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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.
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
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, w
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 co
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 sh
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 d