Distribution of Eigenvalues of Highly Palindromic Toeplitz Matrices

TL;DR

This paper investigates the eigenvalue distribution of highly palindromic Toeplitz matrices, revealing a universal distribution with fatter tails than previously observed, using the method of moments and Diophantine equation analysis.

Contribution

It extends prior work by analyzing matrices with multiple palindromic structures, showing they have a distinct universal spectral distribution with heavier tails.

Findings

Eigenvalue moments converge to a new universal distribution.

The new distribution has a fatter tail than known spectral measures.

Multiple palindromes influence the spectral measure's tail behavior.

Abstract

Consider the ensemble of real symmetric Toeplitz 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 [BDJ,HM] showed that the limiting spectral measures (the density of normalized eigenvalues) converge weakly and almost surely to a universal distribution almost that of the Gaussian, independent of p. The deficit from the Gaussian distribution is due to obstructions to solutions of Diophantine equations and can be removed (see [MMS]) by making the first row palindromic. In this paper, we study the case where there is more than one palindrome in the first row of a real symmetric Toeplitz matrix. Using the method of moments and an analysis of the resulting Diophantine equations, we show that the moments of this ensemble converge to an universal distribution with a fatter tail than any previously seen limiting spectral measure.