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Register a new userA note on inhomogeneous percolation on ladder graphs
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Bernardo Nunes Borges de Lima
Publication date
2019
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English
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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.
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We consider oriented long-range percolation on a graph with vertex set $mathbb{Z}^d times mathbb{Z}_+$ and directed edges of the form $langle (x,t), (x+y,t+1)rangle$, for $x,y$ in $mathbb{Z}^d$ and $t in mathbb{Z}_+$. Any edge of this form is open with probability $p_y$, independently for all edges. Under the assumption that the values $p_y$ do not vanish at infinity, we show that there is percolation even if all edges of length more than $k$ are deleted, for $k$ large enough. We also state the analogous result for a long-range contact process on $mathbb{Z}^d$.
A bootstrap percolation process on a graph G is an infection process which evolves in rounds. Initially, there is a subset of infected nodes and in each subsequent round every uninfected node which has at least r infected neighbours becomes infected and remains so forever. The parameter r > 1 is fixed. We consider this process in the case where the underlying graph is an inhomogeneous random graph whose kernel is of rank 1. Assuming that initially every vertex is infected independently with probability p > 0, we provide a law of large numbers for the number of vertices that will have been infected by the end of the process. We also focus on a special case of such random graphs which exhibit a power-law degree distribution with exponent in (2,3). The first two authors have shown the existence of a critical function a_c(n) such that a_c(n)=o(n) with the following property. Let n be the number of vertices of the underlying random graph and let a(n) be the number of the vertices that are initially infected. Assume that a set of a(n) vertices is chosen randomly and becomes externally infected. If a(n) << a_c(n), then the process does not evolve at all, with high probability as n grows, whereas if a(n)>> a_c(n), then with high probability the final set of infected vertices is linear. Using the techniques of the previous theorem, we give the precise asymptotic fraction of vertices which will be eventually infected when a(n) >> a_c (n) but a(n) = o(n). Note that this corresponds to the case where p approaches 0.
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 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.
Consider Bernoulli bond percolation a locally finite, connected graph $G$ and let $p_{mathrm{cut}}$ be the threshold corresponding to a first-moment method lower bound. Kahn (textit{Electron. Comm. Probab. Volume 8, 184-187.} (2003)) constructed a counter-example to Lyons conjecture of $p_{mathrm{cut}}=p_c$ and proposed a modification. Here we give a positive answer to Kahns modified question. The key observation is that in Kahns modification, the new expectation quantity also appears in the differential inequality of one-arm events. This links the question to a lemma of Duminil-Copin and Tassion (textit{Comm. Math. Phys. Volume 343, 725-745.} (2016)). We also study some applications for Bernoulli percolation on periodic trees.
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