Self-avoiding walk in five or more dimensions I. The critical behaviour
Self-avoiding walk in five or more dimensions I. The critical behaviour
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在五个或更多维度中的自我回避行走 I. 关键行为
DOI：
10.1007/bf02099530
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发表时间：
1992
影响因子：
2.4
通讯作者：
G. Slade
中科院分区：
文献类型：
--
作者：
T. Hara;G. Slade
利用蕾丝展开法研究了超立方晶格中标准自避行走的问题。我们证明了n步自回避行走的数目cn满足cn~Aμn，其中μ是连接常数（即γ=1），并且均方位移在步数（即v=1/2）上是渐近线性的。得到了cn(x)的一个界，cn(x)是结束于atx的n步自避免行走的次数。相关长度渐近发散为（μ−−Z）1/2。临界两点函数的衰减速度至少与<s:2> × × <e:2>−2一样快，其傅里叶变换渐近于k−2的倍数→0（即η=0）。在布朗运动的分布收敛的意义上，我们也证明了尺度极限是高斯的。构造了无限自我回避行走。本文在假设蕾丝展开的收敛性的前提下证明了这些结果。在另一篇论文中证明了蕾丝展开的收敛性。
We use the lace expansion to study the standard self-avoiding walk in thed-dimensional hypercubic lattice, ford≧5. We prove that the numbercn ofn-step self-avoiding walks satisfiescn~Aμn, where μ is the connective constant (i.e. γ=1), and that the mean square displacement is asymptotically linear in the number of steps (i.e.v=1/2). A bound is obtained forcn(x), the number ofn-step self-avoiding walks ending atx. The correlation length is shown to diverge asymptotically like (μ−−Z)1/2. The critical two-point function is shown to decay at least as fast as ⋎x⋎−2, and its Fourier transform is shown to be asymptotic to a multiple ofk−2 ask→0 (i.e. η=0). We also prove that the scaling limit is Gaussian, in the sense of convergence in distribution to Brownian motion. The infinite self-avoiding walk is constructed. In this paper we prove these results assuming convergence of the lace expansion. The convergence of the lace expansion is proved in a companion paper.