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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. Saff
  • Sylvia Serfaty
Article
  • 102 Downloads

Abstract

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.

Keywords

Riesz gases Gibbs measure Large deviation principle Empirical measures Minimal energy 

Mathematics Subject Classification

Primary 82D10 82B05 Secondary 31C20 28A78 

Notes

Acknowledgements

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

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Douglas P. Hardin
    • 1
  • Thomas Leblé
    • 2
  • Edward B. Saff
    • 1
  • 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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