Dualities in random matrix theory

TL;DR

This paper reviews duality identities in random matrix theory, highlighting their structure, applications, and the mathematical tools like Jack polynomials and loop equations used to analyze them across various ensembles.

Contribution

It provides a comprehensive overview of duality identities in random matrix theory, including their derivation, mathematical framework, and applications to correlation functions and spectral analysis.

Findings

Duality identities relate averages of characteristic polynomials across different matrix ensembles.

Jack polynomial theory is essential for analyzing beta-generalized ensembles.

Loop equations are key tools for studying dualities in spectral moments.

Abstract

Duality identities in random matrix theory for products and powers of characteristic polynomials, and for moments, are reviewed. The structure of a typical duality identity for the average of a positive integer power $k$ of the characteristic polynomial for particular ensemble of $N \times N$ matrices is that it is expressed as the average of the power $N$ of the characteristic polynomial of some other ensemble of random matrices, now of size $k \times k$. With only a few exceptions, such dualities involve (the $\beta$ generalised) classical Gaussian, Laguerre and Jacobi ensembles Hermitian ensembles, the circular Jacobi ensemble, or the various non-Hermitian ensembles relating to Ginibre random matrices. In the case of unitary symmetry in the Hermitian case, they can be studied using the determinantal structure. The $\beta$ generalised case requires the use of Jack polynomial theory, and in particular Jack polynomial based hypergeometric functions. Applications to the computation of the scaling limit of various $\beta$ ensemble correlation and distribution functions are also reviewed. The non-Hermitian case relies on the particular cases of Jack polynomials corresponding to zonal polynomials, and their integration properties when their arguments are eigenvalues of certain matrices. The main tool to study dualities for moments of the spectral density, and generalisations, is the loop equations.