A cutting process for random mappings

In this paper we consider a cutting process for random mappings. Specifically, for 0 < m < n, we consider the initial (uniform) random mapping digraph Gn on n labelled vertices, and we delete (if possible), uniformly and at random, m non-cyclic directed edges from Gn. The maximal random digraph consisting of the uni-cyclic components obtained after cutting the m edges is called the trimmed random mapping and is denoted by Gn . If the number of non-cyclic directed edges is less than m, then Gn consists of the cycles, including loops, of the initial mapping Gn. We consider the component structure of the trimmed mapping Gn . In particular, using the exact distribution we determine the asymptotic distribution of the size of a typical random connected component of Gn as n,m →∞. This asymptotic distribution depends on the relationship between n and m and we show that there are three distinct cases: (i) m = o( √ n), (ii) m = β √ n, where β > 0 is a fixed parameter, and (iii) √ n = o(m). This allows us to study the joint distribution of the order statistics of the normalized component sizes of Gn . When m = o( √ n), we obtain the PoissonDirichlet(1/2) distribution in the limit, whereas when √ n = o(m) the limiting distribution is Poisson-Dirichlet(1). Convergence to the Poisson-Dirichlet(θ) distribution breaks down when m = O( √ n), and in particular, there is no smooth transition from the PD(1/2) distribution to the PD(1) via the Poisson-Dirichlet distribution as the number of edges cut increases relative to n, the number of vertices in Gn.