Asymptotics for Coulomb gases in the plane and geometry

Coulomb gases in the complex plane or on the real line, sometimes called log-gases, are interesting and important objects from many points of view, statistical mechanics, potential theory, random matrix theory, orthogonal polynomials and more.

The purpose of the proposal is to investigate asymptotical properties of planar Coulomb gases confined to a curve or a region and how they relate to the geometry of the curve or the region.

The background to the research proposal is the recent unexpected discovery in a preprint of mine that the asymptotics of the partition function of a Coulomb gas confined to a Jordan curve in the plane contains a contribution which is equal to a quantity called the Loewner energy of the curve.

The  Loewner energy was introduced recently in a completely different context related to Schramm-Loewner evolutions (SLE). Curves with finite Lowener energy are called Weil-Petersson quasicircles and have many different characterizations. They appear for instance in the theory of the universal Teichmüller space.

These unexpected connections between confined Coulomb gases and the Loewner energy raises many questions on the relation between the asymptotics of these partition functions and the geometry of the curve.

The proposal contains several projects of interest, e.g. studying the partition function on Jordan arcs instead of closed curves. What happens to the asymptotics when the curve is not a Weil-Petersson quasicircle, e.g. has corners?