We study the boundary of the liquid region L in large random lozenge tiling models defined by uniform random interlacing particle systems with general initial configuration, which lies on the line px, 1q, x P R " BH. We assume that the initial particle configuration converges weakly to a limiting density φpxq, 0 ď φ ď 1. The liquid region is given by a homeomorphism WL : L Ñ H, the upper half plane, and we consider the extension of W´1 L to H. Part of BL is given by a curve, the edge E , parametrized by intervals in BH, and this corresponds to points where φ is identical to 0 or 1. If 0 ă φ ă 1, the non-trivial support, there are two cases. Either W´1 L pwq has the limit px, 1q as w Ñ x non-tangentially and we have a regular point, or we have what we call a singular point. In this case W´1 L does not extend continuously to H. Singular points give rise to parts of BL not given by E and which can border a frozen region, or be "inside" the liquid region. This shows that in general the boundary of BL can be very complicated. We expect that on the singular parts of BL we do not get a universal point process like the Airy or the extended Sine kernel point processes. Furthermore, E and the singular parts of BL are shocks of the complex Burgers equation.

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