Large Deviation Analysis for Canonical Gibbs Measures

TL;DR

This paper develops a large-deviation theory for functionals of canonical Gibbs processes, extending understanding of their probabilistic behavior under various interactions and boundary conditions.

Contribution

It introduces coupling constructions for analyzing large deviations in Gibbs processes, covering bounded, unbounded, and hard-core interaction models.

Findings

Established large deviation principles for empirical fields of Gibbs processes.

Extended results to unbounded and non-negative increasing local observables.

Provided bounds for tail probabilities of unbounded observables.

Abstract

In this paper, we present a large-deviation theory developed for functionals of canonical Gibbs processes, i.e., Gibbs processes with respect to the binomial point process. We study the regime of a fixed intensity in a sequence of increasing windows. Our method relies on the traditional large-deviation result for local bounded functionals of Poisson point processes noting that the binomial point process is obtained from the Poisson point process by conditioning on the point number. Our main methodological contribution is the development of coupling constructions allowing us to handle delicate and unlikely pathological events. The presented results cover three types of Gibbs models - a model given by a bounded local interaction, a model given by a non-negative possibly unbounded increasing local interaction and the hard-core interaction model. The derived large deviation principle is formulated for the distributions of individual empirical fields driven by canonical Gibbs processes, with its special case being a large deviation principle for local bounded observables of the canonical Gibbs processes. We also consider unbounded non-negative increasing local observables, but the price for treating this more general case is that we only get large-deviation bounds for the tails of such observables. Our primary setting is the one with periodic boundary condition, however, we also discuss generalizations for different choices of the boundary condition.