On multiple circuit chains with a countable infinity of states

Denumerable rth order circuit chains (r> 1) are defined as genuine denumerable Markov chains of order r with transition law expressed in terms of the weights of a denumerable class of overlapping directed circuits in the plane. Recurrence and stationarity of such chains are studied in connection with a suitable planar motion through a directed network with r-series-connected points as nodes. In particular a generalization of Pólya's theorem concerning random walks on multi-dimensional lattice-points is derived. We also show a relationship between the approach to denumerable Markov chains with multiple states defined by weighted circuits and diffusion of electrical current through a network.