First-order global asymptotics for confined particles with singular pair repulsion

Chafäı, Djalil; Gozlan, Nathaël; Zitt, Pierre-André

doi:10.1214/13-aap980

Cited by 89 publications

(163 citation statements)

References 55 publications

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“…The leading-order contribution to (2.28)-(2.31) has recently been derived, along with the corresponding large-deviation principle, in [18] under the assumptioň n 1 log n in our units. In this regime one may use either EOE ˇ or EOE 0 for leading-order considerations, but whenˇ/ n 1 it is necessary to use the former.…”

Section: Main Results: Finite Temperature Casementioning

confidence: 99%

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Higher‐Dimensional Coulomb Gases and Renormalized Energy Functionals

Rougerie

Serfaty

2015

Comm Pure Appl Math

183

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We consider a classical system of n charged particles in an external confining potential, in any dimension d ≥ 2. The particles interact via pairwise repulsive Coulomb forces and the coupling parameter is of order n −1 (mean-field scaling). By a suitable splitting of the Hamiltonian, we extract the next to leading order term in the ground state energy, beyond the mean-field limit. We show that this next order term, which characterizes the fluctuations of the system, is governed by a new "renormalized energy" functional providing a way to compute the total Coulomb energy of a jellium (i.e. an infinite set of point charges screened by a uniform neutralizing background), in any dimension. The renormalization that cuts out the infinite part of the energy is achieved by smearing out the point charges at a small scale, as in Onsager's lemma. We obtain consequences for the statistical mechanics of the Coulomb gas: next to leading order asymptotic expansion of the free energy or partition function, characterizations of the Gibbs measures, estimates on the local charge fluctuations and factorization estimates for reduced densities. This extends results of Sandier and Serfaty to dimension higher than two by an alternative approach.

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Section: Main Results: Finite Temperature Casementioning

confidence: 99%

“…w.x y//d n .x/d n .y/ any dimension [18,72]. This is also of interest in the more elaborate settings of complex manifolds, cf., e.g.…”

mentioning

confidence: 99%

Higher‐Dimensional Coulomb Gases and Renormalized Energy Functionals

Rougerie

Serfaty

2015

Comm Pure Appl Math

183

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“…These systems have recently received much attention, as attested for example by the works of Sandier, Serfaty and their coauthors (see [19] for a review), and also, closer to our point view, by the work of Chafaï, Gozlan and Zitt in [6], based on large deviations. In these works, very general results about the existence of the minimisers and their properties are stated, but only few minimisation problems are explicitly solvable, and even less when a constraint is added.…”

Section: Introductionmentioning

confidence: 67%

On the Douglas-Kazakov phase transition

Lévy¹,

Maïda²

2015

ESAIM: Proc.

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Abstract. We give a rigorous proof of the fact that a phase transition discovered by Douglas and Kazakov in 1993 in the context of two-dimensional gauge theories occurs. This phase transition can be formulated in terms of the Brownian bridge on the unitary group U(N ) when N tends to infinity. We explain how it can be understood by considering the asymptotic behaviour of the eigenvalues of the unitary Brownian bridge, and how it can be technically approached by means of Fourier analysis on the unitary group. Moreover, we advertise some more or less classical methods for solving certain minimisation problems which play a fundamental role in the study of the phase transition.

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“…The classical Sanov theorem corresponds formally to this regime when we turn off the pair interaction by taking g = 0. This regime is considered in particular in [14,9,8,18,31,2].…”

Section: General Planar Coulomb Gasesmentioning

confidence: 99%

Macroscopic and edge behavior of a planar jellium

Chafäı

García-Zelada

Jung

2020

Journal of Mathematical Physics

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We consider a planar Coulomb gas in which the external potential is generated by a smeared uniform background of opposite-sign charge on a disc. This model can be seen as a two-dimensional Wigner jellium, not necessarily charge neutral, and with particles allowed to exist beyond the support of the smeared charge. The full space integrability condition requires low enough temperature or high enough total smeared charge. This condition does not allow at the same time, total charge neutrality and determinantal structure. The model shares similarities with both the complex Ginibre ensemble and the Forrester-Krishnapur spherical ensemble of random matrix theory. In particular, for a certain regime of temperature and total charge, the equilibrium measure is uniform on a disc as in the Ginibre ensemble, while the modulus of the farthest particle has heavy-tailed fluctuations as in the Forrester-Krishnapur spherical ensemble. We also touch on a higher temperature regime producing a crossover equilibrium measure, as well as a transition to Gumbel edge fluctuations. More results in the same spirit on edge fluctuations are explored by the second author together with Raphael Butez.Contents P →, convergence in law and in probability, respectively, and we annotate X ∼ µ to mean that the law of the random variable X is given by the probability distribution µ. We denote by P(C) the set of probability measures on C, equipped with the topology of weak convergence with respect to continuous and bounded test functions, and its associated Borel σ-field. This topology is metrized by the bounded-Lipschitz metric

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First-order global asymptotics for confined particles with singular pair repulsion

Cited by 89 publications

References 55 publications

Higher‐Dimensional Coulomb Gases and Renormalized Energy Functionals

Higher‐Dimensional Coulomb Gases and Renormalized Energy Functionals

On the Douglas-Kazakov phase transition

Macroscopic and edge behavior of a planar jellium

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