Monte Carlo study of weighted percolation clusters relevant to the Potts models

Sweeny, M.

doi:10.1103/physrevb.27.4445

Cited by 124 publications

(86 citation statements)

References 20 publications

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“…Since the connectedness, a non-local property, must be determined with every iteration to know δC lk , it is important to find faster algorithm to evaluate δC lk . In order to quickly determine the connectedness of large clusters, we have implemented an algorithm [33] using an auxiliary data structure based on the BKW construction [26].…”

Section: The Metropolis Algorithm Of the Generalized Potts Modelmentioning

confidence: 99%

Numerical study for the c-dependence of fractal dimension in two-dimensional quantum gravity

Kawamoto

Yotsuji

2002

Nuclear Physics B

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We numerically investigate the fractal structure of two-dimensional quantum gravity coupled to matter central charge c for −2 ≤ c ≤ 1. We reformulate Q-state Potts model into the model which can be identified as a weighted percolation cluster model and can make continuous change of Q, which relates c, on the dynamically triangulated lattice. The c-dependence of the critical coupling is measured from the percolation probability and susceptibility. The c-dependence of the string susceptibility of the quantum surface is evaluated and has very good agreement with the theoretical predictions. The c-dependence of the fractal dimension based on the finite size scaling hypothesis is measured and has excellent agreement with one of the theoretical predictions previously proposed except for the region near c ≈ 1.

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Section: The Metropolis Algorithm Of the Generalized Potts Modelmentioning

confidence: 99%

Numerical study for the c-dependence of fractal dimension in two-dimensional quantum gravity

Kawamoto

Yotsuji

2002

Nuclear Physics B

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“…In practice, one can generate random subgraphs according to W 1 with importance sampling, as explained above, and according to W q using the very efficient cluster algorithm of Chayes and Machta [4]. This algorithm allows for simulation for arbitrary values of q, similar to other approaches [19,23,24].…”

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confidence: 99%

Calculation of Partition Functions by Measuring Component Distributions

Hartmann

2005

Phys. Rev. Lett.

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A new algorithm is presented, which allows to calculate numerically the partition function Zq of the d-dimensional q-state Potts models for arbitrary real values q > 0 at any given temperature T with high precision. The basic idea is to measure the distribution of the number of connected components in the corresponding Fortuin-Kasteleyn representation and to compare with the distribution of the case q = 1 (graph percolation), where the exact result Z1 = 1 is known. As application, d = 2 and d = 3-dimensional ferromagnetic Potts models are studied, and the critical values qc, where the transition changes from second to first order, are determined. Large systems of sizes N = 1000 2 respectively N = 100 3 are treated. The critical value qc(d = 2) = 4 is confirmed and qc(d = 3) = 2.35 (5) The partition function is a quantity of fundamental importance, because it describes completely the behavior of any statistical physics model. Unfortunately, in finitedimensions, only few models are analytically tractable [1]. Hence, Monte Carlo (MC) simulations [2,3] are usually applied. The standard approach to obtain the partition function is to measure the free energy by thermodynamic integration of the specific heat, i.e. the fluctuations of the energy. Since this approach is based on measuring fluctuations, it is not very efficient, hence limited to small sizes. One can speedup simulations for certain types of systems by applying cluster algorithms [4,5,6], multi-histogram methods [7], multicanonical simulations [8,9] or transition-matrix Monte Carlo [10], but the general problem of the strong fluctuations remains. To overcome this problem, recently Wang and Landau introduced [11] a simple yet very efficient method to obtain the partition function. The key idea is to measure the density of states by performing a biased random walk in energy space via spin flips. It works well for unfrustrated systems, e.g. the standard q-state Potts model [12], which has become a standard testing ground for Monte Carlo algorithms. The Potts model is of profound interest, because, for dimensions d larger than one, it exhibits order-disorder phase transitions [13], which are of second order for q smaller than a critical value q c (d), while they are of first order for q > q c (d). It is analytically proven [14] that q c (2) = 4, but e.g. for d = 3, the exact value of q c is not known. From various analytical work [15,16,17] and simulations of moderate-size systems [18,19,20], 2 < q c (3) < 3 seems likely. Unfortunately, since the Wang-Landau method is based on spin flips, it works only for integer values of q, hence the partition function for 2 < q < 3 cannot be obtained for large systems in this way.In this letter, an algorithm is presented, which allows to calculate numerically the partition function Z q of the d-dimensional q-state Potts models for arbitrary real values q > 0 at any given temperature T with high precision for large systems. The basic idea is to measure the distribution of the number of connected components in the correspond...

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“…On the other hand, the use of arbitrary large moves (such as increasing the maximum displacement in a fluid or moving a large number of particles randomly) can lead to a significant decrease in the acceptance probability. One approach to overcoming these problems has been the development of cluster acceleration methods such as the Sweeny 16 and Swendsen-Wang (SW) 17,18 algorithms. The SW algorithm is based on a mapping of the Ising model to a percolation model by C.M.…”

Section: Cluster Accelerationmentioning

confidence: 99%

“…The Kronecker delta in Equation 16 corresponds to ferromagnetic coupling, that is, the energy is minimized when nearest-neighbor spins have the same value. In the SW algorithm, bonds are created between all neighboring spins with the same value with probability p = 1 -exp(-J/k B T), thus leading to a set of bond clusters.…”

Section: Cluster Accelerationmentioning

confidence: 99%

The Monte Carlo method in science and engineering

Amar

2006

Comput. Sci. Eng.

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Monte Carlo study of weighted percolation clusters relevant to the Potts models

Cited by 124 publications

References 20 publications

Numerical study for the c-dependence of fractal dimension in two-dimensional quantum gravity

Numerical study for the c-dependence of fractal dimension in two-dimensional quantum gravity

Calculation of Partition Functions by Measuring Component Distributions

The Monte Carlo method in science and engineering

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