Truncations of random unitary matrices

We analyse properties of non-Hermitian matrices of size M constructed as square submatrices of unitary (orthogonal) random matrices of size N > M , distributed according to the Haar measure. In this way we define ensembles of random matrices and study the statistical properties of the spectrum located inside the unit circle. In the limit of large matrices, this ensemble is characterized by the ratio M / N . For the truncated CUE we analytically derive the joint density of eigenvalues and all correlation functions. In the strongly non-unitary case universal Ginibre behaviour is found. For N - M fixed and N to the universal resonance-width distribution with N - M open channels is recovered.