Abstract
In this paper, the Hierarchical Model is studied near a non-trivial fixed point φɛ of its renormalization group. Our analysis is an extension of work of Bleher and Sinai. We prove the validity of the ɛ-expansion for φɛ. We then show that the renormalization transformations around φɛ have an unstable manifold which is completely characterized by the tangent map and can be brought to normal form. We then establish relations between this result and the critical behaviour of the model in the thermodynamic limit.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Baker, G.A.: Ising model with a scaling interaction, Phys. Rev. B5, 2622 (1972)
Van Beyeren, H., Gallavotti, G., Knops, H.: Conservation laws in the hierarchical model. Physica78, 541 (1974)
Bleher, P.M., Sinai, Ja. G.: Investigation of the critical point in models of the type of Dyson's hierarchical models. Commun. math. Phys.33, 23 (1973)
Bleher, P.M., Sinai, Ja. G.: Critical indices for Dyson's asymptotically hierarchical models. Commun. math. Phys.45, 347 (1975)
Crandall, M.G., Rabinowitz, P.H.: Bifurcation from simple eigenvalues. J. Funct. Anal.8, 321 (1971)
Dunford, M., Schwartz, J.T.: Linear operators, Part I. New York: Interscience Publishers 1958
Dyson, F.J.: Existence of a phase-transition in a one-dimensional Ising ferromagnet. Commun. math. Phys.12, 91 (1969)
Dyson, F.J.: Non existence of spontaneous magnetization in a one-dimensional Ising ferromagnet. Commun. math. Phys.12, 212 (1969)
Dyson, F.J.: An Ising ferromagnet with discontinuous long-range order. Commun. math. Phys.21, 269 (1971)
Gallavotti, G., Knops, H.: The hierarchical model and the renormalization group. Rivista Nuovo Cimento5, 341 (1975)
Hamilton, R.S.: The inverse function theorem of Nash and Moser. Preprint Cornell University (1974);
Hörmander, L.: The boundary problems of physical geodesy. University of Lund Report No. 9 (1975);
Schwartz, J.T.: Nonlinear functional analysis. New York: Gordon and Breach 1969;
Sergeraert, F.: Un théorème de fonctions implicites sur certains espaces de Fréchet et quelques applications. Ann. Sci. Ecole Norm. Sup.5, 599 (1972);
Zehnder, E.: Generalized implicit function theorems with applications to some small divisor problems. Commun. pure appl. math.27, 91 (1975);29, 49 (1976)
Hirsch, M.W., Pugh, C.C.: Stable manifolds and hyperbolic sets, in Global analysis (Proc. Symp. Pure Math., Vol. 24, Berkeley, Calif. 1968), pp. 133–163. Providence R.I.: Amer. Math. Soc. 1970
Jona-Lasinio, G.: The renormalization group: a probabilistic view. Il Nuovo Cimento26B, 99 (1975)
Kato, T.: Perturbation theory for linear operators. Berlin-Heidelberg-New York: Springer 1966
Lanford, O.E. III: In Statistical mechanics and mathematical problems (1971 Battelle Rencontres). Lecture notes in physics, Vol. 20, (ed. A. Lenard), pp. 1–113. Berlin-Heidelberg-New York: Springer 1973
McBryan, O.A., Rosen, J.: Existence of the critical point in φ4 field theory. Commun. math. Phys.51, 97 (1976)
Nelson, E.: Topics in dynamics. I. Flows; Mathematical notes. Princeton: Princeton University Press 1969
Ruelle, D.: Statistical mechanics; Mathematical physics monograph series. New York: W. A. Benjamin, Inc. 1969
Wegner, F.J.: Corrections to scaling laws. Phys. Rev. B5, 4529 (1972)
Author information
Authors and Affiliations
Additional information
Communicated by E. Lieb
Rights and permissions
About this article
Cite this article
Collet, P., Eckmann, J.P. The ɛ-expansion for the Hierarchical Model. Commun.Math. Phys. 55, 67–96 (1977). https://doi.org/10.1007/BF01613151
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01613151