Averages of Exponentials from the point of view of Superintegrability

TL;DR

The paper tackles the problem of computing Gaussian averages of exponentials $\langle {\rm Tr}_R e^{\lambda X} \rangle$ in Hermitian matrix models by exploiting the superintegrability structure, which reduces these averages to Schur-based expressions involving Laguerre calculus. The author derives a general triangular decomposition: $\sigma_R = \sum_{Q\le R} K_{RQ} e^{\frac{1}{2}\mu_Q\lambda^2} P_Q(\lambda^2)$, where $K_{RQ}$ are Kostka-type coefficients and $\mu_Q = \sum_a k_a^2$ for the partition $Q$, with the polynomials $P_Q(\lambda^2)$ constructed from Laguerre data and traces of noncommuting matrices $\mathcal A_{k\lambda}$. Mnemonic computer experiments confirm the structure and yield explicit low-level forms, illustrating how $P_Q$ can be built from Laguerre polynomials and matrix traces, and revealing nontrivial combinatorics and ordering effects. The results connect Schur averages, Laguerre calculus, and Kostka matrices in a programmable framework, while highlighting open questions about explicit $N$-dependence, time-variable labeling, and a rigorous Kostka interpretation for the coefficients $K_{RQ}$.

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.