Supercritical percolation on finite transitive graphs I: Uniqueness of the giant component

Let (Gn)n≥1=((Vn,En))n≥1 be a sequence of finite, connected, vertex-transitive graphs with volume tending to infinity. We say that a sequence of parameters (pn)n≥1 in [0,1] is supercritical with respect to Bernoulli bond percolation PpG if there exists ε>0 and N<∞ such that P(1−ε)pnGn(the largest cluster contains at leastε|Vn|vertices)≥ε for every n≥N with pn<1. We prove that if (Gn)n≥1 is sparse, meaning that the degrees are sublinear in the number of vertices, then the supercritical giant cluster is unique with high probability in the sense that if (pn)n≥1 is supercritical, then limn→∞PpnGn(the second largest cluster contains at leastc|Vn|vertices)=0 for every c>0. This result is new even under the stronger hypothesis that (Gn)n≥1 has uniformly bounded vertex degrees, in which case it verifies a conjecture of Benjamini (2001). Previous work of many authors had established the same theorem for complete graphs, tori, hypercubes, and bounded degree expander graphs, each using methods that are highly specific to the examples they treated. We also give a complete solution to the problem of supercritical uniqueness for dense vertex-transitive graphs, establishing a simple, necessary, and sufficient isoperimetric condition for uniqueness to hold.