2003/07/31 by Włodzimierz Bryc, Amir Dembo, Tiefeng Jiang · 6 citations
Mathematics · #Advanced Algebra and Geometry #Geometry and complex manifolds #Random Matrices and Applications #math.CO #math.PR #math.ST #msc:15A52 #msc:60F10 #msc:60F99 #msc:62H10 #stat.TH
paper · pdf · doi:10.1214/009117905000000495
published as Annals of Probability 2006, Vol. 34, No. 1, 1-38 · Published at http://dx.doi.org/10.1214/009117905000000495 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2006/01/01 · arxiv created 2006/02/27 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Xk of unit variance, and for symmetric Markov matrices generated by i.i.d. random variables Xijj>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, nonrandom, symmetric distributions γH, γM and γT of unbounded support. The moments of γH and γT are the sum of volumes of solids related to Eulerian numbers, whereas γM 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 Xijj>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 Mn scaled by √(2nlog n) converges almost surely to 1.