Coulomb gas on a Jordan domain

Consider a Coulomb gas restricted to a Jordan domain in the complex plane. How does the asymptotic expansion of the free energy depend on the geometry of the domain, as the number of particles tends to infinity? I will explain how this problem is related to the Grunsky operator -- a classical tool in complex analysis -- and how this in turn reveals a close connection to the Loewner energy. I will further discuss the effect of corners,  which turns out to be universal in a certain sense. This is joint work with Kurt Johansson (KTH).