Pólya’s Theorem on Random Walks via Pólya’s Urn

This is Polya’s theorem referred to in the title. Theorem 1 motivates the definition of a transient graph. Let G be a graph with vertex set V and edge set E . We will often write x ∼ y to mean that {x, y} ∈ E . The degree of a vertex v, denoted deg(v), is the number of edges containing v. A graph is locally finite if all vertices have finite degree. We assume throughout this paper that all graphs are connected and locally finite. The simple random walk on G moves at each unit of time by selecting a vertex uniformly at random among those adjacent to the walk’s current location. The graph G is transient if there is a positive probability that the simple random walk on G never returns to its starting position. The graph G is recurrent if it is not transient. (Since G is assumed to be connected, transience and recurrence do not depend on the starting vertex.) For more on transience and recurrence, see, for example, [14, 20, 9]. We will prove Theorem 1 by the method of flows, which we now describe. Let G be a connected graph with vertex set V and edge set E . An oriented edge is an ordered pair (x, y) such that {x, y} ∈ E . We will sometimes write xy to denote the oriented edge (x, y). A function θ defined on oriented edges is antisymmetric if θ( xy) = −θ( yx) for all oriented edges xy. For a function θ defined on oriented edges, the divergence of θ at v is defined as