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Asymptotic expansion for transport maps between laws of multimatrix models

David Jekel, Evangelos A. Nikitopoulos, Félix Parraud

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.

Asymptotic expansion for transport maps between laws of multimatrix models

Abstract

We study the large- behavior of random matrix tuples with joint density proportional to for some convex function in non-commuting variables satisfying certain bounds on its second derivative. We give an asymptotic expansion in powers of of the trace of noncommutative smooth functions of . We also give an asymptotic expansion for a family of maps that transport the law of a tuple of independent GUE random matrices to the law of and, as a consequence, show strong convergence for the multimatrix models . 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 . 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.

Paper Structure

This paper contains 20 sections, 30 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Standard definitions
  3. Free probability
  4. Random matrix theory
  5. Noncommutative smooth functions and derivatives
  6. Preliminary definitions and an important example
  7. The space of noncommutative trace polynomials
  8. Example of $\mathcal{C}^k$-functions
  9. Stability of the noncommutative smooth functions under various operations
  10. The asymptotic expansion
  11. Analysis of a key stochastic differential equation
  12. An integral equation
  13. Existence and uniqueness for the stochastic differential equation
  14. Estimate on the large-time asymptotics
  15. Construction of transport maps
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $V \in\mathcal{C}^{16k+23}(X_1,\dots,X_d)$ be a $(4k+4)$-regular potential (Definition definition:kregular), $f\in\mathcal{C}^{4k+4}(X_1,\dots,X_d)$, and $Y^N$ be a multimatrix model with potential $V$ (Definition def:randompot). There exist functions $f_i\in\mathcal{C}^{k-i+1}(X_1,\dots,X_d)$ ( where $\lVert*\rVert{f}_{\mathcal{L}^k,K} \coloneqq \sup \{ \lVert*\rVert{f}_{\mathcal{C}^k,K} e^{-

Theorems & Definitions (50)

  • Theorem 1.1: Asymptotic expansion of expectations
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Proposition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 3.2
  • ...and 40 more