On the critical behavior of the magnetization in high-dimensional Ising models. (English) Zbl 0629.60106

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 \(d\geq 4\) dimensions the magnetization is continuous at \(T_ c\) and its critical exponents take the classical values \(\delta =3\) and \(\beta =\), with possible logarithmic corrections at \(d=4\). The continuity, and other explicit bounds, formally extend to \(d>3\). Other systems to which the results apply include long-range models in \(d=1\) dimension, with \(1/| x-y|^{\lambda}\) couplings, for which 2/(\(\lambda\)-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.


60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B27 Critical phenomena in equilibrium statistical mechanics
82B05 Classical equilibrium statistical mechanics (general)
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