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Instability of Set Recurrence and Green’s Function on Groups with the Liouville Property

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Abstract

Let μ and ν be probability measures on a group Γ and let G μ and G ν denote Green’s function with respect to μ and ν. The group Γ is said to admit instability of Green’s function if there are symmetric, finitely supported measures μ and ν and a sequence {x n } such that G μ (e, x n )/G ν (e,x n ) →0, and Γ admits instability of recurrence if there is a set S that is recurrent with respect to ν but transient with respect to μ. We give a number of examples of groups that have the Liouville property but have both types of instabilities. Previously known groups with these instabilities did not have the Liouville property.

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References

  1. Benjamini, I.: Instability of the Liouville property for quasi-isometric graphs and manifolds of polynomial volume growth. J. Theor. Probab. 4(3), 631–637 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  2. Benjamini, I., Schramm, O.: Harmonic functions on planar and almost planar graphs and manifolds, via circle packings. Invent. Math. 126(3), 565–587(1996)

    Article  MATH  MathSciNet  Google Scholar 

  3. Kaimanovich, V., Woess, W.: Boundary and entropy of space homogeneous Markov chains. Ann. Probab. 30(1), 323–363 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  4. Karlsson, A., Ledrappier, F.: Linear drift and Poisson boundary for random walks. Pure Appl. Math. Q. 3(4, Part 1), 1027–1036 (2007)

    MathSciNet  Google Scholar 

  5. Lyons, T.: Instability of the Liouville property for quasi-isometric Riemannian manifolds and reversible Markov chains. J. Differ. Geom. 26(1), 33–66 (1987)

    MATH  MathSciNet  Google Scholar 

  6. Pittet, C., Saloff-Coste, L.: On random walks on wreath products. Ann. Probab. 30(2), 948–977 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  7. Woess, W.: Random walks on infinite graphs and groups. In: Camb. Tracts Math., vol. 138, 334 pp. (2000)

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Authors and Affiliations

  1. Weizmann Institute, Weizmann, Israel

    Itai Benjamini & David Revelle

Authors
  1. Itai Benjamini
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  2. David Revelle
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Correspondence to Itai Benjamini.

Additional information

Research partially supported by an NSF postdoctoral fellowship.

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Benjamini, I., Revelle, D. Instability of Set Recurrence and Green’s Function on Groups with the Liouville Property. Potential Anal 34, 199–206 (2011). https://doi.org/10.1007/s11118-010-9189-6

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  • DOI: https://doi.org/10.1007/s11118-010-9189-6

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