On the truncated anisotropic long-range percolation on \(\mathbb Z^2\)
The following infinite-range bond percolation process on \(\mathbb Z^2\) is studied. Each \(x\in \mathbb Z^2\) is connected to both nearest neighbours in the vertical direction with probability \(\varepsilon >0\); each \(x\in \mathbb Z^2\) is connected to each vertex \(x{+\atop -}(i,0)\) with probability \(p_i\geq 0\), \(i\geq 1\); all different connections are independent. Given \(\varepsilon >0\) and let the truncated process \(K\) be defined from the original one by replacing \(p_i\), \(i> K\), by zero. The main result says that, if \(p_i\geq \frac {1}{i\ln i}\) for \(i\) sufficiently large, then for every \(\varepsilon >0\) there is a \(K\) such that the probability of the open cluster of the origin being infinite is positive for the corresponding truncated process. The method of proof is a reduction to anisotropic nearest neighbour Bernoulli percolation process with \(p_V+p_H>1\), where \(p_V\) corresponds to the vertical and \(p_H\) to the horizontal connections. The reduction uses some renormalization procedure in the horizontal direction. The study is motivated by the open problem for infinite-range percolation processes in \(\mathbb Z^d\) asking if the condition \(\sum _{y\in \mathbb Z^d} p_{\langle x,y\rangle}=\infty \) for every \(x\in \mathbb Z^d\) is sufficient for the existence of infinite open cluster of the origin for some truncated processes.
scientific article; zbMATH DE number 1554319