The Alexander-Orbach conjecture holds in high dimensions

• 成果类型: Article
• 署名作者: Kozma, Gady; Nachmias, Asaf
• 署名单位: Microsoft; Weizmann Institute of Science
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-009-0208-4
• 发表日期: 2009
• 页码: 635-654
• 关键词: incipient infinite cluster; quenched invariance-principles; mean-field criticality; simple random-walk; oriented percolation; critical-behavior; brownian-motion; phase; inequalities; exponents

摘要

We examine the incipient infinite cluster (IIC) of critical percolation in regimes where mean-field behavior has been established, namely when the dimension d is large enough or when d > 6 and the lattice is sufficiently spread out. We find that random walk on the IIC exhibits anomalous diffusion with the spectral dimension d(s) = 4/3, that is, p(t)(x, x) = t(-2/3+o(1)). This establishes a conjecture of Alexander and Orbach (J. Phys. Lett. (Paris) 43:625-631, 1982). En route we calculate the one-arm exponent with respect to the intrinsic distance.