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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 further assume that the site percolation is insertion-tolerant and a.s.~there exists a unique infinite 0-cluster, then a.s.~there are no infinite 1-clusters. The proof is based on the analysis of a class of delicately constructed interfaces between clusters and contours. Applied to the case of i.i.d.~Bernoulli site percolation on infinite, connected, locally finite, transitive, planar graphs, these results solve two conjectures of Benjamini and Schramm (Conjectures 7 and 8 in cite{bs96}) in 1996.
We study infinite ``$+$ or ``$-$ clusters for an Ising model on an connected, transitive, non-amenable, planar, one-ended graph $G$ with finite vertex degree. If the critical percolation probability $p_c^{site}$ for the i.i.d.~Bernoulli site percolat
We consider the discrete Boolean model of percolation on graphs satisfying a doubling metric condition. We study sufficient conditions on the distribution of the radii of balls placed at the points of a Bernoulli point process for the absence of perc
In this article we study the sharpness of the phase transition for percolation models defined on top of planar spin systems. The two examples that we treat in detail concern the Glauber dynamics for the Ising model and a Dynamic Bootstrap process. Fo
Let $mathbb{G}=left(mathbb{V},mathbb{E}right)$ be the graph obtained by taking the cartesian product of an infinite and connected graph $G=(V,E)$ and the set of integers $mathbb{Z}$. We choose a collection $mathcal{C}$ of finite connected subgraphs o
We consider different problems within the general theme of long-range percolation on oriented graphs. Our aim is to settle the so-called truncation question, described as follows. We are given probabilities that certain long-range oriented bonds are