By G. Jona-Lasinio (auth.), Jürg Fröhlich (eds.)
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33, 23 (1973). G. JonaLasinio, CODUll. Math. Phy~41, 301 (1975) and independently in Ya. G. Sinai, Theory of Prob-:-and its Appl. XXI, 64 (1976) . terminology: the physicist concept of universality corresponds in probability to that of domain of attraction. G. JonaLasinio, Advances in Physics 27, 913 (1978) where one can find additional references . Spencer, "Some Recent Rigorous Results in the Theory of Phase Transitions and Critical Phenomena" Seminaire Bourbaki, 34e annee, 1981/82, nO 586. /6/ Results on large deviations for Gibbs random fields are still very scanty.
We now need some more mathematical formalism to develop these ideas : Let A be an arb itrary, finite sublattice of Zd with trivial homology. A site in A is called a a-cell, an or iented bond b a I-cell, an or iented plaquette a 2-cell, etc. 20) where c -1 denotes the k-cell obtained from c k by reversing the k orientation. 23) With respect to th is inner product, d and 0 are adjoints. 1 1) 0 2 = d 2 = a . 2) If a is a k-form such that oa=O then there exists a (k+l)-form S such that a = oS . ) Similar statements hold with 0 replaced by d .
Although the simplicity of the picture provided by Landau theory i s appealing, its quantitative predictions, concerning critical exponents, for example, turn out to be incorrect in dimension d ~4 ; (superstructures used to "deri veil it, 1ike catastrophe theory, cannot change thi s fact). A proof that many of the predictions of Landau theory (mean field theory) are correct for certain la ttice spin systems in dimension d ~5 forms the contents of other review articles in this volume. If and where 31 applicable the renormalization group, in particular the £-expansi on, makes predictions which are in excellent agreement with numerical data and experiments.