Global and local scaling limits for the $\beta = 2$ Stieltjes--Wigert random matrix ensemble

TL;DR

This paper studies the eigenvalue distribution of a generalized $eta=2$ random matrix ensemble related to the Stieltjes--Wigert polynomials, deriving exact moments, global density, and edge scaling limits, connecting to known kernels like the Airy kernel.

Contribution

It provides an exact expression for moments of the ensemble, computes the global density, and shows the transition of the correlation kernel to the Airy kernel in a specific limit.

Findings

Exact moments expressed via little $q$-Jacobi polynomials

Global eigenvalue density derived from large $N$ analysis

Edge scaling limit reduces to the Airy kernel in a certain limit

Abstract

The eigenvalue probability density function (PDF) for the Gaussian unitary ensemble has a well known analogy with the Boltzmann factor for a classical log-gas with pair potential $- \log | x - y|$, confined by a one-body harmonic potential. A generalisation is to replace the pair potential by $- \log |\sinh (\pi (x-y)/L) |$. The resulting PDF first appeared in the statistical physics literature in relation to non-intersecting Brownian walkers, equally spaced at time $t=0$, and subsequently in the study of quantum many body systems of the Calogero-Sutherland type, and also in Chern-Simons field theory. It is an example of a determinantal point process with correlation kernel based on the Stieltjes--Wigert polynomials. We take up the problem of determining the moments of this ensemble, and find an exact expression in terms of a particular little $q$-Jacobi polynomial. From their large $N$ form, the global density can be computed. Previous work has evaluated the edge scaling limit of the correlation kernel in terms of the Ramanujan ($q$-Airy) function. We show how in a particular $L \to \infty$ scaling limit, this reduces to the Airy kernel.