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

Todd Kemp ()
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Todd Kemp: University of California, San Diego

Journal of Theoretical Probability, 2017, vol. 30, issue 2, 397-451

Abstract: Abstract This paper studies the empirical laws of eigenvalues and singular values for random matrices drawn from the heat kernel measures on the unitary groups $${\mathbb {U}}_N$$ U N and the general linear groups $${\mathbb {GL}}_N$$ GL N , for $$N\in {\mathbb {N}}$$ N ∈ N . It establishes the strongest known convergence results for the empirical eigenvalues in the $${\mathbb {U}}_N$$ U N case, and the first known almost sure convergence results for the eigenvalues and singular values in the $${\mathbb {GL}}_N$$ GL N case. The limit noncommutative distribution associated with the heat kernel measure on $${\mathbb {GL}}_N$$ 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 $$L^p$$ L p estimates for even integers p.

Keywords: Heat kernel analysis on Lie groups; Empirical eigenvalue distributions; Random matrices; Free probability; 60B20; 46L54 (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/s10959-015-0643-7

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