Numerical computation of the free additive convolution of measures

Abstract: In this talk we propose an algorithm to compute the free additive convolution of two given measures. 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). A key ingredient of our method is rewriting the quantities needed in the computation 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.