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

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.