Averages of Exponentials from the point of view of Superintegrability

TL;DR

This paper uses superintegrability to compute Gaussian averages of matrix exponentials, providing explicit Schur average expressions involving Laguerre polynomials and complex combinatorial sums.

Contribution

It introduces a novel approach leveraging superintegrability to derive explicit formulas for matrix exponential averages in terms of Laguerre polynomials.

Findings

Explicit formulas for Schur averages expressed via Laguerre polynomials.

Involves a triangular sum over partitions with exponential and polynomial factors.

Some formula components are not fully general, indicating scope for future research.

Abstract

We calculate Gaussian averages of arbitrary exponentials of the matrix variable $X$ with the help of superintegrability, which provides explicit expressions for Schur averages. As in the simpler cases the answer is expressed in terms of Laguerre polynomials, but in a somewhat sophisticated way. It involves triangular sum over partitions, with simple exponential factor and a complicated polynomial prefactor. Some ingredients of the formula are not found in full generality and there is still a room for further work.