Inequalities for critical exponents
Abstract

Nonrigorous renormalizationgroup calculations predict meanfield critical behavior for systems above their upper critical dimension d _{ c } (e.g. d _{ c } = 4 for shortrange Isingtype models). For systems below their upper critical dimension (e.g. d = 3), RG methods predict exact scaling laws relating critical exponents, and give reasonably accurate numerical predictions of individual critical exponents (and other universal quantities).^{1}

Rigorous mathematical analysis has given a proof of (some aspects of) meanfield critical behavior for (certain) systems above their upper critical dimension (e.g. shortrange Ising models for d > 4). For systems below their upper critical dimension, much less is known. Often one half of a scaling law can be proven as a rigorous inequality. Likewise, rigorous upper or lower bounds on individual critical exponents can in many cases be proven.
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