  We study random plane partitions with respect to volume measures with
periodic weights of arbitrarily high period. We show that near the vertical
boundary the system develops up to as many turning points as the period of the
weights, and that these turning points are separated by vertical facets which
can have arbitrary rational slope. In the lozenge tiling formulation of the
model the facets consist of only two types of lozenges arranged in arbitrary
periodic deterministic patterns. We compute the correlation functions near
turning points and show that the point processes at the turning points can be
described as several GUE-corners processes which are non-trivially correlated.
  The weights we study introduce a first order phase transition in the system.
We compute the limiting correlation functions near this phase transition and
obtain a process which is translation invariant in the vertical direction but
not the horizontal.
