  We study the joint asymptotic behavior of spacings between particles at the
edge of multilevel Dyson Brownian motions, when the number of levels tends to
infinity. Despite the global interactions between particles in multilevel Dyson
Brownian motions, we observe a decoupling phenomenon in the limit: the global
interactions become negligible and only the local interactions remain. The
resulting limiting objects are interacting particle systems which can be
described as Brownian versions of certain totally asymmetric exclusion
processes. This is the first appearance of a particle system with local
interactions in the context of general $\beta$ random matrix models.
