The 2D two-component Coulomb gas is a planar system of positive and negative point charges interacting via a logarithmic potential. At a certain temperature, opposite charges bind together into dipoles. As the temperature decreases, the system exhibits an infinite sequence of phase transitions driven by the clustering of dipoles into multipoles. This sequence accumulates at the "Berezinskii-Kosterlitz-Thouless" temperature.

I will discuss joint work with Sylvia Serfaty which establishes these transitions rigorously. Our proof develops a new framework based on adaptive cluster expansions and electrostatic estimates.