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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Boundary-connectivity via graph theory


Author: Ádám Timár
Journal: Proc. Amer. Math. Soc. 141 (2013), 475-480
MSC (2000): Primary 05C10, 05C63; Secondary 20F65, 60K35
Published electronically: June 21, 2012
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Abstract: We generalize theorems of Kesten and Deuschel-Pisztora about the connectedness of the exterior boundary of a connected subset of $ \mathbb{Z}^d$, where ``connectedness'' and ``boundary'' are understood with respect to
various graphs on the vertices of $ \mathbb{Z}^d$. These theorems are widely used in statistical physics and related areas of probability. We provide simple and elementary proofs of their results. It turns out that the proper way of viewing these questions is graph theory instead of topology.


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Additional Information

Ádám Timár
Affiliation: Hausdorff Center for Mathematics, Universität Bonn, D-53115 Bonn, Germany
Email: adam.timar@hcm.uni-bonn.de

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11333-4
PII: S 0002-9939(2012)11333-4
Received by editor(s): March 25, 2010
Received by editor(s) in revised form: February 21, 2011, July 1, 2011, and July 5, 2011
Published electronically: June 21, 2012
Communicated by: Richard C. Bradley
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.