Directed percolation near a wall

TL;DR

This study uses series expansion methods to analyze directed percolation clusters near a wall, revealing that while the percolation threshold remains unchanged, critical exponents for cluster probability and size are significantly different from the bulk, with some exponents remaining unaffected.

Contribution

The paper provides the first detailed analysis of directed percolation near a wall, showing how critical exponents differ from bulk values and proposing an exact value for the cluster length exponent.

Findings

Percolation threshold remains the same as bulk.

Critical exponents for cluster probability and size differ from bulk.

The cluster length exponent is conjectured to be exactly 1.

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 $\beta_1 = 0.7338 \pm 0.0001$ and $\gamma_1 = 1.8207 \pm 0.0004$ respectively. On the other hand the exponent $\Delta_1=\beta_1 +\gamma_1$ characterising 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 $\nu_{1\parallel} = 1.7337 \pm 0.0004$ and is also unchanged. The exponent $\tau_1$ of the mean cluster length is related to $\beta_1$ and $\nu_{1\parallel}$ by the scaling relation $\nu_{1\parallel} = \beta_1 + \tau_1$ and using the above estimates yields $\tau_1 = 1$ to within the accuracy of our results. We conjecture that this value of $\tau_1$ is exact and further support for the conjecture is provided by the direct series expansion estimate $\tau_1= 1.0002 \pm 0.0003$.