gaunt-Fisher configuration}, in honor of \cite{Gaunt1965}, and abbreviated as GFc.
|
|
|
|
|
The GFc's are constructed from the incorrect particles, and are chosen to be thick enough so that, in the effective GFc model, pairs of GFc's only interact via a hard-core repulsion.
|
|
|
|
|
In addition, GFc's will retain information on the particles inside them, and on the index of the close-packing outside the GFc.
|
|
|
|
|
This will allow the mapping between configurations and GFc's to be one-to-one.
|
|
|
|
|
@ -554,7 +557,7 @@ We first prove that ground states must be periodic, which will be useful in prov
|
|
|
|
|
|
|
|
|
|
\begin{proof}
|
|
|
|
|
We prove this by contradiction: suppose that for every linearly independent family $k_1,\cdots,k_d$, $\mathcal L^\#$ is not invariant under $k_i$ translations.
|
|
|
|
|
Choose an infinite family of $k_i$ that are translations of $\lambda_\infty$.
|
|
|
|
|
Choose an infinite family of $k_i$ that are translations of $\Lambda_\infty$.
|
|
|
|
|
In this case, there would be an infinite number of ground states, obtained from $\mathcal L^\#$ by translating by $k_i$, which contradicts Item \ref{asm:finitely_many_ground_states} of Assumption \ref{assumption}.
|
|
|
|
|
\end{proof}
|
|
|
|
|
|
|
|
|
|
@ -641,7 +644,7 @@ The first is that the local density of every ground state is constant.
|
|
|
|
|
.
|
|
|
|
|
\label{rhomax_avg_loc}
|
|
|
|
|
\end{equation}
|
|
|
|
|
Now, taking a limit in which $X\to\mathcal L^\#$ such that $\Lambda_X\Uparrow\Lambda_{\infty}$ in the sense of Van Hove (there are many senses in which this limit can be taken; see, for instance, Definition \ref{def:distance-between-configurations}), we find that
|
|
|
|
|
Now, taking a limit in which $X\to\mathcal L^\#$ such that $\Lambda_X\Uparrow\Lambda_{\infty}$ (there are many senses in which this limit can be taken; see, for instance, Definition \ref{def:distance-between-configurations}), we find that
|
|
|
|
|
\begin{equation}
|
|
|
|
|
\lim_{X\to\mathcal L^\#}\frac1{\abs{X}}
|
|
|
|
|
\sum_{x\in X}\rho_{\mathcal L^\#}(x)^{-1}=
|
|
|
|
|
@ -717,7 +720,7 @@ Thus, provided that
|
|
|
|
|
\mathcal R_2\ge \mathcal R_0,
|
|
|
|
|
\label{ineq_R2_1}
|
|
|
|
|
\end{equation}
|
|
|
|
|
all the particles in $A_{j}$ whose Voronoi cell intersects $\partial^{\textrm{in}}A_{j}$ are $(\#_j,\mathcal R_2)$-correct for the same $\#_{j}\in\mathcal{G}$.
|
|
|
|
|
all the particles in $A_{j}$ whose Voronoi cell intersects $\partial^{\textrm{in}}A_{j}$ are $(\#_j,\mathcal R_2)$-correct and all these particles have the same $\#_{j}\in\mathcal{G}$ (using Lemma \ref{lemma:unique_Rcorrect}).
|
|
|
|
|
Similarly, if $A_j=\mathrm{ext}(\gamma_i)$, then we consider the exterior boundary of $A_j^c$, which is simply connected, and apply the same reasoning to find that the interior boundary of $A_j$ is lined with $(\#_j,\mathcal R_2)$-correct particles.
|
|
|
|
|
The labeling function $\mu_{\gamma_{i}}:\set{\mathrm{ext}(\gamma_{i}),\mathrm{int}_{1}(\gamma_{i}),\cdots,\mathrm{int}_{N}(\gamma_{i})}\rightarrow\mathcal{G}$ assigns this ground state $\#_{j}$ to $A_{j}$.
|
|
|
|
|
\end{enumerate}
|
|
|
|
|
@ -730,9 +733,9 @@ See Figure \ref{fig:GFc} for an example.
|
|
|
|
|
|
|
|
|
|
\caption{
|
|
|
|
|
An example of a GFc associated to a particle configuration for the 3-staircase model; see Section \ref{subsec:n_staircases}.
|
|
|
|
|
The red and yellow particles (the two outermost rings) are in one ground state and the green and cyan particles (the innermost) are in another ground state.
|
|
|
|
|
The blue and light blue particles (the two outermost rings) are in one ground state and the red and light orange particles (the innermost) are in another ground state.
|
|
|
|
|
For this model, $\mathcal R_2=3$.
|
|
|
|
|
The green and yellow particles are $\mathcal R_2$-incorrect, and thus are in the GFc.
|
|
|
|
|
The light blue and light orange particles are $\mathcal R_2$-incorrect, and thus are in the GFc.
