 
Summary: Digital Object Identifier (DOI) 10.1007/s0044000404072
Probab. Theory Relat. Fields 133, 117 (2005)
David Aldous · GrŽegory Miermont · Jim Pitman
Weak convergence of random pmappings
and the exploration process of inhomogeneous
continuum random trees
Received: 19 February 2004 / Revised version: 26 October 2004 /
Published online: 14 July 2005 c SpringerVerlag 2005
Abstract. We study the asymptotics of the pmapping model of random mappings on [n] as
n gets large, under a large class of asymptotic regimes for the underlying distribution p. We
encode these random mappings in random walks which are shown to converge to a functional
of the exploration process of inhomogeneous random trees, this exploration process being
derived (AldousMiermontPitman 2004) from a bridge with exchangeable increments. Our
setting generalizes previous results by allowing a finite number of "attracting points" to
emerge.
1. Introduction
We study the asymptotic behavior as n of random elements of the set [n][n]
of mappings from [n] = {1, 2, . . . , n} to [n]. Given a probability measure p =
(p1, . . . , pn) on [n], define a random mapping M as follows: for each i [n], map
i to j with probability pj , independently over different i's, so that
