PHASE TRANSITION FOR PERCOLATION ON A RANDOMLY STRETCHED SQUARE LATTICE

• 成果类型: Article
• 署名作者: Hilario, Marcelo R.; Sa, Marcos; Sanchis, Remy; Teixeira, Augusto
• 署名单位: Universidade Federal de Minas Gerais; Instituto Nacional de Matematica Pura e Aplicada (IMPA)
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/22-AAP1887
• 发表日期: 2023
• 页码: 3145-3168
• 关键词: dimensional contact-processes; critical-points; survival; sharpness; BEHAVIOR; dilute; MODEL

摘要

Let {& xi;i}i & GE;1 be a sequence of i.i.d. positive random variables. Starting from the usual square lattice replace each horizontal edge that links a site in the ith vertical column to another in the (i + 1)th vertical column by an edge having length & xi;i. Then declare independently each edge e in the resulting lattice open with probability pe = p|e| where p & ISIN; [0, 1] and |e| is the length of e. We relate the occurrence of a nontrivial phase transition for this model to moment properties of & xi;1. More precisely, we prove that the model undergoes a nontrivial phase transition when E(& xi;1 & eta;) < & INFIN;, for some & eta; > 1. On the other hand, when E(& xi;1) = & INFIN;, percolation never occurs for p < 1. We also show that the probability of the one-arm event decays no faster than a polynomial in an open interval of parameters p close to the critical point.