Existence of a percolation threshold on finite transitive graphs

TL;DR

This paper establishes the existence of a percolation threshold on finite transitive graphs by characterizing when such thresholds occur, linking it to the absence of certain dense graph subsequences, and uses advanced probabilistic techniques.

Contribution

It provides a necessary and sufficient condition for the existence of a percolation threshold on finite transitive graphs, extending the understanding of phase transitions in these structures.

Findings

Percolation threshold exists iff no dense pathological subsequences are present.

Uses an adaptation of Vanneuville's proof for phase transition sharpness.

Connects the threshold existence to graph subsequence properties.

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