Large deviation principle for empirical fields of Log and Riesz gases

Thomas Leblé, Sylvia Serfaty

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)645-757
Number of pages113
JournalInventiones Mathematicae
Issue number3
StatePublished - Dec 1 2017


  • 15B52
  • 82B05
  • 82B21
  • 82B26

ASJC Scopus subject areas

  • General Mathematics


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