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

Barbara Dembin, Dor Elboim, Ron Peled

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.

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

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 which is covered by infinite geodesics starting at the origin is at most an inverse power of . This result is obtained without explicit limit shape assumptions.
Paper Structure (38 sections, 24 theorems, 262 equations, 6 figures)

This paper contains 38 sections, 24 theorems, 262 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Coalescence of geodesics
  4. The influence of edges
  5. Highways and byways
  6. Many sides to the limit shape
  7. Attractive geodesics
  8. Overview of the proofs
  9. Discussion, extensions and open problems
  10. Coalescence of geodesics
  11. The influence of edges
  12. Highways and byways
  13. Limit shape properties
  14. Related models
  15. Higher dimensions and minimal surfaces
  16. ...and 23 more sections

Key Result

Theorem 1.1

Suppose $G$ satisfies eq:assumption i, eq:assumption ii and $\mathop{\mathrm{Sides}}\nolimits(\mathcal{B}_G)>32$. There exists $C>0$ (depending only on $G$) such that for each $0<\epsilon \le1/17$, each $\delta\ge0$ and all $y\in\mathbb{Z}^2$ with $\|y\|\ge 2$, where $p_1\triangle p_2$ is the set of edges belonging to exactly one of the paths $p_1, p_2$.

Figures (6)

  • Figure 1: A computer simulation of the geodesics from $(-1000,0)$ to $(1000,0)$ (blue and purple) and from $(-1000,30)$ to $(1000, 30)$ (red and purple) in first-passage percolation on $\mathbb Z ^2$ with weight distribution uniform on $[0,1]$. The pictures depict the geodesics in independent samples of the environment. Theorem \ref{['mainthm2']} states that nearby geodesics coalesce with high probability. The geodesics in the fourth simulation did not coalesce and, moreover, were far from each other for most of the way. This is compatible with our results as Proposition \ref{['prop:attractive geodesics intro']} shows that geodesics which stay close to each other for a significant amount of time have a very high probability to coalesce.
  • Figure 3: Illustration of the proof of Theorem \ref{['mainthm2']}
  • Figure 4: Illustration of the proof of Theorem \ref{['prop:probedgebulk']}
  • Figure 5: Illustration of the proof of Theorem \ref{['mainthm2']}
  • Figure 6: Illustration of the proof of Claim \ref{['claim:limit']}
  • ...and 1 more figures

Theorems & Definitions (66)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Proposition 1.7: Attractive Geodesics
  • Proposition 1.8
  • Lemma 2.1
  • Lemma 2.2
  • ...and 56 more