Tatyana Turova: Phase Transitions in the One-dimensional Coulomb Gas Ensembles

Abstract: We consider the system of particles on a finite interval with pair-wise nearest neighbours interaction and external force. This model was introduced by Malyshev (2015) to study the flow of charged particles on a rigorous mathematical level. It is a simplified version of a 3-dimensional classical Coulomb gas model. We study Gibbs distribution at finite positive temperatureextending recent results on the zero temperature case (ground states). We derive the asymptotics for the mean and for the variances of the distances between the neighbouring charges. We prove that depending on the strength of the external force there are several phase transitions in the local structure of the configuration of the particles in the limit when the number of particles goes to infinity. We identify 5 different phases for any positive temperature.

The proofs rely on a conditional central limit theorem for non-identical random variables, which may have an interest on its own.