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The boundary of a square tiling of a graph coincides with the Poisson boundary.
- Source :
-
Inventiones Mathematicae . Mar2016, Vol. 203 Issue 3, p773-821. 49p. - Publication Year :
- 2016
-
Abstract
- Answering a question of Benjamini and Schramm (Ann Probab 24(3):1219-1238, ), 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, ). 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. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00209910
- Volume :
- 203
- Issue :
- 3
- Database :
- Academic Search Index
- Journal :
- Inventiones Mathematicae
- Publication Type :
- Academic Journal
- Accession number :
- 113084118
- Full Text :
- https://doi.org/10.1007/s00222-015-0601-0