Probability Theory and Related Fields

, Volume 97, Issue 1–2, pp 81–101 | Cite as

On the construction of the three dimensional polymer measure

  • Erwin Bolthausen
Article

Summary

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.

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References

  1. 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. 2.
    Brydges, D.C., Fröhlich, J., Sokal, A.: A new proof of the existence and nontriviality of the continuum φ24 and φ34 quantum field theories. Commun. Math. Phys.91, 141 (1983)Google Scholar
  3. 3.
    Dynkin, E.B.: Regularised self-intersection local times of planar Brownian motion. Ann. Probab.16, 58–74 (1988)Google Scholar
  4. 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 1985Google Scholar
  5. 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. 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 1969Google Scholar
  7. 7.
    Westwater, J.: On Edwards' model for polymer chains. Commun. Math. Phys.72, 131–174 (1980)Google Scholar
  8. 8.
    Westwater, J.: On Edwards' model for polymer chains. III. Borel summability. Commun. Math. Phys.84, 459–470 (1982)Google Scholar
  9. 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. 10.
    Zhou, Z.H.: The intersection local time for the Westwater process. Probab. Theory Relat. Fields91, 375 (1992)Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Erwin Bolthausen
    • 1
  1. 1.Institut fúr Angewandte MathematikUniversitát ZúrichZúrichSwitzerland

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