### Abstract

We consider spread-out models of self-avoiding walk, bond percolation, lattice trees and bond lattice animals on ℤ^{d}, having long finite-range connections, above their upper critical dimensions d = 4 (self-avoiding walk), d = 6 (percolation) and d = 8 (trees and animals). The two-point functions for these models are respectively the generating function for self-avoiding walks from the origin to x ∈ ℤ^{d}, the probability of a connection from 0 to x, and the generating function for lattice trees or lattice animals containing 0 and x. We use the lace expansion to prove that for sufficiently spread-out models above the upper critical dimension, the two-point function of each model decays, at the critical point, as a multiple of |x|^{2-d} as x → ∞. We use a new unified method to prove convergence of the lace expansion. The method is based on x-space methods rather than the Fourier transform. Our results also yield unified and simplified proofs of the bubble condition for self-avoiding walk, the triangle condition for percolation, and the square condition for lattice trees and lattice animals, for sufficiently spread-out models above the upper critical dimension.

Original language | English |
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Pages (from-to) | 349-408 |

Number of pages | 60 |

Journal | Annals of Probability |

Volume | 31 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 1 2003 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

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## Cite this

*Annals of Probability*,

*31*(1), 349-408. https://doi.org/10.1214/aop/1046294314