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Journal of Statistical Physics

, Volume 174, Issue 2, pp 259–275 | Cite as

On Properties of Optimal Paths in First-Passage Percolation

On Propertie of Optimal Paths
  • Shuta NakajimaEmail author
Article
  • 31 Downloads

Abstract

In this paper, we study some properties of optimal paths in the first passage percolation on \(\mathbb {Z}^d\) and show the following: (i) the number of optimal paths has an exponentially growth if the distribution has an atom; (ii) the means of intersection and union of optimal paths are linear in the distance. For the proofs, we use the resampling argument introduced in van den Berg and Kesten (Ann Appl Probab 3:56–80, 1993) with suitable adaptions.

Keywords

Random environment First passage percolation Random geometry Geodesics 

Mathematics Subject Classification

Primary 60K37 Secondary 60K35 82A51 82D30 

Notes

Acknowledgements

The author would like to express his gratitude to Masato Takei for introducing him the idea of Theorem 2 in [12]. He is also indebted to Hugo Duminil–Copin for introducing [4] prior to its publication.

Funding

The author has received research Grants from JSPS KAKENHI 16J04042.

Compliance with Ethical Standards

Research involving Human Participants and/or Animals

This article does not contain any studies with human participants performed by any of the authors.

Informed Consent

Informed consent was obtained from all individual participants included in the study.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Research Institute in Mathematical SciencesKyoto UniversityKyotoJapan

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