Smallest gaps of the two-dimensional Coulomb gas

TL;DR

This paper investigates the smallest gaps between particles in the two-dimensional Coulomb gas, showing they are typically of order n^{-3/4} and that the rescaled gaps converge to a Poisson process, extending previous results beyond quadratic potentials.

Contribution

It establishes the order of smallest gaps and their limiting distribution for the Coulomb gas with general potential, generalizing prior quadratic potential results.

Findings

Smallest gaps are of order n^{-3/4}.

Rescaled gap process converges to a Poisson point process.

Limiting density of the k-th smallest gap is proportional to x^{4k-1}e^{-rac{ extstyle ext{ extbackslash J}}{4}x^{4}}.

Abstract

We consider the two-dimensional Coulomb gas with a general potential at the determinantal temperature, or equivalently, the eigenvalues of random normal matrices. We prove that the smallest gaps between particles are typically of order $n^{-3/4}$, and that the associated joint point process of gap locations and gap sizes, after rescaling the gaps by $n^{3/4}$, converges to a Poisson point process. As a consequence, we show that the $k$-th smallest rescaled gap has a limiting density proportional to $x^{4k-1}e^{-\frac{\mathcal{J}}{4}x^{4}}$, where $\mathcal{J}=\pi^{2}\int \rho(z)^{3}d^{2}z$ and $\rho$ is the density of the equilibrium measure. This generalizes a result of Shi and Jiang beyond the quadratic potential.