PUB - Publikationen an der Universität Bielefeld

We investigate the spectral properties of the product of $M$ complexnon-Hermitian random matrices that are obtained by removing $L$ rows andcolumns of larger unitary random matrices uniformly distributed on the group${\rm U}(N+L)$. Such matrices are called truncated unitary matrices or randomcontractions. We first derive the joint probability distribution for theeigenvalues of the product matrix for fixed $N,\ L$, and $M$, given by astandard determinantal point process in the complex plane. The weight howeveris non-standard and can be expressed in terms of the Meijer G-function. Theexplicit knowledge of all eigenvalue correlation functions and thecorresponding kernel allows us to take various large $N$ (and $L$) limits atfixed $M$. At strong non-unitarity, with $L/N$ finite, the eigenvalues condenseon a domain inside the unit circle. At the edge and in the bulk we find thesame universal microscopic kernel as for a single complex non-Hermitian matrixfrom the Ginibre ensemble. At the origin we find the same new universalityclasses labelled by $M$ as for the product of $M$ matrices from the Ginibreensemble. Keeping a fixed size of truncation, $L$, when $N$ goes to infinityleads to weak non-unitarity, with most eigenvalues on the unit circle as forunitary matrices. Here we find a new microscopic edge kernel that generalizesthe known results for M=1. We briefly comment on the case when each productmatrix results from a truncation of different size $L_j$.