Abstract
The last two lectures go back to the source of the notion of " strong spectral convergence ", and discuss the spectrum of large random Hermitian matrices of size NxN, drawn according to a Gaussian distribution (GUE model). The seminal work of Haagerup-Thorbjørnsen is described, which states the strong convergence of independent Gaussian Hermitian matrices A^{N}_1, ... A^{N}_r to a free semicircular family, when N is large. For a family of independent random matrices, the " Pisier linearization trick " reduces the problem of strong convergence (nonlinear in matrices) to comparing the spectrum of linear combinations of these matrices. In a now classic way, we use the Stieltjes transform to compare the spectra.
References
- U. Haagerup, S. Thorbjørnsen. A new application of random matrices: Ext(C∗red(F2)) is not a group. Annals of Mathematics, 162 (2005), 711-77
- G. Pisier, On a linearization trick. L'Enseignement Mathématique (2) 64 (2018), 315-326
- R. Froese, D. Hasler, W. Spitzer, Transfer matrices, hyperbolic geometry and absolutely continuous spectrum for some discrete Schrödinger operators on graphs, Journal of Functional Analysis, Volume 230, Issue 1, 1 January 2006, Pages 184-221
