On deciding whether a surface is parabolic or hyperbolic

In our book [3], Laurie Snell and I tell how a method from the classical theory of electricity called Rayleigh’s short-cut method can be used to determine whether or not a person walking around at random on the vertices of a given infinite graph is certain to return to the starting point. In this paper, I will present an application of Rayleigh’s method to the classical type problem for Riemann surfaces, which is really just the same random walk problem, only now instead of walking around on an infinite graph our walker is diffusing around on a surface. My aim will be to convince you that if you want to figure out whether or not a random walker gets lost, you should use Rayleigh’s method. The problem to which we will apply Rayleigh’s method was raised by Milnor [5]. Milnor considered infinite surfaces that are rotationally symmetric about some point p, and showed that if the Gaussian curvature is negative enough then the surface is conformally hyperbolic, i.e., can be mapped conformally onto the unit disk, while if the curvature is just a little less negative then the surface is conformally parabolic, i.e., can be mapped conformally onto the whole plane. In probabilistic terms, this means that if the curvature is negative enough a particle diffusing around on the surface will eventually wander off and never come back, while if the curvature is just a little less negative the particle is bound to come back near where it started, no matter how far off it may have wandered.