TY - JOUR

T1 - Hölder continuity of cumulative distribution functions for noncommutative polynomials under finite free Fisher information

AU - Banna, Marwa

AU - Mai, Tobias

N1 - Funding Information:
This work has been supported by the ERC Advanced Grant NCDFP 339760 held by Roland Speicher. The authors would like to thank the anonymous referee for his/her careful review, remarks and suggestions.
Publisher Copyright:
© 2020 Elsevier Inc.

PY - 2020/11/1

Y1 - 2020/11/1

N2 - 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 Hölder continuous cumulative distribution functions with an explicit Hölder 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 Hölder exponent [Formula presented], 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 [8]. We further provide a very general criterion that gives for weak approximations of measures having Hölder 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 [21] from convergence in distribution to convergence in Kolmogorov distance; in the GUE case, we even provide explicit rates, which quantify results of [23,24] in terms of the Kolmogorov distance.

AB - 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 Hölder continuous cumulative distribution functions with an explicit Hölder 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 Hölder exponent [Formula presented], 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 [8]. We further provide a very general criterion that gives for weak approximations of measures having Hölder 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 [21] from convergence in distribution to convergence in Kolmogorov distance; in the GUE case, we even provide explicit rates, which quantify results of [23,24] in terms of the Kolmogorov distance.

KW - Free Fisher information and entropy

KW - Hölder continuity

KW - Noncommutative polynomials

KW - Random matrices

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U2 - 10.1016/j.jfa.2020.108710

DO - 10.1016/j.jfa.2020.108710

M3 - Article

AN - SCOPUS:85088031415

VL - 279

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 8

M1 - 108710

ER -