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

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

Extended Defects in the Potts-Percolation Model of a Solid: Renormalization Group and Monte Carlo Analysis

We extend the model of a 2$d$ solid to include a line of defects. Neighboring atoms on the defect line are connected by ?springs? of different strength and different cohesive energy with respect to the rest of the system. Using the Migdal-Kadanoff renormalization group we show that the elastic energy is an irrelevant field at the bulk critical point. For zero elastic energy this model reduces to the Potts model. By using Monte Carlo simulations of the 3- and 4-state Potts model on a square lattice with a line of defects, we confirm the renormalization-group prediction that for a defect interaction larger than the bulk interaction the order parameter of the defect line changes discontinuously while the defect energy varies continuously as a function of temperature at the bulk critical temperature.Comment: 13 figures, 17 page

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

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

Center clusters in the Yang-Mills vacuum

Properties of local Polyakov loops for SU(2) and SU(3) lattice gauge theory at finite temperature are analyzed. We show that spatial clusters can be identified where the local Polyakov loops have values close to the same center element. For a suitable definition of these clusters the deconfinement transition can be characterized by the onset of percolation in one of the center sectors. The analysis is repeated for different resolution scales of the lattice and we argue that the center clusters have a continuum limit.Comment: Table added. Final version to appear in JHE

Percolation on the average and spontaneous magnetization for q-states Potts model on graph

We prove that the q-states Potts model on graph is spontaneously magnetized at finite temperature if and only if the graph presents percolation on the average. Percolation on the average is a combinatorial problem defined by averaging over all the sites of the graph the probability of belonging to a cluster of a given size. In the paper we obtain an inequality between this average probability and the average magnetization, which is a typical extensive function describing the thermodynamic behaviour of the model

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

Potts-Percolation-Gauss Model of a Solid

We study a statistical mechanics model of a solid. Neighboring atoms are connected by Hookian springs. If the energy is larger than a threshold the "spring" is more likely to fail, while if the energy is lower than the threshold the spring is more likely to be alive. The phase diagram and thermodynamic quantities, such as free energy, numbers of bonds and clusters, and their fluctuations, are determined using renormalization-group and Monte-Carlo techniques.Comment: 10 pages, 12 figure

Poisson approximations for the Ising model

A $d$-dimensional Ising model on a lattice torus is considered. As the size $n$ of the lattice tends to infinity, a Poisson approximation is given for the distribution of the number of copies in the lattice of any given local configuration, provided the magnetic field $a=a(n)$ tends to $-\infty$ and the pair potential $b$ remains fixed. Using the Stein-Chen method, a bound is given for the total variation error in the ferromagnetic case.Comment: 25 pages, 1 figur