Critical Exponents for Long-Range Interactions

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 lsing (and more generally 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 Jx = 1/Ixl a+~ with 0 < a~< d/2, then the exponents ~, 6, ? and A 4 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 try> 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.