Spectral measure of large random Hankel, Markov and Toeplitz matrices

Abstract

Abstract. We study the limiting spectral measure of large symmetric random matrices of linear algebraic structure. For Hankel (Hn = [Xi+j]1≤i,j≤n) and Toeplitz (Tn = [X |i−j|]1≤i,j≤n) matrices generated by i.i.d. random variables {Xk} of zero mean and unit variance, and for symmetric Markov matrices (1) generated by i.i.d. random variables {Xi,j}j>i of zero mean and unit variance, scaling the eigenvalues by n we prove the almost sure, weak convergence of the spectral measures to universal, non-random, symmetric distributions γT and γH, γM of unbounded support. We show that γH is not unimodal and that γT is not normal. The moments of γT and γH are the sum of volumes of solids related to Eulerian numbers. The distribution γM has a bounded smooth density given by the free convolution of the semi-circle and normal densities. For symmetric Markov matrices (1) generated by i.i.d. random variables {Xi,j}j>i of mean m and finite variance, scaling the eigenvalues by n we prove the almost sure, weak convergence of the spectral measures to the atomic measure at −m. If m = 0, and the fourth moment is finite, we prove that the spectral radius of Mn scaled by √ 2n log n converges almost surely to one. 1. Introduction and