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## Markov Random Fields and Percolation on General Graphs (1999)

Venue: | Adv. Appl. Probab |

Citations: | 26 - 3 self |

### Citations

1254 |
Interacting Particle Systems.
- Liggett
- 1985
(Show Context)
Citation Context ...(X(o) = +1)s1 2 + " for all n. It is also well known that the existence of more than one Gibbs measure is increasing in fi. This was originally proved using so-called Griffiths inequalities (see =-=e.g. [33]-=-); the modern approach based on the random-cluster model will be indicated in Section 3.4. The following result is an immediate consequence. Theorem 2.2 For any G 2 G, there exists a critical value fi... |

355 |
Gibbs measures and phase transitions, de Gruyter
- Georgii
- 1988
(Show Context)
Citation Context ... random assignment of +1's and \Gamma1's to the vertices of G. It was introduced in the 1920's as a model for ferromagnetism, and is today the most studied of all Markov random field models; see e.g. =-=[31, 17] for-=- introductions and some history. Take G = (V; E) 2 G. A probability measure �� on f\Gamma1; 1g V is said to be a Gibbs measure for the (ferromagnetic) Ising model on G at inverse temperature fi ? ... |

158 | Probability on Trees and Networks,
- Peres
- 2012
(Show Context)
Citation Context ....4 and Proposition 2.3 as in the proof of Lemma 6.2. 2 7.1 Galton--Watson trees A tree is a graph with no cycles. A Galton--Watson tree is a random tree T = (V T ; E T ) obtained as follows (see e.g. =-=[40, 37]-=- for more extensive discussions). Let L be a nonnegative integer-valued random variable, and set p k = P(L = k) for each k. We call fp k g 1 k=0 the offspring distribution for T . The so-called root a... |

157 |
On the random-cluster model. I. Introduction and relation to other models, Physica 57, 536–564. Explicit isoperimetric constants 27
- Fortuin, Kasteleyn
- 1972
(Show Context)
Citation Context ...on theory on one hand, and the issue of Gibbs state multiplicity in Markov random fields on the other. Examples of such connections are the Fortuin--Kasteleyn representation of Ising and Potts models =-=[16, 1, 20, 23]-=-, the disagreement percolation technique for establishing Gibbsian uniqueness [5, 6], and the equivalence between spin percolation and Gibbs state multiplicity for Ising and Potts models on the square... |

154 |
Random walks and percolation on trees,
- Lyons
- 1990
(Show Context)
Citation Context ...o obtain a more explicit structural chararcterization, e.g. of graphs with p bond c ! 1. However, a general result of this kind appears to be fairly remote. For trees, p bond c = p site c , and Lyons =-=[36]-=- characterized the critical value in terms of a rather explicit quantity known as the branching number of the tree; in particular, p site c ! 1 if and only if the branching number is strictly greater ... |

139 |
Domination by product measures.
- Liggett, Schonmann, et al.
- 1997
(Show Context)
Citation Context ...mas: Lemma 6.1 G b BP ` G b SP Lemma 6.2 G b SP ` G b WR Lemma 6.3 G b SP ` G b BM Lemma 6.4 G b BM ` G b BP The proofs of Lemmas 6.1 and 6.4 use the following result of Liggett, Schonmann and Stacey =-=[34]-=-. For G = (V; E) 2 G, consider some probability measureson f0; 1g V , and a corresponding f0; 1g V -valued random element X. We say thatsis 1-dependent if X(A) and X(B) are independent for all finite ... |

124 | The random geometry of equilibrium phases,
- Georgii, Haggstrom, et al.
- 2001
(Show Context)
Citation Context ...on technique for establishing Gibbsian uniqueness [5, 6], and the equivalence between spin percolation and Gibbs state multiplicity for Ising and Potts models on the square lattice [15, 13]; see also =-=[18]-=- for a general introduction to such ideas. Here we shall focus on the two basic percolation models (bond percolation and site percolation) and on three different Markov random field models (the Ising ... |

81 | Percolation beyond Z d , many questions and a few answers
- BENJAMINI, O
- 1996
(Show Context)
Citation Context ...lue in terms of a rather explicit quantity known as the branching number of the tree; in particular, p site c ! 1 if and only if the branching number is strictly greater than 1. Benjamini and Schramm =-=[3]-=- conjectured that a Cayley graph of an infinite finitely generated group has p site c ! 1 unless it is a finite extension of Z. Theorem 1.1 stresses the importance of this conjecture. The rest of this... |

