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 $N\times N$ random matrices $\hat{ A}$ of such a type, corresponding to systems with broken time-reversal invariance. Deviations from unitarity are characterized by rank $M\le N$ and a set of eigenvalues $0<T_i\le 1, i=1,...,M$ of the matrix $\hat{ T}=\hat{ { \bf 1}}-\hat{ A}^{ \dagger}\hat{ A}$. 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.
Categories: nlin.CD cond-mat