Statistical mechanics of permanents, real-Monge-Ampere equations and optimal transport

Abstract: We give a new probabilistic construction of solutions to real Monge-Amp`ere equations in R^n satisfying the second boundary value problem with respect to a given target convex body P) which fits naturally into the theory of optimal transport. More precisely, certain beta-deformed permanental (bosonic) N-particle point processes are introduced and their empirical measures are, in the large N-limit, shown to converge exponentially in probability to a deterministic measure whose potential satisfies the real Monge-Amp`ere equation in question. In particular, this allows us to represent the solution as a limit, as N tends to infinity of explicit integrals over the N-fold products of R^n and it also leads to explicit limit formulas for optimal transport maps to the convex body P. Connections to the study of K\"ahler-Einstein metrics on complex algebraic varieties, and in particular toric ones, are briefly discussed.