Abstracts Category : Other

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

by Deepan Basu

This thesis generalizes and interprets Kesten's Incipient Infinite Cluster (IIC) measure in two ways. Firstly we generalize Jrai'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.Advisors/Committee Members: Sapozhnikov, Artem (advisor), Heydenreich, Markus (referee), Sapozhnikov, Artem (referee).