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

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 $n\to \mathrm{\infty }$, $$R_n(z_0+\frac{t}{\sqrt{2n\mathrm{\partial }\bar{\mathrm{\partial }}Q(z_0)}}\nu (z_0))=n\mathrm{\partial }\bar{\mathrm{\partial }}Q(z_0)\frac{\mathrm{erfc}t}{2}+\sqrt{n\mathrm{\partial }\bar{\mathrm{\partial }}Q(z_0)}C(z_0;t)+\mathcal{O}({\log }^{3}n)\mathrm{.}$$ Here $t$ is a real parameter, $z_0$ is a regular boundary point of the (connected) Coulomb droplet and $\nu (z_0)$ 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.