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March 2018 First-passage percolation on Cartesian power graphs
Anders Martinsson
Ann. Probab. 46(2): 1004-1041 (March 2018). DOI: 10.1214/17-AOP1199


We consider first-passage percolation on the class of “high-dimensional” graphs that can be written as an iterated Cartesian product $G\square G\square\dots\square G$ of some base graph $G$ as the number of factors tends to infinity. We propose a natural asymptotic lower bound on the first-passage time between $(v,v,\dots,v)$ and $(w,w,\dots,w)$ as $n$, the number of factors, tends to infinity, which we call the critical time $t^{*}_{G}(v,w)$. Our main result characterizes when this lower bound is sharp as $n\rightarrow\infty$. As a corollary, we are able to determine the limit of the so-called diagonal time-constant in $\mathbb{Z}^{n}$ as $n\rightarrow\infty$ for a large class of distributions of passage times.


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Anders Martinsson. "First-passage percolation on Cartesian power graphs." Ann. Probab. 46 (2) 1004 - 1041, March 2018.


Received: 1 October 2015; Revised: 1 April 2017; Published: March 2018
First available in Project Euclid: 9 March 2018

zbMATH: 06864078
MathSciNet: MR3773379
Digital Object Identifier: 10.1214/17-AOP1199

Primary: 60C05
Secondary: 60K35 , 82B43

Keywords: First-passage percolation , high dimension , hypercube , power graph , Time constant

Rights: Copyright © 2018 Institute of Mathematical Statistics


Vol.46 • No. 2 • March 2018
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