Critical probabilities for site and bond percolation models

• 成果类型: Article
• 署名作者: Grimmett, GR; Stacey, AM
• 署名单位: University of Cambridge
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1998
• 页码: 1788-1812
• 关键词: random-cluster processes; potts models

摘要

Any infinite graph G = (V, E) has a site percolation critical probability p(c)(site) and a bond percolation critical probability p(c)(bond). The well-known weak inequality p(c)(site) greater than or equal to p(c)(bond) is strengthened to strict inequality for a broad category of graphs G, including all the usual finite-dimensional lattices in two and more dimensions. The complementary inequality p(c)(site) less than or equal to 1 - (1 - p(c)(bond))(Delta - 1) is proved also, where Delta denotes the supremum of the vertex degrees of G.