Two-dimensional volume-frozen percolation: deconcentration and prevalence of mesoscopic clusters

Abstract

htmlabstractFrozen 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 +nite 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→∞), 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 θ(p) and the characteristic length L(p) as p ↘ pc.