Invaded Cluster Algorithm for Equilibrium Critical Points

Machta, Jonathan; Choi, Yongsoo; Lücke, Andreas; Schweizer, Teia M.; Chayes, Lincoln

doi:10.1103/physrevlett.75.2792

Cited by 78 publications

(114 citation statements)

References 11 publications

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“…In order to find the value of p c numerically, we employ a tree-based invasion algorithm similar to the invaded cluster algorithm used to find the percolation point in Ising systems [33,34]. This algorithm can calculate the entire curve of average cluster size versus p in time which scales as L log L [35].…”

Section: Percolationmentioning

confidence: 99%

Scaling and percolation in the small-world network model

Newman¹,

Watts²

2011

The Structure and Dynamics of Networks

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In this paper we study the small-world network model of Watts and Strogatz, which mimics some aspects of the structure of networks of social interactions. We argue that there is one non-trivial length-scale in the model, analogous to the correlation length in other systems, which is well-defined in the limit of infinite system size and which diverges continuously as the randomness in the network tends to zero, giving a normal critical point in this limit. This length-scale governs the cross-over from large-to small-world behavior in the model, as well as the number of vertices in a neighborhood of given radius on the network. We derive the value of the single critical exponent controlling behavior in the critical region and the finite size scaling form for the average vertex-vertex distance on the network, and, using series expansion and Padé approximants, find an approximate analytic form for the scaling function. We calculate the effective dimension of small-world graphs and show that this dimension varies as a function of the length-scale on which it is measured, in a manner reminiscent of multifractals. We also study the problem of site percolation on small-world networks as a simple model of disease propagation, and derive an approximate expression for the percolation probability at which a giant component of connected vertices first forms (in epidemiological terms, the point at which an epidemic occurs). The typical cluster radius satisfies the expected finite size scaling form with a cluster size exponent close to that for a random graph. All our analytic results are confirmed by extensive numerical simulations of the model. 05.50.+q, 05.70.Jk, 64.60.Fr

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Section: Percolationmentioning

confidence: 99%

Scaling and percolation in the small-world network model

Newman¹,

Watts²

2011

The Structure and Dynamics of Networks

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“…However, the ensemble of the IC algorithm is not necessarily clear, and it has a problem of "bottlenecks", which causes the broad tail in the distribution of the fraction of the accepted satisfied bonds. 14) In contrast, it is guaranteed that we approach the canonical ensemble in our PCC algorithm.…”

Section: )mentioning

confidence: 99%

Probability-Changing Cluster Algorithm: Study of Three-Dimensional Ising Model and Percolation Problem

Tomita

Okabe

2002

J. Phys. Soc. Jpn.

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We present a detailed description of the idea and procedure for the newly proposed Monte Carlo algorithm of tuning the critical point automatically, which is called the probabilitychanging cluster (PCC) algorithm [Y. Tomita and Y. Okabe, Phys. Rev. Lett. 86 (2001) 572]. Using the PCC algorithm, we investigate the three-dimensional Ising model and the bond percolation problem. We employ a refined finite-size scaling analysis to make estimates of critical point and exponents. With much less efforts, we obtain the results which are consistent with the previous calculations. We argue several directions for the application of the PCC algorithm.

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“…However, the original method and its developments (both classical and quantum) [6][7][8] are essentially non-local schemes, and we are not aware of any exception from this rule.…”

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confidence: 99%

“…Acceptance ratios are given by Eqs. (6)(7)(8) where N b is understood as one of the numbers describing the bond state. An estimator for the winding number (which has exactly the same meaning as in worldline quantum Monte Carlo [11]) of the CP configuration is given by the difference between the bond numbers…”

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confidence: 99%

Worm Algorithms for Classical Statistical Models

2001

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We show that high-temperature expansions may serve as a basis for the novel approach to efficient Monte Carlo simulations. "Worm" algorithms utilize the idea of updating closed path configurations (produced by high-temperature expansions) through the motion of end points of a disconnected path. An amazing result is that local, Metropolis-type schemes may have dynamical critical exponents close to zero (i.e., their efficiency is comparable to the best cluster methods). We demonstrate this by calculating finite size scaling of the autocorrelation time for various universality classes.PACS numbers: 75.40. Mg, 75.10.Hk, 64.60.Ht Metropolis scheme [1] is usually the most universal and easy to program approach to Monte Carlo simulations. However, its advantages are virtually canceled out near phase transition points. It is believed that any scheme based on local [2] Metropolis-type updates connecting system configurations into Markovian chain is inefficient at the transition point because its autocorrelation time, τ , scales as L z , where L is the system linear dimension and z is the dynamical critical exponent which is close to 2 in most systems [3,4].An enormous acceleration of simulations at the critical point has been achieved with the invention of cluster algorithms by Swendsen and Wang [5]. However, the original method and its developments (both classical and quantum) [6][7][8] are essentially non-local schemes, and we are not aware of any exception from this rule.In this Letter we propose a method which essentially eliminates the critical slowing down problem and yet remains local. The corner stone of our approach is the possibility to introduce the configuration space of closed paths. Closed-path (CP) configurations may be then sampled very efficiently using Worm algorithm (WA) introduced in Ref. [9] for quantum statistical models in which closed trajectories naturally arise from imaginarytime evolution of world lines. In classical models the CP representation derives from high-temperature expansions for a broad class of lattice models (see, e.g. Ref.[10]). In 2D, another family of WA may be introduced by considering domain-wall boundaries as paths.We note, that our approach is based on principles which differ radically from cluster methods and, most probably, has another range of applicability. For one thing, the CP representation is most suitable for the study of superfluid models by having direct Monte Carlo estimators for the superfluid stiffness (through the histogram of winding numbers [11]) which are not available in the standard site representation.In what follows we first recall how high-temperature expansions work by employing Ising model as an example (still, trying to keep notations as general as possible). We then explain how WA is used to update the path configuration space. Next, we discuss specific implementations of WA for |ψ| 4 -, XY-, and q = 3 Potts models, and comment on the special property of 2D models which allows an alternative CP parameterization of the configuration space. The ef...

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Invaded Cluster Algorithm for Equilibrium Critical Points

Cited by 78 publications

References 11 publications

Scaling and percolation in the small-world network model

Scaling and percolation in the small-world network model

Probability-Changing Cluster Algorithm: Study of Three-Dimensional Ising Model and Percolation Problem

Worm Algorithms for Classical Statistical Models

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