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

TL;DR

This paper proves that in high-dimensional percolation, the critical cluster functions converge to those of integrated super-Brownian excursion, confirming the critical exponent delta equals 2.

Contribution

It extends the analysis of the incipient infinite cluster in high-dimensional percolation by establishing convergence to ISE and determining the critical exponent delta.

Findings

Two-point and three-point functions converge to ISE

Probability of cluster size n scales as n^{-3/2}

Critical exponent delta equals 2

Abstract

For independent nearest-neighbour bond percolation on Z^d 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-\epsilon} with \epsilon >0. This is a strong statement that the critical exponent delta is given by delta =2.