On properties of optimal paths in first passage percolation
Probability
2021-03-31 v3
Abstract
In this paper, we study some properties of optimal paths in the first passage percolation on and show the followings: (1) the number of optimal paths has an exponential growth if the distribution has an atom; (2) the means of intersection and union of optimal paths are linear in the distance. For the proofs, we use the configuration--flipping argument introduced in [J. van den Berg and H. Kesten. Inequalities for the time constant in first-passage percolation. Ann. Appl. Probab. 56-80, 1993] with suitable adaptions.
Keywords
Cite
@article{arxiv.1709.03647,
title = {On properties of optimal paths in first passage percolation},
author = {Shuta Nakajima},
journal= {arXiv preprint arXiv:1709.03647},
year = {2021}
}
Comments
14 pages, 6 figures