Random truncations of Haar distributed matrices and bridges

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.

References in corpus (5)

• Integration with respect to the Haar measure on unitary, orthogonal and symplectic group
• The Dirichlet Markov Ensemble
• On polynomial integrals over the orthogonal group
• Spectral gap of doubly stochastic matrices generated from equidistributed unitary matrices
• Limiting Empirical Singular Value Distribution of Restrictions of Unitary Matrices

Cited by in corpus (1)

• The eigenvectors of Gaussian matrices with an external source