High-precision Dynamic Monte Carlo Study of Rigidity Percolation

TL;DR

This work introduces a dynamic pebble game algorithm to study rigidity percolation on a triangular lattice as bonds are added sequentially, achieving $O(V^{1.15})$ scaling by using persistent cluster data and a let-go-of-the-largest merger optimization. It uncovers temporal self-similarity in rigidity dynamics, with clean dynamic distributions for gap and merged-cluster statistics and well-defined event-based pseudo-critical points that yield high-precision estimates of $p_c$, $1/ u$, and $d_f$. The results show that cascade, multi-cluster mergers near criticality drive substantial finite-size corrections and explain the anomalous static distributions observed with traditional ensembles. The approach provides a robust framework for high-precision critical phenomena in RP, with potential extensions to site RP and higher dimensions, and highlights the role of large-scale cascade events in RP universality.

Abstract

Rigidity percolation provides an important basis for understanding the onset of mechanical stability in disordered materials. While most studies on the triangular lattice have focused on static properties at fixed bond~(site) occupation probabilities, the dynamics of the rigidity transition remain less explored. In this work, we formulate a dynamic pebble game algorithm that monitors how rigid clusters emerge and evolve as bonds are added sequentially to an empty lattice, with computational efficiency comparable to the standard static pebble game. We uncover a previously overlooked temporal self-similarity exhibited in multiple quantities, including the cluster size changes and merged cluster sizes during bond addition, as well as the number of simultaneously merging clusters. We identify large-scale cascade events in which a single bond addition triggers the merger of an extensive number of clusters that scales with system size with inverse correlation-length exponent. Using an event-based ensemble approach, we obtain high-precision estimates of the critical point $p_c = 0.660\,277\,8(10)$, the inverse correlation-length exponent $1/ν= 0.850(3)$, and the fractal dimension $d_f = 1.850(2)$, representing substantial improvements over existing values.

Figures (15)

• Figure 1: Illustration of a multi-cluster merger event in rigidity percolation on the triangular lattice. Different colors represent distinct rigid clusters before the addition of the new bond (shown as a dashed red line), with each green bond representing a single-bond rigid cluster consisting of one bond and two sites. Upon adding this bond, nine rigid clusters—including the rigid cluster formed by the new bond itself—merge into a single large rigid cluster. Notably, the merger involves clusters (such as the green and purple clusters) that are not directly connected to the newly added bond, exemplifying the non-local characteristic of rigidity percolation. To identify all merging clusters, one must traverse not only the clusters directly connected by the new bond (blue and black clusters), but also their neighboring clusters (green clusters), and iteratively check the neighbors of those clusters (purple cluster), adding newly discovered merging clusters to the traversal queue until no unvisited clusters remain.
• Figure 2: Finite-size scaling of the pseudo-critical points $t_G$ (top) and $t_{\chi}$ (bottom). Main panels show $t_c - t_{\alpha}$ vs. $L$ on a log-log scale, with $t_c = 0.660\,277\,8$. Dashed lines represent a power-law behavior $\sim L^{-0.85}$. Insets show the fluctuations of pseudo-critical points $\sigma_{t_{\alpha}}$ vs. $L$. The dashed lines in the insets for large $L$ correspond to $\sim L^{-0.85}$. For $L < 128$, the fluctuation $\sigma_{t_G}$ is well described by $\sim L^{-0.5}$ (blue dashed line).
• Figure 3: Comparison of the finite-size scaling behavior for pseudo-critical points $t_K$ (black squares) and $t_{\chi}$ (red circles). The plot shows $t_c - t_K$ and $t_c - t_{\chi}$ versus system size $L$ on a log-log scale. The dashed lines indicate power laws with exponents $-1.09$ and $-0.85$, respectively. For small to moderate system sizes, $t_K$ appears to converge faster, but this reflects the dominance of a subleading correction term rather than the true critical scaling. Notably, the data demonstrate that $t_K$ at $L=512$ is even closer to $t_c$ than $t_{\chi}$ at $L=8192$.
• Figure 4: Examples of cluster merging patterns and cascade dynamics. Red dashed lines indicate newly added bonds that trigger the merging process. (a) The elementary triadic merger, where three clusters merge simultaneously. (b) A five-cluster merger motif demonstrating more complex irreducible merger patterns. (c) An eye-shaped demo illustrating the cascade effect, where the addition of a single bond (red dashed line) can initiate a continuous sequence of triadic mergers. (d) A detailed cascade process showing how seven initially independent clusters (including the newly added bond as a rigid cluster) progressively merge through a series of triadic mergers, ultimately forming a single rigid body.
• Figure 5: The mean number of merged clusters, $K(t) = \langle \mathcal{K}(t) \rangle$, as a function of time step $t$ for various system sizes $L$. The data for different $L$ collapse well, exhibiting a sharp peak at the critical point $t_c = 0.660\,277\,8(10)$, indicated by the dashed orange line. The inset shows the total number of clusters per site, $n$, versus $t$. This quantity decreases monotonically, with its fastest decline occurring around $t_c$ and showing minimal finite-size effects.