Matrix models and eigenvalue statistics for truncations of classical ensembles of random unitary matrices

Abstract

We consider random non-normal matrices constructed by removing one row and column from samples from Dyson's circular ensembles or samples from the classical compact groups. We develop sparse matrix models whose spectral measures match these ensembles. This allows us to compute the joint law of the eigenvalues, which have a natural interpretation as resonances for open quantum systems or as electrostatic charges located in a dielectric medium. Our methods allow us to consider all values of $\beta >0$, not merely $\beta =1,2,4$.

arXiv comment36 pp, to appear in Comm. Math. Phys

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• Analogies between random matrix ensembles and the one-component plasma in two-dimensions
• Extreme eigenvalues and the emerging outlier in rank-one non-Hermitian deformations of the Gaussian Unitary Ensemble
• Multiplicative non-Hermitian perturbations of classical $\beta$-ensembles