Random stochastic matrices from classical compact Lie groups and symmetric spaces

TL;DR

This paper investigates the spectral properties of random stochastic matrices derived from classical compact Lie groups and symmetric spaces, revealing universality in their spectral statistics akin to well-known random matrix ensembles.

Contribution

It introduces a framework for analyzing spectral statistics of matrices from Lie groups and symmetric spaces, and demonstrates universality with classical random matrix ensembles using Weingarten functions.

Findings

Spectral statistics resemble GOE for symmetric matrices

Spectral statistics resemble Ginibre ensemble for non-symmetric matrices

Connections established with permutation enumeration problems

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 associated symmetric spaces. We observe numerically that, for large dimensions, the spectral statistics of $M$, discarding the Perron-Frobenius eigenvalue $1$, are similar to those of the Gaussian Orthogonal ensemble for symmetric matrices and to those of the real Ginibre ensemble for non-symmetric matrices. Using Weingarten functions, we compute some spectral statistics that corroborate this universality. We also establish connections with some difficult enumerative problems involving permutations.