Analysis/Mathematical Physics Seminar

Loop-erased random walk (LERW) is a random self-avoiding curve obtained by erasing the loops of a random walk according to chronological order. Studying LERW on the two-dimensional integer lattice, Schramm introduced a model of one-parameter planar random curves known today as Schramm-Loewner evolution (SLE(\kappa)). Subsequently, in their seminal work Lawler, Schramm and Werner proved that LERW on the two-dimensional integer lattice converges to SLE(2) as the mesh tends to zero. Their proof uses the lattice structure, and the question arises whether a similar result holds also for perturbed lattices or other graphs. We generalize their result, showing that the scaling limit of LERW on a planar irreducible graph G, so that the random walk on G converges to Brownian motion, is SLE(2). Joint work with Ariel Yadin.