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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 ISSN 0020-9910.
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Official URL: http://dx.doi.org/10.1007/s00222-015-0601-0
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 | ||||||||||
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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: |
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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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