A Phase Transition for the Metric Distortion of Percolation on the Hypercube

Let H_n be the hypercube {0,1}^n, and let H_{n,p} denote the same graph with Bernoulli bond percolation with parameter p=n^-\alpha. It is shown that at \alpha=1/2 there is a phase transition for the metric distortion between H_n and H_{n,p}. For \alpha<1/2, asymptotically there is a map from H_n to H_{n,p} with constant distortion (depending only on \alpha). For \alpha>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.