|
|
|
|
|
Taking the union of the supports of their Voronoi cells, we find the support of the GFc, which is delineated by a thick line.
|
|
|
|
|
\label{fig:GFc}
|
|
|
|
|
}
|
|
|
|
|
@ -740,7 +743,7 @@ See Figure \ref{fig:GFc} for an example.
|
|
|
|
|
|
|
|
|
|
Hence, each GFc is a triplet $\gamma:=(\bar\gamma,X_{\gamma},\mu_{\gamma})$, but these are not arbitrary: there is a compatibility condition that ensures that a collection of GFc's corresponds to a particle configuration.
|
|
|
|
|
For one, the labels $\mu_{\gamma}$ must be compatible: if a GFc lies inside another, their labels must match up.
|
|
|
|
|
In addition, the hard-core repulsion imposes an a priori constraint on the $X_{\gamma}$, but we will now dispense with it by assuming that $\mathcal R_2$ is large enough, which prevents the $X_{\gamma}$ from interacting with each other.
|
|
|
|
|
In addition, the hard-core repulsion imposes an a priori constraint on the $X_{\gamma}$, but we will now dispense with it by assuming that $\mathcal R_2$ is large enough, which prevents the particles in different GFc's from interacting with each other.
|
|
|
|
|
|
|
|
|
|
\begin{lemma}
|
|
|
|
|
\label{lem:R_large_valid_configuration}
|
|
|
|
|
@ -777,6 +780,7 @@ For each GFc $\gamma$, the canonical configuration $\xi_{\gamma}$ has $\gamma$ a
|
|
|
|
|
\end{proposition}
|
|
|
|
|
|
|
|
|
|
The proof of this fact is simple, though writing it out is a touch tedious, so we postpone it to Appendix \ref{appx:canonical_configuration_gfc_correspondence}.
|
|
|
|
|
\bigskip
|
|
|
|
|
|
|
|
|
|
Let us now define the set of configurations of GFc's.
|
|
|
|
|
|
|
|
|
|
@ -888,7 +892,7 @@ To prove the Peierls condition from Lemma \ref{lem:cost}, we first need to defin
|
|
|
|
|
|
|
|
|
|
\begin{definition}[effective volume]
|
|
|
|
|
\label{def:effective_volume}
|
|
|
|
|
Define the effective volume of a GFc $\gamma=(\bar{\gamma},X_{\gamma},\mu_{\gamma})$ by starting with $\bar\gamma$ and removing the weights due to the particles that are outside the GFc but whose Voronoi cell intersects its support (see Definitions \ref{def:R_correct} and \ref{def:effective_particle} for notation):
|
|
|
|
|
Define the effective volume of a GFc $\gamma=(\bar{\gamma},X_{\gamma},\mu_{\gamma})$ by starting with $\bar\gamma$ and removing the weights due to the particles that are outside the GFc but whose Voronoi cell intersects its support (see Definitions \ref{def:R_correct} and \ref{def:effective_particle} for the notations):
|
|
|
|
|
\begin{equation}\label{eqn:effective_volume}
|
|
|
|
|
\norm{\bar{\gamma}}:=\sum_{\lambda\in\bar{\gamma}}\left[1-\sum_{\#}\sum_{x\in \mathcal{C}_{\#}^{(\mathcal R_2)}(X):\ \sigma^{\#}_{x}\ni\lambda}v^{\#}(\lambda)\right],
|
|
|
|
|
\end{equation}
|
|
|
|
|
@ -896,7 +900,7 @@ where $X$ is any configuration of which $\gamma$ is a GFc.
|
|
|
|
|
We note that it is not difficult to verify that \eqref{eqn:effective_volume} is independent of the choice of $X$.
|
|
|
|
|
\end{definition}
|
|
|
|
|
|
|
|
|
|
For an example, see the GFc in Figure \ref{fig:GFc}, in which the sites on $\bar\gamma$ that contribute less than 1 are the uncovered sites that neighbor the thick black line.
|
|
|
|
|
For an example, see the GFc in Figure \ref{fig:GFc}, in which the sites in $\bar\gamma$ that contribute less than 1 to $\norm{\bar{\gamma}}$ are the uncovered sites that neighbor the thick black lines.
|
|
|
|
|
|
|
|
|
|
The following simple bound on the effective volume will be useful.
|
|
|
|
|
|
|
|
|
|
@ -1748,15 +1752,15 @@ See Figure \ref{fig:staircase}.
|
|
|
|
|
|
|
|
|
|
We will prove that the $3$- and $4$-staircase models satisfy Assumption \ref{assumption}, and, therefore, crystallize at high fugacities.
|
|
|
|
|
The arguments can be extended to the general case of $n$-staircases.
|
|
|
|
|
In fact, the analog of Lemma \ref{lem:n_staircases_local_configurations} is proved in Appendix \ref{appx:n_staircases}.
|
|
|
|
|
In fact, the analog of Lemma \ref{lem:34_staircases_local_configurations} is proved in Appendix \ref{appx:n_staircases}.
