Brownian bridge asymptotics for random mappings

Abstract

This chapter reviews Brownian bridge asymptotics for random mappings, first described in 1994 by Aldous and Pitman. The limit distributions as n→∞, of various functionals of a uniformly distributed random mapping from an n element set to itself, are those of corresponding functionals of a Brownian bridge. Similar results known to hold for various non-uniform models of random mappings, according to a kind of invariance principle. A mapping M _n : [n] → [n] can be identified with its digraph {i → M _n (i), i ∈ [n]}, as in Figure 1.