On the moments of the moments of the characteristic polynomials of random unitary matrices

Denoting by PN(A, θ) = det (I- Ae-iθ) the characteristic polynomial on the unit circle in the complex plane of an N× N random unitary matrix A, we calculate the kth moment, defined with respect to an average over A∈ U(N) , of the random variable corresponding to the 2 βth moment of PN(A, θ) with respect to the uniform measure dθ2π, for all k, β∈ N. These moments of moments have played an important role in recent investigations of the extreme value statistics of characteristic polynomials and their connections with log-correlated Gaussian fields. Our approach is based on a new combinatorial representation of the moments using the theory of symmetric functions, and an analysis of a second representation in terms of multiple contour integrals. Our main result is that the moments of moments are polynomials in N of degree k2β2- k+ 1. This resolves a conjecture of Fyodorov and Keating (Philos Trans R Soc A 372(2007):20120503, 2014) concerning the scaling of the moments with N as N→ ∞, for k, β∈ N. Indeed, it goes further in that we give a method for computing these polynomials explicitly and obtain a general formula for the leading coefficient