Abstract
We define a hierarchical Gaussian field in a way that is motivated by the finite-range decomposition of the Gaussian free field. The hierarchical Gaussian free field is a hierarchical field that has comparable large distance behaviour to the lattice Gaussian free field. We explicitly construct a version of the hierarchical Gaussian field and verify that it has the desired properties. We define the hierarchical |φ|4 model and state the main result proved in this book, which gives the critical behaviour of the susceptibility of the 4-dimensional hierarchical |φ|4 model. In preparation for the proof of the main result, we reformulate the hierarchical |φ|4 model as a perturbation of a Gaussian integral.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
A. Abdesselam, Towards three-dimensional conformal probability. p-Adic Numbers Ultrametric Anal. Appl. 10, 233–252 (2018)
A. Abdesselam, A. Chandra, G. Guadagni, Rigorous quantum field theory functional integrals over the p-adics I: anomalous dimensions. Preprint (2013). https://arxiv.org/abs/1302.5971
M. Aizenman, R. Graham, On the renormalized coupling constant and the susceptibility in \(\phi _4^4\) field theory and the Ising model in four dimensions. Nucl. Phys. B 225(FS9), 261–288 (1983)
G.A. Baker Jr., Ising model with a scaling interaction. Phys. Rev. B5, 2622–2633 (1972)
R. Bauerschmidt, D.C. Brydges, G. Slade, A renormalisation group method. III. Perturbative analysis. J. Stat. Phys. 159, 492–529 (2015)
P.M. Bleher, Ya.G. Sinai, Investigation of the critical point in models of the type of Dyson’s hierarchical models. Commun. Math. Phys. 33, 23–42 (1973)
P.M. Bleher, Ya.G. Sinai, Critical indices for Dyson’s asymptotically-hierarchical models. Commun. Math. Phys. 45, 247–278 (1975)
D.C. Brydges, J.Z. Imbrie, End-to-end distance from the Green’s function for a hierarchical self-avoiding walk in four dimensions. Commun. Math. Phys. 239, 523–547 (2003)
D.C. Brydges, J.Z. Imbrie, Green’s function for a hierarchical self-avoiding walk in four dimensions. Commun. Math. Phys. 239, 549–584 (2003)
D. Brydges, S.N. Evans, J.Z. Imbrie, Self-avoiding walk on a hierarchical lattice in four dimensions. Ann. Probab. 20, 82–124 (1992)
P. Collet, J.-P. Eckmann, A Renormalization Group Analysis of the Hierarchical Model in Statistical Mechanics. Lecture Notes in Physics, vol. 74 (Springer, Berlin, 1978)
F.J. Dyson, Existence of a phase transition in a one-dimensional Ising ferromagnet. Commun. Math. Phys. 12, 91–107 (1969)
J. Fröhlich, On the triviality of \(\varphi _d^4\) theories and the approach to the critical point in d ≥ 4 dimensions. Nucl. Phys. B200(FS4), 281–296 (1982)
K. Gawȩdzki, A. Kupiainen, A rigorous block spin approach to massless lattice theories. Commun. Math. Phys. 77, 31–64 (1980)
K. Gawȩdzki, A. Kupiainen, Triviality of \(\phi _4^4\) and all that in a hierarchical model approximation. J. Stat. Phys. 29, 683–698 (1982)
K. Gawȩdzki, A. Kupiainen, Non-Gaussian fixed points of the block spin transformation. Hierarchical model approximation. Commun. Math. Phys. 89, 191–220 (1983)
K. Gawȩdzki, A. Kupiainen, Massless lattice \(\varphi ^4_4\) theory: rigorous control of a renormalizable asymptotically free model. Commun. Math. Phys. 99, 199–252 (1985)
T. Hara, A rigorous control of logarithmic corrections in four dimensional φ 4 spin systems. I. Trajectory of effective Hamiltonians. J. Stat. Phys. 47, 57–98 (1987)
T. Hara, H. Tasaki, A rigorous control of logarithmic corrections in four dimensional φ 4 spin systems. II. Critical behaviour of susceptibility and correlation length. J. Stat. Phys. 47, 99–121 (1987)
T. Hara, T. Hattori, H. Watanabe, Triviality of hierarchical Ising model in four dimensions. Commun. Math. Phys. 220, 13–40 (2001)
H. Koch, P. Wittwer, A nontrivial renormalization group fixed point for the Dyson–Baker hierarchical model. Commun. Math. Phys. 164, 627–647 (1994)
J.R. Norris, Markov Chains (Cambridge University Press, Cambridge, 1997)
C. Wieczerkowski, Running coupling expansion for the renormalized \(\varPhi ^4_4\)-trajectory from renormalization invariance. J. Stat. Phys. 89, 929–945 (1997)
C. Wieczerkowski, Construction of the hierarchical ϕ 4-trajectory. J. Stat. Phys. 92, 377–430 (1998)
C. Wieczerkowski, Rigorous control of the non-perturbative corrections to the double expansion in g and \(g^2\ln (g)\) for the \(\phi ^4_3\)-trajectory in the hierarchical approximation. Helv. Phys. Acta 72, 445–483 (1999)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Singapore Pte Ltd.
About this chapter
Cite this chapter
Bauerschmidt, R., Brydges, D.C., Slade, G. (2019). The Hierarchical Model. In: Introduction to a Renormalisation Group Method. Lecture Notes in Mathematics, vol 2242. Springer, Singapore. https://doi.org/10.1007/978-981-32-9593-3_4
Download citation
DOI: https://doi.org/10.1007/978-981-32-9593-3_4
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-32-9591-9
Online ISBN: 978-981-32-9593-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)