In this note we establish the large $N$ asymptotics of the positive integer moments of the partition function of the C$\beta$E field, or equivalently the moments of the moments of the characteristic polynomial of the C$\beta$E ensemble. This is done by making use of symmetric function theory.
On the logarithm of the characteristic polynomial of the Ginibre ensemble
Exchangeable arrays and integrable systems for characteristic polynomials of random matrices
Multiplicative chaos and the characteristic polynomial of the CUE: the $L^1$-phase
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