Abstract
This paper is the second 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 method is set within a normed algebra \(\mathcal {N}\) of functionals of the fields. In this paper, we develop a general method—localisation—to approximate an element of \(\mathcal {N}\) by a local polynomial in the fields. From the point of view of the renormalisation group, the construction of the local polynomial corresponding to \(F \in \mathcal {N}\) amounts to the extraction of the relevant and marginal parts of \(F\). We prove estimates relating \(F\) and its corresponding local polynomial, in terms of the \(T_\phi \) semi-norm introduced in part I of the series.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bauerschmidt, R., Brydges, D.C., Slade, G.: Critical two-point function of the 4-dimensional weakly self-avoiding walk. Commun. Math. Phys. (2014). 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. (2014). 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
Brydges, D.C., Imbrie, J.Z., Slade, G.: Functional integral representations for self-avoiding walk. Probab. Surveys 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. IV. Stability analysis. J. Stat. Phys. (to appear). doi:10.1007/s10955-014-1166-9
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
de Boor, C., Ron, A.: The least solution for the polynomial interpolation problem. Math. Z. 210, 347–378 (1992)
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, where part of this work was done. We thank Benoît Laslier for insightful comments which led to important corrections, and also to simplifications in the proofs of Propositions 1.5 and 1.11. We also thank an anonymous referee for numerous pertinent suggestions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Brydges, D.C., Slade, G. A Renormalisation Group Method. II. Approximation by Local Polynomials. J Stat Phys 159, 461–491 (2015). https://doi.org/10.1007/s10955-014-1164-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-014-1164-y