Spectra of Random Contractions and Scattering Theory for Discrete-Time Systems

2000/05/29 by Yan V. Fyodorov, H. -J. Sommmers · 1 citation
Physics and Astronomy · #nlin.CD #cond-mat

paper · pdf · doi:10.1134/1.1335121

published as JETP Lett. 72, 422--426 (2000) · Complete solution of the problem discussed in earlier e-preprint nlin.CD/0002034

arxiv created 2000/05/29 · arxiv updated 2009/11/30

Abstract

Random contractions (sub-unitary random matrices) appear naturally when considering quantized chaotic maps within a general theory of open linear stationary systems with discrete time. We analyze statistical properties of complex eigenvalues of generic N× N random matrices A of such a type, corresponding to systems with broken time-reversal invariance. Deviations from unitarity are characterized by rank M≤ N and a set of eigenvalues 0<T_i≤ 1, i=1,...,M of the matrix T=\bf 1-A^†A. We solve the problem completely by deriving the joint probability density of N complex eigenvalues and calculating all n- point correlation functions. In the limit N>>M,n the correlation functions acquire the universal form found earlier for weakly non-Hermitian random matrices.

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• Random Matrix Theory for Chaotic Wave Scattering and Transport