61 |
Nonuniqueness of measures of maximal entropy for subshifts of finite type.
- Burton, Steif
- 1994
(Show Context)
Citation Context ...graphs G 2 G with the property that there exists somes? 0 for which the Widom--Rowlinson model on G has more than one Gibbs measure. 2.6 Beach model The beach model was introduced by Burton and Steif =-=[9]-=- as an example of a so-called subshift of finite type which has more than one measure of maximal entropy despite having strong irreducibility properties. The following formulation is slightly differen... |

54 |
Nearest neighbor and hard sphere models in continuum percolation,
- Haggstrom, Meester
- 1996
(Show Context)
Citation Context ... it will be evident how to generalize it to higher dimensions. The key ingredient (besides Theorem 7.1) of the proof is a simple renormalization argument, similar to one used by Haggstrom and Meester =-=[24] in a-=- different context. Note first that by scaling, the probability in (22) is independent of the choice of , so we are free to chooses? 0 as we wish. Set " = 1\Gammap c (G 0 ) 3 , where G 0 is the s... |

54 |
The Ising model and percolation on trees and tree-like graphs
- Lyons
- 1989
(Show Context)
Citation Context ...on of [37, Chapter 3, Prop. 6]. We remark that it is only the inclusions T 2 GWR and T 2 GBM that are new results; the other inclusions T 2 GBP , T 2 G SP and T 2 G I are clear from the work of Lyons =-=[35, 36, 37]. One may ask whethe-=-r T 2 GBP " G SP " G I " GWR " GBM holds for any tree T with branching number greater than one (see any of [36, 40, 37] for the definition of branching number), but the answer is n... |

52 |
Remarks on the FKG inequalities,
- Holley
- 1974
(Show Context)
Citation Context ...h �� and j have positive ��-probability, we can move from �� to j through single-site flips without passing through any element of zero ��-probability. The following result, essentiall=-=y due to Holley [29]-=-, will play a key role in most of the rest of this paper. The proof is the same as Holley's original proof (which he gave under slightly stronger conditions); see e.g. [18]. Theorem 3.7 (Holley) Let X... |

52 |
A new model for the study of liquid-vapor phase transitions.
- Widom, Rowlinson
- 1970
(Show Context)
Citation Context ...ple Gibbs measures. The graph G supports phase transition for the Ising model if and only if fi c ! 1, and we define G I = fG 2 G : fi c (G) ! 1g : (2) 2.5 Widom--Rowlinson model The Widom--Rowlinson =-=[44]-=- model was originally introduced as a (two-type) point process in R d . The following discrete variant was soon thereafter studied in [43] and [32]. Each vertex of a graph G = (V; E) takes values in f... |

51 | Critical percolation on any nonamenable group has no infinite clusters - Benjamini, Lyons, et al. - 1999 |

39 |
A uniqueness condition for Gibbs measures, with application to the 2-dimensional Ising antiferromagnet,
- Berg
- 1993
(Show Context)
Citation Context ...he other. Examples of such connections are the Fortuin--Kasteleyn representation of Ising and Potts models [16, 1, 20, 23], the disagreement percolation technique for establishing Gibbsian uniqueness =-=[5, 6]-=-, and the equivalence between spin percolation and Gibbs state multiplicity for Ising and Potts models on the square lattice [15, 13]; see also [18] for a general introduction to such ideas. Here we s... |

39 | Percolation on transitive graphs as a coalescent process: relentless merging followed by simultaneous uniqueness.
- Haggstrom, Peres, et al.
- 1999
(Show Context)
Citation Context ...e Z d in ds2 dimensions; see [19] for an introduction to percolation theory with emphasis on the Z d case. Recently, there has been an upsurge of interest in percolation beyond this setting; see e.g. =-=[3, 2, 26]-=-, where the focus is mainly on Cayley graphs and other quasi-transitive graphs, which still have some structure that can be exploited in various ways. In this paper we basically drop all such structur... |

