  We consider n-point sticky Brownian motions: a family of n diffusions that
evolve as independent Brownian motions when they are apart, and interact
locally so that the set of coincidence times has positive Lebesgue measure with
positive probability. These diffusions can also be seen as n random motions in
a random environment whose distribution is given by so-called stochastic flows
of kernels. For a specific type of sticky interaction, we prove exact formulas
characterizing the stochastic flow and show that in the large deviations
regime, the random fluctuations of these stochastic flows are Tracy-Widom GUE
distributed. An equivalent formulation of this result states that the extremal
particle among n sticky Brownian motions has Tracy-Widom distributed
fluctuations in the large n and large time limit. These results are proved by
viewing sticky Brownian motions as a (previously known) limit of the exactly
solvable beta random walk in random environment.
