# Moments of moments of characteristic polynomials of random unitary matrices and lattice point counts

## TL;DR

This paper provides a combinatorial proof for the asymptotics of moments of characteristic polynomials of random unitary matrices, relating them to lattice point counts and Gelfand-Tsetlin patterns.

## Contribution

It introduces a new explicit formula for the leading coefficient in the asymptotics, connecting random matrix theory with lattice point enumeration.

## Key findings

- Asymptotic behavior of moments is characterized without computational methods.
- Explicit volume formula involving Gelfand-Tsetlin patterns is derived.
- Connection established between random matrix moments and lattice point counting.

## Abstract

In this note we give a combinatorial and non-computational proof of the asymptotics of the integer moments of the moments of the characteristic polynomials of Haar distributed unitary matrices as the size of the matrix goes to infinity. This is achieved by relating these quantities to a lattice point count problem. Our main result is a new explicit expression for the leading order coefficient in the asymptotic as a volume of a certain region involving continuous Gelfand-Tsetlin patterns with constraints.