Coalescence of Geodesics and the BKS Midpoint Problem in Planar First-Passage Percolation

Barbara Dembin, Dor Elboim, Ron Peled*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We consider first-passage percolation on 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.

Original languageEnglish
Pages (from-to)733-797
Number of pages65
JournalGeometric and Functional Analysis
Volume34
Issue number3
DOIs
StatePublished - Jun 2024

Funding

FundersFunder number
NCCR Catalysis
Princeton University2.14
European Research Council101002733, 851565, 678520
Israel Science Foundation1971/19
Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung175505

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