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

TL;DR

This paper combines Stein's method with heat kernel techniques to analyze the distribution of traces of powers of elements in compact Lie groups, establishing a normal limit with error bounds even as the power grows with the group size.

Contribution

It introduces a novel combination of Stein's method and heat kernel techniques to handle growing powers in the trace distribution of compact Lie group elements.

Findings

Trace of the jth power converges to a normal distribution with error of order j/n.

Method applies to U(n,C), USp(n,C), and SO(n,R).

Useful for studying eigenfunction value distributions.

Abstract

Combining Stein's method with heat kernel techniques, we show that the trace of the jth power of an element of U(n,C), USp(n,C) or SO(n,R) has a normal limit with error term of order j/n. In contrast to previous works, here j may be growing with n. The technique should prove useful in the study of the value distribution of approximate eigenfunctions of Laplacians.