Large Deviation Principles for Hypersingular Riesz Gases

Douglas P. Hardin, Thomas Leblé, Edward B. Saff, Sylvia Serfaty

Research output: Contribution to journalArticlepeer-review


We study N-particle systems in Rd whose interactions are governed by a hypersingular Riesz potential | x- y| -s, s> d, and subject to an external field. We provide both macroscopic results as well as microscopic results in the limit as N→ ∞ for random point configurations with respect to the associated Gibbs measure at scaled inverse temperature β. We show that a large deviation principle holds with a rate function of the form ‘β-Energy + Entropy’, yielding that the microscopic behavior (on the scale N-1/d) of such N-point systems is asymptotically determined by the minimizers of this rate function. In contrast to the asymptotic behavior in the integrable case s< d, where on the macroscopic scale N-point empirical measures have limiting density independent of β, the limiting density for s> d is strongly β-dependent.

Original languageEnglish (US)
Pages (from-to)61-100
Number of pages40
JournalConstructive Approximation
Issue number1
StatePublished - Aug 1 2018


  • Empirical measures
  • Gibbs measure
  • Large deviation principle
  • Minimal energy
  • Riesz gases

ASJC Scopus subject areas

  • Analysis
  • General Mathematics
  • Computational Mathematics


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