Macroscopic and edge behavior of a planar jellium

TL;DR

This paper studies a two-dimensional Coulomb gas model with a smeared background charge, revealing phase behaviors and fluctuation properties similar to random matrix ensembles, with insights into edge fluctuations and phase transitions.

Contribution

It introduces a non-neutral planar jellium model with unique edge and bulk behaviors, connecting Coulomb gases to random matrix theory and analyzing phase transitions.

Findings

Equilibrium measure is uniform on a disc in certain regimes.

Farthest particle exhibits heavy-tailed fluctuations.

Transition to Gumbel edge fluctuations at higher temperatures.

Abstract

We consider a planar Coulomb gas in which the external potential is generated by a smeared uniform background of opposite-sign charge on a disc. This model can be seen as a two-dimensional Wigner jellium, not necessarily charge neutral, and with particles allowed to exist beyond the support of the smeared charge. The full space integrability condition requires low enough temperature or high enough total smeared charge. This condition does not allow at the same time, total charge neutrality and determinantal structure. The model shares similarities with both the complex Ginibre ensemble and the Forrester--Krishnapur spherical ensemble of random matrix theory. In particular, for a certain regime of temperature and total charge, the equilibrium measure is uniform on a disc as in the Ginibre ensemble, while the modulus of the farthest particle has heavy-tailed fluctuations as in the Forrester--Krishnapur spherical ensemble. We also touch on a higher temperature regime producing a crossover equilibrium measure, as well as a transition to Gumbel edge fluctuations. More results in the same spirit on edge fluctuations are explored by the second author together with Raphael Butez.