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Titel: Hölder continuity of cumulative distribution functions for noncommutative polynomials under finite free Fisher information
VerfasserIn: Banna, Marwa
Mai, Tobias
Sprache: Englisch
Titel: Journal of Functional Analysis
Bandnummer: 279
Heft: 8
Verlag/Plattform: Elsevier
Erscheinungsjahr: 2020
Freie Schlagwörter: Free Fisher information and entropy
Noncommutative polynomials
Hölder continuity
Random matrices
DDC-Sachgruppe: 500 Naturwissenschaften
Dokumenttyp: Journalartikel / Zeitschriftenartikel
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 con jugate 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 2 3 , and get Lipschitz continuity in the case of Lipschitz conjugate variables. In particular, our results guarantee that such polynomial evaluations have fi nite 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 cumu lative 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.
DOI der Erstveröffentlichung: 10.1016/j.jfa.2020.108710
URL der Erstveröffentlichung: https://doi.org/10.1016/j.jfa.2020.108710
Link zu diesem Datensatz: urn:nbn:de:bsz:291--ds-419763
hdl:20.500.11880/37562
http://dx.doi.org/10.22028/D291-41976
ISSN: 0022-1236
Datum des Eintrags: 30-Apr-2024
Fakultät: MI - Fakultät für Mathematik und Informatik
Fachrichtung: MI - Mathematik
Professur: MI - Keiner Professur zugeordnet
Sammlung:SciDok - Der Wissenschaftsserver der Universität des Saarlandes

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