|
|
|
|
|
Let us mention that in the general case, even and odd $n$ behave differently, which is why we discuss both examples $n=3,4$.
|
|
|
|
|
|
|
|
|
|
\begin{lemma}\label{lem:n_staircases_local_configurations}
|
|
|
|
|
\begin{lemma}\label{lem:34_staircases_local_configurations}
|
|
|
|
|
If $n=4$, then the density of any local configuration $X$ is maximized if and only if
|
|
|
|
|
\begin{equation}
|
|
|
|
|
\textstyle
|
|
|
|
|
\set{\pm(2,2),\pm(-4,2),\pm(2,-4)}\subseteq X
|
|
|
|
|
\label{n_staircase_local_even}
|
|
|
|
|
\label{34_staircase_local_even}
|
|
|
|
|
\end{equation}
|
|
|
|
|
and if $n=3$, the density is maximized if and only if
|
|
|
|
|
\begin{equation}
|
|
|
|
|
@ -1764,7 +1768,7 @@ and if $n=3$, the density is maximized if and only if
|
|
|
|
|
\set{\pm(2,1),\pm(-3,2),\pm(1,-3)}\subseteq X
|
|
|
|
|
\quad\mathrm{or}\quad
|
|
|
|
|
\set{\pm(1,2),\pm(-3,1),\pm(2,-3)}\subseteq X
|
|
|
|
|
\label{n_staircase_local_odd}
|
|
|
|
|
\label{34_staircase_local_odd}
|
|
|
|
|
\end{equation}
|
|
|
|
|
see Figure \ref{fig:staircase_close_packings}.
|
|
|
|
|
A configuration that does not include these has, for $n=3,4$,
|
|
|
|
|
@ -1795,13 +1799,13 @@ A configuration that does not include these has, for $n=3,4$,
|
|
|
|
|
\end{subfigure}
|
|
|
|
|
\captionsetup{font=small,width=0.8\textwidth,singlelinecheck=off}
|
|
|
|
|
\caption[]{Maximal-density local configurations in the $3$- and $4$-staircase models on $\Z^{2}$.
|
|
|
|
|
The Voronoi cell of the central (cyan) particle consists of the support of the particle along with the green sites.
|
|
|
|
|
Each green site carries a weight of $\frac13$.
|
|
|
|
|
The Voronoi cell of the central (blue) particle consists of the support of the particle along with the light green sites.
|
|
|
|
|
Each light green site carries a weight of $\frac13$.
|
|
|
|
|
\begin{enumerate}
|
|
|
|
|
\item If $n=3$, there are exactly two maximal-density local configurations; the one shown in Figure \ref{fig:staircase_close_packings_odd_n} and its reflection across the line $y=x$.
|
|
|
|
|
The local density of the cyan particle is $\frac17$.
|
|
|
|
|
The local density of the blue particle is $\frac17$.
|
|
|
|
|
\item If $n=4$, there is exactly one maximal-density local configuration as shown in Figure \ref{fig:staircase_close_packings_even_n}.
|
|
|
|
|
The local density of the cyan particle is $\frac1{12}$.
|
|
|
|
|
The local density of the blue particle is $\frac1{12}$.
|
|
|
|
|
\end{enumerate}
|
|
|
|
|
}
|
|
|
|
|
\label{fig:staircase_close_packings}
|
|
|
|
|
@ -1839,7 +1843,7 @@ A configuration that does not include these has, for $n=3,4$,
|
|
|
|
|
\rho_X^{-1}(\mathbf 0)\ge 12
|
|
|
|
|
.
|
|
|
|
|
\end{equation}
|
|
|
|
|
Thus, by \eqref{local_density_staircase}, the configurations \eqref{n_staircase_local_even}-\eqref{n_staircase_local_odd} maximize the local density.
|
|
|
|
|
Thus, by \eqref{local_density_staircase}, the configurations \eqref{34_staircase_local_even}-\eqref{34_staircase_local_odd} maximize the local density.
|
|
|
|
|
|
|
|
|
|
\begin{figure}[h]
|
|
|
|
|
\centering
|
|
|
|
|
@ -1855,41 +1859,41 @@ A configuration that does not include these has, for $n=3,4$,
|
|
|
|
|
\caption{}
|
|
|
|
|
\label{fig:4staircase_max}
|
|
|
|
|
\end{subfigure}
|
|
|
|
|
\caption{For $n=3$, at least one of the magenta, one of the green, and one of the yellow sites must be left uncovered.
|
|
|
|
|
For $n=4$, at least {\it two} of the magenta, two of the green, and two of the yellow sites must be left uncovered.}
|
|
|
|
|
\caption{For $n=3$, at least one of the blue, one of the light green, and one of the light blue sites must be left uncovered.
|
|
|
|
|
For $n=4$, at least {\it two} of the blue, two of the light green, and two of the light blue sites must be left uncovered.}
|
|
|
|
|
\label{fig:staircase_max}
|
|
|
|
|
\end{figure}
|
|
|
|
|
|
|
|
|
|
Let us now check that these are the only ones.
