A LOWER BOUND ON THE TWO-ARMS EXPONENT FOR CRITICAL PERCOLATION ON THE LATTICE

• 成果类型: Article
• 署名作者: Cerf, Raphael
• 署名单位: Universite Paris Saclay
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/14-AOP940
• 发表日期: 2015
• 页码: 2458-2480
• 关键词: infinite cluster; probability; uniqueness

摘要

We consider the standard site percolation model on the d-dimensional lattice. A direct consequence of the proof of the uniqueness of the infinite cluster of Aizenman, Kesten and Newman [Comm. Math. Phys. 111 (1987) 505-531] is that the two-arms exponent is larger than or equal to 1/2. We improve slightly this lower bound in any dimension d >= 2. Next, starting only with the hypothesis that theta(p) > 0, without using the slab technology, we derive a quantitative estimate establishing long-range order in a finite box.