A formula for the edge density $\sqrt{n}$-correction for two-dimensional Coulomb systems

TL;DR

The paper addresses the edge-density correction of a planar Coulomb gas in a confining potential $Q$ with a connected droplet $S$. By leveraging Charlier's subleading off-diagonal kernel and a fluctuation-based expectation formula for linear statistics, the authors identify the parameters governing the subleading edge behavior and derive an explicit expression for the coefficient $C(z_0;t)$ in the edge density expansion. They prove that, under appropriate regularity (connected, smooth boundary) and growth conditions on $Q$, the 1-point function satisfies $R_n(z)=n\Delta Q(z_0) \frac{\operatorname{erfc} t}{2}+\sqrt{n\Delta Q(z_0)}\,C(z_0;t)+\mathcal{O}(\log^3 n)$ with $t$ in the edge scaling, where $C(z_0;t)$ is written in terms of $\partial_n L(z_0)$, $\partial_n L^S(z_0)$, $\kappa(z_0)$, and $\partial_n\Delta Q(z_0)$. The work yields a Hele-Shaw universality limit and connects the edge correction to fluctuations of linear statistics, also clarifying the structure of the subleading kernel and its diagonal. These results enhance understanding of edge fluctuations in 2D Coulomb systems and provide precise asymptotics for edge densities in a broad class of potentials.

Abstract

In connection with recent work on smallest gaps, C. Charlier proves that the 1-point function of a suitable planar Coulomb system $\{z_j\}_1^n$, in the determinantal case with respect to an external potential $Q(z)$, admits the expansion, as $n\to\infty$, $$R_n\bigg(z_0+\frac t {\sqrt{2n\partial\bar{\partial} Q(z_0)}}ν(z_0)\bigg)=n\partial\bar{\partial} Q(z_0)\frac {\operatorname{erfc} t}2+\sqrt{n\partial\bar{\partial} Q(z_0)}\,C(z_0;t)+\mathcal{O}(\log^3 n).$$ Here $t$ is a real parameter, $z_0$ is a regular boundary point of the (connected) Coulomb droplet and $ν(z_0)$ is the outwards unit normal; the coefficient $C(z_0;t)$ has an apriori structure depending on a number of parameters. In this note we identify the parameters and obtain a formula for $C(z_0;t)$ in potential theoretic and geometric terms. Our formula holds for a large class of potentials such that the droplet is connected with smooth boundary. Our derivation uses the well known expectation of fluctuations formula.

Theorems & Definitions (20)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3
• Theorem 1.4
• Remark
• Remark
• Remark
• Lemma 2.1
• proof
• Lemma 2.2