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Register a new userBoolean Percolation on Doubling Graphs
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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 percolation, provided that the retention parameter of the underlying point process is small enough. We exhibit three families of interesting graphs where the main result of this work holds. Finally, we give sufficient conditions for ergodicity of the discrete Boolean model of percolation.
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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 percolation on $G$ is less than $frac{1}{2}$, we find an explicit region for the coupling constant of the Ising model such that there are infinitely many infinite ``$+$-clusters and infinitely many infinite ``$-$-clusters, while the random cluster representation of the Ising model has no infinite 1-clusters. If $p_c^{site}>frac{1}{2}$, we obtain a lower bound for the critical probability in the random cluster representation of the Ising model in terms of $p_c^{site}$.
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 of $G$ and consider a model of Bernoulli bond percolation on $mathbb{G}$ which assigns probability $q$ of being open to each edge whose projection onto $G$ lies in some subgraph of $mathcal{C}$ and probability $p$ to every other edge. We show that the critical percolation threshold $p_{c}left(qright)$ is a continuous function in $left(0,1right)$, provided that the graphs in $mathcal{C}$ are well-spaced in $G$ and their vertex sets have uniformly bounded cardinality. This generalizes a recent result due to Szabo and Valesin.
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 open; assuming that the sum of these probabilities is infinite, we ask if the probability of percolation is positive when we truncate the graph, disallowing bonds of range above a possibly large but finite threshold. We give some conditions in which the answer is affirmative. We also translate some of our results on oriented percolation to the context of a long-range contact process.
We consider the Bernoulli Boolean discrete percolation model on the d-dimensional integer lattice. 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 percolation, provided that the intensity of the underlying point process is small enough. We also study a Harris graphical procedure to construct, forward in time, particle systems with interactions of infinite range under the assumption that the corresponding generator admits a Kalikow-type decomposition. We do so by using the subcriticality of the boolean model of discrete percolation.
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