Point-to-Point Distance in First Passage Percolation on (Tree) ×Z

By Itai Benjamini, P. Maillard. 2013

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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.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/2dd623fc4fa5cae7aae0d2cd55108239094b56b7