Large deviations for empirical measures generated by Gibbs measures with singular energy functionals

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 bn and an energy functional that is the sum of a (possibly singular) interaction and confining potential. Under fairly general assumptions on the potentials, we establish LDPs both with speeds (bn)/(n) ® ¥, in which case the rate function is expressed in terms of a functional involving the potentials, and with the speed bn =n, when the rate function contains an additional entropic term. Such LDPs are motivated by questions arising in random matrix theory, sampling and simulated annealing. Our approach, which uses the weak convergence methods developed in ``A weak convergence approach to the theory of large deviations", establishes large deviation principles with respect to stronger, Wasserstein-type topologies, thus resolving an open question in ``First-order global asymptotics for confined particles with singular pair repulsion". It also provides a common framework for the analysis of LDPs with all speeds, and includes cases not covered due to technical reasons in previous works.