2D Coulomb Gases and the Renormalized Energy

By Etienne Sandier, Sylvia Serfaty. 2012

143 citations recorded by the source corpus.

We study the statistical mechanics of classical two-dimensional “Coulomb gases” with general potential and arbitrary β, the inverse of the temperature. Such ensembles also correspond to random matrix models in some particular cases. The formal limit case β=∞ 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 β→∞, 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.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/df0d83610f8b810a0f0b98dc1187ce7d1f1d1438

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Bogoliubov Spectrum of Interacting Bose Gases (2012, background): “They proved in [51] that for the classical 2D Coulomb systems, the ground state energy behaves like NeH − 1 4 logN + c+ o(1)”
Higher‐Dimensional Coulomb Gases and Renormalized Energy Functionals (2013, methodology): “annot prove it in general because it is most certainly not true: were it true, this would mean that Wfis the complete Γ-limit (at next order) of Hn. But in dimension 2 at least, it has been proven in [SS4] that the corresponding Γ-limit is Wf. By uniqueness of the Γ-limit, this would mean that R W d”
Large deviation principle for empirical fields of Log and Riesz gases (2015, background): “e behavior of the Hamiltonian H N and of its minimizers has been understood at the next order and at the microscopic scale where the points become wellseparated, i.e. N 1=d. First, it was remarked in [SS12a] (for the case (1.3)), [SS15] (for the case (1.2)), [RS13] (for all the Coulomb cases), and [”
Hyperuniform states of matter (2018, unknown): “[182] S. Serfaty, Ginzburg-Landau Vortices, Coulomb Gases, and Renormalized Energies, J. Stat.”