Asymptotic expansion for transport maps between laws of multimatrix models

TL;DR

This paper develops an asymptotic expansion framework for transport maps and trace functions in large random matrix models, providing insights into their convergence and behavior.

Contribution

It introduces a systematic method for asymptotic expansions of transport maps and trace functions in multimatrix models with convex potentials.

Findings

Derived an $1/N^2$ expansion for traces of noncommutative functions of matrix tuples.

Established strong convergence results for multimatrix models using transport maps.

Unified polynomial and smooth function approaches for asymptotic analysis.

Abstract

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 $1/N^2$ 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 $(S^{N}_t)_{t \geq 0}$. 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.