Negative moments of characteristic polynomials of random GOE matrices and singularity-dominated strong fluctuations

TL;DR

This paper computes negative moments of characteristic polynomials for GOE matrices, confirming a conjecture related to singularity-dominated fluctuations, and providing the first proof of such predictions.

Contribution

It provides the first rigorous proof of a conjecture connecting negative moments of GOE characteristic polynomials with singularity-dominated fluctuation theory.

Findings

Negative moments computed and matched with conjecture

Confirmed nontrivial predictions from singularity theory

Established a new link between random matrix theory and fluctuation phenomena

Abstract

We calculate the negative integer moments of the (regularized) characteristic polynomials of N x N random matrices taken from the Gaussian Orthogonal Ensemble (GOE) in the limit as $N \to \infty$. The results agree nontrivially with a recent conjecture of Berry & Keating motivated by techniques developed in the theory of singularity-dominated strong fluctuations. This is the first example where nontrivial predictions obtained using these techniques have been proved.