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
 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

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.


References

1. R.J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, London, 1982.

2.  H.A. Bethe, Zur theorie der metalle. I. Eigenwerte und eigenfunktionen der linearen atom kette, Z. Phys. 71 (1931) 205–226.

3.  R. Brak, J. Essam, and A.L. Owczarek, Exact solution of N directed non-intersecting walks interacting with one or two boundaries, J. Phys. A 32 (1999) 2921–2929.

4.  M.E. Fisher, Walks, walls, wetting and melting, J. Stat. Phys. 34 (1984) 667–729.

5.  P.J. Forrester, Probability of survival for vicious walkers near a cliff, J. Phys. A 22 (1989) L609–L613.

6.  P.J. Forrester, Vicious random walkers in the limit of large number of walkers, J. Stat. Phys. 56 (1989) 767–782.

7.  P.J. Forrester, Exact solution of the lock step model of vicious walkers, J. Phys. A 23 (1990) 1259–1273.

8.  I.M. Gessel and X. Viennot, Determinants, paths, and plane partitions, 1989, preprint.

9.  I.M. Gessel and X. Viennot, Binomial determinants, paths, and hook length formulae, Adv. Math. 58 (1985) 300–321.

10.  A.J. Guttmann, A.L. Owczarek, and X.G. Viennot, Vicious walkers and young tableaux i: Without walls, J. Phys. A 31 (1998) 8123–8135.

11.  S. Karlin and G. McGregor, Coincidence probabilities, Pacific J. Math. 9 (1959) 1141–1164.

12.  S. Karlin and G. McGregor, Coincidence probabilities of birth-and-death processes, Pacific J. Math. 9 (1959) 1109–1140.

13.  E.H. Lieb, Residual entropy of square ice, Phys. Rev. 162 (1967) 162–172.

14.  B. Lindström, On vector representations of induced matroids, Bull. London Math. Soc. 5 (1973) 85.

15.  A.L. Owczarek and R.J. Baxter, Surface free energy of the critical six vertex model with free boundaries, J. Phys. A 22 (1989) 1141–1165.

16.  F.Y. Wu, Remarks on the modified potassium dihydrogen phosphate model of a ferroelectric, Phys. Rev. 168 (1968) 539–543.


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