Large components of bipartite random mappings

A bipartite random mapping TK,L of a finite set V = V1 ∪ V2, |V1| = K and |V2| = L , into itself assigns independently to each i ∈ V1 its unique image j ∈ V2 with probability 1/L and to each i ∈ V2 its unique image j ∈ V1 with probability 1/K . We study the connected component structure of a random digraph G(TK,L) , representing TK,L, as K → ∞ and L → ∞. We show that, no matter how K and L tend to infinity relative to each other, the joint distribution of the normalized order statistics for the component sizes converges in distribution to the Poisson-Dirichlet distribution on the simplex ∇ = {{xi} : ∑ xi ≤ 1, xi ≥ xi+1 ≥ 0 for every i ≥ 1}.