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Universal microscopic correlation functions for products of truncated unitary matrices

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