Analyticity for locally stable hard-core gases via recursion

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 $e\hat{I}{”}_{\ddot{I}}•({\hat{I}}^2{)}^{-1}$, where $\hat{I}{”}_{\ddot{I}}•({\hat{I}}^2)\in (0,C_{\ddot{I}}•({\hat{I}}^2)]$ 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 $e^{2-2W(e{A}_{\ddot{I}}•({\hat{I}}^2)/\hat{I}{”}_{\ddot{I}}•({\hat{I}}^2))}\hat{I}{”}_{\ddot{I}}•({\hat{I}}^2{)}^{-1}{e}^{-({\hat{I}}^{2}C+1)}$, where $C\ge 0$ is the local stability constant, $W(\cdot )$ the Lambert $W$-function, $A_{\ddot{I}}•({\hat{I}}^2)$ the contribution from the attraction in the pair potential to the temperedness constant, and $\hat{I}{”}_{\ddot{I}}•({\hat{I}}^2)\in [A_{\ddot{I}}•({\hat{I}}^2),C_{\ddot{I}}•({\hat{I}}^2)]$ 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 $C_{\ddot{I}}•({\hat{I}}^2{)}^{-1}{e}^{-({\hat{I}}^{2}C+1)}$ by at least a factor of $e^2$.

Major overhaul with improved results. 31 pages