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

Understand

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.