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Inequalities for critical exponents

  • Roberto Fernández
  • Jürg Fröhlich
  • Alan D. Sokal
Part of the Texts and Monographs in Physics book series (TMP)

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.

Keywords

Ising Model Critical Exponent Critical Isotherm Bubble Diagram Full Neighborhood 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Roberto Fernández
    • 1
  • Jürg Fröhlich
    • 1
  • Alan D. Sokal
    • 2
  1. 1.Institut für Theoretische PhysikETH HönggerbergZürichSwitzerland
  2. 2.Department of PhysicsNew York UniversityNew YorkUSA

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