A simple condition implying rapid mixing of single-site dynamics on spin systems

Hayes, Thomas P.

doi:10.1109/focs.2006.6

Cited by 60 publications

(108 citation statements)

References 15 publications

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“…There is some overlap between Theorem 1.1 and a result of Hayes [19,Proposition 14] for q = 2, which was generalised to arbitrary q by Ullrich [31, Corollary 2.14]. Ullrich showed that when the inverse temperature β satisfies β ≤ 2c/ for some 0 < c < 1, then the Glauber dynamics is rapidly mixing on graphs of maximum degree .…”

Section: Comparison With Related Results and Phase Transitionsmentioning

confidence: 95%

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Mixing of the Glauber dynamics for the ferromagnetic Potts model

Bordewich

Greenhill

Patel

2014

Random Struct. Alg.

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ABSTRACT:We present several results on the mixing time of the Glauber dynamics for sampling from the Gibbs distribution in the ferromagnetic Potts model. At a fixed temperature and interaction strength, we study the interplay between the maximum degree ( ) of the underlying graph and the number of colours or spins (q) in determining whether the dynamics mixes rapidly or not. We find a lower bound L on the number of colours such that Glauber dynamics is rapidly mixing if at least L colours are used. We give a closely-matching upper bound U on the number of colours such that with probability that tends to 1, the Glauber dynamics mixes slowly on random -regular graphs when at most U colours are used. We show that our bounds can be improved if we restrict attention to certain types of graphs of maximum degree , e.g. toroidal grids for = 4.

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Section: Comparison With Related Results and Phase Transitionsmentioning

confidence: 95%

“…Fix ≥ 3 and let κ ∈ 1, 2 . Suppose that β is defined by (19) and let q ≥ 2 be an integer which satisfies (16). Let If B ≤ 0 then for all x ∈ (0, 1), the second factor on the right hand side is positive and the first factor is negative, establishing (ii).…”

Section: Then the Conductance Of The Markov Chainmentioning

confidence: 92%

Mixing of the Glauber dynamics for the ferromagnetic Potts model

Bordewich

Greenhill

Patel

2014

Random Struct. Alg.

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“…It is well known that the Glauber dynamics achieves the optimal mixing rate τ (ǫ) = O n log n ǫ under the Dobrushin's condition for the decay of correlation [15,31]. By a standard coupling argument, we show that the LubyGlauber algorithm achieves a mixing rate τ (ǫ) = O ∆ log n ǫ under the same condition, where ∆ is the maximum degree of the network.…”

Section: Our Resultsmentioning

confidence: 76%

“…The chromatic-scheduler-based parallelization of Glauber dynamics was studied in [28]. This parallel chain is in fact a special case of systematic scan for Glauber dynamics [17,18,31], in which the variables are updated according to a fixed order. Empirical studies showed that sometimes an ad hoc "Hogwild!"…”

Section: Related Workmentioning

confidence: 99%

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What Can be Sampled Locally?

Feng

Sun

Yin

2017

Proceedings of the ACM Symposium on Principles of Distributed Computing

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The local computation of Linial [FOCS'87] and Naor and Stockmeyer [STOC'93] concerns with the question of whether a locally definable distributed computing problem can be solved locally: more specifically, for a given local CSP whether a CSP solution can be constructed by a distributed algorithm using local information. In this paper, we consider the problem of sampling a uniform CSP solution by distributed algorithms, and ask whether a locally definable joint distribution can be sampled from locally. More broadly, we consider sampling from Gibbs distributions induced by weighted local CSPs, especially the Markov random fields (MRFs), in the LOCAL model.We give two Markov chain based distributed algorithms which we believe to represent two fundamental approaches for sampling from Gibbs distributions via distributed algorithms. The first algorithm generically parallelizes the single-site sequential Markov chain by iteratively updating a random independent set of variables in parallel, and achieves an O(∆ log n) time upper bound in the LOCAL model, where ∆ is the maximum degree, when the Dobrushin's condition for the Gibbs distribution is satisfied. The second algorithm is a novel parallel Markov chain which proposes to update all variables simultaneously yet still guarantees to converge correctly with no bias. It surprisingly parallelizes an intrinsically sequential process: stabilizing to a joint distribution with massive local dependencies, and may achieve an optimal O(log n) time upper bound independent of the maximum degree ∆ under a stronger mixing condition.We also show a strong Ω(diam) lower bound for sampling: in particular for sampling independent set in graphs with maximum degree ∆ ≥ 6. Independent sets are trivial to construct locally and the sampling lower bound holds even when every node is aware of the entire graph. This gives a strong separation between sampling and constructing locally checkable labelings.

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A Note on Degenerate and Spectrally Degenerate Graphs

Alon

2012

Journal of Graph Theory

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A simple condition implying rapid mixing of single-site dynamics on spin systems

Cited by 60 publications

References 15 publications

Mixing of the Glauber dynamics for the ferromagnetic Potts model

Mixing of the Glauber dynamics for the ferromagnetic Potts model

What Can be Sampled Locally?

A Note on Degenerate and Spectrally Degenerate Graphs

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