Abstract
A celebrated problem in numerical analysis is to consider Brownian motion originating at the centre of a \(10\,\times \,1\) rectangle and to evaluate the probability of a Brownian path hitting the short ends of the rectangle before hitting one of the long sides. For Brownian motion this probability can be calculated exactly (The SIAM 100-digit challenge: a study in high-accuracy numerical computing. SIAM, Philadelphia, 2004). Here we consider instead the more difficult problem of a self-avoiding walk (SAW) in the scaling limit and pose the same question. Assuming that the scaling limit of SAW is conformally invariant, we evaluate, asymptotically, the same probability. For the SAW case we find the probability is approximately 200 times greater than for Brownian motion.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
It is the only process compatible with both conformal invariance and the so-called domain Markov property.
References
Abramowitz M, Stegun IA (1965) Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover publications
Beffara V (2012) Probability and statistical physics in two and more dimensions, Clay Mathematics Institute Proceedings, vol 15, p 467
Bornemann F, Laurie D, Wagon S, Waldvogel J (2004) The SIAM 100-digit challenge: a study in high-accuracy numerical computing. SIAM, Philadelphia
Borwein J (2005) The 100 digit challenge: an extended review. Math Intell 27:40–48
Clisby N (2010) Efficient implementation of the pivot algorithm for self-avoiding walks, arXiv:1005.1444. J Stat Phys 140:349–392
Dyhr B, Gilbert M, Kennedy T, Lawler G, Passon S (2011) The self-avoiding walk spanning a strip. J Stat Phys 144:1–22
Guttmann AJ (ed) (2009) Polygons, polyominoes and polycubes., lecture notes in physics, vol 775, Springer, Dordrecht
Kennedy T, Lawler G F (2011) Lattice effects in the scaling limit of the two-dimensional self-avoiding walk, arXiv:1109.3091v1
Kennedy T (2012) Simulating self-avoiding walks in bounded domains. J Math Phys 53:095219
Lawler G, Schramm O, Werner W (2004) On the scaling limit of planar self-avoiding walk, fractal geometry and applications: a Jubilee of Benoit Mandelbrot, Part 2, 339–364, Proc Sympos Pure Math 72, Am Math Soc, Providence, RI (arXiv:math/0204277v2)
Löwner K (1923) Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I Math Ann 89:103–121
Madras M, Slade G (1993) The self-avoiding walk, probability and its applications. Birkhäuser Boston Inc, Boston
Riesz F, Riesz M (1916) Über die Randwerte einer analytischen Funktion. Quatrième Congrès des Mathématiciens Scandinaves, Stockholm, pp 27–44
Trefethen LN (2002) SIAM News, 35, No. 1 Jan/Feb. Also (2002) SIAM News, 35, No. 6, July/Aug
Acknowledgments
AJG would like to thank Jon Borwein, for introducing him to the original problem, and Nathan Clisby, for a discussion that spawned the current work. This work was supported by the Australian Research Council through Grant DP120100939 (AJG).
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to the memory of Milton van Dyke. Truly a scholar and a gentleman.
Rights and permissions
About this article
Cite this article
Guttmann, A.J., Kennedy, T. Self-avoiding walks in a rectangle. J Eng Math 84, 201–208 (2014). https://doi.org/10.1007/s10665-013-9622-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10665-013-9622-0