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

TL;DR

This paper proves that the distribution of traces of certain functions on the unitary group under the heat kernel measure converges to a Gaussian distribution as the matrix size grows, with explicit covariance formulas.

Contribution

It establishes a central limit theorem for trace functions on the unitary group under the heat kernel measure, providing explicit covariance formulas and connecting to known results for uniform measures.

Findings

Gaussian fluctuations of trace functions as N→∞

Explicit covariance formula of order N^{-1}

Covariance convergence to Diaconis-Evans result at infinite time

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.