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

We address the local spectral behavior of the random matrix $\Pi_1 U^{\otimes k} \Pi_2 U^{\otimes k *} \Pi_1$, 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 $k=1$.