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.
Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:15590 |
Date | 08 March 2017 |
Creators | Basu, Deepan |
Contributors | Sapozhnikov, Artem, Heydenreich, Markus, Universität Leipzig |
Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
Language | English |
Detected Language | English |
Type | doc-type:doctoralThesis, info:eu-repo/semantics/doctoralThesis, doc-type:Text |
Rights | info:eu-repo/semantics/openAccess |
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