|
|
|
|
|
Consider $n=3$ first.
|
|
|
|
|
(To follow this discussion, it may be helpful to draw the particles on a sheet of graph paper as they are added.)
|
|
|
|
|
Choose which of the two yellow sites is covered.
|
|
|
|
|
Choose which of the two light blue sites is covered.
|
|
|
|
|
Without loss of generality, let us assume it is $(2,1)$.
|
|
|
|
|
Since the gray sites are neither yellow, nor green, nor magenta, they must be covered to maximize the local density.
|
|
|
|
|
Notice that the gray sites must be covered to maximize the local density.
|
|
|
|
|
Having placed a particle at $(2,1)$, the gray site $(3,0)$ can be covered in only one way.
|
|
|
|
|
Having placed this latest particle, the green site $(2,-1)$ cannot be covered, so the other two green sites must be covered, which can only be done in one way.
|
|
|
|
|
Once this is done, the magenta site $(-1,1)$ can no longer be covered, so the other magenta site $(-1,2)$ must be covered, which can only be done in one way.
|
|
|
|
|
Having placed this latest particle, the light green site $(2,-1)$ cannot be covered, so the other two light green sites must be covered, which can only be done in one way.
|
|
|
|
|
Once this is done, the blue site $(-1,1)$ can no longer be covered, so the other blue site $(-1,2)$ must be covered, which can only be done in one way.
|
|
|
|
|
This then leaves a unique way of covering the remaining gray site $(0,3)$.
|
|
|
|
|
Thus, the maximal-density local configuration is unique, once we have chosen which yellow site is to be covered.
|
|
|
|
|
Thus, the maximal-density local configuration is unique, once we have chosen which light blue site is to be covered.
|
|
|
|
|
|
|
|
|
|
The argument for $n=4$ is similar.
|
|
|
|
|
(Again, it is recommended to follow along with graphing paper.)
|
|
|
|
|
Let us first try to place a particle in one of the yellow sites that is not $(2,2)$, say $(3,1)$.
|
|
|
|
|
In this case, the other two yellow sites $(2,2)$ and $(1,3)$ are left unoccupied.
|
|
|
|
|
However, $(2,3)$ will also be left unoccupied, so $(2,2)$ and $(1,3)$ will each only neighbor two particles, so their contribution to $\rho^{-1}$ will be at least $\frac12$ each, so the density will not be minimal.
|
|
|
|
|
Therefore, the only yellow site that can be occupied in the maximal density configuration is $(2,2)$.
|
|
|
|
|
Having fixed $(2,2)$, there are two ways of covering the gray site $(4,0)$, but one of these will leave the yellow site $(3,1)$ with only two neighbors, and will thus not be the maximal density.
|
|
|
|
|
Let us first try to place a particle in one of the light blue sites that is not $(2,2)$, say $(3,1)$.
|
|
|
|
|
In this case, the other two light blue sites $(2,2)$ and $(1,3)$ are left unoccupied.
|
|
|
|
|
However, $(2,3)$ will also be left unoccupied, so $(2,2)$ and $(1,3)$ will each only neighbor two particles, so their contribution to $\rho^{-1}$ will be at least $\frac12$ each, so the density will not be maximal.
|
|
|
|
|
Therefore, the only light blue site that can be occupied in the maximal density configuration is $(2,2)$.
|
|
|
|
|
Having fixed $(2,2)$, there are two ways of covering the gray site $(4,0)$, but one of these will leave the light blue site $(3,1)$ with only two neighbors, and will thus not be the maximal density.
|
|
|
|
|
There is then just one possibility left to cover the gray site $(4,0)$.
|
|
|
|
|
Having placed this particle, the green site $(3,-1)$ must be left uncovered.
|
|
|
|
|
If we tried to cover the green site $(1,-1)$, then $(3,-1)$ would only neighbor two particles, and the density would not be maximal.
|
|
|
|
|
Therefore, the green site $(1,-1)$ must be left uncovered, and so the green site $(2,-1)$ must be covered, which can only be done in one way.
|
|
|
|
|
We repeat this argument to cover the gray site at $(0,4)$ and the magenta site $(-1,2)$.
|
|
|
|
|
Having placed this particle, the light green site $(3,-1)$ must be left uncovered.
|
|
|
|
|
If we tried to cover the light green site $(1,-1)$, then $(3,-1)$ would only neighbor two particles, and the density would not be maximal.
|
|
|
|
|
Therefore, the light green site $(1,-1)$ must be left uncovered, and so the light green site $(2,-1)$ must be covered, which can only be done in one way.
|
|
|
|
|
We repeat this argument to cover the gray site at $(0,4)$ and the blue site $(-1,2)$.
|
|
|
|
|
At this point, there is a unique way of covering the remaining sites at $(-1,0)$ and $(0,-1)$.
|
|
|
|
|
|
|
|
|
|
Thus, for $n=4$, there is a unique way to maximize the local density.
