## The Annals of Probability

### First-passage percolation on Cartesian power graphs

Anders Martinsson

#### Abstract

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.

#### Article information

Source
Ann. Probab., Volume 46, Number 2 (2018), 1004-1041.

Dates
Revised: April 2017
First available in Project Euclid: 9 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.aop/1520586274

Digital Object Identifier
doi:10.1214/17-AOP1199

Mathematical Reviews number (MathSciNet)
MR3773379

Zentralblatt MATH identifier
06864078

#### Citation

Martinsson, Anders. First-passage percolation on Cartesian power graphs. Ann. Probab. 46 (2018), no. 2, 1004--1041. doi:10.1214/17-AOP1199. https://projecteuclid.org/euclid.aop/1520586274

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