First order global asymptotics for confined particles with singular pair repulsion
Zitt, Pierre-André; Gozlan, Nathael; Chafaï, Djalil (2014), First order global asymptotics for confined particles with singular pair repulsion, The Annals of Applied Probability, 24, 6, p. 2371-2413. http://dx.doi.org/10.1214/13-AAP980
TypeArticle accepté pour publication ou publié
External document linkhttp://hal.archives-ouvertes.fr/hal-00818472
Journal nameThe Annals of Applied Probability
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Abstract (EN)We study a physical system of N interacting particles in Rd, 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.
Subjects / KeywordsEquilibrium measure; Coulomb interaction; Riesz kernel; Large deviations; Potential theory; Mean field limit; Interacting particle systems
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