Matrizes estocásticas aleatórias associadas a grupos de Lie clássicos e espaços simétricos

Abstract

We consider random stochastic matrices M with elements given by $M_{ij} = |U_{ij}|2$, with U being uniformly distributed on one of the classical compact Lie groups or some of the associated symmetric spaces. We observe numerically that, for large dimensions, the spectral statistics of M (discarding the Perron-Frobenius eigenvalue) are similar to those of the Gaussian Orthogonal ensemble for symmetric matrices and to those of the real Ginibre ensemble for nonsymmetric matrices. We compute some spectral statistics using Weingarten functions and establish connections with some difficult enumerative problems involving permutations.