FIRST-PASSAGE PERCOLATION ON CARTESIAN POWER GRAPHS

Journal article, 2018

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...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,...,v) and (w, w,...,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 ->infinity. As a corollary, we are able to determine the limit of the so-called diagonal time-constant in Z(n) as n ->infinity for a large class of distributions of passage times.