ON THE INFLUENCE OF EDGES IN FIRST-PASSAGE PERCOLATION ON Zd

• 成果类型: Article
• 署名作者: Dembin, Barbara; Elboim, Dor; Peled, Ron
• 署名单位: Swiss Federal Institutes of Technology Domain; ETH Zurich; Institute for Advanced Study - USA; Tel Aviv University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/24-AOP1715
• 发表日期: 2025
• 页码: 544-556
• 关键词: 1st passage percolation; fluctuations; distance; variance

摘要

We study first-passage percolation on Z(d), d >= 2, with independent weights whose common distribution is compactly supported in (0,infinity) with a uniformly-positive density. Given & varepsilon;>0 and v is an element of Z(d), which edges have probability at least & varepsilon; to lie on the geodesic between the origin and v? It is expected that all such edges lie at distance at most some r(& varepsilon;)r(& varepsilon;) from either the origin or v, but this remains open in dimensions d >= 3. We establish the closely-related fact that the number of such edges is at most some C(& varepsilon;)C(& varepsilon;), uniformly in v. In addition, we prove a quantitative bound, allowing & varepsilon; to tend to zero as parallel to v parallel to & Vert;v & Vert; tends to infinity, showing that there are at most O(& varepsilon;-(2d)/d(-1)(log & Vert;v & Vert;)(c)) such edges, uniformly in & varepsilon; and v. The latter result addresses a problem raised by Benjamini-Kalai-Schramm (Ann. Probab. 31 (2003) 1970-1978). Our technique further yields a strengthened version of a lower bound on transversal fluctuations due to Licea-Newman-Piza (Probab. Theory Related Fields 106 (1996) 559-591).