Lebowitz Inequalities for Ashkin-Teller Systems

We consider the Ashkin-Teller model with negative four-spin coupling but still in the region where the ground state is ferromagnetic. We establish the standard Lebowitz inequality as well as the extension that is necessary to prove a divergent susceptibility.Comment: Ams-TeX, 12 pages; two references added, final version accepted for publication in Physica

Cardy's Formula for Certain Models of the Bond-Triangular Type

We introduce and study a family of 2D percolation systems which are based on the bond percolation model of the triangular lattice. The system under study has local correlations, however, bonds separated by a few lattice spacings act independently of one another. By avoiding explicit use of microscopic paths, it is first established that the model possesses the typical attributes which are indicative of critical behavior in 2D percolation problems. Subsequently, the so called Cardy-Carleson functions are demonstrated to satisfy, in the continuum limit, Cardy's formula for crossing probabilities. This extends the results of S. Smirnov to a non-trivial class of critical 2D percolation systems.Comment: 49 pages, 7 figure

Convergent Sequences of Dense Graphs I: Subgraph Frequencies, Metric Properties and Testing

We consider sequences of graphs and define various notions of convergence related to these sequences: ``left convergence'' defined in terms of the densities of homomorphisms from small graphs into the graphs of the sequence, and ``right convergence'' defined in terms of the densities of homomorphisms from the graphs of the sequence into small graphs; and convergence in a suitably defined metric. In Part I of this series, we show that left convergence is equivalent to convergence in metric, both for simple graphs, and for graphs with nodeweights and edgeweights. One of the main steps here is the introduction of a cut-distance comparing graphs, not necessarily of the same size. We also show how these notions of convergence provide natural formulations of Szemeredi partitions, sampling and testing of large graphs.Comment: 57 pages. See also http://research.microsoft.com/~borgs/. This version differs from an earlier version from May 2006 in the organization of the sections, but is otherwise almost identica

On the rate of convergence for critical crossing probabilities

For the site percolation model on the triangular lattice and certain generalizations for which Cardy’s Formula has been established we acquire a power law estimate for the rate of convergence of the crossing probabilities to Cardy’s Formula

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

Monte Carlo study of the Widom-Rowlinson fluid using cluster methods

The Widom-Rowlinson model of a fluid mixture is studied using a new cluster algorithm that is a generalization of the invaded cluster algorithm previously applied to Potts models. Our estimate of the critical exponents for the two-component fluid are consistent with the Ising universality class in two and three dimensions. We also present results for the three-component fluid.Comment: 13 pages RevTex and 2 Postscript figure

Vertex Models and Random Labyrinths: Phase Diagrams for Ice-type Vertex Models

We propose a simple geometric recipe for constructing phase diagrams for a general class of vertex models obeying the ice rule. The disordered phase maps onto the intersecting loop model which is interesting in its own right and is related to several other statistical mechanical models. This mapping is also useful in understanding some ordered phases of these vertex models as they correspond to the polymer loop models with cross-links in their vulcanised phase.Comment: 8 pages, 6 figure