The number of geodesics in planar first-passage percolation grows sublinearly
Abstract
We study a random perturbation of the Euclidean plane, and show that it is unlikely that the distance-minimizing path between the two points can be extended into an infinite distance-minimizing path. More precisely, we study a large class of planar first-passage percolation models and show that the probability that a given site is visited by an infinite geodesic starting at the origin tends to zero uniformly with the distance. In particular, this show that the collection of infinite geodesics starting at the origin covers a negligible fraction of the plane. This provides the first progress on the `highways and byways' problem, posed by Hammersley and Welsh in the 1960s.
Keywords
Cite
@article{arxiv.2208.11576,
title = {The number of geodesics in planar first-passage percolation grows sublinearly},
author = {Daniel Ahlberg and Jack Hanson and Christopher Hoffman},
journal= {arXiv preprint arXiv:2208.11576},
year = {2022}
}
Comments
49 pages, 3 figures. A video summary may be found at: https://youtu.be/vzrO1iKIen4