Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models

• Takashi Hara ^[1] ; Gordon Slade ^[2] ; Remco van der Hofstad ^[3]
  1. [1] Nagoya University

     Nagoya University

     Naka-ku, Japón

     
  2. [2] Delft University of Technology

     Delft University of Technology

     Países Bajos

     
  3. [3] University of British Columbia

     University of British Columbia

     Canadá

     

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• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 31, Nº. 1, 2003, págs. 349-408
• Idioma: inglés
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• Resumen

  • We consider spread-out models of self-avoiding walk, bond percolation, lattice trees and bond lattice animals on Zd, 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∈Zd, 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.