Lamplighter graphs do not admit harmonic functions of finite energy
HTML articles powered by AMS MathViewer
- by Agelos Georgakopoulos PDF
- Proc. Amer. Math. Soc. 138 (2010), 3057-3061
Abstract:
We prove that a lamplighter graph of a locally finite graph over a finite graph does not admit a non-constant harmonic function of finite Dirichlet energy.References
- Mohammed E. B. Bekka and Alain Valette, Group cohomology, harmonic functions and the first $L^2$-Betti number, Potential Anal. 6 (1997), no. 4, 313–326. MR 1452785, DOI 10.1023/A:1017974406074
- Itai Benjamini and Oded Schramm, Harmonic functions on planar and almost planar graphs and manifolds, via circle packings, Invent. Math. 126 (1996), no. 3, 565–587. MR 1419007, DOI 10.1007/s002220050109
- Sara Brofferio and Wolfgang Woess, Positive harmonic functions for semi-isotropic random walks on trees, lamplighter groups, and DL-graphs, Potential Anal. 24 (2006), no. 3, 245–265. MR 2217953, DOI 10.1007/s11118-005-0914-5
- Donald I. Cartwright and Wolfgang Woess, Infinite graphs with nonconstant Dirichlet finite harmonic functions, SIAM J. Discrete Math. 5 (1992), no. 3, 380–385. MR 1172746, DOI 10.1137/0405029
- Warren Dicks and Thomas Schick, The spectral measure of certain elements of the complex group ring of a wreath product, Geom. Dedicata 93 (2002), 121–137. MR 1934693, DOI 10.1023/A:1020381532489
- Reinhard Diestel, Graph theory, 3rd ed., Graduate Texts in Mathematics, vol. 173, Springer-Verlag, Berlin, 2005. MR 2159259
- Reinhard Diestel and Imre Leader, A conjecture concerning a limit of non-Cayley graphs, J. Algebraic Combin. 14 (2001), no. 1, 17–25. MR 1856226, DOI 10.1023/A:1011257718029
- Anna Erschler, On drift and entropy growth for random walks on groups, Ann. Probab. 31 (2003), no. 3, 1193–1204. MR 1988468, DOI 10.1214/aop/1055425775
- Anna Erschler, Generalized wreath products, Int. Math. Res. Not. , posted on (2006), Art. ID 57835, 14. MR 2276348, DOI 10.1155/IMRN/2006/57835
- Rostislav I. Grigorchuk and Andrzej Żuk, The lamplighter group as a group generated by a 2-state automaton, and its spectrum, Geom. Dedicata 87 (2001), no. 1-3, 209–244. MR 1866850, DOI 10.1023/A:1012061801279
- V. A. Kaĭmanovich and A. M. Vershik, Random walks on discrete groups: boundary and entropy, Ann. Probab. 11 (1983), no. 3, 457–490. MR 704539
- Anders Karlsson and Wolfgang Woess, The Poisson boundary of lamplighter random walks on trees, Geom. Dedicata 124 (2007), 95–107. MR 2318539, DOI 10.1007/s10711-006-9104-x
- Franz Lehner, Markus Neuhauser, and Wolfgang Woess, On the spectrum of lamplighter groups and percolation clusters, Math. Ann. 342 (2008), no. 1, 69–89. MR 2415315, DOI 10.1007/s00208-008-0222-7
- Russell Lyons, Robin Pemantle, and Yuval Peres, Random walks on the lamplighter group, Ann. Probab. 24 (1996), no. 4, 1993–2006. MR 1415237, DOI 10.1214/aop/1041903214
- Steen Markvorsen, Sean McGuinness, and Carsten Thomassen, Transient random walks on graphs and metric spaces with applications to hyperbolic surfaces, Proc. London Math. Soc. (3) 64 (1992), no. 1, 1–20. MR 1132852, DOI 10.1112/plms/s3-64.1.1
- Guia Medolla and Paolo M. Soardi, Extension of Foster’s averaging formula to infinite networks with moderate growth, Math. Z. 219 (1995), no. 2, 171–185. MR 1337213, DOI 10.1007/BF02572357
- C. Pittet and L. Saloff-Coste, On random walks on wreath products, Ann. Probab. 30 (2002), no. 2, 948–977. MR 1905862, DOI 10.1214/aop/1023481013
- Ecaterina Sava. A note on the Poisson boundary of lamplighter random walks. Monatshefte für Mathematik. 159 (2010), 329–344.
- Paolo M. Soardi, Potential theory on infinite networks, Lecture Notes in Mathematics, vol. 1590, Springer-Verlag, Berlin, 1994. MR 1324344, DOI 10.1007/BFb0073995
Additional Information
- Agelos Georgakopoulos
- Affiliation: Technische Universität Graz, Steyrergasse 30, 8010, Graz, Austria
- MR Author ID: 805415
- ORCID: 0000-0001-6430-567X
- Received by editor(s): August 12, 2009
- Published electronically: May 14, 2010
- Additional Notes: The author was supported by FWF grant P-19115-N18
- Communicated by: Jim Haglund
- © Copyright 2010 Agelos Georgakopoulos
- Journal: Proc. Amer. Math. Soc. 138 (2010), 3057-3061
- MSC (2010): Primary 05C25
- DOI: https://doi.org/10.1090/S0002-9939-2010-10279-4
- MathSciNet review: 2653930