Combinatorially random curves on surfaces

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.