Duality in random matrix ensembles for all Beta

TL;DR

This paper explores duality relations in Gaussian and Chiral Beta-Ensembles of random matrices for all Beta values, revealing new integral representations and connections to orthogonal polynomials using Jack polynomials.

Contribution

It establishes duality relations for Beta-Ensembles valid for all Beta, extending known cases and linking to matrix integrals and orthogonal polynomials.

Findings

Duality relations {Beta,N,n} <--> {4/Beta,n,N} hold for all Beta>0.

Derived matrix integrals of Airy (Kontsevich) type at the spectrum edge.

Discussed implications for multiple orthogonal polynomials and matrix model partition functions.

Abstract

Gaussian and Chiral Beta-Ensembles, which generalise well known orthogonal (Beta=1), unitary (Beta=2), and symplectic (Beta=4) ensembles of random Hermitian matrices, are considered. Averages are shown to satisfy duality relations like {Beta,N,n} <--> {4/Beta,n,N} for all Beta>0, where N and n respectively denote the number of eigenvalues and products of characteristic polynomials. At the edge of the spectrum, matrix integrals of the Airy (Kontsevich) type are obtained. Consequences on the integral representation of the multiple orthogonal polynomials and the partition function of the formal one-matrix model are also discussed. Proofs rely on the theory of multivariate symmetric polynomials, especially Jack polynomials.