RMF accessibility percolation on oriented graphs

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 \(\mathbb{L}^2_{alt}\).