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The Liouville and the intersection properties are equivalent for planar graphs
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Benjamini, Itai, Curien, Nicolas and Georgakopoulos, Agelos. (2012) The Liouville and the intersection properties are equivalent for planar graphs. Electronic communications in probability, Vol.17 . Article no. 42. ISSN 1083-589X
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Official URL: http://dx.doi.org/10.1214/ECP.v17-1913
Abstract
It is shown that if a planar graph admits no non-constant bounded harmonic function then the trajectories of two independent simple random walks intersect almost surely.
| Item Type: | Journal Article |
|---|---|
| Subjects: | Q Science > QA Mathematics |
| Divisions: | Faculty of Science > Mathematics |
| Library of Congress Subject Headings (LCSH): | Random walks (Mathematics), Graph theory |
| Journal or Publication Title: | Electronic communications in probability |
| Publisher: | University of Washington. Dept. of Mathematics |
| ISSN: | 1083-589X |
| Date: | 2012 |
| Volume: | Vol.17 |
| Page Range: | Article no. 42 |
| Identification Number: | 10.1214/ECP.v17-1913 |
| Status: | Peer Reviewed |
| Publication Status: | Published |
| Access rights to Published version: | Restricted or Subscription Access |
| References: | [1] O. Angel and O. Schramm. Uniform infinite planar triangulation. Comm. Math. Phys., 241(2- 3):191–213, 2003. MR-2013797 [2] I. Benjamini and N. Curien. Ergodic theory on stationary random graphs. arXiv:1011.2526. [3] I. Benjamini and O. Schramm. Harmonic functions on planar and almost planar graphs and manifolds, via circle packings. Invent. Math., 126(3):565–587, 1996. MR-1419007 [4] I. Benjamini and O. Schramm. Lack of sphere packing of graphs via non-linear potential theory, arXiv:0910.3071 [5] A. Georgakopoulos. Lamplighter graphs do not admit harmonic functions of finite energy. Proc. Am. Math. Soc., 138(9):3057–3061, 2010. MR-2653930 [6] O. Gurel-Gurevich and A. Nachmias. Recurrence of planar graph limits. Annals Math. (to appear). [7] M. Krikun. Local structure of random quadrangulations. arXiv:0512304. [8] R. B. Richter and C. Thomassen. 3-connected planar spaces uniquely embed in the sphere. Trans. Amer. Math. Soc., 354(11):4585–4595, 2002. MR-1926890 [9] P. M. Soardi Potential theory on infinite networks. Springer-Verlag, 1994. MR-1324344 [10] R. Thomas. Recent excluded minor theorems for graphs. In Surveys in combinatorics, 1999 (Canterbury), volume 267 of London Math. Soc. Lecture Note Ser., pages 201–222. Cambridge Univ. Press, Cambridge, 1999. MR-1725004 |
| URI: | http://wrap.warwick.ac.uk/id/eprint/52018 |
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