Existence of a percolation threshold on finite transitive graphs

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 \[ \begin{split} \lim_{n \to \infty} \mathbb{P}_{(1+\varepsilon)p_n}^{G_n} \left( \left\lVert K_1 \right\rVert \geq α\right) &= 1 \quad \text{for some $α> 0$, and} \lim_{n \to \infty} \mathbb{P}_{(1-\varepsilon)p_n}^{G_n} \left( \left\lVert K_1 \right\rVert \geq α\right) &= 0 \quad \text{for all $α> 0$}. \end{split}\] 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].

Theorems & Definitions (24)

• Definition 1
• Theorem 2
• Theorem 3: Bollobás, Borgs, Chayes, Riordan 2010
• Corollary 4
• Proposition 5
• Definition 6
• Theorem 7: Easo and Hutchcroft, 2021
• Lemma 8
• proof
• Lemma 9