This is the first of two papers on the critical behavior of bond percolation models in high dimensions. In this paper, we obtain strong joint control of the critical exponents η and δ for the nearest neighbor model in very high dimensions d ≫ 6 and for sufficiently spread-out models in all dimensions d > 6. The exponent η describes the low-frequency behavior of the Fourier transform of the critical two-point connectivity function, while δ describes the behavior of the magnetization at the critical point. Our main result is an asymptotic relation showing that, in a joint sense, η = 0 and δ = 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 ℝd 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 neighbor model in dimensions d ≫ 6.
All Science Journal Classification (ASJC) codes
- Statistical and Nonlinear Physics
- Mathematical Physics