A generalized plasma and interpolation between classical random matrix ensembles

TL;DR

This paper extends the understanding of eigenvalue distributions in classical random matrix ensembles by deriving a Pfaffian form for multi-point correlations in generalized plasma models that interpolate between different ensembles.

Contribution

It provides a new Pfaffian expression for the (k_1+k_2)-point correlation function in generalized plasma models interpolating between classical ensembles.

Findings

Derived the general (k_1+k_2)-point correlation function in Pfaffian form.

Established a Vandermonde determinant identity in terms of Pfaffians.

Confirmed a perfect screening sum rule in the generalized plasma.

Abstract

The eigenvalue probability density functions of the classical random matrix ensembles have a well known analogy with the one component log-gas at the special couplings \beta = 1,2 and 4. It has been known for some time that there is an exactly solvable two-component log-potential plasma which interpolates between the \beta =1 and 4 circular ensemble, and an exactly solvable two-component generalized plasma which interpolates between \beta = 2 and 4 circular ensemble. We extend known exact results relating to the latter --- for the free energy and one and two-point correlations --- by giving the general (k_1+k_2)-point correlation function in a Pfaffian form. Crucial to our working is an identity which expresses the Vandermonde determinant in terms of a Pfaffian. The exact evaluation of the general correlation is used to exhibit a perfect screening sum rule.