4. Structure of geodesics
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 $y$.-
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 [MR2016607].The problem has been solved by Ahlberg and Hoffman [arXiv:1609.02447] for a large class of first-passage percolation models on $\mathbb{Z}^2$.
Problem 4.1.
[Jean-Christophe Mourrat] Does $\mathbb{P}(0\in{\rm Geo}(-n{\bf e}_1,n{\bf e}_1))\to0$ as $n\to\infty$? -
While some results regarding the existence of a bigeodesic in a given direction have been obtained by Licea and Newman [MR1387641], Damron and Hanson [arXiv:1512.00804], and Ahlberg and Hoffman [arXiv:1609.02447], even the following has not yet been fully settled.
Problem 4.2.
[Christopher Hoffman] Prove that for a given unit vector $v$, there is almost surely no bigeodesic with one end directed in that direction. -
When it comes to one-sided infinite geodesics the state of the art says that on $\mathbb{Z}^2$ there are at least four geodesics originating from the origin [MR2462555], [MR3152744]. It is further known that every one-sided geodesic has an ‘asymptotic direction’ in a generalized sense: For any geodesic $g=(v_0,v_1,\ldots)$ the set of limit points of the set $\{v_k/|v_k|:k\ge1\}$ is either a point or an interval associated to a flat piece of the asymptotic shape ${\rm Ball}$, see [MR3152744], [arXiv:1609.02447]. However, whether there always exists a geodesic whose asymptotic direction is a point remains unknown.
Problem 4.3.
[Michael Damron] Show that there exists a geodesic $g=(v_0,v_1,\ldots)$ and a unit direction $v$ such that $v_k/|v_k|$ tends to $v$ as $k\to\infty$. -
Let $\sigma_v$ denote the shift operator along the vector $v\in\mathbb{Z}^2$, and denote by $\omega=\{\omega_e\}$ a realization of the edge weights. Given an infinite geodesic $g=(v_0,v_1,\ldots)$, consider the empirical measure as seen along the geodesic $g$, e.g., $\frac{1}{n}\sum_{k=0}^{n-1}\sigma_{v_k}\omega$.
Problem 4.4.
[Christopher Hoffman] Prove that this empirical measure converges (weakly) to some probability measure. Is this measure singular with respect to the original law on the edge weights? -
The following two problems are related to the above problem. Here $|{\rm Geo}(x,y)|$ denotes the length of the geodesic between $x$ and $y$.
Problem 4.5.
[Michael Damron] Show that $\lim_{n\to\infty}\frac{1}{n}|{\rm Geo}(0,n{\bf e}_1)|$ exists almost surely. -
Problem 4.6.
[Antonio Auffinger] Let $(e_1,e_2,\ldots,e_m)$ be an enumeration of the edges of the geodesic from the origin to $n{\bf e}_1$. How does $\max_{1\le k\le m}\omega_{e_k}$ scale with $n$? -
A topic that has received much attention over the past decade is the effect of small perturbations of the realization of the process. For first-passage percolation the following question is relevant.
Problem 4.7.
[Jean-Christophe Mourrat] How does the geodesic ${\rm Geo}(0,n{\bf e}_1)$ change when each edge weight is resampled independently with probability $\varepsilon>0$? -
In higher dimensions much less is known about the structure of geodesics.
Problem 4.8.
[Firas Rassoul-Agha] Prove (or disprove) that in sufficiently high dimensions the limit $\lim_{n\to\infty}\big[T(0,n{\bf e}_1)-T({\bf e}_1,n{\bf e}_1)\big]$ does not exist.
Cite this as: AimPL: First passage percolation, available at http://aimpl.org/firstpercolation.