MiS Preprint Repository

We show that for almost every translation surface the number of pairs of saddle connections with bounded virtual area has asymptotic growth like $c{R}^2$ where the constant $c$ depends only on the area and the connected component of the stratum. The proof techniques combine classical results for counting saddle connections with the crucial result that the Siegel-Veech transform is in $L^2$. In order to capture information about pairs of saddle connections, we consider pairs with bounded virtual area since the set of such pairs can be approximated by a fibered set which is equivariant under geodesic flow. In the case of lattice surfaces, small virtual area is equivalent to counting parallel pairs of saddle connections, which also have a quadratic growth of $c{R}^2$ where $c$ depends in this case on the given lattice surface.