Journal of Statistical Physics

, Volume 86, Issue 1–2, pp 37–55 | Cite as

Hypergeometric series in a series expansion of the directed-bond percolation probability on the square lattice

  • M. Katori
  • N. Inui
  • G. Komatsu
  • K. Kameoka


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,dn,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 thatdn,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.

Key Words

Directed bond percolation percolation probability asymmetry series expansion correction terms hypergeometric series 


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Copyright information

© Plenum Publishing Corporation 1997

Authors and Affiliations

  • M. Katori
    • 1
  • N. Inui
    • 2
  • G. Komatsu
    • 2
  • K. Kameoka
    • 2
  1. 1.Department of Physics, Faculty of Science and EngineeringChuo UniversityTokyoJapan
  2. 2.Department of Mechanical and Intelligent EngineeringHimeji Institute of TechnologyHimeji, HyogoJapan

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