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Improved perturbation expansion for disordered systems: Beating Griffiths singularities. (English) Zbl 0574.60098
We introduce a new expansion to prove exponential clustering of connected correlations in a large class of disordered systems. Our expansion converges for values of the temperature and magnetic field where standard cluster expansions diverge, due to the presence of Griffiths type singularities. It is organized inductively over an infinite sequence of increasing distance scales. In each induction step one redefines what is meant by the ”unperturbed system”, a procedure somewhat reminiscent of K.A.M. theory. Our techniques may be useful in dealing with the so-called large-field problem in real-space renormalization group schemes.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B05 Classical equilibrium statistical mechanics (general)
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[1] Griffiths, R.B.: Nonanalytic behavior above the critical point in a random Ising ferromagnet. Phys. Rev. Lett.23, 17 (1969); · doi:10.1103/PhysRevLett.23.17
[2] S?to, A.: Weak singularity and absence of metastability in random Ising ferromagnets. J. Phys. A15, L749-L752 (1982) · doi:10.1088/0305-4470/15/12/016
[3] Benfatto, G., Cassandro, M., Gallavotti, G., Nicol?, F., Olivieri, E., Presutti, E., Scacciatelli, E.: Some probabilistic techniques in field theory. Commun. Math. Phys.59, 143 (1978); Ultraviolet stability in Euclidean scalar field theories. Commun. Math. Phys.71, 95 (1980) · Zbl 0381.60096 · doi:10.1007/BF01614247
[4] Gawedzki, K., Kupiainen, A.: A rigorous block spin approach to massless lattice theories. Commun. Math. Phys.77, 31-64 (1980); Renormalization group for a critical lattice model. Commun. Math. Phys.88, 77-94 (1983); Block spin renormalization group for dipole gas and (??)4. Ann. Phys.147, 198-243 (1983); Lattice dipole gas and (??)4 at long distances: decay of correlations and scaling limit. Commun. Math. Phys.92, 531-554 (1984) · doi:10.1007/BF01205038
[5] Balaban, T., Imbrie, J., Jaffe, A.: Exact renormalization group for gauge theories. In: Progress in gauge field theory, Carg?se 1983. Jaffe, A., Lehman, H., Mitter, P., ’t Hooft, G. (eds.): New York: Plenum Press (to be published)
[6] Fr?hlich, Spencer, T.: Absence of diffusion in the Anderson tight binding module for large disorder or low energy. Commun. Math. Phys.88, 151 (1983) · Zbl 0519.60066 · doi:10.1007/BF01209475
[7] Kolmogorov, N.: Dokl. Akad. Nauk.98, 527 (1954);
[8] Moser, J.: Nachr. Akad. Wiss. G?ttingen. Math. Phys. Kl.11a, 1 (1962);
[9] Arnol’d, V.: Russ. Math. Surv.18, No. 5, 9-36 (1963) · Zbl 0129.16606 · doi:10.1070/RM1963v018n05ABEH004130
[10] Balaban, T., Gawedzki, K.: A low temperature expansion for the pseudoscalar Yukawa model of quantum fields in two space-time dimensions. Ann. Inst. H. Poincar?, Sect. A36, 271-400 (1982) · Zbl 0509.46059
[11] Imbrie, J.Z.: Phase diagrams and cluster expansions for low temperature 180-1(?)2 models. I and II. Commun. Math. Phys.82, 261-304 (1981) (See, in particular, pp. 284-288) · doi:10.1007/BF02099920
[12] Seiler, E.: Gauge theories as a problem of constructive field theory and statistical mechanics. In: Lecture Notes in Physics, Vol. 159. Berlin, Heidelberg, New York: Springer 1982
[13] Berretti, A.: J. Stat. Phys. (to appear)
[14] Glimm, J., Jaffe, A., Spencer, T.: The particle structure of the weakly coupledP(?)2 model, and other applications of high temperature expansions. In: Constructive quantum field theory. Lecture Notes in Physics, Vol. 25. Velo, G., Wightman, A.S. (eds.). Berlin, Heidelberg, New York: Springer 1973
[15] Roepstorff, G.: The Peierls-Griffiths argument for discussed Ising systems. J. Math. Phys.22, 3002 (1981) · doi:10.1063/1.524865
[16] Fisher, D.S., Fr?hlich, J., Spencer, T.: The Ising model in a random magnetic field. J. Stat. Phys.34, 863-870 (1984) · doi:10.1007/BF01009445
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