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

van den Berg, Jacob; Kiss, Demeter; Nolin, Pierre

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 4, pp. 1017-1084.

La percolation gelée sur l'arbre binaire a été introduite par Aldous [1], inspiré par les transitions sol-gel. Nous étudions une version de ce modèle sur le réseau triangulaire, pour laquelle les composantes connexes arrêtent de croître (« gèlent ») dès qu'elles contiennent au moins N sommets, où N est un paramètre (typiquement grand).

Pour le processus dans certains domaines finis, nous prouvons une « séparation d'échelles », et nous l'utilisons pour démontrer une propriété de déconcentration. Ensuite, pour le processus dans tout le plan, nous établissons une comparaison précise avec le processus dans des domaines finis adéquats, et nous obtenons qu'avec grande probabilité (lorsque N→∞), l'origine appartient, dans la configuration finale, à une composante connexe mésoscopique, c'est-à-dire, une composante qui contient un grand nombre de sommets, mais beaucoup moins que N (et qui est donc non-gelée).

Pour ce travail, nous développons de nouvelles propriétés intéressantes de la percolation presque-critique, en particulier des formules asymptotiques faisant intervenir la probabilité de percolation θ(p) et la longueur caractéristique L(p) quand p↘pc.

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→∞), 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.

Classification : 60K35, 82B43.
Keywords: Frozen percolation, near-critical percolation, deconcentration inequalities, sol-gel transitions, pattern formation, self-organized criticality.
Mot clés : Percolation gelée, percolation presque-critique, inégalités de déconcentration, transitions sol-gel, formation de motifs, criticalité auto-organisée.

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