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.
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Notes
It is the only process compatible with both conformal invariance and the so-called domain Markov property.
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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).
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Dedicated to the memory of Milton van Dyke. Truly a scholar and a gentleman.
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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
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DOI: https://doi.org/10.1007/s10665-013-9622-0