SELF-AVOIDING WALK IN 5 OR MORE DIMENSIONS .1. THE CRITICAL-BEHAVIOR

We use the lace expansion to study the standard self-avoiding walk in the d-dimensional hypercubic lattice, for d greater-than-or-equal-to 5. We prove that the number c(n) of n-step self-avoiding walks satisfies c(n) approximately A-mu(n), where mu is the connective constant (i.e. gamma = 1), and that the mean square displacement is asymptotically linear in the number of steps (i.e. v = 1/2). A bound is obtained for c(n)(x), the number of n-step self-avoiding walks ending at x. The correlation length is shown to diverge asymptotically like (mu--1 - z)-1/2. The critical two-point function is shown to decay at least as fast as absolute value of x-2, and its Fourier transform is shown to be asymptotic to a multiple of k-2 as k --> 0 (i.e. eta = 0). We also prove that the scaling limit is Gaussian, in the sense of convergence in distribution to Brownian motion. The infinite self-avoiding walk is constructed. In this paper we prove these results assuming convergence of the lace expansion. The convergence of the lace expansion is proved in a companion paper.