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arXiv · 1709.03647

On properties of optimal paths in first passage percolation

Shuta Nakajima
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Abstract

In this paper, we study some properties of optimal paths in the first passage percolation on $\Z^d$ 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.

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Shuta Nakajima. 2017-09-12. On properties of optimal paths in first passage percolation. https://doi.org/10.1007/s10955-018-2179-6

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