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

Understand

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.