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Let Pp be the probability measure on the configurations of occupied and vacant vertices of a two-dimensional graph N, under which all vertices are independently occupied (respectively vacant) with probability p (respectively l p ) . Let p~ be the critical probability for this system and W the occupied cluster of some fixed vertex w o. We show that for many(More)
Abstract Shnerb et al. (2000), (2001) studied the following system of interacting particles on Z: There are two kinds of particles, called A-particles and B-particles. The A-particles perform continuous time simple random walks, independently of each other. The jumprate of each A-particle is DA. The B-particles perform continuous time simple random walks(More)
We prove that the critical probability for bond or site percolation on Z is asymptotically equal to 1/(2d) as d → ∞. If the probability of a bond (respectively site) to be occupied is γ/(2d) with γ > 1, then for the bond model the percolation probability converges as d → ∞ to the strictly positive solution y(γ) of the equation y = 1− exp(−γy). In the site(More)
We consider critical site percolation on the triangular lattice, that is, we choose X(v) = 0 or 1 with probability 1/2 each, independently for all vertices v of the triangular lattice. We say that a word (ξ1, ξ2, . . . ) ∈ {0, 1}N is seen in the percolation configuration if there exists a selfavoiding path (v1, v2, . . . ) on the triangular lattice with(More)
The uniform spanning forest (USF) in Z is the weak limit of random, uniformly chosen, spanning trees in [−n,n]. Pemantle (1991) proved that the USF consists a.s. of a single tree if and only if d ≤ 4. We prove that any two components of the USF in Z are adjacent a.s. if 5 ≤ d ≤ 8, but not if d ≥ 9. More generally, let N(x, y) be the minimum number of edges(More)