Accessibility percolation on n-trees

TL;DR

This paper studies accessibility percolation on n-trees, deriving exact probabilities for accessible paths and identifying thresholds based on the growth rate of branching and bias in random variables.

Contribution

It provides an asymptotically exact expression for accessibility probability in n-trees and reveals phase transitions related to branching growth and bias.

Findings

Probability tends to 1 if branching grows faster than linearly with height.

Probability tends to 0 if branching grows slower than linearly.

A finite bias threshold induces a percolation transition.

Abstract

Accessibility percolation is a new type of percolation problem inspired by evolutionary biology. To each vertex of a graph a random number is assigned and a path through the graph is called accessible if all numbers along the path are in ascending order. For the case when the random variables are independent and identically distributed, we derive an asymptotically exact expression for the probability that there is at least one accessible path from the root to the leaves in an $n$-tree. This probability tends to 1 (0) if the branching number is increased with the height of the tree faster (slower) than linearly. When the random variables are biased such that the mean value increases linearly with the distance from the root, a percolation threshold emerges at a finite value of the bias.