Annals of Combinatorics

, Volume 3, Issue 2–4, pp 451–473 | Cite as

Adsorbing staircase walks and staircase polygons

  • Buks van Rensburg


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.

AMS Subject Classification

05A15 82B20 82B23 82B41 


staircase walks dyck paths adsorption staircase polygons 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 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.Google Scholar
  2. 2.
    R. Brak, A.L. Owczarek, and T. Prellberg, Exact scaling behaviour of partially convex vesicles, J. Stat. Phys.76 (1994) 1101–1128.Google Scholar
  3. 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.Google Scholar
  4. 4.
    K. De'Bell and T. Lookman, Surface phase transitions in polymer systems, Rev. Mod. Phys.65 (1993) 87–114.Google Scholar
  5. 5.
    M.P. Delest and G. Viennot, Algebraic languages and polyominoe enumeration, Theor. Comput. Sci.34 (1984) 169–206.Google Scholar
  6. 6.
    E. Eisenreigler, Dilute and semidilute polymer solutions near an adsorbing wall, J. Chem. Phys.79 (1983) 1052–1064.Google Scholar
  7. 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.Google Scholar
  8. 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.Google Scholar
  9. 9.
    W. Feller, An Introduction to Probability Theory and its Applications, Wiley, 1968.Google Scholar
  10. 10.
    M.E. Fisher, Walks, walls, wetting, and melting, J. Stat. Phys.34 (1984) 667–729.Google Scholar
  11. 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.Google Scholar
  12. 12.
    P.J. Forrester, Probability of survival for vicious walkers near a cliff, J. Phys. A: Math. Gen.22 (1989) L609-L613.Google Scholar
  13. 13.
    I.M. Gessel, A probabilistic method for lattice path enumeration, J. Stat. Planning and Inference14 (1986) 49–58.Google Scholar
  14. 14.
    B.R. Handa and S.G. Mohanty, On a property of lattice paths, J. Stat. Planning and Inference14 (1986) 59–62.Google Scholar
  15. 15.
    W.H. McCrea and F.J.W. Whipple, Random paths in two and three dimensions, Proc. Royal Soc. Edinburgh60 (1940) 281–298.Google Scholar
  16. 16.
    C. Michelletti and J.M. Yeomans, Adsorption transition of directed vesicles in two dimensions, J. Phys. A: Math. Gen.26 (1993) 5705–5712.Google Scholar
  17. 17.
    H. NiederHausen, The enumeration of restricted random walks by Sheffer polynomials with applications to statistics, J. Stat. Planning and Inference14 (1986) 95–114.Google Scholar
  18. 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.Google Scholar
  19. 19.
    V. Privman, G. Forgacs, and H.L. Frisch, New solvable model of polymer chains adsorption at a surface, Phys. Rev. B37 (1988) 9897–9900.Google Scholar
  20. 20.
    V. Privman and S. Švrakić, Directed models of polymers, interfaces, and clusters: Scaling and finite-size properties, Lecture Notes in Physics, Vol. 338, Springer-Verlag, 1989.Google Scholar
  21. 21.
    L. Takács, Some asymptotic formulas for lattice paths, J. Stat. Planning and Inference14 (1986) 123–142.Google Scholar
  22. 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.Google Scholar
  23. 23.
    H.S. Wall, Analytic Theory of Continued Fractions, Chelsea, 1967.Google Scholar
  24. 24.
    S.G. Whittington, A Directed walk model of copolymer adsorption, 1998, preprint.Google Scholar

Copyright information

© Springer-Verlag 1999

Authors and Affiliations

  • Buks van Rensburg
    • 1
  1. 1.Department of Mathematics and StatisticsYork UniversityTorontoCanada

Personalised recommendations