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

TL;DR

This paper explains a recent theorem linking the Monge--Ampère equation to probabilistic models, showing how large deviation principles and optimal transport describe the behavior of certain particle systems.

Contribution

It provides an accessible exposition of Berman's theorem and introduces an alternative proof that handles all temperature cases simultaneously.

Findings

Probabilistic interpretation of the Monge--Ampère boundary value problem

Explicit rate function using optimal transport

Unified proof approach for all temperature regimes

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.