January 01, 1970
We prove a simple, explicit lower bound on the radius of a zero-free disk for Gibbs point processes defined by finite-range, repulsive multi-body interactions. Our lower bound improves on those previously known, and we demonstrate that it is essentially sharp in the generality with which our arguments apply. The key ingredient is a multi-body generalization of integral identities for point densities of Gibbs point processes in the spirit of earlier work of Michelen and Perkins.
Many of the simplest models of statistical mechanics of interacting particles arise by imposing constraints (or other interactions) on Poisson point processes. A well-known example is the two-body hard sphere gas. In this case, the Poisson process \(X\) has points in \(\mathbb{R}^{d}\), and the interaction is a pure constraint: it restricts \(X\) to configurations such that spheres of radius \(r>0\) about each point in \(X\) are disjoint. Equivalently, no point in space is within distance \(r\) of two distinct elements of \(X\). Despite the simplicity of this model, relatively little is rigorously understood; see, e.g., the introductions of [1], [2] or [3].
This paper is motivated by the following question. Modify the two-body hard sphere gas to obtain a three-body hard sphere gas. That is, subject \(X\subset \mathbb{R}^{d}\) to the constraint that no point in space is within distance \(r\) of three distinct elements of \(X\). For a fixed a priori intensity of the underlying Poisson process, it seems clear that if the two-body hard sphere model is in a dilute (gaseous) phase, then so must be the three-body hard sphere model. Unfortunately, known criteria on the intensity that provably ensure a model is in a dilute phase do not behave in the expected manner: they yield a larger dilute phase region for the two-body hard sphere gas than the three-body hard sphere gas. We discuss this further in 1.2.1 below. In the preceding sentences the phrase ‘dilute’ should be interpreted as a synonym for analyticity of the pressure of the models as a function of the activity of particles (intensity of the point process).
The central issue seems to be that while analyticity criteria for multi-body continuum particle systems have been explored (see, e.g., [4]–[8]), known results have been primarily concerned with existential statements. In particular, existing treatments can be analytically inconvenient and/or inefficient – they guarantee the existence of activities small enough that analyticity is true, but without easy control of precisely how small. Motivated by this, the present paper develops a simple criteria that readily verifies the intuitively clear statement above. That is, we give explicit domains of analyticity for the pressure of multi-body repulsive finite-range particle systems.
Despite the somewhat frivolous question that initiated this work, we think our results reveal some important questions and perspectives. We defer our discussion of these matters, as well as connections to related literature, until after precisely describing our main results.
While our results are more general, for notational ease in this section let \(\mathbb{X}= \mathbb{R}^{n}\) be \(n\)-dimensional Euclidean space. A potential \(\phi\) is a collection of (measurable) functions \((\phi_k)_{k \in \mathbb{N}}\), \(\phi_{k}\colon \mathbb{X}^{k} \to \mathbb{R}\cup \{\infty\}\). Given \(\mathbf{x}\in \mathbb{X}^{k}\), we write \(\phi(\mathbf{x})=\phi_{k}(\mathbf{x})\). We will always assume that \(\phi\) is symmetric, i.e., for each \(k \ge 2\), \(\phi_{k}\circ \pi = \phi_{k}\) for all permutation \(\pi\) of the \(k\) coordinates \(\mathbf{x}\in \mathbb{X}^{k}\). We say \(\phi\) is repulsive if \(\phi_k\geq 0\) for all \(k \in \mathbb{N}\).
Given \(\mathbf{x}\in \mathbb{X}^{k}\) and \(S\subset [k]\mathrel{\vcenter{:}}= \{1,2,\dots, k\}\), let \(\mathbf{x}_{S} = (x_{i})_{i\in S}\). Any \(\phi\) defines a Hamiltonian by \[H(\mathbf{x}) \mathrel{\vcenter{:}}= \sum_{S\neq\emptyset} \phi(\mathbf{x}_{S}),\] where the sum ranges over non-empty subsets of \([k]\) if \(\mathbf{x}\in \mathbb{X}^{k}\). For \(\lambda\in \mathbb{C}\) and \(\Lambda \subseteq \mathbb{X}\) bounded and measurable the associated partition function is \[\label{eq:PF} Z_{\Lambda,\phi}(\lambda) \mathrel{\vcenter{:}}= \sum_{k \in \mathbb{N}_0} \frac{\lambda^k}{k!} \int_{\Lambda^k} e^{-H(\mathbf{x})} \nu^{k}(\mathrm{d}\mathbf{x}),\tag{1}\] where \(\nu^{k}\) denotes the \(k\)-fold product of Lebesgue measure \(\nu\) on \(\mathbb{R}^{n}\) with itself. The parameter \(\lambda\) is called the activity. The assumption that \(\phi\) is repulsive ensures that \(Z_{\Lambda,\phi}\) exists, and moreover, \(|Z_{\Lambda,\phi}|\leq \exp( |\lambda| \nu(\Lambda))\). In particular, for \(\lambda \ge 0\), one obtains a point process \(\mu\) in \(\Lambda\) by defining an expectation \[\mathbb{E}_{\mu}[f] \mathrel{\vcenter{:}}= \frac{1}{Z_{\Lambda, \phi}(\lambda)} \sum_{k \ge 0} \frac{\lambda^k}{k!} \int_{\Lambda^k} f(\delta_{x_1} + \dots + \delta_{x_k}) e^{-H(\mathbf{x})} \nu^{k}(\mathrm{d}\mathbf{x}).\] to every measurable function \(f\colon \mathcal{N} \to \mathbb{R}_{\ge 0}\) on the space of locally-finite counting measures \(\mathcal{N}\).
We write \(\mathsf{diam}(x_1, \dots, x_k) \mathrel{\vcenter{:}}= \max_{i, j} d(x_i, x_j)\) for the diameter of a tuple of points, and we say a potential \(\phi\) has range (at most) \(R\) if \(\phi(\mathbf{x}) = 0\) whenever \(\mathsf{diam}(\mathbf{x}) > R\). Our main result is the following.
Theorem 1. Suppose \(\phi\) is repulsive with finite range \(R\), and let \(B_{R}\) denote that volume of a ball of radius \(R\). There is a \(C>0\) such that if \(\left\lvert \lambda \right\rvert < (eB_{R})^{-1}\), \[\left\lvert \log(Z_{\Lambda, \phi}(\lambda)) \right\rvert \le C \cdot \nu(\Lambda).\]
1 implies that the limiting pressure \[p(\lambda) \mathrel{\vcenter{:}}= \lim_{\Lambda\uparrow \mathbb{X}} \frac{1}{\nu(\Lambda)} \log Z_{\Lambda,\phi}(\lambda)\] is an analytic function on the open disk \(|\lambda| < (e B_{R})^{-1}\) whenever the limit exists (e.g., if \(\mathbb{X}=\mathbb{R}^d\), \(\phi\) is translation invariant,1 and we consider a van Hove sequence of regions [9]). Correspondingly, there are no phase transitions for the particle system associated to the potential \(\phi\) when \(|\lambda|<(eB_{R})^{-1}\).
Example 1. The \(k\)-body hard sphere gas* has only the single non-zero term \[\phi_{k}(x_{1},\dots, x_{k}) = \begin{cases} \infty &\text{ if } \exists y \in \mathbb{X}: \{x_1, \dots, x_k\} \subseteq \mathbb{B}_y(r) \\ 0 &\text{ otherwise} \end{cases},\] where \(\mathbb{B}_y(r)\) denotes the closed ball with radius \(r\) and center \(y\). Thus, \(\phi\) forbids \(k\) spheres of radius \(r\) from overlapping at a single point in space. 1 gives the same disk of analyticity \(|\lambda|\leq (eB_{2r})^{-1}\), irrespective of the value of \(k\).*
Prior results have generally imposed either restrictive criteria (e.g., requiring a hard-core pair potential [5], [6]), imposed analytic requirements on potentials for technical purposes [4], [8], or some combination of both. The imposition of a hard-core is a natural criteria that removes one of the central difficulties in the analysis of multi-body potentials, see 1.2.3 below.
The criteria in [8] are rather difficult to apply if one wishes to obtain explicit estimates, even in the case of repulsive potentials. The criteria in [4] (see also [10]), however, are relatively simple for potentials of the form considered in the present paper, as the condition of regularity used in [4] is satisfied by repulsive potentials. This leads to bounds \(|\lambda|\leq (2 e B_{R})^{-1}\), which is only slightly worse than the bound \((e B_{R})^{-1}\) the Kirkwood-Salsburg method gives for two-body potentials. Our improvement of removing this factor of two is relatively modest. However, we think our method more transparently shows the difficulties in obtaining stronger results.
We prove 1 by establishing a recursive integral identity for (a modified version of) the point densities of finite-volume Gibbs point processes with repulsive multi-body potentials. See 2. Provided that this integral identity is a contraction on a rich enough class of potentials and \(|\lambda|\) is small enough, this yields uniform bounds on the modified point densities. 1 then follows by expressing \(\log(Z_{\Lambda, \phi}(\lambda))\) in terms of modified point densities.
Our approach is inspired by the work of Michelen and Perkins, who derived a similar integral identity for repulsive pair potentials [2], [11], [12]. As discussed by Michelen and Perkins, their approach was inspired by Weitz’s tree recursion for discrete spin systems [13]. In a similar way our identity can be viewed as an adaptation of a tree recursion for spin systems on hypergraphs [14], [15] to the continuum setting of Gibbs point processes.
While our integral identity (2) is derived using ideas very similar to those applied in the case of two-body potentials, it should be noted that the derivation in [2] uses the classical point density functions associated to a Gibbs point process. In contrast, our identity concerns a modified version of these densities, which allows for a finer control over the terms that appear at deeper levels of the recursion. In fact, the modified point densities used in our identity can be seen as lying midway between the classical point densities and a type of thermodynamic ratio that was studied for pair potentials in a largely unnoticed paper by Meeron [16]. We note, however, that although the recursive identity in [16] has a similar flavor to that in [2], it remains unclear if Meeron’s analysis has a useful extension to the multi-body setting.
While for simplicity we have used our method only to obtain disks of analyticity, our method is not intrinsically restricted to such domains. One could attempt to find complex domains that contain larger segments of the positive real axis on which our integral identity is a contraction. Such a program has been carried out for two-body potentials [2], as well as for the hypergraph tree recursion for discrete spin systems that our identities mimic [15].
