Local properties of a random mapping model Jennie C. Hansen ∗ and Jerzy Jaworski † ‡ July 10, 2007 Abstract In this paper we investigate the ‘local’ properties of a random mapping model, T ˆ D n , which maps the set {1, 2, ..., n} into itself. The random mapping T ˆ D n was introduced in a companion paper [?] is con- structed using a collection of exchangeable random variables ˆ D 1 , ...., ˆ D n which satisfy ∑ n i=1 ˆ D i = n. In the random digraph, G ˆ D n , which repre- sents the mapping T ˆ D n , the in-degree sequence for the vertices is given by the variables ˆ D 1 , ˆ D 2 , ..., ˆ D n , and, in some sense, G ˆ D n can be viewed as an analogue of the general independent degree models from ran- dom graph theory. By local properties we mean the distributions of random mapping characteristics related to a given vertex v of G ˆ D n - for example, the numbers of predecessors and successors of v in G ˆ D n . We show that the distribution of several variables associated with the local structure of G ˆ D n can be expressed in terms of expectations of sim- ple functions of ˆ D 1 , ˆ D 2 , ..., ˆ D n . We also consider two special examples of T ˆ D n which correspond to random mappings with preferential and anti-preferential attachment, respectively, and determine, for these examples, exact and asymptotic distributions for the local structure variables considered in this paper. * Actuarial Mathematics and Statistics, Heriot–Watt University,Edinburgh EH14 4AS, UK. E-mail address: J.Hansen@ma.hw.ac.uk † Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umul- towska 87, 61-614 Pozna´ n, Poland. E-mail address: jaworski@amu.edu.pl ‡ J.Jaworski acknowledges the generous support by the Marie Curie Intra-European Fellowship No. 501863 (RANDIGRAPH) within the 6th European Community Framework Programme. 1