Coalescence of geodesics and the BKS midpoint problem in planar first-passage percolation

TL;DR

This paper proves that in planar first-passage percolation with certain conditions, geodesics tend to coalesce significantly, leading to a quantitative solution of the BKS midpoint problem and related geometric properties.

Contribution

It establishes coalescence of geodesics under specific shape assumptions, resolving the BKS midpoint problem quantitatively and analyzing geodesic coverage without explicit shape assumptions.

Findings

Geodesics with nearby endpoints coalesce significantly.

Probability of a geodesic passing through a given edge decays as a power of distance.

Expected coverage of the grid by infinite geodesics is inversely proportional to a power of n.

Abstract

We consider first-passage percolation on $\mathbb Z^2$ with independent and identically distributed weights whose common distribution is absolutely continuous with a finite exponential moment. Under the assumption that the limit shape has more than 32 extreme points, we prove that geodesics with nearby starting and ending points have significant overlap, coalescing on all but small portions near their endpoints. The statement is quantified, with power-law dependence of the involved quantities on the length of the geodesics. The result leads to a quantitative resolution of the Benjamini--Kalai--Schramm midpoint problem. It is shown that the probability that the geodesic between two given points passes through a given edge is smaller than a power of the distance between the points and the edge. We further prove that the limit shape assumption is satisfied for a specific family of distributions. Lastly, related to the 1965 Hammersley--Welsh highways and byways problem, we prove that the expected fraction of the square $\{-n,\dots ,n\}^2$ which is covered by infinite geodesics starting at the origin is at most an inverse power of $n$. This result is obtained without explicit limit shape assumptions.