The scaling limit of the incipient infinite cluster in high-dimensional percolation. II. Integrated super-Brownian excursion

Abstract

For independent nearest-neighbor bond percolation on Zd with d≫6, we prove that the incipient infinite cluster’s two-point function and three-point function converge to those of integrated super-Brownian excursion (ISE) in the scaling limit. The proof is based on an extension of the new expansion for percolation derived in a previous paper, and involves treating the magnetic field as a complex variable. A special case of our result for the two-point function implies that the probability that the cluster of the origin consists of n sites, at the critical point, is given by a multiple of n−3/2, plus an error term of order n−3/2−ε with ε>0. This is a strong version of the statement that the critical exponent δ is given by δ=2.