Dualities in random matrix theory
Peter J. Forrester
TL;DR
This paper surveys dualities in random matrix theory, showing that averages of powers and products of characteristic polynomials for $N\times N$ ensembles can be recast as averages in dual ensembles of size $k\times k$ (and vice versa). It unifies classical invariant ensembles ($eta=1,2,4$) and their $\beta$-generalizations via Jack polynomials, with extensions to non-Hermitian and circular ensembles; the core tools include determinant identities, Toeplitz/Hankel structures, and Jack-polynomial based hypergeometric functions. The work also connects these dualities to large-$N$ scaling limits, deriving global densities, soft/hard edge behaviors, and moments, using loop equations and analytic continuations. Overall, the dualities provide a powerful framework for computing correlation functions and distribution limits across a broad class of random-matrix models, including both Hermitian and non-Hermitian settings, with explicit links to universal scaling forms such as the Wigner semicircle, Marchenko–Pastur/Wachter laws, and Airy-type soft edges.
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 $β$ 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 $β$ 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 $β$ 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.