Accessibility Percolation with Rough Mount Fuji labels
Diana De Armas Bellon, Matthew I. Roberts

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 times its distance from the root . That is, we label vertex with . We say that accessibility percolation occurs if there is an infinite path started from 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 such that there is accessibility percolation with positive probability if and only if . We also give more explicit bounds on the value of . The lower bound holds for a much more general class of trees. When the graph is the lattice for , we show that there is a non-trivial phase transition and give…
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