On the asymptotic density in a one-dimensional self-organized critical forest-fire model

Consider the following forest-fire model where the possible locations of trees are the sites of $bold$. Each site has two possible states: `vacant' or `occupied'. Vacant sites become occupied at rate $1$. At each site ignition (by lightning) occurs at ignition rate $lambda$, the parameter of the model. When a site is ignited, its occupied cluster becomes vacant instantaneously. In the literature similar models have been studied for discrete time, finite (but large) volume and finite (but large) speed at which the fire spreads out. The most interesting behaviour seems to occur when the ignition rate goes to $0$, as this allows clusters to grow very large before being hit by lightning. It has been stated by Drossel, Clar and Schwabl (1993) that then (in our notation) the density of vacant sites (in equilibrium) is of order $1 / log(1 / lambda)$. Their proof uses a `scaling ansatz' and is not rigorous. We give, for our version of the model, a rigorous and mathematically more natural proof. Our proof shows that regardless of the initial configuration, already after time of order $log(1 / lambda)$ the density is of the above mentioned order $1 / log(1 / lambda)$. We also point out how our proof can be modified for the model studied by Drossel et al

The lowest crossing in 2D critical percolation

We study the following problem for critical site percolation on the triangular lattice. Let A and B be sites on a horizontal line e separated by distance n. Consider, in the half-plane above e, the lowest occupied crossing R from the half-line left of A to the half-line right of B. We show that the probability that R has a site at distance smaller than m from AB is of order (log (n/m))^{-1}, uniformly in 1 <= m < n/2. Much of our analysis can be carried out for other two-dimensional lattices as well.Comment: 16 pages, Latex, 2 eps figures, special macros: percmac.tex. Submitted to Annals of Probabilit

High-dimensional graphical networks of self-avoiding walks

We use the lace expansion to analyse networks of mutually-avoiding self-avoiding walks, having the topology of a graph. The networks are defined in terms of spread-out self-avoiding walks that are permitted to take large steps. We study the asymptotic behaviour of networks in the limit of widely separated network branch points, and prove Gaussian behaviour for su#ciently spread-out networks on in dimensions d&gt;4

Infinite volume limit of the Abelian sandpile model in dimensions d >= 3

We study the Abelian sandpile model on Z^d. In dimensions at least 3 we prove existence of the infinite volume addition operator, almost surely with respect to the infinite volume limit mu of the uniform measures on recurrent configurations. We prove the existence of a Markov process with stationary measure mu, and study ergodic properties of this process. The main techniques we use are a connection between the statistics of waves and uniform two-component spanning trees and results on the uniform spanning tree measure on Z^d.Comment: First version: LaTeX; 29 pages. Revised version: LaTeX; 29 pages. The main result of the paper has been extended to all dimensions at least 3, with a new and simplyfied proof of finiteness of the two-component spanning tree. Second revision: LaTeX; 32 page

Minimal configurations and sandpile measures

We give a new simple construction of the sandpile measure on an infinite graph G, under the sole assumption that each tree in the Wired Uniform Spanning Forest on G has one end almost surely. For, the so called, generalized minimal configurations the limiting probability on G exists even without this assumption. We also give determinantal formulas for minimal configurations on general graphs in terms of the transfer current matrix.Comment: 16 pages; the introduction has been expanded and minor corrections have been mad

Outlets of 2D invasion percolation and multiple-armed incipient infinite clusters

We study invasion percolation in two dimensions, focusing on properties of the outlets of the invasion and their relation to critical percolation and to incipient infinite clusters (IIC's). First we compute the exact decay rate of the distribution of both the weight of the kth outlet and the volume of the kth pond. Next we prove bounds for all moments of the distribution of the number of outlets in an annulus. This result leads to almost sure bounds for the number of outlets in a box B(2^n) and for the decay rate of the weight of the kth outlet to p_c. We then prove existence of multiple-armed IIC measures for any number of arms and for any color sequence which is alternating or monochromatic. We use these measures to study the invaded region near outlets and near edges in the invasion backbone far from the origin.Comment: 38 pages, 10 figures, added a thorough sketch of the proof of existence of IIC's with alternating or monochromatic arms (with some generalizations