1. Main open problems
The following set of problems was the ones distinguished by the participants of the workshop as of particular importance. They concern topics like the asymptotic shape, shape fluctuations and scaling exponents, and the existence of bigeodesics.-
Subadditive ergodic theory foresees the existence of a convex and compact set ${\rm Ball}$, with non-empty interior, which describes the large scale behavior of balls in the first-passage metric. However, we know very little else about this shape, apart having to preserve the obvious lattice symmetries. While a precise formula for the asymptotic shape may be somewhat irrelevant, there are various problems in first-passage percolation that have been found to be fundamentally linked to certain properties of ${\rm Ball}$, such as strict convexity and differentiability of its boundary. For that reason, the following problem has come to be one of the more important problems in the area.
Problem 1.1.
[Charles Newman] Prove, for some reasonable class of edge weights, that the asymptotic shape is strictly convex. Is the boundary of the asymptotic shape differentiable?-
Remark. The word ‘reasonable’ could here be taken to mean continuous distributions satisfying some moment condition. However, proving this for any distribution, say exponential, would already be a major breakthrough. A recent simulation study, with suggestive conclusions for the shape, can be found in [MR3406703].
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Dating back to the work of Kardar, Parisi and Zhang [karparzha86], two exponents $\chi$ and $\xi$ have been proposed to describe the order of fluctuations of travel times around their mean and transversal fluctuations of geodesics away from a straight line. While it is unclear what the correct definition of $\chi$ and $\xi$ should be, the heuristic picture is that $T(0,ne_1)-\mathbb{E}[T(0,ne_1)]$ is order $n^\chi$ and that the vertical displacement of ${\rm Geo}(0,ne_1)$ is order $n^\xi$, with high probability. The following conjectured behavior has been verified for certain models closely related to first-passage percolation, but remain here a mystery.
Problem 1.2.
[Kenneth Alexander] Prove the conjectured asymptotics for two dimensional first-passage percolation predicted by the KPZ equation. That is, show that $\chi=1/3$ and $\xi=2/3$, and that travel times (properly rescaled) converge to a Tracy-Widom distribution. -
By todays techniques our best bounds on the scaling exponents give the very modest $0\le\chi\le1/2$ and $0\le\xi\le1$. Obtaining strict inequalities is considered a major step forward.
Problem 1.3.
[Michael Damron and Firas Rassoul-Agha] Prove that, in two dimensions, $0<\chi<1/2$ and $1/2<\xi<1$. -
In higher dimensions there is no consensus for how $\chi$ and $\xi$ should behave.
Problem 1.4.
[Christopher Hoffman] Come up with a convincing heuristic for the behaviour of $\chi$ and $\xi$ for higher dimensions, and support the heuristic with rigorous bounds. -
The study of infinite geodesics was pioneered by Newman [MR1404001] in the mid 1990s. Our limited understanding of the structure of geodesics is closely related to our poor understanding of the asymptotic shape. The possible existence of bigeodesics, i.e. doubly infinite paths for which each finite segment is a geodesic, is a particularly important question as it is related to the number of ground states in the disordered Ising ferromagnet.
Problem 1.5.
[Charles Newman] Rule out the existence of bigeodesics for $d=2$, or for $d$ small. That is, prove that the probability that the origin is visited by a bigeodesic is zero.
Cite this as: AimPL: First passage percolation, available at http://aimpl.org/firstpercolation.