Phase transition for accessibility percolation on hypercubes

Li Li

Published 2015-02-26Version 1

In this paper, we consider accessibility percolation on hypercubes, i.e., we place i.i.d.\ uniform random variables on vertices of a hypercube, and study whether there is a path (possibly with back steps) 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, Brunet, and Shi (2014). Our work on accessibility percolation is motivated by the NK fitness model in biological evolution. Indeed, placing i.i.d.\ random variables on the hypercube amounts to the special case for NK fitness model for $N = K$. A second result of our work concerns the global maximum for NK fitness model, and proves that in the case whe fitness variables are Gaussian it is asymptotically equivalent to the maximum of i.i.d.\ variables if and only if $K\to \infty$ as $N\to \infty$.