Random truncations of Haar distributed matrices and bridges

TL;DR

This paper studies the asymptotic behavior of randomly truncated Haar-distributed matrices, showing convergence to Gaussian processes and extending previous results on deterministic truncations.

Contribution

It introduces a model with random truncation of Haar matrices and proves convergence to Gaussian processes, generalizing earlier deterministic truncation results.

Findings

Convergence of the two-parameter process to a Gaussian process.

Extension of previous results from deterministic to random truncations.

Identification of other interesting convergence phenomena.

Abstract

Let $U$ be a Haar distributed matrix in $\mathbb U(n)$ or $\mathbb O (n)$. In a previous paper, we proved that after centering, the two-parameter process \[T^{(n)} (s,t) = \sum_{i \leq \lfloor ns \rfloor, j \leq \lfloor nt\rfloor} |U_{ij}|^2\] converges in distribution to the bivariate tied-down Brownian bridge. In the present paper, we replace the deterministic truncation of $U$ by a random one, where each row (resp. column) is chosen with probability $s$ (resp. $t$) independently. We prove that the corresponding two-parameter process, after centering and normalization by $n^{-1/2}$ converges to a Gaussian process. On the way we meet other interesting convergences.