TY - JOUR

T1 - Two-Dimensional volume-frozen percolation

T2 - Deconcentration and prevalence of mesoscopic clusters

AU - van den Berg, Jacob

AU - Kiss, Demeter

AU - Nolin, Pierre

PY - 2018/7/1

Y1 - 2018/7/1

N2 - 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 Òseparation of scalesÓ and use this to prove a deconcentration property. Then, for the full-plane process, we prove an accurate comparison to the process in suitable +nite domains, and obtain that, with high probability (as N→), the origin belongs in the nal con+guration 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 asymp-totic formulas involving the percolation probability θ(p) and the characteristic length L(p) as p → pc.

AB - 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 Òseparation of scalesÓ and use this to prove a deconcentration property. Then, for the full-plane process, we prove an accurate comparison to the process in suitable +nite domains, and obtain that, with high probability (as N→), the origin belongs in the nal con+guration 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 asymp-totic formulas involving the percolation probability θ(p) and the characteristic length L(p) as p → pc.

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U2 - 10.24033/asens.2371

DO - 10.24033/asens.2371

M3 - Article

AN - SCOPUS:85055169092

SN - 0012-9593

VL - 51

SP - 1017

EP - 1084

JO - Annales Scientifiques de l'Ecole Normale Superieure

JF - Annales Scientifiques de l'Ecole Normale Superieure

IS - 4

ER -