Geodesics and Bounded Harmonic Functions on Infinite Planar Graphs

It is shown there that an infinite connected planar graph with a uniform upper bound on vertex degree and rapidly decreasing Green's function (relative to the simple random walk) has infinitely many pairwise finitelyintersecting geodesic rays starting at each vertex. We then demonstrate the existence of nonconstant bounded harmonic functions on the graph. Let g be an infinite, simple, connected, planar graph, g also denotes the vertex set of the graph. If two vertices x and y are connected by an edge, we write xEy. For a vertex x, the degree of x is d(x) = \{y £ g: yEx}\, and we assume: (1) S = sup d(x) < oo. x£g A finite [infinite] walk y is a sequence (y(0), ... , y(n)) [(y(0), y(l), ...)] of elements of g such that y(k)Ey(k + 1) for all 0 < k < « 1 [for all k > 0]. We say that y starts at y(0) and, in the first case, ends at y(n) and has length « . Since g is connected, we may define a metric: d(x, y) = inf{« : « is the length of a finite walk from x to y }. A path is a walk whose vertices are distinct. A geodesic y is a path such that d(y(m), y(n)) = \m-n\ for all possible m and « . For x € 9, T(x, «) is the set of geodesies that have length « and start at x ; F(x) is the set of geodesies that have infinite length and start at x . The following propositions are useful; the first is easy to prove by a diagonal type argument. Proposition 1. For all x £ q, F(x) ^ 0. Proposition 2. Given x, y £ q and y £ F(x), there exists a y £ F(y) such that y and y eventually coincide. Received by the editors October 10, 1989 and, in revised form, September 26, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 60J45; Secondary 60J15, 05C38.