RMF accessibility percolation on oriented graphs

TL;DR

This paper investigates the conditions under which the Rough Mount Fuji (RMF) accessibility percolation occurs on specific graphs like hypercubes and 2D lattices, extending percolation theory inspired by evolutionary biology.

Contribution

It determines the critical values of the drift parameter for RMF accessibility percolation on hypercube and 2D lattice graphs, advancing understanding of this percolation model.

Findings

Identifies drift thresholds for hypercube graphs.

Establishes conditions for percolation on 2D lattices.

Extends accessibility percolation theory to new graph classes.

Abstract

Accessibility percolation is a new type of percolation problem inspired by evolutionary biology: a random number, called its fitness, is assigned to each vertex of a graph, then a path in the graph is accessible if fitnesses are strictly increasing through it. In the Rough Mount Fuji (RMF) model the fitness function is defined on the graph as $\omega(v)=\eta(v)+\theta\cdot d(v)$, where $\theta$ is a positive number called the drift, $d$ is the distance to the source of the graph and $\eta(v)$ are i.i.d. random variables. In this paper we determine values of $\theta$ for having RMF accessibility percolation on the hypercube and the two-dimensional lattices $\mathbb{L}^2$ and $\mathbb{L}^2_{alt}$.