zbMATH — the first resource for mathematics

A renormalisation group method. V. A single renormalisation group step. (English) Zbl 1317.82016
Summary: This paper is the fifth in a series devoted to the development of a rigorous renormalisation group method applicable to lattice field theories containing boson and/or fermion fields, and comprises the core of the method. In the renormalisation group method, increasingly large scales are studied in a progressive manner, with an interaction parametrised by a field polynomial which evolves with the scale under the renormalisation group map. In our context, the progressive analysis is performed via a finite-range covariance decomposition. Perturbative calculations are used to track the flow of the coupling constants of the evolving polynomial, but on their own perturbative calculations are insufficient to control error terms and to obtain mathematically rigorous results. In this paper, we define an additional non-perturbative coordinate, which together with the flow of coupling constants defines the complete evolution of the renormalisation group map. We specify conditions under which the non-perturbative coordinate is contractive under a single renormalisation group step. Our framework is essentially combinatorial, but its implementation relies on analytic results developed earlier in the series of papers. The results of this paper are applied elsewhere to analyse the critical behaviour of the 4-dimensional continuous-time weakly self-avoiding walk and of the 4-dimensional \(n\)-component \(|\varphi|^4\) model. In particular, the existence of a logarithmic correction to mean-field scaling for the susceptibility can be proved for both models, together with other facts about critical exponents and critical behaviour.

82B28 Renormalization group methods in equilibrium statistical mechanics
82B27 Critical phenomena in equilibrium statistical mechanics
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
PDF BibTeX Cite
Full Text: DOI arXiv
[1] Abdesselam, A., Chandra, A., Guadagni, G.: Rigorous quantum field theory functional integrals over the \(p\)-adics I: anomalous dimensions. Preprint (2013). arXiv:1302.5971
[2] Adams, S., Kotecký, R., Müller, S.: Strict convexity of the surface tension for non-convex potentials (2015, in preparation)
[3] Bałaban, T; Fröhlich, J (ed.), Ultraviolet stability in field theory. the \(ϕ ^4_3\) model, (1983), Boston
[4] Bauerschmidt, R, A simple method for finite range decomposition of quadratic forms and Gaussian fields, Probab. Theory Related Fields, 157, 817-845, (2013) · Zbl 1347.60037
[5] Bauerschmidt, R; Brydges, DC; Slade, G, Scaling limits and critical behaviour of the \(4\)-dimensional \(n\)-component \(|φ |^4\) spin model, J. Stat. Phys., 157, 692-742, (2014) · Zbl 1308.82026
[6] Bauerschmidt, R., Brydges, D.C., Slade, G.: Structural stability of a dynamical system near a non-hyperbolic fixed point. Annales Henri Poincaré (2014). doi:10.1007/s00023-014-0338-0 · Zbl 1347.37041
[7] Bauerschmidt, R., Brydges, D.C., Slade, G.: Critical two-point function of the 4-dimensional weakly self-avoiding walk. Commun. Math. Phys. (to appear). arXiv:1403.7268 · Zbl 1320.82031
[8] Bauerschmidt, R., Brydges, D.C., Slade, G.: Logarithmic correction for the susceptibility of the 4-dimensional weakly self-avoiding walk: a renormalisation group analysis. Commun. Math. Phys. (to appear). arXiv:1403.7422 · Zbl 1318.60049
[9] Bauerschmidt, R., Brydges, D.C., Slade, G.: A renormalisation group method. III. Perturbative analysis. J. Stat. Phys. (to appear). doi:10.1007/s10955-014-1165-x · Zbl 1319.82008
[10] Benfatto, G., Gallavotti, G.: Renormalization Group. Princeton University Press, Princeton, NJ (1995) · Zbl 0830.58038
[11] Benfatto, G; Cassandro, M; Gallavotti, G; Nicolò, F; Oliveri, E; Presutti, E; Scacciatelli, E, Some probabilistic techniques in field theory, Commun. Math. Phys., 59, 143-166, (1978) · Zbl 0381.60096
[12] Benfatto, G; Cassandro, M; Gallavotti, G; Nicolò, F; Oliveri, E; Presutti, E; Scacciatelli, E, On the ultraviolet stability in the Euclidean scalar field theories, Commun. Math. Phys., 71, 95-130, (1980) · Zbl 0427.60098
[13] Brydges, D.C.: Lectures on the renormalisation group. In Sheffield, S., Spencer, T., (eds.) Statistical Mechanics. American Mathematical Society, Providence, vol. 16, pp. 7-93. IAS/Park City Mathematics Series (2009) · Zbl 1186.82033
