Mean-field behavior of cluster dynamics

Persky, N.; Ben-Av, Radel; Kanter, Ido; Domany, Eytan

doi:10.1103/physreve.54.2351

Cited by 19 publications

(31 citation statements)

References 12 publications

(22 reference statements)

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“…In other words, mixing becomes so efficient that equilibration of the global spin correlations is observed even before all spins of the systems are flipped. One concludes, that performance of the irreversible scheme is at least as good as the one of the cluster algorithms [9,10] tested on the spin cluster model [11,12]. (We note, however, that direct comparison of the two algorithms is not straightforward, as the cluster algorithm flips many spins at once and therefore its convergence is normally stated in renormalized units.…”

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confidence: 91%

Irreversible Monte Carlo algorithms for efficient sampling

Turitsyn

Chertkov

Vucelja

2011

Physica D: Nonlinear Phenomena

117

188

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Equilibrium systems evolve according to Detailed Balance (DB). This principe guided development of the Monte-Carlo sampling techniques, of which Metropolis-Hastings (MH) algorithm is the famous representative. It is also known that DB is sufficient but not necessary. We construct irreversible deformation of a given reversible algorithm capable of dramatic improvement of sampling from known distribution. Our transformation modifies transition rates keeping the structure of transitions intact. To illustrate the general scheme we design an Irreversible version of Metropolis-Hastings (IMH) and test it on example of a spin cluster. Standard MH for the model suffers from the critical slowdown, while IMH is free from critical slowdown. [3,4,5] for a sample set of reviews in physics and computer science.) If one formally follows the letter of the original MCMC suggestion one ought to ensure that the Detailed Balance (DB) condition is satisfied. This condition reflects microscopic reversibility of the underlying equilibrium dynamics. A reader, impressed with indisputable success of the reversible MCMC techniques, may still wonder if the equilibrium dynamics is the most efficient strategy for sampling and evaluating the integrals? In this letter we argue that typically the answer is NO. Let us try to illustrate the ideas on a simple everyday life example. Consider mixing sugar in a cup of coffee, which is similar to sampling, as long as the sugar particles have to explore the entire interior of the cup. DB dynamics corresponds to diffusion taking an enormous mixing time. This is certainly not the best way to mix. Moreover, our everyday experience suggests a better solutionenhance mixing with a spoon. Spoon steering generates an out-of-equilibrium external flow which significantly accelerates mixing, while achieving the same final result -uniform distribution of sugar concentration over the cup. In this letter we show constructively, with a practical algorithm suggested, that similar strategy can be used to decrease mixing time of any known reversible MCMC algorithms.There are two main obstacles which prevent fast mixing by traditional MCMC methods. First, the effective energy landscape can have high barriers, separating the energy minima. In this case mixing time is dominated by rare processes of overcoming the barriers. Second, slow mixing can originate from the high entropy of the states basin (too many comparably important states) providing major contribution to the system partition function. In the later case mixing time is determined by the number of steps it takes for reversible (diffusive) random walk to explore all the relevant states. MCMC algorithms are best described on discrete example. Consider a graph with vertices i = 1, . . . , N each labeling a state of the system and edges (i → j) corresponding to "allowed" transitions between the states. For instance, an Ndimensional hypercube corresponds to a system of N spins (with N = 2 N states) with single-spin flips allowed. An MCMC algorithm can be described in ...

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confidence: 91%

Irreversible Monte Carlo algorithms for efficient sampling

Turitsyn

Chertkov

Vucelja

2011

Physica D: Nonlinear Phenomena

117

188

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“…Suffice it to observe once again that while this bound is close to sharp (and possibly even sharp modulo a logarithm) for the two-dimensional Potts models with q = 2, 3, 4, it is clearly far from sharp in the three-and four-dimensional Ising models. Our data for the three-dimensional Ising model yield z SW ≈ 0.46, compared to α/ν ≈ 0.1756 [41]; and for the four-dimensional Ising model it is generally believed that z SW = 1 [16,[21][22][23][24], compared to α/ν = 0 (× log 1/3 ). Clearly, some other physical mechanism, beyond the one exploited in the Li-Sokal proof, must be principally responsible for the critical slowingdown in these latter models; the central open problem is to identify this mechanism and to determine theoretically the dynamic critical exponent.…”

Section: Discussionmentioning

confidence: 99%

Dynamic critical behavior of the Swendsen?Wang algorithm for the three-dimensional Ising model

Ossola

2004

Nuclear Physics B

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We have performed a high-precision Monte Carlo study of the dynamic critical behavior of the Swendsen-Wang algorithm for the three-dimensional Ising model at the critical point. For the dynamic critical exponents associated to the integrated autocorrelation times of the "energy-like" observables, we find z int,N = z int,E = z int,E ′ = 0.459 ± 0.005 ± 0.025, where the first error bar represents statistical error (68% confidence interval) and the second error bar represents possible systematic error due to corrections to scaling (68% subjective confidence interval). For the "susceptibility-like" observables, we find z int,M 2 = z int,S 2 = 0.443 ± 0.005 ± 0.030. For the dynamic critical exponent associated to the exponential autocorrelation time, we find z exp ≈ 0.481. Our data are consistent with the Coddington-Baillie conjecture z SW = β/ν ≈ 0.5183, especially if it is interpreted as referring to z exp .

