  This paper establishes a universality result for scaling limits of uniformly
random lozenge tilings of large domains. We prove that whenever a boundary of
the domain has three adjacent straight segments inclined under 120 degrees to
each other, the asymptotics of tilings near the middle segment is described by
the GUE--corners process of random matrix theory. An important step in our
argument is to show that fluctuations of the height function of random tilings
on essentially arbitrary simply-connected domains of diameter $N$ have smaller
magnitude than $N^{1/2}$.
