Accessibility percolation and first-passage site percolation on the unoriented binary hypercube

Inspired by biological evolution, we consider the following socalled accessibility percolation problem: The vertices of the unoriented ndimensional binary hypercube are assigned independent U(0, 1) weights, referred to as fitnesses. A path is considered accessible if fitnesses are strictly increasing along it. We prove that the probability that the global fitness maximum is accessible from the all zeroes vertex converges to 1 − 1 2 ln ( 2 + √ 5 ) as n → ∞. Moreover, we prove that if one conditions on the location of the fitness maximum being v̂, then provided v̂ is not too close to the all zeroes vertex in Hamming distance, the probability that v̂ is accessible converges to a function of this distance divided by n as n→∞. This resolves a conjecture by Berestycki, Brunet and Shi in almost full generality. As a second result we show that, for any graph, accessibility percolation can equivalently be formulated in terms of first-passage site percolation. This connection is of particular importance for the study of accessibility percolation on trees.