2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/67873Let 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^-α. It is shown that at α=1/2 there is a phase transition for the metric distortion between H_n and H_{n,p}. For α<1/2, asymptotically there is a map from H_n to H_{n,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.11 pagesProbabilityCombinatorics60C05, 05C80, 68R10text