Coulomb gas ensembles and Laplacian growth

Abstract

We consider weight functions Q : ℂ→ℝ that are locally in a suitable Sobolev space and impose a logarithmic growth condition from below. We use Q as a confining potential in the model of one-component plasma (2-dimensional Coulomb gas) and study the configuration of the electron cloud as the number n of electrons tends to infinity, while the confining potential is rescaled: we use mQ in place of Q and let m tend to infinity as well. We show that if m and n tend to infinity in a proportional fashion, with n/m→t, where 0<t<+∞ is fixed, then the electrons accumulate on a compact set St, which we call the droplet. The set S_t can be obtained as the coincidence set of an obstacle problem, if we remove a small set (the shallow points). Moreover, on the droplet St, the density of electrons is asymptotically ΔQ. The growth of the droplets St as t increases is known as the Laplacian growth. It is well known that Laplacian growth is unstable. To analyse this feature, we introduce the notion of a local droplet, which involves removing part of the obstacle away from the set S_t. The local droplets are no longer uniquely determined by the time parameter t, but at least they may be partially ordered. We show that the growth of the local droplets may be terminated in a maximal local droplet or by the droplets' growing to infinity in some direction (‘fingering’).