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Register a new userTruncated Connectivities in a highly supercritical anisotropic percolation model
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Bernardo Nunes Borges de Lima
Publication date
2013
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English
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We consider an anisotropic bond percolation model on $mathbb{Z}^2$, with $textbf{p}=(p_h,p_v)in [0,1]^2$, $p_v>p_h$, and declare each horizontal (respectively vertical) edge of $mathbb{Z}^2$ to be open with probability $p_h$(respectively $p_v$), and otherwise closed, independently of all other edges. Let $x=(x_1,x_2) in mathbb{Z}^2$ with $0<x_1<x_2$, and $x=(x_2,x_1)in mathbb{Z}^2$. It is natural to ask how the two point connectivity function $prob({0leftrightarrow x})$ behaves, and whether anisotropy in percolation probabilities implies the strict inequality $prob({0leftrightarrow x})>prob({0leftrightarrow x})$. In this note we give an affirmative answer in the highly supercritical regime.
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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.
In this work we study the Poisson Boolean model of percolation in locally compact Polish metric spaces and we prove the invariance of subcritical and supercritical phases under mm-quasi-isometries. In other words, we prove that if the Poisson Boolean model of percolation is subcritical or supercritical (or exhibits phase transition) in a metric space M which is mm-quasi-isometric to a metric space N, then these phases also exist for the Poisson Boolean model of percolation in N. Then we apply these results to understand the phenomenon of phase transition in a large family of metric spaces. Indeed, we study the Poisson Boolean model of percolation in the context of Riemannian manifolds, in a large family of nilpotent Lie groups and in Cayley graphs. Also, we prove the existence of a subcritical phase in Gromov spaces with bounded growth at some scale.
We consider dynamical percolation on the $d$-dimensional discrete torus of side length $n$, $mathbb{Z}_n^d$, where each edge refreshes its status at rate $mu=mu_nle 1/2$ to be open with probability $p$. We study random walk on the torus, where the walker moves at rate $1/(2d)$ along each open edge. In earlier work of two of the authors with A. Stauffer, it was shown that in the subcritical case $p<p_c(mathbb{Z}^d)$, the (annealed) mixing time of the walk is $Theta(n^2/mu)$, and it was conjectured that in the supercritical case $p>p_c(mathbb{Z}^d)$, the mixing time is $Theta(n^2+1/mu)$; here the implied constants depend only on $d$ and $p$. We prove a quenched (and hence annealed) version of this conjecture up to a poly-logarithmic factor under the assumption $theta(p)>1/2$. Our proof is based on percolation results (e.g., the Grimmett-Marstrand Theorem) and an analysis of the volume-biased evolving set process; the key point is that typically, the evolving set has a substantial intersection with the giant percolation cluster at many times. This allows us to use precise isoperimetric properties of the cluster (due to G. Pete) to infer rapid growth of the evolving set, which in turn yields the upper bound on the mixing time.
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 $mathbb{Z}^d$.
We study the critical probability for the metastable phase transition of the two-dimensional anisotropic bootstrap percolation model with $(1,2)$-neighbourhood and threshold $r = 3$. The first order asymptotics for the critical probability were recently determined by the first and second authors. Here we determine the following sharp second and third order asymptotics: [ p_cbig( [L]^2,mathcal{N}_{(1,2)},3 big) ; = ; frac{(log log L)^2}{12log L} , - , frac{log log L , log log log L}{ 3log L} + frac{left(log frac{9}{2} + 1 pm o(1) right)log log L}{6log L}. ] We note that the second and third order terms are so large that the first order asymptotics fail to approximate $p_c$ even for lattices of size well beyond $10^{10^{1000}}$.
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