At the edge of a one-dimensional jellium
Chafaï, Djalil; García-Zelada, David; Jung, Paul (2020), At the edge of a one-dimensional jellium. https://basepub.dauphine.psl.eu/handle/123456789/22352
TypeDocument de travail / Working paper
External document linkhttps://hal.archives-ouvertes.fr/hal-03049591
Series titleCahier de recherche du CEREMADE
MetadataShow full item record
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Institut de Mathématiques de Marseille [I2M]
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.
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Bordenave, Charles; Chafaï, Djalil; García-Zelada, David (2021) Article accepté pour publication ou publié