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2 Made a more detailed answer.

Aldous and Pitman have a paper on "Brownian bridge asymptotics for random mappings", which describes a setting in which Brownian bridge shows up as a limit object and is most naturally thought of as indexed by a circle rather than by an interval. There are two follow-up papers (one, two) by Aldous, Miermont and Pitman, the first of which "give[s] a conceptually straightforward argument which both proves convergence and more directly identifies the limit" (as well as extending the results to more general kinds of random mappings).

The basic idea is that mapping, i.e. a function $f$ from $[n] = \{1,\ldots,n\}$ to $[n]$ can be represented in terms of "basins of attraction". Create a digraph by joining $i$ to $j$ if $f(i)=j$. In this digraph, each connected component will contain a unique directed cycle, and each vertex $i$ of the cycle will be the root of a tree all of whose edges are oriented towards $i$. It is then possible to code the structure of the cycle-plus-trees in terms of a lattice path, with height corresponding to distance from the cycle.

When the underlying mapping $f$ is a uniformly random mapping, the resulting lattice path, suitably interpreted and after rescaling, converges to (the absolute value of) Brownian bridge.

1

Aldous and Pitman have a paper on "Brownian bridge asymptotics for random mappings", which describes a setting in which Brownian bridge shows up as a limit object and is most naturally thought of as indexed by a circle rather than by an interval. There are two follow-up papers (one, two) by Aldous, Miermont and Pitman, the first of which "give[s] a conceptually straightforward argument which both proves convergence and more directly identifies the limit" (as well as extending the results to more general kinds of random mappings).