January 2023 Absence of backward infinite paths for first-passage percolation in arbitrary dimension
Gerandy Brito, Michael Damron, Jack Hanson
Author Affiliations +
Ann. Probab. 51(1): 70-100 (January 2023). DOI: 10.1214/22-AOP1588

Abstract

In first-passage percolation (FPP), one places nonnegative random variables (weights) (te) on the edges of a graph and studies the induced weighted graph metric. We consider FPP on Zd for d2 and analyze the geometric properties of geodesics, which are optimizing paths for the metric. Specifically, we address the question of existence of bigeodesics, which are doubly-infinite paths whose subpaths are geodesics. It is a famous conjecture originating 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 under no unproven assumptions. Our main result is that geodesic graphs, introduced in a previous paper of two of the authors, constructed in any deterministic direction a.s. do not contain doubly-infinite paths. As a consequence, 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.

Funding Statement

The research of M. D. is supported by an NSF CAREER grant.
The research of J. H. is supported by NSF Grant DMS-161292, and a PSC-CUNY Award, jointly funded by The Professional Staff Congress and The City University of New York.

Acknowledgments

The authors are grateful for the feedback of an anonymous reviewer that resulted in an improved presentation.

Citation

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Gerandy Brito. Michael Damron. Jack Hanson. "Absence of backward infinite paths for first-passage percolation in arbitrary dimension." Ann. Probab. 51 (1) 70 - 100, January 2023. https://doi.org/10.1214/22-AOP1588

Information

Received: 1 December 2020; Revised: 1 February 2022; Published: January 2023
First available in Project Euclid: 22 November 2022

MathSciNet: MR4515690
zbMATH: 07628798
Digital Object Identifier: 10.1214/22-AOP1588

Subjects:
Primary: 60K35
Secondary: 82B43

Keywords: bigeodesics , First-passage percolation , geodesic measures

Rights: Copyright © 2023 Institute of Mathematical Statistics

Vol.51 • No. 1 • January 2023
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