Accessibility percolation with backsteps

Consider a graph in which each site is endowed with a value called fitness. A path in the graph is said to be “open” or “accessible” if the fitness values along that path is strictly increasing. We say that there is accessibility percolation between two sites when such a path between them exists. Motivated by the so called House-of-Cards model from evolutionary biology, we consider this question on the L-hypercube {0, 1} where the fitness values are independent random variables. We show that, in the large L limit, the probability that an accessible path exists from an arbitrary starting point to the (random) fittest site is no more than x1/2 = 1− 2 sinh −1(2) = 0.27818 . . . and we conjecture that this probability does converge to x1/2. More precisely, there is a phase transition on the value of the fitness x of the starting site: assuming that the fitnesses are uniform in [0, 1], we show that, in the large L limit, there is almost surely no path to the fittest site if x > x1/2 and we conjecture that there are almost surely many paths if x < x1/2. If one conditions on the fittest site to be on the opposite corner of the starting site rather than being randomly chosen, the picture remains the same but with the critical point being now x1 = 1 − sinh−1(1) = 0.11863 . . .. Along the way, we obtain a large L estimation for the number of self-avoiding paths joining two opposite corners of the L-hypercube. 2000 Mathematics Subject Classification: Primary 60J80; Secondary 60G18