  At a typical cusp point of the disordered region in a random tiling model we
expect to see a determinantal process called the Pearcey process in the
appropriate scaling limit. However, in certain situations another limiting
point process appears that we call the Cusp-Airy process, which is a kind of
two sided extension of the Airy kernel point process. We will study this
problem in a class of random lozenge tiling models coming from interlacing
particle systems. The situation was briefly studied previously by Okounkov and
Reshetikhin under the name cuspidal turning point.
