  We study topological properties of random closed curves on an orientable
surface $S$ of negative Euler characteristic. Letting $\gamma_{n}$ 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 on a finite generating set, then
with probability converging to $1$ as $n$ goes to infinity, (1) the point in
Teichm\"uller space at which $\gamma_{n}$ is length-minimized stays in some
compact set; (2) the self-intersection number of $\gamma_{n}$ is on the order
of $n^{2}$, the minimum length of $\gamma_{n}$ taken over all hyperbolic
metrics is on the order of $n$, and the metric minimizing the length of
$\gamma_{n}$ is uniformly thick; and (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 $\gamma_{n}$ admits a simple elevation
(which we call the $\textit{simple lifting degree}$ of $\gamma_{n}$) grows at
least like $n/\log(n)$ and at most on the order of $n$.
  We also show that these properties are $\textit{generic}$, 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.
