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

TL;DR

This paper investigates whether percolation persists in long-range models when bonds beyond a certain range are truncated, providing conditions under which percolation remains positive, extending previous results.

Contribution

It offers new conditions ensuring positive percolation probability after truncation in long-range percolation models, generalizing earlier findings.

Findings

Identifies conditions for positive percolation probability after truncation.

Extends previous results to broader classes of long-range percolation models.

Provides partial answers to open questions in the field.

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 $\Z^2$. We give some conditions in which the answer is affirmative. One of these results generalize the previous result in [Alves, Hil\'ario, de Lima, Valesin, Journ. Stat. Phys. {\bf 122}, 972 (2017)].