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

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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 \(N\in {\mathbb {N}}\). 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 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.

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Acknowledgments

The author wishes to thank Bruce Driver for many helpful and insightful conversations, particularly with regard to Section 4.1, and Brian Hall for careful proofreading and clarifying remarks regarding Theorem 1.3.

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  1. Department of Mathematics, University of California, San Diego, La Jolla, CA, 92093-0112, USA

    Todd Kemp

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  1. Todd Kemp
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Correspondence to Todd Kemp.

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Supported by NSF CAREER Award DMS-1254807.

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Kemp, T. Heat Kernel Empirical Laws on \({\mathbb {U}}_N\) and \({\mathbb {GL}}_N\) . J Theor Probab 30, 397–451 (2017). https://doi.org/10.1007/s10959-015-0643-7

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