Heat Kernel Empirical Laws on Uₙ and GLₙ

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