The scaling limit of fair Peano paths

Abstract

We study random Peano paths on planar square grids that arise from fair random spanning trees. These are trees that are sampled in such a way as to have the same (if possible) edge probabilities. In particular, we are interested in identifying the scaling limit as the mesh-size of the grid tends to zero. It is known \cite{lawler-schramm-werner2002} that if the trees are sampled uniformly, then the scaling limit exists and equals ${\rm SLE}_8$. We show that if we simply follow the same steps as in \cite{lawler-schramm-werner2002}, then fair Peano paths have a deterministic scaling limit.