Circular law and arc law for truncation of random unitary matrix

Abstract

Let V be the m × m upper-left corner of an n × n Haar-invariant unitary matrix. Let λ1, …, λm be the eigenvalues of V. We prove that the empirical distribution of a normalization of λ1, …, λm goes to the circular law, that is, the uniform distribution on $\lbrace z\in \mathbb {C};\, |z|\le 1\rbrace${z∈C;|z|≤1} as m → ∞ with m/n → 0. We also prove that the empirical distribution of λ1, …, λm goes to the arc law, that is, the uniform distribution on $\lbrace z\in \mathbb {C};\, |z|=1\rbrace${z∈C;|z|=1} as m/n → 1. These explain two observations by Życzkowski and Sommers (2000).