Loading…

Incipient Infinite Percolation Clusters in 2d

We study several kinds of large critical percolation clusters in two dimensions. We show that from the microscopic (lattice scale) perspective these clusters can be described by Kesten's incipient infinite cluster (IIC), as was conjectured by Aizenman. More specifically, we establish this for i...

Full description

Saved in:
Bibliographic Details
Published in:The Annals of probability 2003-01, Vol.31 (1), p.444-485
Main Author: JARAI, Antal A
Format: Article
Language:English
Subjects:
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:We study several kinds of large critical percolation clusters in two dimensions. We show that from the microscopic (lattice scale) perspective these clusters can be described by Kesten's incipient infinite cluster (IIC), as was conjectured by Aizenman. More specifically, we establish this for incipient spanning clusters, large clusters in a finite box and the inhomogeneous model of Chayes, Chayes and Durrett. Our results prove the equivalence of several natural definitions of the IIC. We also show that for any k ≥ 1 the difference in size between the kth and (k + 1)st largest critical clusters in a finite box goes to infinity in probability as the size of the box goes to infinity. In addition, the distribution of the Chayes-Chayes-Durrett cluster is shown to be singular with respect to the IIC.
ISSN:0091-1798
2168-894X
DOI:10.1214/aop/1046294317