  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.
