SEARCH

SEARCH BY CITATION

Keywords:

  • random mappings;
  • Poisson-Dirichlet distribution;
  • component structure

Abstract

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 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 Gmath image. If the number of noncyclic directed edges is less than m, then Gmath image consists of the cycles, including loops, of the initial mapping Gn. We consider the component structure of the trimmed mapping Gmath image. In particular, using the exact distribution we determine the asymptotic distribution of the size of a typical random connected component of Gmath image as n, m [RIGHTWARDS ARROW] . This asymptotic distribution depends on the relationship between n and m and we show that there are three distinct cases: (i) equation image , (ii) equation image , where β > 0 is a fixed parameter, and (iii) equation image . This allows us to study the joint distribution of the order statistics of the normalized component sizes of Gmath image. When equation image , we obtain the Poisson–Dirichlet(1/2) distribution in the limit, whereas when equation image the limiting distribution is Poisson–Dirichlet(1). Convergence to the Poisson–Dirichlet(θ) distribution breaks down when equation image , and in particular, there is no smooth transition from the equation imageD(1/2) distribution to the equation imageD(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