Bridges and random truncations of random matrices

Abstract

Let U be a Haar distributed matrix in 𝕌(n) or 𝕆(n). In a previous paper, we proved that after centering, the two-parameter process [Formula: see text] 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, in which each row (respectively, column) is chosen with probability s (respectively, 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.