Large Deviation Principle for Empirical Fields of Log and Riesz Gases

We study a system of N particles with logarithmic, Coulomb or Riesz pairwise interactions, confined by an external potential. We examine a microscopic quantity, the tagged empirical field, for which we prove a large deviation principle at speed N. The rate function is the sum of an entropy term, the specific relative entropy, and an energy term, the renormalized energy introduced in previous works, coupled by the temperature. We deduce a variational property of the sine-beta processes which arise in random matrix theory. We also give a next-to-leading order expansion of the free energy of the system, proving the existence of the thermodynamic limit.Comment: 80 pages, final version, to appear in Inventiones Mat

The two-dimensional one-component plasma is hyperuniform

We prove that at all positive temperatures in the bulk of a classical two-dimensional one-component plasma (also called Coulomb or log-gas, or jellium) the variance of the number of particles in large disks grows more slowly than the area. In other words the system is hyperuniform. We obtain a non-sharp but quantitative bound on the number variance's growth rate, which is the first mathematical justification of an old prediction in the physics literature about "suppression of charge fluctuations". We introduce an argument of approximate conditional independence for well-separated sub-systems and a trick using "isotropically averaged localized translations" in order to control the expectation of non-smooth linear statistics

DLR equations and rigidity for the Sine-beta process

We investigate Sine$_\beta$, the universal point process arising as the thermodynamic limit of the microscopic scale behavior in the bulk of one-dimensional log-gases, or $\beta$-ensembles, at inverse temperature $\beta>0$. We adopt a statistical physics perspective, and give a description of Sine$_\beta$ using the Dobrushin-Lanford-Ruelle (DLR) formalism by proving that it satisfies the DLR equations: the restriction of Sine$_\beta$ to a compact set, conditionally to the exterior configuration, reads as a Gibbs measure given by a finite log-gas in a potential generated by the exterior configuration. Moreover, we show that Sine$_\beta$ is number-rigid and tolerant in the sense of Ghosh-Peres, i.e. the number, but not the position, of particles lying inside a compact set is a deterministic function of the exterior configuration. Our proof of the rigidity differs from the usual strategy and is robust enough to include more general long range interactions in arbitrary dimension.Comment: 46 pages. To appear in Communications on Pure and Applied Mathematic