34 | Nonmonotonic behavior in hard-core and Widom-Rowlinson models.
- Brightwell, Haggstrom, et al.
- 1999
(Show Context)
Citation Context ...from the work of Aizenman et al. [1], although they stated their results only in a Z d setting. An example of a graph which is in G SP but not in GWR can be found in Brightwell, Haggstrom and Winkler =-=[8]-=-. It would of course be desirable to obtain a more explicit structural chararcterization, e.g. of graphs with p bond c ! 1. However, a general result of this kind appears to be fairly remote. For tree... |

31 | Conformal invariance of Voronoi percolation
- Benjamini, Schramm
- 1998
(Show Context)
Citation Context ...--Voronoi tessellations in R d with ds2. We restrict the discussion to a quick definition and our main result. An extensive discussion of Poisson--Voronoi tessellations can be found in [38]; see also =-=[4]-=- for a treatment of percolation on such tessellations. Let X 1 ; X 2 ; : : : be the points of a homogeneous Poisson process in R d with intensitys? 0. These points will be the nuclei in a Voronoi tess... |

30 |
Disagreement percolation in the study of Markov
- Berg, Maes
- 1994
(Show Context)
Citation Context ...he other. Examples of such connections are the Fortuin--Kasteleyn representation of Ising and Potts models [16, 1, 20, 23], the disagreement percolation technique for establishing Gibbsian uniqueness =-=[5, 6]-=-, and the equivalence between spin percolation and Gibbs state multiplicity for Ising and Potts models on the square lattice [15, 13]; see also [18] for a general introduction to such ideas. Here we s... |

30 |
The analysis of the Widom–Rowlinson model by stochastic geometric methods
- Chayes, Chayes, et al.
- 1995
(Show Context)
Citation Context ...s discovered independently by several different research groups and which arises as a random-cluster representation of the original continuum model of Widom and Rowlinson [44]; see e.g. Chayes et al. =-=[12]-=-. Let G = (V; E) and f n g 1 n=1 be as before. The wired site-random-cluster model OE n;p;q WR forsn with parameters p 2 [0; 1] and q ? 0 is defined as the probability measure on f0; 1g V which to eac... |

28 |
Percolation processes. Lower bounds for the critical probability,
- Hammersley
- 1957
(Show Context)
Citation Context ... then goes through as for the Widom--Rowlinson case, although it remains to show that we can pick p 0 as in (15), i.e. to show that p site c (G 00 ) ? 0. This, however, is immediate from Hammersley's =-=[27]-=- result that any graph whose degree is bounded by some \Delta has p site cs1 \Delta\Gamma1 ; we get p site cs1 11 . 2 Proof of Lemma 4.8: Let G = (V; E) be as in the proof of Lemma 4.5, with k i = log... |

26 | Amenability and phase transition in the Ising model
- Jonasson, Steif
- 1999
(Show Context)
Citation Context ... for h = 0 implies uniqueness for all h 6= 0, so that G I;h ` G I . This inclusion is strict, also for bounded degree graphs, as examplified e.g. by the usual Z d lattice with ds2. Jonasson and Steif =-=[30]-=- make interesting progress towards characterizing G I;h . Multitype Widom--Rowlinson model. Similarly to the Potts generalization of the Ising model, the Widom--Rowlinson model has been extended to a ... |

24 | Nearest-neighbor walks with low predictability profile and percolation in 2 + ǫ dimensions.
- Haggstrom, Mossel
- 1998
(Show Context)
Citation Context ...conditions, one may ask whether p site c ! 1 and transience of simple random walk together form a sufficient condition, but this is probably not the case; the graph considered in the final section of =-=[25]-=- is almost certainly a counterexample. Y. Peres has conjectured that a somewhat stronger property is necessary and sufficient, namely that there exists a p ! 1 such that bond percolation with paramete... |

23 |
New results on measures of maximal entropy.
- Burton, Steif
- 1995
(Show Context)
Citation Context ...the five other properties in Theorem 1.1. Multitype beach model. A q-type variant of the beach model, analogous to the Potts model and the q-type Widom--Rowlinson model, was introduced and studied in =-=[10]-=-. We get a random-cluster representation of the extended model by replacing 2 by q in the beach-random-cluster model. Using this, and following the arguments in Sections 3.4, 4 and 6, we get the follo... |

22 |
Random-cluster representations in the study of phase transitions
- Häggström
- 1998
(Show Context)
Citation Context ...on theory on one hand, and the issue of Gibbs state multiplicity in Markov random fields on the other. Examples of such connections are the Fortuin--Kasteleyn representation of Ising and Potts models =-=[16, 1, 20, 23]-=-, the disagreement percolation technique for establishing Gibbsian uniqueness [5, 6], and the equivalence between spin percolation and Gibbs state multiplicity for Ising and Potts models on the square... |

