## Abstract

We study N-particle systems in 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→ ∞ 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 language | English (US) |
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Pages (from-to) | 61-100 |

Number of pages | 40 |

Journal | Constructive Approximation |

Volume | 48 |

Issue number | 1 |

DOIs | |

State | Published - Aug 1 2018 |

## Keywords

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

## ASJC Scopus subject areas

- Analysis
- Mathematics(all)
- Computational Mathematics