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 \[ \begin{ split} \lim_{ n \to \infty} \mathbb{ P}_{ (1+\varepsilon)p_n}^{ G_n} \left( \left\lVert K_1 \right\rVert \geq \alpha \right) &= 1 \quad \text{ for some $\alpha > 0$, and} \lim_{ n \to \infty} \mathbb{ P}_{ (1-\varepsilon)p_n}^{ G_n} \left( \left\lVert K_1 \right\rVert \geq \alpha \right) &= 0 \quad \text{ for all $\alpha > 0$}. \end{ split}\] 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 [Van22] together with our recent work with Hutchcroft on the uniqueness of the giant cluster [EH21].
Categories: math.PR math-ph math.CO math.MP