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Generalizations and Interpretations of Incipient Infinite Cluster measure on Planar Lattices and Slabs

This thesis generalizes and interprets Kesten\'s Incipient Infinite Cluster (IIC) measure in two ways. Firstly we generalize Járai\'s result which states that for planar lattices the local configurations around a typical point taken from crossing collection is described by IIC measure. We prove in Chapter 2 that for backbone, lowest crossing and set of pivotals, the same hold true with multiple armed IIC measures. We develop certain tools, namely Russo Seymour Welsh theorem and a strong variant of quasi-multiplicativity for critical percolation on 2-dimensional slabs in Chapters 3 and 4 respectively. This enables us to first show existence of IIC in Kesten\'s sense on slabs in Chapter 4 and prove that this measure can be interpreted as the local picture around a point of crossing collection in Chapter 5.

Identiferoai:union.ndltd.org:DRESDEN/oai:qucosa.de:bsz:15-qucosa-223724
Date25 April 2017
CreatorsBasu, Deepan
ContributorsUniversität Leipzig,, Prof. Dr. Artem Sapozhnikov, Prof. Dr. Markus Heydenreich, Prof. Dr. Artem Sapozhnikov
PublisherUniversitätsbibliothek Leipzig
Source SetsHochschulschriftenserver (HSSS) der SLUB Dresden
LanguageEnglish
Detected LanguageEnglish
Typedoc-type:doctoralThesis
Formatapplication/pdf

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