A new proof of the sharpness of the phase transition for Bernoulli percolation on Zᵈ

We provide a new proof of the sharpness of the phase transition for nearest-neighbour Bernoulli percolation on Z^d . More precisely, we show that In [DCT], we give a more general proof which covers long-range Bernoulli percolation (and the Ising model) on arbitrary transitive graphs. This article presents the argument of [DCT] in the simpler framework of nearest-neighbour Bernoulli percolation on Z^d .