Geodesics Toward Corners in First Passage Percolation

TL;DR

This paper constructs an example in two-dimensional first passage percolation where geodesics only exist in axis directions, especially at corners of the limit shape, challenging previous assumptions about geodesic directions.

Contribution

It provides a novel example with an octagonal limit shape where geodesics are limited to axis directions, highlighting new phenomena at shape corners.

Findings

Geodesics only in four axis directions

Multiple geodesics in each axis direction

Limit shape is an octagon

Abstract

For stationary first passage percolation in two dimensions, the existence and uniqueness of semi-infinite geodesics directed in particular directions or sectors has been considered by Damron and Hanson (Commun. Math. Phys., 2014), Ahlberg and Hoffman (preprint, 2016), and others. However the main results do not cover geodesics in the direction of corners of the limit shape $\mathcal{B}$, where two facets meet. We construct an example with the following properties: (i) the limiting shape is an octagon, (ii) semi-infinite geodesics exist only in the four axis directions, and (iii) in each axis direction there are multiple such geodesics. Consequently, the set of points of $\partial \mathcal{B}$ which are in the direction of some geodesic does not have all of $\mathcal{B}$ as its convex hull.