Abstract
Random contractions (subunitary 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  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 $$\hat T = \hat 1 - \hat A^\dag \hat A$$ . 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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