1 concerns general multi-body potentials, and the domain of analyticity determined does not depend on either the minimum nor maximum (if it exists) of the arities \(k\) such that \(\phi_{k}\neq 0\). One might wonder if a hypothesis that \(\phi_{k}=0\) for all \(k\leq K\) can lead to an improvement (in the dependence on range) of bounds. Somewhat surprisingly, improved bounds are not possible in generality: in 6 we give an example that shows the disk of analyticity in 1 is (essentially) sharp. To go beyond our results (or those of [4]) requires exploiting geometric properties of the model being considered.
The impossibility of improved bounds is inspired (and established) by analogous results in the discrete setting of hypergraphs. The discrete analog of the hard sphere gas is the hard-core model (or independent set model). We recall the relevant definitions in 6; they are not needed for the present discussion. On a usual graph (edges all of size \(2\)) the optimal zero-free disk for the hard-core partition function over graphs of maximum degree \(\Delta\) is known and has connections to the Lovász Local Lemma from combinatorics [17], [18]. For \(k\)-uniform hypergraphs of maximum degree \(\Delta\) (each edge of size \(k\)), \(k \ge 3\), one might expect an improved bound; it has been proved recently that the zero-free disk is at least as large as that for graphs [15], [19], [20], but the disk need not be much larger [21]. This is in spite of the fact that a dilute phase persists for a much larger range of positive activities \(\lambda\) [22]–[24]. The analogy between the hard-core model on hypergraphs and particles interacting via a \(k\)-body interaction is not entirely accurate, however. Since only one particle is allowed per vertex in a discrete system, there is an implied short-range \(2\)-body hard-core interaction which need not be present in the continuum system.
Relatedly, and recalling our motivating discussion in 1 and Example 1, it is natural to wonder if the domain of analyticity for \(3\)-body hard spheres is strictly greater than for \(2\)-body hard spheres. While improvements by detailed calculation may be feasible, a softer argument that applies to \(k\)-body hard spheres would be preferable. It would be even more interesting to have a non-perturbative argument that establishes that for \(k\geq 3\), the first point of non-analyticity on the positive real activity axis for \(k\)-body hard spheres cannot occur before the corresponding point for \((k-1)\)-body hard spheres. That is, \(k\)-body hard spheres do not undergo a phase transition before \((k-1)\)-body hard spheres.
One might also wonder if the strength of the potential can be exploited to obtain improved bounds. To explain this by example, consider the \(k\)-body \(\alpha\)-soft sphere model, given by the pure \(k\)-body interaction which has the single non-zero potential \[\phi_{k}(x_{1},\dots, x_{k}) = \begin{cases} \alpha &\text{ if } \exists y \in \mathbb{X}: \{x_1, \dots, x_k\} \subseteq \mathbb{B}_y(r) \\ 0 &\text{ otherwise} \end{cases},\] for some \(\alpha \in (0,\infty)\). Intuitively, the low-density phase of the \(k\)-body \(\alpha\)-soft sphere model should occur in a strictly larger domain than for the \(k\)-body hard sphere model, and one might expect the same is true for domains and/or disks of analyticity. Our results do not establish this, however. The fundamental difficulty is that there are configurations of particles in which there are many particles at essentially the same location, and the potential created by these particles can approximate the \(k\)-body hard sphere potential arbitrarily well. It seems difficult to exploit the fact that such configurations are entropically unlikely (or that there are associated helpful \(1\)-body potentials created as well) with our method.
It is perhaps worth remarking that in the special case of a \(2\)-body interaction, it is well-known that a soft interaction does lead to an improved disk of analyticity via the temperedness constant. See, e.g., [2]. In this special case the improvement is straightforward since the potentials created by particles are \(1\)-body potentials, which only have the effect of decreasing the activity parameter. A method for analyzing \(k\)-body potentials which clearly exhibited similar improvements would be interesting.
Let \((\mathbb{X}, d)\) be a complete, separable metric space, and let \(\mathcal{B}\) be the Borel algebra generated by \(d\). Equip \((\mathbb{X}, \mathcal{B})\) with a reference measure \(\nu\), which we refer to as volume. We will always assume \(\nu\) is locally finite, i.e., assigns finite volume to bounded Borel sets.
As indicated earlier, the basic example to have in mind is \(\mathbb{X}=\mathbb{R}^{n}\), \(d\) the Euclidean metric. For more general \(\mathbb{X}\) we will need to impose one “smoothness” assumption that Euclidean space possesses. For every \(x \in \mathbb{X}\), let \(\nu_x\) denote the push-forward of \(\nu\) under the map \(y \mapsto d(x, y)\), i.e., for \(U \subseteq \mathbb{R}\) measurable, \(\nu_x(U) = \nu(\{y \in \mathbb{X}\mid d(x, y) \in U\})\).
Assumption 1. We assume \((X,d,\nu)\) has the property that for all \(x\in \mathbb{X}\), the measure \(\nu_x\) is absolutely continuous with respect to Lebesgue measure on \(\mathbb{R}\).
In this generality we define potentials and partition functions exactly as in 1.1; the only change is the interpretation of the metric and volume measure. 2 below will describe how 1 will be used.
Given a potential \(\phi\) and a point \(x \in \mathbb{X}\), we define the potential with pinning \(x\), denote \(\phi(\cdot \mid x)\), by \[\phi(\mathbf{y} \mid x) \mathrel{\vcenter{:}}= \phi(\mathbf{y}) + \phi(\mathbf{y}, x), \quad \mathbf{y} \in \mathbb{X}^m, m \in \mathbb{N}.\] In this equation we have slightly abused notation; \(\phi(\mathbf{y},x) = \phi( (\mathbf{y},x))\) where \((\mathbf{y},x)\) is interpreted as an element of \(\mathbb{X}^{m+1}\). We extend this modification to tuples of points \(\mathbf{x}\in \mathbb{X}^{k}\) by \[\phi(\mathbf{y} \mid \mathbf{x}) \mathrel{\vcenter{:}}= \sum_{S \subseteq [k]} \phi(\mathbf{y}, \mathbf{x}_{S}),\] where we have again abused notation slightly. Denote by \(H(\cdot \mid \mathbf{x})\) the Hamiltonian associated with \(\phi(\cdot \mid \mathbf{x})\) and write \(Z_{\Lambda, \phi}(\lambda \mid \mathbf{x})\) for the partition function on \(\Lambda\) with activity \(\lambda\in \mathbb{C}\) and modified potential \(\phi(\cdot \mid \mathbf{x})\).
Provided \(Z_{\Lambda,\phi}(\lambda)\neq 0\), define modified (complex) \(k\)-point densities by \[\label{eq:mkptdensity} \kappa_{\Lambda, \phi, \lambda}(\mathbf{x}) \mathrel{\vcenter{:}}= \lambda^k \cdot \frac{Z_{\Lambda, \phi}(\lambda \mid \mathbf{x})}{Z_{\Lambda, \phi}(\lambda)} ,\tag{2}\] where we allow ourselves to omit \(\Lambda, \phi\) or \(\lambda\) from the notation if they are clear from the context. Using a telescopic product, modified \(k\)-point densities for \(\mathbf{x} = (x_1, \cdots, x_k)\) can be written as products of modified one-point densities: \[\begin{align} \label{eq:telescoping} \kappa(\mathbf{x}) = \prod_{j = 1}^{k} \lambda \cdot \frac{Z_{\Lambda, \phi}(\lambda \mid x_1, \dots, x_{j})}{Z_{\Lambda, \phi}(\lambda \mid x_1, \dots, x_{j-1})} =\prod_{j = 1}^{k} \kappa(x_j \mid x_1, \dots, x_{j-1}), \end{align}\tag{3}\] where \(\kappa(x_j \mid x_1, \dots, x_{j-1})\) are the modified one-point densities at \(x_j\) under the potentials \(\phi(\cdot \mid x_1, \dots, x_{j-1})\). The \(j=1\) term is to be understood as \(\kappa(x_1)\).
Remark 1. For \(\lambda\geq 0\), the modified densities are related to the classical \(k\)-point densities (sometimes also called correlation functions) via \(\rho(\mathbf{x}) = e^{-H(\mathbf{x})} \cdot \kappa(\mathbf{x})\), see [9].
The goal of this section is to establish 2, an integral identity for modified point densities. The correct formulation of the identity requires several preliminary definitions.
Fix an arbitrary \(z\in \mathbb{X}\). Given a tuple of points \(\mathbf{x} = (x_1, \dots, x_k) \in \mathbb{X}^{k}\), define \[\label{eq:Dchoice} D(\mathbf{x}) \mathrel{\vcenter{:}}= \sum_{j \in [k]} d(z, x_j).\tag{4}\] The specific choice of \(D\) in 4 is not crucial; what is important are the following properties of the restriction \(D_{k}\) of \(D\) to \(\mathbb{X}^{k}\). The proof of the lemma will be deferred to 4.
Lemma 1. If \((\mathbb{X}, d, \nu)\) satisfies 1, then for every \(k\in\mathbb{N}\):
(1) \(D_k\) is symmetric and measurable.
(2) The push-forward of \(\nu^k\) under \(D_k\) is absolutely continuous with respect to the Lebesgue measure on \(\mathbb{R}_{\ge 0}\).
(3) For every \(k \in \mathbb{N}\) and \(\nu^k\)-almost all \(\mathbf{x} \in \mathbb{X}^k\), it holds that \(D\) induces a strict total order on subsets of \([k]\) by setting \(T < S\) iff \(D(\mathbf{x}_T) < D(\mathbf{x}_S)\).
The significance of part [lemma:ordering:continuity] of the preceding lemma is the following.
Lemma 2. Under 1, for each \(k\geq 1\), there is a family of measures \(\{\nu_{t}^{k}\}_{t \in \mathbb{R}_{\ge 0}}\) such that
\(\nu^{k}_{t}\) assigns measure zero to the set \(\{\mathbf{w}\in \mathbb{X}^{k} \mid D(\mathbf{w})\neq t\}\), and
there is a density \(g_{k}\) on \(\mathbb{R}_{\ge 0}\) such that \(\int_{\mathbb{X}^{k}} h(\mathbf{w}) \nu^k(\mathrm{d}\mathbf{w}) = \int_{\mathbb{R}_{\ge 0}} \int_{\mathbb{X}^{k}} h(\mathbf{w}) \nu^k_t(\mathrm{d}\mathbf{w}) g_k(t) \mathrm{d}t\) for all \(h\colon \mathbb{X}^{k} \to \mathbb{C}\) such that the integral exists.
