Random Delaunay triangulations and metric uniformization

We present a new connection between the discrete conformal geometry problem of disk pattern construction and the continuous conformal geometry problem of metric uniformization. In a nutshell, we discuss how to construct disk patterns by optimizing an objective function, which turns out to be intimately related to hyperbolic volume. With the use of random Delaunay triangulations, we then average this objective function to construct an objective function on the metrics conformal to a fixed one. Finally, using this averaged objective function, we may reprove the uniformization theorem in two dimensions.