Gaussian unitary ensemble in random lozenge tilings

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