22 |
Phase transitions in binary lattice gases
- Lebowitz, Gallavotti
- 1971
(Show Context)
Citation Context ... (2) 2.5 Widom--Rowlinson model The Widom--Rowlinson [44] model was originally introduced as a (two-type) point process in R d . The following discrete variant was soon thereafter studied in [43] and =-=[32]-=-. Each vertex of a graph G = (V; E) takes values in f\Gamma1; 0; 1g, where \Gamma1 and 1 should be thought of as two types of particles with a mutual hard-core exclusion, and 0 as an empty location. F... |

20 |
Discontinuity of the magnetization in one-dimensional 1=jx \Gamma yj Ising and Potts models
- Aizenman, Chayes, et al.
- 1988
(Show Context)
Citation Context ...on theory on one hand, and the issue of Gibbs state multiplicity in Markov random fields on the other. Examples of such connections are the Fortuin--Kasteleyn representation of Ising and Potts models =-=[16, 1, 20, 23]-=-, the disagreement percolation technique for establishing Gibbsian uniqueness [5, 6], and the equivalence between spin percolation and Gibbs state multiplicity for Ising and Potts models on the square... |

15 |
Probability on Trees: An Introductory Climb, École d’Été de SaintFlour XXVII
- Peres
- 1999
(Show Context)
Citation Context ....4 and Proposition 2.3 as in the proof of Lemma 6.2. 2 7.1 Galton--Watson trees A tree is a graph with no cycles. A Galton--Watson tree is a random tree T = (V T ; E T ) obtained as follows (see e.g. =-=[40, 37]-=- for more extensive discussions). Let L be a nonnegative integer-valued random variable, and set p k = P(L = k) for each k. We call fp k g 1 k=0 the offspring distribution for T . The so-called root a... |

14 |
Percolation and phase transitions in the Ising model
- Coniglio, Nappi, et al.
- 1976
(Show Context)
Citation Context ...agreement percolation technique for establishing Gibbsian uniqueness [5, 6], and the equivalence between spin percolation and Gibbs state multiplicity for Ising and Potts models on the square lattice =-=[15, 13]-=-; see also [18] for a general introduction to such ideas. Here we shall focus on the two basic percolation models (bond percolation and site percolation) and on three different Markov random field mod... |

12 |
Lack of monotonicity in ferromagnetic Ising model phase diagrams
- SCHONMANN, TANAKA
- 1998
(Show Context)
Citation Context ... we can have that G is "larger" than another graph G 0 , yet G 2 G n GWR , G 0 2 GWR . The significance of graphs with this kind of "dead end decorations" was first discovered by S=-=chonmann and Tanaka [42]-=-. Proof of Lemma 4.7: Let G be as in the proof of Lemma 4.6; we need to show that for any M the beach model on G with parameter M has a unique Gibbs measure. The proof proceeds similarly as the proof ... |

11 |
Percolative problems, Probability and Phase Transition
- Grimmett
- 1994
(Show Context)
Citation Context |

10 | Staggered phases in diluted systems with continuous spins
- Chayes, Koteck´y, et al.
- 1997
(Show Context)
Citation Context ...-Rowlinson model on Z d in which nonuniqueness of Gibbs measures corresponds, not to a breakdown of the particle symmetry, but rather to a breakdown of an even-odd lattice symmetry; see e.g. [41] and =-=[14]-=-. This is similar to what happens in the hard-core model discussed below. Nevertheless, for bounded or unbounded degree graphs, the disagreement percolation condition of van den Berg [5] can be used t... |

10 |
On phase transitions for subshifts of finite type
- Häggström
- 1996
(Show Context)
Citation Context ... M ! M c we have that the beach model on G with parameter M has a unique Gibbs measure whereas for M ? M c there are multiple Gibbs measures. For G = Z d , this result was first obtained by Haggstrom =-=[21]-=-; in Section 3.4 we shall prove the full result using the random-cluster approach. We define GBM = fG 2 G : M c (G) ! 1g : In this language, the main result in [9] says that Z d 2 GBM for ds2. Alterna... |

