Abstract We study the distribution of geometrically and topologically nearly geodesic random surfaces in a closed hyperbolic 3‐manifold . In particular, we describe invariant measures on the Grassmann bundle which arise as limits of random minimal surfaces. We show that if contains at least one totally geodesic subsurface then every topological limiting measure is totally scarring (i.e., supported on the totally geodesic locus), while we prove that geometrical limiting measures are never totally scarring.
Kahn et al. (Sun,) studied this question.