Probability Surveys

Functional integral representations for self-avoiding walk

David C. Brydges, John Z. Imbrie, and Gordon Slade

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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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Probab. Surveys, Volume 6 (2009), 34-61.

First available in Project Euclid: 11 August 2009

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Zentralblatt MATH identifier

Primary: 81T60: Supersymmetric field theories 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]


Brydges, David C.; Imbrie, John Z.; Slade, Gordon. Functional integral representations for self-avoiding walk. Probab. Surveys 6 (2009), 34--61. doi:10.1214/09-PS152.

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