2011/10/24 by Artem Sapozhnikov, Sapozhnikov, Artem
Mathematics · #60K35 #82B27 #82B43 #82C43 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60K35 #msc:82B27 #msc:82B43 #msc:82C43
paper · pdf · doi:10.48550/arxiv.1110.5269
6 pages
arxiv created 2011/10/24 · arxiv updated 2011/10/25
This note is motivated by results in arXiv:math/0608132 and arXiv:0806.2425 about global relations between the invasion percolation cluster (IPC) and the incipient infinite cluster (IIC) on regular trees and on two dimensional lattices, respectively. Namely, that the laws of the two objects are mutually singular, and, in the case of regular trees, that the IIC stochastically dominates the IPC. We prove that on two dimensional lattices, the IIC does not stochastically dominate the IPC. This is the first example showing that the relation between the IIC and IPC is significantly different on trees and in two dimensions.