Hierarchical models of rigidity percolation

We introduce models of generic rigidity percolation in two dimensions on hierarchical networks and solve them exactly by means of a renormalization transformation. We then study how the possibility for the network to self organize in order to avoid stressed bonds may change the phase diagram. In contrast to what happens on random graphs and in some recent numerical studies at zero temperature, we do not find a true intermediate phase separating the usual rigid and floppy ones.

Figure 1

An example of a graph with $N=5$ sites. Before the addition of the dashed bond, there are $M=6$ bonds, none of which is redundant. The number of floppy modes is 4: two global translations, one global rotation, and the additional rotation of the one-coordinated site. The dashed bond is redundant: the subgraph isolated by the dashed curve has $N_{sub}=4$ sites and $M_{sub}=6>2N_{sub}-3$ bonds. The whole graph with the dashed bond also has four floppy modes, in agreement with Eq. (2).

Figure 2

Four examples of hierarchical graphs. On the left, the graphs at step $t=0$, with two sites and one bond; on the right, the graphs at $t=1$. We call the $t=1$ graph the “elementary cell.” (1) is an elementary cell with five sites and nine bonds. This is the hierarchical graph we will mostly consider in the following. (2), (3) and (4) are elementary cells with, respectively, four sites and five bonds, seven sites and 12 bonds, and six sites and 12 bonds. At step $t$, each bond of step $t-1$ is replaced by a subgraph identical to the $t=1$ graph.

Figure 3

Renormalization of an elementary cell, and counting of redundant bonds. Top left: sites $A$ and $B$ are rigidly connected so that the elementary cell is renormalized into a bond; there is one redundant bond. Top right: sites $A$ and $B$ are rigidly connected, no redundant bond. Bottom left: sites $A$ and $B$ are not rigidly connected, no redundant bond. Bottom right: sites $A$ and $B$ are not rigidly connected, one redundant bond.

Figure 4

(Color online) The red curve $y=\phi \left(p\right)$ (thick line), together with the blue $y=p$ straight line (thin line). The critical point defined by $\phi \left(p^{\ast }\right)=p^{\ast }$ is indicated by an arrow.

Figure 5

(Color online) Examples of functions $\phi \left(p,\beta \right)$, as a function of $p$; from left to right, $\beta =0,1,5$ (thick lines, respectively red, black, and light blue curves). The intersection with the $y=p$ straight line (blue thin line) is the critical point. A low temperature shifts the critical point toward high $p$ values.

Figure 6

(Color online) The mean connectivity of the network $x$ as a function of $p$. From left to right, $\beta =0,1,5$ (respectively, red, black, and light blue curves). The critical points are also indicated.

Figure 7

(Color online) The number of redundant per site is plotted as a function of the mean connectivity $x$ for $\beta =0,1,5$ (from left to right, respectively, red, black, and light blue curves). The crosses indicate the critical points; the filled circles are results of Metropolis Monte Carlo simulations, for $\beta =1$, using the pebble game algorithm, on hierarchical graphs with $t=5$ (22 145 sites). They are in perfect agreement with the analytical results.

Figure 8

(Color online) Top panel: the self organization indicator $s=\left(p-3x/8\right)/p$ as a function of the mean connectivity $x$. Bottom panel: fraction of isostatic elementary cells as a function of the mean connectivity. In both panels, $\beta =0,1,5$ from bottom to top (respectively, red, black, and light blue curves), and the crosses indicate the critical points. Note that $s$ is identically 0 when $\beta =0$.