Crystalline ordering and large fugacity expansion for hard core lattice particles

TL;DR

This paper proves phase transitions and sublattice orderings in certain hard core lattice particle systems using an extension of Pirogov-Sinai theory, and demonstrates the convergence of the Gaunt-Fisher expansion for pressure.

Contribution

It extends Pirogov-Sinai theory to a broad class of hard core lattice particles and establishes the convergence of the pressure expansion in inverse fugacity.

Findings

Proves phase transitions for systems with finite close packed configurations.

Shows the Gaunt-Fisher expansion has a nonzero radius of convergence.

Includes many previously unproven cases of sublattice ordering.

Abstract

Using an extension of Pirogov-Sinai theory we prove phase transitions, corresponding to sublattice orderings, for a general class of hard core lattice particle systems with a finite number of close packed configurations. These include many cases for which such transitions have been proven. The proof also shows that, for these systems, the Gaunt-Fisher expansion of the pressure in powers of the inverse fugacity (aside from an explicit logarithmic term) has a nonzero radius of convergence.

Figures

Equations

$${\Xi }_{\Lambda }(z)=\sum _{X\subset \Lambda }{z}^{|X|}\prod _{x\ne x^{\prime }\in X}\phi (x,x^{\prime })$$

$$|X|⩽N_{\max }⩽|\Lambda |$$

$$p_{\Lambda }(z):=\frac{1}{|\Lambda |}\log {\Xi }_{\Lambda }(z)$$

$$p(z):=\underset{\Lambda \to {\Lambda }_{\infty }}{\lim }{p}_{\Lambda }={\rho }_m\log z+f(y)$$

$${\Xi }_{\Lambda }(z)=\sum _{k=0}^{N_{\max }}{z}^k{Z}_{\Lambda }(k).$$

$$p_{\Lambda }=\sum _{k=1}^{\infty }{z}^k{b}_k(\Lambda )$$

$$b_k(\Lambda ):=\frac{1}{|\Lambda |}\sum _{n=1}^k\frac{(-1{)}^{n+1}}{n}\sum _{{k_1,\cdots ,k_n⩾1}_{k_1+\cdots +k_n=k}}{Z}_{\Lambda }(k_1)\cdots Z_{\Lambda }(k_n).$$

$$Q_{\Lambda }(k):=Z_{\Lambda }(\frac{|\Lambda |}{2}-k)$$

$${\Xi }_{\Lambda }(z)=2z^{\frac{1}{2}|\Lambda |}\sum _{k=0}^{\infty }\left(\frac{1}{2}{z}^{-k}{Q}_{\Lambda }(k)\right)$$

$$p_{\Lambda }=\frac{1}{|\Lambda |}\log 2+\frac{1}{2}\log z+\sum _{k=1}^{\infty }{y}^k{c}_k(\Lambda )$$

$$c_k(\Lambda ):=\frac{1}{|\Lambda |}\sum _{n=1}^k\frac{(-1{)}^{n+1}}{n{2}^n}\sum _{{k_1,\cdots ,k_n⩾1}_{k_1+\cdots +k_n=k}}{Q}_{\Lambda }(k_1)\cdots Q_{\Lambda }(k_n).$$

$$p=\log \left(\frac{1+\sqrt{1+4z}}{2}\right)=\frac{1}{2}\log z+\log \left(\sqrt{1+\frac{1}{4z}}+\frac{1}{2\sqrt{z}}\right).$$

$$R⩽|\xi |⩽{\tilde{R}}^{-1}.$$

$$f_{\Lambda }(y):=p_{\Lambda }-{\rho }_m\log z+o(1)$$

$$f_{\Lambda }(y)=\sum _{k=1}^{\infty }{y}^k{c}_k(\Lambda ),|c_k(\Lambda )|<{\tilde{R}}^k,\underset{\Lambda \to {\Lambda }_{\infty }}{\lim }{c}_k(\Lambda )=c_k$$

$$\frac{1}{|\Lambda |}{Z}_{\Lambda }(k_1)\cdots Z_{\Lambda }(k_n)$$

$$b_2(\Lambda )=\frac{1}{|\Lambda |}\left(Z_{\Lambda }(2)-\frac{1}{2}{Z}_{\Lambda }(1{)}^2\right).$$

$$z^{16}(1+4y+10y^2+8y^3+y^4)$$

$$z^{16}(1+6y+18y^2+48y^3+43y^4+13y^5+y^6).$$

$$\bar{\rho }=z\frac{\partial p}{\partial z}=-y\frac{\partial p}{\partial y}={\rho }_m+\sum _{j=1}^{\infty }k{c}_k{y}^k.$$

