Edge Behavior of Two-Dimensional Coulomb Gases Near a Hard Wall

Abstract

We consider a two-dimensional determinantal Coulomb gas confined by a class of radial external potentials. In the limit of large number of particles, the Coulomb particles tend to accumulate on a compact set S, the support of the equilibrium measure associated with a given external potential. If the particles are forced to be completely confined in a disk \({\mathcal {D}}\) due to a hard-wall constraint on \({\partial }{\mathcal {D}}\subset {\text {Int}}S\), then the equilibrium configuration changes and the equilibrium measure acquires a singular component at the hard wall. We study the local statistics of Coulomb particles in the vicinity of the hard wall and prove that their local correlations are expressed in terms of “Laplace-type” integrals, which appear in the context of truncated unitary matrices in the regime of weak non-unitarity.