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“…(10) in Ref. [27]] (4) where m 0 is the value of m after a sweep in which the active group contains the largest cluster, and t is the deviation from the critical temperature. Clearly, q 2 is a special case because the coefficient of the linear term equals 1: we have 1=2 and z 1, and it is clear from the derivations [16,27] that z is actually = .…”

Section: H Y S I C a L R E V I E W L E T T E R S Week Ending 3 Augustmentioning

confidence: 99%

“…[27]] (4) where m 0 is the value of m after a sweep in which the active group contains the largest cluster, and t is the deviation from the critical temperature. Clearly, q 2 is a special case because the coefficient of the linear term equals 1: we have 1=2 and z 1, and it is clear from the derivations [16,27] that z is actually = . For 1 q < 2, by contrast, both the statics and dynamics are in the percolation universality class with 1 and z 0: small perturbations from equilibrium relax exponentially with a finite autocorrelation time exp;m q= log q= 2q ÿ 2 that diverges as q " 2.…”

Section: H Y S I C a L R E V I E W L E T T E R S Week Ending 3 Augustmentioning

confidence: 99%

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Critical Behavior of the Chayes–Machta–Swendsen–Wang Dynamics

et al. 2007

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We study the dynamic critical behavior of the Chayes-Machta dynamics for the Fortuin-Kasteleyn random-cluster model, which generalizes the Swendsen-Wang dynamics for the q-state Potts model to noninteger q, in two and three spatial dimensions, by Monte Carlo simulation. We show that the Li-Sokal bound z = is close to but probably not sharp in d 2 and is far from sharp in d 3, for all q. The conjecture z = is false (for some values of q) in both d 2 and d 3. [3,4]: the autocorrelation (relaxation) time diverges as the critical point is approached, most often like z , where is the spatial correlation length and z is a dynamic critical exponent. For local algorithms, one usually has z 2. This effect severely limits the efficiency of Monte Carlo studies of critical phenomena in statistical mechanics and of the continuum limit in quantum field theory.An important advance was made in 1987 with the invention of the Swendsen-Wang (SW) cluster algorithm [5] for simulating the q-state ferromagnetic Potts model [6,7] at positive integer q. The SW algorithm is based on passing back and forth between the Potts spin representation and the Fortuin-Kasteleyn (FK) bond representation [8,9]. This algorithm does not eliminate critical slowing-down, but it radically reduces it compared to local algorithms. Much effort has therefore been devoted, for both theoretical and practical reasons, to understanding the dynamic critical behavior of the SW algorithm as a function of the spatial dimension d and the number q of Potts spin states [10]. Unfortunately, it is very difficult to develop a physical understanding from the small number of ''data points'' at our disposal: second-order transitions occur only for d; q 2; 2 , 2; 3 , 2; 4 , 3; 2 , and 4; 2 [11]. A further advance was made in 1998 by Chayes and Machta (CM) [12], who devised a cluster algorithm for simulating the FK random-cluster model [8,13]-which provides a natural extension of the Potts model to noninteger q-at any real q 1. The CM algorithm generalizes the SW algorithm and in fact reduces to (a slight variant of) it when q is an integer. By using the CM algorithm, we can study the dynamic critical behavior of the SW-CM dynamic universality class as a function of the continuous variable q throughout the range 1 q q c L , where q c L is the maximum q for which the transition is second-order on the lattice L [14]. This vastly enhances our ability to make theoretical sense of the numerical results.In this Letter, we report detailed measurements of the dynamic critical behavior of the CM algorithm for twodimensional random-cluster models with 1 q 4 [15] and for three-dimensional models with q 1:5, 1.8, 2, and 2.2 [16]. Among other things, we find strong evidence against the conjecture z = recently proposed by two of us [10], which had seemed plausible from the data for integer q.The FK random-cluster model with parameter q > 0 is defined on any finite graph G V; E by the partition functionwhere A is the set of ''occupied bonds'' and k A is the number of connected components ('...

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Mean-field behavior of cluster dynamics

Cited by 19 publications

References 12 publications

Irreversible Monte Carlo algorithms for efficient sampling

Irreversible Monte Carlo algorithms for efficient sampling

Dynamic critical behavior of the Swendsen?Wang algorithm for the three-dimensional Ising model

Critical Behavior of the Chayes–Machta–Swendsen–Wang Dynamics

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