I will discuss the local structure of the boundary at the rough-smooth (or liquid-gas) interface in the two-periodic Aztec diamond. At the frozen-rough interface we have a well-defined boundary path which converges to the Airy process. At the rough-smooth smooth boundary the situation is more complicated although here also we expect to have an Airy process. Which geometric structure converges to the Airy process? At this boundary we see both local and long range structures. There should be a last path among these long range structures that converges to the Airy process as can be seen in pictures. We will not prove this but discuss some results in this direction. This is joint work with Vincent Beffara and Sunil Chhita.