2D Coulomb gases and the renormalized energy

Abstract

We study the statistical mechanics of classical two-dimensional “Coulomb gases” with general potential and arbitrary $\beta$, the inverse of the temperature. Such ensembles also correspond to random matrix models in some particular cases. The formal limit case $\beta=\infty$ corresponds to “weighted Fekete sets” and also falls within our analysis.

It is known that in such a system points should be asymptotically distributed according to a macroscopic “equilibrium measure,” and that a large deviations principle holds for this, as proven by Petz and Hiai [In Advances in Differential Equations and Mathematical Physics (Atlanta, GA, 1997) (1998) Amer. Math. Soc.] and Ben Arous and Zeitouni [ ESAIM Probab. Statist. 2 (1998) 123–134].

By a suitable splitting of the Hamiltonian, we connect the problem to the “renormalized energy” $W$, a Coulombian interaction for points in the plane introduced in [ Comm. Math. Phys. 313 (2012) 635–743], which is expected to be a good way of measuring the disorder of an infinite configuration of points in the plane. By so doing, we are able to examine the situation at the microscopic scale, and obtain several new results: a next order asymptotic expansion of the partition function, estimates on the probability of fluctuation from the equilibrium measure at microscale, and a large deviations type result, which states that configurations above a certain threshhold of $W$ have exponentially small probability. When $\beta\to\infty$, the estimate becomes sharp, showing that the system has to “crystallize” to a minimizer of $W$. In the case of weighted Fekete sets, this corresponds to saying that these sets should microscopically look almost everywhere like minimizers of $W$, which are conjectured to be “Abrikosov” triangular lattices.