Probability Surveys

Functional integral representations for self-avoiding walk

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

Full-text: Open access


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.

Article information

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

First available in Project Euclid: 11 August 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

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.

Export citation


  • [1] Aragão de Carvalho, C., Caracciolo, S., and Fröhlich, J. (1983). Polymers and g|ϕ|4 theory in four dimensions. Nucl. Phys. B 215 [FS7], 209–248.
  • [2] Berezin, F. (1966). The Method of Second Quantization. Academic Press, New York.
  • [3] Brydges, D., Evans, S., and Imbrie, J. (1992). Self-avoiding walk on a hierarchical lattice in four dimensions. Ann. Probab. 20, 82–124.
  • [4] Brydges, D., Fröhlich, J., and Sokal, A. (1983). The random walk representation of classical spin systems and correlation inequalities. II. The skeleton inequalities. Commun. Math. Phys. 91, 117–139.
  • [5] Brydges, D., Fröhlich, J., and Spencer, T. (1982). The random walk representation of classical spin systems and correlation inequalities. Commun. Math. Phys. 83, 123–150.
  • [6] Brydges, D. and Imbrie, J. (2003a). End-to-end distance from the Green’s function for a hierarchical self-avoiding walk in four dimensions. Commun. Math. Phys. 239, 523–547.
  • [7] Brydges, D. and Imbrie, J. (2003b). Green’s function for a hierarchical self-avoiding walk in four dimensions. Commun. Math. Phys. 239, 549–584.
  • [8] Brydges, D., Járai Jr., A., and Sakai, A. (2001). Self-interacting walk and functional integration. Unpublished document.
  • [9] Brydges, D. and Muñoz Maya, I. (1991). An application of Berezin integration to large deviations. J. Theoret. Probab. 4, 371–389.
  • [10] Brydges, D. and Slade, G. Papers in preparation.
  • [11] Dynkin, E. (1983). Markov processes as a tool in field theory. J. Funct. Anal. 50, 167–187.
  • [12] Fernández, R., Fröhlich, J., and Sokal, A. (1992). Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory. Springer, Berlin.
  • [13] Gennes, P. de (1972). Exponents for the excluded volume problem as derived by the Wilson method. Phys. Lett. A38, 339–340.
  • [14] Greub, W., Halperin, S., and Vanstone, R. (1972). Connections, Curvatures and Cohomology. Vol. I. Academic Press, New York.
  • [15] Imbrie, J. (2003). Dimensional reduction and crossover to mean-field behavior for branched polymers. Ann. Henri Poincaré 4, Suppl. 1, S445–S458.
  • [16] Le Jan, Y. (1987). Temps local et superchamp. In Séminaire de Probabilités XXI. Lecture Notes in Mathematics #1247. Springer, Berlin, 176–190.
  • [17] Le Jan, Y. (1988). On the Fock space representation of functionals of the occupation field and their renormalization. J. Funct. Anal. 80, 88–108.
  • [18] Madras, N. and Slade, G. (1993). The Self-Avoiding Walk. Birkhäuser, Boston.
  • [19] McKane, A. (1980). Reformulation of n0 models using anticommuting scalar fields. Phys. Lett. A 76, 22–24.
  • [20] Mitter, P. and Scoppola, B. (2008). The global renormalization group trajectory in a critical supersymmetric field theory on the lattice Z3. J. Stat. Phys. 133, 921–1011.
  • [21] Parisi, G. and Sourlas, N. (1980). Self-avoiding walk and supersymmetry. J. Phys. Lett. 41, L403–L406.
  • [22] Rudin, W. (1976). Principles of Mathematical Analysis, 3rd ed. McGraw–Hill, New York.
  • [23] Salmhofer, M. (1999). Renormalization: An Introduction. Springer, Berlin.
  • [24] Seeley, R. (1964). Extensions of C functions defined on a half space. Proc. Amer. Math. Soc. 15, 625–626.
  • [25] Symanzik, K. (1969). Euclidean quantum field theory. In Local Quantum Field Theory, R. Jost, Ed. Academic Press, New York.