Postprocessing techniques for gradient percolation predictions on the square lattice

In this work, we revisit the classic problem of site percolation on a regular square lattice. In particular, we investigate the effect of quantization bias errors on percolation threshold predictions for large probability gradients and propose a mitigation strategy. We demonstrate through extensive computational experiments that the assumption of a linear relationship between probability gradient and percolation threshold used in previous investigations is invalid. Moreover, we demonstrate that, due to skewness in the distribution of occupation probabilities visited the average does not converge monotonically to the true percolation threshold. We identify several alternative metrics which do exhibit monotonic (albeit not linear) convergence and document their observed convergence rates.

Figure 1

Convergence of the percolation threshold $p_c^{*}$ to the known limit $p_c$ for site percolation on a square lattice is apparently linear. However, for small probability gradients (inset), the behavior is nonlinear and even nonmonotonic.

Figure 2

Uncertainty in ${\mu }_{MD2}$ after $5.12\times {10}^{10}$ steps (50 000 simulations of width 2048 or 25 000 simulations of width 4096).

Figure 3

Proportionality between $\sigma$ and $\nabla p^{3/7}$ for all distribution fitting procedures.

Figure 4

Variation of ${\mu }_{\mathrm{MoM},j}$ with $p_0$ for a large probability gradient, $\nabla p={12}^{-1}$.

Figure 5

Variation of ${\mu }_{\mathrm{MoM},j}$ with $p_0$ for a slightly smaller probability gradient, $\nabla p={40}^{-1}$.

Figure 6

Variation of ${\mu }_{MD2,j}$ with $p_0$ for a large probability gradient, $\nabla p={12}^{-1}$.

Figure 7

Distribution of occupation probabilities of sites visited during all simulations with $\nabla p={16}^{-1}$. Asymmetric tails result in a leftward shift of the mean relative to the median.

Figure 8

Reduction in skewness as $\nabla p\to 0$ is proportional to $\nabla p^{3/7}$.

Figure 9

Convergence of ${\mu }_{\mathrm{MoM}}$ to ${\mu }_{\mathrm{Med}}$ as $\nabla p\to 0$ is approximately linear.

Figure 10

Variation of error $|\mu \left(\nabla p\right)-p_c|$ with $\nabla p$ on the square lattice with von Neumann connectivity.