On the random-cluster model II. The percolation model

Fortuin, C.M.

doi:10.1016/0031-8914(72)90161-9

Cited by 124 publications

(57 citation statements)

References 8 publications

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“…Unfortunately, however, they are less general applicable. We therefore consider first only the Ising model, where the prescription for cluster update algorithms can easily be read off from the equivalent Fortuin-Kasteleyn representation (Kasteleyn and Fortuin 1969, Fortuin and Kasteleyn 1972, Fortuin 1972a, 1972b,…”

Section: Cluster Algorithmsmentioning

confidence: 99%

Monte Carlo Simulations of Spin Systems

Janke

1996

Computational Physics

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Abstract. This lecture gives a brief introduction to Monte Carlo simulations of classical O(n) spin systems such as the Ising (n = 1), XY (n = 2), and Heisenberg (n = 3) model. In the first part I discuss some aspects of Monte Carlo algorithms to generate the raw data. Here special emphasis is placed on non-local cluster update algorithms which proved to be most efficient for this class of models. The second part is devoted to the data analysis at a continuous phase transition. For the example of the three-dimensional Heisenberg model it is shown how precise estimates of the transition temperature and the critical exponents can be extracted from the raw data. I conclude with a brief overview of recent results from similar high-precision studies of the Ising and XY model.

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Section: Cluster Algorithmsmentioning

confidence: 99%

Monte Carlo Simulations of Spin Systems

Janke

1996

Computational Physics

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“…Random-cluster models have proved to be the natural setting for many arguments of value, as well as being objects worthy of study in their own right. The relationship between random-cluster models and ferromagnetic systems was first described by Fortuin and Kasteleyn, (8,9,10,11,19) and more recently by Edwards and Sokal. (7) For recent accounts of the theory of random-cluster models, see Refs.…”

Section: Introductionmentioning

confidence: 97%

Comparison and disjoint-occurrence inequalities for random-cluster models

Grimmett

1995

J Stat Phys

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×ØÖØº A principal technique for studying percolation, (ferromagnetic) Ising, Potts, and random-cluster models is the FKG inequality, which implies certain stochastic comparison inequalities for the associated probability measures. The first result of this paper is a new comparison inequality, proved using an argument developed in Refs. 2, 6, 14, and 21 in order to obtain strict inequalities for critical values. As an application of this inequality, we prove that the critical point p c (q), of the random-cluster model with cluster-weighting factor q (≥ 1), is strictly monotone in q. Our second result is a 'BK inequality' for the disjoint occurrence of increasing events, in a weaker form than that available in percolation theory.

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“…The strategy of this paper is firstly to establish phase transition results for anisotropic percolation models, in the following Section 2, and then in Section 3 to derive results for the Ising model by use of the famous Fortuin-Kasteleyn random cluster representation [6,4,5]. In conclusion, Section 4 describes some illustrative simulations and discusses prospects for further work.…”

Section: Plan Of Papermentioning

confidence: 99%

Ising models and multiresolution quad-trees

Kendall

Wilson

2003

Adv. Appl. Probab.

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We study percolation and Ising models defined on generalizations of quad-trees used in multiresolution image analysis. These can be viewed as trees for which each mother vertex has 2 d daughter vertices, and for which daughter vertices are linked together in d-dimensional Euclidean configurations. Retention probabilities and interaction strengths differ according to whether the relevant bond is between mother and daughter or between neighbours. Bounds are established which locate phase transitions and show the existence of a coexistence phase for the percolation model. Results are extended to the corresponding Ising model using the Fortuin-Kasteleyn random-cluster representation.

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On the random-cluster model II. The percolation model

Cited by 124 publications

References 8 publications

Monte Carlo Simulations of Spin Systems

Monte Carlo Simulations of Spin Systems

Comparison and disjoint-occurrence inequalities for random-cluster models

Ising models and multiresolution quad-trees

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