The phase transition in a general class of Ising-type models is sharp

For a family of translation-invariant, ferromagnetic, one-component spin systems-which includes Ising and φ{symbol}⁴ models-we prove that (i) the phase transition is sharp in the sense that at zero magnetic field the high- and low-temperature phases extend up to a common critical point, and (ii) the critical exponent β obeys the mean field bound β≤1/2. The present derivation of these nonperturbative statements is not restricted to "regular" systems, and is based on a new differential inequality whose Ising model version is M≤βhχ+M³+ βM²∂M/∂β. The significance of the inequality was recognized in a recent work on related problems for percolation models, while the inequality itself is related to previous results, by a number of authors, on ferromagnetic and percolation models.