Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models
Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models
复制标题
关键两点函数和展开高维渗滤及相关模型的花边展开
DOI：
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发表时间：
2000
期刊：
影响因子：
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通讯作者：
G. Slade
中科院分区：
文献类型：
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作者：
T. Hara;R. Hofstad;G. Slade
我们考虑${mathbb{Z}^d}$上的自避免行走、键渗透、格树和键格动物的扩展模型，具有长的有限程连接，在它们的上临界维数d=4$（自避免行走），d=6$（渗透）和d=8$（树和动物）以上。这些模型的两点函数分别是{mathbb{Z}^d}$中从原点到$x的自避免行走的生成函数、从0到x的连接的概率以及包含0和x的格树或格动物的生成函数。我们使用花边展开证明，充分展开模型以上的临界尺寸，两点函数的每个模型衰减，在临界点，作为倍数的$|X| ^{2-d}$ as $x 不需要钱。我们用一种新的统一方法证明了花边展开式的收敛性。该方法基于x空间方法而不是傅立叶变换。我们的结果也产生统一和简化的证明泡沫条件的自我避免行走，三角形条件的渗流，和正方形条件的格树和格子动物，充分展开的模型以上的上临界尺寸。
We consider spread-out models of self-avoiding walk, bond percolation, lattice trees and bond lattice animals on ${mathbb{Z}^d}$, 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 $x in {mathbb{Z}^d}$, 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 $x o infty$. 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.