The Polynomial Method for Random Matrices |
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Authors: | N Raj Rao Alan Edelman |
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Institution: | (1) MIT Department of Electrical Engineering and Computer Science, Cambridge, MA 02139, USA;(2) MIT Department of Mathematics, Cambridge, MA 02139, USA |
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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, i.e., 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 cases.
Algebraicity 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 form.
In 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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Keywords: | Random matrices Stochastic eigenanalysis Free probability Algebraic functions Resultants D-finite series |
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