O. H AGGSTR OM, "The random-cluster model on a homogeneous tree," Probability Theory and Related Fields 104 (1996), pp. 231--253.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Finally assign the spin i to all vertices of in nite clusters. Then X has distribution i . We identify f1; 1g with f; g. An immediate consequence of Lemma 5. 3 is that 6= i p;2 w gives rise to in nite cluster(s) In the special case when G = T r , it is known (see [7]) that when p= 2 p) 1=r, i.e. when p 2= 1 r) then p;2 w is just iid bond percolation with retention parameter p= 2 p) One consequence is that p = 2= 1 r) is the critical value for percolation for p;2 w and another is that as p # 2= 1 r) the probability for a given edge to be ....

O. H aggstr om, The random-cluster model on a homogeneous tree, Probab. Th. Rel. Fields 104 (1996), 231-253.

....model on nonamenable groups, it would be useful to know whether Theorem 1.1 also extends to the FK random cluster model for q 2; see, e.g. Grimmett (1995) for definitions and background. For q 2, the (wired) random cluster model on a regular tree has infinite clusters at criticality; see Haggstrom (1996). Combining Theorem 1.1 with a result of Haggstrom and Peres (1998) gives that for nonamenable Cayley graphs, the function (p) P p [K(o) is infinite] is continuous for all p 2 [0; 1] One may wish for quantitative versions of this result. For example: Problem 5.1: Bound the modulus of ....

H aggstr om, O. (1996). The random-cluster model on a homogeneous tree, Probab. Th. Rel. Fields 104, 231--253.

....model on nonamenable groups, it would be useful to know whether Theorem 1.1 also extends to the FK random cluster model for q 2; see, e.g. Grimmett (1995) for de nitions and background. For q 2, the (wired) random cluster model on a regular tree has in nite clusters at criticality; see H aggstr om (1996). Combining Theorem 1.1 with a result of H aggstr om and Peres (1998) gives that for nonamenable Cayley graphs, the function (p) P p [K(o) is in nite] is continuous for all p 2 [0; 1] One may wish for quantitative versions of this result. For example: Problem 5.1: Bound the modulus of ....

H aggstr om, O. (1996). The random-cluster model on a homogeneous tree, Probab. Th. Rel. Fields 104, 231-253.

....combination of the q Gibbs states obtained from one spin boundary conditions. It is also proved that on the regular tree, T n , with q 1 and p close enough to 1, there is unique random cluster measure despite the presence of more than one infinite cluster. This partly proves Conjecture 1. 9 of [11]. 1 Introduction The purpose of this paper is two fold: To introduce a technique that overcomes the difficulties involved in explicitly constructing random cluster measures on a general infinite graph with bounded degree and to give a characterization of nonamenability for transitive graphs in ....

....model. An infinity wired random cluster measure is defined in the usual Dobrushin LanfordRuelle spirit in such a way that all infinite clusters are considered as one, i.e. connected to each other at infinity. The idea of regarding all infinite clusters as one was introduced by Haggstrom in [11] where the random cluster model on a homogeneous tree is considered. The precise definition will be given in Section 2, Definition 2.2. In Section 2 we also introduce the promised method for finding such measures on a general infinite graph. Theorem 1.2 Let G = V; E) be an infinite graph. a) ....

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O. H AGGSTR OM, The random-cluster model on a homogeneous tree, Probab. Th. Rel. Fields 104 (1996), 231-253.

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Haggstrom, O. (1996) The random-cluster model on a homogeneous tree, Probab. Th. Rel. Fields 104, 231-253.

....the present paper. It seems reasonable to expect that (28) extends to all quasi transitive graphs (except those for which the critical value is 1) For q = 1, this was conjectured by Benjamini and Schramm [5] The situation for WRC seems to be more complicated. For instance, as shown in [10] and [17], when G is the regular tree T n with n 2, we get that the WRC c ;q probability of seeing an in nite cluster is 0 or 1 depending on whether q 2 [1; 2] or q 2. 5.2 Simultaneity statements For quasi transitive graphs, the famous nite energy argument of Newman and Schulman [26] shows ....

Haggstrom, O. (1996) The random-cluster model on a homogeneous tree, Probab. Th. Rel. Fields 104, 231-253.

....the present paper. It seems reasonable to expect that (28) extends to all quasi transitive graphs (except those for which the critical value is 1) For q = 1, this was conjectured by Benjamini and Schramm [5] The situation for WRC seems to be more complicated. For instance, as shown in [10] and [17], when G is the regular tree T n with n 2, we get that the WRC G p wired c ;q probability of seeing an in nite cluster is 0 or 1 depending on whether q 2 [1; 2] or q 2. 5.2 Simultaneity statements For quasi transitive graphs, the famous nite energy argument of Newman and Schulman [26] ....

Haggstrom, O. (1996) The random-cluster model on a homogeneous tree, Probab. Th. Rel. Fields 104, 231-253.

.... cluster at p c (q) It is known that there is none for q = 1 on nonamenable quasi transitive unimodular graphs [4, 5] On the other hand, it is known that there can be in nitely many in nite clusters for q 2 on the Cayley graphs T n for n 2 with respect to the wired random cluster measure; see [13, 27]. While we do not have a criterion that settles the question completely, we have the following partial results: Theorem 3.1. Let G be a quasi transitive nonamenable unimodular graph and q 1. Then there is no in nite cluster FRC G p free c (q) q a.s. Also, the following are equivalent: i) ....

....Proposition 5.2] and the inequalities (9) and (10) it is not hard to show that one can take q to be slightly larger than 1 in all such examples. For an example where the inequality in (22) is strict, we can simply take G to be the regular tree T n with n 2 and q 2 (see, e.g. H aggstr om [27]) or take any nonamenable regular graph with q suciently large (this follows from [31, Theorem 1.2(a) in combination with (8) Finally, for an example where (23) is strict, we refer to Section 4. The inequalities (20) 23) say nothing about the relation between p wired u and p free c . Here ....

Haggstrom, O. (1996) The random-cluster model on a homogeneous tree, Probab. Th. Rel. Fields 104, 231-253.

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O. H AGGSTR OM, "The random-cluster model on a homogeneous tree," Probability Theory and Related Fields 104 (1996), pp. 231--253.

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O. H AGGSTR OM, "The random-cluster model on a homogeneous tree," Probab. Theory Related Fields 104 (1996), pp. 231--253.

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