Conformal Invariance of the Loop-Erased Percolation Explorer

Conformal Invariance of Voronoi Percolation

It is proved that in the Voronoi model for percolation in dimension 2 and 3, the crossing probabilities are asymptotically invariant under conformal change of metric. To deene Voronoi percolation on a manifold M , you need a measure , and a Riemannian metric ds. Points are scattered according to a Poisson point process on (M;), with some density. Each cell in the Voronoi tessellation determined...

Conformal invariance of planar loop-erased random walks and uniform spanning trees

We prove that the scaling limit of loop-erased random walk in a simply connected domain D $ C is equal to the radial SLE2 path. In particular, the limit exists and is conformally invariant. It follows that the scaling limit of the uniform spanning tree in a Jordan domain exists and is conformally invariant. Assuming that ∂D is a C1 simple closed curve, the same method is applied to show that th...

Conformal Invariance in Two-dimensional Percolation

The word percolation, borrowed from the Latin, refers to the seeping or oozing of a liquid through a porous medium, usually to be strained. In this and related senses it has been in use since the seventeenth century. It was introduced more recently into mathematics by S. R. Broadbent and J. M. Hammersley ([BH]) and is a branch of probability theory that is especially close to statistical mechan...

Loop-Erased Random Walk

Anisotropic limit of the bond-percolation model and conformal invariance in curved geometries.

We investigate the anisotropic limit of the bond-percolation model in d dimensions, which is equivalent to a (d-1) -dimensional quantum q-->1 Potts model. We formulate an efficient Monte Carlo method for this model. Its application shows that the anisotropic model fits well with the percolation universality class in d dimensions. For three-dimensional rectangular geometry, we determine the crit...