### Abstract

We study a system of N particles with logarithmic, Coulomb or Riesz pairwise interactions, confined by an external potential. We examine a microscopic quantity, the tagged empirical field, for which we prove a large deviation principle at speed N. The rate function is the sum of an entropy term, the specific relative entropy, and an energy term, the renormalized energy introduced in previous works, coupled by the temperature. We deduce a variational property of the sine-beta processes which arise in random matrix theory. We also give a next-to-leading order expansion of the free energy of the system, proving the existence of the thermodynamic limit.

Original language | English (US) |
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Pages (from-to) | 1-113 |

Number of pages | 113 |

Journal | Inventiones Mathematicae |

DOIs | |

State | Accepted/In press - Jun 7 2017 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Inventiones Mathematicae*, 1-113. https://doi.org/10.1007/s00222-017-0738-0

**Large deviation principle for empirical fields of Log and Riesz gases.** / Leblé, Thomas; Serfaty, Sylvia.

Research output: Contribution to journal › Article

*Inventiones Mathematicae*, pp. 1-113. https://doi.org/10.1007/s00222-017-0738-0

}

TY - JOUR

T1 - Large deviation principle for empirical fields of Log and Riesz gases

AU - Leblé, Thomas

AU - Serfaty, Sylvia

PY - 2017/6/7

Y1 - 2017/6/7

N2 - We study a system of N particles with logarithmic, Coulomb or Riesz pairwise interactions, confined by an external potential. We examine a microscopic quantity, the tagged empirical field, for which we prove a large deviation principle at speed N. The rate function is the sum of an entropy term, the specific relative entropy, and an energy term, the renormalized energy introduced in previous works, coupled by the temperature. We deduce a variational property of the sine-beta processes which arise in random matrix theory. We also give a next-to-leading order expansion of the free energy of the system, proving the existence of the thermodynamic limit.

AB - We study a system of N particles with logarithmic, Coulomb or Riesz pairwise interactions, confined by an external potential. We examine a microscopic quantity, the tagged empirical field, for which we prove a large deviation principle at speed N. The rate function is the sum of an entropy term, the specific relative entropy, and an energy term, the renormalized energy introduced in previous works, coupled by the temperature. We deduce a variational property of the sine-beta processes which arise in random matrix theory. We also give a next-to-leading order expansion of the free energy of the system, proving the existence of the thermodynamic limit.

UR - http://www.scopus.com/inward/record.url?scp=85020265417&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85020265417&partnerID=8YFLogxK

U2 - 10.1007/s00222-017-0738-0

DO - 10.1007/s00222-017-0738-0

M3 - Article

SP - 1

EP - 113

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

SN - 0020-9910

ER -