Finite free convolutions of polynomials
Adam W. Marcus, Daniel A. Spielman, Nikhil Srivastava
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.
Topics & Concepts
MathematicsMultiplicative functionConvolution (computer science)Orthogonal polynomialsDiscrete orthogonal polynomialsPure mathematicsFree probabilityDifference polynomialsMacdonald polynomialsClassical orthogonal polynomialsInvariant (physics)Context (archaeology)Discrete mathematicsAlgebra over a fieldMathematical analysisComputer scienceMathematical physicsMachine learningPaleontologyBiologyArtificial neural networkRandom Matrices and ApplicationsAdvanced Combinatorial MathematicsAdvanced Algebra and Geometry