Uniqueness of the infinite cluster in Bernoulli line percolation

Determine whether Bernoulli line percolation on the integer lattice Z^d almost surely has a unique infinite cluster whenever an infinite cluster exists; equivalently, establish uniqueness of the infinite cluster in the supercritical regime for Bernoulli line percolation.

Background

The paper studies uniqueness of the infinite cluster for a broad class of dependent site-percolation models generated by monotone automata applied to stochastically increasing particle configurations, covering models such as the Abelian sandpile, activated random walk, and bootstrap percolation.

Classical uniqueness proofs (e.g., Burton–Keane) rely on insertion tolerance or finite energy, properties often absent in dependent models. The authors develop a new approach to prove uniqueness for their class of models despite the lack of insertion tolerance.

They note that for some dependent percolation models outside their framework, uniqueness remains unresolved. In particular, for Bernoulli line percolation, the question of whether the supercritical phase has a unique infinite cluster is, to their knowledge, open.