A Combinatorial Approach to Mixed Ratios of Characteristic Polynomials

TL;DR

This paper derives an asymptotic formula for mixed ratios of characteristic polynomials over the unitary group using combinatorial methods, extending previous proofs and providing insights into eigenvalue sums.

Contribution

It introduces a combinatorial derivation of a general asymptotic formula for mixed ratios of characteristic polynomials, generalizing prior work by Bump, Gamburd, Conrey, Forrester, and Snaith.

Findings

Derived an asymptotic formula for mixed ratios of characteristic polynomials.

Extended combinatorial proof techniques to more general ratios.

Provided an explicit formula for sums over eigenvalues of random unitary matrices.

Abstract

We provide a combinatorial derivation of 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. One application of this formula is an asymptotic expression for sums over zeros of a random characteristic polynomial from the unitary group, which we call an explicit formula for eigenvalues in an analogy to what is called an explicit formula in the context of $L$-functions.