  A Gelfand-Tsetlin scheme of depth N is a triangular array with m integers at
level m, m=1,...,N, subject to certain interlacing constraints. We study the
ensemble of uniformly random Gelfand-Tsetlin schemes with arbitrary fixed N-th
row. We obtain an explicit double contour integral expression for the
determinantal correlation kernel of this ensemble (and also of its
q-deformation).
  This provides new tools for asymptotic analysis of uniformly random lozenge
tilings of polygons on the triangular lattice; or, equivalently, of random
stepped surfaces. We work with a class of polygons which allows arbitrarily
large number of sides. We show that the local limit behavior of random tilings
(as all dimensions of the polygon grow) is directed by ergodic translation
invariant Gibbs measures. The slopes of these measures coincide with the ones
of tangent planes to the corresponding limit shapes described by Kenyon and
Okounkov in arXiv:math-ph/0507007. We also prove that at the edge of the limit
shape, the asymptotic behavior of random tilings is given by the Airy process.
  In particular, our results cover the most investigated case of random boxed
plane partitions (when the polygon is a hexagon).
