Swendsen‐Wang algorithm on the mean‐field Potts model

Galanis, Andreas; Štefankovič, Daniel; Vigoda, Eric

doi:10.1002/rsa.20768

Cited by 28 publications

(35 citation statements)

References 28 publications

(73 reference statements)

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“…For β ∈ (β s , β S ), Swendsen-Wang was shown in [9] to slow down to t mix exp(c √ n) (extending the lower bound at β = β c due to Gore and Jerrum). Analogously, for the related Glauber dynamics for the mean-field FK model (see §2 for precise definitions) with q > 2, Blanca and Sinclair [1] proved that t mix ≥ exp(c √ n) whenever λ = np is in the critical window (λ s , λ S ).…”

Section: Introductionmentioning

confidence: 70%

“…An update of the dynamics, started from σ, first samples, independently for every i = 1, ..., q, a configuration G i ∼ G(|V i |, p) on the subgraph of V i , forming an FK configuration ω as Figure 1. The mixing times of mean-field Potts Glauber (green) and Swendsen-Wang (red) dynamics when q > 2 as β varies; the dashed line represents the previous lower bound [9,13] when β ∈ (β s , β S ) the union of the G i 's; then, it assigns an i.i.d. color X C ∼ Uni({1, ..., q}) to each cluster C in ω, and for every x ∈ C, sets σ x = X c in the new state σ of the Markov chain.…”

Section: Introductionmentioning

confidence: 99%

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Exponentially slow mixing in the mean-field Swendsen–Wang dynamics

Gheissari¹,

Lubetzky²,

Peres³

2020

Ann. Inst. H. Poincaré Probab. Statist.

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Swendsen-Wang dynamics for the Potts model was proposed in the late 1980's as an alternative to single-site heat-bath dynamics, in which global updates allow this MCMC sampler to switch between metastable states and ideally mix faster. Gore and Jerrum (1999) found that this dynamics may in fact exhibit slow mixing: they showed that, for the Potts model with q ≥ 3 colors on the complete graph on n vertices at the critical point βc(q), Swendsen-Wang dynamics has tmix ≥ exp(c √ n). The same lower bound was extended to the critical window (βs, βS) around βc by Galanis et al. (2015), as well as to the corresponding mean-field FK model by Blanca and Sinclair (2015). In both cases, an upper bound of tmix ≤ exp(c n) was known.Here we show that the mixing time is truly exponential in n: namely, tmix ≥ exp(cn) for Swendsen-Wang dynamics when q ≥ 3 and β ∈ (βs, βS), and the same bound holds for the related MCMC samplers for the mean-field FK model when q > 2. arXiv:1702.05797v2 [math.PR] 2 May 2017 1 It was shown in that work that tmix = O(log n) for β / ∈ [βs, βS), whereas tmix n 1/3 at β = βs.

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Section: Introductionmentioning

confidence: 70%

Section: Introductionmentioning

confidence: 99%

Exponentially slow mixing in the mean-field Swendsen–Wang dynamics

Gheissari¹,

Lubetzky²,

Peres³

2020

Ann. Inst. H. Poincaré Probab. Statist.

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“…Prior to this work, the study of the SwendsenWang algorithm and related cluster dynamics is focused on special graphs, such as the complete graph (the "mean-field" situation) or the two-dimensional lattice Z 2 . For complete graphs, the mixing time is very well understood for all q ≥ 1 [20,1,11]. For Z 2 , for all q ≥ 1, the dynamics is fast mixing at all temperatures other than the critical one [26,27,2].…”

Section: The Weight W(σ) Of Configuration σ Is β M(σ) Where M(σ) Is Tmentioning

confidence: 99%

Random cluster dynamics for the Ising model is rapidly mixing

Guo¹,

Jerrum²

2017

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

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“…or that every pair of vertices (x, l), (y, l) ∈ ∂ R connected in ξ satisfy (14). Consequently, all other connections through ξ between vertices (…”

Section: Defining the Blocks For The Block Dynamicsmentioning

confidence: 99%

Random-cluster dynamics in $\mathbb{Z}^{2}$: Rapid mixing with general boundary conditions

Blanca¹,

Gheissari²,

Vigoda³

2020

Ann. Appl. Probab.

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The random-cluster model with parameters (p, q) is a random graph model that generalizes bond percolation (q = 1) and the Ising and Potts models (q ≥ 2). We study its Glauber dynamics on n × n boxes Λ n of the integer lattice graph Z 2 , where the model exhibits a sharp phase transition at p = p c (q). Unlike traditional spin systems like the Ising and Potts models, the random-cluster model has non-local interactions. Long-range interactions can be imposed as external connections in the boundary of Λ n , known as boundary conditions. For select boundary conditions that do not carry long-range information (namely, wired and free), Blanca and Sinclair proved that when q > 1 and p = p c (q), the Glauber dynamics on Λ n mixes in optimal O(n 2 log n) time. In this paper, we prove that this mixing time is polynomial in n for every boundary condition that is realizable as a configuration on Z 2 \ Λ n . We then use this to prove near-optimalÕ(n 2 ) mixing time for "typical" boundary conditions. As a complementary result, we construct classes of non-realizable (non-planar) boundary conditions inducing slow (stretched-exponential) mixing at p ≪ p c (q).We say e is a cut-edge in (Λ n , S t ) if the number of connected components in S t ∪ {e} and S t \ {e} differ. Under a boundary condition ξ, the property of e being a cut-edge is defined with respect to the augmented graph (Λ n , S ξ t ), where S ξ t adds external wirings between all pairs of vertices in the same element of ξ. This Markov chain converges to π Λn,p,q by construction, and we study its speed of convergence.A standard measure for quantifying the speed of convergence of a Markov chain is the mixing time, which is defined as the time until the dynamics is close (in total variation distance) to its stationary distribution, starting from a worst-case initial state. We say the dynamics is rapidly mixing if the mixing time is polynomial in |V |, and torpidly mixing when the mixing time is exponential in |V | ε for some ε > 0.The corresponding Glauber dynamics for the Ising/Potts model (which updates spins one at a time according to the spins of their neighbors), is by now quite well understood on finite regions of Z 2 . In the high-temperature region β < β c (corresponding to p < p c ) the Glauber dynamics has optimal mixing time Θ(n 2 log n) on boxes Λ n [28, 7, 2, 1]; moreover, this same asymptotic bound of Θ(n 2 log n) holds for every fixed boundary condition. These bounds follow as a consequence of the exponential decay of correlations of the model in the high-temperature regime, which holds even near the boundary for arbitrary Ising/Potts boundary conditions; this property is known as strong spatial mixing. In the low-temperature region β > β c the mixing time is exponential in n for free and periodic (toroidal) boundaries [35,6,17]. The more general problem of understanding the mixing time of the Glauber dynamics for other boundary conditions at low temperatures is a long-standing open problem, e.g., see [25,29].The FK-dynamics is quite powerful since the self-...

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Swendsen‐Wang algorithm on the mean‐field Potts model

Cited by 28 publications

References 28 publications

Exponentially slow mixing in the mean-field Swendsen–Wang dynamics

Exponentially slow mixing in the mean-field Swendsen–Wang dynamics

Random cluster dynamics for the Ising model is rapidly mixing

Random-cluster dynamics in $\mathbb{Z}^{2}$: Rapid mixing with general boundary conditions

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