For infinite radially symmetric trees, the p-modulus of all root-to-infinity paths is (sum_k (sigma_k |S_k|)^{-q/p})^{-p/q}, and a critical p-value separates positive from zero modulus.
The homogenous tree as an electric network
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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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$p$-Modulus on radially symmetric trees
For infinite radially symmetric trees, the p-modulus of all root-to-infinity paths is (sum_k (sigma_k |S_k|)^{-q/p})^{-p/q}, and a critical p-value separates positive from zero modulus.