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

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