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Critical Slowing Down of Cluster Algorithms for Ising Models Coupled to 2-d Gravity

Mark J. Bowick, Marco Falcioni, Geoffrey Harris, Enzo Marinari

Published Nov 23, 1993• 4 citations

Abstract

We simulate single and multiple Ising models coupled to 2–d gravity using both the Swendsen-Wang and Wolff algorithms to update the spins. We study the integrated autocorrelation time and find that there is considerable critical slowing down, particularly in the magnetization. We argue that this is primarily due to the local nature of the dynamical triangulation algorithm and to the generation of a distribution of baby universes which inhibits cluster growth. SU–HEP–93–4241–560 SCCS 568 hep-lat/9311036 Considerable work has been devoted to the study of the performance of cluster algorithms in reducing the critical slowing down of many statistical models. The Swendsen-Wang [1] and the Wolff [2] algorithms have proven very effective in beating the critical slowing down (CSD) exhibited by these models when simulated with a standard Metropolis [3] update. Reviews on cluster algorithms and CSD may be found on refs. [4]. In recent years, the numerical study of Potts models coupled to 2–d gravity has received much attention [5, 6, 7], aided as well by increased analytical understanding of these models [8]. Cluster algorithms have proved useful in saving computational effort on the update of the Potts variables. The present study is motivated by the fact that there is little understanding of the actual extent of the improvement achieved in these simulations. It is worthwhile, therefore, to measure CSD in the case of simple Ising spins coupled to a dynamical lattice in order to quantify the performance of cluster algorithms. We find that there is considerable CSD, especially in the magnetization, and we relate this to the dynamics of cluster formation on a random lattice.

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Acta Physica Sinica•Jan 1, 1993•0 citations

We simulate single and multiple Ising models coupled to 2–d gravity using both the Swendsen-Wang and Wolff algorithms to update the spins. We study the integrated autocorrelation time and find that there is considerable critical slowing down, particularly in the magnetization. We argue that this is primarily due to the local nature of the dynamical triangulation algorithm and to the generation of a distribution of baby universes which inhibits cluster growth. SU–HEP–93–4241–560 SCCS 568 hep-lat/9311036 Considerable work has been devoted to the study of the performance of cluster algorithms in reducing the critical slowing down of many statistical models. The Swendsen-Wang [1] and the Wolff [2] algorithms have proven very effective in beating the critical slowing down (CSD) exhibited by these models when simulated with a standard Metropolis [3] update. Reviews on cluster algorithms and CSD may be found on refs. [4]. In recent years, the numerical study of Potts models coupled to 2–d gravity has received much attention [5, 6, 7], aided as well by increased analytical understanding of these models [8]. Cluster algorithms have proved useful in saving computational effort on the update of the Potts variables. The present study is motivated by the fact that there is little understanding of the actual extent of the improvement achieved in these simulations. It is worthwhile, therefore, to measure CSD in the case of simple Ising spins coupled to a dynamical lattice in order to quantify the performance of cluster algorithms. We find that there is considerable CSD, especially in the magnetization, and we relate this to the dynamics of cluster formation on a random lattice.

In the present work, we present an extensive Monte Carlo simulation study on the dynamical properties of the two-dimensional random-bond Ising model. The correlation time τ of the Swendsen-Wang and Wolff cluster algorithms is calculated at the critical point. The dynamic critical exponent z of both algorithms is also measured by using the numerical data for several lattice sizes up to L=512. It is found for both algorithms that the autocorrelation time decreases considerably and the critical slowing-down effect reduces upon the introduction of bond disorder. Additionally, simulations with the Metropolis algorithm are performed, and the critical slowing-down effect is observed to be more pronounced in the presence of disorder, confirming the previous findings in the literature. Moreover, the existence of the non-self-averaging property of the model is demonstrated by calculating the scaled form of the standard deviation of autocorrelation times. Finally, the critical exponent ratio of the magnetic susceptibility is estimated by using the average cluster size of the Wolff algorithm.

We have measured the dynamical critical exponent z for the Swendsen-Wang and the Wolff cluster update algorithms, as well as a number of variants of these algorithms, for the q=2 and q=3 Potts models in two dimensions. We find that although the autocorrelation times differ considerably between algorithms, the critical exponents are the same. For q=2, we find that although the data are better fitted by a logarithmic increase in the autocorrelation time with lattice size, they are also consistent with a power law with exponent z\ensuremath{\approxeq}0.25, especially if there are non-negligible corrections to scaling.

The simulation of spin models close to critical points of continuous phase transitions is heavily impeded by the occurrence of critical slowing down. A number of cluster algorithms, usually based on the Fortuin-Kasteleyn representation of the Potts model, and suitable generalizations for continuous-spin models have been used to increase simulation efficiency. The first algorithm making use of this representation, suggested by Sweeny in 1983, has not found widespread adoption due to problems in its implementation. However, it has been recently shown that it is indeed more efficient in reducing critical slowing down than the more well-known algorithm due to Swendsen and Wang. Here, we present an efficient implementation of Sweeny's approach for the random-cluster model using recent algorithmic advances in dynamic connectivity algorithms.

Monte Carlo cluster algorithms are popular for their efficiency in studying the Ising model near its critical temperature. We might expect that this efficiency extends to the bond-diluted Ising model. We show, however, that this is not always the case by comparing how the correlation times τ_{w} and τ_{sw} of the Wolff and Swendsen-Wang cluster algorithms scale as a function of the system size L when applied to the two-dimensional bond-diluted Ising model. We demonstrate that the Wolff algorithm suffers from a much longer correlation time than in the pure Ising model, caused by isolated (groups of) spins which are infrequently visited by the algorithm. With a simple argument we prove that these cause the correlation time τ_{w} to be bounded from below by L^{z_{w}} with a dynamical exponent z_{w}=γ/ν≈1.75 for a bond concentration p<1. Furthermore, we numerically show that this lower bound is actually taken for several values of p in the range 0.5<p<1. Moreover, we show that the Swendsen-Wang algorithm does not suffer from the same problem. Consequently, it has a much shorter correlation time, shorter than in the pure Ising model even. Numerically at p=0.6, we find that its dynamical exponent is z_{sw}=0.09(4).

In order to reduce the critical slowing down in Monte Carlo simulations, a numerical study for the Ising model on 3-D random lattice using Swendsen-Wang algorithm is performed. The critical point obtained isβc=0. 075?. 001. Moreover, to analyse the dynamical property of Swendsen-Wang method, the dynamical critical exponent z is estimated. The result is z =0. 74 ±0. 03. This indicates that the Swendsen-Wang method can greatly reduce the critical slowing down for the Ising model on 3-D random lattice.