ISSN:
1572-9613
Keywords:
Directed bond percolation
;
percolation probability
;
asymmetry
;
series expansion
;
correction terms
;
hypergeometric series
Source:
Springer Online Journal Archives 1860-2000
Topics:
Physics
Notes:
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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF02180198
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