A new proof of the sharpness of the phase transition for Bernoulli percolation on Z(d)

We provide a new proof of the sharpness of the phase transition for nearestneighbour Bernoulli percolation on Z d. 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 in finite cluster satisfies the mean-field lower bound theta (p) >= p-p(c) /1- p(c) ). 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