Two-dimensional volume-frozen percolation: deconcentration and prevalence of mesoscopic clusters

Abstract

Frozen percolation on the binary tree was introduced by Aldous around fifteen years ago, inspired by sol-gel transitions. We investigate a version of the model on the triangular lattice, where connected components stop growing ("freeze") as soon as they contain at least $N$ vertices, for some parameter $N \geq 1$. This process has a substantially different behavior from the diameter-frozen process, studied in previous works: in particular, we show that many (more and more as $N \to \infty$) frozen clusters surrounding the origin appear successively, each new cluster having a diameter much smaller than the previous one. This separation of scales is instrumental, and it helps to approximate the process in sufficiently large (but not too large), as a function of $N$, finite domains by a Markov chain. This allows us to establish a deconcentration property for the sizes of the holes of the frozen clusters around the origin. For the full-plane process, we then show that it can be compared to the process in large finite domains, so that the deconcentration property also holds in this case. In particular, we obtain that with high probability (as $N \to \infty$), the origin does not belong to a frozen cluster in the final configuration. This work requires new properties for near-critical percolation, which we develop along the way, and which are interesting in their own right: in particular, an asymptotic formula involving the percolation probability $θ(p)$ as $p \searrow p_c$, and regularity properties for large holes in the infinite cluster. Volume-frozen percolation also gives insight into forest-fire processes, where lightning hits independently each tree with a small rate, and burns its entire connected component immediately.