Developments in perfect simulation of Gibbs measures through a new result for the extinction of Galton-Watson-like processes

TL;DR

This paper introduces new conditions for perfect sampling of Gibbs measures with infinite range interactions, demonstrating that local modifications do not hinder simulability and optimizing conditions for specific models.

Contribution

It provides novel sufficient conditions for the extinction of Galton-Watson-like processes, impacting perfect simulation methods for Gibbs measures with long-range interactions.

Findings

Local modifications do not affect perfect simulation feasibility.

Optimized conditions for Ising and finite-range spin models.

Complete solutions for specific model classes.

Abstract

This paper deals with the problem of perfect sampling from a Gibbs measure with infinite range interactions. We present some sufficient conditions for the extinction of processes which are like supermartingales when large values are taken. This result has deep consequences on perfect simulation, showing that local modifications on the interactions of a model do not affect simulability. We also pose the question to optimize over a class of sequences of sets that influence the sufficient condition for the perfect simulation of the Gibbs measure. We completely solve this question both for the long range Ising models and for the spin models with finite range interactions.