At the edge of a one-dimensional jellium

TL;DR

This paper studies a one-dimensional Coulomb gas with a background charge, analyzing the edge behavior and fluctuations of particles, revealing non-universal tail behaviors including Tracy-Widom-like distributions.

Contribution

It provides the first detailed asymptotic analysis of edge fluctuations in a non-neutral one-dimensional Coulomb gas with varying background charges.

Findings

Edge fluctuations depend on the background charge, exhibiting a range of tail behaviors.

The system's existence requires the background charge to exceed the number of electrons minus one.

The paper introduces a Renyi-type representation for particle order statistics beyond the background support.

Abstract

We consider a one-dimensional classical Wigner jellium, not necessarily charge neutral, for which the electrons are allowed to exist beyond the support of the background charge. The model can be seen as a one-dimensional Coulomb gas in which the external field is generated by a smeared background on an interval. It is a true one-dimensional Coulomb gas and not a one-dimensional log-gas. The system exists if and only if the total background charge is greater than the number of electrons minus one. For various backgrounds, we show convergence to point processes, at the edge of the support of the background. In particular, this provides asymptotic analysis of the fluctuations of the right-most particle. Our analysis reveals that these fluctuations are not universal, in the sense that depending on the background, the tails range anywhere from exponential to Gaussian-like behavior, including for instance Tracy-Widom-like behavior. We also obtain a Renyi-type probabilistic representation for the order statistics of the particle system beyond the support of the background.