Self-avoiding walks in a rectangle

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• 作者：Anthony J. Guttmann (1)
  Tom Kennedy (2)
  
• 关键词：Brownian motion ; Conformal invariance ; Scaling limit ; Self ; avoiding walks ; SLE
• 刊名：Journal of Engineering Mathematics
• 出版年：2014
• 出版时间：February 2014
• 年：2014
• 卷：84
• 期：1
• 页码：201-208
• 全文大小：532 KB
• 作者单位：Anthony J. Guttmann (1)
  Tom Kennedy (2)
  
  1. Department of Mathematics and Statistics, ARC Centre of Excellence for Mathematics and Statistics of Complex Systems, University of Melbourne, Melbourne, VIC, 3010, Australia
  2. Department of Mathematics, University of Arizona, Tucson, AZ, 85721-0089, USA
  
• ISSN：1573-2703

文摘

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.