| Abstract: | We provide a new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model. The proof applies to infinite-range models on arbitrary locally finite transitive infinite graphs. For Bernoulli percolation, we prove finiteness of the susceptibility in the subcritical regime ({beta < beta_c}), and the mean-field lower bound ({mathbb{P}_beta[0longleftrightarrow infty ]ge (beta-beta_c)/beta}) for ({beta > beta_c}). For finite-range models, we also prove that for any ({beta < beta_c}), the probability of an open path from the origin to distance n decays exponentially fast in n. For the Ising model, we prove finiteness of the susceptibility for ({beta < beta_c}), and the mean-field lower bound ({langle sigma_0rangle_beta^+ge sqrt{(beta^2-beta_c^2)/beta^2}}) for ({beta > beta_c}). For finite-range models, we also prove that the two-point correlation functions decay exponentially fast in the distance for ({beta < beta_c}). |
