The Self-avoiding Walk Spanning a Strip

Ben Dyhr, Michael Gilbert, Tom Kennedy, Gregory F. Lawler, Shane Passon

Research output: Contribution to journalReview articlepeer-review

10 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)1-22
Number of pages22
JournalJournal of Statistical Physics
Issue number1
StatePublished - Jul 2011


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

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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