ABSENCE OF BACKWARD INFINITE PATHS FOR FIRST-PASSAGE PERCOLATION IN ARBITRARY DIMENSION

• 成果类型: Article
• 署名作者: Brito, Gerandy; Damron, Michael; Hanson, Jack
• 署名单位: University System of Georgia; Georgia Institute of Technology; University System of Georgia; Georgia Institute of Technology; City University of New York (CUNY) System; City College of New York (CUNY)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/22-AOP1588
• 发表日期: 2023
• 页码: 70-100
• 关键词: 1st passage percolation; geodesics; coexistence

摘要

In first-passage percolation (FPP), one places nonnegative random vari-ables (weights) (te) on the edges of a graph and studies the induced weighted graph metric. We consider FPP on Zd for d > 2 and analyze the geometric properties of geodesics, which are optimizing paths for the metric. Specifi-cally, we address the question of existence of bigeodesics, which are doubly-infinite paths whose subpaths are geodesics. It is a famous conjecture origi-nating from a question of Furstenberg and most strongly supported for d = 2 that, for continuously distributed i.i.d. weights, there a.s. are no bigeodesics. We provide the first progress on this question in general dimensions un -der no unproven assumptions. Our main result is that geodesic graphs, in-troduced in a previous paper of two of the authors, constructed in any de-terministic direction a.s. do not contain doubly-infinite paths. As a conse-quence, one can construct random graphs of subsequential limits of point-to-hyperplane geodesics, which contain no bigeodesics. This gives evidence that bigeodesics, if they exist, cannot be constructed in a translation-invariant manner as limits of point-to-hyperplane geodesics.