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

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.Comment: Second version added more simulations and a proof of irreducibility. 25 pages, 16 figure

Phase separation in the neutral Falicov-Kimball model

The Falicov-Kimball model consists of spinless electrons and classical particles (ions) on a lattice. The electrons hop between nearest neighbor sites while the ions do not. We consider the model with equal numbers of ions and electrons and with a large on-site attractive force between ions and electrons. For densities 1/4 and 1/5 the ion configuration in the ground state had been proved to be periodic. We prove that for density 2/9 it is periodic as well. However, for densities between 1/4 and 1/5 other than 2/9 we prove that the ion configuration in the ground state is not periodic. Instead there is phase separation. For densities in (1/5,2/9) the ground state ion configuration is a mixture of the density 1/5 and 2/9 ground state ion configurations. For the interval (2/9,1/4) it is a mixture of the density 2/9 and 1/4 ground states.Comment: 14 pages, latex file and 3 postscript figures. Figures are included using the epsf macro