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 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.