Transfinite Random Walks

As with many discussions in this book, the theory of random walks on pristine permissive networks is much simpler than that presented in [34, Chapter 7]. In this case, the principle reason is the following. In [34], the idea of a random walker passing through a transfinite β-node was developed recursively by shorting the branches in a small region around that β-node—thereby achieving a decrease in rank—then applying the theory for a lower rank of transfiniteness, and then allowing the region to contract to obtain in the limit a transition through the β-node. This involved proving that certain node voltages converged as the region was contracted. Such a region was defined in strictly graph-theoretic terms, which made it difficult to prove the needed convergence of node voltages and in fact mandated some complicated arguments and additional severe restrictions on the structure of the graph in that region. In place of that region, we now have a vicinity of the β-node defined in terms of a metric. Moreover, we have a potential that is continuous at the β-node with respect to that metric. Thus, we can invoke the theory of Section 6.7 to conclude that node voltages converge as the vicinity contracts. No longer do we need an extended argument to obtain those convergences.