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Essential Spanning Forests and Electrical Networks on Groups

Author

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  • Rita Solomyak

    (Tel Aviv University)

Abstract

Let Γ be a Cayley graph of a finitely generated group G. Subgraphs which contain all vertices of Γ, have no cycles, and no finite connected components are called essential spanning forests. The set $$Y$$ of all such subgraphs can be endowed with a compact topology, and G acts on $$Y$$ by translations. We define a “uniform” G-invariant probability measure μ on $$Y$$ show that μ is mixing, and give a sufficient condition for directional tail triviality. For non-cocompact Fuchsian groups we show how μ can be computed on cylinder sets. We obtain as a corollary, that the tail σ-algebra is trivial, and that the rate of convergence to mixing is exponential. The transfer-current function ψ (an analogue to the Green function), is computed explicitly for the Modular and Hecke groups.

Suggested Citation

  • Rita Solomyak, 1999. "Essential Spanning Forests and Electrical Networks on Groups," Journal of Theoretical Probability, Springer, vol. 12(2), pages 523-548, April.
    • Handle: RePEc:spr:jotpro:v:12:y:1999:i:2:d:10.1023_a:1021638430098
    DOI: 10.1023/A:1021638430098
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