Teichm\"uller 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 Teichm\"uller space into the space of geodesic currents. The proof depends on a formula for the intersection number $i(C,C')$ of a pair of multicurves, expressed in terms of Dehn coordinates on $ML_g(\mathbb{Z})$.