Counting tangles in coverings of closed 2-orbifolds via ribbon categories

We describe what a typical covering of a closed 2-orbifold looks like on a small scale. Specifically, we prove that a uniformly random covering of degree n contains in expectation $Θ(n^0)$ short closed geodesics, potentially $Θ(n^0)$ pairs of nearby order-2 cone points, in general $Θ(n^{\frac{1}{m}}