  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.
