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Register a new userA note on the phase transition for independent alignment percolation
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We study the independent alignment percolation model on $mathbb{Z}^d$ introduced by Beaton, Grimmett and Holmes [arXiv:1908.07203]. It is a model for random intersecting line segments defined as follows. First the sites of $mathbb{Z}^d$ are independently declared occupied with probability $p$ and vacant otherwise. Conditional on the configuration of occupied vertices, consider the set of all line segments that are parallel to the coordinate axis, whose extremes are occupied vertices and that do not traverse any other occupied vertex. Declare independently the segments on this set open with probability $lambda$ and closed otherwise. All the edges that lie on open segments are also declared open giving rise to a bond percolation model in $mathbb{Z}^d$. We show that for any $d geq 2$ and $p in (0,1]$ the critical value for $lambda$ satisfies $lambda_c(p)<1$ completing the proof that the phase transition is non-trivial over the whole interval $(0,1]$. We also show that the critical curve $p mapsto lambda_c(p)$ is continuous at $p=1$, answering a question posed by the authors in [arXiv:1908.07203].
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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. For both of these models we prove that their phase transition is continuous and sharp, providing also quantitative estimates on the two point connectivity. The techniques that we develop in this work can be applied to a variety of different dependent percolation models and we discuss some of the problems that can be tackled in a similar fashion. In the last section of the paper we present a long list of open problems that would require new ideas to be attacked.
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.
The existence (or not) of infinite clusters is explored for two stochastic models of intersecting line segments in $d ge 2$ dimensions. Salient features of the phase diagram are established in each case. The models are based on site percolation on ${mathbb Z}^d$ with parameter $pin (0,1]$. For each occupied site $v$, and for each of the $2d$ possible coordinate directions, declare the entire line segment from $v$ to the next occupied site in the given direction to be either blue or not blue according to a given stochastic rule. In the one-choice model, each occupied site declares one of its $2d$ incident segments to be blue. In the independent model, the states of different line segments are independent.
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.
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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