THERMOGAMAS - Bootstrap Percolation on Rhombus Tilings (Solène Esnay)

Talk given during the THERMOGAMAS Workshop.

Abstract: In this talk, we introduce dynamical (bootstrap) percolation, where a polygonal tiling of the plane has each tile independently colored 0 or 1 following a measure mu; then the 1 state propagates step by step to tiles adjacent to at least two 1-colored tiles. We are interested in the asymptotic behavior of this system, which is a cellular automaton with a spreading 1 state: is the entire tiling asymptotically invaded by 1s from mu-almost every initial configuration?

After summarizing how this is known on ℤ², we answer this question positively for a wide variety of tilings (all rhombus tilings, with a focus during the presentation on Penrose tilings) and measures (a large class that includes nontrivial Bernoulli measures). We do so with a careful study of what stable non-invading patches of 1s would look like, then use probabilistic arguments to explain why this situation mu-almost never happens.