TWO-DIMENSIONAL VOLUME-FROZEN PERCOLATION: DECONCENTRATION AND PREVALENCE OF MESOSCOPIC CLUSTERS

Frozen percolation on the binary tree was introduced by Aldous [1], 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, where N is a (typically large) parameter. For the process in certain finite domains, we show a "separation of scales" and use this to prove a deconcentration property. Then, for the full-plane process, we prove an accurate comparison to the process in suitable finite domains, and obtain that, with high probability (as N -> infinity), the origin belongs in the final configuration to a mesoscopic cluster, i.e., a cluster which contains many, but much fewer than N, vertices (and hence is non-frozen). For this work we develop new interesting properties for near-critical percolation, including asymptotic formulas involving the percolation probability theta (p) and the characteristic length L (p) as p SE arrow p(c) .