Random truncations of Haar distributed matrices and bridges

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 $$T^{(n)}(s,t)=\sum _{i\le \lfloor ns\rfloor ,j\le \lfloor nt\rfloor }\mathrm{\mid }{U}_{ij}{\mathrm{\mid }}^2$$ 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 $n^{-1\mathrm{/}2}$ converges to a Gaussian process. On the way we meet other interesting convergences.