Spectra of random contractions and scattering theory for discrete-time systems

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.