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

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.