I am seeking to understand the properties of a typical pair of permutations $(\sigma,\tau) \in \mathrm{Sym}(n)^2$ chosen uniformly at random from all pairs such that $\sigma$ and $\tau$ commute. In particular, I would like to prove that the following holds.

$(\ast)$ With high probability, the graph on $\{1,\ldots,n\}$ induced by $\sigma$ and $\tau$ has few 'unnecessary' short cycles (i.e. short cycles which do not correspond to commutators).

It seems that the best way to do this would be to have a straightforward algorithm to generate such a random pair. I have been unable to find such an algorithm in the literature and I would appreciate if someone could tell me what is known about this subject.

I would be equally happy to have a way of establishing $(\ast)$ for a random pair of approximately commuting permutations. To make the last statement precise, for each $\epsilon > 0$ I would like to know that if $n$ is large enough then $(\ast)$ holds for a pair chosen uniformly at random from of the set of all pairs $(\sigma,\tau) \in \mathrm{Sym}(n)^2$ such that $\sigma \tau \sigma^{-1} \tau^{-1}$ has at least $(1-\epsilon)n$ fixed points.