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This paper studies the empirical measures of eigenvalues and singular valuesfor 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 forthe empirical eigenvalues in the $\mathbb{U}_N$ case, and the first knownalmost sure convergence results for the eigenvalues and singular values in the$\mathbb{GL}_N$ case. The limit noncommutative distribution associated to theheat kernel measure on $\mathbb{GL}_N$ is identified as the projection of aflow on an infinite-dimensional polynomial space. These results are thenstrengthened from variance estimates to $L^p$ estimates for even integers $p$.

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