## Boundary-connectivity via graph theory

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- by Ádám Timár PDF
- Proc. Amer. Math. Soc.
**141**(2013), 475-480 Request permission

## 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.## References

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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
- 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
- © Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**141**(2013), 475-480 - MSC (2010): Primary 05C10, 05C63; Secondary 20F65, 60K35
- DOI: https://doi.org/10.1090/S0002-9939-2012-11333-4
- MathSciNet review: 2996951