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A generalised inductive approach to the lace expansion
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  • Published: March 2002

A generalised inductive approach to the lace expansion

  • Remco van der Hofstad1 &
  • Gordon Slade2 

Probability Theory and Related Fields volume 122, pages 389–430 (2002)Cite this article

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  • 42 Citations

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Abstract.

 The lace expansion is a powerful tool for analysing the critical behaviour of self-avoiding walks and percolation. It gives rise to a recursion relation which we abstract and study using an adaptation of the inductive method introduced by den Hollander and the authors. We give conditions under which the solution to the recursion relation behaves as a Gaussian, both in Fourier space and in terms of a local central limit theorem. These conditions are shown elsewhere to hold for sufficiently spread-out models of networks of self-avoiding walks in dimensions d > 4, and for sufficiently spread-out models of critical oriented percolation in dimensions d + 1 > 5, providing a unified approach and an essential ingredient for a detailed analysis of the branching behaviour of these models.

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Authors and Affiliations

  1. Stieltjes Institute for Mathematics, Delft University, Mekelweg 4, 2628 CD Delft, The Netherlands. e-mail: R.W.vanderHofstad@its.tudelft.nl, , , , , , NL

    Remco van der Hofstad

  2. Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada. e-mail: slade@math.ubc.ca, , , , , , CA

    Gordon Slade

Authors
  1. Remco van der Hofstad
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  2. Gordon Slade
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Additional information

Received: 13 September 2000 / Revised version: 16 May 2001 / Published online: 20 December 2001

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van der Hofstad, R., Slade, G. A generalised inductive approach to the lace expansion. Probab Theory Relat Fields 122, 389–430 (2002). https://doi.org/10.1007/s004400100175

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  • Issue Date: March 2002

  • DOI: https://doi.org/10.1007/s004400100175

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Keywords

  • Fourier
  • Detailed Analysis
  • Limit Theorem
  • Central Limit
  • Central Limit Theorem
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