On Large Deviations for Gibbs Measures, Mean Energy and Gamma-Convergence

Robert J Berman

doi:10.1007/s00365-018-9437-3

Robert J Berman

Open Access

https://doi.org/10.1007/s00365-018-9437-3

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Abstract

We consider the random point processes on a measure space (X,mu _{0}) defined by the Gibbs measures associated with a given sequence of N-particle Hamiltonians H^{(N)}. Inspired by the method of Messer–Spohn for proving concentration properties for the laws of the corresponding empirical measures, we propose a number of hypotheses on H^{(N)} that are quite general but still strong enough to extend the approach of Messer–Spohn. The hypotheses are formulated in terms of the asymptotics of the corresponding mean energy functionals. We show that in many situations, the approach even yields a large deviation principle (LDP) for the corresponding laws. Connections to Gamma-convergence of (free) energy type functionals at different levels are also explored. The focus is on differences between positive and negative temperature situations, motivated by applications to complex geometry. The results yield, in particular, large deviation principles at positive as well as negative temperatures for quite general classes of singular mean field models with pair interactions, generalizing the 2D vortex model and Coulomb gases. In a companion paper, the results will be illustrated in the setting of Coulomb and Riesz type gases on a Riemannian manifold X, comparing with the complex geometric setting.

Highlights

Let X be a compact topological space endowed with a probability measure μ0
Given a sequence of symmetric functions H (N) on the N -fold products X N, which are absolutely integrable with respect to the Borel measure μ⊗0 N, the corresponding Gibbs measures at inverse temperature βN ∈ R are defined as the following sequence of symmetric probability measures on X N : μ(βNN )
The ensemble (X N, μ(βNN)) defines a random point process with N particles on X which, from the point of view of statistical mechanics, models N identical particles on X interacting by the Hamiltonian H (N) in thermal equilibrium at inverse temperature βN

Summary

Introduction

Let X be a compact topological space endowed with a probability measure μ0. Given a sequence of symmetric functions H (N) on the N -fold products X N , which are absolutely integrable with respect to the Borel measure μ⊗0 N , the corresponding Gibbs measures at inverse temperature βN ∈ R are defined as the following sequence of symmetric probability measures on X N : μ(βNN ) := Z N ,β e−βN H (N)μ0, assuming that the partition function Z N,βN is finite:Z N,βN := X N e−βN H (N) μ⊗0 N < ∞. Given a sequence of symmetric functions H (N) on the N -fold products X N , which are absolutely integrable with respect to the Borel measure μ⊗0 N , the corresponding Gibbs measures at inverse temperature βN ∈ R are defined as the following sequence of symmetric probability measures on X N : μ(βNN ) := Z N ,β e−βN H (N). Μ0, assuming that the partition function Z N,βN is finite:. We assume that the following limit exists: β lim N →∞ βN ∈] − ∞, ∞]. The ensemble (X N , μ(βNN)) (called the canonical ensemble) defines a random point process with N particles on X which, from the point of view of statistical mechanics, models N identical particles on X interacting by the Hamiltonian (interaction energy) H (N) in thermal equilibrium at inverse temperature βN.

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