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

TL;DR

This paper analytically proves the asymptotic behavior of moments of moments of characteristic polynomials in random symplectic and orthogonal matrices, providing new integral formulas and linking to conjectures on L-functions.

Contribution

It offers an analytic proof of asymptotics for moments of moments, deriving new integral expressions and connecting these results to conjectures on L-functions with symplectic and orthogonal symmetry.

Findings

Derived asymptotic formulas for moments of moments.

Provided alternative integral expressions for leading coefficients.

Linked conjectures on L-functions to the shifted moments conjecture.

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.