Central limit theorem for the heat kernel measure on the unitary group |
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Authors: | Thierry Lévy |
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Institution: | a Département de Mathématiques, Ecole Normale Supérieure, 45, rue d'Ulm, F-75230 Paris Cedex 05, France b Laboratoire de Mathématiques, Faculté des Sciences d'Orsay, Université Paris-Sud, F-91405 Orsay Cedex, France |
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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 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.N. Evans in a previous work on uniformly distributed unitary matrices. Finally, we discuss some combinatorial aspects of our results. |
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Keywords: | Central limit theorem Random matrices Unitary matrices Heat kernel Free probability |
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