AI 中文总结
该研究探讨稀疏随机强化对方格点阵近临界伯努利点渗流的影响，揭示强化点集几何结构对相变临界阈值的调控规律及临界态渗流的矩条件。
AI 中文摘要
我们研究了稀疏随机强化对方格点阵第一象限上近临界伯努利点渗流的影响。强化机制通过随机选取水平线和垂直线，并仅在它们的交点处提高开路概率$p$来实现，因此强化作用支撑在具有长程相关性的稀疏点集上。我们确定了该随机集的几何结构如何影响相变。\n 我们首先研究了强化是否会降低临界阈值。当相邻选取线之间的间距服从几何分布时，我们证明，对于每个高于临界阈值的$p$，即使强化区域外的开路概率$q$严格低于临界值，渗流也会发生。相比之下，如果一个方向上的间距衰减速度慢于指数分布，而另一个方向上的间距服从几何分布，则不会发生阈值偏移：无论$p$取何值，当$q$低于临界阈值时渗流都不可能发生。\n 接下来我们研究了临界情形$q=p_c$。我们表明，间距分布的足够高阶矩意味着临界状态下存在渗流，而不存在阶数为$\alpha\in(0,1)$的分数阶矩则排除了渗流的可能性。在对强化顶点进行额外的独立稀疏化后，这两种情形都保持不变。因此，支撑在随机交点上的强化既可能降低临界阈值，也可能使其保持不变，同时从根本上改变相变的性质。
英文摘要
We study the effect of sparse random reinforcement on near-critical Bernoulli site percolation on the first quadrant of the square lattice. The reinforcement is generated by randomly selecting horizontal and vertical lines and increasing the opening probability $p$ only at their intersections, so that the reinforcement is supported on a sparse set of points with long-range dependence. We determine how the geometry of this random set influences the phase transition. We first investigate whether the reinforcement lowers the critical threshold. When the spacings between consecutive selected lines are geometric, we prove that, for every $p$ above the critical threshold, percolation occurs even if the opening probability $q$ outside the reinforcement is strictly below criticality. In contrast, if the spacings in one direction decay slower than exponentially while those in the other are geometric, then no threshold shift occurs: percolation is impossible for $q$ below the critical threshold, regardless of $p$. We next study the critical case $q=p_c$. We show that sufficiently high moments of the spacing distributions imply percolation at criticality, whereas the absence of a fractional moment of order $α\in(0,1)$ rules it out. Both regimes remain unchanged after an additional independent thinning of the reinforced vertices. Thus, reinforcement supported on random intersections may either lower the critical threshold or leave it unchanged while fundamentally altering the nature of the phase transition.
Comments30 pages, 5 figures
