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Symmetric Function Theory and Unitary Invariant Ensembles

Bhargavi Jonnadula, Jonathan P. Keating, Francesco Mezzadri

TL;DR

This work develops a symmetric-function framework for unitary-invariant Hermitian ensembles (GUE, LUE, JUE), generalizing the role of Schur polynomials to Hermitian matrices through multivariate orthogonal polynomials. By constructing determinant-based multivariate polynomials from Hermite, Laguerre, and Jacobi bases and a generalized dual Cauchy identity, the authors derive exact finite-N formulas for joint moments of traces and characteristic polynomials across the three ensembles. They show that these moments are polynomials in N with integer-root structures tied to partition content, and they establish explicit asymptotic rates for the convergence of Chebyshev-polynomial functionals to Gaussian limits, along with cumulant expansions linked to ribbon-graph enumeration. The results unify and extend previous group-theoretic approaches, providing a powerful toolkit for exact calculations and asymptotics in Hermitian random-matrix models. The framework has potential implications for understanding fluctuations and universality in a broad class of unitary-invariant ensembles.

Abstract

Representation theory and the theory of symmetric functions have played a central role in Random Matrix Theory in the computation of quantities such as joint moments of traces and joint moments of characteristic polynomials of matrices drawn from the Circular Unitary Ensemble and other Circular Ensembles related to the classical compact groups. The reason is that they enable the derivation of exact formulae, which then provide a route to calculating the large-matrix asymptotics of these quantities. We develop a parallel theory for the Gaussian Unitary Ensemble of random matrices, and other related unitary invariant matrix ensembles. This allows us to write down exact formulae in these cases for the joint moments of the traces and the joint moments of the characteristic polynomials in terms of appropriately defined symmetric functions. As an example of an application, for the joint moments of the traces we derive explicit asymptotic formulae for the rate of convergence of the moments of polynomial functions of GUE matrices to those of a standard normal distribution when the matrix size tends to infinity.

Symmetric Function Theory and Unitary Invariant Ensembles

TL;DR

This work develops a symmetric-function framework for unitary-invariant Hermitian ensembles (GUE, LUE, JUE), generalizing the role of Schur polynomials to Hermitian matrices through multivariate orthogonal polynomials. By constructing determinant-based multivariate polynomials from Hermite, Laguerre, and Jacobi bases and a generalized dual Cauchy identity, the authors derive exact finite-N formulas for joint moments of traces and characteristic polynomials across the three ensembles. They show that these moments are polynomials in N with integer-root structures tied to partition content, and they establish explicit asymptotic rates for the convergence of Chebyshev-polynomial functionals to Gaussian limits, along with cumulant expansions linked to ribbon-graph enumeration. The results unify and extend previous group-theoretic approaches, providing a powerful toolkit for exact calculations and asymptotics in Hermitian random-matrix models. The framework has potential implications for understanding fluctuations and universality in a broad class of unitary-invariant ensembles.

Abstract

Representation theory and the theory of symmetric functions have played a central role in Random Matrix Theory in the computation of quantities such as joint moments of traces and joint moments of characteristic polynomials of matrices drawn from the Circular Unitary Ensemble and other Circular Ensembles related to the classical compact groups. The reason is that they enable the derivation of exact formulae, which then provide a route to calculating the large-matrix asymptotics of these quantities. We develop a parallel theory for the Gaussian Unitary Ensemble of random matrices, and other related unitary invariant matrix ensembles. This allows us to write down exact formulae in these cases for the joint moments of the traces and the joint moments of the characteristic polynomials in terms of appropriately defined symmetric functions. As an example of an application, for the joint moments of the traces we derive explicit asymptotic formulae for the rate of convergence of the moments of polynomial functions of GUE matrices to those of a standard normal distribution when the matrix size tends to infinity.

Paper Structure

This paper contains 19 sections, 21 theorems, 197 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Statements and results
  3. Background
  4. Review of symmetric functions
  5. Multivariate orthogonal polynomials
  6. Correlation functions of characteristic polynomials
  7. Correlations
  8. Change of basis between symmetric functions
  9. Moments of Schur polynomials
  10. Moments of characteristic polynomials
  11. Joint moments of traces
  12. Eigenvalue fluctuations
  13. Moments
  14. Second degree
  15. General degree
  16. ...and 4 more sections

Key Result

Lemma 2.1

Let $\varPhi_\mu$ be multivariate polynomials given in eq:mul_poly. Let $p,q\in\mathbb{N}$ and for $\lambda\subseteq (q^p)\equiv (\underbrace{q,\dots,q}_{p})$ let $\tilde{\lambda}=(p-\lambda_q^\prime,\dots,p-\lambda_1^\prime)$. Then

Figures (4)

  • Figure :
  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (36)

  • Lemma 2.1
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Proposition 3.1: Cauchy Identity Macdonald1998
  • Proposition 4.1: Laplace Expansion
  • Proposition 4.2
  • proof : Proof of Lemma. \ref{['lemma:dualcauchy_to_mulmonicpoly']}
  • proof : Proof of Thm. \ref{['thm:correlations_charpoly']}
  • Proposition 5.1
  • ...and 26 more