Coalescence of Geodesics and the BKS Midpoint Problem in Planar First-Passage Percolation
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 a...
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| Published in | Geometric and functional analysis Vol. 34; no. 3; pp. 733 - 797 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
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Springer International Publishing
01.06.2024
Springer Nature B.V |
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| ISSN | 1016-443X 1420-8970 |
| DOI | 10.1007/s00039-024-00672-z |
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| 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. |
|---|---|
| AbstractList | 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. 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. 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 ,…, 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. |
| Author | Peled, Ron Elboim, Dor Dembin, Barbara |
| Author_xml | – sequence: 1 givenname: Barbara surname: Dembin fullname: Dembin, Barbara organization: D-MATH, ETH Zürich – sequence: 2 givenname: Dor surname: Elboim fullname: Elboim, Dor organization: Department of Mathematics, Princeton University – sequence: 3 givenname: Ron surname: Peled fullname: Peled, Ron email: peledron@tauex.tau.ac.il organization: School of Mathematical Sciences, Tel Aviv University, Institute for Advanced Study and Princeton University |
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| Snippet | We consider first-passage percolation on
with independent and identically distributed weights whose common distribution is absolutely continuous with a finite... We consider first-passage percolation on $\mathbb{Z}^{2}$ with independent and identically distributed weights whose common distribution is absolutely... We consider first-passage percolation on with independent and identically distributed weights whose common distribution is absolutely continuous with a finite... |
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| SubjectTerms | Analysis Coalescing Geodesy Mathematics Mathematics and Statistics Percolation |
| Title | Coalescence of Geodesics and the BKS Midpoint Problem in Planar First-Passage Percolation |
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