Limiting Spectral Measures for Random Matrix Ensembles with a Polynomial Link Function

TL;DR

This paper studies the spectral measures of generalized random matrix ensembles with polynomial link functions, showing convergence to universal distributions and how these depend on polynomial coefficients.

Contribution

It extends previous results on Toeplitz and Hankel matrices to polynomial link functions, establishing convergence to universal distributions and their dependence on polynomial parameters.

Findings

Spectral measures for linear polynomial link functions converge to non-semicircular distributions.

Distributions approach the semicircle law as polynomial coefficients grow large.

Sum or difference of polynomials with different degrees yields semicircular spectral measures.

Abstract

Consider the ensembles of real symmetric Toeplitz matrices and real symmetric Hankel matrices whose entries are i.i.d. random variables chosen from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. Previous work on real symmetric Toeplitz matrices shows that the spectral measures, or densities of normalized eigenvalues, converge almost surely to a universal near-Gaussian distribution, while previous work on real symmetric Hankel matrices shows that the spectral measures converge almost surely to a universal non-unimodal distribution. Real symmetric Toeplitz matrices are constant along the diagonals, while real symmetric Hankel matrices are constant along the skew diagonals. We generalize the Toeplitz and Hankel matrices to study matrices that are constant along some curve described by a real-valued bivariate polynomial. Using the Method of Moments and an analysis of the resulting Diophantine equations, we show that the spectral measures associated with linear bivariate polynomials converge in probability and almost surely to universal non-semicircular distributions. We prove that these limiting distributions approach the semicircle in the limit of large values of the polynomial coefficients. We then prove that the spectral measures associated with the sum or difference of any two real-valued polynomials with different degrees converge in probability and almost surely to a universal semicircular distribution.