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VOLUME 72 | ISSUE 8 | PAGE 605
Spectra of Random Contractions and Scattering Theory for Discrete-Time Systems
PACS: 03.65.Nk, 05.45.Mt
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 Ν χ N random matrices A of such a type, corresponding to systems with broken time-reversal invariance. Deviations from unitarity are characterized by rank Af < N and a set of eigenvalues 0 < T* < 1, i — 1,..., Μ of the matrix Τ = 1 A*A. We solve the problem completely by deriving the joint probability density of N complex eigenvalues and calculating all η-point correlation functions. In the limit Ν » M,n the correlation functions acquire the universal form found earlier for weakly non-Hermitian random matrices.