Geodesic length and shifted weights in first-passage percolation

TL;DR

This paper explores the relationship between geodesic length and weight shifts in first-passage percolation, revealing new insights into the shape function's regularity, concavity, and singularities across various distributions and dimensions.

Contribution

It introduces a convex duality framework to analyze geodesic lengths and shape function regularity, extending classical results to all distributions, directions, and dimensions.

Findings

The shape function is strictly concave in the weight shift.

Atoms in the weight distribution cause nondifferentiability points.

A dense set of shifts lead to singularities when multiple atoms are present.

Abstract

We study first-passage percolation through related optimization problems over paths of restricted length. The path length variable is in duality with a shift of the weights. This puts into a convex duality framework old observations about the convergence of the normalized Euclidean length of geodesics due to Hammersley and Welsh, Smythe and Wierman, and Kesten, and leads to new results about geodesic length and the regularity of the shape function as a function of the weight shift. For points far enough away from the origin, the ratio of the geodesic length and the $\ell^1$ distance to the endpoint is uniformly bounded away from one. The shape function is a strictly concave function of the weight shift. Atoms of the weight distribution generate singularities, that is, points of nondifferentiability, in this function. We generalize to all distributions, directions and dimensions an old singularity result of Steele and Zhang for the planar Bernoulli case. When the weight distribution has two or more atoms, a dense set of shifts produce singularities. The results come from a combination of the convex duality, the shape theorems of the different first-passage optimization problems, and modification arguments.