Combinatorial Structures in Random Matrix Theory Predictions for $L$-Functions

TL;DR

This paper explores algebraic combinatorics, specifically a new operation called overlap on partitions, to analyze random matrix theory predictions for the statistical behavior of L-functions, including asymptotic formulas for characteristic polynomial averages.

Contribution

It introduces the overlap operation on partitions, proves new identities for Littlewood-Schur functions, and generalizes formulas for averages of characteristic polynomial ratios over the unitary group.

Findings

Proved two overlap identities for Littlewood-Schur functions.

Derived an asymptotic formula for averages of mixed ratios of characteristic polynomials.

Provided combinatorial tools that may lead to new number theoretic proofs.

Abstract

Our results can be viewed as applications of algebraic combinatorics in random matrix theory. These applications are motivated by the predictive power of random matrix theory for the statistical behavior of the celebrated Riemann $\zeta$-function (and $L$-functions in general), which was discovered by Montgomery (with regard to zeros of $L$-functions) and by Keating and Snaith (with regard to values of $L$-functions). The first results revolve around a new operation on partitions, which we call overlap. We prove two overlap identities for so-called Littlewood-Schur functions. The first overlap identity represents the Littlewood-Schur function $LS _\lambda(X; Y)$ as a sum over subsets of $X$, while the second overlap identity essentially represents $LS_\lambda(X; Y)$ as a sum over pairs of partitions whose overlap equals $\lambda$. Both identities are derived by applying Laplace expansion to a determinantal formula for Littlewood-Schur functions due to Moens and Van der Jeugt. In addition, we give two visual characterizations for the set of all pairs of partitions whose overlap is equal to a partition $\lambda$. The second result is an asymptotic formula for averages of mixed ratios of characteristic polynomials over the unitary group, where mixed ratios are products of ratios and/or logarithmic derivatives. Our proof of this formula is a generalization of Bump and Gamburd's elegant combinatorial proof of Conrey, Forrester and Snaith's formula for averages of ratios of characteristic polynomials over the unitary group. The generalization relies on three combinatorial results, namely the first overlap identity, a new variant of the Murnaghan-Nakayama rule and an idea from vertex operator formalism. We conclude this thesis by explaining how this approach might lead to new number theoretic proofs.