Central limit theorem for the heat kernel measure on the unitary group
Central limit theorem for the heat kernel measure on the unitary group
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DOI：
10.1016/j.jfa.2010.08.005
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发表时间：
2009-05
影响因子：
1.7
通讯作者：
Thierry L'evy;M. Maida
中科院分区：
文献类型：
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作者：
Thierry L'evy;M. Maida
证明了在酉群U(N)上给定时间的适当标度热核测度下（trf1，…，trfn）的分布在N趋于无穷时具有高斯起伏，其协方差为N−1阶，我们给出了一个公式。在时间趋于无穷大的极限下，证明了该协方差收敛于P. Diaconis和S.N. Evans在前人关于一致分布酉矩阵的研究中得到的协方差。最后，我们讨论了我们的结果的一些组合方面。
We prove that for a finite collection of real-valued functions f1,…,fnon the group of complex numbers of modulus 1 which are derivable with Lipschitz continuous derivative, the distribution of (trf1,…,trfn) 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.