The homogenous tree as an electric network

Details The homogenous tree as an electric network The homogenous tree as an electric network

2010-07-26

arxiv.org/abs/1007.4565v1

Mathematics

Probability

12 figures

Scientific paper

Let T be an infinite homogenous tree of homogeneity $q+1$. Attaching to each edge the conductance $1$, the tree will became an electric network. The reversible Markov chain associated to this network is the simple random walk on the homogenous tree. Using results regarding the equivalence between a reversible Markov chain and an electric network, we will express voltages, currents, the Green fuction hitting times, transitions number, probabilities of reaching a set before another, as functions of the distance on the homogenous tree. This connection enables us to give simpler proofs for the properties of the random walk under discussion.

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