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...
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| Veröffentlicht in: | The Annals of probability 2003-01, Vol.31 (1), p.444-485 |
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| container_title | The Annals of probability |
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| creator | JARAI, Antal A |
| description | 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. |
| doi_str_mv | 10.1214/aop/1046294317 |
| format | Article |
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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.</abstract><cop>Hayward, CA</cop><pub>Institute of Mathematical Statistics</pub><doi>10.1214/aop/1046294317</doi><tpages>42</tpages><oa>free_for_read</oa></addata></record> |
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| source | JSTOR Mathematics & Statistics; Jstor Complete Legacy; EZB-FREE-00999 freely available EZB journals; Project Euclid Complete |
| subjects | 60K35 82B43 critical phenomena Critical point phenomena Cylinders Exact sciences and technology Gauge theory incipient infinite cluster Infinity Integers Mathematical lattices Mathematics Open star clusters Percolation Physics Probability and statistics Probability theory and stochastic processes Rectangles Sciences and techniques of general use spanning cluster Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications) Statistical physics, thermodynamics, and nonlinear dynamical systems Terminology Thermodynamics Vertices |
| title | Incipient Infinite Percolation Clusters in 2d |
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