Skip to main content

On the construction of the three dimensional polymer measure


The three dimensional polymer measure was first constructed by Westwater in 1980 with a very complicated proof. We give an alternative construction for small coupling parameter which is based on the approach by Brydges-Fröhlich-Sokal in quantum field theory and Bovier-Felder-Fröhlich, using skeleton inequalities. The main new features are the proof of convergence which had been open in the Brydges-Fröhlich-Sokal construction, and the construction of the measure on the space of paths with fixed time length.


  1. Bovier, A., Felder, G., Fröhlich, J.: On the critical properties of the Edwards and self-avoiding walk model of polymer chains. Nucl. Phys. B230, 119–147 (1984)

    Google Scholar 

  2. Brydges, D.C., Fröhlich, J., Sokal, A.: A new proof of the existence and nontriviality of the continuum φ 42 and φ 43 quantum field theories. Commun. Math. Phys.91, 141 (1983)

    Google Scholar 

  3. Dynkin, E.B.: Regularised self-intersection local times of planar Brownian motion. Ann. Probab.16, 58–74 (1988)

    Google Scholar 

  4. Kusuoka, S.: On the path property of Edwards' model for long polymer chains in three dimensions. In: Albeverio, S. (ed.) Proc. Bielefeld Conf. in infinite dimensional analysis and stochastic processes. (Pitman Res. Notes Math. Sci., vol. 124, pp. 48–65) Harlow: Longman and New York: Wiley 1985

    Google Scholar 

  5. Rosen, J.: A local time approach to the self-intersections of Brownian paths in space. Commun. Math. Phys.88, 327–338 (1983)

    Google Scholar 

  6. Varadhan, S.R.S.: Appendix to “Euclidian quantum field theory” by K. Symanzik. In: Jost, R. (ed.) Local quantum theory. New York London: Academic Press 1969

    Google Scholar 

  7. Westwater, J.: On Edwards' model for polymer chains. Commun. Math. Phys.72, 131–174 (1980)

    Google Scholar 

  8. Westwater, J.: On Edwards' model for polymer chains. III. Borel summability. Commun. Math. Phys.84, 459–470 (1982)

    Google Scholar 

  9. Yor, M.: Complements aux formules de Tanaka-Rosen. In: Azéma, J., Yor, M. (eds.) Seminaire de Probabilites XIX. (Lect. Notes Math., vol. 1123, pp. 332–349) Berlin Heidelberg New York: Springer 1985.

    Google Scholar 

  10. Zhou, Z.H.: The intersection local time for the Westwater process. Probab. Theory Relat. Fields91, 375 (1992)

    Google Scholar 

Download references

Author information

Authors and Affiliations


Rights and permissions

Reprints and Permissions

About this article

Cite this article

Bolthausen, E. On the construction of the three dimensional polymer measure. Probab. Th. Rel. Fields 97, 81–101 (1993).

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI:


  • Polymer
  • Field Theory
  • Stochastic Process
  • Probability Theory
  • Mathematical Biology