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

Abstract

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.

44 pages

References in corpus (4)

• Central limit theorem for linear eigenvalue statistics of random matrices with independent entries
• Second Order Freeness and Fluctuations of Random Matrices: II. Unitary Random Matrices
• Second Order Freeness and Fluctuations of Random Matrices, III. Higher order freeness and free cumulants
• Central limit theorem for linear eigenvalue statistics of orthogonally invariant matrix models

Cited by in corpus (9)

• Strong Convergence of Unitary Brownian Motion
• Free energies and fluctuations for the Unitary Brownian Motion
• Dip-ramp-plateau for Dyson Brownian motion from the identity on $U(N)$
• Heat Kernel Empirical Laws on ${\mathbb{U}}_N$ and ${\mathbb{GL}}_N$
• Random pure quantum states via unitary Brownian motion
• Hierarchical analytical approach to universal spectral correlations in Brownian Quantum Chaos
• Convergence of the empirical spectral measure of unitary Brownian motion
• Wilson loop expectations as sums over surfaces on the plane
• Matrix liberation process II: Relation to orbital free entropy