Perturbation theory for self-avoiding walks in Python

Parisi-Sourlas [1] and McKane [2] have expressed correlation functions for self-avoiding walks in terms of a supersymmetric φ⁴-type field theory. This has been surveyed in [3]. Brydges-Slade have developed a rigorous approach to study this field theory by renormalisation group analysis of which a brief summary is given in [4]. Their method depends on a certain explicit calculation (described in detail in [5, 6]), and estimates that prove that this calculation is correct to third-order (given in [7]). The explicit calculation can be done by hand, for example using Feynman diagram mnemonics, but these calculations can become very tedious. The program below (written in the Python programming language) does these computations automatically.

Perturbation theory for self-avoiding walks in Python (preliminary version) requires Python 2.6 or higher

Its result plays a role in our work in progress about logarithmic corrections of self-avoiding walks.

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 described in [4] or with more details in [5, 6]. The result of its execution gives information about the critical behavior of four dimensional self-avoiding walks.

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, Renormalisation group analysis of weakly self-avoiding walk in dimensions four and higher, Proceedings of the International Congress of Mathematicians. Volume IV (New Delhi), Hindustan Book Agency, 2010, pp. 2232-2257, link (Preprint).
5. D.C. Brydges and G. Slade, A renormalisation group method. II. Approximation by local polynomials, Preprint, 2012.
6. D.C. Brydges and G. Slade, A renormalisation group method. III. Perturbative analysis of weakly self-avoiding walk, Preprint, 2012.
7. D.C. Brydges and G. Slade, A renormalisation group method. IV. Nonperturbative analysis of weakly self-avoiding walk, Preprint, 2012.