Distributional limits of Riemannian manifolds

Suppose that $(M_n)$ is a sequence of closed Riemannian manifolds with pinched sectional curvature and pairwise $K$-quasiconformal to each other. Adapting an argument due to Benjamini and Schramm, I will discuss the possible pointed Gromov-Hausdorff limits of $(M_n,p_n)$ where $p_n\in M_n$ is chosen by random. This is joint work with Hossein Namazi and Pekka Pankka.