Heat Kernel Empirical Laws on $${mathbb {U}}_N$$ and $${mathbb {GL}}_N$$GLN

This paper studies the empirical laws 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 (Nin {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 with 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.