Accessibility percolation and first-passage percolation on the hypercube

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