On the Moments of the Moments of the Characteristic Polynomials of Random Unitary Matrices

Abstract

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.

Moments of moments of characteristic polynomials of random unitary matrices and lattice point counts

“…The aim of this note is to give a new proof, alternative to the one in [1] (which gives a different expression for the leading order coefficient), of the following theorem. Theorem 1.1.…”

“…There is expected to be a freezing transition at kβ 2 = 1 which determines the large β limit. We refer to [11], [1] for further motivation and background on these conjectures. The case k = 1 of Theorem 1.1 is classical, see for example [13], and MoM N 1, β can be calculated explicitly (in fact this can be done for any real β) using the celebrated Selberg integral (see [10]):…”

Moments of Moments and Branching Random Walks

“…As reviewed in Sect. 1 , it is now known that (see [ 8 , 9 , 12 , 16 ]) for , and as for some positive constants 6 and depending only on . Furthermore, for , is a polynomial in N , see [ 9 ].…”

“…For integer , it was proved in [ 9 ] that is a polynomial in the matrix size, N , of degree , in line with ( 7 ).…”

Extreme values of CUE characteristic polynomials: a numerical study

“…As a benchmark for our numerics we compare some low moments with the analytically known values. Analytically one finds [5] M N, We have here introduced as well the reduced moments M red N,β,k . The reduced moments obey M red N,β,k → 1 as N → ∞ and are of order unity for finite values of N .…”