V Relation with general scaling theory

[71.2.2.1] The results of the previous section are closely related with the general classification theory of phase transitions [18, 19, 20, 21, 22, 23], the probabilistic approach in the theory of critical phenomena [34, 35, 36], and finite size scaling theory for the order parameter distribution [7]. [71.2.2.2] The relation with the general classification theory of phase transitions [21, 22] has already been given above. [71.2.2.3] The relation with the probabilistic approach to critical phenomena [35] is that scaling and universality are obtained probabilistically from stability and nonempty domains of attraction for stable distributions. [71.2.2.4] The difference to [34, 35, 36] is that in those works the usual scaling limit of the measure [page 72, §0]    μ⁢φ→,a,L,Π→ is studied instead of the much simpler distributions appearing in the ensemble limit. [72.1.0.1] Similarly the differences with finite size scaling theory of the order parameter distribution [7] arise from the difference between the finite size scaling limit and the ensemble limit.

[72.1.1.1] The relation with the renormalization group scaling theory of critical points [37] is provided by the identification (4.15) relating the thermodynamic fluctuation exponents to the field theoretic correlation exponents, i.e. by hyperscaling. [72.1.1.2] The present theory considers only relevant operators by virtue of the general inequality ϖX>0. [72.1.1.3] Note that marginal operators correspond formally to ϖX→±∞, not to ϖX→0. [72.1.1.4] The influence of irrelevant operators is reflected in the general presence of a slowly varying function Λ⁢x in all scaling relations.

[72.1.2.1] The traditional classification into irrelevant (IO), marginal (MO) and relevant operators (RO) can be extended by three additional distinctions. [72.1.2.2] The first refinement is into equilibrium (ERO) and anequilibrium relevant operators (ARO) according to yE⁢R⁢O≤d for equilibrium relevant operators and yA⁢R⁢O>d for anequilibrium relevant operators. [72.1.2.3] ARO’s are readily constructed from ERO’s and are well known to occur in many models. [72.1.2.4] Examples are non-primary operators in conformal field theory [17], the energy and order parameter in anequilibrium phase transitions [21, 22], high gradient operators in the O⁢n nonlinear σ models [38, 39] or the hierarchical shell number modes in shell models for turbulence [40]. [72.1.2.5] An intriguing formal analogy exists between the random local events building up a multifractal measure and anequilibrium relevant operators [41].

[72.1.3.1] A second refinement of the traditional classification is to distinguish between gaussian and nongaussian relevant operators. [72.1.3.2] A relevant operator X is called gaussian if yX≤d/2 and nongaussian if yX>d/2. [72.1.3.3] By virtue of the duality law [28]

where ζX′=ζX′⁢ϖX,ζX an additional third distinction is expected for operators with yX<2⁢d as compared to those with yX≥2⁢d. [72.1.3.4] The precise nature of this distinction remains to be explored.

[72.1.3.5] The new extended classification of the spectrum of critical operators may (in obvious notation) be summarized by the inequalities

in which the relevance increases from left to right.