The Polynomial Method for Random Matrices

Authors: N Raj Rao Alan Edelman

Publish Date: 2007/12/13

Volume: 8, Issue: 6, Pages: 649-702

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Abstract

We define a class of “algebraic” random matrices These are random matrices for which the Stieltjes transform of the limiting eigenvalue distribution function is algebraic ie it satisfies a bivariate polynomial equation The Wigner and Wishart matrices whose limiting eigenvalue distributions are given by the semicircle law and the Marčenko–Pastur law are special casesAlgebraicity of a random matrix sequence is shown to act as a certificate of the computability of the limiting eigenvalue density function The limiting moments of algebraic random matrix sequences when they exist are shown to satisfy a finite depth linear recursion so that they may often be efficiently enumerated in closed formIn this article we develop the mathematics of the polynomial method which allows us to describe the class of algebraic matrices by its generators and map the constructive approach we employ when proving algebraicity into a software implementation that is available for download in the form of the RMTool random matrix “calculator” package Our characterization of the closure of algebraic probability distributions under free additive and multiplicative convolution operations allows us to simultaneously establish a framework for computational noncommutative “free probability” theory We hope that the tools developed allow researchers to finally harness the power of infinite random matrix theory

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