## 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.

### MSC:

 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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### References:

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