Full text

Crystalline ordering and large fugacity expansion

**for hard-core lattice particles **

Ian Jauslin

School of Mathematics, Institute for Advanced Study

jauslin@ias.edu

Joel L. Lebowitz

Departments of Mathematics and Physics, Rutgers University

Simons Center for Systems Biology, Institute for Advanced Study

lebowitz@math.rutgers.edu

Abstract

Using an extension of Pirogov-Sinai theory we prove phase transitions, corresponding to sublattice orderings, for a general class of hard-core lattice particle systems with a finite number of perfect coverings. These include many cases for which such transitions have been proven. The proof also shows that, for these systems, the Gaunt-Fisher expansion of the pressure in powers of the inverse fugacity (aside from an explicit logarithmic term) has a nonzero radius of convergence.

Dedicated to Ben Widom, on the occasion of his 90th birthday.

1 Introduction

The study of order-disorder phase transitions for hard-core lattice particle (HCLP) systems has a long history (see, to name but a few, [Do58, Ka63, GF65, Ba80, Mc10, RD12, DG13] and references therein). These are purely entropy driven transitions, similar to those observed numerically and experimentally for hard spheres in the continuum [WJ57, AW57, PM86, IK15]. Whereas a proof of the transition in the hard sphere model is still lacking, there are several HCLP systems in which phase transitions have been proved.

One example is the hard diamond model on the square lattice (see figure 1a), which is a particle model on $\mathbb{Z}^{2}$ with nearest-neighbor exclusion. The existence of a transition from a low-density disordered state, in which the density of occupied sites is the same on the even and odd sublattices, to a high-density ordered state, in which one of the sublattices is preferentially occupied, was proved by Dobrushin [Do68], using a Peierls-type construction. This transition had been predicted earlier, using various approximations. In particular, Gaunt and Fisher [GF65] did an extensive study of this model using Padé approximants, obtained from a low- and a high-fugacity expansion for the pressure $p(z)$, to determine the location of a singularity on the positive $z$ axis. They estimated that there is a transition at fugacity $z_{t}=3.8$ and density $\rho_{t}=0.37$, which is in good agreement with computer simulations.

Another example is that of hard hexagons on a triangular lattice. Baxter [Ba80, Ba82] obtained, following earlier numerical and approximate work, an exact solution of this system, and found a transition at $z_{t}=\frac{1}{2}(5\sqrt{5}+11)\approx 11.09$ and $\rho_{t}=\frac{1}{10}(5-\sqrt{5})\approx 0.28$. There was further work, particularly by Joyce [Jo88], which elaborated on this solution. Baxter’s solution provides a full, albeit implicit, expression for the pressure $p(z)$ in the complex $z$ and $\rho$ planes, implying, in particular, that $p(z)$ is analytic for all $z\geqslant 0$ except at $z_{t}$. The transition at $z_{t}$ is of second order, as is expected to be the case for diamonds. For $z>z_{t}$, this system has 3 ordered phases, corresponding to the 3 different perfect coverings.

Yet another HCLP model for which an order-disorder transition was shown to occur, with only a sketch of a proof [HP74], is that of hard crosses on the square lattice (see figure 1b). This model has 10 distinct perfect coverings (see figure 2), and is conjectured [EB05] to have a first order phase transition at $z_{t}\approx 39.5$. At this fugacity the density jumps from $\rho_{f}\approx 0.16$ to $\rho_{s}\approx 0.19$. We shall use this model as an illustration for the type of system to which our analysis applies, and for which we can prove crystalline order at high fugacities, and the convergence of the high-fugacity expansion.

The expansion in powers of $y\equiv z^{-1}$ was first considered by Gaunt and Fisher [GF65] specifically for the diamond model, but has been used later for other HCLP systems [Jo88, EB05]. As far as we know, there has been no study of the convergence of this series, though Baxter’s explicit solution [Ba80] for the hard hexagon model shows it is so for that solvable model. This is in contrast to the low-fugacity expansion of the pressure $p(z)$ in powers of $z$, which dates back to Ursell [Ur27] and Mayer [Ma37]. This expansion was proven to have a positive radius of convergence, in all dimensions, by Groeneveld [Gr62] for positive pair-potentials and by Ruelle [Ru63] and Penrose [Pe63] for general pair-potentials.

