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We consider the empirical eigenvalue distribution of an $mtimes m$ principle submatrix of an $ntimes n$ random unitary matrix distributed according to Haar measure. Earlier work of Petz and Reffy identified the limiting spectral measure if $frac{m}{n}toalpha$, as $ntoinfty$; under suitable scaling, the family ${mu_alpha}_{alphain(0,1)}$ of limiting measures interpolates between uniform measure on the unit disc (for small $alpha$) and uniform measure on the unit circle (as $alphato1$). 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 Chafai, Hardy, and Maida.
We consider the empirical eigenvalue distribution of an $mtimes m$ principal submatrix of an $ntimes n$ random unitary matrix distributed according to Haar measure. For $n$ and $m$ large with $frac{m}{n}=alpha$, the empirical spectral measure is well
This is a brief survey of classical and recent results about the typical behavior of eigenvalues of large random matrices, written for mathematicians and others who study and use matrices but may not be accustomed to thinking about randomness.
Let $U$ be a Haar distributed matrix in $mathbb U(n)$ or $mathbb O (n)$. In a previous paper, we proved that after centering, the two-parameter process [T^{(n)} (s,t) = sum_{i leq lfloor ns rfloor, j leq lfloor ntrfloor} |U_{ij}|^2] converges in dist
We consider the real eigenvalues of an $(N times N)$ real elliptic Ginibre matrix whose entries are correlated through a non-Hermiticity parameter $tau_Nin [0,1]$. In the almost-Hermitian regime where $1-tau_N=Theta(N^{-1})$, we obtain the large-$N$
Let $O(2n+ell)$ be the group of orthogonal matrices of size $left(2n+ellright)times left(2n+ellright)$ equipped with the probability distribution given by normalized Haar measure. We study the probability begin{equation*} p_{2n}^{left(ellright)} = ma