## Abstract

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}~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 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.

Original language | English |
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Pages (from-to) | 101-136 |

Number of pages | 36 |

Journal | Communications in Mathematical Physics |

Volume | 147 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jun 1992 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics