Analyticity for locally stable hard-core gases via recursion

TL;DR

This paper extends the analyticity results of pressure in particle systems to locally stable, tempered, hard-core potentials, providing improved bounds and new recursive methods for high-temperature regimes.

Contribution

It introduces a novel recursive identity and activity modulation approach for locally stable hard-core potentials, extending prior analyticity bounds.

Findings

Pressure is analytic up to a new activity bound involving Lambert W-function.

The new bound improves classical results by at least a factor of e^2 in high-temperature regimes.

Method applies to a broader class of potentials, including hard-core and tempered interactions.

Abstract

In their recent works [Comm. Math. Phys. 399:1 (2023)] and [arXiv:2109.01094], Michelen and Perkins proved that the pressure of a system of particles with repulsive pair interactions is analytic for activities up to $e\Delta_{\phi}(\beta)^{-1}$, where $\Delta_{\phi}(\beta)\in(0,C_{\phi}(\beta)]$ is a constant they called the potential-weighted connective constant. This paper extends their method to locally stable, tempered, and hard-core pair potentials. Our main result is that the pressure of such a system is analytic for activities up to $e^{2-2W(eA_{\phi}(\beta)/\Delta_{\phi}(\beta))}\Delta_{\phi}(\beta)^{-1}e^{-(\beta C+1)}$, where $C\ge0$ is the local stability constant, $W(\cdot)$ the Lambert $W$-function, $A_{\phi}(\beta)$ the contribution from the attraction in the pair potential to the temperedness constant, and $\Delta_{\phi}(\beta)\in[A_{\phi}(\beta),C_{\phi}(\beta)]$ a counterpart of the constant defined by Michelen and Perkins. The main ingredients in the proof include a recursive identity for the one-point density tailored to locally stable hard-core potentials and a corresponding notion of modulations of an activity function. In the high-temperature regime, our result surpasses the classical Penrose-Ruelle bound of $C_{\phi}(\beta)^{-1}e^{-(\beta C+1)}$ by at least a factor of $e^{2}$.