Universality for ensembles of matrices with potential theoretic weights on domains with smooth boundary

We investigate a two-dimensional statistical model of NN charged particles interacting via logarithmic repulsion in the presence of an oppositely charged compact region KK whose charge density is determined by its equilibrium potential at an inverse temperature corresponding to β=2β=2. When the charge on the region, ss, is greater than NN, the particles accumulate in a neighborhood of the boundary of KK, and form a determinantal point process on the complex plane. We investigate the scaling limit, as N→∞N→∞, of the associated kernel in the neighborhood of a point on the boundary under the assumption that the boundary is sufficiently smooth. We find that the limiting kernel depends on the limiting value of N/sN/s, and prove universality for these kernels. That is, we show that, the scaled kernel in a neighborhood of a point ζ∈∂Kζ∈∂K can be succinctly expressed in terms of the scaled kernel for the closed unit disk, and the exterior conformal map which carries the complement of KK to the complement of the closed unit disk. When N/s→0N/s→0 we recover the universal kernel discovered by Lubinsky (2010) in [13].