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Fluctuations of the Stieltjes transform of the empirical spectral distribution of selfadjoint polynomials in Wigner and deterministic diagonal matrices

[HYBRID WORKSHOP] Connections and Introductory Workshop: Universality and Integrability in Random Matrix Theory and Interacting Particle Systems, Part 1 August 23, 2021 - August 27, 2021

August 26, 2021 (09:30 AM PDT - 10:30 AM PDT)
Speaker(s): Mireille Capitaine (Centre National de la Recherche Scientifique (CNRS))
Location: SLMath: Online/Virtual
Tags/Keywords
  • random matrices

  • free probability

  • Empirical spectral distribution

  • Stieltjes transform

  • Fluctuations

  • Linearization

  • Operator-valued subordination

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification No Secondary AMS MSC
Video

Fluctuations Of The Stieltjes Transform Of The Empirical Spectral Distribution Of Selfadjoint Polynomials In Wigner And Deterministic Diagonal Matrices

Abstract

We will present the following result. When the dimension goes to infinity, the recentered analytic process on nonreal complex numbers of nonnormalized traces of resolvents of any selfadjoint noncommutative polynomial in a Wigner matrix and a deterministic diagonal matrix converges to a centred complex Gaussian process whose covariance is expressed in terms of operator-valued subordination functions in free probability theory. This is a joint work with Serban Belinschi, Sandrine Dallaporta and Maxime Février.

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Video/Audio Files

Fluctuations Of The Stieltjes Transform Of The Empirical Spectral Distribution Of Selfadjoint Polynomials In Wigner And Deterministic Diagonal Matrices

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