Elementary Proof for Asymptotics of Large Haar-Distributed Unitary Matrices

TL;DR

This paper presents an elementary proof demonstrating that the scaled top-left submatrix of a large Haar-distributed unitary matrix converges in distribution to a matrix of independent complex Gaussian variables as the matrix size grows.

Contribution

It provides a simplified, elementary proof of a known asymptotic result for Haar-distributed unitary matrices, making the theorem more accessible.

Findings

Scaled submatrices converge to complex Gaussian matrices

Convergence occurs as matrix size n approaches infinity

Result applies to any k×k submatrix of the unitary matrix

Abstract

We provide an elementary proof for a theorem due to Petz and R\'effy which states that for a random $n\times n$ unitary matrix with distribution given by the Haar measure on the unitary group U(n), the upper left (or any other) $k\times k$ submatrix converges in distribution, after multiplying by a normalization factor $\sqrt{n}$ and as $n\to\infty$, to a matrix of independent complex Gaussian random variables with mean 0 and variance 1.