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

• 成果类型: Article
• 署名作者: Hara, T; Van Der Hofstad, R; Slade, G
• 署名单位: Nagoya University; University of British Columbia; Delft University of Technology; Eindhoven University of Technology
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2003
• 页码: 349-408
• 关键词: self-avoiding walk; critical-behavior; triangle condition; branched polymers; infinite cluster; lattice animals; Scaling Limit; trees

摘要

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 E 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 --> infinity. 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.