Proof. This is the disintegration theorem for the Lebesgue measure combined with part [lemma:ordering:continuity] of 1. ◻
For tuples \(\mathbf{x} \in \mathbb{X}^{k}, \mathbf{y} \in \mathbb{X}^{m}\) and \(t \in [0, \infty]\) we write \(\mathbf{x} \prec t\) if \(D(\mathbf{x}) < t\) and \(\mathbf{x} \prec \mathbf{y}\) if \(D(\mathbf{x}) < D(\mathbf{y})\). Moreover, given \(y \in \mathbb{X}, \mathbf{x} \in \mathbb{X}^k, \mathbf{w} \in \mathbb{X}^m\), we use these relations to define the modified potentials \[\begin{align} \phi(\mathbf{x} \mid y_{\prec t}) \mathrel{\vcenter{:}}= \begin{cases} \phi(\mathbf{x} \mid y) &\text{ if \mathbf{x} \prec t} \\ \phi(\mathbf{x}) &\text{ otherwise} \end{cases}, \qquad \phi(\mathbf{x} \mid y_{\prec \mathbf{w}}) &\mathrel{\vcenter{:}}= \begin{cases} \phi(\mathbf{x} \mid y) &\text{ if } \mathbf{x} \prec \mathbf{w} \\ \phi(\mathbf{x}) &\text{ otherwise.} \end{cases} \end{align}\] Note that \(\phi(\cdot \mid y) = \phi(\cdot \mid y_{\prec \infty})\) and \(\phi(\cdot \mid y_{\prec \mathbf{w}}) = \phi(\cdot \mid y_{\prec D(\mathbf{w})})\). We think of these modified potentials as being partial pinnings: the pinning at \(\mathbf{y}\) is imposed on configurations that precede \(\mathbf{w}\) (according to \(D\)).
As in 2.1 for pinned potentials, we extend this notation to Hamiltonians, partition functions and modified point densities. For example, we write \(H(\cdot \mid y_{\prec \mathbf{w}})\), \(Z_{\Lambda, \phi}(\cdot \mid y_{\prec \mathbf{w}})\) and \(\kappa(\cdot \mid y_{\prec \mathbf{w}})\) for the Hamiltonian, partition function and modified point density associated with the potential \(\phi(\mathbf{x} \mid y_{\prec \mathbf{w}})\).
Occasionally, it will be necessary to iterate pinnings/partial pinnings. When this occurs, we separate the modifications we make with a semicolon, and the modifications are made from left to right. For example, \(\phi(\cdot \mid y_{\prec \mathbf{w}}; \mathbf{z}) = \psi(\cdot \mid \mathbf{z})\) is the pinning of \(\psi = \phi(\cdot \mid y_{\prec \mathbf{w}})\) at \(\mathbf{z}\). As before, we extend this notation to Hamiltonians, partition functions, and point densities.
Recall that modified \(k\)-point densities are only well-defined when the associated partition function is non-zero. To be able to discuss \(k\)-point densities for a class of potentials, we define the notion of being \(\delta\)-zero-free.
Definition 1. Let \(\delta > 0\). A collection of potentials \(\Phi\) is \(\delta\)-zero-free* for a bounded and measurable \(\Lambda \subset \mathbb{X}\) and an activity \(\lambda \in \mathbb{C}\) if \(|Z_{\Lambda, \phi}(\lambda)| \ge \delta\) for all \(\phi\in\Phi\).*
Our integral identity will relate \(k\)-point densities to \(k\)-point densities associated with modified potentials. Accordingly, we introduce the following definition.
Definition 2. A collection of potentials \(\Phi\) is closed under modification* if*
(1) for every \(\phi \in \Phi\), \(y \in \mathbb{X}\) and \(t \in [0, \infty]\) it holds that \(\phi(\cdot \mid y_{\prec t}) \in \Phi\), and
(2) for all \(\phi \in \Phi\) and measurable \(\Delta \subseteq \mathbb{X}\), the potentials \(\psi\) with \(\psi_k = \phi_k\) for all \(k \ge 2\) and \[\psi_1(x) = \begin{cases} \infty &\text{ if x \in \Delta}\\ \phi_1(x) &\text{ otherwise } \end{cases}\] are contained in \(\Phi\).
Remark 2. Property (2) is a consistency condition that allows for the imposition of boundary conditions. Property (1) ensures that the \(\Phi\) is closed under (partial) pinnings of a single point, using that, for every \(y \in \mathbb{X}\), \(\phi(\cdot \mid y) = \phi(\cdot \mid y_{\prec \infty})\). This further extends to pinnings of tuples \(\mathbf{y} \in \mathbb{X}^k\) via the recursion \(\phi(\mathbf{x} \mid \mathbf{y}) = \psi(\mathbf{x} \mid y_k) \text{ for } \psi(\mathbf{x}) \coloneq \phi(\mathbf{x} \mid \mathbf{y}_{[k-1]})\).
Sets of potentials that are closed under modification satisfy a useful continuity property, summarized in the next lemma.
Lemma 3. Let \(\Phi\) be a collection of repulsive potentials, and let \(U \subset \mathbb{C}\) be bounded. For every bounded, measurable \(\Lambda \subset \mathbb{X}\), the family of functions \[\{U \to \mathbb{C}, \lambda \mapsto Z_{\Lambda, \phi}(\lambda) \mid \phi \in \Phi\}\] is uniformly equicontinuous. Moreover, if \(\Phi\) is closed under modification and \(\delta\)-zero-free on \(\Lambda\) at some activity \(\lambda_0 \in \mathbb{C}\) for some \(\delta > 0\), then there exists some complex neighborhood \(W\) of \(\lambda_0\), such that the family of functions \[\{W \to \mathbb{C}, \lambda \mapsto \kappa_{\Lambda, \phi, \lambda}(x) \mid \phi \in \Phi, x \in \Lambda\}\] is uniformly equicontinuous.
Proof. Set \(M = \sup\{\left\lvert \lambda \right\rvert \mid \lambda \in U\}\). Note that for \(\lambda_1, \lambda_2 \in U\), \(|\lambda_{1}^{k}-\lambda_{2}^{k}|\leq kM^{k-1}|\lambda_{1}-\lambda_{2}|\). Hence for all \(\phi \in \Phi\) and \(\lambda_1, \lambda_2 \in U\) it holds that \[\left\lvert Z_{\Lambda, \phi}(\lambda_1) - Z_{\Lambda, \phi}(\lambda_2) \right\rvert \le \sum_{k \in \mathbb{N}} \frac{\left\lvert \lambda_1^k - \lambda_2^k \right\rvert}{k!} \cdot \nu(\Lambda)^k \leq \left\lvert \lambda_1 - \lambda_2 \right\rvert \cdot \nu(\Lambda) \cdot e^{M \nu(\Lambda)}.\] This proves the first statement. For the second part, let \(U\) be any bounded neighborhood of \(\lambda_0\). Using the fact that \(\Phi\) is \(\delta\)-zero-free at \(\lambda_0\) and the first part of the lemma, we can find some neighborhood \(W \subseteq U\) such that \[\{W \to \mathbb{C}, \lambda \mapsto Z_{\Lambda, \phi}(\lambda) \mid \phi \in \Phi\}\] is uniformly equicontinuous and uniformly bounded from below by \(\delta/2\). By the definition 2 of the modified point density and the fact that \(\Phi\) is closed under modification, \(\kappa_{\Lambda, \phi, \lambda}(x)\) is (\(\lambda\) times) a ratio of functions in this last set. This shows that \[\{W \to \mathbb{C}, \lambda \mapsto \kappa_{\Lambda, \phi, \lambda}(x) \mid \phi \in \Phi, x \in \Lambda\}\] is uniformly equicontinuous as well. ◻
The main result of this section is the following theorem.
Theorem 2. Suppose \((\mathbb{X}, d, \nu)\) satisfies 1. Let \(\Lambda \subseteq \mathbb{X}\) be bounded and measurable, \(\lambda \in \mathbb{C}\) and let \(\Phi\) be a collection of symmetric and repulsive potentials that are closed under modification. If \(\Phi\) is \(\delta\)-zero-free for these parameters, then for all \(\phi \in \Phi\) the associated modified point densities satisfy \[\label{eq:itegral95identity} \kappa(y) = \lambda \cdot \exp \left(- \sum_{k \in \mathbb{N}} \frac{1}{k!} \int_{\Lambda^k} \left(1 - e^{-\phi(y, \mathbf{w})}\right) \cdot e^{-H(\mathbf{w} \mid y_{\prec \mathbf{w}})} \cdot \kappa(\mathbf{w} \mid y_{\prec \mathbf{w}}) \nu^k(\mathrm{d}\mathbf{w})\right) .\qquad{(1)}\]
To prove 2 we need three lemmas. The first lemma is a fundamental theorem of calculus. We were unable to find a reference for this (surely well-known) result, and hence provide a proof in 5.
Lemma 4. Let \(t > 0\), and let \(f\colon [0, t] \to \mathbb{C}\setminus \{0\}\) be absolutely continuous. It holds that \[\frac{f(t)}{f(0)} = \exp\left(\int_{0}^{t} \frac{f'(s)}{f(s)} \mathrm{d}s \right),\] where the integral should be understood as a Lebesgue integral and \(f'\) is almost everywhere a derivative of \(f\).
The second lemma is elementary.
