Spectra of Random Matrices Close to Unitary and Scattering Theory for Discrete-Time Systems

Abstract

We analyze statistical properties of complex eigenvalues of random matrices $\hat{A}$ close to unitary. Such matrices appear naturally when considering quantized chaotic maps within a general theory of open linear stationary systems with discrete time. Deviation from unitarity are characterized by rank $M$ and eigenvalues $T_i,i=1,...,M$ of the matrix $\hat{T}=\hat{1}-{\hat{A}}^{†}\hat{A}$. For the case M=1 we solve the problem completely by deriving the joint probability density of eigenvalues and calculating all $n-$ point correlation functions. For a general case we present the correlation function of secular determinants.

arXiv comment4 pages, latex, no figures, a few misprints are corrected

Cited by in corpus (4)

• On the mean density of complex eigenvalues for an ensemble of random matrices with prescribed singular values
• Matrix models and eigenvalue statistics for truncations of classical ensembles of random unitary matrices
• Extreme eigenvalues and the emerging outlier in rank-one non-Hermitian deformations of the Gaussian Unitary Ensemble
• Dynamics of a rank-one multiplicative perturbation of a unitary matrix