7 |
Phase transitions and random walks on graphs: a generalization of the MerminWagner theorem to disordered lattices, fractals, and other discrete structures
- Cassi
- 1992
(Show Context)
Citation Context ...to be less common than e.g. in the Ising model. For bounded degree graphs, it is strongly believed that transience of simple random walk is necessary for phase transition in the rotor model; see e.g. =-=[11]-=-. Furthermore, still assuming bounded degree, the disagreement percolation technique of van den Berg and Maes [6] can be exploited to show that p site c ! 1 is another necessary condition. Regarding s... |

7 |
Comparison of atom and bond percolation,
- Hammersley
- 1961
(Show Context)
Citation Context ... prove nor disprove. Some of the implications in the above results are known from previous work. The inclusion G SP ` GBP is immediate from the relation p bond csp site c , which is due to Hammersley =-=[28]-=-. The equivalence GBP = G I is clear from the work of Aizenman et al. [1], although they stated their results only in a Z d setting. An example of a graph which is in G SP but not in GWR can be found ... |

7 |
Lectures on Random Voronoi Tessellations
- M��ller
- 1994
(Show Context)
Citation Context ...ated to Poisson--Voronoi tessellations in R d with ds2. We restrict the discussion to a quick definition and our main result. An extensive discussion of Poisson--Voronoi tessellations can be found in =-=[38]-=-; see also [4] for a treatment of percolation on such tessellations. Let X 1 ; X 2 ; : : : be the points of a homogeneous Poisson process in R d with intensitys? 0. These points will be the nuclei in ... |

7 |
Phase equilibrium and critical behavior in a two-component Bethe-lattice gas or three-component Bethe-lattice solution
- Wheeler, Widom
- 1970
(Show Context)
Citation Context ...G) ! 1g : (2) 2.5 Widom--Rowlinson model The Widom--Rowlinson [44] model was originally introduced as a (two-type) point process in R d . The following discrete variant was soon thereafter studied in =-=[43]-=- and [32]. Each vertex of a graph G = (V; E) takes values in f\Gamma1; 0; 1g, where \Gamma1 and 1 should be thought of as two types of particles with a mutual hard-core exclusion, and 0 as an empty lo... |

6 |
Percolation and ferromagnetism on Z : the q-state Potts cases
- Chayes
- 1996
(Show Context)
Citation Context ...agreement percolation technique for establishing Gibbsian uniqueness [5, 6], and the equivalence between spin percolation and Gibbs state multiplicity for Ising and Potts models on the square lattice =-=[15, 13]-=-; see also [18] for a general introduction to such ideas. Here we shall focus on the two basic percolation models (bond percolation and site percolation) and on three different Markov random field mod... |

4 |
Percolation and the hard core lattice gas model
- Berg, Steif
- 1994
(Show Context)
Citation Context ...f nonuniqueness of Gibbs measures in the hard-core model is the Z d lattice for ds2. In this case, the nonuniqueness manifests itself as a breaking of the odd-even symmetry of the lattice. Results in =-=[7]-=- and [22] suggest that the phase transtion phenomenon is rather non-robust under modifications of the graph structure, in the sense that even a relatively minor perturbation of this lattice symmetry w... |

2 |
Ergodicity of the hard core model on Z with parity-dependent activities, Ark
- Haggstrom
- 1997
(Show Context)
Citation Context ...queness of Gibbs measures in the hard-core model is the Z d lattice for ds2. In this case, the nonuniqueness manifests itself as a breaking of the odd-even symmetry of the lattice. Results in [7] and =-=[22]-=- suggest that the phase transtion phenomenon is rather non-robust under modifications of the graph structure, in the sense that even a relatively minor perturbation of this lattice symmetry will be en... |

1 | Robust phase transition for spherical and other models on general trees - Pemantle, Steif - 1998 |

1 |
Phase transitions of a multicomponent Widom
- Runnels, Lebowitz
- 1974
(Show Context)
Citation Context ...pe Widom--Rowlinson model on Z d in which nonuniqueness of Gibbs measures corresponds, not to a breakdown of the particle symmetry, but rather to a breakdown of an even-odd lattice symmetry; see e.g. =-=[41]-=- and [14]. This is similar to what happens in the hard-core model discussed below. Nevertheless, for bounded or unbounded degree graphs, the disagreement percolation condition of van den Berg [5] can ... |