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 2βth moment of PN (A, θ) with respect to the uniform measure dθ 2π , 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 k 2 β 2 − k + 1. This resolves a conjecture of Fyodorov & Keating [23] 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.

We establish formulae for the moments of the moments of the characteristic polynomials of random orthogonal and symplectic matrices in terms of certain lattice point count problems. This allows us to establish asymptotic formulae when the matrix-size tends to infinity in terms of the volumes of certain regions involving continuous Gelfand-Tsetlin patterns with constraints. The results we find differ from those in the unitary case considered previously.

Following the work of Conrey, Rubinstein and Snaith [11] and Forrester and Witte [16] we examine a mixed moment of the characteristic polynomial and its derivative for matrices from the unitary group U (N ) (also known as the CUE) and relate the moment to the solution of a Painlevé differential equation. We also calculate a simple form for the asymptotic behaviour of moments of logarithmic derivatives of these characteristic polynomials evaluated near the unit circle.DIMA -DIPARTIMENTO DI MATEMATICA VIA DODECANESO,

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We review recent progress relating to the extreme value statistics of the characteristic polynomials of random matrices associated with the classical compact groups, and of the Riemann zeta-function and other L-functions, in the context of the general theory of logarithmically-correlated Gaussian fields. In particular, we focus on developments related to the conjectures of Fyodorov and Keating concerning the extreme value statistics, moments of moments, connections to Gaussian multiplicative chaos, and explicit formulae derived from the theory of symmetric functions.