The homogenous tree as an electric network

2010/07/26 by Alice Vatamanelu, Vatamanelu, Alice · 1 citation
Mathematics · Physics and Astronomy · #Complex Network Analysis Techniques #FOS: Mathematics #Graph theory and applications #Markov Chains and Monte Carlo Methods #Probability (math.PR) #math.PR

paper · pdf · doi:10.48550/arxiv.1007.4565

12 figures

arxiv created 2010/07/26 · openalex publication_date 2010/07/26 · arxiv updated 2010/07/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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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