Lemma 5. Suppose \(A\) is a totally ordered finite set with total order \(<\), and let \(\{x_{a}\}_{a\in A}\) be indeterminates. Then \[\label{eq:orderproduct} \prod_{a\in A}(1+x_{a}) = 1+\sum_{b\in A}x_{b}\prod_{a<b}(1+x_{a}).\qquad{(2)}\]
Proof. Apply the binomial theorem. Collect terms according to the largest \(b\) such that \(x_{b}\) occurs. ◻
Lemma 6. Consider the setting of 2. For all \(t \in [0, \infty]\) it holds that \[Z_{\Lambda, \phi}(\lambda \mid y_{\prec t}) = Z_{\Lambda, \phi}(\lambda) - \sum_{k \in \mathbb{N}} \frac{\lambda^k}{k!} \int_{\Lambda^k} \mathbb{1}_{\mathbf{w} \prec t} \cdot \left( 1 - e^{-\phi(y, \mathbf{w})} \right) \cdot e^{-H(\mathbf{w} \mid y_{\prec \mathbf{w}})} Z_{\Lambda, \phi}(\lambda \mid y_{\prec \mathbf{w}}; \mathbf{w}) \nu^k(\mathrm{d}\mathbf{w}).\]
Proof. Recall that \[Z_{\Lambda, \phi}(\lambda \mid y_{\prec t}) = 1 + \sum_{m \in \mathbb{N}} \frac{\lambda^m}{m!} \int_{\Lambda^m} e^{-H(\mathbf{x} \mid y_{\prec t})} \nu^m(\mathrm{d}\mathbf{x}) .\] Moreover, note that for all \(m \in \mathbb{N}\) and \(\mathbf{x} \in \Lambda^m\) it holds that \[e^{-H(\mathbf{x} \mid y_{\prec t})} = e^{-H(\mathbf{x})} \prod_{S \subseteq [m]: S \neq \emptyset} \left( 1 - \mathbb{1}_{\mathbf{x}_S \prec t} \cdot \left(1 - e^{- \phi(y, \mathbf{x}_S)}\right)\right).\] By part [lemma:ordering:maximum] of 1, for \(\nu^m\)-almost all \(\mathbf{x}\), the set of \(S\subset [m]\) are totally ordered according to whether or not \(\mathbf{x}_{S}\prec \mathbf{x}_{S'}\). Hence by applying 5, \[\begin{align} e^{-H(\mathbf{x}\mid y_{\prec t})} &= e^{-H(\mathbf{x})} - e^{-H(\mathbf{x})}\sum_{B\subset [m]: B\neq \emptyset} \mathbb{1}_{\mathbf{x}_{B}\prec t}(1-e^{-\phi(y,\mathbf{x}_{B})}) \prod_{S\neq \emptyset: \mathbf{x}_{S}\prec \mathbf{x}_{B}} (1-\mathbb{1}_{\mathbf{x}_{S}\prec t}(1-e^{-\phi(y,\mathbf{x}_{S})})) \\ &= e^{-H(\mathbf{x})} - e^{-H(\mathbf{x})}\sum_{B\subset [m]: B\neq \emptyset} \mathbb{1}_{\mathbf{x}_{B}\prec t}(1-e^{-\phi(y,\mathbf{x}_{B})}) \prod_{S\neq \emptyset: \mathbf{x}_{S}\prec \mathbf{x}_{B}} e^{-\phi(y,\mathbf{x}_{S})} \\ &= e^{-H(\mathbf{x})} - \sum_{B\subset [m]:B\neq\emptyset} \mathbb{1}_{\mathbf{x}\prec t}(1-e^{-\phi(y,\mathbf{x}_{B})})e^{-H(\mathbf{x}|y_{\prec \mathbf{x}_{B}})} \end{align}\] by the definition of \(H(\mathbf{x}| y_{\prec \mathbf{x}_{B}})\). Substituting this back into the definition of \(Z_{\Lambda, \phi}(\lambda \mid y_{\prec t})\) yields \[Z_{\Lambda, \phi}(\lambda \mid y_{\prec t}) = Z_{\Lambda, \phi}(\lambda) - \sum_{m \in \mathbb{N}} \frac{\lambda^m}{m!} \sum_{\substack{B \subseteq [m]:\\ B \neq \emptyset}} \int_{\Lambda^m} \mathbb{1}_{\mathbf{x}_B \prec t} \cdot \left(1 - e^{-\phi(y, \mathbf{x}_B)}\right) \cdot e^{-H\left(\mathbf{x} \mid y_{\prec \mathbf{x}_{B}}\right)} \nu^{m}(\mathrm{d}\mathbf{x}) .\] The value of the integral only depends on the cardinality of \(B\), and hence the last term above can be rewritten as \[\sum_{m \in \mathbb{N}} \frac{\lambda^m}{m!} \sum_{k=1}^{m} \binom{m}{k} \int_{\Lambda^k} \mathbb{1}_{\mathbf{w} \prec t} \cdot \left(1 - e^{-\phi(y, \mathbf{w})}\right) \int_{\Lambda^{m-k}} e^{-H(\mathbf{w}, \mathbf{x} \mid y_{ \prec \mathbf{w}})} \nu^{m - k}(\mathrm{d}\mathbf{x}) \nu^{k}(\mathrm{d}\mathbf{w}) .\] Changing the order of summation, setting \(\ell = m - k\), and exchanging summation and integration rewrites this quantity as \[\begin{align} \sum_{k \in \mathbb{N}} \frac{\lambda^k}{k!} \int_{\Lambda^k} \mathbb{1}_{\mathbf{w} \prec t} \cdot \left(1 - e^{-\phi(y, \mathbf{w})}\right) \sum_{\ell \ge 0} \frac{\lambda^{\ell}}{\ell!} \int_{\Lambda^{\ell}} e^{-H(\mathbf{w}, \mathbf{x} \mid y_{\prec \mathbf{w}})} \nu^{\ell}(\mathrm{d}\mathbf{x}) \nu^{k}(\mathrm{d}\mathbf{w}), \end{align}\] and thus \[\begin{align} Z_{\Lambda, \phi}(\lambda \mid y_{\prec t}) = Z_{\Lambda, \phi}(\lambda) - \sum_{k \in \mathbb{N}} \frac{\lambda^k}{k!} \int_{\Lambda^k} \mathbb{1}_{\mathbf{w} \prec t} \cdot \left(1 - e^{-\phi(y, \mathbf{w})}\right) \sum_{\ell \ge 0} \frac{\lambda^{\ell}}{\ell!} \int_{\Lambda^{\ell}} e^{-H(\mathbf{w}, \mathbf{x} \mid y_{\prec \mathbf{w}})} \nu^{\ell}(\mathrm{d}\mathbf{x}) \nu^{k}(\mathrm{d}\mathbf{w}). \end{align}\] The exchanges summation and integration above can be justified by using that \(\phi\) is repulsive. Finally, observing that \[H(\mathbf{w}, \mathbf{x} \mid y_{\prec \mathbf{w}}) = H(\mathbf{w} \mid y_{\prec \mathbf{w}}) + H(\mathbf{x} \mid y_{\prec \mathbf{w}}; \mathbf{w})\] yields \[\sum_{\ell \ge 0} \frac{\lambda^{\ell}}{\ell!} \int_{\Lambda^{\ell}} e^{- H(\mathbf{w}, \mathbf{x} \mid y_{\prec \mathbf{w}})} \nu^{\ell}(\mathrm{d}\mathbf{x}) = e^{- H(\mathbf{w}\mid y_{\prec \mathbf{w}})} \cdot Z_{\Lambda,\phi}(\lambda \mid y_{\prec \mathbf{w}}; \mathbf{w}),\] which concludes the proof. ◻
Proof of 2. Our first step is to obtain an expression for the derivative \(\frac{\partial Z_{\Lambda, \phi}(\lambda \mid y_{\prec t})}{\partial t}\) on \(\mathbb{R}_{\ge 0}\). We will do this by manipulating the formula obtained in 6. By applying 2 for each \(k\), \[\begin{align} &\sum_{k \in \mathbb{N}} \frac{\lambda^k}{k!} \int_{\Lambda^k} \mathbb{1}_{\mathbf{w} \prec t} \cdot \left( 1 - e^{-\phi(y, \mathbf{w})} \right) \cdot e^{-H(\mathbf{w} \mid y_{\prec \mathbf{w}})} Z_{\Lambda, \phi}(\lambda \mid y_{\prec \mathbf{w}}; \mathbf{w}) \nu^k(\mathrm{d}\mathbf{w}) \\ &= \int_{0}^{t} \sum_{k \in \mathbb{N}} \frac{\lambda^k}{k!} \int_{\Lambda^k} \mathbb{1}_{\mathbf{w} \prec t} \cdot \left( 1 - e^{-\phi(y, \mathbf{w})} \right) \cdot e^{-H(\mathbf{w} \mid y_{\prec \mathbf{w}})} Z_{\Lambda, \phi}(\lambda \mid y_{\prec \mathbf{w}}; \mathbf{w}) \nu_s^k(\mathrm{d}\mathbf{w}) g_k(s) \mathrm{d}s . \end{align}\] The exchange of summation and integration is allowed by absolute convergence since the potentials are repulsive (see discussion after 1 ). Substituting this back into 6 and applying the fundamental theorem of calculus for Lebesgue-integrable functions yields that \(t \mapsto Z_{\Lambda, \phi}(\lambda \mid y_{\prec t})\) is absolutely continuous on bounded intervals in \(\mathbb{R}_{\ge 0}\) and almost-everywhere differentiable, with \[\begin{align} \frac{\partial Z_{\Lambda, \phi}(\lambda \mid y_{\prec t})}{\partial t} = - \sum_{k \in \mathbb{N}} \frac{\lambda^k}{k!} \int_{\Lambda^k} \left( 1 - e^{-\phi(y, \mathbf{w})} \right) \cdot e^{-H(\mathbf{w} \mid y_{\prec \mathbf{w}})} \cdot Z_{\Lambda, \phi}(\lambda \mid y_{\prec \mathbf{w}}; \mathbf{w}) \nu_t^k(\mathrm{d}\mathbf{w}) g_k(t). \label{eq:derivative95partition95function} \end{align}\tag{5}\]
Observe that \(\phi(\cdot \mid y_{\prec 0}) = \phi\) and \(\phi(\cdot \mid y_{\prec \infty}) = \phi(\cdot \mid y)\). Moreover, it holds that \(Z_{\Lambda, \phi}(\lambda \mid y_{\prec t}) \to Z_{\Lambda, \phi}(\lambda \mid y_{\prec \infty})\) as \(t \to \infty\). Hence, by continuity of the exponential function and dominated convergence, the conclusion of the lemma follows if, for every \(t \in \mathbb{R}_{>0}\), \[\frac{Z_{\Lambda, \phi}(\lambda \mid y_{\prec t})}{Z_{\Lambda, \phi}(\lambda \mid y_{\prec 0})} = \exp \left(- \sum_{k \in \mathbb{N}} \frac{1}{k!} \int_{\Lambda^k} \mathbb{1}_{\mathbf{w} \prec t} \left(1 - e^{-\phi(y, \mathbf{w})}\right) \cdot e^{-H(\mathbf{w} \mid y_{\prec \mathbf{w}})} \cdot \kappa(\mathbf{w} \mid y_{\prec \mathbf{w}}) \nu^k(\mathrm{d}\mathbf{w})\right).\] Towards this, observe that since \(s \mapsto Z_{\Lambda, \phi}(\lambda \mid y_{\prec s})\) is non-zero and absolutely continuous on bounded intervals, 4 yields \[\frac{Z_{\Lambda, \phi}(\lambda \mid y_{\prec t})}{Z_{\Lambda, \phi}(\lambda \mid y_{\prec 0})} = \exp\left( - \int_0^t \frac{1}{Z_{\Lambda, \phi}(\lambda \mid y_{\prec s})} \cdot \frac{\partial Z_{\Lambda, \phi}(\lambda \mid y_{\prec s})}{\partial s} \mathrm{d}s\right) .\] Moreover, by 5 and the fact that \(\nu_s^k (\mathrm{d}\mathbf{w}) g_k(s) \mathrm{d}s = \nu^k(\mathrm{d}\mathbf{w})\) and \(Z_{\Lambda, \phi}(\lambda \mid y_{\prec s}) = Z_{\Lambda, \phi}(\lambda \mid y_{\prec \mathbf{w}})\) for \(\nu_s^k\)-almost all \(\mathbf{w} \in \Lambda^k\), we get \[\begin{align} & \int_0^t \frac{1}{Z_{\Lambda, \phi}(\lambda \mid y_{\prec s})} \cdot \frac{\partial Z_{\Lambda, \phi}(\lambda \mid y_{\prec s})}{\partial s} \mathrm{d}s \\ & = \int_0^{t} \sum_{k \in \mathbb{N}} \frac{\lambda^k}{k!} \int_{\Lambda^k} \left( 1 - e^{-\phi(y, \mathbf{w})} \right) \cdot e^{-H(\mathbf{w} \mid y_{\prec \mathbf{w}})} \cdot \frac{Z_{\Lambda, \phi}(\lambda\mid y_{\prec \mathbf{w}}; \mathbf{w})}{Z_{\Lambda, \phi}(\lambda \mid y_{\prec s})} \nu^k_s(\mathrm{d}\mathbf{w}) g_k(s) \mathrm{d}s\\ &= \sum_{k \in \mathbb{N}} \frac{1}{k!} \int_{\Lambda^k} \mathbb{1}_{\mathbf{w} \prec t} \left( 1 - e^{-\phi(y, \mathbf{w})} \right) \cdot e^{-H(\mathbf{w} \mid y_{\prec \mathbf{w}})} \cdot \kappa(\mathbf{w} \mid y_{\prec \mathbf{w}}) \nu^k(\mathrm{d}\mathbf{w}) , \end{align}\] where the exchange of integration and summation is justified by dominated convergence, using the fact that \(\mathbf{w} \mapsto \kappa(\mathbf{w} \mid y_{\prec \mathbf{w}})\) is bounded given that \(\Phi\) is closed under modification and \(\delta\)-zero-free on \(\Lambda\) at activity \(\lambda\). ◻
Remark 3. One approach to proving zero-freeness for partition functions is to derive a “connected parts” identity that shows \(\log Z\) can be expressed as a sum over terms indexed by trees (for two-body potentials) or generalized tree-like objects [25]. We view ?? as a variation on this theme: \(\kappa(y)\) is a derivative of \(\log Z\), and our formula locally explores the tree-like structure of the particle at \(y\). Our choice of \(D\) in 4 plays the role of the choice of a partition scheme, see, e.g., [26]. By avoiding explicit sums over tree-like objects we avoid convergence issues related to their enumeration.
