Force-balance percolation

We study models of correlated percolation where there are constraints on the occupation of sites that mimic force balance, i.e., for a site to be stable requires occupied neighboring sites in all four compass directions in two dimensions. We prove rigorously that $p_c<1$ for the two-dimensional models studied. Numerical data indicate that the force-balance percolation transition is discontinuous with a growing crossover length, with perhaps the same form as the jamming percolation models, suggesting that all models belong to the same universality class with the same underlying mechanism driving the transition in both cases. We find a lower bound for the correlation length in the connected phase and that the correlation function does not appear to be a power law at the transition. Finally, we study the dynamics of the culling procedure invoked to obtain the force-balance configurations and find a dynamical exponent similar to that found in sandpile models.

Figure 1

(Color online) The sets for the yellow (light gray) site as defined by the spiral model.

Figure 2

(Color online) Sets for the yellow (light gray) site as by defined by the 24 nearest-neighbor force-balance model.

Figure 3

(Color online) Yellow (light gray) sites denote occupied sites and red-colored (dark gray) sites denote neighboring unoccupied sites. (a) Allowed configuration. (b) Forbidden configuration. While the number of occupied nearest neighbors is greater than 3, the force-balance condition is violated.

Figure 4

(Color online) A second force-balance model with 16 nearest neighbors for the yellow (light gray) site.

Figure 5

(Color online) A force-balance model on a three-dimensional lattice with 26 nearest neighbors for the patterned site. A red (dark gray) site participates in only one of the six sets, a black site in two of the six sets, and a yellow (light gray) site in three of the six sets.

Figure 6

(Color online) $k=6$ NE-SW model.

Figure 7

(Color online) Adjacent clusters of three sites in the NE-SW model.

Figure 8

(Color online) Local configurations of three clusters in the NE-SW model.

Figure 9

(Color online) The force-balance rules can be implemented with (${\text{A}}^{\prime }$ and ${\text{C}}^{\prime }$) or (${\text{D}}^{\prime }$ and ${\text{B}}^{\prime }$) or (${\text{E}}^{\prime }$ and ${\text{B}}^{\prime }$ and ${\text{C}}^{\prime }$) or (${\text{F}}^{\prime }$ and ${\text{C}}^{\prime }$ and ${\text{D}}^{\prime }$) or (${\text{G}}^{\prime }$ and ${\text{A}}^{\prime }$ and ${\text{B}}^{\prime }$) or (${\text{H}}^{\prime }$ and ${\text{A}}^{\prime }$ and ${\text{D}}^{\prime }$) or (${\text{A}}^{\prime }$ and ${\text{G}}^{\prime }$ and ${\text{H}}^{\prime }$) or (${\text{B}}^{\prime }$ and ${\text{E}}^{\prime }$ and ${\text{G}}^{\prime }$) or (${\text{C}}^{\prime }$ and ${\text{E}}^{\prime }$ and ${\text{F}}^{\prime }$) or (${\text{D}}^{\prime }$ and ${\text{F}}^{\prime }$ and ${\text{H}}^{\prime }$) or (${\text{E}}^{\prime }$ and ${\text{F}}^{\prime }$ and ${\text{G}}^{\prime }$ and ${\text{H}}^{\prime }$).

Figure 10

(Color online) Mean culling time as a function of initial occupation density for the 24 NN model, $L=2896$, the largest system size studied. Error bars are smaller than the symbols.

Figure 11

(Color online) Peak of the mean culling time as a function of $L$ for the 24 NN model. The best fit indicated by the solid line has a slope of $1.226\pm 0.027$.

Figure 12

(Color online) Log-log plot of $P\left(s\right)$ for $L=128$ using periodic boundary conditions.

Figure 13

(Color online) Log-log plot of $P\left(s\right)$ for $L=128$ and 512 for comparison. The $L=512$ curves have been shifted downward by a factor of 0.1.

Figure 14

(Color online) The probability of spanning for wired boundary conditions in the 24 NN model.

Figure 15

(Color online) The probability of spanning for periodic boundary conditions in the 24 NN model.

Figure 16

(Color online) $p_c\left(L\right)$ for the 24 NN model, using two definitions of the critical point: (1) the occupation probability at which $P_s=1/2$ and (2) the occupation probability at which the mean culling time peaks.

Figure 17

(Color online) The widths for the 24 NN model with both wired and periodic boundary conditions, on a log-log plot. Both boundary conditions show clear deviations from power laws at large system sizes.

Figure 18

(Color online) The widths for the 24 NN model with both wired and periodic boundary conditions, using a fitting form motivated by the TBF results.

Figure 19

(Color online) Fitting of $p_c\left(L\right)$ as a function of $L$ using the TBF fitting form, for the 24 NN force-balance model with wired boundary conditions. We obtain the best fit with $p_c\left(\infty \right)=0.414\pm 0.008$.

Figure 20

(Color online) Order parameter for the 24 NN model using wired boundary conditions.

Figure 21

(Color online) Correlation lengths for the 24 NN force-balance model as a function of $p$, for $L=100$ and 200.

Figure 22

(Color online) Correlation function for $L=400$, for the 24 NN force-balance model, at $p=0.37$, 0.38, and 0.39. For clarity, the $p=0.37$ correlation function has been shifted up by 1.0, and the $p=0.39$ correlation function has been shifted down by 1.0.

Figure 23

(Color online) Correlation function for $L=200$, $p=0.395$, for the 24 NN force-balance model. The best fit line has a slope of $-1.76\pm 0.03$.

Figure 24

(Color online) Peak of the mean culling time as a function of $L$ for the all three models. For the 16 NN model, the best fit has a slope of $1.240\pm 0.025$. For the spiral model, the best fit has a slope of $1.246\pm 0.027$. Wired boundary conditions are used. The data for the 16 NN model have been shifted upward by unity, and the data for the spiral model have been shifted downward by unity.

Figure 25

(Color online) Log-log plot of $P\left(s\right)$ for the 16 NN model with $L=128$ in the presence of periodic boundary conditions.

Figure 26

(Color online) The probability of spanning for periodic boundary conditions for the 16 NN force-balance model.

Figure 27

(Color online) The widths for all three models with both wired and periodic boundary conditions, on a log-log plot. All curves show clear deviations from power laws at large system sizes.

Figure 28

(Color online) The widths for all three models with both wired and periodic boundary conditions assuming the TBF form.

Figure 29

(Color online) Fitting of $p_c\left(L\right)$ as a function of $L$ using the TBF fitting form, for the 16 NN force-balance model with wired boundary conditions. We obtain the best fit with $p_c\left(\infty \right)=0.497\pm 0.007$.

Figure 30

(Color online) Order parameter for the 16 NN model using wired boundary conditions.