Accessibility percolation and first-passage percolation on the hypercube

Abstract: In this thesis, we consider two percolation models on the n-dimensional binary hypercube, known as accessibility percolation and first-passage percolation. First-passage percolation randomly assigns non-negative weights, called passage times, to the edges of a graph and considers the minimal total weight of a path between given end-points. This quantity is called the first-passage time. Accessibility percolation is a biologically inspired model which has appeared in the mathematical literature only recently. Here, the vertices of a graph are randomly assigned heights, or fitnesses, and a path is considered accessible if strictly ascending. We let and denote the all zeroes and all ones vertices respectively. A natural simplification of both models is the restriction to oriented paths, i.e. paths that only flip 0:s to 1:s. Paper I considers the existence of such accessible paths between and for fitnesses assigned according to the so-called House-of-Cards and Rough Mount Fuji models. In Paper II we consider the first-passage time between and in the case of independent standard exponential passage times. It is previously known that, in the oriented case, this quantity tends to 1 in probability as n tends to infinity. We show that without this restriction, the limit is instead . By adapting ideas in Paper II to unoriented accessibility percolation, we are able to determine a limiting probability for the existence of accessible paths from to the global fitness maximum. This is presented in Paper III.