Title: The asymptotic distribution of the diameter of a random mapping
Author: David Aldous and Jim Pitman 
Date: February 2002
Abstract:
The asymptotic distribution of the diameter of the digraph of a uniformly
distributed random mapping of an $n$-element set to itself is represented
as the distribution of a functional of a reflecting Brownian bridge.
This yields a formula for the Mellin transform of the asymptotic distribution,
generalizing the evaluation of its mean by Flajolet and Odlyzko (1990).