Limit Shapes for the Asymmetric Five Vertex Model

We compute the free energy and surface tension function for the five-vertex model , a model of non-intersecting monotone lattice paths on the grid in which each corner gets a weight r>0 . We give a variational principle for limit shapes in this setting, and show that the resulting Euler–Lagrange equation can be integrated, giving limit shapes explicitly parameterized by analytic functions.