Corrections to scaling for watersheds, optimal path cracks, and bridge lines

We study the corrections to scaling for the mass of the watershed, the bridge line, and the optimal path crack in two and three dimensions (2D and 3D). We disclose that these models have numerically equivalent fractal dimensions and leading correction-to-scaling exponents. We conjecture all three models to possess the same fractal dimension, namely, $d_f=1.2168\pm 0.0005$ in 2D and $d_f=2.487\pm 0.003$ in 3D, and the same exponent of the leading correction, $\Omega =0.9\pm 0.1$ and $\Omega =1.0\pm 0.1$, respectively. The close relations between watersheds, optimal path cracks in the strong disorder limit, and bridge lines are further supported by either heuristic or exact arguments.

Figure 1

Mass $M$ of the watershed (WS site or bond), the main crack (MC), and the bridge line (BL) models as functions of the system size $N$, defined as the number of sites (bonds) in the system, in both two and three dimensions. The error bars are much smaller than the symbols. The lines show the fractal dimensions obtained in this work.

Figure 2

Corrections to scaling $C_L=M/N^{d_f/d}$ of the watershed (WS site and bond), the main crack (MC), and the bridge line (BL) models as functions of the system size $N$, defined as the number of sites (bonds) in the system, in two dimensions. The fractal dimension $d_f\approx 1.217$, consistent with the more precise estimate obtained later, has been chosen such that $C_L$ converges to a constant value for large $N$. The error bars are typically much smaller than the symbols. The lines show fits of truncated versions of Eq. (3) to the data, which is divided here by $a_{00}$ to show the matching of the scaling behavior of the different models for large $N$.

Figure 3

(a) Inverse of the quality $Q$ as a function of ${\Omega }^{\prime }$ for different values of $\alpha$, as obtained from fits of the ansatz Eq. (6) to the reduced mass $M{L}^{-\alpha }$ for the watershed on bonds in 2D with sizes as indicated in Table . The vertical lines give the position of the global minimum in $1/Q$ (solid) and the estimated error (dashed). (b) For the same system as in (a), the minimum value $1/Q_{\mathrm{loc}}$ as a function of $\alpha$ is shown, where $Q_{\mathrm{loc}}$ is obtained from curves $1/Q\left({\Omega }^{\prime }\right)$ for a given $\alpha$, like those shown in (a). The vertical lines highlight the value of $\alpha$ at the global minimum $1/Q_{\max }$ (solid) and the estimate for the error (dashed). The error bars are determined from the width of the minima. The vertical lines show the estimate $d_f=1.2168\pm 0.0005$ for the fractal dimension of the watershed on bonds and the horizontal ones the corresponding leading correction $\Omega =0.95\pm 0.05$. These exponents were obtained from the analysis of a single model (WS bond). By combining the results for different models, we obtain more accurate estimates for the exponents.

Figure 4

Inverse of the quality at the minimum $1/Q_{\mathrm{loc}}$ as a function of $\alpha$ for the different models in 2D. The inset shows for each model the inverse of the quality $Q$ as a function of ${\Omega }^{\prime }$ with $\alpha$ fixed to its value at the global minimum $1/Q_{\max }$. The vertical lines show the averages $d_f=1.2168\pm 0.0005$ and $\Omega =0.9\pm 0.1$ of the estimates for the fractal dimension and for the leading correction, respectively.

Figure 5

Inverse of the quality at the minimum $1/Q_{\mathrm{loc}}$ as a function of $\alpha$ for the watershed on bonds, the watershed on sites, and the bridge line in 3D. The inset shows for each model the inverse of the quality $Q$ as a function of ${\Omega }^{\prime }$ with $\alpha$ fixed to its value at the global minimum $1/Q_{\max }$. The vertical lines show the averages $d_f=2.487\pm 0.003$ and $\Omega =1.0\pm 0.1$ for the estimates for the fractal dimension and for the leading correction.

Figure 6

Estimated leading correction ${\Omega }_{\text{est}}$ as defined in Eqs. (7) and (8) from the mass data of the bridge line, the watershed on bonds (sites), and the main crack in 2D. The value of $\alpha$ is fixed to $1.2168$, the fractal dimension estimated by the fit quality method. For better visibility, the data of each model are shown with connecting lines, and data points with $\Delta {\Omega }_{\mathrm{est}}>1$ have been removed. The values for the main crack are shown for comparison, but without their error bars. The horizontal lines give the value (solid) and error bar (dashed) for $\Omega$ as estimated by the fit quality method, as well as the value for $\omega$ (dotted).

Figure 7

Estimated leading correction ${\Omega }_{\text{est}}$ as defined in Eqs. (7) and (8) from the mass data of the bridge line, the watershed on bonds (sites), and the main crack in 3D. The value of $\alpha$ is fixed to $2.487$, the fractal dimension estimated by the fit quality method. For better visibility, the data of each model are shown with connecting lines, and data points with $\Delta {\Omega }_{\mathrm{est}}>1$ have been removed. The values for the main crack are shown for comparison, but without their error bars. The horizontal lines give the value (solid) and error bar (dashed) for $\Omega$ as estimated by the fit quality method.

Figure 8

Cumulative distribution $P(m_s>m)$ of the masses obtained with the WS site, WS bond, and the BL model for system size $L=128$ in 2D. The tail of the WS site case follows a power law with exponent $-1.8$.

Figure 9

Representative system with $L=3$, where each square cell represents a site of the lattice. The numbers in the lower left corner of each cell give the heights. Letters “B” and “W” indicate that a site is part of the bridge line (i.e., the great wall) and of the site watershed, respectively. Notice that the center site in the bottom row is part of the site watershed (as it is upstream of the central site), but is not part of the great wall, because the wall built at the center site prevents water from flowing there. Arrows indicate the flow of water from sites belonging to the two basins.