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Register a new userA note on truncated long-range percolation with heavy tails on oriented graphs
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Bernardo Nunes Borges de Lima
Publication date
2017
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English
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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 $mathbb{Z}^d$.
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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.
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 directed last-passage percolation on the random graph G = (V,E) where V = Z and each edge (i,j), for i < j, is present in E independently with some probability 0 < p <= 1. To every present edge (i,j) we attach i.i.d. random weights v_{i,j} > 0. We are interested in the behaviour of w_{0,n}, which is the maximum weight of all directed paths from 0 to n, as n tends to infinity. We see two very different types of behaviour, depending on whether E[v_{i,j}^2] is finite or infinite. In the case where E[v_{i,j}^2] is finite we show that the process has a certain regenerative structure, and prove a strong law of large numbers and, under an extra assumption, a functional central limit theorem. In the situation where E[v_{i,j}^2] is infinite we obtain scaling laws and asymptotic distributions expressed in terms of a continuous last-passage percolation model on [0,1]; these are related to corresponding results for two-dimensional last-passage percolation with heavy-tailed weights obtained by Hambly and Martin.
We consider Bernoulli bond percolation on oriented regular trees, where besides the usual short bonds, all bonds of a certain length are added. Independently, short bonds are open with probability $p$ and long bonds are open with probability $q$. We study properties of the critical curve which delimits the set of pairs $(p,q)$ for which there are almost surely no infinite paths. We also show that this curve decreases with respect to the length of the long bonds.
We consider an inhomogeneous oriented percolation model introduced by de Lima, Rolla and Valesin. In this model, the underlying graph is an oriented rooted tree in which each vertex points to each of its $d$ children with `short edges, and in addition, each vertex points to each of its $d^k$ descendant at a fixed distance $k$ with `long edges. A bond percolation process is then considered on this graph, with the prescription that independently, short edges are open with probability $p$ and long edges are open with probability $q$. We study the behavior of the critical curve $q_c(p)$: we find the first two terms in the expansion of $q_c(p)$ as $k to infty$, and prove that the critical curve lies strictly above the critical curve of a related branching process, in the relevant parameter region. We also prove limit theorems for the percolation cluster in the supercritical, subcritical and critical regimes.
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