Truncations of Haar unitary matrices, traces and bivariate Brownian bridge.
Résumé
Let U be a Haar distributed unitary matrix in U(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.
Domaines
Probabilités [math.PR]
Origine : Fichiers produits par l'(les) auteur(s)