Central limit theorem for the heat kernel measure on the unitary group

Details Central limit theorem for the heat kernel measure on the unitary group Central limit theorem for the heat kernel measure on the unitary group

2009-05-20

arxiv.org/abs/0905.3282v2

Journal of Functional Analysis 259, 12 (2010) 3163-3204

Mathematics

Probability

44 pages

Scientific paper

10.1016/j.jfa.2010.08.005

We prove that for a finite collection of real-valued functions $f_{1},...,f_{n}$ on the group of complex numbers of modulus 1 which are derivable with Lipschitz continuous derivative, the distribution of $(\tr f_{1},...,\tr f_{n})$ under the properly scaled heat kernel measure at a given time on the unitary group $\U(N)$ has Gaussian fluctuations as $N$ tends to infinity, with a covariance for which we give a formula and which is of order $N^{-1}$. In the limit where the time tends to infinity, we prove that this covariance converges to that obtained by P. Diaconis and S. Evans in a previous work on uniformly distributed unitary matrices. Finally, we discuss some combinatorial aspects of our results.

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