Truncations of Haar distributed matrices, traces and bivariate Brownian bridges

Details Truncations of Haar distributed matrices, traces and bivariate Brownian bridges Truncations of Haar distributed matrices, traces and bivariate Brownian bridges

2010-07-08

arxiv.org/abs/1007.1366v4

Mathematics

Probability

Random matrices: Theory and Applications (RMTA) To appear (2012) http://www.editorialmanager.com/rmta/

Scientific paper

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.

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