The five vertex model is a probability measure on monotone nonintersecting lattice paths, where each corner of each path gets weight r. It generalizes the lozenge tiling model which is the case r=1. There is a rigorous "Bethe Ansatz" solution, leading to explicit limit shapes for tilings of bounded regions such as the boxed plane partition.

Fluctuations around the limit shape are conjecturally given by a Gaussian free field with a spatially-varying stiffness. We discuss a recent approach to getting local statistics in the model.