Polluted bootstrap percolation with threshold two in all dimensions

• 成果类型: Article
• 署名作者: Gravner, Janko; Holroyd, Alexander E.
• 署名单位: University of California System; University of California Davis; University of Washington; University of Washington Seattle
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-018-0892-3
• 发表日期: 2019
• 页码: 467-486
• 关键词: sharp metastability threshold; DYNAMICS

摘要

In the polluted bootstrap percolation model, the vertices of a graph are independently declared initially occupied with probability p or closed with probability q. At subsequent steps, a vertex becomes occupied if it is not closed and it has at least r occupied neighbors. On the cubic lattice Z(d) of dimension d >= 3 with threshold r = 2, we prove that the final density of occupied sites converges to 1 as p and q both approach 0, regardless of their relative scaling. Our result partially resolves a conjecture of Morris, and contrasts with the d = 2 case, where Gravner and McDonald proved that the critical parameter is q/p(2).