Six-vertex model with rare corners and random restricted permutations

We study limit shapes in two equivalent models: the six-vertex model in the $c\to 0$ limit and the random Mallows permutation with restricted permutation matrix. We give the Euler-Lagrange equation for the limit shape and show how to solve it for a class of rectilinear polygonal domains. Its solutions are given by piecewise-algebraic functions with lines of discontinuities.