  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.
