Truncation of long-range percolation models with square non-summable interactions

Abstract

We consider some problems related to the truncation question in long-range percolation. Probabilities are given that certain long-range oriented bonds are open; assuming that these probabilities are not summable, 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. This question is still open if the set of vertices is Z 2 . We give some conditions under which the answer is affirmative. One of these results generalizes a previous result in [ Alves, Hilário, de Lima, Valesin, Journ. Stat. Phys. 122, 972 (2017)].