Perfect simulation of infinite range Gibbs measures and coupling with their finite range approximations

TL;DR

This paper introduces a perfect simulation algorithm for infinite range Gibbs measures and their finite range approximations, providing a method to accurately sample and quantify errors in approximation.

Contribution

It presents a novel perfect simulation algorithm applicable to general Gibbsian interactions with tail conditions, enabling exact sampling of infinite and finite range measures.

Findings

Algorithm successfully samples from infinite range Gibbs measures.

Provides an upper bound on errors when using finite range approximations.

Applicable to general Gibbs interactions with tail decay conditions.

Abstract

In this paper we address the questions of perfectly sampling a Gibbs measure with infinite range interactions and of perfectly sampling the measure together with its finite range approximations. We solve these questions by introducing a perfect simulation algorithm for the measure and for the coupled measures. The algorithm works for general Gibbsian interaction under requirements on the tails of the interaction. As a consequence we obtain an upper bound for the error we make when sampling from a finite range approximation instead of the true infinite range measure.