arXiv:cond-mat/0701113 · Physical Review Letters

Critical Speeding-Up in the Local Dynamics of the Random-Cluster Model

Youjin Deng, Timothy M. Garoni, Alan D. Sokal

Published Jun 7, 2007• 23 citations

Abstract

We study the dynamic critical behavior of the local bond-update (Sweeny) dynamics for the Fortuin-Kasteleyn random-cluster model in dimensions d=2, 3 by Monte Carlo simulation. We show that, for a suitable range of q values, the global observable S2 exhibits "critical speeding-up": it decorrelates well on time scales much less than one sweep. In some cases the dynamic critical exponent for the integrated autocorrelation time is negative. We also show that the dynamic critical exponent zexp is very close (possibly equal) to the rigorous lower bound alpha/nu and quite possibly smaller than the corresponding exponent for the Chayes-Machta-Swendsen-Wang cluster dynamics.

DOI: 10.1103/physrevlett.98.230602

Related papers

Critical Behavior of the Chayes–Machta–Swendsen–Wang Dynamics

Youjin Deng, Timothy M. Garoni, Jonathan L. Machta +3 more

We study the dynamic critical behavior of the Chayes-Machta dynamics for the Fortuin-Kasteleyn random-cluster model, which generalizes the Swendsen-Wang dynamics for the q-state Potts model to noninteger q, in two and three spatial dimensions, by Monte Carlo simulation. We show that the Li-Sokal bound z >or= alpha/nu is close to but probably not sharp in d = 2 and is far from sharp in d = 3, for all q. The conjecture z >or= beta/nu is false (for some values of q) in both d = 2 and d = 3.

arXiv:0705.2751Physical Review LettersJul 31, 200724 citations

Dynamical critical properties of the random-bond three-state Potts model

Anonymous, Ulvi Kanbur, Mohammad Sadra Najafi +2 more

We study the dynamic critical behavior of the two-dimensional random-bond three-state Potts model using large-scale Monte Carlo simulations with the Wolff and Swendsen-Wang cluster algorithms. At the disorder-dependent critical temperature, we compute integrated and exponential autocorrelation times and extract the dynamic critical exponent z via finite-size scaling. Critical slowing down is significantly weakened by bond randomness, with the dynamic exponent decreasing from z≈0.54 in the pure system to z≈0.24 at strong disorder. From the finite-size scaling of the Wolff cluster size, we obtain γ/ν≈1.73, where γ and ν are the susceptibility and correlation-length exponents, respectively, consistent with the universality class of the two-dimensional three-state Potts model. These results indicate that bond randomness strongly affects the critical dynamics but leaves the underlying static universality class unchanged. Furthermore, both static and dynamic observables, including the specific heat and the autocorrelation times, show a lack of self-averaging.

arXiv:2610.01176Physical review. EAug 13, 20260 citations

Sweeny dynamics for the random-cluster model with small Q

Zirui Peng, Eren Metin Elçi, Youjin Deng +1 more

The Sweeny algorithm for the Q-state random-cluster model in two dimensions is shown to exhibit a rich mixture of critical dynamical scaling behaviors. As Q decreases, the so-called critical speeding-up for nonlocal quantities becomes more and more pronounced. However, for some quantity of a specific local pattern, e.g., the number of half faces on the square lattice, we observe that, as Q→0, the integrated autocorrelation time τ diverges as Q^{-ζ}, with ζ≃1/2, leading to the nonergodicity of the Sweeny method for Q→0. Such Q-dependent critical slowing-down, attributed to the peculiar form of the critical bond weight v=sqrt[Q], can be eliminated by a combination of the Sweeny and the Kawasaki algorithm. Moreover, by classifying the occupied bonds into bridge bonds and backbone bonds, and the empty bonds into internal-perimeter bonds and external-perimeter bonds, one can formulate an improved version of the Sweeny-Kawasaki method such that the autocorrelation time for any quantity is of order O(1).

arXiv:2308.00254Physical review. ENov 20, 20231 citation

Single-cluster dynamics for the random-cluster model

Youjin Deng, Xiaofeng Qian, Henk W. J. Blöte

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.

arXiv:0907.1711Physical Review ESep 30, 20095 citations

Critical relaxation in two-dimensional random-bond Potts models

S. Chen, David Paul Landau

We present an extensive Monte Carlo study of the critical relaxation for the two-dimensional square lattice random-bond Potts model using Swendsen-Wang cluster flipping. The integrated autocorrelation time \ensuremath{\tau} is calculated and the dynamic exponent z is estimated by analyzing the size dependence of \ensuremath{\tau}. We find that z\ensuremath{\approx}0 in agreement with estimates for the two-dimensional Ising model. We also present a study of the size dependence of the dynamic behavior of the pure eight-state Potts model which undergoes a first-order transition. The scaling is describable by the product of an exponential times a power law, a behavior which is quite different from that found in the random-bond case.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsJan 1, 19973 citations

Critical dynamics of cluster algorithms in the random-bond Ising model

Ulvi Kanbur, Zeynep Demir Vatansever

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.

Physical review. EFeb 28, 20242 citations