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High-Dimensional Graphical Networks of Self-Avoiding Walks

Published online by Cambridge University Press:  20 November 2018

Mark Holmes
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2 e-mail: holmes@math.ubc.ca, tetsu@math.ubc.ca, slade@math.ubc.ca
Antal A. Járai
Affiliation:
Centrum voor Wiskunde en Informatica, P. O. Box 94079, NL-1090 GB Amsterdam, The Netherlands e-mail: Antal.Jarai@cwi.nl
Akira Sakai
Affiliation:
Centrum voor Wiskunde en Informatica, P. O. Box 94079, NL-1090 GB Amsterdam, The Netherlands e-mail: Antal.Jarai@cwi.nl
Gordon Slade
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2 e-mail: holmes@math.ubc.ca, tetsu@math.ubc.ca, slade@math.ubc.ca
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Abstract

We use the lace expansion to analyse networks of mutually-avoiding self-avoiding walks, having the topology of a graph. The networks are defined in terms of spread-out self-avoiding walks that are permitted to take large steps. We study the asymptotic behaviour of networks in the limit of widely separated network branch points, and prove Gaussian behaviour for sufficiently spread-out networks on ${{\mathbb{Z}}^{d}}$ in dimensions $d\,>\,4$ .

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2004

References

[1] Brydges, D. C. and Spencer, T., Self-avoiding walk in 5 or more dimensions. Commun. Math. Phys. 97(1985), 125148.CrossRefGoogle Scholar
[2] Duplantier, B., Statistical mechanics of polymer networks of any topology. J. Statist. Phys. 54(1989), 581680.CrossRefGoogle Scholar
[3] Hara, T., Critical two-point functions for nearest-neighbour high-dimensional self-avoiding walk and percolation. In preparation.Google Scholar
[4] Hara, T., van der Hofstad, R. and G. Slade, Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models. Ann. Probab. 31(2003), 349408.Google Scholar
[5] van der Hofstad, R. and Slade, G., The lace expansion on a tree with application to networks of self-avoiding walks. Adv. Appl. Math. 30(2003), 471528.CrossRefGoogle Scholar

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