Supercritical sharpness of percolation

Abstract

We prove that for supercritical percolation on every infinite transitive graph, the probability that the origin belongs to a finite cluster of size at least $n$ decays exponentially in $Φ(n)$, where $Φ$ is the isoperimetric function of the graph.

Figures (2)

• Figure 1: On the left, the blue balls are where the seed layer was revealed: each light blue ball was found to contain at least one closed edge, whereas the dark blue ball was found to be entirely open, forming a seed. The green region is the $p$-cluster of the seed, which connects to infinity and touches the peach-coloured set $S$. On the right, we have repeated the process of finding a seed and exploring its cluster twice more. The pink edges represent the sprinkling used to glue each new green cluster to the existing green clusters: the light pink edges are closed, and the dark pink edges are open.
• Figure 2: The dashed brown line is midway between the peach set $S$ and the green set $K$ in the sense that any $r$-ball centred on this line has high probability to connect to both $\partial^{+}\! S\setminus K$ and to $K$. This line cannot be too short because together with the brown edges making up $\partial S\cap \partial K$, we have a cutset from $S$ to $K$ and hence to infinity. Therefore, even if we are given a not-too-big region $D$ (not displayed), we can find many $r$-balls centred along the dashed line that are disjoint from each other and from $D$. We define $\mathsf {Seeds}(D,K)$ to be the sequence of such balls.

Theorems & Definitions (34)

• Theorem 1
• Theorem 2
• Corollary 1.1
• Proposition 2.1
• proof : Proof of Theorem \ref{['thm: main']} given Proposition \ref{['prop: main']}
• Lemma 4.1
• proof
• Lemma 4.2
• proof
• Lemma 4.3