SUBCRITICAL PERCOLATION WITH A LINE OF DEFECTS

• 成果类型: Article
• 署名作者: Friedli, S.; Ioffe, D.; Velenik, Y.
• 署名单位: Universidade Federal de Minas Gerais; Technion Israel Institute of Technology; University of Geneva
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/11-AOP720
• 发表日期: 2013
• 页码: 2013-2046
• 关键词: ornstein-zernike theory; connectivities

摘要

We consider the Bernoulli bond percolation process P-p,P-p' on the nearestneighbor edges of Z(d), which are open independently with probability p < p(c), except for those lying on the first coordinate axis, for which this probability is p'. Define xi(p,p') := - lim(n ->infinity) n(-1) log P-p,P-p' (0 <-> -ne(1)) and xi(p) := xi(p,p). We show that there exists p(c)' = p(c)'(p, d) such that xi(p,p)' = xi(p) if p' < p(c)' and xi(p,p)' < xi(p) if p' > p(c)'. Moreover, p(c)'(p, 2) = p(c)'(p, 3) = p, and p(c)'(p, d) > p for d >= 4. We also analyze the behavior of xi(p) - xi(p,p)' as p' down arrow p(c)', in dimensions d = 2, 3. Finally, We prove that when p' > p(c)', the following purely exponential asymptotics holds: P-p,P-p' (0 <-> ne(1)) = psi(d)e(-xi p,p'n) (1 + o(1)) for some constant psi(d) = psi(d)(p, p'), uniformly for large values of n. This work gives the first results on the rigorous analysis of pinning-type problems, that go beyond the effective models and don't rely on exact computations.