On the Mixing Time of the 2D Stochastic Ising Model with “Plus” Boundary Conditions at Low Temperature

Martinelli, Fabio; Toninelli, Fabio Lucio

doi:10.1007/s00220-009-0963-5

Cited by 30 publications

(83 citation statements)

References 24 publications

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“…Recall that, for G 2 (s), M(B F (s)) is believed to be rapidly mixing at all temperatures (as it corresponds to M GD ), and its mixing time is known to be at most O(s log s ) for small λ [16]. In contrast, Theorems 3.1 and 3.3 imply that for small λ, the mixing time of M(B) is 2 Ω(s) for every cycle basis B of G p 2 (s).…”

Section: Our Resultsmentioning

confidence: 99%

“…The behavior of the classical single-site update Markov chain M GD , known as Glauber dynamics, has received intense study from the physics and computer science communities. It is believed to be rapidly mixing at all temperatures on the 2-dimensional square grid with a fixed plus boundary condition [16], but is known to require exponential time to mix at low temperatures if the lattice has free or periodic boundary conditions [21,22]. This critical slowing down is closely related to the existence of multiple Gibbs states-where spin configurations are dominated by either plus spins or minus spins-since the local nature of M GD makes it difficult to move between the two regimes.…”

Section: Introductionmentioning

confidence: 99%

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Cycle Basis Markov Chains for the Ising Model

Streib

2017

2017 Proceedings of the Fourteenth Workshop on Analytic Algorithmics and Combinatorics (ANALCO)

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A very challenging problem from statistical physics is computing the partition function of the ferromagnetic Ising model, even in the relatively simple case of no applied field. In this case, the partition function can be written as a function of the subgraphs of the underlying graph in which all vertices have even degree. In their seminal work, Jerrum and Sinclair showed that this quantity can be approximated by a rapidly converging Markov chain on all subgraphs. However, their chain frequently leaves the space of even subgraphs. Our aim is to devise and analyze a new class of Markov chains that do not leave this space, in the hopes of finding a faster sampling algorithm.We define Markov chains by viewing the even subgraphs as a vector space (often called the cycle space) whose transitions are defined by the addition of basis elements. The rate of convergence depends on the basis chosen, and our analysis proceeds by dividing bases into two types, short and long. The classical singlesite update Markov chain known as Glauber dynamics is a special case of our short cycle basis Markov chains. We show that for any graph and any long basis, there is a temperature for which the corresponding Markov chain requires an exponential time to mix. Moreover, we show that for d-dimensional grids with d ≥ 2-those of the most physical importance-all fundamental bases (a natural class of bases derived from spanning trees) are long. For the 2-dimensional grid on the torus, we show that there is a temperature for which the Markov chain requires exponential time for any chosen basis. * Official contribution of the National Institute of Standards and Technology; not subject to copyright in the United States.† ampasco@super.org ‡ nsstrei@super.org

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Section: Our Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Cycle Basis Markov Chains for the Ising Model

Streib

2017

2017 Proceedings of the Fourteenth Workshop on Analytic Algorithmics and Combinatorics (ANALCO)

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“…By Claims 4.4 and 4.5, the first term in (15) is bounded by M (2e −c log 4.5 n + e −c log 9 n ), which is certainly o(1/n 4 ), while the second term is o(1/n 4 ) by Theorem 3.4 and the fact that M n 2 log n (the mixing time of M par n ). Hence the variation distance of the dynamics is o(1/n 4 ), as required in (11).…”

Section: The Single-site Dynamicsmentioning

confidence: 87%

“…More concretely, we conjecture that the proof techniques we develop in this paper may be useful in analyzing the Ising model. (For recent progress in this direction, see [15]. )…”

Section: Introductionmentioning

confidence: 99%

Mixing time for the solid-on-solid model

Martinelli

Sinclair

2009

Proceedings of the Forty-First Annual ACM Symposium on Theory of Computing

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We analyze the mixing time of a natural local Markov chain (the Glauber dynamics) on configurations of the solid-onsolid model of statistical physics. This model has been proposed, among other things, as an idealization of the behavior of contours in the Ising model at low temperatures. Our main result is an upper bound on the mixing time ofÕ(n 3.5 ), which is tight within a factor ofÕ( √ n). The proof, which in addition gives insight into the actual evolution of the contours, requires the introduction of several novel analytical techniques that we conjecture will have other applications.

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“…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-duality of the model on Z 2 implies that it is rapidly mixing in the low-temperature regime where the Ising/Potts Glauber dynamics is torpidly mixing. For the FK-dynamics on Λ n , [4] showed that the mixing time is Θ(n 2 log n) for all q > 1 whenever p = p c (q); see also [15] for recent results concerning the cutoff phenomenon in the FK-dynamics.…”

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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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On the Mixing Time of the 2D Stochastic Ising Model with “Plus” Boundary Conditions at Low Temperature

Cited by 30 publications

References 24 publications

Cycle Basis Markov Chains for the Ising Model

Cycle Basis Markov Chains for the Ising Model

Mixing time for the solid-on-solid model

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

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