Statistics of the critical percolation backbone with spatial long-range correlations

We study the statistics of the backbone cluster between two sites separated by distance $r$ in two-dimensional percolation networks subjected to spatial long-range correlations. We find that the distribution of backbone mass follows the scaling {\it ansatz}, $P(M_B)\sim M_B^{-(α+1)}f(M_B/M_0)$, where $f(x)=(α+ ηx^η) \exp(-x^η)$ is a cutoff function, and $M_0$ and $η$ are cutoff parameters. Our results from extensive computational simulations indicate that this scaling form is applicable to both correlated and uncorrelated cases. We show that the exponent $α$ can be directly related to the fractal dimension of the backbone $d_B$, and should therefore depend on the imposed degree of long-range correlations.

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