Critical values in a long-range percolation on spaces like fractals
The paper considers a long-range percolation process on a general countable discrete space \(X\) where the distinct points \(x\), \(y\) are connected with probability \(p(x,y)\) satisfying \[ \lim_{\rho(x,y)\to\infty} {p(x,y)\over\beta \rho(x,y)^{-\alpha}}=1, \] and \(\rho(x,y)\) is a symmetric function playing the role of the distance. It is also assumed that the space \(X\) together with the function \(\rho\) satisfies a type of volume growth condition, which is used to define ``dimension'' \(D\) of the space \(X\). Under some additional assumptions on the structure of \((X,\rho)\), the author proves that 1. If \(\alpha < D\), then all points in \(X\) are connected with probability 1. 2. If \(\alpha > D\), then there exists \(p\) such that there is no infinite percolation cluster. 3. If \(\alpha > 2D\), then there is no infinite percolation cluster. This parallels results obtained for the long-range percolation on \(\mathbb Z\) [e.g. \textit{M. Aizenman} and \textit{C. M. Newman}, Commun. Math. Phys. 107, 611--647 (1986; Zbl 0613.60097) or \textit{C. M. Newman} and \textit{L. S. Schulman}, Commun. Math. Phys. 104, 547--571 (1986; Zbl 0604.60097)]. The results are then applied to the long-range percolation on some fractals.
scientific article; zbMATH DE number 5127775
scientific article