On the moments of moments of random matrices and Ehrhart polynomials

TL;DR

This paper studies the polynomial behavior and symmetry of moments of moments of characteristic polynomials of random matrices, using Ehrhart-Macdonald reciprocity and lattice point bijections.

Contribution

It classifies the integer roots of these polynomials and proves their symmetry, confirming predictions from Bailey's thesis.

Findings

Classified the integer roots of the moments of moments polynomials.

Proved the symmetry property of these polynomials.

Connected the polynomials' properties to Ehrhart-Macdonald reciprocity.

Abstract

There has been significant interest in studying the asymptotics of certain generalised moments, called the moments of moments, of characteristic polynomials of random Haar-distributed unitary and symplectic matrices, as the matrix size $N$ goes to infinity. These quantities depend on two parameters $k$ and $q$ and when both of them are positive integers it has been shown that these moments are in fact polynomials in the matrix size $N$. In this paper we classify the integer roots of these polynomials and moreover prove that the polynomials themselves satisfy a certain symmetry property. This confirms some predictions from the thesis of Bailey. The proof uses the Ehrhart-Macdonald reciprocity for rational convex polytopes and certain bijections between lattice points in some polytopes.