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 - | x - y|, confined by a one-body harmonic potential. A generalisation is to replace the pair potential by - | (π (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 ∞ scaling limit, this reduces to the Airy kernel.