Local spectrum of truncations of Kronecker products of Haar distributed unitary matrices

Abstract

We address the local spectral behavior of the random matrix [Formula: see text] where U is a Haar distributed unitary matrix of size n × n, the factor k is at most c0 lg n for a small constant c0 > 0, and Π1, Π2 are arbitrary projections on [Formula: see text] of ranks proportional to nk. We prove that in this setting the k-fold Kronecker product behaves similarly to the well-studied case when k = 1.