Essential Spanning Forests and Electrical Networks on Groups |
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Authors: | Rita Solomyak |
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Institution: | (1) School of Mathematical Sciences, Tel Aviv University, 69978 Tel Aviv, Israel |
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Abstract: | Let be a Cayley graph of a finitely generated group G. Subgraphs which contain all vertices of , have no cycles, and no finite connected components are called essential spanning forests. The set
of all such subgraphs can be endowed with a compact topology, and G acts on
by translations. We define a uniform G-invariant probability measure on
show that is mixing, and give a sufficient condition for directional tail triviality. For non-cocompact Fuchsian groups we show how can be computed on cylinder sets. We obtain as a corollary, that the tail -algebra is trivial, and that the rate of convergence to mixing is exponential. The transfer-current function (an analogue to the Green function), is computed explicitly for the Modular and Hecke groups. |
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Keywords: | probability measure Cayley graph spanning forest non-cocompact Fuchsian group |
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