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

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

1999-03-25

arxiv.org/abs/math-ph/9903042v1

J. Statist. Phys., 99 (2000) 1075-1168

Physics

Mathematical Physics

72 pages, 12 Postscript figures, in AMS-LaTeX, with graphicx, epic, and xr packages

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

The scaling limit of the incipient infinite cluster in high-dimensional percolation. I. Critical exponents does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with The scaling limit of the incipient infinite cluster in high-dimensional percolation. I. Critical exponents, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The scaling limit of the incipient infinite cluster in high-dimensional percolation. I. Critical exponents will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-87944