Site Percolation on Planar Graphs

TL;DR

This paper proves new properties of site percolation on certain planar graphs, showing the absence of multiple infinite clusters under specific conditions, and resolves two longstanding conjectures in the field.

Contribution

It establishes that non-amenable, planar graphs with one end cannot have exactly one infinite 0- and 1-cluster simultaneously, confirming two conjectures from 1996.

Findings

No exactly one infinite 1- and 0-cluster coexistence.

Under insertion-tolerance, only zero-infinite clusters can exist.

Results confirm two conjectures of Benjamini and Schramm.

Abstract

We prove that for a non-amenable, locally finite, connected, transitive, planar graph with one end, any automorphism invariant site percolation on the graph does not have exactly 1 infinite 1-cluster and exactly 1 infinite 0-cluster a.s. If we further assume that the site percolation is insertion-tolerant and a.s.~there exists a unique infinite 0-cluster, then a.s.~there are no infinite 1-clusters. The proof is based on the analysis of a class of delicately constructed interfaces between clusters and contours. Applied to the case of i.i.d.~Bernoulli site percolation on infinite, connected, locally finite, transitive, planar graphs, these results solve two conjectures of Benjamini and Schramm (Conjectures 7 and 8 in \cite{bs96}) in 1996.