A Berry-Essen Type Theorem for Finite Free Convolution

We prove that the rate of convergence for the central limit theorem in finite free convolution is of order $n^{1/2}$Comment: 8 pages, published in 201

A rectangular additive convolution for polynomials

We define the rectangular additive convolution of polynomials with nonnegative real roots as a generalization of the asymmetric additive convolution introduced by Marcus, Spielman and Srivastava. We then prove a sliding bound on the largest root of this convolution. The main tool used in the analysis is a differential operator derived from the "rectangular Cauchy transform" introduced by Benaych-Georges. The proof is inductive, with the base case requiring a new nonasymptotic bound on the Cauchy transform of Gegenbauer polynomials which may be of independent interest

On matrices in finite free position

We study pairs $(A,B)$ of square matrices that are in additive (resp. multiplicative) finite free position, that is, the characteristic polynomial $\chi_{A+B}(x)$ (resp. $\chi_{AB}(x)$) equals the additive finite free convolution $\chi_{A}(x) \boxplus \chi_{B}(x)$ (resp. multiplicative finite free convolution $\chi_{A}(x) \boxtimes \chi_{B}(x)$), which equals the expected characteristic polynomial $\mathbb{E}_U \, [ \chi_{A+U^* BU}(x) ]$ (resp. $\mathbb{E}_U \, [ \chi_{AU^* BU}(x) ]$) over the set of unitary matrices $U$. We examine the lattice of algebraic varieties of matrices consisting of finite free complementary pairs with respect to the additive (resp. multiplicative) convolution. We show that these pairs include the diagonal matrices vs. the principally balanced matrices, the upper (lower) triangular matrices vs. the upper (lower) triangular matrices with a single eigenvalue, and the scalar matrices vs. the set of all square matrices

Real roots of hypergeometric polynomials via finite free convolution

We examine two binary operations on the set of algebraic polynomials, known as multiplicative and additive finite free convolutions, specifically in the context of hypergeometric polynomials. We show that the representation of a hypergeometric polynomial as a finite free convolution of more elementary blocks, combined with the preservation of the real zeros and interlacing by the free convolutions, is an effective tool that allows us to analyze when all roots of a specific hypergeometric polynomial are real. Moreover, the known limit behavior of finite free convolutions allows us to write the asymptotic zero distribution of some hypergeometric polynomials as free convolutions of Marchenko-Pastur, reversed Marchenko-Pastur, and free beta laws, which has an independent interest within free probability.Comment: 44 pages, 8 table

Rectangular matrix additions in low and high temperatures

We study the addition of two random independent $M\times N$ rectangular random matrices with invariant distributions in two limit regimes, where the parameter beta (inverse temperature) goes to infinity and zero. In low temperature regime the random singular values of the sum concentrate at deterministic points, while in high temperature regime we obtain a Law of Large Numbers for the empirical measures. As a consequence, we deliver a duality between low and high temperatures. Our proof uses the type BC Bessel function as characteristic function of rectangular matrices, and through the analysis of this function we introduce a new family of cumulants, that linearize the addition in high temperature limit, and degenerate to the classical or free cumulants in special cases.Comment: 68 pages, 2 figures, more references are adde

Cumulants for finite free convolution

In this talk we will explain recent results with Daniel Perales where we define cumulants for finite free convolution. We give a moment-cumulant formula and show that these cumulants satisfy desired properties: they are additive with respect to finite free convolution and they approach free cumulants as the dimension goes to infinity.Non UBCUnreviewedAuthor affiliation: Centro de Investigacion en Matematicas (Guanajuato-Mexico)Facult