On the moments of the partition function of the C$β$E field

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.