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

Takashi Hara, Gordon Slade

Research output: Contribution to journalArticlepeer-review

49 Citations (Scopus)


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 cn of n-step self-avoiding walks satisfies cn~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 cn(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.

Original languageEnglish
Pages (from-to)101-136
Number of pages36
JournalCommunications in Mathematical Physics
Issue number1
Publication statusPublished - Jun 1992
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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