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

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Abstract

We use the lace expansion to study the standard self-avoiding walk in thed-dimensional hypercubic lattice, ford≧5. We prove that the numberc n ofn-step self-avoiding walks satisfiesc n ~Aμ n, 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 forc n(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.

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Author notes

  1. Takashi Hara

    Present address: Department of Applied Physics, Tokyo Institute of Technology, Oh-Okayama, Meguroku, 152, Tokyo, Japan

Authors and Affiliations

  1. Department of Physics, Gakushuin University, Toshima-ku, 171, Tokyo, Japan

    Takashi Hara

  2. Department of Mathematics and Statistics, McMaster University, L8S 4K1, Hamilton, Ontario, Canada

    Gordon Slade

Authors
  1. Takashi Hara
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  2. Gordon Slade
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Additional information

Communicated by M. Aizenman

Supported by the Nishina Memorial Foundation and NSF grant PHY-8896163.

Supported by NSERC grant A9351

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Hara, T., Slade, G. Self-avoiding walk in five or more dimensions I. The critical behaviour. Commun.Math. Phys. 147, 101–136 (1992). https://doi.org/10.1007/BF02099530

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  • DOI: https://doi.org/10.1007/BF02099530

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