Functional integral representations for self-avoiding walk

Details Functional integral representations for self-avoiding walk Functional integral representations for self-avoiding walk

2009-06-04

arxiv.org/abs/0906.0922v2

Probability Surveys 2009, Vol. 6, 34-61

Mathematics

Probability

28 pages. Published at http://www.i-journals.org/ps/viewarticle.php?id=152 in the Probability Surveys (http://www.i-journals

Scientific paper

10.1214/09-PS152

We give a survey and unified treatment of functional integral representations for both simple random walk and some self-avoiding walk models, including models with strict self-avoidance, with weak self-avoidance, and a model of walks and loops. Our representation for the strictly self-avoiding walk is new. The representations have recently been used as the point of departure for rigorous renormalization group analyses of self-avoiding walk models in dimension 4. For the models without loops, the integral representations involve fermions, and we also provide an introduction to fermionic integrals. The fermionic integrals are in terms of anti-commuting Grassmann variables, which can be conveniently interpreted as differential forms.

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