The following lemma has a proof similar to that of 2; the conjunction of the lemmas is what will allow us to establish zero-free disks for partition functions.
Lemma 7. Suppose \((\mathbb{X}, d, \nu)\) satisfies 1, and let \(z\in \mathbb{X}\).2 Let \(\Lambda \subset \mathbb{X}\) be bounded and measurable, \(\lambda \in \mathbb{C}\) and let \(\Phi\) be a collection of repulsive potential that is closed under modification. For every \(\phi \in \Phi\) and \(x \in \mathbb{X}\), define the potential \(\phi^{(x)}\) with \(\phi^{(x)}_k = \phi_k\) for \(k \ge 2\) and \[\phi^{(x)}_1(y) \mathrel{\vcenter{:}}= \begin{cases} \infty &\text{ if } d(z, y) < d(z, x) \\ \phi_1(y) &\text{ otherwise}. \end{cases}\] Note that \(\phi^{(x)} \in \Phi\). If \(\Phi\) is \(\delta\)-zero-free on \(\Lambda\) at activity \(\lambda \in \mathbb{C}\) for some \(\delta > 0\), then, for all \(\phi \in \Phi\), it holds that \[\log(Z_{\Lambda, \phi}(\lambda)) = \int_{\Lambda} e^{-\phi^{(x)}(x)} \kappa^{(x)}(x) \nu(\mathrm{d}x),\] where \(\kappa^{(x)}\) is the modified point density associated with \(\phi^{(x)}\) on \(\Lambda\) at activity \(\lambda\).
Proof. For \(t \in [0, \infty]\) define the potential function \(\phi^{(t)}\) with \(\phi^{(t)}_k = \phi_k\) for all \(k \ge 2\) and \[\phi^{(t)}_1(y) \mathrel{\vcenter{:}}= \begin{cases} \infty \text{ if } d(z, y) < t \\ \phi_1(y) \text{ otherwise } \end{cases}\] for the same \(z \in \mathbb{X}\) as in the definition of \(\phi^{(x)}\). Write \(H^{(x)}\) and \(H^{(t)}\) for the energy functions associated with \(\phi^{(x)}\) and \(\phi^{(t)}\). Moreover, define \(\Lambda_x = \{y \in \Lambda \mid d(z, y) < d(z, x)\}\). We start by observing that, for every \(m \ge 1\), it holds that \[\begin{align} \int_{\Lambda^m} \lambda^m e^{-H^{(t)}(\mathbf{z})} \nu^m(\mathrm{d}\mathbf{z}) &= m \int_{\Lambda} \int_{(\Lambda_x)^{m-1}} \lambda^m e^{-H^{(t)}(x, \mathbf{z})} \nu^{m-1}(\mathrm{d}\mathbf{z}) \nu(\mathrm{d}x) \\ &= m \int_{\Lambda} \lambda e^{-\phi^{(t)}(x)} \int_{(\Lambda_x)^{m-1}} \lambda^{m-1} e^{-H^{(t)}(\mathbf{z} \mid x)} \nu^{m-1}(\mathrm{d}\mathbf{z}) \nu(\mathrm{d}x) \\ &= m \int_{\Lambda} \lambda e^{-\phi^{(t)}(x)} \int_{\Lambda^{m-1}} \lambda^{m-1} e^{-H^{(x)}(\mathbf{z} \mid x)} \nu^{m-1}(\mathrm{d}\mathbf{z}) \nu(\mathrm{d}x), \end{align}\] where the first equality follows from the fact that \(H^{(t)}\) is invariant under permutation of points. Hence \[\begin{align} Z_{\Lambda, \phi^{(t)}}(\lambda) - 1 = \sum_{m \in \mathbb{N}} \frac{\lambda^m}{m!} \int_{\Lambda^m} e^{-H^{(t)}(\mathbf{z})} \nu^{m}(\mathrm{d}\mathbf{z}) = \int_{\Lambda} \lambda e^{-\phi^{(t)}(x)} Z_{\Lambda, \phi^{(x)}}(\lambda \mid x) \nu(\mathrm{d}x). \end{align}\] The last integral can be written as a double integral by 2 in the case \(k=1\). Using the fundamental theorem of calculus for Lebesgue integrable functions then yields that \(t \mapsto Z_{\Lambda, \phi^{(t)}}(\lambda)\) is absolutely continuous on bounded intervals and almost everywhere differentiable with \[\frac{\partial Z_{\Lambda, \phi^{(t)}}(\lambda)}{\partial t} = \int_{\Lambda} \lambda e^{-\phi^{(t)}(x)} Z_{\Lambda, \phi^{(x)}}(\lambda \mid x) \nu_t(\mathrm{d}x) g(t) \mathrm{d}t.\] Further, observe that \(Z_{\Lambda, \phi^{(\infty)}}(\lambda) = \lim_{t \to \infty} Z_{\Lambda, \phi^{(t)}}(\lambda) = 1\) and \(Z_{\Lambda, \phi^{(0)}}(\lambda) = Z_{\Lambda, \phi}(\lambda)\). Applying 4 similarly to the proof of 2, using that \(\Phi\) is closed under modification and \(\delta\)-zero-free, yields \[\begin{align} \log(Z_{\Lambda, \phi}(\lambda)) &= - \int_0^{\infty} \frac{1}{Z_{\Lambda,\phi^{(t)}}(\lambda)}\frac{\partial Z_{\Lambda, \phi^{(t)}}(\lambda)}{\partial t} \mathrm{d}t \\ &= - \int_{\mathbb{R}_{\ge 0}} \int_{\Lambda} \lambda e^{-\phi^{(t)}(x)} \frac{Z_{\Lambda, \phi^{(x)}}(\lambda \mid x)}{Z_{\Lambda, \phi^{(t)}}(\lambda)} \nu_t(\mathrm{d}x) g(t) \mathrm{d}t. \end{align}\] Using that \(Z_{\Lambda, \phi^{(t)}}(\lambda) = Z_{\Lambda, \phi^{(x)}}(\lambda)\) for almost all \(t\) and \(\nu_t\)-almost all \(x\) gives \[\begin{align} \log(Z_{\Lambda, \phi}(\lambda)) &= - \int_{\mathbb{R}_{\ge 0}} \int_{\Lambda} \lambda e^{-\phi^{(t)}(x)} \frac{Z_{\Lambda, \phi^{(x)}}(\lambda \mid x)}{Z_{\Lambda, \phi^{(x)}}(\lambda)} \nu_t(\mathrm{d}x) g(t) \mathrm{d}t \\ &= \int_{\Lambda} e^{-\phi^{(x)}(x)} \kappa^{(x)}(x) \nu(\mathrm{d}x).\qedhere \end{align}\] ◻
We denote by \(\Phi_R\) the collection of all repulsive potentials with range at most \(R\). Our main result is the following.
Theorem 3. Suppose \((\mathbb{X}, d, \nu)\) satisfies 1. Let \(B_r \mathrel{\vcenter{:}}= \sup_{x \in \mathbb{X}} \nu(\mathbb{B}_x(r))\). For \(R\geq 0\) and \(\lambda \in \mathbb{C}\) with \(\left\lvert \lambda \right\rvert < \frac{1}{e B_R}\), there is a \(C \ge 0\) such that for all bounded, measurable \(\Lambda \subseteq \mathbb{X}\) and \(\phi \in \Phi_R\) it holds that \(\left\lvert \log(Z_{\Lambda, \phi}(\lambda)) \right\rvert \le C \cdot \nu(\Lambda)\).
The last ingredient needed for the proof of the theorem beyond 2 and 6 is the following contraction argument.
