Random normal matrices: eigenvalue correlations near a hard wall

TL;DR

This paper analyzes eigenvalue correlations near a hard wall in a Coulomb system with large particle number, revealing slow decay at the edge and providing detailed asymptotics in various regimes.

Contribution

It offers new asymptotic formulas for correlation kernels near a hard wall in Coulomb systems at coupling constant 2, including oscillatory behaviors and interpolating regimes.

Findings

Correlation functions decay slowly near the hard wall edge.

Asymptotic formulas involve oscillatory theta functions.

Different regimes exhibit distinct decay and oscillation patterns.

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 $n\to\infty$ 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 Szeg\H{o} kernels.