Two-dimensional volume-frozen percolation: exceptional scales

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 [14, 6]. 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 [16], 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.