  This paper studies the empirical measures of eigenvalues and singular values
for random matrices drawn from the heat kernel measures on the unitary groups
$\mathbb{U}_N$ and the general linear groups $\mathbb{GL}_N$, for
$N\in\mathbb{N}$. It establishes the strongest known convergence results for
the empirical eigenvalues in the $\mathbb{U}_N$ case, and the first known
almost sure convergence results for the eigenvalues and singular values in the
$\mathbb{GL}_N$ case. The limit noncommutative distribution associated to the
heat kernel measure on $\mathbb{GL}_N$ is identified as the projection of a
flow on an infinite-dimensional polynomial space. These results are then
strengthened from variance estimates to $L^p$ estimates for even integers $p$.
