A phase transition for the metric distortion of percolation on the hypercube

Let Hn be the hypercube {0, 1} n, and let Hn,p denote the same graph with Bernoulli bond percolation with parameter p = n−α. It is shown that at α = 1/2 there is a phase transition for the metric distortion between Hn and Hn,p. For α < 1/2, asymptotically there is a map from Hn to Hn,p with constant distortion (depending only on α). For α > 1/2 the distortion tends to infinity as a power of n. We indicate the similarity to the existence of a non-uniqueness phase in the context of infinite nonamenable graphs.