|
|
|
|
|
|
|
|
|
|
Finally, let us estimate $\epsilon$.
|
|
|
|
|
For $n=3$, the only way to deviate from the construction above is if one had an extra empty site among the magenta, green, and yellow.
|
|
|
|
|
For $n=3$, the only way to deviate from the construction above is if one had an extra empty site among the blue, light green, and light blue.
|
|
|
|
|
This would increase $\rho_X^{-1}$ by $\ge\frac13$.
|
|
|
|
|
(In fact, there exists a local configuration with $\rho_X^{-1}=7+\frac13$, (obtained by placing a particle at $(2,2)$) so this is optimal.)
|
|
|
|
|
|
|
|
|
|
@ -1898,14 +1902,15 @@ A configuration that does not include these has, for $n=3,4$,
|
|
|
|
|
|
|
|
|
|
\end{proof}
|
|
|
|
|
|
|
|
|
|
Using Lemma \ref{lem:n_staircases_local_configurations}, we construct configurations that have a constant maximal local density by extending the local configurations in Figure \ref{fig:staircase_close_packings}.
|
|
|
|
|
To do so, we apply Lemma \ref{lem:n_staircases_local_configurations} to a neighbor of $\mathbf 0$ and repeat.
|
|
|
|
|
Using Lemma \ref{lem:34_staircases_local_configurations}, we construct configurations that have a constant maximal local density by extending the local configurations in Figure \ref{fig:staircase_close_packings}.
|
|
|
|
|
To do so, we apply Lemma \ref{lem:34_staircases_local_configurations} to a neighbor of $\mathbf 0$ and repeat.
|
|
|
|
|
\bigskip
|
|
|
|
|
|
|
|
|
|
We now verify Assumption \ref{assumption}.
|
|
|
|
|
\begin{enumerate}
|
|
|
|
|
\item[\ref{asm:lattice}.] $\mathbb Z^2$ is a periodic graph with coordination number $4$ and it is easy to check (and this is also well known \cite{Timar2013}), the boundary of any connected set is $2$-connected.
|
|
|
|
|
\item[\ref{asm:lattice}.] $\mathbb Z^2$ is a periodic graph with coordination number $4$ and it is easy to check (and also well-known \cite{Timar2013}) that the boundary of any simply connected set is $2$-connected.
|
|
|
|
|
|
|
|
|
|
\item[\ref{asm:finitely_many_ground_states}.] By considering the possible translations of the packings in Lemma \ref{lem:n_staircases_local_configurations}, we find that there are $6\times2=12$ ground states for $n=3$ and $10$ ground states for $n=4$.
|
|
|
|
|
\item[\ref{asm:finitely_many_ground_states}.] By considering the possible translations of the packings in Lemma \ref{lem:34_staircases_local_configurations}, we find that there are $6\times2=12$ ground states for $n=3$ and $10$ ground states for $n=4$.
|
|
|
|
|
|
|
|
|
|
\item[\ref{asm:isometry}.] If $n=4$, the ground states are related by translations.
|
|
|
|
|
If $n=3$, they are related to each other by translations and the reflection $(x,y)\mapsto(y,x)$.
|
|
|
|
|
@ -1914,12 +1919,12 @@ We now verify Assumption \ref{assumption}.
|
|
|
|
|
\item[\ref{asm:density_max_local}.] By Lemma \ref{lemma:rholoc_bound_rho}, $\rho_{\mathrm{max}}$ is smaller or equal to the local density of the configurations in Figure \ref{fig:staircase_close_packings} (which is the maximal local density).
|
|
|
|
|
Since those can be extended to a configuration on all of $\Lambda_\infty$, $\rho_{\mathrm{max}}=\rho_{\mathrm{max}}^{\mathrm{loc}}$.
|
|
|
|
|
|
|
|
|
|
\item[\ref{asm:imperfect_transition}.] Without loss of generality, let us consider a $\#$-correct particle at $\vec{0}$, which, by Lemma \ref{lem:n_staircases_local_configurations}, must then be in one of the local configurations (\ref{n_staircase_local_even})-(\ref{n_staircase_local_odd}).
|
|
|
|
|
\item[\ref{asm:imperfect_transition}.] Without loss of generality, let us consider a $\#$-correct particle at $\vec{0}$, which, by Lemma \ref{lem:34_staircases_local_configurations}, must then be in one of the local configurations (\ref{34_staircase_local_even})-(\ref{34_staircase_local_odd}).
|
|
|
|
|
By Figure \ref{fig:staircase_close_packings}, no site at $\mathrm{d}_{\Lambda_{\infty}}=1$ from $\sigma_{\vec{0}}$ can neighbor any other particle than those already appearing in the local configuration.
|
|
|
|
|
In other words, the neighbors of $\vec{0}$ are precisely those specified in the local configuration.
