Percolation on Grigorchuk Groups

Muchnik, Roman; Pak, Igor

doi:10.1081/agb-100001531

Cited by 22 publications

(11 citation statements)

References 23 publications

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“…Recent advancements include improved upper and lower bounds, solution of the word problem, abstract presentation, bond percolation, etc. (see [5,9,11,12]). We refer to review articles [7,9] for the references.…”

Section: Introductionmentioning

confidence: 99%

On Growth of Grigorchuk Groups

Muchnik

Pak

2001

Int. J. Algebra Comput.

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Section: Introductionmentioning

confidence: 99%

On Growth of Grigorchuk Groups

Muchnik

Pak

2001

Int. J. Algebra Comput.

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“…[4,5,23,20,21,22,37,25,26,27,19,18]. There have been also a few papers dealing with percolation processes on quasi transitive or transitive graphs, including amenable graphs, see e.g.[2], [3], [30]. Roughly speaking, in a transitive graph G any vertex of the graph is equivalent; in other words G "looks the same" by observers sitting in different vertices.…”

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“…[2], [3], [30]. Roughly speaking, in a transitive graph G any vertex of the graph is equivalent; in other words G "looks the same" by observers sitting in different vertices.…”

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Convergent expansions for random cluster model with $q>0$ on infinite graphs

Procacci¹,

Scoppola²

2008

Communications on Pure &Amp; Applied Analysis

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In this paper we extend our previous results on the connectivity functions and pressure of the Random Cluster Model in the highly subcritical phase and in the highly supercritical phase, originally proved only on the cubic lattice Z d , to a much wider class of infinite graphs. In particular, concerning the subcritical regime, we show that the connectivity functions are analytic and decay exponentially in any bounded degree graph. In the supercritical phase, we are able to prove the analyticity of finite connectivity functions in a smaller class of graphs, namely, bounded degree graphs with the so called minimal cut-set property and satisfying a (very mild) isoperimetric inequality. On the other hand we show that the large distances decay of finite connectivity in the supercritical regime can be polynomially slow depending on the topological structure of the graph. Analogous analyticity results are obtained for the pressure of the Random Cluster Model on an infinite graph, but with the further assumptions of amenability and quasi-transitivity of the graph. graphs with no circuits) , see e.g. [4,5,23,20,21,22,37,25,26,27,19,18]. There have been also a few papers dealing with percolation processes on quasi transitive or transitive graphs, including amenable graphs, see e.g.[2], [3], [30]. Roughly speaking, in a transitive graph G any vertex of the graph is equivalent; in other words G "looks the same" by observers sitting in different vertices. In a quasi-transitive graph G there is a finite number of different types of vertices and G "looks the same" by observers sitting in vertices of the same type.Some general results about percolation on general infinite graphs (i.e. not necessarily non amenable and/or quasi-transitive) appeared in [4], [5], and [2], and more recently in [33] (see also references therein). There are also some other works about the Potts model, in particular the antiferromagnetic case, on general finite graphs [38,39] and on amenable quasi-transitive infinite graphs [35]. In this paper we focus our attention on the study of the dependent percolation process known as Random Cluster Model (RCM) on general infinite graphs. The RCM was proposed by Fortuin and Kasteleyn in the early seventies [12] as a generalization of the Potts model. The RCM on a graph G depends on two real parameters: the parameter p ∈ [0, 1] and the parameter q ∈ (0, +∞). The parameter p represents the weight of an edge of G to be open independently of the other edges and it is related to the temperature of the Potts model. The parameter q, when different from 1, introduces a dependence in the percolation process described by RCM, and, when integer greater than 2, it represents the number of colors in the Potts model.Some results on RCM can be proved for all the values of the parameters q and p. In particular, there are results about the logarithm of the total weight of the measure (pressure). Namely, the existence of the pressure of the RCM, its independency on boundary conditions and its differentiability (with respect to p al...

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“…Вопрос об аменабельности группы U остается открытым. Заметка [20], в которой утверждается, что она аменабельна, к сожалению, содержит ошибку.…”

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Universal Groups of Intermediate Growth and Their Invariant Random Subgroups

Benli¹,

Grigorchuk²,

Nagnibeda³

et al. 2015

Funktsional. Anal. i Prilozhen.

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В работе приведены примеры групп промежуточного роста с $2^{\aleph_0}$ эргодическими непрерывными инвариантными случайными подгруппами. Эти примеры - универсальные группы, ассоциированные с некоторым семейством групп промежуточного роста.

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Percolation on Grigorchuk Groups

Cited by 22 publications

References 23 publications

On Growth of Grigorchuk Groups

On Growth of Grigorchuk Groups

Convergent expansions for random cluster model with $q>0$ on infinite graphs

Universal Groups of Intermediate Growth and Their Invariant Random Subgroups

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