arXiv subjects

Kathryn Stewart

Publications and source records attributed to Kathryn Stewart.

3 recordsLinked to original sources

arXiv2019

Eigenvalue rigidity for truncations of random unitary matrices

We consider the empirical eigenvalue distribution of an $m\times m$ principal submatrix of an $n\times n$ random unitary matrix distributed according to Haar measure. For $n$ and $m$ large with $\frac{m}{n}=α$, 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 $\sqrt{\frac{\log(m)}{m}}$ 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.

math.PR↗

arXiv2019

On the eigenvalues of truncations of random unitary matrices

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 $\frac{m}{n}\toα$, as $n\to\infty$; under suitable scaling, the family $\{μ_α\}_{α\in(0,1)}$ of limiting measures interpolates between uniform measure on the unit disc (for small $α$) and uniform measure on the unit circle (as $α\to1$). 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 $μ_α$ 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.

math.PR↗

arXiv2019

Total variation approximation of random orthogonal matrices by Gaussian matrices

The topic of this paper is the asymptotic distribution of random orthogonal matrices distributed according to Haar measure. We examine the total variation distance between the joint distribution of the entries of $W_n$, the $p_n \times q_n$ upper-left block of a Haar-distributed matrix, and that of $p_nq_n$ independent standard Gaussian random variables. We show that the total variation distance converges to zero when $p_nq_n = o(n)$.

math.PR↗