  In this note we investigate the multi-parameter Potential Theory on the
weighted $d$-tree (Cartesian product of several copies of uniform dyadic tree),
which is connected to the discrete models of weighted Dirichlet spaces on the
polydisc. We establish some basic properties of the respective potentials,
capacities and equilibrium measures (in particular in the case of product
polynomial weights). We explore multi-parameter Hardy inequality and its trace
measures, and discuss some open problems of potential-theoretic and
combinatorial nature.
