Abstract
This paper is the fourth in a series devoted to the development of a rigorous renormalisation group method for lattice field theories involving boson fields, fermion fields, or both. The third paper in the series presents a perturbative analysis of a supersymmetric field theory which represents the continuous-time weakly self-avoiding walk on \({{{\mathbb Z}}^d }\). We now present an analysis of the relevant interaction functional of the supersymmetric field theory, which permits a nonperturbative analysis to be carried out in the critical dimension \(d = 4\). The results in this paper include: proof of stability of the interaction, estimates which enable control of Gaussian expectations involving both boson and fermion fields, estimates which bound the errors in the perturbative analysis, and a crucial contraction estimate to handle irrelevant directions in the flow of the renormalisation group. These results are essential for the analysis of the general renormalisation group step in the fifth paper in the series.
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References
Abdesselam, A.: A complete renormalization group trajectory between two fixed points. Commun. Math. Phys. 276, 727–772 (2007)
Bałaban, T.: (Higgs)\(_{2,3}\) quantum fields in a finite volume. Commun. Math. Phys. 85, 603–636 (1982)
Bauerschmidt, R.: A simple method for finite range decomposition of quadratic forms and Gaussian fields. Probab. Theory Relat. Fields 157, 817–845 (2013)
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
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
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
Bauerschmidt, R., Brydges, D.C., Slade, G.: Structural stability of a dynamical system near a non-hyperbolic fixed point. Annales Henri Poincaré (to appear). doi:10.1007/s00023-014-0338-0
Bauerschmidt, R., Brydges, D.C., Slade, G.: Scaling limits and critical behaviour of the \(4\)-dimensional \(n\)-component \(|\varphi |^4\) spin model. J. Stat. Phys. 157, 692–742 (2014)
Benfatto, G., Gallavotti, G.: Renormalization Group. Princeton University Press, Princeton (1995)
Brydges, D., Dimock, J., Hurd, T.R.: The short distance behavior of \((\varphi ^4)_3\). Commun. Math. Phys. 172, 143–146 (1995)
Brydges, D.C., Imbrie, J.Z., Slade, G.: Functional integral representations for self-avoiding walk. Probab. Surv. 6, 34–61 (2009)
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
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
Brydges, D.C., Slade, G.: A renormalisation group method V. A single renormalisation group step. J. Stat. Phys. (to appear). doi:10.1007/s10955-014-1167-8
Dimock, J.: Infinite volume limit for the dipole gas. J. Stat. Phys. 135, 393–427 (2009)
Dimock, J.: The renormalization group according to Bałaban I. Small fields. Rev. Math. Phys. 25, 1330010 (2013)
Dimock, J., Hurd, T.R.: A renormalization group analysis of correlation functions for the dipole gas. J. Stat. Phys. 66, 1277–1318 (1992)
Dimock, J., Hurd, T.R.: Sine-Gordon revisted. Annales Henri Poincaré 1, 499–541 (2000)
Falco, P.: Kosterlitz–Thouless transition line for the two dimensional Coulomb gas. Commun. Math. Phys. 312, 559–609 (2012)
Falco, P.: Critical exponents of the two dimensional Coulomb gas at the Berezinskii–Kosterlitz–Thouless transition. (2013). arXiv:1311.2237
Federbush, P.: Quantum field theory in ninety minutes. Bull. Am. Math. Soc. 17, 30–103 (1987)
Gawȩdzki, K., Kupiainen, A.: Massless lattice \(\varphi ^4_4\) theory: rigorous control of a renormalizable asymptotically free model. Commun. Math. Phys. 99, 199–252 (1985)
Gawȩdzki, K., Kupiainen, A.: Asymptotic freedom beyond perturbation theory. In Osterwalder, K., Stora, R. (eds.), Critical Phenomena, Random Systems, Gauge Theories, Amsterdam, (1986). North-Holland, Les Houches (1984)
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)
Rivasseau, V.: From Perturbative to Constructive Renormalization. Princeton University Press, Princeton (1991)
Salmhofer, M.: Renormalization: An Introduction. Springer, Berlin (1999)
Slade, G., Tomberg, A.: Critical correlation functions for the \(4\)-dimensional weakly self-avoiding walk and \(n\)-component \(|\varphi |^4\) model. (2014). arXiv:1412.2668
Acknowledgments
The work of both authors was supported in part by NSERC of Canada. DB gratefully acknowledges the support and hospitality of the Institute for Advanced Study at Princeton and of Eurandom during part of this work. GS gratefully acknowledges the support and hospitality of the Institut Henri Poincaré, and of the Mathematical Institute of Leiden University, where part of this work was done. We thank Roland Bauerschmidt for numerous helpful discussions.
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Brydges, D.C., Slade, G. A Renormalisation Group Method. IV. Stability Analysis. J Stat Phys 159, 530–588 (2015). https://doi.org/10.1007/s10955-014-1166-9
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DOI: https://doi.org/10.1007/s10955-014-1166-9