## 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

### Keywords:

random matrix; Stein’s method; heat kernel
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