1108.3052 (Christopher D. Sinclair et al.)

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

Christopher D. Sinclair, Maxim L. Yattselev

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 to \beta = 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 \to \infty, 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 \zeta \in
\partial 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 K
to the complement of the closed unit disk. When N / s \to 0 we recover the
universal kernel discovered by Doron Lubinsky in Universality type limits for
Bergman orthogonal polynomials, Comput. Methods Funct. Theory, 10:135-154,
2010.

View original: http://arxiv.org/abs/1108.3052