Coalescence of Geodesics and the BKS Midpoint Problem in Planar First-Passage Percolation
OPEN ACCESS
Loading...
Author / Producer
Date
2024-06
Publication Type
Journal Article
ETH Bibliography
yes
Citations
Altmetric
OPEN ACCESS
Data
Rights / License
Abstract
We consider first-passage percolation on Z² 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,…,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.
Permanent link
Publication status
published
External links
Editor
Book title
Journal / series
Volume
34 (3)
Pages / Article No.
733 - 797
Publisher
Springer
Event
Edition / version
Methods
Software
Geographic location
Date collected
Date created
Subject
Organisational unit
Notes
Funding
175505 - Loops, paths and fields (SNF)
851565 - Critical and supercritical percolation (EC)
851565 - Critical and supercritical percolation (EC)