Let $S_n$ be the symmetric group on $n$ elements, let $d$ be the Cayley distance, and let $m$ be a Haar measure on $S_n$.
Let $s$ denote a random permutation with respect to $m$, i.e., $s$ is an $S_n$-valued random variable with law $m$. Let $\delta := d(s,0)$ be the Cayley distance from $s$ to $0$, i.e., the minimum number of transpositions to shuffle $s$ back to the origin. Since $\delta$ is an $\mathbb N$-valued random variable its expectation and variance are well-defined (and dependent on $n$).
What is the asymptotic behavior of $\mathbb E[\delta]$ and $\operatorname{var}(\delta)$ as $n \to \infty$?