No directed fractal percolation in zero area

We show that fractal (or "Mandelbrot") percolation in two dimensions produces a set containing no directed paths, when the set produced has zero area. This improves a similar result by the first author in the case of constant retention probabilities to the case of retention probabilities approaching 1

Graphical Representations for Ising Systems in External Fields

A graphical representation based on duplication is developed that is suitable for the study of Ising systems in external fields. Two independent replicas of the Ising system in the same field are treated as a single four-state (Ashkin-Teller) model. Bonds in the graphical representation connect the Ashkin-Teller spins. For ferromagnetic systems it is proved that ordering is characterized by percolation in this representation. The representation leads immediately to cluster algorithms; some applications along these lines are discussed.Comment: 13 pages amste

Invaded cluster algorithm for equilibrium critical points

A new cluster algorithm based on invasion percolation is described. The algorithm samples the critical point of a spin system without a priori knowledge of the critical temperature and provides an efficient way to determine the critical temperature and other observables in the critical region. The method is illustrated for the two- and three-dimensional Ising models. The algorithm equilibrates spin configurations much faster than the closely related Swendsen-Wang algorithm.Comment: 13 pages RevTex and 4 Postscript figures. Submitted to Phys. Rev. Lett. Replacement corrects problem in printing figure

The geometry of fractal percolation

A well studied family of random fractals called fractal percolation is discussed. We focus on the projections of fractal percolation on the plane. Our goal is to present stronger versions of the classical Marstrand theorem, valid for almost every realization of fractal percolation. The extensions go in three directions: {itemize} the statements work for all directions, not almost all, the statements are true for more general projections, for example radial projections onto a circle, in the case $\dim_H >1$, each projection has not only positive Lebesgue measure but also has nonempty interior. {itemize}Comment: Survey submitted for AFRT2012 conferenc

Cluster Monte Carlo study of multi-component fluids of the Stillinger-Helfand and Widom-Rowlinson type

Phase transitions of fluid mixtures of the type introduced by Stillinger and Helfand are studied using a continuum version of the invaded cluster algorithm. Particles of the same species do not interact, but particles of different types interact with each other via a repulsive potential. Examples of interactions include the Gaussian molecule potential and a repulsive step potential. Accurate values of the critical density, fugacity and magnetic exponent are found in two and three dimensions for the two-species model. The effect of varying the number of species and of introducing quenched impurities is also investigated. In all the cases studied, mixtures of $q$-species are found to have properties similar to $q$-state Potts models.Comment: 25 pages, 5 figure