Bridges and random truncations of random matrices

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 T-(n)(s, t) = Sigma(i <= left perpendicular ns right perpendicular,j <= left perpendicular nt right perpendicular) vertical bar U-ij vertical bar(2), s, t is an element of[0, 1] 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.