Large deviations for configurations generated by Gibbs distributions with energy functionals consisting of singular interaction and weakly confining potentials

TL;DR

This paper proves large deviation principles for empirical measures of Gibbs distributions with singular interactions and weakly confining potentials, relevant to fields like random matrix theory and simulated annealing.

Contribution

It establishes LDPs under general conditions for a broad class of potentials, including weakly confining ones, using the weak convergence method and stronger topologies.

Findings

LDPs with speed $eta_n/n ightarrow abla$ and rate functions involving potentials.

LDPs with speed $eta_n = n$ including entropic terms.

Addresses cases with non-compact support minimizers, resolving open questions.

Abstract

We establish large deviation principles (LDPs) for empirical measures associated with a sequence of Gibbs distributions on $n$-particle configurations, each of which is defined in terms of an inverse temperature $% \beta_n$ and an energy functional consisting of a (possibly singular) interaction potential and a (possibly weakly) confining potential. Under fairly general assumptions on the potentials, we use a common framework to establish LDPs both with speeds $\beta_n/n \rightarrow \infty$, in which case the rate function is expressed in terms of a functional involving the potentials, and with speed $\beta_n =n$, when the rate function contains an additional entropic term. Such LDPs are motivated by questions arising in random matrix theory, sampling, simulated annealing and asymptotic convex geometry. Our approach, which uses the weak convergence method developed by Dupuis and Ellis, establishes LDPs with respect to stronger Wasserstein-type topologies. Our results address several interesting examples not covered by previous works, including the case of a weakly confining potential, which allows for rate functions with minimizers that do not have compact support, thus resolving several open questions raised in a work of Chafa\"{\i} et al.