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Weak interaction limits for one-dimensional random polymers
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  • Published: April 2003

Weak interaction limits for one-dimensional random polymers

  • Remco van der Hofstad1,
  • Frank den Hollander2 &
  • Wolfgang König3 

Probability Theory and Related Fields volume 125, pages 483–521 (2003)Cite this article

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Abstract.

 In this paper we present a new and flexible method to show that, in one dimension, various self-repellent random walks converge to self-repellent Brownian motion in the limit of weak interaction after appropriate space-time scaling. Our method is based on cutting the path into pieces of an appropriately scaled length, controlling the interaction between the different pieces, and applying an invariance principle to the single pieces. In this way, we show that the self-repellent random walk large deviation rate function for the empirical drift of the path converges to the self-repellent Brownian motion large deviation rate function after appropriate scaling with the interaction parameters. The method is considerably simpler than the approach followed in our earlier work, which was based on functional analytic arguments applied to variational representations and only worked in a very limited number of situations.

We consider two examples of a weak interaction limit: (1) vanishing self-repellence, (2) diverging step variance. In example (1), we recover our earlier scaling results for simple random walk with vanishing self-repellence and show how these can be extended to random walk with steps that have zero mean and a finite exponential moment. Moreover, we show that these scaling results are stable against adding self-attraction, provided the self-repellence dominates. In example (2), we prove a conjecture by Aldous for the scaling of self-avoiding walk with diverging step variance. Moreover, we consider self-avoiding walk on a two-dimensional horizontal strip such that the steps in the vertical direction are uniform over the width of the strip and find the scaling as the width tends to infinity.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. e-mail: rhofstad@win.tue.nl, , , , , , NL

    Remco van der Hofstad

  2. EURANDOM, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. e-mail: denhollander@eurandom.tue.nl, , , , , , NL

    Frank den Hollander

  3. Institut für Mathematik, TU Berlin, Straße des 17. Juni 136, D-10623 Berlin, Germany. e-mail: koenig@math.tu-berlin.de, , , , , , DE

    Wolfgang König

Authors
  1. Remco van der Hofstad
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  2. Frank den Hollander
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  3. Wolfgang König
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Additional information

Received: 6 March 2002 / Revised version: 11 October 2002 / Published online: 21 February 2003

Mathematics Subject Classification (2000): 60F05, 60F10, 60J55, 82D60

Key words or phrases: Self-repellent random walk and Brownian motion – Invariance principles – Large deviations – Scaling limits – Universality

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Hofstad, R., Hollander, F. & König, W. Weak interaction limits for one-dimensional random polymers. Probab. Theory Relat. Fields 125, 483–521 (2003). https://doi.org/10.1007/s00440-002-0248-9

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  • Issue Date: April 2003

  • DOI: https://doi.org/10.1007/s00440-002-0248-9

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Keywords

  • Random Walk
  • Vertical Direction
  • Interaction Parameter
  • Weak Interaction
  • Variational Representation
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