Georgakopoulos, Agelos (2016) The boundary of a square tiling of a graph coincides with the Poisson boundary. Inventiones Mathematicae, 203 (3). pp. 773-821. doi:10.1007/s00222-015-0601-0 ISSN 0020-9910.

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Abstract

Answering a question of Benjamini and Schramm (Ann Probab 24(3):1219–1238, 1996), we show that the Poisson boundary of any planar, uniquely absorbing (e.g. one-ended and transient) graph with bounded degrees can be realised geometrically as a circle, which arises from a discrete version of Riemann’s mapping theorem. This implies a conjecture of Northshield (Potential Anal 2(4):299–314, 1993). Some of our technique apply to the non-planar case and might have further applications. When the graph is also hyperbolic then, under mild conditions, we prove the equivalence of several boundary constructions.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH):	Graph theory , Boundary value problems, Stochastic geometry
Journal or Publication Title:	Inventiones Mathematicae
Publisher:	Springer
ISSN:	0020-9910
Official Date:	March 2016
Dates:	Date Event March 2016 Published 19 May 2015 Available 26 April 2015 Accepted 1 May 2014 Submitted
Volume:	203
Number:	3
Page Range:	pp. 773-821
DOI:	10.1007/s00222-015-0601-0
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Date of first compliant deposit:	9 June 2016
Date of first compliant Open Access:	9 June 2016
Funder:	Fonds zur Förderung der Wissenschaftlichen Forschung (Austria) (FWF)
Grant number:	Grant P-24028-N18

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