Abstract
The asymmetric directed-bond percolation (ADBP) problem with an asymmetry parameterk is introduced and some rigorous results are given concerning a series expansion of the percolation probability on the square lattice. It is shown that the first correction term,d n,1 (k) is expressed by Gauss' hypergeometric series with a variablek. Since the ADBP includes the ordinary directed bond percolation as a special case withk=1, our results give another proof for the Baxter-Guttmann's conjecture thatd n,1(1) is given by the Catalan number, which was recently proved by Bousquet-Mélou. Direct calculations on finite lattices are performed and combining them with the present results determines the first 14 terms of the series expansion for percolation probability of the ADBP on the square lattice. The analysis byDlog Padé approximations suggests that the critical value depends onk, while asymmetry does not change the critical exponent β of percolation probability.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. R. Broadbent and J. M. Hammersley, Percolation processes I. Crystals and mazes,Proc. Camb. Philos. Soc. 53:629–641 (1957).
J. Blease, Series expansions for the directed-bond percolation problem,J. Phys. C: Solid State Phys. 10:917–924 (1977).
E. Domany and W. Kinzel, Directed percolation in two dimensions: Numerical analysis and an exact solution,Phys. Rev. Lett. 47:5–8 (1981).
D. Dhar, M. Barma, and M. K. Phani, Duality transformation for two-dimensional directed percolation and resistance problems,Phys. Rev. Lett. 47:1238–1241 (1981).
D. Dhar, Diode-resistor percolation in two and three, dimensions: I. Upper bounds on critical probability,J. Phys. A. Math. Gen. 15:1849–1858 (1982).
W. Kinzel, Directed percolation, inPercolation Structures and Processes, G. Deutscher, R. Zallen, and J. Adler, eds, (Adam Hilger, Bristol, 1983), pp. 425–445.
R. Durrett, Oriented percolation in two dimensions,Ann. Prob. 12:999–1040 (1984).
J. W. Essam, K. De'Bell, J. Adler and F. M. Bhatti, Analysis of extended series for bond percolation on the directed square lattice.Phys. Rev. B33:1982–1986 (1986).
R. J. Baxter and A. J. Guttmann, Series expansion of the percolation probability for the directed square lattice,J. Phys. A. Math. Gen. 21:3193–3204 (1988).
J. W. Essam, A. J. Guttmann, and K. De'Bell, On two-dimensional directed percolation,J. Phys. A: Math. Gen. 21:3815–3832 (1988).
T. M. Liggett, Survival of discrete time growth models, with applications to oriented percolation,Ann. Appl. Probab. 5:613–636 (1995).
I. Jensen and A. J. Guttmann, Series expansions of the percolation probability for directed square and honeycomb lattices,J. Phys. A: Math. Gen. 28: 4813–4833 (1995).
N. J. A. Sloane,A Handbook of Integer Sequences (Academic Press, New York, 1973).
J. Riordan,Combinatorial Identities (Robert E. Krieger, Huntington, New York, 1979).
M. Bousquet-Mélou, Percolation models and animals,Europ. J. Comb., to appear.
M. Delest, Polyominoes and animals: Some recent results,J. Math. Chem. 8:3–18 (1991).
N. Inui and M. Katori, Catalan numbers in series expansion of the directed percolation probability on square lattice,J. Phys. A: Math. Gen. 29:4347–4364 (1996).
K. Y. Lin, S. K. Ma, C. H. Kao, and S. H. Chiu, Number of anisotropic spiral self-avoiding loops,J Phys. A: Math. Gen. 20:1881–1888 (1987).
A. J. Guttmann and I. G. Enting, The size and number of rings on the square lattice,J. Phys. A: Math. Gen. 21:L165-L172 (1988).
A. J. Guttmann and I. G. Enting, The number of convex polygons on the square and honeycomb lattices,J. Phys. A: Math. Gen. 21:L467-L474 (1988).
K. Y. Lin and S. J. Chang, Rigorous results for the number of convex polygons on the square and honeycomb lattices,J. Phys. A: Math. Gen. 21: 2635–2642 (1988).
M. L. Martins, H. F. Verona de Resende, C. Tsallis and C. N. de Magalhãses, Evidence for a new phase in the Domany-Kinzel cellular automaton,Phys. Rev. Lett. 66:2045–2047 (1991).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Katori, M., Inui, N., Komatsu, G. et al. Hypergeometric series in a series expansion of the directed-bond percolation probability on the square lattice. J Stat Phys 86, 37–55 (1997). https://doi.org/10.1007/BF02180198
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02180198