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August 2022 At the edge of a one-dimensional jellium
Djalil Chafaï, David García-Zelada, Paul Jung
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Bernoulli 28(3): 1784-1809 (August 2022). DOI: 10.3150/21-BEJ1397

Abstract

We consider a one-dimensional classical Wigner jellium, not necessarily charge neutral, for which the electrons are allowed to exist beyond the support of the background charge. The model can be seen as a one-dimensional Coulomb gas in which the external field is generated by a smeared background on an interval. It is a true one-dimensional Coulomb gas and not a one-dimensional log-gas. The system exists if and only if the total background charge is greater than the number of electrons minus one. For various backgrounds, we show convergence to point processes, at the edge of the support of the background. In particular, this provides asymptotic analysis of the fluctuations of the right-most particle. Our analysis reveals that these fluctuations are not universal, in the sense that depending on the background, the tails range anywhere from exponential to Gaussian-like behavior, including for instance Tracy – Widom-like behavior. We also obtain a Rényi-type probabilistic representation for the order statistics of the particle system beyond the support of the background.

Funding Statement

DGZ was supported by the French ANR-16-CE40-0024 SAMARA project.
PJ was funded in part by the National Research Foundation of Korea (NRF) grants NRF- 2017R1A2B2001952 and NRF-2019R1A5A1028324.

Acknowledgments

PJ (respectively DC) thanks the hospitality of Université Paris-Dauphine – PSL (respectively KAIST). Also, all authors thank the hospitality of CIRM at Luminy.

Citation

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Djalil Chafaï. David García-Zelada. Paul Jung. "At the edge of a one-dimensional jellium." Bernoulli 28 (3) 1784 - 1809, August 2022. https://doi.org/10.3150/21-BEJ1397

Information

Received: 1 December 2020; Published: August 2022
First available in Project Euclid: 25 April 2022

Digital Object Identifier: 10.3150/21-BEJ1397

Keywords: Coulomb gas , edge statistics , jellium , one-dimensional model

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Vol.28 • No. 3 • August 2022
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