Gibbs point process approximation: Total variation bounds using Stein's method

TL;DR

This paper develops bounds on the total variation distance between Gibbs point process distributions using Stein's method, with applications to spatial statistics and physics.

Contribution

It introduces a general approach for bounding total variation distances between Gibbs processes via Stein's method, including explicit couplings and Stein factors.

Findings

Bounds for Lennard-Jones processes comparison

Hard core approximation of area interaction process

Approximation of lattice by continuous Gibbs process

Abstract

We obtain upper bounds for the total variation distance between the distributions of two Gibbs point processes in a very general setting. Applications are provided to various well-known processes and settings from spatial statistics and statistical physics, including the comparison of two Lennard-Jones processes, hard core approximation of an area interaction process and the approximation of lattice processes by a continuous Gibbs process. Our proof of the main results is based on Stein's method. We construct an explicit coupling between two spatial birth-death processes to obtain Stein factors, and employ the Georgii-Nguyen-Zessin equation for the total bound.