Quasipolynomial-time algorithms for Gibbs point processes

TL;DR

This paper presents a quasipolynomial-time deterministic algorithm for approximating the partition function of Gibbs point processes with finite-range stable potentials, extending to all activities within a zero-free region.

Contribution

It introduces a novel quasipolynomial-time deterministic approximation algorithm for the partition function of Gibbs point processes, applicable under broad activity conditions.

Findings

Provides a quasipolynomial-time algorithm for all activities with zero-free partition functions.

Improves activity range for repulsive potentials like hard-sphere gases by a factor of at least e^2.

Uses the interpolation method of Barvinok to approximate coefficients of the cluster expansion.

Abstract

We demonstrate a quasipolynomial-time deterministic approximation algorithm for the partition function of a Gibbs point process interacting via a finite-range stable potential. This result holds for all activities $\lambda$ for which the partition function satisfies a zero-free assumption in a neighborhood of the interval $[0,\lambda]$. As a corollary, for all finite-range stable potentials we obtain a quasipolynomial-time determinsitic algorithm for all $\lambda < /(e^{B + 1} \hat C_\phi)$ where $\hat C_\phi$ is a temperedness parameter and $B$ is the stability constant of $\phi$. In the special case of a repulsive potential such as the hard-sphere gas we improve the range of activity by a factor of at least $e^2$ and obtain a quasipolynomial-time deterministic approximation algorithm for all $\lambda < e/\Delta_\phi$, where $\Delta_\phi$ is the potential-weighted connective constant of the potential $\phi$. Our algorithm approximates coefficients of the cluster expansion of the partition function and uses the interpolation method of Barvinok to extend this approximation throughout the zero-free region.