# Monotonicity and phase diagram for multi-range percolation on oriented trees

## TL;DR

This paper investigates the critical conditions for percolation on oriented trees with multiple bond ranges, revealing how the percolation threshold depends on bond length and probabilities.

## Contribution

It introduces a detailed analysis of the critical curve in multi-range percolation on oriented trees and demonstrates its monotonicity with respect to bond length.

## Key findings

- The critical curve decreases as the length of long bonds increases.
- Percolation properties are characterized for different bond probability pairs.
- The study provides insights into phase transitions in multi-range percolation models.

## Abstract

We consider Bernoulli bond percolation on oriented regular trees, where besides the usual short bonds, all bonds of a certain length are added. Independently, short bonds are open with probability $p$ and long bonds are open with probability $q$. We study properties of the critical curve which delimits the set of pairs $(p,q)$ for which there are almost surely no infinite paths. We also show that this curve decreases with respect to the length of the long bonds.