Annals of Combinatorics 3 (1999) 265-275

Partial Difference Equation Method for Lattice Path Problems

R. Brak1, J.W. Essam2, and A.L. Owczarek1

1Department of Mathematics and Statistics, The University of Melbourne, Parkville, VIC 3052, Australia
brak@maths.mu.oz.au, aleks@ms.unimelb.edu.au

2Department of Mathematics, Royal Holloway, University of London, Egham, Surrey TW20 0EX, England
j.essam@vms.rhbnc.ac.uk

AMS Subject Classification: 05A15, 82B41

Abstract. 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. 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.

Keywords: directed paths, interacting walks, lattice paths, difference equations, constant term, Gessel--Viennot theorem, Bethe Ansatz, transfer matrix method

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