Chemical distance in graphs of polynomial growth

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 > pc(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 pc.