### Abstract

We study the statistical mechanics of a one-dimensional log gas or β-ensemble with general potential and arbitrary β, the inverse of temperature, according to the method we introduced for two-dimensional Coulomb gases in Sandier and Serfaty (Ann Probab, 2014). Such ensembles correspond to random matrix models in some particular cases. The formal limit β=∞ corresponds to “weighted Fekete sets” and is also treated. We introduce a one-dimensional version of the “renormalized energy” of Sandier and Serfaty (Commun Math Phys 313(3):635–743, 2012), measuring the total logarithmic interaction of an infinite set of points on the real line in a uniform neutralizing background. We show that this energy is minimized when the points are on a lattice. By a suitable splitting of the Hamiltonian we connect the full statistical mechanics problem to this renormalized energy W, and this allows us to obtain new results on the distribution of the points at the microscopic scale: in particular we show that configurations whose W is above a certain threshold (which tends to minW as β→∞) have exponentially small probability. This shows that the configurations have increasing order and crystallize as the temperature goes to zero.

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

Pages (from-to) | 795-846 |

Number of pages | 52 |

Journal | Probability Theory and Related Fields |

Volume | 162 |

Issue number | 3-4 |

DOIs | |

State | Published - Nov 2 2014 |

### Fingerprint

### Keywords

- 15B52
- 82B05
- 82B21
- 82B26

### ASJC Scopus subject areas

- Analysis
- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Probability Theory and Related Fields*,

*162*(3-4), 795-846. https://doi.org/10.1007/s00440-014-0585-5

**1D log gases and the renormalized energy : crystallization at vanishing temperature.** / Sandier, Etienne; Serfaty, Sylvia.

Research output: Contribution to journal › Article

*Probability Theory and Related Fields*, vol. 162, no. 3-4, pp. 795-846. https://doi.org/10.1007/s00440-014-0585-5

}

TY - JOUR

T1 - 1D log gases and the renormalized energy

T2 - crystallization at vanishing temperature

AU - Sandier, Etienne

AU - Serfaty, Sylvia

PY - 2014/11/2

Y1 - 2014/11/2

N2 - We study the statistical mechanics of a one-dimensional log gas or β-ensemble with general potential and arbitrary β, the inverse of temperature, according to the method we introduced for two-dimensional Coulomb gases in Sandier and Serfaty (Ann Probab, 2014). Such ensembles correspond to random matrix models in some particular cases. The formal limit β=∞ corresponds to “weighted Fekete sets” and is also treated. We introduce a one-dimensional version of the “renormalized energy” of Sandier and Serfaty (Commun Math Phys 313(3):635–743, 2012), measuring the total logarithmic interaction of an infinite set of points on the real line in a uniform neutralizing background. We show that this energy is minimized when the points are on a lattice. By a suitable splitting of the Hamiltonian we connect the full statistical mechanics problem to this renormalized energy W, and this allows us to obtain new results on the distribution of the points at the microscopic scale: in particular we show that configurations whose W is above a certain threshold (which tends to minW as β→∞) have exponentially small probability. This shows that the configurations have increasing order and crystallize as the temperature goes to zero.

AB - We study the statistical mechanics of a one-dimensional log gas or β-ensemble with general potential and arbitrary β, the inverse of temperature, according to the method we introduced for two-dimensional Coulomb gases in Sandier and Serfaty (Ann Probab, 2014). Such ensembles correspond to random matrix models in some particular cases. The formal limit β=∞ corresponds to “weighted Fekete sets” and is also treated. We introduce a one-dimensional version of the “renormalized energy” of Sandier and Serfaty (Commun Math Phys 313(3):635–743, 2012), measuring the total logarithmic interaction of an infinite set of points on the real line in a uniform neutralizing background. We show that this energy is minimized when the points are on a lattice. By a suitable splitting of the Hamiltonian we connect the full statistical mechanics problem to this renormalized energy W, and this allows us to obtain new results on the distribution of the points at the microscopic scale: in particular we show that configurations whose W is above a certain threshold (which tends to minW as β→∞) have exponentially small probability. This shows that the configurations have increasing order and crystallize as the temperature goes to zero.

KW - 15B52

KW - 82B05

KW - 82B21

KW - 82B26

UR - http://www.scopus.com/inward/record.url?scp=84939254340&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84939254340&partnerID=8YFLogxK

U2 - 10.1007/s00440-014-0585-5

DO - 10.1007/s00440-014-0585-5

M3 - Article

AN - SCOPUS:84939254340

VL - 162

SP - 795

EP - 846

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

SN - 0178-8051

IS - 3-4

ER -