First-passage percolation on Cartesian power graphs

TL;DR

This paper studies first-passage percolation on high-dimensional Cartesian product graphs, establishing a lower bound on passage times and characterizing its sharpness, with implications for the diagonal time-constant in large dimensions.

Contribution

It introduces a natural asymptotic lower bound called the critical time and characterizes when this bound is sharp in high-dimensional product graphs.

Findings

Established a lower bound on first-passage times in high-dimensional graphs.

Characterized conditions for the sharpness of the lower bound.

Determined the limit of the diagonal time-constant in z^n for various distributions.

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.