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In this paper we consider independent site percolation in a triangulation of $mathbb{R}^2$ given by adding $sqrt{2}$-long diagonals to the usual graph $mathbb{Z}^2$. We conjecture that $p_c=frac{1}{2}$ for any such graph, and prove it for almost every such graph.
On the square lattice raindrops fall on an edge with midpoint $x$ at rate $|x|_infty^{-alpha}$. The edge becomes open when the first drop falls on it. Let $rho(x,t)$ be the probability that the edge with midpoint $x=(x_1,x_2)$ is open at time $t$ and
In Poisson percolation each edge becomes open after an independent exponentially distributed time with rate that decreases in the distance from the origin. As a sequel to our work on the square lattice, we describe the limiting shape of the component
Bootstrap percolation on a graph is a deterministic process that iteratively enlarges a set of occupied sites by adjoining points with at least $theta$ occupied neighbors. The initially occupied set is random, given by a uniform product measure with
We prove that for a non-amenable, locally finite, connected, transitive, planar graph with one end, any automorphism invariant site percolation on the graph does not have exactly 1 infinite 1-cluster and exactly 1 infinite 0-cluster a.s. If we furthe
Chase-escape percolation is a variation of the standard epidemic spread models. In this model, each site can be in one of three states: unoccupied, occupied by a single prey, or occupied by a single predator. Prey particles spread to neighboring empt