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

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