Conformal invariance and stochastic Loewner evolution predictions for the 2D self-avoiding walk - Monte Carlo tests

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13 Citations (Scopus)

Abstract

Simulations of the two-dimensional self-avoiding walk (SAW) are performed in a half-plane and a cut-plane (the complex plane with the positive real axis removed) using the pivot algorithm. We test the conjecture of Lawler, Schramm, and Werner that the scaling limit of the two-dimensional SAW is given by Schramm's stochastic Loewner evolution (SLE). The agreement is found to be excellent. The simulations also test the conformal invariance of the SAW since conformal invariance implies that if we map infinite length walks in the cut-plane into the half plane using the conformal map z → √z, then the resulting walks will have the same distribution as the SAW in the half plane. The simulations show excellent agreement between the distributions.

Original languageEnglish (US)
Pages (from-to)51-78
Number of pages28
JournalJournal of Statistical Physics
Volume114
Issue number1-2
StatePublished - Jan 2004

Fingerprint

Stochastic Loewner Evolution
Monte Carlo Test
Conformal Invariance
half planes
Self-avoiding Walk
invariance
Half-plane
Prediction
predictions
Walk
pivots
simulation
Conformal Map
Simulation
Pivot
Scaling Limit
Argand diagram
scaling
Imply

Keywords

  • Pivot algorithm
  • Self-avoiding walk
  • SLE

ASJC Scopus subject areas

  • Mathematical Physics
  • Physics and Astronomy(all)
  • Statistical and Nonlinear Physics

Cite this

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AB - Simulations of the two-dimensional self-avoiding walk (SAW) are performed in a half-plane and a cut-plane (the complex plane with the positive real axis removed) using the pivot algorithm. We test the conjecture of Lawler, Schramm, and Werner that the scaling limit of the two-dimensional SAW is given by Schramm's stochastic Loewner evolution (SLE). The agreement is found to be excellent. The simulations also test the conformal invariance of the SAW since conformal invariance implies that if we map infinite length walks in the cut-plane into the half plane using the conformal map z → √z, then the resulting walks will have the same distribution as the SAW in the half plane. The simulations show excellent agreement between the distributions.

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