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:bsz:15-qucosa-223724 |
| Date | 25 April 2017 |
| Creators | Basu, Deepan |
| Contributors | Universität Leipzig,, Prof. Dr. Artem Sapozhnikov, Prof. Dr. Markus Heydenreich, Prof. Dr. Artem Sapozhnikov |
| Publisher | Universitätsbibliothek Leipzig |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | doc-type:doctoralThesis |
| Format | application/pdf |
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