Seminar on Free Probability
and Random Matrices

Fall 2025

I will give a broad overview of the connections between random matrix theory, Hankel/Toeplitz determinants, and their associated systems of orthogonal polynomials. Building on these connections, I will explain how complex-analytic Riemann-Hilbert problems provide a natural framework for deriving asymptotic results on the eigenvalues of random matrices from various ensembles.

This week I will continue the discussion on showing that the norm of a random matrix is large is exponentially small.

This week I will continue the discussion on showing that the norm of a random matrix is large is exponentially small. Notes for week 1

This term the seminar will have a focus on strong convergence of random matrix ensembles. The interest in strong convergence is motivated by the need to understand the largest and smallest eigenvalue of a self-adjoint random matrix. Last January, Felix Parraud gave us an introduction to the subject in his colloquium talk. Recent papers of Chen, Garza-Vargas, and van Handel have produced what the authors describe as a soft approach. I will begin by reviewing some preliminary material from the books of Tao and Vershynin. These will be used to revisit strong convergence for the Gaussian unitary ensemble. The material only uses basic probability theory and linear algebra.