  We consider the domino tilings of an Aztec diamond with a cut-off corner of
macroscopic square shape and given size, and address the bulk properties of
tilings as the size is varied. We observe that the free energy exhibits a
third-order phase transition when the cut-off square, increasing in size,
reaches the arctic ellipse---the phase separation curve of the original
(unmodified) Aztec diamond. We obtain this result by studying the thermodynamic
limit of certain nonlocal correlation function of the underlying six-vertex
model with domain wall boundary conditions, the so-called emptiness formation
probability (EFP). We consider EFP in two different representations: as a
tau-function for Toda chains and as a random matrix model integral. The latter
has a discrete measure and a linear potential with hard walls; the observed
phase transition shares properties with both Gross-Witten-Wadia and
Douglas-Kazakov phase transitions.
