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Scaling limit of the loop-erased random walk Green's function
CUNY Brooklyn Coll, Brooklyn, NY 11210 USA..
Univ Chicago, Chicago, IL 60637 USA..
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. KTH Royal Inst Technol, Stockholm, Sweden.
2016 (English)In: Probability theory and related fields, ISSN 0178-8051, E-ISSN 1432-2064, Vol. 166, no 1-2, 271-319 p.Article in journal (Refereed) Published
Abstract [en]

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 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
2016. Vol. 166, no 1-2, 271-319 p.
Keyword [en]
Loop-erased random walk, Green's function, scaling limit, loop measure, Poisson kernel, Fomin's identity, Schramm-Loewner evolution
National Category
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-306747DOI: 10.1007/s00440-015-0655-3ISI: 000384427700005OAI: oai:DiVA.org:uu-306747DiVA: diva2:1044532
Funder
Swedish Research Council
Available from: 2016-11-03 Created: 2016-11-03 Last updated: 2016-11-03Bibliographically approved

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