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At the edge of a one-dimensional jellium

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

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Type
Article accepté pour publication ou publié
Date
2022
Journal name
Bernoulli
Volume
28
Number
3
Publisher
International Statistical Institute
Pages
1784-1809
Publication identifier
10.3150/21-BEJ1397
Metadata
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Author(s)
Chafaï, Djalil
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
García-Zelada, David
Institut de Mathématiques de Marseille [I2M]
Jung, Paul
Department of Mathematical Sciences, KAIST
Abstract (EN)
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. We first observe that the system exists if and only if the total background charge is greater than the number of electrons minus one. Moreover we obtain a R\'enyi-type probabilistic representation for the order statistics of the particle system beyond the support of the background. Furthermore, 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.
Subjects / Keywords
Coulomb gas; edge statistics; jellium; one-dimensional model

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