Constructive Approximation

, Volume 48, Issue 1, pp 61–100 | Cite as

Large Deviation Principles for Hypersingular Riesz Gases

  • Douglas P. Hardin
  • Thomas Leblé
  • Edward B. SaffEmail author
  • Sylvia Serfaty


We study N-particle systems in \(\mathbb {R}^d\) 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 \(N\rightarrow \infty \) for random point configurations with respect to the associated Gibbs measure at scaled inverse temperature \(\beta \). We show that a large deviation principle holds with a rate function of the form ‘\(\beta \)-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 \(\beta \), the limiting density for \(s>d\) is strongly \(\beta \)-dependent.


Riesz gases Gibbs measure Large deviation principle Empirical measures Minimal energy 

Mathematics Subject Classification

Primary 82D10 82B05 Secondary 31C20 28A78 



The authors thank the referee for a careful reading and helpful suggestions.


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Authors and Affiliations

  • Douglas P. Hardin
    • 1
  • Thomas Leblé
    • 2
  • Edward B. Saff
    • 1
    Email author
  • Sylvia Serfaty
    • 2
    • 3
    • 4
  1. 1.Department of Mathematics, Center for Constructive ApproximationVanderbilt UniversityNashvilleUSA
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  3. 3.Institut Universitaire de FranceParisFrance
  4. 4.UPMC, CNRS, UMR 7598, Laboratoire Jacques-Louis LionsSorbonne UniversitésParisFrance

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