Abstract
We consider the empirical eigenvalue distribution of an $m\times m$ principle submatrix of an $n\times n$ random unitary matrix distributed according to Haar measure. Earlier work of Petz and Réffy identified the limiting spectral measure if $\frac{m} {n}\to \alpha $, as $n\to \infty $; under suitable scaling, the family $\{\mu _{\alpha }\}_{\alpha \in (0,1)}$ of limiting measures interpolates between uniform measure on the unit disc (for small $\alpha $) and uniform measure on the unit circle (as $\alpha \to 1$). In this note, we prove an explicit concentration inequality which shows that for fixed $n$ and $m$, the bounded-Lipschitz distance between the empirical spectral measure and the corresponding $\mu _{\alpha }$ is typically of order $\sqrt{\frac {\log (m)}{m}} $ or smaller. The approach is via the theory of two-dimensional Coulomb gases and makes use of a new “Coulomb transport inequality” due to Chafaï, Hardy, and Maïda.
Highlights
Let U be an n × n Haar-distributed unitary matrix
We prove an explicit concentration inequality which shows that for fixed n and m, the bounded-Lipschitz distance between the empirical spectral measure and the corresponding μα is typically of order log(m) m or smaller
In the case that m = o( n), the truncated matrix is close to a matrix of independent, identically distributed Gaussian random variables; the circular law for the Ginibre ensemble would lead one to expect that the eigenvalue distribution was approximately uniform in a disc, and this was verified by Jiang in [5]
Summary
∞; under suitable scaling, the family {μα}α∈(0,1) of limiting measures interpolates between uniform measure on the unit disc (for small α) and uniform measure on the unit circle (as α → 1). We prove an explicit concentration inequality which shows that for fixed n and m, the bounded-Lipschitz distance between the empirical spectral measure and the corresponding μα is typically of order log(m) m or smaller. The approach is via the theory of two-dimensional Coulomb gases and makes use of a new “Coulomb transport inequality” due to Chafaï, Hardy, and Maïda
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