Central limit theorems for the Brownian motion on large unitary groups (Q2880705)

The author studies the large \(n\) limit of the distributions of linear combinations of the entries of a Brownian motion on the group of \(n\times n\) unitary matrices. The process of such a linear combination is proved to converge to a Gaussian one. Several specific results of this kind are obtained, for various scales of time and various initial distributions (such as the uniform measure on the group of permutation matrices). As an application, a short proof is given for some results by \textit{A. D'Aristotile, P. Diaconis} and \textit{C. M. Newman} [in: K. Athreya (ed.) et al., Probability, statistics and their applications: Papers in honor of Rabi Bhattacharya. IMS Lect. Notes, Monogr. Ser. 41, 97--116 (2003; Zbl 1056.60081)] about the asymptotic normality of linear combinations of the entries of uniform random unitary matrices.