Finite Free Convolution: Infinitesimal Distributions

Finite-free additive and multiplicative convolutions are operations on the set of polynomials with real roots, introduced independently by Szeg\"{o} and Walsh in the 1920s. These operations have regained some interest, in the last decade, after being rediscovered by Marcus, Spielman, and Srivastava as the expected characteristic polynomial of randomly rotated matrices. They converge, as the degree $d$ of the polynomials increases, to the additive and multiplicative convolution of measures from free probability of Voiculescu. In this paper, we investigate the fluctuations of order $1/d$ -- also known as infinitesimal distributions -- related to these two operations and their limiting behavior, providing a detailed description of their convergence. Our approach relies on understanding the infinitesimal moment-cumulant formulas and the corresponding functional relations. We also establish several applications and examples, including instances related to the infinitesimal free convolution of Belinschi and Shlyakhtenko, as well as the computation of infinitesimal distributions after differentiation of polynomials.