Kesten’s Incipient Infinite Cluster

Abstract

We introduce the high-dimensional incipient infinite cluster, which is an infinite cluster at the critical threshold \(p_c\) as constructed by Kesten. Since the results in the previous chapters imply that there is no infinite cluster at the critical point, this construction relies on a limiting argument. We first motivate the incipient infinite cluster, and then discuss its construction and properties in high dimensions. We also discuss its robustness, in the sense that various natural limiting schemes give rise to the same measure.