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

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 $% β_n$ 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 $β_n/n \rightarrow \infty$, in which case the rate function is expressed in terms of a functional involving the potentials, and with speed $β_n =n$, 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\"ı et al.