Uniqueness of the infinite component in a random graph with applications to percolation and spin glasses

Gandolfi, Alberto; Keane, Michaël; Newman, Charles M.

doi:10.1007/bf01274266

Cited by 97 publications

(86 citation statements)

References 26 publications

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“…This is a direct consequence of Theorem 1 from [13], once we show that the limiting measure satisfies the positive finite energy condition. Using a slightly stronger form of the condition than that in [13], we say that a GRC measure µ has positive finite energy if for all bonds b ∈ B(Z d ), we have…”

Section: Uniqueness Of the Infinite Clustermentioning

confidence: 99%

“…In order to prove (3.15), we have to work a bit harder. Let (∆ n ) n≥1 be an increasing sequence of boxes centered at the origin and let 13) and…”

Section: Lemma 52mentioning

confidence: 99%

“…Proof of Theorem 3.3. Since the positive finite energy condition has been established in both relevant cases, the result follows immediately from Theorem 1 in [13]. In order to prove the corollary to Theorem 3.3, we will prove a lemma that states that the specifications µ GRC B are "almost surely quasilocal" in the language of [16,21].…”

Section: Lemma 61 Let µ Be a Translation Invariant Grc Measure That mentioning

confidence: 99%

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Gibbs states of graphical representations of the Potts model with external fields

Biskup

Borgs

Chayes

et al. 2000

Journal of Mathematical Physics

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Abstract. We consider the ferromagnetic q-state Potts model, with each of the q spin values coupled to an external field. We also introduce a generalized random cluster model, which includes both the Potts model in arbitrary homogeneous external fields and the non-integer q random cluster model as special cases. We establish the FKG property, the finite energy condition, uniqueness of the infinite cluster, and Gibbsianness of limit states for this generalized model. Furthermore, we develop the theory of Gibbs states for the Edwards-Sokal representation of the Potts model in a field, and relate the phase structure in this representation to those in the spin and random cluster representations. Finally, we characterize the possible color(s) of the infinite cluster(s) and show that the correspondence between Edwards-Sokal Gibbs states and their random cluster marginals is bijective, once the color of the infinite cluster is fixed.

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Section: Uniqueness Of the Infinite Clustermentioning

confidence: 99%

“…In order to prove (3.15), we have to work a bit harder. Let (∆ n ) n≥1 be an increasing sequence of boxes centered at the origin and let 13) and…”

Section: Lemma 52mentioning

confidence: 99%

Section: Lemma 61 Let µ Be a Translation Invariant Grc Measure That mentioning

confidence: 99%

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Gibbs states of graphical representations of the Potts model with external fields

Biskup

Borgs

Chayes

et al. 2000

Journal of Mathematical Physics

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“…Just as in the ferromagnet, if percolation of a random cluster occurs in the FK representation, the percolating cluster is unique (in each realization of FK spins and bonds) [25,30]. In spite of this, RC percolation alone is no longer sufficient to prove broken spin-flip symmetry in the EA spin glass.…”

Section: B Spin Glass Models; the Trfk Representationmentioning

confidence: 99%

A Percolation-theoretic Approach to Spin Glass Phase Transitions

Machta

Newman

Stein

2009

Progress in Probability

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The magnetically ordered, low temperature phase of Ising ferro-magnets is manifested within the associated Fortuin-Kasteleyn (FK) random cluster representation by the occurrence of a single positive density percolating cluster. In this paper, we review our recent work on the percolation signature for Ising spin glass ordering -both in the short-range Edwards-Anderson (EA) and infinite-range Sherrington-Kirkpatrick (SK) models -within a two-replica FK representation and also in the different Chayes-Machta-Redner two-replica graphical representation. Numerical studies of the ±J EA model in dimension three and rigorous results for the SK model are consistent in supporting the conclusion that the signature of spin-glass order in these models is the existence of a single percolating cluster of maximal density normally coexisting with a second percolating cluster of lower density.

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“…The connected components of this graph are called clusters. If is amenable, then Burton and Keane [5] and Gandolfi et al [8] proved that there is at most one infinite cluster almost surely. A major open conjecture of Benjamini and Schramm [3] is the converse, that if is non-amenable, then, for some interval of p, there are infinitely many infinite clusters almost surely.…”

Section: Introductionmentioning

confidence: 99%