Existence of a Percolation Threshold on Finite Transitive Graphs

Abstract

Abstract Let $(G_{n})$ be a sequence of finite connected vertex-transitive graphs with volume tending to infinity. We say that a sequence of parameters $(p_{n})$ is a percolation threshold if for every $\varepsilon> 0$, the proportion $\left \lVert K_{1}\right \rVert $ of vertices contained in the largest cluster under bond percolation ${\mathbb {P}}_{p}^{G}$ satisfies both $$ \begin{align*} \begin{split}{} \lim_{n \to \infty} {\mathbb{P}}_{(1+\varepsilon)p_{n}}^{G_{n}} \left( \left\lVert K_{1}\right\rVert \geq \alpha \right) &= 1 \qquad \textrm{for some}\ \alpha> 0, \textrm{and}\\ \lim_{n \to \infty} {\mathbb{P}}_{(1-\varepsilon)p_{n}}^{G_{n}} \left( \left\lVert K_{1}\right\rVert \geq \alpha \right) &= 0 \qquad \textrm{for all}\ \alpha > 0. \end{split} \end{align*}$$We prove that $(G_{n})$ has a percolation threshold if and only if $(G_{n})$ does not contain a particular infinite collection of pathological subsequences of dense graphs. Our argument uses an adaptation of Vanneuville’s new proof of the sharpness of the phase transition for infinite graphs via couplings [27] together with our recent work with Hutchcroft on the uniqueness of the giant cluster [15].