In this note, we sketch a proof that the radius of convergence of the high-fugacity expansion is positive for a certain class of HCLP systems in $d\geqslant 2$ dimensions. For details of the proof, see [JL17]. The proof is based on an extension of Pirogov-Sinai theory [PS75, KP84], and implies the existence of phase transitions in these models. Unlike the low-fugacity expansion, the positivity of the radius of convergence does not hold for general HCLP systems: there are, indeed, many examples, in 1 and higher dimensions, in which the coefficients in this expansion diverge in the thermodynamic limit.

1.1 Description of the model

It is convenient, for our analysis, to think of these HCLPs as having a finite shape $\omega$ in physical space $\mathbb{R}^{d}$, and impose the constraint that, when put on lattice sites $x$ and $y$, the shapes do not overlap (for example, for the nearest-neighbor exclusion on the square lattice, $\omega$ could be a diamond, see figure 1a). Equivalently, one can think of each particle as occupying a finite collection of lattice sites. Note that the choice of these shapes, or of the lattice points assigned to each particle, is not unique: two different shapes can translate to the same hard-core interaction. For instance, the nearest neighbor exclusion on the square lattice can be obtained by taking diamonds or disks of radius $r$ with $\frac{1}{2}<r<\frac{1}{\sqrt{2}}$.

Given a $d$-dimensional lattice $\Lambda_{\infty}$ and a finite subset $\Lambda\subset\Lambda_{\infty}$, we define the grand-canonical partition function of the system at activity $z>0$ on $\Lambda$, with some specified boundary conditions, as

[TABLE]

in which $X$ is a particle configuration in $\Lambda$, $|X|$ is the number of particles, and, setting $\omega_{x}\equiv\{x+y,\ y\in\omega\}$, $\varphi(x,x^{\prime})\in\{0,1\}$ enforces the hard-core repulsion: it is equal to 1 if and only if $\omega_{x}\cap\omega_{x^{\prime}}=\emptyset$. Note that, due to the hard-core interaction, the number of particles is bounded:

[TABLE]

where $|\Lambda|$ denotes the number of lattice sites in $\Lambda$. Our aim, in this note, is to prove that, in certain cases, the finite-volume pressure of the system, defined as

[TABLE]

satisfies

[TABLE]

in which $\rho_{m}$ is the maximum density in $\Lambda$, that is, $\rho_{m}=\lim_{\Lambda\to\Lambda_{\infty}}N_{\mathrm{max}}/|\Lambda|$, and $f$ is an analytic function of $y\equiv z^{-1}$ for small values of $y$. The expansion of $f$ in powers of $y$ is called the high-fugacity expansion of the system. Note that, as is well known, $p(z)\equiv\lim_{\Lambda\to\Lambda_{\infty}}p_{\Lambda}(z)$ is independent of the boundary conditions for all $z\geqslant 0$ (see, for instance, [Ru99]). This is not so for the correlation functions, which may depend on the boundary conditions, and can also be shown to be analytic in $y$ with the same radius of convergence as the high-fugacity expansion.

1.2 Low-fugacity expansion

It is rather straightforward to express the pressure $p_{\Lambda}$ as a power series in $z$ (which converges for small values of $z$, thus earning the name ‘‘low-fugacity expansion’’). Indeed, defining the canonical partition function by $Z_{\Lambda}(k)$ as the number of particle configurations with $k$ particles, (1) can be rewritten as

[TABLE]

Substituting (3) into (5), we find that, formally,

[TABLE]

with

[TABLE]

As is well known (see, for instance, [Ru99]), there is a remarkable cancellation that eliminates the $\Lambda$ dependence from $b_{k}(\Lambda)$ when $\Lambda\to\Lambda_{\infty}$. This is readily seen by writing these coefficients in terms of Mayer graphs, which implies that $b_{j}(\Lambda)$ converges to $b_{j}$ as $\Lambda\to\Lambda_{\infty}$, independently of the boundary condition. Furthermore, the radius of convergence $R(\Lambda)$ of (6) converges to $R>0$, which is equal, for positive potentials (like those considered here), to the radius of convergence $R_{\infty}$ of $\sum_{j=1}^{\infty}b_{j}z^{j}$ [Pe63].