Lemma 8. Let \(\Lambda \subseteq \mathbb{X}\) be bounded and measurable and \(\lambda\in \mathbb{C}\) with \(|\lambda| \le \frac{1 - \varepsilon}{e B_R}\) for some \(\varepsilon > 0\). For all \(\phi \in \Phi_{R}\) and all \(f \colon \bigcup_{k \in \mathbb{N}} \mathbb{X}^k \to \mathbb{C}\) with \(|f(\mathbf{x})| \le B_R^{-k}\) for \(\nu^k\)-almost all \(\mathbf{x}\in \Lambda^{k}\) it holds that \[\left | \lambda\exp \left(- \sum_{k \in \mathbb{N}} \frac{1}{k!} \int_{\Lambda^k} \left(1 - e^{-\phi(y, \mathbf{x})}\right) \cdot e^{-H(\mathbf{x} \mid y_{\prec \mathbf{x}})} f(\mathbf{x}) \nu^k(\mathrm{d}\mathbf{x})\right) \right | \le \frac{1 - \varepsilon}{B_R} \,.\]
Proof. We show that, for all \(z \in [0, B_R^{-1}]\) and \(\phi \in \Phi_{R}\), it holds that \[\sum_{k \in \mathbb{N}} \frac{z^k}{k!} \int_{\Lambda^k} \left(1 - e^{-\phi(y, \mathbf{x})}\right) \cdot e^{-H(\mathbf{x} \mid y_{\prec \mathbf{x}})} \nu^k(\mathrm{d}\mathbf{x}) \le 1,\] from which the claim follows. To this end, define a sequence of functions \[G_{N}^{\phi}(z) \mathrel{\vcenter{:}}= \sum_{k = 1}^{N} \frac{z^k}{k!} \int_{\Lambda^k} \left(1 - e^{-\phi(y, \mathbf{x})}\right) \cdot e^{-H(\mathbf{x} \mid y_{\prec \mathbf{x}})} \nu^k(\mathrm{d}\mathbf{x})\] for \(N \in \mathbb{N}\). By taking \(N\to\infty\), it suffices to inductively show that \(G_N^{\phi} \le 1\). The base case \(N = 1\) follows as (i) since \(\phi\) is repulsive, the exponential term can be omitted for an upper bound and (ii) the bounded range assumption gives an upper bound of \(B_{R}\) for the remaining integral.
Now, suppose \(G_N^{\phi} \le 1\) for all \(\phi \in \Phi_R\) and some \(N \ge 1\). The proof will consist of finding a potential \(\hat{\phi}\in \Phi_{R}\) such that \(G_{N+1}^{\phi}(z)\leq G_{N}^{\hat{\phi}}(z)\); the claim then follows by induction. To achieve this we will combine the terms \(k=N\) and \(k=N+1\) in \(G_{N+1}\). The key identity for establishing this is that \[H(\mathbf{x}, w \mid y_{\prec \mathbf{x}, w}) = H(\mathbf{x} \mid y_{\prec \mathbf{x}}) + \phi(\mathbf{x}, y) + \sum_{T \subseteq [k]} \phi(\mathbf{x}_T, w \mid y_{\prec \mathbf{x}, w}),\] which follows from splitting all sub-tuples of \((\mathbf{x}, w)\) based on whether they contain \(w\) or not, and the fact that \(\mathbf{x}_{S}\prec \mathbf{x} \prec (\mathbf{x}, w)\) for \(\nu^{k}\)-almost all \(\mathbf{x}\in \mathbb{X}^{k}\) and \(\nu\)-almost all \(w \in \mathbb{X}\) provided \(S\subsetneq [k]\). Using this, we will show that \[G_{N + 1}^{\phi}(z) = G_{N-1}^{\phi}(z) + \frac{z^N}{N!} \int_{\Lambda^N} \left(1 - e^{-\phi(y, \mathbf{x})} \cdot (1 - \psi(\mathbf{x}, y))\right) \cdot e^{-H(\mathbf{x} \mid y_{\prec \mathbf{x}})} \nu^N(\mathrm{d}\mathbf{x}).\] where \[\psi(\mathbf{x}, y) = \frac{z}{N + 1} \int_{\Lambda} \left(1 - e^{-\phi(y, \mathbf{x}, w)}\right) \prod_{T \subseteq [N]} e^{- \phi(\mathbf{x}_T, w \mid y_{\prec \mathbf{x}, w})} \nu(\mathrm{d}w) .\] Indeed, this holds as \[\begin{align} &\frac{z^N}{N!} \int_{\Lambda^N} \left(1 - e^{-\phi(y, \mathbf{x})}\right) \cdot e^{-H(\mathbf{x} \mid y_{\prec \mathbf{x}})} \nu^N(\mathrm{d}\mathbf{x}) + \frac{z^{N+1}}{(N+1)!} \int_{\Lambda^{N+1}}\left(1 - e^{-\phi(y, \mathbf{w})}\right) \cdot e^{-H(\mathbf{w} \mid y_{\prec \mathbf{w}})} \nu^{N+1}(\mathrm{d}\mathbf{w}) \\ &= \frac{z^N}{N!} \int_{\Lambda^N} \left[ \left(1 - e^{-\phi(y, \mathbf{x})}\right) \cdot e^{-H(\mathbf{x} \mid y_{\prec \mathbf{x}})} + \frac{z}{N+1} \int_{\Lambda}\left(1 - e^{-\phi(y, \mathbf{x}, w)}\right) \cdot e^{-H(\mathbf{x}, w \mid y_{\prec \mathbf{x}, w})} \nu(\mathrm{d}w) \right] \nu^N(\mathrm{d}\mathbf{x}) \\ &= \frac{z^N}{N!} \int_{\Lambda^N} \left(1 - e^{-\phi(y, \mathbf{x})} \cdot (1 - \psi(\mathbf{x}, y))\right) \cdot e^{-H(\mathbf{x} \mid y_{\prec \mathbf{x}})} \nu^N(\mathrm{d}\mathbf{x}) , \end{align}\]
Since \(\phi\) is repulsive with range \(\le R\) and \(z \in [0, B_R^{-1}]\), it holds that \(0 \le \psi(x, y) \le 1\). Since \(\phi(y, \mathbf{x}, w) = 0\) if \(\mathsf{diam}(y, \mathbf{x}) > R\), in fact \(\psi(x, y) \le \mathbb{1}_{\mathsf{diam}(y, \mathbf{x}) \le R}\). Thus, setting \[\hat{\phi}(\mathbf{w}) \mathrel{\vcenter{:}}= \begin{cases} \phi(\mathbf{w}) \text{ if } |\mathbf{w}| \le N \\ \mathbb{1}_{\mathsf{diam}(\mathbf{w}) \le R} \cdot \infty \text{ otherwise} \end{cases}\] we have that \(\hat{\phi}\) is repulsive, and \[1 - e^{-\phi(y, \mathbf{x})} \cdot (1 - \psi(\mathbf{x}, y)) \le 1 - e^{-\hat{\phi}(y, \mathbf{x})}\] for all \(\mathbf{x} \in \Lambda^{N}\), and \(G_{N-1}^{\phi}=G_{N-1}^{\hat{\phi}}\). Hence, we have \[G_{N + 1}^{\phi}(z) \le G_{N-1}^{\hat{\phi}}(z) + \frac{z^N}{N!} \int_{\Lambda^N} \left(1 - e^{-\hat{\phi}(y, \mathbf{x})}\right) \cdot e^{-H_{\hat{\phi}}(\mathbf{x} \mid y_{\prec \mathbf{x}})} \nu^N(\mathrm{d}\mathbf{x}) = G_{N}^{\hat{\phi}}(z)\] as desired. ◻
Proof of 3. Fix some bounded, measurable \(\Lambda \subseteq \mathbb{X}\) and let \(A \subseteq \mathbb{C}\) be the set of all activities \(\lambda\) such that, at activity \(\lambda\), \(\Phi_R\) is \(e^{-\nu(\Lambda)/B_R}\)-zero-free on \(\Lambda\) and \(\left\lvert \kappa_{\phi}(x) \right\rvert \le \frac{1}{B_R}\) for all \(\phi \in \Phi_R\) and \(x \in \Lambda\). Our goal is to show that \(\mathbb{D}(1/e B_R) \subseteq A\), where \(\mathbb{D}(t)\) is the open disk of radius \(t\) around \(0\) in \(\mathbb{C}\). We note that this is equivalent to showing that \(t^*\mathrel{\vcenter{:}}= \sup\{t \ge 0 \mid \overline{\mathbb{D}(t)} \subseteq A\} \geq (e B_R)^{-1}\).
Towards a contraction, suppose that \(t^* = \frac{1 - \varepsilon}{e B_R}\) for some \(\varepsilon > 0\). We first observe that \(t^* \ge 0\), since at activity \(\lambda=0\) the modified point densities are \(0\) and the partition functions are \(1\). Together with the uniform equicontinuity of modified 1-point densities and partition functions given in 3, this implies \(t^{*} > 0\). Further, by a similar continuity argument, we know that \(\overline{\mathbb{D}(t^*)} \subseteq A\).3 However, using the telescoping product in 3 and 2, we note that, for every \(\phi \in \Phi_R\), \(x \in \Lambda\) and \(|\lambda| = t^*\), the modified \(k\)-point densities that appear in the recursion for \(\kappa_{\phi, \lambda}(x)\) given by 2 are bounded by \(B_R^{-k}\). Thus, applying 8 yields that, for every \(\left\lvert \lambda \right\rvert = t^{*}\), all \(\phi \in \Phi_R\) and all \(x \in \Lambda\), it holds that \(\left\lvert \kappa_{\phi, \lambda}(x) \right\rvert \le \frac{1 - \varepsilon}{B_R}\). Again, by uniform equicontinuity, we can conclude that there is some neighborhood \(\mathcal{N}(\lambda)\), uniformly in \(x \in \Lambda\) and \(\phi \in \Phi_R\), such that \(\left\lvert \kappa_{\phi}(x) \right\rvert \le \frac{1}{B_R}\) and \(\Phi_R\) is \(\delta\)-zero-free for some \(\delta > 0\) at all activities in \(\mathcal{N}(\lambda)\). An application of 7 then proves that \(\Phi_R\) is in fact \(e^{-\nu(\Lambda)/B_R}\)-zero-free on \(\Lambda\) at all activities in \(\mathcal{N}(\lambda)\). Since these neighborhoods \(\mathcal{N}(\lambda)\) for activities \(\left\lvert \lambda \right\rvert = t^*\) give an open cover of the compact set \(\overline{\mathbb{D}(t^*)} \setminus \mathbb{D}(t^*)\), we can pass to a finite sub-cover to find some \(t' > t^{*}\) such that \(\overline{\mathbb{D}(t')} \subseteq A\), which contradicts the definition of \(t^{*}\). ◻
TH thanks the Hausdorff Institute for Mathematics for its hospitality, where part of this work was carried out (funded by the Deutsche Forschungsgemeinschaft under Germany’s Excellence Strategy – EXC-2047/1 – 390685813). MP was funded by the HPI Research School on Foundations of AI. WP supported in part by NSF grant DMS-2348743.
