Mixing Time of Critical Ising Model on Trees is Polynomial in the Height

Ding, Jian; Lubetzky, Eyal; Peres, Yuval

doi:10.1007/s00220-009-0978-y

Cited by 22 publications

(25 citation statements)

References 35 publications

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“…A new work of Ding et al [7] gives very sharp bounds on the mixing time of the Glauber dynamics for the Ising model on the complete tree, and illustrates it undergoes a phase transition at the reconstruction threshold. For the case of colorings, recently Hayes et al [14] proved polynomial mixing time of the Glauber dynamics for any planar graph with maximum degree b when k > 100b/ ln b.…”

Section: Introductionmentioning

confidence: 99%

Phase Transition for the Mixing Time of the Glauber Dynamics for Coloring Regular Trees

Tetali¹,

Vera²,

Vigoda³

et al. 2010

Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms

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We prove that the mixing time of the Glauber dynamics for random k-colorings of the complete tree with branching factor b undergoes a phase transition at k = b(1+o b (1))/ ln b. Our main result shows nearly sharp bounds on the mixing time of the dynamics on the complete tree with n vertices for k = Cb/ ln b colors with constant C. For C ≥ 1 we prove the mixing time is O(n 1+o b (1) ln 2 n). On the other side, for C < 1 the mixing time experiences a slowing down, in particular, we prove it is O(n 1/C+o b (1) ln 2 n) andThe critical point C = 1 is interesting since it coincides (at least up to first order) to the so-called reconstruction threshold which was recently established by Sly. The reconstruction threshold has been of considerable interest recently since it appears to have close connections to the efficiency of certain local algorithms, and this work was inspired by our attempt to understand these connections in this particular setting.

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Section: Introductionmentioning

confidence: 99%

Phase Transition for the Mixing Time of the Glauber Dynamics for Coloring Regular Trees

Tetali¹,

Vera²,

Vigoda³

et al. 2010

Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms

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“…The upper bound T 1 (T) T 2 (T) cLT rel (T) was proved in [16,Corollary 1]]. It remains to prove the lower bound and this is what we do now following an idea of [6].…”

Section: Mixing Times: Proof Of Theoremmentioning

confidence: 75%

Mixing time of a kinetically constrained spin model on trees: power law scaling at criticality

Cancrini

Martinelli

Roberto

et al. 2014

Probab. Theory Relat. Fields

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ABSTRACT. On the rooted k-ary tree we consider a 0-1 kinetically constrained spin model in which the occupancy variable at each node is re-sampled with rate one from the Bernoulli(p) measure iff all its children are empty. For this process the following picture was conjectured to hold. As long as p is below the percolation threshold pc = 1/k the process is ergodic with a finite relaxation time while, for p > pc, the process on the infinite tree is no longer ergodic and the relaxation time on a finite regular sub-tree becomes exponentially large in the depth of the tree. At the critical point p = pc the process on the infinite tree is still ergodic but with an infinite relaxation time. Moreover, on finite sub-trees, the relaxation time grows polynomially in the depth of the tree.The conjecture was recently proved by the second and forth author except at criticality. Here we analyse the critical and quasi-critical case and prove for the relevant time scales: (i) power law behaviour in the depth of the tree at p = pc and (ii) power law scaling in (pc − p) −1 when p approaches pc from below. Our results, which are very close to those obtained recently for the Ising model at the spin glass critical point, represent the first rigorous analysis of a kinetically constrained model at criticality.

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“…The second key property of the Isolated-vertex dynamics concerns whether moves (or partial moves) of the dynamics could be censored from the evolution of the chain without possibly speeding up its convergence. Censoring of Markov chains is a well-studied notion [40,13,23] that has found important applications [38,7,8].…”

Section: Isolated-vertex Dynamicsmentioning

confidence: 99%

Swendsen‐Wang dynamics for general graphs in the tree uniqueness region

Blanca

Chen

Vigoda

2019

Random Struct Algorithms

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The Swendsen-Wang dynamics is a popular algorithm for sampling from the Gibbs distribution for the ferromagnetic Ising model on a graph G = (V, E). The dynamics is a "global" Markov chain which is conjectured to converge to equilibrium in O(|V | 1/4 ) steps for any graph G at any (inverse) temperature β. It was recently proved by Guo and Jerrum (2017) that the Swendsen-Wang dynamics has polynomial mixing time on any graph at all temperatures, yet there are few results providing o(|V |) upper bounds on its convergence time.We prove fast convergence of the Swendsen-Wang dynamics on general graphs in the tree uniqueness region of the ferromagnetic Ising model. In particular, when β < β c (d) where β c (d) denotes the uniqueness/non-uniqueness threshold on infinite d-regular trees, we prove that the relaxation time (i.e., the inverse spectral gap) of the Swendsen-Wang dynamics is Θ(1) on any graph of maximum degree d ≥ 3. Our proof utilizes a version of the Swendsen-Wang dynamics which only updates isolated vertices. We establish that this variant of the Swendsen-Wang dynamics has mixing time O(log |V |) and relaxation time Θ(1) on any graph of maximum degree d for all β < β c (d). We believe that this Markov chain may be of independent interest, as it is a monotone Swendsen-Wang type chain. As part of our proofs, we provide modest extensions of the technology of Mossel and Sly (2013) for analyzing mixing times and of the censoring result of Peres and Winkler (2013). Both of these results are for the Glauber dynamics, and we extend them here to general monotone Markov chains. This class of dynamics includes for example the heat-bath block dynamics, for which we obtain new tight mixing time bounds.Thus, it is sufficient for us to show that Kf 1 , Kf 2 νm ≤ f 1 ,f 2 νm .Consider the partial order on Ω m J where (F 1 , σ, C 1 ) ≥ (F 2 , τ, C 2 ) if and only if F 1 = F 2 , C 1 = C 2 and σ ≥ τ . The following property of the matrix K A S * T * will be useful.Claim 23. Suppose f : R |Ω| → R is an increasing positive function. Then,f : R |Ω m J | → R wherê f = K A S * T * f is also increasing and positive.Given ω ∈ Ω m J , let ρ ω = K(ω, ·) be the distribution over Ω m J that results after applying K (i.e., assigning uniform random spins to all marked connected components) from ω. We getGiven ω, the distribution ρ ω is a product distribution over the spin assignments of all the marked connected components in ω. Therefore, ρ ω is positive correlated for any ω ∈ Ω m J by Harris inequality (see, e.g., Lemma 22.14 in [31]). Sincef 1 and f 2 are increasing by Claim 23, we deduce that for any ω ∈ Ω mHence,

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Mixing Time of Critical Ising Model on Trees is Polynomial in the Height

Cited by 22 publications

References 35 publications

Phase Transition for the Mixing Time of the Glauber Dynamics for Coloring Regular Trees

Phase Transition for the Mixing Time of the Glauber Dynamics for Coloring Regular Trees

Mixing time of a kinetically constrained spin model on trees: power law scaling at criticality

Swendsen‐Wang dynamics for general graphs in the tree uniqueness region

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