  We prove that any two finite-area non-compact hyperbolic Riemann surfaces S
and T have finite covers that are arbitrarily close in the normalized
Weil-Petersson metric, where we normalize by dividing the square of the metric
by the area of the surface. In the case where T is the modular surface this
reduces to showing that S has a finite cover with a proper ideal triangulation
where most of the shear coordinates are small; we will construct such a cover
out of a random collection of immersed ideal triangles in S.
