Self-avoiding walk in five or more dimensions I. The critical behaviour

Abstract

We use the lace expansion to study the standard self-avoiding walk in thed-dimensional hypercubic lattice, ford≧5. We prove that the numbercn ofn-step self-avoiding walks satisfiescn~Aμn, where μ is the connective constant (i.e. γ=1), and that the mean square displacement is asymptotically linear in the number of steps (i.e.v=1/2). A bound is obtained forcn(x), the number ofn-step self-avoiding walks ending atx. The correlation length is shown to diverge asymptotically like (μ−−Z)1/2. The critical two-point function is shown to decay at least as fast as ⋎x⋎−2, and its Fourier transform is shown to be asymptotic to a multiple ofk−2 ask→0 (i.e. η=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.