Teichmüller theory via random simple closed curves

We show the map $\sigma :T_g\to C_g$ sending a compact hyperbolic surface $X$ to a random simple closed geodesic on $X$ determines a proper embedding of Teichmüller space into the space of geodesic currents. The proof depends on a formula for the intersection number $i(C,C^{\mathrm{\prime }})$ of a pair of multicurves, expressed in terms of Dehn coordinates on $M{L}_g(\mathbb{Z})$.