Phase Transition for Accessibility Percolation on Hypercubes

In this paper, we consider accessibility percolation on hypercubes, i.e., we place i.i.d. uniform [0, 1] random variables on vertices of a hypercube, and study whether there is a path connecting two vertices such that the values of these random variables increase along the path. We establish a sharp phase transition depending on the difference of the values at the two endpoints and determine the critical window of the phase transition. Our result completely resolves a conjecture of Berestycki et al. (Accessibility percolation with backsteps. Preprint, available at http://arxiv.org/abs/1401.6894).