Lower bounds for boundary roughness for droplets in Bernoulli percolation

• 成果类型: Article
• 署名作者: Uzun, HB; Alexander, KS
• 署名单位: University of Southern California
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-003-0276-0
• 发表日期: 2003
• 页码: 62-88
• 关键词: bond percolation; fluctuations; inequalities

摘要

We consider boundary roughness for the droplet created when supercritical two-dimensional Bernoulli percolation is conditioned to have an open dual circuit surrounding the origin and enclosing an area at least l(2), for large l. The maximum local roughness is the maximum inward deviation of the droplet boundary from the boundary of its own convex hull; we show that for large l this maximum is at least of order l(1/3) (log l)(-2/3). This complements the upper bound of order l(1/3) (log l)(2/3) proved in [Al3] for the average local roughness. The exponent 1/3 on l here is in keeping with predictions from the physics literature for interfaces in two dimensions.