Local Spectrum of Truncations of Kronecker Products of Haar Distributed Unitary Matrices

Abstract

We address the local spectral behavior of the random matrix ${\Pi }_1{U}^{\otimes k}{\Pi }_2{U}^{\otimes k\ast }{\Pi }_1$, where $U$ is a Haar distributed unitary matrix of size $n\times n$, the factor $k$ is at most $c_0\log n$ for a small constant $c_0>0$, and ${\Pi }_1,{\Pi }_2$ are arbitrary projections on ${\ell }_2^{n^k}$ of ranks proportional to $n^k$. We prove that in this setting the $k$-fold Kronecker product behaves similarly to the well-studied case when $k=1$.