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

August 26, 2021
  • random matrices
  • free probability
  • Empirical spectral distribution
  • Stieltjes transform
  • Fluctuations
  • Linearization
  • Operator-valued subordination
  • 15A18
  • 15B52
  • 60B20
  • 60F05
  • 46L54
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 FĂ©vrier.