Inequalities for critical exponents

Abstract

The principal goal of the theory of critical phenomena is to make quantitative predictions for universal features of critical behavior — critical exponents, universal ratios of critical amplitudes, equations of state, and so forth — as discussed in Section 1.1. (Non-universal features, such as critical temperatures, are of lesser interest.) The present status of the theory of critical phenomena is roughly the following:

• 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.