Convergence of loop-erased random walk in the natural parametrization

Understand

Loop-erased random walk, abbreviated LERW, is one of the most well-studied critical lattice models.

• It is the self-avoiding random walk one gets after erasing the loops from a simple random walk in order or alternatively by considering the branches in a uniformly chosen spanning tree.
• This paper proves that planar LERW parametrized by renormalized length converges in the lattice size scaling limit to SLE(2) parametrized by 5/4-dimensional Minkowski content.
• In doing this we also provide a method for proving similar convergence results for other models converging to SLE.