I am studying proper colorings of complete bipartite graphs and I'd like to be able to pick a random proper coloring and the compute some things about it.

Recall that a proper coloring of a complete bipartite graph is precisely a latin rectangle where the $(i,j)$ entry is the color on the edge from vertex $i$ to vertex $j$.

My questions are generally of the form "for a large $m$ and $n$, is there a positive probability that the random proper coloring of $K_{m,n}$ has a copy of the subgraph $H$ which has a specific coloring?"

Thanks for your insights!