Stable directions determine critical thresholds on multigrid dual rhombus tilings

Determine whether, for percolation processes on {0,1}^T where T is a multigrid dual rhombus tiling, the set of stable directions of the process F uniquely determines the classification of the critical percolation threshold p_c into the three regimes p_c=0, p_c=1, or 0<p_c<1, analogous to the known direction-based classification on Z^2.

Background

On Z2, a well-developed theory classifies percolation processes (including U-bootstrap) by their stable directions, yielding a dichotomy between trivial thresholds (p_c ∈ {0,1}) and non-trivial thresholds (p_c>0).

The authors propose extending this direction-based classification to sufficiently regular rhombus tilings, focusing on multigrid dual tilings (canonical cut-and-project rhombus tilings such as Penrose and Ammann–Beenker), where chains are nearly straight and geometric regularity may support analogous results.

References

We similarly conjecture that the non-triviality of the critical threshold on sufficiently regular rhombus tilings is determined by the stable directions. Do the stable directions of $F$ on $T$ determine the trichotomy between trivial percolations thresholds and non-trivial percolation threshold?

— Bootstrap percolation on rhombus tilings  (2409.02520 - Esnay et al., 2024) in Section Open questions