0–1 law for invasion on uniformly repetitive rhombus tilings

Establish that for the adjacency graph G=(V,E) of any uniformly repetitive (uniformly recurrent) rhombus tiling, and for any monotone, freezing percolation process F acting on configurations in {0,1}^V, the invasion event I ⊆ {0,1}^V (the set of initial configurations whose F-dynamics infects every vertex) has Bernoulli probability μ(I) ∈ {0,1} for every Bernoulli product measure μ on {0,1}^V.

Background

The paper proves almost-sure invasion for 2-neighbour bootstrap percolation on adjacency graphs of rhombus tilings, but notes that classical 0–1 laws relying on symmetry and ergodicity generally fail in this non-translationally invariant setting. Appendix examples show percolation processes on certain tilings without a 0–1 law.

To recover a 0–1 law in a broader tiling setting, the authors propose focusing on sufficiently regular rhombus tilings—specifically uniformly repetitive ones—and ask for a 0–1 law for the invasion event under Bernoulli measures and general monotone, freezing percolation dynamics.