Simulations of uniformly random domino tilings of large Aztec diamonds give striking pictures due to the emergence of two macroscopic regions. These regions are often referred to as solid and liquid phases. A limiting curve separates these regions and interesting probabilistic features occur around this curve, which are related to random matrix theory. The two-periodic Aztec diamond features a third phase, often called the gas phase. In this talk, we introduce the model and discuss some of the asymptotic behavior at the liquid-gas boundary. This is based on joint works with Vincent Beffara (Grenoble), Kurt Johansson (Stockholm) and Benjamin Young (Oregon).