Open Access
2009 Functional integral representations for self-avoiding walk
David C. Brydges, John Z. Imbrie, Gordon Slade
Probab. Surveys 6: 34-61 (2009). DOI: 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 anticommuting Grassmann variables, which can be conveniently interpreted as differential forms.


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David C. Brydges. John Z. Imbrie. Gordon Slade. "Functional integral representations for self-avoiding walk." Probab. Surveys 6 34 - 61, 2009.


Published: 2009
First available in Project Euclid: 11 August 2009

zbMATH: 1193.82014
MathSciNet: MR2525670
Digital Object Identifier: 10.1214/09-PS152

Primary: 81T60 , 82B41
Secondary: 60J27 , 60K35

Rights: Copyright © 2009 The Institute of Mathematical Statistics and the Bernoulli Society

Vol.6 • 2009
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