Swendsen-Wang dynamics for general graphsin the tree uniqueness region

Antonio Blanca, Zongchen Chen, Eric Vigoda

Research output: Contribution to journalArticle

Abstract

The Swendsen-Wang (SW) dynamics is a popular Markov chain for sampling from the Gibbs distribution for the ferromagnetic Ising model on a graph G = (V,E). The dynamics is conjectured to converge to equilibrium in O(|V| 1/4 ) steps at any (inverse) temperature β, yet there are few results providing o(|V|) upper bounds. We prove fast convergence of the SW dynamics on general graphs in the tree uniqueness region. In particular, when β < β c (d) where β c (d) denotes the uniqueness/nonuniqueness threshold on infinite d-regular trees, we prove that the relaxation time (ie, the inverse spectral gap) of the SW dynamics is Θ(1) on any graph of maximum degree d ≥ 3. Our proof utilizes a monotone version of the SW dynamics which only updates isolated vertices. We establish that this variant of the SW 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). Our proof technology can be applied to general monotone Markov chains, including for example the heat-bath block dynamics, for which we obtain new tight mixing time bounds.

Original languageEnglish (US)
JournalRandom Structures and Algorithms
DOIs
StatePublished - Jan 1 2019

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Uniqueness
Mixing Time
Graph in graph theory
Maximum Degree
Relaxation Time
Relaxation time
Markov processes
Markov chain
Monotone
Gibbs Distribution
Ising model
Heat Bath
Spectral Gap
Nonuniqueness
Ising Model
Update
Sampling
Upper bound
Denote
Converge

All Science Journal Classification (ASJC) codes

  • Software
  • Mathematics(all)
  • Computer Graphics and Computer-Aided Design
  • Applied Mathematics

Cite this

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abstract = "The Swendsen-Wang (SW) dynamics is a popular Markov chain for sampling from the Gibbs distribution for the ferromagnetic Ising model on a graph G = (V,E). The dynamics is conjectured to converge to equilibrium in O(|V| 1/4 ) steps at any (inverse) temperature β, yet there are few results providing o(|V|) upper bounds. We prove fast convergence of the SW dynamics on general graphs in the tree uniqueness region. In particular, when β < β c (d) where β c (d) denotes the uniqueness/nonuniqueness threshold on infinite d-regular trees, we prove that the relaxation time (ie, the inverse spectral gap) of the SW dynamics is Θ(1) on any graph of maximum degree d ≥ 3. Our proof utilizes a monotone version of the SW dynamics which only updates isolated vertices. We establish that this variant of the SW 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). Our proof technology can be applied to general monotone Markov chains, including for example the heat-bath block dynamics, for which we obtain new tight mixing time bounds.",
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Swendsen-Wang dynamics for general graphsin the tree uniqueness region. / Blanca, Antonio; Chen, Zongchen; Vigoda, Eric.

In: Random Structures and Algorithms, 01.01.2019.

Research output: Contribution to journalArticle

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