### Abstract

We study the statistical mechanics of classical two-dimensional "Coulomb gases" with general potential and arbitrary Β, the inverse of the temperature. Such ensembles also correspond to random matrix models in some particular cases. The formal limit case Β =∞corresponds to "weighted Fekete sets" and also falls within our analysis. It is known that in such a system points should be asymptotically distributed according to a macroscopic "equilibrium measure," and that a large deviations principle holds for this, as proven by Petz and Hiai [In Advances in Differential Equations and Mathematical Physics (Atlanta, GA, 1997) (1998) Amer. Math. Soc.] and Ben Arous and Zeitouni [ESAIM Probab. Statist. 2 (1998) 123-134]. By a suitable splitting of the Hamiltonian, we connect the problem to the "renormalized energy" W, a Coulombian interaction for points in the plane introduced in [Comm. Math. Phys. 313 (2012) 635-743], which is expected to be a good way of measuring the disorder of an infinite configuration of points in the plane. By so doing, we are able to examine the situation at the microscopic scale, and obtain several new results: a next order asymptotic expansion of the partition function, estimates on the probability of fluctuation from the equilibrium measure at microscale, and a large deviations type result, which states that configurations above a certain threshhold of W have exponentially small probability. When Β→∞, the estimate becomes sharp, showing that the system has to "crystallize" to a minimizer of W. In the case of weighted Fekete sets, this corresponds to saying that these sets should microscopically look almost everywhere like minimizers of W, which are conjectured to be "Abrikosov" triangular lattices.

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

Pages (from-to) | 2026-2083 |

Number of pages | 58 |

Journal | Annals of Probability |

Volume | 43 |

Issue number | 4 |

DOIs | |

State | Published - 2015 |

### Fingerprint

### Keywords

- Abrikosov lattice
- Coulomb gas
- Crystallization
- Fekete sets
- Ginibre ensemble
- Large deviations
- One-component plasma
- Random matrices
- Renormalized energy
- Triangular lattice

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Annals of Probability*,

*43*(4), 2026-2083. https://doi.org/10.1214/14-AOP927

**2D coulomb gases and the renormalized energy.** / Sandier, Etienne; Serfaty, Sylvia.

Research output: Contribution to journal › Article

*Annals of Probability*, vol. 43, no. 4, pp. 2026-2083. https://doi.org/10.1214/14-AOP927

}

TY - JOUR

T1 - 2D coulomb gases and the renormalized energy

AU - Sandier, Etienne

AU - Serfaty, Sylvia

PY - 2015

Y1 - 2015

N2 - We study the statistical mechanics of classical two-dimensional "Coulomb gases" with general potential and arbitrary Β, the inverse of the temperature. Such ensembles also correspond to random matrix models in some particular cases. The formal limit case Β =∞corresponds to "weighted Fekete sets" and also falls within our analysis. It is known that in such a system points should be asymptotically distributed according to a macroscopic "equilibrium measure," and that a large deviations principle holds for this, as proven by Petz and Hiai [In Advances in Differential Equations and Mathematical Physics (Atlanta, GA, 1997) (1998) Amer. Math. Soc.] and Ben Arous and Zeitouni [ESAIM Probab. Statist. 2 (1998) 123-134]. By a suitable splitting of the Hamiltonian, we connect the problem to the "renormalized energy" W, a Coulombian interaction for points in the plane introduced in [Comm. Math. Phys. 313 (2012) 635-743], which is expected to be a good way of measuring the disorder of an infinite configuration of points in the plane. By so doing, we are able to examine the situation at the microscopic scale, and obtain several new results: a next order asymptotic expansion of the partition function, estimates on the probability of fluctuation from the equilibrium measure at microscale, and a large deviations type result, which states that configurations above a certain threshhold of W have exponentially small probability. When Β→∞, the estimate becomes sharp, showing that the system has to "crystallize" to a minimizer of W. In the case of weighted Fekete sets, this corresponds to saying that these sets should microscopically look almost everywhere like minimizers of W, which are conjectured to be "Abrikosov" triangular lattices.

AB - We study the statistical mechanics of classical two-dimensional "Coulomb gases" with general potential and arbitrary Β, the inverse of the temperature. Such ensembles also correspond to random matrix models in some particular cases. The formal limit case Β =∞corresponds to "weighted Fekete sets" and also falls within our analysis. It is known that in such a system points should be asymptotically distributed according to a macroscopic "equilibrium measure," and that a large deviations principle holds for this, as proven by Petz and Hiai [In Advances in Differential Equations and Mathematical Physics (Atlanta, GA, 1997) (1998) Amer. Math. Soc.] and Ben Arous and Zeitouni [ESAIM Probab. Statist. 2 (1998) 123-134]. By a suitable splitting of the Hamiltonian, we connect the problem to the "renormalized energy" W, a Coulombian interaction for points in the plane introduced in [Comm. Math. Phys. 313 (2012) 635-743], which is expected to be a good way of measuring the disorder of an infinite configuration of points in the plane. By so doing, we are able to examine the situation at the microscopic scale, and obtain several new results: a next order asymptotic expansion of the partition function, estimates on the probability of fluctuation from the equilibrium measure at microscale, and a large deviations type result, which states that configurations above a certain threshhold of W have exponentially small probability. When Β→∞, the estimate becomes sharp, showing that the system has to "crystallize" to a minimizer of W. In the case of weighted Fekete sets, this corresponds to saying that these sets should microscopically look almost everywhere like minimizers of W, which are conjectured to be "Abrikosov" triangular lattices.

KW - Abrikosov lattice

KW - Coulomb gas

KW - Crystallization

KW - Fekete sets

KW - Ginibre ensemble

KW - Large deviations

KW - One-component plasma

KW - Random matrices

KW - Renormalized energy

KW - Triangular lattice

UR - http://www.scopus.com/inward/record.url?scp=84930840288&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84930840288&partnerID=8YFLogxK

U2 - 10.1214/14-AOP927

DO - 10.1214/14-AOP927

M3 - Article

AN - SCOPUS:84930840288

VL - 43

SP - 2026

EP - 2083

JO - Annals of Probability

JF - Annals of Probability

SN - 0091-1798

IS - 4

ER -