|
|
|
|
|
Now, if any neighbor of $\vec{0}$ is correct, it must too be in a local configuration specified in Lemma \ref{lem:n_staircases_local_configurations}, but it is straightforward to check that the only way this does not cause particles to overlap is if it is in the same local configuration as is $\vec{0}$, so they must be $\#$-correct for the same $\#\in\mathcal{G}$.
|
|
|
|
|
Now, if any neighbor of $\vec{0}$ is correct, it must too be in a local configuration specified in Lemma \ref{lem:34_staircases_local_configurations}, but it is straightforward to check that the only way this does not cause particles to overlap is if it is in the same local configuration as is $\vec{0}$, so they must be $\#$-correct for the same $\#\in\mathcal{G}$.
|
|
|
|
|
|
|
|
|
|
\item[\ref{asm:density_local_density}.] By Lemma \ref{lem:n_staircases_local_configurations}, if a particle $x$ is incorrect, then
|
|
|
|
|
\item[\ref{asm:density_local_density}.] By Lemma \ref{lem:34_staircases_local_configurations}, if a particle $x$ is incorrect, then
|
|
|
|
|
\begin{equation}
|
|
|
|
|
\rho_{X}^{-1}(\mathbf 0)\ge\rho_{\mathrm{max}}^{\mathrm{loc}}{}^{-1}+\epsilon_n
|
|
|
|
|
\end{equation}
|
|
|
|
|
@ -2019,7 +2024,7 @@ Nevertheless, it is worth pointing out that our construction gives explicit valu
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\subsection{Disks of radius $5/2$}
|
|
|
|
|
\subsection{Disks of radius $5/2$}\label{sec:disk}
|
|
|
|
|
|
|
|
|
|
The hard-disk model on $\Z^{2}$ of radius $5/2$ has particles whose support is
|
|
|
|
|
\begin{equation}
|
|
|
|
|
@ -2031,7 +2036,7 @@ This model is the {\it 12th nearest neighbor exclusion}, and is equivalent to th
|
|
|
|
|
\hfil\includegraphics[width=2.4cm]{disk.pdf}
|
|
|
|
|
\hfil\includegraphics[width=2.4cm]{octagon.pdf}
|
|
|
|
|
|
|
|
|
|
\caption{The hard-core model based on the disk of radius 2.5 is equivalent to that with hard-core octagons.}
|
|
|
|
|
\caption{The hard-core model based on the disk of radius 2.5 is equivalent to that with octagons.}
|
|
|
|
|
\label{fig:octagon}
|
|
|
|
|
\end{figure}
|
|
|
|
|
|
|
|
|
|
@ -2091,10 +2096,10 @@ The density of a local configuration is maximal if and only if one of the follow
|
|
|
|
|
\label{fig:disk_max_2}
|
|
|
|
|
\end{subfigure}
|
|
|
|
|
\caption{\ref{fig:disk_neighbors}: the 16 neighbors of the disk.
|
|
|
|
|
\ref{fig:disk_max_1}: the magenta particle covers 3 neighbors of the cyan particle, and the two yellow sites must be left uncovered.
|
|
|
|
|
In addition, the yellow sites cannot neighbor more than 2 particles, so their weight is at least $\frac12$.
|
|
|
|
|
\ref{fig:disk_max_2}: the magenta particles cover 2 neighbors of the cyan particle, and the two green sites must be left uncovered.
|
|
|
|
|
In this case, the green sites can neighbor up to 3 particles, so the weight is at least $\frac13$.}
|
|
|
|
|
\ref{fig:disk_max_1}: the light green particle covers 3 neighbors of the blue particle, and the two light blue sites must be left uncovered.
|
|
|
|
|
In addition, the light blue sites cannot neighbor more than 2 particles, so their weight is at least $\frac12$.
|
|
|
|
|
\ref{fig:disk_max_2}: the light green particles cover 2 neighbors of the blue particle, and the two light yellow sites must be left uncovered.
|
|
|
|
|
In this case, the light yellow sites can neighbor up to 3 particles, so the weight is at least $\frac13$.}
|
|
|
|
|
\label{fig:disk_max}
|
|
|
|
|
\end{figure}
|
|
|
|
|
|
|
|
|
|
@ -2115,7 +2120,7 @@ The density of a local configuration is maximal if and only if one of the follow
|
|
|
|
|
By Item \ref{leave2} again, if there were 6 particles, there would also be at least 6 empty sites, and if two of these have weight $1/2$, the total weight of the empty sites is at least $7/3>2$, and so the density would not be maximal.
|
|
|
|
|
Thus, when 1 particle covers 3 sites, we can only have 5 particles that neighbor $\sigma_{\mathbf 0}$.
|
|
|
|
|
By Item \ref{leave2}, the total number of empty sites is at least 5, two of which have a weight $1/2$, so for the density to be maximal, the remaining 3 must have weight $1/3$, and so must neighbor 3 particles.
|
|
|
|
|
At this stage, there are not very many possibilities: we have 5 particles other than that at $\mathbf 0$, one of which is at $(0,5)$, the other 4 must cover two neighbors of $\sigma_x$, and three of the empty sites that neighbor $\sigma_{\mathbf 0}$ must all neighbor three particles each.
