Sweeny and Gliozzi dynamics for simulations of Potts models in the Fortuin-Kasteleyn representation

Abstract

We compare the correlation times of the Sweeny and Gliozzi dynamics for two-dimensional Ising and three-state Potts models, and the three-dimensional Ising model for the simulations in the percolation representation. The results are also compared with Swendsen-Wang and Wolff cluster dynamics. It is found that Sweeny and Gliozzi dynamics have essentially the same dynamical critical behavior. Contrary to Gliozzi's claim [Phys. Rev. E 66, 016115 (2002)], the Gliozzi dynamics has critical slowing down comparable to that of other cluster methods. For the two-dimensional Ising model, both Sweeny and Gliozzi dynamics give good fits to logarithmic size dependences for the correlation times; for two-dimensional three-state Potts model, their dynamical critical exponent z is 0.49+/-0.01; the three-dimensional Ising model has z=0.37+/-0.02.

Dynamic critical behavior of the Swendsen?Wang algorithm for the three-dimensional Ising model

“…Our data resolve the discrepancies between previous works [9,[14][15][16][17][18][19][20], which can now be understood as arising from corrections to scaling.…”

“…And since there is good reason to believe (see Section 5.2 below) that C 1 does indeed have a significant overlap with the slowest mode in the Swendsen-Wang algorithm, this suggests that Wang's observable is not interpretable within the standard Swendsen-Wang algorithm, but rather represents a new slow mode in the variant algorithm. 7 A pure power-law fit to the raw data of Wang, Kozan and Swendsen [20] yields a decent χ 2 if (and only if) L min ≥ 32. Our preferred fit is L min = 32, and yields z int,E = 0.502 ± 0.012 (χ 2 = 0.440, 1 DF, level = 50.7%).…”

The Monte Carlo method in science and engineering

“…Typically, trial configurations are randomly chosen, as in the standard Metropolis algorithm, so that the trial configuration selection matrix T ij is symmetric. The acceptance probability for a move from state i to state j is then , (18) which implies that the probability of configuration i is given by 1/g(E i ). Because the density of states is g(E i ), this eventually leads to a flat histogram in energy space-that is, a "random walk" in energy space.…”

“…Because the clusters can be arbitrarily large at the critical temperature in this algorithm, the new configuration can differ substantially from the original one. In Monte Carlo simulations of the 2D Ising model performed with this algorithm, 17,18 researchers found that critical slowing down was significantly reduced.…”

“…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.…”