RMF accessibility percolation on oriented graphs

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 $\mathbb{L}^2_{alt}$.