Stein's method, heat kernel, and traces of powers of elements of compact Lie groups

Abstract

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 $C \dot j/n$, with $C$ an absolute constant. In contrast to previous works, here $j$ may be growing with $n$. The technique might prove useful in the study of the value distribution of approximate eigenfunctions of Laplacians.