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 ω ( v ) = η ( v ) + θ ⋅ d ( v ) , where θ is a positive number called the drift, d is the distance to the source of the graph and η ( v ) are i.i.d. random variables. In this paper, we determine values of θ for having RMF accessibility percolation on the hypercube and the two-dimensional lattices L 2 and L a l t 2 .