TWO-DIMENSIONAL VOLUME-FROZEN PERCOLATION: EXCEPTIONAL SCALES

• 成果类型: Article
• 署名作者: van den Berg, Jacob; Nolin, Pierre
• 署名单位: Centrum Wiskunde & Informatica (CWI); Swiss Federal Institutes of Technology Domain; ETH Zurich
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/16-AAP1198
• 发表日期: 2017
• 页码: 91-108
• 关键词: brownian intersection exponents; square lattice; plane; clusters; VALUES

摘要

We study a percolation model on the square lattice, where clusters freeze (stop growing) as soon as their volume (i.e., the number of sites they contain) gets larger than N, the parameter of the model. A model where clusters freeze when they reach diameter at least N was studied in van den Berg, de Lima and Nolin [Random Structures Algorithms 40 (2012) 220-226] and Kiss [Probab. Theory Related Fields 163 (2015) 713-768]. Using volume as a way to measure the size of a cluster instead of diameter leads, for large N, to a quite different behavior (contrary to what happens on the binary tree van den Berg, de Lima and Nolin (2012), where the volume model and the diameter model are asymptotically the same). In particular, we show the existence of a sequence of exceptional length scales.