On the purity of the limiting gibbs state for the Ising model on the Bethe lattice

Bleher, Pavel; Ruiz, Jean; Zagrebnov, Valentin A.

doi:10.1007/bf02179399

Cited by 216 publications

(211 citation statements)

References 7 publications

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“…See [12,13,32]. The reconstruction problem for the CFN model was analyzed in [1][2][3]8,14,27]. In particular, the role of the reconstruction problem in the analysis of the mixing time of Glauber dynamics on trees was established in [2,27].…”

Section: Introductionmentioning

confidence: 99%

Evolutionary trees and the Ising model on the Bethe lattice: a proof of Steel’s conjecture

Daskalakis¹,

Mossel

Roch³

2009

Probab. Theory Relat. Fields

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A major task of evolutionary biology is the reconstruction of phylogenetic trees from molecular data. The evolutionary model is given by a Markov chain on a tree. Given samples from the leaves of the Markov chain, the goal is to reconstruct the leaf-labelled tree. It is well known that in order to reconstruct a tree on n leaves, sample sequences of length (log n) are needed. It was conjectured by Steel that for the CFN/Ising evolutionary model, if the mutation probability on all edges of the tree is less than p * = ( √ 2 − 1)/2 3/2 , then the tree can be recovered from sequences of length O(log n). The value p * is given by the transition point for the extremality of the free Gibbs measure for the Ising model on the binary tree. Steel's conjecture was proven by the second author in the special case where the tree is "balanced." The second author also proved that if all edges have mutation probability larger than p * then the length needed is n (1) . Here we show that Steel's conjecture holds true for general trees by giving a reconstruction algorithm that recovers the tree from O(log n)-length sequences when the mutation probabilities are discretized and less than p * . Our proof and results demonstrate that extremality of the free Gibbs measure on the infinite binary tree, which has been studied before in probability, statistical physics and computer science, determines how distinguishable are Gibbs measures on finite binary trees.

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Section: Introductionmentioning

confidence: 99%

Evolutionary trees and the Ising model on the Bethe lattice: a proof of Steel’s conjecture

Daskalakis¹,

Mossel

Roch³

2009

Probab. Theory Relat. Fields

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“…In particular, as → ∞, for β β 1 the Gibbs measure on T with the above random boundary η, converges (weakly) a.s. to the free measure µ free . Another way to look at β 1 is to say that µ free is an extremal Gibbs measure iff β β 1 (see [4,15,16,2] and, more recently, [24]). Finally β 1 has also the interpretation of the non-reconstruction/reconstruction threshold in the context of "bit reconstruction problems" on a noisy symmetric channel [10,27,26].…”

Section: Introductionmentioning

confidence: 99%

Phase ordering after a deep quench: the stochastic Ising and hard core gas models on a tree

Caputo

Martinelli

2005

Probab. Theory Relat. Fields

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ABSTRACT. Consider a low temperature stochastic Ising model in the phase coexistence regime with Markov semigroup Pt. A fundamental and still largely open problem is the understanding of the long time behavior of δηPt when the initial configuration η is sampled from a highly disordered state ν (e.g. a product Bernoulli measure or a high temperature Gibbs measure). Exploiting recent progresses in the analysis of the mixing time of Monte Carlo Markov chains for discrete spin models on a regular b-ary tree T b , we study the above problem for the Ising and hard core gas (independent sets) models on T b . If ν is a biased product Bernoulli law then, under various assumptions on the bias and on the thermodynamic parameters, we prove ν-almost sure weak convergence of δηPt to an extremal Gibbs measure (pure phase) and show that the limit is approached at least as fast as a stretched exponential of the time t. In the context of randomized algorithms and if one considers the Glauber dynamics on a large, finite tree, our results prove fast local relaxation to equilibrium on time scales much smaller than the true mixing time, provided that the starting point of the chain is not taken as the worst one but it is rather sampled from a suitable distribution.

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“…The first of these, ¬ ¼ ½ ¾ ÐÒ´ ·½ ½ µ, marks the dividing line between uniqueness and non-uniqueness of the Gibbs measure: i.e., the "high temperature" region, in which the Gibbs measure is unique, is defined by ¬ ¬ ¼ [37]. However, in contrast to the model on , there is now a second critical point ¬ ½ ½ ¾ ÐÒ´Ô ·½ Ô ½ µ [7,19], which delimits the region where "typical" boundary conditions exert long-range influence on the root. I.e., there is now an "intermediate" region ¬ ¼ ¬ ¬ ½ in which the´·µ-and´ µ-boundaries exert long-range influence but typical boundaries do not, while in the "low temperature" region ¬ ¬ ½ long-range influence occurs even for typical boundaries.…”

Section: The Ising Model On Treesmentioning

confidence: 99%

“…In the special case of a free boundary and ¼ , part´ µ of Theorem 4.1 was first proved in [7] via a lengthy calculation, which was considerably simplified in [19]. It was later reproved in [4] (for arbitrary boundary conditions) as a consequence of the fact that the spectral gap is bounded in this situation.…”

Section: Verifying Spatial Mixing For the Spectral Gapmentioning

confidence: 99%

“…We stress that, while the tree is simpler in some respects than due to the lack of cycles, in other respects it is more complex: e.g., it exhibits a "double phase transition" (see below). Moreover, the Ising model on trees has recently received a lot of attention as the canonical example of a statistical physics model on a "non-amenable" graph (i.e., one whose boundary is of comparable size to its volume) -see, e.g., [4,6,7,12,19,24,27,39]. In the next subsection, we briefly describe the Ising model on trees before stating our results in more detail.…”

Section: Introductionmentioning

confidence: 99%

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The Ising model on trees: boundary conditions and mixing time

Martinelli

Sinclair²,

Weitz³

44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings.

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We give the first comprehensive analysis of the effect of boundary conditions on the mixing time of the Glauber dynamics for the Ising model. Specifically, we show that the mixing time on an Ò-vertex regular tree with´·µ-boundary remains Ç´Ò ÐÓ Òµ at all temperatures (in contrast to the free boundary case, where the mixing time is not bounded by any fixed polynomial at low temperatures). We also show that this bound continues to hold in the presence of an arbitrary external field. Our results are actually stronger, and provide tight bounds on the log-Sobolev constant and the spectral gap of the dynamics. In addition, our methods yield simpler proofs and stronger results for the mixing time in the regime where it is insensitive to the boundary condition. Our techniques also apply to a much wider class of models, including those with hard constraints like the antiferromagnetic Potts model at zero temperature (colorings) and the hard-core model (independent sets).

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On the purity of the limiting gibbs state for the Ising model on the Bethe lattice

Cited by 216 publications

References 7 publications

Evolutionary trees and the Ising model on the Bethe lattice: a proof of Steel’s conjecture

Evolutionary trees and the Ising model on the Bethe lattice: a proof of Steel’s conjecture

Phase ordering after a deep quench: the stochastic Ising and hard core gas models on a tree

The Ising model on trees: boundary conditions and mixing time

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