Global and local scaling limits for the β = 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 generalization is to replace the pair potential by-log |sinh((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 polynom..