Central limit theorem for the heat kernel measure on the unitary group
Abstract
We prove that for a finite collection of real-valued functions f1,...,fn on the group of complex numbers of modulus 1 which are derivable with Lipschitz continuous derivative, the distribution of ( f1,..., fn) under the properly scaled heat kernel measure at a given time on the unitary group (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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