Proof of 1. For part [lemma:ordering:measurability] symmetry is immediate. Measurability is shown by induction, using \(D_{k+1}(x_1, \dots, x_{k+1}) = D_k(x_1, \dots, x_k) + d(z, x_{k+1})\).
For [lemma:ordering:continuity] we need to show that for all Lebesgue null sets \(U \subset \mathbb{R}\), it holds that \(\nu^{k} \circ D_k^{-1} (U) = 0\) for all \(k\geq 1\). For \(k=1\), this follows from 1. Inductively, if the statement holds for some \(k \in \mathbb{N}\), then \[\begin{align} \nu^{k+1} \circ D_{k+1}^{-1}(U) &= \int_{\mathbb{X}^{k+1}} \mathbb{1}_{D_{k+1}(\mathbf{x}) \in U} \nu^{k+1}(\mathrm{d}\mathbf{x}) \\ &= \int_{\mathbb{X}} \int_{\mathbb{X}^{k}} \mathbb{1}_{D_{k}(\mathbf{x}) \in U - d(z, y)} \nu^{k}(\mathrm{d}\mathbf{x}) \nu(\mathrm{d}y) , \end{align}\] where \(U - d(z, y)\) is the translation of the set \(U\) by \(- d(z, y)\). Since this translate is again a Lebesgue null set, it follows from the induction hypothesis that the inner integral evaluates to \(\nu^{k} \circ D_k^{-1} (U - d(z, y)) = 0\), concluding the induction.
For [lemma:ordering:maximum], we note that the induced ordering is trivially irreflexive, asymmetric, and transitive. It remains to show that, for \(\nu^k\)-almost all \(\mathbf{x} \in \mathbb{X}^k\) and all \(S \neq T\) it holds that \(D(\mathbf{x}_S) \neq D(\mathbf{x}_T)\). We define \[E_k \mathrel{\vcenter{:}}= \{\mathbf{x} = (x_1, \dots, x_k) \in \mathbb{X}^k \mid \text{ \exists S,T \subseteq [k], S \neq T such that D(\mathbf{x}_S) = D(\mathbf{x}_T)}\},\] and prove that \(\nu^{k}(E_k) = 0\) by induction on \(k\). For \(k = 1\), this follows from 1. Suppose the statement holds for some \(k \in \mathbb{N}\). For \(\mathbf{x} = (x_1, \dots, x_k) \in \mathbb{X}^k\), define \[F_{\mathbf{x}} \mathrel{\vcenter{:}}= \big\{y \in \mathbb{X}\mid d(z, y) \in \bigcup_{S, T \subseteq [k]: S \neq T} \{\left\lvert D(\mathbf{x}_S) - D(\mathbf{x}_T) \right\rvert\}\big\}.\] A necessary conditions for \((x_1, \dots, x_{k+1}) \in E_{k+1}\) is that \((x_1, \dots, x_k) \in E_k\) or \(x_{k+1} \in F_{x_1, \dots, x_k}\). Thus (applying a union bound) we have \[\nu^{k+1}(E_{k+1}) \le \int_{\mathbb{X}} \int_{\mathbb{X}^k} \mathbb{1}_{\mathbf{x} \in E_k} \nu^{k}(\mathrm{d}\mathbf{x}) \nu(\mathrm{d}x_{k+1}) + \int_{\mathbb{X}^k} \int_{\mathbb{X}} \mathbb{1}_{x_{k+1} \in F_{\mathbf{x}}} \nu(\mathrm{d}x_{k+1}) \nu^k(\mathrm{d}\mathbf{x}) .\] By the induction hypothesis, the first term is \(0\) and, by 1, the second term is \(0\). ◻
At first glance, 4 appears to follow from taking the logarithm of both sides of the identity and then applying the fundamental theorem of calculus. However, under the conditions of the statement, \(\log f (z)\) might be discontinuous for any branch of the complex logarithm. Hence, the proof of 4 is slightly more technical and requires some auxiliary claims.
To this end, we start with a recap of the basic definitions related to line integrals. For \(U \subseteq \mathbb{C}\) connected and open, a map \(\gamma\colon [a, b] \to U\) is called a rectifiable path in \(U\) if it is continuous and has bounded variation. Given a continuous function \(f\colon U \to \mathbb{C}\), the line integral \(\int_{\gamma} f(z) \mathrm{d}z\) is defined as the limit of sums of the form \[\sum_{j = 1}^n f(\gamma(t_j^*)) \cdot (\gamma(t_j) - \gamma(t_{j-1})) ,\] where \(a = t_0 < \dots < t_n = b\), \(t_j^* \in [t_{j-1}, t_j]\) and the limit is taken as the mesh size \(\max_{j \in [n]} \left\lvert t_j - t_{j-1} \right\rvert\) goes to \(0\). See [27] for a proof of convergence. We note that every absolutely continuous function \(\gamma\colon [a, b] \to U\) is a rectifiable path (see [28] Lemma 3.34). Our first ingredient is the following identity for line integrals over absolutely continuous paths.
Lemma 9. Suppose \(\gamma\colon [a, b] \to U\) is an absolutely continuous path in a connected open set \(U \subseteq \mathbb{C}\) and \(f\colon U \to \mathbb{C}\) is continuous then \[\int_{\gamma} f(z) \mathrm{d}z = \int_a^b f(\gamma(s)) \gamma'(s) \mathrm{d}s,\] where the integral on the right-hand side should be understood as a Lebesgue integral and \(\gamma'\) is almost everywhere a derivative of \(\gamma\).
While we expect 9 is well-known, it is typically presented for piecewise smooth \(\gamma\), in which case the right-hand side can be treated as a Riemann integral. For completeness we give a proof of the more general claim.
Proof of 9. As \(\gamma\) is absolutely continuous, it is almost everywhere differentiable and the derivative \(\gamma'\) is Lebesgue integrable on \([a, b]\) [29]. We must show \[\sum_{j = 1}^n f(\gamma(t_j^*)) \cdot (\gamma(t_j) - \gamma(t_{j-1})) ,\] converges to \(\int_a^b f(\gamma(s)) \gamma'(s) \mathrm{d}s\) as the mesh size \(\max_{j \in [n]} \left\lvert t_j - t_{j-1} \right\rvert\) approaches \(0\). Note that for all \(a \le t_{j-1} < t_{j} \le b\) the second fundamental theorem of calculus for Lebesgue integrals (see [29]) yields \(\gamma(t_j) - \gamma(t_{j-1}) = \int_{t_{j-1}}^{t_{j}} \gamma'(s) \mathrm{d}s\), and hence \[\left\lvert \int_a^b f(\gamma(s)) \gamma'(s) \mathrm{d}s - \sum_{j = 1}^n f(\gamma(t_j^*)) \cdot (\gamma(t_j) - \gamma(t_{j-1})) \right\rvert \le \sum_{j = 1}^n \int_{t_{j-1}}^{t_j} \left\lvert f(\gamma(s)) - f(\gamma(t_j^*)) \right\rvert \cdot \left\lvert \gamma'(s) \right\rvert \mathrm{d}s.\] Note that \(f \circ \gamma\) is uniformly continuous on \([a, b]\) by compactness. Hence for \(\varepsilon > 0\) we may choose \(\max_{j \in [n]} \left\lvert t_j - t_{j-1} \right\rvert\) small enough such that \[\sum_{j = 1}^n \int_{t_{j-1}}^{t_j} \left\lvert f(\gamma(s)) - f(\gamma(t_j^*)) \right\rvert \cdot \left\lvert \gamma'(s) \right\rvert \mathrm{d}s \le \varepsilon \cdot \int_a^b \left\lvert \gamma'(s) \right\rvert \mathrm{d}s\] which implies the result since \(\gamma'\) is integrable. ◻
Lemma 10. Let \(\gamma\colon [a, b] \to U\) be a rectifiable path in a connected open set \(U \subseteq \mathbb{C}\setminus \{0\}\). Then \[\exp \left( \int_{\gamma} \frac{1}{z} \mathrm{d}z \right) = \frac{\gamma(b)}{\gamma(a)} . \qedhere\]
10 says that \(\int_{\gamma} \frac{1}{z} \mathrm{d}z\) is a logarithm of \(\gamma(b)/\gamma(a)\). This is a standard result if \(U \subseteq \mathbb{C}\setminus \{0\}\) is simply connected and \(\gamma\) is piecewise smooth [30]. By approximating rectifiable paths by polygonal paths, we lift this to connected open sets.
Proof of 10. By [27], it holds that, for every \(\varepsilon > 0\), we can find some polygonal path \(\Gamma\colon [a, b] \to U\) with \(\Gamma(a) = \gamma(a)\), \(\Gamma(b) = \gamma(b)\) and \(\left\lvert \int_{\gamma} \frac{1}{z} \mathrm{d}z - \int_{\Gamma} \frac{1}{z} \mathrm{d}z \right\rvert < \varepsilon\). Hence it suffices to show \(\exp(\int_{\Gamma} \frac{1}{z} \mathrm{d}z) = \Gamma(b)/\Gamma(a)\). Let \(\Gamma_{i}\) denote the set of line segments comprising \(\Gamma\). Since each \(\Gamma_{i}\) avoids \(0\), we can find some simply connected open set \(U_i \subseteq \mathbb{C}\setminus \{0\}\) that contains \(\Gamma_i\). Since each \(\Gamma_i\) is smooth, \(\exp(\int_{\Gamma_i} \frac{1}{z} \mathrm{d}z) = \Gamma(t_i)/\Gamma(t_{i-1})\). Since \(\int_{\Gamma} \frac{1}{z} \mathrm{d}z = \sum_{i=1}^{k}\int_{\Gamma_i} \frac{1}{z} \mathrm{d}z\) taking the exponential of both sides and collapsing the resulting telescopic product concludes the proof. ◻
Proof of 4. Set \(\gamma\colon [0, t] \to \mathbb{C}, s \mapsto \frac{f(s)}{f(0)}\). Note that \(\gamma(0) = 1\) and \(\gamma(t) = \frac{f(t)}{f(0)}\). By absolute continuity of \(s \mapsto f(s)\) and the fact that \(f\) avoids \(0\), \(\gamma\) is a rectifiable path in \(\mathbb{C}\setminus \{0\}\). Hence by 10 it holds that \(\exp ( \int_{\gamma} \frac{1}{z} \mathrm{d}z ) = \frac{f(t)}{f(0)}\), and it suffices to prove that \(\int_{\gamma} \frac{1}{z} \mathrm{d}z = \int_{0}^{t} \frac{f'(s)}{f(s)} \mathrm{d}s\). Since \(z \mapsto 1/z\) is continuous on \(\mathbb{C}\setminus 0\) and \(\gamma\) is an absolutely continuous path in \(\mathbb{C}\setminus \{0\}\), this follows from 9 and the fact that \(\gamma' = f'\) almost everywhere. ◻
Call a potential \(\phi = (\phi_m)_{m \in \mathbb{N}}\) a pure \(k\)-body potential if \(\phi_{m}=0\) for all \(m \neq k\). This appendix proves the following proposition.
