Critical exponents for long-range interactions |
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Authors: | Michael Aizenman Roberto Fernández |
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Institution: | (1) Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., 10012 New York, NY, USA;(2) Department of Mathematics, University of Texas at Austin, 78712 Austin, TX, USA |
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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+ with 0<![delta](/content/t2125577m204g03w/xxlarge948.gif) d/2, then the exponents
, , and 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 >-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.Also in the Physics Department. Research supported in part by the National Science Foundation Grant PHY 86-05164. |
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