A fast algorithm for 2D Rigidity Percolation

TL;DR

This work develops a fast, exact algorithm for two-dimensional central-force Rigidity Percolation by uniting the Pebble Game with the Newman–Ziff framework and new rigidity-theory theorems. The method updates rigid-cluster states after each bond activation with near-linear scaling $\sim N^{1.02}$, enabling simulations on systems with over $5\times10^8$ nodes and precise determination of the RP critical point and exponents. The authors establish three key cluster-merge mechanisms—Pivoting, Rigidification, and Overconstraining—along with a Pivot Network to efficiently coalesce rigid clusters and detect wrapping via a Machta-inspired approach adapted for RP. Their results deliver high-precision estimates: $p_c^{\rm RP}=0.6602741(4)$, $D_f^{\rm RP}=1.8423(7)$, $\nu^{\rm RP}=1.1694(8)$, and $\gamma^{\rm RP}=1.928(3)$, demonstrating that 2D central-force RP belongs to a universality class distinct from standard CP. The work provides both a practical numerical tool for large-scale RP studies and new theoretical insight into nonlocal rigidification phenomena that differentiate RP from CP in soft-matter and amorphous systems.

Abstract

Rigidity Percolation is a crucial framework for describing rigidity transitions in amorphous systems. We present a new, efficient algorithm to study central-force Rigidity Percolation in two dimensions. This algorithm combines the Pebble Game algorithm, the Newman-Ziff approach to Connectivity Percolation, as well as novel rigorous results in rigidity theory, to exactly identify rigid clusters over the full bond concentration range, in a time that scales as $N^{1.02}$ for a system of $N$ nodes. We perform extensive numerical simulations with systems larger than $500$ million nodes, far beyond the previous limitations. We obtain new, precise estimates for the critical exponents, $ν=1.1694(8)$ and $D_f=1.8423(7)$, and locate the critical threshold at $p_c = 0.6602741(4)$. Besides opening the way to further accurate numerical studies of Rigidity Percolation, our work provides new rigorous theoretical insights on specific cluster merging mechanisms that distinguish it from the standard Connectivity Percolation problem.

Figures (7)

• Figure 1: Dynamics of pebble searches on the pebble graph. Top: searches of type I. Bottom: searches of type II. To ease visualization, bonds are not triangulated in the bottom panels. (a) Green nodes have two pebbles, blue nodes have one, grey have zero. Pebbles are depicted as black dots. The red undirected bond is the newly activated bond. A pebble search of type I starts at the grey end node of the new bond. The purple path leads to a pebble. (b) The path identified in (a) is reversed, the node where it started has now one pebble, and a second search is started. The search traverses the black edges without finding a pebble. (c) The edges traversed during the previous search are triangulated over the base made by the new bond. (d) Red (blue) nodes have been marked rigid (floppy) during previous searches, while open circles represent unmarked nodes. A pebble search of type II starts at the yellow star. The search traverses the black edges without finding a floppy node. Note that edges originating from rigid nodes are not walked. (e) The nodes visited during the search of panel (d) are marked rigid. The visited edges are triangulated (not shown). A new search starts at the yellow star and, following the blue path, stops when it hits a floppy node. (f) The nodes traversed by the path identified in (e) are marked floppy.
• Figure 2: Stages of the rigidity transition. The blue, orange and green curves represent the probability, as a function of the bond concentration, that the activation of a new bond results respectively in a pivoting event, a rigidification event or an overconstraining event. We indicate the critical bond concentrations $p_c^{CP}=2 \sin (\pi /18)$sykes64 and $p_c^{RP}\approx 0.6602$Jacobs1997AnJacobsThorpe1995 of the CP and RP transition respectively. Curves are shown for $L=2^{10}$ and are observed not to depend on $L$.
• Figure 3: Example of a rigidification process as implemented by our algorithm. The different steps are illustrated both on the triangular lattice (top row) and on the corresponding Pivot Network (bottom row). Each color indicates a different rigid cluster. Top row: pivots are indicated with black dots. At each step, enqueued pivots are highlighted as stars. (a) The blue bond is the newly activated one: its two end nodes, marked with yellow stars, are initially enqueued in the queue of pivots. Each rigid cluster is represented by a different color. Nodes with the same color of their bonds belong to that rigid cluster only, black nodes are pivots. (b) The system state after having checked the rigid clusters led to by the pivots enqueued in panel (a). Red stars indicate pivots that have been processed. The arrows illustrate the Machta displacements $d_p^{r_{\rm large}}$ and $\,d_p^{r_{\rm small}}$, from pivot $p$ to the large and small roots. (c) The system state after having checked the rigid clusters led to by the pivots enqueued in panel (b). The displacement $d_{u_s}^{r_{\rm large}}$ has been updated according to Eq. (\ref{['eq:update_us']}). The arrows show the displacements $d_{p'}^{r_{\rm large}}$ and $d_{p'}^{r_{\rm large}\,'}$ given by Eq. (\ref{['eq:machta_piv_update']}), whose difference allows to detect that wrapping has occurred along the vertical axis. Note that the pivot $p^*$ is never enqueued, but that the green cluster is nonetheless reached via another pivot (yellow star), cf. discussion at the end of sec. \ref{['sec:implementation']}. (d) The system state after having checked the rigid clusters led to by the pivots enqueued in panel (c). Black stars indicate processed pivots leading to floppy clusters. (e) The system state after having checked the rigid clusters led to by the pivots enqueued in panel (d). (f) The final lattice configuration. Bottom row: the pivot network corresponding to each panel of the top row. Nodes correspond to rigid clusters, edges represent pivots. Note that pivots with pivotal class three form triangles in this network; we indicate again the non-enqueued pivot $p^*$. Note that as rigid clusters coalesce, nodes disappear and the network is rearranged. Also, note that the bond representing pivot $p^*$ disappears from panel (i) to panel (l). If it had larger pivotal class, it would still be an edge in the Pivot Network.
• Figure 4: Performance of the algorithm. (a) The average time (in minutes) to complete a simulation as a function of the system size. The blue solid line is the best fit to the log-log of the data and shows the scaling $N^{1.02}$. Error bars are smaller than the markers' size. (b) Average number of pebble searches (of any type) needed at each bond activation, as function of the bond concentration $p$, for different system sizes $N=L^2$. (c) Average time (in seconds) of each bond activation as function of $p$, for the same system sizes as in panel (b).
• Figure 5: Average number of nodes visited during type I searches (blue circles, scaling as $N^{1.02}$), average number of nodes visited during type II searches (red squares, scaling as $N^{1.01}$) and average number of pivots $p'$ over wich we iterate during step 7 of the algorithm (green triangles, scaling as $N^{1.00}$). Data are shown as a function of the system size.