Strict monotonicity for first passage percolation on graphs of polynomial growth and quasi-trees

TL;DR

This paper extends strict monotonicity results for first passage percolation from $Z^d$ to graphs of polynomial growth and quasi-trees, establishing conditions under which the time constant decreases with increased variability of edge weights.

Contribution

It proves a strict monotonicity theorem for a broader class of graphs, including those with polynomial growth and certain quasi-trees, introducing the geometric concept of 'admitting detours' and analyzing measure properties.

Findings

Graphs with detours admit strict monotonicity.

Cayley graphs of virtually nilpotent groups satisfy strict monotonicity.

Graphs without detours do not satisfy strict monotonicity.

Abstract

In 1993 van den Berg and Kesten proved a strict monotonicity theorem for first passage percolation on $\mathbb{Z}^d$, $d \ge 2$: given two probability measures $\nu$ and $\tilde{\nu}$ with finite mean, if $\tilde{\nu}$ is strictly more variable than $\nu$ and $\nu$ is subcritical in an appropriate sense, the time constant associated to $\tilde{\nu}$ is strictly smaller than the time constant associated to $\nu$. In this paper, an analogous result is proven for (not necessarily almost-transitive) graphs of strict polynomial growth and for bounded degree graphs quasi-isometric to trees which satisfy a certain geometric condition we call "admitting detours." It is also proven that if a bounded degree graph does not admit detours, then such a strict monotonicity theorem with respect to variability cannot hold. Large classes of graphs are shown to admit detours, and we conclude that for example any Cayley graph of a virtually nilpotent group which is not isomorphic to the standard Cayley graph of $\mathbb{Z}$ satisfies strict monotonicity with respect to variability, as does any Cayley graph of $F \rtimes F_k$, $F$ a nontrivial finite group and $F_k$ a free group. Moreover, it is proven that for graphs of strict polynomial growth and bounded degree graphs quasi-isometric to trees, if the weight measure is subcritical in an appropriate sense, then it is "absolutely continuous with respect to the expected empirical measure of the geodesic." This implies a strict monotonicity theorem with respect to stochastic domination of measures, whether or not the graph admits detours.