### 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) |
---|---|

Pages (from-to) | 1-22 |

Number of pages | 22 |

Journal | Journal of Statistical Physics |

Volume | 144 |

Issue number | 1 |

DOIs | |

State | Published - Jul 2011 |

### Fingerprint

### Keywords

- Bridge decomposition
- Self-avoiding walk
- SLE
- 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

**The Self-avoiding Walk Spanning a Strip.** / Dyhr, Ben; Gilbert, Michael; Kennedy, Thomas G; Lawler, Gregory F.; Passon, Shane.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 144, no. 1, pp. 1-22. https://doi.org/10.1007/s10955-011-0258-z

}

TY - JOUR

T1 - The Self-avoiding Walk Spanning a Strip

AU - Dyhr, Ben

AU - Gilbert, Michael

AU - Kennedy, Thomas G

AU - Lawler, Gregory F.

AU - Passon, Shane

PY - 2011/7

Y1 - 2011/7

N2 - 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 → βc- 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. (βc 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 βc{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 SLE8/3.

AB - 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 → βc- 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. (βc 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 βc{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 SLE8/3.

KW - Bridge decomposition

KW - Self-avoiding walk

KW - SLE

KW - Strip

UR - http://www.scopus.com/inward/record.url?scp=79959677898&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79959677898&partnerID=8YFLogxK

U2 - 10.1007/s10955-011-0258-z

DO - 10.1007/s10955-011-0258-z

M3 - Article

AN - SCOPUS:79959677898

VL - 144

SP - 1

EP - 22

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 1

ER -