Bonjour à Tou-tes,

Cette semaine, nous écoutons Solène ESNAY parler de percolation dans les
pavages de Penrose, à 11h à la Frumam à Saint-Charles

Bootstrap Percolation on Rhombus Tilings

In this talk, we introduce dynamical (bootstrap) percolation, where a
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?

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 with probabilistic arguments as to
why this situation mu-almost never happens.