### Abstract

We consider nearest-neighbor self-avoiding walk, bond percolation, lattice trees, and bond lattice animals on ℤ^{d}. The two-point functions of these models are respectively the generating function for self-avoiding walks from the origin to x € ℤ^{d}, the probability of a connection from the origin to x, and the generating functions for lattice trees or lattice animals containing the origin and x. Using the lace expansion, we prove that the two-point function at the critical point is asymptotic to const.|x| ^{2-d} as |x| → ∞, for d ≥ 5 for self-avoiding walk, for d ≥ 19 for percolation, and for sufficiently large d for lattice trees and animals. These results are complementary to those of [Ann. Probab. 31 (2003) 349-408], where spread-out models were considered. In the course of the proof, we also provide a sufficient (and rather sharp if d > 4) condition under which the two-point function of a random walk on ℤd is asymptotic to const.|x|^{2-d} as |x| → ∞.

Original language | English |
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Pages (from-to) | 530-593 |

Number of pages | 64 |

Journal | Annals of Probability |

Volume | 36 |

Issue number | 2 |

DOIs | |

Publication status | Published - Mar 1 2008 |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty