  Liouville theory describes the dynamics of surfaces with constant negative
curvature and can be used to study the Weil-Petersson geometry of the moduli
space of Riemann surfaces. This leads to an efficient algorithm to compute the
Weil--Petersson metric to arbitrary accuracy using Zamolodchikov's recursion
relation for conformal blocks. For example, we compute the metric on $\mathcal
M_{0,4}$ numerically to high accuracy by considering Liouville theory on a
sphere with four punctures. We numerically compute the eigenvalues of the
Weil-Petersson Laplacian, and find evidence that the obey the statistics of a
random matrix in the Gaussian Orthogonal Ensemble.