Proposition 4. Fix \(R>0\), \(k\ge 2\). There exists a sequence of complete, separable metric spaces \((\mathbb{X}_n, d_n,\nu_{n})_{n \in \mathbb{N}}\) satisfying 1 and a pure \(k\)-body potential \(\phi^{(n)}\) of range \(R\) such that
\(B^{(n)}_R \mathrel{\vcenter{:}}= \sup_{x \in \mathbb{X}_n} \nu_n(\{y \in \mathbb{X}_n \mid d_n(x, y) \le R\})\) satisfies \(\lim_{n \to \infty} B^{(n)}_R = \infty\).
There are \(\Lambda_{n}\subseteq \mathbb{X}_{n}\) and \(\lambda_n = O\left(\frac{\log B^{(n)}_R}{B^{(n)}_R}\right)\) such that, for every \(n \in \mathbb{N}\), \(Z_{\Lambda_n, \phi^{(n)}}(-\lambda_n) = 0\).
The takeaway of 4 is that \(B_{R}\) is essentially the optimal scaling for the activity if one wants to ensure the analyticity of the pressure in a disk of positive radius. The proof of 4 is by construction of an appropriate sequence of spaces that allow us to make use of a known result about the location of negative roots in the hypergraph hard-core model. We will shortly define what this last object is, but first, we state two lemmas that explains why it is useful.
Lemma 11. For every \(\Delta \in \mathbb{N}\), every \(k\)-uniform hypergraph \(G = (V, E)\) of maximum degree \(\Delta\) and every \(R > 0\), there is a complete, separable metric space \((\mathbb{X}, d)\) with compatible locally-finite volume measure \(\nu\), a pure \(k\)-body potential \(\phi\) of range \(R\) and bounded, measurable region \(\Lambda \subseteq \mathbb{X}\) such that \((\mathbb{X}, d, \nu)\) satisfies 1, \(B_R = \Delta + 1\) and \(Z_{\Lambda, \phi}(\lambda) = Z_{G}(e^{\lambda} - 1)\) for all \(\lambda \in \mathbb{C}\), where \(Z_{G}\) denotes the independence polynomial of \(G\).
Given 11, which is proven below, 4 follows from the following theorem concerning the zeros of the hypergraph independence polynomial.
Theorem 5 ([21]). For every \(k \ge 2\), there is a sequence of \(k\)-uniform hypergraphs \((G_n)_{n \in \mathbb{N}}\) with maximum degrees \(\Delta_n \to \infty\), such that for some sequence of vertex activities \(z_n \in O\left(\frac{\log \Delta_n}{\Delta_n}\right)\) such that \(Z_{G_n}(-z_n) = 0\).
The following definitions concerning hypergraphs and the hypergraph independence polynomial are all that will be needed to prove 11. A hypergraph \(G=(V,E)\) generalizes a graph by allowing for \(E\) to be a set of subsets \(e\subset V\) of arbitrary cardinality \(|e|\geq 2\). It is \(k\)-uniform if \(|e|=k\) for all \(e\in E\). The degree of a vertex \(v\) is the number of hyperedges containing \(v\). A subset \(I\subset V\) is an independent set if \(|I\cap e|<|e|\) for all \(e\in E\). Lastly, \(Z_{G}(z) = \sum_{I\subset V} z^{|I|}\mathbb{1}_{I\text{ independent} }\).
Proof of 11. It will be convenient to assume \(V = [N]\) for some \(N \in \mathbb{N}\). Let \(d_G\) denote the hop distance in \(G\). That is, \(d_{G}(v,v')=k\) if the shortest sequence \(\{e_{i}\}\) of hyperedges such that \(v\in e_{1}\), \(v'\in e_{k}\) and \(e_{i}\cap e_{i+1}\neq\emptyset\) has length \(k\). Set \(\mathbb{X}\mathrel{\vcenter{:}}= \bigcup_{j \in [N]} [2j, 2j+1]\), \[\label{e:UBd} d(x, y) \mathrel{\vcenter{:}}= \begin{cases} \frac{R}{10} \cdot |x-y| &\text{ x, y \in [2j, 2j+1]},\\ \frac{R}{10} \cdot \left( 8d_G\left(\lfloor x \rfloor/2, \lfloor y \rfloor/2\right) + \frac{1}{10} \cdot |x - \lfloor x \rfloor| + \frac{1}{10} \cdot |y - \lfloor y \rfloor|\right) &\text{ otherwise,} \end{cases}\tag{6}\] and let \(\nu\) be Lebesgue measure on \(\mathbb{R}\) restricted to \(\mathbb{X}\). In the right-hand side of 6 , and below, we have made use of our above identification of integers with vertices of \(G\).
We first verify the structural properties of \((\mathbb{X},d,\nu)\) and 1. It is easily checked that \(d\) is indeed a metric on \(\mathbb{X}\). To see that \((\mathbb{X}, d)\) is complete, we note that every sequence that is Cauchy with respect to \(d\) must eventually be contained in some interval \([2j, 2j+1]\) for some \(j \in [N]\). Convergence in \([2j,2j+1]\) is then inherited from the completeness of \((\mathbb{R},|\cdot|)\) since \(d\) is a rescaling of the Euclidean distance. This verifies completeness.
Next, note that the Borel algebra on \((\mathbb{X}, d)\) is inherited from the Borel algebra \(\mathcal{B}(\mathbb{R})\) on \(\mathbb{R}\), i.e., consists of sets \(\{ B\cap \mathbb{X}\mid B\in \mathcal{B}(\mathbb{R})\}\). To see this, note that every Borel subset of \(B \subseteq \mathbb{X}\) in \(\mathbb{R}\) can be partitioned into sets \(B_j = B \cap [2j, 2j+1]\) for \(j \in [N]\). It suffices then to show that \(B_j\) is also in the Borel algebra generated from \(d\), which again follows from the equivalence of metrics on such intervals.
Since \(\nu\) is Lebesgue measure on \(\mathbb{R}\) restricted to \(\mathbb{X}\), we can observe that \(B_R = \Delta + 1\). To see that \((\mathbb{X}, d, \nu)\) satisfies 1, it is easy to check that, for every \(x \in \mathbb{X}\), the map \(r \mapsto \nu(\{y \in \mathbb{X}\mid d(x, y) \le r\})\) is an absolutely continuous function on \(\mathbb{R}\).
We now define \(\phi\). Set, for \((x_1, \dots, x_k) \in \mathbb{X}^k\), \(\phi_k(x_1, \dots, x_k) = \infty \cdot \mathbb{1}_{E}(\{\lfloor x_1 \rfloor/2, \dots, \lfloor x_k \rfloor/2\})\). Otherwise set \(\phi_j = 0\) for all \(j \neq k\). By definition, \(\phi\) is a pure \(k\)-body potential and has range at most \(R\).
It remains to show that \(Z_{\Lambda, \phi}(\lambda) = Z_{G}(e^{\lambda} - 1)\) for all \(\lambda \in \mathbb{C}\). Given \((x_{1},\dots, x_{m})\), let \(I(x_1, \dots, x_m) \mathrel{\vcenter{:}}= \{\lfloor x_i \rfloor/2 \mid i \in [m]\}\) denote the corresponding vertex set in \(G\). Note that \(H(x_{1},\dots, x_{m})=0\) if \(I(x_{1},\dots, x_{m})\) is an independent set in \(G\), and \(H(x_{1},\dots, x_{m})=\infty\) otherwise. Evidently \(|I(x_1, \dots, x_m)| \le m\), with \(|I(x_1, \dots, x_m)| = m\) if and only if each all \(x_i\) are in distinct intervals \([2j, 2j+1]\) for \(j \in [N]\). Recall that for all finite sets \(A, B\) there are \(|B|! \genfrac\{\}{0pt}{}{|A|}{|B|}\) subjective maps from \(A\) to \(B\), where \(\genfrac\{\}{0pt}{}{a}{b}\) denotes Stirling number of the second kind. Using this, observe that for every non-empty independent set \(S\) in \(G\) and every \(m \ge |S|\), it holds that \[\int_{\Lambda^m} \mathbb{1}_{I(\mathbf{x}) = S} \nu(\mathrm{d}\mathbf{x}) = |S|! \genfrac\{\}{0pt}{}{m}{|S|}.\] Combining the observations above, we have \[Z_{\Lambda, \phi}(\lambda) = 1 + \sum_{m \in \mathbb{N}} \frac{\lambda^{m}}{m!} \int_{\Lambda^m} e^{-H(\mathbf{x})} \nu(\mathrm{d}\mathbf{x}) = 1 + \sum_{m \in \mathbb{N}} \frac{\lambda^{m}}{m!} \sum_{\ell = 1}^{m} \ell! \genfrac\{\}{0pt}{}{m}{\ell} |\mathcal{I}_{\ell}|,\] where \(\mathcal{I}_{\ell}\) is the set of independent sets of \(G\) with cardinality \(\ell\). Exchanging the sums and using that \(\sum_{m=\ell}^{\infty} \frac{x^{m}}{m!} \genfrac\{\}{0pt}{}{m}{\ell} = \frac{(e^x - 1)^{\ell}}{\ell!}\), we obtain \[Z_{\Lambda, \phi}(\lambda) = 1 + \sum_{\ell \in \mathbb{N}} |\mathcal{I}_{\ell}| \sum_{m =\ell}^{\infty} \frac{\lambda^{m}}{m!} \ell! \genfrac\{\}{0pt}{}{m}{\ell} = 1 + \sum_{\ell \in \mathbb{N}} |\mathcal{I}_{\ell}| (e^{\lambda} - 1)^{\ell} = Z_{G}(e^{\lambda} - 1).\qedhere\] ◻