Boundary visits of loop-erased random walks


A physics principle asserts that the scaling limit of a critical random model 
on a planar lattice, as increasingly dense lattices approximate a continuum 
domain, is described by a conformal field theory (CFT). The aim of this talk 
is to prove rigorously conformal invariance properties in the scaling limits 
of two closely related models: the loop-erased random walk (LERW) and the 
uniform spanning tree (UST). We study the probabilities of multiple disjoint 
boundary-to-boundary branches appearing between given boundary vertices in 
the UST, as well as the boundary visits of a LERW. The related scaling limits 
are shown to be conformally covariant and to solve partial differential 
equations, as predicted by CFT. This is among the first verifications of 
third-order PDEs of CFT. Using the PDE solutions, we also show that the 
random geometry related to the multiple UST branches converges to a 
conformally invariant law, called the local multiple SLE(2).