Perturbation theory for self-avoiding walk in Python

Parisi-Sourlas [1] and McKane [2] have expressed correlation functions for self-avoiding walks in terms of a supersymmetric |φ|4-type field theory. This has been surveyed in [3]. Brydges and Slade have developed a rigorous approach to study such field theories by renormalisation group analysis. The method depends on a certain explicit calculation (described in detail in [5], see also [4]). The mechanical calculation can be done by hand, for example using Feynman diagram mnemonics, but these calculations can become tedious. The program below (written in the Python programming language) performs these computations in an automated way.

  • Instructions: PDF
  • Source code: TGZ (requires Python 2.6 or higher)

The results play a role in [8], [9] (which rely on [6], [7] for control of the non-perturbative remainder).

Contents

  • wick.py is a collection of Python classes to handle the combinatorics involved in calculating Gaussian integrals and their fermionic analogs. It cannot be executed by itself.
  • sawpt.py utilizes wick.py to compute the specific expressions for the supersymmetric self-avoiding walk field theory as described in [5].
  • phi4npt.py similarly computes the specific expressions for the n-component |φ|4 model as described in [9].

References

  1. G. Parisi and N. Sourlas, N., Self-avoiding walk and supersymmetry, J. Phys. Lett. 41 (1980), L403-L406, link (subscription required).
  2. A.J. McKane, Reformulation of n → 0 models using anticommuting scalar fields, Phys. Lett. A 76 (1980), no. 1, 22-24, link (subscription required).
  3. D.C. Brydges, J.Z. Imbrie, and G. Slade, Functional integral representations for self-avoiding walk, Probab. Surv. 6 (2009), 34-61, link.
  4. D.C. Brydges and G. Slade, A renormalisation group method. II. Approximation by local polynomials, JSP, 2015.
  5. R. Bauerschmidt, D.C. Brydges and G. Slade, A renormalisation group method. III. Perturbative analysis, JSP, 2015.
  6. D.C. Brydges and G. Slade, A renormalisation group method. IV. Nonperturbative analysis of weakly self-avoiding walk, JSP, 2015.
  7. D.C. Brydges and G. Slade, A renormalisation group method. V. A single renormalisation group step, CMP, 2015.
  8. R. Bauerschmidt, D.C. Brydges and G. Slade, Logarithmic correction for the susceptibility of the 4-dimensional weakly self-avoiding walk: a renormalisation group analysis, CMP, 2015.
  9. R. Bauerschmidt, D.C. Brydges and G. Slade, Scaling limits and critical behaviour of the 4-dimensional n-component |φ|4 spin model, JSP, 2015.