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.
 Award ID(s):
 2002826
 NSFPAR ID:
 10432681
 Date Published:
 Journal Name:
 Alea
 Volume:
 18
 Issue:
 2
 ISSN:
 19800436
 Page Range / eLocation ID:
 1585–1623
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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