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

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.

25 pages

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