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

We use the lace expansion to study the standard self-avoiding walk in the d-dimensional hypercubic lattice, for d≧5. We prove that the number c_n of n-step self-avoiding walks satisfies c_n~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 for c_n(x), the number of n-step self-avoiding walks ending at x. 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 {curly logical or}x{curly logical or}^-2, and its Fourier transform is shown to be asymptotic to a multiple of k^-2 as k→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.