Clusters and Recurrence in the Two-Dimensional Zero-Temperature Stochastic Ising Model

We analyze clustering and (local) recurrence of a standard Markov process model of spatial domain coarsening. The continuous time process, whose state space consists of assignments of +1 or -1 to each site in ${\bf Z}^2$, is the zero-temperature limit of the stochastic homogeneous Ising ferromagnet (with Glauber dynamics): the initial state is chosen uniformly at random and then each site, at rate one, polls its 4 neighbors and makes sure it agrees with the majority, or tosses a fair coin in case of a tie. Among the main results (almost sure, with respect to both the process and initial state) are: clusters (maximal domains of constant sign) are finite for times $t< \infty$, but the cluster of a fixed site diverges (in diameter) as $t \to \infty$; each of the two constant states is (positive) recurrent. We also present other results and conjectures concerning positive and null recurrence and the role of absorbing states.Comment: 16 pages, 1 figur

Two-Dimensional Scaling Limits via Marked Nonsimple Loops

We postulate the existence of a natural Poissonian marking of the double (touching) points of SLE(6) and hence of the related continuum nonsimple loop process that describes macroscopic cluster boundaries in 2D critical percolation. We explain how these marked loops should yield continuum versions of near-critical percolation, dynamical percolation, minimal spanning trees and related plane filling curves, and invasion percolation. We show that this yields for some of the continuum objects a conformal covariance property that generalizes the conformal invariance of critical systems. It is an open problem to rigorously construct the continuum objects and to prove that they are indeed the scaling limits of the corresponding lattice objects.Comment: 25 pages, 5 figure

Monte Carlo study of the hull distribution for the q=1 Brauer model

We study a special case of the Brauer model in which every path of the model has weight q=1. The model has been studied before as a solvable lattice model and can be viewed as a Lorentz lattice gas. The paths of the model are also called self-avoiding trails. We consider the model in a triangle with boundary conditions such that one of the trails must cross the triangle from a corner to the opposite side. Motivated by similarities between this model, SLE(6) and critical percolation, we investigate the distribution of the hull generated by this trail (the set of points on or surrounded by the trail) up to the hitting time of the side of the triangle opposite the starting point. Our Monte Carlo results are consistent with the hypothesis that for system size tending to infinity, the hull distribution is the same as that of a Brownian motion with perpendicular reflection on the boundary.Comment: 21 pages, 9 figure

Probability Theory in Statistical Physics, Percolation, and Other Random Topics: The Work of C. Newman

In the introduction to this volume, we discuss some of the highlights of the research career of Chuck Newman. This introduction is divided into two main sections, the first covering Chuck's work in statistical mechanics and the second his work in percolation theory, continuum scaling limits, and related topics.Comment: 38 pages (including many references), introduction to Festschrift in honor of C.M. Newma