Two-dimensional volume-frozen percolation: exceptional scales

TL;DR

This paper investigates a percolation model on a square lattice where clusters freeze upon reaching a certain volume, revealing unique behaviors and exceptional scales that differ from diameter-based models.

Contribution

It introduces and analyzes a volume-based freezing rule in percolation, uncovering new phenomena and exceptional length scales not observed in diameter-based models.

Findings

Existence of a sequence of exceptional length scales.

Volume-based freezing leads to different behavior than diameter-based models.

Contrasts with behavior on binary trees.

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 earlier papers. 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, 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.