Partial Difference Equation Method for Lattice Path Problems

Many problems concerning lattice paths, especially on the square lattice have been accurately 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 themselves. 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. Here, we review this partial difference equation method for the case of one path in a half plane. We then demonstrate that the answer for arbitrary numbers of non-intersecting paths interacting with a wall can be obtained using this method. One reason for doing this is its pedagogical value in showing its ease of use compared to the transfer matrix method. The solution is expressed in a new form as a “constant term” formula, which is readily evaluated. More importantly, it is the natural method that generalizes easily to many intersecting paths where there is inter-path interactions (e.g., osculating lattice paths). We discuss the relationship of the partial difference equation method to the transfer matrix method and their solution via a Bethe Ansatz.