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

  • Neal Madras (a1)
Abstract

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.

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[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. 22322257.
[2] Duminil-Copin, H. and Hammond, A., Self-avoiding walk is sub-ballistic. arxiv:1205.0401
[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, 101136. http://dx.doi.org/10.1007/BF02099530
[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, 235327. http://dx.doi.org/10.1142/S0129055X9200008X
[5] Madras, N. and Slade, G., The self-avoiding walk. Probability and its applications. Birkhäuser, Boston, 1993.
[6] Slade, G., The lace expansion and its applications. Lectures from the 34th Summer School onProbability Theory held in Saint-Flour, July 624, 2004. Lecture Notes in Mathematics, 1879. Springer-Verlag, Berlin, 2006.
[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. 189199.
[8] Vanderzande, C., Lattice models of polymers. Cambridge University Press, Cambridge, 1998
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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