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 $\mathbb{P}_\beta[0\longleftrightarrow\infty]\ge (\beta-\beta_c)/\beta$ 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 $\langle \sigma_0\rangle_\beta^+\ge \sqrt{(\beta^2-\beta_c^2)/\beta^2}$ for $\beta>\beta_c$. For finite-range models, we also prove that the two-point correlations functions decay exponentially fast in the distance for $\beta<\beta_c$.
