One-dimensional long-range percolation: A numerical study

In this paper we study bond percolation on a one-dimensional chain with power-law bond probability $C/r^{d+\sigma }$, where $r$ is the distance length between distinct sites and $d=1$. We introduce and test an order-$N$ Monte Carlo algorithm and we determine as a function of $\sigma$ the critical value $C_c$ at which percolation occurs. The critical exponents in the range $0<\sigma <1$ are reported. Our analysis is in agreement, up to a numerical precision $\approx {10}^{-3}$, with the mean-field result for the anomalous dimension $\eta =2-\sigma$, showing that there is no correction to $\eta$ due to correlation effects. The obtained values for $C_c$ are compared with a known exact bound, while the critical exponent $\nu$ is compared with results from mean-field theory, from an expansion around the point $\sigma =1$ and from the $\varepsilon$-expansion used with the introduction of a suitably defined effective dimension $d_{\mathrm{eff}}$ relating the long-range model with a short-range one in dimension $d_{\mathrm{eff}}$. We finally present a formulation of our algorithm for bond percolation on general graphs, with order $N$ efficiency on a large class of graphs including short-range percolation and translationally invariant long-range models in any spatial dimension $d$ with $\sigma >0$.

Figure 1

Pictorial representation of the LR bond percolation model. The lines represent the bonds with width decreasing as the probability of establishing it decreases (a). An instance of the model together with the clusters generated is shown in (b) (the established bonds are drawn with a bold line).

Figure 2

(a) $S{N}^{\sigma }$ as a function of $C$ for $\sigma =0.25$. The lines refer to increasing (small) sizes $N={10}^3$, $2\times {10}^3$, $4\times {10}^3$, $8\times {10}^3$, and $16\times {10}^3$ denoted by up triangles, diamonds, squares, down triangles, and circles, respectively. Errors are smaller than the size of the symbols and the lines are just a guide to the eyes. In the inset, a zoom of the figure around the crossing point is shown: The curves refer to sizes $N={10}^3$, $2\times {10}^3$, $2^2\times {10}^3$, $2^3\times {10}^3$, $...$, $2^9\times {10}^3$ with the curves becoming steeper as the system size is enlarged. (b) Same as (a) with $\sigma =0$. As it is apparent, no crossing occurs.

Figure 3

(a) Ratio $Q$ as a function of $C$ for $\sigma =0.5$. The lines refer to increasing (small) sizes $N={10}^3$, $2\times {10}^3$, $4\times {10}^3$, $8\times {10}^3$, and $16\times {10}^3$ plotted by up triangles, diamonds, squares, down triangles, and circles, respectively. Errors are smaller than the size of the symbols and the lines are just a guide to the eyes. The inset shows the curves in the vicinity of $C=1$ for all the considered system sizes ($N={10}^3$, $2\times {10}^3$, $2^2\times {10}^3$, $2^3\times {10}^3,...,2^9\times {10}^3$ with the curves becoming steeper as the system size is enlarged) as used to locate the transition. (b) Same as (a) with $\sigma =1$. In the inset a zoom around the $C=1$ point is reported: One can notice that the curves at the level of available precision do not show any crossing.

Figure 4

Critical thresholds $C_c$ as a function of $\sigma$. The dotted line is the rigorous lower bound (2).

Figure 5

Estimated values for $\eta$ as a function of $\sigma$ together with predicted value (blue continuous line) from the RG prediction $\eta =2-\sigma$ [27]. (Inset) Deviation from the RG prediction $\eta =2-\sigma$.

Figure 6

Estimated values for $1/\nu$ (circles) together with predicted values from RG in the classical region ($\sigma <1/3$, blue continuous line) and the results for the nonclassical region ($1/3<\sigma <1$). The red dashed line on the right of the figure is the prediction (12) (dashed red) valid close to the BKT point, while the gray and green lines are obtained by using the effective dimension (9) in the second (dotted gray, bottom) and fourth (dash-dotted green, top) $\varepsilon$-expansion results (11) taken from [54].

Figure 7

(a) Ratio $Q_G$ as a function of occupation probability $p$ for the SR bond percolation on a square lattice. The lines refer to increasing linear sizes $N=32$, 64, 128, and 256 plotted with up triangles, diamonds, squares, and circles, respectively. Errors are smaller than the size of the symbols and the lines are just a guide to the eyes. (b) $S{N}^{\gamma /\nu }$ as a function of $p$ for the same sizes in (a).