We provide a new proof of the sharpness of the phase transition for nearest-neighbour Bernoulli percolation. More precisely, we show that
- for $p<p_c$, the probability that the origin is connected by an open path to distance $n$ decays exponentially fast in $n$.
- for $p>p_c$, the probability that the origin belongs to an infinite cluster satisfies the mean-field lower bound $\theta(p)\ge\tfrac{p-p_c}{p(1-p_c)}$.
This note presents the argument of \cite{DumTas15}, which is valid for long-range Bernoulli percolation (and for the Ising model) on arbitrary transitive graphs in the simpler framework of nearest-neighbour Bernoulli percolation on $\mathbb Z^d$.
