Accessibility Percolation with Rough Mount Fuji labels

TL;DR

This paper analyzes the conditions under which infinite increasing paths exist in a graph with vertices labeled by a combination of random variables and distance-based bias, revealing phase transitions in different graph structures.

Contribution

It provides an exact characterization of the critical bias parameter for accessibility percolation on Galton-Watson trees and establishes phase transition bounds on lattice graphs.

Findings

Exact critical value $ heta_c$ for Galton-Watson trees

Bounds on $ heta_c$ for general trees

Existence of phase transition on $ abla^n$ lattice graphs

Abstract

Consider an infinite, rooted, connected graph where each vertex is labelled with an independent and identically distributed Uniform(0,1) random variable, plus a parameter $\theta$ times its distance from the root $\rho$. That is, we label vertex $v$ with $X_v = U_v + \theta d(\rho,v)$. We say that accessibility percolation occurs if there is an infinite path started from $\rho$ along which the vertex labels are increasing. When the graph is a Bienaym\'e-Galton-Watson tree, we give an exact characterisation of the critical value $\theta_c$ such that there is accessibility percolation with positive probability if and only if $\theta>\theta_c$. We also give more explicit bounds on the value of $\theta_c$. The lower bound holds for a much more general class of trees. When the graph is the lattice $\mathbb{Z}^n$ for $n\ge 2$, we show that there is a non-trivial phase transition and give some first bounds on $\theta_c$. To do this we introduce a novel coupling with oriented percolation.