H\"older Continuity of Cumulative Distribution Functions for Noncommutative Polynomials under Finite Free Fisher Information

TL;DR

This paper proves that evaluations of noncommutative polynomials in free probability have H"older continuous distributions under finite free Fisher information, providing explicit exponents and convergence rates, and partially resolving a conjecture.

Contribution

It establishes H"older continuity of distributions for noncommutative polynomials with finite free Fisher information, including explicit exponents and convergence rates, and extends eigenvalue distribution results.

Findings

H"older continuity with explicit exponents for polynomial evaluations

Optimal H"older exponent 2/3 for linear polynomials

Explicit convergence rates in Kolmogorov distance for eigenvalue distributions

Abstract

This paper contributes to the current studies on regularity properties of noncommutative distributions in free probability theory. More precisely, we consider evaluations of selfadjoint noncommutative polynomials in noncommutative random variables that have finite non-microstates free Fisher information, highlighting the special case of Lipschitz conjugate variables. For the first time in this generality, it is shown that the analytic distributions of those evaluations have H\"older continuous cumulative distribution functions with an explicit H\"older exponent that depends only on the degree of the considered polynomial. For linear polynomials, we reach in the case of finite non-microstates free Fisher information the optimal H\"older exponent $\frac{2}{3}$, and get Lipschitz continuity in the case of Lipschitz conjugate variables. In particular, our results guarantee that such polynomial evaluations have finite logarithmic energy and thus finite (non-microstates) free entropy, which partially settles a conjecture of Charlesworth and Shlyakhtenko [CS16]. We further provide a very general criterion that gives for weak approximations of measures having H\"older continuous cumulative distribution functions explicit rates of convergence in terms of the Kolmogorov distance. Finally, we combine these results to study the asymptotic eigenvalue distributions of polynomials in GUEs or matrices with more general Gibbs laws. For Gibbs laws, this extends the corresponding result obtained in [GS09] from convergence in distribution to convergence in Kolmogorov distance; in the GUE case, we even provide explicit rates, which quantify results of [HT05,HST06] in terms of the Kolmogorov distance.