  Consider an N-dimensional Markov chain obtained from N one-dimensional random
walks by Doob h-transform with the q-Vandermonde determinant. We prove that as
N becomes large, these Markov chains converge to an infinite-dimensional Feller
Markov process. The dynamical correlation functions of the limit process are
determinantal with an explicit correlation kernel. The key idea is to identify
random point processes on Z with q-Gibbs measures on Gelfand-Tsetlin schemes
and construct Markov processes on the latter space. Independently, we analyze
the large time behavior of PushASEP with finitely many particles and
particle-dependent jump rates (it arises as a marginal of our dynamics on
Gelfand-Tsetlin schemes). The asymptotics is given by a product of a marginal
of the GUE-minor process and geometric distributions.
