Concentration for Coulomb gases on compact manifolds

García-Zelada, David

doi:10.1214/19-ecp211

Cited by 15 publications

(13 citation statements)

References 15 publications

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“…The present paper fits in a recent line of works focusing on properties that are valid at all temperatures (and not only for β = 2) and at all scales -from the microscopic/local lengthscale (here ∼ 1) to the macroscopic/global one (here ∼ √ N ). Topics that have been investigated include: lower bounds on the minimal distance between points [Ame18], concentration inequalities for the empirical measure of the particles [CHM18;Gar19b], upper bounds for the local density of points [LRY19], generalizations to gases on Riemannian manifolds [Ber19], etc.…”

Section: Context and Related Resultsmentioning

confidence: 99%

The two-dimensional one-component plasma is hyperuniform

Leblé¹

2021

Preprint

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We prove that at all positive temperatures in the bulk of a classical two-dimensional one-component plasma (also called Coulomb or log-gas, or jellium) the variance of the number of particles in large disks grows more slowly than the area. In other words the system is hyperuniform.We obtain a non-sharp but quantitative bound on the number variance's growth rate, which is the first mathematical justification of an old prediction in the physics literature about "suppression of charge fluctuations".We introduce an argument of approximate conditional independence for well-separated sub-systems and a trick using "isotropically averaged localized translations" in order to control the expectation of non-smooth linear statistics.1 Our argument works for α > 31 32 .

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Section: Context and Related Resultsmentioning

confidence: 99%

The two-dimensional one-component plasma is hyperuniform

Leblé¹

2021

Preprint

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“…for any probability measure µ on R with a continuous density (see the discussion in [5, Page 2304-2305]). Generalization of the concentration of measure inequality 1.14 to general Coulomb (and Riesz) gas ensembles (dP N,β , R N ) in Euclidean R N has been obtained in [32,12] and on compact Riemannian manifolds in [17]. In particular, in the case of the spherical ensemble the inequalities in [17] say that (1.15)…”

Section: (See Section 21)mentioning

confidence: 99%

The spherical ensemble and quasi-Monte-Carlo designs

Berman¹

2019

Preprint

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The spherical ensemble is a well-known ensemble of N repulsive points on the two-dimensional sphere, which can realized in various ways (as a random matrix ensemble, a determinantal point process, a Coulomb gas, a Quantum Hall state...). Here we show that the spherical ensemble enjoys nearly optimal convergence properties from the point of view of numerical integration. More precisely, it is shown that the numerical integration rule corresponding to N nodes on the two-dimensional sphere sampled in the spherical ensemble is, with overwhelming probability, nearly a quasi-Monte-Carlo design in the sense of Brauchart-Saff-Sloan-Womersley (for any smoothness parameter s ≤ 2). The key ingredient is a new explicit concentration of measure inequality for the spherical ensemble.

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“…Coulomb gas on compact Riemannian manifolds. This particle system has been introduced in [12] and further studied in [13]. It consists of a Gibbs measure (1.2)-( 1.3) where the interaction kernel g is the Green function of the Laplace-Beltrami operator on a compact manifold X of dimension n ≥ 3.…”

Section: Local Fluctuationsmentioning

confidence: 99%

Poisson statistics for Gibbs measures at high temperature

Lambert¹

2019

Preprint

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We consider a gas of N particles with a general 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 βN → γ ≥ 0 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 R n for any n ≥ 1, as well as to the edge behavior of β-ensembles on R.

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Concentration for Coulomb gases on compact manifolds

Cited by 15 publications

References 15 publications

The two-dimensional one-component plasma is hyperuniform

The two-dimensional one-component plasma is hyperuniform

The spherical ensemble and quasi-Monte-Carlo designs

Poisson statistics for Gibbs measures at high temperature

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