On the moments of the partition function of the C$\beta$E field

TL;DR

This paper derives a combinatorial formula for the moments of the partition function of the $Ceta E_N$ field, revealing their large N asymptotics in a specific regime using Jack polynomials.

Contribution

It introduces a new combinatorial formula for moments of the $Ceta E_N$ partition function and analyzes their asymptotic behavior with Jack polynomials.

Findings

Derived a combinatorial formula for moments

Established large N asymptotics in the supercritical regime

Utilized Jack polynomials to analyze the moments

Abstract

We obtain a combinatorial formula for the positive integer moments of the partition function of the $C\beta E_{N}$ field, or equivalently the moments of the moments of the characteristic polynomial of the $C\beta E_{N}$ ensemble. We then use this formula to establish the large $N$ asymptotics of these moments in the "moment-supercritical" regime. A key role is played by Jack polynomials.