Supercritical percolation on graphs of polynomial growth

We consider Bernoulli percolation on transitive graphs of polynomial growth. In the subcritical regime ($p<p_c$), it is well known that the connection probabilities decay exponentially fast. In the present paper, we study the supercritical phase ($p>p_c$) and prove the exponential decay of the truncated connection probabilities (probabilities that two points are connected by an open path, but not to infinity). This sharpness result was established by [CCN87] on $\mathbb Z^d$ and uses the difficult slab result of Grimmett and Marstrand. However, the techniques used there are very specific to the hypercubic lattices and do not extend to more general geometries. In this paper, we develop new robust techniques based on the recent progress in the theory of sharp thresholds and the sprinkling method of Benjamini and Tassion. On $\mathbb Z^d$, these methods can be used to produce a new proof of the slab result of Grimmett and Marstrand.