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 $$\Gamma =2$$ 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 $$1/\sqrt{n}$$ 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 $$K_{n}(z,w)$$ K n ( z , w ) as $$n\rightarrow \infty $$ n → ∞ in two microscopic regimes (with either $$|z-w| = \mathcal{O}(1/\sqrt{n})$$ | z - w | = O ( 1 / n ) or $$|z-w| = \mathcal{O}(1/n)$$ | z - w | = O ( 1 / n ) ), as well as in three macroscopic regimes (with $$|z-w| \asymp 1$$ | z - w | ≍ 1 ). For some of these regimes, the asymptotics involve oscillatory theta functions and weighted Szegő kernels.