A cutting process for random mappings

Abstract

AbstractIn 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 labeled vertices, and we delete (if possible), uniformly and at random, m noncyclic directed edges from Gn. The maximal random digraph consisting of the unicyclic components obtained after cutting the m edges is called the trimmed random mapping and is denoted by G. If the number of noncyclic directed edges is less than m, then G consists of the cycles, including loops, of the initial mapping Gn. We consider the component structure of the trimmed mapping G. In particular, using the exact distribution we determine the asymptotic distribution of the size of a typical random connected component of G as n, m → ∞. This asymptotic distribution depends on the relationship between n and m and we show that there are three distinct cases: (i) $m=o(\sqrt{n})$ , (ii) $m=\beta\sqrt{n}$ , where β > 0 is a fixed parameter, and (iii) $\sqrt{n}=o(m)$ . This allows us to study the joint distribution of the order statistics of the normalized component sizes of G. When $m=o(\sqrt{n})$ , we obtain the Poisson–Dirichlet(1/2) distribution in the limit, whereas when $\sqrt{n}=o(m)$ the limiting distribution is Poisson–Dirichlet(1). Convergence to the Poisson–Dirichlet(θ) distribution breaks down when $m=O(\sqrt{n})$ , and in particular, there is no smooth transition from the ${\cal P}$D(1/2) distribution to the ${\cal P}$D(1) via the Poisson–Dirichlet distribution as the number of edges cut increases relative to n, the number of vertices in Gn. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 30, 287–306, 2007