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

Abstract

We consider some problems related to the truncation question in long-range percolation. It is given probabilities that certain long-range oriented bonds are open; assuming that this 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 ${\mathbb{Z}}^2$. We give some conditions in which the answer is affirmative. One of these results generalize the previous result in [Alves, Hilário, de Lima, Valesin, Journ. Stat. Phys. {\bf 122}, 972 (2017)].

References in corpus (2)

• Mean-field driven first-order phase transitions in systems with long-range interactions
• A note on truncated long-range percolation with heavy tails on oriented graphs

Cited by in corpus (1)

• Long-range contact process and percolation on a random lattice