Truncations of Haar distributed matrices, traces and bivariate Brownian bridges

TL;DR

This paper demonstrates that certain sums of squared entries of Haar distributed matrices, when properly centered, converge to a bivariate tied-down Brownian bridge, using second order freeness techniques.

Contribution

It establishes the convergence of truncated sums of Haar matrix entries to a bivariate Brownian bridge, introducing a novel application of second order freeness.

Findings

Convergence of the process to a bivariate Brownian bridge.

Application of second order freeness in matrix truncation analysis.

Provides new insights into the spectral properties of Haar matrices.

Abstract

Let U be a Haar distributed unitary matrix in U(n)or O(n). We show that after centering the double index process $$ W^{(n)} (s,t) = \sum_{i \leq \lfloor ns \rfloor, j \leq \lfloor nt\rfloor} |U_{ij}|^2 $$ converges in distribution to the bivariate tied-down Brownian bridge. The proof relies on the notion of second order freeness.