Large components of bipartite random mappings

A bipartite random mapping T-K,T-L of a finite set V = V-1 U V-2,V- \V-1\ = K and \V-2\ = L, into itself assigns independently to each i is an element of V-1 its unique image j is an element of V-2 with probability 1/L and to each i is an element of V-2 its unique image j is an element of V-1 with p robability 1/K. We study the connected component structure of a random digr aph G(T-K,T-L), representing T-K,T-L, as K --> infinity and L --> infinity. We show that, no matter how K and L tend to infinity relative to each othe r, the joint distribution of the normalized order statistics for the compon ent sizes converges in distribution to the Poisson-DirichIet distribution o n the simplex del = {{x(i)}: Sigmax(i) less than or equal to 1,x(i) greater than or equal to x(i+1) greater than or equal to 0 for every i greater tha n or equal to 1}.(C) 2000 John Wiley & Sons, Inc.