Abstract
We consider first passage percolation (FPP) on \(\mathbb{T}_{d} \times G\), where \(\mathbb{T}_{d}\) is the d-regular tree (d ≥ 3) and G is a graph containing an infinite ray 0, 1, 2, …. It is shown that for a fixed vertex v in the tree, the fluctuation of the distance in the FPP metric between the points (v, 0) and (v, n) is of the order of at most logn. We conjecture that the real fluctuations are of order 1 and explain why.
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- 1.
This is Property (1) in [1]. Properties (2) and (3) are actually not needed, since on page 3 of that article, one can bound the right-hand side of the last inequality by Emin(Z n , Z n ′) + KC and continue from that point on.
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Acknowledgements
We thank an anonymous referee who has spotted some typographical errors.
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Benjamini, I., Maillard, P. (2014). Point-to-Point Distance in First Passage Percolation on (Tree) ×Z . In: Klartag, B., Milman, E. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2116. Springer, Cham. https://doi.org/10.1007/978-3-319-09477-9_3
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DOI: https://doi.org/10.1007/978-3-319-09477-9_3
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