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Green's Function for a Hierarchical Self-Avoiding Walk in Four Dimensions

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Abstract

This is the second of two papers on the end-to-end distance of a weakly self-repelling walk on a four dimensional hierarchical lattice. It completes the proof that the expected value grows as a constant times \({{\sqrt{{T}}\log^{{\frac{{1}}{{8}}}}T {{\left({{1+O{{\left({{\frac{{\log\log T}}{{\log T}}}}\right)}} }}\right)}}}}\), which is the same law as has been conjectured for self-avoiding walks on the simple cubic lattice ℤ4. Apart from completing the program in the first paper, the main result is that the Green's function is almost equal to the Green's function for the Markov process with no self-repulsion, but at a different value of the killing rate β which can be accurately calculated when the interaction is small. Furthermore, the Green's function is analytic in β in a sector in the complex plane with opening angle greater than π.

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Authors and Affiliations

  1. Mathematics Department, University of British Columbia, #121-1984 Mathematics Road, Vancouver, B.C, V6T 1Z2, Canada

    David C. Brydges

  2. Department of Mathematics, University of Virginia, Kerchof Hall, P. O. Box 400137, Charlottesville, VA, 22904-4137, USA

    John Z. Imbrie

Authors
  1. David C. Brydges
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  2. John Z. Imbrie
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Correspondence to John Z. Imbrie.

Additional information

Communicated by M. Aizenman

Research supported by NSF grant DMS-9706166 and NSERC of Canada.

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Brydges, D., Imbrie, J. Green's Function for a Hierarchical Self-Avoiding Walk in Four Dimensions. Commun. Math. Phys. 239, 549–584 (2003). https://doi.org/10.1007/s00220-003-0886-5

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  • DOI: https://doi.org/10.1007/s00220-003-0886-5

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