Partial difference equation method for lattice path problems

Many problems concerning lattice paths, especially on the square lattice have been exactly solved. For a single path, many methods exist that allow exact calculation regardless of whether the path inhabits a strip, a semi-infinite space or infinite space, or perhaps interacts with the walls. It has been shown that a transfer matrix method using the Bethe Ansatz allows for the calculation of the partition function for many non-intersecting paths interacting with a wall. This problem can also be considered using the Gessel-Viennot methodology. In a concurrent development, two non-intersecting paths interacting with a wall have been examined in semi-infinite space using a set of partial difference equations.