Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the moments of characteristic polynomials

Bhargavi Jonnadula, Jon Keating, Francesco Mezzadri

TL;DR

This work refines our understanding of moments of characteristic polynomials in Hermitian random matrices, with a focus on the GUE. Using a novel combinatorial framework built from multivariate orthogonal polynomials and a generalized dual Cauchy identity, the authors derive exact finite-$N$ expressions and reveal a surprising parity dependence: the leading asymptotics differ for even and odd matrix sizes, though averaging over parity recovers the classical BH0-type behaviour. They map the asymptotics near and away from the semicircle centre, connect the results to the semicircular law in the Dyson limit, and apply the formalism to secular coefficients, establishing a broad, parity-aware understanding that extends to Laguerre and Jacobi ensembles. The findings advance the analytical toolkit for moments and related quantities in random matrix theory with potential implications for number theory and quantum chaos.

Abstract

We examine the asymptotics of the moments of characteristic polynomials of $N\times N$ matrices drawn from the Hermitian ensembles of Random Matrix Theory, in the limit as $N\to\infty$. We focus in particular on the Gaussian Unitary Ensemble, but discuss other Hermitian ensembles as well. We employ a novel approach to calculate asymptotic formulae for the moments, enabling us to uncover subtle structure not apparent in previous approaches.

On the moments of characteristic polynomials

TL;DR

This work refines our understanding of moments of characteristic polynomials in Hermitian random matrices, with a focus on the GUE. Using a novel combinatorial framework built from multivariate orthogonal polynomials and a generalized dual Cauchy identity, the authors derive exact finite- expressions and reveal a surprising parity dependence: the leading asymptotics differ for even and odd matrix sizes, though averaging over parity recovers the classical BH0-type behaviour. They map the asymptotics near and away from the semicircle centre, connect the results to the semicircular law in the Dyson limit, and apply the formalism to secular coefficients, establishing a broad, parity-aware understanding that extends to Laguerre and Jacobi ensembles. The findings advance the analytical toolkit for moments and related quantities in random matrix theory with potential implications for number theory and quantum chaos.

Abstract

We examine the asymptotics of the moments of characteristic polynomials of matrices drawn from the Hermitian ensembles of Random Matrix Theory, in the limit as . We focus in particular on the Gaussian Unitary Ensemble, but discuss other Hermitian ensembles as well. We employ a novel approach to calculate asymptotic formulae for the moments, enabling us to uncover subtle structure not apparent in previous approaches.

Paper Structure

This paper contains 11 sections, 7 theorems, 115 equations, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Moments of characteristic polynomials
  4. Asymptotics
  5. Centre of the semi-circle
  6. Away from the centre of the semi-circle
  7. Second moment
  8. Higher moments
  9. Secular coefficients
  10. Asymptotics of ratio of factorials
  11. More on the second moment

Key Result

Lemma 3.1

Let $p,N\in\mathbb{N}$. For $\lambda\subseteq (N^p)\equiv (\underbrace{N,\dots,N}_{p})$, let $\tilde{\lambda}=(p-\lambda_N^\prime,\dots,p-\lambda_1^\prime)$. Then Macdonald1998

Theorems & Definitions (10)

  • Lemma 3.1: Dual Cauchy identity
  • Lemma 3.2
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 4.1
  • proof
  • Proposition 5.1
  • proof
  • Proposition 5.2
  • proof