  We study lozenge tilings of a domain with partially free boundary. In
particular, we consider a trapezoidal domain (half hexagon), s.t. the
horizontal lozenges on the long side can intersect it anywhere to protrude
halfway across. We show that the positions of the horizontal lozenges near the
opposite flat vertical boundary have the same joint distribution as the
eigenvalues from a Gaussian Unitary Ensemble (the GUE-corners/minors process).
We also prove the existence of a limit shape of the height function, which is
also a vertically symmetric plane partition. Both behaviors are shown to
coincide with those of the corresponding doubled fixed-boundary hexagonal
domain. We also consider domains where the different sides converge to $\infty$
at different rates and recover again the GUE-corners process near the boundary.
