Bootstrap percolation on rhombus tilings

Abstract

2-boostrap percolation on a graph is a diffusion process where a vertex gets infected whenever it has at least 2 infected neighbours, and then stays infected forever. It has been much studied on the infinite grid for random Bernoulli initial configurations, starting from the seminal result of van Enter that establishes that the entire grid gets almost surely entirely infected for any non-trivial initial probability of infection. In this paper, we generalize this result to any adjacency graph of any rhombus tiling of the plane, including aperiodic ones like Penrose tilings. We actually show almost sure infection of the entire graph for a larger class of measure than non-trivial Bernoulli ones. Our proof strategy combines a geometry toolkit for infected clusters based on chain-convexity, and uniform probabilistic bounds on particular geometric patterns that play the role of 0-1 laws or ergodicity, which are not available in our settings due to the lack of symmetry of the graph considered.

Figures (10)

• Figure 1: A fragment of a Penrose rhombus tiling.
• Figure 2: In light grey a chain of rhombuses, in darker gray and with the starting tile in bold a half-chain of rhombuses.
• Figure 3: Adjacent tiles along a chain and vector.
• Figure 4: On the left a stable half plane of $1$s (represented in dark gray), on the right a simple stable finite cluster of $1$s.
• Figure 5: A chain $\chi$ touching twice a stable set $S$. We prove that the domain $D$ must be empty.

Theorems & Definitions (49)

• Remark 1
• Lemma 1
• proof
• Lemma 2
• proof
• Definition 1: Chain $\chi$
• Lemma 3: Chain crossing kenyon1993
• Lemma 4: Uniform monotonicity
• proof
• Remark 2: Rhombus and parallelograms