1.3 High-fugacity expansion

The main idea of the high-fugacity expansion, due to Gaunt and Fisher [GF65], is to perform a low-fugacity expansion for the holes of the system. In other words, instead of expressing the pressure $p_{\Lambda}$ in terms of the number of particle configurations with $k$ particles, we express it in terms of the number of hole configurations in the absence of $k$ particles from perfect covering. To make this idea more precise, let us consider the example of nearest-neighbor exclusion (which corresponds to hard diamonds) on the square lattice.

Assume that $\Lambda$ is a $2n\times 2n$ torus, so that $\Lambda$ can be completely packed with diamonds. Note that, by the invariance of the system under translations, there are two perfect coverings, each of which contains $|\Lambda|/2$ particles. Let

[TABLE]

denote the number of configurations that are missing $k$ particles, in terms of which

[TABLE]

(we factor the $2$ out, because $Q_{\Lambda}(0)=2$ and we wish to expand the logarithm in (3) around 1). We thus have, formally

[TABLE]

where $y\equiv z^{-1}$ and

[TABLE]

The first nine terms of this expansion were computed in [GF65, table XIII] for periodic boundary conditions. They found that, as in the low-fugacity expansion, these nine coefficients $c_{k}(\Lambda)$ converge to $c_{k}$ as $\Lambda\to\Lambda_{\infty}$. However, there is no systematic way of exhibiting the cancellations needed for this convergence to hold for general HCLP systems. In fact there are many simple examples where $c_{2}(\Lambda)\to\infty$ as $\Lambda\to\Lambda_{\infty}$. For instance, consider the nearest-neighbor exclusion in 1 dimension (which maps, exactly, to the one-dimensional monomer-dimer model). In the language of this paper, the model is characterized by $\Lambda_{\infty}=\mathbb{Z}$, $\omega=(-1,1)\subset\mathbb{R}$. It is easy to compute, using, for instance, the transfer matrix technique, that the infinite-volume pressure of this model is

[TABLE]

Therefore, $p-\frac{1}{2}\log z$ (note that $\rho_{m}=\frac{1}{2}$) is not an analytic function of $y\equiv z^{-1}$ at $y=0$ (though it is an analytic function of $\sqrt{y}$). For the $n$-nearest-neighbor exclusion in one dimension, $p-\rho_{m}\log z$ is analytic in $y^{\frac{1}{n}}$. Similar effects occur in higher dimensions as well, for instance in systems exhibiting columnar order at high fugacities [GD07].

Note that, for systems whose pressure admits a convergent high-fugacity expansion, the partition function may not have any roots for large values of $|z|$, which implies that the Lee-Yang [YL52, LY52] zeros of such systems are confined within an annulus: denoting the radius of convergence of the low- and high-fugacity expansions by $R$ and $\tilde{R}$, every Lee-Yang zero $\xi$ satisfies

[TABLE]

Furthermore, it can easily be seen (by Krammers-Wannier duality) that systems with bounded repulsive pair-potentials all have a convergent high-fugacity expansion, so their Lee-Yag zeros lie within an annulus.

Here, we prove that, for a class of HCLP systems which we call ‘‘non-sliding models’’ (which include the three models discussed above, that is, the hard diamond, cross and hexagon models), the function

[TABLE]

is analytic in a disk around $y=0$ uniformly in $|\Lambda|$, in which $o(1)\to 0$ as $\Lambda\to\Lambda_{\infty}$. That is,

[TABLE]

for some $\tilde{R}>0$, independent of $|\Lambda|$. We thus prove the validity of the Gaunt-Fisher expansion for non-sliding models. Our method of proof further shows that, for such systems, the high-fugacity phases exhibit crystalline order.

A precise definition of the notion of non-sliding will be given below. An example of a sliding model is the hard $2\times 2$ square model on the square lattice: given a perfect covering, whole columns or rows of particles can slide without forming vacancies (see figure 3). On the other hand, hard diamonds do not slide: the close-packed configurations are rigid. The same is true of hard crosses and hard hexagons, as well as the nearest neighbor exclusion on $\mathbb{Z}^{d}$ for any $d\geqslant 2$.

2 Sketch of the proof

In this note, we will only give a detailed sketch of the proof. The full details can be found in [JL17].

