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.
Categories: math-ph math.MP math.PR