Random Walk on the Speiser Graph of a Riemann Surface

We consider the problem of determining the conformai type—hyperbolic or parabolic—of a covering surface of the Riemann sphere with n punctures. To such a surface there corresponds a graph called the Speiser graph of the covering, and it is natural to ask for a criterion for the type of the surface in terms of properties of the graph. We show how to define a random walk on the vertices of the graph, so that the random walk is transient if and only if the surface is hyperbolic. 1. The type problem. A simply-connected open Riemann surface is conformally equivalent either to the open unit disk or to the entire complex plane [1]. In the first case the surface is said to be hyperbolic, or to have hyperbolic type; in the second case it is said to be parabolic. This dichotomy is extended to multiply-connected surfaces by declaring a surface to be hyperbolic if, like the unit disk, it has finite electrical resistance out to infinity, and parabolic if, like the plane, it has infinite resistance. Equivalently, a hyperbolic surface is one on which Brownian motion is transient, and a parabolic surface is one on which Brownian motion is recurrent [8, 9]. The classical type problem for Riemann surfaces is the problem of determining whether a given open Riemann surface is hyperbolic or parabolic. 2. The Speiser graph of a covering surface. One special case of the type problem that has received a lot of attention is the problem of determining the type of an infinitely-sheeted covering surface of the Riemann sphere with n punctures. Such a covering surface can be represented by a Speiser graph, as I will now describe. Start by drawing a simple closed curve C through the n branch points, as shown in Figure 1. The branch points divide C into n segments, which we label C i , . . . , Cn. The curve C divides the sphere into two parts, which we label Sa and S&. Cutting along the curves that cover C i , . . . ,C n separates the covering surface into an infinite number of pieces, some that cover Sa and some that cover S&. To reconstruct the surface, we must glue each copy of Sa along each of the n curves that form its boundary to one or another of the copies of SbThe Speiser graph gives a recipe for carrying out these gluings. It is an infinite graph with vertices labelled a and b and edges labelled by integers between 1 and n. Each edge joins a vertex labelled a to one labelled b. Each vertex has n edges coming into it, labelled 1 , . . . , n. The vertices labelled a correspond to copies of Sa, and those labelled b to copies of Sb. An edge labelled i indicates Received by the editors June 12, 1984 and, in revised form, June 29, 1984. 1980 Mathematics Svbject OassificatUm. Primary 30F20, 31A15, 60J15. ©1984 American Mathematical Society 0273-0979/84 $1.00 + $.25 per page 371