Truncations of Haar distributed matrices, traces and bivariate Brownian bridges

Understand

Let U be a Haar distributed unitary matrix in U(n)or O(n).

• We show that after centering the double index process $$ W^{(n)} (s,t) = \sum_{i \leq \lfloor ns \rfloor, j \leq \lfloor nt\rfloor} |U_{ij}|^2 $$ converges in distribution to the bivariate tied-down Brownian bridge.
• The proof relies on the notion of second order freeness.