Random Normal Matrices: Eigenvalue Correlations Near a Hard Wall

We study pair correlation functions for planar Coulomb systems in the pushed phase, near a ring-shaped impenetrable wall. We assume coupling constant Γ =2 Γ = 2 and that the number n of particles is large. We find that the correlation functions decay slowly along the edges of the wall, in a narrow interface stretching a distance of order 1/ n from the hard edge. At distances much larger than 1/√n 1 / n , the effect of the hard wall is negligible and pair correlation functions decay very quickly, and in between sits an interpolating interface that we call the “semi-hard edge”. More precisely, we provide asymptotics for the correlation kernel Kₙ(z,w) K n ( z , w ) as n→ ∞ n → ∞ in two microscopic regimes (with either |z-w| = O(1/√n) | z - w | = O ( 1 / n ) or |z-w| = O(1/n) | z - w | = O ( 1 / n ) ), as well as in three macroscopic regimes (with |z-w| 1 | z - w | ≍ 1 ). For some of these regimes, the asymptotics involve oscillatory theta functions and weighted Szegő kernels.