## The Annals of Probability

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

### First-passage percolation on Cartesian power graphs

#### 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**

Received: October 2015

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

**Subjects**

Primary: 60C05: Combinatorial probability

Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B43: Percolation [See also 60K35]

**Keywords**

First-passage percolation power graph high dimension time constant hypercube

#### 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