Truncations of Haar unitary matrices, traces and bivariate Brownian bridge.

Catherine Donati-Martin (1) , Alain Rouault (2)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 LMV - Laboratoire de Mathématiques de Versailles

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]

Catherine Donati-Martin, Alain Rouault. Truncations of Haar unitary matrices, traces and bivariate Brownian bridge.. 2010. ⟨hal-00498758v1⟩