Ultimately, the reasoning behind (15) is similar to that underpinning the convergence of the Mayer expansion (6), so let us first discuss the Mayer expansion, and, in particular, focus on the uniform boundedness of $b_{k}(\Lambda)$ defined in (7). For the sake of simplicity, we will consider periodic boundary conditions, and take $|\Lambda|$ sufficiently larger than $k$. First of all, note that $Z_{\Lambda}(k)$ is a polynomial in $|\Lambda|$ of order $k$ with no constant term, thus,

[TABLE]

is a polynomial in $|\Lambda|$ of order $k-1$. Therefore, in order for $b_{k}(\Lambda)$ to remain bounded in the $\Lambda\to\Lambda_{\infty}$ limit for $k>1$, there must be a significant cancellation. For the purpose of illustration, consider

[TABLE]

In the 1-particle case, the particle can occupy any site in $\Lambda$, so $Z_{\Lambda}(1)=|\Lambda|$. In the 2-particle case, the particles must not overlap. We can therefore write $Z_{\Lambda}(2)$ as the number of unconstrained configurations (excluding the cases in which particles coincide) minus the number of configurations in which the particles overlap. The former is equal to $\frac{1}{2}|\Lambda|(|\Lambda|-1)$, and the latter is proportional to $|\Lambda|$. The $|\Lambda|^{2}$ term thus cancels out. This reasoning can be extended to all $b_{k}(\Lambda)$.

Following [GF65], we construct the high-fugacity expansion in a similar way, but instead of counting particle configurations, we count hole configurations. To that end, we factor out $z^{\rho_{m}|\Lambda|}$ from the partition function, as in (9), thus giving each hole a weight $z^{-\rho_{m}}$. The most significant difference with the Mayer expansion is that the interaction between holes is not simply a hard-core repulsion, since the sites that are not empty must be covered by particles which, in turn, must satisfy the hard-core constraint. In particular, the connected components of the empty space may come in various shapes and sizes, but they are constrained by the fact that the overall empty volume is an integer multiple of the volume of each particle (see figure 4). This implies that different connected components of the empty volume could, in principle, interact strongly, even if they are arbitrarily far from each other (see figure 4b). If that were the case, then the $|\Lambda|^{2}$ terms in $c_{2}(\Lambda)$ would not cancel out, and the high-fugacity expansion would be ill-defined in the thermodynamic limit. This phenomenon will be called sliding.

In this paper, we will only consider models in which there is no sliding, a notion which we will now define precisely. First of all, in order to qualify as a non-sliding model, the system must only admit a finite number of distinct perfect coverings, and must be such that any particle configuration is entirely determined by the location of the holes and the particles adjacent to them. In addition, whenever different locally close-packed phases coexist, the interface between the phases must contain a number of holes proportional to its length (see figure 5). This condition is analogous to the Peierls condition in the standard Pirogov-Sinai theory [PS75, KP84]. This rules out situations similar to figure 4b, in which the interface between the central column and the other two may be arbitrarily long, while having only two holes. More precisely, a model is said to exhibit no sliding if, for every connected particle configuration $X\subset\Lambda$ that is not a subset of a perfect covering of $\Lambda$, and for every configuration $Y\supset X$, there exists at least one empty site neighboring a particle in $X$ (see figure 5). Note that, having fixed $X$, there are many possible connected configurations $Y$ that contain $X$, and we require that every one of them contain some empty space. The notions of connectedness and neighbors are inherited from the lattice structure, and a particle configuration $X$ is said to be connected if the set of lattice sites that are covered by particles is connected.

In the absence of sliding, it is easy to see which holes are correlated. Indeed, consider the connected components of the union of the holes and their neighboring particles (see figure 6). Each such component is called a Gaunt-Fisher (GF) configuration. The space that is not occupied by GF configurations is completely covered by particles, so, since there is no sliding, each of its connected components can be extended to a perfect covering of $\Lambda$. To each such component, we associate an index that specifies to which of the possible perfect coverings the configuration in the component can be extended. The interaction between GF configurations is then rather simple: distinct GF configurations must be disjoint, and each connected component of the complement of each GF configuration must be coverable by particles in a configuration that can be extended to the perfect covering specified by the index of the component. The latter interaction is thus mediated by the indices of the connected components of the complements of the GF configurations. We then completely remove this interaction, as described below.

