二维渗流中等周问题的Wulff晶体在趋近临界点时的极限
Limit of the Wulff crystal when approaching criticality for isoperimetry in 2D percolation
AI总结：
本文研究三角格点超临界点渗流无穷团簇的等周集，通过调整已有证明方法，证明重标度等周集收敛于归一化Wulff晶体的平移，且趋近临界点时Wulff晶体趋近欧氏圆盘，解决了相关点渗流版本猜想。
摘要：
我们研究三角格点上超临界点渗流的无穷团簇的等周集，即给定体积下顶点边界最小的集合。设$p$为渗流参数，$p_c$为临界点。通过调整Biskup、Louidor、Procaccia和Rosenthal[6]针对方格点键渗流等周问题的证明，我们证明了经适当重标度后的等周集几乎必然收敛于归一化Wulff晶体$\u2070W_p$的一个平移。更重要的是，我们证明当$p\to p_c^+$时，$\u2070W_p$趋近于一个欧几里得圆盘。这解决了文献[6]中提出的点渗流版本猜想。证明的一个关键输入是作者近期证明的近临界Bernoulli首次通过渗流的极限形状收敛性。
英文摘要：
We consider isoperimetric sets, i.e., sets with minimal vertex boundary for a prescribed volume, of the infinite cluster of supercritical site percolation on the triangular lattice. Let $p$ be the percolation parameter and let $p_c$ be the critical point. By adapting the proof of Biskup, Louidor, Procaccia and Rosenthal [6] for isoperimetry in bond percolation on the square lattice, we show that the isoperimetric sets, when suitably rescaled, converge almost surely to a translation of the normalized Wulff crystal $\widehat{W}_p$. More importantly, we prove that $\widehat{W}_p$ tends to a Euclidean disk as $p\downarrow p_c$. This settles the site version of a conjecture proposed in [6]. A key input to the proof is the convergence of the limit shapes for near-critical Bernoulli first-passage percolation proved by the author recently.
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