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

This is the first of two papers on the critical behavior of bond percolatio n models in high dimensions. In this paper, we obtain strong joint control of the critical exponents vl and delta for the nearest neighbor model in ve ry high dimensions d much greater than 6 and for sufficiently spread-out mo dels in all dimensions d>6. The exponent eta describes the low-frequency be havior of the Fourier transform of the critical two-point connectivity func tion, while delta describes the behavior of the magnetization at the critic al point. Our main result is an asymptotic relation showing that, in a join t sense, eta = 0 and delta = 2. The proof uses a major extension of our ear lier expansion method for percolation. This result provides evidence that t he scaling limit of the incipient infinite cluster is the random probabilit y measure on R-d known as integrated super-Brownian excursion (ISE), in dim ensions above 6. In the sequel to this paper, we extend our methods to prov e that the scaling limits of the incipient infinite cluster's two-point and three-point functions are those of ISE for the nearest neighbor model in d imensions d much greater than 6.