Chemical distance in graphs of polynomial growth

TL;DR

This paper establishes bounds on the chemical distance in transitive graphs of polynomial growth and Cayley graphs near critical percolation, demonstrating Lipschitz continuity of time constants in supercritical regimes.

Contribution

It proves an Antal-Pisztora type theorem for polynomial growth graphs and extends results to Cayley graphs, also showing Lipschitz continuity of chemical distance time constants.

Findings

Chemical distance is linearly bounded by graph distance with high probability.

Results apply to Cayley graphs of finitely presented groups near p=1.

Time constants are Lipschitz continuous in p above p_c.

Abstract

We prove an Antal-Pisztora type theorem for transitive graphs of polynomial growth. That is, we show that if $G$ is a transitive graph of polynomial growth and $p > p_c(G)$, then for any two sites $x, y$ of $G$ which are connected by a $p$-open path, the chemical distance from $x$ to $y$ is at most a constant times the original graph distance, except with probability exponentially small in the distance from $x$ to $y$. We also prove a similar theorem for general Cayley graphs of finitely presented groups, for $p$ sufficiently close to 1. Lastly, we show that all time constants for the chemical distance on the infinite supercritical cluster of a transitive graph of polynomial growth are Lipschitz continuous as a function of $p$ away from $p_c$.