Limiting spectral distribution of the product of truncated Haar unitary matrices

Abstract

Let [Formula: see text], [Formula: see text], be [Formula: see text] probabilistically independent matrices of order [Formula: see text] (with [Formula: see text]) which are the left-uppermost blocks of [Formula: see text] Haar unitary matrices. Suppose that [Formula: see text] as [Formula: see text], with [Formula: see text]. Using free probability and Brown measure techniques, we find the limiting spectral distribution of [Formula: see text].