|
|
|
|
|
At this stage, there are not very many possibilities: we have 5 particles other than that at $\mathbf 0$, one of which is at $(0,5)$, the other 4 must cover two neighbors of $\sigma_{\mathbf 0}$, and three of the empty sites that neighbor $\sigma_{\mathbf 0}$ must all neighbor three particles each.
|
|
|
|
|
It is straightforward to check that there is only one way to do this, which is the one in Figure \ref{fig:disk_close_packing_2}.
|
|
|
|
|
|
|
|
|
|
Now, if no particle covers 3 sites, then all cover at most 2.
|
|
|
|
|
@ -2150,15 +2155,16 @@ Using Lemma \ref{lem:disk_local_configurations}, we construct configurations tha
|
|
|
|
|
We first note that the configuration in Figure \ref{fig:disk_close_packing_2} cannot be extended: consider the particle at $(3,-4)$, if its local density is to be maximal, then it must be part of a local configuration of the form of Figure \ref{fig:disk_close_packings} and its reflections and rotations, but because of the presence of the particle at $(-3,-4)$, it is clear that this cannot be the case.
|
|
|
|
|
Thus, Figure \ref{fig:disk_close_packing_2} cannot lead to a close-packing configuration.
|
|
|
|
|
|
|
|
|
|
On the other hand Figure \ref{fig:disk_close_packing_1} can be extended: each pink particle in the figure looks locally like the cyan one.
|
|
|
|
|
On the other hand, Figure \ref{fig:disk_close_packing_1} can be extended: each light yellow particle in the figure looks locally like the blue one.
|
|
|
|
|
The completion of the close-packing is unique.
|
|
|
|
|
Indeed, consider the particle at $(2,5)$.
|
|
|
|
|
Because of the presence of the particle at $(5,1)$, it can only have a maximal density if it is surrounded in the same way as the cyan particle in Figure \ref{fig:disk_close_packing_1}, or its rotation by $\pi$.
|
|
|
|
|
Because of the presence of the particle at $(5,1)$, it can only have a maximal density if it is surrounded in the same way as the blue particle in Figure \ref{fig:disk_close_packing_1}, or its rotation by $\pi$.
|
|
|
|
|
But since Figure \ref{fig:disk_close_packing_1} is symmetric under rotations by $\pi$, this construction is unique.
|
|
|
|
|
We can make the same argument for each of the particles surrounding $\sigma_{\mathbf 0}$, which makes the extension of the figure unique.
|
|
|
|
|
|
|
|
|
|
Therefore, taking into account the horizontal and vertical reflections, as well as the $\pi/2$ rotation, there are 6 minimal density local configurations that can be extended to distinct close-packings.
|
|
|
|
|
Therefore, taking into account the horizontal and vertical reflections, as well as the $\pi/2$ rotation, there are 6 maximal density local configurations that can be extended to distinct close-packings.
|
|
|
|
|
Taking into account the translations of the particle at $\mathbf 0$, this yields a total of $18\times6=108$ close-packing configurations.
|
|
|
|
|
\bigskip
|
|
|
|
|
|
|
|
|
|
We now verify Assumption \ref{assumption}.
|
|
|
|
|
Items \ref{asm:lattice}, \ref{asm:density_max_local}, and \ref{asm:imperfect_transition} are the same arguments as for the staircases; see above.
|
|
|
|
|
@ -2203,14 +2209,24 @@ The details are omitted here, as the computation is very similar to the staircas
|
|
|
|
|
\end{subfigure}
|
|
|
|
|
\captionsetup{font=small,width=0.8\textwidth}
|
|
|
|
|
\caption{Two possible maximal-density local configurations for the hard-disk model.
|
|
|
|
|
The Voronoi cell of the central (cyan) particle consists of the support of the particle along with the green and yellow sites.
|
|
|
|
|
The green sites have a weight $\frac13$ and the yellow sites $\frac12$.
|
|
|
|
|
The Voronoi cell of the central (blue) particle consists of the support of the particle along with the light blue and light yellow sites.
|
|
|
|
|
The light yellow sites have a weight $\frac13$ and the light blue sites $\frac12$.
|
|
|
|
|
Thus, since the particle itself covers 21 sites, the local density in these configurations is $\frac1{23}$.}
|
|
|
|
|
\label{fig:disk_close_packings}
|
|
|
|
|
\end{figure}
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\section*{Acknowledgments}
|
|
|
|
|
The authors thank J.L. Lebowitz for many useful discussions.
|
|
|
|
|
I.J. acknowledges support from the Simons Foundation, Grant Number\-~825876.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\section*{Data availability}
|
|
|
|
|
No data was produced for this work.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\newpage
|
|
|
|
|
\bibliographystyle{plain}
|
|
|
|
|
\bibliography{Bibliography.bib}
|
|
|
|
|
@ -2378,7 +2394,7 @@ The proposition follows immediately from Lemmas \ref{lem:canonical_non_GFc_parti
|
|
|
|
|
\section{Analysis of the $n$-staircase model}
|
|
|
|
|
\label{appx:n_staircases}
|
|
|
|
|
|
|
|
|
|
In this appendix, we prove the following generalization of Lemma \ref{lem:n_staircases_local_configurations} to general $n$.
