Inequalities for critical exponents
Non-rigorous renormalization-group calculations predict mean-field critical behavior for systems above their upper critical dimension d c (e.g. d c = 4 for short-range Ising-type 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) mean-field critical behavior for (certain) systems above their upper critical dimension (e.g. short-range 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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