Structure of geodesics · Question 4.1

Workshop introduction

This is a summary of the set of problems discussed at the AIM workshop on first-passage percolation and related models in August 2015. The name of the person who raised the question at the workshop is mentioned within parenthesis for each problem.

In first-passage percolation the edges of the $\mathbb{Z}^d$ nearest neighbor lattice is equipped with non-negative random weights $\{\omega_e\}$, which are usually assumed to be i.i.d. The resulting weighted graph induces a metric $T:\mathbb{Z}^d\times\mathbb{Z}^d\to[0,\infty)$ on $\mathbb{Z}^d$, and it is understanding the large-scale behavior of distances, balls and geodesics in this random metric space which is the primary objective. See Source citation: arXiv:1511.03262 for an extensive recent survey.

Section context

The following is a list of problems related to the structure of large finite and infinite geodesics. Most work in this direction has been made in the case $d=2$, so unless where stated otherwise, these problems should primarily be thought of as in two dimensions. We below let ${\rm Geo}(x,y)$ denote the geodesic between two points $x$ and $y$.

Question context

Above, the question was raised regarding the possible existence of bigeodesics. The following question is a weaker version of that conjecture and concerns the probability that the geodesic between two far away points visit a given point midway between the two. The question has come to be known as the `midpoint problem', and originates from Benjamini, Kalai and Schramm Source citation: MR2016607.

Dated source record, inspected 2026-09-10. Source reports a solution for a large class of two-dimensional models; broader scope is not certified. Source-reported results have not been independently verified.

Status note in the source

The problem has been solved by Ahlberg and Hoffman Source citation: arXiv:1609.02447 for a large class of first-passage percolation models on $\mathbb{Z}^2$.