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arXiv:2102.12716

Monte Carlo algorithm for the random cluster model

Christian R. Scullard

Published Feb 25, 2021• 0 citations
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Abstract

I present a generalization of the Newman-Ziff algorithm, a standard technique in numerical percolation, to the random cluster model. The new method is straightfoward to implement, works for real $q>0$, and is completely free of the critical slowdown problem. By using the method to sample the critical polynomial, accurate estimates of Potts critical points can be found using relatively small lattice sizes. Furthermore, results for all values of $q$ can be found at once within a single simulation. I demonstrate its utility here by computing critical points for non-integer values of $q$ on the square lattice to compare with the exact solution, and on the non-planar square matching lattice.

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I present a generalization of the Newman-Ziff algorithm, a standard technique in numerical percolation, to the random cluster model. The new method is straightfoward to implement, works for real q>0q>0q>0, and is completely free of the critical slowdown problem. By using the method to sample the critical polynomial, accurate estimates of Potts critical points can be found using relatively small lattice sizes. Furthermore, results for all values of qqq can be found at once within a single simulation. I demonstrate its utility here by computing critical points for non-integer values of qqq on the square lattice to compare with the exact solution, and on the non-planar square matching lattice.
We propose a site random-cluster model by introducing an additional cluster weight in the partition function of the traditional site percolation. To simulate the model on a square lattice, we combine the color-assignation and the Swendsen-Wang methods to design a highly efficient cluster algorithm with a small critical slowing-down phenomenon. To verify whether or not it is consistent with the bond random-cluster model, we measure several quantities, such as the wrapping probability Re, the percolating cluster density P∞, and the magnetic susceptibility per site χp, as well as two exponents, such as the thermal exponent yt and the fractal dimension yh of the percolating cluster. We find that for different exponents of cluster weight q=1.5, 2, 2.5, 3, 3.5, and 4, the numerical estimation of the exponents yt and yh are consistent with the theoretical values. The universalities of the site random-cluster model and the bond random-cluster model are completely identical. For larger values of q, we find obvious signatures of the first-order percolation transition by the histograms and the hysteresis loops of percolating cluster density and the energy per site. Our results are helpful for the understanding of the percolation of traditional statistical models.
The two-dimensional Potts models were simulated with the use of a Monte Carlo method. This study is unique in that we simulated the weighted percolation clusters first described by Kasteleyn and Fortuin in 1969. We describe an auxiliary data structure which enables us to determine the connectedness of large clusters very efficiently. Our simulation of the percolation problem was found not to be affected by a critical slowing down. Critical exponents were found by studying the size and shape of large clusters. Our results agree with previous studies done on integer values of qqq, and with the conjecture of den Nijs.
We formulate a single-cluster Monte Carlo algorithm for the simulation of the random-cluster model. This algorithm is a generalization of the Wolff single-cluster method for the q-state Potts model to noninteger values q>1. Its results for static quantities are in a satisfactory agreement with those of the existing Swendsen-Wang-Chayes-Machta (SWCM) algorithm, which involves a full-cluster decomposition of random-cluster configurations. We explore the critical dynamics of this algorithm for several two-dimensional Potts and random-cluster models. For integer q, the single-cluster algorithm can be reduced to the Wolff algorithm, for which case we find that the autocorrelation functions decay almost purely exponentially, with dynamic exponents z(exp)=0.07 (1), 0.521 (7), and 1.007 (9) for q=2, 3, and 4, respectively. For noninteger q, the dynamical behavior of the single-cluster algorithm appears to be very dissimilar to that of the SWCM algorithm. For large critical systems, the autocorrelation function displays a range of power-law behavior as a function of time. The dynamic exponents are relatively large. We provide an explanation for this peculiar dynamic behavior.
We present a self consistent method based on cluster algorithms and Renormalization Group on the lattice to study critical systems numerically. We illustrate it by means of the 2D Ising model. We compute the critical exponents ννν and ηηη and the renormalization group flow of the probability density function of the magnetization. The results, compared to the standard Monte Carlo Renormalization Group proposed by Swendsen [1], are very accurate and the method works faster by a factor which grows monotonically with the lattice size. This allows to simulate larger lattices in reachable computational times.
In this work we apply a highly efficient Monte Carlo algorithm recently proposed by Newman and Ziff to treat percolation problems. The site and bond percolations are studied on a number of lattices in two and three dimensions. Quite good results for the wrapping probabilities, correlation length critical exponent, and critical concentration are obtained for the square, simple cubic, hexagonal close packed, and hexagonal lattices by using relatively small systems. We also confirm the universal aspect of the wrapping probabilities regarding site and bond dilution.
We report numerical simulations of two-dimensional qqq-state Potts models with emphasis on a new quantity for the computation of spatial correlation lengths. This quantity is the cluster-diameter distribution function Gdiam(x){G}^{\mathrm{diam}}(x)Gdiam(x), which measures the distribution of the diameter of stochastically defined cluster. Theoretically it is predicted to fall off exponentially for large diameter xxx, ${G}^{\mathrm{diam}}\ensuremath{\propto}\mathrm{exp}(\ensuremath{-}x/\ensuremath{\xi})$, where $\ensuremath{\xi}$ is the correlation length as usually defined through the large-distance behavior of two-point correlation functions. The results of our extensive Monte Carlo study in the disordered phase of the models with q=10q=10q=10, 15, and 202020 on large square lattices of size $300\ifmmode\times\else\texttimes\fi{}300$, $120\ifmmode\times\else\texttimes\fi{}120$, and $80\ifmmode\times\else\texttimes\fi{}80$, respectively, clearly confirm the theoretically predicted behavior. Moreover, using this observable we are able to verify an exact formula for the correlation length ${\ensuremath{\xi}}_{d}({\ensuremath{\beta}}_{t})$ in the disordered phase at the first-order transition point ${\ensuremath{\beta}}_{t}$ with an accuracy of about 11%1--22%2 for all considered values of qqq. This is a considerable improvement over estimates derived from the large-distance behavior of standard (projected) two-point correlation functions, which are also discussed for comparison.