We will begin with some history of the literature of the number of visited sites problem
(Dvoretsky and Erdos 1950, then Donsker and Varadhan 1979), and try to reach some kind of agreement
regarding the meaning of the known results for a simple random walk in Z^d. Then we will consider the
meaning of a random walk on a graph, and attempt to parse some of the known results from Z^d in this
broader context. As time permits we will consider the idea of the proof, with a few details, of the beginning of
the main theorem of a paper I am currently working on. Since this material will be used in future lectures,
we will take care that everyone understands the ideas as we proceed.