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February 2021 Poisson statistics for Gibbs measures at high temperature
Gaultier Lambert
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Ann. Inst. H. Poincaré Probab. Statist. 57(1): 326-350 (February 2021). DOI: 10.1214/20-AIHP1080

Abstract

We consider a gas of N particles subject to a two-body interaction and confined by an external potential in the mean field or high temperature regime, that is when the inverse temperature β>0 satisfies βNγ0 as N+. We show that under general conditions on the interaction and the potential, the local fluctuations are described by a Poisson point process in the large N limit. We present applications to Coulomb and Riesz gases on Rn for any n1, as well as to the edge behavior of β-ensembles on R.

On étudie le comportement asymptotique d’un gaz à N particules à l’équilibre modélisé par une interaction de type champ moyen, c’est-à-dire dans le régime à haute température où la constante de couplage satisfait βNγ0 quand N+. On démontre que sous des hypothèses générales sur l’interaction à deux corps et le potentiel confinant, les fluctuations locales du gaz sont régies par un processus de Poisson. On discute des applications au gaz de Coulomb et de Riesz sur Rn pour n1 quelconque, ainsi que du comportement au bord des β-ensembles.

Acknowledgements

G.L. is supported by the SNSF Ambizione grant S-71114-05-01. G.L. thanks Trinh Khanh Duy for interesting discussions about the problem studied in this article and the anonymous referee for his helpful comments.

Citation

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Gaultier Lambert. "Poisson statistics for Gibbs measures at high temperature." Ann. Inst. H. Poincaré Probab. Statist. 57 (1) 326 - 350, February 2021. https://doi.org/10.1214/20-AIHP1080

Information

Received: 21 January 2020; Revised: 14 May 2020; Accepted: 22 June 2020; Published: February 2021
First available in Project Euclid: 12 March 2021

Digital Object Identifier: 10.1214/20-AIHP1080

Subjects:
Primary: 60B20, 60F10, 60G55, 60G70, 82B31

Rights: Copyright © 2021 Association des Publications de l’Institut Henri Poincaré

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Vol.57 • No. 1 • February 2021
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