A combinatorial interpretation of the free-fermion condition of the six-vertex model

The free-fermion condition of the six-vertex model provides a five-paramete r submanifold on which the Bethe ansatz equations for the wavenumbers that enter into the eigenfunctions of the transfer matrices of the model decoupl e, hence allowing explicit solutions. Such conditions arose originally in e arly field-theoretic S-matrix approaches. Here we provide a combinatorial e xplanation for the condition in terms of a generalized Gessel-Viennot invol ution. By doing so we extend the use of the Gessel-Viennot theorem, origina lly devised for non-intersecting walks only, to a special weighted type of intersecting walk, and hence express the partition function of N such walks starting and finishing at fixed endpoints in terms of the single-walk part ition functions.