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

We address the local spectral behavior of the random matrix Π_1U^(⊗k)Π_2U^(⊗k∗)Π_1 where U is a Haar distributed unitary matrix of size n × n, the factor k is at most c0lgn for a small constant c_0 > 0, and Π_1, Π_2 are arbitrary projections on ℓ^n^k_2of ranks proportional to n^k. We prove that in this setting the k-fold Kronecker product behaves similarly to the well-studied case when k = 1