重新审视基本增强功能
Essential enhancements revisited
摘要：
1991年，Aizenman和Grimmett声称，晶格上任何“基本增强”的位置或键渗透都会降低临界概率，这是一个具有许多含义的重要结果，例如在适当的晶格对上临界概率之间的严格不等式。他们的证明有两个部分，一个是概率的，一个是组合的。在本文中，我们指出，一个关键的组合引理，他们只提供了一个数字作为证明，是错误的。我们证明了另一种形式的引理，从而增强的结果，在特殊情况下的网站渗流的正方形，三角形和立方晶格，并为键渗流$Z^d$，$d\ge 2$。一般情况下仍然开放，即使是在$Z^d$，$d\ge 4$上的部位渗透。
英文摘要：
In 1991 Aizenman and Grimmett claimed that any `essential enhancement' of site or bond percolation on a lattice lowers the critical probability, an important result with many implications, such as strict inequalities between critical probabilities on suitable pairs of lattices. Their proof has two parts, one probabilistic and one combinatorial. In this paper we point out that a key combinatorial lemma, for which they provide only a figure as proof, is false. We prove an alternative form of the lemma, and thus the enhancement result, in the special cases of site percolation on the square, triangular and cubic lattices, and for bond percolation on $Z^d$, $d\ge 2$. The general case remains open, even for site percolation on $Z^d$, $d\ge 4$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
