A note on truncated long-range percolation with heavy tails on oriented graphs

TL;DR

This paper investigates long-range oriented percolation on a lattice with heavy-tailed connection probabilities, demonstrating percolation persists despite the removal of long edges when certain conditions are met.

Contribution

It establishes that percolation remains possible with truncated long-range edges under heavy-tailed probability distributions, extending understanding of long-range percolation models.

Findings

Percolation persists after removing edges longer than a large threshold k.

Heavy-tailed connection probabilities ensure percolation despite truncation.

Results also apply to long-range contact processes on  lattices.

Abstract

We consider oriented long-range percolation on a graph with vertex set $\mathbb{Z}^d \times \mathbb{Z}_+$ and directed edges of the form $\langle (x,t), (x+y,t+1)\rangle$, for $x,y$ in $\mathbb{Z}^d$ and $t \in \mathbb{Z}_+$. Any edge of this form is open with probability $p_y$, independently for all edges. Under the assumption that the values $p_y$ do not vanish at infinity, we show that there is percolation even if all edges of length more than $k$ are deleted, for $k$ large enough. We also state the analogous result for a long-range contact process on $\mathbb{Z}^d$.