  We relate the counting of rational curves intersecting Schubert varieties of
the Grassmannian to the counting of certain non-intersecting lattice paths on
the cylinder, so-called vicious and osculating walkers. These lattice paths
form exactly solvable statistical mechanics models and are obtained from
solutions to the Yang-Baxter equation. The eigenvectors of the transfer
matrices of these models yield the idempotents of the Verlinde algebra of the
gauged u(n)-WZNW model. The latter is known to be closely related to the small
quantum cohomology ring of the Grassmannian. We establish further that the
partition functions of the vicious and osculating walker model are given in
terms of Postnikov's toric Schur functions and can be interpreted as generating
functions for Gromov-Witten invariants.
