  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.
