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December 2014 First-order global asymptotics for confined particles with singular pair repulsion
Djalil Chafaï, Nathael Gozlan, Pierre-André Zitt
Ann. Appl. Probab. 24(6): 2371-2413 (December 2014). DOI: 10.1214/13-AAP980


We study a physical system of $N$ interacting particles in $\mathbb{R}^{d}$, $d\geq1$, subject to pair repulsion and confined by an external field. We establish a large deviations principle for their empirical distribution as $N$ tends to infinity. In the case of Riesz interaction, including Coulomb interaction in arbitrary dimension $d>2$, the rate function is strictly convex and admits a unique minimum, the equilibrium measure, characterized via its potential. It follows that almost surely, the empirical distribution of the particles tends to this equilibrium measure as $N$ tends to infinity. In the more specific case of Coulomb interaction in dimension $d>2$, and when the external field is a convex or increasing function of the radius, then the equilibrium measure is supported in a ring. With a quadratic external field, the equilibrium measure is uniform on a ball.


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Djalil Chafaï. Nathael Gozlan. Pierre-André Zitt. "First-order global asymptotics for confined particles with singular pair repulsion." Ann. Appl. Probab. 24 (6) 2371 - 2413, December 2014.


Published: December 2014
First available in Project Euclid: 26 August 2014

zbMATH: 1304.82050
MathSciNet: MR3262506
Digital Object Identifier: 10.1214/13-AAP980

Primary: 31B99 , 60F10 , 82C22

Keywords: Coulomb interaction , Equilibrium measure , interacting particle systems , large deviations , Mean-field limit , potential theory , Riesz kernel

Rights: Copyright © 2014 Institute of Mathematical Statistics

Vol.24 • No. 6 • December 2014
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