Nontrivial fixed points and screening in the hierarchical two-dimensional Coulomb gas

Benfatto, G.; Renn, Jürgen

doi:10.1007/bf01049006

Cited by 5 publications

(4 citation statements)

References 8 publications

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“…Notice the following facts: the number of term in the sum is not bigger than 3 , an upper bound for (6.56) is…”

Section: Proof Of Theorem 35mentioning

confidence: 99%

“…Despite its fast improvement, Fröhlich-Spencer method seemed to have some unavoidable limitations: it could not provide the exact power of the correlations fall-off, nor could exclude logarithmic corrections to such decay (which actually were expected along the KT transition line); and it did not provide any useful bound for correlations of integer charges. For this reasons, different authors started developing an RG approach to the model -at the beginning in some approximate form: hierarchical metric, or order by order in perturbation theory; see [2], [30], [12], [33], [24], [3]. Later, Dimock and Hurd, [14], achieved a rigorous construction, under no approximation, of the pressure in a region of the dipole phase that included (β, z) = (8π, 0) but not the rest of the KT transition line; they could not discuss charge correlations, though.…”

Section: Introductionmentioning

confidence: 99%

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Kosterlitz-Thouless Transition Line for the Two Dimensional Coulomb Gas

Falco

2012

Commun. Math. Phys.

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“…Notice the following facts: the number of term in the sum is not bigger than 3 , an upper bound for (6.56) is…”

Section: Proof Of Theorem 35mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Kosterlitz-Thouless Transition Line for the Two Dimensional Coulomb Gas

Falco

2012

Commun. Math. Phys.

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“…to denote the obvious corresponding objects in D. Hierarchical models were introduced by Dyson in statistical mechanics to study a variant of the one-dimensional Ising model [22,23]. Subsequently, the so called Dyson's hierarchical model has led to a lot of activity, see for e.g., [7,44,21,40,8,32] for results about the two-dimensional hierarchical Coulomb gas. However, before [18] not much was known about this model in any dimension greater than 2.…”

Section: 1mentioning

confidence: 99%

Ground states and hyperuniformity of the hierarchical Coulomb gas in all dimensions

Ganguly

Sarkar

2019

Probab. Theory Relat. Fields

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Stochastic point processes with Coulomb interactions arise in various natural examples of statistical mechanics, random matrices and optimization problems. Often such systems due to their natural repulsion exhibit remarkable hyperuniformity properties, that is, the number of points landing in any given region fluctuates at a much smaller scale compared to that of a set of i.i.d. random points. A well known conjecture from physics appearing in the works of Jancovici, Lebowitz, Manificat, Martin, and Yalcin (see [43,46,37]), states that the variance of the number of points landing in a set should grow like the surface area instead of the volume unlike i.i.d. random points. In a recent beautiful work [18], Chatterjee gave the first proof of such a result in dimension three for a Coulomb type system, known as the hierarchical Coulomb gas, inspired by Dyson's hierarchical model of the Ising ferromagnet [22,23]. However the case of dimensions greater than three had remained open. In this paper, we establish the correct fluctuation behavior up to logarithmic factors in all dimensions greater than three, for the hierarchical model. Using similar methods, we also prove sharp variance bounds for smooth linear statistics which were unknown in any dimension bigger than two. A key intermediate step is to obtain precise results about the ground states of such models whose behavior can be interpreted as hierarchical analogues of various crystalline conjectures predicted for energy minimizing systems, and could be of independent interest.

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“…The model proposed above is sometimes called the 'hierarchical Coulomb gas'. The 2D hierarchical Coulomb gas has received considerable attention in the mathematical physics literature [14,15,39,51,58,68]. However, not much is known about this model in dimensions three and higher.…”

Section: 3mentioning

confidence: 99%

Rigidity of the three-dimensional hierarchical Coulomb gas

Chatterjee¹

2017

Preprint

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A random set of points in Euclidean space is called 'rigid' or 'hyperuniform' if the number of points falling inside any given region has significantly smaller fluctuations than the corresponding number for a set of i.i.d. random points. This phenomenon has received considerable attention in recent years, due to its appearance in random matrix theory, the theory of Coulomb gases and zeros of random analytic functions. However, most of the published results are in dimensions one and two. This paper gives the first proof of hyperuniformity in a Coulomb type system in dimension three, known as the hierarchical Coulomb gas. This is a simplified version of the actual 3D Coulomb gas. The interaction potential in this model, inspired by Dyson's hierarchical model of the Ising ferromagnet, has a hierarchical structure and is locally an approximation of the Coulomb potential. Hyperuniformity is proved at both macroscopic and microscopic scales, with upper and lower bounds for the order of fluctuations that match up to logarithmic factors. The fluctuations have cube-root behavior, in agreement with a well-known prediction for the 3D Coulomb gas. For completeness, analogous results are also proved for the 2D hierarchical Coulomb gas and the 1D hierarchical log gas.

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Nontrivial fixed points and screening in the hierarchical two-dimensional Coulomb gas

Cited by 5 publications

References 8 publications

Kosterlitz-Thouless Transition Line for the Two Dimensional Coulomb Gas

Kosterlitz-Thouless Transition Line for the Two Dimensional Coulomb Gas

Ground states and hyperuniformity of the hierarchical Coulomb gas in all dimensions

Rigidity of the three-dimensional hierarchical Coulomb gas

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