Fractional Calderón problem on a closed Riemannian manifold

We study topological properties of random closed curves on an orientable surface S of negative Euler characteristic.Letting γn denote the conjugacy class of the n th step of a simple random walk on the Cayley graph driven by a measure whose support is a finite generating set, then with probability converging to 1 as n goes to infinity,(1) when S is closed, the point in Teichmüller space at which γn is length-minimized stays uniformly close to a certain basepoint determined by the generating set;(2) the self-intersection number of γn is on the order of n 2 , the minimum length of γn taken over all hyperbolic metrics is on the order of n, and the metric minimizing the length of γn has systole length bounded uniformly from zero;(3) when S is punctured and the distribution is uniform and supported on a generating set of minimum size, the minimum degree of a cover to which γn admits a simple elevation (which we call the simple lifting degree of γn) grows at least like n/log(n) and at most on the order of n.We also show that many of these properties are generic, in the sense that the proportion of elements in the ball of radius n in the Cayley graph for which they hold, converges to 1 as n goes to infinity.The lower bounds on simple lifting degree for randomly chosen curves we obtain significantly improve the previously best known bounds which were on the order of log (1/3) n.As applications, we give relatively sharp upper and lower bounds on the dilatation of a generic point-pushing pseudo-Anosov homeomorphism in terms of the self-intersection number of its defining curve, as well as upper bounds on the simple lifting degree of a random curve in terms of its intersection number which outperform bounds for general curves.