On the Ornstein–Zernike Behaviour for the Supercritical Random-Cluster Model on $$\mathbb {Z}^{d},d\ge 3$$ Z d , d ≥ 3

Campanino, Massimo; Gianfelice, Michele

doi:10.1007/s10955-015-1222-0

“…Let us mention that the approach developed by Dima and his collaborators led to a great variety of applications to important problems in equilibrium statistical mechanics, related to the properties of interfaces in planar lattice systems [22,15,23,19,18,28,40,27,26], the effect of a stretching force on self-interacting polymers with or without disorder [29,30,31,33,32,34,25], the asymptotic behavior of more general correlation functions [14,9,12,10,39], etc. 1.2.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Ornstein-Zernike behavior for Ising models with infinite-range interactions

Aoun¹,

Ott²,

Velenik³

2021

Preprint

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We prove Ornstein-Zernike behavior for the large-distance asymptotics of the two-point function of the Ising model above the critical temperature under essentially optimal assumptions on the interaction. The main contribution of this work is that the interactions are not assumed to be of finite range. To the best of our knowledge, this is the first proof of OZ asymptotics for a nontrivial model with infinite-range interactions.Our results actually apply to the Green function of a large class of "self-repulsive in average" models, including a natural family of self-repulsive polymer models that contains, in particular, the self-avoiding walk, the Domb-Joyce model and the killed random walk.We aimed at a pedagogical and self-contained presentation.

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“…These theorems are proven for the independent percolation case in [Campanino and Gianfelice 2011] and for the general case in [Campanino and Gianfelice 2015].…”

Section: 2mentioning

confidence: 94%

“…Here we expose some recent results on the asymptotic behavior of finite connection functions in three or more dimensions that appeared in [Campanino and Gianfelice 2011;2015]. These results refer both to Bernoulli percolation and to the FK random-cluster model in the highly supercritical regime, i.e., when the parameter p related to the occupation probability of a bond is close to 1.…”

Section: Introductionmentioning

confidence: 91%

See 1 more Smart Citation

Some results on the asymptotic behavior of finite connection probabilities in percolation

Campanino

¹

,

Gianfelice

²

2016

Math. Mech. Compl. Sys.

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We review results of two previous papers on the asymptotic behavior of finite connection probabilities in three or more dimensions for Bernoulli percolation and the Fortuin-Kasteleyn random-cluster model. In the introduction, we prove a multidimensional renewal theorem that is needed for these results and previous results on Ornstein-Zernike behavior; the proof is significantly simpler than that originally derived by Doney (1966) and those of other subsequent works on this subject. Communicated by Raffaele Esposito. Campanino and Gianfelice are partially supported by the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) and thank the referee for useful suggestions.

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“…Let us mention that the approach developed by Dima and his collaborators led to a great variety of applications to important problems in equilibrium statistical mechanics, related to the properties of interfaces in planar lattice systems [15,18,19,22,23,[26][27][28]40], the effect of a stretching force on self-interacting polymers with or without disorder [25,[29][30][31][32][33][34], the asymptotic behavior of more general correlation functions [9,10,12,14,39], etc.…”

Section: A Brief Historymentioning

confidence: 99%

Ornstein–Zernike behavior for Ising models with infinite-range interactions

Aoun,

Ott,

Velenik

2024

Ann. Inst. H. Poincaré Probab. Statist.

0

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We prove Ornstein-Zernike behavior for the large-distance asymptotics of the two-point function of the Ising model above the critical temperature under essentially optimal assumptions on the interaction. The main contribution of this work is that the interactions are not assumed to be of finite range. To the best of our knowledge, this is the first proof of OZ asymptotics for a nontrivial model with infinite-range interactions.Our results actually apply to the Green function of a large class of "self-repulsive in average" models, including a natural family of self-repulsive polymer models that contains, in particular, the self-avoiding walk, the Domb-Joyce model and the killed random walk.We aimed at a pedagogical and self-contained presentation.Résumé. Nous prouvons, sous des hypothèses essentiellement optimales sur l'interaction, que le comportement asymptotique de la fonction à 2-point du modèle d'Ising au-dessus de sa température critique prend la forme prédite par Ornstein et Zernike. La contribution principale de ce travail est que nous ne supposons pas l'interaction de portée finie. À notre connaissance, il s'agit de la première preuve du comportement Ornstein-Zernike pour un modèle non trivial avec des interactions de portée infinie. Nos résultats s'appliquent plus généralement à la fonction de Green d'une grande classe de modèle « auto-répulsifs en moyenne », incluant une famille naturelle de modèles de polymère auto-répulsif à laquelle appartiennent, en particulier, la marche aléatoire autoévitante, le modèle de Domb-Joyce et la marche aléatoire tuée.Nous nous sommes efforcés de rendre notre présentation aussi pédagogique et complète que possible.

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On the Ornstein–Zernike Behaviour for the Supercritical Random-Cluster Model on $$\mathbb {Z}^{d},d\ge 3$$ Z d , d ≥ 3

Cited by 4 publications

References 16 publications

Ornstein-Zernike behavior for Ising models with infinite-range interactions

Ornstein-Zernike behavior for Ising models with infinite-range interactions

Some results on the asymptotic behavior of finite connection probabilities in percolation

Ornstein–Zernike behavior for Ising models with infinite-range interactions

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