AI 中文总结
该研究提出有限图伯努利键渗流的一个猜想，可推出d≥2时Z^d上θ(p_c)=0，证明有限平面图情形成立，反例显示连通性结构不可或缺。
AI 中文摘要
我们针对有限图上的伯努利键渗流提出一个猜想。大致而言，该猜想断言：若每个边界顶点都关联着一个对渗流构型单调递增的高概率事件，则在原点与边界相连的条件下，原点很可能与一个其关联事件发生的边界顶点相连。该猜想对所有d≥2的Z^d上的伯努利键渗流都蕴含θ(p_c)=0。我们证明了当原点与边界顶点位于外表面边界上时，有限平面图上的该猜想成立，且有明确界值1−2√ε。该证明结合了左优先深度优先探索与适配到停止时间前单调性的部分FKG不等式。一个反例表明连通性结构是必不可少的：当连通性事件被任意递增事件取代时，类似命题不成立。
英文摘要
We introduce a conjecture for Bernoulli bond percolation on finite graphs. Roughly speaking, it asserts that if each boundary vertex is associated with a highly probable event that is increasing with respect to the percolation configuration, then, conditionally on the origin being connected to the boundary, the origin is likely to be connected to a boundary vertex whose associated event occurs. The conjecture implies a high-probability connectivity statement which is known to imply $θ(p_c)=0$ for Bernoulli bond percolation on $\mathbb Z^d$, for every $d\geq2$. We prove the conjecture for finite planar graphs when the origin and the boundary vertices lie on the outer-face boundary, with the explicit bound $1-2\sqrt{\varepsilon}$. The proof combines a left-first depth-first exploration with a partial FKG inequality adapted to monotonicity up to a stopping time. A counterexample shows that the connectivity structure is essential: the analogous statement fails when the connectivity events are replaced by arbitrary increasing events.
Comments19 pages, 6 figures
