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A Lower Bound for the End-to-End Distance of the Self-Avoiding Walk

Published online by Cambridge University Press:  20 November 2018

Neal Madras
Neal Madras*
Affiliation:
Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON M3J 1P3 e-mail: madras@mathstat.yorku.ca
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Abstract

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For an $N$ -step self-avoiding walk on the hypercubic lattice ${{Z}^{d}}$ , we prove that the meansquare end-to-end distance is at least ${{N}^{4/(3d)}}$ times a constant. This implies that the associated critical exponent $v$ is at least $2/(3d)$ , assuming that $v$ exists.

Keywords

82B4160D0560K35self-avoiding walkcritical exponent

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 57 , Issue 1 , 14 March 2014 , pp. 113 - 118
Copyright
Copyright © Canadian Mathematical Society 2014

References

[1] Brydges, D. and Slade, G., Renormalisation group analysis of weakly self-avoiding walk in dimensionsfour and higher. In: Proceedings of the International Congress of Mathematicians, Volume IV, Hindustan Book Agency, New Delhi, 2010, pp. 2232–2257.Google Scholar
[2] Duminil-Copin, H. and Hammond, A., Self-avoiding walk is sub-ballistic. arxiv:1205.0401 Google Scholar
[3] Hara, T. and Slade, G., Self-avoiding walk in five or more dimensions. I. The critical behaviour. Comm. Math. Phys. 147 (1992), no. 1, 101–136. http://dx.doi.org/10.1007/BF02099530 Google Scholar
[4] Hara, T. and Slade, G., The lace expansion for self-avoiding walk in five or more dimensions. Rev. Math. Phys. 4 (1992), no. 2, 235–327. http://dx.doi.org/10.1142/S0129055X9200008X Google Scholar
[5] Madras, N. and Slade, G., The self-avoiding walk. Probability and its applications. Birkhäuser, Boston, 1993.Google Scholar
[6] Slade, G., The lace expansion and its applications. Lectures from the 34th Summer School onProbability Theory held in Saint-Flour, July 6–24, 2004. Lecture Notes in Mathematics, 1879. Springer-Verlag, Berlin, 2006.Google Scholar
[7] Slade, G., The self-avoiding walk: A brief survey. In: Surveys in stochastic processes, EMS Ser. Congr. Rep., European Mathematical Society, Zurich, 2011, pp. 189–199.Google Scholar
[8] Vanderzande, C., Lattice models of polymers. Cambridge University Press, Cambridge, 1998Google Scholar