The Fortuin-Kasteleyn mapping between the Ising model and the site-bond correlated percolation model is shown to be only one of an infinite class of exact mappings. These new cluster representations are a result of "renormalized" percolation rules correlated to entire blocks of spins. For example these rules allow for percolation on "virtual" bonds between spins not present in the underlying Hamiltonian. As a consequence we can define new random cluster theories each with its own Monte Carlo cluster dynamics that exactly reproduce the Ising model. By tuning parameters on the critical percolation surface, it is demonstrated numerically that cluster algorithms can be formulated for the 2-d and 3-d Ising model that have smaller autocorrelations than the original Swendsen-Wang algorithm.
The Fortuin-Kasteleyn mapping between the Ising model and the site-bond correlated percolation model is shown to be only one of an infinite class of exact mappings. These new cluster representations are a result of "renormalized" percolation rules correlated to entire blocks of spins. For example these rules allow for percolation on "virtual" bonds between spins not present in the underlying Hamiltonian. As a consequence we can define new random cluster theories each with its own Monte Carlo cluster dynamics that exactly reproduce the Ising model. By tuning parameters on the critical percolation surface, it is demonstrated numerically that cluster algorithms can be formulated for the 2-d and 3-d Ising model that have smaller autocorrelations than the original Swendsen-Wang algorithm.
The Fortuin-Kasteleyn (FK) random-cluster model, which can be exactly mapped from the q-state Potts spin model, is a correlated bond percolation model. By extensive Monte Carlo simulations, we study the FK bond representation of the critical Ising model (q=2) on a finite complete graph, i.e., the mean-field Ising model. We provide strong numerical evidence that the configuration space for q=2 contains an asymptotically vanishing sector in which quantities exhibit the same finite-size scaling as in the critical uncorrelated bond percolation (q=1) on the complete graph. Moreover, we observe that, in the full configuration space, the power-law behavior of the cluster-size distribution for the FK Ising clusters except the largest one is governed by a Fisher exponent taking the value for q=1 instead of q=2. This demonstrates the percolation effects in the FK Ising model on the complete graph.
We apply generalizations of the Swendson-Wang and Wolff cluster algorithms, which are based on the construction of Fortuin-Kasteleyn clusters, to the three-dimensional ±1 random-bond Ising model. The behavior of the model is determined by the temperature T and the concentration p of negative (antiferromagnetic) bonds. The ground state is ferromagnetic for 0≤p<p_{c}, and a spin glass for p_{c}<p≤0.5 where p_{c}≃0.222. We investigate the percolation transition of the Fortuin-Kasteleyn clusters as a function of temperature for large system sizes up to N=200^{3} spins. Except for p=0 the Fortuin-Kasteleyn percolation transition occurs at a higher temperature than the magnetic ordering temperature. This was known before for p=1/2 but here we provide evidence for a difference in transition temperatures even for p arbitrarily small. Furthermore, for all values of p>0, our data suggest that the percolation transition is universal, irrespective of whether the ground state exhibits ferromagnetic or spin-glass order, and is in the universality class of standard percolation. This shows that correlations in the bond occupancy of the Fortuin-Kasteleyn clusters are irrelevant, except for p=0 where the clusters are strictly tied to Ising correlations so the percolation transition is in the Ising universality class.
We discuss a new cluster representation for the internal energy and the specific heat of the d -dimensional Ising model, obtained by studying the percolation mapping of an Ising model with an arbitrary set of antiferromagnetic links. Such a representation relates the thermal operators to the topological properties of the Fortuin-Kasteleyn (FK) clusters of Ising percolation and is a powerful tool to get new exact relations on the topological structure of FK clusters of the Ising model defined on an arbitrary graph.
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).
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.
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.