## Abstract

Series expansion methods are used to study directed bond percolation clusters on the square lattice whose lateral growth is restricted by a wall parallel to the growth direction. The percolation threshold p_{c} is found to be the same as that for the bulk. However, the values of the critical exponents for the percolation probability and mean cluster size are quite different from those for the bulk and are estimated by β_{1} = 0.7338 ± 0.0001 and γ_{1} = 1.8207 ± 0.0004 respectively. On the other hand the exponent Δ_{1} = β_{1} + γ_{1} characterizing the scale of the cluster size distribution is found to be unchanged by the presence of the wall. The parallel connectedness length, which is the scale for the cluster length distribution, has an exponent which we estimate to be ν_{1∥} = 1.7337±0.0004 and is also unchanged. The exponent τ_{1} of the mean cluster length is related to β_{1} and ν_{1∥} by the scaling relation ν_{1∥} = β_{1} + τ_{1} and using the above estimates yields τ_{1} = 1 to within the accuracy of our results. We conjecture that this value of τ_{1} is exact and further support for the conjecture is provided by the direct series expansion estimate τ_{1} = 1.0002 ± 0.0003.

Original language | English |
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Pages (from-to) | 1619-1628 |

Number of pages | 10 |

Journal | Journal of Physics A: Mathematical and General |

Volume | 29 |

Issue number | 8 |

DOIs | |

Publication status | Published - 21 Apr 1996 |