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The boundary of a square tiling of a graph coincides with the Poisson boundary

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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

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Official URL: http://dx.doi.org/10.1007/s00222-015-0601-0

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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:
DateEvent
March 2016Published
19 May 2015Available
26 April 2015Accepted
1 May 2014Submitted
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
Funder: Fonds zur Förderung der Wissenschaftlichen Forschung (Austria) (FWF)
Grant number: Grant P-24028-N18

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