DIVERGENCE OF SHAPE FLUCTUATIONS IN 2 DIMENSIONS

• 成果类型: Article
• 署名作者: NEWMAN, CM; PIZA, MST
• 署名单位: University of California System; University of California Irvine
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1176988171
• 发表日期: 1995
• 页码: 977-1005
• 关键词: ising percolation; phase-transitions; directed polymers; random impurities; peierls contours; random-media; MODEL; temperatures; GROWTH

摘要

We consider stochastic growth models, such as standard first-passage percolation on Z(d), where to leading order there is a linearly growing deterministic shape. Under natural hypotheses, we prove that for d = 2, the shape fluctuations grow at least logarithmically in all directions. Although this bound is far from the expected power law behavior with exponent chi = 1/3, it does prove divergence. With additional hypotheses, we obtain inequalities involving chi and the related exponent xi (which is expected to equal 2/3 for d = 2). Combining these inequalities with previously known results, we obtain for standard first-passage percolation the bounds chi greater than or equal to 1/8 for d = 2 and xi greater than or equal to 3/4 for all d.