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

We investigate a two-dimensional statistical model of N charged particles interacting via logarithmic repulsion in the presence of an oppositely charged compact region K whose charge density is determined by its equilibrium potential at an inverse temperature corresponding toβ= 2. When the charge on the region, s , is greater than N , the particles accumulate in a neighborhood of the boundary of K , and form a determinantal point process on the complex plane. We investigate the scaling limit, as 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/s , and prove universality for these kernels. That is, we show that, the scaled kernel in a neighborhood of a point ζ ∈ ∂ 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 K to the complement of the closed unit disk. When N/s → 0 we recover the universal kernel discovered by Lubinsky in [13].