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
- Peter Antal and Agoston Pisztora, On the chemical distance for supercritical Bernoulli percolation, Ann. Probab. 24 (1996), no. 2, 1036–1048. MR 1404543, DOI 10.1214/aop/1039639377
- Eric Babson and Itai Benjamini, Cut sets and normed cohomology with applications to percolation, Proc. Amer. Math. Soc. 127 (1999), no. 2, 589–597. MR 1622785, DOI 10.1090/S0002-9939-99-04995-3
- Jean-Dominique Deuschel and Agoston Pisztora, Surface order large deviations for high-density percolation, Probab. Theory Related Fields 104 (1996), no. 4, 467–482. MR 1384041, DOI 10.1007/BF01198162
- Guy Gielis and Geoffrey Grimmett, Rigidity of the interface in percolation and random-cluster models, J. Statist. Phys. 109 (2002), no. 1-2, 1–37. MR 1927913, DOI 10.1023/A:1019950525471
- Alan Hammond, Greedy lattice animals: geometry and criticality, Ann. Probab. 34 (2006), no. 2, 593–637. MR 2223953, DOI 10.1214/009117905000000693
- Geoffrey R. Grimmett and Alexander E. Holroyd, Entanglement in percolation, Proc. London Math. Soc. (3) 81 (2000), no. 2, 485–512. MR 1770617, DOI 10.1112/S0024611500012521
- Harry Kesten, Aspects of first passage percolation, École d’été de probabilités de Saint-Flour, XIV—1984, Lecture Notes in Math., vol. 1180, Springer, Berlin, 1986, pp. 125–264. MR 876084, DOI 10.1007/BFb0074919
- Harry Kesten and Yu Zhang, The probability of a large finite cluster in supercritical Bernoulli percolation, Ann. Probab. 18 (1990), no. 2, 537–555. MR 1055419
- Gábor Pete, A note on percolation on $\Bbb Z^d$: isoperimetric profile via exponential cluster repulsion, Electron. Commun. Probab. 13 (2008), 377–392. MR 2415145, DOI 10.1214/ECP.v13-1390
- Agoston Pisztora, Surface order large deviations for Ising, Potts and percolation models, Probab. Theory Related Fields 104 (1996), no. 4, 427–466. MR 1384040, DOI 10.1007/BF01198161
- Ádám Timár, Cutsets in infinite graphs, Combin. Probab. Comput. 16 (2007), no. 1, 159–166. MR 2286517, DOI 10.1017/S0963548306007838
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