Dynamic and static properties of the invaded cluster algorithm

Abstract

Simulations of the two-dimensional Ising and 3-state Potts models at their critical points are performed using the invaded cluster (IC) algorithm. It is argued that observables measured on a sub-lattice of size l should exhibit a crossover to Swendsen-Wang (SW) behavior for l sufficiently less than the lattice size L, and a scaling form is proposed to describe the crossover phenomenon. It is found that the energy autocorrelation time τε(l, L) for an l × l sub-lattice attains a maximum in the crossover region, and a dynamic exponent z IC for the IC algorithm is defined according to τε,max ∼ L z IC . Simulation results for the 3-state model yield z IC = .346 ± .002 which is smaller than values of the dynamic exponent found for the SW and Wolff algorithms and also less than the Li-Sokal bound. The results are less conclusive for the Ising model, but it appears that z IC < .21 and possibly that τε,max ∼ log L so that z IC = 0-similar to previous results for the SW and Wolff algorithms.

Equilibriumlike invaded cluster algorithm: Critical exponents and dynamical properties

“…For the lattice sizes considered we obtain ≈ 0.38 and ≈ 1.26 for EIC and IC cases respectively (in terms of the width scaling X EIC ≈ 0.81 and X IC ≈ 0.37). This indicates that the ensemble sampled by the EIC algorithm is indeed canonical, contrary to the one produced by the standard IC algorithm, which appears to be much wider and scales with different exponent, characterizing the algorithm itself [10]. As it shall be seen in the next section, the EIC algorithm can be used to obtain very accurate values for critical exponents, not only the magnetic one related to criticality, but also the thermal critical exponent related to the approach to criticality and implying the validity of fluctuation-dissipation theorem.…”

“…Their method, based on the invasion percolation [9] and the cluster algorithm, appeared to converge much faster than in the Swendsen-Wang (SW) algorithm, with a dynamical exponent z close to zero. In the same time it opened a number of issues [10], [11] related to the fact that the underlying nonequilibrium procedure does not generate an equilibrium ensemble and consequently is not able to reproduce correct critical exponents when a finite size scaling is applied. There were also some other attempts to design a cluster algorithm that would selfregulate to criticality [12,13].…”

Quenched disorder: Demixing thermal and disorder fluctuations

“…It can be seen that this procedure is self-regulating to the point of percolation of the largest cluster. The shortcoming of the IC algorithm is that the fluctuations of p remain excessive in this process, making the procedure a nonequilibrium one [39]. Thus, data for p , which should approximate the a priori set bond probability in the expression (3), have a much wider distribution.…”

Invaded cluster algorithm for critical properties of periodic and aperiodic planar Ising models

“…In contrast to the previous IC studies [12,24,25,26], in which the average of f was determined and β est inferred in the end using (2), we computed β est in each step. This leads to the same results after finite-size scaling, as shown in [29].…”

“…In the thermodynamic limit, it is expected to be equivalent to the latter, but about the finite-size scaling [7] nothing helpful is known. Also, the thermal exponent and equivalent quantities, such as ν and α, are not easily measured with the IC algorithm [26,14].…”