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

Abstract Let $D$ D be a Jordan domain of unit capacity. We study the partition function of a planar Coulomb gas in $D$ D with a hard wall along $\eta = \partial D$ η = ∂ D , $$ Z_{n}(D) =\frac{1}{n!}\int _{D^{n}}\prod _{1\leqslant k < \ell \leqslant n}|z_{k}-z_{\ell }|^{2} \prod _{k=1}^{n} d^{2}z_{k}. $$ Z n ( D ) = 1 n ! ∫ D n ∏ 1 ⩽ k < ℓ ⩽ n | z k − z ℓ | 2 ∏ k = 1 n d 2 z k . We are interested in how the geometry of $\eta $ η is reflected in the large $n$ n behavior of $Z_{n}(D)$ Z n ( D ) . We prove that $\eta $ η is a Weil-Petersson quasicircle if and only if $$ \lim _{n \to \infty } \log \frac{Z_{n}(D)}{Z_{n}(\mathbb{D})} = -\frac{1}{12}I^{L}( \eta ), $$ lim n → ∞ log Z n ( D ) Z n ( D ) = − 1 12 I L ( η ) , where $I^{L}$ I L </jats:inl