Non-coincidence of quenched and annealed connective constants on the supercritical planar percolation cluster

• 成果类型: Article
• 署名作者: Lacoin, Hubert
• 署名单位: Universite PSL; Universite Paris-Dauphine
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-013-0520-1
• 发表日期: 2014
• 页码: 777-808
• 关键词: self-avoiding-walks; disorder relevance; random environment; critical-point; pinning model; lattices; polymers

摘要

In this paper, we study the abundance of self-avoiding paths of a given length on a supercritical percolation cluster on . More precisely, we count , the number of self-avoiding paths of length on the infinite cluster starting from the origin (which we condition to be in the cluster). We are interested in estimating the upper growth rate of , , which we call the connective constant of the dilute lattice. After proving that this connective constant is a.s. non-random, we focus on the two-dimensional case and show that for every percolation parameter , almost surely, grows exponentially slower than its expected value. In other words, we prove that , where the expectation is taken with respect to the percolation process. This result can be considered as a first mathematical attempt to understand the influence of disorder for self-avoiding walks on a (quenched) dilute lattice. Our method, which combines change of measure and coarse graining arguments, does not rely on the specifics of percolation on , so our result can be extended to a large family of two-dimensional models including general self-avoiding walks in a random environment.