On large deviation principles and the Monge--Amp\`ere equation (following Berman, Hultgren)

Abstract

This is mostly an exposition, aimed to be accessible to geometers, analysts, and probabilists, of a fundamental recent theorem of R. Berman with recent developments by J. Hultgren, that asserts that the second boundary value problem for the real Monge--Amp\`ere equation admits a probabilistic interpretation, in terms of many particle limit of permanental point processes satisfying a large deviation principle with a rate function given explicitly using optimal transport. An alternative proof of a step in the Berman--Hultgren Theorem is presented allowing to to deal with all "tempratures" simultaneously instead of first reducing to the zero-temperature case.