Exact sampling from non-attractive distributions using summary states

Propp and Wilson's method of coupling from the past allows one to efficiently generate exact samples from attractive statistical distributions (e.g., the ferromagnetic Ising model). This method may be generalized to non-attractive distributions by the use of summary states, as first described by Huber. Using this method, we present exact samples from a frustrated antiferromagnetic triangular Ising model and the antiferromagnetic q=3 Potts model. We discuss the advantages and limitations of the method of summary states for practical sampling, paying particular attention to the slowing down of the algorithm at low temperature. In particular, we show that such a slowing down can occur in the absence of a physical phase transition.Comment: 5 pages, 6 EPS figures, REVTeX; additional information at http://wol.ra.phy.cam.ac.uk/mackay/exac

Correlations of a bound interface over a random substrate

The correlation function of a one-dimensional interface over a random substrate, bound to the substrate by a pressure term, is studied by Monte-Carlo simulation. It is found that the height correlation , averaged over the substrate disorder, fits a form exp(-(j/b)^c) to a surprising precision in the full range of j where the correlation is non-negligible. The exponent c increases from 1.0 to 1.5 when the interface tension is taken larger and larger.Comment: 7 pages, 5 figure

Green's Functions from Quantum Cluster Algorithms

We show that cluster algorithms for quantum models have a meaning independent of the basis chosen to construct them. Using this idea, we propose a new method for measuring with little effort a whole class of Green's functions, once a cluster algorithm for the partition function has been constructed. To explain the idea, we consider the quantum XY model and compute its two point Green's function in various ways, showing that all of them are equivalent. We also provide numerical evidence confirming the analytic arguments. Similar techniques are applicable to other models. In particular, in the recently constructed quantum link models, the new technique allows us to construct improved estimators for Wilson loops and may lead to a very precise determination of the glueball spectrum.Comment: 15 pages, LaTeX, with four figures. Added preprint numbe

Critical Droplets and Phase Transitions in Two Dimensions

In two space dimensions, the percolation point of the pure-site clusters of the Ising model coincides with the critical point T_c of the thermal transition and the percolation exponents belong to a special universality class. By introducing a bond probability p_B<1, the corresponding site-bond clusters keep on percolating at T_c and the exponents do not change, until p_B=p_CK=1-exp(-2J/kT): for this special expression of the bond weight the critical percolation exponents switch to the 2D Ising universality class. We show here that the result is valid for a wide class of bidimensional models with a continuous magnetization transition: there is a critical bond probability p_c such that, for any p_B>=p_c, the onset of percolation of the site-bond clusters coincides with the critical point of the thermal transition. The percolation exponents are the same for p_c<p_B<=1 but, for p_B=p_c, they suddenly change to the thermal exponents, so that the corresponding clusters are critical droplets of the phase transition. Our result is based on Monte Carlo simulations of various systems near criticality.Comment: Final version for publication, minor changes, figures adde

Local and cluster critical dynamics of the 3d random-site Ising model

We present the results of Monte Carlo simulations for the critical dynamics of the three-dimensional site-diluted quenched Ising model. Three different dynamics are considered, these correspond to the local update Metropolis scheme as well as to the Swendsen-Wang and Wolff cluster algorithms. The lattice sizes of L=10-96 are analysed by a finite-size-scaling technique. The site dilution concentration p=0.85 was chosen to minimize the correction-to-scaling effects. We calculate numerical values of the dynamical critical exponents for the integrated and exponential autocorrelation times for energy and magnetization. As expected, cluster algorithms are characterized by lower values of dynamical critical exponent than the local one: also in the case of dilution critical slowing down is more pronounced for the Metropolis algorithm. However, the striking feature of our estimates is that they suggest that dilution leads to decrease of the dynamical critical exponent for the cluster algorithms. This phenomenon is quite opposite to the local dynamics, where dilution enhances critical slowing down.Comment: 24 pages, 16 figures, style file include

Predictions of bond percolation thresholds for the kagom\'e and Archimedean (3,122)(3,12^2) lattices

Here we show how the recent exact determination of the bond percolation threshold for the martini lattice can be used to provide approximations to the unsolved kagom\'e and (3,12^2) lattices. We present two different methods, one of which provides an approximation to the inhomogeneous kagom\'e and (3,12^2) bond problems, and the other gives estimates of $p_c$ for the homogeneous kagom\'e (0.5244088...) and (3,12^2) (0.7404212...) problems that respectively agree with numerical results to five and six significant figures.Comment: 4 pages, 5 figure

Oriented Percolation in One-Dimensional 1/|x-y|^2 Percolation Models

We consider independent edge percolation models on Z, with edge occupation probabilities p_ = p if |x-y| = 1, 1 - exp{- beta / |x-y|^2} otherwise. We prove that oriented percolation occurs when beta > 1 provided p is chosen sufficiently close to 1, answering a question posed in [Commun. Math. Phys. 104, 547 (1986)]. The proof is based on multi-scale analysis.Comment: 19 pages, 2 figures. See also Commentary on J. Stat. Phys. 150, 804-805 (2013), DOI 10.1007/s10955-013-0702-

Dynamic Critical Behavior of the Chayes-Machta Algorithm for the Random-Cluster Model. I. Two Dimensions

We study, via Monte Carlo simulation, 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 ferromagnet to non-integer q \ge 1. We consider spatial dimension d=2 and 1.25 \le q \le 4 in steps of 0.25, on lattices up to 1024^2, and obtain estimates for the dynamic critical exponent z_{CM}. We present evidence that when 1 \le q \lesssim 1.95 the Ossola-Sokal conjecture z_{CM} \ge \beta/\nu is violated, though we also present plausible fits compatible with this conjecture. We show that the Li-Sokal bound z_{CM} \ge \alpha/\nu is close to being sharp over the entire range 1 \le q \le 4, but is probably non-sharp by a power. As a byproduct of our work, we also obtain evidence concerning the corrections to scaling in static observables.Comment: LaTeX2e, 75 pages including 26 Postscript figure

Rejection-free Geometric Cluster Algorithm for Complex Fluids

We present a novel, generally applicable Monte Carlo algorithm for the simulation of fluid systems. Geometric transformations are used to identify clusters of particles in such a manner that every cluster move is accepted, irrespective of the nature of the pair interactions. The rejection-free and non-local nature of the algorithm make it particularly suitable for the efficient simulation of complex fluids with components of widely varying size, such as colloidal mixtures. Compared to conventional simulation algorithms, typical efficiency improvements amount to several orders of magnitude

Dynamic Critical Behavior of the Swendsen-Wang Algorithm: The Two-Dimensional 3-State Potts Model Revisited

We have performed a high-precision Monte Carlo study of the dynamic critical behavior of the Swendsen-Wang algorithm for the two-dimensional 3-state Potts model. We find that the Li-Sokal bound ($\tau_{int,E} \geq const \times C_H$) is almost but not quite sharp. The ratio $\tau_{int,E} / C_H$ seems to diverge either as a small power ($\approx 0.08$) or as a logarithm.Comment: 35 pages including 3 figures. Self-unpacking file containing the LaTeX file, the needed macros (epsf.sty, indent.sty, subeqnarray.sty, and eqsection.sty) and the 3 Postscript figures. Revised version fixes a normalization error in \xi (with many thanks to Wolfhard Janke for finding the error!). To be published in J. Stat. Phys. 87, no. 1/2 (April 1997