2D COULOMB GASES AND THE RENORMALIZED ENERGY

• 成果类型: Article
• 署名作者: Sandier, Etienne; Serfaty, Sylvia
• 署名单位: Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite Gustave-Eiffel; Universite Paris-Est-Creteil-Val-de-Marne (UPEC); Sorbonne Universite; Universite Paris Cite; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Sorbonne Universite; Universite Paris Cite; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); New York University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/14-AOP927
• 发表日期: 2015
• 页码: 2026-2083
• 关键词: algebraic equations; RANDOM MATRICES; UNIVERSALITY; fluctuations; roots

摘要

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 = infinity 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 -> infinity, 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.