Given an infinite transitive graph (such as the lattice Z^d), consider the random subgraph obtained by independently retaining each edge with probability p. This model undergoes a phase transition as p varies across a critical value p_c that marks the emergence of an infinite component. A classical result states that for each p < p_c, the probability that a given vertex belongs to a component of size at least n decays exponentially in n. Sahar Diskin, Ritvik Ramanan Radhakrishnan, Benny Sudakov, Vincent Tassion, and I have recently proved the analogous result for p > p_c.