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

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, Hilario, de Lima, Valesin, Journ. Stat. Phys. 122, 972 (2017)].