On the critical behavior of the magnetization in high-dimensional Ising models

We derive rigorously general results on the critical behavior of the magnetization in Ising models, as a function of the temperature and the external field. For the nearest-neighbor models it is shown that in d≥4 dimensions the magnetization is continuous at T_c and its critical exponents take the classical values δ=3 and β=1/2, with possible logarithmic corrections at d=4. The continuity, and other explicit bounds, formally extend to d>3 1/2. Other systems to which the results apply include long-range models in d=1 dimension, with 1/|x-y|^λ couplings, for which 2/(λ-1) replaces d in the above summary. The results are obtained by means of differential inequalities derived here using the random current representation, which is discussed in detail for the case of a nonvanishing magnetic field.