Random Lattices, Punctured Tori and the Teichm\"uller distribution

The moduli space of lattices of $\mathbb{C}$ is a Riemann surface of finite hyperbolic area with the square lattice as an origin. We select a lattice from the induced uniform distribution and calculate the statistics of the Teichm\"uller distance to the origin. This in turn identifies distribution of the distance in Teichm\"uller space to the central "square" punctured torus in the moduli space of punctured tori. There are singularities in this p.d.f. arising from the topology of the moduli space. We also consider the statistics of the distance in Teichm\"uller space to the rectangular punctured tori and the p.d.f and expected distortion of the extremal quasiconformal mappings.