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Coulomb gases under constraint : some theoretical and numerical results

Chafaï, Djalil; Ferré, Grégoire; Stoltz, Gabriel (2019-07), Coulomb gases under constraint : some theoretical and numerical results. https://basepub.dauphine.fr/handle/123456789/20181

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1907.05803.pdf (848.2Kb)
Type
Document de travail / Working paper
External document link
https://hal.archives-ouvertes.fr/hal-02184896
Date
2019-07
Publisher
Cahier de recherche CEREMADE, Université Paris-Dauphine
Series title
Cahier de recherche CEREMADE, Université Paris-Dauphine
Published in
Paris
Pages
39
Metadata
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Author(s)
Chafaï, Djalil cc
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Ferré, Grégoire
Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique [CERMICS]
Stoltz, Gabriel
Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique [CERMICS]
Abstract (EN)
We consider Coulomb gas models for which the empirical measure typically concentrates, when the number of particles becomes large, on an equilibrium measure minimizing an electrostatic energy. We study the behavior when the gas is conditioned on a rare event. We first show that the special case of quadratic confinement and linear constraint is exactly solvable due to a remarkable factorization, and that the conditioning has then the simple effect of shifting the cloud of particles without deformation. To address more general cases, we perform a theoretical asymptotic analysis relying on a large deviations technique known as the Gibbs conditioning principle. The technical part amounts to establishing that the conditioning ensemble is an I-continuity set of the energy. This leads to characterizing the conditioned equilibrium measure as the solution of a modified variational problem. For simplicity, we focus on linear statistics and on quadratic statistics constraints. Finally, we numerically illustrate our predictions and explore cases in which no explicit solution is known. For this, we use a Generalized Hybrid Monte Carlo algorithm for sampling from the conditioned distribution for a finite but large system.
Subjects / Keywords
Constrained dynamics; Coulomb gases; Random matrices; Large deviations; Conditioning; Gibbs principle; Numerical simulation

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