The Self-avoiding Walk Spanning a Strip

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

Research output: Contribution to journalArticle

8 Citations (Scopus)

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 → 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
Volume144
Issue number1
DOIs
StatePublished - Jul 2011

Fingerprint

Self-avoiding Walk
Strip
strip
Probability Measure
Half-plane
half planes
Directly proportional
Conditioning
Walk
conditioning
Simulation
Relationships
simulation

Keywords

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

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

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

In: Journal of Statistical Physics, Vol. 144, No. 1, 07.2011, p. 1-22.

Research output: Contribution to journalArticle

Dyhr, B, Gilbert, M, Kennedy, TG, Lawler, GF & Passon, S 2011, 'The Self-avoiding Walk Spanning a Strip', Journal of Statistical Physics, vol. 144, no. 1, pp. 1-22. https://doi.org/10.1007/s10955-011-0258-z
Dyhr, Ben ; Gilbert, Michael ; Kennedy, Thomas G ; Lawler, Gregory F. ; Passon, Shane. / The Self-avoiding Walk Spanning a Strip. In: Journal of Statistical Physics. 2011 ; Vol. 144, No. 1. pp. 1-22.
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