Limiting spectral distribution of the product of truncated Haar unitary matrices

Abstract: Let $A_m^{(1)},\ldots, A_m^{(k)}$ be $m\times m$ left-uppermost blocks of $k$ independent $n\times n$ Haar unitary matrices where $\frac{n}{m}\to \alpha$ as $m\to \infty$, with $1<\alpha<\infty$. Using free probability and Brown measure techniques, we find the limiting spectral distribution of $A_m^{(1)}\cdots A_m^{(k)}$.