Asymptotic expansion for transport maps between laws of multimatrix models

We study the large-$N$ behavior of random matrix tuples $Y^N=(Y_1^N,\dots ,Y_d^N)$ with joint density proportional to $e^{-N^{2}V}$ for some convex function $V$ in non-commuting variables satisfying certain bounds on its second derivative. We give an asymptotic expansion in powers of $1\mathrm{/}{N}^2$ of the trace of noncommutative smooth functions of $Y^N$. We also give an asymptotic expansion for a family of maps $T^N$ that transport the law of a tuple of independent GUE random matrices to the law of $Y^N$ and, as a consequence, show strong convergence for the multimatrix models $Y^N$. Our proof is based on an asymptotic expansion for the heat semigroup associated to the measure, which is expressed in terms of smooth functions of a matrix Brownian motion $(S_t^N{)}_{t\ge 0}$. We introduce spaces of noncommutative smooth functions that unify and generalize the cases of polynomials and single-variable smooth functions and allow the systematic application of asymptotic expansion techniques to multimatrix models with convex interaction.