Large deviation principle for empirical fields of Log and Riesz gases

• 成果类型: Article
• 署名作者: Leble, Thomas; Serfaty, Sylvia
• 署名单位: Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Sorbonne Universite; Universite Paris Cite; Institut Universitaire de France; New York University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-017-0738-0
• 发表日期: 2017
• 页码: 645-757
• 关键词: maximum-entropy principle; thermodynamic limit; statistical-theory; energy-levels; obstacle problem; coulomb gases; free-boundary; monte-carlo; fluctuations; asymptotics

摘要

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.