My motivation for this question comes from the study of synchronizing automata. There is a general consensus that random automata are synchronizing and have short synchronizing words. I am hoping that this can be made precise.

Let $T_n$ be the monoid of all self-mappings of $\{1,\ldots, n\}$. What would are some natural ways to generate a random subset of $T_n$?

I am particularly interested in models which seem intuitive and for which there is some hope to compute/estimate the following.

- The probability the subsemigroup generated by a random set contains a constant map
- The expected minimum length of a constant map.

One possible way to choose a random subset is to flip a coin for each element of $T_n$ and add it to the subset if we get a heads, but I don't really like this method.

A second method might be to first flip $n^n$ coins. Let $k$ be the number of heads. Then choose $k$ transformations in some random way. Perhaps one can choose without repetition $k$ iid transformations according to some distribution on elements of $T_n$. Possible distributions would be the uniform distribution or a stationary distribution of some natural random walk on $T_n$ (side note, what is a natural random walk on $T_n$?). Or one can use the distribution based on the method of generating a transformation by choosing the image of each $i\in \{1,\ldots,n\}$ uniformly at random from $\{1,\ldots, n\}$.

Any thoughts?