Two-dimensional volume-frozen percolation: Exceptional scales

Abstract

We study a percolation model on the square lattice, where clusters “freeze” (stop growing) as soon as their volume (i.e., the number of sites they contain) gets larger than $N$, the parameter of the model. A model where clusters freeze when they reach diameter at least $N$ was studied in van den Berg, de Lima and Nolin [Random Structures Algorithms 40 (2012) 220–226] and Kiss [Probab. Theory Related Fields 163 (2015) 713–768]. Using volume as a way to measure the size of a cluster—instead of diameter—leads, for large $N$, to a quite different behavior (contrary to what happens on the binary tree van den Berg, Kiss and Nolin [Electron. Commun. Probab. 17 (2012) 1–11], where the volume model and the diameter model are “asymptotically the same”). In particular, we show the existence of a sequence of “exceptional” length scales.