 Infinite percolation cluster

An infinite percolation cluster is an open connected component of a graph with infinitely many graph vertices. On a countable translation-invariant lattice, a zero root percolation probability implies that no infinite percolation cluster exists almost surely, by the countable union over all graph vertices. A positive root probability implies almost-sure existence by translation ergodicity of Bernoulli percolation; uniqueness is an additional theorem.