Fluctuations and scaling relations · Question 3.5

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 best current bounds on fluctuations of the first-passage metric says that for some constant $c>0$ we have

where the lower bound holds for $d=2$, see Source citation: MR1283187, Source citation: MR1349159, and the upper bound for $d\ge2$, see Source citation: MR2016607, Source citation: MR3405617.

Question context

An inhomogeneous model for first-passage percolation was introduced in Source citation: MR3485346 in which edges in the left and right half-planes of $\mathbb{Z}^2$ are assigned weights according to different distributions $F_-$ and $F_+$. It was further proven that a shape theorem holds in this setting, and that the asymptotic shape equals the convex hull of the shapes associated with the distributions $F_-$ and $F_+$, intersected with respective half-plane, and a potential additional line segment along the vertical axis. In the presence of a line segment this would produce a `pyramid' on the shape in the vertical direction, and it is known that there exists pairs $(F_-,F_+)$ for which such a pyramid is produced.

Dated source record, inspected 2026-09-10. Source explicitly says the question was answered affirmatively during the workshop. Source-reported results have not been independently verified.

Status note in the source

This question was answered affirmatively by Ahlberg, Bhatnagar, Dey and Ganguly during the workshop.