The six-vertex model, also known as the square-ice model, is one of the central and most studied systems of 2d statistical mechanics. It offers various combinatorial interpretations. One of them involves molecules of water on the square grid; another one deals with non-intersecting lattice paths, which can be also viewed as level lines of an integer-valued height function. Despite many efforts since the 1960s, the limit shapes for the height function are still unknown in general situations. However, we recently found ways to compute them in a degeneration, which leads to a low density of corners of paths (or, equivalently, of horizontal/vertical molecules of water). I will report on the progress in this direction emphasizing various unusual features: appearance of hyperbolic PDEs; discontinuities in densities; connections to random permutations.
