Weak interaction limits for one-dimensional random polymers |
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Authors: | Remco van der Hofstad Frank den Hollander Wolfgang König |
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Institution: | 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;2.EURANDOM, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. e-mail: denhollander@eurandom.tue.nl,NL;3.Institut für Mathematik, TU Berlin, Stra?e des 17. Juni 136, D-10623 Berlin, Germany. e-mail: koenig@math.tu-berlin.de,DE |
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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.
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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