### Abstract

A celebrated problem in numerical analysis is to consider Brownian motion originating at the centre of a (Formula presented.) 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.

Original language | English (US) |
---|---|

Pages (from-to) | 201-208 |

Number of pages | 8 |

Journal | Journal of Engineering Mathematics |

Volume | 84 |

Issue number | 1 |

DOIs | |

State | Published - Feb 1 2014 |

### Fingerprint

### Keywords

- Brownian motion
- Conformal invariance
- Scaling limit
- Self-avoiding walks
- SLE

### ASJC Scopus subject areas

- Mathematics(all)
- Engineering(all)

### Cite this

*Journal of Engineering Mathematics*,

*84*(1), 201-208. https://doi.org/10.1007/s10665-013-9622-0

**Self-avoiding walks in a rectangle.** / Guttmann, Anthony J.; Kennedy, Thomas G.

Research output: Contribution to journal › Article

*Journal of Engineering Mathematics*, vol. 84, no. 1, pp. 201-208. https://doi.org/10.1007/s10665-013-9622-0

}

TY - JOUR

T1 - Self-avoiding walks in a rectangle

AU - Guttmann, Anthony J.

AU - Kennedy, Thomas G

PY - 2014/2/1

Y1 - 2014/2/1

N2 - A celebrated problem in numerical analysis is to consider Brownian motion originating at the centre of a (Formula presented.) 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.

AB - A celebrated problem in numerical analysis is to consider Brownian motion originating at the centre of a (Formula presented.) 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.

KW - Brownian motion

KW - Conformal invariance

KW - Scaling limit

KW - Self-avoiding walks

KW - SLE

UR - http://www.scopus.com/inward/record.url?scp=84956713401&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84956713401&partnerID=8YFLogxK

U2 - 10.1007/s10665-013-9622-0

DO - 10.1007/s10665-013-9622-0

M3 - Article

AN - SCOPUS:84956713401

VL - 84

SP - 201

EP - 208

JO - Journal of Engineering Mathematics

JF - Journal of Engineering Mathematics

SN - 0022-0833

IS - 1

ER -