Limiting spectral distribution of the product of truncated Haar unitary matrices

TL;DR

This paper investigates the asymptotic spectral distribution of products of truncated Haar unitary matrices using free probability and Brown measure techniques, revealing new insights into their limiting behavior as matrix size grows.

Contribution

It introduces a novel analysis of the spectral distribution of products of truncated Haar unitaries using advanced free probability methods.

Findings

Derived the limiting spectral distribution for the product matrices

Applied free probability and Brown measure techniques

Extended understanding of spectral behavior in large random 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 $A_m^{(1)}\cdots A_m^{(k)}$.