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 +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 ppc.
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Journal Annales Scientifiques de l'École Normale Supérieure
van den Berg, J, Kiss, D, & Nolin, P. (2017). Two-dimensional volume-frozen percolation: Deconcentration and prevalence of mesoscopic clusters. Annales Scientifiques de l'École Normale Supérieure, 51(4), 1017–1084. doi:10.24033/asens.2371