The scaling limit of the incipient infinite cluster in high-dimensional percolation. I. Critical exponents

TL;DR

This paper establishes the critical exponents eta and delta for high-dimensional percolation models, providing evidence that the incipient infinite cluster's scaling limit is the integrated super-Brownian excursion (ISE).

Contribution

It extends the expansion method to control critical exponents and links the incipient infinite cluster to ISE in high dimensions.

Findings

Proves eta=0 and delta=2 in high dimensions.

Supports that the scaling limit of the incipient infinite cluster is ISE.

Extends methods to analyze two- and three-point functions for high-dimensional models.

Abstract

This is the first of two papers on the critical behaviour of bond percolation models in high dimensions. In this paper, we obtain strong joint control of the critical exponents eta and delta, for the nearest-neighbour model in very high dimensions d>>6 and for sufficiently spread-out models in all dimensions d>6. The exponent eta describes the low frequency behaviour of the Fourier transform of the critical two-point connectivity function, while delta describes the behaviour of the magnetization at the critical point. Our main result is an asymptotic relation showing that, in a joint sense, eta = 0 and delta = 2. The proof uses a major extension of our earlier expansion method for percolation. This result provides evidence that the scaling limit of the incipient infinite cluster is the random probability measure on R^d known as integrated super-Brownian excursion (ISE), in dimensions above 6. In the sequel to this paper, we extend our methods to prove that the scaling limits of the incipient infinite cluster's two-point and three-point functions are those of ISE for the nearest-neighbour model in dimensions d>>6.