Stability of gas measures under perturbations and discretizations

TL;DR

This paper establishes a probabilistic framework for gas models, providing a diluteness condition that ensures uniqueness, stability, and perfect simulation of the infinite-volume measure, surpassing previous cluster expansion methods.

Contribution

It introduces a weaker, purely probabilistic diluteness condition for gas models that guarantees measure uniqueness, stability, and perfect simulation, improving upon existing cluster expansion criteria.

Findings

Proves uniqueness of the infinite-volume measure under the diluteness condition.

Demonstrates stability of the measure under parameter perturbations and discretizations.

Provides a coupled perfect-simulation scheme for the measure and its perturbations.

Abstract

For a general class of gas models ---which includes discrete and continuous Gibbsian models as well as contour or polymer ensembles--- we determine a \emph{diluteness condition} that implies: (1) Uniqueness of the infinite-volume equilibrium measure; (2) stability of this measure under perturbations of parameters and discretization schemes, and (3) existence of a coupled perfect-simulation scheme for the infinite-volume measure together with its perturbations and discretizations. Some of these results have previously been obtained through methods based on cluster expansions. In contrast, our treatment is purely probabilistic and its diluteness condition is weaker than existing convergence conditions for cluster expansions.