Annals of Combinatorics 3 (1999) 451-473


Adsorbing Staircase Walks and Staircase Polygons

Buks van Rensburg

Department of Mathematics and Statistics, York University, Toronto, Ontario M3J 1P3, Canada
rensburg@boltzman.math.yorku.ca

Received November 3, 1998

AMS Subject Classification: 05A15, 82B20, 82B23, 82B41

Abstract. The adsorption of staircase walks and staircase polygons on the main diagonal in the square lattice is reviewed. We draw attention to the connection between adsorbing random walks in subsets of the integers and the square lattice, and this problem. The generating functions of adsorbing staircase walks and polygons are determined using several techniques, and information about the adsorption transition is found by the calculation of a critical exponent associated with it.

Keywords: staircase walks, dyck paths, adsorption, staircase polygons


References

1.  R. Brak , J.M. Essam, and A.L. Owczarek, New results for directed vesicles and chains near an attractive wall, J. Stat. Phys. 93 (1998) 155–192.

2.  R. Brak, A.L. Owczarek, and T. Prellberg, Exact scaling behaviour of partially convex vesicles, J. Stat. Phys. 76 (1994) 1101–1128.

3.  M.C.T.P. Carvalho and V. Privman, Directed walk models of polymers at interfaces, J. Phys. A: Math. Gen. 21 (1988) L1033–L1037.

4.  K. De'Bell and T. Lookman, Surface phase transitions in polymer systems, Rev. Mod. Phys. 65 (1993) 87–114.

5.  M.P. Delest and G. Viennot, Algebraic languages and polyominoe enumeration, Theor. Comput. Sci. 34 (1984) 169–206.

6.  E. Eisenreigler, Dilute and semidilute polymer solutions near an adsorbing wall, J. Chem. Phys. 79 (1983) 1052–1064.

7.  E. Eisenreigler, Adsorption of polymer chains at surfaces II: Amplitude ratios for end-to-end distance distributions at the critical point of adsorption, J. Chem. Phys. 82 (1985) 1032–1041.

8.  E. Eisenreigler, K. Kremer, and K. Binder, Adsorption of polymer chains at surfaces: Scaling and Monte Carlo analysis, J. Chem. Phys. 77 (1982) 6296–6320.

9.  W. Feller, An Introduction to Probability Theory and its Applications, Wiley, 1968.

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

11.  G. Forgacs, V. Privman, and H.L. Frisch, Adsorption-desorption of polymer chains interacting with a surface, J. Chem. Phys. 90 (1989) 3339–3345.

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

13.  I.M. Gessel, A probabilistic method for lattice path enumeration, J. Stat. Planning and Inference 14 (1986) 49–58.

14.  B.R. Handa and S.G. Mohanty, On a property of lattice paths, J. Stat. Planning and Inference 14 (1986) 59–62.

15.  W.H. McCrea and F.J.W. Whipple, Random paths in two and three dimensions, Proc. Royal Soc. Edinburgh 60 (1940) 281–298.

16.  C. Michelletti and J.M. Yeomans, Adsorption transition of directed vesicles in two dimensions, J. Phys. A: Math. Gen. 26 (1993) 5705–5712.

17.  H. NiederHausen, The enumeration of restricted random walks by Sheffer polynomials with applications to statistics, J. Stat. Planning and Inference 14 (1986) 95–114.

18.  A.L. Owczarek and T. Prellberg, Exact solution of the discrete (1+1)-dimensional SOS model with field and surface interactions, J. Stat. Phys. 70 (1993) 1175–1194.

19.  V. Privman, G. Forgacs, and H.L. Frisch, New solvable model of polymer chains adsorption at a surface, Phys. Rev. B 37 (1988) 9897–9900.

20.  V. Privman and S. Svrakic, Directed models of polymers, interfaces, and clusters: Scaling and finite-size properties, Lecture Notes in Physics, Vol. 338, Springer-Verlag, 1989.

21.  L. Takàcs, Some asymptotic formulas for lattice paths, J. Stat. Planning and Inference 14 (1986) 123–142.

22.  A.R. Veal, J.M. Yeomans, and G. Jug, The effect of attractive monomer-monomer interactions on the adsorption of a polymer chain, J. Phys. A: Math. Gen. 24 (1991) 827–849.

23.  H.S. Wall, Analytic Theory of Continued Fractions, Chelsea, 1967.

24.  S.G. Whittington, A Directed walk model of copolymer adsorption, 1998, preprint.


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