Abstract
In this paper we approach the theory of the so called effective potential from a probabilistic point of view. We consider lattice systems and we clarify the connection between effective potential, fluctuations of mean fields (e.g. the magnetization in a large volume) and preasymptotic behaviour of the renormalization group (corrections to limit theorems). We point out how this approach may provide non perturbative calculational tools based on recursion and we discuss the case of hierarchical models as an example.
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References
L. Landau, E. Lifchitz, “Physique Statistique” MIR, Moscou 1967. A more complete exposition of Landau theory can be found in A.Patashinsky, V.Pokrowsky, “Fluctuation Theory of Phase Transitions” 2nd edition, Nauka, Moscow 1982 (Russian).
The effective potential first appeared in a perturbative context in J. Goldstone, A. Salam, S. Weinberg, Phys. Rev. 127, 965 (1962).
It is a special case of the more general notion of effective action introduced in G. Jona-Lasinio, Nuovo Cimento 34, 1790 (1964) and developed in H.Dahmen, G.Jona-Lasinio, Nuovo Cimento A52, 807 (1967). The physical meaning of the effective action was especially discussed in an appendix to the last paper.
See also S. Coleman “Secret Symmetries” in Laws of Hadronic Matter” ed A. Zichichi, Acad. Press, New York 1975.
P. M. Bleher, “Large Deviations Theorem near the Critical Point of the ψ4-Hierarchical Model” Abstracts of the 1981 Vilnius Conference on Probability Theory and Mathematical Statistics.
I became aware after the completion of the present work that aa somewhat related point of view was proposed by R. Fukuda, Progr. Theor. Phys. 56, 258 (1976) within the usual field theoretic heuristic. I am indebted to K.Symanzik for informing me about Fukuda’s work.
Probabilistic concepts in R.G. theory first appeared in P.M. Bleher, Ya. G. Sinai, Comm. Math. Phys. 33, 23 (1973).
The general connection between the R.G. and limit theorems for random fields and the basic notion of stable (equivalently: automodel, self-similar) random field was introduced in G. Jona-Lasinio, Nuovo Cimento B26, 99 (1975)
G. Gallavotti, G. Jona-Lasinio, Comm. Math. Phys. 41, 301 (1975) and independently in Ya. G.Sinai, Theory of Prob. and its Appl. XXI, 64 (1976). The notion of self-similar random field is the probabilistic counter part of the fixed point in the usual R.G.terminology: the physicist concept of universality corresponds in probability to that of domain of attraction. For an informal exposition of the connection between R.G. and probability see M.Cassandro, G.Jona-Lasinio, Advances in Physics 27, 913 (1978) where one can find additional references. A more recent perspective which includes field theory is in J.Frölich, T.Spencer, “Some Recent Rigorous Results in the Theory of Phase Transitions and Critical Phenomena” Seminaire Bourbaki, 34e année, 1981/82, n° 586.
Results on large deviations for Gibbs random fields are still very scanty. See however S.K. Pogosian, Uspehi, Mat. Nauk. 36, 201 (1981) (Russian) and the interesting preprint by R.Ellis, “Large Deviations and Other Limit Theorems for a class of Dependent Random Variables with Applications to Statistical Mechanics”.
See for example H.D.Dahmen, G.Jona-Lasinio in Ref. /2/.
I.A. Ibragimov, Yu.V. Linnik, “Independent and Stationary Sequences of Random Variables” Groningen, Wolter Noordhoff Publ. 1971.
P.M. Bleher, Works of the Moscow Mathematical Society 33, 155 (1975) (Russian).
A systematic and rigorous discussion of hierarchical models is given in P. Collet, J.P. Eckmann, “A Renormalization Group Analysis of the Hierarchical Model”, Lect. Notes in Physics 74, Springer, Berlin-Heidelberg-New York, 1979.
This trend was initiated by G.Gallavotti and his collaborators and is reviewed by G.Gallavotti in “Quantum Fields — Algebras, Processes” ed. L.Streit, Wien 1980. For more recent developments see K.Gawedsky, A.Kupiainen, contribution to this volume.
L.S. Shulman, J.Phys. A. 13, 237 (1980).
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Jona-Lasinio, G. (1983). Large Fluctuations of Random Fields and Renormalization Group: Some Perspectives. In: Fröhlich, J. (eds) Scaling and Self-Similarity in Physics. Progress in Physics, vol 7. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-6762-6_1
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DOI: https://doi.org/10.1007/978-1-4899-6762-6_1
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