The Intersection of Brownian Paths as a Case Study of a Renormalization Group Method for Quantum Field Theory

  • Michael Aizenman


A new approach is presented for the study of the probability that the random paths generated by two independent Brownian motions in ℝd intersect or, more generally, are within a short distance a of each other. The well known behavior of that function of a-above, below, and at the critical dimension d = 4, as well as further corrections, are derived here by means of a single renormalization group equation. The equation’s derivation is expected to shed some light on the β-function of the λø d 4 quantum field theory.




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  1. 1.
    Kakutani, S.: On Brownian motions in w-space. Proc. Acad. Tokyo 20, 648 (1944)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Dvoretzky, A., Erdös, P., Kakutani, S.: Double points of Brownian paths in w-space. Acta Sci. Math. Szeged 12, 75 (1950)MathSciNetGoogle Scholar
  3. 3.
    Erdös, P., Taylor, S.J.: Some intersection properties of random paths. Acta Math. Acad. Sci. Hung. 11, 231 (1960)MATHCrossRefGoogle Scholar
  4. 4.
    Lawler, G.F.: The probability of intersection of Random walks in four dimensions. Commun. Math. Phys. 86, 539 (1982)MathSciNetADSMATHCrossRefGoogle Scholar
  5. 5.
    Spencer, T.: Private communicationGoogle Scholar
  6. 6.
    Symanzik, K.: Small distance behavior in field theory and power counting. Commun. Math. Phys. 18, 227 (1970) Callan, C.G. Jr.: Broken scale invariance in scalar field theory. Phys. Rev. D2, 1541 (1970) Wilson, K.G.: Anomalous dimension and the breakdown of scale invariance in perturbation theory. Phys. Rev. D2, 1478 (1970)MathSciNetADSMATHCrossRefGoogle Scholar
  7. 7.
    Brezin, E., Le Guillou, J.C., Zinn-Justin, J.: In: Phase transitions and critical phenomena. Domb, C., Green, M.S., (eds.). London, New York, San Francisco: Academic Press 1976 Itzykson, C., Zuber, J.B.: Quantum field theory. New York: McGraw-Hill 1980Google Scholar
  8. 8.
    Symanzik, K.: Euclidean quantum field theory. I. Equations for a scalar model. J. Math. Phys. 7, 510 (1966)MathSciNetADSCrossRefGoogle Scholar
  9. 9.
    Aizenman, M.: Geometric analysis of ϕ4 fields and Ising models. Commun. Math. Phys. 86, 1 (1982)MathSciNetADSMATHCrossRefGoogle Scholar
  10. 10.
    Fröhlich, J.: On the triviality of λϕd4 theories and the approach to the critical point in d > 4 dimensions. Nucl. Phys. B200, [FS4] 281 (1982)ADSCrossRefGoogle Scholar
  11. 11.
    Aragão de Carvalho, C., Caraciollo, S., Fröhlich, J.: Polymers and g\ϕ\4 theory in four dimensions. Nucl. Phys. B215 [FS7], 209 (1983)ADSCrossRefGoogle Scholar
  12. 12.
    Aizenman, M., Graham, R.: On the renormalized coupling constant and the susceptibility in (ϕ44 field theory and the Ising model in four dimensions. Nucl. Phys. B225 [FS9], 261 (1983)MathSciNetADSCrossRefGoogle Scholar
  13. 13.
    Fröhlich, J.: Quantum field theory in terms of Random walks and Random surfaces. Cargèse (1983) lecture notesGoogle Scholar
  14. 14.
    Felder, G., Fröhlich, J.: Intersection properties of simple Random walks: a renormalization group approach. Commun. Math. Phys. 97, 111–124 (1984)ADSCrossRefGoogle Scholar
  15. 15.
    Aizenman, M.: Rigorous studies of critical behavior, to appear in the proceedings of the VIII sitges conference. L. Garrido (ed.), Berlin, Heidelberg, New York: Springer 1984Google Scholar

Copyright information

© Springer-Verlag Berlin, Heidelberg 1985

Authors and Affiliations

  • Michael Aizenman
    • 1
  1. 1.Departments of Mathematics and PhysicsRutgers UniversityNew BrunswickUSA

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