Negative moments of characteristic polynomials of random matrices: Ingham-Siegel integral as an alternative to Hubbard-Stratonovich transformation

TL;DR

This paper introduces a new integral representation for calculating negative moments of characteristic polynomials of GUE matrices, revealing structural similarities with positive moments and implications for random matrix theory and the Riemann zeta function.

Contribution

It presents a compact integral approach using Ingham-Siegel integrals as an alternative to Hubbard-Stratonovich transformation for negative moments of GUE characteristic polynomials.

Findings

Derived asymptotic expressions for large N

Uncovered structural similarities between negative and positive moments

Discussed implications for supersymmetry and Riemann zeta zeros

Abstract

We reconsider the problem of calculating arbitrary negative integer moments of the (regularized) characteristic polynomial for $N\times N$ random matrices taken from the Gaussian Unitary Ensemble (GUE). A very compact and convenient integral representation is found via the use of a matrix integral close to that considered by Ingham and Siegel. We find the asymptotic expression for the discussed moments in the limit of large $N$. The latter is of interest because of a conjectured relation to properties of the Riemann $\zeta-$ function zeroes. Our method reveals a striking similarity between the structure of the negative and positive integer moments which is usually obscured by the use of the Hubbard-Stratonovich transformation. This sheds a new light on "bosonic" versus "fermionic" replica trick and has some implications for the supersymmetry method. We briefly discuss the case of the chiral GUE model from that perspective.