Macroscopic and edge behavior of a planar jellium

Chafäı, Djalil; García-Zelada, David; Jung, Paul

doi:10.1063/1.5126724

Cited by 11 publications

(7 citation statements)

References 66 publications

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“…The condition does not depend on β, which is not a surprise in view of the scaling property in Remark 1.2. This is in contrast with the two-dimensional analogue considered in [CGZJ20].…”

Section: Resultsmentioning

confidence: 70%

At the edge of a one-dimensional jellium

Chafaï,

García-Zelada,

Jung

2020

Preprint

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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 Rényi-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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“…The condition does not depend on β, which is not a surprise in view of the scaling property in Remark 1.2. This is in contrast with the two-dimensional analogue considered in [CGZJ20].…”

Section: Resultsmentioning

confidence: 70%

At the edge of a one-dimensional jellium

Chafaï,

García-Zelada,

Jung

2020

Preprint

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“…We refer to [10] for potential theoretic motivations on such boundary conditions. See also [17,27] for related boundary conditions. This convex combination gives rise to universal point fields which interpolate between the point fields with the 1-point function (1.15) and with (2.1).…”

Section: Discussion Of Main Resultsmentioning

confidence: 99%

Random normal matrices in the almost-circular regime

Byun¹,

Seo²

2021

Preprint

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We study random normal matrix models whose eigenvalues tend to be distributed within a narrow "band" around the unit circle of width proportional to 1 n , where n is the size of matrices. For general radially symmetric potentials with various boundary conditions, we derive the scaling limits of the correlation functions, some of which appear in the previous literature notably in the context of almost-Hermitian random matrices. We also obtain that fluctuations of the maximal and minimal modulus of the ensembles follow the Gumbel or exponential law depending on the boundary conditions.

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“…The idealized total background charge κ N = N would correspond to global charge neutrality of the system, but we ask κ N > N for the law P N to be well-defined without spatial truncation. This choice is discussed [19,Section 2.1] for κ N = N + 1 and also in [8]. The terminology weakly confining is related to the growth of the potential at infinity U µ (z) = log |z| + O(1) (cf.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Universality for outliers in weakly confined Coulomb-type systems

Butez¹,

García-Zelada²,

Nishry³

et al. 2021

Preprint

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This work treats weakly confined particle systems in the plane, characterized by a large number of outliers away from a droplet where the bulk of the particles accumulate in the many-particle limit. We are interested in the asymptotic behaviour of outliers for two families of point processes: Coulomb gases confined by regular background measures at determinantal inverse temperature, and a class of random polynomials. We observe that the limiting outlier process only depends on the shape of the uncharged region containing them, and the global net excess charge. In particular, for a determinantal Coulomb gas confined by a sufficiently regular background measure, the outliers in a simply connected uncharged region converge to the corresponding Bergman point process. For a finitely connected uncharged region Ω, a family of limiting outlier processes arises, indexed by the (Pontryagin) dual of the fundamental group of Ω. Moreover, the outliers in different uncharged regions are asymptotically independent, even if the regions have common boundary points. The latter result is a manifestation of screening properties of the particle system.

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Macroscopic and edge behavior of a planar jellium

Cited by 11 publications

References 66 publications

At the edge of a one-dimensional jellium

At the edge of a one-dimensional jellium

Random normal matrices in the almost-circular regime

Universality for outliers in weakly confined Coulomb-type systems

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