The two-dimensional one-component plasma (OCP), also known as the Coulomb gas, is a system that consists of identical electrically charged particles embedded in a uniform background of the opposite charge, interacting through a logarithmic potential and kept at a fixed temperature. In the 1990s, Jancovici, Lebowitz and Manificat discovered a law for the probabilities of observing large charge fluctuations in the OCP. Mathematically, this law has been fully proved only for one special value of the temperature, corresponding to the eigenvalues of the Ginibre random matrix ensemble. A few years ago, Chatterjee introduced a hierarchical version of the OCP, inspired by Dyson's hierarchical model of the Ising ferromagnet. In this talk, I will sketch the proof of the JLM law for the hierarchical Coulomb model at any finite positive temperature.

This is based on a joint work with Alon Nishry (arXiv:2403.03603).