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

We study the q‐state ferromagnetic Potts model on the n‐vertex complete graph known as the mean‐field (Curie‐Weiss) model. We analyze the Swendsen‐Wang algorithm which is a Markov chain that utilizes the random cluster representation for the ferromagnetic Potts model to recolor large sets of vertices in one step and potentially overcomes obstacles that inhibit single‐site Glauber dynamics. Long et al. studied the case q = 2, the Swendsen‐Wang algorithm for the mean‐field ferromagnetic Ising model, and showed that the mixing time satisfies: (i) Θ(1) for βnormalβc, where βc is the critical temperature for the ordered/disordered phase transition. In contrast, for q≥3 there are two critical temperatures 0

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“…This establishes non-negativity of G 3 (y) for y ≥ −s for all s > 0. This completes the proof of (14) and hence the proof of the lemma.…”

Section: Lemmasupporting

confidence: 62%

“…They provide an analogue of Theorem 1, though their analysis excludes the critical points B = B u and B = B rc . Very recently, Gheissari, Lubetzky, and Peres [14] improved the lower bound on the mixing time in the window B u < B < B rc to exp(Ω(n)), both for the Swendsen-Wang and the Chayes-Machta dynamics.…”

Section: Formentioning

confidence: 99%

Swendsen‐Wang algorithm on the mean‐field Potts model

Galanis

Štefankovič

Vigoda

2018

Random Struct Algorithms

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“…Proof of Theorem 1.5. It was established in [16] that for every q > 2 and every ℓ sufficiently large, there exists an interval (λ s (q), λ S (q)) such that if p ′ = λ ′ /ℓ with λ ′ ∈ (λ s (q), λ S (q)), then the spectral gap at parameters (p ′ , q) satisfies…”

Section: Slow Mixing Under Worst-case Boundary Conditionsmentioning

confidence: 99%

“…The high-level idea is that for sufficiently small p(m) the effect of the configuration away from the boundary is negligible, and so the mixing time of the FK-dynamics on G completely governs the mixing time of FK-dynamics near the boundary ∂Λ n . We can then use known torpid mixing results for the mean-field random-cluster model (the case where G is the complete graph) in its critical window at q > 2 [20,3,14,16].…”

Section: Introductionmentioning

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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“…The SW dynamics for the Ising model is quite appealing as it is conjectured to mix quickly at all temperatures. Its behavior for the Potts model (which corresponds to q > 2 spins) is more subtle, as there are multiple examples of classes of graphs where the SW dynamics is torpidly mixing; i.e., mixing time is exponential in the number of vertices of the graph; see, e.g., [19,14,2,17,3,4].…”

Section: Introductionmentioning

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

Cited by 10 publications

References 20 publications

Swendsen‐Wang algorithm on the mean‐field Potts model

Swendsen‐Wang algorithm on the mean‐field Potts model

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

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

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