Supercritical percolation on finite transitive graphs I: Uniqueness of the giant component
Let $(G_n)_{n \geq 1} = ((V_n,E_n))_{n \geq 1}$ be a sequence of finite, connected, vertex-transitive graphs with volume tending to infinity. We say that a sequence of parameters $(p_n)_{n \geq 1}$ in $[0,1]$ is supercritical with respect to Bernoulli bond percolation $\mathbb P_p^G$ if there exists $\varepsilon >0$ and $N<\infty$ such that \[ \mathbb P_{(1-\varepsilon)p_n}^{G_n} \left( \text{the largest cluster contains at least $\varepsilon |V_n|$ vertices}\right) \geq \varepsilon \] for every $n\geq N$ with $p_n <1$. We prove that if $(G_n)_{n \geq 1}$ 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 $(p_n)_{n \geq 1}$ is supercritical then \[ \lim_{n\to\infty}\mathbb P_{p_n}^{G_n} \left( \text{the second largest cluster contains at least $c|V_n|$ vertices} \right) = 0 \] for every $c>0$. This result is new even under the stronger hypothesis that $(G_n)_{n \geq 1}$ 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.