A Solvable Two-Charge Ensemble on the Circle

TL;DR

This paper introduces a solvable two-charge particle ensemble on the circle, interpolating between classical random matrix ensembles, with explicit scaling limits and a Pfaffian point process structure.

Contribution

It presents a new two-charge ensemble on the circle with a Pfaffian structure, connecting circular unitary and symplectic ensembles through a solvable interpolation.

Findings

Identified the proper scaling of fugacity with N.

Derived the matrix kernel scaling limits as a function of the interpolation parameter.

Established the Pfaffian point process structure of the ensemble.

Abstract

We introduce an ensemble consisting of logarithmically repelling charge one and charge two particles on the unit circle constrained so that the total charge of all particles equals $N$, but the proportion of each species of particle is allowed to vary according to a fugacity parameter. We identify the proper scaling of the fugacity with $N$ so that the proportion of each particle stays positive in the $N \rightarrow \infty$ limit. This ensemble forms a Pfaffian point process on the unit circle, and we derive the scaling limits of the matrix kernel(s) as a function of the interpolating parameter. This provides a solvable interpolation between the circular unitary and symplectic ensembles.