Coulomb gas and the Grunsky operator on a Jordan domain with corners

Abstract

Let $D$ be a Jordan domain of unit capacity. We study the partition function of a planar Coulomb gas in $D$ with a hard wall along $\eta = \partial D$, \[Z_{n}(D) =\frac 1{n!}\int_{D^n}\prod_{1\le k < \ell \le n}|z_k-z_\ell|^{2} \prod_{k=1}^n d^2z_k.\] We are interested in how the geometry of $\eta$ is reflected in the large $n$ behavior of $Z_n(D)$. We prove that $\eta$ is a Weil-Petersson quasicircle if and only if \[ \lim_{n \to \infty} \log \frac{Z_n(D)}{Z_n(\mathbb{D})} = -\frac{1}{12}I^L(\eta), \] where $I^L$ is the Loewner energy, $\mathbb{D}$ is the unit disc, and $\log Z_n(\mathbb{D}) = \log \pi^n/n!$. We next consider piecewise analytic $\eta$ with $m$ corners of interior opening angles $\pi \alpha_p, p=1,\ldots, m$. Our main result is the asymptotic formula \[ \lim_{n\to\infty}\frac 1{\log n} \log \frac{Z_n(D)}{Z_n(\mathbb{D})} =-\frac 16\sum_{p=1}^m \left(\alpha_p+\frac 1{\alpha_p}-2 \right) \] which is consistent with physics predictions. The starting point of our analysis is an exact expression for $\log Z_{n}(D)$ in terms of a Fredholm determinant involving the truncated Grunsky operator for $D$. The proof of the main result is based on careful asymptotic analysis of the Grunsky coefficients. As further applications of our method we also study the Loewner energy and the related Fekete-Pommerenke energy, a quantity appearing in the analysis of Fekete points, for equipotentials approximating the boundary of a domain with corners. We formulate several conjectures and open problems.