In this context a mapping is just a function, represented by its directed graph with edges . An old paper  showed how the asymptotic joint distributions of numerous statistics of a uniform random mapping are identifiable as corresponding joint distributions of reflecting Brownian bridge. Motivated in part by recent work of O'Cinneide and Pokrovskii  we are currently studying the case of -mappings, where for fixed the random variables are i.i.d. with some distribution . Using Joyal's bijection between mapping and trees and our recent theory [8,5] of limits of random -trees, we have analyzed  the ``uniform asymptotic negligibility" case where the limit is Brownian bridge. In work in preparation we study the general case where the limit should be describable in terms of the ICRT (inhomogeneous continuum random tree) limit of general -trees, along lines indicated below.
Our ``continuum tree" approach [4,5] to the stochastic
additive coalescent has recently been complemented by more
direct constructions using Brownian excursion with drift
 and then more general Levy or exchangeable-increment
processes [7,9] such as the following.
For suitable fixed positive parameters
use Brownian bridge
construct a ``bridge" process