Computing free convolutions via contour integrals

Alice Cortinovis, Stanford University
8/30, 2023 at 4:10PM-5PM in 939 Evans (for in-person talks) and

In this talk we propose an algorithm to compute the free additive and multiplicative convolution of two given measures. Free convolution can be used to understand, for example, the asymptotic eigenvalue distribution of the sum of two random matrices. We assume that the measures have compact support, and their density function exhibits a square-root decay at the boundary (for example, the semicircle distribution or the Marchenko-Pastur distribution). A key ingredient of our method is rewriting the intermediate quantities of the free convolution using the Cauchy integral formula and then discretizing these integrals using the trapezoidal quadrature rule, which converges exponentially fast under suitable analyticity properties of the functions to be integrated.