### Abstract

We study N-particle systems in (Formula presented.) whose interactions are governed by a hypersingular Riesz potential (Formula presented.), (Formula presented.), and subject to an external field. We provide both macroscopic results as well as microscopic results in the limit as (Formula presented.) for random point configurations with respect to the associated Gibbs measure at scaled inverse temperature (Formula presented.). We show that a large deviation principle holds with a rate function of the form ‘(Formula presented.)-Energy + Entropy’, yielding that the microscopic behavior (on the scale (Formula presented.)) 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 (Formula presented.), where on the macroscopic scale N-point empirical measures have limiting density independent of (Formula presented.), the limiting density for (Formula presented.) is strongly (Formula presented.)-dependent.

Original language | English (US) |
---|---|

Pages (from-to) | 1-40 |

Number of pages | 40 |

Journal | Constructive Approximation |

DOIs | |

State | Accepted/In press - May 17 2018 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Analysis
- Mathematics(all)
- Computational Mathematics

### Cite this

*Constructive Approximation*, 1-40. https://doi.org/10.1007/s00365-018-9431-9

**Large Deviation Principles for Hypersingular Riesz Gases.** / Hardin, Douglas P.; Leblé, Thomas; Saff, Edward B.; Serfaty, Sylvia.

Research output: Contribution to journal › Article

*Constructive Approximation*, pp. 1-40. https://doi.org/10.1007/s00365-018-9431-9

}

TY - JOUR

T1 - Large Deviation Principles for Hypersingular Riesz Gases

AU - Hardin, Douglas P.

AU - Leblé, Thomas

AU - Saff, Edward B.

AU - Serfaty, Sylvia

PY - 2018/5/17

Y1 - 2018/5/17

N2 - We study N-particle systems in (Formula presented.) whose interactions are governed by a hypersingular Riesz potential (Formula presented.), (Formula presented.), and subject to an external field. We provide both macroscopic results as well as microscopic results in the limit as (Formula presented.) for random point configurations with respect to the associated Gibbs measure at scaled inverse temperature (Formula presented.). We show that a large deviation principle holds with a rate function of the form ‘(Formula presented.)-Energy + Entropy’, yielding that the microscopic behavior (on the scale (Formula presented.)) 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 (Formula presented.), where on the macroscopic scale N-point empirical measures have limiting density independent of (Formula presented.), the limiting density for (Formula presented.) is strongly (Formula presented.)-dependent.

AB - We study N-particle systems in (Formula presented.) whose interactions are governed by a hypersingular Riesz potential (Formula presented.), (Formula presented.), and subject to an external field. We provide both macroscopic results as well as microscopic results in the limit as (Formula presented.) for random point configurations with respect to the associated Gibbs measure at scaled inverse temperature (Formula presented.). We show that a large deviation principle holds with a rate function of the form ‘(Formula presented.)-Energy + Entropy’, yielding that the microscopic behavior (on the scale (Formula presented.)) 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 (Formula presented.), where on the macroscopic scale N-point empirical measures have limiting density independent of (Formula presented.), the limiting density for (Formula presented.) is strongly (Formula presented.)-dependent.

KW - Empirical measures

KW - Gibbs measure

KW - Large deviation principle

KW - Minimal energy

KW - Riesz gases

UR - http://www.scopus.com/inward/record.url?scp=85047108687&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85047108687&partnerID=8YFLogxK

U2 - 10.1007/s00365-018-9431-9

DO - 10.1007/s00365-018-9431-9

M3 - Article

SP - 1

EP - 40

JO - Constructive Approximation

JF - Constructive Approximation

SN - 0176-4276

ER -