Abstract
AbstractWe 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.
Funder
Directorate for Computer and Information Science and Engineering
Directorate for Mathematical and Physical Sciences
Alfred P. Sloan Foundation
Simons Foundation
John D. and Catherine T. MacArthur Foundation
Publisher
Springer Science and Business Media LLC
Subject
Statistics, Probability and Uncertainty,Statistics and Probability,Analysis
Reference30 articles.
1. Walsh, J.: On the location of the roots of certain types of polynomials. Trans. Am. Math. Soc. 24(3), 163–180 (1922)
2. Marden, M.: Geometry of Polynomials, vol. 3. American Mathematical Society, Providence, Rhode Island (1966)
3. Rahman, Q.I., Schmeisser, G.: Analytic Theory of Polynomials. Clarendon Press, Oxford (2002)
4. Mirabelli, B.: Hermitian, Non-Hermitian and Multivariate Finite Free Probability. PhD thesis, Princeton University (2020)
5. Szegö, G.: Bemerkungen zu einem Satz von J H Grace über die Wurzeln algebraischer Gleichungen. Math. Z. 13(1), 28–55 (1922)
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