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

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, Hilario, de Lima, Valesin, Journ. Stat. Phys. {\bf 122}, 972 (2017)].