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Stein’s method, heat kernel, and traces of powers of elements of compact Lie groups. (English) Zbl 1252.60012

Summary: Combining Stein’s method with heat kernel techniques, we show that the trace of the \(j\)-th power of an element of \(U(n,\mathbb{C}), USp(n,\mathbb{C})\), or \(SO(n,\mathbb{R})\) has a normal limit with error term \(C j/n\), with an absolute constant \(C\). In contrast to previous works, \(j\) may be growing with \(n\). The technique might prove useful in the study of the value distribution of approximate eigenfunctions of Laplacians.

MSC:

60B20 Random matrices (probabilistic aspects)
15B52 Random matrices (algebraic aspects)
60F05 Central limit and other weak theorems
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