|
|
|
|
|
In this appendix, we prove the following generalization of Lemma \ref{lem:34_staircases_local_configurations} to general $n$.
|
|
|
|
|
|
|
|
|
|
\begin{lemma}\label{thm:n_staircases_local_configurations}
|
|
|
|
|
If $n$ is even, then the density of any local configuration $X$ is maximized if and only if
|
|
|
|
|
@ -2412,7 +2428,7 @@ which are, respectively, the upper triangular regions to the northeast, northwes
|
|
|
|
|
\begin{figure}[h]
|
|
|
|
|
\centering
|
|
|
|
|
\includegraphics[width=6cm]{5staircase_regions}
|
|
|
|
|
\caption{We study the contribution to the local density at the blue particle from the three upper triangular regions around it, which are shown in light gray.}
|
|
|
|
|
\caption{We study the contribution to the local density at the violet particle from the three upper triangular regions around it, which are shown in light orange.}
|
|
|
|
|
\label{fig:5staircase_regions}
|
|
|
|
|
\end{figure}
|
|
|
|
|
|
|
|
|
|
@ -2538,10 +2554,10 @@ $U_{(n-1,-1)}^{n-1}$ &
|
|
|
|
|
\end{subfigure}
|
|
|
|
|
|
|
|
|
|
\caption{Illustrations of the cases described in Table \ref{tab:nstaircase}.
|
|
|
|
|
The particle at $\vec{0}$ is in blue.
|
|
|
|
|
The particle that intersects one of the upper triangular regions around the particle at $\vec{0}$ is in green.
|
|
|
|
|
Light gray represents an empty site within the upper triangular region which we neglect.
|
|
|
|
|
The subregions that we select are in dark gray.}
|
|
|
|
|
The particle at $\vec{0}$ is in violet.
|
|
|
|
|
The particle that intersects one of the upper triangular regions around the particle at $\vec{0}$ is in orange.
|
|
|
|
|
Light orange represents an empty site within the upper triangular region which we neglect.
|
|
|
|
|
The subregions that we select are in light violet.}
|
|
|
|
|
\label{fig:5staircase_cases}
|
|
|
|
|
\end{figure}
|
|
|
|
|
|
|
|
|
|
@ -2616,8 +2632,8 @@ There exists a unique order-3 automorphism $T$ of $\Z^{2}$ that fixes $U_{(a,b)}
|
|
|
|
|
\begin{figure}[h]
|
|
|
|
|
\centering
|
|
|
|
|
\includegraphics[width=6cm]{upper_triangular_boundary}
|
|
|
|
|
\caption{The upper triangular region $U^{4}_{(0,0)}$ is drawn in gray.
|
|
|
|
|
The point sets $P_{1}(U^{4}_{(0,0)})$, $P_{2}(U^{4}_{(0,0)})$, and $P_{3}(U^{4}_{(0,0)})$ are respectively to the east (magenta), north (green), and southwest (yellow) of $U^{4}_{(0,0)}$.}
|
|
|
|
|
\caption{The upper triangular region $U^{4}_{(0,0)}$ is drawn in green.
|
|
|
|
|
The point sets $P_{1}(U^{4}_{(0,0)})$, $P_{2}(U^{4}_{(0,0)})$, and $P_{3}(U^{4}_{(0,0)})$ are respectively to the east (light green), north (blue), and southwest (light blue) of $U^{4}_{(0,0)}$.}
|
|
|
|
|
\label{fig:upper_triangular_boundary}
|
|
|
|
|
\end{figure}
|
|
|
|
|
|
|
|
|
|
@ -2674,7 +2690,7 @@ Then, $T^{\ell}\vec{x}$ and $T^{\ell+1}\vec{x}$ each contribute at least $1/2$ t
|
|
|
|
|
|
|
|
|
|
Here, we recapitulate the important constants and estimates in our analysis of the high-density behavior of hard-core lattice particle models satisfying Assumption \ref{assumption}.
|
|
|
|
|
|
|
|
|
|
In the proof of Proposition \ref{prop:central_estimates}, we show that the weights of the GFc's that we construct in \S\ref{subsec:gfc} satisfy the Peierls condition
|
|
|
|
|
In the proof of Proposition \ref{prop:central_estimates}, we show that the weights of the GFc's that we construct in Section \ref{subsec:gfc} satisfy the Peierls condition
|
|
|
|
|
\begin{equation}\tag{\ref{eqn:weightestimate}}
|
|
|
|
|
\abs{w_{\mathbf{z}}^{\#}(\gamma)}\le e^{-\tau\abs{\bar{\gamma}}},
|
|
|
|
|
\end{equation}
|
|
|
|
|
|
