Some consequences of exchangeability in random-matrix theory

Abstract

Properties of infinite sequences of exchangeable random variables result directly in explicit expressions for calculating asymptotic densities of eigenvalues ρ_∞(λ) of any ensemble of random matrices H whose distribution depends only on tr (H†H), where H† is the Hermitian conjugate of H. For real symmetric matrices and for Hermitian matrices, the densities ρ_∞(λ) are constructed by summing up Wigner semicircles with varying radii and weights as confirmed by Monte Carlo simulations. Extensions to more general matrix ensembles are also considered.