Two-Dimensional Scaling Limits via Marked Nonsimple Loops

Details Two-Dimensional Scaling Limits via Marked Nonsimple Loops Two-Dimensional Scaling Limits via Marked Nonsimple Loops

2006-04-16

arxiv.org/abs/cond-mat/0604390v1

Physics

Condensed Matter

Statistical Mechanics

25 pages, 5 figures

Scientific paper

We postulate the existence of a natural Poissonian marking of the double (touching) points of SLE(6) and hence of the related continuum nonsimple loop process that describes macroscopic cluster boundaries in 2D critical percolation. We explain how these marked loops should yield continuum versions of near-critical percolation, dynamical percolation, minimal spanning trees and related plane filling curves, and invasion percolation. We show that this yields for some of the continuum objects a conformal covariance property that generalizes the conformal invariance of critical systems. It is an open problem to rigorously construct the continuum objects and to prove that they are indeed the scaling limits of the corresponding lattice objects.

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