Abstract
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 ind⩾4 dimensions the magnetization is continuous atT c and its critical exponents take the classical valuesδ=3 andβ=1/2, with possible logarithmic corrections atd=4. The continuity, and other explicit bounds, formally extend tod>3 1/2. Other systems to which the results apply include long-range models ind=1 dimension, with 1/|x−y|λ couplings, for which 2/(λ−1) replacesd 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.
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Research supported in part by NSF grant PHY-8301493 A02, and by a John S. Guggenheim Foundation fellowship (M.A.).
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Aizenman, M., Fernández, R. On the critical behavior of the magnetization in high-dimensional Ising models. J Stat Phys 44, 393–454 (1986). https://doi.org/10.1007/BF01011304
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DOI: https://doi.org/10.1007/BF01011304