In this context a *mapping*
is just a function, represented by its directed graph with
edges .
An old paper [3] 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
[10] 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 [1]
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
[6] and then more general Levy or exchangeable-increment
processes [7,9] such as the following.
For suitable fixed positive parameters
use Brownian bridge
to
construct a ``bridge" process

Use the

This in turn is the process needed to define, via a continuous analog of Joyal's transformation, the limit process (generalizing reflecting Brownian bridge) of the general random -mapping. Formalizing this is work in preparation.