Random matrix theory and moments of moments of $L$-functions

Abstract

We give an analytic proof of the asymptotic behaviour of the moments of moments of the characteristic polynomials of random symplectic and orthogonal matrices. We therefore obtain alternate, integral expressions for the leading order coefficients previously found by Assiotis, Bailey and Keating. We also discuss the conjectures of Bailey and Keating for the corresponding moments of moments of L-functions with symplectic and orthogonal symmetry. Specifically, we show that these conjectures follow from the shifted moments conjecture of Conrey, Farmer, Keating, Rubinstein and Snaith.