### Abstract

We study systems of (Formula presented.) points in the Euclidean space of dimension (Formula presented.) interacting via a Riesz kernel (Formula presented.) and confined by an external potential, in the regime where (Formula presented.). We also treat the case of logarithmic interactions in dimensions 1 and 2. Our study includes and retrieves all cases previously studied in Sandier and Serfaty [2D Coulomb gases and the renormalized energy, Ann. Probab. (to appear); 1D log gases and the renormalized energy: crystallization at vanishing temperature (2013)] and Rougerie and Serfaty [Higher dimensional Coulomb gases and renormalized energy functionals, Comm. Pure Appl. Math. (to appear)]. Our approach is based on the Caffarelli–Silvestre extension formula, which allows one to view the Riesz kernel as the kernel of an (inhomogeneous) local operator in the extended space (Formula presented.). As (Formula presented.), we exhibit a next to leading order term in (Formula presented.) in the asymptotic expansion of the total energy of the system, where the constant term in factor of (Formula presented.) depends on the microscopic arrangement of the points and is expressed in terms of a ‘renormalized energy’. This new object is expected to penalize the disorder of an infinite set of points in whole space, and to be minimized by Bravais lattice (or crystalline) configurations. We give applications to the statistical mechanics in the case where temperature is added to the system, and identify an expected ‘crystallization regime’. We also obtain a result of separation of the points for minimizers of the energy.

Original language | English (US) |
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Journal | Journal of the Institute of Mathematics of Jussieu |

DOIs | |

State | Accepted/In press - May 29 2015 |

### Fingerprint

### Keywords

- Fekete points
- point separation
- renormalized energy
- Riesz energy

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Journal of the Institute of Mathematics of Jussieu*. https://doi.org/10.1017/S1474748015000201

**NEXT ORDER ASYMPTOTICS AND RENORMALIZED ENERGY FOR RIESZ INTERACTIONS.** / Petrache, Mircea; Serfaty, Sylvia.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - NEXT ORDER ASYMPTOTICS AND RENORMALIZED ENERGY FOR RIESZ INTERACTIONS

AU - Petrache, Mircea

AU - Serfaty, Sylvia

PY - 2015/5/29

Y1 - 2015/5/29

N2 - We study systems of (Formula presented.) points in the Euclidean space of dimension (Formula presented.) interacting via a Riesz kernel (Formula presented.) and confined by an external potential, in the regime where (Formula presented.). We also treat the case of logarithmic interactions in dimensions 1 and 2. Our study includes and retrieves all cases previously studied in Sandier and Serfaty [2D Coulomb gases and the renormalized energy, Ann. Probab. (to appear); 1D log gases and the renormalized energy: crystallization at vanishing temperature (2013)] and Rougerie and Serfaty [Higher dimensional Coulomb gases and renormalized energy functionals, Comm. Pure Appl. Math. (to appear)]. Our approach is based on the Caffarelli–Silvestre extension formula, which allows one to view the Riesz kernel as the kernel of an (inhomogeneous) local operator in the extended space (Formula presented.). As (Formula presented.), we exhibit a next to leading order term in (Formula presented.) in the asymptotic expansion of the total energy of the system, where the constant term in factor of (Formula presented.) depends on the microscopic arrangement of the points and is expressed in terms of a ‘renormalized energy’. This new object is expected to penalize the disorder of an infinite set of points in whole space, and to be minimized by Bravais lattice (or crystalline) configurations. We give applications to the statistical mechanics in the case where temperature is added to the system, and identify an expected ‘crystallization regime’. We also obtain a result of separation of the points for minimizers of the energy.

AB - We study systems of (Formula presented.) points in the Euclidean space of dimension (Formula presented.) interacting via a Riesz kernel (Formula presented.) and confined by an external potential, in the regime where (Formula presented.). We also treat the case of logarithmic interactions in dimensions 1 and 2. Our study includes and retrieves all cases previously studied in Sandier and Serfaty [2D Coulomb gases and the renormalized energy, Ann. Probab. (to appear); 1D log gases and the renormalized energy: crystallization at vanishing temperature (2013)] and Rougerie and Serfaty [Higher dimensional Coulomb gases and renormalized energy functionals, Comm. Pure Appl. Math. (to appear)]. Our approach is based on the Caffarelli–Silvestre extension formula, which allows one to view the Riesz kernel as the kernel of an (inhomogeneous) local operator in the extended space (Formula presented.). As (Formula presented.), we exhibit a next to leading order term in (Formula presented.) in the asymptotic expansion of the total energy of the system, where the constant term in factor of (Formula presented.) depends on the microscopic arrangement of the points and is expressed in terms of a ‘renormalized energy’. This new object is expected to penalize the disorder of an infinite set of points in whole space, and to be minimized by Bravais lattice (or crystalline) configurations. We give applications to the statistical mechanics in the case where temperature is added to the system, and identify an expected ‘crystallization regime’. We also obtain a result of separation of the points for minimizers of the energy.

KW - Fekete points

KW - point separation

KW - renormalized energy

KW - Riesz energy

UR - http://www.scopus.com/inward/record.url?scp=84930167487&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84930167487&partnerID=8YFLogxK

U2 - 10.1017/S1474748015000201

DO - 10.1017/S1474748015000201

M3 - Article

JO - Journal of the Institute of Mathematics of Jussieu

JF - Journal of the Institute of Mathematics of Jussieu

SN - 1474-7480

ER -