Scaling limit of the loop-erased random walk Green’s function
2015 (English)In: Probability theory and related fields, ISSN 0178-8051, E-ISSN 1432-2064Article in journal (Refereed) Published
We consider loop-erased random walk (LERW) running between two boundary points of a square grid approximation of a planar simply connected domain. The LERW Green’s function is the probability that the LERW passes through a given edge in the domain. We prove that this probability, multiplied by the inverse mesh size to the power 3/4, converges in the lattice size scaling limit to (a constant times) an explicit conformally covariant quantity which coincides with the (Formula presented.) Green’s function. The proof does not use SLE techniques and is based on a combinatorial identity which reduces the problem to obtaining sharp asymptotics for two quantities: the loop measure of random walk loops of odd winding number about a branch point near the marked edge and a “spinor” observable for random walk started from one of the vertices of the marked edge.
Place, publisher, year, edition, pages
Fomin’s identity, Green’s function, loop measure, Loop-erased random walk, Poisson kernel, scaling limit, Schramm–Loewner evolution
IdentifiersURN: urn:nbn:se:kth:diva-177752DOI: 10.1007/s00440-015-0655-3ScopusID: 2-s2.0-84940055942OAI: oai:DiVA.org:kth-177752DiVA: diva2:875084
QC 201511302015-11-302015-11-252015-11-30Bibliographically approved