Non-coincidence of quenched and annealed connective constants on the supercritical planar percolation cluster
Non-coincidence of quenched and annealed connective constants on the supercritical planar percolation cluster
复制标题
超临界平面渗流簇上淬火和退火连接常数的不重合
DOI：
10.1007/s00440-013-0520-1
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发表时间：
2012
影响因子：
2
通讯作者：
H. Lacoin
中科院分区：
文献类型：
--
作者：
H. Lacoin
在本文中，我们研究了 $$\mathbb{Z }^d$$Zd 上超临界渗流簇上给定长度的自回避路径的丰度。更准确地说，我们计算$$Z_N$$ZN，即从原点（我们条件是位于簇中）开始的无限簇上长度为$$N$$N 的自回避路径的数量。我们有兴趣估计 $$Z_N$$ZN 的上限增长率，$$\limsup _{N\rightarrow \infty } Z_N^{1/N}$$lim性supN→∞ZN1/N，我们称之为稀晶格的连通常数。证明这个连接常数是 a.s.非随机，我们关注二维情况，并表明对于每个渗滤参数 $$p\in (1/2,1)$$p∈(1/2,1)，几乎可以肯定，$$Z_N$$ZN 的增长速度比其预期值慢。换句话说，我们证明 $$\limsup _{N\rightarrow \infty } (Z_N)^{1/N}{<}\lim _{N\rightarrow \infty } \mathbb{E }[Z_N]^{1/N}$$lim suN→∞(ZN)1/N<limN→∞E[ZN]1/N，其中期望是针对渗流过程而取的。这一结果可以被认为是理解无序对（淬灭的）稀晶格上自回避行走的影响的第一次数学尝试。我们的方法结合了测量变化和粗粒度参数，不依赖于 $$\mathbb{Z }^2$$Z2 上渗透的细节，因此我们的结果可以扩展到一大群二维模型，包括随机环境中的一般自回避游走。
In this paper, we study the abundance of self-avoiding paths of a given length on a supercritical percolation cluster on $$\mathbb{Z }^d$$Zd. More precisely, we count $$Z_N$$ZN, the number of self-avoiding paths of length $$N$$N on the infinite cluster starting from the origin (which we condition to be in the cluster). We are interested in estimating the upper growth rate of $$Z_N$$ZN, $$\limsup _{N\rightarrow \infty } Z_N^{1/N}$$lim supN→∞ZN1/N, which we call the connective constant of the dilute lattice. After proving that this connective constant is a.s. non-random, we focus on the two-dimensional case and show that for every percolation parameter $$p\in (1/2,1)$$p∈(1/2,1), almost surely, $$Z_N$$ZN grows exponentially slower than its expected value. In other words, we prove that $$\limsup _{N\rightarrow \infty } (Z_N)^{1/N}{<}\lim _{N\rightarrow \infty } \mathbb{E }[Z_N]^{1/N}$$lim supN→∞(ZN)1/N<limN→∞E[ZN]1/N, where the expectation is taken with respect to the percolation process. This result can be considered as a first mathematical attempt to understand the influence of disorder for self-avoiding walks on a (quenched) dilute lattice. Our method, which combines change of measure and coarse graining arguments, does not rely on the specifics of percolation on $$\mathbb{Z }^2$$Z2, so our result can be extended to a large family of two-dimensional models including general self-avoiding walks in a random environment.
自我回避行走是亚弹道式的
影响因子：
2.4
作者：
Duminil-Copin H
通讯作者：
Duminil-Copin H