### Abstract

We review the existence of the infinite length self-avoiding walk in the half plane and its relationship to bridges. We prove that this probability measure is also given by the limit as β → β_{c}- of the probability measure on all finite length walks ω with the probability of ω proportional to β^{{pipe}ω{pipe}} where {pipe}ω{pipe} is the number of steps in ω. (β_{c} is the reciprocal of the connective constant.) The self-avoiding walk in a strip {z : 0 < Im(z) < y} is defined by considering all self-avoiding walks ω in the strip which start at the origin and end somewhere on the top boundary with probability proportional to β_{c}^{{pipe}ω{pipe}}. We prove that this probability measure may be obtained by conditioning the SAW in the half plane to have a bridge at height y. This observation is the basis for simulations to test conjectures on the distribution of the endpoint of the SAW in a strip and the relationship between the distribution of this strip SAW and SLE_{8/3}.

Original language | English (US) |
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Pages (from-to) | 1-22 |

Number of pages | 22 |

Journal | Journal of Statistical Physics |

Volume | 144 |

Issue number | 1 |

DOIs | |

State | Published - Jul 1 2011 |

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### Keywords

- Bridge decomposition
- SLE
- Self-avoiding walk
- Strip

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Statistical Physics*,

*144*(1), 1-22. https://doi.org/10.1007/s10955-011-0258-z