Smallest gaps of the two-dimensional Coulomb gas

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.

29 pages, 3 figures