Universal microscopic correlation functions for products of truncated unitary matrices

Abstract

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 U (N + L). Such matrices are called truncated unitary matrices or random contractions. We first derive the joint probability distribution for the complex 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 .

Singular Values of Products of Ginibre Random Matrices, Multiple Orthogonal Polynomials and Hard Edge Scaling Limits

“…A similar determinantal structure holds for the eigenvalues of products of truncated unitary matrices [3]. The determinantal structure opens up the way to a more detailed analysis at the finite n level [3,6].…”

“…A similar determinantal structure holds for the eigenvalues of products of truncated unitary matrices [3]. The determinantal structure opens up the way to a more detailed analysis at the finite n level [3,6]. Very recently, Akemann, Kieburg, and Wei [5] found that the squared singular values of products of complex Ginibre matrices are a determinantal point process on the positive real line.…”

“…This was further extended to the case of products of rectangular Ginibre matrices by Akemann, Ipsen and Kieburg [4]. The correlation kernels in [2,3,4,5,25] are all expressed in terms of Meijer G-functions.…”

How Many Eigenvalues of a Product of Truncated Orthogonal Matrices are Real?

“…We note that the density (4.35) is the same as for products of truncated unitary matrices [5,1]. In fact, as suggested in [5], the validity of (4.35) can be proven using techniques from free probability; figure 1 shows a comparison with numerical data.…”

“…We will now look at asymptotic properties and we will start with the m = 1 case which is the simplest by far. This simplicity, first identified in [24], is due to the simple form of the weights w (1) r and w (1) c . Thus, it follows from (1.10) and (1.14) that w (1)…”