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).