Extreme statistics and index distribution in the classical $1d$ Coulomb gas

TL;DR

This paper analyzes the statistical behavior of a one-dimensional Coulomb gas confined by a harmonic potential, revealing new fluctuation distributions for the rightmost particle, gaps, and index, with analytical and numerical validation.

Contribution

It introduces novel limiting distributions and large deviation functions for the Coulomb gas observables, differing from known log-gas results.

Findings

Distinct distribution for the rightmost particle fluctuations from Tracy-Widom

Explicit large deviation functions for atypical fluctuations

Numerical validation of analytical predictions

Abstract

We consider a one-dimensional gas of $N$ charged particles confined by an external harmonic potential and interacting via the one-dimensional Coulomb potential. For this system we show that in equilibrium the charges settle, on an average, uniformly and symmetrically on a finite region centred around the origin. We study the statistics of the position of the rightmost particle $x_{\max}$ and show that the limiting distribution describing its typical fluctuations is different from the Tracy-Widom distribution found in the one-dimensional log-gas. We also compute the large deviation functions which characterise the atypical fluctuations of $x_{\max}$ far away from its mean value. In addition, we study the gap between the two rightmost particles as well as the index $N_+$, i.e., the number of particles on the positive semi-axis. We compute the limiting distributions associated to the typical fluctuations of these observables as well as the corresponding large deviation functions. We provide numerical supports to our analytical predictions. Part of these results were announced in a recent Letter, Phys. Rev. Lett. 119, 060601 (2017).