Abstract
Long-range components of the interaction in statistical mechanical systems may affect the critical behavior, raising the system's ‘effective dimension’. Presented here are explicit implications to this effect of a collection of rigorous results on the critical exponents in ferromagnetic models with one-component Ising (and more genrally Griffiths=Simon class) spin variables. In particular, it is established that even in dimensions d<4 if a ferromagnetic Ising spin model has a reflection-positive pair interaction with a sufficiently slow decay, e.g. as J x=1/|x|d+δ with 0<δ≤d/2, then the exponents \(\hat \beta \), δ, γ and Δ4 exist and take their mean-field values. This proves rigorously an early renormalization-group prediction of Fisher, Ma and Nickel. In the converse direction: when the decay is by a similar power law with δ>-2, then the long-range part of the interaction has no effect on the existent critical exponent bounds, which coincide then with those obtained for short-range models.
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Also in the Physics Department. Research supported in part by the National Science Foundation Grant PHY 86-05164.
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Aizenman, M., Fernández, R. Critical exponents for long-range interactions. Lett Math Phys 16, 39–49 (1988). https://doi.org/10.1007/BF00398169
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DOI: https://doi.org/10.1007/BF00398169