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Register a new userA remark on monotonicity in Bernoulli bond Percolation
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Bernardo Nunes Borges de Lima
Publication date
2015
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English
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Consider an anisotropic independent bond percolation model on the $d$-dimensional hypercubic lattice, $dgeq 2$, with parameter $p$. We show that the two point connectivity function $P_{p}({(0,dots,0)leftrightarrow (n,0,dots,0)})$ is a monotone function in $n$ when the parameter $p$ is close enough to 0. Analogously, we show that truncated connectivity function $P_{p}({(0,dots,0)leftrightarrow (n,0,dots,0), (0,dots,0) leftrightarrowinfty})$ is also a monotone function in $n$ when $p$ is close to 1.
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We study a dependent site percolation model on the $n$-dimensional Euclidean lattice where, instead of single sites, entire hyperplanes are removed independently at random. We extend the results about Bernoulli line percolation showing that the model undergoes a non-trivial phase transition and proving the existence of a transition from exponential to power-law decay within some regions of the subcritical phase.
We prove that for Bernoulli percolation on $mathbb{Z}^d$, $dgeq 2$, the percolation density is an analytic function of the parameter in the supercritical interval. For this we introduce some techniques that have further implications. In particular, we prove that the susceptibility is analytic in the subcritical interval for all transitive short- or long-range models, and that $p_c^{bond} <1/2$ for certain families of triangulations for which Benjamini & Schramm conjectured that $p_c^{site} leq 1/2$.
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.
In this note we study some properties of infinite percolation clusters on non-amenable graphs. In particular, we study the percolative properties of the complement of infinite percolation clusters. An approach based on mass-transport is adapted to show that for a large class of non-amenable graphs, the graph obtained by removing each site contained in an infinite percolation cluster has critical percolation threshold which can be arbitrarily close to the critical threshold for the original graph, almost surely, as p approaches p_c. Closely related is the self-destructive percolation process, introduced by J. van den Berg and R. Brouwer, for which we prove that an infinite cluster emerges for any small reinforcement.
We prove that the heavy clusters are indistinguishable for Bernoulli percolation on quasi-transitive nonunimodular graphs. As an application, we show that the uniqueness threshold of any quasi-transitive graph is also the threshold for connectivity decay. This resolves a question of Lyons and Schramm (1999) in the Bernoulli percolation case and confirms a conjecture of Schonmann (2001). We also prove that every infinite cluster of Bernoulli percolation on a nonamenable quasi-transitive graph is transient almost surely.
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