Pierre Mergny -- Interpolating between classical and free convolutions
Channel: Matrices Et Graphes Aléatoires GDR CNRS
Date: 2022-02-09

Abstract: The theory of free probability allows one to compute the  limiting spectrum of the sum C = A+B of two symmetric matrices as their dimension goes  to infinity, provided one replaces the notion of independence from classical probability by the notion of freeness from non-commutative probability. The operation to get the spectrum of C is known as the free convolution between the spectrum of A and  the spectrum of B and a natural question is to construct an operation interpolating between the well-known classical convolution and this free convolution. In this talk, I will explain how one can (hopefully) construct such interpolation by taking the additive spherical integral in the so-called « high temperature » regime. If time permits, I will also describe a remarkable connexion between the aforementioned construction and the Finite Free Convolution of Marcus, Spielman and Srivastava.