Directed walks and vertex models: renormalization, multicriticality and beyond

We discuss the exact renormalization of interacting directed random walks (DRW) (using dimensional regularization), and the exact results for the multicritical points. We discuss how these exact results of the DRW problem can give exact critical exponents ofthe ferroelectric five vertex model in all dimensions. The multicritical DRW's lead to a new class of vertex models whose multicritical exponents can be computed exactly. A random vertex model and its connection to the DRW problem in a random medium is also discussed.