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

Yan V. Fyodorov
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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 $\hat{T}=\hat{\bf 1}-\hat{A}^{\dagger}\hat{A}$. 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.

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Yan V. Fyodorov. 2000-02-22. Spectra of Random Matrices Close to Unitary and Scattering Theory for Discrete-Time Systems. https://doi.org/10.1063/1.1358183

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