[14] Brydges, DC; Imbrie, JZ, Green’s function for a hierarchical self-avoiding walk in four dimensions, Commun. Math. Phys., 239, 549-584, (2003) · Zbl 1087.82010
[15] Brydges, D.C., Slade, G.: A renormalisation group method. I. Gaussian integration and normed algebras. J. Stat. Phys. (to appear). doi:10.1007/s10955-014-1163-z · Zbl 1317.82013
[16] Brydges, D.C., Slade, G.: A renormalisation group method. II. Approximation by local polynomials. J. Stat. Phys. (to appear). doi:10.1007/s10955-014-1164-y · Zbl 1317.82014
[17] Brydges, D.C., Slade, G.: A renormalisation group method. IV. Stability analysis. J. Stat. Phys. (to appear). doi:10.1007/s10955-014-1166-y · Zbl 1317.82015
[18] Brydges, DC; Yau, H-T, Grad \(ϕ \) perturbations of massless Gaussian fields, Commun. Math. Phys., 129, 351-392, (1990) · Zbl 0705.60101
[19] Brydges, D; Evans, SN; Imbrie, JZ, Self-avoiding walk on a hierarchical lattice in four dimensions, Ann. Probab., 20, 82-124, (1992) · Zbl 0742.60067
[20] Brydges, DC; Guadagni, G; Mitter, PK, Finite range decomposition of Gaussian processes, J. Stat. Phys., 115, 415-449, (2004) · Zbl 1157.82304
[21] Brydges, DC; Imbrie, JZ; Slade, G, Functional integral representations for self-avoiding walk, Probab. Surv., 6, 34-61, (2009) · Zbl 1193.82014
[22] Chae, S.B.: Holomorphy and Calculus in Normed Spaces. Marcel Dekker Inc, New York (1985) · Zbl 0571.46031
[23] Dimock, J, The renormalization group according to bałaban I. small fields, Rev. Math. Phys., 25, 1330010, (2013) · Zbl 1275.81068
[24] Dimock, J; Hurd, TR, A renormalization group analysis of correlation functions for the dipole gas, J. Stat. Phys., 66, 1277-1318, (1992) · Zbl 0925.82083
[25] Falco, P, Kosterlitz-thouless transition line for the two dimensional Coulomb gas, Commun. Math. Phys., 312, 559-609, (2012) · Zbl 1254.82012
[26] Falco, P.: Critical exponents of the two dimensional Coulomb gas at the Berezinskii-Kosterlitz-Thouless transition. Preprint (2013). arXiv:1311.2237 · Zbl 1087.82010
[27] Feldman, J., Knörrer, H., Trubowitz, E.: Fermionic Functional Integrals and the Renormalization Group. CRM Monograph Series, vol. 16. American Mathematical Society, Providence, (2002)
[28] Feldman, J; Magnen, J; Rivasseau, V; Sénéor, R, Construction and Borel summability of infrared \(Φ ^4_4\) by a phase space expansion, Commun. Math. Phys., 109, 437-480, (1987)
[29] Gawȩdzki, K; Kupiainen, A, Block spin renormalization group for dipole gas and \((∇ φ )^4\), Ann. Phys., 147, 198-243, (1983)
[30] Gawȩdzki, K; Kupiainen, A, Lattice dipole gas and \((∇ φ )^4\) models at long distances: decay of correlations and scaling limit, Commun. Math. Phys., 92, 531-553, (1984)
[31] Gawȩdzki, K; Kupiainen, A, Massless lattice \(φ ^4_4\) theory: rigorous control of a renormalizable asymptotically free model, Commun. Math. Phys., 99, 199-252, (1985)
[32] Gawȩdzki, K., Kupiainen, A.: Asymptotic freedom beyond perturbation theory. In: Osterwalder, K., Stora, R., (eds.) Critical Phenomena, Random Systems, Gauge Theories, Amsterdam, (1986). Les Houches, North-Holland (1984)
[33] Mastropietro, V.: Non-Perturbative Renormalization. World Scientific, Singapore (2008) · Zbl 1159.81005
[34] Mitter, P.K., Scoppola, B.: The global renormalization group trajectory in a critical supersymmetric field theory on the lattice \({\mathbb{Z}}^3\). J. Stat. Phys. 133, 921-1011 (2008) · Zbl 1161.82310
[35] Pöschel, J., Trubowitz, E.: Inverse Spectral Theory. Pure and Applied Mathematics, vol. 130. Academic Press Inc., Boston, MA (1987) · Zbl 0623.34001
[36] Rivasseau, V.: From Perturbative to Constructive Renormalization. Princeton University Press, Princeton, NJ (1991)
[37] Rivasseau, V., Wang, Z.: Corrected loop vertex expansion for \(ϕ ^4_2\) theory. Preprint (2014). arXiv:1406.7428 · Zbl 1275.81073
[38] Salmhofer, M.: Renormalization: An Introduction. Springer, Berlin (1999) · Zbl 0913.00014
[39] Slade, G., Tomberg, A.: Critical correlation functions for the \(4\)-dimensional weakly self-avoiding walk and \(n\)-component \(|φ |^4\) model. Preprint (2014). arXiv:1412.2668 · Zbl 1342.82070
[40] Wilson, KG; Kogut, J, The renormalization group and the \(ϵ \) expansion, Phys. Rep., 12, 75-200, (1974)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.