Conformal Invariance and Stochastic Loewner Evolution Predictions for the 2D Self-Avoiding Walk - Monte Carlo Tests

Abstract

Simulations of the 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 would imply that if we map the walks in the cut-plane into the half plane using the conformal map z -> sqrt(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.

Second version added more simulations and a proof of irreducibility. 25 pages, 16 figures

References in corpus (2)

• Monte Carlo Tests of SLE Predictions for the 2D Self-Avoiding Walk
• A faster implementation of the pivot algorithm for self-avoiding walks

Cited by in corpus (15)

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• Monte Carlo comparisons of the self-avoiding walk and SLE as parameterized curves
• Conformal invariance of the 3D self-avoiding walk
• The Exit Distribution for Smart Kinetic Walk with Symmetric and Asymmetric Transition Probability
• Transforming fixed-length self-avoiding walks into radial SLE_8/3
• Fluctuation force exerted by a planar self-avoiding polymer
• Intersection probabilities for a chordal SLE path and a semicircle
• $SLE(\kappa ,\rho )$ processes, hiding exponents and self-avoiding walks in a wedge
• A non-intersecting random walk on the Manhattan lattice and SLE_6
• Stochastic Loewner Evolution