On Poisson-Dirichlet limits for random decomposable combinatorial structures

We prove a joint local limit law for the distribution of the r largest components of decomposable logarithmic combinatorial structures, including assemblies, multisets and selections. Our method is entirely probabilistic, and requires only weak conditions that may readily be verified in practice. Combinatorics, Probability and Computing (1999) 8, 193–208. Printed in the United Kingdom c © 1999 Cambridge University Press On Poisson–Dirichlet Limits for Random Decomposable Combinatorial Structures R I C H A R D A R R A T I A1†, A. D. B A R B O U R2‡ and S I M O N T A V A R É1† 1 Department of Mathematics, University of Southern California, Los Angeles, CA 90089-1113, USA (e-mail: [email protected] [email protected]) 2 Abteilung für Angewandte Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057, Zürich, Switzerland (e-mail: [email protected]) Received 20 June 1997; revised 16 March 1998 We prove a joint local limit law for the distribution of the r largest components of decomposable logarithmic combinatorial structures, including assemblies, multisets and selections. Our method is entirely probabilistic, and requires only weak conditions that may readily be verified in practice.