发表机构
The Open University of Israel（以色列开放大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对一维长程渗流，通过二阶矩论证等方法得到其临界值$C_c$的显式上界，该界也适用于定向渗流，且与已知数值结果相符。
AI 中文摘要
我们研究一维格点上的键渗流问题，其中距离为$r$的两个位点以概率$p_r=C/r^{1+\boldsymbol{\u03c3}}$相连，这里$0<\boldsymbol{\u03c3}<1$。当$C$超过临界值$C_c(\boldsymbol{\u03c3})$时会出现无限簇，而$C_c(\boldsymbol{\u03c3}}$的精确值未知：现有结果包括Schulman给出的下界$C_c\boldsymbol{\u2265}1/[2\u03b6(1+\boldsymbol{\u03c3})]$以及数值估计值。本文中我们证明了显式上界。我们对一族随机单调路径应用二阶矩论证，这些路径的步具有重尾（Sibuya）分布。两条此类路径仅有限次相遇，其重叠度通过更新理论精确计算。所得结果为闭式界，当$\boldsymbol{\u03c3}\to0$时给出$C_c\boldsymbol{\u2264}\boldsymbol{\u03c3}+O(\boldsymbol{\u03c3}^2)$，且对$\boldsymbol{\u03c3}<0.667$时$C_c$小于1。由于路径是单调的，该界也适用于定向渗流，且确定了定向阈值的渐近行为：当$\boldsymbol{\u03c3}\to0$时$C_c^{\rightarrow}\boldsymbol{\u03b6}(1+\boldsymbol{\u03c3})\to1$。将位点替换为块的粗粒化版本论证，对每个$\boldsymbol{\u03c3}<1$给出显式界$C_c<1$。我们将这些界与$C_c$的已知数值进行了比较。
英文摘要
We study bond percolation on the one-dimensional lattice in which two sites at distance $r$ are connected with probability $p_r=C/r^{1+σ}$, where $0 < σ< 1$. An infinite cluster exists if $C$ exceeds a critical value $C_c(σ)$, which is not known exactly: the available results are the lower bound of Schulman, $C_c\ge1/[2ζ(1+σ)]$, and numerical estimates. Here we prove explicit upper bounds. We apply a second-moment argument to a family of random monotone paths whose steps have a heavy-tailed (Sibuya) distribution. Two such paths meet only a finite number of times, and their overlap is computed exactly by renewal theory. The result is a closed-form bound, which gives $C_c\leσ+O(σ^2)$ as $σ\to0$ and is below $1$ for $σ< 0.667$. Since the paths are monotone, the bound holds also for oriented percolation, and it determines the oriented threshold asymptotically: $C_c^{\rightarrow}ζ(1+σ)\to1$ as $σ\to0$. A coarse-grained version of the argument, with blocks of sites in place of sites, gives an explicit bound $C_c < 1$ for every $σ< 1$. We compare the bounds with the known numerical values of $C_c$.
Comments21 pages, 3 figures, 1 table
