Random Normal Matrices: Eigenvalue Correlations Near a Hard Wall

Abstract

We study pair correlation functions for planar Coulomb systems in the pushed phase, near a ring-shaped impenetrable wall. We assume coupling constant \(\Gamma =2\) 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 \(1/\sqrt{n}\), 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 \(K_{n}(z,w)\) as \(n\rightarrow \infty \) in two microscopic regimes (with either \(|z-w| = \mathcal{O}(1/\sqrt{n})\) or \(|z-w| = \mathcal{O}(1/n)\)), as well as in three macroscopic regimes (with \(|z-w| \asymp 1\)). For some of these regimes, the asymptotics involve oscillatory theta functions and weighted Szegő kernels.