Rigidity percolation and boundary conditions

• 成果类型: Article
• 署名作者: Holroyd, AE
• 署名单位: University of California System; University of California Los Angeles
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• 发表日期: 2001
• 页码: 1063-1078
• 关键词: Existence

摘要

We study the effects of boundary conditions in two-dimensional rigidity percolation. Specifically, we consider generic rigidity in the bond percolation model on the triangular lattice. We introduce a theory of boundary conditions and define two different notions of ''rigid clusters,'' called r(0)-clusters and r(1)-clusters, which correspond to free boundary conditions and wired boundary conditions respectively. The definition of an r(0)-cluster turns out to be equivalent to the definition of a rigid component used in earlier papers by Holroyd and Haggstrom. We define two critical probabilities, associated with the appearance of infinite r(0)-clusters and infinite r(1)-clusters respectively, and we prove that these two critical probabilities are in fact equal. Furthermore, we prove that for all parameter values p except possibly this unique critical probability, the set of r(0)-clusters equals the set of r(1)-clusters almost surely. It is an open problem to determine what happens at the critical probability.