Eigenvalue rigidity for truncations of random unitary matrices

2019/05/06 by Meckes, Elizabeth, Stewart, Kathryn
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.1905.02233

Abstract

We consider the empirical eigenvalue distribution of an m× m principal submatrix of an n× n random unitary matrix distributed according to Haar measure. For n and m large with (m)/(n)=α, the empirical spectral measure is well-approximated by a deterministic measure μ_α supported on the unit disc. In earlier work, we showed that for fixed n and m, the bounded-Lipschitz distance between the empirical spectral measure and the corresponding μ_α is typically of order √((log(m))/(m)) or smaller. In this paper, we consider eigenvalues on a microscopic scale, proving concentration inequalities for the eigenvalue counting function and for individual bulk eigenvalues.

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