Let us consider only the most external GF configurations, that is those that do not lie inside any other GF configuration. To eliminate their interaction, we fix the boundary condition on $\Lambda$ once and for all, so as to fix the index of the perfect covering on the outside. At this point, the constraint on each GF configuration is that it be compatible with the boundary condition, which is independent of the other GF configurations, thus eliminating the interaction between them. Now, the GF configurations may have holes (see figure 6c), and other configurations may lie inside these holes. These would interact with the GF configurations that contain them. In order to eliminate this interaction, we make use of a technique from Pirogov-Sinai theory [PS75, KP84]. Namely, the partition function in each hole is a partition function on a smaller volume, with the boundary condition imposed by the index of the covering of the hole. Now, in general, the partition function in a hole may depend on the boundary condition (see figure 7), but this dependence is weak. Indeed, one can show that the ratio of the partition function with one boundary condition divided by that with another is at most exponential in the size of the boundary (whereas each partition function is exponential in the size of the bulk). Thus, at the price of an exponential factor in the size of the GF configuration, called the flipping term, we can pretend that the partition function in each hole has the same boundary condition as the entire space. At this point, the interaction between GF configuration simply states that they must not overlap. And, since there is no sliding, each GF configuration contains a number of holes which is proportional to its size, thus contributing a factor $z^{-c|\mathrm{size}|}$ to the partition function (from which, we recall, we have factored out $z^{-\rho_{m}|\Lambda|}$ so that each hole contributes $z^{-\rho_{m}}$), which outweighs the flipping term.

We have thus constructed a model of GF configurations, which interact via a purely hard-core potential, and which have a very small effective fugacity. We then use a low-fugacity expansion to express the function $f_{\Lambda}$ (14) as a convergent series, following [KP86]. In addition, the effective fugacity of a GF configuration of volume $|V|$ is equal to the fugacity of the particles inside the configuration divided by $z^{\rho_{m}|V|}$. Furthermore, since the GF configuration contains empty space, the number of particles in the GF configuration is smaller than $\rho_{m}|V|$. Finally, since the volume obtained by removing the GF configuration from the lattice can be covered by particles, $\rho_{m}|V|$ is an integer. Therefore, the fugacity of a GF configuration is an analytic function of $y\equiv z^{-1}$, which implies that $f_{\Lambda}$ is analytic in $y$, uniformly in $|\Lambda|$.

3 Concluding remarks

\point

In this paper, we have focused on the pressure $p$ at large $z$. Other thermodynamic quantities can be computed from $p$, or by a computation similar to that of $p$. We recover the average density $\bar{\rho}$ from the pressure by

[TABLE]

Since the pressure is independent of the boundary condition, it is the same in all phases, which implies that the average density is as well. In order to distinguish between phases, one could consider the local density $\rho(x)$ at $x$, which does depend on the phase. Thus, for the diamonds on the square lattice, at large fugacities, the local density at sites on the even sublattice would be different from that on the odd sublattice: in the even phase, the former would be close to $1$, whereas the latter would be close to [math]. In general, when there are $n$ close-packed phases, there are $n$ sublattices, and, in each phase, the local density at one of the sublattices is close to $1$, while the others are close to [math]. The local density $\rho(x)$ can be expanded in powers of $y$, using similar methods to those described in this paper. Similarly, one can expand higher-order correlation functions, and find that, when the series converges, the truncated correlation functions in a specified phase decay exponentially.

\point

Here, we have only considered HCLP systems that have a single shape. A natural extension would be to consider systems in which several types of particles of different shapes can coexist, provided there is a finite number of perfect coverings, and no sliding. In that case, different particles may have different fugacities, for instance, one might set $z_{\alpha}=\lambda_{\alpha}z$ and expand in $z^{-1}$. The qualitative behavior of the system might depend on the $\lambda_{\alpha}$. Further extensions could be to consider more general pair potentials, by, for instance, allowing for smooth interactions, or for more general hard-core repulsions, such as the Widom-Rowlinson [WR70] interaction.

\point

If, instead of overlap between the particles being forbidden, it were merely discouraged by replacing the hard-core potential by a strongly repulsive potential $J$, then one would expect, using a technique similar to that sketched in this paper, to prove that the pressure is analytic in an intermediate regime $z_{0}<|z|\ll e^{J}$. This was shown for the soft diamond model in [BK73].

\point

The methods described here allow, in some cases, to approach the continuum, but not to reach it. For instance, in the cross model, one can make the lattice finer (or, equivalently, the crosses can be made thicker). However, the radius of convergence vanishes in the continuum limit. New ideas are needed to treat such a case.

Acknowledgements

We are grateful to Giovanni Gallavotti and Roman Kotecký for enlightening discussions. The work of J.L.L. was supported by AFOSR grant FA9550-16-1-0037. The work of I.J. was supported by The Giorgio and Elena Petronio Fellowship Fund and The Giorgio and Elena Petronio Fellowship Fund II.

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