Spectral measure of large random Hankel, Markov and Toeplitz matrices

Abstract

We study the limiting spectral measure of large symmetric random matrices of linear algebraic structure.

For Hankel and Toeplitz matrices generated by i.i.d. random variables {X_k} of unit variance, and for symmetric Markov matrices generated by i.i.d. random variables {X_ij}_j>i of zero mean and unit variance, scaling the eigenvalues by $\sqrt{n}$ we prove the almost sure, weak convergence of the spectral measures to universal, nonrandom, symmetric distributions γ_H, γ_M and γ_T of unbounded support. The moments of γ_H and γ_T are the sum of volumes of solids related to Eulerian numbers, whereas γ_M has a bounded smooth density given by the free convolution of the semicircle and normal densities.

For symmetric Markov matrices generated by i.i.d. random variables {X_ij}_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 norm of M_n scaled by $\sqrt{2n\log n}$ converges almost surely to 1.