Truncated long-range percolation on oriented graphs

TL;DR

This paper investigates the conditions under which long-range oriented percolation retains a positive probability of occurring after truncating long-range bonds, providing new insights into long-range percolation and contact processes.

Contribution

It establishes conditions ensuring positive percolation probability after truncation in long-range oriented graphs, advancing understanding of long-range percolation models.

Findings

Positive percolation probability under certain conditions

Conditions linking infinite sum of probabilities to percolation after truncation

Translation of results to long-range contact processes

Abstract

We consider different problems within the general theme of long-range percolation on oriented graphs. Our aim is to settle the so-called truncation question, described as follows. We are given probabilities that certain long-range oriented bonds are open; assuming that the sum of these probabilities is infinite, 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. We give some conditions in which the answer is affirmative. We also translate some of our results on oriented percolation to the context of a long-range contact process.