  We consider a multilevel continuous time Markov chain $X(s;N) = (X_i^j(s;N):
1 \leq i \leq j \leq N)$, which is defined by means of Jack symmetric functions
and forms a certain discretization of the multilevel Dyson Brownian motion. The
process $X(s;N)$ describes the evolution of a discrete interlacing particle
system with push-block interactions between the particles, which preserve the
interlacing property. We study the joint asymptotic separation of the particles
at the right edge of the ensemble as the number of levels and time tend to
infinity and show that the limit is described by a certain zero range process
with local interactions.
