  In this paper we consider a probability distribution on plane partitions,
which arises as a one-parameter generalization of the q^{volume} measure. This
generalization is closely related to the classical multivariate Hall-Littlewood
polynomials, and it was first introduced by Vuletic.
  We prove that as the plane partitions become large (q goes to 1, while the
Hall-Littlewood parameter t is fixed), the scaled bottom slice of the random
plane partition converges to a deterministic limit shape, and that one-point
fluctuations around the limit shape are asymptotically given by the GUE
Tracy-Widom distribution. On the other hand, if t simultaneously converges to
its own critical value of 1, the fluctuations instead converge to the
one-dimensional Kardar-Parisi-Zhang (KPZ) equation with the so-called narrow
wedge initial data.
  The algebraic part of our arguments is closely related to the formalism of
Macdonald processes. The analytic part consists of detailed asymptotic analysis
of the arising Fredholm determinants.
