Characterization of the threshold for multi-range percolation on oriented trees

TL;DR

This paper characterizes the percolation threshold for a multirange model on oriented trees using polynomial roots and Galton-Watson processes, providing exact critical points for specific cases.

Contribution

It introduces a novel method to determine the percolation threshold as the first positive root of a polynomial using multi-type Galton-Watson processes.

Findings

Exact critical point for k=2 case

Threshold characterized as polynomial root

Method applicable to other multirange models

Abstract

We give a characterization of the percolation threshold for a multirange model on oriented trees, as the first positive root of a polynomial, with the use of a multi-type Galton-Watson process. This gives in particular the exact value of the critical point for the model studied in [2] and [3] for k = 2.