论文标题
关于Bernoulli Percolation的一些关键阈值的注释
A note on some critical thresholds of Bernoulli percolation
论文作者
论文摘要
考虑在本地有限的，连接的图形$ g $上考虑Bernoulli债券渗透，让$ p _ {\ mathrm {cut}} $是对应于“第一矩方法”下限的阈值。 kahn（\ textIt {electron。\ comm。\ probab。\卷8，184-187。}（2003））构建了一个反例，以示例为$ p _ {\ mathrm {cut}} = p_c $的$ p _ {\ mathrm {cut}} = p_c $，并提出了修改。在这里，我们对卡恩的修改问题给出了积极的答案。关键观察结果是，在Kahn的修改中，新的期望数量也出现在单臂事件的差异不平等中。这将问题与Duminil-Copin and Tassion的引理联系起来（\ textit {Comm。Math。Math。Phys。Solume。343，725-745。}（2016））。我们还研究了一些在周期树上伯努利渗透的应用。
Consider Bernoulli bond percolation on a locally finite, connected graph $G$ and let $p_{\mathrm{cut}}$ be the threshold corresponding to a "first-moment method" lower bound. Kahn (\textit{Electron.\ Comm.\ Probab.\ Volume 8, 184-187.} (2003)) constructed a counter-example to Lyons' conjecture of $p_{\mathrm{cut}}=p_c$ and proposed a modification. Here we give a positive answer to Kahn's modified question. The key observation is that in Kahn's modification, the new expectation quantity also appears in the differential inequality of one-arm events. This links the question to a lemma of Duminil-Copin and Tassion (\textit{Comm. Math. Phys. Volume 343, 725-745.} (2016)). We also study some applications for Bernoulli percolation on periodic trees.
