Abstract
We study three convolutions of polynomials in the context of free probability theory. We prove that these convolutions can be written as the expected characteristic polynomials of sums and products of unitarily invariant random matrices. The symmetric additive and multiplicative convolutions were introduced by Walsh and Szegö in different contexts, and have been studied for a century. The asymmetric additive convolution, and the connection of all of them with random matrices, is new. By developing the analogy with free probability, we prove that these convolutions produce real rooted polynomials and provide strong bounds on the locations of the roots of these polynomials.
Highlights
We study three convolutions on polynomials that are inspired by free probability theory
We develop the analogy with Free Probability by proving that Voiculescu’s R and S-transforms can be used to prove upper bounds on the extreme roots of these polynomials
We show that for monic polynomials, the operation +d can be realized as an expected characteristic polynomial of a sum of random matrices
Summary
We study three convolutions on polynomials that are inspired by free probability theory. Instead of capturing the limiting spectral distributions of ensembles of random matrices, we show that these capture the expected characteristic polynomials of random matrices in a fixed dimension. We develop the analogy with Free Probability by proving that Voiculescu’s R and S-transforms can be used to prove upper bounds on the extreme roots of these polynomials. Two of the convolutions have been classically studied. The third, and the connection of all of them with random matrices, is new. We begin by defining the three convolutions and stating the algebraic identities that establish this connection, as well as basic results regarding their real-rootedness properties
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