Annals of Combinatorics 3 (1999) 251-263

From the Bethe Ansatz to the Gessel--Viennot Theorem

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

1Department of Mathematics and Statistics, The University of Melbourne, Parkville, VIC 3052, Australia,

2Department of Mathematics, Royal Holloway, University of London, Egham, Surrey TW20 0EX, England

Received November 30, 1998

AMS Subject Classification: 05A15, 82B41

Abstract. We state and prove several theorems that demonstrate how the coordinate Bethe Ansatz for the eigenvectors of suitable transfer matrices of a generalized inhomogeneous, five-vertex model on the square lattice, given certain conditions hold, is equivalent to the Gessel--Viennot determinant for the number of configurations of $N$ non-intersecting directed lattice paths, or vicious walkers, with various boundary conditions. Our theorems are sufficiently general to allow generalisation to any regular planar lattice.

Keywords: vicious walkers, lattice paths, Gessel--Viennot theorem, Bethe Ansatz, transfer matrix method.


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